Advrinces in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 38
Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Mux-Plunk-Institutfiir Quantenoptik Garching bei Miinchen Germany
Editorial Board P. R. BERMAN University of Michigtrn Ann Arbol; Michigan M. GAVRILA E 0. M. Institure vnor Atoom-en Molecuulfysica Amsterdam, The Netherlands M. INOKUTI Argonne Nutionul Laborciroty Argonne, Illinois W. D. PHILIPS National Institute j b r Standurds and Technology Gaithersburg, Maryland
Founding Editor SIRDAVID BATES
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK
Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANK INSTITUT FUR QUANTENOPTIK MUNICH, GERMANY
Volume 38
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Contents CONTKIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Electronic Wavepackets R. R. Jones and L. D.Noordiim I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
11. Rydberg Wavepackets . . . . . . . . . . ......................... 111. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 35 3s 36
IV. Acknowledgments . . . . . . . . . . . . . ......................... V. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chiral Effects in Electron Scattering by Molecules K . Blum und D. C. Thompson 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Chiral Objects and Their Syniietries: True and False Chirality . . . 111. Definitions and Fundamental Symmetries of Spin-Dependent Amp1
1v. V. v1. VII. VIII. IX. X. XI.
Experimental Observahles: Oriented Molecules . . . . . . . . . . . . . . . . . . . . . . . . Experimental Observables: Randonily Oriented Target Systems . . . . . . . . . . . . Experimental Observahles: Attenuation Experiments . . . . . . . . . . . . . . . . . . . . The Physical Cause of Chiral Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical and Computational Details . . . ......................... Results of Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . Experiniental Results . . . . ........................... Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xu. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40 53 58 62 66
I1
Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium Serguri I. Kunorsky und Antoine Weis 1. lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Structure of the Point Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Implantation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Magnetic Resonance Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88 90 95 97 111
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Contents
VI. ConcludingRemark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References . . . , . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 117
Rydberg Ionization: From Field to Photon G. M. Lankhuijzen and L. D. Noordum I. Introduction . , . , . . . . . . . 11. 111. Ramped Field Ionization . IV. Microwave Ionization . . . . . . . . . . . . . ..........._... V. THz Ionization . . . . . . . VI. VII. VIII.
..............
IX. X. References . . . , . . . . . . . . . . . . . . .
121 126 131 135 141 143 146 150 150 150
Studies of Negative Ions in Storage Rings L. H. Andersen, T. Andersen, and P: Hvelplund I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Lifetime Studies of Negative Ions , . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Electron-Impact Detachment From Negative Ions . . . . . . . . . . . . . . . . . . . . . .
IV. Interactions Between Photons and Negative Ions . . . . , . . . , . . . . , . . . . . . . . . V. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155 158 172 185 188 188
Single-MoleculeSpectroscopy and Quantum Optics in Solids
W E. Moernel; R. M . Dickson, und D. J. Norris I. Introduction . . . . . . . . 11. Physical Principle 111. Methods . . . , . . .
IV. Quantum Optics . . . . . . . . . , . . . . . V. Problems and Promise for Room Temp VI. References . . . . . I
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SUBJECTINDbX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF VOLUMES IN THIS SERIES . , ., , ., .., . . .. .... .. ., ....
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193 196 206 22 1 228 232 237 247
Contri butors Nuinhers in parentheses indicate the pages on which the authors’ contributions begin.
L. H. ANDERSEN (155), Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, DENMARK T. ANDERSEN (155), same address as for L. H. Andersen K. BLUM(401, Institut fur Theoretische Physik 1, Universitat Munster. WilhelmKlemm-Strasse 9,48 149 Munster, GERMANY R. M. DICKSON (193), Department of Chemistry and Biochemistry, University of California, San Diego, 9500 Gilman Drive, Mail Code 0340, La Jolla, CA 92093-0340 P. HVHPI.UND (15.5),Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, DENMARK ROBEIUR. JONES( 1 ), Department of Physics, Jesse W. Beams Laboratory of Physics, University of Virginia, McCormick Road, Charlottesville, VA 2 9 0 1.
SEKGUE~ I. KANORSKY (87), Max-Planck Institut fur Quantenoptik, HansKopfermann Str. 1, D-8.5748, Garching. GERMANY G. M. LANKHLHJZEN (121), FOM Institute for Atomic and Molecular Physics Kruislaan 407, 1098 SJ Amsterdam, THE NETHERLANDS. W. E. MOERNER (1 93), Department of Chemistry and Biochemistry, University of California, San Diego, 9500 Gilman Drive, Mail Code 0340, La Jolla CA 92093-0340
L. D. NOORDAM (1 ) (121), FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, THE NETHERLANDS
...
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Contributors
D. J. NORRIS(193), Department of Chemistry and Biochemistry, University of California, San Diego, 9500 Gilman Drive, Mail Code 0340, La Jolla, CA 92093-0340 D. G. THOMPSON (39), Dept. of Applied Mathematics & Theoretical Physics, Queen’s University at Belfast, Belfast BT7 lNN, N. IRELAND
ANTOINE WEIS(87), Institut fur Angewandte Physik, Universitat Bonn, Wegelerstr. 8, D-53115, Bonn, GERMANY
Advrinces in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 38
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ELECTRONIC WAVEPACKETS
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. RydhcrgWavcpackel\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Monitoring Wavepnckel Dyii;imics . . . . . . . . . . . . . . . . . . . . . . . . . . I . Tiinc-Resolved Photoemissioii and Ahsoi-piion . . . . . . . . . . . . . . . 2. Optical Ramsey Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Impulsive Momen~umK e k i e \ a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Atomic Streak Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Additional Expcrimcntal Ccmsltlcrations . . . . . . . . . . . . . . . . . . B . Rydbcrg Wavepocket\ i i n d Classii~ulCot-upondence . . . . . . . . . . . . . . I . Radial Wavepackcis . . . . . . . . . . . . . 2. Angular or Oriented WiivqixcLcis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Stark Wavepackis . . . . . . . . . . . . . . . ................. 4 Continutiin Wavcpackcrs 111 ;I Strong Static Field . . . . . . . . . . . . . . . . 5. Tw,o-Electron Wa\ep:rckci\ . . . . . . . . . . . . . . . . . . . . . . . . . . C . Wavepacket\ Cre;itcil by Strong I ~ \ c Fields r ................... I . D w k Wavepackel\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Wavepaclds Excited DuIlng h,Zultll on loni/:\tioii . . . . . . . . . . . . . 3 . Wa\epackets Excilrd with Mitl-IR t Pulsea . . . . . . . . . . . . . . . . 4. Wavepackets kxciietl uitli T H / Hall’-(‘ycle Pulses . . . . . . . . . . . . . . . 5. Wavepackets Excitctl with GHz klalt-Cycle Pul\es . . . . . . . . . . . . . . . D. Wavepackel Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Conclulolls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV Acknowlcdgincnts . . . . . . . . . . V. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Recent I y. there has been a great deal of cxpcriinental and theoretical interest in wavepacket studies in atomic, molecular, and condensed matter physics as well as in physical chemistry. Using the most general definition, a “wavepacket” in any quantum system is a “non-stationary” state with time-dependent expectation values for one or more operators. The wavepacket can be described tnathematically as a coherent superposition of non-degenerative, stationary-state wavefunc-
2
R. R. Jones ond L. D.Noordarn
tions, T(7,t ) = 2, u&~&?)e-"'v' a.u. The complex amplitudes, a<,, are constant in the absence of any time-dependent perturbation, but the complex phase of each of the constituent stationary waves, +&?), evolves at a rate proportional to its energy, E,. Therefore, the characteristics of the wavepacket change as different parts of the superposing waves add in or out of phase as time evolves. The energy differences between the stationary states determine the rate at which the wavepacket changes. Since most physical phenomena of interest are inherently time-dependent, it is often more intuitive to study quantum systems in the time domain, by directly monitoring the evolution of the wavefunction in analogy with classical dynamics. This report will concentrate on a small subset of wavepacket research. Specifically, we will review work on the creation and detection of electronic wavepackets within atoms. This work is pertinent to many problems of current interest in atomic physics involving time-dependent interactions, including collisions between atoms and charged particles, the evolution of electrons in atoms exposed to strong laser fields, and collisions between electrons within multi-electron atoms. Such interactions naturally involve transitions from an initial atomic state to one or more final bound or continuum levels at different energies. The collection of final states constitutes a coherent superposition or wavepacket that is nonstationary as time evolves. Important insights into these complicated problems can be gained by monitoring the evolution of wavepackets. Since almost all timedependent interactions between atoms and neutral or charged particles are electromagnetic in nature, essentially any physical situation can be mimicked by using an appropriate sequence of electromagnetic pulses with various properties. The form of the wavepacket that is produced depends critically on the initial state of the atom as well as the power spectrum and spectral phase of the radiation pulse, for these parameters uniquely determine the final electronic state distribution and phases. Moreover, one can imagine creating a specific wavepacket that exhibits certain temporal behavior to alter or control the outcome of subsequent atomic processes or interactions. Creating electronic wavepackets in an atom is actually very straightforward. Assume for simplicity that the atom is initially in a stationary eigenstate. Population amplitude can be transferred to other states in the atom by exposing it to a pulse of electromagnetic radiation. The excitation is coherent as long as the duration of the pulse is short compared to any incoherent relaxation processes, which for isolated atoms is limited to spontaneous emission. A wavepacket is created irrespective of the amplitude and phase relationship between the various spectral components in the pulse. A short bright pulse of noisy white light with randomly phased spectral components will produce some wavepacket in any atom. However, the dynamics of the wavepacket depend critically on the light field characteristics as well as the absorption cross-sections of the atom. The difficult aspect of wavepacket experiments is not the excitation of some wavepacket; instead it is
ELECTRONIC WAVEPACKETS
3
the creation of a pcirticirlcir wavepacket exhibiting specific behavior that can be reproduced on consecutive measurements. Hence, wavepacket experiments require precise control over the time-dependent electric field in the pulse that generates the superposition. Remarkable experimental progress in creating and detecting electronic wavepackets has been made recently (Noordam eta/., 1994; Noordam and Jones, 1997). This work has been made possible due to fantastic technological advances in the generation of coherent broad-band radiation pulses. It is now possible to generate laser pulses with coherent bandwidths greater than 0.1 eV (Taft ot d . , 1995) and peak field strengths much greater than the atonlic unit, 5 X 10” V/cm (Chambaret et d., 1996). Furthermore. using new electro-optic devices it is possible to independently alter the spectral phase and amplitude within these pulses to produce an enormous variety of different time-dependent fields (Weiner et ril., 1990; Fermann et d., 1993). Production of pulses with arbitrary field characteristics is essential for the creation of wnvepackets with specific dynamic qualities and, therefore. is the key to quantum control of atomic and molecular systems (Tannor and Rice, 1985; Brumer and Shapiro, 1986; Warren et a/., 1993). Of course. even with its rapid advancement, current technology still drastically limits the types of wavepackets that can be created and monitored experimentally. Wavepackets are produced when a radiation pulse transfers real population from some initial stationary state to several final states. The dynamics of the wavepacket will depend on how many states are excited, their energy spacings. and the spatial wavefunction of each. Wavepacket excitation can occur via single or multiphoton absorption within the coherent bandwidth of the radiation pulse. As shown in Fig. 1, several final states can be excited if the initial state is coupled to a band of states whose energy spacing is small compared to the bandwidth of the pulse. Alternatively, the initial state can be excited to a “ladder” of energy levels. which lie an integral multiple of the photon energy apart. Furthermore, if intense pulses are used, AC Stark shifts in the atom facilitate the excitation of states that cannot be resonantly excited in weaker fields (Freeman er al., 1987). These energy shifts can exceed the photon energy, allowing for excitation of a vast number of states. With few exceptions, experimentally producing and monitoring wavepackets coniposed of low-lying states in atoms is extremely difficult. First, the coherent bandwidth of available pulses is insufficient for excitation of neighboring levels. A notable exception to this statenicnt is the creation of spin-orbit wavepackets composed of fine or hyperfine structure components of a given principal and orbital angular momentum level. The technique of quantum beat spectroscopy exploits the time-dependent changes in the polarization of the enutted spontaneous emission from these dynamic states to determine lifetimes and level splittings with high resolution (Demtroder, 1982). Second, excitation of ladder systems has been studied, but these experiments are limited by ionization (Broers rt u / . , 1992;
R. R. Jories and L. D. Noordam
4 (A)
(B)
FIG. I . Different ways in which a coherent superposition state is produced when a bi-oad-hand laser pulse interacts with a stalionary state i n an atom. ( A ) Excitation of a band of closely spaced levels within the laser handwidth. ( B ) Excitation of a “lndder” of states spaced at or near multiples of the central frequency of the laser pulse. (C) Excilation of states outside the laser bandwidth via timedependent AC Stark shifts as the intensity of the pulse rises and falls.
Balling rt al., 1994; Jones, 1995b). The energy spacing between the lowest energy levels in most atoms is comparable to the ionization potential. Therefore, in most atoms, only two or three ladder states may be excited below the continuum. Lastly, even though the excitation of many excited states via AC Stark shifts has been observed experimentally (de Boer and Muller, 1992; de Boer and Muller, 1993; Jones r f a/., 1993; Story et al., 1993). observing the dynamic evolution of this superposition has only been possible in a few special cases (Jones, 1995b; Conover and Bucksbaum, 1996). As we will discuss in more detail in a following section, experiments are generally performed with ensembles of atoms. Spatial intensity variations in the experiments combined with the intensity dependence of the excitation probability yields an ensemble of wavepackets whose evolution is spatially dependent. This inhomogeneity smears out any experimentally observable single atom response. However, it should be noted that the evolution of wavepackets in strong fields is currently a topic of great theoretical interest. The dynamic evolution of these non-stationary states is responsible for high-order harmonic generation (Krause rt al., 1992; Salieres ef a/., 1995) and above threshold ionization in intense laser fields (Grobe and Fedorov, 1993; Yang et al., 1993; Kulander rr d., 1995). Experimental observation of the actual motion of such a wavepacket would be extremely important for direct confinnation of these theories. In contrast, highly excited Rydberg atoms are excellent candidates for experimental wavepacket studies. First, the energy splitting between Rydberg states
ELECTRONIC WAVEPACKETS
5
with principal quantum numbers 17 and n + 1 decreases as lh? while the number of states per principal quantum number increases as i f 2 .Therefore, the density and variety of accessible Rydberg states is enommous compared to that of the lowest excited states. Levels with many different quantum numbers and spatial wavefunctions can be used to generate extremely complicated wavepackets. The evolutionary time scales for Rydberg wavepackets can be in the picosecond regime. much longer than the duration of electromagnetic pulses, which are now easily produced. Conversely, the time scales for evolution for the ground and low-lying excited states are on the order of one femtosecond. much faster than even the shortest “shutters” available. Second, the general properties of Rydberg states are exaggerated relative to those of low-lying states in atoms (Gallagher, 1094). and the dynamic evolution of a Rydberg electron is extremely sensitive to relatively weak perturbations. Strong-field effects such as multiphoton absorption and AC Stark shifts in excess of the photon energy are observable without significant ionMoreover, it is straightforward to create Rydberg wavepackets and manipulate their evolution using current technology. Rydberg wavepackets have been produced using short laser pulses with frequencies ranging from gigahertz to petahertz (ten Wolde et al., 1988; Jones, 1996: Conover and Rentz, 1996; Hoogeiuaad r’t trl., 1997). sometimes in combination with additional static or dynamic electric or magnetic fields (Yeazell and Stroud, 1988; Noordam ern/., 1989; Yeazell et 01.. 1993; Broers et NI., 1993; Wals ef t d . , 1994). The evolution of the wavepacket and the underlying physics of the problem can be vastly different for each. I n the remainder of this paper we will discuss methods for creating and detecting Rydberg wavepackets as well as the physical insight that can be gained from these studies. We will begin with a description of some of the more common techniques for monitoring wavepackets experimentally. Next, we will discuss a number of different experiments involving Rydberg wavepackets and comment on their results. We conclude with a discussion of future experiments in this exciting and rapidly growing field of research. For additional information on Rydberg wavepackets, the reader is directed to several previous theoretical and experimental reviews of Rydberg wavepacket research (Alber and Zoller. 190 1 ; Averbukh and Perel’man, 199 I ; Noordam et ill., 1994).
11. Rydberg Wavepackets A. MONITORING WAVEPAWET DYNAMICS
The key to the success of any wavepacket study is the ability to observe the electron dynamics with adequate time resolution. Several techniques are useful in this regard. and it is important to know the limitations of each before its iniplementa-
6
R. R. Jones and L. D. Noordam
tion in a particular experiment. Each method employs a laser or electric field pulse as a fast “shutter” to view the evolution of the electron wavefunction directly. The “fast” criterion is of course relative, and relates to the particular system of interest. If the electronic evolution is to be accurately represented, the duration of the shutter must be less than the time scales for electron motion within the atom. These time scales are determined by the inverse of the energy differences between the states that make up the superposition-the larger the energy difference, the more rapid the motion. Detailed descriptions of the various techniques for monitoring wavepacket evolution are given in a recent review (Noordam and Jones, 1997). In the following sections we will briefly discuss these methods to facilitate our discussions of specific wavepacket experiments. 1. Time-Resolved Photoemission and Absorption After a wavepacket has been excited, it will eventually decay via spontaneous emission to a lower energy level. However, since the spatial extent of the wavepacket changes as a function of time, the characteristics of the spontaneous emission will also be time-dependent. For the case of purely angular motion, the atomic dipole moment is constant in amplitude, but changes direction as the wavepacket evolves. As a consequence, the total fluorescence signal is time-independent while the polarized fluorescence varies at the wavepacket beat frequencies (Demtroder, 1982). If the wavepacket exhibits radial oscillations, the magnitude of the dipole moment is also time-dependent, leading to oscillations in the total emission rate as well (Alber et al., 1986). For optical frequency emission from Rydberg states, the final state of the decay is tightly bound and its wavefunction is localized near the nucleus. Therefore, the time-resolved fluorescence signal is only sensitive to changes in the Rydberg wavepacket near the nucleus. Unfortunately, measuring the time-resolved fluorescence from a Rydberg wavepacket can be considerably more difficult than ordinary quantum beat spectroscopy. First, the dynamical time scales for commonly studied Rydberg wavepackets are on the order of lo-’’ to lo-“’ sec. Therefore, a photodetector with an extremely fast response time is required to observe any signal modulation. Second, the large spatial extent of the Rydberg wavefunction ensures that the spontaneous decay rates for these states are considerably smaller than those of the low-lying excited states. The photoemission probability per nuclear passage is on the order of lo-’. Therefore, there are very few photons in the required short time window, producing extremely poor signal-to-noise ratios. Consequently, this technique is rarely implemented. Alternatively, wavepacket evolution can be studied via photoionization using a short probe pulse. It is not readily apparent that photoabsorption from a large Rydberg state to a continuum level of infinite extent will give any information on
ELECTRONIC WAVEPACKETS
7
the position of the wavepacket. After all, the continuum wavefunction has nonnegligible amplitude over the entire volume occupied by the Rydberg wavefunctions. However, the absorption of an optical frequency photon will excite the Rydberg state to an energy of greater than 1 eV in the continuum. There is, then, a great mismatch in the frequencies of the radial wavefunctions for the Rydberg and continuum states except at very small distances from the nucleus. (GiustiSuzor and Zoller, 1987; Alber and Zoller, 199 1). In other words, photoionization of Rydberg states at optical frequencies is only sensitive to the amplitude of the wavefunction near the nucleus. In classical terms, the Rydberg electron cannot absorb a photon without the heavy nucleus nearby to help conserve momentum. Therefore, by monitoring the number of photoelectrons produced by a short probe pulse, the probability for finding the wavepacket near the nucleus can be measured directly. As in the case of fluorescence detection, changes in the total ionization rate indicate radial oscillations while variations in the angular distribution of ejected electrons indicate angular motion. The temporal resolution of this method is determined by the duration of the probe pulse. Like time-resolved fluorescence detection, the pump-probe technique does suffer from several limitations. First, at any given time, it is very unlikely that there will be significant wavefunction amplitude near the nucleus. The wavepacket photoionization probability is necessarily small, often smaller than that of the ground state of the atom. Therefore, the considerable background signal due to direct ground state ionization can lead to relatively poor signal-to-noise ratios. Second, since short-pulse probe ionization is only capable of monitoring changes in the wavepacket amplitude near the ionic core, it cannot detect the evolution of the wavefunction in other regions of space where the wavepacket spends most of its time.
2. Optical Rumsey Method Another technique, known as the optical Ranisey method ( O M ) or bound-state interferometry, circumvents both of the limitations of short-pulse photoionization (Scherer et (if., 1990; Scherer ct al., 1991; Noordam et al., 1992). To the lowest order in perturbation theory, two idcntical, temporally delayed wavepackets are produced by exposing the atom to two phase-coherent pulses. By measuring the degree of interference between the initial ‘pump’ wavepacket and the delayed ‘probe’ wavepacket, the time evolution of the initial wavepacket in the atomic potential can be followed. The first wavepacket evolves in time and the degree of interference with the delayed probe pulse is determined by the spatial overlap of thc two wavepackets. If the two wavepackets overlap in space and are in phase, they interfere constructively: The amplitudes add up and the coherent total Rydberg population will be four times as high. If the wavepackets overlap but are out of phase, the amplitudes cancel and there is no resulting Rydberg population. If
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R. R. Jones and L. D. Noordctnt
there is no spatial overlap between the ‘pump’ and ‘probe’ wavepacket they will not interfere, irrespective of the phase difference: The resulting incoherent Rydberg population after two pulses is twice the population after a single pulse. The Ramsey wavepacket detection technique is based on measuring the interference in the Rydberg population. By varying the phase of the probe pulse, while keeping the delay fixed, and monitoring the variations in the total Rydberg population, the overlap between the wavepackets is measured. No population variations indicate no spatial overlap, and the initial or ‘pump’ wavepacket has evolved such that its shape no longer resembles its shape at t = 0. Large variations in the resulting population indicate that the evolved wavepacket is very similar to its initial conditions. The optical Ramsey method has several advantages over pump-probe ionization or time-resolved fluorescence detection. First, it can be implemented using any two, identical field pulses with arbitrary temporal characteristics (Jones et al., 1995).These two pulses are easily produced by sending the pulse that will excite the wavepacket through a Michelson interferometer. The relative time or phase delay between the two pulses is then adjusted by changing the optical path length in one arm of the interferometer. Second, “the signal” using ORM is just the total Rydberg population. By using ramped field ionization, this signal is virtually background-free, yielding a substantially higher signal-to-noise ratio. Unfortunately, ORM also has severe limitations. Most notably, only certain types of wavepackets can be studied using this technique. Even though the optical Rainsey method makes it possible to compare wavepackets at times when they are far from the ionic core, the electron must pass by the core at some point during the laser pulse. Furthermore, in the “weak-field” limit discussed above, the detected signal is completely insensitive to the phases of the constituent states in the wavepacket (Jones rt al., 1995). ORM only detects how siinilar the wavepacket is to its initial configuration after some time delay. It cannot be used to determine the spatial distribution of an arbitrary wavepacket as time evolves. Instead, ORM gives the dynamics for the wavepacket which would be excited if the atom were exposed to a laser pulse with the same frequency spectrum as the actual pulse, but with identical spectral phase at each frequency. It is often the case that this particular wavepacket is never actually excited. The optical Ramsey method can be used to characterize the evolution of a fundamental class of wavepackets, but is not particularly useful for monitoring the true dynamics for all superposition states.
3. lmpulsivr Momenturn Rrtrievul Up to now, we have only discussed wavepacket probing using short opticlil pulses. Optical pulses are limited in that they only interact with a Rydberg electron near the nucleus. However, other probing techniques which do not rely on
ELECTRONIC WAVEPACKETS
9
optical pulses have been developed recently. One of these methods, impulsive momentum retrieval (IMR), utilizes unipolar electric field pulses, commonly referred to as half-cycle pulses (HCPs) (You et al., 1993; Jones et NI., 1993a), to nionitor the full momentum-space probability distribution of the wavepacket as a function of time (Jones, 1996). The method is straightforward to understand from the framework of classical mechanics. If the duration of the HCP is negligible compared to the time scale for variations in the electron’s position and momentum, then the electron receives a momentum “kick” or inipulse from the HCP. The impulse also induces a change in the electron’s total energy, which depends only on the amplitude of the kick and the component of the electron’s initial momentum in the direction of the HCP field (Jones at id., 1993a; Reinhold rt Li1.. 1993; Bugacov at al., 1995). If the energy gained by the electron exceeds its initial binding energy, the atom is ionized. So by measuring the field dependence for HCP ionization at a specific time, the distribution of monientum in the direction of the field kick can be determined (Jones, 1996).If multiple pulses with orthogonal polarizations are used, the full 31) momentum distribution of the wavepacket can be measured as a function of time. The method works equally well for any type of wavepacket provided it is bound to the atom and its evolutionary time scales are long compared to the duration of the HCP. However, since the probability distribution is measured, not the momentum-space wavefunction, the spatial distribution of the wavepacket canriot be ascertained from IMR alone.
4. Atomic S t r e d Cornera Several different methods for monitoring the dynamics of hound wavepackets have been discussed in the previous sections. It is also interesting to study the dynaniics of electrons leaving the atom. A loosely bound Rydberg electron can escape from its parent ion in several ways. For example, during field ionization the electron can travel over (or through) the saddle point of the potential in the presence of the externally applied electric field. Alternatively, the electron may exchange energy with an excited inner- electron and escape the nuclear attraction via autoionization. To investigate ~ h r nthe electron leaves the atom, optical pump-probe techniques are not appropriate, since these techniques probe the dynamics near the ion core and not near the border of the potential. In other words, optical measurements cannot distinguish an electron that has escaped the atom from an electron that is not near the nucleus but has not yet escaped. IMR is not useful for monitoring continuum wavepackets since the method requires that the wavepacket be bound. Using a recently developed Atomic Streak Cameras (Latikhuijzen and Noordam, 1996a) the electron emission can be probed with picosecond resolution. Fig. 2 shows the principle of the atoniic strcak camera used to measure the electron
10
R. R. Jones und L. D. Noordam
escape over the saddle point. Atoms in a static electric field are excited to ionizing Rydberg states by a short laser pulse. The time of ionization of the Rydberg atom determines the shape of the ejected electron pulse. The electrons are accelerated by the electric field, pass through a slit, and enter the deflection region. The voltage of the upper plate is swept (2.5 kV in 500 ps) so that the defection of the electrons depends on their time of arrival between the deflection plates (Lankhuijzen and Noordam, 1996b). As a result, the temporal profile of the ejected electron pulse is transformed into a spatial profile on the detector. The atomic streak camera is a valuable addition to bound-state probe techniques, providing a method for probing the dynamics of free electrons as they leave the atom.
5. Additional Experimental Considerations It is important to note that in all of the wavepacket experiments discussed here, it is implicitly assumed that we are studying the response of one or more electrons in a single atom. While it is quite easy to study single atoms theoretically, it is not very practical experimentally. Instead, an ensemble of atoms in an atomic beam or vapor cell is actually used. To a first approximation each atom in the ensemble can be considered to be identical. However, we must consider the experimental conditions carefully to understand the limitations of this assumption. First, the atoms can interact with each other through collisions. These collisions cause a dephasing of the wavepackets within the colliding atoms, relative to atoms that have not collided. Within a time 7,all of the atoms will have undergone a colliphosphor screen
fs pumo wise/
CCD camera
delayed switch Duke
FIG. 2. Principle of the atomic streak carnera. Using a short laser pulse, rubidium atoms are excited in a strong electric field. Once the electrons have escaped over the saddle point, they are accclerated and pass through a slit entering the dellection region. The voltage over the deflection p l a l e is swept in t h e (typically 2.5 kV/500 pb), tranhforming the temporal protile of the electron pulse into a spatial profile, which is measured using a position sensitive detector.
ELECTRONIC WAVEPACKETS
11
sion and no phase relationship between the electronic motion in different atoms remains. Therefore. there is no detectable wavepacket motion for the eiisemble. This dephasing is the time domain analog of collisional broadening in spectroscopy (Denitroder, 1982). Another limitation occurs because the experiments take place at finite temperatures, and the ensemble of atoms has a velocity distribution. Since observation of electronic motion requires some type of first shutter, there must be a well-defined time interval, T,between the wavepacket's creation and detection. As we discussed in the preceding sections, this shutter is typically a probe pulse or a gated fluorescence detector. Consider an atom that moves a distance, z, toward the probe pulse or florescence detector during the time interval, 7.The time interval between creation and detection of the wavepacket in the moving atom is different from that for a stationary atom by a time AT = d c , where c is the speed of light. Therefore, the distribution of velocities over the ensemble produces a spread in the observation times for the different wavepackets. Assume that the wavepacket has some characteristic evolution period, T, and that the width of the velocity distribution of the ensemble of atoms along the axis of detection is Ail:. Under these circumstances, the temporal resolution of the experiment will become too poor to resolve the characteristic motion after a delay of T,,,:,, = TdAv,. It should be noted that the only true time variations detected in ORM (see Sec. lI.A.2)are near the fundamental oscillation period of the laser field. The low-frequency modulations that reflect the evolution of the Rydberg part of the wavepacket are caused by beating of the higher, optical transition frequencies between the ground and excited states. Therefore, the characteristic period in an ORM experiment is the laser period, not the fundamental period of the Rydberg wavepacket, resulting in a much more severe restriction on the maximum allowable observation time. The previous two paragraphs describe limitations in the maximum observation time of the wavepacket. There are also technical limitations as to the smallest time interval that can be resolved using an ensemble of atoms. As discussed in the previous paragraph, time resolution is diminished if the wavepackets in different atoms have a different time interval. 7,between creation and detection. This restricts the geometry for the radiation beams used for excitation and detection of the atoms. Assume that the excitation and detection beams leave the sample of atoms at an angle of 8 with respect to each other. The width, Ax, of the sample of atoms along an axis orthogonal to the average direction of propagation of the two beams will determine the distribution of values of T for the ensemble, AT = hu sin( 8/2)/c.The best resolution is obtained for co-propagating beams where A7 = 0 for any spatial extent of the ensemble, and the worst is obtained with counterpropagating beams. Of course, adequate resolution can be obtained with any geometry by limiting the spatial extent of the sampled atoms Ax. If strong radiation pulses are used to excite the wavepacket, the amplitudes and phases of the final states are often intensity dependent. If the exciting beam
12
R. R. Jones and L. D. Nonrdam
has a spatial intensity variation, for example near a focus, then the wavepackets excited at different locations in the ensemble of excited atoms will behave differently. The degree to which this spatial non-uniformity limits one’s ability to monitor the temporal wavepacket evolution depends a great deal on the experimental geometry, laser intensity, and specific system under study. In some cases, limiting the spatial extent of the sampled atoms to a size much smaller than any appreciable intensity variation is possible. In any case, it is important to recognize the limitations introduced by strong-field effects. In addition to the experimental limitations caused by studying ensembles of atoms, there are dephasing mechanisms that are important even for single atoms. First, spontaneous emission between different states in the wavepacket cause it to lose coherence. Second, incoherent initial state superpositions, population in different ground hyperfine levels for instance, will cause dephasing at a rate comparable to the energy differences between the initial states. Lastly, for highly excited states, background black-body radiation and stray electric fields can induce incoherent transitions, which will destroy the wavepacket coherence (Gallagher, 1994). B. RYDBERG WAVEPACKETS AND CLASSICAL CORRESPONDENCE The first discussions of Rydberg wavepackets appeared in the Literature approximately ten years ago (Parker and Stroud, 1986; Alber et al., 1986; Yeazell and Stroud, 1987). Since that time an enormous amount of theoretical and experimental effort has been devoted to comparing and contrasting the dynamic evolution of wavepackets to that of classical electrons. Experimentally, the basic notion is to use a short, broad-band laser pulse to coherently excite a group of closely spaced Rydberg levels. Depending on the presence of additional static or time-varying electric or magnetic fields, the rapid excitation of the Rydberg states creates a wavepacket that is localized in one or more dimensions (not necessarily spatial dimensions). The dynamic evolution of the localized wavepacket can be monitored and compared to classical expectations. By altering the experimental conditions, dynamics ranging from regular to chaotic can be observed. In these experiments, the excitation of the wavepacket does not significantly deplete the initial state and most of the probability density in the atom still remains in the ground state. Therefore, when one discusses the evolution of the electronic wavefunction it is implicitly assumed that it is the Rydberg or wavepacket yurt. In the following sections, several experiments probing the dynamics of localized Rydberg wavepackets will be discussed, including radial, angular, Stark, and two-electron wavepackets. 1. Rudial Wuveyackets
Perhaps the most familiar type of Rydberg wavepacket is known as a “radial” wavepacket (Parker and Stroud, 1986; Alber et al., 1986; ten Wolde et a/., 1988). In a spherical coordinate system, radial wavepackets are characterized by oscilla-
ELECTRONIC WAVEPACKETS
13
tions i n the radial coordinate only, with no variation in either the polar or azimuthal angle coordinates. Radial wavepackets are commonly produced via short laser pulse excitation of Rydberg states from the atomic ground state or some other tightly bound level. We first consider the process in the time domain. In essence, the short pulse projects the iiutial state wavefunction, which is localized near the nucleus onto the Rydberg states at higher energies. As long as the laser pulse is much shorter than the classical Kepler period at the excited energy, the wavefunction has insufficient time to expand radially and fill the energetically accessible volume during the laser pulse. After the laser pulse, the localized wavepacket moves radially outward, reflects from the Coulomb potential at large radius, and, for a time, executes periodic radial motion in analogy to a classical electron in a Coulomb potential. The fundamental period of the motion is, T~ = 27~N’,where N is the average principal quantum number of the constituent states. This is identical to the period of a classical electron with an energy, E = - 1/2N2. In the frequency domain, the fact that the duration of the optical pulse is short compared to the classical Kepler period means that the coherent bandwidth of the pulse is sufficient to simultaneously excite several Rydberg states with different principal quantum numbers, 11. Since the initial state of the excitation usually has very low angular momentum, dipole selection rules restrict the angular monientum of the wavepacket to low quantum numbers as well, so that n >> L. This superposition of Rydberg states with identical phases and angular momentum is a radial wavepacket localized near the nucleus. The wavepacket “moves” as the complex phase of each Rydberg state evolves at a rate equal to its binding energy. The shape of the wavepacket changes drastically as it moves in the effective radial potential as shown in Fig. 3. During the excitation of the wavepacket, probability amplitude is deposited in the Rydberg states near the inner turning point of the classical potential. Since the depth of the potential energy well is greatest at this radial position, the part of the wavepacket excited on the leading edge of the pulse moves away from the nucleus at a velocity much greater than the avrrN , ~ Pradial velocity of a classical electron with the same energy and angular momentum (Gallager, 1994). Therefore, the radial extent of the wavepacket at the end of the exciting laser pulse is usually considerably larger than that of the initial eigenstate. The amount of time it takes a classical Rydberg electron in a low angular momentum orbit to move from its inner turning point to a radius halfway to the outer turning point, r,,,,,= 2n’. is roughly only one-tenth of the Kepler period. T~ = 2mi’. Therefore, if a 100 fsec pulse is used to excite a wavepacket with an average principal quantum number N = 20, then the radial extent of the wavepacket will be about 400 a.u. when the laser turns off. The N = 20 wavepacket shown in Fig. 3 was excited with a 300 fsec (FWHM) laser pulse. Its radial extent is = 3/4 r,)”,when the laser turns off. Because of their different radial positions, the head and tail of the wavepacket have very different radial velocities. As the wavepacket moves toward the outer turning point, the tail rapidly catches up with the more slowly moving head. By the time the wavepacket
R. R. Jones and L. D. Noordam
14
B
C
D
1.0T A
E
_ _b _ _ .
0
0 T
200 400 600 800 1000
Radius (a.u.) Rci. 3. Snapshots of the time-dependent radial probability distribution calculated for a radial wavepacket composed of hydrogenic np states centered at n = 20. In the simulation, the wavepacket was excited from the ground state using a 300 fscc laser pulse with a Gaussian temporal profile. ( A ) Shows the radial probability ror the 20p eigenstate for comparison. In (B)-(F), the radial probability lor the wavepacket is plotted at different times after the laser pulse. The times listed at the right of each t r x e arc in units of T = I .2 psec. the classical Kepler period of the 20p state. (€3) shows the radial spread in the wavepacket as it first leaves the nucleus. (C) shows the localization of the wavepacket far from the nucleus a1 one-half the Kcpler period. (D) shows the first “return” o f the wavepacket to the ion core. (E) shows the radial probability during a “fractional” revival where two distinct wavepackets can be seen. (F) shows an “integer revival” where the shape of the wavepacket rescmbles its original form after its firs1 excursion to large r .
reaches the outer turning point, its radial extent is only a small fraction of its initial size. Therefore, the wavepacket oscillation between its inner and outer turning points is accompanied by a periodic “breathing” of its radial size. Because the radial wavepacket has no well-defined energy, it is not really appropriate to compare its motion to that of a single classical electron with a well-
ELECTRONIC WAVEPACKETS
15
defined energy. Instead, it corresponds more directly to an ensemble of classical electrons with an energy spread equal to that of the quantum wavepacket and distributed about the atom according to the quantum probability distribution. This classical ensemble will exhibit the same spatial breathing as it moves in the radial potential. In addition, the wavepacket will spread as it oscillates due to energy dispersion. The period of motion for the higher-energy electrons in the ensemble is greater than that of the lower-energy electrons, so after some time the energy of the electrons in the ensemble is “chirped” radially. High-energy electrons are grouped near the tail of the bunch while the low-energy electrons form the head. Eventually, the head of the bunch overtakes the tail and the ensemble fills the classically allowed volume. If the energy distribution of the classical ensemble is continuous, localization is lost forever and the classical packet will never reform. However, to better approximate the quantum problem, we can forni the classical distribution out of a set of electrons with discrete quantum energies rather than a classical continuum. In this case, at certain periods of time, each electron in the ensemble will have undergone an integer number of round trips in the potential (not necessarily the same integer number). At these times of “integer revival,” the classical ensemble resembles its original form as shown in Fig. 3F (Parker and Stroud, 1986; Yeazell et ul., 1990). Not unexpectedly, the quantum wavepacket also exhibits spreading due to energy dispersion as well as integer revivals at specific times after its creation. However, there are aspects of the quantum wavepacket evolution that cannot be described classically (Averbukh and Perel’man, 1989; Nauenberg 1989). When the head and tail of the quantum packet overlap in space, interference occurs due to the wave nature of the packet. This interference creates interesting patterns in the radial shape of the wavepacket between the times of integer revival. In particular, the wavefunction will assume the form of an integer number of “mini” wavepackets orbiting in the radial potential at different radii. These non-classical, “fractional revivals” as well as other features of radial wavepackets have been observed experimentally using pump-probe ionization and ORM as discussed in Secs. II.A.1 and II.A.2, respectively (Yeazell and Stroud, 1991; Wals et ul., 1994). Fig. 4 shows the complex revival structure for a radial wavepacket in rubidium obtained using O M . Dispersion as well as integer and fractional revivals is not limited to radial wavepackets. These features of wavepacket motion are characteristic of discrete levels in any anharmonic potential where the allowed energy levels are not equally spaced. In these systems, there is no well-defined spacing between adjacent states, and therefore, no well-defined wavepacket oscillation period. A larger spread in energy differences between adjacent states causes more rapid “collapse” of the wavepacket. The excitation strength to each level as well as the spacing between levels determines the details of the wavepacket motion.
16
R. R. Jones and L. D.Noordani
A
B
0
delay T (ps)
FIG. 4. Evolution of an np radial wavepacket in rubidium with a mean principal quantum ninnher. N = 45, and consisting of approximately 7 states. (A) shows the evolution measured using ORM and (B) shows theory. Integer as well as several different orders of fractional revivals are clearly visible (I‘rom Wals ct ( I / . , 1994).
2. Angular or Oriented Wavepackets Rydberg wavepackets localized in other spatial coordmates have also been observed experimentally. Yeazell and Stroud (1988) used a short laser pulse in combination with a circularly polarized rf field to excite “angular” wavepackets oriented at particular values of the polar and azimuthal angles in a spherical coordmate system. Specifically, they were able to distinguish the orientation of the wavepacket by ionizing it using nanosecond field pulses. Changes in the field ionization signal were observed as a flunction of the phase of the rf field during the Rydberg state excitation. The angular evolution of the wavepacket, which is expected to take place over msec time scales due to precession about the non-hydrogenic sodium core, was not observed. 3. Stark Wavepackets
In hydrogen, any non-zero electric field lifts the degeneracy of the l-states corresponding to a given principal quantum number, n, producing a manifold of equally spaced energy levels, which are not angular momentum eigenstates (Bethe and Salpeter, 1977). Instead, these Stark states have permanent dipole moments whose projection on the field axis is a conserved quantity with corresponding quantum number, k. The energy spacing of the adjacent Stark states within a manifold is given by LIE = 3Fn.
ELECTRONIC WAVEPACKETS
17
The dynamics of a Rydberg electron in a static electric field strongly depend on the strength of the field. We will consider two extreme cases. In this section we describe wavepacket dynamics in weak to moderate electric fields, in which the Stark states of adjacent n-manifolds do not mix. In the next section we describe electron dynamics in the strong field limit where the electron is classically free, and escapes from the atom on a picosecond time scale. Consider the excitation of a manifold of k-states via single-photon absorption from a ground s-state in an alkali atom in the presence of a static electric field. The non-zero quantum defects of the low angular momentum states in non-hydrogenic atoms ensure that the initial state is unaffected by the relatively small fields required to cause significant Stark splittings in the Rydberg states. Dipole selection rules insist that only p-wave character in the Stark states can be directly excited from the ground state. Because angular momentum is not a conserved quantity in the presence of a static field, the excited p-wave will precess into d, J and high angular momentum as time evolves. However, if the laser pulse is significantly shorter than this angular momentum precession period, the electron will be in a pure p-state when the laser turns off. Subsequently, the angular momentum will oscillate from its initial value to the maximum value of I = n - 1 and back to 1 = 0. The period of the oscillation is inversely proportional to the spacing between the excited Stark states rs,,,,l = 2 d 3 F n . For an n = 20 “Stark wavepacket” the oscillation period is 42 ps in a field of 300 Vkm. Stark wavepackets are localized in angular momentum as opposed to radial or angular coordinates. In the frequency domain, the laser pulse that creates the wavepacket must have a coherent bandwidth greater than the spacing between adjacent Stark states in order to see some angular momentum precession. However, the frequency spectrum of the pulse must cover the entire manifold of states to obtain good localization in a single I-state. An interesting feature of Stark wavepackets is that to lowest order, the excited energy levels are equally spaced. Therefore, the “potential well” in which the wavepacket oscillates is harmonic, and the wavepacket does not disperse in time. The dynamics of Stark wavepackets have been studied using the optical-pump, photoionization-probe technique (Noordam et al., 1989). Because the large centrifugal barrier excludes lugh angular momentum electrons from the small radius region near the nucleus, the photoionization probability of a Rydberg state drops dramatically as the angular momentum increases. Therefore, photoionization of a Stark wavepacket is most likely when its angular momentum is low, and the ionization signal reflects the low angular momentum character of the wavepacket, which oscillates with a period, Ty,r,,.k.This periodic ionization property of Stark atoms has been used to create a dark wavepacket as discussed in Section 1I.C.1 (Jones and Bucksbaum, 1991). Stark wavepackets can be easily manipulated. For instance, modulation of the applied field results in an oscillation of the wavepacket period (Noordam and
18
R. R. Jones and L. D. Noordam
Gallagher, 1991). Applying a microwave field of a few GHz results in an external control over the wavepacket dynamics during its evolution. Furthermore, sufficiently short laser pulses can be used to excite several Stark manifolds simultaneously, thereby producing a wavepacket that oscillates radially as well as in angular momentum.
4. Continuum Wavepackets in n Strong Stark Field Simultaneous radial and angular momentum oscillations can also be observed in strong static fields where the Stark manifolds from adjacent n levels overlap. Optical excitation by a pulse with a relatively small bandwidth creates a wavepacket that exhibits a double quantum oscillation (Broers et al., 1993; Broers et al., 1994). Such a wavepacket has a slow angular momentum modulation as well as a fast, radial oscillation, caused by the beating of Stark states of different n-manifolds. As one might expect, complicated temporal wavepacket evolution is associated with the complex energy level structure of many, nearly degenerate states in the overlapping Stark manifolds. A strong electric field tilts the potential energy surface on which the wavepacket moves, forming a saddle point on one side of the atom. A Rydberg electron with an energy larger than E, = - 2
19
ELECTRONIC WAVEPACKETS Laser polanzarion parallel to E-field
7Experimental 2 0 kV/cm t=-1
32
40 Theory : 1.985 kVlcm E=-1.30 4.6 ps pulse duration
- 8 4 -8 2 -8 0 - 7 8 Energy ( 1 0 a u )
no 0
20
40
60
Time (ps) FK;. 5. Coinparison between iiieastired w a v c p a c k e ~ decay with thc atninic streak ciiiiicra (Lankhtiij~enarid Noordam. 199hh). and a MQDT ciilctilatioii (Kohichcaux and Shaw. 1996) of the emitted electron Hux of ;I ntbidiuin wavcpnckct. Thc streak coinera has a I picmecontl tinic resoliitinn. The inset iii the uppel- graph shows the classical boundary of the Rydberg electron iil this scaled encrzy. The inset in the lnwer graph shous thc ahsorption \pectrum of rtihidiiiiii and the specti-tini 0 1 the optical pulse.
5. Tuw Electron Wuvepackrts
So far we have discussed the excitation of wavepackets in atoms with a single optically active electron. Wavepackets have also been studied in atoms with two active electrons. These wavepackets are inherently more complex due to the additional degrees of freedom in the problem. Two-electron systems have been studied extensively in the frequency domain. The elcctron4ectron Coulomb interaction induces off-diagonal couplings between independent particle configurations, so that the eigenstates of the system are linear superpositions of several different configurations. This nixing destroys angular momentum and energy conservation for each electron separately and is the mechanism responsible for autoionization. It is perhaps more intuitive to consider these systems in the time domain, where one can speak of collisional angular momentum and energy exchange between electrons at specific points in time as opposed to mixed configuration two-electron eigenstates. In fact, descriptions of configuration interaction in two electron atoms have often
20
R. R. Jones and L. D.Noordam
relied on this time domain picture, but Zoller and co-workers (Henle et al., 1987) were the first to specifically discuss the influence of a second electron on Rydberg wavepackets. Recently, Schumacher et al. (1996) have observed the scattering of two electrons between degenerate bound state configurations in Ba. Specifically, they studied the excitation of wavepackets in the 6snd Rydberg series in Ba in the vicinity of a 5d7d perturbing level. The eigenstates near this perturber have mixed configuration wavefunctions of the form $,, = a, 1 5d7d > + 6,,16std >. The 5d7d coefficients, a,,, are non-negligible over an energy range of approximately 30 cm-' centered about the 6s27d level. This energy width is inversely proportional to the amount of time required for two electrons in the 5d7d state to collide into the 6snd configuration, T, = 1/2 psec. In the experiment, a short laser pulse transfers population from the 6s6p level to several mixed configuration eigenstates. The duration of the laser pulse is 200 fsec, which is much less than the configuration mixing time T ~In . the absence of configuration interaction, the 5d7d level cannot be directly excited by single photon absorption from the 6s6p state. Therefore, immediately after the laser pulse, only a pure 6snd wavepacket exists. This single mode excitation in a mixed system is similar to the excitation of a pure angular momentum eigenstate using a short pulse in the presence of a static field as discussed in Sec. ILB.3. In analogy with the electric field case, the two-electron wavepacket evolves into another configuration, the 5d7d, after some time. As time evolves the wavepacket periodically scatters back and forth between 6snd and 5d7d modes. The 5d7d character of the wavepacket is easily probed using short-pulse photoionization. Since both electrons in the 5d7d configuration reside near the core, the photoionization probability from this state is significantly greater than that from the 6snd Rydberg states. So by monitoring the photoionization signal, the amount of 5d7d character in the wavepacket can be determined. In this way, scattering between the 6snd and 5d7d modes has been monitored as a function of time (Schumacher et al., 1996). Wang and Cooke (1991; 1992) discussed an elegant way of creating autoionizing Rydberg wavepackets in two-electron atoms. Starting from the ground state, one electron is excited to a stationary Rydberg state. Subsequently a short laser pulse drives a bound-bound transition of the other, core electron. As a result, the total energy of the system exceeds the first ionization threshold, and autoionization can occur through collisional energy exchange between the two electrons. npically, the inner electron is still tightly bound, so collisions (and autoionization) only occur if the Rydberg electron is near the nucleus. Consequently, a fraction of the Rydberg wavefunction near the nucleus during the sudden core excitation will ionize. The rest of the wavefunction remains in a Rydberg state. However, probability amplitude will move from large radius toward the nucleus
ELECTRONIC WAVEPACKETS
21
to fill the “dent” which has been created via autoionization. As a consequence, the dent propagates away from the ion core at a velocity equal to that of a classical electron at the same energy and radius. As long as fresh Rydberg wavefunction is supplied from the outer regions of the Rydberg orbit, the ionization nrte is constant. However, once the dent has spread over the entire wavefunction. the partly depleted wavefunction is all that remains to feed the autoionization process, and the autoionization rate drops to a lower value. This evolution continues, producing a stepwise decay of the autoionization rate. The duration of the steps is determined by the radial oscillation period of the Rydberg electron. In the frequency domain, this stepwise decay can be understood as the result of coherent broad-band excitation of the non-Lorentzian line-shape, which is characteristic of isolated-core transitions (Cooke ef a/., 1978). Hanson and Lambropoulos (1995) considered the effect of exciting the second electron in a two-electron atom using an intense, long laser pulse that turns on suddenly after the first electron has been prepared in a Rydberg wavepacket. At a certain laser intensity, the Rabi flopping time of the inner electron transition exactly matches the period of the Rydberg wavepacket. Under these conditions, autoionization can be frustrated. Assume that the inner electron is in the excited state when the Rydberg wavepacket is at its outer turning point. Autoionization is prohibited because the overlap of the two electrons is vanishingly small. When the wavepacket reaches the nucleus, the inner electron has been driven back to the ground state, and there can be no autoionization. Since the Rabi and Kepler periods are matched, the situation will be identical on each roundtrip (as long as wavepacket spreading is not too severe) and the two-electron system is stable against autoionization. Experimental work on isolated core excitation (ICE) of two-electron wavepackets is sparse. Van Druten and Muller (1996) observed that starting from a stationary Rydberg state, adjacent Rydberg states are populated when the Rabi frequency of the core transition exceeds the Rydberg spacing. This redistribution of population indicates the production of a wavepacket. Story ef al. (1993a) have investigated ICE in magnesium using two short-pulse lasers. The first laser excites a bound Rydberg wavepacket, and the second drives the ICE of the inner electron. They observe that if the core laser is tuned off resonance, the inner electron can only be excited when the Rydberg wavepacket is near the nucleus so that it can carry away (or supply) the excess energy in the shake-up (down) process. Fig. 6 shows the autoionization yield as a function of the delay between the excitation of the Rydberg wavepacket and the core excitation. Maxima in the ojjf-resonance excitation probability are observed at times when the wavepacket is near the nucleus. Conversely, recent experiments in Ca show that the probability for on-resonnnce ICE decreases when the outer electron is near the nucleus (Jones, 1996a) due to broadening of the resonance.
22
R. R. Jones and L. D. Noordurn
Relative Timing (ps)
FIG. 6 . Time-resolved excitation spectrum of a Mg autoioniring wavepacket (Story et al., 1993a). The initial Mg wavepacket is centered around ti = 54 and the core laser is detuned lo the bluc of the ionic transition. The excitation of the autoionizing state is enhanced when the time delay between the Rydberg excitation laser and the core dressing laser is equal to an integral number of roundtrips of the Rydberg electron. For t i = 54 the classical orbit period is 24 ps.
C. WAVEPACKETS CREATED BY STRONG LASERFIELDS In Section II.B, the excitation of wavepackets using relatively weak laser fields was discussed. Lowest order perturbation theory is adequate to describe the creation of wavepackets in this regime. Only a negligible amount of population is transferred from the initial state to produce the wavepacket, and the energy level structure of the atom is not significantly affected by the driving field. In strong fields, the situation can be much more complicated. First, significant population can be transferred out of the initial state, affecting the excitation of the wavepacket. In the case of initial state depletion, the entire wavefunction, not just a small part of it, moves as a function of time. Second, the energy levels of the atom shift in the presence of a strong excitation field so that the number and characteristics of the states excited by a strong field can differ greatly from those populated in a weak field. In this section we will discuss the creation of several different types of wavepackets using strong, non-perturbative radiation fields.
1. Dark Wavepackets In Section II.B.l we described how a radial wavepacket can be created by rapidly exciting a wavefunction from a tightly bound state to a Rydberg state, producing a localized bundle of probability density, which undergoes quasi-periodic motion in the radial potential. The inverse of a localized radial wavepacket, an “anti-” or “dark” wavepacket, can be produced using a strong laser field (Stapelfeldt ef al.,
23
ELECTRONIC WAVEPACKETS
1991; Jones and Bucksbaum. 1991; Noordam et id., 1992). A dark radial wavepacket is essentially a probability hole or notch that propagates back and forth in the radial potential through an otherwise stationary wavefunction. This anti-wavepacket represents a region of space in which there is little probability for finding the electron. A dark wavepacket is produced when a stationary Rydberg state is exposed to a pulse of high-intensity optical frequency radiation whose duration is short compared to the classical Kepler period of the stationary state. Since the frequency of the optical radiation is much greater than the energy spacing between adjacent Rydberg states of the same angular momentum, simultaneous conservation of energy and momentum dictates that only that part of the electronic wavefunction near the nucleus can exchange energy with the optical radiation field. When the laser pulse turns on, probability amplitude near the nucleus can be excited to the continuum or deexcited to any lower lying bound states that are at (or near) single or multiphoton resonance. Since the laser pulse is intense, the wavefunction near the nucleus is rapidly depleted and amplitude from larger radius flows toward the nucleus to fill the “hole.” However, as in the case of rapid core excitation of an autoionizing state (Sec. ILBS), the rate at which the wavefunction can move toward the nucleus is limited to the velocity of a classical electron at that energy and radius. So if the pulse is much shorter than the Kepler period, the part of the wavefunction far from the ion core has insufficient time to move during the laser pulse. Therefore, a hole or notch in the wavefunction remains near the nucleus after the laser pulse has turned off. The wavefunction resembles that of a localized radial wavepacket ridded to a stationary Rydberg state with a 7~ phase shift. The radial packet part of the wavefunction moves as described previously while the stationary state part remains motionless. The resulting probability distribution is that of a stationary state with a radially oscillating hole in it. In the frequency domain, the anti-wavepacket is formed by stimulated absorption and re-emission of photons through real or virtual continuum or bound states (Radmore and Knight, 1984). These Raman transitions transfer population from the initial state to neighboring Rydberg levels to produce the required coherent superposition as shown in Fig. 7. Note that the laser intensity must be great enough to deplete the initial state via multiphoton transitions and its bandwidth must be great enough so that the Rainan process can excite neighboring Rydberg levels and still conserve energy. The characteristic redistribution of population from a single Rydberg state to its neighbors by a strong optical frequency pulse was first observed by Noordam rt al. (1992a). The dark wavepacket was formed from several mixed configuration Rydberg states near a 5d7d perhuber in barium. As outlined in Sec. II.B.5, the ei enstates near this perturber have wavefunctions of the form, GI, = u, 5d7d > + b, 6 s d >. Subsequent experiments observed the redistribution (see Fig. 8) as well as the coherent evolution of the dark wavepacket using a strong-field version of ORM (Duncan
I
7
24
R. R. Jones arid L. D. Noordam
FIG. 7. Possible roiitt's for redistrihution of Rydhcrg population during thc creation of a dark wavcpacket in Ba. Stimulated Ranian trnrisitions through real or virtual hound and continuum stiitcs transfer population from a single stationary Rydberg states to other nearby lcvels within the handwidth of tlic short laser pulse (froin Noordam C I d., IC)Y?a).
and Jones, 1995). The 5d7d valence character in these Rydberg states has a continuum coupling significantly hgher than that of pure Rydberg levels. Therefore, the 5Lnd part of these states mediates the production of the observed dark wavepackets in Ba (Hoogenraad et a/., 1994; Vrijeii et al., 1995; Duncan and Jones, 1995). The evolution of dark wavepackets in potassium has also been observed (Jones rt ul., 1993b). In those experiments, efficient redistribution among nf Rydberg levels occurred through resonant coupling of the initial Rydberg state to the 3d states. Although it has been predicted theoretically (Burnett et al., 1993), the creation of a dark Rydberg wavepacket has never been observed when a pure Rydberg state is coupled solely to a flat continuum, even at intensities greater than 10'4Wlcm'. The creation of dark wavepackets is a mechanism through which excited states become less susceptible to ionization, and therefore, is important to studies of atomic systems in strong fields (Burnett et ul., 1993).
2. Wuvepuckets Excited During Multiphoton Ionization The excitation of intermediate resonance plays an important role in the multiphoton ionization of atoms (Freeman et ul., 1987). Typically, only chance multiphoton resonances between bound states in the atom exist at low laser intensities.
ELECTRONIC WAVEPACKETS
25
FIG. 8. Time of Right (TOF) clectron aignal showing the population redistribution from the nominal 5d7d level lo other Rydberg states during the creation of a dark wavepacket in Ba. From hottom to top. the energy of the laser pulse that creates the dark wavepacket is 0.0. 0.05, 0.2. and 0.8 mJ, for an estimated peak laser intensity of 0.0. 0.6, 2.5, and 10 TWlcm'. respectively. At low intensity. only the initial eigenstate is populated. while at higher intensity population is detected in many adjacent Rydherg states. The traces are offset vertically from each other and the true signal level at the left side of the plot is zero for each trace. The uppermost trace wab also taken using a peak intensity of 10 TWlcm' and its amplitude has been inultiplied by a faclor of 8 relative to the other scans (from Duncan and Jones. 1995).
However, AC Stark shifts in the atom due to a pulsed laser field can produce transient resonances as the laser intensity rises and then falls during the pulse as shown in Fig. 9 (Story et al., 1993; Vrijen et al., 1993; Jones, 1995a). These Stark shifts can be greater than the photon energy so that many states may be populated during a single laser pulse. Several recent experiments have explicitly measured the population left in excited states after exposing a ground state atom to an intense laser pulse (de Boer and Muller, 1992; de Boer et ul., 1993; Jones et ul., 1993; Story et al., 1993). The constituents of the resulting wavepacket are not limited to high-lying Rydberg states and can include levels of drastically different principal and angular momentum quantum numbers. The evolution of the bound superposition state that remains after an intense pulse ionizes ground state atoms can be studied theoretically. The combination of large Stark shifts and multiphoton excitation makes it possible to coherently populate a great number of stationary states to produce wavepackets with extremely fast evolutionary time scales. Experimentally observing the dynamics is a real challenge because the evolutionary time scales of the wavepacket can be much
26
H.R. Jones and L. D. Noorclam
Time (ps)
FIG. 9. Plot of the dressed states in potassium as a l'unclion of time in the presence of an intense 590 nni laser pulse. The 4s ground state dreased by two photons shifts through the nd Rydherg serics, producing avoided level crossings between the ground and excited states during the rising and falling edges of the laser pulse. Population transfer to the Rydberg states occurs via Landau-Zener transitions at these avoided level crossings (from Story er ( I / . % 1993).
less than the optical period of the laser field. However, fast wavepackets may ultimately be of practical importance in the amplification of ultra-short laser pulses (Noordam et nl., 1990) and strong-field coherent control of photoprocesses in atoms and molecules. In a recent experiment, a 150 fsec, 777 nm laser pulse was used to four-photon ionize Na atoms (Jones, 1995b). Transient bound state resonances dominate the ionization process and a coherent superposition of several bound states is produced by the pulse. Fig. 10 shows the probability for ionizing the Na atom as a function of delay between two identical pulses. There is only a small amount of ionization during the first pulse, and the dramatic changes in the two-pulse ionization yield are due to ionization of the wavepacket produced by the first pulse as it evolves in time. The two pulses do not overlap in time, so the maximum laser intensity seen by the atom is identical to that with a single pulse. Since the two laser pulses are phase coherent, the oscillation of the wavepacket has a definite phase relationship with the second pulse. This mutual coherence can be exploited to enhance the multiphoton ionization yield. At certain delays between the two laser pulses, the two-pulse ionization yield is nearly a factor of twenty greater than that with a single pulse alone. In addition, Fourier analysis of the two-pulse ionization signal can be used to identify the bound states that are populated. The Fourier transform of the ionization signal is also shown in Fig. 10. The clear resonances correspond to one-, two-, and three-photon transitions between the 3s,
ELECTRONlC WAVEPACKETS
27
F ~ G .10. (A) Temporal interlkrogranl 01’ Na ionization signal vs. the time delay hetween two. IS0 fsec, 0.1 TWlcm’ laser pulses. The oscillation\ iti-c due to Rainsey interference in the excitation of high-lying states during the two pulaca. Below s:ituration. the signal minima and maxima are approximately 4 and I8 times higher than single pulse yield. rebpectively. ( B ) Discrete Fourier translorm 01 the temporal interlcrogr-am ahown i n ( A ) . Notc the three distinct peaks that comespond to one-photon ) , three-photon ( 3 a - 7 ~transitions ) (from Jones, 199Sb). (3s-7p). two-photon ( ~ s - J sand
4s, and 7p levels in Na. Apparently, monitoring non-Rydberg wavepackets can give insight into the evolution of atonlic systems subsequent to their exposure to intense laser fields. 3. W~ivepucketsExcited with Mid-IR FEL Pulses
Rydberg wavepackets excited from a low-lying, stationary, Rydberg state using short-pulse FIR radiation can be quite different from those produced from the ground state using optical frequencies. The size of the ground-state wavefunction limits the area in which population can be transferred to the Rydberg states to about ~ i , as , discussed in Section 1I.B. 1. In the far-infrared, however, the overlap between two Rydberg states extends farther, and one expects that larger areas of the wavefunction should participate in the population transfer. One can estimate the effective radius contributing to the population transfer by inspecting the matrix elements. Assuming that An << n, the semiclassical FIR matrix elements reduce to
(n + An 1 r 1 n ) = Cn’(An)-”’.
(1)
Since the wavefunctions of the two states deviate most in the remote region far from the nucleus, the net contribution to (11 + An r n ) from large r is very small. After normalizing Eq. (1) to the size of the Rydberg wavefunction I?, we find that
I I
28
R. R. Jones and L. D. Noordam
the fraction of the radial wavefunction contributes to the matrix elements. For example, if we excite the n = 30 level to states near n = 40, the innermost 2 percent (20 ao) of the initial wavefunction contributes to the population transfer. Clearly, FIR Rydberg transitions still take place in a restricted area near the core. However, the size of that area (tens of ao) is substantially larger than for excitation from the ground state (ao). In a recent experiment Hoogenraad et al. (1997) excited a radial wavepacket using FIR radiation and measured the excited population distribution. In the experiment rubidium Rydberg states are exposed to picosecond FIR pulses from the free electron laser FELIX. The Rydberg atoms are initially prepared in vacuum using a tunable dye laser. After exposing the Rydberg atoms to the FIR pulse, a ramped electric field is used to analyze the final-state distribution.Fig. 11 shows the population redistribution after the FIR pulse creates a wavepacket around n = 4.0 from the 21d level. The bandwidth (SMh = 5%) corresponds to 10 cycles within the 218 cm-' FIR pulse. An attractive feature of creating a wavepacket starting from a Rydberg state, instead of the ground state, is that we can manipulate the initial state. In the presence of a moderate electric field, the initial Rydberg state can be oriented so that most of the wavefunction is located on one side of the nucleus. After FIR photoabsorption we will have an oriented wavepacket, as discussed in detail in Sec. II.B.2. Excitation from high-lying states also gives more flexibility for the excitation of wavepackets with interesting new properties. For instance, the dynamics of a wavepacket change as the excitation matrix elements of the individual states in the wavepacket change. This effect is dramatic for excitation near a minimum in the cross-section, where even the sign of the matrix elements changes. Strong variations in photoexcitation cross-sections are not uncommon in non-hydrogenic atoms near a Cooper minimum or Fano resonance, for example. Such a Cooper
1
1
,
'
'
FEL p h o n = 218 cm-'
._ +
-0
,
a
u
p
"
,G
0
,: ', .. . : , .
0
Q I
-300
I
,
I
D
U 0
U
N
21d/5
,
,
,
I
.
I
,
I
I
I
I
4
-200 -100 Final state energy (cm-')
FIG. I I . Rydberg state distribution as probed with state-elective field ionization after excitation of the rubidium 21d state to higher lying states using 218 cm-'FIR radiation. The dotted line shows scaled spectra of the initial state as obtained from reference shots without FIR radiation.
ELECTRONIC WAVEPACKETS
29
minimum occurs for the s + p Rydberg far-infrared transitions in lithium (Hoogenraad e t a / . , 1995). The interference between states that is required for the formation of a wavepacket is severely changed due to the sign change of the matrix elements that accompanies the minimum. For short-pulse excitation of a typical radial wavepacket, the radial matrix elements between the initial and final states are approximately constant, and the wavepacket fills a fraction of the Kepler orbit proportional to the ratio of the pulse duration to the Kepler orbit time T,,.The wavepacket is excited near the core and moves out during the laser pulse. The wavepacket spectrum created by a Gaussian pulse, centered around qIis:
with I' the power bandwidth of the pulse and C the approximately energy independent common matrix element. On the other hand, if the central frequency of the pulse coincides with a Cooper minimum, the excitation matrix elements scale linearly with the detuning. The effective excitation spectrum of a Gaussian pulse changes therefore to
with C'a prefactor of the matrix elements. The temporal shape of the wavepacket can be determined by squaring the Fourier-transformed amplitude spectrum. Both excitation spectra (Eqs. ( 2 ) and (3)) have the same shape in the frequency and time domain. In both cases, the pulse shape resembles the effective power spectrum (the square of Eqs. (2) and ( 3 ) ) .A bandwidth-limited pulse with a Gaussian frequency spectrum (Eq. ( 2 ) ) is again Gaussian, and narrows as the frequency spectrum broadens. Likewise, the pulse shape resulting from Eq. (3) is doubly-peaked, like its power spectrum. The time between the two peaks is equal to the original excitation pulse length, 2n-A'. The single, Gaussian, pulse (Eq. (2)) excites one well-localized wavepacket. The doubly-peaked eflwfivr pulse (Eq. ( 3 ) )excites wavepackets at both the rising and the falling edge of the retil pulse. Meanwhile, the excited population moves out radially, so that spatially the excited wavepacket resembles the effective temporal pulse shape. Fig. 12(b) shows the calculated evolution of a wavepacket excited from the 22d state to near the 461) level in Li using hydrogenic matrix elements. Alternatively, for excitation from the 22s-state, the center of the frequency spectrum coincides with a Cooper minimum. so that Eq. (3)applies and a Cooper minimum wavepacket is created. This novel wavepacket shows a douhlv-peaked structure, which is clearly visible in Fig. 12 when the wavepacket is near its outer turning point at 7psec and at 15 ps when the wavepacket is near the ion core. The split return near the hydrogenic recurrence corresponds to two wavepackets that are excited by the two peaks in the effective time profile of the laser pulse. At times
30
R. R. Jones and L.D.Noordam 0
4000
2000
6000
50
40
30 20 10
0
0
2000
0
2000
4000
6000
R (A. U.) 4000
6000
50
50
40
40
30
30
20
20
10
O
10
h
0 0
2000 R (A.
4000
6000
U)
I
I'
FIG. 12. The evolution of the radial wavefunction rZ $fr) after exciting a wavepacket around the 4611 state in lithium exciled with a 3 ps FIR-pulse from (a) the 22s state and (b) the 22d statc. The (hydrogcn-like) wavepacket excited from the 22d state starts at the core, and returns as a whole after each roundtrip time of I S ps. The wavepacket excited through the 22s - np Cooper minimum, however, is broken into two parts. At an integer limes the roundtrip time, no wavefunction is present at the core. The light intensity is shown as gray intensity to the left on the same time axis.
when the wavepacket return is classically expected, the two parts of the wavepacket interfere destructively. Similar double wavepacket structure has been observed in the excitation of perturbed, two-electron wavepackets in Ba due to a similar minima in the excitation cross-section (Schumacher et al., 1996). 4. Wavepackets Excited with THz Half- Cycle Pulses Up to this point, we have considered the excitation of wavepackets using electromagnetic pulses with bandwidths considerably smaller than the central frequency of the pulse. For instance, the central frequency of a 100 fsec, 800 nm laser pulse is 12500 cm-' while its bandwidth is only 150 cm-'. The pulse has approximately 40 complete electric field cycles. While the shortest optical frequency pulses to date still have several full electric field oscillations (Taft et al., 1995), it is possible to produce pulses with less than one optical cycle at frequencies less than a few THz. In particularly, recent work on electro-optic
31
ELECTRONIC WAVEPACKETS
switches has shown that it is possible to produce subpicosecond pulses characterized by a unipolar or “half-cycle” electric field (You et al., 1993). Because of their unique unipolar nature, these half-cycle pulses (HCPs) can be used to create novel types of wavepackets in Rydberg atoms. Consider the interaction of a broad-band HCP with a stationary Rydberg state whose Kepler period is long compared to the duration of the pulse. At low field strengths we can calculate the probability amplitude transfer to other bound states using perturbation theory. The spectrum of the HCP is extremely broad and covers a frequency range from DC to several THz. To lowest order in perturbation theory, only dipole allowed, single-photon transitions between the initial state and other states within the HCP bandwidth occur (Tielking and Jones, 1995). The states with the largest excitation cross-section from the initial state will have the same principal quantum number (see Eq. 1). Therefore, a superposition state composed of the initial state n, 1 > as well as n, 1 - 1 > and n, 1 + 1 > is produced at very low fields. At slightly higher fields, population continues to move up and down in angular momentum but remains in the same principle quantum number. At intermediate fields, single and niultiphoton transitions to states with higher and lower principle quantum numbers and all possible angular momentum become likely (Tielking and Jones, 1995). Finally, in strong fields, transitions to continuum states occur with up to 100 percent efficiency (Jones et al., 1993a). If we are interested in wavepacket dynamics, it is more illuminating to consider the Rydberg atom/HCP interaction in the time domain. Unlike multi-cycle laser pulses, the unipolar nature of the HCP field, makes it possible for a Rydberg electron to exchange energy and momentum with the field at any distance from the nucleus. Classically speaking, the HCP gives the electron a momentum kick or impulse. In quantum mechanical terms, this kick corresponds to a translation of the electron’s momentum space probability distribution in a direction opposite to the HCP electric field. The dynamics of the wavepacket can be described as an oscillation of the probability distribution along one dimension in momentum space with an associated motion along the conjugate coordinate in configuration space. It would be extremely difficult to predict this novel, but simple, onedimensional oscillation from the frequency domain picture, especially when one considers the enormous number of stationary states that are excited by the broadband HCP. In fact, most of the real insight into the interaction between atoms and these pulses (outside of the weak-field, perturbative regime) has come from classical analyses. The evolution of Rydberg wavepackets created by HCPs in the weak and intermediate field regime has been studied using a second HCP as a probe (Tielking and Jones, 1995). By monitoring the population in a particular Rydberg state (not necessarily the initial state) as a function of the delay between the pump and probe HCP, a strong-field interferogram is obtained (Jones er a/., 1993b). As in ordinary ORM, interference occurs because coherent population transfer occurs
I
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I
32
R. R. Jones and L. D. Noordam
at two different times. The interferogram verifies the coherent wavepacket excitation and labels the Rydberg states that make up the coherent superposition. Using this technique, the transition from single- to multiphoton dynamics within a single HCP has been identified. At high fields, the number of final states that contribute to the wavepacket can be extremely large. The combination of transition frequencies between all of the interacting states blur together in the interferograms and eventually no oscillations are distinguishable from the noise inherent in the measurement. At these field strengths, the near continuum of quantum phases in all of the populated discrete and continuum states make the system extremely difficult to describe quantum mechanically. However, these same features make it possible to describe the ionization dynamics classically. The dynamics of the wavepacket created when a HCP kicks a stationary Rydberg state have also been studied using the IMR method discussed in Section II.A.3 (Jones, 1996). First, a HCP with a field amplitude just below that required to ionize the atom produces a wavepacket. Next, the probability for ionizing the wavepacket using a second HCP is measured as a function of the amplitude and delay of the second pulse. This ionization data is then used to reconstruct the time-dependent momentum distribution of the initial wavepacket in the direction of the field in the second pulse. The results of this reconstruction are shown in Fig. 13. Clear modulations in the one-dimensional momentum distribution can be seen as the wavepacket oscillates back and forth in the direction of the initial kick. Therefore, the one-dimensional motion predicted by the intuitive time domain picture has been experimentally verified using IMR.
FIG. 13. Ttiiiporal evolution trf the iiioiiienluin distrihution foI B wavepacket produced by “kickKlWtrOll iii the 25d cigenstate i n N a wiih a 2.5 kV/cni, 0.5 psec HCP. Oscillation\ iii the oncdimensional distrihution arc clearly visible in this IMR reconstruction. The time interval hetwecn adjacent traces is 230 h e c (froin Jones, 19%). ing” the
ELECTRONIC WAVEPACKETS
33
5. W
D. WAVEPACKET CONTROL Recently, an enormous amount of research effort in atomic and molecular physics as well as physical chemistry has been devoted to studying the possibilities for optical control of photoprocesses in atoms and molecules. One method that makes such control possible involves the creation of a specific wavepacket which evolves in time from one configuration to another (Tannor and Rice, 1985; Warren rt ul., 1993). At a specific time, the wavepacket has developed into some
34
R. R. Jones and L. D. Noordarn
desired target state that can be further (de)excited to a final product state. There are two major difficulties in this coherent control scheme. The first is the identification of the coherent electromagnetic field that is capable of exciting the optimum initial wavepacket. The second is the generation of this specific time-dependent field once it is identified. In the following sections we discuss methods that have been used to produce novel electromagnetic fields to excite novel Rydberg wavepackets . As we discussed in section I, the key to producing a specific wavepacket is the construction of the appropriate radiation pulse that can be used to excite it. Clearly, it would be possible to produce any wavepacket in an arbitrary quantum system if one had complete control over the amplitude and phase of each frequency component in a pulse with a perfectly white spectrum. Although coherent white light pulses are not currently available, pulses with coherent bandwidths > 0.1 eV can be produced (Taft et al., 1995). When we consider the effect that a strong broadband pulse has on an atom, including AC Stark shifts and multiphoton absorption, it is clear that amplitude and phase control over the 0.1 eV (or even 100 cm-’) wide spectrum in a strong pulse would allow the production of a variety of different wavepackets in atoms. Several different methods have recently been developed to control the amplitude and phase of individual spectral components in a coherent broad-band pulse (Weiner et al., 1988; Weiner rt al., 1990; Fermann et al., 1993). All of the methods rely on spectral phase and amplitude filtering. This type of filtering can be accomplished by first dispersing the different frequency components of the pulse in space. The dispersed pulse is then directed through a spatially varying phase and/or amplitude mask that alters the phase and amplitude of each frequency component individually. The spectral features are then recombined to form a “shaped” pulse. The phase and amplitude masks can have a fixed spatial pattern (Weiner et al., 1988) or be composed of a set of variable polarization rotators such as liquid crystals (Weiner et al., 1990) or acousto-optic modulators (Fermann et al., 1993). Several groups have demonstrated that an enormous variety of temporal field shapes can be produced by “tailoring” short optical pulses in this manner (Wefers and Nelson, 1993). Tailored light pulses have been used to create odd versions of the more familiar types of Rydberg wavepackets (Jones rt al., 1993a; Schumacher et al., 1995) as well as vibrational wavepackets in molecules (Kohler er al., 1995). Shaping of the frequency spectrum of coherent light pulses is also possible in the time domain. Using interferometric methods, multiple phase-coherent pulses with independently controllable amplitude and relative delay can be combined to form a single time-dependent field. Control over the amplitude and phase of the field as a function of time implies phase and amplitude control of the spectrum as well. Multiple pulses have been used to excite tailored radial wavepackets similar to the ones produced using frequency-domain shaping (Noel and Stroud, 1996). Furthermore, in Section II.C.2 we discussed the use of two coherent pulses to drive a coherent superposition in phase, thereby enhancing the multiphoton
ELECTRONIC WAVEPACKETS
35
ionization yield. It has been predicted that a train of coherent pulses can be used to greatly aid the transfer of population through a multi-state ladder system by periodically driving the dipole moment of the wavepacket in phase with the optical field (Jones, 1995b). So far, our discussion of wavepacket control has been limited to the shaping of light fields to produce a particular electronic superposition that will exhibit some specific dynamic motion. Alternatively, we can consider building the electronic motion directly, without explicit consideration of the complex amplitude of each bound state in the wavepacket. Ultra-short half-cycle pulses can be useful in this regard. One can imagine launching an electronic wavepacket in almost any orbit by a sequence of short kicks with different amplitudes and directions. A first pulse kicks the wavefunction in a certain direction, and after some amount of timc, the orbit is adjusted with one or inore kicks in other directions. The final orbit is created by periodic adjustments and refinements of the initial motion. Although these types of experiments are purely speculative at this point, experiments utilizing niultiple HCPs polarized in diff'erent directions are currently underway (Bensky ri d., 1996).An orbit that should be straightforward to produce with only two pulses is that of a circular wavepacket. One pulse causes the wavefunction to oscillate along a single axis. When the wavepacket is at its maximum distance from the nucleus (primarily on one side of the nucleus) a second pulse kicks the wavepacket in an orthogonal direction, sending it into a circular orbit.
111. Conclusions The creation of wavepackets by any non-trivial time-dependent process or interaction is unavoidable. Technological advances have made it possible for experimentalists to create and probe wavepackets to study a variety of problems. In particular, Rydberg atoms have provided a convenient laboratory for studying many different types of electronic dynamics and have stimulated the development of techniques for monitoring the evolution of arbitrary wavepackets. Wavepackets are the connection between classical and quantum physics. Observation of electron dynamics in atoms makes it possible for us to use classical insight to understand these quantum systems. Ultimately, it may be possible to control the evolution of practically interesting quantum systems using specially designed wavepackets.
IV. Acknowledgments The authors would like to thank P. H. Bucksbaurn, C. W. S. Conover, D. I. Duncan, T. F. Gallagher, J . H. Hoogenraad, G. M. Lankhuijzen, H. B. van Linden van den Heuvell, D. W. Schumacher, and W. J. van der Zande for valuable discussions. RRJ gratefully acknowledges support from the U.S. Office of Naval
36
R. R. Jones and L. D. Noordam
Research, the U.S. Air Force Office of Scientific Research, and the Packard Foundation. LDN is supported by the Foundation for Fundamental Research on Matter, financially supported by the Netherlands Organization for Advancement of Research.
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A1)VANCR.S IN Al'OhllC'. MDI.RC'I'I.AR. AND OPTICAL PHYSICS VOI . 38
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
D . G. THOMPSON
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Chiral Objectb and Their Symmetries . True and False Chirality . . . . . . . . . . . . 111. Delinitions and Fund;iinerital Syiiiiiietrie\ of Spill-dependent Amplitudes . . . . A . Application of Symmetry Principle\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Chiral Properties o f g and R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Exainples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Expcriinental Ohscrvahles: Oriented Molecules . . . . . . . . . . . . . . . . . . . . . . A . Genelul Expressions and Exmriplcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Classitication of Chird Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Expcriniental Observahles: Rand~)iiilyOriented Target Systems . . . . . . . . . . . A . Chiral Moleculea . Electron Circular Dichroim and Optical Activity . . . . . B. Randomly Oriented Non-chiral Systems . . . . . . . . . . . . . . . . . . . . . . . . . C . Moleciiles with Time-odd (false) Chirality . . . . . . . . . . . . . . . . . . . . . . . . D . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Experimental Observables: Attenuation Experiments . . . . . . . . . . . . . . . . . . . . A . Structure 01' the M-inatrix for Forward Scattering . . . . . . . . . . . . . . . . . . B . Relation to the Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Theory of the Attenuation Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . The Physical Cause of Chiral Elfccts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction to the Scattcring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Spin-orbit Interactioii Involving the Incident Electron . . . . . . . . . . . . . . . . C . Spin-orbit Interaclion Involving the Molcculai Electi.ons . . . . . . . . . . . . . . D . The Spin-other-orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI11 . Theoretical and Coiiiputational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . The Attenuation Experiment and the Asymrne(ry of the T Matrix . . . . . . . C . Attenuntion Experiment: Behavior a s k + 0 . . . . . . . . . . . . . . . . . . . . . .
30
40
44 49 49 51 52 53 53 54 56 58
58 61 hl
62 62
62 63
64 66 66 68 60 70 71 71 73 74
40
K. Blum and D. G . Thompson IX. Results of Nurnerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Oriented Molecules. . . . . . . . . . . . . . . . . . ..................... B. Randomly Oriented Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Experiinental Rcaulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. XIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75 IS 76 81 84
84 X5
I. Introduction The concept of chirality plays an important role in many domains of science. Chiral properties can be found in elementary particles, in molecules, in crystals, and in the macroscopic world of plants and animals. The occurrence of chiral objects on different dimensional scales has been treated in the past in mutually independent frameworks, but in the last few decades, clues have begun to emerge that connect chirality on different levels. The study of chiral phenomena has now become an important meeting point between the physical and life sciences (Garay, 1987; Hegstrom and Kondepurdi, 1990). The basic concepts were introduced by Louis Pasteur in 1850. He found that a certain salt of tartaric acid formed two types of crystals, one of which rotated the plane of linearly polarized light clockwise, the other one counterclockwise. Pasteur was the first to reveal that this “optical activity” on the molecular level is due to a structural dissymmetry, and that the two types of molecules are related as non-superposable mirror-image forms. These two forms of a dissymmetric system are called enantiomers, or optical isomers. They are usually called laevo (from left) and dextro (from right) and are often designated in the biological literature by the labels L and D respectively. Pasteur’s term “dissymmetric” was later replaced by Kelvin’s “chiral”, meaning handedness, from the familiar analogy of the mirror-image relation between left and right hands. Another chiro-optical effect, the differential absorption of left- and right-circularly polarised light by chiral molecules, circular dichroism, was first discovered for crystals by Fresnel in 1825, subsequently by Cotton in isotropic solution in 1895. This phenomenon provides another basis for studies of the interaction between molecules and light. These discoveries were fully corroborated by many investigators, and Pasteur’s ideas about the mechanisms of chiro-optical effects were fully developed. It has been shown that in chiral molecules the asymmetrically distributed static charges gave rise to a helical potential field that distorts the molecular orbits (Rosenfeld, 1928). Thus, during photon absorption, the rearrangement of the orbits has a helical character that gives rise to parallel or anti-parallel electric and magnetic transition moments. Chiral molecules therefore absorb circularly polarized light with opposite handedness differently.
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
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These investigations of chiro-optical effects have more recently been enormously expanded in depth, leading to many new developments in optical and electronic technology, and they have become increasingly important in stereo-chemistry and bio-chemistry. For an introduction to this subject, its history and recent development we refer to the monographs of Barron (1982) and Mason (1982). Let us now consider interactions between molecules and polarized electrons (or other particles with spin) instead of’ discussing interactions with polarized light. Longitudinally polarized electrons exhibit chirality, their right- or lefthanded nature dependent upon the respective parallel or anti-parallel relation between their polar momentum and axial spin vectors (as will be explained in more detail in section 11). The question arises then whether “electro-chiral” effects exist. Can helical electrons distinguish between molecules of opposite chirality? For example, will left-handed electrons interact differently with left- and right-handed molecules, showing a preference of one enentionier over the other‘? Or on the other hand will left- and right-handed electrons be scattered differently from one and the same optical isomer’! This would be an analogue of circular dichroism. Electron beams can also be transversely polarized. The rotation of a transverse spin polarization about the beam axis would be the analogue of optical activity. As pointed out by Farago (1980), the existence of such effects can be expected in principle on general symmetry arguments. Similar problems have also been investigated in the context of neutron scattering (Kadir et (11.. 1974; Cox and Richardson, 1977; Harris and Stodolsky, 1979; Gazdy and Ladik, 1982). The importance of such work lies in its exploration of new methods for investigating details of the structure of chiral molecules and their interactions. There is a further motivation for studying the interaction between chiral molecules and polarized electrons. Although a chiral ob,ject and its mirror image differ from each other. there is no a-priori reason that one forni should be superior to the other. Yet, the real world does display a preference and this is particularly striking in the case of living organisms. Living organisms contain a large number of chiral constituents, but only L-amino acids are present in proteins, and only D-nucleotides in nucleic acids. This happens in spite of the fact that the formation of right- and left-handed forms has an equal probability in a non-chiral environment. However, only one of the two occurs in nature, and the particular enantiomorphs involved in life processes are the same in humans. animals, and plants. In fact, it is generally believed that the fomiation of chiral molecules was a necessary step toward the origin of life. Pasteur believed that if he could discover how nature introduced asyminetry into organic compounds he would be close to the secret of life itself. The origin of the homochirality of bioniolecules is still a puzzle today. Several hypotheses have been put forward. For ;Idetailed discussion of these theories we refer to the reviews by Mason (1988), Goldanskii and Kozmin (1989), and Keszthelyi (1995). Here we will only briefly consider one possible cause of homochirality, which was proposed after parity-nonconservation had been discovered in 1956. It was shown that i n P-decay, which is caused by the weak
42
K. Blum and D.G. Thowipson
interaction, the emitted electrons are preferentially left-handed, with their spinvector oriented anti-parallel to their momentum vector. The preference is nearly total when the emitted electrons move almost at the velocity of light. Righthanded electrons with velocity near the velocity of light were never observed. Thus, similar to life, one handedness is preferred over the other, and the fascinating question arises whether there is a connection between the asymmetry at the level of elementary particles and optical isomerism. Vester ef (11. (1959) were the first to suggest such a link. They assumed that at the beginning of evolution a racemic mixture of L- and D-structures, capable of self-reproduction, was present. Vester et ul. proposed that the molecular dissymmetry may have originated through preferential destruction of one enantiomer of a racemic mixture by spinpolarized electrons arising from @-decay, or circularly polarized bremstrahlung produced during their deceleration (@ radiolysis). The produced slight asymmetries were then assumed to be amplified by autocatalytic processes during the long periods of prebiotic evolution. This idea is known as the Vester-Ulbricht hypothesis of biological homochirality (Ulbricht and Vester, 1962). Since then many theoretical and experimental papers have been published, dealing with the problems of whether relativistic electrons can preferentially destroy one of the optical isomers. However, no reproducible experimental results have been reported from such investigations, either using relativistic electrons, positrons, or protons, produced by particle accelerators, or emitted by @-nucleotides,or by studying bremstrahlung effects. For a critical assessment of this work we refer to the reviews by Keszthelyi (1995) and Bonner (1991). A new development in the investigations of possible links between spin-polarized electrons and chiral molecules started with Farago (1980). In view of the complexity inherent in destruction and ionization reactions, Farago proposed to start with the simplest possible process, namely elastic collisions, between polarised low-energy electrons and handed molecules, and to try to understand this process in depth as a starting point for further research of more complex cases. As it has turned out, the study of elastic electron-molecule collisions is ideal for obtaining deeper insights into more fundamental aspects of chirality. It is the aim of the present article to give a review of this research, which followed from Farago’s proposition. The following section I1 is devoted to a qualitative introduction to the subject. Using simple examples, the basic definition of chirality will be introduced and illustrated. Symmetry principles are then employed in order to obtain a deeper description of chirality than usually encountered in stereo-chemistry. It was first pointed out by Barron (1986) that, besides the spatial transformation properties, time-reversal is essential for any attempt to classify chiral objects and their properties. Barron introduced the notions of “frue” and “jizlse” chirality. This distinction has proved to be important for a proper understanding of the structure and properties of handed molecules and the factors included in their synthesis and trans-
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
43
formations (Barron, 1991). Furthermore, the achieved sharper perception of the word “chiral” provided productive analogies between the physics of chiral molecules and elementary particles. As shown in section 11, prototypes of truly chiral systems are stationary, spinless chiral molecules (e.g., finite helices), or longitudinally polarized electrons, or circularly polarized photons. A simple examples of a system exhibiting false chirality is a spin-polarized diatomic molecule with strong spin-axis coupling. There is a further important aspect that must be taken into account. It is well known that typical chiral effects can be observed in processes with non-chircrl, hiit oriented inolecules for certain orientations (see for example the monograph by Barron, 1982). This has first been shown experimentally for crystals of silver gallium sulphide by Hobden (1967), and for certain nematic liquid crystals by Williams (1968).We will give some examples in section 11 and show that a necessary condition for observing chiral effects in reactions with non-chiral molecules is that a screw sense is defined by the geometry of the experiment. These qualitative considerations will then be put on a firm formal basis in section 111, which is the central part of this review. We will concentrate on elastic collisions between polarized electrons and closed-shell molecules. The aim is to give a unified treatment, including chiral and non-chiral molecules. Based on fundamental symmetry principles, a quantitative meaning of the concepts of true (or time-even) and false (or time-odd) chirality will be given, resulting in a clear definition and classification of chiral effects. The abstract theory will be illustrated with examples from atomic and molecular physics. Parity-violating interactions will not be taken into account, and consequently all chiral effects are due to either the handedness of the target molecules, or to a screw sense defined by the geometry of the experiment, or both. Sections IV-VI are devoted to the thcory of relevant experiments. General equations are derived which relate the initial and final electron polarizations. The initial polarization vector can be arbitrary, allowing us to consider both longitudinal and transverse po1,arization. It is important that the results allow us to identify chiral and non-chiral contributions to the experimental observables, and to separate true (or time-even) and false (or time-odd) chiral influences. Using these equations we can deduce what experiments must be perfornied to determine the various efTects. From such experiments, or the corresponding numerical calculations, one can expect to obtain a deeper insight into the nature of the chiral processes. In section V we will consider collisions with randomly oriented molecular ensembles. Applying symmetry arguments it follows that all “falsely” chiral contributions vanish identically in processes with closed-shell molecules. The special case of attenuation experiments, closely related to forward scattering, requires a separate treatment. The necessary forinal background is derived in section V1. The electro-chiral analogues of circular dichroism and optical activity are discussed in some detail.
44
K. Blum and D. G. Thompson
Several dynamical mechanisms have been put forward to explain the spin-depeiident interaction between electrons and molecules. They include the spin-orbit interaction between the electrons, both incident and molecular, and the nuclei, and also a spin-other-orbitinteraction involving all the electrons. In section VII we attempt to give a unified treatment of these dynamical models. One of the models (Thompson, 1996; Thompson and Blum, 1997) is described in more detail in section VIII and the numerical procedures are outlined; examples of numerical calculations are given. Detailed numerical calculations have been performed for an oriented closed-shell model molecule. Several observables of experimental interest have been calculated in order to obtain estimates for the various effects. In particular, contributionsof true and false ckality for various observables are identified, and calculated separately, in order to get some insight into their relative importance. The model calculations show that the c h i d effects can be quite large (up to 25 percent) for oriented molecules containing at least one heavy atom, if the collision energy is close to a resonance. This result demonstrates explicitly for the first time the enhancement due to resonances. The values obtained for randomly oriented molecules are much smaller. In particularly model calculations for attenuation experiments given results for spin asymmetry parameters of the order of magnitude of lop4,increasing to lo-' in resonance regions. Calculations have also been carried out for some small real non-chiral oriented molecules, and results of magnitude lo-? and greater have been obtained; they are discussed in detail in section IX. The present experimental situation is reviewed in section X. Here, important progress has been achieved by the Munster group. Spin-dependent attenuation of low-energy electron beams transmitting through the vapor of chiral molecules was investigated. For several molecules containing heavy atoms, spin asymmetries of the order of were found. These results represent the first clear experimental evidence for electro-chiral effects giving some credence to the UlbrichtVester hypothesis of biological homochirality.
11. Chiral Objects and Their Symmetries. True and False Chirality Chirality manifests itself in the distinction between left and right. Objects that cannot be superposed on their mirror image are termed chiral, or handed. (Since the mirror-image operation is equivalent to spatial inversion plus a rotation we can also define chiral systems as being distinguishable from their image under spatial inversion.) Systems that exist in non-superposable mirror-image forms are said to exhibit enantiomorphism. A simple example is the model molecule of type A H , shown in Fig. 1, where the three bond lengths A H , , AH,, A H , are different from each other. The three unit
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
l a
45
I b
FIG. I. Modcl iiiolcctilc ( A ) atid mirror iinage ( B ) . Thc dotlcd line iiidicates ltic iiiii-rcit- pl~iiie
vectors el, e,, e, are mutually orthogonal and define a molecule-fixed coordinate system. The two enantiomers in Fig. 1 appear as an object and its mirror image and cannot be transformed into each other by a rotation. The geometrical properties of the objects, shown in Fig. 1. show fundamental similarity to the properties of idealized left and right hands. This is the origin of the word “chiral,” which stems from the Greek work for “hand”. The chirality of molecule 1 is exhibited by the left-handed (counterclockwise) or right-handed (clockwise) sequence of the groups A H , and A H , when viewed in the direction A H , . The given sequence defines an axial vector e,xe, (or a “sense of rotation”) and when multiplied with the polar vector e3 gives the pseudoscalar le,xe,I .e,. Pseudoscalar quantities are of central importance in the theory of chiral phenomena. We recall that the reflection of a right-handed screw in a mirror appears to be ii left-handed screw but there is no way of rotating or reorienting a screw that changes its handedness. This behavior is reflected by the properties of the pseudoscalar [e,xe,].e,: It changes its sign under reflections (and spatial inversions) but remains invariant under rotations. All molecules of a given isomer have therefore the same value of [e,xe2].e,. independent of the molecule orientation in space. The pseudoscalar defines a ”screw sense” that is positive for molecule 1, and negative for its optical antipode. A finite helix provides another example, since reflection reverses the screw sense. We note that chiral systems are not necessarily devoid of all symmetry elements since they may have one or more proper
46
K. Blum and D.G . Thompson
rotation axes. For example, a finite helix has a twofold rotation axis through the midpoint of the coil perpendicular to the helix axis. The definition of chirality is a little more involved for moving objects. As a simple example consider a stationary sphere. This is clearly identical with its mirror image and is therefore non-chiral. Even if the sphere is spinning, its mirror image can be transformed back into the original object by a rotation (by turning it upside down). In fact, only a “sense of rotation” is defined in this case (which can be expressed in terms of the axial angular momentum or spin vector), but no screw sense. If the sphere is moving along its rotation axis, then the mirror image cannot be superposed on the original. Let us consider some examples. At rest, spinning elementary particles such as electrons or positrons are non-chiral. But if the spinning particle is moving in either direction of its axis, it becomes chiral (Fig. 2). The right- or left-handed chirality of a particle depends on the respective parallel or antiparallel relation between the polar momentum vector p and the axial spin vector S. In Fig. 2(A) the spinning particle behaves as a right-handed screw, and in Fig. 2(B) as a lefthanded screw. The scalar product p.S remains unchanged under rotations, but under spatial inversion we have p + -p and S -+ S. Hence p.S transforms as a pseudoscalar and defines a “screw sense”. Transversely polarized electrons are not chiral since their mirror-image forms can always be brought back into the original object by a rotation. The prototype of an elementary chiral particle is the photon in a circularly polarized light beam. This photon is in a spin eigenstate with its spin vector parallel or antiparallel to the direction of motion. In contrast to electron states, spin vectors perpendicular to the momentum vector are not possible. This result follows from the fact that photons have rest mass zero. Similar results hold for neutrinos and antineutrinos, where only one chiral state exists. Another type of chiral object is a diatomic heteropolar molecule with strong spin-axis coupling, so that the molecular spin S is directed parallel or antiparallel
2a
2b
FIG.2. Chiral states of spin 1/2 particlc. (A) right-handed; (B) left-handed. The spin vector is drawn as a double arrow. its corresponding “rotation sense” is indicated.
CHIRAL EFFECTS IN ELECTRON SCATTEMNG BY MOLECULES
47
to the internuclear axis n. The product n.S transforms as a pseudoscalar. However, there is an important difference to the cases discussed above. Under time reversal we have p + -p. S + -S, n + n. Hence, p.S transforms as a time-even pseudoscalar, and n.S as a time-odd pseudoscalar. This example shows that besides spatial inversion, time-reversal must also be taken into account in order to classify chiral objects. Let us clarify these concepts further by considering the spin-axis polarized molecule in more detail. A spatial inversion transfornis the original molecule (Fig. 3A) into Fig. 3B. The space-inverted version is clearly not superposable on the original. Hence, the molecule exists in the two enantiomeric states (3A) and (3B). However, time-reversal followed by a rotation through 180" about an axis perpendicular to n transforms Fig. 3B back into the original Fig. 3a. The optical isomers (3A) and (3B) can therefore be interconverted by time-reversal combined with a rotation. Hence the enantiomorphism established by spin-axis polarized molecules (characterized by the pseudoscalar n.S) is non-invariant under time-reversal, as distinct from that of longitudinally polarized electrons or photons. Barron was the first to draw attenlion to the importance of time-reversal for a classification of chiral objects. He introduced the concept of true (or time-invariant) and false (or time-noninvariant) chirality (Barron, 1986, 1991): True chirality is established by systems that exist in two ertantionieric stutes that are interconverted by spuce inversion, but not by time rer~ersrilcombined with unv spatial rotcrtion. Longitudinally polarized electrons exhibit true chirality; the spin-axis polarized molecule is an example of B system with false chirality. Another example is obtained by considering a magnetic field B collinear with an electric field E. This is a chiral system because distinguishable parallel and antiparallel arrangements are interconverted by space inversion; the polar vector E is reversed by the par-
3a
3b
3c
3d
FIG. 3. Translomalions of Ihe spin-axis polariLed molecule: ( A ) original position: ( A ) + (B) spatial inversion: (€3) + ( C ) rime-reversal: ( C ) + ( D ) rotalion.
48
K. Blum and D.G. Thompson
ity operation, the axial vector B is not. However E is time-even and B is timeodd, which means that parallel and antiparallel arrangements are also interconverted by time-reversal. Collinear electric and magnetic fields define therefore a system of false chirality. Banon has used these concepts to obtain a deeper insight into chiral effect in stereo-chemistry and optics, and clarified many aspects in these fields. Barron introduced the terms “true” and “false” chirality in order to draw attention to the importance of time-reversal for any classification of chiral effects. However, he did not intend this to become standard nomenclature; rather, he suggested that the word “chiral” be reserved in future for the truly chiral systems (Barron, 1991).Johnston ef al. (1993) have made another suggestion. They introduced the notion of “time-even” and “time-odd‘’ for “true” and “false” chirality respectively. We will adopt this notation here and use it throughout this paper. We will further develop these concepts in section 111 and we will in particular give them a quantitutive meaning; i.e., we will relate time-even and time-odd chirality with particular amplitudes describing electron-molecule spin-dependent scattering. This will allow us to disentangle time-even and time-odd contributions in collision processes, to calculate or measure them separately, and discuss their relative importance for particular observables. A collision between longitudinally polarized electrons or photons with chiral molecules can be considered as a simple prototype of a reaction between two “screws”. However, as stated in the introduction, chiral structures are not necessarily required in order to produce chiral effects. A non-chiral but oriented molecule can appear as part of a helix to an electron for certain directions of its momentum. In such a case a handedness is not defined by the structure of the molecule but by the geometry ofthe experiment, i.e., by the directions of initial and final electron momenta relative to the molecular orientation axis. More precisely, either an axial vector a and the polar vector b must be defined by the experimental arrangement (so that the scalar product a.b is non-vanishing), or three polar vectors a, b, c in such a way that the pseudoscalar axb.c is different from zero. Consider for example polarized electrons colliding with a spinless heteronuclear diatomic molecule. In the rest frame of the projectile electrons the molecule moves around the electron with axis n fixed in space (assuming that the molecular orientation does not change during the collision). In this case a “screw sense” is defined not by the structure of the molecule but by the geometry cfthe e.uperirnenr, namely by the initial wave vector k,,, the final wave vector k,, and the molecular axis n. If n lies outside the scattering plane ((k, - k,)-plane) then a reflection in this plane will transform n into its mirror image n’, and the three vectors k,, k, and n’ cannot be transformed back into the initial position k,,, k,, and n by a rotation. Hence, k,, k, and n define a chiral system. (In fact, a screw sense is defined by the pseudoscalar [k,,xk,].n). The spin of the electron “sees”
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
49
this handed structure and interacts with it, and chiral effects can be produced. We will return to this case in more detail in section 111. The same principle applies also to other cases. Chiral effects can be expected in any process, if a screw sense is defined by the geometry of the experiment; a well studied example is the photoionization of non-chiral but oriented molecules (Cherepkov, 1982; Dubs et al., 1985; Cherepkov and Kuznetsov, 1987; Chandra, 1989; Westphal et al., 1989), or photoionization of excited atoms possessing an asymmetric charge distribution (Cherepkov and Kuznetsov, 1989). In order to discuss these and related phenomena we first have to develop a general theoretical framework. Besides longitudinally polarized electrons, transverse polarization has also to be taken into account. Furthermore, the theory has to be developed in such a way that chiral molecules, or non-chiral but oriented molecules, are treated on the same footing. That is, the theory must apply to both cases, whether a screw sense is defined by the structure of the molecule, or whether a screw sense is given by the geometry of the experiments. Furthermore, we have to give a precise definition of a “chiral effect”, and a classification of these effects. We will do this in the following sections concentrating on elastic electron-molecule collisions.
111. Definition and Fundamental Symmetries of Spin-DependentAmplitudes A.
APPLICATION OF SYMMETRY PRINCIPLES
As stated above, it is one of the aims of the present section to give quantitative meaning to the terms “time-even (true)” and “time-odd (false)” chirality which will then allow us to disentangle these effects in collisions. In order to do this, we have to consider first collisions between electrons and oriented (chiral or non-chiral) molecules. It will be assumed throughout this paper that the molecules are spinless, and in A , ground states (Le., the axis-polarized molecule discussed above is not included in the following analysis). We will also assume that the molecule axis direction remains fixed during the collision. Following Farago (1980) we introduce a space-fixed coordinate system by the three orthogonal unit vectors
where k,, and k, are wave vectors of initial and scattered electrons respectively. A set of molecule fixed axes is specified by three orthogonal unit vectors el, e,, e, (see e.g., Fig. 1). Let us denote by m,, and m , spin components of initial and scattered electrons respectively relative to a suitably chosen quantization axis. Let us further denote
50
K. Blum and D. G. Thompson
byf(k,m,, k,m,,; eij the corresponding scattering amplitude for a collision with an oriented molecule, assuming that the orientation does not significantly change during the scattering. The orientation of the molecule is fixed by giving the relations between the molecule fixed axes e, = el, e,, e,, and the axes (3.1). We define an operator M = M(k,, k,, e,) in spin space by the condition (m1 I M I m& = flklm,, k , ~ , )ei),
(3.2)
The spin scattering matrix M contains all information on the elastic collision and has been used extensively in atomic collision physics (see e.g., Kessler, 1985). The scattering of electrons by atoms and molecules is dominated by electromagnetic interactions. Contributions from weak interactions which are parity violating can be neglected. It follows that the electron-molecule collision system is invariant under rotations, spatial inversion, and time-reversal; that is, M must be transformed as a proper scalar. Talung this symmetry condition into account it has been shown by Johnston et al. (1993j that the most general form of M is given by the expression
M = g,,l
+ g,n,.a+ g2n,.u + g,n,.a
(3.3)
where the four (complex) functions g, ( i = 0, , . . , 3) are functions of k,, k,, e,: (3.4)
R, = ~ [ ( k ,k,,, , ei>
and (T = (o,, a2,a3) denote the Pauli matrices. The transformation properties of some relevant vectors are shown in Table 1. Note in particular that time-reversal transforms k, into -k,, and k, into -k,, and that spin vectors are reversed. From the fundamental condition that M must transform as a proper scalar, and from the transformation properties of the basic vectors (see Table 1) we obtain the following symmetry requirements of the g, functions (Johnston et al., 1993). Under spatial inversion we have
g,(-kl, -ko, -ei)
=
g,(k,, k,,, e,) j
=
0, 3
g,(-kl, -k,, -ei) = -g,(k,, k,, e,) j = 1, 2 TABLE 1
TRANSFORMATION PROPERTIES OF S O M E VECTORS. THE Pl.US ( M I N U S ) SIGN DENOTES E V ~ (ODD) N BEHAVIOR OF ’ r H b VECTORS UNDER THE
RELEVANT OPERATION.
Spntial inversion
Time-reversal
+
(3.5) (3.6)
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
51
and for time reversal (k, t)-k,,):
s,(-k,,, -k,, e,) = g,(k,, k,,, ei) j
=
0, 1, 3
g?(-k,,, -k,. ei) = -g2(kl, k,, ei>
(3.7)
(3.8)
Under rotations of the total system. k,,, k,, e,, all g, must remain invariant. We note that M must also remain invariant under a symmetry operation of the molecule alone since all physical properties remain unchanged. (More precisely, a molecular symmetry operation can only change the overall phase of M ,which is physically unimportant and can be put equal to + 1.) As a consequence. the g,-functions remain invariant. If the molecular symmetry operation transform e, into el then we must have
s,(k,.k,, e,) = s,(k,,k,,, el) B.
C H I R A L PROPERTlES OF g, AND
(3.9)
sz
It follows from eqs. 3.5 to 3.8 that go and ‘q3 transform as proper scalar functions (invariant under rotations, inversion, and time-reversal), while g, and gz transform as time-even and time-odd pseudoscalars respectively. g, and g, will therefore be central for the study of chirality. Let us consider this property from a different point of view. Consider a collision between an electron and a non-chird molecule and assume that no screw sense is de$ned hv the geome t q qffrhe collision (for example, a diatomic molecule with its axis in the scattering plane). This means that the image under spatial inversion (-kl, -k,), -ei) of the vectors k,, k,, e, can be brought, by a rotation, into a position (k,, k,,, ei), which is physically indistinguishable from the original one. Under these operations s, and g2 transform in the following way:
g,(k,, k,, ei)
=
-g,(-k,, -k,,, -ei)
(3.10)
-
-R,(kI, k,. ei)
(3.11)
k,,, ei)
(3.12)
= -g,(k,,
for i = 1,2. The first equality (3.10) describes the effect of spatial inversion (eq. 3.6). The second equality (3.11) follows from our assumption that no screw sense has been defined. The last equality (3.12) follows from condition 3.9. Thus g , and g, vanish for non-chiral systems. As an example. consider a diatomic molecule lying in the scattering plane. As discussed in Section 11, no screw sense is defined in this geometry. Assume that the e3 axis is chosen along the internuclear axis. Since the choice of el and e2 is arbitrary we can assume that el is perpendicular to the scattering plane. Under spatial inversion (k,,, k,, e,) and transformed into (-k,, -k,, -ei). A rotation of T about the el axis transforms (-k,,, -k,) into their original positions (k,, k,), and transforms the molecule into a position (-e,, e2, e,) physically indistin-
K . Blum und D. G. Thompson
52
guishable from its original one. A reflection of the molecule alone in the e2 - e3 plane leaves the molecule unchanged but brings (-e,, e2,e,) into (el, e,, e,). In conclusion a necessary condition for g, and g2to be different from zero is that a screw sense is defined, either by the structure of the molecule, or by the geometry of the experiment, or both. These concepts will be further developed in Section 4.2.
C. EXAMPLES In order to illustrate the general results we will give some examples. Since the set gJ must transfer as scalars under rotations of the total collision system, they can only depend on scalar products constructed from $,k, and the three vectors e,.From equations 3.5 to 3.8 it follows that g,, and g, can depend only on proper scalar functions, g , can depend only on time-even pseudoscalars such as ((k, + $).e,)((k+k,).e,), and g2only on time-odd pseudoscalars like @&,).el. For elastic scattering from spinless atoms in their ground state the only scalar combinations available are k; = k: = k2 and kl.kowhich are invariant under spatial inversion and time-reversal. No pseudoscalars can be constructed from k, and k,. From equation (3.6) it follows that g,(k2, kl.ko) = -g,(k2, kl.ko) j
= 1,
2
Hence g, = g2 = 0 because of the invariance of the interaction under spatial inversion. Although this is a sufficient condition, it can be shown that invariance under time-reversal requires that g, = 0 and invariance under a combination of spatial inversion and time-reversal requires that g, = 0. Equation (3.3) then reduces to the well-known expression found in electron-atom scattering:
M
= g,,l
+ g3n3.g
(3.13)
As a second example let us briefly consider interactions violating parity invariance, assuming again isotropic target states. In this case M will not remain invariant under spatial inversion and equations (3.5) and (3.6) will not hold. However, assuming that time-reversal symmetry applies, equations (3.7) and (3.8) will still be valid. The spherical symmetry of the target dictates that the gJ functions depend again on the proper scalars h? and k l . b and from equation (3.8) it follows that g2(k2,k,&) = -g,(k2, k,.ko)
Hence time reversal invariance requires g2to vanish. The most general form of
M that allows for parity violation but is invariant under time reversal is given by
M
= g,l
+ g,n,.a + g,n,.a
(3.14)
if the target possesses spherical symmetry. This case has been discussed in detail by Kessler (1985).
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
53
As a final example consider electron collisions with an oriented diatomic hetero-nuclear molecule (for example. a molecule absorbed at a surface). As discussed in section I1 a screw sense is defined by the geometry of the experiment (that is, by the vectors k,, k,, and the molecular axis n) for certain orientation n. In this simple case it is easy to give a more explicit forni for the scattering matrix M, and for the pseudoscalar functions gI and g2.We write (3.15) R?
=
(n3.n)gi
(3.16)
where now g ; and gi are proper scalars and remain invariant under spatial and time inversion and rotations of the total system. In fact. n,.n - [k,xk,,J.ntransforms as a time-odd pseudoscalar, and n1.n as a time-odd scalar. Hence, equations (3.14) and (3.15) explicitly give the syninmetry properties (3.6) to (3.8) of gI and g_..The matrix M can be written in the form
M
= g,,1
+ g;(n,.n)(n,.n)n,.cr+ gi(n3.n)n2.u+ g3n3.rr
(3.17)
We can read off from equations (3.15) and (3.16) the essential geometrical conditions that must be satisfied in order to observe chiral effects. If n lies in the scattering plane, then n3.n = 0 and g, and g, vanish identically. No screw sense is defined by the geometry of the collision, and the system exhibits no chirality. Furthermore, g , vanishes also if n,.n = 0, that is, if n lies in the n2 - n3 plane (in particular, if n is perpendicular to the scattering plane). This is an example of an oriented system that exhibits time-odd but no time-even chirality (see section IVB). In the present case chirality is defined by the relationship between the vectors k,, k,, and n. The “isomer” is obtained by spatial inversion, or more simply by a reflection, for example in the scattering plane, which transfornis n into its mirror image n’.The functions go, g ; , gi, g, remain invariant under this operation because of their scalar character. Hence, the spin scattering matrix M , describing the “antipodal system”, is simply obtained by substituting n‘ for n in equation (3.12). T h s result illustrates the well-known fact that the dynamics (expressed by g,,, g ; , gi, g,) remain unchanged when a chiral system is transformed into its mirror image.
IV. Experimental Observables: Oriented Molecules A. GENERAL EXPRESSIONS A N D EXAMPLES
Suppose that an electron beam of Larbitrarypolarization P scatters from a medium composed of identically oriented molecules. The resultant differential crosssection I and the polarization P’ of the scattered electrons are given by the expressions (Kessler, 1985):
54
K. Blum and D.G. Thompson
where 1 denotes the identity matrix. Using the identities (A and B denote arbitrary vectors)
(a.A)(o.B) = A.B
+ iu(AxB)
(4.3)
and
rr[(A.a)cr]= 2A
(4.4)
and inserting equation 3.3 for M into equations 4.1 and 4.2 yields, after some algebra, 3 3 3
3
J.k= 1 3
.j= I
1 eukIm(gig;)+ 1{(2~e(g,gr) + I go 1’6, - C ~ ~ ! ~ ~ ~ m (+g ,I g, , g I’S,I} ; ) p.q
IP‘.n, = 2~e(g,g;) -
(4.6)
k= I
where eijkis the totally antisymmetric tensor and ‘Re’ and ‘Im’ denote the real and imaginary part respectively. These equations describing the general dependence of I and P’ on P are complex compared with the atomic case for which the pseudoscalars g , and g2 must equal zero according to the discussion of section IIIC. One important property is taken over from atoms. If I P I = I, it follows that 1 P‘ 1 = 1. It is clear from physical considerations that this must be the case. The change in polarization due to scattering is a function of molecular orientation. If all molecules are identically oriented, each scattering event changes the spin vectors of the electrons in the same way, so that a beam initially in a pure state will remain so after scattering.
B. CLASSIFICATION OF CHIRAL EFFECTS We will now use the results of sections IIIA and B and give a definition and classification of chiral effects with regard to electron scattering. An observable will be called chiml if (and only if) it depends linearly on g,, or g 2 , or both. A further classification of chiral observables can be obtained by considering the transformation properties under time-reversal, equations 3.7 and 3.8. Developing these concepts, and adapting them to our present case of interest, we obtain the following hierarchy of spin-dependent effects.
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
55
(i ) A spiticillv inverted system cun be transjorniecl back into its original position bv rotution (that is, the systeni is non-chirul). In this case we have in general gll
f
0,
g , = 0,
Si
f
0
g? = 0
as has been shown in section IIIB. (ii) A spatially inverted systcvn cut1 be trunsformed back into its originul position bv time-reverscil followed by a rotution. In this case we have in general the relations
so f 0,
s3
# 0,
92 f
0
XI = 0
That g , is necessarily zero follows from eqs. 3.5 to 3.8. Applying successively spatial inversion (eq. 3.6), time-reversal (eq. 3.7), and then a rotation that transforms all vectors back into their original position gives
g,(k,, k,,. ei)= -sl(-kl,
-k,, -eiJ
where in the last step eq. 3.9 has been taken into account, similar to the derivation of eq. 3.12. Hence gI necessarily vanishes. Note that it is sufficient to consider the transformation of b,k, and the axes e, since the g, functions are independent of electron spin. (iii) A time-reversed system can be trtrnsformed back to its original position by orily a rotution. It has been shown by Johnston et al. (1993) that closed-shell systems exhibiting only time-even chirality can exist. Assume that the system is indistinguishable from its image under tiinereversal, followed by a rotation. In this case we have from equation 3.8 gz(kl,k,,, e,) = -g2(-k,,, -kl, ei) =
-gz(ki, k,, e,)
(4.8)
where again eq. 3.9 has been taken into account. Hence g2 vanishes but g, can be different from zero. (iv) Nrither (i), (ii) nor (iii) i s possible. In this case all four g, are different from zero. We will call an observable “titne-add” if it depends linearly on gz, but not g , , and we will speak of a “time-even” observable if it depends linearly on gI and not g 2 .As shown by equations 4.5 and 4.6, it is of course possible for an observable to contain time-odd and time-
56
K. Blirrn and D. G. Thonipson
even components. Time-even and time-odd chirality correspond to “true” and “false” chirality in Barron’s notation, as follows from our discussion in section 11. C. EXAMPLES
In section IVB we have related time-even and time-odd chirality to g , and g2 respectively. A determination of these parameters allows us to obtain detailed information on time-even and time-odd effects, and their relative importance for particular processes. As shown by equations 4.5 and 4.6 the observables (cross-sections and polarization) depend on both g,and gz.Hence collisions with oriented (or partially oriented) molecules allows us to study the relative importance o j time-even arid time-odd e#ects f o r cnllisiorzs. By performing several experiments with different initial polarization, and detecting the electrons with different final polarization, time-even and time-odd amplitudes g , and g? can be extracted separately from the measurements. Some cases have been discussed by Johnston et a/. (1993). Here we will give a few examples of possible experimental interest. Let us consider a beam of initially unpolarized electrons. For collisions with atoms in isotropic states the in-plane components Pi and Pi will be zero. These results are well-known from atomic collision physics (see e.g., Kessler, 1985) and follow from the fact that the total system must be invariant under reflection in the scattering plane. For chiral target systems the collision plane is in general no longer a symmetry plane, and in-plane components P ; and Pi can be produced as shown by eq. 4.6. With regard to the in-plane components it is more common to deal with the longitudinal, PI,,and transversal, P , polarization components with respect to k,. Expressing k, in terms of n, and n2 (Fig. 4) we obtain
k,
=
n1cos 812 + nz sin 812
where 8 is the scattering angle. This yields for initially unpolarized electrons
Now consider an incident beam that is completely longitudinally polarized parallel or antiparallel to k,. The asymmetry is defined by (4.10)
where I , and 1- denote the differential cross-sections for initially parallel or antiparallel longitudinal polarization. Using
k,, = n, cos 812 - nz sin 912
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
57
(Fig. 4 ) we obtain from equation 4.6
A
=
2[Rr(g,,g;) cos 812 -
Re(,y,,g;)sin 012
-
Im(g,g;) sin 0/2
+ Im(g2g;) cos 6/21/I: I g, 1’
(4.11)
Comparing equations 4.9 and 4.1 1 we see that Pi and A differ from each other in the sign of the g,-dependent terms. Pi and A vanish identically if the system exhibits no chirality, that is, if g , and g2 vanish simultaneously. We note that Pliis the sum of a term describing time-even chirality and a term describing time-odd chirality, while A is constructed from the difference of these terms. Hence, the contributions of time-even and time-odd chirality can be obtained by summing and subtracting experimental results of Piland A . It is interesting to express Pi and A in the helicity representation. Let us denote by a(+ -) the differential cross-section when the initial electrons have “spin down” with respect to the initial momentum k,,, and when the scattered electrons have “spin up” with respect to the final momentum k,, and similarly for the other combinations. We obtain
’PI;= [a(++)- a(--)] + [a(+-)- a(-+)]
(4.12)
and the asymmetry A is obtained by reversing the sign in front of the second bracket. It can be shown that the first term of equation 4.12 vanishes if g , = 0, and the second term vanishes if gz = 0. Time-even chirality is therefore responsible for the difference between the non-flip cross-sections, and if the system exhibits time-odd chirality then the helicity flip terms will differ.
tz
FIG. 4. Collision gcometry.
58
K. BIum and D.G. Thompsoii
V. Experimental Observables: Randomly Oriented Target Systems Let us now consider electron collision with an ensemble of isotropically distributed molecules. The expressions for differential cross-section and spin polarization are obtained from equations 4.5 and 4.6 by taking the average over all molecular orientations. Certain terms will vanish as a consequence of the averaging process. Which terms vanish is dictated by symmetry. We will demonstrate this for non-chiral systems, and for time-even and time-odd chiral systems, and at the end of this section we will summarize our main conclusions. A. CHIRAL MOLECULES. ELECTRON CIRCULAR DICHROISM AND OPTICAL ACTIVITY Let us first consider collisions between electrons and (truly) chiral molecules. We will consider the transformation properties of g , and g, under certain symmetry operations that will leave the handedness of the molecules unchanged. Since these functions are independent of the spin we will have only to consider the transformation of k,, k, and the molecular vectors e,. A possible sequence of operations is shown in Fig. 5, where the circle represents the isotropic molecular ensemble. Time-reversal transforms the experimental arrangement 5A into position 5B. Applying the two rotations shown in Fig. 5C and 5D we bring the system 5B back into position 5D,which is indistinguishable from the original situation 5A. The
5a
5b
5c
5d
FIG. 5 . Sequence of symmetry operations: (A) + (9)time-reversal; (9)+ (C) rotation of 7~ Hahoul the y-axis, which brings ( - k , ) into the position of the original k, vector. (C) + (D) rotalion of T about the z-axis, which hrings k; into the position of the original k,.
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
59
molecular system remains unchanged as a whole. Any individual molecule is in general transformed into another orientation, say, the initial vectors e, are brought into a final orientation el. But because of the isotropy of the molecular orientations this does not change the overall appearance of the target system. Any time-odd pseudoscalar or scalar function must vanish under the sequence of symmetry operations, 5A --+ 5D. In particular we obtain for g? g2(k,.k,,, e,) = -g&,
(5.1)
k,,, ei)
when we apply condition 3.8 and use that g, remains invariant under rotations. el denotes the final molecular position in Fig. 5D. However, we also note that under these operations g,&, k,,, e,) = gi(k,,k,,, el)
for
i
f
2
(5.2)
Equations 5.1 and 5.2 can be interpreted in the following way: For any molecular orientation ei we can find a molecule in position ef in such a way that the contributions of these molecules to g2g:, i f 2 cancel each other if the average is taken. Hence, all terms in equations 4.5 m d 4.6 linear in g 2 vanish ifthe averccge
(5.3)
(5.4)
(5.5)
(5.6) Although the symmetry of the target is higher than for a collection of oriented molecules, the symmetry is still low enough to allow effects that would be symmetry-forbidden in the scattering of electrons from isotropic atoms. These effects are characterized by interference terms in the time-even pseudoscalar gI with the proper scalars go and gi. Using the classification of chirality of section IVB we note that this corresponds to time-even chirality. Only time-eve11 chirality is exhibited by ensembles of randomly oriented closed-shell molecules. This result, obtained for electron collisions, is a special case of more general theorems derived by Evans, who applied group theoretical methods for statistical niechanics (Evans, 1988, 1989).
60
K. Blum and D. G. Thompson
Suppose that equations 5.3 to 5.6 give the observables for collisions with a collection of molecules of one enantiomeric form. A reflection in the k, - k, plane leaves the wave vectors invariant, the orientation of the molecules will change in general (say, vectors ei are transformed into orientation e y), and the handedness of the molecules is inverted. Since the reflection is equivalent to spatial inversion followed by a rotation, which returns k, and k,, to their original positions, we can use equations 3.5 and 3.6 to obtain
where the function on the right-hand sides describes the optical isomers of the original molecules. Any bi-linear combination (. . .) occurring in equations 5.3 to 5.6, which is linear in g,. changes its sign when relation 5.7 is inserted. Hence, we deduce that the equations fiw molecules of the opposite hundedness are obtained by reversing the sign ofg, in equations 5.3 to 5.6. For racemic mixtures all terms depending linearly on g , vanish therefore. Let us consider some examples. First, we note that from equations 4.9 and 4.11 we obtain for randomly oriented molecules of one enantiomeric form
Note that (Pi) and ( A ) are equal in this case. It is evident from equation 5.9 that (PI,)and ( A ) change their sign if the handedness of the molecules is reversed, and both parameters vanish for a racemic mixture. Equation 5.9 shows that unpolarized electrons can obtain a longitudinal polarization when interacting with the target system, and left- and right-handed electrons are scattered with different intensity. Both effects are analogous to optical phenomena. When unpolarized light passes through an optically active medium, the emerging beam is generally circularly polarized. In addition, a beam of right-handed circularly polarized light is absorbed differently from a left-handed beam. This phenomena is called “circulardichroism” in optics. The analgous effect for electrons, described by equation 5.9, might be considered as the electronic analogue of circular dichroism. A related phenomena in optics is “optical activity”. Here, the chiral medium causes the polarization axis of linearly polarized light to be rotated in a plane perpendicular to the direction of propagation. The analogous effect in “electron optics” is the rotation of the transverse polarization vector. Assume for example that the initial electrons are polarized in the x-direction of the lab system (see Fig. 4). From equation 5.4 we obtain that the emerging electron beam has obtained a polarization component in the y-direction, corresponding to Pi.Expressing P.n, and P.n, in terms of unit vectors along the x- and z-directions respectively (see Fig. 4). we can rewrite equation 5.4 in the form
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
61
where H is the scattering angle. The tirst term describes the production of a polarization component perpendicular to the scattering plane, which is not a chiral effect, and is well-known from atomic collision physics (Kessler, 1985).The second and third terms can be interpreted as a rotation of the initial polarization vector out of the initial s direction towards they direction perpendicular to the incoming beam. This is truly c k a l effect (since it depends linearly on gI), which vanishes for a racemic mixture. The three effects described by equations 5.9 and 5.10 were first proposed by Farago (1980). It can be deduced from equations 5.5 and 5.6 that a rotation of a polarization vector, initially parallel to the x-axis, in the scattering plane toward the z-direction, is not a chiral effect since the corresponding terms do not depend linearly on g,.In fact, rotation of P in the collision plane has been observed in atomic physics and is described by the so-called U-parameter (Kessler. 1985).
B. RANDOMLY ORIENTED NON-(*HIRAI.MOL~CULES The resulting equations follow immediately from equations 5.7 and 5.8. Since the molecules are non-chiral the k, - k,, plane is now a symmetry plane of the total system. A reflection in this plane now gives the relation
(5.11) instead of equation 5.7, where we have used condition 3.9. Condition 5.1 1 can be interpreted in the following way: The contributions to g , of molecules in positions e, and e‘; cancel each other. Hence. if the average over all molecular orientations is taken, it follows that till terms in eqwtions 5.3 to 5.6 linear in g, iniist vanish. The resulting equations are similar to those obtained for electron scattering from spinless atoms in isotropic states. In-plane and out-of-plane components of P are decoupled. The only remaining evidence of the extended M-matrix required for molecules is in the square moduli terms I g , 1’ and I g 2 1.’ C. MOLECULES WITH TIME-ODD (FALSE) CHIRALITY
Finally, let us consider electron collisions with target systems having time-odd (or “false”) chirality. Let us choose for example the spin-axis polarized molecule discussed i n section 11. The arguments from section VA leading to equation 5.1 do not apply, since time-reversal would change the handedness of the molecule. However, the combined operation of spatial inversion and time-reversal leaves the molecular chirality invariant. y , will change its sign under the coinbined operation, the other three g-functions will not. Hence, by performing a similar sequence of operations as shown in Fig. 5 it can be shown that all terms, lineur
K. Blurn and D. G. Thompson
62
in g,, will vunish in equation 4.6 ifthe average overall orientations is taken. Contributions from g 2 will survive so that time-odd chiral effects can be observed in this case. This may be an interesting new chiral effect.
D. SUMMARY We have obtained the following results for collisions with randomly oriented molecules: (a) For non-chiral molecules (or a racemic mixture of chiral molecules), all terms in equations 4.5 and 4.6 linear in the pseudoscalar functions g ! and g, vanish. (b) The prototype of “chiral” experiments, performed so far with photons or electrons, are collisions from randomly oriented molecules with time-even chirality. In this case, all terms in equations 4.5and 4.6, linear in the timeodd pseudoscalar function g,, vanish and only time-even (or “true”) chiral effects can be observed. The results for molecules with the opposite handedness are obtained by inverting the sign of g , . (c) Collisions with isotropically distributed time-odd chiral systems allow time-odd chiral effects to be experimentally observed, and all terms in equations 4.5 and 4.6 linear in g , will vanish.
VI. Experimental Observables: Attenuation Experiments A.
STRUCTURE OF THE M-MATRIX FOR
FORWARD SCATTERING
We will first develop the general theory of these effects for electron scattering. The situation is more complicated than in the previous sections since now we have to consider forward scattering and thus the interference between scattered and unscattered beams. For non-forward scattering we used the coordinate system defined by the unit vectors 3.1, and expanded M in terms of this set (eq. 3.3). For collisions in the forward direction (k, = k, = k) the set 3.1 is not defined and one has to apply a different approach. It was shown by Blum and Thompson (1989) that the M matrix for forward collisions from molecules with fixed axes can be expressed in the form 3
M
= g,,l
+ 2 g,’(kxei).cr+ gJ[(e,xe,).e,]k.a
(6.1)
r=l
Here el, e2, e3 span the molecular coordinate system as in the preceding sections. The dynamics are contained in the functions g,’ ( i = 1, . . . , 5), which are proper scalars that depend on any scalar functions that can be formed from the set ei and
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
63
k; i.e., all five functions are invariant under rotations, spatial inversion, and timereversal. As shown by Blum and Thonipson (1989) there are only four independent functions. It is convenient, however, to work with the full expression 6.1. We stress the importance of the pseudoscalar [e,xe,].e, in equation 6.1. A term like g(k.a) (where g is a proper scalar) is not invariant under space inversion and could therefore only occur in M if the interaction itself would violate parity conservation. Neglecting parity violating effects, a contribution -k.a can therefore only occur if a pseudoscalar like [e,xe,].e, can be formed from strucrz~rulelements of the target. The importance of pseudoscalars for a description of chiral phenomena has been stressed in previous sections. All molecules of a given isomer have the same value of [e,xe2].e3,independent of the molecular orientation in space. The sign of the pseudoscalar differs for the two optical antipodes. TO THE SCATTERING AMPLITUDES B. RELATION
We have already discussed that, in general, the spin scattering matrix M is characterized by four parameters g,. This number is of course related to the number of possible spin-dependent processes. In the experinients under discussion (with k and the molecular orientation fixed) there are four possible processes, two nonflip ones with t -+ 2,and two spin-flip processes ? + T . The corresponding four sattering amplitudesfim,, wzO) are related to M by equation 3.1:
It is particularly convenient to choose k as the quantization axis and we will do this throughout this section. Choosing the direction of k as the z axis of a laboratory system we have k.cr cr- and the terms kxe, are combinations of u, and CT,.The parameter g4 depends then only on the non-flip processes and g , , g 2 and g 7 are related to the spin-flip processes only. By taking the corresponding matrix elements of A4 we obtain from 6.1 :
-
+
g,, is the spin-independent part of the amplitudesf’( 1/2, + 1/2) andf( - 112, - 112) and g4the spin-dependent part. The flip amplitudes are related to g , , g,, and gl. The explicit relation depends on the direction of k relative to the molecule and will not be derived here. However, assuming that the k is purallel to e, we obtain from 6.1 and 6.2: f(1/2, -1/2)
fC-1/2,
=
-is, - g2
1/2) = ig,
since the vector product kxe, vanishes.
-
g2
(6.5) (6.6)
K. Blurn arid D. G. Thornpsorz
64
For the following it is more convenient to use the amplitude representation and we will write the M matrix in the form: f(1/2, 1/21 ,f(1/2, - 1/21 f( - 1/2, 1/2) f ( - 1/2, - 1/2
(6.7)
C. THEORY OF THE ATTENUATION EXPERIMENT In order to describe the attenuation of electron beams we have to take the interference between scattered and unscattered beams into account. Let us consider an arbitrarily polarized electron beam incident in the i-direction, on a slab of gas with width Az, containing n molecules per unit volume. We introduce the matrix
(6.8)
N=l+iyAzM
where y = 2mdk. The matrix describes how the incident beam is affected by the medium, and takes interference into account (see e.g., Mott and Massey, 1965). We will only sketch the theoretical developments here: for more details see Thompson and Kinnin (1995) and Thompson and Blum (1997). Let I,, and Po be the initial intensity and polarization of the electron beam. The intensity I , and polarization PI after passing through the slab can be obtained from the expressions
+ P,.a)N' I,P, = (I,,/2> tr[N(l + P,,.a))N'a] I , = (I,J2) tr[N(l
(6.9) (6.10)
similar to equations 4.1 and 4.2. The further analysis requires some algebra but is straightforward. Taking the limit Az + clz, and neglecting terms of second order in the amplitudes, we obtain a set of four coupled linear first-order differential equations for I , IP,, IP,,, and IP.. These equations, which are necessary for the study of attenuation in oriented molecules, have been derived by Thompson and Kinnin (1995) and by Fandreyer (1991), and they are formally similar to equations obtained for attenuation of light beams (Baron, 1982, chapter 3 ) . The equations can be simplified for randomly oriented molecules by noting that the spin-flip amplitudes, averaged over all molecular orientations, vanish. The four equations reduce to two sets of two coupled equations, one set for I and 4 the other for IP, and IPS,and solutions can be easily obtained (cf. also Farago, 1981). The equations for I and IP. become dlldz =
-
yl Im<.f,,+ f 2 J
4l~-)ld= z - yl Im(fll - f2.,
-
r(1c)WfII - f Z 2 )
(6.11)
Im<.f,,+ ,f2?)
(6.12)
- rCZPJ
CHIRAL EFFECTS IN ELECTRON SCATTERING B Y MOLECULES
65
Adding, we obtain d(l + lP:)/dz = -2*l
+ lP.)lti~,/',,
and subtracting,
d(1 - lP.)/dz
=
-2*I
~
/P.)ll?l,j&
which are easily integrated to give /
+ 1p-== /"(I + p(.'),/-:l~~JG
I
-
(6.13)
Ip- = /"( 1 - P)'),/ TI/"'/??
(6.14)
where zI = 2 yz = 4 m d k . Thus the final intensity 1 becomes I = I()[,/-:I/JWll + p - - l l r l l / ? ? ] + I~lp![p~:i/JU/,l = ,/ -:11JJI/?> --I
(6.15)
Froin equations 4.10 and 6.9 the asymmetry becomes
A = p(I[,/-:I/IU/(~ -
:(IIJl/??]/[,/T{/JJI/\\
,
= -P! tanhlz, lrn(,f',
=
,
-P!,7,
-
+
-1
C--~llJl/7,
(6.16) (6.17)
.fz2)/2I
(6.18)
InlCf;, - f 2 2 ) / 2
provided If; - .f2? I << 1, which is what we would expect. There are several alternative, but equivalent, expressions in the literature. From the optical theorem we can express the total cross-section, Q, as ~~
Q = (4rr/k)lmj
so that 21 = I"( 1 + p!),/-Wl+ I"( 1 - P!),/
"-Q??
(6.19)
and
A
Defining Q = ( Q , ,
=
-P'! tanh[nz(Q,, - Q22)/21
(6.20)
==
-P~.hz(QII- Qz2)I2
(6.21 )
+ Q,,)/2 -_ we have A
where X = ( Q , , - Qr2)/(Qll
X
=
rz
-P!Qn?X
(6.22)
+ Q,,). __ Again using the Optical theorem we have lm( j',, - .f?z)/Itn(f',l + .f2?)
(6.23)
If I(0) is the final intensity for an unpolarized beam we note that A = P'l ln[.l(0)/ll~]X
(6.24)
K. Blum and D. G. Thompson
66
In experiments it is usual to have
I PI‘
ln[1(0)/1,~1 = 1
so that A = X . We have noted that there is a possible second experiment in which the final longitudinal polarization is measured for an initially unpolarized beam. We note from equations 6.7 and 6.8 that the induced polarization can be obtained from 21p* = [“[(l+ p‘;)e--l””~ll- (1 - p‘!)e-‘l’?’] If P!
=
(6.25)
0 then p- =
[e-:l‘”dll
= tanh[z,
-
~-:lll’l/~~]/[e-:llll~l,
Mf,I
+
p--llnl/,?
--I
(6.26) (6.27)
- f22)/21
The two electron dichroism experiments are related by A = -P‘!P-
(6.28)
We can also obtain expressions for the electron activity effect. We have noted above that for randomly oriented molecules equation 6.4 will give two coupled equations for IP, and IP,. These can be easily shown to be d(IP,)/dz = -y(IP,) I d f I I
+ f?2)y(IPI)M
J I I
-
d ( F ~ Y d= z -HIP,) Mf,,- fir) - W P , ) INflI+ f2J
(6.29) (6.30)
from which we easily obtain q p , + i p , ) = p ( p ; + jpt‘)e-’I/nVIl+ /22V2erW11 Choosing initial transverse polarization Py angle 0 from tan0 = P,/P, giving
= 0 and PI) =
-f2$
(6.31)
1 we obtain the rotation
0 = z I Wfl - f2$2
(6.32)
VII. The Physical Cause of Chiral Effects A. INTRODUCTION TO THE SCATTERING MODEL
Throughout this review we have been considering chiral effects produced in the elastic scattering of electrons by fixed, rigid molecules with closed-shell electronic configurations. We have discussed the properties and structure of the scattering matrix in terms of four functions g,and seen the necessary requirement for g, and/or gz to be non-zero and discussed the observables of experimental interest. In this section we turn to the dynamical mechanism and nature of the electron-molecule interaction that is responsible for the chiral effect. Several differ-
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
67
ent mechanisms have been suggested but we will discuss them from within a single theoretical framework. We first note that the interaction between the electron and the molecule must be spin-dependent. This can be seen from the relation between the g , functions and the scattering amplitudes (ml I M I m,,) of equation 3.2. Assuming the n3 axis of equation 3.1 to be the quantization axis it is straightforward to show (Johnston rt ~ l . 1993) , that 2g,, = (112 I M 1 112) + (- 112 I M 1
112)
(7.1)
112)
(7.2)
2i& = (-112IMl 112) - ( 1 / 2 l M l - 1/2)
(7.3)
2g,
(7.4)
2g,
=
=
( 1 / 2 l M l - 112)
-
+ (-112IMl
( 1 1 2 l M l 1/2) - ( - 1 1 2 I M l
-
112)
Using spin-independent interactions in elastic scattering by closed-shell, spinzero molecules, the two non-flip amplitudes will be equal and the spin-flip amplitudes will be zero. The target cannot differentiate between electrons of different spin and cannot change the spin of the continuum electron. Thus, from equations 7.1 and 7.2, we will have g , = 0 = gz for spin-independent interactions and there will be no contribution to the chiral effect. To see how we can incorporate a spin-dependent interaction we look in detail at a particular scattering model (Thompson, 1996), which leads to equations for the continuum electron function F(r, (T). We consider the molecule to have N electrons, spatial and spin coordinates s,= I , , ui and M nuclei, coordinates Rj and charges Z,. The total wavefunction for the system of molecule plus electron, q ( x , , . . . , xN , can be expanded in terms of the molecular eigenstates, @, in the usual way. As a first approximation let us retain only the ground state, @,,, in this expansion; i.e., we take (dropping the suffix) Ni I
*(XI,. . . X N + , ) =
y (-1)'+'@(i-9F(x,) I=
(7.5)
I
where (i- I ) represents all coordinates x,, . . . x N, except x,. Assuming @ is antisymmetric for interchange of electrons the form of equation 7.5 ensures that v' is also antisymmetric. Equations for F, the scattered electron function, can be obtained from l @ * ( x , . . . X , ~ ) [ H.(.~. , N
+ 1) = F ] W ( X ~
Since the Hamiltonian H is synimetric for interchange of electrons this reduces to
0
=
/@([A'
+ 11 - ' ) * [ H I , . . . , N + I ] -
[@([A'+ I I
~
' )F(x,,)
-
F]
N@(N I )F(xN)dxI
K. BIum mid D. G. Thompson
68
Write the Hamiltonian as
H(l . . . N
+ 1) = H,,,,,,(1 . . . N) + V,,,,(I, . . . , N + 1) - 1/2v;,
(7.8)
where H,,,,,, is the molecular Hamiltonian and V,,, is the interaction potential between the electron and the molecule. V,,,, is the sum of a spin-dependent term V,,, discussed below, and a spin-independent term Vel: M
N
Vel
=
-c L Ir, I=
-
r N i l) - I
+ 1Z j / l R, - rN.,II
(7.9)
j-- I
I
Let us assume that @ is given by a single determinant of single-electron functions u J r , a). @(xl, . . . x,) = N-"'
E,,p
r
q 4 . . ur(N)
%(11
1
(7.10)
where ect equals 1 if a . . . 7~ is an even permutation of 1 . . . N , equals - 1 for odd permutations, and equals zero otherwise. Assuming H,,,,,,(l,. . . , N ) @ ( x , . . . x,) E
=
El @(xl . . . x N )
=
El
+ k'I2
(7.11) (7.12)
and neglecting V,(l and any spin-dependent terms in H,,,, equation 7.7 reduces to the usual Static-Exchange or Hartree-Fock approximation. [(-1/2)V'
- (112)k'
+ V,,(r)]F(r)=
K(r, s, a)F(s, a)ds d u
(7.13)
1Z,/ I Rj- r I
(7.14)
The static potential is
VJr)
=
2 J' u:(s, a)I r - s
(-I
u,,(s, a)& d a +
ru
/
and the exchange kernel is
K(r, s, d =
1uX(s, a ) I r
-
s I-' uJr, (T)
(7.15)
0
We stress again that this equation alone will not give any chiral effect. We now discuss several ways we can include spin-dependent potentials in order to obtain a positive effect. B.
SPIN-ORBIT INTERACTION INVOLVING THE INCIDENT
ELECTRON
Consider the spin-orbit interaction between the incident electron and the nuclei. For simplicity let the interaction be with just one nucleus, assumed to have a large value of 2 if this nucleus is at the center of our coordinate system "\d
with 5(r) = ZI2c'r'.
= K r ) I,+ I .SN I I
(7.16)
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
69
This interaction has been used in the calculations of Blum and Thompson (1989). Fandreyer et (11. (1990), Johnston et trl. (1993). Greer and Thompson (1995). and Smith rt ( I / . (1997). Appreciable chiral effects are obtained showing that this spinorbit term is a significant contribution to the required spin-dependent interaction. We obtain an interesting pictorial insight into this effect by looking at it in t e r m of a simplified independent atom model. Assume that an unpolarized beam (with wave vector k,,) is scattered by a hetero-polar oriented diatomic molecule AB as shown in Fig. 6. Assume that B is a heavy atom, and that A is sufficiently light so that is does not influence the spin. Assume that the electrons :ire first scattered by B. and consider that peat of the electron beam scattered in the direction of A. The collision with B produces a spin polarization P perpendicular to the k, - n plane (denoted by @ in the figure). Collision with the light atom A does not noticeably change the direction of P. Hence, the polarization of the outgoing electrons is still perpendicular to the k,, - n plane, and if k, lies outside this plane the electrons will possess a longitudinal spin component. Production of a longitudinal effect is a chiral effect as discussed in previous sections. In conclusion it is the combination of the helical movement of the electron, defined by the three vectors k,,, n, k,, and the usual spin-orbit interaction which creates chiral effects in this model. Numerical calculations within the independent atom model have been performed by Hayishi (1988).
c. SPIN-ORBIT INTERACTION I N V O L V I N G THE Mt)Lt:CLILAR
EIKTRONS
To be consistent, we should also include the spin-orbit interaction between the molecular electrons and the nuclei. i.e., there should be an additional spin-orbit term in the molecular Hamiltonian H,,,,,,.The effect of such a term is to change the form of the niolecular orbitals uck.Remember that we are considering a closedshell system that contains subshells of two electrons. Neglecting spin-orbit these electrons have the .same spatial wavefunction but opposite spins. In a collision electron exchange can only take place if the spin is unchanged and the interaction
d"
70
K. Blum and D. G . Thompson
will be the same whether the incident electron has spin “up” or spin “down”. However, when spin-orbit is included the electron wave functions contain contributions from both spins. The perturbation is asymmetric; in the collision, exchange of an electron with unchanged spin is different for the “up” and “down” states and the spin of the incident electron can now be altered by exchange. Another way of looking at this is to consider the two-electron wavefunction for the subshell. Without spin-orbit it has singlet total spin; inclusion of spin-orbit leads to a wave function with a mixture of singlet and triplet spins. Exchange of the incident electron with one of the subshell electrons during the collision is spin-dependent. In summary if we calculate the exchange term, equation 7.15, with orbitals perturbed by the spin-orbit interaction we will obtain a spin-dependent potential that will give a non-zero contribution to the chiral effect. Rich et ul. (1982) discussed this contribution for the elastic scattering of electrons by closed-shell targets and Hegstrom (1982) also considered positronium formation in positron collisions and ionization in electron collisions. There were no detailed calculations in any of these cases but it was suggested that this mechanism would be a major factor in the calculation of chiral effects for molecules containing heavy atoms since the effect varies as 2* where Z is the nuclear charge. Hegstrom referred to this treatment as “the bound helical electron model”, following the ideas of the “helical electron gas model” of Garay and Hrasko (1975). He also introduced the concept of “helicity density”. For a two-electron subshell this is defined as (7.17) When spin-orbit is neglected the contribution from the two electrons is equal and opposite and h is zero. With spin-orbit included h still vanishes for an atom or molecule with center of inversion; it is non-zero for oriented molecules and the average overall orientations is zero for non-chiral molecules. h can therefore be used as a measure of the chirality of a bound system. This includes even atoms if the electro-weak interaction is included (Hegstrom, 1991).Hegstrom obtained approximate expressions for positronium and ionization asymmetries in terms of h.
D. THESPIN-OTHER-ORBIT INTERACTION Gallup (1994) has pointed out that the incident electron causes considerable molecular distortion and the electron spin can interact with these induced currents through a spin-other-orbit potential. We can include this suggestion in our scattering model by considering more terms in our eigen function expansion of the total wavefunction. Inclusion of other states (either physical or pseudo-states) allows for the effect of molecular distortion and leads to the solution of coupled integrodifferential equations for the scattered functions F,(r, v).Formally the procedure
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
71
is equivalent to adding an "optical potential" to V,, in equation 7.13, which may be non-local and complex. For incident energies below the first excitation threshold this extra term, which can be viewed as producing distortion through "virtual excitations", gives a potential that varies as I-' as I , the incident electron coordinate, tends to infinity, and has a size proportional to the polarizability of the molecule. A simpler way of dealing with this term is to use perturbation theory to obtain the distortion of the molecule due to the incident electron and then to calculate the effect of this distortion back on the incident electron. We replace @ in the second line of equation 7.7 by @ 0' where Qi ) is the perturbation correction to the ground state. will have the form
+
@"' =
'
1@, I V,",I % ) 4 4 - E,)
(7.18)
i- I
where the sum is over all excited moleculnr states and V,,,,is the spin-independent part of the electron interaction with the molecule. There will be an extra direct "polarization" potential, (Q,, I V,,,,10") and ) a new spin-dependent term (aoI V,,,,,1 @'I)) where V,,,,, is the spin-other-orbit correction to the Hamiltonian. (There will also be extra exchange terms.) The spin-other-orbit term has the form
1
('1
I=
I Vim I Q,X@[,I vwo I Q ~ ) / ( E( , E,)
(7.19)
I
Gallup has pointed out that this expression has similarities to the form obtained from optical activity theories. The first integral contains matrix elements involving the electric dipole operator and the second matrix elements involving the magnetic dipole operator. Gallup approximated the above ideas in a resonance model and carried out a calculation for propylene epoxide (C,H,O), obtaining a chiral effect of the order of 10 '. This value is surprisingly large, remembering that tne molecule contains only light atoms, and that the spin-other-orbit interaction is known to be very small. In conclusion, we have shown here how the various dynamical theories proposed so far are related to each other. In addition we have outlined a mathematical framework that could be used as the basis for detailed numerical calculations.
VIII. Theoretical and Computational Details A. THESCATTER~NG AMPLITUDE
Some of the equations discussed in section VII have been solved by expanding the continuum function F(r, v) in an uncoupled basis of angular and spin functions:
K. Bluiri und D.G. Thompson
72
Coupled (integro-) differential equations can be obtained for the!;,,,,,,,,, having the form
Explicit expressions for the various contributions to the V matrix elements from static, exchange and spin-dependent interactions have been given by Thompson (1996) and Thompson and Blum (1997) (and in part by Greer and Thompson, 1995). We discuss their general structure later in this section. Rather than use the coordinate axes defined in equation 3.1, it is more convenient in computational work to introduce a laboratory fixed frame, x~?, which the electron is incident along the z-axis and scattered in the xz plane. The orientation of the molecule is now given by the relation between the molecular axes e, and xyz; we can consider eito be obtained from xyz by a rotation through Euler angles spy). Let the initial and final spin functions be x,,,~, and xMl(quantization axis z ) and let the scattering amplitude for spin transition M(, to MI be f ( M , , M,,, k,; 0);the asymptotic form of F(r, g)is then
a(=
F(r, u)4 ekgr xMo+ X J M , , M,,,k,;
a)xMld"/r
as
r -+
a
(8.3)
MI
With the introduction of the new frame xyz the relation between the g , functions and the scattering amplitudes now becomes (Johnston et al., 1993; note that in that paper the scattering is into the yz plane)
2g, = f ( 1/2, 1/2) t f ( - 1/2, - 112) 2g,
=
[f(1/2, 1/21 -j(-1/2,
(8.4)
-1/2)1 cos el2
+ i [ f (1/2, - 1/2) - f(- 1/2, 1/2)] sin 192
(8.5)
2g2 = 1-f( 1/2, 1/2) t ,f(- 1/2, - 1/2)] sin 812 - i [ f ( - 1/2, 1/2) - f (1/2, - 1/2)] cos 1312
(8.6)
2g7 = -f(- 1/2, 1/2) - f(1/2, - 112)
(8.7)
The equations for f, ( a = lmm,) provide a set of linearly independent solutions which we will label with p(= App,,). The S matrix is obtained in the usual way from the asymptotic form of the!&: -,(A,
-lv/?j
% -, '%p
el(kt-17r/21
Greer and Thompson (1995) have shown that the scattering amplitude of equation 8.3 can be expressed as
f ( M , , M,,;ii,,R) =
1(l/ik)[T(21 + l)]Il?i'-A Y@,) A
(YO'/
D;l,,,*(n,D;p(~>D;l,,,,*(o)Dgwp)7-,lj,y (8.8)
73
CHIRAL EFFECTS IN ELECTRON SCATTERING B Y MOLECULES
where T = S - 1 and the D are rotation matrices (cf. Brink and Satchler. 1993). The quantization axis for n2 and tn\ ( p and p s is ) the z-axis of the body-fixed coordinate system ei. We note immediately that if the electron-molecule interaction is spin-independent then T/ml/2.Ap!12 -
(8.9)
I12,Ap- I/Z
(8.10) (8.11) from which it follows that (as noted already in section VII) ,f(l/2, 112) =.f(-1/2, -1/2)
(8.12)
f(l/2, - 1/2) = 0
(8.13)
j(-l/2, 1/2) = 0
(8.14)
and there are no chiral effects. To obtain a non-zero contribution to the chiral effect the interaction must be such that the potential matrix elements have the property "/~~~,A~III, '/J,!,!Ij.,\p
~111,
' '
'Iiii-
m,Ap
-HI,,
(8.15) (8.16)
These conditions are satisfied by the spin-orbit interaction between the continuum electron and the molecule; however, they are not satisfied by the static potential and are only satisfied by the exchange terms when spin-orbit is included in the molecular Hamiltonian. In this latter case the molecular orbitals are expanded in a similar fashion to the continuum function u , = r-l
1
";:J,,,l,(y)y;"(;')~J,l~(~)
(8.17)
llllm,
and not as for the spin-independent molecular Hamiltonian when the expansion is (8.18)
B. THE ATTENL~ATION EXPERIMENT AND THE ASYMMETRY OF THE T MATRIX We have seen (equation 6.18) that the asymmetry in the attenuation experiment depends on the difference between the two non-flip amplitudes. Thompson and Bluni (1997) show, using equation 8.8, that this difference, averaged over all molecular oricntations, can be written as
(f(1/2, 1/2) - f ( - 1/2. - 1/2)) =
2 P,,jT,jo 43
(8.19)
K. Blirm and D.G. Thompson
74 where
P,, = (1/2ik)[(21
i(: ^o
+ 1)(2A + l)]”2j’-A(-
A) i
l
-in
h p
) (!iT\ &)
l M,
Interchanging the dummy indices a and and I I - A 1 = I gives
(,f(1/2, 1/2) -f(-1/2,
6”’
p. and noting that P,,
= (-
-1P,,T,,
-1/2)) =
(8.20)
P,, (8.21)
UP
We have immediately that the attenuation asymmetry is zero if the T matrix is symmetric. Thompson and Blum (1997) have shown that T is symmetric for any molecule containing a reflection plane and that the chiral effect is zero for molecules with a center of inversion because the T matrix elements are zero for 1 1 A I = 1. It is interesting to note that chiral molecules can have C,, (or D,,) rotational axes of symmetry; Thompson and Blum (1997) show that for a molecule with a C, axis of symmetry, chosen to be the x-axis,
+
Since in equation 8.20 1 1 - A 1 = 1 and m m, = p + p\,we have T,, = -Tctp for the matrix elements of interest. In scattering calculations it is usual to introduce a K matrix related to S by S = (1 - iK)-‘ (1 iK). Since S is unitary it is easy to show that K is Hermitian. In most scattering calculations K is real, from which it can be shown that S and T are symmetric. However, this is not the case for electron scattering by chiral molecules using the uncoupled (Imm,)basis discussed here. Greer and Thompson (1995) have shown that the potential matrix elements in this basis are Hermitian, but not in general real.
+
C. ATTENUATION EXPERIMENT: BEHAVIOR AS K
+0
We have noted that the asymmetry factor is closely related to (cf. equation 6.22)
X =
h [ f (1/2, 1/2) - f ( - 1/2, - 1/2)] Im[f(112, 112) + fC- 1/2, - 1/2)]
(8.23)
Fandreyer et ul. (1990) discussed the behavior of X as k + 0 within a distorted wave treatment and showed that X -+ const. We can verify this result for the scattering model being discussed here. Let us assume that as k + 0 the expression for (f( 1/2, 1/21 - f(- 112, - 1/21) is dominated by a single T,, element with ( I , A) = (0, 1) or (1,O). The difference varies as Im(T,,)/k. Since
S
=
(1 - iK)-’(l
+ iK) = (1 + K’)-’(l - K’ + 2iK)
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
75
the difference varies as Re K,,,lk, that is, k . The denominator of equation 8.23 varies as k and thus we have our result X+const
as k + O
Thus theoretically we might expect appreciable attenuation asymmetries for very low electron energies; however the experimental evidence is ambiguous. We note that if Re K,,, = 0 the variation of the numerator will be a higher power of K and X + 0 as k + 0. This occurs, for example, for chiral molecules with a C2 axis of symmetry.
IX. Results of Numerical Calculations Several calculations have been carried out using the scattering model described in section 7.2, i.e., the inclusion of the spin-orbit interaction between the continuum electron and the nuclei of the molecule. Various approximations have been made to the scattering equations discussed in section 8. A. ORIENTED MOLECULES
(a) Johnson ef ril. (1993) carried out calculations for a model niolecule BiH,,having the form of Fig. 1, with the Bismuth atom at the center of the coordinate system and the hydrogen atoms along the three coordinate axes. Bismuth, with Z = 83, was chosen to maximize the spin-orbit effect. The bond lengths AH,, AH, and AH, were chosen to be 2 , 3 and 4 a.u. respectively. The scattering equations were solved using a distorted wave approximation and neglected exchange. An interesting point is to find out the importance of time-even and time-odd effects in collisions with chiral molecules. Johnston et ril. considered the Pi,and A of equations 4.9 and 4.11; they calculated (lR)(Pi ? A ) where the ' f ' gives a timeeven quantity and the '-' a time-odd quantity. Their results are shown as a function of molecular orientation 4. The initial orientation of the molecules ( 4 = 0) is as follows: the el axis is perpendicular to the scattering plane and parallel to the n, axis, e2 is parallel to n,. and e, is parallel to n,. The scattering angle is kept fixed and the molecule rotated through 4 about the n2 axis (e,) axis (Fig. 7). Figs. 8-10 show results for different scattering angles and energies. The values of (Pi;t A)(1/2) vary considerably with molecular orientation. Their size is in general rather small. of the order lo-', though much larger values can be found, particularly in two cases: (i) If the cross-section becomes small then Pi 2 A can increase in magnitude significantly. Enhancement of polarization effects at minima in the differential cross-section is well known in atomic physics. A comparison of Fig. 8 8 with Fig. 9B shows that for electron energy 10 eV and scattering angle 90" I is much smaller than for scattering angle 30". Consequently the cor-
76
K. Bluin and D. G. Thompson
\
*3
FIG. 7. Groinelry used in numerical calculations.
responding values of Pi and A , and of their time-even and time-odd chiral parts (Fig. 8A), are considerably larger than in Fig. 9A. (ii) Again, as in atomic physics, large polarization effects can be expected in resonance regions. It was noted in Fandreyer et al. (1990) that there are resonances at electron energies of approximately 0.5 eV for BiH3. Fig. 10 illustrates this case and shows that in the vicinity of resonances chiral parameters can attain considerable values. (b) Greer and Thompson (1995) have solved the full exchange equations plus spin-orbit interaction (involving heavy atom only) for the planar molecules H,O and H2S. If the molecules are in the xz plane (the scattering plane) or the y z plane (perpendicular to the scattering plane), the chiral effect is zero. An intermediate orientation, R = (45,45,0) was chosen. Results for Pi and A, as functions of scattering angle, are shown for H2S in Fig. 11. Since the value of 2 for S is only 16 the chiral effects were not expected to be large. In general, values of were obtained but again values were greatly enhanced near cross-section minima. The broad resonance structure in H I S (cf. Greer and Thompson, 1994) may influence the results slightly, enhancing them at 5 eV compared to 10 eV, but the effect is not so pronounced as for BiH, where the resonances are much narrower.
B. RANDOMLY ORIENTED MOLECULES This section, of course, only applies to chiral molecules. (a) Fandreyer er al. (1990) used the model molecule approach described in section LXA above to obtain values for the attenuation asymmetry discussed in
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
77
PIC;. 8. Scattering 01' 10 eV electrons with xattcring anglc 30". ( A ) (P,:+ A ) / ? anlid linc) and (IJ: A)/' (dotted line) are plotted as a lunctlon o l the niolecular orientation C/J defined in thc text: ( R ) Diffcrcntial cross-section as a fuunctlnn o f cb. (From Johnston er ( I / . . 1993.) -
K. Blirm and D. G. Thompson
78
4
0
50
t 0I 0
1
1 50
1
'
I
I
1 I00
1
I
I
I
'
I
150
' '
'
200
300
250
150 ZOO $(degrees)
100
I
'
1
I
'
250
I
'
I
360
I
300
1
1
1.' 050
$(degrees)
Frc;. 9. Scattering of 10 eV electrons with scattering angle 90". ( A ) as for fig. 8A: ( B ) as for Fig 8B. (From Johnston cf ( I / . , 1993.)
CHIRAL EFFECTS IN ELECTRON SCATTERING B Y MOLECULES
79
1
c
20
1-
I
t FIG. 10. Scattering of 0.5 eV electrons with scartel-ingangle of30". ( A ) as Ihr Fig. 8A: (B) as for Fig. 8B. (From Johnston e t n l . , 1993.)
80
K. BIurn arid D. G. Thompson H2S 5 OV
t
H2S 5 eV
Oo's DO1
j,
I
Frc;. 1 I . Pi nnd A for clectron suttcring hy H I S at orientation ( u , 0. y ) = ( d 4 , d.1.0): A,Pi: H. A . ( A ) 5 eV; (B) 9eV. (From Grecr and Thompson, 1995.)
section VI. Their results for the X of equation 6.23 (which is closely related to the asymmetry A (equations 4.9 and 6.22)) for the model molecule BiH,are given in Table 2. The values are much smaller than for oriented molecules, even with the large Z of Bismuth, except for the low-energy region where we have already noted there is sharp resonance structure for this scattering model at 0.5 eV. X also
TABLE 2 REsLII,TS ITOR THE
Energy in eV
x . 10''
0 1
-72.7
.I
-76.3
X OF EQIIA'I'ION 6.23 FOR THE MODELMOLEeiiLE BiH, .2 -64.0
.5
-44.3
1.0
-10.3
2.0
3.0
2.13
1.64
4.0
6.0
.I3 - . 5 8
8.0
10.0
p.44
p.47
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES TABLE 3 REWLISFOR THE X o~ EQUATION 6.23
FOR
81
N,.Y,. ~~
Energy
x .lo’
in
eV
I ,O
2.0
2.0
4.5
-
4.0
6.0
1.6
- 1.4
8.0
.I1
10.0
-54
varies considerably with energy, even changing sign several times, from which we can infer that this type of experiment cannot be used to identify the handedness of a chiral molecule. (b) Smith et al. (1997) have reported preliminary results for electron scattering by H,S,, a chiral molecule with a C2 symmetry axis. They have extended the Continuum Multiple Scattering Method to include the spin-orbit interaction between the incident electron and each nucleus. Coordinate space is divided into regions by surrounding each nucleus with a sphere; the scattering problem is solved with simple potentials within each sphere and between the spheres, solutions being joined on the sphere boundaries by constraints of continuity. Exchange was neglected. Results for the parameter X are shown in Table 3; values are small, of the order of lo-’, which may increase for molecules containing heavier atoms. The variation with energy is very similar to the results for the model molecule. An approximate version of this technique has been used by Hayashi ( 1988) for a model molecule.
X. Experimental Results As discussed in section VI, spin-dependent attenuation of electron beams in the vapor of chiral molecules gives information on chiral interactions. Measurements of the asymmetry A , defined in equation 4.10 (see also section VIC) have been performed by several groups in recent years. Campbell and Farago (1985) did observe an asymmetry in the transmission through camphor vapor of a 5 eV electron beam with 28 percent longitudinal polarization. They found an asymmetry of A lo-’ and concluded that they had observed electron circular dichroism. However, the theoretical interpretation caused problems. As shown in sections VII and VIII, most calculations show that sufficiently heavy nuclei in the molecule are required in order to obtain noticeable effects. Camphor, however, is composed of relatively light nucei. In fact, a repitition of the measurements by the Miinster groups (Meyer and Kessler, 1995) and by the Nebraska group (Trantham et ul., 1995) gave the result that A was zero for electron energies between 0.9 and 10 eV with an experimental accuracy of much better than lo-‘. The great impor-
-
K. Blum arid D. G. Thompson
82 3 2
. .-
go1 7
F 0
z E E, -1
a
-2
-3
1
2
3
4
5
6
7
8
9
10
Electron Energy (eV)
. o
(-)-Dibmmocarnphor
(+)-Dibromocamphor
t
-2 -31
"
1
"
2
"
3
"
4
"
5
"
6
"
7
"
8
"
9
"
10
1
Electron Energy (eV)
FIG. 12. (A) Asymmetry A measured for D-bromocamphor (filled symbols) and L-bromocainphor (open symbols) as a function or electron encrgy E. The error bars indicafe the statistical uncertainty. Where no error bars are given they are smaller than the symbols denoting the measured quantities. (From Mayer et d ,1996.)(B) As for (A): Dibromo-camphor. (From C. Nolting and J. Kcasler, unpublished.)
tance of Campbell and Farago's experiment lies in the fact that it stimulated theoretical and experimental work in this area. Finally, the Miinster group succeeded in demonstrating experimentally electron dichroism. They used targets with heavy nuclei like Yb(hfc), (with Ytterbium 2 = 70 as the heavy nucleus), Bromo-camphor (with 2 = 35 for Bromine), Pr(hfc), (Zp,= 59), and Eu(hfc), (ZEff= 63) (Mayer and Kessler, 1995; Mayer et al, 1996"). Results are shown in Figs. 12 and 13. The results show that A changes sign when the handedness of the molecules is inverted. A is largest in resonance regions as follows from a separate experimental study of the resonances. It is *Note that in both papers the sign of the asymmetries must be inverted in the published figures
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
31
-4L
2 1
83
li,
I
P
0
(+)-Yb(hfc),
4
1 0 -1 -2 1
2
3
4
5
6
7
8
9
10
Electron Energy (eV)
FIG. 13. ( A ) As for Fig. 12(A): Pr(hCcJ,.(Frtrni C. Nolting and J. Kcsdcr, unpuhlishcd.) ( B ) As for Fig. 121A):Eu(hfc),. (FI.OIII C. Nolting and J . Kcsslcr, ~inpuhlishetl.)(C) As Cor Fig. IZ(A): Yh (hlc)+.(From Mayer et al.. 1906.)
interesting that in all cases the values of A are of the same order of magnitude. Bromo-camphor was not anticipated to show an electron dichroism as large as measured for Yb(hfc), because the atomic number of bronune is much lower than for Ytterbium. The experimental results indicate that in addition to the heavy atoms causing spin-orbit interaction, the structure of the molecule is of importance too. Mayer et al. (1996) speculated that in the case of Yh(hfc), the large number of light atoms by which the heavy atom is surrounded may be detrimental to the effect. However, further investigation is now required to clarify the role of the molecular structure for chiral effects, and to interprete the experiments. In conclusion, the experiments of the Munster group represent the first clear evidence for chiral effects in electron-molecule collisions, and for the existence of “electron-dichroism”. The results give credence to the Ulbricht-Vester hypothesis, mentioned in the introduction, and may therefore be of importance for further attempts to solve the problem of biological homochirality in nature.
84
K. Blum and D. G . Thompson
XI. Conclusions and Outlook In the present paper we have reviewed theoretical and experimental results on spin-dependent collisions between electrons and molecules. Because of its inherent simplicity, low-energy electron-molecule scattering is an excellent case for explaining and exploring the fundamental aspects of chirality, including even the more subtle aspects like time-even and time-odd (true and false) chirality. Numerical model calculations have shown that the spin asymmetries are of the order of 10-3-10-' for randomly oriented molecules, but that much larger values can be expected for oriented molecules, particularly near resonances, and if the molecules contain at least one heavy atom. On the experimental side, important progress has been achieved by the Munster group, resulting in the first unequivocal proof of the existence of the proposed chiral effects. These results open new and interesting possibilities for investigations of spindependent electron-molecule collisions. One possible point of departure could be to develop some deeper understanding of the present data. For example, it is very interesting that the values obtained for the asymmetry parameters for randomly oriented molecules are of the same order of magnitude in nearly all numerical and experimental studies performed so far, although the molecules used have completely different structures. At present the underlying physical basis for this is unclear. Furthermore, more numerical studies on real molecules are required. This requires the development of efficient computer codes for electron-molecule collisions including spin-dependent interactions, starting perhaps from collisions with oriented diatomics. With regard to possible tests of the Ulbricht-Vester hypothesis, these studies should be extended to ionization and fragmentation. On the experimental side, it might be worthwhile to extend the present studies to collisions with oriented molecules (e.g., molecules adsorbed on surfaces, or laser-excited molecules in the gas phase). Here, larger spin-dependent effects can be expected than for randomly oriented target systems. In addition, besides timeeven, time-odd chiral effects, which vanish for randomly oriented (closed-shell) molecules, can also be studied. Of course, these experiments are much more involved than measurements with unoriented molecules, but the qualitatively new information that can be obtained from them justifies the effort. Hence, further theoretical and experimental investigations are needed in order to fully explore these new avenues of research, and to get a deeper understanding of the underlying physics.
XII. Acknowledgements We are grateful to J. Kessler and C. Nolting for interesting discussions on the experimental aspects, and for permission to use their unpublished data. We would also like to thank A. Busalla for his help in preparing the figures. This work has
CHIRAL EFFECTS IN ELECTRON SCATTERING BY MOLECULES
85
been supported by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 216 "Polarisation and Correlation in Atoinic Collision Complexes"), and by EC-HCM Network Contract Number ERB CHRX CT 920013, and by Nato Collaborative Research Grant CRG 930056.
XIII. References
86
K. Blum und D. G. Thompson
Mason, S. F. (1982). “Molecular Optical Activity and Chiral Discrimination.” Cambridge University Press (Cambridge). Mason, S. F, (1988). C/iem.Soc.Rev. 17, 347. Mayer. S. and Kessler, J . (1995). Phys.Ret:Lett. 74, 4803. 29, 3497. Mayer. S., Nolting, C., and Kesalcr, J. ( 1996).J.Phvs.B:At.Mo/.Opt.P/rv.s. ~ ,S . W. (1965). “Theory of Atomic Collisions” 2nd ed. Oxford University Mott, N. F. and M ~ s s KH. Press (Oxford). Nolting, C. (1997). Dissertation. Universitlt Miinster. Unpublished. Rich, A,, van House, J., and Hegstrom, R. A. ( 1 982). P/i?;.s.Rrv.Lett.48, I34 I. Rosenfeld, L. (1928). Z.Phy.v. 52, 161. Sinith, I., Thoinpson, D., and Bluin. K. (1997). To be puhlishcd. Thompson, D. G. (1996). J.Cm.P/iys. In press. Thompson, D. and Kinnin, M. (1995). J . P / i ~ , s . B : A f . M o / . O / ~ r . P /28, i y s .2473. Thompson, D. G . and Bluni, K. (1997). “Proceedings of PECAM I1 Conference. Belfast 1996” (P.G. Burke and C. Joachain, Eds.). Plcnum (New York). 28, L543. Trantham, K.W., Johnston, M.E., and Gay. T. J. (1995). J.Phys.B:At.Mol.O~~t.Ph~s. Ulbricht, T. and Vester, F. ( 1962). Terrrrhedrm. 18. 629. Vester, F., Ulbricht. T., and Krauss, H. (1989). Nufur. Westphal, C.. Bausinamm, J., Gelztaff, M., and Schonhcnse, G . (1989). P/fys.Rev.Lrtt. 63, 1.51. Williams, R. (1968). Phvs.Rev.Lett. 2 I, 342.
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY OF POINT DEFECTS IN CONDENSED HELIUM
1. Intrduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I 1 . Structure 01' the Point Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
III. I v.
V.
v1. V1I .
nowballs a n L l B u h b l e ~ . . . . . . . . . . . . . . . . . . . . . . . R . Atomic Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impl:intation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Eleclron Buhbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Liquid Matrice\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Solid Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R . Excited States of Hr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Atoms w i t h Negative Chemical Polerilial . . . . . . . . . . . . . . . . . . . . . . . D . Atoms with Positive Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . I . Recombination Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Laser-Induced Fluoresccnce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Nonradiativc Pi-ocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Atonis and Molec~ile~ in and on H r Clusters . . . . . . . . . . . . . . . . . . . . . . Magnetic Resonance Spectro\copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Electron Rubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . P;iram;ignetic A t o m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Optical Pumping and Magnetic Resonatice i n Zeeman Multipletts . . . . 2 . Relaxation Time\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Hyperline Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Matrix Elfecta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S . EDM Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Reinark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refercnces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8X 90 90
92 YS
97 97 9X 100 101 10'2 102 104 107 109 10') 111
111 112 1I ? 111 114
115 116
117 117
88
Srrguei I. Kunorsky and Antoine Weis
I. Introduction The trapping of atomic and molecular impurities in matrices of condensed helium is a particular case of the matrix isolation technique-a method that was developed in the early 1950s for the study of free radicals and other unstable or reactive species. Earlier attempts to trap such particles have used organic glasses, alkali halide discs, or polymer films cooled to liquid nitrogen temperature. When liquid He became available at reasonable costs these studies were extended to solid heavy noble gas (NG) matrices. Such matrices offered a number of advantages, the most important ones being the possibility of preparing samples with a well-determined dilution and stabilizing even the most reactive species. The term ”matrix isolation” itself first appeared in 1954 in the work of E. Whittle et (11. (1954). In the following ten years the method was widely used not only for spectroscopic studies of reactive species, but also in electron spin resonance (ESR) studies of NG-matrix isolated paramagnetic atoms and free radicals. The invention in the 1960s of alternative spectroscopic techniques and in particular the use of supersonic jets allowed the study of low-temperature spectra in a collision-free environment. This method was free from the “matrix effects” inherent to the matrix isolation technique, thus significantly lowering its usefulness, without, however, making it completely obsolete; a certain revival of the activity in the field can even be observed in the last ten years. Traditionally, matrix isolation is the means to trap the species of interest in a chemically neutral environment for subsequent optical or magnetic resonance (MR) investigations. The obvious disadvantage of the method is that the matrix, as a rule, perturbs the intrinsic properties of the particle under investigation. With heavy NG matrices the problem is particularly severe because of the formation of multiple trapping sites. The results obtained with this method are therefore often considered only as a starting point for the planning and/or analysis of high-resolution gas phase investigations. At the same time, the trapped particles can serve as microscopic probes, which provide valuable information about the structure and properties of the host matrix, its interaction with the impurity, and perturbations of the structure caused by the impurities. In this respect matrix effects, which for a long time have been considered as a main drawback of the technique, have often become the main subject of investigations. The field of conventional matrix isolation spectroscopy has been reviewed by several authors (Andrews and Moskovits, 1989; Clark and Hester, 1989; Bondybey rt ul., 1996). In the present paper we concentrate on novel aspects that have emerged from the extension of matrix isolation spectroscopy to the quantum host matrices of superfluid and solid ‘He. The liquid and solid phases of ‘He have many unique and outstanding features, which are described for example by Wilks (1 967). One of the most prominent properties of condensed ‘He is the fact that it becomes superfluid below 2.17K, thus
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
89
showing macroscopic quantum coherence. and that it stays liquid down to T = OK. The de Broglie wavelength of He atoms in the condensed phase is larger than the interatomic separation, and the quantum properties arise as a consequence of the strong overlap of the atomic wave functions. The large zero point vibrations of the He atoms and their mutual hard-core type repulsion at short distances are responsible for the large molar volume of condensed He and the fact that helium can be solidified only when pressurized above 35 atmospheres. The different phases of condensed ' H e relevant for the present discussion are shown in Fig. 1. He1 and He11 denote the normal and superfluid liquid components, separated by the so-called A-line. Three crystalline phase are known for solid He. A face-centered cubic (f.c.c.) phase characteristic of all other noble gas crystals exists only above 1000 bar for ' H e . The favored crystalline structure at low pressures is the hexagonal close-packed (h.c.p.) structure; a small island of a body-centered cubic (b.c.c.) structure in the phase diagram was only discovered in 1961 (Vignos and Fairbank, 1961). Recently the symmetries of the b.c.c. and h.c.p. phases-the former is isotropic, while the latter is uniaxial-have started to play important roles in the spectroscopy of spin-polarized foreign atoms isolated in solid He matrices. He is the least polarizable atom in the periodic table and solid He is the monoatomic solid with the weakest binding energy. The interaction of any implanted foreign atom, ion, or molecule with neighbouring He atoms will thus in general be stronger than the interaction between adjacent He atoms. As a result, the defect particles will create their own trapping sites, be it by repelling or by attracting surrounding He atoms. The atomic or molecular properties will be affected in a homogeneous manner. This behavior differs radically from the situation encountered in any other solid-state host matrix, in which the guest species is localized in a variety of different trapping sites in the matrix, leading to inhomogeneous line broadenings.
29 30
i
h.c.p.
A
FIG. I . Phase diagram of condensed 'He
90
Serguei
I. Knnorsky rind Antoine Weis
In most matrices the symmetry of the local trapping site determines the property of the guest atordmolecule, as is evidenced for example by the structure of ESR spectra. As in H e matrices, the guest-host interaction is in general stronger than the host-host interaction, it is the guest particle that will impose its own symmetry on the local environment. In this way atoms with spherically symmetric electronic configurations will reside in spherical cavities, the center of which will be, free from local fields and field gradients. The absence of spin-local field interactions has the most important consequence that a high degree of spin polarization can be created in trapped species, that the spin polarization will be exceptionally long-lived, and that highly sensitive magnetic resonance experiments can be performed with such samples. We shall review such experiments and discuss the dramatically different results obtained when performing the experiments either in HeII, b.c.c., or h.c.p. solid 'He.
11. Structure of Point Defects A. CHARGED DEFECTS: SNOWBALLS AND BUBBLES Since the late 1950s, the first point defects in condensed He to have their properties extensively studied were positive ions and excess electrons. These early investigations concentrated mainly on the measurements of ionic transport properties (Padmore, 1972) and gave first insights into the interaction of the particles with the matrix atoms. This field has been reviewed in great detail by several authors (e.g., Meyer, 1960; Schwarz, 1975). tn recent years the study of ionic mobilities received renewed attention when it was shown that the ionic motion can be directly visualized through the spatially resolved observation of the light emitted during the recombination of positive ions with electrons (Tabbert rt al., 1995b). In the early experiments the ionic mobilities in liquid 'He were found to be orders of magnitude smaller than, for example, the mobility of 'He atoms in liquid "He. Below the lamda-point, the temperature dependencies of the ionic mobilities reflected the temperature dependence of the roton density, which allowed the inference that collisions with these elementary excitations were the main mechanism limiting the ionic motion. These early observations on He ions have recently been confirmed to also hold for Nu', Ag', Au', and Ba' (Giinther et d., 1996). The small absolute values of the experimental mobilities indicated that the masses of the charged defects were much larger than the ones of the free ions, and that their collisional cross-sections exceed the geometrical ones. Note that later the effective masses of positive and negative ions were measured directly in a resonance experiment on ions trapped near the surface of liquid He by the combined forces of an external field and their image charge (Poitrenaud and Williams, 1972, 1974). The experiments yielded m,lm,, = 44(2) and m h , = 243(5). This lead
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
91
to the conclusion that the ions do not move as free particles in the helium matrix, but that their interaction with the surrounding He atoms forms larger entities whose motion is hindered by collisions with rotons. Positive and negative species form different structures in condensed ‘He as evidenced by the fact that the mobilities of positive ions are approximately 50 percent larger than those of negative ions. Positive ions are produced by ionizing the He fluid proper and consist of He;,. n being a small integer with a predominance of He: (Atkins, 1958). In condensed helium they form what is usually referred to as “snowballs”. The responsible process is electrostriction: The polarizable dielectric He bulk surrounding the positive ion is pulled into the high field region of the ionic Coulomb field. This explanation was first advanced by Atkins ( 1958). He showed that inside of a sphere of 6.3A radius the He density becomes so high that it solidifies. which explains the name given to the structure. The large mass (40 He atoms) of the snowball and its large radius and hence large collisional cross-section with rotons are responsible for the observed low mobilities. The structure formed by negative ions-xcess electrons in condensed He-has a completely different origin. The shift of thc photoelectric threshold in liquid helium (Woolf and Rayfeld, 1965) and the study of the energy loss of injected electrons (Sommer, 1964) had led to the conclusion that electrons have to overcome a potential barrier of approximately 1 eV in order to penetrate into the He bath. This negative electron affinity is mainly due to the Pauli exclusion principle. The He atom, having a closed S-shell with two tightly bound electrons, exerts a strongly repulsive potential on approaching s-wave electrons. In contrast to positive ions, which attract surrounding He atoms, inmersed electrons repel the latter and form
--. 4 ?‘O
I3
electron bubble
j 1.0 0.5
-.____-’
bubble radius
i
0 000
05
10
IS
2 0 (00)
radial distance from ion [nm]
05
1.0
1.5
,’
2.0
2.5
radial distancc from ion [mn]
FIG. 2. Density 11 of He atoms around the structures formed by positive (left) and negative (right) ions in liquid ‘ H e . P ~ ~ is , ( ,the ~ density of the H e hulk far away from the defect\.
92
Serguei I. Kanorsky und Antoine Weis
small cavities from which He atoms are expelled. These structures have been given the name of “electron bubbles” and have radii ranging from 1.7 nm at SVP (saturated vapor pressure) to 1.1 nm in liquid He pressurized to 25 atmospheres. The existence of bubbles was first suggested by Ferrell (1957) in order to explain the observed long lifetime of positronium in liquid He, and has since found many confirmations. Atkin’s snowball-model was later extended by Cole and Bachman, who showed that positive ions, such as alkaline-earth ions, may also form bubble structures, as do electrons (Cole and Bachman, 1977). The property that in condensed He the electrons form localized states (bubbles) strongly contrasts the situation encountered in other condensed noble gas matrices, in which electrons move as delocalized waves that interact only weakly with the matrix atoms. General criteria for the formation of localized electron states in other nonpolar fluids have been developed by Springett et al. (1968). In recent years, experimental evidence for the autolocalization of electrons in liquid Ne matrices has been obtained by Sakai et ul. (1992).
B. ATOMIC DEFECTS The structure of the trapping site for a guest atom in a 4He matrix differs in many respects from the one in other NG matrices but bears a close resemblance to the structures formed by charged defects in 4He. The extreme softness and compressibility of condensed helium, the strong delocalization of helium atoms (zeropoint vibrations with amplitudes comparable to the interatomic separation), and the strong overlap of their wave functions make a modeling of the He matrix in terms of continuous medium very suitable for the description of the formation and properties of the trapping sites. In heavy NG (Ne,Ar, Kr, Xe) matrices the implanted guest atom (X) is mainly trapped in interstitial or substitutional sites produced by removing one or several host atoms from the lattice. In the helium matrix the weak He-He binding is readily overridden by the perturbation from the impurity ( X ) , so that the geometry of the trapping site is predominantly determined by the He-X interaction and not. by the bulk structure of the host matrix. Moreover, the coarse structure of the trapping site is preserved under solidification of the helium matrix. Depending on the symmetry and the strength of the He-X interaction, a variety of trapping site geometries may exist. Their general structure may be guessed following a very simple rule formulated by Dupont-Roc (1995). In this approach condensed helium is treated as a continuous compressible medium described by a local equation of state (Sherill and Edwards, 1985) with a local pressure modified by the interaction with the impurity (Cole and Bachman, 1977). By minimizing the total energy of the point defect with respect to the surface of the cavity occupied by the impurity atom, Dupont-Roc has shown that this surface, in a first approximation, coincides with the surface on which the adiabatic pair poten-
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
93
tial of the “He-X interaction vanishes. Helium is thus expelled from regions where a significant electronic density gives rise to a strong exchange repulsion that ovemdes the attraction due to polarization effects. The most simple situatioii occurs for the ground states of ulkuli and ulktrlineeurth atoms. The valence electrons of these atoms are in S-shells and their interaction with the closed S-shell configuration of He atoms is isotropic and is characterized by a strong hard-core type of repulsion at sniall interatomic distances and a shallow (few Kelvin deep) attractive van der Waals minimum at intermediate distances. Dupon-Roc’s rule of thumb then allows a guess at the geometry of the trapping site for such atoms: a spherical cavity with a radius of about 10 atomic units, the so called atomic. hihhle. Its radius and the width of the transition region can be determined using the approach introduced by Hiroike et al. (1965) for electron bubbles in liquid helium. In this approach the helium density around the point defect is described by a spherically symmetric radial trial function. p(R; R,,, a ) =
{
-
+
- RJI exp(-a(R - R,,))}
,R < R,,. , R > R,,
(1)
The density vanishes within a sphere of radius R,, around the impurity and asymptotically approaches the bulk value as R + x. Note that this ansatz treats He as an irzcnrnpressible medium as the density never exceeds the bulk value. The total energy of the point defect is determined as the sum of all He-X pair interactions V,,,,
El,,,=
(1’ R ptR)V,,,,(R)
plus the bubble energy
+ El,,,. The shape parameters a and R,,are found by minimizing the total energy El,,,hbk, For atomic species this approach was first introduced by Hickman et (11. (1975) to study the excited states of He atoms in liquid ‘He. Later the bubble model became very popular and has since been used by a number of groups to describe the structure of the trapping sites for the implanted atoms (Bauer et d., 1990; Beijersbergen et NI., 1993; Kanorsky et al., 1994b; Kinoshita rt ul., 1995a). The behavior of a single massive atonuc impurity in bulk liquid helium was studied by Kiirten and Clark (1985) using a variational Monte Carlo method based on a Jastrow ground-state wavefunction for the host-impurity complex. Numerical results for the chemical-potential difference, partial radial distribution functions, and partial structure functions were obtained for single Xe and CS impurities. According to these results Cs atoms fonn bubbles with a radius of 7 A in agreement with the prediction of the bubble model.
94
Serguei I. Kunorsky and Antoine Weis
Recently DeToffol et al. (1996) have presented new results for the structure of a point defect formed by an alkali atom in liquid helium. In their approach liquid ‘He is described by the phenomenological density functional introduced by Dupont-Roc et cil. (1990), the impurity electronic structure is described by the density functional scheme of Kohn and Sham (1965) where the valence electron, represented by the wave function q(r), interacts with the frozen core via the ab-initio pseudopotentials of Bachelet et al. (1982), and the impurity-He interaction is given by the sum of a short-range repulsive potential arising from the overlap of the electronic wavefunctions and a long-range van der Waals attraction. The equilibrium He density and the wavefunction of the valence electron of the impurity are found by minimizing the total energy of the point defect. Note that here no a priori assumptions of the geometry of the trapping site are used. A comparison of the results of these different approaches shows that for atoms with an S-ground-state wavefunction all methods give approximately the same result. Being the most simple, the bubble model has become the most popular approach for the treatment of this problem. The situation becomes a little more elaborate for the description of excited states of alkaline earth atoms. The pair impurity-He potentials become anisotropic reflecting the symmetry of the excited P-state of the impurity. In the bubble model this is taken into account (Hickman et ul., 1975; Bauer et al., 1990) by allowing for a quadrupolar bubble deformation:
where the additional parameter P is also found by a variational minimization of the total energy of the point defect. One of the main features of the bubble model in its simplest form outlined above is the assumption of an incompressible helium matrix, which is valid as long as the depth of the attractive part of the impurity-He potential does not significantly exceed the He-He interaction. The first excited states of light alkali atoms are examples where this approximation fails. This point is discussed in more detail in section IV.D.3. The problem of heavier NG impurities in bulk helium has been addressed in Kiirten and Clark (1983, Gordon et al. (1989, 1993). According to Kurten and Clark (1985), Xe impurity atoms form XeHe,, ( n = 14,15) complexes, for which the formation enthalpy was calculated to be -300 K. In Gordon et al. (1993) it was shown that the localization of helium atoms around the heavy impurities in van der Waals clusters results in the freezing out of such clusters and in the formation of an impurity-helium solid phase. The suppression of the zero point vibration amplitude of helium atoms in this phase makes it stable up to temperatures as high as 7-8 K and pressures up to 1 atm.
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
95
111. Implantation Techniques The fact that helium remains liquid at any temperature down to T = 0 makes the traditional implantation technique used in matrix isolation spectroscopy-condensation of the initially prepared host + impurity mixture on a cold finger-not applicable. For metal atoms the implantation into condensed He is further hindered by a significant (- 1 eV) potential barrier at the liquid helium interface. For a long time this was the main problem that prevented the use of helium as a host for the purpose of matrix isolation. The first impurities, besides excited He atoms, to be studied in liquid helium were ions, which could be produced above the liquid and then accelerated into it with an appropriately shaped electric field. With respect to the ease of implantation, impurities can be divided into two classes according to the sign of their chemical potential inside the helium matrix. Particles with a positive chemical potential-practically all metal atoms with strongly delocalized valence electrons-tend to form bubbles, while atoms with a negative chemical potential-heavy noble gases, nitrogen, and some molecules-favor the formation of snowballs. Particles of the first type are therefore repelled by the interface to the helium matrix, while the others are soluble in liquid helium and can in principle be implanted into the matrix through the liquid interface. Quite naturally, the soluble species were studied first. In 1981 a team in Chernogolovka (Gordon et NI., 198 1) developed an implantation technique suitable for such impurities. A jet of helium gas with a small admixture of the species to be implanted was directed onto the liquid helium interface. Impurities transported to the interface by the gas flow stick to it and are partially dissolved in the liquid. In the 1980s a technique for doping large helium clusters with atomic and molecular impurities was developed. First demonstrated by Gough ef al. (1985) for Ar clusters it was later extended to He clusters by Scheidemann et id. (1990). These experiments are closely related to the experiments in macroscopic He matrices. A collimated supersonic jet of huge (N,*, = 10') helium clusters traverses a pickup cell in which impurities are created in the gas phase by heating. The helium clusters are loaded with the impurities and are studied downstream by optical methods. Species soluble in liquid helium will reside insidr of the clusters, while impurities with a positive chemical potential will be trapped on the outer surface of the clusters. As the loading of liquid He with metallic impurities through the liquid interface does not work, the first experiments 011 H e isolated metal atoms were based on the electrostatic acceleration of metal ions into the liquid (Himbert et d . ,1985; Bauer rt d., 1985). The revolutionary step made by the Heidelberg group (Bauer rt d . , 1989) was to let the implanted atomic ions recombine inside the liquid with electrons produced by field emission from a metal tip. In this way atomic impurities
96
Serguei I. Kaiiorsky tind Aiifoine Weis
could be produced in situ. The method is quite universal (since, practically, ions of any species can be produced above the liquid interface by laser ablation), and works equally well for normal and superfluid helium. It does not only allow the creation of neutrul atomic impurities in liquid He-the spectroscopic analysis of the light emitted during the recombination process yields valuable information on the properties of the implanted species. The major practical disadvantage of the technique is the limited residence time-on the order of 300 ms ( h d t et a/., 1993)-of the implanted species. The convection flow accompanying the thermal gradients created by the implantation process will remove most of the produced atoms within a few seconds from the observation region. Although this time is sufficient for most optical studies, much longer trapping times are needed if high resolution ESR studies are to be performed in condensed He. In 1993 Arndt et ul. (1993) and Fujisalu et af. (1993) independently discovered that atomic species can also be efficiently implanted into liquid helium by laser ablation from a solid target placed inside of the liquid matrix. By focusing a pulsed beam from a frequency-doubled Nd:YAG laser onto a metal target, up to 10"' atoms could be dispersed into the surrounding helium. The physics of the ablation process inside liquid helium has not been studied in detail so far, but can be described qualitatively in the following way. When the laser beam hits the target a bubble of helium gas is formed above the hot spot. The ablated material is evaporated into this gas region and spread inside the liquid by convection after the collapse of the gas bubble. The superfluidity of helium is essential for the success of this technique, a plausible explanation being that the large heat conductivity of superfluid helium prevents the helium from boiling during the ablation process and thus from defocusing and scattering the implantation beam by gas bubbles. However, single isolated atoms constitute only a small fraction of the ablated and implanted material, which consists mainly of clusters and microscopic particles. In order to increase the atomic density, Fujisaki et nl. (1993) and Beijersberger et (11. (1993) have used a second pulsed laser beam, which hits the floating particles implanted by the first beam, thus further dissociating them. The Tokyo group has observed that although the implanted atoms escape from the observation volume within a few seconds due to convection and cluster formation, larger particles can reside in the He bath for up to 30 minutes (Beijersbergen et a/., 1993). This allows repeated production of atoms inside the He matrix by dissociating these particles with a pulsed laser without the need of implanting new material from the target. This laser destruction of microscopic particles also works in normal helium. Practical advantages of the described implantation method are its simplicity and universality. Relatively high concentration of atomic species (up to 10"' cm-3 can be produced in the region of the laser focus with this technique. Nevertheless, this concentration drops rapidly by several orders of magnitude within the first few milliseconds mainly due to convection caused by the large amount of heat
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
97
deposited during the implantation. In Arndt et al. (1993) it was nevertheless reported that atomic concentrations on the order of lOo-IO' c m ~ can be observed in liquid helium 10 s after the implantation. The method of direct laser ablation inside the matrix was a major step toward extending He isolation spectroscopy toward solid He matrices (Kanorsky Pt (11.. 1994a; Arndt et ul., 19951). Note that due to the facts that helium solidifies only at a pressure exceeding 25 bar and that the solid and gas phases cannot coexist, none of the methods that rely on loading the impurities through the helium surface can be applied in this case. Atoms can nevertheless be implanted into a solid helium matrix using the following modification of the direct laser ablation technique (Amdt et lil., 1995a). A chunk of the material to be implanted is placed at the bottom of a pressure cell filled with a helium crystal. Energetic laser pulses (typically 20 mJ. 10 ns, 10 Hz repetition rate) are focused onto the target. The absorbed heat liquefies a portion of the helium crystal above the target and simultaneously the focused beam ablates material off the target surface. Atoms, clusters. and even microscopic particles can be generated and convectively spread inside this molten region. After resolidification the frozen species can be further destroyed by an appropriately focused laser beam, thus producing atoms at concentrations of 10x-lO" cni '. Due to the absence of convective flow (the main loss mechanism of impurities in liquid helium) in the solid, the observed residence time of the implanted atoms in solid helium is much longer: several hours for Btr atoms and 10 to 15 minutes for C.7 atoms. The main loss mechanism for Cs atoms is presumably cluster formation. The latter can again be dissociated by further laser pulses, thus allowing maintenance of a high atomic concentration over a practically unlimited period of time. This implantation method has been the key technique that enabled the highresolution ESR studies of paramagnetic impurities in solid helium described below. Long observation times are a prerequisite for such experiments. So far a great variety of atomic and molecular species have been implanted into superfluid He and studied therein, whereas solid matrices were only used with Cs and Bci atoms.
'
IV. Optical Spectroscopy A. ELECTRON BLiBBLES The steepness of the potential at the bubble boundary makes the electron bubble a textbook-like experimental realization of a particle in a spherically symmetric square-well potential of approximately 1 eV depth. The bubble model has been very successful in describing the optical properties of such localized electrons. Valuable predictions can already be obtained from its simplest form One assumes that the electron resides in an infinitely deep spherically symmetric square-well potential of radius R. The radius is determined by the equilibrium of the inward-
Serguei I. Kanorsky and Anmine
98
Weis
directed pressures exerted by the helium bulk @He,) and the bubble surface tension won one side and of the outward-directed pressure due to the confinement (localization) of the wavefunction of the electron on the other side. For a bubble with a radius of 1.7 nm (corresponding to a practically vanishing He bulk pressure), the pressure exerted by the zero point motion of the electron is 4 bar. First bubble model calculations were performed by Jortner et al. (1963, who used a pseudopotential formalism that was further refined in a variational calculation by Hiroike el al. (1965). The first calculation of the resonant optical absorption properties of electron bubbles in the frame of the bubble model was performed by Fowler and Dexter (1968). The authors present the energies of the Is, lp, 2s, and 2p states in configuration coordmate diagrams. This treatment was extended later by Miyakawa and Dexter, who included a pressure-dependent potential depth (Miyakawa and Dexter, 1970). In both papers a pressure-dependent surface tension (Amit and Gross, 1966) was assumed. Grimes and Adams found, however, that their experimental results are best fitted by a model assuming the potential depth and the surface tension to be pressure independent (Grimes and Adams, 1990). Their calculation is shown as the solid line on the low-pressure side of Fig. 3. Electron bubbles form the simplest color centers and it is quite natural to use the powerful technique of optical spectroscopy to study their structure. The measurements of resonant optical excitation spectra of electron bubbles in condensed He matrices and the determination of their pressure dependence has confirmed the bubble hypothesis in a beautiful quantitative way. 1. Liquid Matrices
The first optical spectroscopic study of electron bubbles in liquid 'He was performed by Northby and Sanders (1 967), who observed the photoionizing transition to the continuum at a wavelength near 1 pm. In this, as well as in subsequent
h
$ 3
4
0.3
superfluid He 4
."u
.- 0.2 -
i
&
2
0.1
c
0.1
.". .
,
..c . . . I
I
.
.
,
I
,
10 He pressure (atm) 1
:.
,
.
7
..,.I
100
FIG. 3 . Pressure dependence of the Is-lp transition in electron.
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
99
experiments, the optical absorption was detected by the recording of a change in the ionic mobility (drift current). These photoconductivity measurements rely 011 the fact that below 1.3K electron bubbles tend to create vortex rings to which they are bound (Springett and Donnelly. 1966). The ionizing transition liberates the electron from the core of the vortex ring, thus temporarily increasing the electron mobility. In the relaxation process following a transition to a bound state the electron bubble is thermally excited from the vortex line. The first observation of a resonant transition from the ground state ( I s ) to a bound state (2p) near 2.5 ,urn was reported by Zipfel and Sandars in 1968 (Allen r t a/., 1968), erroneously interpreted as the 1 s-lp transition (Miyakawa and Dexter, 1970). The first experiment revealing the transition to the first excited state ( l p ) was done in 1990 by Grimes and Adams (1990) and by Parshin and Pereversev (1990). The latter team directly measured the absorption (typical signals AZ/l = - 10 - 5 ) of fixed frequency radiation at 6.7 p i by electron bubbles. Resonances were detected by pressure scanning the transition frequency. The photoabsorption detection has the advantage that it can be used at arbitrary pressures and temperatures, whereas the photoconductivity technique is restricted to p < 18 bar and T < 1.6K. Grimes and Adams used tunable infrared radiation and tneasured resonances in the photocurrent from a field emission tip when the wavelength of the radiation was tuned over the 1 s-lp transition. The pressure dependence of the transition frequencies agreed perfectly with the predictions from a square-well potential model assuming a pressure-independent surface tension and a pressure-independent well depth. This experiment yielded the first spectroscopic detemiination of the pressure dependence of the bubble radius for pressures ranging from saturated vapor pressure to the liquid-solid phase transition (25 bar). A comparison with earlier, non-spectroscopic determinations of the electron bubble radius was made by the authors (Grimes and Adams, 1990). In 1992, Grimes and Adams repeated their experiment (Grimes and Adams, 1992) over a wider range of pressures and temperatures, this time using the resonant photoabsorption technique pioneered by Parshin and Pereversev (1990). The same year, Parshin and Pereversev also reported quantitative results of the pressure dependence of the Is-lp transition (Parshin and Pereversev, 1992). Fig. 3 shows a compilation of the pressure dependence of the 1s-lp transition energy as measured by different authors. The open circles represent the data obtained by Grimes and Adams (1990, 1992) and Parshin and Pereversev ( 1992) in liquid He. The solid line through the low-pressure data is the prediction by the bubble niodel assuming a pressure-independent surface tension (Grimes and Adams, 1990). The experiments observed line-widths ranging from 0.02 to 0.04 eV, of which 0.01 eV could be attributed in early estimations to breathing mode and quadrupolar bubble oscillations (Grimes and Adams, 1990; Celli et ul., 1968). Recently Lerner Ct al. have reanalyzed the contribution from quadnilpolar fluctuations to the
Serguei 1. Knnorsb mid Antoitie Weis
I00
line-shape in a model taking the Franck-Condon principle for the optical transition into account, and found good agreement with the observed widths (Lerner et al., 1993). In their 1990 paper, Grimes and Adams speculated for the first time that the trapping on vorticity may deform the bubble shape thus lifting the m-degeneracy of the energy levels (Grimes and Adams, 1990). Neither the optical absorption spectra nor the ESR spectra of electron bubbles have so far given experimental evidence (such as line splittings, appearance of forbidden transitions) of such deformations. Note, however, that recently several observations in the magnetooptical spectra of atoms in solid helium matrices have led to an interpretation in terms of static and dynamical bubble deformations (section V.B.4).
2. Solid Matrices Experiments in the 1960s had revealed that the mobility of electrons in solid He matrices was about 5 orders of magnitude lower than in the heavier solid noble gas matrices (Nr, Kr, X e ) and that the mobility in He had a positive temperature coefficient in contrast to other solid noble gases, where it was negative. Furthermore the observations of Shalnikov (1965) had evidenced a strong similarity of the motion of electrons in liquid and in solid He. These observations led Cohen and Jortner (1969) to propose that excess electrons in solid helium also form stable localized states for which structural changes of the local environment are energetically favored, thus leading to the formation of a cavity, in analogy to the electron bubble observed in liquid helium. This localized trapping site for excess electrons in a solid is quite unique, as electrons in solids are usually delocalized as quasi-free charges in band structures. Cohen and Jortner (1969) estimated a bubble radius of 9 an optical transition energy of 0.5 eV (at 30 bar), and a stability of the bubble for pressures up to 4000 bar. These theoretical considerations were first unambiguously confirmed in experiments by Golov and Mezhov-Deglin, who extended the optical absorption experiments on the Is-lp transition in electron bubbles in liquid 'He to hexagonal close-packed (h.c.p.) solid 'He (Golov and Mezhov-Deglin, 1992) and to bodycentered cubic (b.c.c.1 solid 'He (Golov and Mezhov-Deglin, 1992, 1994) matrices. In these experiments electrons were injected into the matrix by ionization of Hr atoms with radiation from a p-emitter. Tunable infrared radiation from a lamp-prism nionochromator source was sent through the sample and the absorption detected by a bolometer. Typical absorptions of were observed with electron densities of 4 . 10" cm-' (Golov, 1995). In 4He the absorption cross-section was found to have a constant value of 12(3) A2 over the whole range of investigated pressures (40-100 atm). The pressure dependence of the resonance frequencies found by Golov and Mezhov-Deglin are shown in Fig. 3 as solid circles. The data are perfectly well described by the predictions from a bubble model based on the Wigner-Seitz method (Cohen and Jortner, 1969) shown as a solid
A,
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
101
line in Fig. 3 . As in liquid matrices. a bubble model assuming a pressure-independent bubble surface tension gives the best fit to the data.
B. EXCITEIJ STATESot-'Hf; The introduction in 1969 by Dennis of trl. of the electron beam excitation technique rendered possible the first spectroscopic investigations of the excited states of condensed helium (Dennis et ( I ! . , 1969; Doake and Gribbon, 1969; Hill ei d., 1971; Ihas and Sanders, 1971: Keto P r d., 1972). In this method liquid helium was excited with an energetic electron bemi (220 e V ) penetrating the container with the liquid through a thin metal foil window, and the fluorescence was analyzed by a grating spectrometer. The observed spectra were identified as originating from excited states of Hr atoms and of He? molecules. However, the obtained data were not consistent with a model that assumed an unperturbed distribution of' Hr atoms around the point defect formed by the excited impurity: Such a distribution would greatly perturb the wavefunctions of the excited species, resulting in much larger spectral line shifts than were observed experimentally. Moreover. the emission spectra of molecular He! impurities exhibited rotational and vibrational structures similar to the one found in free molecular species. In their seminal papers Hickman and Lane (Hickman and Lane, 197 I ; Mickman rt d., 1975) have explained these facts by introducing the bubblc model of the trapping site to the excited impurity described in section I1.B of this article.
c. ATOMSWITH NEC;&I'IVE
CHEMICAL POTtNTI.41.
Optical studies of atomic impurities in liquid helium matrices were tirst performed by Gordon el ill. (1981). The authors have studied the spectra of nitrogen afterglow at helium temperatures: A jet of gaseous helium with a small admixture of' molccular nitrogen was condensed onto a surface of liquid helium after passing a region of microwave discharge. Apart from the fluorescence emitted from the gas phase crhovt. the liquid, the authors observed the formation of a light-emitting condensate irisirk the liquid helium. The comparison with emission spectra of nitrogen deposited on cooled substrates and of solid nitrogen bombarded by fast electrons made it possible to attribute the observed fluorescence to the enussion of metastable N('D) atoms formed in thc dischurgc and stabilized i n the liquid helium matrix. As we have discussed in section 111, this implantation method is suitable for atoms that have negative chemical potentials with respect to liquid helium. The observed luminescence of metastable N atoms. the so called a-group. corresponds to the doubly forbidden ' D -+' S transitioii at 520 nm with radiative lifetime of 4.4 . 10' s. It can only be observed whcn the close proximity of heavy perturber in the helium matrix partially breaks thc intrinsic atomic symmetry. The authors have therefore concluded (Boltnev et d., 1994) that N. molecules are also
102
Serguei I. Kanorsky and Atitoine Weis
present in the core of helium clusters formed during the condensation of the gas flow on the liquid surface. These studies have recently been extended to Ne impurities (Boltnev et ul., 1994, 1995) in condensed helium. The lifetime of the Ne metastable 3 ~ [ 3 / 2 ] ~ ( ~ P J state was long enough to enable laser-induced fluorescence (LIF) studies of the transitions originating from this level. In these studies the 3s[3/2]Y(’P2) + 3p’[1/2], and 3s[3/2];(’P2) + 3p’[3/2I2 transitions were pumped and the fluorescence was detected on the 3p’[1/2], -) 3s’[1/2];‘ and 3p’[3/2I2 + 3s’[1/2]; transitions. The observed excitation spectra reveal remarkably narrow lines (the resolution was limited by the line-width of the pulsed dye laser = 0.18 A). Moreover, within the experimental accuracy the excitation lines were not shifted with respect to the free atomic lines. These results are surprising: The electronic structure of Ne atoms in the metastable state with one outer s-electron is analogous to that of alkali atoms in the ground state, and for the latter a pronounced broadening and shift of the excitation lines have been observed, as will be discussed below.
D. ATOMSWITH POSITIVE CHEMICAL POTENTIAL The study of the optical properties of atomic impurities with positive chemical potentials with respect to the helium matrix started in 1989 (Bauer et ul., 1989) with the invention of a “neutralization” technique suitable for such atoms. Due to its universality this method of implantation has enabled the study of a wide variety of atomic species. The number of studied impurities was further increased after the development in 1993 (Arndt et ul., 1993; Fujisalu et al., 1993) of the implantation technique of “direct laser ablation”. This last method has allowed studies to extend to matrices of solid helium. Two types of spectra of atomic impurities-recombination fluorescence (RF) and laser-induced fluorescence (LIF)-have been studied with these implantation techniques. The recombination spectra give valuable information about the process of ion-electron recombination and about the relaxation of the excited atomic states inside the helium matrix. From the LIF studies information about the geometry of the trapping site for the ground-state atoms and its dynamic in the optical excitation-emission cycle can be extracted. We shall review these two types of spectroscopic investigations separately. 1. Recombination Fluorescence
The neutralization implantation technique is ideally suited for the study of RF spectra as excited states are very efficiently populated by the recombination process. It has, however, been shown (Fujisaki et al., 1993; Beijersbergen et al., 1993) that it is also possible to observe RF with the alternative “direct ablation” method. In this case the recombining ion-electron pairs are created by the intense
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
103
radiation of a second "dissociation" laser pulse and the RF can be observed during a few ps following this pulse. Table 1 gives an overview of elements for which the recombination fluorescence has been observed in helium matrices. All observed recombination spectra share the following common features: The identification of the observed lines is rather straightforward and unambiguous. As a general rule the line shifts do not exceed I nm with respect to the corresponding lines of free atoms, which indicates only a minor distortion of the electronic wavefunctions of the impurities by the helium matrix. The studied RF spectra give a strong evidence for the existence of radiationless relaxation processes for certain excited atomic states. The most impressive manifestation is the failure to observe recombination emission from alkali metals such as Li, Nrr. and K (Tabbert (It ol., 1995a). The recombination spectra of IB group elements Cu, Ag, and Au also indicate a radiationless relaxation of several levels. For example. the 6p'P,,? term of gold shows no emission to the ground 6s'S,,, state, although this term is definitely populated as is evidenced by the observation of optical lines originating from higher-lying levels and ending on this level. The resonance D, line is also quenched for A g , although D, lines could be identified for both Ag T.ABLE I ATOMS AND IONS FROhl WHIC Ii RtC'OhlBINAl I O N I;LIIORESC'ENCF HAS BEEN OHSkRL 111) IN IiRLIIIM MATRICES.
I A
B
A
v1
Ill
11
I3
A
B
A
Ion\
B
104
Serguei I. Kanorsky and Aiitoine Weis
and Au. Both resonance lines do not show up in the recombination spectra of Cu, although the excited levels of these transitions are definitely populated. We shall address this phenomenon in section IV.D.3. None of the observed transitions lines starts from an excited state lying closer than 1.8 eV to the ionization limit of the free atom. The last fact was explained by Beau et al. (1996) in a tunneling model of the ion-electron recombination process, as follows: The electron and ion approach each other under the action of their Coulomb attraction until the electron tunnels from its bubble into the electrostatic potential of the positive ion, thus forming the neutral atom. This process is accompanied by an energy dissipation in the form of phonons, rotons, and vortices by the moving charged particles. The initial ionic bubble is too small to accommodate the newly formed neutral atom. As a consequence, the point defect relaxes to a new configuration determined by the size of the excited atom, thereby releasing energy in the form of phonons. The total energy dissipated during the ion-electron recombination process was estimated to be on the order of 1.8 eV and to be rather independent of the nature of the recombining ion (Beau et al., 1996). Energy conservation is thus responsible for the observed absence of lines originating from levels which lie closer than 1.8 eV to the ionization limit. Recently Hui et al. (1995) have tried to observe fluorescence from high-lying levels of Sr, Cu, and Ag under direct laser excitation of these levels. The excited levels from which the authors could observe emission were 1.3 eV for S r and 1.4 eV for other elements below the ionization limit. Fluorescence was observed not only from the directly populated levels, but also from a number of other neighboring levels, suggesting the existence of an effective nonradiative relaxation following the atomic excitation. Above this energy region a broad absorption band was observed. The excitation of these absorption bands resulted in typical “recombination fluorescence” emission spectra. Here again, no lines starting from levels in this region could be identified in the observed spectra. This fact was qualitatively explained by Takami (1996) as an autoionization of the excited atom: If the kinetic energy of the optical electron on the boundary of the atomic bubble immediately after the excitation exceeds I eV, the electron can penetrate the bubble wall and for a short time reside in a separate bubble. The produced “free” electron will eventually recombine with the ion and emit the usual recombination fluorescence.
2. Lser-Induced Fluorescence The implanted neutral atoms may be further excited on a selected transition with resonant laser radiation and the emitted radiation can be analyzed with a spectrally selective detection system. By scanning the excitation laser frequency with the detecting spectrometer adjusted to the peak of the corresponding emission line, one
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
105
can record the e.ucitution spectru. Scarining the monochromator with the excitation laser set to the maximum of the absorption gives the emission spectrum. In conibination with short laser pulse excitation and time-resolved fluorescence detection this method allows determination of excited state lifetimes of matrix isolated atoms. Most studied with this technique are the alkaline-earth elements (Bauer et a/., 1990; Beijersbergen er al., 1993; Kanorsky C t d . , 1994b; Kanorsky et ul., 1995) and the heavy alkalis Rh (Takahashi et d.,1993; Kinoshita e t a / . , 1995a; Kinoshita et uI., 1996b) and Cs (Takahashi ef d . ,1993; Kinoshita rt NI., 1995a, 199%. 199621;Lang er d., 1996). The implantation method of direct laser ablation has further allowed study of the optical properties of atoinic species in pressurized and even solid helium (Kanorsky rt L I ~ . ,1994a, 1995; Kinoshita et LII., 1995a, 199%; Lang et nl., 1996). As with the recombination fluorescence spectra, the LIF spectra of all studied atoms show common features: The excitation lines are strongly blue shifted with respect to the free atoinic transitions and strongly broadened with a characteristic asymmetry of the line; The positions and the line-shapes of the emission lines exactly coincide with the lines observed by the recombination fluorescence method. They are much less shifted and broadened. As in recombination studies no emission has been reported so far from the laser-excited states of the light alkalis (Li.K , N i l ) . Nevertheless, the emission of both D, and D? lines have been observed for Cs and Rh. These general properties of the LIF spectra can be understood on the basis of the bubble model of the trapping site for the implanted atoms described in section 1I.B. This model should give a reasonable description of the trapping site for at least the ground states of alkali and alkaline-earth atoms. The absorption takes place inside the bubble formed by the atom in its ground state and occurs, according to the Frank-Condon principles, without a change of this equilibrium bubble configuration. Since the wavefunction of the optical electron in the excited state extends to larger distances from the atomic core, the corresponding excited atomic level is strongly perturbed by the helium surrounding. which explains the relatively large shift of the excitation lines in helium matrices. The radiative lifetime of the excited s t a t e s - o n the order of a few ten ns-is long enough for the surrounding helium atoms to relax to a new configuration that reflects the symmetry and size of the excited atomic state. An experimental upper bound on this relaxation time T, 5 l r z s was set (Hui et NI., 1995a) in an observation of laser quenching of the metastable 6s6p3f': state of Yb. The emission then takes place in this relaxed bubble, which as a rile has a larger volume to accommodate the more extended electronic wavefunction of the excited electron. This explains the fact that the emission lines are less affected by the helium surrounding.
106
Serguei I. Kanorsky and Antoine Weis
This simple qualitative picture may be put on a quantitative basis (Hickman et al., 1975; Bauer et al., 1990; Kanorsky et al., 1994b; Kanorsky et al., 1995). Here we discuss merely excitation lines since these are most sensitive to the matrix effects of interest. Once the equilibrium structure of the point defect is determined using the methods described in Section II.B, the corresponding shift of the optical line can be readily found from Fermi’s golden rule as the difference of the atom-helium interaction energies in the initial and final states.
Only the foreign atom-helium interaction energy contributes to the shift, as the bubble contribution to the point defect energy does not change during the optical transition due to the Frank-Condon principle. The problem of calculating the line-shapes is much more involved, since it demands the inclusion of the dynamics in the model. To our knowledge this has not been properly done so far. One popular approach to this problem for atomic impurities was introduced by Bauer et al. (1990) in a configuration coordinate model of the optical excitation. In this model the line broadening originates from oscillations of the bubble radius around its equilibrium value. The amplitude of these oscillations is found by fixing the bubble shape parameters a and p defined in section 1I.B. to their stationary values and considering the shape parameter R,, as the configuration coordinate of the system. The total energies of the point defect for the impurity atoms in the ground and the excited states are calculated as functions of this coordinate. The energy curve obtained in this way for the ground-state atom is then treated as the potential for an effective “bubble oscillator”. By quantizing the motion of this oscillator one determines the probability distribution I+(R,# for the bubble radius R,, and the spectrum is obtained by projecting this probability function onto the energy curve of the excited state. This model gives the right qualitative description of the observed spectra-a large blue shift and asymmetry of the excitation lines and a relatively smaller shift and broadening of the emission lines. The weak points of this model in its simplest form are that it considers only the lowest order (breathing mode) of the bubble oscillations and that it uses an ill-defined parameter, the effective mass of the bubble oscillator, in the quantization procedure. The expression usually used for this parameter was obtained (Fowler and Dexter, 1968; Lerner et ul., 1993) by the assumption of an incompressible He matrix around the point defect, and is definitely only a poor approximation of the real situation. This is probably the reason for the observed discrepancies between the predictions of this model and experimental results: The configuration coordinate model regularly underestimates the observed broadening of the excitation lines. Recently Kinoshita et (I/. (1995b, 1996a) have further refined the model by allowing for quadrupole oscillations, that is, oscillations in the two-dimensional
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
107
parameter space spanned by R,, and ,3. The allowance for distortions of the equilibrium bubble shape lifts the three-fold degeneracy of the excited P-states, giving rise to an additional line broadening and an asymmetry via the Jahn-Teller effect. This extended model improves the agreement with experiment in the case of the CSD, line, but the theoretical line-width is still smaller than the experimental one by a factor of almost two. An alternative approach to the problem of the calculation of the spectral lineshapes in He matrices has been introduced by Hickman et (11. (1975) in their analysis of the spectra of excited states of liquid helium and more recently used by Kanorsky et ul. (1995) for the interpretation of the excitation spectra of Ba atoms in pressurized and solid helium. Within this approach the line broadening is assumed to originate from fluctuations of the helium density in the vicinity of the point defect. These fluctuations are assumed to be of the same type as for an ideal gas (see Hickman et d . ,1975; Kanorsky and Weis, 1996 for the discussion of the validity of this approximation). The spectrum is found as the Fourier transform of the optical dipole d ( t ) autocorrelation function C(T):
where C(T) = ( d * ( f ) d (+ f
T)),
The perturbation by helium atoms shortens the autocorrelation time and thus broadens the line. The autocorrelation function is found from the known equilibrium density of helium around the point defect p ( R ) as
where Am is the shift of the transition frequency by a single He atom at position
2 with respect to the impurity atom.
The line-shapes produced by this approach showed a good agreement with experimental findings for the BN singlet excitation line in pressurized liquid and even solid helium matrices (Kanorsky r f ( I / . , 1994b, 1995). thus showing that the atomic bubble model of the trapping site in liquid helium can also be used for the solid phase of the helium matrix.
3. Norirdiative Processes
A first evidence for the existence of radiationless relaxation processes in a helium matrix was inferred from the analysis of recombination spectra discussed above. Recently an extensive study of the relaxation process in Sr atoms was performed by the Tokyo group (Takami, 1996) using the method of selective multi-step exci-
108
Serguei I. Kanorsky crrid Antoine Weis
tation. It was shown that the lower excited states are stable in the sense that only LIF originating from these levels was observed. However, the excitation of higher lying states results in an emission from a manifold of neighboring levels, suggesting an efficient nonradiative relaxation following the atomic excitation. The origin of this relaxation is the breaking of the intrinsic atomic symmetry due to the formation of an aspherical bubble by the excited atoms. In Section 1I.B it was pointed out that the symmetry of the trapping site reflects the symmetry of the electronic wavefunction of the trapped atom. Only for atoms in spherically symmetric states, such as IS(,2S,/2 , and *P,,,, will the atomic bubble be spherical, thus preserving the central symmetry of the atomic Hamiltonian. The formation of distorted bubbles for higher angular momentum states will break the central field approximation and mix neighboring atomic states. This effect will be more pronounced in the region of higher energies due to the higher density of energy levels. Another evidence for the existence of radiationless relaxation has come from measurements of the lifetimes of the excited states of Mg in liquid helium (Gunther et al., 1995). Measurements of radiative lifetimes of a number of atoms (Hui et al., 1995b) have shown that the observed values are close to those of free atoms, while the metastable ' P , state of Mg has shown a lifetime of 15 ms, which is three times longer than the free atomic state. In Gunther et al. (1995) this anomaly was attributed to the instantaneous repopulation of this state from the neighboring levels via nonradiative relaxation. Similar effects have been observed for the resonance levels of Rb and Cs (Kinoshita et al., 1995a, 1995b), where the excitation of the D2 line results in fluorescence emitted on both D, and D, resonance lines. The most striking manifestation of nonradiative processes is the quenching of radiative transitions in light alkalis. The D, line of Rb is not quenched in liquid He at saturated vapor pressure, but the lifetime of the 5P,,, state is considerably reduced when the pressure is raised to 25 bar (Kinoshita et al., 1996b). The Cs D, line remains practically unquenched up to 25 bar. All situations listed above have the common feature that the atomic states, from which the radiationless transitions originate, have a P-symmetry. It is for these states that the bubble model of the trapping site in its standard form is not applicable: Neither the assumptions of continuity and incompressibility nor the form of the trial function (Eq. 1) is applicable for the excited states of these atoms. This fact is best illustrated for the Nu'(3P) state in liquid He (Dupont-Roc, 1995; Kanorsky et al., 1995). In contrast to the ground state sodium atom, the Nu'(3P)-He adiabatic pair potential (Pascale, 1983) is strongly anisotropic: A deep attractive minimum exists in the nodal plane of the P-state electronic $-function. Several helium atoms are localized in this 650 K deep potential well in a ring of -4 8, diameter, thus forming a molecular complex Nu*(3P)He,,. In Kanorsky et al. (1995) the number of helium atoms in this complex was estimated by minimizing the total energy of the complex with the result n = 5. Five helium atoms
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
109
localized at a distance of 2 A from the sodium atom cause the 3s and 3P levels to cross and thus quench the excited atomic state. DeToffol et crl. (1996) got similar results using a density functional formalism. Recently Persson et al. (1996) have shown that a similar situation occurs for IIb group atoms. After excitation of the D1 line, silver atoms in liquid helium form a linear He-Ag*-He exciplex trapped in a inicrocavity in bulk helium. This exciplex formation was evidenced by the strong characteristic red shift and broadening of the emission D , line. The authors also predict that a similar C s ' H e , exciplex should be formed after D, excitation of the Cs impurity.
E. MOLECULES Data on the spectroscopic properties of tizolmilor impurities in He matrices are not as abundant as for atomic impurities. Emission spectra of Agl and As, have been reported by the Heidelberg group (Tabbert rt NI., 1995a). A band consisting of seven broad peaks in the range from 430 to 470 nm was identified as emission from Ag dimers formed in the liquid during the recombination process. Another band around 380 nm was not resolved and was tentatively assigned to the emission of A X . LIF spectra of Liz and Nu1 molecules were observed by Takahashi et (11. (1993). The observed excitation and emission bands of these molecules could not be identified. Recently Perrson et ul. (1995) have succeeded in observing complete excitation and emission spectra of the 's:, - 'c: transitions in Ca, and Cii? dimers. Both spectra show blue shifts on the order of 100 cm- relative to the free molecular transitions, and the vibrational constants were found to be only slightly affected by the helium matrix. Nevertheless, the surrounding Hr causes a very rapid vibrational relaxation so that no emission could be observed from the excited vibrational levels.
'
F.
ATOMS AND
MOLECULES IN AND ON HE CLLISTEKS
More complex organic molecules have not been studied in bulk helium so far. Existing implantation methods are not well suited for these species. Nevertheless, significant progress has been achieved in past years in the study of molecules trapped in the bulk or on the surface of large helium clusters. The first observation of excitation spectra of S F , molecules embedded in helium clusters was reported by Goyal et NI. (1992). Due to the low resolving power of the spectroscopy lasers used (0.3 cm-'), the rotational and vibrational structure of the spectrum could not be resolved in the first experiments. Later, rotationally resolved spectroscopy of the S F , molecules in liquid helium clusters was perfornied by Hartmann ef al. The observed spectra were interpreted in terms of a spherical top Hamiltonian, although the resulting rotational constant was consid-
Sergrtei 1. Kaizorsky and Antoine Weis
110
erably smaller than in the free SF, molecule. This fact was attributed to the influence of the helium surrounding. By fitting the experimental spectra, the temperature of the host cluster was determined as 0.37(5) K. Recently these studies were further extended to include the spectroscopy of glyoxal (C,H,O,) molecules embedded in helium clusters (Hartmann et al., 1996). The excitation spectrum of the intravalent (S, t So) electronic transition recorded in the range of 21730 to 22820 cm-' consists of four very sharp zeru phonon lines accompanied by weak broader features on their blue sides, which the authors have ascribed to be phonon wings. These wings reflect the excitation of phonons in the He droplet during the photon absorption. The analysis of the structure of this phonon wing has revealed remarkable features: No phonon signal is observed in a gap of 4 cm-' near the zero phonon line. The first peak of the phonon structure is separated by 5.8 cm-' (8.1 K) from the zero phonon line. This energy corresponds to the excitation threshold of rotons in superfluid helium and the observation may thus be interpreted as the first experimental evidence for superfluidity in He clusters. T h s observation needs to be further confirmed by other cluster isolated species. The species that are unsoluble in liquid helium, for example alkali atoms, will not penetrate into the He cluster, but will rather be bound to its surface due to the van der Waals attraction. In Ancilotto et al. (1995) the profile of the cluster surface in the vicinity of the captured alkali atom was calculated using the density functional approach. These results were used by Stienkemeier er al. (1996) for the interpretation of the observed excitation and emission spectra of Li, Na, and K atoms attached to large He clusters using an approach that treats the trapped atom-cluster complex as a diatomic molecule, where the helium cluster plays the role of the second atom. The results of these model calculations were in good qualitative agreement with the experimental findings. By varying the density of particles in the pick-up cell one can control the number ( n = 2, 3, . . .>of alkali atoms picked up by the helium cluster. The pickedup atoms will eventually recombine and form dimers, trimers, or larger clusters. In this way weakly bound complexes can be synthesized in a controlled way. Using this technique the group at Princeton University has studied the vibrational structure of the 1 t1 transition in the sodium dimer (Stienkemeier et ul., 1995). The observed transitions were found to be only slightly shifted with respect to the free Nu, molecule. The fact that the cluster surface has no effective mechanism that induces spinchanging transitions in the captured atoms enables use of such clusters for the preparation of high spin aggregates. This possibility has been recently realized by Higgins et ul. (1996), who for the first time have observed the formation of quartet spin 3/2 states of alkali trimers on the surface of He clusters, and have studied their excitation and emission spectra. The analysis of the LIF spectra has revealed that upon electronic excitation the quartet trimers undergo intersystem crossing to
'xi
"x:
OPTICAL AND MAGNETO-OPTICAL SPECTROSCOPY
111
the doublet manifold, followed by dissociation of the doublet trimer into an atom and a covalently bound singlet dinier. This work has demonstrated that the alkali aggregates synthesized on the surface of Hr clusters can be used for optical studies of fundamental chemical dynamics processes such as nonadiabatic spin conversion, change of bonding nature, and unimolecular dissociation.
V. Magnetic Resonance Spectroscopy He matrix-isolated particles present, besides their unconventional structure, another outstanding feature-their magnetic properties. In 197 1 it was anticipated that electron bubbles should have an extremely long spin-lattice relaxation time (Huang, 1971). This is a direct consequence of the diamagnetic nature of the matrix atoms: neither the closed electronic S-shell, nor the spinless nucleus of ‘He carry magnetic moments and one expects no first order guest-host spin-spin interaction. Experiments indeed revealed spin-lattice relaxation times in excess of 100 ms for electron bubbles in Hell. In 1993 Amdt et nl. speculated that the same property should hold for paramagnetic atoms immersed in condensed ‘He. The same authors later demonstrated relaxation times in excess of 1 second (Amdt rt ul., 199%). In paramagnetic atoms a high degree of spin polarization can be created by optical pumping. Spin-polarized atoms in condensed He combine the unique features of long storage and hence observation times, long relaxation times due to non-magnetic, isotropic trapping sites, and highly sensitive detection of magnetic resonance by optical means. High-resolution spin physics experiments with such atoms can hence be applied for a number of experiments such as the study of atoniic diffusion by spin-echo techniques, radiation detection of optical pumping, investigation of hyperfine anomalies, or the challenging search for P (parity) and T (time reversal invariance) violating permanent electric dipole moments of atoms (Arndt ef d., 1993; Weis et al., 1995).
A. ELECTRON BUBBLES The ground state of the electron bubble has an S,,>-stateconfiguration and magnetic dipole transitions can be driven in this ideal 2-level quantum system when the degeneracy is lifted by an external uniform magnetic field. Experiments in the 1970s used conventional electron-spin-resonance (ESR) techniques to study the magnetic properties of electron bubbles in liquid ‘He and ’He as well as in ‘He-’He mixtures. Reichert and Dahm (1974) succeeded in the first observation of the ESR transition in a field of 4.8 kG (transition frequency 13.6 GHz). Polarized electrons wcre implanted in superfluid ‘He by field emission from a ferromagnetic tip (iron whisker) Gleich et al., 1971). The observed resonance line-width of 45(10) mG
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was later found to be of instrumental origin and did not yield a quantitative determination of spin relaxation times. In subsequent experiments the line-width (HWHM) could be lowered to 5.0(4) mG (Zimmermann et al., 1977) and then to 0.9 mG (Reichert et al., 1979). The electron g-factor was found to coincide with its vacuum value at a level of 10 ppm and the (longitudinal) spin-lattice relaxation time T , was estimated to be greater than 100 ms (Zimmermann et al., 1977). Reichert and coworkers extended the measurements to ’He matrices (Reichert et al., 1979), for which they found a strong increase of the integrated ESR signal intensity with increasing He pressure, while in ‘He matrices the signal decreased with He pressure. This observation was tentatively interpreted as a consequence of the dipolar coupling of the bound electrons to the ‘He nuclear spins. The ESR spectra of positive ions in ‘He, ’He and ‘He-’He mixtures were studied by Reichert and Herold (1984). In contrast to electron bubbles the observed linewidths of the positive ions, mainly He:, in ‘He-’He mixtures, showed a strong temperature dependence, varying from 1 to 200 mG as the temperature was raised from 1 to 4 K. This behavior was explained again in terms of the relaxation induced by the hyperfine coupling to the nuclei of ’He atoms in the snowball’s frozen shell, in which the ’Hel‘He ratio is strongly temperature dependent. B. PARAMAGNETIC ATOMS The conventional ESR spectroscopy of electron bubbles requires many hours of signal averaging in order to yield reasonable signahoise ratios due to the low concentration of the spins and their small (< 1%) degree of polarization. Paramagnetic atoms can be implanted into He matrices at densities comparable to the ones achieved with electron bubbles. Magnetic resonance experiments on these species have the big advantage that optical methods can be used to build up a very large degree of (nuclear and electronic) spin polarization and that the same optical fields can be used to detect the magnetic resonance transitions in double-resonance experiments, for details see review article by Suter and Mlynek (1991). The build-up of polarization is achieved by means of optical pumping, a process in which the angular momentum of circularly polarized light is transferred to the atomic sample by successive absorption-reemission cycles. The polarized atoms are in a non-absorbing (dark) state with respect to the light field, and the magnetic resonance induced by depolarizing radio frequency or microwave radiation is detected via a resonant reappearance of fluorescence. The homogeneous optical absorption linewidth r of He-isolated atoms is approximately 6 orders of magnitude larger than in free atoms, and the peak scattering cross-section is suppressed by the same factor. However, the long electronic spin relaxation times T , , on the order of one second, anticipated for paramagnetic atoms in H e matrices mean that the atoms can be optically pumped with high efficiency, even with moderate cw laser intensities. This can be seen from
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the saturation parameter CI'T,K of the optical pumping process, where R is the optical Rabi frequency. The creation of a substantial degree of spin polarization in foreign atoms isolated in heavier noble gas matrices (Ne, Ar, etc.) is impeded by the strong depolarization due to the coupling of the defect's spins to local crystalline fields present in multiple anisotropic trapping sites. To our knowledge, no optical pumping signals from atoms in such matrices have been reported in the literature so far. Recently the possibility of optically pumping molecules in condensed He has also been discussed (Takami, 1996).
1. Optical Pumping and Magnetic Resonance in Zeemun Multipletts The first optical pumping and magnetic resonance signals from atoms in super$hid He were observed in 1994 by Yabuzaki and coworkers at Kyoto University (Kinoshita et d.,1994; Takahashi rt al., 1995). In that experiment, spin polarizations in excess of 50 percent were created in "Rb, "Rb, and '33Csatoms by optical pumping with circularly polarized light. Magnetic resonance transitions induced by pulsed radio frequency fields in static magnetic fields of 30-50 G were observed. From the resonance magnetic fields and their Breit-Rabi shift when pumping with crf or cr- polarized light, the authors deduced g,-factors of 2.12(2), 2.13(4), and 2.10(8) as well as hyperfine coupling constants of 1.13(6), 3.25(2), and 2.37(11) GHz for X5Rb,X7Rb,and i33Csrespectively. These values agree, within the experimental errors, with the corresponding vacuum values. Because of signal fluctuations due to turbulences induced by the sputtering process (section III), strong r.f. pulses were needed in the experiment, resulting in r.f. power-broadened magnetic resonance line-widths on the order of 500 mG, corresponding to an effective transverse spin relaxation time T2in the psec range. The difficulties related to the short observation times in liquid matrices was overcome by the development of an efficient implantation technique for atoms into solid He. The first observation of optical pumping, optically detected magnetic resonance and magnetic level crossing signals from i33C.yatoms in solid 'He, was reported in 1995 (Arndt et ul., 19951.3; Weis et al., 1995). These experiments yielded resonance line-widths of a few mG, which were limited by magnetic field inhomogeneities. With an improved magnetic field control a resonance line-width (HWHM) of 10Hz, corresponding to 30 pG, could recently be achieved in a driven spin precession experiment with phase-sensitive detection (Kanorsky ef ul., 1996). The signal had an equivalent magnetometric sensitivity of approximately 30 nG with a 1 second integration time. In that experiment it was further shown that even with an optical sample thckness of only lo-', magnetic resonance in He-isolated atoms can also be detected by monitoring the power of the transmitted laser beam.
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2. Relu..xution Times
The optical scattering process used to detect the magnetic resonance affects the lifetime of the sample magnetization and thus contributes to the line-width of the magnetic resonance. The observed line-widths thus do not reflect the intrinsic spin relaxation time of the sample. No measurements of the spin relaxation times of polarized atoms in liquid He have been reported so far. However, both longitudinal T , (Arndt et al., 1995b) and transverse T2 (Kanorksy et id., 1996) electronic spin relaxation times of Cs atoms in the cubic phase of solid He have been measured by the technique of relaxation in the dark (Franzen, 1959). In this threestep process, optical pumping is first used to create a longitudinal/transverse spin polarization, which is then allowed to relax in the absence of light, and whose surviving degree of polarization is finally detected optically. T , values of approximately 1 second were measured (Arndt et al., 1995b) and were found to be independent of the external magnetic holding field for strengths varying from a few pG to a few G. A first measurement of the transverse relaxation time of the same sample yielded T2 = 100 msec (Kanorsky et al., 1996). This value could recently be pushed to 300 msec (Lang et al., 1997), but has still to be considered as a lower bound for the intrinsic T2.The recent observation of a spin echo from Cs atoms in b.c.c. ‘He supports the assumption that part of the dephasing time still originates from magnetic field inhomogeneities. The observed T2 time of 300 msec is equivalent to an average variation of the magnetic field of 2.5 pG over the sample volume of 1 cm’. So far, no quantitative theoretical model for the atomic spin relaxation has been developed and one can only speculate about the exact nature of the depolarization process. Depolarization by a dipolar coupling to the paramagnetic nuclei of the 3He contamination of “He, as well as dipole4ipole interactions between Cs atoms can be estimated to give a negligible contribution to the relaxation rate. The two most likely depolarizing scenarios are spin-spin interactions with paramagnetic Cs clusters, which may be present in large quantities in the sample, or spin couplings to dynamically deformed atomic bubbles. As discussed below, a quadrupolar bubble deformation can couple to the electronic spin by the combination of an electric quadrupole interaction and the hyperfine interaction. A future comparison of the relaxation times of different atomic species, or preferably, of different isotopics of the same species, should shine more light on this open question.
3. Hyperfine Transitions The first experimental determination of the hyperfine coupling constant of Cs in condensed He was performed by the Kyoto team in a microwave-optical double resonance experiment (Takahashi et al., 1995) in superjuid He under saturated
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vapor pressure. The coupling constant was determined to be A,, = 2.3127( 1) GHz, which is 0.63 percent larger than the vacuum value. The first investigation of the hyperfine structure of Cs in a solid (h.c~.c.He 1nntri.v was performed by Lang et crl. (1995) and yielded a blue shift of the transition frequency by 2.1 percent (197 MHz) with a pressure shift rate of 1.4 MHAbar. Unlike the situation encountered in heavier noble gas matrices (Jen er d., 1962; Goldsborough and Koehler, 1964; Coufal et ul., 1984), only a single hypertine resonance line was found in solid helium, thus supporting the bubble model assumption of identical trapping sites for all atoms. The observed blue shift of the hypertine frequency induced by the He matrix can be traced back to the Pauli principle, by which the valence electron wave function Ur(r) is compressed onto the nucleus, thereby increasing the coupling constant proportional to IUr(0)l'. Takahashi er d . ( 1995)calculate an increase of 0.6 percent for lq(0)l' in the liquid phase, which agrees well with the experimental finding. Kanorsky and Weis (1996) obtain a shift of 0.54 percent for Cs in liquid He and a shift of 1.53 percent for Cs in solid He at 27 bar. Both the experiment in the liquid and in the solid phase recorded the IF = 4, M = 24) + IF = 3, M = 43) hyperfine Zeeman component. In the liquid phase a lowest line-width of 10 mG (corresponding to about 25 kHz) was recorded in a free-induction decay experiment (Takahashi et ul., 1995), whereas in the solid matrix a line-width of approximately 100 kHz was found (Lang et ul., 1995). It is interesting to note that the latter result was obtained under the same experimental conditions that yielded a 10 Hz line-width (Kanorsky rr ul., 1996) for the magnetic resonance transitions within the F = 4 niultiplett. It is likely that radial (breathing mode) bubble oscillations and the induced modulation of the hyperfine coupling constant are responsible for this extraordinary broadening of the hyperfine lines. No quantitative calculation of this effect has been performed so far.
4. Mutrix Eflects The control of the matrix pressure and temperature allows implanting the foreign atoms into either the cubic (b.c.c.) or the hexagonal (h.c.p.) phase. So far only spin-polarized Cs atoms have been investigated in both phases, and the spin properties were found to have some very distinct phase dependent features (Weis P t nl., 1996; Lang rt ul., 1996). As a general rule, the efficient build-up of a large and long-lived atomic polarization can only be achieved in the isotropic cubic phase. In the hexagonal phase, local fields depolarize the atomic sample and the longitudinal spin relaxation times are in the msec range with a pronounced magnetic field dependence. As a consequence the magnetic resonance lines are orders of magnitude broader than in b.c.c. The magnetic resonance spectra show further forbidden AM = 2,3 magnetic transitions, which indicate that the cylindrical symmetry imposed by the external magnetic field is broken. These observations can be explained by assuming that in h.c.p. the atoms reside in slightly deformed
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bubbles with random orientations, The following mechanism is likely to explain the coupling of the electron spin to the bubble axis: In lowest order the deformation of the bubbles has a quadrupolar symmetry with a major axis oriented along the c-axis of the local hexagonal matrix environment. The repulsive potential exerted by the bubble wall on the Cs valence electron then leads to a slightly deformed electronic orbit, which can be described quantum mechanically as a small admixture of D-orbital to the ground state S-orbital. The D part of the atomic wavefunction produces an electric field gradient at the nucleus, which couples to the nuclear electric quadrupole moment and hence to the nuclear spin. The latter couples via the hyperfine interaction to the electron spin. The strength of this spin-bubble coupling can be inferred, for example from the magnetic field dependence of the longitudinal spin relaxation time T , . With Cs in a low-pressure h.c.p. matrix a typical decoupling frequency of 10 kHz was observed. The spinbubble coupling also lifts the Zeeman degeneracy of the defect atoms in the absence of a magnetic field, thus enabling the observation of a zero-field magnetic resonance spectrum with a dominating resonance line at 10 kHz. In principle all the observations described above can be described in terms of a single parameter-the static bubble deformation parameter p, defined in Section 11.Band the future quantitative modeling of the effects should enable its experimental determination. Recently we have also obtained some experimental evidence for dynamical bubble deformations when measuring the ratio g,/gl of the nuclear and electronic g-factors. This ratio showed a surprisingly strong magnetic field dependence at low fields, which can be traced back to quadrupolar bubble oscillations. Here again a precise modelling of the effect should enable the measurement of the amplitude of such oscillations. 5. EDMSearch
Experiments searching for permanent electric dipole moments (EDM) of atoms are borderline experiments in which the precision measurement of atomic properties is used to test modern elementary particle theories. The existence of a particle EDM is forbidden by parity conservation (P symmetry) and time reversal invariance (T symmetry). Atomic EDMs can only arise as a consequence of weak interaction processes within the atom and are a sensitive tool for searching for physics beyond the standard model of weak interactions (Ramsey, 1995). The search for an EDM in He-matrix isolated paramagnetic atoms is one of the most challenging applications of spin-polarized atoms in He matrices. It is in fact the outlook toward a possible atomic EDM experiment that has triggered our own interest in these samples some 5 years ago. Sensitive EDM experiments call for samples with extremely long electronic spin relaxation times and very large electrical break-down voltages, both conditions that are met by He-matrix isolated
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atoms. The anticipated sensitivity of EDM experiments in cryogenic samples has been discussed in the literature (Weis, 1995, 1996).
VI. Concluding Remark In the past ten years the traditional field of matrix isolation spectroscopy in solid noble gas matrices has been successfully extended to superfluid and solid He matrices in which isolated atoms and molecules exhibit some quite unique and outstanding features not encountered in any other liquid or solid matrix. So far the properties of the local trapping sites have been investigated in optical experiments that have allowed inference of their gross structure, while magnetic resonance experiments now open the way to study their static and dynamic deformations. As a consequence of the long trapping times and the isotropy of the local trapping sites encountered with many species, ultra-high-resolution magnetic resonance experiments have become feasible. In this review we have discussed the wealth of information gained in the past years by the few, mainly experimental, groups who are developing this novel interdisciplinary branch of spectroscopy. We coiiclude by expressing our hope that in future years the field will attract more attention from both the low-temperature and the traditional matrix isolation community, and that fruitful collaborations will arise from the exchange of cross-fertilizing ideas between these communities.
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Tabbert, B., Beau, M., Giinther, H. Haualer, W.. Hiinninger, C., Meyer, K., Plagemann, B., and zu Putlitz, G. (1995a). 2.Phys. B97.425. Tabbert, B., Beau, M.,Foerste, M.,Giinther, H., Hiinninger, C., Lust, H., Layer, K., zu Putlitz, G., and Schumacher, T. (1995b). 2.Phys. B98,399. Takahashi, Y., Sano, K., Kinoshita, T., and Yabuzaki, T. (1993). Phys. Rev. Lert. 71, 1035. Takahashi, Y.,Fukuda, K., Kinoshita, T., and Yabuzaki, T. (1995). 2.Phys. 898,391. Takami, M. (1996). Comments At. Mol. Phys. 32,2 19. Vignos, J. H. and Fairbank, H. A. (1961). Phys. Rev. Leu. 6,265. Weis, A. (1995). “Proceedings ofthe Xxyth Rencontre de Moriond.” B. Guiderdoni, G . Greene, E. Hinds, and J. Van,Tran Thanh, Eds. Editions Frontieres (Gif-sur-Yvette). Weis, A. (1996). “Proceedings of the 12th International Congress LASER 95.” W. Waidelich, H. Hugel, H. Opower, H. Tiziani, R. Wallenstein, and W. Zinth, Eds. Springer (Heidelberg). Weis, A., Kanorsky, S. I., Arndt, M., and Hansch, T. W. (1995). Z. Phys. B98, 359. Weis, A., Lang, S., Kanorsky, S. I., Arndt, M., Ross, S. B., and Hhsch, T. W. (1996). “Laser Spectroscopy XXII.” M . Inguscio, M. Allegrini, and A. Sasso, Eds. World Scientific. (Singapore). Whittle, E., Dows, D. A., and Pirnentel, G. C. (1954). J. Chem Phys. 22, 1943. Wilks, J. (1967). “The Properties of Liquid and Solid Helium.” Clarendon Press (Oxford). Woolf, M. A. and Rayfeld, G. W. (1965). Phys. Rev. Lett. 15,235. Zimrnermann, P. H., Reichert, J. F., and Dahm, A. J. (1977). Phys. Rev. B15.2630.
ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS VOL 38
RYDBERG IONIZATION: FROM FIELD TO PHOTON G. M.LANKHUIJZEN AND L. D.NOORDAM FOM Institute for Atomic and Molecular Phy.rics Kruislaan 407, 1098 SJ Amstemhm, the Netherlands tel :020-6081234 email :NOORDAM@AMOLENL (September 27, 1996) 1. Introduction .................................................. A. Properties of Rydberg Atoms. . . . . . . . . . . , . . . , . . . . . . . . . . . . . . . . . . B. Rydberg Ionization: From Field to Photon. . . . . . . . . . . . . . . . . . . . . . . 11. DC Field Ionization . . . . . . . . . . . . . . . . . . . . . . . . . , . . . , . . . . . . . . . . , . , . . A. Stark States. . . . . . . . , B. Wavepacket Decay in an Electric Field . . . . . . . . , . . . . . . , . . . . . . . . . 111. Ramped Field Ionization . . . . . . . . . . . . . . . ............... A. Ionization by Ramped Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Ionization by Half-Cycle Pulses , . . . . . . . . , . , . . IV. Microwave Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Regime I : w < l/n" , B. Regime I1 : o l/n3... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . , , . V. THz Ionization . . . , . . . , . . . . . . . , . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . VI. Far Infrared Ionization . . . . . . . . . . , . . . , . . . . . . . , . . . , , . . . . . . . . . . . . . . A. Far Infrared Dipole Matrix Elements. . . . . . . . . . , . . . . . . . . . . . . . . . . . . B. Multiphoton Ionization of Rydberg Atoms Bypassing a Cooper Minimum. VII. Optical Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . , . . . , . . B. Inner Electron Excitation and Ionization . . . . . . . . . . . . . . . . . . . . . . . . . C. Rydberg States as Population Trap in Multiphoton Processes. . . . , . . . . , . VIII. Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . IX. Acknowledgment . . . . . .............................. X. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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121 122 123 126 127 128 131 13 1 134 135 136 139 141 143
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I. Introduction Atoms where one of the electrons is in a highly excited state, the so-called Rydberg atoms, have proven to be a useful tool for studying the atom-radiation interaction. In the last decade a large variety of mhation sources have been used to study the ionization mechanisms of Rydberg atoms. The physical mechanism of ionization depends on the radiation frequency. For very low frequency thejeld amplitude determines the ionization, while for higher fiquencies the photon energy is most
121
Copyright 0 1997 by Academic Press. Inc. All rights of reproductiw in any form reserved ISBNQ1243lt38-2
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G . M . Lnnkhuijzen and L. D.Noordam
essential. In this article we will review the ionization mechanism of Rydberg atoms exposed to radiation ranging from DC fields to optical frequencies. In the interaction of Rydberg atoms with a radiation field the Rydberg electron can gain enough energy to escape from the ionic potential: The atom ionizes. Important parameters to be considered in the ionization process are: (1) the ratio between the radiation frequency, w , and the binding energy of the Rydberg electron, E,,; (2) the ratio between w and the energy spacing between the Rydberg levels, AE,,; and (3) the coupling interaction between the Rydberg states. By studying the interaction with Rydberg atoms of different principal quantum number n at a given radiation frequency, these ratios vary. Hence to some extent frequency and principle quantum number are interchangeable (see Fig. 1). We consider the ionization of Rydberg atoms in the range of 10 < n < 100. In this section we introduce some properties of Rydberg atoms, and highlight some of the ionization mechanisms when these Rydberg atoms are exposed to various kinds of radiation. In subsequent sections we discuss in more detail the mechanism of Rydberg atom ionization in the different radiation frequency regimes.
A. PROPERTIES OF RYDBERG ATOMS For an extensive overview of Rydberg atoms we refer to the excellent book by T. F. Gallagher (1994). The binding energy of the Rydberg electron in the potential of a singly charged ion is given by
E
= n,‘
1
2(n
-
6,)Z’
where n is the principal quantum number, and 6, the quantum defect of the angular momentum state 1 (atomic units are used unless stated otherwise). This correction term (6,) on the binding energy arises from the presence of the core elec-
FIG. I . The frequency range that will be discussed in this chapter is plotted in the top of the figure. The lower two bars show the principal quantum number n for which the energy spacing between n states (AE,, = 11,i’) arid the binding energy (AE,, = - l/2n2) coincides with the corresponding photon energy of the radiation.
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123
trons. The low angular momentum states will penetrate the (2 - I ) core electrons giving rise to a higher effective potential. This results in a inore deeply bound energy for the low angular momentum states. For instance, the s-state of rubidium has a quantum defect of a,, = 3. I . Theoretically, the hydrogen atom is the easiest to tackle, due to the lack of core electrons and, as a result, a lack of coupling between the Rydberg states. In the experimental study of Rydberg atoms hydrogen has also been used in various experiments. However, alkali metal atoms. which can be considered one-electron atoms, are preferred experimentally because the production of a dilute gas of these atoms is rather simple. State sclective production of Rydberg atoms has become possible with the invention of tunable dye lasers. The Rydberg series of the alkali atoms lies in the operation range of the tunable dye lasers. Furthermore, interesting physics arises from the existence of the core electrons, which introduce coupling between the Rydberg slates. As we will see, this coupling can play a crucial role in the ionization mechanism of the Rydberg atom. especially for the cases of pulsed electric fields and microwave fields. The energy spacing of adjacent Rydberg states in hydrogen is given by
This energy spacing is an important property when resonant transitions between Rydberg states become possible and when the frequency and intensity of the radiation field induce energy shifts of this order. The frequency of the radiation for these transitions to occur lies in the far-infrared regime.
B. RYDBERG IONIZATION: FROMFIELD TO PHOTON We will now briefly mention some highlights in the ionization mechanisms organized in terms of the radiation frequency, covering the range where w << Ell,AE till w >> E,l,AE. The mechanisms will be discussed in more detail in the subsequent paragraphs.
I . DC-Field Ionization When an atom is placed in a static electric field, the atomic potential is altered by the presence of the field (see Fig. 2 ) . The tilting of the potential gives rise to a lower ionization threshold. Rydberg states with energy below the threshold can only ionize by tunnelling. We will see that the states that lie above the saddle point energy still have a finite lifetime. This is demonstrated by experiments that use short optical pulses to excite wavepackets above the saddle point energy. The evolution of the short-lived electron can be probed in two ways, either near the
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FIG. 2. The Coulomb potential experienced by a Rydberg electron without (full curve) and with (dotted curve) an external electric field applied. For DC field ionization the potential is lowered on one side, enabling electrons with energies exceeding the saddle point energy to escape from the parent ion. For optical ionization photons are absorbed by the electron leading to ionization.
core by an optical probe or near the saddle point by an atomic streak camera. It is found that the lifetime as measured by an optical techntque is not the same as the time it takes the electron to leave the atom, as measured by the atomic streak camera. 2. Ramped Field Ionizution The ionization mechanism of a Rydberg atom exposed to ramped electric field pulses in the MHz-GHz regime (rise time of the pulses ranging from microseconds to nanoseconds) is already different from the case of DC-field ionization. During the ramp of the electric field the energy of the Rydberg state will change. The change of energy depends strongly on the coupling between the Stark states, giving rise to several ionization threshold fields. Furthermore the lifetime of the “quasi continuum” states that lie above the saddle point energy plays an important role in the ionization dynamics when half-cycle pulses are used. When the lifetime of the continuum state is shorter than the pulse duration, the ionization will be suppressed. 3. Microwave lonizution
Further increasing the radiation frequency brings us into the regime of microwave radiation (GHz). For the case where the GHz frequencies are smaller than the energy spacing between the Rydberg states, Stark states and Landau-Zener tran-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
125
+
sitions are used to describe the ionization. Transitions from n + ti I become possible due to the DC shift of the energy levels by the electric field, bring the levels close together. The amplitude of the field required for ionization is much lower than in the case of DC-tield ionization. For high 17 states the frequency of the radiation field can become comparable to the energy spacing between the Rydberg levels, AE,,, giving rise to multiphoton type transitions.
4. TH: Ioni7ution The production of THz radiation has recently stimulated a lot of experimental and theoretical work on the ionization of Rydberg atoms by THz radiation. Experiments have been performed with essentially unitary half-cycle pulses (HCP) with a pulse duration < 1 ps, giving a large \pectral bandwidth with a central frequency in the THz regime. Because the pulse duration is much shorter than the Kepler period of a Rydberg electron (2.5 ps for 11 = 2 5 ) , the Rydberg electron IS frozen on the time scale of the interaction with the HCP. The ionization is described using a model where the electron experiences a momentum kick from the HCP. The atom ionizes when the energy gained by the electron is larger than its original binding energy. 5. Fur hlfi-urrd Icitiizntion
Increasing the radiation frequency even further. the photon energy of the radiation becomes comparable to the binding energy of Rydberg electrons. Using far infrared radiation, transitions between Rydberg states where An >> 1 have been studied. Starting from a Rydberg state the ionization occurs via a few resonant intermediate states and ionization is very efficient. A surprising exception is the ionization of lithium Rydberg atoms. The two-photon ionization cross-section of a lithium ns ( n - 17) state is very small due to a Cooper minimum in the lithium bound-bound ns -+ n’p transition. Therefore, ionization only proceeds in a complicated manner. The Rydberg electron is first deexcited by one photon to the lower lying np state, and then climbs up the ladder (nd + n’p + continuum) bypassing the s -+ p Cooper minimum in the bound-bound transitions.
6. Optictrl Ioni:ation In the optical regime, ionization can occur by absorbing a single photon ( w >> E,,) (see Fig. 2). Although in lowest order perturbation theory the frequency determines the ionization cross-section, we will see that the field aniplitude can still play an important role at these high frequencies. Several mechanisms of stabilization of the Rydberg electron against ionization will be discussed depending on the laser intensity and laser pulse duration. Furthermore, at specific optical frequen-
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G. M . knkhuijzen and L. D.Noordam
cies the inner electrons can also be excited, giving rise to autoionizing mechanisms. Finally, Rydberg states can act as population traps in the multiphoton ionization process of ground-state atoms.
11. DC Field Ionization When an atom is placed in a static electric field, the potential experienced by a Rydberg electron is given by 1 V = - - -r F z
(3)
where F is the electric field strength in the -z direction (see Fig. 3). In the +z direction the potential is lowered by the external electric field, giving rise to a saddle point in the potential surface at z = l/
F=-
1 16n4
(4)
In an experiment by Littman et al. (1976) this classical threshold was clearly observed. Lithium atoms in a static electric field were excited using a tunable
FIG. 3. Potential-energy surface of an electron in a combined Coulomb and elcctric field ( V = - I/r - Fz). The classical ionization energy is lowered by the external electric tield to E, = - 2 d F . An electron with energy larger than E,. can escape over the saddle point.
RYDBERG IONIZATION: FROM FIELD TO PHOTON
I27
laser. By scanning the laser frequency, the effective quantum number n of the excited Rydberg state was varied, showing a sharp ionization threshold when the binding energy became larger than the saddle point energy. Above E, resonances in the ionization cross-section were observed, showing that the Rydberg states still have a non-zero lifetime. Furthermore, quantum calculations and experiments on hydrogen showed the existence of bound states above the saddle point energy (Bailey et al., 1965; Koch and Mariani, 1981). From these observations it is clear that the classical description presented above is not sufficient to describe the dynamics of the electronic states in an electric field.
A. STARK STATES When a hydrogen atom is placed in an electric field the degeneracy of the angular momentum states is lifted by the field. As the angular momentum ( I ) is no longer a conserved quantity in the field we use a new quantum number, k. which labels the parabolic Stark states in the electric field. The energy of these Stark states is given to first order by
E,,,, =
1
-7 t
2n-
3 -Fnk 2
(5)
where F is the electric field strength and k ranges from -n + 1, -11 + 3, . ., n - 1. From Eq. (5) we see that the energy of the Stark state is linearly depending on the electric field strength, F, showing that each k-state has a fixed electric dipole moment, independent of the electric field. States with k > 0 are called blue states because their energy increases as a function of the electric field, and states with k < 0 are called red states. The blue Stark states are located on the uphill side of the potential and the red states are on the downhill side (see Fig. 3). In the case of hydrogen there is no coup h g between the Stark states. Because the blue (uphill) Stark states are located far away from the saddle point, they can only ionize via tunnelling to the continuum. The tunnelling rates of these blue states have been calculated (Bailey et al., 1965) and measured (Koch and Mariani, 1981). The lifetimes can be so long (> /AT) that the states are quasi stable even though their energy exceeds the saddle point energy in the potential. For non-hydrogenic atoms the Stark states are coupled by the presence of the core electrons. The blue states can now ionize on a much faster time scale through coupling with the red continuum states. In an experiment by Littman et (11. ( 1978) the ionization rate of the sodium ( n , k , m ) = ( I 3 , 3 , 2 ) state in the region of an avoided crossing with the rapidly ionizing red (14, - 11, 2 ) state was measured by scanning the electric field strength. A sharp increase in the ionization rate was observed (from lo6 s-' to >lox s ') at the avoided crossing between the two states, showing the effect of the coupling between the slowly ionizing and the fast ionizing state.
G. M. Lmkhuijzrn arid L. D.Noordum
128
The lifetime of these “quasi continuum” states is also reflected in the width of the resonances as observed in an absorption spectrum. In Fig. 4 an absorption spectrum is shown of rubidium in an electric field of 4.3 kV/cm (Lankhuijzen and Noordam, 199%). The complex structure of observed resonances can be attributed to Stark states belonging to several rz-manifolds. Close to El we observe sharp resonances, limited by the laser resolution of 0.2 cm-‘, indicating longlived states. For increasing energy of the exciting laser we observe broader peaks in the spectrum, indicating lifetimes on the order of a few picoseconds. In a similar experiment by Freeman er u1. (1978) a resonant structure above the zero-jield ionization limit was observed in rubidium in a static field of 4.335 kV/cm. The externally applied electric field induces shape resonances above the zero-field ionization limit. These resonances have also been observed in hydrogen (Glab and Nayfeh, 198% 198%; Rottke and Welge, 1986) and are more pronounced due to the lack of couping with the red continuum states.
B.
WAVEPACKET
DECAYIN AN ELECTRIC FIELD
The ionization dynamics of Stark states above the saddle point have been studied using short optical pulses to excite a coherent superposition of the “quasi continuum” states. Using optical techniques the decay of the wavepacket is monitored by measuring the amount of wave function returning to the core region as a func-
I
parallel laser polarization
-
1
E d 297
~
298
--
-
299
Wavelength (nm)
FIG. 4. loni~ationyicld of rubidium in an electric lield of 4.3 kV/cm 21s a fuiiction of the waveluiipth. The laser polarimtioii wah chosen parallel to the electric field. The upper graph shows a inagiiificd vicw of the spectrum from 299.5-300.5 nin (froin Lankhuijzen and Noordam, 19OSb).
RYDBERG IONIZATION: FROM FIELD TO PHOTON
129
tion of time. In an experiment by Broers et al. ( I 993, 1994; Christian et nl., 1993) a technique called Ramsey interferometry (Noordam et al., 1992a) was used to measure the wavepacket decay of rubidium atoms in a static electric field. In these experiments a picosecond optical pulse excites a wavepacket above the saddle point energy. After a delay, 7,.a second identical pulse is applied to probe the amount of wave function that has returned to the core region. In this way the evolution of the wavepacket can be observed. Equivalently, in the frequency domain the absorption spectrum of rubidium in an electric field has been measured. It was shown (Lankhuijzen and Noordam, 1995b) that by Fourier transforniing the relevant part of the frequency domain spectrum, the recurrence spectrum as measured with the Rariisey technique is obtained. A disadvantage of using optical techniques to study the ionizcition dynamics is the fact that the dynamics of the Rydberg electron can only be probed near the core. In the study of the ionization dynamics of atoms one would prefer to measure time resolved at the esccipe of the electron from the ionic potential, as this defines the ionization event. Using an atomic streak camera (Lankhuijzen and Noordam, 1996a),the escape of the electron has been measured directly in the time domain (Lankhuijzen and Noordam, 1996b). The atomic streak camera nieasures the escaping electron flux from the atom with picosecond time resolution. In the experiment, rubidium atoms were excited in a static electric field of 2.0 kV/cm by a short optical pulse. The excitation energy was chosen to be above the saddle point energy, thus creating an autoionizing wavepacket. By measuring the time-dependent electron flux with an atomic streak camera the ionization dynamics were resolved. In Fig. 5(a) a time resolved ionization spectrum of rubidium excited with a 4 ps pulse around 0.67 E,.is shown, where E, is the saddle point energy. Inspection of the figure shows that instead of observing an exponential decaying type of wavepacket, the main ionization was surprisingly delayed 12 ps. The polarization of the laser used to excite the wavepacket was chosen perpendicular to the electric field. The wavepacket will therefore be located perpendicular to the electric field. Because the electron can only escape in the +z direction it is still bound in the direction perpendicular to this axis. The wavepacket needs to reorient itself in the direction of the saddle point to escape. This reorienting can occur by scattering from the core electrons. The scattering in turn depends on the average value of the angular momentum of the wavepacket. For this particular case, the excited wavepacket makes an oscillation in angular momentum in 6 ps. Excited as a low angular momentum state, the wavepacket begins lo increase its angular momentum. When the wavepacket is i n a high angular momentum state it will be located far away from the core electrons (r,,,,,,(l) I ( I + 1j j , making it impossible to scatter. From Fig. 5(a) we observe that the wavepacket ionizes dominantly at the second angular recurrence. In Fig. 5(b) the corresponding recurrence spec-
-
(JX
G. M . Lunkliuijzen and L. D. Noordam
130
Rb in 2.0 2 0 kVlcm
I excitation at 0.87 0 87 E, I
-
-m
;lo 0
-
-B-
-
a,
g05 0 00
1
10-
0
'
1
'
1
'
40
20
(b) Recurrence
A
....
-
'05
00
-20
0
20 Time (ps)
40
60
FIG. 5. Comparison between time-rcsolvetl ionization spectrum as measured by the atomic streak caniei-a (a) and recurrence spectrum (b). The excihtion is at an encrgy of 0.87 E , . The laser polarization is perpendicular to the electric field of 2.0 kV/cm. In the upper spectrum the electron is prohcd at thc saddle poini, whereas in the lower spectrum the electron is probed near the core (fromLankhuijzen and Noordam. 1996b).
tmm is plotted. This spectrum is a measure for the amount of wave function that comes back to the 2 = 1 state as a function of time. We see that due to dispersion in the wavepacket, the amplitude is rather low after the first oscillation in angular momentum at 6 ps. The second angular recurrence, however, gives a high amplitude indicating that a larger fraction of the wavepacket returns to low angular momentum states. This causes the scatter event with the core electrons leading to the large ionization at 12 ps observed in Fig. 5(a). These spectra have been reproduced by a Multilevel Quantum Defect Theory (MQDT) calculation by Robicheaux and Shaw (1996). In their calculation they could observe the scatter event by observing the transfer of population from a closed channel (bound states) into an open channel (ionizing states) when the wave function returned to the core region. From the comparison of the recurrence spectra with the streak spectra the conclusion can be drawn that the lifetime as measured by an optical technique is not the same as the time it takes the electron to leave the atom. This conclusion can be drawn by probing the Rydberg electrons at different locations in the potential well. The Rydberg electron can be far away from the core, invisible for optical techniques, but still be captured in the attractive force of the parent ion. When the electron does not pass the core before ionizing, the electron appears to be ionized for an optical technique, but in fact has not yet escaped from the atom.
-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
131
111. Ramped Field Ionization A. IONIZATION BY RAMPED ELECTRIC FELIX In the previous section we saw that an externally applied electric field will lower the ionization threshold of a Rydberg atom. We will now discuss the case where the electric field is not stationary, but has a pulsed character. How Rydberg states evolve from zero-tield angular momentum states to Stark states that reach the ionization threshold can be best understood in terms of a Stark energy level diagram (see Fig. 6). In this figure the energy levels of rubidium are plotted as a function of the external electric field. Inspection of the figure shows a few important features: First, every n has its own manifold of states, composed of the so-called parabolic k-states. The upper states within a manifold, blue states, increase their energy as a function of the applied electric field and are located uphill in the potential (see Fig. 3). while the red states decrease their energy and are located on the downfield side. Second, the low angular momentum states have a different zero-field energy than the high angular momentum manifold states due to their quantum defect (see Eq. 1 ). Third, the Stark states exhibit avoided crossings due to the coupling between the states.
Rb Stark rnaD -62
-63
-64
-67
-68
-69
o
5
in
15
zn
25
30
35
40
Electric Field (V/cm)
FIG. 6. Energy levels of ruhidium around I I 41 a\ ii tbnction of lhc static elcctric lield. Thc low angular nioiiicntuiii states ( I 5 2) h a w a dilfererir .xro-lield energy than the higher angular momenturn states due to their quantum defect. The dotted trajeclory hhows the adiabatic ticld ioniiation path of the 42tl statc. 7
G. M. Laiikhuijzen and L. D. Noordam
132
When the electric field is ramped, these avoided crossings, also called LandauZener crossings, can be traversed in two different ways depending on the speed at which the crossing is taken (see Fig. 7). How the crossings are traversed determines at what field strength the atom will ionize. The probability of malung a diabatic transition is well approximated by the Landau-Zener transition probability (Landau and Lifshitz, 1977),
where AE is the energy splitting at the avoided crossing and dEldt can be written as dE/dF X dFldt where dE/dF is the field-dependent Stark shift between the two levels, and dF/dt is the slew rate of the electric field. From Eq. (6) we see that for slow field ramps (i.e., dE/dt << AE'), all the population follows the adiabatic path, whereas for fast field ramps (i.e., dE/dt >> AE'), all the population follows the diabatic path (see Fig. 7). If we for instance start in the 42d state and slowly switch on the electric field, the population follows the dashed adiabatic trajectory of Fig. 6 and reaches the ionization threshold at the F = 1/16n4. In this case the energy of the state has hardly changed (it can clearly be distinguished from the 41d or 43d). When the sweep speed is increased, part of the population will follow the diabatic path at the intersection of the n = 40, n = 41 manifold and may continue to make diabatic traversals of avoided crossings further down in energy, eventually ionizing at a higher field strength of F = 1/9n4.The decreasing energy of the Stark states implies that a higher field strength is needed to suppress the saddle point energy sufficiently. This adiabatiddiabatic ionization behavior has
x
P
5
Electric Field
*
FIG.7. A I a d a u - Z e n e r croasirlg can hc travcrscd in two ditfrrenl ways: For fast swcep rates of the tield the crossing is mversed diahatically (dotted linc), not changing the character of the slate. whereas for slow sweep rate5 the crossing is traversed adiabatically (full line) changing the chacactcr of the state froin 1 I ) to 12).
RYDBERG IONIZATION: FROM FIELD TO PHOTON
133
been demonstrated very clearly by Neijzen and Donszelmann ( 1982) (see Fig. 8). I n their experiment the 66/1 Rydberg state of indium was exposed to an electric field pulse with a varying slew rate. For slow slew rates mainly adiabatic ionization occurred at the F = 1/16n" threshold ( ( I ) in Fig. 8). When the slew rate was increased the 66p population makes one adiabatic crossing as it joins the adjacent IZ manifold. From then on it stays on the diabatic path ( 2 ) in Fig. 8, giving rise to a higher threshold field at F = 1/9n". For even higher slew rates another interesting ionization mechanism was observed. The slew rate is now fast enough to make a diabatic crossing at the encounter of the ad,jacent manifold. Blue states are populated, which follow an upward path in energy as the field is ramped: path ( 3 ) in Fig. 8. Because these blue states are located on the uphill part of the potential their ionization rate only becomes fast enough when the field strength is about three times higher than the classical ionization limit. The lifetime of the "quasi continuum" states now plays an important role in the ionization. From Fig. 8 we see that for even faster slew rates (lowest graph) the ionization appearing from the third path is shifted to even higher field strength. The population survives the electric fields ranging from 2 0 4 0 V/cm because the lifetime in that range is longer than the time the population spends in that region.
FIG. 8. The ioiiiiatiori signal ot the 66p 'P \talc u\ ;I ltinction of the increasing electric liclil for v;irioiis s k u I-ates ol'tlie lield pulsc. The ionization signal i s given i n arhitrary units and the iirca iindcr the ctirve lins not hecn normalized. The tipper par1 of h e lipire \cheni:itically eriplnin\ the oi-!gin 0 1 the ohservcd peak5 (from Neijren and Dbnsrclniann. I YX? 1.
134
G. M.Lunkhiiijzen and L. D.Noordum
A very useful application of ramped field ionization is state-selective field ionization. This technique is widely used to determine the Rydberg population of an atom state selectively (Gallagher, 1994). The Rydberg atoms are exposed to a slowly increasing electric field, enabling the electron to follow the adiabatic path to the F = 1/16n4threshold leading to ionization. By measuring the electrons or ions as a function of time, and knowing the time characteristics of the electric field pulse, the binding energy of the Rydberg population can be retrieved, thus determining the originally populated states. The technique can also be used to determine the angular momentum distribution of the Rydberg population confined to a single manifold. The zero-field energy spacing between the angular momentum states is too small to be resolved, but the higher angular momentum states will follow a diabatic route, giving rise to a higher ionization threshold that can be resolved more easily.
B. IONIZATION B Y HALF-CYCLE PULSES A natural continuation of the ramped field experiment is to switch the electric pulse off as well. By turning the electric field on and off very quickly, the Rydberg atoms are exposed to a short unidirectional electric field pulse, which brings the population into the “quasi continuum” for a short time before returning it back to zero field again. We will discuss here the interaction of these so-called halfcycle pulses with Rydberg atoms in the regime where the pulse duration is much longer than the Kepler orbit time of the Rydberg electron: T,, >> T,,. For a discussion on T,, << T,, we refer to section V. In an experiment by Kristensen et ul. (1997), rubidium Rydberg atoms were exposed to half-cycle pulses of different rise/fall times, duration, and amplitude. In Fig. 9 the remaining Rydberg population after exposure to two different halfcycle pulses is shown as a function of the scaled field k = F/n*4.In this experiment the amplitude of the half-cycle pulse is kept constant, but the laser exciting the Rydberg population is scanned, therefore scanning the effective quantum number n*.The k = 1/16 and F = 1/9 thresholds indicate the adiabatic and diabatic ionization thresholds respectively. For the pulse with the slowest rise time (9 ns rise time) we see that the onset of ionization occurs at the adiabatic F = 1/16 threshold. For the fast pulse (700 ps rise time, 1 ns duration) we observe a sharp threshold at the diabatic F = 119 threshold in agreement with the Landau-Zener model. For the fast pulse the diabatic path is followed, populating red Stark states dominantly. As soon as these states reach the ionization threshold at F = 1/9 they will ionize rapidly. For the pulse with a slow rise time, both red and blue states are populated and will reach the continuum by different paths. The fact that more blue states are also populated and reach the ionization threshold is reflected in the ionization behavior. An indication of this can be observed in Fig. 9, where beyond 5 = 0.2 there is more surviving population after the longer pulse than after the shorter pulse. Thls counterintuitive obser-
135
RYDBERG IONIZATION: FROM FIELD TO PHOTON
119
0.7 ps. 50 Vicm 9 ns, 50 Vicm
1
F 0.6
a: .-
I
‘\
m
E
0.2
u
\
. 0.0
~~
0.06
0.08
0.1
~
0.2
-
- -.__
~
0.4
0.6
0.8
1
Field in units of n-4
FIG. 9. Remaining Rydberg populniion a\ ii llinction 01‘ waled lield F after exposure to a hall‘. cycle pulse having a 700 ps rise time (full curve) and a hall-cycle pulhe having a 9 11s risc lime (dotted c~irve).Both ptilses have an amplittidc of SO V/cm. The vertical lines indicate the F = 1/16 and t h e F = ID threshold (from Kristenxn ct ( I / . . 1997).
vation might be explained in terms of the lifetimes of the “quasi continuum” states that are populated. In an experiment by van de Water rt al. (1984) the surviving Rydberg population of excited triplet helium atoms, after exposure to a square half-cycle pulse, showed a remarkable non-monotonic behavior as a function of the pulse amplitude. The Rydberg atoms ( n = 19) experienced a sharp field ramp and then a constant field for 200 ns. For ionization to occur the populated states during the constant field of the pulse need to couple with rapidly ionizing red Stark states. For certain values of the electric field the ionization was strongly reduced due to the lack of red Stark states, giving rise to the observed non-monotonic behavior in the ionization yield. These measurements show clearly that not only the amplitude of the electric field pulse, but also the duration and slew rate are of great importance in the ionization process of the Rydberg atom.
IV. Microwave Ionization In this section the ionization of Rydberg atoms when exposed to microwave radiation will be discussed. First we discuss the regime where the microwave radiation frequency, w, is less then the energy spacing between the Rydberg levels, given by AE,, = Un’. The field amplitude required for ionization will be much
I36
G . M. Lmkhuijzen atid L. D. Noordum
less than the classical field ionization limit deduced from the saddle point energy. The ionization process can be described as a sequence of half-cycle steps at the manifold crossings. Second, in experiments were high n states are populated the microwave frequency can become comparable to the energy spacing, giving rise to multi-photon type transitions leading to a different ionization mechanism. In section 1V.B we will see that the frequency of the radiation starts to play a more important role in the ionization process, in particular for microwave ionization of hydrogen.
A. REGIME I : w < l/n’
As discussed in the previous section, when a non-hydrogenic Rydberg atom is placed in a time-varying electric field, the evolution of the population can be described using Stark states and Landau-Zener transitions. This picture is also applicable to microwave ionization for w < l/n’ (Gallagher, 1994).In Fig. 10 the Stark energy levels in the electric field are schematically plotted. The levels are mirrored around the zero-field axis as the microwave radiation is bipolar. Also shown is the classical ionization limit given by F = 1/16n4. Starting from a particular Rydberg state and applying the many-cycle microwave pulse, the popula-
-50
..-‘E
-.=
m
-100
-150
t w
-200
-250
-1000
-500
0
500
1000
Electric Field strength (Vlcrn)
FIG. 10. Schematic graph of the Stark energy levels a s ii function of the electric tield. For microwave amplitudes exceeding the F,, value the population can follow the trajectory intlicatcd by the arrows to the ionirotion limit. Note that high up i n the Rydberg ladder trnnsitions with h i > I become possible.
RYDBERG IONIZATION: FROM F E L D TO PHOTON
137
tion will evolve through the Start states. For microwave fields below the critical value E,, (see Fig. 10) the population will spread out over the n = 20 manifold, but is unable to reach any other manifold: Ionization will not occur. When the microwave field amplitude is high enough to reach the intersection of the adjacent ( n + 1) manifold, population can be transferred to the higher manifold in a single cycle of the electric field (Lankhuijzen and Noordam, 1995a). The population cannot be transferred down to the n - 1 manifold because the field amplitude is not large enough to reach that intersection. On subsequent cycles of the microwave field the Rydberg population can be transferred to even higher n-manifolds. If the atom interacts with the microwave radiation for a long enough time, the population will climb up the Rydberg ladder until the field reaches the classical ionization threshold at F = 1/1611':The atom ionizes. We see that the first step in the ionization process-the traversal of the first anticrossing with the adjacent manifold-is the rate limiting step for long microwave pulses. The field at which the manifolds intersect can easily be calculated and is given by 1
F,, = 311'
(7)
For n > 6 this so-called Inglis-Teller limit is smaller than the F = 1/16n4value. In an experiment by Pillet et 01. (1984). sodium Rydberg atoms were ionized with long (-0.5 ps) pulses of 3.15 GHz microwave radiation. For the Jrnl 5 1 components the onset of ionization occurred at the F,., value, but for the Irnl = 2 states the ionization occurred at the much higher field of F = 1/9n4. The coupling between the Iml = 2 states of sodium is so small that all the avoided crossings are traversed completely diabatically, making it impossible for the population to climb the Rydberg ladder. Therefore the ionization occurred at the classical limit of the red states. This has also been studied by Mahon et (11. (1991) as a function of the microwave frequency ranging from 10 MHz-I 5 GHz. In an experiment by Hettema r t al. (1990), it was found that the ionization rate is frustrated when Stark states that lie in the middle of the manifold are excited. During the oscillations ofthe microwave field these middle Stark states do not change their energy, and the coupling with the reddest members of the higher manifold is frustrated. The diabatic ionization threshold has also been observed for hydrogen (Bayfield and Koch, 1974; van Leeuwen et L i I . , 1985). If we only take into account the first order Stark effect, the slope of the Stark states, dE/dF, is constant. When a particular Stark state is put in the microwave field the Rydberg orbit does not change because the dipole moment is constant. A middle Stark state, which will remain a middle Stark state after a reverse of the field amplitude, will ionize at a much higher field then the reddest Start state ofthat manifold due to the lack of coupling with the red continuum states. The second order Stark effect is needed to describe the observed ionization threshold at F = 1/9n4. When the second order is taken into account dEldF is not constant any more, but is decreasing as a
138
G. M. Linkhuijzen and L. D. Noordatn
function of the field. On the reversing of the microwave field each Stark state will therefore be projected onto several other Stark states. The population will be diffused through the Stark manifold and ionize once the field is large enough to ionize one of the Stark states, that is, the reddest Stark state. From these observations we see that Landau-Zener transitions are possible whenever the frequency, o,is in the range where a partially diabatic transition (see Eq. 6) is possible. The role of the frequency of the microwave radiation is rather limited, as it only determines at which of the two amplitude thresholds the ionization will occur. In order to reach the ionization limit, many oscillations of the microwave field are needed. In an experiment by Gatzeke et al. (1994), short microwave pulses consisting of as little as 25 oscillations of the electric field were used to ionize Na n = 24-33 Rydberg states. Instead of the sharp thresholds observed at F = 1/31? for the long microwave pulses, the thresholds became broader, indicating that the ionization was frustrated by the limited number of oscillations of microwave radiation field. By state selective field ionization they observed an enhanced population of higher Rydberg states after exposure to the shortest microwave pulses (25 cycles). Recently Watluns et al. (1996) used even shorter microwave pulses, down to only 5 oscillations of the microwave field at 8 GHz to ionize Na n = 32-44d Rydberg states (see Fig. 11). Not only did they observe a broader threshold in the ionization signal as a function of the microwave amplitude, but for the shortest pulses (6 cycles) the threshold shifted to the classical diabatic ionization limit at F = 1/9nJ. In this case the number of steps needed to reach the ionization threshold for a microwave ampli-
--c 3000 cycles
0
100
200
300
Microwave field amplitude (Vicm)
F a . I I . Surviving Rydberg population after expostire of the 44d state in sodium to inicrowavc pulses of various duration. For the longest pulse ionization occurs at the F = 1/3n5threshold. For the shortest pulse the ladder climbing is frustrated and ionization takes place at the F = 1/9n4 threshold (from Watkins et a/., 1996).
RYDBERG IONIZATION: FROM FIELD TO PHOTON
139
tude corresponding to F,T= 1/3n5 is not sufficient. Using state selective field ionization after the short microwave pulse was applied to the n = 24s state, they were able to determine that the population was trapped in higher and lower Rydberg states (n = 23-30) due to the limited number of cycles. We have seen that, for microwave ionization of Rydberg atoms following the mechanism of climbing the Rydberg ladder, Landau-Zener transitions are needed to transfer the population up the Rydberg ladder. The transition should lie in the intermediate regime between a fully diabatic and adiabatic transition, putting some constraints on the frequency of the microwave radiation. B. REGIME 11 : w
- 1/n3
When the frequency of the microwave radiation becomes comparable to the zerofield Rydberg spacing, the corresponding slew rate of the field becomes so large that the Landau-Zener picture is not applicable any more and microwave ionization is described as a combination of photon transitions to hgher Rydberg states followed by field ionization of this higher Rydberg state. These photon transitions have been observed in hydrogen. In an experiment by Bayfield and Koch (1974) hydrogen ti 65 (23 GHz spacing between neighboring states) atoms were exposed to microwave radiation of different frequencies. For frequencies of 30 MHz and 1.5 GHz the diabatic ionization threshold was found to occur at the field value of F = 1/9n4. Because the Stark states are not coupled in hydrogen, the population transfer to higher n-manifolds was not possible, giving rise to the classical ionization limit. However, in the case of 9.9 GHz microwave radiation the required field for ionization was lowrer than this classical limit. In an experiment by van Leeuwen rf ul. (1985), the ionization of hydrogen by 9.9 GHz was studied by varying the initial Rydberg n-state. For the case where w << lln' the ionization occurred at the classical F = 1/9n' limit, but approaching the regime w I/n' the ionization threshold dropped well below this classical threshold. This was well explained by the mechanism of multiphoton transitions to higher lying n-states and subsequent ionization of that state at the F = 1/9(n An)" threshold field. In an experiment by Galvez et [ I / . (1988) the regime from w < 1/n3to w > l/n3 was investigated for microwave ionization of hydrogen by varying the principal quantum number 11 (see Fig. 12). The threshold fields for ionization lie well below the classical limit (n:,F = 1/9) in this regime and exhibit well defined structure. The observed structure is explained in terms of rational fractions of n:,w at 1/1, 2/1, 5/2, etc., indicating a single-photon excitation to the II I state, a two-photon excitation to the IZ 1 state, and a five-photon excitation to the 12 + 2 state respectively. There are many more states involved in the ionization process than in excitation to adjacent manifold states, making it more difficult to observe these resonances. However, in an experiment by Richards rt 01. (1989), a resonance in the
-
-
+
+
+
G. M.Lankhuijzen and L. D. Noordam
140
n,3 w (scaled freq. for w / 2 n
= 36.02
GHz)
FIG. 12. Ten percent ionization threshold lield for microwave ionization of hydrogen by 36.02 GHz a s a function of ti. Also plotted are 3D classical calculations (cro5s and diamond) (from Galvez el ctl., 19x8).
ionization yield was observed (see Fig. 13). In their experiment they measured the n = 38 ionization yield of hydrogen resulting from a microwave field of 9.92 GHz. By increasing the amplitude of the microwave radiation a non-monotonic behavior of the ionization yield was observed. At a particular field strength of the microwave field, resonant transitions occur as a result of the AC Stark shift 1.0-
0.8:
0.6 PI 0.L:
0.2 1
0 0.10
0.12
0.1 1,
0.1 6
F
_. ... , .~ ..: FIG. 13. Measured ionization fraction of the rI = JX state ot nyarogen exposeo t o a i i l i u u w a v c licld of9.92 GHz as a function of the microwave amplitude (curve (c)). Also plotted arc a one-dimensional quantum calculation (curve ( I )), one-dimensional classical calculation ( f d l dots), three-dimensional classical calculation (open dots) (froiii Richards et u/., 1989). I
RYDBERG IONIZATION: FROM FIELD TO PHOTON
141
induced by the microwave field shifting the states into resonance. There is no clear understanding of the states responsible for the resonance because of the large amount of states involved in the problem.
V. THz Ionization In recent years the production of THz radiation has stimulated a lot of experimental and theoretical work on the interaction of this type of radiation with Rydberg atoms. THz radiation can be generated by illuminating a biased GaAs wafer with a femtosecond optical pulse (You p r a/., 1993). The radiation, produced by accelerating charge carriers in the wafer, is an almost unitary half-cycle of the electromagnetic field with a temporal shape given by the envelope of the optical pulse. Because the duration of this optical pulse is -0.5 ps, the generated radiation has frequency components well into the terahestz regime. In an experiment by Jones and Bucksbaum ( 1993), the ionization threshold of sodium nu' Rydberg atoms after exposure to such a half-cycle pulse (HCP) has been measured. From these measurements a scaling law was deduced: The amplitude, F,,, of a HCP needed to ionize a Rydberg atom scales as FIIx Iln.
(8)
This scaling law, which is very different from the scaling laws presented in the fonner sections (see Eqs. 4, 7), was explained by Reinhold rt al. (1993). In the regime where the radiation frequency w >> AE,!,or equivalently the pulse duration, T,,. is much less than the classical round-trip time of the electron, T,!,the ionization can be described using the impulse kick model. In this model it is assumed that the Rydberg electron does not move during the pulse, but gains momentum from the THz pulse in the form of a momentum kick given by t .,
Ap
=
Fl,
,
u(t)dt,
(9)
where F,, is the field amplitude and u ( t )is the normalized temporal profile of the HCP. The classical momentum of a Rydberg electron can be calculated from its binding energy El, = - 112n' and is given by plI = = Iln. The change in total energy of the Rydberg electron after excitation with the half-cycle pulse is then given by
AE
2 To
=
($' -
'T;'@
=
. A$
+ A$'/2
(10)
= + A$. We see that for Ap >> po the change in energy is given by AE -- AS;)'/2. When the change in energy exceeds the binding energy of the Rydberg electron (AE> 1/2n'), the atom can ionize, giving the relation Ap F,, > lln. In Fig. 14 the calculated scaled fields for 10 percent ionization of hydrogen are plot-
where
G. M. Lankhuijzen and L. D. Noordam
142
loo UP
lo-'
lo'
3" loo
(Jones et al 1993)
FIG. 14. Scaled field (upper figure) and scaled inomenturn transfer (lower figure) for 10 percent ionization threshold of H(n, 1 = 2, r n = 0 ) atoms as a function of the scaled pulse duration: classical resulta for inverse parabolic (full curve) and rectangular (dotted curve) pulses, quantum mechanical result for ionization of the 10d (full triangle) and Sd (inverted full triangle) states for rectangular pulses and experimental data by Jones and Bucksbaum (1993) for Na(nd) atoms, inultiplied hy a factor 2 3 (opcn squares) (from Reinhold et u / . ?1993).
ted as a function of the scaled time q, = ?,IT,,(full curve). From this figure we clearly see the transition from the short-pulse regime to the long-pulse regime. The long-pulse regime, that is, the adiabatic regime, has been discussed in section 111, giving rise to the F = lh4 scaling law. For the short-pulse regime significantly stronger pulses are needed to ionize the Rydberg atoms. Both classical and quantum calculations give the same result, indicating that the classical impulse-kick model is valid in the short pulse regime. The characterization of these HCPs in amplitude and pulse shape is still rather difficult. Because the pulse is freely propagating through space, the integral of the electric field over time has to be zero, showing that the pulse can never be unipolar. The main HCP peak is followed by a long negative tail with low amplitude. The effect of this negative tail following the HCP can substantially alter the ionization probability (Tielking et al., 1995). To circumvent this problem an experi-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
143
ment was performed by Frey et ul. ( I 996) where extremely high n-states were ionized by nanosecond unitary half-cycle pulses. The production of these nanosecond electrical pulses could be done in a much more controlled fashion giving a complete characterization of the pulses. In the experiment, n-states around n 388 and n 520 were exposed to half-cycle pulses with durations ranging from T, = 2 ns to 110 ns. The corresponding classical round trip time of these n-states is T,, - 9 ns and T,, 21 ns respectively, enabling the researchers to study the transition from the short-pulse (T, << T I )to the long-pulse CT,, >> T,J regime. They found perfect agreement with the theory by Reinhold rf al. (1993).
-
-
-
VI. Far Infrared Ionization In this section we discuss the radiation frequency regime where photon-transitions between Rydberg states with An >> 1 can occur. The transitions to higher lying states and subsequent ionization are determined by the resonance frequency of the radiation and the dipole matrix elements between the Rydberg states. The ionization with far infrared radiation (FIR) is examined using blackbody radiation sources (Beiting et al., 1979; Gallagher and Cooke, 1979; Figger et id., 1980) and CO? lasers (Burkhardt et ~ 1 . 1993). . At 300 K the energy density of the blackbody radiation peaks at a wavelength of 9.6 pm. However, the stimulated absorption and emission rates of the Rydberg states depend on the photon occupation number, which drops rapidly as a function of the photon energy. In an experiment by Beiting et (11. (1979), the redistribution of the xenon 26j Rydberg state to higher states was measured for various exposure times to the blackbody radiation. They found that after a 15.5 ps exposure time Rydberg states with n > 30 were populated. The states were detected using state-selective field ionization (see section 111). The spectra showed clear peaks in the spectrum, indicating that the populated states were low angular momentum states. This showed that photon transitions (A1 = 2 1 ) to the higher lying states were indeed responsible for the population redistribution. Ionization by blackbody radiation only occurs for very high n states and long exposure times and is strongly frustrated by de-excitation to lower lying states. In an experiment by Burkhardt et nl. (1993) the ionization from the 12s state in sodium by irradiation with laser light coming from a COz laser (a few lines around 10pm) was studied. By changing the electric field high Rydberg states that lie just below the ionization threshold shifted into resonance, giving rise to a large enhancement of the two-photon ionization yield. A disadvantage of using blackbody radiation to ionize an atom is the fact that the bandwidth of the incoherent radiation is enormous, making it very difficult to drive transitions between two specific Rydberg states. Recently an intense farinfrared free-electron laser has become operational (Oepts et d.,1995). This laser has a much narrower bandwidth (ANA = 0.1% - 10%) and a large tuneability
144
G. M.Lnnkhuijzeri aiid L. D. Noordm
(A = 6 - 100 pm), malung it possible to drive photon transitions between Rydberg states where An >> 1 (Hoogenraad, 1996). Multiphoton ionization is efficient because the excitation via intermediate states is resonant within the bandwidth of the FIR laser pulse. For instance, in rubidium a resonant four-photon ionization process is possible from the 14cl state. When this state is irradiated with 46-pm radiation, the ionization process resonantly follows the 14d + 18p + 24s -+ 6Op + e / d ladder (Hoogenraad, 1996; Hoogenraad et al., 1996). A. FARINFRARED DIPOLE MATRIX ELEMENTS
In this section we will introduce some analytical results on the matrix elements between Rydberg states. The dipole matrix elements between loosely bound states of hydrogen can be approximated semiclassically with high accuracy. The following derivation has been taken from Delone et al. (1994) and Hoogenraad and Noordam (1996). From the correspondence principle it follows that dipole matrix element equals the Fourier component at the transition frequency w of the classical radial coordinate r(t) of the Rydberg electron along its Kepler orbit, as follows:
I
111
(n’(r(n)=
T,,
r(t)cos wfdf.
(11)
I1
T,, is the period of the classical orbit time (T,! = 2 m ’ ) . For transitions between near-lying states, the transition frequency can be expressed as 0 = (n’ - n)/n’. In the half of the orbit where the electron moves away from the core, the trajectory of an out-going 1 = 0 orbit can be written as
1 r(t) = -(61)~/~ 2 Note that in this equation IZ is absent: The orbiting times of higher n states increase as n3,while the outer turning points (the maximal value of r) scale as n2. Substituting the half-orbit time in Eq. 12 yields the correct n dependence of the outer turning point. By substituting Eq. 12 into Eq. 11 and using the relation 7’,, = 2m’, the matrix element is given by
where the integral takes values between 0.5 and 1 and slowly converges to ( r ( 2 / 3 ) / d 3 The . two main characteristics of matrix elements between Rydberg states are present in this formula: First, the matrix elements are normalized to the density of states (n- I ) , so that an integral over an interval of the spectrum is normalized. Second, the matrix elements depend on the transition frequency as @-?
RYDBERG IONIZATION: FROM FIELD TO PHOTON
145
Transitions between nearby-lying states are favored over high-frequency transitions. Equation 13 can be further generalized to remove the asymmetry of exchanging the initial n and final n’. Reformulating the results of Goreslavski et (11. (1982) in terms of the binding energy of the states, w,, = 1/2n2, and the exact transition frequency w = Iqj,- q,l yields
The matrix element consists of three parts: The densities of states in the Rydberg series at states n and n‘, a general frequency dependence, and a prefactor C = 0.4108. This prefactor is only valid for 1 = 0 states. It can, however, be replaced by a frequency and angular-momentum dependent function (Hoogenraad, 1996; Berson, 1982; Goreslavski; ~t ul., 1982). In doing so the C factor stays of the same order but shows a higher value for the A1 = 1 transition compared to the A1 = - 1 transition, and for transitions where the energy increases, in agreement with the Bethe rule.
+
B. MULTIPHOTON IONIZATION OF RYDBERC ATOMSBYPASSING A COOPER MINIMUM The multiphoton ionization rate of a Rydberg state depends strongly on the dipole matrix elements between the intermediate states. In general, these dipole matrix elements are rather large, making the atom very susceptible to far-infrared radiation excitation, but there are exceptions. For instance, a Cooper minimum (Cooper, 1962) exists in the bound-bound ns - n‘p transition in lithium, giving rise to a very low excitation probability (Hoogenraad et al., 1995; Hoogenraad, 1996). In Fig. 15 the relevant energy levels of lithium are plotted. The left part of the figure shows the matrix elements for the 23s to np transition, with the Cooper minimum occurring around the 50p state. When the 23s state was exposed to farinfrared radiation of A = 62.5 pm (- 160 cm-’), the measured ionization yield was much higher than expected on the basis of the presence of the Cooper minimum. By scanning the frequency of the far-infrared radiation, an enhanced ionization yield was measured at the frequency corresponding to the resonant transition from the 23s down to the 17p state. If only the direct 2-photon ionization (23s 4 50p -+continuum) was the relevant ionization mechanism, the ionization yield would be a smooth function of the frequency, because the bandwidth of the laser pulses is much larger than the Rydberg spacing around n = 50. The resonance indicates that another path dominates the ionization process. Within the bandwidth of the laser, resonant transitions follow the 23s -+17p + 22d + 50p + edcf continuum, apparently the dominant pathway in the ionization process. Due to the Cooper minimum, two-photon ionization is frustrated and a higher-order process is favored, starting with stimulated emission of a FIR photon instead of absorbing photons.
146
G. M. Lnnkhuijzen and L. D. Noordam
Binding Energy (cm
-I)
r
-20
-30
0
-40 -50
-200
-60
-400
-600
0
1
2
3
Angular Momentum Frc. IS. (a) Squared and normalized matrix clemen~snear the Cooper minimum in the 23s -+ np series. The normalized matrix elements from the 23s to the 171, state are 3.8 X 10” ( c I J R ~ ) (b) ’ . The frustrated direct two-photon path (starting with the dashed line) and alternative higher-order multiphoton paths to the continuum, bypassing the Cooper minimum (from Hoogenraad et a/., 1995).
VII. Optical Radiation The energy of an optical photon (0.1 p m < A < 1 pm) exceeds the binding energy of Rydberg states, and ionization can occur by the absorption of a single photon. Although in lowest order perturbation theory the frequency determines the ionization cross-section, the field amplitude can still play an important role at these high frequencies. It has been found that for high amplitudes of the optical radiation the ionization cross-section actually decreases (an effect known as adiabatic stabilization, which will be discussed in section VI1.A). At high frequencies the excitation of inner electrons also becomes possible (section VI1.B). The energy exchange of the excited inner electron with the Rydberg electron can lead to autoionization. In the final section V1I.C we will see that in the process of multiphoton ionization of ground state atoms, Rydberg states can be a “dead end” en route to the ionization continuum.
RYDBERC IONIZATION: FROM FIELD TO PHOTON
147
Let us start with the ionization rate, R, and its physical implications. Starting from Eq. 14, it follows that the single-photon ionization rate for a low angular momentum Rydberg state (I << n ) is given by (Goreslavskii e / trl., 1992). R ,,-?w-l~l/'I (15) Here we find the expected linear intensity (I) dependence. The decreasing rate with increasing photon energy ( w ) reflects the decreasing overlap of the continuum wavefunction with the Rydberg state. In order to conserve momentum, photoabsorption occurs near the atomic core (Giusti-Suzor and Zoller, 1987). Hence the absorption rate is proportional to the time the electron spends near the core and is thus inversely proportional to the Kepler orbit time TI,= 27171'.
A. STAH~LIZATION Several mechanisms of stabilization can prevent the Rydberg atom from completely ionizing despite the fact that the fluence of the pulse, based on Eq. 15, is sufficient for depletion. In this section we discuss three types of stabilization (for a more detailed review see Muller and Fedorov, 1996).
1. Transient Stabilizution As tnentioned above, only the fraction of the extended Rydberg wavefunction near the core can absorb an optical photon. For laser pulse durations (T,,)shorter than the Kepler orbiting time ( T J , the fraction of the wavefunction that passes the core during the pulse and hence can be ionized is at niost T,,lT,l.This simple picture is confirmed both by quantum calculations (Hoogenraad and Noordam, 1993) and experiment (Hoogenraad et al., 1994) (see Fig. 16). This type of stabilization is called transient because more intense pulses will not ionize the atom (in contrast to Eq. 15), while longer pulses ionize the atom completely, as all the Rydberg wavefunction will pass the core during such a longer pulse. The short laser pulse drills a hole in the stationary wavefunction and as a result a nonstationary anti-wave packet is formed. The states adjacent to the initial Rydberg state are coherently populated by a Raman transition via the continuum (Yeazell and Stroud, 1991). Both frequencies required for the Raman transition are within the large bandwidth of the short laser pulse. The redistribution of Rydberg population by short laser pulses has been observed by several groups (Noordam ef al., 1992b; Duncan and Jones, 1996).
2. Interjkrence Stabilization In the caEe of interference stabilization (Fedorov and Movsesian, 1989) the coherent superposition of adjacent Rydberg states is not due to the short laser pulse and the resulting large bandwidth, but rather the high intensity of the laser.
148
G. M. Lunkhuijzen and L. D. Noordariz
0.6 Y
r
vl
\
?
e
0.4 -
c P)
B 0.2
c
1
J
t '
Short
0
1 2 3 Pulse duration (ps)
FIG. 16. The measured photoionization yield of the 6s27d'D2 state in barium as a function of laser pulse duration for a constant Huence of 7.8 Jkm?. The full curve is a calculation based on the mechanism described in the text (from Hoogenraad et nl., 1994).
The Rabi flopping time of the Rydberg state to the continuum becomes comparable to the Kepler orbit time (or equivalently in the frequency domain: the lifetime broadened Rydberg state overlap in energy). The transferred amplitudes out of the different Rydberg states interfere destructively in the continuum. 3. Adiabatic Stabilizution
In the case of adiabatic stabilization the ionization is suppressed at high intensities, irrespective of the pulse duration, and adjacent Rydberg states play no role. Because the photon energy is large compared to the binding energy of the Rydberg electron, the oscillations of the optical field are rapid compared with any electronic motion. In this high-frequency limit, the electron experiences the timeaveraged potential of the core plus the laser field. Calculations show that the large reduces the interacelectron excursion due to the field (amplitude q,= tion with the core and thereby the photoionization probability (Pont and Gavrila, 1990; Vos and Gavrila, 1993). Indeed, experiments show a lower ionization cross-section at higher intensities (de Boer et al., 1993a; van Druten et al., 1996).
m)
B.
INNER
ELECTRON EXCITATION AND IONIZATION
Optical excitation of an ionic transition with the remote Rydberg electron as a spectator is known as isolated core excitation. When the Rydberg electron passes the core it can exchange energy with the excited core and autoionize. Autoion-
RYDBERG IONIZATION: FROM FIELD TO PHOTON
149
ization spectroscopy in which the core is excited with a narrow bandwidth lowintensity laser pulse is a field in its own right (see reviews by Sandner, 1987. and Gallagher, 1994). If the intensity of the core excitation is high enough, the Rabi frequency of the inner electron transition becomes comparable to the Kepler orbit frequency of the outer Rydberg electron. Several authors (van Druten and Muller, 1995, 1996; Robicheaux, 1993; Grobe and Haan, 1994; Hansen and Lambropoulos, 1995, 1996) have predicted that the autoionization dynamics are strongly influenced by the Rabi flopping of the inner electron. Suppose the Rydberg electron orbits as a wavepacket around the core and the inner electron is flopping with the same frequency from the ground state to the excited state. Autoionization only occurs when the Rydberg electron passes the core while the inner electron is excited. So if the core is in the ground state when the Rydberg electron passes by and the Rabi and Kepler frequency are the same, the ion will be again in the ground state at the next passage of the Rydberg wavepacket: The doubly excited state is relatively stable against autoionization. In contrast, when the Rydberg electron is in a stationary eigenstate while the pulse duration of the core excitation is short, there is a sudden turn on of the autoionization of the Rydberg atom. Once the core is excited, the incoming Rydberg electron flux penetrating the core region can autoionize. As long as fresh wavefunction penetrates the core the autoionization rate is constant. After one Kepler orbit time all the wavefunction has passed the core once and the ionization rate drops to a lower value. This stepwise decay of the autoionization as described by Wang and Cooke (1991, 1992) is very similar to the stepwise decay of a Rydberg atom when exposed to a laser field that is suddenly turned on and kept constant thereafter (Hoogenraad and Noordam, 1993). It has also been shown that off-resonant excitation of the inner electron leading to autoionization can be accomplished if the outer electron is near the core at the time of excitation. In this situation the energy m i s match of the core excitation is absorbed by the energy of the outer electron. The outer electron gives or takes energy to or from the inner electron (Story et NI.,1993). As discussed above, Rydberg atoms are difficult to photoionize with pulses that are short compared to the Kepler orbit time. However, such short pulses can ionize inner electrons in a non-resonant inultiphoton process. In two independent experiments (Stapelfeldt eta/., 1991; Jones and Bucksbauni, 1991) it was found that for a Brr Rydberg atom, exposed to short intense radiation, it was not the loosely bound Rydberg electron but rather the deeply bound 6s electron (10 eV) that was ionized. The Rydberg state of the neutral atom is projected on several Bat Rydberg states (Vrijen, 1997; Vrijen and Noordam, 1996; Vrijen et nl., 1996) by suddenly kicking out the inner electron. Ionizing inner electrons while the Rydberg electron remains attached to the ionic core creates an alternative scheme for an x-ray laser (Vrijen and Noordam, 1996; Vrijen et nl., 1996). For instance one can start with a gas of lithium Rydberg atoms ( n = 8). Photoionization of the inner electrons results in a hollow atom: Li" ( n = 14). Using a second laser pulse the ti = 14 population can be stimulated down to I I = 5, resulting in a population inversion on the S y -+ 1s transition in Li' ' at 10.5 nni.
G. M.Lankhuijzen und L. D.Noordrrm
150
C. RYDBERG STATES AS POPULATION TRAPIN MULTIPHOTON PROCESSES Using sub-picosecond laser pulses, Freeman et al. (1987) demonstrated that Rydberg states play an important role in the multiphoton ionization process of groundstate atoms with optical radiation. Until recently it was believed that these intermediate Rydberg states enhance the ionization rate and no excited state population was left over after the intense pulse. Due to the large AC Stark shift (-1 eV) of the Rydberg states during the intense pulse, the states shift in and out of N-photon resonance. In fact, transfer to the subsequent Rydberg states, shifting in resonance, can deplete the ground state during the rising edge of the short pulse, (Vrijen et al., 1993; Story et al., 1993b). Recently de Boer et al. (1992, 1993b) have demonstrated that these states are indeed populated during the pulse, but that the ionization rate out of the Rydberg states (see Eq. 15) is so low that a large fraction of the population is trapped in these states. The role of Rydberg resonances in multiphoton ionization processes was recently reviewed by Vrijen et al. (1994).
VIII. Open Questions Despite the vast amount of investigation on Rydberg ionization in the different frequency regimes, as summarized in this contribution, many problems are not yet solved. We will address a few problems that require further investigations, both theoretical and experimental. The dynamics of an electron excited above the saddle point of the combined Coulomb static field is particularly well studied near the core. As demonstrated by Lankhuijzen and Noordam (1996b), the behavior near the core does not disclose when the electron is actually ejected. As the stability of highly excited electrons in external fields is of practical relevance in plasma environments, this field deserves a more detailed study. The mechanism of microwave ionization ( w < l / d ) with long pulses is well established. However, how the quantum diffusion proceeds if the Rydberg atom is only exposed to a few cycles is less clear. The transfer per cycle from n to n + 1 is well below 100 percent and coherent addition of amplitude, transferred during subsequent cycles, will definitely play a role. Well-defined experiments investigating the coherence effects in microwave ionization are rare, and the large amount of quantum states involved requires a major computational effort. Another open question is how high in n must the electron climb before the last half cycle can ionize the atom. Half cycle ionization with (sub)nanosecond pulses suggests that the threshold field amplitude is well beyond F = 1/9n4.To treat this problem theoretically, both the impulsive kick approximations, only applicable to very short pulses, and (a)diabatic sweeping up to the threshold description, fail, the latter because states above threshold are too long lived to assume direct ionization.
+
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Ionization with half-cycle THz pulses is a new and exciting field with many unsolved problems. What is the influence of the core electrons (in other words, is there a difference between sodium and hydrogen)? Does the core play a role in the time of ejection of the electron from the atom? How does ionization proceed for circular states (rn = I = I? - I), that is, is there a polarization dependence in the ionization behavior of circular states? Single-photon ionization with far infrared radiation provides the electron with little excess energy. Infrared photoionization of a polarized Rydberg atom in a static field is at the unexplored border of optical ionization and a n -+ n + 1 transition. Absorption of optical photons occurs near the core, and the rate of ionization strictly depends on the amount of initial state wavefunction near the core. For a n -+I I 1 transition bluehphill n-states couple best with bluehphill n 1-states, due to their large overlap in wavefunctions. A much larger fraction of the initial state wavefunction contributes to the transition probability. Stabilization of Rydberg atoms against photoionization with optical radiation, being one of the exciting new theoretical findings in the field of atoms in strong laser fields, has been verified experimentally only in a limited fashion. There is no experimental evidence of interference stabilization; adiabatic stabilization has been studied in some detail (deBoer et id., 1993a; van Druten et al., 1996), demonstrating that the atom is not completely ionized at high intensities. However, there is no explicit evidence for a reduced rate at very high intensities. Finally, transient stabilization has been experimentally confirmed (Hoogenraad ef nl., 1993), but the connection with adiabatic stabilization, that is, ionization with very short and intense pulses, is unexplored, both theoretically and experimentally.
+
+
IX. Acknowledgment The authors would like to acknowledge T. F. Gallagher and his group for the productive collaborations over many years resulting in some of the work summarized in this paper. We thank D. I. Duncan and M. J. J. Vrakking for carefully reading the manuscript. Some of the work described in t h s paper is part of the research program of the Stichting Fundamenteel Onderzoek van de Materie (Foundation for Fundamental Research on Matter) and was made possible by the financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Research).
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Lankhuijzen. G. M., and Noordarn. L. D. (19%). Phvs. Rev. Left. 74, 355. Lankhuijzen, G. M.. and Noordam. L. D. (1995h). Phvs. Rev. A. 52, 2016. Lankhuijzen. G. M.. and Noordam, L. D. (1996a). Upt. Conirnun. 129, 361. Lankhuijzen, G. M.. and Noordam. L. D. (1996h). Phys. Rev. Lett. 76, 1784. Littman. M. G., Zimmerman, M. L.. and Kleppner, D. (1976). Phys. Rev. Lett. 37,486. Littman, M. G., Kash. M. M., and Kleppner, D. (1978). Phys. Rev. Lett. 41, 103. Mahon, C. R., Dexter, J. L., Pillet. P., and Gallagher. T. F. (1991). Phys. Rev. A. 44, 1859. Muller, H. G., and Fedorov, M. V. (1996). Super-lntense Lrrser-Atom Physics IV, 1st ed. Kluwer Academic Publishers (Dordrecht/Boston/London). Neijzen. J. H. M., and Donszelmann, A. (1982). J . Phys. B: At. Mol. Opt. Phys. 15, L87. Noordam, L. D., Duncan, D. I., and Gallagher. T. F. (1992a). Phys. Rev. A. 45, 4734. Noordam, L. D.. Stapelfeldt. H., Duncan. D. I., and Gallagher, T. F. (1992b). Phys. Rev. Lett. 68, 1496. Oepts. D.. van der Meer. A. F. G., and van Amersfoort, P. W. (1995). Infrared Phys. Techno/. 36,297. Pillet, P., van Linden van den Heuvell. H. B., Smith. W. W., Kachru, R., Tran, N. H., and Gallagher, T. F. (1984). P h y . Rev. A. 30.280. Pont, M., and Gavrila, M. (1990). Ph.w. Rev. Left. 65, 2362. Reinbold, C. 0.. Melles, M., Shao, H.. and Burgdorfer, J. (1993). J. Phys. B: A f . Mol. Opt. Phys. 26, L6S9. Richards, D., Leopold, J. G.. Koch, P. M., Galvez, E. J., van Leeuwen. K. A. H., Moorman, L., Sauer, B. E., and Jensen, R. V. (1989). J. fhys. B : A / . Mu/. Opf. Phvs. 22, 1307. Robicheaux. F. (1993). f h y s . Rev. A. 47, 1391. Robicheaux, F., and Shaw, J. (1996). Submitted to Phvs. Re)).Lerf. Rottke, H., and Welge, K. H. (1986). fhys. Rev A. 33, 301. Sandner, W. (1987). Comrn. At. Mol. Plivs. 20, 171. Stapelfeldt, H., Papaioannou, D. G., Noordam, L. D., and Gallagher, T. F. (1991). Phys. Rev. Left 67,3223. Story, J . G., Duncan, D. I., and Gallagher, T. F. (1993a). f h v s . Rev. Left. 71, 3431. Story, J. G., Duncan, D. I., and Gallagher, T. F, (1993b). fhys. Rev. Left. 70, 3012. Tielking, N. E., Bensky, T. J., and Jones. R. R. (1995). Phys. Rev. A. 51, 3370. van de Water, W., Mariani, D. R., and Koch, P. M. (1984). Phys. Rev. A . 30,2399. van Druten, N. J., and Muller, H. G. (1995). Phys. Re\! A. 52, 3047. van Druten, N. J., and Muller, H. G. (1996). J . Phy's. 8;A f . M d . Opt. Phys. 29, 15. van Druten. N. J., Constantinescu, R. C., Schins, J. M.. Nieuwenhuize, H., and Muller, H. G. (1997). P h p . Rev. A. 55, 622. van Leeuwen, K. A. H., Oppen, G. V., Renwick, S.. Bowlin, J. B., Koch, P. M., Jensen, R. V.. Rath, O., Richards, D., and Leopold, J. G . (1985). Phys. Rei! Left. 55, 2231. Vos, R. J.. and Gavrila, M. (1993). Phys. Rev. A. 48.46. Vrijen, R. B.. Hoogenraad, J . H., Muller, H. G., and Noordam. L. D. ( I 993). Phys. Rev. Left. 70,30 16. Vrijen, R. B., Hoogenraad, J. H., and Noordam, L. D. ( 1994). Modern Physics LeffersB. 8, 205. Vrijen, R. B., and Noordam, L. D. (1996). J . Opt. Soc. Am. B. 13, 189. Vrijen, R. B., van Ingen, M., and Noordam, L. D. in H. C. Muller and M. V. Fedorov (Eds.), SuperIntense Laser-Arom Physics IV Kluwer Academic Publishers (DordrechtiBostonLondon, 1996). pp. 23-35. Vrijen, R. 8 . (1997). Ph.D. thesis, University of Amsterdam. (The Netherlands). Wang, X., and Cooke, W. E. (1991). Pliys. Rev. Lett. 67,976. Wang, X., and Cooke, W. E. (1992). Phys. Rev. A. 46,4347. Watkins. R. B., Vrijen, R. B., Griffith, W. M., and Gallagher, T. F. (1996). Submitted to Phys. Rev. A. Yeazell, J. A., and Stroud Jr., C. R. (1991). Phys. Rev. A. 43, 5153. You. D., Jones, R. R., Bucksbaum, P. H., and Dykaar, D. R. (1993). Opt. Left. 18,290.
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A I I V 4 N C E S IN ATOMIC. MOLI.('tILAR. A N D O P f I C A L PIHYSICS VOL. .3X
STUDIES OF NEGATIVE IONS IN STORAGE RINGS L. H. ANDERSEN, T ANDERSEN, AND P: HVELPLUND Institute of Physics irtitl Astrononn: Utiri~c~rsi!s qf'rltrrliirs DK-8000 Arirhus C. Dentnrirk 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Lifetime Studies of Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Atomic Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . 'The He (ls2s2p'P) Ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Alkaline-Earth Ions Br . Cn and Ba- . . . . . . . . . . . . . . . . . . . . . 9. Molecular Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. The He;(%,) I o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Cluster Ions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Fullerene Lifetimeh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lifetimes of Small Carbon Clustcn . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Electron-Impact Dctachment from Negative Ions . . . . . . . . . . . . . . . . . . . . . .
A. Atomic Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Classical Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Semiclassical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Quantum Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Doubly Charged Negative Atomic Ions. . . . . . . . . . . . . . . . 9. Molecular Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Resonances in the Detachment Cross Section . . . . . . . . . . . .... IV. Interactions Between Photons and Negative tons . . . . . . . . . . . . . . . . . . . . . . . A. Atomic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I , ResonanccsinH~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Cluaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Photodetachment of C,, and C,,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Acknowledgements . . . . . . . . . . . ............................... V1. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
158 I58 158
162 I65
167 172 172 I75 I76
183 185 I85 18.5 186 I86 188 188
I. Introduction Ever since the pioneering studies in the 1950s of negative ions in the gas phase (Branscomb, 1962),progress in the exploration of the properties of negative ions has been closely linked to the development of new experimental techniques. Branscomb's studies were stimulated by the recent introduction of ultrahigh-vacuum and crossed-beams techniques. Subsequently, electron scattering on atoms and molecules, collisions between negative ions and neutral atoms, and laserspectroscopic techniques were introduced. Within the last decade, storage rings, which have served as a tool in nuclear and particle physics for many years, have Is.5
Copyright 0 1997 by Academic Pres.;. Inc All right\ ot reproduction 111 any form resewed. ISBN n- I 2-(x)3x38-2
also attracted the attention of atomic and molecular physicists. A small number of ion storage rings, mainly located in Europe, are now in operation and dedicated to research in these areas. Negative ion studies represent a special niche in the physics of atomic interactions due to the strong electron-electron correlation effects and the fact that the outer electron in a negative ion “feels” a potential that asymptotically decays as r-‘. In general, this potential gives rise to only one bound state. The absence of the Rydberg series strongly limits the application of experimental techniques developed for structural studies of neutral or positively charged atoms or molecules in negative ion studies. Negative ions of the rare gases ( H e - , Ne -,A r - ) or of the lightest alkali earth ions ( B e - , M g - ) , which may be created by adding an extra electron to a closed s’ or p h shell, are not stable, but exist as metastable autodetaching ions with lifetimes ranging from ns to ms. The binding energies as well as the lifetimes of such metastable negative ions are well suited for testing the role of electron-electron interaction in the negative ion wavefunction. The two types of information are complementary since they test different parts of the wave function. A heavy-ion storage ring is a device in which a beam of charged particles can be confined in an ultra-high vacuum system by a number of magnetic focusing and steering elements. In most cases, the lifetime of stored particles is limited by collisions with the background gas. Some beams may be stored for several hours and others only for a few seconds, sufficient time for the particles to circulate lo5 times or more in the ring. This makes it possible for the ions to react repeatedly with “dilute” targets such as laser light or free electrons. It also allows the stored species to be studied over time scales that are longer than the intrinsic times for most atomic and molecular systems, and much longer than what can be measured in standard single-pass experiments. The large majority of atomic and molecular physics experiments at these rings (Larsson, 1995; Mokler and Stohlker, 1996) have mainly concentrated on positive ions. However, storage rings may also be suitable for the study of negative ions. The present chapter describes such studies performed within atomic, molecular, and cluster physics at the Aarhus STorage RIng Denmark (ASTRID) (Mqiller, 1991), so far the only ring adapted to negative ion studies. ASTRID has a circumference of 40 m, two bending magnets in each of the four corners, 16 quadrupole magnets, and 16 correction dipoles (see Fig. I). The negative ions are created outside the ring, either directly in an ion source or by charge exchange in an alkali-metal vapor. Ions are injected with an energy of typically 10-100 keV. In the simplest mode of operation, ions are injected into ASTRID and stored at the injection energy. The count rate of neutral particles detected behind one of the magnets is then recorded as a function of the time after injection. This provides an easy way to measure lifetimes of negative, metastable atomic, molecular, and cluster ions that decay by electron emission or frag-
-
STUDIES OF NEGATIVE IONS LN STORAGE RINGS
Mirror
Current transformer
I57
Dipole
mag nets
Electron cooler
Detector chamber
,
FIG. I . The storage ring ASTRID. Shown arc wnie of thc cbsential features for the expenmenth with negative ions. The I6 quadrupolcs i d correction dipolcs are not h w n .
mentation. The maximum magnetic rigidity of ASTRID is 1.93 Tm, which yields a maximum storage energy given by
E,,,;,,= 931.5 . (d0.3744q'
+ M:
-
M , ) MeV.
(1)
where 4 is the ion charge in units of e, the elementary charge, and M , is the mass of the stored ion in amu. Light, singly charged ions can be accelerated by an R F cavity and stored at a lunetic energy of several MeV. This permits a significant Doppler shift of applied laser light. It also allows the ions to be merged with a beam of electrons of about equal velocity and thus the study of low-energy ion-electron collisions. Singly charged cluster ions such as C, can also be stored, but only at keV energies owing to their large mass. For the electron-impact detachment studies, the ions were accelerated to about 5 MeV before being merged with the electron beam of the electron cooler (see Fig. 1). The acceleration was normally accomplished within a few seconds. The electrons were delivered by an electron cooler, which can produce -mA electron beams with energies between -50 and 2000 eV (Andersen ef ul., 1990). At a given kinetic energy E,, of the electrons in the laboratory frame, the relative electron energy E is determined from
L. ff. Andersen, I: Andersen, and I? Hvelplund
158
where m is the electron mass, v, and v,,are the laboratory velocities of the ions and the electrons, respectively, and E, is the ion-beam energy. The electron beam is essentially uniform, with a density of the order of lo7cm-3. In the studies of electron-impact phenomena, in particular threshold behavior and resonances, it is crucial that the electron velocity distribution in the experiment is known. The properties of electron coolers, where the electron beam is guided by a longitudinal magnetic field of typically several hundred Gauss, have been discussed in several papers (see, e.g., Poth et al., 1989; Andersen et al., 1990; Calabrese and Tecchio, 1991). For more details about the experimental conditions in the electron-impact detachment measurements, the reader is referred to Vejby-Christensen et al. (1996).
11. Lifetime Studies of Negative Ions Negative ions, which can retain their extra electron for an indefinite period of time, are considered stable, and additional energy is required to remove the electron. Stable ions with binding energies below -150 meV will, however, be destroyed in less than 100 ms at room temperature due to the interaction with black-body radiation. Ions, which are not stable but hold on to their extra electron s, a period of time making them suitable for beam experifor more than ments, are considered metastable. Negative ions with shorter lifetimes may manifest themselves as resonances that usually decay via electron emission.
-
A. ATOMIC NEGATIVE IONS This section describes studies of the simplest metastable negative ion, H e - , as well as the alkali earth ions Be-, Ca-, and Ba-. Its purpose is to illustrate the potential of the storage-ring technique, but also to point out some limitations. I . The He-(Is2s2p ‘P)Ion With only three electrons, He- is the simplest metastable negative ion and as such has received a great deal of attention (Blau et al., 1970; Mader and Novick, 1972; Bunge and Bunge, 1979; Peterson et al., 1985). It is formed in the ls2s2p ‘P state, bound by only 77 meV relative to the first excited state, ls2s 3S,of helium. The fine-structure splitting is shown in Fig. 2. The 4P ion cannot decay by Coulomb autodetachment to the He ground state, nor by radiative decay. Therefore it is metastable, and the decay of the three fine-structure components is due to higher-
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
-
(159850 cm-’)
1s’
---.
(1 59229 ern.') He’
159
.._ -*
-0.0572cm-’
1s
BBZzm He (0 cm-l)
FIG.2. Schematic energy-level diagram of Hr and Hcz- showing the tine structure o l H e ( ls2s2p ’f,).
order effects such as autodetachment induced by spin-orbit or spin-spin coupling. Due to the spin-orbit coupling between the JP,,2,,,2and 2P3i7,,,2levels of the ls2s2p configuration, the lifetimes of the ‘P,,? and 4P,,2levels will be shorter than for 4P,,2since coupling to the ‘ P levels can provide fast autodetachment. The decay mode for JP,,2is entirely via autodetachment induced by spin-spin interaction. Because the interaction Hamiltonian is explicitly known, the calculation of the lifetime of the JP,,2 level is simpler than that for the two other fine-stmcture levels. The J = 5/2 lifetime is therefore well suited to test the theoretical description of the initial wave function (ls2s2p “P,,z in H e - ) and of the final wave function (1s’ ‘ S in He + an electron). The lifetime of the J = 5/2 level has been a theoretical challenge (Estberg and La Bahn, 1968; Laughlin and Steward, 1968; Estberg and La Bahn, 1970; Davis and Chung, 1987; Brage and Froese Fischer, 1991). The calculated values ranged from 266 ps (assuming single-state initial configuration wavefunction) to 550 ps obtained for configuration-ionization calculation based on the golden-rule formula. The best calculations for the J = 5/2 lifetime were considered to be those of Estberg and La Bahn (1970) and Davis and Chung (1987), yielding 455 ps and 497 ps, respectively. The latter group had been able to reproduce the binding energy of the “P state with good accuracy. This indicates that the outer part of the wavefunction, which is essential for the binding-energy calculation, was well understood. But to account for the lifetime, the inner part of the wave function should also be properly modelled. The most accurate measurement for J = 5/2 was performed by Blau et (11. (19701, utilizing a single-pass setup with a 10 m long beam line. However, the uncertainty of the result, 345 2 90 ps, was so large that all types of calculations could claim agreement with experiment. This exper-
160
L. H. Andersen, T Andersen, and I? Hvelplund
imental value had been unchallenged for two decades when the first lifetime experiments at ASTRID were initiated in 1991. For negative ions with binding energies above -0.2 eV, the storage time in ASTRID is limited by ion-rest-gas collisions to the range of seconds (Andersen et al., 1991). Lifetimes of the metastable ions in the range -10 ps-100 ms or longer can thus be measured, the lower limit being set by the revolution time in the ring. For light ions such as He-, it is approximately 10-15 ,us and for Ba- -75 ps at the injection energies available. Figure 3 shows a typical neutral atom count rate versus time for ‘He-, obtained at ASTRID (Andersen et al., 1993b). The figure shows that after an initial period of 1 ms, during which the particle detector is saturated due to a high count rate, the decay can be described by a single exponential. After a few revolutions in the ring, only particles that are well confined remain circulating. Slit scattering, which can lead to significant systematic errors on direct beam lines, is practically nonexistent after a couple of revolutions. Because the decay curve can be followed out to 10 lifetimes, it is possible to obtain the lifetime with low statistical uncertainty. By varying the gas pressure, beam energy, or beam intensity, and by studying the reproducibility of the data from one run to another, it is possible to check the integrity of the data. However, two effects need attention: the influence of the magnetic fields in the ring, and the effects of black-body radiation.
-
0
2
4
6
Storage Time (ms)
FIG. 3. Typical neutral atom signal from the tandem channel-plate detector as a function of time following ion henin injection o l H r - (Andersen or d., 199%).
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
161
The field in the dipole magnets is strong enough to mix the magnetic substates from the different, but close-lying, fine-structure components with the same A4, quantum number (see Fig. 2). In first-order perturbation theory, the nixing amplitudes will be proportional to the magnetic field strength (Zeeman coupling); thus the decay rate is expected to depend quadratically on the field strength or linearly on the beam energy at sufficiently low energies. Because the mixing induced by the magnetic field depends on the MJ value, the various sublevels are depleted at different rates and the corresponding components could, in principle, be observed in the decay curve. For J = 512, the lMJl = 512 states should exhibit a much longer lifetime than states with lMJl = 312 or 112. However, this separation is eliminated by the action of the in-plane component of the field in the quadnipole magnets on the straight sections of ASTRID. A simple estimate shows that the transverse field component will be large enough to effectively randomize the population over the magnetic sublevels with respect to the direction of the stronger fields of the bending magnets. A general account of the field-mixing effect requires diagonalization of the Zeeman Hamiltonian and a dipole treatment of the nonadiabatic effects (Andersen rf al., 199%). The effective decay rate for the J = 512 state in He- may be written
r5,?,
and r,/are black-body-radiation-induced and natural decay rates, where r,,, respectively, with ZJ(E) representing the fractional transient population of the other fine-structure components ( J = 312, 1/2) due to the Zeeman mixing in the dipole field. The relation Z,,,(E) < Z,,,(E) holds for He- due to the much smaller fine-structure splitting between J = 312 and 512 than between J = 112 and 512 (see Fig. 2 ) . The experimental data were fitted to the equation above, with neglect of the contribution from J = 112, and rH,( determined independently. The resulting lifetimes were T(J = 5/2) = 350 2 15 ps and TCJ = 312) = 12 f 2 ps. The influence of black-body radiation at room temperature on weakly bound atomic systems is well established, leading to ionization of' high-lying Rydberg states in neutral atoms or detachment of weakly bound electrons (electron affinities below -0.15 eV) from negative ions. In order to test the effects of black-body radiation on the lifetimes of He (ls2s2p 'f,,), the temperature dependence of the apparent lifetimes was tested. By heating one half of the storage ring to 120°C. the observed lifetimes were consistently reduced at all beam energies. The reduction was approximately 20 percent. Since the recently calculated cross-sections for photodetachment of He- (Saha and Compton. 1990) are in excellent agreement with the most accurate experimental photodetachment cross-sections (Pegg et (11.. 1990), it is possible to obtain information about the contribution to the lifetime from black-body radiation. By folding the Planck radiation distribution with the theoretically estimated photodetachment cross-sections, a numerical analysis can yield the contribution. The comparison between the observed lifetime shortening and the theoretical prediction was good (Andersen rt al., 1993b).
162
L. H. Andersen, T Andersen, and I? Hvelplund
Within the experimental accuracy, the storage-ring lifetimes for the He- ( 1 ~ 2 . ~ 2 ~ ‘Pg12 and 4P,,,) levels are in good agreement with the most accurate of the previous experimental values (Blau ef al., 1970). However, using a storage ring, the accuracy for J = 5/2 was improved by a factor of six, now allowing a detailed test of the theoretical predictions. The 350 ? 15 ps lifetime value clearly eliminates a series of calculational approaches so far considered to be among the most reliable for prediction of negative ion properties (Estberg and La Bahn, 1970; Davis and Chung, 1987). The storage-ring lifetime stimulated new theoretical research (Miecznik et al., 1993). In an attempt to find the reasons for the discrepancies between the previous calculations and the new experimental value, three factors were considered: the autoionization mechanism, the correlation effects in the initial state, and correlations in the final state. The selection rules for the Breit-Pauli operators applied to HeC single out only the two-body spin-spin operator, and therefore the main decay mode for the ‘P5/?component is via two-body interaction. Correlations play an important role for the binding of the 2p electron in H e C . In the multiconfiguration Hartree-Fock approximation, correlation is accounted for by mixing in other configurations with the original ls2s2p 4Pwavefunction. Most important are the contributions from the ls2p3d4Pand 2 ~ 3 . 4P ~ 2configurations. ~ The strongest perturbation is in fact introduced via the interaction between the ls2s2p and 2s3s2p configurations, which brings about a radial correlation to the I s orbital and significantly reduces the lifetime. The correlation effects in the final state are negligible because the dominant contribution comes from the main 1s’ ‘S component. The final theoretical value of 345 ps is in good agreement with the experimental result from the storage ring. The advantages of using a storage ring rather than a single-pass beam to study negative ion lifetimes in the ps-100-ms time regime are: 1. Data can be extracted over a much greater range-out to several lifetimes-with a good signal-to-noise ratio. 2. Slit scattering is essentially eliminated in the ring. 3. The ultrahigh-vacuum conditions render collisional quenching entirely negligible. 4. The intrinsic time structure of the ring allows rapid data acquisition and makes the results quite insensitive to low-frequency instabilities.
The magnetically induced mixing effects caused by the quadrupoles may be considered a disadvantage for the ring technique with a view to obtaining the lifetime of the 4P5,2level, but in return, these effects can be utilized to obtain information on the lifetime of the other fine-structure components, in cusu of 4P,,2.
2. The Alkaline-Earth Ions Be-, Cu-, and Bu Up to 1987, it was generally accepted that the alkaline earth elements were unable to form negative ions in the ground state, but they existed as negative metastable
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
I63
ions in the nsnp' "P state (n = 2 for Be, 4 for Cu, and 6 for Ba). The observation of Pegg et (11. (1987) that Ccr- did indeed exist in the 4.~~41, ' P stable ground state as a rather weakly bound ion, finally established by Petrunin er ul. (1996) to have a binding energy of only 24.55 2 0.10 meV, initiated a great interest in these ions. The first of these to be studied at ASTRID was Be-(2s2p' ' P ) (Balling et a/., 1992). Its " P state is bound by 290 meV with respect to the Be(2s2p ' P ) state (Kristensen et ul., 1995),and thus it is unaffected by black-body radiation at room temperature. In addition, the fine-structure splittings are nearly equal and of the order of 0.6 cm-I. Thus the Zeeman mixing, playing such an important role for H e - , is less noticeable. The lifetime of the long-lived Be-(4P,,z) ion, decaying by autodetachment induced by spin-orbit coupling to the Be(lS) state, was determined to be 45 2 5 ps, which was nearly a factor of 30 shorter than predicted at that time (Aspromallis et ul., 1985, 1986; Brage and Froese Fischer, 1991).The lifetimes of the two shorter-lived components, J = 5/2 and J = 1/2, could not be determined at ASTRID-only a relationship between their lifetimes could be obtained. However, benefitting from the knowledge that the Be-(JP,,2)ion lives much longer than the other two fine-structure components and the unique existence of two metastable states in the Be- ion, 2s2p' ' P and 2p' "S, connected by an optical transition, Andersen et trl. (1996a) have developed a state-selective stepwise two-photon detachment technique that has made it possible to obtain the lifetimes for Bep("P,,2)= 0.73 t 0.08 ps and Be-('PP,,,) = 0.33 2 0.06 ps. The lifetimes for all the Be -('P) fine-structure components are in good agreement with recent theoretical calculations by Aspromallis et ul. (1996). In 1989 Hanstorp et ul. reported the existence of a long-lived Ca ( 4 . ~ 4 '~P' ) ion, in addition to the stable CK(4s'4p ' P ) ion observed by Pegg et ul. (1987) a few years earlier. The C K ( " P ) ion should have a lifetime of the order of 300 ps. Pegg et ill. used Li vapor to produce C i ( ' P ) from Cai , whereas Hanstorp et u1. used Cs. The different charge-exchange materials should account for the different final products. Using ASTRID, Haugen et (11. ( 1992) studied the origin of the claimed metastable Ca-('P) ion. By using different charge-exchange materials, it was possible to optimize the production of ClC ions present either in the ' P ground state or the ' P metastable state. Figure 4 illustrates the neutral-atom signal versus time for Cu- at ASTRID. The same decay pattern was observed, independent of charge-exchange materials, and the lifetime for survival in the ring could be determined to be close to 500 ps. It was notable that the entire beam was eliminated, indicating that it consisted of only one component, the stable ' P ground state or the ' P metastable state. By varying the temperatures of the ring, it was possible to show that the storage ring lifetime was due to black-body-radiation-induced detachment of the electron in the weakly bound ground state. Very recently, Kristensen et al. (1996) measured the lifetimes of the Cap('P) finestructure components from absorption experiments. The lifetimes are in the range 10 ns-10 ps, a factor of -10 shorter than predicted theoretically (Miecznik et id.,
L. H. Andersm, I: Atzdersen, nnil P Hvelplund
164
1o4 1o3 1o2 10'
1oo
0
2
4
6
Storage time (rns)
FIG. 4. Seiiiil~~~~ii-ilhiiiic plot oi the delachnicnt yield w r . w time following injection of a weak (pA) 100-keV Ctr beam into ASTRID. The straight line reprcaents the cxperiiiiental lit. The rapid Iluctuation oi the dctachruent yield is due to beterron oscillations in the alofiigc ring (Haugen ~t d., 1992).
1992) and four to five orders of magnitude shorter than originally assumed. The Cu- study also showed that weakly bound, negative ions may be identified rather easily at storage rings due to their decay via interaction with black-body radiation. For the BaC ion (Petrunin ef ul., 1995), this facet was utilized to gain information about the stable 6 ~ ~ 6 '1P) ground state. Figure 5 shows the presence of two Bu- fine-structure components, decaying with lifetimes of 1.66 ms and 10.8 ms. The storage ring data show that the population ratio between the two components is 2:1, consistent with a statistical population of the Ba-('P) ground state. By increasing the temperature of the storage ring, it was possible to change the pattern of black-body radiation. On the basis of decay times measured at different temperatures, it was possible to estimate the fine-structure splitting to be -50 meV by assuming that the lifetimes can be related to the total number of photons available above thresholds. This result is consistent with the more precise value of 55.02 -+ 0.09 meV obtained from resonant ionization spectroscopy (Petrunin et d . , 1995). However, in addition to the stable ' P ground state, Bu- also possesses a metastable 5t16s6p 'FF,,?state with a lifetime exceeding 5 ps, yet shorter than could be studied in the first attempts at ASTRID (Petrunin el a/., 1995). A new and smaller storage ring (ELISA) is now under construction, which will make it possible to measure the lifetime for metastable states such as Ba-('F,,J.
STUDIES OF NEGATIVE lONS IN STORAGE RINGS 1o5 h
.-u) C
165
" '
4-
3
1o4
4
(d
Y
z!
a
1o3
F 102
10'
100
0
20
40
60
Time after injection (ms) FIG.5. Schematic plot of thc dctnchment yicld I'CT.SII.\ time follow)inp injcctiori ol' ii nA 100-kcV Ha inlo ASTRID. The straight lines reprehciil a lu.o-exponentii~1fil ( Pelrunin c/ ( I / . , 19')SJ.
B. MOLECULAR NEGATrVE IONS The storage ring technique has also proven valuable for lifetime studies of molecular ions, positively as well as negatively charged. Andersen et a1. demonstrated this for the CO" molecule undergoing ulow, spontaneous dissociation ( 1993a). and for the metastable H e , ion (1994).
I . The H e , ('lZJ Ion This molecule was discovered experimentally in 1984 (Bae et al., 1984). Frgure 6 shows the relevant potential energy curves for He2 and He:. Only the 11 = 0 vibrational state is located below He,(u'C ,I,), and it is therefore assigned responsibility for the long-lived metastable H e ; state, which decays to the ground state by emission of an electron. Because the interaction caused by rotation of the niolecule is considerably more important than the spin-orbit interaction for molecular systems such as Hey, a rather detailed analysis is needed to assign the origin of the singleexponential decay curve observed by Andersen et ( I / . (1994), yielding a lifetime of 135 'I 15 ps. For increasing rotational quantum number, the interaction between the rotation of the nuclei and the orbital angular momentum will cause a splitting of the '1lK
166
L. H. Andersen, 7:Andersen, and f! Hvelplund
-
0.8
1.0
1.2
1.4
1.6
1.8
2.0
FIG. 6. Potential energy curves for He,(ci ’Xr:) and He;(‘nn,) together with the autodetachment transition for the He? ion (Andersen et d., 1994).
‘n;.
component into two A-doubling components, ‘II; and The ‘fl; component may be rather short-lived, as it is coupled strongly via rotation to the ‘2; state responsible for the origin of the A doubling and located only -0.2-0.3 eV above the 411gstate (and thereby also above the limit for autodetachment to the He,(3Xl) state). The ‘II; component is most likely the long-lived component of He; as it decays only via the much weaker spin-orbit coupling to the ‘2; state or via the ?A; continuum. So far, no reliable calculation exists for the lifetime of the He-(‘n,) state. C. CLUSTER IONS In recent years, the closed-cage structured fullerene molecules, among which the spherical C,,, buckminsterfullerene is the most prominent example, have attracted a great deal of interest. The stability of the fullerene molecules made them obvious objects for collisional studies because they can survive collisions in the 100eV regime and heating to more than 1000°C.These molecules were first discovered in the experimental work of Kroto, Smalley, and coworkers in 1985 (Kroto et ul., 1985). In 1990, a method for production of macroscopic amounts of C,,, and other fullerenes was found by Kratschmer, Huffman, and co-workers (Kratschmer et ul., 1990), and this discovery started an explosive interest in the physics and chemistry of these soccerball-shaped objects. Several excellent
STUDIES OF NEGATIVE IONS 1N STORAGE RINGS
167
review articles and monographs treat the properties of fullerenes, and the reader is referred to these for further details (Smalley, 1990; Huffman, 1991; Kumer et al., 1992; Dresselhaus et ul., 1996). The buckminsterfullerene molecule C,,,consists of 60 carbon atoms, all in equivalent positions at the corners of a truncated icosahedron. Commercially available fullerene powder contains predominantly C,,, and C,(,,but also smaller amounts of heavier fullerenes. When this material is heated to around 400"C, the vapor pressure is 1 mtorr, and the fullerene vapor can easily be introduced into an ion source. The ion source used at ASTRID is a plasma source (Almtn and Nielsen, 1957) capable of delivering nA beams of both positive and negative fullerene ions (Yu et al., 1994a,b). Just as for positively charged fullerene ions, the negative ions appear predominantly at even numbers in the mass spectra. Fullerene anions containing from 48 to 84 carbon atoms can be produced in sufficient quantities from the plasma source to allow storage in the ring. Smaller carbon clusters consisting of 2-9 carbon atoms, presumably in the form of chains (Watts and Bartlett, 1992), have been produced in a sputter ion source (Andersen et ul., 1 9 9 6 ~ ) . Total cross-sections for destruction of negative carbon clusters in collisions with H , have been measured in single-pass experiments (Shen et nl., 1996). We assume that electron detachment,
-
C,, + H z + C,, + H1 + e,
(4)
is the dominant process in these collisions. If collision with rest-gas atoms in the storage ring, mainly HZ, is the only loss mechanism, then the time evolution of the number of stored particles can be expressed as N ( t ) = N,,r
where the collision lifetime
7,can
'"~,
(5)
be expressed as
Here, (T is the cross-section for collisional destruction, P the average ring pressure, and E and M the kinetic energy and mass of the stored ions. Figure 7 shows the decay rate for 50-keV C& A fit to the data yields a lifetime of 7 s, which corresponds to a destruction cross-section of about 2 X lo-'' cm'.
I . Fullerene Lijetimes Only if the injected cluster ions are stable against all forms of particle decay will the evolution of the number of stored particles be described by the simple exponential time dependence (Eq. 5 ) . If clusters possess internal energy, other decay channels such as unimolecular fragmentation and themionic emission may open up. Figure 8 shows the decay of stored C,, ions in the millisecond range after sub-
168
15.H . Andersen, i? Andersen, and P Hvelplund
15
10
Storage Time
20
(s)
Fic;. 7 . The yield of neutral particles at the detector as a function of time after injection of SOkeV &,. The average pressure in the ring was 2 X I O - " mbar. and the collision-induced lifetime is found to be -7s.
0
20
40
60
80
loo
time [msl FIG. 8. Rate decay by therinionic emission of a stored Chi,beam. A contribution from collisions with the rest gas has heen subtracted. The curve through the data points corresponds to fits with 7,. = 4.3 ms and n = 7.6, and the other curve to a f dependence.
~'
STUDlES OF NEGATIVE IONS IN STORAGE RINGS
169
traction of the nearly constant contribution from rest-gas collisions, illustrated in Fig. 7. The rapid decay on the millisecond time scale is most likely caused by electron autodetachment, since the electron affinity is much smaller than the activation energy for unimolecular fragmentation. The functional form of the decay rate at short times carries information about the internal state of the clusters and the development of that state over time. In the atomic case, where all or a fraction of the ions could be in a well-defined metastable state (see the previous section about H e - ) , the decay is described by a single exponential. If several metastable states are populated, the decay function will be more complicated, containing exponentials with different lifetimes. However, in the limit where many states are populated, with a broad distribution of lifetimes 7, the decay function again becomes simple. If we assume that the density of states is so large that the excitation energy E may be treated as a continuous parameter and denote the initial distribution in excitation energy by the total decay rate may be written as an integral,
If rtE) is a steeply decreasing function and g ( E ) is slowly varying, the integrand will peak sharply at an energy E = E,,,. The two first factors in the integrand may be written as exp( -log7 - t / T ) , and by setting the derivative of this exponent with respect to E equal zero at E,,,, we obtain d E J = t. Thus the maximum value of the integrand is roughly proportional to t I . The width of the peak can be estimated by a second order expansion of the exponent, corresponding to approximation of the peak by a Gaussian, exp(-lopT - t/T) t 'exp( - ( E - E , , f / 2 & ) , with cr = (-d/dE log.r)-'. This logarithmic derivative will normally change much more slowly with E than does T , and hence also the integral in Eq. 7 will be approximately proportional to t- . This description may be expected to apply to negative fullerene ions stored in ASTRID. In the ion source, the clusters are bombarded with electrons, and the extracted negative ions have a broad distribution in excitation energy. Furthermore, even at moderate excitation energies, the vibrational level density is enormous. As a result, we might expect to observe a decay due to thermionic emission nearly proportional to t - ' . However, as seen in Fig. 8, the experimental result is quite different, and the rate decreases nearly exponentially in the millisecond range. We have interpreted this as evidence for radiative cooling of the hot clusters. Just as for the negative atomic ions with very small binding energy, the interaction with the radiation field can have a strong influence on the decay rate, but here it is a quenching of the decay due to rniissioti of radiation. We now turn to the details of the statistical description of the decay and cooling of the stored ions. As discussed in Andersen c>t til. (1996b), there are strong theoretical arguments and experimental evidence for the description of electron detachment from excited C;, ions as a thermally activated process analogous to
-
'
I70
L. H . Andersen, T. Anderseiz, and P. Hvelplund
thennionic emission from a hot filament, and the decay of molecules with excitation energy E may be expressed as d N(E) = v exp dt
--
The preexponential favor v is of order 10” s-’, corresponding roughly to a geometrical cross-section for the inverse process, attachment of thermal electrons. The effective barrier Sor emission, E,, = 2.92 eV, corresponds to the measured electron affinity, 2.67 eV, with an additional contribution from an angular momentum barrier associated with the absence of s-wave attachment. The effective emission temperature T, is calculated in two steps. First the microcanonical temperature T for a molecule with excitation E is estimated from the relation between the average vibrational energy and the canonical temperature in a thermal equilibrium. For high temperatures (T > 1000 K), one obtains to a good approximation a linear relation, E = 7.4 i- C(T - 1000) eV, where the heat capacity is C = 0.0138 eV/K. Next, a correction is applied for the finite size of the vibrational heat bath. It follows from a simple statistical argument that the effective temperature for electron emission rate corresponds to the average energy of the heat bath before and after the emission, and one obtains T , = T - EJ2C. Without cooling, the distribution N(E) is depleted from the high-energy side by electron emission, with an upper cut-off corresponding approximately to the temperature,
4
with G
T: = L, kG
=
log(vt).
(9)
Most of the decays at time t stem from molecules in a narrow interval around tlus temperature, which decreases slowly due to the depletion. For a flat initial distribution N(E), the total emission rate will be nearly proportional to t - ’ , as discussed above. Cooling by radiation will quench the electron emission when the reduction of the emission rate becomes significant on the time scale given by the emission rate i.e. according to Eqs. 8 and 9 this means for times larger than a characteristic time 7, given by 1
d
7,.
dt
- = -G-1ogT
for
T=Tf,
where the time differentiation refers to cooling. Owing to the large heat capacity, the finite magnitude of photon energies can be neglected to a first approximation, r,, the rate and the cooling can be treated as a continuous process. For t decreases roughly proportionally to exp(- t/7J the depletion by electron emission becomes insignificant and instead, the distribution N ( E ) moves down in energy (or temperature).
-
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
171
The curve through the data points in Fig. 8 is a fit calculated from the statistical model of the competition between electron emission and cooling. The radiation intensity, as obtained from Eq. 10 and the heat capacity C, is about 190 eVIs. In Andersen er al. ( 1996bh this very strong radiation was modeled by a dielectric description, in which the radiation stems from strongly damped plasma oscillations. But such a model is apparently inconsistent with the existence of a large HOMO-LUMO (Highest Occupied to Lowest Unoccupied Molecular Orbital) gap in C,,,. It was therefore suggested that the influence of the gap is reduced at high temperatures owing to the strong coupling to vibrations. Measurements have been made for fullerene anions C,, with n = 48-76, and the characteristic times for cooling derived from analyses analogous to that discussed above are shown in Fig. 9. The observed more rapid cooling of fullerenes with less pronounced shell structure supports the model, although it is surprising that the effect of shell structure is much stronger for I I = SO than for n = 70 (Woo ef al., 1993). However, as discussed in Andersen rt "1. (1996b), the contribution to the radiation intensity from the additional electron in the anion can be important, and it will depend on the energy and strength of the transitions available for this electron. We should note also that the electron affinities determining the effective barriers E,, for emission are different, being lower for C, and C,,, by -1.5 percent than the values of -3.2 eV for the other fullerenes. It is clear that the differences observed in Fig. 9 are not caused by this variation: The affinity is nearly the same for C,, and C,,, and similarly for the pair C,, and Cia, and yet the decay times are quite different.
t
40
8
50
60
70
80
L"
FIG. 9. Thc characteristic lime
T,
for cooling of C,, derived from iits to decay curves.
172
L. H.Andersen, T Andersen, and I? Hvelplund
2. Lijetimes of Small Carbon Clusters Small carbon clusters were already predicted to be linear molecules in the early calculations by Pitzer and Clementi (1959). This has later been confirmed by more refined calculations (Smalley, 1990; Watts and Bartlett, 1992) and by experiments, although some controversy still remains concerning the structure of the even-numbered molecules (Weltner and van Zee, 1989). The qualitative picture of the electronic structure of the linear molecule C,, is very simple. Of the 4n valence electrons, 2n-2 electrons are accommodated in u bonding orbitals, 4 electrons occupy u orbitals at the two ends of the chain, and the remaining 2n-2 electrons occupy o orbitals that are all fourfold degenerate (including spin). For odd n, the neutral molecule therefore has a closed-shell electronic structure, whereas the HOMO orbital is only half occupied for n even. This results in a greater stability for odd-n but larger electron affinity for the even-n molecules, as has been confirmed by photoelectron spectroscopy (Arnold et al., 1991; Yang et al., 1988). Metastable anions are probably formed in excited electronic states and stabilized by a high spin because the energy stored in vibrations is insufficient for electron detachment from the ground state at moderate temperatures. The simplest explanation for metastability of the odd-n clusters is excitation to a quartet state. According to the simple scheme discussed above, the ground state has spin 1/2, with one electron in the LUMO of the neutral molecule. If an electron is excited from the filled HOMO, the two electrons can combine to spin 1 and further combine with the spin of the hole in the HOMO to give spin 3/2. This, in fact, represents a doubly excited state, which can decay by an Auger process, possibly assisted by vibrational deexcitation. The fast component for the odd-n molecules in Fig. 10 are clearly not exponential but could result from a superposition of decays from several vibrational states (Andersen et al., 1996~).For the even-n molecules with n > 2, there is no fast component at millisecond times, and one can find a qualitative explanation in the fact that because the lowest quartet state for n even would be only singly excited, there is no electronic excitation available for electron emission. In addition, the electron affinities are larger for the even-n systems. The molecule C ; is a special case with an exponential fast component, which also has been suggested to stem from vibrationally assisted electron emission from a quartet state close to the continuum (Brink et al., 1996).
111. Electron-Impact Detachment from Negative Ions Electrons are excellent particles for probing atomic and molecular systems. They are easy to produce and may be accelerated to essentially any energy that is appropriate on the scale of atomic and molecular physics. Collisions between free electrons and atoms, molecules, and their ions are important from a fundamental
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
173
1
100
l I I l L C i
t
t
FIG. 10. Decay rate of' stored aniona of small ciirbon clusters. The crosses indicate the fast c o n ponent. ohrained alter suhtraction of a (nearly) constanl contibution lrom collisions with the rest gas, and the yield has been normalized to this contrihution.
scientific point of view and in areas such as plasma physics, laser technology, atmospheric research, and the chemistry of the interstellar medium. Experimentally, the field has advanced considerably with the development of intense electron beams with high-energy resolution and the ability to produce specific targets of atoms and ions. From a theoretical point of view, the basic physical processes are to a large degree understood and can be calculated with the ever-improving computational facilities (Gianturco, 1989). However, the recent experimental progress has presented new challenges to theory. Electron-impact ionization of atoms and ions has been studied since the early days of quantum mechanics, and much of our understanding of the structure of atoms emerged from scattering experiments with electrons (Massey and Burhop,
174
L. H. Andersen, 'I: Andersen, and I? Hvelplund
1985; Miiller, 1991). There have been significantly fewer experimental studies on electron-negative ion collisions, and only a few elements have been studied. Tisone and Branscomb (1966, 1968) and Dance et al. (1967) measured the detachment cross-section of H- from around 10 eV to 500 eV, and Tisone and Branscomb also made an investigation of 0-. Later, Peart et al. (1970) studied H- and other systems such as C-, 0-, and F- (Peart et al., 1979a, 1979b). Recently, detachment from D- and 0- was studied by Andersen et al. (1995) and by Vejby-Christensen et al. (1996). In these measurements, which will be discussed in some detail in the present paper, the cross-section was measured from 0 to about 30 eV. On the theoretical side, there have been a number of works dealing with electron-impact detachment from negative atomic ions. Several groups have performed Born-type calculations on H- (McDowell and Williamson, 1963; Smimov and Chibisov, 1966; Inokuti and Kim, 1968; Bely and Schwartz, 1969; John and Williams, 1973), but the results differ significantly, in particular in the region around the cross-section maximum where the requirement of orthogonality of the initial and final wavefunctions is crucial (Inokuti and Kim, 1968). The situation around threshold cannot be described by a usual born calculation because the incoming electron interacts with the target over a long time. Alternatively, one can apply an adiabatic picture where the loosely bound electron is squeezed ouf opposite to the approachmg electron at an opening in the binding potential over the barrier, or escapes via a non-classical tunneling process through the barrier (Smimov and Chibisov, 1966; Demkov and Drukarev, 1965). More recently, semiclassical calculations including tunneling were performed by Ostrovsky and Taulbjerg (1996), and dlstorted-wave calculations by Pindzola (1996). When electrons interact with molecules and molecular ions, a number of processes may take place, such as excitation of the rotational, vibrational, or electronic motion and dissociation, in combination with ionization, electron attachment, electron detachment, or recombination. Non-resonant vibrational and rotational excitation with neutral molecules is only important for interactions with low-energy electrons because typically the electron moves on a time scale much shorter than characteristic times for rotations and vibrations (Burke, 1989). In the case of electron-negative-ion collisions, the incoming electron is slowed down by the Coulomb repulsion as it approaches the negative molecular ion. Consequently, such time considerations may become less relevant in electron-negative ion collisions. In the last coupled of years, heavy-ion storage rings have been used in a number of studies involving collisions between free electrons and positive molecular ions. Primarily, dissociative recombination and dissociative excitation processes have been considered (Zajfman et al., 1996). To our knowledge, the electron impact detachment measurement with C; (Andersen et al., 1996d) is the only experiment involving electron-negative molecular ion collisions that has been
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
175
carried out so far. We know of no theoretical work on this subject, and it is interesting to note that the electron detachment process involving molecular ions is often ignored even in review papers despite the fact that the cross-section may be as large as ~ o - ' ~ - I o cm'. -'~ From an experimentalist's point of view, negative ions are difficult to work with owing to large collisional destruction cross-sections, which cause negatively charged beam particles to neutralize. This, in turn, gives rise to a background of atoms that must be distinguished from atoms resulting from the detachment process
X-
+e
-+X"
+ 2e-,
(11)
where X - represents an atomic or molecular negative ion. The recent experimental achievements with negative ions at the ASTRID laboratory (Andersen rf al., 1995; Vejby-Christensen et ul., 1996; Andcrsen rt al., 1996d) were made possible because ASTRID has an appropriate dense electron source (the electron cooler), can provide high-energy negative ions suitable for merged-beams experiments with electrons from the electron cooler, has good vacuum conditions that ensure relatively long storage times, and has efficient detectors for counting of the resulting neutral particles. It has not been possible to study electron-cluster ion interactions because the heavy cluster ions may be stored only at low energy and hence are disturbed too much by the space-charge field of the electron beam to remain stored in the ring. We first discuss data for electron impact detachment from negative atomic hydrogen and oxygen ions and compare the data with various theoretical approaches. Then we discuss electron impact dissociation of the molecular ion C , and the possible formation of doubly charged negative ions. Atomic units are used unless otherwise indicated.
NEGATIVE IONS A. ATOMIC Electron impact detachment from negative ions involves an incident electron, an outer target electron, a neutral core of the negative ion, and the interactions between these constituent particles. Such interactions involve both the long-range Coulomb interaction and some short-range interactions with the core. At low energies E = 1/2v', where v is the electron velocity in the ion rest frame, the classical minimum distance of approach in the Coulomb collision, Dil = I/E, is large compared to the extension of the wavefunction of the bound electron. Exchange can therefore be neglected, and the problem can be treated as detachment under the influence of an external force. If the angular momentum of the incoming electron is large, a semiclassical description may be applied where the electron is moving along a classical trajectory. Typical impact parameters are of order D,, or larger, and hence a sufficient condition for a semiclassical description is that the Bohr parameter K = 2/v is large, which is well fulfilled near threshold. On the
176
L. H. Andersen, Z Andersen, and J? Hvelplund
other hand, the binding of the outer electron by the neutral atom and the detachment of the electron under the influence of the external field is a quanta1 problem for which a classical treatment is basically not justified. Still, important features can be illuminated by simple classical considerations, such as the adiabaticity of the collision near the threshold for detachment.
1. Classical Calculations Negative ions, atomic as well as molecular, typically have binding energies E of a few eV. However, to have a sizeable detachment cross-section, the kinetic energy of the incident electron must be significantly larger than E . This is seen from the following classical arguments. The active target electron is bound in a short-range potential with the asymptotic l/r4 dependence. Here we assume a simple binding potential with the correct asymptotic behavior of the form (we consider one dimension only relevant close to threshold)
where a is a measure of the size of the ion (-4-5 for H , D- and -3 for 0 - ) (Smirnov, 1982). The perturbing force from the incident electron on the target electron must exceed the maximum binding force of the potential V,(x) to result in adiabatic detachment. The binding force is -aV,ldx and the maximum perturbing force is approximately l/Di = E', where D,, is the distance of closest approach and E is the initial kinetic energy of the incident electron. By balancing the binding and perturbing forces, a detachment threshold energy may be obtained as (Vejby-Christensen ef al., 1996).
E,, = G.
(13)
The same classical over-the-barrier threshold may be derived by requiring that the work on the target electron done by the incident electron equals the binding energy at threshold (Solov'ev, 1977). With realistic values ( E 1 eV, a 4), the classical threshold energy E,/,is typically 2-3 times the binding energy. For energies between E and E,,,, the detachment process is purely quantum-mechanical and caused by tunnelling through the potential barrier created by the binding potential and the perturbing potential of the incident electron. Tunnelling is, however, also important for E > E,, since some collisions at large impact parameters may still not result in escape over the barrier. In fact, it is to a large extent the impact parameter and distance of closest approach that determine the detachment probability rather than the impact energy itself. Assuming that the incoming electron experiences a purely repulsive Coulomb potential, the distance of closest approach as a function of the impact parameter p is given by
-
-
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
I77
D ( p ) = 112 D,, + v’(112D,,)? + p’.
14)
If it is assumed that detachment takes place (with probability p ) only when the inconling electron gets inside a reaction volume with radius R, the cross-section may be expressed as
where R is related to the threshold energy as R = l/&. The reaction volume is defined as the region in space in which the perturbing force exceeds the binding force and thus over-the-barrier transitions are allowed (Andersen et al., 1995). The threshold behavior given by Eq. 15 (with p = 1 ) is well known from nuclear and molecular collisions. As seen in Figs. 1 I and 12, this classical “reaction model”, labelled TI, reproduces the general behavior of the experimental data remarkably well when the
0
0
4 LT,
C
r .&
D
2 T5
0 0
___ 10
20
30
FIG. I I Electron-impact dctachnicnt cross-sxtion\ (I! I t and D . The cxpcrimentnl data i~re from Aiidcraeii r t u / . ( 199.5)(solid SquaresJ. Tisotie and B r a i r ~ ~ w n( h1968) (open tinangl ((1. (1967) (open circles), and Walton ef a/.(1970) (open SqtiaresJ. The theoretical c ~ i w e aare: Thc reactim iiiodel with 1) = 0 2 and R = 14.5 (Andcrzen C I u/., 199.5: Vejhy-Christensen V I u/.. 19961 (TI J, tunnelling model hy Smirnov and Chibisov ( 196hJ(TZ). tunnelling model including polnri~nlion by Sniirnov and Chibisov ( 1966)(T3).the claa\ical calculation by Oatrovaky and Taulhjerg without tunnelling ( 1996) (T4). the calculation by Ostrovaky and Tmlbjerg including tunnelling ( 1906) (T5). nnd llic distorted-wave theory by Pindzola i 19%) ( T ~ J .
178
L. H. Andersen, T Andersen, and t? Hvelplund - .
10
Detachment from 0.
N -
€
0
6
E
0 Y - 4 D
2
n
FIG. 12. Electron-impact detachment cross-sections of 0-. The experimental data are from Vejby-Christensen e t a / . (1996) (solid squares), Tisone and Bransconib (1968) (open circles), Peart et a / . (1979) (open squares). The theoretical curves are: The reaction model withp = 0.12 and R = 8 (Andersen ef al., 1995) (TI), tunnelling model by Smimov and Chibisov (1966) (T2), and the distorted-wave theory by Pindzola (1996) (T3).
reaction radius R is 14.5 (Efh= 1.9 eV) for D- and 8 (E,,, = 3.4 eV) for 0-. According to Eq. 13, this yields a = 5.8 for D- and 3.4 for 0-, values that are not unreasonable. The good agreement is to a large extent obtained by the choice of p , which in the model is equal for all impact parameters, probably fortuitous as the tunneling mechanism is ignored. To understand the need for the probability parameter p (10-20%), one must consider the dynamics of the process in greater detail. One may include the escape dynamics by considering the classical decay rate l?,lorsrc,,l(t), and obtain the detachment probability as m
and the cross-section cc Q
=2
~ pP,(p)dp. /
(17)
0
Solov'ev (1977) and later Ostrovsky and Taulbjerg (1996) have worked with an improved classical model. Ostrovsky and Taulbjerg considered the classical period of the radial motion of the target electron and took into account the angu-
STUDIES OF NEGATIVE IONS IN STORAGE KINGS
179
lar opening in the potential and the time available for escape through the opening. As expected, the resulting classical cross-section is significantly smaller than that of the reaction zone model (with p = 1) and smaller than the experimental data as well, as seen in Fig. 11.
2. Svnziclassicul Calculutions The problem of electron impact detachment at low impact energy is closely related to the problem of a negative ion in a constant electric field (Demkov and Drukarev, 1965), as realized by Smirnov and Chibisov (1966). Smirnov and Chibisov considered escape via tunnelling and assumed that the negative ion is associated with a three-dimensional well with negligible extension. This is a good approximation when the field from the projectile electron is weak. The tunnelling-decay probability per unit time W for a Coulomb field was calculated as
where A’ is 2.65 for D- (Smimov and Chibisov, 1966) and 1.35 for 0- (Smimov, 1982).The Coulomb field at the negative ion is F = (l/RIJ(p,t))’, where R,(p, r) is the radial distance between the incoming electron and the negative ion as a function of time and impact parameter. The decay probability as a function of impact parameter may be expressed as
and the cross-section is 1
u = 2?r/
PiPjPdP,
(20)
0
which is calculated numerically. The decay probability as a function of impact parameter P i p ) turns out to be significantly smaller than one €or many impact parameters smaller than the classical reaction radius, thus explaining why p is smaller than one. The model accounts reasonably well for the onset of the crosssection near threshold; however, the shape of the threshold region is not perfect, as seen in Figs. 11 and 12. The theory of Smirnov and Chibisov (1966) describes detachment from s states and is consequently not strictly applicable to 0-. Furthermore, the value of A’ used for 0 - is somewhat uncertain, as discussed by Esaulov ( 1986), and the absolute value found in the model may thus be in error.
180
L. H. Andersen, ?: Andersrn, and P. Hvelpllrnd
Polarization can be included in the model of Smirnov and Chibisov (1966) by replacing the binding energy with E, (r/(2r4),where a is the polarizability of the negative ion, and Y is the distance to the incident classical electron. The inclusion of polarization reduces the cross-section because the binding energy increases, and thus the tunneling probability decreases. As seen from Fig. 1 I , the cross-section T3 is reduced by a factor of two to three in the near-threshold region. The contribution from tunnelling was also considered by Ostrovsky and Taulbjerg (1996). The classical decay rate in Eq. 16 is replaced by the total decay rate, which includes tunneling:
+
I'rotn,
=
r,I<,\,,t
(il
+
r,,i,,r,i
(21)
It is seen from T4 and T5 in Fig. 11 that detachment via classical over-the-barrier transitions as well as tunneling are important at around threshold as well as at high energy. The actual magnitude of the cross-section, as calculated by Ostrovsky and Taulbjerg, depends on the parameters used to describe the potential as well as those describing the escape dynamics. 3. Quuntum Culculaticms
Pindzola (19961 has recently calculated the detachment cross-section using a standard quantum-mechanical treatment based on lowest-order distorted-wave theory. The method has previously been used to calculate ionization cross-sections for atoms and positive ions. He found that the inclusion and choice of the polarization potential for the continuum distorted waves was important. As polarization potential Pindzola used
where rc is the cutoff radius, taken to be the mean radius of the outermost orbital. The cross-sections for H - (D-) corresponds to the 1s' IS + 1s 'S transition, with a = 4.5 and r, = 1.5, as shown in Fig. 11. There is a reasonably good agreement between the calculation and the experiment. In the case of negative oxygen, there are three final terms of the ground-state configuration of oxygen to consider ' D , and IS), each with different threshold energies. Pindzola (1996) also applied the distorted-wave theory for this target, used a = 4.9 and rc = 1.2, corresponding to the polarizability and outer mean radius of the neutral oxygen, and took all final terms into account in the calculation. As seen in Fig. 12, the agreement with the experimental data is excellent. This is perhaps somewhat fortuitous, as the importance of polarization may indicate that higher-order perturbation terms might also be important (Pindzola, 1996).
STUDIES OF NEGATLVE IONS IN STORAGE RINGS
181
4. Douhlv Churgrrl Nrgutive Atomic Ioiz,s A number of papers have discussed the possible existence of doubly-charged atomic negative ions (Simon, 1974, 1978; Spence et ul., 1982; Esaulov, 1986; 1994). Lieb, 1984; Yannoni et nl., 1991; Compton, 1997; Robicheaux et d., Experiments have indicated that ions such as H ' - , 0' , F' -, C1'- exist in free space with lifetimes of the order of 10-" s to lo-' s. The most extreme case is H ' - , which belongs to the three-electron isoelectronic series H'-, He-, Li, Bet, etc. It contains a very large amount of Coulomb repulsion with three negative charge units and only one positive. Walton ef ul. (1970a.b) and Peart and Dolder (1973) reported on two resonances in the collisions e - + H - -+e e - H" at 14.5 eV and 17.2 eV, respectively, each with a width of about 1 eV (equivalent lifetime of 10- I s s). These resonances were attributed to short-lived states of H '-. The existence of such states at an energy slightly uhoi~ethe threshold for complete breakup (0.75 eV + 13.6 eV) of the four-particle system was indeed exceptional. The states were attributed to the (2.~1'2~ ('P) and (2p)' ('P, ' D ) configurations by Taylor and Thomas (1972, 1974) for the low- and high-energy resonance, respectively. Despite this agreement between early theory and experiment, Robicheaux et nl. (1994) questioned the existence of such resonance states. Robicheaux et (11. presented two independent types of ub inirin calculations for the three electrons in the field of the proton, and neither showed any evidence for the earlier proposed resonances. It was also argued (Robicheaux et nl., 1994) that the observed cross-section variation violated unitarity of the scattering matrix or flux conservation. Recently, Yang et nl. (1995) used hyperspherical coordinates to consider triply excited states of the H' system with 'Psymmetry. They found no trace of resonances for this particular symmetry. The data by Peart et ul. (1970) were not confirmed in the experiment at ASTRID (Andersen at al., 1995).As seen in Fig. 11, there is no structure related to the existence of short-lived H 2 - resonance states. Thus, scattering on ground-state H - apparently results in no resonances. A new calculation of the (2p3)4Sstate of H'- was performed by Somnierfeld et af. (1996), who found that this particular state has a lifetime of 3.8 X 1 0 - l " ~ (a width of 1.6 eV) and a total energy of -0.063 a.u., which is below the threshold for total breakup of the system. An experimental verification of the ' 9'resonance state would involve scattering on H in the excited ( ~ P ' ) state, ~ Pwhich is a very difficult task. Because 0'- has an Ne-like closed shell structure, it might be expected that this system in particular should have a lifetime sufficiently long to provide detectable structures in the electron-impact detachment cross-section. Herrick and Stillinger (1975) performed variational calculations and predicted an 0'- resonance at 5.38 eV with a width of 1.3 eV. Gadzuk and Clark (1989) calculated a resonance energy of 8.8 eV, and Huzinaga and Hart-Davis (1973) obtained 7.68 eV. The latter is close to the value of 7.2 eV predicted from an extrapolation of
+
-
~
+
182
L. H. Aizdersert, I: Andersen, and l? Hvelplund
the ls’2s22ph (IS,,)-state energy of the isoelectronic sequence for Sijt, A1” , Mg”, Na’, Ne, and F (Robicheaux et al., 1994). In the experimental work by Peart et al. (1979c), two resonances were found at approximately 19.5 eV and 26.5 eV. It seems unlikely that any of these resonances are due to the ground-state configuration of 0 2 - As . seen from Fig. 12, there are no structures in the new ASTRID data (Vejby-Christensen et al., 1996) that may be related to the existence of doubly charged, negative oxygen ions.
B. MOLECLILAR NEGATIVE IONS A measurement of the cross-section for electron impact detachment from C, was recently performed at ASTRID (Andersen et al., 1996d). Three reactions, which all lead to neutral particles, were considered in the experiment, namely electron detachment e-
+ C, + C? + 2e-,
(23)
electron detachment and dissociation
e-
+ Cl-+ C + C + 2e-,
(24)
and electron impact dissociation e-
+ C ; + C + C- + e-.
(25)
Internal degrees of excitation were not measured. To distinguish between the different reactions, a solid-state detector was used, by which events where a single C atom was produced could be distinguished from those where two C atoms were produced (either as C + C or C?).The ratio between the C C and C, channel was determined by insertion of a grid with known transmission probability in front of the solid-state detector. The cross-sections for the reactions of Eqs. 23-25 are shown in Fig. 13. The pure detachment cross-section is by far the dominant one. The detachment crosssection of C, has a magnitude comparable to that of 0-.If no resonances were present, a smooth energy dependence is expected (see Eq. 15). In the case of detachment from D- and 0-, the energy dependence was described by (T = v,,( 1 - $,/E), where a(,is a constant and E,, is the apparent threshold energy. The dashed curve in Fig. 13 is a fit with a function of this type to the cross-section cm’ and E,,, 7 eV. As with the atomic above 10 eV, which yields a,, 7 negative ions, E , , is about 2-3 times the binding energy, which for C , is 3.27 eV (Ervin and Lineberger, 1991). Unlike the situation with atomic negative ions, a structure is observed in the C j data. This structure, discussed below, may be a sign of the formation of a C:resonance state. This would then be the smallest dianion ever observed.
+
-
1
-
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
183
FIG. 13. Cross-sectians ti,r electron-impact cletnchment, electron-impact dissociation. ;tnd electron-impact detachment plus dissociation as a tunction of cnergy for c': (Andersen r i [ I / , . 199hd). Thc dashed line is the tit described in Ihc text.
I . Resonances in the Detnchmerzr Cmss-section When electrons scatter on neutral atoinic and molecular systems, they do, at short range, become a part of a transient negative ion whose short-lived quantum states may produce resonance structures in the scattering cross-section (Lengyel et ctl., 1992). With negative ions and electron energies of the order of the threshold energy E,,,, the incident electron never gets close to the negative target system because of the Coulomb repulsion. At E = 3 eV, for example, the distance of closest approach is 9 a.u. Thus, in the threshold region, the cross-section is expected to be a smooth function of energy without resonances. This, however, may not be the case when the energy is increased and the incident electron gets close enough to interact with the core. With C;, a structure in the form of' an enhanced cross-section is observed in the pure detachment channel between about 8 and 11 eV (see Fig. 13).It is important to note that the structure is not likely to be related to the opening of the other channels (8.3 eV for dissociation and 9.6 eV for dissociation and detachment) since these channels are associated with small cross-sections at this energy. It is possible that the structure is due to the formation of a doubly-charged negative molecular ion:
L. H. Andersen, T. Andersen, and P: Hvelplund
184
This interpretation is supported by the potential energy curve of C:; which was estimated by adding the Coulomb repulsion to the well-known ground-state potential of the isoelectronic N2 molecule (Edwards et ul., 1993). The idea is to superimpose the chemical bond onto the repulsive Coulomb potential. The method is not expected to provide an exact potential, yet some qualitative features may be obtained. The obtained Cf-potential well has an equilibrium distance R,, = 2.3 a.u. (1.2 A); it traps 21 quasibound vibrational levels and has w, = 1875 cm-' (0.23 eV) (Bjerre, 1996). Figure 14 shows the relevant ground-state potential curves of Ci; C , and Cz. It is seen that the energy needed to produce the lowest vibrational states of C:-from the vibrational ground-state of C, is about 10 eV, which corresponds to the observed resonance position. The lifetimes of the lowest vibrational levels of the electronic ground-state potential of C i are on the scale of seconds or longer (Rosmus and Werner, 1984). Thus, the beam may contain some C, molecules in the lowest, vibrationally excited levels ( u 4 2 ) , and this may consequently make the structure start somewhat below the C y ( v = 0) to Ci-(v = 0) transition energy. The structure should persist until an energy is reached that corresponds to a transition from the u = 0 level of the electronic ground-state of C ; to the highest accessible vibrational level of the Ci- state. According to the Franck-Condon principle, we expect a
10 -
5-
co+ co:
,,-
.,
.__.--
,I'
W
0-
.10 0
,
c,
-
'
I
I
c- + c--
,
j',::'
'
'
"
5
'
'
'
"
10
'
'
'
'
15
FIG. 14. The potential energy curves of C;,-Ci, and C2.The C;,~curve was obtained froin the isoclcclronic N? potential energy cuive (Edwards et u / . , 1993). The C; and Cz potential curves were obtained from Mead el ul. ( 1985).
STUDIES OF NEGATIVE IONS IN STORAGE RINGS
185
structure with a width of a few eV. which is in good agreement with the experimental data. The inclusion of rotation changes the Ci potential at the equilibriuin distance by less than 0.4 eV when the rotational quantum number is less than 40. Rotation changes the initial C; state in 21 similar way. and the only expected significant effect is a reduction of the potential barrier height of the C:-state. Due to the considerable electronic Coulomb energy, C;- lies in the electronic continuum of C2and C,. It also lies in the C + C dissociation continuum. Most of the Ci- vibrational levels are, however, essentially stable against dissociation into CC- because of the deep potential well. This. in turn, originates from the very deep potential well of N?. Only the few vibrational states with v > 19 have a lifetime against tunnelling shorter than 10- “’s. It is therefore concluded that the system decays almost exclusively by electron emission (autodetachment). Thus the observed resonance width may also be influenced by the autodetachment rate. The measured structure i n the C2 channel is hence associated with either a one-step process with the emission of two electrons, or a sequential process involving autodetaching states of C;. ~
+
IV. Interactions between Photons and Negative Ions A. ATOMICSYSTEMS Due to the lack of singly excited states in atomic negative ions, the interaction with photons has mainly been applied to study of the structural properties of the bound ground-state by photodetachment studies, or to exploration of the properties of doubly excited states that maniiest themselves as resonances in the photodetachment cross-section. Taking the properties (position, width) of resonances into account, many such studies can be performed at small accelerators, using the single-pass technique, but for systems such as H , the fundamental negative ion, it is an advantage to apply storage-ring techniques to be able to measure the width of the narrow resonances predicted for the ion. Such a study is currently in progress at ASTRID.
1. Rrsontrrices
it1
H
Resonances in H are located in the regions near the n = 2, 3. 4, etc. thresholds of the neutral H atom. The properties for the H resonances have already been studied by Bryant and coworkers (Bryant ~t ul., 1977, 1983; MacArthur et id., 1985; Harris ct al. 1990) in a series of experiments performed at LAMPF at Los Alamos. using an 800-MeV H - beam overlapped under a variable angle by a visible or ultraviolet laser beam. These experiments have covered an impressive range of photon energies from below threshold to above the limit for two-electron ~
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L. H. Andersen, 7: Andersen, and I? Hvelplund
ejection, but the resolution has been Doppler-limited to -7 meV. This is an important constraint, especially in the study of the resonances located near n = 2 in H since some of these have been predicted to be as narrow as -40 peV (Broad and Reinhard, 1976; Ho, 1991; Sadeghpour et al., 1992; Tang er al., 1994; Lindroth, 1995). The region first under investigation at ASTRID (Balling et al., 1996) is near the n = 2 threshold of the neutral H atom. This region is dominated by two pronounced structures: a narrow 'P resonance below the n = 2 threshold, which is commonly called the Feshbach resonance, and a broad structure above the threshold, the 'P-shape resonance. The study of this region is complicated by the need for photon energies near 11 eV, for which a new approach to Doppler-tuned spectroscopy was developed. VUV light of fixed wavelength (1 18 nm) was obtained from pulsed lasers, combined with nonlinear optical conversion and utilized together with a collinear ion-beam geometry to minimize the Doppler broadening. The laboratory photon energy is kept fixed, and the effective photon energy is varied by adjusting the velocity of the H - beam stored in the heavy-ion storage ring. This approach gives a comparatively high resolution, which has permitted an accurate determination of the position of the Feshbach resonance at 10.9243(3) eV, which is in good agreement with recent theoretical calculations (Ho, 1991; Tang et al., 1994; Lindroth, 1995), but deviates significantly from previously reported experimental results obtained at LAMPF (Bryant et al., 1983; MacArthur et al., 1985). The technique used to determine the position of the Feshbach resonance represents a 20-fold improvement in resolution compared to previous investigations, and further improvement by electron-cooling of the H beam is currently in progress in order to allow a determination of the width of the Feshbach resonance, commonly denoted 2{0}7 'PI,using the ,*{u}1 notation (see Sadeghpour et al., 1992). Here n and m represent the principal quantum numbers of the inner and outer electrons, and u is the angular correlation between the electrons, since it is the binding vibrational quantum number of the three-body rotor formed by the nucleus and the two electrons. The quantum number A = - 1 corresponds to the electrons being radially out of phase when approaching the nucleus as in the asymmetric-stretch mode. By increasing the energy of the H- beam from about 1 MeV, as used in the first study (Balling et al., 1996), it will also be possible to explore the resonances located at the higher-lying thresholds in the H atom.
B . CLUSTERS
I . Photodetachment
of' C,, and
C,,
One of the most important and well defined quantities for negative ions is the electron binding, that is, the electron affinity EA of the neutral system. For an ion in the ground state, EA can be measured as the minimum energy required for pho-
STUDLES OF NEGATIVE IONS IN STORAGE RINGS
187
todetachment, the photodrtachment threshold, However, for molecular ions with internal excitation, this minimum energy tnay be shifted, = EA - El,, where EI, is the part of the internal energy that is liberated in the detachment process. One obvious way of getting rid of internal energy is simply to let the cluster ions radiate for a sufficient period of time. In ring experiments, the photodetachment threshold can be measured as a function of the time elapsed after injection, and hence information on its temperature dependence can be obtained. For the photodetachment measurements, a beam of C,,,or C70ions circulating in the ring was merged with a photon beam obtained from a dye laser pumped by the third harmonic of a Q-switched Nd:YAG laser (7 ns pulse width, 10 Hz repetition rate). The detachment signal was recorded on a gated neutrals detector as a function of (i) time after injection into ASTRID and (ii) laser wavelength. Background signals were recorded by counting in a time window where the laser beam was turned off. Figure 15 shows detachment data obtained at times 50 ps, 0.5 s, and 5.5 s after exit of the ions from the source. The 50-ps curve was observed in a single-pass experiment conducted at a conventional accelerator, and a threshold value of 2.49 eV (497 nm) is obtained. The observations in the storage-ring experiments have been interpreted as a superposition of two curves, corresponding to direct photodetachment and to detachment assisted by liberation of internal energy, respectively. The thresholds of both curves are well above the value obtained in the single-pass experiment, that is, the detachment threshold for very hot ions. The time evolution of the
0
0
I
460
.
.
11'
, 480
,
0
500
Wavelength (nm)
Fiti. IS. Variation of photodetachment cross-scction for C'(,,, with laser wavelength 7 = SO ps refers to a single-pas5 cxperimcnt, whereas T = 0.5 s and T = 1.5s refer fo different times after injection of C,,, into ASTRID.
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two thresholds obtained in the ring experiments is such that the lower value of E,,,,, shifts toward higher energies as the storage time increases, while the higher value of E,,,,, remains constant. These observations are consistent with the interpretation of the higher, constant value of E,,,,, as the threshold for photodetachment without liberation of excitation energy and lead to a value EA = 2.666 t 0.001 eV for the electron affinity. This result is in good accord with the earlier reported value of 2.65 5 0.05 eV (Wang rt ul., 1991). For C,, an electron affinity of 2.676 t 0.001 eV has been obtained in a similar manner. The shift with time of the lower threshold IS interpreted as a result of cooling, which especially reduces the excitation of the high-frequency vibrations. It may also be noted that closely related sidebands, due to excitation of vibrations, have been observed in photoelectron spectroscopy on cold C& (Gunnarsson el ul., 1995).
V. Acknowledgments The collaboration with staff members at the Institute of Physics and Astronomy and at the Institute for Storage Ring Facilities, University of Aarhus, as well as with graduate students, post-graduate students, and guests is gratefully acknowledged. The negative ion research at the storage ring ASTRID is currently supported by the Danish National Research Foundation through the research center ACAP and has also been supported by the Danish Natural Science Research Council.
VI. References AlinCn, O., and Nielscn. K. 0. (1957).N i r d . Iristntrri. Merliotl.~.I , 302. Andcrscri. H. H.. Balling. P.. Petrunin, V. V., and Andersen T. (1996a)..I. PIiys. B: At. M o / . Opt Plij..~. 29%L41.5. Andcrsen. J. U., Brink, C.. Hvclplund, P., Larsson, M . O., Nielsen. 6 . B., and Shen, H. ( l996h). Piiys. Rel! LPU. 77. 399 I . Antlersen. J. U.. Brink. C.. Hvclplund. P., Larsson. M. 0.. arid Shen, H. (1996~). Z. Pliysik D. ( i n press). Aiidersen, L. H., e l d.( 1993a). P h p . Re\: L(,rr. 7 I . 18 12. Andcracri. L. H.. Andcrsen. T., Haugcil, H. K.. Hcrkl, N., Hvclplund, P., Mbllcr-. S. P.. and Siniiti, W. W. (1991). Plrys. Lett. A. 162. 336. Anderwn. 1.. H.. Bolko. J.. and Kvi*(gaartl, P. (1990). Plrys. Rev. A . 41, 13-93. Anderaen, I-. H., Hvclplund, P., Kella, D., Moklcr. P. H.. Pedcrsen. H. B., Schmidt, H. T., and VejhyChrisiensen. I,. ( 1906d). .I. Pliys. B: Aiom. Molec. o)Jf.Phys. 29, 2643. Anderscn. LA.H., Mathur. TI., Schmidt, H. T., and Vejby-Christensen, L. (1995).Pliys. K a . .Lett. 74. 892. Aiidcrscn, T.. Anderscn. L. H., Balling, P., Huugen,H. K.. Hvrlplund. P., Smith, W. W., and Taulbjerg K . ( I993bl. Piiys. Rei: A. 47. 890. Andcrsen, 7:. Anderscri. I-.H., Bjeri-e. N., Hvelplund. P., and Posrhumus, J. H. (1994). J. Plr\.s. R: At. Mid. Opt. Ph.vs. 27. I 135.
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Arnold. D. W.. Bradford, S . E.. Kitsopulos. T N.. iiiid Neiiliiark. D. M. ( 1991 ). J . C / I ~ I.So<.. ~ I . 95, 8753. Aspromallis. G.. Nicolaideb. C. A.. arid Beck. D. K. ( 1986). ./. P/ry.\ B; Arotrl. M<,[.Phj,s. 19, 1713. Aspi-omallia, G., Nicolaides, C. A., aiid Koiiiiiiof. Y. 19x5). J. P/I~.\.B: Atorn. M f j / , Plry.~. 18. 1.545. .4bproma~lis.G.. Sinanis. C., iind Nicol;iitles. C, A. ( 1996). .J. P/J!J. 8:At. M f , / .q , t . p/lT.s.29. ~ 1 , Bac, Y. K.. Coggiola, M. J.. and Petcrsoii. J . R . ( I YX4). Plr?~ Rei: Lctr. 52, 747. Balling. P. e/ rr/. (1996). P/ry.r. R n : Len. 77. 2905. Belling. P..Anderscn, I,. H.. Andci-sen. T., Haugen. H. K.. tlvelpl\lnd. P.. und Tatilhjclg. K . (1992). Plnu. Rei: Lrtt. 69. 1041. B ~ l y 0.. , nlid Schwiirtz, S. B. 1969)../. Plry,~.R: A I I ) J UMole<.. P/ry.\ 2. 159. Bjerre, N. ( 1996). private coinmtiiiicalioii. Blau. L. M.. Novick. R., and Weinflash. D. (1970).Plru. K P I : L m . 24. 1268. Bragc. T., and Frocse Fischer. C . 11991).P/I!\. ,?(,I: .4. 44. 71. Branscoinb. L. M.. ( 1962). "Atomic. md M o l r c . u / ~ l Pro(. v (New York), 17. 100. Brink. C.. Peterhen, H. B.. Andei-sen. I,. H.. Hvelpluiitl. P.. Kella. D., iincl Shen, H. i1996).to be published. Broad. J . T.. arid Reinhard. W. P. ( 1976). P / I ~ \R. Bryant. H . C. er ( I / , ( I 977). P/rJ,.r.R ~ TLert. 38, 228. Bryant. H. C. ef ol. ( 1983).PIrjx Rri: A. 27. 3889. Bunge. A . V.. and Runge. C. F, (1979).Phw. KOI: A. 19. 152. Burke. I? G. ( 1989). "Co//i.siorfT/irr'r,r-y~or..4torlr,\ ciritl Molec.rdc.s" (F. A. Gianturco. Ed.) Plenum Press, (New York and London) N.4TO AS1 serie\ H. Physics; I96 p. 1 I Calahrese. R., and Tecchio. L. (Eds.) ( I99 I ). "Elec-rrorr Coding ~rndN ~ C'oolirq M 7 k h n i q ~ t e . s . "World Scientific (Singapore). Con1pton. R. N. ( 1997) "Ne,ycifi\vI n t i s . " (V. A. Eaaulov. Ed.) Cambridge University Prcab (England). Dance. D. F., Harrison. M. F. A.. and Rundel. R. D. ( 1967) P,nc R. Soc. A299, 525. Davis. B. F.. and Chting, K. T. (1987). Pliw Rci: .A. 36. 1948. Dcmkov. Y. N., and Dnikarev. G. 1- (I965). S o l : P l r ~ c .JET/? . 20. 614. Dresselhaus. M . S., Dre\selhaus, G.. and Eklund. P. C. ( 1996). "Sc.iww o j ' F ~ r l / ~ n v r cc ~u .ds C(rr.horI Nmotuh~,.~." Academic Press (Sari Dicgo 1 , Edwards S., Roncin, J . Y., Launay. F.. and Ro. . F. 1 1993).J. Mol, S/,ec.tt: 162. 257. Ervin. K. M..and Lincbcrger, W. C. ( 190 I ). J . Plr\\. Clrcwi. 45. I 167. Eaaulov. V. A. ( 19x6). A m Pliys. R: I I . 393. Estherg. G N.. and L;I Bahn. R. W. ( 1968). Ph,v.s. L m . A. 38. 420. Esthcrg. G. N.. and La Bahn. R. W. (1970).P h y ~Rei: L w . 24. 1265. Gadiuk, J . W.. and Clark. C. W. (IY89).J. Chr~rrr.Phy\. 91. 3174. f o Aiorris ~ t i r i d Mo/rc~i/r.r" Plenum Press (New York Gianturco. E A . (Ed.) (I989). "Cd/isiorr T/WJ~:\ and London) NATO AS1 series B. Physics: 196. Gunnarasori, 0.. Handachuh, H.. Bechrhold. P. S., Kessler. B.. Gantefiir, G., and Eherhardt. W. ( I W S ) . P/ry.s. Re\>.Larr. 71, 1875. Hanatorp. D., Devynck. P., Graham. W. G., and Petcr\on. J . ( 1989). Ph!.s. Rev. Lett. 63, 368. Harris, P. G.. Bryant. H . C.. Mohaghcghi, A . H.. Kcetlei. K.A., Tang, C. Y.. Donahtie, J. B.. a i d Quick. C. K. 11990). Pliy.~.R w A. 42. 6443. Haugcn. H . K., Andcrben. L. H., Andersen, T.. Ballill$. P., Hertel. N.. Hvclpund. P., and Malle~-, S. P. ( 1992). P/I~.Y. Rci,. A. 46, R I . Hei-rick. 11. R. and Stillinper. E H. (1975).J. Clrcwr. f / / y . s . 62. 3360. Ho, Y. K. ( 199 I).Chirtr. J . Phya. 29. 327. Huffinanri, L). R. ( 1991 ). Phyics f i x i c r ~ .Nov., p. 22. Hurinaga. S.. and I-lart-Davis. A . (1973). Pl7v.7. Rei.. A. 8. 1734. Inokuti. M.. and Kiln. Y-K. (1968). Phys. Rev. 173. 154. John. T. I*.. and Williams. B. (1973).J . P l r y ~B: . Aforri. Mdrc. Phy.s. 6. L,381.
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ADVANCES IN ATOMIC. MOLECULAR, AND OPnCAL PHYSICS VOL. 38
SZNGLE-MOLECULESPECTROSCOPY AND QUANTUM OPTICS IN SOLIDS BY U? E. MOEmER, R. M.DICKSON, AND D. J. NORMS Department of Chemistry and Biochemistv University of California, San Diego 9SOa Gilman Drive, Mail Code 0340. La Jolla. California 92093-0340 Phone: 619-822-0453 Far: 619-534-7244 Email:
[email protected]
I. Introduction . . . . . . . . . . . 11. Physical Principles and 0 A. General Considerations ........................... B. Spectral Selection Using C. Peak Absorption Cross-Section . . D. Other Important Requirements for Single-MoleculeSpectroscopy. . . . . . . E. Materials Systems and Structures . . . . . . . , . . . . . . . . . . . . . . . . . . . . HI. Methods . . . . ... . . . . . A. GeometricalConfigUrations for Focusing,Fluorescence Collection, Microscopy B. Detection Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Direct Absorption (Frequency-Modulation Spectroscopy). . . . . . . . . . . 2. Fluorescence Excitation Spectroscopy . , . . . C. Examples of Experimental Measurements . . . . . . . . . . . . . . . . . . . . . . . 1. Single-MoleculeLineshape. . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . 2. Single-Molecule Imaging in Frequency and Space . . . . . . . . . . . . . . . . 3. Spectral Shifting:Trajectories, Correlation, and Lineshape Distributions IV. QuantumOptics ............................................... A. Photon Bunching and Antibunching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. QuantumJumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Optical (AC) Stark Effect . . . . . . . . . . . . . D. Single-MoleculeCavity Quantum Electrod V. Problems and Promise for Room Temperature A. Extension of Single-MoleculeStudies to Ambient Temperatures . . . B. Beginnings of Room Temperature Single-MoleculeQuantum Optics. C. Summary and Prospects for the Future . . . . . . . . . . . . . . . . . . . . . , . . . . VI. References . . . . . . . . . . . . . . . . . ............................
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I. Introduction Over the past few years, the power of optical spectroscopy with high-resolution laser sources has been extended into the fascinating domain of individual impurity molecules in solids. Using techniques described in this article, exactly one molecule hidden deep within a solid sample can now be probed by tunable laser radiation (Moerner and Kador, 1989; Kador et al., 1990; Orrit and Bernard, 193
Copyright 0 1997 by Academic Ross. Inc. All righta of reproduction in any form reserved.
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1.1! E. Morrner; R. M. Dickson, und D..I Norris .
1990). This represents detection and spectroscopy at the ultimate level of sensimoles of material, or 1.66 yoctomoles!* tivity-1.66 X Optical spectroscopy in this new regime is generating much interest for a variety of reasons. Most importantly, single-molecule measurements completely remove the normal ensemble averaging that occurs when a large number of molecules are probed simultaneously. Thus, the usual assumption that all molecules contributing to the ensemble average are identical can now be directly examined on a molecule-by-molecule basis, Information may be obtained about the basic optical properties of the molecule itself or its interactions with the surrounding solid. In this latter case, each individual molecule acts as an exquisitely sensitive probe of the immediate local environment (the “nanoenvironment”), and microscopic dynamical theories can be directly tested due to the absence of ensemble averaging (Reilly and Skinner, 1993, 1995). Single-molecule studies are also appealing because this regime has previously been unexplored, and new physical and chemical behavior may be observed. Several examples of this type of discovery are discussed in this review. First, however, we wish to mention that single-molecule spectroscopy (SMS) in solids is related to, but distinct from, the well-established field of spectroscopy of single electrons or ions confined in electromagnetic traps (Dehmelt et al., 1990; Diedrich et d., 1988; Itano et d., 1987). The vacuum environment and confining fields of an electromagnetic trap are quite different from the environments experienced by single molecules in solids and liquids. The trap experiments must deal with micromotion in the confining trap potential, and to date, no single molecule has been cooled sufficiently to be bound by an electromagnetic trap. In SMS, however, the interactions with the lattice act to constrain the molecule, hindering or preventing molecular rotation. At the same time, the single molecule is continuously bathed in the phonon vibrations of the solid available at a given temperature, and can interact with the electric, magnetic, and strain fields of the nanoenvironment. These interactions become very large at room temperature, thus the ability to obtain high-resolution spectral information becomes limited. Nevertheless, SMS techniques facilitate studies of motion and photophysical dynamics in both physical and biological systems (Barbara and Moerner, 1996; Bascht et al., 1997). Useful comparison may also be made to another important field, the direct probing of atoms or molecules on surfaces with scanning tunneling microscopy (or STM, see Binnig et al., 1986) or atomic force microscopy (or AFM, see Binnig et d., 1986; Rugar and Hansma, 1990). In contrast to the invasive natures of “Since a single molecule is the ainallest unit o f a molecular substance, a more appropriate unit in this case would hc the guurntnole, which is the quantily of moles exactly equal to the inverse of avocado’s number (Mocmer. 1995).(With apologies to Aniadeo Avogadro.)
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STM and AFM, SMS usually operates noninvasively in the optical far-field with a corresponding loss in spatial resolution to a value on the order of the optical wavelength (1 pm), but with no loss in spectral resolution. Moreover, single molecules can be studied below the surface in the body of the sample, and different single molecules can be selected by simply changing the excitation wavelength. The first single-molecule spectra were obtained i n 1989 (Moerner and Kador, 1989) and 1990 (Omt and Bernard, 1990) by utilizing low-temperature, high-resolution methods. In 1993, single-molecule imaging was achieved with near-field optical techniques with 100 nin resolution at rooni tenzperature (Ambrose rt ul., 1994; Betzig and Chichester, 1993). These near-held optical studies at room temperature and other near-field studies at liquid helium temperatures (Moerner et ul., 1994b) have been described in a recent review (Xie, 1996). Room-temperature far-field SMS measurements have also been performed with great success with results ranging from spectral manifestations of different local fluctuating environments (analogous to the low-temperature experiments, see Macklin et nl., 1996; Trautman and Macklin, 1996: Xie, 1996) to biophysical studies of molecular motors (Funatsu et al., 1995; Vale ef ~ l . 1996). , This article presents an overview of the physical principles and methods of high-resolution spectroscopy on single impurity centers in low-temperature solids and on single molecules at room temperature. The presentation will concentrate on far-field excitation methods, in which the spatial resolution is limited by diffraction effects to beam diameters on the order of 1 pm. While many aspects of this topic have been discussed in a series of reviews (BaschC et al., 1997; Kador, 1995; Kador et al., 1990; Moerner, 1991, 1994a, 1996; Moerner and BaschC, 1993a, 1993b: Moerner and Kador, 1989; Orrit and Bernard, 19901, here we emphasize the very recent quantum optical experiments performed on single molecules. Experiments in this area (perhaps surprisingly) show that single molecules are particularly suited to quantum optics. Despite their apparent complexity in terms of molecular vibrations, rotations, and interactions with the host material, single-molecule systems have exhibited extremely simple “two-level” and “three-level’’ behavior. This has allowed several quantuni optical measurements to be performed on a single molecule. One of the most appealing aspects of these experiments is that the surrounding solid traps the single molecule such that extended measurements on the same molecule are possible. Normal transit time or nucroniotion effects present in trap or beam experiments are absent. In Section IV we review the quantum optical experiments that have been performed on single molecules at low temperatures. Section V concludes by briefly presenting the problems and promise for room temperature quantum optical studies. First, however, we describe the fundamental requirements for high-resolution SMS in Section I1 (Moerner, I994b). Section 111 outlines the various experimental methods used to achieve SMS in solids, with selected examples.
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W. ELMoertirc R. M. Dicksnn, atid D. J. Norris
11. Physical Principles and Optimal Conditions A. GENERAL CONSIDERATIONS One may ask: How is it possible to use optical radiation to isolate a single impurity molecule hidden deep inside a host matrix? To answer this question concisely, single-molecule optical spectroscopy is accomplished by (a) selecting experimental conditions such that only one molecule is in resonance in the volume probed at a time and (b) ensuring that the signal from one molecule is detectable. A combination of small probing volume, low concentration of the impurity molecule of interest, and spectral selection is necessary to ensure that only one molecule is pumped by the laser beam. At the same time, it is important to select a fairly stable host-guest combination and detection technique such that the detected optical signal from one molecule can be observed with sufficient signal-to-noise in a reasonable averaging period. To proceed from one mole of material (6.02 X I@’ molecules) to only one molecule, many orders of magnitude must be spanned. The spatial coherence available from modem laser sources facilitates the probing of a small volume by providing focal spot sizes on the order of one to a few pm in diameter. It is best to use small sample thicknesses no larger than the Rayleigh range of the focused laser spot (3-10 pm). Thus, in most experiments a small volume of sample on the order of 10-100 pm3 is probed. This action alone represents an effective reduction of the number of molecules potentially in resonance by some 11 to 12 orders of magnitude, depending on the actual molar volume of the material. Single-molecule experiments generally utilize samples in which the molecule of interest is present as a dopant or guest impurity in a transparent host matrix. Obviously, then, if the concentration of the guest is sufficiently small, only one molecule of interest will be in the probing volume. For experiments at room temperature where no spectral selection method is available, it is indeed necessary to reduce the concentration of the impurity dramatically to lo-” moles/mole or lower, and to be very sure that no other unwanted impurity in the probed volume is capable of producing a signal that would overwhelm the signal from the single guest molecule of interest. All room-temperature single-molecule studies to date work precisely in this regime-by extreme concentration reduction, one and only one guest molecule is allowed at a time in the volume pumped by the laser. However, for high-resolution SMS at low temperatures, such extreme reductions in concentration are not generally required, and samples are usually doped with the guest at concentrations in the range lo-’ to lo-’ moles/mole. This additional 7 to 9 orders of magnitude reduction in the number of potentially resonant molecules is insufficient to guarantee that only one impurity molecule in the probed volume is in resonance with the laser at a time. The requisite additional selectivity on the order of a factor of lo4to lo5is provided by spectral selection, which involves carefully
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OPTICS IN SOLIDS
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selecting the guest and host and using the well-known properties of inhomogeneously broadened absorption lines in solids, to be described next.
B.
SPECTRAL SELECTION USING ZERO-PI (ONON AND
LINES
INHOMOGENEOUS BROADENING
There is an important physical effect that facilitates the optical detection and spectroscopy of single molecules in solids at low temperatures, known as inhomogeneous broadening. This effect occurs most clearly when the optical transition pumped by the laser is a purely electronic, zero-phonon line (ZPL),froin the lowest vibrational level of the electronic ground state to the lowest vibrational level of the electronically excited state. (We assume that the placement of the guest molecule in the solid effectively hinders rotation of the molecule.) In the visible, such a so-called (0-0)transition at energy h v shown in Fig. 1 often has a very long lifetime. because the deexcitatioii to the ground state manifold requires a large number of phonons to be emitted, and such a high-order emission process is improbable. At zero temperature, such a transition could have a lifetime-litnited transition width of tens to a few hundred MHz (for an electric-dipole-allowed transition in the visible, even smaller for a partially forbidden transition). At finite but low temperatures, the width of the ZPL can still be close to the lifetime-limited value. One may wonder why the time-varying perturbations due to the phonons of the solid do not broaden such transitions dramatically. First, considering linear coupling to the phonons, it is in fact the extremely high frequency of the phonons compared to the radiative width of the ZPL that places the fluctuations of the optical transition frequency in the motionally-narrowed regime
Fit;. I . Scheiiiutic of ihe electronic cnei-gy levels of a niolectile showing the ground singlet hiate .S,,. the lir\i excited singlci state S,,and ihc Ioucst triplet state 7 , .For cach clcctronic state, hcvcral level\ i n ihe vibrational progression are shown. Laser excitation pumps the (0-0) iransition with cncl-gy I I I J .'The intcraystcm crossing rate froin the singlet nianifold to the triplcta i s k < $ ,and . the triplet decay rate i h X , = ( T , ) - ' . Fluorescence emission shown as clotted line\ originates li-on1 S , and terininates o n vai-ious vibrationally excited levels o f .$,.
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W E. Moerner; R. M. Dickson, and D. J. Norris
(Kubo, 1969), so no broadening of the transition occurs. From the point of view of recoil, the ZPL is often regarded as the optical analog of the Mossbauer line, so that the entire solid sample recoils during optical absorption (Silsbee, 1962). As a result of these considerations, a ZPL transition can only dephase and broaden by second-order coupling to the phonons, that is, by phonon scattering (Zphonon processes, see McCumber and Sturge, 1963). At liquid helium temperatures, few phonons are present to produce phonon scattering, so the homogeneous width yH of zero-phonon optical transitions in crystals often approaches the lifetime-limited value of some tens of MHz mentioned above. Because the optical transition frequency is near 500 THz, the Q or quality factor of such a narrow transition is very large, typically 107-10x.In amorphous materials, other low-frequency excitations arising from two-level systems are present, and the homogeneous width is somewhat larger than the lifetime-limited value (Phillips, 1981j, but still far narrower than at high temperatures. It is the extremely high Q of single-molecule lines in solids that leads to exquisite sensitivity to nanoscopic changes-very weak perturbations produced by electric, magnetic, or strain fields in the nanoenvironment can easily produce a detectable shift in the single-molecule absorption. Now consider what happens for the collection of guest molecules located in the probed volume. If the sample were a perfect crystal and all local environments were identical, the optical absorption would be a single narrow Lorentzian line of width yH. (A normalized Lorentzian absorption profile centered at wo of fullwidth at half-maximum yH is given by ( y H / 2 r ) / ( ( 0- o,,)’ - (yd2)’)). However, these conditions are seldom met in real solid samples. The optical absorption spectrum that is actually measured for such an assembly is far broader than yH. The extreme narrowness of the ZPL for each of the molecules is obscured by a distribution of center frequencies for the various members of the ensemble, and the resulting overall profile is termed an inhomogeneously broadened line (Rebane, 1970; Stoneham, 1969) (see Fig. 2). The distribution of resonance frequencies is caused by dislocations, point defects, or random internal electric and strain fields and field gradients in the host material. Such imperfections are generally always present, even in crystals, and will be observed as long as yH is smaller than the inhomogeneous distribution. In the simplest case of inhomogeneous broadening, the overall line profile of width r, is caused by an approximately Gaussian (normal) distribution of center frequencies for the individual absorbers that is broader than the homogeneous lineshape. Inhomogeneous broadening is a universal feature of high-resolution laser spectroscopy of guest molecules or ions in solids (Skinner and Moerner, 1996; Wiersma, 1981; Yen and Selzer, 1981) and of other cases where ZPLs are probed, such as Mossbauer and magnetic resonance spectroscopy. It is for this reason that methods such as spectral hole-burning (Jankowiak et al., 1993; Moerner, 1988), coherent transients (Allen and Eberly, 1975), and most often, photon echoes (Narasinlhan et al., 1990), have been utilized to learn about the homogeneous line-width hidden under such inhomogeneous spectral profiles.
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Fortunately, the normally troublesome phenomenon of inhomogeneous broadening facilitates the spectral selection of individual molecules for SMS. In effect, the spread of center frequencies means that different guest molecules have different resonance frequencies, so if the total concentration is low enough, one simply uses the tunability of a narrowband laser to select different single molecules. This spectral selection must be done in a region of the frequency spectrum where the number of molecules per yH is on the order of or less than one. In general, this may be accomplished in three different ways as shown in Fig. 2: (a) by tuning out into the wings of the inhomogeneous line, (b) by using a sample with a very low
Inhomogeneous Line
Inhomogeneous Line
/
c.
t
Inhomogeneous Line
h
Flc;. '2. Schematic showing an inhomogeneous line at IOU Icinperatures and several wnys t o LIX spectral selection 10 achieve single-niolecule detection i n solids. The entire line is fornicd as a superposition of (generally Lorelmian) homogeneous protile\ of the individual abhorhers. with a distribution of center resonance frequencies caused by rmdoin \trains and impel-fectioiis which can he norinally dihtribtited (Gaussian). In the upper right inscl. several dopant inoleculcs ai-e sketched a h rectangles with different local cnvironnients produced by .;(rains, local electric fields. and other imperfections in the host iiiatrix. The laser line-width is iiegligiblc on the scale shown. (a) Selection olsingle molecules by wing wavelengths in the wings of the rnhoinogeneous line. (b) Selection of' single molecule by lowering the concentration of absorbers. l c ) Selection by drainatic increase in the value of for example by increasing the strain on the smiplc.
r,,
W E. Moerner; R. M . Dickson, and D . J. Norris
200
r,.
doping level, or (c) by using a sample with a very large A useful analogy is provided by the problem of tuning in a radio station when one is out in the country where only a few stations can be received. As the receiver is tuned, mostly static is received (no signal) until the exact frequency of a distant station is reached. Similarly, when inhomogeneous broadening causes the different single molecules in the probed volume to have different resonance frequencies, and the molecules are spaced apart by more than yH on average, single molecules can be pumped selectively, one at a time, simply by tuning the laser. To provide a slightly more realistic picture of the inhomogeneous broadening phenomenon, Fig. 3 shows a simulated inhomogeneous lineshape produced by the superposition(summation) of varying numbers of individual homogeneous Lorentzian absorption profiles. In this simulation, T,/y,, was chosen to be 10 for simplicity. (In many physical systems, the ratio r,IyHis much larger, often in the range lo3 to lo‘.) In trace (a), only ten centers were superposed to produce the entire “inhomogeneous” line, and the identification of structures corresponding to single molecules is clearly evident. This is equivalent to the situation depicted in Fig. 2(b). The other traces show the result of successively increasing the number of centers N in the probed volume. The “spectral roughness” called stutisticul fine strucfure (Moerner and Carter, 1987)on the peak of the profile is clearly evident, although it is decreasing in relative magnitude as U f i . In real experiments, statistical fine structure is
0 -4
I
I
I
I
-2
0
2
4
Frequency (units of I-, ) FIG. 3. Simulated absorption spectra with different total numbers of absorbers N.using Lorentzkin profiles for the individual ahsorhers and a Gaussian random variable to select center frequencies in (he inhornogcneoua line, Traces (a) through (d) correspond to N values of 10. 100. IO(j0, and 10.000. respectively, und the traces have been divided by the Factors shown. For clarity, yw is taken to he one-tenth of lor this simulation. (Alter Moemer and Basche, 1993b).
r,
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OPTICS IN SOLIDS
201
often observed before proceeding to the SMS regime, in order to optimize the detection system. Indeed, the early observation of statistical fine structure for pentacene in p-terphenyl crystals (Carter et ul., 1988b; Moerner and Carter, 1987) provided useful encouragement that the single-molecule regime could eventually be reached. With difficulty, statistical fine structure has also been observed for ions in inorganic materials (Brocklesby el a/., 1989; Carter rr 01.. 1988a).
C. PEAKABSORPTION CROSS-SECTION A central concept crucial to single-molecule spectroscopy in solids is the role of the peak absorption cross-section of the guest molecule o;,. Study of this parameter helps to answer the question: Why is the signal from a single molecule large enough to be detected above background in a reasonable period of time'? As will be shown below, the signal-to-noise ratio (SNR) for SMS depends crucially on maximizing the value of a,,,no matter which technique is used for detection. For focal spots of area greater than or equal to the diffraction limit (approximately A', with A the optical wavelength), the rate at which the resonant optical transition is pumped is given by the product of the incident photon flux [in photons/(s cm')] and a,, (in cm'). Stated differently, the probability that a single molecule will absorb an incident photon from the pumping laser beam is just ?,/A where A is the cross-sectional area of the focused laser beam. High a,,means that the photons of the incident light beam are efficiently absorbed and background signals from unabsorbed photons are minimized. To understand what controls the value of q,, we recall that for molecules with weak electron-phonon coupling with appreciable oscillator strength in the lowest purely electronic transition, the optical homogeneous linewidth yH becomes very small at low temperatures. The key point to remember is that due to well-known sum rules on optical transitions, the peak cross-section depends inversely on yH, so that the narrow line-width of a ZPL translates into a very large peak absorption cross-section. For allowed transitions of rigid molecules, the value of 0;)becomes extremely large, approaching the ultimate limit of A'. Although the importance of the peak absorption cross-section has been recognized since the first SMS experiment (Moerner and Kador, 1989), it is useful to review how a/,can be estimated using sun-rule techniques. The standard integrated absorption sum rule (Hilborn, 1082; Moerner et al., 1983; Moerner rt al., 1984; Sievers, 1964) may be written
where e is the electronic charge, m is the electron mass, the local field factor is F = [(n' + 2)/3]', n is the index of refraction, c is the speed of light, N,,,, is the number density of absorbers producing the integrated absorption S (units cm -'),
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W E. Moerner; R. M.Dickson, and D. J. Norris
f is the oscillator strength, Fo is the frequency at the center of the band in
wavenumbers (cm-I), h is Planck’s constant, and is the transition dipole moment magnitude. Because the oscillator strength (or dipole moment) is generally independent of temperature, the ratio is a constant that may either be evaluated for the homogeneous band at room temperature or for a single Lorentzian profile at low temperature. Applying Eq. (1) to the single-molecule Lorentzian absorption profile of width ACH = (.rrcT,)-l = yH/(2rrc),where T, is the dephasing time, and identifying the peak absorption divided by the number density of absorbers producing that absorption as the peak cross-section (per center) yields the standard formula (Moerner, 1994b)
If the integral to determine S in Eq. ( I ) is performed for a room-temperature liquid solution, the resulting value of oscillator strength f applies to the entire electronic transition, including all the phonon and vibrational sidebands of the ZPL. To obtain the oscillator strength of the ZPL only,f,,,, for use in Eq. ( 2 ) ,the value of fmust be multiplied by FFcFDw,where FFcis the Franck-Condon factor for partitioning between the ZPL and vibronic sidebands, and F,, is the Debye-Waller factor for partitioning between the ZPL and the phonon sidebands. Similar considerations apply to the dipole moment and integrated absorption strength S. Thus, with measurement of the low-temperature line-width (or T,) and either the oscillator strength, dipole moment, or the value of S/Nl(,lfor the transition, q,can be determined. (For an equivalent alternative approach based on radiative lifetimes, see Rebane and Rebane, 1993.) For molecules like pentacene, perylene, or similar rigid aromatics with a strongly allowed lowest electronic transition (see Section II.E for structures), the net result is that at low temperatures where T, is large (tens of ns), the peak absorption cross-section increases to levels as high as lo-” cm2, approximately 4,000 times the (van der Waals) area of a single molecule! Thus, even though any dimension of a single molecule is much smaller than the optical wavelength, the effective area of the molecule for optical absorption is not nearly so small, and a zero-phonon optical transition of a molecule in a solid can be made to absorb light quite efficiently if the sample is cooled to low temperatures. To give a specific example for the case of pentacene, the low-temperature homogeneous width is 7.8 MHz (Patterson et al., 1984), the measured ZPL dipole moment without orientational averaging is 0.7 Debye (de Vries and Wiersma, 1979) where one Debye is defined as 3.33 X C-m, and thus the value of IT,, is estimated to be approximately 9 X lo-” cm2, not far from that for the allowed transitions of an isolated sodium atom.
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OPTICS IN SOLIDS
D.
OTHER IMPORTANT REQUIREMENTS FOR
203
SINGLE-MOLECULE SPECTROSCOPY
In addition to large u,,,several additional requirements are necessary to ensure that the signal from a single molecule dominates over all background signals (for full details see Moerner and BaschC, 1993a and Moerner, 1994b). In the case of fluorescence detection, the quantum yield for photon emission per absorption event & should clearly be high, as close to unity as possible. Extreme care should be taken to minimize scattering backgrounds, which may arise either from Rayleigh scattering, or from Raman scattering from the sample and the substrate. A further requirement on the absorption properties of the guest molecule stems from the general fact that although higher and higher laser power generally produces more and more signal, the optical transition must not be saturated (see Sect. III.B.2). When saturation occurs, further increases in laser power generate more background rather than signal, and this is true for both fluorescence and absorption detection methods. The saturation intensity is maximized when absorbing centers are chosen that do not have strong bottlenecks in the optical pumping cycle (see Fig. 1). In organic molecules, intersystem crossing (ISC) from the singlet states into the triplet states represents a common bottleneck, because both absorption of photons and photon emission cease for a relatively long time equal to the triplet lifetime when ISC occurs. This effect results in premature saturation of the emission rate from the molecule and reduction of the absorption cross-section 0;) compared to the case with no bottleneck (Ambrose rf al., 1991). For later reference, the saturation intensity I,s for a molecule with a triplet bottleneck may be written (de Vries and Wiersma, 1980; Plakhotnik rf al., 1995).
where T , = l/k2, is the inverse of the rate of direct decay from S , to S,,, k,,, is the rate of intersystem crossing as shown in Fig. 1, and k , is the total decay rate from the triplet back to Scj.The factor outside the parentheses is the 2-level saturation intensity if there were no triplet bottleneck, which represents an upper limit for I,. Thus, minimizing the triplet bottleneck means small values of k,,L and large values of k,, requirements that may be easily satisfied for rigid, planar aromatic dye molecules. A final requirement for SMS is the selection of a guest-host couple that allows for photostability of the impurity molecule and weak spectral hole-burning, where by spectral hole-burning we include any fast light-induced change in the resonance frequency of the molecule caused either by direct photochemistry of the molecule or by a photophysical change in the nearby environment (Jankowiak et ul., 1993; Moerner, 1988). For example, most fluorescence detection schemes with overall photon collection efficiency of 0.1% to 1% require that the quantum
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W E. Moerner; R. M.Dickson, and D. J. Norris
efficiency for hole-burning be less than lo-” to lo-’. This is necessary to provide sufficient time averaging of the single-molecule signal before it changes appreciably or moves to another spectral position Qumps out of resonance). It should be noted that the additional requirements for SMS stated in this section represent a “best-case” in order to produce the highest possible signal-tonoise in a high-resolution experiment. If some loss in signal-to-noise or spectral resolution can be tolerated, these requirements can be weakened accordingly. Specific forms of the SNR for various detection methods will be presented below.
E.
-
MATERIALS SYSTEMS AND STRUCTURES
Thls section briefly lists the most common guest-host materials systems in which high-resolution, low-temperature SMS studies have been performed (BaschC er al., 1997).To date, the guest impurity molecules have been selected exclusively from the class of rigid conjugated hydrocarbons with specific cases shown in Fig. 4: (a) pen-
FIG. 4. Structures of some ofihe molecules that havc hcen studied hy SMS. (a) Pentacenc, (b) peryh e , (c) terrylene, (d) tetra-(f-hutyI)-tenylene(TBT), (e) diphenyloctatctraenc (DPOT), rf) 7.8, IS.16dihenzotenylene (DBT). and (g) 2.3,8.9-dihcnzanthanIhl.ene(DBATT).
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tacene, (b) perylene. (c) tenylene, (d) tetra-0-buty1)-tenylene (TBT), (e) diphenyloctatetraene (DPOT), (f) 7.8,15.16-dibenzoterrylene(DBT), and (g) 2.3,8.9-dibenzanthanthrene (DBATT).These molecules have strong singlet-singlet absorption, excellent emission properties, and weak triplet bottlenecks. They also feature the weak Franck-Condon distortion necessary to guarantee a strong (0-0) electronic transition. In most cases, workers have concentrated on fairly large aromatic hydrocarbons (AHCs) (a)-(d), (0,(g) in order to place the lowest electronic transition in the mid-visible. This allows standard tunable single-frequency dye lasers to be utilized for pumping the transition. The one exception, DPOT (e), represents a special case that was pumped either with two-photon excitation in the near-IR (Plakhotnik et d., 1996), or with doubled light at 444 nm from the output of a cw Ti-sapphire laser. The choice of the host material is generally dictated by the need to maintain a weak electron-phonon coupling and to prevent high-efficiency spectral hole-burning. The latter requirement has so far prevented hydrogen-bonded matrices from being successfully utilized for high-resolution, low-temperature SMS. The actual host-guest combinations fall into three categories: AHCs in crystals, AHCs in polymers, and AHCs in Shpol’skii matrices. Table I lists a variety of host-guest combinations that have been studied, along with the position of the ZPL, the single-molecule linewidth (FWHM, generally at -2K), and leading references. The favorite single crystal hosts have been p-terphenyl, naphthalene, and anthracene, all of which can be sublimed to form clear micron-thick samples. Generally, single molecules in these hosts are relatively stable, with occasional spontaneous spectral diffusion driven by defects in the host crystal (see section III.C.3). The polymer hosts have been selected from poly(ethy1ene) (PE), poly(iso-butylene) (PIB), poly(viny1 butyral) (PVB), poly(styrene) (PS), and poly(methy1 methacrylate) (PMMA). Although single-molecule absorption lines are not as stable in these hosts as in the crystals, we include them here for reference. Single molecules in polymers are characterized by highly dispersive spectral shifting phenomena, both spontaneous and light-driven. In Table 1, a range of line-widths indicates a distribution, and the original references should be consulted for more detail on the shape of this distribution. Finally, various Shpol’skii matrices such as frozen hexadecane, nonane, and tetradecane have also been exanined. These convenient polycrystalline hosts yield relatively stable single-molecule spectra, with a slow, lightdriven spectral shifting observed in several cases. To date, the total number of systems that have been studied by high resolution SMS techniques is approximately 15-20. Work is in progress on new materials, such as other AHCs and additional Shpol’skii matrices. It is to be expected that some laser dye molecules in appropriate hosts will also have the strong fluorescence, weak triplet bottlenecks, and near-absence of spectral hole-burning required for SMS studies.
u! E. Moernec
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R. M.Dickson, and D.J. Norris
TABLE I HOST-GUESTCOMBINATIONS STUDIED BY SINGLE-MOLECULESPECTROSCOPY
Host AHC: single crystals
pentacene p-terphenyl
terrylene
AHC: polyiners
DBT perylene terrylene
TBT AHC: terrylene Shpol’skii matrices perylene DPOT DBATT
ZPL Wavelength Line-width References (nm) (MHz) 592.32 (0,) 7.8 (0,) 592.18 (0,)
(Ambrose ef al.. 1991 ; Amhrose and Moerner, 199I ; Moerner and Kador, 1989; Orrit and Bernard,
naphthalene 602.8 p-terphenyl 580.4 (X,), 578.5 (XJ, 578.3 (X3), 577.9 (X,) anthracene naphthalene 757.7 PE -442
29 48.8 (X,)
1990) (Kummer et al., 1996) (Kummer et ul.. 1994)
50-500 25-35 52-142
(Kozankiewicz et a/., 1994) (Jelezko et al., 1996) (Bascht and Moerner, 1992)
PE PVB PMMA PS PE PIB hexadecane
569 562 557 566 567.6 57 1.9
60- 150 200-2000 200-2200 600-4000 50-320 40-370 40
(Omt er ul., 1992) (Kozankiewicz ef a/., 1994) (Kozankiewicz e t a / . , 1994) (Kozankiewicz et a/.. 1994) (Kettner et al., 1994) (Kettner e t a / . , 1994) (Moerner et al., 1994a; Plakhotnik eta/., 1994)
nonane tetradecane
-440 444 ( I -photon) 589.1
30
(Pirotta et al., 1996) (Plakhotnik et ul., 1996)
12-30
(Boiron e f a/., 1996)
hexadecane
111. Methods A. GEOMETRICAL CONFIGURATIONS FOR FOCUSING, FLUORESCENCE COLLECTION,
MICROSCOPY A variety of different configurations have been used for achieving the required focusing of radiation in a small sample at liquid helium temperatures. If a direct absorption method is used, one need only collect the transmitted light and direct it to the detector. In the case of fluorescence detection, it is necessary to carefully collect the emitted fluorescence photons from the single molecule over as large a solid angle as possible, without collecting unwanted transmitted pumping radiation or scattered light. Because of the large number of experiments utilizing fluorescence detection, all configurations described here refer to the fluorescence method.
SINGLE-MOLECULE SPECTROSCOPY AND QUANTUM OPTlCS IN SOLIDS
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One useful experimental setup is shown in Fig. 5(a), the “lens-parabola” configuration. All the components shown can be immersed in superfluid helium. The focal spot from the pumping laser is produced by a small lens of several mm focal length placed directly in the liquid helium. Generally, some provision must be made for adjusting the focus at low temperatures. In one solution to this (Ambrose et d.,1991), the lens is mounted on a thin stainless steel plate, which can be flexed by a permanent magnet M/electromagnet C pair to adjust the focal position. The sample is mounted on a transparent substrate, ideally an alkali halide, whose center of symmetry prevents first-order Raman scattering by the substrate. After passage through the sample, the transmitted pumping radiation is blocked by a small beam block. The emitted fluorescence is collected by a paraboloid with numerical aperture (NA) near 1.0, and directed out of the cryostat. Using a standard 2-element achromatic lens, a spot size of 3-5 p m diameter can be produced. The size of the focal spot is limited by the distortion and aberrations produced by cooling a lens designed for operation at room temperature. With optimized optics able to tolerate liquid helium temperatures, smaller spot sizes should be achievable with this configuration. One feature of this configuration is
5
M
C
sample Ihidiness-8um
FIG. 5. Various experimental aii-angements tor SMS in tluoremxce excitation: (a) lens-piiraholoid (reprinted with permission from Amhrose et d., 1991. copyright 1991 Arnerican Iilstittite of‘ Physics). L-leiis, P-paraboloid. S-sample, B-bcam block, M-magnet. C-coil electromagnet, ih) fiber-paraholoid (first used in Orrit and Bernard, 1990),( c ) pinhole (reprinted from Pirotta et d.. 1993 with kind permission from Elsevier Science-NL, Sara Burgcrhartstraat 2.5, 10.5.5 K V Amsterdam, ‘The Nctherlands.). (d) paraboloid focus and collection (reprinted from Fleury c’t a/.. I995 with kind permission from Elscvier Science-NL, Sara Burgerhartstraat 15. 105.5 KV Amsterdam, The Netherlands.).
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U! E. Moernel; R. M . Dickson, and D. J . Norris
the ability to scan the position of the focal spot at low temperatures across the sample by tilting of the incident pumping laser beam (Ambrose et al., 1991). A second useful configuration (termed “fiber-parabola”) used in the first fluorescence excitation SMS (Onit and Bernard, 1990) takes advantage of the small spot size produced automatically by the core of a single mode optical fiber as shown in Fig. 5(b). Here the thin crystalline or polymeric sample is attached to the end of the optical fiber using epoxy or index-matching gel and held by capillary action. The fiber end with the sample is again placed at the focus of a high NA paraboloid for collection of the emission. The spot size in this case is controlled completely by the fiber core diameter, and mode diameters of 4 pm are common. This configuration avoids the need to adjust the position of the focus, but does not allow any change of the volume probed by the laser after cooldown. In addition, some strain can be introduced into the sample as a result of the gluing process. A third experimental configuration involves the use of a small pinhole aperture (Pirotta et al., 1993) in a thin metal plate, as shown in Fig. 5(c). The sample is mounted directly against a thin stainless steel plate with a 5 pm pinhole in it. The pinhole is illuminated with the laser beam from the opposite side. The fluorescence is collected with small lenses or a 0.85 NA microscope objective placed in the liquid He (not shown). While this method is relatively easy to implement, the sample can be easily strained during mounting, resulting in very broad inhomogeneous lines. A more fundamental problem results from the strong oscillations in the local intensity of the laser beam produced by the nearby conducting aperture (Campillo et al., 1973). In general, it is more difficult to determine the laser intensity at the position of the single molecule with this approach. In order to study the vibronic structure of the single molecule, researchers have spectrally dispersed the emission by collecting the emitted light from the fiber-parabola setup and focusing it on the slit of a spectrometer (TchCnio et al., 1993b). A liquid-nitrogen-cooled CCD with very high quantum efficiency is used to detect the spectrally dispersed photons. The result is actually a resonance Raman experiment on a single molecule (Myers et al., 1994), which contains useful information about possible molecular distortions produced by the local environment. In a recent enhancement of this technique shown in Fig. 5(d), a highquality diamond-turned A1 paraboloid was used both to focus the emitting radiation, as well as to collect the emission in an epi-fluorescence configuration (Fleury et al., 1995). The sample was placed at the focus of the paraboloid by flexing a thin metal plate. The focal spot diameter was estimated to be less than 2 pm, and the total sample volume probed as small as 10 pm’. For experiments in which microwave and magnetic fields are required to pump transitions between spin sublevels, a sophisticated enhancement of the lensparabola design has been described (van der Meer et al., 1995) that allows lowtemperature positioning of the sample along two axes without use of electro- or permanent magnets.
SINGLE-MOLECULE SPECTROSCOPY A N 1 QUANTUM OPTICS IN SOLIDS
209
Far-tield fluorescence microscopy using two spatial dimensions has been demonstrated by researchers at ETH Zurich using a microscope objective and other optics in the liquid helium to efficiently collect emission from a large-area sample (Guttler ef d . ,1994; Jasny rt ol., 1996). This powerful method can allow the behavior of many single molecules to be followed simultaneously.
B. DETECTION TECHNIQUES
I . Direct Absorption ( F r e y u r t i c ~ - ~ ~ ) ~ ~ i ~Spectroscopy) lutiorr The first single-molecule spectra were recorded in the pentacene in p-terphenyl system in 1989 using a sophisticated zero-scatteringbackground absorption technique, frequency modulation (FM) spectroscopy (Bjorklund, 1980: Bjorklund rt L I ~,. 1983). combined with either Stark or ultrasonic modulation of the absorption line (Kador rt d., 1990; Moemer and Kador, 1989). Rather than describe the complete details of the FM technique here, it is more useful to describe the basic characteristics of the method, and the reader may consult the literature (Kador rt (11.. 1990; Moerner and BaschC, 1993a) for more infomiation. Referring to Fig. 6, using an electro-optic modulator (EO). the singlc-frequency cw dye laser (DL) at (carrier) frequency y.is phase-modulated at the local oscillator (LO) radio frequency of,?, which produces two sidebands on the laser carrier at w, + w,,,and at a,. - w , , with ~ opposite phase. Any amplitude modulation of the laser beam exiting the sample is detected using a fast photodiode (e.g., an APD) and a phase-sensitive radio frequency lock-in (mixer M) driven at w,,,. If no narrow spectral features are present, only a dc signal appears at the detector output, since a perfect phase-modulated laser beam has no amplitude modulation. The background noise level at the 1 port of the mixer is produced by detector noise and by the laser noise at w,,,. For LO frequencics above about 1 MHz. the limiting noise source can be quantum (or shot) noise if no excess noise is introduced by the detector. Generally, the laser power at the detector must be above a certain minimum value for shot noise to dominate over other noise sources (typically Johnson noise). If a narrow spectral feature is present with line-width on the order of or less than w,,,, the unbalancing of the two sidebands will convert the phase-modulated laser beam into an amplitude-modulated beam that produces a strong oscillating photocurrent at w,,,at the detwtor output. More precisely, the detected signal at the 1 port of the mixer (in the absorption phase) is proportional to [a(w( + o,,,) - a(q - w,,,)]L where a is the absorption coefficient and L is the sample thickness. Thus the FM signal nieasures the difference in d at the two sideband frequencies. For a spectral feature narrower than w,,,,two copies of the absorption line appear, one positive and one negative, as each of the two sidebands is swept over the absorption. In actual practice, it is generally difficult to produce absolutely pure phase modulation of the laser beam, and a frequency-dependent interfering signal called
210
W E. Moernel; R. M. Dickson, and D. J. Norris
FIG. 6. Schematic of frequency-modulation spectroscopy with Stark secondary modulation (FMStark). The top of the figure shows the light spectrum at the output of the dye laser (DL), after the electro-optic phase modulator (EO), and after the cryostat (C, from left to right), the arrows indicating the relative amplitudes and phases of the electric light fields. The rf source drives the EO modulator and produces the local oscillator signal (LO) for the mixer M. The output of the avalanche photodiode (APD) drives the R port of the mixer. The simple FM signal is present at the I port of Lhe mixer. For secondary Stark modulation, a high-voltage source (HV) at reference frequencyfproduces a time-varying electric field across the sample. A lock-in amplifier detects the output of the I port of the mixer, and the resulting signal is averaged on a digital oscilloscope (DS). (Reprinted from Carter et a/., 1988a with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 10% KV Amsterdam. The Netherlands.).
residual amplitude modulation (RAM) is often present. While many methods have been proposed to eliminate RAM (Gehrtz et al., 1985; Whittaker ef al., 1985), internal secondary modulation of the spectral features by Stark-shifting or ultrasound have proved most useful in SMS experiments (Kador er al., 1990). For the FM-Stark approach illustrated in Fig. 6 , an oscillating high-voltage (HV) source driven at an audio frequency f impresses a time-varying electric field across the sample. This will produce a periodic shifting of the spectral feature. Because the RAM does not oscillate at frequencyf, the RAM may be removed by detection of the mixer I-port output with a lock-in amplifier (LIA) driven atf(for a linear Stark shift) or at 2f(for a quadratic Stark shift). The resulting line-shape will appear as the appropriate spectral derivative of the FM signal (see Moerner and Kador, 1989). The signal-to-noise ratio for a single molecule detected by FM spectroscopy has been described in detail (Moerner and BaschC, 1993a, 1993b). Because this
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OPTICS IN SOLIDS
2I 1
is an absorption technique, clearly single molecules with higher absorption crosssections lead to larger FM signals, and detectors with internal gain such as APDs are helpful in reducing the effect of detector noise. In contrast to fluorescence methods, Rayleigh scattering and Raman scattering are unimportant. Only the shot noise of the laser beam contributes to the background, assuming RAM and detector noise are properly controlled. Figure 7 shows examples of the optical absorption spectrum from a single molecule of pentacene in p-terphenyl using the FM-Stark method. Although this early observation and similar data from the FM-ultrasound method served to stimulate much further work, there is one important limitation to the general use o f FM methods for SMS. As was shown in the early papers on FM spectroscopy (Bjorklund, 1980; Bjorklund rr ul., 1983). extremely low absorption changes as small as lo-’ can be detected in a I s averaging time, but on/v (f large laser powers on the order of several mW can be delivered to the detector to reduce the relative size of the shot noise. This presents a problem for SMS
- kP-4 0.4
>
v
-0 C cn .ln
-6 c 0
0
-3 U
I
-Dm 2
a
-0.4
-0.8 0
200
400
600
Laser Frequency (MHz)
FIG. 7. The tirst single-molecule optical spectrn. showing use of the PM-StaIk technique for pentacene i n pterphenyl. ( a ) Simulation of absorption line with (powcr-broadened) line-width of 6.5 MHI. (b) Simulation of FM spectiuin for (a). w,,, = 7.5 MHz. (c) Simulation of FM-Stark line-shapc. (d) singlemolecule spectra at 502.423 nrn, 5 12 averages, 8 traces overlaid, bar shows value of 2w,,, = 150 MHz. (el Average 0 1 traces in (d) with fit to the in-focus inolecule (smooth curve). (I) Signal f i r off line at 597.514 iini. ( g )Tmces of SFS at the O2 line center, 502.186 iiiii. (After Moemer arid Kador. 1989).
212
M! E. Moenrer; R. M. Dicksnn, and D. J. Nor‘s
in the following way. Because the laser beam probing the sample must be focused to a small spot, the power in the laser beam must be maintained below the value that would cause power broadening {saturation) of the sitigle-molecule lineshape. As a result, it is quite difficult to utilize laser powers in the mW range for SMS of allowed transitions at low temperatures-in fact powers below 100 nW are generally required. This is one reason why the SNR of the original data on single molecules of pentacene inp-terphenyl in Fig. 7 was only on the order of 5 . (The other reason was the use of relatively thick cleaved samples, which produced a larger number of out-of-focus molecules in the probed volume. This problem has been overcome with much thinner sublimed samples in modern experiments.) If either materials with higher saturation intensity or squeezed light beams with reduced shot noise become easily available in the future, the utility of the FM method will improve.
2. Fluorescence Excitution Spectroscopy In 1990, Orrit et a/. also began experiments on the pentacene in p-terphenyl system and demonstrated that fluorescence excitation spectroscopy produces highquality signals if the emission is collected efficiently and the scattering sources are minimized (Orrit and Bernard, 1990; Orrit rt al., 1993). Most subsequent experiments have used this technique, in which a tunable narrowband single-frequency laser is scanned over the absorption profile of the single molecule, and the presence of absorption is detected by measuring the fluorescence emitted. A longpass filter is used to block the pumping laser light and Rayleigh scattering, and the fluorescence shifted to long wavelengths is detected with a photon-counting system, usually a photomultiplier and discriminator. The detected photons generally cover a broad range of wavelengths, because the emission from the ground vibrational level of the electronically excited state terminates on various vibrationally excited (even) levels of the electronic ground state as shown in Fig. 1. In fluorescence excitation, the detection is usually background-limited and the shot noise of the probing laser is only important for the signal-to-noise of the spectral feature, not the signal to background. For this reason, it is critical to efficiently collect photons (as with a paraboloid or other high numerical aperture collection system), to reject the pumping laser radiation, and to hope for low Raman scattering from the host matrix. To illustrate, suppose a single molecule of pentacene in p-terphenyl is probed with 1 mW/cm’, near the onset of saturation of the absorption due to triplet level population. The resulting incident photon flux of 3 X photonsls-cm’ will produce about 3 X lo4 excitations per second. With a fluorescence quantum yield of 0.8 for pentacene, about 2.4 X lo-‘emitted photons can be expected. At the same time, 3 X 10’ photons/s illuminate the focal spot 3 p m in diameter, Considering that the resonant 0-0 fluorescence from the molecule must be
SINGLE-MOLECULE SPECTROSCOPY AND QUANTUM OPTICS IN SOLIDS
2 13
thrown away along with the pumping light, rejection of the pumping radiation by a factor greater than lo5 to 10’ is generally required, with minimal attenuation of the fluorescence. This is often accomplished by low-fluorescence long-pass glass filters or by holographic notch attenuation filters. The attainable SNR for single molecule detection in a solid using fluorescence excitation can be approximated by the following expression (BaschC et d., 1992aj:
s, (noise)<,,,\
__ -
(D4,.u,,P,).r)l(Ahu ) V (D&u,,P,,~)l(Ahv)+ C,,P,,T f N<,T
(4)
where the numerator, S , , is the peak detected fluorescence counts from one molecule in an integration time 7, is the fluorescence quantum yield, a/, is the peak absorption cross-section on resonance as defined above, P,, is the laser power, A is the focal spot area, h v is the photon energy, N,, is the dark count rate, and C, is the background count rate per watt of excitation power. The factor D = I)&,E;; F; describes the overall efficiency for the detection of emitted photons, where qu is the photomultiplier quantum efficiency, is the fraction of the total emission solid angle collected by the collection optics, 5 is the fraction of emitted fluorescence that passes through the long pass filter, and F, is the total transmission of the windows and additional optics along the way to the photomultiplier. The three noise tenns in the denominator of Eq. 4 represent shot noise contributions from the emitted fluorescence, background, and dark signals, respectively. For a detailed discussion of the collection efficiency for a single molecule taking into account the dipole radiation pattern, total internal reflection, and the molecular orientation, see Plakhotnik et al. (1995). Assuming the collection efficiency D is maximized, Eq. 4 shows that there are several physical parameters that must be chosen carefully in order to maximize the SNR. First, as stated above, the values of 4 , and q7should be as large as possible, and the laser spot should be as small as possible. The power cannot be increased arbitrarily because saturation causes the peak absorption cross-section to drop from its low-power value o;,according to
+,;
5,
e,
where I is the laser intensity and I, is the characteristic saturation intensity (Moerner and Ambrose, 1991). The effect of saturation in general can be seen in both the peak on-resonance emission rate from the molecule R ( I ) and in the single-molecule line-width AV(l) according to (Ambrose et ctl., 1991):
R(I) = R ,
[
Ilt, 1 + 1Il.J
____
A v ( I ) = Au(O)[l
+ (I/I,s)J”’
(7)
214
W E. Maernel: R. M. Dickson, mid W . J. Norris
where for the three-level system in Fig. 1, the maximum emission rate is given by
Equations 5 and 7 show that the integrated area under the single-molccule peak falls in the strong saturation regime. However, at higher and higher laser power, the scattering signal increases linearly in proportion to the laser power, so the difficulty of detecting a single molecule increases. The dependencies of the maximum emission rate and line-width on laser intensity in Eqs. 6 and 7 have been verified experimentally for individual single molecules (Ambrose et al., 1991). To illustrate graphically the tradeoffs inherent in Eqs. 4 and 5, parameters for the model system pentacene in p-terphenyl will be used, for which (PF = 0.78 (de Vries and Wiersma, 1979) and D 0.01 in the lens-parabola geometry. Measured values for the background scattering level, dark count rate, and other parameters have also been reported (Moerner and BaschC, 1993a). Equations 4 and 5 then yield a relationship between the SNR, P,,, and A . Assuming 1 s integration time, Fig. 8 shows the S N R verms laser power and beam area. It is clear that for a fixed laser spot size, an optimal power exists that maximizes the tradeoff between the saturating fluorescence signal and the linearly (with P,,) increasing background signal. For fixed spot size, the S N R at first improves, because the signal increases linearly with laser power and the shot noise from the power-dependent ternis in the denominator only grow as the square root of the laser power. As saturation sets in, however, the SNR falls because the signal no longer increases. Another relationship illustrated in Fig. 8 is that the best-case SNR at smaller and smaller
FIG. 8. Signal-to-noise ratio for fluorescence excitation of pentacene in p-terphcnyl versus probing laser power and laser beam cross-sectional area. (After Moerner and Baschk, 1993h).
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OPTICS IN SOLIDS
2 1S
beam areas levels off (the flattening of the ridge at small beam area). This is due to the effect of saturation and shot noise-at smaller and smaller areas the power must be reduced eventually to the point where the SNR is controlled by the shot noise of the detected signal (first term in the denominator of Eq. 4). in any case, SNR values on the order of 20 with 1 s averaging time are quite useful for spectroscopic studies. However, more information about dynamical effects can be obtained if the SNR is increased further, which is one continuing challenge to the experimenters in this field. From another point of view, improvements in the SNR would allow probing at lower laser power so that materials with non-optimal photophysics (such as higher hole-burning quantum efficiency) may be studied.
c. EXAMPLES OF EXPERIMENTAL MEASUREMENTS I. Single-Molecule Lineshupr To provide an example of specific experimental spectra using the fluorescence excitation method, we again turn to the model system of pentacene in p-terphenyl for simplicity. Figure 9 shows fluorescence excitation spectra at 1.5 K for a 10 p m thick sublimed crystal of p-terphenyl doped with pentacene using the lensparaboloid setup (Moerner, 1994a). The 18 GHz spectrum in Fig. 9(a) (obtained by scanning a 3 MHz line-width dye laser over the entire inhomogeneous line) contains 20,000 points; to show all the fine structure usually requires several meters of linear space. The structures appearing to be spikes are not noise; all features shown are static and repeatable. Near the center of the inhomogeneous line, the statistical fine structure (SFS) characteristic of N > I is observed. It is immediately obvious that the inhomogeneous line is far from Gaussian in shape and that there are tails extending out many standard deviations from the center both to the red and to the blue. Figure 9(b) shows an expanded region in the wing of the line. Each of the narrow peaks is the absorption profile of a single molecule. The peak heights vary due to the fact that the laser transverse intensity profile is bell-shaped and the molecules are not always located at the center of the laser focal spot. Even though these spectra seem narrow, they are in fact slightly power-broadened by the probing laser. Upon close examination of an individual single-molecule peak at lower intensity [Fig. 9(c)], the lifetime-limited homogeneous line-width of 7.8 t 0.2 MHz can be observed (Moerner and Ambrose, 1991). This line-width is also termed “quantum-limited” because the optical line-width has reached the minimum value allowed by the lifetime of the optical excited state. This value is in excellent agreement with previous photon echo measurements using large ensembles of pentacene molecules (de Vries and Wiersma, 1979; Patterson et cil., 1984). Wellisolated, narrow single-molecule spectra such as those in Fig. 9 are wonderful for the spectroscopist: Many detailed spectroscopic studies of the local environment
216
W E. Moerner; R. M.Dickson, and D. J. Norris 15000
10000
5000 A
.o
Y
MHz
FIG. 9. Fluorescence excitation spectra for pentacene in p-terphenyl at I .S K measured with a tunable dye laser of line-width 3 MHz. The laser dctuning frequency is referenced to the line center at 592.32 1 nm. (a) Broad scan of the inhomogeneously broadened line; all the sharp features are repeatable sfructures. (h) Expansion of 2 GHz spectral range showing several single molecules. (c) Lowpower scan of a single molecule at S92.407 nin showing the IiCetime-limited width of 7.8 MHz and a Lorentzian fit. (Reprinted with permission from Moerner, I994a. Copyright 1994 American Asaociation for the Advancement of Science).
can be performed, because such narrow lines are much more sensitive to local perturbations than are broad spectral features. It is instructive at tlus point to compare the signal-to-noiseratio for SFS ( N >> 1) to that for one single molecule (Q. 4). Defining N,, as the number of molecules with resonance frequency within one homogeneous width of the laser frequency, and recalling that the SFS signal excursions scale as the square root of the number of molecules in resonance (see Fig. 3),
where the last approximation assumes that background and dark counts are negligible. In this limit, the SNR is independent of the number of molecules in resonance! Therefore, when detection of SFS has been accomplished, the SNR at that point is a good estimate for the SNR for one single molecule, which degrades only when large background and dark counts are present. It is also interesting that the SNR for SFS scales as the inverse square root of the beam area. Of course,
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OmICS IN SOLIDS
2 I7
since fluorescence excitation is not a zero-background method, the SFS signal is still a small signal with a relritivr size I N % , which must be detected on a large background. Thus laser amplitude drifts and low-frequency noise must be minimized in order to see the fine structure. 11 is precisely this last point that stimulated the original SFS experimenters (Moerner and Carter, 1987) to utilize FM spectroscopy for SFS.
2. Single-Molrcwle lmtrging in Frequency mcl Sprite With the ability to record high-quality single-molecule absorption line-shapes such as those in Fig. 9. it becomes useful to acquire spectra as a function of the position of the laser focal spot in the sample and to obtain microscopic images in two spatial dimensions. A single molecule should be localized in space as well as in resonance frequency. Space-frequency “pseudo-images’’ of single molecules of pentacene in p-terphenyl were obtained by scanning a laser beam across the sample in one dimension and recording fluorescence excitation spectra at each position (Anibrose and Moerner, 1991).The resolution of such images in the spatial dimension is clearly limited by the several pm-diameter laser spot; in fact, the single molecule is actually serving as a highly localized nanoprobe of the laser beam diameter itself. However, in the frequency dimension the features are fully resolved (with typical cw laser line-widths of -3 MHz). The extension of this concept to nlicroscopy using two spatial dimensions was first reported in 1994 (Giittler el id., 1994; Jasny e t a / . , 1996). Microscopic imaging has the advantage of following many diffraction elements of the sample at the same time, which can help in finding single molecules when the concentration is low. Using a microscope objective immersed in the superfluid helium and a highquantum efficiency Si CCD detector, we have obtained microscopic images of a single tenylene molecule in a p-terphenyl crystal at 1.5 K as shown in Fig. 10.
FIG. 10. Microscopic image of the eiiiissioii lkoni a \inglc molecule of [eriylcne i n p-terphenyl at I .5 K . The measured fluorescence excitation signal (:-axis) collccted by ii liquid nitrogen coolcd CCD detector during ii I s integration is shown a$ a function of t - \ position ovet-:i t-itiig~60 piii X 65 pin.The peak rcpresents 7S.ON) detected photons. Lam wavclength = 578.478 nim Spatial rceaolution = I .8 pm.
218
W E. Moerner; R. M. Dickson, and D. J. Norris
The z-axis is the detected long-wavelength emission, and the spatial range in the x and y dimensions is 65 and 60 pm, respectively. The strong single molecule is
clearly evident as a large, isolated mountain, and the dim ridges in the figure result from cracks in the sublimed sample. 3. Spectral Shifring: Trajectories, Correlation, und Line-shape Distributions
When a new regime is first opened for study, new physical effects can often be observed. In the course of the early SMS studies of pentacene in p-terphenyl, an unexpected phenomenon appeared: resonance frequency shifts of individual pentacene molecules in a crystal at 1.5 K (Ambrose and Moerner, 1991), called “spectral diffusion” by analogy to similar shifting behavior long postulated for amorphous systems (Friedrich and Haarer, 1986). Here, spectral diffusion means changes in the center (resonance) frequency of the guest molecule due to configurational changes in the nearby host that affect the frequency of the electronic transition via guest-host coupling. In the pentacene in p-terphenyl system, two distinct classes of single molecules were identified: class I, which have center frequencies that are stable in time, and class 11, which showed spontaneous, discontinuous jumps in resonance frequency of 20-60 MHz on a 1-420 s time scale. In the ensuing work, it has become clear that spectral shifts of single-molecule lineshapes are common in many systems, including crystals, polymers, and even polycrystalline Shpol’slui matrices (Moerner et al., 1994a; Plakhotnik et al., 1994).Spectral diffusion effects have been described in detail in several locations (Brown and Orrit, 1997; Skinner, 1997); here only the principal experimental methods for studying this behavior will be briefly mentioned. Again talung pentacene in p-terphenyl as an example, Fig. 1l(a) shows a sequence of fluorescence excitation spectra of a single molecule taken as fast as allowed by the available SNR. The laser was scanned once every 2.5 s with 0.25 s between scans, and the hopping of this molecule from one resonance frequency to another from time to time is clearly evident. One useful method for studying such behavior is the measurement of the spectral trujectory o,,(t) (Ambrose and Moerner, 1991). By sequentially acquiring hundreds to thousands of fluorescence excitation spectra and utilizing the power of digital processing to retain a record of the resonance frequency position of each such spectrum, a trajectory or trend of the resonance frequency versus time w,,(t) can be obtained as shown in Fig. ll(b) [for the same molecule as Fig. 1 l(a)]. For this molecule, the optical transition energy appears to have a preferred set of values and performs spectral jumps between these values that are discontinuous on the 2.5 s time scale of the measurement. The behavior of another molecule is shown in Fig. ll(c) at 1.5 K and in Fig. ll(d) at 4.0 K. This molecule wanders in frequency space with many smaller jumps, and both the rate and range of spectral diffusion increase with temperature, suggesting a phonon-driven process.
SINGLE-MOLECULE SPECTROSCOPYAND QUANTUM OPTICS IN SOLIDS
2 19
Laser deruning (MHz)
1
-400 0.0
100.0
200.0
300.0
400.0
500.0
0
2
P
-400 0.0
500.0
1000.0
500.0
1000.0
1500.0
200 0
-200
'
-400 0.0
2000.0
I
I
1500.0
20000
Time (s)
FIG. I 1. Examples of single-molecule spectral dilfusioii for pentacene i n p-terphenyl at 1 .S K . (a) A series of fluorescence excitation spectra cach 2.5 s long spaced by 0.25 s showing discontiniious shifts in resonance frequency. with zero detuning = 592.546 tiin. (b) Trend or trajectory of [he resonance 1-equency over a long time scale for the rnolecule in (a). ( c ) Rchonance frequency trcnd for a different molecule at 592.582 nni at 1.5 K and at (d) 4.0 K. (Reprinted with permission liom Moerner. 19941. Copyright 1994 American Association lor the Advancement of Science).
The first question that should be asked when such behavior is observed is this: Is the effect spontaneous, occurring even in the absence of the probing laser radiation, or is it light-dnven, that is, produced by the probing laser itself! To answer this, it is usually necessary to observe the spectral shifting behavior as a function of the probing laser power. Ln the case of the type I1 pentacene molecules in p-terphenyl, the spectral diffusion appeared to be a spontaneous process rather than a light-induced spectral hole-burning effect (Moerner and Ambrose, 1991). but other materials have shown light-induced shifting behavior (Bascht and Moemer, 1992; Plakhotmk er LII., 1994; Tchtnio et al., 19934, which may be regarded as the single-molecule analog of the nonphotochemical spectral hole-burning process (Hayes et ul., 1981). Because the optical absorption for pentacene in p-terphenyl is highly polarized (Giittler et al., 1993) and the peak signal from the molecule does not decrease
220
W E. Moernel; R. M . Dickson, and D.J . Norris
when the spectral jumps occur, it is unlikely that the molecule is changing orientation in the lattice. As the resonance frequency of a single molecule in a solid is extremely sensitive to the local strain field, the conclusion from these observations is that the spectral jumps are due to internal dynamics of some configurational degrees of freedom in the surrounding lattice. The situation is analogous to that for amorphous systems, which are postulated to contain a multiplicity of local configurations that can be modeled by a collection of double-well potentials (the two-level system or TLS model) (Phillips, 1981j. The dynamics result from phonon-assisted tunnelling or thermally activated barrier crossing in these potential wells. One possible source for the tunnelling states (Ambrose rf al., 1991) could be discrete torsional librations of the central phenyl ring of the nearby p-terphenyl molecules about the molecular axis. The p-terphenyl molecules in a domain wall between two twins or near lattice defects may have lowered barriers to such central-ring tunneling motions. Thus, the spectral shifting for type I1 molecules of pentacene in p-terphenyl is a result of disorder, which is consistent with the experimental observation that all the type I1 molecules are located in the wings of the inhomogeneously broadened line. Spectral trajectories contain much information about the stochastic behavior of the single molecule. If a simple measure of the average time scale of spectral shifts is required, it is useful to calculate the autocorrelation of the spectral trajectory, CJt), given by CJT) =
j
w,,(t)%(t
+ 7)df
(10)
which is the Fourier transform of the power spectral density of the frequency fluctuations. More complex measures can also be computed, such as the survival probability for the resonance frequency to stay at its initial value (Reilly and Skinner, 1993, 1995; Skinner, 1997). Although the measurement of the spectral trajectory in principle contains all the dynamical information about the system, there are practical limitations to the time scale for which information can be obtained. The principal shortcoming of the spectral trajectory measurement results from the time required to scan the absorption line with sufficient SNR to determine the resonance frequency. In many systems, this minimum scanning time is limited by the photon emission rate to times on the order of 10-100 ps, which means that dynamical behavior on faster time scales is not adequately represented. A partial solution to this problem is provided by direct time correlation measurements on the emission signal itself (Brown and Orrit, 1997). Nevertheless, the direct observations of the dynamics of a nanoenvironment of a single molecule by the spectral trajectory method have sparked fascinating new theoretical studies of the underlying microscopic mechanism (Reilly and Skinner, 1993, 1995; Skinner, 1997; Zumofen and Klafter, 1994). It is worth noting that such spectral trajectories cannot be obtained when a large ensemble of molecules is in resonance. The individual jumps are gener-
SINGLE-MOLECULESPECTROSCOPY AND QUANTUM OPTICS IN SOLIDS
22 1
ally uncorrelated, thus the behavior of an ensemble-averaged quantity such as a spectral hole would only be a broadening and smearing of the line. A final manifestation of spectral shifting is the appearance of a measured linewidth for the single-molecule absorption that vanes from molecule to molecule and, in addition, depends on averaging time (BaschC and Moerner, 1992). This is a result of spectral shifting of the line position on a time scale much faster than the time required to scan over the absorption. To characterize such behavior, experimenters have generally measured a distribution of line-widths as a histogram (Fleury ef nl., 1993). This distribution can be compared with the prediction of a theoretical model for the dynamics. It is clear that spectral shifting behavior in amorphous hosts is so complex that all three methods (trajectories, correlation, and line-width distributions) are often necessary to obtain the clearest physical picture of the phenomenon. I n the rest of this article, however, we concentrate on the relatively stable single molecules found in well-ordered crystalline hosts, as stability of the absorption frequency is generally required for the observation of quantum optical phenomena.
IV. Quantum Optics While many interesting quantum optical effects had been predicted for individual quantum systems and had been observed in atomic beams and trapped ions, it was not obvious that such measurements would be possible in single niolecule systems complicated by molecular vibrations, phonons, and interactions with the host matrix. In this section we review progress in addressing this issue-the possibility and potential of single molecule quantum optics.
A.
PHOTON BUNCHING AND ANTIBUNCHING
One property that clearly exhibits the inherent quantum nature of a single molecule is its emitted photon stream. The photons emitted from a single molecule contain information about the emission process encoded in the photon arrival times. Experimentally, such information can be obtained by fixing the pumping laser frequency in resonance with ;1 single molecule and counting photons. While the spectral trajectories discussed above reveal single-molecule dynamics on a relatively long time scale, the photon stream can reveal much shorter time scale information. However, even in this “short time” domain, two distinct behaviors are expected for a typical single molecule-photon hiinching and photon anfibunching. Figure 12(a) shows this behavior schematically for a single molecule with a dark triplet state, here taken to be pentacene (see Fig. 1). While cycling through the singlet states S,, --;r S,+ S,,,photons are emitted until intersystem crossing occurs. Because the triplet yield is 0.5% (de Vries and Wiersma, 1979), on aver-
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I I I 1 1 1 I i i I I 1 1 1 1 1 1 I I I 1 1 I I I (_II I 1 1 I
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T i m e (ns) FIG. 12. (a) Schematic o f the temporal behavior of photon emission from a single molecule showing bunching on the scale of the triplet lifetime (upper half) and antibunching on the scale of the inverse of the Rabi frequency (lower half). (Reprinted with permission from Moerner, 1994a. Copyright 1994 American Association for the Advancement of Science). Panels (b) and (c) demonstrate photon antibunching for a single pentacene molecule with a Rabi frequency (x)of I 1.2 and 68.9 MHz, respectively. The solid lines are fits to the data with ,y = (b) 1-10 MHz and (c) 71.3 MHz. (Data from Basch6 rt a/., 1992b).
age 200 photons are emitted before a dark period occurs with an average length equal to the triplet lifetime, 7r. This causes bunching of the emitted photons as shown in the upper half of Fig. 12(a). Even though the actual length of time in the triplet state is a random variable with an exponential distribution, the bunching can be easily detected by measuring the autocorrelation of the photon emission signal S(t), which is defined by CS(7)=
J S(t)S(t + 7)dt
(11)
Autocorrelation analysis has long been recognized as a useful method for statistical study of stochastic dynamical processes which may be obscured by noise (Elson
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and Magde, 1974; Magde rt al., 1974). By definition, the autocorrelation measures the similarity (overlap) between the function and a copy of itself delayed in time by T. Because the noise component of the signal is uncorrelated, its contribution to Cs(7)decays quickly, leaving information about the average time-domam correlations of S(t). In practice, when photon counting is used, a commercial digital correlator is employed to measure CY(7)by analyzing the arrival times of many photon pairs. With modem correlators a logarithmic scale covering many decades in time can be obtained, a feature that is very useful for studying the dispersive dynamics characteristic of amorphous systems. Note, however, that while correlation measurements can extract information about the single molecule on much shorter time scales (down to the millisecond range) than can the spectral trajectories described above, the dynamical process must be stationary. The dynamics must not change during the relatively long time (many seconds) needed to record enough photon arrivals to generate a valid autocorrelation. In addition, because the laser frequency is held fixed, when the molecule hops out of resonance with the laser, all information about the new resonance frequency of the single molecule is lost. The first observation of photon bunching in single molecule emission was reported by Orrit and Bernard (1990). who measured the decay in the autocorrelation of the emitted photons for pentacene in p-terphenyl. Differences from molecule to molecule in the triplet yield and triplet lifetime (Bernard et al., 1993) were observed with this method. Such changes occur due to distortions of the molecule by the local nanoenvironment. In the nanosecond time regime the emitted photons from a single molecule can provide still more useful information. On the time scale of the excited state lifetime, the statistics of photon emission from a single quantum system are expected to show photon antibunching (Carmichael and Walls, 1976). As shown in the lower half of Fig. 12(a), the photons “space themselves out in time”, that is, the probability that two photons arrive at the detector at the same time is small. This uniquely quantum-mechanical effect, first observed for sodium atoms in a lowdensity beam (Kimble er a/., 1977). is measured by computing the second-order correlation, g‘”(r). The function g‘”(r), which is simply the normalized form of the intensity-intensity correlation function C.s(~), shows a drop below the uncorrelated value of unity when antibunching is present (Loudon, 1983). For a single molecule, photon emission implies that the system has returned to its ground state and cannot immediately emit a second photon. Therefore, antibunching can be interpreted as the time that must elapse before the probability to emit a second photon is appreciable. This time is on the order of the inverse of the Rabi frequency, x-’.In fact, at sufficiently high laser intensity, Rabi oscillations should be observed as the laser coherently drives the single molecule into and out of the excited state before emission occurs. To overcome limitations caused by the dead time of photomultipliers, in practice two identical detectors are used as in the classical Hanbury Brown-Twiss experiment
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(Loudon, 1983). Such an experiment measures N(T),the distribution of time delays between the arrival of consecutive pairs of emitted photons. By measuring N(T), which is directly related to &t), the expected antibunching in single-moleculeemission was first observed for pentacene in p-terphenyl at the IBM Atmaden Research Center (Basch6 et al., 1992b). The obtained N(T), shown in Figure 12(b), clearly shows a strong decrease in the photon correlation at zero delay. In addition, at high intensity the expected Rabi oscillations in the correlation function are observed [Fig. 12(c)]. From careful study of the shape of the correlation function in the nanosecond time regime both T , and T2for a single terrylene molecule in p-terphenyl have been determined (Kummer et al., 1995). Of course, if more than one molecule is emitting, both the bunchmg and antibunching effects quickly disappear since the various resonant molecules emit independently. The observation of high-contrast antibunching remains one of the strongest proofs that the observed spectral features are those of single molecules. The observation of antibunching also strongly demonstrates that quantum optics experiments can be performed for single molecules in a solid host, in spite of the complexity of a single molecule compared to a single ion. We note that the emitted antibunched light is multicolor, in that it has the spectrum of the emission from the molecule terminating on various vibrationally excited levels of the electronic ground state. The reason antibunching correlations are preserved stems from the extremely short (ps) lifetimes of these excited vibrational levels due to vibrational relaxation. B. QUANTUM JUMPS While the influence of the triplet state on single molecule emission is observed indirectly in the autocorrelation function CS(7),a second type of quantum optical experiment can detect transitions into the triplet level directly. These experiments are based on a technique originally proposed by Dehmelt for single atoms (Dehmelt, 1975). He considered a three-level atomic system with a ground state and two excited levels in which one of the ground-state transitions is strong while the other is weak. By monitoring the emission from the allowed state, a transition into the long-lived dark state can be detected with near unit probability, even if this event is extremely rare. Such spectroscopic amplification occurs because the emission from the strong transition ceases while the system is shelved in the dark state. However, this method is more than just an extremely sensitive spectroscopic technique (Arecchi et al., 1986; Cook and kmble, 1985; Schenzle and Brewer, 1986; Schenzle et al., 1986). It also reveals the nature of quantum measurement. The experiment allows the detector (e.g., the human eye) to directly observe the quantumjump into (or out of) the dark state. Quantum jumps were first demonstrated ten years ago for single ions isolated in radio frequency traps (Bergquist et al., 1986; Nagourney et al., 1986; Sauter et al., 1986). More recently, quantum jumps have been observed in single terrylene mole-
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cules doped into a p-terphenyl crystal (Basche ~t nl., 1995). Figure 13 shows single tenylene emission in which the “random telegraph” signal typical of quantum jump behavior is clearly evident. In the single molecule case, the metastable triplet level TI acts as the dark state (see Fig. 1). A quantum jump signifies intersystem crossing (emisbetween the singlet and triplet manifolds: S , 4 TI (emission stops) or T , -+S,, sion resumes). In contrast to the atomic case, thesejumps occur due to radiationless transitions. By determining intersystem crossing rates for terrylene in p-terphenyl (Kummer et ul., 1994), BaschC rt d.have shown that the data in Fig. 13 are consistent with the expected quantum jump statistics (Basche rt al., 1995). In a more general sense, single molecules have also been used to detect quantum jumps in the surrounding matrix (Moerner and Ambrose, 1991). As discussed above, the nearby environment of the molecule can influence its resonant frequency (see Fig. 1I). In the TLS model (Phillips, 1981) this behavior is explained by the interaction of the molecule with various configurational degrees of freedom in the matrix, each of which is modeled by a simple double-well potential.
FIG. 13. Fluorescence couiits detected for a single terrylciie molecule as a function of’ time. ‘l’hc sample h a s cooled LO I .4 K a n d excited ;it a h e r aavelength of 578.403 n i n wilh ail intensity o l 3 W/cin2. The Huorcscence counts were I-ecoi-dcdwith 0. I ins aarnpling intervals and no delay hetween wcces\ivc‘ hina. (lowel-)A 400 ma section o l the fill1 I .04 s data b e t . (upper)A 38 ins region c x l ~ m d e d fui-lhcr tu show individual quantum jumps. (Reprinted with periiiissioii from Baschi- or (//.. 1995. Copyright 1995 Macniillan Magazines. 1-td.).
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In several single molecule systems [perylene in polyethylene (Bascht et al., 1992a), terrylene in polyethylene (Fleury et al., 1993; Zumbusch et al., 1993), and terrylene in hexadecane (Moerner et al., 1994a)l an extreme case of this model has been observed in which the molecule is coupled to only a single TLS. The molecule then changes reversibly between only two resonant frequencies. Therefore, by observing the single molecule emission with fixed excitation laser frequency, a random telegraph signal denotes quantum jumps in the TLS. Although the exact origin of the TLS is unknown, a quantum jump in the combined molecule/matrix system is nonetheless observed. (AC) STARK EFFECT C. OPTICAL The effects discussed so far can be characterized as “linear” quantum optics. With the exception of Fig. 12(c), where Rabi oscillations are observed at high excitation power, the experiments are performed with weak optical fields. When strong fields are applied to single molecules, “nonlinear” effects may appear. For example, in simple two-level atomic systems the strong field can induce an AC Stark effect on the optical transition, which causes line-width broadening and a frequency shift (Shen, 1984). At sufficiently high power, splittings in the atomic optical spectrum are also predicted (Mollow, 1969, 1972, 1975) and have been observed in both resonant emission as the Mollow triplet (Grove et al., 1977; Schuda et al., 1974; Walther, 1975; Wu et al., 1975) and in absorption as the Autler-Townes effect (Autler and Townes, 1955; Wu et al., 1977). This behavior occurs because a strong photon field is no longer a small perturbation to the atomic level structure. Solutions to the entire quantum system must then be considered. Physically, this leads to a picture where the atom is “dressed” by the photon field (Jaynes and Cummings, 1963; Meystre and Sargent 111, 1991). The AC Stark effect then results from the strong coupling between the atom and field. In order to observe these effects in a single molecule system, the molecule must be extremely stable even under strong pumping conditions. This requires the absence of photo-induced decomposition as well as spectral diffusion. In addition, the triplet yield should be sufficiently small so that the molecule effectively acts as a two-level system. These requirements have been satisfied in several single molecule systems. Thus, the simplest nonlinear effect, saturation broadening of the molecular linewidth, has been observed for a number of single molecules, including pentacene in p-terphenyl (Ambrose et al., 199l), terrylene in p-terphenyl (Kummer et al., 1994), DBATT in n-hexadecane (Boiron et al., 1996), and DBT in naphthalene (Jelezko et al., 1996). These systems all show promise for more complicated quantum optical experiments. For terrylene in p-terphenyl, the field-induced shift of the molecular frequency has also been measured directly with a pump-probe experiment. Utilizing the high spectral stability of the X , site in this system, Tamarat et al. (1995) observed
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a linear light shift of the terrylene transition as a function of the intensity of a strong pump field, as shown in Fig. 14. In this experiment the pump frequency is detuned from the single molecule transition while a weak probe beam is scanned over the single molecule resonance. The light shift was found to be linear in the inverse pump detuning, as expected from theory. The experimenters also performed the same measurement with the pump directly in resonance with the molecule. In this case Autler-Townes-like structures were observed due to the spiitting of the molecular transition by the strong pump field. The splitting was shown to be roughly equal to twice the pump Rabi frequency, again consistent with theory.
D. SINGLE-MOLECULE CAVITY QUANTUM ELECTRODYNAMICS The success of these single molecule measurements also indicates that more complicated quantum optical experiments may be possible. One potential area to investigate is the interaction between a single molecule and an optical cavity. In this case cavity quantum electrodynamics (QED) (Berman, 1994) predicts that the molecule will exhibit novel optical behavior due to the modification of the vacuum field inside the cavity. One simple consequence is a change in the spontaneous emission rate of the single molecule, which can be strongly enhanced (or
150,
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i/I, FIG. 14. Light shift of the zero-phonon line ol'a single terrylene molecule i n ;I p-tcrphcnyl crystal at I.8 K. The shift varies linearly with the intensity of the near-resoiiitnt pump beam, dctuned by 300 MH7 to the blue trom the molecular line. The inset shows (for a different molecule a1 I .4 K ) the unshilted molecular line (a) recorded with the probe heam alone, and the light-shifted line ( h ) with the pump detuned by 190 MHz to the blue side of the line. (After Tamamt P I a/., 1995).
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W. E. Moerner, R. M. Dickson, and D. J. Norris
inhibited) if the molecular transition is in (or out of) resonance with a low loss mode of the cavity. Purcell predicted in 1946 (Purcell, 1946) that the enhancement of the spontaneous emission rate for a two-level system in resonance with the cavity is proportional to QA’/V, where Q is the quality factor of the cavity, V is its modal volume, and A is the transition wavelength. Therefore, by coupling a single molecule to a high-Q optical microcavity, a significant enhancement of the emission rate is expected. The enhancement of spontaneous emission along with many other predictions of cavity QED was first verified in atomic beams (for reviews see Berman, 1994; Haroche and Kleppner, 1989). Enhancement has also been demonstrated at room temperature for ensembles of molecules (Barnes et a / . , 1996; De Martini et al., 1987; Yokoyama etal., 1991) and ions (Lin et al., 1992). While the earliest experiments utilized Fabry-Perot microcavities, more recently the resonances of small spherical cavities have been used either in the form of liquid droplets or dielectric spheres. Spherical cavities are particularly convenient in that they form highQ resonators with low modal volume, they are easy to fabricate, and their optical properties are well known (Barber and Chang, 1988; Chang and Campillo, 1996). They could provide a convenient cavity for single molecule investigations. The prospect of performing such experiments on a single molecule is extremely interesting in that it extends cavity QED measurements to a single quantum system embedded in a solid. The single molecules described above are particularly suited to such experiments since they are extremely photostable and have high Q, typically lo7 (see Table I). The Purcell formula predicts that a microcavity with Q comparable to the molecule might lead to significant enhancement effects. However, the potential also exists to observe many other novel properties. In the limit the Purcell formula no longer applies. The molecule enters the Q,,,,,,, >> QNIOICLllle “strong coupling” regime and new behavior should arise (Goldstein and Meystre, 1995). The molecule has the possibility to interact with its own emitted photon and a single molecule microlaser may be observed (An et al., 1994). Thus, the potential in this area is very strong and several groups are actively pursuing such experiments.
V. Problems and Promise for Room Temperature A. EXTENSION OF SINGLE-MOLECULE STUDIES TO AMBIENT TEMPERATURES Recently, room temperature detection of single molecules has also become possible (for a review, see Xie, 1996). In contrast to low-temperature SMS, roomtemperature (RT) single-molecule measurements are ultimately limited by photochemical bleaching. While photodecomposition has not been observed at low temperature for the rigid aromatics even after more than 10” photons have been
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emitted (although spectral shifts can occur), a typical single molecule studied at RT emits only -10' photons before decomposing. The exact mechanism that causes the RT bleaching is not known, but it presumably involves molecular oxygen immobilized in solids at liquid helium temperatures. The end result is that many fewer photons are available for RT single molecule studies. Despite this limitation, room-temperature single molecule studies are beginning to yield information about the molecules and their environments. Because electronic dephasing times are very short at room temperature, absorption profiles are very broad and, in contrast to low-temperature studies, the technique's environmental sensitivity is limited. The single molecule has a significant homogenous line-width ( -yH) and only large-scale changes in the nanoenvironment can be monitored. Large y,, also implies that the molecule must be excited with higher laser intensity, thus necessitating extremely efficient background reduction methodologies such as confocal and total internal reflection microscopies. Through applications of these techniques, several authors have observed gross spectral shifts in the emission of single dye molecules immobilized on glass surfaces as shown in Fig. 15. Interpretations in specific cases have ranged from effects such as proximity to a polymedair or polymer/glass surface, dipolar reorientation, or electron transfer to an indium tin oxide electrode (Trautman and MacMin, 1996; Trautman et al., 1994; Xie, 1996; Xie and Dunn, 1994). This
FK;. IS. Emission spectra of a single sullolhodamine I01 molecule on ii silica substrate taken scquentially with 532-nin excitation at 170 nis intervah. Sigiiilicant spectral shifts are evident during the c o u w o l nieawreiixnt. (Reprinted with permission From Lu and Xie i1997). Copyright 1997 Macini 1Ian Mapazi nes. Ltd. 1.
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active research area has focused on the spectral and temporal manifestations of the molecule-environment interaction. Although several such systems are well understood at low temperatures, the mechanisms of large-scale spectral shifts at RT remain to be elucidated. Room-temperature methods do have several advantages, however, over their low-temperature counterparts. They require less complexity and allow observation of three-dimensional motion in real space, as shown in Fig. 16. (Dickson et al., 1996). Initial extensions of these techniques to biophysical and analytical studies have already been performed, yielding some intriguing results (Fundtsu et al., 1995; Vale et al., 1996). When coupled with the increased need for background reduction in room-temperature experiments, bleaching has thus far imposed the most severe limitations in observing quantum optical behavior. More efficient signal enhancement techniques need to be developed.
FIG. 16. (a) SO pni X 50 p m image of individual nile red molecules moving in an I8 percent polyacrylamnide gel at rooin temperature. Molecular motion in three dimensions could he followed in time as a result of the evanescent lield excitation resulting from total internal reflection (see Dickson et d , 1996). A molecular spatial trajectory generated from IS successive I-second exposures at 0.4 Hz is shown in (b), while a different molecular trajectory is plotted in (c) with 100 msec exposures at I .S Hz.
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B. BEGINNINGS OF ROOM-TEMPERATURE SINGLE-MOLECULE QUANTUM OPTICS Low-temperature studies have provided a strong foundation on which roomtemperature experiments are being built. Analogous to the low-temperature twophoton fluorescence spectra for DPOT in hexadecane (Plakhotnik et ul., 1996), Webb and coworkers have observed two-photon single molecule signals at room temperature (Mertz el al., 1995). This technique has the great advantage of efficient background reduction resulting from the laser frequency being very different from the molecular absorption and emission. The added benefits of enhanced spatial resolution resulting from the non-linear nature of the excitation and reduced photochemical decomposition of out-of-focus molecules significantly increases the appeal of this technique. Although such nonlinear optical experiments have been successful, only very recently have the beginnings of room temperature quantum optical experiments been performed on a single molecule. An elegant set of experiments has been performed over the past 10 years by studying molecules in levitated microdroplets (Barnes et al., 1995). Enhancements of molecular fluorescence and energy transfer efficiencies have been observed relative to those in bulk solutions of the same concentrations, and the source attributed to QED effects (Arnold and Fohn, 1989; Barnes etcil., 1994; Fohn et ul.. 1985). These techniques have been refined to the point of being able to observe single molecules in microdroplets (Barnes et al., 1993), but so far, quantum optical effects in droplets have only been reported on ensembles of molecules. Interestingly, using a parallel plate microcavity, subPoissonian light was recently reported for a single Oxazine-720 molecule in an ethylene glycol solution (De Martini rt d., 1996). In a very different set of experiments, Ambrose, et a1. studied multiple molecules on surfaces and added their signals to observe photon antibunching of a series of individual molecules at room temperature (Ambrose et al., 1997). Individual molecular antibunching signals were obtained until the molecule bleached, then a new molecule was selected and averaged with the others. This process was perfomled on about 20 molecules to generate the data in Fig. 17. Although in essence a bulk measurement, a clear antibunching signature was observed.
C. SUMMARY A N D PROSPECTS FOR THE FUTURE Although the field of single molecule spectroscopy is quite young, many fascinating experiments have already been performed. True quantum optical effects on single molecules continue to be observed at low temperatures, which should lead to the possibility of future applications and further insights into the physical nature of the interaction between the molecule and the radiation field. Roomtemperature single molecule quantum optics is still in its infancy, needing signal
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FIG. 17. Multiple-individual-molecule-averaged photon pair distributions. Data were obtained for (a) higher coverage and (b) lower coverage surfaces. (a) is offset by 50 counts. The sampling interval was (47.3 ? 0.1) ps, which was later convolved to 236 ps. The solid lines show model calculations using photophysical parametem obtained from saturation data. (Reprinted from Ambrose et d., 1997 with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.).
enhancement techniques or more robust systems to be developed for further advancement. With the large body of single moleculehon and quantum optical experiments being continually expanded, prospects for room-temperature single molecule quantum optics look reasonably good. The appealing fact that these experiments represent the ability to work with a single molecule as a very tiny quantum mechanical source of light with dimensions on the order of 0.1 nm on a side should continue to stimulate further research.
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Elson. E. L.. and Mag&, D. (1974). Biopolynzers 13, 1.
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Index A
Autoionization, 146 Autoionization rate, stepwise decay of, 21 Autoionization spectroscopy, 149 Autler-Townes effect, 226, 227
Aarhus STorage RIng Denmark (ASTRID), 156, 175 AC Stark shifts, 3, 4, 25, 33 Adams, A., 98,99, 100 Adiabaticldiabatic ionization behavior, 132 Adiabatic regime, 142 Adiabatic stabilization, 146, 148 Alkali atoms excited states of, 94 ground states of, 93 Alkaline-earth ions, 162-64 Arnbrose, W. P., 231 Amplitude controlling, 34
B Bachelet, G. B., 94 Bachman, R. A,, 92 Barron, L. D., 41,42,47, 48 BaschC, T., 225 Bauer, H., 106 Bayfield, J. E., 139 Beau, M., 104 Beijersberger, J. H. M., 96 Beiting, E. J., 14.3 Bernard, J., 223 Biniolecules, origin of homochirality of, 41 Black-body radiation
field, 121
Amplitude filtering, 34 Ancilotto. F., 110 Andersen, H. H., 163 Andersen, J. U., 169, 171 Andersen, L. H., 165, 174 Amdt, M., 96, 97, 1I1 Aspromallis, G., 163 ASTRID, 156, 175 Atkins, K. R., 91 Atkm’s snowball model, 92 Atomic bubble, 93 Atomic defects, 92-94 Atomic force microscopy, 194-95 Atomic negative ions, 175-82 Atomic optical spectrum, splittings in, 226 Atomic streak camera, 9-10, 18, 129 Atornic systems, 185-86 collisions between, 10 Atoms
influence of, I61 sources of in far infrared ionization, 143
Blau, L. M., 159 Blue states, 127, 131, 133, 134 Blum, K., 62, 69. 72, 73, 74 Bonner, W. A,, 42 Bound helical electron model, 70 Bound-state interferometry, 7-8 Bound superposition state, evolution of, 25 Branscomb, L. M., 155, 174 Breathing, of radial size of wavepacket oscillation, 14 Breit-Pauli operators, selection rules for, 162 Broad-band pulse, 34 Broers, B., 129 Bryant, H. C., 185 Bubble deformations, experimental evidence for, 116 Bubble model, 93. 94, 98 Bubble oscillator, potential for, 106 Buckminsterfullerene, 166 Bucksbaurn, P. H., 141 Burkhardt, C. E.. 143 Busalla. A,, 84
on helium clusters. 109-1 1
Rydberg. 121, 122-23, 123, 125 Attenuation, equations necessary for study of, 64 Attenuation experiments, 43, 62-66, 73-75 Attenuation symmetry, values for, 76 Autocorrelation analysis, 222 Autocorrelation of spectral trajectory, 220
237
INDEX
238 C
Campbell, D. M., 81, 82 Camphor, 81 Carbon clusters, lifetimes of small, 172 Charged defects, 90-92 Chibisov, M. I., 179, 180 Chiral effects classification of, 54-56 in electron-molecule collisions, 83 non-zero contribution to, 73 physical cause of, 66-71 i n reactions with non-chid molecules, 43 Chirality, 41 concept of, 40 definition of for moving ohjects, 46 false. 42-43. 44-49, 61-62 time-even, 55, 59 true and false, 42-43, 44-49 Chiral molecules, 58-6 1 spin-polarizcd electrons and. 42 Chiral object and mirror image, 41 and their symmetries, 44-49 C h i d properties, 40, 5 1-52 Chung, K. T., 159 Circular dichroism, 40, 60 electron, 58-61 Clark, C. W., 181 Clark, J. W., 93, 94 Classical correspondence, Rydberg wavepackets and, 12-21 Clementi, E., 172 Closed-cage clustered fullerene molecules, 166 Closed-shell molecules, elastic collisions with polarized electrons, 43 Closed-shell systems, with time-even chirality, 55 Cluster ions, 166-72 Clusters, 186-88 Cohen, M. H., 100 Coherence effects, in microwave ionization, 150 Coherent broad-band radiation pulses, generation of, 3 Coherent light pulses, shaping of, 34 Coherent transients, 198 Cole, M. W., 92 Collisions, between atoms, 10 Condensed helium, phases of, 89
Configuration coordinate model, of optical excitation, 106 Conover, C. W. S., 33 Continuum Multiple Scattering Model, 8 1 Cooke, W. E., 20, 149 Cooling, by radiation, 170 Cooper minimum, 28,29 bypassing, 14546 Co-propagating beams, 1I Correlation, 218-21 Cotton, 40
D Dahm, A. J., 111 Dance, D. E, 174 Dark wavepackets, 22-24 Davis, B. E, 159 DC-field ionization, 123-24, 126-30 de Boer, M. P.,150 Debye-Waller factor, 202 Decay probability, as function of impact parameter, 179 Dehnielt, H., 224 Delone, N. B., 144 Dennis, W. E., 101 Dephasing, 11, 12 Detachment cross-section, 182 resonances in, 183-85 DeToffol, G . , 94, 109 Dexter, D. L., 98 Diahatic ionization threshold, 137 Diatomic heteropolar molecule, with strong spin-axis coupling, 46 Dichroism, electron, 82, 83 Differential equations, coupled, 72 Dipole matrix elements, 145 Direct absorption, 209-12 Direct laser ablation, 97, 102 Dissociative excitation processes, 174 Dissociative recombination, 174 Distorted-wave theory, 180 Dolder, K. T., 181 Donszelmann, A., 133 Doping, of large helium clusters, 95 Doubly-charged negative atomic ions, 181-82 Doubly-peaked effective pulse, 29 Dupont-Roc, J., 92,94 Dupont-Roc’s rule of thumb, 93
INDEX
E Effective power spectrum, 29 Eigenstate, pin, 46 Elastic collisions, 42, 43 Elastic scattering, from spinless atoms in their ground state, 52 Electric dipole moments, 116-17 Electromagnetic fields, novel to excite novel Rydberg wavepackets, 34 Electromagnetic interactions, 50 Electron binding, 186 Electron bubbles, 90-92,95,97-101, 111-12 spectroscopy to study structure of, 98 Electron circular dichroism, 9 - 6 1 Electron collisions. with oriented diatomic hetero-nuclear molecule, 53 Electron detachment, 169, 182 Electron dichroism, 82, 83 Electron ejection dynamics, of rubidium Rydberg wavepacket, 18 Electron-electron Coulomb interaction, 19 Electron-impact detachment, from negative ions. 172-85 Electron impact detachment measurement, 174 Electrons cnergy of, IS cxcess. 90 excitation of inner, 146. 148-49 incident, 68-69. 176 longitudinally polarized, 41. 47 polarized colliding with spinless heteronuclear diatomic molecule, 48 spin-orbit interaction involving molecular: 69-70 unpolarized obtaining longitudinal polarization. 60 Electron scattering chiral effects with regard to, S4-% left and right-handed. 60 Electrostriction, 91 Emission spectra, 105 Enantiomers, 40 Enantiomorphism, 44,47 Energy level splittings, between Rydberg states, 33 Esaulov. V. A,, 179 Estberg. G . N., 159 Evans, M. W., 59 Evolutionary time scales, for Rydberg wavepackets, 5
239
Excitation from high-lying states, 28 of ladder systems, 3 lines of, 106
ofl'kxonance. 2 I spcctra of, 105 Excited state lifetimes, of matrix isolated atoms, 105 Excited states of nlkali atoms, 94
olheliurn, 101 Experimental results, 8 1-83
F Fabry-Perot microcavities, 228 False chirality, 42-43, 44-49, 61-62 Fandreyer, R., 64, 69, 74, 76 Fano resonance, 28 Farago, P. S.,4 I , 42.49.61, 8 1, 82 Far-tield fluorescence microscopy, 209 Far infrared dipole matrix elements. 144-45 Far infrared ionization, 125, 143-46 Far infrared radiation, ionization with, 15 I Fermi's golden rule, 106 Ferrell, R. A,, 92 Feshbach resonance, 186 Fiber parabola, 208 Field amplitude, 121 Fluorescence collection, contiguration for, 206-09 Fluorescence excitation spectroscopy, 212-15 FM-Stark approach, 210-212 Focusing, configurations for, 206-09 Forward scattering, structure of M-matrix for, 62-63 Fourier transformation of ionization signal, 26 0 1 photoabsorption spectrum, I8 Fowler, W. B., 98 Fractional revivals, IS Franck-Condon principle, 100, 105, 106, 202 Freeman, R. R., 128, 150 Frequency-modulation spectroscopy, 209- 12 Frequency spectrum, shaping of, 34 Fresncl, 40 Frey, M. T., 143 Fujisaki, A,, 96 Fullerene lifetimcs, 167-7 1
INDEX
240 G
Gadzuk, J. W., 181 Gallup, G. A., 70, 71 Galvez, E. I., 139 Gated fluorescence detector, 11 Gatzke, M., 138 Gaussian pulse, wavepacket spectrum created by, 29 GHz half-cycle pulses, wavepacket excited with, 33 Goldanskii, V. I., 41 Golov, A. I., 100 Gordon, E. B., 94, 101 Goreslavski, S. P., 145 Cough, T. E., 95 Greer, R., 69, 72, 74,76 Grimes, G. G., 98.99, 100 Ground states, of alkali and alkaline-earth atoms, 93 Guest-host interactions, 90 Guest-host materials systems, 204
H
I
Half-cycle pulses, 9, 15I Hanbury Brown-Twiss experinlent, 223-24 Hanson, L. G., 21 Hanstorp, D., 163 Hart-Davis, A,, 181 Hartmann, M., 109 Hartree-Fock approximation, 68, 162 Haugen, H. K., 163 Hayishi, S., 69, 81 HCPs, multiple polarized in different directions, 35 Head and tail, overlap of, 15 Heavy-ion storage ring, 156 Hegstrom, R. A., 70 Heidelberg group, 95, 109 Helical density, concept of, 70 Helium excited States of, 101 superfluidity of, 96 Helium ion, 165-66 Hermitian, 74 Herold, G. S., 112 Herrick, D. R., 181 Hettema, J. M., 137 Hickman, A. P., 93, 101, 107
Higgins, J., 110 Hiroike, K., 93, 98 Hobden, M. V., 43 Homochirality, origin of, 41 Hoogenraad, J. H., 28, 144 Huffman, D. R., 166 Hui, Q., 104 Huzinaga, S., 181 Hyperfine transitions, 114-15
I Implantation techniques, 95-97 Impulse luck model, 141 Impulsive momentum retrieval, 8-9 Impurity molecules, individual, 193 IMR method, used to study dynamics of wavepacket when HCP kicks a stationary Rydberg state, 32 Incident electron kinelic energy of. I76 spin-orbit interaction involving, 68-69 Inglis-Teller limit, 137 Inhomogeneous broadening, 197-201 Inner electrons, excitation of, 146, 1 4 8 4 9 Integer revival, times of, 15 Intense far-infrared free-electron laser, 143 Interference, 15 Ionization DC field, 123-24, 126-30 far infrared, 125, 143-46 four-photon. 26 by half-cycle pulses, 134-35 microwave, 124-25, 135-41, 150 multiphoton. 145-46 optical, 125-26 physical mechanism of, 12 1 ramped field, 124. 131-35 Rydberg, 123-26 THz, 125, 141-43 Ionization limit, 138 alkaline-earlh, 162-64 atomic negative, 175-82 Ions cluster, 166-72 negative, 91, 155, 158-85 positive, 90, 9 I Isolated core excitation (ICE), 21, 148
24 1
INDEX
J Jahn-Teller effect, 107 Johnston, C . , 48, 50, 56. 69. 75 Jones, R. R., 141 Jortner, J., 98, 100
K Kanorsky, S. I., 107, 108, 115 Kelvin, 40 Kessler, J., 52, 84 Keszthelyi, L., 41, 42 Kinetic energy, of incident electron. I76 Kinnin, M., 64 Kinoshita, T., 106 Koch, P. M., 139 Kohn, W., 94 Kozniin, V., 41 Kratschmer, W., 166 Kristensen, P., 134, 163 Kroto, H. W., 166 Kiirten, K. E., 93, 94 Kyoto team, 114
L La Bahn, R. W., 159 Laboratory fixed frame, 72 Lambropoulos, P., 21 Landau-Zener crossings, 132 Landau-Zener transition probability, 132 Landau-Zener transitions, 124-25, 138 Lane, N. F., 101 Lang, S.. 115 Lankhuijzen, G. M.. 150 Laser ablation, 96 Laser-induced fluorescence, 102, 104-07 Laser period, in ORM experiment, 11 Lens-parabola configuration, 207 Lenier, P. B., 99 Linuted residence time, of implanted species, 96 Line-shape distributions. 218-21 Line-shapes, calculating, 106 Liquid helium, atomic impurity in, 93 Liquid matrices, 98-100 Littman, M. G., 126, 127 Living organisms, 4 I Long-pulse regime, transition to from shortpulse regime. 142
Low-energy electron-molecule scattering, 84 Low-temperature spectra, study of in collision-free environment, 88
M Magnetic fields, influence of in the ring. I 60-6 1 Magnetic resonance spectroscopy, I 1 1-17 Mahon, C . R..137 Mason, S . F., 4 I Matrix effects, 115-16 Matrix isolation spectroscopy, 88 Matrix isolation technique, 88 Mayer, S . , 83 Metastable anions, 172 Mezhov-Deglin, L. P., 100 Microscopy, configurations for, 206-09 Microwave ionization. 124-25, 135-41. IS0 Miyakawa, T., 98 Mlynek, J., 112 Molecular linewidth, saturation broadening of, 226 Molecular negative ions, 165-66. 182-85 Molecules, 109 collisiotis with oriented. 84 on helium clusters, 109-1 I inleracttons wiih polarized clectrons. 41 non-chiral oriented. 48 oi-icnted,S3-57. 75-76 randomly oriented. 61. 76-81 w i t h tiinc-odd chirality, 61-62 Mollow triplet, 226 Mossbauer line, 198 Muller. H. G . , 21 Multi-channel quantum defect theory calculations. 18 Multilevel Quantum Defect Theory calculation, 130 Multiphoton ionization, 145-46 wavcpackets excited du1-ing.24-27 Multiphoton ionization yield, enhancing, 26 Multiphoton transitions, 139 Miinster group, 44, 81, 82, 83, 84
N Near-field optical studies, 195 Nebraska group, 8 1
INDEX
242 Necessary condition, 52 Negative chemical potential, 95 atoms with. 101-02 Negative ions, 91 atomic, 175-82 electron-impact detachment from, 172-85 interactions with photons, 185-88 lifetime studies of, 158-72 molecular, 165-66, 182-85 properties of, 155 Negative ion studies, 156 Neijzen, J. H. M., 133 Neutral atom count rate, versus time, 160 Nolting, C., 84 Non-chiral oriented molecule, 48 Nonradiative processes, 107-09 Noordam, L. D., 23, 144, 150 Northby, J. A,, 98
0 Observables, 56 experimental, 53-57, 58-66 Off-resonance excitation, 2 1 On-resonance ICE, 21 Optical activity, 58-61 Optical control, of photoprocesses in atoms and molecules, 33 Optical ionization, 125-26 Optical isomers, 40 Optical pulses, tailoring short, 34 Optical pumping, 112-13 Optical-pump photoionization-probe technique, 17 Optical radiation, 146-50 stabilization of Rydberg atoms with, 151 Optical Ramsey method ( O W ) , 7-8 Optical spectroscopy, 97-1 11 Optical (AC) Stark effect, 226-27 Orrit, M., 212, 223 Oscillation, of wavepacket period, 17-18 Ostrovsky, V. N., 174, 178, 180
P Paramagnetic atoms, 112-17 Parity invariance, interactions violating, 52 Parshm, A. Y., 99 Pasteur, Louis, 40
Pauli exclusion principle, 91, 11.5 Peak absorption cross-section, 201-02 Peart, B., 162, 174, 181 Pegg, D. J., 163 Pereversev, S. V., 99 Periodic ionization property, of Stark atoms, 17 Persson, J. L., 109 Perturbation theory, to obtain distortion of molecule due to incident electron, 71 Petrunin, V. V., 163 Phase, controlling, 34 Phonon antibunching, 221-24 Phonon bunching, 221-24 Phonon wings, 110 Photoabsorption spectrum, Fourier transforming, 18 Photodetachment, 186-88 Photoionization, studying wavepacket evolution via, 6 Photon echoes, 198 Photon energy, 121 Photon occupation number, 143 Photons in circularly polarized light beam, 46 interactions with negative ions, 185-88 Photon stream, emitted, 221 Photon transitions, 139 Pick-up cell, varying density of particles in, 110 Pillet, P., 137 Pindzola, M. S., 174, 180 Pin eigenstate, 46 Pinhole aperture, 208 Pitzer, K. S . , 172 Point defects, structure of, 90-94 Polarization, 180 transverse, 49 Polarization component, perpendicular to scattering plane, 61 Polarization effects, enhancement of, 75 Polarization vector, rotation of initial, 61 Polarized electrons elastic collisions with closed-shell moleculea. 43 interactions with molecules, 41 Positive chemical potential, 9.5 atoms with, 102-09 Positive ions, 90, 91 Potential well, 17
INDEX Prefactor, 145 Probability hole, 23 Probe pulse, 11 Pseudoscalar quantities, 45 Pump-probe technique, 7 Purcell, E. M., 228 Purcell formula, 228
Q Quantum beat spectroscopy, 3 Quantum jumps, 224-26 Quantum optics, 221-28 rooin-teinpemturehingle-molecule. 13 1 Quasi continuum states, 127-28
R Rabi flopping time, 21, 149 Radiation blackbody, 143. 16 1 cooling by. I70 l'ar infrared. I 5 1 optical. 146-50 Radiationless relaxation processes, existence of, 107-08 Radiation pulse, construction of appropriate, 34 Radiative lifetimes, measurements of, 108 Radiative transitions, quenching of, 108 Rarnan transitions, 23, 147 Ramped tield ionization, 124, 131-35 Ramsey interferometry, 129 Randomly oriented molecular ensembles, 43 Randomly oriented non-chiral molecules, 61 Randomly oriented target systems, 58-62 Recombination fluorescence, 102-04 Recurrence spectra, comparison with streak spectra, 130 Red states, 127, 131, 134 Reichert. J . F., 111, 112 Reinhold, C. 0.. 141, 143 Relaxation times, I14 Residual amplitude modulation, 2 10 Resonances in detachruent cross-section. 183-85 Fano, 18 Fcshbach. 186 in H-.185-86
243
i n the ionization yield. 139-40 Revival structure, for radial wavepacket, 15 Rich, A,, 70 Richards, D., 139 Robichcaux, F., 130, 181 Room-temperature far-tield SMS measurements, 195 Room-temperature single-molecule measurements, 228-32 RT bleaching, 229 Rydberg atom/HCP interaction in the time domain, 3 1 Rydberg atoms, 121, 124, 125 properties 01'. 122-23 Rydberg population, determining, 134 Rydberg states, 150 characteristic.; of matrix elements between. I44 measuring cncrgy-level splittings between, 33 Rydberg wavepackets and classical correspondence, 12-2 I creation of autoionizing in two-electron atoms, 20 electron ejcction dynamics or, I8 excited with mid-IR FEL pulses, 27-30 exciting novel. 34 memiring timc-resolvcd fluorescence froin, 6 time scales for. 5
S Saddle point, 18 Sakai, Y..92 Sampled atoms, limiting spatial extent of, 12 Sanders, T. M., 98, 99 Saturation, 213, 214 Saturation broadening, of molecular linewidth, 226 Scaling law, 141 Scanning tunneling nucroscopy, 194-95 Scattering of electrons between degenerate hound state configurations, 20 lnrward. 62-63 non-forward. 62 Scattering amplitudes, 63. 71 Scattering model introduction to, 66-68 numerical calculations using. 75-81 Scheidcmann, A., 95
244 Schumacher, D. W., 20 Screw sense, 45,48 defined, 52 Selective multi-step excitation, 107-08 Shalnikov, A. I., 100 Sham, L. J., 94 Shaw, J., 130 Short-pulse excitation, of typical radial wavepacket, 29 Short-pulse regime, transition to long-pulse regime from, 142 Short pulses, ionization of inner electrons by, 149 Shot noise, 212, 215 Signal-to-noise ratio, 210, 216 Single-molecule cavity quantum electrodynamics, 227-28 Single-n~oleculeimaging, 217-218 Single-molecule lineshape, 215-17 Single-molecule optical spectroscopy, 194, 196 requirements for, 203-04 Slit scattering, 160 Smalley, R. E., 166 Smirnov, B. M., 179, 180 Smith, I., 69, 81 Snowballs, 90-92, 95 Solid matrices, 100-101 Solov-ev, E. A., 178 Soluble species, implantation technique suitable for, 95 Sommerfeld, T., 181 Spatial non-uniforniity, 12 Spectral diffusion, 218 Spectral hole-burning, 198,203 Spectral phase, 34 Spectral selection, 196-201 Spectral shifting, 218-21 Spectral trajectory, measurement of, 218 spectroscopy frequency-modulation, 209- I 2 magnetic rcsonance, I 1 1-1 7 matrix isolation, 88 optical, 97-1 I I single-molecule, 194 Spin asymmetries, 84 Spin-dependent amplitudes, 49-53 Spin-dependent effects, hierarchy of, 54-56 Spin-dependent electron-molecule collisions, 84
Spin-dependent interaction, 44.67 Spin down, 70 Spin-lattice relaxation time, of electron bubbles, 111 Spin-local field interactions, absence of, 90 Spin-orbit interaction between continuum electron and molecule, 73 involving incident electron, 68-69 involving molecular electrons, 69-70 Spin-other-orbit interaction, 70-7 1 Spin-polarized electrons, and chiral molecules, 42 Spin up, 70 Springett, B. E., 92 Stabilization, 147-48 adiabatic, 146, 148 Standard integrated absorption sum rule, 201 Stark energy level diagram, 131 Stark states, 124, 127-28, 131 above the saddle point, 128 middle, 137 Stark wavepackets, 16-18 State-selective field ionization, 134 Static-Exchange approximation, 68 Statistical fine structure, 200-201, 215 Stepwise decay, of autoionization, 21, 149 Stienkenieier, F., 110 Stillinger, F. H., 181 Storage rings, 155 heavy-ion, 156 lifetime of, 162 Stored ions, decay and cooling of, 169 Story, J. G., 21 Streak spectra, comparison with recurrence spectra, 130 Strong-field effects, limitations introduced by, 12 Strong-field interferograni, 31 Strong laser fields, wavepackets created by, 22-33 Superfluidity of helium, 96 Suter, D., 112 Symmetry principles, application of, 49-5 1
T Takahashi, Y.,109 Takami, M., 104 Tamarat, P., 226
INDEX Taulbjerg, K., 174, 178, 180 Taylor, H. S., 181 Thomas, L. D., 181 Thompson, D., 62, 64, 69.72, 73, 74. 76 Threshold energy, 177 THz ionization, 125, 141-43 Time-dependent electric field, control over, 3 Time domain, 19 Time-even effects, 48, 49, 56-57 i n collisions with c h i d molecules, 75 Time-even observable, 55 Time interval, limitations on observable, I I Time-odd effects, 48,49, 56-57, 61-62 i n collisions with chiral molecules. 75 Time-odd observable, 55 Time-odd pseudoscalar function. vanishing under sequence of symmetry operations, 59 Time-resolved fluorescence, measuring, h Time resolved ionization spectrum. 129 Time-resolved photoemissions and absorption, 6-7 Time-reversal, 42, 47 in classification of chiral effects, 48 Tisone, G., 174 Tokyo group, 96, 107 Trajectories, 218-21 Transformation properties of some vectors, SO table Transition from ground state to bound state, first observation of, 99 Transition to first excited state, 99 Transverse polarization, 49 Trapping site geometries, 92 True chirality. 42-43, 44-49 Tunable dye lasers, 123, 193 Tunneling, 176 contribution froin, 180 Tunneling-decay probability. 179 Tunneling model, of ion-electron recombination process, 104 Tunneling process, 174 Tunneling states, possible source for, 220 Two-electron wavefunction. 70
U Ulbricht-Vester hypothesis, 83, 84 U-parameter, 61
245 V
van de Water. W., 135 van Druten, N. J., 21 van Leeuwen, K. G . H., 139 Vejby-Christensen, L., 174 Vester, E, 42 Vester-Ulbricht hypothesis of biological homochirality, 42 Vrijen. R. B.. IS0
W Walton, D. S.. 181 Wang, X., 20, 149 Watkins, R. B., 138 Wavepackets. See a1.m Rydberg wavepackets, Stark wavepackets Sc17d character of. 20 angular, 16 changing characteristics of. 2 circular. 35 continuum i n strong static field. I8 control of, 33-35 created by strong laser fields, 22-33 created when HCP kicks a stationary Rydherg state, 32 crcation and detection of electronic, 2 crcation of yn-orbit, 3 dark. 22-24 decay of in an electric field. 128-30 delinilion of. 1 evolution of dark in potassium. 24 excitation of, 3 excited during multiphoton ioni~ation.24-27 excited with GHr half-cycle pulses. 33 excited with Mid-IR FEL pulses, 27-30 excited with THz half-cycle pulses. 3&32 hst. 26
limitations in niaxiniuni ohsewation time of: I I oricntetl. 16 oscillation of. 14. 17-18 radial. 12-15 revival structure for radial, IS Rydberg, 5, 6 . 12-2 I , 27-30. 34 spreading duc to energy dispersion. I S Stark, 16-18 techniques for monitoring evolution of, 6-12 two-electron. 19-22 Webb, W. W., 231
LNDEX
246 Weis, A. R., 115 Whittle, E., 88 Wigner-Seitz method, 100 Wilks, J., 88 Williams, R., 43
Y Yabuzaki, 113 Yang, X., 181
Z Zeeman component, hyperfine, 115 Zeeman Hamiltonian, diagonalization of, 161 Zeeman multipletts, 113 Zero-field Rydberg spacing, 139 Zero-phonon lines, 197-201 Zipfel, 99 Zoller, P., 20
Contents of Volumes in This Serial Volume 1
Volume 3
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules. C. G. Hull and A . T Amos Electron Affinities of Atoms and Molecules. B. L. Moiseiwitsch Atomic Remangement Collisions, B. H. Brurzsden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takuyanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H . Pauly und J. P Toennies High-Intensity and High-Energy Molecular Beams, J. B. Anderson, R. P Andres, und J. B. Fen
The Quanta1 Calculation of Photoionization Cross Sections. A. L. SteNmt Radiofrequency Spectroscopy of Stored Ions I: Storage, H . G. Dehnielr Optical Pumping Methods in Atomic Spectroscopy, B. Bitdick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments. H . C. WOK
Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Wads Scattering, E. C/ianoch Beder Reactive Collisions between Gas and Surface Atoms, Heivy Wise and Berriurd J. Wood
Volume 4
Volume 2
The CPlculation of van der Waals Interactions, A. Dulgarno and W D. Davison Thermal Diffusion in Gases, E. A. Moson, R. J. M u m , and Fruncis 1.Smirh Spectroscopy in the Vacuum Ultraviolet. W R. S.Ccrrmn The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A . R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Syqtems, E J . cle Heer Mass Spectrometry of Free Radicals, S. N . Foner
H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular lon, D. R. Butes and R. H. G Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckirighmn and E. Gal Positrons and Positronium in Gases, P A Fruser Classical Theory of Atonuc Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L. Moiselwirsch Resonances in Electron Scattering by Atoms and Molecules, P G. Burke Relativistic Inner Shell Ionizations, C. 5.0. Mohr 247
248
CONTENTS OF VOLUMES THIS SERIAL
Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions. D. W 0. Heddle and R. G. W Keesing Some New Experimental Methods in Collision Physics, R. E Stebbings Atomic Collision Processes in Gaseous Nebulae, M. 1. Seaton Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space, R. L. E Boyd
Volume 5
Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, E C. Fehsenfeld, and A. L. Schrneltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelr The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R. J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s?s"'py, C. D. H. Chisholm, A. Dulgarno, and E R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Volume 6
Dissociative Recombination, J. N. Bardsley and M. A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman
The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu ltikawu The Diffusion of Atoms and Molecules, E. A. Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston
Volume 7
Physics of the Hydrogen Master, C. Audoin, J. P: Schennann, and I? Crivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules-QuasiStationary Electronic States, Thomas F: 0 'Malley Selection Rules within Atonlic Shells, B. R. Judd Green's Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylol; and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J. Greenfield
Volume 8
Interstellar Molecules: Their Formation and Destruction, D. McNully Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. k: Chen and Augustine C. Chen
CONTENTS OF VOLUMES THIS SERIAL Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I. Schoen The Auger Effect, E. H. S.Burhop and V! N. Asaad
Volume 9
Correlation in Excited States of Atoms. A. W Weiss The Calculation of Electron-Atom Excitation Cross Sections, M. R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oku The Differential Cross Section of LowEnergy Electron-Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy,Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10
249
Electron Impact Excitation of Positive Ions, M. J. Seaton The R-Matrix Theory of Atomic Process, I? G. Burke and N! D. Robb Role of Energy in Reactive Molecular Scattering: An Infonnation-Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Grieni Chemiluminescence in Gases, M. E Golde and B. A. Thrush Vnlume 12
Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atonuc Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, I? Lumbropoulos Optical Pumping o f Molecules, M. BroyeK G. Goudedard, J. C. Lehmann, and J. WguP
Relativistic Effects in the Many-Electron Atom, Llovd Armstrong, Jr: and Serge Feneuille The First Born Approximation, K. L. Bell and A. E. Kingston Photoelectron Spectroscopy, W C. Price Dye Lasers in Atomic Spectroscopy, W Lunge. J. LutheK rind A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fuwrett A Review o f Jovian Ionospheric Chemistry. Wesley T Huntress. Jr. Volume I 1
The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Percival and D. Richards
Highly lonized lons, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Willielm Ruith Ion Chenustry in the D Region, George C. Reid Vnlume 13
Atomic and Molecular Polarizabilities-A Rcview of Recent Advances, Thomas M. Miller und Benjamin Bederson Study of Collisions by Laser Spectroscopy, Pad R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. G: Hertel and V! Sroll Scattering Studies of Rotational and Vibrational Excitation of Molecules. Manfred Faubel and J. Peter Toennies
250
CONTENTS OF VOLUMES THIS SERIAL
Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W B. Somerville Volume 14
Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster; Michael J. Jamieson, and Ronuld E Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and ' h o Electron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. K Bobashev Rydberg Atoms, S. A. Edelstein and I: E Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree
Volume 15
Negative Ions, H. S. W Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. E Stebbings Theoretical Aspects o f Positron Collisions in Gases, J. W Humberston Experimental Aspects of Positron Collisions in Gases, I: C. G r i ~ t h Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer CoIIisions at Low Energies, J. B. Hasted
Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, D. W; 0. Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, P. G. Burke
Volume 16
Atomic Hartree-Fock Theory, M. Cohen and R. I? McEachrun Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren Sources of Polarized Electrons, R. J. Celotfu and D. I: Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M. H. Key and R. J. Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experment, E. N. Fortson and L. Wilets
Volume 17
Collective Effects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D. S. E Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M. J? H. Schuurmans, Q. H. E Vrehen, D. Polder, a i d H. M. Gibbs
CONTENTS OF VOLUMES THIS SERIAL Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G. Payne, C. H. Chen, G. S. Hurst, arid G. U? Foltz Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P L. Digton and A. E. King.\ton
Volume 18
Theory of Electron-Atom Scattering in a Radiation Field. Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppiln Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Nonnand, untl G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions. A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. U? Norcross and L. A. Collins Quantum Electrodynanuc Effects in FewElectron Atonic Systems, G. W E Drake
Volume 19
Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K. Janev Interactions of Simple Ion-Atom Systems, J. Z Park High-Resolution Spectroscopy of Stored Ions, D. J . Winelanti, Wayne M. frano. and R. S. Van Dvck, JK
25 1
Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. Bluin and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, E Jen? The Vibrational Excitation of Molecules by Electron Impact, D. G . Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Fuitbel Spin Polarization o f Atomic and Molecular Photoelectrons, N. A. Cherepkov
Volume 20
Ion-Ion Recombination in an Ambient Gas, D.R. Bates Atomic Charges within Molecules, G . G. Hall Experimental Studies on Cluster Ions, 7: D. Mark and A. U? Castleman. JI: Nuclear Reaction Effects on Atomic InnerShell Ionization, U? E. Meyerhof and J.-E Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and Duvid A. Amstrong On the Problem of Extreme UV and X-Ray Lasers, I. I. Sobel’man and A. V Vino~radov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J. M. Ralmond Rydberg Atoms: High-Resolution Spectroscopy and Radiation InteractionRydberg Molecules, J. A. C. Gallas, G. Leuchs, H. Walthel; and H. Figger
Volume 21
Subnatural Linewidths in Atomic Spectroscopy, Dennis P O’Erien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jirrrgen
252
CONTENTS OF VOLUMES THIS SERIAL
Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M. R. C. McDowell and M. Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More
Volume 22
Positronium-Its Formation and Interaction with Simple Systems, J. II! Humberston Experimental Aspects of Positron and Positronium Physics, I: C. GrifJith Doubly Excited States, Including New Classitication Schemes, C. D. Lin Measurements of Charge Transfer and IoniLation in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Pearl Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Could Continued-Fraction Methods in Atomic Physics, S. Swain
Volume 23
Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the ReIativistic Theory of Atomic and Molecular Structure, Ian I? Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Arnouk and M. Klapisch
Photoionization and Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, E J. Wuilleurnier; D. L. Ederer, and J. L. Picque
Volume 24
The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N . C. Adams Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. J. Smith and C. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J. Knize, Z. Wu, and W Happer Correlations in Electron-Atom Scattering, A. Crowe
Volume 25
Alexander Dalgarno: Life and Personality, David R. Bates arid George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McE1ro.y Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He’ -He Collisions at KeV Energies, R. E Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-lnduced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu Model-Potential Methods, G. Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen
CONTENTS OF VOLUMES THIS SERIAL Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H. C. Reid Elcctron Impact Excitation, R. J. W. Hen? and A. E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes. Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H. Bransden and D. I?Dewangan Relativistic Random-Phase Approximation, W R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W F Drake und S.P. Goldman Dissociation Dynamics of Polyatoniic Molecules, 7: Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirbv and Ewine E van Dishoeck The Abunddnces and Excitation of Interstellar Molecules, John H. Black
Volume 26
Coniparisons of Positrons and Electron Scattering by Gases, Walter E. Kaupjda arid Tulbert S. Stein Electron Capture at Relativistic Energies, B. L. Moiseiwitsrh The Low-Energy, Heavy Particle Collisions-A Close-Coupling Treatment, Mineo Kitnuro and Neal ?I Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weinel; Frunqoise Masnou-Sweeuws. and Annick Ciusti-Snzor On the p Decay of '"Re: An Interface of Atomic and Nuclear Physics and Cosniochronology, Zonghau Chen, Leonurd Rosenberg, and Lar? Sprrrch
253
Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Muva and R. Lugu.shenko Volume 27
Negative Ions: Structure and Spectra. David R. Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachirn Kesskr Electron-Atom Scattering, I. E. McCarthJJ and E. Weigold Electron-Atom Ionization, 1. E. McCarthy ond E. Weigold Role o f Autoionizing States in Multiphoton Ionization o f Complex Atoms, V 1. Lengyel und M . I. Huysuk Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28
The Theory of Fast Ion-Atom Collisions, .I. S. Briggs and J. H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W Milonni and Sitrendru Singh Squeezed States of the Radiation Field, Kliulid Zaheer and M. Suhuil Zubuiq Cavity Quantum Electrodynamics, E. A. Hind7 Volume 29
Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin ond L. W Anderson Cross Sections for Direct Multiphoton Ionionization of Atoms, M. V Ammosov, N. B. Delone, M . YI~. Ivuno~:I. I. Bondal; and A. F Masalov Collision-Induced Coherences in Optical Physics, G. S. Aganval Muon-Catalyzed Fusion, Joliunn Rafelski and Helga E. Rafelski
254
CONTENTS OF VOLUMES THIS SERIAL
Cooperative Effects in Atomic Physics, J. I? Conneracie Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J. H. McGuire
Volume 30
Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinohu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J . C. Nickel The Dissociative Ionization of Simple, Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, P: S. Julienne, A. M. Smith, and K. Bumett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in IonAtom Collisions, Derrick S.E Crothers und Louis J. Dub6
Volume 31
Energies and Asymptotic Analysis for Helium Rydberg States, G. W E Drake Spectroscopy of Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Wulfher Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Diiren, and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Michele Lamoureux
Volume 32
Photoionization of Atomic Oxygen and Atomic Nitrogen, K. L. Bell and A. E. Kingston
Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H. Bransden and C. J. Noble Electron-Atom Scattering Theory and Calculations, P: G. Burke Terrestrial and Extraterrestrial H;, Alexander Dalganro Indirect Ionization of Positive Atomic Ions, K. Dolder Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G. W E Drake Electron-Ion and Ion-Ion Recombination Processes, M. R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H. E. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I. P: Grant The Chemistry of Stellar Environments, D. A. Howe, J. M. C. Rawlings, and D. A. Williams Positron and Positronium Scattering at Low Energies, J. W Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D. S. E Crothers Electron Capture to the Continuum, B. L. Moiseiwitsch How Opaque Is a Star? M. J. Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow-Langmyir Technique, David Smith and Patrik Spang1 Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain Atoms in Cavities and Traps, H. Wulther Some Recent Advances in Electron-lmpact Excitation of n = 3 States of Atomic Hydrogen and Helium, J. E Williams and J. B. Wang
CONTENTS OF VOLUMES THIS SERlAL Volunie 33
Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical ‘Techniques.A. R. Filippelli, Clnin C. Lin. L. W Andersen. and J. W McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons. S. Trujmar arid J. W McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R. W Crompfon Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbodv The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Bur? I. Srhneider Analytic Representation of Cross-Section Data, Mitio lnokufi, Mine0 Kiinuru. M. A. Dillon, Isuo Shirnumuru Electron Collisions with N?, O2and 0: What We Do and Do Not Know, Kikikuzii Ifikawa Need for Cross Sections in Fusion Plasma Research, Hugh F! Sunimers Need for Cross Sections in Plasm Chemistry, M. Cupirelli, R. Celiberto. und M. Curciafore Guide for Users of Data Resources, Jeun W Gullugher Guide to Bibliographies. Books, Reviews, and Compendia of Data on Atomic Collisions, E. W McDuniel and E. J. Munskv
Volume 34
Atom Interferometry, C S Adums, 0 Carnal. ond J Mlvnek OptiLal Tests of Quantum Mechanics, R Y Chiao. P G Kwrur, a n d A M Sternberg
255
Classical and Quantum Chaos in Atomic Systems, Dorninique Delande and Andreas Buchleitner Measurements of Collisions between LaserCooled Atoms, Thud Wulker utid Puul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas. J. E. LawIer and D. A. Dough@ Polarization and Orientation Phenomena in Photoionization of Molecules, N. A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E. C. Monrenegro, W E. MeyerhoJ and J . H. McCuire lndirect Processes in Electron Impact Ionization of Positive Ions, D. L. Moores mid K. .I. Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bares
Volunie 35
Laser Manipulation of Atoms, K. Sengstock and W Erfmer Advances in Ultracold Collisions: Experiment and Theory, J. Weirier Ionization Dynamics in Strong Laser Fields, L. F: DiMauro and F! Agosfirii Infrared Spectroscopy of Size Selected Molecular Clusters, U.Buck Femtosecond Spectroscopy of Molecules and Clusters, 7: Baumer und G. Cerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. 7: Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium lsoelectronic Sequence, W R. Johnson, D. R. Plunte, cind J. Sapirsfein Rotational Energy Transfer in Small Polyatomic Molecules, H. 0. Everift and E C. De Lucia
256
CONTENTS OF VOLUMES THIS SERIAL
Volume 36
Complete Experiments in Electron-Atom Collisions, Nils Overgaard Andersen, and Klaus Bartschat Stimulated Rayleigh Resonances and RecoilInduced Effects, J.-K Courtois and G. Grynberg Precision Laser Spectroscopy Using Acousto-Optic Modulators, W A. van Wijngaarden Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You
Volume 37
Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan P. Dowling and Julio GeaB ana c1oche Optical Lattices, I? S. Jessen and I. H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert F: Krause and Sheldon Datz
Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N.J. van Druteti Nonclassical States of Motion in Ion Traps, J. I. Cirac, A. S. Parkins, R. Blatt, and P: Zoller The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, P: H. Mokler and Th. Stohlker
Volume 38
Electronic Wavepackets, Robert R. Jones and L. D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D. G. Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G.M. Lankhuijzen and L. D. Noordam Studies of Negative Ions in Storage Rings, L. H. Andersen, T Andersen, and P. Hvelplund Single-Molecule Spectroscopy and Quantum Optics in Solids, W E. Moerner; R. M. Dickson. and D. J. Norris
