Atomic Physics Methods in Modern Research: Selection of Papers Dedicated to Gisbert Zu Putlitz on the Occasion of His 65th Birthday

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K. Jungmann J. Kowalski I. Reinhard F. Tfiiger (Eds.)

Atomic Physics Methods in Modern Research Selection of Papers Dedicated to Gisbert zu Putlitz on the Occasion of his 65th Birthday

~

Springer

Editors Klaus Peter Jungmann Joachim Kowalski Irene Reinhard Physikalisches Institut, Universit~it Heidelberg Philosophenweg 12 D-6912o Heidelberg, Germany Frank Tr~ger Fachbereich Physik, Universit~t Kassel Heinrich-Plett-Strasse 4o D-34132 Kassel, Germany

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Atomic physics methods in modern research : selection of papers dedicated to Gisbert zu Putlitz on his 65th birthday / K. E Jungmann ... (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Pads ; Santa Clara ; Singapore ; Tokyo : Springer, 1997 (Lecture notes in physics ; 499) ISBN 3-540-63716-8

ISSN oo75-845o ISBN 3-540-63716-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved,whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Vertag.Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement,that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by the authors/editors Cover design: design&production GmbH, Heidelberg SPIN: 10643850 55/3144-543210 - Printed on acid-free paper

Preface

Many of the significant advances in the course of the development of atomic physics were associated with newly invented scientific methods and experimental tools. Today these techniques are successfully employed in a wide spread variety of highly active areas in modern research, which extend from investigations of fundamental interactions in physics to experiments related to applied issues and technical aspects. With increasing importance they are found in areas outside of classical atomic physics in fields such as nuclear and particle physics, physics of condensed matter and surfaces, physical chemistry, chemistry, medicine and environmental research. The spectrum of methods includes among others optical and microwave spectroscopy, molecular beams, spin resonance, spin echo, particle trapping and tunneling microscopy. Laser spectroscopy is one example of a widely used technique: The fundamental process of light interacting with single atomic particles can be investigated especially profitably. Laser spectroscopy is essential in many high precision experiments for determining most accurate values of fundamental constants and for deriving conclusions on basic interactions which are complementary to results obtained in high energy physics. Optical properties of molecules, small clusters and bulk solid state material can be investigated both for revealing elementary processes and for studying, for example, new concepts of optical data storage. Processes in combustion devices can be characterized. Remote sensing of environmental pollution can be carried out with high sensitivity. Laser optical pumping of noble gases nowadays yields novel opportunities for nuclear magnetic resonance imaging in medical diagnostics. In February 1996 an international symposium on Atomic Physics Methods in Modern Research was held in Heidelberg on the occasion of the 65th birthday of Professor Gisbert zu Putlitz. In his scientific work atomic physics with its great diversity of facets has played an essential role with numerous significant contributions being highly esteemed by the community. The nature of the event inspired the authors of this volume, which is dedicated to Gisbert zu Putlitz. It comprises invited lectures and articles selected to give an overview of the manifold of developments in this area.

VI The editors would like to thank all authors for their articles and W. BeiglbSck for publishing this volume. The assistance of T. Katzenmaier, C. Kr~imer, E. Nowak and M. Zinser of the Physikalisches Institut der Universit~it Heidelberg in preparing this volume is gratefully acknowledged. Financial support for bringing the authors together was provided by the Stiftung Universit~it Heidelberg and the companies ABB, BASF, Fibro, Friatec, Lambda Physics, Spectra Physics and B. Struck. We are grateful to all of them. Heidelberg, August 1997 K. Jungmann J. Kowalski I. Reinhard F. Tr/iger

Contents

T w o - P h o t o n M e t h o d for M e t r o l o g y in H y d r o g e n B. Cagnac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

H i g h Precision A t o m i c S p e c t r o s c o p y of M u o n i u m a n d Simple M u o n i c A t o m s V. W. Hughes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

M u o n i u m A t o m as a P r o b e of Physics Beyond the Standard Model L. Willmann and K. Jungmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

C a n A t o m s T r a p p e d in Solid He Be Used to Search for P h y s i c s B e y o n d t h e S t a n d a r d M o d e l ? A. Weis, S. Kanorsky, S. Land and T.W. H~insch . . . . . . . . . . . . . . . . . . . . .

57

g-Factors of S u b a t o m i c Particles B.L. Roberts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

L a s e r S p e c t r o s c o p y of M e t a s t a b l e Antiprotonic Helium Atomcules T. Yamazaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Polarized, C o m p r e s s e d SHe-Gas a n d Its A p p l i c a t i o n s E. Often . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

M e d i c a l N M R Sensing w i t h Laser Polarized 3He a n d 129Xe W. Happer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

Test of Special R e l a t i v i t y in a H e a v y Ion Storage R i n g G. Huber, R. Grieser, P. Merz, V. Sebastian, P. Seelig, M. Grieser, P. Grimm, T. Kiihl, D. Schwalm and D. Habs . . . . . . . . . . . . . . . . . . . . . . .

131

R e s o n a n c e F l u o r e s c e n c e of a Single Ion J.T. Hb'ffges, H.W. Baldauf, T. Eichler, S.R. Helm#led and H. Walther . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

R e s o n a n c e R a m a n Studies of t h e R e l a x a t i o n of P h o t o e x c i t e d Molecules in Solution on t h e Picosecond Timescale W.T. Toner, P. Matousek, A . W . Parker and M. Towrie . . . . . . . . . . . . .

151

F o u r - Q u a n t u m R F - R e s o n a n c e in t h e G r o u n d S t a t e of an Alkaline A t o m E.B. Alexandrov and A.S. Pazgalev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t59

VIII

Hard Highly Directional X - R a d i a t i o n E m i t t e d B y a C h a r g e d P a r t i c l e M o v i n g in a C a r b o n N a n o t u b e V. V. Klimov and V.S. Letokhov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

Quasiclassical A p p r o x i m a t i o n in t h e T h e o r y of S c a t t e r i n g of Polarized A t o m s E.L Dashevskaya and E.E. Nikitin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

I o n B e a m I n e r t i a l Fusion R. Bock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211

Spin-Echo E x p e r i m e n t s w i t h N e u t r o n s and with Atomic Beams G. Schraidt and D. Dubbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231

A N e w G e n e r a t i o n of Light Sources for Applications in S p e c t r o s c o p y M. Inguscio, F.S. Cataliotti, C.- Fort, F.S. Pavone and M. Prevedelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

R e m o t e Sensing of t h e E n v i r o n m e n t Using Laser R a d a r Techniques M. Andersson, E. Edner, J. Johansson, S. Svanberg, E. WaUinder and P. Weibring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257

A p p l i e d Laser S p e c t r o c o p y in C o m b u s t i o n Devices V. Sick and J. Wolfrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271

T h e Surface of Liquid H e l i u m - an U n u s u a l S u b s t r a t e for U n u s u a l C o u l o m b S y s t e m s P. Leiderer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283

A s p e c t s of Laser-Assisted Scanning T u n n e l i n g M i c r o s c o p y of T h i n Organic Layers S. GrafstrSm, J. Kowalski and R. Neumann . . . . . . . . . . . . . . . . . . . . . . . . .

295

O p t i c a l S p e c t r o s c o p y of M e t a l Clusters M. VoUmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311

N e w C o n c e p t s for I n f o r m a t i o n Storage B a s e d on Color C e n t r e s A. Winnacker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335

E x c i t o n s a n d R a d i a t i o n D a m a g e in Alkali Halides K. Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

P o l a r i z a t i o n of Negative M u o n s I m p l a n t e d in t h e Fullerene C60: Speculations A b o u t a Null R e s u l t A. Schenck, F.N. Gyax, A. Amato, M. Pinkpank, A. Lappas and K. Prassides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367

P o s i t r o n i u m in C o n d e n s e d M a t t e r Studies w i t h Spin-Polarized P o s i t r o n s J. Major, A. Seeger, J. Ehmann and T. Gessmann . . . . . . . . . . . . . . . . . . .

381

IX Light-Induced Liberation of Atoms and Molecules f r o m Solid S u r f a c e s F. T r @ e r

...........................................................

On the Shoulders of Giants E a r l y H i s t o r y of H y p e r f i n e S t r u c t u r e Spectroscopy. For Gisbert zu Putlitz P. Br/x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423

439

Two-Photon Method for Metrology in Hydrogen Bernard Cagnac Laboratoire Kastler-Brossel, Ecole Normale Sup~rieure et Universitfi Pierre et Marie Curie, 75252 Paris Cedex 05, France

1

Introduction

The history of two-photon transitions ( E 2 - E 1 = 2hw) starts with the beginning of quantum mechanics, during this fascinating period around nineteen thirty when all the modern physics was born. The calculation of the twophoton processes was one of the first applications of the time dependent perturbation theory. It was pub!ished in Annalen der Physik in 1931 as the thesis of Maria GSppert-Mayer "Uber Elementarakte mit zwei Quantenspriingen" [1] at the University of GSttingen in Germany. At the end of the paper she thanks professor Born and Weisskopf. Owing to the relatively small probability of such processes, the experimental realization requires a high intensity of the electromagnetic wave. This is the reason why the first experimental observations where done in the radiofrequency range [2,3]. In the optical range, the ruby laser opened in 1961 the era of multiphoton excitation with the experiment of Kaiser and Garret [4] between braod bands in a crystal. The first precise experiment between narrow atomic levels was done by Abella in Cs vapour using thermal tuning of the ruby laser [5]. From 1968 on, the realization of the really tunable dye laser permitted easier experiments [6] and was followed by an explosion of the number of experiments in atomic physics.

2

Two-Photon Method

Figure 1 represents the energy diagram for the process of two-photon absorption, showing the energy defect hAwr = (Er - El) - h~ of the intermediate relay level Er. The square of this energy defect (z~wr) 2 appears in the denominator of the transition probability, calculated in the perturbation theory to 2nd order. (Strictly speaking, it is necessary to carry out the summation over all other levels, but often only one Er has an important contribution). One must compensate the big term (Awr) 2 (the energy defect) in the denominator by a high intensity of the light beam. For that, you can enclose the atoms between two mirrors forming a Fabry Perot cavity. Then the atoms are exposed to two light beams travelling in opposite directions: forward and

Bernard Cagnac

ENERGX

ATOI v -

FRAM :)

E:-'I, BACh(WARD Fig. 1. Energy diagram of a two-photon transition

backward. If you calculate in the rest frame of one particular atom, and if you take into account the Doppler shift of the light frequency due to the velocity v of this atom (the component on the common axis of the two light beams) the apparent frequencies of the two light beams are symmetrically shifted (w + kv), where k is the wave vector, inverse to the wavelength. If the atom absorbs one photon of each oppositely travelling wave, the total amount of the absorbed energy is 2hw independent of the velocity. And the two-photon transition will be Doppler-free. This analysis was done in 1970 by Professor Chebotaev and coworkers in Novosibirsk [7]. In our laboratory, we came later on the same problems [8], but we were interested to study the real feasibility of the two-photon method: the detailed and numerical calculation of the probability showed that the experiment was possible with the low power of dye lasers with very high spectral purity. We analysed the selection rules. We were also able to show that it was possible to overcome the problem of light shifts. We benefited by the vicinity of Professor Cohen-Tannoudji, who had first understood the origin of the light shifts [9], and had the good formulas for their evaluation [10]. The first experiments applying these ideas were carried out almost simultaneously in three groups: in Paris [11], in Harvard [12] and in Stanford [13] in 1974. A review of early experiments can be found in [14]. The proposal for the 1S-2S transition in hydrogen, we gave at this moment [8], was made also independently by Baklanov and Chebotaev [15]. Everybody knows the success obtained later with that experiment by the team of Professor Hgnsch in Garehing [16]. The difficulty was to produce enough light power at the corresponding wavelength of 243nm (two times the L y m a n - a wavelength of 121.5 nm; cf. energy diagram of Fig. 2a showing recent experiments). The 243 nm radiation was produced by frequency doubling pulsed light at 486 nm in a lithium formate crystal.

Two-Photon Method for Metrology in Hydrogen ZERO OF ENERGY [ionizedhydrogen}

I'

~9c

~|

Sw2

Z~ .... M

3/2

D3/2 D5/2 ~ 12=3

~/2

j

i

P312

S C A L E xl I f ULTRAV'IOLK'T

SCALE

I ~3.10~s hcR

x4

~,mb shift

n=2

n=l

-~ [orbit.} 0 ~

(a)

~"'x' ~'~'~.~ .~.,.~='~' ~.,.- O~ (total.]

i12

1

1

I/2

312

2

3/2

5/2

(c)

Fig. 2. Energy diagram of atomic hydrogen showing the transitions recently studied in the indicated laboratories (corresponding wavelengths are indicated at the bottom. a) full diagram from the ground level n = 1 up to ionisation (zero of energy). b) energy scale multiplied by 4 showing the studied transitions from the metastable 2S level, c) fine structure of levels n = 2 and n = 3 depending on the two quantum numbers orbital l and total j.

But the rigorous control of the light frequency during pulses raised m a n y problems. Improvements of such pulsed experiments were obtained via pulseamplified continuous wave (cw) lasers and filtering of the laser pulses. Nevertheless, it is impossible to avoid frequency shifts during the amplification [17]. A detailed study of this frequency chirping effect [18] shows clearly that high precision experiments must be performed with continuous wave light beams.

4

3

Bernard Cagnac

L o n g i t u d i n a l or C o l l i n e a r I r r a d i a t i o n of an A t o m i c Beam

In Paris, the choice was to work in a more conventional range of wavelength, and that permitted to observe in 1985 the first two-photon transition produced with a cw light b e a m in an atomic beam of hydrogen [19], developing the collinear geometry, which has been adopted now in all two-photon precision experiments: the atomic b e a m is irradiated longitudinally, along its own direction, thus producing the m a x i m u m first order Doppler shift, for each of the two counter-propagating laser beams. It does not matter, as the first order Doppler shift is perfectly cancelled. Surely, this collinear geometry increases the very small residual second order Doppler shift; but it increases also largely the interaction time between atoms and light and reduces the transit time broadening below 10 kHz. T h e more conventional wavelengths were found by using the metastabte 2S level of H as the departure of the transitions (instead of arrival; cf. the magnified energy diagram in Fig. 2b). Figure 3 shows the heart of the experiments in Paris for transitions starting from the metastable 2S level of hydrogen: molecular hydrogen is dissociated in a water cooled radiofrequency discharge and the atoms effuse through a nozzle into a first vacuum chamber. A small fraction of them (10 -7) is excited to the metastable 2S state via electron b o m b a r d m e n t . Some of these atoms, deviated through an angle of about 20 ° as a result of the collisions, pass through an aperture into a second vacuum chamber. Another aperture

I Re~ord~ag I~ \

FeedbacA¢

Towards the pumps

Fig. 3. Experimental set up for two-photon transitions from the metastable 2S level, with the longitudinal or colfinear irradiation of the atomic beam.

Two-Photon Method for Metrology in Hydrogen

5

about 50 cm downstream is followed by a detection region in which an electric field is applied to quench the metastable atoms by mixing the 2S and 2P states of neighbouring energy (cf. Fig. 2c) as in Lamb's experiment: the Lyman-~ intensity emitted from the 2P state, recorded with photomultiplier tubes, is proportional to the number of the atoms in the metastable 2S state. The excitation of the resonance is signalled by the depletion of the population of the 2S state as most of the excited atoms fall down in energy to the ground level directly or through the 2P level, avoiding the metastabte 2S state. The apertures help to define the atomic trajectories, a knowledge of which is essential to the interpretation of the atomic spectra. A continuous wave tunable laser propagates along the central axis of the atomic beam. There are, in fact, two intense laser beams as the entire beam apparatus is located inside a high finesse optical cavity. Resonant coupling of the laser light into a TEM00 mode of this cavity ensures both complete overlap of forward and backward beams and enhancement of the optical power available to excite the atoms, both of which are necessary in order to achieve a decent two-photon excitation rate. In the first experiments [19] a dye laser was used, though now a titanium::sapphire laser has replaced it yielding higher power and better frequency stability. In this way, the energy spectral density in the interaction region is equivalent to powers of up to 100 W in a bandwidth of a few kHz. For experiments starting from the 1S ground level, the electron bombardment does not exist; and the hydrogen atoms prepared in the lateral discharge are guided through a Teflon tube [20] to a nozzle, which is a narrow channel centred on the axis of the laser beam. In the case of the transition to the 2S level, like e.g. in Garching, these metastable atoms are also detected at the end of the beam by electrical quenching [21].

4

Light

Shifts

and

Two-Photon

Line

Shapes

A detailed understanding of the line shape is crucial to the success of these experiments. In fact, the most important problem concerning the lineshape comes from inhomogeneous light shifts. When an atom crosses the laser beam, it experiences a varying light intensity and hence a varying light shift. In some instances, this can produce asymmetrical line shapes. However, one can show that if the two-photon transition is far below saturation (two-photon transition probability p(2) ~ < Fe) the light shifts may be much smaller than the natural width F~ [8]. In such cases it is sufficient to work with different optical powers and to extrapolate to zero power. Close to saturation and in precise metrology, one is forced to calculate the light shifts exactly. This was done for example in the experiment on atomic hydrogen performed in Paris on the 2S-nS/nD transitions. The excitation geometry, taken into account in the computer calculation, is indicated in Fig. 4 (the atomic beam is larger than the laser beam). The lineshapes are catcu-

6

Bernard Cagnac o n e particular atomic trajectory

lStdial:)h.

2mldiaph. ,

Fig. 4. Detailed geometry of atomic trajectories through the Gaussian laser beam, taken into account in the computer calculation.

L

lated by summing the contributions of all possible atomic trajectories crossing the laser beams, taking into account the Gaussian distribution of the light power. Figure 5 shows two examples of such lineshapes for the 2S1/2-10S1/2 and the 2S1/2-10D~/2 transitions. As explained previously, these signals appear in absorption. Both signals have asymmetrical lineshapes which are well modelled by calculations. After adjustment of the theoretical curve with the experimental points, the computer is able to calculate the precise true (i.e. unshifted) position of the line [22]. As an illustration we compare in Fig. 6 for each value of the light power: - this true position of the transition (laser frequency) given by the computer calculation after correction of the light shifts - the "brute" half maximum centre of the experimental line The extrapolation of "brute" centres is not exactly linear, whereas the calculated positions are well independent of the light power, which confirms the validity of the theoretical model.

n(2S) n(2S)

(a)

(b) )

! atomic frequency

i

i

i

I

atomicf~uency

Fig. 5. Experimental recordings of two-photon transitions 2S-10S and 2S 10D showing the red-shift for one and the blue-shift for the other in agreement with theory (points are experimental; full lines are theoretical).

Two-Photon Method for Metrology in Hydrogen

7

H a l f - m a x i m u m center of the line 1

- = =:=:-:-::-

....... ! ....... ....... T/i

....... i .......

position

o

<

Light power Fig, 6. Control of the light shift: for each value of the light power inside the Fabry Perot cavity (horizontal axis) the computer indicates on the frequency vertical axis: * small stars: "brute" half maximum centre of the experimental recording, o small circles: corrected "true" position deduced from the theoretical curve.

In the experiments from the 1S ground level, it is not possible to obtain such high power in the ultra-violet cw light b e a m (some m W only), and the two-photon transitions are far below the saturation, corresponding to negligible light shifts. The two-photon transition probability is also very small, but it is applied to the big number of atoms in the ground level and gives enough signal. In the case of the 1S-2S transition, the theoretical linewidth is so small (some Hz) that the width is determined in practice by the transit time of the atoms in the light wave. The team in Garching has strongly reduced the 1S-2S line width by cooling the nozzle from which the atomic b e a m originates, thus reducing the atomic velocities [23], as illustrated in Fig. 7. The shift and asymmetrical shape observed in Fig. 7 is due to the seco.

-~ ~

S~-3skHzj

( =~K

oE

I

-200

-150

-1GO

-50

0

u'v deluning in kHz

50

1C~

Fig. 7. Recordings of the 1S-2S two-photon transition in Garching [23] in a cooled atomic beam, showing the asymmetry due to the second order Doppler effect (points: experimental; full line: theoretical).

8

Bernard Cagnac

ond order Doppler shift and can be exactly modelled. The width decreases as the atomic velocity v, whereas the second order Doppler shift decreases proportional to v 2, the square of the velocity; this is the reason why this shift seems to disappear at the lowest temperature. In all cases, it is possible to determine the true position of the line centre with a precision which is a small fraction of the linewidth (better than one hundredth; probably one thousandth), permitting metrologic measurements. 5

Two

Types

of Experiments

- Transitions

in an

Integer Ratio In fact two types of experiments can be distinguished in optical spectroscopy of hydrogen: 1) Absolute wavelength or frequency measurements of the transitions by comparison with an optical standard, with the aim of determining the Rydberg constant (see next section). 2) Owing to the simplicity of the Rydberg law, governing the energy levels of hydrogen, it is possible to compare directly the frequencies of some transitions which would be in an integer ratio, measuring without any standard the small difference between the high frequency and the harmonic of the low frequency. This second type of experiments permits to determine experimentally the small corrections calculated in the theory, coming from the quantum electrodynamics (Q.E.D.) or from the electron penetration inside the finite volume of the nucleus. These two corrections are particularly important for S levels (l = 0, with a significant probability density to find the electron inside of the extended attraction centre) and are responsible for the energy difference between S and P levels corresponding to the same total quantum number j (see Fig. 2c): they are the so-called Lamb shifts [24]. The Lamb shift of the 2S metastable level could be determined with precision in radiofrequency measurements of the separation 2S-2P [25], whereas the 1S ground level is isolated and its Lamb shift can be only deduced indirectly from the total energy of the 1S level. Table 1 indicates transitions, whose frequencies are in a ratio 4 and which are investigated in various groups in experiments with the aim of measuring the 1S Lamb shift (see Fig. 2a and 2b). If one compares a two-photon transition with a one-photon transition, the two experimental frequencies are only in the ratio 2. If both transitions are two-photon transitions, their frequencies are also in the ratio 4, i.e. one must realise frequency doubling two times in succession for the comparison of frequencies [28,30]. Figure 8 shows the experimental set-up of the most recent experiment, in Paris, for comparison of the 1S-3S transition to 2S6S/6D. Two ring cavities (astigmatically compensated) are used to enhance

Two-Photon Method for Metrology in Hydrogen ~rsttransifion

Z1

Second transifion

Z2

Place, date

9

Reforence

ITwo-Photon IS-2S 243 nm

I photon 2S-4P (saturation)

486 nm 'Stanford 1980

[261

Two-Photon 1S-2S 243 am

1 photon 2S-4P

486 am Oxford 1992

[27]

(crossed beams)

Two-Photon IS-2S 243 am

2-photon-2S-4S/4D 972 nra Garching 1992 and 1994

[281

Two-Photon IS-2S 243 nm

1 photon 2S-4P

486 nm Yale 1995

[291

2-photon 2S-6S/6D 820 nm Paris 1995

[301

(crossed beam)

Two-Photon IS-3S 205 nm

Table 1. Comparison of wavelengths in some hydrogen transitions.

the doubling efficiency in the two successive doubling crystals LBO (from IR to blue) and BBO (from blue to UV). The apparatus for the 2S 6S/6D transition is the same as the one shown in Fig. 3; the apparatus for the 1S3S transition uses a beam source analogous to the one in Garching, but the detection must utilise the red Ha line emitted by the atoms excited to the 3S state and is more noisy (sensitive to parasite light). The values of the 1S Lamb shifts obtained in the most recent experiments are in agreement within the error bars:

L1s --8 172.860 (60) MHz Lls ----8172.827 (51) MHz Lls =8 172.798 (46) M n z

[28] [29] [30]

The interpretation of these values has benefited of the recent advance in theoretical calculations (see section 8). Previously, these values of the 1S Lamb shift seemed to be in favour of the smaller value of the r.m.s, radius of the proton V / ~ = 0.805fm [31]. With the recent theoretical advance in the Q.E.D. calculation, the larger (and more recent) value ~ = 0.862fm [32] is in better agreement. The difficulty in interpreting the Lamb shift value arises from the fact that it mixes two independent corrections (Q.E.D. and nuclear size). One solution would be to work with another nucleus; this is the interest of the project now developed in Yale University to measure the Lamb shift in the He + single ion, where the nucleus (the a particle) has a well known radius [33]. Nevertheless, it is not clear if the effect of the penetrating electron in a composite nucleus can be represented with a single parameter x/(r~).

~0

Bernard Cagnac

UV photomullipliers ~ 82Ohm ~]

I TiS:lLase( [

e~

0 41{)am

.~LBO

v

]

,,~,s,

2S-6S/6D apparatus

205 nm ~_ It trY6Inn Filler--~ Pholomulliplier--~J

1S-3S apparat.s Fig. 8. Experimental set-up for direct comparison of the two-photon transitions 2S-6S/6D (~ = 820nm) and 1S-3S (A = 205nm).

6

F r e q u e n c y Calibration - T h e R y d b e r g C o n s t a n t

In the classical spectroscopic measurements, you compare the wavelength of your particular transition with the well known wavelength of an optical standard using an interferometric apparatus (Miehelson or Fabry Perot). Everybody knows the current optical standards used in the laboratories, i.e. laser stabilised by saturated absorption on a Doppler-free molecular line: I2 near 633nm, or CH4 near 3,39ttm with a He-Ne laser; OsO4 near 10/zm with a CO2 laser. Another secondary standard used in these experiments is the Te2 molecule near 486nm (wavelength of the Balmer-fl line) with a Coumarine dye laser [34]. The absolute measurement of wavelengths or frequencies (it comes to the same thing, as now the light velocity c is fixed) of the hydrogen transitions permits the determination of the Rydberg constant. But one must not forget that, in order to deduce the Rydberg constant from the measured transition frequencies, a theoretical interpretation is necessary, which comprises many small corrections (see table 2) As we have seen in the preceding section, these corrections are particularly important in the S levels (quantum number l=0) which are involved practically in all experiments. In the case of transitions starting from the metastable 2S level, it is possible to use the experimental value of the 2S-2P Lamb shift [25] and to reduce the theoretical calculations to the small corrections in P or D levels; these transitions are more convenient for the Rydberg determination. The transitions from the 1S ground level are better suited for the measurement of the 1S Lamb shift and the deduction of the proton radius, as explained in the preceding section. Figure 9 compares the values of the Rydberg constant determined since the last adjustment of fundamental constants [35]. The horizontal scale of

Two-Photon Method for Metrology in Hydrogen I0¢I237.31~00 200 I I

400 '

cod,,~ I

•

"r~O-eHOVONaS.~n0DI

:',

O/~.pI.IOTON[2S-3P

"~

]

L2~--4p

T'WO-PHOTON

m.zs,fl L

(:~-1 a x o ~

[3s]

I Paz~.sl ~ :

I

[

Yale Yale

t I

•!

1986 1987

i

~- ~ i,#,

Paris 1988

. . . . . -,, -...

_,~....... ~__," ,ff "--..~ vwo-P~ovoNm-zs ~ I

; 1

TWO-~0TON

~-es~

1 -

I

1992_ D6]

i •

I

tl i _,,,: - , , :

Garching199")Faro1988 c o r r e c t e d

[]el Fig. 9. Values measured for the Rydberg constant since 1986. The horizontal scale (in cm -1) is multiplied C39| two times: by 5 and by 8, showing [41 ] the increasing precision with time (the numbers in brackets refer to the references). [4tl ~7I [38I

I Seaz~rd t ~

!

iI_

TWO-PHOTON 2S-SD/10DII2D x8

D

600 800 i,, I i)

Z ,., , ~' x5

19~

11

,

8Ds/z

18D3/z , ~ ~ 2 ("

[4~1

Mean .J

the Rydberg constant is magnified twice (× 5 and x 8) from the top to the b o t t o m of the figure, demonstrating the increasing precision with time. When you exceed the level of precision of 10 -1°, it becomes very difficult to take into account exactly the corrections unavoidable in the interferometric measurement comparing different wavelengths: the reflective phase shift, due to different penetration depths of the evanescent waves inside the coating of the mirrors; the Fresnel phase shift associated to different curvatures of the wavefronts. These effects can be almost negligible only if the two wavelengths are very close to each other. Therefore, for the highest level of precision, it is better to perform direct frequency comparisons with heterodyne techniques. This was the case in the latest measurement in 1993 [44]: one mixes in a lithium iodate crystal (LiIO3) the hydrogen transition frequency (2S-8S/8D, A = 778 nm i.e. u = 385.2 THz) and a standard He-Ne laser stabilised on CH4 (A = 3,39#m i.e. u = 88,4THz), producing the frequency sum 473,6THz, which falls in the vicinity of the frequency of the standard He-Ne laser stabilised on iodine at A = 633 am. The gap of 89GHz between this standard and the frequency sum can be bridged by a triple mixing of the two optical frequencies and the microwave of a Gunn generator in a Schottky diode. These measurements utilised at the beginning the frequency value of the iodine standard determined for the first time in 1983 at the U.S. National Bureau of Standards (NBS) [45]. Its precision has been improved with a new determination in the Observatory of Paris in 1992 [46]. The good agreement between the results obtained in Paris and in Garching using different standards gives confidence in the value obtained for the Rydberg constant: R ~ = 109 737.315 684 1 (41) R ~ = 109737.3156834 (24)

[42] [44]

12

Bernard Cagnac 5D $/2

5Sl/2(Ft=l)-5D~(Fc=3) BTRb T']8 n m

.....

5 P+a / 5

778nm /

P1//

420 nm

P 5Sm+l ._~ V~ 85Rb

S7Rb

1MHz I

!

(a)

I

I

fr~lU~-y

,fits--> Z ~3 +~z.--->1 + 2

(b)

Fig. 10. Principle of two-photon standard in rubidium: a) energy diagram ; b) experimental recording: UV fluorescence at 420 nm versus the laser frequency (points: experimental - full line: fitted Voigt profile).

The precision of the experiment in Paris is limited by the precision of the iodine standard (the width of the Doppler-free line of iodine is rather large, around 5 MHz).

7

D e v e l o p m e n t of a N e w Optical Standard

The problem is to find a new optical standard with a smaller linewidth than iodine, and a wavelength better fitted to the ones used in the hydrogen experiments. The two-photon method has given such a standard with the rubidium atom. Figure 10a shows the energy diagram of the lower levels of the rubidium a t o m with the two-photon transitions. Numerous two-photon transitions have been observed in rubidium, at the beginning of the two-photon method, reaching excited levels with n = 11 to 32 [47] and then up to n = 124 [48]. The lower excited level could not be attained with the dye lasers at that time. Now the Ti::Sapphire laser or the laser diodes working in the near infra-red permit to observe the lower levels. The 5S-5D transition at )~ = 7 7 8 n m is particularly intense because the laser frequency is close to the resonances with 5P1/2 (795 nm) or better 5P3/2 (780 nm). T h a t permits to produce the two-photon transitions with low intensity, without focusing the laser b e a m and to reduce the light shifts to a negligible level. The detection is very easy by collecting the fluorescence light at A = 420 nm in the near UV, emitted in the cascade from the 5P level. Figure 10b shows a particular line chosen inside the hyperfine multiplet [49]; the experimental points are fitted with a Voigt profile. This typical recording is obtained with 100 m W light power each way in an area rt w 2 = 10 m m 2. The width of 500 kHz is ten times smaller than the iodine lines and the signal-to-noise ratio is of the order of 400-500. We have verified

Two-Photon Method for Metrology in Hydrogen ECL

13

~-~merphic plum

Fig. 11.

~eJding

to lock-in ampllfler

Experimental set-up of the two-photon standard (ECL: Extended Cavity laser diode with grating for wavelength selection).

that the position of the line is quasi-independent of the Rb vapour pressure (from 10 -4 Torr to 2.10 -5 Torr by varying the cold point of the Rb cell from 90 °C to 50 °C) and to the change of the cell. The only significant shift is the light shift: in this case, it is much smaller than the line width; and, working with increasing light power up to 600 mW, we have verified that it can be linearly extrapolated. The residual light shift in the case of 100 m W (case of Fig. 10b) is 2 kHz only and can be stabilised with the light intensity. The second step was to build an autonomous standard [50] following the scheme of Fig. 11: The laser diode is mounted in an extended cavity with a grating in order to assure a single mode oscillation. The shape of the light beam is corrected by an amorphic prism before irradiating the Rb cell through 2 or 3 Faraday isolators preventing a spurious feedback to perturb the laser diode. A lateral lens with big aperture collects the UV photons emitted in the return cascade from the 5P level. The two-photon resonance could be observed by simply reflecting the laser beam back onto itself with a mirror without the usual Fabry Perot cavity; but the cavity is better to assure a perfect coincidence of the incoming and return beams. The stabilization is obtained with two servo loops: the fast one reacts on the laser diode current while the slow one controls the piezoelectric transducer (PZT) supporting the grating (Fig. 11). In a set of preliminary measurements the frequency stability has been controlled; the square root of the relative Allan variance is: a(2, r ) / u = 3.1013/V/~

up to

r = 1000sec .

(1)

It is ten times better than a He-Ne laser stabitised on iodine. A preliminary calibration has been done [49], relative to the iodine standard, using the same frequency chain as for the hydrogen experiment [44]. The precision of some kHz is limited by the iodine standard. A new calibration is in development in the Laboratoire Primaire des Temps et Fr6quences (L.P.T.F.) in the Observatory of Paris, using the fact that the rubidium frequency is close to the 13th harmonic of the CO2 laser stabilised on OsO4.

]4

Bernard Cagnac

An optical fiber (less than 3 km) between the Observatory and our university permits to exchange the lightwaves of our Rb standards and to control their stability; it will permit to transfer the calibration without displacement of any standard. The calibration of the hydrogen transition 2S-8S/8D will be obtained by the triple mixing on a Schottky diode of the Ti::Sapphire light wave used for hydrogen, the rubidium light wave, and a 40GHz klystron adjusted to bridge the gap. Surely, that will permit to surpass the precision of 10 -11 . Up to 10 -12 ? - We will see. Anyway, except for the particular transition 1S-2S, taking into account the natural line widths of the hydrogen levels, it seems hardly likely that the experimental precision can be pushed largely beyond 10 -1~. Surely, the highly excited Rydberg levels (particularly "circular" states with l = n - t) are long living; but they are also very sensitive to residual parasitic fields and not adapted to metrology. 8

Confrontation

with

Improvements

of the

Theory

Admitting that the experimental precision can be pushed in the future to the 10 -12 level, these measurements can be interpreted only if the theory reaches the same level of precision, which is not yet the case. Where has the theory arrived now? All calculations start from the formula obtained by Dirac from his relativistic equation [51] modified with the reduced mass/~ = m/(1 + re~M) in place of the true mass m of the electron (M is the proton mass). This modified formula has no exact theoretical justification, as the classical centre of mass has no sense in relativity; nevertheless, it permits to take into account to first order the problem of the nuclear motion. Therefore the binding energy E,~j of the level of quantum numbers n and j is given at zero order by the formula: EnJ :

#c2Z2~ 2

I,.gC2

1

V with

l j(

= J + 2 -

J+

- (Za)2

~

,zo,2

2j + 1

(3)

Ze is the charge of the nucleus (e the elementary charge). (~ = e2/4aotic ,~ 1/137 is the fine structure constant, which gives the order of magnitude of the ratio v/c of the electronic velocity v to the light velocity c. In fact the relativistic calculation and the nuclear motion corrections are intimately intricated and one has to perform an expansion simultaneously in powers of two independent variables m / M and Z a . T h a t explains the

Two-Photon Method for Metrology in Hydrogen

15

five lines in the top of table 2; the well known terms of first order in m/M or in ( Z a ) 2 do not appear in this table, as they are taken into account inside the Dirac formula. This table is intended for non specialists; and for simplification, when we give the order of each term in the expansion, we choose as term of zero order, i.e. as unity, the binding energy E,j (Eq. (2)), which is measured in experiments. (The specialists choose as zero order term the mass energy mc2; i.e. each term must be multiplied by (Za) 2 if you compare with the theoretical formulas, indicated by their reference numbers in some theoretical review papers [52-55].) The radiative corrections, coming from the quantum electrodynamics, necessitate an other expansion in power of c~, which occupies the central part of the table 2. Among these corrections one distinguishes the self-energy (S.E.), the vacuum polarization (V.P.) and the effect of the anomalous magnetic moment, which affects the spin orbit constants in the P and D levels. The most i m p o r t a n t terms correspond to the self-energy in the S states (l = 0) in which the electron has a significant probability density in the nucleus; that raises the degeneracy in l, characteristic for the Dirac formula and produces the so called L a m b shift between the $1/.~ and P1/2 levels (see Fig. 2c). It is yet necessary to calculate the crossed corrections, i.e. radiative corrections to the recoil terms, or recoil corrections to the L a m b shift. The term of lowest order in (m/M)o~(Zo~)2 does not appear in table 2, because it is taken into account by multiplying all radiative corrections with the factor (p/rn) 3 which represents the modification of the wave function at the origin. The terms of higher orders are small but not negligible, and their calculations is not yet quite clear [63,64]. Finally, one must add the correction due to the finite size of the nucleus, which is particularly important for the S levels (l = 0; important electron probability density in the nucleus) and depends on a badly known p a r a m e t e r the r.m.s, proton radius x/(r2). All these calculations use the values of the two parameters rn/M and a. But, since the last adjustment of the fundamental constants [35], the precision has been strongly increased on these two parameters: the electron to proton mass ratio m/M in Penning trap experiments [65], the fine structure constant a in measurement of (g-2) [66] and progresses in its calculation [67]. The present uncertainty on these parameters does not raise problems on the calculation of the energy levels before the 10 -12 level of precision. The limitation comes from the high order necessary in the expansion. The number of terms (or Feynman diagrams) which appear in the expansion increase strongly with the order of the expansion; when you come to the fifth order it can approach a hundred. The calculation is not easy and all terms to the fifth order gave rise in the past years to controversies; the most of which seem now to be settled, owing to very recent advances since two years. These most recent papers are referenced in table 2. The problem arises from the fact that the coefficients found inside some of these terms can be as big as 100, and then these terms are equivalent to

16

Bernard Cagnac

Physical effec~ involved

[S2l 'Order in d~e

~paeston

On;/m"o f

ma~

a~,! -7.=1

n:duced rna~

1

1

E ~ ....

Review- P a l m {531 {5,,1 Sa~tei~ y~ 1990

M~ra" 1995

~admda 1995

(39) (40)

, (19)

(41)

(34)

w,et,k

(2-.4)

(2-7) ~(2-3)

Rela~t~"

(55]

pb.oeon

and

,.,o ~wo

Recoil

(24)

a~xerams

(2-9)

photon

(2-73

(43)

OC.gnms

o~e

loop Radiam,e

~ (Za)~

I0-~

-a(Za)'

to+

~(z~)"

1o-9

(2-10) t2-11)

(C-t)

(2-5) ),a,b.¢

(44)

(36)

(62) • (45) r (50)

S.F_ {21) (23)

$.E.

~' V,P.

[,t61 [4'71 v.P.

(2n

[.561

[5"71

{ss]

[5[[ [S3I oml.~.-tlo~

-a(Za) ~ fc

two Q.E.D.

[59]

to-n

2

(2-1'2)

( ~ ) (Za) z

10-9

(~)~(Za) J

10-|0

(29)

(2-13, (2-153 r (61)

loop diagrams

01)

[61]

(32) (33)

[62]

(37) ~ (38)

{~l [6,t}

(40)

[~]

: Crossed oor~c~ons

m a Za)* (~)~'(

(2-8)

I0"'

(ReL only)

radiative and mvoil m 2~. g l x :

Fimte s g e of nucleus

(2-30)

(2-11)

(64)

(40) I

Table 2. Summary of the power expansion ia the calculation of energy levels.

the terms of preceding order. This was the case in particular for the term in ( a / l r ) 2 ( Z a ) 3 (two loop diagram in the radiative correction) which, after exact calculation, was found ten times bigger than expected, and modified consequently the interpretation of the 1S L a m b shift (see section 5). It remains yet a controversy on the crossed correction of order (re~M)(a/Tr)(Zoo)3 and most of the sixth order terms are unknown. As a consequence, one can now hardly await the 10 -11 precision from the theory.

Two-Photon Method for Metrology in Hydrogen

17

Another limitation comes from the large uncertainty on the proton radius, ( V / ~ , which appears in the nuclear finite size correction (last line of table 2). In fact, the finite size correction and the Q.E.D. corrections are intricately mixed in the exact determination of S levels; and the advances in the Q.E.D. correction will permit to deduce a value of the proton radius. If both theoreticians and experimentalists attain in the future the level of precision of 10 -12 on the totM energy (or a few 10 -7 on the 1S Lamb shift) the proton radius could be determined with a precision better than 1%. On the other hand, the determination of the Rydberg constant can be in some respects independent of the proton radius (and partially of the Q.E.D. corrections) if one utilises some particular combinations of transitions, using the fact that the finite size correction scales as 1/n 3 with successive n levels with the same orbital quantum number I (that is the same for a big part of the Q.E.D. corrections). As it was noticed in [62], for example, the combination (Lls - 8L2s) of the 1S Lamb shift and the 2S Lamb shift is small and quite independent of the proton radius; and the present limit of the theory could nevertheless permit a determination of the Rydberg constant at the 10 -12 level, if the experimentalists continue their advances.

9

Conclusion

At the present time, experimentalists and theoreticians have attained the same level of precision of a few 10 -11 in the determination of the energy levels of hydrogen. One proof is given by the coincidence at this level of precision of the values obtained for the Rydberg constant R ~ from different transitions. Taking into account the big progresses which have been accomplished in the past few years, in experiments and in theory it is possible to hope for further advances in the next years up to a level of 10 -12 with its consequence for the knowledge of the proton radius. The comparisons with other hydrogenic systems (ionic He +, muonium, positronium, ...) surely will help to improve the knowledge of the Q.E.D. corrections and participate to this advancement; these problems are explained in other papers. Nevertheless, it is not sure that it will be possible to pass beyond the experimental limit of 10 -12. Certainly, the exceptional precision of the particular transition 1S-2S will be used in the future for an optical frequency standard with very slow atoms from an hydrogen trap [68]. But the theoretical interpretation, depending on two parameters, R ~ and X / ~ , requires two independent measurements with the same level of precision (supposing that c~ and m / M will follow the same advance...). Anyway, the Rydberg constant R ~ , determined with 10 -12 precision, will remain a corner stone of the adjustments of fundamental constants, far in advance compared to other constants. R ~ is a combination of three fundamental constants m, h and e; two of them h and e are linked by the fine

18

Bernard Cagnac

structure constant a with a precision, better than 10-8; but a third relation at the same level of precision is missing; and m, h and e are known individually with a precision hardly better than 10 -6. Will some advance in the Josephson effect (depending on h/e) be able to bridge this gap? Let me conclude with a personal feeling of perplexity when looking at table 2, which summarizes the calculations: is it not surprising that any correspondence with the reality is obtained after so numerous pages with complicated integrals in the scientific journals and after so many hours of abstract computer work? Nevertheless it works! All right, we believe that physics explains the world and provides us with an understanding of reality i.e. reduces apparent complexity to simplicity. But in what sense is it possible to speak of explanation and understanding when we compare these long and tedious calculations with the spontaneous functioning of any simple electron in all water molecules of the ocean, or in all atoms of the intergalactic clouds?

Note added in proof." The experiments with the new Rb standard and the optical fiber (described in section 7) obtained very recently a new result at a level of 10 -11 for the Rydberg constant R ~ = 109737.3156859 (10) [69].

References [1] Ghppert-Mayer M., Ann.Phys., Lpz 9, 273 (1931) [2] Hughes V.W. et Grabner L., Phys. Rev. 79, 314 and 819 (1950) [3] Brossel J., Cagnac B. et Kastler A., C.R.Acad.Sci.Paris 237, 984 (1953), and J.Physique 15, 6 (1954); Kusch P., Phys.Rev.93, 1022 (1954) and 101, 1022

(1956) [4] Kaiser W., Garrett C.G.B., Phys.Rev.Lett. 7, 229 (1961) [5] Abella I.D., Phys.Rev.Lett. 9,453 (1962) [6] Bonch-Brnevich A.M., Khodovoi V.A., Khronov V.V., JETP Lett. 14, 333 (1971); Agostini P., Ben Soussan P., Boulassier J.C., Opt. Comm. 5,293 (1972) [7] Vasilenko L.S., Chebotaev V.P. and Shish£v A.V., JETP Lett. 12, 113 (1970) [8] Cagnac B, Grynberg G. and Biraben F., J.Physique 34, 845 (1973) [9] Cohen-Tannoudji C., Ann.Phys. 7, 423 et 469 (1962) Alexandrov E.B., BonchBruevich A.M., Kostin N.N., Khodovoi V.A., JETP Lett. 3, 53 (1966) [10] Cohen-Tarmoudji C., Dupont-Roc J., Phys. Rev.A 5, 968 (1972) [11] Biraben F., Cagnac B. et Grynberg G., Phys.Rev.Lett. 32, 643 (1974) [12] Levenson M.D., Blembergen N., Phys.Rev.Lett. 32, 645 (1974) [13] H£nsch T.W., Harvey K., Meisel G. and Schawlow A.L., Optics Comm. 11, 50 (1974) [14] Grynberg G., Cagnac B. and Biraben F., in: "Coherent Non Linear Optics" (Springer) Topics in Current Physics, vol. 21, p 111 (1980) [15] Baklanov E.V. and Chebotaev V.P., Opt.Comm. 12, 312 (1974) [16] H£nsch T.W., Lee S.A., Wallenstein R. and Wieman C., Phys. Rev.Lett. 34, 307 (1975); Lee S.A., Wallenstein R., H/insch T.W., Phys.Rev.Lett. 35, 1262 (1975); Wieman C., H£nsch T.W., Phys.Rev.A. 22, 192 (1980) [17] Tr~hin F., Biraben F., Cagnac B. and Grynberg G., Opt.Comm. 31, 76 (1979)

Two-Photon Method for Metrology in Hydrogen [18] [19] [20] [21]

19

Danzmann K., Fee M.S. and Chu S., Phys.Rev. A 39, 6072-3 (1989) Biraben F. and Julien L., Opt.Comm. 53,319 (1985) Walraven J.T.M. and Silvera I.F., Rev. of Sci.Instr. 53, 1167 (1982) Zimmermann C., Kallenbach R. and H~izlsch T.W., Phys.Rev.Lett. 65, 571-4

(1990) [22] Oarreau J.C., Allegrini M., Julien L. and Biraben F.J., Physique 51, 2263, 2275 and 2293 (1990) [23] Schmidt-Kaler F., Leibfried D., Seel S., Zimmermann C., K6nig W., Weitz M. and HKnsch T.W., Phys.Rev. A 51, 2789 (1995) [24] Lamb W.E. Jr. and Retherford R.C., Phys.Rev. 72, 241-3 (1947) [25] Lundeen S.R. and Pipkin F.M., Metrologia 22, 9 (1986); Hagley E.W. and Pipkin F.M., Phys.Rev.Lett. 72, 1172 (1994) [26] Wieman C.E. and H~insch T.W., Phys.Rev. A 22, 192-205 (1980) [27] Thompson C.D., Woodman G.H., Foot C.J., Hannaford P., Stacey D.N. and Woodgate G.K., J.Phys.B.: At.Mol.Opt.Phys. 25, L1-4 (1992) [28] Weitz M., Schmidt-Kaler F. and H£nsch T.W., Phys.Rev.Lett. 68, 1120-3 (1992); Weitz M., Huber A., Schmidt-Kaler F., Leibfried D. and H~insch T.W., Phys.Rev.Lett. 72, 328 (1994) [29] Berkeland D.J., Hinds E.A. and Boshier M.G., Phys.Rev.Lett. 75, 2470 (1995) [30] Bourzeix S., de Beauvoir B., Nez F., Plimmer M.D., de Tomasi F., Julien L., Biraben F. and Stacey D.N., Phys.Rev.Lett. 76, 384 (1996) [31] Hand L.N., Miller D.G. and Wilson R., Rev.Mod.Phys. 35, 335 (1963) [32] Simon G.G., Schmitt C.H., Borkowski F., Walther V.H., Nucl.Phys. A 333,

381 (1980) [33] Sick I., Phys. Lett. B 116, 212 (1982) [34] Barr J.R.M., Girkin J.M., Ferguson A.I., Barwood G.P., Gill P., Rowley W.R.C. and Thompson R.C., Opt. Comm. 54, 217 (1985) [35] Cohen E.R. and Taylor B.N., Rev.Mod.Phys. 59, 1121 (1987) [36] Biraben F., Garreau J.C. and Julien L., Europhys.Lett. 2, 925 (1986) [37] Zhao P., Lichten W., Layer H.P. and Berquist J.C., Phys.Rev.A 34, 5138

(1986) [38] Zhao P., Lichten W., Layer H.P. and Berquist J.C., Phys.Rev.Lett. 58, 1293 (1987) [39] Beausoleil R.G., McIntyre D.H., Foot C.J., Hildum E.A., Couillaud B. and H£nsch T.W., Phys.Rev./k 35, 4878 (1987) [40] Boshier M.G., Baird P.E.G., Foot C.J., Hinds E.A., Plimmer M.D., Stacey D.N., Swan J.B., Tate D.A., Warrington D.M. and Woodgate G.K., Nature 330, 463-5 (1987) [41] Biraben F., Garreau J.C., Julien L. and Allegrini M., Phys.Rev.Lett. 62,621 (1989) [42] Andreae T., KSnig W., Wynands W., Leibfried D., Schmid-Kaler F., Zimmermann C., Meschede D and H£nsch T.W., Phys.Rev.Lett. 69, 1923-6 (1992) [43] Nez F., Plimmer M.D., Bourzeix S., Julien L., Biraben F., Felder R., Acef O., Zondy J., Laurent P., Clairon A., Abed M., Millerioux Y. and Juncar P., Phys.Rev.Lett. 69, 2326-9 (1992) [44] Nez F., Plimmer M.D., Bourzeix S., Julien L., Biraben F., Felder R., Millerioux Y. and De Natale P., Europhys.Lett. 24, 635 (1993)

20

Bernard Cagnac

[45] Jennings D.A., Pollock C.R., Petersen F.R., Drullinger R.E., Evenson K.M., Wells J.S., Hall J.L. and Layer H.P., Opt.Lett. 8, 136 (1983) [46] Acef O., Zondy J.J., Abed M., Rovera D.G., G6rard A.H., Clairon A., Laurent P., Mill6rioux Y. and Juncar P., Opt. Comm. 97, 29 (1993) [47] Kato Y. and Stoicheff B.P., JOSA 66, 490 (1976) [48] Stoicheff B.P. and Weinberger E., Can.J.Phys. 57, 2143 (1979) [49] Nez F., Biraben F., Felder R. and Millerioux Y., Opt.Comm. 102,432 (1993) [50] Millerioux Y., Touhari D., Hilico L., Clairon A., Felder R., Biraben F. and de Beauvoir B., Opt.Comm. 108, 91 (1994) [51] Dirac P.A.M., Proc.Roy.Soc. A 117, 610 (1928) [52] Erickson G.W., J.Phys.Chem. Ref.Data 6, 831 (1977) [53] Sapirstein J.R. and Yennie D.R., in "Quantum Electrodynamics", edited by T. Kinoshita (World Scientific, Singapore, 1990) [54] Mohr P.J., "Fundamental Physics" in Atomic, Molecular and Optical Physics Reference Book, Drake (ed.) (American Institute of Physics, 1996) [55] Pachucki K., Leibfried D., Weitz L., Huber A., Kgnig W. and H£nsch T.W., Lecture at NATO Advanced Study Institute (Edirne, Turkey) - September 1994 [56] Khriplovich I.B., Milstein A.I. and Yelkhovsky A.S., Physics Scripta T 46, 252 (1993); Fell R.N., Khriplovich I.B., Milstein A.I. and Yelkhovsky A.S., Phys.Lett. A 181, 173 (1993) [57] Pachucki K. and Grotch H., Phys. Rev. A 51, 1854 (1995) [58] Pachucki K., Ann.Phys. (N.Y.) 226, 1 (1993) [59] Molar P.J., Phys.Rev. A 46, 4421 (1992) [60] Eides M.I. and Grotch H., Phys.Lett. B 301,127 and B 308,389 (1993); Eides M.I., Karshenboim S.G. and Shelyuto V.A., Phys.Lett. B 312, 358 (1993); Eides M.I. and Shelyuto V.A., JETP Lett. 61, 478 (1995) [61] Pachucki K., Phys.Rev.Lett. 72, 3154 (1994) [62] Karshenboim S., JETP Lett. 79, 230 (1994) [63] Bhatt G. and Grotch H., Phys.Lett. A 58, 471 (1987) and Arm.Phys. (N.Y.) 178, 1 (1987) [64] Pachucki K., Phys.Rev. A 52, 1079 (1995) [65] Van Dyck R.S., Farnham D.L. and Schwinberg P.B., I.E.E.E. Trans.Instr.Meas. 44, 546 (1995) [66] Van Dyck R.S. Jr., Schwinberg P.B. and Dehmelt H.G., Phys.Rev.Lett. 59, 26 (1987) [67] Kinoshita T. and Lindquist W.B., Phys.Rev. D 42, 636 (1990); Kinoshita T., Phys.Rev.Lett. 75, 4728 (1995) [68] Cesar C.L., Fried D.G., Killian T.C., Polcyn A.D., Sandberg J.C., Doyle J.M., Yu I.A., Greytak T.J. and Kleppner D., Communication to Fifth Symposium on Frequency Standard and Metrology - Woods Hole, October 1995 [69] de Beauvoir B., Nez F., Julien L., Cagnac B., Biraben F., Touahri D., Hilico L., Acef O., Clairon A. and Zondy J.J., Phys.Rev.Lett. 78, 440 (1997); details can be found in the thesis of B6atrice de Beauvoir (Paris, unpublished).

High Precision Atomic Spectroscopy of Muonium and Simple Muonic Atoms Vernon W. Hughes Yale University, Physics Department, J.W. Gibbs, New Haven, CT 06520 USA

Over a period of about 30 years Gisbert zu Putlitz and his colleagues have studied - rapidly and often one by one - some 1015 m u o n i u m atoms, which is equivalent to the number of hydrogen atoms in a bottle of H2 gas with a volume of 0.01 m m 3 at a pressure of 1 atm.

1

Introduction

It's a great pleasure to be able to celebrate Gisbert's 65th birthday with this S y m p o s i u m on Atomic Physics Methods in Modern Research at this great university and in this beautiful city. I first met Gisbert here in Heidelberg in about 1960 when we held the annual Brookhaven Molecular Beams Conference organized by Bill Cohen of BNL which met first here at Heidelberg with Hans K o p f e r m a n n as our host and then at Bonn with Wolfgang Paul as our host. At Heidelberg there were two very competent young associates of Professor Kopfermann who handled the slide projector and related matters. One was Gisbert and the other was Ernst Otten. My next significant meeting with Gisbert was 5 or 6 years later when he visited Yale and we discussed the possibility of his coming to Yale to do research in atomic physics. T h a t Heidelberg meeting and Gisbert's later joining the Yale research faculty was surely one of the most fortunate occasions in my life. It was the beginning of a close scientific collaboration which continues after 30 years. The central theme of our scientific collaboration, which appealed greatly to both of us, was muonium, the #+e- atom, and other related topics in m u o n physics. During Gisbert's memorable stay at Yale research on m u o n i u m was very active for us at the Columbia University Nevis Synchrocyclotron Laboratory where muonium had been discovered in 1960 (Hughes et al., 1960; Hughes et al., 1970). We were involved principally in measurement of the hfs of m u o n i u m in its ground state by microwave magnetic resonance spectroscopy (Thompson et al., 1969; Crane et al., 1971)(Fig. 1). A quite different activity to which Gisbert contributed i m p o r t a n t l y was the organization at Yale of an International Conference on Atomic Physics

22

Vernon W. Hughes

Fig. 1. Experimental setup at Nevis in 1969 showing the Ar gas target and Navy barbettes for shielding.

(ICAP) held at NYU in New York City (ed. by Hughes et al., 1969). Fig. 2 shows Gisbert and L. Wilets at the Conference. Even during the Conference it was not clear that a second ICAP would occur. During a memorable boat trip around Manhattan Kim Woodgate and Pat Sandars agreed to host ICAP 2 at Oxford in 1970. Four years later in 1974 Gisbert hosted ICAP 4 in Heidelberg (ed by zu Putlitz et al., 1975). ICAP has become the principal international conference covering basic atomic physics broadly. The 15th ICAP has just been held in Amsterdam where celebration of the centennial year for the discovery of the Zeeman effect was part of the ICAP program. After the Nevis Synchrocyclotron Laboratory was closed down in the early 1970's and following a short period of muonium research at SREL in Williamsburg, Gisbert and I pursued our muonium experiments at the new meson factories, LAMPF and SIN, where eventually intensity increases of 103 to 104 compared to Nevis were achieved. Fig. 3 shows a three-dimensional electromagnetic coil system - built personally by expert machinist Gisbert zu Putlitz - and used in the study of muonium formation in gases, which was the first published physics research from LAMPF (Stambaugh et al., 1974). The topic I shall talk about is broader than, but includes, muonium. It is high precision atomic spectroscopy of muonium and simple muonic atoms. Several of us: Gisbert and Klaus Jungmann from Heidelberg and Malcolm Boshier and I from Yale have been writing a little article on this topic (Boshier et al., 1996a). My talk includes much of the material we have developed together, but is updated and also includes additional material.

Muonium and Simple Muonic Atoms

F i g . 2. Photograph taken at ICAP in 1968.

Fig. 3. Three-dimensional electromagnetic coil system at LAMPF.

23

24

2 2.1

Vernon W. Hughes

Precision Tests of Q E D Introduction

The principal scientific goal of high precision atomic spectroscopy is to test and study quantum field theory or, more specifically, quantum electrodynamics, the unified electroweak theory and quantum chromodynamics - all now encompassed within the modern standard theory of particle physics. In addition to testing fundamental theory, precision atomic spectroscopy also determines values of fundamental constants including particle masses and magnetic moments, R ~ , ~ and others. Real particles and atoms involve simultaneously the electromagnetic, weak and strong interactions and this often limits the sensitivity of the experimental tests of the theory. Thus for quantum electrodynamics effects of strong interactions or of hadronic structure are at present limiting importantly the QED tests in the simplest one- and two-electron atoms of H and He. With the purely leptonic atoms muonium and positronium the hadronic structure effects are avoided entirely. With simple muonic atoms the hadronic effects can be measured, and hence together with electronic atoms more sensitive tests of QED can be made. As examples we consider briefly the two classic and most important low energy tests of QED - the electron anomalous magnetic moment or g-2 value and the Lamb shift in hydrogen. 2.2

Electron A n o m a l o u s Magnetic M o m e n t and t h e F i n e S t r u c t u r e C o n s t a n t oL

The anomalous g-value a~ = (9 - 2)/2 has been measured to a precision of 3.4ppb (Van Dyck, Jr., 1990) and the QED radiative corrections (Kinoshita, 1990) have been calculated to an even higher precision as a result of a recent improved evaluation of the 6th and 8th order corrections (Kinoshsita, 1996). (This recent evaluation changed the earlier value (Kinoshita, 1995) for radiative corrections by about 50 ppb which is well outside the expected uncertainty.) In order to compare theory and experiment for ae a precise value for the fine structure constant a is needed. Figure 4 shows a plot of the most accurate determinations of a (Kinoshita and Lepage, 1990). In addition to the value recommended by CODATA in the "1986 Adjustment of Fundamental Constants" (Cohen and Taylor, 1987), these include muonium hyperfine structure (Hughes and zu Putlitz, 1990; Mariam et al., 1982), the neutron de Broglie wavelength (Kriiger et al., 1995), the ac Josephson effect (Williams, et al., 1989) (in combination with the gyromagnetic ratio of the proton in water), the quantized Hall effect (Cage et al., 1989), and finally the value obtained from a~ by equating the experimental value to the theoretical expression (Van Dyck, Jr., 1990; Kinoshita, 1996). Determination of a from the neutron de Broglie wavelength is an attractive new method. It involves

Muonium and Simple Muonic Atoms

25

measuring the ratio of Planck's constant to the mass of the neutron (h/mn) from its velocity and de Broglie wavelength and then obtaining cr using the Rydberg constant R ~ and mass ratios and has no theoretical ambiguities. For comparing a~(theor) with a~(expt) the quantized Hall effect now provides the best value with a precision of about 24 ppb. As shown in Fig. 4 agreement of the various ~ values is not too satisfactory. Such a plot is very useful to test our understanding of the different approaches.

CODATA

muonium

his

~

o

I

~

o

Inutlnm

o

i

~

, t

t ac,l & ~'p'

quantum

Hall

t

o

t

a~

-0.40

I

I

I

I

I

-0.30

-0.20

-0.10

0.00

0.10

( a "1 - 1 3 7 . 0 3 6 0 )

0.20

x 104

F i g . 4. R e c e n t d e t e r m i n a t i o n s of t h e fine s t r u c t u r e c o n s t a n t c~.

For a test of QED, rather than rely upon a determination of a involving condensed matter theory it would be preferable to determine a by a method involving the simplest theoretical assumptions such as that from h/mn, or at least from a simple atom which can be treated within QED. The approach involving hiM in which M is the mass of an atom is attractive if adequate precision can be obtained (Weiss et ah, 1993; Martinos e t a h , 1994). For hydrogen the hfs interval Av in its ground state is known experimentally (Ramsey, 1990) with a precision of better than 1 part in 1012 and At, is proportional to cz2. However, the theoretical value for At, has a contribution from proton structure and polarizability of about 30 ppm and an associated uncertainty of about 1 ppm (Sapirstein and Yennie, 1990). Results of a recent evaluation of the proton size and polarizability contribution to At, are shown in Fig. 5 (Unrau, 1996). For muonium ( p + c - ) Au is known experimentally to 36 ppb (Hughes and

26

Vernon W. Hughes Av~ = 1 420 405 751.766 7(9) Hz (1 part in 10~2) AV~

= AVF(I + ~ + ~ ) AVF = 16aZcR I% 3 " #~o ~=

~+

(uncertainty _" 50 ppb)

~ghero~rmr~

(uncem~inty<_50ppb)

2x

rigid)+~(po~nzabmty) = -35.1(7)x104+~pol) ~,=Protonrecoiland smacatre term e e

P ¢,/l//////:vl~o p Two-PhotonExchangeContributionto ~ Due to ProtonSpin Stmcmm 8p(pol)is determin~ by proton spin structure functions Gt(v,Q2) and G~(v,Q2) Recent evaluation (1996) based on present knowledge of Gl and G2 gives ~l~(pol) = (0 -+ 0.4) ppm

Av(exp0 - Av(theor)= (2.4 + 0.7) ppm Fig. 5. Theoretical value of hydrogen hyperfine structure interval Au, indicating contributions of proton structure and polarizability.

zu Putlitz, 1990; Mariam et al., 1982) and the theoretical value including QED and electroweak contributions has been evaluated to 2 0 p p b and no hadronic uncertainties are involved (Kinoshita and Nio, 1994; Kinoshita and Nio, 1996; Karshenboim, 1996). A value for a accurate to 164ppb is obtained, which is limited by our present knowledge of the magnetic m o m e n t ratio which itself is determined from the muonium Zeeman effect. An experiment is in progress at Los Alamos Meson Physics Facility (LAMPF) (Boshier et al., 1995) to improve the value of Au and of and thus determine a to the level of 30 ppb. This on-going experiment will be discussed in more detail later in Section 3.

Pu/Pp,

p~/#v

3Ps

The fine structure of helium in its ls 2s, states is a promising candidate for an accurate determination of a because the 3 p states have a relatively long radiative lifetime of 10-Ts and the fine structure intervals are as large as 30 GHz. (Pichanick and Hughes, 1990) The leading theoretical term is of course proportional to a2Ry. An early determination of a to 0.8 p p m was reported (Pichanick et al., 1968; Lewis et al., 1970; Kponou et al., 1981;

Muonium and Simple Muonic Atoms

27

Frieze et al., 1981). Other recent experiments, particularly those using new precision laser sources, may make possible measurements of the fine structure intervals to a precision of about 50ppb (Shiner et al., 1994; Inguscio et al., 1995; Inguscio et al., 1996; Prevedelli et al., 1996). Calculation of the theoretical values for the fine structure intervals has been made through radiative correction terms of order a T m c ~ In a (Drake and Yan, 1992; Zhang, Yan and Drake, 1996). In order to achieve a theoretical accuracy sufficient to determine a to 25 ppb the order a Z m c 2 term must be calculated. 2.3

T h e L a m b S h i f t in H y d r o g e n

The second basic low energy test of QED is the Lamb shift in H. The interval H ( 2 S 1 / 2 - 2 P 1 / 2 ) has been measured by microwave spectroscopy with an uncertainty of about 9 ppm, and it appears likely that a still higher precision value can be obtained (Pipkin, 1990). Proton structure contributes at the level of 100 ppm, and present uncertainty in the root m e a n square radius < r~ >1/2 of the proton, which has been determined from elastic e p scattering, (Simon et al., 1980; Hand, Miller and Wilson, 1963) results in an uncertainty of 10 ppm in the theoretical value of the Lamb shift. Precise measurements (Weitz et al., 1994; Berkeland et al., 1995; Boirieux et al., 1996) by laser spectroscopy of the 1S-2S transition in H and D determine the Lamb shift in the 1S state and are similarly limited by uncertainty in proton structure. Figure 6 indicates that the Lamb shift measurements can be considered to determine < rp2 > 1 / 2

"small"proton

"big"proton I Lnndeen& Pipkin (2S-2PRF) Hag/ey& Pipkin (2S-2PRF)

~---------_ _ ~ - - - r - - - - - ~ I I ;

I

I I I

Weirs

et

al

(IS Laser)

~------ I-------~

B o ~

et

al

(1S Laser)

I I

"l'nis work

..--.A

m

-30

-20

- 10

I I I

0

(IS Laser)

10

20

30

Deviation from measured Lamb shift (ppm) Fig. 6. Indicating the determination of proton size, < rp2 >1/2 from H spectroscopy. "small" proton < r v2 >x/~ ___0.805(11)fm, "big" proton < rv2 >a/2 = 0.862(12)fm.

28

VernonW. Hughes

There are two approaches to improving the test of QED from the Lamb shift which involve muonium and muonic hydrogen. For muonium this same 1S-2S transition has been observed by laser spectroscopy (Chu et al., 1988; Maas et al., 1994) and the 1S Lamb shift has been determined to 0.8%. It should be possible in the future to achieve very high precision in this measurement where hadronic structure is not involved. The M(2S1/2 - 2P1/2) Lamb shift interval has been measured to about 1% by microwave spectroscopy (Oram et al., 1984; Badertscher, 1985; Hughes and zu Putlitz, 1990; Woodle et al., 1990) and this accuracy could also be substantially improved. Secondly, the Lamb shift in p-p (22S1/2 - 22p1/2) might be studied by laser spectroscopy. (It has been attempted, but so far unsuccessfully (von Arb et al., 1986)). The theoretical level shifts have been evaluated (DiGiacomo, 1969; Pachucki, 1996). In this atom the proton structure effect is very large and an accurate value of < 7~ >1/2 could be measured and then used to improve the theoretical value of the Lamb shift in hydrogen. 3

Current

Muonium

Experiment

at LAMPF

As mentioned in Section 2.1, an experiment is in progress at the Los Alamos Meson Physics Facility (LAMPF) to determine with high precision the hyperfine structure interval Au (to ,-~ 10 ppb) and the muon to proton magnetic moment ratio Pu/#p (to --~ 60ppb) in the ground state of muonium (#+e-). These precision goals correspond to increases in precision for Au and for Pu/Pp by about a factor of 5 compared to present knowledge. The general method of the experiment (Hughes, 1966; Hughes and zu Putlitz, 1990) is microwave magnetic resonance spectroscopy as applied to muonium. It relies on parity nonconservation in the 7r+ --+ #+uu decay to produce polarized p+ and in the p+ -+ e+iuue decay to indicate the spin direction of/~+. The most recent LAMPF experiment provides the present experimental values (Mariam, 1982):

Aue~p = 4 463 302.88(16) kHz (36ppb); Pu/Pp = 3.183 346 1(11) (360ppb) The theoretical expression for Au can be written as: Au(theory) = Av(binding, rad) + Au(recoil) + Au(rad - recoil) + Au(weak) Except for the small term Av(weak) coming from the weak interaction, and a small known contribution arising from hadronic contributions to the photon propagator, this expression arises solely from the electromagnetic interaction of two point-like leptons of different masses in their bound state. The present theoretical value is (Kinoshita and Nio, 1994, 1996; Kinoshita, 1996):

Auth = 4 463 302.38(1.34)(0.04)(0.17)kHz (0.3 ppm) The principal error of 1.34 kHz arises from the uncertainty of Pt,/Pv. The second uncertainty arises from that in c~ based on the electron g-2 experiment,

Muonium and Simple Muonic Atoms

29

and the third is the estimate of the theoretical error in the latest QED calculation (Karshenboim, 1996). Weak neutral current effects associated with Z exchange in the e - p interaction contribute -0.065kHz or 15 ppb and are included in Auth. The experimental and theoretical values for Au agree well:

z~Vth -- Z~Vexp = --0.50(1.4)kHz ; AUth -- A ~ , ~ p = - - ( 0 . 1 1 ± 0 . 3 ) p p m . Allexp

The Breit-Rabi energy level diagram for ground state muonium is shown in Fig. 7. The history of the various determinations of Au and of IzF,/pp is shown in Fig. 8.

H(kG)

0

2 3 4 5 6 7 8 9 101112131415161718192021.22232425 I I I I I I I I I 111111111111111.J

98 5

+,

(F,Mp)

-3 -

(o,o) ~ - - - . . . . . . ~ t / 2 , - i / 2 )

-5 -6 -7

-8-9 H=

(-I/2,1/2)

0

v31

3

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8 X

9

10

11

12

13

14

15

~ -~ a -~ Ip.J +~t~gj J. H-~-

I~ p, ~Ip.H -~ Stng

AV_VI2+V3

2~g;H

16

4 4" AV [-X "4- ( l + X 2 ) trz]

alh = Av = AW/h=4463 M l l z --Y

F=J+

--)

ip. 1

WF=I±I MF 2 2'

= - -AW

4

V. ,

1

- ILBg~tMFH + - ~ A w ~ ] I + 2 M F x

+ X1

Fig. 7. Breit-Rabi energy level diagram for muorfium in its 12S1/2 ground state in a magnetic field.

The new experiment improves on the earlier one (Mariam, 1982) in several ways. First, the p+ b e a m intensity is now larger by a factor of 3 and is 1.107#+/s, with a duty factory of 6 to 9% and has greater purity, achieved principally by use of an E x B separator to reduce the e + background in the beam. Second, the magnetic field from a commercial Magnetic Resonance

30

Vernon W. Hughes

Av {HICaOO-19TO t ~ * t ' ) [*.eppm) 1O0

I

BERK[L[Y*IgTZIFSR. HZO)

I

I

I

YJLE" HEIOELBERG-1977 ( ~ ' , ' 1 [I.4 ppm|

•

SIN - t 9 7 8 ( ~ W . l r z l (O.Sppm)

•

SIN-1981 l ~ n . e . t l (O.5$Ppml

I

._= YALE-NEIO[LS[RG-Ig@2IF.,-) (0.36ppm| LzJ

'oI ; ~ ' ' , ; [ ~ I p p I I0 ll- 3 1 e 3 3 4 0 |

0

22 ppb

2.2 ppbf 196~

1966

)970

197q

1978

1980

1984

1990

Fig. 8. History of muonium Av and

p,,/pp

measurements.

Imaging superconducting magnet system operating in persistent mode provides a field with a homogeneity of better than 1 p p m over the active region of the microwave cavity and with a stability of 0.01 p p m to 0.1 p p m / h r . Third, an electrostatic chopper in the muon beam line provides a muon beam with an on-period of 4 ps and an off-period of 10#s, which allowed the observation of a resonance line from muonium atoms which had lived longer than the 2.2#s mean lifetime v~ of inuons (Boshier, et al., 1995). Such resonance lines can be narrower than the natural linewidth determined by 7~, and indeed narrower than a line obtained by the conventional method by factors of up to 3. The experimental setup is shown in Fig. 9. The longitudinally polarized #+ b e a m of about 26 MeV/c is stopped in a microwave cavity contained in a pressure vessel filled with krypton at a pressure between 0.5 and 1.5 atm. The entire apparatus is in a region of a strong magnetic field of 1.7T. Muonium is formed in an electron capture reaction between #+ and a Kr atom. The e + from p+ decay is detected in a scintillator telescope. Figure 10 shows the MRI solenoid and indicates the characteristics of this precision magnet which is operated in persistent mode. Resonance lines for transitions /'12 and /'34 are observed by sweeping the magnetic field with fixed microwave frequency.

Muonium and Simple Muonic A t o m s

~//////~//////////.Z/~///////~ $hlmCoils

Shielding

ModulationCoil'1"""'"~*a~Pressm'eV~I,~.

]IIL

/ 3 rail MylwW'~ndow~At-Collimllo¢

!,

PosiU-onT,lcscope.,.~Bij~mmr~

~

!L t.

p*

'-T-'

F i g . 9. E x p e r i m e n t a l setup in 1994/95.

• • • • • •

Manufacturer: Operating Field: Field StabiliLy: Length: Clear Bore 9 : Homogeneity: Modulation Coil:

Oxford Mag'aet Technology (OMT) 1.7 Tesla 10 ppb/hr 2.26 m 1.05 m

shimmed to I ppm over 20 cm dsv to scan field over +_50 G

F i g . 10. L A M P F superconducting solenoid 2 T MRI magnet.

31

32

•

Vernon W. Hughes

Idea: Observationofdecayiogs ~ s

actimesT> C.~nven~on~ method

CO~veNr,o.AL ~.ev.oo

I

..,o0 LPt.f~fi..l'3~J'~l~l~]J

I

] ¢

u,CROW,VES e* ¢,aTc e" CO,.C,OeNCE ~

"oLo" . u o . , u . . E T ~ O 0

e ] ''2 ~Vm = (natural)= Z = 145 kHz Old muonium 8Vm (y, Ib~T) I and I~ <~

;qf~,J [ 6-T--',

I

I

~

.,c~ow~,,Es

e*~,T~

~ " co,,c,oc,cE

Svm = 21bt < ~ v m (natural) x

[b~isproportional to microwavemagnetic field

~SeC

Fig. 11. The conventional and old muonium methods of observing resonance lines and the associated linewidths.

The conditions and lineshapes for the conventional and "old muonium" resonances are given in Fig. 11. Observed resonance curves are shown in Fig. 12. It can be seen that a narrower resonance line is obtained for muonium atoms which have lived a longer time. Also wings develop for cases where older muonium atoms are observed and larger signal amplitudes are obtained. During runs in 1994 and 1995 with a total beam time of about 3 months, data were taken with stopping Kr gas pressures of 0.5, 1.0 and 1.5 atm. Some 2,000 sweeps of the resonance line were obtained. Additional data-taking followed in 1996 when resonance lines were obtained by varying the microwave frequency with fixed magnetic field. These data should provide an important check for systematic errors (Boshier et al., 1996b). The scientific importance of a more precise determination of At, and # , / p p includes the following: 1. Most precise test of QED for the two lepton bound state and of the behavior of the muon as a heavy electron. 2. Precise determination of the fundamental constants: muon magnetic moment and muon mass. Also a value of the fine structure constant a accurate to about 30 ppb will be obtained; this accuracy is comparable to that from condensed matter determinations. 3. Precise values of m , and p , are very important for the BNL muon g-2 experiment (Hughes, 1994) and in the determination of the muon neutrino mass m~, (Particle Properties, 1996).

Muonium and Simple Muonic Atoms

~ " 2 2 • 'v12 ~' m',t')CanvlnmOt (l ~H0 321789f215) .~~u ~ ~.0 20 hl o=-81(0.03)G x'/DoF t.o6 DoF 125

'

'

(a )

1~ ~

30

'°

14

6O5) F Z-/{:~ 0.9o

!

-

33

Jr~¢ 6 ? ,1~I,,

FWHM 1121r.~4~ 2(] 15

12

l

10 8

6 200

250

350

300

400

2OO

450

v~-720~kHz) C

4"02-~'97tt'~ec

HO 321479(510)Hz.~1 , ttl IJOG • x'/DoF0.98 f ~llVc~ ~. DoF t26 40 FWHM811d~

( )

"~50

J

3OO

350 4OO 450 v -72000~kHz)

F"'vl2 " ~ 1 3 ~ ) $,92_-6.87$Lu~:

-J 4 °.)

250

' "

'

Ho 32152~ssz~z ~,~ hi I.~G SO 128-3(2.2) / ~ t -/D~ O.99

5O

(d)

3~ 20 2~ 10 0 , I

2OO

~'60

v12

50 4O

250

~

30O

(l~an'o) '

-10

'

350 400 450 v~-72000(kHz) ~

) 7,g2-8.77~t,se~ H0 32229~s3o~tu8 / SO 105.9(3.2) I~1] x'/Dof 1.00 IF1. ,

DoF tins F'WHM 49kHz

'

'"

~

200

250

300

5(

(e)

8.~-9,T/~sec

HO

• hi

~

,I , l f ~ . ~

350 400 450 v~,:72000(kHz)

30

30

2O

2O

10

322292f1336)Hz IJOG

[

, ~:• t.L)

FWI.IM45kI~

)

I0 0

10

-10 2OO

250

30O

350

40O

450

v~,-72000fkI-Iz)

2~

~

3001

3~ v

~

450

:72OO0(kHz)

Fig. 12. Old muonium resonance lines. Observed u12 resonance fines obtained by magnetic field scan w i t h a 1.Satm Kr target. The signal -= (No~+n - N ~ ; ) / N ~ [ % ] in which No~+ (No~n +) is the number of decay e + counts with microwaves on (off). The magnetic field is measured by the proton NMR frequency U~MR. The fitting parameters are the line center Ho, the microwave field amplitude H1 and a scaling factor So. The F W H M for a line obtained by a frequency sweep would be 1.7 times the FWHM in units of u. . . . The solid line fits have been made using measured magnetic field and muon stopping distributions as well as the microwave magnetic field distribution for the cavity mode. (a) Conventional resonance line without the/J+ beam chopped and with a microwave power of 10 W in the cavity. (b)-(f) Old muonium resonance lines with a microwave power of 4 W and different observation times measured from the beam-off time.

34

Vernon W. Hughes

HYDROGEN

MUONIUM F 26 ~ z

F

F F

10 922MFIz |

10 969 ~

,?:-I

2sl/2-

7 T 1o48 MNZ

zP,,, l

I

I ~ . IMHZ I

r lz16

X

/t

lSll2 o

F i g . 13. Energy levels in the n = l and n = 2 states.

Finally, to emphasize the similarity of muonium to hydrogen, Fig. 13 shows their energy levels in the n = l and n=2 states. From the viewpoint of tests of fundamental theory muonium has the distinction of being free of hadronic effects, apart from the very small and well known modification of the photon propagator associated with virtual processes. The energy intervals shown can be measured for both H and M, but for M the measurements are more difficult primarily because few atoms are available, and the precision of the measurements is lower at present.

4 4.1

Laser Spectroscopy of M u o n i u m and M u o n i c Atoms E x p e r i m e n t s D o n e thus Far

The pioneering early experiment was a measurement at CERN on the muonic helium ion, (4He p - ) + , of the 2S to 2P transitions (Bertin et al., 1975a; Bertin et al., 1975b; Carboni et al., 1977). Negative muons were stopped in He gas at a pressure of 40 atm. The negative muons were captured by He into a highly excited state of muonic helium in an Auger process, and after subsequent Auger and radiative capture processes several percent of the stopped /tformed the 2S1/~ state of the muonic helium ion ( 4 H e # - ) + . The decay rate 2S1/~ --+ 1S1/2 through spontaneous two-photon radiative decay is 1.1.105 s- 1 which corresponds to a mean lifetime of 9.1 #s. The experimenters determined

Muonium and Simple Muonic Atoms

35

2Pj/~ 0.1S eV

b)

1.37 eV

tim6"S-*15~"

K

-~ 7o

6 a

t

.2261,eV

I z

~t~

I ,s ,

,

f

~,,

I. •

,

,

.

i ',

' f

"

r'

'

~

-

Fig. 14. a) Energy levels of (4Hep-)+ in n=2 state, b) Observed 2Sl/2 -+2Pi/2 resonance signal.

that the collision quenching rate for the 2S1/2 state in He at 40 atm was less than 8.103 s -1 atm -1 (Carboni et al., 1973). Using a pulsed dye laser at about 8120/~, which was triggered at the time of an individual incoming p - , the transition 2S1/2 -+2P3/2 was induced and then the subsequent 8.2keV Xray from the 2P3/2 --+1S1/2 radiative decay was detected in a NaI detector. Unambiguous resonance lines were observed as the pulsed dye laser was tuned (Fig. 14). Subsequently the 2S1/2 --+2P1/2 resonance was observed in a similar manner. Using the observed resonance wavelengths for these transitions, the dominant QED term, which is due to vacuum polarization, was determined to about 0.1% in agreement with the theoretical prediction. Alternatively, the measurements were used to determine the most precise value for the root mean square radius of the 4He nucleus, < r24He )1/2. Subsequently a fascinating puzzle has arisen about the expected lifetime of the 2S1/2 state of (4He p - ) + in helium at 40 atm pressure, which was the condition for the above experiment. Theoretical estimates indicated that the 2S1/2 state would be rapidly quenched in collisions with He atoms (Mueller et al., 1975). Furthermore several experiments concluded (yon Arb et al., 1984; Eckhause et al., 1986; Rosenkranz et al., 1990) that the lifetime of the 2S1/2 state of (4He#-)+ with respect to quenching in a He collision would be much shorter than 2 ps at 40 atm of He. This called into question how the 2S-+2P transition could have been observed. A possible explanation was

36

Vernon W. Hughes

provided by calculations indicating that a cluster of He atoms centered on the 2S1/=(4He/_t-) + ion might be formed in which the 2S state was stable, although later work showed that this mechanism did not in fact explain the observation (Cohen, 1982). More recently, this idea has developed into a suggestion that at sufficiently high pressures the cluster could form in longlived excited states in which quenching of the 2S state is inhibited (Baracci and Zavattini, 1990). Of course in this case, to test QED one must then know the exact energy levels of the muonic helium ion within the cluster. An unsuccessful attempt was also made recently to observe again the 2S1/p. to 2P1/2 transition at Paul Scherrer Institut (PSI) in Switzerland (Hauser et al., 1992). Hence the puzzle about the lifetime and precise energy levels of ( 4 H e p - ) + in He gas remains. The second laser spectroscopy experiment has been on the muonium 1S-+2S transition. This transition is induced by a two-photon Doppler-free process and detected through the subsequent photoionization of the 2S state in the laser field. The key to success in this experiment was the production of muonium into vacuum from the surface of heated W or of SiO2 powder. The discovery experiment (Chu et al., 1988) was done at the KEK facility in Japan with a pulsed muon beam and an intense pulsed laser system. A subsequent experiment (Maas et al., 1994) done with the pulsed p+ beam at RAL and a similar pulsed laser has improved the signal substantially and has achieved a precision of about 10 - s in the 1S -+ 2S interval, thus determining the Lamb shift in the IS state to about 1% accuracy (Fig. 15). The precision of this experiment should be greatly improved in a new experiment now underway at RAL. This experiment will provide a precise determination of the muon mass from the muonium-hydrogen isotope shift in the transition since this shift is primarily due to reduced mass.

4.2

A d d i t i o n a l P o s s i b l e E x p e r i m e n t s o f Interest

A substantial number of important energy intervals in the simple muonic a t o m s - muonium, muonic hydrogen ( p - p ) , muonic deuterium ( p - d ) and in the muonic helium ions, (4He # - ) + and (3He # - ) + - should be measurable by the method of laser spectroscopy (Jungmann, 1992). The intervals include the 1S-2S interval, fine structure and Lamb shift intervals, and hyperfine structure intervals. They arise from fundamental QED processes and include the fundamental constants of particle masses and magnetic moments as well as the nuclear radii of p, d, 3He and 4He and nuclear polarizabilities. At a more sensitive level electroweak effects appear as well. For electronic atoms these intervals have been measured with high precision over the past 50 years. They have provided fundamental data out of which quantum electrodynamics has been developed and tested. Our knowledge of many fundamental constants and of properties of particles and nuclei also comes from these data.

Muonium and Simple Muonic Atoms

~ = ( a 4 ~ - _ ~ ) USz

,a'~ : ; ~ o o

:"

~o0

.

?;0 8 ; o

37

/

~ o ,d00";,oo u_-v,.CMHz]

Transition Frequency M ( I S - 2 S ) ~ r = 2, 455,529,002(33X46) M]-]z M(1S-2S)QED = 2, 455,528,934(3.6) M]']Z L a m b shifts

[L,s-/-~s] Exrr = 6,98g(33X46) MHz [Lls -/v.s] QED= 7,056.I(1.0) MHz agrees

with QED at the 0.8% level.

Fig. 15. Observed 1S-2S transition in muonium by two-photon laser spectroscopy.

Table 1 lists a number of important transitions in muonium and muonic atoms which might be measured by laser spectroscopy. The physics interest and precision potential are also given, as is a possible pulsed laser system. (The precision potential does not take into account #+ beam intensity limitations and hence in some cases is at present unrealistic.) We note that the size of muonic atoms and their magnetic moments are smaller than for electronic atoms by the mass ratio r n e / m u so that electric dipole and magnetic dipole matrix elements are smaller by m e / m u and hence high laser power is required to induce the transitions. A pulsed high power laser system is required for these experiments.

5

Sources of Pulsed Muons

The ideal experimental facility for future laser spectroscopy measurements on muonium, on muonic hydrogen and on the important muonic helium ions is an intense pulsed low energy muon source, where the pulse length is small compared to the muon lifetime and the repetition rate is similar to that of pulsed high power lasers. Such a facility is in operation with the 800MeV proton synchrotron at Rutherford Appleton Laboratory (RAL) (Cox, 1990). At LAMPF, its high intensity 800 MeV proton linac feeds a proton storage ring which converts the long duty factor (6 to 10%) proton linac beam into a pulsed proton source. A pulsed muon beam could be developed from this

48.4 TIIz (6.2 tun)

43.9 TIIz (6.8 ~ n )

1.6 TI Iz ( 188 pro)

2S-2P

IS, F=0-F=I

3D-3P

muonic hydrogen (H P')

282.7 TIIz, 365.1 Tllz (1.06 ~ n , 822 nm)

(9.85 ~,m)

30.4 Tllz

2S-2P

3D-3P

2S-2P

2,457 Tllz + 819 Tllz 0 2 2 n m + 355 nm)

IS-2Pcontinuum

Frequency-tripled Alexandrite

Laser System (Pulsed)

Free electron laser

Difference frequency generation

Carbon monoxide or nonlinear mixing

Ti:Sapphiro

Carbon dioxide

YAG-pumped Ti:Sapphtre with fourwave mixing in Kr/Ar + doublin~ in BBO 369.6 TIIz. 334.2 Tilz Dye or Tl:Sapphire (812 rim, 898 rim)

2 x1,228.5 Tllz (2 x2,~ nm)

Frequency (Wavelength)

I S-2S

Transition

muonic 3lie (W Slle) *

muonic 'tile ion (it 'tile)+

muonium, M ([a* e )

System

QED vacuum polarization, (insensitive to nuclear structure)

proton charge radius and polarizability

I. proton charge radius and polarizability 2. QED vacuum polarization

3lie nuclear size and poladzahiUty

QED vacuum polarization, (insensitive to nuclear structure)

I. QI-:I) vacuum polari:,arinn 2. ft.-particle charge radius

I. Lamb shift without nuclear structure 2. QED recoil, reduced mass 3. }.t"mass Ultra-slow H" beam

Physics Inlerest

Current Statu!g

Potential (line split to I part in 10 ~)

Approved at PSI

An uncertainty of 10 Gilz improves current knowledge of (r~) 'n (2% level)

Under discussion at PSi. An uncertainty as large as 20 GI Iz improves current knowledge of 2 Ill . (r~)

No transition observed at PSI Test experiment at BNL did not see a clear signal.

to 0.2%

100 (]llz uncertainty (CF~RN): tests vacuum polarization at 0.2% level or determines ~r:~ 'n

Potential for a 6 ppm test of vacuum polarization, all improvement of 1000 Oil cnrrent tests.

Potential for five orders of magnitude improvement beyond this 10 Gtlz.

(~),/2 t o a few partsin 10'

PSI experiment aims for eventual uncertainty of 500 Mllz. giving

Measurement widt uncertainly of 1/5 of linewidth would provide new test of vacuum polarization. Potential for furdrer improvement b~' a factor of 200. Similar to muonic 4lie ion

At least another two orders of magnitude.

60 MIIz uncertainty (RAL) Potential for attolher five orders tests QED to 0.8% and gives m.u of magnitude beyond current to 5 ppm. Order of magnitude result. improvement expected in next RAL experiment Under construction at KEK

T a b l e 1: Possibilities for laser spectroscopy of muonie atoms

m

©

On

GO

Muonium and Simple Muonic Atoms

39

Table 2: Parameters of the positive and negative muon channels at the Rutherford Appleton Laboratory

(RAL), and of future pulsed muon sources proposed at the Los Alamos Meson Physics Facility Proton Storage Ring (LAMPF PSR), the Japan Hadron Project (JHP) and the European Spallation Source (ESS).

Intensity (IMs)

Momentum spread Ap/p

Pulse structure

Spot size

RAL (g÷)

RAL (ix)

3xlO 6

2xlO 5

10%

82ns FWHM at 50Hz

1.2cm x 2cm

JHP(g °)

ESS(p. ÷)

l.TxlO ~

4.5x107

4.5x 107

5% at 40 MeWc

10%

10%

10%

82 ns FWHM at 50 Hz

279 ns FWHM

300 ns

300 ns

FWHM

FWHM

at 40 Hz

at 50 Hz

at 50 H z

3cm x 2cm

1.2cm x 2cm

1.2cm x 2cm

4cm x ,k:m

L A M P F PSR

proton source which would be 1 order of magnitude brighter than the RAL beams (White, 1992). Recent plans in Japan for a Japan Hadron Project (JHP) and in Europe for a European Spallation Source (ESS) (G. Eaton, private communication) could also provide powerful new pulsed muon sources. These machines are discussed in more detail below. At present the Rutherford Appleton Laboratory in England operates a fast-cycling proton synchrotron with 800MeV energy and an average proton current of 200#A. It is used primarily to provide a pulsed spallation neutron source but it also provides a pulsed source of low energy (surface) muons produced from a thin target. Six muon channels are presently available. Measurement of the 1S--+2S transition in muonium is in progress with a pulsed muon beam having the characteristics shown in Table 2. A second source of pulsed muons with a considerably higher intensity could be provided from the Proton Storage Ring (PSR) which is fed by the LAMPF 800 MeV proton linac at Los Alamos. Extraction of the proton beam from the PSR is already provided. A transport line for the proton beam, a target and a muon channel would be needed. The characteristics of the pulsed muon source which would be possible with a muon channel consisting of iron electromagnets are also given in Table 2, which also includes parameters for the possible muon sources at J H P and ESS. Finally, we note one other interesting possibility for the future. Design studies at Brookhaven National Laboratory of a muon injection system for a possible high-energy p+p- collider (R. Palmer, 1996) suggest that very high efficiencies are possible (0.3 muons/proton), indicating that this system may also be useful as a source of pulsed muons. Research supported in part by the U.S. Department of Energy.

40

Vernon W. Hughes

References von Arb H.P. et al., (1984): Phys. Lett. 136 B, 232. von Arb H.P. et al., (1986): Proceedings o] the Workshop on Fundamental Muon Physics: Atoms, Nuclei and Particles, ed. C.M. Hoffman, V.W. Hughes and M. Leon, LA-10714-C, p. 103. Badertscher A. et al., (1985): Atomic Physics 9, ed by R.S. van Dyck, Jr. and E.N. Fortson, (World Scientific) p. 83. Baracci L. and Zavattini E. (1990): Phys. Rev. A 41 2352. Berkeland D.J., Hinds E.A. and Boshier M.G. (1995): Phys. Rev. Lett. 75, 2470. Bertin A. et al., (1975a): Phys. Lett. 55B, 411. Bertin A. et al., (1975b): Atomic Physics 4, ed. G zu Putlitz, E.W. Weber and A. Winnacker (Plenum, New York) p. 141. Boirieux T., et al., (1996): Phys. Rev. Lett. 76, 384. Boshier M.G. et al., (1995): Phys. Rev. A 52, 1948. Boshier M.G., Hughes V.W., Jungmann K., zu Putlitz G. (1996a): Comments Atomic and Molecular Physics 33, 17. Boshier M.G. et al., (1996b): Abstracts ZICAP, Amsterdam, The Netherlands, ThK3. Cage M.E. et al., (1989): IEEE Trans. Instrum. Meas. 38, 284. Carboni G. et al., (1973): Nuovo Cimento 6,233. Carboni G. et al., (1977): Nucl. Phys. A 278 381. Crane T. et al., (1971): Phys. Rev. Lett. 27, 474. Chu S. et al., (1988): Phys. Rev. Lett. 6{}, 101. Cohen J. (1982): Phys. Rev. 25, 1791 and references therein. Cohen E.R. and Taylor B.N. (1987): Rev. Mod. Phys. 59, 1121. Cox S. (1990): Proceedings o.f the Pulsed Muon Workshop (Los Alamos National Laboratory). DiGiacomo A. (1969): Nucl. Phys. B l l , 411; B23,641. Drake G.W.F. and Yan Z.-C (1995): Phys. Rev. A46, 2378. Eckhause M. et al., (1986): Phys. Rev. A 33, 1743. Frieze W. et al., (1981): Phys. Rev. A24, 279. Hand L.N., Miller D.B. and Wilson R. (1963): Rev. Mod. Phys. 35,335. Hauser P. et al., (1992): Phys. Rev. A 46, 2363. Hughes V.W., McColm D.W., Ziock K. and Prepost R. (1960): Phys. Rev. Lett. 5 63. Hughes V.W. (1966): Arm. Rev. Nucl. Sci. 16,445. Atomic Physics, ed by Hughes V.W., Bederson B., Cohen V.W., Pichanick F.M.J., (1969): (Plenum Press, NY,). Hughes V.W. et al., (1970): Phys. Rev. A1, 595; A2, 551. Hughes V.W. and zu Putlitz G. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p 822. Hughes V.W. ((1994): A Gift o] Prophecy, ed. E.C.G. Sundarshan (World Scientific, Singapore) p. 222. Inguscio M. et al., (1995): Proceedings of the IEEE Frequency Control Symposium. Inguscio M. et al., (1996): Submitted to Physiea Scripta, Special issue in honor of Prof. G. Series. Jungmann K., (1992): Z. Phys. C56, $59.

Muonium and Simple Muonic Atoms

41

Karshenboim S.G. (1996): Z. Phys. D 36, 11. Kinoshita T. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p. 218. Kinoshita T. and Lepage G.P. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p. 81. Kinoshita T. and Nio M. (1994): Phys. Rev. Lett. 72, 3803. Kinoshita T. (1995): Phys. Rev. Lett. 75, 4728. Kinoshita T. (1996): Rep. Prog. Phys. 59, 1459. Kinoshita T. and Nio M. (1996): Phys. Hey. D 53, 4909. Kponou A. et al., (1981): Phys. Rev. A24, 264. Kriiger E., Nistler W. and Weirauch W. (1995): Metrologia 32, 117. Lewis S.A. et al., (1970): Phys. Rev. A2, 86. Maas F.E., et al., (1994): Phys. Lett. A 187, 247. Mariam F.G. et al., (1982): Phys. Rev. Lett. 49,993. Martinos J.M., Nahum M. and Jensen H.D. (1994): Phys. Rev. Lett. 72,904. Mueller R.O. et al., (1975): Phys. Rev. A l l , 1175. Oram C.J. et al., (1984): Phys. Rev. Lett. 52, 910. Pachucki K. (1996): Phys. Rev. A53, 2092. Palmer R. 1996: (private communciation) "Review of Particle Properties" 1994:, Phys. Rev. D50, 1173. Pichanick F.M.J. et al., (1968): Phys. Rev. 169, 55. Pichanick F.J.M. and Hughes V.W. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p 905. Pipkin F. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p 696. Prevedelli M. et al., (1996): Optics Communciations 125 231. Atomic Physics 3, ed by zu Putlitz G., Weber E.W. and Winnacker A. (1975): (Plenum Press, NY) Ramsey N.F. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p 673. Rosenkranz J. et al., (1990): Ann. Phys. 47, 667. Sapirstein J.R. and Yennie D.R. (1990): Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore) p 560. Shiner D., Dixson R. and Zhao P. (1994): Phys. Rev. Lett. 72, 1802. Simon G.G. et al., (1980): Nucl. Phys. A333 381. Stambangh R.D. et al., (1974): Phys. Rev. Lett. 33 568. Thompson P.A. et al., (1969): Phys. Rev. Lett. 22, 163. Unrau P. (1996): Ph.D. Thesis, Spin Dependent Nucleon Structure at Intermediate Q2 Massachusetts Institute of Technology. Van Dyck, Jr R.S., (1990): Quantum Electrodynamics, ed. by T. Kinoshita (World Scientific, Singapore) p. 322. Weiss D.C., Young B.C. and Chu S. (1993): Phys. Hey. Lett. 70, 2706. Weitz M. et al., (1994): Phys. Rev. Lett. 72, 328. White H. (1992): LAMPF Pulsed Lepton Source, Los Alamos National Laboratory. Williams E.R. et al., (1989): IEEE Trans. Instrum. Meas. 38, 233. Woodle K.A. et al., (1990): Phys. Rev. 41, 93. ZhangT, Yan Z.-C and Drake G.W.F. (1996): University of Windsor preprint.

T h e M u o n i u m A t o m as a P r o b e of P h y s i c s beyond the Standard Model L. Willmann* and K. Jungmann Physikalisches Institut der Universit/it Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

1

Introduction

Precision measurements on atomic systems have played an important role in the course of the development of modern physics. In many cases they have lead to discoveries which had significant impact on the understanding of the physical laws of nature. The explanation of the carefully measured electromagnetic spectrum of atomic hydrogen by the Schr6dinger equation was a great success for quantum mechanics. The observed fine structure was included in the solutions to the Dirac equation which demonstrated the necessity of a relativistic description of the atomic structure. Precise investigations of the hydrogen Balmer-a line revealed a faint nearby line [1] which was the discovery of deuterium through its spectroscopic isotope shift. A small deviation of the measured hyperfine splitting in hydrogen [2] from the value predicted in Fermi's theory on the 0.1% level could be explained by the anomalous magnetic moment of the electron. This discovery together with the observation of the Lamb-shift (22S, - 22Fx ) in hydrogen [3] has initiated and boosted the development of t]ae modern field theory of quantum electrodynamics (QED). The unification of the weak and electromagnetic interactions in the electroweak standard model was strongly supported by the observation of parity violation in precise spectroscopic measurements in heavy atoms. Today electroweak processes examined both at high energies [4] at the LEP electronpositron storage ring collider at CERN, Geneva, Switzerland, and in atomic parity violation experiments in heavy atoms, e.g. in cesium and thallium [5, 6], have ascertained the power of the unified electroweak theory which is valid over a range of 10 orders of magnitude in momentum transfer. Today the standard model appears to be a very successful effective description of all known interactions between particles and no significant deviation from it could be established so far. Its predictions are subject to high precision experiments which allow to extract a set of intrinsic parameters including the masses in the leptonic and the quark sectors and mixing angles between different quarks. However, there still remain unresolved questions within this sophisticated theoretical framework like the number of interactions, the number of lepton and quark generations or the nature of parity violation. Partic-

7

44

L. Willmann and K. Jungmann

ularly the question of lepton number conservation is investigated by various experiments, since no underlying symmetry could be discovered to be associated with it yet. The muonium atom (M=It + e - ) , the bound state of a positive muon It+ and an electron e - , can be considered a light hydrogen isotope. This fundamental system is ideally suited for investigating bound state quantum electromagnetic theory and it renders the possibility to test fundamental concepts of the standard model. The spectroscopic measurements of electromagnetic transitions like the hyperfine interval in the ground state or the ls-2s energy splitting are generally considered precise tests of QED and are used to infer accurate values of fundamental constants [7]. In addition, they may be used to extract information on fundamental symmetries. For example, the latest measurement of the ls-2s energy interval [7, 8] can be regarded as the best test of the charge equality of leptons from different particle generations at a level of 10 - s relative accuracy [9]. The system offers further unique possibilities to search for yet unknown interactions between leptons, in particular, since the close confinement of the bound state allows its constituents a rather long interaction time which is ultimately limited by the lifetime r u = 2.2Its of the muon.

2

Test

of Lepton

Number

Conservation

A spontaneous conversion of muonium into antimuonium (M = # - e +) would violate additive lepton family (generation) number conservation by two units. This process is not provided in the standard model like others which are intensively searched for, e.g. It --+ e7 [10], It --4 eee [11], It - e conversion [12] or the muon decay mode It+ - 4 e + + u u +-P7 [13]. However, in the framework of many speculative theories, which try to extend the standard model in order to explain some of its not well understood features, lepton number violation is a natural process and muonium to antimuonium conversion is an essential part in several of these models (Fig. 1) [14-19]. Traditionally, muonium-antimuonium conversion is described as an effective four fermion interaction with a coupling constant GM~ which can be measured in units of the Fermi coupling constant of the weak interaction GF [20]. Many of the speculative models would allow a strength of the interaction as large as the experimental bound at the time they were created. In minimal left-right symmetric theory muonium and antimuonium could be coupled through a doubly charged Higgs boson A++. In this case even a lower bound has been predicted for GM~, provided the muon neutrino mass m,~, were larger than 35 keV/c 2 [14]. With the present experimental limit of m , , <_ 170 keV/c 2 [21] the coupling constant GM~ should be larger than 2- 10 -4 GF. This figure would even increase for an improved bound on m ~ . For neutrinos being Majorana particles a coupling between muonium and antimuonium is possible by an intermediate pair of neutrinos. A limit on

The Muonium Atom as a Probe of Physics beyond the Standard Model #+

45

#-

A++

Wa.. '.. WL

~+

e

(a)

(b)

#+

e+

#+

.

~N m

IX++ #-

e

#-

(c)

e-

¢+ (d)

Fig. 1. Muonium-antimuonium conversion can be described in various theories beyond the standard model. The interaction could be mediated by (a) a doubly charged Higgs boson A ++ [14], (b) heavy Majorana neutrinos [15], (c) a neutral scalar ~b~v [16], which could be for example a supersymmetric r-sneutrino P~ [18] or (d) a dileptonic gauge boson X ++ [19].

the effective coupling can be estimated based on the Majorana mass limit of the electron neutrino which has been deduced from experimental searches for neutrinoless double fl-decay [22] to GM~ < 10-hGF [15]. Supersymmetric theories allow an interaction to be mediated by a 7-sneutrino PT, the supersymmetric partner of the 7.-neutrino. The predictions are G M ~ < 10-2GF for a mass value rn~r of 100 G e V / c 2 [18]. Some of these speculative theories need to introduce neutral scalar bosons to explain the mass spectrum in the leptonic sector. These models predict a coupling strength of 10-2GF which is in the range of the sensitivity of present experimental search [16]. In the framework of grand unification theories (GUT) muonium-antimuonium conversion could be explained by the exchange of a gauge boson X ++ which carries both an electronic and a muonic lepton number. B h a b h a scattering experiments at P E T R A storage ring at DESY in Hamburg, Germany, bounded the mass of this dileptonic particle to mx++/g3t >_ 340 G e V / c 2, where g3l depends on the particular s y m m e t r y and is of order unity [23]. This can be translated into GM~ _< 10-2GF.

3

The Conversion Process

Muonium and antimuonium are neutral atoms which are degenerate in their energy levels in the absence of external fields. In 1957 Pontecorvo suggested the possibility of a muonium to antimuonium conversion process even before the a t o m had been formed for the first time by V.W. Hughes and his coworkers in 1960 at the NEVIS cyclotron of Columbia University, New York, USA [24]. He proposed a coupling by an intermediate neutrino pair state in analogy to the K ° - K ° oscillations, which were discovered at that time [25].

46

L. Willmann and K. Jungmann

Any possible coupling between m u o n i u m and its a n t i a t o m will give rise to oscillations between the two species. For atomic s-states with principal q u a n t u m number n a splitting of their energy levels =

8GF

GM~-

(1)

v%2~a03 @ is caused, where a0 is the Bohr radius of the atom. For the ground state equals 1.5-10 -12 e V - ( G M ~ / G r ) which corresponds to 519 Hz for G M ~ = GF. We note that this value is three times larger than the uncertainty reported for the best measurement of the m u o n i u m ground state hyperfine structure interval [26]. Therefore, the interpretation of precise measurements of the hyperfine structure must include considerations on how such a process would affect the accuracy under the particular experimental conditions [27]. A system starting at time t -- 0 as a pure state of m u o n i u m could be observed in the antimuonium state at a later time t with a probability of (Fig. 3)

(~_~.) e-A~,t

pM~(t) = s i n 2 ~t

(~t~ 2

~' \ 2 h )

-

"¢ ~,t,

(2)

where A, = 1 / % is the muon decay rate. The approximation is valid for a weak coupling as suggested by the known experimental limits on GM~. In this case the process should be considered a conversion rather than an oscillation. The m a x i m u m of the probability for a decay as a n t i m u o n i u m is found at tm~x = 2/A,, while the ratio of antimuonium to m u o n i u m continuosly increases with time. The total conversion probability integrated over all decay times is

PM~=2-5610 -5

\--~F J

"

(3)

This demonstrates the advantage of experiments in which the system is allowed an extended time interval (a duration of order r , or longer) for developing a conversion. The degeneracy of corresponding states in the a t o m and its a n t i a t o m is removed by external magnetic and electric fields which can cause a suppression of the conversion and a reduction of the probability PMM"T h e influence of an external magnetic field depends on the interaction type of the process. The reduction of the conversion probability has been calculated for all possible interaction types as a function of field strength (Fig. 3) [28, 29]. In the case of an observation of the conversion process the coupling type could be revealed by measurements of the conversion probability at two different magnetic field values. The conversion process is strongly suppressed for m u o n i u m in contact with matter, since a transfer of the negative muon in a n t i m u o n i u m to any other a t o m is energetically favored and breaks up the s y m m e t r y between

The Muonium Atom as a Probe of Physics beyond the Standard Model

47

0.5

e "Xgt

0.45

0.4 0.35 "~ 0.3 0.25 0.2 0.15 0.1 0.05 0

xlO 8 for P'-~MM= 1.8

• 1 0 -2

= 1000

0

1

2

3

4

5

"'"'"" 4..

6

7

8

9 10 time [ps]

Fig. 2. Time dependence of the probability to observe an antimuonium decay for a system which was initially in a pure muonium state. The solid line represents the exponential decay of muonium in the absence of a finite coupling. The decay probability as antimuonium is given for a coupling strength of GM~- = 1000 by the dotted line and for a coupling strength small compared to the muons decay rate (dashed line). In the latter case the maximum of the probability is at 2 muon lifetimes. Only for strong coupling several oscillation periods could be observed.

m u o n i u m and antimuonium by opening up an additional decay channel for the antiatom only. In gases at atmospheric pressures the conversion probability is about five orders of magnitude smaller than in vacuum [30] mainly due to scattering of the a t o m s from gas molecules. In solids the reduction amounts to even 10 orders of magnitude. Therefore a sensitive experiment will benefit largely from employing muonium atoms in vacuum.

4

History

of Experimental

Search

There is a strong connection between experimental searches for nmonium to antimuonium conversion and the development of efficient sources of m u o n i u m atoms. In the earliest experiment in 1967 at the NEVIS cyclotron [31] muons were stopped in a Ar noble gas target of pressure 1 a t m with a technique similar to the one which was used in the discovery of nmonium. A large fraction of the muons forms m u o n i u m by electron capture. A conversion process would be indicated by K s X-rays originating from an argon a t o m after the transfer of the negative muon from antimuonium. A sensitivity of G'M~ < 5800 GF (95% C.L.) could be reached which was mainly limited by the strong suppression of the conversion in gases. One year later M¢ller scattering was investigated at the Princeton-Stanford electron storage rings at Stanford, USA. An analysis of the channel

48

L. Willmann and K. Jungmann

"

0.61 ~

.

.~. . . . .i. . . .0. . . . .". . . . 8. . . . .~. . . . . . . . . . . . . . . . . .

~0.4 F 0

effective Hamiltonianform: ~ (V-+A)×(VT-A)or (STP)x(ST-P)

l-

0.2 ~-

~

l-~

10

10

-2

GM~< 1 G F

~ ' ~

(v~A)~(v+~)or(S~PNs-+P)

_ ~_i~....p~,~ .... --ss ss

-3

"",

10

........ ,

-1

1

........ ,

10

......... ,

10

........ ,

2

.\ ..,.~-~

3

........

4

10 10 10 magnetic field [G]

5

Fig. 3. The muonium to antimuonium conversion probability depends on external magnetic fields and the coupling type. Recent independent calculations were performed by Wong and Hou [16] and Horikawa and Sasaki [29].

e- + e- --+ # - + p - , which is essentially the same physical process as muonium-antimuonium conversion, yielded nearly two orders of magnitude higher sensitivity on the coupling strength [32]. 1 Significant increases in sensitivity could be achieved after 1980 by taking advantage of newly developed sources of muonium in vacuum. At the Los Alamos Meson Physics Facility (LAMPF) in Los Alamos, USA, muonium in vacuum could be produced by a beam foil technique from thin aluminum foils. The velocity-resonant nature of the electron capture process causes typical kinetic energies of the atoms of a few keV. The corresponding high velocities and finite dimensions of the apparatus restricted the time interval available for the conversion. Antimuonium could have been discovered by secondary electrons and muonic X-rays from a bismuth catcher foil. The coupling constant GM~ could be limited to below 7.5 GF (90% C.L.) [33]. The discovery of muoniurn formation in fine grain SiO~ powders [34, 35] with about 60 % efficiency [36], after stopping muons from a surface beam stimulated experimental work at the Tri University Meson Facility (TRIUMF) in Vancover, Canada, in the early 80's. The signature for antimuonium was the detection of X-rays after capturing the negative muon in a calcium host atom. The data were analyzed under the assumption that the muonium atoms escape from the grains into the intergranular voids and yielded GM~ _< 42 GF (95% C.L.) [37]. With a more complete understanding of the behaviour of muonium atoms inside of a powder target a reanalysis of the Today e- + e - scattering experiments would have to run for approximately 1 year at LEP beam energies and luminosities to reach a sensitivity similar to a modern muonium-antimuonium conversion experiment in medium energy laboratories.

The Muonium Atom as a Probe of Physics beyond the Standard Model

49

data limited the coupling constant GM~ to less than 20 GF (95% C.L) [38]. The observation of a few percent of the muonium atoms leaving SiO2 powder target surfaces with thermal energies at TRIUMF [38] and at at the Paul Scherrer Institut (PSI) in Villigen, Switzerland [39], was a major breakthrough in the mid 80's. It has boosted experimental efforts searching for muonium to antimuonium conversion and has been employed in all new approaches since. At TRtUMF an experiment using thermal muonium in vacuum requested a signature consisting of X-rays generated by the transfer of the negative muon to a host atom and followed by the delayed decay of a radioactive tantal nucleus created by nuclear muon capture. A sensitivity of GM~ ~ 0.29 GF (90% C.L.) could be reached [40]. The thermal kinetic energy of the muonium atoms corresponds to a velocity of 7.4(1) mm/ps [39]. This assures that the atoms will stay in a small volume of about 100 cm 3 for several natural lifetimes T, and allows for long times for the conversion to antimuonium. At the Phasotron accelerator in Dubna, Russia, another experiment has been carried out using muonium in vacuum from a SiO2 target. The only signal required was the observation of a single energetic electron from the negative muon's decay in a narrow momentum band of 6.3 MeV/c right below the maximum possible momentum 53 MeV/c of the decay electron in muon decay. A limit of GM~ < 0.14GF (90% C.L.) was deduced after one single event has been observed to fulfill the weak required criterion [41].

5

Coincidence

Signatures

of the Atom's

Decay

Major progress was achieved using a new powerful and clean signature requesting the coincident detection of both constituents of the antimuonium atom in its decay. This method was developed and applied for the first time in an approach at LAMPF, where a magnetic spectrometer was used to search for an energetic electron from the p - decay. The positron, which is expected to be left behind from the atomic shell with a mean kinetic energy corresponding to the system's Rydberg energy [42], could be electrostatically extracted from the interaction region and registered on a microchannel plate (MCP) detector. A limit of GM~ < 0.16 G~- (90% C.L.) could be established [43]. The latest experiment at PSI (Fig. 5) [44, 45], implemented major improvements over the LAMPF setup. The solid angle for the detection of the energetic electron was increased by three orders of magnitude to 70 % of 4zr by using a cylindrical magnetic spectrometer equipped with five concentric proportional chambers and a 64-fold segmented hodoscope which was constructed from the former SINDRUM I detector. The atomic positron is electrostatically accelerated and guided in a momentum selective transport system parallel to the magnetic field lines to a position sensitive MCP with resistive anode readout [46]. The tracks of these particles can be traced back

50

L. Willmann and K. Jungmann

Fig. 4. Top view of the apparatus at PSI. The observation of the energetic electron from the p - decay in the antiatom in a magnetic spectrometer with a magnetic field strength 0.1 T is required in coincidence with the detection of the positron, which is left behind from the atomic shell of the antiatom, on a MCP and at least one annihilation photon in a CsI calorimeter.

to the interaction region for reconstructing a decay vertex providing an additional suppression of background. Further the annihilation radiation of the positrons can be observed in a 12-fold segmented undoped, highly pure CsI crystal calorimeter (Fig. 5). One of the design goals for the setup was to achieve as high as possible s y m m e t r y for the detection of both, antimuonium and muonium, in order to reduce the systematic uncertainties arising from corrections for efficiencies and acceptances of the detector subsystems. The production of the a t o m s has been monitored by reversing all electric and magnetic fields regularly every few hours for half an hour. The setup at PSI has a significantly higher sensitivity for observing the decay of muonium atoms in vacuum than to those decaying inside of the production target, as it allows the coincident detection of a fast and a slow particle after the decay. Therefore a new determination method for the muonium production yield could be uniquely exploited. It is based on a model established in independent dedicated experiments [39], which assumes that

The Muonium Atom as a Probe of Physics beyond the Standard Model

51

the atoms are produced inside of the SiO2 powder at positions given by the stopping distribution of the muons. A one dimensional diffusion process describes the escape of the muonium a t o m s into vacuum where their velocities follow a Maxwell-Boltzmann distribution. The distribution of time intervals between a stop of a muon and the detected decay of a m u o n i u m a t o m in vacu u m includes full information on the a t o m ' s production rate (Fig. 5). Using an effective diffusion equation for the movement of atoms inside the target parallel to an axis (y) orthogonal to the target surface the time distribution is derived for a diffusion length l = ~ which is small compared to the target thickness a

2-v~.

DM

• exp ( • /0 t dr' fo ~ dy S(y)x/~Tg (a -- y)

(a - - y)2

4T~MM :7, )

,

(4)

with DM the diffusion constant, fM the fraction of muons stopped with a density S(y) inside of the target and forming muonium. There is no muonium in vacuum at t = 0. The m a x i m u m of r ~ M v a c ( t ) is approximately at 1.5 ps which is about the average time of diffusion for the muonium atoms in the target. The spectra contain a small exponentially decaying background arising from muon decays within the target which can be associated with the release of a secondary electron from the target material. Numerical fits (Fig. 5) typically yield a few percent of n m o n i u m a t o m s in vacuum with respect to the incoming muons. For a flat stopping distribution S(y) and target thickness a large compared to the diffusion length l an analytical integration results in ngv~c(t) = f g

exp(--Aut)

D~

(5)

The expression contains the diffusion constant DM only as a part of the normalization factor. An intermediate result from the PSI experiment, which is carried out in a two step approach, is available [45] on the basis of an effective measurement time of 210 hours during which 1.4(1) • 10 ~ muonium atoms decayed inside of the fiducial volume of diameter 9 cm and length 10 cm. No decay of an antimuonium a t o m was observed. There are no entries in a 20 ns wide window around the expected time of flight of 70 ns for the positrons from the atomic shell (Fig. 6). The apparent structure around tTOF--te×p~ct~a = --50 ns arises from the allowed rare decay mode #+ --+ e+e+e-ueP£ in which one of the positrons is released with low kinetic energy, while the electron is detected in the magnetic spectrometer. Due to their significantly higher initial m o m e n t a positrons from these processes arrive at earlier times at the M C P and can be significantly distinguished from possible antimuonium decays. A small part of this observed background signal is due to positrons from normal muon

52

L. Willmarm and K. Jungmann

~100

0

2000

4000

6000 tdecay

8000

[ns]

Fig. 5. In the setup at PSI the detection of decays of muonium atoms in vacuum is favored compared to decays of muons inside of the target. The distribution of time intervals between the detection of an incoming muon in the beam counter and the observation of the decay of an atom in vacuum carries information on the efficiency of the muonium production. The dashed line represents an exponentially decaying background.

decay which experience B h a b h a scattering in any structural component in the target region. The probability for a conversion in a 0.1 T magnetic field is PM~(0.1 T) < 2.8-10 -9 (90% C.L.), where corrections have been applied to account for differences in detection efficiencies while measuring the m u o n i u m production yield and while searching for antimuoniuna. For (V =t=A) × (V =k A) type interactions, the conversion probability is suppressed in a magnetic field of 0.1 T to 35% of the zero field value. This leads through Eq.(3) to an upper limit of GM~ _( 1.8. 10 -2 GF (90% C.L.). For diteptonic gauge bosons X ++ in G U T models a tight new mass limit of Mx++/93t > 1.1 T e V / c 2 (90% C.L.) can be extracted. This bound exceeds significantly the one deducted from high energy Bhabha scattering [23]. With these results from the first step of the PSI experiment models with dilepton exchange [19] as well as models with heavy leptons and radiative generation of lepton masses appear to be less attractive [16]. This is a nice examples for contributions of research using atomic objects for solving problems in the domain of particle physics.

6

O u t l o o k for F u t u r e E x p e r i m e n t s

The measurements in the second stage of the experiment at PSI promise further advances. Among the m a j o r improvements are a detector for positrons with four times enhanced efficiency [46] and a beam line (~Eh) with 5 times

The Muonium Atom as a Probe of Physics beyond the Standard Model

(a) events with abs(troF-texp,~d ) _<10PS

15

53

(b) events with abs(Rd, ~) _< 12mm

~4 i ~"

10

2

o~

..... :, '

..

, o

..

no~ "

'

Rdc a

°1oo

[Clll]

tl

nnnn_~,',

~.~ n n n . n .

o

loo

tTOF -texpected

Ins]

Fig. 6. The number of events with identified energetic electron and slow positron as a function of (a) the distance of closest approach Rdc~ between the electron track in the magnetic spectrometer and the back projection of the position measured at the MCP and (b) the difference of the positron's time of flight tTOF to the expected arrival time texpect,a. The signal at earlier times is due to the allowed decay channel/z --+ 3e2v and Bhabha scattering. It is smeared out because of the different acceleration voltages used. No event satisfied the required coincidence signature. The dotted and dashed curves correspond to a simulated signal for GM~ = 0.05 GF-

higher muon flux. D a t a have been collected for some 1300 hours and a preliminary result [47] is available which sets in (V ± A) × (V + A) coupling an upper limit of GM~ _< 3.2 - 10 - a GF (90%C.L.). This provides an even more stringent test for speculative extensions to the standard model, in particular to the left-right symmetric models predicting a lower bound on GM~ [14]. In order to increase the sensitivity of detecting a possible conversion process of m u o n i u m into antimuonium considerably in the future a new experimental approach will be required. The present setup has come close to the limits imposed by the available muon fluxes at present meson factories, realistic durations of running times and the rate capabilities of available proportional wire chambers. The number of accidental coincidences is expected to become a serious problem for higher beam rates. At present, possible improvements to the existing setup at PSI appear to promise only marginal progress. However, at future highly intense pulsed muon sources one could take advantage of the time evolution of the conversion signal which increases quadratically in time (Eq. 2). Therefore a detection scheme could be envisaged which again uses the powerful coincidence detection of both constituents of antimuonium and which starts to look for antimuonium decays a few muon lifetimes after the formation of the system. It should be noted that final state interaction in muonium decay could mimic an antimuonium decay, when energy is transfered from the positron of the #+ decay to the electron in the atomic shell (internal B h a b h a scattering). An energy transfer of more than 10 MeV while the positron remains with less than 0.1MeV kinetic energy has a probability of well below 10 -11 [44]. However, this process can be distinguished from an antimuonium decay by

54

L. Willmann and K. Jungmann

the characteristic energy spectra of the detectable particles as in the case of potential background from the allowed p --+ 3 e 2 u decay. Therefore, there is no principle limitation which could prevent much more sensitive searches beyond the present bounds.

Acknowledgements We are indebted to Prof. G. zu Putlitz for his constant support, advice and encouragement during our work on this appealing subject in the framework of international collaboration. * Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, MA. 02139

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L. Willmann and K. Jungmann

[34] G.M. Marshall, J.B. Warren, D.M. Garner, G.S. Clark, J.H. Brewer and D.G. Fleming, Phys.Lett. A65,351 (1978). [35] Cabot Corporation, "Cab-O-Sil Properties and Functions", technical report, Cab-O-Sil Division, Tuscola, IL (1988). [36] R.F. Kiefl, J.B. Warren, C.J. Oram, G.M. Marshall, J.H. Brewer, D.R. Harshmann, and C.W. Clawson, Phys.Rev. B26, 2432 (1982). [37] G.M. Marshall, J.B. Warren, C.J. Oram and R.F. Kiefl, Phys.Rev. D25,

1174 (1982). [38] G.A. Beer, G.M. Marshall, G.R. Mason, A. Olin, Z. Gelbark, K.I:{. Kendall, T. Bowen, P.G. Halverson, A.E. Pifer, C.A. Fry, J.B. Warren and A.R. Kunselman, Phys.Rev.Lett. 57, 671 (1986). [39] K. Woodle, K.-P. Arnold, M. Gladisch, J. Hofmann, M. Janousch, K.P. Jungmann, H.-J. Mundinger, G. zu Putlitz, J. Rosenkranz, W. SchMer, G. Schitt, W. Schwarz, V.W. Hughes and S.H. Kettell, Z.Phys. D9, 59 (1988), see also: A.C. Janissen, G.A. Beer, G.R. Mason, A. Olin, T.M. Huber, A.R. Kunselman, T. Bowen, P.G. Halverson, C.A. Fry, K.IR. Kendall, G.M. Marshall and J.B. Warren, Phys.Rev. A42, 161 (1990). [40] T.M. Huber, A.R. Kunselman, A.C. Janissen, G.A. Beer, G.R. Mason, A. Olin, T. Bowen, P.G. Halverson, C.A. Fry, K.R. Kendall, G.M. Marshall, B. Heinrich, K. Myrtle and J.B. Warren, Phys.Rev. D41, 2709 (1990). [41] V.A. Gordeev, A.Yu. Kiselev, N.P. Aleshin, E.N. Komarov, O.V. Miklukho, Yu.G. Naryshkin, V.A. Sknar, V.V. Sulimov, [.I. Tkach, V.M. Abazov, V.A. Baranov, A.N. Bragin, S.A. Gustov, N.P. Kravchuk, T.N. Mamedov, I.V. Mirokhin, O.V. Savchenko and A.P. Fursov, JETP Lett. 59, No. 9, 589 (1994). [42] L. Chatterjee , A. Chakrabarty, G. Das and S. Mondal, Phys.Rev. D46, 46

(1992). [43] B.E. Matthias, H.E. Ahn, A. Badertscher, F. Chmely, M. Eckhause, V.W. Hughes, K.P. Jungmann, J.R. Kane, S.H. Ketell, Y. Kuang, H.-J. Mundinger, B. Ni, H. Orth, G. zu Putlitz, H.R. Sch~ifer, M.T. Witkowski and K.A. Woodle, Phys.Rev.Lett. 66, 2716 (1991). [44] K. Jungmann, B.E. Matthias, H.-J. Mundinger, J. Rosenkranz, W. Sch/ifer, W. Schwarz, G. zu Putlitz, D. Ciskowski, V.W. Hughes, R. Engfer, E.A. Hermes, C. Niebuhr, H.S. Pruys, R. Abela, A. Badertscher, W. Bertl, D. Renker, H.K. Walter, D. Kampmarm, G. Otter, R. Seeliger, T. Kozlowski and S. Korentschenko, PSI proposal R-89-06 (1990). [45] R. Abela, J. Bagaturia, W. Bertl, R. Engfer, B. Fischer yon VVeikersthal, A. Grossmann, V.W. Hughes, K. Jungmann, D. Kampmann, V. Karpuchin, I. Kisel, A. Klaas, S. Korentschenko, N. Kuchinsky, A. Leuschner, B.E. Matthias, R. Menz, V. Meyer, D. Mzavia, G. Otter, T. Prokscha, H.S. Pruys, G. zu Putlitz, W. Reichart, I. Reinhard, D. Renker, T. Sakelashvilli, P.V. Schmidt, R. Seeliger, H.K. Walter, L. Willmann and L. Zhang, Phys.Rev.Lett. 77, 1950 (1996). [46] P.V. Schmidt, L. Willmann, R. Abela, J. Bagaturia, W. Bertl, B. Braun, H. Folger, K. Jungmann, D. Mzavia, G. zu Putlitz, D. Renker, T. Sakelashvilli and L. Zhang, Nucl.Inst.Meth. A376, 139 (1996). [47] P.V. Schmidt, doctoral thesis, Heidelberg, unpublished (1997); see also: H.P. Wirtz, doctoral thesis, Zfirich, unpublished (1997).

Can A t o m s Trapped in Solid H e l i u m Be U s e d to Search for Physics B e y o n d the Standard Model? A. Weis, S. Kanorsky, S. Lang and T.W. H/insch Max-Planck-Institut fiir Quantenoptik, Hans-Kopfermann-StraBe 1, 85748 Garching, Germany

Introduction Since the 1960's experiments using atomic physics techniques have played an ever increasing role in testing fundamental elementary particle theories. In particular studies of atomic processes in which the fundamental symmetries P (mirror inversion, parity) and T (time reversal) are violated have yielded physical insights which are complementary to the information gained from nuclear and high energy physics experiments. One of the most sensitive ways to test T-symmetry is the search for permanent electric dipole moments of elementary particles which cannot exist if P and T are good symmetries. In the past four decades experiments aimed at the detection of such moments have been carried out on the neutron and on various atoms (Ramsey 1994). While reaching spectacular sensitivities all of these experiments have so far measured values of the dipole moments which are compatible with zero. In quest for a radically new experimental approach to the dipole moment search in atoms we have been led to consider paramagnetic atoms trapped in condensed 4He as a promising new sample. The scope of the present paper is to review the knowledge we have gained in the past years on the properties of this novel sample and to discuss its applicability to an experiment searching for T-violation. In order to motivate the interest in atomic electric dipole experiments we will start by giving a brief general account of the role played by atomic physics experiments in the investigation of the discrete symmetries P and T. We will show that the search for permanent electric dipole moments is a search for physics beyond the standard model of electroweak interactions. After reviewing the state-of-the-art of the experimental results in this field we will outline the general principles common to all electric dipole experiments. The major part of the paper will be devoted to a detailed discussion of the novel approach to the search for electric dipole moments pursued by our team at the Max-Planck-Institut fiir Quantenoptik (MPQ) in Garching since a couple of years.

58

Can Atoms Trapped in Solid Helium Be Used ... ?

Violations of Discrete Symmetries Parity Violation In 1957 several experiments observed that parity is violated in ~-decay, i.e. in processes which are governed by the weak interaction. The subsequent development of the standard model of electroweak interactions - a unified description of electromagnetic and weak interactions - in the 60's and the experimental observation of weak neutral currents in neutrino scattering experiments in the early 70's gave rise to the speculation that atomic properties should also be affected by the weak interaction and that parity violation should be observable in atomic physics. In a seminal paper C. and M.-A. Bouchiat showed in 1974 (Bouchiat and Bouchiat 1974) that heavy atoms are particularly sensitive to contributions from weak interactions (Z 3 law), and that modern laser spectroscopic techniques would have the sensitivity required to measure such effects. In the two decades since, parity violation was measured in various heavy atoms. All experimental results are in perfect agreement with the predictions from the standard model, the most precise experiments reaching an accuracy of 2.5% (for a review see e.g. Bouchiat 1991). The origin of parity violation in atoms is a quantum-mechanical interference of two (neutral current) amplitudes describing the interaction of the electron with the nucleus; one being the electromagnetic Coulomb interaction mediated by virtual photons, and the second being the weak interaction mediated by the neutral intermediate vector boson Z°. The fascinating aspect of the atomic parity experiments is the fact that table top experiments allow to observe effects induced by the Z° at q2 values (q = four-momentum transfer between photon and electron) which are ten orders of magnitude below the ones used in accelerator experiments. The next generation of atomic parity violation experiments aims at pushing the sensitivity to the 1% level and even beyond (Bouchiat 1991, Moriond 1994). This level of precision will allow the study of a number of new phenomena, such as nuclear spin dependent P-violating effects , the nuclear anapole moment (a P-violating nuclear property), as well as radiative corrections to the standard model. Atomic parity violation is thus expected to provide further tests of the standard model. Time Reversal Invariance While the parity symmetry and the various manifestations of its violation are nowadays well understood in the frame of the standard model, time reversal symmetry remains rather obscure. So far only a single experimental evidence for T-violation was found (decay of the long-lived component of the K ° meson). In the 50's Purcell and Ramsey had already pointed out (Purcell 1950) that there was no a priori reason to a s s u m e the conservation of discrete symmetries and that such a belief must be based oil experiment. In 1950 they proposed to search for an electric dipole moment (EDM) of the neutron as a test of P-conservation. After the fall of parity in 57 they argued further that T conservation was still a strong argument against the existence of permanent EDM's in elementary particles. The search for P and T violating electric dipole moments in atoms and the

EDM Experiments

59

neutron is now widely recognized to be one of the most sensitive ways to test T-symmetry (Ramsey 1994). The standard model can accomodate T-violation through a complex phase factor (Kobayashi-Maskawa mixing angle). However, the magnitudes of the electron and neutron electric dipole moments predicted by this naeehanism are 8 - 9 orders of magnitude below the current experimental sensitivities. When considering that, averaged over the past 30 years, the experimental sensitivity has been lowered by of one order of magnitude every 8 years, there is little hope that the experiments wilt ever be able to test standard models predictions. On the other hand there are several reasons to believe that the standard model is not the ultimate theory of electroweak interactions 1. For this reason several extensions of the standard model or alternative theories have been developed in the .past decade, some of them predicting values for the electron or neutron EDM's which are well within the reach of modern EDM experiments or their future extensions (Barr 1994, Bernreuther and Suzuki 1991). In this sense the hunt for the EDM of an elementary particle is probably the most promising way to search for physics beyond the standard model, or, as Steven Weinberg, one of the fathers of the standard model has put it: "... may be that the next exciting thing to come along will be the discovery of a neutron 07" atomic or electron electric dipole moment. These electric dipole moments . . . s e e m to me to offer one of the most exciting possibilities for progress in particle physics." 2. Because of the smallness of the standard model predictions any non-zero result in an EDM experiment would be a clear indication for physics beyond the standard model.

EDM Experiments State-of-the-art The neutron EDM has been chased by N. Ramsey and his collaborators for more than 40 years, first with neutron beam magnetic resonance techniques and more recently with ultracold neutrons trapped in a neutron "bottle" (Smith et al 1990, Ramsey 1994). This series of experiments, which counts among the longest ongoing experimental effort in the history of modern physics, has recently been resumed at the ILL in Grenoble after a long shut-down of the reactor. Atomic physics has entered the scene of T-violation experiments in the 60's, with a first experiment that set an upper limit on the EDM of the laaCs atom using a conventional Ramsey magnetic resonance method. More and more refined techniques have been developed since, in particular by applying optical methods to spin polarize the samples and to detect the magnetic resonance. Nowadays the sensitivities of EDM experiments in atoms and the neutron have now reached comparable levels. In Table 1 we compare the present upper limits for the EDM of the neutron and of various atoms, as well as the inferred EDM's of the electron and the proton. t For a discussion of arguments see e.g. the review papers by Barr (Barr I994, Barr 1993). 2 cited in (Barr 1994)

60

Can Atoms Trapped in Solid Helium Be Used ... ?

T a b l e 1. Experimental upper limits for permanent electric dipole moments taken from a recent compilation (Ramsey 1994). Numerical values ai-e given in traits of 10-2r e cm. neutron

t99Hg

129Xe

2OST1

133Cs

system EDM

< 90

< 0.63

< 210

< 2 300

< 20 000

inferred electron EDM

-

<44

< 2104

<3

<90

T h e atomic experiments can be seen to fall in two categories, viz. experiments on diamagnetic atoms (Hg, Xe) and experiments on paramagnetic atoms (Cs, T1). The former give lower limits on the atomic EDM from which mainly information on T-violating processes within the nucleus or in the electron-nucleus interaction can be deduced, while the latter are more sensitive to contributions from an EDM of the electron. Atomic EDM's Electric dipole moments d are measured via their interaction - d . E with a static electric field E . The acceleration of charged particles by strong electric fields therefore precludes the direct measurement of the EDM of particles such as the proton or the electron. If one tries to counteract this acceleration by other forces, as in bound systems of charged particles such as atoms, the total electric field at the particle's location will vanish and the individual particles will no longer experience a dipole interaction. This simple fact, which was first pointed out by Ramsey and Purcell in the 50's was generalized in the 60's by Schiff (Schiff 1963) in a form usually referred to as the Schiff Theorem. The i m p o r t a n t point of Schiff's paper was however the remark that the above considerations no longer hold when the motion of the particles is relativistic, when other than electrostatic forces act on the particles or when the charge distribution within the particles differs from the EDM distribution, conditions which are met in heavy atoms. Sandars (Sandars 1966) was the first to calculate the atomic dipole m o m e n t due to a dipole nmment of the electron and found that for heavy paramagnetic atoms the above discussed screening is actually an antiscreening, usually referred to in terms of the enhancement factor R = datom/detection. Note that this phenomenon is closely related to the well-known Sternheimer correction which describes the screening/antiscreening of external electric filed gradients experienced by the nucleus in an atom. Typical values of R are 120 for 133C8and -600 for 2°STl. The atom as a whole can only have a significant EDM contribution from the electron EDM if the atom has a non-zero electronic angular m o m e n t u m J. In diamagnetic atoms with J = 0 (such as 199Hg or 129Xe) the main EDM contribution comes from nuclear EDM's, which are not enhanced, due to the nonrelativistic motion of the nucleus. This explains why paramagnetic atoms are more sensitive to electron EDM's (cf Table 1).

Atoms in Solid 4He

61

G e n e r a l principles of E D M e x p e r i m e n t s All EDM experiments share several common principles which rely on the fact that if a particle has both magnetic/~ and electric d dipole moments, these have to be oriented parallel or antiparallel to the particle's angular momentum J . The spin precession frequency of the particle when placed in parallel/antiparallel magnetic and electric fields B and E is then given by

where WL--

tt.B h

is the Larmor frequency and

WE --

d.E h

its electrical

equivalent; the relative orientation of the fields is given by ~ = / ) • ~J. The various EDM experiments differ by the techniques used to detect the precession frequency and its EDM induced change. Most experiments use magnetic resonance techniques, while some recent atomic EDM experiments use magnetic level crossing techniques. The experimental sensitivity is determined by the phase angle ¢ = COprr accumulated during the coherent spin precession time r. Sensitive experiments thus call for large electric fields and long interaction times (spin relaxation times) r, i.e. narrow magnetic resonance lines. Experiments in diamagnetic atoms use nuclear magnetic resonance (NMR), while experiments on paramagnetic atoms detect electron spin resonance (ESR). Nuclear spin polarized samples are known to have spin relaxation times which are orders of magnitude larger than electronic spin relaxation times. This explains why experiments on diamagnetic atoms usually yield lower limits for atomic EDM's.

A t o m s in Solid 4He A t o m s in H e I I In 1991 one of us (S.K.) proposed to use spin polarized paramagnetic atoms implanted into superfluid 4He (HelI) as a radically novel sample to search for atomic EDM's. This proposal was motivated by the perspective that liquid He could serve as a trap for the implanted atoms, where long storage times in an isotropic, non magnetic environment would allow long coherent spin precession times and hence very narrow electronic magnetic resonance lines. In addition superfluid He was known to hold electric field strengths in excess of 1O0 kV/cm and presented furthermore the advantage of being ideally suited for the implementation of superconducting magnetic field shields which allow to isolate the sample from magnetic perturbations from the laboratory enviromnent. Moreover, the matrix itself has vacuum-like properties for the electromagnetic fields used in the experiments, be they static E or B fields, optical fields in the visible or near-infrared spectral regions, or radio-frequency and microwave fields. The idea of using matrix-isolated atoms was triggered by the reading of a series of

62

Can Atoms Trapped in Solid Helium Be Used ... ?

papers in which a team from the University of Heidelberg described a technique which had allowed them for the first time to implant foreign atoms into superfluid Helium and to observe recombination fluorescence, as well as laser-induced fluorescence from a variety of atomic species. The team was directed by G. zu Putlitz. In 1991 we started at MPQ in Garching a project aimed at the investigation of magneto-optical effects in paramagnetic atoms in condensed Helium. In the past the main interest of the only team working in the field (Heidelberg) was concentrated on the development of implantation techniques and on the study of optical properties of implanted species (Reyher et al. 1986, Tabbert et al. 1995). At the time it was not known whether atoms in HeII could be spin-polarized by optical pumping nor whether it would be possible to observe magnetic resonance in such atoms. Inspired by the Heidelberg work we designed a double bath cryostat which became operational in 1992. In a first series of experiments we studied optical spectra of Ba atoms in superfluid He. These studies have yielded two important results. On one hand the study of the pressure shift of the Ba resonance lines allowed us to infer quantitative informations on the structure of the atomic trapping sites in He (Kanorsky et al. 1994a, Kanorsky and Weis 1996). On the other hand we discovered a novel and simpler implantation technique for foreign atoms. An energy of approximately leV is required for an atom to penetrate into superfluid He. For this reason atoms cannot be implanted into liquid He from an atomic beam hitting the liquid-vapor interface of a He bath. The Heidelberg technique consisted in creating atomic ions above the He bath either by a discharge or by laser ablation. We chose the laser sputtering method in which ablated positive ions are accelerated electrostatically into the superfluid volume, where they are made to recombine with electrons extracted by field emission from a tunneling microscope tip. During our experiments we found that when the He level in the inner chamber rose above the target, the ablation process worked equally well, and that even when no voltage was applied to the field emission tip, a strong atomic LIF signal could be detected (Arndt et al. 1993). A similar technique was developped independently at the same time by a team at the university of Kyoto (Fujisaki et al. 1993). This important discovery thus opened the road to extend the experiments from the liquid phase of He to its solid phase (Kanorsky et al. 1994b). Starting in 1993 we therefore focussed our efforts first on pressurized superfluid and afterwards exclusively on solid He matrices. The latter offers the great advantage that it allows to obtain atomic trapping times ranging from minutes (Cs) to hours (Ba). Implantation

o f a t o m s i n t o solid H e

The original experiments were done in a double bath cryostat, in which the temperature is lowered to 1.5 K by evaporative cooling. Fig. 1 shows a schematic diagram of the cryostat used in our recent experiments. The major features are a single bath and a pressure cell with optical access fi'om 5 sides. The cell has optical access through five fused silica windows. Soft copper gaskets instead of In wire rings were used to seal the windows in order to avoid distortions of the magnetic fields applied with three pairs of superconducting Hehnholtz coils (not shown ill Fig.i) located outside of the cell. Two pairs

Atoms in Solid 4He

63

capillary

to pump

f Nd-YAG (2¢o)

~ I ~- ' \ - - - - ~ ~

J !iiii:~ ~ ~spectroscopy -' ']'~ F'--- cloudwith implantedatoms

Fig. 1. Schematic diagram of the cryostat used for the spectroscopy of atoms in solid 4He (left) and blown-up view of the pressure cell (right). The ablation laser can be focussed either onto the target for the implantation process or into the atomic cloud in order to dissociate clusters.

of coils inside the cell were used to produce the oscillatory magnetic fields. A/zmetal ( C R Y O P E R M 10) cylinder closed on one side shields laboratory magnetic fields at a level of raG. Optional additional features of the cell are resistive temperature and pressure sensors. At a temperature of 1.7 K a pressure of 27 bar has to be applied in order to solidify He (Fig.2). The required pressure is controlled by a room temperature pressure tank connected to the cell by a capillary. The ablation laser (frequency doubled Nd-YAG) is focussed through the top window onto the metal target located at the b o t t o m of the cell. By the deposited heat (typically 40 mJ per pulse at a rate of 10 Hz) the solid helium melts in the vicinity of the target. At the same time atoms are ablated into this liquid region and trapped therein in a conically shaped region after resolidification of the helium. Details of this implantation technique have been given elsewhere (Arndt et al. 1995a).

64

Can Atoms Trapped in Solid Helium Be Used ... ?

31 3O -~

29

"~

28 27 26 25

1.4

1.5

1.6

1.7

1.8

1.9

T (K) Fig. 2. p-T phase diagram of condensed 4He The spectroscopy laser 3 is sent through the cloud of implanted a t o m s and the fluorescence light - collected at right angles with respect to the direction of the laser b e a m - is filtered by an interference filter and detected with a cooled photonmttiplier. We observe a decrease of the fluorescence intensity on a time scale of minutes which we attribute to atomic diffusion and cluster formation. An efficient way to regenerate the atomic signal is to occasionally focus a few pulses of the ablation laser into the region where the atomic cloud and the spectroscopy laser overlap. Optical properties

o f a t o m s in c o n d e n s e d 4 H e

The optical properties of immersed foreign a t o m s can be described the so-called bubble model (Bauer et al. 1990). We only briefly review here its basic features; an extensive discussion can e.g. be found in (Kanorsky and Weis 1996). The interaction of tile implanted foreign a t o m with the m a t r i x a t o m s is characterized by a strong (hard-core type of) repulsive potential 4 and a weak van der Waals attractive potential. The repulsion expels He a t o m s from the location of the atomic defect, thus creating a bubble-shaped cavity. The shape of this bubble reflects the s y m m e t r y of the atomic wave function and its size is determined by the balance of the Pauli repulsion on one side and the bubble surface tension and pressure-volume work required for the bubble creation on the other side. The optical absorption-emission cycle (Fig.3) can now be understood as follows: Consider an a t o m with a spherically symmetric ground state located in a spher3 depending on the implanted species we used dye lasers, a Ti:Sapphire laser or diode lasers. 4 due to the Pauli principle

A~oms in Solid ~He

65

ically symmetric cavity. During the optical absorption process the shape of the cavity does not change as a consequence of the Franck-Condon principle.

~/. ....

.HI,

S S I ~ J S] H ~ t S ~ N I ~ . , , .

1p|'~¢~;~{~;~

.

®

................

6pl/;

......

.

.

.

.

.

.

.

.

.

...........

,[,

........... ISo

.

® iiil

.. . . . . .. . . .

.

.

.I S o .

.

.

.

.

.

.

.

.

""" ......

681/2

Fig. 3. Schematic representation of the bubble evolution during the absorptionemission cycle of Ba (a) and Cs (b) atoms in condensed He The atomic wave function of the excited state occupies a larger volume and thus exerts a stronger pressure on the cavity wall. The energy of the system is minimized by a shape relaxation of the bubble, which adopts to the symmetry of the atomic wave function. It becomes oblong for the aspherical 1P1 state of excited Ba and stays spherical in the case of the excited 2P1/2 state of Cs. This shape relaxation process is fast (ps) as it involves the displacement of He atoms by a few /1~at the speed of sound in He. The atom then decays in the relaxed bubble, and again, the shape does not change during this emission process. The cycle is completed by a relaxation of the bubble towards its initial state. As a consequence the optical excitation line is strongly perturbed and shows a large blue shift (40 nm for the Cs D1 line) and a strong broadening (several nm), while the emission line is less shifted and less broadened. From an experimental point this large Stokes shift is very convenient as it allows to efficiently discriminate against scattered laser light in laser induced fluorescence experiments. The above qualitative arguments can be put into a quantitative form and give a reasonable description e.g. of the lineshapes of excitation lines in pressurized He (Kanorsky et al. 1994a, Kanorsky and Weis 1996).

66

Can Atoms Trapped in Solid Helium Be Used ... ?

O p t i c a l p u m p i n g a n d spin r e l a x a t i o n t i m e The observation that the bubble model gives a satisfactory description of the optical spectra of Ba atoms in solid helium let us expect that the repulsion exerted by atoms in spherically symmetric ground states is stronger than the helium binding energy. As a consequence we expected that the symmetry of the trapping site to be determined by the symmetry of the atom-helium interaction rather than by the crystalline structure of the matrix 5. The absence of local crystalline field gradients resulting from the spherical symmetry of the cavity, together with the diamagnetic character of the matrix (neither the He nucleus nor its electron shell carry magnetic momenta) are necessary conditions for the creation of a large degree of spin polarization of implanted atoms. We tested this hypothesis in optical pumping experiments on implanted paramagnetic Cs atoms. Cs was the atom of choice for these studies, on one hand because it has a readily accessible resonance line in the near infrared with a large oscillator strength, and on the other hand because it is a promising candidate for a future EDM experiment. Optical pumping is a very efficient method for the creation of spin polarization in atoms. Due to the large homogeneous linewidth (3~optical 400 A~h.].,.) of the optical absorption line the hyperfine structure/-3~hJ.s, of the transition remains completely unresolved. When Cs atoms are illuminated with circularly polarized light resonant with the Dl-transition (6S1/2, F = 3,4 -+ 6P1/2, F = 3,4), the optical field couples to all the Zeeman sublevels of the ground state with the exception of the IF=4, MR=4) state, which is thus a dark state for circularly polarized light. After several absorption-emission cycles most of the population will be transferred to the dark state; in which both nuclear and electronic spins are fully polarized as IF=4, ME - - 4 ) - - I J = l / 2 ,

Mj = 1/2)1I=7/2, M r = 7/2).

The build-up of polarization, i.e. the increase of dark state population can be conveniently detected by the observation of a decrease in fluorescence. We have measured the longitudinal spin relaxation time T1 by the method of "relaxation in the dark". The technique consists of first polarizing the sample by optical pumping, and then turning off the exciting light, upon which the spins are allowed to relax "in the dark". The fluorescence level found immediately after turning the pump light back on after a variable dark time is a measure of the polarization which has survived the dark period. The results showed that the spin relaxation times were indeed very long. Values for T1 the range of 1 - 2 seconds were found (Arndt et al. 1995b) with magnetic holding fields as low as 10 mG (Fig. 4). The T1 values showed no significant magnetic field dependence up to 7 G, but scattered slightly from sample to sample. We assume tha.t impurities, such as paramagnetic Cs clusters are responsible for the observed depolarization rate of 1 s -1. These relaxation times are among the longest ever measured for electronic spin polarization. However, these long relaxation times could only be observed with a.toms implanted in the isotropic b.c.c, phase of the matrix. In 5 Deviations from this expectation will be discussed below

Atoms in Solid 4He

67

the (anisotropic) h.c.p, phase almost 1000 times smaller T1 times were observed (Fig. 4) indicating additional relaxation processes in that phase (see below).

1.0 IN,

'

{

'

t

'

0.8~-{-~-~

0.6:

I

'

I

--

holt mamx: bcc l--

"4

/

°4 i 0.2

(a)

0.0

,

l

0.0

,

{

0.5

t 1.0~

,

I

,

{

1.0

'

,

{

1.5

2.0

(s) '

J

0.8

,

I

'

1

{

host matrix: hcp

=

(4)

m

0.6-

L

"~ 0.4 "~, e-

_

0.0

(,b) I

0

2

,

I

,

4

I

6

,

I

8

,

I

10

tdark(ms)

Fig, 4, Measurement of longitudinal spin relaxation times in b.c.c, matrix (a) and h.c.p, matrix (b). Solid lines represent fits with exponential decay curves.

Optically detected magnetic resonance Having shown that a large degree of long-lived spin polarization can be created in Cs atoms isolated in solid 4He we have investigated magnetic resonances in the Cs ground state. In these experiments a longitudinal magnetic holding field B0 stabilizes the polarization build up by optical pumping. A linearly polarized magnetic field oscillating at the frequency wrf resonantly destroys spin

68

Can Atoms Trapped in Solid Helium Be Used ... ?

polarization when ¢zrf = ~0, where w0 is the Larmor precession frequency in the static field. The magnetic resonance is detected via the resonant increase of the fluorescence. In our early experiments (Arndt et al. 1995b, Weis et al 1995) we have observed in this way magnetic resonance lines with typical widths of a few kHz, corresponding to effective values for the transverse relaxation time T2 of 100 #s. T h e relatively large linwidths were determined by magnetic field inhomogeneities. In a recent rebuilding of the experimental chamber we have shielded the pressure cell from laboratory fields and have improved the homogeneity of the applied fieldS. The results were very satisfying and have yielded resonance linewidths which are 2-3 orders of magnitude narrower than in the previous experiments. Fig. 5 shows a typical example of such resonances.

i

i

i

i

i

i

i

.-_ 0.28 e-I

.~ 0.24

: FWHM = 15 Hz

~9

=

0.16

\

/

0.20

\

\

J

F=4 40

-20

F=3

0' , 2t0 , 4O , , 60 , , 8~ , v - v (Hz) rf

0

Fig. 5. Optically detected electron spin resonance in Cs atoms in b.c.c. 4He. The magnetic field Bo was 42.5 mG, corresponding to a resonance frequency Vo of 14.89 kHz. The line splitting is due to the nuclear magnetic moment.

The linewidth in this particular case is 15 Hz and still does not reflect the intrinsic transverse relaxation time, but is rather due to power broadening from tile optical p u m p i n g process. In order to infer the intrinsic T2 value we have performed a magnetic resonance experiment in the time domain. Fig. 6 shows a typical free-induction (FID) decay signal obtained by a sudden switching of the orientation of the longitudinal holding field by 90 degrees. Note that Fig.5 is basically the Fourier transform of Fig. 6. The FID decay time is of course also shortened by the optical p u m p i n g process and is on the order of 10 ms for the particular case shown in Fig.6. The FID technique allows however to perform an " F I D in the dark" type of experiment analogous to the one used to measure T1. Here again the laser is switched off a few FID periods after the flipping of the magnetic field so that the decay occurs in the dark. By varying the dark time one

Atoms in Solid 4He

69

can thus measure the intrinsic transverse relaxation time. In this way we have obtained T2 times exceeding 100 ms. The reason why T2 is'approximately one order of magnitude shorter than Tt is not clear at present, but may probably be traced back to magnetic field inhomogeneities at the level of a few #G. Planed spin echo experiments will probably help to clarify this point.

I

0.4

~

--~ -

!

- long

!

y~-Z

s

0.3

0.2

0.1

0.0 I

0.00

i

I

,

0.01

l

0.02

L

0.03

t (s) Fig. 6. Free induction decay of Cs ground state spin polarization in transverse magnetic field

Magnetometrie

sensitivity

One of the experimental problems regarding a possible EDM experiment is the fact that the atomic signal degrades as a consequence of cluster formation, thus posing severe normalization problems if signals recorded with electric fields of opposite polarities are to be compared. A possible way to circumvent this problem is the use of a dispersively-shaped resonance signal in an active feed-back loop which keeps the detuning u0 - ur] nulled. In this way any signal change affects only the gain in the feed-back loop and does not show up as a signal change. Flecently we have performed first tests with such an arrangement. The technique is based on a spin precession driven by a if-field as shown in Fig 7. Here the static magnetic field is oriented at 45 ° with respect to the optical pumping direction and a linearly polarized r.f. field resonantly drives the spin precession. The fluorescence is detected with a phase sensitive amplifier tuned to the r.f. frequency. This scheme has the further advantage that high modulation frequencies can be used, thus suppressing low-frequency noise in the sytem. A typical spectrum recorded with this arrangement is shown in Fig. 8.

70

Can Atoms Trapped in Solid Helium Be Used ... ?

fluorescence detection

X

y/

z

~0

.'"'~

1

lock-in

Q circ. pol. light

Brf(t)

S

Fig. 7. Experimental arrangement used to observe driven spin precessmn

I

I

i

I

I

[

i

0.4 0.3 0.2 ..a

0.1

z5

0.0 -0.1 -0.2 -0.3 -40

t

i

-30

-20

,

I

,

-to

I

,

0

I

lo

,

I

20

,

I

30

,

40

r e - v ° (Hz) Fig. 8. Driven spin precession recorded with integration time of 1 second/point. Inset shows data scatter near center of resonance (to = 4 0 k H z ) .

Tile analysis of the curve shows that we can detect frequency changes of approx. 10 mHz with an integration time of 1 second. This corresponds to magnetic field changes of 28 nG. Note that this is a preliminary first result obtained

Atoms in Solid 4He

71

without optimizing system parameters. It is nevertheless very promising in view of an EDM experiment as we will discuss below. T o w a r d s a n E D M e x p e r i m e n t w i t h Cs a t o m s in s o l i d 4 H e

Statistical considerations. The noise in the data of Fig.8 is predominantly determined by instabilities of the magnet current supply and fluctuations of the imperfectly shielded laboratory magnetic fields. The planed implementation of a superconducting magnetic shield and the integration of the current supply in a feed-back loop should allow to suppress these noise sources and hence to improve the magnetometric sensitivity AB. Using Eq. 1 the sensitivity ._4d to electric dipole moments can be directly expressed in terms of A B through

Ad = l i A B l E Using E = lOOkV/crn and I*/h = 1.4MHz/G we get Ad = 1.6. l0 -21 e c m with an integration time of 1 second. Typical total integration times in EDM experiments are 100-200 hours. If we assume that the sensitivity B d scales with the integration time T as Ad(T) = Ad(lsecond) T -t/2 we obtain

Ad(lOOhours) = 2.7.

10 - 2 4

ecm.

from the present results. When comparing this value to the upper limit of the EDM in paramagnetic atoms set in the past by the Amherst (Cs) (Murthy et al. 1989) and Berkeley (T1) (Commins et al. 1994) teams Ad(laacs) < 9.10 .24 and Ad(2°STl) < 2 . 1 0 -24 ecru we see that our present apparatus is already quite competitive. We stress once more that there are several realistic possibilities to further increase the sensitivity. Besides the above mentioned reduction of the magnetic noise, the signal can be further enhanced by optimization of the fluorescence detection system. We anticipate that an improvement by one to two orders of magnitude seems to be quite realistic. Systematic effects. Tile statistical considerations presented above were based on the assumption that the EDM experiment can be carried out in an electric field of 100 k V / c m . So far we have not tried to apply any electric field to our samples. While condensed He is known to hold electric fields in excess of 100 kV/cm the question whether this is also true for our doped He samples remains open and will definitely require an experimental answer. Several systematic effects that can mimic an EDM have to be considered. Any effect which has the same signature as an EDM (as e.g reversal of sign under a reversal of E) will contribute to the systematics, but may be suppressed by using further reversals of system parameters, such as the magnetic field or the handedness of the pumping light. Motional fields. Compared to beam experiments EDM experiments on trapped species offer the great advantage that sytematic effects arising from the motional magnetic field seen by atoms moving through tile electric field are absent.

72

Can Atoms Trapped in Solid Helium Be Used ... ?

hnperfect field reversal. EDM's are inferred from a change of the Larmor precession frequency under a reversal of the electric field.: A change A E of the field magnitude upon reversal mimics an EDM. For an ultimate limit of Ad = 10--95e em the maximum tolerable value is A E = Adios(~) = 1 m V / c m , where a(2) is the static electric tensor polarizability of the ground state, which, for S-states arises only in third order of perturbation. The T1 EDM experiments have shown that this effect can be dealt with (Commins 1994). Leakage currents. Leakage currents between the field electrodes produce magnetic fields, which may affect the spin precession. As these fields reverse with the plate voltage, they can also mimic an EDM. As we have no experience with electric fields yet we cannot make a quantitative statement about this effect at present. Gravitational effect. The atom is held within the He bubble by its electron cloud. Due to gravity the nucleus of the atom will therefore be displaced from the center of mass of the electron cloud, and thus give rise to an atomic electric dipole m o m e n t dgravity whose magnitude can be estimated by

dg,.~,ity= ( S ez S} = 2M~tomge ](PIzIS)I2 E, - Ep If we consider only the admixture of the closest P state we obtain for Cs dg .... ity ~ 2. lO-~3ecm. This value is approximately two orders of magnitude larger than the ultimately sought sensitivity to EDM's. It will therefore put severe constraints on the control of the electric field orientation. On the other hand the gravity induced dipole moment is ideally suited to test our apparatus, as it has a well known orientation and is of a magnitude which can already be accessed with our present sensitivity. Matrix effects. We remind that in the pressure and temperature range of our experiments solid helium has two crystalline structures (Fig.2): an isotropic body cubic-centered (b.c.c.) phase and an anisotropic uniaxial hexagonal close-packed (h.c.p.) phase. In the h.c.p, matrix the lattice constants are in the ratio of 1.6:1. An interesting question, not addressed so far by theorists, is whether the Pauli repulsion between implanted atom and the He matrix is strong enough for the atom to impose a spherical shape on the bubble, or whether the bubble shape is influenced by the anisotropy of the matrix far from the local trapping site. We have several experimental evidences that the bubbles in h.c.p, are indeed deformed. On one hand we have observed (in h.c.p.) A M F = 2, 3 forbidden magnetic resonance lines which indicate a breakdown of the cylindrical s y m m e t r y imposed by the magnetic field and thus evidence the existence of all additional local axis. Further evidence is that the spin relaxation times are two to three orders of magnitude smaller than in b.c.c, which again may be explained by a

Atoms in Solid 4He

73

'?

",:-.:-"::..::......

i':-:

Fig. 9. Electric dipole moment induced by gravity.

spin coupling to local fields. The strongest evidence is our recent observation of magnetic resonances in zero magnetic field, which we attribute to a quadrupolar splitting of the hyperfine Zeeman sublevels due to the interaction of the nuclear electric quadrupole moment with electric field gradients present in deformed bubbles. The analysis of these effects is still in progress and should allow to infer from the quadrupole splitting the strength of the field gradients and, together with model calculations, to determine the size of the bubble deformation. First estimates show that the deformations are small compared to the bubble size. The complete absence of all the effects discussed here in the b.c.c, phase should allow us in the future to ascertain with very high accuracy the degree of sphericity of the bubbles in that phase.

Summary In this paper we have reviewed tile information gained in the past three years in a novel approach to high resolution spin physics in matrix-isolated atoms. We hope to have convinced the reader that these samples - besides being interesting by themselves - have very promising applications in view of an experiment testing fundamental symmetries. To the best of our knowledge our laboratory has remained so far the only one investigating foreign atoms in solid lie matrices. We do hope that in future more researchers will be attracted by these most f~scinating novel samples.

74

Can Atoms Trapped in Solid Helium Be Used ... ?

Acknowledgement s The authors acknowledge the contributions of M. Arndt, who, during his Ph.D. work has set up the low-temperature lab at MPQ and who was in charge of all the experiments in the early stage of the project. We also thank R. Dziewior, S. Liicke, and S.B. Ross who have provided valuable contributions at different stages of the experiments reported above.

References Arndt M., Kanorsky S.I., Weis A., and Hansch T.W. (1993): Can Paramagnetic Atoms in Superfluid Helium be Used to Search for Permanent Electric Dipole Moments?, Phys.Lett. A174, 298 Arndt M., Kanorsky S.I., Weis A., and Haensch T.W. (1995a): Implantation of Metal Atoms into Solid Helium, Z.Phys. B98, 377 M. Arndt, S.I. Knnorsky, A. Weis, and T.W. Haensch (1995b): Long Electronic Spin Relaxation Times of Cs Atoms in solid 4 He, Phys.Rev.Lett. 74, 1359 Burr S. (1993): A Review of CP Violation in Atoms, Int.J.Mod.Phys. A8, 209 Barr S. (1994): Atomic Electric Dipole Moments and CP Violation, in XXIXth Rencontres de Morion& (Editions Frontieres) Bauer H., Beau M., Friedl B., Marchand C., Miltner K. and Reitner H.J. (1990): Laser Spectroscopy of Alkaline Earth Atoms in HeII, Phys. Lett. A146, 134 Bernreuther W. and Suzuki M. (1991): The Electric Dipole Moment of the Electron, Rev.Mod.Phys. 63, 313 Bouchiat M.A. and Bouchiat C.C. (1974): Weak Neutral Currents in Atomic Physics, Phys.Lett. 48B, 111 Bouchiat M.A. (1991): Atomic Parity Violation, a Low Energy Test of the Electroweak Standard Model, in it Atomic Physics 12 ed. by R.R. Levis and J.C. Zorn, p 399 Commins E.D., Ross S.B., DeMille D., and Regan B.C. (1994): Improved Experimental Limit on the Electric Dipole Moment of the Electron, Phys.Rev.A50, 2960 Fujisaki A., Sano K., Kinoshita T., Takahashi Y., and Yabuzaki T. (1993): Implantation of neutral atoms into liquid helium by laser sputtering, Phys.Rev.Lett. 71, 1039 Kanorsky S.I., Arndt M., Dziewior R., Weis A., and H~inseh T.W. (1994a): Pressure Shift and Broadening of the Resonance Line of Barium Atoms Trapped in Liquid Helium, Phys.Rev.B 50, 6296 Kanorsky S.I., Arndt M., Dziewior R., Weis A., and H/insch T.W. (1994b): Optical Spectroscopy of Atoms Trapped in Solid Helium, Phys.Rev.B49, 3645 Kanorsky S. and Weis A. (1996): Atoms in Nano-Cavities, Les Houches Summer School Quantum Optics of Confined Systems, (Kluver). Moriond (1994): see various contributions in XXIXth Rencontres de Moriond, (Editions Frontieres } Murthy S.A., Nrause D., Li Z.L., and Hunter L.R. (1989): New Limits on the Electron Electric Dipole Moment from Cesium, Phys.Rev.Lett. 63,965 Purcell E.M. and Ramsey N.F. (1950): On the Possibility of Electric Dipole Moments for Elementary Particles and Nuclei, Phys.Rev. 78,807 Ramsey N.F. (1994): Electric Dipole Tests of Time Reversal Symmetry, in Atomic Physics 14, ed. by D.J. Wineland, C.E. Wieman, and S.J. Smith, pp 3-17 Reyher M.J., Bauer H., Huber C., Mayer R., SchMer A., and Winnacker A. (1986): Spectroscopy of Barium Ions in [IeII, Phys.Lett. A l l S , 238

References

75

Sandars P.G.H. (1966): Enhancement Factor for the Electric Dipole Moment of the Valence Electron in an Alkali Atom, Phys.Lett. 22, 290 Schiff L.I. (1963): Measurability of Nuclear Electric Dipole Moments, Phys.Rev. 132, 2194 Smith K.F. et al. (1990): A Search for the Electric Dipole Moment of the Neutron, Phys.Lett. B234, 191 Tabbert B. et al. (1995): Atoms and Ions in s~tperfluid helium. L Optical spectra of atomic and ionic impurities, Z.Phys. B97, 425 Weis A., Kanors.ky S.I., Arndt M., and H£nsch T.W. (1995): Spin Physics in Solid Helium: Experimental Results and Applications, Z.Phys. B98, 359

g-Factors of S u b a t o m i c Particles B. Lee Roberts Department of Physics, Boston University, Boston, MA 02215, U.S.A.

1

Introduction

The measurements of magnetic moments of subatomic particles have provided fundamental information about particle structure, as well as information on the nature of the underlying theories of QED and QCD. While all of the stable spin ½ particles have anomalous magnetic moments, i.e. their g-factors are not exactly equal to 2, only the electron and muon appear to be pointlike. The anomalous moments of these two leptons are dominated by the lowest order radiative correction 2-'~, which is a small correction to the Dirac moment of 1. The proton, neutron and other baryons have large anomalous moments, which we have come to realize are related to their internal structure. Classical electromagnetic theory predicts that a magnetic dipole moment will be produced by a current loop. If the current is caused by a charged particle such as an electron orbiting about a nucleus, the dipole moment tL can be related to the angular momentum L through e_L_L .

/~ ----gl 2m

(1)

The proportionality constant gl, which is called the "gyromagnetic ratio", is the ratio of the magnetic moment to the angular m o m e n t u m and is equal to unity for an orbital angular momentum. For an intrinsic spin angular momentum, where we assume that the magnetic moment is proportional to the spin, 1 the moment is given by ~,, = g , ( ~ ) S

.

(2)

The g-factor for the electron was first measured in 1922 by Stern and Gerlach [1, 2] with a beam of silver atoms. In 1927 Phipps and Taylor [3] used the Stern-Gerlach technique on hydrogen, and clearly demonstrated that the effect was a result of the spin of the electron, and not an atomic effect. The value of g determined from these experiments was consistent with g -- 2. The relativistic Dirac formalism [4] of 1928 predicted that g is exactly 2. Since the g-factor need not be exactly 2, one can write g as the sum of two parts, g = g0 + gl (3) 1 The subscript on g, is almost always omitted.

78

B. Lee Roberts

where 90 = 2 is the Dirac value, and gl represents a correction/addition to the simplest theory. In terms of the 9 factors, the magnetic m o m e n t can be written as

(go + g~) eh P -

2

2m

(4)

where m is the particle's mass. For a positive particle, this is written as eh p = ( 1 + a) 2m

(5)

where the first term is called the Dirac m o m e n t and the second is the Pauli or anomalous moment. The first term is the charge in units of the electron charge, (q/e = -l-l, 0), and the Pauli m o m e n t can be written as a = (

) -=- ~- .

(6)

The quantity (eh/2m) is called a magneton; (eh/2rne) = liB is called the Bohr magneton, and (eh/2mp) = ]2u is called the nuclear magneton. Baryon magnetic moments are usually quoted in nuclear magnetons. In addition to these two magnetons, the Particle D a t a Tables [5] also use the muon magneton (eh/2mu). The first indication that g might not be exactly equM to two came from the proton, not the electron. In 1933, well in advance of the discovery of an anomalous 9 factor for the electron, O. Stern and co-workers [6] showed that the proton had a magnetic m o m e n t of (2.5 -4- 10%) PN, which we now know to be 2.79284739(6)/~N [5]. Thus 9p = 5.58 rather than the expected value of 2. Stern et al. also measured the deuteron m o m e n t , Pd : 0 . 8 , which suggested to some people that perhaps the neutron m o m e n t p,~ was ~ --2/2N. In 1940 L. Alvarez and F. Bloch [7] showed that the neutron had a magnetic m o m e n t of (-1.935 =t=0.02)#N rather than zero, which might have been expected naively. In the conclusion to their paper they say: "The fact alone t h a t pp differs from unity and pn differs from zero indicates that, unlike the electron, these particles are not sufficiently described by the relativistic wave equation of Dirac and that other causes underly their magnetic properties." [7]. These large anomalous m o m e n t s remained inexplicable until the 1970s. While it must have become clear that these anomalies could result from internal structure of the proton and neutron, a quantitative calculation was impossible. As late as 1967 in a discussion about the Schwinger term (see below) and the large proton anomaly, J.J. Sakurai states "However, if we consider these particles are complicated objects surrounded by virtual meson clouds, the failure of the simple prescription does not seem surprising." [8]. Even with the development of unitary s y m m e t r y [9] in 1961 which made it possible to derive relationships among the magnetic m o m e n t s of the baryon

g-Factors of Subatomic Particles

79

Fig. 1. The lowest order radiative correction to g.

octet [10], it was not clear if quarks were genuine constituents, or simply convenient mathematical devices. Only in 1969, with the deep inelastic e - p scattering experiments at SLAC [11] did the community begin to take the quark-parton model seriously. We now know that hadrons are made of quarks, and these large anomalous moments are the result of the internal quark structure of the proton and neutron. One of the early successes of the SU(6) model was the prediction that tLp /~n

_

3 2

(7)

which is experimentally found to be -1.460. In the next section we return to the topic of hyperon moments. In 1948 Kush and Foley [12] showed that g of the electron was not exactly 2, but was g~ : 2(1.001 19 ~ 0.000 05) . (8)

This difference from two was explained by the lowest order radiative correction (Fig. 1) to go, which was first calculated by Schwinger [13]. He obtained gl-

~r

-- 2(0.001 16...) ,

(9)

where a is the fine-structure constant, in excellent agreement with experiment. This correction is now usually couched in terms of Feynman diagrams, and the Schwinger term is represented by the diagram in Fig. 1. Schwinger's calculation represented one of the first in the new field of quantum electrodynamics, which has now become a calculational industry. The QED calculations of gl of the electron have now been carried out to tenth-order by Kinoshita [14]. When the electron g-factor was measured by Kush and Foley, the value was determined to be slightly greater than 2. The electron anomalous moment is now measured to an experimental accuracy of a few parts per billion [15], and is well described by quantum electrodynamics calculation [14]. To the

80

B. Lee Roberts

level of measurement, only photons and electrons contribute, and gl ~- 1 × 10 -3. There is no evidence to date, both from g-factor measurements or e+e scattering, to indicate that the electron has any internal structure. To conclude this section, we briefly describe two phenomena which have been exploited in the measurement of magnetic moments: precession in a magnetic field, and fine-structure splitting in atomic energy levels. If a magnetic dipole with dipole moment/~ is placed in a magnetic field, it will experience a torque "r=~xB . (10) This fact has been exploited in the measurement of most magnetic moments. We detail how it permitted measurement of hyperon g-factors. The magnetic moment is along the spin direction, and will precess in a magnetic field. One starts with a polarized beam of hyperons which is injected into a magnetic field. After the region of magnetic field, hyperon decays are observed. For these two-body weak decays, the distribution of the decay baryons relative to the hyperon spin is given (in the CM frame) by (1 + a P c o s O )

,

(11)

where P is the beam polarization and a is the decay asymmetry parameter. From the distribution of decay products one can tell how far the spin has precessed. The magnetic moment manifests itself in atomic spectra through finestructure splitting. In the simplest picture of the hydrogen atom, the atomic electron sees a proton orbiting about it which produces a magnetic field like a current loop. The magnetic moment of the electron interacts with this magnetic field with the interaction energy

Hint : -/~ " B .

2

2.1

(12)

E x o t i c A t o m M e a s u r e m e n t s of t h e ~ and ~ g-Factors Exotic Atoms

Hydrogen-like atoms with a single p - or 7r- captured into atomic states were postulated in 1947 [16], in response to the experiments of Conversi Pancini and Piccioni [17]. The first clear observations of x rays from such an atom was made by Camac, McGuire, Platt and Schulte with a beam of zr- from the Rochester cyclotron [18]. The first detailed study of x-rays from muonic atoms was carried out by Fitch and Rainwater at the Nevis cyclotron [19]. By the late 1960s,/3 and K - atoms had been observed and studied also. The principle is simple. A beam of the negative particles from an accelerator is degraded in energy and brought to rest in a target. When a negative particle slows down to roughly the velocity of the outer atomic electrons, it

g-Factors of Subatomic Particles

81

is captured into a high atomic level with principal quantum number given by n ,., X / ~ / m ~ . It then cascades towards the atomic Is state, first by Auger emission, and then by radiative transitions. Experimentally, one looks for x rays in coincidence with a stopped beam particle. Since the 1960s, most such experiments have used solid-state Ge(Li) photon detectors, which typically have a (FWHM) resolution - ~ ,-~ 7 x 10 -3 in the energy range 100 to 700 keV, and absolute photopeak efficiency of 10 - 20%. More recently, bent crystal spectrometers, which have even better resolution, have been employed to study low-Z pionic, muonic and kaonic atoms. Since the Bohr radius of this exotic atomic system n2

r,~ -

(13)

mZa

depends inversely on the orbiting particle's reduced mass, the heavy negative particle is soon inside of the cloud of atomic electrons and the modification of the Coulomb potential of the nucleus by these electrons becomes a very small correction. This system is very close to a hydrogen-like atom. For nuclei heavier than He, the orbiting particle can have a substantial overlap with the nucleus. The cascades of ~'-, K - and i5 atoms with nuclear charge Z > 2 are observed to terminate before the K~ x ray is observed. If one calculates the fine-structure splitting for a hydrogen-like atom of nuclear charge + Z e , and includes a Pauli moment term, the energy splitting is given by [2(1] • (ZoO 4 m (14)

A E n ' t = ( 2 +gl) ~n3

e(e+l)

where n is the principal quantum number and ~ is the orbital angular momentum. This calculation is done in the "Pauli approximation" and is accurate up to (and including) terms of relative order (f~/c) 2, where ~ is the expectation value of the electron velocity. 2 The total angular m o m e n t u m is J = L + S, and the total angular mo1 For (g0/2 + gl) < 0 the state with mentum quantum number is j = ~ + ~. highest total angular momentum j lies highest in energy. The difference in weighting between the two terms in Eq. 14, (go + 291), arises from the famous factor of two from Thomas precession [21], which subtracts off an amount from the fine-structure splitting which is exactly half that expected from the Dirac moment term. This fine structure splitting should be compared with the Bohr energy m(Z~)2 n2

En -=- -

(15)

For g = n - 1, n >_ 2, the splitting relative to the total energy is AE,,,_I

(Za) 2 _~

En

(16)

2n2(u - 1)

2 The mean velocity O/c is approximately Zc~/n.

82

B. Lee Roberts

(9,8)

/ /

fi2, i0

TM

(-}

t

Fig. 2. A portion of the atomic cascade. The energy separations are not to scale. In the blow up, the fine-structure states are labelled + for j = g + ½, - for j = g - !2 ~ and the Dirac and Pauli moments are both assumed to be negative.

where the Pauli m o m e n t contribution has been neglected. For the 2p state of h y d r o g e n this ratio is 6.6 × 10 -6. T h e basic features of the a t o m i c cascade are given in Fig. 2. T h e population of the circular (those with ~ = n - 1) energy states tends to become enhanced over the statistical 2 l + 1 weighting as the cascade proceeds towards lower q u a n t u m states. T h e circular transitions, i.e. those between circular states, tend to d o m i n a t e the experimental spectrum. This can be understood by considering the three E1 transitions shown in Fig. 2 which are labelled c~, /~ and 7, where a is the circular transition. T h e radiative transition probability depends on ~3 times the dipole m a t r i x element. T h u s the A n = --2 transition labelled fl is preferred over the "noncircular" A n = - 1 transition labelled 7A s s u m i n g a statistical p o p u l a t i o n of the fine-structure states, the spin flip transition b can be shown to be weak for transitions between two adjacent levels of large principal q u a n t u m n u m b e r [22]. T h e intensity ratio for the three c o m p o n e n t s of a circular transition (i.e. from (n, ~) = (n -4- 1, n) -+ (n, n - 1)) is given by a : b : c = 2n 2 + n -

1 : 1 : 2n 2 - n -

1

(17)

where n is the principal q u a n t u m n u m b e r of the lower state. For n = 10 this ratio is 209 : 1 : 189, where we assume t h a t the state of highest j lies highest. If the a n o m a l o u s m o m e n t were the opposite sign to the Dirac m o m e n t , these two c o m p o n e n t s could reverse in energy (see Eq. 14). Experimentally one

g-Factors of Subatomic Particles

83

would observe two x-ray lines separated in energy by an amount equal to the difference in fine-structure splitting of the upper and lower states.

2.2

Measuring g-Factors with Exotic Atoms

The first measurements of the hyperon magnetic moments were carried out at the Brookhaven AGS and the CERN PS in the late 60s and early 70s. Since the hyperon energies were low, as was the average polarization, these experiments were quite difficult, and did not yield precise measurements. Furthermore, the magnetic moment of the S - , which has a decay asymmetry parameter (cr = -0.068 + 0.008), see Eq. 11, was impossible to measure using these low energy beams. The development of constituent quark models motivated by QCD [2329] permitted theorists to make definite predictions for the moments of the baryons in the SU(3) flavor octet, usually with the proton moment as input to the calculation. Thus in the mid 1970s, the measurement of hyperon magnetic moments became an important testing ground for these models of the quark structure of the baryons. Motivated by the desire to measure the Z - moment, a group working at the Brookhaven AGS developed a new technique for measuring the magnetic moments of the antiproton (p) and ,U-. Since the traditional precession technique did not work for either of these particles (the p does not decay) the exotic atom technique was invented. From Eq. 14 we can see that the x-ray lines observed from # - , p and Z'- will exhibit fine-structure splitting and one can measure the magnetic moment by measuring this splitting. In practice, this technique is useful only for particles with a large anomalous moment, which excludes a precision measurement of the muon moment by this technique. Since the fine-structure splitting goes as ( Z a ) 4 n -5, one wishes to look at the lowest transition not affected by the strong interaction, and in the highestZ target possible. One must choose targets with no static quadrupole moment, i.e. nuclear spin 0 or ½, to eliminate hyperfine structure which would interfere with a measurement of the fine-structure. Even with high-Z nuclei, the finestructure splitting is almost never experimentally resolved, so one has to be very careful to measure the detector resolution and lineshape independently under the same conditions as the data are collected. A deformed nucleus such as 238U, which does not have a static quadrupole moment, can still cause dynamic E2 effects [30]. While the total centroid for the/3 energy levels is shifted, the individual fine-structure components are shifted together to within 8 eV, which was negligible and could be included as a small correction in the analysis. The experimental resolution is such that except for the/5(n = 11 --+ n = 10) transition in uranium, which is shown below, the two components are not well resolved. On the other hand, since the shape and intensities of the two components are known, the chi-square of a fit with no splitting is larger than

84

B. Lee Roberts

I DiracOnly 12.5

(10,9,+)

(,o,s,+)

(

8 '°.°t-

N .~ 5 . 0 -

All Corrections

~ 2.5 0.0

Pauli Moment

-

(Q)

(b)

(c)

Fig. 3. Relative positions of the energy levels of the Pb ,U-(n = t0,~ = 9) and (n = 10, g = 8) states. The states are labelled by (n,e,:J=) where

1 (a) shows the pure Dirac levels, the -4- gives the total j = £ 4- ~. (b) shows the inclusion of all corrections except the anomalous moment, and (c)

shows the fully corrected levels assuming that p ( E - ) = -1.11PN

the m i n i m u m value, which occurs at some finite splitting, by as much as 100 for combined d a t a sets. For small subsets it is larger by at least 10 units as will be shown below. To determine the intensities of the non-circular transitions relative to the circular ones, it is necessary to carry out a simulation of the exotic a t o m cascade [31]. Since the An = - 2 transitions are experimentally observed, their observed intensities relative to the An = - 1 transitions can be reproduced by the cascade calculation by adjusting the initial population and the strong interaction parameters. In addition to the Pauli m o m e n t term, other corrections such as vacuum polarization [32], nuclear polarization [33] and electron screening [34], were included by integrating the Dirac equation [35]. In Fig. 3 we show the effect of these corrections to the Dirac energy levels for S - Pb. Note that the degeneracy of states with the same total angular m o m e n t u m is lifted by the inclusion of these corrections. It has been pointed out that the g-factor of a bound particle is decreased relative to its free value [36]. This effect has been demonstrated [37] to be considerable for a p - bound in the l s level of a high-Z muonic atom. In the experiments discussed below, the lowest principal q u a n t u m state considered was the n = 10 state, where the effect was estimated to be smaller than 0.1% and was neglected. The first magnetic m o m e n t measured with this technique was the p moment measurement carried out at the Brookhaven AGS in the s u m m e r of 1972

g-Factors of Subatomic Particles

85

300

o~ I-- 2 0 0 o o

I00

0~

I

366

I

I

:570

I

I

374

ENERGY (keV)

Fig.4. The/5(n = 11 ~ n --- 10) transition in depleted U. The two fine-structure components can clearly be seen. The non-circular contribution to this transition is negligible.

[38]. In Fig. 4 the #(n = 11 -+ n = 10) transition is shown for a depleted uranium target. For this transition the non-circular intensity is negligible. The solid curve through the d a t a is a free fit of two Gaussians plus background to the data. The measured fine-structure splitting is consistent with a magnetic m o m e n t of p(p) = -2.79, the value expected from the C P T theorem which requires particles and antiparticles to have magnetic m o m e n t s with the same magnitude but opposite in sign. The intensity ratio obtained from the fit is consistent with the expected ratio of 209 : 189. The combined # m o m e n t obtained from the two early experiments [38, 39] was p(#) = (--2.795+0.018)#N. The current value of the 15 magnetic m o m e n t is dominated by an experiment [40] which utilized the unique capabilities of the L E A R facility at CERN to measure the fine-structure splitting in the /5(11 --+ 10) transition in Pb. T h a t experiment obtained the result (-2.800 + 0.0090)pg. While the value of the # m o m e n t is firmly predicted by the C P T theorem, the ~ - m o m e n t was scarcely predicted at all when this technique was developed. Since the Z - lifetime is too short to produce a b e a m and bring it to rest, it was necessary to use a stopped K - beam to produce hyperons through the reaction K - + N -+ ,U- + rr, where N is a nucleon in the target nucleus. Although weak ~ - x rays had been observed in the spectra from low-Z kaonic atoms, Z - x rays had never been seen in high-Z atoms, which would be necessary to measure the magnetic m o m e n t . The first observation of ~ - x rays from high-Z atoms was reported at Brookhaven in 1973 [41] where a limit was placed on the magnetic m o m e n t . With more data, it was possible to measure a value for the magnetic m o m e n t [42], which

86

B. Lee Roberts

7°°0 F

,

6°°° I-

t

=+

soool-f

+

T

f

4oo01.- l I

=

~

|r,

® t

°u .... r, lljL. 1000 -

T+°+ i + ++-

h

~~,~

0~

100

"+

~

t

_~

t

£

I

I

200

300

400

,500

600

ENERGY ( keY} Fig. 5. The x-ray spectrum from kaons stopping in a laminar W target.

was p(~U-) = ( - 1 . 4 8 + 0.37)#N. A similar result was obtained in a second experiment by a Columbia-Yale collaboration [43]. A follow-up experiment at Brookhaven (E723) used a laminar target with sheets of Pb or W placed in a liquid hydrogen bath. A K - beam was stopped in this target, and the reaction K-

+ p -4 ~-

+ ~r+

B . R . = 46%

(18)

was used to produce Z - in the target. The monoenergetic 7r+ was stopped in a range telescope and the 7r+ -4 p+ -4 e + decay chain was observed to tag 2 - production. The x-ray spectrum in coincidence with stopped K - is shown in Fig. 5, and the tagged spectrum is shown in Fig. 6. The Z - (11 -4 10) transition in W is shown in Fig. 7. The tagging improves the 2 - x-ray signalto-noise by a factor of 20 to 30 over the first experiments. To combine all the data, the maps of ~2 vs. fine-structure splitting for the two targets and three detectors were added together. The resulting map showed a one standard deviation preference for the negative sign. The final result [44] was p ( ~ - ) = (-1.105 ± 0.029 + 0.010)#N, where the first error is statistical and the second systematic. Of course, the biggest difference between the first and second generation experiments is signal-to-noise, whose importance cannot be underestimated in such a difficult measurement. With the discovery of polarization in the hyperon beam at Fermilab [45], it became possible to perform precision magnetic moment measurements there, and eventually the S - moment was measured by the precession technique in the same time frame as E723. The current world average [5] is # ( S - ) = (-1.160-t-0.025)pN. The difficulty of this particular measurement is reflected in the fact that the world average includes a scaling up of the combined error

g-Factors of Subatomic Particles

87

200

IZ :::)

0

loo

T

50

,

r

1

-

g

~, t

t

I

i

100

200

300

400

riO(

700

ENERGY(keY) F i g . 6. The tagged spectrum from kaons stopping in a laminar W target.

by a scale factor of 1.7, since the three most precise experiments (E723 and the two most recent Fermilab experiments) do not agree well with each other. All of the hyperon magnetic moments have now been measured well [5]. The models have been further refined. While a specific calculation may come closer to predicting one of the magnetic moments better than the other calculations, the proper test is to look at how one model predicts the whole set. With that criterion, the agreement is only at the 10% level.

3

A N e w E x p e r i m e n t to M e a s u r e t h e M u o n g - F a c t o r

For the muon and electron, the strong interaction contribution to the magnetic moment is very small. The QED calculations of their anomalous moments represent a calculationat tour de force. While the g-factor of the electron has provided a testing ground for QED, the anomalous magnetic moment of the muon has provided an even richer source of information. The muon's mass of 105.7 MeV is quite large compared to the electron's mass of 0.511 MeV, and heavier virtual particles can contribute in a measurable way to its anomalous moment. The relative contribution from heavier particles to the muon anomaly scales as (m~/m~) ~, and in a series of three elegant experiments [46-48] virtual muons and quarks (pions) have been shown to contribute at measurable levels. The current experimental accuracy is not sufficient to observe the W and Z ° gauge boson contributions. At present there is good agreement between experiment and theory [49], and there is no indication of any substructure to the muon.

88

B. Lee Roberts

:1 5ot 160.

°:

150. 140

10

130

-z

.t5

-i

0

-o.s

6.s

i

3O4

1.5

3O6

~{£'1 (Nuclear Magnetons}

(a)

(b)

Fig. 7. (a) A X2 map from a fit to a tagged sample of the W - S - ( 1 1 --+ 10) transition. There are 116 degrees of freedom. (b) The best fit to the observed S transition, corresponding to the X2 minimum. Both the circular and noncircular fine-structure doublets are shown separately. This figure contains ,-- 7% of the total data taken.

Unlike the short lived hyperons, the muon's long lifetime permits a b e a m to be stored in a storage ring, and the spin rotation frequency can be measured. While the decay #+ --4 e+~,ue is a three-body decay, the highest energy positrons go along the electron spin direction, and the parity violation in the weak decay permits one to determine the direction of the spin vector. A charged particle moving in a uniform magnetic field will execute cyclotron motion with the orbital cyclotron frequency we =

eB

(19)

m7

The spin precession frequency in a magnetic field is given by ~

=

geB 2m

eB + (1 - ~ / ) - m"/

,

(20)

with the Larmor and T h o m a s precession terms explicitly separated. Thus the spin vector of a charged particle moving in a uniform magnetic field will precess, relative to the m o m e n t u m vector, with a frequency wa, which is given by the difference between the orbital cyclotron frequency wc and the spin precession frequency we. ~za :

~v~ - w~ :

e

--aB

(21)

m

is directly proportional to the anomalous m o m e n t and independent of the particle's m o m e n t u m . For particles with a Pauli m o m e n t of the same sign

g-Factors of Subatomic Particles

89

as the Dirac moment, the spin vector will lead the m o m e n t u m vector. Experimentally one measures the precession frequency of a particle's spin in a known magnetic field, and determines the magnetic moment. In a real experiment, the field B in Eq. 21 is the average field seen by the ensemble of muons in the storage ring. Vertical focussing must be provided to keep the muon beam stored, which can be accomplished with magnetic multipoles, or with an electrostatic quadrupole field. However, if magnetic multipoles are used, it is difficult to know the average B field to the accuracy needed for a precision measurement of a~. In a region in which both magnetic and electric fields are present, the relativistic formula for the precession is given by [50]

~oa-

dOR _ e dt ~n

[ (') a,B-

a,

72-

1

/3xE

1

'

(22)

where OR = (s,/3) is the angle between the muon spin direction in its rest frame and the muon velocity direction in the laboratory frame. The other quantities refer to the laboratory frame. If the muon beam has the "magic" value of 3' = 29.3, the electric field does not change the relative orientation between the spin and momentum vectors. Thus the precession of the spin relative to the momentum is determined entirely by the magnetic field, and one can use electrostatic quadrupoles for vertical focussing. The magnetic field can be a pure dipole field and can be determined very accurately. This technique was used in the third CERN experiment [48] and will be used in the new experiment at Brookhaven. The famous CERN experiments measured # = 1.001 165 9230 (85) ( e h / 2 r n u ) , a precision of -t-7.3 parts per million (ppm) for a~. While this result tested QED to a high level, and showed for the first time the contribution of virtual hadrons to the magnetic moment of a lepton, this sensitivity was not sufficient to observe the contribution of the W and Z ° gauge bosons. A new experiment [51], is being constructed to measure ( 9 - 2 ) of the muon to better than =k0.35 ppm in order to measure the electroweak contribution to ( 9 - 2). The first data collection run will take place in 1997. This experiment has been described in some detail elsewhere [52, 53], and only a few comments will be made here. The goal of the experiment is to verify the standard model prediction, and to search for physics beyond the standard model. The theoretical value of a u consists of contributions from QED [49], virtual hadrons [54, 55], and virtual electroweak gauge bosons [56-58]. Taking the value of (~ from the electron (9 - 2) experiment [59], the total QED contribution is a QED = 116 584 706(2) × 10 -11. The QED part is calculated to a precision of a few parts per billion, and the agreement between the calculated and measured (g - 2) values for the electron gives us great confidence ill the QED calculation. The hadronic correction cannot be calculated from first principles but can be calculated using dispersion theory and data from e+e - --4 hadrons.

90

B. Lee Roberts

The total hadronic contribution is tL# _Had . ~ 6882(154) x 10 -11. The two recent evaluations [54, 55] agree that the current error on this contribution is ~ 5=1.3 ppm. The latter authors estimate that this error can be reduced to (,-~ 5=59 x 10 -11) or ,.o +0.5 p p m after CMD2, the new Novosibirsk experiment, has finished the analysis of their data. To further reduce this theoretical error, additional e+e - d a t a from x/~ = 1.4 GeV (the m a x i m u m energy for CMD2) to above the J / ¢ threshold will be needed, or perhaps theoretical calculations in this energy region can further reduce this error [60]. The theoretical limit seems to be set by the contribution from hadronic "light by light" scattering [61], aHad(lbl) = --52 (18) × 10 -11, which has an uncertainty of 5=0.15 ppm. This contribution cannot be estimated from data, but might be improved by a lattice QCD calculation. While the single W and Z loop calculations have been available for some time [56], recent higher order calculations which include both fermionic [57] and bosonic [58] two-loop contributions, obtain a higher order contribution which turns out to be surprisingly large. The first order weak contribution of 195(4) x 10 - i t is reduced to 151(4) x 10 -11 (1.3 p p m of au) when the second order terms are included. Since the ability to calculate loop diagrams is intimately tied to the renormalizeability of the theory, this measurement will provide an important test of the renormalization prescription. = 116 591 739 (154) ×10 -11 The total theoretical prediction is a Th~orv u where the theoretical error of 5=1.3 p p m is dominated by the uncertainty on the hadronic vacuum polarization. Since (g - 2)u is very sensitive to W and p substructure, as well as supers y m m e t r y in the mass region below 130 GeV or any SUSY model with large tan/3, a result which agrees with the standard model will place significant new limits on physics beyond the standard model [52, 62]. The experimental goal is nominally stated as a precision of +0.35 ppm, a factor of twenty better than the C E R N experiment. With the option of direct muon injection into the ring, which requires a full aperture kicker to store the muon beam, one m a y be able to do much better. If the estimates of the incident muon flux and capture efficiency into the storage ring for direct muon injection are correct, then the final result for a t could reach a statistical sensitivity of 5=0.12 p p m at the end of the d a t a collection. This statistical error of 5=0.12 p p m happens to be equal to the current estimate of the experimental systematic errors, and the collaboration is studying ways of reducing systematics further. While this statistical error would represent a ten standard deviation measurement of the electroweak contribution, the uncertainty in the current evaluation of the hadronic contribution is much larger, and will need to be improved to realize the full potential of the final experimental result.

g-Factors of Subatomic Particles

4

91

Conclusions

The anomalous g-factors of the muon and electron have played an i m p o r t a n t role in the development of QED, and in our understanding of the nature of these leptons. The current precision of the electron anomaly makes it one of the most precisely measured quantities in nature, and one could argue that QED is so well understood that this measurement can be used to provide the best measurement of the fine-structure constant. The muon g-factor has measurable contributions from virtual hadrons, and soon we hope to measure the W and Z ° gauge boson contributions. This new measurement also opens a window for the discovery of new physics beyond the standard model. For a full interpretation of this experiment, additional data, and calculations, will be needed to permit a more precise evaluation of the hadronic vacuum polarization contribution to a , . Baryon magnetic m o m e n t s is another subject altogether. Because of the difficulties of calculating low energy phenomena in QCD, it is necessary to resort to QCD inspired phenomenological models, which describe the situation to about 10%, decades away from the spectacular agreement between theory and experiment for the leptons. Nevertheless, when the constituent quark models were being developed in the 70s, hyperon magnetic m o m e n t s played an important role in testing their validity. Perhaps some day, computing power will increase to the point where lattice calculations of the static baryon properties can reach an accuracy which would permit a meaningful comparison with experiment. Acknowledgement. I have learned much about exotic atoms from the collaborators listed in the references, especially M. Eckhause, J. Miller, and It. Welsh, as well as from C.3. Batty, M. Blecher and M. Leon who are not listed. The early exotic atom work described here was done in friendly and beneficial competition with C.S. Wu and her collaborators. I am grateful to K. Johnson and S. Glashow for interesting discussions on magnetic moments and baryon structure. T. Kinoshita and W. Mareiano have been invaluable resources on the theory of the muon g-factor. I wish to thank D.W. Hertzog and Y.K. Semertzidis for their helpful comments on this manuscript.

References [1] O. Stern, Z. Phys. 7 (1921) 18. [2] W. Gerlach and O. Stern, Z. Phys. 8 (1922) 110, Z. Phys. 9 (1922) 349, Z. Phys. 9 (1922) 353 and Ann. Phys. 74 (1924) 45. [3] T.E. Phipps and J.B. Taylor, Phys. Rev. 29 (1927) 309. [4] P.A.M. Dirac, Proc. Roy. Soc. 117 (1928) 610, and 118 (1928) 341. [5] Particle Data Group, Review of Particle Properties, Phys. Rev. Dh0 (1994) 1173. [6] R. Frisch and O. Stern, Z. Phys. 85 (1933) 4, and I. Estermann and O_ Stern, Z. Phys. 85 (1933) 17.

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[7] Luis W. Alvarez and F. Bloch, Phy. Rev. 57 (1940) 111. [8] J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, Reading Massachusetts (1967), p. 110. [9] See M. Gell-Mann, C.I.T. Report CTSL-20 (1961) unpublished, and Y. Ne'eman, Nucl. Phys. 26 (1961) 222, which are reprinted in the classic collection of articles The Eightfold Way, M. Gell-Mann and Y. Ne'eman ed., Benjamin, New York, 1964. [113] S. Col.eman and S.L. Glashow, Phys. Rev. Lett. 6 (1961) 423. [11] E.D. Bloom, D.H. Coward, H. DeStaebler, J. Drees, G. Miller, L.W. Mo, R.E. Taylor, M. Breidenbach, J.I. Friedman, G.C. Hartmann and H.W. Kendall Phys. Rev. Lett. 23 (1969) 930, and M. Breidenbach, J.I. Friedman, H.W. Kendall, E.D. Bloom, D.H. Coward, H. DeStaebler, J. Drees, G. Miller, L.W. Mo, R.E. Taylor, Phys. Rev. Lett. 23 (1969) 935. [12] P. Kush and H.M. Foley, Phys. Rev. 74 (1948) 250. [13] J. Schwinger, Phys. Rev. 73 (1948) 416, and Phys. Rev. 76 (1949) 790. [14] A review of these calculations is given in T. Kinoshita, Quantum Electrodynamics (Directions in High Energy Physics, Vol. 7), T. Kinoshita ed., World Scientific 1990, p 218. These calculations are constantly being extended. [15] R.S. Van Dyck, Jr., P. Schwinberg, and H. Dehmelt, Phys. Rev. Lett. 59 (1987) 26 and in Quantum Electrodynamics, T. Kinoshita ed., World Scientific, 1990, p. 322. [16] E. Fermi, E. Teller and V. Weisskopf, Phys. Rev. 71 (1947) 314, and E. Fermi and E. Teller, Phys. Rev. 72 (1947) 399. [17] M. Conversi, E. Pancini and O. Piccioni, Phys. Rev. 71 (1947) 209. [18] M. Camac, A.D. McGuire, J.B. Platt and H.J. Schulte, Phys. Rev. 88 (1952) 209. [19] V.L. Fitch and J. Rainwater, Phys. Rev. 92 (1953) 789. [20] H.A. Bethe and E. Salpeter, Quantum Mechanics of One- and Two- Electron Atoms, (Springer-Verlag, 1957). The discussion of magnetic moments and freestructure splitting begins in section 10. [21] L.H. Thomas, Nature 107 (1926) 514. [22] See Bethe and Salpeter, op. cit. p. 273. [23] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [24] T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [25] N. Isgur and G. Karl, Phys. Rev. D21 (1980) 3175. [26] G.E. Brown, M. Rho and V. Vento, Phys. Lett. 84B (1979) 383. [27] S. Theberge and A.W. Thomas, Nucl. Phys. A393 (1983) 252, and references therin. [28] J. Franklin, Phy. Rev. D30 (1984) 1542. [29] Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 55 (1985) 1839. [30] M.Y. Chen, Y. Asano, S.C. Cheng, G. Dugan, E. Hu, L. Lidofsky, W. Patton, C.S. Wu, V. Hughes and D. Lu, Nucl. Phys. A254 (1975) 413. [31] The most modern code, which includes the strong absorption of the orbiting hadrons is by M. Leon and R. Seki, Phys. Rev. Lett., 32 (1974) 132. The code for muonic atoms written by J. H/iffner, Z. Phys. 195 (1966) 365, was widely used in the 1960s. The basic atomic physics is given in Y. Eisenberg and D. Kessler, Nuovo Cim. 19 (1961) 1195. They discuss strong absorption

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of orbiting hadrons in Phys. Rev. 123 (1961) 1472 and Phys. Rev. 130 (1963) 2352. The most recent review of cascade calculations is by C.J. Batty and R.E. Welsh, Nucl. Phys. A589 (1995) 601. [32] J. Blomqvist, Nucl. Phys. B48 (1972) 95. [33] T.E.O. Ericson and J. H/finer, Phys. Lett. 40B (1972) 459. [34] P. Vogel, At. Data Nuc]. Data Tables 14 (1974) 599 and P. Vogel eta]. Phys. Lett. B70 (1977) 39 and references therein. [35] E. Borie, Phys. Rev. A28 (1983) 555 and references therein. We are grateful to E. Borie for a copy of her code. [36] H. Margenau, Phys. Rev. 57 (1940) 383, and K.W. Ford, V.W. Hughes and J.G. Wills, Phys. Rev. 129 (1963) 194. [37] T. Yamazaki, S. Nagamiya, O. Hashimoto, K. Nagamine, K. Nakal, K. Sugimoto and K.M. Crowe, Phys. Lett. 53B (1974) 117. [38] J.D. Fox, P.D. Barnes, R.A. Eisenstein, W.C. Lain, J. Miner, R.B. Sutton, D.A. Jenkins, R.J. Powers, M. Eckhause, J.R. Kane, B.L. Roberts, M.E. Vislay, R.E. Welsh, and A.R. Kunselman, Phys. Rev. Lett. 29 (1972) 193. The final analysis of these data is reported in B.L. Roberts, Phys. Rev. D17 (1978) 358. [39] E. Hu, Y. Asano, M.Y. Chen, S.C. Cheng, G. Dugan, L. Lidofsky, W. Patton, C.S. Wu, V. Hughes and D. Lu, Nucl. Phys. A254 (1975) 403. [40] A. Kreissl, A.D. Hancock, H. Koch, Th. KShler, H. Poth, U. Raich, D. Rohmann, A. Wolf, L. Tauscher, A. Nilsson, M. Suffert, M. Chardalas, S. Dedoussis, H. Daniel, T. yon Egidy, F.J Hartmann, W. Kanert, H. Plendi, G. Schmidt and J.J. Reidy, Z. Phys. C37 (1988) 557. [41] J.D. Fox, W.C. Lain, P.D. Barnes, R.A. Eisenstein, J. Miller, R.B. Sutton, D.A. Jenkins, M. Eckhause, J.R. Kane, B.L. Roberts, R.E. Welsh, A.R. Kunselman, Phys. Rev. Lett. 31 (1973) 1084. [42] B.L. Roberts, C.R. Cox, M. Eckhause, J.R. Kane, R.E. Welsh, D.A. Jenkins, W.C. Lam, P.D. Barnes, R.A. Eisenstein, J. Miller, R.B. Sutton, A.R. Kunselman, R.J. Powers and J.D. Fox, Phys. Rev. Lett. 32 (1974) 1265, and Phys. Rev. D12 (1975) 1232. [43] G. Dugan, Y. Asano, M.Y. Chen, S.C. Cheng, E. Hu, L. Lidofsky, W. Patton, C.S. Wu, V. Hughes and D. Lu, Nucl. Phys. A254 (1975) 396. [44] D.W. Hertzog, M. Eckhause, K.L. Giovanetti, J.R. Kane, W.C. Phillips, W.F. Vulcan, R.E. Welsh, R.J. Whyley, R.G. Winter, G.W. Dodson, J.P. Miller, F. O'Brien, B.L. Roberts, D.R. Tieger, R.J. Powers, N.J. Colella, R.B. Sutton, and A.R. Kunselman, Phys. Rev. Lett. 51 (1983) 1131, and Phys. Rev. D37 (1988) 1142. [45] G. Bunce, R. Handier, R. March, P. Martin, L. Pondrom, M. Sheaf[, K. Heller, O. Overseth, P. Skubic, T. Devlin, B. Edelman, R. Edwards, J. Norem, L. Schachinger and P. Yamin, Phys. Rev. Lett. 36 (1976) 1113. [46] G. Charpak, F.J.M. Farley, R.L. Garwin, T. Muller, J.C. Sens and A. Zichichi, Nuovo Cim. 37 (1965) 1241. [47] J. Bailey, W. Bartl, B. von Bochmann, R.C.A. Brown, F.J.M. Farley, M. Giesch, H. JSstlein, S. van der Meer, E. Picasso and R.W. Williams, Nuovo Cim. 9A (1972) 369. [48] J. Bailey, K. Borer, F. Combley, H. Drumm, C. Eck, F.J.M. Farley, J.H. Field, W. Flegel, P.M. Hattersley, F. Krienen, F. Lange, G. Leb~e, E. McMillan,

94

[49] [50]

[51]

[52] [53]

[54] [55] [56]

[57] [58] [59] [60] [61] [62]

B. Lee Roberts G. Petrucci, E.Picasso, O. Runolfsson, W. yon Riiden, R.W. Williams and S. Wojcicki, Nucl. Phys. B150 (1979) 1. T. Kinoshita and W.J. Marciano in Quantum Eleetrodynamics (Directions in High Energy Physics, Vol. 7), T. Kinoshita ed., World Scientific 1990, p. 419. V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett. 2 (1959) 435, are generally given credit for this formula. As noted by J.D. Jackson in ClassicM Electrodynamics, (John Wiley & Sons, New York, 1975), p. 556, Thomas published an equivalent equation in 1927. Brookhaven National Laboratory AGS E821: D.H. Brown, R.M. Carey, E. Efstathiadis, E.S. Hazen, F. Krienen, J.P. Miller, O. Rind, B.L. Roberts*, L.R. Sulak, W.A. Worstell, J. Benante, H.N. Brown, G. Bunce §, J. Cullen, G.T. Danby, C. Gardner, J. Geller, L. Jia, H. Hseuh, J.W. Jackson, R. Larsen, Y.Y. Lee, R.E. Meier, W. Meng, W.M. Morse*, C. Pai, I. Polk, S. Rankowitz, J. Sandberg, Y.K. Semertzidis, R. Shutt, L. Snydstrup, A. Soukas, A. Stillman, T. Tallerico, F. Toldo, K. Woodle, T. Kinoshita, Y. Orlov D. Winn, A. Grossmann, K. Jungmann, G. zu Putlitz, P.T. Debevec, W. Deninger, D.W. Hertzog, S. Sedykh, D. Urner M.A. Green, U. Haeberlen, P. Cushman, S. Giron, J. Kindem, D. Maxam, D. Miller, C. Timmermans, D. Zimmerman, L.M. Barkov, D.N. Grigorev, B.I. Khazin, E.A. Kuraev, Yu.M. Shatunov, E. Solodov, K. Nagamine, K. Endo, H. Hirabayashi, S. Ichii, S. Kurokawa, Y. Mizumachi, T. Sato, A. Yamamoto, K. Ishida, S.K. Dhawan, F.J.M. Farley, M. GrossePerdekamp, V.W. Hughes*, D. Kawall, R. Prigl, and S.I. Redin (*Spokesmen, §Project Manager). B.L. Roberts, Z. Phys. C56 (1992) S101. V.W. Hughes, et al., Proc. 10th International Symposium on High Energy Spin Physics, Nagoya, 9-14 November 1992, T. Hasegawa, N. Horikawa, A. Masaike, S. Sawada ed., p. 717. S. Eidelman and F. Jegerlehner, Zeit. Phys., C:67 (1995) 585. W.A. Worstell and D.H. Brown, (1996), Phys. Rev. D54 (1996) 3237, and Muon (g-2) technical note # 220 (1995). W.A. Bardeen, R. Gastmans and B Lautrup, Nucl. Phys. B46 (1972) 319; R. Jackiw and S. Weinberg, Phys. Rev. D5 (1972) 157; I. Bars and M. Yoshimura, Phys. Rev. D6 (1972) 374. A. Czarnecki, B. Krause and W.J. Marciano, Phys. Rev. D52 (1995) R2619. A. Czarnecki, B. Krause and W.J. Marciano, Phys. Rev. Lett. 76 (1996) 3267. T. Kinoshita, Phys. Rev. Lett. 75 (1995) 4728. W. Marciano, private communication. M.Hayakawa, T. Kinoshita and A.I. Sanda, Phys. Rev. Lett. 75 (1995) 790 and Phys. Rev. D54 (1996) 3137. See the review by Kinoshita and Marciano, oF). cit. for a more complete discussion.

Laser Spectroscopy of Metastable Antiprotonic Helium Atomcules Toshimitsu Yamazaki CERN, CH-1211 Geneva, Switzerland and Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo, 188 Japan

1

Discovery

of Long-lived

Antiprotons

in Helium

The lifetime of an exotic atom/molecule sets a natural time window which constrains the precision of spectroscopy through the natural width. The high precision studies of muons, muonic atoms and muonium have been facilitated by their long enough lifetime of 2.2 ps. Before 1991 such a favourable situation had not been conceived for antiprotonic atoms. We knew well that even the positron and positronium are short-lived in matter and that the antiproton is destined to annihilate in matter in a few picoseconds via the strong interaction. A surprising situation emerged in 1991, when the University of Tokyo group discovered at KEK that about 3 % of antiprotons stopped in liquid helium survive for microseconds [1], 106 times longer than usually believed. This experiment had been triggered by a preceding one in which they encountered accidentally the longevity of K - mesons in liquid helium [2] while they were searching for £7 hypernuclei [3]. The longevity of negative mesons in liquid helium was suggested by Condo [4] based on their anomalously large free decay components. Condo proposed the formation of metastable exotic atoms X - e - H e 2+ with large principal and orbital quantum numbers (n, l), and this peculiar atom was studied theoretically by Russell [5]. Although Russell calculated the lifetime of antiprotonic helium atoms to be in the microsecond region, it was hardly believed that such metastable atoms are stable in high density medium like liquid helium. So, this extremely interesting subject had remained untouched nearly for two decades till the discovery in 199I. Immediately, a new experimental group (called PS205) was formed to study this phenomena comprehensively by using the Low Energy Antiproton Ring (LEAR) of CERN, which provides a superb beam for this purpose. This monoenergetic low energy antiproton beam made experiments even with gas targets very efficient and productive. In the first period (1991-93) they measured delayed annihilation time spectra (DATS) precisely in various phases of helium and also studied the effect of admixture of other atoms and molecules [6, 7, 8, 9, 10]. Although the information obtained from systematic studies of DATS indicated that the longevity is due to the formation of metastable atoms ~e- He 2+ (=~He +), it is rather "macroscopic" and "integral", and no

96

Toshimitsu Yamazaki

direct proof of the proposed atom was obtained. We needed a new "microscopic" and "differential" method with which individual states and transitions of the metastable atoms can be identified. In 1993 a new breakthrough emerged. The PS205 group succeeded in the first laser resonance experiment on the antiprotonic helium atom [11] based on the proposal by Morita et al. [12]. This success opened a wide field and as of 1995 seven resonance transitions have been found [13, 14, 15, 16, 17]. Parallel to these experiments, theoretical treatments of this atom advanced [18, 19, 20]. In this way the initial curiosity-oriented research of the PS205 group is turning toward high-precision frontier of fundamental physics.

2

Antiprotonic

Helium

Atomcules

The metastability of the antiprotonic helium atom is well understood by now. In the metastable ~He + atom the electron stays in the ls orbit while the antiproton with large-(n,l) undergoes a nearly classical orbital motion (see Fig. 1). The slowly-moving p polarizes the electron in the opposite direction, which helps retard the electric dipole transition. This effect was interpreted as being similar to the nuclear core polarization effects in terms of ls-np configuration mixing [18]. Another approach is to regard the ~ and He 2+ as the two centers of a molecule [19]. The large-(n,l) states of ~He + can be metastable because the Auger transitions to ionized states are suppressed due to the large ionization potential. As shown in Fig. 1, the whole levels are well divided into the radiation-dominated metastable zone and the Augerdominated short-lived zone. Each metastable state is expected to proceed as (n,/) --+ (n - 1 , / - 1) with a radiative lifetime of about 1.5 #s. The typical level spacing (transition energy) is around 2 eV, which is in the range of visible light. The metastable states are hardly destroyed via Stark mixing in collisions with surrounding helium atoms because the degeneracy in I is removed (about 0.3 eV energy difference between (n,l) and (n,l - 1)). Since this atom is neutral and has one electron, characterized as a kind of hydrogenic atom, it does neither stick to nor penetrate into surrounding He atoms. Since this atom possesses a dual character as an atom and a molecule in itself, it is often called "atomcule". Each state (n, l) is interpreted as a member of vibrational states with a vibrational quantum number v. A unique correspondence between (n,l) and (v, J) holds: J = l and v = n - l - 1. It is interesting that this behavior and correspondence result as the n and l become large in a two-body atom. Whether such large-/states can be formed in the p capture by a He atom or not is essential. Yamazaki and Ohtsuki [18] considered the non-statistical distribution of angular momenta brought in by the ~ in a naive way to understand the delayed fraction of 3 %. Korenman [21] made a more realistic calculation. It seems to be interesting that the maximum angular m o m e n t u m

Laser Spectroscopy of Metastable Antiprotonic HeLium Atomcules

97

H e+ A t o m c u l e

:p

E n e r g y (a.u.) He+

-2.0

Ionized ~He++ .... Io = 0.90 a.u. (24.6 eV)

1

He°

.................... ................

- - - :": - ~ ~

::: ::: - - - _

........ :.'.'.":-': - - - ~ . . . .

---==!

._. ; - ; : : : ::: ~ _ ~

_

~ "--" ~ " _

_

~

39

~

~ , 0

~;

42

=~/~-38

...::: .... - _ . H , -3.0

outra, P-H+o 30 Fig. 1. (Upper) The p and e - orbital distributions in a typical pile + atomcule state. (Lower) Level scheme of the ~He + atomcule. The bold, broken and dotted Lines represent metastable, short-lived and ionized states, respectively. From [18].

b r o u g h t in the capture process roughly corresponds to the m a x i m u m I ~ 37 when the ~ is b o u n d m o s t likely to a state with n ,~ ( M * / m ¢ ) 1/2 "~ 38. T h e appearance of metastability in antiprotonic a t o m s is a c c o m m o d a t e d by the above independent reasons jointly. There is no possibility in other elements. So, the antiprotonic helium a t o m c u l e is a miraculous existence of an antiparticle coupled to n o r m a l particles. Thus, the a t o m c u l e serves as a unique interface between m a t t e r and a n t i m a t t e r domains.

98

Toshimitsu Yamazaki 1993 v=3 ( 3 9 , 3 5 ) - > ( 3 8 , 3 4 ) r

•

•

i

.

.

.

1994 v=2 (37,34)->(36,33)

.

i

0.010

. . . .

i

•

•

0.010

o cl

o.ooi

0.001 I 1

. . . . Time

v

2.5

I . . . . 2 (,us)

o

1.5

v.

1.0

1 Time

597.259(2)3

2.0

"~

<

~ 470.724(2)~

?

-

~

21

LI 1 0.008

.

0.5 0

0.0 597.25 Vacuum Wavelength

597.30 (nm)

2 (p,s)

A A '^ 470.70 Vacuum

.

.

.

.

.

.

.

^~^^ 470.75 Wavelength (nm)

Fig. 2. The first two laser resonances observed in 1993 [11] and 1994 [14].

3

Laser

Spectroscopy

Morita et al. [12] proposed a laser resonance method to identify individual states. Since all the states are deeply bound with ionization energy of 25 eV, the laser ionization technique does not work. Instead, we can induce stimulated transitions from a metastable state to a short-lived state and detect the resonance from the response in the annihilation rate. Specifically, one can apply a pulsed laser and detect a spike in DATS. In 1993 and 1994 they succeeded in observing laser resonances in the (39, 35) --+ (38, 34) [11, 13] and (37,34) --~ (36, 33) [14] transitions, respectively. The resonance spikes and resonance profiles observed are shown in Fig. 2. Hayano et al. [13] succeeded in using two successive laser pulses to obtain information on the lifetime and population of the resonating state. They determined the decay rate of the (39,35) state to be 0.72 + 0.02 #s -1, which turned out to be nearly equal to the predicted one. In other words, this experiment confirmed the predicted retardation effect due to the e - - ~ (anti)correlation [18, 19]. The above method applies only to a pair of metastable and short-lived states. Hayano et al. [16] created a double resonance method in which a metastable --+ metastable --+ short-lived sequence can be studied. They identified the (38, 35) --+ (37, 34) up on the (37,34) 3tate. In 1995 the PS205 group succeeded in observing the laser resonances in Av = 2 "unfavoured transitions", (n, l) -+ (n + l, l - 1) [17], which have very small transition probabilities. The experiment was performed in a strong

Laser Spectroscopy of Metastable Antiprotonic Helium Atomcules -0.5

i .... (a)

-1

i ....

i ....

99

i",

k=726.096 nm (37,35) -~ (38,34)

,5

i (c)),=713.588 nm i (37,34) --> (38,33)

i-5 °~

"2.5

[l,,,l,.,,l 0.9

1

....

-6 ~=~lj,,ll

1,

1,1

1

'

I

'

I

'

I ....

I.

1.1

1.2

time ()as)

time (~ts) 300

....

0.9

12

I

'

~1

'

I

'

I

'

I

20

"~ 100

E

I

~

I

,

i

,

0.085 0.09 0.095 0.1 0.105 wavelength - 7 2 6 (nm)

0.5 0.55 0.6 0.65 wavelength - 7 1 3 (rim)

Fig. 3. The 726 nm and 714 n m " unfavoured" resonance spikes in analog-DATS and their resonance profiles, a) Analog DATS showing a resonance spike at 726.096 nm. b) Resonance profile of the 726 nm transition. The vertical bars indicate variances of ADATS peak intensities after repeated measurements and a single Gaussian function is used to determine the central wavelength, c) Analog DATS showing a resonance spike at 713.588 nm. d) Resonance profile of the 714 nm transition.

pulsed ~ b e a m by employing the analogue counting method. The resonance spikes revealed in "analogue" DATS and the resonance profiles obtained are shown in Fig. 3.

4

Comparison between Experiment and Theory

The search for laser resonance transitions was very much helped by theoretical predictions of the wavelengths. Without such theoretical guides it would have been impossible to find resonances. As of 1993-94 two or three different theoretical values were available [18, 19, 22, 23], but they are distributed as much as 1000 ppm, none of the theories at that time claimed they had precision better than 1000 ppm. In 1995 a theoretical breakthrough emerged; Korobov [20] calculated the energies of the atomcule by the molecular-expansion variational method and showed that they succeeded in reproducing the experimental values then available within 50 ppm. This triggered extensive experimental searches for unknown transitions. The successes in experiments after 1995 mentioned above were all guided by this theory. In Fig. 4 we summarize the present state of art in the comparison between

100

Toshimitsu Yamazaki L= 33

34

35 n = 39

....... ~ , , ~ 5 9 7 . 2 5 g

~470.724

-150

36

(kth- ;%xp) / ~'exp (ppm) -100 -50

5O i

t,v=O

i

I

v=3

I

I ,

v--3 7

IA (39,35)-* (38,34)J

• Korobov(1995) non-relativistic • Korobov (1996)

relativistic

-

• •

-

AV=2

v=2 v=21 (:38,35)--*(37,34)14He v=2 v=21 (37,34)--)(36,33)l v=3 v--3 7 (38,34)-~(37,33)J v=2 v=21 3He (36,33)-~(35.32)_J V=2 V=4 7 I-- --I (37,34)-~(38'33)1 )t=l v--31'He I- ~(37,35)-*(38,34)~

Fig.4. (Upper) Partial level scheme established by the present laser resonance spectroscopy. The observed resonance wavelengths are shown. (Lower) Comparison of the experimental wavelengths of various transitions with Korobov predictions (closed squares without [20] and closed triangles with [24] relativistic corrections). The upper part is for Av = A(n -- l -- 1) = 0 intraband transitions and the lower part is for Av = 2 interband transitions. The error bars are the experimental ones.

experiment and theory. The experimental wavelengths have high precisions ( ~ 2 ppm). The theoretical values of Korobov [20] are all close to the experimental ones, but with a close look we observe that the theoretical values are systematically longer than the experimental ones. Korobov further took into account possible missing effects; the most important m a y be the relativistic correction for the motion of the bound electron. Since the electron binding is correlated with the antiproton q u a n t u m numbers, the relativistic correction is state dependent and brings a sizable correction on the transition energy. The theoretical values including the relativistic effect by Korobov [24] are surprisingly close to the experimental

Laser Spectroscopy of Metastable Antiprotonic Helium Atomcules

101

values, as shown in Fig. 4. The remaining discrepancies are now only within several ppm. From this agreement we can say that the antiprotonic Rydberg constant is equal to the proton Rydberg constant to the precision of several ppm.

5

Hyperfine

Structure

The hyperfine structure associated with the interaction of the electron spin with the ~ magnetic moment (/,(~) = [9, (p)s + 9, (~)l]/.tN) will provide unique information on the e l e c t r o m a g n e t i c property of the 13. This is a new type of hyperfine structure, but similar to those of hydrogen and muonium and thus is expected to have a splitting of the order of GHz. On the other hand, the "fine structure" splitting (j = l + 1/2) associated with the one-body 1-s interaction is much smaller and thus is neglected at the present stage. Therefore, each state (n, l) is split into doublet states with F + = l + 1/2 and F - = l - 1/2. The unique feature of this hyperfine structure is that the "nucleus" here is not a proton, but an antiproton which has a negative charge. So, we expect that the F + state lies lower than the F - state. We also expect that because of the large 1 the parent doublet states with F + = l + 1/2 and F - = I - 1/2 proceed (without crossing) to daughter doublet states with F '+ = l ' + 1/2 and F ' - = l' - 1/2, respectively (see Fig. 5). This means that when the hyperfine splitting is nearly the same for the parent and daughter states the splitting in the laser resonance profile must be too small to be observed. So, we have to choose a good candidate transition which has sufficiently different hfs values between the parent and the daughter state. In 1995 this new type of hyperfine splitting was evaluated theoretically by Kartavtsev [25], Bakalov et hi. [26] and Korobov [27]. From these studies we learned that the best candidate for observing a doublet structure is the unfavoured transition (37, 35) --+ (38, 34), whose 1-2 GHz separation can be marginally observed by a high resolution laser with a band width of about 1 GHz. Fortunately, this transition is what was observed in 1995 (see Fig. 3). This will be one of the experimental programs to be carried out soon. As the splitting can only just be optically resolved by our present lasers, the resolution for the hyperfine energy (.~ several GHz) is poor (.-. 1 GHz). Therefore, our aim is to determine the hfs energy of the (37,35) state directly via microwave resonance with ~ 1 MHz resolution. However, how can we detect the microwave resonance without having polarization/asymmetric decay as in muonium spectroscopy? We propose first to induce an asymmetric population of the doublet states by a laser pulse tuned to the F + = l + 1/2 --~ F '+ = l' + 1/2 transition, then to turn the electron spin (i.e. invert the population asymmetry) by applying the microwave source for a suitable period, and finally to detect the inverted population with a second laser pulse, as illustrated in Fig. 5.

102

Toshimitsu Yamazaki

unfavoured transition (37,35)-+(38,34) •,'~,-

F_ = L- 1/2

LmewKI1h= 1.2GHz

" ~ h v =L,1t2 F

A

B

V

V_

.jay.,, :

F.' = k' + 1/2

..=_ ¢~ 0.04 ,.~ 0.03

t

~ ~

i

i

~ v

<

i

t2

J

>

2rdf~o

• O

0.02 O

L°"(v I / "-,,'.

0.01

.

.

.

.

"

$cn "~

."/ -,,. . . . . . . .,,.,~.... . . . . .--.,.,,...~,.-..~ . .

~ 01

.5

r 2

I;"(v ) , 2.5

i 3 t

i 3.5

4

(~)

Fig. 5. Hyperfme structure and the laser transitions involving the hyperfine doublets. The 1st laser tuned to u_ causes asymmetric populations of the F + and F states, which can be detected by the 2rid laser. Modulation of the 2rid resonance intensities occurs on the microwave resonance of the parent state with a Rabi period of about 1/~s (principle of 2-laser-microwave triple resonance).

6

Future

Scope

We started this series of studies from very strong curiosity. Gradually, our understanding about the structure of this atomcule advanced, but we hardly believe these would contribute to fundamental physics. With the advent of laser spectroscopy, however, we now hope that high precision spectroscopy of this unique three-body system will yield fruitful information on the basic aspects of the antiparticle, if theory develops as well. The laser resonance method has also played an important role as a microscopic probe of the interactions of the atomcule with neighbouring helium atoms or foreign atoms/molecules. The atomcule is a hydrogen like atom involving many metastable states which can be tagged by laser resonances and thus is expected to serve as a unique probe to study various chemical reactions microscopically [28, 29].

Laser Spectroscopy of Metastable Antiprotonic Helium Atomcu]es

7

103

Acknowledgement

I would like to thank Professor Gisbert zu Putlitz for his continuous interest and stimulation in exotic atom spectroscopy for many years. The present work is based on the collaborative experiments of the PS205 group at CERN. I would like to thank all the members. This work is supported by Grant-inAid for Specially Promoted Research of the Japanese Ministry of Education, Science and Culture.

References [1] [2] [3] [4] [5]

M. Iwasaki et al., Phys. Rev. Lett. 67 (1991) 1246. T. Yamazaki et at., Phys. Rev. Lett. 63 (1989) 1590. R.S. Hayano et at., Phys. Lett. B231 (1989) 355. G.T. Condo, Phys. Lett. 9 (1964) 65. J.E. Russell, Phys. Rev. Lett. 23 (1969) 63; Phys. Rev. 188 (1969) 187; Phys. Rev. A1 (1970) 721, 735, 742. [6] T. Yamazaki et at., Nature 361 (1993) 238. [7] S.N. Nakamttra et al., Phys. Rev. A49 (1994) 4457. [8] E. Widmann et al., Phys. Rev. A51 (1995) 2870. [9] E. Widmarm et al., Phys. Rev. A53 (1996) 3129. [10] B. Ketzer et al., Phys. Rev. A53 (1966) 2108. [11] N. Morita et at., Phys. Rev. Left. 72 (1994) 1180. [12] N. Morita, K. Ohtsuki and T. Yamazaki, Nucl. Instr. Meth. A330 (1993) 439. [13] R.S. Hayano et al., Phys. Rev. Lett. 73 (1994) 1485; 73 (1994) 3181(E). [14] F.E. Maas et at., Phys. Rev. A52 (1995) 4266. [15] H.A. Torii et at., Phys. Rev. A53 (1996) R1931. [16] R.S. Hayano et al., Phys. Rev. A55 (1997) 1. [17] T. Yamazaki et at., Phys. Rev. A, in press [18] T. Yamazaki and K. Ohtsuki, Phys. Rev. A45 (1992) 7782. [19] I. Shimamura, Phys. Rev. A46 (1992) 3776. [20] V.I. Korobov, Phys. Rev. A54 (1996) R1749. [21] G.Y. Korenman, Hyperfine Interactions 101/102 (1996) 463. [22] O. Kartavtsev, private comm. (1994). [23] Y. Kino, private comm. (1994). [24] V.I. Norobov, to be published. [25] O. Kartavtsev, Hyperfine Interactions 103 (1996) 369. [26] D. Bakalov et at., Phys. Lett. A211 (1996) 223. [27] V.I. Korobov, to be published. [28] T. Yamazaki et al., Chem. Phys. Lett. 265 (1997) 137. [29] B. Ketzer et al., Phys. Rev. Lett. 78 (1997) 1671.

Polarized, Compressed 3He-Gas and its Applications E. Otten Institut fiir Physik, Johannes Gutenberg-Universit/it Mainz, Germany

1

Introduction

The history of polarizing 3He gas traces back to the early sixties, when two different pumping techniques were developed almost simultaneously to tackle the problem. Colegrove, Schearer and Waiters [1] started from a gas discharge at low pressure (,-~ 1 mbar) in which the metastable 3S1 state is populated with a relative concentration of 10-6. This state can be efficiently pumped by the infrared circularly polarized resonance lines (~ = 1.0832 pm) to the u p states. The nuclear polarization built up in the 3S 1 state by hyperfine coupling is then transferred to ground state atoms by metastability exchange collisions which occur with a very large cross section of the order of 10-15 cm 2. In these early experiments in Houston most of the interesting physics of this polarization technique was carefully investigated and understood. Relaxation studies showed that surface collisions and surface diffusion at glass walls are the dominant processes, limiting relaxation times to a couple of hours depending on the particular glass or metal surface in question. Also first nuclear reaction experiments on this low density polarized target were performed already in these years. The alternative method is due to Bouchiat and Carver [2]. They investigated spin exchange of 3I-te with optically pumped rubidium vapour which occurs with a much smaller cross section of the order of 10 - 2 4 c m 2 by hyperfine coupling of the SHe nuclear spin to the polarized rubidium valence electron during collisions. Although this method is disfavoured from the side of the exchange cross section, most of this disadvantage can be compensated by much higher densities of the exchange partners. The advantage of the latter method lies, indeed, with the fact that it works almost independently of the 3He pressure: It takes the same time (~ 10 h) to polarize a sample at a few mbar as at a few bar of pressure. The method has been developed to its present perfection and successes by the efforts of the groups of Happer in Princeton [3] and Chupp [4] in recent years. When scattering experiments of polarized electron beams at the Mainz microtron (MAMI) came into sight by the end of the eighties my colleague W. Heil and myself decided to build a polarized 3He target in collaboration with the group of Mich~le Leduc at the l~cole Normale Sup6rieure (ENS),

106

E. Otten

Paris, following the Houston technique which is also called the direct pumping method. The main obstacles which had prevented up to then the polarization of large quantities of 3He were the lack of sufficient laser power at the desired wave length and the difficulty of compressing the polarized gas to reasonable density. The first problem was solved in these years at the ENS by developing a series of different laser types which culminated finally in the arc-lamp pumped LNA laser which delivers more than 5 Watt of laser power at reasonable cost [5]. The compression of polarized 3He was pioneered much earlier already by the work of Daniels and Timsit [6] in Toronto. They demonstrated that a Toepler pump driven with liquid mercury offers in principle a viable solution to the problem. In fact, they observed a few percent polarization in the compressed phase limited by the weak pumping power of the spectral lamps which were still in use at that time. This and other technical difficulties stopped this target project in the pilot phase before being used at accelerators, unfortunately. We decided to take up again Daniels' idea of the Toepler pump, but to develop in parallel also a piston compressor for its larger versatility. The target based on the Toepler compressor has reached now a steady state polarization of 50 % in a volume of about 100 cm 3 at a pressure of 1 bar [7]. It was exposed for many 100 hours to the 870 MeV electron beam of MAMI at a current of 5 to 10 #A. This experiment served for a measurement of the neutron electric form factor as described in section 3.

2 2.1

Polarizing

and

Compressing

aHe

Basic of Direct 3He Pumping

After recalling the basics of 3He pumping I will confine myself to the compression technique by pistons. Figure 1 shows the basic set-up and parameters of direct 3He pumping. The before mentioned laser delivers more than 5 Watt circularly polarized laser light which corresponds to a photon flux of the order of 1020 quanta/sec. The light is passed through an absorption cell containing about 1020 3He atoms, excited to a fraction of about 10 -6 to the metastable 3S1 state by a weak discharge. The cell has a volume of about 31. It is important to provide a long absorption length to make maximum use of the available light. Therefore, we use cells of 1 m length with a dichroic mirror at the end which reflects the laser light back into the cell but transmits red fluorescence light of the 1D2 -4 1P1 transition. The circular polarization of this fluorescence serves to determine the nuclear polarization of the sample, since it is transferred to the electron angular m o m e n t u m through hyperfine coupling after excitation [8]. To this end, the degree of circular polarization of the fluorescence light is analyzed. Figure 2 shows details of the optical pumping scheme. The laser is usually tuned to one of the two hyperfine components ( F --- 3/2 -4 F ' = 1/2)

Polarized, Compressed 3He-Gas and its Applications dichroic mirror

cell, l= ,m,,ID'=6 cm /a~ ~ ~3mbarl ~ ~

=

102_.°a

-T'-

l

orfo

o

--T-

Laser 1.08 jim 5 Watt =10z° hv/s circularly pol.

1 m

Pol. filter

X/4 / I.F. s

~

~

rlv' ~ q L°c~''' ,

~ \

IDz ~

107

v

,

z

IPI fluorescence Pol. monitor

Fig. 1. Principal design and parameters of direct aHe pumping via the aSx state in a low pressure discharge. On the right: Polarization monitoring through measuring the degree of circular polarization of the i D1 1 P1 fluorescence light.

and (F = 1/2 --+ F' = 1/2) of the 3S 1 -+3p0 transition, which are traditionally called C9- and Cs-transition, respectively. Choosing e.g. righthanded C9-pumping light, absorption takes place from the Zeeman sublevels ( m e = - 3 / 2 ) and (mR = - 1 / 2 ) with relative transition probabilities 3:1. Due to its orbital angular momentum the 3p0 state suffers almost complete relaxation by collisions not only between its own two Zeeman levels but within all states of the 3p multiplet. Consequently, the fluorescence repopulates each sublevel of the 3S1 state with equal probability. Setting up rate equations it turns out that an initially unpolarized sample gains in this pumping scheme 5/4h of angular momentum for each absorbed c~+-photon on the average. In this starting phase about 25% of the light is absorbed resulting in a gain of more than 1019 polarized spins/sec. With increasing polarization, however, the efficiency of the pumping process drops dramatically since the absorbing states are getting depopulated and the sample will no longer interact with the pumping light. 2.2

P u m p i n g D y n a m i c s a n d Spin E x c h a n g e

Very important is the fast exchange of the 3S1 excitation energy from one atom to the next by collisions which occur at a rate of typically 106/sec under the given conditions. These collisions conserve the total angular momentum of the colliding system and its projection, since spherical atomic S-states are involved only; but they will exchange electronic and nuclear spin polarization due to hyperfine coupling in the initial and final excited states. Under usual conditions this spin exchange rate is the fastest process in the rate equations. Hence, it leads to a spin exchange equilibrium, characterized by population numbers

N(mF) N(mD

-

exp(

(mr-

=

(1)

108

E. Otten

Is2p ~Po

F=i/2

P~

T

°/l _

n,-

(3) a' a (1)

uz

exitation transfer collisions Isz ISo

collisional mixing '~ isotropic °/lreemission

-3/2

F=I= 112

-1/2

Z2..au

+ I/2

i .... l -I/2 +i/2

+ 3/2

m F

my= ml

Fig. 2. Level diagram of the pumping scheme indicating also the principal processes. The height of the columns represent the relative population of Zeeman levels in spin exchange equilibrium at P1 = 33 %.

The case x = 2, which is plotted in Fig. 2 by the height of the columns on each m F sublevel, corresponds to a nuclear polarization Ps = (N(]') - N ( $ ) ) ( N ( $ ) + N($)) = 33% .

(2)

At Ps = 85 % (about the m a x i m u m which has been achieved, so far) z has risen to 12.3 . At that point the absorbing levels are depopulated so much already, that the pumping efficiency has dropped to about 1% of its initial value. These features of the pumping dynamics are illustrated in Fig. 3 which is an on-line recording of the built-up of polarization in the pumping cell in the first 200see, followed by the relaxation in the plasma after the laser has been turned off. Within the first 25see the polarization rises already to 65 %. The pumping efficiency at that point is indicated by the tangent to the slope; it corresponds to a gain of about 3 × 10 is polarized spins/see. The final rise to the saturation level of Ps = 85 % takes 10 times longer. Hence, this saturation point is not economic anymore under conditions where one has to maximize a quality factor defined as the number of atoms times the polarization squared. From the right half of Fig. 3 we read a relaxation time of the gas of Ta ~ 5 rain. It is caused by exciting the atoms to higher levels in the plasma where the nuclear polarization couples to orbital angular m o m e n t u m which is then dissipated by fluorescence and gas kinetic collisions. The stronger the plasma, the faster the relaxation! On the other hand, one needs a m i n i m u m discharge level in order to maintain a sufficient density of metastable atoms. The o p t i m u m is usually found for a very soft, homogenous radio-frequency discharge.

Polarized, Compressed 3He-Gas and its Applications

90

~

g

......

_P: :85~o

laser

beam

109

"o1"["

70

so

"~

30

~

20

running

~ = 5.6cm I = 100 cm

3He :I m bar

~0

0 t 0

,

I

SO

,

I 100

'

I

1~

I

,

200

*

,I

I

2~

300

350

Time lsecl Fig. 3. Built-up and decay of 3He polarization in the optical pumping cell. After 200 sec of pumping the laser was shut off. The tangent to the curve at P = 65 % -4 corresponds to a polarization rate of 3.1018 3He/s.

2.3

Relaxation by Molecular Ions and Questions of Gas Purity

Another source of relaxation is the formation of 3He+ molecules which are formed in the discharge bY 3-body collisions. They are paramagnetic and have a strong hyperfine coupling of the nuclear spin to the electronic one. The latter is again weakly coupled to the orbital angular momentum of the molecule which relaxes very fast in gas kinetic collisions. As their formation rate rises with the square of the gas pressure it starts to dominate relaxation above 1 mbar and causes the equilibrium polarization to drop. The relaxing power of these molecular ions is enhanced furthermore by a kind of catalytic process which consists in exchanging atoms during collisions. This process is particular dangerous in high pressure targets under irradiation of an accelerator beam. Under these conditions a single molecular ion may cause a thousands of relaxations until it is quenched by neutralization [9]. Therefore, it is important to spike the helium with some quenching gas. An admixture of 0 . 1 % of N2, e.g., suffices already for that purpose. Therefore, in our system spiking of the compressed, polarized 3He gas is provided by a needle valve. On the other hand, any kind of impurity of the helium gas in the optical pumping cell is fatal because it also quenches the essential metastable 3He atoms and, moreover, lowers the electron temperature in the plasma to a level which prohibits the excitation of helium completely. Therefore, spectral purity of the 3He (corresponding to about 1 ppm impurity level) is mandatory in this region. For that purpose the helium is passed through a NEC-getter and a LN2-trap before it enters the pumping cell (see Fig. 4). Also the connection between the pumping cell and the compressor is secured by a LN2-trap in order to suppress back diffusion of rest gas from the compressor into the pumping region.

110

E. Otten

;H-'e-Filling-Station ,,it. 3He-Reservoir , SI.... 4 optical

~LN:

Pumping-cells [ ~

4.J W !~/I = 1083nm

i -,-.4,,

OpL

Pol.-Detek'tion

t%

,Fie.Container

HelmhoRzguiding'field

2.4

Design and Performance

Fig. 4. Schematic design of the 3He polarizer and compressor.

of the Piston Compressor

Our latest version of the piston compressor has two stages with a buffer cell in between (Fig.4) [10,11]. They are made of titanium for the excellent mechanical and vacuum properties of this material. Note that any ferromagnetic material has to be avoided for reasons of relaxation through magnetic field gradients. Even austhenetic stainless steel cannot be tolerated. The stroke of the pistons is bridged by a titanium bellows in order to guarantee tightness to the outside. The pistons are sealed with respect to the cylinders by special Quad-rings and fluorinated grease. After the second stage a pressure of up to l o b a r is reached in the target cell. The cell can be detached from the apparatus by the help of a glass cock and a flange. It may then be transferred for use in some remote experiment. During transport a pair of Helmholtz coils guarantees a sufficiently homogenous guiding field of a few Gauss. After use the cell is reconnected to the system and evacuated, whereas the depolarized 3He gas is sluiced back through the aforementioned getter into the optical pumping cell. In the optical p u m p i n g region which actually consists of a battery of four 1 m long cells we reach at present a polarization between 60 and 70 %, distinctly smaller than in sealed off cells. The spectrum of the discharge shows that the latter seem to be cleaner. The best result obtained in the compressed phase so far is PI = 52 % at p = 6 bar. The system polarizes up to 10 m b a r l / m i n . The reasons for polarization loss during compression are in principal clear and more or less under control. Since relaxation occurs predominantly at

Polarized, Compressed 3He-Gas and its Applications

1] ]

surfaces, the relaxation time is proportional to the ratio of the volume to the surface. This ratio is getting very unfavourable during the last phase of compression. Moreover, the dead volume in the transfer region between the compressor and the storage cell which includes in particular the valve is very dangerous. There, the gas is stored for at least one compression cycle which is 10seconds for the first and a couple of minutes for the second compressor. Also in this region the volume to surface ratio is unfavourable. In the original version of the piston compressor [12] these transfer losses were much more serious than at present after we have redesigned the valves and the transfer region. There are reasons to expect that also the residual transfer losses will be overcome in the course of further development work.

2.5

Relaxation Studies with Glasses and Surface Coating

For a volume to surface ratio of (V/S) = 1 cm, corresponding to a spherical cell with ~ ~ 6 cm one measures a relaxation rate of about 1 hour for Pyrex glass and roughly 10 hours for an alumino-silicate glass, like Corning 1720 or the Supremax glass from Schott. These latter glasses are supposed to have a tighter surface which is less penetrated by diffusion. Moreover, the paramagnetic purity of the glass is decisive. First experiments with iron free Supremax glass show substantial improvement with T1 ranging from 20h to 60h. The numbers given apply to carefully cleaned, baked and highly evacuated vessels. T1 may drop to several minutes for an untreated vessel which has just been pumped down to rough vacuum. Using a detached cell in a remote experiment asks for particularly long relaxation times. A great variety of materials had been tested already in early experiments, e.g. by Timsit et al [6]. They inserted bulk samples into glass vessels which were then sealed off. None of the materials tested reached relaxation times longer than those obtained by the best glasses. Therefore, we decided to evaporate in situ very pure, volatile metals onto the inner surface of Supremax vessels. Our primary choices were bismuth and caesium for the following reasons: Bismuth is known to be a diamagnetic metal, having only minimal pockets of paramagnetic band structure. Caesium on the other hand shows the usual paramagnetism of a conduction band. But it is known, that it practically does not bind any helium to its surface (in fact, it is the only surface which is not wetted by superfiuid helium) [13]. The suspicion that an alkali metal might be good arose also from the observation that groups using rubidium exchange pumping reported consistently longer relaxation times than obtained by direct pumping in a clean 3He atmosphere. Both our expectations were fulfilled. With bismuth we reached relaxation times of 50 hours in Supremax vessels, for caesium up to 117 hours were observed, (Fig.5), [14]. Detachable ceils coated with bismuth or caesium showed no evidence for increased relaxation after several refillings [14], [10]. Still, one has to add the caveat that these very long relaxation times were not reproduced

112

E. Otten

50

z.O

ci

p=2.17bar I10cm3

~

r30 o 'K. 20 o

.o

-5

a_

10

T~" 117h = 30rain

:12 oJ

- " ~

i

0

100

200

300

~0

Time [hi

Fig. 5. Relaxation of compressed 3He in a detached Cs coated cell with 0 ~ 6cm.

in every of these experiments; but they seem to correlate positively to the effort invested in the preparation of the surfaces. 3 Determination of the Electric Form Factor of the Neutron i n D o u b l y P o l a r i z e d a H e ( 7 , e' n ) - S c a t t e r i n g The internal structure of hadrons has always been a central issue in particle physics since R. Hoffstadter discovered in the fifties that charge and magnetism of the proton are spread over a certain volume. Nowadays the form factors of the nucleons are cornerstones for checking effective quark models and QCD-calculations. In this respect the electric form factor of the neutron is particularly interesting, since it should vanish identically in any first order quark model where the quarks occupy the same spatial wave function. Indeed, it is so small that it escaped precise measurements so far, except for its mean squared charge radius which - surprisingly enough - could be measured with high accuracy in thermal neutron scattering to be [15] 2 ( r ch}n = - 0 . 1 1 6 e fm 2 .

(3)

Charge distribution and magnetization of a neutron enter the elastic cross section of electron scattering through their Fourier-transforms, the elastic form factors G~ (Q2), G ~ (Q:) which are functions of the square of the four momentum transfer Q2. They modify the M O T T cross section of a point-like charge by do-(Q ~) dX?

=

(d~(Q~)) • [a G ~ ( Q 2) + b G~u(Q2)] \ dc~ MOTT

(4)

where a and b are kinematical factors known from the Rosenbluth formula. At Q2 = 0 the two form factors yield the total charge (i.e. zero for the neutron)

Polarized, Compressed 3He-Gas and its Applications

] 13

and magnetic moment, respectively. The neutron problem is characterized by the fact that for all Q

GE,~(Q2) _< 10-2G2

/O2~1

(5)

holds. Thus, it is extremely difficult to extract its electric form factor from a measurement of (4) reliably with the experimental and systematic errors of such experiments given. The problem can only be solved if one manages to enhance the effect of electric scattering by interference with the magnetic one. This situation occurs in spin polarized scattering, since the electric scattering amplitude is insensitive to the relative orientation of spins whereas the magnetic one is. Thus, one searches for an asymmetry of the scattering cross section with respect to positive and negative helicity of the electron beam given by A --

cr+--cr-

-- Pe

Pn±CGE,nGM,n+PlnldG2Mn = A± +All '

.

(6)

Pe is the polarization of the electron beam. Thanks to the efforts of E. Reichert and his group [16], we dispose now at MAMI of an excellent polarized electron (~+) - source, delivering 10 to 15 pA at a polarization of 80 % to 50%. PnJ- and pie are the components of the neutron polarization parallel and perpendicular to the momentum transfer which it got by the electron. The coefficients c and d are two more known kinematical factors. We see that the asymmetry measured for perpendicular neutron polarization A ± is indeed proportional to the interference term between electric and magnetic scattering whereas A II is of purely magnetic origin. Measuring the ratio of these two asymmetries gives directly the electric form factor in terms of the known magnetic one:

GE,n (O2) = (c Al /d AII) GM,n(Q ~) .

(7)

How to get a target of free polarized neutrons of sufficient density? We can only dream of it. But a polarized 3He target is a realistic substitute since its spin and magnetic moment is ahnost entirely carried by the unpaired neutron (see Fig. 6). To make sure that it is the neutron and not the proton from which the electrons are scattered quasi-elastically one asks for coincidence between the scattered electron and the neutron. For the case of parallel neutron polarization Fig. 7 shows the electron spectrum for positive and negative electron helicity, respectively. In the quasielastic peak on the right hand one recognizes the large asymmetry of this scattering. After having analyzed 50 % of the total data, the decisive ratio of asymmetries in (7) could be determined to be A 1e x p Al!xp

=

0.146 (13)

(8)

114

E. Otten

,tire

t,, ~

~:~, n-defector

P°I

P

-detecfor

up dow n mognefisofion

C:J

Fig. 6. Principle of determining neutron form factors by exclusive doubly, polarized, 3 -4

quasi-elastic He ( e , e ' n ) scattering.

from which we determine an electric neutron form factor of GE,n (Q2 = 9 fro-2)

=

0.045 (±) 0.005

(preliminary) .

(9)

Figure 8 shows this value together with some earlier measurements which also dealt with polarization variables. The point at 8 fm -2 stems from our pilot experiment [17]. Note that this experiment was not performed with the offline target cells described above, but still used the on-line Toepler compressor which yielded in a target of 100 cm 3 a pressure of 1 bar at P t = 50 %. Figure 8 shows in addition a few theoretical curves for GE,n including the first recent QCD-prediction by O. Nachtmann [18] which seems to fit quite well. [ , i , , . i

6O0

A~xp=- ll' °/°

i:iii-- G-

ii =-16 oVo At.It

5OO

_

LO0 I/% ,.l,..

r-3OO o i,..J 20O

-1~ inelostiC quas'ci |mr evenl's

\

tO0 . ~ - i

0

2O

,

I

i

I

i

I

=

60 80 100 120 Energy (a. u. )

140

Fig. 7. Electron spectra of 3He ( e , e ' n ) scattering obtained by a Pb-glass detector array. The asymmetry with respect to negative (upper curve) and positive (lower curve) electron helicity shows up in the quasi-elastic peak. The 3He spin is oriented parallel to the momentum transfer.

Polarized, Compressed 3He-Gas and its Applications

115

0.12 Dl~,e'fi)

/ d

aK

f

c~

I Wetimir~ry }

1a-~ \ 3

"°°°°°°°°*'*'*°°. . . . . . . . . . . . . . . . .

He I~,e'nl

IJL 0

~

'

I(}

2O

(12 (fm-2l Fig. 8. Results of GE,n (Q2) measurements in doubly polarized electron scattering experiments together with various theoretical curves. The point at Q2 _ 8 fm -2 is the most recently pubfished value, obtained in the pilot phase of the experiment at Mainz [17]; the point at Q~ = 9fm -2 stems from a preliminary evaluation of 50% of the full data set obtained in that experiment. The curve with points and short dashes results from a recent QCD-calculation [18].

4

Neutron

Spin

Filter

A very promising application of samples of dense polarized 3He opens up for thermal and epithermal neutron beams. Neutrons with spin opposite to that of 3He are absorbed with a cross section of 6000barn; this enormous cross section applies for neutrons with a wavelength of 1 ~ and decreases in proportion to the wavelength. The parallel component, on the other hand, is hardly attenuated by elastic scattering with a cross section of a few barn only. Thus, polarized 3He can serve as a neutron spin filter for a broad band of energies and for any direction of the neutron m o m e n t u m . This is a great advantage over traditional neutron polarizers or analyzers which accept only a very limited phase space, like Bragg-reflexes or total reflection from magnetized materials. First successful a t t e m p t s to polarize a neutron b e a m by a 3He spin filter have been undertaken with a sample, polarized by the Rb spin exchange method [19]. Figure 9 shows polarization and transmission of a neutron spin filter as a function of its opacity which is proportional to the product of its pressure, its length and the wave length of the neutrons; it is given in practical units: bar ~ cm. The contrast of the filter rises of course with the SHe polarization. Two curves are plotted for PI = 20 % and 52 %, respectively. The lower value, 20 %, was achieved in our first test experiment at the Mainz T R I G A reactor in 1994 [10,11]. The second test with 52 % SHe polarization by the end of last year yielded a neutron polarization of 84 % at a total transmission

116

E. Otten

IOO 1995

80

F~

60. 40.

20-

1

•

~"

T m

00

I0

20

Pile * L

30 *

40

50

60

70

I [bar */~ * cm ]

Fig. 9. Polarization Pn and transmission Tn of a 3He neutron spin filter as a function of its opacity which is the product of pressure, absorption length and neutron wave length. Plots are given for two different 3He polarizations as indicated. The pilot data points result from an experiment, performed at the Mainz TRIGA reactor [10,11].

of 20 %. This is not too far anymore from the ideal values of PI = 100 % at 50 % transmission. For this experiment the target was polarized at the piston compressor in the Institut fiir Physik and then transferred to the broad band neutron b e a m from the T R I G A reactor on campus. The polarization was analyzed by the Bragg-reflex from a magnetized cobalt iron single crystal. At present we are building in our laboratory in collaboration with the Institut Laue-Langevin another two-piston 3He polarizer and compressor. Similar instruments are planned at the National Institute of Standards and Technology (NIST) in Washington, at the Hahn-Meitner Institut in Berlin and for the Research Reactor in Munich. 3He spin filters would be particularly useful as polarizers and analyzers for pulsed neutron beams from spallation sources (Fig. 10). T i m e of flight analysis and a large detector array would allow simultaneous measurement of doubly differential cross sections regarding scattering energy and angle. A 3He analyzer close to the scatterer could analyze the polarization of the scattered neutrons simultaneously at full solid angle of scattering. A single 3He cell would thus replace the extremely expensive arrangement of an array of super mirrors otherwise used for polarization analysis of scattered neutrons. 5

3He Tomography

of the

Human

Lung

Last year W. H a p p e r ' s group at Princeton demonstrated in cooperation with the magnetic resonance imaging group at Duke University in a seminal paper

Polarized, Compressed 3He-Gas and its Applications

1 17

-+

Fig. 10. Possible scheme of using 3He spin filters for polarizing a pulsed broad band neutron beam from a spauation source and analyzing the polarization of scat+ tered neutrons; a polarized 3He counter may integrate the analyzing and detection function.

the possibility of imaging lung tissue filled with hyperpolarized noble gas [20]. The term "hyperpolarized" indicat,es that the noble gas has been polarized (by means of optical pumping) to a degree far beyond the ordinary Boltzmann equilibrium P ~ o l t z m a n n= exP ( - p B / k T ) (lo) which is of order to under usual nuclear magnetic resonance (NMR) conditions. The first pilot experiments were performed with small samples of lZ9Xe and later 3He polarized by rubidium spin exchange. The gas was inserted into the lungs of small test animals and NMR-pictures taken. The enormous gain in polarization degree by 5 orders of magnitude as compared to conventional NMR outweighs not only the loss in spin density between tissue and gas which is of order 1000, it also has enough signal capacity in order to feed the many hundred consecutive NMR pulses which each pixel suffers in the process of imaging taking. For that purpose the magnetization is tilted only by a few degrees out of the field axis by each radio-frequency pulse, such that the longitudinal magnetization is effected marginally per pulse. The induction signal is still large enough! This procedure is enforced by the circumstance that the hyperpolarization can be used up only once during image taking, once destroyed it will recover only to the tiny value of the Boltzmann equilibrium (10) which is useless at these small densities. The large interest raised in the medical community by this method stems from the fact that porous tissue like the lungs is difficult to image by conventional magnetic resonance imaging (MRI). Also X-ray or nuclear radiation methods do not give truly satisfactory results for that task. With our technique of compressing polarized 3 ~ into e detachable and transportable cells we found ourselves well prepared for entering this new field

1~8

E. Otten

Fig. 11. Sagittal slice of a human thorax imaged with conventional MRI (left) and MRI on hyperpolarized 3He, inhaled in the lungs (right) [21].

of research. In collaboration with the Radiological Institute of the University of Mainz and with the Deutsche Krebsforschungszentrum in Heidelberg we made a first try to image the human lung in vivo by inhaling a liter of 3He polarized to a degree of 46 %. The result is shown in Fig. 11 which shows two sagittal slices of the volunteer's thorax, on the left taken with conventional proton MRI, on the right with 3He MRI. Where proton MRI leaves a large, black hole, indicating no structure, 3He gives a bright signal revealing the lung in any detail. The in-plane resolution of this picture is (1.2mm) 2 [21]. In parallel, also the Princeton-Duke-collaboration has produced an in vivo 3He picture of the human lung [22]. 6

Conclusions

Optical pumping of 3He has been subject of studies in atomic physics for very many years. With the advent of powerful pumping lasers it became possible and worthwhile to start a research and development program towards production of large quantities of spin polarized 3He. Polarization preserving compression techniques as well as storage cells with relaxation times exceeding several days were developed. Production rates of order I bar liter/h were achieved at a polarization of about 50 %. Such large amounts of spin polarized 3He gas are necessary and useful for interdisciplinary applications. To that end we have engaged ourselves into three very different topics: 1. In Particle Physics: Determination of the electric form factor of the neu---)-

__+

tron in 3He ( e , e n) - scattering. 2. In Neutron Physics: Establishing 3He spin filter for polarizing and analyzing thermal and epithermal neutron beams for scattering experiments. 3. In Clinical Research: 3He MR-tomography of the human lung.

Polarized, Compressed aHe-Gas and its Applications

119

References [1] [2] [3] [4]

[5] [6]

[7] [8] [9] [10] [11]

[12]

[13] [14] [15] [16]

[17] [18] [19] [20] [21]

[22]

F.D. Colegrove, L.D. Schearer and G.K. Waltes, Phys. Rev. 132 (1963) 2561 M. Bouchiat, T.-R. Carver and C.M. Varnum, Phys. Rev. Lett. 5, (1990) 463 W. Happer, this volume J.R. Johnson, A. K. Thompson, T.E. Chupp, T.B. Smith, G.D. Cates, B. Driehuys, H. Middleton, N.R. Newbury, E.W. Hughes and W. Meyer, Nucl. Instr. and Meth. in Phys. Research A 356 (1995) 148; also reviewed e.g. by T. Chupp in Proc. Int. Workshop on Polarized Beams and Polarized Gas Targets, Cologne, June 1995, eds. H. Paetz gen. Schieck and L. Sydow, World Scientific Co., Singapore, 1996, p. 49-62 C.G. Aminoff, C. Larat, M. Leduc, B. Viana and D. Vivien, J. Luminescence 50 (1991) 21. R.S. Timsit, J.M. Daniels, E.T. Denifing, A.K.C. Kiang and A.D. May, Can. J. Phys. 49 (1971) 508; R.S. Timsit, W. Hilger and J.M. Daniels, Rev. Sci. Instrum. 44 (1973) 1722 G. Eckert, W. Heil, M. Meyerhoff, E.W. Otten, R. Surkau, M. Werner, M. Leduc, P.J. Nacher and L.D. Schearer, Nucl. Instrum. Meth. A 320 (1992) 53 N.P. Bigelow, P.J. Nacher and M. Leduc, J. Phys. II,1 France, 2 (1992) 2159 K.D. Bonin, T.G. Walker and W. Happer, Phys. Rev. A 37 (1988) 3270 R. Surkau, Doctoral Thesis, MaJnz, 1995 J. Becker, M. Ebert, T. Grossmann, W. Heil, H. Humblot, M. Leduc, E.W. Otten, D. Rohe, M. SchMer, K. Siemensmeyer, M. Steiner, R. Surkau, F. Tasset and N. Trautmann, Journal of Neutron Research 5 (1996) 1 J. Becker W. Heil, B. Krug, M. Leduc, M. Meyerhoff, P.J. Nacher, E. W. Otten, Th. Prokscha, L.D. Schearer and R. Surkau, Nucl. Instrum. Meth. A ~46 (1994) 45 P.J. Nacher and J. Dupont-Roc, Phys. Rev. Lett. 67 (1991) 2966 W. Heil, H. Humblot, E.W. Otten, M. 8chMer, R. Surl~u and M. Leduc, Phys. Lett. A 201 (1995) 337 H. Leeb and C. Teichtmeister, Phys. Rev. C 48 (1993) 1719 E. Reichert in Proc. Int. Workshop on Polarized Beams and Polarized Gas Targets, Cologne, ~lune 1995, eds. H. Paetz gen. Sehieek and L. Sydow, World Scientic Co., Singapore, 1996, pp. 285 M. Meyerhoff et al., Phys. Letters B 327 (1994) 201 O. Nachtmann, Lectures given at the First ELFE Summer School, July 1995, Cambridge, UK, to be published K.P. Coulter, A.B. Mc Donald, W. Happer, T.E. Chupp and M. Wagshul, Nucl. Instr. and Meth. in Phys. Res. A 288 (1990) 463 M.S. Albert, C.D. Cates, B. Driehuys, W. Happer, B. Saam, C.S. Springer Jr. and A. Wishnia, Nature, Vol. 370 (1994) 199 M. Ebert, T. Groflmann, W. Heil, E.W. Often, R. Surkau, M. Ledue, P. Bachert, M.V. Knopp, L.R. Schad, and M. Thelen, LANCET 347 No. 9011, p. 1297 Reported in Laser Focus World, Nov. 1995.

Medical N M R Sensing with Laser-Polarized aHe and 129Xe William Happer Department of Physics, Princeton University, Princeton, NJ 08544, USA I want to review how some interesting new applications of atomic physics to medical imaging have been influenced by work clone at Heidelberg by Professor Gisbert zu Putlitz and his predecessors over the past two centuries. I will begin my story in the year 1860 when two Heidelberg scientists Robert Bunsen, then aged 50, and 36 year-old Gustav Kirchhoff aspirated the residue of forty tons of evaporated Diirkheim mineral water through a Bunsen burner and were the first persons to see the beautiful, sky-blue spectral line of a new alkali metal which they called "cesium" from the Latin adjective "blue-gray", a typical Latin usage being " c a e s i o s o c u l o s G e r m a n o r u m - the blue eyes of the Germans." A year later in 1861 Bunsen and Kirchhoff observed a dark red spectral line from a few grains of the mineral lepidolite, the first discovery of another new alkali metal, which they named "rubidium" from the Latin adjective r u b i d u s - red or blushing. Rubidium, especially, is going to play an important role in the medical applications that are the subject of this paper. In 1894, seven years after Kirchhoff's death, Lord Rayleigh and William Ramsay discovered the noble gas argon. A year later, in 1895, Ramsay discovered helium, fossil alpha particles as we know now, by outgassing the uranium mineral c l e v i t e . Four years later, in 1898 Ramsay and Travers discovered the other noble gases, neon, krypton and xenon. Bunsen died in 1899, living just long enough to see the discovery of all of the stable noble gases. Noble-gas atoms and alkali-metal atoms are very closely related. For example, a rubidium atom has a single valence electron bound to a krypton-like core. The single valence electron of the alkali-metal atom is extremely efficient in absorbing or emitting light, and that is one of the reasons Bunsen and Kirchhoff were so successful in their first application of spectroscopic analysis to the alkali-metal atoms, which make beautiful colors in the flame of a Bunsen burner, or in the fireworks displays that Rector zu Putlitz was so fond of arranging above the Heidelberg Castle. It remained for the scientists of this century to begin to work out the improbable physics that makes magnetic resonance imaging possible: the discovery of isotopes, the development of quantum mechanics and its spectacularly successful application to atoms and nuclei. I want to mention a major contribution of Heidelberg during this period, the studies of atomic hyperfine structure by Professor Hans Kopfermann [1] and his research group. Of the many outstanding students trained by Kopfermann, Gisbert zu Putlitz was not the least distinguished. Much of what we know today about atomic hy-

122

William Happer

Fig. 1. One of the first images of a human lung [2], that of Dr. Cecil Charles of Duke University, taken with laser-polarized 3He gas. perfine structure was discovered in Heidelberg. The hyperfine interactions of nuclear moments with their surrounding electrons play a crucial role in the applications of spin polarized 3He and 129Xe to medical imaging. The first application of physics to medical imaging was made in Germany 100 years ago in the fall of 1895 by Wilhelm RSntgen with his discovery of x-rays. Many other applications of physics to medical imaging have occurred since RSntgen's discovery, for example, the use of radioactive isotopes, positron emission tomography, ultrasound, etc. Nevertheless, RSntgen's xrays have remained the most widespread imaging method in medicine. However, a fairly recent entry into the medical imaging field, magnetic resonance imaging or MRI, is giving x-rays serious competition for the first time. Probably most of you know a little bit about MRI already, but a recent, novel variant of MRI is displayed in Fig. 1, which shows the first MRI picture of a human lung, obtained with laser-polarized 3He gas in a collaboration of our Princeton group with researchers at Duke University [2]. The very first images with laser polarized gas, a crude image of the excised lungs of a mouse [3], were obtained in a collaboration between our Princeton group, led by Gordon Cates, and a group at the State University led by Arnold Wishnia. Unlike x-rays, which scatter preferentially from the higher-Z elements of the body - the calcium of bones for example, conventional magnetic resonance images are of the protons in the watery and fatty tissues of the body. For this

Medical NMR Sensing with Laser-Polarized aHe and 12°Xe

123

reason the soft tissue, which is full of protons, shows up very clearly with lots of structure, while the same tissue would be almost featureless when observed with x-rays. Until recently, the lung, has stubbornly resisted all attempts to produce good images - with MRI, x-rays or any other modality. The reason is that the lung is designed to be filled with air. Consequently, it has a very low proton density and a complicated three dimensional structure of tissue and gas spaces, with magnetic susceptibility gradients that produce artifacts in conventional MRI images. There is little to scatter x-rays in a healthy lung, but well-advanced pathologies like the calcified lesions of tuberculosis do show up clearly. So the ability to make detailed lung images like those of Fig. 1 has been a welcome innovation, and I want to review some of the curiosity-driven basic research that has made it possible. Nuclear magnetic resonance (NMR), which was discovered in 1946 by E. M. Purcell, H. C. Torrey and R. V. Pound [4], and independently by F. Bloch, W. W. Hansen and M. Packard [5], makes use of the small voltages induced in a pickup coil by the changing magnetic flux from nuclear spins, which are precessing about a magnetic field. In NMR experiments, the signal voltage is proportionM to the static magnetization density of the nuclei, which can be written as M = NpIP , (1) where N is the number density (spins/cma), P l is the magnetic moment of a nucleus of spin I, and the polarization P measures how nearly the spins are lined up with the magnetic field. For simple spin-l/2 nuclei like those of ordinary hydrogen atoms, 3He or 129Xe atoms, the thermal-equilibrium polarization P at a temperature T and in a magnetic field B is P=tanh(~T

B)

(2)

Because the nuclear moment # I is so small, the nuclear polarization (2) is very small at room temperature, even in very large magnetic fields. For example, for a proton in a magnetic field of 1 T, the polarization is P = 3.4 x 10 -6. However, the number density N of (1) is often so large (for example N = 6.7 x 1022 c m - 3 in water) that this weak thermal polarization is adequate to produce good signals. The lung images of Fig. 1 were made possible by using lasers to increase the small thermal polarization described by (2) by some five orders of magnitude, to values on the order of P ~ 0.3. Therefore, the signal per aHe nucleus in the lung of the volunteer of Fig. 1 was 100,000 times stronger than the signal per proton, and this enormous signal enhancement m o r e than made up for ~, 2,000 times smaller density of 3He nuclei in the lung compared to the density of protons in human tissue like muscle or the brain. Among the many applications of NMR, magnetic resonance imaging (MRI) is perhaps the most striking to the average person. In 1973 Paul Lauterbur

124

William Happer

1 Slowly processing

proton spin

Distance

Rapidly pre~ essing proton spin

RF signal

intensity

Fig. 2. Basic idea of magnetic resonance imaging [6]. The patient is placed in an inhomogeneous magnetic field and the protons in the body are tipped with appropriate radiofrequency pulses. As the protons precess about the field, the more slowly precessing protons in the patient's foot produce lower-frequency radio waves, and the more rapidly precessing protons of the patients head produce higher-frequency radio waves. In practice, very elegant pulse sequences are used to make the best images for diagnostic purposes. [6] pointed out that it should be possible to use magnetic resonance in an inhomogeneous magnetic field to produce images of the distribution of nuclei in a three-dimensional object. Rather than struggling to eliminate all magnetic field inhomogeneities, one applies carefully controlled inhomogeneities to map spatial information onto the free induction decay transients and spin echos of classical NMR. The essence of Lauterbur's idea is sketched in Fig. 2. The high polarization of the 3He nuclei used to make the image of Fig. 1 has its origins in photons of light and spin-exchange collisions. In 1948, Albert Kastler [7] showed that it is possible to transfer much of the spin polarization of circularly polarized photons to the electron and nuclear spins of atoms. Kastler called this process optical pumping. The great majority of optical pumping work is done with the alkali-metal vapors, mercury vapor or metastable helium atoms [8]. Although the polarized 3He gas used to make the image of Fig. 1 was produced by spin exchange with optically pumped rubidium vapor, I should mention that the research team of Professor Ernst OLten [9], a classmate of Gisbert zu Putlitz, has developed such fantastic equipment for pumping metastable 3He, that they make equally good images of human lungs [10] with this alternate method. Fittingly enough, the first images of the Otten

Medical NMR Sensing with Laser-Polarized 3He and 129Xe

]25

group were obtained in Heidelberg, in collaboration with the a group led by Mich~le Leduc and Pierre-Jean Nacher of the Ecole Normale Sup~rieure in Paris. Atoms that can be efficiently optically pumped have several features in common. The polarized atoms have spherically symmetric pumped states, 2S states for the alkali-metal atoms, a 1S state for mercury atoms, and a 3S state for metastable helium atoms. Such electronic states are exceptionally resistant to electron or nuclear spin depolarization during collisions because their electric charge distributions are nearly perfectly round. Furthermore, all of these systems can be pumped with near-visible light, and they have appropriate atomic number densities to absorb most of the pumping light at convenient temperatures. Containing the gases and vapors presents manageable materials problems. In our group at Princeton, we have mostly made use of the rubidium atoms discovered at Heidelberg by Bunsen and Kirchhoff. A typical arrangement for optically pumping rubidium vapor is sketched in Fig. 3. Not shown is the source of the static magnetic field H, which is often only a few Gauss produced by Helmholtz coils and designed to prevent spin depolarization by stray 60 Hz fields or the ambient field of the earth. A simple oven is used to keep the cell, containing a few droplets of rubidium metal, at a constant temperature, usually greater than 80 °C and less than 150°C. This ensures that the density of Rb atoms in the cell is adequate to absorb the light and to collide frequently enough with 3He or 129Xe atoms to efficiently transfer spin to their nuclei. The number density of Rb atoms is typically 1011 to 1014 atoms cm -3. More intense pumping light is needed to handle the higher number densities. The sample cell also contains the noble gas which is to be polarized by spin-exchange and a chemically inert quenching gas, normally nitrogen, to prevent reradiation of light from the excited atoms. Since the reradiated light would be nearly unpolarized and can be absorbed by the Rb atoms, it would optically depump the atoms. The damage done by the reradiated photons becomes more severe at high Rb vapor densities where the reradiated photon can be scattered several times and depolarize several atoms before escaping from the cell. The detailed atomic physics of optical pumping for the arrangement of Fig. 3 is shown in Fig. 4. We consider an imaginary alkali-metal atom for which the nuclear spin is zero. All real alkali-metal atoms have non-zero nuclear spins, but the basic mechanisms of optical pumping and spin exchange are qualitatively similar. Circularly polarized D~ resonance light is incident on the atoms. The light can excite the atoms from the 2S1/2 ground state into the 2P!/2 first excited state. In both the ground state and the excited state, the electronic angular momentum of the atom can point up or down. This is represented by two Zeeman sublevels, which are split by their respective Larmor frequencies when the atom is in an external magnetic field. Since the atom must absorb both the energy and the spin angular momentum of the photon, transitions are only possible from the spin-down ground-state sub-

126

WilliamHapper n

Illuminated

Dark Volume

Magnetic Field

Volume ~ .................

~ "- .

~-~

(•)Rb @ Xe

sI

JJ

C3N

2

Pumping Light Sample Cell Fig. 3. Optical pumping of an optically thick vapor of rubidium atoms (see text).

level to the spin-up excited-state sublevel. The noble gas and the quenching gas collisionally transfer atoms between the sublevels of the excited state. When a quenching collision finally deexcites the atom, both ground state sublevels will be repopulated with nearly equal probability. Since the atom had - 1 / 2 units of spin angular momentum before absorbing the photon and 0 units of spin angular momentum after being quenched from the excited state, each absorbed photon deposits 1/2 a unit of spin angular momentum in the vapor. The precise frequency of the pumping light is not important, so lasers or lamps with very broad spectral bandwidths can be used for pumping. Even though the alkali-metal vapor is often many optical depths thick, intense circularly polarized laser light can still illuminate most of the cell [11], as indicated in Fig. 3. In the illuminated part of the cell most of the alkali-metal atoms are pumped into the non-absorbing +1/2 sublevel of the ground state and the spin polarization is nearly 100%. In the dark volume where no light penetrates, the alkali-metal spin polarization is nearly zero, but some small spin polarization may be maintained by spin exchange with the polarized nuclei of the noble gas. Because of its long relaxation time, the spin polarization of the noble gas is nearly the same throughout the cell. The boundary between the illuminated and dark volumes of the cell is about one optical depth thick. In a well-designed system, the gas composition, the cell temperature and the laser intensity are matched to ensure that most of the cell is illuminated.

Medical NMR Sensing with Laser-Polarized 3He and 120Xe

]27

Collisional Mixing

-1/2~

1/2

50%

2p1/2

50%

Quenching by N 2

/ [_ -1/2

2S112

1/2

Fig. 4. Optical and collisiona] transitions involved in the optical pumping of an alkali-metal atom with D1 fight at high gas pressure (see text).

A key step forward was the discovery by T. R. Carver, M. A. Bouchiat and C. M. Varnum [12] in 1963 that a substantial fraction of the electron spin angular m o m e n t u m of optically pumped alkali-metal atoms could be transferred to the nucleus of 3He by spin exchange collisions. In 1978 B. C. Grover [13] and his colleagues at Litton Industries reported the extension of the spin exchange method from 3He to isotopes of xenon and krypton. Extensive studies of the physics of spin exchange between the electrons of alkali-metal atoms and noble-gas nuclei were subsequently carried out by our group at Princeton University [14]. As indicated in Fig. 5, spin exchange from the alkali-metal atomic electron to the nucleus of the noble gas can occur either in a three-body collision (Fig. 5A), for which a third body carries away the binding energy of the van der Waals molecule which is formed, or in a simple binary collision (Fig. 5B). The main observable difference between the two types of collisions is that the relaxation and spin transfer caused by the van der Waals molecules, while extremely efficient, can be suppressed with external magnetic fields of a few hundred to a few thousand Gauss. The relaxation due to binary collisions is hardly affected by magnetic fields which are readily available in the laboratory [14]. The relaxation due to van der Waal molecules also depends nonlinearly on the gas pressures in the cell [14]. The three-body collisions can account for much of the spin exchange for the heavier noble gases, Kr, Xe or Rn,

128

Wilfiam Happer B

Fig. 5. Polarization by spin-exchange collisions of optically pumped Rb atoms with the nuclei of noble gases. Three-body collisions (A), leading to the formation of weakly-bound van der Waals molecules, are very important for spin-exchange polarization of 129Xe. Binary collisions (B) are the sole polarization mechanism for 3He, and they are also very important at high gas pressures for 129Xe.

but binary collisions are the dominant spin exchange mechanism for the light gases He and Ne, which have particularly weak van der Waals attraction for alkali-metal atoms. The transfer of angular m o m e n t u m from the electron spin S of the alkalimetal a t o m to the nuclear spin I of a noble-gas a t o m during a collision is mediated by the Fermi contact interaction [15] aS-I

.

(3)

Bernheim [16] first pointed out that a large fraction of the spin angular m o m e n t u m S of the alkali-metal atoms is lost to the rotational angular mom e n t u m N of the alkali-metal a t o m and its collisional partner through the spin-rotation interaction 7S-N

.

(4)

The fraction [14] e of angular m o m e n t u m which is collisionally transferred to the nucleus of the noble gas is ot2 _

(5) +

Medical NMR Sensing with Laser-Polarized aHe and 12~Xe

129

For example [14], for the pair 129Xe Rb, the root mean squared values of the coupling coefficients are a / h = 38 MHz and "/N/h = 121 MHz so the efficiency of spin exchange would be about 4%, that is, 1/7/= 25 photons are needed to produce each fully polarized t29Xe nuclear spin. Especially for the lighter noble gases 3He and 21Ne, the spin-exchange probability per collision is relatively small, and it is advantageous to operate with as high a number density of alkali-metal atoms as possible to speed the growth of polarization in the noble-gas nuclei. At high densities, there are frequent spin exchange collisions between alkali-metal atoms A and B A(I") + B(J,) -+ A($) + B ( t ) .

(6)

The up and down arrows indicate the orientations of the electron spins. The cross section for electron-electron spin exchange collisions [17] are on the order of 10 -14 cm -2, so the exchange rates can be on the order of 104 s -1 at alkalimetal atomic number densities of 1014 cm -3. Electron-electron exchange is much faster than any other spin relaxation rate, but since such collisions conserve the total spin of the vapor they do not interfere with the polarization of noble-gas nuclei. Unfortunately, a small fraction of the collisions between alkali-metal atoms transfer spin angular to the translational angular momentum of the vapor [18] in spin-dependent collisions of the form A(~) + B($) --+ A($) + B($) .

(7)

The angular m o m e n t u m losses are most pronounced for vapors of the heavy alkali metal cesium, less for rubidium, and still less for potassium [19]. For the important case of aHe gas, in which the electron-nucleus spin exchange interaction (3) and the spin-rotation interaction (4) are both very small, it is spin-destroying collisions between alkali-metal atoms like those of (7) that set the practical upper limit on the alkali-metal vapor pressure and on the speed with which the 3He can be polarized. To date, the basic physics of the loss mechanism (7) has remained a mystery, with current theories predicting much smaller loss rates than those that are experimentally measured. Although Kastler and his contemporaries did the first optical pumping experiments with a few hundred microwatts of useful resonance light from lamps, it is now straightforward to get watts of useful laser light for optical pumping experiments. A watt of 7947 A Rb resonance radiation is 4 x l0 Is photons per second. Since one unit of angular m o m e n t u m h is added to the electron spin of the vapor of alkali-metal atoms for every two photons absorbed, 4 x 1018 fully polarized electron spins can be produced per second. In favorable cases, about 10% of the electron spin polarization can be transferred by spin exchange t o nuclear spin polarization of noble gases in the same container with the alkali-metal vapor. The spin can be accumulated for a time period on the order of T1, the longitudinal spin relaxation time of the noble gases. 2"1 can be hours or longer for spin-l/2 species like 129Xe or 3He,

130

William Happer

so some 10 ~1 highly polarized spins can be produced. The exceptionally long values of T1 for these gases are essential to the success of gas imaging, but they are the least understood part of the physics and most in need of further basic research. With more powerful tunable lasers, for example, diode arrays with tens or even hundreds of watts of power, it would be possible to produce many grams of solid 129Xe with nearly complete nuclear polarization. Like so many other useful new technologies, magnetic resonance imaging of lungs resulted from years of curiosity-driven, basic research, much of it done or inspired by zu Putlitz and his talented colleagues at Heidelberg. As we are about to move into the twenty-first century, let us hope that society will continue to find ways to support basic research for all the benefits it brings, intellectual, moral and practical.

References [1] H. Kopfermann, Nuclear Moments, Academic Press, New York (1958). [2] James R. MacFall et al., Radiology 200, 553 (1996). [3] M.S. Albert, G.D. Cates, B. Driehuys, W. Happer, B. Saam, C.S. Springer, Jr. and A. Wishnia, Nature 370, 199 (1994). [4] E.M. Purcell, H.C. Torrey and R.V. Pound, Phys. Rev. 69, 37 (1946). [5] F. Bloch, W.W. Hansen and M. Packard, Phys. Rev. 70, 474 (1946). [6] P.C. Lauterbur, Nature 242, 190 (1973). [7] A. Kastler, J. Phys. Radium 11, 225 (1950). [8] G.K. Walters, F.D. Colgrove and L.D. Schearer, Phys. Rev. Lett.8,439 (1962). [9] J. Becker et al. Nuclear Instruments and Methods in Physics Research A 346, 45 (1994). [10] P. Bachert et al., Magnetic Resonance in Medicine 36, 192 (1996). [11] A.C. Tam and W. Happer, Phys. Rev. Lett. 43, 519 (1979). [12] M.A. Bouchiat, T.R. Carver and C.M. Varnum, Phys. Rev. Lett. 5, 373 (1960). [13] B.C. Grover, Phys. Rev. Lett. 40, 391 (1978). C.H. Volk, T.M. Kwon, and J.G. Mark, Phys. Rev. A 21, 1549 (1980). [14] N.D. Bhaskar, W. Happer and T. McClelland, Phys. Rev. 49, 25 (1982); N.D. Bhaskar, W. Happer, M. Larsson and X. Zeng, Phys. Rev. Lett. 50, 105 (1983); W. Happer, E. Miron, S. Schaefer, D. Schreiber, W. A. van Wijngaarden and X. Zeng, Phys. Rev. 29, 3092 (1984); X. Zeng, Z. Wu, T. Call, E. Miron, D. Schreiber and W. Happer, Phys. Rev. 31, 260 (1985); S.R. Schaefer, G.D. Cates, Ting-Ray Chien, D. Gonatas, W. Happer and T.G. Walker, Phys. Rev. 39, 5613 (1989). [15] E. Fermi and E. Segrr, Z. Physik 82, 729 (1933). [16] R.A. Bernheim, J. Chem. Phys. 36, 135 (1962). [17] W. Happer, Rev. Mod. Phys. 44, 169 (1972). [18] N.D. Bhaskar, J. Pietras, J. Camparo and W. Happer, Phys. Rev. Lett. 44, 930 (1980). [19] R.L. Knize, Phys. Rev. A 40, 6219 (1989).

Test of Special Storage Ring

Relativity

in a Heavy

Ion

G. Huber 1 , R. Grieser l, P. Merz 1 , V. Sebastian 1, P. Seelig 1, M. Grieser 2, R. Grimm 2, D. Habs 2, D. Schwalm 2, T. K/ihl 3 1 Institut fiir Physik, Universit~it Mainz, D-55099 Mainz, Fed. Rep. Germany 2 Max Planck Institut ffir Kernphysik Heidelberg, D-69029 Heidelberg, Fed. Rep. Germany 3 Gesellschaft fiir Schwerionenforschung, D-64220 Darmstadt, Fed. Rep. Germany

1

Introduction

Experiments can be described in any inertial frame using the space time transformation of the special theory of relativity (SRT) as far as gravitational effects can be neglected. The existence of a preferred cosmological frame of reference [1] would violate the underlying symmetries of the theory. The natural candidate for such a preferred frame would be the the 3 K cosmic background radiation horizon with a relative velocity to the solar system of v' = 3 0 0 k m / s [2, 3]. All known experiments are in agreement with the special theory of relativity. Local deviations due to the general theory of relativity are beyond the the limitations of present experiments. Nevertheless, it is an experimental challenge to establish new upper limits for any deviation from the special theory of relativity. In a historical key experiment in 1938 Ires and Stilwell [4] have determined the Doppler shifted spectrum of a fast hydrogen beam as a function of its kinetic energy. In this experiment the time dilation factor

sRT = (1 -

,

(1)

where ~ = v/c is the atoms' velocity v measured in units of the speed of light c, was tested for any additional terms which could be present as a result of a hypothetical deviation from the Lorentz transformation between two inertial reference frames. Mansouri and Sexl [5] have developed a test theory for deviations induced by the presence of any preferential frame with relative velocity/3~ with respect to the laboratory and a moving clock with relative velocity/3 in the laboratory frame. These small and hypothetical deviations can be expressed in general as a function f(/3,/3') which modifies the time dilation factor to 7 = (1 - j32) - 1 / 2 . (1 + a - f ( / 3 , / 3 ' ) ) where a is a parameter measuring this deviation ( a = 0 for SRT).

(2)

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G. Huber et al.

In the original experiment a could be determined to < 0.01. Modern modifications have used highly relativistic hydrogen beams. At the Los Alamos Meson Physics Facility (LAMPF) in New Mexico, USA, a limit of a < 1.9.10 -4 was found for/3 = 0.84 [6]. At the NASA Jet Propulsion Laboratory an upper limit of a < 1.8.10 -4 could be established with/3 = 0.0016 [7]. Among the most accurate tests, MSt3bauer rotor experiments using fast moving disks yielded a < 10 -5 [8]. The space born experiment "Gravity Probe A", employing highly stable hydrogen maser clocks, determines from the frequency residuals between experiment and theory an upper limit of 0" < 2.1.10 -6 [9]. It also produced today's most stringent test on gravitational red shift [1]. In a laser spectroscopy experiment using the 3s[~]2-4d 30 , [3]3 5 0 transition in 2°Ne atoms at/3 = 3.6-10 -3 the limit for the existence of any preferred frame of reference was set to 0" < 1.4.10 -6 [10]. An atomic two-photon transition is induced by two counter-propagating laser beams. This technique is considered "Doppler free" spectroscopy, since the first order terms (1 +/3) and (1 - / 3 ) cancel and the resonance condition is given through u0 = 2 7//LaserWith the realization of heavy ion storage rings like the Test Storage Ring (TSR) in Heidelberg a new generation of experiments became feasible. A relativistic 7Li+ ion beam with well defined velocity, i.e. small m o m e n t u m spread, can be used for high resolution laser spectroscopy at the TSR. These ions had been investigated before at many places with high accuracy as they are fundamental two electron systems, e.g. in the laboratory of G. zu Putlitz [11]. The 7Li+ ions are stored as fast moving clocks at 13.3MeV particle energy corresponding to/3 = 0.064 . Two transitionswith frequencies vl and v2 in the rest frame are used in the experiment, as explained in Sec. 2. The first one is observed with a laser beam parallel to the ion beam, the second one with an antiparallel laser beam. In the laboratory frame, the resonance frequencies are Doppler-shifted to Ua and ~v, respectively. They are compared with two clocks at rest, realized by single-mode lasers stabilized to calibrated transitions in molecular 12712, in order to determine the frequencies ua and u v on an absolute scale. According to SRT they are related to the resonance frequencies vl and v2 in the particle rest frame by .p = va3`(1 - / 3 ) va = v27(1 +/3)

(3)

In the special theory of relativity the relation between the clock rates in the moving system and the laboratory system is -avp = v1~'2

(4)

since 3,2(1-/3)(1 +/3) = 1, independent of the clock's velocity. Any deviation of the time dilation factor from Eq. (1) may be written as '~ = 3,SRT ( I -'1- 0"/3 2 "l- higher order corrections ) .

(5)

Test of Special Relativity in a Heavy Ion Storage Ring

133

This could be observed experimentally through a violation of Eq. (4) which can be extended to read [5]

(6)

v.vp = vlv2 (1 + 2~/3 2 + ...)

A detailed analysis in the framework of Mansouri's and Sexl's test theory taking into account the kinematic situation in the storage ring experiment has been worked out by Kretzschmar et al. [12], resulting in a modification of Eq. (4) (7)

vavp : vlv2 ( 1 + 2a(/3 2 q- 2/3~' cos(K2)) )

where/2 is the angle between the velocities/3 and/3~. Usually/3r is identified with the velocity relative to the 3 K background. Evidently, an increase in/3 (at the T S R / 3 = 0.064 corresponds to v ,-* 19000km/s) always increases the sensitivity to the parameter a according to Eq. (7).

2

Result

of a Doppler

Effect Experiment

at t h e T S R

The experimental procedure is based on saturation spectroscopy with 7Li+ ions in a collinear excitation [t3]. The 7Li+ ion has a helium like spectrum and, similar to helium, only the triplet system with its high lying metastable 3S1 state is suitable for optical laser spectroscopy. Due to the hyperfine interaction all states in the triplet system exhibit a large hyperfine splitting, as can be seen in Fig. 1 with the relevant l s 2 s 3S1 --+ 3 P 2 line. The transitions F = 5/2 ++ F r --- 5/2 ++ F --- 3/2 are used for saturation spectroscopy. With their common upper level they form a "A" system which shows strong fluorescence when both transitions are driven in the same velocity class of the ions. The 7Li+ ions are partially excited into the metastable 3S1 state in the stripper of the Heidelberg TANDEM accelerator. After injection into the T S R the ion beam is cooled by Coulomb interaction with an isochronously cooled electron beam. The 7Li+ ion beam exhibits a narrow, well defined velocity distribution and is overlapped with two co- and counterpropagating laser beams from a single mode argon ion laser at A = 515nm and a single mode dye laser at )~ = 585 nm to excite the A resonance. This laser beam passes through a single mode fiber into the experimental area. Special expansion optics is needed to direct the beam through the 10 m long experimental section of the T S R where it is back reflected to reenter the optical fiber. The laser frequencies are determined by two auxiliary saturation spectroscopy setups for recording precisely calibrated resonances in molecular 12712. The argon ion laser is tuned to a wavelength standard, where the optical frequency is given by [14] and the wavelengths of the 12712 resonances close to the frequency where the dye laser excites the A resonance together with the argon

134

G. Huber et al.

eV

Ionization Limit

v5.8 ~ " / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / F 7/2 6t,3

sD n43 s_~~ [11.7758(05)GHz "' I ~-',i 9"6087(;~0) ~-H; \I

;k=545.5 nm

5.2035(05) _CH_~

.....

I-

.....

/_

v=546466.91879 4.0) GHz /

_ .5/2 3/2

-

- i/2

/

59.0

~

\

11.890018(40)C n z

k____1~So

3/2 1/2 3/2

Fig. 1. Transitions in 7Li+ relevant for the experiments at TSR. The closed two level system F = 5/2 --+ F ' = 7/2 and the A-system transitions F : 5/2 -+ F ' : 5/2 and F = 3/2 --+ F' = 5/2 are indicated. Most of the data are given by [11].

ion laser have been calibrated at the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig relative to the 12712 stabilized helium-neon laser to a---~-~= 1.3- 10 - l ° [15]. These reference signals are recorded simultaneously with the A resonance as shown in Fig. 2. The resonance position of the maximum was determined to ua = 512 667 592.4 (3.1) MHz

(8)

where the error reflects the uncertainty of systematic contributions. The observed linewidth of this resonance is twice the value resulting from various broadening effects present in the laser excitation. The further broadening seems to be mainly due to wave front distortions in the laser beams, as can be shown in a numerical solution of the optical Bloch equations for ions moving in the T S R at typical parameter settings of the ion beam optics. An uncertainty of 2.7 MHz for the position of the line center has to be taken into account. Sources for systematic uncertainties are possible light shifts and recoil-induced shifts, the method for determining the laser frequency and the uncertainty in the angular alignment between lasers and ion beam respectively. All these add up to a total uncertainty of Au = +3.1 MHz .

Test of Special Relativity in a Heavy Ion Storage Ring

I

,---~3000

I

I

I

I

135

i

N

2500-

~2000 I

1500 0

1000 500 0

0 -300

I

I

100 200 -200 -100 0 Detuning f r o m i - c o m p o n e n t

300 400 [MHz]

Fig. 2. Spectrum of the A-resonance with the simultaneously detected 12T12fines.

The measured frequency ua can be compared with the frequency b'a,SRT calculated according to Eq. (4) with the known frequencies. The value U~,SRT = 512 667 588.2(8) MHz

(9)

differs from the experimental value u~ by 4.3(3.2) MHz. Within two standard deviations the result is compatible with zero, and with Eq. (7) the corresponding limit of a < 8 - 10 -7 is deduced. 3

Laser Cooling

and Bunching

for Precision

Experiments

So far, the test of special theory of relativity at the TSR has been performed with an electron cooled ion beam with a longitudinal momentum spread of Ap/p = 5.10 -5 and a beam of diameter 1 mm consisting of ions in both singlet and triplet states of 7Li+. With additional laser cooling, the longitudinal momentum spread of the stored lithium beam has been reduced significantly to Alp/p = 1- 10 -6 at the TSR [16, 17]. The applied cooling scheme using an Ar + laser and an induction accelerator allows in a natural way for a separation of the singlet and triplet ions in phasespace. Moreover, it is possible to cool a radio frequency (RF) bunched ion beam with a laser [18]. In this case, the particle velocity and phase are known with high precision, which leads to the expectation of a significant increase in sensitivity for the test experiment.

136

G. Huber et al.

0.00020

,

I

I

I

I

I

I

I

I

I

-'2

-'1

(3 A¢

I

2

0.0001,5 0.00010

-

0.00005 0.00000 -0.00005

-

-0.00010 -0.00015 -0.00020

-3 22 -'i

-3

&¢

3

Fig. 3. Phasespace trajectories for RF-bunched ion beams. On the left a hot beam is bunched resulting in synchrotron oscillations. On the right additional laser cooling is applied, leading to a damping of the ion's motion. Ap and A¢ correspond to the momentum- and phase deviation of the particle's trajectory relative to the RF phase. The calculation has been carried out with a RF amplitude corresponding to the maximum laser cooling force.

The principle of laser cooling of a bunched ion beam is shown in Fig. 3. Without laser the particles are subjected to the voltage U of the RF cavity resulting in a net force which depends on the particle's phase relative to the phase of the RF frequency :

dp = eU sin(A¢)/2~:ns dt

(10)

with the radius Rs of the 'sollbahn' and the particle's phase difference A¢ relative to the phase of the oscillating voltage. The corresponding trajectories in phasespace can be calculated [19] and the result is shown on the left side of Fig. 3. The influence of an additional collinearly aligned laser is illustrated on the right of Fig. 3. The additional force of the cooling laser is [20] dp _ hk r

dt

s

(11)

2 I+S

with the absolute value of the wave vector k =J k J, the natural decay rate F of the optical transition and the velocity dependent saturation parameter

S

=

n2/2 (n~ + (r12)~)

(12)

Test of Special Relativity in a Heavy Ion Storage Ring

137

where $2 denotes the Rabi frequency of the optical transition and z~ ~- w0 --WLaserT(1 -- t3) Due to the velocity dependence of the laser cooling force (Eq. (11)) a damping of the longitudinal synchrotron oscillation in the ion b e a m occurs. In this simplified picture, the bunch length is strictly related to the longitudinal temperature. In a more sophisticated approach space charge forces of the b e a m that are limiting the bunch length significantly have to be taken into account [18]. At the T S R this bunched laser cooling has been observed with 7Li+ by time resolved detection of the fluorescence light. The spatial resolution of the ion position inside the bunched ion b e a m is achieved with a time resolved signal registration. The fluorescence of the laser cooled and bunched ions is detected. With this scheme, which is phase synchronized to the RF frequency, a time resolved spatial resolution of 0.17m in the moving ion b e a m is obtained. Moreover, the demultiplexing of the fluorescence light allows for additional background suppression by a factor of more than 5. With this device the integration time necessary for the detection of one single ion reduces to a fraction of a second.

800 ..~ 600

4oo

~oo

~_.~..%.~

~

.

~

z

~

~

Fig. 4. Recorded longitudinal phase space distribution function of laser cooled and bunched 7Li+ ions at the TSR. The position AS in the confining RF potential is measured by a time resolved and synchronous signal detection relative to the RF phase. The momentum deviation is calibrated during the scan of an additional dye laser using the Doppler effect. At AS = 0 a constant count rate for every value of Ap/p is visible. These photons are produced by the fixed frequency cooling laser.

138

G. Huber et al.

The velocity distribution function is determined with the optical Doppler effect. With an additional scanning laser the m o m e n t u m distribution Ap/p can be estimated from the detuning of the scanning laser. Since the optical transition frequency in the 7Li+ rest system is known with sufficient accuracy, the mean velocity can be calibrated with a precision of Av/v < 10 -7. A typical spectrum recorded with a 7Li+ beam at the T S R is shown in Fig. 4.

4

Outlook

A bunched and cooled ion beam represents an ensemble of clocks moving at fixed velocity and phase, which can be used to probe the structure of local space time. Similar to the quality of the ion beam the interacting laser beams will have to meet highest phase front performances. The precision of the frequency measurement will be increased with a new laser spectrometer which is presently being developed at the Universit/it Mainz. Due to the preparation of the laser cooled and bunched ion beam, the AC stark shift and the photon recoil can be precisely controlled and a sensitivity of a _< 1.5 10 -7 can be expected. Of course, a dramatic increase in sensitivity to the value of a can be expected by performing this experiment at the Experimental Storage Ring (ESR) at the Gesellschaft fiir Schwerionenforschung (GSI) in Darmstadt. With an ion velocity of/3 = 1/3, va is shifted to va = vo/v/2 (,.~ 776 nm) and up = 2va. In this case one single frequency standard can be used. Its fundamental frequency can serve as reference for the antiparallel excitation and for parallel excitation the frequency can be compared to a frequency doubled beam generated from the light of the same reference source. Using the velocity dependence of the correction term in Eq. (7), the overall sensitivity to violations of special theory of relativity can be increased by a factor of about 30. -

References [1] C.M. Will, 'Theory and experiment in gravitational physics', Cambridge University Press (1993) [2] A.A. Penzias, R.H. Wilson, Astrophys. Journ. 142, 419 (1965) [3] G.F. Smoot, M.V. Gorenstein, Phys. Rev. Lett. 39, 898 (1977) [4] H.E. Ires and G.R. Stilwell, J. Opt. Soc. Am. 28, 215 (1938) [5] R. Mansouri und R. Sexl, Gen. Rel. and Grav. 8(7), 497, 515, 809 (1977) [6] D.W. MacArthur, K.B. Butterfield, D.A. Clark, J.B. Donahue, P.A.M. Gram, H.C. Bryant, C.J. Harvey, W.W. Smith, G. Comtet, Phys. Rev. Lett. 56, 282

(1986) [7] T.P. Krisher, L. Maleki, G.F. Lutes, L.E. Prima.s, R.T. Logan, J.D. Anderson, C.M. Will, Phys. Rev. D. 42, 731 (1990) [8] D.C. Champeney, G.R. Isaac, A.M. Khan, Phys. Lett. 7, 241 (1963)

Test of Special Relativity in a Heavy Ion Storage Ring

139

[9] R.F.C. Vessot, M.W. Levine, E.M. Mattison, E.L. Blomberg, T.E. Hoffman, G.U. Nystrom, B.F. Farrel, R. Decher, P.B. Eby, C.R. Baugher, J.W. Watts, D.L. Teuber, F.D. Wills, Phys. Rev. Lett. 45, 2081 (1980) [10] E. Riis, L.-U. A. Andersen, Nis Bjerre, Ove Poulsen, S. A. Lee, J. L. Hall, Phys. Rev. Lett. 60, 81 (1988) [11] J. Kowalski, R. Neumann, S. Noethe, K. Scheffzek, H. Suhr, G. zu Putlitz, Hyperf. Interact. 15/16, 159 (1983) 70, 251 (1993) [12] M. Kretzschmar, Z Phys. A 342,463 (1992) [13] R. Grieser, R. Klein, G. Huber, S. Dickopf, I. Klaft, P. Knobloch, P. Merz, F. Albrecht, M. Grieser, D. Habs, D. Schwalm, T. Kiihl, Appl. Phys. B 59, 127 (1994) [14] Documents concerning the New Definition of the Metre, Metro/ogia 19, 163 (1984) Comit~ International des Poids et M6sures (CIPM), 8U session (1992), Recommendation 3 [15] R. Grieser, G. BSnsch, S. Dickopf, G. Huber, R. Klein, P. Merz, A. Nicolaus, H. Schnatz, Z. Phys. A 348, 147 (1994) [16] S. SchrSder, R. Klein, N. Boos, M. Gerhard, R. Grieser, G. Huber, A. Karafilfidis, M. Krieg, N. Schmitt, T. Kiihl, R. Neumann, V. Balykin, M. Grieser, D. Habs, E. Jaeschke, D. Kr~imer, M. Kristensen, M. Music, W. Petrich, D. Schwalm, P. Sigray, M. Steck, B. Warmer, A. Woff, Phys. Rev. Lett. 64, 2901

(1990) [17] W. Petrich, M. Grieser, R. Grimm, A. Gruber, D. Habs, H.-J. Miesner, D. Schwalm, B. Wanner, H. Wernoe, A. Wolf, R. Grieser, G. Huber, R. Klein, T. Kiihl, R. Neumann, S. SchrSder, Phys. Rev. A. 48, 2127 (1993) [18] J.S. Hangst, J.S. Nielsen, O. Poulsen, P. Shi, J.P. Schiffer, Phys. Rev. Lett. 74, 4432 (1995) [19] J. LeDuff, in CERN 94-01 (1994) [20] C. Cohen-Tannoudji, 'Atomic motion in laser light', in 'Fundamental systems in quantum optics', J. Dalibard et al. (eds.), North Holland (1992)

R e s o n a n c e F l u o r e s c e n c e of a Single Ion J. T. HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther Max-Planck-lnstitut f/it Quantenoptik and Sektion Physik der Universitiit Miinchen 85748 Garching, Fed. Rep. of Germany Resonance fluorescence of atoms is a basic process in radiation-atom interactions, and has therefore always generated considerable interest. The methods of experimental investigation have changed continuously due to the availability of new experimental tools. A considerable step forward occurred when tunable and narrow band dye laser radiation became available. These laser sources are sufficiently intense to easily saturate an atomic transition. In addition, the lasers provide highly monochromatic light with coherence times much longer than typical natural lifetimes of excited atomic states. The laser light, when scattered off atoms from a well collimated atomic beam, leads to an absorption width being practically the natural width of the resonance transition, and it became possible to investigate the frequency spectrum of the fluorescence radiation with high resolution. However, the spectrograph used to analyze the reemitted radiation was a Fabry-Perot interferometer, the resolution of which did reach the natural width of the atoms, but was insufficient to reach the laser linewidth, see e.g. Hartig et al. (1976) and Cresset et al. (1982). More recently, ion traps have allowed the study of the fluorescence from a single laser cooled particle which is practically at rest, thus providing the ideal case for spectroscopic investigation. The other essential ingredient for achievement of high resolution is the measurement of the frequency spectrum by heterodyning the scattered radiation with laser light. Such an optimal experiment with a single trapped Mg+ ion is described in this paper. The measurement of the spectrum of the fluorescent radiation at low excitation intensities is presented. Furthermore, the photon correlation of the fluorescent light has been investigated under practically identical excitation conditions. The comparison of the two results shows a very interesting aspect of complementarity since the heterodyne measurement corresponds to a "wave" detection of the radiation whereas the measurement of the photon correlation is a "particle" detection scheme. It will be shown that under the same excitation conditions the wave detection provides the properties of a classical atom, i.e. a driven oscillator, whereas the particle or photon detection displays the quantum properties of the atom. Whether the atom displays classical or quantum properties thus depends on the method of observation. The spectrum of the fluorescence radiation is given by the Fourier transform of the first order correlation function of the field operators, whereas

142

J . T . HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther

the photon statistics and photon correlation is obtained from the second order correlation function. The corresponding operators do not commute, thus the respective observations are complementary. Present theory on the spectra of fluorescent radiation following monochromatic laser excitation can be summarized as follows: fluorescence radiation obtained with low incident intensity is also monochromatic owing to energy conservation. In this case, elastic scattering dominates the spectrum and thus one should measure a monochromatic line at the same frequency as the driving laser field. The atom stays in the ground state most of the time and absorption and emission must be considered as one process with the atom in principle behaving as a classical oscillator. This case was treated on the basis of a quantized field many years ago by Heitler (1954). With increasing intensity upper and lower states become more strongly coupled leading to inelastic components, which increase with the square of the intensity. At low intensities, the elastic part dominates since it depends linearly on the intensity. As the intensity of the exciting light increases, the atom spends more time in the upper state and the effect of the vacuum fluctuations comes into play through spontaneous emission. An inelastic component is added to the spectrum, and the elastic component goes through a maximum where the Rabi flopping frequency 12 -- Fly/'2 (F is the natural linewidth) and then disappears with growing 12. The inelastic part of the spectrum gradually broadens a s / 2 increases and for 12 > F/2 sidebands begin to appear. For a saturated atom, the form of the spectrum shows three well-separated Lorentzian peaks. The central peak has width F and the sidebands which are each displaced from the central peak by the Rabi frequency are broadened to 3F/2. The ratio of the height of the central peak to the sidebands is 3:1. This spectrum was first calculated by Mollow (1969). For other relevant papers see the review of Cresser et al. (1982). The experimental study of the problem requires, as mentioned above, a Doppler-free observation. In order to measure the frequency distribution, the fluorescent light has to be investigated by means of a high resolution spectrometer. The first experiments of this type were performed by Schuda et al. (1974) and later by Walther et al. (1975), nartig et al. (1976) and Ezekiel et al. (1977). In all these experiments, the excitation was performed by singlemode dye laser radiation, with the scattered radiation from a well collimated atomic beam observed and analyzed by Fabry-Perot interferometers. Experiments to investigate the elastic part of the resonance fluorescence giving a resolution better than the natural linewidth have been performed by Gibbs et al. (1976) and Cresset et al (1982). The first experiments which investigated antibunching in resonance fluorescence were also performed by means of laser-excited collimated atomic beams. The initial results obtained by Kimble, Dagenais, and Mandel (1977) showed that the second-order correlation function g(2)(t) had a positive slope

Resonance Fluorescence of a Single Ion

143

which is characteristic of photon antibunching. However, g(2)(0) was larger than g(2)(t) for t ~ ~ due to number fluctuations in the atomic beam and to the finite interaction time of the atoms (Jakeman et al. 1977; Kimble et al. 1978). Further refinement of the analysis of the experiment was provided by Dagenais and Mandel (1978). Rateike et al. used a longer interaction time for an experiment in which they measured the photon correlation at very low laser intensities (see Cresset et al. 1982 for a review). Later, photon antibunching was measured using a single trapped ion in an experiment which avoids the disadvantages of atom number statistics and finite interaction time between atom and laser field (Diedrich and Walther 1987). As pointed out in many papers photon antibunching is a purely quantum phenomenon (see e.g. Cresser et al. 1982 and Walls 1979). The fluorescence of a single ion displays the additional nonclassical property that the variance of the photon number is smaller than its mean value (i.e. it is sub-Poissonian). This is because the single ion can emit only a single photon and has to be re-excited before it can emit the next one which leads to photon emissions at almost equal time intervals. The sub-Poissonian statistics of the fluorescence of a single ion has been measured in a previous experiment (Diedrich and Walther 1987 and also Short and Mandel 1983 for comparison). The trap used for the present experiment was a modified Paul-trap, called an endcap-trap (Schrama et al. 1993) (see Fig. 1) which produces strong confinement of the trapped ion. Therefore, the number of sidebands, caused by the oscillatory motion of the laser cooled ion in the pseudopotential of the trap, is reduced. The trap consists of two solid copper-beryllium cylinders (diameter 0.5 mm) arranged co-linearly with a separation of 0.56 mm. These correspond to the cap electrodes of a traditional Paul trap, whereas the ring electrode is replaced by two hollow cylinders, one of which is concentric with each of the cylindrical endcaps. Their inner and outer diameters are 1 and 2 mm respectively and they are electrically isolated from the cap electrodes. The fractional anharmonicity of this trap configuration, determined by the deviation of the real potential from the ideal quadrupole field is below 0.1% (see Schrama et al. 1993). The trap is driven at a frequency of 24 MHz with typical secular frequencies in the xy-plane of approximately 4 MHz. This required a radio-frequency voltage with an amplitude on the order of 300 V to be applied between the cylinders and the endcaps, and with AC-grounding of the outer electrodes provided through a capacitor. The measurements were performed using the 32S1/2- 32P3/2 transition of the 24Mg+-ion at a wavelength of 280 nm. The natural width of this transition is 42.7 MHz. The exciting laser light was produced by frequency doubling the light from a rhodamine 110 dye laser. The laser was tuned slightly below resonance in order to Doppler-cool the secular motion of the ion. All the men-

144

J . T . HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther

% U1

Uu

U2

UD

? U 0 + U cos Fig. 1. Electrode configuration of the endcap trap. The open structure offers a large detection solid angle and good access for laser beams testing the micromotion of the ion. Micromotion is minimized by applying dc voltages: U1, U2, Uu, Up.

surements of the fluorescent radiation described in this paper were performed with this slight detuning. For the experiment described here, it is important to have the trapped ion at rest as far as possible to minimize the light lost into motional sidebands. There are two reasons which may cause motion of the ion: the first one is the periodic oscillation of the ion within the harmonic pseudopotential of the trap and the second one is micromotion which is present when the ion is not positioned exactly at the saddle point of the trap potential. Such a displacement may be caused by a contact potential resulting, for example, by a coating of the electrodes by Mg produced when the atoms are evaporated during the loading procedure of the trap. Another reason may be asymmetries due to slight misalignments of the trap electrodes. Reduction of the residual micromotion can be achieved by adjusting the position of the ion with DC-electric fields generated by additional electrodes. For the present

Resonance Fluorescence of a Single Ion

145

experiment they were arranged at an angle of 120 ° in a plane perpendicular to the symmetry axis of the trap electrodes. By applying auxiliary voltages (U1 and U2) to these electrodes and Utr and UD to the outer trap electrodes (Fig. 1), the ion's position can be adjusted to settle at the saddle point of the trap potential. The micromotion of the ion can be monitored using the periodic Doppler shift at the driving frequency of the trap which results in a periodic intensity modulation in the fluorescence intensity. This modulation can be measured by means of a transient recorder, triggered by the AC-voltage applied to the trap. There are three laser beams (lasers 1-3 in Fig. 2) passing through the trap in three different spatial directions which allow measurement of the three components of the micromotion separately. By adjusting the compensation voltages Ua, U2, Utr and UD the amplitude of the micromotion could be reduced to a value smaller than )~/8 in all spatial directions. The amount of secular motion of the ion resulting from its finite kinetic energy cannot be tested by this method since the secular motion is not phase coupled to the trap voltage. However, the intensity modulation owing to this motion can be seen in a periodic modulation of the photon correlation signal. For all measurements presented here, this amplitude was on the order of )~/8. This corresponds to a temperature of the ion of 1 mK determined from the kinetic energy. This means that the vibrational sidebands of the trapped ion are populated up to n = 7 which results in less than 50 % of the fluorescence energy being lost into the vibrational sidebands. The heterodyne measurement is performed as follows. The dye laser excites the trapped ion with frequency WL while the fluorescence is observed in a direction of about 54 ° to the exciting laser beam (see Fig. 2). However, both the observation direction and the laser beam are in a plane perpendicular to the symmetry axis of the trap. Before reaching the ion, a fraction of this laser radiation is removed with a beamsplitter and then frequency shifted (by 137 MHz with an acousto-optic modulator (AOM)) to serve as the local oscillator. The local oscillator and fluorescence radiations are then overlapped and simultaneously focussed onto the photodiode where the initial frequency mixing occurs. The frequency difference signal is amplified by a narrow band amplifier and then further mixed down to 1 kHz so that it could be analyzed by means of a fast Fourier analyzer (FFT). The intermediate frequency for this mixing of the signal was derived from the same frequency-stable synthesizer which was used to drive the accousto-optic modulator producing the sideband of the laser radiation so that any synthesizer fluctuations are cancelled out. An example of a heterodyne signal is displayed in Fig. 3, where Aw is the frequency difference between the heterodyne signal and the driving frequency

146

J . T . HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther

Photo-~l diode,, ................I0 Detector~ ~ 137.001MHz /i/ ~ / / Local t : I "~ Oscill. o

o L- 137 MHz ,//Laser 1

| kk _/ U2 k\ Laser 3 " ~

Fig. 2. Scheme of heterodyne detection. The trap is omitted in the figure with only two of the compensation electrodes shown. Laser 3 is directed at an angle of 22 ° with respect to the drawing plane and Laser 2.

of the AOM. Frequency fluctuations of the laser b e a m cancel out and do not influence the linewidth because at low intensity the fluorescence radiation always follows the frequency of the exciting laser while the local oscillator is derived directly from the same laser beam. The residual linewidth results mainly from fluctuations in the optical p a t h length of the local oscillator or of the fluorescent beam. Both beams pass through regular air and it was observed that a forced motion of the air increased the frequency width of the heterodyne signal. The frequency resolution of the F T T was 3.75 Hz for the particular measurement. The heterodyne measurements were performed at a

~2[2 of 0.7, where A is the laser detuning. saturation p a r a m e t e r s = a2+(r~/4) In this region, the elastic part of the fluorescent spectrum has a m a x i m u m (Cohen-Tannoudji et al. 1992). The signal to noise ratio observed in the experiment is shot noise limited. The signal in Fig. 3 corresponds to a rate of the scattered photons of about 104s -1 which is an upper limit since photons were lost from detection due to scattering into sidebands caused by the secular motion of the ion. In order to reduce this loss as much as possible, a small angle between the directions of observation and excitation was used. Investigation of photon correlations employed the ordinary Hanbury-Brown and Twiss setup with two photomultipliers and a b e a m splitter. The setup

Resonance Fluorescence of a Single Ion

I

I

I

I

147

I

I0

S

::> ..,

6 Hz

6

D

4

-200

- 100

0

100

200

Ao ( Hz ) Fig. 3. Heterodyne spectrum of a single trapped 24Mg+-ion for s = 0.7, A _ --2.5F, /2 ~= 3.9F. Integration time: 267 ms.

was essentially the same as described by Diedrich and Walther (1987). The pulses from the photomultipliers (RCA C31034-A02) were amplified and discriminated by a constant fraction discriminator (EG&G model 584). The time delay t between the photomultiplier signals was converted by a time-toamplitude converter into a voltage amplitude proportional to the time delay. A delay line of 100 ns in the stop channel allowed for the measurement of g(2)(t) for both positive and negative t in order to check the symmetry of the measured signal. The output of the time-to-amplitude converter was accumulated by a multichannel analyzer in pulse height analyzing mode. Two typical measurements, each with weak excitation intensities but with different detunings, are shown in Fig. 4. The generalized Rabi-flopping frequency I2~ = ~ +/22 for the respective measurements are given in the figure caption. For small time delays (< 20 ns) the nonclassical antibunching effect is observed, superposed with Rabi oscillations which are damped out with a time constant corresponding to the lifetime of the excited state. Our experimental results are reasonably well reproduced by the theory for weak excitation (Loudon 1980): = 1 +

xp(-2rt) - 2cos(,at)

xp(-rt).

148

J . T . HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther

(a)

I

$ 0 ~at)

(b)

2

-30

-20

-I0

0

I0

20

30

40

T i m e ( ns ) F i g . 4.

Antibunching signals of a single 24Mg+-ion. (a) s = 0.7, A = --1.2F, ~ ' = 1.9/'. Integration time: 95 min. (b) s ----0.4, A _-- --0.5F, g2' = 0.8F. Integration time: 220 rain. The sofid line is a theoretical fit, see text for details.

A measurement of g(2) (t) with an averaging time of hours and time delays up to 500 ns resulted in no visible micromotion effects when the compensation voltages U1, U2, Uu and UD were correctly adjusted. Micromotion results in a periodic modulation of the photon correlation at the driving frequency of the t r a p (compare Diedrich and Walther 1987). The stray-light counting rate was so low that there was no need to correct the measurement shown in Fig. 4 (b) for accidental counts. There was actually not a single count in the t = 0 channel within the integration time of 220 min. In conclusion, we have presented the first high-resolution heterodyne measurement of the elastic peak in resonance fluorescence. At identical experimental parameters we have also measured antibunching in the photon correlation of the scattered field, Together, both measurements show that, in the limit of weak excitation, the fluorescence light differs from the excitation radiation in the second-order correlation but not in the first order correla-

Resonance Fluorescence of a Single Ion

149

tion. However, the elastic component of resonance fluorescence combines an extremely narrow frequency spectrum with antibunched photon statistics, which means that the fluorescence radiation is not second-order coherent as expected from a classical point of view. This apparent contradiction can be explained easily by taking into account the quantum nature of light, since first-order coherence does not imply second-order coherence for quantized fields (Loudon 1980). The heterodyne and the photon correlation measurement are complementary since they emphasize either the classical wave properties or the quantum properties of resonance fluorescence, respectively. In a recent treatment of a quantized trapped particle (Glauber 1992) it was shown that a trapped ion in the vibrational ground state of the trap will also show the influence of the micromotion since the wavefunction distribution of the ion is pulsating at the trap frequency. This means that a trapped particle completely at rest will also scatter light into the micromotion sidebands. Investigation of the heterodyne spectrum at the sidebands may give the chance to confirm these findings. It is clear that such an experiment will not be easy since other methods are needed to verify that the ion is actually at rest at the saddle point of the potential .

Acknowledgements We would like to thank Roy Glauber for many discussions in connection with his quantum treatment of a trapped particle. We also thank Girish S. Agarwal for many discussions. We dedicate this paper to Gisbert zu Putlitz on the occasion of his 65th birthday. A major part of his scientific work was pursued using the resonance fluorescence of atoms. We hope that the described new experiments will find his interest

References Cresser J.D., Hgger J., Leuchs G., Rateike F.M., Walther H. (1982): Resonance Fluorescence of Atoms in Strong Monochromatic Laser Fields. Dissipative Systems in Quantum Optics, edited by Bonifacio R. and Lugiato L. (Springer Verlag) Topics in Current Physics 27, 21-59 Cohen-Tannoudji C., Dupont-Roc J., Grynberg G. (1992): Atom-Photon Interactions (J. Wiley & Sons, Inc.) Diedrich F., Walther H. (1987): Non-classical Radiation of a Single Stored Ion. Phys. Rev. Lett. 58, 203-206 Gibbs H.M. and Venkatesan T.N.C. (1976): Direct Observation of Fluorescence Narrower than the Natural Linewidth. Opt. Comm. 17, 87-94 Glauber R. (1992): Proceedings of the International School of Physics "Enrico Fermi", Course CXVIII Laser Manipulation of Atoms and Ions, edited by Arimondo E., Phillips W.D., Strumia F. (North Holland) 643

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J . T . HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther

Hartig W., Rasmussen W., Schieder R., Walther H. (1976): Study of the Frequency Distribution of the Fluorescent Light Induced by Monochromatic Excitation. Z. Physik A278, 205-210 Heitler W. (1954): The Quantum Theory of Radiation, (Oxford University Press, Third Edition) 196-204 Jakeman E., Pike E. R., Pusey P.N., and Vaugham J.M. (1977): The Effect of Atomic Number Fluctuations on Photon Antibunching in Resonance Fluorescence. J. Phys. A 10, L257-L259 Kimble H. J., Dagenais M., and Mandel L. (1977): Photon Antibunching in Resonance Fluorescence. Phys. Rev. Lett. 39, 691-695 Kimble H. J., Dagenais M., and Mandel L. (1978): Multiatom and Transit-Time Effects in Photon Correlation Measurements in Resonance Fluorescence. Phys. Rev. A 18, 201; Dagenais M., Mandel L. (1978): Investigation of Two-Atom Correlations in Photon Emissions from a Single Atom. Phys. Rev. A 18, 22172218 Loudon R. (1980): Non-Classical Effects in the Statistical Properties of Light. Rep. Progr. Phys. 43, 913-949 MoUow B.R. (1969): Power Spectrum of Light Scattered by Two-Level Systems. Phys. Rev. 188, 1969-1975 Schrama C. A., Peik E., Smith W.W., and Walther H. (1993): Novel Miniature Ion Traps. Opt. Comm. 101, 32-36 Schuda F., Stroud C., Jr., Hercher M. (1974): Observation of the Resonant Stark Effect at Optical Frequencies. J. Phys. BT, L198-L202 Short R. and Mandel L. (1983): Observation of Sub-Poissonian Photon Statistics. Phys. Rev. Lett. 51, 384-387, and in Coherence and Quantum Optics V, edited by Mandel L. and Wolf E. (Plenum, New York) 671 Walls D.F. (1979): Evidence for the Quantum Nature of Light. Nature 280,451-454 Walther H. (1975): Atomic Fluorescence Induced by Monochromatic Excitation. Laser Spectroscopy, Proceedings of the 2nd Conference, Meg~ve, France, ed. by Haroche S., Reborg-Peyronla J.C., H£nsch T.W., Harris S.E., Lecture Notes in Physics (Springer) 43, 358-369 Wu F. Y., Grove R.E., Ezekiel S. (1977): Investigation of the Spectrum of Resonance Fluorescence Induced by a Monochromatic Field. Phys. Rev. Lett. :15, 1426-1429; Grove R.E., Wu F. Y., Ezekiel S. (1977): Measurement of the Spectrum of Resonance Fluorescence froma Two-Level Atom in an Intense Monochromatic Field. Phys. Rev. A 15, 227-233

R e s o n a n c e R a m a n S t u d i e s of the R e l a x a t i o n of P h o t o e x c i t e d M o l e c u l e s in S o l u t i o n on t h e Picosecond Timescale W.T. Toner ~, P. Matousek ~, A.W. Parker 2 and M. Towrie 2 1 Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, UK 2 Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire O X l l 0QX, UK

1

Introduction

Following the photoexcitation of a molecule in solution, any excess vibrational energy is redistributed a m o n g its vibrational modes and transferred to the solvent as heat, and the excited state molecule assumes its new equilibrium geometry and chemical relationship with its solvent neighbours. These four related processes take place on overlapping timescales extending from a hundred femtoseconds to a few picoseconds. Stilbene has been a particular object of study because it is one of the simplest molecules to undergo a fifth non-radiative process on a similar timescale, relaxation to the ground state via a twisted configuration which gives nearly equal yields of the trans and cis isomers from either excited state. The lifetimes of the excited state trans and cis isomers are --~ 70 ps and ~ 1 ps, respectively, at room temperature. It is natural to study this complex evolution of vibrational populations and structure through vibrational spectroscopy. Several groups have used the time resolved resonance R a m a n scattering technique, since measurements of band frequency, width and intensity are a rich source of structural and dynamical information, particularly in relation to the bonds involved in the dipole coupling to the higher electronic state which is in resonance. But as this paper illustrates, a rather complete set of d a t a is required before one can make full use of the information without ambiguity.

2

Method

The time resolved resonance R a m a n spectrum of a photoexcited molecule is measured using a probe laser pulse tuned to resonance with a higher electronic state, incident at various delay times after photoexcitation by a first (pump) laser pulse. Until recently, dye lasers have been used to generate the tunable probe beams and the p u m p b e a m has been supplied by a harmonic of the laser used to excite the dye, or by a harmonic of the dye laser itself. The molecules studied have therefore been restricted to those whose ground and excited state transitions can both be matched in this way. High average

152

W.T. 'loner, P. Matousek, A.W. Parker and M. Towrie

laser power is required to give the statistical precision needed to extract the Raman scattering signal from fluorescence and other backgrounds which are frequently very strong in comparison. But it is necessary to avoid fluences resulting in saturation and for measurements on the picosecond timescale, peak intensities which could give rise to non-linear artefacts must also be avoided. In practice, a compromise is made between high repetition rate and high peak power. The laser system used for the work described in the first part of this paper had a repetition rate of five kHz and produced tunable probe beams in the wavelength range from 550 to 700nm with up to five microjoules per five picosecond pulse [1]. Pump beams were obtained by frequency doubling with typically 10 to 20 % efficiency. The beams were focused on the surface of a jet of the sample solution flowing from a nozzle and photons scattered through 90 ° were collected and analysed in a high resolution spectrometer equipped with a cooled CCD detector. The strong enhancement of the Raman scattering signal resulting from working close to resonance also varies strongly with probe wavelength. Measurements of the probe wavelength dependence of the scattering, called Raman excitation profiles, are therefore necessary before reliable detailed interpretations of band intensities can be made. A lack of independence of pump and probe tuning prevents the measurement of these profiles under constant photoexcitation conditions. A new laser system based on the amplification of portions of a single white light continuum source in two independently tunable Optical Parametric Amplifiers (OPAs) has now been developed [2] to overcome this limitation. It also gives very broad spectral coverage and a time resolution extending down to ~-, 150 fs. A spectral filter of variable resolution produces probe beam pulses whose width is set by the transform limit of the spectral resolution chosen for any particular experiment. A high repetition rate (40kHz) avoids the fluence and peak power problems discussed above, at some cost to the statistical precision for a given data acquisition time. This system was used for the measurements on quaterphenyl described at the end of the paper.

3

Stokes Spectrum of $1 Trans-Stilbene

Measurements of the resonance Raman spectrum made with picosecond time resolution show that several bands in the Stokes spectrum of S1 trans-stilbene (SITS) shift significantly to the blue and decrease in width in the 25 picoseconds immediately following photoexcitation [3,4]. An example is shown in Fig. 1. The magnitudes of these mode-specific and solvent-dependent changes increase monotonically with the energy of the exciting photon [4,5] and timeindependent changes of a similar kind can be induced by changes in the temperature of the solvent bath, although the ratios of the shifts to the width changes are quite different in the static and dynamic cases [4]. Similar (but

Resonance Raman Studies of the Relaxation of Photoexcited Molecules

153

qD

1600

1400

1200

wavenumber/cm" 1 Fig. 1. Stokes resonance Raman spectra of $1 trans-stilbene at delays of 10ps (solid line) and 80ps (dashed line) after photoexcitation at 285 nm. The probe beam wavelength was 570 nm. From Ref. [4].

much smaller) mode-specific temperature dependencies are seen in the ground state resonance Raman spectrum [6]. In the dynamic case, the changes were taken to represent the cooling of a molecule in approximate internal equilibrium following an initial intramotecular vibrational relaxation cascade too fast for observation. They were related to changes in the absorption spectrum with time [7]. The dynamic and static (temperature induced) changes were attributed by us to the same basic cause, which may be called a temperature dependent solvent shift. Some authors have proposed a connection with the isomerisation process [8]. We will not discuss here the various models of such shifts, but rather concentrate on relating the empirical Raman observations to each other. Further work showed that very similar dynamic shifts and linewidth narrowing effects in the Stokes spectra were exhibited by many molecules, for example, dimethoxystilbene and quinquiphenyl [9]; biphenylyl-phenyl-oxadiazol, bisbiphenylyl-butylbiphenyl-oxadiazol, and biphenylyl-butylbiphenyl-oxadiazol [10]. Quaterphenyl and tetra-t-butyl-p-sexiphenyl showed another kind of behaviour [10], with the disappearance of some bands at late times suggesting that the early time spectra might include components from a short lived isomer. In all these molecules, the excited states are thought to be subject to twists of conjugated chains of C-C and C = C bonds, as in the case of Stilbene.

4

Anti-Stokes Spectrum of $1 Trans-Stilbene

The first measurements of the anti-Stokes spectrum of photoexcited transstilbene [11] up to the band arising from the C = C olefinic stretching mode at 1600 cm -1, shown in Fig. 2, appeared to contradict the idea of approximate internal equilibrium on the picosecond timescale: the intensities of several

154

W.T. Toner, P. Matousek, A.W. Parker and M. Towrie

0 ps

Stokes

300

500

700

900

1100

Wavenumber/cm

1300

1500

1700

-1

Fig. 2. Anti-Stokes resonance Raman spectra of $1 trans-stilbene at delays of (a) 0ps and (b) 50ps after photoexcitation taken with 8ps time resolution. Stokes spectra at 20ps delay are shown in (c) for comparison. Photoexcitation was at 305 nm and the probe beam wavelength was 610 rim. From Ref. [11].

bands decrease markedly with time over c. 10 ps, relative to their neighbours (in the case of the c. 1600cm -1 band, by a factor of at'least 20). Others have reported anti-Stokes spectra showing similar changes [12]. These d a t a present a difficult problem of interpretation. Although the establishment of equilibrium populations in stilbene in solution has been shown [13] to take much longer than the 50 to 500fs commonly supposed, taking the early/late ratios of the band intensities as a direct measure of early time population relative to Boltzmann in this case would imply an unreasonably large difference from equilibrium persisting to the few picosecond timescale. The spectra are also remarkable in a second respect: the ratios of the anti-Stokes to the Stokes intensities of all the bands in the 1100 to 1600 cm -1 region are very much higher than would be expected on the basis of a Boltzm a n n population distribution (in the case of the c. 1530cm -1 band, by a factor of ,-~ 150), even at times as late as 50ps, when time-dependent spectral changes have ceased and thermal equilibrium must be fully established. Since the initial state for anti-Stokes scattering is vibrationally excited, resonance with the v = 0 level in the higher electronic state occurs at a lower probe frequency than for Stokes scattering, but it would require a very narrow resonance to produce such a large change through the ratio of the Stokes and anti-Stokes resonance denominators, even if it was assumed that only one term was dominant. It was not possible to make a quantitative interpretation of these results in the absence of R a m a n excitation profile data.

Resonance Raman Studies of the Relaxation of Photoexcited Molecules

155

5 Time-Resolved Resonance Raman Excitation Profile of Quaterphenyl The new laser system described above makes it possible to vary the probe b e a m wavelength independently of the p u m p and we present here a preliminary analysis of spectra which contain the first information to be reported [14] on the R a m a n excitation profile of an excited state molecule. Quaterphenyl was chosen for the first trials of the new system in view of its very large cross-section. The dye was dissolved in dioxane at millimolar concentration and the p u m p b e a m had a wavelength of 277 nm. T h e resolution of the probe b e a m spectral filter was set at 20 cm -1, giving a 700fs wide probe pulse. Resonance R a m a n spectra were observed to rise from zero to full intensity between -1 and +1 ps p u m p - p r o b e delay. Stokes spectra at delays of 2 and 50 ps following photoexcitation are shown in Fig. 3 for seven probe wavelengths between 593 and 647nm. There are marked changes in intensity for very small changes in probe wavelength. Figure 4 shows the probe wavelength dependence of the intensities of the two bands having the largest changes. The excitation profile of the c. 1515cm -z band can be fit to a resonance centered at a probe b e a m frequency of ,-- 16250cm - I having a width ,-, 500cm -z (FWHM). We have remarked above on the need for such a sharp resonance in stilbene to account for the anti-Stokes spectrum. In the case

2 ps

50 ps

647nm 635nm 620nm 610nm 605nm 602nm 593nm

t 1200

I 1600

I

~

r 1200 Raman shift, crn "t

I

I 1600

Fig, 3. Stokes resonance Raman spectra of $1 Quaterphenyl for various probe beam wavelengths at 2 and 50ps after photoexcitation at 2 7 7 n m .

156

W.T. Toner, P. Matousek, A.W. Parker and M. Towrie (b) 1616 band

(a) 1616 band at 2 ps 120 I

4~4~/ ~ '

40 t 0 15000

~ 16000

i

120

~

40

f 17000

/

0 15000

at 60 ps

\

: 16000

Probefrequency,cm "~

Probefrequency,cm "~

(c) 1686 band at 2 pa

(d) 1585 band at 50 ps

i"

17000

ll

i° l 0

150OO

[

16000

Probefrequency,cm "~

I

17000

0

i

16000

i

17000

Probe ~ ' ~ l ~ n c y , c m "

Fig. 4. Band intensities versus probe beam frequency for the data of Fig. 3. The values shown for the c. 1585 cm -1 band are sums of components which were not clearly resolved.

of the c. 1585cm -1 band, the profile has the appearance of an interference of the same resonance with a non-resonant background. This requires the involvement of a second electronic state which is distant. Interference of this kind has been observed in ground state resonance R a m a n scattering [15]. Changes of intensity with time delay can also be seen in Fig. 3, particularly for probe wavelengths of 610 and 620 nm ( 16390 and 16130 c m - 1). An increase in the intensity of the c. 1515 band at late time for a probe wavelength of 6 1 0 n m can also be seen in a spectrum of better quality in reference [10]. The excitation profiles in Fig.4 suggest that this m a y be due to a small but significant shift to the blue of the frequency of the resonance, and not to any change in intrinsic strength. This would imply coupling between the electronic and vibrational degrees of freedom during the relaxation. Work is in progress to confirm and extend these preliminary observations. Measurements of the t e m p e r a t u r e dependence of the ground state absorption spectrum of quaterphenyl were made which show that the separation of the broad and structureless absorption bands centered at c. 295 and c. 2 1 0 n m increases by several percent for a cooling of 500. The c. 210 nm band is likely to be due to one of the higher states responsible for the shape of the excitation profile so that this result suggests a connection between the static t e m p e r a t u r e dependence and the cooling of the photoexcited molecule which parallels that previously observed for the vibrational frequencies of stilbene [4,6].

Resonance Raman Studies of the Relaxation of Photoexcited Molecules

6

157

Conclusions

The experiments described above show the great sensitivity of time-resolved resonance Raman spectroscopy to the changes taking place during the relaxation of photoexcited molecules in solution. During the time that excess energy is transferred to the solvent, there are changes in vibrational frequencies and bandwidths, in level populations and also, if the quaterphenyl results are confirmed, in electronic energies. These dynamic effects all have parallels in the changes which take place on slow timescales with variations in solvent temperature, but the static temperature dependencies and dynamic changes also differ. The sharpness of the quaterphenyl excitation profiles and the possibility that the resonance parameters may change with time confirms the need for measurements of this kind of the anti-Stokes spectrum of stilbene to determine the degree to which non-equilibrium populations contribute to the observed changes in band intensities.

7

Acknowledgements

The vibrational spectroscopy of molecules which have at least 25 atoms too many and are in intimate contact with a solvent, which is the subject of this paper, is far from the main field of this proceedings. But techniques draw together scientists from different fields, and the authors have found it very stimulating and enjoyable to work with Gisbert zu Putlitz and his team on the laser spectroscopy of an atom not included in the periodic table, whose ground state lifetime is 2.2 microseconds. We wish him good health and continued success in research and in the building of bridges between cultures, both national and scientific. We thank our collaborators in the early stages of this work, R.E. Hester, D.L. Faria and J.N. Moore for their participation and many useful discussions, and M. Scully for making the ground state absorbance measurements. This work was carried out at the Central Laser Facility, Rutherford Appleton Laboratory with support from the EPSRC.

References [1] P. Matousek, R.E. Hester, J.N. Moore, A.W. Parker, D. Phillips, W.T. Toner, M Towrie, I.C.E. Turcu and S. Umapathy, Meas. Sci. Technol. 4 (1993) 1090 [2] P. Matousek, A.W. Parker, P.F. Taday, W.T. Toner and M. Towrie, Opt. Comm. 127 (1996) 307 [3] W.L. Weaver, L.A. Houston, K. Iwata and T.L. Gustafson, J. Phys. Chem. 96 (1992) 8956; K. Iwata and H. Hamaguchi, Chem. Phys. Letters 196 (1992) 462 [4] R.E. Hester, P. Matousek, J.N. Moore, A.W. Parker, W.T. Toner and M. Towrie, Chem. Phys. Letters 208 (1993) 471

158

W.T. Toner, P. Matousek, A.W. Parker and M. Towrie

[5] 3.N. Moore, P. Matousek, A.W. Parker, W.T. Toner, M. Towrie and R.E. Hester in Time - Resolved Vibrational Spectroscopy VI (1994), Springer Proceedings in Physics 74, p.89 [6] W.T. Toner, R.E. Hester, P. Matousek, J.N. Moore, A.W. Parker and M. Towrie, ibid, p.115 [7] B.I. Greene, R.M. Hochstrasser and R.B. Weisman, Chem. Phys. Letters 62 (1979) 427; F.E. Doany, B.I. Greene and R.M. Hochstrasser, Chem. Phys. Letters 75 (1980) 206. [8] H. Hamaguchi, Proc. XVII Intl. Conf. on Photochemistry, paper IN8 (1995), to be published. [9] R.M. Butler, M.A. Lynn and T.L. Gustafson, J. Phys. Chem. 97 (1993) 2609; D.L. Morris, Jr. and T.L. Gustafson, Appl. Phys. B 59 (1994) 389; J. Phys. Chem. 98 (1994) 6275. [10] M. Towrie, P. Matousek, A.W. Parker, W.T. Toner and R.E. Hester, Spectrochimica Acta A 51 (1995) 2491 [11] A.W. Parker, P. Matousek, W.T. Toner, M. Towrie, D.L.A. de Faria, R.E. Hester and J.N. Moore, Proc. XIV Int. Conf. on Raman Spectroscopy (1994), Extra booklet, p.E-9; P. Matousek, A.W. Parker, W.T. Toner, M. Towrie, D.L.A de Faria, R.E. Hester and J.N. Moore, Chem. Phys. Letters 237 (1995) 373 [12] J. Qian, S.l. Schultz and J.M. Jean, Chem. Phys. Letters 233 (1995) 9. [13] R.J. Sension, A.Z. Sarka and R.M. Hochstrasser, J. Chem. Phys. 97 (1992) 5239. [14] A.W. Parker, P. Matousek, P.F. Taday, M. Towrie, W.T. Toner and R.H. Bisby, Proc. New Developments in Ultrafast Time-resolved Vibrational Spectroscopy, Tokyo (1995). [15] G.E. Galica, B.R. Johnson, J.L. Kinsey and M.O. Hale, J. Phys. Chem. 95

(1991) 7994

Four-Quantum RF-Resonance S t a t e of an A l k a l i n e A t o m

in t h e G r o u n d

E.B. Alexandrov and A.S. Pazgalev Vavilov's State Optical Institute, St. Petersburg, Russia

1

Introduction

The topic of this paper goes back to the times of the golden age of Optical Pumping in 1950-1960. In those days a lot of refined studies of radiofrequency (RF) spectroscopy of atoms were performed. In particular, multiplequantum transitions were observed and interpreted by the team of A. Kastler when they were studying RF-spectra of optically pumped sodium atoms [1] and by P. Kusch who applied an atomic-beam technique to potassium atoms [2]. These and many other investigations were summarized in an important paper [3] of J. Winter (see also the review of A. Bonch-Bruevich and V. Khodovoi [4]). Later on the interest in multiple-quantum processes shifted from the RF-domain to the optical range where double-quantum transitions in counter-propagating light beams became very popular, because of their ability to suppress Doppler broadening of spectral lines [5]. In the RF-domain, multiple-quantum resonances did not find any further application in research or metrology in spite of their obvious advantage: an n-quantum transition is n times narrower than the single-quantum one (in the limit of low driving field power). But this advantage is heavily depreciated by a rather strong dependence of the resonance frequency on the driving field of the power. Indeed, from the most general point of view the use of the multiple-quantum processes seems to be inexpiable because of the necessity to perturb the system by the much stronger driving field. Nevertheless, it looks like we have found a particular case of RF-induced multiple-quantum transitions in the ground state of an alkaline atom which could be of interest for low magnetic field metrology. We regard a four-quantum resonance m R = 2 ~ m F = --2 in an atom with nuclear moment I = 3/2 (see inlet in Fig.l). Two features of this transition attract attention: (i) the energy of the unperturbed transition is strictly linear with respect to the magnetic field strength H unlike any other transition and (ii) the frequency of this transition seems to be almost unaffected by the driving field H1 again unlike all other multiple-quantum transitions. The latter feature follows from estimates based on a perturbation approach [4]: under the influence of the field H1 both states m R = ±2 are expected

160

E.B. Alexandrov and A.S. Pazgalev r~

1.9

+2 +1 0 -1 -2

F=2

Signal (a.u.) 1.8 1.7

I

/

1.(~

4F=1~.~Hz (='K) -1

0

+I

1.,=

1.4 1.3 1.2

I

,

I

.

I

,

I

,

I

349000 349500 350000 350500 351000

i

Frequency

(Hz) Fig. 1. RF-spectra of the ground state double-resonance signal of 39K at four different values of the driving field Ha strength (in units ofg = fjHl: 1)g = 1; 2)g = 20; 3) g -- 80; 4)g = 190. Inlet: a level diagram displaying the ground state magnetic splitting of an alkaline atom with nuclear moment I = 3/2.

to shift almost equally. This prediction needs to be confirmed by accurate calculations which is the main aim of the following consideration. In the past multiple-quantum processes were theoretically analyzed mainly in the framework of a perturbation approach - as higher terms of a power expansion of the transition probability. Apart from Bloch's well known exact solution of the resonance problem in two-level systems only a three-level system interacting with either a single- or a two-frequency field has been thoroughly analyzed in m a n y publications (see for instance [6-9] and references therein). In principle, using the approximation of rotating waves the exact stationary solution can be obtained analytically for any k-level system. It is noteworthy, however, that for k > 4 it is a highly troublesome task with extremely cumbersome and practically uninterpretable results. But nowadays there exist powerful computers and advanced software which make it possible to get easily numerical or even analytical solutions for a many-level problem. They are valid for any power of the driving field. Below we present the result of a computer simulation of 4 - q u a n t u m RF resonances in optically p u m p e d alkaline atoms.

Four-Quantum RF-Resonance in the Ground State of an Alkaline Atom

2

161

Formulation of t h e P r o b l e m

The frequencies fro,m-1 of the RF-transitions between adjacent magnetic sublevels m and m - 1 of the ground 2S1/2 state obey the well-known Breit-Rabi formula. They are presented below as a power expansion over the magnetic field strength H:

I2,1= L,o = Io,-I = AI,-2 z

a+H - 3bH 2 + cH 3a+H - bH 2

-

cH

3 +

a+H + bH 2 - cH 3-

. .. ...

...

a+H + 3bH 2 + cH 3 + ...

(1)

This results in set of 4 lines within the state F = 2 and of two lines for F -- 1: fl,o = a_H

f0,--1 ---- a _ H

+ bH 2 - cH 3 -

...

- bH 2 + cH 3 + ...

(2)

The coefficients a+ are very close to -t=7G H z / T : a+ = fi ± L ~ ~fY

(3)

where fi = 9 i p B / h , f j = ( g j - 9 i ) # B / ( 4 h ) , 9i and g j are nuclear and electron g-factors, II8 and h are the Bohr magneton and the Plank constant, b

=///Ahfs

c -= 6 b f j / A h f ~

(4)

with Ahfs being the hyperfine splitting of the ground state. The spectrum of the RF-induced transition, at sufficiently low power of the driving field H1 comprises the 6 above lines if the RF-field induced width f j H 1 of the transitions remains much smaller than the frequency difference between the adjacent lines. The same is true for the intrinsic width F i j of the line. Under such conditions the RF-field interacts with the multiple-level system like with an ensemble of independent two-level systems. But as the field grows, the lines are getting broader, their resonance positions shift, and additional two-photons peaks appear with resonance frequencies fJ2) ~ ( f j + l - f j - 1 ) / 2 , where f j is the frequency of level j. Upon further increase of H1 similar tripleand four-quantum transitions appear within the sublevels Of the F = 2 state. It should be mentioned that four-quantum transitions have never been observed experimentally so far, probably, because it could not be distinguished from the background of the overlapping broadened neighbouring lines. To make it detectable, it is necessary to achieve a great sharpness of the spectrum, i.e., a large ratio of the line splittings to the line widths. In the subsequent calculation we will assume the intrinsic line width to be extremely small - 1 Hz - consistent with our recent experimental results [10]. Let us

162

E.B. Alexandrov and A.S. Pazgalev

consider the optically pumped atomic vapour in a so called Mz- configuration, which means that the vapour is being pumped by a circular polarized resonant light beam along the magnetic field H and the magnetic resonance is observed as a change of the absorption ~ of the pumping light:

(5)

t~ ~- E p i w i

where Pi and wi are the population and the absorption probability of the sublevels i. The problem is to calculate the population distribution under the action of optical pumping and a driving RF-field. We search for a steadystate solution of the density matrix equation pjk with phenomenological terms to describe the optical pumping and relaxation processes. The equation for off-diagonal terms reads as follows:

ihOpsk/Ot =

[Ho + V ( t ) , P]jk -

ih(rp)jk

(6)

where the Hamiltonian H0 with the diagonal matrix describes the atom in a constant magnetic field, the operator V(t) takes into account the effect of the RF-field H1 and the matrix Fp describes the relaxation of the coherence Pjk due to thermal processes and optical excitation. We assume, for simplicity, that in the absence of excitation by light all elements pjk relax with the same rate F. Pumping light shortens the life time of atoms and thus additionally broadens the level k by Fk. The coherences pjk relax with the rates Fjk =

r + (rj + r~)/2. Equation (6) neglects the coherence transfer in the course of the optical pumping which corresponds to the buffer gas optical pumping. Dealing with the populations pj - pjj it is necessary to add to Eq. (6) terms describing the optical pumping. In the absence of pumping all populations relax to the same value p = 1/8. With optical pumping the atom at the level j is excited to the states # and after spontaneous decay goes to the sublevel k of the ground state with the probability Bkj = bkjIp which is proportional to the pumping intensity Ip. Finally, the equations for the populations pjj become: i

Opjj/Ot = - ~ [ H 0 + V(t), p]jj - pjj(F + Fj) + F/8 + E pkkBjk

Q

(7)

The probabilities bkj and wi are listed in the paper of Franzen and Emslie [11]. In the rotating wave approximation we have V = V exp(iwt), where w is the driving field frequency. Seeking for the steady-state solution, we will find the coherences pjk(t) in the form pjk(t) = pjk e x p [ i ~ ( j - k)t]. We will also keep in mind that the RF-field does not produce microwave coherences between sublevels belonging to different hyperfine states. It has also non-zero matrix elements Y)k for j - k = +1 only. Presenting pjk as pjk = xjk+ivjk, we can reduce the equations (6) and (7) to a set of 34 linear algebraic equations for 8 populations (steady-state conditions imply Opjj/Ot = 0) and 26 values for Xjk and Yjk.

Four-Quantum RF-Resonance in the Ground State of an Alkaline Atom

Signal

163

1.8

(a.u.) 1.7 1.6 1.5 1.4 34996O

*" 350'000

'

350040'

'

Frequency

(Hz) Fig. 2. The four-quantum resonance contour at g = 190.

3

Results

The realistic case of 39K in the average earth magnetic field of about 50 ttT has been simulated 1. T h e pumping light intensity has been chosen such that the main (strongest) single-photon resonance f2,1 became 1.5 times broader compared to its "dark" width F. This corresponds (more or less) to the condition of getting m a x i m u m resolution for the single-photon R F - s p e c t r u m [12]. The set of 34 equations has been solved numerically and a n u m b e r of graphs to(w) has been plotted for different values of H1. Figure 1 shows a family of graphs n(w) for fjH1 from 1 to 190 in units o f f which was assumed to be 1 Hz. The strength of H1 was selected in each case to maximize the sharpness of each n - q u a n t u m resonance in the region from n = 1 to n = 4. Only very weak traces of transitions within the sublevels of the state F = 1 can be found in two of four spectra. Figure 2 presents in more detail the R F - s p e c t r u m in the vicinity of a four-photon resonance. The rather flat background of the resonance is related to the deeply saturated two-photon resonance mF = +1 mR : - 1 at ahnost the same frequency. Figure 3 displays the m a x i m a l steepness S,n (H1) = max[dS/dw], where S is the signal strength. The signal steepness characterizes the accuracy of the resonance peak location, playing the leading role in the evaluation of the resonance line for a real application. Figure 3 compares the steepness of the main ordinary resonance with that of the four-quantum transition. The steepness of the last one is about 7 times higher. Finally, Fig. 4 shows the resonance shift induced by the driving field for the single- and four-quantum lines. For each resonance the field H1 was chosen in the vicinity of its optimal value. One can see t h a t field induced shifts are of the same order being in fact negligible.

i a+/a_ = (7.004666/--7.008639) GHz/T; b = 106.327 GHz/T~; c = 9681 G H z / T 3

E.B. Alexandrov and A.S. Pa~.galev

164

a

a 0.030

-o.o35

dS/dg

Shift

(a.u.) 0 025

( H z ) -oo3o ~0.025

0.020

-.0,020 0015 -0.015 0.010 -0.010 0.005

-0.005

0.000

i 1

, 2

, 3

A 4

O00O

1

2

3

4 g(Hz)

g(Hz) b

..o.o12

Shift

d S l d g 020

(Hz)

(a.u.)

-O.OLO

OA5 -0.0(]8 O.lO -0.004

o o5

~].002 0.00

'

150

200

250

300 g(Hz)

Fig. 3. The maximal steepness of the single-quantum (a) and four-quantum (b) resonances (arbitrary units) vs the driving field strength.

4

15o

200

250

300

g(Hz)

Fig. 4. The frequency shifts for the single-quantum (a) and four-quantum (b) resonances (in Hertz) induced by the driving field in units of fj Hj.

Conclusion

The results of a detailed computer simulation support our initial supposition about attractive features of the four-quantum resonance F = 21 m E = --2 ~ ',, m R = +2 : its quality (frequency discriminating ability) surpasses that of the ordinary resonances and its position is not affected even by a fairly strong driving field. It appears that the four-quantum resonance can compete with the slngle-photon transition in magnetometric application. The strictly linear dependence of these resonance frequencies on the magnetic field strength should also be pointed out. With the optimal intensity of the driving field this resonance can be easily distinguished from adjacent broadened resonances.

Four-Quantum RF-Resonance in the Ground State of an Alkaline Atom

5

165

Acknowledgements

This work has been supported by International Science Foundation. E. Alexandrov is grateful also to the Alexander yon Humboldt-Stiftung. Many thanks to Prof. Dr. G. zuPutlitz with whose group the author cooperated in 1994 when the main idea of this work appeared.

References [1] [2] [3] [4] [5]

J. Brossel, B. Cagnac et A. Kastler, C.R. Acad. Sci., Paris, 237, 984 (1954). P. Kusch, Phys.Rev. 93, 1022 (1954). J.M. Winter, Ann. Phys. 4, 745 (1959). A.M. Bonch-Bruevich and V.A. Khodovoi, Sov. Phys. Uspekhi 8, 1 (1965). See for example V.P. Chebotaev, and V.S. Letokhov, Nonlinear laser spectroscopy, Springer-Verlag, Berlin, Heidelberg, New York (1977). [6] V.S. Butylkin, A.E. Kaplan, Yu.G. Khronopulo, and E.I. Yakubovich, Resonance interaction of matter with light, Moscow, Publishing house "Nauka"

(1977). [7] [8] [9] [10]

Th. H~nsch, and P. Toschek, Z. Physik 236, 213 (1970). C. Feuillade, and P.R. Berman, Phys. Rev. A29, 1236 (1984). M. Schubert, I. Siemers, and R. Blatt, Phys. Rev. A39, 5098 (1989). E.B. Alexandrov, V.A. Bonch-Bruevich, and N.N. Yakobson, Sov. J. Opt. Teehnol. 60, 754 (1993). [11] W. Franzen and A.G. Emslie, Phys. Rev. 108, 1453 (1957). [12] E.B. Alexandrov, A.K. Vershovskii, and N.N. Yakobson, Soy. Phys.-Tech. Phys. aa, 654 (1988)

Hard Highly Directional X - R a d i a t i o n E m i t t e d by a Charged Particle Moving in a Carbon Nanotube V.V. Klimov 2 and V.S. Letokhov 1 1 Institute of Spectroscopy, Russian Academy of Sciences, 142092 Troitsk, Moscow Region, Russia, and Optical Sciences Center, University of Arizona, Tuscon, USA 2 p. N. Lebedev Physical Institute, Russian Academy of Sciences, 53 Leninsky Prospect, 117294 Moscow, Russia

1

Introduction

C o n s i d e r a b l e advances have been made in recent years in the synthesis of the so-called carbon nanotubes being important objects for use in nanotechnologies [1-6]. These wonderful objects, a mere nanometer in diameter, may have quite a macroscopic length of up to a few centimeters. Being hollow, nanotubes seem to be natural candidates for transporting both neutral and charged particles in various nanodevices. On the other hand, charged particles propagating in nanotubes interact with their walls and thus can generate a coherent electromagnetic radiation which can be of special interest. The radiation due to charged particles channeling in crystals has now been well studied [7-10]. Such a radiation has a wide range of properties making it of practical use. The investigation of charged particles channeling in nanotubes was started recently [11, 12]. The aim of the present paper is to continue the investigation of the electromagnetic effects taking place in the course of propagation of positively charged particles in single-layer nanotubes. In our view, the most important advantages of nanotubes are due to their large diameter (compared with that of channels in ordinary crystals). The equipotential lines for a particle channeling in a nanotube and in a diamond channel are shown in Fig. 1. In this figure one can see the high azimuthal symmetry of the nanotube potential. This is due to the so called Hermann theorem [13]. Within this potential positrons can oscillate with large amplitudes. The typical trajectory of a channeling positron is shown in Fig. 2. Electrons can channel near the cylindrical nanotube surface. Such channeling has no analogues in usual crystal channeling. The typical trajectory of a channeling electron is shown in Fig. 3. Besides, in a system consisting of parallel nanotubes [1], the ratio between the cross-sectional area of the walls and the total cross-sectional area is small, so that the system can effectively capture sufficiently wide beams and hence emit radiation effectively.

168

V.V. Klimov and V.S. Letokhov

-6 -6

-4

-2

0

2

4

Fig. 1. The equipotential lines for a nanotube of diameter 11/k and for a diamond channel (in the central box).

6

x[£]

The plan of this paper is as follows. In Sect. 2 we will consider the structure of the nanotube and find the expression for the potential acting on a charged particle within the tube. In Sect. 3, the classical picture of motion of a positively charged particle near the nanotube axis will be examined along with the characteristics of the resultant radiation. In Sect. 4 we will consider the motion and radiation of a single electron within classical as well as quantum approaches. In the Conclusion, the results obtained will be summed up and avenues of further investigations outlined.

2

The

Structure

Acting

of the

on a Charged

Nanotube Particle

and Inside

the

Potential

it

The specific electromagnetic effects in the nanotube are entirely due to its structure, and so let us consider this structure in more detail. Best developed today are methods of synthesizing single-layer nanotubes from carbon atoms [1-4]. A nanotube in this case can be visualized as a hexagonal graphite layer rolled into a tube with a diameter of the order of i nm. Generally speaking, one can imagine a number of ways of rolling a hexagonal graphite layer into a tube [5, 6]. Here, for simplicity we consider the case when an elementary cell of four atoms (see Fig. 4) is arranged on the surface of the nanotube so that its side with a period of ax/~ is oriented in the direction of the perimeter of the tube (strictly across the tube axis) and that with a period of 3a along the tube axis. The radius R of the single-layer tube thus obtained can easily be found from the elementary relation associated with the conservation of the carbon

Hard X-Radiation Emitted by a Charged ParticLe

xl

o<:

F i g . 2. Trajectory of a positron in a carbon nanotube.

x!

o4 t,..-d

~q

-2

"bv

F i g . 3. Trajectory of an electron in a carbon nanotube.

169

170

V.V. Klimov and V.S. Letokhov

Z

Fig. 4. Nanotube development. ¢p

bond length:

Nav~ = 2~rR --~ R - Nay/3 27r

(1)

where N is the number of elementary periods present along the tube perimeter and a = 1.42/~ is the carbon bond length. Since the nanotube diameter is usually around 11/~ [1], it is not very difficult to estimate the number of elementary cells arranged along the perimeter of the tube: N = 14. Thus, any a t o m in the nanotube can be defined by three discrete parameters: p - the serial number of the a t o m in the elementary cell, k - the number of translations along the tube axis, and q - the number of translations along the tube perimeter. In a cylindrical coordinate system (p, ~, z) with the z-axis directed along the nanotube axis, the coordinates of an a t o m will have the form:

rp,k,q= ( R , N (2qW~(P)),3a(k ~-~!P))) = [0,1,1,

(2)

0],

,

p = 1..4, k = - o c . . c ~ , q = 0..N - 1 With the nanotube structure defined, the question concerning the interaction between a charged particle and the nanotube arises. The main type of interaction is the Coulomb interaction between a particle and the charge of a nucleus, partially screened by the electron shell of the atom. If retardation is disregarded, the interaction potential between an a t o m with the nuclear charge Z2e located at the point rl and a particle with the

Hard X-Radiation Emitted by a Charged Particle

charge Zle located at the point [14, 71:

1-2

171

may be approximated by the expression

where a* is the screening parameter and a B = h2/me2is the Bohr radius. The total particle-nanotube interaction energy can be obtained by summing (3) over all the atoms of the nanotube:

(4) If the nanotube radius is large enough in comparison to the interatomic distance, the expression for the interaction potential can in a first approximation be averaged over the periodic coordinates ( z ,cp) to get the expression

where the dimensionless part of the potential is described by the equation

In expression (6)' use is made of the dimensionless quantities:

7 In the case of a typical nanotube of diameter 11 A, we have R = -

K&lE = 0.07. Fig. 5 shows the potential well for a positron (Z1= 1)and for an electron (21 = -1) interacting with the carbon nanotube (22 = 12).

V.V. Klimov and V.S. Letokhov

172

e +

j

I0C

f

.J

50

T e-

-100

,

P

-6

-6

-4

,

-2

,

,

,

o

2

,

o

r[A]

3

~

Fig. 5. Potential well and energy levels (l = 0) for electrons and positrons with a longitudinal energy of 10 MeV.

M o t i o n and R a d i a t i o n of a P o s i t i v e l y C h a r g e d P a r t i c l e in a N a n o t u b e

Consider now the motion of a charged particle along a nanotube in the potential given by equation (6). Channeling near the surface of the nanotube is possible for negatively charged particles. In the case of positively charged particles, the situation is more advantageous for propagation inside the nanotube, because a particle coming close to the surface of the nanotube experiences a repulsive force due to the incomplete screening of the positively charged atomic nuclei. Considering what has been said above, in this section we will examine a relatively fast (relativistic) positive particle entering the nanotube at a small angle with respect to its axis and not too close to its surface. In the general case, the trajectory will be complex and nonperiodic. We will consider the case where a positron enters a nanotube in direction parallel to its axis. The motion trajectory will in that case be flat, and the motion along the nanotube axis will be quasiuniform, while oscillations develop in the transverse direction with an amplitude equal to the distance of the particle's point of entry from the axis of the nanotube. The linearly polar-

Hard X-Radiation Emitted by a Charged Particle

173

ized radiation generated in this case will be similar to the radiation emitted by a plane undulator. The equation of motion in the transverse plane may be written in the usual form d 0U

Op

d~ bmVA -

(7)

where 7 = 1 / 4 1 - (vz/c) 2 is . a relativistic . . . . whereto . . radial . factor the oscillation velocity is taken to be negligible in comparison to the velocity vz along the z-axis and m is the mass of the particle at rest. In the case where the particle enters the nanotube close to its axis, one may use the harmonic approximation for the potential U= 5o+U2p ~

g2

5~ = 4 N Z 1 Z 2 e 2

(3a) 3

(8)

(/~2 + g'2) 2

Within this approximation, the equation of motion in the radial direction, (7), can easily be solved:

p = p0 cos (~pt)

(9)

where P0 is the radius at the point of entry of the particle into the nanotube and ~2p, the radial oscillation frequency of the positron, is given by F 2 ~ - 252 _ 8 N Z I Z 2 e 2

g2

(10)

mr(3o) 3 In the case of a typical nanotube with a diameter of 11•, we have, instead

of (10), 2.8. 1015

~ ~ - v~

[s -1]

(11)

which for positrons with an energy of 1 GeV gives g2p/2~r = 10 la Hz. The total radiation power P of a particle moving with an acceleration along a curvilinear trajectory is defined by the well-known formula (see e.g., [151): 2e2 4.2 2e 2 C 74 (12) P = ~c37 v 3 R-~2 -

-

where R* is the instantaneous radius of curvature of the trajectory. In our case, the period-averaged inverse square of the radius of curvature is given by the relation 4 2 1 _ 1~2~p0 (13) R .2 2 c4

174

V.V. Klimov and V.S. Letokhov

Inserting this relation into (12), we get the final expression for the radiation power of the particle:

p -- 4p~oe2 0~72 = ~ 64e6 (NZIZ2) 2 3c.m (3a)4 3c3m 2

g4

(14)

where iS0 = po/(3a) is the dimensionless radius of the point of entry of the particle into the nanotube. Accordingly, for the loss per unit length, I, we have I

P

=c'

_~ 64e 6 (NZ1Z2) 2

g4

- -

0

47"/2 .

(15)

In the case of a typical nanotube with N = 14 (11]~ in diameter), the expression for the loss by radiation per unit length is as follows: I ~ P027210 - 1 4 [erg]

(16)

Lcm, For positrons with an energy of Ez = 1 GeV, 7 = 2000, we have, at fi0 = 1 (P0 ~ 4.3/~-), I -- 4-10 -s erg/cm = 25 keV/cm. In that case, the transverse oscillation energy and velocity are E; ~ 10eV and vp ~ 4.2. 106 cm/s, so that the maximum angle of deviation of the particle from the nanotube axis is around 10 -4 rad. Correspondingly, if the incoming beam of positrons has a divergence of the same order of magnitude, the overwhelming fraction of the particles will be captured by the nanotube and will effectively emit radiation. To analyze the spatial-temporal structure of the resultant radiation in detail, use should be made of the well-known formula for the radiation energy in a unit frequency interval emitted into a unit solid angle [15]: 2

dld~(W)__ 41r2ce 2 i/nx(l=_~:n) -j[(n-/7)× j] ei•{t-[nr(t)/cl}dt

(17)

where n is the unit vector in the observation directionand /7 = vlc. Using the fact that the radial oscillationvelocity is small in comparison to the longitudinalvelocity,one can reduce expression (17) to the form dI(w)

e2

d~

- 47r2~ ~

(PO~v) ~

1

[

(sinOcos~) 2

IFI 2

( 1 - flz cos O)2 1 - 72 ( 1 - flz cos 0) 2 2~

f cosCei(,¢_ucos¢)d¢,n - L ~ F - - sin ~ m(~vn) ~ 27rc o

(18)

Hard X-Radiation Emitted by a Charged Particle

]75

L denotes the nanotube length, n the total number of radial oscillations the particle executes while traversing the nanotube, v = w (1 -/~z cos 0), wpo # = sin 0 cos ~ and 0 and ~ are angles in the spherical coordinate system C

(r, 0, ~) with the polar axis coinciding with the nanotube axis. The analysis of the spatial-temporal structure of the radiation shows that the maximum radiation energy is emitted in the 0-direction if u ~ 1, i.e., at the frequency w (0) -

~p 1 - ~z cos 0

(19)

where 0 is the angle between the nanotube axis and the observation direction. It can be seen from (19) that with the angle 0 fixed, radiation is emitted at a perfectly definite frequency. Radiation at the maximum frequency Wm~× is emitted in the forward direction, i.e., at 0 = 0. The frequency is

(1 ~p

Wr,~× -

L

4e

+

-

/2NZaZ2 z/2 V

'

i.e., it rises nonlinearly as the energy of the particle increases. In the case of a typical nanotube of diameter 11/~, we get from (20): Wmax ~-~5.6-1015"f 3/2 Is -1]

(21)

For a single positron channeling near the nanotube axis with an energy of Ez= 1 GeV , 3' = 2000, the maximum energy of the quanta can easily be found from (21) to be hwm~× ~ 0.33MeV, i.e., hard X-quanta are emitted. Note that the maximum frequency of the quanta emitted by positrons (with the same energy) channeling in ordinary crystals is much higher than that given by (21). This is due to the smaller distances between the crystal planes and hence larger electrostatic forces acting on the positrons and resulting in higher oscillation frequencies. The radiation linewidth is mainly governed by the number of oscillations n the particle executes while passing through the nanotube, i.e., by the factor sin ( ~ (1 -,8~ cos0) 7rn) sin ( ~ (1 - / ) z cos 0) rr)

(22)

in (18). Simple calculations show that the linewidth at half-maximum is defined by the expression 2x* aw =

w (o)

(23)

7rn

where z* = 1.39 is found by solving the equation sin ( x ) / x = 1/v'2. For a single positron with an energy of E~ = 1 GeV propagating in a 1 mm long nanotube (n = 33), the linewidth of the quanta emitted in the forward direction can easily be found from (23) to be h a w ~ 0.01 MeV.

176

V.V. Klimov and V.S. Letokhov

0.2

F i g . 6. Radiation energy emitted into a unit solid angle in a unit frequency interval by a positron with an energy of 1GeV propagating in a nanotube l l A in diameter and l m m long at a distance of 4.3A from its axis as a function of the radiation energy E and the angle 8 (normaliza-

0.4

e r g y fA~r . o.] o.~2-'~--..3~ t*vle v ]

0.34

e 2

0

tion to 4rr2c (t -/3z) 2 )"

Fig. 6 d i s p l a y s t h e r a d i a t i o n energy e m i t t e d into a unit solid angle in a u n i t frequency interval as a f u n c t i o n o f the r a d i a t i o n frequency at 8 = 0, while Fig. 7 shows the t o t a l r a d i a t i o n e n e r g y e m i t t e d into a unit solid angle as a f u n c t i o n of t h e angle tg. T h e a n a l y s i s of these figures clearly shows t h e c o n c e n t r a t i o n of h a r d X - r a d i a t i o n in t h e b e a m p r o p a g a t i o n d i r e c t i o n w i t h i n t h e l i m i t s of angles t? ~ 1/7. T h e r a d i a l m o t i o n is q u a n t i z e d , a n d this b r i n g s u p t h e question w h a t t h e c o n d i t i o n s are u n d e r which a classical d e s c r i p t i o n is possible [16]. It is evident from qualitative considerations that the motion of particles entering the n a n o t u b e n o t t o o close to its axis (and such p a r t i c l e s are in the m a j o r i t y ) is a l w a y s quasielassical, since even at n o n r e l a t i v i s t i c l o n g i t u d i n a l velocities x l 0 -s

F i g . 7. Total radiation energy emitted into a unit solid angle by a positron with an energy of 1 GeV propagating in a nanotube l t / ~ in diameter and l m m long at a distance of 4.3A from its axis as a function of the angle O (normalization to

1

0.5

°:"

;

Oy

"'

e2~* 4rr2c(1 - / 3 z )

2) "

Hard X-Radiation Emitted by a Charged Particle

]77

many de Broglie wavelengths are present along the nanotube radius. For particles with low transversal energy, the de Broglie wavelength is always large, so that their motion must be described by quantum equations, but the fraction of such particles decreases as their longitudinal energy is raised. In order to estimate the number of particles that can be described in a quasiclassical manner upon their entry into the nanotube, one should find the expression for the critical entry point radius pc of a particle in the nanotube, i.e., the radius where a single de Broglie wavelength is present along the nanotube diameter (taking into account the relativistic increase in mass): N

-

2R / - h 3'b" (Pc) ~, 1

2R )~dB

(24)

Using a harmonic approximation for estimating the potential energy, i.e., formulas (8) and (10), one can get the following estimate for the critical entry-point radius: h (25)

Pc~ 2Rm~/(2

For a positron propagating in a nanotube of diameter 11 ~, we obtain 0.4 Pc ~ ~ [~]

(26)

One can see from the above expression that the overwhelming majority of particles is described in a quasiclassical fashion, and the analysis performed in the preceding section therefore proves to be justified.

4

Motion and Radiation Particle in a Nanotube

of a Negatively

Charged

In the case of positrons propagating close to the axis of the nanotube, the solution of (7) was greatly simplified by virtue of the fact that the potential energy could be approximated by a quadratic potential. In the case of electrons captured by the nanotube surface, the situation is more complicated, so that potential (5) near the nanotube surface cannot be approximated well by a harmonic potential even for small oscillation amplitudes. In that case, we will use the linear approximation of the potential in the neighborhood of the well bottom:

U~-4NZIZ2e2 } 3 { g-a ~-~ R =

(27)

which is illustrated in Fig.8. As can be seen from the figure, this approximation increases the potential with the result that all the characteristic frequencies are too large.

178

V.V. Klimov and V.S. Letokhov

> ¢o ¢o -lOG

"=a

;

i

i

a

i

a

~

i

r[h] Fig. 8. Potential of equation (5) (solid line) and linear approximation (27) (dotted line) as a function of the entry-point radius for an electron in a nanotube of diameter n£. The classical motion in the potential (27) is easy to find: It is a periodic motion a quarter of a period of being described by the parabola

~2t2~

a/5 = a/5o (1 -

2-~0/

(28)

In expression (28), 6/5o = /5 - /~ is the dimensionless amplitude of the electron oscillations around the equilibrium position (nanotube surface) and is the characteristic frequency given by ~;2 _ 4 N ZI Z2e2 (3a) 3 m3'k

(29)

For further analysis, it is convenient to approximate (28) by the harmonic function rr~ a/5 = 6/50 cos ( & t ) , & = (30)

242a/5o

The most important feature of (28) and (30) is the fact that the frequency of the electron oscillations depends essentially on their amplitude, whereas for the positron (at least in a harmonic approximation), the oscillation frequency is independent of the amplitude. Another important feature of (28) and (30) is that the oscillation frequency depends on the energy of the particle, while in the ordinary undulators it depends on the geometry of the wiggler magnets. Let us illustrate the characteristics of the radiation emitted by an electron with an energy of 1 GeV and an oscillation amplitude corresponding to the

Hard X-Radiation Emitted by a Charged Particle

179

quantum-mechanical ground state of transverse motion with an energy of E0 ~ - l l 6 e V . In this case, one m a y use (27) to estimate the oscillation amplitude: 5/30 = g

(

Emin - E o ) ,~ 3.4 × 10 -3 •

mTn

(31)

In (31), Emin = -124eV is the well depth for the electron (see Figs. 5, 8). Substituting this value into (30), we obtain for the oscillation frequency of a 1 GeV electron: J2e = 1.775 x 1016s-1, hl2e = l l . 7 e V .

(32)

This value is much higher than the oscillation frequency of the positron and is due to the electron's motion in the immediate proximity to the carbon nuclei. The spatial oscillation wavelength L = V~27r/l-2~ corresponding to (32) amounts to some 0.1 p m , and therefore the electron executes around 104 oscillations while traversing a nanotube which is 0.1 cm long. Because (30) is similar to (9) it is sufficient to substitute J2v --4 12~ in the preceding section to obtain results for the radiation properties of an electron, propagating in a nanotube. For the total radiation power we have

p = 6poe 2 2 12e 4 4 3c 3

72

(33)

7

For the loss per unit length in the given case of plane motion of the electron, we have I

P

=3-=

2 2 Be4 74 5P°e

3c3

e2

-(3a)2

r4(4N) 2

263

(Z1Z2e27h 2

(34)

A remarkable feature of (34) is the fact that the loss here is independent of the oscillation amplitude of the electron. It is not very difficult to see that this fact is associated with the singularity of the potential at p = R and the oscillation frequency being dependent of the oscillation amplitude. In the case of a typical nanotube with N = 14 (11 ~ in diameter), the loss by radiation per unit length of an electron with an energy of 1 GeV amounts to some 2 G e V / c m , which also differs substantially from that in the case of positrons. Radiation at the m a x i m u m frequency Wmax is emitted into the forward direction, i.e., at 0 = 0. This frequency is expressed as

Wmx -- (l--Z)

212~72

The m a x i m u m radiation frequency in the case of a 1 GeV electron is h~ma× 94 MeV. The radiation linewidth is mainly governed by the number of oscillations n, which the particle performs while traversing the nanotube. In case of a nanotube of length 0.1 cm ( n -- 104) it is very narrow: h a w ~ 8.3keV.

180

V.V. Klimov and V.S. Letokhov

So far we have argued within the framework of the classical theory, and the results obtained are represented in graphical form and are mainly of qualitative character. To obtain more exact results, use should be made of the quantum theory. In our case the transverse motion of the electron is quantized, which has to be described by a nonrelativistic Schrbdinger equation where the mass experiences a relativistic increase due to the longitudinal motion [17]. Insofar as the scalar potential is independent of the z-coordinate, the eigenfunction # with a certain energy of E -- Ez + e can be represented in the form

=

i

e-

i

(p, E)

(35)

where e is the transverse-motion energy and p, E = ~/c2p 2 -}- rn2c 4 are the longitudinal m o m e n t u m and energy, respectively. The Schrbdinger equation describing the transverse oscillations of particles in the nanotube is the following: - - - ~ - - A ± + U (p) ~ = c~

(36)

wherein the part of the effective mass is played by the longitudinal energy Ez. Thus, the quantum equation of transverse motion is, up to within small terms U / E ~ , of nonrelativistic character in full accordance with the classical equation (7), the coupling between the longitudinal and transverse motions manifesting itself in the parametric dependence of the energy levels and transverse-motion wave functions on the longitudinal momentum Pz (energy Ez). Since the potential in our case is axially symmetric, the radial and angular variables in equation (36) are separated, and the transverse-motion quantum numbers are the radial (np) and azimuthal (1) quantum numbers. The radial part of the Schrbdinger equation after the usual substitution

(37) takes the following form:

+

+ 5 (p)

=

(3s)

In the electron case the bound state is localized near the nanotube surface so the boundary conditions may be approximated for example by ~5(0) = ~5(p = 2R) = 0 .

(39)

The Schrbdinger equation with the exact potential (5) was solved numerically by the finite difference method [18] using a grid of 2000 (l =fl 0) and 4000 (I = 0) intervals. The lower-level wave functions obtained by diagonalizing the Hamiltonian are presented in Fig. 9. The transverse-motion spectrum of an electron with a longitudinal-motion energy of 1 GeV is shown in Fig. 10.

Hard X-Radiation Emitted by a Charged Particle

i

181

f

£

~0

L~ .,oa

"%

~

4

5

6

7

r[J,] Fig. 9. Electron potential energy (solid line) and squared transverse-motion wave function of an electron of energy 1 MeV. The dotted line with a single maximum corresponds to the ground state (1 = O, rip=0), while the second dotted line is for the first excited state (l --- O, no=l).

From the standpoint of quantum mechanics, radiation generated by a particle passing through a nanotube results from spontaneous transitions among the transverse-motion energy levels of the particle, which also entails a change in its longitudinal energy. The relationships between the characteristic emission frequencies of the particle and its energy can be obtained on the basis of simple considerations of the conservation of its energy and m o m e n t u m [17]: hw (0) =

{~ (E) - {I (E)

(40)

where the subscripts i and f refer to the initial and final state, respectively, and p and E = X/p2+ m2c ~ are the initial longitudinal m o m e n t u m and energy, respectively. One can easily see that expression (40) for ultrarelativistic particles is equivalent to (19), the classical oscillation frequency corresponding to the difference between the initial and final energy levels at a fixed longitudinal energy. Fig. l l and 12 present the maximum radiation frequencies (0 = 0) found in accordance with (40) for the transitions (np= 2 , / = 2) --+ ( n p = 1,1 = 1) and (np= 1,1= 2) --+ ( n p = 1,1 = 1). It can be seen from these figures that in the case of an electron the radiation frequency in the transition ( n p = 2,1 = 2) -+ (np -- 1,1 = 1) (87MeV) is close to the frequency found in the classical approximation {94MeV). The

182

V.V. Klimov and V.S. Letokhov

LL] 1(~

C=TTV~7~=V~VVC:$VCV$$222==$ i C===VTTOOTTS=T22222~C=~=:$2= I

1

I Orbital Number

Fig. 10. Energy levels of an electron in a nanotube at a transverse energy of 200 MeV. In this case the increase of energy with the orbital number is rather

small.

radiation in the classical case is somewhat harder which arises from the stiffness of the linear approximation in (27). As to the transition probabilities and radiation intensity, these are essentially associated with the process of capture of particles by the nanotube and require individual studies.

5

Conclusion

In this work radiation is considered which is emitted by a charged particle propagating inside a carbon nanotube and interacting without retardation with the screened charges of the nanotube nuclei. Analytical expressions are obtained for the total radiation power and the relationship between the radiation frequency and the entry angle of the particle. It is shown that even in the case of positrons with an energy of 1 GeV, the channeling of a b e a m with a divergence of the order of 10 -4 tad is possible. In that case, in the region of small entry angles there emission of hard X-quanta occurs with an energy of around 0.33..2MeV and a width of some 10 keV (for a 0.1 cm long nanotube). In the framework of classical as well as q u a n t u m mechanical approaches it is shown t h a t if electrons with an energy of 1 GeV (or 200 MeV) are captured by the surface of a nanotube for small entry angles radiation is emitted consisting of quanta with an energy of the order of 94 MeV (6 MeV) and with a linewidth of some l0 keV (0.35 keV). Note that the development of such a source of monochromatic and highly directional "),-radiation would be of interest in the implementation of selective

Hard X-Radiation

Emitted by a Charged Particle

183

10a Tran~uou (rip -.2, 1 - 2 ) - > ( % - l , I - I)

10o

~

J I0

a

101

10;

I0~

Electron Energy [McVl Fig. 11, Transition frequencies

(.p=2,1=2)-~

(n.=l,Z=l)

of a n e l e c t r o n as a f u n c t i o n of t r a n s v e r s e e n e r g y

.

10":

;> Tnm.~iou (% - 1,1 - 2)- >(up - 1, I - I) >..,, 10-3

t"

.o 10~

10 ~

.

10'

10:

,

I0~

Electron Energy [MeV] Fig. 12, Transition frequencies (n o = 1, l=2)--~ ( n . = l , l = l )

of a n e l e c t r o n as a f u n c t i o n of t r a n s v e r s e e n e r g y .

photonuclear reactions [20]. The main points associated with the propagation of a charged particle in a nanotube and the resulting radiation are considered, a number of important points being left out in the analysis. For example, when we consider the beam of particles, the total radiation linewidth will broaden due to the anharmonicity of the oscillations. This and other effects will be considered elsewhere [21].

184

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V.V. Klimov and V.S. Letokhov

Acknowledgments

The authors thank the Russian Basic Research Foundation (Grant 96-0219753) and the US Department of Defense (mediated by the University of Arizona) for their financial support of the work.

References [1] R.E. Smalley. From Balls to Tubes to Ropes: New Materials from Carbon, Presentation, American Institute of Chemical Engineers, South Texas Section, January Meeting in Houston, t996; T.W. Ebbesen, Phys.Today, June 1996, 26. [2] S. Iijima and T. Ichihashi. Nature, 363 (1993), 603. [3] D.S. Bethune, C.H. Klang, M.S. de Vries, G. Gorman, R. Savoy, J. Vazquez and R.Beyers. Nature, 363 (1993), 605. [4] T. Guo, P. Nikolaev, A. Thess, D.T. Colbert, R.E. Smalley. Chem. Physics Lett., 243 (1995), 49. [5] M.S.Dresselhaus, G.Dresselhaus, P.C.Eklund. Science of Fullerenes and Carbon Nanotubes. Academic Press,1996. [6] J.W.Mintmire and C.T.White.Carbon, 33, (1995), pp.893-902. [7] M.A.Kumakhov and F.F.Komarov. Radiation of Charged Particles in Solids (Minsk, University Press), 1985. [8] B.Berman et al. Channeling Radiation Experiments Between 10 and 100 MeV. In: A.Carrington, J.A.Ellison (eds). Relativistic channeling. NATO ASI Series B, Physics 165, Plenum Press,1986. [9] F.Fujimoto, K.Komaki. Channeling Radiation Experiments Between 100 and 1000 MeV In: A.Carrington, J.A.Ellison (eds). Relativistic channeling. NATO ASI Series B, Physics 165, Plenum Press, 1986. [10] H.Andersen, L.Rehn,(eds). Beam Interaction with Materials and Atoms. Nuclear Instr. And Meth. B, 119, October 1996. [11] V.V. Klimov, V.S.Letokhov. Physics Letters A, 222 (1996), pp. 424-428. [12] V.V. Klimov, V.S.Letokhov. Physics Letters A, 226 (1997), p.244-252. [13] C.Hermann, Zs.Kristallogr., 89 (1934), 32. [14] J. Lindhard. Mat. Fys. Medd. Dan. Vid. Selsk., 34 (1965), No.14. [15] J. Jackson. Classical Electrodynamics, John Wiley & Sons, Inc. New York London, 1962. [16] B.Kagan, Yu.V.Kononets, ZHETF, 58 (1970), 226; 63 (1973), 1041. [17] V.A.Bazylev, N.K.Zhevago. Rad. Eft., 54 (1981), p.41. [18] N.N. Kalitkin. Numerical Methods (Publ. House "Nanka", Moscow), 1978. [19] Algard et al., Nucl. Instr. And Moth., 170 (1986), 7. [20] V.S. Letokhov. Conclusion Talk on Nobel Symposium on Femtochemistry and Femtobiology (Bjorkborn, Sweden, September 9-12, 1996). [21] V.V. Klimov, V.S.Letokhov. Physica Scripta,(1997) , (in print) .

Quasiclassical Approximation in the Theory of Scattering of Polarized Atoms E.I. Dashevskaya and E.E. Nikitin Department of Chemistry, Technion, Haifa, 32000, Israel

1

Introduction

The quasiclassical approximation in quantum mechanics constitutes a quite general method of attacking numerous problems and has been treated in many advanced text on Quantum Mechanics (see e.g. [1]). With the advent of efficient and fast numerical techniques, the quasiclassical approximation gradually looses its significance as a method of getting reliable theoretical results, but still retains its role as a simple and transparent means for interpretation of the underlying physics. This is related to the fact that the quasiclassical approximation uses classical quantities (such as trajectories and the action integrals along these trajectories), and incorporates in a simple manner the superposition principle of quantum mechanics. In the theory of atomic collisions, the quasiclassical approximation was successfully and widely used in its simplest form, when the nuclei were assumed to move classically along a rectilinear trajectory (the so-called impact parameter approximation), or, more generally, along a trajectory governed by a certain single potential (a common trajectory approximation) [2]. Within this picture, the electronic states of a quasimolecule evolve according to the non-stationary Schrhdinger equation the time dependence of which is governed by the motion of nuclei. This approximation which completely neglects the effect of the motion of electrons on the motion of nuclei performs well for not too slow collisions when the kinetic energy of nuclei exceeds noticeably the energy spacing between different potentials of a quasimolecule. For slow collisions, one should abandon the notion of a common trajectory and use a more general treatment. The latter should properly account for the fact that the motion of nuclei in different electronic states is governed by different potentials and, besides, it is also affected by the coupling between electronic states. If the coupling between states can be described using a proper representation (e.g. by adopting different Hund coupling cases [3]) in a form of localized interactions, a general quasiclassical method becomes simple enough to be helpful in deciphering the complicated dynamics. Recently, a great deal of attention was paid to the study of scattering of polarized atoms in slow collisions [4-6] and to the high-frequency oscillations in the differential cross section, in particular in connection with atomic

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interferometry [7-9]. The specificity of this event is a recoupling of the electronic angular momentum, in the course of a collision, from a direction fixed in space (determined by the condition of the experiment) to the axis of a quasimolecule, and back. The final result of this recoupling depends on the rotation angle of a molecular axis in a collision; in turn, the rotation angle of the axis depends on the potentials, acting on the atoms, and therefore on the populations of different electronic states. The interplay between these two effects results in a complicated pattern of the differential cross sections for the scattering of polarized atoms. The aim of this contribution is to present a quasiclassical picture of the inelastic scattering accompanied by the recoupling of relevant angular momenta with the help of the so-called locking approximation [10-14]. The plan of the paper is as follows: In Sect. 2 we consider the constraints imposed on the polarization-transfer cross sections by the symmetry of collision events. In Sect. 3 the so-called locking approximation is introduced and briefly discussed. Sect. 4 and 5 are devoted to the discussion of general features of the integral and differential cross sections, respectively. Sect. 6 provides examples which demonstrate the use of the quasiclassical approximation in the interpretation of the angular m o m e n t u m recoupling in a collision: collisional creation of polarization, high-frequency oscillations in the differential cross section for the scattering of unpolarized atoms, and large right-left scattering asymmetry in collisions of helicopter-oriented atoms. Neither of these events can be understood in the framework of the common trajectory approximation. Finally, in conclusion we summarize the main features of the quasiclassical approach to the scattering of polarized atoms.

2

Symmetry Cross

Constraints

on the

Polarization-Transfer

Sections

As an example, we consider the scattering of an atom A in a degenerate electronic states (initial and final angular momenta j and j~ ) on a spherically symmetric atom B in a collision: A ( d ~ ) + B(:S0) --+ A(d',~,) + B(1S0)

(1)

Here ~ and ~',~, are the initial and final spherical polarization moments (with r specifying the rank of the tensor and s its spherical projection) of the atom A (for the discussion on state moments, see, e.g. [15]). The relevant cross sections for a process (1) are the polarization-transfer cross sections; in a general case, this transfer is accompanied by a transformation of one type of polarization into another. A change of a polarization moment in collisions is described by the relevant cross sections of the Waldmann-Snider type [16,17] for a ~ -+ ~,~, transition. The set of cross sections actually constitute a matrix with each element connecting initial and final states. In case the collision events have some ,t

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187

symmetry, there will be certain symmetry relations for the cross sections. In particular the cross-section matrix can be diagonal in certain quantum numbers. To emphasize this fact, the relevant quantum numbers will be placed in the superscript of the cross section and will not be repeated twice. Basically, there are three different types of experiments for processes described by Eq.(1): experiments in the bulk measuring the integral cross sections, experiments in a beam measuring the integral cross sections, and experiments in a beam measuring the differential cross sections. For the experiments in the bulk, there is no preferential orientation which is related to two-body collisions. Therefore, the quantities measured in bulk experiments, are the integral cross sections averaged over all orientations of the initial collision direction (the wave vector of the relative motion, k) with respect to a space-fixed frame in which polarization moments are defined. In these so-called isotropic collisions, the only possible process is a transfer of polarization between states j and j~ and without any change of the tensorial property of polarization (transition j, r, s --+ f , r, s) or the loss of polarization in the same j state [17,18]. The relevant cross sections are the relaxation cross sections ~ (j', j) ; they do not depend, in view of the spherical symmetry, on the projection s of the state moment. The formal symmetry of the scattering event in the bulk is Kh. For experiments in beams measuring the integral cross sections, there is a preferable orientation which is specified by the vector k. If the space-fixed quantization axis is directed along k the integral cross sections describe the polarization transfer ~ , --4 d : s which occurs, due to the axial symmetry, without chan~e of the projection s. The relevant (integral) cross sections are ~ , r ( j I, j; k). The presence of the unit vector k in the latter expression emphasizes the fact that the quantization axis specifying the initial and final states coincides with this vector. The symmetry of this event is C ~ . . For experiments in beams measuring the differential cross sections, there are two orientations connected with a collision: the initial and the final wave vectors, k and k ' . The relevant (differential) cross sections are qr,s,;rs (jl, j; ~:, ~:, and a most convenient quantization axis fi is parallel to the vector k x ~1, fi c~ x l~I. The symmetry of this event is Cs, with the reflection plane coinciding with the collision plane which in turn is normal to n. The hierarchy of the cross sections

qr',';~,(J

-1

,3;~:, ~:1) --4 er,.q ~(3' 1 3,- . ~¢) --4 ~r"(j',j)

(2)

carries, in the sequence indicated, a decreasing amount of information on the collision dynamics, and provides a convenient way to discuss collisions of polarized atoms. Note, that the cross section q~,,;~, (jl, j; ~, i~1) integrated over all scattering angles yields the quantity 6~,,r x ~r~,~ ' (3.i , 3,.. ~:) , and the averaging of ~ , r (J', J;k) over s yields the quantity 5~,~ x ~ (j', j). The appearance of the Kronecker deltas in this expressions illustrates clearly, how the information content in the cross section is lost when one passes to simpler and simpler

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experiments. It is interesting, that the quasiclassical methods in application to the scattering of polarized atoms also exhibit a kind of hierarchy being less and less demanding for calculating more and more averaged cross sections.

3

Locking, Slipping and Slippage in the Angular Momentum Recoupling

The important quasiclassical concepts used nowadays in scattering theory are locking, slipping and slippage of atomic orbitals with respect to the rotating molecular axis. These concepts have arisen in the attempt to provide a description of the interplay between two basic interactions in a diatomic quasimolecule: the coupling of the electronic angular momentum to the molecular axis which is due to the electrostatic forces and the decoupling of the electronic momentum from the axis due to the Coriolis force. Since the Coriolis force is very weak compared to typical electrostatic force at the point of closest approach of the colliding atoms, the former can affect the collision dynamics only at large interatomic distances where the latter is also very weak. This happens in the locking region, where the frequency of electronic transition between molecular states is comparable with the angular velocity of rotation of the collision axis. It can be shown that under quasiclassical conditions the relative kinetic energy of colliding atoms, when they cross the locking region, is substantially higher than the splitting between potential energy curves in this region [20], and therefore the approximation of a common trajectory holds in this region. With further approach of atoms, the difference in the interaction potentials might become comparable with the initial kinetic energy, and the approximation of a common trajectory becomes invalid in this region. However, here the electronic angular momentum gets locked to the molecular axis, and each potential can be assumed to drive the system along the respective potential curve. The quasiclassical description in this region is based on different trajectories that originate from a single initial trajectory defined by a certain impact parameter b and collision velocity v. This situation is exemplified in the upper part of Fig. 1 which refers to a half-collision of an atom A in an adiabatically isolated state j = 1 with a spherically symmetrical atom B. For this system, there are two potential curves Ua (R) differing in the projection of j onto the molecular axis, Uo(R) and U1 (R). For definiteness, we assumed that both interaction are repulsive, and Uo(R) is more repulsive than U1 (R). The locking region is schematically shown by the cross-hatched area that separates the atomic and molecular regions. In the former, the good quantization number for the electronic angular momentum is the projection o f j onto any space-fixed axis, while in the latter, the good quantization number is the projection of j onto the molecular axis. Two important parameters of the locking region are the location of its center RL and its characteristic width ARL; both of them depend on the impact parameter b and the collision velocity v.

Quasiclassical Approximation in the Theory of Scattering

189

Q

slippclge Fig. 1. Quasiclassical scattering in the locking approximation for the case j = 1. Shown are the locking region (cross hatched) with the locking distance RL(b) and ARL width, incoming wave front (wavy line), and regions of tight and loose locking. The upper part of the figure is appropriate for calculating the integral cross section: it shows two trajectories with different deflection angles r/o and r/1 originating from one trajectory with impact parameter b and the locking angle a(b). The lower part is appropriate for calculating the differential cross section: it shows two trajectories with different impact parameters, b0 and bl, and different locking angles cr0(~9) and a1(69) deflected through the same scatterting angle O.

Within this picture, the qualitative dynamics of the collision is as follows: First the system moves in the atomic region along a common trajectory (in the case under discussion this common trajectory is assumed to be a rectilinear trajectory). Here the interaction between the atoms is very weak, and can be calculated within the Hund coupling cases d or e. Then the system crosses the locking region. Here, the motion is still described by a rectilinear common trajectory, but electronic states of colliding atoms change due to the change of the good quantization axis. With further decrease of the interatomic distance, the system evolves adiabatically along two molecular potentials; each of them deflects the trajectory under different angles. Two points of closest approach mark the end of the half-collision. The second half of the collision proceeds in a similar, though reversed, fashion; the only difference being that finally, at very large interatomic distances, there will be two trajectories deflected under two different angles, ~0(b) and 711(b). An important attribute of the scattering within this picture, is the phase difference accumulated during the motion of the colliding system between two locking regions (one on the way of the incoming motion, and the other for

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E.I. Dashevskaya and E.E. Nikitin

the outgoing motion). If this phase difference is large enough one speaks of the tight locking; otherwise, the locking is said to be loose. The loose locking is also called slippage, and asymptotically, at very large b, the slippage means just an unperturbed evolution of electronic states of colliding atoms. The above nomenclature of locking, slipping and slippage is consistent with what was introduced previously [4,5,12,21]. The concept of locking is equivalent to the notion of "orbital following" [22] and the locking distance is the same as the matching distance [10]. A great deal has been learned on the mechanism of locking for low values of j [20, 23-26]. An alternative description of the collision is illustrated in the lower part of Fig. 1 where again two trajectory are shown, but this time both are deflected through the same scattering angle 0 ; therefore they correspond to different impact parameters, b0 and bl . These two alternative descriptions are appropriate for a calculation of the integral and differential cross sections respectively (see Sect. 4 and 5). An important property of the tight locking is that the two regions of the recoupling of angular momenta (on the incoming and outgoing ways) are well separated by the molecular region where the good quantum number is the projection, 12 , of the electronic angular momentum onto the molecular axis (the so-called R-helicity quantum number). On the other hand, a very helpful quasiclassical representation of the overall scattering matrix S is the J-helicity representation, when the good quantum number is a projection, u, of the electronic angular momentum onto the classical conserved vector of the total angular momentum J (the J-helicity quantum number). (We note in passing that the electronic R-helicity quantum number 12 is not an exact quantum number. However, in the limit R --+ oc, 12 becomes an exact quantum number which will be called w. The quantum number w is simply related to yet another exact quantum number, the P-helicity quantum number [2]) Let the "single way" locking event be represented by the locking matrix C, which describes the recoupling of the angular momenta and the change of representation from the J-helicity to R-helicity. Then the scattering matrix S in the J-helicity representation can be written in the form: s =

CtrsMc

(3)

where the molecular scattering matrix S M and the locking matrix C can be calculated independently of each other [2,12] and it is typically diagonal in j. In a number of interesting cases, the molecular scattering matrix can be assumed to be diagonal in 12 . Then, the scattering is just the succession of two J-helicity - R-helicity recouplings separated by the adiabatic evolution of states in the molecular region. In the region of tight locking, the most important parameters of the locking event are the locking distance RL (or the locking angle a(b), see Fig. 1) and the locking time rL which is proportional to ARL. If VL can be considered to be vanishingly short, one uses a simplified version of tight locking,

Quasiclassical Approximation in the Theory of Scattering

191

\

0\ \ \ \ \ \

\ \

-e. \

(y

ARt.

VH'HH,

I/ -- Ft

Rt. Fig. 2. Rotronic adiabatic correlation diagram of the energy levels for a rotating diatom composed of an atom A in a state j = 1 and a spherically symmetric atom B. The extreme right corresponds to separated atoms; the extreme left to the energy levels of a diatomic molecule. The intermediate region close to RL corresponds to the transition from the Hund coupfing case d or e to case b or c. Sudden change in the pattern of the energy levels in this region mimics the stronger R-dependence of the electrostatic interaction compared with that for the Coriolis interaction. The adiabatic energy levels are labeled by the exact (reflection) quantum number ~r = + or - , and by the good quantum numbers u and Ig?I on both sides of the transition region of width ARL. The wavy arrows indicate a nonadiabatic transitions with the probability 1/2 - s.

the so-called sudden locking. A deviation of the locking event from its sudden limit is ascribed to the slipping phenomenon. T h e slippage can be regarded as a limiting f o r m of slipping when the tight locking changes over into the loose locking. T h e notion of the locking and slipping is illustrated in Fig. 2 which shows the adiabatic correlation d i a g r a m of rotronic (rotational+electronic) states of an a t o m i c pair A ( j = 1 ) + B ( j = 0) [27] for a passage between free a t o m states (extreme right) and the molecular states. T h e three energy levels shown are the eigenvalues of the r o t a t i n g electronic H a m i l t o n i a n for a fixed value of the total angular m o m e n t u m J. T h e y are labeled with exact reflection q u a n t u m n u m b e r cr (c~ = + or - ) which characterizes the reflection s y m m e t r y of the electronic state in the plane perpendicular to the vector J [27]. T h r e e energy levels at the right h a n d side of the d i a g r a m correspond to the Coriolis splitting of a free a t o m i c j = 1 state as seen from the molecular frame before. T h e frequency of this splitting corresponds to the frequency of r o t a t i o n o f the molecular frame: a free a t o m i c state observed f r o m the molecular f r a m e will exhibit a precession o f the electronic angular m o m e n t u m vector j a b o u t the total angular m o m e n t u m vector J. T w o energy levels at the extreme

192

E.I. Dashevskaya and E.E. Nikitin

left correspond to two electronic energy states, f2 = 0 and S? = +1, when the electronic angular m o m e n t u m is strongly coupled to the molecular axis. A week splitting of the I ~ l = l energy levels slightly to the right from the extreme left is the familiar A-doubling phenomenon. The rest of the diagram is constricted by applying the noncrossing rule for the states of the same reflection symmetry. The adiabatic picture presented in Fig. 2 describes the recoupling of the angular momenta from J-helicity to the R-helicity limits in the course of infinitely slow change of the interatomic distance R. In a real collision, the radial motion induces nonadiabatic transitions between the adiabatic states of the same reflection symmetry. These transition are localized in a region the position of which (the locking distance) corresponds to the wavy arrow. In the limit of sudden locking, the transition probability between two positive states is equal to 1/2 ; it means, e.g. that the population of the upper J-helicity level t,=-i is shared equally between ~2 = 0 and ~2 = 1 molecular states when they pass the recoupling region. It may be surmised from Fig. 2 that the finite energy splitting between two positive levels at the transition region will produce a deviation from the sudden limit, making the transition probability P smaller that its sudden limit 1/2. The deviation of P from 1/2 is called by the slipping probability s. Therefore, in general, the fraction of the population of the upper J-helicity level is transferred to the lower adiabatic molecular state while the survival probability to remain in the upper adiabatic state is 1/2+s. As an example we present here the elements of the locking matrix for an isolated state j = 1 which corresponds to the correlation diagram shown in Fig. 2. The 3 ® 3 matrix C splits into two blocks each corresponding to the definite reflection symmetry. The J-helicity quantum numbers are u = 1, t, = - 1 (c~ = + if the atomic state is odd) and fit, = 0 (o" = - if the atomic state is odd), while the R helicity quantum numbers are 0 + (for ~2 = 0 ), 1+ (for I~21 -- 1) and 1- (for I~1 = 1). The elements of the locking matrix for two positive states and one negative state are:

1/2-s

(Cl+hv) = (~,/1/2+sexp(-icr) -V~-

X/I/2 - s e x p ( i a ) ) + s exp(ia) ;1¢21 -- 0, 1; v = 1 , - 1 s exp(-ia) ~1/2

(4)

Cl~l, ~ = 1;

ts?l = 1 ; t , = 0

where rows are numbered like If2[, ~r, and columns by u. Here, a and s are the functions of impact parameter b or total angular momentum J + 1/2 = In the representation adopted, the locking angle a is positive (see Fig. 1). On the other hand, the slipping probability s is a signed quantity: it is positive if to the left of the locking region the C2 = 0 potential curve is above the ~2 = 1 potential curve (as in Fig. 2), and negative otherwise. The sign of s is established on the ground of the rotronic [27] correlation diagrams (s is positive when the state with the lowest value of v adiabatically correlates with the state of the lowest [~21 value).

kb.

Quasiclassical Approximation in the Theory of Scattering

193

In the approximation of sudden locking s = 0 , and the only difference in columns in the matrix C comes from discrimination between two magnetic atomic states ~, = q-1 and L, = - t which is linked to the right-left asymmetry of the rotation. Finally in the adiabatic approximation we have a = 0 .

4

Integral

Cross

Section

for Polarization

Transfer

Let the quantization axis coincide with the direction of the initial wave vector of relative motion k. The integral cross section, c~,,~(3 ~ "' , g,'" k) " , for the polarization transfer is expressed via self-explanatory bilinear combinations of the scattering matrix elements in the j, m, l, n representation [2]: ~ ' ~ ( J ' , J ; k ) = ~Z

~

m' - ~ ' s

-

×

(5)

rn I ~rr~~ ' J ,~"

where t, n are the quantum numbers of angular relative motion, and [ i i i] are Clebsch-Gordan coefficients. There is a physical significance associated with the two summations in Eq.(3): The sum over m', m, ~", ~ arises as a result of the transformation of the initial and final density matrices in the Zeemanstate representation, PJ~,m and PJ~-',m" to their spherical counterparts, ~ and ~',~,; this step is necessary since only the spherical representation exhibits a simple axial s y m m e t r y of the polarization-transfer cross sections. The sum over t, ~ describes the interference between different initial orbital states, the sum over ~' accounts for all final orbital states; there is no interference between final orbital waves since the corresponding terms vanish when integrated over all scattering angles. There is no summation over n', n, W, since n = 0, ~ = 0 for the chosen quantization axis, and n ~ --- m - m ~ and = ~ - ~ . Note also that under the quasiclassical conditions we have the relation: £, ~, g' > > j~, j. Due to the conservation of total angular momentum, the quantum numbers g and ~ vary within a relatively narrow interval centered at gr while the latter changes from zero to infinity. The cross sections ~r~,,~(j', j; 1¢) are not necessarily real-valued quantities; in the case j' = j and

r' = r, c~~',~(j,j) , are negative, and they correspond to the total loss of ~ due to transfer of this component to other states. We now perform the quasiclassical analysis of Eq.(3) which will reveal the significance of the g, ~ interference. To this end we turn to the total angular momentum representation of the scattering matrix with R-helicity elements S]~,;jo~. The important property of the quasiclassical S-matrix is that each element of it can be represented as a linear superposition of terms which show quite specific behavior with respect to dependence on the total angular

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E.I. Dashevskaya and E.E. Nikitin

momentum J: each term can be represented as a product of a rapidly varying exponential and slowly varying preexponential factor. Explicitly, -

i ~

exp

(6)

-7

Now, expressing the S matrix in the J, m, t, n representation through the S matrix in the j,w, J representation, and substituting this into Eq. (4), one arrives at a double sum over total angular momenta, say J and J. This double sum stems from the interference of initial states for different relative angular momenta l and t . Since £ and ~ are close to each other, so are J and ~ . Therefore, when calculating the quantities like (S...)(S...) J 7 • one can account for the different values of J and J only in the exponent of each term in the sum (6) in the first order with respect to A J : J and J , and assume that J and J and which enter into the preexponentials are the same. The final result of the analysis [11,12] yields the cross section in Eq.(5) as the quasiclassical counterpart:

~rr,,~(j 8

"I

-

^

, j ; k ) -- 2~r

/0

,5

-I

,j;£,b)bdb •

(7)

where the transition probability under the integral over the impact parameters is P~'r (J', J; k, b) =

z

.

.

.

(8)

x

.

Eqs.(7) and (8) contain no more large angular momenta which are now incorporated into the classical quantity, the impact parameter b. The objects S... (b) that enter into these equations are related to the partial contributions to the quasiclassical scattering matrix (6) as

Sj,m,;jm(b)~

= Z

Rm',o~'J' (qj,~,j~(b))'Y

• sj,~, "~ j~(b) exp[2i~],~,,j~(b)]J~:m

(9)

~jOJ I

j'

where /~,~, ~,(fl) are the elements of the Wigner matrix for rotation by an

j,

-,

angle fl around the y axis, R,~,,,~, (8) = DJ,~,,~, (0, fl, 0) . Note that quantities ~

Sj,m,;jm (b) are not derived from the scattering matrix as a whole, but rather are synthesized from the partial contributions to the respective elements of the quasiclassical scattering matrix and elements of the rotation matrix. The impact parameter b in the exponents and preexponential factors in the r.h.s. of Eq.(9) replaces the total angular m o m e n t u m J in Eq.(6) via the quasiclassical relation J + 1/2 =- kb , and the deflection angle ~lj~'~,,jw (b) that enters

Quasiclassical Approximation in the Theory of Scattering

195

into the element of the rotation matrix is defined by the usual quasiclassical relation: 2 O[~,~,j~ (b)] (10) b rlJ"~';J"( ) = k cOb A peculiar character of Eq.(9) is that the rotation angle r/ is different for different exit states which is a signature of the multiple-trajectory description of the collision event. If the common trajectory approximation is adopted, the deflection angle is assumed to be independent on the exit state, and Eq.(9) would imply a simple rotation of the quasiclassical S-matrix: through the deflection angle ~ : ~

Sj'm,;jm(b)

j~

~f~R.v,,~,(~] ) .r

_

(11)

03 1

Finally, in the impact parameter approximation when r/is taken to be zero, Sj,,~,jm (b) becomes simply the quasiclassical S matrix expressed via the impact parameter:

~,~,jm(b) = s/,~,,~ I

(12) J=kb--l[2 w~tn

The probability and the cross section for polarization transfer in isotropic collisions can be expressed by formulae which are obtained from Eq.(8) by averaging over all projections s. After this averaging, the dependence of the cross sections and transition probabilities on k disappears, and the quantization axis for electronic angular momenta may be any axis fixed in space. The appropriate expressions are:

~r'(j',j) = 2~r

#

W(j',j;b)bdb

(13)

where

P'(j',j;b)

(-1)J'-~'

-=

"+~

j' j '

j j

rrtl~ms~S,~s

{~,m,;~m(b) S*~,~,;~(b) - 6~,;j 6~,~6,,,,,,,

}

(14)

We now incorporate the locking approximation into the J-helicity scattering matrix. In order to see most clearly the effect of locking, we assume that the molecular scattering matrix is diagonal in ~ and j (no transition occur in the molecular region), and that locking is sudden. Then the S-matrix will be diagonal in j, and it can be represented in the form:

JJ

exp [2i~ (J)]

(15)

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E.I. Dashevskaya and E.E. Nikitin

where ~2 is now identical with % This equation identifies the coefficients in Eq.(6) as

g2

= ~j,jRJ,,~[a(J)]RJ,sl[a(J)]

(16)

Yet another common approximation, used in the quasiclassical calculation of the integral cross section is the so-called random phase approximation. It is based on the observation that the interference terms which are present in Eq.(8) and which contain differences in the phase shifts in bilinear combinations of the scattering matrix for different paths ~ and ~ ' do not provide a noticeable contribution to the integral cross section since they virtually vanish when integrated over impact parameters. The random-phase approximation, (P~,,,.(j', j; ~:, b)), to the transition probability (8) reads: (P:,,~ (j', j; k, b)) = -(fj,,j(f~,,~+

=

~

RJ

r s~

s,"'[rl)'~',J~(b)](-1)J'-5'-J+~

j ' j'

J -J

(17) r'

s'

-~s'

. sja%, j=(b) sj,=,,j~(b) s~. This expression does not contain rapidly oscillating terms, but includes contributions that vary slowly with the deflection angle. Coupled with approximation (14), it describes the probability of the polarization transfer in terms of deflection angles for different trajectories that originate from a single trajectory. The latter is characterized by impact parameter b, collision velocity u and by the elements of the locking matrix. In the sudden locking approximation, the only parameter of the locking matrix is the locking angle a. Some examples of application of the above formulae will be discussed in Sect. 6. 5

Differential

Cross

Sections

for Polarization

Transfer

Application of quasiclassical methods to calculations of differential cross sections for polarization transfer is not as straightforward as calculations of integral cross sections. There are two basic approaches: the eikonal approximation which is valid for the scattering through small angles ((9 << 1) and the approach which is valid for scattering by classical angles, i.e. angles larger than the angle of diffraction scattering 6~d ((9 >> ~d)- Since under quasiclassical conditions [~d is extremely small, two approaches possess a common region of applicability, Od << (9 << 1, thus allowing to construct quite general quasiclassical description. In this section we discuss scattering by classical angles. By now, there are not too many experiments of this kind which directly refer to process (1). We mention here the measurements of the total (summed over all magnetic quantum numbers) differential cross section of excited Na(~Pj) atoms on Ne [33],

Quasiclassical Approximation in the Theory of Scattering

197

the study of the left-right asymmetry of -polarized excited alkali atoms on noble gases [34] and the scattering of the excited metastable Ne* atoms on Ne [35] and Ar [36]. In the case of scattering of alkali atoms, close-coupling quantal calculations are available [35,37], and quasiclassical approximation have been used to provide the physical interpretation of numerical data [38,39]. The quasiclassical theory of scattering by classical angles [12,39] is based on the J-helicity representation which takes advantage of the fact that for large relative angular momenta the classical J vector is either parallel or antiparallel to the vector fi which defines the collision plane and which can be taken as a space-fixed quantization axis Z. In other words, in this approximation the inelastic scattering occurs in a fixed plane. A schematic quasiclassical picture of the scattering is represented in Fig. 3. The wave front of incoming particles, impinging onto the target placed in the origin of the SF X Y Z frame (called the natural collision frame, NCF) is subdivided in the scattering plane into two parts which define the ensembles of the right and left atoms. These parts, both of which refer to large values of J, are separated by a "wall" that corresponds to small J (a cross-hatched region in the lower part of Fig. 3). For small values of J the quasiclassical description is not valid; however, under quasiclassical conditions this region makes a small contribution to the scattering amplitude, and this contribution can be safely neglected if one considers scattering angles not very close to zr. In general, the scattering direction is specified by two angles, the polar angle 6}(0 _< 6} < 7r) and the azimuthal angle 9(0 > p >_ 27r) in a SF frame X Y Z . When one considers scattering in the plane, the scattering to the right and to the left corresponds to the same value of 6}, but to two different values of ~v (say ~ = zr/2 and = 37r/2 ); alternatively, one can speak about scattering to the right and to the left by the same angle 6}. Of course, in general, both right and left atoms contribute to the scattering to the right; and both contribute to the scattering to the left. The scattering to the right and to the left are separated by the region of small (diffraction) angles (a cross-hatched region in the upper part of Fig. 3) in which the quasiclassical description is not valid. As an example, Fig. 3 shows the scattering to the right, and the two contributions to the scattering amplitude originating from the right and left atoms. The scattering amplitudes to the right and to the left from the initial state [j, n) to a final state ]j', n') in the NC frame, li~,right j , , , , ; j , , (6}) and i~left , j , , , , ; j , , (t:}] ~v I, can be expressed via the positive and negative J-helicity (JH) scattering amplitudes, fJ+,~';L~(6}) and f~,~,;j,~(6}). Here, n, n' are projections of j,j', onto the Z axis (NC frame), while u, u' are the projections o f j , j ~, onto the J vector (JH) frame. The relation between quantum numbers n and u for the right and left atoms is illustrated in Fig. 3 [12,39]. The appropriate formulae for JH and NC scattering amplitudes read:

f)+ ~,;j,~(O) = - i exp(-iTr/4)

f0 c° e x p ( - i O J )

Sj,,.,;j,,~(J)v/-ffdJ

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E.I. Dashevskaya and E.E. Nikitin

ba

bI

Fig. 3. Quasictassical description of scattering of an atom A(j = 1) on a spherically symmetric atom B positioned in the origin of the natural collision frame XYZ. Shown are two trajectories deflected by the same scattering angle O; trajectory 1 corresponds to repulsive scattering (right atoms are scattered to the right), trajectory 2 to attractive scattering (left atoms are scattered to the right). Total angular momenta J1 and J1 corresponding to these trajectories are shown by vectors originating from the points of closest approach for the respective trajectories. The cones at the initial points of trajectories, each of height n and slant height j, illustrate schematically the polarization state ]j, n) of incoming atoms. The hatched regions in the X Y plane correspond to small impact parameters and small scattering angles where the quasiclassical description is not valid.

--i exp(irr/4) f~,:,,;j,,(O) = -~2-~.kCs~nO

f0 °°

(18)

exp(iOJ) Sj, u,;j,u(J)v/JdJ

and

t~__exp[in,(O F ~,n~;j,n j r ~'-~ i! g h t

~r)]-

[fj+':,,;j,u(O)l ~-~,,-F

(19)

1

Fj~O~t t,n';j,n t~'~ t" ] = exp [-in'(O + ~)][ (-1)J'+U[',.';J, .(°)1J2--:,

" (0 + " (--l'J'+Jf-" "~',"';~,"

"=" )1.,_.,

]

Eq.(18) shows that JH scattering amplitudes have the same s y m m e t r y properties as the JH scattering matrix: it is s y m m e t r i c with respect to q u a n t u m numbers of the initial and final state, and it vanishes for transition between different reflection states. The former s y m m e t r y property disappears in NC scattering amplitudes, Eq.(19), since the s y m m e t r y of the scattering event in a space-fixed frame is lower that the s y m m e t r y of this event in a body-fixed

Quasiclassical Approximation in the Theory of Scattering

199

frame. The meaning of the JH positive and negative scattering amplitudes can be seen by referring to elastic scattering and by evaluating the integrals in the stationary phase approximation. One finds then, that f + (O) describes the scattering of atoms under the action of the repulsive forces (right atoms are scattered to the right, and left atoms to the left), while f - ( 0 ) describes the scattering under the action of the attractive forces (right atoms are scattered to the left and vice verse). left /~'~ The manifold of scattering amplitudes Fj~,igh,~j,~(O), ;, and F j,,,v;j,,~t':") , can be used for calculation of differential cross sections which describe the collisional transfer of polarization moments of an atom. If the polarization state of the atom in NC frame is specified by the set of irreducible spherical.state moments ~ s , the cross section of the polarization transfer , ~ s -4 ~,~,1~21~ = JiJ~qAma~2p52s, 3 192, n = -t-2, Qjr,s,;jrs(O), is constructed as a ClebschGordan contraction of NC scattering amplitudes. Now, if one goes back to the X ' Y ' Z t frame, and specifies the polarization state of the atom by the set of polarization moments Pjrp in this frame, the appropriate polarization transfer cross sections qjr,~;j~ (0, ~) can be obtained from the rotation transformation of the matrix of the cross sections Oi~,~,;j~(O) : = exp

- p)]

•

zX;

(20)

where A;a = R;~(rr/2) [40]. The polarization-transfer cross sections qjr,p,;j~p(O; T) are not axially-symmetric and are complex-number quantities; their very simple dependence on azimuthal angle is due to the fact that these cross sections relate irreducible spherical components of the polarization tensor. Had one considered a scattering of an atom whose polarization state is a linear superposition of irreducible spherical components, one should have dealt with objects that are linear combination of qjr'p';jrp(O, ~). The ~-dependence of the latter can be quite complicated. Two examples of the cross sections which are expressed in a simple way via NC scattering amplitudes are the total population-transfer cross section for the scattering of unpolarized atoms, qtot;unpol(O, j -4 jr) , and the right-left difference cross section qaify;h~u~ (0, j, n -4 j') for the scattering of helicopteroriented atoms:

qtot,,,npodO,j -4 j') = (2j + 1) - I E = (2j + 1)

qdiff;helic(O,j, It -+ j') = ~ n t

b~right i'O~ 2 ~J ,v,j,~, ,

(21)

teJ, :

it2)~ ~wright J n';Jn ~v ll 2 -- E

•it;,ieft .,;in(o)?

(22)

n t

The next step in the quasiclassical approximation consists in recovering the JH scattering matrix from the classical attributes of adiabatic motion

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E.I. Dashevskaya and E.E. Nikitin

across the molecular potential curves and from the parameters of the matrices of non-adiabatic transitions. We exemplify this by way of scattering of an atom in an isolated state j = 1 (j = j~ -- 1) for a repulsive interaction in both molecular states (see Fig. 1). The two deflection functions, q0 (b) and 7/1(b), can be used to recover the phase-shifts for elastic scattering from the differential relation (10), 60(J) and 61(J). Other quantities which will enter into the JH scattering matrix are the locking angle L(J) and the slipping probability s(J). The ultimate result of rather simple calculations according to the JH counterpart of Eq.(13) yields the JH scattering matrix which decomposes into two blocks, one with u, u ~-- 4-1 for positive reflection symmetry and the other with u, u ~-- 0 for negative reflection symmetry:

S~I~I =

(1/2)[exp(2i6o 4- 2ia)(1 4- 2s) + exp(2i~l J: 2ia)(1 ~ 2s)]

(23)

So0 = exp(2i61) where all the quantities are functions of b or J. According to Eq.(18), each element of the JH scattering matrix generate two (positive and negative) JH scattering amplitudes. Each amplitude will contain two terms that correspond to the scattering at ~2 -- 0 and ~2 -- 1 molecular potentials, Uo(R) and U1 (R). Finally, if the quasiclassical approximation is pushed further, one solves the integrals over J in the stationary phase approximation; this will express the scattering amplitudes via classical scattering cross sections, appropriate phases, locking angles and two slipping probabilities for different branches of a particular deflection function. All these attributes are generated in the same way as one calculates the quasiclassical elastic scattering cross section from the single deflection function. The only difference being that besides the classical elastic cross section here one has to consider the locking and slipping events, and also the interference between waves scattered elasticity by different potentials. Some examples of application of this approach will be discussed in Sect. 6. For a simple repulsive scattering depicted in Fig. 1 (in this case there is only one branch for each deflection function) all negative JH scattering amplitudes vanish, and four positive JH scattering amplitudes f+(6}), f + (6}), f -+1 - 1 ( 6} ) , f+-1(6}) = f + n ( O ) are expressed as a linear combination of two amplitudes for elastic scattering on $2 -- 0 and ~2 = 1 molecular potentials, £(6}) and f1(6}), with the coefficients depending on locking angles and slipping probabilities, co(6}), a1(6}), and so(O), sl (6}). The elastic scattering amplitudes f a (6}) are expressed via elastic scattering cross sections qa(6}) and phases ¢s~(O) as fa(6}) = ~

exp (i4~a(6}))

(24)

and the meaning of two locking angles, an(6}) , is illustrated in the lower part of Fig. 1. In the approximation of the sudden locking

1[

f+1+1(0) = -~ fo(O)exp(4-2iao(6})) + fl(6})exp(zt:2ial(6}))

]

,

(25)

Quasiclassical Approximation in the Theory of Scattering

20]

1

f+1~:i(6)) = f+1+1(0) = ~ [f0(69) - fi(/9)] , f~(O) = f1(8) . A simple relation of the scattering amplitudes in Eq.(25) to the scattering matrix in Eq.(23) is evident.

6

Case Studies of the Recoupling of Electronic Angular Momentum in Collisions

In this section, we discuss three types of simple collision events which illustrate the ability of the quasiclassical description to elucidate the intimate dynamical features of the recoupling of the electronic angular momentum in a collision. Neither of these events can be described within the common trajectory approximation.

Creation of polarization in the beam scattering of unpolarized atoms. When atoms collide under the axially-symmetric conditions, their electronic shell acquires polarization [15], in particular alignment. The creation of polarization from an unpolarized state in a transition j -4 j~ with j~ > j can be easily understood from the simple consideration that the number of magnetic sublevels in the final state are larger than in the initial (unpolarized) state. Therefore, it is quite improbable that the initiM statistical distribution over Zeeman states will result, after a collision, in a statistical distribution over the Zeeman final states. The case of the same j state is different: the question whether the same initially unpolarized state can be collisionally polarized or not, requires a more thorough study. It was proven that a cross section for the creation of the polarization calculated in the impact parameter approximation is zero [41]. Since the impact parameter treatment of relaxation of the state moments via Eq.(13) and (14) with Eq.(12) yielded nonzero cross sections and was believed to provide an accurate quasiclassical limit of the appropriate quantum cross section, it was argued that the above proof is general so that the j-conserving collisions can not create polarization. However, this conclusion is not correct: the vanishing cross section for the creation of polarization in the impact parameter approximation is an artifact related to the superfluous symmetry of the collision event introduced by this approximation. In order to see limitations of the impact parameter treatment we consider a collision as in Eq.(1) taking j = y . Adopting the sudden locking approximation, we substitute sjn,~,j~(b) from Eq. (16) into Eq. (17) and yield for the probability of transfer of the state polarization moment from rank

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E.I. Dashevskaya and E.E. Nikitin

zero (r = 0) to rank r' for the axially-symmetric component (s = s' = 0):

(P°,,o(j, j; 1¢, b)) = E{Pr,(cos[2a(b) +

Tin(b)] ) - £ , , 0 } ( - 1 ) j - n

[,a9 -~2,

$2

(26) where P~,(...) is the Legendre polynomial of order r'; r' can assume values from zero '(in this case the probability corresponds to the loss of population in the initial state) , to 2j. The meaning of dynamic parameters in the r.h.s. of Eq.(26), the locking angle a(b) and the deflection angles r/0 (b) and 7/+1(b) is explained in the upper part of Fig. 1 for the case when both molecular potentials, Uo(R) and U±I (R) are repulsive, and the former is stronger than the latter (therefore) r/0(b) > q+l(b)). For r' = 0, Eq.(26) yields zero probability since the first factor under the sum in the r.h.s, of Eq.(26) vanishes. This simply corresponds to the fact that the total population of the state is not affected by the collision. For odd values of r' (r' = 1 corresponds to the creation of orientation), Eq.(26) also yields zero probability because of the cancellation of contributions to the sum from positive and negative a') (a symmetry property of the Clebsch-Gordan coefficients). This corresponds to the fact that the final polarization component ( r' odd) and the initial component (r = 0) possesses opposite symmetry with respect to a reflection in a plane through the symmetry axis. For even values of r' (r' = 2 corresponds to the creation of alignment), Eq.(26) yields, in general, a nonzero probability. However, if the first factor under the sum in the r.h.s, of Eq.(24) does not depend on a'2 ( the common trajectory approach, the impact parameter approximation in particular) it becomes zero because of the orthogonality relation for the Clebsch-Gordan coefficients. It is thus clear that the vanishing probability of creation of polarization is related to the superfluous symmetry of the problem in this approximation: a common trajectory description of a collision introduces an additional symmetry into the collision picture that actually does not exist when one incoming trajectory generates two trajectories, running on different potentials The integral cross section for the creation of polarization are related to the asymmetry in the Zeeman cross sections for transitions m --+ m' and m' --+ m. For instance, the cross section for creating alignment in a state with j = 1, ~7°,o(l, l; k), determines the difference between two Zeeman cross sections: o'°,0(1, 1;1¢) =

2 [Oh,o(l,l;k:)- O'o,i(1,i;]~)]

(27)

Eq.(27) provides an example when Zeeman cross sections for transitions m --+ m' and m' -+ m are different. Of course, the relation ~5,,.~ (j, j) :~ crS,.~, (j, j) does not contradict the detailed balance principle which requires that the cross sections for a direct, m --+ m ' , and the reverse, - m ' -+ - m , transitions

Quasiclassical Approximation in the Theory of Scattering

203

be equal, c~r,,,n (j, j) = trim,_ m, (j, j) , However, for isotropic collisions, the Zeeman cross sections for transitions m ~ m' and m' --+ m are the same. The importance of the multiple-trajectory description for the collisional creation of polarization poses the question to what extent this effect will show up in the values of relaxation cross sections for isotropic collisions. Consider a cross section a r ( j , j ) , and represent it as a sum of a cross section calculated in the impact parameter approximation, o'r'ip(j,j), and the correction Aar(j, j). This correction is associated, via Eq.(13), with the respective correction to the probability of relaxation Ap"(j,j; b). The expression for A P ~(j, j; b) reads [42]:

AP"(j,j;b)-

~

{R~0 (qjo(b)

+ 2c~(b))- R~oo(2o~(b))}

Jl

(28) -1- (2r q- 1~-----)

- - ~ 2S2

{ -2t2,20

(rlj.q(b) -t-

-- Rr--2a,2~

The correction to the transition probability is due to the fact, that the angles of rotation of the molecular axis in the region where the electronic angular momentum is locked to this axis, r r - Oja(b)- 2a(b), are different from the angle of rotation when the system moves along a rectilinear trajectory, 7r- 2~(b). For scattering off a hard core with small impact parameters, when the molecular axis virtually does not rotate at all, the relaxation probability is very small. In this case, the correction Ap~(j, j; b) is almost opposite to P~,iP(j,j;b). We also see from Eq.(26) that the correction Apr(j,j;b) does not vanish when one passes from multiple trajectory description (different deflection angles qjo(b)) to the common trajectory description (a single deflection angle r/different from zero). Of course, Apr(j,j;b) vanishes for rectilinear trajectories, 77= 0.

Interference pattern of the differential scattering of unpolarized atoms. Measurements of the differential cross sections for scattering of excited sodium atoms on Ne at the collision energy 0.15 eV reveals an oscillating structure [33] which can not be explained by interference of waves scattered by the attractive and repulsive portions of the same potential [43]. The closecoupling calculations with realistic potential functions reproduce the structure and suggest that the reason for it is the interference of the waves scattered by two different molecular potentials that arise from the P state of Na [36]. Since the interference requires participation of coherent states one can ask the question how these states can arise from incoherent superposition of atomic magnetic substates. The quasiclassical study of this question reveals that the oscillatory pattern that is due to the interference of the waves scattered by different potentials strongly depends on the locking angles, and that the coherence is created in the locking event.

204

E.I. Dashevskaya and E.E. Nikitin

For Na*-Ne collisions, the approximation of a fixed electronic spin is appropriate [44], and the total differential cross section for the scattering in a single fine structure state, 2 p j , j = 1/2 or 3/2, turns out to be equal to the total differential cross section for the scattering of a spinless atom in a P state: qtot,~,,~pot(O;2 Pj) = qtot;,,npot(O; P) (29) The quasiclassieal expression for qtot,,,npot(O; P) can be analyzed in terms of contributions from the scattering at ~ and H potentials with proper account for interference and locking. First we separate the slow- and rapidly-varying contributions to the cross section

qtot;u.vo,(O, P) -~- qtstow rapid .[172, . . . 1"1 ot,unpol ..-. 1(72,p) + qtot,unpol

(30)

The explicit expressions for slow and rapid contributions depends on the number of classical trajectories that are deviated by the same scattering angle O. For Na*-Ne scattering, there are, in general four trajectories; one repulsive trajectory for the ~ potential (Z,rep) one repulsive trajectory for the / / trajectory (H,rep), and two attractive trajectories for the rainbow scattering at the rr potential. For the scattering angles close to the primary rainbow, two attractive trajectories are very close to each other, and can be collapsed in a single trajectory (//,rain). Accordingly, we have three amplitudes for elastic scattering, f.~,rep(O), and f//,rep(O) and fFI,rain(O). slow t ~ p) is just the contribution from The slow-varying portion qtot,unpoltV, elastic scattering by a repulsive E' potential, qt/,rep(O) , from elastic scattering by the repulsive part of a / / p o t e n t i a l , qH,rep(O) , and from elastic scattering by the rainbow part of a / / p o t e n t i a l , qH,rain(O) :

qtot,unpol[O , s. . .l . o p) = (1/3)[q2,rep(O) + 2qr,rep(O) -t- 2q~r,rain(O)]

(31)

We also note that, if the scattering to the right is considered, the first and second term in the r.h.s, of Eq.(31) come from the right atoms, while the third term comes from the left atoms. The rapidly-varying portion is due to interference between waves scattered by the repulsive and rainbow branches of the H potential and between waves scattered by the Z repulsive potential and the rainbow branches of the H potential:

qrapid tot;unpol (O, P) = (2/3)\/qII, repqH,rain Cos( ArN)Fl,rep;Fl,rain) " [2 - sin2(an,~p + an,~in)

(32)

--(2/3),~/qH,repqH,rain Cos(A4:ibH,rep;H,rain) sin2(Otll,rep "4- O'H,rain)] The significance of the locking can be seen from this expression: If the adiabatic approximation were assumed to be valid at all internuclear distances, one would set here all the locking angles equal to zero (the locking of the electronic angular momentum to the collision axis occurs already at infinitely

Quasiclassica] Approximation in the Theory of Scattering

205

r,~pia " " P) in Eq.(32) becomes a familiar large distances). In this case qtot;~,,~pot[~; high-frequency contribution to the elastic scattering by the H potential, and the sum in Eq.(30) a weighted sum of cross sections for independent elastic scattering on the ~ and / / potentials. This is the so-called elastic approximation to the scattering cross section of unpolarized atoms; it is this approximation which failed to explain experimental findings [43]. On the other hand, with proper account of the locking phenomenon, Eq.(30) together with Eqs.(31) and (32) reproduce both the quantum-mechanical strong coupling calculations and the experimental results [44]. Due to the simplicity of the quasiclassical expressions, it helped to correct an earlier inconsistency of the semiclassical description [37] and to reveal the role of spin-decoupling in the collision event. We note also that the amplitude of the oscillations which are due to interference of waves scattered at different potentials serves as a measure of the difference of the locking angles for scattering at these potentials. Right-left asymmetry in the scattering of helicopter-polarized atoms. Differential scattering of unpolarized atoms to the right and to the left by the same angle 69 is characterized by equal cross sections. This evident property of scattering can be related to the mirror symmetry of the initial state with respect to the plane that separates right and left atoms. If atoms are polarized, and the initial state is not symmetrical with respect of reflection in this plane, and one would expect the right-left asymmetry of the scattering. Normally, the reflection of the polarization state of atoms in this plane will change the distribution of the electron density of atoms with respect to the incident velocity vector. As a result, left and right atoms with the same values of the impact parameters will populate different molecular states to a different extent. Now, since different molecular states cause different deflection of trajectories, it is not at all surprising that the scattering of polarized atoms will exhibit a certain right-left asymmetry. One can say, that the rightleft scattering asymmetry can be traced back to the different distribution of electron density for right and left atoms with respect to the molecular axis at the moment of locking. However, there exists one type of polarization for which the mirror reflection does not change the distribution of the electron density. This is the so-called polarization when the atoms are oriented perpendicular to the collision plane. For this type of polarization, the right and left atoms possess exactly the same distribution of the electronic density with respect to the molecular axis. Therefore, the scattering asymmetry should be due to the interaction which is not invariant under the reflection in the mirror plane. As the only difference between the right and left atoms at the moment of the locking is the relative orientation of the electronic angular momentum j to the total angular momentum J (their J-helicity quantum numbers are just opposite) the interaction in question is the Coriolis interaction. Since

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E.I. Dashevskaya and E.E. Nikitin

the Coriolis interaction is very week it would be interesting to see how much asymmetry can be induced by this interaction. Experimentally, the right-left scattering asymmetry has been observed in the scattering of excited -polarized alkali atom on rare gases [34] and for helicopter-polarized Ne(2p53s, 3 P2) atoms in collisions with Ne [35] and Ar [36]. The latter system provide a very interesting case for testing the locking approximation. This system differs in many respects from the case of scattering excited alkali atoms. First, all the potentials of the Ne*(2p53s, 3 P2)-Ar pair are governed mainly by the exchange interaction of electrons of the valence closed shell of Ar with the diffuse 3s orbital of the excited electron of Ne* [45] and therefore are believed to be repulsive. Second, the lifting of degeneracy in the 3 P 2 state of Ne* occurs as a result of core-core interaction, and therefore locking takes place when the trajectory already deviates from the rectilinear path. Third, for low collision energies, there is no appreciable fine structure transition, and therefore one can not speak about decoupling of spin and electronic angular m o m e n t u m . A lengthy analysis [46] yields an approximate expression for the cross section difference A(O) of helicopter-polarized atoms scattered to the right and to the left:

A(O) = --[3q(0)/4] sin4[a0(O) -- al(O)] sin 2A¢0~(O)

(33)

= A,~a~(O) sin 2z5~o1(O) where q is the cross section for scattering of unpolarized atoms, and locking angles a0(O) and a l ( O ) correspond to the scattering off the molecular potentials and that arise from the state j = 2. From Eq.(33) we arrive at an expression for the ratio R(O) of the envelope A,~a,(O) to the cross section q: R(O) -

Ama (O) = 3 s i n 4 [ a 0 ( O ) -q(O) 4

(34)

The difference in the locking angles Aa01 = a 0 ( O ) - a l (6~) depends on the details of the interaction potentials which we do not consider here. However, it is clear from the meaning of the locking angle, that over a range of the scattering angles which are well outside of the backward scattering, Acr01(O) increases starting from zero. Then R(O) should pass a maximum at a certain scattering angle O = ~,,, reaching a value of 0.75; at this scattering angle, the difference in the locking angles equals 7r/8. We turn now to Fig.4. At small angles (say, below Otab = 5°), the asymmetry cross section is much smaller than the total cross section. We interpret this as an indication about the threshold behavior of the asymmetry cross section: for small A a , it increases proportionally to Aa. As seen from Fig. 4, the ratio R increases with increasing O, reaches a maximum of 0.6 at O = 11 °, and then decreases down to 0.3 at O = 21 ° . We identify this maximum with

Quasiclassical Approximation in the Theory of Scattering

207

6 I

*4 A

q d •I

0

"*t '

o! Fig. 4. Experimental data on the total differential scattering cross section q, and the envelope of the cross section difference Arnax for scattering of unpolarized and helicopter-polarized Ne*(2p53s, 3 P2, n = 4-2) atoms on Ar at collision energy of 64 meV (after [36]).

6rnax

f

I

I

I

0

I 10 °

I

2

I

a

i

20"

Olab

the m a x i m u m of function R(O) from Eq.(34) and ascribe the difference between the theoretical prediction 0.75 and the experimental value 0.6 to the contribution of other terms neglected in approximate Eq.(34). Finally, we note that the value of Ac~01 at the m a x i m u m , A(~01(Om) = r / 8 , is quite reasonable when one takes into account that the range of variation of individual locking angles is 0 - ~r/2. We can thus say that the behavior of the O-dependent cross section difference for the system Ne*-Ar demonstrates a transition from slippage to locking, and a rather large scattering asymmetry of left- and right helicopter-polarized a t o m s in this case is ultimately due to the fact that the initially prepared atomic states correspond to the largest-possible J-helicity states, ~ = 2 and v = - 2 .

7

Conclusion

In this paper, we have shown that a quasiclassical analysis of the scattering of atoms indeed provides a simple insight into collision dynamics. This is related to the fact that under slow quasiclassical conditions the total inter-

208

E.I. Dashevskaya and E.E. Nikitin

Table 1. Different levels of quasiclassical approximation in the description of scattering of polarized atoms T y p e of the process

Lowest level of quasielassical description

Isotropic collisions in the bulk Transfer of polarization moments. Integral cross sections (rr (j', j).

Impact parameter (common trajectory) approximation with sudden locking.

Anisotropic collisions in beams. Transformation of polarization moments. Integral corss sections o'~.,,.(j', j; k ).

Multiple trajectory description; sudden locking. Common trajectory description; non-sudden locking.

Multiple trajectory description; Anisotropic collisions in beams. Transformation of polarization moments. non-sudden locking Differential cross sections q,-, ~;r~.(j', j; k, ~:).

action occurring in a collision can be split into different contributions, which can be analyzed separately. One of these, which is important for the polarization phenomena, is the recoupling of the electronic angular m o m e n t u m from a space-fixed to the body-fixed quantization axis, accompanied by a partial breakdown of the L S coupling in free atoms. This type of reeoupling is most conveniently described in terms of transient Hund coupling cases. The approach along this line was first formulated within a semiclassical picture [3], and later generalized for quantum mechanical formulation [47,48]. An analysis of recoupling of angular momenta allows to identify the interactions which are responsible for specific effects in the scattering of atoms in degenerate electronic states. For instance, in the scattering of unpolarized atoms, the interference structure in the differential scattering is strongly affected by the long-range Coriolis interaction, and the creation of alignment is critically dependent on the difference of the deflection angles for trajectories, corresponding to the same value of the impact parameter. Different levels of the quasiclassical approximation in the description of scattering of polarized atoms are summarized in Table 1. We indeed see that for some processes a rather crude quasiclassical approximation suffices to provide a reasonable result. On the other hand, we believe that the sophisticated quasiclassical multiple-trajectory description of a collision that includes locking and slipping is able to reproduce all the essential features of the polarization transformation in collisions provided the scattering process occurs in the range of classical angles. In conclusion, quasiclassical analysis of scattering, complemented with additional simplifications related to small values of the ratio j / J , provides a very useful tool in the interpretation of both accurate numerical results and

Quasiclassical Approximation in the Theory of Scattering

209

experimental findings. This is in line with growing interest in semiclassical methods in Postmodern Quantum Mechanics. It is a pleasure to dedicate this paper to Prof. Gisbert zu Putlitz on his 65th birthday. Our work on the theory of scattering of polarized atoms has been closely connected with the activity of zu Putlitz' group in Physikalishes Institut, Universit/it Heidelberg, and we benefited much from the seminars and discussions there. Finally, we will never forget friendly ties with Gisbert during last, sometimes uneasy, twenty five years.

8

Acknowledgment

We acknowledge very constructive discussions with F. Masnou-Seeuws, J. Baudon and F. Perales. This work was supported by the Technion V.P.R. Fund - Promotion of Sponsored Research, and by the Giladi program.

References [1] L.Landau, and E.Lifshitz, Quantum Mechanics (Oxford, Pergamon Press, 1977). [2] E.E.Nikitin, and S.Ya.Umanskii, Theory of Slow Atomic Collisions (BerlinHeidelberg, Springer, 1984). [3] E.E.Nikitin, in Atomic Physics 4. Edited by G. zu Putlitz, E.W. Weber and A. Winnacker (N.Y., Plenum, 1975), p.529. [4] I.V.Hertel, H.Schmidt, A.BSktring, and E.Meyer, Rep.Prog.Phys., 48, 375.(1985). [5] E.E.B.Campbell, H.Schmidt, and I.V.Hertel, Adv.Chem.Phys., 75, 37 (1988). [6] J.Baudon, R.Diiren, and J.Robert, Adv.At.Mol.Opt.Phys., 30, 141 (1993). [7] O.Carnal, and J.Mlynek, Phys.Rev.Lett., 66, 2689 (1991). [8] P.W.Keith, C.R.Ekortom, Q.A.Gurchette, and D.E.Pritchard, Phys.Rev.Lett., 66, 2693 (1991). [9] J.Robert, Ch.Miniatura, S.Le Boiteux, J.Reinhardt, V.Bocvarski, and J.Baudon, Eur.Phys.Lett., 16 , 29 (1991). [10] E.I.Dashevskaya, and N.A.Mokhova, Optika i Spektr. 33, 817 (1972). [11] E.E.Nikitin, Khimicheskaya Fizika, 3, 1219 (1984). [12] E.E.Nikitin, Khimicheskaya Fizika, 5, 15 (1986). [13] E.I.Dashevskaya, and E.E.Nildtin, Optika i Spektr., 68, 1006 (1990). [14] E.I.Dashevskaya, and E.E.Nikitin, J.Chem.Soc. Faraday Trans., 89, 1567

(1993). [15] E.B.Alexandrov, M.P.Chaika, and G.I.Khvostenko, Interference of Atomic States (Berlin-Heidelberg, Springer, 1993). [16] L.Waldmarm, Z.Naturforsch. B, 13, 609 (1958). [17] R.F.Snider, J.Chem.Phys 32, 1051 (1960). [18] A.Omont, J.Phys. 26, 26 (1965). [191 M.I.D'yakonov, and V.I.Perel., ZhETF, 48, 405 (1965). [20] A.Berengolts, E.].Dashevskaya, and E.E.Nikitin, J.Phys., B26, 3847 (1993).

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E.I. Dashevskaya and E.E. Nikitin

[21] P.Hansen, L.Kocbach, A.Dubois, and S.E.Nielsen, Phys.Rev.Lett. 64, 2491

(1990). [22] [23] [24] [25] [26]

C.T.Rettner, and R.N.Zare, J.Chem.Phys., 75, 3636 (1982). B.PouiUy, and M.H.Alexander, Chem Phys., 145, 191 (1990). J.Grosser, Z.Phys.D 3, 39 (1986). L.J.Kovalenko, S.R.Leone, J.B.Delos, J.Chem.Phys., 91, 6942 (1989). A.Berengolts, E.I.Dashevskaya, E.E.Nikitin, and J.Troe, Chem.Phys. 195, 271 (1995). [27] E.E.Nikitin , and R.N.Zare, Molec.Phys., 82, 85 (1994) [28] E.I.Dashevskaya, E.E.Nikitin, and S.Ya.Umanskii, Khimicheskaya Fizika 3, 627 (1984). [29] E.E.Nikitin, Optika i Spektr. 58, 964 (1985). [30] V.N.Rebane, and T.K.Rebane, Optika i Spektr., 33, 219 (1972). [31] A.G.Petrashen', V.N.Rebane, and T.K.Rebane, Optika i Spektr., 35, 408 (1973). [32] E.I.Dashevskaya, and N.A.Mokhova, Chem.Phys.Lett., 20, 454 (1973). [33] G.Caxter, D.Pritchard, M.Kaplan, and T.Ducas, Phys.Rev.Lett., 35, 1144 (1975). [34] R.Diiren, and E.Hesselbrink, J.Phys.Chem., 91, 5455 (1987). [35] J.Baudon, F.PeraJes, Ch.Miniatura, ].Robert, G.Vassilev, J.Reinhardt and H.Haberland H , Chem.Phys. 145, 153 (1990). [36] F.Perales, Effets de Polarisation dans des Collisions aux Energies Thermiques Impliquant des Atomes Metastables de Neon, Thesis, Universit$ de Paris XI, 1990. 37. F.Masnou-Seeuws, M.Phihppe, E.Roueff, and A.Spielfield, J.Phys.B, 12, 4065 (1979). [37] E.I.Dashevskaya, R.Diiren, and E.E.Nikitin, Chem.Phys. 149, 341 (1991). [38] E.I.Dashevskaya, F.Masnou-Seeuws, and E.E.Nikitin, J.Phys.B, 29, 395 (1996). [39] D.A.Varshalovich, A.N.Moskalev, and V.K.Khersonskii, Quantum Theory of Angular Momentum (Singapore, World Scientific), 1988. [40] A.G.Petrashen', V.N.Rebane, and T.K.Rebane, Zhurn. Eksp.Teor.Fiz., 67, 147 (1984) [41] E.l.Dashevskaya, and E.E.Nikitin, Optika i Spektr., 62, 742 (1987). [42] C.Bottcher, J.Phys.B, 9, 3099 (1976). [43] E.I.Dashevskaya, F.Masnou-Seeuws, and E.E.Nikitin, J.Phys.B, 29, 415 (1996). [44] H.Kukal, D.Hennecart, and F.Masnou-Seeuws, Chem.Phys., 145, 163 (1990). [45] E.I.Dashevskaya, E.E.Nikitin, J.Baudon, and F.Perales, (to be published). [46] V.Aquilanti, and G.Grossi, J.Chem.Phys 73, 1165 (1980). [47] V.Aquilanti, S.Cavalli, and G.Grossi, Z.Phys.D, 36, 215 (1996).

Ion B e a m Inertial Fusion R. Bock Gesellschaft fiir Schwerionenforschung, D-64291 Darmstadt, Germany

1

Introduction

The development of thermonuclear fusion to a future energy source is one of the outstanding objectives of present research, its realization one of the great challenges for our scientific community. Consequently, in the present situation where crucial problems of the confinement concepts are not solved yet, all possible options still have to be taken into consideration. The research programs of the United States and of Japan, for example, cover both alternatives of fusion energy research, magnetic and inertial confinement, whereas the European effort is concentrated exclusively to magnetic confinement. Magnetic confinement fusion (MCF), having been studied with an enormous world-wide effort for nearly four decades, is obviously the more advanced concept as compared to inertial confinement (ICF) with respect to its technological and conceptual achievements. With respect to power generation, however, each of the two concepts have their specific advantages and will exhibit their specific problems. It is this long-term perspective, the potential for an economical and environmentally attractive power generation, which requires a continued investigation of both concepts and an evaluation of the two approaches on a comparable basis. For this reason inertial confinement fusion deserves more systematic investigation. Three different driver options have been studied during the last two decades for inertial confinement: Laser beams, light ion and heavy ion beams. The specific advantage of ion beams is based on two outstanding features: (a) the coupling of ion beams to the target is well understood and exhibits a 'classical' behavior, the deposition of the ion energy on the target is nearly 100% and is not impaired by the plasma or any other medium, such as magnetic fields, surrounding the target. (b) the efficiency of heavy ion and light ion drivers is high (around 25%) and the repetition rate for heavy ion accelerators is excellent. Both properties are the key issues for energy generation, which - in the case of the heavy ion accelerator- fulfill the requirements of a power plant already now. The high pulse intensities and the short pulse lengths requested for ignition have not been achieved yet with ion beams. This is a matter of technical development and needs further research. For the heavy ion accelerator [1,2], extrapolations from the operation of existing large facilities indicate that the

212

R. Bock

specifications for a reactor driver can be reached. Moreover, a large experienced accelerator community gives confidence in a competent handling of the necessary development programs. For light ion beams [3] the development of pulsed-power technology has made considerable progress during the last decade and the achievements concerning specific deposition power are promising. Some key issues, however, in particular the high repetition rate of light-ion diodes, are not solved yet. According to our present understanding, the various driver options for inertial confinement fusion can be characterized as follows: Based on the enormous effort for the development of laser technology, ignition will, most probably, first be demonstrated with the powerful single-shot Nd-glass laser facilities now under construction, the National Ignition Facility (NIF) in the US and the Mega-Joule Facility in France. They will, however, not meet the requirements for a reactor driver with respect to repetition rate and efficiency. Whether the KrF gas laser will meet these conditions is doubtful. Light ion beam facilities have achieved high deposition power and will provide significant results on beam-target interaction. Both facilities, laser and light ion beams are necessary and indispensable for the fundamental investigations on target performance for both, directly and indirectly driven targets, independent of their reactor-driver capability. The heavy ion accelerator, however, with its excellent repetition rate and high efficiency offers the superior prospects for a reactor driver.

2

The Physics of the Target

Energy generation by inertial confinement fusion is based on the following concept [4,5]: The deuterium-tritium (DT) fuel enclosed in a small spherical shell of some millimeter radius, the 'pellet' or 'target', is compressed isentropically by ablation of the shell and heated up to ignition conditions. The energy necessary for this procedure is supplied by short and adequately shaped pulses of intense laser or particle beams. As in magnetic confinement (MCF), the ignition temperature of about 100 million degrees has to be reached and the Lawson Criterion has to be fulfilled in order to achieve substantial burn. Different from MCF, the confinement time which can be obtained by the inertia of the imploding matter is extremely short, in the sub-nanosecond time scale, so the fuel density at ignition needs to be correspondingly higher by many orders of magnitude. The Lawson Criterion for a shock compressed sphere can be expressed by the relation Pi " Ri > 3 g / c m 2, where Pi and R i designate fuel density and fuel radius at ignition. Consequently, the fuel mass in the pellet and its density to be reached at ignition are inversely related: The smaller the mass of the fuel, the higher is the compression needed. For a pellet explosion to be handled in a reactor vessel, the fuel mass is limited for mechanical reasons to an order of some milligrams, and the corresponding fuel density to be achieved at ignition is about 103 . The beam intensity

Ion Beam Inertial Fusion a) Direct Drive

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b) Indirect Drive

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Fig. 1. Principle of inertial confinement. (a) Direct drive: A hollow sphere of 5-10 mm diameter, filled with cryogenic DT fuel, is heated by heavy ion beams and evaporates material by ablation. By the evaporation a high radial pressure is produced by which the fuel is compressed and finally heated up to ignition temperature by shock waves. (b) Indirect drive: The kinetic energy of the heavy ion beam is converted into electromagnetic radiation which is confined by the outer eIlipsoidal high-Z casing (hohlraum radiation) which compresses the inner shell.

required to about 6 MJ in less than as bismuth,

obtain such a compression by a direct heating of the pellet is and the implosion dynamics requires this pulse to be delivered 10 ns (direct drive, see Fig. la). For a beam of heavy ions, such this corresponds to 3 - 1018 ions per pulse.

In order to achieve an isotropic compression and to avoid dynamical instabilities, a high degree of azimuthal symmetry of the incident beam intensity is necessary. In case that the symmetry requirements can not be reached by direct drive with a reasonable number of beams, another concept, indirect drive, has been proposed and is at present widely investigated by computer simulations. In this case the pellet is enclosed in a larger casing of high-Z material with two converters on opposite sides (Fig. lb). The kinetic energy of the heavy ion pulse is deposited in the converters which radiate, according to Stefan-Boltzmann's law, soft x-rays into the casing. By the arrangement of converters and shields inside the casing, this radiation which drives the implosion can be made very isotropic. At a specific deposition power of 1016 W / g the conversion efficiency of the kinetic ion energy into electromagnetic radiation is predicted to be about 90%, and the hohlraum radiation inside the casing is reaching a temperature of 300 eV, sufficient to drive the implosion. Research in Europe on key issues of target design and target dynamics [4-7], such as Rayleigh-Taylor instabilities, radiation symmetrization, conversion of beam energy into radiation, radiation confinement and beam-target interaction, has been carried out by theory groups at Frascati [4], Frankfurt [6], Garching [5] and Madrid [7] with significant results. Recently, Russian groups from Moscow and Arzamas [8] with a long-lasting experience in

214

R. Bock

joined these activities. Experimentally, some work on beam-target interaction at a low power level is going on at the Gesellschaft fiir Schwerionenforschung (GSI), Darmstadt, with heavy ion beams [9] and at the Forschungszentrum (FZ) Karlsruhe with light ions beams [3]. A group of the Max-Planck Institut fiir Quantenoptik (MPQ) in Garching is participating in experiments at the high-power laser facilities in Japan [10] and in France [11]. Japan, with the 30kJ laser facility G E K K O XII in Osaka, has strong activities in this field, and the record of fuel compression, 600 times normal fluid density of hydrogen, has been achieved at this facility some years ago [12]. The main activities in the field of target physics, however, are located outside Europe, partially in classified areas. Most of these - previously classified - results achieved in the US at the high-power laser facilities and with classified computer programs are now becoming published, after the new declassification policy, decided in 1993 by the US government, is effective. Target physics evolved to an interesting world-wide activity with new ideas and concepts for both fusion and basic research. A new ignition concept was suggested recently, the fast ignitor [13], and is presently pursued with great enthusiasm [14]. Different from the usual concept in which the ignition spark is created by shock waves in the very center of the fuel, fast ignition is achieved by a picosecond high power laser pulse (> 1018 W / c m 2) which produces a relativistic electron beam strongly focused in forward direction

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Ion Beam Inertial Fusion

215

and drilling a hole through the pre-compressed fuel. Compression of the target is attained by the heavy ion pulse, ignition by the laser pulse. The most important quantity of the DT-filled target is its gain, the ratio of the produced fusion energy divided by the energy of the beam pulse. According to simulations with hydrodynamic codes [15], ignition and breakeven of an indirectly driven pellet is predicted at a beam pulse energy of about 1-2 MJ. A gain of 100, the working regime for a reactor, needs a pulse energy of about 6 M J, as exhibited in Fig. 2. Existing driver facilities are still far below this energy. With the next generation of laser facilities now under construction, e.g. the National Ignition Facility (NIF), a Nd-glass laser in Livermore with 192 beams and designed to reach 2 M J/pulse, it is the goal to demonstrate ignition. Apart from its key role for energy generation, the target is a fascinating object of basic research. The physics of matter at extreme pressures, densities and temperatures can be studied - in regimes not accessible otherwise - thus opening an exciting perspective for future research on dense and non-ideal plasmas.

3

Reactor

and

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The ICF power plant has some intriguing features. The clear separation between the reactor chamber and the reactor driver and the absence of large installations for high magnetic fields as in Tokamaks facilitate greatly the design of the reactor chamber, its operation and maintenance. The realization of a liquid protective wall for the first structural wall of the reactor chamber is one of the great advantages of an ICF reactor concept. This protective wall consists of FLiBe or LilTPbs3, an eutectic alloy with extremely low vapor pressure, which allows a high repetition rate of about 5 Hz for a reactor chamber. Systems studies carried out in the early stage of our research program in 1980/81 by KfK Karlsruhe, GSI Darmstadt, MPQ Garching, Giessen University and the University of Wisconsin resulted in a conceptual design study for a power reactor, HIBALL [16], driven by an if-linear accelerator with storage rings. The main goal of this study was to demonstrate the feasibility of such a concept and, in particular, to show whether a heavy ion accelerator can meet the technical and economical requirements of a power reactor. It was based on 20 beams of Bi +, an ion energy of 10 GeV (50 MeV/nucleon) and a pulse energy of 5 MJ. An improved design, HIBALL II, published in 1985 [17], is a concept consisting of 4 reactor chambers operated by the same driver accelerator. With this design a power of 3.6 GW~ was achieved at a reasonable cost level for the produced electricity. In subsequent investigations carried out in the US and in Japan several heavy ion driven reactor concepts have been studied for the two existing types of heavy ion driver accelerators, the induction linac and the rf-linac. They are beyond the scope of this paper. Two new studies, however, carried

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out under the auspices of the US Department of Energy and published in 1992, should be mentioned: OSIRIS [18], and Prometheus H [19], two 1 GWe power plant designs with an induction linac driver. The reactor cavity design of OSIRIS (Fig. 3), for example, has some attractive technological, environmental and safety features. For the first wall and the support structures of the chamber low activation carbon fabric material is used. The reduction of the radioactive inventory after shutdown by orders of magnitude as compared to fission reactors (Fig. 4), but also much better than magnetic fusion designs, is one of the greatest advantages of ICF reactors. The relative cost of the naain components of the OSIRIS plant (i.e. reactor chamber 32%, driver 37%, conventional equipment 22%) are similar to those of HIBALL II. Based on the estimated total cost of 3.1 G$ the resulting cost of electricity is about the same as for other ICF plants and slightly better than for the Tokamak designs. HIBALL fits favorably into these numbers (Fig. 5). For a light-ion driven reactor concept systems studies were carried out in a collaboration between FZ Karlsruhe and the University of Wisconsin. The LIBRA design and improved modifications LIBRA-LiTE and LIBRA-SP [20] have the advantage of a simple modular design, of low cost and of small size reactor units (1 GWe or below). The three design concepts differ mainly by the beam injection concept, which is one of the serious problem areas of a light ion reactor: Channel transport for LIBRA, ballistic focusing for LIBRALiTE and self pinched transport for the SP version. Other problems to be

Ion Beam Inertial Fusion 1011

Fig. 4. Radioactive inventory (30 years operation) as a function of time after shutdown for an ICF reactor (Cascade, LLNL). Comparison with a fission reactor shows that the radioactivity for the ICF reactor (including target materiM and tritium) is smaller by more than a factor of 100.

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solved are the repetition rate of light ion diodes and the pulse shaping of beams. Attractive are the safety features and, compared to heavy ions, the low cost of the driver. In conclusion, existing conceptual design studies of ion inertial fusion power plants exhibit a promising perspective. The separation between driver and reactor chamber allows an optimization of both m a j o r components of an ICF power plant, resulting in the realization of advanced technical concepts with specific advanced low-activation materials and with an easy maintenance. In present design studies, m a n y features remain preliminary, because they can not be investigated on an adequate level of funding. In future research programs this has to be an area of increased activities. 0

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4 4.1

T h e H e a v y I o n A c c e l e r a t o r as I C F D r i v e r General Remarks

In a worldwide frame, two different types of heavy ion driver concepts are investigated. Whereas in the US the induction accelerator [21] is pursued, both a linear and a re-circulator mode [22], research in Europe is concentrating on the combination of the rf linear accelerator with storage rings [1, 23]. Both activities are complementary. At the present stage of modest expenses, research on both concepts should be continued until a clear advantage of one or the other will become evident. The induction linac is a single-pass accelerator with parallel beams through all the induction modules (64 beams conceived for the driver). At the frontend electrostatic focusing, for the rest of the accelerator magnetic focusing is considered (Fig. 6). Research on the induction linac is carried out at the Lawrence Berkeley National Laboratory (LBNL) and the Lawrence Livermore National Laboratory (LLNL), USA. The European approach is based on the tradition of more than 40 years of accelerator research and development for nuclear and particle physics. A large accelerator community is involved in the research and in operations of such facilities. High-current acceleration is one of the main directions of ongoing accelerator research, and the heavy ion driver is one of potential applications. Modern particle accelerator facilities consist of a combination of different accelerating and beam handling modules: linear accelerators, synchrotrons and storage rings, combined with sophisticated diagnostics for beam manipulations and beam control. In principle, most of the components of a driver accelerator already exist, and a wealth of advanced driver technology, such as ultra-high vacuum technology, superconductivity, fast kickers, beam control etc. has been developed for existing machines. The operations of such facilities have achieved a high standard for long-term stability and reliability, and are based on the experience of a large community of physicists and

Ion Beam Inertial Fusion

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technicians. The technical specification of a fusion driver accelerator is partially beyond the achievements reached so far, because highest beam quality is required. High currents and ultra-short beam pulses with excellent phase space density are necessary in order to satisfy the needs for the ignition of the pellet. Progress in the development of specific components has been made during the last years, but much more effort is necessary to reach these goals. 4.2

The rf Linac/Storage Ring Concept

During the last decade various concepts of reactor drivers were investigated. The basic structure is about the same (Fig. 7): Acceleration is achieved by a linear accelerator consisting of various types of if-structures, such as rfquadrupole (RFQ), WiderSe, Interdigital-H (IH) and Alvarez structures, delivering a continuous beam of singly charged Bi + ions of between 100 and 400 mA with an energy of 10 GeV (50 MeV/nucleon). The linac beam is injected into a system of storage rings, between 10 to 20 turns each, in order to achieve the necessary current multiplication and the formation of the required bunch structure. Because of space charge limits at low ion velocities, the front-end of the linac consists of 16 parallel channels which are combined by successive funneling with frequency doubling at each step, into a single beam. The rest of the linae, its major part up to a length of about 5 kin, consist of Alvarez structures. New if-structures, such as the RFQ and IH structures were developed in recent years for high-current acceleration, and satisfy the requirement of the front-end part of a driver accelerator. Ion sources for singly charged very heavy ions with the required specification as to current and emittance have been developed at GSI and elsewhere.

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The current multiplication of 105 , necessary from the source to the final bunch at the target will be achieved by different techniques: (1) funneling at the front-end, (2) multi-turn injection into the storage rings, (3) bunch merging and (4) final bunching. During acceleration and all beam manipulations, constraints on emittance growth and momentum spread determine the crucial design parameters for the driver facility [24] (Fig. 8). In addition, since the beams in the storage rings are space charge dominated, there is a limitation of storage time in order to avoid instabilities, particularly the longitudinal micro-wave instability. Theoretically, these processes and their influence on beam dynamics and beam quality are well recognized and they are subject of many investigations by computer simulations. Specific experimental investigations on space charge dominated beams are in progress at existing accelerators [25], in particular, at the GSI two-ring facility consisting of a heavy ion synchrotron (SIS) and a storage and cooler ring (ESR), (Fig. 9). According to our present knowledge on beam dynamics and accelerator technology the driver concepts so far envisaged meet the required specification of 6 MJ on the target, for both direct and indirect drive. In case of indirect drive, however, more stringent conditions are requested for focusing and bunching in order to reach the specific deposition power of 1016 W / g in the converter. If further improvement of beam quality should be necessary, non-Liouvillean techniques have to be applied, such as laser cooling in the storage rings [27], telescoping of successive buncher, and the change of the charge state at injection into the storage rings by photoionisation with laser techniques [28].

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Fig. 9. Layout of the GSI Heavy Ion Facility, a fast-cycling synchrotron of 18 Tesla-meter (SIS) with UNILAC as an injector, and a storage and cooler ring (ESR). Intensity upgrade of the UNILAC to be in operation by 1999 will deliver an increase by a factor of 100 for the beam intensity of the heaviest ions [26]. 4.3

High Energy Exp. Areo

K e y Issues and P r e s e n t A c c e l e r a t o r R e s e a r c h

The present driver concept is based on the experience with existing accelerators and on computer simulation of beam dynamics at high intensity and high phase space density. Many problems have been studied during the last decade, some of them have made substantial progress or are considered to be solved. Among the remaining problems a number of key issues need further investigations, such as: high-current performance of linac structures emittanee growth by funneling emittance growth by multi-turn injection - instabilities in storage rings with space charge dominated beams, in particular the longitudinal microwave instability fast bunching and resonance crossing in storage rings fast kickers - beam losses, in particular at injection and extraction - final focusing and repulsive forces between beamlets near the target

-

-

-

-

-

Some of them are under investigation at various existing accelerators, mainly at GSI, at CERN, at Frankfurt University and at Rutherford Appleton Laboratory. An experimental program for the systematic investigation of some of these issues is in progress at GSI. These investigations will be intensified

222

R. Bock

in 1999 after completion of the high-current injector facility now under construction, which will increase the intensity of the heaviest ions by a factor of hundred [26]. Some preliminary results of ongoing experiments on driver relevant issues are quite remarkable and shall be briefly mentioned: - Experiments at the ESR in Darmstadt as well as at the T S R storage ring at the Max Planck Institut (MPI) in Heidelberg and at the Low Energy Antiproton Ring (LEAR) of CERN have shown that beams in storage rings remain stable up to a factor of 10 beyond presently assumed stability limits (Keil-Schnell limit). Bunched beams, now under investigation, are stabilized by the tails of their Landau distribution. - For the fast crossing of an integer resonance in a ring due to the increase of space charge it was demonstrated at the CERN proton synchrotron that only a small increase of emittance occurs. Experiments at GSI have reached 5-101° Ne 5+ ions equal to 50 Joule. Extrapolation to Bi + give evidence that design parameters for a driver accelerator can be reached. - Other experimental activities with high-intensity beams are progressing at GSI in the field of plasma physics [9], in particular measurements of the stopping power and of charge-exchange cross sections for heavy ions in plasma, showing a considerable increase as compared to the cold gas. The most specific feature for heavy ion beams will be the possibility of generating dense plasmas by heating an extended volume of dense matter in a well defined geometry. The investigation of these plasmas, their expansion, cooling and decay, transport properties, measurement of opacities and equation of state, predicted phase transitions to metallic states (metallic hydrogen), are interesting research objectives, with respect to inertial fusion as well as to astrophysical applications. 4.4

A S t u d y G r o u p 'Heavy Ion Ignition Facility'

a) A c t i v i t i e s a n d P a r t i c i p a t i o n . Results in heavy ion inertial fusion research during the last decade, in particular the progress in accelerator technology, have greatly increased our confidence, that the heavy ion accelerator is the superior choice among the driver candidates for a power reactor. In the early Nineties the European Inertial Fusion Community had a series of meetings and workshops in which these achievements and the future strategy and prospects were discussed. It was realized that - after more than a decade of exploratory research in Europe - it is now timely to establish a coherent European program. In a final workshop at CERN in the middle of 1993 with participants from several European countries, the concept of a dedicated facility aiming at ignition was defined as the next logical step. In a proposal submitted to the European Union [29] a Study Group was proposed to elaborate a preliminary design of such a facility. It was submitted to the European Union fusion authorities by ENEA Frascati, DENIM

Ion Beam Inertial Fusion 200

1

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Madrid, FZ Karlsruhe and GSI Darmstadt. Other research institutes, in particular CERN, Rutherford Appleton Laboratory, MPQ Garching and several university institutes from Germany and France participate with their special knowledge in one of the related research areas in this program. b) Scientific Goals. The Study Group started working in March 1995 and had a series of workshops, mainly on driver issues. The study is concerned with the critical issues of the design of the heavy ion driver, of targets and the means of their production, and of the required reaction chamber with the goal to develop a coherent set of parameters. According to present knowledge ignition would require a pulse energy of about 2 MJ [15] (Fig. 10), delivered on a target of a few mm diameter within about 6 ns, the specific deposition power being of the order of 104 TW/g. For the suggested driver, a 6 - 10 GeV linac and a number of storage rings, scenarios and sets of parameters have been discussed in recent workshops. During the last year scenarios for the injector linac, injection into the storage rings, the final bunch compression and bunch synchronization and the final transport to the reaction chamber were developed. Bunch telescoping and laser cooling in the longitudinal phase space have been proposed as nonLiouvillean techniques for the improvement of beam quality. In the field of target physics, the implosion symmetry and hydrodynamic instabilities represent the key issues for igniting targets. Indirect drive which is accepted as the most appropriate approach to heavy ion inertial fusion, relies on radiation symmetrization inside the target casing to ensure spherical implosion of the fusion pellet. Extensive numerical studies are being carried out [4-6] in order to determine the parameters necessary for the accelerator design. New target designs for an ignition driver have been proposed by the Russian group [8]. It is our understanding that the proposed study shall result in a feasibility report for the next logical step on the route to a driver facility. Whether the goal to achieve ignition can be realized in one step or whether an intermediate step is necessary and useful, depends of the result of the study, especially as

224

R. Bock

the accelerator has a larger potential for the investigation of a number of ICF related issues. The facility to be built should be an optimum choice for addressing the various aspects of an ICF development plan. Obviously, it is the great advantage of the heavy ion approach to fusion that the accelerator technology of an ignition driver is identical to that envisaged for a final power reactor. Heavy ion beams, therefore, offer a direct route towards fusion energy production. 5 5.1

Inertial

Confinement

with

Light

Ion Beams

General Remarks

Since the stopping power for protons is smaller by about a factor of 103 as compared to the very heavy ions discussed before, their energy must be smaller by the same factor in order to meet the target requirements. Consequently, the reduced kinetic energy of the ions has to be compensated by an increase of beam current by the same factor. Therefore, light ion beam currents need to be as high as mega-amperes in order to reach the pulse energy of 6 MJ necessary for ignition. Beam intensities of this order of magnitude can not be handled with the accelerator technology described before. Concept and technology of the light ion driver is completely different from heavy ion accelerator techniques. Light ion beam devices for high currents consist of two main components: (1) The energy is provided by a pulsed-power device, an electrical capacitor array (Marx generator) with a pulse forming line, delivering a short electrical pulse to (2) a diode in which the proton or lithium beam of very high intensity is produced and focused (Fig. 11). This beam pulse is transported to and injected into the reactor chamber to the target. Recent research is concentrating on improvements of diode performance, in particular to ions heavier than protons and at higher voltage. As mentioned before, ongoing research is done using single-shot devices at the Sandia Laboratories in the USA, e.g. with the powerful Particle Beam Fusion Accelerator PBFA II [30], a multi-beam facility, and in Europe with the Light Ion Facility KALIF at the Karlsruhe Research Center [3]. Upgrades to more powerful installations are in progress at both laboratories. The most attractive features of this technology with respect to the inertial fusion application are their high e~ciency and low specific cost. The problems are located mainly in the diode performance, its beam quality and its repetition rate capability, and in the beam transport to the target. Repetitive operation of pulsed-power generators is supposed to be achievable with available technology. 5.2

Present Research: Achievements and Problem Areas

Without going into any technical details, some ideas about achievements and directions of present research shall be given.

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Considerable progress has been made in the understanding of diode physics, both by sophisticated diagnostics and by the application of 3-D simulation codes. The beam divergence is one of the key issues which need further investigations. The achieved value of 17 mrad needs to be reduced to about 10 for an ignition facility and to 5 for a reactor driver. The main sources of beam divergence were identified to be caused by inhomogeneities of the anode plasma and by instabilities of the free electron sheath at the virtual cathode. Two-stage diodes now under consideration may improve existing deficiencies. Great progress is also achieved with the diode voltage and the power density in the target. With a lithium current of 1 MA as high as 10 MV have been reached at Sandia. With these performance parameters a specific power deposition of 1000 T W / g has been achieved in a target, resulting in a

226

R. Bock

plasma temperature of 65 eV, to be compared to 300 eV needed for ignition. With voltage generators of the 1 TW class, such as KALIF in Karlsruhe, 200 T W / g have been reached, an order of magnitude which is already of interest for target investigations. One of the problem areas is the final beam transport. Several transport schemes are considered for a reactor concept. Most promising are ballistic transport combined with solenoidal focusing and self-pinched transport. In the first scheme a background gas provides charge and current neutralization, a disadvantage being the location of the solenoidal lens rather close to the target. No transport device is required for the self-pinched transport scheme. Both schemes need further theoretical and experimental investigation, in particular the self-pinched transport, where little experimental work has been done so far. In conclusion, the light ion approach has the advantage, that already now considerable power densities in targets have been achieved with facilities in operation. They open significant opportunities for investigations in target physics, related to ICF as well as to basic research (Fig. 12). The field of pulsed-power technology is well advanced. The area of diode performance needs a continued research effort, with respect to the required intense lowdivergence beams and the repetition rate capability. Beam transport in the target chamber is another key issue which needs increased investigation.

6

Concluding

Remarks

and Outlook

The investigation of many issues of ion beam fusion, both with light and heavy ion beams is well in progress at several European laboratories. Funding in Europe, however, is by far not sufficient for a balanced program which addresses the key issues of ICF adequately. Light ion activities at the Karlsruhe Research Center have achieved remarkable results with respect to both a conceptual design study for a light ion beam reactor facility and experimental investigations on the application of pulsed-power technique to inertial fusion and to problems of materials research. Collaborations with US laboratories have achieved design studies on light ion beam reactor concepts which have some attractive features, but obviously some essential problems of such concepts are still far from being solved. A collaboration with Sandia Laboratories opens access to larger pulsed-power facilities dedicated to inertial confinement research. In the field of h e a v y ion b e a m s several European accelerator facilities open opportunities for the investigation of problems relevant to heavy ion inertial fusion. In particular, GSI with its intensity upgrading program will after its completion in 1999 - enable an interesting research program on specific driver related issues of beam handling and beam dynamics as well as on problems of plasma physics (Fig. 12). -

Ion Beam Inertial Fusion

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103

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On the route to the physics of dense plasmas, heavy ion beams are excellent tools for the generation of volume-heated plasmas at solid state density and above, and they provide a new technique for the investigation of properties of m a t t e r under extreme conditions (and are in some way complementary to those with high-power lasers): beam target interaction, processes in nonideal plasmas, the equation-of-state and hydrodynamics of dense plasmas, probably the discovery of phase transitions. After the injector upgrade at GSI, plasma temperatures of up to 10 eV are expected. A further reasonable step to higher energy density in the plasma with a new facility would be an increase of t e m p e r a t u r e by a factor of 10. At about 100 eV a new regime of phenomena would become accessible: radiation physics - conversion and transport phenomena, opacities - with its relevance to astrophysics and to

ICE" targets. On the route to energy, the European Study Group has started a new effort for a systematic approach to a driver facility and to the relevant problems of targets and systems. The concept of an ignition facility or, alternatively, a staged procedure to achieve this goal will be worked out. The heavy ion accelerator presently investigated by the Study Group is - specific to European accelerator expertise - based on an experienced scientific c o m m u n i t y and - based on a well developed European collaboration. Depending on the results of the study a coherent European research program could be established with a reasonable effort which would allow to place ICF adequately into the context of an international fusion energy strategy.

228

R. Bock

References [1] R.Bock, Status and Perspectives of Heavy Ion Inertial Fusion, Proceedings of the Internat. School of Physics 'Enrico Fermi', Varenna 1990, Course CXVI, p.425-447, North Holland, Amsterdam 1992; C.Rubbia, Proc. of the IAEA Techn.Comm.Meeting on Drivers ]or Inertial Confinement Fusion, (D.Banner and S.Nakai, Eds.), Osaka 1991, p.23; Nucl.Physics A553 (1992) 375-395; Nuovo Cim. 106A (1993) 1429-44; I.Hofmann in Advances of Accelerator Physics and Technology (H.Schopper,Ed.) World Scient.Publ.Co., Singapore 1995, p.348-61; R.Bock, Proc. of the EPS Conference on Large Facilities in Physics, Lausanne 1994 (M.Jacob and H.Schopper, Eds.) World Scient. Publ.Co., Singapore 1995, p.348-61 [2] R.O.Bangerter and RM.Bock in Energy from Inertial Fusion, IAEA, Wien 1995 p.111-135 [3] H.Bluhm and G.Kessler in Physics o] Intense Light Ion Beams, Annual Report 1995 FZKA 5840 (H.Bluhm, Ed.) [4] S.Atzeni, Proc.of the Internat.Symposium on Heavy Ion Inertial Fusion, Frascati 1993, (S.Atzeni and R.A.Ricci,Eds.) Nuovo Cim. 106A (1993) 1429-1995; S.Atzeni in Physics with Multiply Charged Ions, (D.Liesen,Ed.) Plenum Press, New York 1995, p.319-356; Fus.Eng.Design 32/33 (1996) 61-71 [5] J.Meyer-ter-Vehn, The Physics of Inertial Fusion, Proc. of the Internat. School of Physics 'Enrico Fermi', Vareima 1990, Course CXVI, North Holland, Amsterdam 1992, p.395-423; M.Murakami and J.Meyer-terVehn, Nucl.Fusion 31(1991)1315 and 1333; J.Meyer-ter-Vehn, J.Ramirez and R.Ramis, Proc. of the Symp. on Heavy Ion Inertial Fusion, Princeton 1995, Fus.Eng.Design 32/33 (1996) 585; A.M.Oparin, S.I.Anisimov and J.Meyerter-Vehn, Nucl.Fusion 36 (1996) 443-452 [6] K.J.Lutz, J.A.Maruhn, R.C.Arnold, Nuc]. Fusion 32 (1992) 1609; K.H.Kang, K.J.Lutz, N.A.Tahir and J.A.Maruhn, Nucl.Fusion 33 (1993) 17 [7] J.M.Martinez-Val et al., Nuovo Cim.106A (1993) 1873; G.Velarde et al., Particle Accel.37 (1992) 537-542 [8] Yu.A.Romanov, Nuovo Cim 106A (1993) 1913; M.Basko, Nucl.Fusion 33 (1993) 615 and Phys.Plasmas 3 (1996) 4148; Yu.A.Romanov and V.V.Vatulin, Fus.Eng.Design 32/33 (1996) 87-91; V.V.Vatulin et at., Fus.Eng.Design 32/33 (1996) 603 and 609; M.M.Basko, M.D.Churazov and D.G.Koshkarev, Fus.Eng.Design 32/33 (1996) 73-85 [9] D.H.H.Hoffmann et al., Phys.Rev.Letters 65 (1990) 2007; 66 (1991) 1705; 69 (1992) 3623; 74 (1995) 1550; M.Stetter et al., Fus.Eng.Design 32/33 (1996) 503; M.Dornik et al., Fus.Eng.Design 32/33 (1996) 511; R.Bock (Ed.), Annual Reports High Energy Density in Matter produced by Heavy Ion beams 1985-95, GSI Reports, in particular GSI-95-06 and GS1-96-02 [10] R.Sigel et at., Phys.Rev.Letters 65 (1990) 587; T.Loewer et al. Phys.Rev.Letters 72 (1994) 3186 [11] D.Batani, M.Koenig, T.Loewer, A.Benuzzi and S.Bossi, Europhysics News 27 (1996) 210 [12] C.Yamanaka, Proc. of the IAEA Techn.Comm.Meeting on Drivers ]or Inertial Confinement Fusion (D.Banner and S.Nakai,Eds.), Osaka 1992, p.1

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229

[13] M.Tabak et al., Phys.Plasmas 1 (1994) 1626-34; A.Caruso, Proc.of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion, (J.Coutant, Ed.), Limeil-Valenton 1995, p.325-39 [14] A.Pukhov and J.Meyer-ter-Vehn, Phys.Rev.Letters 76 (1996) 3975 [15] J.D.Lindl, Nuovo Cim. 106A (1993) 1467-34 [16] B.Badger et al., HIBALL, A Conceptual Heavy Ion Beam Driven Fusion Reactor Study, KfK 3202/VWFDM-450 (1981); D.Boehne et al., Nucl.Eng.Design 72 (1982) 195 [17] HIBALL II, An Improved Conceptual Heavy Ion Beam Driven Fusion Reactor Study, KfK-3840/FPA 84-4/VWFDM-625 (1985) [18] W.R.Meier et al., OSIRIS and SOMBRERO Inertial Confinement Fusion Power Plant Designs, WJSA-92-01 and DOE/ER[54100-1 (2 Vols.), 1992 [19] L.M.Waganer et al., Prometheus L and Prometheus H Inertial Fusion Energy Reactor Design Studies, DOE/ER-54101 and MDC 92E0008 (3 Vols), 1992 [20] G.L.Kulcinski et al., Proceedings of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion, (J.Coutant,Ed.), Paris 1994, p.49-56 [21] R.O.Bangerter, Nuovo Cim. 106A (1993) 1445; R.O.Bangerter, Fus.Eng.Design 32/33 (1996) 27-32 [22] A.Friedman et al., Fus.Eng.Design 32/33 (1996) 235-46; J.J.Barnard et al., Fus.Eng.Design 32/33 (1996) 247-58 [23] I.Hofmarm, Proc.of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion Osaka 1992 [24] I.Hofmann, Nuovo Cim.1O6A (1993) 1457; Fus.Eng.Design 32/33 (1996) 33; Proc. 5th Europ.Part.Accel.Conf. (EPAC'96),Sitges 1996, p.255; I.Hofmann et al. Proc. of the 15th Internat.Conf on Plasma Physics and Controlled Fusion (Seville 1994), IAEA Vienna 1995, Vo.2, p.709-14 [25] U.Oeftiger and I.Hofmann, Fus.Eng.Design 32/33 (1996) 365-70; U.Oeftiger, I.Hofmann and P.Moritz, Proc.of the 5th European Particle Accel.Conf, (EPAC'96) p.1099 [26] B.Franzke, Proc.of the 3rd European Particle Accel.Conf. (EPAC'92) Berlin 1992 p.367-71; K.Blasche und B.Franzke, Proc.of the 4th European Particle Accel.Conf. (EPAC'94) London 1994; N.Angert, Status and Development of the GSI Accelerator Facility Proc.of the 5th European Particle Accel.Conf. (EPAC'96), Sitges 1996, p.125 [27] J.S.Hangst et al. Phys.Rev.Letters 74 (1995) 4432; D.Habs and R.Grimm, Annual Rev.Nucl.Part.Sci. 45 (1995) 391 [28] C.Rubbia, Nuclear Inst. Meth. A273 (1989) 253-265; C.Rubbia, Proc.of the 3rd European Particle Accel.Conf. (EPAC'92) Berlin 1992 p.35 [29] A Study Group 'Heavy Ion Ignition Facility' GSI-Report 95-03 (1995) Proposal to the Commission of the European Union 1994 (G.Plass, Project Manager) [30] A.B.Filuk et aJ., Proc. of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion (J.Coutant, Ed.), Limeil-Valenton 1995, p.233

Spin-echo E x p e r i m e n t s with N e u t r o n s and with Atomic Beams Christian Schmidt and Dirk Dubbers Physikalisches Institut der Universit~it Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

1

Introduction

Spin-echo spectroscopy was invented in the fifties [1] as a new tool in nuclear magnetic resonance (NMR). Over the years, NMR spin-echo has steadily evolved towards the powerful multiple pulse, multiple frequency, and magnetic gradient NMR-techniques now in standard use in physics and chemistry to probe the various couplings of the resonating nuclei to their atomic neighbours in real space. "In-flight" spin-echo was developed as a high-resolution method in polarized neutron scattering during the seventies [2]. It turned out to be a powerful tool for the investigation of the entangled molecular movements within complex materials. Although conceptually similar to NMR spin-echo, the information content of neutron spin-echo (NSE) is quite different. In-flight spin-echo, too, has much evolved in recent years. The technical aspects of this evolution is well covered by the existing literature. Still, the field is not easily accessible, due to its mathematical intricacies. The present survey tries to fill this gap by discussing the various ramifications of in-flight spin-echo, using only a small number of physical arguments. Finally, an astonishingly simple quantitative description of the spin-echo technique as a magnetic birefringence phenomenon will be presented. In many scattering experiments in physics, the energy and momentum transfer onto the sample is measured via the change in energy and momentum of the scattered beam. In inelastic neutron scattering, the energy state selection of the beam before and after scattering is usually done by Bragg reflection in three-axes spectrometers, or by time-of-flight measurements in pulsed neutron machines. The momentum state selection is accomplished by the choice of the scattering angle. All these scattering methods have in common that, due to the necessity of state selection, high resolution and high intensities are mutually exclusive. What is learned from a conventional neutron scattering experiment? The most important information on the system under study is given by the correlation function G(r, t), which gives the probability that a particle is found at position r at time t when there was a particle at position r = 0 at time t = 0. This correlation function covers all cases of interest, from the structure of ordered crystals or disordered liquids to the study of diffusion and of phonon and other

232

Christian Schmidt and Dirk Dubbers

excitations. (The description via G(r, t) however, is not complete when higher order correlations which depend on the position of more than two particles come into play.) The overall result of neutron scattering theory is that, for a given energy transfer hw and momentum transfer hq onto the sample, the intensity S of the scattered neutrons is the Fourier transform of the correlation function S(q,w) =

2

space-time Fourier transform of G(r, t).

(1)

Classical Neutron Spin-Echo (NSE)

In-beam spin-echo has the specific feature that it allows measurements with very high resolution but with no penalty in intensity. In its simplest possible configuration (which is only rarely realized), spin-echo works as following: Neutrons, for example, in a spin-polarized beam are scattered from a sample into a given direction and are detected after spin analysis. Along the incoming beam, a magnetic field of typically up to B0 = 50 m T is applied, over a typical beam length of 2 m. In this field, transversally polarized cold neutrons of average velocity 500 ms -1 make about 6000 revolutions with Larmor precession frequency WL. As the neutrons travel with various velocities, their transverse polarization is lost after a few mm of flight within the magnetic field. Along the outgoing beam, a magnetic field of equal size, but pointing into a direction opposite to the direction of the first field, is applied. In this second field, the neutron spins will turn back by full - 6 0 0 0 revolutions; that is, at the exit of this second field the neutron polarization is fully recovered, giving what is called a spin-echo signal. The full spin-echo signal, however, is only recovered if the neutron velocity after scattering is the same as before scattering. If, after scattering, the velocity decreases by as little as 10 -5, then the polarization of the outgoing beam is off by an angle of ¢ ~ 20 °, which is easily detectable in a polarization analyser. This explains the extreme sensitivity of spin-echo spectrometers to changes in neutron energy as small as a few neV. In comparison, time-of-flight or three-axes neutron scattering instruments have a resolution of only 0.1 to 1 meV. As all neutrons contribute to the spin-echo signal, independent of their velocity, one does not have to pay a penalty in intensity due to excessive state selection. What is the connection between this spin-echo signal and the observables of condensed matter? During scattering on the sample, the neutrons create or absorb an internal excitation of energy hco, and therefore leave the sample with a higher or a lower energy ("up-scattering" or "down-scattering"). The relation between the physical quantity ~ and the response of the apparatus then is simply: ¢ = wT.

(2)

Here, r is an instrumental quantity of dimension time, called the spin-echo time, given by Ernagn w -- E tTOF C( Bo, (3)

Spin-echo Experiments with Neutrons and with Atomic Beams

233

where tTOF is the time of flight of the neutron through the first arm of the 1 instrument (in our example: ~ 4 ms). Here Emagn _- ghWL = 60neV.B0(Tesla) is the neutron magnetic energy in the magnetic field B0, and E is the initial kinetic energy of the neutron, in our example about 1 meV. Hence, for cold neutrons, a typical spin-echo time is ~- = 20 ns. Relation (3) is derived for instance in [3]. An alternative and physically more appealing derivation is given in section 5 of the present article. In order that the relation ¢ = w~- be applicable, the width of the velocity distribution should not exceed about 20 %. In the following, the connection between the spin-echo signal and the physics of the sample under study will be given only for the simplest case of quasi-elastic neutron scattering, where the energy transfer hoJ is small compared to the quasiparticle excitation energies of the sample. In quasi-elastic scattering there is an equal number of up-scattered and of down-scattered neutrons, and therefore no overall shift of the spin-echo signal is expected. Instead, the size of the signal will decrease. When this decrease is measured as a function of spin-echo time 7-, (i.e. of magnetic field B0), and of scattering angle 0, then one obtains [3]: Spin-echo signal = transversal polarization P(~, B0) = "intermediate scattering function" I(q, T) = spatial Fourier transform of G(r, T) = frequency Fourier transform of

S(q, w).

(4)

The time dependence of the correlation function G of the system under study can thus directly be measured by simply varying the size of the magnetic field along the neutron beam. In contrast to the usual scattering methods, neutron spin-echo does not require a Fourier transform to obtain this time dependence. Instead, the spin-echo instrument performs the Fourier transform itself. Further, the method does not require excessive state-selection. In contrast to NMR spin-echo, in-beam spinecho provides no local probe in real space, but operates in momentum space of the reciprocal lattice. Therefore, by measuring the time dependence of the correlation function for various momentum transfers q, one is able to study the time response of the system separately on different scales of length. An example of a NSE measurement [4] is given in figure 2. The measurement shows the time dependence of the correlation function of a polyethylen-propylen copolymer. According to the reptation model of de Gennes, each chain moves in a "tube" defined by the positions of the neighbouring chains. At large momentum transfer (the lower curves in both plots), one observes the movement on a small scale, which is a fast lateral movement limited by the tube walls. At small momentum transfer (the upper curves of both plots) one sees the movement on a larger scale, which is a slow reptation movement along the tubes. Further, at higher temperatures one finds a larger effective diameter of the tubes. The solid lines in the plot are derived from a model calculation. A real NSE spectrometer is somewhat more complicated than described above. Usually, in both arms of the spectrometer, the magnetic fields point into the same direction in order to

234

Christian Schmidt and Dirk Dubbers

: d

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Fig. 1. Neutron spin-echo results on PEP copolymer, taken at two different temperatures. In each plot, the upper curves are taken at small momentum transfer and show the slow large-scale movement of the polymer chains along a "reptation" channel. The lower curves are taken at large momentum transfer and show the fast lateral small-scale movement of the chain between the channel walls (from [4]).

avoid a zero field region on the neutron trajectory in between. Then, to obtain the neutron spin-echo signal, the same tricks have to be applied as in N M R spin-echo. Before the neutron enters the large B0 field of the first arm of the spectrometer, its spin is turned by ~ in a small static magnetic field B1, which is applied at right angles to B0 over a short distance of the beam. Near the sample, another B1 field over twice the length (or twice as strong) as the first BI field "refocusses" the spins via a ~r-flip, much as in N M R spin-echo. At the end of the flight path, another ~ -flip is applied to the neutrons before they enter the analyser. Over the years, the NSE method has seen many sophisticated extensions [3]. Spin-echo methods have been developed for truly inelastic, incoherent, paramagnetic, ferromagnetic, and antiferromagnetic neutron scattering.

3

Zero-Field Neutron Spin-Echo (NRSE)

Several years ago an interesting proposal appeared [5] which claimed t h a t in a spin-echo experiment no B0 fields at all are necessary along the neutron beam, if the various static ~ - and ~r-flip coils are replaced by N M R flip coils. In [5], this claim was proved mathematically, and the proof will not be repeated here. Instead, two different explanations will be given which provide a better physical

Spin-echo Experiments with Neutrons and with Atomic Beams

235

understanding of the phenomenon. The first explanation is for the reader familiar with NMR, the second explanation is intended for the non-specialist. The principle of this neutron resonance or "zero-field" spin-echo (NRSE) can be derived directly from conventional NSE by going into the frame, attached to the neutron, which rotates at Larmor frequency CJL about the strong B0 field. In this rotating frame, the B0 field is simply transformed away and reappears, with opposite sign, at the position of the static ~- and 7r- flip coils. In the new frame the fields B1 in the flip coils wilt rotate with --COL,that is, the static flip coils have transformed into ordinary NMR flip coils. The spin-echo signal does not change under this transformation. Hence, NRSE is equivalent to NSE. This is the shortes possible proof, but it does not really enhance the basic understanding of the zerofield spin-echo trick. Therefore we give another, more simplistic explanation. NSE, basically, is a time-of-flight method. In everyday life, there are two methods to measure time of flight. One method is to watch one's wrist watch while flying from one place to another, and to read the time difference. This is the method of conventional NSE, where the neutron measures time via its personal Larmor precession clock. Another method is to look at the stationary airport clock at take-off, to remember this time of departure, and to look again at the airport clock upon arrival. One trusts that both clocks are in phase, and calculates the time difference. This is the method of NRSE: After the ~-flip about the momentary direction of the rotating B1 field, the neutron spin displays the time of departure on a dial which lies in the plane of B1. During its flight through the zero-field region, the neutron remembers this time of departure via the fixed direction of its spin. The B1 field of the second flip-coil rotates in phase with the field of the first flip-coil. Via the 7r-flip about the momentary direction of B1 in the second coil, the neutron records the time of arrival and calculates the time difference. Therefore, the neutron spin indicates the time of flight through the first spectrometer arm, in the same way as it does in conventional NSE. In the second arm of the spectrometer the polarization is then refocussed in the usual way. In this context, a word on the celebrated Ramsey method is in order. While, in conventional spin-echo, timing is done by the moving particle's clock, in resonance spin-echo timing is done via the stationary radio-frequency clock. In the Ramsey method, both kinds of time taking are combined, and the moving particle's private clock is compared with the stationary clock of the radio-frequency generator. In an atomic clock, this comparison is done to stabilize the radiofrequency generator with the atomic signal. When applied to basic physics, the comparison is done to search for small anomalies in the particle's clock, as is done for instance in the search for an electric dipole moment of the neutron. For NRSE, too, time has given birth to several extensions of the original method, see [6], and references therein. For instance, it was shown that when not three but four NMR flip coils are used, two 7r-flip coils in the first arm and two in the second arm, then the resolution of the instrument can be doubled for the same size of B0 in the NMR coils. When one uses not four single but four pairs of NMR coils and applies what is called the boot-strap trick [7], resolution

236

Christian Schmidt and Dirk Dubbers

can be doubled again. Still, a legitimate question is: If NRSE, in principle, is equivalent to NSE, why take the pain and invest in this new method? There are several distinct advantages to NRSE. Firstly, no large volume B0 fields are necessary. Instead, one only needs simple mu-metal tubes to shield the zero-field region. (On the other hand, it must be reminded that the small NMR coils used in NRSE required a considerable amount of development work.) Another advantage of NRSE is that the system is very insensitive to external perturbations coming, for instance, from moving steel constructions like cranes in an experimental hall. Further, in NRSE, with no Bo field along the main neutron path, there are no problems due to field inhomogeneities, which, in NSE, require fine tuning of numerous correction coils. The geometrical path length variations due to beam divergence, however, remain the same. Conventional NSE also is plagued by the fringe fields of the large coils, which must extend out to the sample position. When the angle between the ingoing and outgoing arms of the spectrometers is changed, the resulting fringe fields change as well and must be newly corrected. The main advantage of zero field spin-echo, however, is that the use of multidetectors for spin-echo work is easily possible. If there is no magnetic field then there is no preferred direction, and the spin-echo trick can be applied to all outgoing neutrons simultaneously, independent of their scattering angle. With conventional spin-echo this is not easily possible, as the field in the second arm usually singles out one scattering direction. While there exist several schemes to introduce large solid angle detection also in NSE, none has been successful yet. Another point in favour of NRSE may appear rather technical, but has important implications. In truly inelastic scattering, spin-echo can be used to measure the lifetimes of the excitations under study. However, a dispersion curve ca(q) usually has a non-zero slope, and a mediocre q-resolution will spoil the very high resolution in energy ha;. It has been shown [3] that when the direction of the B0 field is tilted away from the beam axis by a certain angle, then the dispersion curve can be crossed under right angles, and ca resolution decouples from q resolution. In NSE, to tilt the large volume B0 field is rather akward. In NRSE, to tilt the small NMR coils, on the other hand, is no problem at all. Tilted fields in NRSE can also profitably be used to do small angle elastic scattering [8]. Again, this method combines very high resolution, this time in momentum transfer q, with high overall intensities, as all incoming angles of a divergent beam can be used. This means that the spin-echo trick can be applied also to the momentum variables (instead of the energy variables), and then gives a direct measure of the spatial correlation function G(r). The first prototype of a zero-field neutron spin-echo instrument was developed in our group [9]. In this work we were able to draw on our experience with polarized neutrons in various static, rf, and zero-magnetic field configurations, used in experiments on Berry phases, dressed neutrons, neutron-antineutron oscillations and others, see also the review [10]. At present, two larger NRSE instruments have been developed and are being installed. One instrument was developed in Garehing [11] and is installed at the Orph6e reactor in Paris. The

Spin-echo Experiments with Neutrons and with Atomic Beams

237

other instrument was developed in Heidelberg and Garching [12] and is temporarily being installed at ILL, Grenoble. The two instruments were built for different purposes, each with its specific technical problems. In the GarchingParis installation, the main emphasis is to build a high-resolution instrument with tiltable coils, but only applicable to monodetector use. In the HeidelbergILL installation, the main emphasis is on a multideteetor system with a large angle of acceptance for each individual coil. As a first application, simple diffusion problems will be measured with our instrument, but later on the main interest is the study of phase transitions, see also next section.

4

Spin-Echo with Atomic Beams

In neutron spin-echo the bulk of a crystal is investigated as a whole, and the method is hardly applicable to surface problems. For surfaces, on the other hand, there exist many sophisticated methods to determine structures (tunnel microscopy, electron diffraction, and others), and also some methods for time dependent studies (mainly laser spectroscopy). What is really wanted, however, is a high resolution method for the combined study of time and space dependent processes on surfaces, in order to obtain the full correlation function G(r, t). It turns out that helium atomic beams can be for surface studies what neutrons are for bulk studies. Like neutrons, slow helium atoms have de Broglie wavelengths of atomic size, and, at the same time, kinetic energies comparable to the typical excitation energies of condensed matter. In fact, both helium diffraction and helium inelastic scattering from surfaces are well established fields now. Inelastic helium scattering [13] uses a chopped supersonic helium beam for time-of-flight measurements. Its energy resolution is limited to roughly 0.1 meV. If one wants to do spin-echo with a helium atomic beam, then one must use 3He, which has a nuclear spin one half. 3He spin-echo would permit an energy resolution orders of magnitude better than the energy resolution obtained with time-offlight methods. Recently, such a 3He spin-echo machine was developed in our group [14]. In this instrument, shown in figure 4, the simplest version of spin-echo is used, i.e. two magnetic fields of opposite sign are applied. A simple trick was used to prove the neV resolution capabilities: The whole apparatus was ramped by an angle of six degrees towards the horizontal, so that the 3He beam had to fly upwards in the earth's gravitational field. Hence, in the second coil, the 3He atoms had lost a kinetic energy of 33 neV, as compared to their energy in the first coil. This energy loss could easily be resolved with the apparatus, see figure 4. One main problem of a 3He spin-echo instrument, as compared to a neutron spinecho instrument, is the requirement that no obstacles like the ~ - or zr-flip coils, customary in neutron spin-echo, are allowed in the atomic beam. This problem was solved by installing zero magnetic field regions between the various regions of high magnetic field. In this way, one can change the direction of the magnetic fields from one place to the other non-adiabatically with respect to the spin

238

Christian Schmidt and Dirk Dubbers scattenng chamber

i

-.

:

~-±

----9

vallable scattering angle

B1

0- 90= spin-echo coil 1

\

\9 ~ / ]

B2

Stem-Gerlach-

3 He supersonm

quadrupole

beam source

polarizer

at T-1.3 K

spin-echo coil 2

Stern-Gedachhexpole analyzer 2350 mm

e-- bombardment detector

Fig. 2. The Heidelberg 3He atomic beam spin-echo apparatus for the study of slow motions on surfaces.

direction of 3He. Another problem was to polarize and to analyze the 3He beam efficiently. To this end, the 3He beam was cooled to a temperature near 1 Kelvin and polarized in a conventional magnetic quadrupole field. An alternative would be to use the transportable high pressure 3He source developed by Hell and Otten, see also the contribution by E.W. Otten in this volume. The new atomic beam spin-echo instrument is now being adapted for surface studies. The instrument is so sensitive that even the residual magnetic fields of antimagnetic inox steel are inadmissibly high. Therefore, titanium was used for the scattering chamber and its accessories. In spin-echo, in general, high energy resolution really means sensitivity to slow motion. In 3He spin-echo, the movement for instance of large molecules or of their aggregates on a surface, or of biological membranes and of their adsorbates can be studied, cases which are of interest to the organic chemist or the biologist. For the physicist, universality in second order phase transitions is a field of high general interest. Within a given universality class, critical behaviour seems to be the same for all systems. For a given symmetry of the order parameter (scalar, vector, etc.), one universality class is distinguished from the other only by the spatial dimension of the system under study. Therefore, also for two spatial dimensions the dynamics of structural phase transitions should be tested as well as possible. But the experimentalist faces the problem that universality holds only asymptotically close to the critical temperature, where the system's movements are very slow ("critical slowing down"). Therefore, to test universality one must be able to study very slow motions. Hence, one main aim of atomic beam spin-echo will be the study of the dynamics of second order phase transitions in two dimensions.

Spin-echo E x p e r i m e n t s w i t h N e u t r o n s a n d w i t h A t o m i c B e a m s

-40

~---7

1.0

-30

-20

-10

0

10

20

30

239

40

Polarization Product

Max.

0.90 _* o.o,

0.6 0.4 i.•

cO .DN

!i I.,.

0.(1 0.2 ~ ~ / i i i -0.2

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- - ! . . . . 4.... J. . . . ......-: . . . .

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-30

-20

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........ x . . . . . . . .

:

0

.

10

.

.

.

.

.

.

.

.

.

.

.

20

30

40

20

30

40

0.4 -[

J'Bldl = 9.36 mr.m

.

,~

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tO =

o.o-1

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• ~ ~T 1'

~" -O.2 -! I -0.4

t!,

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-30

-20

-10

0

"10

Magnetic Field Integral Detuning [~T-m]

F i g . 3. aHe a t o m i c b e a m spin-echo curves t a k e n w i t h t h e a p p a r a t u r s s h o w n in figure 2, for a s t r a i g h t b e a m , r a m p e d by an angle of six degrees. T h e lower curve is s h i f t e d to t h e right due to t h e 33 neV energy loss of t h e 3He a t o m s in t h e e a r t h ' s g r a v i t a t i o n a l field.

240

Christian Schmidt and Dirk Dubbers

The first physics problem to be studied with the new 3He spin-echo apparatus is the dynamics of a "lattice gas", consisting of flat circular molecules moving on a smooth surface, like a puck on the ice. Further variants of this atomic beam spin-echo technique are possible and are under study. 5

Spin-Echo

In-Beam

as a Birefringence

Phenomenon

As has been pointed out earlier [15, 16], Larmor precession in flight can be regarded as a birefringence phenomenon. In the following, we want to show that, based on this picture, the spin-echo time (3) can be understood as the time difference accumulated by two states that show different group velocities in a magnetic potential on the way to the investigated probe. Because these partial states are thus scattered at different times, the spin-echo signal is naturally found to measure correlations on this time scale [17]. When a beam of spin ~1 particles of mass m and gyromagnetic ratio 7, travelling along the axis z, traverses a region with a stationary magnetic field B(z), then, in the laboratory frame, the Hamiltonian is p2 1 -B(z). H = ~mm - 2 7h a

(5)

When B(z) is sufficiently smooth so that particle reflections on the magnetic potential can be neglected then the WKB approximation gives the two spin states

¢±:exp(-hEt)

exp(h~oZp±(()d~) ,

(6)

where E=p~/2m is the total energy, P0 is the particle's momentum outside the field region, and p±(() = (p2o =kmh'ylB(<)l)½ (7) are the kinetic momenta of the two spin states within the field region. Usually, the magnetic energy :

is small compared to

h lBI

(8)

E, and

( i0/zl)

f9/

with phase

¢(z) = ± mh~ f0 z .mZBzz 2p0 IB(¢)ld¢ = + 2p0

(10)

B is the average field along the trajectory. The spin polarization resulting from (9) then is P(z) =

sine(z) 0

.

(11)

Spin-echo Experiments with Neutrons and with Atomic Beams

241

When the initial beam shows a spread dn/dpo in momentum P0, then

The results (9) to (12) can be interpreted as follows: The polarization (11) is time-independent, and forms a standing helical wave within the magnetic potential. T h a t is, Larmor precession in-beam is a precession in space and not in time. As such, Larmor-precession in beam does not differ from any other standing wave phenomenon of a particle in a potential well. For a finite momentum velocity spread, the standing helical wave (12) is strongly damped. The spin-up state ¢+ is subject to a negative magnetic potential, whereas the spin-down state ¢ _ feels a positive magnetic potential. So the kinetic momentum of the spin-up state in the magnetic field is enhanced, whereas it is reduced for the spin-down state. Classically this can be visualised as spinup particles being accelerated when entering the region of negative potential, and spin-down particles being decelerated in the positive potential. Quantum mechanically the two waves move with different velocities through the region of magnetic potential. In optics such effect is known as birefringence. So in-beam Larmor precession can be interpreted equally as a birefringence phenomenon. If we now focus the attention to the sample, this means that the spin-up part will reach the sample first and thus be scattered first, whereas the spindown part will reach the sample delayed by some time interval At and wilt be scattered later. The scattered waves then enter the second spin-echo coil, where the magnetic field direction is inverted with respect to the first one. There the two states change role, spin-up becomes spin-down and vice versa. 1 At the exit of the spin-echo configuration the time delay between the partial states will be completely compensated at the spin-echo point. So there is no need to worry about the two waves not to overlap at the analyzer. It is precisely the spinecho configuration that brings the two coherent states to spatially overlap at the analyzer, however far they might have run apart in the sample region. The magnetic configuration that causes maximum overlap and thus polarisation is called the spin-echo point. For a quasielastic scattering processes this occurs at equal field integrals, whereas for truly inelastic scattering the spin-echo point is found at detuned fields. Mathematically it is defined by the requirement for stationary phase d¢/dp0 = 0. Any scattering process results in an additional phase and an additional momentum. But, as long as the phase relation between spin-up and spin-down is not disrupted by the scattering process that is as long as they suffer the same change in phase, we still expect to measure maximum polarization at the spinecho point. This will certainly be the case if the target system does not undergo any change during the interval At. A dynamical change of the target configuration during this time interval means that the spin-down state will be scattered 1 Inversion of field direction is essentially a ~-flip.

242

Christian Schmidt and Dirk Dubbers

by a slightly modified potential as compared to the potential seen by the spin-up state. The phases and momenta of the two states will in general be altered differently by different potentials, so that they loose their correlation. Polarization at the spin-echo point will therefore be decreased as compared to the signal of a static target. This is because the more or less statistic changes in phase relation of the contributing coherent states are mirrored in statistically altered spin orientations at the detector. We conclude that polarization at the spin-echo point is a measure of time correlations in the investigated target 2. Polarization will be high if either the target is static during a time interval At, or if the target periodically resumes its original potential with a period of integral fraction of At. The interrogated correlation time At can be calculated via the group velocities of the contributing wave functions (9) d H _ p± (13) v i - dp± m

m

1

v±

--

p±

1 q

-

-

po

mhT]B I T

-

(14)

-

2pao

The resultant time difference at the target is found to be

1 1 / d~ At = ~0L( v_-(~) v~(~)

- -

//~2h~ p3 ~0L IB(~)ld~= tToFE~ gn -

-

- -

7-

,

(15)

where L is the length of the magnetic field region, and tTOF = Lm/po. T h a t is At is the spin-echo time T as introduced in (2). The spin-echo technique thus interrogates time correlations on the timescale of the spinecho time T. The larger the field integral, the longer the timescale in question. In practice, timescales of up to 10 -s s can be realized. A scan of polarization as a function of spin-echo time 7-, i.e. of magnetic field strength, gives direct insight into dynamical processes in the target. The function sampled is the intermediate scattering function I(q, 7-) as will be shown in the following. Calculation of the expected polarization comes down to adding up all possible spin orientations weighted by the probability that the incoming beam includes a state of kinetic energy E as well as the probability that a change in energy hw occurs while scattering. The former probability is given by the distribution in energies dn/dE, whereas the latter is defined as the scattering function S(q, w). Thus polarization can be calculated as

dn

P = f dw f dE~-ES(q,w)exp(z¢(E,w)). In this notation Px is the real part, and part.

Py

(16)

(analyser at 90 °) is the imaginary

2 Polarization depends on the crossterms of the spin-density matrix, i.e. the product of ¢+ and ~b_

243

Spin-echo Experiments with Neutrons and with Atomic Beams

For elastic scattering the time delay will, of course, be compensated by equal field integrals. With quasielastic scattering this is true only in the mean, that is the spins suffer upscattering with equal probability as downscattering. For equal field integrals we find from (10) ¢(E,w):.y/)L(2-~)½{1-

(1---~)-½}.

(17)

When ¢ is expanded into a Taylor series about the mean kinetic energy E of the incoming beam and the mean energy transfer hco = 0, then

{

3hw

3AE

15(BE)

2

15AEhw

5 ( ~ _ ~ ) 2}

(18) When hw and AE = E - E are sufficiently small compared to E, then the polarization measured as a function of r oc B is

P ,.~ f dwS(q,w)ei~°r=

I(q,r),

(19)

which is the Fourier transform of S(q, w) as predicted in (3). Additionally, we would like to mention that polarization is a multiplicative quantity. That is, any further systematic effect that might reduce polarization at the spin-echo point can be included in the theory as a multiplicative factor. So the apparative resolution function comes in as a mere multiplicative function of spin-echo time and possibly scattering angle. It can be determined by scanning the rigid, i.e. cold target, where no movement is expected. For this reason, the data can be corrected by pure division. No unfolding procedure is necessary. In conclusion, the method of in-beam spin-echo is evolving strongly and will, so we hope, in the future become a standard technique also in surface science. References [1] E.L. Hahn, Phys. Rev. 80 (1950) 580 [2] F. Mezei, Z. Physik 255 (1972) 146 [3] F. Mezei, Springer Lecture Notes in Physics 128 (1980) 3 [4] D. Richter, Physica B 180 &= 181 (1992) 7 [5] R. G/ihler and R. Golub, Z. Physik B 65 (1987) 269 [6] R. Gghter, R. Golub, T. Keller, Physiea B 180 gz 181 (1992) 899 [7] R. Golub, H. Yoshiki, R. Gghler, Nucl. Instr. Meth. A 284 (1989)16 [8] R. Pynn, Springer Lecture Notes in Physics 128 (1980) 159 [9] D. Dubbers, P. E1-Muzeini, M. Kessler, J. Last, Nucl. Instr. Meth. A 275 (1989) 294 [10] D. Dubbers, Progr. Part. Nucl. Phys. 26 (1991) 173 [11] T. Keller, P. Zimmermann, R. Golub, R. Giihler, Physica B 162 (1990) 327 [12] U. Schmidt, Dissertation Technical University Munich, 1995 (unpublished) [13] J.P. Toennies, Springer Series in Surface Science 21 (1991) 111

244

Christian Schmidt and Dirk Dubbers

[14] M. DeKieviet, D. Dubbers, C. Schmidt, D. Scholz, U. Spinola, Phys. Rev. Lett. 75 (1995) 1919 [15] F. Mezei, Physica B 151 (1988) 74 [16] R. Golub, R. G~hler, T. Keller, Am. J. Phys. 62 (1994) 779 [17] C. Schmidt, Dissertation Universit£t Heidelberg, 1996 (unpublished)

N e w Generation of Light Sources for Applications in Spectroscopy M. Inguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli European Laboratory for Non-Linear Spectroscopy (LENS) and Dipartimento di Fisica dell'Universitg di Firenze

1

From

Spectral

Discharge

Lamps

to Solid State Lasers

Thirty years of impressive progress in atomic physics have been made possible by the discovery of many new types of laser sources and their application to the development of a large variety of spectroscopic techniques. Still in the late sixties only spectral discharge lamps were widely available for the study of optical spectra. Significant limitations arose from the Doppler limited resolution, the poor tunability, the sparse spectral coverage and the low brightness. Today, on the contrary, we have access to tunable and highly monochromatic laser sources in most of the spectral regions important for atomic and molecular spectroscopy. As an illustration of this exciting story, let us consider the problem of the anomaly in the intensity ratios of doublet lines in heavy alkalis. This was discussed in 1969 by Prof. G. zu Putlitz [1] in the first volume of "Comments on atomic and molecular physics". Starting from some very early speculations of E. Fermi [2] , zu Putlitz computed for the 6 S1/2 -+ 8 P1/2 Cs transition at 389 nm an anomalously small oscillator strength of 3 × 10 -4, i.e. three orders of magnitude weaker than the corresponding resonance transition at 894 nm. In the spectral lamps era, this Cs transition could be excited by W. Happer and coworkers [3] by taking advantage of its accidental coincidence with one of the lines emitted by a microwave excited He lamp. Those were the years of the measurements of nuclear effects on atomic levels and indeed the hyperfine structure of the 8 P1/2 level was measured by means of microwave-optical double resonance spectroscopy [4]. Nowadays ultraviolet radiation at 389 nm, in excess of 10 m W and with a frequency jitter below 1 MHz, is available from a Ti:Sapphire laser operating at 778nm and frequency doubled in a nonlinear crystal. This provides the possibility not only to detect the 6S1/2 ~ 8 P1/~., but also to avoid the Doppler broadening by means of H~nsch-Bord~ type saturation spectroscopy [5] and to resolve the hyperfine structure [6] as shown in Fig. 1. The narrow natural linewidth (less than 480kHz) would make Doppler-free excitation of the 6 $1/2 --+ 8 P1/2 attractive for ultra precise spectroscopy. It is worth noting that the necessary 778 nm radiation is almost coincident With that used

246

M. Inguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli 82~72

F = 4

170MHz

I

F=3

6ZStn

]

<

F=4 9192.631770 MHz F=3

a) arb, units 62S 1/2,F=4 - 82P1/2

62S 1/2,F=3 - 82P1/2

180

160 #

140

120

100

/

°

# o L~._**g °

~:

.*%

o

°•

ca

2~Hz

° ~

V

b)

arb. units

cross over resonance

180

160

F-----4- F----4 '

F=4 - - F'=3 "

~

-

,

~

-

•

.~

140

120

---" ---X_

1 O0 v

62S 1/2 F=3 - 82P1/2

c) F i g . 1. Direct excitation of the anomalously weak (zu Putlitz, 1969) Cs transition at 389 nm. Coherent radiation is provided by a frequency doubled Ti:Sapphire laser. In b) the Doppler broadened recording allows to resolve only the hyperfine structure in the ground state while in c) the resolution of the much smaller hyperfine structure of the 8 PII2 is achieved by means of saturation spectroscopy.

New Generation of Light Sources for Applications in Spectroscopy

247

for sub-Doppler excitation of the 5S1/2 --+ 5D5/2 two photon transition in Rb. The absolute optical frequencies of the Rb hyperfine components of this transition were measured [7] with an uncertainty of 5 kHz (i.e. 1.3 x l0 -11) taking advantage of the close coincidence with the difference in frequency between the iodine stabilized HeNe laser and the methane stabilized HeNe, at 473THz (633nm) and 8 8 T H z (3.39#m) respectively. As a consequence the "Rb optical standard" could be eventually used for a direct measurement of the Cs transition at 3 8 9 n m following the scheme that we successfully introduced in [8] for the measurement of the closely coincident transition in Helium.

2

T h e I m p a c t of S e m i c o n d u c t o r D i o d e Lasers

The two photon transition of Rb can now be excited using a simple semiconductor diode laser (SDL) source available around 778 nm, as shown in Fig. 2 for a portion of the hyperfine structure. This is a further demonstration of how scientific achievements in atomic spectroscopy follow the development of new laser sources. In this respect the new generation of SDL sources are really showing a revolutionary impact. Although they were introduced m a n y years ago they were not used much in atomic spectroscopy until recently [9]. I m p o r t a n t characteristics which are relevant for SDL sources in our field are: spectral coverage, tunability, spectral linewidth and amplitude stability [10]. The emission wavelength depends on the semiconductor band gap and, in principle, it should be possible to construct l ~ e r s at all wavelength by choosing the proper stoichiometric abundance for the dopant elements. In practice, laser diodes are commercially available only at sparsely distributed emission wavelengths and the experimentalist soon faces the problem of tuning the

87Rb

5S1/2 ~

5D5/2 F=2 ~ F=4

F=2 --~ F=3"

~ l NI-h

Fig. 2. Doppler-free recording of the Rb 5S1/~ --+ 5Ds12 two-photon transition at 778 nm. Radiation is provided by a semiconductor diode laser which can serve as an optical frequency standard for the Cs transition of Fig. i at 389 = 778/2 nm.

248

M. Inguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli

laser precisely to a atomic or molecular absorption line. An elegant solution to the problem is the operation in an extended cavity (EC) configuration, where the feedback from a grating also provides line narrowing down to few hundred kHz [11]. This scheme provides a compact and relatively cheap source with an output power ranging from few m W to about 100 m W in the visible to near infrared spectral range. In particular the resonance transitions of all the alkalis except Na are accessible to high resolution investigations. Indeed the recording of Fig. 2 was obtained using a low power (~5 m W ) SDL "grating tuned" to resonance. Fine tuning was then achieved by sending a r a m p to the injection current and to a piezoelectric ceramic to change the length of the extended cavity [10]. It is worth noting that the linewidth of each resolved component in Fig. 2 is about t MHz, almost twice the natural width, possibly because of non perfect compensation of the Doppler effect but certainly unaffected by the laser width which is only few hundred kHz. 3

Saturation

Spectroscopy

and

Optical

Pumping

The versatility of EC SDL sources is also illustrated by saturation absorption spectroscopy of Cs on the resonant D~ transition at 852 nm. In this case the J = 3/2 value for the upper level offers a larger number of hyperfine components than for the transition in Fig. 1. For instance we show in Fig. 3 a saturation spectrum of the 6 2S1/2, F = 3 --~ 6 2P3/~ transition. L a m b dips are recorded for all allowed components and the crossover signals. This Cs absorption spectrum constitutes the basis for frequency locking of SDL lasers that is now widely used for the production of ultra cold atoms, as, for instance~ F=3 - F'=2 F=3 - F'=3 F=3 - F'=4

-4

Fig. 3. Saturation spectrum of the 6 S 1 / 2 , F = 3 --+ 6Pa/2 Cs transition at 852nm ; an extended cavity grating tuned semiconductor diode laser is used.

O

o

e~ <

I

500

i

I

1000

Frequency (MHz)

h

New Generation of Light Sources for Applications in Spectroscopy Helmholtz coils

rnPrD~o~

249

~..mirror mirror

beamsplit~ Ffg. 4. Experimental apparatus for the observation of velocity selective optical pumping phenomena in Cs atoms. The light coming from an EC SDL source is divided in two parts by a beam-splitter. A system with a polarizing beam-splitter cube and a )~/4 plate can control the beam polarizations. The signal is the difference between the light incident on the two photodiodes PDI and PD2. Tiffs scheme allows one to subtract the Doppler background from the saturated absorption signals.

done by our group in [12] for double resonance experiments in a magneto-optical trap (MOT). However, in the present work, we want to illustrate some of the somehow anomalous results which one can obtain when saturation spectroscopy is combined with optical pumping. The latter is a phenomenon widely applied in atomic physics since the original proposal by A. Kastler [13] and many exciting experiments were performed using discharge spectral lamps [14]. Lasers and in particular SDL make opticM pumping easily to be observable as a change of the absorption of polarized light. In addition, the narrow laser linewidth and the counterpropagating beam configuration can combine optical pumping phenomena with sub-Doppler resolution. A typical experimental scheme for the observation of velocity selective optical pumping of Cs is shown in Fig. 4. When the polarizations of both the counterpropagating laser beams are chosen to induce transitions with the same A M selection rule, one can detect the usual increase of the atomic transparency, of course with sub-Doppler resolution, as shown in Fig. 5, for the same transition of Fig. 3. When, on the contrary, the two beams induce transitions with opposite A M selection rule, the recorded signal offers a rather anomalous behavior. In particular, one of the hyperfine resolved components is "inverted", hence showing an increase of the absorption instead of an increase of transparency. This is caused by the increase of population of the Zeeman sublevel which then is responsible for the absorption of the analysis beam. This phenomenon

250

M. lnguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli (I + - C~+ F=3

"-"x

- F=3

(I+- a-

~ F=3 - F'=3

F=3 - F'-2 ;

i F=3 - F=2

Fig. 5. Saturated absorption signal (the Doppler-background is electronically subtracted) of the 6 S 1 / 2 , F = 3 --4 6P3/2 transitions using two a + beams as pump and probe light.

Fig. 6. Saturated absorption signal for the 6 $1/2, F = 3 --+ 6P3/2 transitions. In this case pump and probe beams have opposite circular polarizations and (as explained in the text) the effect of optical pumping, evident on the "closed" transition F = 3 --4 F j = 2, is to reduce the transmission.

is evident in Fig. 6 for the component F = 3 --4 F ' = 2 for which the efficiency of Zeeman pumping is very high since the transition is "closed". For the other components there is instead a competition with hyperfine optical pumping [15]. Indeed the slightly reduced transmission only causes a decrease of the signal to noise ratio. The situation is quite different for the F = 4 --4 F ' -- 5 transition (Fig. 7). In the case, when the two beams have the same circular polarization, we observe an increase in absorption compared to the case of opposite circular polarization (~r+, c~-). This behavior can be explained if we take into account the interaction time, that is a serious limitation in optical pumping. Because of the finite time of the interaction, the population is not completely transferred in the M R ----4 Zeeman sublevel. The effect of the optical pumping is to increase the absorption transferring population in sublevels that, in this case, are more strongly coupled to the light (this is evident calculating the ClebschGordan coefficient of the different Zeeman transition). When a magnetic field Bz ¢ 0 is added, a differential optical pumping of the different ground levels occurs [16] and causes a dispersive lineshape for the F = 4 -4 F ' = 5 component. This can be seen in the recordings of Fig. 8. An increase of the laser field intensity will cause a stronger optical pumping making these effects more and more pronounced.

New Generation of Light Sources for Applications in Spectroscopy &-

AFiF5

a+

F=4 - F'=3

251

Bz~O F--4 - F--4 ;

l.....

~ + - O-

G + - G-

F=4 - F=5

Bz~0

F=3 - F=4 F=-3 - F'=3

F--4- F=4

~F=3 - F=2 F i g . 7. Saturated absorption signal obtained when the laser frequency is tuned across the 6 S 1 / 2 , F = 4 --+ 6 PzI~ transitions. In this case, changing from a + - cr+ to cr+ - or- configuration we observed an increase in the transmission signal for the F =- 4 ~ F ' = 3 hyperfine transitions. This is due to optical pumpingselecting weaker transitions.

4

F i g . 8. We report the signal observed for the two different transitions starting from 6 S l l 2 , F = 3, F ~- 4 --4 6P312 levels, increasing the pump beam intensity and adding a magnetic field. In this case optical pumping is highly efficient and its effects are more evident.

Injection Locking and Frequency Doubling

O n e of t h e l i m i t a t i o n s o f E C S D L can be t h e r e l a t i v e l y low power (up to 10 r o W ) . T h i s rules o u t t h e p o s s i b i l i t y of p e r f o r m i n g e x p e r i m e n t s involving n o n l i n e a r processes such as frequency d o u b l i n g or s u m frequency g e n e r a t i o n . High p o w e r (up to 100 m W ) S D L are now c o m m e r c i a l l y available, b u t t h e i r frequency c h a r a c t e r i s t i c s are q u i t e poor. I n j e c t i o n locking has been d e m o n s t r a t e d to be an easy a n d effective way to o v e r c o m e these l i m i t a t i o n s : a low

M. Inguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli

252

PZTI Master ~asor

PC

To WZT2

~_ 12

Anamorphic prisms

cylindrical lens

k/4

Isolator ]otis

M3 lens

~

~

~

LBO

~'2 M2

Slave

Anamo~hic

laser

prisms

Isolator

f=300 ram f=30 mm

MI

M,~

'z~

Fig. 9. Experimental apparatus for the injection locking and the doubting of a SDL. The high power SDL (slave) is injected by an EC SDL (master). The slave is frequency doubled in an LBO crystal placed in an enhancement cavity in order to obtain higher efficiency.

power EC SDL acts as master oscillator injecting a high power SDL slave. In this way one can easi]y have the best of both worlds. With this kind of setup we were able to produce up to about 1 m W of UV radiation at 397 nm by frequency doubling SDL at 794 nm. The experimental scheme is shown in Fig. 9. High doubling efficiency was obtained using a LBO crystal in an enhancement cavity resonant with the fundamental radiation [17]. This system provides UV radiation from a compact all solid state source, making accessible experiments that until now required more complicated lasers.

5

Frequency Control and Precision S p e c t r o s c o p y of Helium

In the already mentioned first volume of Comments in Atomic and Molecular Physics in 1969, V.W. Hughes [18] critically discussed the advances in knowledge of the value of the fine structure constant c~. In particular, he was

New Generation of Light Sources for Applications in Spectroscopy

253

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Laser--

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e

3 cm

partial refl. m i r r o r ~ilter

:5


1.o L

J

I'

'

|

|

'

ID

'

|

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Fig. 10. Experimental setup for line narrowing of DBR diode lasers at 1083 nra by means of optical feedback from a partially reflecting mirror. The spectrum in the lower part shows the beat note between two identical lasers.

200kHz

"O

._=_ E <

_3

0.0 -5.0

I

I

I

I-

-I

i

I

-3.0 -1.0 1.0 3.0 Freq. Offset (MHz)

i

5.0

suggesting the measurement of the fine structure intervals in the 2 3 p j state of helium as one of the most promising possibilities for determining a. In those years the helium intervals had been measured [19] with a precision of about 2 ppm, while the calculations for a two-electrons atom, obviously necessary to extract a value of a, seemed to Hughes "a tractable problem with modern digital computing machines". Indeed, nowadays theoretical predictions are likely to reach a precision of parts in 108 [20]. At the same time, the new generation of SDL is now providing the possibility of an eventually similar accuracy for the measurements. Distributed Bragg Reflector (DBR) diode lasers are available at 1.08 #m, i.e. around the 23S1 -+ 23p multiplet of helium [21]. By means of feedback from a partially reflecting mirror, the spectral width of these lasers (Fig. 10) can be reduced to only few hundred kHz [22]. This is one order of magnitude narrower than the natural linewidth of the helium transitions which can then be investigated with s u b - D o p p l e r resolution. As an example, we show in Fig. i I the recordings corresponding to the 2 3S1 -+ 2 3p1 and 2 3SI --~ 2 3P0 components. The experiment is performed in a collimated metastable atomic b e a m with a residual Doppler width of about 60 MHz. The interaction with the counterpropagating laser beams allows one to detect a saturation dip in fluorescence with good signalto-noise ratio. Laser radiation is frequency modulated and phase sensitive detection is used. After a fitting procedure, the line center can be obtained with an accuracy of below 25 kHz. The signals recorded in Fig. 11 occur at

254

M. Inguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli

3O

•

~

23S1-23p0

25

~

Fig. 11. Derivative signals for the 23S1 --+ 23P0 (trace (a)) and 23S1 --~ 2Sp1 (trace (b)) transitions in helium. The scan width is 100 MHz. The shift necessary to superimpose the two scans is v0 -~ 29617 MHz.

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T-23p~

"~

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--

Ilo

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-S0

-40

-30

-20

-10

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0

10

20

30

40

50

60

(MHz)

laser frequencies separated by the fine structure splitting of the 2 3p levels which then can be determined. In this respect the possibility of a fast control of the SDL frequency through the injection current is a crucial feature of these lasers. Indeed it has been possible to "phase-lock" one to the other two independent identical diode lasers [22] with a frequency difference up to 40 GHz and with an Allan variance or(r) <0.2 Hz for 1 s < r <100 s. These results are summarized in Fig. 12 where both the beat note between the two phase-locked lasers and the results for the Allan variance are shown. A straightforward application in high precision spectroscopy of helium is to stabilize the frequency of one of the two lasers to a reference resonance and then perform "frequency calibrated" spectroscopy using the other. This work is indeed in progress and a preliminary value for the 2 3p0 --+ 2 3P1 splitting has been obtained [22], [23] with an accuracy of 9 parts in 107. This is likely to be further improved in the near future.

6

Conclusions

All the above considerations should have convinced the reader of the importance of semiconductor diode lasers for atomic physics. They can potentially cover the visible, the near infrared and also the UV by means of frequency doubling. Optical feedback techniques can be used to tune to the proper wavelength and to drastically reduce the linewidth to a level which allows very narrow transitions to be directly investigated. Coherent phase-locking of independent lasers allows the high resolution to be combined with high precision spectroscopy and a direct determination of the frequency of the optical transitions. Available powers can be further increased by means of injection techniques. However, the extremely reduced amplitude fluctuations, typically 10 dB above the shot noise limit, are making high sensitivity measurements possible even with low power lasers. Typical applications can be

New Generation of Light Sources for Applications in Spectroscopy I

0 rn

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I

I

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I

0

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< -40 I

f

I

-1

0

,

-2

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1

2

Fig. 12. The upper part of the figure shows the beat note between two phase-locked diode lasers at 1083rim at a 10kHz resolution bandwidth. The beat frequency can be set anywhere from 30MHz up to 40 GHz. In the lower part, the Allan variance a(r) of the beat note is plotted. The data can be fitted by a(r) = 0.2/r Hz in the 1 s-100 s interval.

Freq. Offset (MHz) 100

-1-

,

i,i,ll~!

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;

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I

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101

102

|

,,,,,,

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~(s)

found in the detection of overtone molecular spectra and, more recently, of O~ on the forbidden b l Z + ( v ' = 0) +-- X 3 Z ~ (~," = 0) band at 760nm [24]. Complex experimental schemes can be simplified by the use of SDL and experiments which could not even be conceived before can now be performed. Progress in the fascinating world of laser spectroscopy have always depended on the development of new sources. We are sure that this new generation of lasers will allow new important chapters to be written in the field of atomic physics. This work has been performed in the frame of the ECC contract G E l * C T 9 2 0046.

References [1] [2] [3] [4]

G. zu Putlitz: Comm. At. Mol. Phys. 1, 51 (1969) E. Fermi: Z. Phys. 59, 680 (1969) C. Tai, R. Gupta, and W. Happer: Phys. Rev. A 8, 1661 (1973) for a review of the measurements of hyperfme structure of alkali atoms see: E. Arimondo, M. Inguscio, and P. Violino: Rev. Mod. Phys. 49, 31 (1977) [5] For an updated discussion of the modern techniques of Laser Spectroscopy a useful reference can be: Frontiers in Laser Spectroscopy ed. by T. W. H~isasch and M. Inguscio (North-Holland, Amsterdam 1994)

256

M. lnguscio, F. S. Cataliotti, C. Fort, F. S. Pavone and M. Prevedelli

[6] F. S. Cataliotti, C. Fort, F. S. Pavone, and M. Inguscio: submitted to Z. Phys. [7] F. Nez, F. Biraben, R. Felder, and Y. Millerioux: Opt. Comm. 102,432 (1993) [8] F. S. Pavone, F. Marin, P. De Natale, M. Inguscio, and F. Biraben: Phys. Rev. Lett. 73, 42 (1994) [9] J. C. Camparo: Contemp. Phys. 26,443 (1985) [10] See the lecture of M. Inguscio: "High-Resolution and High-Sensitivity Spectroscopy using Semiconductor Diode Lasers", p. 41, in [5] [11] C. E. Wieman, and L. Hollberg: Rev. Sci. Instrum. 62, 1 (1991) [12] C. Fort, F. S. CatMiotti, P. Raspollini, G. M. Tino, and M. Inguscio: Z. Phys. D 34, 91 (1995) [13] A. Kastler: J. Phys. Paris 11,255 (1950) [14] Historical reviews can be found in: J. Brossel: "Pompage Optique", in Quantum Optics and Electronics, ed. by C. De Witt, A. Blandin, and C. CohenTannoudji, pp. 189-327 (1965); C. Cohen-Tannoudji, and A. Kastler " Optical Pumping", in: Progress in Optics 5, ed. by E. Wolf, pp. 1-81 (1966) [15] K. Ernst, P. Minguzzi, and F. Strumia: Nuovo Cimento 51 B, 202 (1967); K. Ernst, and F. Strumia: Phys. Rev. 170, 48 (1968) [16] Y. Nafcha, D. Albeck, and M. Rosenbluh: Phys. Rev. Lett. 67, 2279 (1991) [17] M. de Angelis, G. M. Tino, P. De Natale, C. Fort, G. Modugno, M. Prevedelli, and C. Zimmermann: Appl. Phys. B 62, 333 (1996) [18] V. W. Hughes: Comm. At. Mol. Phys. 1, 5 (1969) [19] F. M. J. Pichanick, R. D. Swift, C. E. Johson, and V. W. Hughes: Phys. Rev. 169, 55 (1968) [20] Z. C. Yang, and G. W. F. Drake: Phys. Rev. Lett. 64, 4791 (1995) [21] A. Arie, P. Cancio Pastor, F. S. Pavone, and M. Inguscio: Opt. Comm. 117, 78 (1995) [22] M. Prevedelli, P. Cancio, G. Giusfredi, F. S. Pavone, and M. lnguscio: Opt. Comm., 125, 231 (1996) [23] P. Cancio Pastor, G. Bianchini, G. Giusfredi, F. S. Pavone, M. Prevedelli and M. Inguscio: "High Precision Spectroscopy of the Helium Atom at 1083nm", in: Proceedings of 5th Symposium on Frequency Standards and Metrology, ed. by J. C. Bergquist, World Scientific Publishing, Singapore (1996) [24] C. Corsi, M. Gabrysch, and M. Inguscio: Opt. Comm.128, 35 (1996)

Remote Sensing of the Environment using Laser Radar Techniques Mats Andersson I , Hans Edner 1 , Jonas Johansson 1, Sune Svanberg 1, Eva Wallinder 2 and Petter Weibring 1 I Department of Physics, Lund Institute of Technology, P.O. Box 118, S-221 00 Lund, Sweden Lighten AB, Ideon Research Park, S-223 70 Lund, Sweden

1

Introduction

Laser techniques have had a strong impact on chemical sensing. Applications include the fields of analytical chemistry (Letokhov 1990, Radziemski et al. 1987), combustion diagnostics (Eckbreth 1987), medical diagnostics (Pettit and Waynant 1996, Svanberg 1989, Svanberg 1996a) and environmental monitoring (Measures 1988, Sigrist 1994, Svanberg 1991). Laser radar monitoring of the environment is an application of timeresolved laser spectroscopy. Differential optical absorption as well as laserinduced fluorescence can be used for this type of remote sensing. Apart from providing range-resolved data, the use of an active illumination source provides a more accurate assessment than if just the ambient passive radiation is employed. However, by necessity a limited monitoring range is imposed by the use of an artificial source. An overview of atmospheric pollution monitoring and vegetation status assessment using laser radar techniques will be given with illustrations from work at the Lund Institute of Technology. We will start with monitoring of industrial and urban air pollution and will continue with measurements of geophysical gas emissions. Applications of fluorescence lidar to vegetation and water are then discussed and, finally, an outlook for the future is given.

2

Monitoring

of Urban

and

Industrial

Air Pollutants

The differential absorption lidar (dial) technique (see e.g. Svanberg 1994 for a review) provides three-dimensional mapping of gas distributions in the atmosphere. Pulses from a tuneable laser are transmitted into the atmosphere and photons, elastically back-scattered from aerosols and major constituents, are collected by an optical telescope giving rise to an electrical transient after detection in a photomultiplier tube. An example of a lidar curve for a range up to 4.5 km obtained with a vertically aimed lidar system (Wallinder et al. 1996) is given in Fig. 1. An excimer laser transmitter was used, providing

258

Andersson, Edner, Johansson, Svanberg, Wallinder and Weibring.

pulses at 317 nm after stimulated R a m a n conversion of 248 nm KrF radiation. Since the laser b e a m is t r a n s m i t t e d from a point displaced by about 0.3 m from the receiving telescope axis, the signal starts at a low value and reaches a m a x i m u m when the transmission and receiving fields of view overlap. Then a 1 / R 2 fall-off is observed, reflecting the normal illumination law. At a distance of about 2.1 k m the increased backscatter from a dual-layered cloud can be seen. In the x 100 magnification the optical attenuation by the cloud reducing the backscattered radiation from above the cloud can also be seen. While the relative distribution of aerosol particles can be displayed by a curve such as the one shown in Fig. 1, molecular concentration measurements require the recording of curves at two closely spaced laser wavelengths.

250000

200000

150000 ~9 t-

¢100000

50000

L_ 1000

x100

2000

3000

4000

5000

Distance (m)

Fig. 1. Lidar signal obtained in vertical sounding using an excimer lidar. Initially the signal displays a gradual increase as the transmitted beam and the telescope fields of view start to overlap. Then a fall-off, basically with a 1/R 2 dependence, follows. An increased back-scatter from thin clouds at a height of about 2 km is observed.

One is chosen corresponding to a strong absorption of the gas under study while the other one, off-resonance, is used for reference purposes. By dividing the "on"-resonance curve by the " off' -resonance curve the influence of aerosols is eliminated and the range resolved gas concentration is calculated

Remote Sensing of the Environment using Laser Radar Techniques

259

from the slope of the divided "dial" curve. An urban measurement scenario is shown is Fig. 2, illustrating our mobile laser radar system during urban pollution monitoring in the city of Prague. The laser beam is transmitted and backscattered radiation is received from the roof-top rotatable dome. The construction of the lidar system is described by Edner et al. (1987). A measurement along a street in Prague is shown in Fig. 3 (Engst et al. 1995). A laser wavelength pair around 226 nm is chosen to yield a vertical mapping of NO as the laser beam is scanned under computer control. The two concentration maxima, spaced by about 100 m, correspond to two street crossings.

Fig. 2. Photograph of the Swedish fidar system during measurements in Prague, the Czech Republic. The Swedish mobile lidar system has recently been fully reconstructed and modernised. While still using the same dye laser transmitter, waiting for replacement by an optical parametric oscillator system, all electronics, computers and software have been replaced (Andersson et al. 1996). In particular, graphical software routines written in the industriM standard, LabVIEW are used and an improved concentration precision and uncertainly assessment is obtained by statistical treatment of the lidar signals. While reliable concentration values are the most interesting ones in urban monitoring, industrial measurements rather focus on flux determinations. Then the wind velocity component, perpendicular to the vertical lidar scan through the plume and/or

260

Andersson, Edner, Johansson, Svanberg, Wallinder and Weibring.

[] [] [] []

o-lo 10.20 20.30 3040 40-50 [ ] 50-60 [ ] 60.70 [] >70

Height (m) 200

100

Fig. 3. Results from measurements of NO along a street in Prague, showing increased concentrations at two subsequent street crossings. Concentration values are given in/Jg/m ~. (Adapted from Engst et al. 1995).

diffuse emissions is needed apart from the concentration values. A video camera plume monitoring technique, incorporating temporal cross-correlation of the plume features has been implemented and geometric angles and distances are inferred by lidar shots. The wind data are obtained simultaneously with the concentration mapping, allowing near real-time flux monitoring. A first example, from a campaign at the NymSlla paper plant, is given in Fig. 4. Here SO2 is measured using a wavelength pair close to 300 nm, and a flux of 140 kg/h was determined in very good agreement with in-stack monitoring values provided by the plant.

3

Measurements

of Gases

of Geophysical

Origin

While pollutants of anthropogenic origin are abundant there are also substantial emissions into the atmosphere from geophysical sources related to geothermal energy, ore deposits and volcanic activities. Among several gases Hg and SO2 could be especially mentioned. Measurement scenarios in a geophysical environment are shown in Fig. 5 (Svanberg 1991). Emissions of atomic mercury follow geothermal steam extraction and mercury could be considered as a tracer gas for geothermal energy. Since free atoms rather than mercury compounds are emitted, a spectroscopic sensi-

Remote Sensing of the Environment using Laser Radar Techniques

i:

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Fig. 4. Sulphur dioxide emission from a Swedish paper mill. The measurements were performed with the upgraded Swedish lidar system, employing LabVIEW software and automatic wind measurements by video cross correlation (From Weibring et al. 1996a).

tivity increase of more than a factor of 10 a is obtained allowing the optical measurement of concentrations down to the order of the Atlantic background value 1 n g / m 3. Lidar techniques for the mapping of Hg using the absorption line close to 254 nm are described by Edner et al. (1989), where measurements of industrial emissions from a chlor-alkali plant are also reported. Fig. 6 shows the results of a vertical lidar scan lee-ward of a cooling tower at the largest geothermal power plant in Europe, Larderello (Tuscany) (Edner et M. 1992a). Results from point monitors employing atomic-absorption measurements on mercury collected as amalgam on gold foils are included in the figure and show good agreement with the dial data. Measurements at the mercury mine at Almad6n, Spain (Ferrara et al. 1996), following studies at the smaller., abandoned mine at Abbadia S. Salvatore (Italy) (Edner et al. 1993), revealed atmospheric mercury fluxes of about 1 kg/hour. A mapping of the diffuse mercury plume from the Almad6n area is shown in Fig. 7 (Ferrara et al. 1996).

:

Andersson, Edner, Johansson, Svanberg, Wallinder and Weibring.

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/

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Remote Sensing of the Environment using Laser Radar Techniques

263

Fig. 7. Distribution of atomic mercury in the valley downwind from Almaden, Spain (From Ferrara et al. 1996).

Very large emissions of SO2 occur from active volcanoes. We have performed two measurement campaigns on board the Italian research vessel Urania to monitor the emissions from Etna, Stromboli and Vulcano (Edner et al. 1994a, Svanberg et al. 1996b, Weibring et al. 1996b). By firing the laser beam vertically while passing under the volcanic plume it was possible to measure the integrated concentration, and also the flux by combining with wind data. A photograph showing the Urania with the lidar on its aft deck and with the island of Vulcano in the background is presented in Fig. 8. Lidar data were compared with differential optical absorption (dons) results obtained by a spectral analysis of the overhead sky radiation (passive monitoring). Measurement results for Mt. Etna obtained during the most recent cruise are shown in Fig. 9. The uncorrected dons data are almost a factor of 3 higher than the lidar results (35 tonnes/h). The reason for the large discrepancy is scattering within the plume, strongly affecting the passive dons data. The scattering will now be modelled to allow the correct evaluation also from the more simple passive measurements, which can also be performed with COSPEC correlation spectroscopy.

4

Fluorescence Monitoring of Water and Vegetation

When a pulse of ultraviolet radiation from a lidar system is impinging on a target, fluorescence is induced. The fluorescence can be collected by the lidar receiving telescope and be spectrally dispersed and detected on a CCD detec-

264

Andersson, Edner, Johansson, Svanberg, Wallinder and Weibring.

Fig. 8. Photograph of the research vessel Urania with the Swedish lidar system, off Lipari with the island of Vulcano in the background. Etna, Sept. 10, 1994 Flux: 35 t/h (LIDAR), 99 tJh (DOAS)

g C 0 (.I rE

0

0

(D

Horizontal position (km) Fig. 9. Sulphur dioxide data from traverses with a ship-borne remote-sensing system under the volcanic plume of Mt. Etna, Italy. The integrated over-head SO2 load as inferred from dial and doas monitoring are presented (From Weibring et al. 1996b).

Remote Sensing of the Environment using Laser Radar Techniques

265

tor preceded by an image intensifier, which is gated in synchronism with the arrival of the laser-induced signal. In this way the whole fluorescence spectrum can be detected for every laser shot. Scenarios for water and vegetation monitoring are shown in Fig. 10 (Svanberg 1994). Measurements from a vehicle can be used in preparation for airborne monitoring. In Fig. 10a, water monitoring is illustrated (Edner et M, 1992b), The spectrum features a broad blue-green distribution from DOM (Dissolved Organic Matter) or Gelbstoff, A weak signal due to chlorophyll in algae is seen in the red spectral region. Oil on the water surface would result in a very strong blue fluorescence emission (Reuter 1991). Vegetation monitoring (Svanberg 1995) (Fig. 10b) yields prominent chlorophyll peaks at 690 nm and 735 nm, the relative intensity of which depends on the chlorophyll concentration (Lichtenthaler and Rinderle 1988). In addition, blue-green fluorescence is obtained relating to cell structures and the leaf surface layer. The spectrum can carry information on stress conditions. For efficient excitation of chlorophyll and less stringent eye safety requirements, a wavelength just below 400 nm should be chosen. This can be done by using the Nd:YAG laser third harmonic at 355 nm and then shifting the radiation to 397 nm by stimulated Stokes Raman scattering in deuterium, Examples of single-shot and multishot spectra from different trees are given in Fig. 11 (Andersson et al. 1994).

DOWNSTREAM OF FLORENCE

_

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(b)

•

-

.

REMOTE BEECH SPECTRAA

/L

WAVELENGTH( ~ l

Fig. 10. Overview of hydrospheric and vegetation monitoring using remote laserinduced fluorescence. Typical fluorescence spectra are included (From Svanberg

1994).

266

Andersson, Edner, Johansson, Svanberg, Wallinder and Weibring.

.S

o . . . . .

500 550 600 650 700 c) Plane-tree: tO0 shots. 210m

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700

750

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600

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700

750

Fig. 11. Examples of point monitoring of vegetation fluorescence for different species and different distances (From Andersson et al. 1994).

Multi-spectral fluorescence imaging can be performed by expanding the laser beam to cover a certain area that is then imaged onto a two-dimensional CCD detector with simultaneous recording of the image in several suitably chosen spectral bands (Edner et al. 1994b). The area cannot, however, be increased in an unrestricted way since the induced fluorescence has to compete with the ambient light in day-light monitoring. Then the laser radiation can instead be spread out in a streak using cylinder-lens optics, and the fluorescence along the lines can be captured. When the line is swept over e.g. a tree from the root to the top, image lines can be detected sequentially and a full image can be built up in the computer. Such imaging is shown in Fig. 12 where a spruce tree was illuminated at a distance of 60 m (Johansson et al. 1996). From individual colour registrations ratio images can be formed as shown in the figure. In this way spectral anomalies may be enhanced while still eliminating influences by topography and uneven illumination.

5

Discussion

Optical remote-sensing techniques using lasers for the monitoring of the environment illustrate how methods, primarily developed for more fundamental laboratory spectroscopy can be brought to practical applications. Frequently,

Remote Sensing of the Environment using Laser Radar Techniques

267

Picea abies 60 m distance single scan mode 480 nm

685 nm

740 nm

,_-

685nm

"t

lower

740nm

~"]t:::: ] oc

needlside~ e

upperneedle)~ide __ 500

600

700

800

LL

560 66o r6o 86o wavelength (nm) Fig. 12. Fluorescence images of a spruce tree at a distance of 60 m. An image obtained by dividing images recorded at 735 and 690 nm, respectively, is also shown. Typical fluorescence spectra (point monitoring) from spruce are included (From Johansson et al. 1996).

the most advanced techniques are required for front-line atomic physics research, and new methods and instruments can then be taken over in the applied field. I m p o r t a n t applications can then provide additional motivation for fundamental research. Laser monitoring of the environment is developing quickly and is likely to be used more routinely in the near future. This is because realistic and reliable laser sources based on all-solid-state technology are now becoming readily available, and very powerful and cost-effective computers can now handle large amounts of d a t a in near real time. Decreasing size and weight also allows practical installation arrangements for laser-based sensors. Spacebased lidar systems have thus been developed and have been successfully tested.

268

Andersson, Edner, Johansson, Svanberg, Wallinder and Weibring.

Acknowledgements This paper is dedicated to Professor G. zu Putlitz on the occasion of his 65 th birthday. This work was supported by the Swedish Space Board and the Swedish National Sciences Research Council.

References Andersson M., Edner H., Johansson J., Ragnarson P., Svanberg S. and Wallinder E. (1994): in Physical Measurements and Signatures in Remote Sensing, Proceedings of the ISPRS Symposium, Val d'Is~re, January 1994, p. 835. Andersson M. and Weibring P. (1996): A user friendly lidar system based on LabVIEW, Lurid Reports on Atomic Physics LRAP-201 (Lund Institute of Technology). Eckbreth A.C. (1987): Laser Diagnostics for Combustion Temperature and Species, (Abacus Press, Turnbridge Wells). Edner H., Fredriksson K., Sunesson A., Svanberg S., Un~us L., Wendt W. (1987): Mobile remote sensing system for atmospheric monitoring, Appl. Opt. 26, 4330. Edner H., Faris G.W., Sunesson A. and Svanberg S. (1989): Atmospheric atomic mercury monitoring using differential absorption lidar techniques, Appl. Opt. 28, 921-930. Edner H., Ragnarson P., Svanberg S., Wallinder E., De Liso A., Ferrara R. and Maserti B.E. (1992a): Differential absorption lidar mapping of atmospheric atomic mercury in Italian geothermal fields, J. Geophys. Res. 97D, 3779-3786. Edner H., Johansson J., Svanberg S., Wallinder E., Cecchi G., and Pantani L. (1992b): Fluorescence lidar monitoring of the Arno River, EARSeL (Eur. Assoc. Remote Sens. Lab.). Adv. Remote Sens. 1, 42-45. Edner H., Johansson J., Svanberg S., Wallinder E., Bazzani M., Breschi B., Cecchi G., Pantani L., Radicati B., Raimondi V., Tirelli D. and Valmori G. (1992c): Laser-induced fluorescence monitoring of vegetation in Tuscany, EARSeL (Eur. Assoc. Remote Sens. Lab.). Adv. Remote Sens. 1, 119-130. Edner H., Ragnarson P., Svanberg S., WaUinder E., Bargagli R., Ferrara R. and Maserti B.E. (1993): Atmospheric mercury mapping in a cinnabar mining area, Sci. Total Environ. 133, 1-15. Edner H., Ragnarsson P., Svanberg S., Wallinder E., Ferrara R., Cioni R., Raco B. and Taddeucci G. (1994a): Total fluxes of sulphur dioxide from the Italian volcanoes Etna, Stromboli and Vulcano measured by differential absorption lidar and passive differential optical absorption spectroscopy, J. Geophys. Res. 99, 18,

827. Edner H., Johansson J., Svanberg S. and WaJlinder E. (1994b): Fluorescence lidar multi-color ima9in 9 of ve9etation, Appl. Opt. 33, 2471. Engst P., Janouch F., Keder J., Andersson M., Edner H., Hessman M., Svanberg S. and Wallinder E. (1995): Co dokd~e LIDAR: Strudn6 zprdva z m ~ e n l ovzduM v Ceskd repulbice; What Warrant the LIDAR: A short report from the air pollution control in Czech Republic, (~eskoslovenski¢ ?zasopis pro fyiku 45, 216-228.

Remote Sensing of the Environment using Laser Radar Techniques

269

Ferrara R., Maserti B.E., Andersson M., Edner H., Ragnarson P., Svanberg S. and Hernandez A. (1996): Atmospheric mercury concentrations and fluxes in the Almadgn district (Spain), Atmos. Env., in press. Johansson J., Andersson M., Edner H., Mattsson J. and Svanberg S. (1996): Remote fluorescence measurements of vegetation spectrally resolved and by multi-colour fluorescence imaging, J. Plant Physiology, 148,632-637. Letokhov V.S. (ed.) (1990): Laser Analytical Spectrochemistry, (Hilger, Bristol). Liehtenthaler H.K. and Rinderle U. (1988): The role o] cholorophyll fluorescence in the detection of stress conditions in plants, CRC Crit. Rev. Anal. Chem. 19,

Suppl. 1, $29-$88. Measures R.M. (ca.) (1988): Laser Remote Chemical Analysis, (Wiley, New York). Pettit G. and Waynant R.W. (eds.) (1996): Lasers in Medicine, (Wiley, New York), in press. Radziemski L.J., Solarz R.W., Paisner J.A. (eds.) (1987): Laser Spectroscopy and its Applications, (Dekker, New York). Reuter R. (1991): Hydrographic applications of airborne laser spectroscopy, in Martellucei S. and Chester A.N. (eds.): Optoeleetronics for Environmental Science, (Plenum, New York) pp. 149-160. Sigrist M.W. (ed.) (1994): Air Monitoring by Spectroscopic Techniques, (Wiley, New York). Svanberg S. (1989): Medical applications of laser spectroscopy, Physica Scripta T26, 90. Svanberg S. (1991): Environmental monitoring using optical techniques, in W. DemtrSder and M. Inguscio (eds.): Applied Laser Spectroscopy, (Plenum, New York) pp. 417. Svanberg S. (1994): Differential absorption lidar (DIAL), in Sigrist M.W. (ed.) : Air Monitoring by Spectroscopic Techniques, (Wiley, New York) Chap. 3. Svanberg S. (1995): Fluorescence lidar monitoring o] vegetation status, Physica Scripta T58, 79. Svanberg S. (1996a): New developments in laser medicine, Physica Scripta, in press. Svanberg S., Anderson M., Andersson P., Edner H., Johansson J., Ferrara R., Maserti E., Cecchi G., Pantani L., Mazzinghi P., Alberotanza L., Cioni R. and Caltabiano T. (1996b): Laser monitoring of the environment, in Inguscio M., A1legrini M. and Sasso A. (eds.): Laser Spectroscopy, (World Scientific, Singapore) pp. 423. Wallinder E., Edner H., Ragnarson P. and Svanberg S. (1996): Vertically sounding ozone lidar system based on a KrF excimer laser, Physica Seripta 55(6), 714. Weibring P., Andersson M., Edner H. and Svanberg S. (1996a): Emission measurements of industrial gases with lidar and wind videography, 18th International Laser Radar Conference, Berlin, July 22-26, 1996. Weibring P., Andersson M., Edner H., Svanberg S., Cecchi G., Pantani L. and Caltabiano T. (1996b): to appear.

Applied Laser Spectroscopy in Combustion Devices Volker Sick, Jiirgen Wolfrum Physikalisch-Chemisches Institut, Uaiversitgt Heidelberg Im Neuenheimer Feld 253, 69120 Heidelberg, Germany

1

Introduction

Experimental possibilities for studying complex chemical systems, such as combustion devices, have expanded quite dramatically in recent years as a result of the development of various laser sources with high temporal, spectral and spatial resolution. In addition to illuminating microscopic details, laser spectroscopic methods are especially important for nonintrusive measurements in practical systems in which chemical kinetics is coupled with transport processes. Data gained from such experiments are the basis for comparison with detailed mathematical modeling of laminar and turbulent reactive flows including heat and species transport. Mathematical simulation models can help to find optimal conditions for the various combustion parameters to lower pollutant formation and fuel consumption. On the other hand in situ laser diagnostics offers also new ways for the active control of industrial combustion systems in order to maintain the optimum process conditions. 2

Imaging of NO O t t o Engines

Concentrations

and Temperatures

in

Among the most widely applied laser-spectroscopic techniques for combustion diagnostics is laser-induced fluorescence (LIF). Many combustion related molecules, like OH, CH, NH, O~ and NO have absorption spectra in the ultraviolet spectral region, where they can be conveniently excited with commercially available laser sources. Detection limits in the ppm range are easily obtained. However, the quantitative determination of concentrations is often complicated, due to the lack of information about eollisional effects on the excited species. The measured fluorescence intensity ]LIF can be described by ILm = NoIBBEL

Teff ~

.r

g(r,) ---c ~? v Tra d 471"

(1)

where No is the number density of molecules in the electronic ground state, the quantity one is interested in, fB denotes the Boltzmann fraction of the

272

Volker Sick, J/.irgen Wolfrum

level where the excited transition starts, B is the Einstein coefficient, EL the laser pulse energy, 9(u) is the spectral overlap integral of the molecular absorption line with the laser emission. The ratio of the effective lifetime ref~ to the radiative lifetime rr~a accounts for fluorescence losses due to radiationless deactivation of excited molecules by collisions, a~/4rc gives the solid angle from which fluorescence photons are collected with the detection optics. accounts for losses of fluorescence photons due to absorption in the collection optics and the filters that are used, whereas ~ gives the photoelectron efficiency of the detection system and V is the laser illuminated volume that is imaged onto the detector. The application of LIF for measurements of NO using the A-X (0,2) absorption band near 248 nm [1] in an internal combustion engine is shown in Fig. 1. Narrowband laser pulses of this wavelength with pulse energies of more than 200 mJ can be generated with a tunable KrF excimer laser and expanded to a thin light sheet for planar excitation of NO inside the engine. The engine under study allowed optical access to the entire combustion chamber through cylinder walls made from synthetic quartz [2]. Perpendicular to the light sheet an image intensified camera (ICCD) was mounted to detect the fluorescence signals. On the opposite side of the engine, a second camera detects Rayleigh scattering, induced with the same laser pulse as the LIF signals. The Rayleigh signals/Rayleigh c a n be converted into temperature

I

Top View:

_j~[~." "Jsparkl light Valves ~ p-'~-ug-~"~sheet

Tunable KrF ExcimerLaser

J

JJ

.,,O~--~Filter ~ Piston ~ ..... ~--['/' !~UU .L;arnera........ y (Hayielgnbcattermg)

// //

Filter~lCCD I__lI Camera ,¢~1~~1 ........ ~~.~-~ U'~u-t-'rJ

/

//

///.

"

.

~ l]~ Data Acquisitien

Delay CrankAngle ~'.-~./2 Encoder

~

Fig. 1. Experimental setup for simultaneous measurements of nitric . . oxide and temperature distributions in an optieally transparent Otto engine (Daimler-Benz AG)

Appfied Laser Spectroscopy in Combustion Devices

273

fields using the expression $2 /Rayleigh = N ° ' e f f E L ~

e/] V

(2)

Since the Rayleigh scattering signal is proportional to the number density N and the effective scattering cross section eefr, temperatures can be deduced using the ideal gas law, provided that c%ef is known, cr~er is the summation over all mole fraction-weighted Rayleigh cross sections from all species present in the gas mixture. For combustion conditions there is a sudden change in ~ f f between unburnt and burnt gases. The Rayleigh images must then be analyzed with spatially varying scattering cross sections. Absolute determinations of e and ~ are difficult and usually a reference image at ambient pressure and room temperature is recorded to eliminate e and r1 from the data reduction. Because of the resonant character of Rayleigh scattering reflections of the laser beam at surfaces must be avoided. To account for small amounts of reflected light, background images that are subtracted from the actual data later, are usually recorded with the measurement area flooded with helium. Helium has a very small ~r and thus the measured signal can be attributed to undesired reflections [3]. For the analysis of NO-LIF signals measured in a running engine, the variation of g (u) must be carefully investigated. The NO linewidths vary from 0.5 cm-1 at atmospheric pressure to more than 10 cm-1 under engine conditions. Then all rotational lines blend and the resolved rotational structure found at atmospheric pressure is lost. Only a detailed modeling of the absorption spectrum allows the evaluation of the overlap integral. For each rotational transition a Voigt profile with a pressure- and temperature-dependent width and frequency-shift is calculated. Spectra, measured in a high pressure burner in m e t h a n e / a i r flames [4] and corresponding model calculations are shown in Fig. 2.

lbar

LIF

=

Me o emont

,

I

=

Model I

40325

I

40345 Frequency [cm-1]

40365

40325

40345 Frequency [cmq ]

I

40365

Fig. 2. Measured and calculated NO A-X(0,2) LIF spectra. Data were taken in methane/air flames.

Volker Sick, Jiirgen Wolfrum

274

Collision quenching leads to a decrease of refr by radiationless deactivation of excited molecules and thus to a decrease of the fluorescence intensity. The quenching rate scales linearly with pressure, but is usually a sensitive and complicated function of t e m p e r a t u r e and gas composition. It turns out, however, that fortunately under combustion conditions where the NO is formed, the quenching rates are nearly independent of t e m p e r a t u r e and the gas composition. Only the variation with pressure must then be included in the d a t a analysis [5]. To achieve a high detection sensitivity, the optical system used must cover a large solid angle w/47r to collect as m a n y photons as possible. For the NO measurements, specially designed achromatic lens systems with f # = 2 were used. Additionally, the optical system must have a high transmission efficiency ¢, anti-reflection coatings for the lenses and high efficiency broadband reflection filters. Finally, the detector efficiency r/must be included in the d a t a analysis. To be able to fully calculate the fluorescence signal from a given concentration or vice versa, all of these factors must be known very precisely, including the spatial overlap of excitation and detection volume V. In practical applications it is therefore usually easier to calibrate these factors by adding known amounts of nitric oxide to the engine and determine a calibration factor for the measurement of absolute nitric oxide concentrations in the engine. In a combustion environment, especially at high pressures when the lines are broadened a laser tuned to an NO absorption line can also excite other

2

0

1

3

I

i

I

3

4

I

I

4 I

7 I

Filter

9

8 I

I

N2

H20

h$

I.S

5 I 10

I

NO (0,v")

Oz (o,v") Raman

l o .=_ m .1

215

0

i

225

235

265 245 255 Emission wavelength lnml

275

285

295

Fig. 3. Emission spectrum measured in a NO doped CH4 /air flame at 40 bar excitation at 247.95 nm). The bandpass filter used for the selective detection of NO-LIF signals was designed to transmit the A-X(0,0) and (0,1) bands between 220 and 240 nm.

Applied Laser Spectroscopy in Combustion Devices

2 .

I

1.5

'

275

I

J

4000

~ . ~ m

•

3000 ,..]

0

x 1016 N O molecules/cm 3

3 o

300

T [K]

/W

2000

w,

3000 0.5 ©

"../i -2

•

4 cm

-1 0 1 Distance from Spark Plug [cm]

2

>

Fig.4. Corresponding NO and temperature distributions measured in apropane-fueled Otto engine with laser induced fluorescence and Rayleigh scattering. The profiles show a horizontal cut through the images, as indicated by the arrows.

species. A carefully selected combination of excitation and detection wavelength is necessary for specific detection of the desired species. In the case of NO, the laser is tuned to the O12 bandhead. Here many transitions of NO are excited simultaneously and the absorption spectrum of molecular oxygen has a local minimum. The NO fluorescence signals, which are emitted in the A-X (0,0) and (0,1) bands at shorter wavelengths are detected (Fig.3). This scheme provides a possibility to reject interference with fluorescence from molecular oxygen and Stokes-shifted Raman signals. Additionally, the short laser wavelengths used in the experiments can excite a variety of intermediates, found during the combustion of hydrocarbons. These compounds (unsaturated hydrocarbons, aromatics etc.) have strongly red-shifted emissions, which are cut off with the filter used for the detection of nitric oxide [1, 4]. Figure 4 shows an example of an image pair with corresponding nitric oxide concentrations and temperature fields, measured simultaneously in the transparent engine. It can be seen that the formation of nitric oxide occurs in high temperature areas. While the overall spatial distribution of nitric oxide and temperature is strongly correlated, the profiles in Fig. 4 show that the temperature distribution is more uniform than the NO concentration distribution. This is an important result for the comparison with mathematical models which are developed for engine design. Due to the strong nonlinear temperature dependence of the NO production rate, careful control and homogenization of the combustion conditions should allow significant reduction of the primary NO formation.

276

3

Volker Sick, Jfirgen Wolfrum

Imaging of Fuel Distributions in Otto Engines

Lean-burn engines are under development to decrease the formation and release of pollutants from automobiles. To ensure reliable ignition of the fuel/air mixture in a lean-burn engine, the goal is to stratify the fuel inside the cylinder to ensure an ignitible mixture near the spark plug. LIF-imaging of the fuel distribution can be used to visualize the mixing and the fuel distribution. This was done in a four cylinder engine (a slightly modified series engine from Peugeot SA, Fig. 5) with LIF of (1 - 5 %) diethyl-ketone, which was used as fluorescence marker for the fuel. Diethyl-ketone is well-suited for this purpose, since its boiling point matches that of iso-octane and the fluorescence intensity is proportional to the concentration, independent of t e m p e r a t u r e and pressure over a wide range [6]. The measurements (Fig. 6) show the fuel distribution as a function of crank angle. Combustion is marked by the disappearing LIF signals.

Intake Valve g tlet Valve

E

.ight Sheet

or

gnal

4

Fig. 5. Optically accessible Otto engine. The engine is mounted at an angle of 300 as it is used in cars. Quartz windows in the crank case and the piston allow laser-based measurements of in-cylinder processes.

Thermography and Laser-assisted Combustion Control S y s t e m for Waste Incinerators

A novel type of combustion control system for municipal waste incinerators is possible using infrared thermography to obtain information about the temperature distribution in the furnace interior. A scanner camera operating in

Applied Laser Spectroscopy in Combustion Devices

)++~ :,"+~';+: i . }!:: : : . . . . . . . . . . . . .

:

277

unburnt burnt

350.6 deg c.a.

l/

355.4 deg c.a.

U : +~ i ,. ,:+, : °:.,.,+•,~: 353.0 deg c.a.

354.2 deg c.a.

• +

359.0 deg c.a.

Fig. 6. Start of combustion and flame propagation in an iso-octane fueled Otto engine, visualized with laser-induced fluorescence of diethyl-ketone.

361.4 deg c.a.

the mid-infrared was installed at the waste-to-energy plant in Coburg (Germany) (Fig. 7). Spectral selection of the emission wavelength interval around the 3.9pro region allows direct imaging of the fuel bed through the overlying flame and flue gas atmosphere and the determination of t e m p e r a t u r e distributions within seconds. These d a t a can be used for fine tuning of the total underfire air (UFA) flow. Local imbalances in fuel bed t e m p e r a t u r e are evaluated to redistribute UFA between individual grate zones. The influence of variations in the calorific value of the waste on the position of the main zone of combustion can easily be measured, thus allowing the sensitive adaptation of UFA distribution to local requirements. Incorporation of the camera system results in an improved overall consistency of the combustion process, as reflected by a reduction of excess air fluctuations and of t e m p e r a t u r e imbalances across the grate. Measurements in the Coburg plant have shown that flue gas as well as residue burn-out quality was improved. Compared to conventional control, emissions of carbon monoxide, unburnt hydrocarbons and the fraction of unburnt m a t t e r in the residue were significantly reduced [7]. In addition a Laser In-Situ A m m o n i a - m o n i t o r (LISA) based on a 13CO2waveguide laser was developed to establish an on-line control of the Selective Non-Catalytic NO~ Reduction (SNCR) in an industrial environment by measuring in-situ the excess a m m o n i a behind the reduction zone [8]. The determination of the NH3 concentration by differential absorption techniques is based on the Lambert-Beer law. Letting Ul, u2 denote the aaco2-1aser emission lines coinciding with the NHa absorption m a x i m u m and m i n i m u m and l the absorption p a t h length, the ratio of the transmitted laser intensities is

l(ul) _

f(v2)

lo(ut) exp(_ONH3NNH3/) t0Cv+)

(3)

Volker Sick, Jiirgen Wolfrum

278 -- -- 7

',CCS Il

IR-scanner-

~

camera

i 13C02.laser

i

I

[7

I LISA

I

M

I

I

I t. .

I .

.

.

.

.

.

.J

1000°C

500°C

fuel bed temperature distribution

Fig. 7. Schematic view of the thermography system with an example of a temperature field. Also shown, is the LISA system for non-catalytic reduction of nitric oxide by ammonia injection.

with the collision broadened ammonia absorption coefficient O~NH3 . If I R (ul), IR(u2) denote the intensities at a reference detector, the NH3 concentration is given by -1

In [/(/-'1) In(u2)]

(4)

NNH3 -- O~NH3-""~ [1(/-'2) IR(Pl) With this differential absorption spectroscopy sensitivities in the order of several p p m . m of NH3 at atmospheric pressure have been achieved even in heavily dust laden flue gases. The system allowed a 90 % reduction of the NO emissions from the waste-to-energy plant and a significant decrease of the residual ammonia level after the NO reduction zone.

5

In S i t u A l k a l i C o n c e n t r a t i o n M e a s u r e m e n t s Pressurized Fluidized-Bed Coal Combustor

in a

The release of alkali species is a factor of considerable uncertainty in evaluating advanced solid fuel conversion systems. New combined cycle schemes for

Applied Laser Spectroscopy in Combustion Devices

279

electricity generation promise net efficiencies on the order of 50 % for pressurized coal fired power plants. However, hot gas cleaning, i.e. the removal of particulates and detrimental trace gases such as evaporated alkali compounds at suitable gas turbine inlet temperatures, is still a problem to solve if such efficiencies are to be realized. For the entrance to turbines, corrosion damage due to alkali compounds is feared even at ppb concentrations. So far, classical collective sampling techniques have frequently been used due to their common availability, but require sampling/analysis times of more than 1/2 hour, in pressurized systems often of many hours. Transient behaviour, e.g. upon changes of combustion conditions, cannot be detected in this way, nor can different forms of alkali be clearly discriminated. A suitable on-line method for detecting gas phase alkali concentrations is excimer laser induced fragmentation fluorescence (ELIF)[9, 10]. Alkali compounds (MX) were photolyzed at 193 nm by a broadband ArF excimer laser using energy densities of several m J / c m 2 whereby simultaneously excited Na(32p) and K(42P) atoms (M*) are formed. MX --+ M* + X

(5)

Fluorescence from M* was detected at 589 nm (sodium) and 768 nm (potassium), respectively. A calibration factor Cdet can be determined using the alkali number densities in a calibration cell obtained from thermodynamic data and the following relation for the fluorescence signal IELIF /ELIF ~-~ CdetELNMxO%bsq

(6)

where EL is the effective laser energy in the reactor, NMX the number density of the alkali salt MX, a~bs its UV absorption cross section for production of M*, and q describes the fluorescence quenching effect: q = A / ( A + Q), Q = E

nivi~i

(7)

i

with the Einstein coefficient A of the excited alkali atom and Q the quenching rate. The value of Q was determined using measured quenching cross sections ai, the number densities ni of the quenching partners i and the relative velocities vi of i and M*. The most important quenching partners in the flue gas were N2, CO2, 02 and H 2 0 with typical mole fractions of 0.76, 0.14, 0.05, and 0.04, respectively. The pressurized fluidized bed reactor used for the measurements has a rating of about 100kW at 10 bar. It comprises a lower, bubbling fluidizedbed combustion zone of 1.5 m bed depth and an upper, extended freeboard zone (Fig. 8). After leaving the reactor, part of the ash is removed from the flue gas in a cyclone located immediately in front of the optical access port for the alkali measurements. To allow operation at 10 bar, the reactor head, flue gas pipe, cyclone and optical access were designed using heat resistant steel. Care was taken to optimize the N2-flushing of the windows to minimize

280

Volker Sick, Jiirgen Wolfrum

f

Trace heated hot gas duct Extended freeboard Pressure shell Heating jacket

Cooling water - outlet

N2 ushngi xaaner K

JCooling jacket

# ~ C o o l i n g water inlet Ash offtake pipe - -

Electric start-up heater

Inlet primary air 500mm

~

l

~

~ m N.d. filters I I Atomic line filter

~Na

Fig. 8. Pressurized fluidized bed reactor and ELIF setup for alkali detection.

dust deposition. Figure 9 shows representative results for experiments under various conditions. In each case, the temporal course of potassium and sodium concentrations is shown together with the corresponding history of the flue gas temperature. The general concentration level for both alkali compounds was below 10 ppb for the hard coal, but in the range 300-800 ppb for the lignite. A major reason for the large difference in measured concentrations for the two coals is probably their very different mineral composition. Analysis of the ash samples shows a very high proportion (,~ 30 % wt) of alumosilicates in the hard coal, which are known to retain alkali compounds strongly at the temperatures of FB-combustion. In comparison, the lignite ash contains only ,~ 3 % wt alumosilicates, even though the amount of ash is about the same (~ 5 % wt) for the two coals and the proportion of alkali is significantly higher in the hard coal. On-going developments and refinements of the ELIF-method will now allow reliable and detailed investigations of alkali release and conversion in pressurised coal combustors in actual power plants. Furthermore, the extension of the ELIF method to heavy metal compounds appears feasible.

Applied Laser Spectroscopy in Combustion Devices

8OO '-z

~-

.=

~

10OO

820

800

8oo 0~

75o ~

g6

•,~

~4

780 g a" 760

6oo

~ 400

8

650 ~

~2

"<

I Hard coal[ 0 , : ', : : : ', : ', : ', : ,, 600 9 10 11 12 13 14 15 Time [hours]

d

281

_

740 2"

2oo

720 ?

I

0

I

11

.i,e I I

i

I

13 12 Time [hours]

700

4

14

Fig. 9, Sodium and Potassium concentrations in the pressurized fluidized-bed combustor, measured with ELIF.

6

Acknowledgement

The work presented here was funded by the B u n d e s m i n i s t e r i u m fiir Bild u n g und Forschung, contract No. 13N6283 3, the Commission of the European Communities, contract Nos. CT92-0037 and C T 95-0010, Land BadenWiirttemberg, contract No. 0 1 B W l 0 0 1 , Martin G m b H and Peugot SA and Daimler-Benz AG.

References [1] C. Schulz, B. Yip, V. Sick, J. Wolfrum, Chem. Phys. Letters 242, 259-264

(1995) [2] C. Schulz, V. Sick, J. Wolfrum, V. Drewes, M. Zahn, R. Maly, 26th Symp. (Int.) on Combustion, Naples, Italy (1996) [3] A. Orth, V. Sick, J. Woffrum, R. R. Maly, M. Zahn, 25th Syrup. (Int.) on Combustion, 143-150 (1994) [4] C. Schulz, V. Sick, J. Heinze, W. Stricker in Laser Application to Chemical and Environmental Analysis, Vol.3, OSA Technical Digest Series 1996, (Optical Society of America, Washington DC), 133-135 (1996) [5] A. Brgumer, V. Sick, J. Woffrum, V. Drewes, R. R. Maly, M. Zahn, SAE Paper No. 952462, Toronto (1995) [6] F. Grofimann, P. B. Monkhouse, M. Ridder, V. Sick, J. Woffrum, Appl. Phys. B 62, 249-253 (1996) [7] F. Schuler, F. Rampp, J. Martin, J. Wolfrum, Comb. Flame 99, 431-439 (1994) [8] W. Meienburg, J. Woffrum, H. Neckel, 23rd Syrup. (Int.) on Combustion, 231236

(199o)

[9] R. C. Oldenburg, S. L. Baughcum, Anal. Chem. 58, 1430-1436 (1986) [10~ F. Greger, K. T. Hartinger, P. B. Monkhouse, J. Wolfrum, H. Baumann, B. Bonn, 26th Syrup. (Int.) on Combustion, Naples, Italy (1996)

The. Surface of Liquid H e l i u m - an U n u s u a l S u b s t r a t e for U n u s u a l C o u l o m b S y s t e m s P. Leiderer Fakult~it fiir Physik, Universit£t Konstanz, D-78434 Konstanz

1

Introduction

The study of basic physical problems has always benefited from a choice of systems which are as simple as possible. In atomic physics such a system is apparently the hydrogen atom. I would like to discuss in this lecture an example in the field of condensed matter physics, which in its simplicity, but moreover also with respect to some of its particular features bears some resemblance with the hydrogen atom. The topic are electrons, trapped at the surface of liquid helium [1]. Imagine an electron, supplied by a suitable source in the vapor space above the helium surface, which is pulled towards the liquid by an externally applied weak electric field. Close to the surface the potential for the electron, given by the image charge resulting from the polarization of the liquid, is

1)

V(z)

--

4z(e+l)

(1)

where z and e are the distance from the surface and the dielectric constant of helium, respectively. For z < 0, i.e. inside the liquid, the potential is about + 1 eV. The electron is therefore trapped above the surface in a potential well, acting perpendicular to the surface, whereas parallel to the surface it is (nearly) free to move. The potential well given by Eq. (1) is in its z-dependence identical to the radial dependence of the hydrogen potential. Hence we obtain similar energy levels for such a surface state electron (SSE) on helium as for the hydrogen atom, yet with a pre-factor which is four orders of magnitude smaller, because the image charge is only a small fraction of the elementary charge. The ground state, for example, instead of being 13.6 eV is only 1 meV below the vacuum level. The corresponding effective "Bohr radius", describing the average distance of the electron from the surface, is 76A, large compared to the interatomic spacing of the helium surface, so that this "substrate" appears essentially smooth for the electron 1. Experimentally, the existence of 1 apart from quantized surface waves (ripplons) from which the electrons can be scattered as discussed below

284

P. Leiderer

1~2 D CO {Z

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O

I

I

I

I

I

I

/

I

I

[

2

t,

6

8

I0

t2

!/,

16

18

20

POTENTIAL DIFFERENCE ACROSS

CELL (VOLTS}

Fig. 1. Experimental trace showing the absorption derivative due to surface state electrons on liquid hefium versus voltage across the experimental cell. The incident microwave frequency was 160 GHz and the temperature 1.2 K. The linear Stark effect was utilized to tune the splitting between bound electronic surface states to resonance with the incident radiation. The 1--+2 and 1--+3 transitions are analogous to the Lyman-a and -/3 transitions of hydrogen. (From Ref. [2]).

the hydrogen-like energy levels of SSE was confirmed by Grimes and Brown using a spectroscopic measurement in the microwave regime [2]. (See Fig. 1). If we put more than one electron into the surface potential well, we have to consider the Coulomb interaction a m o n g the charges, and also the Fermi nature of the electrons. The latter aspect, however, will only become important if the thermal de Broglie wavelength /~th is comparable to or larger than the interparticle distance r. For Ath < r the electrons are expected to behave classically in their motion parallel to the surface. Since most of the experiments to be discussed here were carried out in the t e m p e r a t u r e range around 1 K, corresponding to /~th ~' 103 ]k, the electrons can be treated as a classical 2-dimensional Coulomb system up to densities of 101° e / c m 2.

2

Some Basic E x p e r i m e n t s with Surface State Electrons

In a way, the electron sheet on liquid helium represents the most simplified and abstract realization of a physisorbed film on an inert substrate. It turns out, nevertheless, that in spite of this simplicity it provides a rich variety of physical phenomena: As already indicated the electrons are essentially free to move in directions parallel to the surface. Scattering by ripplons and, at temperatures above ~ 1 K, by atoms in the gas phase limits the mobility. In addition

The Surface of Liquid Helium - an Unusual Substrate

285

the Coulomb repulsion between the charges can lead to strong correlations in the electronic motion. The latter effect depends on the plasma p a r a m e t e r F, which is the ratio between the Coulomb energy and the kinetic energy of the particles. As long as F is below unity, the electron sheet can be considered as a 2-dimensional gas. For Y ) 1 the correlations lead to a liquid-like behavior, as observed by Zipfel and coworkers [3]. ii) Collective excitations in the SSE sheet are expected to be 2-dimensional plasmons (density waves), and these plasmons have also been found experimentally. At electron densities of l0 s e / c m 2 and wavelengths on the order of a few m m the frequency of these plasmons is in the MHz range [4] (Fig. 2). dAldN

f

Fig. 2. Experimental traces of the derivative with respect to electron surface charge density of If absorption plotted versus frequency. The upper trace displays a plasmon standing wave resonance having Q ~ 4 centered at a frequency of 52 MHz. The lower trace was taken with 1 dB greater rf drive power where heating of the electrons occurs near the center of the standing wave resonance and causes structure to appear in the lineshape. (From Ref. [4]).

iii) Magneto-transport investigations in B-fields up to 2 0 T have revealed a surprisingly complex behavior, already if it comes to understanding the contributions of the two scattering mechanisms of the surface state electrons mentioned above. Besides, many-electron effects have been shown to play an i m p o r t a n t role [5]. iv) A series of experiments has also been devoted to the tunneling of the electrons out of the potential well at the surface, which might take place if for a short time the external electric field, usually applied to draw the electrons towards the surface, is reversed [6]. It could be shown that at temperatures below 0.3 K tunneling through the resulting potential barrier is the dominating mechanism indeed, whereas for higher T thermal activation over the barrier determines the electron lifetime in the surface states. These SSE experiments have contributed to the general under-

286

P. Leiderer

standing of tunneling, in particular in the presence of a magnetic field, and have revealed, e.g. the existence of a "correlation hole" in the 2-D Coulomb system, left by the tunneling electron, which contributes to the potential relevant for the tunneling process. v) One of the most intriguing phenomena in the SSE sheet, which has attracted considerable attention, is the phase transition from an electron liquid to a 2-D solid at plasma parameters F > 130. As demonstrated by Grimes and Adams in a pioneering experiment [7], the electrons are then localized on the helium surface in a lattice, which has triangular symmetry. This was derived from the sequence of resonances observed in the absorption spectrum of the SSE system as the temperature was lowered and the phase transition from the liquid to the solid state took place (see Fig. 3). In subsequent studies the phase transition was found to be of the Kosterlitz-Thouless type (i.e. a continuous phase transition, where the loss of crystalline order upon heating is due to the dissociation of dislocation and disclination pairs) [8]. Since the electron solid is characterized by a finite shear modulus, one obtains an additional branch in the plasmon excitation spectrum, namely transverse plasmons. Near the phase transition the transverse plasmon frequency displays a pronounced softening, which also is in quantitative agreement with the model of a Kosterlitz-Thouless transition [9] (Fig. 4). i

oc

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x

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I

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~

I

,

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Fig. 3. Experimental traces displaying the sudden appearance of coupled plasmon-ripplon resonances with decreasing temperature. The resonances only appear below 0.457 K where the sheet of electrons has crystallized into a triangular lattice. (From Ref. [7]).

40

F(MHz) In the classical regime of the SSE system the average kinetic energy of the electrons is given by their thermal energy, equal to kBT for the two translational degrees of freedom in a 2-D system, whereas the Coulomb energy per particle is proportional to the average electron distance and hence to n 1/2 (where n is the electron density). The transition from the electron liquid to the solid should therefore follow a curve n oc T 2. It is to be expected that this dependence for the classical system changes once quantum corrections due to Fermi statistics come into play. Since the Fermi energy in 2 dimen-

The Surface of Liquid Helium - an Unusual Substrate

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TEMPERATURE |inK)

sions is proportional to n, the Fermi character should eventually dominate completely, and the system should exhibit another phase transition, from the electron crystal to the degenerate 2-D Fermi gas, if a certain electron density is exceeded. The electrons are then no longer confined to lattice sites, but should be in a delocalized state comparable to the situation of the 2-D electron gas at semiconductor interfaces like M O S F E T S and heterostructures. Such a transition was predicted already six decades ago by E. Wigner for 3D Fermi systems 2, whence the name "Wigner crystal" for the electron solid was coined. The total phase diagram expected on this basis is shown in Fig. 5 [10]. The phenomenon of " q u a n t u m " melting at high densities, where the loss of crystMline order is caused by q u a n t u m statistics rather than by thermal excitations, is a very interesting phase transition, which has been looked for in various 2-D systems, also in the context of the q u a n t u m Hall effect in semiconductor structures. We shall briefly come back to it later in this lecture.

3

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It would appear as a challenging experiment to image the electrons in the Wigner crystal by some suitable method and thus obtain direct information about details of its structure, like defects, grain boundaries etc.. Such an experiment has so far not been carried out. I can, however, present a picture of a macroscopic counterpart of the microscopic Wigner crystal, shown in Fig. 6. In this photograph the white dots do not represent individual electrons, but 2 To be more precise, Wigner considered the phase transition in the opposite direction, arguing that the degenerate Fermi gas of the electrons in a metal should crystallize if its density could be reduced sufficiently.

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/

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a c c u m u l a t i o n s of ~ 106 e, which are localized in m a c r o s c o p i c d i m p l e s on t h e h e l i u m surface. T h i s " d i m p l e c r y s t a l " forms as a result of the c o u p l i n g between the m o d e s of the S S E s y s t e m and t h e liquid surface, l e a d i n g to an e l e c t r o - h y d r o d y n a m i c i n s t a b i l i t y of the surface [11]. T h e i n i t i a l l y h o m o g e neous charge d i s t r i b u t i o n is t h e n broken u p into a r e g u l a r a r r a y of electron clusters w i t h a l a t t i c e c o n s t a n t equal to 27ra, where a is t h e c a p i l l a r y l e n g t h of t h e h e l i u m surface 3. By v a r y i n g t h e a p p l i e d electric field, which pulls t h e electrons a g a i n s t the surface, t h e s y s t e m can reversibly be cycled b e t w e e n the h o m o g e n e o u s electron s y s t e m a n d the d i m p l e c r y s t a l . Since the d o m i n a t i n g

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3 This instabifity is the reason why the high-density regime of the phase diagram in Fig. 5 is not readily accessible with electrons on He. The density limit on bulk helium is nc ~ 2 • 109e/cm 2.

The Surface of Liquid Helium - an Unusual Substrate

289

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Fig. 7. Calculated electron density n(r) and surface profile z(r) of a dimple containing 5 * l0 s electrons in an electric field E = 3400 V/cm. The 4He temperature is 2.5 K. (From Ref. [12]).

interaction in this macroscopic array is the Coulomb repulsion as for the microscopic Wigner crystal, these two lattices, in spite of their vastly different length scales, should be quite analogous. One might even think of studying the (classical) melting of the dimple crystal as the temperature of the helium substrate is increased, in order to learn about the underlying mechanisms of such a phase transition. In analogy to the microscopic Wigner crystal one would expect this transition to be of the Kosterlitz-Thouless type. As an attractive experimental advantage the dimple system would allow directly the observation of dislocation and disclination pairs, which are essential features of this kind of phase transition. One can easily convince oneself, however, that the melting temperature would be higher than 10 3 K, due to the strong Coulomb interaction of the dimples. Hence thermodynamic melting phenomena are unfortunately not accessible with this macroscopic system. The charged dimples can exist on the liquid helium surface not only in the form of a regular array, but also as individual objects. Figure7 shows the calculated charge distribution and the corresponding surface deformation of such an isolated dimple. If the external electric field is raised, the local depression of the surface increases in depth, and the charges are concentrated more and more in the centre [12]. Eventually, the local pressure in the dimple can become so large that it is no longer compensated by the restoring forces due to buoyancy and surface tension, and the electrons punch through the surface in the form of a charged bubble. The size of this bubble is determined by the balance of surface tension and the Coulomb repulsion of the roughly 106 electrons at the inner bubble surface, and is typically some ten pm. In contrast to the one-electron bubbles in liquid helium [13], which are very stable, the multi-electron bubbles are quite "soft" and prone to instability. As they move through the helium under the influence of the external electric field, they pick up energy which is transformed to bubble oscillations and from there is dissipated into the surrounding liquid. Due to this internal degree of freedom, data for the drift velocity of the bubbles do not follow

290

P. Leiderer

~t

Fig. 8. Track of a fissioning multi-electron bubble, moving from the top of the picture towards the anode pin at the bottom. The illuminating laser beam was chopped at a frequency of 370 Hz, causing the bubble track to appear interrupted.

a simple Stokes law, and moreover exhibit pronounced scatter. The oscillations can even be excited to such a large amplitude that fission of the bubble occurs, as it is shown in Fig. 8 [14]. 4

Phase

Diagram

of the

SSE

System

Let us now come back to the phase diagram of the surface state electrons in Fig. 5. From the total charge of a multi-electron bubble and its small surface area it follows that this system, which locally can be considered as being 2-dimensionM, has an electron density well above the regime of the classical Coulomb gas. Rather, the electrons should form a Wigner crystal, and even the quantum melting of the Wigner crystal into a degenerate Fermi gas should become accessible. It therefore appears quite interesting to study these multi-electron bubbles in more detail. For that purpose it would be advantageous to trap an individual bubble in a suitable field configuration so that it can be investigated over extended periods of time. It has in fact been shown already that it should be possible to set up such a trap using alternating electric fields [14]. (We have also considered a combination of electric and magnetic fields, but the effective mass of the bubbles is so large that tremendous B-fields would be necessary to obtain the desired effect). Investigations of the eigenmodes of such trapped bubbles, i.e. their frequencies and damping, should reflect the mobility and hence also the thermodynamic phase of the bubble electrons. Whereas such experiments on the properties of the electrons in the bubbles still have to be carried out, we have chosen an alternative route to obtain high electron densities and approach the quantum regime in the phase diagram of the 2-D electron system. The electro-hydrodynamic instability,

The Surface of Liquid Helium - an Unusual Substrate

291

which as described before prevents the accumulation of a SSE density beyond 2 , 1 0 9 e / c m 2, can be suppressed by additional stabilizing forces acting on the helium surface. Such forces are provided, e.g. by the van der Waals interaction, if instead of bulk liquid helium a thin helium film on a solid surface of another material is used. It was found indeed that a helium film with thickness of about 100/~ can support electron densities up to 1011 e / c m 2 [15], well inside the range where Fermi effects should be important. For such a system the response in an electric field at microwave frequencies (9 GHz) has been studied [16]. From the position and the width of the resonance of the microwave cavity one obtains the conductivity ~r and susceptibility X of the SSE. For the case of a polymer substrate underneath the helium film results are plotted in Fig. 9. Starting with small electron densities, one is first in a linear regime, where the electrons behave like a classical Coulomb gas, as in the Drude model, and hence both the conductivity and susceptibility are proportional to the electron density. When the electron density is sufficiently large (~., 1.8,10 9 in that case) that the Coulomb interaction outweighs the thermal energy, Wigner crystallization sets in, leading to a kink both in the conductivity and the susceptibility curve, because in the Wigner state the electrons are no longer free to move as before. Figure 10 shows a similar measurement, now with a helium film on a quartz substrate, where due to the larger dielectric constant of this material the Coulomb interaction of the electrons is more strongly shielded, so that Wigner crystallization only sets in around 1011 e / c m 2. Upon increasing the electron density further a and X again remain essentially constant, until at 9 * 1011 e / c m 2 a second, very pronounced feature is observed, where both quantities display a j u m p towards larger values. This behavior, which implies a sudden increase of the electron mobility, could be an indication of the long-sought quantum melting of the Wigner crystal. At this stage clearly more quantitative measurements are required to identify this phase transition unequivocally. Nevertheless, electrons on He films appear as a very promising model system to trace out a large range of the theoretical phase diagram of 2-D electrons as plotted in Fig. 5. Let me close with a suggestion which might seem somewhat speculative, yet bears interesting physics: Since the electrons are repelled by the helium surface if they approach too closely, the same should be true in the reverse situation, where a helium atom approaches a metal surface with the electron wave functions extending far beyond the lattice sites of the corresponding ions. Such surfaces are provided by some alkali metals, in particular by Cs. There the wave functions spill out several/~ngstrSms, leading to a repulsion of the helium atom at distances much larger than for most other surfaces, including a helium substrate itself. As a result, helium does not wet certain alkali surfaces at sufficiently low temperatures [17]. Vice versa, it should be possible to trap alkali atoms at a helium surface above the liquid. First investigations in this direction, with helium clusters as the substrate, have recently been reported [18].

292

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.

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EJektronendichte [xl 0~cm "2]

Fig. 10. Absorption 1/Q and relative susceptibility of a microwave cavity containing a sheet of surface state electrons, like in Fig.9, however with SiO2 (¢ ~ 5) as solid support. Notice the different scale for the electron density. (From Ref. [16]).

T h e examples presented in this lecture can only give a rough impression of the m a n y aspects which make electrons on helium such a unique and fascinating topic. I have focussed here on the properties of the 2-D electron system; other interesting subjects, like the application of SSE as a probe for studying the helium surface, or ions inside the liquid helium [19] could not be addressed in the limited amount of time. I hope that in spite of this restriction the value of a simple, yet appeMing model system for condensed matter physics could be demonstrated.

References [1] See, e.g., Cole, M.W., Rev.Mod.Phys. 46 (1974) p.451; Grimes, C.C., Surf.Sci. 73 (1978) p. 379; Edelman,V.S., Sov.Phys. Usp. 23 (1980) p. 227; Williams, F.I.B., Sur].Sci. 113 (1982) p. 371; Monazkha, Yu.P., and Shikin, V.B.,

The Surface of Liquid Helium - an Unusual Substrate

293

Sov.J.Low Ternp.Phys. 8 (1982) p. 279; Leiderer, P., Physica 1 2 6 B + C (1984) p. 92; Studart, N., and Hipolito, O., 1986, Revesta Brasileira de Fisica 16, p. 194; Shikin,V.B., and Monarkha,Yu.P., Two-dimensional charged systems in liquid helium, (Editorial board for physical and mathematical fiterature Moscow, 1989); Leiderer, P., J.Low Temp.Phys. 87 (1992), p. 247 [2] C.C. Grimes and T.R. Brown, Phys.Rev.Lett. 32,280 (1974) [3] C. L. Zipfet, T. R. Brown and C.C. Grimes, Phys.Rev.Lett. 37, 1760 (1976) [4] C.C. Grimes, Sur].Sci. 73,379 (1978) [5] M.J. Lea, P. Fozooni, P.J. Richardson, and A. Blackburn, Phys.Rev.Lett. 73, 1142 (1994) [6] Andrei, E.Y., Yiicel, S., and Merma, L., Phys.Rev.Lett. 67 (1991) p. 3704 [7] C.C. Grimes and G. Adams, Phys.Rev.Lett. 42,795 (1979) [8] R. Mehrotra, B.M.. Guenin, and A.J. Dahm, Phys.Rev.Lett. 48, 641 (1982) [9] G. Deville, A. Valdes, E. Andrei, and F.I.B. Williams, Phys.Rev.Lett. 53, 588 (1984) [10] F. Peeters and P.M. Platzmann, Phys.Rev.Lett. 50, 2021 (1983) [11] Warmer, M., and Leiderer, P., Phys.Rev.Lett. 42, 315 (1979); Leiderer, P., and Wanner, M., Phys.Lett. 73A, 189 (1979); Ebner, W., and Leiderer, P., Phys.Lett. 80A (1980) p. 277 [12] Leiderer, P., Ebner, W., and Shikin, V.B., Sur].Sci. 113 (1982) p. 405; Physica 107B (1981) p. 217 [13] For a review, see Fetter, A.L., in The Physics of Liquid and Solid Helium, eds. Bennemann, K.H., and Ketterson, J.B. (Wiley, New York, 1976), Vol. 1, p. 207 [14] Albrecht, U., and Leiderer, P., Europhys.Lett. 3 (1987) p. 705 [15] Etz, H., Gombert, W., tdstein, W., and Leiderer, P., Phys.Rev.Lett. 53 (1984) p. 2567 [16] Gfinzler, T., Bitnar, B., Mistura, G., Neser, S., and Leiderer, P., Surf.Sci. 361/362 (1996) p. 831; T. Giknz]er, Ph.D. Thesis (Konstanz, 1994) [17] Cheng, E., Cole, M.W., Saam, W.F., and Treiner, J., Phys.Rev.Lett. 67 (1991) p. 1007; Nacher, P.J., and Dupont-Roc, J., Phys.Rev.Lett. 67 (1991) p. 2966 [18] Stienkemeier, F., Higgins, J. Ernst, W.E., and Scales, G., Phys.Rev.Lett. 74 (1995) p. 3592 [19] See, e.g., Proc. of the Int. Symposium on Ions andAtoms in SuperfluidHelium, eds. H. Gfinther, G. zu Putlitz and B. Tabbert, Z.Phys.B 98 (1995) p. 29

Aspects of Laser-Assisted Scanning Tunneling Microscopy of Thin Organic Layers S. GrafstrSm 1, J. Kowalski 1, and R. Neumann 2 1 Physikalisches Institut, Universit£t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany Gesellschaft fiir Schwerionenforschung, Planckstr. 1, D-64291 Darmstadt, Germany

1 1.1

Introduction Historic Remarks

Tunneling spectroscopy on samples with macroscopic dimensions was performed in early experiments by Giaever [1]. He observed electron tunneling across an insulating aluminum oxide layer less than 50 ~ thick between two metallic A1 electrodes. Searching for the energy gap in a superconductor as postulated by the BCS theory, Giaever later replaced one of the A1 electrodes by lead and measured the current-to-voltage characteristics at liquid-helium temperature [2]. He observed that the tunneling current started to flow above a certain threshold voltage, which clearly demonstrated the existence of an energy gap in the lead superconductor. At even lower temperatures (1.2 K), where A1 too becomes superconducting, an additional structure appeared in the I - V diagram with a negative differential resistance due to the presence of two superconducting electrodes with different energy gaps [1]. These famous experiments, for which Giaever was awarded the Nobel prize in 1973, can be considered as the first example of tunneling spectroscopy, where discrete energy states of the solids involved appear as steps in the I - V spectrum. Bardeen gave a theoretical description of the tunneling process by treating the electron transition between the electrodes by means of time-dependent perturbation theory [3]. The current is determined by the square of the transition matrix element of the electron and by the product of the density of states of the two electrodes involved. Tunneling spectroscopy attracted renewed interest after the invention of the scanning tunneling microscope (STM) by G. Binnig and H. Rohrer. As a matter of fact, their original idea was to perform spectroscopy locally on a fixed area with less than 100 ~ diameter [4] in order to study inhomogeneities of thin oxide layers grown on a metal surface. The idea to use a positionable tip and control the distance via the tunneling current while scanning the tip across the surface opened up the possibility not only of performing local spectroscopy but also of obtaining topographic images: the STM was born.

296

S. GrafstrSm et al.

During the rapid development of the field of scanning tunneling microscopy the question how the energy states of an atom or molecule enter into the observed tunneling current has always been important. In the usual application of the STM the surfaces or surface-adsorbed molecules are in their electronic ground state when being imaged. The question arises, whether molecular or atomic excited levels can be populated by shining intense laser light with suitable wavelength into the tunneling contact while imaging with the STM, and whether the excitation causes a change of the tunneling current. In this way, a characterization and identification of the adsorbed atoms or molecules might become possible. The combination of scanning tunneling microscopy (STM) with irradiation of light into the tunneling region has been applied by several groups in the past under different aspects. On the one hand, the aim was to locally detect light-induced effects by means of the STM. These investigations include the resonant excitation of surface plasmons in thin metal films and their interaction with the STM tip, photovoltaic effects in semiconductors, and the dynamic behaviour of thermal effects arising when modulated laser light is used. On the other hand, the interaction of the optical radiation with the tip-sample system itself was studied, which exhibits a nonlinear electric response. These experiments concerned in particular the mixing and rectification of optical frequencies in the infrared region at the nonlinear tip-sample antenna system. In the STM studies of surface plasmons, laser light is coupled to a metal film via a prism in the Kretschmann configuration [5, 6, 7, 8, 9]. Correct adjustment of the light k-vector leads to resonant generation of surface plasmons, which are scattered by the STM tip. This leads to a change of the reflected light intensity recorded by a photodiode. Scanning the tip across the sample surface yields a nanometer-scale map of the interaction strength between tip and surface plasmons, which is correlated to the topography of the surface and of adsorbates [8]. In another experiment rectification of the optical near-field accompanying the surface plasmons in the tunneling contact was observed [6]. When semiconductors are irradiated by light, the generation of a surface photovoltage (SPV) is one of the most dominating effects. It appears as a shift of the current-to-voltage characteristic of the STM. As an important example the Si(111)-7×7 surface covered with submonolayers of Ag was studied with a STM [10]. The photovoltage produced by the light of a He-Ne laser was monitored with spatial resolution. However, even directly over small, widely separated Ag islands no variation of the photovoltage compared to the surrounding clean Si(lll) surface was observed. The lack of spatial variations of the SPV was attributed to the finite surface conductivity of the Si(lll)7x7 surface. The SPV data from these measurements were independent of the magnitude and direction of the tunneling current. On the other hand, an analogous investigation of the Si(001) surface showed a dependence on the

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injected tunneling current [11]. Spatial variations were observed and were attributed to charging of the surface at specific defects by the tunneling current. The same effect was observed at the S i ( l l l ) - 2 × 1 surface [12]. Thermal effects are unavoidable when shining intense laser light on the tunneling junction of the STM. Intensity modulation of the laser light leads to periodic heating of the tip-sample region causing a modulation of the distance and therefore of the tunneling current. A study of the dynamic behaviour of the current amplitude I versus modulation frequency u of the light gives insight into the heat transfer properties of tip and sample [13]. Material and geometry of the tip as well as the structure of the sample determine the frequency response. We have performed detailed studies of these phenomena, some of which wilt be described below. The tunneling junction of a STM has a nonlinear I - V characteristic very similar to, e.g., a metal-insulator-metal point contact diode. Irradiation of laser light leads to an alternating voltage between tip and sample at the laser frequency and, as a consequence of the nonlinearity, to rectification. A dc voltage appears across the tunnel gap, which depends on the laser intensity. If two laser beams with different optical frequencies are used, frequency mixing occurs, leading to a signal at the beat frequency of the lasers. The laserinduced current amplitudes are proportional to d2I/dV 2, which is a measure of the nonlinearity. A series of experiments have been performed by Krieger et al. [14] using two single-mode CO2-1asers at 9.3 tim wavelength. STM images with atomic resolution were obtained by recording the dc signal caused by rectification, or the mixing signal in the case, where two lasers are used. These experiments aim at an optical spectroscopy of adsorbed molecules on an atomic scale. The amplitude of the measured heterodyne signal is expected to depend on optical resonances of adsorbed molecules or surface excitations like plasmons or excitons. Processes in a certain sense inverse to those described above have been studied by Gimzewski et al. [15, 16, 17], who analyzed the light emitted from the tunneling region. The tunneling electrons excite surface plasmons in the metallic sample, which are scattered into radiative states, whose decay occurs partly via photon emission. Besides the high-energy threshold of the observed spectrum, given by the tunnel voltage according to hv = eU, different resonances occur which depend on the sample material. STM investigations of the wide-band-gap semiconductor CdS yield an additional narrow band structure in the luminescence spectrum d u e to transitions between levels close to the conduction band and the valence band [18]. 1.2

Our Approach

Molecular monolayers on atomically fiat substrates play a central role in current efforts to realize an electronics on a molecular scale. In this context the knowledge of the electronic properties of such layers and their modification by the presence of the substrate and by the interaction within the layers is

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of utmost importance. Scanning tunneling microscopy as a new tool for the analysis of surfaces and adsorbed molecules still lacks the capability to identify adsorbed molecules or constituents of them. It is the aim of this work to realize a local spectroscopy, which combines the high spatial resolution of the STM with the high energy resolution inherent in optical spectroscopy. A laser serves as an intense light source, whose wavelength is tuned to the absorption band of the molecules imaged by the STM. Resonant electronic excitation of the adsorbate is expected to open up an additional path in the tunneling process, leading to a change of the tunneling current. For a sensitive detection of this effect the laser intensity is modulated and the resulting modulation of the tunneling current is detected by means of lock-in technique. Light absorption by tip and sample, however, leads to a periodic thermal expansion resulting in a modulation of the tunneling gap width and hence a modulation of the tunneling current, which exceeds by far any possible resonant contribution. This thermal background and its active compensation by a second laser are discussed in Section 2. Extensive work has been devoted to the preparation of thin organic films containing chromophoric molecules, which can serve as model samples for laser-assisted tunneling microscopy. Several adsorbate-substrate systems are described in Section 3. In first laser-assisted measurements, presented in Section 4, we used films containing two molecular species with different optical properties. The two species can be deposited simultaneously on an atomically flat substrate and form self-organized ordered domains, which should appear with different contrast in the laser-assisted STM image. 2 2.1

Laser-STM

Setup

Thermal Background

The most prominent effect, when intense light is focussed on the tunneling junction, is heating of tip and sample. It leads to a change of the tunneling gap width because of thermal expansion and thus perturbs the tunneling current. Detection of weak light-induced effects, such as the current change resulting from resonant excitation of an adsorbate, requires the usage of modulation techniques to ensure a high enough sensitivity. In the case of a straightforward power modulation, however, thermal expansion leads to a periodic variation of the tunneling gap width and hence to a strong current modulation, which represents a severe obstacle for the detection of weaker light-induced effects. The magnitude of this modulation depends on a number of parameters, such as modulation frequency, focus diameter, material and geometry of tip and sample, etc.. In many cases the contribution of the tip dominates, since the heat produced in the tip by the laser beam can only escape through the shaft in an essentially one-dimensional flow, causing a thermal expansion along its way that contributes fully to the modulation of the tip-sample distance. In contrast, heat conduction in the sample takes place in three dimensions, so

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that the heat deposited in the sample is much more efficiently sunk to the surroundings in a both longitudinal and lateral flow with respect to the tip. Therefore, the temperature rise of the tip may well be larger than that of the sample although the sample absorbs much more light, and the effective thermal expansion of the tip may even by far exceed that of the sample. We measured the thermal gap width modulation as a function of the modulation frequency for different combinations of tip and sample. Figure la depicts the result for a P t / I r tip on a gold sample. The frequency response is characterized by an overall decrease of the amplitude with increasing frequency above a characteristic cut-off frequency, which is approximately 1 Hz in the present case. As a theoretical analysis shows, the cut-off occurs when the thermal diffusion length becomes smaller than the total length of the tip. It is thus directly connected to the length and the thermal constants of the tip [13]. Similarly, the detailed shape of the frequency response at higher frequencies is determined by the geometrical shape of the tip apex, as the fraction of the tip probed by the thermal wave becomes shorter with increasing frequency. The problem can be treated theoretically in a one-dimensional approach, which assumes that the temperature depends only on the coordinate z in the direction of the tip and takes into account the continuous variation of the cross section with z. This model represents an extension of the treatment presented in [13]. Figure lb shows the result of a numerical evaluation for a conically shaped P t / I r tip. The behaviour agrees well with the experimental finding, confirming that the thermal expansion observed in the experiment may be attributed mainly to the tip in this case. The photoinduced thermal expansion can be used as a method for producing a gap-width modulation as required for the measurement of the local apparent barrier height. The photothermal technique offers some advantages at high frequencies where a conventional piezoelectric modulation is ham-

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pered by mechanical resonances causing problems, such as cross talk between longitudinal and lateral tip movements. The photothermal images, obtained by monitoring the modulated tunneling current in this mode of operation, exhibit contrast both due to variations of the apparent barrier height and due to geometrical effects. The latter effects produce a topography-related image with enhanced contrast on corrugated surfaces [19].

2.2

C o m p e n s a t i o n of N o n r e s o n a n t Background

In the context of laser-assisted STM for the detection of resonant adsorbate excitations, the thermal effects represent a strong background signal which must be suppressed efficiently. To some extent the choice of material and geometry of tip and substrate as well as modulation frequency may prevent excessive thermal effects. For example, W is a markedly better heat conductor than P t / I r and hence more favourable as a tip material. A high modulation frequency gives a smaller thermal modulation, but in practice, amplifier noise and bandwidth limit the useful frequency range and make necessary a trade-off. After all, a strong thermal current modulation persists despite the measures mentioned, and an additional compensation is indispensable. With the compensation setup realized in our laboratory, we take advantage of the wavelength dependence of a possible resonant light-induced tunneling current by modulating the frequency of the light instead of its power. A fast frequency modulation of a single laser is however necessarily very limited in amplitude and not suitable for probing broad spectral features as expected for adsorbed molecules. Therefore, rather than performing a true frequency modulation, we switch between two frequencies by power modulating two lasers of different wavelengths with a mutual phase shift of 180 °. With a proper adjustment of the modulation amplitudes, both lasers together produce a constant heating power. In this way the thermal gap width modulation produced by the first laser is exactly cancelled by the second one. More generally, the amplitudes and the relative phase of the two power modulations may be set for complete suppression of the total nonresonant background signal, including also effects, such as thermo- and photovoltage generation or rectification of the laser field in the tunneling junction. The two wavelengths are chosen such that one of them lies inside the absorption profile of the adsorbate under study while the second one is far apart, so that the current modulation caused by the resonant optical excitation is not affected by the compensation system and may be extracted with a lock-in amplifier. The scheme outlined above is realized with a setup that uses electro-optic modulators to produce well-defined and stable modulations of a dye laser and an Ar + laser in a closed-loop configuration [20]. The system suppresses the thermal current modulation by a factor of 4 × 10 -3 at frequencies of up to 50 kHz, powers exceeding 50 roW, and a focal-spot diameter of 15 #m. At these powers, the expected temperature rise is still rather moderate and lies in the range of 15 K for a graphite substrate and 50 K for a typical W tip. It

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30]

should however be stressed that a sufficiently large heat conductance of the sample is important in order to avoid thermal damage, which precludes the use of thin metal films on insulating substrates in these experiments.

3 3.1

Ordered

Molecular

Layers

General Remarks

The aim of studying the optical excitation of physisorbed chromophoric molecules by laser irradiation of the tunneling contact of an STM requires a careful choice and thorough preparation of an appropriate adsorbate-substrate combination. Ordered submonolayers of dye molecules surrounded by a matrix of optically transparent molecules represent a system, which should allow one to monitor a resonant optical excitation of the dye molecules via an increase of the tunneling current with high spatial resolution and sufficient signal-to-noise ratio with respect to a nonresonant environment [20]. The electrical and optical behaviour of physisorbed thin molecular layers is influenced by a variety of characteristics, such as the binding sites, orientation, and formation of domain structures on the substrate surface as well as the anchoring mechanisms and bond strengths, being effective between adsorbate and substrate and between neighbouring molecules. In addition, also substrate properties including electrical and thermal conductivity, and the creation of photo-induced currents and voltages, must be taken into account. Possible applications in molecular electronics represent a major aspect of the studies under consideration here. With the reasonable assumption that a future molecular electronics will have close links to the conventional semiconductor-based microelectronics, semiconducting substrates are especially important. Furthermore, from a practical point of view, semiconducting materials have, in comparison with metals, the advantage of a reduced interaction with the electrons of the adsorbate and therefore less quenching of optically excited molecular states [21]. In the course of the present work, layered crystals, in particular MoS2 and highly oriented pyrolytic graphite (HOPG) - the latter material is a semi-metal -, have proven to be especially suitable substrates. A clean surface, atomically flat o v e r hundreds of nanometers, can be easily prepared by cleaving the crystal parallel to its lattice planes by means of an adhesive tape, immediately prior to depositing the molecular adsorbate. 3.2

C o - A d s o r b e d 8CB a n d P T C D A

We chose the adsorbate combination of optically transparent n-alkylcyanobiphenyl (nCB) liquid-crystal molecules (finally preferring the n=8 type) and perylene-tetracarboxylic-dianhydride (PTCDA) dye molecules. The ordering of nCB liquid crystals on HOPG has been investigated in detail in the past,

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Fig.2. (a) STM image of a PTCDA-island, surrounded by 8CB-liquid crystal on a graphite substrate. The image size is 200 × 200 _~2. (b) High-resolution image of the boundary region between a PTCDA-domaln and an 8CB-domain on MoS2 (image size 101 × 101 /~2).

and can be well explained by the registry of the alkyl chains with the surface lattice of the substrate [22, 23, 24, 25]. The 8CB molecule, normally used in our studies, has an alkyl chain with eight carbon atoms. In most cases, unit cells of eight molecules are formed which exhibit a slight sawtooth-like staggering of subsequent cells. Submonolayer films and small islands of P T C D A were created by evaporating the dye at 2 5 0 - 3000 C in high vacuum [26]. The herringbone pattern of vapour-deposited PTCDA-molecules on graphite and MoS2 can easily be identified and has been studied intensively [27, 28]. The dye films were embedded in an 8CB-matrix by depositing a liquid-crystal droplet in ambient air. The STM images the physisorbed P T C D A islands by "ploughing" with its tip the superposed thick liquid-crystal layer. A STM micrograph of a PTCDA-island surrounded by 8CB is shown in Fig. 2a. There is a clear indication that the liquid crystal has "undermined" the dye island with an 8CB-monolayer which is now acting as a spacer, probably lowering the interaction between the dye molecules and the substrate surface. The structural order of the lowest liquid-crystal layer adsorbed on the substrate is influenced by the features of the substrate surface. Thus, the 8CB pattern on MoS2 is very different from that on H O P G . It consists of parallel rows, the molecules being located at right angles to the row direction and alternately reversing their orientation. This behaviour is displayed in Fig. 2b. The image also visualizes with excellent resolution the transition region between a PTCDA- and a 8CB-domain.

Laser-Assisted STM 3.3

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A New Adsorbed 12CB-Phase on HOPG

When imaging pure 12CB-films adsorbed on H O P G we discovered a 12CBsurface phase not known before [26]. It has a unit cell consisting of ten molecules as compared to the structure with eight molecules per unit cell, which was already described earlier by other authors [23]. The 12CB-films were prepared by means of two different methods: by heating of a liquid droplet beyond the transition temperature leading to the isotropic phase and subsequent deposition on the substrate, and alternatively by evaporation under high-vacuum conditions (base pressure 10 -6 mbar). The preparation method did not influence the observed structures. The sample was kept at a temperature of at least 29 ° C during the STM imaging procedure. Figure 3 compares the well-known arrangement of eight 12CB-molecules (a) per unit cell with the new phase (b). The latter shows the same type of unit cell, but with a fifth pair of molecules. Each molecule can be identified by the elongated bright region which represents its phenyl head group, whereas the alkyl chains cause only a weak contrast in the darker areas of the image. Both phases were found with the same probability, in some cases even on the same sample. However, due to the large size of their domains, no borders separating the two phases could be found. The width of the unit cell, being defined as the distance of equivalent points and measured at right angles to the orientation of the rod-shaped molecules, is 26 ]k for the smaller and 32 for the larger cell.

Fig.3. STM images of an ordered monolayer of 12CB on HOPG, illustrating two possible surface phases. In (a) each unit cell contains 8 molecules (image size 100 × 100 ~2), whereas in (b) the unit cell contains an additional pair of molecules (image size 140 × 140/~2).

304 3.4

S. GrafstrSm et al. Phthalocyanine

on HOPG

The 8CB-liquid crystal is also a suitable solvent for the phthalocyanine (Pc) molecule. The same procedure as used for P T C D A provides a Pc-film on H O P G , consisting of domains with a typical diameter of 20 nm. Within the domains one observes a two-dimensional lattice of molecules which are obviously lying flat. A STM image of this structure is shown in Fig. 4a. When imaged by STM, the Pc-molecule on H O P G looks different from the closely related Cu-Pc which was studied by several groups with STM on various metallic substrates and on GaAs [29, 30]. Each molecule appears as a fourleaved clover with a dark center in a molecular lattice which is commensurate with the graphite substrate [26]. The distance of the four bright spots from the center of the molecule is about 2.2 ~. They coincide approximately with the positions of the four inner N-atoms. The imaging of the molecules is not influenced by the liquid-crystal layer which covers them. This finding holds also for PTCDA. The Pc-lattice shows a hexagonal symmetry with one molecule per unit cell. We found that the lattice vectors enclose an angle of 600 + 2 °, and that adjacent molecules have a distance of (13.0-i-0.4) ~. The correlation between the Pc-lattice and the graphite lattice underneath can be determined from Fig. 4b. In the upper part, one can see the Pc-lattice, whereas the graphite substrate appears in the lower part due to an intentional decrease of the tunneling gap during the scanning procedure. The two lattices are rotated against each other by 300 + 1°. Since all Pc-molecules exhibit a similar shape and brightness, we conclude that they all occupy equivalent points of the

Fig.4. (a) Ordered layer of phthalocyanine on HOPG, imaged with intramolecular resolution (image size 59 x 62 £2). (b) In the upper part of the image (total size 63 x 66/~2) the molecular Pc-lattice is resolved, whereas the lower part shows the underlying substrate lattice. See text for details.

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graphite lattice. The exact position of the center of the molecule relative to the atoms of the unit cell of graphite cannot be determined from Fig. 4b, since the molecules appear heavily distorted in this image, caused by a convolution with the shape of the tunneling tip. 3.5

The Diazo Dye D2

Further STM studies concerned the red diazo dye 4-[4-(N, N-dimethylamino) phenylazo] azobenzene, designated as D2 by the manufacturer BDH Chemicals (Poole, England) with the summation formula C20H19Ns. The dichroic molecule, also used in coloured liquid crystal displays, is rod-shaped and consists of three phenyl rings connected by two azo bonds and a dimethylamine group attached to one phenyl ring. It has a size of about 21 x 6 ~2. Ordered mono- and submonolayers of D2 on freshly cleaved H O P G were prepared by molecular-beam deposition in high vacuum (base pressure < 10 -5 mbar) using oven temperatures of 100-120 ° C and keeping the substrate at room temperature. Alternatively, dye films of similar structure could also be produced by dissolving the dye in 8CB and depositing a heated droplet (70 o C) on the substrate. Figure 5a shows two domains separated by a ditch-like zone of uncovered substrate with a width of about 24 A which is somewhat larger than the length of the D2-molecule. From this width and the specific image structure we conclude that the molecules in the ordered layer extend from the middle of a bright stripe to the middle of the neighbouring dark stripe. The molecules are oriented at an angle of about 78 o with respect to the stripes.

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Fig.5. The diazo dye D2, adsorbed on HOPG (image size 232 × 232 A2). A missing-row defect runs through the image (a), whose upper part is shown schematically in (b).

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The upper part of the ditch in Fig. 5a is depicted schematically in Fig. 5b, the open circles indicating the positions of the phenyl rings with the attached dimethylamine head group, and the closed circles mapping the positions of the other phenyl rings. The STM micrographs lead us to the conclusion that the molecules form double-rows with a head-to-head and tail-to-tail arrangement. A closer analysis of the images provides a detailed model of the structure of the adsorbed film [31]. An especially peculiar and interesting feature is shown in Fig. 6: a wedge-shaped domain whose double-rows are shifted by one D2-row against the surrounding domain. As can be concluded from the narrower bright stripes at the left and right end of this domain, the rows are formed by a head-to-tail rather than a head-to-head arrangement of the molecules, resulting in a shift of the bright stripes by one molecule length. The wedge may origin from the initially independent growth of the small domain, which was then enclosed by the larger one, or from a local perturbation of the substrate surface. 4

First Laser-Assisted

Measurements

The coadsorbed PTCDA/8CB-films on graphite and MoS2 discussed in the previous section represent a promising model system for the local detection of optical excitations by scanning tunneling microscopy. They consist of isolated islands whose dye molecules possess a light absorption profile in the visible region around )~=500 nm, surrounded by the liquid crystal film, which is transparent at these wavelengths. Hence, an image of the resonant lightinduced current should exhibit contrast between the two species. However, one has to be aware that the thermal background may also produce contrast between the different molecular regions, despite the fact that we expect a uniform temperature on the length scale of the molecular structures, which are small compared with the thermal diffusion length at the modulation frequencies used. As thermal expansion is predominantly a bulk effect,

Fig.6. Defect in a D2-monolayer, forming a wedge-shaped domain (image size 360 × 360 A2).

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the resulting gap width modulation is also not expected to vary spatially on a homogeneous substrate, but differences in the apparent barrier height may nevertheless cause spatial variations of the thermal current modulation. They must be thoroughly discriminated against the true resonant contrast. The criterion for a perfect compensation of the thermal background is that the light-induced current modulation should vanish completely on the 8CB-covered areas, which we thus use as a reference for the adjustment of the compensation system. This adjustment is, however, limited in accuracy by short-term noise. In order to determine more precisely the point of vanishing thermal background, we take a series of images at different settings of the relative phase and amplitude of the two power modulations. We then extract the current modulations iscs and iPTCDA observed on the two species by averaging the measured signal across the respective regions, which considerably improves the signal-to-noise ratio. The currents i8CB and ipTCDA (which represent complex numbers, characterized by an in-phase and an outof-phase part and measured with a two-phase lock-in amplifier) show a linear mutual correlation when the two power modulations are detuned against each other, provided that the involved effects are linear with respect to the light power. Hence, by interpolating the data by a linear fit, we can find the point iscB = 0, representing vanishing thermal background, and determine the corresponding iPTCDA, which we may identify as the resonant signal. A first evaluation of this kind was performed for a series of images taken on coadsorbed P T C D A and 8CB on MoS2 [32]. Figure 7 shows a topographic image (a) together with a simultaneously recorded image of the

Fig.7. (a) Topographical STM image of a layer of co-adsorbed PTCDA and 8CB on MoS2 (image size 500 × 500 ]ks). (b) Simultaneously recorded map of the laser-induced tunneling current, measured with an intentional mismatch of the two power modulations.

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laser-induced current modulation (b). In the topographic image two PTCDAdomains and three differently oriented 8CB-domains can be distinguished. The laser-induced tunneling current displays a rather clear contrast between the two molecular species, which however is primarily non-resonant and due both to thermal effects and photovoltage generation by the semiconducting substrate. The data analysis as described above shows that when the current modulation within the 8CB area vanishes there is a residual modulation on the PTCDA-covered area of (95 + 54) farms. In this situation, the power modulations of the two lasers were 9 mWpeak-peak at A = 514 nm and 6.6 mWpeak-peak at A = 590 nm, respectively. The residual modulation observed on the P T C D A represents a first evidence for a laser-induced tunneling current produced by resonant excitation of adsorbed molecules. Further efforts are necessary to improve the signal-to-noise ratio, which is a prerequisite for systematic studies of the dependence of the signal on various parameters, such as tunneling voltage and laser frequency.

5

Conclusions

Our work displayed in this article started from the premise that a conceivable future molecular electronics, though intended to operate on a nanometer scale by means of supramolecular structures or even single molecules, will nevertheless need close links to the presently existing, very successful semiconductor electronics. From the numerous problems which must be taken into account and studied in detail, in order to gradually sound the potential and approach the realization of a molecular electronics, we chose the aspect of optical excitation of chromophoric molecules adsorbed on a substrate, and the detection of this excitation with high spatial resolution, i. e., with STM. As a consequence, besides the development of an appropriate setup combining a STM with a laser-optical arrangement, a major emphasis was put on the preparation of submonolayers of ordered dye molecules on suitable substrates. In addition to the results concerning the main aim of this work, we were rewarded for the effort with a number of spin offs such as: (i) an improved knowledge of the thermal behaviour of a tunneling contact under laser irradiation, (ii) experience in the preparation of ordered monolayers of dye and liquid-crystal molecules, (iii) the observation of domain formation and registration, and of peculiar defects in those layers, (iv) the discovery of a previously unknown unit cell of the liquid-crystal 12CB. Finally, we would like to point out the interdisciplinary character of these studies. Laser-assisted scanning tunneling microscopy of thin ordered layers of dye molecules, adsorbed on the surface of a crystalline substrate, combines several fields: Solid state physics is represented by a variety of problems related to surfaces. Chemistry contributes in particular the aspects of chromophoric molecules and supramolecular aggregation. Atomic physics participates with questions such as the probability of electronic excitation as well

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as the subsequent radiative and nonradiative decay of molecular states under the condition of physisorption on a surface. All these subjects mentioned before are embedded in a frame consisting of high-resolution microscopy and laser spectroscopy.

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S. GrafstrSm et al.

[19] O. Probst, S. Grafstr6m, J. Fritz, S. Dey, J. Kowalski, R. Neumann, M. WSrtge, G. zu Putlitz, Contrast mechanisms in photothermal scanning tunneling microscopy, Appl. Phys. A 59, 109 (1994) [20] S. GrafstrSm, O. Probst, S. Dey, J. Freund, J. Kowalski, R. Neumarm, M. WSrtge, G. zu Putlitz, STM studies on dye molecules embedded in ordered liquid crystal structures and an approach for laser assisted scanning tunneling microscopy, in Advances of DNA Sequencing Technology, Richard A. Keller, Editor, Proc. SPIE 1891, 56 (1993) [21] M. Stavola, D.L. Dexter, R.S. Knox, Electron-hole pair excitation in semiconductors via energy transfer from an external sensitizer, Phys. Rev. B 31, 2277

(1985) [22] J.S. Foster and J.E. Frommer, Imaging of liquid crystals using a tunnelling microscope, Nature 333, 542 (1988) [23] D.P.E. Smith, H. HSrber, Ch. Gerber, and G. Binnig, Smectic Liquid Crystal Monolayers on Graphite Observed by Scanning Tunneling Microscopy, Science 245, 43 (1989) [24] D.P.E. Smith, J.K.H. HSrber, G. Binnig, and H. Nejoh, Structure, registry and imaging mechanism of alkylcyanobiphenyl molecules by tunnelling microscopy, Nature 344, 641 (1990) [25] D.P.E. Smith, W.M. Heckl, and H.A. Klagges, Ordering of alkylcyanobiphenyl molecules at MoS2 and graphite surfaces studied by tunneling microscopy, Surf. Sci. 278, 166 (1992) [26] J. Freund, O. Probst, S. GrafstrSm, S. Dey, J. Kowalski, R. Neumarm, M. WSrtge, G. zu Putlitz, Scanning tunneling microscopy of liquid crystals, perylene-tetracarboxylic-dianhydride, and phthalocyanine, J. Vac. Sci. Technol. B 12, 1914 (1994) [27] C. Ludwig, B. Gompf, W. Glatz, J. Petersen, W. Eisenmenger, M. MSbius, U. Zimmermann, and N. Karl, Video-STM, LEED and X-ray diffraction investigations of PTCDA on graphite, Z. Phys. B 86, 397 (1992) [28] C. Ludwig, B. Gompf, J. Petersen, R. Strohmaier, and W. Eisenmenger, STM investigation of PTCDA and PTCDI on graphite and MoS2. A systematic study of epitaxy and STM image contrast, Z. Phys. B 93, 365 (1994) [29] P.H. Lippel, R.J. Wilson, M.D. Miller, Ch. WSll, S. Chiang, High-Resolution Imaging of Copper-Phthalocyanine by Scanning-Tunneling Microscopy, Phys. Rev. Lett. 62, 171 (1989) [30] R. MSller, R. Coenen, A. Esslinger, B. Koslowski, The topography of isolated molecules of copper-phthalocyanine adsorbed on GaAs(ll0), J. Vac. Sci. Technol. A 8, 659 (1990) [31] J. Fritz, O. Probst, S. Dey, S. GrafstrSm, J. Kowalski, R. Neumann, G. zu Putlitz, Scanning tunneling microscopy studies of diazo dye monolayers on HOPG, Surf. Sci. 329, L613 (1995) [32] O. Probst, S. Dey, J. Fritz, S. GrafstrSm, T . Hagen, J. Kowalski, R. Neumann, G. zu Putlitz, Laser-assisted STM studies of thin ordered molecular layers, in O. Marti and R. MSller (Eds.): Photons and Local Probes, NATO ASI Series E, pp. 269-274, Kluwer Academic Publishers, Dordrecht, 1995

Optical Spectroscopy of Metal Clusters Michael Vollmer Physikalische Ingenieurwissenschaften, Fachhochschule Brandenburg, Magdeburgerstr. 50, 14770 Brandenburg, Germany

1

Introduction

Optical spectroscopy is one of the most successful okt fields of physics, which helped a lot to investigate the structure of the microscopic world of atoms, molecules and even nuclei as well as the macroscopic world of stars and galaxies. Application of this experimental tool to the relatively new state of matter in the form of dusters can yield valuable information on the electronic and optical properties of these species which are intermediate between atoms and molecules on the one hand and the solid state on the other. In general clusters are defined as a number of unspecified objects gathered together or growing together, a defimtion which does not include an inherent size scale either for the objects or for the clusters. In the context of the present work, we define clusters as particles composed of a certain number N of atoms with 3 < N <~ 107 thus considering dimers as molecules rather than as clusters. Hence, a sodium cluster may consist of N = 10, 100 or even 1 000 000 atoms sticking together. The thus-defined cluster region covers a rather wide size-range, therefore it is useful to define several size regions [I]. For very small clusters (N<20) the cluster surface cannot be distinguished from the inner volume. Large clusters, on the other hand are defined for N>500. These species have well defined inner volume versus cluster surface. In between are small clusters with 20
312

pronounced changes occur across the largest size range of all known duster effects. To illustrate the complex variations of optical properties of dusters with size, Fig. la-e (top to bottom) gives an overview of experimental spectra of the optical response of sodium from the atom to the solid. Single atoms have the well-known D-lines doublet at 589.0 and 589.6 nm in resonance fluorescence. They correspond to the transition from the 2S 1/2 ground state to the first excited states 2P3/2 and 2P1/2 respectively (Fig. la). As dusters grow (lb - d), the distinct lines disappear and a single broad absorption feature evolves. Whereas the sodium trimer still shows distinct spectral features (Fig. lb), the optical response of a free cluster with N -- 8 atoms (Fig. lc) is mostly dominated by one single strong resonance, a response quite similar to the one appearing in dusters with 105 or 106 atoms. Figure ld depicts this absorption of large Na clusters in a rocksalt matrix. When bulk sodium is finally approached in the form of a thin film, the spectrum becomes structureless in the visible spectral range and can be explained by bulk optical functions for free-electron metals (Fig. le). It should be mentioned, however, that in the intermediate size regions, both between Fig. lc and ld and between Fig. ld and le, more complex spectral features are observed. For the spectra of large dusters in Fig. 1 electrodynamic theory can be applied using bulk optical constants (extrinsic size effects). For small sizes, however, the

Na D-Lines

(a) i

"E i

i

.., ~

--

(b)

.~-"~

,-

_

(c) (/)

g I

o <

(d)

Na Clust~.t

(e) 40O

50O

600

700

Wavelength [nm]

Fig. 1. Optical properties of sodium. From top to bottom~ a) Spectnma of atomic sodium b) Two photon ionization spectrum of Na 3 [2] c) Beam depletion spectrum of Na 8 [31 d) Absorption of large Na clusters in NaCI [4]. The size was below the resolution of optical microscopes and probably ranges around 10nm e) Transmission of a thin film of bulk sodium (thickness 10 nm).

313

optical functions become size-dependent (intrinsic size effects). The line spectra of the very small clusters can be described by quantum mechanical calculations. Intimately related to optical excitation are photodissociation processes. In general the interaction of a duster with a photon does not only tead to dectronic and/or vibrational excitations, but may also result in fragmentation or evaporation processes of the duster. It was shown recently that this photofragmentation is not a hindrance in studying optical spectra, but in fact ought to be regarded as a key tool for spectroscopy of free metal dusters [3,5]. The present knowledge on stabilities, quantum size effects, electronic and optical properties of metal dusters has been summarized in a number of detailed review articles [1, 6-11]. The state of the art is described in conference proceedings of the International Symposium on Small Particles and Inorganic Clusters [12].

2

2.1

Optical R e s p o n s e o f Metal C l u s t e r s : T h e o r e t i c a l C o n c e p t s General Remarks

In principle dusters can be generated either by nucleation from atoms/molecules or by cleavage from macroscopic bulk solids. Accordingly, on the one hand, one has to deal with very small clusters, i.e. with molecule-like structures like trimers or pentamers, on the other hand, one encounters large clusters which are regarded as tiny, size-limited solids. Similarly two different kinds of theoretical concepts have been developed for dusters which differ markedly for the various size regimes. Very small clusters have been successfully treated with molecular quantum-chemical ab initio methods (Sect. 2.3). In the near future, extension of these all-electron calculations to small clusters seems possible, at least for dements with low atomic number For large clusters solid-state physics techniques combined with classical dectrodynamics have been applied (Sect. 2.2). In between these extremes, models with simplifying assumptions for the geometric and electronic duster structure have been developed and were applied to the small and very small cluster size range. Examples are the potential box model and the jellium model (Sect. 2.3). For small and very small clusters spherical- and ellipsoidal-jelhum models can account qualitatively for many prominent experimental features. In particular, they were applied successfully to the monovalent alkali metals. The dominant features of the optical spectra of metal dusters are those associated with the largest oscillator strengths. It is convenient to divide melalduster spectra into features related to single-electron and to collective electron excitations, the latter representing most of the oscillator strength of the delocalized conduction electrons. The resonances which dominate the spectra of small and large metal clusters reflect the collective oscillations of dectrons in the dusters, the dipole modes being related to the giant dipole resonances in atoms or nuclei. According to the way of excitation (by electrons or photons) these collective

314

oscillations of the conduction dectrons are denoted either as surface plasmons or as surface-plasmon polaritons. In this paper, only linear optical properties of single clusters are treated. Cluster-duster-interactions, problems of cluster matter or nonlinear optical effects etc. are treated in [1].

2.2

Classical Electrodynamic Theory for Large Clusters

Mie Theory for Spherical Clusters. The interaction of spheres of arbitrary material with electromagnetic waves was treated already by Gnstav Mie in 1908 [13] by solving Maxwell's equations with appropriate boundary conditions using a multipole expansion of the incoming electromagnetic field (see also [1, 14]). Input parameters are the refractive indices or the dielectric functions of the particle and of the surrounding medium. The boundary conditions are defined by the electron density, which is assumed to have a sharp discontinuity at the surface of the duster with radius R. Following e.g. Bohren and Huffman [14] the extinction, scattering, and absorption cross sections can be calculated analytically to yield

t~

2~ k2 ~ (2L + 1)Re{a L + bL}

(7c°

~

kS

(2L + 1

L

+ ~LI

(1)

L-!

where k is the wavevector and a L and bL are coefficients, containing Bessel and Hankd functions which depend on the complex index of refraction of the particle, the real index of refraction of the surrounding medium and the size parameter x=kR. The general problem consists in the calculation of the coefficients a L and b L with recurrence relations for the Bessel functions or series expansions. The sum index L gives the order of the vector spherical harmonic functions which enter the expressions for the electric and magnetic fields and thus describes the order of spherical mnltipole excitations in the dusters. L=I corresponds to dipole fields, L=2 to quadrupole and L=3 to octupole fields. Far away from the duster the waves are identical to those coming from equivalent mulfipoles. Often these excitations due to the electromagnetic field are simply called plasmons since they are interpreted as collective excitations of the conduction electrons. It is convenient to introduce the quasi-static regime for dusters of sizes R/N < 0.01 (for visible light this refers to duster sizes R _< 5 nm). In the quasi-static regime phase shifts, i.e. retardation effects of the electromagnetic field over the duster diameter are negligible (Fig. 2). This means that the multipolar excitations of Mie's theory are restricted to the dipolar electric mode, i.e. to the lowest order term al:

315

(r=(co)

= 18z

e~.'~

e ,(co)

V~

z

(2)

+ ,(coY

V 0 denotes the particle volume, em and r(co) = ei(co) + i e2(co ) denote the dielectric functions of the medium and particle, respectively. Extinction is only due to the dipole absorption being proportional to the particle volume whereas scattering is negligible in this case. The cross section has a resonance whose position and shape are governed completely b y the dielectric functions. The dipole resonance is determined by the condition el(~0 ) = - 2 rm provided E2(co ) is not too large and does not vary much in the vicinity of the resonance. Steep el(t0) spectra yield narrow resonances whereas low de l((O)/do and large e2(co ) tend to smear out the resonances, sometimes past recognition [15]. The resonance condition also implies that it is possible within certain limits to choose a matrix material for embedded spheres such that the plasma frequency cop shifts to a desired wavelength range. In the Drude/I~rentz/Sommerfeld model the dielectric function for free dectron metals (e.g. the alkali metals) is governed by transitions within the conduction band. It is calculated from the equation of motion of a free electron of mass m e and charge e subject to an external electric field. Using a phenomenological damping constant F for a system of free electrons the condition (o >>F gives " ~,(co)

I -

2 COY

~

CO

,

~, (co) =

60

2

- - 7P r

(3)

CO

For this case the resonance position and shape can be easily determined by inserting (3) into (2). In the vicinity of the resonance the Iineshape is then described by a Lorentzian. In other metals a substantial amount of interband transitions from lower lying bands into the conduction band or from the conduction band into higher unoccupied levels is possible, which alters the simple form of (3) [16]. For the alkali metals the interbaad threshold given by excitations of conduction band electrons to higher levels lies at about 0.64 EFermi, however, their contributions to the dielectric function are small. In contrast, the relevant interband transitions

Quasi-static case:

Z>>2R

E(t=to)

Homogeneouspolarization: dipole excitation

General

case:k ~ 2R

E(t=t0)

Phase shifts in the particles: multipole excitation

Fig, 2. The interaction of light with dusters can be described in a simple way in the quasi static regime (~. >> 2R). In the general case phase shifts of the electromagnetic wave in the particles complicate the optical response.

316

for the noble metals are due to excitation of d-band electrons into the conduction band. They give a positive contribution de 1 to e l, Consequently the dipole resonance frequency given by c I = - 2 em is shifted to lower frequencies, in the case of some noble metals beyond the low frequency interband transition edge. For other metals these effects complicate the optical spectra appreciably and in fact, only a few materials like the alkali and the noble metals as well as aluminum exhibit sharp resonances. With (I, 2) the optical response of metal spheres to incident electromagnetic waves can be calculated. As input parameters the Mie theory uses the phenomenologically introduced dielectric function e (c0) for the dusters. It should be stressed that the Mie theory gives no insight whatsoever to the microscopic excitation mechanisms in the particle material. These are exclusively contained in the applied e((o). Any strong deviations from free electron behavior make it essentially impossible or at least very difficult to derive e(to) from a microscopic theory. Fortunately a huge amount of experimental optical material functions for bulk solids which incorporate all electronic effects is available (e.g. [17]). Figure 3 shows an example of the absorption cross section of single sized sodium dusters as a function of wavelength for fixed radii between 20 um and 100 urn. Input parameters were size independent dielectric functions e (o)) (from [18]). For small duster sizes, i.e. within the quasistatic regime, only a single resonance, the dipolar plasmon polariton is visible. With increasing size, higher order modes come into play. Simultaneously, a red shift of the dipolar mode is observed due to retardation effects (for more details, see e.g. [1, 19]). In comparison, Fig. 4 shows size dependent dielectric functions and the resulting optical absorption spectra within the quasistatic regime. Obviously, the most dramatic change occurs for the damping, since e2(~0) increases for decreasing size. This so called limited mean free path effect accounts for the fact that the duster

t

~"

i~

.

n=2O,,m

31- " ~ V y \ / its, ~, /', " ~o I- L J'-" , ".

FJ/ I!i,:

....... ..........

R. ~ R-~OOm

,. kx.,,, 400

500

600

700

WavetenOt~ (nml

Fig. 3. Absorption cross section as function of wavelength for monodisperse free Na dusters in vacuum with mean sizes R = 20, 40, 60, 80, and 100 urn.

317 boundary will impose an additional scattering, i.e. damping mechanism for the electrons if the mean free path of electrons in the cluster exceeds the cluster size. Concerning the optical properties extrinsic (Fig. 3) and intrinsic (Fig. 4) size effects can now be readily attributed to different parts of the theoretical description: For clusters larger than about 10 nm diameter, the optical material functions e(c0) are size independent, having the values of bulk material. The change of the spectra with size is dominated by retardation effects of the electric field across the dimension of the particle which can cause huge shifts and broadening of the resonances (extrinsic size effects). For smaller clusters the optical material functions do no longer have the values of bulk material, but vary as a function of particle size. This is an intrinsic cluster size effect, as the material properties give rise to a change of the optical response. Still, the Mie theory result (Eq. 2) can be a good description, if a proper dielectric function is used. In summary; the position of the dipolar surface-plasmon resonances of small spherical metal clusters is defined by the condition e 1(~0) = - 2 e m which translates into el(to)=- 2 for spheres in vacuum. Larger particles suffer a peak shift due to phase retardation of the electromagnetic waves and the influence of higher multipoles. For clusters in matrices the dielectric surrounding leads to additional shifts usually towards the red - with regard to clusters in beams. The Mie theory gives a constant width for the dipole resonance, yet, additional damping effects show up, described for metallic clusters by the limited mean free path effect. For larger clusters, damping due to retardation causes broadening with increasing size.

,

-2 I

ElO.)

,

,

,

,

,

,

=2R

2.5 2R=25 2.0

-4

2R=3.1nm 1.5

-6

1.0 0.5

-8

I 380

I 400

I 420

I 440 wavelengthInto]

bulk I l_ I I 380 400 420 440 wavelength[rim]

360

400

440

wavelength

480 [nm]

Fig. 4. Size dependence of dielectric functions e 1(~.)and e2(L) for silver clusters computed from bulk optical constants by including the limited mean free path effect (after [20]). The resulting Mie absorption spectra clearly illustrate the broadening for decreasing size.

318

Extension of Mie Theory: Other cluster shapes. Mie's theory was developed in 1908. Only a few extensions of the theory were performed latex They can be classified as dealing with other geometrical particle shapes, core-shell particles, shape dependent substrate effects, diffuse electron density boundaries, non local optical effects, and duster-cluster interactions in duster matter (for more details see [1]). At this point only nonspherical cluster shape effects on the spectra will be briefly discussed. For shapes differing from spheres, dectrodynamic calculations are much more tedious and closed-form expressions for the cross sections are available mostly in special cases like the quasi-static approximation. For this case ellipsoids, cylinders, cubes and other geometries have been treated [1]. For larger sizes, numerical results were obtained. All of the general new features are already present within the quasi-static approximation. As an example, Fig. 5 depicts the result for ellipsoids with three different axes (after 14]).

::3

e6

f,

" AI-Sphere ( 2R < 10 am ) =t

¢0

0 tO

E~

2

A

Ai

0u~ JD


L-~'~ 4

AEllips°id

Fig. 5. Theoretical absorption spectra of aluminum spheres ' and randomly oriented ~' ~-2:~--~3~ I ellipsoids. CDE corresponds to 8 12 16 a continuous distribution of Photon Energy[eX0 eUipsoidal eccentricities.

For randomly distributed ellipsoid orientations in samples with many dusters, the absorption spectrum is characterized by three distinct peaks of approximately equal magnitude provided the widths of the peaks are small enough. They are due to the three different polarizabilities (classically speaking, the restoring forces for electrons in terms of spring constants which determine the resonance frequencies) along the three axes. According to Fig. 5, both a red and a blueshift with respect to the dassical Mie resonance of a sphere are possible. A continuous cluster-shape distribution leads to a smearing out of such distinct peaks and gives rise to the curve labeled CDE (continuous distribution of ellipsoids) in Fig. 5. In samples of aligned ellipsoids, the three single resonances can be selected using suitably polarized light. Using spheroids, i.e. rotational ellipsoids with only two principal axes consequently results in two resonance peaks.

319

2.3 Quantum M e c h a n i c a l Theories for S m a l l and Very S m a l l C l u s t e r s

The Mie theory needs a dielectric function of the cluster material as input parameter in order to calculate the optical response of the clusters. This e(c0) may be calculated from microscopic theories. Direct determination of the duster polarizability is a competing method, being successful for small and very small clusters. The interaction of electromagnetic radiation with small metal clusters leads to excitations via transitions between different energy states of the whole cluster. As a first approximation (Sect. 2.1), the electronic excitations can be divided into single-electron and collective electron excitations. In small clusters, the single-electron transitions directly reflect allowed transitions between different individual electronic levels. For larger sizes, the additional collective excitations of the metal electrons constructed from the singleelectron transitions are the plasma or Mie resonances. Since small and very small clusters correspond to 2R << 3., collective excitations refer here to the dipole resonance alone. Currently, the question of how collective excitations emerge from molecular excitations is an intensely discussed topic in cluster physics. Quantum-mechanical studies of very small and small metal clusters can be done either with quantum-chemical methods or with approaches borrowed from solid-state quantum theory. Although successful in studying ground-state electronic properties in particular of very small clusters, quantum-chemical all-electron calculations of somewhat larger clusters are limited by available time and capacity of computers, an even more difficult case when considering excited-state configurations. The latter prove much more difficult to calculate than ground states, since in principle all electrons of a coupled system change their state upon excitation of even one single electron. In contrast, the much simpler methods, which use common solid-state concepts like the jdlium approximation, give reasonable results for large and small clusters yet conceptual limitations become apparent for the very small clusters. Contemporary methodical concepts can thus be distinguished as follows: 1) collective versus single-dectron-hole excitations 2) jeUium and quantum-box models versus quantum-chemical ab initio methods The various theoretical concepts for the very small clusters deserve a review article by themselves, here, instead, a very brief description of various approaches will be given. Further information can be found in the articles by Brack [10] and Kresin [8] for thejdlium models and by Koutecky and Bonacic-Koutecky et al. for quantum-chemical concepts [21]. As an example of a jellium calculation Fig. 6 depicts the electron charge density, level structure, and the self-consistent one particle potential for the ground state of Na20-dusters (R ~ 0.565 nm) as well as the optical absorption spectra resulting from such an energy-level scheme (after [22]) in the form of the imaginary part of the frequency-dependent polarizability (z(0~). The solid line represents the corresponding classical Mie resonance, i.e. the result obtained from Mie theory in the quasi-static limit without any size effect

320

10 2

"•

:>,

OQ

10 0

- -

o:

uJ- 0.4

Na2°

i

ion cluster

Na20(rs-4) 0

!~.1~, :~'

\

I

electron

cluster

_E 10"2

, \ ] ~ . - radius[a.u.] 10 ~ 2 0 10-4

/-,. //

potential

I

I

I

0.~,

1.0

1.5 O)/O)Mie

Fig. 6. Left: Ion and electron charge densities, energy levels and self consistent effective one electron potential for Na20-cluster calculated with a spherical jellium model (rs:Wigner-Seitz radius). Right: Imaginary part of the dynamical polarizability for Na20-clusters (dotted line) from self-consistent spherical jellium model. The frequency is scaled in units of the classical Mie frequency for metal spheres. corrections. The central feature in the dotted spectrum around to/toMie = 0.9 (indicated with arrow) was attributed to the redshifted surface plasmon. Small and very small dusters of simple metals usually exhibit red shifts compared to the classical Mie frequency due to soft boundaries (spill out). In addition, a number of peaks were interpreted as excitations of single electron-hole pairs. The spherical jellium models have been extended to account for spheroidal or ellipsoida! shapes [9]. Once the most stable duster shape is known, the components of the polarizability along its three principal axes are calculated giving rise to the three surface-plasmon frequencies. After duster-orientation averaging to simulate a realistic many-cluster system, the absorption cross section can be written as the sum of three Lorentzians, similar to the Mie theory results. In contrast to jellium calculations, ab initio quantum chemical approaches for very small dusters [23] give various possible geometric structures with different ground state energies. The spectra of optically allowed transitions help to identify the duster geometry if compared to experiment (see Fig. 15). The energy dissipation - giving rise to the width of the resonances - is not yet fully understood.

3 Experimental Methods 3.1 General Remarks Optical spectroscopy has been extremely successful in investigating the electronic structure of atoms and small molecules. Depending on the system under

321

investigation and the experimental questions a great number of different spectroscopic techniques have been employed utilizing resonance fluorescence, resonant two photon ionization or e.g. optical pumping or double resonance schemes. The direct optical spectroscopy of large molecules and clusters, however, was often impossible since the large number of degrees of freedom inside a large molecule or duster after the absorption of a photon usually led to radiationless deexeitation, finally resulting in fragmentation of the cluster. For example, two photon ionization spectroscopy was only successful for dimers or trimers (see review [1]) before using ultrashort laser pulses [24, 25]. Hence, spectroscopic methods can be either destructive or nondestructive, i.e. the duster is either undergoing permanent changes upon interaction with a photon or not. The spectroscopic techniques furthermore depend on the type of duster sample. In principle, dusters may be embedded in matrices, supported on surfaces, or free in a beam (a trap). The general set up of nondestructive scattering, fluorescence, and extinction spectroscopy experiments is shown in Fig. 7. It consists of a light source, the sample, and detectors for measurement of the incident, transmitted and scattered light. In contrast, Fig. 8 depicts the schematics for destructive spectroscopies. A duster absorbs a photon of energy lrv. The excitation leads to at least two different reaction products, electrons, ions, or neutrals from the original duster, which can be detected to yield information on the electronic structure of this primary duster. The most simple technique is depletion spectroscopy [3,5] where a metal duster from a beam absorbs a photon and subsequently emits an atom. This results in a decrease of the duster count rate, measured e.g. with a mass spectrometer behind a ~aphragm.

S ~ Iscatl.

detector

detector

S

L

[,,oh, eo.rc. I

detector

S ~ linty,. ]-

Fig. 7. Experimental set up for classical scattering and extinction spectroscopy. Light source and detectors may be combined with monochromators.

322

~

m~

hv

Position and time sensitive detector or diaphragm for

beam depletion

unaffected cluster beam

M= m 1 + m 2 m2

Retardation time

Drift times

Fig. 8. Schematic experimental arrangement for beam depletion spectroscopy.

4 Selected E x p e r i m e n t a l Results 4.1 General Remarks Most optical experiments on large dusters have been performed with metal dusters in matrices and on supports [1, 26, 27]. In these systems interpretations of optical spectra are complicated by particle-particle and particle-matrix interactions. In addition, such samples often show broad distributions of particle sizes which mask the interesting effects. Optical studies of supported dusters were important for the investigation of granular and island films. It was also quite recently demonstrated that large supported metal clusters show interesting dissociation effects along with the capacity to manipulate size and shape distributions on supports [28-30]. Already a cursory inspection of the literature reveals a strong preference of only a few elements, in particular alkali and noble metals. For the very small dusters in beams Na (and less pronounced other alkalis and Ag) dominate whereas experiments with supported or embedded dusters are mostly done with Ag or Au. The reasons are that these metals exhibit pronounced resonances within the visible spectral range. In addition the noble metals have the advantage of being chemically nearly inert. In contrast, the very reactive alkali metals require clean conditions: they have mostly been studied in beams or on supports in vacuum. In the following a brief survey of selected experimental results is given. Many more experiments are discussed in [1] or in conference proceedings [12]. In particular one extm'iment each of dusters in matrices, on supports and in beams will be discussed separately as well as an attempt to directly compare these three kinds of specimen in a single experiment.

323

4.2 Clusters Embedded in Matrices

There is a huge amount of experimental work published on metal dusters in matrices of solid rare gases, CO, glass etc. As one example Fig. 9 depicts extinction spectra of dusters of the noble metals Ag, Au and Cu, demonstrating the influence of the onset of interband transitions on surface plasmon absorption. Whereas silver has a dearly resolved surface plasmon resonance, the interband transitions have lower thresholds in Au and Cu and extend to the resonance frequencies. A detailed analysis of these and similar spectra allow to extract resonance positions and widths as a function of duster size. In particular one can study the influence of the matrix material on the resonance, which can result in blue as well as red shifts. The interpretation of peak shifts can be ambiguous and is far from being a simple task. With respect to the peak width, the data can in general be divided into two groups: 1) Systems which dearly follow a (l/R) size dependence, i.e. a limited mean free path effect, yet with slope parameters depending on the surrounding material. Characteristic for these samples is that the matrices are only weakly interacting with the dusters (e.g. solid rare gases). 2) Systems with strongly increased damping but without any clear correlation to the duster size. These samples are characterized by rather strong interface interactions due to high iomcity of the matrix. In liquid colloidal solutions

Au-clusters:

Ag-clusters: 2R(nm): A 1.0

P

I

'

2R(n'm): '5:6'

Cu-clusters: '-

'

9.2

5.~ 3.~

~

a~o' 46o ' 4 ~ , 0

1.6K

I

'

'

1

400

L

L

500

I

I~"~-

600

"

' 400

'

5( ) 0 '

6 (30'

wavelength(nm)

Fig. 9. Measured extinction spectra of Ag, Cu, and Au dusters of various sizes in a glass matrix (after [20, 31, 32]). The spectra of Cu and Au are dearly resolved due to the high value of em which shifts the resonances away from the interband transition threshold.

324

chemical reactions including charge transfer to or from the dusters have proven to change the width strikingly. Theoretical interpretations of the shifts and broadenings are discussed in [1,15], also with respect to cluster-matrix and cluster-duster interactions.

4.3 Supported Clusters Optical Spectra: Concerning optical spectra of dusters on supports much work was done on discontinuous island films in the past decades. Interpretation of experiments have to deal with the problem of sample characterization, vacuum conditions, duster-cluster interactions, topology effects and shape dependent cluster-substrate interactions. One experiment will be discussed in more detail, which illustrates a way of duster shape determination from optical spectra. Large sodium, potassium and silver dusters on LiF and quartz surfaces were studied in ultra high vacuum with optical extinction spectroscopy [19, 30, 33-35]. Fig. 10 (bottom) depicts normal incidence spectra originating from annealed silver clusters on LiE For sizes above R ~ 40 nm, a shoulder on the small wavelength side develops which reflects the growing importance of the qlmdrupole resonance (L=2). Fig. 10 (top) shows corresponding theoretical extinction spectra for duster size distributions with a FWHM of 50% which are in quite good agreement with experiment. The quantitative differences conceming the widths might reflect deviations from the spherical duster shape. This was investigated by preparing different metal dusters at various substrate temperatures. Clusters on surfaces prepared by nucleation at low temperatures usually grow as oblate spheroids, however, they may be transferred into spheres by annealing at elevated temperatures. The example of Fig. 10 reflects such a sample, the Ag dusters being annealed at 650 K. In contrast Fig. 11 (left) gives an example of extinction spectra of Na clusters, recorded at T = 100 K for s as well as p polarization at an angle of incidence of 500 [34]. Differences in the spectra can readily be explained by oblate cluster shapes and geometrical eousidemtions (Fig. 11 righ0. Analyzing Fig. 16 gives axial ratios of the dusters. Changing the temperature shifts the ratios to larger values, i.e. leads to more spherical shapes [36].

Application" As an example of the application of optical spectra, the novel effect of surface plasmon induced desorption of metal atoms from metal clusters [2829] was studied, using the same Na, K and Ag clusters on LiF surfaces. Ousters are formed by nucleation on a cooled LiF surface and characterized using inelastic atom scattering, optical extinction, and thermal desorption [19, 37]. Typical number densities are 108 to 109 clusters/cm2 and mean sizes range from R= 5 to 100 nm. The supported clusters are irradiated with light of an argon or krypton ion laser and desorbing atoms are detected with the quadrupole mass spectrometer. The kinetic energy of desorbing atoms is determined with time-offlight measurements using a chopped laser beam with pulses of 2 ys duration.

325 (~ext

11o6 A21 / ~ ' ~ 45 nm 0 n\ m ' ~

\\\~

20 n m 500

300

700

900

Wavelength lnml

8O

E

i ° 20 0

j

300

il 0

500 700 Wavelength Into|

900

Fig. 10. Theoretical (top) and experimental (bottom) extinction spectra for eamealed silver dusters of various sizes on a [,iF (100) substrate.

p-Polarization ~

Substrate

'

~,o

,~

s-Polarization I

Fig. 11. (a) Extinction spectra for sodium dusters prepared at T=100 K on a LiF (100) subslrate recorded for an angle of incidence of 50 °. (b) Scheme of an optical experiment for the study of supported metal dusters, Surface plasmon excitation can be accomplished along different axes of the spheroidal dusters with s or p polarized light, indicated by circles and arrows, respectively.

326

A desorption signal can easily be observed as shown in Fig. 12 which depicts the rate of desorbing Na atoms from large sodium dusters (R = 50 nm). Mainly atoms but only a small fraction of diatomic sodium molecules is desorbed with the laser. The desorption rate depends linearly on the light intensity over a range of almost four orders of magnitude with no threshold being observed. Desorption of atoms can be detected even with light intensities as low as 40 mW/cm 2. Time-of-flight measurements indicate that the translational energy of the desorbed atoms is nonthermal. Depending on" the laser wavelength and on the particle size the desorption yield can amount up to 80% of the total coverage. The quantum efficiency of the desorption process is on the order of 10.4. Most importantly, the desorption rate depends on the partide size and on the laser frequency [28]. This is illustrated in Fig. 13 which depicts the imtial desorption signal for fixed duster radius and laser intensity as a function of incident photon energy. A resonance is found, centered at 3. z 500 nm, for particles with 50 am mean radius. The LiF substrate being transparent, the metal particles are responsible for the absorption of light, Comparison of the experimental results (Fig.13) with the calculated absorption cross section spectra of Mie theory for a particle size distribution with FWHM of 50% shows good agreement. The absorption is in the same spectral range and has a similar width. An additional test for the influence of Mie resonances on the desorption consisted in the influence of the polarization of the light incident on oblate clusters. As shown in Fig. 1 1, the extinction spectra of s and p polarized light indicate that sodium spheroids with axial ratios of ~--O.4are present on the surface. The corresponding desorption spectra follow the optical extinction spectra [34].

4000 w

i

2000 Fig.

I

100 Irradiation

I

200 Time [s]

12. laser induced desorption signal (atoms per second) of large Na dusters supported on a LiF (100) surface. The mean cluster radius was about 50 am. The laser was turned on and off in 10 s intervals. It was operated with 22 W/cm2 at an excitation wavelength of 514 nm (after [28]).

327

1000

rr

.~ 500

2.2

2.6 Photon Energy leVI

3.0

Fig. 13. Dependence of laser induced desorption signal from Na dusters supported on a LiF (100) surface on photon energy. The mean duster radius was 50 nm and the data were taken at a laser intensity of I = 22 W/cm 2,

From the above observations it was concluded that sm'faee plasmon excitation is essential for the laser induced desorption of atoms, but the nonthermal desorption/ dissociation process can only be explained using the concept of nonlocal optics. In brief, additional absorption in the form of electron-hole pairs in the surface region of the particle is important. These excitations can be antibonding in nature causing desorption. The surface plasmons only act catalytically insofar the field enhancement at the particle surface, i.e. the near fields determine the number of electronhole pair excitations. The interplay between thermal and nonthermal desorption as well as the desorption of dimers has meanwhile been investigated more thoroughly and bimodal energy distributions have been found [51]. In summary these experiments constitute an important example of nonthermat dissociation from large metal dusters where quenching was often assumed to be so effective that bond breaking by electronic excitation is prohibited. Obviously, even on a metal surface there is not necessarily complete relaxation of the dectronic excitation on a time scale in which photodesorption can occur. 4.4 Free Clusters

Opdcal spectroscopy experiments on large free dusters in beams and smoke have been performed so far for alkali metals and the noble metals copper, silver and gold. Extended work on smokes has been done by Broida and coworkers [3840]. Free silver and sodium dusters have later been reinvestigated (e.g.[41]); the characterization of the beams/smoke was, however, only tried by interpreting the optical spectra themsdves and has thus to be regarded with caution. The incandescem radiation of dusters in the visible spectra range [42] was recently also suggested for a duster lamp. In this section, however, only some experiments on very small free clusters in beams will be addressed. The principle of beam depletion spectroscopy was outlined in Section 3 (see Fig. 8) [3, 5, 43-44]. In the experiments performed so far, mostly alkali and noble

328

20 A=

8

"z

7

599 nm

R

6(11

5-

O)

4-

g

0

3-

Number of atoms per cluster

Fig. 14. Sodium cluster mass spectra with (shaded) and without laser illumination at a wavelength of ~. = 599 am. Obviously the beam depletion strongly depends on cluster size (after [3]). metal dusters were investigated. A duster beam is illuminated with a laser beam and the counting rate at fixed masses of Na dusters is measured as a function of time. Fig. 14 shows an example of two mass spectra, one without (upper curves) and another with illumination (shaded), for dusters with 2 to 21 atoms at a wavelength of 599 nm. Clearly the depletion for Na 8 and Na20 is much less than for those clusters in between, in particular Na 10 and Na 12. This means that photodepletion (a measure for the photoabsorption cross section) depends strongly on size and wavdength. The photodepletion cross sections can be deduced from the measured beam depletion. Figure 15 shows a comparison of experimental cross

oa

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.o

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g

o

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500

600

700

8OO

wavelength I[nm]

Fig. 15. Experimental photoabsorption cross sections for free Na8 dusters (after [45]) and theoretical positions and oscillator strengths of optical traasitious (vertical bars) as calculated with an at) initio molecular orbital approach (after [46]). The inset shows the assumed theoretical geometrical structure of the electronic ground state.

329

sections with the quantum chemical calculations. Originally, lower resolution spectra of Na 8, which is a spherical duster in the jellium model, were interpreted as single peak spectra corresponding to collective electron excitations. The high resolution spectrum of Na 8 [45], however, indicates that in addition to the very strong resonance around 500 nm, there appears to be some shallow structure around 600 nm. Fig. 15 includes the positions of the optically allowed transitions as calculated with the ab initio approach. The height of the lines corresponds to the relative oscillator strength. In the top right comer, the corresponding geometrical ground state structure of Na8 is shown. The overall agreement between theory and experiment is very good. On the other hand, the simple jellium prediction of spheroidal 10-electrondusters which would result in split collective resonances already seems to provide a reasonable first order description for such dusters (see Fig. 16). The usefulness of beam depletion spectroscopy either in collinear [3, 5, 43] or perpendicular [44] ali£nment was recognized at once by several other groups, Meanwhile experiments using the same technique but denoting it, e.g. as photoevaporation or -dissociation spectroscopy have been performed on many different duster systems, in particular the size range was extended up to large dusters. A compilation of a number of photoabsorption spectra of different materials and duster sizes can be found in [1].

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'

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,

,. 6 Energy [eV]

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2

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6 Energy [eV]

Fig. 16. Overview of optical spectra of very small 10-dectron metal dusters in beams: Nal0 [3, 43], Nall+ [47], Ag11+ andAg 9- [48].

330

Finally, pico- and femto-second spectroscopies like two photon ionization were applied to dusters. This opened up the new and wide field of time resolved spectroscopy of dusters, i.e. gives direct access to the dynamics of cluster relaxation and fragmentation [2425]. On the basis of the experimental results, it was concluded that for very small sodium dusters with n < 21, molecular excitations and properties prevail over surface plasmon like properties although collective interactions are expected to dominate for larger dusters [49]. These ps and fs spectroscopy pump and probe experiments are very promising and just the starting point for a detailed understanding of the ultrafast processes involved in duster relaxation and/or fragmentation.

4.5 Comparison of Cluster Spectra in Beams, Matrices Supports in a Single Experiment

and on

Recently, results of an experiment comparing large Ag dusters in beams, deposited on a subslxate, mad embedded m a dielectric matrix were published. This experiment [15] incorporates all three investigafon techniques in one and thus allows direct comparison between the spectra of the same dusters in a beam, on a support and in a matrix. A high temperature nozzle source produced silver dusters in a supersonic expansion operated either with or without Ar cartier gas. Depending on the source temperature (between 1600 and 2300 K) silver clusters with mean sizes of mound 2R = 2nm (N~250) were produced with sufficiently high beam densities to allow direct extinction measurements in the beam. Spectra were recorded in a fast mode using fiber optics connected to a diode array spectrometer. The effective optical path length of the beam was extended to ~ 0.9 m by use of a multiple beam crossing set up. The free dusters were subsequently either embedded in a SiO 2 matrix or deposited on SiO 2 substrates, located at variable distances from the duster source. The spectra of these samples were also investigated, allowing to compare the optical response of the free, deposited, and embedded clusters. Cluster sizes and size distributions were determined by eleclron microscopy. Examples of the recorded absorption spectra for the free dusters, those deposited on a SiO 2 substrate and dusters embedded in a coevaporated SiO 2 matrix are depicted in Fig. 17. The free dusters exhibit a surprisingly narrow Mie peak. It is blue-shifted by about 0.15 eV compared to the theoretical expectations based upon measured dielectric functions [50]. A measured peak shift may result from a superposition of a multitude of effects [1]. The respective red and blue shifts may even partially cancel each other and it is not even clear whether they are all independent from each other. Therefore, we just list up several effects which seem more probable than others for the explanation of the blue shift. • Band edge effects - as observed for Au dusters could yield blue shifts by a narrowing of the 4d-band due to a reduced mean coordination number. As a cousequence, the optical 4d-5sp interband transition edge would be shifted towards the blue, pulling also the Mie resonance in this direction.

331

(a)

2.0xlO-~.

(b) 250x 10-s

Ag c l u s t e r s in v a c u u m

Ag clusters o n SiO2 substrate

.... Mie-calculation; 2R=2.0nm A1=1.0 Aa=0.25 (shifted 0.15eV)

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~ 1_5- _ _ experiment

o= )04

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~

~

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60

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:

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I

4.0 energy leVI

I

4.5

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I

r

2.5

3.0

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I

3.5 4.0 energy [eV]

4.5

Fig. 17. Absorption spectra of Ag dusters produced as a free beam: a) Measured in the free beam and compared to Mie calculations performed with E(c0) after size correction (parameter A) due to limited mean free path effect [15]. b) Measured after deposition on a SiO2-glass substrate with different coverages increasing from bottom to top. c) Measured after embedding in a coevaporated SiO2-matrix with filling factors increasing from bottom to top (results of three rtms with slightly different sizes are shown).

The 5sp electron spill-out would cause the Mie band to shift into the opposite direction, i.e., towards the red. However, the spill-out is drastically reduced for energetically lower lying electrons. In fact, mainly 5sp electrons with momentum parallel to the duster surface and energy near the Fermi energy are outside the ionic duster size whereas 4d electrons do not contribute. The consequence is a blue shift for the Mie peak of Ag dusters.

332

° The free spherical duster surface may suffer a larger lattice contraction than the flat bulk surface, causing an enhanced 5s electron density and consequendy a blue shift of the plasmon. Concerning the Ag duster spectra of Fig. 17, it surprises that the deposited or embedded Ag dusters show no blue shift compared to the Mie daculations. This means, that the 0.15 eV blue shift of the free duster is cancelled upon deposition or embedding. This may be due to permanent charge transfer to the surrounding or chemical interface damping [15].

5 Conclusions Optical properties of metal spheres (R~10nm) can be described by classical dectrodynamics, i.e. Mie theory. The corresponding excitations are commonly called plasmon polaritons or simply plasmons. Application of this theory to large metal dusters is successsful, although for small clusters with less than 100 atoms it makes more sense to use the notation collective rather than plasmon excitations. The transition from molecular like to collective features apparently lies in the size region of about 8 to 20 atoms for free electron metals, however, unequivocal identification is difficult. Finally it should be mentioned that plasmon excitations m metal dusters have interesting properties like e.g. surface plasmon induced desorption processes which - similar to photoelectron spectroscopy or Surface Enhanced Ramar~ Scattering - are strongly enhanced in the vicinity of resonance frequencies since they sensitively depend on the electric near field intensities.

Acknowledgements. Part of this work was supported by the Deutsche Forsehuagsgemeinschaft. I greatly profited from the collaboration with Uwe Kreibig [Univ. of Aachen), the fruitful period while working as a postdoc with Walter Knight (Univ. of California, Berkeley), and the time with Frank Tr'ager (Univ. of Heidelberg and Kassel). Obviously, many students were involved in our experiments, in particular Wemer Hoheisel (Bayer, Leverkusen). Finally I wish to express my special thanks to Gisbert zu Putlitz who has introduced me to the fascinating realm of optical spectrosocpy in physics and who has supported my scientific career permanently from the very beginning.

6 References [1] U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Springer Ser. Mat. Sci. 25, (1995) [2] G. Delacr6taz, L. W6ste, Surf. Sci. 156, 770 (1985) [3] K. Selby, M. Vollmer, V. Kresin, J. Masni, W.A. de Heer, W.D. Knight, Phys. Rev. B 40, 5417 (1989)

333

[4] E. Mollwo, GiStt. Nachr., Math.-Phys. Klasse, Heft 3,254 (1932) [5] W.A. de Heer, K. Selby, V. Kresin, J. Masui, M. Vollmer, A. Ch~tdain, W. D. Knight, Phys. Rev. Lett. $ 9, 1805 (1987) [6] WA. de Heer, WD. Knight, M.Y. Chou, M.L. Cohen, Solid State Physics 40,93(1987) [7] J. Kowalski, T. Stehlin, F. Tr'~er, M. Vollmer, Phase Transitions 24-21, 737 (1990) [8] V. Kresin, Phys. Rep. 220, 1 (1992) [9] WA. deHeer, Rev. Mod. Phys. 65,611 (1993) [10] M. Brack, Rev. Mod. Phys. 6 5,677 (1993) [11] U. Kreibig, in Handbook of Optical Properties Vol. II, CRC Press Inc., Boca Raton, Eds. P. WiBmann, R. Hummel (1996) [12] Proc. First Int. Sympos. on Small Part. and Inorg. Clusters (ISSPIC I), Eds.: J.-P Borel, P. Joyes, J. Farges, B. Caband, J. de Phys. 38, C2 (1977); Proc. ISSPIC II Eds.: J.-P Borel, J. Buttet, Surf. Sci. 106 (1981); Proc. ISSPIC III, Ed. K.H. Bennemann, J. Koutecky, Surf. Sci. 156 (1985); Proc. ISSPIC IV, Eds,: C. Chapon, M.E Gillet, C.R. Henry, Z. Phys. D 12 (1989); Proc. ISSPIC V, Eds.: E. Recknagel, O. Echt, Z. Phys. D 19/20 (1991); Proc. ISSPIC VI, Ed.: S. Berry, J. Burdett, A.W Castleman, Z. Phys. D 2 1 (1993) ; Proc. ISSPIC VII, Surf. Rev. Lett. (Japan) 3 (1996) [13] G. Mie, Ann. Phys. 25,377 (1908) [14] C.E Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles Wiley 1983 [15] H. H6vel, S. Fritz, A. Hilger, U. Kreibig, M. Vollmer, Phys. Rev. B 4 8, 18178 (1993) [16] H. Ehrenreich, H.R. Philipp, Phys. Rev. 128, 1622 (1962) [17] J.H. Weaver, C. Krafka, D.W. Lynch, E.E. Koch, Physics Data." Optical Properties of Metals, Parts 1,2, FIZ Karlsnthe (1981) [18] T. Inagaki, L.C. Emerson, E.T. Arakawa, M.W Wiliams, Phys. Rev. B 1 & 2305 (1976) [19] W. Hoheisel, U. Schulte, M. Vollmer, E Tr~iger, Appl, Phys. A 5 1, 271 (1990) [20] U. Kreibig, Z. Phys. 234, 307 (1970) [21] J. Konteeky, P. Fantucci, Chem. Rev. 86, 539 (t986); V. Bonaeic-Koutccky, P. Fantucci, J. Koutecky, Chem. Rev. 91, 1035 (1991) [22] W. Ekardt, Phys. Rev. Lett. 52, 1925 (1984), Phys. Rev. B 3 l, 6360 (1985) [23] V. Bonacic-Koutecky, P. Fantucci, J. Koutecky, Chem. Phys. Lett. 16 6, 32 (1990) [24] T. Baumert, R. Thalweiser, V. Weiss, G. Gerber, Z. Phys. D 2 1, I31 (1993); see also Phys. Rev. Lett. 6 9, 1512 (1992) [25] H. KtihIing, K. Kobe, S. Rntz, E. Schreiber, L. W6ste, Z. Phys. D 21, 33 (1993); see also J. Phys. Chem. 98, 6679 (1994) [26] J.A.A.J. Perenboom, P. Wyder, E Meier, Phys. Rep. 78, 171 (1981) [27] U. Kreibig, L. Genzel,Surf. Sci. 156,678 (1985)

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[28] W. Hoheisel, K. Jungmann, M. Vollmer, R. Weidenauer, E Tr~tger, Phys. Rev. Lett. 60, 1649 (1988); see also Appl. Phys. A 52, 445 (1991) [29] M. Vollmer, R. Weidenauer, U. Schulte, W. Hoheisel, E Tr~iger, Phys. Rev. B 40, 12509 (1989) [30] W. Hoheisel, M. Vollmer, E Tr~iger,Phys. Rev. B 48, 17463 (1993) [31] U. Kreibig, J. Phys. F 4,999 (1974) [32] U. Kreibig, Habilitationsschrift, Saarbr/ickcn (1977) [33] W. Hoheisel, M. Vollmer, E Trager, Appl. Phys. A 5 2,445 (1991) [34] T. GOtz, M. Vollmer, E Tr~iger, Appl. Phys. A 57, 101 (1993); see also Z. Physik D 33, 131 (1995) [35] M. Vollmer, U. Kreibig, p. 216 in Nuclear Physics Concepts in the Study of Atomic Cluster Physics, Springer Lecture Notes in Physics 404 (1992), R. Sclmaidt, H.O. Lutz, R. Dreizler, (Eds.) [36] T. G6tz, M. Vollmer, E Tr~iger, in Laser Ablation: Mechanisms and Applications II, American Institute of Physics Conf. Proc. 2 2 8, 32 (1994) [37] M. Vollmer, E Tr~iger, Z. Phys. D 3, 291 (1986); Surf. Sci. 187, 445 (1987) [38] D.M. Mann, H.E Broida, J. Appl. Phys. 44,4950 (1973) [39] J.D. Eversole, H.P. Broida, Phys. Rev. B 1 5, 1644 (1977) [40] J. Hecht, J. Appl. Phys. 50, 7186 (1979) [41] S. Mochizuki, R. Ruppin, J. Phys.: Cond. Matter 5,135 (1993) [42] R. Scholl, B. Weber, in The Physics and Chemistry of Finite Systems: From Clusters to Crystals, Eds.: P. Jena, S. Khanna, B. Rao, Nato Asi Series C 374, Kluwer (1992) [43] K. Selby, V. Kresin, J. Masui, M. Vollmer, W.A. de Heer, A. Scheidemann, W.D. Knight, Phys. Rev. B 4 3, 4565 (1991) [44] M. Vollmer, K. Selby, V. Kresin, J. Masui, M. Kruger, W. D. Knight, Rev. Sci. Instr. 59, 1965 (1988) [45] C.R. Wang, S. Pollack, M.M. Kappes, Chem. Phys. Lett. 166, 26 (1990); also J. Chem. Phys. 93, 3787 (1990); J. Chem. Phys. 94, 2496 (1991) [46] V. Bonacic-Koutecky, M.M. Kappes, P. Fantucci, J. Kontecky, Chem. Phys. Lett. 17 0, 26 (1990) [47] C. Br~chignac, P. Cahuzac, E Carlier, M. de Frutos, J. Leygnier, Chem. Phys. Lett. 189, 22 (1992) [48] J. Tiggesb~iumker, L. K611er, H.O. Lutz, K.H. Meiwes-Broer, in Nuc/ear Physics Concepts in the Study of Atomic Cluster Physics, Eds.: R. Schmidt, H.O. Lutz, R. Dreizler, Springer Lecture Notes in Physics 404 (1992); see also Surf. Rev. and Lett. (Japan) 3,509 (1996); Chem. Phys. Lett., in press [49] T. Baumert, G. Gerber, Adv. Atom., Mol. Opt. Phys. Vol 35, 163 (1995) [50] P.B. Johnson, R.W. Christy, Phys. Rev. B 6,4370 (1972) [51] F. Tr'ager, contribution in this book; see also Appl. Phys. A, in print

N e w C o n c e p t s for I n f o r m a t i o n S t o r a g e B a s e d on Color C e n t e r s A. Winnacker Institute for Material Science, Department of Electronic Materials, University of Erlangen, Martensstra~e 7, 91058 Edangen, Germany

1 Introduction A color center in a solid is a point defect which absorbs and emits light like an atom. Since the pioneering work of Richard Pohl and his scholars in GSttingen in the twenties and thirties, color centers have strongly attracted the interest of physicists. They were considered to be prototypes of defects centers in solids. At the same time, being some kind of probes for their crystalline environment, they helped to understand the structure of solids in general. Some of these centers show a photochromic effect: After absorbing a photon they undergo some structural change resulting in a complete change of their absorption spectrum. It has long been understood that this effect could have an interesting application in optical information storage [1]: The information is written by a light beam inducing absorption changes in each "pixel" of the recording medium. "Change of absorption spectrum" could stand for a logic "1", no change for "0" when the information is read. In many cases the photochromic effect is reversible which means that the information can be erased optically, and rewritten many times. This application of color centers has for a long time been considered a fairly remote possibility, sometimes not taken too seriously because of the overwhelming success of other information storage schemes. So it may have gone quite unnoticed that color centers now play an important role in existing storage systems, like the "X-ray storage phosphors", or keep staying on the agenda of research and development due to their unique potential, like persistent spectral hole burning. A review of the situation in these fields is the topic of this paper. Color centers are systems on the borderline between atomic physics and solid state physics. Many terms describing their properties originate from atomic physics: Ground and excited states, level schemes, homogeneous and inhomogeneous broadening of spectral lines and bands, level splittings due to Zeeman- and Stark-effect etc. Many spectroscopic methods used for their investigation have been a tool in atomic physics as well, in particular fairly recent ones like high resolution laser spectroscopy or single atom spectroscopy.

336

A. Winnacker

In spite of this close relationship fundamental differences exist between color centers and free atoms or ions which basically are due the fact that the color center is incorporated in a solid. The main differences are the following

P]:

a) The color center is embedded in the crystalline field which lowers the symmetry of the system and thus gives rise to additional splittings of levels and spectral lines of the type which are familiar to the atomic physicist from the Stark- and the Zeeman effect. b) Lattice phonons can contribute to the absorption and emission process resulting in a broadening of the lines, because the phonon energies may be added to or subtracted from the energy hv of the photon. If no phonon is involved we talk of a "zero phonon line". c) The color center is not embedded in an ideal but a real crystal having all kinds of defects like vacancies, interstitials, dislocations, strain fields etc.. This means that the crystalline field is not the same for each center resulting in slightly different energy levels, and wavelengths of optical transitions, for each one. This constitutes the dominant broadening mechanism for zero phonon lines. So the inhomogeneous line broadening is of a very different nature compared to the one of free atoms which is mostly due to the Doppler effect. d) The spectral lines of a color center in general show a Stokes shift, i.e. the wavelength of the emission line is longer than the one of the corresponding absorption line. This is due to "lattice relaxation": The electronic wave function of the final state of an optical transition is different from the one of the initial state, in other words: The electronic charge distributions are different. Because the relative positions of atoms or ions in a solid are the result of the electric interaction between them, this change of charge distribution is followed by a change of the positions of the surrounding atoms. The surrounding lattice moves towards a new equilibrium configuration. This process of lattice relaxation is usually visualized in the "configuration coordinate model" [2]. e) An excited state of a color center can decay to a lower state not only by emitting photons (i.e. by a "radiative transition"), but also by emitting phonons into the lattice ("non-radiative transitions"). Consequently the lifetimes of excited states may be much shorter than the purely "optical lifetime" would be. It may well happen that an excited state which is occupied via an optical absorption process does not emit at all due to strong radiationless transitions to lower states. The lifetimes of such states can be as short as picoseconds. f) Last but not least a big difference exists between color centers and free atoms concerning the process of ionization. Because of its important role in the context of this paper this will be treated in some more detail in the following.

New Concepts for Information Storage Based on Color Centers

A I

E /eV 4f5d

3

Conductionband

[ gating light

--5D2

-

337

5D0

[ green

.~

501

I burninglight ca.7.5eV 7F0[red, 688nm

l

0

Valence band

Fig. 1. Level scheme of Sm2+ in BaFC1, showing the narrow 4f7 states as well as the broad 4fSd states. The energy positions relative to the conduction band are taken from [8].The two transitions indicated by arrows show the process of"gated hole burning" (Section 4). The trap for the photoionized electron is not shown.

2

Change

of Charge

States

of Color

Centers

in Solids

In discussing differences between color centers in solids and free atoms, it has to be remembered that a color center in a crystal is part of a much larger and more complicated electronic system, namely the crystal as a whole, which is characterized by its energy bands. This fact has a number of i m p o r t a n t consequences. The one that is particularly interesting in our context concerns the process of ionization. To fix ideas let us consider a rare earth ion like S m 2+ in a crystal, e.g. a Sm 2+ occupying the site of the Ba2+-ion in BaFC1. The energy levels of this ion are shown schematically in Fig. 1. These levels originate from spin orbit and crystal field splittings of the 4f-shell and the 4f5d-eonfigurations of this rare earth ion. In order to remove an electron from the ion and transform it into a Sm 3+, one electron has to be liked not to the vacuum like in a free ion, but into the conduction band, where it will form part of an electronic continuum. It is obvious that the energies of the transitions from the bound states to the continuum do not have anything to do with the ionization energies of free ions but are determined by the relative positions of the electronic levels of the ion relative to the conduction band of the host crystal. This process of ionization is crucial for the applications of defect centers for information storage discussed below. Not every color center in a

338

A. Winnacker

particular solid can undergo this process. In many cases only one charge state of an impurity ion in a particular host is energetically stable, other charge states would immediately recapture an electron or hole from the lattice. It should be noted in passing that this capability of changing charge states is, of course, also crucial for the second great application of defect centers in solids, i.e. their role as donors and acceptors in semiconductors. It is this capability 'that makes them "electrically active" by releasing electrons into the conduction band or capturing electrons from the valence band. It is often not clearly understood (though trivial) that talking about the energy level of a defect center in the energy gap always implies the potential for changing that charge state, because this energy level just indicates the energy required to induce this change. For that reason it is not quite precise to talk about e.g. the "energy level of Sm 2+ in BaFC1 relative to the conduction band" (Fig. 1), but rather about "the energy of the "Sm2+a+-level '', because it indicates the energy 3 E required to transform Sm 2+ into Sm 3+ by lifting an electron into the conduction band. Photoionization occurs if the color center is optically excited from a lower electronic state into an upper state which falls into the conduction band (Fig. 1). For completeness it should also be mentioned that in solids the charge state can also be changed by excitation of an electron out of the valence band to the electronic level ("charge transfer transition"). For instance, in Fig. 1 a Sm3+-ion could be transformed into Sm 2+ in this way.

3

Spectral Hole Burning : High Resolution Laser Spectroscopy in Solids

As pointed out under c) in the introduction, the zero phonon optical transitions in solids are usually inhomogeneously broadened due to lattice imperfections (Fig. 2). If the bandwidth of an exciting laser beam is much smaller than this inhomogeneous width, only a small fraction of the color centers are excited, namely those whose lattice environment gives them the proper transition frequency. An interesting situation arises if the optical transition results in a complete change of the absorption spectrum of the color center, which is the case for "photochromic centers" (see introduction). In this case the centers "hit" by the excitation frequency are removed from the absorbing ensemble with the consequence that a subsequent scan of the absorption line reveals a "spectral hole" at the "burning" frequency (Fig. 2). Different mechanisms exist for producing such persistent spectral holes [3]: Often impurity molecules in a solid undergo a configurational change altering their absorption spectrum (" photochemical hole burning"). Or the excited electron can be removed completely by a second excitation step resulting in photoionization ("photophysical hole burning"). Finally it may happen that after optical excitation the configuration of the surrounding lattice undergoes a permanent change due to the change in Coulomb interaction between the color center

New Concepts for Information Storage Based on Color Centers

339

after burning

before buming burning frequency

spectral hole

)

frequency ~

frequency

Fig. 2. Spectral hole burning. The hole is burned at a fixed frequency of the narrow band laser (left side), and subsequently monitored by scanning the absorption line (right side).

and the surrounding ions, a mechanism often encountered in glasses [4]. In any of these cases a spectral hole results in the inhomogeneously broadened line. Its linewidth is determined by the homogeneous width of the transition. The homogeneous linewidth may be very small at low temperature (of the order of 10-4 to 10-z cm-1) compared to the inhomogeneous linewidth of the order of 10 - 1000cm -1. In analogy to the methods of "Doppler-free spectroscopy" in atomic physics persistent spectral hole burning can be used as a tool for high resolution spectroscopy in solids, where the resolution is increased by the ratio of inhomogeneous to homogeneous linewidth. An example [5] is shown in Fig. 3, showing the shift of the F0 - Do transition (687.8 nm) of Sm 2+ in BaFC1 in a magnetic field: A persistent hole has been burnt into this transition at T : 2 K by photoionization hole burning (see next section) at zero field. The width of that hole is about 25 MHz, sitting within an inhomogeneously broadened line of about 13 GHz FWHM (see Fig. 4; this figure mainly refers to the process of "gating" treated in Section 4). This hole is subsequently scanned at different fields, thus revealing the Zeeman shift. The magnetic field is increased by equal steps in Fig. 3, so that the quadratic shift is seen immediately. The results can be compared to calculations based on the 4re-wave function of the Sm2+-ion. In the context of this paper we only want to demonstrate the enormous gain in spectral resolution achieved by the method of hole burning: One view at the horizontal axis of Fig. 3 reveals what the situation would be like if spectroscopy would be based on the inhomogeneous line with its width of 13 GHz. From the point of view of application spectral hole burning has attracted the attention due its basic potential for high density information storage [6].

A. Winnacker

340

.

.

.

BsCIF:Sm2+

o

. r 7D0~5F0 H°J-C,, H0(kG)4eo ~/~//'~

,.o

ProbeLaserFrequency(GHz)

,.o

Fig. 3. Shift of a spectral hole burned into the F0 - Do transition of Sm ~+ in BaFC1 [5]. The curvature of the inhomogeneous line profile (see Fig. 4) is only weakly seen on this horizontal scale.

4

~

8aCIF;sm2÷ ~..~,\\\ ~~~-U07 \\\ F 5D

;

LaserFreQuencyOffset (GHz) Fig. 4. "Gated" spectral holes in BaFCI:Sm 2+ . a) inhomogeneous line. b) central section of the inhomogeneous fine. c) after burning a "0" and "-220MHz" with a laser beam of 2W/cm 2 for 2000s. Holes are barely visible, d) Hole burned in 3s by addition of a "gating beam" of 514nm and 20W/cm 2, showing the dramatic effect of "gating". e) Multiple successively burned holes.

Spectral Hole Burning for High Density Information Storage

The storage density of optical data storage seems at first sight to be limited by diffraction: The writing and the reading light b e a m can not be focused to much better than its wavelength, i.e. to the order of 1 # m or slightly less. This gives an upper limit to the storage density of some 108cm -2. Actually storage densities close to that number are achieved in present optical storage systems like CD-ROM (Compact Disk Read Only Memory) or the reversible E-DRAW (Erase Direct Read After Wright) magneto-optical m e m ories. Frequency domain spectral hole burning (FDSHB) offers the possibility to increase this density by writing spectral holes into the absorption band in each "pixel" of the storage medium, presence of a hole at a particular wavelength meaning e.g. a "1", absence a "0" (Fig. 5). Because in some cases up to 1000 holes can be written into an absorption line, the storage density

New Concepts for Information Storage Based on Color Centers

341

cO 0 ..Q

Bit

1

0

1

0

0

1

0 >

frequency

Fig. 5. Application of persistent spectral hole burning to optical information storage.

could in principle be increased by this factor to about 10 xl cm -2. If, in addition, use were made of the coherence of the light, volume holograms could be written with storage densities up to 1014 cm -3 surpassing the one in the human brain (10Z3cm-3). As a matter of fact holographic memories have been demonstrated where more than 100 holograms were stored in thin films in a frequency interval of only 1 era-1 [7, 8], and more than 1000 all together. A crucial issue with regard to practical application of the concept of FDSHB in data storage is the one of"reading without erasing". It is clear that, while the information is read by scanning the laser frequency across the absorption line, the whole line is continuously burnt down until finally the information is lost. A solution to this problem is the two photon- or "gated" hole burning, which refers to the fact that the information can be written only in the presence of a second "gating" light beam, while it is read by one light beam only. Gated hole burning was demonstrated for the first time in the system BaFChSm [9]. The scheme is explained in Fig. 1: The spectral holes are burnt e.g. into the ZFo-SDotransition (688nm, red). These holes are made permanent by exciting the electron out of the excited state SD0 into the conduction band with a second photon. A crucial point is that the energy of that second photon has to be larger than the one of the first (red) photon, about 2 eV or more. The consequence is that the conduction band can not be reached by the red photons alone, but a second (green or blue) light beam must be present at the same time for persistent hole burning. After reaching the conduction band the electron has to be captured in a thermally stable trap, for details on this crucial point we refer to [9]. In order to use persistent spectral hole burning for data storage commercially it probably has to be performed at room temperature. This, however,

342

A. Winnacker

provides severe problems so far unsolved. At high temperature the intensity of the zero phonon line decreases relative to the one of the phonon replica, and the homogeneous linewidth (i.e. the hole width) dramatically increases, with the result that either no spectral hole can be burnt for lack of zero phonon line, or that the homogeneous linewidth essentially equals the total linewidth of the v.ero phonon line, which means that no gain in optical storage density is achieved any more. Some progress has been achieved in room temperature hole burning in recent years by proceeding along the following lines: 1) In order to have a considerable strength of the zero phonon line even at room temperature, one has to use rare ions as color centers, which, thanks to the good shielding of the inner 4f-shells, have a sufficiently small electron-phonon coupling. 2) Given the fact that the strong increase of the homogeneous linewidth with temperature is unavoidable, one has to turn to systems with an excessive inhomogeneous linewidth, like mixed crystals or glasses, in order to be able to burn a reasonable number of spectral holes (say 50 or more) into the inhomogenous line. Due to the large local disorder in these materials the inhomogeneous linewidth is much larger than usual. In our group we observed, e.g., an inhomogeneous linewidth of the VFo-SD2 -transition of Sm 2+ of a) b) c) d)

0.45cm -1 in a single crystal of BaFBr, 1.9cm -1 in BaFBr powder, 3 0 c m -1 in BaFClo.sBr0.s powder, and 76 cm-1 in a borate glass.

Spectral hole burning at room temperature was indeed achieved for the first time [10] in a mixed crystal SrFC10.sBr0.s:Sm and in the even "mixeder" crystal Mgo.sSro.sClo.sBr0.s:Sm [11]. The multiplexing factor (i.e. the ratio of inhomogeneous to homogeneous linewidth) is of the order of 10 in these systems at room temperature. A similar result was obtained recently in a borate glass doped with Sm [12]. A question which has to be considered separately from burning spectral holes at room temperature is the stability of these holes once they are burnt. Stability of holes requires two properties: i) The holes as discussed above are produced by photoionization. In order for the hole to be stable the trap of the "lost" electron must be thermally stable. ii) Each frequency within the inhomogeneously broadened absorption line corresponds to some particular configuration of the disturbed lattice. This configuration must be stable at room temperature, otherwise the hole would disappear due to "spectral diffusion". Both requirements are fulfilled for BaFCI:Sm and the related mixed crystals. BaFCI:Sm actually has been the first system ever reported which showed stable holes at room temperature [9]. Only very recently [13] a second case

New Concepts for Information Storage Based on Color Centers

343

of a crystalline solid with room temperature stability of spectral holes has been demonstrated: Spectral holes burnt into the internal transition of V 4+ in SiC at low temperature "survive" temperatures of at least 320 K. Whether information storage based on room temperature persistent spectral hole burning will at some time become a reality is still an open question [14]. But surely work will go on. In spite of all its difficulties the method has the fascinating and convincing aspect to be the only known concept that can beat the diffraction limit of optical recording.

5

Spectral Hole Electronics"?

Burning

as a Concept

for "Molecular

Looking further into the future, spectral hole burning is sometimes mentioned in the context of "molecular electronics". There is a certain connection between FDSHB and molecular electronics in the sense that the inhomogeneous broadening of an absorption band in a solid offers the possibility to address a single impurity atom or molecule. If this impurity absorption can be bleached by a photochemical or photophysical reaction - as in persistent spectral hole burning - the process can be visualized as part of a concept of data storage on a molecular level. A few numbers may illustrate this possibility: Let us assume that the absorbing species is present in the host in a concentration of 10 t4 cm -3, corresponding to about 10 -9 - 10 - s mol/mol. If a laser beam of 1 # m diameter is focused on a film of 10 #m thickness, the observation volume is 10 -11 cm -a, so some 1000 impurities are present in that volume. If the ratio of homogeneous to inhomogeneous linewidth is 103, a sufficiently narrow laser beam will address only individual impurities. The fluorescence of such a single impurity can be monitored, as is familiar from the spectroscopy of single atoms of ions confined in electromagnetic traps or on surfaces. To achieve a sufficient signal-to-noise ratio a large oscillator strength is required which guarantees at the same time a strong absorption of the laser light (favorable for a high separation of fluorescence- and scattered light!) and a short lifetime of the excited state, so that many photons can be emitted in a short time. For that reason single molecule detection has first been realized for organic dye molecules in an organic matrix, pentacene in p-terphenyl [15]. Single molecule detection offers fascinating possibilities for solid state spectroscopy like [16] but the aspect of "molecular electronics" will certainly keep pushing the imagination of scientists. One way to look at this possibility is the following: It is known from atomic physics that single atoms or ions can be monitored by light. The barrier to use this for high density data storage is given by the diffraction limit for the focusing of light beams. Due to the inhomogeneous broadening of spectral lines in solids, however, the large number of ions or molecules within the focus of the beam are labeled by their specific absorption frequency and can be addressed individually.

344

6

A. Winnacker

X-Ray Storage Phosphors

A medium for information storage based on color centers, which is already widely used, are "image plates" made of X-ray storage phosphors. The reference to X-rays in this notion points towards the fact that, while information can be written into some of these materials by visible and UV-light as well [17], the most important application is the one for the storage of X-ray images. The underlying effect is the following: In some materials X-rays as well as other ionizing radiation produce defect centers which subsequently can be optically excited giving rise to luminescence. The intensity of this "photostimulated luminescence (PSL)" is proportional to the number of centers created by the radiation, i.e to the received dose. This effect can be used as indicated in Fig. 6: The film or "image plate", usually a layer of storage phosphor powder imbedded in some organic resist, is exposed to X-rays and the photostimulable centers are produced. Subsequently the film is scanned point by point by a laser beam which in each point stimulates the luminescence proportional to the dose received in that point. This PSL-intensity of each point is recorded in a photo tube, digitized and transferred to a computer which subsequently reconstructs the image. The underlying physics is described in the lower part of the figure. X-ray quanta produce electron-hole pairs in a solid. In the classical X-ray phosphors used on X-ray screens, these electron-hole pairs recombine spontaneously under emission of light. In the storage phosphors, however, a large fraction of them are trapped at certain electron and hole traps, respectively. The left lower part of Fig. 6 schematically shows both processes, spontaneous recombination as well as trapping. Consequently the image is stored in form of trapped electrons and holes. The physics of the reading process is shown in the right lower part. The laser beam lifts the electron out of the electron trap into the conduction band from where it migrates to the trapped hole and recombines under emission of light. The advantages of image plates relative to the conventional film/screen system (a photographic film covered by a X-ray phosphor) are: a) The dynamic range, covering more than 6 orders of magnitude, is much higher than the one of the conventional film, which reduces the probability of wrong exposure time and facilitates the detection of otherwise undetected features in the image. b) The image is digitized as an inherent part of the system. So modern methods of image processing, pattern recognition and digital data storage can be applied. Images can easily be transferred within and between hospitals. c) Image plates are reusable. As follows from the reading mechanism presented in Fig. 6 the image is erased by reading (as a matter of fact it is only partly erased during the reading process, because complete erasure would require too much reading time, and completely erased subsequently by light of an intense lamp).

New Concepts for Information Storage Based on Color Centers

345

Scanning

Exposure

x-RayT u ~ ~

c.b.

R?L

..K~ Lasefoeam

"'r'°~:(-V~y" " PM

c.b.

.o,o _ -~:=c~mt;neSC i f PhotonR~--ppmg t.'~ ~ ~te%T2

Cumins=.

_T:.o:_f-v.b.

;r ox v.b.

Fig. 6. Process of X-ray image recording via X-ray storage phosphors. "c.b." stands for conduction band, "v.b." for valence band.

Two types of color centers are involved in the storage process: An electron trap and a hole trap. The spectroscopic properties of these traps determine the performance of the system. This will be discussed in the following for the case of the most widely used storage phosphor BaFBr doped with Eu [18, 19]. i) Nature and properties of the electron trap. Because the PSL is produced by optically removing the electron out of its trap, the PSL excitation spectrum is identical to the absorption spectrum of the electron trap. Figure 7 shows the excitation spectrum of the PSL in BaFBr:Eu. It turns out that this spectrum is identical to the absorption spect r u m of the bromine F-center in this crystal (i.e. an electron at a bromine vacancy). In this way the "active" electron trap in BaFBr:Eu has been identified to be the bromine F-center. For the practical application it is important that the stimulation spectrum falls into a spectral region accessible by commonly available lasers, which, in the case of BaFBr:Eu, is provided by the He-Ne-laser (633 rim). It would be even more favorable for economic reasons if a semiconductor laser could be used for readout. Given the fact that efficient and low cost semiconductor lasers are available at 670 nm a stimulation spectrum shifted somewhat to the red would be desirable. This is a little exercise in "material engineering". It has been known since the early work of Mollwo [20] that the wavelengths of the F-band in different materials can be understood in terms of the simple model of an electron in a potential well, the wavelength being the longer the wider the potential well, i.e. the larger the lattice constant. The lattice constant of the BaFBr-lattice can be increased by adding iodine to the lattice in order to form BaFBrl_=I=. Figure 8 shows the PSL stimulation spectrum in this mixed crystal for x = 0.16 compared to the one of BaFBr. It is shifted to the red, indeed, and nicely includes the wavelength of 670rim. In this way the spectroscopic properties of the

346

A. Winnacker PSL-lmensity

1.0

~

~

B a F B r : Ce

1,2 [ PSi.

lath.

IJni|s

BaF~r"Eu

1.o ~- - -

R/~Bt" TI ,,

/-

I:~d :U , .75

i\

! .50

0,4

....

~

.

.

.

.

.

.

.

.

.

.

i O.2

.

.

.

.

.

.

.

.

.

.

.

.

.

~---

.2E

/

\

-

, ,

,

Wg~zlcnl~lh

Fig. 7. Excitation spectra of the PSL in the storage phosphors BaFBr:Eu, RbBr:T1 and RbI:T1. As discussed in the text they are identical to the F-center absorption bands in these materials, the one in BaFBr to the one of the bromine F-center. Appropriate laser hnes for reading are also shown.

.

500

.

.

.

600

700

,

,

~00 nrn

,

~00

Wavelength

Fig. 8. Comparison of the PSL excitation spectrum in BaFBr and BaFBr0.s410.1s .

electron trap can be optimized to meet the specifications of application. The red shift of the crystal with the larger lattice constant is also seen in Fig. 7 by comparing RbBr and RbI. ii) Nature and properties of the hole trap. The electron released by the reading laser b e a m recombines with the trapped hole. This implies that the emission spectrum of the PSL is identical to one or more emission lines of the hole-trap. Figure 9 shows the PSL spectrum of BaFBr:Eu together with the fluorescence spectrum of Eu 2+ in the same crystal after direct UV excitation [21]. The spectra are identical and correspond to the well known transition of the 4f5d-excited state to the ground state. So Eu 2+ is the hole trap. It should be noted in passing that trapping of the hole does not seem to result in formation of Eu 3+, because never experimental evidence for the formation of Eu 3+ under X-ray irradiation in this material has been observed. Rather a Eu2+-hole complex is formed. T h e wavelength of PSL-emission is around 390 a m . For the application it is i m p o r t a n t that it is clearly distinct from the stimulation wavelength (Fig. 4) which is the case. In this way reading light and PSL can be spectrally separated without loss of detection efficiency. Figure 10 summarizes the physics of the trapping and the reading process: When the reading (Helium-Neon- or semiconductor-) laser b e a m scans the image plate, electrons trapped as F(Br-)-centers are excited. The excited F-center relaxes (see d) in the introduction) into its "relaxed excited state

347

New Concepts for Information Storage Based on Color Centers

....~/~ .....

ConductioBand n &E = 35meV f

/ ~ E u 2+- Emission

1.0-

A-

0,8

--~

0.6

tunneling ..... ~_fSf~p~Sevi_~ ..... direct ~ , : i t a ~

2.sev / ] ,, 2.,ev

of E.uZ+

~ m )

F(Br')-center 1.2eV

0.4 0.2 OY

0

correlated Eu2*/hole-center --

350

400 nm 450 ~vek.ag=h---->

J.2eV x,,..

uncorrelated Eu2*/hole-center

Valence Band

Fig. 9. Comparison of the PSL emission spectrum with the Eu 2+ - luminescence excited directly by UV-light (300 nm). Both spectra are identical as a consequence of Eu being the hole trap.

Fig. 10. Storage and readout process in BaFBr:Eu, for explanation see text.

(RES)" 35 meV below the conduction band. The electron then is released thermally into the conduction band, migrates towards the Eu2+/hole complex and recombines under emission of the characteristic Eu 2+-light at 3.2 eV corresponding to 390 nm. As indicated in Fig. 10 also a tunneling process between the RES and the Eu2+/hole-center can occur [19]. The hole traps that are reached in a tunneling process are called "correlated centers" (in the sense of a spatial correlation), the one reached via the conduction band "uncorrelated" centers. Figure 10 should make clear that the understanding of the storage process in the X-ray storage phosphors implies a detailed knowledge of color center properties. A very important property of the hole trap center is the lifetime T of the excited state, because from this lifetime results a lower limit for the reading time of an image, as seen in the following: When the laser beam scans over the image plates it reads the information pixel by pixel by inducing the PSL. The PSL has a certain decay time r. It is obvious that the laser beam should proceed to the next pixel only when the PSL of the previous one has decayed to a large extend, otherwise the image would be washed out and its spatial resolution be diminished. To avoid this, a time of about 3r has to be provided for each pixel. With N pixels in the image this results in a minimum reading time of :

N

(1)

W h a t determines the PSL decay time r ? In principle this is a complex issue because the PSL excitation process involves several steps: Lifting the electron out of its trap, migration towards the hole trap, recombination and optical emission to the ground state. For all storage phosphors studied by us it turned

348

A. Winnacker PS¢ lwb. un~!

lexc. =590 nm O" 0.20.4 0.6V

~ - 95 ns 1410nm)

0.81.0500 ns/div

Fig. 11. Decay of PSL after pulsed (20 ns) excitation with a flash lamp. The decay time is 750 ns for BaFBr:Eu with its emission at 390 nm, and 95 ns for BaFBr:Ce. The times correspond to the excited states lifetimes of Eu 2+ and Ce 3+ , respectively, see text.

out that at room temperature the last step is the slowest one [19, 22], the contribution of the others being negligible. So 7- = T, and it can be stated t h a t the lifetime T of the optical emission of the trap center to the ground state defines a m i n i m u m for the reading time according to (1). 7- and T in BaFBr:Eu turn out to be 750ns [21]. For a medical image with 4000 x 4000 pixels we have N -- 1.6.107. Inserting these numbers into (1) this results in a m i n i m u m reading time of 36 s, which is acceptable for most purposes. If faster sequences of pictures have to be taken, a hole trap with shorter decay time T of the excited state has to be used. Ce 3+ has T -- 95 ns, and we could show that indeed BaFBr:Ce has a PSL decay time of t h a t size (Fig. 11) resulting in a minimum reading time of 4.6s instead of the 36s for BaFBr:Eu. It turns out that the mechanism described above for BaFBr:Eu is essentially the same for other storage phosphors as well like RbBr:T1 [23] and RbI:X [22] (with X standing for T1, In, Pb or Eu) in the sense t h a t the electron traps are F-centers and the hole traps are the " a c t i v a t o r ' - i o n s like Eu, T1 etc., and in the sense that the model given in Fig. 10 can basically be applied. It is obvious that the performance of the storage phosphors, like in the previously discussed case of photoionization spectral hole burning, critically depends on the capability of the color centers involved to change their charge state, because electrons and holes are trapped, released, and are transferred from one center to the other. In that sense much of the underlying physics of image plates is closely related to that of the other information storage schemes like the ones based on photochromic effects, spectral hole burning or the photorefractive effect. All these involve questions like trapping cross sections, position of levels in the band gap, thermal stability of traps etc..

New Concepts for Information Storage Based on Color Centers

7

Conclusion

349

and Acknowledgements

The progress of information storage via color centers as discussed in this paper will proceed synchronously with further progress in material research, and will be promoted by new instrumental possibilities like the progress of semiconductor lasers, micro-optics, scanning- and near field microscopy. The driving force behind this development will be the ever increasing demand for storing and handling information. I thank my colleagues Dr. R. Macfarlane (IBM), Dr. H. yon Seggern (Siemens AG) and Dr. M. Thoms for years of cooperation and discussion.

References [1] K. Schwartz: The Physics of Optical Recording, Springer, Berlin, Heidelberg 1093 [2] W. B. Fowler: The Physics of Color Centers, Academic Press N.Y., London 1968 [3] Persistent spectral hole burning: Science and Application (W.E. Moerner ed.): Springer, Berlin 1988

[4] Macfarlane, R. M. Shelby: J. Lum. S6, 179 (1987) [5] Macfarlane, R.M. Shelby, A. Winnacker: Phys. Rev. 33, 4207 (1986) [6] G. Castro, D. Haarer, R.M. Macfarlane, H.P. Trommsdorf: US Patent 4101976, 1978 ("Frequency selective optical data storage") [7] U.P. Wild, A. Rebane, A. Renn: Adv. Mater. 3, 453 (1991) [8] C. DeCaro, A. Renn, U.P. Wild: Appl. Opt. 30, 2890 (1991) [9] A. Winnacker, R.M. Shelby, R. M. Macfarlane: Opt. Lett. 10, 350 (1985) [10] R. Jaaniso, H. Bill: Technical Digest Spectral Hole Burning: Science and Appllcation, Vol. 16, Opt. Soc. of America, p. 146 (1991) [11] K. Homday, C. Wei, M. Croci, U.P. Wild: see [10] p. 118 [12] K. Hirao, S. Todorold, D.H. Gho, N. Soga: Opt. Lett. 18(19), 1586 (1993) [13] R. Kummer, C. Hecht, A. Winnacker, submitted to Opt.Lett. (1997) [14] R. Ao, L. Kiimmerl, D. Haarer: Adv. Mat. 7 No. 5, 495 (1995) [15] W.E. Moerner, L. Kador: Phys. Rev. Lett. 62, 235 (1989) [16] W. E. Moerner: Science 265, 46 (1994) and references therein [17] J. Lindmayer: Solid State Techn., August 1988, p. 137 [18] K. Takahashi, K. Kohda, M. Miyanodai, Y. Kamemitsu, A. Amitani, S. Shionoya: J. Lumin. 31~32, 266 (1984) [19] M. Thoms, H. von Seggern, A. Winnacker: Phys. Rev. B44, 9240 (1991) [20] E. Mollwo: Nachr. Ges. Wiss. GSttingen II Math. Physik. K1. 97 (I931) [21] M Sonoda, M. Takano, J. Miyahara, H. Kato: Radiology 148, 833 (1983) [22] M. Thorns, H. von Seggern, A. Winnacker: J. Appl. Phys. 76 (3), 1800 (1994) [23] H. yon Seggern, A. Meijerink, T. Voigt, A. Winnacker: J. Appl. Phys. 66(9),

4418 (1989) [24] D.M. De Leeuw, T. Kovats, S.P. Herko: J. Electrochem. Soc. 134, 491 (1987)

Excitons and Radiation Damage in Alkali Halides K. Schwartz Gesellschaft flit Schwerionenforschung, Planckstr. 1, D-64291 Darmstadt, Germany

1

Introduction

The quantum mechanical description of atomic spectra was the beginning of a new era of quantum physics which today is the theoretical basis of nuclear and solid state physics, chemistry, and biochemistry [1,2]. Quantum mechanics opened also a new look to solid state physics which will be illustrated by the development of the conception of excitons [3]. One of the basic quantum mechanical problems of solid state physics is the electron-phonon interaction which makes the description of electronic excitations more complicated than in atomic physics. The electron-phonon interaction participates in any energy conversion process in solids. It manifests itself in luminescence (Stokes shift), defect creation, polaron formation, etc. [5,6]. An analysis of these problems was started in the 1920s [8] and the first theoretical conceptions were developed by Frenkel [4] and Peierls [7]. In the early 30s, Frenkel proposed a new concept of electronic excitations in insulating solids, the exciton states [4]. Within the quasi-classical interpretation, the exciton is an excited electronic state of the lattice determined by a coupled electron-hole pair whose energy level structure is similar to that of the hydrogen atom. Nevertheless, excitons are collective electronic excitations of the lattice where the local electron and hole interacts with a diverse number of the surrounding atoms. Such a concept originates from atomic physics and is nowadays widely used to describe the energy conversion and electron-phonon interactions in solids. Excitons can migrate through the crystal and by interaction with the atoms of the regular lattice create a polaron. The polaron of a free exciton is a self-trapped state with a longer lifetime than for the free exciton. Thus the main interaction of excitons with the surrounding lattice occurs from self-trapped states. Self-trapped excitons can relax through photon emission (luminescence), defect creation, or non-radiative transitions (i.e., by conversion of the electronic excitation energy into lattice vibrations)

[5,6]. The and the Frenkel citons).

interaction radius of the exciton is determined by both the atomic electronic structures of the lattice and varies from one (small radius excitons) to several atomic spheres (large radius Wannier-Mott exThe excitons in alkali halides are intermediate between Frenkel and

352

K. Schwartz

Wannier-Mott excitons [5,6]. The basic theoretical concepts of various exciton states in crystals were developed in the 30s by Frenkel, Peierls, Mott, and Wannier [4,7,10,11]. A sharp rise of theoretical and experimental investigations of excitons, as well as the exciton induced processes in different classes of solids occurred only in the 50s [5,6,11,13-15]. A further development of excitons is presented in [19,26,33]. This article is mainly focused on the role of excitons in radiation damage creation processes in alkali halides. After a brief historical survey of excitons in alkali halides, the radiation damage mechanism in these crystals and the peculiarities by irradiation with heavy ions are discussed.

2

E x c i t o n s in Alkali Halides

Alkali halide crystals, first studied as minerals (rock salt NaC1, KC1, etc.), had an important role in the formation of modern solid state physics. In the 20s Pohl et al. developed the main concepts of electronic excitations, photoconductivity, point defects, luminescence, etc. in ionic crystals [6,9,11,16-18,33]. Pohl also studied the fundamental absorption spectra of alkali halides and observed a sharp absorption peak in the ultraviolet spectral region near the band gap [17]. Later, this absorption peak was correlated with the low energy Frenkel exciton and described by the Hilsch-Pohl formula [11]. Extensive experimental and theoretical studies of excitons in alkali halides and other crystals (ionic and covalent dielectrics, semiconductors, molecular crystals) were started in the 50s [11]. A new step in the development of polaron states in dielectrics was the observation of self-trapped holes in LiF and other alkali halides (Table 1) [22]. These self-trapped holes (covalent dihalide molecules X 2 replacing two regular anions in the lattice) were the first polarons observed in insulating crystals in agreement with the theoretical predictions by Frenkel soon after introducing the exciton conception [4,13]. At the same time, the self-trapped excitons in alkali halides were definitively detected by their luminescence [6,44,45]. Further investigations showed that various selftrapped electronic states (electrons, holes, excitons) can exist in dielectrics and semiconductors, and their stability is determined by the atomic and electronic structure of the solid [6,37]. In alkali halides free holes and excitons can only relax to a self-trapped state (Fig. 1), and the electron polarons are absent [5,6]. The interaction of the self-trapped excitons or polaron states with the lattice is much stronger than that for free excitons or free electrons and holes. The main steps in the discovery of exciton processes in alkali halides are summarized in Table 1. Free excitons in alkali halides can be created via optical excitation and by irradiation with X-rays or charged particles. By optical excitation excitons with a definite energy are created, whereas by irradiation with X-rays or charged particles excitons with various excitation energy are produced (from high energy K-shell excitons up to low energy excitons with the energy (E,~)

Excitons and Radiation Damage in Alkali Halides

353

Table 1. The development of excitons and exciton induced processes in alkali ha]ides year 1920s

subject or phenomenon photoconductivity of irradiated and additively colored NaCl spectroscopic studies of the fundamental absorption of ionic crystals 1931- 1932 theoretical models for transformation of electronic excitations into heat 1931- 1937 excitons, polaxons, seK-trapped excitons 1957 self-trapped holes (VK-centers) in alkali halides 1957 luminescence of self-trapped excitons 1960 Frenkel defect creation (F- and Hcenters) in alkali halides by X-ray excitation at T --= 4 K 1964 Frenkel defect creation by optical excitation in the fundamental absorption band of NaCl 1966 exciton relaxation models of radiation damage creation in alkali halides

author and reference R~ntgen [9] Hilsch, Pohl [6,17]

Frenkel [4]; Peierls [7]

Frenkel [4]; Peierls, Wannier, Mott [10,11] Castner, K£nzig [22] Teegarden, Knox [45,46]; Kabler, Paterson [20] Rabin, Klick [12]

Lushchik et al. [6] Hersh [24]; Pooley [25]; Lushchik et al. [6]

close to the band gap (Eg)). Photons at the energy h~, ~ Eg induce anion excitons (e °) with the energy E~: ~_ Eg ~ 10 eV. These free excitons interact with the lattice due to luminescence (activator, exciton), self-trapping, or free electron (e) and hole (h) generation. The main path (l) of free excitons (e °) depends on both the atomic and the defect structure of the crystal. The magnitude of I varies from 10 to 104 lattice constants (large l are typical for alkali iodides) and the relaxation time is of the order 10-14 _ 10-13s [6,33]. The self-trapped anion exciton can relax from the triplet state (lscr~) by a ~r-luminescence emission with a lifetime up to several milliseconds. The 1rluminescence has a large Stokes shift (up to 5 eV) which demonstrates the strong conversion of the electronic excitation energy into atomic vibrations (Fig. 2). There is also a ~r-luminescence from singlet self-trapped anion exciton states (2SC~g) with a shorter lifetime of 10 -9 - 10 -6 s [14]. The efficiency of the self-trapped exciton luminescence below the thermal quenching temperature is about 0.1%. A free exciton luminescence in alkali iodides and a non-relaxed ("hot") luminescence from the self-trapped excitons was observed [6]. The efficiency of the non-relaxed luminescence emission is still smaller than that of the self-trapped excitons from the equilibrium state.

354

K. Schwartz

( ( ( (a)

(b)

(c)

Fig. 1. Three configurations of self-trapped excitons in insulating crystals: a) atomic self-trapped exciton; b) - molecular self-trapped exciton; c) - self-trapped exciton with a strong intrinsic electron-hole interaction [29].

The exciton luminescence is thermally quenched at T > 50 K (the quenching temperature depends on the composition of the crystal) [14]. The self-trapped exciton states can be transformed into free electrons and holes (e ° --+ e + h) and vice versa self-trapped excitons can be produced by electron-hole recombination (VK + e --+ e°). An alternative process to the exciton luminescence and non-radiative decay is the Frenkel defect creation from exciton states which will be discussed in §3.

3

Radiation D a m a g e in Dielectrics

Radiation effects in solids were observed long before the discovery of X-rays by R6ntgen and radioactivity by Becquerel [41-43]. Such effects are coloration of rock salt (NaC1) [43] and CaF2 [42], thermoluminescence of minerals [41], and chemical etching of uranium fission product tracks in various minerals with radioactive trace elements [67]. In the 20s, systematic investigations of radiation damage and color centers in ionic crystals were started in G6ttingen by Pohl et al. [33]. A more extensive experimental and theoretical study in various solids began in the 50s stimulated by the advanced technology of nuclear reactors and nuclear weapons. Seitz and Koehler developed a theory of defect creation by elastic atomic collisions under irradiation with fast particles [39]. Such elastic collisions are the main damage creation process in metals and some dielectrics by irradiation with charged particles and fast neutrons [68]. The radiation damage studies in alkali halides, especially Frenkel defect creation by optical excitation in the fundamental absorption band and by X-ray irradiation, demonstrate, however, that radiation defects in these ionic crystals are not created by elastic collisions. Thus the energy loss of charged

Excitons and Radiation Damage in Alkali Halides

'\

\

ASTE

FE

/

'M

/

\

"]!l!illtlll I I',1i'1 \

rE

355

I

I

~MSTE

/MSTC II l

I llll I i~ii!!!l~','ill iii!',',iiiiiii ',1

ii 1'ii! i!lI i',', / Ill

~ i!llii I ',1

,

I!l',jl!l @!1 II!',i~lt IJl',ill I

ill II ~111L Ifll I illl ill Ii

~ g

1II

O.

III I

o

Q,,

o~

o "

i', '',, o,,

Fig. 2. Energy level diagrams of excitons in alkali halides: g - a crystal without excitons; FE - free exciton; MSTE - molecular self-trapped exciton; ASTE - atomic self-trapped exciton. The arrows show transitions of absorption (left.) at T = 0 K and emission (right). The arrows with the broad lines correspond to the relaxed luminescence and arrows with thin lines correspond to the non-relaxed luminescence. qA a n d qM are the activation energies for the atomic and molecular exciton creation

(e ° ~ eD [6].

particles (electrons, protons, etc.) in these and other dielectrics are determined by Coulomb interaction, the transformation of electronic excitations of the lattice into defects becomes a topical problem. T h e irradiation of ionic crystals and other solids is accompanied by a strong heating. Only a small part of the absorbed energy (few percent or less) is transformed into lattice defects [38-40]. A general analysis of the radiation induced heating in condensed systems was first performed by Dessauer in the 20s and qualitatively explained as the conversion of the electronic excitation energy into atomic vibrations [8]. Such an energy conversion was theoretically discussed in the early papers of Frenkel [4] and Peierls [7] and further extended for exciton induced processes in dielectrics and semiconductors [5,6,26-36]. A p a r t from the exciton luminescence and non-radiative decay, the Frenkel defect creation from self-trapped exciton states is an alternative process of the exciton energy conversion. Excitons in details explain the interaction of electronic excitations with phonons in insulators and semiconductors. Today, the exciton mechanism of radiation d a m a g e creation in alkali halides and other dielectrics is well understood [5,6].

356

K. Schwartz

In insulating crystals the energy transfer from fast charged particles to the lattice occurs via ionization of atoms if the velocity of the charged particle v exceeds the Bohr velocity of the atomic electrons %. The electronic energy losses lead to the primary ionization of lattice atoms followed by the creation of secondary d-electrons: primary ionization --->d-electrons --+ electronic excitations of the lattice (1) The energy loss from the charged particle takes place within a time scale of ~ 10-17s (primary atom ionization) to ,-~ 10-14s (thermalization of delectrons) which is much shorter than the relaxation time of lattice excitations and the non-elastic defect creation time. The energy transfer from heavy ions to the lattice is described by the TRIM code [50], and for the lateral energy distribution the models of Katz and Waligorski are used [51,52]. The magnitude of linear energy loss (dE/dx) of various charged particles (electrons, protons, heavy ions) in solids varies on a large scale from 10 -6 e V / ~ (electrons of ~ 1MeV) to 103 e V / ~ (heavy ions of 10 MeV/u) [53]. In such large scale of dE/dx (over 9 orders of magnitude!), the radiation damage creation mechanism can be different at various excitation levels of dE/dx [5,49,50]. Nevertheless, for alkali halides, as demonstrated by Perez et al. [53] and Balanzat et al. [54], the defect creation (at T ~ 15K and at room temperature) and the exciton luminescence (at T ~ 15K) are similar both by irradiation with X-rays and heavy ions. This means that the exciton induced luminescence and defect creation dominate within an extremely large scale of excitation energy density, and the elementary defect creation mechanism by the decay of self-trapped excitons in alkali halides is independent of the type of radiation. Nevertheless, at high magnitude of dE/dx, collective electronic excitations (i.e., superposition of elementary excitations) can modify the exciton defect creation mechanism I34,35]. Various excited electronic states, induced by either charged particles or optical excitation in the fundamental absorption band, rapidly relax to the lowest electronic excitations of the lattice with the energy close to the band gap Eg (self-trapped excitons (e°~),free electrons and holes (e and h)). Only these lowest electronic excitations of the lattice can produce Frenkel defects [5,6,33,36]. The relaxation of high electronic excitations of the lattice is a complicated process where one high energy electronic excitation (E~t > Eg) is converted into several low electronic excitations (Eex ,~ Eg). Such multiplication of electronic excitations is accompanied with a strong electron-phonon interaction, and only one third of the initial energy of E~I(E > Eg) is transformed into excitons and electrons and holes while two thirds are transformed into atomic vibrations [36]:

E~t(E > Eg) --+E~: + E(e) + E(h)

(2)

where E ~ is the energy of the self-trapped exciton, and E(e) and E(h) are the energy of the created electron and hole, respectively. Thus the effective mass

Excitons and Radiation Damage in Alkali Halides

357

of holes in Mkali halides is larger than that of the electrons, holes determine the range of the secondary interaction in the lattice. The energy of the created electrons and holes, usually, exceeds the equilibrium kinetic energy of the charge carriers in the conduction and valence band, i.e., hot electrons and hot holes are produced. In alkali halides, the hot holes either relax to a selftrapped state (within a time of ~ 1- 10ps with a mean free path o f / ~ 10 nm, and a diffusion length of L ~ 100 rim) or can be captured by lattice defects [6,33]. The mean free path and diffusion length of electrons is larger than that e h for holes clue to their effective mass relation (rnef f < meff). The electrons can be trapped by lattice defects or can recombine with a self-trapped hole and create a self-trapped exciton e °. The migration and interaction of excitons, electrons, and holes in solids lead to a remarkable increase (up to 1000/~ and more) of the initial lateral range, determined by the energy transfer from the fast particles. The multiplication of electronic excitations in alkali halides was studied in detail by Lushchik et al. using synchrotron radiation [6,33-35]. The presence of various elementary electronic excitations was observed: primary excitons, secondary low energy excitons, hot electrons and holes, double excitons (2e°), double electron-hole pairs (2(e + h)), etc. [34,35]. The efficiency of Frenkel defect creation by various superpositions of low energy electronic states is different. In KBr the defect creation efficiency is the highest by a superposition of a self-trapped exciton and an electron-hole pair ( e Os, e + h) [34]. The Frenkel defect creation by self-trapped excitons is a relaxation of the lattice in which the electron and the hole are converted into spatially separated defects. Such relaxation can be described as a non-elastic collision, and this is possible if the interaction of the electron and the hole with the surrounding lattice is stronger than their intrinsic interaction in the exciton state (Fig. 1 and Fig. 3). The dominating primary Frenkel defects in alkali halide crystals are the F- and H-centers (Fig. 3). During the further interaction with the lattice most of the F- and H-centers are annihilated, and only a small part of them are separated to stable color centers. A peculiarity of alkali halides is the different chemical binding of the host lattice (ionic) and of various hole centers (covalent). The covalent binding of hole centers is determined by the electronic structure of halogen atoms (ns2pb). Various hole centers are combinations of X~, and the halogen products of radiolysis are covalent molecules X2 [5,6,40,68]. The main radiation damage creation in alkali halides takes place in the anion sub-lattice: the electron centers (F-centers and their aggregates) are created on the anion vacancies; hole centers are created by replaced anions (Fig. 3 and Fig. 4). Nevertheless, cation vacancies are also produced from exeitons states [6,33]. The creation of elementary Frenkel defects occurs from molecular exciton states having a higher probability of electron and hole center separation than that of the atomic self-trapped exciton (Fig. l b and Fig.3) [5,6,33]. Two

358

K. Schwartz

)

(a)

(b)

Fig. 3. The structure of an atomic exciton (a) and of a Frenkel pair (b) with separated F- and H-centers in alkali halides [5]

different Frenkel pairs can be produced: e sO - + v a + I

or

e Os ~ F + H

(3)

where e~° is the self-trapped exeiton, and (F, H) and (Va,I) are the induced Frenkel pairs (Fig. 4). A defect creation similar to (3) can also take place by the recombination of an electron (e) with a self-trapped hole (hs). A conversion of exeitons into electron-hole pairs with a following self-trapping and vice versa is possible (e ° ~ e + h). Thus, Frenkel defects can be created ° e + h) . by the reaction (3) from both electronic excitations ( e ~, As mentioned above, by the decay of excitons various Frenkel pairs can be created (equation (3)). The relation of the concentration of (va + I) and (F + H) pairs in various alkali halides is different [5,6]. Nevertheless, the Frenkel pair (Va + I) corresponds to a positively charged anion vacancy Va and negative interstitial ion X~t , and the probability to capture an electron or a hole by the charged v~ and I centers is high. At higher temperature these defects are transformed into other more stable lattice defects [5]. In alkali halides the H-centers are stable at T <_ 30 K (in LiF at T < 60 K). Above this critical temperature the H-centers become mobile and can interact with the lattice atoms and defects. In the temperature range 30 K < T < 110 K the neutral H-centers are transformed into self-trapped holes VK ~Fig.-4). These self-trapped holes at T >_ l l 0 K are mobile which leads to an increase of their interaction radius with the surrounding lattice. Vg-centers can create self-trapped excitons by means of recombination with electrons (VK + e ---+e°). At higher temperature Vg-eenters disappear into more stable hole centers [33,68]. The F-centers produced at low temperatures (T = 4 K) are stable up to higher temperatures (for LiF up to T > 500 K) [68]. At high irradiation doses and higher temperatures, F-center aggregates (F~-centers with n < 4) and colloid centers (macroscopic aggregates of F-centers) are created [57,58].

359

Excitons and Radiation Damage in Alkali Halides F

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-

-

+

-

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-

+

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+

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+

-

,+

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+

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+ ~']

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r'--'l

•

I--'~

* ' - L -' - J ,2e, +

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--/f÷

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+ ,H, ' ~

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+

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[]

.1

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--

.,-

Vx:X~.lX-X_ ] H=X2"IX-]

Aggregote Hole

centers

Fig. 4. Various types of defects and color centers in alkali halides: va, vc - anion and cation vacancies; F, F', FA, M (F2) - electron color centers; H, VK - hole centers; nF - aggregates of F-center which correspond to the initial step of colloid center formation [16].

The kinetics of elementary Frenkel pair creation in alkali halides was studied by two photon pulsed laser spectroscopy in a large temperature range (from 4 K up to the melting point) [32]. A relaxation time (r) for stable (F-H) pair creation of r --- 10ps was found. This demonstrates that the Frenkel defect creation time v is much longer than the lattice vibration period rvib~ "~ 10 -13 s. The efficiency of primary Frenkel defect creation (7/) has a complicate dependence on time and temperature. After a short time interval (At ~ 50ps) following the excitation with the laser pulse, the primary efficiency of defect pair creation (r/0) is increasing with the temperature (e.g., in KC1 r/o ~ 1 at T = 880K, and r/0 ~ 0.16 at T ~ 4K). These primary Frenkel defects, however, are rapidly annealed especially at high temperatures, reducing the efficiency 7/of stable Frenkel pairs at optimal conditions to only 5 - 6 %. The dependence of the efficiency of Frenkel defect creation on temperature by optical excitation [32] determines also the efficiency of radiation damage creation at various temperatures. From a general point of view radiation damage creation in alkali halides and in other insulating crystals depends on the irradiation dose, dose rate (i.e., excitation energy density), and temperature. In alkali halides at room temperature the stable Frenkel defects are F- and Vs-centers (V3-centers have a micro structure of v c h X ~ ) [34]. At high irradiation doses and at temperatures above room temperature, complex electron and hole centers are produced (microscopic and macroscopic clusters of F-, X~-- centers, vacancy clusters, covalent halogen molecules X2, etc.) [55,57,58]. At low temperature the defect creation and the exciton luminescence in various alkali halides have a different dependence on the temperature which determines also the efficiency of the Frenkel defect creation at various temperatures [5, 6, 14]. The defect creation efficiency can be described by the

360

K. Schwartz

energy to create one Frenkel pair (AEF). Two different classes of alkali halide crystals exist [12]. For the crystals of the first class (NaCl, NaBr, KI, etc.) at low temperature, AEF is approximately several hundred thousand electron volt and for crystals of the second class (LiF, KC1, KBr, NaF, etc.) A~EF is only several thousand electron volt [6]. At higher irradiation temperatures, the efficiency of defect creation increases, and at room temperature in both classes of alkali halide crystals, the energy to create one Frenkel pair AEF is about 1000 eV. In crystals of the first class, the temperature dependence of the exciton luminescence is anti-correlated with the efficiency of Frenkel pair creation. The magnitude of A E F strongly depends on the excitation energy density (dE/dx). By irradiation with heavy particles (protons, a-particles, heavy ions), AEF increases as the value of dE/dx increases [53, 54, 59, 60]. Such increase of AEF is determined by two processes: (1) recombination (annihilation) of the primary Frenkel defects; (2) aggregation of F-centers to macroscopic metal colloids (see §4). The defect creation by relaxation ofexciton states in dielectric materials is possible if excitons (or electrons and holes) are self-trapped and if the energy of this state Eel: exceeds the energy Ed,e required for defect creation [33,66]. The defect creation energy by the decay of electronic states Ed,e for various dielectrics is about 10 eV which is smaller than the energy of defect creation by elastic atomic collisions (Ed ~ 25 eV) [15, 33, 39, 68]. The defect creation in various solids by electronic excitations of the lattice was analyzed in detail by Itoh [5,26-31], Lushchik [6,33], etc. It was found that dielectrics without self-trapped states have a small probability for Frenkel defect creation, and therefore these materials are resistant under irradiation (Table 2). If the energy of self-trapped excitons in dielectrics is smaller than the defect creation energy (Eej: < Ed,e), Frenkel defect creation from single excitons is not possible (A1203, SiO2, etc., Table 2). Nevertheless, in these materials defects can be created by a superposition of single electron excitations. Itoh has analyzed the exciton damage creation processes in SiO2 and found that the self-trapped excitons in SiO2 correspond to a configuration where the strong intrinsic interaction of the electron and hole prevent their separation and defect creation (Fig. lc) [26,48]. Radiation damage creation by the decay of double excitons was observed in various dielectrics [26,27]. In alkali halides double excitons can produce

Table 2. Self-trapped excitons and radiation damage efficiency of dielectrics [5,6] Type Examples

Self-trapping

1

MgO, ZnO

no

2 3

c-SiO2,A1203, Y203 yes, E ~ < Ea,~ LiF, NaC1 and other alkali halides yes, Eem > Ed,~ AgCI, AgBr etc. CaF2 etc.

Sensitivity to irradiation low low high

Excitons and Radiation Damage in Alkali Halides

361

halogen molecules (2e ° -+ X2) in the bulk or at the surface [27, 29-31, 63-65]. Nevertheless, the efficiency of Frenkel defect creation by the relaxation of two electronic excitations is smaller than by the decay of single self-trapped excitons.

4

Radiation Damage and Heavy Ion Track Formation in Ionic Crystals

Heavy ion induced radiation effects opened a new sphere of radiation damage creation processes in solids [46-49]. The excitation density dE/dx of heavy ions with a specific energy of about 10 MeV/u is several orders of magnitude higher than by conventional irradiation with 7-rays or electrons (of 1 MeV). For such ions the energy transfer to the target electrons is determined by electronic losses [50-52]. Usually only at the end of the ion path (where the velocity of the ion (Vio,~) is below the Bohr velocity (vo) of the electrons in the target atoms) defects are created via elastic collisions (nuclear loss). Thus, the main radiation induced phenomena in solids by heavy ion irradiation occur under an extremely high electronic excitation level. Under heavy ion irradiation, the collective defect creation processes play an important role and these collective excitations are similar in metals and dielectric materials where the defect creation via single electronic excitations is impossible (Table 2) [46-48]. In alkali halides irradiated with heavy ions, however, the simple exciton mechanism of radiation damage creation was demonstrated by Perez et al. [53] and Balanzat et al. [54]. It was shown that the defect creation energy for LiF, NaC1, and KBr crystals under heavy ion irradiation at room temperature and at T ~-, 15K was close to that observed by X-ray excitation. Also the yield of the exciton luminescence in the range of 15 - 200K was the same as that for X-ray excitation. Perez at. al. demonstrated that the main electronic centers of the ion track in LiF crystals irradiated with Ne, Ar, Kr, and Xe ions at room temperature are F- and F2-centers while the presence of macroscopic F-center aggregates was not detected [53]. Nevertheless, aggregates of F-centers and Li colloids were observed in LiF when irradiated with high doses at room or higher temperatures (thermal neutrons, ion implantation, etc.) [57-59,64-66]. Young observed uranium fission tracks in LiF crystals by chemical etching [55]. Gilman and Johnson showed that only macroscopic aggregates of F-center (various Li colloids) can be chemically etched in irradiated LiF crystals [58]. Therefore, we initiated experiments intended to understand the track damage morphology in heavy ion irradiated LiF crystals using optical spectroscopy, small angle X-ray scattering (SAXS), and chemical track etching [59-60]. In LiF crystals irradiated with various heavy ions (U, Au, Pb, Bi, Xe, Se, and Zn) chemical track etching was observed if the value of dE/dx exceeds a critical magnitude of (dE/dx)e,crit ~ 1.2keV/•. SAXS studies demonstrated that the heavy ion

362

K. Schwartz

induced track etching is correlated with a cylindrical damage region with a radius of 10 - 20 A. The radius of the cylindrical track damage region is increasing with dE/dx > (dE/dx)~,cr,t) [60]. Such macroscopic cylindrical damage region is determined by large F-center aggregates (Li-colloids). The radius of the observed track damage region is in good agreement with the lateral track radius for 50 % energy loss (rt~t ~ 10 £) [70]. The heavy ion induced defect structure around the ion path is complicated. Secondary electronic and atomic migration processes in the lattice leads to an extension of the cylindrical radiation damage region around the ion path. Such processes are the relaxation of excitons, electrons, and holes. In alkali halides holes with a larger effective mass than that of electrons, determine the extension of the primary electronic energy loss lateral radius up to rt~t > 1000 ~ [69]. In the extended lateral track damage region single point defects (electron and hole color centers) are dominating, whereas in the central track core region (rtat ~ 20A) defect aggregates and local phase transitions prevail. The aggregates (colloid centers) in the central track core determinate also the chemical etching of the ion track [59,60]. Such large defect aggregates can be created only at an extremely high excitation level. In LiF crystals irradiated with Zn, Se, Xe, Au, Pb, and U the energy to create one Frenkel pair A E F correlates with the magnitude of dE/dx of the ion [60,61]. The increase of AEF at higher excitation energy density leads to the recombination (annihilation) of primary Frenkel pairs which is in good agreement with various experiments [48,54,68]. The increase of AEF can be explained by a higher recombination efficiency of primary Frenkel pairs at high dE/dx, as well as by a more efficient aggregation process from single F-centers to metallic colloids. Such process was observed in LiF crystals irradiated with Se and Xe heavy ions where the initial value of dE/dx at the external surface of LiF was below the critical threshold for chemical etching (dE/dx)~,~it. Under these conditions etchable tracks were produced only in LiF crystals irradiated with Se and Xe ions through a polycarbonate filter which leads to an increase of the magnitude of dE/dz above the threshold value (dE/dx)~,¢~,t [59,60]. The observed chemical etching effect correlates with a decrease of the concentration of F-centers for several times (and a corresponding increase of AEF). These results demonstrate that the formation of colloidal centers in the track core influences the concentration of single F-centers around the ion path (i.e., in the lateral region with a radius rt~t >> 20 A). Nevertheless, the colloid centers and single point defects are created in various track regions by different mechanisms. F-center aggregates and metal colloids in LiF can be created only under definite conditions: (1) the concentration of F-centers (NF) must exceed a critical value (NF > NF,crit,); (2) the hole centers must be spatially separated from the electron centers to prevent their annihilation; (3) thermal or radiation enhanced diffusion of single color centers can realize the aggregation

Excitons and Radiation Damage in Alkali Halides

363

(1) and separation (2) process [33, 59, 60]. Under heavy ion irradiation the formation of F-center aggregates (colloids) at room temperature can be explained as an effect of diffusion of F-centers or by the generation of extremely high local concentration of F-centers (see Fig. 4). Nevertheless, thermal diffusion of F-centers in LiF crystals occurs only at T > 500K, whereas the hole centers can migrate at much more lower temperatures. For both models the hole centers must be spatially separated. Probably, the primary H-centers at the critical" excitation density rapidly migrate to the lateral track region and, therefore, the aggregation of single primary F-centers to large aggregates is possible. The observed critical value of (dE/dx)e,crit for etchable damage creation in LiF crystals demonstrates the important role of the excitation density and a high local irradiation dose in the ion track. Nevertheless, it is difficult to distinguish the role of the dose and the excitation density for the aggregate (colloid) center creation [60]. To understand the elementary mechanism of colloid center formation in heavy ion tracks in LiF crystals, additional experiments at low temperatures are necessary.

5

Conclusion

In alkali halides the excitons are responsible for any energy conversion processes by optical excitation in the fundamental absorption band or by irradiation with charged particles or X-rays with various excitation energy densities. Exciton processes determinate also both luminescence and defect creation under heavy ion irradiation with an extremely high excitation energy density of 103eV/•. The heavy ion track in LiF crystals consist of lithium atom aggregates in the central part (with a radius of rtat < 20 ~) and single color centers in the extended lateral track region. Acknowledgement I am very thankful to Prof. Ch.B. Lushchik (Tartu) and to O. Geit3 (GSI, Darmstadt) for many fruitful discussions and remarks to this review.

References [1] H. Wimmel. Quantum Physics ~ Observed Reality. A critical interpretation of quantum mechanics World Scientific, Singapore, 1992 [2] J.T. Cushing. Quantum Mechanics. Historical contingency and the Copenhagen hegemony University Chicago Press, Chicago, 1994 [3] Ed. W. Neuser, K. Neuser-von Oettingen. Quantenphilosophie Spektrum, Berlin, 1995 [4] J. Frenkel. On the transformation of light into heat in solids I Phys. Rev. 37 (1931) 17 - 44; On the transformation of light into heat in solids//Phys. Rev. 37 (1931) 1276 - 1294 [5] N. Itoh and K. Tanimura. Formation of interstitial-vacancypairs by electronic excitations in pure ionic crystals J. Phys. Chem. Solids 51 (1990) 717 - 735

364

K. Schwartz

[6] Ch.B. Lushchik. Creation of Frenkel Pairs by Excitons in Alkali Halides In: Physics of Radiation Effects in Crystals. Ed. R. A. Johnson, A. N. Orlov. Elsevier Science Publ., Amsterdam, 1986, 473- 525 [7] R.E. Peierls. Zur Theorie der Absorptionsspektren fester KSrper Arm. Physik 13 (1932) 905- 952 [8] H. Dessauer. Uber einige Wirkungen yon Strahlen IV. Zs. Phys. 20 (1923) 288 - 298 [9] W.C. R&ntgen. Uber die Elektrizitfftsleitung in einigen Kristallen und iiber den Einflufl einer Bestrahlung daraufAnn. Phys. 64 (1921) 1 - 195 [10] G. H. Wannier. The structure of electronic excitation levels in insulating crystals Phys. Rev. 52 (1937) 191 - 199 [11] R.S. Knox. Theory of Excitons. Solid State Physics Vol.5, Ed. F. Seitz, D. Turnbull, Academic Press, New York, 1963 [12] H. Rabin and C.C. Klick. Formation ofF-centers at low and room temperature Phys. Rev. 117 (1960) 1005 - 1010 [13] A.L. Shluger and A.M. Stoneham. Small polarons in real crystals: concepts and problems. J. Phys. Condens. Matter 5 (1993) 3049 - 3086 [14] R.T. Williams and K.S. Song. The self-trapped exciton J. Phys. Chem. Solids 51 (1990) 679 - 716 [15] M. Klinger, Ch. Lushchik, T.V. Mashovets, G.A. Kholodar, M.K. Sheikman, M. Elango Defect formation in solids by decay of electronic excitations Sov. Phys. Usp. 28 (1985) 994 - 101 [16] K. Schwartz. The Physics of Optical Recording Springer Verlag, Berlin - Heidelberg, 1993 [17] R. Hilsch and R.W. Pohl. Uber die ersten ultravioletten Eigenfrequenzen einiger einfaeher Kristalle Z. Physik 48 (1928) 384 - 396 [18] A. Smakula. Uber die Verfiirbung der Alkalihalogenidkristalle dutch ultraviolettes Licht Zs. f. Physik 63 (1930) 762 - 770 [19] A. S. Davydov. Theory of Molecular Excitons Plenum Press, New York, 1971 [20] M.N. Kabler and D.A. Paterson. Evidence for a triplet state of the self-trapped exciton in alkali halide crystals Phys. Rev. Lett. 19 (1967) 652 [21] R.A. Kink, G.G. Liidja, Ch.B. Lushchik, and T.A. Soovik. Izv. SSSR, ser. fiz. 31 (1967) 1982 [22] T.G. Castner and W. K/inzig. The electronic structure of V-centers J. Phys. Chem. Solids 3 (1957) 178 - 195 [23] Ch.B. Lushchik, G.G. Liidja, and M.A. Elango. Fiz. Tverd. Tela 6 (1964) 2256 (Sov. Phys. Solid State 6 (1965) 1789) [24] H.N. Hersh. Proposed excitonie mechanism of eolour center formation in alkali halides Phys. Rev. 148 (1966) 928 - 932 [25] D. Pooley. Defect creation mechanism by exeitons Proc. Phys. Soc. 87 (1966) 245 [26] N. Itoh. Self-trapped exciton model of heavy-ion track registration Proc. Intern. Conf. on Radiation Damage, Italy, 1995 - in print [27] N. Itoh and T. Nakayama. Electronic excitation mechanism of sputtering and track formation by energetic ions in the electronic sputtering regime Nucl. Instr. Meth. B 13 (1986) 550- 555 [28] N. Itoh, K. Tanimura, A.M. Stoneham, and A.H. Harker. The initial production of defects in alkali halides: F and H centre production by non-radiative

Excitons and Radiation Damage in Alkali Halides

365

decay of self-trapped exciton J. Phys. C: Solid State Physics 10 (1977) 4197 4209 [29] N. Itoh, K. Tanimura Radiation effects in ionic solids Rad. Eft. 98 (1986) 269 - 287 [30] N. ltoh. Sputtering and dynamic interstitial motion in alkali halides Nucl. Instr. Meth. 132 (1976) 201 - 211 [31] N. ltoh and T. Nakayama. Mechanism of neutral particle emission from electron-hole plasma near solid surface Physics Lett. 92 A (1982) 471 - 484 [32] R.T. Williams, J.N. Bradford, and W.L. Faust. Short-pulse optical studies of exciton relaxation and F-center formation in NaCl, KCI, and NaBr Phys. Rev. B 18 (1978) 7038- 7057 [33] Ch.B. Lushchik, A.Ch. Lushchik. Decay of Electronic Excitations with Defect Formation in Solids Nauka, Moscow, 1989 - in Russian [34] A. Lushchik, I. Kudrjavtseva, Ch. Lushchik, and E. Vasil'chenko. Creation of stable Frenkel defects by vacuum uv radiation in KBr crystals under conditions of multiplication of electronic excitations Phys. Rev. B 52 (1995) 10069- 10072 [35] A. Lushchik, E. Feldbach, R. Kink, and Ch. Lushchik. Secondary excitons in alkali halides Phys. Rev. B 53 (1996) 5379 - 5387 [36] R.C. Alig, S. Bloom. Electron-hole pair creation energies in semiconductors Phys. Rev. Lett. 35 (1975) 1522 - 1525 [37] A. Sumi. Phase diagram of an exeiton in phonon field J. Phys. Soc. Japan 43 (1977) 1286- 1294 [38] F. Seitz. The motion of charged particles through solid matter Disc. Faraday Soc. 5 (1949) 271 - 289 [39] F. Seitz, J.S. Koehler. Displacements of atoms during irradiation Sol. Stat. Phys. 2, ed. F. Seitz and D. Turnbull, N.Y., Acad. Press, London, 1956 [40] F. Seitz. Color centers in alkali halides II. Rev. Mod. Phys. 26 (1954) 1 - 102 [41] H. Oldenburg. Thermoluminescence of fluorites Phil. Trans. Roy. Soc. London 3 (1705) 345 [42] T.J. Pearsall. On the effects of electricity upon minerals which are phosphorescent by heat J. Royal Inst. 1 (1830) 77, 267 [43] K. Przibram. Irradiation colors and luminescence Pergamon Press, London, 1956 [44] K. J. Teegarden. Luminescence of potassium iodide Phys. Rev. 105 (1957) 1222 - 1227 [45] R.S. Knox, and K.J. Teegarden. In: Physics of Color Centers ed. W. B. Fowler, Academic Press, New York, 1968, p. 1 [46] A. Dunlop, D. Lesser, P., H.. Effects induced by high electronic excitations in pure metals: a detailed study in iron Nucl. Instr. Meth. B 90 (1994) 330 - 341 [47] A. Meftah, F. Brisard, J.M. Constantini, E. Dooryhee, M. Hage-Ali, M. Hervieu, J.P. Stoquert, F. Studer, and M. Toulemonde. Track formation in Si02 quartz and the thermal-spike mechanism Phys. Rev. B 49 (1994) 12457 12463 [48] A M. Stoneham. Radiation effects in insulatorsNucl. Instr. Meth. A 91 (1994) 1 - 11 [49] E. Balanzat. Heavy ion induced effects in materials Rad. Eft. 126 (1993) 97 101 -

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[50] J.F. Ziegler, J.P. Biersack, U. Littmark. TRIM 89. The stopping and ranges of ions in solids Pergamon Press, New York, 1985 [51] M.P.R. Waligorski, R.N. Harem, R. Katz. The radial distribution of dose around the path of a heavy ion in liquid water Nucl. Tracks Rad. Meas. 11 (1986) 309- 319 [52] R. Katz, K.S. Loh, L. Daling, and G.-R. Huang. An analytic representation of the radial distribution of dose from energetic heavy ions in water, Si, LiF, NaI, and Si02 Rad. Eft. 114 (1990) 15 - 20 [53] A. Perez, E. Balanzat, J. Dural. Experimental study ofpoint-defect creation in high-energy heavy-ion tracks Phys. Rev. B 41 (1990) 3943 - 3950 [54] E. Balanzat, S. Bouffard, A. Cassimi, E. Dorothyee, L. Protin, J.P. Grandin, J.L. Doualan, and J. Margerie. Defect creation in alkali halides under dense electronic excitations: Experimental results on NaCl and KBr Nucl. Instr. Meth. B 91 (1994) 134- 139 [55] D.A. Young. Etching of radiation damage in lithium fluoride Nature 183 (1958) 375- 378 [56] C.3. Delbecq, P. Pringsheim. Absorption bands in irradiated LiF J.Chem. Phys. 21 (1953) 794 - 800 [57] K.K. Schwartz, A.J. Vitol, A.V. Podins. Radiation effects in pile-irradiated LiF crystals Phys. Status Solidi 18 (i966) 897 - 909 [58] J.J. Gilman and W.G. Johnson. Dislocations, point-defect clusters, and cavities in neutron irradiated LiF crystals J. Appl. Phys. 29 (1958) 877 - 888 [59] K. Schwartz. Electronic excitations and defect creation in LiF crystals Nucl. Instr. Meth. B 107(t996) 128 - 132. [61] K. Schwartz, C. Trautmann. Heavy ion induced radiation damage in LiF crystals GSI Nacrichten GS[ 09-95 (1995) 13- i5 [60] K. Schwartz, C. Trautmarm, and T. Steckenreiter. Ion tracks in LiF crystals GSI Jahresbericht 1995 [6i] D. Albrecht, P. Armbruster, and R. Spohr. Investigation of heavy ion produced defect structure by small angle scattering Appl. Phys. A 37 (1985) 37 - 44 [62] A.T. Davidson, J.D. Comins, T.E. Derry, and F.S. Khumalo. The production of defects and colloids in litthium fluoride crystals by implantation with rare gas ions !Ra& Eft. 98 (1986) 305 - 312 [63] N. Seifert, S. Vijayalakshmi, Q. Yan, A. Barnes, R. Albridge, H. Ye, N. Tolk, and W. Husinsky. Optical absorption spectroscopy of defects in halides Rad. Eft. 128 (1994) 15 - 26 [64] A.E. Hughes. Metal colloids in ionic crystals Adv. Physics 28 (1979) 717 - 828 [65] J.R.W. Weerkamp, J.C. Groote, J. Seinen, and H.W. den Hartog. Radiation damage in NaC1. I - I V P h y s . Rev. B 50 (1994) 9781 - 9801 [66] F. Agullo-Lopez, C.R.A. Callow, P.D. Townsend. Point Defects in Materials Academic Press, London, 1988 [67] G.A. Wagner, P. Van den Haute. Fission-track dating Ferdinand Enke Verlag, Stuttgart, 1992 [68] M. Elango. Elementary inelastic radiation-induced processes American Institute of Physics, New York, 1991 [69] M. Elango. Hot holes in irradiated ionic solids Rad. Eft. 128 (1994) 1 - 13 [70] M. Kr/imer and G. Kraft. Calculations of heavy ion track structure Radiation Environment Biophys. 33 (1994) 91 - 109

Polarization of Negative Muons Implanted in the Fullerene C60: Speculations about a Null Result A. Schenck 1, F.N. Gygax 1, A. Amato 1, M. Pinkpank 1, A. Lappas 2, K. Prassides 2 Institute for Particle Physics of ETH Z/irich, CH-5232 Villigen PSI, Switzerland School of Chemistry and Molecular Sciences, University of Sussex, Falmer, Brighton BN1 9Q J, United Kingdom

1

Introduction

Since the discovery of parity violation in the rr - p - e decay chain it is well known that negative muons (p-) implanted in graphite will preserve about 1/6 of their initial polarization when cascading down to the lowest Bohr orbital (12S1/2) after capture by a C-atom [1]. The loss of polarization during the cascade is due to spin-orbit coupling in a given muonic level E(ng, j = 1 + s) where n is the main quantum number and g the angular momentum. Spin-orbit coupling may be understood to produce a very strong field at the p - causing its spin to turn. Depolarization will happen if the lifetime 1--1 of this state is long enough to allow t h e / J - spin to rotate by a measurable degree, i.e. /7 << A, where A is the fine structure splitting. In the initial phase of the cascade this condition is not fulfilled due to rapid decay by Auger transitions and hence no depolarization is occurring. Once, however, the/~- has reached the first state with F0 << Al0 (with main quantum number no depending on the host atom) one finds a population of the fine structure levels with j0 = 20 + 1/2 of 2j0 + 1 wj - 2(220 + 1)

(1)

and residual polarizations in these states of

PJ= ( 3 + j o ( j o + l ) _ t o ( t o + l ) ) 3j0(j0 + 1)

2 P0

,

(2)

which, for t0 large, assume wj = 1/2 and Pj = ~1 [2]. The condition F0 << Ao implies that the state decays by a radiative transition. In the further radiative cascade down to the muonic K-shell the polarization in the 20 + 7l state is preserved in each step since in this case the dipole transition leaves the projection of the p - spin on the quantization axis unchanged. This is not

368

A. Schenck et al.

true if the transition starts from the ~0 - ~1 level in which case after a number of transitions all polarization will effectively be lost. On the reasonable assumption that muon capture will populate dominantly states with t0 near its m a x i m u m (no - 1) [2] one estimates a residual polarization for the p - in the lowest 1 s orbital of

1 1

Pre~ ~- ~" 5Po

1 = ~P0

(3)

This is indeed what is found in graphite and many other non-magnetic elements (notably metals) with zero nuclear spin [3]. It implies that in these cases no other depolarizing processes take place after the p - has reached its lowest level in the K-shell. Additional depolarization may be possible if the electronic shell is left in a paramagnetic state with non zero electronic angular momentum, either during the cascade (the loss of Auger electrons may leave the atom, at least for short times, in a highly ionized state) or after the cascade. The absence of additional depolarization indicates that the paramagnetic state may be too short-lived to cause any substantial p--spin rotation by hyperfine coupling to the electronic angular m o m e n t u m or that the latter undergoes some extremely rapid relaxation, suppressing effectively the hyperfine coupling. However, in many organic and inorganic compounds, both liquid and solid, the residual polarization is found to amount to less then (1/6) P0 but rarely less than ,-~ 0.5 (1/6) P0 [3] (we are only considering here zero nuclear spin systems). A notable exception is Pd, where Pros -~ 1/3 of Pres in graphite [4]. It is also observed that Pr~ may vary with temperature as, e.g., in Si [5]. In all these cases the residual polarization is observed to show no measurable time dependence. In transverse field spin rotation (pSR) measurements the residual polarization precesses with the "free muon" Larmor frequency, showing the p - to be in a diamagnetic state. The reduced polarization must then be caused by some transient paramagnetic state in which hyperfine effects lead to some further depolarization but the lifetime of this state is not long enough to allow for a complete loss of polarization. It is most likely that the additional depolarization happens after the p - has reached its lowest orbital (for Pd this conclusion follows from the fact that p - in the excited muonic 2p state of Pd carries the expected Pre~ [6]). A long-lived paramagnetic state after termination of the cascade should become visible by characteristic precession frequencies ~hf of the hyperfine coupled y - - e l e c t r o n ( p - e - ) system like in the well studied muonium (p+ e - ) atom. In particular in zero external field it should be possible to observe directly the hyperfine frequency w0 corresponding to the splitting of the F = 1 and F = 0 (F = I t + Se) states. To our knowledge the only indication for a long-lived paramagnetic state may have been found in pure silicon at low temperatures, where in zero field a #SR signal with a frequency of 650 MHz was barely visible [7].

Polarization of Negative Muons implanted in the Fullerene C60

369

In this contribution we wish to discuss some recent results on the residual polarization Pres of # - implanted in the fullerenes C60 and K3C60. While, as pointed out before, /~- in graphite (and also in diamond [5]) display the maximum possible Pre~ in the lowest muonic 1 s level, p - in C60 appear completely depolarized [8]. Yet in metallic K3C60 at room temperature Pre~ is again the same as in graphite. Since it appears unlikely that depolarization during the cascade is any different in the three carbon allotropes: graphite, diamond, C60, the zero P~¢s in C60 must arise from peculiar circumstances, effective in C60 but not in the two other systems, after completion of the cascade. In muonic carbon in its ground state the nucleus appears to the electrons as shielded by one charge unit (Z - 1 = 5), i.e. it acts like a boron nucleus and the electronic configuration will resemble that of a boron atom or ion. On this basis we will try to understand our results. This will be the subject of section 3. In section 2 the experimental facts will be summarized first. Conclusions and a summary are presented in section 4.

2

Experimental

Results

This will be a brief summary of our observations up to now. The powder C60 sample used in this work contained about 10% of C70. 1. Transverse field (TF) #SR in a field of 0.1 T at room temperature did not reveal any long-lived residual polarization at the free muon Larmor frequency. This is most evident when comparing Fourier transformed p S R spectra from C60 and graphite (see Fig. 1). The graphite target was identical in shape and mass with the C60 sample. 2. TF-/~SR in a small field of 0.8 m T between 12 K and 300 K did not reveal any residual polarization neither at the free p - Larmor frequency nor at higher frequencies (up to 250 MHz). In the presence of a tong-lived paramagnetic state (e.g. for a spin 1/2 - spin 1/2 system like muonium) one would expect to see the precession of the F = 1 triplet state with frequency wt ~ 11 MHz (at 0.8 roT) and an amplitude or asymmetry of 1 A~t ~-, Prcs ~- 71 " ~P0 -~ 1.7 %. Stopping instead/~+ in our C60 sample we observed clearly the well known muonium plus muonic radical signal [9] at ,~ 11 MHz as can be seen in Fig. 2 (amplitude ~ 4.4%). Comparing the two Fourier spectra in Fig. 2 it is clear that the amplitude of any p signal at ~ 11 MHz must be far below 1%. 3. In order to decouple the # - spin from some electronic angular momentum, longitudinal-field (LF) repolarization studies were undertaken. Measurements up to 0.8 T at 300 K, however, did not produce any significant repolarization effect (Fig. 3) (For comparison the insert in Fig. 3 shows the repolarization curve in the case of g+ stopped in C60.) Less reliable exploratory data above 0.8 T give a hint that some repolarization may take place above 1 T.

370

A. Schenck etal.

in Graphite

O

o

. . . . . . 8 10

L L 14 16 Frequency ( M H z )

~J ,

.

,

If._ 4

6

4

6

LJ.+

in C6o

8 10 12 14 16 18 20 Frequency (MHz)

.~,

12

.

,

,

.

,

20

1'8

.

,

,?

.

IJin C6o

8

10

12

14

16

18

20

Frequency(MHz)

.A - 8

10

12

14 1'6"" Frequency ( M H z )

18

20

Fig. 1. Fourier transforms of the TF (Bext = 0.1 T)-spectra in graphite and C60 at room temperature. No polarization is seen at the "free" p - Larmor frequency.

Fig. 2. Fourier transforms of the TF (Bext = 0.8 mT) pSR signals of p+ and p - in C60. The tJ+ signal reflects the precession of the triplet F = i state of endohedral muonium and the muonic radical on the surface of a C60 molecule. Nothing similar is seen for p - in C60.

4. T F - # - S R at 0.052 T between ,-, 30 K and room temperature w i t h / ~ stopped in a K3C60 sample (containing about 5 % K4C60) produced a precession signal at the "free" p - Larmor frequency, corresponding to a residual polarization of about 1/6 P0 above 100 K (see Fig. 4). Interestingly below 100 K the residual polarization appeared reduced by about 20 %. The presence of a p - - s i g n a l in K3C60 with nearly m a x i m u m possible asymmetry, at least at high temperatures, provides further evidence that the complete loss of polarization in C60 cannot be associated with the muonic cascade.

3

Discussion

Several routes m a y be followed in trying to understand the results described above. From the outset it is clear that p - in the lowest Bohr orbital of muonic carbon in C60 must be exposed to a strong hyperfine coupling, i.e. some paramagnetic configuration of sufficient lifetime must be formed in order for

Polarization of Negative Muons Implanted in the FuUerene C60

371

1.0 ..........

|..~.|

....

.

:" o,,

0.~ =

0.5

. o:

5

~ 0.6

o.4

o

,<

in C6o

/z + o m,

N ~ 0.4

•

o:2 '0:,

' d~

' oJs ' 0:, rm.o if)

/z-

0.2 0.0

.

.

.

.

.

0.0

.

.

.

.

.

.

0.5

.

.

.

.

.

.

1.0

.

in Cr~

.

.

.

Fig. 3 . Longitudinal-field quenching curves. The insert shows the expected repolarization for p+ in C6o up to 0.8 T (data taken at room temperature).

.

1.5

F,rLD (r)

0-45I o-4o1'f,

l,

o o| 0~'5

Fig. 4. p--precession signal in K 3 C 6 0 at 110 K (Bext = 0.052 T).

{

o.,q

0"0%:0 ' olz 'o14

q(

' o18 ' o'.a ' ,:o ' ,:z TIME

,:4 ' ,.~

,.8 'zo

(~-sec)

the p - to suffer complete depolarization. We assume here (see also below) that the p - spin is coupled to an electronic angular m o m e n t u m J -- ½. The hyperfine Hamiltonian is given by 7-/hf = A I~, - J

(4)

with A = woh the hyperfine coupling constant. This Hamiltonian is the same that describes the hyperfine splitting in, e.g., muonium ( p + e - ) , hydrogen (pe-) and neutral muonic helium ( 4 H e 2 + p - e - ) . Two possibilities will be considered in more detail: (i) the paramagnetic state is stable, at least for a period of several # - lifetimes; (ii) the paramagnetic state is of a transient nature with a lifetime much shorter than the p--lifetime vs. Ad (i): If this condition should hold, the absence of any pSR signal either at wu or some hyperfine splitting frequency Whf can only come about by a fast p - spin relaxation, induced by fluctuations with rate u of the electronic moments. Following Nosov and Yakovleva [10] we find, in the limit

. >>, o(1 +

(5)

372

A. Schenck etal.

Ix = (geiteB+g~,itUB)Bext/hwo, Bext = applied field], the following expressions for the time evolution of the muons polarization P(t) along the direction of the initial it--polarization P0, .

Co02t

1.

Bext-l-Po

:

P_L(t) ~-- exp(----~-) coswut

2.

BextllP0

:

Pll(t) ~ e x p ( - - ~ - )

,

(6)

¢a02t.

(7)

The absence of any observable time dependent polarization implies that 4u/w~ must be smaller than the dead time of our spectrometer which is about 10-Ss. Together with the condition given by Eq. (5), we then have w0(1 + x2) 112 << u ~ 4.10Ss -1

(8)

'

or

l<(l+x2)

0.) 0 1/2<< _ v_ <~ oJo 4. lOSs -1

'

(9)

or

u >>wo >> 4- 10Ss -1

(10)

In muonium (Mu), o~0(Mu) amounts to 27r.4.46 GHz. If neutral pseudo-boron should be formed with the unpaired outer electron in a 2Pu2-state (this state is called itB°), wo can be calculated from the measured electronic hyperfine splitting Aup1/2 = 2.Ap~/2 = 732 MHz [11] (nuclear spin of liB = 3/2, 7XXB/27r ----1.366 kHz/G): w o ( P l l 2 ) = 2zr %` 711B

AgP'z~ -2

2.3-101° rad/s

(11)

If the pseudo-boron atom is not neutral, possible hyperfine frequencies would be even larger. Let us consider the two ionic states [(Cp-)le]4+ls 2S1/2 = #B4+ls 2S1/2 and [(Ce#-)3e] 2+ 2s 2S1/2 = pB 2+ 2s 2S1/2. We estimate the # - hyperfine frequency as follows: 3

w0 = w0(Mu)a 3~'°'(wiu)= w°(Mu)Z3

(12)

a0 is the effective Bohr radius of the Is or 2s electron, which scales like 1/Z, where Z is the charge seen by the outer electron. We obtain: ¢v0(itB4+) = 3.5. 1012s -1 and w0(pB 2+) = 2.5. 1011s -1. Very similar values are obtained when rescaling the known hyperfine fields in l i b (ts22s22p)2p 2P1/2 and ¢Li (ls22s)2s 2S1/2 [11, 12]. The hyperfine coupling in pB 4+ constitutes most likely the maximum for it- captured by carbon. On the other hand if the unpaired electron is not part of the pseudo-boron but located somewhere else on the involved C 6 0 i t - complex the hyperfine coupling will be certainly much smaller. Drawing on the observation of muonic (#+) radicals in C~0 [9], one may expect to encounter hyperfine coupling constants of the order of

Polarization of Negative Muons Implanted in the Fullerene C60 T a b l e 1. List o f parameters and limits state wo(rad/s) Xmax (W0T)min LF,/3 = O.8 T radical

Train(S)

2 • 109

70

"~ 104

5 • 10 -6

2.3- 10 l°

6

,,, 103

4.3.10 -s

,JB+2 2s 2Sll 2 2.5.10 n

0.6

,,* 1 0 3

4 - 10 -9

~B +41s2SlI~ 3.5.10 ~2

4 . 1 0 -2

~103

2.9.10 -1°

(C60D-)"

pB °2p 2Pll 2

373

(V/WO)min,(V/W0)max

10 ,-*1

~ 102 , ,,,10 **

,,~5.10 -1 ~ 1 0 ** 10 -2

~ 10 **

• estimated for Wor -= 104, ** estimates for w0r = 10z (r'/Wo)min decreases, (u/WO)ma,, increases with rising wo~-

w0 = 2 - 109 r a d / s which would also still be compatible with the limits defined by Eq. (10). We call this configuration [C60#-] • All considered w0 are listed in Table 1. So far we have presumed the validity of Eq. (5). If on the other h a n d u << w0(1 + x2) 1/2 one should have seen in T F the precession of the p a r a m a g n e t i c center directly or, at least in LF, the residual p - polarization should have decayed slowly enough to be seen directly, like in the case of endohedral m u o n i u m i n RbaC60 [13]. In the absence of those observations Eq. (5) appears to be well justified. T h e reappearance of a non zero residual polarization in KaC60 implies, according to Eqs. (6) and (7) a vastly increased fluctuation rate u which m a y be induced by spin flip scattering with the free charge carriers in this compound. Since the residual polarization does not display any further relaxation we arrive at the condition w--~°2<< v~-1 ~, 0 . 4 9 . 1 0 6 4u

s -1

(13)

where r u is t h e / t - effective lifetime in C. This leads to the following lower limits for u for the four considered states: ~tB 4+ l s 2S1/~ : u >> 6 . 2 . 1 0 a S s -1

,

pB2+2s2S1/2

: u > > 3 . 2 . 1 0 a 6 s -1

,

# B ° 2p 2P1/~ : u >> 2.7. 1014s -1

,

[C60P-]*

: u >> 2 . 0 - 1012s - 1

Since electronic relaxation rates in solids are never m u c h faster t h a n a b o u t 1014/s it appears t h a t the first three states would have to be excluded. In any case the observation of a s o m e w h a t reduced residual polarization in K3C60 cannot be accounted for in this m o d e l and one is naturally lead to the possibility (ii).

374

A. Schencketal.

Ad (ii): A finite lifetime of the paramagnetic state will allow for some remaining polarization in the subsequent diamagnetic state. The problem is now characterized by three parameters: T, the lifetime of the paramagnetic s t a t e ; u, the electronic fluctuation rate in the paramagnetic state and w0, the effective hyperfine frequency. Formulae for the remaining polarization for t >> r for both the T F and the LF case have been worked out by Ivanter and Smilga [14]. For ncxtllP0 (wor)2(1/2 4- uv) PII = 1 - (1 4- 2uv) 2 4- (~or)2(1-4- uv4- x 2)

(14) '

and for Be×t±P0

.1 B(A + B) } Pz = Re l + z-~VWOAB 2 _ A _ B

(15)

with • 0JOT

4-

2x

(16)

O.;0T

(x is defined as above). Figs. 5 and 6 present plots of PII and P± versus x for various values of u/w0 or w0T, respectively. With respect to P~I we note that the maximum used field of 0.8 T leads to upper values of x as listed in Tab. 1. Inspection of these figures shows that in order to reproduce the zero residual polarization, w0r must be larger than a certain lower limit, (W0r)min, depending on w0. These limits together with the resulting lower limits of the lifetime r are listed in Tab. 1 as well. Bounds on the allowed ranges of u/wo are also shown. These depend on the assumed value for w0r > (w0r)min- Tab. 1 teaches that the radical state [C60p-]' could in principle have been observed directly and consequently possibility (i) should apply. In this case the non-observability of any residual polarization implies a lower limit on u as given by Eq. (5). A similar reasoning also applies to the state pB ° 2p 2P1/2. The derived limits are all physically reasonable and it is not possible on the basis of the available information to determine the correct hyperfine states. However, some additional clue is provided by the results on K3C60. It is reasonable to assume that the hyperfine state formed in this system is not any different from the one in C60. Restoration of the full residual polarization can be achieved by a very much increased electronic fluctuation rate or by a shortened lifetime of the paramagnetic state. Fig. 7 (pertaining to the T F measurements at Be×t -- 0.052 T (see Fig. 4)) shows that in order to push P± to near unity for unchanged lifetime v the fluctuation rate u would have to increase by more than 3 orders of magnitude, which would again push u into an unphysical regime for the two first hyperfine states. On the other hand, if one allows also the lifetime v to become shorter P± will quickly increase. The presence of free charge carriers in K3C60 may indeed shorten 7- as well as increase u. Exchange scattering of the free charge carrier spins off the unpaired local spin will certainly enhance the electronic (Korringa) relaxation.

Polarization of Negative Muons Implanted in the Fullerene C60

P,

,

3~4(5

,.o

~

/ . / . . -_.';--/7"- ~"

~

o.8: , 0.6

_

-4,-J

/

~2/q/o

0.4-

"

,,

,

/ //

o~= 1o

0 . 4 ~-" '73/2

:--~--:~

0.4: ~

-1

/

,' I ,' /~o~= l~ /i ,~-o;~-lo ,/0 /1 /-2- 2 /-1 110 / ./ --/;'/2 / l ,, /

/ 3-3

/

l

-0.5

0

0.5

1

1.5

2

i

1.01 0.80.6

0 :5

, oL__~

o81.--~----;,-/-/--/-/_-" /

~- / 00, / 1 "1 .,,'-1

0'

'

,/

,

~'~---?--7

012.5 -1 -0.5

-4 ." /

J

,- ,' ,'<~o~=,°~

°~1- .:4-// °

0.

o.e~

,

,

I

1.o.-~,5

,

375

3

.4, " - - - ~

2.5-1

-0's

.

1i5

2'

2.5

Y: ~ .

0.81 4 ,,'" 0.6 0.4

1'

.

t

/

/

-3/ / .

I

o

.

.

.

-2/ / .

.

.,(Do.~_ 10 4 -1,' ,' -/,~ /-" ,*

.

7

I

0:5 1 log (x)

I

. . . . .

,:5

!

2

25

,_.~=

,,,,,-'" .4," -3

5

o.2

.." . . . .

0 "2.1, -I -0.5

: : 3 _ ~3 _

0

/

/

1' .-

_=_~__ _~ _ -_~# _

015 1 log (X)

Fig. 5. Residual polarization PII in LF as function of external field 70 S (x = (gelJ~ + gMJB)B, xt/wo) for different w0r (r = lifetime of paramagnetic state). The different curves are labeled with their log(wov) values where v is o the electronic spin fluctuation rate (cal~_ culated with Eq. (14)). :2 2.5

I.'5

J

It is not so clear what the role of the free electrons could be in shortening r. In the case of the ionic states pB 4+ and/~B 2+ one m a y envision electron capture leading quickly to a diamagnetic spin-singlet configuration. The neutral paramagnetic as well as the radical state m a y become intrinsicMly unstable in the presence of conduction electrons like in a real metal. We expect that these mechanisms are largely t e m p e r a t u r e independent. This m a y be different in the insulating compounds K4C60, K6C60 with small band gaps, in which indeed thermally excited electrons have been shown to have a pronounced effect on the endohedral muonium state [15]. The really i m p o r t a n t piece of information derives from the reduced a s y m m e t r y of the TF-signal

376

A. Schenck et al.

p,

5),,3

1.0

' 1.0

2 0.8

5:4

. . . . . . . . . . . . . . . . . .

3

0.8

1

0.6

-5- .......

0.6

COot= 10

0.4

~ ° t = 102 ~"

0.4

O.

0.2

-_-2

-~;:3

. . . . -1

-0.5

1.0

0

0.5

-2 -1 0

-3

I

I

i

1

1.5

2

2.5 -1

0.8

~

i

I

i

i

i

-0.5

0

0.5

1

1.5

2

1.0 . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

2.5

_ ..........

0.8

3 030~ =

0.6

a}Ot = 10 4

10 3

0.6

0.4

3.4~

.......

0

"A''Y

2 ......

3

_

0.2 .

.

.

.

.

.

.

.

0 -1

.

.

L .........

ol

-o'.5

6

0:5

~

1:5

2.5 -1

-0'5

~.o.-~,?.-3.-; o 0.5 ,

~is

;

2.5

l o g (x)

1.0 0.8

5

. . . . . . . . . . . . . . . . . . . . . .

(001: = 105

0.6 0.4

Fig. 6. Residual polarization P.L in TF as function of external field calculated with Eq. (15). See caption of Fig. 5.

4

0.2 3 0 /

-1

-0.5

2. i,0,- lt-2,-3,~-4 0 0.5 1 log (x)

1.'5

2

2.5

in K3C60 below 100 K (recall that this signal is not displaying relaxation). It is in fact providing direct evidence for a finite lifetime (< 10 - s s) of some paramagnetic state since depolarization ceases as soon as this state is terminated. Assuming that the lifetime v is similarly limited also above 100 K (no t e m p e r a t u r e dependence, as conjectured above) we can immediately rule out the radical state [C60#-]* (now w~- ~ 20 in contradiction to the required (~0V)min in Tab. 1). The neutral paramagnetic state is still marginally allowed but is also less probable. Hence, the ionic states are left as the most probable configurations.

Polarization of Negative Muons Implanted in the Fullerene C6o PZ 1.0

"

- "

X = 3 . 1 0 "3" 1.0

-~-c..--

~,1 \2 '\3 ',, XX '

\4 ',5 \

0.5

0

O:

-2 1.0

(3

I

2

4

' . ~" \ " "", ~ ' x " " ~

~

X\

',

0.5

I

--

{3

8

. ----

377 =0.4

,, ~ ,?~ \4\ ,2,,

,1

,,:\,,

\

,,

,, \ x \ x

^ _.....-xx%..-.+.. .2.~---~ _~ +

_

-2 1.0

x = 0.04

""1 \\,2 \~\3~4'",5

', '\ "x \ ",

0.5

-1--~\,

X X \

0.5

',

,4,\', ', ',, \ 0

0

-2

8

-2

8

Fig. 7. Residual polarization P.L in TF as function of w0r for different x, chosen to correspond to the values expected for the four considered states in Bext = 0.052 T, used in the TF-measurement on K3C60.

4

C o n c l u s i o n s and S u m m a r y

Of course, the most interesting question is what renders pure C60 so special in comparison to graphite and diamond (and other materials as well). In all the carbon allotropes p - capture leads to the formation of a pseudoboron a t o m or ion. In graphite and diamond the pseudo-boron must be in a diamagnetic state. Any possible paramagnetic precursor state must be so short lived that no depolarization can occur. In particular highly electron stripped configurations /~B"+ (4 > n > _ 2) must be refilled so quickly as to be ineffective. The neutral state pB ° m a y not be stable either since it m a y be energetically favourable to promote the 2p electron into a delocalized state. This would leave us with the diamagnetic singly charged ion pB 1+ 1S0. In C60 in contrast these considerations cannot apply in view of the results. We suggest that the pseudo-boron ion (or atom?) is displaced t o the centre of the Cs0 cage due to the recoil transmitted to the pseudo-boron in the muonic 2 p - ls transition (E.~ = 75 keV) during the cascade providing a recoil energy of 0.25 eV. The remaining disturbed Cs9 m a y rearrange to form the more stable Css. The recoil energy m a y be compared with the binding

378

A. Schenck et al.

energy of 0.6 eV of a C-atom in the C60 molecule [16]. If a C-atom is replaced by a boron atom the binding energy is expected to be smaller since B has only 3 valence electrons so that one double bond is broken [17]. It thus seems that there may be just enough recoil energy to allow the pseudo-boron to leave the 659B net-work. If finally placed inside the cage the pseudo-boron is well shielded from the outside world, enabling it to survive for a certain time relatively undisturbed in whatever state it initially was in. In particular refilling of empty states of the ionic configurations may be sufficiently slowed down. On the other hand in graphite and diamond, where no protective cages are available, the formed pseudo-boron suffers a different fate leading quickly to the mentioned diamagnetic state. This leaves us with the results on K3660. It is reasonable to assume that the final (ionic) pseudo-boron state in this compound is the same as in C60. The presence of free charge carriers is likely to increase u via spin exchange processes (as discussed above) and at the same time to kill the ionic, paramagnetic states by electron capture with the effect that a finite residual polarization may become observable. The fact that the residual polarization in K3C60 depends on temperature is an indication that this conjecture may be right, since temperature may have an effect in particular on v (see, e.g., the result on the electronic fluctuation rate in endohedral muonium in Rb3C60 [13]). Alternatively one may conjecture that the (ionic) pseudo-boron is situated outside of the cage. This situation, however, may have more in common with the situation in graphite or diamond where it is exposed "unshielded" to its neighbours and would thus be expected to transfer very quickly into a diamagnetic state. Unfortunately the available information is too limited to allow to decide which state may actually be formed. What would the next steps have to be in order to unravel the various possibilities? The curves in Fig. 5, 6, 7 teach that one should try to change 7- and u in a systematic way. The few results on K3C60 suggest that detailed studies of temperature dependencies could provide the missing clues. It would be in particular interesting to study K3C60 below the superconducting transition temperature of 19 K, below which temperature the number of unpaired conduction electrons should be reduced. Other metallic and insulating C60 compounds should be tried as well. The measurement of LF-decoupling curves over an extended field range appears also promising provided that x can be made larger than 1. This excludes the state/~B 4+ ls 2S1/2. For instance extending the field range to 5 T would increase the available x to 3.5 for pB ~+ 2s 2S1/2 and to 36 for pB ° 2p 2P1/2. Inspecting in Fig. 5 the plots for w0v ~ 103, one finds appreciable repolarization if U/Wo < 10 -1. The mentioned tentative results above 1 T are very encouraging in this respect and will motivate further measurements. In summary the observed zero residual polarization of # - - i m p l a n t e d into C60 both in T F and LF-measurements has been discussed on the basis of a model which comprises the formation of a paramagnetic state with a strong

Polarization of Negative Muons Implanted in the Fullerene Cs0

379

hyperfine coupling and rapid spin fluctuations of the unpaired electron spin. It is concluded that the paramagnetic state is of a transient nature. This state is susceptible to the presence of free charge carriers as in K3C60. The drastic and unique difference between the p - residual polarization in graphite and diamond on one side and in C60 on the other side seems to point to a rather well shielded paramagnetic state in the latter compound which suggests that the formed pseudo-boron atom is located at the center of a rearranged Css-cage possibly in the ionic configurations [(C60/J-)ls]4+(2S1/2) or

References [1] R.L. Garwin, L.M. Ledermann and M. Weinrich, Phys. Rev. 106 (1957) 1415 [2] I.M. Shmushkevich, Nucl. Phys. 11 (1959) 419 R.A. Mann, M.E. Rose Phys. Rev. 121 (1961) 293 [3] V.S. Evseev, in Muon Physics III, Chemistry and Solids ed. by V.W. Hughes and C.S. Wu (Academic Press, New York, 1975) p. 235 V.S. Evseev, T.N. Mamedov, V.S. Roganov, Negative Muons in Matter (Energoatomistat, Moscow, 1985) (in russian) [4] T. Yamazaki et al., Phys. Rev. Lett. 42 (1979) 1241 [5] Th. Stammler et al., phys. stat. sol. (a) 137 (1993) 381 [6] R. Abela et al., Nucl. Phys. A395 (1983) 413 [7] M. Koch et al., Hyperfine Interact. 65 (1990) 1039 [8] A. Schenck et al., Hyperfine Interact. 86 (1994) 831 [9] E.J. Ansaldo et al., Z. Physik B 86 (1992) 317 [10] V.G. Nosov and I.V. Yakovleva, Soy. Phys.-JETP 16 (1963) 1236 [11] G. Wessel, Phys. Rev. 92 (1953) 1581 [12] P. Kusch and H. Taub, Phys. Rev. 75 (1949) 1477 [13] R.F. Kiefl et al., Phys. Rev. 70 (1993) 3487 [14] I.G. Ivanter and V.P. Smilga, Sov. Phys-JETP 27 (1968) 301, ibid 28 (1969) 796 [15] R.F. Kiefl et al., Phys. Rev. Lett. 69 (1992) 2005 [16] D. Bakowies, W. Thiel, J. Am. Chem. Soc. 113 (1991) 3704 [17] see e.g.W. Andreoni et al., Chem. Phys. Lett. 190 (1992) 159

Positronium in Condensed Matter with Spin-Polarized Positrons

Studied

Jgnos Major 1,2, Alfred Seeger 1,2, JSrg Ehmann 1, and Thomas Gessmann 1'2 1 Max-Planck-Institut ffir Metallforschung, Institut ffxr Physik, HeisenbergstraBe 1, D-70569 Stuttgart, Germany 2 Unlversitgt Stuttgart, Institut fftr Theoretische und Angewandte Physik, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

1 Introduction

and general

background

The positron, the positively charged antiparticle of the electron, had been seen but not recognized in cloud-chamber observations of the cosmic radiation for several years when in 1932 Anderson (1933) identified certain cloudchamber traces as being due to a positively charged particle with a mass of the same order of magnitude as the electron mass. He proposed the name positron for the "new" particle. Its identification with the antiparticle of the electron as predicted by Dirac's relativistic theory of the electron (Dirac 1930) became certain when Klemperer (1934) detected the pairwise emission of the two 511 keV photons resulting from the annihilation of a positron (e +) and an eIectron (e-) according to

e+ + e- ~ 27

(1.1)

By measuring the angular correlation of the annihilation radiation (ACAR) of positrons from the fl+-deeay of 64Cu (obtained by bombarding natural Cu with deuterons) Beringer and Montgomery (1942) confirmed the theoretical prediction, based on the conservation of energy and momentum, that the two 7-quanta in (1.1) should be emitted in opposite directions. For a full account of the discovery and identification of the positron the reader is referred to the book of Hanson (1963). As early as 1934 St. Mohorovi~i~ (1934) suggested that a positron and an electron may form a bound entity, (e+e -), that is analogous to the hydrogen atom but unstable because of the overlap of the e+ and e- wavefunctions and hence of a finite probability of e+e - annihilation. In a discussion of optical and other properties of this "atom", Ruark (1945) proposed the name posiZronium. The "chemical" symbol Ps for positronium appears to have been introduced by McGervey and DeBenedetti (1959; see also McGervey 1905). Independently of Ruark, Wheder (1946) worked out binding energies and annihilation rates for what he called "polyelectrons", viz. entities consisting of small numbers of e+ and e-. Making use of Dirac's (1930) expression for

382

the cross-section of the 27-annihilation of e + and e - , he obtained for the 27annihilation rate of the "bi-eleetron" (e+e - ) in the 1S0 state with principal q u a n t u m number n

lr2 = c ( oe2 2

I

'

(1.2)

where c denotes the speed of Hght in vacuum, me and e mass and electrical charge of the electron, Po := 1 47r. 10- 7 N. A - 2 the permeability of the vacuum, and ~P(0) the value of the e + wavefunction at the e - location in the ground state of Ps. With h : Planck's constant, we may write fo~ the e + density at the e - position ~ . 2 ~.omee2C2 ~ 3

\

(1.3)

/

As will be discussed later, in (1.3) t¢ is a parameter that takes into account the modification of the wavefunction of positronium in matter. The solution of the SchrSdinger equation for Ps in vacuum corresponds to i¢ : 1. As is the case for hydrogen atoms, the hyperfine interaction (in the present case between the e + and e - magnetic moments) splits the IS ground state into a 13S1 (triplet) state, with e + and e- spins parallel, and into a 11So (singlet) state with antiparallel spins of the two particles. In analogy to ortho- and parahydrogen, Ps atoms in the 13S1 state are referred to as orthopositronium (o-Ps), those in the 1 1S 0 state as parapositronium (p-Ps). In the lowest non-vanishing order of perturbation theory ( : fourth-order terms in powers of Sommerfeld's fine-structure constant a : : I~oce2/2h 1/137) the energy of o - P s is larger than that of p - P s by 14

2

2

the so-called hyperfine splitting 2. In (1.4) /z~ = 9 . 2 8 4 . 1 0 -24 J T -1 is the magnetic moment of the electron. In addition to the interaction between the magnetic moments familiar from the theory of the hyperfine splitting of hydrogen and muonium, Eq. (1.4) contains a term of comparable magnitude due to the spin dependence of the virtual e+e - annihilation and pair production. 1 := and ----:mean that the symbol on the side of the colon is defined by the quantity on the side of the equality sign. 2 W e prefer the notation AEo_p for the hyperfine splitting of the 1S state of Ps to the conventional notation AE hf because it is unambiguous. Generally speaking, for Ps the distinction between the fine structure and the hyperfine structure does not play the same r61e as for "ordinary" atoms, since the magnetic moment of the Ps "nucleus" (the e +) differs only in sign from that of the e-. In the present paper, P, D, etc. states of Ps need not be considered since owing to the very small overlap of the e + and e- wavefunctions in these states the annihilation rate is much smaller than the rate of optical transitions to lower-lying states.

Positronium in Condensed Matter If we insert for Pe the Bohr magneton PB := eh/2me,h := m a y be written in terms of the fine-structure constant as AEo_p =

7r,,a4mec2

383

h/2,r,Eq. (1.4) (1.5)

For the corrections of (1.51 due to higher-order terms in a and the comparison with the experimental vacuum value

AEo_p = 0.8412 meV

(1.6)

the reader is referred to a recent summary by Yennie (1992). The agreement between the predictions of quantum electrodynamics (QED / and the (more accurate / experimental value is within the estimated uncertainty of terms that have not yet been calculated. The energy difference (1.6) between the o-Ps and p--Ps is approximately equal to the energy equivalent of 10 K. Since i¢ is not expected to be appreciably larger than unity, in the absence of a magnetic field the ratio of o-Ps and p-Ps formed at not too low temperatures will be given by the statistical weights of the two states; if the electron spin polarization is zero, the ratio is thus equal to three. (This is likely to be true even below 10 K, since Ps formation in condensed matter occurs on such a short time scale that the spins do not yet "feel" the temperature of the host material. This question deserves more detailed attention, however. For the case of applied magnetic fields see Sect. 2.) Although the overlap of the wavefunctions ofe + and c- in the 1381 and in the 11S0 states is the same, it has been known since the early work on Ps that the annihilation rates for o-Ps and p-Ps are very different (Wheeler 1946). Wolfenstein and Ravenhall (1952) showed that this is a consequence of the facts (i) that the interaction mediating the annihilation of e+ and e-, the electromagnetic interaction, is invariant under the so-called charge-conjugation operation, which replaces all particles in a system by their antiparticles while preserving their spins and momenta, and (ii) that the Hamiltonian of selfconjugated systems, i.e. systems that consist of equal numbers of particles and their antiparticles plus an arbitrary number of self-conjugated particles (e.g., photons), does not change under charge conjugation; hence in such systems eigenstates of the Hamiltonian are also eigenstates of the charge-conjugation operator C. The eigenvalues of C are C : (-1) s+z, where S denotes the total spin and I the orbital momentum in units of h. Hence the C eigenvalue of the 1So states is (-110+0 = 1; that of 3S1 is (-1) 1+° = -1. Charge conjugation implies the change of sign of the dectrical charges and the magnetic moments of the particles (and in the case of fermions also of their inner parity) and therefore the reversal of the directions of the electrical and magnetic fields. From this it may be deduced that the C eigenvalue of ~ photons is (-11". It follows from the conservation of C in self-conjugated systems that a Ps "atom" in a 1S0 state can annihilate only into an e v e n number of photons

384

and that in a 3S1 state it can annihilate only into an odd number of photons. Since momentum conservation requires the participation of at least two photons (the probability of one-photon annihilation, with balance of energy and momentum maintained by surrounding matter, is extremely small at the low kinetic energies of e + in Ps; see Heitler 1954) this means that the minimum number of photons resulting from the annihilation of Ps in 3S1 states is three. Prom the standard perturbation treatment of electromagnetic processes it follows immediately that the ratio of the 3-photon annihilation rate to the 2-photon annihilation rate should be of the order of magnitude of Sommerfeld's fine-structure constant a. The detailed calculations of Ore and Powell (1949) gave for the 3~f-annihilation rate of the Ps 3S1 triplet states with principal quantum number r~ the result 2~" 2 91m°c2. ~-~ 3-/'37 : 9-~(~" -h rt3

(1.7/

or, expressed in terms of 1/"27 (Eq. 1.2), 4a" 2 9) 1 r ~ v = 1 .1F2 7 3/'3.r = ~--~(z" • 111---4

(1.8)

Eq. (1.2) predicts for the mean lifetime (= the inverse of annihilation rate) of parapositronium in vacuum 7"p_Ps = 124.5 ps

(1.9)

This time is too short for a high-precision determination. Direct measurements of Tp-ps are therefore not suited for a critical test of QED by comparing them with the QED calculations carried to higher orders in a (Berko and Pendleton 1980). An indirect value for :~p-Ps has been obtained by A1Ramaclhan and Gidley (1994) making use of the admixture of the 11So state of Ps to the 13S1(m = 0) state (cf. Sect. 2). The ratio of the 47-decay rate of the 11So state of Ps to the 27decay rate has recently been determined experimentally by two different groups as aF4.y/1F2.y = [1.48 ± 0.13(statistical) + O.12(systematic)J 10-6 'I

(Adaehi, Chiba, Hirose, Nagayama et al. 1994) a n d 1F47/1/'t27 :

[1.50 ±

O.OT(statistical) ± 0.09(systematic)] • 10 -6 (yon Busch, Thirolf, Ender, Habs et al. 1994). Both results agree with each other and with the QED predictions by various authors (see Adachi et al. 1994). 1F47/IF27 is too small to have an influence on the lifetime of p-Ps. At present, the best value for the mean lifetime of parapositronium in vacuum is rp-ps =: 1/~p_p, = 125.164 ps

(1.10)

obtained from detailed QED calculations (Khriplovich and Yelkhovsky 1990 and references therein) which include the a21na term as highest-order term.

Positronium in Condensed Matter

385

The intrinsic orthopositronium lifetime is in a convenient range for direct precision measurements. However, here one encounters the difficulty that the positrons in triplet Ps m a y undergo 27-annihilation with electrons of opposite spin which they "pick up" from their environment. This so-called pick-off annihilation of Ps m a y reduce the "true" lifetime of o-Ps by orders of magnitude compared to its "intrinsic" value as determined by the 37annihilation referred to above. The dependence of the pick-off annihilation and hence of the o-Ps lifetime on the electron density of the environment allowed M. Deutsch (1951a, b) to demonstrate the formation of o-Ps in gases. Making use of the fact that the jS+-decay of 22Na is accompanied by the emission of a 1.27 M e V ~,-quantum virtually simultaneously with that of a positron and determining from the delays between the recording of a "prompt" 1.27 M e V photon and one of the annihilation photons the individual lifetimes of the positrons (now called positron age, following the usage of MacKenzie and M c K e e 1976; see Sect. 4), Dcutsch (1951a) showed that the admixture of a small amount of nitric oxide to pure N2 gas resulted in a significant decrease of the number of annihilation photons belonging to positron ages that exceed about 100 ns. Measurements of the energy distribution of the annihilation photons with a Nal scintillation detector revealed that the addition of N O increased the number of quanta in the 511 kcV photon peak but decreased the occurrence of photons of lower energies. Both experimental results wcrc interpreted as a reduction of 37-annihilation events due to a rapid conversion of orthopositronium into parapositronium induced by the exchange of electrons between Ps atoms and N O molecules (cf. Sect. 3.2). In further work on e+ annihilation in gaseous CCI2F2 (dichlordifluoromethane = freon) Deutsch (1951b) observed that at pressures above about 4.104 Pa all annihilation events belonging to positron ages that exceeded 80 ns were due to 37-decays. By extrapolating the pressure dependence of the long e+ ages in freon above 4.104 Pa to zero pressure he determined the "intrinsic" o-Ps lifetime and found it to be in agreement with the calculated 37-annihilation rate of o-Ps, thus establishing the existence of Ps "atoms". Deutsch's extrapolation method is stillbeing used to determine the intrinsic lifetime of o--Ps in vacuum. According to an estimate of Labellc, Lepage, and Magnea (1994) of the not yet completely calculated a 2 term in the correction factor to (1.7), quantum electrodynamics gives for the lifetime of o-Ps in vacuum ro-Ps =: I/,~o-es = 142.038 ns

(1.11)

This is to be compared with the most recent experimental values, viz. those of Nico, Gidley, and Rich (1990) and Nico, Gidley, Skalsey, and Zitzewitz (1992) (~'o-Ps = (141.880 + 0.032) ns), and of Asai, Orito, and Shinohara (1995) (~'o-Ps = (142.15 + 0.03 + 0.05) ns). In the latter value the first error limit arises from the statistics of the lifetime spectrum and the time calibration, the second one from the statistics of the photon-energy measurements and

386 systematic errors. The error limits of the two experimental values appear to be incompatible with each other. It may therefore be too early to form a final judgement on whether quantum electrodynamics predicts the intrinsic o-Ps lifetime correctly or not, clearly a question of fundamental importance. In the abovementioned investigations of Beringer and Montgomery (1942) and Deutsch (1951a, b) as well in most later work on positron annihilation, irrespective of whether on fundamental or applied problems, the positrons were obtained from the fl+-decay of neutron-deficient radioactive nuclides 3 that were produced in accelerators or nuclear reactors. The prediction of Lee and Yang (1956) that processes such as fl-decay that are mediated by the Weak Interaction violate invariance under space reflection (= conservation of parity) was verified first by the experiments of Wu, Ambler, Hayward, Hoppes et al. (1957) on the fl--deeay of 6°Co and subsequently by those of Garwin, Lederman, and Weinrich (1957) and of Friedman and Telegdi (1957) on the decay chain lr+ --,/~+ -~ e ÷. The non-conservation of parity has the consequence that positrons and electrons emitted in fl-decay as well as muons (/~±) obtained from the decay of pi-mesons (~r+) are spin-polarized. In dealing quantitatively with spin-polarized positrons or electrons, it is useful to introduce two pseudoscalar quantities, viz. the helicity 7g and the pseudoscalar spin polarization 7~. In the literature they arc sometimes insufficiently distinguished. The helicity of s p i n - l / 2 particles is defined as 7~ := 2 ( s - p / p )

,

(1.12)

where s is the (axial) spin vector, measured in units of h, and p the (polar) momentum vector. (...) denotes ensemble averages. The pseudoscalar spin polarization (in the following abbreviated to "polarization") of particles with spin s is defined as := ( s - i ' ° ) / s

,

(1.13)

where t0 is a unit vector in a fixed space direction. If we choose this as the z-direction of a Cartesian coordinate system, IP is the z-component of the spin-polarization vector P . The modulus P = IPI satisfies P > 7~. Whereas the pseudoscalar spin polarization 9v is determined by the ensemble average of the projection of the spin onto a given direction of space, the helicity 7~ is a measure of the ensemble average of the projection of the spin onto the momentum direction of the spin carriers. We see that 7~ and 9v can coincide only for a collimated beam of particles whose polarization vector P is parallel or antiparallel to the direction of the beam. (Such a beam is often referred to as longilltdinally polarized.) For non-collimated beams we have 9v < 7~. Whereas spin-carrying particles at rest may possess a non-vanishing polarization, helicity can only be assigned to moving particles. As a consequence of a For the conditions a nuclide must satisfy to be an e+ emitter see, e.g., MayerKuckuk (1994).

Positronium in Condensed Matter

387

the A-V Universal Fermi Interaction, the helicity of e+ emitted with velocity v from a/~+-active source is x :

(1.14)

For positrons with a kinetic energy of 1 MeV Eq. (1.14) gives us 7~ = 0.94. The polarization of a non-collimated positron beam obtained from a/~+ source may be calculated from (1.14) by integrating over the velocity distribution of the beam. For a practical example see Gessmann, Harmat, Major, and Seeger (1997a). If we wish to exploit the polarization of positrons in condensed-matter studies, we must be able to determine it at the time of annihilation of the positrons. This is by no means straightforward, since the 27-annihilation process, being mediated by the electromagnetic interaction, is invariant under space reflection and hence does not provide information on the e+ spin direction. In this respect positrons are radically different from positive muons (/~+). Since the decay of/z + into a positron, a neutrino and an antineutrino is mediated by the Weak Interaction, the emission probability of the e +, which serves as indicator of the decay and thus plays a r61e similar to that of the 511 keV photons in e+e - annihilation, is asymmetric with respect to the muon spin direction. The various/~+SR techniques ( = m u o n spin rotation, relaxation, resonance) are based on the information on the p+ spins at the time of the/~+ decay contained in the direction distribution of the emitted e + (see, e.g., Chappert 1984, Smilga and Belousov 1994). "Positron polarimeters" can nevertheless be designed by employing a magnetic field B that is strong enough for its direction to establish a quantization axis for the e + spins. If we then identify the B direction with 1'° (Eq. 1.13) and denote by N+ and N_ the numbers of e + with spins parallel or antiparallel to this direction, the polarization of an e + ensemble is given by _ N+ - N_ N++N_

(1.15)

The two main possibilities for determining experimentally the right-hand side of (1.15) are: (i) Modification of the positvonium states by the external magnetic field (Page and Heinberg 1957, Bisi, Fiorentini, Gatti, and Zappa 1962, Dick, Feuvrais, Madansky, and Telegdi 1963; for further references see Berko and Pendleton 1980 and Consolati 1996). (it) Modification of the momentum distribution of the electrons involved in the e+e - annihilation by reversing the magnetization of the sample (Hanna and Preston 1957, Akahane and Berko 1982, Seeger, Major, and Banhart 1987, Banhart, Major, and Seeger 1989). This has been used to study the relaxation of the e + polarization due to the interaction of the e + magnetic moments with spatially varying internal magnetic fields in ferromagnets and the trapping of e + in paramagnetic eentres in additivdy coloured KCI (Lauff, Major, Seeger, Stoll et al. 1993, Deckers, Ehmann, Greif, Keuser et al. 1995).

388

In analogy to the acronym p+ SR for " m u o n spin relaxation", this technique has been dubbed e+SR (= positron spin relaxation) (Seeger et al. 1987). Although both possibilities have been known for a long time, they have only rarely been employed in condensed-matter studies in spite of the fact that their/~+SR counterparts have provided us with very valuable information. The reason for this is clearly that, in contrast to p+SR, investigations making use of the e + spin polarization can be performed only on samples which may serve as polarimeters. The classes of materials that may be investigated are thus severely restricted, particularly in the case of magnetically ordered materials (see Seeger et al. 1987). This is in marked contrast to p+SR, for which the sample requirements are much less restrictive. Nevertheless, since both approaches to measuring e + polarization mentioned above can provide us with information that is hard to obtain otherwise, it is certainly worthwhile to develop them further. The present contribution is one of a series of three papers (see also Gessmann et al. 1997a, Gessmann, Harmat, Major, and Seeger 1997b) that review recent progress in the study of positronium in condensed matter with spinpolarized posilrons. It will be seen that this field is an excellent example for the transfer of atomic-physics methods to other areas of physical research. On the other hand, it is hoped that some of the new results to be reported will have an impact on the atomic-physics research on positronium. Concentrating on the theoretical aspects, the present paper discusses in Sect. 2 the properties, in particular the annihilation characteristics, of positronium in a magnetic field. They are well known for 1S-Ps in vacuum; the present discussion emphasizes the assumptions that have to be made when the results obtained for Ps in vacuum are to be applied to Ps in matter. A new (in our opinion improved) nomenclature for the energy eigenstates of 1S-Ps in a magnetic field will be introduced. Sect. 3 uses the density matrix approach for calculating the evolution of the Ps states when polarized e ÷ form positronlum in the presence of a magnetic field. The treatment includes, in addition to the self-annihilation of Ps, the so-called pick-off annihilation and the spin-exchange processes. Again, some of the topics treated may already be found in the literature. Instead of putting together results obtained by different authors, including ourselves, and trying to harmonize the various notations, we have preferred to give a unified treatment which is believed to be in several respects more straightforward and more transparent than earlier accounts. In Sect. 4 we survey the experimental aspects of the field, including a short account of the recently improved Stuttgart set-up for studying the relaxation of e + spin polarization (e+SR) in matter described in greater detail elsewhere (Gessmann et al. 1997a). Finally, Sect. 5 treats the data analysis and illustrates it by an experimental example obtained with the Stuttgart set-up. The Laplace-transformation treatment on which Sect. 3 is based will be seen to be particularly well adapted to the analysis of the e+SR experi-

Positronium in Condensed Matter

389

ments on Ps. For further details and results the reader is referred to one of the two accompanying publications mentioned above (Gessmann et al. 1997b).

2 P o s i t r o n i u m in a m a g n e t i c field The following discussion of the influence of a static magnetic field B on the ground state properties of positronium and on positronium formation in condensed matter is based on two fundamental assumptions: (a) As in the absence of external fields (sec Appendix A), the total Ps wavefunction m a y be separated into a spatial part and into a spin part. The spatial part is not affected by the magnetic field; the infiuencc of the surrounding matter is taken into account by the parameter ~ (Eq. 1.3), which in general differs from its vacuum value unity. (b) The spin Hamiltonian differs from that of vacuum Ps in zero magnetic field only by the Zeeman term -/~P8 • B, where/~p~ is the positronium magnetic moment. This assumption implies that, apart from the Zeeman term, the hypetfine interaction term remains isotropie even in the spin-one states. This is not trivial, since in general spin-one particles do possess an electric quadrupole moment, which may interact with electric-field gradients at the Ps site. The corresponding interaction term may be absorbed in the spin Hamiltonian and gives then rise to an anisotropic hyperfine interaction. However, in self-conjugated systems such as isolated Ps this term is strictly zero (Baryshevsky and Kuten 1977). Therefore, a detectable anisotropic hyperfine interaction for Ps in matter can only arise if, due to a strong interaction with the electronic system of the host, the eJfeclive masses of the positrons and electrons forming positronium are sui~ciently different (Baryshevsky 1984, Bondarev and Kuten 1994). This might be the case if, e.g., Ps atoms play a similar r61e as hydrogen atoms in hydrogen bridges. One approach to the problem of a possible anisotropic hyperfine-interaction term in the spin Hamiltonian is to assume that the hypetfine interaction is isotropic (i.e., that it differs from that of Ps in vacuum only by the normalization factor i¢), to work out the consequences of this assumption, and to compare the i¢ value required to fit the experimental data with the i¢ value that may be obtained from precise measurements of the lifetime of para-Ps (cf. Eq. 1.2). A discrepancy between the "lifetime i¢" and the "hyperfine s¢" will indicate that the assumption of an isotropie hyperfine interaction was inadequate, since there can be little doubt that assumption (a) is valid for the magnetic fields used in laboratory experiments (see Appendix A). An alternative approach to the anisotropy problem is to study Ps in single crystals at different crystallographic orientations of the magnetic field. In the absence of a magnetic field the eigenstates of the spin Hamiltonian are also eigenstates of the S 2 (square of the total spin) and Sz (projection of the total spin onto the axis of quantization) operators. They are the singiet state 1So (total spin zero)

390

Io, o> =

1

and the triplet states 3S 1 (total spin

11, o) =

1

(I T£>-IH>)

(2.1)

one) (I

[ I , I > = I TT> , II,-I) = l

÷ I (2.2)

On the left-hand sides of (2.1) and (2.2) the first number gives the total spin, the second one the so-called magnetic quantum number m. In the symbols on the right-hand side the first one of the up-or-down arrows refers to the e + spin, the second one to the e- spin. As discussed in Sect. 1, the hyperfine interaction between the e + and the e- spins leads to the "hyperflne splitting" AEo_p between the 11S0 and the 13S1 state. If the hyperfine interaction is isotropic, as will be supposed for the remainder of the paper (cf. assumption (b) above), in zero magnetic field all three states (2.2) are degenerate. Neither the 1S0 nor the 3S1 states of positronium carry a magnetic moment, as may be seen in the following. In the singlet state there is no preferred spin direction, hence the magnetic moment must be zero. In the triplet states the magnetic moments of the two particles are opposite, hence there is no total magnetic moment either 4. The vanishing of the magnetic moment of Ps in the absence of an external field means that there cannot be a linear Zeeman effect. The following arguments (Wolfenstein and Ravenhall 1952) show that the 3S 1 states with m : +1 are, in fact, not split at all by a magnetic field B . A rotation by 180 ° around an axis perpendicular to B will interchange the levels m : +1 and reverse the sign of B . By subsequently applying the charge conjugation operation, which leaves the Ps Hamiltonian invariant, the magnetic field can be brought back to its original direction. Hence the two energy levels must be equal. Furthermore, since the sum of the two energies is not affected by the magnetic field, the m : +1 levels must be independent of B. An alternative way to derive the preceding result is the following. If we identify the direction of the magnetic field with the quantization axis, the spin Hamiltonian retains its rotational symmetry around this axis even in the presence of a magnetic field. Therefore, the projection of the total spin remains a good quantum number irrespective of the magnetic field strength. Since the magnetic moments of e + and e- are equal and opposite, the total magnetic moments of the states m : +1 are exactly zero even in large magnetic fields; hence the energies of these two states are not changed by a magnetic field of any strength. 4 In the case of Ps the conventional designation of the eigenvaines of Sz as "magnetic" quantum number is thus somewhat misleading.

Posltronlum in Condensed Matter

391

An analogous reasoning is clearly not applicable to the m : 0 states (2.1) and (2.2), since in the presence of a magnetic field the lotal spin is not a good quantum number and since the "magnetic" quantum number m, which is preserved, is the same for both states. In a finite magnetic field the eigenstates of the Hamiltonian may acquire non-vanishing magnetic moments. The Hamiltonian of a system of two spins 1/2 interacting isotropically with each other in an external magnetic field is known as the Breit-Rabi Hamiltonian (Breit and Rabi 1931). The "Breit-Rabi diagram" (energy eigenvalues E vs. magnetic field, apparently established for the first time by Darwin 1928) of Ps may be obtained from the general case (different gyromagnetic ratios of the two spins) by specialization. Suitable dimensionless variables are E/AEo_p and z ::

4geB/AEo-p

(2.3)

Or

:= Arcsinh z

(2.4)

The use of~ instead o f z has several advantages. It results in more transparent mathematical expressions. At small fields ~ = z + O(~ 3) holds, so that the difference between linear and quadratic Zeeman effects is maintained in E-vs.plots. In large fields, however, ~ grows only logarithmically with increasing B; hence in ~ plots a wider B range can be covered than in z plots. Fig. 2.1 shows the Breit-Rabi diagram for positronium. The zero on the energy scale has been chosen in such way that the sum of the four energy values (which is independent of B) is zero. As discussed above, the energies of the states m = -4-1 are independent of B; with the normalization just mentioned they are

(2.5)

E(fft = ±1) = ZIEo-p/4 The other two energy eigenvalues, given by /

E(fft = 0 ) - : --(AEo_p/4) [1 =t=2~¢/1+z 2) : - - ( A E o _ p / 4 ) ( 1 + 2Cosh~)

show a quadratic Zeeman effect.

,

(2.6)

392

s[T] 0 2

1 ,

2 ,

4

8

16

32 + 1.5

"meikt

+ 1.0 ,13Sl

?

~

odho-Ps (m=.t 1)

+0.5 0.0

LU

t-,--t

>

E

-0.5 -

1.0

-1.5 -2 0.0

I

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 2.1. Energy eigenvalues vs. magnetic field (Breit-Rabi diagram) for 1Spositronium. The energy E is measured in units of the energy difference between o-Ps and p-Ps; the zero of the energy scale is such that the magnetic-field independent sum of the four energy eigenvalues is zero. The right-hand scale corresponds to ~ = 1. For the definition of the dimensionless variable ~ see (2.3) and (2.4), for the nomenclature of the positronium energy eigenstates in finite magnetic fields see Appendix B. The eigenvectors of the m = + 1 states are not changed by the application of the magnetic field; they are given by (2.2). The eigenvector that for B --* 0 reduces to the singlet state (2.1) is [0, O) + 2 - 1 / 2

--

I1, O)

= [Cosh(U2). to, o> + SinhIU2). I', 0)[ = 2-1/2 [exp(~/2) • I t,~) - exp(-~/2) • I ,LT)] v / - S - ~

,

(2.7)

that reducing to the m = 0 triplet state is

-2-1/211-(l+z2)-i/2jr|11/210,0)+

2-1/~[1

+ (1 +z2)-1/2]

1/2 11,o)

= [ - Sinh(~/2). I0,0)+ Cosh(~/2). 11,0)] S ~

= 2-1/2 [exp(-U2). I T~>+ exp(~/2). I~T>[S ~ L

J

(2.8)

Positronium in Condensed Matter

393

The left-hand sides of (2.7) and (2.8) give the decomposition of the eigenvectors into the triplet and singlet contributions, whereas the exp(~/2) terms on the right-hand sides correspond to the dominant terms in the high-field regime. Owing to our restriction to magnetic fields that do not affect the spatial part of the Ps wavefunetion (cf. Appendix A), we may argue intuitively that the annihilation rates of the states with m = +1 are equal to that of the triplet states and that the annihilation rates of the states (2.7) and (2.8) are determined by the relative weights of the ringlet and triplet states in them and by the annihilation rates of these states. If for simplicity we restrict ourselves to the principal quantum number n = 1, this reasoning gives us for the annihilation rate of the state (2.7)

)~pp-ps = [Cosh2 (~/2))~p- ps + Sinh2 (~/2)Xo-Ps] Sech~ 1

1

= ~(~p-P8 + ~o-Ps) + ~(~p-P~ - ~o_p~)Sech~

(2.9)

and for that of the state (2.8)

~m-Ps---- [Sinh2(~/2))~p-p~ + Cosh2(~/2)~o-Ps] Seeh~ 1 15 = ~(J~p-Ps -}- ~o-Ps) -- ~( p-Ps -- )~o_Ps)Sech~

(2.10)

The lifetime of the state (2.7) becomes -1

pPs

psi1 Seth,Sinh,,J2(1 Ps]opsj m

"Fp--ps 1 - Sech~ Sinh2(~/2)

(2.11)

'

that of the state (2.8) becomes

Tm-Ps----ro-Ps

[

[1 + Seeh~Sinh2(~/2)(v°-p----k -Tp-Ps

1

+ Sech~ Sinh2(~/2)] -1

To---Ps

Tp_ Ps

J

-1 1)]

(2.12)

(The notations )~pp-v~, )~m-e~, rpp-V~, and rm-ps will be justified below.)

394

B[T] 0

1

2

4

8

16

32

10 4

(a)

~ o r t h o - P s (m =.+ 1)

10 -e

10 3

10 .7

~

{ 102

101

s" (m = O)

10 8 10 .9

"plesiopara-Ps" (m 0 )=~

~

10 o 11S 0 0.0

~

10-1o

para-Ps

J

L

I

I

I

I

0.5

1.0

1.5

2.0

2.5

3.0

B [mT] 0 104

200

400

600

I

I

I

~//

13Sl

(b) ,/ortho-Ps (m = +- 1)

103

10 -8 10 .7

102

"meikto-Ps" (m = 0)

101

10-8 10-9

"plesiopara-Ps" (m = 0)

100

101 o X11S 0

0.00

I

I

I

0.05

0.10

0.15

0.20

X Fig. 2.2. The lifetimes of positronium energy eigenstates (left-hand scale: norrealized by the pazapositronium lifetime rp_p~; right-hand scale: unnormalised for = 1) for two different ranges of the magnetic field. (Top scales: magnetic field for = 1; bottom scales: reduced magnetic fields - for definitions of z and ~ see (2.3) and (2.4)) a) wide B range, b) small magnetic fields (z ~ ~). For the states (2.7) and (2.8) Dick et al. (1963) introduced the n a m e s pseudo-singlet (pseudo 1So) or pseudo-triplet (pseudo 3S1), respectively. T h e y are widely used in the literature (see, e.g., Berko a n d Pencl]eton 1980, Du-

Positronium in Condensed Matter

305

pasquier 1981). As argued in Appendix B, we do not consider this to be a good nomenclature. From a practical viewpoint the most important property of these states is the dependence of their lifetimes on the magnetic field. It is determined by the decomposition of the states into the slowly annihilating triplet state I1, 0) and the rapidly annihilating singlet state t0, 0) as exhibited on the left-hand sides of (2.7) and (2.8). For very large magnetic fields the lifetimes of both states approach the value 2rp-Ps. (We neglect here and in the following terms of the order Tp_Ps/To_Ps ~ ]0-3.) However, this limiting case is far outside the regime of terrestrial experiments. At small or moderate magnetic fields the lifetimes of these two states exhibit very different dependences on the magnetic field strength, as is illustrated in Fig. 2.2a and Fig. 2.2b. We may easily deduce that a magnetic field which increases rpp-Ps compared to its zero-field value rp-ps (eq. 2.11) by /3Tp_ps (/3 > 0) leads to a decrease of rm-ps compared to its zero-field value ro-ps (eq. 2.12) by ~To--Ps~p--Ps/~o--Ps ~ 103~To-Ps. To illustrate this result, consider B = 1 T. In the case t¢ = 1 this corresponds to z = 0.276, ~ = 0.273. According to (2.11), rpp-ps is about 1.02rp_Ps, i.e., only by a few picoseconds longer than the zero-field value rp_ps, whereas rm_ps is smaller by a factor of 20 than the zero-field value ro-Ps. We see that the "mixed" character of the m = 0 states (from the point of view of the annihilation modes) makes itself strongly felt in the lifetime of that state that for B = 0 reduces to one of the triplet states, whereas for not too large B the lifetime of the other m = 0 state is almost that of parapositronium. In keeping with the Greek origin of the prefixes ortho and para (op~6s = straight, ~rc~p& = alongside) we propose to call the long-lived m = 0 state meiktopositvonium (m-Ps) and the short-lived one plesioparapositvouium (pp--Ps) 5. As discussed above, an external magnetic field does not affect the 13S1 (m = ±1) states, hence their "intrinsic" lifetime is that of orthopositronium. It is therefore appropriate to refer to them as o-Ps even in strong magnetic fields. Perhaps the most obvious consequence of the "mixing" of the m = 0 states in a static magnetic field is the reduction of the ratio of 37-annihilations (from the triplet states) to that of the 2~,-annihilations, which for isolated Ps in zero magnetic field is 3. This effect is known as the "magnetic quenching of o--Ps". As early as 1951 it was employed by Deutsch and Dulit (1951) in measurements on Ps in freon gas (CC12F2), for the first experimental determination of AEo_p. An early theoretical treatment of the magnetic quenching of o-Ps, including resonance effects due to alternating magnetic fields, was given by Halpern (1956). The fact that the intrinsic lifetime Tm_Ps is significantly shorter than l"o-ps already in rather small magnetic fields has been used by AI-Ramadhan 5 For the etymological justification of the proposed nomenclature the reader is referred to Appendix B.

396

and Gidley (1994) to determine 7"p_Ps experimentally. They determined the m - P s lifetime in a gas mixture of N2 and isobutane in a magnetic field of 0.4 T. The obtained value for the mean lifetime of rp_ps [cf. Eq. (2.12)], rp-ps = (125.142 ± 0.026) ps

,

(2.13)

is in good agreement with the theoretical value (1.10). In condensed matter, the lifetime of o-Ps is always limited and that of m - P s often limited not by the intrinsic annihilation processes discussed so far but by the so-called pick-off annihilation (cf. Sect. 1) or by spin-exchange processes (Ferrell 1958). If this is the case, it may be difficult to distinguish experimentally between o-Ps and m-Ps. A common name for them might then be useful. Since they have in common that they survive at positron ages at which pp-Ps has virtually disappeared, we propose to designate them operationally as leipopositronium (1-Ps).

3 Density m a t r i x description of the spin states of positronium The temporal evolution of the occupation numbers of the four substates 10, 0), I1, 0), I1, 1), and I1,-1) (cf. Eqs. (2.1) and (2.2)) of the positronium ground state (1S) may be described by a 4×4 spin density matrix p(t) obeying the LiouviUe equation of motion

ihdP( ) : np(t) - p ( 0 S + dt

(3.1)

In (3.1) H = Hhf --//Ps" B - ih~/2

(3.2)

is the "magnetic Ps Hamiltonian", where Hhf describes the hyperflne interaction, - / ~ p s . B is the Zeeman term, and H + the adjoint of H. The annihilation term - i h ~ / 2 , to be discussed below, causes the Hamiltonian (3.2) to be nonHermitian. In Subsection 3.1 we shall take into account, in addition to the selfannihilation of Ps discussed in Sects. 1 and 2, the effect of pick-off annihilations. The influence of spin-exchange processes (Ferrell 1958, Mills 1975) on the elements of the spin-density matrix is discussed in Subsection 3.2. It is shown that as long as the spin polarization of the Ps-forming electrons may be neglected, we may derive first-order rate equations governing the dependence of the diagonal elements of the e+ spin-density matrix and hence of the populations of the four Ps substates on the positron age t. Subsection 3.3 solves these equations, making full use of the Laplace-transformation technique. As will be shown in Sect. 5, from these results the quantity required for the comparison with experiment may be easily obtained.

Positroninm in Condensed Matter

397

3.1 P i c k - o f f p r o c e s s e s If in addition to self-annihilation pick-off annihilation processes are taken into account and if a static magnetic field B is applied in the z-direction, the matrix representation of the Ps magnetic Hamiltonian in the basis {J0, 0), ]1, 0), Jl, 1), }1,-1)} reads 1 /-

H=~

0 0 0

~ 0 0

+ ;po -1~

0 0 ~.,' 0 0 ~

2~eB

0 0 0

-|0

0 0 0

o

o

o

0

lo_p~ + Apo

0

0

0

Xo-Ps + J~po

) "

(3.3)

In (3.3) we have introduced hw := AEo_p (see Eq. (1.4)) and the pick-off rate Apo (= rate of 27 annihilation of the e + in Ps with a "foreign" e- of opposite spin). If all annihilation terms are neglected, the diagonalization of (3.3) gives us the energy eigenstates and the energies in a magnetic field as specified in (2.5) - (2.8). The annihilation rates (2.9) and (2.10) of pp-Ps and m - P s are obtained if in (3.3) the pick-off term Apo is neglected and the remaining annihilation terms are treated as small perturbations (this is justified since Ap-ps/~ -~ 5 . 1 0 - 3 ) . The initial condition p(O) required for the solution of (3.1) may be derived as follows, e + and e- are particles with spin 1/2, hence their spin density matrices are given by (Blum 1981) Pe-,e+ -----~

Pio'i

1+

,

(3.4)

where Pi is t h e / - t h component of the spin polarization vector, 1 is the unity matrix, and ~ri are the Pauli matrices. If we choose the direction ore + emission from the/3+-active source as z-axis ( P = (0, 0, 7~)), the spin density matrix of the positrons is given by 1(1+~ Pc+ = ~ 0

0 ) 1 - 7'

(3.5)

In this subsection we assume that the e- involved in the Ps formation are not spin-polarized; hence their density matrix is 0e-=2

0

1

In the basis {I tT), I TJ~), I ~T), I ~J.) } the initial spin density matrix is given by the direct product Pc+ ® Pc-. From this the initial condition in the

398

basis { I0, 0), ll, 0), ll, 1), ll,-1) } is obtained by an orthogonal transformation, giving us 1

p~oj:~,,

1

0

0

0

o l+p

0 o 1-7~

(3.7)

The Liouville equation (3.1) with the Hamiltonian (3.3) leads to a set of coupled first-order differential equations for the components p0(t). Treating the annihilation term A in the Hamiltonian as a small perturbation, the firstorder perturbation-theory solution for the initial condition (3.7) gives us the

followingnon-vanishingcomponents of p(t): P11(t) : - ~ ~

Tanh2~ cos (w t Cosl~)7~ exp(-A t)

+ ¼Sech~ Cosh2 (~/2) (1 + 7~Tanh~) exp(-All t) + ¼Sech~ Sinh 2 (~/2) (1 - 7~Tanl~) exp(-A22 t) ,

(3.8a)

p2~(t) : -~~ T a n h 2 ~

cos(w t Cosh~)7~ exp(-A t) + ¼Sech~ Cosh ~ (~/2) (1 - 7~Tanh~) exp(-X~2 t) + ¼Sech~ Sinh 2 (~/2) (1 + P Tanh~) exp(-A11 t)

,

(3.8b)

p12 (1~) : ¼ [Sech~ cos(w t Cosh~) + i sin(w t Cosh~)] 7~ Sech~ exp(-A t)

1Tanh~ [(1 - 7~Tanh~) exp(-~22 t) m

(1 + "PTanh~) exp(--~ll t)]

,

(3.8C)

P21(t) = ¼ [Seche cos(,~, Coshe) - i sin(,~ t Cosh~)] ~Seche e~p(- A t)

+ 1Tanh~ [(1 + P ~ n l ~ ) exp(-All t)

(3.8d)

- (1 - ~'Wanh~) e x p ( - X ~ t)] p~3(t) = ¼(1 + ~') exp(-~3~ 0 p44(t) = ¼(1 - ~') e x p ( - ~ 0

(3.8e) (3.8f)

, ,

where ~11 :~--- Xpp-Ps -~ "~po ,

X22 := Xm-p, + Apo , X33 := Xo-ps + Xpo ,

(3.9a) (3.9b) (3.9c)

and 1X

1X

(3.10)

Positronlum in Condensed Matter

399

For later use we note that - - in contrast to All and ,~2 - - A33 and A, and therefore also ~11 +~22, are independent of the magnetic field. The parameter characterizing the magnetic field has been defined in (2.3) and (2.4). The diagonal elements of p(t) are the occupation numbers of the basis states { [0, 0), [1, 0), [1, 1), [1,-1) } and hence the quantities determining the annihilation characteristics of Ps. For instance, the fraction of Ps annihilating at a time t by 27 self-annihilation is given by ~p-PsPll. From (3.8a - b) it can be seen that the decay of the occupation numbers of the states 10, 0) and I1, 0) is described by two exponentials with the annihilation rates All or A~2. This is a consequence of the fact that, as discussed in Sect. 2, in a magnetic field pp-Ps and m - P s and not the states I0' 0) and I1, 0) are eigenstates of the Hamiltonian. The "magnetic" elgenstates (2.7) and (2.8) will in the following be denoted as Iepp_ps) and Iem-ps). In the basis { [¢pp_p~), Iem_ps), ll, 1), [1,-1) } the HamUtonlan (3.2) is diagonal. For the diagonal elements of the spin-density matrix the Liouville equation (3.1) reduces to the rate equations ape(t) _ dt

]kii pB(t)

(3.11)

where ~ii stands for the annihilation rate of the i-th basis state (~33 = ~44). The general solution of (3.11) reads

p (t) : p (o) exp(- , t)

(3.12)

The initial values p~(0) may be deduced from (3.7) by means of the orthogonal transformation p~(O) : Mp(O)M t , (3.13) where

M

=

v/se-a~ Cosh(~/2) -v/'~--~Sinh(~/2) 0 0

v / ~ - - ~ Sinh(~/2) V/-S--d~-~Cosh(~/2) 0 0

0 0 1 0

0 0 0 1

) (3.14)

transforms the states 10,0) and I1,0) into the Ps energy eigenstates in a magnetic field, ]@pp_p~) and ]~m-Ps), and where the superscript t denotes the transposed matrix. We thus obtain 1 pB(O) --

1 + ~ rranl~ "P Sech~

~P Sech~ 1 - "P Tanh~

o

o

1+

o

0

0

0

1-~

The only non-vanishing

pB12(*)

0 0

non-diagonal elements

0 0

\

)

are given by

= p~; (t) = plB2(o) e x p ( - i w f Cosh~) e x p ( - A t)

400

where the superscript * denotes the complex conjugate. They are oscillating with the same circular frequency w Cosh~ (_~ 1.3 • 1012 s -1) as the oscillatory terms in (3.8a - d). Since these oscillatory terms cannot be observed experimentally, they may be neglected. The great advantage of the use of the basis { [~pp-Vs), I~m-vs), I1, 1), [1,-1) } is that, in contrast to the nondiagonal elements in the basis { 10,0), I1,0), I1, 1), I1,-1)}, the non-diagonal elements pB2 and pB1 consist of an oscillating term only. Therefore only the diagonal 'elements of the spin density matrix have to be taken into account in the further treatment. This leads to a great simplification of the Liouvil]e equation if spin-exchange processes are to be included (see Subsect. 3.2). Then the spin density matrix in the basis { [0, 0), 11, 0), 11, 1), [1,-1)} may be calculated from (3.12) and (3.16) making use of the transformation

p(t) = M t p B ( t ) M

(3.17)

If the non-diagonal elements (3.16) are disregarded, we o b t a i n - except for the unobservable oscillatory terms - - the same results as in (3.8a - f). The preceding discussion shows that from a practical standpoint the set of tale equations for the diagonal elements of the density matrix in the basis of the eigenstates of Ps in a magnetic field used by Stritzke (1991) give the correct result for the occupation numbers of the four Ps states [0, 0/, 11, 0/, [1, 1), and [1,-1), i.e., the quantities determining the annihilation characteristics (cf. Sect. 2). The subject will be taken up again in Subsect. 3.2, where we shall derive a more general solution that includes spin-exchange processes. 3.2

Spln-exchange processes

The situation becomes more complicated if Ps spin-exchange processes are to be included. A necessary condition for spin-exchange processes to occur is that the system in which Ps is formed contains unpaired electrons, since otherwise the exchange of the electrons in Ps with electrons of opposite spin in the matrix is forbidden by Pauli's exclusion principle. Spin-exchange processes were first observed by Deutsch (1951a) through the decrease of the probability of 37 annihilations due to the conversion of o-Ps into p-Ps in N2 gas to which paramagnetic NO molecules had been added (cf. Sect. 1). The first correct theoretical treatment of spin-exchange processes is due to Ferre]] (1958). Spin-exchange processes (also known as "spinconversion") may play an important r61e in Ps-forming organic liquids containing paramagnetic additions. Well studied examples are methanol and benzene containing 4-hydroxy-2,2,6,6-tetra methylpiperidine-l-oxyl (HTEMPO) (Billard, Abb6, and Duplgtre 1991, Schneider, Seeger, Siegle, Stoll et al. 1993, Major, Schneider, Seeger, Siegle et al. 1995, Castellaz, Major, Schneider, Seeger et al. 1996). The diffusion-reaction problems arising in this context have been treated by Wiirschum and Seeger (1995). We confine ourselves to paramagnetic materials. The spin polarization of the unpaired e- in a magnetic field B is then given by

Positronium in Condensed Matter

(#eB'~

Pc- = Tanh k, kB T ]

401 (3.18)

'

where kB is Boltzmann's constant and T the absolute temperature. The probabilities that the spin of the unpaired e- is paralld (T) or antiparallel (,[) to the magnetic field are (1 + ;De-)/2 and (1 - :Pc-)/2, respectively. The spin-density matrix of the unpaired electrons, Pue-, is given by (3.5) with replaced by :Pc-. Following Ferrell (1958), let us picture the individual spin-exchange processes as "collisions" between Ps atoms and unpaired electrons with spin either up (T) or down (1). Since there are four possible Ps states, we have to distinguish eight different eases. As an example, consider collisions of electrons with spin up with Ps in the state I1, 0). The collisions change the state of the system "unpaired electron + Ps" from

I T) I~, o) = I T) ~ (I T 1) + I ~ T)) before the collisions to aex

aex

(gd -- T )

aex

I t) [1, 0) + ~

aex

aex

[ ]) [1, 1) + ~-- I T) I0' 0)

(3.19)

after the collisions, where ad is the "direct" and aex the "exchange" amplitude of the collision. The minus sign of the last term on the left-hand side of (3.19) is due to the fact that this term describes the exchange of two electrons with parallel spins (hence with a symmetrical spin wavefunction) and that this must result in an antisymmetrical spatial part of the wavefunction. The remaining seven possible cases may be handled in analogy to (3.19). Spin-exchange processes can formally be described by a "spin-exchange operator" U which transforms the states before a collision into those after the collision. E.g., U] T)I1, 0) corresponds to (3.19). The spin-density matrix after the collision, ~=on, is then related to that before the collision, ~, by the transformation

~o. = u ~ u+

,

(3.20)

where the superscript + denotes the adjoint operator. The superscript ^ indicates that in (3.20) the spin density matrices are taken in the basis { 1¢1) = 1'I')1o, o),1¢5)= I I)lo, o), I¢a)= I T)ll, O),..., 1¢8) = I I)l 1,-1)}. Since before a collision the spin of the unpaired electron and the Ps spin state are not coupled, ~ is the direct product of the Ps spin-density matrix p and the spin-density matrix Pue- of the unpaired electrons. The matrix elements of ~:on are given by

(0~1~:o. lea = (0~1u ~ u+ lea = ~(¢dUCk)(¢kl.~lCz)(U¢,l¢./) kl

,

(3.21)

402 where the completeness relationship ~-~k [ ~ ) ( ~ k l : 1 has been used. The right-hand side of (3.21) can be calculated from the known effects of U on the wavefunctions I~i) and on the matrix elements of ~. Because in the experiments the spins of the unpaired electrons are not controlled, we have to sum over their different orientations in order to obtain a relationship between the elements of the Ps spin-density matrix before and after the collision. This gives US pcol111: [O,d __ ~a.ex[12 Pll ÷ (~adl* aex -- lI¢/'ex[2) "Pe- PI2 1 ad aex * -- lla-exl 2) ~Oe- P21 ÷ ( ~"

÷ 41 [aex[2 [P22 + (1--'Pc-) P33 + ( l + ' P c - ) P 4 4 ] ,

(3.22a)

p~oo,, : I-d - ~-oxll ~ p ~ + (~adl .ox. - ¼1~.1 ~) ~'e- Pl~ 1 *

+ ( ~ d .ox - ¼1~oxl~) ~'o- p~l +¼1a~x12[p11 + ( 1 - ' P ~ - ) p 3 3 + (1-t-'P~-)p44] coll 1 2 PI2 : lad -- 5aex] P12 +

1

]aex] 2

[

,

(3.225)

(1 -- " ~ e - ) P33

L

(1 -'f- ~e-)

P44 -I- P21] 1 * + (5-d,~o~ - ¼1,~o~l~) ~'o- pll

+ ( ~1 " a " c*x - ¼laexl ~) ~ ' ~ - p ~ ,

(3.22c)

pcoll 1 2 [ 21 :--- lad - ~a0,t ,021 + ¼ la0xl ~ (I

- -

~ 0 - ) .3~

(1 ÷ ' P c - ) P44 ÷ P12] 1

*

+ (~=d =0~- ¼1-~xl ~) ~'~:- pll 1 * ~ox - 11.o~1~) ~,o- p ~ , + (~-~

(3.22d)

p~'~ -- ½ [I.d--.0~l ~ (1 + T~e- ) + lad[ 2 ( 1 - ~ - )

]p33

+ ¼ laexl 2 (1 ÷ ~e-) [Pll ÷ P22 ÷ P21 ÷ P21] , 044 : ~ l a d - a ~ l ~ ( 1 - ' P e - )

÷ ladl 2 ( 1 + 7 ~ e - )

÷ ¼ [aex[2 (1 -- Pc-) [Pll ÷ P22

-

-

P21 -- P21]

(3.22e)

P44

(3.22f)

For e+ polarized in the z-direction, the only non-vanishing non-diagonal elements of the density matrix are P12 and P2I (see (3.7)). Since spin-exchange processes with unpaired e- that are either unpolarized or polarized in zdirection cannot lead to a net polarization in the z- or y-direction, P12-c°nand p~]U will be the only non-vanishing non-diagonal elements of the density matrix after the collision. The conservation of the total number of Ps "atoms" during spin-exchange processes leads to the following relationship between the traces:

Positronium in Condensed Matter Tr(p) := E

p" : Tr(pC°'z)

403 (3.23)

ii

Together with the postulate of random phases, i.e., the fact that the terms a~ aex and ad aex vanish when we average over many spin-exchange collisions, inserting (3.22) into (3.23) gives us ladl 2 + laexl 2 = 1

(3.24)

If laal 2 and the mixed products a~ a~x and ad aex are eliminated from (3.22) by making use of (3.24) and the random-phase postulate we find that the quantities Pij _con -Pij, i.e. the average change of Pij effected by one "collision", is proportional to la~xl2. This suggests that we may obtain the rate of change of Pij by exchange processes, to be denoted by [d Pij/dt]ex, by multiplying pcjol]- pij with the "collision frequency" Ucon. The product UcoUlaexl 2 may be identified with the rate constant kex of spin-exchange processes. This gives us finally

[dpH/dt]ex : hex [ - - ~3P l l -~- ~022 1 -~- 1 (1 - "Pe-) 033 _~.~1 (1 + '~e-) 044 -- 41"Pe- (012 "4- 021)] 3

1

+ 1 (1 + ~ e - )

1 (1 --

,

(3.2 a)

,

(3.25b)

7~,_ ) 033

044 - ~P¢- (012 + 021)

]

[dp12/dl[]e x : kex -- ~'0123 "4- ~'0211 "4- ~'1 (1 - Pe-) 033

,

1 (l_{_~e_) 044 _ i ~ e _ (011 -4-022)

4

[dml/dt]ox=

41 (1 +

[d 033/d $]ex =/¢ex

-f- ~

[dp44/dt]ex

]

(3.25c)

021 + ¼Pl + 1 ( 1 - ~De-) 033

~e-) 044

-- ~ ~lDe- (011 @ 022)

(3.25d)

_ ~1 (1 - ~ e - ) 033

011

022

012

021

l%x

The spin-exchange terms (3.25) may be inserted into the Liouville equation "by hand". For temperatures above 100 K and magnetic fields not exceeding 1 T the spin polarization of the unpaired electrons, P~-, is less than

404 0.01. In many applications we are thus allowed to assume that the two spin directions of the unpaired electrons have equal probability and to set ~eequal to zero. The following calculations make use of this simplification. The fact, demonstrated at the end of Subsect. 3.1, that without spinexchange processes the solution of the Liouville equation can be easily obtained when we choose the energy eigenstates of Ps in a magnetic field as basis functions suggests that it will be advantageous to express the additional spin-exchange terms in terms of the elements of the spin-density matrix in the basis { ICpp_ps), ICm_Ps), I1, 1), I1,--1)}. Transformation of (3.25) to this basis by means of the orthogonal transformation M (see (3.13) and (3.14))" gives US

[dPlB1/dt]ex : _ 1 kex [(2 + Sech2~) plB1 -- p~2 Sech2~ -

(P~2 + P~I) Tanh~ Sech~

- (1 + T a n h O p ~ - (1 - T a n h ~ ) p ~ ]

,

(3.26a)

,

(3.26b)

[dpB2/dt]ex = -¼ k~x [(2 + Sech2~) pB2 - P~I Sech2~ + (P~2 + P~I) Tanh~ Sech~ - (1 - Tanh~)p~3 - (1 + Tanh~)pB4]

[dpBl2/dt]~: _1/%~

[(4 - Sech2~)P~2 - (PRB1 -- P~2)Tanl~ Sech~

-- pB 1Sech2~ - (p3B - pB4) Sech~]

,

(3.26c)

[dp~l/d,]o~ : -¼ k~ [ ( 4 - Sech~0p~l - (p~ - p~2) Tanh~ Sech~ - Pls2Sech2~ -(P~3 - PL)Sech~J

[dpB3/d t] ex ~-----41 k~x [2P3B

(3.26d)

,

- (1 + Tanh~) PlB1

- ( 1 - Tanh~)pB2 - (pB2 + pBi)Sech~]

,

(3.26e)

[dp4B4/d/]~x = - I k e x [ 2 a S - ( 1 - Tanh~)plB1

- (1 + Tanh~) pB2 + (pB2 + pB) Seeh~]

(3.26f)

Eqs. (3.26) show that spin-exchange processes couple the diagonal elements to the non-diagonal elements PP2 and p2B1.However, since in the absence of spinexchange processes these non-diagonal elements oscillate with the circular frequency ~Cosh~ (see (3.16)), we may assume that on time scales large compared to (w Cosh~) -1 _~ 8 . 1 0 -13 s, there is no net effect of transitions between diagonal and non-diagonal elements. Neglecting these transitions then gives us the following rate equations for the diagonal elements of the spin-density matrix:

Positroniurn in Condensed Matter

405

~1---[~11 + lkox(2+ Sech~0].r, + ~ k~x p~ S~ch~ + (1 + T ~ ) +

p~3 ~ + (1 - Tanh~) p~, ,(3.27~)

+

+ ¼k~ [plB1Sech2~+ (1 - Tanh~) p3B3+ (1 + Tanh~) p4B4] ,(3.27b) 1 ~--- - -

P33

1

+ ~kex [(1 -I- Tanl~) PlB1+ ( 1 - - Tanh~) pB2] ,

(3.27c)

+ ¼kex [ ( 1 - 2"~.nh~)p~1 + (1 +Tanh~)P~2] .

(3.27d)

3.3 G e n e r a l s o l u t i o n o f t h e r a t e e q u a t i o n s

Our further strategy is as follows. We shall transform the system (3.27) into a system of 4 linear equations for the Laplace transforms OO

£B(s) := f pB(t) exp(-st)dt

(3.28)

0

These equations may easily be solved for £~i(s). The orthogonal transformation (3.17) then gives us the Laplace transform of the diagonal elements of the density matrix in the basis { 10, 0), [1, 0), 11, 1/, 11, -1 / }, the appropriate basis for determining the Ps fraction annihilating by 2~/self-annihilation and thus responsible for the "narrow components" of the 511 keV photon line (cf. Sect. 5). In the e+SR set-up referred to in Sect. 1 and briefly described in Sect. 4 it is this component that is used to measure the e+ spin relaxation. The set-up integrates over all positron ages. This has the consequence that we do not have to invert the Laplace transforms in order to compare experiment and theory but that it suffices to consider the quantities £~(0). Using the rdationship

f b~(t) exp(-st)dt =

S ~Bii($)

- p~(O) ,

(3.29)

0

we obtain by the procedure just outlined £1B1

+ PTanh~) a2

t

+kex (s+X33+kox) S e c h ~ ,

J

(3.30~)

406

1:~(,) - al - -"~]GeXa2{ [2 (s+)t33)(s+ All)+h~x (2s+ A33+All)] (1-7~Tanh~) +kex (s+X33+ke~) Sech2~,

(3.30b)

J

+ 2~(~ + ~11)(~+ ~ ) T~ ( ~ - ~1~) ko~(~ + k~) Tan~](~ + ~') %

(3.3Oe)

+ kex(al + k~x) (. + a + ko~)Sech~ / with the abbreviations a 1 := 2s + 2~33 -4- kex ,

(3.31a)

a2 := [a1(25-~- 2~tll 3I- kex)- ke2x] [a1(25-4- 2~t22 + kex)- ke2x]

+ kex(al + kex)[al(2s + 2A + kex)- ke2x]Sech2~

(3.31b)

The Laplace transforms of the diagonal dements of the density matrix in the basis { 10,0), I1,0), I1, 1), I1,-1)}, £ii(s;~), are obtained from (3.30) by the orthogonal transformation (3.17). This results in al + kex [(a, + kox)$ + k~, X33] (1 + ~'Tanh~ Se~h~) a2 t

+ ~5 (a, + k~x) Seeh2~ + al Seeh~ [~11Sinh2(~/2) (1 - 7) Tanh~) + )t22Cosh2(~/2) (1 + :P Tanl~) ] /

'

(3.32a)

£22(s;~)- al + kex ~ [(al + /%~)s + k~x:~Z3]( 1 - 3VTanl~ Sech~) a2 (

+ ~ (al + k~,) SeCh~ + a, SeCh~ [~1, Cosh~(~/2) (~ - ~ Tan~)

+ )t~.~.Sinh2(~/2) (1 + 79 Tanh~) ] / •33($; ~) = £B3(8 ] ~) , £44(s; ~) = £B4(S;~)

'

(3.32b) (3.32C) (3.32d)

The time evolution of the diagonal elements of the spin-density matrix in the basis { 10, 0), [1, 0), I1, 1), I1,-1) } is obtained by Laplace-inverting (3.32).

Positronium in Condensed Matter

407

To do this one has to find the zeros of the fourth-order polynomial a2 : as(s). In general these four zeros are distinct. This leads to a representation of the population numbers pii(t) of the four Ps states in terms of four exponentially decaying functions. The quantities £ii(O; 4) required for the interpretation of age-integrating experiments [cf. Eqs. (5.1)] may be easily obtained from (3.32) and (3.30). 3.4 E x p l i c i t s o l u t i o n s for s p e c i a l cases Although, as repeatedly emphasized, for the comparison of the theory developed in this paper with age-integrating experiments Eqs. (3.32) need not to be transformed to the time domain, on occasions it may nevertheless be desirable to have explicit expressions for the age dependence of the populations of the various positronium states. As remarked at the end of Sect. 3.3, this requires finding the roots of an algebraic equation of fourth order. In general this is best done numerically. However, there are special eases in which the equation factorizes into algebraic equations of degree two or one, so that simple explicit solutions may be obtained. These cases are as follows: (1) Very large magnetic fields (Sech2~ = 0) (2) No magnetic fields (Tanh2~ = 0) (3) Negligible spin-exchange processes (kex : 0) From these special solutions, approximate solutions in the neighbourhood of the limiting cases may be obtained by perturbation theory. (1) Limit of very large magnetic fields (Seth2~ = 0) In the limit of high magnetic fields (tteB >> AEo_p) terms proportional to Sech2~ = (1 + z2) -1 may be replaced by zero. Then the fourth-order polynomial a2(s), whose zeros give us the decomposition of £ii(s; 4) into partial fractions "(cf. Eqs. (3.31) and Eqs. (3.32)) and hence the representation of £1i(g) as a sum of exponential functions of t, factorizes into two identical second-order polynomials. Their zeros follow from al(2s+2A+kex)=

ke~x ,

(3.33)

where the high-field relationship 2~11 = 2X22 = 2A was used (see (2.9), (2.10), and (3.10)). With (3.31a), Eq. (3.33) becomes (2s + kex) 2 + 2(A + )t33)(25 -~- kex ) -4- 4 A ,)t33

-- ke2x :

0

(3.34)

If we denote by rj (j = 1, 2, 3, 4) the negatives of the roots of a2(s) = 0, i.e. the quantities satisfying a2(-rj) = 0 we obtain from (3.34)

,

(3.35)

408 2rl

2r~ = A + ~33 + kox ± , / ( A - A33)~ + k~x

2r 3 = 2r 4

(3.36)

J Eqs. (3.32) simplify to $ -t- ~t33 + kex

Cl,(S; oo) = £~2(s; oo) = 4(s + rl)(s + r3) C3~(s; oo) }

s + A + kex

'

(3.37a)

(1 +

(3.37b)

C44($; OO) -- = 4(8--'~- ff)"~ T-r3)

Laplace-inversion of (3.37) gives us

o11(0 : p~(0 : ~ + (1 \

-

1 + ~/~-_ ~

¥~x

)

A - ~33 - kex

V/~-~--~333-~ ~ ~¢e2x) e x p ( - - r 3 t ) )

o~p(-r~0 ,

.4 - A33 + k~× + (1 + V/~-~-~333~ T ~e2x) exp(-r3 t) /

(3.38a)

(3.38b)

k

The fact that in the present hmit p11(t) is 7~-independent means that for z >> 1 we cannot obtain information on the e + spin relaxation from observing the 27 self-annihilation. (2) Limit of vanishing magnetic fields (Tanh2~ = O) For arbitrary ~, a2(s) = 0 may be written as tt~

[4(s+ A)(s+ A+/%x)-(Ap-p~- Ao_ p~)2] +4alkex(S+ A33)(s+ A+/%x) =

We see immediately that if ( = 0, (3.39) is satisfied by is solved by -,~ = -(~

a~

= o; hence (3.35)

+ ko~/2)

(3.40)

The remaining zeros of a2(s) have to obey the third-order equation 0,1{($ -~- a ) ( $ --I- A + ~ex) - (A - ~33) 2 }

+ k~x(S + A33)(s + a +/%x) -- 0 or

(3.41a)

Positronium in Condensed Matter

(O.1 + ~ex){($ + A)($ + A + kex) - (A - A33 + k e x / 2 ) ( A - A33)} : 0

409

(3.41b)

Eq. (3.41b) is satisfied by (al + kex) : 0; hence

--T3 -- --(J~33 "}- kex)

(3.42)

is a further solution of (3.35). It follows from (3.41b) that the remaining two roots of a2(s) = 0, - r l and - r 2 , have to obey the quadratic equation (s + A) 2 + kex(S + A) - (A - A33)(A - A33 + kex/2) = 0

(3.43)

We thus obtain 2rl / 2 11/2 2 r 2 , = 2A + kex ± [(Ap-p, - Ao-P, + kex/2) 2 + 3k J 4 1

. (3.44)

Since - r 3 is a zero of the denominators of (3.30) and (3.32), for ~ = 0 the Laplace transforms simplify to s + A2~ + kex £11($; 0) = 4(s + rl)($ -~- T2)

'

£22(8; O) =

,

$ "+ All -}" kex 4(s + P1)($ -}- r2)

£33($; O) "~ _-/+(s) £44($; 0) J 8($ + 7'1)($ + r2)($ --~ 1'4)

(3.45a)

(3.45b) (3.45c)

with

f+($)----- [2($+ ~11)($+)~22) -4- kex(2$+ A + ~ 3 3 ) ] ( 1 + ~ ) + kex($ + A + hex)

(3.46)

Performing the inverse Laplace transformation on (3.45) gives us for ke× # 0 1 {1 - (Ap-ps - Ao-ps) + ~ex Pll(t)---~ ~ A } exp(-rlt)

1 (~p-Ps -- ~to-P,) + kex + ~{1 + A } exp(-r2 t),

(3.47a)

1 (J~p-Ps -- Jlo-Ps) -- kex p22(t) : ~{1 + A } e x p ( - r l t) + ~{11 _ (Ap-Ps

-- ~o-PS)A-

kex } exp(-r2 t),

(3.47b)

p33(t) } f+(-ra) e x p ( - r ] t) p44(t), : 4A(Ap_p-~ - ~ o - P s + A) _

f+ ( - r 2 ) exp(-r2 t) 4A(Ap_ps - Ao-Ps - A)

+

f+ ( - r 4 ) 2 [(

p-ps -

o-Ps) 2 - A2]

exp(-r4 t)

(3.47c)

410

with 2 A := [ ( ~ p - P s - Ao-ps q- kex/2) 2 q- 3kex/4 j11/2

(3.48)

In contrast to (3.45), Eqs. (3.47) are not in a convenient form for obtaining the limiting case ]%x : 0. This case is included in the remarks of Subsect. 3.4(3).

(s) Sph-ex h nge processes neg"g ble ( ox : 0) If the spin-exchange processes may be neglected (]%x : 0) the fourth-order equation cz2(s) : 0 separates into four linear equations with three distinct roots - r l , -~'2, -T3 : - e 4 . They are given by gq. (3.9) with ri : J~id ( i : 1 , 2, 3). As it should, the result for the density matrix p(t) agrees with (3.8) if there the terms oscillating with the angular frequency wCosh~ are neglected.

4 Experimental techniques The experimental techniques available for studying e+SR in condensed matter are modifications of the techniques estabfished for e + annihilation studies that disregard the e + spin polarization or use unpolarized e +. The information that may be derived from the e+e - annihilation characteristics perrains either to the positron annihilation rates (i.e., to the overlap of the e + and e - wavefunctions) or to the momentum distribution of the annihilating e+e - pairs. The three "classical" techniques are (see, e.g., Seeger and Banhart 1990) (i) e + lifetime spectroscopy, (ii) angular correlation of the annihilation radiation (ACAR), (iii) Doppler broadening of the 511 keV annihilation line (AE-r). The techniques (ii) and (iii) have in common that they both measure the momentum distribution but differ in the momentum components that they investigate (transverse to the direction of observation in the case of ACAR, along this direction in the measurements of the Doppler shift AE~) and in their experimental characteristics and hence in the information they can provide. ACAR has the advantage of a much higher resolution compared with AE~ measurements, but at the price of a lower data accumulation rate. In AE~ measurements the energy of only one of the annihilation photons has to be recorded. This has the advantage not only of much faster data accumulation compared with ACAR but also of offering the possibility to use the second photon from 27 annihilations for a correlated second AE~ measurement or for determining the e + age of the e + whose annihilation led to the observed Doppler shift. The "positron age" of an individual positron is the time interval between the "zero" of the lifetime measurement and the annihilation event; it is approximately equal to the time the positron has spent in the sample or, in the case of Ps formation, the time interval between Ps formation and annihilation.

Positronium in Condensed Matter

411

Lifetime spectroscopy gives us the distribution function of the positron ages. For the "death signal" one of the annihilation photons has to be used. The "birth signal" m a y either be obtained by the "classical n p r o m p t - p h o t o n technique referred to in Sect. 1 (usually based on a 22Na positron source) or by recording the passage of the e + through a scintillator detector located between the e + source and the sample. The latter method, known a s / 3 + 7 technique in contradistinction to the "/'y technique deriving the birth signals from p r o m p t photons, has the advantage of a higher count rate for a given source strength. However, full use of its potential for excellent time resolution can only be made if positrons of relativistic energies are available (for details see Schneider et al. 1993). Since these are also required for achieving a high e + spin polarization, the/3+'), technique is clearly the method of choice for e+SR lifetime experiments. The techniques (i) and (iii) m a y be combined in the so-called A M O C ( = a g e - m o m e n t u m correlation) technique ~. As indicated above, it uses the annihilation photon not required for the AE.~ measurements to measure simultaneously the age of the annihilating positrons and thus to establish the correlation between one of the m o m e n t u m components of the annihilating e+e - pairs and the positron age 7. The e + spin polarimeters based on the modification of the positronium states by a magnetic field B (i.e., possibility (i) of Sect. 1) all make use of the fact t h a t the population of the m = 0 substates of 1S positronium depends on whether the spin polarization is parallel or antiparallel to the magnetic field. Positrons with spins opposite to B are preferentially captured in the 13S1(m = 0) substate at the expense of the 11S0 substate, and vice versa 8 Since e + in Ps "atoms" that without magnetic field would have annihilated 6 In principle an analogous combination of (i) and (ii) is also possible; in practice, however, simultaneous high-resolution measurements of lifetime and ACAR face considerable technical difficulties. 7 Professor Innes MacKenzie of Guelph University informed us in a letter dated August 21, 1996, that when the experimental discovery of positronium in gases by M. Deutsch became known (cf. Sect. 1), Professor W. Opechowski, a theoretical physicist at the University of British Columbia, remarked that one should try to measure time and momentum spectra simultaneously, and that this remark was the initiation for the attempts of Innes MacKenzie and Barry McKee to establish age-momentum correlations when large Ge(Li) detectors became available. The last (unpublished) age-momentum correlation measurements at Guelph University based on the classical ~f~f age measurements, using large CsF scintillators, produced 100 coincidences/s at a time resolution of 290 ps and an energy resolution of 1.3 keV. This is to be compared with about 1.2-10 a coincidences/s, a time resolution of 230 ps, and an energy resolution of 1.3 keV that can be obtained with a 100 mCi 2~Na source at the fl+-y AMOC set-up of the Max-Planck-Institut f ~ Metallforschung in Stuttgart. s This has been used by A1-Ramadhan and Gidley (1994) to increase the measuring effect in their indirect determination of rp-p~ (cf. Sect. 2.).

412

with the characteristics of o-Ps annihilate in the presence of a magnetic field with the characteristics of p-Ps ("magnetic quenching of o-Ps", cf. Sect. 2), any technique that distinguishes p--Ps annihilation from o-Ps annihilation m a y be used as an indicator of the e+ polarization. (Since the Zeeman energy is very small compared with the Ps binding energy, we m a y assume that the formation of positronium "atoms" is not affected by laboratory magnetic fields; cf. Appendix A.) Measuring the ratio between 37- and 2v-annihilation events and its dependence on B might appear to be the simplest way to determine the e + polarization. However, the field strengths required to shift the occupation ratio of the two m = 0 substates of 1S-Ps appreciably are so large that virtually all annihilations in these substates occur by 2"r-processes (cf. Sect. 2) whereas the fraction of 3"y-annihilations from the ]m] = 1 substates is unaffected not only by the strength but also by the direction of the magnetic field. When Page and Heinberg (1957) demonstrated for the first time, by ACAR measurements on various mixtures of argon and propane (C3Hs) gas, that the e + from/3 + decay (in their case of 22Na) had the expected right-handed helicity, they made use of the following effects. Under the experimental conditions chosen the Ps atoms "thermalize" so slowly that their momenta at the average age at which p p - P s annihilates (rpp_p~) cause the corresponding ACAR distribution to be rather wide. At ages of the order of magnitude of rm_ps(_ ~ 3- 10 -9 s in typical magnetic fields), however, the Ps thermalization is virtually complete, so that the 27-annihilations of m - P s give rise to a very narrow ACAR distribution. By measuring the dependence of the 1800 coincidence rate on the direction of the magnetic field the e + polarization could thus be determined. The method of Page and Heinberg is not applicable to Ps in condensed matter because of the much shorter thermalization times. Here the principal possibilities to distinguish p--Ps annihilations from o-Ps annihilations are the following. (1) The great majority of Ps "atoms" annihilating in condensed matter have thermal kinetic energies. Their self-annihilation gives rise to a much narrower momentum distribution than the pick-off annihilation or the annihilation of unbound e + , in which the wide momentum distribution of the host electrons involved in the annihilation characteristic for e- in condensed matter dominates the momentum distribution of the annihilating pairs. Since virtually all p - P s undergo self-annihilation whereas self-annihilation is negligible for o-Ps, the narrow component of the momentum distribution as seen by ACAR measurements is a quantitative indicator of p - P s annihilation. E.g., Greenberger, Mills, Thompson, and Berko (1970) and Herlach and Heinrich (1972) used ACAR to demonstrate the magnetic quenching of orthopositronium in a-quartz and KC1. (2) By applying a large enough magnetic field, the lifetime of m - P s may be lowered below that associated with pick-off annihilation. (In order to reduce

Positronium in Condensed Matter

413

Tm_ps t o 2 flS, we reqnire B : 1.8 T i f f : 1, and B : 1.3 T i f f ; : 0.7.) With the help of the expressions given in Sect. 2, measurements of Tm_ps permit the determination of the ratio of o-Ps to p-Ps annihilations. In discussing the pioneering work of Page and Heinberg (1957), Lundby (1960) pointed out that the modification of the lifetime spectrum due to the magnetic quenching of o-Ps may be used to investigate the e + spin polarization. This possibility was realized by Bisi et al. (1962) as wen as by Dick et al. (1963) and later applied extensively to the study of Ps in a wide range of substances by the group at the Politecnico Milano (for a review see Consolati 1996). The two techniques for distinguishing p-Ps from o-Ps referred to above have in common that they both rely on coincidences. Since because of this obtaining good statistics takes a long time, it is dimcnlt to detect by these methods, let alone study quantitatively, the formation of small fractions of Ps. In the present context, lifetime spectroscopy has the additional drawback that the unavoidable magnetic fields interfere with the operation of the photomnltiplier tubes in the photon detectors. This requires the use of fairly long light guides, which reduce the achievable time resolution. As a result, it is not only difficult to determine the lifetime rpp-Ps with an accuracy that allows a reliable determination of the electron-density parameter ~ (of. . Eq. (2.11)) but also often impossible to separate Tp_ps from the lifetime of those e + that do not form Ps. The presence of spin-conversion processes (cf. Sect. 3) and chemical reactions such as complex formation or oxidation (Billard et al. 1991, Castellas et al. 1996) may complicate the identification of ~'m-Ps further. As a non-coincidence technique, Doppler-broadening measurements can be performed with better statistics than ACAR or lifetime measurements. The chances to detect small fractions of Ps-forming e+ are therefore much better. An additional advantage of the Doppler-broadening technique is that, without impairing the lineshape determination, the annihilation photons not required for this measurement may be used for AMOC or, at least, ageselected measurements. The z~ET-based e+SR-measurement set-up built in Stuttgart some years ago (Seeger et al. 1987) and used to study e+ spin relaxation in c~-iron (Banhart 1988, Banhart et al. 1989) as well as positronium formation (Major, Seeger, and Stritzke 1992) has recently been modernized (Gessmann et al. 1997a). It utilizes spin-polarized positrons from a radioactive 6SGe/6SGa source (maximum kinetic energy of the emitted e + 1.88 MeV, average kinetic energy 0.81 MeV) and measures the broadening of the 511 keV annihilation photon line in external magnetic fields of different magnitudes up to 2.6 T that are applied parallel or antiparallel with respect to the e + spin polarization. Reversal of the magnetic field is equivalent to a change of sign of 7~. An improved data-taking system allows the quantitative detection of small fractions of Ps-forming positrons. A practical example will be discussed in Sect. 5.

414

5 Data

analysis

and

results

In condensed matter studies by all three methods (i - iii) mentioned at the beginning of Sect. 4, the most important quantity is the ratio of the number of counts due to 27 self-annihilations to that due to annihilations by pickoff (in the following called yields Ip-ps and Ipo, respectively). According to (3.28) we have oo ~p-Ps(~) = Z~Ps ~p-Ps / P 1 1 ( ~ , ~ ) d ~ o --'~ Nps "~p-Ps ~11(0; ~) ,

(5.1a)

flO

Ipo(~) = ~¢p. ~po / [pl~(~, ~) + p22(~, ~) + p~3(~, ~) + p44(~, ~)] at 0 4 = "NP' ~P° [ E L:ii(O;~)] i-----1

(5.1b)

Since for typical pick-off rates in condensed matter (2 • 10 s - 10 9 s -1) the 3"y-annihilation may be neglected, Nps --- Ip-Ps(~) -}- Zpo(~)

(5.2)

holds. From the Doppler-broadened 511 keV lines as measured at the two opposite fidd directions (indicated by -4-~) we may derive the quantity

p* :

Nps

(5.3)

where K is a dimensionless proportionality factor that depends on the experimental set-up and on the data-handling procedure. Eq. (5.2) holds independently of the sign of the applied magnetic field; hence it follows that

Zpo(-~) - Ipo(+~) = zp_p.(+~) - Zp_p.(-~)

(5.4)

Making use of (5.4) and (5.1), P* may be written as p*

= • ~p_p. [~11(0; +~) - ~1~(0;-~)]

(5.s)

Inserting Eq. (3.32a) into (5.5) and using (2.4) gives us finally p* - 8 K ' ~ ) i p _ P s ) t o ( ~ ° + k e x ) 2a~ C0 ~- C2Z2

with

(5.6)

Positronium in Condensed Matter

415

c0=2(~o-~-kex)(2~o'~-kex) [(4~o + kex)~p + 3~o~ex],

(5.7&)

c~ = [(2~o + k~,)(~, + ~o) + 2ko,~o] ~

(5.7b)

0.06

0 1 2 i/...,.......

¢b.

EL

4

B[T] 8

16

32

I

I

(a)

~Z,po=10B1 S-1

'L

0.04

nv: um

0.02 I ~ , ~ ~lpo=l'O7.s-1

....

0.00 0.0

0.06

0.5

0 1 2

1.0

4

1.5

2.0

B[T] 8

2.5

16

32

(b)

Zpo=108 s"1 0.04

3.0

I kex=101° s-1

¢b.

PS in vacuum/

EL 0.02

~[/~.~. " ' / " f " ..... ~ . . . '/./.k.x=lO 8 s"1, Zpo = 0~"~ r.,~4,

0.00 0.0

0.5

..

,

,kex=l01° s~ Zpo=0~"'--.. 1.0

1.5

2.0

2.5

3.0

Fig. 5.1. The dependence of the quantity P * / K 7 ~ on the magnetic field as calculated from (5.6) for ~ = 1. The influence of the pick-off annihilation rate Apo is illustrated for kex = 0 in Fig. 5.la, that of kex for different pick-off rates in Fig. 5.lb.

416

In (5.7) ~p : = ~p-Ps -~- ~po and ]o : = ~o-Ps -~- )(po = )133 are the total annihilation rates of p-Ps and o-Ps, respectively. The field parameter z has been defined in (2.3); Note that through ZlEo_p it depends on ~. P* as given by (5.6) was also derived earlier by Stritzke (1991). The functional form of its dependence on the applied magnetic field agrees with the result of Mills (1975). Fig. 5.1 shows the field dependence of (5.6) as calculated for different values of :~po and kex. (If, in addition, reactions between Ps and "free" positrons are taken into consideration, co, c2, and the factor ~o()io + ]%x)2 in (5.6) have to be modified; for explicit results see Gessmann et al. 1997b.) ~p is usually much larger than either )Lpo and k~x. Then (5.6) and (5.7) may be simplified to

CO = 2(~o "f- /¢ex)(2,~o + ]¢ex)(4~o "}- ]¢ex))Ip-P. 2 C2 : (2)(o "~- kex) 2 '~p-Ps ,

,

(5.8a) (5.8b)

and p. =

8K:PAo ()io + kex) 2 z 2~o + kex 2(go + kex)(4:~o + ]¢ex) + (2)io -~- kex)~p-Ps 22

.

(5.9)

If the spin-exchange rate kex is much larger than the pick-off rate, we have the further simplification p . = 8K7~

~poZ 2kex + ~ p - P s ~ 2

(5.10)

Eq. (5.6) shows that P* is proportional to the initial positron polarization 7~ with a proportionality factor that depends on the annihilation rates, the spin-exchange rate, and the applied magnetic field. Apart from a proportionality factor, P* may be interpreted as the residual polarization of the positrons in Ps at the moment of their annihilation. This is a relative measure of the positron spin relaxation, provided we know the dependence of K and 7~ on the magnetic field (cf. Gessmann et al. 1997b). The maximum of P*/K7 ~ as a function of the magnetic field occurs at z = ~max = (C0/C2) 1/2. If Eqs. (5.8) are applicable, we have

[2(,~o + ~ex)(4,~o -}- kex)] 1/2 Zmax---- L (-2~o ~ ~e--~p----P: J

(5.11)

The maximum is then given by

(P*/KT~)max =

( k2 o +

kox 3/2 [2(4~o + kex))~p-ps] 1/2

(5.12)

We see that as long as the condition )tpo << ~p-Ps is fulfilled, a large pickoff rate increases P* whereas large spin-conversion rates lead to a decrease.

Positronium in Condensed Matter

417

Fig. 5.1 illustrates this behaviour. Fig. 5.2 shows the dependence of P* on the applied magnetic field measured on teflon at room temperature. r

T

1

T

1

r

1

~L 0.01 ""-''"--. ............................... 0.00

l 0.0

0.5

"1.0

I__£

1.5

2.0

1. 2.5

3.0

B [T] Pig. 5.2. The quantity P* in teflon (polytetrafluorethylene) at room temperature as derived from measurements in the Stuttgart e ÷ SR set-up. The solid line is a fit to the data, the dotted line corresponds to Ps in vacuum. As expected from (5.6), P* increases linearly at small magnetic fields from its zero-field value P* : 0. At large magnetic fields P* approaches zero since in the high-field limit the direction of the magnetic field no longer affects the population numbers of the energy eigenstates of Ps (cf. Sect. 3.4). An analysis of the experimental d a t a on teflon leads to a density parameter : 0.74 ± 0.04, a pick-off rate Apo : (0.77 :k 0.07) • 109s - 1 , and a fraction r = 0.32 -6 0.01 of Ps-forming positrons. The technique is very sensitive against positronium formation. Positronium fractions of the order of magnitude of a few percents may be detected.

Appendix A On the separation of the spatial and the spin wavefunction of p o s l t r o n l u m in t h e p r e s e n c e o f a m a g n e t i c fie]d The influence of an external magnetic field B on the spatial wavefunction of positronium is expected to be negligible if the Zeeman energy -/~ps " B is small compared with the positronium binding energy, 112mee4c41c/(16]g2). The m a x i m u m value the o f positronium magnetic moment is close to twice the Bohr magnetic moment, hence pPs ~ 2pB : eh/me. (We disregard the small deviation of the electron g-factor from 2.) The above criterion m a y thus be written as

418

(A.1) In (A.1) ~ m : = h / e : 4.136.10 -15 Wb denotes the magnetic flux quantum and meC 2 : 8.187- 10 -14 J the energy equivalent of the electron mass. Inserting these values into (A.1) leads to the conclusion that the separation of the positronium wavefunction into a spatial part that is unaffected by an external magnetic field B and into a spin part is justified if B (< 6 . 1 0 4 T.

(A.2)

This inequality is fulfilled for virtually all conceivable laboratory experiments. It may be grossly violated under astrophysical conditions, e.g., in the strong magnetic fields of pulsars.

Appendix B What's

in a name

?

" W h a t ' s in a name? T h a t which we call a rose by any other name would smell as sweet." (W. Shakespeare, Romeo and Juliet, 2.2.23.) Juliet may have been right on roses but scientific nomenclature is a different matter. It should have some relationship to the scientific idea to be conveyed, and it should certainly not be misleading. The Greek word "~bev6Qs" means false. Indeed, according to Webster's New World Dictionary (Neufeldt 1988) the adjective "pseudo" may be used for "sham", "false", "spurious", "pretended", or "counterfeit". The prefix "pseudo-" has analogous meanings. In our opinion there is nothing "pseudo" about the positronium states referred to as "pseudo singlet" and "pseudo triplet" in the literature. Rather, in non-vanishing magnetic fields B the two m = 0 states of 1S positronium are m i x t u r e s of parapositronium and one of the orthopositronium states. The most important indicator of these mixed states is their intrinsic mean lifetime. For magnetic fields of interest in terrestrial science the lifetime of what usually has been named "pseudo singlet" (i.e., the state reducing to parapositronium if B --, 0) is so close to that of parapositronium that the name "plesiopara-Ps" (pp-Ps) appears appropriate (Greek adjective 7r~71~ios = near, almost). On the other hand, the reduction of the intrinsic lifetime of the "pseudo triplet" from that of the 1 3S1(m = 0) orthopositronium state, to which it reduces for B --* 0, becomes noticeable at rather small magnetic fields, so that its "mixed" character is evident in many experiments. We therefore propose to call Ps in that state "meikto-Ps" (m-Ps), derived from the Greek word #eLt~T6S = mixed. Finally, it appears desirable to have a common name for the states that for B --, 0 reduce to ortho-Ps. In typical laboratory magnetic fields a characteristic property of these states is that they are much longer lived than plesiopara-Ps, irrespective of whether they decay intrinsically (in the case of

Positronium in Condensed Matter

419

m = +I by 37 annihilation, in the case of m = 0 by 27 as well as 37 annihilation) or by pick-off annihilation. In the lifetime spectrum of positronium these states survive at positron ages at which plesiopara-Ps has died out. Making use of the Greek verb Ad~rw (= to leave behind) we propose to call them "leipo-Ps" (l-Ps).

Acknowledgements Helpful discussions with Dr. L. Schimmele (Stuttgart) are gratefully acknowledged. The authors are indebted to Professor H.-D. Carstanjen (Stuttgart) and Professor R. Grynszpan (Paris) for their competent advice regarding Appendix B, and to Professor I. K. MacKenzie (Guelph) for critically reading the manuscript.

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Neufeldt, V. (ed.) (1988): Websters New World Dictionary o] American English; Third College Edition (Websters New World, New York, N. ¥.), p. 1084 Nico, J. S., Gidley, D. W., Rich, A., and Zitzewit~, P. W. (1990): Phys. Key. Letters 65 1344 Nico, J. S., Gidley, D. W., Skalsey, M., and Zitzewitz, P. W. (1992): Mat. Sci. Forum 105-110 401 Ore, A. and Powell, J. L. (1949): Phys. Rev. 75 1696 Page, L. A. and Heinherg, M. (1957): Phys. Rev. 106 1220 Ruark, A, E. (1945): Phys. Rev. 68 278 Schneider, H., Seeger, A., Siegle, A., Stoll, H., Billard, I., Koch, M., Lauff, U., and Major, J. (1993): J. Physique IV $ C4 69 Seeger, A., Major, J., and Banhart, F. (1987): phys. stat. sol. (a) 102 91 Seeger, A. and Banhart, F. (1990): Helv. Phys. Acta 65 403 Smilga, V. P. and Belousov, Yu. IV[. (1994): The Muon Method in Science (Nova Science Publ., Commack, N. Y.) Stritzke, O. (1991): Dr. rer. nat. thesis, Universit~t Stuttgwrt yon Busch, H., Thirolf, P., Ender, Ch., Habs, D., K~ck, F., Schulze, Th., and Schwalm, D. (1994): Phys. Letters B 825 300 Wheeler, J. R. (1946): Ann. N. ¥. Acad. Sci. 48 221 Wolfensteln, L. and Ravenhall, D. G. (1952): Phys. Rev. 88 279 Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D., and Hudson, R. P. (1957): Phys. Rev. 105 1413 Wfirschum, R. and Seeger, A. (1995): Z. Phys. Chem NF 192 47 Yennle, D. B. (1992): Z. Phys. C 56 813

Light-Induced Liberation of Atoms and Molecules from Solid Surfaces Frank Tr'ager Fachbereich Physik, Universitnt Kassel, Heinrich-Plett-StraBe 40, D-34132 Kassel, Germany

I

Introduction

Atoms located at the surface of solid materials continuously try to realize their full genuine identity by escaping from the condensed state into the gas phase. Owing to thermal vibration, each atom makes about 1012 attempts per second to desorb from the surface potential. Stlange enough, its neighboring fellow prisoners, though also trying to escape, contribute significantly to the height of the potential barrier that keeps all of them in captivity. From time to time an atom is lucky enough to gain sufficient kinetic energy for bond breaking thus encouraging the others to keep trying. However, instead of simply letting the imprisoned atoms suffer their undeserved fate, one can offer help, for example, by irradiating the surface with light. Absorption of light, in particular laser radiation, increases the surface temperature together with the average kinetic energy with which the atoms oscillate around their equilibrium positions in the surface potential. As a result, the probability for bond breaking is greatly enhanced and the escape is called "thermal desorption". Furthermore, surface atoms can surmount the "fence" keeping them in confinement if the light excites anti-bonding electronic states localized in their immediate spatial vicinity. This causes photochemical bond breaking, i.e. desorption without thermalization of the absorbed photon energy. In fact, such non-thermal liberation processes are particularly interesting and, as will be described in the course of this paper in more detail, allow one to treat the surface atoms individually. This means that those of them located at certain binding sites of the surface can be liberated selectively by appropriately choosing the wavelength of the incident light. The course of non-thermal desorption reactions, however, turns out to be quite complicated, since the atoms remaining behind on the surface do their very best to jointly recapture the refugees: eleclron-electron and electron-phonon coupling quench the induced repulsive electronic excitation on an ultrafast time scale and force most of the escaping atoms back into the captivity of the solid. As a consequence, only a single atom is liberated per about 104 stimulated anti-bonding electronic excitations. Nevertheless, large desorption rates are readily achieved since the optical absorption cross sections can be quite high and lasers easily provide fluences in excess of 1016 photons per second.

424

Frank Trfiger

2 Thermal and Non-Thermal Desorption Processes along the lines of the two scenarios outlined above, i.e. thermal and nonthermal bond breaking, have been studied extensively in the past with particular emphasis on the mechanisms of photochemical desorption. Thermal desorption is characterized by an exponentially growing signal together with an increase of the average velocity of the detached species as a function of laser fluence. Conversely, the desorption rate increases linearly with laser fluence in non-thermal reactions and the kinetic energy is independent of fluence (as long as multi-photon processes can be excluded). The understanding of these desorption mechanisms is an issue of central importance for basic science as well as a large variety of applications. Thus, desorption of molecular species from many different surfaces and removal of substrate atoms from semiconductors, metals and insulators by laser radiation has been studied (see e.g. [1]). It turns out that metal atoms like Au, Ag, AI, K or Na can even desorb in a non-thermal reaction [2-15], thermal evaporation only coming into play for high fluences [12-15]. In view of the large body of publications on laser-induced desorption one is tempted to believe at first sight that the classical process of thermal evaporation does not attract much interest any more and our understanding of non-thermal surface reactions should be advanced enough to explain all of their features in a microscopic picture. Closer examination of these issues, however, reveals that essential details regarding the rupture of the surface bond are still unresolved and call for novel experimental and theoretical studies. For example, one might ask • from which binding sites desorbing atoms come off preferentially, • whether or not they make joint attempts to escape causing detachment of dimers, trimers, etc., • if the mass distribution of the desorption products depends upon the heating rate, or • if evaporation is influenced by the duration of a certain temperature rise, to mention only few of a large variety of open questions. Thus, despite a large number of experimental and theoretical investigations on light-induced desorption to be found in the literature, our present understanding of such reactions is still rather limited. In order to clarify in more detail how atomic or molecular species are liberated from a surface as a result of heating or electronic excitation, experiments along the following lines are of particular value: • The incident light fluence should be kept as low as sufficient signal-to-noise ratio permits so that only subtle changes of the surface morphology are induced. Desorption studies under such conditions of low reaction rate open the door to correlate the desorption behavior with details of the electronic and geometric structure of different binding sites of the surface. Furthermore, light intensities not sufficient to remove large quantities of material per laser pulse ensure that gas phase collisions of the detached species can be avoided and their genuine kinetic energy distribution determined. • It has been found in previous experiments that atoms or molecules are preferentially desorbed from "defects" of the surface, i.e. from sites with particularly low coordination numbers [3,16]. For this reason the preparation of surfaces with the largest

Light-Induced Liberation of Atoms and Molecules from Solid Surfaces

425

possible number of such binding sites is essential. In other words, the surface under study should exhibit a pronounced but reproducible atomic "roughness" to provide a detectable desorpfion signal even at low photon fluences. The present paper gives an overview of recent desorption studies performed with the objective to elucidate thermal as well as non-thermal desorption processes in more detail. It particularly focuses on the question which role the surface structure plays on a microscopic scale. Surfaces with large roughness, i.e. atoms with different binding sites of low coordination number, have been prepared by the deposition of metal atoms on dielectric single crystal surfaces held at low temperature. For this purpose sodium adsorbed on quartz served as a model system. The deposited atoms form small particles, with the surface defects acting as nucleation centers, a process known as Volmer-Weber growth mode in thin film epitaxy [17]. If the deposition is continued further and further, the clusters finally grow together into a thin film. As will be described in more detail below, laser-induced desorption of atoms from small particles is particularly useful for the investigations reported here, since their surfaces in fact offer the above mentioned considerable number of sites with low coordination numbers from which desorption occurs preferentially. Another essential argument is that the relative number of such "defect" sites can be varied intentionally by changing the size and shape of the particles. Furthermore, formation of small particles on the surface of dielectric substrates turns out to ensure well-defined conditions in the sense that the roughness and therefore the desorption signals detected in subsequent experiments are reproducible. Figure 1 visualizes the sample preparation and the topography of the metal surfaces prepared along the lines indicated above. Whereas the upper part of the figure illuslrates deposition of atoms from the gas phase and subsequent growth of small particles, the lower half provides a magnified image of the cluster surface. The figure displays adatoms and steps on plane terraces, examples of binding sites with low coordination number that can be regarded as individual building blocks of surface roughness. As shown schematically in Figure 2 electronic excitations that are localized at these sites can be driven by an incident light field and it is intuitively clear that such excitations do not only play an essential role in promoting desorption but are also associated with distinct resonances. Their positions and widths reflect the properties of each binding site of the surface. Of course, other electronic excitations, particularly (non-localized) surface plasmon polaritons, can also be stimulated. In fact, the localized resonances play their most essential role if they spectrally overlap with the surface plasmon. In such a case the localized single-electron excitations are driven with exceptionally high rate because of the well-known pronounced enhancement of the electric field at the particle surface that is associated with the collective resonance [18,19]. At this point the question naturally arises how desorption experiments can distinguish between detachment of atoms from different binding sites of the surface, for example from those of Fig. 1, and how their electronic properties can be determined. For this purpose, the total number of desorbed atoms or molecules, the kinetic energy dist-

426

Frank Tr/iger

ributions of the ejected species and the dependence of the desorption signal on the irradiation time or the number of feed laser pulses can be measured as a function of laser wavelength and fluence. Furthermore, preparation of the surface as described above allows to increase the surface roughness intentionally by depositing an increasing number of atoms or diminish it by annealing at different substrate temperatures. beamof metalatoms

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Fig. 2. Schematic representation of electronic excitations (shaded areas) localized at the defect sites shown in Fig. 1.

Light-Induced Liberation of Atoms and Molecules from Solid Surfaces

3

427

Experimental

The experimental arrangement used for desorption studies is shown schematically in Figure 3. It basically consists of an ultrahigh-vacuum system with the sample, two lasers for stimulating desorption and photo-ionizing the desorption products, and a time-of-flight mass spectrometer. A quartz crystal was used as a substrate for small Na particles from the surface of which desorption was stimulated. Being mounted on a manipulator, it could be cooled to 80 K and heated to about 750 K for cleaning. A thermal atomic beam of Na atoms with well-defined constant flux was generated by a Knudsen cell and directed onto the substrate in order to deposit a predetermined coverage of atoms onto the surface held at 80 K. The flux of the atomic beam was on the order of 1012 atoms per second and cm 2. It was measured with a quartz crystal microbalance. The deposited atoms assembled into small particles by surface diffusion and nucleation and, at large coverages, formed a continuous, rough metal film. As mentioned above, the experiments reported here were carried out at low coverage, i.e. under conditions where the generated clusters on the surface are well separated and far from growing together. The defects of the quartz surface being "decorated" during the nucleation, the number density of the clusters remained essentially constant during their growth. Consequently, the coverage, i.e. the number of deposited atoms per cm 2, and the average particle size are unequivocally related to each other. In the following the mean cluster radius and the metal coverage will therefore often be used synonymously. For example, at a typical defect density of the quartz surface of 1010/cm2 the coverage of 5.6x 1014 atoms/cm2 (about one monolayer of Na) corresponds to an average cluster radius of 8 nm. As mentioned above, Na adsorbed on quartz served as model system for the desorption studies described below. Sodium is an ideal candidate for such measurements since the coverage, after completion of an experiment, can be evaporated completely at a temperature of only about 300 K. This made it possible to use the same substrate repeatedly and thus ensured reproducible preparation of the sample as well as reproducible desorption signals. Furthermore, surface plasmon excitation in Na clusters can be stimulated readily with visible light and opens up the possibility to characterize them with regard to their size and shape by optical spectroscopy [19,20]. After preparation and characterization the samples were irradiated with the fundamental or higher harmonics of a Nd:YAG laser at k = 1064, 532, 355 and 266 nm in order to stimulate desorption from the surface of the Na particles. The angle of incidence was 50 ° with respect to the surface normal and the pulse duration about 7 ns at a repetition rate of 10 Hz. Laser tluences ranging from 0.1 up to 150 mJ/cm 2 were used. At a distance of 20 mm in front of the sample the detached species were ionized with the light of an excimer laser operating at ~. = 193 or 248 nm. Its beam was focussed with a magnesiumfluoride lens to a diameter of 100 ~tm in the ionization region. The generated ions passed the time-of-flight spectrometer for identification of the mass and were finally detected with a secondary electron multiplier. The time dependent ion signal was processed with a boxcar integrator and stored in a computer. By

428

Frank Tr~iger

UHV

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Fig. 3. Experimental arrangement used for the study of laser-induced desorption of atoms and molecules under ultrahigh vacuum conditions. Desorption is stimulated by shining light of a Nd:YAG laser with ~. = 266, 355, 532 or 1064 nm onto the sample surface. The desorbed species are ionized by the beam of an excimer laser with ~. = 193 or 248 nm that is fired after a variable delay time. The mass distribution of the desorption products is determined by a time-of-flight mass spectrometer.

varying the delay time between the two laser pulses used for desorption and ionization the time-of-flight distributions I(t) of the desorption products could be determined. Kinetic energy distributions f(E) were derived from the time-of-flight spectra by using the relation f(E) = C I(t) t 2 where C denotes a constant and t is the variable delay time between the two laser pulses [21]. Data were taken for different fluence, frequency and polarization of the light used for desorption but also in dependence of the metal coverage, i.e. for different surface roughness.

4

Desorption of Na Atoms

As an example, Fig. 4 displays a series of three kinetic energy distributions obtained by desorbing atoms with laser light of ~ = 355 rim. The laser fluence was held constant at 6.6 mJ/cm 2 and the substrate temperature was kept at T = 80 K, the value at which the sample had been prepared. Measurements were performed for increasing

Light-Induced Liberation of Atoms and Molecules f~om Solid Surfaces

429

Na coverage on the quartz substrate, i.e. different surface roughness. Surprisingly, at very low integral coverage of 2.2x 1014 atoms/cm 2, which corresponds to an average particle size of Ray = 5 nm, two distinct maxima appear in the kinetic energy distribution. They are located at Ekin = 0.16 and 0.33 eV (Fig. 4a). The distribution can be approximated by two Gaussians, the sum of which is represented by the solid line. If the coverage is increased to 3.8x1014 atoms/cm 2 (average cluster size of Ray = 7 nm) the 200

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Frank Trfiger

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integral desorption rate grows. Simultaneously, the amplitudes of the two maxima change (Fig. 4b). Their positions, i.e. the most likely values of the kinetic energies, however, remain constant. The same development continues upon further growth of the coverage to 1.1xl015 atoms/cm 2 (Ray = 10 nm cluster size). At this point the maximum at Ekin = 0.16 eV is by far predominant in comparison to the second one at Ekin = 0.33 eV (Fig. 4c). In addition to the above measurements, the wavelength of the light stimulating desorption has been varied. For this purpose the second (~ = 532 nm) and fourth harmonic (Z, = 266 nm) of the Nd:YAG laser were also used to irradiate the metal particles. The experimental findings are as follows. For both wavelengths, i.e. ~ = 266 and 532 nm, only atoms with kinetic energies of around Ekin = 0.16 eV are detected. An example illustrating the results of these measurements is shown in Figure 5. The single maximum found in the distributions is identical to the low-energy signal recorded for ~, = 355 nm at the same value of Ekin. The bimodal distribution described above is not observed here. As can also be seen from Fig. 5, the position of the maximum is independent of coverage. Furthermore, it does not depend upon the fluence and polarization of the laser light. As for ~, = 355 nm the integral desorption rate depends linearly on the fluence of the light stimulating desorption. Totally different results are obtained if the measurements are repeated with laser light of ~, = 1064 nm, i.e. by shining in the fundamental of the Nd:YAG laser. For this wavelength desorption can only be observed for large fluences above approximately 80 mJ/cm 2 and for large coverages exceeding 9.0x1015 atoms/cm2. Above the "threshold" value of 80 mJ/cm 2 the desorption rate increases exponentially. Furthermore, the measured kinetic energy distributions do not show maxima.

Light-Induced Liberation of Atoms and Molecules flom Solid Surfaces 431 As mentioned above the surface roughness has been gradually reduced by heat treatment of the samples. For this purpose the substrate temperature was raised slowly by using an electrical heater incorporated in the sample holder. Following a predetermined temperature increase, the sample was cooled down to the initial value of 80 K. Subsequently, desorption measurements were carried out. The results of these studies can be summarized as follows. If the temperature rises by about 30 K, the peak at Ekin = 0.16 eV disappears; if the increase is chosen to be larger than 50 K the maximum at Ekin = 0.33 eV also vanishes. Obviously, the two maxima exhibit different "annealing" behavior.

5

Desorption of Na-Dimers

As mentioned above, an essential issue is to investigate if atoms can more successfully escape jointly from the surface potential than on their own. In order to detect dimers and possibly even larger liberated aggregates, "soft" ionization must be accomplished to prevent dissociation of the desorbed molecules and unambiguously identify their masses. This has been realized here by using laser light with ~ = 248 nm for ionization. The corresponding photon energy lies only little above the dissociation threshold of Na 2 and guarantees minimal fragmentation. 0.10

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432

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Tr~ger

In our measurements the total coverage and the laser fluence were varied systematically. At low laser fluences of ~ = 355 nm a maximum is observed for all kinetic energy distributions of the desorbed dimers. The data are compiled in Fig. 6. It displays the positions of the maxima of the distributions for five values of the laser fluence and four different coverages. The most essential result is that the positions of these maxima at Ekin = 0.06(1) are identical within the experimental error and neither depend on the particle size nor on the laser fluence. Furthermore, the integral desorption rate depends linearly on the laser fluence (~, = 355 nm) as long as it is kept at low values, Fig. 7. Above approximately ~ = 10 mJ/cm 2, however, the rate grows exponentially. At the same time, the shape of the kinetic energy distribution changes from a curve exhibiting a maximum to an exponential decrease. If light with ~, = 532 or 1064 nm is used to stimulate desorption, the measured distributions fall off rapidly, do not show maxima and change as a function of fluence. Moreover, the total number of dimers desorbed per laser pulse rises sharply with increasing fluence and follows an exponential.

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Light-Induced Liberation of Atoms and Molecules from Solid Surfaces

6

433

Discussion

6.1 Liberation of Atoms We first conclude from the experimental data, that desorption of atoms for the three wavelengths of ~. = 266, 355 and 532 nm and low fluences obviously constitutes a non-thermal reaction [12]. This is supported by the shape of the kinetic energy distributions of the detected atoms, the absolute energies of 0.16 and 0.33 eV as well as by their independence on the laser fluence. Furthermore, the Na particles only experience a moderate temperature rise under the conditions quoted above. Also, the transition from non-thermal to thermal desorption can be followed and is reflected in the transition from a linear to an exponentially growing dependence as a function of laser fluence. Further, the kinetic energy distribution changes. For laser light with ~, = 1064 nm, on the other hand, the measured kinetic energy distributions are always characteristic of thermal bond breaking and desorption is only observed if the fluence is about two orders of magnitude larger than for the other wavelengths. Secondly, we notice that the measured kinetic energy values of Ekin = 0.16 and 0.33 eV are independent of coverage, i.e. particle size, and, as mentioned above, of the light intensity. The two energies actually remain constant under variation of all experimental parameters. Only the desorption efficiency of the two channels changes if, for example, the total coverage of metal atoms on the substrate and therefore the surface topography varies. We conclude that both values of the released kinetic energies OfEkin = 0.16 and 0.33 eV obviously constitute fingerprints of the metal surface. The different annealing behavior further indicates that desorption of atoms takes place from well defined and different binding sites. They predominantly exist at low temperature. The variable relative amplitude of the maxima at Ekin = 0.16 and 0.33 eV reflects a change of their concentrations as the metal coverage on the dielectric substrate is increased. Since both non-thermal signals disappear upon heating, these sites must have particularly low coordination numbers. The electronic wavefunctions not being totally delocalized at these "defects", the electron-phonon-coupling is reduced in strength, even though prevailing, making possible desorption with finite probability. As mentioned already, light with ~. = 355 nm stimulates desorption with a bimodal kinetic energy distribution whereas a single maximum is observed if the cluster surface is irradiated with light of ~. = 532 nm. This indicates that desorption with the two energies is characterized by different wavelength dependencies and provides first experimental evidence of specific electronic properties of different binding sites. Unfortunately, the resonances associated with the defect sites of the metal surface are not known and the electronic and geometric signatures of special surface sites embedded in the kinetic energies of the desorbed atoms cannot be readily observed with standard surface science tools like photocmission. Also, these sites from which desorption is possible cannot be identified by the classical method of temperature programmed desorption (TPD): before the temperature is reached where evaporation of the particles commences, the sites responsible for non-thermal desorption have disappeared because of annealing. Also, the modern scanning probe microscopies cannot (yet) be

434

Frank Trfiger

utilized to characterize surfaces with such binding sites in detail. Thus, the sites responsible for desorption cannot be identified unambiguously at present. Still, measurements with different polarization of the laser light provide some hints from where the atoms could be detached. For example, p-polarized radiation with ~, = 532 nm stimulates desorption more efficiently than s-polarized fight. This indicates that p-polarized laser radiation predominantly interacts with a dipole in the direction normal to the surface of the oblate clusters. It could therefore be that the maximum at Ekin = 0.16 eV is due to adatoms detached from terraces of the particle surface (see also [22,23]). The experimental results allow us to suggest a desorption mechanism [12] along the fines of the Menzel-Gomer-Redhead scenario [24,25]. Laser fight is absorbed and, depending on the applied wavelengths of k = 266, 355 or 532 nm, initially populates different eleclxonic levels. The energy balance of the desorption reaction indicates that each of these excitations is followed by rapid relaxation into a lower-lying intermediate state, i.e. the absorbed photon energies are only partially converted into the kinetic energies of the liberated atoms or needed to supply the binding energy of a surface atoms of about 0.7 eV [26]. The intermediate state being antibonding, the atoms are accelerated away from the surface and gain kinetic energy E A. The value of E A depends upon the lifetime of the repulsive state which in turn determines the released kinetic energy. For k = 1064 nm, on the other hand, the antibonding state cannot be reached. As a result, the absorbed photon energy can only be converted into heat causing thermal desorption.

6.2 Escape of Na-Dimers Laser-induced desorption of Na dimers is not only readily detectable but also takes place with surprisingly large rate. Under certain experimental conditions the rate of dimers desorbed per laser pulse can by far surmount the rate of atoms. For laser fight of k = 532 nm the measured kinetic energy distributions are exponential and characteristic of thermal bond breaking. Furthermore, the experimental data prove that desorption of Na 2 with ~, = 355 and low fluences (~ < 8 mJ/cm 2) is non-thermal. This is supported by the shape of the kinetic energy distributions of the detected dimers, the linear fluence dependence as well as by the constant value of the most likely kinetic energy OfEkin = 0.06(1). This value turns out to be independent of coverage, i.e. particle size, and, as mentioned above, of the fight intensity. Furthermore, the Na particles only experience a moderate temperature rise under the conditions quoted above. Also, the transition from non-thermal to thermal desorption as a function of laser fluence can be identified, Fig. 7. It is reflected in the transition from a linear to an exponential dependence of the integral desorption rate and the simultaneous change of the kinetic energy distributions. Table 1 summarizes the results obtained for the applied wavelengths and fluence ranges, gives the type of reaction and compares the observed desorption rates of Na atoms and Na 2 molecules. Further details can be found in ref. [14] and [15].

Light-Induced Liberation of Atoms and Molecules from Solid Surfaces

435

fluence regime

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< 15 mJ/cm 2

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thermal

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wavelength

532 nm

1064 nm

Table 1. Synopsis of laser-induced desorption of Na atoms and Na 2 dimers for different wavelengths and fluence ranges of the applied laser radiation

The comparable numbers of dimers and atoms detached in the non-thermal channels (Tab. 1) indicates that the dimers constitute highly stable subunits of the metal surface that can be released in a photochemical reaction without breaking into individual atoms. This seems to imply that the dimers are oriented on the surface in such a way that the axis connecting the two Na atoms is not parallel to the substrate surface but rather points into the direction of the surface normal. The repulsive electronic excitation preceding desorption must therefore be localized in the region between the dimer and the remaining substrate atoms, Fig. 1. On the other, the dimer rate is even larger than the rate of desorbed atoms in thermal desorption. In contrast, the number of removed Na dimers only amounts to about 5 - 10 % of the rate of desorbed monomers in a classical thermal desorption experiment (TPD) with low heating rate (10 K/s) [27]. This can be understood as follows. Before the temperature is reached where evaporation of the particles in TPD commences, the sites responsible for laserinduced thermal desorption have disappeared because of annealing. Laser heating, on the other hand, with elevated temperatures only persisting for at most several microseconds obviously does not last long enough for mass transport by diffusion and annealing to occur. This gives rise to the unusual observation that the composition of the desorption products depends upon the heating rate. We have also found recently [28] that surface defects are not only essential in promoting non-thermal desorption of Na atoms but also play a decisive role in thermal desorption of Na-dimers. In fact, two special binding sites from which the dimers are removed can be distinguished experimentally.

436

Frank Tr~ger

8 Conclusions The experiments summarized in this article have shown that atoms and dimers can be liberated from solid surfaces in a variety of ways and it is particularly interesting that conditions exist under which joint attempts of atoms to leave their prison in the surface potential, i.e. desorption of dimers, can by far be more successful than escapes of individual atoms. Applications might consist in a novel method to generate beams of Na-dimers. Surface defects have been found to play a decisive role in bringing about non-thermal as well as thermal desorption. Two such special surface sites can be distinguished in photochemical bond-breaking of atoms. This is reflected in well-defined kinetic energies carried by the liberated atoms after escaping into the gas phase. In future experiments the non-thermal desorption reaction as well as the interplay between thermal and non-thermal bond-breaking will be studied in more detail. So far, only laser wavelengths that lie relatively far apart have been used to stimulate desorption. In order to characterize the localized electronic resonances preceding nonthermal desorption in more detail, use of continuously tunable laser radiation is planned. This step is also essential in order to fully exploit the selectivity of photochemical bond-breaking to deplete certain predetermined surface sites. Further, the angular distribution of the desorbed atoms and dimers should give further information on the desorption mechanism. A very interesting question is also, whether or not desorption of aggregates X n with n > 2 can be stimulated. In order to address this problem, use of longer ionization wavelengths is essential to prevent photofragmentation. Furthermore, extension of the experiments to other metals is under way.

Acknowledgements The author would like to thank all his coworkers and colleagues for a very fruitful and pleasant collaboration. Part of the work presented here was supported by the Deutsche Forschungsgemeinschaft and the Fond der Chemischen Industrie.

References [1] [2] [3] [4] [5] [6]

Proc. 6 th International Workshop Desorption Induced by Electronic Transitions, DIET VI, Nucl. Instnma. Meth. Phys. Res. B 101 (1995) P.D. Brewer, M. Sp~ith, M. Stuke: Mat. Res. Soc. Symp. Proc. 334, 245 (1994) W. Hoheisel, M. VoUmer, F. Tr~iger. Phys. Rev. B 48, 17463 (1993) W. Hoheisel, K. Jungrnann, M. Vollmer, R. Weidenauer, F. T~ger: Phys. Rev. Lett. 60, 1649 (1988) M.J. Shea, R. N. Compton: Phys. Rev. B 47, 9967 (1993) I. Lee, J.E. Parks II, T.A. Callcott, E.T. Arakawa: Phys. Rev. B 39, 8012 (1989)

Light-Induced Liberation of Atoms and Molecules from Solid Surfaces [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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I. Lee, T.A. CaUcott, E.T. Arakawa: Phys. Rev. B 47, 6661 (1993) A.M. Bonch-Bruevich, T.A. Vartanyan, Yu.N. Maksimov, S.G. Przhibel'skii, V.V. Khromov: Soy. Phys. JETP 70,993 (1990) A.M. Bonch-Bruevich, T.A. Vartanyan, Yu.N. Maksimov, S.G. Przhibel'skii, V.V. Khromov: Surf. Sci. 307-309, 350 (1994) H.S. Kim, H. Helvajian: J. Chem. Phys. 95, 6623 (1991) F. Balzer, R. Gerlach, J.R. Manson, H.-G. Rubahn: J. Chem. Phys. 106, 7995 (1997) T. G6tz, M. Bergt, W. Hoheisel, F. Tr/iger, M. Stuke: Appl. Phys. A 63,315 (1996) T. G6tz, M. Bergt, W. Hoheisel, F. Tr/iger, M. Stuke: Appl. Surf. Sci. 96-98, 280 (1996) J. Viereck, M. Stuke, F. Tr~iger: Surf. Sci. 377-379,687 (1997) J. Viereck, M. Stuke, F. Tr~iger: Appl. Phys. A 64, 149 (1997) N. Nishi, H. Shinohara, T. Okuyama: J. Chem. Phys. 80, 3898 (1984) J.A. Venables: Surf. Sci. 299/300,798 (1994) C.F. Bohren, D.R. Huffman: Absorption and Scattering of Light by Small Particles, Wiley (1983) M. Vollmer, U. Kreibig: Optical Properties of Metal Clusters, Springer Set. Mat. Sci. 25 (1995) T. G/Jtz, W. Hoheisel, M. Vollmer, F. Tr~iger: Z. Physik D 33, 133 (1995) G. Scoles: Atomic and Molecular Beam Methods, Oxford University Press (1988) M.D. Thompson, H. B. Huntington: Surf. Sci. 116, 522 (1982) N.D. Lang: Surf. Sci. 299/300,284 (1994) D. Menzel, R. Gomer: J. Chem. Phys. 41, 3311 (1964) P.A. Redhead: Can. J. Phys. 42, 886 (1964) M. Vollmer, F. Tr~ger: Z. Physik D 3,291 (1986) T. GOtz, M. Bergt, W. Hoheisel, F. Tr~iger, M. Stuke: Appl. Surf. Sci. 96-98, 280 (1996) J. Viereck, F. Stietz, M. Stuke, T. Wenzel, F. Tr~ger: Surf. Sci. 383, L749 (1997)

On t h e S h o u l d e r s of G i a n t s E a r l y H i s t o r y of H y p e r f i n e S t r u c t u r e Spectroscopy. For G i s b e r t zu P u t l i t z Peter Brix Kastellweg 7, D-69120 Heidelberg, Germany Ten years ago, the University of Heidelberg celebrated its 600th anniversary. The "Festschrift" of six thick volumes starts with an introduction by the Rector magnificus (Fig. 1). He wrote: More than in other fields it holds for science that new insights can only be developed from the knowledge of the past . . . . This is expressed by the motto of the anniversary: "From Tradition into the Future". In the spirit of this motto, I was asked to say something about the tradition in the life of the physicist Gisbert zu Putlitz. He did not wish to be praised, said the organizers. I should talk about the environment from where he started his scientific career, his teacher and the history of the field. The field is hyperfine structure spectroscopy. Isaac Newton said: I have seen farther than others because I stand on the shoulders of giants. This familiar quotation - which I chose to make you curious - is usually connected with Newton's name. It can be tracked back, however, to at least the 12th century. The aphorism illustrates that every generation of scientists needs the work of former ones. Newton separated the colours of the sunlight and introduced the word spectrum into physics. On his shoulders stood the giant Joseph Fraunhofer. He measured and used the dark lines in the spectrum of the sun. The famous German poet Goethe disliked them: Nature is beautiful and simple, he said, the physicists seek these artificial things. I t took 45 years till, in 1859, Gustav Robert Kirchhoff recognized the origin of the Fraunhofer lines: It could, e.g., not happen by chance that 60 of these coincided with 60 iron emission lines measured in the laboratory. Kirchhoff and the chemist Robert Wilhelm Bunsen jointly developed the m e t h o d of spectrochemical analysis started here in Heidelberg. Spectroscopy became a rapidly growing field. However, the meaning of all the lines was not known until 1913, when Niels Bohr formulated his model of the atom. A textbook of 1897 states: 'The spectroscope not only informs us about the chemical constitution of the farthest stars but also about their velocity in the line of sight'. Nobody could know that optical spectroscopy had also begun to detect fine details deep in the center of atoms. T h a t was in 1892, when Albert Abraham Michelson at Clark University in Worcester, Massachusetts

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Fig. 1. Reproductions from the "Festschrift" of the 600th anniversary of the Ruprecht-Karls-UniversitKt Heidelberg.

discovered and measured the hyperfine structures (hfs) of some spectral lines with his interferometer. For the green resonance line (5351~) of thallium, Michelson's Fourier analysis of his "visibility curve" yielded the his shown in Fig. 2. The two strong components belong to T1 205, the others to TI 203 (relative abundances 7:3). Both isotopes have nuclear spin 1/2 (theoretical intensity ratios 3:1) and nearly the same nuclear magnetic moment. The magnetic hfs splitting is that of the upper (7s) level, the 6p2P3/2 -splitting being very small. The obvious isotope shift (IS) between 203 and 205 originates from slightly different nuclear radii. Had someone told Michelson all that, he would have shrug his shoulders. Most of the words were completely meaningless at that time. This example shows that, occasionally, signals of something new may be visible long before they can be understood. In this case, it took 39 years.

On the Shoulders of Giants

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Dated March 10, 1931, two months before Michelson died, the interpretation just given was published by Hermann Schiller (Astrophysics Observatory Potsdam) together with the English student I.E. Keyston. They had made new hfs measurements with the liquid air cooled hollow cathode which Schiller had introduced into hfs spectroscopy in 1928. Hermann Schiller (1894 - 1964) stood on the shoulders of the spectroscopist Friedrich Paschen. Independently and practically at the same time (March 28, 1931), the discovery of the IS of heavy elements (as different from the mass dependent shift in light elements) was announced by Hans Kopfermann in Berlin-Dahlem. He demonstrated the IS of lead with radiogenic isotopes. In the Fabry-Perot photographs of the PbII line 4245.~ (Fig. 3), the strong component for ordinary lead (left) is due to Pb 208, the exposure with uranium lead (right) identifies Pb 206. The IS between 206 and 208 measures a 0.2% increase in nuclear rms charge radius (12 am): fascinating, isn't it? An extensive theoretical treatment of the IS "on the hypothesis of small changes in nuclear radii" was published already in 1932 by Gregory Breit in New York. m i m

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Kopfermann's life stations are shown in Fig. 4. His continuing interest in the IS led, in 1947, to the discovery of intrinsic nuclear quaclrupole moments in post-war G6ttingen. The main tool was again a Fabry-Perot interferometer, the standard instrument for hfs studies at that time. Figure 5 shows the hfs of the Sm I line 6589/~ ( A S is 0.153 "mK'). We got the idea that the "jump" in the IS between Sm 150 and 152 (first noticed by Schiller and Schmidt in 1934) arose from an increase in nuclear deformation: A crazy assumption for I--0 nuclei at that time. A good idea is one thing, but the main thing is the test whether the idea is good. For this we had to evaluate the increase in nuclear rms radius from the measured hfs. That was very difficult because of the complicated spectrum of the rare earth element samarium. To cut a long story [6] short: I measured the IS in about 100 Sm-lines, made assumptions about screening effects on closed s2-shells and applied semi-empirical methods. Finally we could believe that our interpretation was correct, because the postulated intrinsic deformations of Sm 150 and 152 agreed with the deformations of Eu 151 and 153 as derived from the known spectroscopic quadrupole moments.

On the Shoulders of Giants

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By this, I wanted to stress that all the hfs spectroscopists stood on the shoulders of atomic spectroscopists who had previously measured and analyzed the spectra. This should not be forgotten. I single out Heinrich Kayser (1853 - 1940), the "master of spectroscopy". It is a pity that the unit "Kayser"(K) for the wave number is now obsolete. Kayser stood on the broad shoulders of the giant Hermann yon Helmholtz. The magnetic hfs was first interpreted by Wolfgang Pauli (1924) and Back and Goudsmit (1927). The importance of the measured nuclear spins and magnetic moments became evident through the well known diagram published in 1937 by Theodor Schmidt (1908 - 1986) with the "Schmidt-lines". The spin orbit coupling of the valence nucleons in a single particle model as read from his diagram played an important role for the shell model of nuclei by Maria GSppert-Mayer and Haxel, Jensen and Silss (1949). Kopfermann was engaged in the discussions, and the relation between nuclear moments and shell structure fascinated him throughout his life. Figure 6 is a Schmidt diagram compiled by Kopfermann in 1954. For 52 of the 60 nuclei included, the spins had been first determined from hfs-patterns obtained with optical high resolution spectrographs - many by himself and his students. Collecting precious stones may, as in this case, be quite important, if, in the end, they yield a wonderful mosaic. Spectroscopic nuclear quadrupole moments were discovered at the Potsd a m astrophysicM observatory. There a collection of rare earth samples existed which had been used to study line spectra of stars. In the spring of 1935, Schiller and Schmidt put europium into their hollow cathode. The hfs of three resonance lines showed that both isotopes 151 and 153 had spins 5/2. But the measured intervals ("gemessen" in Fig. 7) did not follow the interval rule exactly ("theoretisch" in Fig. 7). The deviations were larger for 153 (broken lines in Fig. 7) with the smaller magnetic moment. T h a t excluded

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level perturbations: Schiller and Schmidt had found a deviation of the nuclear charge 9 distribution from spherical symmetry. Their paper is dated March 2, 1935. The quantum mechanical evaluation of the nuclear electric quadrupole moments was published already June 1, 1935 by Hendrik Casimir in Leiden. Pure nuclear quadrupole resonances in solids were found 1949 in Kopfermann's institute in G6ttingen by Hans Dehmelt and Hubert Krfiger. Later, optical pumping gave another important new access for studying quadrupole moments. Otto Haxel and Hans ]ensen succeeded in persuading Kopfermann to come to Heidelberg University in 1953. He followed Walther Bothe as director of the "I. Physikalisches Institut'.

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In 1953, the 22 years old physics student Gisbert zu Putlitz also came to Heidelberg. He has told how he managed to join Kopfermann's institute: " T h r e e candidates for a 'Diplomarbeit' were to be distributed on three institutes. Kopfermann did this by drawing lots with matches of different lengths. He did not reMize that I could see their mirror image in a window. I chose him and have never regretted that decision". Please allow me to quote further from zu Putlitz' warmhearted words about his teacher, spoken when the Hans-Kopfermann-Street was named in Garching at the MPI ffir Quantenoptik: "Even as a young physicist, working for my diploma, I had the opportunity to explain my experiments all on m y own to :James Franck or Isidor Rabi, Frisch and Kuhn, and many others. There were always permanent guests in our laboratories. Through the presence of Bothe, Maier-Leibnitz, Gentner, Haxel and :Jensen in Heidelberg we also had the pleasure to participate in the scientific talks of their guests. International Conferences in Heidelberg (Kopfermann's 60th birthday 1955, 7th Brookhaven Conference 1959, Conference on Optical Pumping 1962) gave us young physicists a chance to establish connections with the "wide world". In 1960, the main research activities in Kopfermann's "I. Physikalisches Institut der Universit£t Heidelberg" were (according to his article on the history of physics in Heidelberg since 1945): 1. Nuclear moments by studies of the hfs in atomic spectra. 1.1 Hfs of electric dipole transitions by optical interferometry (Photography and photoelectric registration). Isotope shifts. 1.2 Magnetic dipole transitions between hfs-levels of neutral atoms by HF-methods (Atomic beams). 1.3 Hfs of short lived excited states of neutral atoms by double resonance: Nuclear quadrupole moments. 2. Photonuclear reactions with a 35 MeV betatron.

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Peter Brix

Hans Kopfermann's opus magnum, the second edition of his book "Kernmomente (Nuclear Moments)" (first edition Kiel 1940) appeared 1956. The new chapters had usually been thoroughly discussed in seminars. The last reviews which Kopfermann gave and published inform us about his interests in those years: Uber den heutigen Stand der Kernmomentenforschung; 1954 Nuclear properties obtained from high resolution atomic spectroscopy; 1954 La r~sonance quadrupolaire nucl~aire; 1956 Uber GrSBe und Gestalt der Atomkerne; 1959 Uber optisches Pumpen an Gasen; 1960 Zur Geschichte der Heidelberger Physik seit 1945; 1960 Die Bedeutung der Atomstrahlresonanzmethode ffir die Atomspektroskopie; 1961 A few words about the last talk, given in Vienna 1961: The publication contains the photograph of the "Heidelberger atomic beam resonance apparatus". With that, an old dream had been fulfilled. Kopfermann had always admired the skillful experiments of Isidor Rabi and collaborators. Kopfermann's last publication bears the title: "lJber das Quadrupolmoment des Sr 87-Kernes" (with H. Bucka and G. zu Putlitz) 1962. It used the optical double resonance method of Kastler, Brossel and Bitter. Kopfermann had been fascinated by the pioneering experiments when he was a guest at the MIT in 1950. A review of the method as of 1964 was later written by zu Putlitz [8]. Hans Kopfermann's unexpected death in January 1963 was a great shock. Gisbert zu Putlitz had lost his teacher two month after his doctoral examination. His colleagues and he were suddenly responsible for themselves. We all know how remarkably well they managed. A question often raised is about Kopfermann and the invention of the laser. It is difficult to say something after a third of a century has passed. Instead I quote the obituary of E.E. Schneider in Nature, Nov. 1963. Dr. Schneider, the translator of the "Nuclear Moments" into English, knew Kopfermann well. Hans Kopfermann "worked for his doctorate under the creative guidance of James Franck. His thesis on sensitized fluorescence of lead and bismuth vapour was published in 1924 . . . . " His "first post-doctoral appointment led him to the Kaiser Wilhelm Institut in Berlin-Dahlem... as assistant of Ladenburg . . . . In careful experiments on the dispersion around several neon lines . . . he established (in 1928) the existence of the effect of stimulated emission . . . I n retrospect this work appears as an important landmark on the long road from Einstein's conception of the idea of stimulated emission in 1919 . . . to crystal masers and finally, ending not far from the starting point, to the helium-sensitized neon gas laser with its immense possibilities for research and technologies" [9].

On the Shoulders of Giants

447

James Franck studied in Heidelberg during his first year. Here he met his lifelong friend Max Born. When Born was offered the chair of theoretical physics at G6ttingen University in 1921, he accepted on the condition that Franck be offered a second chair of experimental physics as director of the "II.Physikalisches Institut". It was this chair that Kopfermann later held, and he tried to lead the institute in the spirit of his highly respected teacher. James Franck (1882 - 1964) stood on the shoulders of Emil Warburg (1846 - 1931). Warburg also began his studies in Heidelberg, hearing Bunsen, Kirchhoff and Helmholtz. A bit simplified it can be said that he later stood on the shoulders of Hermann yon Helmholtz (1821 - 1894). Helmholtz worked in Heidelberg from 1858 to 1871. In a speech on the occasion of the 500th anniversary of our university, he asked: "Is it by chance that from these green hills mankind looked for the first time into the unthinkable depth of the universe with an insight into the chemical nature of the stars?" - I have found my way back to Kirchhoff, Bunsen and the tradition of the Ruprecht Karls University. The "green hills" are here, close to the Philosophenweg, the physics institute. Dear zu Putlitz: Your scientific connections embrace the world. But here are your roots. Here, in the tradition and spirit of your teacher Hans Kopfermann, you have continued to study and teach atomic physics. You apply atomic physics methods in modern research. Others now stand upon your shoulders, especially the organizers of this very fine symposium. You stand on Kopfermann's shoulders together with Ernst Often and Herbert Walther close by, and with many others, including the Nobel laureates Wolfgang Paul and Hans Dehmelt. Couldn't these have been your words: Not to have stepped upon the shoulders of such a predecessor would have been an inexcusable r i s k . . . ("nicht auf die Schultern eines solchen Vorg/ingers getreten zu sein, w/ire ein unverzeihliches Wagestfick gewesen . . . ' ) ? T h a t quotation is taken (without any alteration!) from Georg Christoph Lichtenberg (1742 - 1799), professor of physics in G6ttingen. Toward the end of his life, Lichtenberg noted in his "Sudelbuch" (L 935): " . . . Hauptbemfihung der Physiker Nutzen ffir das menschliche Geschlecht. . . . W i r m~issen ...ffir kfinftige Physiker zu arbeiten suchen . . . " (The main effort of physicists should be benefit(s) for mankind. We must try to work for future physicists.) Your central aims as physicist cannot be described better. Since I am not supposed to praise, let me close with a few personal words of thanks. These can never come early enough. As a colleague, I have followed your academic life through four decades. You were never afraid to accept responsibilities and to make necessary decisions. After the glamour of the rectorate, you lectured for beginners. You train future physicists by modern research, bring them into contact with the 'wide world' of science and with industry. T h a t basic and applied research belong together, that is your deep conviction. You came

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to the university with a completed training as mechanic (with honours); one feels that not only the scientists are close to your heart. You have done much for science in Germany, for Heidelberg, the Ruperto Carola, our faculty, the Physics Institute, your group. Please continue to do so. T h a n k you, and all good wishes. (This is a slightly condensed version of a talk that was given with additional pictures) References [1] Michelson, A., Phil. Mag. S. 5, 34, 280 and P1. VI (1892); see also Schiller, H. and J.E. Keyston, Z. Physik 70, 1 (1931). [2] Kopfermann, H., Z. Physik 75, 363 (1932). [3] Brix, P. and H. Kopfermann, Z. Physik 126, 344 (1949). [4] Kopfermann, H., Proc. Rydberg Centenn. Conf. on Atomic Spectroscopy, Lund 1954, p. 43. [5] Schiller, H. and Schmidt, Th., Z. Physik 94, 457 (1935). [6] Brix, P.: Fifty Years of Nuclear Quadrupole Moments. Z. Naturforsch. 41a, 3 (1986) [7] Kopfermann, H.: Kernmomente. Frankfurt a.M. 1956. - Nuclear Moments. New York 1958. [8] zu Putlitz, G.: Determination of Nuclear Moments with Optical Double Resonance. Erg. exakt. Naturwiss. 37, 105 (1965). [9] Schneider, E.E.: Prof. Hans Kopfermann, Nature 200, 403 (1963). [10] Weisskopf. V.: Hans Kopfermann (1895 - 1963). Nucl. Phys. 52, 177 (1964) (With a list of Kopfermann's pubhcations).

References for the figures ([1] to [5]) are followed by a few selected relevant publications written in English.