Advances in Electronics and Electron Physics, Volume 23

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 23

CONTRIBUTORS TO THISVOLUME Eugene R. Chenette P. A. Grivet W. C. Livingston E. A. Lynton W. L. McLean L. Malnar H. Motz C. J. H. Watson

Advances in

Electronics and Electron Physics EDITEDBY L. MARTON National Bureau of Standards, Washington, D.C.

Assistant Editor CLAIREMARTON EDITORIAL BOARD

T. E. Alliborie H. B. G. Casimir L. T. DeVore W. G. Dow A. 0. C. Nier

E. R. Piore Jf. Ponte A. Rose 1,. 1’. Smith F. K. Willenbrock

VOLUME 23

1967

ACADEMIC PRESS

New York and London

C O P Y R I G H T @ 1967, BY ACADEMICP R E S S INC. ALL RIGHTS RESERVED. NO PART O F T H I S BOOK MAY B E REPRODUCED I N ANY FORM, B Y PHOTOSTAT, MICROFILM, OR ANY O T H E R MEANS, W I T H O U T W R I T T E N PERMISSION FROM T H E PUBLISHERS.

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United Kin dom Edition ublished by ACAIIEMfC PRESS IRC. ( L O N D O N )’ LTD. Berkeley Square House, London W.l

LIBRARYOF CONGRESS CATALOG NUMBER:49-7504

P R I N T E D I N T H E U N I T E D S T A T E S O F AMERICA

CONTRIBUTORS Numbers in parentheses refer to the pages on which the authors’ contributions begin.

EUGENER. CHENETTE*(303), Electrical Engineering Department, University of Minnesota, Minneapolis, Minnesota P. A. GRIVET(39), University of Paris, Institut d’Electronique Fondamentale, Orsay, France

W. C. LIVINGSTON (347), Kitt Peak National Observatory, Tucson, Arizona E. A. LYNTON(l), Department of Physics, Rutgers-The New Brunswick, New Jersey

State University,

W. L. MCLEAN(l), Department of Physics, Rutgers-The versity, New Brunswick, New Jersey

State Uni-

L. MALNAR,C.F.S. (39), Dept. de Physique Appliqub, Corbeville par Orsay, France H. MOTZ(153) , Department of Engineering Science, Oxford University, Oxford, England C. ,J. H. WATSON(153), Merton College, Oxford, England

* Present address: Bell Telephone Laboratories, Allentown, Pennsylvania. V

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FOREWORD For the first time in this serial publication I am in a position to present two reviews of a subject which we failed to treat earlier: superconductivity. The two reviews contained in this volume cover both some theoretical (Lynton and McLean) and some practical aspects (Laverick). Because of the direction of development, I hope to come back to different aspects of this subject in future volumes. I n the foreword to Volume 21, I mentioned that while four reviews on plasmas appeared in Volumes 20 and 21, in view of the importance of the subject, further reviews were planned. The article by Mota and Watson covers an important phase of plasma research. Although three special volumes (12, 16, and 22) were devoted to photoelectronic imaging devices and techniques, it seemed advisable to present a slightly dissident viewpoint by Livingston. It has been a long time since the subject of magnetic field measurements was treated here (Volume 4). The review by Grivet and Malnar couples this classical subject with a very modern technique: magnetic resonance. Last, but not least, the review by Chenette covers a n equally neglected subject in these series: noise in semiconductor devices. The last review on this subject appeared in Volume 4 (van der Ziel). As done in previous volumes, I would like to list the titles and authors of future reviews: Cooperative Phenomena Progress in Microwave Tubes Application of Group Theory t o Waveguides Optimization of Control Thermal Energy Ion-Molecule Reaction Rates Novel High Frequency Solid State Ultrasonic Devices The Analysis of Dense Electron Beams Ion Waves and Moving Striations The Electron Beam Shadow Methods of Investigating Magnetic Properties of Crystals vii

J. L. Jackson and L. Klein 0. Doehler and G. Kantorowicz 1).Kerns A. Blaquiere

E. E. Ferguson

N. G. Einspruch K. Aniboss N. L. Oleson and A. W. Cooper A. E . Curzon and N. D. Lisgarten

viii

FOREWORD

Ion Beam Bombardment and Doping of Semiconductors Nuclear and Electronic Spin Resonance Josephson Effect and Devices Linear Ion Accelerators Electron Spin Resonance: A Tool in Mineralogy and Geology Linear Ferrite Devices for Microwave Applications Reactive Scattering in Molecular Beams Thermionic Cathodes Radio Wave Fading Photoelectric Emission from Solids Dielectric Breakdown The Hall Effect and Its Applications Electrical Conductivity of Gases Progress in Traveling Wave Devices Electromagnetic Radiation in Plasmas Millimeter and Submillimeter Wave Detectors Luminescence of Compound Semiconductors The Statistical Behavior of the Scintillation Counter: Theories and Experiments Radio Backscatter Studies of Thin Polycrystalline Films by Electron Beams Gas Lasers and Conventional Sources in Interferometry Application of Lasers to Microelectronic Fabrication Theory of the Unrippled Space-Charge Flow in General Axially Symmetric Electron Beams Study of Ionization Phenomena by Mass Spectroscopy Recent Advances in Circular Accelerators Image Formation a t Defects in Transmission Electron Microscopy Quadrupoles as Electron Lenses Resolution in the Electron Microscope Nonlinear Electromagnetic Waves in Plasmas Ion Bombardment Doping of Semiconductors Space-Charge Limited Corona Current Molecular Reactions in Glow Discharges

D . B. Medved E. R. Andrew and S. Clough J. E. Mercereau and D. N. Langenberg E. L. Hubbard

w. Low W. H. von Aulock and C. E. Fay S. Data P. Zalm M. Philips F. Allen N. Klein S. Stricker J. M. Dolique W. E. Waters J. R. Wait G. I. Haddad

F. E. Williams E. Gatti and V. Svelto M. Philips C. W. B. Grigson

K. D. Mielenz M. I. Cohen and J. P. Epperson W. E. Waters H. M. Rosenstock

J. P. Blewett S. Amelinckx P. W. Hawkes E. Zeitler

J. Rowe V. S. Vavilov A. Langsdorf, Jr. R. A. Hartunian

I n addition, we expect to publish our third supplement volume soon: “Narrow Angle Elect>ronGuns and Cathode Ray Tubes” by H. MOSS.

FOREWORD

ix

It is my pleasure to announce that Dean F. I(. Willenbrock, of the State University of New York at Buffalo, has joined our Editorial Board, filling the gap created by the death of Professor W. B. Nottingham. Washington, D. C. May, 1967

L. MARTON

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CONTENTS LIST O F CONTRIBUTORS .

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FOREWORD . .

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V

vii

Type I1 Superconductors

.

E . A LYNTON A N D W . L . MCLEAN

I . Introduction . . . . . . . . . . . . . I1. Basic Properties of Superconductors . . . . . . I11. London Equation . . . . . . . . . . . IV . Quantization of Flux . . . . . . . . . . V. The Ginzburg-Landau Equations . . . . . . VI The Interphase Surface Energy . . . . . . . VII . The Static Properties of the Mixed State . . . . VIII . Surface Superconductivity . . . . . . . . . I X Dynamic Effects . . . . . . . . . . . . Appendix . . . . . . . . . . . . . References . . . . . . . . . . . . .

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1 1 4 5 6 10 13 23 25 33 35

Measurement of Weak Magnetic Fields by Magnetic Resonance P . A . GRIVETA N D L . MALNAR I . Introduction . . . . . . . . . . . . . . . . . . I1. Order of Magnitude and blain Characteristics of Natural Fields . . I11. Nuclear Resonance . . . . . . . . . . . . . . . .

.

IV Optical Detection of an Electron Nuclear Resonance . . . V. An Example of Design: The Cesium Vapor Magnetometers . VI . Superconducting Interferometers as Magnetometers . . . References . . . . . . . . . . . . . . .

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40 45 55 76 111 143 146

The Radio-Frequency Confinement and Acceleration of Plasmas H . MOTZA N D C . J . WATSON

. . . . . . . . . . . . . . . . Introduction 1 . Single Particle Motions . . . . . . . . . . . . . 2. The Theory of Radio-Frcquenry confinement of Plasma . . . . 3. Theory of Combined Radio-Frequency and Magnetostatic Confinement Plasma . . . . . . . . . . . . . . . . . . 4 . Stability Theory . . . . . . . . . . . . . . . 5. Application to Fusion Reactors . . . . . . . . . . 6. Experiments Related to Radio-Frequency Confinement . . . . 7. The Theory of Radio-Frequency Acceleration of Plasma . . . . 8 . Experiments on Radio-Frequency Acceleration of Plasma . . . References . . . . . . . . . . . . . . . . . xi

. 154 . 159 . 194 of

. 223 . 227 234

. 241 . 264 . 283

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298

xii

CONTENTS

Noise in Semiconductor Devices

EUGENE R . CHENETTE

I . Introduction . . . . . . . . . . . . . I1. Theory of Noise in Semiconductor Devices . . . . I11. Experimental Verification of the Theory . . . .

. . . I\'. Practical Low-Noise Amplifiers . . . . . . . . V. Summary . . . . . . . . . . . . . . References . . . . . . . . . . . . .

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303 304 319 329 340 345

Properties and Limitations of Image Intensifiers Used in Astronomy W . C. LIVINGSTON

I . Introduction . . . . . . . . . . . . . . . . . . 347 I1. Qualitative Comparison Between the Photographic Plate and the Image Tube . . . . . . . . . . . . . . . . . . . . 349

I11. IV . V. VI .

Quantitative Evaluation of Image Tubes . . . . . . . . . 352 Description of Tubes and Results . . . . . . . . . . . 354 Prospects for Future Developments . . . . . . . . . . . 372 Summary and Conclusions . . . . . . . . . . . . . 380 References . . . . . . . . . . . . . . . . . 381

AUTHOR INDEX . . SUBJECTI N D E X .

. . . . . . . . . . . . . . . . . 475 . . . . . . . . . . . . . . . . . 485

Type I1 Superconductors E. A. LYNTON

AND

W. L. AIcLEAN

Department of Physics Rulgers-The State University New Brunswick, New Jersey

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Properties of Superconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . London Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantization of Flux.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ginzburg-Landau Equations. . . . . . . . . . . . . . The Interphase Surface Energy.. . . . . . . . . . . . . . . The Static Properties of the Mixed State.. . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Superconductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Effects .......................................... A. Steady-State Flux Flow.. . . . . . . . . . . . . . . . . . .. B. Vortex Motion., . . . . . . . . . . . . . . . . . . . . . . . . . .. C. Vortex Waves.. . . . . . . . . . . ........................... D. The Surface Ba,rrier. . . . . . ........................... E. Vortex Motion in Thin Films.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11. 111. IV. V. VI. VII. VIII. IX.

I

1 4 5 13 23 25 25

28 30 31 32 33 35

I. INTRODUCTION During the past few years, a great deal of fundamental experimental and theoretical research in superconductivity has concentrated on socalled Type I1 superconductors. This work is of interest both because it has brought a renewed interest in macroscopic quantum phenomena as well as in certain basic electrodynamic problems and because of its relevance to the application of superconductivity to magnets and related devices. We shall, therefore, review the principal aspects of this work after a brief sumniary of basic superconducting properties. 11. BASIC.PROPERTIES OF SUPERCONDUCTORS' A superconductor is a substance which below some well-defined critical temperature, T,, has zero electrical resistance for low-frequency currents 1 References to most of the original work which is summarized in this chapter can be found in Lynton (1). 1

2

E. A. LYNTON AND W. L. MCLEAN

and in which, therefore, the electric field E vanishes. At this time, 24 metallic elements are known to become superconducting, with values of T , ranging from 11.2"K down to about 0.01"K. I n addition, there are more than 500 superconducting compounds and alloys, with T , as high as 18"I<, including even semiconducting materials. A superconductor is not only a perfect conductor, it also has a property which is sometimes described as perfect diamagnetism. When a small external magnetic field is applied to a superconductor, or when a superconductor is cooled in such an external field to temperatures below Tc, the magnetic induction inside the material vanishes, i.e.,

B

=

H

+ 4aM = 0

so that M = -H/4a. This occurs regardless of whether the substance becomes superconducting before or after the external field is applied, in contrast to what would be expected with a perfect conductor. The application of a magnetic field to a perfect conductor would induce eddy currents a t the surface which would prevent the magnetic flux inside the conductor from changing. If the field were already applied before the transition to perfect conductivity occurred, the flux would remain locked in the perfect conductor. If the external source of magnetic field were removed, eddy currents would be induced in the perfect conductor keeping the flux constant. It should be noted that a metal which has lost its electrical resistance through the disappearance of the mechanism causing the electron scattering is not a "perfect conductor" in the sense used above. The diffusion of the conduction electrons throughout the metal would lead to a gradual dying away of the eddy currents which had been induced by applying an external field. I n the case of a typical metal, the magnetic flux would enter with a time constant of the order of seconds [Pippard (S)]. A superconductor, unlike a merely perfect conductor, undergoes a reversible transition from the normal to the superconducting state in the presence of a magnetic field. This has the great virtue of allowing us to apply simple thermodynamic arguments. The exclusion of the magnetic flux from the interior of the superconductor raises its magnetic free energy. As the external magnetic field is gradually increased, the positive magnetic contribution to the total free energy rises until it just counterbalances the negative condensation energy, that is, the free energy difference between the normal and superconducting phases in the absence of the magnetic field. At this value of the external field, called the critical field, the normal and superconducting phases exist together in equilibrium. A further increase of the external field causes a complete transition to the normal state which now has a lower free energy than would the superconducting state.

3

T YP E I1 SUPERCONDUCTORS

The critical magnetic field depends on the temperature and below To varies, to a very good approximation, parabolically according to Hc(T) = Ho[l

- (T/TJ21

where H o = H,(O) (-1OOT, Oe for typical superconductors). The reversibility of the normal-to-superconducting transition a t H,( T) allows us to treat this as a conventional phase transition and to equate the free energies2: Per unit volume and for an ellipsoidal3 specimen, we can write Gn(0)

-

Gn(Hc)

=

Ga(Hc) = G,(O)

-

loHo M ( H ) dH

=

G,(O)

+ Hc2/8~,

since the normal metal susceptibility is generally negligibly small. Hence,

G,(O) - GB(0)= HO2/87r. We then immediately see that, since the entropy density S Sn(0) - &(O)

=

=

-(dG/d7')~,~,

-Hc(dHc/dT)/4~.

At all T < T,, Hc > 0 and dHc/dT < 0, so th a t Sn(0) > S,(O), except a t T = T, and T = 0, where S,(O) = X.(O). Thus we see that the superconducting phase is one of higher internal order than the normal one. From the absence of any change in lattice characteristics a t T,, it is clear that the ordering which takes place must occur in the electronic system. From this in turn it follows that the ordering is in momentum rather than in configuration space, since it is difficult to conceive of a spatial condensation of the electrons inside a metal analogous to the condensation of liquid from its vapor that would not be prevented by the electrical repulsion between like charges. However, considering the fact that a t very low temperatures the conduction electrons of a metal already form a highly ordered system in momentum space, the nature of this ordering is not immediately apparent. From the solution of the Schroedinger equation for the electrons in a superconductor proposed by Bardeen et al. (S), we now know that it involves the condensation of electron pairs into the lowest momentum state, in agreement with the much earlier predictions by London (4). * T h e Helmholtz free energy F and the Gibbs free energy G are defined in the appendix. a T h i s stipulation is made in order to ensure t h a t the magnetic intensity H is related t o the external magnetic field He and the magnetization M by the simple relation H = H e - 4 m M , where n is the demagnetization coefficient for the ellipsoid. In what follows, we shall restrict ourselves t o samples for which n << 1, to avoid an unnecessarily complicated discussion.

4

E. A. LYNTON AND W. L. MCLEAN

Before discussing London’s work, we note that a thermodynamic treatment of superconductivity is possible without knowing the microscopic nature of the ordered state. One can describe it macroscopically in terms of an order parameter #. This changes from zero in the regions which are in the normal phase to a maximum in those parts of the superconducting regions farthest from the normal regions and not near to the surface of the sample. Starting ab initio, we might define the order parameter in terms of the entropy density. Alternatively, if we regard the superconductor a t a finite temperature as containing a mixture of superconducting and normal electrons, the order parameter could be defined as the fraction of electrons which are superconducting. We will see in Section V that in the approach of Ginzburg and Landau the parameter is not defined initially, other than as a convenient quantity in terms of which the free energy can be expanded.

111. LONDON EQUATION The full implications of the ordering mechanism in superconductivity were first recognized by London (4) thirty years ago with a degree of perception, the extent of which has become apparent only in recent years. Realizing that an electrodynamic treatment based solely on Maxwell’s equations in the limit of zero resistivity would always predict the irreversible behavior of the perfect conductor and not the reversible diamagnetism of the superconductor, London introduced an additional equation. The form of this equation can’be justified in various ways, such as, e.g., by minimizing the free energy of the current and field distribution [DeGennes (6)] or by supposing that the superconducting wave functions are perfectly rigid against the application of an external field [London ( d ) ] ,but for our purposes i t is sufficient to consider it as an intuitive hypothesis fully justified by its success. The equation proposed by London is (4?rX2/c)V X J

+B =0

where J is the current density, B the flux density, and X2 = mc2/4nnq2, m, q being the mass and charge of superconducting carriers and n the density of these carriers. With the use of Maxwell’s equation v X B = 47rJ/c, the London equation can also be written

B + X2V X V X B

=0

or

B - X2V2B = 0.

The solution of this equation for a superconducting region of linear dimensions much greater than X is B(z) = B(0) exp (-z/X), where B(z) is the flux density at a depth z below the surface. Using a typical value for

5

TYPE I1 SUPERCONDUCTORS

-

the electron density n and the values of q and m for a free electron, we find X 500 8. The London equation provides a very basic clue as to the nature of the superconducting order. Writing it in terms of the vector potential A where V X A = B,choosing a gauge such that V * A = 0, and considering a superconductor which is singly connected, we obtain as the London equation: (47rX2/c)J A = 0.

+

In the presence of a vector potential, the canonical momentum of the charged particles is given by P = Z p = Z(mv qA/c). The average momentum per particle can be written as p = (q/c)[(47rX2/c)J A] = 0. According to this, then, the superconducting order is due to the condensation of all the participating carriers into the lowest momentum state P = 0. From the uncertairity principle we deduce a correspondingly infinite range of spatial order or infinite “coherence” and a n inability of the system of electrons to be affected by localized variations of field and configuration.

+

+

IV. QUANTIZATION OF FLUX Consider now a multiply connected superconductor, such as, for example, a ring or any piece containing a hole. On the assumption that the long range order just mentioned is a basic characteristic of superconductors and exists in this case as wcll, one must then require th a t the wave functions describing the superconducting carriers be single valued along any closed path enclosing the hole. By analogy with the electronic wave functions in an atomic orbit one can then apply the Bohr-Sommerfeld quantization rules and require that for each of the superconducting carriers $p * dl = wh, where w is an integer and h is Planck’s constant, the line integral being evaluated along any path enclosing the hole. Substituting the London equation we obtain

+ $A

$(47rX2/c)J dl

dl

=

whc/q.

-

-

With Stokes’ theorem, $A dl = JJB dS = 9, where the double integral is over the surface enclosed by the path and 9 is the total magnetic flux threading the contour. Thus, (47rX2/c)$J

*

dl

+9

=

whc/q.

The left-hand side of this equation was called aJlutoid by London, and leads to the general statement that for any path enclosing a hole in a superconductirig material, the total flux threaded plus the path integral of (47rX2/c)J q u a 1 an integral multiple of hc/q. If this path is taken a t a

6

E . A. LYNTON AND W. L. MCLEAN

distance from the hole large compared to the penetration depth A, J is zero and the above reduces to @ = whc/q =

W&

where $0 = hc/q. Thus a superconductor displays the fascinating characteristic of quantization of JIux. The existence of this phenomenon has been during the past years amply demonstrated, and the quantum of flux has been shown to be

-

$0

-

-

2X

G-cm2

SO that q 3.2 x 10-19 C 2e, where e is the electronic charge. This shows that the superconducting carriers have a double electronic charge, suggesting strongly that they are pairs of electrons. Indeed this has been shown to be the case by the successful microscopic theory developed by Bardeen et al. (3).

V. THE GINZBURG-LANDAU EQUATIONS The London equation, as has been mentioned, implies that the characteristics of the superconducting state remain rigidly unchanged under the application of an external field and, furthermore, that the superconducting current density at a point is determined by the vector potential a t that point. This local relationship is a reasonable approximation when the field penetration depth X is large compared to all other relevant distances. Nonlocal modifications allowing for the long range of coherence are beyond the scope of the present review. The implied rigidity of the superconducting characteristics, such as in particular some suitable parameter describing the degree of order, precludes the use of the London equation in situations in which this parameter has a spatial variation. I n particular, the London equation by itself is inadequate to treat surface phenomena and the behavior of interphase boundaries between normal and superconducting regions, where there is a gradual change from the superconducting to the normal phase over a small but finite distance. The mention of this limitation, however, should not obscure the great usefulness of the London equation in treating the macroscopic electrodynamic response of bulk superconductors. A phenomenological theory of great power and versatility in treating spatial variations of the superconducting order parameter was introduced by Ginzburg and Landau (6) in 1950. It is based on Landau’s treatment of second-order phase transitions (7) and is meant to apply near to the transition temperature. A microscopic derivation of the results of the Ginzburg-Landau theory has been given by Gor’kov (8).

TYPE I1 SUPERCONDUCTORS

7

Considering first a superconductor in the absence of an external field and with uniform order throughout, we may express the free energy as a power series in even powers of tlhe order parameter +(r), which in the present case may be a complex number. Thus

Fs

+

+

+

Fn 8PllC.[* * * * where a ( T ) < 0 for T < T,, CY(T,)= 0, and ( d a / d T ) ~#, 0; P > 0 for =

T 6 T,. Assuming CY and to be continuous functions of temperature and bearing in mind that we shall restrict ourselves to J + J 2 << 1, we may take a = constant X ( T - T , ) / T cand p = p , its value a t T = T,. For T < T,, the mininium value of F , will occur for II, = +o(T) where l+o(T)12= -a/P 0: ( T , - T ) and the corresponding energy is F , = F , - a2/2/?. From our earlier discussion we see that a 2 / 2 @= HC2/87r. Consider now, still in the absence of any external field, a variation of +(r)from point to point. Arguments similar to those used in deriving the initial expansion of F, indicate that, providing the variation is sufficiently slow that third and higher order spatial derivatives of +(r) may be neglected, further terms proportional to la+/az12, JC3+/ayl2, ld+/C3~1~must be added to the free energy.4 The free energy per unit volume for a n isotropic system thus becomes Fa = F n

+

+ 8PI+14 + (h2/2m)IV+12-

aI+12

The constant of proportionality has been expressed in terms of a parameter m,not necessarily the electronic mass, so that the last term has the form of the kinetic energy density of quantum mechanics. Suppose now that a magnetic field is applied producing a flux density B(r) in the superconductor. Besides adding the thermodynamic field energy term B2/87rto Fa, Landau and Ginzburg replaced the term in V+ by (1/2m)I( -ihV - qA/c)+I2, in the same way that the kinetic energy in quantum mechanics is modified to include the effects of a magnetic field. Such a replacement ensures gauge invariance, providing that when the gauge of A is changed, so also is the phase of +(r)-again in the same way that the phase of the wavefunction in quantum mechanics has to be changed. I n the case where +(r) is uniform, this term reduces to (1/2m) (y2/ c2)A21+J2,which may be recognized as the kinetic energy density of the supercurrent (Appendix). The free energy is therefore

Fa = Fn

+

a1+I2

+ +PI+I' + (l/8r)B2 + (h2/2m)I(-iihV - 4A/c)+l2.

To find the equilibrium conditions in the presence of the magnetic field, it is now necessary to minimize the total free energy of the superconductor

$*Vv

Terms like lead, when integrated over t h e whole volume, to contributions of the same kind as those coming from the term lV$*l.

8

E. A. LYNTON AND W. L. MCLEAN

SF. d3r with respect to arbitrary variations of the complex order parameter

+(r) and the vector potential A(r). The result is the so-called GinzburgLandau equations: a+

and

+ P1+I2+ + (1/2m)(-ihV - qA/c12$

(c/4?r)V X V X A = J(r) = -(qhi/2m)(+* V+ -

=

0

+ V+*)

- ((r2/mC)++*A(r>*

The first of these fundamental equations describes the equilibrium spatial variation of the order parameter; the second, the current distribution, i.e., the diamagnetic response of the superconductor to the external field. The Ginzburg-Landau equations give rise to two fundamental lengths characteristic of a given superconductor. One of these we have already encountered: the penetration depth A. In a weak field, arid to first order in B, [$I2 can be replaced by its equilibrium value in the absence of a field, $02, which is independent of position. To this order, the second G-L equation reduces to one formally equivalent to the London equation:

V X J = -(q2/mc)+02B, leading to an exponential decay of an applied field with a characteristic penetration depth X2(T) = m ~ ~ / 4 ? r q ~ +0:~ l/(To ~ ( T )- T). The other characteristic length follows from the appearance of the gradient term in the first G-L equation, which prevents, by making it too costly in energy, any rapid spatial variation of +(r). The scale on which this variation occurs can be seen as follows: We write the first G-L equation in the absence of a field for a semi-infinite slab with its surface lying in the plane z = 0; we assume that depends on z only,

+

+ may be taken to be real if the field and current are zero. We introduce and

( ( T ) having the dimensions of length. With this, the equation reduces to -[(T)2(d2f/dZ2)- f f 3 = 0. Clearly ((7') is the natural unit for the spatial variation of f(z); and from

+

the temperature variation of

(Y

and its value in terms of microscopic

TYPE I1 SUPERCONDUCTORS

9

quantities, one can show that

F(T) = 5oITe/(Tc

- T)11/2,

where 50 = 0.18hv~/k~T~, V F is the Fernii velocity, and k s is the Boltzmann constant. tois the so-called coherence length of a superconductor a t absolute zero [see Goodman (9)]. Note that both X(T) arid ( ( T )diverge as T -+ Tc,but that their ratio K = X ( T ) / [ ( T )= rn~p/qh(27r)'/~ is, in the same limit, a constant. This is the so-called Ginzburg-Landau parameter of a superconductor, to which we will refer in a later section. The existence and even the approximate magnitude of the coherence length follows from quite general phenonienological considerations [Pippard ( l o ) ] .It is, as was just stated, the minimum length over which variations can occur in the order parameter. Specifically this means that the smallest natural size of a superconducting region surrounded by normal material is l o . Pippard pointed out that the finite and in fact considerable value of this size follows from the extreme sharpness of the superconducting transition. For example, in a well annealed sample of tin, the transition is observed to occur within less than a millidegree. The small limit to the statistical fluctuations, inferred from this result, indicates that superconductivity nucleates in a region with a diameter of approximately cm. The same magnitude of toalso follows from a simple argument based on the uncertainty principle. The coherence length can be regarded as characterizing the spatial definition of the superconducting electrons, related to the uncertainty in their momentum by

50 AP

-

h.

But the electrons involved in the superconducting condensation are those within an energy kBTc of the Fermi surface, so that and

-

For most superconducting elements, to 10-5-10-4 cm. For transition metals such as niobium and tantalum, however, the Fernii velocity VF is very low (due to the high density of electron states a t the Fermi surface) and T , is uriusually high. Thus for these substances tois only of the order of cm. As a result K = A/[ for transition niet,als is close to unity, whereas, e.g., K 0.01 for aluminum. Both the range of coherence and the penetration depth vary with the normal electronic mean free path. As this becomes shorter with decreasing

-

10

E. A. LYNTON A N D W. L. MCLEAN

purity of the metal, X increases and [ decreases. Thus, K increases with decreasing mean free path. We will discuss below how the basic characteristics of a superconductor with small K values differ in a fundamental way from those of one with large K values. The change from one type of behavior to the other can be produced by alloying a superconducting element, thereby decreasing its mean free path.

VI. THEINTERPHASE SURFACE ENERGY The gradual spatial variation of the superconducting order parameter together with the finite penetration depth give rise to a contribution to the specimen energy for every unit area of surface separating superconducting and normal material [Pippard (II)]. Consider a unit area of interphase boundary in a plane normal to the plane of the paper. The order I

I

H

G M FIQ.1. Variation of the superconducting order parameter

+(T)

and the flux density

B a t the boundary between normal and superconducting regions.

parameter +(r) decreases from its equilibrium value in the superconducting material down to zero in the normal material over a distance of the order of [. The flux density falls to zero from its value H , in the normal material over a distance of the order of A. Representing the gradual variation of and B by abrupt changes, it is as if the configurational or ordering boundary occurred a t C and the magnetic boundary at M (see Fig. 1). Thus in the diagonally shaded volume one loses the advantage of superconducting order, and there is a consequent increase in energy equal to this volume times the energy difference per unit volume between normal and superconducting material, Hc2/87r.On the other hand, the field is not excluded from the crosshatched volume, leading to a decrease of the energy equal to the product of this volume times the energy per unit volume owing to the exclusion of flux of density H , H2/8n.When the field in the normal region equals H,, we thus have

+

Increase in energy due to loss of order Decrease in energy due to field penetration Net increase in energy per unit area

---

[Hc2/8r XHC2/8?r ( 4 - X)HO2/8?r

TYPE I1 SUPERCONDUCTORS

11

Thus there is a net contribution per unit area of interphase boundary, the sign and magnitude of which depend on the relative sizes of the coherence length [ and the penetration depth A. It is evident that the sign of this surface energy determines whether or not it is energetically favorable to form normal inclusions in the superconducting matrix below the field H , a t which the volume energies of the two phases become equal. Thus there will be very different magnetic behavior for superconductors with positive and for those with negative surface energy, and these are accordingly differentiated by being called, respectively, type I and type I1 superconductors. Let us use the same simple approach to investigate a t which exterior field a normal inclusion becomes energetically favorable. Consider such an inclusion in the form of a thin normal thread, at the center of which we

-€FIG.2. Variation of the order parameter + ( T ) and the flux density B in a cylindrical normal region surrounded by superconducting material.

have a flux density equal to the external field, H , and a vanishing order parameter # = 0. The former decreases over a distance A, the latter increases over a distance [. The inclusion becomes favorable when the energy due to flux exclusion becomes equal to or greater than the energy decrease due to the condensation into the more ordered state, i.e., when i.e., Thus, there are two cases: (a) Type I. [ A : The field a t which isolated inclusions become possible exceeds H , ; thus the material is entirely diamagnetic for H 6 H,, as indicated in Fig. 3 by the dash lines. (b) Type 11. f ,< A : Normal regions appear a t

>

Hci= ( f / A ) H c 6 H c

12

E. A . LYNTON A N D W. L. MCLEAN

and by the same token, superconducting regions persist to fields greater than H,, as the negative surface energy compensates for the greater volume energy. Hence the magnetization curve for type I1 is as shown by the solid line. The limiting field H,2 for the persistence of a mixture of normal and superconducting material can be found from the G-L equation. For this purpose we take the z-axis perpendicular to the field and suppose that I) depends on one coordinate, z, only. It may easily be verified that, if I) varies with x or y, the value of Hc2 obtained in that case is not higher than the value obtained here. Also, we consider either an infinite medium or parts of the superconductor much further than t: from the surface so that there is no need to take the surface into account. I n the normal state, the field is uniform and so in fields with B = H >/ Hc2, a suitable vector

FIG.3. The magnetization curves for type I (dashed lines) and type I1 (solid curve) superconductors.

potential is A = (zH,O,O). The G-L equation can thus be written in terms of the zero field equilibrium order parameter # o and the quantities K and X in the form

Near Ho2, it is reasonable to assume + ( H ) <<&, so that the equation reduces to a linear one

+ 5X [ 1 - &]

9 dz2

I) = 0.

This equation has solutions I)(z) bounded a t z = & 00 only if

4 2 K H , / H = 2(n + $), where n is a positive integer or zero. The highest field for which this is satisfied is clearly H,2 = 4 2 KH,.

TYPE I1 SUPERCONDUCTORS

13

Thus we note that H C zis greater or less than H , depending on whether K is greater or less than 1/43. This is the value distinguishing type I from type I1 superconductors. I n a n earlier section we saw that K = h ( T ) / f ( T ) .The transition from type I to type I1 behavior is thus again seen to occur approximately when

-

Hcz > H , for h > f. It may be noted that the relation between HCzand H , is independent of the assumption that # depends on z only, and that a semi-infinite slab is not a necessary requirement for the result obtained. The more general derivation of this result is mathematically equivalent to the calculation of the Landau levels of an electron in a uniform magnetic field given by Dingle (1.2). h(T)

f(T),

and

VII. THESTATIC PROPERTIES OF THE MIXEDSTATE From the preceding section it is apparent that for external fields H such t ha t Hcl < H < Hc2, a type I1 superconductor is in a state which is neither entirely superconducting nor entirely normal. The detailed nature of this state, called the mixed state, can, in principle, be deduced from a solution of the G-L equations under appropriate conditions. The nonlinearity of these equations makes it necessary to use certain approximations, as was first done by Abrikosov ( I S ) . He deduced the nature of the mixed state near H C zby an iterative procedure, substituting into the nonlinear G-L equation the solution to the linearized one, modified by a small additive function. This yielded the remarkable result that in a plane normal to the applied field the order parameter is a doubly periodic function, varying from a zero value on a lattice of points, as shown in Fig. 4, to a maximum value midway between neighboring zeros. Near the zeros, the symmetry is circular. There is a gradual transition to an almost square pattern at the boundary of the primitive cell centered on a zero. I n three dimensions the pattern of field penetration into the mixed state is th a t of a grid of normal filaments, each surrounded by surfaces of equal order which also correspond to sheets along which the supercurrents flow, perpendicular to the direction of the field. Near to each of the normal filaments, the current density varies in a similar fashion to the velocity near a vortex in a classical fluid. Thus, in the mixed state the interior of the superconductor contains a set of parallel current vortices. From our discussion in Section VI, i t is evident that each such vortex, being topologically equivalent to a superconductor with a cylindrical filamentary normal hole along the axis, must carry an integral number of flux quanta &. A remarkable feature of this vortex structure predicted by Abrikosov for the mixed state is that the superconducting order is finit,e everywhere

14

E. A. LYNTON A N D W. L. MCLEAN 2

312

I

1/2

0

112

I

312

2

FIG.4. Contours of equal [$I2 (which are also lines of current flow, the tangent a t any point giving the direction of J . a t t h at point) according to the Abrikosov model. [From A. A. Abrikosov, Phys. Chem. Solids 2, 199 (1957).]

FIG.5. Lincs of magnetic flux, current flow, and the variation of order parameter and flux density near the center of a vortex.

TYPE I1 SUPERCONDUCTORS

15

except alorig the filaments of negligible volume. The mixed state can, therefore, he considered as everywhere superconduct ing and can be characterized by an average order parameter. An early verification of this came from thermal conductivity measurements, the results of which were successfully analyzed in ternis of such an average [Dubeck el al. (BG)]. Let us first look at the properties of an isolated vortex, following essentially the treatment of deGennes (5), and making the simplifying assumption that K >> 1, i.e., that X >> E. The description of a vortex is as follows : The order parameter and the effective density of superconducting carriers rises from zero a t the center to its equilibrium value over a distance of the order of f . The flux density B is maximum at the center, and it extends over a distance of the order of A, being screened by circular current loops also extending over a distance A. Neglecting the energy of the “normal core” of radius E, we have for the vortex energy per unit length

+

21rr d ~ [ B ( r ) ~ / 8 1 r$ n , m ~ ( r ) ~ ]

where n,

=

constant for r

> f . From Maxwell’s equations

V - B= O

V XB so that, putting

XL2

=

=

47rJ/c

=

4an,ev/c

mc2/4~n,e2, we get 27rr dr[B(r)2

= J7>E

+ XL2(V X Bl21/8*.

The condition that this be a minimum yields

B

+ XL2V X V X B

= 0.

This is again the London equation, showing that as expected the field penetrates a dist,ance hL. However, this equation can be expected to be valid only where there is nonvanishing order, i.e., not a t the center. The equation applicable throughout the whole region is

B

+ XL’ V X V X B

=

+oS(r),

where 6 ( r ) is the two-dimensional delta function, arid +o is a vector in the field direction, of magnitude equal to the quantum of flux. The validity of this form of the equation can be easily verified by integrating this equation over the surface bounded by a circle of radius T and encircling the vortex axis in a plane normal to the axis. The general solution of the equation is B ( r ) = +oK0(r/XL)/2aXL2where KO is a Bessel function of imaginary argument and of order zero. The

16

E. A. LYNTON AND W. L. MCLEAN

asymptotic values are, for t B(r)

arid for r >> XL,

B(r)

=

< T << XL, =

ln(XL/r)/2nh1,~

r # ~ ~

r#~(nXL/2r)'/~exp( -r/XL)/4XL2.

One can put this back into the free energy expression and integrate it to find, for the energy per unit length,

F

=

( 9 0 / 4 n X ~ln(XL/S). )~

We have not yet taken into account any contribution to the vortex energy from the core region T < [. Caroli et al. (14) have shown th a t for K >> 1 this core can be considered essentially as a normal region and that this contributes a very small additional amount to F . The total energy can be written F = ( ~ O / ~ ~ X L ) ~ ~ ( X-I-/ E€1,) where a 0.1. It is interesting to note that the energy varies as the square of 9 0 . Thus to double the amount of flux in the specimen, it is energetically more favorable to double the number of vortices rather than to have two flux quanta in each vortex. The interaction between two parallel vortices can be determined in much the same way, except that now the magnetic flux distribution is determined by

-

B

+ X L ~ VX V X B = +,[6(r - r,) + 6(r - r2)l,

where r1 and r2 are the position vectors of the two vortices. The solution of this is

+ B2(r),

B(r)

=

Bdr)

B4r)

=

( $ J ~ / ~ T X ~~ )r i K l/X~ L ) .( J ~

where

As before, this is substituted into the expression for the energy, yielding in addition to the self-energy of each filament an energy of interaction per unit length pi2 = 4oBi2/4T, where BI2= Bl(r2) = B2(rl) = (+o/4~~2)Ko(lrl - r21/xL). ~ ' ~* 1 = 2 This is a repulsive energy which varies as exp ( - ~ ~ 2 / X ~ ) / r ~ 2for Irl - r21>> X and as ln(X/rlz) for E < r12 << XL. With these expressions for the self energy and the interaction energy of vortex lines, i t is now possible, a t least in principle, to derive in detail the magnetic behavior of a type I1 superconductor. The Gibbs free energy

TYPE I1 SUPERCONDUCTORS

17

per unit volume of a specimen containing n (>>1) vortex lines per unit area perpendicular to the lines is G = nF Zij Fij - HB/47r. As every vortex carries a single flux quantum c $ ~ , we have for the average flux density,

+

B

= n+o.

The first question of interest is at what field H c l it is energetically favorable to have flux penetration. At this field the line density will be low, so that we neglect the interaction term and write

G

-

nF - B H / 4 n

-

B ( F / & - H/47r).

The dependence of G on B changes when H = 47rF/&,. For H < 4nF/&, G increases with B, and the lowest energy is obtained for B = 0, i.e., B = 0, the so-called Meissner state. For H > 47rF/&, G decreases with increasing B, so that flux penetration becomes advantageous. Clearly, therefore, Hcl = 47rF/d0 = ( 4 0 / 4 r X ~ ~ln(AL/E). ) For K = X/E > > 1, Hcl << H,.

As soon as vortex lines begin to penetrate, their interaction must be taken into account. This interaction, as we have seen, falls off exporientially, according to exp ( - r 1 2 / X L ) , and thus a t HO1lines will very quickly move to within a distance -XL of each other without much cost in energy. It is clear, and can be demonstrated formally, that this produces a n infinite value of the slope of th e magnetization curve a t H c l . The exact nature of the transition a t Hcl from the Meissner to the mixed state has been the subject of much discussion. If the line interaction were indeed of finite range, i.e., became zero a t r12 3 XL, the magnetization would change discontinuously and the transition would be of first order and there would be a latent heat. No evidence for this has been found in specific heat measurements. Recent, very careful investigation of the magnetization of pure niobium [Serin (15)Jin fact suggests that the transition is of the so-called lambda type, with the magnetization near Hcl varying logarithmically with field. As the external field increases beyond H c l , the vortices will form a regular array so as to minimize their interaction energy. Detailed calculations of the total energy have shown that, throughout the mixed state, the most advantageous arrangement is a triangular one. This has been verified in a most direct and convincing manner by Cribier and co-workers (16), who compared thc diffraction of very slow neutrons by a specimen in the mixed state with that, by the same specimen in its normal phase. The neutrons are scattered because of the interaction of their magnetic moment with the field in the vortex line. Th e scattering angle is very

18

E. A . LYNTON A N D W. L . MCLEAN

nearly equal to the ratio of the neutron wave length to the vortex spacing. Even with the slowest available neutrons the former is no more than about 5 8, while the latter is more than 100 times larger. Thus, the scattering angle is only of the order of minutes of arc, so that the peak is very difficult to observe. The more the external field increases beyond Hcl,the closer the vortex lines move together. This is evident from the expression for the free energy: the term -BH/4?r makes it increasingly advantageous to increase B as H increases. This corresponds to a magnetic pressure resulting in a n increasingly tight packing of the lines and a corresponding growth of their inter action energy. For fields well above H c l , the simple model used thus far breaks down, for one can no longer ignore the contribution of the core regions to the energy. One is thus forced to go back to the Ginzburg-Landau equations. The following treatment is based on the approach of deGennes (27). When H is just a little less than Hc2,the values of which satisfy the complete, nonlinear G-L equation must be very close to values +L which satisfy the linearized equation

+

(1/2m)[-iihV

where V X A.

=

- 2eAo/cJ2+L=

-&L,

H,z. Th at is, we assume +L satisfies

+ P ~ + I , ~ ~++ L(1/2m)(-ihV

- 2eA/c)%

a + ~

= 0.

It is possible to make certain general statements about +L without a knowledge of its detailed form. Multiplying by +L* and integrating over the volume of the superconductor, we get a m

+ Pm + (1/2m)](-ihv

- 2eA/c)+~l% = 0,

where 8 = J X d 3 r /Jd%, and we have used partial integration to rearrange the last term on the left-hand side. We now want to simplify this by making use of the requirement that +L satisfies the linearized equation. This can be done by letting

A

=

Ao iAil

where AD is the vector potential corresponding t o the uniform induction Hc2,and A1 is the small modification due to the slight variation in the flux density due to the existence of supercurrent vortices. T o first order in A,, and using the linearized equation, the above becomes where

Pm - (l/c)Al - jL jL = (e/m)[+L*( -ihV - 2eAo/c)+L

=

0,

+ complex conjugate].

19

TYPE I1 SUPERCONDUCTORS

This is the current associated with the solution

v X A1, v X b,

bl =

=

+L.

Defining

44/c

and integrating by parts, we obtain

- iFf5/47r

PIJ.LJ4

=

0.

It follows from the definitions that

bl

H - He2 + b,.

=

(2)

Furthermore, we have stated in the earlier description of vortices, and it can be shown formally, that the current sheets of jL coincide with the surfaces J $ L ( T ) \ ~ = constant, i.e., jL = (eR/m)k x vJ+LJt where k is a unit vector parallel to the magnetic field. Hence, since

v x b,

=

4&/c,

we have

b,

=

- (4?reh/mc)&,J2.

Combining this with Eqs. (1) and ( 2 ) reduces the normalization condition to

Pm 4-

(eh/mc)1gLIz[H- H C z - (4?reh/mc)(+~J~] = 0.

Again introducing reduced variables by setting I+LI

=

f+o

and using the defining equation K

=

(P/2~)l/~mc/2eh

yields F(1 - 1/2K2)

- p(1 - H/H,2)

0.

This equation holds very near Hc2 independently of the form of +L and in particular independently of the nature of the array formed by the vortex lines. This result was also obtained by Abrikosov ( I S ) using a less general form of $. This allows us to write down the values of a number of important quantities near H c 2 . The average flux density is the average of the microb,, i.e., scopically varying B = H

+ B = H +& = H

- (4?reh/mc)m ( H , / d z K)F.

=H -

The average free energy per unit volume is

F

=

- (P/Z)lJ/L14

+ ( B 2 / 8 ~ =) - (Hc2P/8a)+ ( B 2 / 8 ~ ) .

20

E. A. LYNTON AND W. L. MCLEAN

This can be reduced to

f

=

[ ( 8 ) ’ / 8 ~ ]- {(He’

- @‘/[I

f”/(F)z

+ (2K’ - ~ ) P L ] ) ,

where P L = is a numerical constant close to unity which depends on the particular forms of the solution, but not on 8. From this and the relation d p / d B = H/47r it follows that &? =

and

(8- H)/4T

= (H

- Hc2)/P~(2K2- 1)

d M / d H = 1/@~(2K’- 1).

A simple model of the mixed state has recently been proposed by Rothwarf (18). It is based on the consequences of assuming the angular momentum of each of the pairs of electrons which enter into the theory of superconductivity to be one atomic unit, i.e., h. Each pair has associated with this orbital angular momentum a magnetic moment equal to the Bohr magneton eh/2mc. I n the presence of a n externally applied field, the fraction of dipole moments aligned a t a given instant parallel to the field is different from the fraction aligned opposite to the field. The maximum diamagnetic moment per unit volume is then the dipole density 4 2 , where n, is the electron density, multiplied by the moment of each dipole, i.e.,

where X L ~= mc2/47rn,e2.Therefore, the maximum external field which can This should be be screened from the interior is -47rMmnx = I$o/47rX~2. compared with the expression for Hcl obtained above for deGennes’ vortex model and with Abrikosov’s result,

Hcl = ( ~ + 0 / 4 r X ~ ~K) (4-l n0.08). Above Hcl, Rothwarf assumes that normal cylinders of radius 5 are formed inside the superconductor, each cylinder containing flux I$. The number of cylinders per unit area normal to the field is B/t$, so that the volume of diamagnetic material is reduced by a factor ( 1 - TF~B/+). The normal cylinders are assumed not to contribute to the magnetic moment, so that the average magnetization is A? = - ( H c 1 / 4 r )(1 - 7 r E 2 B / ~Using ). B =H 4 r M , this gives = -(Hcz - H)/47r[(Hcz/Hel)-1] where H,z = 4 / 4 ~ 5 Assuming ~. M = - H / k for H < H c l and the above result for H G 1< H 6 HC2yields

+

TYPE I1 SUPERCONDUCTORS

21

The expression for the magnetization becomes M = ( H - H C z ) / 4n(2X2 - l), where it has been assumed that C$ = C$o and X = XL/[. This should be compared with the result obtained by Abrikosov from the Ginzburg-Landau equations which was mentioned earlier. Although the more detailed theory of Abrikosov deals thoroughly with certain aspects which are either assumed or ignored in this model, it is interesting that the general results can be obtained in such a straightforward way. Rothwarf has recently extended the model to include magnetic effects arising from the motion of the centers of mass of the pairs and there is good agreement between the results and some experiments by Cardona et al. (19) on the barrier to flux movement through the surface of the superconductor. We have seen that the G-L parameter K characterizes the magnetic behavior of a type 11 superconductor in a number of different ways. It determines the relation of H C 1to H , and that of HOzto H,, and it also enters into the expression for the slope of the magnetization curve near HCz. Near T,, the values of K obtained for a given sample from these three relations should all be the same. Indeed this has been verified by careful magnetization measurements of well annealed alloy samples [Kinsel et al. (20), Boil Rlardion et al. (21)],which also confirmed the basic shape and the reversibility of the magnetization curve predicted by the Abrikosov theory. Furthermore, the values of K determined in this fashion also equal what might be considered to be the fundamental K value characteristic of the given sample, namely that given by the ratio of the penetration depth to the coherence length. This ratio can be related to quantities describing the behavior on a microscopic scale of the superconductor. The phenonienological Ginzburg-Landau treatment is strictly applicable only in a narrow range of temperatures just below T,. This is true for a number of reasons. First of all, the simple form of the coefficients a! and /3 in the expansion of the free energy in terms of the order parameter is valid only very near TC.I n addition, the range of variation of the order parameter must be large compared to the range of coherence, [ ( T )>> to, and again this holds only near T,. Finally, a local relation between superconducting current density and the vector potential can be used only when the latter varies slowly over distances of the order of to,i.e., if A(!)’ >> to, which happens near T,. I n recent years the theory has been extended to lower temperatures yielding different temperature [Gor’kov (22); deGennes (23); Rlaki (24)], dependences for K values deduced from various features of the magnetization curve. It is useful to follow nlaki in defining

22

E. A . LYNTON AND W. L. MCLEAN

K ~ ( Tas) the value deduced froni d@/dH near H,z, and K ~ ( Tas) that

obtained from H,l(T). In the limit of low electronic mean free path, theory predicts that both K~ and K 2 rise slowly with decreasing temperature by unequal but similar amounts. This has been verified by experiments on alloys. However, in the

0000 0000000

I I 0000000 0000

0000 a9 000000 0000!3~po0000 (b)

00000000000 00000000 00000 FIG. 6. The relation of the vortices to the thin film used by Parks et al. (28) (a) At low fields the vortices are too large to fit into the narrow arm joining the two more extensive parts of the film. (b) At a certain field the vortices can just fit inside the arm and do so to reduce the free energy of the system.

pure limit, there is no such agreement. Both calorimetric and magnetic measurements on pure niobium [McConville and Serin (26), Strnad and Kim (SS)]show that ~ ~ ( 5 "increases ) by about as T -+ 0, whereas the calculation predicts a behavior much closer to that in the impure limit, in which K ~ T( ) increases by only about 20 %. Furthermore, ~ 2 T( ) is experimentally found to increase with lowering temperature also in the pure case, whereas it is predicted to decrease.

+

TYPE I1 SUPERCONDUCTORS

23

We conclude this section by noting that current vortices can also occur in type I superconducting materials under special conditions : namely, when the superconductor is in the form of a thin film normal to a magnetic field of magnitude slightly less than that required to destroy completely the superconductivity. Tinkham (27) has found that fluxoid quantization plays a dominant role in this situation and has carried out a n analysis using the Ginzburg-Landau theory, predicting the variation of the critical field with angle of inclination to the surface and the dependence on temperature of the critical field when the field is perpendicular to the film. Parks et al., (28) have recently carried out experiments in which the size of the vortices in a thin film perpendicular to a strong field was observed to have an effect on the transition temperature of part of the film. The film was of uniform thickness but was of the shape shown in Fig. 6. An increase in the transition temperature of the bridge joining the two more massive parts was observed when the magnetic field reached a value a t which the vortices were of small enough diameter to fit into the bridge, presumably because the free energy of the system was reduced once the vortices were able to spread into the bridge. We shall return later to discuss further experiments on vortices in thin films th a t have considerably clarified our understanding of the mixed state. VIII.

SURFACE SUPERCONDUCTIVITY

For many years experimenters have observed that in certain metals superconductivity, detected by resistance measurements, persisted in magnetic fields higher than the field required to restore the diamagnetic moment of the sample to its normal state value. This phenomenon was generally ascribed to strains or inhomogeneities which cause the metal to behave in a way different from that of a pure system. However, SaintJames and deGennes (29) have shown that when the externally applied field is parallel to the surface of a superconductor, a layer of thickness -l a t the surface remains superconducting up to the field Hc3 = 1.69 4 KH,.This result was obtained by a treatment similar to th a t described in Section VII for obtaining Hcz = 4 9 KH,,the maximum field a t which superconductivity can nucleate from the normal metal. Surface effects were not considered in that derivation. However, the solution of the Ginzburg-Landau equations near the surface is quite different from in the interior, and if the magnetic field is parallel to the surface, superconductivity can nucleate at a maximum field Hc3. Numerous experiments have since been carried out verifying the existence of the surface layer of superconducting material and in many cases confirming the quantitative estimate for H c 3 .I t was found by Rosenbluni and Cardona (SO)that surface superconductivity could also occur in type I superconductors since al-

24

E. A . LYNTON AND W. L. MCLEAN

though in these K < 1/42, it is possible for 1.69 4 3 K to be greater than 1, i.e., 4 3 K H , < H , < 1.69 4 3 KH,. If the magnetic field makes an angle 0 with respect to the surface, the superconducting layer is destroyed at a field lower than Hc3.When 0 = 90"' the critical field is H,z. Saint-James (31) has extended the solution of the Ginzburg-Landau equations in the vicinity of the surface to cover 0 6 0 6 90". Tinkham's (2'7) treatment of the transition in thin films, mentioned in the last section, agrees well with this exact solution only when the film thickness is much less than E . I n experiments carried out on cylindrical samples with their axes parallel to the magnetic field, it might be thought that the annular region of superconducting material surrounding the cylinder would act like a perfectly conducting shield and prevent magnetic flux from entering the interior of the rod until the external field had reached Ho3. A simple calculation shows that the positive magnetic contribution to the free energy because of flux exclusion is much greater than the negative contribution from the condensation energy of the surface layer ( H O z< H < H c 3 )and so exclusion of the flux in the way envisaged would be thermodynamically unstable. We are thus lead to the concept of a maximum or critical net current that can be carried by the surface sheath. If the flux density were the same on both sides of the layer-in the interior of and outside the cylinder-there would be diamagnetic currents shielding the inside of the layer from the magnetic field, but no net current. A gradual increase of the external field would induce a net current, proportional to the difference between the external and internal flux densities, but the current eventually would reach its critical value whereupon the flux inside the cylinder would increase. Except for the very careful measurements of Sandiford and Schweitzer (32), bulk magnetic moment measurements on samples which should have had superconducting surface layers have never revealed any shielding by the surface layers. The shielding would manifest itself by an obvious hysteresis pattern. I n the mixed state with HC1< H < Hc2, the absence of vortices from the region of thickness 5 near the surface implies that there should be small shielding effects there just as in the range H,z < H < Hc3. I n practice these are obscured in bulk magnetic moment measurements by flux trapped in the interior which dominates the hysteresis. Rothwarf (33) has developed the simple model mentioned in Section VII, which treats the orbital motion of electron pairs about their centers of mass differently from the motion of their centers of mass, and he has predicted that the high-frequency surface impedance should vary linearly with field between HCzand Hc3.Such behavior has recently been observed

TYPE I1 SUPERCONDUCTORS

25

in the surface rwctancc at 2 Nc/se(. by Carlson (34) arid in the microwave surface resist ariw by Gittlemuii and Rosenblum (35). A linear variation in the surface impedanc’e has also been predicted by Malti (36), from a solution to the Gor’lcov equations (8)’ for the surface sheath of very impure superconductors. Even the most impure systems studied so far appear to be relatively too pure for the theory to apply.

EFFECTS IX. DYNAMIC The phcnoriiena discussed in previous chapters have pertained to the equilibrium states of a type I1 superconductor-the order parameter has been considered to vary spatially but not with time and the appropriate free energy of the system minimized to determine the state of thermodynamic equilibrium. In this chapter, we consider effects in which there is also a temporal variation of the order parameter. Ideally one would like to obtain from the microscopic theory not only the solution given by Abrikosov, discussed in Section VII, which applies to the system in equilibrium, but also solutions applicable when the equilibrium is disturbed by the application of external fields or which would describe the transient behavior when the magnetic ficld is changed from one static value to another. Although some advances have been made from the standpoint of the microsvopic theory, a clearer physical picture of dynamic effects is emerging from a hydrodynamic treatment of the motion of a vortex through the superfluid. Experimental results are complicated in many cases by structural defects in the metal. We shall restrict our discussion mainly to situations where there is a clear-cut and significant agreement or disagreement with the predictions of the vortex model.

A . Steady-State Flux Flow I t has been found that when a steady current is passed through a type I1 superconductor in the mixed state, electrical resistance may appear, in spite of the large fraction of the material that is superconducting [Kim et al. (37)l. T he interpretation of this result in terms of the motion of magnetic flux has been important in the development of models of vortex motion [Anderson and Kim (38)l. Figure 7 shows the potential drop across the type I1 superconductors NbsoTasoand Pbs31n17as a function of current for various values of the magnetic field. At low currents there is no measurable resistance. As the current is increased a potential drop appears, indicating an electric field inside the superconductor. This field is generated by electromagnetic induction owing to the movement of vortices-either individually or in linked groups. The vortices are driven by what has been misleadingly called the “Lorentz force,’’ which has the form J X +o/c per unit length

26

E. A. LYNTON A N D W. L. MCLEAN

of the vortex, where J is the superimposed or transport current density and the flux in a vortex. This expression has been derived by a thermodynamic argument by Friedel et al. (39). It may be noted that this force is in the opposite direction from the reaction to the Lorentz force per unit

+,,

I (amp)

FIG.7. Flux-flow resistance of two different type I1 superconductors, after Kim et al. (37).

volume J X B/c which acts on the current carriers passing through the flux of the vortex. That some caution is needed in using the simple classical treatment of a charged particle moving in a magnetic field should be apparent from a n attempt to consider the motion of the electrons in a

TYPE I1 SUPERCONDUCTORS

27

single vortex. Some force besides the Lorentz force must be invoked in order to explain the motion of the current in circular loops. We return to the nature of the forces in the next section. At low currents, the “Lorentz force’’ is insufficient to overcome the forces which “pin” the vortices to imperfections in the crystal lattice. The probability of the vortices leaving their potential wells is increased by raising the current and also by raising the temperature. At high currents, the trapping mechanism has little effect and the vortices “flow,” i.e., move through the metal with a speed limited by a viscous force that we discuss later. Motion of the flux occurs at intermediate values of the current by the “flux creep” process in which the vortices spend part of their time “flowing” and the rest of it trapped by imperfections. I n addition to the characteristics which stand out in Fig. 7, it has been found that the resistivity varies with field according to p / p , = H/Hc2. From the Abrikosov theory, H B &,/d2, where d is the average distance between vortices, while Hcz c # Q / ~ .so ~ , that H/H,2 = E2/d2 (the fraction of the volume that is in the normal state). This assumes, as has been demonstrated b y Caroli et al. ( 1 4 , th at the vortex core can be regarded as a cylinder of normal material of radius [. The field variation of the resistivity is thus proportional to the fraction of normal metal, and leads to the surprising conclusion th at the current flows uniformly through the normal and superconducting parts-instead of avoiding the normal cores and passing through the superconducting parts only. Microwave surface resistance measurements [Rosenblum and Cardona (do)]have also supported this conclusion. We return to a discussion of this result in the next section. The fact that the electrical resistance in type I1 superconductors apparently arises from electromagnetic induction has been a considerable source of confusion and discussion. Th e experiments referred to above were carried out under steady-state conditions so that the flux through the measuring circuit did not change with time. On the other hand, there was a movement of flux through the superconductor, from which a n electric field arose within the superconductor. The clearest general explanation of the paradox has been given by Josephson ( & ) , who has shown th a t in the steady state the difference in the electrochemical potential between two points in a superconductor is equal to the rate at which flux crosses a line in the superconductor joining the two points. This treatment is based on the assumption that the driving force for the current in a superconductor is proportional to the electric field and to the gradient of the chemical potential pc, i.e., that

- --

28

E. A. LYNTON AND W. L. MCLEAN

+

where p = po eV is the electrochemical potential and V the scalar electric potential. This assumption appears to lead to a correct explanation of the observed thermoelectric and magnetoelectric behavior of superconductors [see also Luttinger (4S)], one which is in accord with the theory mentioned in the last section of this chapter. It may be noted th a t the motion of flux in quantized amounts through the superconductor does not imply that flux enters at one side and leaves at the other side of the superconductor. When a vortex approaches the boundary, its flow pattern is no longer circular since no current can flow across the boundary. Just as boundaries may be taken into account in electrostatics and magnetostatics by the method of images, the actual flow pattern in the superconductor can be synthesized by the superposition of the patterns of two undistorted vortices, the first with its center a t the center of the real vortex, the second the image of the first in a mirror coincident with the boundary of the superconductor. As the center of the vortex approaches the boundary, there is an overlap between the flow patterns of the two auxiliary vortices. Since their senses of rotation are opposite, they tend to cancel each other. Finally, the first vortex is completely annulled by the second when they both reach the boundary. The flux bundle does not pass out of the superconductor but merely dies away.

B. Vortex Motion Many of the conclusions drawn from the experiments above have been derived from a hydrodynamic treatment of the motion of a single vortex moving through the superfluid of electrons [Bardeen and Stephen (43); NoziBres and deGennes (&)I. The starting point in these theories has been the application of Euler’s equation, p[(av/at) v VV]= the total force density, to the superfluid which has been assumed to follow London’s equation (see Section 111) a t every point, except in the vortex core. The self-consistent internal electromagnetic field is in this case the analog of the pressure gradient which makes the important contribution to the force density in an uncharged, nonviscous classical fluid. The situation studied has been a single vortex with a transport current superimposed upon it. From the Euler equation, the fields in the superfluid have been related to the vortex motion: the field and current in the vortex core have been connected with the fields in the surrounding superfluid by use of continuity conditions a t the core boundary. The first of the two treatments cited above arrives at the unexpected result mentioned in the previous section that the applied or transport current density is the same inside the core as outside it: the current flows equally through both normal arid superconducting parts. This equality is assumed in the second treatment as the only quantitatively siniple assumption that will lead to

+

TYPE II SUPERCONDUCTORS

29

resistance of the same order as that observed in practice. The dependence of the flux flow resistance on magnetic field is also explained. We summarize here the theoretical predictions regarding the actual motion of the vortex under the driving force produced by the transport current and a viscous force arising from power dissipation in and around the core. Bardeen and Stephen have obtained V L ~ ~=T V L ~ tan T CY = V T H / H , ~ where U L ~ ~and T uLIT are the components of the velocity of the vortex line and V T is the drift velocity of the transport current. The relative orientations of these two velocities and the velocity of the electrons in the core, zr, are shown in Fig. 8. Also, E, is the uniform electric field in the core of

FIG. 8. The relative orientations of the “Lorentz force,” PI,,the electric field E, and the drift velocity v, in a vortex core, the velocity of the vortex line n,and the superimposed superfluid flow velocity VT (parallel to the transport current JT),according to the analysis of Bardeen and Stephen (43).

the vortex, J T is the transport current density (parallel to VT), and F L is the “Lorentz force.” CY is the Hall angle for the normal core and is given by tan CY = wcr, where wc = eB/m and 7 is the relaxation time for the normal state. I n the very impure case, wcr << 1 and the line moves perpendicular to the current in the direction of the “Lorentz force.” At the other extreme when wc7 >> 1, the line moves with the transport current, as expected by analogy with the classical fluid. A solid cylinder placed in an originally uniform flow of fluid of velocity v1, with its axis perpendicular to the fluid flow direction, is subject to a force along a direction mutually perpendicular to the cylinder axis and ul. This force is called the Magnus force and its magnitude is pkvl per unit length of the cylinder, where k = $v. dl is the circulation of fluid around the cylinder-the contour

30

E. A . LYNTON AND W. L. MCLEAN

of integration being just outside the cylinder-and p is the fluid density. If the cylinder is in motion with velocity VL, the Magnus force becomes p k ( ( ~1 V L ) ~ I. n the steady state, the total force is zero so the cylinder moves with the fluid, with VL = vl. Similarly the core of a vortex in a classical fluid may be treated as a solid cylinder and the same behavior deduced for the motion of a vortex line. It should be noted that the Magnus force is of the same nature as the centripetal force in circular motion and has to be supplied by some physical means. I n the classical fluid case, it arises from the gradient in pressure. I n the superconductor it is produced by the action of the electromagnetic field. The analogy with the classical fluid suggests that, in a pure type I1 superconductor, it should be possible to detect the Hall effect in the mixed state, with a large Hall angle [Vinen (46)l. (The Hall angle in a type I superconductor is zero.) The theoretical investigation mentioned above predicts the Hall angle to be that of the normal metal in a magnetic field equal to the field in the core. Recent observations in type I1 superconductors [Niessen and Staas ( 4 6 ) ; Reed et ul. (47)] have shown the Hall angle to be large but not as large as in the normal state, possibly because of pinning effects preventing free motion of the vortices under the action of the driving and viscous forces. No detailed comparison with theory of the complicated results reported in the former reference has yet been made. C . Vortez Wuves

A taut string plucked aside from its equilibrium position undergoes vibrations which can be described in terms of waves traveling along the string. Similarly, a vortex line disturbed from its equilibrium position in a magnetic field, by, for instance, passing a localized current impulse near one part of the vortex, is expected to undergo a precession about the steady field direction which can be described in terms of circularly polarized waves traveling along the vortex. A derivation of the wave equation has been given by deGennes (6). A search for such waves in the mixed state of type I1 superconductors has been carried out in many laboratories without success [see, for instance, Borcherds et al. (48)].The analyses of NoziBres and deGennes (44), and of Bardeen and Stephen (43), have indicated that the criterion for the propagation of these waves is W ~ T>> 1; where wC is the cyclotron resonance frequency, eB/m, at an induction B equal to the flux density in the cores of the vortices in the mixed state; and 7 is the relaxation time in the normal state of the metal. I n all the cases studied so far, W ~ Thas been much less than unity (in many cases the type I1 materials were alloys with low T ) , mainly because the highest field at which the mixed state can be studied, Nc2, is relatively small in com-

TYPE I1 SUPERCONDUCTORS

31

parison with the values with which the magnetic properties of normal metals are observed. Consideration of the motion of a vortex is of course a convenient model for analyzing the time variation of the order parameter in the mixed state. Nozihres arid deGerines (49) have also treated the motion of a “flux-tube”-such as could be formed in the intermediate state of a type I superconductor [cf., e.g. Faber (50)], and have predicted th a t the phase boundary in that situation moves in such a way that again circularly polarized waves can propagate with the dispersion formula q2 = w / h 2 w 0 . Here q and w are the wavevector and the angular frequency of the waves respectively, XI,^ = m c 2 / 4 ~ n e and 2 , w o is the cyclotron resonance frequency a t a flux density H , (the critical field), regardless of the external field H , providing H 6 H,. This formula is similar to the dispersion formula for helicon waves (51) in the normal metal except that w o is replaced by the cyclotron frequency a t a field H . The methods of excitation and detection of such waves are similar to those for helicons. Circularly polarized waves with a dispersion which depends on the external field in the manner predicted by NoziBres and deGennes have been observed in very pure indium by Hays (52) and more recently by Kushnir (53),and by Maxfield and Johnson (54).The damping of these waves was much less than might be expected from a set of unconnected cylindrical superconducting regions, suggesting that a filamentary structure, of the type observed in aluminum by Faber (50), forms also in indium in the intermediate state. An elegant deinoristration of the existence of circularly polarized or precessional modes in the intermediate state has been given by Haenssler and Ririderer (55) who observed that when a field was applied normal to a superconductirig indium disk which had been sprinkled with fine diamagnetic niobium powder, a spiral pattern of flux entry (or flux escape when the field was reduced) was formed. A similar demonstration has been given by DeSorbo (66) from moving photographs taken of a surface just above which was a cerous phosphate plate of high Faraday coefficient. Linearly polarized light has its plane of vibration rotated by an amount which depends on the density of flux emerging from the surface of the superconductor immediately below.

D. The Surface Burrier It had been noted in experiments on type I1 superconductors th a t there was considerable irreversibility-for instance, the magnetization depended not only on the value of the magnetic field and the temperature but also on previous values of the magnetic field; i.e., whether it was being increased or decreased. Bean and Livingstori (57) suggested that apart from the effects of structural imperfections, there was a n intrinsic

32

E. A . LYNTON A N D W. L. MCLEAN

irreversibility caused by a potential energy barrier for vortices a t the metal surface. As a vortex approaches the boundary, it becomes distorted as discussed earlier and is attracted towards the boundary (there is an attraction between the two counterflowing vortices-the undistorted vortex and its image). However, if the superconducting order parameter is nonzero, the magnetic induction falls off below the surface, providing a magnetic pressure gradient which repels the vortex from the boundary. The second of these two forces is dominant except when the vortex is close to the boundary. deGennes arid Matricon (58) have considered the barrier by taking into account the presence of the surface in solving the Ginzburg-Landau equations. Their result is that, although it may be energetically favorable for a vortex to form in the body of the material once H is greater than Hcl, nucleation cannot occur a t the surface until H = H,, the thermodynamic critical field. A convincing verification of this result has been obtained by DeBlois and DeSorbo (59). From this model we might expect that near the surface there would always be a lower vortex density than in the body of the superconductor. I n other words, the order parameter to a depth approximately E near the surface should not be depressed as it is in the vicinity of the vortex core, although the body of the material may be closely packed with vortices. This is borne out by detailed calculations from the Gineburg-Landau equations [Fink (60)l.

E. Vortex Motion in Thin Films As has already been mentioned in Section VII, vortices can form not only in type I1 superconductors but also in thin films of type I superconductors, with the strong magnetic field perpendicular to the plane of the film. An interesting connection between vortex motion and the more fundamental theoretical formulation of dynamic effects in terms of the timedependence of the order parameter +(r) has been deduced from experiments on thin film bridges [Anderson and Dayem ( S l ) ] .According to the microscopic theory of superconductivity [Gor’kov (S)], the energy gap function A, and hence the order parameter for the electrons in a superconductor, contains a time-varying phase factor exp ( - 2 i p t / h ) , where p is the electrochemical potential. I n many circumstances, the phase has no observable effect since it is usually quantities like \+I2 or $* V+, etc., which govern the behavior of the measurable properties of a system. A case where the phase factor does matter is in the Josephson effect in the tunneling of a current from one superconductor to another through a n insulating layer separating the two (62). In addition to other effects, if a potential difference V is established between the two superconductors

TYPE I1 SUPERCONDUCTORS

33

across the insulating layer, an alternating current of frequency f,given by hf = 2eV, passc.s lhrough the insulating barrier. The thin film bridge also evidently act,s in the same way as the insulating layer in a tunneling experiment, allowing weak coupling of the two more massive pieces of superconductor on either side. When a current is passed across the bridge, a potential difference may arise, as it does in the flux-flow experiments, through motion of the vortices in a direction that has a component perpendicular to the superimposed current flow. I n Anderson and Dayem’s experiment, the superposition of an oscillating current of frequency f caused a resonancepresumably with the Josephson-type oscillationwhich was detectable in the dc voltage-current curves a t voltage intervals 8V given by hf = 2e SV. Anderson and Dayem have suggested that such effectsmay be understood in terms of what they call the “other” GinzburgLandau equation

+.

which explicitly gives the time variation of Here AD is the Debye screening length, and is related to the London penetration depth XL and the Fermi velocity V F by AD = X L u p / d $ c. A phenomenological derivation of this equation has recently been given by Anderson et al., (65). A similar result has been obtained from an extension of the Gor’ltov (8)theory by Abrahams and Tsuneto (64). APPENDIX

The Free Energies of a Superconductor in a Magnetic Field Magnetism problems are usually formulated in terms of either magnetic poles or Amperian currents. In the latter approach, the magnetic material is hypothetically replaced by a nonmagnetic medium in which there is a distribution of currents which produces the same flux density a t every point in space as exists in the actual case. If the magnetic moment per unit volume of the magnetic material is M(r), the Amperian current density is given by J(r) = curl M(r). Both the Amperian currents and the conduction currents (of density Jeond.) are coupled to the magnetic flux so that curl B = 4s(J Jcond.). The magnetic properties of superconductors arise from the fields generated by the supercurrents which flow in the superconductor. These must not be counted both as Amperian and conduction currents. Here we prefer to consider them as Amperian currents and to relate a density of magnetization to them by the equation given above. To avoid the complications arising from demagnetization effects, we

+

34

E. A . LYNTON A N D W. L. MCLEAN

shall restrict ourselves to specimens of negligible demagnetizing coefficient, such as long thin cylinders with the external field parallel to their axes. We now derive first, of all the Helmholtz free energy per unit volume, F , using the result that the change in F is equal to the work done on the system during an isothermal change. Prom Poynting’s theorem, the rate of flow of energy across a closed surface just outside the boundary of the superconductor is

dW/dt

= - (c/4n)JE = (1/4s)J(B.

.

X B n dS aB/at E aE/at) d3r

+

+ JE . J d3r.

Assuming that the electric field exists only during the transient stage of applying the magnetic field, we get for the work done on the superconductor

/ (F(B,T) - F ( 0 , T ) ) d3r / d3r { (1/4s) B - dB + 1:- E - J dt} / d3r { B 2 / 8 r + /:- E - J dt}, =

/oB

=

where the upper limit of integration B is the steady flux density that has been reached a t t = 0 in the volume element d3r. Thus we may take5

F ( B , T ) - F ( 0 , T ) = B2/8n

+ 1:- E

J dt.

The second term represents the density of kinetic energy of the supercurrent, being the work done in setting up the current. The thermodynamic potential whose minimum gives the condition for equilibrium in the presence of a fixed external field is the Gibbs free energy

G

=

-

F - H B/4n

where H is the flux density of the external field alone or the magnetic intensity. Thus

G(B,T) - G ( 0 , T ) = ( B 2 - 2 H . B)/87r

+ kinetic energy density.

The part of this energy that the system would have if it were nonmagnetic, that is if B = H , is not of interest and we subtract it off, leaving

G(B,T) - G ( 0 , T ) = (B - H)2/8a

+ kinetic energy density.

The kinetic energy density can be expressed in a number of different forms. For instance, if the supercurrent satisfies the London equation A aJ/at = E, where A = 4 ~ X ~ / then c~,

JE* J dt = JA(aJ/at) * J dt = J(a/dt)(+AP) dt = +AJz. The energy of a system in general cannot be considered to be localized in particular parts of the system [see Heine (6741. 6

35

TYPE I1 SUPERCONDUCTORS

Using J

=

nev arid A

m/ne2, we get

=

+hJ2 = n(+mv2).

Alternatively, from the other London equation,

chJ

=

- A , +AJ2 = ne2A2/2mc2.

This should be compared with the expression on p. 7, recalling that in the Ginzburg-Landau theory, n = J$12. Finally we show the relation between

dW

=

- + JE - J dt] d3r

J[(1/4s)B dB

and a more usual forin for dW.

dW

=

= = = =

- +

J d3r [(1/4s)B dB JE curl M dt] J d3r [(1/4n)B dB + JM curl E dt] J d3r [(1/4s)B * dB - JM aB/dt dtl J d 3 r (1/4s)(B - M) . d B J d3r (1/4s)H dB

+ J dt Jn

*

E X M dS

-

(M = 0 over the surface of integration which is outside the superconductor.) Again, subtracting off the work that would be done if the superconductor were nonmagnetic, we get for the contribution per unit volume H dM/4s. ACKNOWLEDGMENTS We are very grateful to the persons mentioned in thc list of references who have privately communicated results of their work prior t o publication. We wish to thank Mr. D. E. Carlson for help with the drawings. We are grateful to Professor E. Abrahams and Professor P. R. Weiss for pointing out some errors and obscurities in the original manuscript.

REFERENCES 1. E. A. Lynton, “Superconductivity,” 2nd ed. Methuen, London, 1964. 1. A. B. Pippard, The Dynamics of Conduction Electrons, in “Low Temperature Physics” (C. DeWitt, B. Dreyfus, and P. C . deGennes, eds.). Gordon and Breach, New York, 1962. 9. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). 4 . F. London, “Superfluids,” Vol. I. Wiley, New York, 1950. 6. P. G. decennes, TroisiPme Cycle Notes, “MBtaux et Alliages Supraconducteurs,” Vol. 11. Paris, 1962-1963. Now available in P. G. deGennes “Superconductivity of Metals and Alloys.” Benjamin, New York, 1966. 6 . V. L. Cinzburg and L. D. Landau, Soviet Phys. J E T P 20, 1064 (1950). 7. L. D. Landau and E. M. Lifshitz, “Statistical Physics.’’ Pergamon Press, Oxford, 1958.

36

E. A. LYNTON AND W. L. MCLEAN

L. P. Gor’kov, Soviet Phys. J E T P 9, 1364 (1959); 10, 998 (1960). B. B. Goodman, Rev. Mod. Phys. 36, 12 (1964). A. B. Pippard, Proc. Roy. SOC.A216, 547 (1953). A. B. Pippard, Proc. Cambridge Phil. SOC.47 Pt. 3, 617 (1951). R. B. Dingle, Proc. Roy. SOC.A211, 500 (1952). A. A. Abrikosov, Soviet Phys. J E T P 6, 1174 (1957); Phys. Chem. Solids 2, 199 (1957). 14. C. Caroli, P. G. deGennes, and J. Matricon, Phys. Letters 9, 307 (1964). 16. B. Serin, Phys. Letters 16, 112 (1965). 16. D. Cribier, B. Jacrot, B. Farnoux, and L. Madhav Rao, 11th Conf. on Magnetism and Magnetic Materials, San Francisco, 1965. J . A p p l . Phys. 37, 952 (1966). i7. P. G. deGennes, Troisiiime Cycle Notes, “M6taux et Alliages Supraconducteurs,” Vol. IV. Paris, 1963-1964. Now available in P. G. deGennes “Superconductivity of Metals and Alloys.” Benjamin, New York, 1966. 18. A. Rothwarf, Phys. Letters 16, 217 (1965). 19. M. Cardona, J. Gittleman, and B. Rosenblurn, Phys. Letters 17, 92 (1965). 20. T. Kinsel, E. A. Lynton, and B. Serin, Rev. Mod. Phys. 36, 105 (1964). 2i. G. Bon Mardion, B. B. Goodman, and A. Lacaze, Phys. Chem. Solids 26, 1143 (1965). 22. L. P. Gor’kov, Soviet Phys. J E T P 10, 593 (1960). 23. P. G. deGennes, Phys. Condensed Matter 3, 79 (1964). 24. K. Maki, Physics ( N . Y . )1, 127 (1964). 26. T. McConville and B. Serin, Phys. Rev. 140, A1169 (1965). 26. A. R. Strnad and Y. B. Kim. Private communication, preprint 1965. 27. M. Tinkham, Phys. Rev. 129, 2413 (1963); Rev. Mod. Phys. 36, 268 (1964). 28. R. D. Parks, J. M. Mochel, and L. V. Surgent, Jr., Phys. Rev. Letters 13, 331a (1964). 29. D. Saint-James and P. G. deGennes, Phys. Letters 7, 306 (1963). 30. B. Rosenblum and M. Cardona, Phys. Letters 9, 220 (1964). 31. D. Saint-James, Phys. Letters 16, 218 (1965). 32. 1).J. Sandiford and D. G. Schweitzer, Phys. Letters 13, 98 (1964). 33. A. Rothwarf, Private Communication, 1965. 34. D. E. Carlson, Private Communication, 1965. 36. J. Gittleman and B. Rosenblum, Private Communication, 1965. 36. K. Maki, On surface superconductivity in the sub-critical region, Publ. COO-264. Univ. of Chicago, Chicago, Illinois, 1964. 37. Y. B. Kim, C. F. Hempstead, and A. R. Strand, Phys. Rev. 139, A1163 (1965). 38. P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39 (1964). 39. J. Friedel, P. G. deGennes, and J. Matricon, A p p l . Phys. Letters 2, 119 (1963). 40. B. Rosenblum and M. Cardona, Communicated a t Conf. Phys. Type I1 Superconductivity, Cleveland, 1964. 42. B. D. Josephson, Phys. Letters 16, 242 (1965). 42. J. M. Luttinger, Phys. Rev. 136, A1481 (1964). 43. J. Bardeen and M. J. Stephen, Phys. Rev. 140, A1197 (1964). 44. P. Nozihres and P. G. deGennes, “Magnus Force and Flux Flow in Superconductors.” Private communication, preprint, 1964. 46. W. F. Vinen, Rev. Mod. Phys. 36, 48 (1964). 46. A. K. Niessen and F. A. Staas, Phys. Letters 16, 26 (1965). 47. W. A. Reed, E. Fawcett, and Y. B. Kim, Phys. Rev. Letters 14, 790 (1965). 48. P. H. Borchcrds, C. E. Gough, W. F. Vinpn, and A. C. Warren, Phil. Mag. 10,349 (1964). 8. 9. 10. 11. 12. IS.

TYPE I1 SUPERCONDUCTORS

37

49. P. N0eiPrt.s and P. G. deGennes, Phys. Letters 16, 216 (1965). 50. T. E. Faber, PTOC. Roy. SOC.A248, 460 (1958). 61. For review papers on this topic, see “Plasma Effects in Solids,” Proc. 7th Intern. Conf. Phys. Semicond., Paris, 1964. Academic Press, New York, 1964. 58. 11. A. Hays, Private Communication, 1965.

63. A. J . Kushnir, Private Communication, 1965. 54. B. W. Maxfield and E. J . Johnson, Phys. Rev. Letters 16, 677 (1965).

66. F. Haenssler and L. Itinderer, Phys. Letters 16, 29 (1965). 66. W. DeSorbo, Phil. Mag. 11, 853 (1965). 67. C. P. Bean and J. D. Livingston, Phys. Rev. Letters 12, 14 (1965). 68. P. G. deCennes and J. Matricon, Rev. Mod. Phys. 36, 45 (1964). 69. R . W. DeBlois and W. DeSorbo, Phys. Rev. Letters 12, 499 (1964). 60. H. J. Fink, Phys. Rev. Letters 14, 853 (1965). 61. P. W. Anderson and A. H . Dayem, Phys. Rev. Letters 13, 195 (1964). 68. B. 11. Josephson, Phys. Letters 1, 251 (1962); Rev. Mod. Phys. 36, 216 (1964). 63. P. W. Anderson, N. 12. Werthamer, and J . M. Luttinger, Phys. Rev. 138, A1157 (1965). 64. E. Abrahams and T. Tsuneto. Phys. Rev., to be published. 65. V. Heine, Proc. Cambridge Phil. SOC.62, 546 (1956). 66. L. Dubeck, P. Lindenfeld, E. A. Lynton, and H. Rohrer, Phys. Rev. Letters 10, 98 (1963).

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Measurement of Weak Magnetic Fields By Magnetic Resonance P. A. GRIVET t!Jnaversily of Paris Institut d’Eleetronique Fondamenlale Orsay (gl-Essonne), I’’rance AND

L. MALNAR C.F.S. Depl. & Physique Appliquie Corbeville par Orsoy (91-Essone),France

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A . Outline of This Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B . Comparison with Nonresonant Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 C . Scalar and Vector Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 13 . Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 I1. Order of Magnitude and Main Characteristics of Natural Fields . . . . . . . . . . 45 A . Geomagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 B . Interplanetary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 I11. Nuclear Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A . Prepolarization and Free Nuclear Precession . . . . . . . . . . . . . . . . . . . . . . . . . 55 B . Overhauser Polarization and Spin Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 64 IV. Optical Detection of An Electron Nuclear Resonance ( A m p = f 1) . . . . . . . . 76 A . Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 B . Zeeman Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 C . Experimental Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 1) . Helium Magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 V . An Example of Design : The Cesium Vapor Magnetometers ............... 111 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 B. The Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 C. Optimum Working Conditions and Limitations in Use . . . . . . . . . . . . . . . . 129 D . Examples of Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 VI . Superconducting Interferometers as Magnetometers . . . . . . . . . . . . . . . . . . . . . 143 A . Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 B. The First Practical SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 39

40

P. A . GRIVET AND L. MALNAR

I. INTRODUCTION

A . Outline of This Study I n 1946 the discovery of nuclear magnetic resonance by Purcell and Bloch opened a new era in the domain of accurate magnetic field measurement. Indeed, the new phenomenon links accurately and linearly the value

rt+

FIG.1. Free precession of a magnetic moment ing to the law d M / d t = r(M X B),for 7 > 0.

M around a magnetic field B accord-

of a field B to a circular frequency w , by introducing a new physical constant, the gyromagnetic ratio: w =

rB

(1)

where w is the angular velocity of precession of the magnetic moment M of the specimen around the vector B (Fig. l), and y characterizes the substance in which the resonance is observed or, eventually and with some minor corrections, the nucleus, the magnetic moment of which is put into resonance.

M E A S U R E M E N T OF W E A K M A G N E T I C F I E L D S

41

At high fields, protons are ordinarily provided by such convenient liquids as water or benzene, and then one has (cgs) (mks)

y = 2.67513 X lo4gauss sec-I y = 2.67513 X lo8 tesla sec-I

for pure deoxygenated water. The passage to y p , the value characterizing the bare protons of the theory, would involve a few local field corrections, of the order of a few parts per million in relative value. They are of no importance for the type of measurements involved here but their small magnitude explains why the y of any diamagnetic liquid is very near the proton’s value, so that for measurements which do not seek an accuracy better then the choice of the specimen is not critical and the resonance method is a very convenient one. The number of significant figures given for y shows that the resonance is a very sharp one, and that Hipple el al. (1) and later Bender and Driscoll ( 2 ) had to resort to the most painstaking care in their classical measurements of B a t the National Bureau of Standards in order to obtain y with such a n accuracy of a few parts in As long as B is steady during some 10 sec, the measurement of w presents no difficulty. The only limiting condition for this technique to maintain all its advantages is that the field be sufficiently high-higher than 500 gauss, to state a definite limit. For lower fields the resonance frequency still obeys Eq. (1) and the line remains sharp, but the signal-to-noise ratio deteriorates, and progressively impairs the accuracy. Considering, for simplicity, the case where the sample used is of constant volume, the signal decreases with B2.Moreover, when the 10-gauss region is reached, the frequency enters the 2O-kc/sec range and flicker noise appears. Geophysical fields are still an order of magnitude lower, near 0.5 gauss, in the middle latitude countries, and the direct detection of a nuclear resonance signal succeeds only in exceptionally good “natural” conditions (see Section 11). This tour de force could be accomplished only very recently in the geophysical observatory a t Jussy, near Geneva, by the specialized team of B&nBel al. (3, 4). In 1957, clever use of the Overhauser effect led Abragam et al. ( 5 ) to introduce a new technique of “double irradiation” in nuclear resonance, and so obtain a signal which decreases as the first, power of B . It will be shown in Section 111 that, by this method, a fair signal-to-noise ratio is produced, of the order of 100 or more, for fields of geophysical interest. In the same year, 1957, the flight of the first Sputnik focused attention on interplanetary conditions. In the nearest region of space, and in first approximation, the earth acts as a magnetic dipole, and its field decreases rapidly as ( r / R ) - 3 with the distance r to the center, R being the earth

42

P. A. GRIVET AND L. MALNAR

radius ( R N 6400 km). So, at some 30 earth radii, the influence of the earth practically disappears, reaching the 20-pG (microgauss) level, and one reaches the “interplanetary level”; in this region the sources of the field are largely unknown. Today it is only known that it is characterized by a very low amplitude of some 50 p G and that the currents due to ionic winds are the probable cause of the interplanetary field. In this range, the linear law in B, characteristic of “double irradiation” in nuclear resonance, becomes unfavorable too, and moreover the resonance frequency lies in a very low and also unfavorable range: 0.2 cps for 50 pG. Then, as explained later, the spin of the space ship interferes badly with the spin of the nuclei: nuclear resonance in liquids is no longer useful. TABLE I

*6Rb ‘33Cs ‘He 3He

Resonant frequency for B = 1 gauss in cps

Frequency a t the residual space level of 50 pG in cps

4 . 6 7 X 106 3 . 5 x 106 2 . 8 X lo6 3.8 X 106

23 17 140 190

But in 1957, “optical detection” of electron’ resonance in an alkali metal vapor, as proposed sometime before by the discoverer of “optical appeared ), for the first time as feasible. This was pumping,” Kastler2” (‘i‘ due to the important improvements suggested by Dehmelt (8) and implemented by Bell and Bloom (9) at Varian Associates. For the simple free electron resonance ye would be 660 times larger than y p . In fact, here, for the coupled electron-nucleus resonance, the so-called Amp = 5 1 transition (see Section IV), y is of the same order of magnitude or even larger; for example, see Table I. Remark. As regards frequency, helium magnetometers are comparable to the alkali metal vapor type and are included in the table. The 4He nucleus is devoid of any nuclear spin and the metastable level2 2%’ is split into three equidistant mJ = 0, _+1levels, so that the magnetic transitions AmJ = k 1 all have the same frequency. For 3He which has a nuclear spin of +, the metastable 23S1 level is split in two sublevels 1 Here the phrase “electron resonance” is a n oversimplification: in a classical vector (6, Chapter 8) model, the vector precessing around B would not be a pure electron spin moment, but the resultant of both an electron and a nuclear moment. 2 See Leighton (6, pp. 204, 205) for a clear explanation of atomic level symbolism. 2 0 Nobel prize in physics, 1966.

MEASUREMENT OF WEAK MAGNETIC FIELDS

+

43

F = and F = Q by hyperfine coupling. Two magnetic transitions are available. The one mentioned in Table I relative to F = Q is the highest in frequency. The other transition is less favorable: it pertains to the F = 8 level, and occurs a t half the frequency of the first. Table I shows clearly that the resonance frequency still amounts to usable values and that 3He offers the most favorable conditions in this respect. Of still greater importance is a general advantage of Kastler’s optical method valid for all its variants: the signal intensity is independent of B. This is a “trigger” method: one does not observe directly the quantum emitted in the magnetic transition, which would be hw = ZLyB, e.g., proportional to B. Instead, one receives in a photocell an optical quantum hvg, V D being the frequency of the D line of the alkali metal, of constant energy. The emission of the big and constant optical quantum is “triggered” by each3 appearance of a small magnetic quantum irrespective of the latter’s size, which is proportional to B. This high sensitivity a t very low fields is the chief advantage of the “optical pumping method.” The efficiency of optical magnetometers would only wane for extremely low fields in the neighborhood of 100 pG. This is an order-of-magnitude estimate of the line width of the optical transition, when translated from the ordinary frequency domain (Av = Aw/2a) to the magnetic field scale by use of law (1). I n this review, we shall concentrate on the two types of low-field magnetometers which we have presented in this historical introduction. But before starting with the main subject, we shall recall briefly, in Section 11, the characteristics of the natural magnetic fields, which bear on the accuracy of the measurements. Indeed, geomagnetic, like interplanetary, fields are not steady and their rates of change influence the technique of measurement. On the other hand, the study of their variation in time is of high interest. The conditions in geophysics are very different from those which prevail for high field, in the laboratory: here as soon as the physicist has accomplished some noticeable progress in accuracy, he uses this knowledge in determining the residual fluctuations of the field. He is then able to compensate them by some kind of feedback process: by so doing, he is paving the way for new progress in the measuring art. I n the natural field domain, fluctuations are both a factor limiting the accuracy of the measuring process and also an interesting object of study. They cannot be suppressed and for this reason a rough knowledge of the natural and often erratic variations of the natural fields seems in order before studying the instruments and methods of measurement. 8

Neglecting relaxation “leaks.”

44

P. A . GRIVET A N D L. MALNAR

B. Comparison with Nonresonant Methods In the past decade, older methods were greatly perfected too, and for some applications look more attractive then the resonance method, even today. I n these paragraphs, we indicate some references in order to offer modern points of comparison with resonance. 1. Induction. Induction coils equipped with ferrite cores have high sensitivity for fast variations of the field: they are used for the study of the geomagnetic spectrum in the range 1-50 cps, as was done, for example, by Stefant (10). The modern integrating fluxmeter, for example, makes the value of B directly available, instead of dB/at as shown by Grivet et al. (11). In space probes, the natural spin of the space ship may be put to good use to rotate the sensing coil and to obtain high sensitivities, a s shown in the equipment of Pioneer I :

Sensor, a coil with 30,000 turns of n”40 wire on a ferronickel core Angular velocity in space, 2 cps. Dynamic range, 6 HG-12 mG. Bandwidth of the useful spectrum, 1.25-2.5 cps. Threshold of sensitivity, 6 pG. Source power, 25 mw. (For other data see 12-15.> 2. Hall Efect. Hall plates associated with a ferrite static field concentrator reach a threshold of 1 pG in sensitivity, for a signal-to-noise ratio of 10 db. Further details on this device can be found in the review by Grivet (16) and in Epsteiri et al. (17‘). 3. Fluxgate. This method is also known as that using the “second harmonic.” This form of signal is induced in a coil wound around a magnetic yoke, by nonlinear addition. The combined effects of the unknown field B and of an auxiliary sinusoidal field of frequency w , produce a signal, proportional to B , of frequency 2w. This technique has been fully developed since World War I1 (18) and its modern applications were described in Vol. 9 of this series by Melton (19). An elaborate example is offered today by the type of apparatus which complemented an optical magnetometer on the Interplanetary Monitoring Platform, IMP-I. This satellite, also known as Explorer 18, was launched in November 1963. The monoaxial fluxgate had the following characteristics (10): dynamic range, k400 HG;sensitivity, k 2 . 5 pG. (For a description of other recent devices of this type see 21-23.) It is worthwhile to remark here that the fluxgate principle was trans-

45

MEASUREMENT OF WEAK MAGNETIC FIELDS

posed to the resonance domain in one recent proposal for a vector magnetometer (24).

C . Scalar and Vector A4easurements The magnetic field is an axial vector, B, and in order to determine this vector completely, measurements of the intensity of three noncoplanar components must be made. This is necessary when one is using resonance magnetometers, which give “scalar” measurements; indeed, formula (1) links the frequency to the inlensity of the field vector, or, in other words, the measured frequency is completely independent of the orientation of the magnetometer with respect to the field. The angular positioning of the magnetometer reacts on the amplitude of the signal only. One must then resort to the classical methods using compensating coils in order to obtain the three components. Such methods are discussed in the literature for fixed stations, for example, Minitrack stations (25), and the recent proposal (26) embodies a high degree of automation. The same scheme is actually used on mobile observatories and on satellites (27) * D. Units A traditional unit in geomagnetism is the “gamma” (y), which in the recent past represented roughly the smallest measurable variation of the field B. In terms of the B vector, we have 1 gamma

=

10 microgauss (UEMcgs)

= =

lop6 gauss tesla (mks)

=

1 picotesla

tjhetesla being the internationally adopted name for the weber per square meter. I n terms of the H vector one has 1 gamma = 10 microoersteds = 0.79 milliampere-turn/meter (mks)

The use of the gamma is inconvenient here, because we shall also have to deal often with y, the gyromagnetic ratio. We shall therefore adopt the microgauss (pG) as the usual unit, which fits the accuracy of present measurements well.

11. ORDER

MAIN CHARACTERISTICS NATURAL FIELDS

O F MAGNITUDE AND

OF

A . Geomagnetic Field

I. Normal Conditions (Quiet Days). a. Spatial variations. We describe here the average characterist,ics of the earth field; they are easily observnhle, since the pioneer work of Gauss, Weber, arid Schuster (see Ruricorri

46

P. A. G R I V E T A N D L. M A L N A R

28, for a short history), during the many days when variations with time are largely absent and which for this reason are specified as “quiet days.” Intensity, direction; dipolar approximation (Gauss) : When geographical variations are averaged out, the field is nearly the same as would be produced by a suitable dipoleaa located a t the center of the earth; its axis is contained in the meridian plane defined by longitude 69’W, and is slightly inclined on the geographical south-north axis, by 11.5’; in other words the magnetic north pole is located in northern Canada, a t a point

radii -

I FIG.2. The earth as a magnetic dipole.

defined by latitude 78.5’N, longitude 69’W (Greenwich). It should be remarked that the south pole of the equivalent dipole points to the north direction (Fig. 2). Uniformity: The field is spatially highly uniform, even very near the surface of the earth; heterogeneity of the magnetic properties of the soil, as well as telluric currents, may spoil this homogeneity locally; raising the magnetometer probe some 5 meters over the ground does away with a good part of the perturbation in magnetically smooth regions. On the other hand, anomalies may be very strong. They can be divided into two categories : interesting geological anomalies, pointing, e.g., to the existence of interesting ores; or, on the contrary, artifacts due, for example, to the *a If M is the magnitude of the magnetic moment and a the earth’s radius, Ma-a is a magnetic field equal t o the intensity at the magnetic equator or one half of t h a t at the magnetic north pole. Between 1962 and 1965, one had Mav3 = 311,110 [miwooerstedt] and the secular variation of the quant.ity was of thc order of 160 [ r o c ] pcr year (28~).

MEASUREMENT OF W E A K MAGNETIC FIELDS

47

residual magnetization by currents orire produred by ancient lightning striking the soil. Near latitude 45"N (F = 0, 45 gauss), the average degree of uniformity of the field may be characterized by the following values of the gradient: 200 pG per kilometer in altitude; 50 pG per kilometer in latitude. Nomenclature: I n order to make reference to the introductory books on the earth's field (29-31) and to the literature easier, we collect the standard notation in Table 11. TABLE I1 Standard letter symbol Total field Horizontal component Vertical component Declination (angle between H and the geographical north direction) Inclination or dip (angle between F and the horizontal plane)

P

Average value near latitude 45"N (gauss)

H

z

0.45 0.15-0.25 0.4

D

Depends on location

Z

60-70" ( F pointing down)

b. Variations in time. Nowadays the average value of the earth's field is thought to stem mainly from currents in the earth's interior; indeed, the core is highly conducting (as much as the sea a t a depth of 700 meters) owing to the high temperature (1500°C at 700 meters) (68). On the contrary, fluctuations in time of the field vector are 75% due to external currents in the ionosphere; only 25% is linked directly to internal currents. A coupling exists between both kind of currents; but the qualitative duality of sources*makes it easy to understand that the fluctuation in time may be either slow, and very slow, or fast. In fact, one observes: (a) Secular variations at a rate of 300 pG per year. (b) Diurnal variations; the intensity F decreases rapidly during the early morning, soon after sunrise, reaches a minimum a t noon, and increases during the afternoon and night. Th e amplitude of this daily oscillation is a t its highest in June and reaches k250 p G ; its minimum value is in January, when it amounts to f 2 5 pG. The direction of the vector fluctuates also, as indicated b y the correlated variation of H and Z which was recorded, for example, in the U. S. Coast and Geodetic Charts 30772 and 3077H. We need not go deeper into this subject here since resonance methods are only sensitive to the variation of H . 4 The duality of sources may he proved by measurements on the earth's surface only, as shown theoretically by Gauss and first measured by Schuster. On this point, see Chapman (29, p. 54) or Massey and Boyd (30, p. 186).

48

P. A . GRIVET A N D L. M A L N A R

(c) Fast variations; actually, there are a few types of well-characterized quick variations. Known sincc a long time are the Eschenhagen oscillations :amplitude:5 to 50 &, period :25 sec, total duration: a few minutes. All the newly discovered ones (4, 10, 32, 33, 33a) seem to be linked with ionospheric activity and are then the indirect result of solar activity. Their common characteristics are a very small amplitude and a relatively high frequency; for example, see the accompanying tabulatJion. Frequency (cps) Amplitude ( N G ) Hydromagnetic “pearls” Oscillation of the cavity between earth and ionosphere

0.3-3

0.2

5-40

0.02

They are most often observed by induction (10) and some of them lie a t the limit of sensitivity of resonance apparatus. They offer a wide choice of natural signals useful to test the resonance magnetometer, as shown, for example, by Arnold et al. (34). 2. Exceptional Variations; Magnetic Storms. Magnetic storms happen several times in a year: their occurrence is linked with solar activity and they occur after strong eruptions a t the surface of the sun in the form of sunspots (35). The associated magnetic variations on earth are of a n order of magnitude of 5 to 15 mG, e.g., 100 or 1000 times larger than the interesting details in a magnetic exploration. Thus the occurrence of magnetic storms precludes any other magnetic study than that of the storm itself. Magnetic storms have been studied since the beginning of the 19th century and are of widely varying intensities. The one th a t occurred in 1859 was so strong as to be visible by looking at a magnetic needle; it involved variation of F of the order of 10%. This by far exceeds the dynamic range of standard magnetometers and the study of storms requires special instruments. Optical (pumping) magnetometers are a n exception to this rule; when equipped with digital registration they show a dynamic range of 0.5 gauss.

3. Magnetic Exploration with Mobile Magnetometers. Magnetic exploration is done in a variety of ways and magnetometers are carried by automobiles, ships, or planes. The movement of the carrier transforms the spatial variations into time variations: the useful variation of the signal is then mixed with parasitic variations: some of them arise from the regular geographical variation of the averaged field; others stem from the irregular time variations. Corrections are necessary and in the second case can only be done by comparison with a stationary station operating at the same time.

MEASUREMENT OF WEAK MAGNETIC FIELDS

49

The procedures, as well as the categories of signal strength which may occur, are numerous. Two very different examples may illustrate the variety of cases. One is a deep sea signal recorded in the course of the magnetic search for the submarine Thresher's hull with a proton precession magnetometer (36) (Fig. 3). The other is given by an airborne magnetometer during the broad-scale magnetic exploration of the Arctic, which resulted in the characterization of two different halves of the sub-

cn

E-

60-

5040 -

30 -

Time

0430

0445

0500

FIG.3. The deep sea signal given by a proton precession magnetometer on the spot of the Thresher's accident.

merged continental shelf (Fig. 4). Further illustrations may be found in Raff (37) and King et al. (38).

4. Geophysical Observatory. Since 1830, when Gauss initiated highaccuracy measurement of the geomagnetic field, this aspect of the science of magnetism developed steadily. Now, in many countries, there are several geomagnetic observatories, as free as possible from any man-made magnetic disturbance. This implies mainly the following conditions: (1) A wooden building, where all metallic; parts including the small ones (nails, wires, etc.) are copper or aluminum, and verified to be nonmagnetic material (for example, brass, which should be nonmagnetic, nevertheless often is magnetic, owing to the practice of recasting old scrap, possibly nickel plated). (2) A safe distance (40-60 km) from the nearest high-voltage line, usually an electric railway line.

50 (3) line.

P. A. GRIVET AND L. MALNAR

A suitably screened bifilar or coaxial feed system for the power

Apart from large and well-known national laboratories such as that near Fredericksburg, Virginia (see also 38a), universities may build smaller but efficient laboratories a t relatively low cost, such as the Jussy laboratory, which has been described by the Geneva physicists (39).

Pi

/

I

*4

/

-Central

magnetic z o n e 1

Wrangel

+=

I

a B

F I ~4.. Aerial magnetic exploration of the Arctic shelf.

I n such a location, the gradient of F is as low as 1 pG per meter; a n experienced geophysicist may claim to measure F with an accuracy of 0.1 pG. He should work during a quiet day and have a t his disposal a n auxiliary continuously operating survey apparatus: he may then be able to choose the most favorable period (thour perhaps) for the measurement. More often, he niay be happy to reach the microgauss limit only, and this is now feasible with any conimerrial resonance magnetometer. 5. Ordinary Laboratories. Compensating Natural and Man-Made Fluctuations. If one turns to ordinary physical laboratories, the conditions are by far worse. An unforseeable value of the spatial gradient is produced by the ferromagnetic girder embedded in the concrete, or by open beams

MEASUREMENT OF W E A K MAGNETIC FIELDS

51

(elevators): values ranging from 1000 to 5000 pG per meter are not exceptional. On the other hand, sinusoidal or transient fields a t the mains frequency and its harmonics also reach a level of some lo3 pG and are very disturbing. It was long thought that such an unfortunate location simply makes any experimentation on high-accuracy magnetometers impossible. But recently successful attempts have been made to quiet the field fluctuations with time, so it may become profitable to further improve the conditions by compensating the static gradient to first order. The proposals are all based on a high-sensitivity pick-up system, which measures the fluctuations : this signal is linearly amplified, and then fed back with the proper phase to a pair or quadrupole of compensating coils of large dimensions. Three types of pick-ups were successfully used:

FIQ.5. Automatic balancing of the fluctuations of the apparent earth field with an helium magnetometer ( 4 2 ) .

induction coils (40), fluxgate (41), and a helium resonance magnetometer (49). The induction system has a large bandwidth (10-10000 cps) and reduces ac hum by a factor which may be better than 120. But the lower cut-off frequency is between 10 and 5 cps, and the device is rather inefficient against low-frequency fluctuations. These may be reduced by careful choice of the room in the building. Fluxgate and resonance magnetometer show complementary qualities; they are efficient from 10 cps down to zero frequency. For resonance studies, it is sufficient to stabilize F in magnitude: in other words, fluctuations in the direction of F are of secondary importance. Figure 5 shows the efficiency of such a device described by Schearer (42). Wolff (41) described a more complete system regularizing the three components of the vector, bracketed within +lo0 pG. More recently, a simpler technique was used with success: a large multiple magnetic screen surrounds the system to be studied (43-45a). One meets the following difficulties:

52

P. A . GRIVET A N D L. MALNAR

(1) Economic considerations limit the volume to roughly 1 m3 or less. (2) One has to demagnetize the screen carefully, and this operation is necessary after each important change in the external magnetic environment (moving of iron masses) ; permanent demagnetizing coils are a necessary part of the screen. (3) Inside the screen, the field is limited in intensity to some 10 gauss for Helmholtz coils (44) and to 100 gauss for solenoids (46, 46b).

This solution looks very promising and will take a still more convenient form with forthcoming improvements in the art of magnetic

I2 mG

v = 2025 cps B- 916 mG

I

FIG.6. A faint component of the nuclear resonance spectrum of POBH2, which would remain hidden in external noise in the absence of a magnetic shield around the low-field spectrograph.

joints; Fig. 6 shows a faint nuclear line, which could only be observed with such a protection.

B, Interplanetary Field (46a) 1. Order of Magnitude. Three main types of measurements have been actively made for the past few years: (1) Survey of the “cIassica1” geomagnetic field in space, by means of low-altitude satellites. One hopes for a more thorough, fast and accurate survey; the magnetic conditions remain similar to those a t the surface of the earth. The main interest lies in the small Gaussian departures from the dipolar field. The altitude is between 300 and 800 km, where the magnetic conditions are better than a t sea level because of the smoothing

53

MEASUREMENT OF WEAK MAGNETIC FIELDS

by distance of a good part of the influence of geographical heterogeneity. On the other hand, one has t o face the general difficulties which occur in spatial experimentation and which are described later. The recent program is ably described by Heppner et al. (46-49). (2) Near zone exploration, which extends from 1000 km to 15 earthradii, and includes the region of the Van Allen belts. There, the field is mainly the dipolar field and its intensity decreases, following a ( T / R ) - ~ law,:reaching some 3 mG at 5R and 100 pG at 20R. In this range one is

4

6

8 -~

--

10

12

Earth radii [RE]

FIG.7. The earth field in the “near zone” as measured by Pioneer V.

interested in the departures due to the intense ionic currents (Fig. 7) produced by the ionic belts, such as the one discovered a t 8R by Pioneer V. It produced a “bump” of some 200 pG on the dipolar field curve. Reviews of the magnetic data for this zone are given by Harrison (60) and Cahill (61). (3) Farther then 20R, the conditions are less accurately known and Pioneer V data (1961) are now considered as inaccurate. The most recent experiment [IMP.I (20)]provides more reliable data and indicates a field of 50 pG located in the ecliptic plane. I n this region many spatial and time variations of an intensity of some 10 pG are related to interesting aspects of the solar ionic wind; a discussion of the difficulties encountered in the exploration of this far region is to be found also in Ness et al. (20).

54

P. A . GRIVET A N D L. MALNAR

2. Spatial Requirements. This brief summary of the results shows that magnetic measurements in space are of a very delicate nature; moreover, the space ship is moving a t high speed (some 2 km per second) and genuine time variations are mixed with artificial ones, produced by the movement through heterogeneous ion clouds and currents. We refer the reader to the last-mentioned references and to Scull and Ludwig (52) for a discussion of these interesting problems. We shall only summarize here the final requirements which actually shape the construction of a spatial resonance magnetometer: (a) Sensitivity, 0.1 pG. (b) Frequency band of the order of 10 cps. (c) Weight, 2.5 pounds (sensor); 10 pounds total with converters and container. 2 watts (thermal control). (d) Power, 6 watts (e) Insensitive to the spinning of the satellite. (f) Insensitive to the “attitude” of the satellite, e.g., to the orientation of an internal reference direction with respect to the astronomical reference frame.

+

The last two requirements need some explanation. (1) Modern space probes are stabilized in direction; for magnetic measurements, the preferred solution to this problem is to have the craft spin a t moderate speeds (22.3 rpm for IMP.1). One ensures by this means that the axis of rotation keeps a constant orientation with respect to the astronomical reference frame. The consequence for the magnetic precession magnetometer is important. Indeed, it ma.y be proved generallya detailed analysis is given by Heims and Jaynes (5S)-that a rotation of the laboratory defined by the rotation vector SL has the same action on the magnetometer as an additional fictitious field

b

= Q/y

(2)

would have on the same apparatus supposed a t rest : if y is small as in the pure nuclear resonance, the error field b is high.6 For example, for proton resonance, an angular speed of 1 turn per second, when and B are parallel, means an error of 1 part in 2000 on the earth’s field a t sea level and middle latitudes, (blreaching the value 600 pG. Such a systematic error is unacceptable for the low space fields, and pure nuclear resonance is ruled out for this reason. Optical resonances correspond to y’s lo6 times higher and are perfectly suitable. 6

Only in the exceptional case 0 IB, would the error be negligible.

MEASUREMENT OF WEAK MAGNETIC FIELDS

55

(2) I n space, the most interesting contribution to the field, due to ionic currents, shows no preferred direction. On the other hand, the intensity of the signal due t o magtietic precession depends on the direction of the measured field with respect to the magnetometer. The dependence law, as will be explained later, leaves one with a dead angle of some & 15’ with respect to B. The only present solution to this difficulty is to usc “double” optical magnetometers and to add complementary magnetometers: in IMP.1, a monoaxial Rb resonance magnetometer is coupled

FIG.8. A nondirectional, triple-head, helium resonance magnetometer for satellite exploration (courtesy of Texas Instr. Co.).

with two fluxgates (20),and Fig. 8 shows a triple system of helium magnetometers for spatial use. 111. NUCLEAR RESONANCE

A . Prepolarization and Free Nuclear Precession 1. Method. I n 1953, Packard arid Varian (64) disclosed a free-precession experiment performed in the earth’s field and started the development of modern resonance magnetometers. Moreover, the first commercial model proved to be so efficient in the field that it still remains of great practical value in nearly its original form. I n order to rcach the low-field domain, the itiveritors separated the free-precession experiment already known for high fields (55) into two successive processes (Fig. 9). For this purpose, they took advantage of the long nuclear relaxation time of such ordinary liquids as water (2’1 ‘v 3 sec) or benzene (7’1 10 sec) when purified from the dissolved oxygen.

56

P. A . GRIVET A N D L. MALNAR

I n a first step, the sample is strongly polarized in a n intense field B, of the order 100 to 800 gauss; the 250-ml water sample then acquires a sizable bulk magnetic moment M which will only slowly decrease in magnitude during the 10 see necessary for an experiment. During this lapse of

FI-J Fostswitch polorizing position

DC. power

( N o t to scale)

FIG.9. Timing of the main steps in the Packard-Varian proton precession niagnetometer.

time the moment decreases following an exponential law e--l’T1.The establishment of polarization itself requires a time of a few times TI, e.g., a few seconds, to be fully accomplished. The second step begins with the abrupt breaking of the current pro-

MEASUREMENT OF WEAK MAGNETIC FIELDS

57

ducing the polarizing field B,; the direction of B, is chosen approximately a t right angles to the unknown B.Then, a t the instant that B, disappears, the magnetization M = xB, begins to precess around B (Fig. 10) inducing a signal in the pick-up, coils a t the frequency w = rB. I n fact the switching out of the polarizing current and the brealtdown of the field B, proceed continuously during a certain time, say 1 psec, during which the total field vector quickly, but continuously, evolves from the state B, B = B p to B. If this change is not made swiftly enough the magnetization M will follow smoothly (“adiabatically”), the

+

13.

‘i-/

FIG.10. Free precession of the magnetic moment M during the measuring period.

field remaining a t every time nearly aligned with it; and no precession will be observed. I n order that M will surely be left back, the change in the B direction must occur in a time short compared to the time constant of the movement of M.As is known from Bloch’s equations and as has already been known for a long time from the experiments of Rabi’s school, this is adequately measured by the period of precession of M in the actual field a t the time t. A safe limit is the period in the polarizing field B,; for B, = 100 gauss the period is near 2 psec and the cut-off must occur in 1 or 2 rsec; these were the approximate conditions in the first experiment. As the magnetizing current reaches the 5- to 15-amp range (feeding 200-500 turns of 135 w.g. wire wound around a 500-ml bottle), this is an exacting condition on the current breaker. Today (66,

58

P. A . GRIVET A N D L. M A L N A R

67) one softens this condition by first decreasing the cwrrent slowly (in a few tenths of a second) to an intermediate value, 0.5 amp for example; a t this stage, B,’ = 6 gauss and is still much larger then the earth’s field; M remains aligned on Bp’. At this instant one only has to cut the smaller current of 0.5 amp rapidly; moreover one now has more time a t his disposal, the new value of the limit being 10 times larger; indeed it is the precession period in the B,’ field, e.g., 50 Fsec. 2. Frequency Measurement. The gain in intensity produced by the prepolarization process is B p / B , ranging from 200 to 400 in the earth’s field for B < 100 < B, < 200 gauss; this is large enough to get a good signal-to-noise ratio during, maybe, 10 sec. Indeed, the signal decreases following an exponential law exp(-t’T2*);Tz* is of the same order of magnitude as TI, if the field is highly homogeneous spatially, but frequently Tz*< T1.This inequality expresses the lack of uniformity of the field; in fact,

l/r Tz* N (6H2)”2

(3)

where (6H2)1’2is the rms value of the spatial variation of the field in the volume of the sample; in an ordinary laboratory, T2* may range from 0.1 to 1 sec for a 500-em3 sample. A simple calculation (cf. IS) shows that the optimum observation time for the minimum error in frequency, taking account of the decrease of the signal-to-noise ratio at the end of the process, is approximately given by Tz*. Measuring the average period over 1 sec with an electronic counter permits one easily to reach with relatively good accuracy; the sample is water with a practical limit for Tt* given by the natural Tz,e.g., 2.1 sec for distilled and 3.1 sec for deoxygenated water; for benzene, Tz is 14 sec. I n the latter case the accuracy is increased by nearly a n order of magnitude if the field remains steady for 10 sec. One can easily show, as is done, for example, by Blaquiere et al. (58), that the error in B, f 6B, is given b y

6B

= l/pyO

(4)

where 0 is the measuring time, and p the signal-to-noise ratio. Counting the number of periods of the signal for a few seconds is considered nowadays as a rather crude manner of using the precious information contained in this continuous signal. Italian scientists (59) have tried to remedy this situation; they were looking for feeble, but predictable, variations of F , such as those produced by the perturbation of the ionosphere that occurs during an eclipse or due to an atomic explosion. They sought to increase substantially the accuracy of their precession magnetometer by processing the signal more carefully. The block diagram of their device is shown in Fig. 11. The method makes

59

M E A S U R E M E N T O F W E A K MAGNETIC F I E L D S

use of synchronous detection. This mode of detection is of current use in radiospectroscopy, for example, where one knows in advance both the frequency and phase of the signal. Here, however, the frequency is only approximately known; but this is not an insuperable difficulty if one has t o ;deal with slow and slight deviations of the frequency. On the other hand, the knowledge of the initial phase of the signal is assured if one prepares the initial state not by fast removal of the polarizing field only, SSB Signal Modulator A *in iw+)

SSB Filter

-

Detector integrator

Error signal

-

t FIG. 11. Block diagram of the method of Faini and Svelto for optimal processing of the information of the free precession signal.

ti

FIG.12. A scheme of the accurate timing necessary in the optimal processing.

but b y a 7r/2 “Hahn pulse” (60, 6 l ) , following a rather slow break of current (Fig. 12). It appears possible, then, to heterodyne the signal a t frequency w by mixing it with an auxiliary oscillation a t frequency 00. The chief experimental difficulty is to generate this heterodyne oscillation with an adjustable frequency and still keep the same high stability of the frequency as in a quartz clock. This makes for most of the rather great complication of the system. In order to measure w one has to adjust w o so that Q = w - w o will be as near zero as possible; this occurs when the output of the integrator is zero; Fig. 13 shows the error signal. The time of integration should be chosen between 1 and 10 sec, so as to be

60

P. A . GRIVET A N D L. MALNAR

much longer then the coherence time of the incoming signal, which is equal to l/Aw, Aw being the bandwidth of the sensor’s circuit at the input. Under these conditions the theory of the “synchronous detector” still applies. (See, for example 66, pp. 96-100, and 63, pp. 389-393.) The detector-integrator set combination cuts the rma value of the noise by a factor of the order of [l/€3Aw]1/2.An exact theory, which may be found in Blaquiere et al. (68) and Faini et al. (69),gives the improvement factor

FIG.13. The error signal for a good adjustment in the Faini-Svelto method.

SB’/SB, 6B’ being the error over B in the new method, 6B that in the old one; one has

6B’

(&) 6B 1/2

=

One gains practically an order of magnitude in accuracy for 8

=

10 sec,

Aw = 2s X 20, reaching the 0.1-pG range. In fact, the indirect effect

of the eclipse on 15 February 1961 could not be detected with this equipment because the residual magnetic activity was not “quiet” enough (64). On the other hand, H-bomb explosions in the atmosphere are now known to be much easier to observe; one does not require such elaborate means, and the diagram on Fig. 14 was registered on 9 July 1962 a t Chambon-la-Forkt with a simple but sensitive induction head (65). The chief interest of this analysis is to show that notable improvements in accuracy are possible but at the expense of a considerable sophistication

M E A S U R E M E N T O F W E A K MAGNETIC F I E L D S

61

of the instrumentation arid of a slowing down of the measuring process (one measurement every 15 sec). 3. Modern Realizations. The gist of the Varim-Packard method rests in an original mixture of simplicity and accuracy, so that most modern versions keep to the original scheme and try to adapt it to various

M

Detector

4

Loudspeaker

FIG.15. A double-head free precession “gradiometer” (68).

specialized functions. We have already mentioned in Section 1,C the methods used to apply it to make a vector magnetometer; we may indicate a few additional uses: (1) Accurate data on the standard realization may be found in Waters and Francis (66), Klose (57), Faini and Svelto (66), and in the numerous references cited by Blaquiere et al. (58). (2) Gradiometer: feeding the same amplifier with two heads in series connection (Fig. 15) and mixing the two signals gives a beat frequency

62

P. A . GRIVET AND L. MALNAR

proportional to the scalar product of the gradient of the field and the vector distance of the two heads; the sign of the gradient is determined in an auxiliary experiment by adding a known gradient produced b y a coil; details are to be found in Rikitake and Tanoka (67) and Aitken and Tite (68). (3) Check of the accuracy by comparison with other magnetometers (69, 70). (4) Rocket and satellite magnetometers. The system has been successfully used on rockets and satellites, €or example, in Vanguard 111. Mansir (71) describes the device, which shows the following performance: (a) B, = 600 gauss. (b) Dynamic range on Vanguard 111: 0.07-0.375 gauss corresponding to altitudes of 510 to 3750 km. (c) Sensitivity: 30 pG. (d) Energy: 200 joules per measuring cycle. (e) Spin of the satellite: less then 0.09 cps.

4. Use of Coupled Spins in Very Low Fields. The Varian-Packard prepolarization scheme provides a law linear in B for the amplitude S of the signal. The decrease of S a t very low field is due to Faraday’s law of induction : a constant magnetic moment precessing with a n angular speed w induces an emf proportional to w and nature shows no exception to this law. But one may add to the very feeble unknown field B a constant and accurately known auxiliary field B,. The resultant B B, is now intense enough to be measured accurately, and one gets B back by difference, measuring B, in an auxiliary experiment. The process would be a safe one if B, were very steady. No man-made field easily meets this last requirement, but atomic fields would. Along this line of thought, Thompson and Brown (72) proposed, and realized in 1961, a free precession experiment using two6 coupled spins P and H, in HPO(OH)2,instead of one. I n the absence of any external field, one may rather loosely think of each of the two nuclei P and H as precessing in a field indirectly produced by the other through “scalar coupling.” When a low external field B is introduced, it splits the resonance and changes the frequency. Quantum-mechanically, the system offers the simplest example of coupled spins and in the general magnetic case, e.g., in the presence of a n external magnetic field, its wave function and energy levels are easily calculated, as shown, for example, by Pople et aE. (73, pp. 119-121). The diagram

+

6 Owing to the rapid exchange between the H of the (OH)z radical and the solvent water, these two protons play no role; their field averages out.

63

MEASUREMENT O F W E A K MAGNETIC F I E L D S

of the levels is shown in Fig. 16, where the zero-field transition is indicated by an arrow. It is of the same famiIy as the AF = 1, Am = 0 transition used in atomic clocks,’ occurring here between one of the triplet states

,

, E/J

J = 695 cps

0 75

FIG. 16. The Breit-Rabi level diagram for the two coupled spins H and P in HPO(OH)2.

(mp = 0) and the singlet state (mp = 0); its frequency is coupling energy between spins is written as

x = JII .

I2

v3

=

J if the (6)

It is a parallel transition, and in a weak external field B it couples best

to a coil whose axis is parallel to B. When the receiving coil axis is at a right angle to B, it is sensitive to the ordinary transitions, which are two in nuniher here, Amp = f l (broken arrows in Fig. 16); their fre-

7 Figure I6 shows t h e same genwal shape as that of H atoms, except for the crossing of levels rnp = -1 and m p = 0, which does not occur for H. The reason is the big difference between the values of the y’s for the P nucleus and for the lone electron in H.

64

P. A. GRIVET AND L. MALNAR

quencies are

+

and v 1 - v 2 = V H V P , where we designate by vIf and V P the ordinary resonant frequencies of isolated atoms H and P in a field B. When the coil axis is a t an intermediate angle to B, all three signals are induced simultaneously; the low frequency v 1 - v 2 appears as a modulation on the slightly damped carrier at frequency v3. A rigorous theory of this free precession experiment (74) explains it in detail. Thus the signal remains strong a t very low fields because the induction law is used a t the constant and convenient carrier frequency v 3 = 695 cps; the frequency measuring the field appears as an amplitude modulation. At present the limit of this technique lies a t some 10 MGwhen the period of the modulation becomes long compared with the damping time T2*of the signal. Nowadays this method appears interesting for the solution of an actual problem: to check how accurately one can realize a field-free space, inside a screen or by dynamic compensation.

B. Overhauser Polarization and S p i n Oscillator 1. Variant of the Overhauser Efect. Since 1955 (75) the Overhauser effect [see the introductory treatments (76, 77) and the reviews (78-80)] has provided a means to obtain continuously, in a liquid sample, a considerable enhancement of nuclear polarization. Practically, in the earth’s field, -... M=-4r&B

-

+

B

FIG.17. The opposite directions of the vectors B and M characteristic of “negative polarization” and “negative susceptibility.”

the gain obtained in this way is of the same order of magnitude as that achieved by the discontinuous Varian technique. The chief advantage of the new process rests in the possibility of continuously monitoring the value of the field. Nevertheless it should be remarked that the nuclear resonance frequencies involved remain of the same low value for the geophysical fields: the accurate determination of the frequency still requires a time 0 (0 of the order of 0.1 to 1 sec). Each point of a continuous plot of B ( t ) measures in fact the average value of B(t), l?(t)el over 0;-in other words, the bandwidth of the device is still limited to 10 cps upward. Moreover, in his 1955 analysis, Abragam showed how to obtain a negative polarization, e.g., a vector M pointing in the direction opposite to that of the measured field B (Fig. 17). This situation still provides, at these acoustical frequencies, the possibility of building a self-oscillator

MEASUREMENT O F WEAK MAGNETIC FIELDS

65

of the maser type: its frequenry is given with a good accuracy by Eq. (1). Indeed, riot only does the analysis of the process made ordinarily for centimeter waves remain valid [see Singer’s review in this series (81) or Gordon et al. (82) and Shimoda et al. (SS)], but also the threshold condition is quantitatively unchanged as it does not include any factor, that would be strongly frequency dependent: nuclear magnetic masers are good, sturdy oscillators. The simple layout of the experiment is shown in Fig. 18. The sample is a water solution and the solute, vehicle of the Overheuser effect, is a paramagnetic ion: the effect is obtained by exciting the paramagnetic

I Generator I

I

/

I

/ 0 E=7Ogauss

-

-

Scanning

FIG. 18. The scheme of a “double-resonance” spectrograph for the study of the Overhauser effect in free radicals.

resonance of the ion so strongly as to saturate the transition. Practically saturation is realized with a moderate power of some 5 to 10watts, because one chooses an ion with narrow resonance line, such as the semiquinone ion of tetrachlorohydroquinone. Such an ion shows a simple resonance spectrum, with a single line only, a t a frequency w,/27r given by we =

lr,lB

lye/ =

( 2 ~ )X 2.80453 X 106gauss sec-’

(8)

the so-called “free electron” resonance frequency.8 By such a choice of a single line ion, one obtains what m a y b e called a “simple” Overhauser effect. When one simultaneously observes the nuclear resonance of the proton in the water molecule of the solvent, this signal appears as strongly enhanced. One obtains a multiplication of y s is negative, which means that vectorially o = - Irs[B,and B and the angular speed vector are antiparallel.

68

P . A . GRIVET AND L. MALNAR

the natural nuclear magnetic susceptibility x0 by an iniportjant but constant magnification factor m :

y e is

the negative (ye = -lye/) gyromagnetic ratio of the free electron, and y p the gyromagnetic ratio of the proton. Changing X O to mxo one gets a bulk nuclear magnetic moment per cubic centimeter of

Mo is still proportional to B, although by a larger proportionality factor. Therefore one preserves the classical and unfavorable law for the signal amplitude which varies as B 2 . I n 1957 this situation was improved and it was shown possible to restore a law linear in B. I n fact, Abragam et al. ( 6 ) could make an efficient use of an idea put forward by Kittel (84). Some ions show not a single line, but a multiplet. This is the case for the peroxylamine disulfonate ion, ON(SO3) z2-, whereg the splitting is due to a “hyperfine” interaction between the ionic magnetic moment and the nuclear moment of 14N.This nucleus has a spin I = 1 and can take three different orientations with respect to the measured field B : parallel, perpendicular, and antiparallel. In each case, the magnetic moment of 14N,by what is known as an “indirect action’’ (see, for example, 73, p. 184 et seq.) slightly changes the field experienced by the magnetic moment of the ion, and this by a quantity which is dependent on the orientation of the nucleus. The result is conspicuous when one is observing the paramagnetic resonance spectrum of the ion: a t high field, one observes three lines, the central one at the frequency defined by Eq. (10) and the two satellites at frequencies us _+ fi with fi/27r = Q x 54.7 Mc/sec; in usual spect,rometers B = 3000 gauss and the splitting is 13 gauss (Fig. 19). The interest of this kind of splitting here is that it survives a t low fields, as shown in Fig. 20; the frequency of the central component decreases indeed to zero, but the two others coalesce in a doubly degenerate single line a t fi/27r = 54.7 Mc/aec. In the earth’s field, near 0.5 gauss, these two components are well separated and may be separately saturated to induce the Overhauser effect; in the original experiment ( 5 , 8 5 , 8 6 ) ,the transition w1 between levels (F = 3, mF = i)and (f = +, mF = 6) was used. What will be the enhancemerit of the nuclear signal under these last conditions? Multiplying ye and y p by B in Eq. (8) one puts this relation 9 Long-term stability of peroxylamine sulfate is not satisfactory; new suitable radicals (84a)are highly stable and allow long storage.

MEASUREMENT O F W E A K MAGNETIC FIELDS

67

3000 gauss (8414[Mc/secl)

ON(S0,);-

FIG. 19. The hyperfine splitting in the high-field electron resonance spectrum of the peroxylaniine sulfate ion. I

05 F=3/2

1 I

1

-[

F=V2

-I

FIG.20. The Zeernan levels of the peroxylarnine sulfate ion in a magnetic field.

68

P. A. GRIVET A N D L. MALNAR

in the form:

Following Kittel’s analysis (84), one then remarks that Eq. (11) is the appropriate form, from a physical point of view. In fact, we and up are the relevant parameters for a description of the Overhauser effect. Indeed, it consists essentially in a peculiar regime of exchange of energy between three thermodynamic systems, the protons of the solution, the ions, and the saturating radiation field. I n all these processes, energy is exchanged by elementary quanta, nu, or hup, and it is then necessary that the law (11) allow these units of energy to stand in the final result of the detailed analysis of the energy balance. Looking at formula (Il), in the low-field region, where W . nears its zero-field limit Q,one might hope to obtain an enhancement of the order of &‘2/wP which would amount to about 11,500 in a field of 0.5 gauss. This much is not achieved in reality, because the multiplet structure offers numerous channels for energy exchange and the analysis of the complicated situation (86, 8‘7) changes formula (11) to m =

4 Q --

27 u p

+ 330

‘v

3880

for B

=

0.5 gauss

(12)

If we adopt an image currently used for masers, dynamic polarization by they?verhauser effect is a “pumping” process and ordinary relaxation acts as ‘a “leak” for the pumped energy. When multilevel ions are used, the chinnels for relaxation are also multiplied in number and it is not possible to reach the limit calculated for m by Eq. (12). Indeed Eq. (12) gives the result for a “relaxation-free” model; in fact, one obtains high values of the order of 1000 to 1500 for m, when B = 0.5 gauss. An essential property appearing in formula (12) is that now the magnification factor varies approximately as l/wP = 1/B, and finally the signal varies linearly as B. This advantage remains for fields much lower then 0.5 gauss. I t may be inferred from hhe data in (87) that when the two electronic lines w1 and w 2 coalesce (their width amounts to 0.7 Mc/sec and their difference in frequency is Aw = -ye B/3), one may safely saturate simultaneously both transitions without impairing the efficiency. 2. Maser Oscillations. An important property of the coupled ion-proton system appears also in formulas (11) and (12) through the minus sign: it implies a negative value form and therefore for Mo’due to relation (10). The significance of the minus sign is that the moment Mo’points in the direction opposite to that of B,the measured field: one obtains a “negative” or “inverted” moment, as in Fig. 17. Such a relative disposition

MEASUREMENT O F WEAK MAGNETIC FIELDS

69

of the vectors Mo and B is not an exclusive feature of the Overhauser method: the same state could be obtained discontinuously, by the Packard-Varian technique, simply by orienting t,he prepolarization field B, just in the direction opposite to B (Fig. 21). It could also be obtained

FIG.21. A possible Varian process for obtaining negative polarization.

FIG.2'2. Scheme of the nuclear maser of Ahragam et

a). (86, 86,83).

continuously by a liquid flow method due to Benoit (88), which will be examined later. Irrespective of the choice of the preparation method, the inverted state offers the opportunity to build simply a maser oscillator whose frequency, determined by Eq. ( I ) , is proportional to B. The schematic of the devire is shown i n Fig. 22. This possibility stems

70

P. A . GRIVET A N D L. MALNAR

from the “conditional radiation instability” of the negative by this term one understands the following properties:

Mo’state;

(1) If the inverted sample is left alone, it is stable and Mo’decreases in the ordinary way, e.g., exponentially following the law exp( - t/T1) just as a positive moment would do. (2) If the inverted sample is contained in a resonant radio circuit tuned to w = vPB and if the electrical quality Q of this circuit is high enough, then the negative Mo’ is “radiation instable” and the strongly coupled system (Mo’ tank circuit) starts oscillating a t a frequency nearly equal to w. If Mo’ is produced by the Overhauser effect continuously and a t a rate high enough to overcome the radiation losses, e.g., to provide the Joule energy lost in the tank, a steady state of oscillation is established and offers new possibilities for measuring accurately o and B. The threshold of instability is determined by Townes’ condition (89)

+

(UEMcgs)

(13)

where all symbols have been previously defined except r ] , a numerical coefficient less then 1, the filling factor, which measures the degree of geometrical coupling between coil and sample; r] = 1 if the liquid fills all the coil volume [note that formula (13) is valid in the mksa system, provided t.he right-hand side is multiplied by 4 , ] . l o There are many ways to prove relation (13). The easiest to understand is the original quantum-mechanical demonstration given b y Townes (89). One remarks that a proton in a field B is a two-level quantum system (Fig. 23) with an upper level, 2, of higher energy and a lower level, 1, of smaller energy; by Boltamann’s law, in thermal equilibrium the number n2 of protons in level 2 is a little less than the number nl in level 1. According to Langevin’s theory of paramagnetism (see 81, p. 96) Mo is then positive. I n the steady state of Overhauser pumping the proton system is not in a state of thermal equilibrium. I n this mode the populations of the levels are “inverted,” e.g., n2 < nl and instability by radiation appears intuitively to be more possible. It is just the peculiar mechanism available to this peculiar system then so prepared as to be in a high-energy nonequilibrium state; it is natural that it follows a n ordinary evolution to a state of minimum energy. Macroscopic reasoning also leads to condition (13) but in a much less lo If Q is smaller than the limit expressed in Eq. (13), then notwithstanding the presence of the circuit, R negative Mo’ is stable and decreases exponentially without oscillations.

71

M E A S U R E M E N T O F WEAK MAGNETIC F I E L D S

transparent manner. Onc cwnibincs the (airwit equation with the Bloch’s equatioris as first done by Blocnibergen nrid I’oiirid (90) and one looks for the evolution of a solutiori dcfiricd by m,, m,, nz,. These quantities represent small departures froin the ordinary (91)expoiieritial evolution from an iriitlial state M, = M , = 0, Mz,: M , = M,, exp(--t/TJ 1- M o . Such a theory was first published by Vladimirsky (92) arid was also developed later by Soloinori (93)and Combrisson (94). Thermol equilibrium

Inverted or “pumped” state c

c FIG.23. The relevant nuclear magnetic lcvels and their populations, in the normal and inverted states.

Finally the complex susceptibility formulation a s applied to the transverse nuclear susceptibility (76, 91) offers a short route (95, 96) to Eq. (13): Bloch’s equations are used to express xL’ = xII jxItr, which appears as simply proportional to xo; one has for xL” a t exact resonance

+

xl’l

=

&xowTz*

(14)

It then appears as natural that a negative xo involves negative values for xLI and xL”. On the other hand, it is well known (see 67 or 76) that a positive xl” entails the appearance of a positive resistance yn in the coil of a resoriarit vircuit coupled to the sample: r n = 4sqx,”Lw

(15)

where L is the self-inductance of the coil. There a negative xL” produces a negative resistance rn and the threshold of oscillation is reached when the negative resistance just compensates the positive ohmic resistance of the coil (r = Lw/Q); if we take into account relations (l), (14), and (15), the condition rn r < 0 leads directly to Townes’ conditions (13).

+

11

The occurrence of instability was overlooked in the first analysis by this method.

72

P. A . GRIVET A N D L. M A L N A R

Experiments beautifully check these predictions in the case of the nuclear maser (see 5, 85, 86, 88). 3. Pulling h’flect. The sole important drawback of the nuclear maser results directly from the threshold condition: the Overhauser effect does not provide a sufficiently high value of Mo to enable one to reduce Ql under a value of 150. The Q of the circuit must reach practically a value twice as high as the minimum limit Ql arid Q must approach values of the order of 300. Under these conditions an error appears because the tank circuit cannot be tuned to the precession frequency w = y p B with sufficient accuracy. Practically it is adjusted a t w,, different from w , and for this reason the maser oscillates a t a frequency wm which differs both from w and wc. It is given by am

-w

= (w, - w ) / q

(16)

where q = wT2*/2Q. Q is the quality of the circuit and Q > Q I ;Qt is the value of Q given by formula (13) when it reduces to an equality:

Exact tuning cannot be achieved for two reasons: (1) lack of a suitable criterion; ( 2 ) necessity to adjust the tuning in order t o follow the variations in intensity of the natural fields. Relation (16) may be established in a simple manner by interpreting the buildup of the oscillations in the following way: lying along -B) there exist small micro(1) I n the initial state (Ma scopic transverse components isotropically distributed so as to have a zero resultant. (2) The noise level of oscillation in the circuit produces a small rotating transverse field B,, and B, gives rise in the usual manner (following Bloch’s steady-state solution) to a small transverse macroscopic moment M , rotating with angular velocity w ; this occurs because B, introduces order in the distribution of microscopic transverse components, e.g., it “phases” their precession: M , induces an emf e in the tank and produces a current i;i generates a rotating field B , which couples to M,. This may be shown by the following feedback diagram: M , -+ e -+ i -+ increase in M,; for this phasing process to be most successful the sum of the phase differences along the feedback loop must be a multiple of 27r. This condition is achieved a t exact tuning12; in the case

+B,

12

The calculation in this case is easy.

M E A S U R E M E N T O F WEAK M A G N E T I C FIELDS

73

of a slight mistuning, the maser still oscillates; therefore the phase difference between e arid i in the circuit, 64c, must be compensated by an opposite phase shift between B , and M,, 6 4 N ; this condition reads 84N

+ 64C

=

0

(18)

and by reference to the Bloch and circuit equations one has

64= ~

(W

- w,) Tz*

6dc

=

2Q (urn - w C ) / W C

(19)

Combining Eqs. (18) and (19) one obtains Eq. (16). A discussion of the practical values of the parameters leads to the conclusion that owing to this “pulling” error, the nuclear maser has too small a dynamic range; it would be useful only if provided with an automatic compensation system for the variation of B.

4. Spin Oscillator. Automatic compensation of the 6Bc’ is a possible solution. A

__

FIG.24. The basic schemc of a nuclear spin oscillator using decoupled Bloch coils.

ti

pair of magnetically

Bonnet (98). They avoided use of the negative sign of M Oand the maser effect. From the Overhauser effect they retained the enhancement of the polarization’s absolute value; in other words, they utilized the high value of m in Eq. (12) but no longer its negative sign. They still kept the continuous oscillator principle but now resorted to a spin-coupled tube oscillator. Indeed, in his first disclosure of the discovery of “nuclear induction” F. Bloch stressed the advantages of electromagnetically decoupled “Bloch coils.” These are two coils with perpendicular axes and a fine electrical adjustment in ordcr to further reduce the residual coupling. Salvi made such a set of highly decoupled coils for frequencies in the neighborhood of 1900 cps (suitable i n the earth’s field) ; the residual coupling may be kept a t a value as low as below 80 dB in a wide-band frequency. If one then connects a pair of such coils to the input and output of a n amplifier (Fig. 24) the system does not oscillate, even if the phase balance is properly adjusted: the coil’s coupling is too low. This state of

74

P. A. GRIVET A N D

L.

MALNAR

affairs changes drastically if a polarized sample is put inside the input coil and if an external field B is adjusted so that the precession frequency w of the moment is located in the middle of the amplifier's passband: the rotating moment efficiently couples the two coils a t frequency w , and a steady oscillation occurs a t this frequency. This spin oscillator also shows threshold; the threshold condition is similar to th a t of the maser but includes as a new and very useful parameter the voltage gain G of the amplifier. It reads

and the product QG of the quality of the circuit and the gain G replaces the single parameter Q in the old formula (13). One can adopt a very low Q and still reach threshold by increasing G to a big value. Under these conditions the pulling error, proportional to Q, may be made negligible. Practical orders of magnitude are Q = 5 and Qr = 100. No other form of pulling occurs if the linear amplifier is of the wide-band type and shows a phase shift independent of frequency near w (cf. 58). The practical limit to the possible high values of G is imposed by the residual electromagnetic coupling: the device should not be able to oscillate in th e absence of B field, by sole virtue of this residual coupling. If one expresses it by the ordinary coupling coefficient k,, one can show that the spin coupling may also be represented by a similar coefficient k,; the value of k , is

xo'

= mXo and ql,q2 are filling factors for sensor and exciting coils. The condition for good performance then reads k,>>k,; it can be easily realized; k , may reach values of the order (for Tz*= 1 sec; w = 27r.2000 cps; m = 1500; XO' = 1500 X 3.3 X lO-'O; UEMcgs; and k , may be reduced to Spin oscillators were used earlier in the high-field domain; they were introduced by Schmelzer at CERN in Geneva in 1952 (99), and the theory was also developed by Kurochkin (100). The industrial resonance magnetometer built in France by Sud Aviation actually works along this principle; the original apparatus built by Salvi had the following characteristics:

(1) Sensitivity, 0.2 pG. (2) Dynamic range, 0.4-0.5 gauss (cut in 10 ranges). (3) Bandwidth, 0 - 1 cps.

75

M E A S U R E M E N T O F W E A K M A G N E T I C FIELDS

Overhauser hf power, 2.5 watts. ( 5 ) Easily transformed to be a gradiometer.

(4)

5. Benoit’s Liquid Plow Method. As mentioned before, the Overhauser effect is not the sole method available to obtain a marked enhancement of the nuclear resonance signal in a liquid. Benoit (88) devised a very efficient procedure, where a liquid, tap water for example, is strongly polarized in the strong field of an electromagnet and then flows to the sensor; this system may be designed as a maser or as a spin oscillator. Indeed, between polariser and sensor one may-at will-invert the moment, by “fast passage” through an auxiliary hf coil, exciting the Pump

r2ateLf‘ow

I

Bt

field r

l ...

Inversion by adiabatic fast passage

I

lTzz3l magnet

Maser’s COI I

FIG.25. Principle of Benoit’s expcrimcnt for continuously obtaining prepolarieation and inversion by the usc of liquid flow.

inversion in the inhomogeneous leakage field of the magnet. The mechanism of the device is explained at length in the review (16),where one may find a complete bibliography on this system. The enhancement factor W L may in practice reach very high values of the order of 10,000-20,000, and with a spin oscillator sensor one obtains a very sensitive, robust, and reliable magnetometer. But the device, by necessity involves the use of a rather bulky and heavy piece of apparatus: the polarizing electromagnet. For this reason it can be used only in stationary observatories and even there one needs to carefully adjust the position of the sensor with respect to that of the polarizer, so as to make the leakage field of the polarizer inoperative; this can be achieved in a systematic manner, as shown by Hennequin (101). The inconvenient necessity of this procedure has until now hindered the development of the device as a commercial magnetometer. But the experiment shown schematically in Fig. 25 is a simple and efficient one, and offers the simplest and most flexible demonstration of the properties of masers and spin oscillators.

76

P. A. GRIVET AND L. MALNAR

IV. OPTICALDETECTION OF AN ELECTRON NUCLEAR RESONANCE (Amp = +1) A . Optical Pumping 1 . Preliminary Survey. a. Optical and magnetic levels. The vivid expression “optical pumping,” coined by Prof. Kastler, calls to mind a rather accurate picture of this kind of experiment. To understand it, one may, for example, look at the usual pumping scheme, from the bottom of a valley, to a high-altitude lake, and back. Describing the process in the more abstract terms of energy transformation, one could then say: Rb vowi cel I Photocell

.

-

.

~

.

Osci Iloscope

FIG.26. The basic setup for studying an optical Zecman spectrum by “optical pumping” of an alkali vapor.

Energy is pumped from a low level to a high level, stored for a while, and then allowed to flow down to another low level. This description may look like a rather dry one, in everyday life; its advantage is that it describes optical pumping equally well; in this process the energy is that of a few cubic centimeters of alkali metal vapor contained in a bulb at low pressure and subjected to a magnetic field of 1 gauss. The pump takes the unusual form of a circularly polarized beam of light, illuminating the alkali metal bulb (Fig. 26). The useful levels of energy are those of the alkali atoms and here they are many in number. The levels appear to fall in one or the other of two classes when the energies involved in possible transitions between them are considered : (1) Optical transitions correspond to energies of the order of 500,000 Gc/sec (gigacycle = lo8 cps) when expressed in frequency units1*; 18

The equivalence between the frequency unit of energy and the mechanical one = E, which results in: 1 cps = 6.625. ergs. Other

is given by Planck’s law hv

M E A S U R E M E N T O F W E A K MAGNETIC F I E L D S

77

(2) Zeeman or magnetic transitions. Ordinarily each optical level is “degenerate”; for a given energy El the level contains a few states which one is justified in describing as different because their detailed properties are different; this is especially clear in theory, because they belong to different #-wave functions. As regards energy in field-free space, this multiplicity is hidden and does not appear on the diagram, as long as one does not consider the effect of a magnetic field on the sample. I n this case the field “removes” the degeneracy and instead of the single level line Eithere now appear di distinct levels, Ei,,with energy differences proportional to the field; this magnetic splitting is described by a Zeeman diagram, Fig. 27a, which shows accurately the energy dependence with respect to the field intensity, in the low-field domain. A more extended graphical description, Fig. 27b, shows in general that the splitting obeys different laws in the low-field region (under 1 gauss), in the intermediate region (from 1 gauss upward to a few hundred), and in the high-field region (from 1000 gauss upward). Optical pumping can occur generally in the low-field region, where the splitting is in equal steps of value Avi given by Kastler (102, IOWa, 102b):

Ti

gFye

= -

2

where yi is the apparent gyromagnetic ratio of the level, ye is the gyromagnetic ratio of the free electron considered in the first part, and appearing with a small correction in the electron paramagnetic resonance of a free radical (such as DPPH), and g F is linked with the multiplicity of the level (it is also called for historicI4 reasons the “Land6 factor”). To the same approximation, one may draw straight lines on the diagram. The magnetic transitions generally, and this more markedly in the low-field domain, are characterized by the order of magnitude of their frequencies, which lie in the radioelectric domain (more specifically, they range from 100 kc/sec to a few megacycles per second; they are typically useful equivalences are detailed in table 7a-2, p. 7.4, of the American Institute of Physics Handhook. The introduction of the frequency unit is convenient for measuring Zeeman energies later on. It may still he worth mentioning that the relation E = kT introduces the correspondence: 1°K 420,835 Gc/sec. Finally, as visible lines of the optical spectruni are known by their wavelength, the wave number I = 1/h [cm-I] is a useful intermediate; I is related to frequency by I = v/3 . 10’0: 1 cm-1 c) 30,000 Gc /sec. I* Historically, the genuine “Land6 factor” described the degeneracy for L-S coupling, ignoring the nuclear spin.

78

P. A. GRIVET A N D L. MALNAR

“radioelectric” frequencies).I6 This low order of magnitude has an important consequence: the do sublevels16of the “optical ground” state (i.e., the lowest of the energy levels in the absence of an external magnetic field) are equally populated, by thermal excitation; indeed, the average thermal energy per particle and per translational degree of freedom (+lcT) at 300°K amounts to a much higher value of energy, to some 4200 Gc/sec, in frequency units; under such a strong thermal excitation all the sublevels of the ground state are equally populated in the ordinary equilibrium. On the other hand, all the other optical levels and sublevels of interest here are located some 500,000 Gc/sec higher, and are practically empty in the thermal equilibrium state. A last important order of magnitude concerns the pumping light. The source is an ordinary high-frequency discharge; it must emit strongly on 16 Some frequencies occur in the microwave spectrum; such are the AF = 1, Amp = 0 transitions used in atomic clocks (105),but they play no role in magne-

tometry, their frequencies being insensitive to B,to the first order. 18 We call d the number of degenerate levels; the standard notation in statistical physics is g, which could be confused with the Land6 factor here.

D lines

Optical pumping frequency 509,000 Gc/sec

I

7

1772 Mc/sec

Zeeman effect for low fields

(a)

FIG.27. (a) The splitting of the ground state in the Na spectrum: the two hyperfine levels F = 1 and F = 2 and t)heir %reman splitting in low fields. The upper

MEASUREMENT O F W E A K MAGNETIC FIELDS

79

the frequenvy of the transition to be pumped. For this aim, one uses the same alkali metal i n thc lam]) as for the pumped sample and, eventually, filters out of the spectrum thc rclcvant single line corresponding to the transition studied in the absorber. But the conditions in the source (full Doppler effect, higher pressures, strong hf excitation) are very different from those in the absorption bulb: the line is broad (1000 to 3000 Mc/sec), much broader than the total width of the magnetic splitting occurring in the corresponding levels of the absorbers; therefore in the presence of a magnetic field the relevant sublevels of the absorber are all subjected to the same optical excitation by the pumping light. These simple conditions occur in the first absorbing layers of the sample: the very process of absorption itself rapidly changes the line shape of the excitation light whenithe beam has penetrated deeper into the absorber. Effects of this kind will be examined later.

E

Zeeman transitions

F=i

LIOCK

transitions

I 1772 Mc/se\ I

B

F=

transitions

Fields

Intermediate Region

Fields

(b) “optical” levels are indicated syrnholically only. (b) The three typical regions in the full Zeeman spectrum: in the low-field region only, the splitting is linear in B.

80

P. A. GRIVET AND L. MALNAR

b. The e1cment:u.y processes in the mechanism of pum1)ing. To summarize, in the simplest conditions one pumps with a broad line, and this illuminates evenly the atoms in the sublevels of the two relevant optical states. Nevertheless, equal illumination corresponds to diff ererit rates of excitation in the sublevels of the ground state for two reasons: (1) The light is circularly polarized; it is, for example, a+ light for which the E vector rotates in the positive direction around the direction of propagation. For such u+ light, the total transition probabilities from these sublevels are different; therefore the diff erent sublevels empties a t unequal rates. (2) By the same token, the magnetic sublevels of an excited optical level (the “upper” levels in the pump image) are populated a t still different, but also unequal, rates. The elementary probability of transition is relative to a pair of sublevels, one in the optical ground state, the other in the optical excited states: these “transition probabilities” are accurately given by the quantum theory (8,104). The total probabilities out of (or to) any sublevel are obtained by summing the elementary probabilities for all the allotted transitions stemming from that level (or ending a t it). The rules defining the allotted transition depend on the polarization of the beam (u+ light, AmF = 4-1) and on the quantum state considered (elementary rules AF = 0, + l ) ; later on, as in Fig. 31, relative total probabilities for ‘T+light are indicated by numbers on each level. Each of the atoms “pumped” to an excited state remains in this upper state for a short while and then returns to a lower state essentially by fluorescence; this radiative process belongs partly to “stimulated emission” under the action of the pumping beam and partly and mostly to “spontaneous emission.” The distinction is important: the first process is the strict inverse of the pumping action and involves the same selection rules and probabilities; in contrast, spontaneous emission obeys different rules and opens new downward channels.” The two processes are pictured in Fig. 28 by a type of level diagram that is very convenient to describe such processes and will be linked to the features of the Zeeman effect in alkali atoms later. For the present it is sufficient to know that energies are measured along a vertical axis, and that magnetic sublevels are characterized by their magnetic quantum numbers mF, which are plotted along a horizontal direction: the big 17 This may be easily understood by considering spontaneous radiation as stimulated by zero-point energy fluctuations of the vacuum; this radiation shows no preferred direction or polarization.

MEASUREMENT O F W E A K MAGNETIC FIELDS

81

energy difference between two “optical” levels appears clearly; the much smaller one between magnetic sublevels is also shown but not to scale (current pravtice in the specialized literature is t o neglect it on such diagrams). The scheme here is purely hypothetical and does not pertain

L T Zeeman Av

\

Relaxation

-2

-I

0

+I

+2

“F

FIG. 28. Hrisenberg’s form of the Zeenian diagram: it is convenient for understanding the pumping action of polarized light; it shows the action of I+ light, of spontaneous emission, and of relaxation; stimulated emission is proved to be quantitatively negligible here.

to any real alkali metals, but its simplicity lets some important features of the pumping scheme appear clearly:

+

(a) Absorption. The selection rule AmF = 1characterizes the pumping effect of U + light: by absorbing it, atoms are transferred from level a to level b‘ a t a rate of Nab! per second, Nab, is given b y the product of the relevant probability of transition U/’ob’ by the actual population n, of the initial sublevel a. This is a big population because a is a sublevel of the ground state and its Zeenian energy is sniall compared with the average thermal energy (z kT). Stirnulat,ed emission. This process is completely negligible here. (b) Iridccd the pumping beam may also induce the reverse transition b’ + a and by the principle of “microreversibility” wbla = w&. But the “optical” level b‘ is so high in energy compared with the thermal excitation that, by

82

P. A . GRIVET AND L. MALNAH.

Boltzman’s formula, it remains practically empty a t equilibrium; pumping does not change these conditions significantly because spontaneous emission empties the upper sublevels quickly. Finally, the rate of transfer by stimulated emission Nbta = Wb’anb’ remains negligible as n b r . (c) Spontaneous emission. I n contrast, spontaneous emission equally well links upper level b’ with any of the low levels a, b, c ; it is an important effect, which determines the order of magnitude of the lifetime in the upper level, 7 between 10 and 100 nsec. (d) Relaxation. Practically, spontaneous emission does not link any pair of Zeeman’s neighbors b’ to a‘ or c’ to b’; the probabilities of transition in such cases still exist but are very low because they are proportional to ( A v ) ~and the Zeeman A d s are so small in comparison with the optical A d s that this theoretical possibility may be safely neglected in practice. Then relaxation transitions become important; they are due to mechanical collisions between atoms or of the atoms on the wall. Very conspicuous is the efficiency of relaxation between Zeeman sublevels of the upper optical levels : relaxation can equalize the population of the sublevels very quickly if the pressure in the bulb is higher than mm of mercury, owing to a higher temperature or to the presence of an inert “buffer” gas; this offers certain advantages as explained later. This description of the main elementary processes in the pumping scheme is sufficient to introduce the origin of the practical classification of the modes of the pumping. In practice one distinguishes between:

(1) Kastler pumping. The bulb contains alkali vapor only a t the mm Hg). right temperature to maintain a very low overall pressure By the choice of a low pressure and the use of a favorable coating on the walls of the bulb, relaxation between the sublevels of the upper optical level is minimized as far as possible. One can then understand the pumping effect by considering two processes only: absorption of the u+ light and spontaneous emission from the upper level. This mechanism is shown for the levels a, b’, b, c on Fig. 28; one sees that the population of level a tends to migrate to the right in levels b and c. This is a general trend for any level of the ground state: for each of them, three arrows move the atoms to the right and only one backward to the original level. Practically, this useful effect may be a feeble one; nevertheless, after a buildup time a permanent regime is established where the levels a t the right, in the ground state, are significantly more populated then those a t the left of the diagram. The buildup time measures the overall efficiency of the scheme and ranges around sec or more. Dehmelt pumping. The bulb contains alkali vapor a t the same (2) pressure as before, because technological reasons impose adoption of the

MEASUREMENT OF WEAK MAGNETIC FIELDS

83

same working temperature, but in principle it could be higher. An inert (nonparamagnetic) buffer gas fills the bulb at a well-chosen pressure (a few millimeters of mercury; see later discussion); the result is th a t relaxation operates efficiently between the Zeeman sublevels of the upper optical level, but not between those of the ground level. The populations are equalized by relaxation in the sublevels of the excited state, but they remain unequal in the sublevel of the ground state. This is the result of the inequalities in the probabilities of emptying the different sublevels of the ground state; theory shows that these probabilities decrease when one goes from the left to the right on the diagram of Fig. 28. The drawing itself enables us to foresee that the probability for emptying sublevel d is zero, because one has no upper level available to draw a u+ arrow, starting from d. The details of the mechanism of pumping are evidently different in both cases, but happily enough both schemes may be made very efficient in practice as regards magnetometer construction, because in this kind of application one is interested in the sublevels of the ground state only. I n any case, the process of pumping appears as continuous and the atoms circle quickly around a “closed cycle” of states. When the permanent regime is established (this requires a buildup time in the range of a fraction of a second), one observes strong inequalities in the populations of the sublevels, a t least in the ground state. c. Detection of the “orientation” in the pumped state. The occurrence of strong inequalities in the populations of the magnetic sublevels represents the essential result of the “pumping cycle.” An important problem is to determine the efficiency of the process, i.e., to measure the permanent differences in populations: magnetometry will appear as a n interesting by-product of the solution. This is possible in a variety of ways and we begin by evoking two methods which are not used here, although they are very popular in other domains, where one deals with condensed matter; here they are ruled out because the sample is a vapor a t low pressure. (1) Enhanced paramagnetism. The alkali atoms in the states considered here have a magnetic moment, and the artificial inequalities in the ni produced by pumping are the cause of strongly enhanced paramagnetism. But the direct measurement of paramagnetic resonance is made difficult by the low density of the vapor. (2) Detection of the “orientation” by nuclear methods. The strongly enhanced paramagnetism explains the expression “oriented state” often used to describe the pumped state. Indeed in the magnetic field B , the magnetic moments and correspondingly, the atomic spins are oriented in

84

P. A . GRIVET A N D L. MALNAR

the direction of the field: the degree of orientation may be high if all the atoms are pumped to a single sublevel of the ground state. As will appear later, this is nearly possible. Such a “completely pumped” sample could be a useful oriented target in nuclear physics (106) if the density of the vapor could be increased; this has been recently achieved, in the closely related case of optically oriented 3He (206)(the reference offers an accurate discussion of this case). (3) Maser action. The differences in population between sublevels may show one sign or the other; a priori, it does not look impossible to choose two levels that show an inverted population, e.g., a negative paramagnetism, and to try to obtain maser oscillations by a suitable coupling with a resonator. Practically, this experiment is very difficult for Zeeman transitions a t radio frequencies, because the alkali vapor is a t too low a pressure: the inequalities in population may be very strong, but the total number of atoms available per cubic centimeter is too low in a vapor a t a pressure of some mm Hg. This kind of experiment was first attempted for microwave clock transitions for which it is essential t o reach the regime of self-oscillation; success was achieved only recently with great difficulty (107, 107a). Nevertheless 3He was shown very recently t o offer an exceptionally favorable case; a nuclear maser was operated successfully with optically pumped 3He (108); the nuclear relaxation time is very long and this permits a n efficient “motion narrowing” of the nuclear resonance (see 91, p. 57) ; the result is a high value of T2*, which in formula (13) overcompensates the influence of a low M o , owing t o the low gas density. I n the general case, Kastler’s discovery of the indirect optical methods offers easy solutions to these difficulties. Only two of these procedures are used in the magnetometer technique. The principle of these “trigger methods,” whose great sensitivity was stressed before, is the following: (a) One submits the sample to a secondla and stronger excitation, a t a radio frequency corresponding to a Zeeman transition between two sublevels, for example, i and j of the ground state; this is called the “double resonance” method; (b) The result is “saturation” of the transition (see 109, p. 168); in other words, there is a trend toward equalizing the populations of the sublevels i and j . Correlatively, it markedly perturbs the pumped regime and changes the permanent populations of all the sublevels; this is often called a “reorientation” of the atoms. 18 For this reason this type of experiment is often called “double resonance,” e.g., optical and radio resonance.

MEASUREMENT O F W E A K MAGNETIC FIELDS

85

The optical consequences are numerous and varied; generally speaking, the radio excitation changes (1) The polarization arid intensity of the fluorescent light, which one may observe to be emitted sideways. The determination of the polarization of the fluorescent light was widely used by Kastler’s school (7) for scientific research. It is not popular as regards magnetometers for two reasons: (a) Experimentally, it is complicated, arid (b) fundamentally, it may be shown that differences in polarization arise from unequal populations of the sublevels of the optically excited state. These are p states for alkali metals; the pelectron

Sodium

/u

FIG.29. Probing the effect of Zeeman rf excitation on a pumped vapor by the use of an auxiliary crossed beam : the secondary beam is amplitude modulated a t the rf Zeeman frequency.

cloud is geomet>ricallyasymmetrical, assuming a n elongated shape and showing an orbit8almagnetic moment. For this reason, the p states are very sensitive to mutual atomic collisions which disturb the orbital magnetic moment, and by “fine structure” and “hyperfine structure’’ coupling induce transitions between the sublevels; one says that the sublevels are “mixed” or “reorient’ed” and naturdly this averaging of the populations strongly deteriorat.es the efficiency of the method a t high vapor pressures or i n the prescnce of “buffer gases” (ot,herwise int,erest.ing for highinterisit y effects) . ( 2 ) The optical absorption of the bulb. One may observe the absorption on the pumping beam itself, receiving it finally, as proposed by Dehnielt ( I I O ) , on a photomultiplier. One may also use an auxiliary beam directed a t right angles to the first (Fig. 29) and much less intense (in

86

P. A . GRIVET AND L. MALNAH

order that it will exert a negligible “pumping” action); this case is very advantageous because, as discovered by Dehmelt (11I), the beam appears as amplitude modulated a t the Zeeman frequency. The absorption depends on the populations of the sublevels of the ground state only. These substates are spherically symmetrical S states with zero orbital magnetic moment: thus an important link between the orientation of the magnetic moment (which in this case originates in the electron spin or in the coupled electron and nuclear spin) and the perturbation during an encounter is mostly suppressed. The mixing of sublevels of the ground state does not occur easily and useful variation of absorption under the action of Zeeman saturation still occurs at the highest pressures in use. The process is simple in principle: “optical pumping” by its very nature empties selectively some sublevels of the ground states; this effect diminishes the over-all absorption of the bulb, which is proportional to

c ij

niwij

ni being the population of sublevel i of the ground state in the permanent regime and wij the transition probability to a sublevel j of the excited state. Saturating a Zeeman transition changes the values of the ni in a complicated manner: experiments and theory (9) show that in general it increases the absorption. This result is nearly evident in the simple and easily realizable case when the Zeeman equalization occurs between a level i included in the summation and a level k, which does not figure in summation (24). Level lc does not participate in optical absorption because of a prohibition expressed by the so-called “selection rules”; such a level automatically fills up because the pump cannot empty it (the relaxation “leaks,” neglected here, will be considered later on; in fact, in a few cases, they can be made of minor importance). In this important case, Zeeman excitation appears clearly as a “depumping action.” Such a case is easily achieved for any alkali metal by a suitable choice of the pumping light, as first proved by Franzen (112). This simple case will be examined in more detail in the next paragra.ph. For the general case, the reader may find a more detailed introduction in recent reviews on “optical pumping” in Kastler (102, 10da), and Bloom et al. (113-118), and a detailed treatment in the references cited there, and more especially in Brossel (119, 120, 12Oa) and Cohen-Tannoudji (121). 2. Alkali Atoms: the Simplified Case of N a . Actually, rubidium and cesium exclusively are used in commercial magnetometers; this choice results from technological advantages of Cs and R b that will appear later

87

MEASUREMENT OF WEAK MAGNETIC FIELDS

on. I n principle, any member of the first column of Mendeleev’s table (222) would work equally well in a pumping cycle. At first sight, one might think that the member in the first row, hydrogen, would prove a simple and typical example. Experimentally, this is not true because the appropriate line, called Lyman a, which corresponds to the transition from the ground state, n = 1 (n radial quantum number), to the first excited state, n = 2, is situated in the far ultraviolet (X = 1215.7 A), where the construction of sources and lenses meets big difficulties. Moreover, t,he natural species is the Hz molecule, which must be first dissociated in H atoms by some kind of Langmuir arc; the experiment was attempted only very recently (123). On the other hand, the theory would appear both singular and oversimplified; it would be of little help in understanding the other cases. Indeed, the hydrogen atom is the only one where a pure Coulomb force acts between nucleus and electron. For this reason, the n levels all show a peculiar degeneracy, the S state (1 = 0) and the P state ( I = 1) having the same energy. As a result, the Lyman a line is a singlet (neglecting the very small unresolved splittings corresponding to the Dirac relativity correction and to the Lamb shift), corresponding to the “radial” transition n = 1 + n = 2. All the other members of the first row are true alkali metals and the structure of the relevant levels are markedly different. I n these cases, one still deals with an unpaired electron; but its lowest level corresponds to n = 2 a t least for Li and n = 3 for Na; we will take Na as an example, because it is the simplest*g of the experimentally studied alkali metals. Between the shell before the lone electron and the nucleus other shells completely filled with electrons are interposed. I n the present problem, the presence of these complete shells is of no importance, except that the central force is no longer Coulombic, and now states with the same n, but different Z’s, have significantly different energies. Consequently the ground state remains an S state (n = 3, 1 = 0 for Na) but the next excited state is a P state still with n = 3 but with Z = 1 (for Na); the relevant optical transitions depend essentially on the azimuthal quantum number (AZ = f l ) , not on n, and as the various alkalis differ essentially in n and the radial part of $, this explains why the optical transitions are similar in all of them. The spin of the lone electron shows its existence by the spin-orbit coupling, which splits this P state (I = 1) into two substates labeled by the quantum number J : (J = Z 8 =

+

+

1 9 The only abundant isotope of Na is *aNa, with the smallest value for the nuclear spin (I = I);87Rb shows the same nuclear spin and for this reason the same set of levels; but natural Rb is a mixture of 87Rb(27%) and 86Rb(73%) with ( I = $). Pure 87Rb is available and is used in magnetometers, giving strong signals; 23Na for technological reasons gives poor signals.

88

P. A. GRIVET A N D L. MALNAR

and J = 1 - & = 6 ) ; this “fine structure” splitting is the origin of the well-known yellow doublet (01, XI = 5986 A; Dz, XZ = 5890 A) of the sodium light; similar doublets occur in the visible spectrum of all the alkali metals as shown in Table 111. TABLE Isotopic abundance

ZaNa JBK 86Rb *7Rb 1JJCS

100 93.1 72.8 27.2 100

Nuclear spin

a3 H

T2 1

1115

DI(A) P1/2 -+ SI/Z 5896 7699 7948 7948 8943.5

-+

S1/2

5890 7665 7800 7800 8521

a The Li doublet at 6708 A (Ah = 0.14 A) is not resolved by current optical spectrographs (these data are excerpts from H. G. Kuhn, “Atomic Spectra.” Longmans, Green, New York, 1962).

High-resolution optical spectroscopy allows a still finer splitting of the levels to appear, the “hyperfine” splitting. As explained by Leighton ( 6 ) and Mockler ( I O S ) , this arises from the dipole coupling between the intrinsic magnetic moments of the electron and of the nucleus. It exists in its simplest form in the H spectrum, where I = 9 for the proton; it is more complicated for z3Na ( I = i) and still more so for 86Rb (for *6Rb, I = 8; for 87Rb, I = 8) and Cs (for 133Cs,I = 8). Each of the previously considered J levels is split into many components corresponding to the values of the “total” quantum number F (vectorially F = I J ; F , the projection of F on the z axis, runs through all the integers between I J and II - J J ;the resulting spectrum may be quite complicated, as shown by Skalinski (164)for Cs. But the mechanism of pumping is not essentially changed because one pumps with a beam of circularly polarized light shining along the B direction and called U+ illumination. For this kind of polarization the selection rules of importance are the same for F (AmF = +1) as for J (AmJ = +1). Therefore, the discussion and the drawings can be simplified by ignoring the hyperfine structure and discussing an approximate model where all the hyperfine levels are lumped together at their center of gravity. Figure 30a shows a model for Na; it may be considered as a typical one for the alkali metals because the principal quantum number n plays no role in the process. The selection rules invoked here are simply expressions of the conservation of angular momentum: for u+ illumination (the E vector

+

+

89

MEASUREMENT OF WEAK MAGNETIC FIELDS

rotating in the same sense as a magnetizing current producing B ) , each photon carries a quantum of angular momentum h and can only disappear by absorption in a collision th at induces a transition in which the atom gains a unit of angular momentum h : these permitted transitions are defined by mF=+1 m J = +1

at

This conservation law is of general validity and applies to the simplified model, as well 3s to the real situation. Fine structure

b, pumping far 23Na (simplified)

Zeeman levels

/

B

\

DI light

D21ight

I

-312

-112

112 mi

(0 )

( b)

3 2 -312

1/2

-1/2

I

312

mi (Cl

FIG.30. The action of the DI and D2 components in u+ pumping, explained on a simplified level diagram : hyperfine structure is ignored. (a) Zeeman diagram; (b) Heisenberg diagram, D1 action; (c) Heisenberg diagram, D2 action (114).

Looking then at the Zeeman diagram for the model in the presence of

B, it is convenient to arrange the levels to correspond to a given value of B, in the manner shown in the right-hand part of Fig. 30. For improved clarity, there, the sublevels all correspond to the given B, and those situated on the same vertical line correspond to the same value of mJ. Therefore, the only allowed transitions for U+ excitation according to the rule (Am., = +1) are pictured by arrows inclined toward the right. It

90

P. .4. GRIVET AND L. MALNAR

then appears that if pumping is done with the D1 line only (filtering out the Dz line), there is no transition to empty the sublevel mJ = & of the ground state; this level then fills up because the atoms pumped to the mJ = sublevel of Pljz fall down to the mJ = 8 sublevel of SliZ by the phenomenon of “spontaneous emission’J20(broken arrow, Fig. 30b). Figure 30c displays the pumping scheme for D z light. This is a more complicated case and since there is a pump transition emptying the mJ = $, S1/2level, the result can no longer be foreseen so simply. Pumping produces a more complicated nonequilibrium population pattern which

+

-3 -2

-I

0

I

2

3

mE

FIG. 31. Complete Heisenberg diagram for Na, including hyperfine structure: the pumped transition using u+, DI light.

can be calculated by rate equations, using the transition probabilities of quantum theory. This is shown in many of the above-mentioned references, especially Franzen and Emslie (112)and Alley (126). It is easy to show that the simplicity and efficiency of (01, u+) pumping is retained when hyperfine splitting is taken into account: the full Na sublevel structure is shown in Fig. 31. One may check that the P l p and S ~ /levels Z each split into the same number of mF sublevels because the relevant values of F are the same in both cases; e.g., F = 2 or F = 1 or more generally F = I +. For this reason the sublevel now labeled by mF = 2, F = 2 , S l / z ,is not pumped, the selection rule AmF = 4-1 here taking the form of a prohibition. 10 As shown by Einstein, the probability of spontaneous emission gains rapidly in importance as the frequency increases (proportional to the cube of the frequency). For this reason ‘k.pontaneousemission” is negligible in the microwave and radioelectric range of frequencies but important for visible light (see Siegman, 109, p. 212).

MEASUREMENT OF WEAK MAGNETIC FIELDS

91

This (bl,u+) process is also very inl,eresting because it demonstrates t,hat as the pump depletes all the available sublevels, absorption of the pumping light strongly decreases : the bulb becomes much more transparent as soon as all atoms are collected in the insensitive WLF = 2 sublevel. The efficiency of the pumping scheme can easily be checked by measuring the decrease in absorption; for example, one may observe an absorption coefficient of 50% when pumping is switched on and it, decreases in a second of time to 40% or less when the pumping cycle is fully established. A correlative and important aspect of this mechanism explains why the entire volume of the bulb is pumped: during the buildup of the cycle, only the first layers are pumped; they soon become transparent and allow the source to pump progressively deeper layers. With this last argument, it appears that pumping with mixed D1 and Dz light could occur by the same simple mechanism: the Dz lightz1has a transition starting from the level m F = 2S1,2,soon filled up by D1. Consequently, it still suffers strong absorption after the sample becomes transparent for D1. D z would be absorbed in a first thin part of the sample; the remaining volume of the bulb would be pumped by D I as described above. This would explain an apparent contradiction: the DID2 pumping appears experimentally efficient even in those cases where theory would predict zero efficiency. It is always possible to filter out D z by interference filters for R b and Cs, the interval between DZ and D1 being sufficiently large (see tabulation).

86Rb 87Rb 133cs

148 148 422

Finally, Franzen’s mechanism also shows clearly the efficiency of radioelectric “depumping.” I n the pumped state, level mF = 2 is full, and saturating the transition AmF = - 1 by a hf magnetic excitation transverse t o B sends atoms to the neighboring sublevel, m F = 1, from which they make transitions to other sublevels and thus restore absorption.

B. Zeeman Excitation 1 . Zeeman Resonance Frequency. The pumping beam therefore suffers a strong absorption when the hf excitation is tuned to the transition *1 Moreover D Ialone suffers twice as much absorption as D1 alone in the same sample.

92

P. A. GRIVET A N D L. M A L N A R

frequency. T he latt’er’svalue is, in general, v = 2.800/(21

+ 1)

[Mc/sec]

(25)

I being the nuclear spin value for the alkali metal chosen. The order of magnitude of v is shown in Table IV. TABLE IV

Isotopic abundance Number of linesa 21 Au(cps)*

+1

A B (rG)’

+

nLevel F = Z J = I For B = 0.5 gauss.

100

4 138 198

93.1 4 531 760

6.9 4 100 144

72.8 6 36 77

27.2 4

36 52

100 8 6.7 19

+a.

Formula (25) would be very accurate for B < 0.1 gauss, but in the earth’s field i t is only approximate, about 0.1%; this accuracy is sufficient for ‘measuring variations in B around a mean value, such as for the earth’s field. If one looks for higher accuracy and absolute measurements in the same range (=0.5 gauss), corrections to formula (25) are necessary and proceed from the Breit-Rabi formula, as explained in detail by Parsons and Wiatr (126). It means that on the Zeemen diagram the vertical B = 0.5 gauss is already located in the “intermediate” region where the curves are no longer rigorously straight lines. The intersection points are no longer equidistant with AV given by Eq. (25). The so-called Back-Goudsmit effect begins to bend the curves slightly, and this phenomenon is accurately described by the theoretical Breit-Rabi formula (127): by its use an accuracy of one part in lo6 for absolute measurements may be reached, if the various component lines are sufficiently narrow to be resolved. The factor determining line width will be discussed in the next paragraph: it will then appear that complete resolution is possible for R b under carefully chosen conditions: among others, a few are relative to external parameters: a very low Zeeman excitation is necessary in order to avoid any saturation broadening; a very uniform field B is required. Commercial magnetometers and ordinary observations do not meet these conditions: in order to get a good signal-to-noise ratio, a strong signal is obtained, a t the expense of broadening, by using an intense excitation. Under these conditions the various lines corresponding to different pairs of sublevels AmF = 1, in the F = I J and F = I - J levels, overlap, and in effect a center of gravity peculiar to the apparatus is used; the

+

MEASUREMENT OF WEAK MAGNETIC FIELDS

93

value of the absolute accuracy must be determined experimentally in each case. These possible differences appear clearly when the signals of "Rb and 87Rb are compared in the same apparatus or when the same field is measured with Cs and R b magnetometers. Practically (49), the frequency of the center of gravity of the aggregate of lines is given by formulas such as the following: v =

466744B f ( K ) 359B2

v = 699585B f ( K ) 216B2

for 86Rb for 87Rb

(26)

where K is an empirical coefficient experimentally determined, which characterizes a given instrument; K < 0.4 generally. The minus sign in this formula is the proper one when B points in the direction of propagation of the pumping light; the plus sign is used when B and the propagation vector arc antiparallel. lcull resolution was achieved early by Bender in his pioneer work on the subject [see, for example, Bender et al. (128)],and a theory of line width was given by Bloom (129). Bender (130) analyzed the practical aspcvts of magnetometers; a detailcd analysis is to be found in Bloom's now classical description of the technology of the Rb magnetometer (131); data on Cs magnetometers will be given in the last chapter of this review. The quantitative aspects of the problem will become clear if one considers the following: (1) The number of magnetic lines. Each of the two hyperfine levels oftheopticalgroundstate,F = I + J =I + + a n d F = I - J = I - + , splits into 2F 1 magnetic levels, e.g., in (21 2 ) and 21 levels, respectively, arid therefore there are (21 1) and (21 - 1) lines (rule AmF = 5 1). ( 2 ) The spacing of the lines on the frequency scale. In the intcrmediate region the curves on the Zeeman diagram can be considered as slightly parabolic (the first-order correction to linearity) ; then the second-order difference of the energy terms is constant: it means that the component lines differ in frequency by constant amounts; these are to be compared to the line width. Table IV is an excerpt of Bloom's calculation (131) for the F = I 4 level. It appears that the components of Cs show the least differences in frequencies and for this reason its use is advantageous as regards absolute measurements.

+

+

+

+

Shifts of a different origin but probably of a smaller order of magnitude (a few cycles per second) (132) are known to exist in theory. They appear in the clock transition in the microwave range and are easier to measure in this case; they correspond to small perturbations of the energy levels due to the pimping light it,self (an effect, similar to the Bloch-Siegert

94

P. A . GRIVET AND L. MALNAR

effect in magnetic resonance) or to the collisions between atoms, or with the “buffer” gas molecule or the bulb walls; we refer the reader to the review by Bender (133) and to Arditi and Carver (134) for information on these phenomena, which gain importance in absolute measurements; see the detailed theory due to Barrat and Cohen-Tannoudji (135).The 4Hetype of apparatus was thoroughly studied and shows a notable defect (a few tens of cycles per second) of this general kind (136). The 3He nuclear resonance magnetometer shows an effect of the same nature but peculiar to the exchange of spin states which characterizes this type of instrunlent (137, 138). 2. Linewidth; ReZaxation. a. Pump leaks; mixing of sublevel populations. For clarity, let us consider the simple case of ( D l , u+) pumping on the simplified Na diagram of Fig. 30; the essential results are as follows:

(1) Pumping alone results in filling up the sublevel mJ = & of the ground state, emptying all the others. (2) Switching on radio excitation first populates the adjacent level mJ = 0 a t the expense of mJ = +, and afterward the combined action of the optical pump and of the radio excitation redistributes the population among all levels of the ground state; this general change in population indirectly produces the optical signal.

+

Any other cause transferring atoms from the privileged mJ = level to others will perturb the scheme and diminish the signal; it will act as a leak for an ordinary pump. This appears clearly when one examines the initial effect of radio excitation: if a parasitic “relaxation” process partially fills the level m = 0, the first action of radio excitation will be impaired. More generally relaxation is due to transitions occurring in collisions between atoms and with the walls; these processes belong to thermal excitation and their trend is to restore the populations occurring a t thermal equilibrium, a state where Zeeman resonance is completely obscured by noise. These difficulties made the initial experiments on optical pumping very difficult; for example, pumping mercury was considered to be impossible for several years. An obvious remedy would be to pump with brighter sources, but the physics of the source is complicated and progress is slow in this domain. Moreover, powerful laboratory lamps are rather bulky and would not fit into portable instruments. Nevertheless, the study of sources is active, and the recent ones, in which a 10-cm cw magnetron powers a hf discharge in a flat, thin tube, are efficient; such lamps are described by Cagnac et al. (139-144). A second way is to diminish the relaxation scrambling of population

MEASUREMENT OF WEAK MAGNETIC FIELDS

95

between levels. Progress of great practical importance was the proof that the mixing of the sensitive sublevels of the excited optical states was of little consequence in the (01, u+) pumping scheme; quantum theory is used to calculate both cases, and Franaen has shown (112) that a t most a factor of 2 in efficiency is lost; consequently, the buildup time of the oriented state is longer, as shown in Fig. 32. The chief relaxation process for the ground state, as explained before, now resides in collisions of atoms with the walls; mutual collisions are inefficient. Bloom (113) states the problem in these vivid terms: each atom suffers some 10,000 collisions per second with the wall a t the low

I

i

Time in unit of Vw, wt : total probability of obsorption per unit time

FIG.32. Fransen’s calculation of two extreme cases in u+, D1 pumping: negligible or complete mixing caused by relaxation in the sublevels of the upper optical level.

pressure in use (10-6 mm Hg); each collision may have a depumping action. On the other hand, the most powerful illumination system provides only 1000 useful photons per second to hit and pump this atom; the pump is largely overcome. A first remedy was found in Kastler’s laboratoryzz(145): Adding to the metal vapor a magnetically inactive “buffer gas” such as argon (with zero atomic magnetic moment), a t relatively high pressure (a few millimeters or centimeters of mercury), replaces the straight path of the alkali atoms by a complicated zig-zag trajectory; encounters with the wall seldom occur and the numerous collisions with Ar atoms are largely innocuous, entailing no magnetic effect. This clearing of magnetic defects 2z

The interesting story of this discovery and of its development is told in Bloom

(113) and Carver (114).

96

P. A. GRIVET AND L. MALNAR

was made so eficient by Dehmelt (146-148) that a residual effect of the glass wall appeared. Dehmelt and others (149-151) successfully suppressed it by coating the wall with long-chain hydrocarbons or silicones. It seems that under the best conditions the only relaxation process still active is the collision with very small metal droplets on the cold parts of the wall. A detailed theory of these processes and an extensive bibliography is to be found in Bernheim and others (162-154). b. Line width. Relaxation processes not only scramble the populations but also significantly disturb the energy levels, usually broadening them. The resonance frequency becomes less sharp, or in other words the line is

-a-

_---.

i-----.

Coated glass wall

/

I --

FIG.33. Dicke’s simplifiedmodel of the buffergas or coating action, in reducing the broadening of lines by the Doppler effect.

broadening. The use of a buffer gas and of coatings is beneficial in this respect but experimentally the advantage would not even appear if these two processes did not have yet another virtue: they dramatically suppress the chief cause of broadening, the Doppler effect (which of course does not count as a relaxation process). This is Dicke’s theoretical discovery (155), and was proved experimentally by his students (156, 157). We analyze here the basic mechanism using a simple linear and classical model introduced by Dicke. Consider an atom flying along the z axis and bouncing back and forth between two glass walls of separation a (Fig. 33). An observer located on the positive part of the axis x receives a wave which is frequency modulated. One assumes complete efficiency of the wall coating: the radiation mechanism is not perturbed at all by the collision. Moreover, the wall in the drawing may symbolize the real wall

97

MEASUREMENT OF WEAK MAGNETIC FIELDS

of the bulb or the average argon atom of the buffer gas in a highly simplified model which is accurate enough for our purpose. The classical theory may be used to obtain the frequency modulation. The time diagram of the apparent frequency radiated by the atom is simple: it is made up of alternately positive and negative rectangular pulses; the height of the positive crests is + S v , and the height of the negative

Y(I

- v/c)

V(l+V/Cl

Emitted frequency

FIG.34. Splitting due to frequency modulation in Dicke’s model.

crests is -6v. Taking the unmodified frequency as the reference, 6 v is the elementary Doppler shift value ( 6 v ( = v / c v , where v is the velocity of the atom, c the velocity of light, and v the frequency. The Fourier spectrum of such a frequency-modulated wave (Fig. 34) is well known, if rather complicated. Around the central line (“carrier”) a t frequency v , we observe satellites (sidebands); the two first are centered on v 6v and v - 6 v ; these two satellites show an amplitude J ~ ( T G v )which , may be compared with J o ( T 6 v ) for the central component; T is the period of

+

98

P. A . GRIVET A N D L. MALNAR

the modulation (T = 2a/v), 6v the elementary Doppler shift, and J o and J1 are standard Bessel functions of order 0 and 1. For small values of the argument, J I / J o ‘v +T 6 v = a 6v/v = a/X, where X is the wavelength of the unmodified line (the last step obtained by taking into account Doppler’s law, 6 v / v = v/c). A small relative amplitude of the satellites with respect to the central line thus depends on the quotient a/X, being equal to it for small values; considering the values of the parameter a/X, very different but typical conditions appear. When the bulb cont.ains alkali vapor only, a is simply the diameter of the bulb, which we take as 5 cm; with a buffer gas, a is the mean free path length L; at high pressures of the buffer gas, L becomes much smaller. A specific example is argon a t pressure of 1 mm Hg, as used in Arditi’s clock experiment (158) with a = 0.1 mm. Typical cases are then: (1)

X

Magnetometer transition in the earth’s field; X

‘v

100 meters;

>>> a for any conditions. It makes no difference whether or not a buffer

gas is used: a/X is always very small and the average continuous spectrum reduces to the central component. This “motional narrowing’’ results in very thin lines and sharp resonances. The best results were obtained by Bender (169,160, 160a, 160b) taking great care to eliminate external disturbances; he obtained a line width of 2.3 cps for Rb. Using the extremely favorable conditions at the Fredericksburg observatory for measuring the lowest-frequency transition in 87Rb, he obtained

F = 2 mF=-2 v = 699585B - 216B2

F = 2 mp=-l ( v in cps, B in gauss)

(27)

(2) Optical clocks; X ‘v 10 cm. The narrowing effect would essentially disappear for a coated bulb (X/a = 2), but the use of a buffer gas restores the narrowing, and with L ‘v 0.1 mm, and X/L = 100, one reaches line widths of the order of 20 cps for Na and 40 cps for Rb and Cs. The narrowness is essential here, because the clock’s longitudinal transition is a weak line in comparison with ordinary Zeeman lines arising from transverse transitions. (3) Optical transitions ( D lines for example). The wavelength, X ~ 0 . p5, is so small that any narrowing would be difficult to obtain, requiring a high pressure for the buffer gas.

3. Transverse Modulation. a. Bloch’s equations and density matrix theory. In the previous paragraphs, the role of the radio field was described in a rather crude manner: it was considered as inducing transitions between pairs of adjacent magnetic levels, changing their populations. For example, the same basic process occurs in a nuclear resonance experi-

MEASUREMENT OF WEAK MAGNETIC FIELDS

99

ment with water protons: here, the level structure is the simplest one with two levels, and -$; the consideration of populations and of selection rules would lead to a good description of the fundamentals of the nuclear resonance. But many experiments, such as adiabatic fast passage, Varian-Packard free precession, and Hahn’s echo method, do not find an easy explanation that way: for all these processes, involving the “phasing” of transverse atomic components of the magnetic moment and the “coherence” properties resulting in the creation of a bulk magnetic moment transverse to the polarizing field B, Bloch’s equations are of great value. How to bridge the two points of view is also known now and was due, in the field of nuclear resonance, largely t o the efforts of Bloch himself (see 161 for the most “introductory” treatment), who succeeded in a precise but difficult theory by means of the “density matrix.” I n the domain of optical pumping, the historical evolution followed the same path: “coherence” effects embodied in the movement of components of the moment M , perpendicular to the steady field B were discovered by Dehmelt (111) and explained by Bell and Bloom on the basis of equations of the Bloch type (9) before the establishment of an accurate density matrix theory by Barrat and Cohen-Tannoudji

++

(121, 162, 163).

Recognizing the simplest features of the Bloch equations in nuclear resonance helps to understand the modulation of a transverse beam: the basic property of a macroscopic moment M is to exist as a definite physical entity. I n other words, it possesses definite properties irrespective of the method employed t o produce it: for example, in the PackardVarian experiment one may produce the initial state M’, in the direction perpendicular to the earth’s field B b y prepolarizing in a static transverse field B, (this scheme is easily described in population language), or use the more complicated process where one changes the orientation from longitudinal to transverse by a suitable hf pulse ( r / 2 pulse), or use other dynamical processes such as adiabatic fast passage: this last procedure is easy to explain starting with Bloch’s equations, but more difficult to understand when one is working with populations (see 16). These many possible choices are all equivalent in the end, and the moment M at a given stage obeys the Bloch equations linking its movement to the actual value of the total magnetic field. Distinguishing transverse components M, and longitudinal ones may be very convenient for the clarity of the description as is the separation of the field into Bo = B11 and B,, but these analytical distinctions do not imply any difference in basic properties. On the other hand, the Bloch equations are linearz3and distinguishing 23

Saturation is not considered here and the amplitudes are assumed very small.

1‘00

P. A . GRIVET A N D L. MALNAR

various components in B in order to separate the various possible excitations show that their effects add linearlyz4;interference (164) terms do not appear in most instances. A well-founded example of this linear addition appears in Rloch’s initial treatment of resonance: he simply adds to the effect of the hf coherent excitation those of the stochastic fields responsible for relaxation: d 1 1 1 -- M = r(M X B) r(M X BI) iM kM (28) jM dt 1

+

+

bf excitation

static term

+

+

relaxation

Applying these results to nuclear resonance, one may predict the effect of radio excitation by a field rotating about the z axis in the plane xy with angular velocity w = y i B ; acting on a sample pumped by u+ along the z axis, it moves the moment Iaway from its z’z direction in a spiral path, inducing it to rotate around the axis z, in synchronism with the hf field. At any instant during the process, M remains endowed with the same properties as in the initial position M,; when M is collinear with the y‘y axis and points in the positive direction, the sample will strongly absorb the light of an auxiliary (u+) beam that is collinear with the y‘y axis and used to optically probe the sample; the absorption of the beam will vary in intensity in proportion to the magnitude of the component of M along the direction of the probing beam My.24aThe effect is that of a radio-frequency modulation of the amplitude of the light a t the precession frequency of M.This is a low frequency compared with that of the light. This “adiabatic transformation’’ does not have any other influence on light emissions but hf modulation. The success of this kind of theory, well verified by experiment, should not lead one to think that density matrix theory represents a superfluous complication. Indeed, the conditions in the optical experiment are original ones: fur example, the magnetic moment in the optical experiment is a .composite one (F = I J = I L S) and this vectorial addition is made flexible through the Back-Goudsmit effect: the conditions are markedly different from those obtained with a rigid nuclear moment I > and the Back-Goudsmit transitions may show slightly different frequencies. For these reasons it is very interesting to use a complete theory; here the density matrix calculations are notably simpler than in statistical mechanics: the reader may judge by himself, consulting Cagnac (I%), Cohen-Tannoudji (166), and Winter (166), who introduce one to the complete treatment found in Cohen-Tannoudji (121). Recently, Professor Carver has given a new formulation to this theory,

+

+ +

+

24 For small excitations, of course; disregarding any saturation phenomenon, for example. 24a When M, < 0 the auxiliary beam is absorbed as a (u-) excitation.

101

MEASUREMENT OF WEAK MAGNETIC FIELDS

which is both clear and efficient, and he has presented it in a pedagogical form, very suitable for a first study (121~). b. Experimental aspects. This modulation is used for measuring w through the construction of a spin-coupled oscillator. In principle it is simple to transform the light into an electrical oscillation with a photomultiplier, t o amplify it, and feed it back to the excitation coils with the proper phase to obtain a self-oscillator of the spin-coupled type. The output of the photomultiplier occurs a t frequency w and not a t 2w because the depth of modulation amounts to a few percent. The optical spin oscillator is very similar to the nuclear Bloch coil oscillator, except for -!-90° Phase shift Field to light

Interference filter (pass

A,= 7948)

Circular polar izer

Rubidium vapor cell

RF coil

b Phase shift

output

signal

FIG.35. Scheme of the one-beam optical spin oscillator. one-point: the photocurrent is directly proportional to M , a t variance with the induced emf which is proportional to dM,/dt. This introduces a x/2 phase shift in the feedback loop that must be compensated by one means or another. If one looks a t the scheme of Fig. 29 where the pumping beam (along 0 2 ) and the probing beam (along Oy) are especially separated one could simply “cross” the axis of the hf coils with the y beam as in the Bloch coil arrangement for resonance. This is not possible in practice: the general setup is always simplified by using the same light beam for pumping and probing (Fig. 35). It must then be inclined a t 45’ to the field t o be measured, and the simplest mechanical arrangement of the coils leads t o alignment of their axes along the beam; for this rigid geometry then/2 phase shift must be compensated by external electrical means. The x/2 phase shift is of importance in geomagnetic exploration: for example, when the observation plane turns back the magnetometer,

102

P. A . GRIVET A N D L. MALNAR

the field B is inverted, and this negative value of B, amounts to a change in sign of w , the angular speed in the precession of M ;for the electrical signal too, it is equivalent to a change in sign of the circular frequency w , the various phase shifts keeping their sign. An equivalent electrical situation is obtained, keeping w positive and changing the sign of the phase shift: this operation is of no importance for the small parasitic shifts, but the ad hoe s / 2 phase shifter must be commuted so as t o produce + s / 2 instead of Ts/2. This inconvenience is suppressed if a single pump lamp feeds two magnetometers symmetrically disposed back to back on the

FIQ. 36. Elimination of the s / 2 phase shifter by a double-head arrangement due t o Bloom (131): the phase shifter is the source of troublesome effects due to rotation

in mobile equipment.

same axis (131). The connection shown on the scheme makes the two s / 2 phase shifts relative to each half of the apparatus cancel one another (Fig. 36). This setup has another marked advantage: the Breit-Rabi formula, developed to second order as in Eq. (26), gives terms linear in R and also quadratic terms; inverting B changes the magnitude of v because the quadratic terms do not change sign, but v and B do; this fact is expressed in ordinary frequency formulas, which link the absolute values of v and B, by double signs such as those of Eq. (26). This double valuedness would be a source of errors in practice; it is removed in the double magnetometers that oscillate on the medium frequencies given by formula (26), where one cuts off the quadratic terms.

MEASUREMENT O F WEAK MAGNETIC FIELDS

103

C . Experimental Orders of Magnitude The choice made in practice for the chief parameters of optically pumped magnetometers often results from the imperatives that appeared in the preceding paragraphs. For example, the usefulness of “coatings” made of straight chain hydrocarbons, with low melting points, favors the selection of R b or Cs as an active medium: these two metals alone offer a high enough vapor pressure at sufficiently low temperatures, as shown by the following tablez6 giving the temperature in degrees Celsius at, which the vapor pressure is mm Hg: 7Li 304

23Na 126

39Rb 63

*‘Rb 39

87Rb 39

la3Cs 22

The intensity of the pumping light may be kept a t a relatively low level in order to reduce the source power and simplify its construction, especially its stabilization mechanism; this is possible because optical detection is highly efficient. For the same reason, when using the hf modulation principle it is also possible to maintain a low level of oscillation (hf field of the order of 100 p G ) , and this precaut>ioneliminates the possibilities of spurious double quantum jump resonances (reviewed in 166) and assures narrow Zeeman lines (Av = 50-100 cps). On the other hand, polarizers are standard componentsz6and quarterwave plates for this favorable part of the spectrum may be made easily from mica or cellophane sheet. Less well determined is the choice between steady absorption and self-oscillator systems for detection: it seems now commonly accepted that the former system is better suited for stationary and aerial observation, and the latter for rocket and satellite experimentation: a n autooscillator a t a high frequency leads more readily to short overall time constants. Further details will be given later in the section on the Cs magnetometer and are also to be found in Bender (130).z7 Still undecided is the choice between use of a buffer gas and of a coating alone, as well as between R b and Cs. Such a comparison will not be attempted because the over-all performances are of the same quality, and result more from careful engineering than from anything else. We will conclude this analysis by describing the recognized characteristics of two types of R b magnetometer, a laboratory one and a commercial one, both using a buffer gas. The last part of this review will give a detailed description of a Cs magnetometer, planned and constructed by 25 26

27

An excerpt of the extensive data given by It. E. Honig, RCA Rev. 18, 195 (1957). Marketed by the Polaroid Company. The reasons are set forth in Section VB.

104

P.

A.

GRIVET AND L. MALNAR

one of us (Malnar). Thorough discussion of the technology is to be found in Colegrove et al. (106) and of the sensitivity in Bender (130) and Bloom (131). It should be remarked that the physics of optical magnetometers is much more complicated than that of nuclear resonance; the relaxation processes are not completely understood or measured. For these reasons, one cannot actually make as clear-cut an analysis as for the nuclear magnetometer. Interference

Photocell

I

Circular polarlzer

Recorder

I

"

Frequency counter

Phose modulator

__

I t -

Lormor frequency oscl I lot or

Low

frequency oscillotor

Frequency control signal

d

FIG.37. Scheme of tho automatic control of the frequency of an rf oscillator for locking it to the center frequency of an Zeeman line: phase modulation is used to scan the line, but a pure carrier is retained for accurate frequency measurement without any filtering.

P. L. Bender has summarized his experiments (130) on a R b magnetometer of the dc type in the following data: (1) (2) (3) (4) (5) (6)

Line width, 15-20 cps or 20-30 pG for B = 0.5 gauss. Absolute accuracy, 12 parts in Line width a t low field, 3 cps or 3 pG for B = 50 mG. Bandwidth for commercial apparatus, 1 cps. Practical sensitivity, 0.1 pG. Frequency, 700 kc/sec per gauss.

Figure 37 shows the electronics used by Bell and Bloom for locking the frequency of an oscillator on the Zeeman frequency (131). I n Ref. (131), A. L. Bloom describes the HF modulation magnetometer of the portable type:

MEASUREMENT OF WEAK MAGNETIC FIELDS

105

(1) Buffer gas, Neon, p = 3 cm Hg. (2) D1 light, A 1 = 7948 A; ( A 2 = 7800 A reject,ed by an interference filter). (3) Average photocurrent, 100 pa. (4) Average signal current, 1 pa. (5) Signal-to-noise ratio, 600. (6) Ultimate sensitivity, 0.03 pG. (7) Line width, 100 cps for 140 pG. (8) Bandwidth, linked with sensitivity, 1 cps for 0.01 pG. (9) Amplitude orientation dependence (double sensor), sin 28.

The low-field limit may be taken approximately as the line width, e.g., 100 pG. The technology of space magnetometers of this type is described in Heppner et al. (167) and Ruddock (168).

D . Helium Magnetometer 1. Helium-4. Helium-4 is a two-electron atom, appearing in column 0 and in the first row of Mendeleev’s table; it has no nuclear moment. Its optical spectrum generally speaking is very different from that of an alkali metal. Its energy terms fall into one or the other of two clear-cut categories: for para helium the spins of the two electrons cancel (S = 0), while for ortho helium they add, giving S = 1. Helium atoms cannot change by a radiative process from one of these categories of states t o the other; in other words, ortho helium and para helium behave markedly as distinct atoms. For example, the ground state ‘So belongs to para helium and the first excited state to ortho helium. Under electric excitation in a hf discharge (a nonradiative process), an atom may reach the 3S1 state; it remains in this “metastable” excited state a long time, of the order of 10 msec, because it cannot return to the ground state by emitting radiation; this would violate both selection rules A1 = +1 and A S = 0. These rules express the conservation of angular momentum and the prohibition is a strong one: the life of the metastable state is long sec). The metastable state 3S1may be considered as a pseudo-ground state for ortho helium: it is obtained by a mild H F discharge in He a t 1 to 5 mm pressure. The density of metastable ortho helium is of the order of 5 X lolo per cubic centimeter, comparable to that of the alkali metal vapor a t 10-6 mm. The spectrum of ortho helium is similar to th a t of alkali meta.ls except that the P levels are three in number, corresponding to S = 1, ms = +1, 0, - 1, and there are in principle three D lines, DO, D , D2. But D I and D 2 are separated by Av = 1800 Mc/sec (e.g., Ah = 0.091 A only) and for simplicity may be considered here as unre-

106

P . A. GRIVET AND L. MALNAR

solved. The aggregate DIDZ is called D3 and the spectrum when reduced to D oand D3 is very similar to the alkali case, except that D ois eight times less intense than D3 in theory and three times in practice; the detailed analysis is given by Franken and Colegrove (169) ; the diagram is shown in Fig. 38. The physical conditions remain also nearly the same as for the alkalis; the pumping light is a line in the near infrared (X ‘v 10,830 A), where standard optical equipment is available; lead sulfide photosensitive conductors or silicon photodiodes are used in the dc type mi

+I

0

-I

-2

/

Ortho helium

16x1OScm-’ I

I

I

4He

/ ,is,,

/ I

( I cm-’

= 30 Gclsec 1

Para helium

FIG.38. Zeeman diagram for ordinary ‘He: the 2% level of orthohelium is SO long sec). lived that it is called “metastable” (life:

of magnetometer; for the hf modulation type an S-1 response phototube works well. There is no need for the addition of a buffer gas; indeed the ground state para helium gas (at a few millimeters Hg pressure, p < 30 mm Hg) plays the role of a buffer gas for the active metastable ortho helium atoms. Coatings are superfluous (170) too, because the line width of the Zeeman transition is broadened to a notable extent b y a secondary effect of the hf auxiliary excitation which produces the metastables; this broadening would make illusive any advantage hoped from a coating. The broadening is the direct consequence of the finite lifetime of the metastables, due to the uncertainty relatiorl, Ah’ T N 1, which links level

107

MEASUREMENT O F W E A K MAGNETIC FIELDS

width AE measured in cycles per second and lifetime 7 . T h e lifetime of “metastability” itself, e.g., of the whole level 3S1,is long, a few tens of milliseconds; but, as shown in (160) the lifetimes of the individual magnetic sec only; sublevels are significantly shorter and amount to some this is natural for several reasons, among others, because the AmJ = 1 transitions are “allotted” ones. The result is a line width of the order of 3000 cps or 1 mG. The main qualities of this type of instrument are simplicity and ruggedness. Extreme simplicity may be achieved by pumping with natural light; one may even pump without lenses or polarizers, by putting the source in contact with the absorbing bulb. This possibility B R F oscillotor

High voltoge RF

l-n- l

Voriable RF signal generator

We0 k electrodeless helium discharge

1

CRO

FIG.39. Pumping metastable ortho helium is efficient, even when using natural unpolarieed light: a simple device for displaying a Zeeman line on a cathode ray oscilloscope.

arises here because the total transition probabilities, away from the sublevels mJ = 1 and mJ = -1, are some 3% lower than for the mJ = 0 level; by this “intensity pumping” the atoms accumulate significantly in the two mJ = k 1 sublevelsZ8(Fig. 39). The possibility is not used in actual commercial instruments because ordinary pumping (171) produces a stronger orientation, e.g., a better signal-to-noise ratio ( X 40), and also a slightly sharper Zeeman resonance. The second main advantage is of higher value, especially in space research: there is no need, as with the alkali vapors, for temperature regulation, which takes considerable power and is a source of troubles

(.w.

The helium magnetometer seems very attractive and was built in a very practical omnidirectional form (49), embodying three sensors around 28 It is possible to show by a symmetry argument that this procedure produces “alignment” and not “polarization”; in other words, the gas is not made paramagnetic, because there are as many aligned atoms pointing in the positive as in the negative direction.

108

P. A . GRIVET AND L. MALNAR

one source; its chief defects are high-frequency shift (some 10 cps) under variation in light intensity, and large line width, which corresponds to a high value for the low-field limit of space. Typical characteristics are the following: (1) (2) (3) (4) a t the (5) (6)

Frequency, 2.8 Mc/sec per gauss. Signal-to-noise ratio, 500. Sensitivity, 0.1 pG. Bandwidth, 1 cps (the bandwidth may be increased to 104 cps expense of sensitivity). Minimum width, 700 pG. Temperature range, -50” to +50” C.

2. Helium-3. The rare but stable isotope 3He is the final product of tritium’s radioactive transformation ; it is now commercially available at a price which makes it suitable for magnetometer construction. The change from 4Heto 3He would be very similar to that from the simplified Na model to the real Na. Indeed, the only peculiarity of 3He is its nuclear spin and its nuclear magnetic moment. Replacement of 4He by 3He would offer no great advantage: The presence of the nuclear spin would be only a complicating factor in the Zeeman diagram (Fig. 40); the multiplication of levels would not improve pumping efficiency; the apparent gyromagnetic ratio could be increased by a factor 4 by choosing the magnetic resonance of the F l i z sublevel; it is reduced by Q for the F3/2 level. But 3He may be used more cleverly as shown by Colegrove et al. (106, 172) to observe purely nuclear resonance, of the nuclear magnetic moment of 3He, which shows a high and accurately known gyromagnetic ratio:

+

- p= ~ 2.03795 ~

X lo4gauss-’ sec-I

This new possibility is opened by the “spin exchange” process. This phenomenon has been much studied since 1956, when it was shown (173) to be responsible for the occurrence of the celestial 21-cm hydrogen emission line. The spin temperature of H atoms in emitting clouds reaches 50”K, some 40” higher than the average radiation temperature in ordinary regions in deep space; through “spin exchange” H atoms are able to gain magnetic internal energy at the expense of the kinetic energy of H atoms in these peculiar clouds (see 122). Spin exchange has been the subject of much research and the source of important discoveries such as that (174) of Dehmelt. Generally the object of these studies is the transfer of orientation from one species of atom to another by collision. The theory of the process is very interesting too; an introductory treatment may be found in Kastler (102) and in Carver (114).

MEASUREMENT OF WEAK MAGNETIC FIELDS

109

Spin exchange occurs here when, for example, an 3He atom in the ground state S = 0 (para helium), mz = collides with another 3He atom in one of the substates of the oriented metastable state S = 1, ms = 1 ; after the encounter the spin states of both atoms are exchanged: the para helium atom is in the state X = 0, mr = +$; and the metastable, in the S = l , m.9 = 0 state; the probability, and correspondingly

-+

Ionization b

-I cm-’

4r

9;

cm-‘

2 ‘So€

F =3/2

- 3/2

’He ( I cm”= 30Gc/sec)

FIG.40. Zeeman diagram of 3He, with the hyperfine structure due to coupling between orbital electron and nucleus ( I = +).

angular momentum. Pumping with polarized light favors the accumulation of metastables in the X = 1 substate, and ultimately results in a population of para helium atoms, mostly in the mr = state. In other words, the lower nuclear state is more populated than the upper one; the corresponding bulk nuclear moment is not only enhanced in magnitude by pumping and exchange but also inverted. These are the proper conditions for obtaining maser oscillation at the nuclear resonance frequency.

-+

110

P. A . GRIVET A N D L. MALNAR

Such a maser oscillator has recently been built and operated with full success (108) a t Harvnrd. The amazing aspect of the experiment is the high degree of nuclear orientation obtained, around 40% (106). This occurs because of the high efficiency of optical pumping and the high mobility of the gas atoms: between two encounters, the pump is able to restore the proper orientation ms = l in the metastable atom; it moves rapidly and may reverse the nuclear spins of many para helium atoms. The conditions are similar to those occurring in the Overhauser effect of a dilute water solution of a paramagnetic radical. As regards magnetometry, a notable progress is achieved toward absolute measurement: frequency shifts are induced by the pumping light only in the levels involved in the optical pumping transitions. The E

Circulatory polarized

u+

-t--

Pumping light

Fro. 41. The extremely long life of nuclear orientation of 3He (some 10 sec) permits the mechanical transport of the orientated nuclei by diffusion over distances of a few inches: one can separate the bulb in two distant parts, one for optical orientation, the other for resonance.

nuclear levels of the ground state para helium are free from this direct disturbance. Unfortunately, a residual perturbation still exists: these para helium atoms need activation, e.g., during the short time they are coupled to the ortho helium metastables the resonance frequency measured is not purely that of the free para helium-3. It can be shown that a small correction of the order of but depending on the rate of activation, e.g., on the intensity of the pumping light, must be introduced in the frequency law (1). The Harvard experiment has not yet been fully exploited for magnetometers, but it affords an effective remedy for the latter defect: nuclear resonance is not induced in the pumped bulb itself, but in an auxiliary one a few centimeters away; the two bulbs (Fig. 41) are connected by a pipe, and the para helium atoms diffuse into the second bulb, without losing their nuclear polarization: the nucleus is so well protected by the S = 0 electronic shell that it does not feel the collisions with the wall during diffusion for some 10 minutes. On the other hand, the enhancement of

MEASUREMENT OF WEAK MAGNETIC FIELDS

111

the polarization magnitude is large enough to reach a signal levcl better than that of the protons in a benzene sample of t,he same volume. This is a very promising experiment. Finally, pumping 3He with an 4Helamp has appeared recently ( 1 7 4 ~ ) to be more efficient than the use of an “e lanip; the strong unresolved component in the 4He light (3X1-3P2 and 3S1-3P1) overlaps with one of the 3S1-3P0transitions ( F : Q -+ 8) of 3He only and pumps it very strongly.

v. AN EXAMPLE OF

DESIGN: THECESIUM

V A P O R RIACNETOUETERS

A . Introduction The object of this section is to give some idea of the practical problems met with in designing optically pumped magnetometers. Questions of Lens

‘:I

Resononce cell

\

Photoelectric cell

4

Circular polarizer

0

FIG.42. Practical optical pumping arrangement.

sensitivity and technological problems are dealt with for the controlled system and for the self-oscillator system in Sections V, B, 2 and V, B, 3, respectively. Certain characteristics of these instruments are concerned with the very principle of optical pumping. Special utilization conditions result therefrom. These matters are considered in Section C. The magnetometers described below utilize cesium vapor. Various commercial versions of these instruments are a t present in use in Europe. With respect to their principles they fall into the general class of instruments described in Section IV, so that the matters dealt with below are qualitatively common to all optically pumped magnetometers. But they differ in regard to one feature mentioned below. As shown in Fig. 42, which illustrates schematically a practical arrangement of optical pumping, cesium vapor magnetometers do not make use of an interference filter for separating the D1 component; indeed,

112

P. A . GRIVET AND L. MALNAR

light is used for pumping, but with the two components D1 and DZ; the effects of relaxation are avoidcd by the use of a paraffin coating only (153, 175, 176) no buffer gas (145, 146, 153, 177) is used, as explained in Section IV, B, 2, b. So Kastler’s mode of pumping (as described in Section IV, A, 1, b) is used. It is worthwhile to mention that, for this type of pumping, experiment appears to offer conclusions that are a t variance with theoretical prediction. All that has been said about orientation is valuable, but, as regards detection, theory states (178) that it is impossible to show a difference of population by a variation of light absorption if the intensities are equal on D1 and Dz. But in practice, experiment shows th a t with bulbs containing no foreign gas (residual pressure being less than mm Hg) and merely coated with paraffin it is preferable to use the two lines simultaneously. An explanation of this discrepancy may be brought forward as already mentioned; the first vapor layers through which light has passed provide an effective selection of D1 which is even more efficient than the use of an interference filter, especially when losses inherent in these devices are taken into consideration. Although this point has not been fully cleared up, the cesium magnetometers described here use the two lines of equal over-all intensity. o+

B. The Magnetometers 1. Principles of Magnetometers. Considered in a very simple way, the magnetic resonance phenomenon detected optically can be described as follows (Fig. 43). The pumping light orientates the magnetic moments of the cesium atoms contained in the resonance cell in the direction of the external field B. This orientation aff ects light transmission through the bulb, because absorption is substantially proportional to the magnetic moment in the direction of the light beam. If a magnetic field 2B1 vibrating at frequency v is applied simultaneously a t right angles by means of coils, the dipoles are acted upon and precess also at frequency v about B ; their movement locks on one of the rotating components29 B1 of the oscillating field. Consequently, the moment M, along B decreases, and light transmission also. T h e effect is a maximum a t resonance, i.e., when w = 2nv = riB, where yi is a proportionality constant defined by Eq. (23). Figure 44 shows the variation of the light intensity I transmitted against the frequency vg of the applied hf field. As shown on Fig. 1, the active component rotates in the negative sense around B is positive and in the positive sense if y is negative.

29

if

113

MEASUREMENT OF WEAK MAGNETIC FIELDS

As explained in Section IV, two techniques are possible for monitoring the magnetic resonance frequency automatically. I n the first case, the component M , parallel to the field B is used. This component passes through a minimum a t resonance; the minimum B

Rotation frequency v ) .

-

'-

@_c

I -'A1

FIG.43. Schematic description of optical pumping.

I

AI- I/IOO

i i b

U

vq

FIG.44. Optical detection of magnetic resonance.

is used for driving a generator, thus leading to the so-called L'controlled'' magnetometer. In the second case, the M , perpendicular component is used to modulate the light in the manner explained in Section IV, B, 3. 2. The Controlled Magnetometer. a. Principle. The pumping light is provided by a source consisting of a cell containing cesium vapor raised

114

P. A . GRIVET AND L. MALNAR

to luminescence by high-frequency excitation. This light is focused by lenses and circularly polarized (u+). The light then passes through a second bulb, known as the resonance cell, which also contains cesium vapor and is subjected to a rf magnetic field provided by a n adjustable frequency generator. The transmitted light is then collected by a photoelectric cell. As already mentioned, the amount of light absorbed, proportional to M,, depends on the difference between the frequency supplied by the generator and the resonance frequency. The principle of control is shown in Fig. 45. B

c

Rotation frequency

vg +Avg sinfit

Circulor

amplifier

AF Oscillator

1

h J

Synchronous detector

FIG.45. Principle of the controlled magnetometer.

Figure 46 explains the elaboration of the error signal; the intensity of the light transmitt,ed by the resonance cell varies with the frequency vg supplied by the generator. The latter is frequency modulated a t a low frequency so as to scan the resonance line about its mean value. Owing to this scanning, the light transmission coefficient of the resonance cell is also modulated and a signal of frequency s2 is then collected a t the photoelectric cell. The phase of the detected signal with respect to the initial modulating signal depends on the position of the mean generator frequency with respect to the middle of the resonance line. When the two low-frequency signals are applied to a synchronous detector the latter restitutes a

MEASUREMENT O F WEAK MAGNETIC FIELDS

115

signal s = dI sin 4, where dI is the amplitude of the fundamental frequency of the detected signal and 4 is its phase. The signal s shows the proper behavior in order to constitute the error signal. This is used in a control loop, to maintain the generator frequency vg coincident with the resonance frequency v, hence the term “controlled magnetometer.”

s = dI s h y ,

f

FIG.46. Principle of detection by frequency modulation.

b. Limit sensitivity. The problem has been thoroughly discussed by Malnar (179).It will be approached here in a slightly different way. The main considerations are the following: (1) It is assumed that the center of the resonance line does not depend on physical parameters other than the magnetic field. Some reservations have to be made as regards this statement; they are examined later on. The problem merely amounts to marking the middle of the resonance line, and sensitivity will be limited by the precision with which this can be done. (2) Sensitivity is defined as the minimum value of the detectable field variation (root-mean-square value) ; it is readily deduced on the

116

P. A. GRIVET AND L. MALNAR

control principle. Near the center of the line one has (Fig. 47)

6B

N

= __

ds/dB

where ds/dB is the differential of the signal s with respect to the field, and N is the sum of all background noise which accompanies the signal s. (It is immaterial whether one considers the frequency or the field, since

1

AI-I 1100

I

--B

FIG.47. Determination of the center of a resonance line.

both are related by the relation 21rv microgauss.)

= yiB

where

y, =

0.35 cps per

Looking a t Fig. 47, one sees that the slope ds/dB is proportional to S I A B , S being proportional to A I / A B , and hence

AB2 6B--B A1 but AB, A I , and N depend on the quantity of light I , on the hf field Bi, and on the values of the frequency and amplitude modulation used. As regards the amplitude of modulation, it can be chosen with the help of theory, it shows that maximum slope corresponds to an appreciable widening of the resonance line by modulation (180, 181); a compromise should be adopted.

MEASUREMENT OF WEAK MAGNETIC FIELDS

117

A calculation of the optical pumping cycle would be necessary to make S and AB optimum, but it is as yet not generally possible, especially for Cs owing to its complicated structure ( I = z ; see Fig. 48). What can reasonably be done is to plot experimentally the evolution of AI and AB against I and B1 to determine the optimum values. E

I,

F=2

F F.3 =4

I

J

Excited state. The Zeemon splitling is not shown.

*B FIG.48. Eriergy levels of the fundamental state and of the excited state of cesium.

As regards background noise, it is possible to identify the different causes and to assess the respective contribution of each (1'79). The following conclusion is finally attained. The intensity of the resonance line AI and the width of the line AB are simple functions of the light intensity I and of the rf field B I .Minimum theoretical detectable 6B can be determined entirely from the two parameters I and B1. Experimental measurements permit plotting the useful quantity d s / d B that appears in formula (29) and which in a way represents the slope of the discriminator constituted by the resonance line (see Fig. 47). This slope ds/dB is represented by the curve of Fig. 49, plotted a t constant light against 2B1.

118

P . A. GRIVET AND L. M A L N A R

-I FIQ.49. Variation of the slope ds/dB against the hf field 28,.

1

FIG.50. Variation of line width against light intensity.

The top of the curve shows the optimum working point which permits choosing the value of field 2B1, for a given amount of light I. At'this point it is possible t o give- a simple expression for the quantity ds/dB against the light intensity only, for line widths plotted against I lie substantially along a straight line (Fig, 50). A straight line is also obtained for the curve that represents A 1 as a function of I 2 (Fig. 51).

MEASUREMENT OF WEAK MAGNETIC FIELDS

119

These results permit writing the following empirical law of evolution for the quantity A Z / A B , which is proportional to ds

I2

dB cv

(ABo

+LUI)~

If i t is assumed that the sole cause of noise is the ultimate one, e.g., luminous shot effect, noise N can be written in the form

N cv

(Idf)1/2

where df is the information passband used, i.e., the passband of the final recorder.

t

FIG.51. Absorption A I near’and at resonance against light intensity.

The minimum detectable field then takes the following simple expression :

As for any measuring instrument, one may use this result to define a figure of merit,, Q , for the magnetometer: Q

6B

(ABo

=-rv

(df)1/2 -

+(YI)~

I3/2

Q may be made a minimum by differentiating it with respect to I ; a very “broad” minimum is obtained for =

3ABo

120

P. A. GRIVET AND L. MALNAR

This shows that the amount of light is not very critical and that there is no need to exceed certain light intensities. When adjusted to satisfy these optimum conditions, the cesium magnetometer would provide a theoretical minimum figure of merit :

Such a n extremely good quality would be difficult to assess in practice. In actual fact this limit is never attained, because actual background noise is substantially higher than the shot noise contribution by the photocell; the input of the amplifier affords an important contribution (low-frequency flicker noise). It should also be noted that in order to avoid certain rotation effects, magnetometers are never adjusted to optimum point, but to higher values of the field B I . This point is explained later (Section C, 1). For these various reasons, sensitivities observed in practice are of the order of 0.1 pG with 1-cps passbands; this order of magnitude appears as quite sufficient for current geophysical problems. c. Associated electronic problems. (1) Requirements. I n practice, optimum pumping conditions, described in the preceding section, are readily obtained. Rut it is difficult to attain theoretically predictable sensitivities. Experience shows that performances obtained are most usually limited by the quality of the associated electronics. It therefore seems useful to give it some consideration and to examine the engineering problems raised in the design. One must weigh the following considerations :

(i) The control error, which is unavoidable, has to be a minimum since it brings about a systematic error in absolute magnetometers. I n the case of mobile instruments the control error may introduce additional rotation effects on account of gain variations related to the orientation (this point will be further gone into in the next paragraph). (ii) The dynamic range has to be sufficient to avoid excessively frequent adjustments; for example, a magnetometer used for measuring the earth’s field has to cover one octave (from 200,000 to 700,000 pG). (iii) The frequency generator has to cover this range by means of a simple control; it is therefore, in principle, not very stable, and control has to be sufficiently effective to correct this instability. (iv) The closed-loop cutoff frequency that finally determines the information passband has to be compatible with long storage in open loop. This storage is necessary because a controlled instrument requires a search system during the starting phase so as to bring about coincidence

MEASUREMENT O F WEAK MAGNETIC FIELDS

121

between the generator frequency and the resonance frequency. This search demands a fairly long time, about, 1 to 2 minutes. I n operation, it may happen th at the control loop fails temporarily for one reason or another; in this case the generator returns to its resting frequency with a time constant th at is th at of open loop control. If this time constant is sufficient to ensure th at the generator frequency remains within the line width for the duration of the interruption, the system will automatically resume operation as soon as normal conditions are re-established. I n this way the loss of time in the search system will be obviated. The type of control system th at would best correspond to these conditions would undoubtedly be an integrator system. But in practice these systems demand rotary machines and their drawback is that they

G(Bi-B)

1

1

Bi-B

FIG.52. Control system.

are magnetic : this excludes the possibility of a compact design. Moreover, for reasons of inertia and friction they possess a threshold and fail to respond to too weak electrical action. This excludes any fine control, unless the machine is associated with a second electronic control acting as vernier. This combination may be a very useful compromise when the electronics can be separate from the actual magnetic probe. I n the design example that will now be described, the solution adopted uses a purely electronic system with a double time constant; this solution, which somewhat approaches ideal integration, is analyzed below. (2) Control analysis. (i) The servo’s equation: Figure 52 shows the control circuit. The frequencies are converted to a magnetic field in accordance with the fundamental law (23) : w =

~ T V=

7iB

Using it, we may define various values of B which will also be useful later on: B, corresponds to the resting generator frequency, Bi is the field read on the instrument, and B is the field to be measured. G is the

122

P . A . GRIVET AND L. MALNAR

gain on open loop; it can be written in the following form:

Go is the total gain of the loop on direct current, taking into account both the actual amplifier gain and the transfer coefficient of the optical pumping system as a whole. The term (1 j w ’ ) relates to the phase lead network rc = 7 ’ ) whose role will be explained later. The term in the denominator takes account of the dependence of gain on frequency; the factor (1 j u ~ relates ) ~ to the two time constants RC actually inserted in the loop; they are of the order of 60 seconds. The product IIi relates to all the time constants inherent in the actual circuit; the main ones are those of magnetic resonance (relaxation time) of the photoelectric cell and of the low-frequency amplifier. These various time constants are short compared with the information passband chosen, so that the circuit can be analyzed while neglecting them. Finally, the expression for the gain becomes

+

+

and the magnetometer response may be written

These expressions describe the behavior of the loop as a function of gain Go as well as of the time constants chosen. The main results are the following. (ii) Static control error: The static control error is the difference between the values read and the real values:

This error is a maximum when Bi is at one end of the range; if a total dynamic range of 2 X 250,000 p G is chosen, the gain Go required in order to maintain an error of the same order as the minimum detectable difference has to be greater than 2.5 X 10”: Go

> 2.5 X

lo6

(31)

(iii) Generator’s frequency noise: This noise is due to fluctuation of the generator’s mean resting frequency ; it is obtained by differentiating

MEASUREMENT OF WEAK MAGNETIC FIELDS

123

formula (30) with respect to B,. I t s rms value is

or where

R appears as a reduction factor giving the amount of attenuation of generator noise by the control.

FIG.53. Reduction factor R of generator noise.

The curves of Fig. 53 show the values of R against frequency for two special values of 7’. Curve (a) corresponds to T’ = 0; in this case there is no phase correction. The reduction factor shows a sharp maximum for the value WT = Ordinarily the maximum corresponds to unstable conditions in which generator noise is actually amplified. This defect can be avoided by a suitable choice of 7’. Curve (b) corresponds to the special value of T’ = 7(2/G0) 1 / 2 which is the minimum value of T’ that will ensure th a t the reduction factor is always less than 1, i.e., that it actually produces a reduction. R then takes the simple form

R =

[

++

1 + [l 2Go

w

““I-’ ~

T

~

]

~

(34)

124

P. A. GRIVET AND L. MALNAR

The attenuation exceeds 20 db when wr is less than (Go/10)1'2.In practice generator noise is negligible at frequencies that satisfy this inequality : WT

< (Go/10)'/2

(3.5)

(iv) Open-loop storage: A precise meaning can be given to storage by defining i t in the following way; it is the time taken by the generator to sweep across the width of the resonance line when left to itself after a sudden break in the control chain. The law of evolution of the generator frequency is then of the form

(This is the law of evolution of a system with two time constants subjected to a sudden variation.) The most unfavorable case occurs when the generator stands a t the end of the range. I n the case Bi(0) - B, = 250,000 pG and since the line width a t resonance is of the order of 500 pG, Bi(tl) - &(O) has to be not more than 500 pG if storage is to be a t least equal to t l . I n view of the law of evolution a third condition results for 7 , which is T

> 15 t i

(36)

(v) Passband: This band is the information passband, i.e., that over which the instrument remains accurate. It is equal to the value of w a t which the closed loop gain drops by 3 db or, practically,

1 y(;{w)I

=

0.707

As a result of the calculations, the frequencies transmitted with a n attenuation less than 3 db have to satisfy the inequality: w

< 2 Go"2

(37)

It will be noticed that condition (35) is more stringent then condition (37). As a n illustration, the following is a practical example of a control system of the type discussed above. To ensure a dynamic range of 2 X 250,000 pG with a control error less than 0.1 pG and a storage of 4 seconds, the following must obtain:

Go = 2.5 X lo6 r = 60 seconds 7' = 60 msec Generator noise is