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 then negligible up to a frequency f = 1.5 cps (condition (35)), and the useful passband is about 6 cps (condition (37)).

MEASUREMENT O F WEAK MAGNETIC FIELDS

125

d. An example of practical design. (1) General scheme. Figure 54 shows the circuit diagram of a complete, controlled magnetometer including, in addition to the control electronics, the auxiliary circuits, such as the starting arrangements, the light source excitation circuits, and the thermal regulation circuits. These various components will be briefly considered. (9) Control. (2) Generator: In order to cover a wide dynamic range, the generator mnsists of two oscillators whose mutual beat frequency is collected.

FIG.54. Complete controlled magnetometer arrangement.

The frequencies t o be obtained extend from 70 kc/sec t o 250 kc/sec and the oscillator frequencies are around 2 Mc/sec. The use of two identical oscillators practically avoids fluctuations of external origin. The frequency of each oscillator is controlled by varicaps which receive either a linearly variable voltage during the search phase, or an error voltage during the normal working phase. (ii) Modulator: One needs to scan the line in the resonance hull, and also to measure accurately the central frequency. For this purpose, the system adopted is the phase modulation type. This permits two separated channels, at the output of the carrier's generator; one feeds the frequency counter directly; the second leads to the resonance cell,

126

P. A . GRIVET AND L. MALNAR

through a low-frequency phase modulator. One then gets for measurement a signal free of the modulation required for control. This last kind of modulation would not be troublesome in the case of analog discrimination, for which it is possible to apply filters. But it may introduce a n error when the frequency is counted digitally, for if we write

1 for the expression of a signal modulated a t frequency F with a n excursion Av, a digital computer gives the number of maxima of a sinusoidal function during the counting time At, The frequency read is then V +

Av sin 27rF At 27rF At

The second term represents an error which depends both on the modulating frequency and on counting time. (iii) Amplifier chain and synchronous detector: To avoid unwanted phase rotation in the control it is useful to provide wide passbands, but such passbands emphasizes noise in the receiver. For example, fastchanging magnetic fields of industrial origin are captured a t the resonance cell and are liable to cause troublesome interference a t the output of the synchronous detector. This error may be important if the modulating frequency is in a simple ratio to the interference frequencies, and compromise is a difficult matter. (3) Starting arrangements. For starting purposes, a single sweep causes the generator to cover the whole of the dynamic range of frequencies; when it crosses the resonance line a signal of frequency twice the modulating frequency appears. This signal stops the sweep and closes the control loop. The start of the sweep is controlled by the temperature of the light source, for it is that temperature that requires the longest time for establishment. (4) Light source. The light source is the most important part in systems utilizing optical pumping. The main problems that arise are the following: (a) The firing of the lamps is aided by strong electric fields, but the discharge is more stably sustained by high-frequency excitation of the magnetic type. (b) The impedance of a gas discharge tube is quite different in the hot and in the cold states; and since it is impossible to insert the lamps directly in the circuits of the excitation oscillators on account of the disturbing magnetic fields, a return loop has to be provided.

MEASUREMENT OF WEAK MAGNETIC FIELDS

127

(c) Basically, better results are obtained by controlling the lamp temperature by heating independently of the high-frequency excitation. (d) The heating arrangements have to ensure heat distribution such that drops of alkaline metal will not mask the light in front of the output pupil. This is specially important in the case of mobile equipment, which may take almost any configuration in the gravity field. ( e ) The light spectrum emitted depends strongly on the optical paths covered and on the distribution of liquid masses inside the lamp. But this distribution may vary with time or with the orientation, and so on. Most of these problems have t o be solved by cut and try methods. Various solutions have been described. Gourber (144) described that which has been adopted for the cesium magnetometer. It enables lamps to be produved with a life of several thousand hours, noise not exceeding the minimum theoretical level, viz., the Schottky noise. Temperature control is one of the conditions of lamp stability. Further, detevted signals pass through a maximum at a given temperature of the resonauw (.ell, a temperature which is the result of a compromise between a sufficiently high vapor pressure and somewhat long relaxation times. For feeding the thermostats it is convenient to use variable power oscillators controlled by thermistor bridges, for they are progressive and cause no magnetic disturbance provided some elementary circuit design precautions are taken. 3. Self-Oscillator System. a. Principle. Figure 55 shows the circuit diagram of the system. Here, in principle, a second light beam is set a t right angles to the first and the variation of a component M , of the magnetization is observed on this beam. But this component oscillates at the frequency of the signal injected in the coil and the amplitude of oscillation passes through a maximum a t resonance. The signal collected in the photoelectric cell is amplified, its phase is shifted, arid it is reinjected in the coils. The system then oscillates at the resonance frequency. This is the principle of the self-oscillator magnetonieter; in theory it appears simply as an optical kind of “spin coupled oscillator”; the use of an optical sensor in the feedback loop does away with the presence of residual cwupling by induction, which is troublesome in other spin oscillators. The optical spin oscillator is a very good one. I n practice it is possible in this system to use a single light beam set at 45” to the direction of the field to be measured. I n this case the components involved are the components of B in the direction of the light beam and the perpendicular direction. The rf field B1 may be applied in a direction perpendicular or parallel to the light beam,

128

P. A. GRIVET A N D L. MALNAR

for the active component is perpendicular to B. But for reasons of mechanical symmetry it is preferable to place the resonance coil parallel to B. b. Sensitivity. This point has been discussed by Bloom (131). I n practice the sensitivity reaches the same value as in the case of controlled systems, e.g., some 0.1 pG for an information passband of 1 cps. c. Electronic problems. (1) Amplifier equivalent scheme. The singlebeam self-oscillator magnetometer can be described schematically as f 0110ws. The subensemble of the system comprised of the excitation coil, the precessing spins, the probing beam, and the photocell may be simulated

r-

Photoelectric cell

,,/

1

\I+AIcos2swt

'-

/ 0-

1

Amplifier

\Circular

2 8,sin 2swt

polarizer

Lamp

FIG.55. Principle of self-oscillator magnetometer.

by a simple series resonant circuit; this is inserted in the return path of the feedback loop, connecting output and input of a wide-band amplifier (Fig. 56). To complete the reaction circuit a black box represents the whole of the auxiliary functions met with in the physics of the self-oscillator system, in particular the + ~ / 2phase shift, which appears depending on whether the angle between the magnetic field and the light beam is greater or less than 90". (2) Requirements. The system represented by Fig. 56 has to oscillate at the proper frequency of the oscillating circuit as nearly as possible.

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

129

Two conditions have to be satisfied: (a) The amplifier gain has to be sufficient to ensure oscillation. This inequality condition is readily obtained. (b) The phase of the amplifier has to be strictly equal to + ~ / 2 over the whole of the frequency band used; otherwise the system will oscillate a t t,he frequency at which the closed-loop phase shift is zero (provided the gain is adequate). This is a n equality condition which must be satisfied with a great accuracy. Noise

FIG.56. Equivalent circuit for the self-oscillator magnetometer.

Indeed, the error in the magnetic field due to the phase error is approximately given by the following expression :

where AB is the line width and d 4 the phase error. I n view of the line width used, about 500 pG, phase errors would have to be less than 0.02" to ensure errors of measurement less than 0.1 pG. I n practice amplifiers30can be constructed with a phase error of about 1" over a frequency band of one octave. Beyond this limit important difficulties appear owing to reproducibility problems.

C . Optimum Working Conditions and Limitations in Use 1. Mobile Equipment. This class includes airborne magnetometers or those carried in vehicles or drogues for detecting anomalies in geomagnetic surveys, and space magnetometers for measuring extraterrestrial fields. The importance of dynamic range in this type of instrument is obvious. But in most cases absolute precision is of secondary importance; what is really required is to compare magnitudes a t different points, SO that sensitivity is the chief quality. But since magnetometers may assume varying configurations with respect to the direction of the magnetic field, secondary effects may 30 Constant phase amplifiers are less common than the constant gain or constant delay types, but the same principle applies for obtaining maximum flatness of the phase characteristic ; phase compensating equalizers are used.

130

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

become important. In particular, the instrument’s sensitivity may depend on its orientation with respect to the field to be measured; in addition, the measurement of the field can be affected by rotating the instrument, which brings about systematic errors related either to the orientation or to the speed of rotation. a. Dynamic range. This is the range of magnetic field over which the instrument is capable of taking measurements without readjustment. As explained before (see Sections I, A and IV, A, 1,c), optically pumped magnetometers are unaffected by a weakening of the field observed under magnetic resonance until one reaches the low level of 100 pG (e.g., one line width). I n practice, however, the electronic system will limit the dynamic range. For the case of the controlled system this limit, which is fixed by control error conditions, has already been discussed in the preceding section. I n the case of the self-oscillator system, the dynamic range is related to the width of the passband in which constant phase can be secured. As soon as the phase differs from the required value, a systematic error is obtained in the absolute value measured. The consequences of this effect are thus identical with those in the controlled system. I n both ca.ses the dynamic range can be defined only by fixing the error which can be tolerated in the measurement of the absolute value of the field. Taking this condition into consideration, it is technically much easier to secure a wide dynamic range with the controlled system. For example, if it is required to cover the band of 200,000 to 700,000 pG, the control error is easily held a t a value of 1 p G with a controlled magnetometer, whereas with a self-oscillator magnetometer it is difficult to secure a phase error corresponding to a setting error of 10 pG. b. Dependence of sensitivity on orientation. The theory of the magnetometer, given in Section IV, is based on the assumption that the pumping light is transmitted in the direction of the magnetic field B and that the rf field BIis applied at right angles to this direction. If such is not the case, light and the rf field are involved only in proportion to their components in the ideal directions. On this account the values of the detected signals, and the line widths, vary with the orientation of the magnetometer, and the same applies to sensitivity. These variations may be predicted from the results given in Section V, B, 2. I n particular in the case of the controlled magnetometer a n emf proportional to the signal is easily measured a t twice the modulation frequency near resonance. In the notations of Section B, 2, this emf is substantially proportional to ds/dB, s being the signal. It can therefore be used for measuring sensitivity. An experimental diagram of sensitivity

MEASUREMENT OF WEAK MAGNETIC FIELDS

131

against orientation obtained in this way is shown in polar coordinates by the curve of Fig. 57, where the modulus of the vector in direction 0 is proportional to the signal, e.g., to ds/dB. This diagram refers to rotation about a fixed axis perpendicular to B. This is the most unfavorable case in practice since the field B1is taken as very strong in the falling part of the curve of Fig. 49. I n particular the diagram shows that the detectable variation in strength does not exceed twice the ideal minimum in a cone of 70" apex angle. This permits using a single magnetometer in the airborne version over the whole B

' 270" FIG. .57. Ihpendence of sensitivity on orientation in the case of a controlled magnetometer.

region of the globe where the field inclination exceeds 50", without appreciably affecting performance. The diagram of the self-oscillator system would be of the same type if it included two light beams at right angles, one for pumping and the other for detection. Since in practice a single beam is used, giving optimum sensitivity a t 45" to the field, the sensitivity diagram becomes zero a t 0" and 90" from the field. The experimental curve of Fig. 58 gives a general idea of this diagram. The space diagram is symmetrical about B. Consideration of the difference between Figs. 57 and 58 helps in choosing between the two systems for any specific application. For example, the use of a single self-oscillator as an airborne magnetometer appears unfavorable, unless an orientable system is used.

132

P. A. GRIVET A N D L. MALNAR

On the other hand, in the case of space applications, in which the “attitude” of the carrier craft is unknown, the probability of correct operation is the same with both types of diagrams. This point has been discussed by Bloom (131). I n actual fact, this probabilit,y is immensely greater in the case of the self-oscillator, which oscillates instantly as soon as it returns to a working zone, whereas the controlled system demands a nonnegligible search time, as has already been seen. Naturally, with both systems it is possible to cover the whole space by using several magnetometers set up in different directions. The installation of such systems raises some rather considerable practical I

90”

270°

FIG.58. Dependence of sensitivity on orientation for a single-beam self-oscillator magnetometer.

problems, for, while the total bulk must be kept reasonably small, coupling between magnetometers, arising through the resonance coils, has t o be avoided. c. Turnover effects. One of the qualities required of magnetometers is that they be isotropic, i.e., they must provide a measurement that is independent of their orientation with respect to the magnetic field. All effects related t o anisotropy are generally designated by the term turnover effects and constitute a severe limitation in their use, especially in the case of mobile instruments. These effects are systematic errors which depend either on the magnetometer’s orientation in the field, or on the speed of rotation as considered below. (1) Purely magnetic effects. (i) Dc fields: If the materials used in the construction of the probes contain either remanent or induced magnetiza-

MEASUREMENT OF WEAK MAGNETIC FIELDS

133

tion, they are the sources of a field b which is added vectorially to the field B to be measured; the sum of the two (b B) depends on the orientation, thus producing a rotation effect. The problem, which is common to all magnetometers, is far from being negligible, though the necessary precautions are clear. I n the case of optical pumping magnetometers special attention has to be given to secondary circuits for temperature control. (ii) Ac fields: A more interesting effect occurs when an alternating field adds vectorially to the dc field that is to be measured. If the frequency of the ac field is higher then the passband of the magnetometer, one observes an error due to the nonlinearity of vector addition. If the ac field shows a component b, cos w t at right angles to the dc field B, one measures effectively the mean value, equal to B[1 ( Z I , / ~ B ) to ~] first order. This correction may be a troublesome error. On the other hand, comparing the measures for two orientations of the magnetometer, one for which the two fields are parallel and the other for which they are perpendicular, one gets an accurate measure of small ac fields by this second-order effect. (2) Line asymmetry. The apparent resonance line of an alkaline atom shows, in weak fields, arid asymmetry that depends on the polarization of the pumping light. This asymmetry is related to the Back-Goudsmit effect which disturbs the linearity of the energy levels with respect to the magnetic field, as explained in Section IV, B, 1. The energy of each level shown on Fig. 48 as a function of the magnetic field is given by the Breit-Rabi formula:

+

+

where AEo is the hyperfine difference; I is nuclear spin (in this case, 4 ) ; p~ is the Bohr magneton; g J and gr are the Land6 factors which represent the magnetic, electronic, and nuclear moments with respect to the Bohr magneton; arid x is the quantity [(gJ - g ~ ) p & ] / A E e . Nonlinearity is introduced by the term under the radical in formula (38). One of the consequences of this fact is that the frequencies of the different Zeeman components are riot coincident. For example, in the case of cesium the difference between lines is 19 pG in a field of 500,000 pG. This difference is less than each line width, so that the spectrum is not resolved. What is actually observed is an over-all line composed of all the elementary lincs. Rut optical pumping with a+ light8has thc advantage of populating levels with the highest value of m. This produces inequality between the different components (Fig. 59), and asymmetry in the over-all line to the

134

P. A. GRIVET A N D L . MALNAR

right. Population is reversed toward negative values of m if the light is and asymmetry also has its sense changed. Since changing the sense of the polarization and the sense of the magnetic field are equivalent operations, an identical effect is obtained if the magnetometer is turned around in the field to be measured. This produces a rotation effect. If this effect were pure, the Breit-Rabi formula would enable its upper limit to be calculated. This upper limit would be reached in the extreme case for which the over-all resonance line is coincident with the strongest of the elementary lines.

U-,

Fro. 59. Combination of lines of unequal amplitudes.

Transformation of the Breit-Rabi formulas shows that in the case of weak fields the relative difference between the extreme frequencies 2 X (vmax - vmin)/(vmax vmin) is independent of the value of nuclear spin and depends only on the Zeeman parameter x;

+

2 x

Vmsx Vmsx

-

+

Vmin Vmin

- 2(gJ

- g1)PBB

=

2x

AEo

The relative difference of fields measured before and after rotation of 180" would then have the following value for the various alkaline metals : Cs 6 X 10-4B 87Rb 8.2 X 10-4B 86Rb 18.6 x 10-4B K 220 x 10-4~ where-B is in gauss. This extreme difference would be encountered in the limit for an infinite light intensity I of the optjical pumping and for a near zero value

13Fj

MEASUREMENT OF WEAK MAGNETIC FIELDS

of the Zeeman field B1; for the very principle of optical detection of magnetic resonance suggests competition between the light intensity, which tends t o accumulate the atoms in a single level, and the Zeeman field, which tends, on the contrary, to equalize the population in all the levels. These antagonistic effects are also apparent in the shape of the resonance lines. I n particular, one would expect, more symmetrical lines and so smaller rotation effects as the light intensity drops or as the field B1 increases. c

30

20

10

--c

28,

FIG. 60. Line asymmetry as a function of the field B , and resulting field displacement.

This tends to be confirmed by curve (D) of Fig. 60, which shows line asymmetry against the field B1. Thus an asymmetry parameter D is defined as the difference between the frequencies a t the top and a t the center of the line, this difference being referred to half the line width. The changes in the value of the indicated magnetic field, dB, corresponding to this asymmetry are shown on the same graph. Curve (b) is quite compatible with rotation effects actually encountered with the cesium magnetometer, which are of the order of 10 to 20 pG in a field of 450,000 pG. When these results are compared to the previsions of the theory resumed in the preceding table it appears th a t asymmetry effects are not the only ones taking part. (3) Residual effects. The frequency shifts related to optical pumping (121)may contribute to the discrepancy although they are not a t present

136

P. A . GRIVET A N D L. MALNAR

known quantitatively. Variations in the light source are linked with the orieiitation on account, of the diff ererit distribution of the unvaporized masses of metal within the lamp. This effect becomes marked in selfoscillator niagnetonieters, which require very accurate definition of the optical axis; it somewhat restricts the efficacy of compensating devices aimed a t suppressing rotation effects by the use of opposite optical beams obtained from the same light source. The above-mentioned effects depend only on the direction of the optical pumping with respect to the magnetic field. There is a further effect related tto 0,the speed of rotation of the magnetometer about B (see Section I , A, a), which can be simply described as follows, with reference to the diagram of Fig. 43. The vapor’s magnetization M is caused to precess by a rotating field whose angular speed is increased (or decreased) by the rotation of the magnetizing coils. In accordance with Eq. ( 2 ) , this produces the following relative error:

dB/B

=

s1/2?rv

is the resonance frequency corresponding to B. This effect is not easily observed with optical resonance magnetometers in which the relevant frequencies are very high as compared with the values of 52 reached in practice. 2. Resettability; Stability. a. Resettability. Often, the interest in absolute precision is not very great. But it is still important to have available an instrument which can be substituted without having to make a fresh calibration. From this point of view resettability can be defined as the ability of several instruments of the same type placed in the same conditions to supply the same readings. It can be numerically stated by the relative difference between these readings. Cesium magnetometers of the controlled type possess a resettability of about 10 pG. The following table offers an example of the results obtainable when checking resettability by comparing the readings of five different cesium magnetometers in the same location. The present experiments (181) were made in the French Geomagnetic Observatory of Chambon-la-For&, in a field of approximately 0.45 gauss; the five apparatuses were later used for establishing a new magnetic map of France during the year 1964, by aerial exploration. Many series of readings were made successively, and for ease of comparison, the readings were reduced to the common value B = 0.45 gauss; here is a randomly chosen reading: (cps]

f

No. 1 157441

No. 2 157442

No. 3 157443

No. 4 157439

No. 5 157439.5

MEASUREMENT OF WEAK MAGNETIC FIELDS

137

The conclusion of the trials for this set of apparatus was expressed as a frequency versus field law: [cps] f = 3.49869 X lo6 X B

gauss

b. Long-term stability. The stability of an instrument can be defined as its ability t o retain its initial calibration. So the problem of stability is the problem of long-term drift, which may be caused by a variation of the instrument in time due to aging or through the action of external factors such as temperature, pressure, and humidity. I n the present state of experience, the chief external cause of drift is temperature. It makes its effect felt through the electronic equipment whose characteristics are liable to vary: (1) Temperature causes variations in the intensity of the high-frequency excitation of the light source, or in the amplitude of the Zeemari field; these variations cause frequency shifts. But these effects are very slight and are negligible compared to the following ones. (2) I n the controlled magnetometer, the most important cause of drift is related to the very principle of the determination of the center of the resonance line by means of a synchronous detector. The latter delivers an output signal by comparing not only the fundamental frequencies of the modulation, but also the various harmonics if they exist in both channels. But they are also present in the return channel, a t least in principle, and may occur in the reference channel if the modulating signal is distorted owing to stray coupling and so on. This is therefore the origin of a false error signal which may vary with temperature. This effect is a t present responsible for drift of about 10 pG for a temperature change of 40°C. (3) I n the self-oscillator systems the dominant effect is phase rotation in the amplifier, caused by variation of the chararteristics of reactive components with temperature. This effect causes drift of about 10 p G for a temperature change of 10°C.

All these causes of drift can be considerably reduced by thermal regulation of the electronic components. 3. Limits of the Passband. Magnetometers are essentially continuous instruments, i.e., designed for measuring slowly varying phenomena. They are gener:dly used with time constants greater than 0.1 sec. More rapidly varying phenomena could be measured with probe coils whose sensitivity increases with the frequency to be measured. Nevertheless, these devices measure only the field components, so it may be useful to know the limit passband of optical pumping magnetometers.

138

P. A. GRIVET AND L. MALNAR

Controlled magnetometers possess a passband limited by the control system itself. As shown by the analysis of Section B, 2, c, it is difficult in practice to obtain a cut-off frequency higher than a few cycles per second for closed-loop control; this will therefore be the cut-off frequency for the magnetometer itself. I n the self-oscillator system the only limitation is the width of the resonance line. One can expect, a t the instrument output, a passband approximately equal to half the line width, measured in cycles per second. This is confirmed by experiment, as shown by the following table, which shows the cut-off frequency of a self-oscillator for different amplitudes of the applied sinusoidal field. Amplitude ( p G ) Cut-off frequency (cps)

350 75

800 70

1700

65

4500 55

10000 50

This problem is discussed theoretically in Ref. (1.31).

D. Examples of Designs The following figures illustrate the preceding sections by a few examples of design. Figures 61, 62, and 63 relate to a magnetometer of the controlled type used for geophysical measurements. The same instrument can be used either fixed or as a mobile instrument in its drogue version. Figure 61 shows the complete instrument. I n the foreground is the actual magnetometer, which is basically an epoxy resin cylinder 1.5 meter long; on one end is the control electronic unit; a t the other the detector probe, i.e., the optical pumping unit. This lengthwise arrangement permits enclosing the system in a bird-shaped envelope, and so forms a magnetometer which can be towed by an aircraft or a helicopter. Further back in Fig. 61 is a unit comprising the power supply and the counter system. The latter can convert the received frequency, which is proportional to the magnetic field, either to a coded signal which can be stored on a magnetic tape, or to an analog signal tto be recorded graphically. The counter is also provided with a high-stability clock for synchronizing two stations separated from one another, so as to obtain simultaneous records. This unit (counter and power supply) is connected to the magnetometer itself by a single coaxial cable which may be 100 meters long. This cable is used as the towing cable when the magnetometer is converted to a drogue version. Figure 62 shows the drogue under a towing aircraft, on the ground, and Fig. 63 shows the same instrument towed by a helicopter, in its normal use.

MEASUREMENT O F WEAK MAGNETIC FIELDS

139

Figure (54shows one example of simultaneous recording in absolute value of the magnetic field measured by two independent magnetometers. These recordings are shown on the same graph. The two magnetometers were 30 meters apart.

FIG.61. The geophysical magnetometer.

FIG.62. The drogue on its adjustable nonmagnetic checkout fixture.

Figure 65 gives a similar recording for two magnetometers 300 km apart. Figure 66 illustrates the measurement method used in the operation of the French magnetic map, with instruments similar to th a t showed by Fig. 62. An aerial magnetometer is flown along a predetermined axis and records both the magnetic profile and the time-dependent fluctuations.

140

P . A . GRIVET AND L. MALNAR

Simultaneously a ground station records these fluctuations only. The difference between the bwo records shows only the magnetic profile. The smoothing effect obtained can be observed in the figure. Figure 67 shows a self-oscillator magnetometer for space measurements. It consists esseritially of the measuring probe (optical pumping

FIG.63. The magnetometer in flight.

unit) and two air-tight boxes containing the electronic unit and the power supply. The frequency-measuring system, which may be placed on the ground, is not shown. Finally Fig. 68 is an example of a record of geomagnetic fluctuations obtained with an instrument of this type. Table V summarizes the typical characteristics of the cesium vapor magnetometers.

141

MEASUREMENT O F WEAK MAGNETIC FIELDS

t

I

2

.

[min]

FIG.64. Example of simultaneous absolute recording at same place.

45283

--

45273

452631

45191

Absolute value Salles Curan May 5.1964

1

I1

45181

-Absolute vclue Condom May 5,1964

45171 -

142

-

P. A . ORIVET A N D L. MALNAR

2

Air record

Ground stotton

FIG.66. Simultaneous aerial and ground record.

FIG.67. The self-oscillator magnetometer.

t

8 [gammas]

-0I t

I

2

3

4

[min]

FIG.08. Example of high sensitivity relative recording.

143

MEASUREMENT OF WEAK MAGNETIC FIELDS

TABLE V Controlled type

Auto-oscillating type --__

Sensitivity (for 1 cps bandwidth) Absolute accuracy in relative value Stability (for A T = 40°C) Possible bandwidth Dynamic range Temperature range Turn-over effect (one head)

0.1 pG 10-6

10 pc: 5 cps 0 . 5 gauss - 20” to 40°C 10 pG max

0 . 1 pG Not measured 20 pG 50 cps 0. .5 gauss - 20” to 40°C +30 pG max

VI. SUPERCONDUCTING INTERFEROMETERS AS MAGNETOMETERS A . Principle A very interesting effect was discovered recently by Jaklevic, Mercereau et al. (18.2, 183, 184) in the domain of superconductivity. They were able to build a device which may be called a n “interferometer in the time domain for de Broglie’s electron waves,” and the structure of the “interference pat,t,ern” is highly sensitive to the intensity and direction of the ambient magnetic field. The magnetic sensor of this device is simply a small loop of superconducting wire of area A . The ultimate sensitivity of the measurement 1 A H I is linked, order-of-magnitude-wise, to the quantum of flux by the relation IAHI

p1 (4ol-4)

[UEfiI]

(39)

where 40 given by %

=

h/2e = 2.1 10-7

[fi4axwell]

(40)

is very small ( h Planck’s constant, e charge of the electron). Moreover, the numerical factor p may reach values of the order of 10 to 100; it measures the accuracy obtained in determining electronically the location of a minimum in a “dark fringe” of the interference pattern; in fact, the interference appears as a beat frequency and evolves in time. Actually the accuracy is limited by noise and stability problems and the useful values for the surface of the loop do not exceed 1 em2; the speed of response is good. In these conditions one may hope to conveniently reach the G level. This gain would be specially valuable for interplanetary fields, which are very low (20 to 50 pG). On the other hand, miniature helium refrigerators have been already developed for use on rockets and satellites

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P. A . GRIVET A N D L. MALNAR

arid an experiment combining both devices aboard a satellite seems promisingly possible. Already, today, a laboratory prototype has been constructed in the same place as the initial discovery, thoroughly tested and compared with optical magnetometers, and shows at least the same quality (183). The device is named SQUID by its inventors (Superconducting Quantum Interference Device) and its properties will be briefly described in the following paragraph. No attempt will be made to survey the basic theory of the interferometer, as it is masterfully expounded in the third volume of Feynman’s lectures on physics (185, lecture 21) and brilliantly too in the reviews and articles (186, 187, 184).

FIG.69. SQUID construction.

However the following point should be stressed : The superconducting ring sensor of the SQUID, shown in Fig. 69, is not simply a homogeneous wire bent in a circle. On the contrary it includes two “weak links.” Until now, the theory considered only one type of such “weak link,” the so-called Josephson’s diode. On the contrary, in practice the “weak link” is simply made of a point contact, under moderate elastic pressure, and its diode characteristic differs notably from the curve for Josephson’s diode.

B . The First Practical Squid The ring sensor shown On Fig. 69 is made of vanadium wire (or niobium). When the pressure is properly adjusted, one observes the characteristic V - I shown on Fig. 70. Starting with I = 0, one moves on the axis V = 0 in the first superconductive region a, but a current of

MEASUREMENT OF WEAK MAGNETIC FIELDS

145

the order of 1 mA is sufficient to partially destroy the superconductivity and to progressively restore resistance: one crosses the useful region b ; finally one reaches the normal region c. The remarkable point is th a t in the useful region, the position of the curve along the I axis depends on the magnetic flux threading the aperture of the sensor: if the flux equals fn90/2, n being an even integer, one describes branch I of the curve;

FIG.70. Voltage-current characteristics for a typical SQUID.

I

-2e0

-4

I

0

+%

+ 2Q0

FIG.71. Voltage versus magnetic flux characteristics for a proper choice of biasing current.

if the flux is changed to one of the values + n a 0 / 2 , n being an odd integer, it describes branch 11. For intermediate value of the field, the figurative point describes the “load line,” oscillating back and forth between points A l and N as shown by the arrows. The absolute value of the slope of the load line is the resistance of the diode in the normal state (region c ) . Biasing the current in the useful region b, the signal V is a smooth oscill+tory function of the magnetic flux 9 shown in Fig. 71. The magnetometer works on the null principle and a servoloop regulates

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P. A. GRIVET AND L. MALNAR

Bias current

amplif ier

Synchroncus detector

Squid Field coil Bolancing+locol scanning

10 KC oscillator

FIG.72. Lock-on SQUID magnetometer block diagram.

a compensating field, so as to lock the figurative point on a position very near from a minimum on diagram 71. The block diagram of the system is given in Fig. 72. The obtained performances for the first SQUID (October 1965) are the following: hlodulation frequency for local scanning of the diode SQUID diameter Short term noise Dynamic range Feeding power Possible long term drift

10 kc

6 or +‘B in. lo-’ (G) for a 4 cps bandwidth 0.01 G to 1 mG 1.5 W anything to 0.01 mG

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The Radio-Frequency Confinement and Acceleration of Plasmas H . MOTZ Department of Engineering Science Oxford University. Oxford. England AND

C . J . H . WATSON Merton College Oxford. England Introduction . . . . . . . . . . .. . . . . . ... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 1. Single Particle Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B. Guiding Center Theory for Arbitrary Radio-Frequency Ficlds . . . . . . . . . . 163 C . Radio-Frequency Plus a Uniform Magnetostatic Field . . . . . . . . . . . . . . . . . 166 D . Radio-Frequency Plus a Nonuniform Magnetostatic Field . . . . . . . . . . . . . . 168 E . The Cyclotron Resonance . (i) Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . 173 F . Particle Motion in a Traveling Electromagnetic Wave . . . . . . . . . . . . . . . . . 187 G . The Cyclotron Resonance . (ii) Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . 190 2 The Theory of Radio-Frequency Confinement of Plasma . . . . . . . . . . . . . . . . . . 194 A. Derivation of the Self-consistent Field Equations . . . . . . . . . . . . . . . . . . . . . 194 B. The Energy-Momentum Tensor Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 C . One-Dimensional Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 D . Infinite Cylindrically Symmetric Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 E. Three-Dimensional Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 3 . Theory of Combined Radio-Frequency and Magnetostatic Confinement of Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A. Derivation of the Self-consistent Field Equations . . . . . . . . . . . . . . . . . . . . . 223 B . One-Dimensional Equilibria with a Uniform Magnetostatic Field . . . . . . . . 225 C . Low Pressure Plasma Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4 . StabilityTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5. Application t o Fusion Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 6 . Experiments Related to Radio-Frequency Confinement . . . . . . . . . . . . . . . . . . . 241 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 B . Single Particle Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 C . Electron BeamFocusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 D . Direct Evidence of Radio-Frequency Confinement of Plasma . . . . . . . . . . . 250 E . Indirect Support for the Quasi-Potential Concept from Breakdown Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7. The Theory of Radio-Frequency Acceleration of Plasma . . . . . . . . . . . . . . . . . . 264 A. Purely Radio-Frequency Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 B Acceleration Using Combined Radio-Frequency and Magnetic Fields . . . . 276 8 . Experiments on Radio-Frequency Acceleration of Plasma . . . . . . . . . . . . . . . . . 283 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 153

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H. MOTZ AND C. J. H. WATSON

INTRODUCTION Most of the work described in this review was inspired by one or the other of two unattained goals of modern physics and engineering-the control of nuclear fusion and the acceleration of high density matter to relativistic velocities. The common feature of these two apparently unrelated projects is that in both cases the most hopeful working substance is a fully ionized plasma, which has to be stably confined in isolation from the material walls of the apparatus. The conventional approach is to use magnetostatic or quasi-magnetostatic forces for this purpose. As early* as 1956 however, the Soviet physicist Veksler (1956) pointed out that the force exerted on a plasma by a radio-frequency electromagnetic field could be quite substantial, and he suggested that a suitable rf field configuration might be capable of simultaneously confining a plasma and accelerating it. The following year, Knox (1957a,b) independently proposed that the rf fields which can be set up in a spherical resonant cavity should be used to confine a thermonuclear plasma. In the years immediately following this, a number of theoretical physicists in the Soviet Union, Britain, and America explored these possibilities in some detail. In the first instance, very crude models were used to represent the behavior of the plasma. On the one hand, it was taken to be a perfectly conducting fluid with sharp boundaries, at which all transverse electric fields had to vanish; on the other hand, it was supposed that the plasma could be treated as a bounded uniform dielectric medium of fixed dielectric coefficient (the value chosen being 1 - wp2/w2, where w was the frequency of the rf field and up some average plasma frequency). Soon, however, these “quasi-metallic” and “quasi-dielectric” models were replaced by an approximate magnetohydrodynamic theory. I t was shown that, provided the electric field gradient remained everywhere relatively small, it was possible to write down approximate expressions for the charge and current densities in the plasma, in terms of the local value of the electric field strength, and hence to obtain a set of self-consistent equations which determined the plasma and rf field configurations. At this time (19581960), attention was focused upon those solutions of these self-consistent equations which tended to confirm the validity of the quasi-metallic model, for the plausible reason that this model predicted the existence of confined equilibria and was significantly simpler than the quasi-dielectric model. Accordingly, stability analyses of these equilibria were carried out on the assumption that the use of this model had been justified-i.e., that

* There had indeed been earlier proposals, circulated as classified reports-e.g. Good (1953)-but Veksler appears to have priority of publication.

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the force which maintained the equilibrium was the radiation pressure of the magnetic component of the rf field alone. The results of this work were very discouraging to those with thermonuclear interests. It was shown that many of the confined plasma configurations, including the spherical configuration of Knox (1957a,b) and the cylindrical configuration of Boot et al. (1958), were unstable against certain deformations of shape. Furthermore, it was pointed out that the rf field strengths required to confine a thermonuclear plasma were so large that the dissipation of energy in the walls of the cavity would exceed the thermonuclear output, except possibly if the frequency were kept very low or the plasma density very high. The calculations of Boot et al. in this respect, which as we shall see were overoptimistic, indicated that a positive net power output could only be obtained if the value of up2/w2 was greater than about lo7. Unfortunately, as Weibel (195813) pointed out, for such very large values of up2/uzthe electric field gradient a t the plasma-radiation boundary, which is proportional to up,becomes sufficiently steep to invalidate the assumption underlying the self-consistent MHD theory upon which the calculations were based, and consequently (he implied) such configurations could not exist. As a result of this theoretical work, interest in the thermonuclear applications of rf confinement declined rapidly, and the few experiments which had been started were discontinued before any extensive results had been obtained. Fortunately the subject was kept open by those whose interests lay in the direction of acceleration. Accelerator physicists have never been deterred by a lavish consumption of rf power and, as a group of accelerator theorists (Levin et al., 1959; Askaryan et al., 1961) showed, it is by no means true that all confined plasma configurations are unstable on the quasi-metallic model. Consequently, in 1961 some experiments were started a t the Physics Institute of the Academy of Sciences in Moscow to test the principle of rf acceleration. One effect of these experiments, which have already produced some quite promising practical results, has been to reawaken the interest of a number of theoreticians in problems connected with the interaction of strong electromagnetic fields with plasmas. Perhaps the most important recent advance has been the demonstration by Silin that it is possible to solve the Vlasov equation for a plasma in a strong but spatially uniform rf electric field. As one of the present authors has shown, his approach can be extended to give an approximate kinetic theory of plasma in slowly varying nonuniform rf fields. This theory, which is described in Section 2, draws on the earlier work of Miller (1958~)on the motion of individual partides in rf fields, and it confirms the validity of the self-consistent MHD equations for plasmas with Maxwellian distributions. Examination of the conditions

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H. MOTZ AND C. J. H. WATSON

under which these are valid however, shows that this equation can never be used to justify the quasi-metallic model of a plasma confined by rf fields. Essentially this is because the gradient of the electric field is related, through Maxwell’s equations, to the strength of the magnetic field; so whenever the magnetic field is so large compared with the electric field that the latter can be neglected (the condition which is assumed to hold at the plasma boundary on the quasi-metallic model), the electric field gradient is so large that the self-consistent field theory is inapplicable. Conversely, whenever the electric field gradient is sufficiently small for the theory to be applicable, the electric field penetrates into the plasma to a significant extent, in the manner qualitatively described by the quasi-dielectric model. An immediate consequence of this is that all stability analyses based on the quasi-metallic model are suspect, since a stability theory cannot be more reliable than the theory of the equilibrium upon which it is based. In the view of the present authors it is now an open question whether there exist configurations in which a plasma with some given plasma frequency is confined by an rf field of very much lower frequency and whether, if they exist, they are stable. Weibel’s argument (1958b) shows only that one cannot use the approximate self-consistent field equations to determine the equilibrium. As we shall see, however, there are qualitative arguments, based on the behavior of individual particles in the presence of steeply sloping electric fields, which suggest that such equilibria may exist but that the width of the boundary layer between the plasma and the radiation has to be rather greater than that predicted by the approximate self-consistent field theory. This possibility has led us to reconsider the feasibility of a thermonuclear reactor working on this basis. I n Chapter 4 we show that, contrary to the optimistic conclusions of Boot et al. (1958), whatever assumptions one makes about the ratio wp2/w2, the rf losses in the cavity walls would preclude a positive power balance at any plasma density which could be confined by attainable rf field strengths, even if the cavity is made of a highly conducting metal such as silver. However, the position has been transformed by a recent technological breakthrough: it has been established that rf cavities can be constructed from superconducting metals, kept at a temperature below their critical point, and that the cavity losses can then be reduced by a factor of over 106. Research in this field is still in a very preliminary state, and there are theoretical indications that even greater reductions in the losses might be obtained; our calculations indicate that although the achievements so far are not quite sufficient, a 50-fold increase in the product of the surface admittance of the superconductor and its temperature of operation might make possible a reactor with positive power balance. In such a reactor both the brems-

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strahlung and the rf dissipation (both in the plasma and in the walls of the cavity) would be negligible compared with the refrigeration effort required to keep the walls superconducting ! The technological problems connected with the design of such a reactor would of course be formidable, but the same is true of any realistic proposal for a reactor using magnetic confinement. I n addition to proposals for purely rf confinement and acceleration, it has been suggested th at the rf fields should be combined with a stationary magnetic field. I n this context a distinction has to be made between two alternatives, which we shall describe as the “resonant” and “nonresonant” approaches respectively, which differ in the relationship chosen between the rf frequency w and the electron cyclotron frequency Q. If Iw - Ql is greater than a quantity which is difficult to estimate precisely but is roughly of order wvo/c, where v g is the thermal or directed velocity of the particles in the plasma, the oscillation of the rf field and the gyrations of the electrons in the magnetic field do not remain in phase for a significant length of time. Under these “nonresonant” conditions, if the magnetic field is uniform, the behavior of the plasma is qualitatively the same as in the rf field only, but the magnitude of the force exerted b y the rf fields on the plasma is amplified by a factor w / ( w - a) and it changes sign when Q becomes greater than (J.The application of this amplification effect in the theory of plasma confinement is discussed in Section 3, and its application to acceleration in Section 7. Its usefulness is unfortunately restricted by the fact that, as the plasma reaches high temperatures or velocities, the amplification factor decreases. A further difficulty, mentioned in Section 4,is that the presence of a magnetic field considerably increases the tendency for microinstabilities to develop in the plasma, the consequences of which for confinement are rather difficult to predict. Nevertheless, the reality of this amplification effect has been demonstrated in the Russian experiments on plasma acceleration. If the magnetic field is nonuniform, it is possible to use the (likewise amplified) rf forces t o improve the confinement properties of the magnetic field alone. Thus, for example, Johnston (1960) has suggested that one might hope t o block up the loss cones of a mirror machine in this way. The chief difficulty with this proposal is that the magnetic field in any confinement device of thermonuclear interest is rather strong (-10 kg) and the associated cyclotron frequency is of order 10“ sec-’. Thus the components of the rf electric field perpendicular to the static magnetic field (which alone are subject to the amplification effect) only enhance confinement if the rf frequency is somewhat greater than this: i.e., if its wavelength is less than about 1 cm. This raises both geometric problems and a very serious difficulty in supplying enough rf power to be useful.

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H. MOT2 AND C . J. H. WATSON

When the value of Iw - QI becomes small enough for particles to remain in resonance for a significant period of time, the approximate selfconsistent field theory of the behavior of the plasma and the fields breaks down. Under these circumstances it is exceedingly difficult to give an adequate account of the motion of even a single particle, let alone a whole plasma, except in situations of unrealistically simple geometry. In Section 1 we summarize the special cases in which the equations of motion of a single particle have been solved, either exactly or in an approximation whose validity can be assessed, and in Section 7 we indicate the way in which these restricted results have been used to predict the behavior of a plasma accelerator based on this “resonance” principle. I n Section 8 we describe the experiments, on resonant confinement and acceleration, conducted by Consoli and his co-workers at Saclay; some of these have been strikingly successful and offer considerable prospects of further development. The practical applications of the ideas which we have outlined above are many and various. Apart from the reopened question of the possibility of an rf thermonuclear machine, rf confinement may well prove useful in machines for the direct conversion of thermal energy into electricity. Rf plasma accelerators may be used to investigate statistically improbable fundamental particle transformations, which require very high beam densities if they are to be observed; or to inject hot plasma into a more conventional thermonuclear machine ; or for space propulsion ; or even [as suggested by Consoli (1963a)I to create very high vacua, by ionizing the residual gas and then accelerating it out of the cavity. But even if it should turn out that there are practical obstacles to all these applications, the theory of the interaction of rf fields with plasma may prove important in the understanding of certain astronomical phenomena, for example, in radio-stars. It would not be the first time that human inventive activity has suggested mechanisms which explain cosmic phenomena. Finally, the forces exerted by rf fields on solid state plasmas have hardly been discussed in the existing literature. In the present review we do not attempt an exhaustive treatment of the whole literature. In particular, we have restricted ourselves to the interaction of natural rf modes with a plasma. For such modes the electric and magnetic field strengths, averaged over the space within which the mode is set up, are equal. It is of course possible to excite a cavity with forced oscillations, imposed by alternating currents in the cavity walls, such that the magnetic field strength is everywhere much larger than the electric field. Such “quasi-magnetostatic” rf modes exert forces on the plasma which cannot be analyzed by the methods described in this review, and we have accordingly excluded this case from our discussion.

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1. SINGLE PARTICLE MOTIONS

-4.Introduction The exact motion of a charged particle in a n arbitrary time-dependent electromagnetic field is one of the notoriously insoluble problems of classical dynamics. Until recently, however, the amplitudes of electromagnetic fields which either were known to arise in nature or could easily be generated in the laboratory were relatively small, and one could plausibly use a linearized approach: i.e., expand the actual particle orbit as a power series in the amplitude of the electric field E about some unperturbed orbit-rectilinear or (in the presence of a uniform magnetic field) spiral. This approach clearly becomes inapplicable if the maximum velocity acquired by the particle under the influence of the field during one cycle, which can be cstimated as V E eE/mw, becomes comparable with its unperturbed velocity. The practically realizable values of VE have risen sharply as a result of recent developments in microwave and laser technology. The present position might be summarized by saying that, for an electron velocities v E c are attainable a t frequencies u p to about 10"; and even at w = 1015, the field strength from a ruby laser gives UE 105 cm/sec. For such field strengths a nonlinear approach is required. Fortunately, in recent years a number of analytical techniques have been developed which yield approximate nonlinear solutions of specifiable precision in all but the most intractable cases. For the most part, these techniques are variants of the method of averaging, first popularized in mathematical physics by Bogolyubov (1955). This method is in fact of very wide applicability; however, 011 the principle that approximate solutions are more credible if they reproduce exact solutions in special cases, we shall begin by considering a few cases where the symmetry properties of the fields make it possible to solve the equations of motion exactly. We shall therefore discuss the motion of a particle of charge e in the field of a plane electromagnetic wave which is derivable from a vector potential A1(z, 1) lying in the z,y plane. Since in this case the Hamiltonian

-

-

-

x

=

(p - eA,(z, t ) ) 2 / 2 w

is independent of the coordinate x and y, we immediately obtain two constants of the motion, p , and p,, related to the dynamic variables by pL = nzv,

+ eA,.

(2)

This determines the motion in the z,y plane. In the z direction we have

p,

=

nzz

= --ax/az.

(3)

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H. MOTZ AND C. J. H. WATSON

At this point it becomes necessary to decide whether A, represents a traveling or standing wave. This distinction is unimportant in linearized theory, where indeed it is customary to use a complex representation in which no such distinction is made; in nonlinear theory the difference is fundamental, as may be seen from the fact that for a traveling wave the quantity A* averaged over one cycle d2is constant, whereas for a standing wave it is a function of z. A t this stage we shall consider only standing waves, since the traveling wave case can only be discussed adequately in a relativistic framework, which would obscure the argument at this point. We shall return to it later. A second decision which is required concerns the state of polarization of the wave. We shall later develop a method which makes it possible to consider quite arbitrary polarizations; here we shall consider only plane or circularly polarized waves and shall write

Al(x,

(4)

t ) = A(z)[cos at, Y sin wt, 01,

where Y = 0, +1 for plane and circularly polarized waves, respectively. Equation (3) then gives

+ v p , sin w t ) a - - (e2A2/2nr)(cos2 wt + v 2 sin2w t ) . 82

e dA m& = - - ( p z cos wt m az

(5)

This equation simplifies significantly if we choose pL = 0. The physical significance of this choice becomes clear if we average Eq. (2) over one cycle: pI is the average particle momentum in the z,y plane. (It is tempting to suppose that in consequence one can always choose pL = 0 simply by = 0. However, such a transforming to a frame of reference in which transformation also changes A, in such a way that the form of (5) is unaltered.) If in addition we consider circularly polarized waves (Y = l ) , Eq. (5) becomes exactly soluble and gives

Qmi2

+ e2A2(x)/2m= & = const;

(6)

that is, J/ = esA2/2nt = e2E2/2mu2acts as a potential well which tends to confine the particle in the neighborhood of a node of the wave. To consider the resulting motion in more detail, let us expand the vacuum wave

E

=

Eosin k z [cos wt, fsin

wt,

01,

k

= w/c,

(7)

about one of its nodcs. We obtain J/ = & v n ( e E ~ / ~ n c=) ~ +nawJ2z2 z~

Thus the particle oscillates harmonically with a frequency W J (vE/c>w.

(8) =

eEo/mc =

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161

If on the ot,her hand we consider a plane polarized wave (v = 0) and again consider the motion in the neighborhood of a node, Eq. ( 5 ) becomes 8

+ 2 0 J 2 z sin2wt = 8 + wJ2(1 - cos 2wt)z = 0

(9)

(where W J has the same value as before if E , is interpreted as the rms value). Equation (9) is Matthieu’s equation, and we can take over its known solutions. A general property of these solutions, demonstrable by Floquet’s theorem, is that they can be written in the form z = P ( w t ) exp(fipt),

where P is a periodic function of wt and p is a (possibly complex) constant. If p has an imaginary part, the solution is described as unstable; the physical significance of this in the present context is that, whatever the initial energy of the particle, it is able to move arbitrarily far from the node (at least until the expansion of the field strength about the node breaks down). Conversely, if p is real the particle remains trapped. The stability and instability regions for a given value of the parameter w.,/w can be obtained from standard tables; in particular, the solution is stable for 0 p w J / w 5 0.83. The nature of the solutions is discussed in (for example) Erdelyi (19,53) (Section 16-2), who show that for small U J / W , p ‘v W J and P(wt) = 1 ( w j 2 / w 2 ) P ’ ( w t ) . Thus, in this limit the solution is strikingly similar to the exact solution obtained above in the case of circularly polarized waves; the only difference is a small rapidly oscillating correction of order w J 2 / w 2 .As this parameter increases, however, the correction term likewise increases and the motion becomes significantly nonsinusoidal. Finally, when W J / W 0.83, x begins to grow exponentially with time and the particle escapes. We shall now show that for small W J / W all the properties of this exact solution can be obtained by the method of averages. We attempt a solution of (9) of the form z = zoaC zs, where zoscrepresents the oscillatory motion on the time scale 3 a / w and z8 represents the smooth motion on the much longer time scale 2?r/wJ. Treating zoscas <
+

-

+

zs

+ w.72zs = 0,

2 0 s ~= W J 2 z B

cos 2Wt.

(11) (12)

In integrating (12) we can ignore the weak implicit time dependence of zs and obtain Zosc =

-(

cos

W J ~ / ~ W ~ ) ~2wt ,

(13)

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H . MOTZ AND C . J. H . WATSON

and hence z = x o exp( +i‘wJt)(l - UJ2/4W2cos 2wt),

(14)

a solution which is of the form (10) as required. Having determined the motion in the z direction, we can calculate the motion in the x,y plane, for by ( 2 )

vL

=

p L / m - eA/m

V,

=

@ O J Z sin wt

or

in the present case. I n computing v, for small W J / W , we can ignore the small contribution of zoscand hence V, E

.\/Z

WJZ.

sin wt.

Thus v, is of the same order as v,; the maximum displacement of the particle in the x,y plane is, however, much smaller than its maximum displacement in the x direction, for 2 = ( W J / W ) Z , cos ot.

FIG.1

Let us form a physical picture of the rcsulting motion of the particle. I n the x direction (i.e., in the direction of V E 2 )there is predominantly a smooth motion of the “guiding center” of the particle under the influence of the quasipotential II. = e2E2/2mw2. In the x direction (i.e., the direction of E) there is predominantly a rapid oscillatory motion of large peak velocity eE/mw but small amplitude eE/mw2.These two dominant motions are complementary; where .is is large, j. is small, and vice versa. Indeed, the quantity T, = Qmx2is constant, and one can think of the quasi potential as operating by converting smooth motion along its gradient into oscillatory motion in the direction of the electric field. I n addition to these two dominant motions, there are also “microosci1lations”-the small correction t o this dominant motion resulting from the term zosc.Since this correction is of order ( V E / C ) ~and is very difficult to calculate except in the special case we have been considering, we shall neglect it in most of this review. I n Fig. 1 we represent schematically the actual orbit of a particle in the case when W J / W A.

+

-

RF CONFINEMENT AND ACCELERATION O F PLASMAS

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A s the value of V E / C (or equivalently w J / w ) increases, the amplitude of these LLmicrooscillationsll increases. It should, however, be noted that the particle orbit remains bounded even when the microoscillations cease to be negligible; the particle only escapes from the quasi potential well when VE 0 . 8 3 ~(i.e., when the amplitude of the “microoscillations” becomes comparable with xo, the maximum displacement of the guiding center in the potential well). This escape phenomenon can thus be regarded as roughly analogous to the process by which a quantum mechanical particle ‘Lleal<sout” of a potential well. An alternative point of view, which helps to explain why this escape phenomenon does not occur in the case of a circularly polarized wave, is that the term e 2 A 2 / 2 min t.he Hamiltonian corresponds t o a potential well which (in the case of a plane polarized wave) rises and collapses to zero twice in each cycle of the rf field. Thus, a particle which can penetrate far enough into a “forbidden” region of spare during each of the quarter cycles in which the well has collapsed to zero cannot be confined in the well.

-

B. Guiding Center Theory for Arbitrary Radio-Frequency Fields We shall now consider the generalization of Section A t o allow for rf fields of arbitrary, even varying, direction of propagation. It is clear from Section A that the appropriate procedure is an expansion in the small parameter v/c; some caution is required, however, in considering the permissible values of the other dimensionless parameter which occurs in the equations, a! = c / w L ~ ,where LE is the scale length for the nonuniformity of the electric field. We can rewrite a! = Lv/LE, when Lv is the length scale for rf waves of the same frequency in vacuo. The method used in this section is valid only if a! 5 1. This condition is fortunately satisfied by almost all the types of rf wave which we shall consider in this review; the few exceptions will be discussed separately as they arise. The analysis which we shall now present is essentially that of Miller (1958) ; in order to make the algebra as transparent as possible we shall work in a “natural” system of units in which e = m = 1 and the fields E and B both have the dimensions of acceleration. The results can be converted into the m k s system used elsewhere in this review by replacing E by (e/nz)E and putting c = 1. I n this system of units the equations of motion are

i:

=

E(r, 0

+ (v/c) X B(r, 0 ,

(15)

and it follows from the assumption a! ,< 1 that B ,< E. As in Section A, we write r as the sum of a smoothly varying guiding center motion R and a rapidly oscillating motion about this, e, and we seek a solution in which p is of order (v/c)LE.Substituting r = R p into (15), expanding for

+

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H . MOTZ AND C. J. H. WATSON

small p and separating into smooth and oscillating parts, we obtain

+

(fc X W,

(17)

where A is a quantity obtained from A by averaging its explicit time dependence over a time 23r/w. I t is clear that the first term on the right of Eq. (17) is of lower order in v/c than the remaining terms, which can therefore be neglected. Furthermore, the error made in ignoring the implicit time dependence of E (through R(t)) when integrating (17) is also of lower order in v/c. Thus, if we write E(R, t ) = E exp(iwt) E* exp( -id) we have

+

Q =

E exp(iwt) - E* exp( -id)

’

iw

E(R, ‘)

e = - W 2 *

(18)

On inserting these into (16) and using Maxwell’s equation

(l/c)B(R, t )

=

-V X

E(R, t’) dt’,

(19)

we obtain

R

= =

+

(-1/w2)[E VE* E* VE -VE. E*/w2 = -V$. 9

+ E X V X E* + E* X V X E] (20)

Thus the smooth guiding center motion is derivable from a scalar potential J.; in order to avoid implying any judgment on priorities in a field where

Soviet, British, and American candidates all possess claims (Gaponov and Miller, 1958a; Boot and Shersby Harvie, 1957; Weibel, 1958), we shall refer to $ as the “quasi-potential” for the guiding center. The first integral of the motion, reexpressed in dimensional form, is E = +mR2

+ e21EI2/mw2= const.

(21)

The apparent discrepancy in numerical factor between this and various other expressions for the quasi-potential to be found in the literature is due to different definitions of the relationship of the complex quantities E and E* to the real magnitude of the electric field, which may be represented by its rms or peak value. In terms of the peak value E,, $ = e2Ep2/4mu2; in terms of the rms value E,,,, = e2E:,,/2mw2. In this review we shall use the RMS value throughout, since, with this choice, we can also write J. = e2@/2mw2 where E(R, t ) is the instantaneous value of the electric ~

RF CONFINEMENT AND ACCELERATION OF PLASMAS

165

field at the guiding center. An additional factor of one-half arises whenone comes to impose thc condition of quasi-neutrality in a plasma, as we shall see in Section 2. It will be noted that the quasi-potential obtained in this section reduces to the expression $ = +mw.&2 which was derived in Section A for the case in which a particle is confined near a node of a plane wave. The form for $ given here represents the generalization of this which is appropriate when the particle motion leads it to regions in which A can no longer be represented as a linear form in z. I n particular, it shows that particles can fail to be trapped in a quasi-potential well if their energy is such that they can fly over it, or “tunnel” through in the manner already described. Incidentally, it is rlear that tunneling will become easier for particles near the summit of the barrier. The question of the shape of thc potential well caused by a given electromagnetic field is in principle trivial, given the expression for $, but can be nontrivial in practice, since the fields have to satisfy Maxwell’s equations with appropriate boundary conditions. A rather complete analysis of the quasi potentials set up by arbitrary modes of excitation of square and circular section waveguides is given by Miller (1958~).His analysis of standing waves in such waveguides needs little interpretation and the reader is referred to it for the details; roughly speaking, he shows that one can create wells in one, two, or three dimensions simply by a suitable choice of frequency and mode type (or types, since quasi-potentials combine linearly provided that the frequencies are incommensurable). His discussion of traveling, or partially traveling, waves needs some comment, however. On the theory given in this section, a traveling wave creates no quasi-potential barrier in its direction of propagation. This result is in fact correct (in the present approximation) for waves whose phase velocity is of the order of c, as we shall see later in this section. It is (as we shall again see) incorrect for waves with phase veIocity v4 << c. However, such waves are nonuniform on a length scale LE = v4/w, which is much less than c / w and hence the theory of the present section does not apply. Such low phase-velocity waves are not to be confused with “nearly standing” waves; i.e., standing waves whose amplitude is a slowly varying function of time, for example E = 4 3 E(z - vot) sin wt (where v o << c). The motion of a particle in such a field is correctly described with the help of a slowly time-dependent quasi-potential $(R,t ) = e2Ez(R,t)/2mw2. Such time-dependent quasi-potentials underlie Gaponov and Miller’s (1958b) proposed device for plasma acceleration and Aslcaryan and Rabinovich’s (19,58a,b) proposal for the combined compression and heating of a plasma by a resonant modulation of the amplitude of the quasipotential.

166

H. MOTZ AND C. J. H. WATSON

C. Radio-Frequency Plus a Uniform Magnetostatic Field The effect of a uniform magnetostatic field on the rf quasi potential was discussed in outline in the first paper of Miller (1958). His subsequent extension of the discussion to the case of nonuniform magnetostatic fields is (as we shall see) incorrect, but it explains in more detail the steps in the analysis for the uniform case, and the account given below is based on it. We defer until later the serious complexities which arise if one requires a theory which is valid for a nonuniform magnetostatic field. I n the system of units of the preceding section, the equation of motion can now be written as

where = e B / m is the vector cyclotron frequency for a particle in the magnetostatic field, which we take to lie in the z direction. The separation or r into an oscillatory and slowly varying part r = R p is now complicated by the fact that there are two kinds of oscillatory motion with frequencies w and 52 even to lowest order in v/c, and the first-order corrections lead to the generation of combination frequencies as well. I n consequence, one cannot obtain an equation for R simply b y averaging (22) over a time 2r/w. Instead, one has to split up (22) into a set of coupled equations by collecting together separately terms with different explicit time dependence. Fortunately, no ambiguities arise in carrying out this separation by inspection in equations correct to first order in v/c, since the combination frequencies do not contribute in this order, provided that l / ( w f 52) is not so large that it approaches the time scale on which R varies l/(wv/c). If we exclude this “resonant” case (we will return to it in Section E), we obtain the separated equations

+

where the overbar now has the significance of eliminating those contributions to the term in question which have any explicit time dependence exp( f i n w t ) , exp( f i n g t ) , exp( f i ( w c! 52)t). Equation (24) possesses a particular integral of time dependence exp(iw2) and a complementary function of arbitrary amplitude with time dependence exp(i52t). We are not entitled to ignore this complementary function in describing the exact particle motion-however, i t is seen th a t to first order in v/c it makes no contribution to the R motion. T o solve (24) it is convenient t,o work with the well-known complex vector system

ItF CONFINEMENT AND ACCELERATION O F PLASMAS 0,

167

defined i n terms of the unit vectors e,, eul8, by T~~ =

(e, -t

ie,)/z/z,

(25)

e0 = 8,.

These complex vectors satisfy the relations eM*

5LV

a*”,

=

0”

x a,

=

0,

0,

x 0-, =

-iveo, Q,

for p , v

=

X

T~ =

ivzy, (26)

1. Given an arbitrary vector A we can define

0,

A,

A -e,,

=

and using (26) we can readily show that A = A,.t-,, where we have used the Einstein summation convention for repeated suffixes, which we assume in what follows except where the contrary is indicated. We can also readily show that A B = A,B-,. Some care is required in taking complex conjugates-we have (Av)* = (A*)-,. Thus the operations of conjugation and projection onto the 0”do not commute; in what follows we shall adopt the convention that projection precedes conjugation except when the converse is explicitly indicated by brackets. Using these properties of the 0, we see that (24) can be written as

py

+ ivQp,

=

E,r(R, t )

-

esy= =

E , exp(iwt) E , exp(iwt)

+ ( E * ) ,exp(-id) + E?, exp( -id)

and hence

. py

E?, exp( -id)

E , exp(iwt) = i(w

+

VQ)

+

-i(w

+

VQ)

+ complementary function

and P Y

=

- --

E-*, exp( - i d )

E , exp(iot) +

(w

-

VQ)

(27)

+ complementary function.

Equation (23) now gives

Now by the vector identity of Eq. (20) T-, *

VA

+

T-,

X (V X A)

- (A * V)T-, - A X (V X T-”) = VA-,

= V(A * T-“)

(29) since in the present section the magnetic field is uniform and hence the e vare constant vectors [though in Miller (1959a,b) it is incorrectly assumed

168

H. MOTZ AND C. J. H. WATSON

that the result is still V A - , even when they are not constant vectors]. Thus 1 EYV(E*)-, E?,VE-,

R - R X P = - -w =

-v

[

1

(0

+ v-9 + +

(E,12/W(W

w

-

VQ)

=

1 -v+.

VQ

(30)

V

Hence $R2

+ + = const = E. Reverting to the mks system we have

and we see that a uniform magnetostatic field affects the particle motion in two ways-it modifies the small amplitude oscillatory motion by adding a component of arbitrary amplitude (determined by initial conditions) with frequency Q and it changes the quasi potential from

mw2

A consequence of the resonant denominators l / ( w 5 Q ) is that a given amplitude of rf field exerts a greatly enhanced force if w is near Q, but not so close that (w - Q ) / w = u/c. To anticipate a great deal of tedious algebra we may remark here that in the following section it will be shown that if the magnetic field is nonuniform, but the length scale of the nonuniformity is large compared with the Larmor radius, Eq. (31) is trivially modified by the addition of the term p B o , where I.( is the adiabatically invariant magnetic moment of the particle resulting from its gyro motion.

D. Radio-Frequency Plus a Nonuniform Magnetostatic Field The equation of motion (22) of the previous section is unaltered if the magnetostatic field is spatially nonuniform, except that P becomes P(r). The problem now contains two independent length scales L, and La, associated with the nonuniformity of Ed and P, respectively. It was previously necessary to assume c/wL, 5 1;for the same reason we now have to add c/wL n ,< 1. However, this is not sufficient, since even in the absence of an rf field, Eq. (22) is only soluble for arbitrary P(r) if the Larmor radius i./Q is much smaller than L n. Thus there are now two dimensionless parameters in the problem, a = i/QLn and p = i / c , both of which must be treated as small. It is clear that the results of the preceding section are correct to first order in p and zero order in a ;however, as is well known, the physically interesting features of the motion of a particle in a nonuniform magnetic field only appear if one works to first order in a,and we shall accordingly do so, though we shall treat terms of order ap as second order.

RF CONFINEMENT AND ACCELERATION OF PLASMAS

169

The equations for R arid p now become

R @

=

=

+ (e

R X Q(R)

*

V)E(R, t )

+ (g/c)

X B(R, 1)

+ 0, x (P . V ) W ) eX +E ( W X ( p . XQ V + + (Q/c) X B(R, 0 + @ X (p

(32)

B(R, t )

Q

*

(33)

V)Q(R),

where the overbar has the same significance as before, and d = A - A. Since all the terms containing p in (32) also contain V or l / c and hence are automatically of first order in a or 0, it might appear that it is sufficient to solve (33) correct to zero order in these parameters, and this is indeed the approach adopted in the analyses of Aliller (1959b) and Fessenden (1963). However, if one considers the case in which Err = 0, it is clear that this procedure must be incorrect, since to zero order Eq. (33) then states that the magnitude of the velocity of the particle is constant, whereas we know that to first order it changes adiabatically, the adiabatic invariant being g = v,2/Bo. The error consists in assuming without proof th a t e can be expanded, as in perturbation theory, p = p(O) a p ( ’ ) with p(1) bounded everywhere; for adiabatically varying quantities such a n expansion is only valid locally. T o see how this adiabatic behavior arises, let us consider the equation 6 = C, X Q(R) using the same complex vector representation as in the preceding section. Here, however, we start with an orthogonal curvilinear coordinate system; i.e., we select the set of orthogonal unit vectors ez, er, 8, so that 8, = Q/Q is everywhere directed parallel to the magnetic field. I n consequence, the r v are no longer constant vectors, though they still satisfy the relationships of Eq. (26). If we project v = pj onto the v v the equation of motion becomes

+

8,

Writing k-,,

=

+ v,(re,’ k-,)

v

a

Vr-,

6,

If we define wv

=

wv =

=

!&,TV

= U X T - ~* Vr-,,

+

iVfhv

’

($-,,

x

$0)

=

-iuov,.

we have

= -v2.’hucZv *

(7-A

’ V)T-,.

v, exp(ivl9 dt’), -wu)Xw,,ru

0

(7-A

- v)s-,,exp[i(v - X - p)JS].

(34)

From their definition, the wv represent the components of the magnitude of the velocity in the direvtions p:irallcl and perpendicular to the direction of the magnetic field. To lowcst order in LY the right-hand side of (34) vanishes, since VT-, = 0 in this order, and we have w, = constant. T o first order, w, can be split up into two parts: a small oscillatory part, deriving from the terms for which Y - p - X # 0 , and a n adiabatically

170

H. MOT2 A N D C. 3. H . WATSON

varying part, of zero-order amplitude, deriving from the terms with v =X p. Using the identities (20) and (26) and the near symmetry of (34) with respect to interchange of the indices p and X, it is not difficult to transform the equation for this part:

+

W, =

-wAw,,~~(Q-A * V ) T - ,8 v , ~ + p

( v not summed)

(35)

into w v = iVwvwO('5v *

x

v

- '50 v

'5-v

and hence (by respectively taking v summing) t o derive dvll/dt = (~,'//a)

V

70,

x 70)

+ X 2 6vO(wAw-A/2) v

0 and multiplying b y

=

dvJdt = - ( ~ l l ~ J 2 )V

' '50,

V ~ W -and ~

.7o,

(36)

which are the Bogolyubov equations, from which the adiabatic invariance of the magnetic moment p can be shown. Our present point, however, is that the adiabatic behavior is a result of the nonlinearity of Eq. (34) ; a linear term of first order in a would only lead to a first-order correction to the cyclotron frequency. Thus, we can obtain a solution for e which is adequate for our purposes (i.e., correct to first order as regards its adiabatic behavior, but with the terms which remain of first order everywhere neglected) by omitting from (33) those first-order terms which are linear in p or independent of it. We can, in fact, omit two further terms in (33) which are not excluded on these grounds, if we observe that only those components of p which have frequency w or Q contribute to (32), since the selection process represented by the overbar eliminates all other terms. The last two terms in (33), however, cannot contribute to these components of .p except in second order and can therefore be neglected. Thus (33) can be reduced to @ = Q X Q E(R,t ) (37)

+

which we can write in the Wv

=

T,

framework as

-

-wAw,T~(o-A V)O-, exp[i(v

-X

- p)JQ] - yo].

+ E , exp[iJw + vQ] + ( E " ) ,exp[-iJw

(38)

For the purposes of Eq. (32) we only need the components of w,which have either no explicit time dependence or frequencies k (W k v Q ) ~so we can write wv = x v y y exp[iJo VQ] z, exp[-iJw - YQ]. (39)

+ +

+

On substituting (39) in (38) we obtain (v not summed) kv =

- zAyp[TO ' ('5-X

yv Z,

' v)'5A-v]

- ~ A ~ Y - X [ ' ~ V' ('5-X

+ i ( w + vQ)yV

- i(u -

'

v)T-pI

avo,

- yvzp[sO

-

(7-A

v)'5-p]6 v O

-

E v - YAZO[T~('S-A W 7 o - yp~o[.ev * '50 VT-J, v ~ ) x ,= (E*), - XASO[C,(Q-A * V)T, - Z , Z O [ T , 20 * VS-,]. =

(40) (41) (42)

171

R F C O N F I N E M E N T A N D ACCELERATION O F PLASMAS

Equations (41) and (42) are linear equations for y v and xv. Thus they do not introduce adiabatic behavior, and we can ignore the first-order corrections, obtaining

+ vQ);

yv = EV/i(w

x V = (E*),/-i(w - vQ).

This confirms that in first order, no adiabatic effects arise in the part of the pv with time dependence exp( k i w t ) and hence for this part we can still use the zero-order solution given in Eq. (27) of the preceding section. A certain adiabatic effect does however arise from the fact that to obtain p we have to combine these with the adiabatically varying T ~ Equation . (40), on the other hand, remains nonlinear; on comparing it with (34), however, we see that for the components perpendicular to the local direction of the magnetic field (v = i - 1 ) (which alone have a n oscillatory behavior and hence contribute to (32)), it is identical with the equation of motion in the absence of the rf field. Thus the magnetic moment p = +mv,2/Bo is still an adiabatic invariant, though v I must now be interpreted as the magnitude of the velocity of the cyclotron motion only; the rf motion is linearly superimposed. Similarly, the contribution of pa to the term O, X p VSLin (32) gives the well-known force -VpBO. Thus we can write (32) as

.

+- E&” + vo X v x E*

E*UT-V

w -

w

+

[& +iEVETpvQ)(w

-

VL?

iE?,E, w(w

pQ)

XVXE]

- vQ)(w

+ pa)

x T P U x Cp* VSL.

1

(43)

This equation differs from (28) in two respects: the vectors T ” are no longer constant and there is an extra term which results from the nonuniformity of SL. We must now undertake the rather heavy algebra required to show that these two effects cancel exactly. We may begin b y assembling the various vector identities which we shall need. The last three equations of (26) can be reduced to Tp

x T, = i[v

6p.--uTo

+

/I 6”0lP

- v 6,oe,I.

(44)

If we define a vector operator D which occurs in (43) by

A - DB

=

A *VB + A X V X B,

the identity of (20) can be expressed as

(A * D)B + (B* D)A

= V(A

a

B).

(45)

172

H. MOT2 AND C. J. H. WATSON

Hence in particular (c, * (c, (c,

D)A = VA, - (A* D)c, D)cY= 0, D)c,= - ( 7 , * D)c,.

Using (44), we have (-0

-

[cf. Eq. (as)]

(46) (47) (48)

D)(c,X 7 y ) = i [ p 6,o(c0 D)cp- v

+

= -i(p = -i(p

v)(6,0

+

6,0(c0 * D)cYl 6,0)(c, D)cv

+ v)(c,- D)cY.

(49)

Using the vector identity (which canrbe proved in a Cartesian system)

A X (B V)C - B X (A.V)C

=

combined with Maxwell’s equation (49), we have

v

bL = 0 and the identities (46) and

-

- 7, X (c, V)sL = - (c, X T,) D P (T, X cy)Q1 Qco * D(T,X -iv 6,,-,~st - i Q ( p v)(T, * D)c,.

T,, X (T,

V)P = -V[co =

.

(V C)A X B - (A X B) DC

+

+

(50)

We are now in a position to simplify ,(43), which we can write as EU(c-, D)E*

+ E?,(c-,- vwD)E

R-RXP+VpBo=

*

w

x[

( c ,

*

D)c-,

w+vn

+

(7-, *

w

D)c,

- pst

where we have successively interchanged the dummy variables p and v, used (46) and (SO), interchanged p and v, and used (48).Taking the scalar

173

R F CONFINEMENT AND ACCELERATION O F PLASMAS

product of (51) with R and reverting to dimensional variables, u-e have

Y

=

const.

(52)

Some comment should be made on the relationship of this analysis to those of Miller (1959a)b) and Fessenden (1963). It differs from both in showing that the magnetic moment of the particle resulting from its cyclotron motion remains adiabatically invariant to first order in a and p even in the presence of a strong rf electric field. This result is naturally only valid if the frequency o is sufficiently different from D everywhere. Equation (52) also differs from the result of Miller, since it does not contain the additional term of the form pl VSL which occurs in his Eq. (22). This is essentially the part of the last term of the first equality of our Eq. (51) with p # V, and we have shown that it is exactly canceled b y the contributions made by the curvature of the to the first two terms, which Miller neglected. As regards the E-dependent term in (52) our result coincides with that of Fessenden, who, however, worked in an arbitrary Cartesian coordinate system. I n this case (52) becomes

-

+mR2

+ pB0 + (ez/mw2)E*

T

E

(53)

where T is a Hermitian tensor with components T

=

i

-b,b,

+ ib,

-b,bz- bx2 ib,

-b,b, - ib, 1 - bU2 - b,b, ib,

+

+

- b,b, ib, -b,bz - ib, 1 - bz2

1

where b = Q/w. It is not difficult to show that Eqs. (52) and (53) are equivalent. Finally, something should be said about the significance of R. It is clear that in dividing (22) into the two equations (32) and (33), one has raised the order of the equation and has hence made it possible for spurious solutions to occur. I n particular, since (32) contains the term k X Ja, it will possess solutions for which R is oscillatory with period l / D , whereas we have assumed in the analysis that R is slowly varying. We are thus both entitled and required to select only those solutions of (51) for which R is slowly varying, and to first order in a this implies that R is parallel to SL.

E’. The Cyclotron Resonance. (i) Standing Waves In Sections C and D we explicitly excluded the case where w & D becomes very much less than w. No precise criterion was given, however, for the value of w D at which the quasi-potential theory might be

+

174

H . MOT2 AND C. J. H. WATSON

expected to break down, and no discussion was given of the theory which might replace it in the neighborhood of the cyclotron resonance. I n this section we shall outline the rather modest progress which has been made to date towards answering these questions; so far, apart from the substantial body of literature on conventional cyclotron theory (where the rf field only acts over a very small fraction of the particle orbit and hence accelerates it without making a large change in its orbit), all the work published on this topic is restricted to the case of a plane standing wave combined with a uniform or nearly uniform magnetic field directed along the wave axis. We shall consider first the case of a circularly polarized standing wave derivable from the vector potential A,(z, t ) = A(z)[cos wt, +sin wt, 01 combined with a spatially uniform magnetic field. Our approach is essentially that of Kulinsky. Since at one point in the analysis the use of nonrelativistic equations is open to doubt, we shall retain relativistic precision until this proves unnecessary. To represent the magnetostatic field Bo we need to add an appropriate vector potential. Twopossible choices, which differ only in the gauge assumed, are

A.

=

BoxP

and

A.

=

-Boy$,

(54)

where 2 and 6 are unit vectors in the directions concerned. The corresponding relativistic Hamiltonians are

x1= c[(P1- eA, - eBoxQ)>2 + m ~ ~ c ~ ] ~ ’ ~ , (55) xZ= c[(Pz - eA, + eBoyL)2+ mozc2]1/2. (56) Since x1is independent of y and xzof ~t:we have as constants of the motion

P I , = mu, P o , = mu,

+ eA, + eBox = const, + eA, - eBoy = const.

(57) (58)

Since a gauge transformation cannot affect any observable quantity, the relations (57) and (58) must hold simultaneously even though they are derived from different Hamiltonians. Differentiating (57) and (58), or using the canonical equations, we obtain (with s2 = eBo/na)

p,

=

ad (mu,) = -eA, + ~ p , ,

RF CONFINEMENT AND ACCELERATION OF PLASMAS

175

We now introduce our particular choice for A, which we shall write in the form A, = A(x)a(l).The relative simplicity of the final result depends upon the following properties of a:

a . 8 = 0, 8 . 8 = w2, = --2&, a x n = f(n/w)8, 8 x 0 = kwDa.

a2 = 1,

(62)

We can write (59)-(61) in the form

pL = -eAa

- eA8 + (p, x a),

pz$z = eAp,

- a.

(63) (64)

By (62) and (G3), we have

d

dt (p,. a) = -eA

*

w + Q +p, w

8,

By (64) and (G5)

Hence by (GG) and (64)

(

”)

- ~-

dt

w

f D dt

-

-deA -

w(w

f D)

?$)

=

-W(W

f D)+ f wDeA

or

ddt ( 2 3)+ w _+ Q dt

(W

f Q)4= kQeA.

I n the nonrelativistic limit, D is a constant of the motion, and Eq. (68) is a straightforward (though rather intractable) fourth-order nonlinear differential equation for 2. O n the other hand, in view of its singularity a t w f D = 0, we should perhaps exercise some caution in taking the nonrelativistic limit near this resonance. Relativistically, however, D is a variable which depends on both the parallel and the perpendicular motion, so we require an equation of motion for Q or, since Q = Q , J c ~ / ~for , X. We obtain this as follows:

Equations (68) and (69) form a closed set of equations for the z motion;

176

H. MOTZ A N D C. J. H. WATSON

once this has been found, the other dynamic variables can be obtained from (57) and (58). We first show that sufficiently far from resonance we can rederive the guiding center theory from (68). If we neglect the first term, the other two give QeA p.pz e2AA $)= = f(70) eA w + Q and hence we2A2 + 2(0 f Q)

+

2 [$

which is the result of guiding center theory. The conditions under which (71) is valid can be found b y inserting the value for 4 given in (70) into the neglected term, which gives (nonrelativistically) f[ Q / ( w L- Q)2] d2(eA)/dt2.Since by hypothesis this must be much smaller than k Q e A , we obtain the condition z/(u f Q ) L E << 1, where La is the length scale for the nonuniformity of A . If we consider a vacuum wave, this gives ( w _+ Q / U ) ~ >> ( i / c ) ' . We may note in passing that if we use the guiding center approximation for q5 in Eq. (69), we obtain (nonrelativistically again)

so +mu2f

e2A2 - -& & Q)2 2nz

wQ (0

=

const.

Combining this with (71), we obtain

&nzvL2

=

w2 (w

f Q)2

e2A2 21n

+ const.

(73)

Thus as we approach resonance, the perpendicular energy of the particle rises more rapidly than the parallel energy declines. Nevertheless, in this approximation, as (72) shows, the motion is conservative; a potential energy function can be defined which does not depend upon the previous history of the particle's motion. We shall now introduce an approach, due essentially t o Kulinsky (1963), which is valid very close to resonance. We can regard (68) as an inhomogeneous equation for 4, with k a e A as a known source term. The exact solution of the homogeneous equation is easily seen to be

177

RF CONFINEMENT AND ACCELERATION OF PLASMAS

where P and Q are constants. Consequently, we can construct the Green's function for (68) and write down the formal solution as

fQ)l

4 = i [2e x p ( - - i J " w -

exp (i

fQeAexp(i/I'wiQ)dt'

1 l w

f Q)

fQeA exp (-i J"'w f Q) dt']

+ P exp (i / I w

f

u> +

Q exp (-i

s"

a),

w

(74

and hence

+ exp(ijfw +

Q)l

-+neAexp(-ifw

f Q)

+ i ( w ~ ~ ) [ ~ e x p ( i ~ -w~ e~x Q p ( )- i J " w

+Q)].

If we insert this in (69), we obtain f

Q -= 3Ck

-_-

WC'

it1

+QeA exp (i

J"'.

fiQeA[Pexp(irw

or

&dt."[/I

t

= - -i

[

-

+QeA exp (i

e A ~ ~ P2 exp (i

x

J"

w

s"'

w

i-w )

k Q)

dt'

1'

in) - Q e x p ( - i J " w + n ) ] f 0 ) dt' -

1'1

Q exp (-i

J" + a)]. w

(75)

Equations (74) and (75) are integro-differential equations which, like the differential equations (68) and (69) from which they were derived, in principle determine the motion. We shall first show that where A is slowly varying on the time scale 2 s / w f 8, they again reproduce guiding center theory. I n this case, (74) gives

4 =

+-wne-4 + Q

+ Pexp(ifw

5

Q)

+Qexp(-iru

f Q),

(76)

and we obtain Eq. (70))apart from an oscillatory correction which vanishes on averaging over a time 21/w f Q. If required, this can be used t o calculate the high-frequency correction to the smooth motion in the x direction, analogous to (and reducing to) the oscillatory correction obtained

178

H. MOT2 A N D C. J. H. WATSON

in Section A in the case Q relativistic limit)

$ (;

my2 f

=

0. Likewise Eq. (75) gives (in the nonieAw -( P exp[i(w f Q ) t ]

- =

,

(w

k Q ) 2 2m

- Q exp[-i(w f Q ) t ] ) (77) which reproduces (72), apart from a small oscillatory correction. The importance of (74) and (75), however, is due to the fact that they simplify when w f Q + 0, since in this limit exp(iJtw i- Q ) becomes a slowly varying quantity compared with A (2). We then obtain approximately (we shall discuss the validity of the approximation below)

4 = Pexp(i/'w

+Q)+Qexp(-i/'w

and d -[xi-&lJu dt -

1

[

i- QeA dt'

ieAwC2 x P exp

(i

i-Q)

(78)

If: Q ) ] .

(79)

1'1

/'

w

k 0 ) - Q exp (-i

/'w

From (78) we derive

$ ($ + y ) A [P exp (i J' =

w

fQ)

+ Q exp (-i

1'

w

f a)]

and hence

== const

= GII.

(81)

In the limit w f Q = 0, this becomes exact and is Kulinsky's equation (18). With the help of Eqs. (67) and (74), the constants P , Q, and Ell can be interpreted in terms of the initial values of v and A.The most interesting feature of (81), however, is that when w f Q = 0 the motion in the z direction becomes conservative again; the quantity = (eZA2/2m) (eA/nz)(P &) acts as a potential function, and the qualitative nature of the motion can again be inferred from the potential profile and the initial parallel energy Ell. In particular, if the injection conditions are such that > 0 everywhere the particle will be repelled from regions of high

+

+

+

RF CONFINEMENT AND ACCELERATION OF PLASMAS

179

field strength or, if GIIis small enough, may be trapped in the neighborhood of a node of the wave. We now consider Eq. (79), which determines the variation in the total energy of the particle. In the nonrelativistic limit a t resonance, it can be written

x - xo= awe J.'A

dt'+

g[J.'A dt']'

where a = - i ( P - &)/nz. This shows that although the parallel motion becomes conservative a t exact resonance, the perpendicular motion does not. I n the case where E is spatially uniform, Eq. (82) reduces to X

- Xo = ev,O. Eot

+ (e2/2m)Eo2t2,

(83)

the well-known expression for the variation in the energy of a particle a t exact cyclotron resonance in a uniform electric field. Equation (82) is thus the appropriate generalization of (83) for nonuniform standing waves; taken in conjunction with Eq. (81) which determines z ( t ) , it shows th a t for positive J/ the z motion is such as to limit the (asymptotically) irreversible pickup of energy described by (83)) by removing the particle from regions of high field strength. Nevertheless, the energy acquired even during one transit of the potential well # can be quite large; in one particular case discussed by Kulinsky, for instance, X - Xo == 2(Bo/E,,,)2mko2. The limitations of this calculation for practical purposes would probably be the assumption that the wave remains plane over the whole (rapidly expanding) Larmor orbit, though if this condition remained satisfied, the relativistic shift of Q would eventually lead to values of w f Q sufficiently large for (81) t o be invalid. Finally, let us consider the very intractable problem of intermediate values of w f 0,when this is too small for the guiding center theory to apply, but too large for Kulinsky's approximation to be valid. This arises when A ( z ) varies on a time scale of order 27r/(w k Q ) . Since the z motion is conservative in both limits, it would be reasonable to expect a smooth transition between the two. However, the situation is in fact more complicated, as can be seen from the one case where Eq. (68) can be solved exactly: A(z) = Aoz = (m/e)oJz. With this choice (68) becomes

which is a linear fourth-order differential equation first obtained by Ulrich and discussed in more detail by Fessenden. Writing z = zo exp(ivt),

180

H . MOT2 AND C . J. H. WATSON

we obtain V4

and hence (with Y2

- [ U J 2 + (W e = w

+

f fi)2]V2

W(w

fl)wJ2 = 0

(85)

- 4fdJ20e]1/2).

(86)

& fl)

= &(WJ2

+ e2 k

[(bJJ2

+ e2)’

For (w +_ 3 ) = e >> wJ, this gives v2 = (w f Q)2 or w J ~ c ~ J / ( w Q ) , which is just what guiding center theory predicts. For e << O J , it gives u2 = W J or w(w i-Q ) . The first of these roots is the one predicted by Kulinsky theory. The second, which vanishes in the limit w k fl = 0, is rather puzzling, until one uses the initial conditions to determine the amplitudes of the various solutions. Fessenden’s general expression for these in the ~ we) gives limit W J >> e (and also U J << sin(we)’4

+ -1 sin uJt - -a c0s(we)~/2t WJ

WJ

where vo and a are constants specifying the initial conditions. I n the limit 0, the terms in (we)’l2t disappear and the motion is as described by Kulinsky’s theory. For small nonzero L, however, this result is only valid ; longer times, the terms in (we)’12, which are for a time of order ( ~ e ) - ~ ’ ~for neglected in the approximation underlying (78) and (79), become important. The most interesting case arises when e W J : Eq. (86) shows that for positive e the gap between e << WJ and e >> WJ is not bridged smoothly, since the two roots degenerate when ( U J ~ e2)2 = 4WJ2Ue, i.e., approximately when e = w J2/4Wor (40UJ2)ll3. For the corresponding critical values of V, say v,, the solution becomes e =

-

+

z = zO(1

+ at)[@exp(iv,t) + y exp(-iv,t)]

(88)

where a, @, y are constants. Between these two critical values of e, v is complex; i.e., in this narrow frequency range the amplitude of the slow (v, << w ) oscillations in the z direction grows with time, linearly or exponentially. The physical significance of this can be seen by comparing it with the result of the discussion of the Matthieu equation derived in Section A where it was shown that when O J , the frequency of oscillation of the particle in the potential well $, becomes comparable with some external driving frequency (in that case a), the particle suddenly “leaks out” of the well. In the present case, when the frequency in the well W J ( W / ~becomes ) ~ / ~ comparable with the driving frequency e, the same escape phenomenon is observed. In this case, however, the upper edge of

~

RF CONFINEMENT AND ACCELERATION OF PLASM.4S

181

the first region of instability is reached without violating any physical conditions. For negative E , one pair of values of v is always real and the other pair always pure imaginary. The significance of the imaginary pair can be seen by taking the limit W J << E ; it simply reflects the fact that for negative e the quasi potential I/. = +mwJ2z2(w/e)acts as a repulsive barrier. I n this case the transition to Kulinsky theory is smooth; it is interesting to note that the maximum repulsive effect of the barrier is achieved when E = ( ~ ~ w J ~ was ) ~may ’ ~ , be seen by differentiating v 2 with respect to e; for smaller values of a the effect of the barrier declines in a manner which can be inferred from Eq. (87) if we note that in this case ( u E ) is ~ ’a ~ n imaginary quantity. To investigate the perpendicular motion in t,his intermediate frequency range, Eq. (75) though formally correct is not particularly useful. It is better to return to (69) which for A a z gives

k

=

wx d -m - - (2 e

dt

+ wJzz)

an equation which can be integrated to yield

x = +m(w/e)(iZ- 2x2 - wJ2x2) + const.

(90)

To obtain x explicitly as a function of time, we need to insert the solution for z. The algebra is tedious and the result unenlightening except to confirm what is in any case clear from (90)-that for positive e, so long as v remains real, x oscillates in time with a frequency of order e, so th a t in the limit e >> W J , we recover (72) on averaging. Where v becomes complex however, x grows exponentially, a result which reflects the fact that for A a z the instability in the x motion drives the particle into regions of ever increasing field strength. The limit to this growth is set by the breakdown of the linear expansion of A @ ) . For negative e the particle energy grows exponentially for all values of e ; this reflects the fact that it monotonically picks up energy as it is repelled by the quasi-potential barrier in this case. Having thus a t some length dealt with the case of a uniform magnetic field, we shall now consider how the conclusion must be modified if it is nonuniform. We shall restrict ourselves to the one case discussed in the literature, in which a plane wave is orientated parallel to the symmetry axis of a nearly uniform magnetic field, with a weak spatial variation in the z direction; this problem has been considered by Consoli and Hall (1963a) and Pilia and Frenkel (1964a,b). I n order to be consistent with Maxwell’s equations we must allow some variation of B in the z,yplane; a simple representation which is appropriate in the neighborhood of the

182

H. MOTZ AND C. J. H. WATSON

symmetry axis is

B

Bo], (91) where Bo = Bo(z)and Bo’ = dBo/dz. This satisfies V B = 0 identically, and, if Bo’ is small, very nearly satisfies V X B = 0 near the symmetry =

[( -~/2)Bo’, (-y/2)Bo’,

-

axis as well. It can conveniently be derived from the vector potential = Bo(x$ - y8)/2. Inserting this and A1(z, t) = A ( z ) & ( w t ) into the Hamiltonian, we obtain the equations of motion

A0

+ 5 2 +~ -h2 y,

j;. =

-eA,(z, t )

J =

-eA,(x, t ) - 52x - - 2,

6 2

where Q(z) = eBo(z)/mand Q’ = d52/dz. Writing xI = these into a form very similar to (63) and (64) :

lh

X,-*,Xsz---(x,XSL) 2Q

2 =

( 5 , y),

(92)

=

eA’ 1 Q’ - (t ’ a) - - -- (%I x x,) m 252

we can put

.p.

(93)

where P = Q2. It is not difficult to recover guiding center theory from these equations, using the methods of Section D. In outline, one proceeds by solving (92) in the WKB approximation (which is simplified by the fact that when Eq. (92) is projected onto the complex vectors sY, it splits into two independent linear equations for the quantities x v ) . The solution as before consists of a particular integral with frequency w and a complementary function of the form

where a y and pv are constants of integration. On inserting this solution into (93) and isolating the nonperiodic terms, we obtain for the case of plane polarization discussed by Consoli and Hall

_ - _a+_ az

(94’)

as required by Eq. (51). This result is, however, a t variance with certain statements made by Consoli and Hall, and some comment should be made a t this point on the first three sections of their paper. There is first a trivial discrepancy between their Eqs. (2) and (4) ; according to (2) the denominator w2 - 522

RF CONFINEMENT A N D ACCELERATION OF PLASMAS

183

does not appear under the gradient sign, whereas in (4)it (correctly) does. In (4) it is correctly stated that the total force acting is the sum of two terms, as in our Eq. (94); but in the subsequent discussion it is incorrectly stated that 1.1 = W J B o where W , depends upon the electromagnetic field. Since they do not give any derivation of their equations, i t is not clear how this error arose; it makes the remaining discussion of their Section 3 incorrect. However, this is unimportant, since it is clear from (94) that, as one approaches resonance, initially a t least the term a$/ax dominates over 1.1 d i l / a x . Indeed, if we expand the force exerted b y the circularly polarized component of the plane wave which is in phase with the particle

it is clear that as we approach resonance the second term dominates over the first unless the gradient of the magnetic field is very much less than the gradient of the electromagnetic field. A simple physical interpretation can be given t o this, if we remember that this term arises when we substitute the particular integral of Eq. (92) into the second term of Eq. (93). Thus it expresses the interaction of an additional dipole moment Nrf induced by the electromagnetic field with the nonuniformity in the magnetic field. As one approaches resonance, this moment prf increases as [l/(w and dominates over the force exerted by the gradient of A z . It is therefore clear that any attempt to discuss the nonresonance motion in a nonuniform magnetic field which is based on the theory of nearresonance motion in a uniform magnetic field is doomed to failure: any nonuriiformity in the magnetic field, almost however small, has a dominating influence on the x motion in this limit. I n particular, it is quite incorrect t o drop the second term in (93) and derive an equation similar to (68). The procedure proposed b y Consoli and Hall to discuss motion through the resonance region is to use the exact theory of motion in a uniform magnetic field t o determine the value of o - 2 ! a t which guiding center theory becomes inapplicable, and then suppose that the particle “tunnels through” the region of space within which o - Q is less than this value, with no change in its parameters. This latter supposition is made most implausible by the analysis given above of the rapid and irreversible transfer of energy from the field to the perpendicular motion of the particle near resonance, not to mention the instability in the z motion which occurs close to resonance. However, even the assumption that the limiting values of w - f2 can be taken over from uniform theory seems rather doubtful because of the quadratic denominator multiplying an/az. I n any case, they are interested in the case where the resonance occurs a t a local maxi-

184

H . MOTZ AND C. J. H. WATSON

mum of the electric field (so that dA2/dz = 0 ) , whereas they take the results of the exact analysis given above for a field A a z to determine the critical value of w - 3. A different and rather more successful attempt to assess the effect of magnetic field nonuniformity has been made by Pilia and Frenkel (1964a,b). They start from equations very similar to (92) and (93), but with A spatially uniform. They then solve (92) by the WKB method, taking into account the variation of with z. I n order to carry out the relevant integrals, however, it is necessary to know 3 as a function of time and hence to know z(t). Since they have not yet solved (and cannot yet solve) (93), they assume for the purposes of (92) that the z motion is that which would occur if the rf field were absen-i.e., adiabatic motion in the nonuniform magnetic field. Having obtained xL(t) in this way, they then obtain the correction in the z motion from (93). This approach is clearly only applicable if Avll, the change in V I I during the passage through resonance, is small compared with 811. However, it is possible to calculate Avll and so to check the validity of this assumption. The opposite case, where A q is large, has been discussed by Canobbio (1966). Pilia and Frankel considered a particular magnetic field profile Q(z) = w(l - z/L). This has the advantage that the adiabatic equation can be integrated exactly to give of motion v11 = (& 3(t) = E / p

- p(wl/2L)2,

(96)

where t has been set equal to zero a t the turning point of the adiabatic motion. However it is clear that most of their argument is valid for any mirror field configuration 3 ( z ) and only uses the fact that with the above choice of time origin 3( - t ) = Q(t). We consider an electric field E = Eo(cos at, - sin wt, 0). If we project all vectors onto the complex vectors 7 , of Section C, we obtain

x, + iv3x,

+ +ivhx, 2

Eo exp(ivwt), = ivQ’x-,x,. =

v = fl,

Factoring the operator of (97) correct to terms of order h

(97) (98)

RF CONFINEMENT AND ACCELERATION OF PLASMAS

185

and integrating, we obtain (correct to terms of order fi)

where we have chosen an arbitrary initial time -T and have identified the constants of integration of (99) in terms of the magnetic moment p of the particle in the absence of the electric field; $0 is the phase of the particle at t = 0. From (101) we can construct xi’12

= X”XL

+ Eo21 j f T exp (i lot’ Q - u) dt‘ lz}-

(102)

The values of the integrals in this expression depend critically on whether

t is such that the particle has passed through the resonance region. If it has, they can be evaluated approximately by the method of steepest descents. A certain mathematical simplification is obtained if one takes t = +T (i.e., such that the particle has undergone one mirror reflection and two transits through the resonance region). We then have, using the fact that $;Q is an odd function of t

(XL2 - 2pQ)

A, =

{

IT

-T

2 2p’/2Q’/2Eocos 40

+ Eo2 [

/-TT

(

IfT (lo”1’}

cos /ot’ (0

cos

(Q

d t ” ) dt’

W)

9

d t ” ) dt’

(103)

an equation which is strikingly reminiscent of Eq. (81). Likewise, using (100) and (101) in (98), we have

186

H. MOTZ AND C. J. H. WATSON

and on integrating from -T to T by parts, All

E

+

( i 2 2110)

If we again use the theory of integration by steepest descents (i,e., t h a t the major contribution to the integral comes from the values of t' where the exponent is varying most slowly, for which Q = w ) , we obtain approximately

[

All = 2 2 w 1 / 2 p 1 / 2 Ecos C

= A,

[-TT

cos

lot'

(Q -

w)

dt'

- A,

where we have written the first term as A, in order to emphasize that i t differs from A, only in that Q T , the value of s2 a t t = T , is replaced by the resonance value w. If we now take T to be sufficiently small for QT = w but sufficiently large for the particle to have left the resonance region, All << AL. Thus in a first approximation the effect of a passage through the resonance region is to cause a finite change in the perpendicular energy of the particle and a negligible change in the parallel energy. This is what happens from the point of view of an observer situated as close as possible to the resonance. An observer at some point z1 displaced by a finite amount from the resonance, however, would reach a different conclusion, since some of the additional perpendicular energy would then have been converted to parallel energy. To compute this, we observe that the motion before and after encountering resonance is determined by = pa,

21,2

+

-

+ = E. E + w 6 p . Thus if the particle starts

21/12

VL2

At resonance, p -1.1 6p and E from a point with given Q # w , passes through the resonance, and returns to a point with the same value of Q, we have AvL2 = Q 6 p , Av112 = 6 E - AvL2 = ( w - Q) 6 p ,

and

AvL2/Avl12 = Q / ( w - D),

RF C O N F I N E M E N T A N D A C C E L E R A T I O N OF P L A S M A S

1S7

a result which reduces to Pilia and Frenkel’s Eq. (20) under the conditions where the latter is valid. The numerical value of &L can be calculated from (103) in the particular case where Q = w ( l - z / L ) since theintegral is then expressible in terms of the Airy function. It will be observed that for sufficiently small Eo,the term proportional to cos t$o dominates; under these circumstances the passage through the resonance region can be regarded as a weak collision, as a result of which the perpendicular energy of the particle changes by an amount proportional t o cos 40 where for ~ be regarded as a random variable. I n consestatistical purposes c $ can quence, a diffusion process in velocity space takes place, which can be described by a Fokker-Planck equation. For the details, the reader is referred to Pilia and Frenkel’s papers (1964a,b).

F . Particle Motion in a Traveling Electromagnetic W a v e We shall consider first the motion in the absence of a static magnetic field. It proves simpler to carry out the analysis from a relativist,ic standpoint and then take the nonrelativistic limit where appropriate, since in certain cases the distinction between a Lorentz transformation and a Galilean transformation turns out to be important. For simplicity we shall consider the case where there is a single direction of propagation, which we shall take as the z direction, though the generalization of some of the results to (for example) spherical traveling waves would not be difficult. The motion of a particle of charge e in such a field can be derived from the relativistic Hamiltonian

x

=

+

c [ ( p - eA(z,, t ) ) 2 mo2c2]1/2

where = k2 - wt and w / k = v+ is the phase velocity of the wave. One very useful consequence of the canonical equations of motion in the case of traveling waves, which has no analog in the case of standing waves, is

which can be integrated to give X - Xo = v+(p, - p z o ) .

(107)

This result, that the energy and momentum of a particle in the direction of propagation of the wave vary in proportion, is the classical equivalent of the quantum mechanical statement that in any photon-electron interaction, both energy and momentum must be conserved. The actual content of (107) is not as simple as its form might, suggest, since X is a function of p , as well as of the other dynamical variables. Writing it out

188

H. MOT2 AND

C. J. H. WATSON

explicitly, we have

p , ~-

kxo

k2C2 -eA,) w2

p,

+ w2 [(p,z - eA,)z + e2Az2+ mo2c2]k2C2

(p.0

-

%)'. (108)

This equation can be solved exactly for p , ; it is more enlightening, however, to consider two limits. First, if v g fi c the first term is negligible, and we obtain p , - pzo =

+

C ~ P ,- e A d 2 - ( P ~ O eAld2 e 2 A 22 [ x 0 e&c - cp,01

+

e2AY(109)

a result obtained in the case of plane waves by Landau and Lifschitz (1951), using Hamilton-Jacobi theory, and by Brown and Kibble (1965), using covariant quantum electrodynamics. It will be observed that unless eA, or pzO are of order moc, Eq. (109) shows that the variation of p , is of order pL2/m0c:i.e., a wave whose phase velocity is close to c has very little effect on the z motion of a nonrelativistic particle. This confirms the essential correctness of the result of guiding center theory in this casethe effect described by (109) is one order in v/c too small to be picked up in that approximation-and gives confidence in the correctness of its account of the motion in the z,y plane for this case. If, on the other hand v4 << c, we can neglect 1 compared with k 2 c 2 / a 2 and obtain approximately

+m(v, - v 4 ) 2

+ (pL- eAJ2/2m

- v4eA,

= & = const.

(110)

It may be noted that the only approximation involved in deriving this equation is that of taking the nonrelativistic limit, as may be seen from the fact that it is easily derived from the nonrelativistic Hamiltonian. Using the canonical equations to reexpress p, in terms of v,, and making a Galilean transformation t o the frame in which the wave is at rest vj = v, - v4, 2' = z - vgt, we obtain +m(v,'2

+ vL2) - v4eA, = E'

= const.

(Naturally, as v+ + c it becomes impermissible to make a Galilean transformation, but in this limit we can always use (108).) This equation shows that the energy of the particle is the same as it would possess in an electrostatic potential 4 = -v4A, set up in the rest frame of the wave. It does not of course follow that the particle motion is the same, since the magnetic field of the wave affects the motion'but not the energy, and the problem of determining the motion does notappear to have been solved in the general case. I n two special cases, however, the problem is soluble.

R F CONFINEMENT AND ACCELERATION OF PLASMAS

189

If the field configuration is such that the magnetic field remains small, in spite of the fact that v+ << c, it becomes permissible to calculate the motion as well as the energy from the formal analogy with the electrostatic problem. A situation of this kind is mentioned in Section 6. On the other hand, if we consider the case of a plane wave, pL is a constant of the motion, since X is independent of x,, and we can then write (110) in the form

+m(vz - v+)2

+ $4')=

E,

(111)

and the motion can then be determined by quadrature. The 1c. appearing in (111) should not be mistaken for the quasi potential obtained with standing waves, in spite of its similarity of form. This confusion is particularly tempting in the case of pL,A, = 0, since the profiles of y5 in (111) and that of the quasi-potential for the equivalent standing wave are then formally identical; the standing wave quasi-potential, however, creates a barrier in the laboratory frame, whereas the traveling wave J. creates the same barrier in the wave frame. Thus, a standing wave tends to trap particles which have a sufficiently low energy in the laboratory frame, whereas a traveling wave traps particles which have a low energy in the waveframe and hence an energy of order +mug2in the laboratory frame. This, incidentally, is the reason why a traveling wave with a phase velocity v+ c has little effect on nonrelativistic particles in the x direction, as guiding center theory predicts. Conversely, (110) is quite compatible with guiding center theory, since a wave with phase velocity v+ << c is necessarily nonuniform on a length scale much shorter than that of a vacuum wave of the same frequency, and hence guiding center theory is inapplicable. We now turn to the case where a static magnetic field is superimposed on the traveling wave. This problem does not appear to have been analyzed in general form; the case in which the static magnetic field is uniform and parallel to the direction of propagation of the wave can be tackled by an extension of the analysis given above. We first note that if we represent the magnetic field by the vector potential A0 = Box$ or -Boy$ the Hamiltonian still only depends upon z and t through the combination wt - kz, and hence Eq. (106) still holds and can be integrated as in (107). Consequently (108) is modified only in that A, must be replaced by A, A,; with this modification (109) and (110) still hold and the effect of a high phase velocity wave on the z motion of a nonrelativistic particle is still small. Now, however, even in the case of a plane wave, (p, - eA,')z is a function of the dynamic variables, and the motion in the z direction for small v+ is not in any way determined by (110). I n any case, as we approach the cyclotron resonance, the perpendicular motion becomes of more interest than the z motion. Provided that we are sufficiently N

+

190

H. MOT2 AND C. J. H. WATSON

-

far from resonance, and if v+ c, we can use guiding center theory. For low phase velocity, however, no theory exists in the general case; for the particular case of a circularly polarized traveling wave, an exact theory, which is in fact valid for all phase velocities and all relations between w and Q, has recently been given and this will be described in the next section. Finally, some comment should be made about the radiation correction to the theory discussed in this section. Conventionally, this is discussed in a quantum electrodynamic framework; since, however, the principal eff ect-Compton scattering-is one which possesses a classical limit (Thompson scattering), it should be possible to derive it by classical methods. This can be done as follows. The assumption upon which the above analysis is based is that the radiation field A is a given external field. Strictly speaking, it should be the sum of the external field and the field generated by the motions of the particle. Since this latter field is small, it can be obtained approximately by computing the radiation of the particle as a result of its motions under the influence of the external field only. This scattered radiation is proportional to the amplitude of the external field, and hence it is possible t o derive a scattering cross section uT which relates the two quantities. For a spatially uniform field, the analysis was first given by Thompson; for the nonuniform wave discussed above, the problem has been treated by Vachaspati (1962). This scatbering also affects the motion of the particle; the simplest procedure for calculating this is that given by Landau and Lifschitz, who show that it experiences an additional force f = uTWno,where W is the average energy density, and nois a unit vector in the direction of propagation. This force is smaller than that due to quadratic terms in the unperturbed Hamiltonian by a factor of wr./c, where re = e2/mc2is the classical radius of the electron.

G . The Cyclotron Resonance. (ii) Traveling Waues We consider a circularly polarized traveling wave derivable from the vector potential

AL(z, t )

=

A(cos(wt - Icz), +sin(wt - Icz), 0) = Aa(d

-

kz)

propagating parallel to a uniform magnetic field Bo.We shall first give a simple derivation of a nonlinear differential equation for the energy of the particle, which was obtained in a somewhat involved manner by Roberts and Buchsbaum (1964). Our analysis parallels rather closely that given for standing waves in Section E. Indeed, Eqs. (55)-(61) are unaltered, except in the significance of A,. Equation (61), however, as we showed in the previous section leads a t once to Eqs. (106) and (107).

191

R F CONFINEMENT AND ACCELERATION O F PLASMAS

Our basic equations are therefore

If we now use the properties of a

=

($)2

a.

a.B=O,

a2=1,

w2,

a

xa

=

z

-d2a _at2

aa

- ki

w

___-

at'

T --, f2 d a

aa X P = +wQa,

and

at

-da,

at

(115)

in conjunction with ( l l a ) , we obtain Cl

& (p1.a) = d - (p, dt

lzi f a

w -

- g)

= -w[(w

da

(116)

PL Z f

- kz)eA

+

(W

- k i -t O)pL.a].

(117)

Hence by (114) (W

- k i f Q ) x=~-eAwc2

dtd

(p,

a)

(118)

and d

x $ dt . ? - ( ~= k) =

X&e3eAwc2[(w- 1ci)eA

+ (w - kd + Q)pL a] *

(119)

-

We now solve (118) and use the solution to eliminate pI a from (119). To do this, me note that by (113) 0

- lci f !l= w

-

+ ~,IXO x x

kc2pz ~

~

I&

f Q.)

&+

(1

-

y) = w

w

- d'1s20XoX

dlQ

+ dzw.

- dzwX2/2,

(120)

192

H. MOTZ AND C. J. H. WATSON

where have

o

is a constant of integration; on substituting this into (119), we

X k - (Xk)= - w 2 k [ ( r 1 r 2 f r3)X03+ (r12 d dt

+ $rldzXo(X - X d 2 + d 2 ( X - X 0 > ~ / 2 l

where rl

+ rzdz)Xo2(X - XO)

+ dzw)/w,

= (dlQo

r2

= -(a

+ e2A2c2/Xo2),

and

(122) Qo e2A2c2

r3

-

=w Xo2

Integrating (122) once more, we obtain

+ + + rzd2)Xo2(X - XO)' + rldzXo(X - Xd3 + td2YX - X O ) ~ I

x2k2 w2[2(r1r2 5

~ ~ ) x O ~ ( X

xO)

(r12

= const

E

r42w2X04. (123)

This equation can be written in the form k

2

+ V(X) = 0,

and is the basic result of Roberts and Buchsbaum's paper. It will be observed that as a consequence of Eq. (113), it is equivalent to an equation for p,, and this in turn determines vL2, since X = mc2 = moc2/(1 - uz/c2)l I 2 . We may note first that when dz # 0, V ( X )3 00 as X 3 k 00 (if dz = 0, V -+ wr12, and the limit rl + 0 has to be taken rather carefully). Thus V can be regarded as a potential well within which X oscillates as a function of time, starting a t t = 0 with X = Xoand k o= r4. For small t, the term in V which is linear in X - Xo dominates; but as X increases, the quadratic and higher terms become important and finally limit the energy a t a value determined by V(X) = 0. This relation can conveniently be written in terms of U = (X - Xo)/Xoas

It will be seen that an important parameter is rddz =

(W

- kio k Q O ) / U . 1 - k2C2/W2

The numerator is the initial displacement from cyclotron resonance (the term kio represents the Doppler shift in the frequency due to its initial motion), the denominator measures the displacement of the phase velocity of the wave from c. We must therefore distinguish two cases: if we consider near-vacuum waves, so that rl/dz is large compared with U , we obtain a quadratic equation with solution

U,,,

= {

-(rlr2

+

f r3) f [(raz & r 3 ) 2 r12r42]1/2)/r12. (125)

R F CONFINEMENT AND ACCELERATION OF PLASMAS

193

The magnitJudeof this can be estimated as % ‘ 3 / r ’ l 2 and it grows resonantly as r1 -+ 0. For d2 = 0 (i.e., a pure vacuum wave), Eg. (125) is exact and the maximum energy attained near resonance is given by

-

(126)

For dz small but finite, (126) remains valid only until U,,, rl/d2. Thus it correctly est,imates the maximum energy provided that rI3 >> d27-3, i.e., (w - lczo - Q0)3

>> Q 0 3 Boi (1 E’2

-

>.

k2C2

The corresponding time period, for arbitrary rl, can be found from the fact that in this limit Eq. (123) is formally equivalent to the equation for the radial motion of a Keplerian planet. hlaking thc appropriate correspondences, we obtain the time period:

-

Hence if we are well away from resonance (rl I), provided that the field strengths and injection energies are nonrelativistic (7-2, r 3 << l ) , r 1/w and the particle energy oscillates rapidly. I n this region, guiding center theory is presumably applicable. As resonance is approached, however, T begins to grow and for very small r l goes to infinity as (~/2)r3/lr1)~w. In the limit r1 = 0, the time dependence is nonperiodic: its form is given in Roberts and Buchsbaum as (asymptotically)

-

X

-

Xo(9r3/2)”2(~t)2’3.

(128)

This should be compared with the expression obtained from linear theory in this limit: X = XO ev,o Eat e2E2t2/2m (129)

+

+

(a result which can easily be derived from (123) by neglecting quadratic and higher terms in V ) . This shows that for vacuum waves initially in exact resonance, the energy of the particle continues to grow indefinitely though not as fast for large time as linear theory predicts. This remarkable result is due to the fact (first pointed out by Davydovskii), that, for vacuum wavcs, the v )( B,f term exactly compensates for the relativistic variation in a, and hence the particle remains in resonance for all time. This can be seen from Eq. (120): for d 2 = 0 if initially d l = 0, U-kif.0 =O

for all time. I n practice, of course, field nonuniformities or the radiative correction would eventually spoil this synchrony.

194

H . MOT2 A N D C. J. H. WATSON

We finally turn to waves which are riot nearly vacuum waves, i.e., for which dt is not too small. In this case wc have to work with a fourth-order V(X) and only the case of exact initial resonance (rl = 0) has been analyzed. If the field strength arid injection energy are not too large, there is a useful approximate solution, valid under the conditions [Roberts and Buchsbaum (1964)]

We then have

x,,, - xo= 2x05 W

[E/(1

-

7)

(130)

and a corresponding t.ime period for oscillations

It is interesting that these expressions agree apart from numerical factors with the results given in a paper by Consoli and Mourier (1963) for the case of a uniform electric field, which is based upon equations which are unintelligible to the present authors. In that paper they contrast the expression obtained with the results of certain experiments, in which were obtained higher maximum energies, which they attributed to collisions or magnetic field nonuniformities. It seems worth pointing out that a more natural explanation would be electric field nonuniformities; as (130) shows, the case k = 0 corresponds to a minimum in the maximum energy reached. 2. THE THEORY OF RADIO-FREQUENCY CONFINEMENT OF PLASMA

A . Derivation of the Self-Consistent Field Equations In Section 1 we assumed that the reaction of the motions of the charged particles upon the fields themselves could be ignored. Such an assumption is reasonable if one is considering the motion of essentially single particles; if these are replaced by a plasma of significant density, however, the reaction is no longer negligible and we have to consider the fields not as given functions of space and time but as self-consistent fields, obtained from Maxwell's equations with source terms determined by the plasma motions. As we shall see, the plasma motions have two effects; they give the space in which the rf wave is set up a certain quasi-dielectric property

R F CONFINEMENT AND ACCELERATION O F PLASMAS

195

and they generate new electric fields as a result of charge separation. T h a t such a charge separation must occur can be seen from the form of the rf quasi potential; since this is smaller for ions than for electrons by a factor m,/mi, the ions are free to penetrate into regions of space which are inaccessible to electrons of the same energy. Such a freedom would lead to a large charge separation, but for the development of electrostatic fields near the edge of the electron distribution which serve to retain the ions. Thus a sheath develops, similar to, but with opposite polarity from, the sheath around an unconfined plasma. It will be observed that the electric field required for this purpose is irrotational, and it has a time dependence which is on a scale much longer than 27r/o. Thus, provided th a t its spatial nonuniformity is on the same length scale as the quasi-potential, its effect on the particle motions can be represented by adding an electrostatic potential & e+ to the quasi potential, when e is the magnitude of the charge of the particle and the f sign determines its sign. Logically, there is no reason why nonlinear particle-wave interaction processes should not generate other fields with different time dependences again; however, we shall see that a wide range of plasma-radiation equilibria exist in which there is only an rf electric field E and a stationary electrostatic field E, = -V+, and in this review we shall only consider such equilibria. I n the absence of a stationary magnetic field, the equation of motion of the guiding center for a particle of charge + e and mass m, is therefore integrable t o give

+mkR2 where ,$,

=

+ +*

e+ = &, = const,

e2E2/2mio2. Remembering that r

e

=

(fe/rrz,)

/ L

1”E(R, t”)

=

R

+ p and

dt” dt’,

we can reexpress &, in terms of the coordinates (r, V) of the particle itself in phase space:

[

+mrt v f (elm,)

/’ E dt’] + +i

[r T (elmi)

1‘/”’ E dt” dt’]

f e4

= 8,.

When written in this form &, is a constant of the motion apart from terms of order v 2 / c 2 . Indeed, correct to this order, we can ignore the quantity e/mJtJt‘Ewhich occurs in the argument of +* and 4. For a collisionless plasma, the distribution functions f, for the ions and electrons respectively satisfy the Vlasov equation and hence can be chosen t o be any arbitrary function of the constants of the motion. By the same token, functions of quantities such as E,, which are constants of the motion in a certain approximation, are solutions of the Vlasov equa-

196

H. MOT2 AND C. J. H. WATSON

tion in the same approximation. I n what follows, we shall assume that the j4are functions of E, only. This excludes certain (e.g., rotating) equilibria but will be sufficiently general for our purposes. I n particular, we can consider systems in thermodynamic equilibrium by selecting f* to be Maxwellian functions of E,. Although we shall not everywhere take them to be Rlaxwellians-as we shall see, this is under some circumstances unduly restrictive-for most purposes this is as plausible a choice as any; it has the advantage that the resulting equations then coincide with the equations obtained from a magneto-hydrodynamic description of the plasma, and indeed some of the equations discussed below were first obtained in this way. Given the particle distribution functions j4, we can now construct the charge and current densities p and j from P = e.r(f+ - s-1 d3v j = eJv(f+ - f-) d3v.

We insert these in Maxwell's, equations which can for our purposes be represented by 1 a aE v X v X E = -- -++/E, C 2 a t (at and

)

-

where we have continued to use our convention that E is that part of the electric field which has the t,ime dependence exp(iwt). Using this property and normalizing our distributions frtso that the particle density is unity at points where # = + = 0, we obtain from (1) and (2)

where wii = no+e2jtom+, w:e = no-e2/eOm-, and no*are the ion and electron densities at points where the potentials vanish. It is clear from (4)that p does not have a high-frequency component [exp(iot)] so in the present approximation we must have V E = 0. Thus if we write (4) in the dimensionless form

-

- V 2 ( e $ / T ) = Ick+J.f+ - kgJf--,

(5)

where T is the temperature associated with the distributionsf, and ICD* are the reciprocal Debye-lengths for the two species, and use our assumption that the variation of + is on the same length scale as that of the

RF CONFINEMENT AND ACCELERATION OF PLASMAS

197

quasi-potential ( - c / o ) , it is seen th at the ratio of the left-hand side to either of the two right-hand terms is of order (o2/wp2)vT2/c2,where VT is the thermal velocity associated with T. Thus except for very low density plasmas we can obtain a solution to (5), which is correct to the same order in v/c as the kinetic theory upon which it is based, by taking V2+ = 0. This is known as the approximation of (‘quasi-neutrality” ; it implies that the space charge is everywhere very small, though the electrostatic potential 4 need not also be small. Indeed we can now obtain the magnitude of C#I from Eq. ( 5 ) , which gives lJ+(+mv2

+ $+ + e+) == Jf-($mvz + $- - eC#I>.

(6)

Since this must hold everywhere, if J+and f- are the same function of their (different) arguments we have 2e4 = -9 ;, a similar expression can be obtained if j + and f- are different but specified functions. For simplicity we will restrict the discussion to the case where they are the same function. We can now rewrite (3) as a nonlinear vector differential equation for E; neglecting terms of order m,/mi compared with unity, we obtain

+-

The vector E appearing in this equation still has the time dependence exp(iwt) ; however, this time dependence can be factored out, and the rms amplitude of the electric field vector is then seen to satisfy the same equation. I n what follows, E will be taken to represent this rms amplitude. Equation (7) is the basic self-consistent field equation upon which most of the results of the present section will be based; some comment should be made about the conditions under which it is valid. Since e is now a function of E 2 only, we obtain a rather simple constraint upon E by performing the operation V upon (7), which gives ae/dE2(E V)E2= 0; 8

(8)

hence if there is any plasma present a t all (so that ae/aE2 # 0 ) , E * V E z = 0. This implies th at E 2 is constant along electric field lines and hence that the plasma density and pressure are constant along these lines. Now if the field configuration is such that plasma is confined in all three dimensions, there must exist closed nested surfaces of constant plasma density and it follows that these surfaces are electric surfaces (i.e., envelopes of electric field lines). Hence by a theorem of Poincar6, the surfaces must have toroidal topology. (The analogous application of the theorem to magrietohydrodynamic equilibria with scalar pressure is well known.) This result is in apparent contradiction with the result mentioned in the previous section-that one can excite vacuum modes for which the

19s

H. MOTZ A N D C. J. H. WATSON

quasi-potential has an absolute minimum a t some point in space, about which the surfaces of constant J. have the topology of spheres, which can confine individual particles in stable equilibrium. There is no real discrepancy here: for very low density plasmas ( w p 2 / w 2 x v T 2 / c 2 )the approximation of quasi-neutrality breaks down. Consequently e is not a function of E 2only, so E 2 need not be constant along field lines and surfaces of constant E 2 need not coincide with surfaces of constant plasma density. A more interesting, and much more difficult, question is whether relatively high density (wp w ) confined plasmas can only have surfaces of toroidal topology. The argument just given shows that this is necessary if p has no high-frequency component and if the nonuniformity of is everywhere on the length scale c / w ; if, however, there were a small high-frequency correction t o p , due to terms of order v2/c2, or if there were regions where deviations from quasi-neutrality existed on a much smaller length scale, so that the &* were no longer constants of the motion, E would no longer necessarily vanish, and confined plasmas with simpler surface topology might be possible. This question does not yet appear to have received detailed consideration. Fortunately, for most of the configuration which we shall discuss in detail E V E 2 = 0, so this difficulty does not arise. Before considering these configurations, however, one further general approach should be mentioned.

-

B. The Energy-Momentum Tensor Approach The problem of finding even a first integral of Eq. (7) is by no means trivial, and it is therefore useful to have a different approach which, though formally equivalent to the self-consistent field theory, gives a rather different perspective. As is well known, it is possible to construct for any system of particles and fields a covariant tensor of the second rank, T N Vknown , as the energy-momentum tensor, such that the equations

a

- T,, =

ax"

0,

(9)

are equivalent t o Maxwell's equations and the equations of motion for that system of particles and fields. The form of T,, for a plasma is most readily obtained by summing the first velocity moments of the Vlasov equations for the ion and electron distribution functions f*, which gives

RF CONFINEMENT AND ACCELERATION OF PLASMAS

199

On integrating the last term by parts and usiiig the relationship [proved in, for cxample, Landau and Lifschitz (1951)]

where T:/Y) is the energy-momentum tensor for the fields alone, we obtain for the relevant components:

Equation (9) is a set of four scalar equations with p = 1, 2, 3, 4. The 4 equation expresses conservation of energy and is not particularly interesting. The other three components express conservation of the total linear momentum of the plasma plus radiation; if it is obvious from the symmetry of the problem which one direction is physically significant, the conservation of momentum in that direction can have useful consequences. For example, if we consider a plane electromagnetic wave, SO that all quantities are functions of the x coordinates only, the z component of Eq. (9) gives p =

This equation is exact if the fk are; if, however, we take the approximate expressions obtained for f + arid f- above, and for simplicity assume them to be thc same function of the integrals of motion &, (so that[* = nof(&*)), we obtain, on averaging (10) over a time 2n/o, as

[&(t082 +

pon2)

+ 2noi,i-

S I jL2d3v

=

0,

(W

which can be integrated at once to give B(eoR2

+

poP)

+p

=

const,

(11)

where p is the local plasma pressure. We shall consider the full significance of this pressure-balance equation shortly ; roughly speaking, however, it

200

H. MOTZ AND C. J. H. WATSON

states that in a confinement situation the radiation pressure outside must equal the plasma pressure inside. The energy-momentum tensor approach also proves useful in the discussion of infinite cylindrically symmetric equilibria (see below). It also sheds some qualitative light on the effect of electric fields which have a component in the direction of v E z . TO see this we note that if in some region one can set up a local Cartesian coordinate system such that the fields and the plasma pressure in t h a t neighborhood depend only on one coordinate xi,the time averaged version of Eq. (9) reads

a p o R 2 ) / 2 - eo8i2 - p0lfi2 axi [(d2

+

+ p ( E ) + n(E)m;i*] = 0

(12)

where pi is the component of the oscillatory part of the particle velocity in the i direction. Thus whereas the transverse components of E and H give a positive contribution to the radiation pressure, the longitudinal components Ei and Hi give a negative contribution, and in addition the high-frequency particle motions modify the expression for the plasma pressure. Thus confinement must depend upon pi, Ei, and Hi remaining suitably bounded. Having summarized the main conclusions that one can draw about rf confinement in general terms, we now consider a number of configurations with particular geometry.

C . One-Dimensional Equilibria For plane waves in the x direction, Eq. (7) takes the form d2E/dz2

+ (w2/c2)rE = 0

(13)

where E = (E,, E,, 0 ) . I n general, this is a pair of coupled equations for the components E,, E,,. However, if we take the case of plane or circularly polarized waves, so that E(z, t ) = E(z)e(wt)where e is a periodic function of w t such that 2 = 1, Eq. (13) reduces to an ordinary differential equation for the amplitude E(z):

+

d2E/dz2 (w2/c2)eE= 0.

(14)

If one further chooses f to be Maxwellian, this can be written as d2E ~

dz2

+ 2 E = +E exp( -e2E2/4mw2T), w2

w 2

C

an equation which has been derived by many authors and has been the subject of considerable controversy in the literature. In what follows, however, we shall not immediately specify f. To solve (14), we multiply

RF CONFINEMENT AND ACCELERATION OF PLASMAS

by d E / d z arid rearrange, using the fact t,hat d E / d z

=

20 1

-wB(x), obtaining

or, integrating by parts over v,,

which is the equation that we previously obtained by the energy-momentum tensor approach. Equation (15) is a first-order differential equation for E ( z ) , so to complete the solution we need to perform one further quadrature and then insert the electric field distribution obtained into the expression for the plasma density distribution :

+

n(z) = noJf(?j1)2u2e2E2/4nzw2) d3v,

(16)

n(z) = noexp( -e2E2/41n02T)

(17)

i.e.,

for a Maxwellian. It is at once clear that the Maxwellian distribution has the property that no finite field is sufficient to confine it. The physical reason for this is that R Maxwelliari includes particles of arbitrarily high energy, which can pass over the barrier of any rf quasi-potential, however large. Under practical conditions, such particles may or may not be present, depending on the relationship of the injection time and collision time t o the time of observation. If,for example, the conditions are such that all these particles are lost before the establishment of the observed equilibrium and are not regenerated by collisions, the distribution function will be truncated a t an energy Em,, corresponding to the maximum height of the pseudopotential barrier. Thus we need to know the solutions of (15) for both typical and mathematically Rlaxwellian and truncated distributions. TWO convenient, truncated distributions are the monoenergetic and Fermi distributions f ( E ) = 6(& - E o ) and j ( 6 ) = [l - St(& - E o ) ] , where 6 is the Dirac 6 function and St the Heaviside unit step function, and E0 is a constant which may but need not coincide with.,,&, On performing the integration 1dJv,we find both these distributions give a density profile of the form n = no(l - E2/E02))Ol for

E2

Eo2

and n = 0 for E2 > E02, (18)

where EO is defined by e2E02/4mw2T= Eo, and a = ?j and Q for the monochromatic and Fermi distributions, respectively.

202

H. MOT2 AND C. J. H. WATSON

It is convenient a t this stage t o introduce dimensionless variables in place of z and E ; measuring z on the scale of c / o and E on the scale of (4mw2T/e2)1'2, we can rewrite (15) as

E'2 where $ ( E ) = X 2

+ $ ( E ) = const = C2,

+ (op2/02)exp(-E2)

(19)

in the Maxwellian case and

in the case of the truncated distributions considered. I n each case, the general form of $ ( E ) is seen to depend in a similar manner upon the value 1 it has a single minimum at E = 0, but for larger of oP2/w2-for wp2/w2 values it has two minima and a maximum a t E = 0, and this property is readily shown from (16) to hold for all distribution functions which are functions of & only. To complete the solution we need to integrate (19). This topic has been discussed by Volkov (1959a), Sagdeev (1959), Weibel (1957), Self (1960), Cushing and Sodha (1959) and R'lotz (1963a,b), but none of these authors gives a wholly satisfactory account of it. Sagdeev considered only truncated distributions and only one special value of the constant C. The others were restricted by their M H D treatments to considering Maxwellian distributions. Of these, Volkov gives the most nearly comprehensive account, though his analysis of the case C2 = wp2/w2 is incomplete, and his scheme of dimensionless parameters obscures the role played by the ratio wp2/wz. Cushing and Sodha's paper (1959) is justly criticized by Self (1960), who, however, bases his treatment of the rather unhelpful parameter Em,, and gives little physical interpretation. Mots (1963), considers only the application to plasma confinement and in this context draws conclusions about plasma leakage which need modification in the light of the kinetic analysis, but he introduces the following mechanical analogy. Equation (19) is analogous to the first (energy) integral of the equation of motion of a classical mechanical particle in the potential well $ ( E ) .The qualitative features of the motion thus follow by inspection from the form of the potential well, once the particle energy (corresponding to Cz) is given. I n Fig. 2 we represent the shape of $Jfor the two cases: (i) wP2/w2 < 1, (ii) wp2/w2 > 1. The dotted lines represent the contribution of the second term and consequently give a measure of the plasma density corresponding to that value of E. In case (i) the results are relatively simple. The minimum value of C2 = C12 gives us E = 0 everywhere. For C2 slightly larger than CI2,

RF CONFINEMENT A N D ACCELERATION OF PLASMAS

203

E 2 1 throughout the motion, and we can use the linearized equation, obtaining E 1 (C2 - C12)1’2 sin(1 - o , , ~ / w ~ ) ~ / ~ z . (20) In this limit the plasma density remains constant everywhere. As C2 increases, w e move out of the region where the linearized equation holds everywhere. The solution remains periodic, but the wavelength decreases and the waveform ceases to be sinusoidal, becoming most severely altered near the nodes [the analysis and a figure are given by Volltov (1959a)l. For C2 >> 1, the presence of the plasma (now confined to the neighborhood of the nodes) becomes unimportant in the wave equation, and we approach the vacuum solution E = C sin ( U / C ) Z almost everywhere.

FIG.2

I n case (ii) ( O ~ , ~ / W>~ l), we have to distinguish four classes of solution, corresponding to the four distinct choices for C2-(a) C2 > CI2, (b) C2 = C12, (c) C I 2 > C2 > C Z 2 , (d) C2 = Cz2,~ h c r e is C the ~ ~minimum possible choice of C2, given by 1 log wp2/w2 for a Maxwellian and by 1 - [ a / ( l C U ) ] ( W ~ / W , , ~for )~/~ a truncated distribution. With choice (a) the solution is periodic arid qualitatively resembles the solution for large C2 in the case w P 2 / w 2 < 1. Here, however, as C2 is decreased toward C12 the wavelength increases to infinity. That this must happen follows a t once from the mechanical analogy: for small C2 - C12,the kinetic energy of the particle near the peak at E = 0 becomes small, and the period of its motion in the well increases rapidly, reaching infinity when C2 = C12, since i t then comes to rest at the point E = 0. Mathematically, this follows from the linearized treatment, valid locally near E’ = 0, which gives

+

+

E

(C2 - C12)1/2 sinh[(wP2/wo2)- 1]1/2z

1

(21)

(we are restricated to this hyperbolic function since by hypothesis there exists a point where E = 0), showing that the value of z for which E becomes of order unity (which givcs a lower bound to the wavelength) approaches infinity as C2 - C12 tends to zero.

204

H. MOTZ AND C . J. H. WATSON

Choice (b), C2 = CL2, has the particular significance that for this value of C2 only is it possible for both E and E' to vanish simultaneously a t some point or points. Since E' is proportional to B , the electromagnetic field vanishes altogether a t these points, leaving pure plasma. Since, as we have seen, the wavelength is in this case infinite, there can a t most be two such points. If there are two, they must be a t x = f 0 0 , and we have the case considered by Volkov-two semi-infinite plasmas separated by (i.e., confining) a slab of radiation. If there is only one such point, we have the case considered by Sagdeev-a semi-infinite plasma separated from a semi-infinite domain of vacuum radiation by a single boundary layer. We may note that in the former case the width of the slab is indeterminate. The field profile in the boundary layer(s) grows exponentially away from the point(s) where the field vanishes until it leaves the linear region, when it continues to grow nonlinearly up to the value Em,, determined by the equation +(Emax) = C12. Although the full width of the boundary layer is infinite, its effective width is of order c/(wpz - w2)1/2. For still lower values of C2, i.e., choice (c), (C12 > C2 > CzZ),we once more obtain periodic solutions, but the oscillations are now between two positive (or negative) values of E, Emin, and Em,,. The sign is not significant, as is clear if we remember that E represents only the amplitude of the electric field; the field itself is E multiplied by a sinusoidal function of time. Finally, for the choice (d), (C2 = CzZ),we have E' = 0 everywhere and hence the plasma is of constant density and the electric field everywhere has the same amplitude and sign, given by & E,, where +(E,) = Cz2. The above discussion classifies the qualitatively different solutions of the nonlinear wave equation (14),using only two parameters C2 and w p 2 / 0 2 which , characterize the field amplitude and plasma density, respectively. The classification turns out to be applicable to any distribution function; the effect of different choices of distribution function only becomes apparent when one considers the distribution in space of the plasma corresponding to these various classes of solution. Since the plasma density is given by Eq. (17), we see that for a Maxwellian, although the plasma is never wholly confined to a finite region of space, its density is exponentially small if (momentarily reverting to dimensional variables) (eoE2/ 4noT)(wp2/w2)>> 1. If wp2 2 w2 there is no restriction on the maximum value of E2, so the plasma can always be localized by making E 2 large enough, the bulk of the plasma sitting at the nodes of the wave, which is periodic but not exactly sinusoidal. For w p 2 > w 2 the position is more complicated and in order to interpret it physically we need to know the relationship between C2 and the maximum value of E 2which occurs in a solution specified by C2. It proves con-

R F C O N F I N E M E N T A N D ACCELERATION O F PLASMAS

205

venient to work with a quantity M = ( w 2 / w p 2 ) E Z ;in dimensional form M = enE2/4nnT and is a measure of the electric pressure. B y (19) we have that both the maximum and minimum values of E2, corresponding to a solution specified b y given C2 and wP2/w, are determined by the transcendental equation

The qualitative features of the solution of this equation are most readily seen graphically. For the periodic solutions with alternating signs, C2 > Cl2 = W , , ~ / W ~ SO , there is no upper bound on M and the plasma density can again be made exponentially small a t the antinodes of the wave simply by taking E:,,, large enough. For the (critical) nonperiodic case C2 = C12, we have M exp( -wP2M/w2) = 1, a transcendental equation whose solution is always fit' l and hencen = noexp(-wp2/w2). Thus for this class of solutions the plasma density is only exponentially small if wpz/w2 >> 1. For the periodic but'*nonalternating solutions Cl2 > C2 > C2, Eq. (22) has two solutions M,,, and M,,, (as may be seen by plotting the left-hand side graphically) corresponding to the maximum and minimum values of E2. Since there now exists a minimum value of E2, wp can no longer be interpreted as the maximum local value of the plasma frequency, which is given instead by the smaller quantity w:,,, = up2exp[- (wp2/w2)Mmln]. This is nevertheless still larger than w 2 , as follows from the fact that Cz > CZ2 = 1 log(wp2/w2),which combined with the definition of upmRx and (22) gives

+

-

+

4max ~W2

log&

w2

=

1

+ C2 - c22> 1

(23)

and hence w: mRx > w2. Incidentally, = wP2 exp[- (wP2/w2)M,,1, which is in fact given by the smaller of the two roots of this equation, is less than w2. Finally for C2 = Czz, upma,and wpmln coincide and equal w . The above analysis applies to Maxwellian plasmas; we should consider briefly what changes one should expect for a truncated equilibrium. It is not difficult to show that apart from the unimportant changes in the values of C1 arid C p , the above analysis goes through unaltered, so t'he spatial distribution of E in the Riaxwelliari and truncated cases is essentially the same. However, there are major differences in the plasma distribution because of the different dependence of the plasma density on E. Since for the truncated distributions n ( E ) vanishes for E 2 En, it follows that any solution for which l3 > Eo anywhere will exclude the plasma from the regions concerned. In particular, if Eo < EmRx is chosen, the periodic solutions will have separate plasma slabs located near the nodes (for the

206

H. MOTZ AND C. J. H. WATSON

solution with alternating sign) or near the minima (for those with fixed sign) and the aperiodic solutions will consist of plasma half-spaces, fully confined by the radiation. The extension of the above analysis to cover waves of arbitrary polarization is mathematically elementary but leads to a rather unexpected conclusion. In general, we can always write

E

=

fi[E,(z) sin(wt + &),

E, sin(ot

+ 4,),

where we have added a normalization constant d2 so that 8 Equation (13) now gives

d2Ei dz2

+ w2

-- ~i c2

= C2

~i

[

exp -

C

01,

2

=

+

EZ2 EU2.

e2Ei2

i

or, in the appropriate dimensionless variables,

E:!

+ E~ =

(

exp -

(W,~/W~)E~

2

~ ~ 2 ) .

i

This system of equations possesses an integral (which is in fact its Haniilt onian)

Writing E , = A cos 8, E, = A sin 8, we have

+ A2eQ + A 2 + (wP2/w2)

X = Af2

exp(-A2).

Since X is independent of e we have at once the analog of conservation of angular momentum, L = A20' = constant, and X = At2

+ ( L 2 / A 2 )+ A 2 + (wp2/w2) exp(-A2)

=

AI2

+ $(A2),

where J. now possesses a centrifugal barrier a t A = 0 except when L = 0, a condition which is easily interpreted as requiring a plane or circularly polarized wave. I n consequence, A can only oscillate between maximum and minimum values of the same sign; i.e., the alternating periodic and aperiodic solutions become impossible, and the field cannot vanish anywhere within the plasma. Transforming back to the variables E, and E , we obtain an oscillatory but nonperiodic motion of these variables precisely analogous to the motion of a classical particle in a non-Keplerian central field of force. The physical significance of these nonperiodic solutions may be seen from the observation that they could be Fourier-analyzed into a continuous spcctrum of periodic waves. Thus they represent what, might be described as n monochromatic turbulent state of the plasma.

R F C O N F I N E M E N T A N D ACCELERATION O F PLASMAS

207

Before leaving one-dimensional equilibria, some discussion should be given of the way in which such equilibria might be established. Under normal experimental conditions the plasma is either injected before switching on the rf power; or it is injected into a preexistent standing wave; or it is created by the rf fields, ionizing neutral gas in a resonant cavity. I n each case, the electromagnetic fields will initially have a time dependence which is more complicated than the dependence sin ut which has been assumed in the above discussion, though it might be expected to settle down into one of the equilibria eventually. So far, no analysis of the transient behavior of the fields appears to have been attempted and the actual behavior is at best a matter for speculation. It is particularly difficult to be certain what would happen if rf power were applied to a plasma such that up2> u2and the power level corresponded to a periodic but nonalternating equilibrium. On linear theory, such radiation would not penetrate into the plasma, and it is not clear what conclusion one should draw from the fact that a nonlinear equilibrium exists in which the radiation has penetrated.

D.Infinite Cylindrically Symmetric Equilibria We shall begin with the energy-momentum tensor approach. The representation of the divergence of a Cartesian tensor in a curvilinear coordinate system (such as cylindrical polar coordinates) requires some caution. The correct expression is given in (for example) McConnell (1957, p. 313). The interesting component is that which expresses conservation of radial momentum. This gives

i a

- -rTrr

r ar

a 1 a f -rl ae Tre - -r Tee +

T,,

+ a Trt = 0.

(24)

If we take all quantities to be functions of r only and average over time we obtain for the TMo mode at cutoff 1 (eoEZ2- p o H e 2 ) 2r

+ dZP = 0

or

and for the TEo mode at cutoff, I d

- - r(coEe2

21. d,.

or

+

poHg2)

+ 2;;1 (eoEe2 -

poHz2)

+ dP = 0

(27)

208

H . MOTZ AND C. J. H. WATSON

where p = 2nm~v2Jd3v.With the help of Maxwell’s equations, one can show that Eqs. (26) and (28) (with f chosen to be Maxmellian for simplicity), respectively, reduce to

and

which are also the equations obtainable from (7). We shall deal with the TMo mode first, since several discussions of it are to be found in the literature; in particular, Boot et al. (1958), Weibel (1958), and Clauser and Weibel (1959) have undertaken numerical solutions of Eq. (29) for certain ranges of values of the parameters op2/wz and Ex (r = 0); Weibel has criticized the earlier work on the grounds that many of the solutions are incompatible with the assumptions upon which (29) is based, a criticism which is in our view well founded. We shall therefore consider the solutions in some detail, introducing an approximate analytic method of solution which puts the detailed numerical solutions into perspective. It will be seen that Eq. (26) differs from the otherwise analogous Eq. (lOa) in that it does not make the quantity $(eoE2 p o H 2 ) p a constant of the motion. However, an analogous constant of the motion can be obtained by integrating (26) from 0 to r ; this gives

+

+(eo~,2

+ p o H e 2 ) + p + p o /01 (He2/r’) dr’

+

= const.

(31)

If we use the Maxwell equation aEz/ar = -aBe/at to eliminate He, transform to dimensionless variables (Ex being measured on the scale (4mw2T/e2)1/2and r on the scale c/o), and drop the subscript z on El Eq. (31) becomes El2

+ E 2 + (wP2/o2) exp(-E2) + 2 J6 ( E t Z / r ’ )dr’

=

C2.

(32)

In selecting the lower limit of integration as r = 0, it is necessary to confirm that the integrand is well behaved at this point. However, if we solve (29) (in dimensionless variables) for small r by a power series expansion in r, we obtain

E = Eo(1 - t [ l -

+

(op2/oz) exp( -Eo2)]r2

+ O(r4))

(33)

and hence E’ = r O(r2),showing that E t 2 / r vanishes a t r = 0 as required. Equation (32) has again a mechanical analog which enables us t o

209

R F CONFINEMENT A N D ACCELERATION OF PLASMAS

classify its various types of solution; if we replace E by x and r by t d Z / m we obtain t +mu2 +(x) = E - m (vZ/t’> dt’; (34)

+

Jo

+

+

i.e., i t is the motion of a particle in the potential well = x 2 (upz/ w 2 ) exp(--s2) subject to a dissipative force proportional t o its velocity, but for which the drag coefficient is inversely proportional to time. Thus, qualitatively, the particle starts from some point on the edge of the well (at t = 0, v = 0, and Jb(v2/t’) dt’ = 0) and accelerates across the well, dissipating energy a t a rate mv2/t as it does so, and after performing a number of oscillations of different periods and peak amplitudes finally comes t o rest a t the bottom of the well.

As in the onedimensional problem of the previous section, the potential well has one minimum at the origin if wpz/wz 5 1 and two minima displaced from the origin if wp2/wz > 1 (see Fig. 3). I n the former case, only one class of (oscillatory) motion exists, which is qualitatively similar (as regards the maximum and minimum values of E ) t o the Bessel function Jo(r) and indeed goes over to J o as wp 3 0. For op2/w2 > 1, a large number of distinct classes of motion exist, examples of which are illustrated in Figs. 3(i) to 3(iii). Class (i) corresponds to the case C2> CI2 in the onedimensional problem above, but differs from it in that after a finite number of oscillations with alternating sign the particle must either come to rest a t the origin or transfer to oscillations without alternation of sign. Class (ii) corresponds to the case C2 = C12 (though clearly in the present context C2will have to be larger than w p z / w z to allow for the dissipation of “energy” en route) ; t,he equilibrium described is a radiation column confined by a plasma pressure which decreases as one moves inwards, initially exponentially but with a nonlinear cutoff as E approaches Em,,. Class (iii) corresponds t o the case C2 < CI2. Here, however, there are two distinct types of solution, depending on whether Eo = E (T = 0) is the larger or smaller solution of the equation +(Eo)= C2. The two possibilities are illustrated in the regions (a) and (b) of Fig. (iii). It will be seen that in

+

210

H. MOTZ AND C. J. H. WATSON

case (a) the plasma is excluded from the origin (since it is the point where E is maximum) and it is therefore concentrated in concentric shells at varying radii, the distinctness of which gets blurred as r increases, merging eventually into a uniform plasma of density no exp(-EW2). I n case (b), on the other hand, the plasma density is maximum on the axis, though there is again an infinite set of concentric shells which eventually merge into the same density. All the computed solutions described in the literature appear to describe equilibria of class (iii). Both authors, however, seek to avoid the awkward fact that these have plasma and a constant field amplitude at infinity instead of the combination of vacuum radiation and no plasma which would enable one to insert the perfectly conducting walls of a waveguide at some suitable radius. They do so by cutting off the plasma at some radius where its density is exponentially small and then matching the radiation into a solution of the vacuum field equations at that radius, These solutions are not true equilibria, since for a Maxwellian distribution the plasma density nowhere vanishes, but the leakage which would occur can be made negligible if wp2/02 >> 1. Naturally, if the distribution function were non-Maxwellian, and such that the plasma density were exactly zero at some radius, there would be no approximation involved in performing this matching operation. Graphs obtained by the numerical integration of Eq. (29) are given in the papers of both authors; Weibel considers only a solution of class (iii) (b) in which the matching operation is performed at the first minimum of the plasma density, whereas Boot et al. plot solutions of class (iii) (a) or (b), with the matching performed at the first or second minimum. (In interpreting their paper, it is necessary to read U = E, no/nc = (wP2/ w2) exp(-Eo2), and n/n, = (wp2/w2) exp( - E 2 ) . ) Some of the graphs which they obtained are illustrated in Fig. 4.They also give a chart showing the plasma radius, maximum electric field, and “fractional tuning” (a measure of the displacement of the wall of the surrounding waveguide required by the presence of the plasma) for a large spectrum of values of no/noin the range no/n, > 1; i.e., in our notation W , ~ / W ~ exp(-Eo2) > 1. Inserting this condition into the series solution for small r [Eq. (33)] we see that this describes a mode in which E increases from r = 0, and hence it identifies the solution as being of class (iii) (b). The qualitative features of their chart can all be derived from the mechanical analogy. Both Weibel and Boot et al. comment on the fact that for the case wP2/w2 > > 1, E, << 1, the numerical solutions show that the density profile is nearly rectangular, falling very suddenly at a critical radius to virtually nothing. The magnetic field reaches its maximum value just outside this radius, whereas the electric field is negligible within it and reaches its

R F CONFINEMENT A N D ACCELERATION O F PLASMAS

211

maximum a t a larger radius. As Weibel points out, this is exactly what one would expect if the plasma were a highly conducting metal. Some discussion should be given of this observation at this stage, since it turns out to be crucial to the theory of the stability of these equilibria. We have seen that they are equilibria of class (iii) (b), the constant C2 of Eq. (32) being chosen such that Emin<< 1. For such equilibria we can match the solution into the vacuum after one half-cycle of variation of E with z since Em,, wp/w >> 1. I n consequence, the role played by the “dissipative” term 2J7,Et2/r’dr’ is rather unimportant; it somewhat reduces

-

(Cl

FIG.4

the maximum values of E’ and E for given C2 but does not affect the qualitative features of the solution. If for simplicity we neglect it, we can use (32) to determine the maximum value of E’ and the value of E a t which this occurs:

(E’):,, = C2 - 10g(wp2/w2) - 1 EcZ = log(wp2/w2),

B

C l

and hence B,2 - -EC2

c 2

- 10g(wp2/w2) - 1 log(wP2/w2)

wp2/w2

log(wp2/w~) *

Thus although the maximum values of E and B are both of order wp/w, the maximum in B is reached first. The corresponding density profile can be

212

H. MOTZ AND C. J. H. WATSON

inferred from the fact that for Eo << 1 the density at the point when B is maximum is given by no = no exp( -EC2)= no exp[ -1og(wP2/w2)]. Thus for u p 2 / w 2>> 1, n, << no, and we see that the plasma is largely confined by magnetic pressure, since most of it lies inside a region where B >> El and that the electric field is small both at its surface and within it. This confirms analytically what Weibel and Boot et al. showed numerically-that for w P 2 / w 2>> 1, the equilibria are quasi-metallic. The same argument shows, however, that as w p 2 / w 2 - + 1 the equilibrium loses this quasimetallic character. The boundary can still be quite sharp, as a result of the nonlinear variation of E as it approaches E,,,, and the radiation can still be quite effectively excluded from the center of the plasma (if C2 is suitably chosen), but E and B are of the same order of magnitude everywhere. For example, if EO= 0 and w p 2 / w 2 = 10, (Bc/Ec)2= 2.9, but EL,, = 10 and at this point n = noe-lO. Such equilibria are radically different from those considered by Weibel and Boot et al. and might appropriately be described as “quasi-dielectric.” It is important to note that strictly speaking the self-consistent field theory developed in this chapter is only valid for these quasi-dielectric equilibria, since for large w p 2 / w 2the length scale LE for the nonuniformity of E near the plasma boundary can be estimated from the above expression as

cWwE,‘ = [ $ / l o g 3 1

2

-112

C/O

which becomes much less than the length scale c / w for vacuum radiation and hence invalidates the method of averaging upon which the concept of the quasi-potential depends. Since no alternative technique has been developed, what happens as w p 2 / w 2is increased is a matter for speculation. It is unfortunately very important that we should express a view on this question, however, for, as we shall see, the feasibility or otherwise of an rf thermonuclear reactor turns precisely upon it. I n the early papers on the applications of rf confinement to thermonuclear fusion (e.g., Boot et al.), reactor designs were proposed in which w p 2 / w 2was as large as lo7,These designs were criticized by Weibel on the grounds that they depended on a theory of the rf equilibrium which was invalid for such large values of wp2/w2. This objection is, as we have seen, well founded; it does not, however, follow that there cannot exist high density confined equilibria which are qualitatively similar to those obtained from this (strictly inapplicable) theory. Some insight into the question can be obtained by considering the exact theory of particle motions in a plane rf wave, discussed in Section 1. We saw then that particles in the neighborhood of the node of a plane wave Eo sin k z remain confined if e 2 E 0 2 k 2 / m 2 w < 2 0.7; that is, if ( v E / c )

RF CONFINEMENT AND ACCELERATION O F PLASMAS

213

( k c l w ) < 0.83. Thus, although the averaging theory upon which the quasi-potential concept depends is valid only if (z~E/c)(Icclw) << 1, confinement is not lost until much larger values of this quantity are reached. If we apply this criterion in the case of a plane wave in a plasma, and take

we obtain a n upper limit on up given by

$/ log

w2

‘v

O.~C~/VE~.

This estimate is, however, unreliable, for it assumes that the electric field distribution is given, independent of the plasma distribution. I n practice, if the plasma failed t o be confined for this reason, the electric field distribution would change as a result of the plasma motions in such a way that its gradient was diminished, and confinement might again become possible. The present authors do not see any way in which the resulting self-consistent plasma field distribution could be calculated other than by numerical computation. It is, thus, premature to assume that such configurations cannot exist; equally, it is dangerous to assume that they would closely resemble the equilibria discussed above. Nevertheless, it seems reasonable to guess that they would differ significantly from the configurations based on quasi-potential theory only in those regions of space where the electric field gradients would on that theory become excessively large; i.e., the breakdown of that theory would simply lead to a widening of the boundary layer between the plasma and radiation. If this reasoning is correct, it follows that the general characteristics of the equilibria obtained above are still relevant. One particular characteristic, which is of interest to the designer of a thermonuclear machine, is the volume of plasma confined. One reason for the very high values of wp2/w2 selected by Boot et al. was the fact that the equilibria which they obtained by numerical computation had volumes which decreased rather sharply as w p p / w 2increased, so that it was only a t these very high values of wpZ/wz that the total number of particles confined was sufficient t o give a useful thermonuclear output. There is not, however, any necessary connection between the value of wP2/w2 and the volume of plasma confined. Reference t o Pig. 3(iii) (b) shows that for w p z / w 2 > 1 there exist equilibria in which the electric field on the axis can be made arbitrarily small. These correspond to B choice of the constant of integration C2 of Eq. (32) very slightly less than ~ , , ~ / wFrom ~ . the analogy of the motion of a particle it is clear that the duration of the first transit of the particle across the well increases sharply as (op2/w2) - C2 tends to zero and hence that the

214

H. MOTZ AND C. J. H . WATSON

radius of plasma over which the radiation fields remain small (and hence the plasma density stays nearly constant) increases correspondingly. Nevertheless, the maximum field amplitude eventually reached is as large as is compatible with Eo fi 0, and it is of order wD2/w2; so, provided that this is somewhat larger than unity (say -10) we can take the plasma density to be exponentially small at the first maximum of E and match on to a vacuum mode at this point. As we have seen, we cannot trust the quasi-potential theory of the structure of the boundary layer for values of wp2/w2 much greater than 10, but there is still hope that such configurations, consisting of a large cylinder of plasma isolated from a conducting wall by a relatively thin rf layer, might exist for much larger values of wP2/w2. These equilibria, in which the ratio of the plasma volume to the volume of space filled with rf can in principle be made indefinitely large, were overlooked by Boot et al., presumably because they only exist for a narrow range of values of the constant C2. Indeed, one practical problem which they raise is the delicate adjustment of the rf power level needed to maintain them, and some feedback mechanism would presumably be required. One feature of Weibel’s numerical calculations should be mentioned. Throughout the above discussion we have assumed that quasineutrality is maintained not only in the region containing the bulk of the plasma (where it undoubtedly is) but also in the low density tail (where it is not). Weibel has integrated the more exact field equation, with

and Poisson’s equation

concurrently by an iterative numerical method and shown that an ion sheath develops in the low density tail, as one would expect. We now turn to the cutoff cylindrical TEomode where the wave equation is given by Eq. ( 3 0 ) . As before, we can obtain a first integral from the energy-momentum tensor theorem; in the (by now) familiar dimensionless unit we have

Et2

+ E2 + (wp2/w2) exp( -E2) + 2 Jd ( E 2 / r ’ )dr’

=

C2

(35)

in analogy with Eq. (32). As before we can show by series solution for small r that E vanishes as r for small r, and hence that the integral is well behaved near r = 0 , and indeed that the contribution of this term to the

RF CONFINEMENT AND ACCELERATION OF PLASMAS

215

left-hand side of (35) goes to 0 with r . Thus it again acts as a kind of pseudofriction (to draw on the mechanical analogy once more), though the rate of dissipation of energy is now proportional to the square of the displacement of the particle instead of the square of its velocity. It is clear that this makes only minor quantitative differences to the actual motion of the representative particle; however, we must now start the motion from points where E = 0, so the classification of solutions becomes different. Examples of the various classes are illustrated in Fig. 5 for the case w p 2 / w 2> 1. The physically interesting class is (iii): b y choosing C2 slightly larger than ( w p 2 / w 2 )we can again construct a solution in which there is a plasma of large radius with a very small trapped rf field, and we can match irito the vacuum a t the first field maximum.

We can rapidly dismiss cutoff cylindrical waveguide modes of lower symmetry; since these necessarily have azimuthal nodes, the corresponding quasi potentials do not possess absolute minima arid consequently cannot be used to confine plasma. Combinations of such modes could be used, but the need to use two generators of different frequency and the extra mathematical complexity of the theoretical analysis render this an unattractive approach.

E. Three-Dimensional Conjineinent Three classes of three-dimensional confinement configuration which rely entirely upon rf confinement appear to have been proposed in the literature, and several more have been proposed in which the rf field supplements a magrietostatic field. The simplest of the former is a configuration in which one of the cutoff TE, or TRIOmodes discussed in the preceding section is hcrit around into a torus. No detailed calculations appear to have been made, hut, it seems probable that for sufficiently small torus aspcct ratio the cy lin d rid theory would be approximately applicable. If this were correct, thc dificwlty raised by the condition (E* V ) E Z= 0 need not arise, sinve, as we have seen, Poiricar6’s theorem shows that this condition can be met on surfaces of toroidal topology. For all the other

216

H. MOTZ A N D C . J. H. WATSON

equilibria discussed below, the use of the equation

V XV

)(

E

=

(w2/c2)eE

(7)

is open to objection for the reasons we gave in Section A. Nevertheless it seems plausible that this might give an approximate representation of the field configuration which would arise if the condition (E V)E2 = 0 were relaxed as a result of one of the effects discussed above. The two further classes of three-dimensionally confined plasmas involve rf modes in cylindrical cavities of finite length and spherical cavities respectively. For such equilibria, the energy-momentum tensor approach used above is of little value, since two or more components of the divergence of the tensor have nontrivial content. One is therefore forced to work with Maxwell’s equations in their fully differential form. This has the unappealing consequence that one has to solve Eq. (7), a nonlinear, partial differential equation in at least two independent variables and (usually) two or more dependent variables, or some equivalent equation or set of equations. The number of independent variables is irreducible in the nature of the problem. (It can be shown from V B = 0 that no single mode exists for which the fields have spherical symmetry.) Attention has therefore been directed towards reducing the number of dependent variables, if possible down to a single scalar function from which all the fields can be derived. For vacuum modes there exists a considerable literature on this subject [see, for example, Nisbet (1955) for a comprehensive discussion, or Stratton (1941) or Panofsky and Phillips (1962) for simpler accounts]. Unfortunately, the methods developed in that context are of little value when e is a (nonlinear) function of E, and the most that they can supply is a classification of the vacuum modes t o which the solutions of (7) might go over in the limit w,, + 0. However, as we shall see, there may be solutions of (1) which cannot exist in the limit up --f 0 (e.g., if they only exist for wp2 > u 2 ) or for which the topology of the fields changes discontinuously when up2> w2, so such classifications could be misleading. We shall first consider equilibria in cylindrical cavities. In cylindrical polar coordinates, the wave Eq. (7) has the form (with a/a@ = 0)

-

It is clear from these equations that Eg is decoupled from the E, and E, components, but that the latter are conditionally coupled, in the sense

R F C O N F I N E M E N T AND ACCELERATION O F PLASMAS

217

that E, can exist without E, only if a2Ez/ar az = 0 and E, can exist without E , only if (l/r)(a/ar) 1’ aE,/az = 0. The former condition is met in the TMomode a t cutoff, a case which we have already discussed; the latter requires E, = A ( z ) / r , which excludes such modes for an empty cylinder but not in a coaxial line, for which it gives the electric field of the T E M (Lecher) mode. This raises the question as to whether a plasma column can act as the central conductor for a TEA4 mode. The answer is negative if a T E M mode is defined as one for which E, = 0, for if E, = A ( x ) / r is substituted in (36) it is seen that no such solution exists, since c = e ( ~ ,2). Nevertheless, it does appear possible that a TEA1 mode, in the sense of a mode possessing no frequency cutoff but having both E , and E, components, might exist; it would be analogous to a vacuum TEA4 mode in a coaxial line with a bumpy inner cylinder. Equation (38) is the equation for the total electric field of the TE, modes. Thus, to summarize, all the fields of a TEomode a t or above cutoff can be derived from the scalar Eg which satisfied (38) ;the fields of a TMo mode at cutoff can be derived from the scalar E , which satisfies (37) with E , = 0; but neither the T M o modes above cutoff nor the T E M modes are determined by a single component of El and for both of these one needs the coupled equations (36) and (37). This is not to say that it is in principle impossible to find a scalar function for these modes, and indeed, as Mot2 has shown, the quantity Hg can under some circumstances be used in this way to derive the T M omodes. To see this we note that the wave equation for the magnetic field

V X (l/t) V X H for the Tillo mode

=

(d/c2)H

(39)

(H= HB) gives

a -i _a (rHo)+---+-H.j a iaHe _ ar tr ar az az

w2

~2

=O.

This equation is unfortunately of very little analytic value for two reasons. First, it has singularities at points where c = 0. This means th a t considerable mathematical difficulties arise for any confined plasma whose maximum density is greater than critical. (It does not follow that such plasmas cannot exist; indeed the converse is demonstrable, for the singularity in (40) is still present if the mode is excited a t cutoff, whereas for this case we have already shown that confined supercritical plasma equilibria exist.) Secondly, e is not an explicit function of He, but is determined by the intractable implicit equation E

=

w z

1 - +exp w

218

H. MOTZ AND C . J. H . WATSON

Nevertheless, (40) is a very convenient form for a nunie~icalintegration, provided that the plasma remains subcritical. The results of such an integration will be outlined shortly. This difference between the TE, and TATomodes-that the former can and the latter in general cannot be derived from a single component electric field-is related to another difference, first commented on by Miller: the quasi potentials of the vacuum fields have quite different topological propert,ies in the two cases. The TE quasi-potential $TE = R(r) sin2 Icz has a factor R(r) which has a minimum a t r = 0. However, since $ T ~vanishes a t the planes where sin kz = 0, it does not possess any absolute minimum and hence cannot confine single particles or low density plasmas. The TM quasi-potential, however, is $TM = Rl(r) sin2 lcz Rz(r) cos2 kz, since the two electric field components E , and E, are out of phase, and hence it does possess an absolute minimum.

+

TABLE I For

W,,~/W~

= 0,920 825

0.079 0.318 0.564 0.753 0.875 0.942 0 .973 0.986 0.991

17.5 351 830 057 505 354 457 441 456

0.561 0.652 0.768 0.860 0.922 0.95.5 0.977 0.987 0.991

113 824 057 625 462 815 910 013 024

0.874 243 0 . 9 0 0 640 0 . 9 3 2 928 0.957 063 0 . 9 7 2 292 0.981 688 0.987 263 0.990 306

0.986 0.988 0.991 0.992 0.993

909 516 031 854 854

Self has investigated the shape of this minimum numerically and shown that it is of optimum depth for single particle confinement u hen the ratio of the radius to the length of the cylindrical cavity is 0.6. However, the presence of a plasma of significant density must alter the shape and depth of the minimum; a numerical calculation of this effect has been made by hlotz, who computed solutions to (40) and (41) for this optimum cavity shape by an iterative procedure, obtaining the modified cavity resonance frequency and plasma arid field distributions for a number of subcritical plasmas. (As previously indicated, (40) is invalid for supercritical plasmas, so no results for such plasmas were obtained.) The solution for one particular set of initial conditions is given in Table I, which gives the value of E at various points of a discrete lattice, with rows and columns corresponding to the z and r directions respectively. The resonance frequency obtained was u 2 = 0.06453c2/a2 as against the vacuum resonance frequency of 0.06437c2/a2,showing that a high Q cavity would be significantly detuned by even this amount of plasma. The consequent need to retune the oscillator during the build-up of the plasma is a matter of some experimental

RF CONFINEMENT AND ACCELERATION OF PLASMAS

219

difiiculty a t the present time. It is clear that in this case, arid indeed for all subcritical plasmas, the ratio of the plasma volume to the volume filled with rf field is very unfavorable. No attempt appears to have been made, however, to compute supercritical equilibria, in which the ratio might be much more acceptable, on the basis of Eqs. (36) and (37). The supercritical radially confined plasma equilibria obtained from Eq. (38) a t cutoff (see above) give grounds for hope that a TEO mode sufficiently above cutoff might confine a supercritical plasma even though it cannot confine single particles; i.e., the plasma would have to make its own field minimum a t the center of the cavity. Ultimately, it would be necessary to test this suggestion by the numerical solution of (38) ; that a positive result is likely can be made plausible by the following considerations. For small T , we can solve (38) approximately by expanding E+ = 2, E2rE, where the E, are arbitrary functions of z. This a t once gives E, = 0, i < 1. Thus Eo vanishes on the axis, and hence for sufficiently small r , E = 1 - wp2/w2. However, under this assumption (38) becomes separable, giving (with E , = R(r)Z(z)and in suitable dimensionless units)

d2R dr2 ~

R + -r1-dR + aR = 0, dr r2 d2Z + bZ = 0, dz2 -

~

+

(42)

(43)

where a b = 1 - wp2/w2. The separation constants a and b are determined by the boundary conditions, which unfortunately have to be applied outside the region of validity of (42) and (43). However, it seems probable that for sufficiently large wp2/w2, b can be negative, which corresponds to an absolute minimum in E 2 at the center of the cavity, since by symmetry d E / d z = 0 there. It is clear that a certain difficulty would arise in the experimental realization of such an equilibrium, since the corresponding vacuum mode does not trap single particles, and hence the plasma could not be built u p gradually. One possible technique would be to create the plasma a t the cavity center with the help of a laser. Another alternative would be to confine the plasma in the z direction during the build-up phase by means of an electrostatic potential, giving the cavity end plates a negative charge. Unlike the final equilibrium, which does not appear to have been investigated numerically, the characteristics of such temporary confinement have been computed by Rlotz. The calculation assumed that the electrons of the plasma would distribute themselves in the z direction in a manner determined by the electrostatic potential due to the charged end plates, and that in a time which is short compared with the ambipolar diffusion time the ions would be trapped by the space charge created by the elec-

220

H. MOTZ AND C. J. H . WATSON

trons, and it was shown that supercritical plasma distributions could be confined in this way. We now turn to equilibria set up in spherical cavities. The wave equation (7) in spherical polar coordinates with 8/84 = 0 becomes a ---i (444 r2 sin e ae ae

ae As in the cylindrical case, we see that E+is decoupled from E , and Eel but now these latter are unconditionally coupled to each other and no solutions exist in which any of the field components are independent of 8. Thus there is no analog of the cylindrical cutoff modes, which have an additional symmetry not possessed by the more general modes. Consequently, no analytic solutions of these equations have been found, and no numerical solutions appear to have been attempted. True, a modified version of Eq. (44c) exists in the literature [Consoli et al. (1964b)l which would be analytically soluble if it were correct, but in its derivation it is unjustifiably assumed that E.+= E,(r) sin 0. One can see from the fact that B is an unfactorizable function of r and 0 that Eq. (44c) is not separable. Thus all that can be said about spherical equilibria at the present time has to be derived from the known solutions of Eqs. (444 to (44c) when e = constant. I n what follows we shall assume that e = 1, though the modification for other, including negative, values of t would not be difficult and might indeed be used to give a qualitative discussion of supercritical solutions, in the manner indicated above for finite cylindrical modes. Just as in cylindrical cavities, the vacuum modes can be classified as TE, TM, or (presumably, although the literature does not seem to mention them) TEM modes if there is an inner sphere. The vacuum fields for the TE and T M modes in a cavity of radius a are given in the tabulation of Eq. (45) [Panofsky and Phillips (1962)l.

E.+ =

a

-ikzi( k r ) Yim(0, ae

+)

RF CONFINEMENT AND ACCELERATION OF PLASMAS

22 1

In Eq. (45), Ic = kn,l is defined so that ka is the n th zero of the Bessel function .I[, z1 is the spherical Bessel function J1+1,2(x)/x1/21and Y p are the spherical harmonics. From these we can a t once derive the quasipotential created by any given mode. I n particular, for the m = 0 modes (a/a4 3 0) we have for the T E modes

and for the T M modes

Expanding the Bessel function for small

T,

we obtain

where Pl(cos 8) are Legendre polynomials. Since the spherical harmonics for 1 2 1 all have radial nodes, we see that the quasi-potentials f i cannot ~ ~ confine single particles. On the other hand, the expression in square brackets in $TM is always positive and nonzero. However, for 1 = 1, the dominant term in $TM is independent of r, and the next term is negative. Thus in this case #TM has a maximum a t the origin. For all higher modes, however, there is a minimum which could confine particles. I n view of these facts it is somewhat strange that all the proposals in the literature, Ilnox (1957s,b), Butler et ul. (1958), and Consoli (1962), are based on the T E modes, using a combination of two or more such modes with different frequencies or azimuthal variation in order to achieve a confining configuration. The explanation would appear to be that these proposals were made before the quasi-potential theory had been developed, a t a time when the plasma was regarded as a metallic conductor, upon which the radiation exerted pressure. K ~ O proposed X the superposition of three TE,lo modes with their orientations chosen to give a spherical over-all configuration; Butler et al. (1958) proposed a combination of TEI10 and TEl,, modes. Consoli et ul. proposed, and have investigated experimentally, a TEllo mode alone, accepting a certain plasma loss along the axis which, they claim, would be reduced by the fact that the oscillations of the particles in the rf field about the position of their guiding centers would be such that the loss cone for particles might be less than one would infer from guiding center theory alone.

222

H. MOTZ A N D C. J. H . WATSON

Before leaving the subject of three-diniensional confinement, we should make some mention of a few papers which discuss this subject from an oversimplified viewpoint and reach conclusions which are, in our view, erroneous. There is first a paper by Johnston (1960) which purports to show that a high density plasma (for which w p 2 / w 2 > 1) cannot be confined a t an absolute minimum of E2. Johnston takes as a model of the plasma a system consisting of two particles, of masses m and M , interacting through a force - wp2z, where z is the separation between them and is taken t o be in the direction of VZ2. He then shows by a trivial extension of the methods of Section 1 that the center of mass of such a system would move according to

where VIIand V, are the components of V parallel to and perpendicular to the direction of E,respectively. It follows from (47) that although a configuration possessing an absolute minimum in E2 would always confine a plasma with wp2 < w 2 and would confine a plasma with up2> w2 provided that E V E 2 = 0 everywhere, the presence of a component of E parallel to V E 2 would lead to a loss of confinement in this direction for plasmas with up2> w2. This state of affairs has a certain plausibility, in the sense that we have only actually succeeded in proving above that three-dimensional confinement is possible for low density plasmas or if E vE2 = 0. However, it will be observed that Johnston’s choice of model is a highly arbitrary one: he assumes without any convincing justification that the system shows plasmalike properties only in the direction of V E 2 (his argument in this context-that one only expects space charge effects to be important in the direction of V n , which is parallel to VE2-would if correct show that a uniform plasma could not sustain plasma oscillations!). We are therefore in no way compelled to accept his conclusion, and the extreme crudity of the model, which makes no allowance for changes in the plasma density resulting from the (near the hypothetical resonance) very large forces acting on it, makes it doubtful whether much significance should be attached to the result. A slightly more plausible model has been proposed by Asaltaryan (1958) and Gildenburg and !Miller (1960), who treat the plasma as a rigid dielectric sphere of radius much less than the wavelength of the confining field, having a dielectric coefficient c = 1 - w p 2 / w 2 . Since we believe that this model may be relevant to the theory of plasma acceleration, we discuss it in some detail in Section 7; here we shall simply state their conclusionthat the center of mass of such a sphere situated in a standing rf field

-

-

RF CONFINEMENT AND ACCELERATION OF PLASMhS

223

should experience a n acceleration

+

where wo2 = wp2/3 (the factor reflecting the spherical geometry of the plasma) arid y is the damping coefficient for plasma oscillations due to collisions or radiation. It will be seen that this theory predicts a resonance, and a reversal in the direction of action of the quasi-potential force as w passes through wo, whether E V E 2 = 0 or not. Thus (48) is absolutely incompatible with the self-consistent field theory of the equilibrium of a high density plasma confined by rf fields as discussed in this chapter. In the present authors’ view, this resonance is a spurious effect resulting from the inadequacy of the plasma model adopted. Finally, there is a paper by Knox (1961) which undertakes to show, on the assumption that the self-consistent field equations remain valid where E * V E 2 # 0, that in this case absolute confinement is impossible if w p 2 / w 2 > 1. Essentially his approach is to consider a particular configuration for which E has a component parallel to VE2-a TR4o mode in a finite cylinder closed a t one end by a plasma. He starts from our Eq. (40) for this mode, which he simplifies by assuming that e is a given function of z instead of being a self-consistently determined function of z and T , and he chooses the functional form of E(Z) so that the equation is soluble analytically in the neighborhood of the point where e = 0. He shows that the general solution is a sum of a singular solution and a well-behaved solution but that the boundary conditions require an admixture of the singular solution and hence the component of E parallel t o V E 2 becomes much larger than the perpendicular component and hence the dominant force is deconfining. I n the present authors’ view, this result is due to his simplifying assumption that E is a given function of z alone; it seems reasonable to expect that, if t can vary in a self-consistent manner, a nonsingular (and hence confined) solution can exist in the present, as in the cutoff, case.

.

3. THEORY OF

COMBINED

RADIO-FREQUENCY AND

nfhGNETOSTATIC

CONFINEMENT OF I’LA4SMA

A . Ijerivation o j the Selj-Consistent Field Equations We proceed by urialogy with the analysis given in the preceding section for pure rf coiifinemcnt. I n the section on single particle motions, we showed that the quantities p+ = Qinhpct2/Boand

H. MOTZ AND C. J. H. WATSON

224

are constants of the motion to first order in a and p. If we generalize this by including a n electrostatic potential r$ and we remember that f = fc en eW and that fc is parallel to 51 and en perpendicular to it, we can write p, and E, in terms of the particle coordinates as

+

+

- @wd2/Bo

P, =

and E, = Qmdv

+ $* + e9,

-

(1)

where to first order in (Y and p we can take $, and 4 to be functions of the actual particle coordinates. We now consider distribution functions f,(&,, p*); such functions can only describe plasmas in which to first order in a and p no stationary currents are flowing, and hence cannot be used t o consider situations in which the presence of the plasma modifies the stationary magnetic field, but they are sufficiently general for our present purposes. We now have, as in the preceding section, j

=

eJv(f+

- j-> d3v

e[bw+no+Jj’+ d3v - ew-no-Jj- d34,

=

(2)

+

wherej& = J+(+nz&v2 $* k eqj, + m ~ v L 2 / Band O ) is normalized to unity a t some reference point. If as before we write

E(r, t )

=

E exp(iwt)

+ E* exp(-ht),

the wave equation becomes

vxvx(E”4

This equation can be put into the same form as Eq. (7) of the previous section:

V XV X E

=

(w~//c~)EE, €

=

(1 - [w,’/w(w

+

YQ)]Jj-d”v>LT,

(4)

if the scalar e appropriate there is replaced by the tensor r which is diagonal (though not a multiple of the unit tensor) only in the ( s y representation. We can as before use the assumption of quasi-neutrality to eliminate r$ from Eq. (4),by using

+ + e9, P+> d3v

Jj’+(4m+v2 $+

=

J,f-(!m-v2

+ $- - e4, P-> d3v,

(5)

but this only leads to an explicit expression for 4 if we make definite assumptions about the functional form of the j*.I n what follows we shall for the most part assume for simplicity t,hat the ,f& are independent of p*

225

RF CONFINEMENT AND ACCELERATION OF PLASMAS

and are the same function. We then obtain e+ = &($- - $+) as before. absence of an rf field, such an assumption is inadequate, since if j , is to describe a confined (and hence nonuniform) plasma, some p dependence is essential. This is reflected in the fact th a t a plasma confined in a mirror machine has a distribution possessing a “loss cone” in the velocity space. If an rf field is present, however, the plasma can be confined by i t even iff+ has no p dependence, and calculations are simplified if this assumption is made. We shall now give certain particular applications of Eqs. (4). 111 the

B. One-Dimensional Equilibria with a Uniform Magnetostatic Field It is clear that the solution of (4) for arbitrary directions of propagation with respect to a uniform magnetostatic field cannot be simpler than the corresponding linear problem, which already has a certain complexity. The most interesting case is the one in which the direction of propagation is parallel to the magnetic field, which we shall take to be the x direction, and we shall therefore concentrate on this particular case. We then obtain the two coupled equations:

These equations were first obtained by Volkov, using a magnetohydrodynamic approach. If we take 3- to be Maxwellian, and work with the usual dimensionless variables, Eq. (6) becomes

d2E+ - ( 1 dz2

2 w(wu pk

[

-=)I]

(A

2 1 IF + n/w l2 n) exp - I

+ 1 - o/w

E,.

(7)

Sirice the complex conjugate quantities E,* satisfy the same equation, if we multiply (7) by dE,*/dz and its conjugate by dE,/dz and add, we obtain

+

+

&lE+’12 JE+I2 IE-’I2

+ IE-12

+g e x p

[ f (m P+I2 + x - n / w) const, ](8) -

1

=

an equation which can at once be recognized as the dimensionless form of the equilibrium condition derivable from the energy-momentum tensor theorem :

226

H. MOT2 A N D C. J. H. WATSON

To complete the solution of the problem we have to obtain expressions for the four independent quantities Eh and E,*. We can in principle still draw on the mechanical analogy used in the preceding section to discuss the qualitative features, but since the motion of a particle in a fourdimensional potential well is not easily visualizable, it is convenient a t this point to restrict attention to either plane or circularly polarized waves. I n the former case, E+ = E- = E ; in the latter, either E'+ or E- = 0, and we can take the nonvanishing quantity to be real. In this case, the equivalent mechanical problem becomes one dimensional, and the arguments of the proceeding section can be taken over almost unaltered. One interesting difference arises, however: the effective potential seen by the particle representing the electric field is

For w > !ilthe resulting motion is largely unaltered, though the nonlinear regime is reached a t much lower values of E* as w approaches 3. For w < 3, however, the topology of the effective potential $ ( E ) is altered; it no longer has a central hump, whatever the value of w p 2 / w 2 .This reflects the fact that a strongly magnetized plasma can transmit even linear waves a t frequencies below the plasma frequency. Under these conditions the only effect of the nonlinearity of the fields is to distort the wave form and alter the wavelength of the waves. The effect on the plasma distribution is striking, however; it is now concentrated a t the antinodes of the wave. One feature which is obvious from Eq. (8) and not from Eq. (3) is that the condition that the reaction of the plasma back on the rf field distribution should be negligible everywhere does not depend upon the presence or absence of the magnetostatic field, and it is (as one might expect) &E2 >> noKT. It is clear that the tensor character of E in Eq. (4) seriously complicates the form of the equation when the rf field geometry is anything other than one dimensional. I n what follows, we shall evade this complexity by considering low pressure plasmas (for which +eoE2>> noKT and hence E = I) and shall discuss the plasma distributions which are obtained with various combinations of rf magnetostatic fields. C . Low Pressure Plasma Distributions The assumption that the plasma pressure is negligible in comparison with the rf pressure enables us to calculate the rf field configuration with the vacuum field equations, satisfying the appropriate boundary conditions a t the walls of the cavity in which it is set up. From this, given the

R F CONFINEMENT A N D ACCELERATION O F PLASMAS

227

geometry of the magnetostatic field, the quasi-potentials

Y

can be calculated and the electrostatic field and the resulting plasma distribution can then be derived from Eq. (5). Now the advantage of combined rf-magnetostatic confinement over pure magnetostatic confinement is that it is in principle possible to confine an isotropic velocity distribution in this way and hence to avoid loss cone instabilities and other anisotropy instabilities. To achieve this, we must takef* = j ( & ) orily and this, as we have seen, implies that e+ = +($+ - $-) and a density distribution n(r) = noJf(+7nv2 &$-) d3v.

+

Thus, contours of constant n then coincide with contours of constant *, and, to ensure confinement, we require a set of nested surfaces of constant # with everywhere positive and increasing outwards. This requirement is most easily met if w > Q everywhere within the region of confinement of the plasma. The required increase in yi can then be achieved either by an increase in one or more of the (E,I toward the periphery or by an increase in Bo toward the periphery in such a way that D approaches w . Naturally, this second alternative economizes in rf power; it should be remembered, however, that we have assumed in this section th a t this power level is high enough for the rf field distribution to be unaffected by the plasma. For any given rf and magnetostatic field distribution, the # contours can be calculated from Eq. (52) or (53) of Section 1.

+

4. STABILITY THEORY

The literature on the stability of rf confined plasmas makes rather depressing reading, both because of the heaviness of the algebra and the uniformitywith which instabilityis reported (Knox, 1957a,b; Weibel, 1957; Whipple, 1959; Sagdeev, 1959; Yanliov, 1959; etc.). The cumulative effect of this theoretical work has been a serious discouragement to experiment in this field. However, all the stability analyses to which we have referred are based on the assumption that the quasi-metallic model of the plasma provides an adequate description both of the equilibrium and of the perturbed configurations. We saw in Section 2, however, that the use of this model of the plasma is never justified. The assumption th a t transverse electric fields vanish a t the plasma boundary only corresponds to the field configuration derived from the approximate self-consistent field equation

V XV X E

=

(w2/c2)eE,

(1)

228

H. MOT2 AND C. J. H. WATSON

where wo2/w2 >> 1. However, whenever wp2/w2 is sufficiently large to justify this neglect of the electric field at the plasma boundary, the electric field gradient obtained from (1) is so large as to invalidate the assumption upon which it was based. True, we argued in Section 2 that there may exist rf confined configurations for which wp2/w2 is much greater than unity, but for such configurations the depth of penetration of the rf into the plasma would have to be much greater than the depth given by (l),and again the use of the quasi-metallic model could not be justified. Furthermore, in such equilibria, if they exist, one could not distinguish the time scale for the motion of the plasma particles in the boundary layer from the time scale for the oscillations of the rf field, so the concept of radiation pressure could not be used to describe the confining force or to analyze the stability of the configuration. (It may be noted that these remarks do not in any way invalidate the use of Eq. (1) or the concept of radiation pressure to describe the interaction of an rf wave with a true metal, since, in that case, lattice forces ensure the confinement of the electrons.) I n this chapter, therefore, we shall ignore all stability analyses based on the quasi-metallic model and shall deal exclusively with the stability of those equilibria which satisfy Eq. (1) and for which that equation is in fact valid. We shall begin by exploiting the interesting, but by no means perfect, analogy between rf and magnetostatic confinement of plasma. The subject of magnetostatic confinement is complicated by the fact that the pressure is usually not scalar; for the purposes of the analogy, however, we shall restrict ourselves to the case where it is. It then follows, as is well known, that an equilibrium is only possible if there exist nested surfaces formed by magnetic field lines (so-called magnetic surfaces) on which the plasma pressure is constant. As we have seen, exactly the same is true (to first order in v/c) of the rf confinement of quasi-neutral plasma: we analogously have electric surfaces on which the plasma pressure is constant. A second common feature of the two confinement systems is the existence of flux tubes formed by lines of force, which are continuous except at singular points where E = 0 and contain a constant flux across any cross section. (This follows from V E = 0, the equation of quasi-neutrality.) I n the electric case, however, this constancy of the flux results trivially from the fact that the tubes are of constant cross section. A third common feature is the fact that in both systems the plasma particles are undergoing highfrequency low amplitude oscillations about the position of their guiding centers, intermediate-frequency oscillations of their guiding centers in the quasi-potential well created by the nonuniformity of the confining field and (though we have not actually proved it for the electric case) a slow precession or drift motion as well. [Higher order terms in the motion of a charged particle in an rf field are discussed in Litvak et al. (1962).]

-

R F CONFINEMENT AND ACCELERATION OF PLASMAS

229

Consequently, we might expect to find an analogous series of instabilities -low-frequency interchange or drift instabilities, intermediate-frequency instabilities due to resonance with the well frequency or improper velocity distributions, and high-frequency resonance instabilities. As regards the low-frequency instabilities, all the detailed analyses carried out to date have (as we see above) been based on a n inadequate model of the equilibrium. However it appears to be difficult to give a more satisfactory treatment. There is a temptation to provide a general stability criterion for interchange stabilities along the lines of the $dl/B criterion of Rosenbluth and Longmire (1957) for magnetostatic confinement. It is easily shown however that one cannot simply take over their argument, replacing magnetic flux tubes by electric flux tubes, since for a n rf confined plasma, the plasma is not “frozen” to the electric field, so there are no grounds for excluding nonconvective interchanges. Furthermore, the field energy within a flux tube cannot be expressed in terms of E2 alone. An alternative approach has been suggested by Fowler (1962), who pointed out that insofar as one can replace the rf field by an equivalent fixed quasi-potential J., his proof of the stability of any distribution function fo($rnv2 J.), which is a monotonically decreasing function of its argument, would be applicable. Since, for such a distribution function, the condition for plasma confinement requires that the electric field configuration should have the minimum E property, Fowler’s argument appears to provide the analog of Taylor’s proof of the stability of certain minimum B configurations. However, as we saw in Section 2, a satisfactory treatment of the self-consistent rf field within a plasma requires that we should take the distribution function to be a function of &m(v - (e/nz)JEdt)2 J. and it is not clear that Fowler’s proof can be generalized to cover such distributions. We now turn to the theory of intermediate- and high-frequency instabilities. Here, as in the case of magnetically confined plasmas, almost all the theory has been given for spatially uniform equilibria, it being plausibly maintained that the results can be taken over in some WKB sense for weakly nonuniform plasmas. The danger of this assumption in t,he magnetic case has already been pointed out (Watson, 1964) but one has to start somewhere. The basic kinetic theory of a uniform plasma containing a uniform rf field Eo sin w0t has been given by Silin (1965) who considered its application to the theory of high-frequency instabilities, and it has been used to discuss intermediate-frequency (streaming) instabilities by Gorbunov and Silin (1965). I t s application to high-frequency instabilities in a uniformly magnetized plasma has been given by Aliev et al. (1966). The following discussion is based on the mathematical techniques of this last paper, from

+

+

230

H. MOTZ AND C. J. H . WATSON

which the results of the other two papers follow as special cases. T h e equilibrium configuration of a plasma in the presence of a uniform rf field and a uniform constant magnetic field B is obtained by solving the Vlasov equation for each species:

which gives .fo = fo(v - v0(t))where f o is an arbitrary function and vo satisfies dvo/dt = (e/m)[Eo vo X Bo], an equation which can be solved exactly since Eo and Bo are spatially uniform. The velocity vo depends upon the sign of the charge and the magnitude of its mass; consequently, for Bo E 0 the equilibrium contains a uniform distribution of ions and electrons streaming past each other in an oscillatory manner; on this may be superimposed a net drift of ions with respect to electrons or of electrons past each other by choosingfo suitably. The effect of Bo is to superimpose a Larmor motion in addition to the above motions. Since Eo is assumed uniform, no drift of guiding centers occurs and no deviations from neutrality are required in the equilibrium. To discuss its stability we linearize the Vlasov equation about this equilibrium; the resulting linear equation can be Fourier-transformed in space (since the equilibrium is uniform) but not in time (since the equilibrium is time dependent). However, since the equilibrium is a periodic function of time with period 27r/w0, an analog of Floquet’s theorem shows that the solution can be expanded,

+

C m

fl

=

n=-m

jln exp[i(nwo

+

w

~

(3)

.

It proves convenient to work with each Vlasov equation in the (oscillating) frame in which the species concerned is a t rest in the equilibrium. If one considers only perturbations in which the perturbed electric field is derivable from a scalar potential, one obtains the following set of coupled equations for the n th components (in the above sense) of the perturbed ion and electron charge densities in the moving frames, pin' and p?), respectively,

-

where J,(a) is a Bessel function of order r and argument a = k L, L being the maximum displacement of a particle from its equilibrium position

R F CONFINEMENT AND -4CCELERATION OF PLASMAS

23 1

during one cycle of the rf field; arid & i ( w , k), &,(w, k) are the ion and electron contributions to the linear. dielectric coefficient appropriate to the electrostatic oscillations of a uniform plasma in the absence of the rf field (the total dielectric coefficient then has the form E = 1 8 ~ i a€,). Thus, in particular, for a cold unmagnetized plasma, 86, = - w i , / w 2 and 6Ei = - w $ / w 2 , where up,, up, are the plasma frequencies for the two species. If we define quantities Rln),RP) b y

+ +

Eq. (4) can be reduced to

and the spectrum of the perturbed osc*illationscan formally he obtained from the determiriaiital consistency condition for ( 5 ) , IM - 11 = 0. I n practice, however, some approximation has to be made; to show that this ispossible let us consider Eq. ( 5 ) . If we restrict attention to the case where wn is of order up, or higher (i.e., to those values of wa for which quasi-potential theory is valid), we see that all the quantities Rim’are much less than unity (by a factor of order m,/mi) except possibly for RI”, whose value depends upon o,the perturbed oscillation frequency. If this is large (w 2 wo ) , R!’” is also <> 1, the remaining Rk’) being -1. In this case, p:) >> pd’), (1 # n ) , and Eq. (5) approximates to

If, on the other hand, w << wa, the term in the summation with nz = 0 dominates and we obtain (on multiplying by J n and summing over n)

We will refer t o (6) and (7) as the high- and low-frequency dispersion relations respectively. It will be observed that if we switch off the external rf signal, so that J , -+ La, both dispersion relations reduce to the form 1 = RiR, or 1 6ti 6ee = 0 as required (in (6), however, the argument of the B E is w n w o ; this reflects the fact, which is evident from the exact Eq. ( 5 ) , that w is only defined up to an additive constant noo). The solutions of the high-frequency dispersion relation (6) fall into two classes, “resonant” and “nonresonant,” depending on the value of w o in relation to the parameters up, and 8,. This is due to the fact that for a = 0

+ + +

232

H. MOTZ AND C. J. H . WATSON

there exist certain frequencies or bands of frequency within which linear waves can propagate in the plasma. Thus, for example, if Bo = 0 and hence Q = 0 the plasma can sustain oscillations with frequency w 2 = w:, (3k2/2)vT2; for Bo # 0 but T = 0

+

w2

= + {W E e

+

Qe2

f [(WE,

+

Qe2)2

- 4w;,9,2

cos28 ] ” 2 ) ,

where 0 is the angle of propagation of the wave with respect to the direction of Bo. If wo approaches any such characteristic frequency of the plasma in the absence of rf, the external signal resonantly excites these oscillations. This phenomenon has been called by Silin a “parametric resonance,” and it almost invariably leads to instability. That such an effect should occur is clear from the form of Eq. (6) : in a first approximation, the characteristic frequencies 0 for linear oscillations without the rf signal are given by 1 6a,(G, k) = 0; (8)

+

and hence as nuo+ 0 the quantity R:) + f 0 0 , so the right-hand side of Eq. (6) passes through almost all values in this neighborhood, including the value 1. That an instability should result only follows from a detailed analysis which shows that

where q is a real quantity of order unity which is a measure of the displacement from exact resonance. For both values of the f sign, there exist values of q for which w turns out complex; the maximum growth rate is seen to be of order [ ( w : ~ / w ~ ~The ) Jcorresponding ~ ~ ] ~ ~ ~expression . for an unmagnetized plasma is obtained by taking the limit Q, + 0. I n this case there is only one band of resonant frequencies for given n ; the band is of width kmarvO(where k,,, is the upper limit on k set by Landau damping) and is bounded below by cope. The band structure is more complicated if BO # 0 ; the details are discussed in Aliev et al. (1966). A regrettable feature, however, is that one of the unstable bands is LO:, I wo2 I w i e 8,’. If wo is sufficiently (in practice, very slightly) larger than the maximum characteristic frequency 0 (which is of order i2= w;, fie2), this resonant high-frequency instability disappears. There is, nevertheless, a nonresonant instability derivable from the high-frequency dispersion relation if the distribution function fo which appears in the quantities 6e has, for example, a two-stream character, though the stability criterion is not in general the same as in the absence of the rf field. This question has been partially explored by Aliev and Silin (1965) who have considered the case of a cold unmagnetized plasma in the limit w o >> up,. If there exists a cur-

+

+

R F CONFINEMENT AND ACCELERATION O F PLASMAS

233

rent in the equilibrium, so th at the electrons are moving with respect to the ions with velocity v, the dispersion relation becomes approximately (the terms with n 2 1 are negligible)

and the instability criterion becomes

Thus, although this streaming instability still exists, the range of k-values which are unstable is reduced by the rf field, and it can be shown that the maximum growth rate is also reduced. An analogous analysis (by Gorbunov and Silin, 1965) of the case of two neutral plasmas streaming past each other, however, shows that in this case the field does not exercise a stabilizing role; nor does it do so if two electron beams stream past each other in a stationary neutralizing background. The implications of Eq. (6) for non-Maxwellian distributions do not appear to have been analyzed in the limit uo < upein any detail, though it can, for example, be shown that an rf field can stabilize an anisotropic distribution in a uniform magnetic field against the Harris instability (resonance between the electron plasmaand ion cyclotron-frequencies). Much remains to be done in this region, but the general impression appears to be that apart from the parametric resonance the rf field does not have a strong destabilizing influence on plasmas which would be stable in its absence. As regards the low-frequency dispersion relation, the picture is rather less clear. Equation (7), which can be solved for a cold magnetized plasma for all angles of propagation except the narrow cone around e = ?r/2 (which causes trouble even in the absence of an rf field), gives approximately

where A(wo,

k, =

T:

n'oo'(n2uo' (%ZWO2

-

- De2)J,'(a) - 6-2)

W+2)(n2W02

and Wrt are the roots of Eq. (8), which determines the characteristic frequencies for electron wave propagation. The expression for an unmagnetized plasma is again obtained by taking the limit D -+ 0. It is clear from (12) that the stability condition is A L 0. This is a criterion which, however, is not too easy to state as a necessary condition upon w 0 , though a sufficient condition is clearly wO2 > fie2, Wrt2; in the limit, a << 1, it is

234

H. MOTZ AND C. J. H. WATSON

easily shown that a sufficient condition for stability is w o > cope (though the actual threshold is substantially lower than this). It will be observed that A has singularities as n2w02-+ O*2. This reflects the fact that both the high-frequency and low-frequency dispersion relations approach the same expression in the neighborhood of the parametric resonances. Except in this resonant region, the growth rate for the low-frequency instabilities never exceeds a quantity of order wpi. To summarize, the current position as regards high and intermediate frequency instabilities of a plasma in an rf field would appear to be that there exists a dangerous, but avoidable, parametric resonance instability when the rf frequency wo approaches one of the frequencies a t which the plasma could propagate waves in the absence of the rf field, but that there are no clear signs that otherwise the situation is any more dangerous than for a plasma confined in any other way. The low-frequency instability regime has scarcely been investigated effectively, but such indications as we have are reasonably encouraging. 5. APPLICATION TO FUSION REACTORS

The motive behind much of the early work on rf confinement was the hope that it might be made the basis of a thermonuclear reactor. This hope was attacked as early as 1957 by Osovets, who pointed out that the rf cavities available a t that time seldom had a quality factor Q better than a few thousand and that, taking this figure, the dissipation of rf energy in the walls of the cavitywould be prohibitively large. Indeed, not onlywould this loss exceed the thermonuclear power output, but for any reasonable size of machine it would exceed the power which was available from any existing rf generating device. This conclusion was confirmed by the calculations of Weibel (1957) and, in consequence, interest in rf confinemen t declined. The position has recently improved in three important respects. First, the upper limit on the rf power available a t any given frequency has increased by several orders, so that, for example, megawatts of cw power are available in the 10-cm waveband. Second, the Q factors of cavities have been raised by over five orders of magnitude by constructing them from superconducting metals kept a t 2°K. Third, as we saw in Section 2, we now know that there exist equilibria in which the ratio of the volume filled by plasma to the volume filled by rf can in principle be made very large. We must therefore re-examine the question of the feasibility of an rf thermonuclear reactor. We may begin by considering what factors determine the viability of any given react.or design. There is necessarily an initial investment of energy, equal to 3nT per unit volume, where n is the

RF CONFINEMENT AND ACCELERATION OF PLASMAS

235

mean ion or electron density and I' is the temperature. If there is either a steady particle loss mechanism or if confinement breaks down after some finite time, there will exist an effective confinement time t for this energy, and one can therefore regard the initial investment as equivalent to a steady dissipation of energy at a rate Pi = 3nT/t. I n addition to this initial investment, there are running costs: the plasma radiates by bremsstrahlung at a rate PB,and the confining fields must be maintained at a level that ensures confinement, which involves dissipating energy at a rate P, in the cavity walls and at a rate energy P, in the plasma itself. Provided that confinement is maintained, however, thermonuclear reactions generate power at a rate P F and there is the possibility of a positive net output Po. These six quantities Pi,PB, Po, P,, P F , and POare functions of the parameters of the reactor n, T, w , etc. Our first step will be to show that practical considerations rapidly narrow down the interesting range of variation of these parameters. A useful reactor must have a reasonably large power output. Conventional reactors have a power density of around 100 W/cm3; this is perhaps uncomfortably high, but certainly a power density below 1 W/cm3 would make a reactor of reasonable total output (say 109 W) unmanageably large. This immediately sets a lower limit to the plasma density and pressure. The fusion power generated by a Maxwellian plasma is given by

P F = n2(av)&/4 where u is the reaction cross section, (av) the appropriate average of uv over the distribution function [evaluated by, for example, Thompson (1957)] and &R the energy liberated per reaction. Taking the D.T reaction for which &R = 17.6 MeV, the graph of (UV)as a function of T given by Post (1956), [confirmed by more recent work: Eder and Motz (1958), Wandel et al., 19581 shows that the maximum value of P F is 7.02 x 1 0 - 2 8 n2 W/cm3 (where n is measured in particles/cm3) and that the corresponding temperature is GO keV. Thus the minimum plasma density (giving P F = 1 W/cm3) is n = 3.78 X 1013 ~ m - ~The . corresponding plasma pressure, which must be balanced by electromagnetic pressure, is p = 2nT

=

7.25 X lo6 dyn/cm2 = 7.2 atm.

This pressure can be reduced slightly by working at a somewhat lower temperature: to determine the minimum plasma pressure we can use the fact that, for temperatures below about 20 keV, it is possible to use the analytical approximation of Gamow to represent the reaction rate (Thompson) :

( a )= 3.7 X 10-12T-2'3exp( -19.9T-1'3)

cm3 sec-';

236

H. MOT2 AND C. J. H. WATSON

(with T in keV). One can then write the plasma pressure at which any given level of fusion power PF is maintained in the form p = (PF/(UU)&)’”%” 0: T4’3 e~p(19.9T-l/~/2), and this has a minimum when T = (19.9/8)3 = 15.3 keV. At this temperature, PF = 1.42 10-2%2 W/cm3 and this reaches 1 W/cm3 when n = 8.42 X 1013~ m - the ~ ; corresponding pressure is p = 4.16 X lo6dyn/cm2. The electric field strength a t which &E2 = p is E = 3.06 X lo6 V/cm; such field strengths, though large, are almost within the reach of current rf technology. However, it would clearly be undesirable to work a t a higher plasma pressure than is absolutely necessary to achieve a reasonable PF. We shall therefore take n = 8.42 X l O I 3 ~ m and - ~ T = 15.3 keV as the basic design parameters for an rf thermonuclear machine, though we may note that there is the possibility of raising the power level of operation if the maximum field strength could be increased. In the above discussion we assumed a N‘axwellian distribution. As is well known, such a distribution is particularly suitable for thermonuclear purposes: the tail wags the thermonuclear dog. It is therefore very important that a t least a reasonable fraction of the tail should be confined. It will be seen from the graphs in Eder and Motz that particles with energies up to about 150 keV should be confined if a 15-keV plasma is to react a t a rate which approaches that derived on the assumption of a Maxwellian distribution. This condition in turn sets a lower limit on the ratio wP2/w2. To see this, we may refer back to Section 2, where we showed that (for an equilibrium computable by the quasi-potential approach a t least) the density distribution given by n

=

no exp( -e2E2/4mw2T) = noexp( -op2tOE2/w24n0T)

-

and that, for oP2/o2 > 1,the maximum value of E2in a confining cylindrical mode is such that eoE2/4noT 1. Thus, for any given wp2/w2, particles are confined which have an energy less than the maximum quasi-potential (wP2/u2)T;so if we are to confine 150-keV particles in a 15-keV plasma, wp2/w2 must be a t least 10. By the same token, if wpz/w2 > 230, we will confine the 3.5-MeV He4 reaction products as well. We shall see shortly that in practice it is essential for wp2/w2 to be larger than this; we should emphasize, however, that whereas quasi-potential theory undoubtedly remains valid for a 15-keV plasma with wp2/w2 = 10, for much larger values of wP2/w2, the plasma-radiation boundary cannot be described by this theory, and the uncertainty discussed in Section 2 hangs over such equilibria. For a plasma with n = 8.42 X 1013,up = 5.16 X 10”sec-l. If we take wP2/o2 = 10, this gives o = 1.63 X 10’’ sec-I, or a wavelength X = 1.16

R F CONFINEMENT A N D ACCELERATION O F PLASMAS

237

cm. For several reasons, this wavelength is unacceptably small. I n the first place, the rf power at present available in the 1-cm waveband is of the order of hundreds of watts (cw) only, whereas (as we shall see) an rf power which is not enormously smaller than the fusion power is certainly required. Secondly, it implies a very thin isolating layer of rf between the plasma and the cavity wall (for reasons connected with the near-degeneracy of modes in a large cavity, this layer cannot be more than a few wavelengths thick), and hence small amplitude oscillations of the plasma about its equilibrium position, even though stable, would tend to cause loss of plasma to the walls. Thirdly, the rf losses in the cavity walls, which increase as w3'2 for conventional conductors and as w 3 for superconductors, 10" sec-I. As we shall see, are in either case prohibitively high at w these problems become much less intractable for X > 20 cm: in this waveband 1 MW of cw per klystron is available, the isolating layer is of reasonable dimensions, and the cavity losses might just be brought under control. 1000, which stretches quasi-potential This, however, implies wp2/w2 theory beyond its limit so we cannot describe the equilibrium with any precision. In what follows, however, we draw only on the qualitative features of the quasi-potential equilibria. We shall first show that at the chosen temperature and density, the bremstrahlung is totally negligible and that the initial investment can be made so, given a confinement time of a few seconds. Spitzer gives the Born approximation :

-

-

PB = (26e6/3hm,c3) ( 2 ~ T / r n , ) ~ ~ ~ Z ~=n5.35 i n , X 10-31n2T1/2 W/cm3 ( T in keV). Various corrections to this have been proposed (see, for example, Wandel et al.) but none make any significant difference at 15 keV. For n = 8.42 X 1013crn4 we obtain PB= 1.74 X

W/cm3

=

0.017 PF.

On the other hand,

Pi

=

3nT/t = 0.62/t

W/cm3.

I n view of the inefficiency of most plasma heating processes, it would clearly be desirable to keep t reasonably long, say 100 sec; but if this were not possible one might hope to improve the efficiency with which this energy was recovered instead. We shall now consider the heating of the plasma which results from the penetration of the rf fields into it. There does not appear to exist any theory of the effects of strong electromagnetic fields on collision processes in a plasma or of the heating which might be expected to result; we shall therefore begin by exploring the consequences of the theory appropriate

238

H . MOTZ AND C. J. H. WATSON

to weak rf fields in a plasma with screened Coulomb collisions. This has been given by (for example) Silin and Ruchadze (1961),who show that the dielectric coefficient of the plasma acquires an imaginary part: Im E = ( w P 2 / w 2 ) ( v , f f / w where ) veff = 2 ( 3 2 ~ / 9 m ) ~ / ~ ( e ~ /log T t 'A,~ )log n A being the usual Coulomb logarithm which we take to equal 20. If we note that n, which appears explicitly in veff and implicitly in up,should be taken as the local value, i.e., n = no exp[ -eo(wP2/w2) (E2/4noT)], we see that the power dissipated in the plasma is given by

s

E 2 d3r = 4vonoT

J

= 4vonoT

1 up2eoE2

- - -exp 4 w 2 noT

)

w toE2 ( - 2 2 - d3r w2 4noT

E 2 exp( -2E2) d3r

where we have reverted to the dimensionless electric field of Section 2 and have written vo = v,ff (n = no).With T in keV,

P,

=

1.17 X 10-26

no2log A

J

T"2

E 2 exp( -2E2) d3r,

in watts, and, with the chosen values of n and T ,

P, = 4240JE2exp(-2E2) d3r. I n order to compare this figure with PB = 1 W/cm3, we need to know the precise plasma configuration-i.e., the plasma volume V , and the value of the integral $E2exp( -2E2) d3r. Even for those configurations for which quasi-potential theory is valid, the integral $E2 exp( -2E2) d3r can only be performed numerically or by elaborate approximation. Since for wp2/02 1000 quasi-potential theory cannot be better than a first approximation and must break down badly in the region of space which contributes most to this integral, we shall simply estimate the integral as 47rr2sa where r is the radius of the plasma (assumed spherical for simplicity), s is the depth of penetration of the rf into the plasma (a depth of order c / w , according to quasi-potential theory but presumably here somewhat greater), and a is the maximum value of the integrand, numerically 0.183. Thus, if we define the volume of the boundary layer V , = 4748,

-

P,

= 4240aVs/V, = 704V,/V, W/cm3.

It may be remarked that this (on any reasonable assumption about V , / V , ) very large quantity, deduced from linear theory, probably enormously exaggerates the heating effect, since (as the theory of electron runaway shows) the viscous drag of a plasma on an electron moving in a strong electric field, such that V B / V B wp2/w2 >> 1 , is very much less than

-

239

RF CONFINEMENT AND ACCELERATION O F PLASMAS

the drag on a thermal electron. Thus, although on the above linear theory the plasma heating would exceed the fusion power and would in consequence require a very efficient heat recovcry procedure at the end of the reaction cycle, it is probable that in practice P, has a negligible effect on the power balance, though it remains a serious burden on the rf generator. Finally, let us consider the rf losses in the cavity walls. Since these depend upon geometric factors and are roughly proportional to the surface area of the cavity, it is necessary to decide upon its shape. It is clear that the most favorable arrangement would be a large spherical plasma, isolated from the cavity walls by a thin rf layer of thickness d = X/2. As the radius r of the plasma tends to infinity (though practical considerations would probably dictate an upper limit of say r 101) the rf configuration in this layer would asymptotically approach that of a plane wave incident on a plane conducting surface. The energy loss resulting from the finite conductivity u of this surface is given by Panofsky and Phillips (1962) as

-

d&/dt =

JNd S

-

-

= w6$+poR2n d S = w 6 + E o ~ ~ , , $ n d S = 2w6noTS

where N is the Poynting vector, J d S is a surface integral over the cavity wall, n is a unit vector in the direction of N (which in the plane wave limit is normal to the wall), and 6 = ( 2 / p ~ o a ) ~is/ ~the depth of penetration of the rf into the wall as a result of its finite conductivity. I n deriving the last two expressions, we have used the fact that for a plane wave + p B 2 is maximum at the wall and equals +eoE2 at one-quarter wavelength inside the wall, where E = E,,,, and have used the result of quasi-potential theory to replace +coE;,, by 2noT. For a spherical plasma of radius r we can express the surface area S = 4s(r d ) 2 in the form S = (3/r)(l d / r ) 2 V , and hence we can obtain the power per unit volume of plasma P, required to maintain the cavity losses:

+

+

=

1.25

(2) [

f(1

+

W/cm3.

The expression in square brackets, which depends simply on the size of the configuration, cannot for practical reasons be indefinitely small. If we take &as its lower limit, we see that for positive power balance ( P P> P o ) , there is a very stringent upper limit on w 6 / d 16. If we take d = X/2, we obtain w26 < 1.5 x 10l2. For copper, 6 = 1 6 . 5 / ~ " ~and , we obtain a maximum frequency w = 2.0 X lo7sec-' and, consequently, wp2/w2 lo*. Frequencies as low as this could not be used to excite the natural modes of a cavity of reasonable size; it would therefore be necessary to use forced

-

-

240

H. MOTZ AND C. J. H. WATSON

oscillations, and the electromagnetic field distribution would be quasimagnetostatic. Such an approach, though possibly feasible, could not be discussed by any of the methods developed in this article, and we shall not w / Q , this argument is essenconsider it further. Since for d = X/2, w 6 / d tially the same as the one which led Osovets to conclude that there was no possibility of using rf confinement as the basis of a thermonuclear reactor. If we allow the possibility of using superconducting cavity walls, the position at once looks more attractive. The theory of superconducting rf cavities, and some very promising experimental results, are described in an article by Schwettman et al. (1964). Since the falloff of the electric field in a superconductor is not exponential, one cannot use the analysis given above to calculate the power loss, but from measurements of the Q of normal and superconducting cavities excited in the same mode, it is possible to determine an experimental effective 6. In the experiment quoted, the Q of a superconducting lead cavity at 1.5"K excited in a TEoll mode a t 2856 Mc/sec was a factor 1.2 X lo6better than that of a copper cavity at room temperature, and even this was a factor of 25 lower than might be expected theoretically. This gives an effective 6 of 10-9 cm, or w 6 / d = 3.4 sec-I. Furthermore, theoretically, the effective skin depth should increase linearly with w, so w6/X a w3, and by working with frequencies even slightly lower than that quoted (for which X = 10 em) a dramatic improvement should be obtained. Electric field strengths of lo6V/cm have already been realized; these correspond to magnetic field strengths of 300 G a t the cavity walls, and the limit appears to have been set by the magnetic field at which the superconducting property was lost. The rather low temperature of operation was likewise dictated by the low electron pairing energy in lead. Thus there is hope that both the critical field strength and the operating temperature could be raised somewhat by a judicious choice of superconducting materials. The question of the operating temperature is very important, since the rf energy which is dissipated in the walls must be pumped away if they are to be kept superconducting, and this involves a refrigeration effort which is by no means trivial. Carnot's theorem shows that there is a limit to the efficiency with which this can be done, equal to T 1 / ( T z- T I )where T1 is the temperature of the superconductor and Tzis room temperature. At 15°K this is already a factor of 1/200, and existing refrigerators do not approach even this low efficiency, though refrigerators acting on the Stirling cycle will soon be available which approach the Carnot efficiency. Thus the energy which must be supplied to maintain the superconductivity is qPc,where q 2 200 at 1.5"K,although it would be reduced substantially if one could use superconductors with a higher critical temperature. Thus if we take X = 10 cm, d = 5 cm, r = 100 cm, and q = 250, we obtain

-

R F CONFINEMENT AND ACCELERATION O F PLASMAS

24 1

P, = 53.8 W/cm3. This figure is still unfavorable, but improvements in the working temperature of the superconductor, or even the realization of the theoretical effective skin depth a t 1.5"K, might bring this loss down to the break-even point, and a reduction in frequency (with a consequent scaling up of the size of the reactor, if we keep d / r fixed) would then give a positive net output Po. Finally, some comment should be made on the permissibility of maintaining a superconductor close to a thermonuclear plasma, since the above calculations have been made on the assumption that the cavity is entirely lined with superconductor and contains no absorbent material other than plasma within it. This assumption is, as it stands, quite unrealistic, since although the charged reaction products remain confined, the neutrons and bremsstrahlung would strike the cavity walls. Althoughit might be feasible to allow the neutrons to pass through a thin superconducting layer, the bremsstrahlung would almost inevitably be absorbed, and the need to pump its energy away would again make the reactor unworkable. Two alternative approaches to this problem suggest themselves. (i) Insert a cooled heat shield of some rf transparent material between the plasma and the cavity wall. For cavity walls of low conductivity such a shield, if made of a low loss dielectric such as quartz or titanium might even enhance the oxide (for which the loss tangent tan A Q value, in the manner discussed by Walker and Hyman (1958); for a superconductor, however, it would cause an unacceptably large loss (this can be seen from the fact that it would give the part of the rf filled volume occupied by it an imaginary dielectric coefficient I m r = Re e tan A which was much larger than that of the plasma). (ii) Break up the cavity surface into a finite number of filamentary superconductors. The theoretical problems connected with such an approach have still to be investigated. We may conclude that the advent of superconductors, though it reopens what appeared to be a closed issue, by no means solves the problem of designing an rf thermonuclear reactor.

-

6. EXPERIMENTS RELATED TO RADIO-FREQUENCY CONFINEMENT

A . Introduction One of the striking features of the"subject of rf confinement is the very limited extent to which any of the (by now) very extensive body of theory has been tested experimentally ; furthermore, the measurements made in the few experiments which have been reported are in almost every case very incomplete. This state of affairs is perhaps hardly surprizing in view of the doubts which have been felt about the practicability of devices based upon rf confinement, but it seriously complicates the task of the

242

H. MOTZ AND C. J. H. WATSON

present authors in giving a straightforward account of the extent to which the theory has been confirmed, since the evidence is for the most part indirect and ambiguous and was in several cases collected for other purposes. We shall begin by reviewing the experiments which establish that, in plane and cylindrical geometrics at least, rf waves do set up a quasi-potential barrier = e @ / 2 m d which is capable of confining or repelling single particles. We shall then describe the experiment which shows that a cylindrical barrier can be used to focus electron beams by balancing the dispersion space charge forces, and the very few experiments which provide direct evidence for at least partial confinement of a quasi-neutral plasma. Finally we shall mention an experiment which provides indirect evidence for the existence of a quasi-potential relief.

+

B . Single Particle Conjinenaent The most convincing quantitative experiments on the confinement of single particles were performed by Bravo-Zhivotovsky et al. (1959) of

FIG.6

Gorky University. The first experiment, which was designed to test the ability of the quasi-potential barrier to cut off an electron beam, used a rectangular resonance cavity 2.85 X 1.25 cm in cross section, irismatched to a waveguide feed and tuned by a plunger in the manner indicated in Fig. 6. The cavity was excited in TElon(n = 1,r2, . . . , 5) modes at a frequency w = 6 X 1 O l o sec-'. This configuration has a quasipotential maximum at the center of the cavity. The height of the barrier is related to the power P supplied to the cavity according to

RF CONFINEMENT A N D ACCELERATION OF PLASMAS

243

where Q is the quality factor of the cavity, V its volume, and a a factor relating the maximum value of 82 to the average value over the cavity, which in the present experiments had the value 0.47. Electrons from a gun mounted in a 6-mm-diameter (cutoff) cylinder in the broad face of the cavity could be shot across the cavity to be collected by an electrode at the end of a similar cutoff metal tube in the opposite face. A weak focusing magnetic field of between 10 and 100 G was used to reduce dispersal of the beam in transit. For each of the modes referred to, the rf power P required to cut off the beam was measured as a function of the voltage V , on the electron gun. The results for n = 1 and n = 5 are shown in Fig. 7, together with the measured values of Q in the two cases. The power level is in watts for n = 1 and kilowatts for n = 5; the solid line gives the theoretical value obtained from the above formula. It will bc

FIG.7

seen that the experimental results agree with the theory to within experimental error (-7%). Another experiment carried out by this group involved a cylindrical quasi-potential well, established by exciting a helical slow wave structure of diameter 0.59 cm and pitch 0.03 cm with a 10-cm traveling wave. The use of such helical structures leads to a great economy in the use of rf power, as we shall see shortly; their disadvantage, from the point of view of conducting an experiment designed to test the quasi-potential concept, is that they create slow traveling waves. Such waves, as we saw in Section 1 have a spatial nonuniformity on a length scale much shorter than the vacuum length scale, and, in consequence, the analysis given there does not prove that the quasipotential concept is applicable in this case. However, as shown by Litvac et al. (1962), it is still applicable provided that the velocity of the particle not only remains much less than c (the condition assumed in Section 1) but also much less than the phase velocity of the slow wave v+. (Essentially this is because, under this stricter condition, the force exerted by the magnetic field is still much smaller than that

244

H. MOTZ AND C. J. H . WATSON

of the electric field and the particle still executes high-frequency oscillations of small amplitude about a guiding center.) I n this experiment v4 c/30 and the electron velocities were always less than, and for the most part much less than, v4 so the experiment does provide a test of quasi-potential theory. Electrons from a gun were fired along the axis of the spiral. Over a short section of their path (1 = 3 cm) they were subjected to a weak perpendicular magnetic field ( B 1 G), as a result of which they acquired a directed perpendicular motion vL = eBol/m that could be varied in the range lO6-lO9 cm/sec and was always significantly greater than the random perpendicular velocity of the electrons resulting from imperfect beam collimation and space charge effects. I n the absence of an rf field on the helix, this perpendicular motion led to a loss of electrons to the walls of the tube, and no electron current was detected. An rf field on the helix, however, creates electric fields

-

-

E , = EoIo(yr)exp[i(wt - kllz)] and

E , = (ikli/r)EoZl(rr)exp[i(wt - kiiz)l. Hence a quasi-potential well given by

+

$ = ( e 2 E ~ 2 / 2 ~ n o 2 ) [ l ~ 2 (11~(yr)l rr>

(2)

is set up, where lo,I l are modified Bessel functions; y is the perpendicular wave number, given by y2 = k112 - k2 = ( w 2 / v m 2 > ( 1- vO2/c2);and Eo is the electric field strength on the axis, a quantity which is related to the power supplied to the helix P by Eo2 = aP,where a is a constant expressible in terms of the dimensions of the helix. [Expressions for the spatial variation of the fields and the impedance of the helix are to be found in, for example, Pierce (1950).]It was found that a sufficiently large quasipotential prevented loss of electrons and led to the detection of the full beam current at the far end of the helix; the power needed to achieve this was shown to be proportional to the magnitude of the perpendicular energy V given to the beam by the uniform magnetic field until the latter was so large that the condition vL << v+ ceased to be met. Some typical experimental points are shown in Fig. 8. In determining the conditions for beam confinement on the basis of Eq. (2), one should of course bear in mind the fact that although the particle guiding center moves under the influence of the quasi potential, the particle itself executes oscillations about this. Thus, particle loss occurs where the turning point of the guiding center lies inside the spiral by an amount eE,/mw2.However, the difference is negligible whenever vL << v4, and the solid line in Fig. 8 gives the constant of proportionality obtained from (2) if one simply requires the

245

R F CONFINEMENT AND ACCELERATION OF PLASMAS

guiding center turning point to lie inside the helix. It will be seen that there is good agreement between theory and experiment until V = 60 eV (i.e., vL = 0 . 4 6 ~ ~ ) . Another experiment on single particle confinement by a cylindrical quasi-potential well has been described by Weibel and Clark (1961). This used the quasi potential of a fast standing wave in a cavity which was designed to create a field distribution approximating that of the TEol mode in an infinite cylindrical waveguide excited at cutoff. I t is no trivial

P (watts)

FIQ.8 RF power from magnetron

4 ) 1 21fcm

Scope

4

--

Spreading beom without confining field

d c m

-12.1cmr \ [\Pierce

- gun

Probe

Confined beam

End resonator

FIG.9 matter to realize such a field configuration; a cutoff mode can only exist in an infinite waveguide, and the equivalent mode of a cylinder of finite length shows considerable variation in the z (axial) direction. To minimize this, Weibel and Clark used a cavity consisting of two cylindrical sections of slightly supercritical radius joined by a long cylindrical waveguide of slightly subcritical radius. Weibel has shown that, provided the bore of this latter tube is constant, to 1 part in lo4, the field strength along it is constant to within 3.5 % and closely resembles the TEol mode at cutoff in an infinite waveguide. The dimensions of the cavity are shown in Fig. 9.

246

H. MOTZ AND C. J. H. WATSON

The quasi-potential in such a cavity is given by

#

=

e2E2/2mw2= (e2B02/2nau2) JI2(ur/c)

where Bo is the rms magnetic field strength on the axis and J 1 is a Bessel function. In these experiments the cavity was fed with 250 kW of rf power at 9.29 kMc in pulses of 2-psec duration. The loaded Q of the system was of order lo4,and hence Bo = 212 G and the peak electric field 52 kV/cm. Thus the quasi-potential maximum was 400 eV. Electrons were injected into the cavity at one end from a Pierce gun delivering 0.3 mA at a voltage of 100 V. This corresponded to a beam ~ ;such densities, as Weibel and Clark show, the density of order lo8~ m - at effect of space charge is negligible compared with the transverse velocity possessed by the electrons as a result of the imperfect collimation of the &. beam from the gun; geometric considerations indicate that v,/vll An axial collector probe of diameter 3 mm was situated at the other end of the cavity. In the absence of rf, virtually no electrons reached this probe, as a result of beam dispersal. During the rf pulses, however, about 90 % of the electrons reached the probe, though small fluctuations were observed in the current, possibly indicating oscillations of the confined electron beam. I n view of the small role played by space charge forces, one can calculate the expected beam diameter on single particle theory, obtaining 1.26 mm. Thus albhough a quiescent beam of this diameter would be entirely collected by a correctly aligned probe of 3-mm diameter, finite oscillations of the column could easily account for the observed fluctuations.

-

C. Electron Beam Focusing

In the above experiments the circumstances were such that the effects of space charges were negligible. One possible application of rf confinement, however, is the use of quasi-potential forces to balance the dispersive space charge forces in an electron beam. Only one experiment on this application has been reported-that of Birdsall and Rayfield (1964). This used a helical slow wave structure qualitatively similar to that used by Bravo-Zhivotovsky et al. (1959), but instead of deliberately creating the perpendicular motion as in the earlier experiment, they carefully collimated the beam in such a way as to minimize this. Under these conditions, the spreading of the beam is determined by space charge forces and it is possible to work at much lower rf power levels. A second difference from the earlier experiment was that by varying the voltage on the gun, electrons could be injected with velocities either larger or smaller than the phase velocity of the wave. A schematic representation of their apparatus is shown in Fig. 10. An electron gun giving a 1-mA beam of

RF CONFINEMENT AND ACCELERATION OF PLASMAS

247

diameter 0.18 cm was collimated and accelerated by the anodes A1 and A2 and injected into a helix 0.53 cm in diameter and 20 cm in length, wound in such a way as to give a phase velocity of 1.4 X 109 cm/sec (i.e., equivalent to an electron energy of 550 eV) when excited at 1.2 knlc/sec. The beam energy could be varied in the range 100-1 100 eV. The current in the beam focused by the helix was measured by means of a collector probe designed to capture all electrons reaching the far end. I n the absence of rf, no current was registered at this collector. For ve << IJ+, the current registered rose rapidly as the power supplied to the helix was increased, reaching a maximum of about 0.4 when the power level was 4 W. Higher power levels reduced the current reaching the collector, a phenomenon which they attribute to “overfocusing.” The loss of 60% of the beam even under optimum conditions is not explained ; it may conceivably have been Pin match

Stream ‘envelope no rf

stream envelope with rf

FIG.10

due to a slight misalignment of the gun with respect to the helix, since as Weibel and Clark’s calculations show, a relatively small directed v L can dominate over space charge effects, or it may have been due to the dishrbance of the rf field configuration by the presence of the injection anode. Experiments with an earlier variant of the apparatus, having a larger diameter helix, showed transmission of up to 87% of the beam, the percentage transmission varying with the magnitude of ve and the direction of propagation of the wave. As v, approached v+, the focusing action fell off rapidly, becoming negligible for (v, - v+)/v+ 5 +-.For v, >>v+, however, a focusing action was again observed; higher power levels were apparently required, but their experiments did extend up to values of v, high enough to measure the focusing in the limit v, >> v+ (the electron gun used worked poorly at higher voltages). In discussing these experiments, we shall begin by considering how the interpretation given by Birdsall and Rayfield, based on the analysis given in the theoretical part of their paper, is related to quasi-potential theory. As we have seen, for v+ >> v,, the results of quasi-potential theory should be applicable and hence the cylindrical potential well given by (2) should be set up. In the absence of perpendicular motion resulting from imperfect

245

H. MOTZ AND C. J. H. WATSON

collimation, the condition for beam focusing is that the quasi-potential force at the edge of the beam should balance the electrostatic force on an electron there. This force is given by (see the paper of Weibel and Clark for a detailed discussion) F = eE, = (eZ/roto)J2n(r)rdr where n is the electron density at the point r and ro is the radius of the beam (a quantity which is perfectly well defined, since by hypothesis there is no thermal spread). This can be re-expressed in terms of the total beam current l o o and its velocity voo as F = e l 0 0 / 2 a r ~ t ~Hence v ~ ~ . the quasi-potential force balances the space charge force a t the boundary if

For voo 2 vd, the quasi-potential approach becomes inapplicable, since the Taylor series expansion of E(R e) about E(R) fails. However, as Birdsall and Rayfield point out, an approximate method of solution can still be developed, since for the slow waves set up within a helix, the magnetic field strength is very small. This is most easily seen by observing that, apart from quantities of order vd2/c2, the electric field components quoted above can be derived from a scalar potential

+

4 = --+Ar = V l l o ( y r ) cos(wt - kllx). Hence V X E = 0 apart from terms of order vp2/c2, and B vanishes in the same approximation. Thus, as we showed in Section 1, by making a Galilean transformation to the frame in which the wave is at rest, one can demonstrate that the actual particle motion is approximately the motion of a particle in an electrostatic potential field 4 = VlI0(yv) cos(kl1z’). However, the problem of the motion of a charged particle in a periodic electrostatic field has been discussed by several authors (e.g., Clogson and Heffner, 1954; Tien, 1954); they show that the field has a focusing effect on a particle moving parallel to the symmetry axis. I n Tien’s analysis, which is more general than Clogson and Heffner’s and gives their results as a special case, a beam of electrons carrying a current lois considered to move along the symmetry axis in an electrostatic potential V , sin fix, with a velocity v. which is modulated as a consequence of the electrostatic potential about an unperturbed velocity vo. The modulations are treated as small (Iv, - vo1 << vo), and hence the space charge force on a peripheral electron can be taken as F = elo/2arotovo.Expanding the equations of motion about a rectilinear orbit at the radius yo and solving the inhomogeneous Matthieu equation obtained by means of a Fourier series in kip, he obtains the condition that such a periodic field should maintain the

249

RF CONFINEMENT AND ACCELERATION OF PLASMAS

beam at a mean radius ro in spite of the space charge forces as

This expresses the condition of equilibrium in the wave frame; to obtain the corresponding condition in the laboratory frame, we need to use the transformation v o = voo - u+ and Ioo/voo= I O / V Owhere , l o o and voo are the beam current and velocity in the laboratory frame. Thus (4) becomes

Equation ( 5 ) is strikingly similar to Eq. (3), and, indeed, using E it is clear that we can write ( 5 ) as

=

-V4,

Thus, for c >> vo >> voo, we recover quasi-potential theory, as we should since with this ordering of velocities both the quasi-potential and electrostatic focusing approaches are valid. For voo >> v+, on the other hand, the electrostatic theory shows that the quasi-potential force is diminished by a factor ( ~ + / v O o ) ~Provided . that this force is sufficient to maintain confinement, the assumptions underlying ( 5 ) remain valid. As voo tends toward v+, however, the assumption that the electrostatic field only modulates predominantly rectilinear orbits becomes invalid, and it is clear that the particles become free to escape in the radial direction at each of the nodal planes of the electrostatic potential. It will be seen that the predictions of both these approaches, which coincide when their conditions of applicability overlap, are in qualitative agreement with the experimental results obtained by Birdsall and Rayfield. In view of the partial beam loss which occurred even under optimum conditions, it does not appear to be possible to claim quantitative confirmation. In their summary, these authors make some remarks about the power requirements of this method of focusing, as compared with focusing by fast traveling waves, which are somewhat misleading. They point out that the power required (-15 W) to confine their beam was smaller than the power used by Weibel and Clark to confine an “almost identical” beam using fast waves by a factor of 16,000, and they speculate about the possibility of turning this power saving to good account in a fusion machine. However, this factor of 16,000 arises from a combination of two circum-

250

H. MOTZ AND C. J. H. WATSON

stances. In the first place, they arranged by beam collimation for the random perpendicular motion of the electrons to be negligible (i.e., they used an essentially “cold” electron beam), whereas in the experiments of Weibel and Clark, the thermal motion was the dominant cause of beam dispersal and hence required a substantially deeper quasi-potential well. Thus, the beams were not “almost identical” in the relevant respect. I n the second place, it is characteristic of a helical slow wave structure that it can be excited a t a much lower frequency than a waveguide of the same dimensions. I n the experiment of Weibel and Clark, for example, the frequency was 9.3 times higher than in that of Birdsall and Rayfield. The power required is, of course, sensitively dependent on the frequency of operation, and this is the main reason why slow waves are a more economical means of focusing electron beams than fast waves. This dependence of t.he power required upon frequency is a combination of two factors: the power loss for a given electric field strength varies as w 3 / 2for normal conductors and as w 3 for superconductors (as we saw in Section 5), whereas the mean square field strength required to maintain a given height of quasi-potential barrier increases as w2. Thus the power required to confine 1 normal ~ single particles of given transverse energy increases as ~ ‘ for metals and as w6 for superconductors. A factor of (9.3)’12 already accounts for most of the difference between the power consumptions in the two experiments. However, in a fusion machine, it is necessary to confine not single particles but a high density plasma, and its pressure has to be balanced by electromagnetic pressure, so the extra factor of w2 associated with the quasi potential does not arise. Nevertheless, the possibility of using a helical structure instead of a waveguide in a fusion machine needs examination.

D. Direct Evidence oJ Radio-Frequency Confinement of Plasma There has been a tendency in the literature to interpret all observations of plasma confinement in which an rf field is present as resulting from the establishment of a quasi-potential well. Thus, for example, Thompson’s dark zones in electrodeless discharges, the confinement of plasmoids between the plates of a condenser fed with rf, and the experiments of Birdsall and Lichtenberg on plasma confinement on the axis of a 3-cm helix fed with rf power in the 3- to 25-Mc/sec frequency range have all been discussed from this point of view. An interesting paper by Butler and Kin0 (1963) points out the wrongness of these interpretations; in each of these cases, the wavelength of the rf is so large compared with the dimensions of the apparatus that the rf fields are not even approximately natural modes for the system. Thus the field nonuniformity is on a length scale which is so short compared with the vacuum length scale

RF CONFINEMENT AND ACCELERATION OF PLASMAS

251

that the quasi-potential concept becomes inapplicable. True, we have just seen that it may still be used t o discuss the action of a helical slow wave structure provided that c >> u+ >> u,; however, in the experiment of Birdsall and Lichtenberg (1959) u+ is so much smaller than the thermal velocity of the electrons that not only is the quasi-potential concept inapplicable, but the electrostatic periodic focusing would be negligible as well. An alternative explanation of the plasma confinement in this and other experiments, in terms of the formation of a positive ion sheath, was suggested by Sturrock (1959); his theory has been extended and experimentally corroborated by Butler and Kino. Measurement of plasma concentration

To

I electromaanetic field I

L To the vacuum system

Nevertheless, in the view of the present authors, a number of experiments have confirmed the possibility of confining a quasi-neutral plasma in a quasi-potential well. The earliest of these are the Russian experiments reported at the Geneva and Salzburg conferences by Glagolev and his co-workers (see Vedenov et al., 1959; Arsenev et al., 1961). Their Geneva paper describes an experiment in which a uniform magnetic field was used to confine a cylindrical plasma in the radial direction, and two rf cavities were used to ensure confinement in the axial direction. A schematic representation of their apparatus is given in Fig. 11. The two rf resonant cavities, which had cutoff openings to receive the quartz tube within which the plasma was created, were excited in the TElol mode by pulses of up to 400 kW of rf power of duration 120 psec supplied by a magnetron operating in the 10-cm waveband. The peak rf magnetic field was 60 G, whereas the uniform magnetic field was in the rangc 0-2000 G. Thus tlhe electron cyclotron frequency could be made greater or less t,hari the rf

252

H. MOT2 AND C . J. H. WATSON

frequency, though the significance of this for confinement is not discussed in the experimental part of the paper. The plasma was created within the quartz tube (of length 40 cm and diameter 1.5 cm) by means of the rf pulse subsequently used to confine it. The ionization process was observed to initiate a t the ends of the tube and to spread down it to the middle. The plasma density was determined by measuring the frequency shift of the measuring resonator shown and by measuring the cutoff frequency for propagation of rf waves with wavelengths in the range 3 to 0.8 cm. The behavior of the apparatus depended on the gas pressure and rf power used. For example, with argon a t 3 X mm Hg a plasma of density 1013ern+ was formed with an rf pulse giving a maximum magnetic field of 30 G ; and under these conditions the confining resonators were highly detuned during part of the pulse, indicating penetration of the plasma into them. When the rf field strength was increased to 60 G, thc plasma density reached a t this gas pressure was unaltered whereas the detuning of the resonators was reduced almost to zero, though a t higher gas pressures, a substantial detuning associated with a higher plasma density was again observed. Similar results were obtained with other gases. To assess the significance of these observations, they measured a t low rf power the detuning of the cavity produced by a measured density of plasma in it, and they established that a plasma of density above 10''' shifted the resonator frequency outside the recaption band for the rf supply system. Thus, their experiments showed that a 60-G rf field could confine a plasma of density 1013cm+ in such a way that the density in the cavities did not exceed 1O1O cm-2. The temperature of the plasma was estimated from the duration of the afterglow as being of order 5 eV. This gives a plasma pressure 80 dyn/cm2. The rf pressure was estimated from the condition which would hold in plane geometry:

+

+e08~+ p o R 2 = + p o 8 &

=

70 dyn/cm2.

Thus, the condition for pressure balance is seen to be satisfied to within experimental error. I n their second (Salzburg conference) paper they described two further experiments. The first was so similar to the experiment described a t the Geneva conference that it is not clear what motivated it. The chief differences were t,he use of a single resonant cavity in place of the two used previously and the employment of a separate plasma injector instead of using the same rf field both to create the plasma and confine it. This latter feature had the advantage that it was possible to vary the plasma density independently of the rf field strength and hence to give a more unambiguous demonstration of confinement. Its disadvantage was that the maximum plasma density attained was an order of magnitude lower. I n this experi-

RF CONFINEMENT AND ACCELERATION OF PLASMAS

-

253

ment, a TElll mode was used, and the maximum field strength was raised to 250 G ( E 100 kV/cm) and the duration of the pulse extended to 1 msec. Furthermore, the plasma density within the resonator was determined by exciting it simultaneously in a low amplitude TMolomode and measuring the frequency shift. I t was shown that at full rf power a plasma of density 10l2 ern+ could be expelled from the cavity, the leakage of plasma back into the cavity not exceeding 0.01 %. The temperature of the plasma is unfortunately not quoted, but it was presumably of the same order as in the previous experiment, so the published measurements on this apparatus do no more than confirm the interpretation given to the measurements made previously. Their second experiment was primarily related to the theory of rf stabilization of plasmas confined by magnetic fields, a subject which has received considerable attention at the Kurchatov Institute (Volkov, 1959; Osovets, 1959; Volkov and Kadomtsev, 1962), but which lies outside the scope of the present review, since the rf frequency or field configuration are chosen in such a way that the dominant force acting on the plasma is due to the magnetic component of the rf field. I n the experiments described above, the rf field was used to maintain confirlenient in one direction only, that of the magnetostatic field. A more ambitious experiment has been undertaken by Consoli’s group a t Saclay, with the object of demonstrating three-dimensional confinement exclusively by rf fields and by rf fields in combination with a magnetic mirror field. In their experiments a spherical cavity was used; in their original design of the experiment it was intended that this should be used to realize the field configuration proposed by Butler et al. (1958)-the rotating mode obtained by exciting the TElll and TEllo modes in quadrature. However, they encountered experimental difficulties in achieving this at high frequencies, and the work reported in the literature describes almost exclusively the effect of a TEllomode in combination with a magnetic mirror field orientated along the symmetry axis of the rf mode. I n 1962, Consoli and Le Gardeur (1962b) reported their first experiment on combined rf magnetic mirror confinement; they used a plasma created by a discharge passing through a spherical resonator supplied with rf from power triodes at 1135 Mc/sec, which is the resonant frequency of a sphere of diameter 38 cm excited in the TEllo mode. I n a regime with pulses of 100 psec. the power level was 20 kW while in the continuous regime the power level was 200 W. The apparatus is shown in Fig. 12. The plasma from the discharge is sketched in, as well as field coils for a static magnetic field hnd a probe which measured the radial flux of particles escaping from the resonator. The arc discharge was run at 100 A in the pulsed regime and at 8 A for

2ti4

H . MOT2 AND C. J. H. WATSON

continuous operation. The density in the center region was measured by microwave interferometry (2-mm wave equipment) and reached 7 X 10ls electrons per cubic centimeter. The ion temperature was measured spectroscopically by observation of the H a line to be -2eV. Gases used were Hz, He, Ar. In the pulsed regime, the following evidence for confinement was obtained. Application of the rf increased the density by a factor of 2 to 3 and increased the decay period of the density when it was switched on n

-

Triode generator 5 0 kW Dulsed

-

2

Particle collector

.. Insulation

Variable feadback

, /

Cathode

H.E voltage

of gas

7

P-

P

-r---t 1 2 0 0

I

FIQ.12

during the afterglow of the discharge. The density increase could not be ascribed to ionization by the rf because at pressures of 4 x mm even full ionization would only give an increase of 10%. I n the continuous regime, the particle flux measured by the probe decreased when the rf was applied. The experiment was repeated for various values of the magnetic field, and Fig. 13 shows the results registered on an oscilloscope with a persistent screen so that the various deflections form a cont,inuous curve. The uppermost trace shows the particle flux in the presence of the magnetic field but without rf. The trace below shows the reduction in flux obtained by the application of rf. The lowest trace shows the ioniza-

RF CONFINEMENT AND ACCELEHATION OF PLASMAS

255

tion obtained by the rf alone when the discharge was switched off. High ionization was achieved when the rf frequency approached the cyclotron frequency, which happens near 400 G. The effect of this ionization tends, of course, to diminish the reduction in flux due to confinement.

I#

= f(B) escaping flux p = 3 x 10-4mm

Gas Argon

Without

Id = 4 a m p

RF field

r h 50

JU

Reversal due to the resononce

.-ICn L

30 a

-

10

a

+ ionization due to RF

-.i B

(gouss)

FIG.13

In order to discuss the significance of these results, we need to calculate the confining pressure due to the rf by means of the formula

where P is the rf power. Q is the quality factor of the cavity, and V is the volume; the constant (Y relates the maximum field strength to the volume integral of the squared field, i.e., a E i a x V = J E 2d3r. P’or the empty cavity Q may be ralwlated to be 6 x lo4 for the TI3110 mode. The paper docs not indicate a measured value. Inserting this valuc of Q we find $€&” = 50/a dyn/cm2 while the particle pressure on thc axis amounts to n ( T , T,) = 224[(T, T,)/eV]. The factor a is approximately and although the ion temperature Ti was 2 eV the electron temperature T , may be higher than this.

+

+

+

256

H. MOT2 AND C. J. H. WATSON

We first consider the pulsed regime. The confining pressure is -100 dyn/cm2 while the plasma pressure is 500 dyn/cm2 or higher. It is therefore not surprising that the plasma escaped. I n the afterglow, however, when the plasma density has decayed to 5th of the original density, equilibrium might have been reached which explains the increase in the decay time. I n the continuous regime, however, the rf pressure is only -1 dyn/cm2 and is negligible compared with the plasma pressure. We therefore have to explain why the rf reduces the particle flux at, say, 250 G. A qualitative explanation may perhaps be given if we consider the quasipotential acting on single particles, which as we have seen is increased by the magnetic field by a factor (1 - Q / W ) - ~ , which is 2.8 for the present experiment. The quasi potential in this case is -1 eV without magnetic field and 2.8 eV a t 250 G. Thus, if we assume that the electron temperature was of order 2 eV, the quasi-potential of the vacuum field in the cavity would be sufficient to confine single particles. We know, however, that a plasma of pressure greater than the rf pressure must eventually modify the vacuum field configuration in such a way that it can escape; but it seems reasonable to expect this process to occur more slowly than the free escape of individual particles, as the experimental results indicate. It is clear, however, that such experiments in the continuous regime do not demonstrate plasma confinement. The same objection applies to similar experiments b y Consoli et al. (1962b) reported in the same year. Further experiments with plasma created in a similar spherical cavity by the rf itself, also in the presence of a dc magnetic field, were reported a t the International Colloquium held at Saclay in 1964 (Consoli et al. 1964b). The power had been increased to 80 kW, the pressure was 2 x mm. Photographs of the appearance of the plasma luminosity were taken by means of an image converter which could be switched on for a very short time; they are shown in Figs. 14 and 15. The oscillograms above each photograph show three traces. The uppermost one shows the pulse that switched on the image converter a t a time which varies progressively in relation to the start of the rf pulse; the middle one shows a quantity corresponding to the escaping particle flux; and the lowest one shows the rf pulse, which is seen to be fairly reproducible except in two cases and which in every case cuts off as a result of cavity detuning due to plasma buildup in it. I n Fig. 14 the power was 80 kW and the pressure 2 X 10-4 mm, while in the case of Fig. 15 the power was 20 kW and the pressure 2 x 10-3 mm. Figure 14 shows, according t o the authors, that the plasma is confined during the rf pulse, but it expands somewhat until, after the pulse, it is no longer confined and subsequently decays. Looking a t Fig. 15 we see

R F CONFINEMENT AND ACCELERATION OF PL.4SM.46

257

that 110 confinement was obtained with the lower rf and higher gas pressure. Unfortunately, no data concerning the temperature and plasma density are given in the paper. It is hard to base conclusions on the observation of the distribution of luminosity alone because this merely indicates the

-

Time

FIG.14

regions where conditions favorable for the generation of visible light exist. In conjunction with the particle loss measurements, however, the photographs provide reasonable evidence for confinement and we can use the rf pulse trace to draw further conclusions. When breakdown is produced in a gas by microwaves, a t a power level such that the rf pressure is not enormously larger than the plasma pressure, the ionization must cease once the plasma frequency exceeds the microwave frequency, because the rf can no longer penetrate into the cavity. The electron density

258

H. MOTZ AND C. J. H. WATSON

could not exceed 1-2 X 1010/cm3under these circumstances, and the plasma (T/eV) dyn/cm2. At the power pressure 2nT could not exceed 3.8 X level of 80 kW, however, the rf pressure is 400 dyn/cm2. We shall see later that the temperature is likely to be of the order of 10 eV, so that rf, should be able to localize the plasma near the axis. Indeed, a t this power level, it would be possible to achieve an equilibrium with a 1000-fold increase of

-

Time

FIG.15

plasma density a t the axis. It is certain however, that the cavity detuned before such densities were reached and more elaborate frequency tracking would be needed to reach this equilibrium, I n fact, one can see from Table I that in cylindrical geometry a t least a detuning of 0.5 % is reached when wp2/w2 is as low as 0.9. Consoli et al. indicated a t the colloquium that their rf supply system could track u p to about 0.5% frequency. If the contraction of luminosity is due to plasma confinement it must be assumed that, a t the higher pressure of 2 X 10+ despite the lower power, the rate

RF CONFINEMENT AND ACCELERATION O F PLASMAS

259

of ionization is so fast that luminosity appears almost simultaneously throughout the cavity. The breakdown mechanism is different at the higher pressures as we shall see below. This would explain why Fig. 15 shows no confinement of luminosity to a central region. The authors state that no light output was observed with the lower pressure for a time of 7 psec after the onset of the rf pulse, and the first photograph of Fig. 14 shows the beginning of the light output. It appears that the collisions leading to light output were most frequent, in the axial region where the particles oscillating in the quasi-potential well have maximum energy. We do not know whether the light output is larger than that observed in Fig. 15 throughout the pulse. The buildup and decay of rf fields in a cavity follows a law exp( & ( u / Q ) t ) = exp { (t/8.5 X 1 0 P ) } in UQCUO in the present case. Looking a t the photographs of Fig. 14, we see that the buildup time is of the order of 6 psec, indicating merely that the loa.ded Q is in fact lower than 6 x lo4 but the decay time is even shorter, i.e., only 2 psec. I n Fig. 15 the buildup time is shorter than in Fig. 14 and the decay time is equal to the buildup time. This indicates that losses in the plasma due to ionization, excitation, and heating are of the order of the copper losses. The sharp onset of the decay in Fig. 14 indicates that the cavity is b y this time filled with plasma exceeding the critical density and the decay time is even shorter than in the case of Fig. 15. If there is confinement, the changed field configuration also lowers Q, because the volume accessible t o the fields is lower. But we may merely see enhanced collision losses. I n yet another experiment by Consoli et al. (1964c), a n axial and an equatorial electron collector allowed the determination of the time dependence of the escaping flux which was carried out a t gas pressures from 1 to 1.7 X 10P mm. The setup was presumably similar to that of the previous experiment. The particle flux arrived a t the collector after a time delay t, with respect to the onset of the rf pulse. When the electrons arrive a t the collector, the rf ceases to flow into the cavity and the rf pulse measured by a loop inserted into the cavity stops. The authors interpret this as the detuning of the cavity by the plasma accumulating in the cavity. Photographs of the image converter which is switched on during the pulse show a cigar-shaped luminosity, elongated along the axis (where, we recall, the TEllo mode has a node and so does not confine in the axial direction); the luminosity develops just before the rf pulse cuts off. The authors have measured the rf pulse length f, as a function of the gas pressure and find a law

260

H. MOTZ AND

C. J. H. WATSON

To explain this result they proposed a somewhat unsatisfactory ad hoc model. They calculate the time dependence of the ionization and find, for the time needed to reach a density of 10" by volume ionization from an initial density of 2 X lo8, l/t, = O.16(pSve - l / ~ ) ,

(8)

where S is the ionization cross section, T the lifetime of electrons, and v8 their velocity. They conclude from the form of the experimcntal law that the buildup does indeed occur by ionization in the cavity volume and, by identifying the theoretical and the experimental expressions, determine T = 5 X sec and Sue = 2 X 109 sec-l/Torr. They state that this cross section corresponds to electrons of 30 eV. On the other hand, the energy of the electrons collected is of the order of 10 eV. The initial density of 2 x 108 is assumed because this is the density required for quasineutrality in the quasi-potential field. Some other mechanism is made responsible for generating this initial density. The present authors think that a more appropriate breakdown theory for these very low pressures is that of Self and Boot (1959) reported in the next section. The quasi-potential for the experiment under discussion is 400 V. As the electrons oscillate in the quasi-potential well they soon acquire energies sufficient for ionizing collisions. The ionization probably builds up exponentially with time but the phenomena are complicated by the plasma accumulation at the center. At higher pressures, the diffusion controlled mechanism of breakdown involving outward particle flux takes over, as seen in the next section.

E. Indirect Support for the Quasi-Potential Concept from Breakdown Measurements Experiments on breakdown of gas in microwave cavities were carried out by Self and Boot (1959) and their theoretical interpretation adduces an interesting confirmation of the quasi-potential concept. They find results similar to those of Brown and Macdonald (1949) within the pressure range used by the latter authors, i.e., within the range of validity of their theory. A t lower pressures, Self and Boot get very different results which must be explained by quasi-potential theory. In the experiments of Brown and Macdonald, the dominating electron producing process is ionization by collision in the cavity volume, while the principal process of electron removal is diffusion to the boundary and subsequent recombination at the cavity walls. Secondary processes at the walls are unimportant unless the cavity dimensions are very small compared to the free space wavelength A, in which case secondary electron resonance (multipactor) occurs. In the cases considered by Brown, the

RF CONFINEMENT AND ACCELERATION OF PLASMAS

26 1

electron energy of oscillation is significantly less than the energy required for ionization. Electrons acquire energy from the rf field sufficient for ionization by a stochastic process involving collision with the neutral molecules. According to Brown and Macdonald, under these circumstances breakdown occurs as soon as the rate of electron production exceeds the rate of removal by diffusion. Stipulating as the breakdown condition that these two rates are equal, they arrive at the equation

V2Dn

+ Fin = 0,

(9)

where Vi is the mean ionization rate, i.e., the average rate of production of new electrons per electron, and D is the diffusion constant for electrons. The solutions of this equation, subject to the boundary condition th a t the electron density n vanishes a t the cavity walls, form a set of eigenfunctions with eigenvalues for Fi/D. The smallest eigenvalue is written

Vi/D

=

1/A2

(10)

and corresponds to the breakdown condition. If the field is uniform, and D are independent of h',so that (9) becomes V2n

+ (Fi/D)n = 0.

Vi

(11)

The smallest cigenvalue for a cylindrical cavity with radius a and length 1 becomes

A is a characteristic diffusion length for the cavity which in the case of a uniform field we shall call the geometrical diffusion length Ag. A high frequency ionization coefficient t = Vi/DE2may be defined, and dimensional considerations show th at breakdown will depend on three variables which may be chosen as EA, E/p, and pX, where p is the gas pressure. We can write [ = l / i P A ) i.e., with A given by (12). From breakdown measurements, [ has been determined as a function of E / p and pX for various gases, using short cylindrical cavities with all > 15 for which the field may be assumed uniform. For nonuniform fields, (9) may be written Vz+

+

+ tE2+

=

0,

(13)

with 4 = D n satisfying = 0 a t the walls. Since [ is experimentally known as a function of E for uniform fields, if we assume th a t for nonuniform fields t: is determined by the local value of E , it is known as a function of position in the cavity, and, in principle, (13) may be solved as an eigenvalue problem for the maximum value of El E,. Boot and Self have shown that the maximum field required for breakdown in nonuniform fields, e.g.,

262

H. MOTZ A N D C. J . H. WATSON

in a cavity with length equal to radius, is always greater than the field that mould be required if the field were uniform in the same cavity. Certain (but, as we shall see, not all) limits of the theory sketched above, the diffusion theory, were discussed by Brown and Macdonald and plotted as limit, lines in a p h , pX plane. The limits which are relevant for our further discussion are the mean free path limit and the uniform field limit. The field is no longer uniform when A/X is no longer small, and Brown and Macdonald consider the theory to be valid when A/A < 1/2a = 0.159. The measurements of Boot arid Self were carried out in a cavity with h/X = 0.135, and since this case they can also show that A = A, the fields are sufficiently uniform for diffusion theory to be valid. Another limit of validity of the diffusion theory is reached when the pressure becomes so low that the electron mean free path becomes equal

-a+-

Electric field c Magnetic field 0 , -

(a 1

-a+

(b)

FIG.16

to A. Brown and Rfacdonald, assuming that the average electron energy is of the ionization energy (for hydrogen the ionization energy = 15.6 eV) obtain for this limit p h = 0.02 mm Hg cm. Boot and Self have carried out experiments with microwaves of free space wavelength X = 3 cm and a cavity for which A = 0.405, so that the mean free path limit is reached and diffusion theory is not applicable for pressures smaller than -0.05 mm. They used two different field modes, TRiIolnand TILloll. The field configurations for these modes are shown in Figs. 1Ga and 16b. It is seen that the TRllollmode has a valley in the center, while the TMolomode has a field maximum near the axis. Detuning of the cavity and consequent decay of the cavity field indicates breakdown in these experiments. The maximum fields in the cavity for breakdown obtained experimentally by Boot and Self are plotted against pressure in Fig. 17. Above 0.3 mm pressure, where the diffusion theory is certainly valid, the measured breakdown fields for the TMoln mode agree with those of Brown and RIacdonald (suitably scaled) obtained a t = 10 em with a TR’Inlo cavity of similar dimensions. The breakdown fields in the case of

+

263

RF CONFINEMENT AND ACCELERATION O F PLASMAS

TRlollare higher, and Boot and Self are able to show that A is less than A, in this mse.

For low pressures the results differ radically for the two cavities. Breakdown could not be reached with the fields available in the experiment of Boot and Self (31,000 Vjcm) in Lhe case of the Thlolo cavity, while in the case of the TAIolLcavity breakdown occurred a t fields lower than 21,000 Vjcm. For low pressures, the minimum cavity field for breakdown asymptotically reached a value, corresponding to a well depth of the quasi p o t e n t d V calculated from the field value, which is equal to

-

v'.15,6 ev

___

E

Y

3

10

_---------

P

-X--X-X\

V.15.6

eV

w 9

+EOM

x

cavity

,,E, cavity Brown and MacDonald's results. Scaled from A = 10.0cm

I

lo-2

I

I

I

10.'

I

10

Pressure p

I lo2

(mrn Hg)

FIG.17

the ionization potential 15.6 eV. (The height V of the hill of quasi-potential in the case of the TMo,o mode becoines equal to 15.6 eV for a somewhat lower maximum field, which has also been marked on the figure). These results are clearly in accordance with quasi-potential theory. In the case of a quasi-potential well, trapped electrons will describe orbits which eventually will lead to collisions with neutral molecules. If the well depth is equal to the ionization energy or greater, the probability of ionizing collisions will become considerable. In the case of a quasi-potential hill initiatory electrons are accelerated by the quasi potential and are lost t o the wall before they have an appreciable chance of an ionizing collision. According to the mean free path criterion, however, diffusion theory should be valid a t pressures below 0.3 mm-in fact down to 5 X mm. This discrepancy may be explained by pointing out that diffusion theory assumes the oscillation energy of the electrons to be small compared to the

264

H. MOTZ A N D C. J . H. WATSON

ionization energy. When the quasi-potential hill or valley can accelerate the electrons to energies approaching the ionization potential this assumption is violated.

7. THE THEORY OF RADIO-FREQUENCY ACCELERATION OF PLASMA We have seen above that a plasma in an rf field undergoes internal microscopic motions, as a result of which it acquires certain quasi-dielectric or quasi-metallic properties. I n consequence, the rf fields can exert macroscopic forces upon it, which can qualitatively be attributed to “radiation pressure,” and can under some circumstances confine it in stable equilibrium. The possibility naturally suggests itself that one might use the same forces to accelerate the plasma as a whole. I n addition to this purely rf acceleration procedure, it has been proposed that one should use the enormous enhancement of the effectiveness of a n rf field induced by the presence of a not-quite-resonant stationary magnetic field to increase the rate of acceleration or that one should adapt the exactly resonant principle of operation of conventional particle accelerators to accelerate the plasma in a resonant manner. I n each case, the suggestion is easily made but is difficult to discuss in a precise quantitative manner, and the rather primitive state of the existing theory of both the nonresonant and resonant acceleration methods will be evident in what follows.

A . Purely Radio-Frequency Acceleration We shall begin by considering the main proposals which rely exclusively on rf fields t o accelerate the plasmn. These differ with respect t o the type of radiation used and the time scale on which it is proposed that the acceleration should take place. Decisions 011 these two questioris cannot be taken independently. A plasma in its natural state is uniform and of infinite extent; to accelerate a finite amount of it as a whole, it is necessary either to do so very quickly, so that the “plasmoid” does not have time to disperse, or to supply fields other than those strictly necessary for acceleration in order to preserve its coherence. i\’lethods of the former type, which might be called “impulsive,” are subject to the difficulty that one must ensure that no component of the necessarily very large accelerative forces has a tendency to break up the plasmoid; the latter type, however, which we shall call “continuous,” raise all the problems of confinement, and of stability of confinement, which we considered in Sections 2 and 4. This decision on the time scale for acceleration affects the model of the plasma which it is appropriate to use in designing the accelerator. I n continuous acceleration devices, t,he plasmoid will have time to adjust its volume and shape to the contours of the force fields being used to confine it; one should therefore use a self-consistent quasi-equilibrium theory of

RF CONFINEMENT -4ND ACCELERATION OF PLASMAS

265

the plasma, not unlihe the theory given in Section 2. I n impulsive devices, such a model would be inappropriate; ideally one should use the kinetic theory of an expanding plasma, but this theory is only beginning to be developed [see, for example, Cheremisin (1965)], arid in all the discussions of impulsive accelerators in the literature, it is assumed that the plasma is a rigid object of fixed shape which happens to possess a dielectric coefficient c = 1 - wp2/w2. Both the mechanical and the electrical aspects of this assumption are rather problematical. On the mechanical side, it is clear that if one attempts to accelerate a fluid object such as a plasmoid by exerting external forces upon it, the initial effect will be to cause internal motions, and only if arid when internal equilibrium has been restored will the plasmoid accelerate as a whole. For the rigid dielectric model to be applicable, it must be supposed that such a readjustment has successfully taken place; for consistency, therefore, one should a t least show that the surface force acting on it (given by Fi = T , k n k where n is a unit normal to the surface and T,r is the electromagnetic stress tensor) is everywhere of a suitable direction arid magnitude to maintain adequate coherence during acceleration. On the electrical side, the assumption of a dielectric coefficient e = 1 - wpZ/wz presupposes that the electron density is high enough, and the duration of the acceleration long enough, for collective electron oscillations to be important, and it neglects all nonlinear effects of the (by hypothesis) very large electric fields on the dielectric coefficient. As we have seen, for a stationary rf confined plasma, large electric fields completely alter the effective dielectric coefficient of a plasma; their effect 011 its noriequilibrium properties are a t best a matter for speculation. We shall now consider the various types of rf wave that have been proposed for acceleration purposes. Although. as we have seen, one should use the appropriate mode of plasma behavior to evaluate them, it is useful t o start by considering their effect on single charged particles. (i) Standzng Waves. In this case, the theory given in Section 1 shows that the rf field exerts a force F = -Ve2E2/2mw2 = -V#. Thus if one were to inject a particle at a point where E = Em,, arid withdraw it a t a point where E = Em,,,it would have increased in energy by an amount A&

=

e 2 ( E i R X- E",,,)/2mw2 = A#.

Qualitatively speaking, if the single particle were replaced by a quasineutral rarified plasmoid, each electron would gain this energy and would share it with the associated ion, and thc velocity of ejection of the whole plasnioid would be

v

=

( 2 A#.-/vI,+)~'*.

(2)

266

H . MOTZ AND C. J. H. WATSON

This method, first introduced for quasi-neutral plasmoids by Asltaryan (1659), is applicable only over one quarter wavelength of a standing wave, or over one e-folding length of an exponentially decaying wave, and is accordingly only suitable as an impulsive accelerator, On single particle theory a t least, it should be possible to insure coherence of the plasmoid in the direction a t right angles to the axis of acceleration by ensuring that $ has a minimum on the axis, but it is not possible to confine the plasmoid along the axis. For a hydrogen plasma, the maximum velocity of ejection is given by v/c = 800E,,,/w where I3 is measured in volts per centimeter and w in cycles per secoiid, and, as we shall see subsequently, it is doubtful whether (2) is valid when wp 2 w. Thus the method is characterized by a regrettable inverse relationship between the maximum density and maximum velocity attainable. Possible techniques for extending the applicability to continuous acceleration have, however, been proposed by Askaryan (1959) and Cildenburg and l\liller (1960). (ii)Neady Standing Waves. If one modulates the amplitude of a standing wave in such a way that the wave form moves with respect to the laboratory frame, but at a velocity very much less than the phase velocity of the wave, the quasi-potential theory given in Section 1is still applicable. One can therefore imagine trapping a particle or a plasmoid in a threedimensional qunsi-potential well and then accelerating this well with respect to the laboratory. This method, first, described by Gaponov and Miller (1958b), is clearly a continuous method and its applicability depends upon the realization of stable confinement. A suitably modulated wave could be obtained by propagating two traveling waves of slightly different frequency in opposite directions along the accelerator axis. There is an upper limit to the rate of acceleration set by the requirement that the plasmoid remains trapped; for a plasma of thermal velocity v8 confined near the node of a wave, so that its particles oscillated with a frequency wJ, the upper limit on the acceleration would be of order wJv8. (iii) Slow Traveling Waves. If one sets up a traveling wave in a delay line of varying characteristics, so that the phase velocity of the wave increases along the axis, one can inject a particle with a velocity close to the minimum phase velocity of the wave, which is then trapped in the manner indicated a t the begirining of Section 1, F, and accelerated. This is the principle of ordinary linear accelerators and will therefore not be considered in this review. (iv) Fast Traveling Waves. For such waves, unlike the slow traveling waves just considered, the rioriuriiformity of the field is on a length scale of order c/o, so quasi-potential theory should be applicable. On this theory,

RF CONFINEMENT AND ACCELERATION O F PLASMAS

267

as we havc seen, there is no force on a partivle in the direction of propagation of the wavc, in apparent contradiction to the well-known result that an electron in a traveling wave causes Thompson scattering and therefore experiences a force. As we mentioned in Section 1,however, this is asecondorder effect; the force exerted by a plane traveling wave is

FT = O ~ E ~ E ' ~ ~ ,

(3)

where n is a unit vector in the direction of propagation and uT is the Thompson scattering cross section; this is smaller than the force due to the quasi potential of a standing wave of the same amplitude and frequency by a factor row/c, where ro is the classical radius of the electron. I n view of the smallness of the effect, it might be supposed that the force exerted by a traveling wave can only be important in, for example, stellar interiors where the radiation density is sufficiently high. Such a conclusion is however, incorrect, as can be seen by considering the implications of the fact that a traveling light wave can be scattered by, and can exert a measurable force on, a thin piece of metal foil. For, if we were to regard this as an assembly of independent electrons of density ~ l O * ~ p a r t i c l e s / c r n ~ , it would be virtually transparent, since the Thompson cross section is of sq cm. I n fact, a metal is opaque to any radiation whose freorder quency is sigriificantly below its plasma frequency. As is well known, this is interpreted to mean that the electrons in it respond collectively to the incident radiation arid in consequence havc a much higher effective cross section than they have individually. Unfortunately, this interpretation is based on linear theory, and on the assumption that the density distribution of the plasma is maintained uniform in a finite region of space by solid state forces, and we are therefore left to speculate as to what an adequate nonlinear theory of the interaction of traveling waves with a plasma not confined in any other way would predict. The position is complicated by the fact that it would be unrealistic to assume that ordy a traveling wave was present, except initially. In any steady state there would be a t least partial reflection of the incident wave, and hence a t least a certain admixture of a standing wave, whose nonlinear effects could be predicted on quasi-potential theory. It is indeed possible that the most significant nonlinear effects of a traveling wave are exerted in this way. However, the feeling remains that the collective electron motions, which as we have seen play such an important role in the linear regime a t a metal surface, might also play an important part. The one paper in the literature which gives a plausible discussion of this question is that of Gildenburg and Miller (1960), to which we referred briefly in Section 2. We shall therefore give a fairly detailed account of this paper, drawing on various others for the justification of certain assertions made in it. [It

268

H. MOTZ AND C. J. H . WATSON

should be remarked that its chief results coincide with those of Askaryan (1958, 1959) who, however, did not give any clear derivation of them.] I t will be assumed that any plasma accelerator based on fast traveling waves must work impulsively. Certainly, no suggestion has been made in the literature that such a wave would tend to confine a plasmoid along the axis of acceleration, although radial focusing could certainly be achieved by quasi-potential forces. We shall therefore adopt the rigid dielectric model, with all the possible defects of that model outlined above. For simplicity we shall assume a spherical plasmoid, though discussions of ellipsoidal plasmoids have been given by Levin et al. (1959) and Veksler et al. (1963). We shall begin by calculating the electromagnetic response of such a dielectric sphere to an incident wave, which, with a view to maintaining maximum generality, we shall not specify to be either traveling or standing. This response turns out to be related to the fact that such a sphere possesses a set of natural modes of electromagnetic oscillation. The theory of these is given in Stratton (1941, p. 554); in outline, the analysis consists of an expansion of the electromagnetic fields at any point outside the sphere in terms of the vector spherical harmonics which represent an outgoing wave satisfying Maxwell’s equations in vacuo, and a similar expansion of the fields inside the sphere in terms of vector spherical harmonics which represent a wave which is well behaved at the origin and satisfies Maxwell’s equations for a medium of dielectric coefficient E . On specifying the boundary conditions which must be met at the dielectricvacuum interface, it becomes clear that these pose an eigenvalue problem, and there consequently exists a set of eigenfrequencies wm,,, which are the natural modes of electromagnetic oscillation of this system. All these modes turn out to be very heavily damped (i.e., all the possess imaginary parts which are of the same order as the real parts); this is to be expected, since in such a mode of oscillation, energy is being radiated away from the central sphere. These modes are qualitatively similar to the radiation fields which would be set up by the oscillations of certain charge distributions at the origin, and they can accordingly be classified by type (transverse electric or magnetic) and by the character of the multipole (dipole, etc.) whose oscillations would generate them. I n consequence, if at some initial time we were to switch on an rf field external to such a dielectric sphere we would expect initially to excite oscillations both at the external frequency w and at the natural frequencies urn,,,,the relative amplitudes of these being determined by the boundary condition at the surface of the sphere. After a short time, however, these natural oscillations would decay, leaving only the forced oscillations. Since the force exerted upon the sphere depends upon the total rf field acting, we need to determine the steady state of this system. T o solve this

R F CONFINEMENT A N D ACCELERATION O F PLASMAS

269

one proceeds as before, but now expanding the field outside thc sphere as a sum of two components, the given external field (assumed to be generated by sources which are not affected by the scattered radiation) and the scattered field. Thus, symbolically,

(for the significance of the vector spherical harmonics M,, 1941) arid Edielectric

=

N, see Stratton,

2 ahd’MLd’+ bf)NAd’, n

where we havc put the superscript (d) on thc vector spherical harmonics inside the dielectric to indicate that they are the eigenfunctions appropriate lo the dielectric medium and are well behaved a t the origin. On imposing the boundary conditions at the surface, we find that the expansion coefficients u(nd),6Ld), ujlsc’, and b r ) are all determined in terms of the expansion coefficients a y t ’ and byt’. Thus, for example, for an incident plane traveling wave, one obtains: = j n ( ~ l ~ z P ) b j d P ) Y- .in(P)[€1’2Pjn(€1/4P)1, =

D7lb)

(4)

where j 8 , ( x )is a spherical Bessel function, p = w a / c , and = 0 is the dispersion relation for the eigenfrequencies of the natural oscillations of the sphere. We may note that as w approaches urn,,,the amplitude of the scattered wave for that mode increases resonantly; since the urn,,, are all complex, however, whereas w is by hypothesis a real quantity, the expansion coefficient never actually becomes infinite. The above procedure is perfectly general and leads to precise expressions for the total electromagnetic field after all the initially excited natural modes have died away. However, for any given external electromagnetic field, even the calculation of its expansion coefficients a r t ) b‘,“xt’ is by no means trivial, and the resulting expression for the total field is very unwieldy. It is therefore desirable to introduce an approximation. The most useful approximation is to assume that the radius of the sphere, a, is much less than the wavelength of the radiation both in uacuo ( c / w ) and in the dielectric ( ~ / w d / ~ )I. n this case, provided that w e are not too close to a resonance we can use the expansion for small p: = -

i(€

-

i)p2r1+3

+ 1)(2n + 3)[(2n+ 1)!!12 i ( e - l)p2”+’ /)y= (en + .n + l ) ( % + 1)[(2n - l ) ! ! ]+ i(€ - l)(n + l)p2’r+l (an

1

(5)

270

H. MOTZ AND C . J . H. WATSON

(Gildenburg and Kondratev, 1963). The largest coefficient is clearly (apart from resonances)

Using the fact that in this approximation the external field is uniform a t the location of the sphere, one can easily show that the scattered radiation field is identical with that of a point dipole of moment

an expression which differs from the well-known result of Raleigh for the scattering of radiation by atmospheric particles [Landau and Lifschitz (1951) Electrodynamics of continuous media, p. 3801 only by the presence of a small but physically important correction in the denominator. If we writle e = 1 - wD2/w2, E = Eoexp(iwt), w o = q,/dZ, y = (2 /9) (wD2/w)p3 = (2/3) (Ne2w2/nzc3), we see that p can be written in the form

p

=

- ( N e 2 / n z ) E / ( ~ 2- mu2 - i y w )

(8)

and satisfies the differential equation p

+ woZp - (y/w2)p

=

(Ne2/~~2)E,

(91

an equation which is the starting point for the discussion of Askaryan (1958) and Gildenburg and Miller (1960). An interesting property of Eq. (9) is obtained by taking the limit in which the number of particles is reduced to one electron-ion pair; in this case we can write p = er and we obtain r wo2r - (y/w2)Y = (e/m)E (10)

+

where y / w 2 = %e2/mc3,which is precisely the equation from which one derives the theory of Thompson scattering [Landau and Lifschitz (1951), Classical theory of fields, p. 2641. Thus by retaining the small imaginary term in the denominators of (7) and (8), it would appear that we are working in an approximation in which Thompson scattering is included ; and we see that for a plasma describable by a real dielectric coefficient e, the Thompson scattered wavelets of different particles add coherently, a t least for the fundamental (dipole) mode. If the dielectric coefficient were made complex, so as to include the effcct, of collisions or thermal motion, the above expression for y would have to be modified accordingly.

RF CONFINEMENT AND ACCELERATION OF PLASMAS

271

If we now assunic that the external electromagnetic fields are nonuniform but slowly varying and that their frequency is sufficiently far from that of any multipole resonance, we can describe the motion of the plasmoid in very much the same manner as we discussed the motion of single particles in Section 1. If r is the location of the center of the plasmoid we have

N(m+

+ m-)r

=

.

+ r X H(r)/c.

(p v ) E ( r )

(11)

If we impose the restriction that the electrons should not execute motions on a scale comparable with the dimensions of the plasmoid (lpl << eNa) we can write r = R pw pu, and solve by Taylor expansion and average, obtaining for the guiding center motion:

+ +

fi =

-

ez(w2 - wo2) 2 7 ? 2 + 7 7 L - [ ( ~ d ~-

VIP

+

(JO')~ (J~')'~]

The first term FV is easily recognizable: taking the limit wo -+ 0 we obtain precisely the expression we derived above for the acceleration of a rarified quasi-neutral plasmoid using the quasi-potential approach. The modification for finite coo, however, and particularly the resonance near = w o are not immediate consequences of the quasi-potential approach, which indicates only that the presence of the plasmoid should alter the rf configuration somewhat. We have already in Section 2 expressed the view that as regards the equilibrium of a plasma which is free to expand until pressure is balanced by electromagnetic pressure, this resonance is a spurious effect, resulting from the inadequacy of the underlying model of thc plasma. I t is much less clear, however, th a t this model is inapplicable in the case of a plasmoid which is being accelerated impulsively. Under these conditions the plasma may not have time to expand, and it is possible that, it is permissible to treat it as a rigid dielectric for processes occurring on a time scalc much shorter than the time scale on which readjustments of shape and size can take place, provided that the accelerative forces are not such as t o tear the plasmoid apart on this short time scale. The second term in (12), Fp, is the one which results from coherent Thompson scattering and absorption processes (if any). If we take the case of an incident plane traveling wave with w >> w 0 , we obtain the force on a lossfree plasmoid: (J

272

H. MOTZ A N D C. J. H. WATSON

where UT is the Thompson cross section for a single electron. Thus we see that the force per particle increases linearly with N , the number of particles in the plasmoid, instead of remaining constant, as it would if the electrons responded independently. This result, first stated in the literature by Veksler (1956), is the basis of his “coherent” principle of plasma acceleration. The relative merits of this principle, which uses traveling waves, and the quasi-potential principle, which uses standing waves, have been discussed by Gildenburg and Miller (1960) (the proposers of the latter method). It is readily shown that the ratio of F , / F p for waves of given amplitude is of order w y / ( w 2 - coo2). Since y << w for all cases for which the theory given above is valid, the coherent method can only give a larger instantaneous acceleration if one works close to resonance. However, it is precisely the physical reality of this resonance in any practical situation which is rendered doubtful by the discussion given above. This suggests that acceleration methods based on standing or nearly standing waves should always be superior to those based on traveling waves. This may be correct for plasmoids of small radius (ka << 1) since in this case the scattered radiation is of amplitude much less than that of the incident radiation (by a factor of order ( k ~ ) ~and ) hence a n incident traveling wave does not, on scattering, establish a significant standing wave. For large plasmoids, however, the forces exerted by the standing wave which results from the scattering of the incident traveling wave will become important. Under these conditions the distinction between traveling wave and standing wave acceleration becomes unreal. No satisfactory analysis of this case has been given, however, though a first approximation can probably be obtained by simply ignoring the unreflected traveling wave and using the theory of nearly standing wave acceleration given below. Having now outlined the main proposals which have been made for the nonresonant rf acceleration of plasma, we shall discuss in more detail the only proposal which allows a reasonably precise theoretical analysis a t the present time-the continuous acceleration of a plasma by a nearly standing wave. Before giving this theory, however, we should mention t,he chief difficulty in realizing this method of acceleration-the problem of injection. I n principle, one could achieve this either by creating the plasma suddenly a t the instantaneous position of some node of the nearly standing wave (e.g., by means of a laser) or by programming the rf fields in such a way that one could create the plasma in a stationary standing wave and then start the process of acceleration. Both of these methods have so far seemed technologically unattractive, and attention has therefore been concentrated on a much less satisfactory variant. This is based on the notion that if one creates a plasma a t some point in a waveguide (e.g., by means of a discharge) and then propagates a traveling wave of frequency

RF CONFINEMENT AND -4CCELERATION OF PLASMAS

273

less than the plasma frequency towards it, this will be totally reflected by the plasma and will consequently set u p a nearly standing wave behind it, and this will in turn then accelerate the plasma down the waveguide. As we shall see below, the nonlinear t,lieory of nearly standing waves offers little encouragement t o the belief that this procedure should work in the manner supposed. Unfortunately, all the experimental work 011 now resonant rf plasma acceleration reported so far has been based OII this approach. We shall now briefly outline a self-consistent theory of the motion of plasma confined and accelerated by nearly standing rf waves. I n the quasipotential approximation, the motion of the guiding center for a particle in an electromagnetic field for which

(15)

Introducing the coordinate 5, which measures the displacement of the guiding center in a frame moving with velocity uo, by f

=;

R

+ lo' vo dt',

we have

We shall assume that the acceleration of the franie is sniall and shall . thus obtain a constant of the motion: neglect the term ~ V O We

If we add an electrostatic field derivable from the scadar potential (6(f) and express R and i. in terms of the particle coordinates in phase space, we obtain an expression analogous to that of Section 2 , w-hicsh is the appropriate generalization for a moving system:

+ fi*(r + vat) L- e+(r f

vat) = const.

(18)

We (winow construct ion and electron distribution functions as in Section 2. We may note that whereas the expression for the particle density is unaltered in form by the motion (since the displacement in velocity space disappears on integration), the expression for the plasma pressure acquires

274

H. MOT& AND C. J . H. WATSON

a new term of the form mnvo2.This has led Gurevich arid Silin (1965) to state that the conditions for plasma-radiation equilibrium become altered as a result of the plasma motion. This is not in fact the case, a t least to first order in v/c. That the equilibrium conditions (expressed in terms of quantities evaluated at a plasma-radiation boundary) cannot be altered as a result of a uniform translation of the plasma follows a t once from an invariance principle-that a system in equilibrium must appear to be in equilibrium when viewed from any Galilean frame of reference. In particular, the expression mnvo2cannot occur in a statement of the equilibrium conditions since it is not invariant under translation. A direct proof that the conditions are in fact unaltered can easily be obtained, in the case of a nearly standing plane wave moving with velocity vo in the x direction, from the energy-momentum tensor theorem, which gives

If we note that

it is clear that the third and last terms in this equation exactly cancel; that is, the effect on the equilibrium of the additional pressure (m+ m-)nvoz which the plasma appears to possess when viewed from the laboratory frame is exactly cancelled by the rriomentunt (m+ 7n-)nvo carried by the plasma. Bearing in mind that for u o << c, the term

+

+

a (E X HI,

-

dt

c

is of order vo/c compared with

we see that to this order the equilibrium condition is still

a

az (gEoBz(E)

+

+ p ) = 0,

+/.LOB2(E)

RF CONFINEMENT AND ACCELERATION OF PLASMAS

275

where p is the pressure of the plasma in the frame in which it is a t rest. The correction term (a/at)(E x H),/c, which averages to zero if the plasma is at rest, reflects the fact that a nearly standing wave does possess a slight electromagnetic momentum, which makes a small (on our assumptions negligible) modification to the conditions for equilibrium. It follows from the preceding discussion that to obtain the self-consistent plasma configuration in a nearly standing, slowly accelerating wave, it is quite justifiable to use the theory of stationary equilibria described in Section 2 to obtain the configuration in the frame in which the plasma is a t rest and then to transform all quantities to the laboratory frame. By way of example, we shall apply this procedure to the situation of experimental interest, which we shall idealize as a semi-infinite plasma separated by a moving boundary layer from a half-space containing a nearly standing plane wave. The conditions for the existence of such an equilibrium are obtained from the discussion of plane wave equilibria given in Sectioii 2 ; it clearly corresponds to the critical case (ii)(b), in which the constant C2which measures the amplitude of the field in vucuo equals CI2. I n terms of physical quantities, this is equivalent t o the condition :

+ kPomt)l(+-*

[ ~ e o W )

=

[p(t)It++*.

(21)

It follows from the theory given in Section 2 that if the radiation pressure exceeds or is less than this critical quantity, the only possible equilibria have plasma (ionfigurations which are periodic in space and have an electromagnetic field which has penetrated right through it. It does not follow that such equilibria would be established immediately; there would certainly be a time during which the plasma underwent internal motions, such that no frame existed in which the plasma was everywhere a t rest. This process of readjustment would continue for a time which would depend on the amount by which the radiation pressure differed from that required by Eq. (21) and on the temperature and density of the plasma. Thus there does perhaps exist the possibility of operating a n impulsive accelerator on this basis. Before the readjustment process led to equilibrium, however, a number of other effects might become important; e.g., the plasma surface might become unstable and break up, allowing the radiation to pass through, or the radiation might lose its coherent character as a result of the development of short wavelength instabilities and the plasma might become turbulent. Finally, it should be pointed out th a t the relevance of all this to the actual experiments depends on the assumption that the plasma would continue to reflect the incident traveling wave in such a way as to maintain a nearly standing wave. Whether it would in fact do so on a proper nonlinear theory of traveling waves incident on a plasma is an open question.

276

H. MOTZ AND C. J. H. WATSON

B . Acceleration Using Corn bined Radio-Frequency and Magnetic Fields The presence of a static magnetic field only significantly alters the role played by the rf field in an accelerator if the rf frequency o and the electron cyclotron frequency are in a certain sense in resonance. However, there exists an important distinction between accelerators in which lo - Ql remains sufficiently large for quasi-potential theory to remain applicable everywhere and those in which there at’least exists a region of space where Iw - 81 becomes smaller than this. The former might be described as conservative,” since the motion of the particles is conservative, in the sense that it is derivable from a scalar quasi-potential function ((

as described in Section 1,D; the latter we shall call “nonconservative,” since the particles undergo an irreversible energy change as they pass through the resonance region, as we saw in Sections l,F and G. It will be clear that conservative accelerators do not differ in any essential way from those considered in Section 7,A, above, except that the presence of the magnetic field amplifies the quasi-potential by a factor W / ( W f D). There are important differences in detail, however, which emerge when one calculates the actual particle orbits. As resonance is approached, the force acting on the guiding center

dominates over the forces

e2 Vh’2 mw(w

*

Q)

arid

p

VB,,

and it becomes directed along the gradient of Q (i.e., approximately along the field lines). This highly desirable focusing effect on the accelerating force, which tends to minimize loss of plasma to the walls of the accelerator, is, however, largely canceled when one takes into account the motion about the guiding center. As is clear from Eq. (73) of Section 1, the gyratory motion about the guiding center increases rapidly as the resonance is approached, and this entails a large Larmor radius of gyration which becomes comparable with the wavelcngth of the rf mode and tends to ca,use loss of plitsma to the walls of the waveguide. When we turn t o nonconservative resonant accelerators, we are faced with the difficulty that it is not possible to give an analytic treatment of

RF CONFINEMENT AND ACCELERATION OF PLASMAS

277

the motion of even a single particle in resonant fields of arbitrary spatial configuration, let alone to give a self-consistent treatment of the coupled motion of the plasma and the fields. We are thus forced t o extrapolate from the few special cases that can be solved exactly, which were discussed in Section 1, and to make reasonable guesses about the way in which the presence of the plasma might modify the fields. To carry out this program we may begin by summarizing the main results given in Section 1 on resonant motion. The motion a t right angles to the direction of the magnetic field is qualitatively similar whether the rf field is a traveling or standing wave : the transverse energy of the particle increases irreversibly as (e2E2/2m)t2 throughout the resonant motion, a t least until relativistic velocities are reached. The motion parallel to the magnetic field depends upon the wave type; traveling waves communicate momentum to the particle a t a rate which is related to the energy pickup by Eq. (107) of Section 1 and hence becomes relatively rapid as resonance is approached. Standing waves, on the other hand, exert an effect which depends upon the location of the resonance in relation to the rf and magnetic field profiles. We showed that this could be computed in a few special cases; if Bo is uniform, the parallel motion can be calculated a t exact resonance for arbitrary plane circularly polarized rf profiles [E= E(z)(cos Qt, fsin Qt, O ) ] and at arbitrary displacements from resonance for the particular field [E= Eoz(cos wt, +sin wt, O)]. If Bo is nonuniform, the parallel motion can be calculated in the WIiB approximation if the electric field can be treated as spatially uniform throughout the resonance and as sufficiently weak to cause only a small change in the longitudinal energy during the transit of the particle. It will be seen that these results provide a rather inadequate basis for drawing reliable general conclusions. Only in one of the special cases (eE/m = w J z ) can one follow through the transition from resonant theory to quasi-potential theory; in that case, as w e saw, the limit to the applicability of quasi-potential theory is given by (W

f Q)eritical

-

(wJ~w)~’~,

(22)

a result which tempts one to conclude that in general the limit is reached when the frequency associated with the quasi-potential motion [ w J 2 0 / ( w f Q ) ] ’ / 2 hecomes comparable with the difference frequency w _+ Q. However, as we saw in Section 1, any magnetic field nonuniformity, almost regardless of how small, gives rise to forces which (according to quasi-potential theory) dominate over forces resulting from nonuniformity in the rf field as resonance is approached, so it is b y no means clear that one is entitled to draw this conclusion. A somewhat different criterion has been proposed by Consoli and Hall, as we shall see shortly.

278

H. MOT2 AND C. J. H. WATSON

Another question which one would like to settle by reference to the exact solutions is whether, if the resonance region is rather narrow, one is entitled to ignore the effect of the rf field on the parallel motion within it. This assumption is made in the paper of Consoli and Hall (1963a) (who, however, assume that one can also ignore the effect on the perpendicular motion); it is in agreement with the results of the analysis of Pilia and Frenkel(1964a,b) in this case, but as we have seen, their analysis assumes that the rf field is sufficiently weak and uniform for the effect on the parallel motion to be negligible. Kulinsky’s analysis, on the other hand, which assumes a wide resonance region (since Bo is taken as uniform), shows that in that limit the particle can be trapped in, or repelled by, the rf field a t resonance, depending on the conditions of injection. I n short, the present state of analytic theory does not offer any definite answer to this question. In addition to the analytic results referred to, however, there are a number of papers that give particular solutions, obtained by numerical methods, to the exact equations of motion. These have apparently played an important role in guiding the Consoli group in the design of their experiments, but very little critical discussion of them is given, and their significance is difficult to assess. Numerical solutions of equations exhibiting resonant behavior are of course notoriously unreliable unless special precautions are taken. Most of the detailed designs of nonconservative resonant accelerators have come from the school of Consoli and are variants on one of the following two basic concepts. An rf wave is superimposed upon a magnetic field which slopes monotonically in such a way that there exists a plane parallel to the wavefront at which the rf and magnetostatic fields are in exact resonance. If the rf wave is a traveling wave, it propagates in the direction of decreasing B o ;if it is a standing wave, the resonant plane is arranged to coincide with an antinode of the wave. The two basic schemes are represented in Fig. 18. I n the case of the traveling wave, no significant force is exerted on a particle until it reaches the resonance region; within this region the particle acquires both momentum in the z direction and transverse energy (or, which is the same thing, a magnetic moment), and hence it interacts with the magnetic field, experiencing a force - p VBo that continues to act after the particle has left the resonant region, and converts part of its acquired transverse energy into longitudinal energy. The amplitude of the traveling wave is attenuated somewhat by absorption in the resonance region. I n the case of the standing wave method there are three phases. The particle is injected from the left, where Ez/(w - Q) = 0, and moves conservatively down the slope of 9 until it reaches the resonant region; it then resonantly picks up transverse energy and experiences a change in its parallel motion about which one can only

R F CONFINEMENT A N D ACCELERATION O F PLASMAS

279

speculate; if, however, it successfully “tunnels through” the resonant region, it then continues to move down the slope of $, the sense of which is unaltered since both w - D and V E 2change sign as one passes through the resonance. As in the case of the traveling wave, the additional transverse energy acquired in the resonant region is partially converted to parallel energy as a result of the subsequent motion.

Troveling

wave

I I

I I

Standing

VE‘
1

wave

VE2>0

I I w-9>0 I

I 1 FIG.18

It is possible to give a somewhat crude mathematical treatment of the acceleration to be expected in these two cases. The standing wave case has been discussed in some detail by Consoli and Hall (1963a), and we shall give a slightly elaborated version of their analysis. We consider the case of a standing wave E = EOcos Icx (cos wt, sin wt, 0) superimposed on a magnetic field B,, = (mw/e)(l - x / L ) , arid we shall label quantities with subscripts indicating the point at which they are evaluated: x1 is the point of injection, 2 2 arid 2 3 the front and rear edges of the resonance region, and xq the outlet from the accelerator. During the first phase a particle starts with energy +172vl2 (which for simplicity we take to be parallel energy) and accelerates down the quasi-potential slope, acquiring an increased parallel

280

€I MOTZ . AND C.

J. H . WATSON

energy a t the front edge of the resonance region given by

+mu;, = +mvlz

+ e2E:es/2mw(Q2- (J) =

+nzvlz

+ +mvB2w/(Rz

- w),

(23)

where we have written V E = eE,,,/mo = eEo/mw. During the second phase we shall neglect the change in v11 occurring during the passage through resonance, since we have 110 means of calculating it. Fortunately the calculations of Pilia and Frenkel gives grounds for the hope that it is much less than the change in perpendicular energy A+mvL2,which is given by A+mvL2 = e2E~es~2/21n; is the transit time across the resonance region, provided that the transit time is sufficiently short for relativistic vclocities not to be reached, but sufficiently long for this term to dominatc over the phase-dependent term of (83). The quantity T can be related to the values of w - 9 a t which the resonance region begins and ends, since the width of the latter is given by

T

and hence

so which by (23) gives

Since this is an irreversible change in the transverse energy of the particle, we see that it acquires a magnetic moment p = A+mvL2/Bres. In the third phase, the particle slides down the quasi-potential slope, acquiring a final longitudinal energy given by

+mvi4 = +n~v\,

+ + m u 2 w / ( w - 93) + p(Rres - B 4 ) .

Using our assumption that

vll3

=

v112, we have

RF CONFINEMENT AND ACCELERATION OF PLASMAS

281

So f a r we have made no attempt to determine the values of x 2 and za, the coordinates of the edges of the resonant region. If we use the criterion ( 2 2 ) , la2

- W(

=

IW

- Qal

= W(vE/c)2’3.

= (CdJ2W)1/3

Consoli and Hall give a different criterion; they argue that one should take the value of w - Q for which this equals the reciprocal of the transit time across the resonance. This gives ~JJ

-

n2,3

=

1 / T = tJ112W/L(fi2 - 3 3 ) .

Taking the resonance region to he symmetrically located about the exact resonance we have (w - n,>z = Vl[ZWj2L, and substituting from ( 2 3 ) , (w

- no)’)-

(w2v12/4L2)(w

- Q,)

= w3vg2/4L2,

an equation which gives approximately, for small injection velocities, (W

- Qc)/W

= (vE2/4L2W2)1/6

-

(2)E/c)2/5;

i.e., their criterion gives a rather wider resonance region. On this criterion, ( 2 5 ) gives =

gmvlz

+

4L2w2

+mvE2

115

4L2w2

2/6

(B,,, -

[‘(F) +( 7) B,,, B4)]’

(26)

This expression differs from that of Consoli and Hall only in that the contribution of the term pB in the quasi-potential has been evaluated explicitly, instead of being estimated by an argument which we showed in Section 1 to be incorrect. It will be seen th at for reasonably large values of (Bra - B4)/B,,, this latter is the dominant term. On theoretical grounds, it seems difficult to choose between the two criteria for the onset of resonance given above. There does, however, appear to be experimental evidence which possibly supports the choice of Consoli and Hall. As regards traveling wave accelerators, the above discussion needs modification, since the rf field gradient does not make a significant contribution to the quasi-potential force acting in this case. Thus unless the particles are injected with an initial magnetic moment, they experience no force until they enter the resonant region. Provided th a t the injection velocity is sufficiently large to carry the particle through the resonance before it reavhes relativistic velocities, Eq. (106) of Section 1 shows that

282

H. MOT2 AND C. J. H. WATSON

so we are justified in neglecting the changc in v1j during this phase, and Eq. (24) remains valid. It is clear that the criterion of Consoli and Hall for the width of the resonant region is inapplicable in this case; if we adopt the criterion l(w - h2)/wl = ( V E / C ) ~ / we ~ , obtain

cc=--

A+mvL2 - + ~ V ~ ~ ~ L ~ ~ ~ ( V E / C ) ~ / ~ 9

Bre,

Vi2Bres

and hence an expression for the final directed energy:

It is doubtful whether this expression would be valid in the case where the particles were created in the accelerator by rf ionization: in this case, the value of V I would probably be too low to carry the particle out of the resonance region before relativistic effects became important. I n this case, one could possibly ignore the nonuniformity of the magnetic field and use the (otherwise) exact theory given in Section 1,G. Finally, some comment is required on the effect of the ions upon the motion of the plasma. I n the analysis given by Bardet et al. (196513) it is assumed that, since charge flow is unrestricted along magnetic field lines but restricted across them, the ion and electron densities averaged over any plane perpendicular to the field will remain equal (i.e., mean quasi neutrality is maintained), but that cha,rge separation may take place within any such plane. Thus, mean quasi-neutrality requires an electrostatic field & in the z direction, and hence the equations of motion of the ions and electrons in the x direction are M dOjz - = ZeE , dt

-e& - d* -7

&z

m-= dt

dz

where 5 is the velocity averaged over the x,y plane and Z is the number of charges on the ion. However, by the equation of continuity, likewise averaged over the x,y plane, afi/at

+ a$az

=

0,

it is clear that t,o maintain mean quasi-neutrality one must also have O,, = Biz = vll; hence

1

dt

m[l

+ M/(mZ)] dx -

Thus to obtain the exit velocity of the plasma in a resonant accelerator one should replace vx2 by vE2mZ/M in Eq. (26). It should be noted that this assumes that the electrostatic drag on the electrons lengthens their

R F CONFINEMENT AND ACCELERATION O F PLASMAS

283

transit time through the resonance and hencc modifies the estimate of the width of the resonance based on w - D = 1/r accordingly. Thus the nonresonant contribution to the acceleration gives

This agrees, apart from differences of notation, with Eq. (18) of Consoli and Hall, who, however, consider only the case 2 = 1. Bardet el al. (1964d), on the other hand, incorrect1y:‘generalise Eq. (18) for 2 # 1, giving v[14 = 1.84 x 1 0 7 ( v E / c ) 4 ~ 6 ~ 1 ’ 2 ~ ~ - - 2 ’ 6 with no units specified. On their assumptions, this equation should read v114 =

1.84 X 109(VE/C)4’5(2/M)2/5cm/sec.

8. EXPERIMENTS ON RADIO-FREQUENCY ACCELERATION OF PLASMA I n this chapter we shall adopt the same classification of accelerators as was used in the preceding section. We shall begin by considering the work on purely rf accelerators, most of which has been carried out in the accelerator laboratory of the Physics Institute of the Academy of Sciences in Moscow, under the direction of M. S. Rabinovich.* Three essentially different approaches have been tried. The simplest of these is straightforward application of the quasi-potential method: a waveguide with a relatively narrow side arm is excited in such a way that an exponentially decaying rf field is set up in the side arm. Plasma is then injected so that it starts a t the junction of the waveguide with the side arm, where the electric field is maximum, and it accelerates down the quasi-potential slope in the side arm. On single particle theory the gain in energy would be given by Eq. (1) of Section 7 ; if it is permissible to regard the plasma as a rigid dielectric, however, the appropriate expression would be

where A& is the change in energy per ion, m is the electron mass and w 0 = ~ , , / ( 3 ) ~as ’ ~follows , from Eq. (12) of Section 7 in the limit where collisions and the radiation correction are negligible. Thus the experiment would in principle be capable of establishing whether (1) is correct under the impulsive conditions of operation of such an accelerator. The experimental layout which was adopted is shown in Fig. 19. A 10-cm wave generated by the pulsed generator 12 was propagated down * See Gekkrr et al. (lY6Sa-c), Veksler et al. (1963), Kononov et al. (1965).

284

H. MOT2 AND C. J. H. WATSON

the waveguide 11 to the absorber 9, setting up an exponentially decaying field in the side arm 10 of diameter 1.2 cm. The spark gun 6 normally injected plasma in the direction of the waveguide, though it could be reversed so as to project plasma straight towards the diagnostic probes 4, from which a direct determination was made of its density (lolo cm-9, peak velocity (200 eV), and the total number of particles (7 X 1O'O). I n normal operation, the spark gun was fired 2 psec before the rf pulse was switched on; the rf pulse was of varying intensity, the electric field strength ranging from 0 to 4 kV/cm. One rather unsatisfactory feature of the design

I

I

I

FIG.19

of the apparatus was the location of the plasma injector. I t was deliberately placed some distance from the waveguide side arm junction (in order to minimize the perturbation of the rf electric field), and the delay in firing the rf signal was chosen in such a way as to allow a substantial amount of plasma to flow up to this junction. In consequence, the plasma was by no means stationary at the point of maximum field strength at the time of switching on the rf pulse; rather, it was distributed over the length of the side arm and it moved with a velocity which depended on several factors, including the possibility of reflection from the far wall of the waveguide. In view of this, it is not surprising that, the energy spectrum of the plasma ions, which was measured by a multielectrode electrostatic probe 4 at the far end of the side arm, showed a considerable energy spread. The form of

R F C O N F I N E M E N T AND ACCE L E RAT I O N O F PLrlSMAS

285

the measured distributions for 4 rf power levels 0, 2.7, 3.3, and 4 kV/cm is shown in Pig. 20. It will be seen that they all have a two-peak character. It is very difficult to know how they should be interpreted. Gekker e t al. (1965~)assumed that the plasma associated with the upper peak is that which was optimally situated for ucceleration in accordance with (1) a t the time of switching on the rf pulse. The difficulty with this interpretation is that either the mean or the maximum energy of the upper peak is a quantity much larger than can be explained by (1) unless one takes the plasma resonance very seriously indeed. For w o = 0, Eq. (1) gives A& = 1.2E2eV where E is measured in ltV/cm. Thus, in order to explain

6-

) .

4= 2-

4 kV/cm

FIG.20. Encrgy distribution of accelerated particles. U,energy in electron volts; f, number of particles, in units of 10' particles per electron volt.

-

the upper peak, it is necessary to suppose that w o 0 . 9 6 ~ for the measurements carried out at high power levels. Quite apart from the question as to whether dissipative processes (radiative or collisional) would limit the validity of (1) before this point, the value of w p required implies a plasma density of order 1 0 1 2 cm-3, which exceeds by two orders of magnitude the measured density in the absence of rf. The authors attempt to explain this by postulating supplementary ionization a t the junction due to the rf electric field; if this were correct, however, one would expect a much narrower distribution of accelerated ions in energy space. I n fact, it seems much more likely that some instability, possibly even the parametric resonance instability described in Section 4, is responsible for the particle acceleration. Thus the results cannot be said to give strong experimental support for Eq. (l), the theoretical basis of which is, as we have seen, weak.

286

n.

MOTZ AND

c.

J. H. W A T S O ~

The other two experiments carried out by the Rabinovich group are attempts to realize the conception of acceleration by traveling or nearly standing waves described in the preceding chapter. I n both cases, plasma is injected into a waveguide at some point and an rf traveling wave is then propagated towards it. The subsequent developments, when the rf wave front reaches the plasma, depend on the relationship between up and w. If w > up, almost any theory of the interaction would predict that it should continue to propagate, though possibly with a certain amount of transient reflection before the plasma motions have built up. I n consequence, the plasma should at least experience the (very slight) acceleration which results from individual Thompson scattering, and, if the theory of coherent Thompson scattering proposed by Veksler (1956) is correct, the acceleration should be much larger than this and possibly given by the term FP in Eq. (12) of Section 7. Furthermore, the wave sets up a quasipotential relief in the cross section of the waveguide and, depending on the mode, this might tend either to focus the plasma onto the axis of t,he waveguide or to repel it to the walls. In particular, in a cylindrical waveguide, the TEolmode has a quasi-potential minimum on the axis, whereas the TEll mode has a maximum. These two effects are to some extent antagonistic; it is of little value to focus the plasma onto the axis if there is no electric field on the axis which can bring about acceleration. Conversely, an electric field maximum, while ideal for acceleration purposes, leads to a rapid dispersal of the plasma to the walls and hence limits the achievable acceleration. The ideal arrangement would be a combination of wave types; so far, however, these two effects have been studied separately, and one machine operates on a TEoland the other on a TEll mode. In both machines, a supplementary static magnetic field has been made available as an optional feature, which can be used to diminish loss of plasma to the walls. When up > w , on the other hand, linear theory, at least, predicts that the traveling wave should be reflected and hence that a standing wave should be set up behind the plasma. The quasi-potential associated with this standing wave would initially tend to accelerate the plasma; on a longer time scale, it would only continue to do so if the radiation pressure exactly balanced the plasma pressure. Otherwise, nonlinear penetration of the rf field into the plasma (or vice versa) would tend to occur, in the manner discussed in Section 7. As regards the behavior of the plasma in the radial direction, it is difficult to make precise predictions. Eventually, it might settle down into one of the nonlinear confined cylindrical equilibria described in Section 2, provided that the power level were such that such an equilibrium existed. Initially, it seems plausible that the quasi-potential profile of the rf field unperturbed by the plasma motions would determine

RF CONFINEMENT .4ND ACCELERATION O F PLASMAS

287

the plasma behavior and it would again be lost to the walls in the TEll mode, but not in the TEol.The effect of a magnetic field on the radial motion would depend upon its direction. I n some of the experiments, a uniform longitudal field was used; as we saw in Section 1, provided that this is strong enough to make the Larmor radius small, it has the effect of tying the particles to the field lines, so that only motion along the field lines is significantly affected by the quasi-potential force. As resonance is approached, however, the actual particle orbit again becomes large arid this beneficial effect is lost. In other experiments, quadrupole or hexapole fields, set up by appropriately connected Joffe bars, were used. A discussion of the motion of plasmoids in such a field configuration has hccn given by Delone and Savchenko (1966).

The TEol machine as originally constructed is schematically represented in Fig. 21. The rf generator 1, which delivered up to 8000 W/cm2 of 10-cm rf power in pulses of 8 psec, was coupled to the cylindrical accelerating waveguide 4 through a ferrite isolator 2 and a King-type transformer from TEFo-+ TEso through a vacuum-tight window 11. The plasma was injected into the waveguide by a Bostick spark gun 5, mounted on a radial arm designed so as to minimize the perturbation of the rf fields. An optional longitudinal magnetic field of up to 300 G could be created by means of the windings 6. The diagnostics consisted of a n electrostatic probe mounted at 9, observation windows a t 10, and rf detector heads a t 12, which detected the transmitted signal and reflection from the plasma. Any rf power reaching the end of the waveguide was absorbed a t 8, and the vacuum pump 1 3 lwpt the background pressure a t 10-6 mm Hg. The waveguide was madc of stainless steel of thickness 1 mm; this allowed nmgnetic field penetration, but kcpt rf losses acceptably low (attenuation 0.2 dB). Initially, the spark gun was operated a t its maximum rate (1.8 1tA for 0.3 psec) giving a total of 1016-1016 ions (c 50% H) a t a density estimated as 1OI2 ~ r n -and ~ with an initial mean velocity 5 x lo6 cm/sec. The rf pulse was fired a variable riumher of psec after the spark

288

H. MOT2 AND C. J. H. WATSON

gun, and the percentage reflection of the rf signal and the number of particles accelerated beyond the initial mean energy were measured. At full rf power, acceleration to velocities exceeding los cm/sec was observed and a total number of accelerated particles not less than 10l2, a number which proved surprisingly independent of the amount of plasma injected. The rf reflection in some cases exceeded 90%; the plasma appeared to “close off” the waveguide almost completely. The plasma injection scheme described above was felt to be unsatisfactory: the plasma was dirty and had an unsatisfactorily high initial spread of velocities, tending to mask the acceleration, and the injector interfered somewhat with the rf configuration. Accordingly, when the

FIG.22

second machine, operating with a TEll mode, was constructed, the spark injector was replaced b y a discharge tube set into the waveguide at right angles in the manner indicated in Fig. 22. Here 1 is the rf generator, 2 a ferrite isolator, 3 the King-type transformer, 4 the vacuum window, 5 the waveguide, 6 the Joffe windings creating a hexapole magnetic field if required, 7 the discharge tube (with the anode and cathode marked A and K ) . The discharge was set up by injecting gas a t a pressure loe2 mm into the anode and cathode areas and was isolated from the walls by a longitudinal magnetic field; neut,ral gas was kept out of the waveguide by the delay chambers 8. The sequence of operation was to fire the plasma discharge, creating a plasma of density ~ m - ~pulse , on the static magnetic field (1000 G ) , which isolated the plasma from the walls, and then fire the rf transmitter. Nearly complete reflection of the rf signal was again observed. The acceleration was detected by probes a t 30 and 70 cm from the point of injection; these registered 110 particles a t all in the absence of

RF CONFINEMENT AND ACCELERATION OF PLASMAS

289

an rf pulse. The ion energy distribution observed at>half and full power are indicated in Fig. 23. Although the density falls off rapidly above 14 keV, particles with energies up to 50 kcV were observed at full power. Although the power level was only some three times larger than in the exponentially decaying wave experiments described above, the energies observed are impressively larger. It is not clear, however, that this reflects a real difference between the two cases, since no precise measurements were made of the fraction of the initially injected plasma which was successfully accelerated, and it is possible that Fig. 23 represenk no more than the high energy tail. In spite of the fact that, at the maximum power level, the energy peak shown substantially exceeds that which would be expected on single-particle quasi-potential theory, the authors felt obliged to discuss the reasons why the average energy was so low.

0.6

0.4

0.2

-

0

2

4

6

8 1 0 1 2 1 4

keV

FIG.23. Ion flux at 1-full

power, 2-half

power.

Recently these experiments have both been modified. Some results obtained with the reconstructed apparatus were described at the Belgrade conference last year and should be briefly summarized. Both systems now operate with a TEll mode. The machine which has the gas discharge tube as plasma injector has now been given new magnetic field windings which has made it possible to produce a uniform longitudinal magnetic field of strength up to the value at which the electron cyclotron resonance takes place. A complicat,ingfeature which has been established by measurement on this apparatus is that it now appears that two plasmoids are formed, whether or not an rf pulse is fired. This presumably results from the program of firing the discharge tube and the main and supplementary windings of the magnetic field structure. The “fast” plasmoid, containing 1011 particles moving at 6 x 106 cm/sec, is followed by a slow plasmoid of particles at lo6 cm/sec, so that when the rf is switched on, the waveguide has already been filled for a length of about 1 meter with plasma of varying density and velocity. The rf pulse (4 lrV/cm) then

290

H. MOTZ AND C. J. H . WATSON

accelerates the fast plasmoid to around lo8 cni/sec and the slow one to -8 X lo6. As the magnetic field is increased, both the ion flux and the mean directed energy of the ions increase; the behavior in the neighborhood of resonance is indicated in Figs. 24 and 25. It will be seen that the slow plasmoid is relatively unaffected by the resonance. Unfortunately, the resolution in the measurements in the immediate neighborhood of

I

0.25

I

0.50

1.

I

1.00

0.75

I

1.25

FIG.24

T

,Ot-

/

/Elow

plasmoid

B BC

FIG.25

resonance does not permit one to draw any conclusions about the way in which quasi-potential theory breaks down there. T he apparatus which formerly operated with a Bostick spark gun has now been modified as indicated in Fig. 26. The plasma generator 6 is now of the parallel plate discharge type, with the discharge occurring along a plexiglass surface and alow power Bostick gun used to inject it. The voltage across the discharge can be varied in such a way that the density of

RF CONFINEMENT AND ACCELERATION OF PLASMAS

29 1

plasma can bc adjusted by an order of magnitude. In this way, plasmas with wp < w and wp > w can be injected. The plasma beam is also carefully collimated, in such a way that it is possible to detect drift to the walls under the influence of the quasi potential when this occurs. This loss of plasma is indeed observed in the absence of a magnetic field, and the reduction in plasma density, computed from %,/no = 1/cosh2 WJT where 7 is the duration of the rf pulse, appears to correspond reasonably well with the observed figures, a t least for low rf powers; for larger powers,

FIG.26

there is a marked deviation, possibly because the formula from which this expression was derived,

is rather a crude representation of the effect of the quasi-potential. To summarize the results of this Russian work: it is clear that the results so far are of a preliminary nature. The rf mode types have not been selected in such a way as to optimize acccleration and minimize radial loss, the plasma injection procedures are all open to criticism on one ground or another, and the diagnostics do not really give a clear picture of what is happening. Nevertheless, the qualitative results seem interesting and encouraging. We now turn to the work of the group of Consoli at Saclay on nonconservative resonant rf accelerators. Three major devices have been constructed, named respectively Circe, Pleiade, and Icare. Figure 27 shows the magnetic and rf field configurations in each case. The first to be constructed, and the only machine for which extensive results are available, is Pleiade, a resonant standing wave accelerator of the type described in the preceding chapter. The main features of Pleiade 111, the accelerator assembly as it was in 1963, are indicated in Fig. 28. Plasma was created by a P I G discharge in section I (ths cathodes are labeled C1, Cz and the anode, A ) and was allowed t o diffuse into the resonant acceleration section 11,

292

H. MOTZ AND C. J. H. WATSON

which was a cylindrical cavity, excited for periods of u p to 300 p sec in the TErll mode by a magnetron delivering u p to 10 kW a t 10.5 cm. An alternative operating procedure was to allow the rf pulse to create its own plasma by ionizing the residual gas (background pressure lop4Torr). The magnetic field decreased across I1 from 1150 to 850 G in such a way that exact resonance occurred a t the midplane of 11,where the rf field was Circe

I

==ElFIQ.27

9 Spectrometer

FIG.28

maximum. Various diagnostic devices were located in sections I11 and IV. Two photomultipliers PM1 and PA12 separated by 1 meter were used to measure the time of flight of the accelerated plasma (to be precise, the electrons, since these are responsible for the luminosity). An electrostatic analyzer or a magnetic mass spectrometer located in section IV was used in conjunction with a measurement of the time of arrival of the ions a t the analyzer to determine the mass, energy, and charge distribution of the ions. Various gases were used, ranging from hydrogen to argon, and the final mean plasma velocity for a given rf power level was plotted as a function of the mass to charge ratio. The results obtained using the various

RF CONFINEMENT AND ACCELERATION O F PLASMAS

293

techniques for measuring the plasma velocity are summarized in Fig. 29. It will be observed that the velocities obtained by different diagnostic methods are in reasonably good agreement and that the slope of the line gives V f a ( M / Z ) * ' 6where M is the atomic number and Z the ionic charge. The constant of proportionality is -4 X lo7 cm/sec. It will be observed that this dependence upon M is in agreement with that predicted by the theory of Consoli and Hall (1963a) described in Section 7; the 2 dependence agrees with that which (as we have seen) they should have derived by generalizing that theory but not with the incorrect expression which they in fact give. The absolute magnitude of V f cannot be compared

I

0 W

c

6

-

I

% c

._ 0

-

P

I

2.

H'

Hi

4 6 810 2 He' k A*";

4 A'

6 810'

2

4

6 8Id

Masslcharge ratia for the ions ( M / Z )

FIG.20

with theory since they do not specify the rf field strength in the cavity; however, it is in reasonable agreement with the quoted maximum power level -10 kW if one assumes that the Q of the cavity is -lo3. The mean ion velocities quoted correspond to energies of order 1 keV for hydrogen and 2 keV for argon. Graphs of the experimentally determined velocity distributions are also given and have the form of single-peak distributions with an energy spread of the same order as the mean. More recently, the experimental arrangement has been substantially modified. Essentially, two accelerators of the Pleiade I11 design have been mounted in line, with their outlets facing each other, in the manner indicated in Fig. 30. It having been established in Pleiade I11that it is adequate to use rf ionization to create the plasma, the discharge apparatus has been replaced by quart,z capillary tubes capable of injecting gas a t a pressure of

294

H. MOTZ AND C. J. H. WATSON

10-3 t o 10-1 Torr into the resonance cavities I and 11,which are kept at a pressure of Torr. The new apparatus can be operated in two modes, accelerative or accumulative. I n the accelerative mode, the magnetic field profile is as represented by the lower line in the graph on the bottom of Fig. 30. With this profile, only the right-hand cavity functions as a n accelerator. I n the accumulative mode, the profile is arranged to correspond t o the upper line; in this case, both cavities act as accelerators and the accelerated plasma accumulates in the mirror machine so formed. The diagnostic equipment has been extended somewhat, and in addition to,the photomultipliers and electrostatic probes used previously, a movable pyrometric probe P and a radially movable Langmuir probe L have been

b /Ii

I

::

probe

Pump

ietric Movable calarin calorimetric probe

I5001

~

1000

500

Accumulation 2

I

I Acceleration

I

1500 'Oo0

BGsu~~

500

0

FIQ.30

introduced into section 111. The pyrometric probe measures ZnvW where n is the density, v the velocity, and W the energy of particles that are intercepted by a tungsten grid, which they heat and whose temperature is determined by optical pyrometry. The Langmuir probe was used to determine the radial variation in the plasma potential. The main results obtained with this modified apparatus are as follows. I n the accelerative mode, attention has been concentrated on the effect of working with cw rf power instead of the pulsed fields used previously. The much lower power level (-500 W) paradoxically somewhat increases the acceleration (i.e., the mean energy for hydrogen now being -13 keV). This is presumably because the transit time across the resonance region 7 = ( l / w ) ( c / z ~ ~ increases )~~~ as V E decreases and hence the last term of Eq. (26) becomes dominant. The much larger value of (B,,, - B4)/B,,, occurring in the modified experiment then compensates for the reduction in the rf field strength. The ion flux was found to be ~ 1 0ions/cm2/sec. ' ~

R F CONFINEMENT AND ACCELERATION OF PLASMAS

295

Taken in coiijection with the mcnn velority, this corresponds to a density of order lo7 ions/cm3, i.e., the plasma frequenry is sufficiently lower than that of the rf to justify the use of a single particle description of the accelerator action. One consequence of the very low power level is that a very significant fraction of it is absorbed by the resonant electrons; a t residual gas pressures of 5 X Torr as much of 25% of the rf energy is transferred to the electrons. Measurements of the radial distribution of flux and plasma potential showed that the core of the beam was positively charged and the surface negatively, the potential difference being as niuch as 300 V

Fro. 31

at maximum density. This confirmed a picture of the general behavior of the ions and electrons which had previously been obtained by analog computer, a sketch of which is given in Fig. 31. I n the accumulation mode, experiments have been carried out in which either both cavities were fed with CW rf power or one was fed with CW and one with pulsed power. In the latter case, plasma can be injected through the pulsed cavity, reflected and given supplementary acceleration by the CW cavity, and passed back through the original cavity (in which the rf field has by then decayed) into the spectrometer beyond it. I n this way plasma with an energy of order 3 keV was reflected and further accelerated to 5 keV. When both cavities are continuously operated, the pyrometer reading increased dramatically, which were taken as evidence of plasma accumulation. I n a typical experiment, helium a t 2 x Torr was accelerated through one cavity in the acceleration mode and

296

H . MOT2 AND C. J. H . WATSON

gave a pyrometer reading of 800". When the magnetic mirror a t the other cavity was set up, but the rf power not fed to the cavity, the pyrometer reading increased to 1300", from which one can infer by Stefan's law that the density in the central section increased 12-fold. When the second cavity was fed with rf power, the temperature increased to 2400°, corresponding to an 81-fold total increase in density. The second machine, Circe, operates on the traveling wave acceleration principle described in the preceding chapter. Like Pleiade, it was originally designed t o function as an accelerator only, but has since been modified to allow for operation in the accelerative or in the accumulative mode.

P (torr)

FIG.32

The accelerator waveguide is fed with 10-cm circularly polarized rf power of 300 W cw or 3.5 kW pulsed. The gas is injected through a specially mounted capillary, designed to give axial injection without affecting the polarization of the rf fields. The acceleration and fluxes were measured by the same diagnostic methods as in Pleiade. The results so far do little more than establish that the accelerator works; ion energies of up to 200 eV have been obtained in argon and fluxes of 2 x 1O1O ions/cm2 measured by means of the pyrometric probe. A rather odd feature is the dependence of the ion energy on the pressure of the injected gas, as shown in Fig. 32. I n the accumulative mode, argon deusities of 1.4 X 10'2 ions/cm3 a t a temperature of 120 eV have been obtained and confinement maintained for about 40 reflections, the limitation being charge exchange. This density approached the cutoff density a t which o = up. The third machine, Icare, can only operate in the accumulative mode, since it is a single resonance cavity, excited in the TEll2 mode in such a

Pump

I

,,

U

FIG.33

Electrostatic

canalyzer

298

H. MOTZ AND C. J. H. WATSON

way that the plasma is trapped in the neighborhood of the node after being accelerated in through the resonance regions established a t the antinodes. I n addition to these arrangements for confinement along the magnetic field, a radially increasing magnetic field gradient can be established by six Joffe bars. The experimental layout is shown in Fig. 33; no measurements are yet available.

ACKNOWLEDGMENTS It is a pleasure to make a number of acknowledgments in connection with this work. It was projected as a result of visits made by one of us (HM) to the Saclay Laboratory of the Commissariat a 1’Energie Atomique (Paris) and by one of us (CJHW) to the Lebedev Institute (Moscow) under the auspices of the Royal Society, and owes much to discussions which we had there, particularly with Mr. T. Consoli, Professor V. P. Silin, Yu. M. Aliev, and I. It. Gekker. We would both like to thank our respective host laboratories for their hospitality. We also benefited from valuable discussions with Professors R. E . Peierls and G. Schmidt. One of us (CJHW) has enjoyed the support of an I C I research fellowship in the Department of Theoretical Physics a t Oxford University and research facilities a t Culham laboratory during the course of the work. Finally, the preparation of this report has been assisted by the Air Force Office of Scientific Research under Grant AF 65-37 with t h e European Office of Aerospace Research (OAR) United States Air Force, and it has been typed by t h e ever-patient Mrs. L. Hill, to whom we are most grateful.

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R. Bardet, T. Consoli, R. Geller, L. Dupas, L. Leroy, and F. Parlange (1965~).Proc. Zntern. Conj. Ionization Phenomena Gases, 7th, Belgrade, 1965. Gradevinska Knjiga, Belgrade, 1966. C. K. Birdsall and A. J. Lichtenberg (1959). Phys. Rev. Letters 3, 163. C. K. Birdsall and G. W. Rayfield (1964). J. Electron. Control 17, 601-622. N. N. Bogolyubov and Yu. A. Mitropolski (1955). “Asymptotic Methods in the Theory of Nonlinear Oxcillations.” Gos. Izd. Fiz. Mat. Lit., Moscow. H. A. IT. Boot and R. S. R. Shersby Harvie (1957). Nature 180, 1187. H. A. H. Boot, S. A. Self, and R. S. R. Shersby Harvie (1958). J . Electron Control 4, 434 (see also S. A. Self, 1959). D. M. Bravo-Zhivotovsky, B. G. Yeremin, E. V. Zagryadsky, M. A. Miller, and S. B. Mochenev (1959). Radiojizika 2, No. 1, 94. L. S. Brown and T. W. B. Kibble (1965). Phys. Rev. 138B,740. S. C. Brown and A. D. McDonald (1949). Phys. Rev. 76, 1629. H. S. Butler and G. S. Kino (1963). Phys. Fluids 6, 1346. J. W. Butler. A. J. Hatch, and A. J. Ulrich (1959). Proc. Intern. Conf. Peaceful Uses At. Energy, dnd, Geneva, 1958, 32, 324. M. Cadart, T. Consoli, L. Dupas, J. Leroy, and F. Parlange (1965). Proc. Inter. Conj. Zonization PhenomenaGases, 7th Belgrade, 1965. Gradevinska Knjiga, Belgrade, 1966. E. Canobbio (1966). Compt. Rend. 262, 996. F. G. Cheremisin (1965). Dokl. Akad. Nauk SSSR, 163, 315. M. U. Clauser and E. S. Weibel (1959). I’roc. Intern. Conf. Peaceful Uses At. Energy, Ind, Geneva, 1968 32, 161. A. M. Clogson and H. Heffner (1954). J. Appl. Phys. 26, 436. T. Consoli (1963). Phys. Letters 7, 254. T. Consoli (1965a). Compt. Rend. 260, 4163. T. Consoli (1965b). Proc. 2nd Conj. Plasma Physics C . T . R. Culham. I.A.E.A., Vienna, 1966. T. Consoli and R. B. Hall (l963a). Nucl. Fusion 3, 237. T. Consoli and R. B. Hall (1963b). Compt. Rend. 267, 2804. T. Consoli and It. Le Gardeur (1962a). Compt. Rend. 264, 3178. T. Consoli and R. Le Gardeur (1962b). Compt. Rend. 264, 3323. T. Consoli and G. Mourier (1963). Phys. Letters 7, 247. T. Consoli. G. Ichtchenko, and M. Weill (1962a). Compt. Rend. 266, 2394. T. Consoli, R. Le Gardeur, and L. Slama (1962b). Rf plus B confinement. N d . Fusion 2, NO.3-4, 148-154. T. Consoli, G. Ichtchenko, and M. Weill (1963a). Compt. Rend. 266, 626, 903, 1090. T. Consoli, R. Le Gardeur, G. Mourier, P. Vial, and R. Roux (1963b). Rf plus B confinement. Phys. Letters 4, 89-91. T. Consoli, R. Le Gardeur, G. Mourier, P. Vial, and R. Roux (1963~).Phys. Letters 4, 106. T. Consoli, R. Le Gardeur, B. Jacquot, and L. Slama (1964a). Colloq. Intern. sur l’lnteraction des Champs avec un Plasma, Saclay, Gij-sur-Yvette, 1964, p. 184. Presses Universitaires de France. T. Consoli, G. Mourier, L. Slama, R. Roux, and P. Vial (1964b). Colloq. Intern. sur I’Interaction des Champs avec un Plaswsa, Saclay, Gif-sur-Yvelle, 1964, p. 149. Presses Universitaires de France. T. Consoli, G. Mourier, R. Roux, and P. Vial (1965). Compt. Rend. 261, 3314-3317. T. Consoli, G. Mourier, R. Roux, and P. Vial (1964~)Saclay, Gif-sur-Yvette, 1964, p. 168. Presses Universitaires de France. V. S. Cushing and M. S. Sodha (1959). Phys. Fluids 2, 494. Errata 3, 142 (1960).

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V. P. Silin (1965). Zh. Eksperim. i Teor. Fiz. 48, 1679. V. P. Silin and A. A. Ruchadee (1961). “Electromagnetnie Svoistva Plaemi i Plasmopodobnikh Sred.” Gosatomiedat, Moscow [see also Aliev et al. (1966); Gorbunov and Silk (1965); Gurevich and Silin (1965)l. J. A. Stratton (1941). “Electro Magnetic Theory.” McGraw-Hill, New York. P. A. Sturrock (1959). Microwave Lab. Rept. No. 638 AF (638)-415. P. K. Tien, J . A p p l . Phys. 26, 1281 (1954). W. B. Thompson, Proc. Phys. SOC.(London) 70, 1 (1957). Vachaspati (1962). Phys. Rev. 128, 664. A. A. Vedenov, T. F. Volkov, L. I. Rudakov, R. Z. Sagdeev, V. 1LI. Glagolev, G. A. Yeliseyev, and V. V. Khilil (1959). Proc. Intern. Conf. Peaceful Uses At. Energy, Geneva, 1968, 32, 239. V. I. Veksler (1956). CERN Symp. High Energy Accelerators Pion Phys., Geneva, 1966, Proc. p. 80. V. I. Veksler (1957). At. Energ. ( U S S R ) 2, 427. V. I. Veksler et al. (1963). Fiz. Instit. Akad. Nauk Rept. N T D 6317 It (preprint of paper read a t Intern. Acc. Conf., l h b n a , 1963. V. I. Veksler et al. (1965). At. Energ. ( U S S R ) 18, 13. T. F. Volkov (1959a). Plasma Phys. Probl. Controlled Thermonvcl. Reactions 3, 395. Leontovich, ed., Eng. transl., Pergamon, Oxford. T. F. Volkov (1959). Zh. Eksperim. i Teor. Fiz. 37, 422. T. F. Volkov (1960). Zh. Tekhn. Fiz. 30, 497. T. F.Volkov and B. B. Kadomtsev (1962). At. Enwg. ( U S S R ) 13, 429. G. B.Walker and J. T. Hyman (1958). Proc. Znst. Elec. Engrs. (London),Pt.B 106,73. C. F. Wandel, T. Hesselberg Jcnsen, 0. Kofocd Hanuen, A. H. Sillesen (1968). Riso Rrpt. No. 2, Hoskilde, Ilcnmnrk. C. J. H. Watson (1964). “The Electrostatic Oscillations of a Non-Uniform Plasma.” Thesis, Oxford Univ. (see also Yu. M. Aliev el a2.).

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Noise in Semiconductor Devices EUGENE R. CHENETTE* Electrical Engineering Department University of Minnesota Minneapolis, Minnesota

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Theory of Noise in Semiconductor Devices.. . . ................... A. Sources of Noise.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Noise in Junction Diodes. . ..................... C. Noise in Bipolar Transistor D. Noise in Field Effect Transistors.. . . . . . . . . . . . . . . . . . . . . . . 111. Experimental Verification of the Theory, . . . . . . . . . . . . ............... A. Experiments on Semiconductor Diodes. . . . . . . . . . . ............... B. Experiments on Bipolar Transistors. , ...................... C. Experiments on Field Effect Transistors,. . . . . . . . . . . . . . . . . . . . . . . . . . . D. Experiments on Flicker Noise.. . . . . . ...................... IV. Practical Low-Noise Amplifiers. . . . . . . . . .......... A. Bipolar Transistor Amplifiers. . . . . . . . . . . . . . . . . . . . . . . . . . B. Field Effect Transistor Amplifiers., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Comparison of Bipolar and FET Amplifiers.. . . . . . . . . . . . . . . . . . . . . . . . D. Tuned Amplifiers.. . . . . . . . . . . . . . . . . . . . . . . . . . . ................. E. Noise in Diodes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Summary.. . . . . . . . . . . . . . . . . . . . . . . . . . . ......................... Appendix. Noise Representation, . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . .........................

303 304

319 320 320 326 328

334 337 338 340

I. INTRODUCTION There are at least two important facets to the subject of noise in semiconductor devices. The first is that noise results from basic physical processes which make a device possible; a study of device noise can lead to increased understanding of device physics. The second is that noise sets the ultimate limitation of the usefulness of a device as a linear amplifier; it is important to the engineer using a device that he be able to optimize the design for maximum signal-to-noise performance commensurate with other requirements. This paper presents a summary of the theory of noise in semiconductor devices with emphasis on junction diodes, bipolar transistors, and field effect transistors. The results of various workers showing the experimental

* Present address : Bell Telephone Laboratories, Allentown, Pennsylvania. 303

304

EUGENE R. CHENETTE

verification of the theory will be summarized. Finally, sonie practical lownoise circuits and “state-of-the-art” low-noise devices will be discussed. Uniform notation, agreeing with IEEE standards, will be used throughout the review. The goal is to make the theory and its experimental verification easily available and useful to as many readers as possible. Emphasis throughout this paper is on physical modeling of the device noise. The point of view being promoted is

“. , . if available information about the physics of the noise in a particular device suggests introducing, inside the two-port, appropriate noise generators or fluctuations (spectral densities) with no frequency dependence, or with a simple frequency dependence, it may be more advantageous to specify the device in terms of these physically suggestive parameters.” ( 1 ) It will be shown that the noise performance of most semiconductor devices is described by a relatively simple, well-behaved set of “internal” noise generators. 11. THEORY O F NOISE I N SEMICONDUCTOR DEVICES

A . Sources of Noise

It is important for the understanding of noise in semiconductor devices to have some understanding of the basic sources of noise in semiconductor materials. Excellent reviews are available on this subject (2-4). The terms “shot noise” and “thermal noise” will be used frequently in this paper. However, they are more popular than precise! A more basic physical picture of the noise sources is stimulated by the names transition noise and transport noise (4). As an example of transition noise consider generation-recombination noise. It results from fluctuations in generation rates, recombination rates, trapping rates, etc. It is the result of fluctuations in interband transitions, i.e., transitions between energy levels or between impurity levels and the valence or conduction band. Thermal noise is an example of noise caused by intraband transitions which result in the scattering of carriers. The noise of intraband transitions stems from velocity fluctuations. Transport noise results because the carrier transport mechanism, whether drift or diffusion, is a random process. In a temperature-limited vacuum diode this transport noise has long been called “shot noise.” The name is frequently applied, as in this paper, to noise resulting from transport fluctuations even though the transport mechanism may not be drift.

NOISE IN SEMICONDUCTOR DEVICES

305

The most characteristic thing about transport noise, for our purposes, is that the magnitude of the noise resulting is, a t least a t low frequencies, about the same as th at of the shot noise of a temperature-limited vacuum diode. Modulation noise refers to noise which is not caused by transition or transport fluctuations directly but which, rather, is caused by some modulation mechanism. A modulation mechanism located a t the device surface probably gives rise to most flicker noise in transistors. Flicker noise will be discussed only briefly. The understanding of this noise with its characteristic 1/J dependence of the spectral density is not very complete. There has been a lot of work on flicker noise because of its importance in limiting the low-frequency usefulness of semiconductor devices but the mechanism or mechanisms which cause it are not well understood.

B. Noise in Junction Diodes The careful study of the development of the theory of noise in p-n junctions (and in transistors) provides an interesting journey into technical literature. One of the most interesting features revealed by this study is the excellence of the earliest, almost intuitive, explanations of noise in junction devices. Later workers have increased the rigor of the understanding of the limitations of the theory but have added little of experimental importance. Two basically different approaches have been followed in deriving theoretical expressions for the noise of p-n junctions. They are commonly referred to as (1) the corpuscular approach, and (2) the collective approach. The collective approach is probably the more scientifically correct and intellectually satisfying. The corpuscular approach seems to work! Any claim to rigor must rest in the fact th at the results obtained agree with those of the collective approach. Van der Ziel has compared the two methods in some detail ( 5 ) . 1. Corpuscular Approach. The corpuscular method depends on being able, on some basis, t o divide the various groups of carriers crossing a p-11 junction into well-defined-and physically correct-groups. Figure 1 shows a representation of the various groups of current carriers in a forward biased junction. T o keep the figure as simple as possible the picture is that of an idealized diode with the only currcnt carriers of consequenc’e being holes. The only electrons iniportaiit to the operation of the device shown in Fig. 1 are those which recombine with the trapped holes of group 3 or are thermally detrapped with the holes of group 4. The holes of groups 3 and 4 are those responsible for the Sah-Noyce-Shocltley current of the junction (6). The addition of electrons crossing the junction to the

306

EUGENE R. CHENETTE

sketch of Fig. 1 would have no influence on the main points of the argument to follow. The holes of groups 1 and 2 are responsible for the diffusion current. If they were the only carriers, the diode would be described precisely by the diode equation I D =

I&xp ( q V / k T ) - 11.

The holes of group 1 carry the current ID of group 2 carry the current -10.

(1)

+ loacross the junction; those

FIG.1. Groups of current carriers (holes) in a p-n junction.

The holes of groups 3 and 4 give rise to the recombination current

where 1~~is a slow function of the volt-drop across the junction, V (6). T h e holes of group 3, together with the electrons with which they recombine after being trapped in the transition region, transport the current IR IRO across the junction; those of 4 are responsible, with their electrons, for the current -I R ~ . According to Schottky’s theorem, if each currcrit carrier contributes a current pulse of area Q to the external circuit and if the crossing of the junction by each carrier is an independent random event, each of these four components of current should show full shot noise. Thus the fluctuations in the dc current through the junction could be represented by a shot noise generator connected in parallel with the small-signal conductance of the junction. Figure 2 shows the rcsultiiig noise equivalent circuit. The mean-squared value of the noise current generator in a narrow fre-

+

quency interval is

+

NOISE IN SNMICONDUCTOR DEVICES

307

i2 = 2 q ( I ~3- 210 -k I R -k 2 1 ~ 0 df ) 2 d O df,

(3)

=

where I = I D IR is the total diode current. The conductance of the diode is the sum of the conductances of the diffusion current and the recombination current. = [gD -k gR1 = (q/kT)[ID-k = ( q / k T )[ID -k

l a -k ( I R -t- I R O ) / ~ ] IR/2].

(4)

This equation can be combined with Eq. (3) to yield the expression

i2 = 4kTg df - 2 q ( I / m ) df;

m = 1 if the recombination current is negligible and in bination current dominates the diode (7).

t

(3%) =

2 if the reconi-

YDicdr

FIG.2. Noise equivalent of a p-n junction in the shot noise frequency range.

So far no mention has been made of the holes of group 5 shown in Fig. 1. These holes are emitted into the n region but are returned to the p region without recombining. Thus they have no effect on dc current and no effect on the low-frequency noise. They are important a t high frequencies. Their effect may be included formally b y replacing the g of Eq. (3a) with the real part of the diode high-frequency conductance. T o emphasize that this substitution has been made one can write the expression as

i2 = 4kT Re(y) df - 2 q ( I / m ) df.

(3b)

The basic arguments of the above model have been presented by several workers. Weisskopf was the first to apply the picture t o point contact diodes (8); Uhlir used it for high-frequency noise in diodes (9) ; van der Ziel and Becking used it and ,applied the approach to transistors (10); it was, apparently, the basis of Giacoletto’s model ( 2 1 ) for noise in diodes. Schneider and Strutt (7) were the first to present Eq. (3b) with the m

308

EUGENE R. CHENETTE

factor included to account for the effect of SNS current on noise in silicon diodes. 2. Collective Approach. As was mentioned above, the collective approach to the theoretical modeling of noise in junction devices is much more sophisticated than the corpuscular model. It requires solution of the differential equations describing carriers within the material subject to the boundary conditions imposed by the junctions and contacts. Fortunately the main points of the performance of a junction device can be included in a one-dimensional model. One-dimensional collective theories were first published by Petritz (12) and by van der Ziel (13). Solow (14) extended Petritz’ solution to the three-dimensional p-n junction. Polder and Baelde (16) have recently presented a theoretical derivation which displays the limitsationsof the simple collective model (and probably of the corpuscular theory as well) most clearly. They used an elegant “equivalent circuit” argument to obtain the final expressions for the magnitudes of the noise generators and their correlation functions, as well as using a formal mathematical procedure. Time does not permit reproducing their argument in detail. However, the basic assumptions and the final result will be stated. The basic assumptions of the junction noise theory as presented by Polder and Baelde are: (1) The concentrations of majority carriers are so high that ohmic volt-drops due to majority currents can be neglected. (2) Injection levels are low everywhere compared with majority carrier concentrations. (3) There is no recombination in the junction space-charge layer. (4) Bulk recombination of carriers is proportional to the excess concentration of minority carriers. (5) Generation-recombination noise at recombination centers consists of the sum of the shot noise of a recombination current and a generation current independently.

Except for these assumptions, the device was of general nature. The model allows a built-in drift field because of variations in doping. The material properties, lifetime, surface recombination velocity, etc., can all be functions of position. The device may be of arbitrary shape and have N junctions to the base region in addition to an ohmic contact. The noise behavior is completely specified by N external noise generators (and their N 2 correlation functions) between the N external leads to the junctions and the base contact. The final result is that the spectral

NOISE I N SEMICONDUCTOR DEVICES

309

densities of these N 2 correlation functions are (15) Sij

=

2kT(Yij

+ Y,,*) - 24Ij,

(5)

where I, is the dc current entering through lead j. Y,, is the transadmittance from the i th to the j t h lead. Y,, is not necessarily equal to YJ Zsince this may be a n “active” device. It is apparent that this result is the same as that of Eq. (3) for a single junction with negligible recombination current. It will be seen below th a t this result is exactly that yielded by the corpuscular model for a bipolar transistor. This is probably a good place to reiterate assumption number ( 2 ) above. The theory is valid only at low injection levels. Van Vliet (16) has considered a general model of a p-n junction which allows for high injection effects. His result at low injection levels agrees precisely with Eqs. (3) and ( 5 ) . At high injection levels his expression for the noise current magnitude becomes

i2 = 4kT Re(y)[(l

+ b)/b] df - 2pI df,

(61

where b is the ratio of the carrier mobilities, i.e., b = (pn/pp). Van Vliet is extending these calculations to the bipolar transistor.

C . Noise in Bipolar Transistors The major changes in the theory of noise in junction transistors since van der Ziel published his classical survey of noise in semiconductor devices (17) [see also van der Ziel (S)] have been in understanding the effect of SNS current on the noise performance and in the problems of high injection levels (7, 18-20). The noise theory below includes the effect of recombination current. More will be said later about high injection effects. The corpuscular approach can be used to derive the basic noise model of a junction transistor so as to identify the effects of recombination current. It is necessary to identify the various groups of carriers in a n operating transistor. The Ebers-Moll equations (21) can be used as a guide in identifying the various groups in much the same way the Shockley diode equation was used above to guide the choice of groups of carriers in a forward biased diode. The use of the Ebers-Moll notation results in a more accurate definition of groups and their contribution to the terminal currents. The notation is slightly different than that published previously by van der Ziel and Becking (10).It will be seen below that the results of this derivation are identical to that obtained with a straightforward application of Eq. (5).

310

EUGENE R . CHENETTE

The derivation begins with the E-M equations for the intrinsic p-n-p transistor. I E = IC =

IES[exp (ql’EB/kT) - 11 - aRICS[exp (qVCB/kT) - 11, (7) -aFIES[exp (qVEB/kT) - 11 ICS[exp (qVCB/kT) - 11. (8)

+

The saturation currents, IES and ICS,and the forward and reverse current gains, LYF and a R , are related by the expression

~FIE =S ~RICS.

(9)

The current which results from the trapping and recombination of current carriers in the junction transition regions is given by the formal expressions I R E = IRES[exp (qVEB/2kT) - 11 (10) and I R C = IRCS[exp (qVCB/2kT) - 11. (11) and IRC are the recombination currents of the emitter and collector transition regions respectively; I R E S and IRcs are the corresponding

IRE

I

IE!

FIQ.3. Groups of current carriers in an idealized p-n-p transistor. See Table I for the contributions each of these groups makes to the dc current.

slightly voltage dependent “saturation currents.” Compare these with Eq. (2) for the recombination current of a single diode. For normal small-signal operation, exp [qVc,/kT] << 1. If this is taken into account and Eqs. (11) and (12) are added to Eqs. (8) and (9) the result is I E = IES

eXp (qVEB/kT) - I E S ( 1 - a F )

+

IRES

exp (qVEB/akT) - IRES

(74

and

Ic

= -aFIEs

exp (qVEB/kT) - Ics(1 - a n ) - IRCS.

(84

N O I S E I N SEMICONDUCTOR DEVICES

311

Some insight into the significance of each of the terms of these two equations can be gained from a brief examination of the various groups of charge carriers in a transistor biased for normal small-signal operation. Figure 3 shows a sketch of the paths followed by the various groups of carriers. (The argument could be generalized to a nonideal p-n-p transistor (one with electrons contributing to the current) but the sketch of charge groups becomes too cluttered for each use. There would be no change in the final result.) Table I gives a brief summary of the various groups of current carriers shown in Fig. 3. The theoretical noise genarators for the transistor are obtained by the use of Schottky’s theorem. The fluctuations in the current resulting TABLE I Contribution to dc current

Group of carriers

1

a p I m exp

(l

(~VEBI~T)

- aYF)’Es

exp

Description of the carriers Holes emitted from the emitter into the base region and collected by the collector. This is the most important group! Holes emitted from the emitter into the

(qVEBIk?o base region and combining there.

Holes, generated in the base region, which are emitted “backwards” into the emitter. Holes, generated in the base region, which are collected by t h e collector.

5

IRES exp (QVEB/%rr)

Holes emitted from the emitter towards the base b u t which are trapped in the transition region. They combine with the electrons of group 6 .

6

IRES

Holes thermally detrapped in the emitter transition region. The electrons move into the base.

7

Holes thermally detrapped in the collector transition region. The electrons return t o the base.

8

0

Holes emitted into the base region but which return t o the emitter. They contribute to the high-frequency behavior of the junction.

9

0

Holes which are trapped in t h e emitter transition region but which are detrapped and return to the emitter.

312

E U G E N E R. CHENETTE

from the emission of carriers across a potential barrier can be represented by a shot noise current generator in parallel with the admittance of that emission. Figure 4 shows the basic noise equivalent circuit. It consists of the common-base physical-“ (22) with two noise current generators added in parallel with the admittance of the emitter-base and the collector-base junctions. I n addition, a noise emf has been added to account for the noise of the base region. The value of this base noise emf has

n

FIG,4. Noise equivalent circuit for a junction transistor using the common-base physical-T small-signal equivalent circuit. usually been taken to be the full thermal noise of a single lumped base resistance, i.e. ( l a ) ,

-

eb2 =

4 k T r ~ df. b

(12)

Current transport across the emitter-base junction results from the carriers of groups 1, 2, 3,5, and 6. Groups 8 and 9 must also be included. The magnitude of the noise current generator is therefore

-

+ +

iI2 = 2q ~ ~ [exp I E(QVEBIJCT) s IRES exp ( q V ~ d 2 k T ) I E S(~ ~F)IE -S (YF) ( ~ I R E S ] 4kT(ge - gee) df ( 1 3 ) = 2q d f [ l -k ~ 2 1 ~ s ( l- LYF) ~ ~ R R -k S ] 4lcT(g, - &o) df. (13a)

+

+

The final term of expression (13) results from the carriers of groups 8 and 9. This is a (‘thermal noise’’ term. ge is the junction conductance and gBOis the low-frequency value. This equation can also be written as

-

iI2= 4kT Re(y,) df

- 2q IE/n%Edf.

(13b)

313

NOISE I N SEMICONDUCTOR DEVICES

The rioisc cwrrent generator iz rcsults from fluctuations in the carriers of groups 1, 4, and 7. Summing the shot noise of each of these groups yields i2' = 2q ~ ~ " ( Yexp FIE ( QsV E B / ~ T ) Ics(1 - a ~ ) IRCS](14) = 2q df Ic. (144

+

+

The correlation between il and iz results from the carriers of group 1. Thus a t low frequencies

-

ii*& = 2Q df[QFIEs exp ( q V ~ n / k T ) ] .

(15)

If one introduces the low-frequency transconductance gceO = a l C / a v E B

Eq. (9) can be rewritten as

=

[q/kT]aFIES

exp (qVEB/kT),

(16)

il*iz = 2kTgc80 df.

At highyfrequencies the same processes which cause the frequency dependencebf the transfer admittance cause the frequency dependence of Thus the correlation term can be written formally as

zx.

or

It can be shown that Eqs. (13), (14), and (17) are exactly the results which would have resulted from a straight,forward application of Eq. ( 5 ) to tJhebipolar transistor.

x

-

FIQ.5. Noise equivalent circuit for a junction transistor using the common-emitter hybrid-pi equivalent circuit. 2 and 2 are shot noise sources; 2 is a l/f noise source.

An alternative noise equivalent circuit which can be very useful is shown in Fig. 5. It was first presented by Giacoletto (11).T h e noise current generators and the thermal noise of the base resistance are superimposed on the hybrid-pi equivalent circuit. The magnitudes of the generators and their correlation may be determined by comparison with

314

EUGENE R. CHENETTE

Fig. 4. Or Schottky’s theorem can be applied directly to determine the fluctuations in the dc currents. If the latter scheme is followed, the meansquared values, at “low frequencies,” are

-

+ +

i ~ =’ 2q @[(I - ~ F ) I E exp S ( Q V E B I ~ T ) IEs(1 - LYF) I R E S exp (qVEB/2kT) -E 2 q I ~df, i C 2 = 29 df[aFIEB exp ((IVEB/kT)I - 2 p I c df, iA2 = 2qdf[lcs(l - ad IRCS] z 0.

+

+

IRES]

(18)

(19) (20)

The approximate forms are valid for modern high gain, low leakage current transistor at normal bias points. The noise of the base resistance is still given by Eq. (12). At high frequencies it is necessary to account for the carriers of groups 7 and 8 and the induced-base noise resulting from the transit of the carriers of group 1 through the base region. It is the author’s opinion that at the extremely high frequencies (near fa) where these effects become important it is more convenient to use the physical-?’ circuit (Fig. 4). The main usefulness of the hybrid-pi is in the range where frequency independent values of the various parameters describe the small-signal behavior. At best this is no higher than about 0.3 fa. It may very well be required to include the effects of packaging parasitics on the equivalent circuit even in this frequency range. This does not change the internal noise generators however.

D. Noise in Field E f e c t Transistors The sources of noise in FET’s are not as well understood as in bipolar transistors. It appears that the ultimate limitation is the thermal noise of the conducting channel (25-25).However, many FET’s of both junction and MOS types show excess noise. All FET’s show excess noise with the familiar l/f spectral distribution; this noise varies greatly in magnitude from unit to unit and is an important limitation at low frequencies. In addition many FET’s show excess noise with a 1/(1 ~ ~ frequency 7 ~ dependence (26-28, 64). This is apparently the result of the trapping of carriers in the gate-charge region. I t is also common to find a “white” excess noise which must arise from a very fast trapping mechanism, or possibly from thermal noise of the substrate. Figure 6a shows an equivalent circuit of an FET with three noise current generators. iEsis placed in parallel with ggs and cgs, the gate-source conductance and capacitance; i g d parallels the gate-drain conductance ggd and capacitance cgd; and ids parallels the drain-source conductance.

+

)

315

NOISE I N SEMICONDUCTOR DEVICES

The circuit of Fig. 6b is equivalent to that of Fig. 6a if ig = i,, id

+

igd

and

- - igd.

= ids

ig2is the mean-squared value of the noise appearing a t the gate with the drain ac grounded. zFis the noise appearing a t the drain with the gate ac grounded.

---I--%

0)1 FIG.6. (a) Noise equivalent circuit for a field effect transistor. (b) This equivalent circuit for noise in an FET is the same as Fig. 6(a) if i, = (igS ind) and

+

id = (id,

- igd).

The most satisfactory derivation of the effect of the magnitudes of these generators has been obtaiped by Shoji (29) who treated the RiIOS transistor as a nonlinear transmission line. His results verify those obtained by van der Ziel using an approximat>eanalysis (24, 25). Van der Ziel's original discussions mere coricerricd mainly with junction FET's.

316

EU G EN E R. CHENETTE

Jordan and Jordan (30) used a similar technique to calculate the drain noise of an enhancement type insulated gate FET with very similar results. It seems there are only small numerical differences between the magnitudes of the noise sources in junction and in RIOS FET’s. An excellent survey of noise in MOS FET’s was presented by Johnson in his book (31). Here is a brief summary of the derivation of the magnitude of the drain-source noise current generator as first presented by van der Ziel. The basic model of the junction F E T is that presented by Shockley (32). Figure 7 shows a sketch of the longitudinal cross section of such an FET with dc bias voltages applied and with both the gate and drain ac shortcircuited to the source. The conducting channel is assumed to be p-type

FIG.7. Sketch of the longitudinal cross section of a junction FET. The incremental resistance between zo and (zo Az) gives rise to a thermal noise emf

+

-

ADZ = 4 k T ( A r ( s , ) ) df

and of uniform conductivity ao. Field dependence of the mobility is neglected. The length of the channel is L ; its thickness is 2 a and it is of unit width. The depletion layers of the gate junction reduce the effective thickness of the channel to 2 6 ( x ) at a point x units from the source contact. The resistance of an element of the channel between x o and x o Ax is

+

According to Shockley’s model, b ( x ) and v(z), the volt-drop across the gate junction, are related by

b(x)

=

41 - ( v ( ~ ) / V n o ) ” ~ l ,

(22)

where Voo is the bias required to cut off the channel. Figure 8 shows an approximate sket,ch of the v(z) as a function of position from the source

317

NOISE I N SEMICONDUCTOR DEVICES

(x = 0) to the drain (x = L). v ( x ) varies from v(0)

=

+ q) to

(VCS

v(L)= (VCS+ cp - VDS).9 is the diffusion potential of the gate junction. x

The drain current, I D , is related to the gate volt-drop at x L by the expression

=

0 and

=

I~

=

gcvoo[y- x - j

( p 2

- ~3/2)1

(23)

whereg, = 2a . a o / Lis the conductance of the open channel, y = [v(L)/V001 is the normalized gate volt-drop at the drain, and z = [v(0)/VOO] is the normalized gate volt-drop a t the source. Thus, the transconductance is g m = - (aID/dVCS) = gc(y'l2 -

2'12) ;

(24)

and the output conductance is

When VDS = 0; v(0)

gd

=

dTD/dVDS = gc(1

=

v(L)or y gd =

gdO

= x =

- y1l2).

(25)

and

gc[l - 21'2].

(26)

At saturation y approaches unity and lim gm = gm(max) = gc[l -

w+

2'121 = gdO.

(27)

1

The magnitude of 2 is calculated as follows: The basic noise source is the thermal noise of the channel. Thus, by Nyquist's theorem, the noise of a narrow region between xo and xo Ax can be represented by the noise emf of the resistance Ar(xO), Eq. (21) :

+

Av(x0)*= 4kT Ar(xo) df

=

AV 4kT df ID

Here A V is the change in volt-drop across the element of resistance Ar(xo) because of the drain current. The perturbation in the channel voltage a t 50produces a perturbation in the channel current Aid (and also a perturbation Ai, in the gate current). A sketch showing the effect of a small emf in the channel a t xo is shown in Fig, 8. This small jump in v ( x ) at x o must cause a small jump in the drain current. It can be shown that Aid and Av(x) arerelated by the expression A i d = gc[l - V0(~0)/V001'21A D ( ~ G ) , (29) and hence = gca[l - ( V , ( S O ) / V O O ) ' / ~ ] ~ A Z (30)

z2

318

E U G E N E R. CI-IENETTE

The total fluctuation in the drain current is obtained by combining Eqs. ( 2 2 ) , (28), and (30) and integrating over the length of the channel. The result can be written as c

ids2 =

41cTgm(max)df Q(VGS,VDS)

(31)

where Q(VGS,VDS) is a complicated function of the gate and drain biases. It lies between and Q for a junction FET and is equal to Q for an MOS FET at normal bias. This result is amazingly close to the most simple guess for the drainsource noise current. At very low drain and source voltages one would

+

VGS’

X

FIG.8. Approximate sketch of the volt-drop across the gate junction as a function of distance from the source. A noise emf a t zocan give rise to a perturbation as shown. This “jump” Av(zo) must cause a jump in the drain current Aid.

expect the drain-source conductance to show full thermal noise,‘i.e., one would expect i d , 2 = 4kTgdodf for zero bias. The simple theory shows (Eq. (27)) g d O = g,(max). The result of Eq. (31) is very close to this value. A similar calculation can be made to show the effect of the coupling of the thermal noise emf through the gate junction capacitance into the gate circuit. Calculating the from each incremental noise emf of the channel and then integrating over the length of the channel yields

where gll = (gm circuited and

+ q,J

is the input conductance with the output shortFET ( 2 4 forfor aanjunction MOS FET. 1

P(VGS,VDS)

=

319

NOISE I N SEMICONDUCTOR DEVICES

Equation (32) is an excellent agreement with the value for is obtained with Eq. ( 5 ) ,

2 which

-

ig2= 4kTgll df - 2qI0 df.

(33)

Polder and Baelde’s basic assumptions allow for variations with position of the electric field in the “bulk region of a junction” and for the variation of material parameters with position. The presence of drift current in the conducting channel does riot violate these assumptions. At high frequencies Eq. (33) will be dominated by the thermal noise term since 911 increases with ~2 and the gate bias current I G = - 1 ~ 0is usually very small. The gate- arid drain-source noise current generators are partially correlated since they are caused by the same incremental noise emf’s in the conducting channel. Van der Ziel has calculated the correlation coefficient and obtai ried

where 0.393 < (KI < 0.446 for normal bias conditions. It is also convenient to know how to divide i, between i,, and can be shown that

i,d2

arid

-

=

4kTg,d df

-

-

igb2 = ig2- i , d 2

igd.

It

(35) = =

4kI’(yll - grid) df 4kTg,, df.

(36)

Thus the gate-drain and gate-source noise current generators both show about full thermal noise of their corresponding conductances. 111. EXPERIMENTAL VERIFICATION OF

THEORY As the title of this section indicates, there is excellent agreement between theory and experiment for semiconductor devices over a wide range of operating conditions. The problems of comparing theoretical and experimental noise performance of a device can be a formidable one. The reader who wishes to verify the agreement for himself faces a n interesting problem. Methods of noise measurement are discussed in detail in several texts (33, 3). A noise measurement system requires good linear amplifiers. It is helpful if the system noise is negligible with respect to the noise of the device under test or at least constant so it can be easily subtracted. True quadratic or square-law detectors are also useful and precise noise calibration standards are essential. It is important that noise measurements be made over a wide freTHE

320

EUGENE R. CHENETTE

quericy range to guard against experimental errors-such as are caused by excess noise in the device under test or by systematic problems. Detailed experimental noise studies are often tedious and time consuming because of the care required to verify the precision of the measurements. Another problem in attempting to compare theoretical and experimental performance is designing the experiment so that it yields maximum information about the noise of the device. Simple noise factor measurements, attractive as they are to the engineer applying a device, seldom are the most useful experiment. The reason for this is that noise factor depends not only on the noise generators but also on the smallsignal performance of the device under test. If a discrepancy occurs it is difficult, in most cases, to determine whether the reason is an error in the noise theory or simply (or not so simply) a problem with the small-signal characterization. This problem will be mentioned again in the discussions of both bipolar and field effect transistors.

A . Experiments on Semiconductor Diodes Studying the noise of junction diodes in the laboratory usually does not allow much freedom in the design of experiments, since there are only two terminals. The noise can be represented either as an emf in series with the impedance of the device or a noise current generator in parallel with the device conductance. The experiment is fairly difficult however because of the low impedance level of the forward biased junction and because the open circuit noise signal of a well-behaved diode is only expected to be about one-half the thermal noise of a resistor of the same impedance. Champlin circumvents this problem with a measurement technique which compares the noise of the diode under test with the thermal noise of a lumped R-C circuit of the same impedance (34). Guggenbuehl and Strutt used transformer coupling between the diode under test and the preamplifier to increase the signal amplitude (35). The results obtained by these workers show good agreement between theory and experiment. Schneider and Strutt found good agreement between theory and experiment for diodes with substantial recombination current (‘7). At high injection levels Schneider and Strutt expected to find junction noise negligible with respect to the “thermal noise of the material of the junction.” Their results were inconclusive because of the large amount of l/f noise a t the high bias currents (18).

B. Experiments on Bipolar Transistors As has been mentioned previously, the problem of comparing the measured noise performance of a transistor with that calculated with the

NOISE IN SEMICONDUCTOR DEVICES

321

theory requires riot only inforliiation on the magnitude of the noise generators but also information 011 the small-signal performance of the transistor. Calculations required for this purpose can be simplified somewhat by converting the noise equivalent circuit of Fig. 4 to the one shown in Fig. 9. The two almost completely correlated noise current generators in Fig. 4 are replaced by an output noise current generator io = iz - ail

I

X

X

FIG.9. Modified noise equivalent circuit for comparison between theoretical and experimental performance.

and a n input noise emf e base current gain is a =

= i12, =

a0

i ~ / y , .The expression for the common-

+.iJ/Im).

exp (,+Kf/.fJl(l

The mean-squared values of the generators are

where and

(37)

322

EUGENE R. CHENETTE

In this author’s opinion the most satisfactory approach to comparing the theoretical and experimental noise performance of bipolar transistor is that followed by van der Ziel and his students (19, 36-38>. The basic experiment is measuring the equivalent noise current a t the output of a transistor with its input ac open-circuited. This yields information almost directly about 2 as long as xc >> Tb%. After this basic verification van der Ziel turns to measurements at the input with source impedance as a parameter. Measurements with Rs ‘v 0 are most sensitive to small errors in .Pand G; the effect of correlation between e and io can be seen by tuning the input for minimum noise (38) and by varying Rs. - It is convenient to define the equivalent saturated diode current of i o 2 . If Eqs. (39) and (41) are combined one can obtain the expression = IC

-

laI2(IE

- IR).

(43)

The equivalent noise resistance is calculated by referring all the noise sources of Fig. 9 to a single apparent noise emf in series with the input. The result is

R,

-

= ejn2/4kT df = R8 rb‘b

+ + + gsllzs + + + Tal

rb‘b

ze

ZSC~~,

(44)

where rsl is the noise resistance of the spectral density of r 2 ; gsl is the equivalent noise conductance of Q referred t o the input; and xsc is the correlation impedance. These parameters can be evaluated by the relations

-

+ 2ze*gea - z e * g ( l E + ~ I E E ) / ( ~ T ) I ~ 29s arei* - [-I + - Z&(IE + 2 I E E ) / ( k T ) ] . 71-1

1

2ec

=

22egeo

2951

z

(46)

(47)

Neglecting Ics(l - a ~ and ) IEE, which amounts to neglecting holeelectron pair generation in the base region, introducing the dc current amplification factor aB by the definition aB = I C / I E ,

and eliminating

I R

(48)

with the help of the expression

I R = 2(ao

- ~B)IE/~o,

(49)

N O I S E I N SEMICONDUCTOR DEVICES

323

one obtains instead of Eq. (43)

I,,

= IE[aB

-

Ja12(-a"

+ 2au)/aol.

(43a)

rhCO,rbl0, and reo of I,,, zSc, ral, arid ze The low-frequency values IeqO, thus become, respectively, IcaO =

IE[aB(l - a B >

+

(a0 - a B ) 2 1 ,

If trapping-recombination effects in the emitter-base transistor region can be neglected, one has I R = 0. It is then convenient to redefine CYB by requiring LYB = ac. If I E E and ICs(1 - aR) are not neglected, one then obt,ains

Since 9.0

= q(IE

+

I,,

IEE)/kT,

= IC -

one obtains

lCI!I2IE

=

((110

-

jaI2)IE

f

ICO,

(43c)

where Icois the collector-saturated current. The expression for I,, and the corresponding expressions for gsl, z,,, and rS1coincide with expressions given earlier by van der Ziel (3, 17). From the above it is clear that a complete calculation of the value of R, requires a knowledge of I,, a~~~ a, Tb'b, and of the source impedance R, seen by the transistor. Figures 10 and 11 show typical results of the measurements of ILq. Figure 12 shows the results of the measurement of R, as a function of operating point. There is good agreement between the theoretical and experimental curves. These results are typical of those obtained by Hanson (36), Chenette (19), and Brunclte (37) with van der Ziel for a wide range of operating conditions. Guggenbuehl and Strutt and Schneider and Strutt have also reported good agreement bet,ween theory and experiment. There is however a small discrepancy between Schneider and Strutt's noise equivalent circuit and van der Ziel's. The reason for it is that,, a t low frequencies, van der Ziel attributes full shot noise to the recombination current as outlined above, while Schneider arid Strutt assign only 4 shot noise to this current. Schncidcr and Strut t's exprcssion should hc valid for frequciwies high

324

EUGENE R. CHENETTE 1000

; i 100

a

5

-i OI

c1

10

t,

!

I

I

I

3

I

lo-‘

I

I

lo-I loo Frequency (Mc)

10’

lo2

FIG.10. Z,,O as a function of frequency.

loo 100

L /,,’,’;.’ /’

,”

,,’ ,/: /’

/‘ ,/

0 Experimental

values

,/

I 10

I

I(

30

FIG.11. Z,,O as a function of bias current. Theoretical curve I includes recombination current; curve I1 neglects it completely; curve I11 uses the high-frequency approximation of Schneider and Strutt.

NOISE IN SEMICONDUCTOR DEVICES

325

with respect to the lifetime of a trapped carrier while van der Ziel’s should hold a t lower frequencies. The effect of this discrepancy is shown on Fig. 11. The theoretical curve attributed to Schneider and Strutt was calculated by converting their expression for the noise factor to the output noise current with input ac open-circuited. T he most important point to be made here is not the small discrepancy between the two theories. The important point is that the experiment which reveals the discrepancy is the measurement of Ieq. Noise factor

I,

(

p amp. 1

FIG.12. Rnoas a function of emitter current in the white noise region for two values of source resistance. Tb‘b was determined from measurements of h i b O .

measurements, as presented by Schneider and Strutt, do not allow one to discriminate between errors in the noise model and in the small-signal equivalent circuit. At very high frequencies it becomes impossible to follow the procedure recommended above. A satisfactory “open-circuit” for the input does not exist, so that accurate measurements of I,, cannot be made. One must, therefore, abandon the above scheme and turn to measurements of noise resistance or noise factor for a much more limited range of source impedances. Fultui (39) and Policky and Cooke (40) have made extensive measurements of the noise performance of transistors in the microwave region. They find essential agreement between theory and experiment as

326

EUGENE R . CHENETTE

long as an adequate characterization of t8heeffect of the transistor header is included in the noise equivalent circuit. This is a tedious procedure! Fukui’s equivalent circuit is the “low-frequency” hybrid-pi circuit shown in Fig. 5. He modifies this circuit to include the effects of the transistor package and then finds excellent agreement between theory and experiment. The most important point of his work is that the simplest low-frequency model for the noise is useful a t microwave frequencies. This must mean that the fa of the transistors studied by Fukui lie well above 1.3 GHz. Schneider and Strutt (18) have studied the effect of high injection effects on noise in transistors. The main modification of their equivalent circuit was to modify the small-signal impedance of the junctions. Their results-again noise figure measurements at nominal source impedanceshow good agreement with the model. Johnson [Johnson et al. (SO)] has made measurements of I,, at high-current densities. Modern high-frequency transistors such as were studied by Fukui and by Policky and Cooke are operating at very high-current densities. Agouridis (66) has investigated the high frequency-high injection level problem and found good agreement between theory and experiment when small signal effects were properly interpreted.

C . Experiments on Field E$ect Transistors Brunclie and van der Ziel (23) have demonstrated that the noise performance of junction FET’s is in good agreement with the thermal noise theory. They find it necessary to take into account that the gate extends only over part of the conducting channel. The inactive part of the channel must be represented by resistances r. and rd in series with the source and drain respectively. The most straightforward measurement to check the validity of the theory is that of measuring the noise current between drain and source with the gate ac short-circuited to the source. This is a direct measurement of G2. Figure 13 shows a comparison of theoretical and experimental values of the spectral density of 2 as a function of frequency. The lowfrequency value of this curve agrees with Eq. (31) ; the increase at high frequencies is the result of the increase in 812 and hence of G. Figure 14 is another of Bruncke and van der Ziel’s results showing the effect of drain volt-drop on the spectral density of 2.The theoretical curve includes the effects of the bias dependent factor Q(Va8, V D S and ) of the extrinsic resistances rs and rd. Unfortunately, many FET’s show excess noise as is shown in the data 0 ~ 7 ~ excess ) noise is apparently associated with of Fig. 15. The 1/(1

+

327

NOISE IN SEMICONDUCTOR DEVICES

f (Mc/s)

FIG. 13. Spectral density if saturation with V C S= 0.

0.I

0.3

3 as a function of

frequency for a junction FET at

3

I

v,

10

(Volts)

in the “white noise” region as a function of VDSwith FIG.14. Spectral density of VGS= 0. The theoretical curve includes the effect of extrinsic source resistance.

trapping of carriers (26-28, 5 4 ) . Shoji (29) has measured the noise of FET’s as a function of temperature. At low temperatures, he has observed a white excess noise which is several times the theoretical thermal noise level. He identified experimentally that this was generation-recombination noise with relatively short time constants. Additional studies of the problem are under way a t the University of Minnesota.

328 I02

10

E U G E N E R. C H E N E T T E

a,

-En l 0

x

M

0.I

\

I,,,(theo) = 63 pamp I

I,,,(theo)

= 31 pomp

I

lo3

102

I

f (Kc/s)

FIG.15. Spectral density of 3 as a function of frequency for two FET’s showing excess noise with a 1/(1 0 2 ~ 2 ) characteristic. The curves approach the theoretical thermal noise level a t high frequencies.

+

D. Experiments on Flicker Noise At low frequencies the noise performance of all semiconductor devices seems to be dominated by noise with a spectral density which varies as I/f. This is commonly referred to as “flicker noise.” Most early experimental studies of noise in diodes and transistors were necessarily studies of the characteristics of the l/f noise sources (41). Modern technology has made possible the fabrication of devices with greatly reduced l/f noise. It will be seen later that l/f noise is relatively more troublesome in modern FET’s than in modern bipolar transistors. Schottky barrier diodes apparently have much less l/f noise than is found in point contact diodes. The theoretical models discussed above have nothing to say about this l/f or flicker noise. It is not possible at the present time t o predict the magnitude of the l/f noise sources in any semiconductor device. Fonger (42) found in studying the flicker noise of diodes and transistors that some of this noise was associated with leakage around a junction

N O I S E I N SEMICONDUCTOR DEVICES

329

and that some was apparently associated with modulation of the normal processes, possibly the result of fluctuation surface recombination velocities. A phenomenological equivalent circuit for l/f noise in a junction transistor is shown in Fig. 16. Experiments by Plumb and Chenette (43, 44) show these two l/f noise generators to be almost completely correlated for a transistor with negligible ohmic leakage a t the collector and that 2 >> 2 for most bias conditions. On the basis of these results

FIG.16. Noise equivalent circuit in the 1/f noise region. In many transistors if2are almost completely correlated and 2 >> 2.

if1

and

the l/f noise of a transistor can be described with a single l/f noise current generator as shown in Fig. 5 . This generator is connected in parallel with 2 and has the formal mean-squared value

if2

=

K I B T f - " df.

(51)

Here y and are often both about unity and K varies greatly from one transistor to another. The only practical way of determining the magnitude of this 1/J' noise generator for any transistor is by measuring the noise in the l/J region. (Y

IV. PRACTICAL LOW-NOISEAMPLIFIERS Since the theoretical niodels of noise in the devices discussed have been shown t o agree well with the results of basic noise studies it is worthwhile to use the noise equivalent circuits to predict the performance of several practical amplifiers. Both bipolar and F E T circuits will be included in the following paragraphs.

330

EUGENE R. CHENETTE

A . Bipolar Transistor Amplifiers An expression was derived above for the equivalent noise resistance of a bipolar transistor using the circuit of Fig. 9. This expression was recommended for comparison with experiment after verifying Icq. The expression can also be used, of course, to predict the noise performance of a practical amplifier. If one is interested in the noise factor, it is given by the relation F = RJR,. (52) Thus Eq. (44)yields (19)

F = l +

rb‘b

+ + rsl

R.

gsl

12s

+

rb’b

f ze

+

&el2*

(53)

This expression is valid in the shot-noise region for common-base, common-emitter, and common-collector connection. It is useful to frequencies near fa (with the sensible inclusion of package parameters). The various terms have been defined above. It was essentially this expression with zeC= 0 which Nielsen used in discussing the behavior of noise figure in transistors (45). The only objection to using this expression for calculating the noise performance of a trarisist,or amplifier is that it is relatively complicated. All of the terms, including ?“b’b, are more or less complicated functions of the operating point and transistor parameters. If one is willing to determine these parameters accurately this expression will yield accurate prediction of the noise performance for a wide range of transistors over a wide range of operating conditions. Such problems as high ICO and the transit time effects which become important when operating near fa are included in the model. It is obvious that the calculations required t o predict the noise performance also require accurate characterization of the circuit in which the transistor is imbedded. If package parasitics affect this circuit, as they do at high frequencies, they must be included. If one assumes in advance that the transistor to be used satisfies the requirements that operation is well below fa and that leakage currents are negligible, it is possible to use the somewhat simpler noise equivalent circuit of Fig. 5 with the mean-squared values of the noise generators (11, 46) i B 2 E2 q I ~ dfi (18) ic2 2qIc d f , (19) iA2 0, ==O, (20) and e b 2 = 4kTr,df. (12)

33 1

N O I S E I N SEMICONDUCTOR DEVICES

The noise factor of the circuit of Fig. 5 is given by the expression

is the mean-squared value of the flicker noise generator (Eq. 51). If this is negligible Eq. (54) can be rewritten as

=

Fo

+ F~(w*).

(54b)

R, is the real part of the source impedance 2,. The hybrid-pi parameters are given by the expressions grn = (q/kT)Ic,

rn =

Po/grn,

and

WT =

grn/(Cr

+ C,).

(55)

PO = hfeo= qlnrr is the low-frequency common-emitter current gain; r. is the base spreading resistance and must be determined by experiment. It is equivalent to the rb'b used earlier. F o is the low-frequency asymptotic value of the noise figure. 14

12

g

10

LL

=

2 0

10'

lo2

lo3

lo4 f -Frequency

lo5

- cps

FIG 17. Noise figure as a function of frequency for a typical silicon transistor (Type 2N3117). (Courtesy of Fairchild Semiconductor Products.)

Even Eq. (54a) is fairly complex. The noise factor is a complicated function of R, and the operating point. However, some insight into the shot-noise performance can be gained by examining the expression. For example, Eq. (54a) shows that the noise figure increases as u2 a t high frequencies. This behavior is seen in the curves of Fig. 17 which show the noise figure as a function of frequency for a typical transistor. The

332

E U G EN E R. CHENETTE

30 1-, Collector current

- pamp

(a)

1-, Collector current - pamp (b)

FIQ. 18. Contours of constant noise figure in the R, - I C plane for several

location of the break to the 0 2 dependence is a function of the operating point and source impedance as well as the WT of the transistor. It is clear that it makes good sense to choose a transistor with a highffr (or, by Eq. (53), a highf,) for good low-noise performance over a wide frequency band. It is a straightforward procedure to calculate the optimum source resistance for a given bias current on the optimum bias current for a

N O I S E I N SEMICONDUCTOH DEVICES

333

I,- Collector current -pamp (C)

Ic- Collector current - ma (d)

frequencies (Type 2N3117). (Courtesy of Fairchild Semiconductor Products.)

given source resistance if one assumes the transistor parameters to be constant. The results are, for the low-frequency shot-noise region (47)

334

EUGENE R . CHENETTE

and when

kT (59) qRs Here h F E = I c / I n is the dc current gain of the transistor. These expressions make clear that low-noise performance requires high h F E and low rz, as well as the large W T . The minimum possible noise factor is [l l / ( h ~ ~ ) % when ] r. is negligible. One of the most useful ways of displaying data about the noise figure is that of showing contours of constant noise figure (46) in the R.-Ic plane as shown in Fig. 18. The 2N3117 is a good example of the transistors currently available for use in low-noise amplifiers a t low frequencies. The typical minimum noise figure of 3.3 d B with R, = 500 Q and I , = 1 ma indicates a base resistance of about 600 a. The contours are in good agreement with this value. - I n the l/f noise region the noise equivalent circuit is dominated by i f 2 . Then the noise figure is given by the expression

Ic =

- (hFE)$‘.

+

F(l/f)

=

1

+ (?/Z?)(Rs + r d 2 + ( K I B ~ / RP)[R, , + r212

(60) unity. This expression has the minimum value for a fixed IC when R, = r.. This agrees with the results of measurements of noise figure contours reported by Gibbons (48). = 1

if y

N

B. Field E$ect Transistor Amplifiers Figure 19 shows a schematic diagram of a typical n-channel junction FET amplifier. The noise figure can be calculated in much the same way

lR9 lRc I

IRL

FIG.19. Schematic diagram of a typical junction FET amplifier.

as it was for the bipolar transistor. The noise equivalent circuitto be used is that given in Fig. 20. The values of the noise generators i,*, i d 2 , arid i,id* are given by Eqs. (31), (32), and (34). A slight modification of

335

NOISE I N SEMICONDUCTOR DEVICES

these values may be necessary to account for the extrinsic source and drain resistances. The noise figure can be calculated by evaluating the short-circuit output noise current of each of the noise sources. The result is

4-id(Ga

Gc F = l + - + G.

+-Yc 4-

yin>/Ytr12

(61)

i.2

where Gc is the real part of the input circuit admittance and G, is the signal source conductance. yi, = gin jbi, is the input admittance of the FET with its output short-circuited and Ytr = ~ Z L ~ 1 = 2 (gtr jbtr) is the net transfer admittance.

+

FIG. 20. Noise i,z = 4kTGcdf are

+

equivalent circuit for the FET amplifier. 3 = 4kTG,df and the result of thermal noise of the source and input circuit

conductances.

This expression may be simplified by dividing i, into two parts, i,' which is fully correlated with id, and i," which is completely uncorrelated with if.

i,

=

ig'

+ i,"

-

-

+

-

ig2= igt2 i:l2.

and

(62)

Equation (61) can then be written as

where gn is the noise conductance of the uncorrelated part of i,. It is defined by - - - 4kT dfgn = i g " 2 = i , 2 - i,'2 = i , 2 - lipid*/$. (63) rn is equivalent noise resistance of

4k!l'rndf

id

=

referred to the input, defined by

g/m.

(64)

Combining Eqs. (64) and (31) yields

r,

=

(gm(max)/lytr12)&(VCs,VDS).

(65)

336

EUGENE R. CHENETTE

yooris the correlation admittance relating the magnitude of i,’ to

id.

It is clear that the noise figure can be tuned to the value F’= 1

Gc gn TnlGs + Gc + gin + gcor12 + -Ge + - GB + Gs

when

(Bc

+ + her) bin

=

0.

(67) (68)

The source conductance can be chosen to yield minimum noise figure. The result from Eq. (67) is Fmin’

for

=

+ 2rn[Gc + gn + goor] + 2[rn(Gc + gn) + rn2*Gc+ gin + goor)z11/2 (69) Gs Gs (opt) [(Gc + gn)/rn + rn2(Go + gin + gcor)z]1’2- (70) 1

=

=

If, as is often the case, (Gc

+ gin + goor)’ << (Gc + gn)/rn,

(71)

the minimum noise figure occurs at GB = GB (Opt) = [(Gc

and has the value

F’ (min)

=

1

The correlation between i, and

id

gn/rnIl/’

+ 2[r,(Gc + gn)]l/z.

(72) (73)

is seen to be of no significance.

3

R,

=IK

2 n Q

I LL

z

I’

0 f - Frequency

FIG. 21. High-frequency noise figure of a 2N3823 junction FET. (Courtesy of Texas Instruments.)

Use of these expressions requires accurate characterization of the FET. An example of the performance which can be expected from FET’s readily available at the present time is shown on Fig. 21 which shows the noise figure of a 2N3823 silicon n-channel FET as a function of frequency

337

NOISE IN SEMICONDUCTOR DEVICES

with R, = 1000 D. Figure 22 shows the low-frequency noise resistance of the same transistor. The high-frequency curve is in good agreement with theory while the curve of r, a t low frequencies lies well above the value expected from Eq. (65).

f -Frequency

FIG. 22. Low-frequency noise resistance for a 2N3823. (Courtesy of Texas Instruments.)

C . Comparison of Bipolar and FET Amplifiers

It is interesting to compare the limiting low-frequency noise performance of bipolar and FET amplifiers. For low source impedance applications it is convenient to compare the short-circuit noise resistances. For the bipolar transistor. lim R,

=

RrtO

+

Rno= rz -I- ( q / 2 k T ) [ l ~ ( r , ) ~Ic(rs

This has the value

+ l/(h~d'/~I

R,o (min)

=

r,[l

Ic

=

(kT/qYz)(hFE)1/2.

+ r,)'].

(74)

(75)

when (76)

Thus the 2N3117 will yield a minimum noise resistance of about 600 D when Ic 0.63 ma. A transistor with rz = 50 52 will yield R,o 'V 50 52 when IC S 7.5 ma.

338

EUGENE R. CHENETTE

The minimum noise resistance for an FET is

Rno = lim FIG, = rn Grt-

=

g,(max>&(Vo, V ~ s ) / l Y t r l ~ .

(77)

I n the frequency range where ytr ‘V g,(max) this becomes

r,

N

&(VGS,VDs>/g,(max>.

(7 7 4

Since & ( V G V ~ ,D Sis ) about 0.7 for an F E T in saturation, a noise resistance of 600 Q requires gm(max) ’v 1400 mhos; a noise resistance of 50 Q requires gm(max) = 14,000 mhos, assuming the F E T fits the thermalnoise model. If there is excess noise in the channel rn often is several times higher than shown by Eq. (77). This and the g,(max) required to obtain a desired r,, is higher than indicated by Eq. (77a). For high source impedance applications it is convenient t o compare equivalent noise conductance of the F E T and the bipolar transistor. For the bipolar transistor

and for the FET

gn is the noise conductance of the uncorrelated part of the input noise current. By the earlier discussion gn N ges 4-gad (q/2kT)I~ where IC is the current of the reverse biased gate junction for a junction FET. The second term of Eq. (79) is the noise of the channel referred to a n input noise conductance. I n most cases this term is negligible with respect to g,. T o illustrate this result, we note that even a transistor such as the 2N3117 with its minimum h p E of 100 at IC = lee a. has a n equivalent input noise conductance by Eq. (78) of G,, = 2 X lo-’ mhos. This is considerably larger than would be expected from the full thermal noise of the low input conductance and shot noise of the gate for a good FET. Thus FET’s are probably superior for most high source impedance applications, while bipolar transistors are often superior for low source impedance applications.

+

D. Tuned AmpliJiers The above discussions are directIy applicable to tuned amplifiers. However, some comments are in order. Equation (53) shows that the minimum noise figure of a bipolar transistor is obtained when the source is tuned so that

xa + ZB + 2 b +

Zsc

= 0.

NOISE IN SEMICONDUCTOR DEVICES

339

The corresponding expression for minimum noise of an FET is Eq. (68). Neither of these requirements yields maximum gain, and tuning for maximum gain never yields minimum noise. This is illustrated by Fig. 23 which shows both gain and noise figure as a function of source reactance for a common-emitter amplifier, The noise figure first drops slightly from its resistive input value but then increases as the amplifier gain increases. Collins (49) found that the use of feedback to make tuning for maximum gain and minimum noise coincide seriously degrades amplifier stability. It seems the most useful approach to the design of low-noise tuned amplifiers is to adjust the tuned input to provide a resistive source

FIQ.23. Noise figure and gain of a tuned input common-emitter amplifier as a function of input tuning inductance.

impedance. The cascade connection is useful because it reduces the stability problems resulting from internal feedback and allows more freedom in adjustment of the input circuit. Near the alpha cutoff frequency the correlation of the noise sources should allow a slight additional improvement in the noise performance of a tuned amplifier. Both FET’s and bipolar transistors can provide very effective lownoise amplifiers at high frequencies. Germanium planar transistors such as those measured by Fukui and by Policky and Cooke have yielded 5 0 4 source noise figures of less than 3 db at 1 gigacycle. The 2N3823 cited earlier is a good example of an n-channel FET capable of excellent lownoise performance. It is to be expected that modern device fabrication

340

EUGENE R. CHENETTE

will permit some continuing improvement in the noise performance of these devices. E. Noise in Diodes The basic shot noise model for noise in a junction diode is directly applicable to the Schottky barrier diode. Advanced device fabrication techniques have made this diode a useful addition to the devices available. The noise performance of these diodes, in practice, is far superior to that I

I

I

I

I

I

3000

10,000

I

I

10‘

g, c

g lo2 E

c

,&Schottky ‘barrier diofe

I,

%

300 pamp

, -*

30 pamp

30

100

300 1000 Frequency (Hz)

FIG. 24. Normalized noise temperature as a function of frequency comparing Schottky barrier diodes and “low-noise” point contact diodes. (Courtesy of HewlettPackard.)

of the point contact diodes so widely used as high-frequency mixers. The reason is apparently that the use of planar fabricat,ion techniques has effectively removed the surface as a serious problem in the limitation of the performance of the device. Figure 24 shows a comparison of the normalized noise temperature of a Schottky barrier diode and a “low-noise” point contact diode. There is little doubt which is the superior device.

V. SUMMARY I n this review the attempt has been made to show the three different aspects of noise in semiconductor devices; theory, experimental investi-

NOISE I N SEMICONDUCTOR DEVICES

341

gation of the theory, and practical application. The basic theory is well enough understood to permit predicting the performance of devices in circuits with precision for a wide range of operating conditions. There are many topics which have not been touched in the course of the review. Space does not permit the inclusion of all the topics which are worthy of review.

ACKNOWLEDGMENT The author wishes to acknowledge the encouragement of A. van der Ziel and K. M. van Vliet in preparing this review and to thank them for many helpful discussions.

APPENDIX.NOISEREPRESENTATION The purpose of this appendix is to present a brief statement of noise theorems and the methods of representing noise which are used in this paper. More detailed discussions may be found in any text on noise and in the references cited.

A . Nyquist’s Theorem (50) Nyquist’s theorem says that the thermal noise of a resistive circuit in equilibrium at temperature T can be represented by noise generators capable of delivering a maximum power in a frequency interval Af of p

=

kT A!,

(All

where k is the Boltzmann’s constant and T is the temperature in degrees Kelvin. The most common circuit representations of this theorem are shown in Fig. A - 1 . This theorem has strong experimental backing and has been used to determine Boltzmann’s constant with better than 1 % accuracy. The theorem holds for any dissipative system in thermal equilibrium and can be applied to the resistive regions of semiconductor devices.

B. Schottky’s Theorem (51) Schottky’s theorem is concerned with the shot noise of temperaturelimited thermionic diodes. It states that the noise of a diode carrying a bias current ID can be represented by a noise current generator i~ in parallel with the conductance of the diode. The mean-squared value of this generator in a frequency interval Af is

-

i2 = 2 q I D Af,

(-42)

where q is the electron charge. This theorem holds for more general situations. Current resulting from the emission of carrier across the potential

342

EUGENE R. CHENETTE

barrier of a p-n junction shows “full shot noise.” The important requirement is that the passage of individual current carrier constitute independent random events and that each carrier contribute a pulse of current of area q to the external circuit.

.

FIG. A-1. Circuit representations of Nyquist’s theorem. The magnitudes of the generators are given by e* = 4kTR df and i a = 4 k T ( l / R ) df.

This theorem also has strong experimental backing and has been used to determine the magnitude of q with less than 1% error.

C. Noise of Two-Terminul Networks The noise of a two-terminal linear network or device in a narrow frequency interval can be represented either by a noise emf in series with the impedance of the network or by a noise current generator in parallel with the admittance of the device. 1. Noise Resistance. The mean-squared value of any noise emf in a narrow frequency interval can be represented by the thermal noise of an imaginary resistor by using Nyquist’s theorem formally. The result is

Re, = 2 / 4 l c T Af,

(A3)

where Reqis called the equivalent noise resistance. 2. Noise Conductance. Nyquist’s theorem is also used to define the noise conductance. The mean-squared value of any noise current genera-

343

NOISE I N SEMICONDUCTOR DEVICES

tor in a narrow frequency interval can be represented by the thermal noise of an imaginary conductance;

G,,

=

2 / 4 k T Af,

(A4)

where G,, is called the equivalent noise conductance. 3. Saturated Diode Current. Another way of representing the spectral intensity of a noise current generator is to apply Schottky's theorem and equate it to the shot noise of ii saturated (or temperature-limited) diode. The result is I,, = in2/2qAf. (A5)

D. Noise in Linear Two-Ports Two partially correlated noise sources are required to represent the noise of a four-terminal network. I n the case of a T-equivalent circuit

I

(b)

FIG.A-2. The general noise equivalent of (a) can be simply reduced to (b).

the most general possible noise representation consists of three noise emf, el, ez, and e o ; one in series with each of the arms of the T (see Fig. A-2). This can be transformed into inhecircuit of Fig. A-2b with the generators el' = el e z and e2/ = e3 ey.

+

+

344

EUGENE R. CHENETTE

1 . Noise Factor (Noise Figure). The noise factor, at a specified input frequency, is defined as “the ratio of ( 1 ) the total noise power per unit bandwidth at a corresponding output port when the noise temperature of the input termination is standard (290’K) to ( 2 ) that portion of (1) caused by the thermal noise of the input termination.” 2. Noise Temperature. The effective input noise temperature of a twoport transducer is defined as the temperature which, when the input was connected to a noise-free equivalent of the transducer, would result in the same output noise power as that of the actual transducer connected to a noise-free input termination. The effective noise temperature in degrees Kelvin is related to the noise factor by the expression

T , = 290(F - 1).

(A61

3. Noise Resistance. There are times when it is convenient to refer all of the noise of a transducer to a single noise emf in series with the input termination. This emf will, in general, depend on the impedance of the input termination. The mean-square value of this single emf can be expressed as an equivalent noise resistance of the two-port.

R , = 2 / 4 k T Af.

(A71

4. Noise Conductance. There are also occasions when the most expedient way of representing the noise of a two-port is with a single noise current generator in parallel with the admittance of the source termination seen by the transducer. The mean-squared value of this noise current generator can be expressed in terms of equivalent noise conductance.

-

GN = ieq2/4kT Af. (A@ It is also possible to express the spectral density of this generator in terms of equivalent saturated diode current,

I,,

=

2 / 2 p Af.

(A91

5. Noise Measure. The noise measure of a transducer has been defined by Haus and Adler as (52)

M =

F-1 1 - l/Ga

(A101

where F is the noise factor as defined above and G, is the available power gain. The noise figure of several transducers connected in cascade is given by the Friis formula (53)

NOISE IN SEMICONDUCTOR DEVICES

345

If we substitute the expression noise measure into this formula, we find F,=l

+ M.

(A121

Haus and Adler have shown that the optimum noise measure of a device is invariant under reciprocal lossless circuit transformations. Thus, for example, common-base, common-emitter and common-collector transistor stages can all be optimized to have the same cascaded noise figure. Also, the parasitics, which seem to effect the optimum noise figure obtainable at high frequencies have no effect on the optimum noise measure.

REFEEENCES 1 . H. A. Haus et al., I R E subcommittee 7.9 on noise, Proc. IRE 48, 69 (1960). 2. K. M. van Vliet, Proc. IRE 46, 1004 (1958). 3. A. van der Ziel, “Fluctuation Phenomena in Semiconductors,” Butterworths, London, 1959.

4. K. M. van Vliet and J. R. Fassett, in “Fluctuation

Phenomena in Solids,” (R. E. Burgess, ed.), Chapter 7. Academic Press, New York, 1965. 6 . A. van der Ziel, I R E Trans. Electron Devices ED-9, 525 (1961). 6 . C. T. Sah, R. N. Noyce, and W. Shockley, Proc. IRE 46, 1228 (1957). 7. B. Schneider and M. J. 0. Strutt, Proc. IRE 47, 546 (1959). 8. V. F. Weisskopf, lion the Theory of Noise in Conductors, Semiconductors, and Crystal Rectifiers,” Natl. Defense Res. Council, N o . 14-133 (1953). 9. A. Uhlir, Proc. IRE 44, 557 (1956). 10. A. van der Ziel and A. G. T. Becking, Proc. IRE 46, 589 (1958). 11. L. J. Giacoletto, in “Transistors I,” p. 296. RCA Labs., Princeton, New Jersey, 1956. 18. R. L. Petritz, Proc. IRE 40, 1440 (1952). IS. A. van der Ziel, Proc. IRE 43, 1639 (1955). 14. M. Solow, Ph.D. Thesis, Catholic Univ. of America, Washington, D.C., 1957. 16. D. Polder and A. Baelde, Solid-State Electron. 6, 103 (1963). 16. K. M. van Vliet, To be published. 17. A. van der Ziel, Proc. IRE 46, 1019 (1958). 18. B. Schneider and M. J. 0. Strutt, Proc. IRE 48, 1731 (1960). 19. E. R. Chenette and A. van der Ziel, IRE Trans. Electron Devices ED-9, 123 (1962). 20. K. H. Johnson, E. R. Chenette, and A. van der Ziel, IEEE Trans. Electron Devices ED-12, 387 (1965). 81. J. J. Ebers and J. L. Moll, Proc. IRE 42, 1761 (1954). 28. R. L. Pritchard, J. B. Angell, It. B. Adler, J. M. Early, and W. M. Webster, Proc. IRE 49, 725 (1961). 83. W. C. Bruncke and A. van der Ziel, IEEE Trans. Electron Devices ED-13, 323 (1966). 84. A. van der Ziel, Proc. IRE 60, 1808 (1962). 86. A. van der Ziel, Proe. IEEE 61, 461 (1963). 26. C. T. Sah, Proc. IEEE 62, 795 (1964). 27. H. E. Halladay and W. C. Bruncke, Proc. IEEE 61, (1963). 88. A. van der Ziel, Proc. IEEE 61, 1670 (1963).

346

E U G E N E R. CHENETTE

89. M. Shoji, IEEE Trans. Electron Devices ED-B, 520 (1966). 30. A. G. Jordan and N. A. Jordan, IEEE Trans. Electron Devices ED-12, 148 (1965). 31. H. Johnson, in “Field Effect Transistors” (J. T. Wallmark and H. Johnson, eds.), p. 160. Prentice-Hall, Englewood Cliffs, New Jersey, 1966. 38. W. Shockley, Proc. IRE 40, 1365 (1952). 33. A. van der Ziel, “Noise.” Prentice-Hall, Englewood Cliffs, New Jersey, 1954. 34. K. S. Champlin, Proc. IRE 46, 779 (1958). 36. W. Guggenbuehl and M. J. 0. Strutt, Proc. IRE 46, 839 (1957). 36. G. H. Hanson and A. van der Ziel, Proc. IRE 46, 1538 (1957). 37. W. C. Bruncke, E. R. Chenette, and A. van der Ziel, IEEE Trans. Electron Devices ED-11, 50 (1964). 38. E. R. Chenette, Proc. IRE 47, 448 (1959). 39. H. Fukui, IEEE Trans. Electron Devices ED-13, 329 (1966). 40. G . J. Policky and H. F. Cooke, NEREM Record 7, 254, (1965). 41. E. Keonjian and J. S. Schaffner, Proc. IRE 40, 1456 (1952). 48. W. H. Fonger, in “Transistors I,” p. 239. RCA Labs. Princeton, New Jersey, 1956. 43. E. R. Chenette, Proc. IRE 46, 1304 (1958). 44. J. L. Plumb and E. R. Chenette, IRE Trans. Electron Devices ED-10,304 (1963). 46. E. C. Nielsen, Proc. IRE 46, 957 (1957). 46. R. D. Thornton, D. DeWitt, E. R. Chenette, and P. E. Gray, “Circuit Limitations of Transistors,” Wiley, New York, 1966. 47. E. R. Chenette, Solid State Design 6, 27 (1964). 48. J. F. Gibbons, IRE Trans. Electron Devices ED-0,308 (1962). 49. L. F. Collins, M. S. Thesis, Univer. of Minnesota, Minneapolis, Minnesota, 1963. 60. H. Nyquist, Phys. Rev. 32, 110 (1928). 61. W. Schottky, Ann. Phys. (Leipzig) 69, 541 (1918). 68. H. A. Haus and R. B. Adler, “Circuit Theory of Linear Noisy Networks.” M.I.T. Press, Cambridge, Massachusetts and Wiley, New York, 1959. 65. H. T. F&, Proc. IRE 32,419 (1944). 64. C. T. Sah, S. Y. Wu, and F. H. Hielscher, IEEE Trans. Electron Devices ED-13, 410 (1966). 66. D. Agouridis, Ph.D. Thesis, Univ. of Minnesota, Minneapolis, Minnesota, 1966.

Properties end Limitations of Image Intensifiers Used in Astronomy W. C . LIVINGSTON Kilt Peak National Observatory* Tucson, Arizona

I. Introduction. . . . ................................................ 11. Qualitative Comparison between the Photographic Plate and the Image Tube 111. Quantitative Evaluation of Image Tubes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Information .........................................

347 349 352 352 354

IV. Description of Tubes and R.esults . . . . . . . . . . . 354 A. Lallemand Tube-Electr B. Cascade Tube., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Transmission Secondary Emission (TSE) Tube. . . . . . . . . . . . . . . . . . . . . . 363 D. Image Orthicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 E. Tubes with Infrared Photocathodes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 V. Prospects for Futu ... 372 A. The Reflective ... 372 B. Microstructure ......................................... 376 C. Fiber Optic Faceplates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 D. The Preformed Photocathode.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 E. Conceptual Tubes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 VI. Summary and Conclusions.. . . . . . . . ... . . . . . . . . . 380 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

I. INTRODUCTION The basis for an interest in image tubes lies in the properties of the photoemissive surface. These properties have become familiar to us through the routine use of the photomultiplier tube. Astronomers have long looked forward to the availability of an image tube which would combine the high quantum efficiency and linear response to light of the photomultiplier with the panoramic image properties of the photographic emulsion. I n 1936 Lallemand (1) suggested the advantages that an image tube would have over conventional photography and point-by-point photometry with a photocell. The property of high quantum efficiency should * Operated by AURA, Inc., under contract with the National Science Foundation. 347

348

W . C. LIVINGSTON

permit greatly reduced exposure times or, of even greater consequence, the observation of fainter objects in the same length of time. Linear response and freedom from saturation are essential because astronomical images contain a wide range of light intensity and quantitative measurement is required. Others (2-6) have since amplified on the advantages to be expected from an image tube. There seems no doubt that the perfection of such a device would be most important for both land-based and space observational astronomy. Since 1936 there have been great advances in electron-optics, particularly for the needs of broadcast television, but also for military applications at low light levels. However, an image tube which fully preserves the excellent properties of the photoemissive surface is yet to be devised. Tubes have been built that are proving to be useful because they reduce exposure times. But the photometric qualities of the original light image are more or less altered. In a g'eneral way, this alteration takes the form of a dependence in response between picture elements. A simple example is scattered light which is permitted by a semitransparent photocathode. A more complicated case is that of coplanar biasing in tubes where the image is integrated by a storage of an electrical charge. This article is intended primarily to aid the investigator who believes that an image tube might be of use to him in a certain physical experiment or observation. Some general characteristics of tubes that are currently being used on telescopes are given. Unfortunately, review of the current literature and of the technical data sheets supplied by tube manufacturers is an inadequate procedure for determining the best tube for the job or even whether such a tube exists. Reports on electronic imaging devices tend to stress the merits and neglect the faults. This is a natural consequence of the preoccupation of tube developers with specific design and processing problems such as resolution or dark background. A single advance in the technology will not result in a universally useful tube. Every practical tube built is a compromise device. For a given application, the compromises must be known and considered before one can judge the usefulness of an image tube. The material that follows is based on several years of experience with a variety of image tubes at Kitt Peak National Observatory. In connection with the design of new telescopes, information was needed on image tube performance. We have explored the use of the image orthicon and the transmission secondary emission (TSE) tube, and we are more familiar with these two devices than with others. Recently, some work at Kitt Peak has been done with a cascade tube made by RCA. However, most of our inforniation on the RCA tube has been derived from discussions with Dr. W. I<. Ford, who works with the Carnegie Image Tube Com-

PROPERTIES AND LIMITATIONS O F IMAGE INTENSIFIERS

349

mittee, and who has been largely responsible for making astronomical tests of the tube. We have no first-hand experience with electronographic devices and will depend entirely on the reports of others. Besides passing along some of our experiences, this article proposes to make an additional point: what is needed is not just a further perfection of existing tubes. To bring about progress in realizing the benefits of the photoemissive surface, new concepts are very much in order. 11. QUALITATIVE COMPARISON BETWEEN

THE

PHOTOGRAPHIC PLATE

AND THE ~ M A G ETUBE

The photographic plate is a thoroughly proven and relatively well understood research tool. It is interesting to note that certain familiar characteristics of the photographic process are not readily carried over into image tubes. I n Fig. 1 the relative advantages of the photographic plate and image tube are diagramatically listed. For the convenience of the reader, the accompanying diagrams of Fig. 1 are numbered as well as labeled. These will be referred to frequently in the following text. The first three advantages listed relate to the image tube; those numbered 4 to 6 refer to the photographic plate. As mentioned, all the advantages of the image tube stem directly from three properties of the photoemissive surface. (1) For this surface the response to light in terms of photocurrent is strictly linear. The dark current from most types of photocathodes can be made negligible ( < 2 * 10-19 amp . cm-2) by cooling. At high light levels overload is set by the resistivity of the photocathode substrate and space charge effects adjacent to the surface. If a 1P21, for example, is operated as a photodiode, linear response is found up to photocurrents of amp. Hence, the latitude of the (5-4)photocathode is about 5 1014. On the other hand, the photographic emulsion has a transfer characteristic of varying slope (gamma-depending on development procedure) and a toe and shoulder separated (the latitude) by one or two decades. The toe is caused by inertia and plate fog; the shoulder, by overlapping grains. (2) The responsive quantum efficiency, or RQE, of a photoelectric surface is simply the ratio of the number of photoelectrons out to the number of incident photons (see Section 111). For a multialkali S-20 surface operating in a reflective mode (see Section V) at a wavelength of 4000 R, we estimate the efficiency to be about 0.45. The detective quantum efficiency, or DQE,' for the Kodak 103a-0 emulsion is reported to be 0.005 [Marchant and Millikan (7)]. These values apply to exposure times the order of 30 min and a density of 0.6. Photographic quantum efficiency depends

-

1 In this case it is appropriate to compare the RQE of a photocathode to the DQE of an emulsion.

350

W. C. LIVINGSTON

Image Tube

Photographic Plate

I. Characteristic curve has toe and shoulder; reciprocity failure.

1. Linear response; reciproci t y obeyed.

LOGII.1,

a

2. Peak detective quantum efficiency = .005 (1030-0).

3.

Cannot be calibrated where exposed.

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

efficient 4. Adaptable optical to systems; e.g. the Schmidt Camero.

,.L

2. Peak responsive quantum efficiency = .45

(5-20 photosurface w i t h oblique reflective enhancement).

3. Can be colibrated.

4.

May be d i f ficult t o match ta high speed optics without vignetting.

FOCUS UA6NET

5. Low back scattering;

5. Photocathode increas-

turbidity and adjacency effects are strictly local.

ing transparent t o red (6G% at 6000A); the returned light i s widespread nnd non-uniform. Ghost images may occur due t o reflected photoelectrons.

6. High spotial stability; spectrum line position may be determined t o 0.7~ (even though the resolution much less).

:I L*07r

6. Spatial position subject to: a) Position i n earth's magnetic field, b) Charging of'.-. surfaces,

c) Coplanar biasing.

IMAGE ORTHICON

, ,:;.-:::'::--~.7.-.+ BEAM BENDING ,

.......__._.--. ---

'~:

,'

FIQ.1. Comparative properties of the photographic plate and image tube.

on exposure time and density [Marchant (S)]. (3) Owing to the permanent and stable nature of a photosurface it can be accurately calibrated for response or sensitivity [Aller and Walker (@I. A similar calibration of a photographic emulsion is impossible because it can be exposed only once. These first three advantages, which are inherent in the image tube by virtue of its photoemissive surface, are fundamental. Less fundamental, perhaps, but so far of great practical significance, are the following assets

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

351

of the photographic process. (4) The photographic plate is readily adaptable to efficient and high-speed optical systems such as the Schmidt camera. The image tube, encumbered by focus magnets or a photocathode actually some distance inside the envelope (as in the Lallemand tube), presents a difficult problem in optical design. ( 5 ) The photographic emulsion rather effectively stops the light. At least scattering and adjacency effect,s are strictly local. I n the case of an image tube with a semitransparent photocathode, multiple reflections take place in the faceplate. Even more serious is the large fraction of light transmitted through the photocathode (see Fig. 2). This transmitted light is largely returned to

I 4000

I 5000

I 6000

I

7000

I

aooo~

FIG.2. Transmission of multi-alkali S-20 photocathodes [after Seachman (56)].

the photocathode with a nonuniform distribution. Returned light creates a nonpredictable dependence in response between image elements. Of completely different origin, but very troublesome, are the interference fringes that develop in the faceplate of an image tube at large F-numbers. The solution for spectroscopic a.pplications can be a contacted 3' wedge prism. (6) The photographic emulsion allows high positional accuracy and long term geometric stability. St,ar and spectrum line positions can be determined to a small fraction of the resolution. For example, on a photographic spectrum which has a limiting resolution of perhaps 10 p, the center of gravity of a sharp line may be found to less than 1 p . I n the case of television-type tubes, the quantization introduced by the raster scan lines influences position measurement. To achieve the same degree of positional precision as is possible in photography, all else being equal,

352

W. C. LIVINGSTON

the raster line separation would have to be 10 times finer than the tube resolution. Geometric stability, to say nothing about image geometry per se, is a special problem in image tubes. Movement of the electronic image during exposure may be caused by magnetic and electrostatic fields both internal and external. The displacement of the image, per stage of length L, voltage V , caused by a transverse magnetic field, H , is proportional to HL2V-”. A long tube is therefore very susceptible to magnetic disturbances. For the same overall length and total accelerating potential, a multistage tube is seen to have an advantage. As an example, if the 5-stage English Electric Valve (EEV) P829 tube, which is 8 in. long and operates at 40 kV overall, is turned completely in the earth’s field (-0.5 G), the image moves 0.25 mm. I n the case of signal generating television-type tubes, the scanning beam alignment is noticeably altered by motion in the earth’s field. A tube mounted on a moving telescope must be magnetically shielded. Magnetic shielding is complicated by the requirements of the tube’s own magnetic focus and also by the need for rather wide openings for light entry. Even where the tube is fixed, as at the coudh focus, the movement of nearby steel structures, such as the telescope dome, has been found by Baum [see Baum et al. (6, 1962)] to produce disturbing effects. Electrostatic forces also contribute to image movement. Any exposed insulating surfaces (e.g., glass or ceramic) are subject to charging and thus may cause a time varying change in the internal potential fields. Presumably this is a problem of proper design. Most formidable, however, are the aberrations produced in signal generating tubes where the multiplied photocharges are stored on some insulating plate or target. The presence of these electrical charges which can “act at a distance” causes an interdependence between every picture element both in regard to their coordinates and also to their individual response function. Collectively, these aberrations are called “coplanar biasing” because the effect is one of a varying potential field across the storage plane. When the reading beam approaches this plane it is deflected locally to produce a light dependent geometric distortion. It is evident that, while the photographic plate may be wasteful of light and otherwise imperfect for photometric purposes, it does have compensating characteristics that are not readily found in an image tube.

111. QUANTITATIVE EVALUATION OF IMAGE TUBES A . The Detective Quantunz Ejiciency ,4 figure of merit by which image receivers can be compared is the detective quantum efficiency (DQE). The properties of this function have

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

353

been discussed by Fellgett ( 3 ) and Jones (10). The DQE of a receiver can be defined by DQE = (s/W20ut/(S/N)2in, where (S/N)i, is the signal-to-noise ratio of the incident image and (S/N),,t is the corresponding signal-to-noise ratio of the transduced image. The great value of the DQE concept lies in its universal applicability to all image receivers. A difficulty in its use lies in being able to specify the signal-to-noise output, considering all the aberrations to which the image tube is susceptible. In general the DQE depends on intensity, exposure time, and the detailed description of the spatial content of the image. In comparing two receivers, it often happens that the first will have the high DQE for one type of image while the second has the higher DQE for a different image. It should be noted that, in the survey type listings of the DQE for various receivers [Jones ( l o ) ] geometric , distortion and resolution have been ignored.

B. Information Rate A practical, working formula for comparing similar tubes has been devised by Baum [see Baum et al. (5, 1962)l. Assuming that the image finally appears in photographic form which can then he microphotometered, we say that the information rate

=

R2/t(AD)2.

R is the resolution in line-pairs per millimeter, t is the exposure time to reach a prescribed density above fog, and AD is the resulting density fluctuation. Normally, the concern is with relative rates; that is, the ratio of the information rates of two devices. The R2factor has particular significance for astronomical applications. Suppose two receivers are identical in regard to exposuredensity gradient and the resulting fluctuation AD, but let the resolution be different in the ratio RI/R2.The receivers are to be fed by a camera of fixed aperture but varying focal length F. This is the usual arrangement for an astronomical spectrograph, the short focal length cameras being used for fainter objects. The resolution on the transduced images will be identical if we make Fz/Fi = But the light flux in the image plane is proportional to 1/F2 for constant aperture. So the relative exposure time to reach the same density will be

h/ti

=

(Ri/Rz)2.

354

W. C. LMNGSTON

In the case of stellar spectra as opposed to nebular spectra, there is no image detail perpendicular to dispersion. Here the information rate is linearly proportioiial to the resolution and riot R2. Suppose for a giveii receiver and application, photon (or shot) noise dominates and dark current, residual plate fog, and other background noise are negligible. Resolution is then no longer a factor because the transduced shot noise per angstrom interval remains the same, independent of the image scale. I n this case, the information rate formula does not apply-

C . Practical Methods As a consequence of the many complications and qualifications connected with the evaluation of image tubes, the research worker may ask : “Will a certain image tube allow me to obtain astronomical data that could not be acquired by the conventional means of photography and/or point-by-point photometry?’’ The answer to this question generally comes only through actual trials and tests on the telescope. Such tests will be described in the next section. OF TUBES AND RESULTS IV. DESCRIPTION

In this section we shall discuss tubes of current interest which have actually been used on telescopes. Each tube will, of necessity, be described only briefly, and the reader will be referred to the literature for more complete information. Advantages and disadvantages of each device will be listed. An attempt is made to confine ourselves to the fundamental aspects and not to developmental problems. Finally, we shall list the astronomical results that have been obtained. To qualify for this list the tube must have been used to gather new astronomical data that, in turn, was published as such.

A . Lallemand Tube-Electronography The Lallemand tube of today is the perfected product of 30 years of work on the part of Lallemand and his associates at the Paris Observatory. The only Lallemand system outside of France is operated by Walker (11) at Lick Observatory. An observational run begins by loading a cassette of Ilford G-5 plates into the tube which is then evacuated and the plates cooled to the temperature of liquid nitrogen. A photocathode, previously prepared and possibly studied for uniformity, is removed from its container and brought into position in the electron optics. Photoelectrons released from the S-11 surface are accelerated to about 40 kV and focused by an electrostatic lens onto the G-5 plate. At this high energy each electron leaves a minute track in the emulsion. After a given exposure a

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

355

manipulation of the cassette brings a new plate into focus. Observations continue until 25 plates are taken. Then the tube is opened for their removal, destroying the photocathode. 1. ADVANTAGES

1. Simplicity of the physical process and inherent high efficiency. This point is questioned by Kron [see Breckinridge et al. ( l a ) ]who indicates that, even for 30 kV electrons, blackening is a statistical process with an efficiency of 50%. See also the micrograph picture of 25 keV electron tracks in Ilford G-5 given in Fig. 3 of Lallemand et al. (IS). 2. Very high resolution. 3. Large storage capacity. The G-5 emulsion shows optical density proportional to the electronic intensity, according to Duchesne and Meallet (14). Both Kron and Wl6rick have stated informally that the density-exposure curve for the G-5 emulsion appears linear to around density = 5.

2. DISADVANTAGES 1. Short photocathode life. Evolution of gas from the emulsion, even at - 196"C, limits the time of maximum photocathode sensitivity to about 2 days. The photocathode is completely destroyed on removal of the exposed plates. 2. Observational inflexibility. The plates cannot be examined immediately after their individual exposure, a procedure which often results in a change of program when an interesting item is found. 3. Difficult optical arrangement. The photocathode lies about 2 in. inside the tube window, and so the tube is not readily adapted to highspeed optics. This is why the Lallemand tube has not been used so far to obtain low dispersion spectra of faint objects. An F1 camera, designed by Bowen for Walker (Ib),may soon remedy this situation.

Effective toe in transfer characteristic. Threshold performance may be limited by fog of the G-5 emulsion. Lallemand states that 250 tracks/ mm2/day are produced by cosmic rays and natural radioactivity.

4.

5. Scattered light effects. Loss of contrast by transmitted light: Grosse and Wlerick (16)have investigated means to reduce this effect. 3. RESEARCH 1. Rosch et al. (17) have taken advantage of reduced exposure time and freedom from the Eberhard effect to obtain new measures of the separation of close double stars.

356 W. C. LIVINGSTON

FIQ.3. Comparative records of star fields taken with the photographic plate, Kron’s electronographic tube, and the image orthicon. A 1 mm thick BG-12 Schott filter waa used throughout. This filter effectively isolated the blue for Kron’s S-9 photocathode, but may have admitted some red 1 1 photocathode of the image orthicon is sensitive. Records A-C were made by light to which the S Kron of M39. Records D-F were made at Kitt Peak of M67 with a type 25294 image orthicon.

PROPERTIES AND LIMITATIONS O F IMAGE INTENSIFIERS

357

2. Wldrick [Bellier et al. (18)] has made a photometric study of the surface of Jupiter and Saturn. Because of atmospheric seeing, even the short half second exposures do not show the planetary details reported by visual observers. However, the high signal-to-noise on their plates has permitted some aperture correction, or spatial sharpening, along microphotometer traces of the planetary images. These corrected traces indicate many of the weak, previously unphotographed divisions and subtleties of brightness. 3. Lallemand et al. (19), and Walker (20)have measured the rotation of the nuclei of M31 and R132. An observation of this kind required the moderately high dispersion of 48 A/mm and large scale along the slit of 1.”9/mni. This was achieved at the coudd focus of the Lick 120-in. reflector. 4.

Aller and Walker (9) made full use of the photometric capability of the Lallemand tube in a study of line intensities in planetary nebulae. I n the case of NGC 6720, the Ring Nebula, the range of line intensity recorded on a single plate was about 500:l. For the first time, reliable intensities were obtained for a network of points across a nebula. Aller derived electron temperatures and densities from the data. 5 . Reduction of exposure time means better time resolution of a spec-

trum variable. Walker (21) has studied AE Aquarii. 4. MODIFIEDLALLEMAND TUBE

The destruction of the photocathode and the subsequent need for reprocessing the tube with each plate loading is an operational disadvantage to the Lallemand tube. Means for preserving the photocathode had been sought by Kron (22), Hiltner (bS), McGee et al. (24), and McGee et al., (26). Kron has introduced a stainless steel valve between the photocathode and the plate holder. Photocathode life is extended one month or even longer. Kron’s tube represents a complete redesign with excellent resolution and image geometry being obtained (26). His principal handicap is that he must make his own photocathodes. Photocathode technology, particularly as applied to the modern multialkali variety, seems to be a proprietary affair. Kron has evaluated his tube on actual star fields. Figure 3(A-C) shows such a test series made by him on the 20-in., F7, astrographic telescope at the Lick Observatory. Kron (12,27) believes that, because of the large storage capacity of the nuclear-track emulsion, it may prove possible to record fainter stars against the sky background by using his electronographic technique than by photographic means. He is continuing work in this direction now at the Naval Observatory, Flagstaff.

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W. C. LIVINGSTON

Hiltner, working with Niklas and the Rauland Corporation, has developed a tube with an electron-transparent foil window between the photocathode and the emulsion. The foil cannot withstand atmospheric pressure, so a vacuum must be maintained in the emulsion compartment. The search for a low maintenance vacuum system led Hiltner to the scheme of cryogenic pumping with liquid hydrogen. It was then found that the cryogenic pump was so effective that the foil could be eliminated, provided the pump remained charged. McGee and his associates at the Imperial College in London have shown that a suitably curved mica window can withstand atmospheric pressure and yet be quite transparent to 35-40 kV electrons. The window configuration required for spectroscopy is long, narrow, and cylindrical in cross section, and thus is inherently strong. The recording electronsensitive emulsion, mounted on a matching cylindrical mandrel, is pressed into contact with the window. The success of this thin mica tube (also called a Lenard window tube or, more recently, the “Spectracon”) depends critically on having a high accelerating potential. Background due to field emission tends to be a problem in a single-stage tube a t 40 kV. McGee’s approach has been to make the tube physically large enough to avoid steep potential gradients. Great length is a disadvantage because of the susceptibility of magnetic disturbances. However, the principle of the tube is sound, and continued progress may be expected. 5. NOTESON POTENTIAL USE

OF

IMAGE TUBEFOR DETECTION OF

FAINTSTARS Considerable attention has been given to this problem of detecting faint stars against the night-sky background and to the role that image tubes might play in advancing the detection threshold. The interested Baum (29),and others presented reader should see papers by Morton (28)) at the 1955 IAU Symposium in Dublin. So far, the image tube in actual tests has generally proved inferior to the photographic emulsion. Expected information gain, arising from the finer grain records obtained with the image tube as compared with astronomical emulsions, has been more than countered by loss of contrast caused by signal induced noise. An aspect of the problem which appears to have been somewhat neglected concerns the dependence of photographic granularity on spatial sample size. Regardless of the kind of photographic material, as the sample area increases from something like the “grain” size upward, the granularity falls, but only to a certain, Jinite value and then increases. This minimum value of granularity may be as low as 1 % for certain materials (see Fig. 4).RhlS granularity is a measure of the random point-to-point fluctuation in density found on an otherwise uniformly exposed plate.

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

359

Against this fluctuating background the already low-contrast star image must be detected. Fine grain emulsions would be expected to extend the detection threshold only if the star image size falls on the high spatial frequency rise of the granularity curve. As progressively finer grain plates

I

.I

.01 .001 SAMPLE AREA (SQUARE m m )

.OOOl

FIQ.4. RMS granularity as a function of sample area for several photographic emulsions and two image tubes. The emulsion was exposed to uniform light and developed 4 min in D19 at 68°F with constant agitation. For each emulsion the resulting density ( D ) is given. The image tube data indicate uniformity effects (and not photoevent statistics).

are considered, or the equivalent technique of larger image scale [Bowen (SO)], a definite and fixed limiting magnitude should be expected. Although the large area RMS granularity of a single plate has a minimum value, one can average many independent exposures together by the technique of composite printing [see Bowen (30) ; also Johnson et al. (SI)]and thus, in principle, reach any desired limiting magnitude. The obstacle to this procedure is that limiting magnitude increases as the

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W. C. LIVINGSTON

square root of the total exposure time. A total exposure time in excess of one-night duration is presumed impractical. Could one reduce this exposure time by means of an image tube and thereby substantially increase the limiting magnitude? The answer would seem to be “No,” because, as indicated in Fig. 4, the image tube shows a rising large area granularity. This granulation would be superimposed on the recording emulsion granularity and would not be random from plate to plate. The data given in Fig. 4 does not specifically cover the photocathode alone or the nuclear-track emulsion. For this reason it will remain of interest to learn of results from Kron’s work on threshold detectivity with the electronographic process.

B . Cascade Tube By cascade tube we refer to the RCA C33011, a two-stage magnetic focus tube specifically developed for astronomical purposes on contracts

H.220 GAUSS

BAR MAGNETS IN CYLINDRICAL HOLDER

FIG.5. Image tube system developed by W. K. Ford and the Carnegie Image Tube Committee.

monitored by the Carnegie Image Tube Committee (6, 1964). Photoelectrons from a 38-mm diam 5-20 photocathode are accelerated through a field of 10 kV and strike a phosphor-photocathode “sandwich” membrane; see Fig. 5. The electron gain across this sandwich is about 30. The second stage is identical to the first, the output being a P-11 phosphor. Distance from the photocathode to output phosphor screen is 4.8 in. Magnification is unity with a small amount of S-distortion and barrel distortion. The phosphor image is transferred to a photographic plate by a relay lens. Resolution is about 40-45 line-pairs/mm, exclusive of loss by the lens and recording emulsion.

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PROPERTIES AND LIMITATIONS O F IMAGE INTENSIFIERS

Figure 10 provides a qualitative comparison between the photographic plate, the cascade tube, and the TSE tube for certain laboratory test objects. 1. ADVANTAGES 1. Good statistics. The high electron gain at the phosphor-photocathode sandwich is a most desirable characteristic because it means that (a) the noise (scattering and dark current) of subsequent stages is less important, and (b) the single-photon event “statistics” will be favorable for photon counting. While the gain of the C33011 is too low for photon counting, more stages can be added. See discussion of pulse counting under the TSE tube heading. 2. Relatively simple to use. This was a special consideration of the Carnegie Image Tube Committee.

2. DISADVANTAGES 1. Dependence on coupling lens. The overall efficiency and resolution depend on the quality of the relay lens. Ford believes that the best arrangement would be an enlargement of 1.5 from phosphor to the IIa-0 emulsion with an acceptance cone of F2. The 1.5 enlargement makes up for the fact that the modulation transfer of the IIa-0 emulsion is about 0.3 a t 40 lp . mrn-’. There does not seem to be a commercially available lens of this aperture that can cover the 40-mm field without vignetting and loss of resolution. 2. Loss of contrast by transmitted light. Additional loss may occur in the second stage, as evidenced by the fact that the contrast of a threestage tube is much poorer than that of the two-stage tube. Figure 6 indicates the photometric performance of the cascade tube system for low contrast images. Gamma is about 0.6. In fairness to the image tube it must be said that much of the degradation is caused by the associated optical system. 3. Uniformity. Two photocathodes and two phosphor screens create a potential problem in uniformity-although the C33011 is excellent in this regard, particularly the small scale granularity (see Fig. 4).

High background. Dark background for this tube is said to be about watt. cm-2 equivalent input at 4200 8. This is equal to 2 lo3 single photoelectrons cm-2 . sec-I which would seem rather high. However, if this background is translated into mainly “ion events” of 30-40 electrons each, then the current is more reasonable. Experience with background effects in the cascade tube at Kitt Peak, based on four tubes, can be summarized as follows. The background con4.

-

[A) SPECTROGRAPH

(B) SPECTROGRAPH

+IMAGE TUBE

X IZERO ICLEAR PLATE)

\I (b)

FIG.6. Photometric response of an image tube system in the case of absorption lines. Source is the sun and an iodine absorption tube at 5890 A and a dispersion of 4 mm/A. (a) Image slicer spectrograph. (b) Same as (a) cylindrical lens C33011 tube F2 transfer lens IIa-0 plate exposed to density 0.95.

+

+

+

+

+

PROPERTIES

AND LIMITATIONS

OF IMAGE

INTENSIFIERS

363

sists of three parts. First, there is a component of bright scintillations that form noticeable starlike, or radial, patterns the exact shape of which depends on the voltage and magnetic setup. Second, there is a component of bright scintillations distinguished by being uniformly distributed over the screen. Third, there is a background of faint scintillations. Components 2 and 3 can be virtually eliminated by cooling but the star pattern remains unaffected. 3. RESEARCH

1. The C33011 has been used by Wilson et al. (32)t o obtain the relative strength of the lithium line at 6707 8 in a large number of stars. The observations were made with high dispersion at the 100-in. telescope coud.4 spectrograph. The tube provided a great saving in time and also permitted the observation of fainter objects than would otherwise be possible.

2. Butslov et al. (33)have used a cascade tube for the photography of galaxies through a narrow band interference filter centered on Ha. Several galaxies were found that showed jet-like features visible only in Ha.

C . Transmission Secondary Emission ( T S E ) Tube Multistage TSE tubes have been made by Wilcock at the Imperial College, Westinghouse, and 20th Century Electronics, but the following description specifically applies to the P829D made by English Electric Valve (34). Photoelectrons from a l-in. diam 5-20 photocathode are accelerated through 4.5 kV and strike a thin potassium chloride surface supported by an electron-transparent aluminum membrane. Focus is by a uniform magnetic field. Interaction of an energetic primary electron with the potassium chloride medium produces an average of five secondary electrons. These secondaries emerge from the back of the dynode where they are accelerated through 4.5 kV to the next membrane. There are 5 such dynodes to produce a total electron gain of about 3000. The output is a P-11 phosphor, and the total light gain, with 36 kV overall, is 5 * lo6. With a relay lens of F5 effective cone, scintillations due to individual photoelectrons can easily be recorded on IIa-0 emulsion. Resolution is normally noise limited, but if slow emulsions are used, 60 lp - mm-1 at center to 40 lp * mm-' at edge can be recorded. Magnification is unity with noticeable S-distortion. Tube length is 8 ins. Figure 10 compares some of the TSE tube and cascade tube characteristics. 1. ADVANTAGES 1. Ability to record single photoelectron events. Gain is sufficiently high so that effects of plate fog are negligible.

364

W. C. LIVINGSTON

2. Low dark current. Dark emission of selected tubes is 10 scintillations . cm-2 * sec-’ a t +20°C. This is probably related to the intrinsic low dark current of the multialkali surface and to the relatively low acceleration potential of 4.5 kV which results in reduced field emission.

Items 1 and 2 combine to give a transfer characteristic linear to lower light levels than any other receiver. 2. DISADVANTAGES 1. Light dependent background. If a single bright line is projected on the photocathode the output image consists of scintillations representing this line, but also there is a distribution of scintillations outside the line. This distribution has been described by Iredale et al. (56). I n part the observed halo of scintillations is due to scattered light. But if we prohibit

OIL-

Fro. 7. Method for delivering light to a semitransparent photocathode without scattered light. Total reflection occurs at the photosurface.

the entry of light into the tube by masks and by the use of prisms contacted on the faceplate (Fig. 7), we find that considerable signal dependent background remains. Evidently electron scattering in one or more forms is prevalent in this type of tube. 2. The pulse height distribution of the output scintillations is exponential in form. See Section 5 on pulse counting. 3. Some evidence (55) indicates that the absolute efficiency of detection of photoelectrons may be only 30%. However, this figure is uncertain. The measurement technique is liable to error due to scattered light and multielectron “ion events,” both of which would tend to reduce the apparent efficiency.

3. RESEARCH 1. The P829D has proven to be of special value at very low light levels ( < 10 photons * sec-’ resolution element-’) and for relatively high contrast images such as emission lines at low dispersion or sharp absorption

W

Ia

n

-I

0

- 1!ii

I!

n

n W

I a

z

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

N

0

rFIG.8. Schematic of image tube system used for spectroscopy with high gain TSE tube.

365

366

W. C. LIVINGBTON

lines at high dispersion. I n this last category are lines whose origin is the cold interstellar gas. Livingston and Lynds (36) have used the tube to resolve the interstellar D lines into multiple components indicating a complex of clouds of different radial velocities (see Fig. 9B). 4. NOTESON

A

TSE TUBESYSTEM FOR SPECTROSCOPY

Figure 8 illustrates features of a system which helps to circumvent some limitations encountered with the TSE tube. The image tube is at

FIG.9. Image tube spectra. (A) Low dispersion spectrum of HD7927, spectrum type F5, 5.7 p.g. mag., Xk 4130-4550, taken by Ford with the C33011 tube. (B) High dispersion spectrum of alpha-Cygni, AX 5887-5891, taken with the P829D tube.

the focus of a 45-ft, F60, vertical spectrograph (37). The spectrum of a star, emerging from the spectrograph, has a width of about 20 mm owing to the presence of a 50 p , 20-element image slicer ($8)forming the entrance slit. A cylindrical lens, contacted with oil to the tube window, narrows this spectrum to about 0.5 mm. The relay lens, with an effective cone of F5, images the intensified image onto a IIa-0 plate. Just in front of the plate is a mask, with about a 0.5 mm width, accurately positioned to fit the spectrum image. Because of the spiral distortion of the tube, the mask is S-shaped. The plate is driven by a motor perpendicular to the dispersion direction during exposure. Resulting spectra are shown in Fig. 9(B).

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

367

FIQ.10. Qualitative comparison in laboratory between the cascade tube (C33011) and the TSE tube (P829D). Records A-D were taken under identical conditions using the electroluminescent light source through a Corning 5-58 filter contained in the test projector designed and described by Baum. Records E and F indicate geometric distortion. Records G-I are of comparable contrast and show response to uniform light of the receivers. To make G, a IIa-0 plate was exposed and brush developed (density = 0.85), and a contact print then made on V-0 (density = 1.0). H and I, which cover the entire unmasked tube area, were recorded directly on V-0 (density = 1.0).

This somewhat devious scheme accomplishes the following: Contrast loss by the light dependent background is greatly by concentrating the image and masking off the spurious noise. Storage capacity on the plate is extended by the widening process. A tilt of the spectrum lines caused by S-distortion is eliminated. (4) Any gradual change of the ambient magnetic field merely tilts the spectrum but does not cause a loss of resolution. (1) reduced (2) (3)

The disadvantage of this system is that narrowing the spectrum causes tube nonuniformities to become more apparent.

368

W. C. LIVINGSTON

5. NOTESON PULSECOUNTING WITH IMAGE TUBES

The clarity with which single photoelectron events are recorded in electronography and with the TSE tube suggests the possibility of image integration by two-dimension scintillation or track counting. This procedure is of special interest for image tubes because it offers, a t least in principle, linear storage and freedom from overload providing the rates are low. The problem of information recovery at low count densities, where plate fog invalidates the usual microphotometer techniques, is a consideration. The feasibility of pulse counting depends on the form of the receiver pulse height distribution. If the shape is exponential, with a preponderance of small amplitude pulses, minute changes in the discriminator setting will result in large changes in the count. Most desirable is a pulse height distribution with a distinct “Poissonian” hump. Providing the discriminator is positioned in the valley between the hump and the low intensity noise, the count is relatively insensitive to change of discriminator position. The fact that photomultipliers with an exponential pulse height distribution can be successfully used for photon counting testifies mainly to the stability of the associated power supplies and discriminator circuits. I n the case of the image tube, however, local nonuniformities cause a point-to-point variation in the gain and thus effectively cause a change in the discrimination setting. Unfortunately, the high gain TSE tube shows an exponential distribution (39).Pulse counting with the EEV P829 results principally in an excellent map of the tube nonuniformity. A multiplying element that does show good gain statistics is the phosphor-photocathode sandwich of the cascade tube [see Rome-et al. (do)].Unfortunately, cascade tubes known to the writer have consisiently shown a high dark background. Whether or not this is an inherent property is not known.

D . Image Orthicon The image orthicon for low light level applications is identical to the standard camera tube of broadcast television except that the former may have a multialkali photocathode and a special target which allows long storage times. The tube has been of interest because (1) it is the one commercially available image tube that has been perfected; (2) under certain conditions of exposure the efficiency is very high [the DQE 0.4 RQE of the photocathode (3, lo)]; and (3) a t a time when other developmental image tubes were in trouble owing to field emission background, the low operating voltage of the image orthicon was considered an advantage.

-

PROPERTIES AND LIMITATIONS

OF IMAGE INTENSIFIERS

369

In the image section, photoelectrons from a 40-mm photocathode fall through a 500-volt field to strike an oxide target. Each photoelectron causes the release of several secondary electrons, leaving a net positive charge. Conduction of this charge through the target then occurs. At the end of the exposure a low-velocity reading beam explores the target from the back side, systematically depositing an equal electronic charge to neutralize the positive image. That part of the beam which is not taken up by the target enters a five-stage pinwheel multiplier. The signal consists of the modulation of this reading beam. A detailed account of how the image orthicon works cannot be given here. Yet it is just this “detail” that is crucial to understanding the performance limitations of the tube. For these details the reader is referred to articles by Weimer (QI), Luedicke et al. (48), and Livingston (43). 1. ADVANTAGES 1. Electrical signal output for remote transmission.

2. Electron redistribution phenomena causes enhancement of star images against sky background as shown in Fig. 3(D-F). 2. DISADVANTAGES 1. The tube has no unique transfer characteristic. I n general, the response at every image point is interrelated to every other point because of the following: (a) Electron redistribution in the image section. The percentage of secondary electrons collected by the target mesh depends on local image potential gradients. Uncollected secondaries fall back on surrounding areas and thus alter the local sensitivity. (b) The target has no fixed electrical capacity. The effective capacity is a function of image detail. (c) The tube tends to have a residual memory of scan and exposure history. 2. The tube has no fixed geometry. During readout the low velocity scanning beam is deflected by local potential fields, causing a variable distortion (beam bending).

Both the response and position of any image point is indeterminate and measurements taken with an image orthicon can, therefore, only be of the very approximate kind. 3. RESEARCH The one job for which the image orthicon seems suited is the rapid, but low precision, photometry of stars (see Sections 4 and 5). Bakos and

370

W. C. LIVINGSTON

Rymer (44) have made use of this characteristic to try and discover very short period variables. A 75-min variable was found in NCC 1893 [see also Hynek and Dunlap (46)l. 4. NOTESON

THE

IMAGE ORTHICON AS

A

FINDING AND

POINTING AID Although its role for data acquisition is in question, experience a t Kitt Peak and elsewhere suggests that the image orthicon might be valuable as an ancillary receiver for finding and positioning faint sources as needed for photometry or spectroscopic study. At the F13.5 focus of a 36411. telescope, 18th magnitude stars are displayed on the television monitor screen for 5-sec exposures. Conventional “off set techniques” can be used which involve positioning on a faint, invisible object by coordinate offset to a nearby bright star. However, even experienced observers prefer to set by direct viewing; the image orthicon with its storage capability makes this possible. Guiding by image orthicon might also be considered to avoid eyestrain. 5. NOTESON OTHER SIGNlL GENERATING TUBES

A general recognition of the deficiencies of the image orthicon plus the increasing need for a television-type transducer that could be used in orbiting telescopes have resulted in a number of new tube ideas. A proposed solution to the problem of beam bending, interelement capacitive coupling, and ot,her aberrations is the mosaic target. The target is divided into microscopic size shielded elements so that the potential field due to a single element is confined to an area the size of the element. See reports on the Optechon (46) (Westinghouse), the Plug Target Image Orthicon (47) (RCA), and also other devices involving microstructure elements such as the Light Scan Camera Tube (48) (Machlett) and the Image Dissecticon (49) (I.T.T. and Bendix). The main defect in the mosaic element approach as found in these tubes is that inherent surface nonuniformity modulates the high light level signal and generates a pattern of noise at low light levels. These troubles could be circumvented if the microstructure element could be preceded by sufficient gain, such as by cascade type amplification, so that the photon noise could override the target noise. Mention should also be made of the vidicon because of its excellent performance in the Ranger and Mariner flights. I n common with the photographic plate, the vidicon is relatively unaffected by internal scattered light and stray magnetic fields. Unfortunately, the DQE of the vidicon is only about 0.001, so that it is not of much interest at low levels. Also it suffers from light dependent geometry because of beam bending.

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

371

E. Tubes with Infrared Photocathodes These devices form a special class because they are not in competition with photographic materials. The S-1 photoemissive surface is useful to about 1.2 p as compared to the lead-sulfide photoconductive target in a vidicon which has sensitivity to about 2 p . S-1 tubes available to date have been of the single-stage, phosphor output variety. Recently, however, RCA has announced a n S-1 version of the two-stage cascade tube (C70056-E), and Lallemand (60) has succeeded in introducing the S-1 photocathode into his tube. The efficiency of the single-stage tube has depended on the optical coupling between the phosphor and the emulsion. For a relay lens system the coupling efficiency (i.e., image/phosphor illuminance) is given approximately by ?rT/4(m

+ 1)2F2,

where T is the transmission of the lens, m the phosphor to image magnification, and F is the relative aperture. Illumination off axis decreases as C O S ~8, where e is the angle between the image point and the optic axis at the lens. A commercial tube is the I.T.T. Industria1 Laboratories FWl67 which has a 40-mm photocathode, P-11 phosphor, and operates at about 10 kV with magnetic focus. The coupling efficiency of an F2 lens, even neglecting transmission loss, is only 5%. Depositing the phosphor screen on a thin mica window and then pressing the emulsion against this (curved) surface during exposure substantially increases the coupling efficiency. The price is some loss of resolution owing to the thickness of the mica and imperfect contact. A oomniercial mica window tube of this kind is the I.T.T. FWlOSA which has a 16-mm photocathode, P-11 phosphor, and uses electrostatic focus with a magnification of 0.68. 1. RESEARCH 1. Volkov et al. (61) have used a mica-window tube to obtain emissionline spectra of gaseous nebulae. They note that line intensities may be disturbed by telluric features. 2. Fredrick (56) has used the I.T.T. tube mica window tube to obtain exploratory spectra of bright stars to 1.2 p. He estimates that at 1.1 1.1 the “information rate” of the image tube relative to the hypersensitized type-Z plate is about 500. 3. Firor and Zirin (63) have used a single-stage tube, with relay lens, operating at 18 kV, to photograph coronal infrared lines representing

372

W. C. LIVINGSTON

5 levels of ionization. Sharp peaks in the distribution of ionization, previously reported by others, were not found. Firor notes that the S-1 image tube used visually aids in the adjustment of the spectrograph and positioning of the chromatic solar image.

4. Kuprevich (64) has made observations of the moon with the infrared vidicon. Pictures taken with an effective wavelength of about 1.6 p show greater contrast than in visual light. This is interpreted as due to decreased luminescence or increased reflectivity. 5. Zirin and Dietz (66), and others, have found that the X 10,830 of

He1 yields unique information on the solar chromosphere. Results suggest that the line originates mainly in the hot spicules which are clearly outlined in the low chromosphere but which appear sparse against the disk. Improved spatial resolution, as a consequence of the reduced exposure time of the S-1 image tube, had shown the patchy origin of the line. Zirin and Howard (unpublished) have also obtained image tube spectroheliograms in the X 10,830 line.

V. PROSPECTS FOR FUTURE DEVELOPMENTS The first goal of image tubes-reduced exposure time with more or less equivalent photographic precision-appears realized. Under certain conditions this reduced exposure can even be translated into the ability to observe fainter objects or to observe bright objects with greater detail. But the full promise of the image tube-precision photometry-is a t present far from being fulfilled. I n view of present deficiencies in available image tubes, we might inquire as to the prospects for a device suitable for two-dimension photometry. As emphasized throughout the preceding discussion, the principal requirement is for complete independence of response between image elements. I n this direction, recent literature contains several pertinent ideas. Reflective photocathodes, use of microstructure elements, fiber optic faceplates, and development of methods for preparing photocathodes external to the tubes-these are advances of major importance. I n this section we shall review these ideas and consider their application.

A . The Rejective Photocathode A reflective photocathode consists of a semitransparent photosurface (e.g., the 5-20) deposited on a highly reflecting substrate. This type of photocathode, while it has never been applied to image tubes, does seem to offer several advantages. These include: (1) No scattered light. (2) Increased quantum efficiency.

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

373

(3) Increased flexibility in the design of the magnetic focus field (see Figs. 17 and 18). Awkwardness of optical image delivery is its disadvantage. The property of no scattered light is of particular importance. Figure 11 indicates how the reflective photocathode differs from the semitransparent in this regard. Of course, the image tube must have a window through which the light enters; but if the image-forming element (mirror) is located in the tube vacuum, no scattering or secondary images result from the multiple reflections at this window. LIGHT

SEMITRANSPARENT SPURIOUS GLASS

’\

IN REFLECTIVE

OPAQUE SUPPORT

FIG.11. Diagram showing that the reflective photocathode is completely free of localized-scattered light.

The reflective photocathode offers the possibility of increased quantum efficiency, particularly in the red. A very interesting analysis of the nature of reflectively enhanced photoemission is contained in a report by Seachman (56) on work done by the Westinghouse Intensifier Engineering Section. Figure 12 is based .on data taken from this report and shows the amount of enhancement that can be obtained as a function of wavelength for two different photocathode configurations. It is not possible to have a maximum of both red and blue enhancement. Seachman shows that optimum efficiency is obtained with a thin (-300 A) photosurface spaced about X/4 from the reflector. This spacer can be a dielectric film, in which case it can be said that multiple reflections occur between the reflector and photocathode, causing almost a total absorption of the light. This “spaced reflective-photocathode” offers even higher red response

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W. C. LIVINGSTON

than indicated. However, unless some way can be found to produce a surface field gradient, the RQE can be no more than 0.5. Another enhancement scheme, applicable to existing tubes with semitransparent photocathodes, has been suggested by Gunter et al. (57). Light is directed into the window, at an angle of about 50" from the normal, by means of a contacted prism. Total internal multiple reflections 60

50

2 w

0

40

E 1

2. 0

z

-

30

LL

,

RED ENHANCED

\

LL

W

5I-

z

20

a

3

0

10

0

FIQ.12. Calculated reflectivity enhanced quantum efficiency for the 5-20 photosurface. Shown for comparison is the unaltered responsive quantum efficiency for the semitransparent 5-20 with and without extended red response [after Seachman (66)].

take place between the front and photocathode surfaces. Photoemission takes place at each photocathode encounter, weakened, of course, because of reduced light intensity. Figure 13 shows a photographic recording of the output of a C33011 image tube for oblique illumination a t 4000 and 8000 A. R4easurement of the relative quantum gain was also made by imaging the phosphor screen onto a distant photomultiplier. I n this case the phosphor screen was carefully masked to exclude all light outside the image points (Fig. 14).

375

P R O P E R T I E S AND LIMITATIONS O F IMAGE I N T E N S I F I E R S MONOCHROMATOR

/ I

f

\/

\

\I

\ZV\/\/V\/\/ S-20 PHOTOCATHODE

U

U

FIG.13. Oblique reflective enhanced photoemission in the C33011 tube.

4

A

t

4000

5000

6000

7000

8000 A

FIG.14. Experimental data oblique reflectivity enhancement in a C33011 tube. (Kitt Peak experimental data.)

376

W. C. LIVINGSTON

Although photocathode enhancement by oblique reflection does not seem to have general application, a special case is spectroscopy with a large F-number system. Figure 15 indicates a proposed scheme for use at the F60 solar telescope.

.

F120

FIG.15. Application of oblique reflectivity enhancement for spectroscopy.

B . Microstructure Elements Following the rapid progress in microelectronic technology, we should expect improvements in what might be called microstructure elements for image tubes. One such device is the Continuous Channel Electron Multiplier. As described by Goodrich and Wiley, (68)this multiplier consists of an array of fine glass tubes with internal diameters of less than 25 1.1. Each tube has an inside wall coated with a semiconductor to form a continuous secondary emission dynode. With 1-2 kV across the tube length, the electron gain is about los. From our viewpoint, this channel multiplier has two important properties. (1) It is the one multiplying element with complete independence between channels, and (2) it is insensitive to external magnetic fields. Another interesting microstructure element is the multiplying emitting electron time, or MEET, disperser [see Nordseth (49)].I n one form the MEET disperser may be described as a monostable electro-optical multivibrator. An input burst of electrons triggers a localized area of a photoconductor-phosphor-photocathode sandwich, which responds by the continued emission, or dispersing, of electrons. After a certain time this dispersing piocess abruptly terminates. The MEET disperser can, therefore, serve as a high-speed two-dimension buffer storage.

PROPERTIES AND LIMITATIONS OF IMAGE INTENSIFIERS

377

C . Fiber Optic Faceplates A number of factors-resolution, vignetting, and physical constraints -combine to place an upper limit to the transfer efficiency at about 3% for lens or mirror relay optical systems. This may be compared with the 60% efficiency of a coherent fiber optic bundle (59,60). The factor of 20 gain in favor of fiber optics is about equivalent to the electron gain of phosphor-photocathode sandwich in the cascade tube. Therefore a singlestage tube, with fiber optic output, should have gain comparable to the two-stage tube with a conventional coupling lens. Another distinguishing feature between the coupling lens and fiber optic scheme is that the latter can be made into an array of several inches in length, appropriate for a spectrum. The problems associated with the fiber optic faceplate have been mainly light leakage at the entrance between fibers and making the fiber diameter small without undue light loss in the interfiber void. For the same reason that the television raster must be about 10 times finer than the limiting resolution, in order to define with equivalent (photographic) precision the wavelength of a spectrum line, the fiber size must also be proportionately small. It is not sufficient that the fiber size merely match the “tube” resolution. Rather it is desirable that the fiber size be onetenth as large to assure accurate spectroscopic line positions.

D. The Preformed Photocathode The geometric design of an image tube is very much complicated by the requirement that the photocathode be formed after the tube is assembled. Allowance must be made for the strategic placement of various evaporator “boats.” Further, the cesium introduced during activation contaminates the entire tube and may be a source of much of the dark emission. Folkes (61) has described a system for preparing the photocathode outside the tube. We also note that the photoconductive faceplates of certain commercial vidicon tubes are made externally and then joined to the final tube with a cold seal. Advances in these techniques are essential for low dark background and to allow the long, special shaped tubes required for spectroscopy.

E. Conceptual Tubes To the writer’s knowledge, no tube has yet been proposed which would fully satisfy, even in principle, the needs of photometry. Several writers have suggested tube designs which make use of one or more of these new techniques and are, therefore, of current interest.

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PROPERTIES AND LIMITATIONS O F IMAGE INTENSIFIERS

379

Nordseth (49)conceives of a device he calls the “Image Dissecticon.” Essentially, the tube consists of four parts: a cascade front end, a microchannel multiplier, the previously described MEET disperser, and a conventional scanning image dissector which sequently samples the buffered electrons. T h e image integration is b y external digital scaling (see Fig. 16). CORRECTION

__

THIN MICA WINDOW O N PRESSURE PLATE

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FIG.17. High speed (F1.03) spectrograph camera-image tube combination [after Seachman (66)]. ROM SPECTROGRAPH REFLECTIVE PHOTOCATHOD

FIG.18. Side view of a tube suitable for long spectra. Length perhaps 12 in. normal to plane of paper.

Seachman (56), following a suggestion by Mestwerdt, describes a tube which uniquely unites an image tube to high-speed optics with low scattered light (Fig. 17). This design makes use of the reflective photocathode. The writer (62) has proposed a tube suitable for the taking of long spectra. At high dispersion, the longer the spectrum the more information

380

W. C. LNINGSTON

acquired in a single exposure. For example, the 144411. camera of the coud6 spectrograph (Kitt Peak National Observatory’s 84411. telescope) takes a 1 X 28-in. plate. No image tube of conventional design approaches this coverage. However, the reflective photocathode admits an arrangement whereby long length becomes feasible. The design is indicated in Fig. 18. VI. SUMMARY AND CONCLUSIONS We have discussed the general properties of all image tubes which have been tried out in obtaining astronomical data, and we have listed the research results in the form of scientific papers. Perhaps the most objective answer to the question ‘(What can the image tube do?” is obtained by a review of these research accomplishments. Of a total of twelve papers, five are based on material obtained by electronography ; and of these, three are by Walker a t Lick Observatory. His work on rapid spectrum variations in AE Aquarii, and on the rotation of the nuclei of M31 and M32, clearly demonstrates the benefits of reduced exposure time. Walker and Aller have made use of the photometric capability of the Lallemand tube in their study of line intensities in planetary nebulae. They find the linear response considerably extends the range of intensity measurement. However, they do not emphasize a gain in precision a t any given intensity over photographic photometry. The Carnegie group has shown, by its work with Wilson on stellar lithium abundance, that the RCA cascade tube also allows substantial reduction in exposure time for equivalent photographic precision. Tubes that do not compete directly with the photographic plate are the high gain TSE tubes (by virtue of their ability to record single photons), the image orthicon (because of its electrical output), and the infrared converters. The observations of interstellar lines (Livingston and Lynds) with the TSE tube reveal the distinctive property of this tube to abstract information when only a few photons are available. The detection of short period variable stars by means of the image orthicon (Bakos and Rymer) brings out the one useful feature of this device, i.e., the ability to transduce star images against a sky background. Because the photographic plate is essentially insensitive beyond 1.1 p, the infrared tubes become invaluable in this spectral domain. Fredrick’s exploratory spectra t o 1.2 p and the Volkov work represent completely new information. The one application of the image tube to solar astronomy is indicated by Zirin and colleagues in the use of the I R converter to observe infrared lines in the corona and the helium 10,830 line on the disk. Less objective, perhaps, are the writer’s own conclusions as to the

PROPERTIES

AND LIMITATIONS

OF IMAGE INTENSIFIERS

381

state of the image tube art. Among available tubes, there seems to be no single type which is best for all applications. The Lallemand tube undoubtedly has superior resolution as well as freedom from overload ; but its threshold detectivity is unproven, and there are the operational disadvantages. I n this last-mentioned regard, we can look forward to Kron’s development. The Carnegie two-stage cascade tube satisfies the need for a compact, easy-to-use device. It gives “photographic type” results with an approximate exposure reduction factor of 10. The main disadvantage is the relatively low gain, making necessary the use of fast coupling optics, which, in turn, degrades the resolution and contrast. The imaging photomultiplier, or TSE tube, excels in having the lowest dark background plus good detectivity for low photon counts. However, light dependent background, leading to reduction in the image contrast, is a serious disadvantage at high light levels or when the input image is of low contrast. All signal generating (TV-type) devices suffer from having a more or less indeterminant transfer characteristic, and they are almost impossible to use for quantitative measurement. The electron redistribution mechanism in the image orthicon does make this device potentially valuable as a pointing and tracking aid on star fields. We have repeatedly emphasized how scattered light and other picture element interactions found in available tubes do compromise their performance. The somewhat meager scientific results achieved so far, despite considerable effort on the part of astronomers to apply these devices, testify to the importance of these degrading effects. The goal of realizing a full, two-dimensional evaluation of the photocurrent remains. New approaches, such as the reflective photocathode and the use of microstructure elements, may open avenues toward this goal. Meanwhile, astronomers will continue to observe mainly with photographic materials and photomultipliers.

ACKNOWLEDGMENTS Besides the many image-tube workers referenced in this article, special thanks must be given t o my colleague Dr. Roger Lynds. Through almost daily discussions (and arguments), his contributions have been many. Credit is also given to Mr. L. A. Doe, who designed and constructed much of the image-tube equipment used at Kitt Peak National Observatory.

REFERENCES 1. A. Lallemand, Compt. Rend. 203, 243-244 (1936). 2. W. A. Hiltner, Image tube developments and the small telescope, in “The Present and Future of the Telescope of Moderate Size” (Frank Bradshaw Wood, ed.), Chapter 1, pp. 11-24. Univ. of Pennsylvania Press, Philadelphia, Pennsylvania, 1958.

382

W. C. LIVINGSTON

3. P. Fellgett, Investigations of image detectors, in “The Present and Future of the Telescope of Moderate Size” (Frank Bradshaw Wood, ed.), Chapter 4, pp. 51-86. Univ. of Pennsylvania Press, Philadelphia, Pennsylvania, 1958. 4. J. D. McGee, Photoelectronic problems in astronomy, in “The Present and Future of the Telescope of Moderate Size” (Frank Bradshaw Wood, ed.), Chapter 3, pp. 31-49. Univ. of Pennsylvania Press, Philadelphia, Pennsylvania, 1958. 6 . W. A. Baum, J. S. Hall, L. L. Marton, and M. A. Tuve, Annual reports by the Carnegie Committee on image tubes for telescopes. Carnegie Znst. Wash. Year Book (1955-1965). 6. N. U. Mayall, Ann. Astrophys. 23, 344-359 (1960). 7. J. C. Marchant and A. G. Millikan, J . Opt. Soe. A m . 66, 907-911 (1965). 8. J. C. Marchant, J . Opt. SOC. A m . 64, 798-800 (1964). 9. L. H. AUer and M. F. Walker, Astrophys. J . 141, 1318-1330 (1965). 10. R. C. Jones, Quantum efficiency of detectors for visible and infrared radiation, Advan. Electronics and Electron Phys. 11, 87-183 (1959). 11. M. F. Walker, Publ. Astron. SOC.Pacific 74, 44-54, (1962). 12. J. B. Breckinridge, G. E. Kron, and I. I. Papiashvili, Astron. J . 69,534-535 (1964). 13. A. Lallemand, M. Duchesne, G. WlBrick, R. Augarde, and M. F. DuprB, Ann. Astrophys. 28, 320-330 (1960). 14. M. Duchesne and M. MBallet, Compt. Rend. 264, 1400-1402 (1962). 16. M. F. Walker, Recent progress in the use of the Lallemand electronic camera in astronomical spectroscopy, Advan. Electronics and Electron Phys. 22B, 761-780 (1966) [“Photoelectronic Image Devices” (J. D. McGee, D. McMullan, and E. Kahan, eds.), PTOC.3rd Symp., Imperial College, London, 19661. 16. G. Wlt?rick and A. Grosse, La camera Blectronique: un rbcepteur d’images sans lumihre difTusBe, Advan. Electronics and Electron Phys. 22A, 465-475 (1966) [“Photoelectronic Image Devices” (J. D. McGee, D. McMullan, and E. Kahan, eds.), Proc. 3rd Symp., Imperial College, London, 19661. 17. J. Rosch, G. Wldrick, and M. F. Duprt?, Compt. Rend. 363, 509-511 (1961). 18. M. Bellier, M. F. DuprB, G. WlBrick, J. Rosch, and J. Arsac, Ilfem. SOC.Roy. Sci. Lihe 7, 522-534 (1 963). 19. A. Lallemand, M. Duchesne, and M. F. Walker, Publ. Astron. SOC.Pacific 72, 76-84, 425 (1960). 20. M. F. Walker, Astrophys. J . 186, 695-703 (1962). 21. M. F. Walker, Sky and Telescope 20, 23-25 (1965). 22. G. E. Kron, A modified Lallemand image tube, Advan. Electronics and Electron Phys. 16, 25-26 (1962) [“Photoelectronic Image Devices’’ (J. D. McGee, W. L. Wilcock, and L. Mandel, eds.), Proc. 2nd Symp., Imperial College, London, 19611. 93. W. A. Hiltner, Image converters for astronomical photography, in “Astronomical Techniques” (W. A. Hiltner, ed.), Vol. 11, Stars and Stellar Systems, Chapter 16, pp. 340-373. Univ. of Chicago Press, Chicago, Illinois, 1962. 24. J. D. McGee, A. Khogali, and A. Ganson, Electron transmission through mica and the recording efficiency of the spectracon, Advan. Electronics and Electron Phys. 22A,31-39 (1966) [“Photoelectronic Image Devices” (J. D. McGee, D. McMullan, and E. Kahan, eds.), Proc. 3rd Symp., Imperial College, London, 19661. 66. J. D. McGee, A. Ganson, A. Khogali, and W. A. Baum, The spectracon-an electronographic image recording tube, Advan. Electronics and Electron Phys. 22A, 11-30 (1966) [“Photoelectronic Image Devices” (J.D. McGee, D. McMullan, and E. Kahan, eds.), PTOC. &d Symp., Imperial College, London, 19661. 26. G. E. Kron and I. I. Papiashvili, Publ. Astron. SOC.Pacific 77, 109-111 (1965).

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27. G. E. Kron, J. B. Breckinridge, and I. I. Papiaahvili, Publ. Astron. SOC.Paci$c 77, 112-114 (1965). See also G. E. Kron and I. I. Papiashvili, Publ. Astron. SOC. Pan’Jc 79, 9 (1967). 28. G. A. Morton, Trans. Intern. Astron. Union 9, 676-681 (1955). 29. W. A. Baum, Trans. Intern. Astron. Union 9, 681-687 (1955). 30. I. S. Bowen, Astron. J . 69, 816-825 (1964). 31. H. L. Johnson, R. F. Neville, and B. Iriarte, [Bull. No. 931, Lowell Obs. Bull. 4, 83-86 (1958). 32. 0. Wilson, W. A, Baum, W. K. Ford, Jr., and A. Purgathofer, Publ. Astron. SOC. Pacific 77, 359-366 (1965). 33. M. M. Butslov, I. M. Kopylov, V. B. Nickonov, A. B. Severnyi, and K. K. Chuvaev, Astron. Zh. 39, 315-322 (1962); Soviet Astron.-AJ 6, 244-252 (1962). 34. P. C. Ruggles and N. A. Slark, IEEE Trans. Nucl. Sci. NS-11, 100-107 (1964). 36. P. Iredale, G. W. Hinder, and D. J. Ryden, ZEEE Trans. Nucl. Sci. NS-11, 139-146 (1964). 56. W. C. Livingston and C. R. Lynds, Astrophys. J . 140, 818-819 (1964). 37. A. K. Pierce, Appl. Opt. 3, 1337-1346 (1964). 38. A. K. Pierce, Publ. Astron. SOC.Pacijic 77, 216-217, 456 (1965). 39. D. L. Emberson, A. Todkill, and W. L. Wilcock, Further work on image intensifiers with transmitted secondary electron multiplication, Advan. Electronics and Electron Phys. 16, 127-139 (1962) [“Photoelectronic Image Devices” (J. D. McGee, W. L. Wilcock, and L. Mandel, eds.), Proc. 2nd Symp., Imperial College, London, 1961). 40. M. Rome, R. V. Gauthier, and J. P. Causse, ZEEE Trans. Nucl. Sci. NS-11, 3, 120-128 (1964). 41. P. K. Weimer, Television camera tubes: A research review, Advan. Electronics and Electron Phys. 13, 387-437 (1960). 42. E. Luedicke, A. D. Cope, and L. E. Flory, Appl. Opt. 8, 677-689 (1964). 43. W. C. Livingston, J . Soc. Motion Picture Television Engrs. 72, 771-786 (1963). 44. G. A. Bakos and K. R. Rymer, Astron. J . 69, 531 (1964). 46. J. A. Hynek and J. R. Dunlap, Sky and Telescope 28, 126-130 (1964). 46. Rept. prepared for Princeton Univ. Obs. Space Telescope Program, Princeton, New Jersey, under Subcontract No. 1, NASA Grant NSG-414, by Westinghouse Elec. Corporation, Electron. Tube Div., Baltimore, Maryland, titled “Study of the Design Problems of an Integrating Television Tube for Astronomical Research,” 1964. 47. S. A. Ochs, RCA Rev. 21, 558-569 (1960). 48. Ward, S. A., and Robbins, C. D., Light scan camera tube, Image Intensifier Symp., Fort Belvoir, Virginia, 1968. O.T.S. No. 151813, pp. 171-184. U. S. Dept. of Commerce, Washington, D. C., 1959. 49. M. P. Nordseth, The image dissecticon, a low-light-level image tube. NAVWEPS Rept. 8220, Naval Ordnance Lab. Corona, California, 1965. 60. A. Lallemand, Perfectionnement de la camera Blectronique-application SL I’infrarouge. Photoelectronic image devices as aids to scientific observation, Advan. Electronics and Electron Phys. ZZA, 1-3 (1966) [“Photoelectronic Image Devices” (J. I). McGee, D. McMullan, and E. Kahan, eds.), Proc. 3rd Symp., Imperial College, London, 19661. 61. I. V. Volkov, V. F. Esipov, and P. V. Shcheglov, Astron. Zh. 39, 323-329 (1962); Soviet Astron.-AJ 6, 253-259 (1962). 62. L. W. Fredrick, [Bull. No. 1141, Lowell Obs. Bull. 6, 149-152 (1961).

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63. J. Firor and H. Zirin, Astrophys. J. 136, 122-137 (1962).

64. N. F. Kuprevich, Astron. Zh. 40, 889-896 (1963); Soviet Astron.-AJ

7, 677-685 (1964). 66. H. Zirin and R. D. Dietz, Astrophys. J. 138, 664-679 (1963); see also R. R. Fisher, Astrophys. J. 140, 1326-1328 (1964). 66. N. J. Seachman, Research on an image intensifier design philosophy. Prepared under Contract AF 33(657)-8613, Westinghouse Elec. Corporation, Electron. Tube Div., Elmira, New York, 1963. 67. W. D. Gunter, Jr., E. F. Erickson, and G. R. Grant, Appl. Opt. 4,512-513 (1965). 68. G. W. Goodrich and W. C. Wiley, Rev. Sci. Instr. 33, 761-762 (1962). 69. N. S. Kapany and D. F. Capellaro, J . Opt. SOC.A m . 61, 23-31 (1961). 60. R. J. Potter and R. E. Hopkins, Fiber optics and its application to image intensifier systems. Image Intensifier Symp., Fort Belvoir, Virginia, October 6-7, 1968. O.T.S.No. 151813, pp. 91-109. U. 5. Dept. of Commerce, Washington, D. C., 1959. 61. J. R. Folkes, Introduction of pre-formed photocathodes into vacuum systems, Advan. Rlectronics and Electron Phys. 16, 325-328 (1962) [“Photoelectronic image Devices” (J. D. McGee, W. L. Wilcock, and L. Mandel, eds.), Proc. gnd. Symp., Imperial College, London, 19011. 61. W. C. Livingston, Appl. Opt. 6, 1335 (1966).

Superconducting Magnet Technology* CHARLES LAVERICK Argonne National Laboratory Argonne, Illinois

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 11. The Concepts of Superconductivity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 A. General Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 B. Historical Outline C. Some Basic Principles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 111. Magnet Materials and Conductors. . . . . . A. General Magnet Development., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 B. Materials Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 C. Early Magnet Problems.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 D. The Effect of Copper Coating on NbZr Wire.. . . . . . . . . . . . . . . . . . . . . 416 E. Cables and Multiple F. Considerations in the G. Conductor Working Stress., . . . . . . . . . . . . . H. Pulsed Superconductors, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 IV. Recent Significant Magnets.. . . A. General Discussion. . B. The ANL67-kG, 7-in. Bore Coil System., . . . . . . . C. The AVCO Model M D. Some Magnets for Thermonuclear Fusion Studies.. . . . . . . . . . . . . . . . . . 437 E. Further Significant Coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Design Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 A. Power Supply Considerations. B. Flux Pumps.. . . . . . . . ....................................... 444 C. Safety Considerations ....................................... 446 D. Persistent Switches. . . . . . . . . . . . . . . . . . . . . . . E. Contacts.. . . . . . . . . . . . . . . . . . . . . F. Refrigeration Requirements. . . . . . G. Predictability of Magnet Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 H. Accessibility ............................................ 453 I. Operation at Temperatures Above and Below 4.2"K. . . . . . . . . . . . . . . . 454 J. Homogeneous Fields.. . K. Radiation Effects. . . . . . VI. Magnet Economics and La 457 A. General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 B. Large Bubble Chamber Magnet Studies.. . . VII. Test Facilities and Instrumentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 VIII. Coil Fabrication Facilities. . , . .................................. 466

* Work performed under the auspices of the TJ. S. Atomic Energy Commission. 385

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CHARLES LAVERICK

IX. Future Developments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General References and Reviews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 469 469

I. INTRODUCTION Superconducting magnets and materials, in ever increasing variety, are being offered for sale throughout the world and have already become an essential tool of modern scientific and technological endeavor. They are being applied on a rapidly increasing scale to problems in diverse fields, and the pace of conductor development and magnet design continues unabated. A large and growing body of literature testifies to the vital importance of this new field, and important new scientific and technical advances are reported frequently. Superconductors and superconducting magnets are not a panacea for all woes. They have their place, not as yet completely determined, in modern electromagnetic research and technology and form a welcome addition to the stock in trade of modern science. Electromagnets had reached a highly sophisticated stage of theoretical and practical development in the 19th century. Kapitaa (1923) produced 500 kG in pulses of a few milliseconds each and Francis Bitter (1939) produced a small bore 100-kG solenoid for continuous operation. Magnetic fields in excess of 1 MG have been produced for a few microseconds at a time using explosives in conjunction with flux compression. Thus, magnetic fields over a wide range of intensity have been available to a few selected people for several decades. The appearance of the supermagnet is almost analogous to the introduction of the Ford car; it makes a wide range of volumes and magnitudes of magnetic field readily available to all laboratories and investigators without the need for large power stations and water pumping stations. It provides mankind, for the first time, with the possibility of possessing and using large volume magnetic fields wherever they are needed at reasonable cost. The pace of development and application over the past five years has been extremely rapid and it is fair to say that the present period represents the outgrowth of a new technology from the studies in physics and applied physics which have preceded it. A 45-kG, 11-in. bore superconducting magnet system has already proven itself operationally in a successful experiment in high energy physics as part of a liquid helium bubble chamber system which was used to take more than 400,000 pictures of interactions involving K- mesons with helium. Small bore 100-kG magnets are now available commercially. A 4-MJ magnet of total length 10 f t , bore 12 in. and transverse central

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field 40 kG has been successfully operated as a demonstration model to prove the feasibility of large superconducting magnets for magnetohydrodynamic power generation. Many medium size superconducting coils have been operated successfully. A 20-kG, 14-ft i.d. hydrogen bubble chamber magnet with a conventional iron return path has been designed and is to be operational in 1969 as part of a $17,000,000 bubble chamber facility. Ambitious programs for the design and construction of substantial superconducting magnets of all shapes and sizes and a t magnetic fields up to and beyond 150 kG are under wa,y. The supermagnet competes favorably with conventional magnet systems in their own limited field range and offers the potential of great economic saving, unprecedented long-term stability, high average current density, greater safety and greater reliability. It is important to appreciate that there is an operating cost to contend with in using supermagnets, even if the power dissipation in the magnet is zero. This is due to the fact that between 500 and 1000 W of room temperature power is needed for each watt of power dissipation or heat input from the surroundings a t 4.2"K. This power either is required to operate a closed cycle refrigeration system or represents a certain loss rate for liquid helium if a liquid helium reservoir is used. Superconductors appear to have 0 resistance to the passage of electric currents below a certain transition temperature which at the present time is in the temperature region below 18°K. Supercurrents can be induced in bulk specimens by external magnetic fields or can be caused to flow by passing current from a power supply through the superconductor. Superconductors can, therefore, be wound into various configurations and used as electromagnets if a suitable low temperature environment and power supply can be provided. A superconducting magnet system, therefore, consists of a superconducting coil with or without ferromagnetic shielding enclosed in a favorable low temperature environment and energized by a suitable electrical power source. The system is usually instrumented so that various important parameters such as magnetic field or coil current are indicated or recorded. Protectmionsystems are usually incorporated to prevent damage to the windings if the system is driven out of the superconducting state. Techniques are available to switch the circulating currents into the persistent mode and so create a permanent magnet system. The object of this paper is to present an up-to-date survey of the present state of superconducting magnet design and development after a preliminary introduction to the language of the subject in a simple account of the ideas underlying the basic theory.

388

CHARLES LAVERICK

11. THECONCEPTS OF SUPERCONDUCTIVITY

A . General Ideas (1) Two main types of pure superconductor exist. The class I materials are driven out of the superconducting state at low magnet field and support low current densities. They exclude flux a t low fields. The class I1 materials remain superconducting at high magnet fields but support low current densities at the high fields. They exclude flux at low fields, allow flux to penetrate up to very high fields, and then, when superconductivity is quenched in bulk specimens, permit surface superconductivity to persist to much higher fields. Under certain conditions, type I materials can exhibit type I1 properties and vice versa. The magnet materials are a subdivision of the class I1 materials and consist of class I1 materials with various types of inhomogeneity or impurity. These materials can support high current densities a t very high magnetic fields. Some prefer to distinguish between pure class I1 materials and the high-field magnet materials. Class I and class I1 materials have reversible magnetization curves, which implies that they can be used with alternating currents without energy dissipation due to ohmic losses. Impure class I1 materials exhibit hysteresis, and energy is dissipated in them when alternating currents flow. They are not suitable for general use a t high alternating current densities, but may be used in certain conditions. The property of superfluidity appears to be a feature of all quantum liquids which obey Bose statistics. Many Fermi liquids form particles of the Bose type at sufficiently low temperatures. This is described as a condensation in momentum space since the total kinetic energy of the system has been reduced to a level a t which a considerable number of particles have been reduced to the lowest possible quantum state where they have almost no momentum. The symmetry properties of the particles in such a state have great influence on the macroscopic properties of the liquid as a whole. The transition to such a condensed state occurs abruptly at a definite temperature. These properties occur in a number of quantum systems at low temperatures. Electrons in metals can form pairs (i.e., Bose particles) a t low temperature leading to superfluidity of the electron “liquid” and consequently to superconductivity since the liquid is charged. The system is in its lowest energy state a t 0°K and is characterized by a set of long wavelength quanta or phonons which have an energy proportional to their momentum. It is now becoming clear that it is the rule rather than the exception for metals to become superconducting at sufficiently low temperature (2).

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More than 300 metals and compounds were found to be superconductors in 1965 alone. I n the temperature region between 0°K and T,, the critical temperature a t which condensation of a substantial number of electron pairs occurs, two completely different states coexist: the superfluid state and the normally conducting state which is formed by the electron gas of conduction electrons. This combined state is the region which is investigated experimentally and its properties are influenced greatly, as may be expected, by both fluids and by temperature. The superfluid is thus a macromolecule which extends over the whole volume of the system and is capable of motion as a whole. Condensation of the electron pairs is complete a t 0°K and all electrons participate in forming this superfluid although only those electrons near the Fermi surface have their motion affected appreciably by the condensation. The electron pairs overlap each other considerably in space and there are also strong pair-to-pair correlations in addition to the correlations between each electron of a pair. These correlations are responsible for a n energy gap in the excitation spectrum of a superconductor, from which, as a consequence, many superconductor properties follow. The surface energy between the superconducting and normal laminar planes is positive in type I materials. I n type I1 materials this surface energy is negative. As the magnetic field is increased, the pure and impure type I1 materials exclude flux up to a magnetic field H,, a t which they assume a mixed state in which a large number of normal filaments in regular triangular array are arranged parallel to the external field and are surrounded by superconducting materials. This mixed state exists up t o a magnetic field H,, at which the superconductivity in the body of the specimen is destroyed. Surface superconductivity continues to exist up t o a magnetic field H,, at which the specimen becomes completely normal. The total flux associated with each normal thread is quantized in units of c$o = hc/2e 2 X lo-’ G-cm2. The number and density of normal threads increase with increasing magnetic field. Vortex lines of superconducting electron pairs are created along each normal thread and these are arranged in a triangular array since this system has the lowest energy. The condensation into the superconducting state a t the critical temperature takes place with no change in the free energy of the electrons associated with either the superconduc.ting or normal states. A discontinuity in the spec.ific heat owurs in a superconductor a t the critical temperature and this is indicative of a change of state. By analogy with the change of state from gas to liquid, we may say that the latent heat of the system does not change a t the critical temperature. Such a change of state is known as a second-order phase transition.

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B . Historical Outline In 1911, Kammerlingh Onnes in Leiden observed that the electrical resistance of lead and mercury vanished at temperatures close to the boiling point of liquid helium (4.2'K) at normal atmospheric pressure. He concluded that the metals had become perfect conductors, and the phenomenon became known as superconductivity. Onnes immediately realized the possibility of using this superconducting property of metals to produce electromagnets with zero power dissipation, but discovered that superconductivity in these materials was quenched a t low current densities or low magnetic fields. Applying Maxwell's equations for a perfect conductor to the behavior of a superconducting sphere or cylinder in an external magnetic field, one could predict the magnetic properties of the specimen on the assumption that its conductivity was infinite. Meissner (1933) found that this was not the case. The transition of the superconductor was found to be reversible, and it excluded flux if cooled down in the presence of an external magnetic field. H. and F. London proposed a phenomenological theory (1935) based upon the assumption that the superconductor acted as a perfect diamagnet, which explained the flux exclusion in bulk superconductors and predicted a penetration depth (-los6 cm) for the magnetic field which was later discovered experimentally. The predicted penetration depth was not correct for the softer class I superconductors such as lead but was correct for class I1 superconductors such as niobium. Pippard (1953), on empirical grounds, proposed a generalization of the London equations which successfully predicted the penetration depth for type I superconductors. His theory is based on the idea of a coherence distance, (, between wave packets (electrons in this case). He suggested that the range of order of the wave functions of the condensed superconducting phase extends over relatively large regions of space (-10-4 cm) in pure materials. Consequently, by these considerations the current density a t a point is determined by the field in a region of -10-4 cm surrounding the point. ( is a parameter approximately equal to the mean free path of the electron for impurity scattering. A complementary explanation of the phenomenon of superconductivity to that advanced by F. and H. London was based upon thermodynamic considerations by Gorter and Casimir (1934). The phenomenological thermodynamic treatment is based upon the assumption that the normal and superconducting states are distinct phases each of which has free energy associated with it. In zero applied field, the normal to superconducting phase transition for a material is of second order since there is no latent heat but there is a discontinuity

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in the specific heats for the two phases at the transition temperature, T,. The energy difference between the normal and superconducting phases eV/atom. Between 0°K and To,two fluids a t 0°K is of the order of are present in the material, that due to the normal conduction electrons and that due to the superconducting electrons. I t can be shown that these two fluid theories are based on energy gap models in which the electrons which are condensed into the superconducting phase at OOK require a finite energy E kT, to excite them. The energy gap models have led to theories which successfully explain many of the experimentally observed facts in superconductivity and thus point the way to considering the phenomena on a microscopic scale. These microscopic theories have to explain why there are few low lying excited states in the superconducting phase. Coulomb and magnetic interactions between electrons and electron-phonon interactions have to be considered and it has been shown (Bardeen and Pines, 1955) why the last are more important. It was soon realized that a superconducting sphere in an external magnetic field would have a critical field, H,, at its equator for which superconductivity should be destroyed when the applied field was +He. It would be contradictory to assume that the equatorial region should become normal in this case. Consequently Peierls (1936) and F. London (1936) suggested that, when this situation arose, some regions of the specimen would be subdivided into a small scale arrangement of alternating normal and superconducting regions with B = H , in the normal regions and B = 0 in the others. This state is known as the intermediate state and its detailed nature was first suggested by Landau (1937), who suggested that the alternate regions had a laminar structure whose width is strongly influenced by the magnitude of the interphase surface energy between the normal and superconducting phases. Ginsburg and Landau (1950) proposed a phenomenological theory of boundary energies based upon a two-fluid model and the concept of an order parameter which gives the effective concentration of superconducting electrons, n.. The order parameter is assumed to change from a temperature dependent equilibrium value on the superconducting side of the boundary to zero on the normal side over a distance of the order of the boundary width. They assume that near the transition temperature, T,, the free energy difference between the superconducting and normal phases may be expanded in a power series of order parameter w , defined so that w = 0 in the normal phase and w = 1 in the superconducting phase a t T = 0°K. Thus, f ( T ) = F,(T) - F,(T) = ~ ( T ) f u +b(T)w2 * * * . The equilibrium value of W>S that which makes f ( T ) a minimum. To a second order w = -a/b and the equilibrium value of f ( T ) per unit volume of

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material is fo(T) = -HC2/S?r = -a2/2b. Ginsburg and Landau identify w with the square of an effective wave function defined so that is equal to the concentration of superconducting electrons, n,. With

+

(J

=

\+Iz

1+p = 1

at T = OOK, n, = no(+I2where no = n, at T = 0°K. = Xo/X(T)2where XO = penetration depth at 0°K Thus, wo(T) = since X2 a l/n,. The Ginsburg-Landau model enables us to treat the superconductor in an external field H , = H,. Several microscopic theories of superconductivity have been proposed through the years. Frohlich (1950) proposed a theory based on electronphonon interactions which would give a contribution to the energy of the system. A phonon is a sound quantum which bears the same relation to a quantized sound wave as does a photon to a quantized electromagnetic wave. Thus, it is a quantum of thermal energy associated with the vibration of the atoms in a crystal lattice. The electron-phonon interaction arises in the following manner. Electrons near the Fermi surface move much more rapidly than the velocity of sound, S. The emission of phonons is a “Cerenkov type” radiation or “bow wave” of a projectile in air moving faster than the velocity of sound. The disturbance is confined to a wake of angle S / V rad, where V is the average electron velocity. The interaction energy of two electrons is zero except in the wake where it becomes positive (repulsive) with a maximum value at the wake boundary. Cooper (1956) showed that if there is a net attraction between a pair of electrons just above the Fermi surface these electrons can form a bound state. This led to the ,Bardeen-Cooper-Schrieffer theory (1957), which is based on the hypothesis that at 0°K the superconducting ground state is a highly correlated one in momentum space, and the normal electron states in a thin shell near the Fermi surface are, to the fullest extent possible, occupied by pairs of opposite spin and momentum. The energy of this state is lower than that of the normal metal by a finite amount which is the condensation energy of the superconducting state and which a t 0°K is equal to HC2/87rper unit volume. The total energy difference between having all paired electrons and a single excited electron is a large multiple of the single pair correlation energy. The theory, therefore, yields an energy gap in terms of the single electron spectrum and has been very successful in explaining many of the phenomena of superconductivity. Shubnikov (1937) suggested that type I1 superconductors in the field region between flux penetration a t H,,and transition to the normal state H,, are in a mixed state of superconducting and normal regions. Abrikosov (1957) used the Ginsburg-Landau equations to analyze correctly the

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magnetic behavior of type I1 superconductors and to describe the mixed state properties of type I1 materials. The meaning of the GinsburgLandau equations is made clear by introducing a dimensionless parameter K defined by K~

=

(2e2/h2c2) Hc2XO4

where Xo2 = r n ~ ~ / 4 ? r e * ~ X ~ ~ .

The subscript 0 denotes the zero field value. Thus, the three parameters of the Ginsberg-Landau theory are K , xo, and H,, in terms of which various field and size effects can be expressed. H , is the bulk critical field. XO is the empirical penetration depth of a superconductor in the weak field limit. It turns out that K = X(T )/((T ).This equation states that both X and ( and therefore K are functions of T. K is temperature independent When K < near T,.For a certain magnetic field H,,, H,,/H, = @ K . l / f l the surface energy between superconducting and normal phases is positive and we have a class I superconductor. When K > l/G the surface energy between superconducting and normal phases is negative and the superconductor is of the second kind. Gorkov (1959) derived an expression for K which is valid when the electronic mean free path is much less than the coherence length (0. Abrikosov considers the mixed state to consist of a matrix of normal material penetrated by cylindrical cores of normal metal of radius (. T h e quantized supercurrent vortex encloses the flux centered along these cores whose large surface area ensures that a large contribution is made to the free energy of the system while the area taken up by the normal filaments is sma.11. As the field is increased, the density of normal filaments increases and overlapping of the current vortices occurs. It is not energetically favorable for two or more flux lines, each containing a single quantum of flux, t o coalesce into a single line. This combination of the work of Ginsburg, Landau, Abrikosov, and Gorkov explains the behavior of class I1 superconductors and the alloy superconductors. It is known as the GLAG theory and has been shown to be consistent with the BCS microscopic theory.

C . Some Basic Principles We can first consider some of the properties of superconductors in applied fields and then a t a later stage consider what happens when transport currents are passed through superconducting wires. For type I superconductors, the variation of critical field with temperature is given in Fig. 1, and the approximate relationship connecting critical field with

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temperature is parabolic, as shown by the relation:

H , = Ho[l

- (T/T,)2].

Above the critical field at a given temperature, the material is normally conducting. A common parameter is the penetration depth A. This is the distance from the sample boundary at which the field drops to l / e of the applied field at the surface of the superconductor. I n the case of type I1 superconductors, the penetration depth is known as the London penetration depth XL. This gives the variation of penetrating field with

FIG.1. Variation of critical field with temperature for a superconductor.

VACUUM

--

SUPERCONDUCTOR

-2

FIQ.2. Variation of field with distance near the boundary of a superconductor in an applied external field ho (h = h,,eZ’hL if 2 >> XL, h = 0. London Equation applies if XL >> €0).

distance from the superconductor boundary, when a slab of superconductor is placed in an applied field. The penetration depth for first kind superconductors is greater than that given by the London equation and the effect has been explained by Pippard who took into account the local variations of field due to the supercurrent. The variation of field with distance from a superconductor boundary in an applied field, Hot is given in Fig. 2. This penetration depth is of the order of lo4A so that in a bulk specimen of type I superconductor we can say that the internal field is reduced to zero in the presence of a weak, applied field because a system of surface currents is induced which cancels the internal field. This is a condition of minimum energy and, as mentioned previously, the behavior of a superconductor in all cases can be predicted by determining the condition for minimum system energy.

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It was meritioned previously that the onset of a superconducting state as the temperature is reduced is indicated by a discontinuity in the specific heat of the material. This discontinuity is shown in Fig. 3, where the variation of electronic specific heat for the superconductor with temperature is shown. At temperatures well below the critical temperature, the specific heat variation is approximately exponential; while beyond the critical temperature, the relationship is linear. Since the phenomenon of superconductivity is associated with obtaining, at each temperature below the

T-

'c

FIQ.3. Variation of electronic specific heat with temperature for a superconductor. F

I

-

T

I I TC

FIG.4. Variation with temperature of free energy/atom for superconducting and normal phases.

critical temperature, the state of minimum energy, and since both normal and superconducting states coexist at all temperatures above O'K, the energy of the system can be represented by the difference between the free energy, F,, in the normal state, and the free energy, F,, in the superconducting state. The situation is described in Fig. 4. At T = 0 , the condensation energy is given by F , - F, = (kBTo)2/EF per atom. Typically, EF lev, and kBTo lou3eV. We now consider another parameter necessary in discussing superconductivity. This is the coherence length, f , and is the distance over

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which the electrons in pairs can influence each other or the distance over which one set of such pairs can influence an adjacent set. A wave packet of plain waves whose momentum has an uncertainty, 6p, has a minimum spatial extent, ax, approximately equal to h / 6 p where h is Plank’s constant. The coherence length is therefore defined as to= hVF/?rA where 6 p = ~ A / V F VF , = pF/m, and A << EFis given by EIF - A < p2/2m < EF A. I n the simple metals or class I superconductors, XL is approximately 300 8, the Fermi velocity is greater than l o 8cm/sec, and t o = lo4 8 for aluminum. Thus, the London equation does not apply; in fact, for this case Xa = 0.62XL2t0,where X > XL and X << to. These are the Pippard superconductors.

+

FIO.5. Schematic of coexistence of superconducting and normal regions in the intermediate state for a superconducting slab in an applied field H o > Hc,.

I n the class I1 materials, such as the transition metals and intermetallic compounds, typical of which are NbrSn, VSGa, and NbZr, XL 2000 8, the Fermi velocity is -lo6 cm/sec, A is proportional to T,, and therefore EO 50 8. The London equation applies, in this case. A general idea of the intermediate state can be obtained from Fig. 5 where the behavior of the superconducting slab in the presence of an applied magnetic field H o > H , is shown (see Fig. 5). The bulk material splits up into laminas of superconducting and normal regions in which the field in the superconducting regions is zero and in the normal regions is H,. The normal regions are much smaller in size than the superconducting regions and both are much less than the thickness of the specimen. This introduces another energy term at the boundaries between the superconducting and normal regions as the flux has penetrated the superconductor, and the energy of the system is greater than that in the purely superconducting state by an amount equal to the

-

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aggregate surface energy a t the boundaries. It is these surface energies which are positive in class I superconductors. In the case of a wire of class I material, in a magnetic field carrying a current, I, more than the surface critical current, the arrangement of superconducting and normal phases is as shown in Fig. 6. The intermediate state core has a radius, R, less than the radius, a, of the wire, and the surface layers are normal.

FIQ.6. Intermediate state condition for wire of radius a carrying a current I h ( R ) = H,.

-4rM

t

/

< I,,

/

H

FIG.7. Magnetization of Types I and I1 superconductors.

It is customary in describing the various forms of superconductivity to consider the magnetization of the specimen. The magnetization curves are a function of the shape of the specimen. I n the case of ideal Oype I and type I1 materials, the magnetization curves are said to be reversible, which means that no energy is dissipated in the superconductor in taking it through a complete cycle of magnetization. The induction external to the sample is given by Be = He 47rM,where M is the magnetization outside the sample. Thus, Be = H external H sample.

+

+

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CHARLES LAVERICK

This is equivalent to saying that the interna.1field can be represented by a magnetization per unit volume given by (-1/4n)HI. The curves for long cylinders of types I and I1 materials are shown in Fig. 7. The induction in flux per square centimeter plotted against applied field for cylinders of types I and I1 superconductors is as shown in Fig. 8. The phase diagram for a long cylinder of Type I1 superconductor is as shown in Fig. 9; thus on the H T surface we first have a complete Meissner effect followed by the Schubnikov phase followed by surface superconductivity.

Hcl tic2

-H

FIG.8.Variation of induction with applied field for types I and I1 superconductors.

-T

Tc

FIG.9.Phase diagram for a long cylinder of type I1 superconductor.

The structure of a Type I1 vortex line is shown in Fig. 10. I n Fig. 10a, the flux is shown threading the normal region parallel t o the 2 axis and is quaatized according to the relationship 4 = $h du = ~40,where 4 0 = ch/2e = 2 X lo-' G-cm2. The vortex of current circles the flux and circles the norma.1region as shown. The situation is plotted in Fig. 10b where the variation of magnetic field H with radius is shown, and it can be seen that the flux falls off to l / e of its value within a distance X which is approximately equal to the radius of the current vortex. When the number of superconducting electrons per cubic centimeter is plotted against radius, the situation is as shown in Fig. 1Oc. At the center of the filament there are no superconducting electrons since it is a normal filament, but a t a distance equal

SUPERCONDUCTING MAGNET TECHNOLOGY

"*

t

W (C)

399

I

2c

FIG. 10. Structure of a type I1 vortex line showing (a) Current vortex due to quantized flux; (b) Distribution of field with radius in the vortex; and (c) Distribution of superconducting electron density with radius.

FIG. 11. Triangular distribution of vortex lines in a superconducting slab gives minimum energy.

to 25, the density of superconducting electrons is back to its maximum value. Thus, the manner in which 5, X, a n d j are associated with the vortex line is demonstrated. I n fact, it has been shown that the arrangement of vortex cores in the bulk superconductor which gives minimum energy is that for which the lattice is triangular (see Fig. 11). The density of vortex states increases with applied field until they overlap.

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The impure type I1 materials contain inhomogeneities in such forms as dislocations in the crystal lattice or impurities in the material which act as pinning sites of normal material in which bundles of flux can be trapped. When transport currents flow in these materials, the vector product of current j and flux B is the Lorentz force j X B which tends to sweep the flux out of the material. Consequently, in pure materials, low current densities suffice to eliminate the superconducting phase; while, in impure materials, the pinning site energies have to be overcome before the flux bundles can move to a new position. Higher current densities can, therefore, be supported at a given magnetic field and the materials are of value in high-field magnets. Energy is dissipated in the material as the flux moves through it and this results in local temperature increases. I n the impure materials, the flux bundles move suddenly and discontinuously to a new position as the Lorentz force reaches some critical value. The momentary temperature increases can be large enough to switch the material out of the superconducting state. The relative severity of the flux jumps appears to depend upon the distribution of the pinning centers; the magnitude of the pinning force; the current density, temperature, shape, and volume of the specimen. Bean (3, 4 ) has developed a phenomenological theory of flux penetration in hard superconductors which premises that there exists a macroscopic superconducting current density &(€€)that a hard superconductor can carry and that any electromotive force, however small, will induce this current to flow. This has led to an understanding of the hysteretic behavior of high-field superconductors and to the mechanism of flux penetration in the superconductor. Several workers have extended these ideas to the problem of highfield magnet wires carrying current in the presence of an external magnetic field. Yasucochi et al. (6) and Livingston and Schadler (6) have independently analyzed theoretically this problem on the basis of supercurrent vortices around flux threads; and the former authors have explained quantitatively for Nb 25% Zr the reduction of magnetization due to the external current and its dependence on the history of the application of current and field. They show that, in the mixed state, the external current brings about a particular state of the critical distribution of flux threads to allow a finite total transport current. The spatial variation of magnetic induction, B, is determined by a net force, F,, resulting from the repulsive interaction between a flux thread and its neighbors counteracted by a pinning force, F,, of inhomogeneities acting on the flux thread. F , is due to the magnetic pressure (B2/8r)/(unit volume), where K is large and is given by F , = -d/& (B2/87r). A critical distribution of flux threads occurs when the two forces balance so that F , = F,. Applying

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Maxwell's cquation yields curl B = ( 4 ~ / c ) j , where , j , is the supercurrent density, to the equation for F,, we have F , = (l/c)j8B, so that the critica,l pinning force is the reverse of this.

111. MAGNET MATERIALSAND CONDUCTORS

A . General Magnet Development The following discussion is intended to give some idea of recent developments in chronological order prior to discussing materials characteristics, and magnet problems. Superconductivity in Nb3Sn with a T, of 18.4"K was reported by Matthias et al. ('7) in 1954 after the announcement of the discovery of superconductivity in NbZr alloys in the previous year (8).The announcement of the high-field properties of Nb3Sn in 1961 led t o an enthusiastic attempt in many quarters to exploit the discovery by making electromagnets which require no power dissipation. The excitement and rapid progress in that year can be discerned by a study of the Proceedings of the 1961 Conference on High Magnetic Fields (9). Yntema (10) first wound a high-field superconducting coil which attained 7 kG using cold worked niobium; a similar coil was wound by Autler (11)who achieved fields of between 4 and 10 kG and who was the first to show that superconducting windings could be used effectively in an iron magnet; and a further advance was made b y Kunzler who showed that 15 kG could be attained in a coil of cold worked molybdenum rhenium wire ( I d ) . At the 1961 meeting on high magnetic fields, a 0.15-in. bore, 55-kG NbZr solenoid was described by Hulm et al. (9a) a0.2-in. bore, 59-kG NbZr solenoid was announced by Hake et at?. ( g b ) , and Kunzler mentioned that a 0.24-in. bore NbsSn magnet had been operated a t fields up t o 70 kG without exceeding its capability. A disadvantage of the NbaSn was th at the conductor contained a powdered core surrounded b y a niobium tube reinforced with a Monel metal shroud. The coil had to be wound, and a final step was to create the brittle Nb3Sn in situ by reacting the mixture of niobium and tin powders a t high temperature for several hours. Consequently, any error in fabrication or design led to loss of the material. Small coils wound from this conductor had the same current-carrying capacity as short samples of the conductor tested in an equivalent external field (13). Niobium, zirconium, and then, later, the niobium titanium alloys when they were produced had lower upper critical fields and temperatures but they were ductile and required no extra heat treatment. Consequently, most early coils were made from these materials. Unfortunately, it was found that the current-carrying capacity of these conductors was considerably less than that anticipated from short sample tests and ap-

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peared to decrease even more as the coil size was increased. These were known as degradation effects (14, 15). In spite of this disappointment, the usefulness of the smaller bore magnets for research became immediately apparent as they were compact and easily obtainable, and required simple power supplies. Complicated conventional coil or magnet systems took longer to obtain. The number of investigations involving studies using magnetic fields began to increase rapidly as investigators who would normally not be able to obtain such facilities because of time, price, or space availability found a ready source available. Studies of the behavior of short samples of material in steady and varying magnetic fields revealed which parameters had most effect on current-carrying capacity (16) and this was confirmed in small coil studies (17). Thin coatings of copper were found to enhance the currentcarrying capacity of coils (18) as did the apparently contradictory approach of encapsulating the turns in thermally insulating epoxy resin (18).The use of a normal conductor surrounding the superconductor was incidental in the “Kunzler” NboSn solenoids which performed as anticipated from the short sample performance ( I S ) . I n retrospect this should have convinced us to copper coat NbZr immediately. The use of interleaved secondaries of copper foil between layers was also found to enhance current-carrying capacity. A problem with the early coils was the need to protect the coil from destruction on transition from the superconducting to t,he normal state at the highest attainable magnetic field and current (19). This was equivalent to injecting a normal region of a few ohms per foot of winding into the conductor. The coil then discharged itself into this region which then spread rapidly throughout the winding. Uncontrolled transitions could therefore lead to the dissipation of too much energy in a short length of conductor with its rapid destruction and incidental destruction of the coil. Alternatively, the high voltages generated during the transition led to destruction of the coil insulation system and again of the coil itself. Stekly was the first to explain the transition mechanism in coils theoretically (20). Techniques to control this discharge were the use of interleaved secondaries to absorb some of the energy on discharge and increase the discharge time constant during the transition period, the use of shunting between each or several layers of the winding so that most of the coil current would be carried by the shunt if a coil section should become normally conducting, and finally the use of thin coatings of copper around the conductor to slow the transition by decreasing the normal resistance of the conductor because of the shunting action of the copper (17, 18). One or all combinations of these systems were used as coils increased in size. The stored energy of the system was eventually removed as heat by

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boiling off liquid helium as the winding temperature returned to 4.2"K after a transition. About 300 liters of liquid helium for each transition at full energy was evaporated during tests on the largest coil to be constructed in this way, and its final test required more than 1500 liters of liquid helium (21). In addition to the prohibitive cost of such tests, the sudden evaporation of each liquid liter of helium results in about 700 liters of gas at 4.2"K. Consequently, the evaporation of large volumes of liquid necessitates attention to the cryogenic system to control and limit pressure increases on discharge. Magnetic field biasing suppresses flux jumps (Z2-24). The bias level required depends upon the thickness and type of superconductor. It appears to be about 25 kG for Westinghouse 0.010-in. diameter, HI-120, NbTi material, and as high as 90 kG for some types of RCA NbtSn. Thus at higher magnetic fields, without the use of copper shunting material, it is possible to attain the short sample characteristic when the material is used in coils. Magnetic field stabilization gives the highest attainable current density in high-field coils; but it must be used with earlier methods of coil and conductor protection since the method affords no protect.ion for coil or conductor when a transition to the normal state occurs. The introduction of stranded cables of copper-coated superconductor at Argonne, with or without extra coatings of copper, and their use in coil designs where the conductor was well cooled and firmly held eliminated most of the effects observed in degraded coils (22, 26, 26). These early cables contained ,approximately 44 % of electroplated copper around the superconductor; since this is the maximum, it is convenient to electroplate around a 0.010-in. wire. The addition of indium impregnation increased the amount of normal material to about 60% of the total cross section. The impregnated cables with 6 superconducting strands cabled around a central copper strand of equa.1diameter were even more stable, at some expense in current density, and contained about 70% copper. The use of additional strands of copper around the superconductor was found to result in magnets which could be operated with normal sections in the superconductors without these normal regions spreading (22, 26,26). The normal region could be eliminated by reducing the coil current. The all-metal coil designs were even more stable since they resulted in higher heat transfer characteristics and even better shunting. One wrap of equal diameter copper wires around a 6-strand Nb 25% Zr cable sufficed to give stability and the short sample characteristic in a well cooled, all-metal design. The amount of copper in the conductor cross section was approximately 86%. A 7-strand cable of this type with 84% copper in the cross section and a length of 20,000 ft was then used

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CHARLES LAVERICK

successfully in the first stable S i n . i.d., 17-kG split coil system designed and operated at Argonne. The AVCO group applied this principle to the stabilization of heat treated NbZr material by increasing the copper content for a highly stable system to more than 95% (27, 28) in a coil which generated 42 kG in a length of 13 in. and bore 54 inches. The average current density in this magnet was about 3000 Amp/cm2. This construction is so stable that the superconducting transition is removed at the same winding current as is necessary to induce it. The copper shunts the additional current without changing the local temperature at that part of the winding. More recently we have been able to stabilize heavy section NbTi superconductors a t Argonne (approximately 0.l-in. diameter) and have operated a stable coil a t more than 9000 Amp/cm2 average current density in the over-all cross section in fields in excess of 55 kG (29). These conductors will facilitate the design, construction, and operation of large high-field magnets up to 100 kG. We have now reached the stage where the principles developed in cooling and ventilating conventional magnets and cryogenic magnets and those developed to provide maximum strength in high-field copper and aluminum coils can be applied, with suitable modification, to superconducting coils. Stable magnets are desirable where they can be used, but unstable systems with enhanced current density will also be used with more confidence as operating experience is gained. Consequently, workers in the field are concerning themselves with problems in heat transfer, predictability of performance, convenience in operating, and applicability to large engineering installations. The technology is beginning to mature.

B . Materials Characteristics The foregoing general discussion leads to some understanding of the processes involved in superconducting magnet performance. Progress in the development of materials continues at a rapid pace and an appreciable number of commercial firms are now engaged in this endeavor. The plateau of critical temperature for superconducting materials at the moment is about 18°K. Some obvious lines of development are to try to obtain materials with higher critical temperatures, higher current densities, higher upper critical fields, greater stability, improved ac response, greater strength, improved ductility, and lower price. 0.010-in. diameter wire was first favored when NbZr alloys were used, since thinner wires were more difficult to handle and broke more easily in manufacture. It was found that higher current densities than could be

SUPERCONDUCTING MAGNET TECHNOLOGY

405

obtained with 0.010-in. diameter wire were obtained with these thinner materials as had been predicted and that higher current densities could also be obtained when they were wound in coils. Operating currents of 11 Amp, at 30 kG, equivalent to approximately 99 Amp, for 0.010-in. diameter wire, has been obtained at ANL (C. Laverick, 1963) with small coils of 0.003-in. diameter NbTi wire. Larger diameter wires were found to have reduced performance, giving approximately 100 Amp (equivalent to 25 Amp in 0.010-in. wire) in a 2-in. bore coil a t 20 kG (C. Laverick, ANL, unpublished). Present-day superconducting magnets use either NbZr, NbTi, or Nb3Sn as the superconducting material although some other alloys have been tried. Mitsubishi of Japan is producing a 3-element system, niobium zirconium titanium and is cladding wires of this type in copper to form 500-Amp conductors. The choice is governed by the magnetic field desired and the economics of the situation. The upper critical fields and critical 60 kG, T , 10°K; temperatures for these materials are: NbZr 120 kG, To 9°K; Nb3Sn 300 kG, T , 18°K. NbZr is NbTi cheapest a t 20-30 kG at present prices, and Nb3Sn is dearest, but the materials supply situation is highly competitive and the economics situation can change overnight. The transition from the normally conducting to the superconducting state for a superconductor is governed by the magnetic field, temperature, or current density. An excess of any one of the quantities or some combination of these is sufficient to cause a rapid transition. It is usual to discuss the current-carrying capacity of a superconductor in terms of its short sample characteristic. This is the current-carrying capacity of a short sample of the material tested in liquid helium and plotted as a function of the magnetic field (SO). These characteristics are normally obtained by increasing the current through the specimen a t given values of field until a transition occurs. They can be obtained by increasing the field through the specimen at certain currents or by varying field and current together. The characteristics obtained by these various methods may not always agree (16). The current-carrying capacities of typical commercial materials as a function of magnetic field at 4.2"K are given in Figs. 12a-12f. Current densities where quoted are for the superconductor and are not averaged over the total cross section of the conductor. Atomics International has recently introduced a new material, Ti-22 at.-% Nb, which is available in stabilized cable and strip. The superconductor is heat treated by a method similar to that described by Vetrano and Boom (51) and sustains useful current densities up to about 70 kG. The J-H characteristics for the material are shown in Fig. 12a. A significant advantage for this conductor, according to Atomics Inter-

-

-

-

-- -

406

CHARLES LAVERICK

national, is that the current density is independent of wire diameter; for example, one 0.05-in. wire can replace twenty-five 0.01-in. wires, with practical and economic advantages. Stabilized cables have been designed by Whetstone [Whetstone et al. (32)]according to: 12R = h AT S where I is the current, R the resistance/cm (copper), h AT = 0.3 to 0.8 watts,/ cm2, and S is the surface cooled/cm in the 200 to 2000 Amp region. The stabilized cables are supplied with spiral insulating wraps to ensure good surface cooling. This provides independent confirmation of recent ANL work, discussed in a later section, with stabilized magnets containing conductors of much heavier section provided by National Research Corporation. The characteristic of Fig. 12d is for a typical 0.090-in. wide RCA ribbon. Supercurrents in superconductive ribbon made by the RCA vapor deposition process are varied by controlling the thickness of the NbrSn deposit, the current being proportional to the amount of Nb3Sn. Larger vapor deposited NbtSn conductors (0.250 to 0.500 in. wide), which are designed to carry 350 to 1000 Amp at 100 kG, are now becoming available. The current-carrying capacities of NbZr wires are enhanced by heat treatment (33))but heat treatment durations in excess of 8 min a t 600°C for NbZr may cause embrittlement of the material even though the

I 0

10

I 20

I

I

30

40

I 50

I 60

H, Kgauss (TRANSVERSE FIELD)

FIQ.12a. Variation of current density with magnetic field for Atomics International Nb 22% at.-wt. Ti.

407

SUPERCONDUCTING MAGNET TECHNOLOGY

-

4 2 OK 0.010" in diam. wire

-

-

-

Kilogauss

FIG.12b. Short sample characteristicsfor superconducting copper coated 0.010-in. diarn Nb 25% Zr and 33% Zr (Supercon Division of National Research Corp.).

-0

20

40

60

80

100 i20

KILO -OERSTEDS

FIG. 12c. Superconducting NbTi short sample characteristics for 0.010-in. diam wire (Supercon Division of National Research Corp.).

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CHARLES LAVERICK

1

4;

- 0

-20 ww .. EE 5 .g -15 -10

100

-

O 0

-

-5

10 20 30 40 50 60 70 80 90 100 Ib o; o; o; 70 00 o; 100 Transverse magnetic field (HI- kilogauss

sb

sb

d% SS €:

? , ?:, :

5 " 2 2 % %

eO n

z; 0 ("e

FIQ.12d. RCA NbsSn 0.090-in. wide vapor-deposited ribbon characteristic.

TESTS AT 4.2OK

100 90 80 70 60 50 40

30

TRANSVERSE MAGNETIC FIELD (KILOGAUSS) SMALL TEST COIL DATA DATA POINTS ARE FOR 1/4"TO 3-1/2" DIAMETER BORE SOLENOIDS ALL AT 4.2OK

b COIL STILL

SUPERCONDUCTING

FIG. 12e. Short sample characteristicsfor brittle General Electric cables of NbaSn formed by heating 6-strand and 49-strand tin-coated 0.003-in. Niobium.

SUPERCONDUCTING MAGNET TECHNOLOGY

409

-0001

100,

u 40

20

FIELD,

60

00

KILOGAUSS

FIG. 12f. Short sample characteristics of General Electric flexible 0.5-in. wide flexible tape.

current-carrying capacity is enhanced. Embrittlement of NbZr wires has occurred with heat treatment, but, where this is carried out with uncoated NbZr wires in a vacuum we have found that embrittlement does not occur. The temperature of the superconductor has a profound effect upon its short sample characteristic (34). Momentary rapid increases of temperature occur in small local regions of a superconducting wire as the magnetic field or current is increased; and these increases are due t o sudden readjustments in the distribution of magnetic flux within the superconductor (35, 36). These flux jumps cause local hot spots which produce large changes in the operating characteristics of a superconducting wire and even result in premature transitions to the normally con-

Field

- H- k G

FIQ. 13a. HIT characteristics for normal Westinghouse Nb 2 5 % Zr (0.010-in. diam wire) (Courtesy of the AVCO Corp.).

410

CHARLES LAVERICK

- -

Field H k G

FIG.13b. H I T characteristics for heat treated Westinghouse Nb 25-% Zr 0.010-in. diam wire (560' for 1 hr) (Courtesy of the AVCO Corp.).

Field - H- kG

FIG.13c. H I T characteristics for 0.010-in. diam normal superconducting Nb 33- % Zr (Courtesy of the AUCO Corp.).

ducting state at low currents. The effect of temperature changes on the short sample characteristics of NbZr are given in Figs. 13a and 13b (34) for ordinary Nb 25% Zr and heat treated Nb 25% Zr wires. A comparison of the two curves gives an idea of the dramatic effect of heat treatment of the wires on the short sample characteristic. Similar curves for ordinary Nb 33% Zr wire are given in Fig. 13c (34).

411

SUPERCONDUCTING MAGNET TECHNOLOGY

Some idea of the degradation which may occur when heat treated NbZr wires are used in unstabilized coils can be gained from Fig. 14 where a hairpin of heat treated wire has been tested under two extreme conditions (16). Similar instabilities are observed in heavy section superconductors. The high current characteristic was obtained by the usual

\I

60-

8

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0

50-

Sample 1 -Early Experimental Wire

.

W v)

X

Swept Field 5 KglMin -Auto Sweep

0

Fixed Field ZU AmpslMln -Auto Sweep

Sample 2 -Later Experimental Wire Swept Field 15 KgfMin -Manual Sweep

-

Ip Fixed Field Approx.

Manual Sweep

ZW AmpslMin

-

Linde Nb58kZr Wire, 0.010" Diameter.

-

2nd Quench 83 Amps. 1st Quench

10 0

5

10 15 20 25 MAGNETIC FIELD-KI LOGAUSS

30

35

FIG.14. Degradation in the short sample characteristic of a heat-treated Nb 25-% Zr wire under swept field conditions, as compared to fixed field conditions.

short sample test technique of holding the sample in a fixed magnetic field while increasing the current gradually until the wire was driven into the normal state. The degraded characteristic was obtained by fixing the wire current a t a given value and increasing the external magnetic field surrounding the specimen. This type of behavior can occur in magnets since the magnetic field is swept through the conductors as the current is increased and is usually much worse than shown for the short sample. The effect is due to the energy liberated as the flux penetrates the super-

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CHARLES LAVERICK

conductor. It does not occur with all materials nor even with different lengths of the same material. Modern magnets, and large magnets in particular, are easier to fabricate if the conductors can carry high currents. An obvious solution to this problem is to use heavier section conductors and a great deal of presentday effort is being expended in this direction. 90

-

80 -

--

60 50 -

Fixed Field Rate 18 AmpslMln

70

Swept Field Rate 10 KglMin Westinghouse Nb2% -0.010' Dia.

8 40-

a I

cz 30-

8 W

a

5

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Holder F r e e

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v Fixed Field

0

b Swept Field

0

0 Swept Fleld -Current Reversed

0 Wire Current Reversed

x Swept Fleld

A

* SweptI Fleld -Field ReversedI 5

I

Flxed Field Fixed Field Reversed Field

-

Fixed Field-0-Full

I

15 20 25 MAGNET1C FI ELD-KI LOGAUSS 10

Value

I 30

:5

FIG.15. The effect of conductor movement on short sample characteristics.

Heavier section wires, even of lower current density than those of 0.010-in. diameter, were found to be more unstable in use and therefore to carry even less current than anticipated from short sample characteristics. This is due to the fact that superconductors are thermal insulators and to the fact that energy is liberated in location sections of the wire during flux jumps. The momentary temperature increases, therefore, tend to be greater in heavier section wires where the energy liberated in a jump is much greater. Good cooling in heavy cross sections can be obtained by choosing favorable surface-to-volume ratios such as are obt.ained in thin super-

413

SUPERCONDUCTING MAGNET TECHNOLOGY

conducting strips. Strip type conductors of heavy current-carrying capacity are being considered for larger magnets. Smaller solenoids have large field gradients whose directions change rapidly. Strips have not been considered seriously for many small coil applications because they are better suited to larger current coils and are in general anisotropic. The short sample characteristics for strips are a function of the angle between 100,

m

90-

0 Fixed Field I0 AmprlMin

00 -

NO Wire Coating Lighl H o t W

6 Swept Field 260 KglMin

Light HDtder

E Thin Wire Coal 01 OULO cement Heivy Mlder

Q u aCemsnl 01 ow0 cement

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Fixed Field Sequence:

Sweot Field 90 KdMin. Wir; i n Frozen ilycerln.

v 12 v

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Axbe H-0 I0

0

5

10

15 20 25 30 MAGNETIC FIELD-KILOGAUSS

35

40

FIG.16.The effect of sweep rate and thermal environment on the transition current for 2 Nb 25-% Zr specimens.

the magnetic field and the normal-to-strip surface. A recent flexible NbpSn strip has been developed which does not show this anisotropy. This is thought to be due to the reduced grain size of the new material. The effect of conductor movement in short samples exposed to changing currents and fields has also been established (16) and usually has an adverse effect on the short sample characteristic (Fig. 15). Similar adverse effects can be induced when the experiments are carried out in a poor thermal environment, such as is obtained if the superconductor is immersed in wax or epoxy resin (16) (Fig. 16).

414

CHARLES LAVERICK

I t is important to note that conductor movement can occur due to slight relative motion of component coils in a system as well as to relative movement of windings within a coil. Dramatic degradation effects can be experienced in either case where copper coated 0.010-in. diameter wires are involved. The effects increase as the wire diameter is increased and decrease as it is decreased. Extra copper reduces the effect due to movement (37, 38),but the use of heat treated wire with its enhanced short sample characteristic can cause degradation where enough copper has been added to stabilize the less sensitive conductor. Thus, the intelligent use of copper with heavy section superconductors, previously thought unworkable, can lead to satisfactory coils. The current density in the superconductor may or may not be reduced as the cross section is increased since there seems to be considerable latitude in the manufacturing process. In higher field, large coils, the use of heavy section conductors operating a t reduced current density may be advantageous in providing adequate strength in the superconductor without the need for auxiliary support systems.

C . Early Magnet Problems The design techniques used in early superconducting magnets are still of value since many laboratory magnets will be constructed in this way in spite of degradation effects. The current density in a 0.010-in. NbZr wire carrying 20 Amp is approximately 40,000 Amp/cm2. Small coils wound with such conductors can have almost 70% of the coil section filled with superconductor (I?’). This fraction, known as packing fraction A, is the ratio of superconductor cross section to total coil cross section. The average current density in a coil is j A where j is the current density in the conductor and in the above example is 28,000 Amp/cm2. It can be seen that while the current may:be lower than hoped for, this average P current density is considerable. I n fact average current densities greater than 40,000 Amp/cm2 have been obtained in small coils (I?’). When the current, and therefore the magnetic field, is increased in a coil, the flux jump intensity in the low-field region increases with current. In badly cooled conductors, which are not shunted, the energy liberated in some flux jump that occurs below the normal critical current for the short sample characteristic is sufficient to cause a temperature rise which is-adequate to drive that section of wire normal. TheFjdisturbance propagates quickly through the coil in times of the order of a few milliseconds for coils which involve a few hundred feet of wire (17).The normal state resistance of the superconducting alloys is of the same order as that of nichrome (a few ohms per foot) and the growth

SUPERCONDUCTING MAGNET TECHNOLOGY

415

of the transition is equivalent to injecting a rapidly increasing resistance into the center of the coil as has been explained previously. Appreciable quantities of energy ('an be switched out of the coil by sensing the onset of the transition and switching a short-circuiting resistor of appropriate vaIue across the whole coil. This reduces the internal protection problems. A recently suggested technique for larger coils of this type is to use a large number of inductively coupled short-circuited turns and to switch off the coil current immediately a transition is detected (59). The field energy is stored momentarily in the secondary circuits until the superconductor recovers and the coil current can be switched rapidly back into the superconductor. The premature transitions a t low current values due to the conductor movement (Fig. 15) were probably due to sudden redistributions of flux in the superconductor causing temporary local hot spots. The premature transitions a t low values of current in early coils and in multiple coil systems were often due to this effect. At Argonne, we were able t o increase the transition current of a split coil system progressively by improving the support system for the coils and windings so as to reduce relative movement of the superconductor with respect to the magnetic field. All early magnets could he driven into the transition region; if the current was increased beyond a certain level, the coil would then immediately discharge as previously described. Techniques for preventing electrical damage were adequate even for systems with considerable stored energy (21, 40). This transition current was usually well below the short sample value in the low-field region but approached the short sample value a t higher fields. Thus, the central 15,000 ft of 6-strand NbTi in the ANL 67-kG system operated on the short sample characteristic with 75 kG a t the wire. These coils are unstable since the appearance of a small normal region in a conductor was enough to cause the whole coil to discharge. It must be emphasized that very large numbers of coils of:this type have given satisfactory and reliable service in spite of this apparent disadvantage. All small bore 100-kG systems in existence operate in this manner. The 45-kG ANL bubble chamber magnet was of this type even though the outer coils were stable and this system operated in a trouble free manner for more than four weeks of continuous operation in a combined bubble chamber-high energy accelerator combination. Completely stable magnets have their place, but in view of recent preoccupation with these types, i t is fair to point out to designers that it is respectable and accepted engineering practice to design and operate systems within their prescribed limits, even where operation beyond these limits would be catastrophic.

416

CHARLES LAVERICK

D. The E$ect of Copper Coating on N b Z r Wire Copper coated NbZr wires were in fact found to carry more current in coils than uncoated wire at some expense in packing factor (17, 19). The central magnetic field in a solenoid is B = jXFG where F and G are geometrical factors dependent upon the shape of the system. Thus, the copper coating while enhancing j may reduce X so much that jX is less than foc some coils with uncoated wire. Reduced packing factors have little effect on larger coils, and the enhanced current-carrying capacity of the superconductor then reduces the initial cost of the system considerably. This is one of the main reasons for employing thick copper coatings around superconductors. The quality of the copper coating and of the electrical bond between the copper and the superconductor are important factors in determining the degree of stability achieved with a given coating thickness. The copper coating enhances the current-carrying capacity of the superconductor by providing improved cooling in the coil and by providing a momentary shunt path for the current if a flux jump should cause a small section of the superconductor to undergo a transition to the normal state. The additional cooling and shunting may permit the superconductor to recover and to operate a t a higher current than formerly. Coils with only an extra 0.00075-in. radial thickness of electroplated copper about the superconductor have transition times which are an order of magnitude larger than those without copper (17). As more copper is added, the propagation velocity of the transition region boundaries continues to reduce until it becomes zero and then negative. In this final case, any induced transition is rapidly damped out by the mass of copper and its electrical shunting effect. The quantity of copper required for any given propagation velocity is reduced as the heat transfer characteristics between the individual conductors and the liquid helium bath are improved, and as the purity of the copper, and consequently its electrical conductivity, are enhanced. Conductors whose thermal environment and copper coating are sufficient to give zero or negative propagation velocities are stable in the sense that they can be operated with normal regions in the superconductor without these regions spreading. The coil current is carried by the copper coating in the region where the superconductor is normal and power is dissipated in the copper. This heat is removed by the thermal environment about the superconductor. I n the most favorable case, each part of the winding is surrounded by liquid helium, and the copper coating will carry a shunt current without significant temperature increase.

SUPERCONDUCTING MAGNET TECHNOLOGY

417

Various figures are used for the heat transfer characteristics; a conservative value for a conductor completely exposed to liquid helium is 0.1 W/cm2 of exposed surface (41). A less conservative figure of 0.4W/cm2 has been suggested (33). The temperature rise in the copper in these cases is not excessive. When this value is exceeded, a vapor barrier is built up between the conductor and liquid helium and the heat transfer characteristic is immediately an order of magnitude worse. The conductor temperature rises quickly thus causing the normal region in the superconductor to spread and the magnetic energy stored in the field to discharge through the resistive portion of the conductor. The new large superconducting magnets utilize stable conductors (22, 25, 27, 28, 37). This saves liquid helium and prevents damage to equipment in the magnetic field since the stored magnetic energy in the coil is retained. In coils containing 1 MJ or less of stored energy, the conductor size is adequate to prevent damage during accidental discharge of the coil and no further protection system is necessary if it is sized on the basis of the above-mentioned heat transfer coefficients. Emergency conditions of operation may require the magnets to operate, at least temporarily, with a portion of the superconductor normal. The cooling passages in a large coil should be arranged to prevent the boiled off liquid from blocking any passage, a condition known as vapor locking. The advantages of closed cycle helium recirculation in such a system is obvious.

E. Cables and Multiple Conductor Configurations We have seen that separate superconducting strands can be copper coated with so much copper that, when the strands are wound into a coil, the propagation velocity of the transition region can be made negative and any disturbances are therefore rapidly damped out. Braided copper coated superconductors provide multiple parallel current paths, a greater amount of copper shunt protection per strand, more eddy current damping for flux jumps, and improved cooling of the superconductor. The heavy conductor is still flexible, and while damage to any one strand impairs the performance of the cable, it is not disastrous as would be the case with single conductors. The heavy current-carrying capacity causes a large reduction in the inductance for a given coil which thus facilitates charging the system and ensures th at designs with favorable thermal characteristics and built-in shunt protection can be considered if necessary. The problem of winding the coils is made easier since fewer turns are involved for a given magnetic field while the conductors are very strong and can be wound a t high tension to produce mechanically stable magnets.

418

CHARLES LAVERICK

More copper is necessary to fully stabilize a superconducting wire a t low magnetic fields than at high magnetic fields because of the higher current-carrying capacity of a given superconductor at lower fields. The ideal conductor should operate at the same current irrespective of magnetic field and, for a given degree of stability, will have an almost constant amount of copper. Thus, the average current density in a coil can be fixed and the cross sectional area of superconductor used can be graded to give the same critical current at any magnetic field. A magnet constructed in this way would require only one power supply and would use the minimum amount of superconductor. In practice, it is not yet possible to grade the amount of superconductor in this way, but the number of superconducting strands used in successive cable lengths can be varied. The current-carrying capacity is limited to that required for operation at themaximum magnetic field if only one length of conductor is used since the number of superconducting strands is then fixed throughout the cable length. This usually occurs in pancake-type constructions or in layer winding where only a single conductor length is used. Many types of multiple strand conductor have been fabricated and tested (37, 42, 43). These are shown in Figs. 17-18. Cable designs using 6 and 7 superconducting NbZr strands with no additional copper wraps result in unstable magnets of good current-carrying capacity (say 200 to 260 Amp a t 35 kG in 2-in. bore) and with average current densities of the order of 15,000 Amp/cm2, where good cooling can be attained with little loss in packing factor. Cables with extra wraps of copper are self-protecting in lengths of up to 10,000 f t when wound into larger coils. They are partially stable and can support normal regions in the superconductor with zero propagation velocity of the transition region if well cooled. They are unstable when fully insulated and badly cooled. They have operated in the stable condition in the ANL 18-in. i.d. coils a t about 5000 Amp/cm2 average current density with cable currents of up to 250 Amp. Average winding current densities are usually lowered although individual cable currents are increased as the amount of copper is increased. Such cables are satisfactory for use with heat treated NbZr strands whose short sample characteristic is now much higher than that of the non-heat treated material in the region below 45 kG. These cables offer considerable economic advantages with safe and reliable operation in large magnets since less high current conductor is required for a given number of ampere turns. Cables of the type shown in Figs. 17.4 and 17.5 produce stable magnets of lower current-carrying capacity than the heavier conductors when used in well-cooled coils. They are unstable when used in long lengths on fully

SUPERCONDUCTING MAGNET TECHNOLOGY

1-0.012"Cu

2-0.02O"Cu

1. S i x superconductinq strands

1-0.02O"Cu

419

I.

2-O.MO"Cu

.

Seven superconducting strands

1-0.010"NbZr 3-0.012"Cu 2-0.012"NbZr 4-0.020"Cu +Cu coat Two suoerconductinq strands

2 1-0.001"Cu

2-0.012"Cu

1-0.0159"C~ 3-0.0156"Al, 2-0.0159"Cu indium coated 4-O.OO8"Nichrome

5. Single superconductor strand

I

. ANL light-weight AC conductor fo stable magnet with Nichrome cable insulation

f

J.

o.o%Tm $c10.1301+

7. AVCO type with wires embedded i n strip

Heavy section superconductor recently stabilized i n ANL coils

0 Superconductor with copper wire

OCopper wire

FIQ.17. Typical wire type composite conductors: (1) 6 superconducting strands; (2) 7 superconductingstrands; (3) 12 superconductingstrands; (4) 2 superconducting strands; (5) 1 superconductingstrand; (6) ANL lightweight ac conductor for stable magnet with nichrome cable insulation; (7) AVCO superRenic strip with wire embedded in strip; and (8) heavy cross section superconductor recently used successfully in ANL coils.

insulated coils, but this type of winding is suitable for medium size magnets where currents of 100 t o 200 Amp are desired. The cooldown costs of large magnets are high and their weight is considerable when large amounts of copper are used. I n space or air-borne applications where full or partial stability is required, the use of copper can be a disadvantage. These difficulties have been recently overcome in the cable design of Fig. 17.6 which uses indium tinned, aluminum wires

420

CHARLES LAVERICK

braided around the individual superconducting strands (44). The particular cable shown uses a central core of braided copper for extra strength but its current-carrying capacity could be enhanced by using an extra superconducting strand with aluminum braided around it. The cable is

FIG.18.. Multiple conductors formed from strip type superconducting materials: (a) Typical-RCA ribbon; (b) Silver-coated Nba Sn ribbon; (c) Copper-stabilized silvercoated NbaSn ribbon; (d) Copper-stabilized multilayer conductor; (e) High-currentdensity unstabilized multilayer conductor; (f) Copper-stabilized multiple-strip configuration for high-field coils of smaller diameter (12 in. or less); (9) General Electric Nb,Sn tape; and (h) General Electric tape with copper shunt.

copper ,wrapped to give mechanical protection to the aluminum and uses nichrome wire braided around it as the insulation. Multiple superconducting strands can also be used in square or rectangular section conductors where the individual strands are rolled or

SUPERCONDUCTlNG MAGNET TECHNOLOGY

42 1

bonded into copper strip (25, 26). The performance characteristics for these conductor types are similar. A typical example of a well stabilized conductor of this type is AVCO supergenic strip (Fig. 17.7) (28) in which 9 strands of heat treated 0.010-in. diameter NbZr wire are rolled into a strip of copper of width 0.5 in. and thickness 0.060 in. Advantages claimed for this type of strip when used in large magnets are that its heat transfer characteristics are determined and it can be easily secured. Disadvantages are its lack of flexibility, and the difficulty of changing its dimensions or the number of superconducting strands in production. Available lengths of these composite conductors are usually short so that many joints have to be wound into the magnet. We have recently been able to stabilize heavy section (approximately 0.1-in. diameter) NbTi conductors a t Argonne (Fig. 17.8) and these offer the possibility of making satisfactory large 100-kG coils with this material. Strip type conductors may be inherently more stable because of the favorable surface-to-volume ratio and of the ease of obtaining good cooling. Some of the configurations being developed a t Argonne for use with RCA Nb3Sn ribbon are shown in Fig. 18. Sta.ndard strips (Fig. 18b) can be stabilized t o various currents by being shunted with different amounts of copper to make a single conductor as in Fig. 18c or multiple sandwiches of stabilized ribbon as in Fig. 18d. The higher field regions in field stabilized coil sections may require high current density windings in which case conductors of the type shown in Fig. 18e may be used; the number of strips being soldered together in the sandwich will depend upon the current-carrying capacity required and the bending radius acceptable. When the strips are displaced from the neutral axis of a composite conductor large strains can be induced a t low bending radii, and these must be limited to values which do not cause damage to the superconductor. I n smaller diameter coils, the construction of Fig. 18f is safer with these materials because of this induced winding strain. The 0.5411. wide, higher current Nb3Sn ribbons, which are now becoming available, will each carry 1000 Amp a t 100 kG and should provide much more economical conductors for these high current density windings. The flexible strip type conductors of Nb3Sn contain a very thin brittle layer of Nb3Sn. I n the case of the RCA material, the Nb3Sn layer is vapor deposited from the respective chlorides onto a stainless steel substrate which provides strength for the conductor. The new General Electric flexible type (Fig. 18g) contains a niobium core surrounded by a thin diffusion layer of Nb3Sn which in turn is covered with tin. Both types of strip ran be stabilized by the addition of extra copper to provide any degree of safety or protection that is required. Extra strength can be

422

CHARLES LAVERICK

provided in the RCA case by vapor deposition on thicker strips of stainless steel. Narrow tapes can be layer wound but wider tapes must be spirally wound into individual pancakes. Satisfactory high-field magnets have been operated by each company at fields in excess of 100 kG and the materials have been successfully used in many laboratories. The diffusion process for NbsSn is also being used by CSF in France to produce ribbons of various widths, currently and in., at thicknesses of 10 or 2 0 p for use in coil designs up to 200 kG. fJ are usually obtained in Low resistance contacts of the order of using these ribbons so that persistent operation usually involves field decay. There is a need to produce zero resistance contacts between lengths of Nb3Sn materials since this is one striking way in which it differs from NbTi and NbZr. The conductors under consideration for the large Brookhaven and Argonne hydrogen bubble chambers are much more conservative than those previously mentioned because of the large scale of the projects. Pancake-type windings of the order of 2 in. wide and 0.1 in. thick with adequate amounts of superconductor bonded in the copper are under consideration. The copper is adequate to support the induced stresses without further support and to conduct the proposed 2000 Amp coil current for reasonable periods of time to permit shutdown and coil discharge in an emergency.

+

+-

F. Considerations in the Choice of Bonding and Shunting Materials (37) Indium was chosen as an impregnating material for cable because its low melting point would have minimum effect upon the superconductor and because it is a malleable material. This gives a flexible cable which can be wound and bent without stressing the individual strands since the various strands can move in relation to each other. The high quality electrical bond between strands is maintained after relative movement of the strands has occurred. A disadvantage of indium is its high price. High purity metals were chosen rather than alloys since alloys have much lower electrical conductivities at low temperature than pure metals. Metals with high melting points with a great affinity for copper were not chosen because they would tend to dissolve the thin layer of electroplated copper around each superconducting strand and thus impair the main electrical bond between superconductor and copper. Cables which are not impregnated with normal metal are more difficult to handle particularly where strands of a more springlike quality are used. The copper coating tends to oxidize in time thus causing a deterioration in the quality of the interconnecting electrical shunts between strands with time. Cadmium was found to be satisfactory for individual copper coated

SUPERCONDUCTING MAGNET TECHNOLOGY

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strands but embrittled the material so that it became impossible to strand lengths of cable without a considerable amount of breakage. We have recently used lead with success on short lengths of cable and this represents a factor of 10 reduction in price over the indium for 99.999% purity lead. Tin is also a good material from the point of view of electrical conductivity, but it may powder and become weak at low temperature. Lead-tin solder has been used with success as an impregnating material with cable and has been found to give a slightly reduced current-carrying capacity in a 20-in. bore, 5-in. long coil using 1000 ft of cable. The solder acted as a better insulator than bare copper and resulted in coils with faster charge times. Since the amount of impregnating material in the cross section of a cable is a small fraction of the whole, high quality impregnants may not be necessary. The problem is being studied. The 0.001-in. thick electroplated copper coating on the 0.010-in. diameter superconducting wire has so far represented 30% of the price of the finished material. Strip type materials are cheaper to plate. The coating has been used to provide the basic high quality electrical bond between the superconductor and copper and has been accepted because it has been the most successful process so far. Other methods of applying normally conducting coatings to the superconductor in almost any thickness and quality at a more reasonable price are being studied. A great range of impregnating materials may be used. Poor quality copper was used in early cables because of its low price and easy availability. A measure of the quality of the copper is the resistance ratio between specimens a t room temperature and at liquid helium temperature. The resistance ratio is affected by the purity of the metal, its state of strain, and the magnetic field. The cross sectional area of the copper required decreases as the resistance ratio increases and thus the packing factor and average current density increases with increasing resistance ratio for a given degree of stabilization. Strains can be imposed on the material by winding, by relative contraction on cooldown to 4.2"K, and by the forces induced in the conductor by the magnetic field. There is little to be gained by using expensive high purity copper conductors where the strains and magnetic fields are high. In early cables we used copper whose final resistance ratio was about 20:l. A t 0 magnetic field, strain free OFHC copper can have a ratio of from about 300:l to 160:l depending on the oxygen content, while 99.999% strain free, high purity copper can have a ratio of about 2000 :1 in bulk and about 40% of this in wire form. Attempts are being made to improve the resistance ratio of the high purity thin wire copper which is degraded during the drawing process. At high magnetic fields of the order of 70 kG and 4.2"K the resistivity of high purity copper is worse than

424

CHARLES LAVERICK

that of OFHC copper because of the magneto-resistive effect (41). Aluminum is preferable to copper since its magneto-resistance coefficient saturates below 30 kG and is much lower than that of copper. It can be easily obtained in the highly pure state and can be annealed a t room temperature. The conclusions are that future superconductors will probably be coated with heavier coats of aluminum or copper using cheaper processes than the present electroplating process. The metal impregnants necessary to interconnect the coated superconductors electrically will thus have less stringent requirements placed upon their degrees of purity, melting points, electrical conductivities, and malleabilities so that large cost reductions will be effected. Heavier section superconductors can be used than were formerly thought possible with the new and heavy coatings of normal conductors. Aluminum coatings are best a t high magnetic fields where maximum stability or protection is necessary in the smallest space. Heavy section conductors for larger magnets may be required to absorb appreciable amounts of energy under emergency conditions and space is then rarely an important consideration. Cheap low purity copper with its high thermal capacity may be the best material for such a purpose and may even be coated with extra shunts of higher specific heat material, such as lead, where necessary.

G. Conductor Working Stress Electromagnets have a long history and many illustrious names are associated with their development. Consequently, it is not surprising that solutions to many of the problems of load and stress for a wide variety of conductor configurations with and without iron are to be found in the literature (46-47). Extensive bibliographies are to be found in Refs. 45 and 46. The radial, F,, and axial, F,, force components respectively are given by dF,/dV = B,B0/4r(ro - T i ) 4 and dF,/dV = B,Bo/4r(ro - ri)4 where

B., -B, are the axial and radial magnetic flux densities; B, is the magnetic flux density a t the center of the coil; ri, ro are the inner and outer coil radii, respectively; V is the conductor volume; a! is the ratio of outer to inner radius ro/ri; B is the ratio of coil half length to inner coil radius. Several techniques of calculation are available for the evaluation of

SUPERCONDUCTING MAGNET TECHNOLOGY

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various types of cylindrical systems since some are homogeneous, others are loosely wound, some externally supported, etc. Usually the results obtained in a given case are approximate. The relation suggested for hoop stress calculations on coils which are loosely wound is uo =

Bo2y(a!- y)/4?r(a!

- l)",

where uo is the circumferential unit stress in the conductor and y is the ratio of turn radius to coil inner radius. No special precautions seem to be necessary in winding and operating small or medium size, high-field magnets to protect them from the working stresses. This is certainly true for the types of NbZr, NbTi, and Nb3Sn in current use. NbZr is one of the strongest materials known with a breaking stress of more than 300,000 psi in 0.010-in. diameter wire form. The material may become more brittle with heat treatment. NbTi has about 30% of the ultimate stress of NbZr. NbsSn is very brittle but has been used most successfully in thin films bonded to substrates of high strength alloys such as Hastalloy. The maximum stresses in the various sections of the ANL 67-kG, 7-in. i.d. system (see Figs. 21 and 22) were within the operating stress levels for the material, and it was not necessary to design additional systems of mechanical protection. The windings withstood the high operating loads imposed by the use of the stainless steel mesh between layers which leads t o radial support by a system of multiple point contacts. I n very large magnets, the situation changes even a t low magnetic fields of 20 kG. The force on a single turn of conductor in a given magnetic field is proportional to Blr a t a given current density. Thus, in comparing the working stresses of a &in. i.d., 100-kG coil with a 14 f t i.d., 20-kG coil, we see that the larger coil has an operating stress level which is =5.6:1 greater assuming the current densities in the conductors to be identical and no other stresses to be imposed by the system of support. I n practice the average current densities in the coils may be equal b u t the current density in the superconductor itself will be higher a t low fields. Usually, in very large magnets in the 20-kG region, attempts will be made to operate the superconductor a t the highest possible current density so as to reduce the amount of superconductor required and therefore the cost,. The stress in the superconductor is proportional t o the current density in the conductor and thus a t high current densities in the superconducting material, it may be necessary to support the superconductor. Large magnet systems with working diameters of several feet have to be wound with superconductors which preferably do not move when energized. The differential temperature on cooldown is approximately

426

CHARLES LAVERICK

300°C. The stresses imposed due to differential contraction of the superconductor with respect to the coil support structure must be kept low. The stresses on both the normally conducting coatings and the superconductors must be kept low. Any strengthening material which is introduced to reduce the operating stresses on the superconductor when composite conductors are used must not introduce high stresses during cooldown. The great difference between normal magnets and superconducting magnets is that the latter have to be cooled down to 4.2”K or less. It is desirable that any differential stresses imposed on the system by these temperature changes should work to the advantage of the working conductor. Reinforcing techniques can easily fail in their purpose if they induce tensile stresses on the conductor during cooldown. I n general, it has been possible so far to wind various coil systems without considering these stress problems in detail, but it is obvious that the higher field systems will profit from the experience gained in refining ordinary electromagnets in the past four decades. When iron is used in conjunction with superconducting windings, the usual considerations apply. The problem can be simplified in small systems where the iron can be cooled down too, but in large systems it is necessary to hold the iron a t room t,emperature and to either devise cryogenic systems to take the loads between coil and iron or position the components so that image force systems leave the windings free of any resultant force. H . Pulsed Superconductors There is a natural interest in the ac characteristics of superconductors and, of course, in pulsed magnets where the field rise times are of the order of 1 sec since such magnets could be of interest for accelerator magnets of the synchrotron type. Niobium is of interest for power transformer applications since its magnetization losses are low and the maximum operating field is less than 12 kG. Some other superconductors such as lead-bismuth may also be of interest for such applications since their critical fields are of the order of 20 kG. The magnetization curves fol“superconductors operated within their limits vary with type (Fig. 7). -The magnetization curves for type I materials are completely reversible and thus they exhibit no loss under conditions of pulsed or ac operation. Unsuitable for magnets, they are suitable for linear accelerators where the operating fields are low, the frequencies high, and the resistive losses negligible. Circuit magnification factors or “Q’s” of lo9 have been attained with lead. Ideal type I1 superconductors have reversible magnetization curves

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and consequently negligible energy loss per cycle of magnetization. A most promising material is pure niobium, which in very thin films has an enhanced upper critical field. I t has been known for some time that thin films of superconductor will operate at higher magnetic fields than the thermodynamic critical field H , (48). Thus, H f i l m = H, 2 l/6 X/d where d is the film thickness of X the penetration depth. If we assume X = 500 b, d = 50 b, H,,= 2500 G, the above equation gives H f i l m = 100 kG (43). In addition to its use for magnetically and thermally operated superconducting switches, it promises to be of use in ac applications for magnetic fields up to and even beyond 20 kG. Power transformers of considerably reduced size and cost can be designed with this material so that more economical power stations and substations are possible. Opinions as yet appear to be divided as to the practicality of medium size superconducting transformers since loss rates would be high if iron cores are used and such losses constitute an appreciable fraction of the total loss. The ductile alloys NbZr and NbTi and intermetallic compounds such as NbaSn have nonreversible magnetization curves and exhibit high energy losses per cycle of magnetization. Such losses obviously depend upon the degree of field penetration. The energy dissipation in these superconductors has been computed to be much higher than that in copper at 60 cps. The losses can be reduced by reducing the superconductor dimensions. It has been computed that the magnetization losses would be reduced to an acceptable level for films of thickness 2 X loe4 in. (50).At Argonne, we have been successful in making material of this type but it has yet to be tested. As the films are reduced to sufficiently small dimensions in combination with copper, there is some possibility that drastic reductions in upper critical field will take place. The problem is being given further study. It has been shown that when an ac field whose amplitude exceeds a certain threshold is superimposed on a dc field regular flux jumps ensue, the threshold being inversely proportional to frequency. At low fresec and identical in quencies the flux jump duration is about 2 X amplitude and duration from 6.3 to 208 cps. These threshold fields caused the bulk sample temperature to rise by no more than O.l°K (61). Similar bulk temperature rises have been measured by other workers (62) a t ac amplitudes of 1 kG and a frequency of 300 cps. At low ac fields, surface currents flow as in soft superconductors with no change in the bulk flux and the losses are low. I n fact NbaSn has been used in rf cavities, and circuit “Q’s” of lo6have been obtained (63) at high frequencies resulting in important improvements in high frequency equipment.

428

CHARLES LAVERICK

Experiments on superconducting to normal transitions in Nb 25% Zr and NbrSn in fast pulsed magnetic fields have revealed that in the range of dH/dZ from lo9 to 10" Oe/sec, values of H,, are less than 25% of the static field values (54). This type of effect has also been observed a t low rates of dH/dZ. The value of H,, under these conditions for test increases with decreasing sample size with rapid increases below 0.04-cm diameter. Copper coated wires under the same conditions of test are heated far above the zero field transition temperatures. A plausible explanation for this lowering of the critical field would be that the flux movement in the superconductor results in high local temperatures a t points in the cross section and in a thermal avalanche which destroys the superconductivity in a manner suggested by Anderson and Kim (65). The final picture is not yet complete but it can be seen from the foregoing general comments that a possible path to the successful development of pulsed superconducting magnets and to high stability in magnets operated in the steady state is to develop thin film superconductors with a minimum amount of normal material. Whether or not pulsed highfield superconducting coils are desirable is another matter. Those preoccupied with room temperature techniques tend to overlook eddy current effects at low rates of rise at low temperatures. These effects become more important because of the rapid decrease in resistivity of pure metals a t the low temperatures. We have pulsed an 8-in. bore, 16-in. long, 6-kG magnet to full field in 0.1 sec, at an average rate of rise of current of 5000 Amp/sec. A small magnet of 1-in. bore was recently pulsed to 50 kG in 1 sec. (66). The losses were not measured in either case but were considerable in the first case where they were indicated by a great increase in helium boil-off rate, however, the results show that pulsed magnets using high-field superconductors can be operated if necessary even with conventional superconductors. Pulsed operation is required for synchrotron magnets and for rapid adjustments of the magnetic field of dc magnets. I n the first case, an AGS type magnet of 150-kG peak field would reduce the circumference of the recentIy considered 1000-GeV machine from 10 miles to 1 mile. Even a t equivalent magnet costs or complete installation costs such a size reduction may be worthwhile in convenience of operation. It must be emphasized that even if such a magnet were available, a complete system study would be required to determine the advisability of using such a device. Estimated costs for such a system are of the order of $lo9. Typical of the second case would be the adjustment of the magnetic field in a cyclotron where convenient operation requires the field change to occur in a few seconds. Such types of operation obviate one of the claimed advan-

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429

tages for superconducting coils which is the elimination of a n expensive power supply. The energy contained in the magnetic field of a large magnet is considerable. Thus, in a cyclotron magnet containing lo7J of stored magnetic energy, a 10% change in magnetic field in 10 sec requires a power supply capable of supplying lo6 J in 10 sec or 100 kW. Assuming reasonable impedance matching, a 9 MW supply would probably be a reasonable choice. If the magnet has to be switched to full field in 5 min, the energy to be delivered requires approximately 30 kw. I n the above case, the power limitation is imposed by the need for the 10% adjustment from full field in 10 sec. The power supply for suitable copper coils for such a magnet would be larger, but not by an order of magnitude. Consequently, the capital cost of the power supply arid its associated auxiliary installations is somewhat reduced but not completely saved by the choice of a superconducting magnet in this case. The case of the synchrotron magnet is even worse since the magnetic field energy has to be forced into the magnet in about 1 sec. Only a small fraction of the power supply losses are associated with the copper coil of a magnet such as the ANL 12.Ti-GeV ZGS synchrotron magnet whose power supply delivers 120 MW peak power, so there would be little power supply savirig in using a superconducting magnet fur such a device. More efficient storage systems have been proposed including zero loss supercapacitors or self-resonant dual synchrotrons in which one machine is a t maximum energy while the other is a t minimum energy so that each coil temporarily stores the energy from the other coil. The energy to be supplied by the power supply is still considerable and little capital savings result from the choice of supermagnet. In such cases other reasons for the adoption of a supermagnet are necessary.

IV. RECENTSIGNIFICANT MAGNETS

A . General Discussion A few recent magnets which have significance in the development of the technology since 1961 are described briefly in this section to illustrate the rapid rate of progress and some design tec4hniques and problems. The first small bore high-field coils were coristructed to gain familiarity with new materials in coils and to increase the maximum central field range as much as possible. The early experimental types have been mentioned previously. Magnets of this type were quickly accepted by those of the scientific. community able to carry out significant experiments a t moderate mngiietic fields of the order of 60 kG in bores of the order of 3 in. Large magnets were under design and construction and these proved

430

CHARLES L l V E R I C K

or disproved various techniques of design and construction depending on their performance. Cabled conductors (September, 1963) were developed and used in experimental magnets in bores ranging from 2 to 6 in. to evaluate cables and solve the various coil design problems which had arisen (25). Several significant 100-kG magnets of small bore were successfully constructed and operated in the 1963-64 period and these had significance in proving materials and design techniques while breaking the “psychological” 100-kG barrier for continuous operation (24, 57, 58).

FIG.19. Voltage current characteristic for the AVCO stabilized magnet.

The first large magnet to operate successfully and thus prove that large high-field supermagnets could be safely designed and operated was the ANL 67-kG, 7-in. bore coil system (October, 1964) (22). The outer sections of this system have provided a 45-kG, 11-in. bore working field for a 10-in. diameter helium bubble chamber system which recently concluded a successful series of experiments in a high energy particle accelerator system. This is the first use of a large superconducting magnet in a significant engineering installation where reliability is a paramount consideration and the loss of time and money due to magnet failure could be disastrous. The first stable superconducting magnet to use heat treated NbZr wire embedded in a substantial copper strip was designed, constructed,

SUPERCONDUCTING MAGNET TECHNOLOGY

43 1

and operated in late 1964 (Fig. 19) by Stekly and Zar (28). Predictable in performance, it served as a prototype for the world’s largest superconducting system to operate successfully to date, which is a large scale model MHD magnet (59). Although this magnet is not to be put to practical use, it reinforces the impressions gained b y the operation of the ANL bubble chamber magnet that large magnets can be built and operated, and it advances the technology a stage further by storing almost an order of magnitude more field energy (-4 MJ) and being capable of operation in the fully stabilized mode. This gain in reliability and ease of operation is obtained at some cost in average current density. Several smaller bore high-field magnets in the range above 100 kG and some model quadrupole focusing magnets with high-field gradients have also been successfully operated as has a superconducting lens in a working electron microscope. The consequence of these successes and of the successful operation of a number of large magnets has been that ambitious plans for the development and construction of a wide range of coil systems have been formulated in laboratories throughout the world.

B. The A N L 67-kG, 7-i7a. Bore Coil Sgstem The first 18-in. i.d. coil of the system was developed after a study of several 6-in. i.d. cabled magnet designs. Incorporating the advanced design ideas developed a t ANL, it was self-protecting, could sustain normal regions without self-propagation of the normal region, was simple to construct and to prepare for test, and incorporated an extra wrap of 0.012-in. copper around the superconductor. The conductor current of 250 Amp or 36 Amp strand a t 20 kG was higher than that obtained in most small bore magncts a t that time. The first S i n . i.d. system was a split coil system made up of 2 of these coils with a gap of 1 in. between them. The compressive loads were carried by the superconductor. Operating with 1 coil, which had been accidentally damaged, the system developed 17-kG central field. This demonstrated th a t the problems of large, split coil systems could be overcome and that stable magnets could be constructed. The way was now clear for the development of a multiple component system in which the outer assemblies were designed for stability and high currents in each superconducting strand while the inner sections were designed to contain the higher loads and operate a t higher current densities with greater packing. The separate coils were developed and there was sufficient confidence in their design and construction for the package to be assembled without individual tests of the coils. The cable types used in the system are shown in Fig. 20, their locations are shown in Fig. 21, and a general view of the nested magnet system is given in

43 2

CHARLES LAVERICK 7 INCH I.D. WINDING 6 STRANDS-HI 120 (.Dl0 DIA. COATED WITH ,002 RADIAL THICKNESS OF COPPER) WOUND ON COPPER CENTER (.014 DIA.) RANDOM NYLON INSULATION LOO1 T H I C K )

18 INCH I.D. WINDING

7 S T R A N D S - N b 2 5 % Z r (.010 DIA. COATED WITH ,0015 RADIAL THICKNESS OF COPPER) AS CENTER-WITH 12 STRANDS (.010 DIA. COPPER) WOUND ON CENTER-INDIUM DIPPEORANDOM NYLON INSULATION t.001 T H I C K )

15 INCH I.D. W I N DING

7 S T R A N D S - N b 2 5 % Zr (.010 DIA. COATED WITH ,0015 RADIAL THICKNESS OF COPPER) RANDOM NYLON INSULATION 1.001 T H I C K )

II INCH I.D. WINDING TWIN L E A D - ( 2 ) 7 STRANDS - HI 120 (010 DIA. COATED WITH ,002 RADIAL THICKNESS OF COPPER) A S CENTER-WITH 12 STRANDS LO12 DIA. COPPER WOUND ON CENTER- INDIUM DIPPED-RANDOM NYLON INSULATION (.001THICK)

Fig. 20. Cable types used in ANL 67 kG magnet. TABLE I DETAILSOF MAGNETPERFORMANCE ~~

Winding designation Inside (7-in.-id.) Inner (11-in.-i.d.) intermediate Outer (15-in.-i.d .) intermediate Outside (18-in4.d.) uncoated Outside (18-in.-i.d.) coated

Wind- Wind- WindField Max. Average ing ing ing No. of con- cur- Packing current i.d., o.d., length, turns, stant, rent, factorh, density, in. in. in. N G/Amp A % Amp/cm2 7.175 10.675 10.875 6225

216

119

15.4

6037

11.08 15.62

4.875 1363

76

283

13.5

5404

15.77 17.44

4.875 1800

88

134

24.26

9188

1952

40

211

7.6

4532

4.625 1846

38

245

7.0

4851

18

24.125 4 . 6

18

24.5

SUPERCONDUCTING MAGNET TECHNOLOGY NiCr W l R E + M Y L A R - - - - 1 , TAPE BINDING

STN. STL. SCREEN .013-.065 THICK SHlM-.002 THlC

STN. STL. SCREE .013-.014 THICK

SHIM-,001 THICK

MYLAR-,002 THICK STN. STL SCREEN ,012 THICK

-MAGNET z

--

I

FIG.21. Schematic diagram of ANL-nested magnet system.

FIG.22. Completed ANL magnet showing nested coils.

433

434

CHARLES LAVERICK

RESERVOIRS

10" BUBBLE CHAMBER

GLASS WINDOW

LIGHT SOURCE

PLASTIC LENS WINOOW

SUPERCONDUCTING MAGNET

N 2 SHIELD

FIQ.23. Superconductingmagnet-bubble chamber assembly.

FIQ. 24. General view of the 10-in. superoonducting helium-bubble chamber in the meson experimental area of the Argonne 12.5 GeV zero gradient synchrotron complex

.

SUPERCONDUCTING MAGNET TECHNOLOGY

435

Fig. 22. The system operated with 67 kG in the 62-in. bore and 45 kG in the ll-in. bore (22). An important point was that the central solenoid, with 15,000 ft of 6-conductor NbTi cable, operated on the short sample characteristic, thus demonstrating that the short sample characteristic can be attained in a large magnet without the use of thick coatings of normal conductor. The performance characteristics of the coil are listed in Table I. The outer windings of the intermediate coil and of the inside coil contained simple lapped joints between separate lengths of cable, thus demonstrating that superconducting cables could be connected electrically and the joints wound into large magnets. The complete bubble chamber assembly is shown schematically in Fig. 23 and in position on the experimental floor in Fig. 24 from which some idea of the complexity of a bubble chamber experiment can be obtained.

C. The AVCO Model M H D System The AVCO magnet has demonstrated successfully th a t large stable superconducting MHD magnets of predictable performance can be built,

FIG.25. Schematic of the saddle shaped AVCO 12-in. I.D. model M.H.D. magnet (Courtesy of the AVCO Corp.).

and it is the world’s largest magnet. The conductor used was a composite consisting of a +-in. wide by 0.40-in. thick copper strip in which were embedded 9 heat treated 0.10-in. diameter wires. The windings were arranged in a saddle shaped configuration (Fig. 2Fj) to produce a central

436

CHARLES LAVERICK

FIQ.26. General view of the AVCO model M.H.D. Magnet assembled for test (Courtesy of the AVCO Corp.).

transverse field of 37 kG with 42 kG at the windings for the current a t which resistance was first detected. The strip current for this condition was 725 Amp and in the maximum resistive condition of operation it was 785 Amp. The stored energy at 37 kG was 3.9 X lo6 J. Portions of the conductor were driven normal by means of a heater with the magnet a t

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full current, and the resistive region did not propagate. The length of the high field region was 5 ft, the bore was 12 in. and the total weight of the assembly was 15,700 Ibs of which 165 Ibs was superconductor and 5000 Ibs was copper. A general view of the magnet is given in Fig. 26. The saddle shaped windings of this magnet (Fig. 25) have general significance since this is the basis of one class of system for producing transverse fields by the use of current sheets. Iron is riot necessary. It is in fact a dipole system and has significance as a bending magnet type in high energy physics for beam transport. The system becomes a quadrupole focusing element by using two sets of current sheets instead of the one shown.

D. Some Magnets for Thermonuclear Fusion Studies An unusual magnet was the minimum B magnet conceived in Livermore by Taylor and constructed and tested a t Argonne using an ANL conductor design (38). The magnet presented some unusual problems and its successful operation on its first test has a special significance in superconducting magnet design. The windings are arranged on a 10-in. diameter spherical surface to follow the approximate path taken by the seam of a baseball. The conductor was designed to operate where a large amount of movement could be expected and has operated at 220 Amp under these conditions. Prior to the development of the cables for the 18-in. i.d. ANL coils, superconductors had to be held rigidly for good performance. Layer winding was not possible in this design and a rugged, flexible conductor was required since the system had to be hand wound. The windings are completely insulated and self-protected but not stable. Transitions do occur in the system but it can be charged quickly. The successful operation of this magnet demonstrated that magnets of unusual shape could be constructed to operate a t satisfactory current levels and t ha t conductor movement could be tolerated. The confining field produced by this magnet had a central minimum of 3 kG with 8 kG a t the outer skin of the 10-in. diameter sphere for a conductor current of 180 Amp and a field in excess of 30 kG a t the winding. The fields a t the outer surface of the sphere and the windings scale up in the appropriate fashion a t the maximum operating current of 220 Amp which was subsequently achieved a t Livermore. Studies in thermonuclear fusion are to continue with more sophisticated equipment using superconducting coil systems. The early Livermore 10-in. internal diameter ‘minimum B’ systems are already reaching an advanced degree of sophistication with improved winding systems and the use of modern advanced conductors. Some models have been tested on the plasma confinenient apparatus and considerably larger spheres are planned.

Pi12

916 D

PT0 9D 16 916 D

PTll

PT0

Pi9

Pi7

D

D

D

018 E

I -

-

D 916 PT3 D -916 P i 4

F ii

w

BBO

Id, 070

BE60

I

92 E1 922 E

069

FIG.27. Mechanical design of the 150-kG, &in. bore magnet (Courtesy of RCA and NASA Lewis Res. Center).

SUPERCONDUCTING MAGNET TECHNOLOGY

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The possibility of applying superconductors in the more conventional type of apparatus where a central coil at a given field is placed between two mirror coils a t about twice this field to confine the plasma has also been the subject of intensive study a t the NASA Lewis Research Center (60,61) in Cleveland, Ohio, as have other types of winding configuration. A preliminary device consisting of a 2-mirror coil, a central field coil, and a quadrupole (Ioffe) field coil has been designed and tested independently of the plasma apparatus, using 50-kG, 4-in. i.d. mirror coils, and a 2 : 1 mirror ratio (62). A 150-kG, &in. bore system has been designed for NASA Lewis by RCA and is under construction. This is a mirror coil of a high-field magnetic bottle system and four 20-in. i d . , 40-kG coils are also being constructed to form the 72-kG1 20-in. bore center coil. The mcchltnical layout of this magnet system is shown in Fig. 27. Designed as a safe, unstable system with a stored energy of 2 M J and an average over-all current density of approximately 10,000 Amp/cm2, the magnet is divided into sections so that most of the high mechanical loads will be supported by the flange system. The most appropriate material, such as high or low field and high or low mechanical strength, is being used for each magnet section. Even higher field systems are under consideration.

E. Further SigniJicant Coils More recently a t ANL we have used aluminum wires successfully as a shunting material in cables and have developed high-field, low current density conductors for large magnets which contain stainless steel reinforcing in addition to shunt aluminum with 42-strand, 0.010-in. diameter NbTi cable. 49-strand cables cwntaining 7 heat treated 0.010-in. diameter NbZr strands, each surrounded by G copper strands have been used in 2 large coils. 10,000 f t of this material with a full nylon insulation operated at 500 Amp in a coil of length 12 in., i.d. 12 in., and 0.d. 20 in. and developed a central field of 30 kG for an average current density of 4200 Anip/cni2 in the winding. The use of a half wrap of nylon so that 50% of the cable surface was exposed resulted in a current of 700 Amp with 5000 ft of the same cable a t a central field of 42 kG in a 7-in. bore. Heavy section NbTi has also been successfully stabilized and operated a t a n average current density of more than 9000 Amp/cni2 a t a central field of 55 kG in a coil of 4-in. bore. This presents a new approach to the problem of designing large high-field magnets of up to 100 kG which simplifies design, construction, and operation. A very small bore, 132-kG coil was constructed by Rosner [Rosner and Benz (SS)] who has also wound p a n c d e of the new General Elecatric ribbon t o form 1-ill. bore 100-kG (#oils.Schindler of RCA has completed

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a 2.94-in. bore, 110-kG magnet and a 2-in. bore, 137-kG magnet (64), and magnets in the 100 to 110-kG in bore sizes of 1 to 3 in. are now commercially available. Sampson of Brookhaven designed and operated a 112-kG, l+inch bore coil constructed from RCA strip (66) in 1965 and has recently obtained 140 kG in l-in. bore (see Fig. 28). This magnet utilizes the coil sections built for the %- magnetic moment experiment (66) which has

FIG.28. The Brookhaven NationalLaboratory,l-in. bore 140-kGmagnet (Courtesy of Brookhaven Natl. Lab.).

recently been deferred or abandoned and the central conical section has been replaced by a l-in. bore cylindrical section wound with the new RCA high-field NbrSn 0.090-in. wide ribbon. The Brookhaven group has also modeled a l-in. bore superconducting quadrupole lens (see Fig. 29) with a field gradient of 10 kG/cm which is 5 times higher than can be obtained with conventional iron quadrupoles (67,68) and has produced these quadrupole fields with both RCA and: G. E.strip type conductors. This confirms the previous ANL findings using cable by demonstrating

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that magnets of unusual shape can also be constructed successfully from tape type conductors. Actual working lenses with higher field gradients and larger bores will soon be available in a number of laboratories (69). These lenses will be used to transport charged particle beams. Conventional magnet systems with copper and iron are used in large numbers in

FIG.29. A Brookhaven-University of Illinois quadrupole field system using current sheets of Nb&n tape t o demonstrate the practicability of high-field gradient focusing magnets for beam transport systems (Courtesy of Brookhaven Nat. Lab.).

conjunction with high energy particle accelerators. Higher field gradients in the lens result in shorter focal lengths and thus shorter beam lengths. The beam length must be as short as possible if particles or states which decay in short path lengths are to be studied. Thus, apart from the fact that superconducting lenses reduce the power cost associated with the

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operation of high energy experiniental facilities and may replace conventional electromagnetic lenses and bending magnets for this reason alone, they may extend the range of experiments possible and reduce the length of many experimental systems. Electron microscopy is developing to the stage where the resolution of separate atoms is becoming a distinct possibility and the dream of reading out and perhaps even changing the code of life, if desired, is within grasp. The low temperature environment now becomes necessary

2 inches

FIG.30. The Fernandez-Moran superconducting objective lens and liquid helium cold-stage assembly for cryo-electron microscope (Courtesy of H. Fernandez-Moran, Univ. of Chicago).

to reduce thermal agitation and superconducting lenses offer possible ways of improving lens stability and reducing focal lengths to the required degree. Dr. Fernandez-Moran of the Department of Biophysics, University of Chicago, has successfully pioneered the application of superconductors to electron microscope lens design (70, 71) arid has recently been able to take pictures a t resolutions of the order of 10 A because of the unique long-term stability of the superconducting lens in the persistent current mode (72). Dr. Fernandez-Moran’s objective lens system is shown in Fig. 30 (73). Intensive studies of this type are being pursued elsewhere.

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A superconducting magnet with iron was used recently in a successful experiment on corpuscular ionography (74) with the CERN synchrocyclotron to produce a transverse field of more than 45 kG in a room temperature gap of rectangular section 0.8 in. by 1.6 in. and length 2 in. A low-field, magnetic shield 6 f t in diameter for the protection of space travelers has been successfully operated and tested, and studies of this type are under way (76).

V. DESIGN PROBLEMS A . Power Supply Considerations One aspect governing the choice of power supply has been considered in the section on pulsed magnets. When electrical current is supplied from an outside source, the problem is to reduce both the heat leak from 300 to 4.2'K and the power diwipation in the input leads to a tolerable minimum. It is necessary to hold the cross sectional area of these leads near to the optimum value and to reduce the number of power leads and instrumentation leads as much as possible. The reason for this is that a heat leak of about 0.7 W into liquid helium evaporates 1 liter of liquid per hour and introduces an operating cost of about $5 per hour if the helium is not recycled. At 0.8 cents per kW hour, this corresponds approximately to the operating cost of the 60 kW power supply. Thus, in small magnets, it is usual to operate at low currents of about 100 Amp or less. In situations where heat leaks must be reduced to an absolute minimum, currents of less than 10 Amp might be considered and the magnet operated as a permanent magnet after being charged. The supply leads can be removed to eliminate heat leak. Such situations are normal for research magnets used in experiments a t 1°K or less. A 100 Amp system can be operated with a heat leak of about 0.2 W into liquid helium. Currents of 200 or 300 Amp are usually acceptable for medium size coils. Closed cycle refrigeration systems may be desirable where continuous use is required, the choice being based on economic considerations. I n large magnets such as those used for heavy engineering installations, no superconducting coils have been used so far. Consideration is being given to this possibility and decisions as to what operating current is best have yet to be made. These magnets would normally use closed cycle refrigeration and may recycle the boiled off helium a t low temperatures, thus eliminating the possibility of using the refrigeration available from the cold gas for cooling the power leads. 1000, 2000, and 10,000 Amp supplies are being considered and the losses to be expected by feeding such currents from room temperature are reasonable and acceptable. The supplies are required to operate a t a maximum of 1 or 2 V where dc magnets are required which do not have

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to be turned on quickly. Rotating machinery such as homopolar generators are probably most convenient for such voltage ranges. Machines of 10,000 t o 30,000 Amp rotating a t 1 to 5 V are now available. Flux pumps are also being considered. These latter devices operate in liquid helium anddeliminate the need for heavy current leads from room temperature to liquid helium. The advantages of 1000 and 2000 Amp coils are that cheap power supplies are available and heat leaks of 1 or 2 W due to the power leads at 4.2'K are attainable. In larger coils, the inductance is high at these currents and favors the use of persistent switches. Larger voltages may be necessary to produce acceptable charging times, but these are readily available. The longer lengths of conductor required a t 1000 or 2000 Amp make i t necessary to use internal contacts between conductors which may reduce reliability and introduce appreciable power dissipation inside the magnet. The smaller conductor may tend to lead to unfavorable mechanical designs with more numerous and much smaller cooling passages. This may increase the possibility of vapor locking if appreciable power dissipation should occur in the coil during operation. A lo8 ,J magnet which must be charged in 104 sec to full energy a t 1000 Amp requires a steady potential of 20 V applied across the coil. This becomes 2 V if the charge time can be increased to lo6sec. The inductance would be 200 H and if the coil had to be discharged a t a constant rate in 1 sec under emergency conditions, the voltage across its terminals would be L di/at = 200,000 v. A 10,000 Amp coil of the same stored energy would result in a better mechanical design than previously for the same packing factor since the conductor would be larger and more robust, as would the cooling passages, while series connections in the windings would be eliminated. A charge time of lo4 sec to lo8 J would require an applied potential of 2 V. The coil inductance would be 2 H and the voltage across the coil terminals if the coil had to be discharged in 1 sec would be 20,000 V. Where high, long-term stability of magnetic field is required with external supplies, a control system will probably be used in which the fringing field outside the magnet is monitored accurately using some system of absolute measurement such as Nuclear Magnetic Resonance. Deviations from the required magnetic field will result in correction signals which readjust the power supply current as required. Alternatively, persistent switches may be used.

B. Flux Pumps Flux pumps are devices which generate current in superconductors without the need for heavy current leads to be led into the 4.2'K environ-

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nient from room teniperature. They achieve this because they operate the low temperature environment in regions of low magnetic field. It may be necessary to use magnetic screening with such devices when they are used to energize large, high-field magnets. They are a t present in their infancy, but the recent successful demonstrations of the practicability of large, high-field superconducting magnets will produce a desire and a need for high current sources. Currents of lo4 and lo6 Amp are being seriously considered for some projects. Several types of flux pump exist and some are older than the present high-field alloys. An early type was proposed in 1933 by Mendelssohn, using a dc transformer. Alternating current transformers operating with superconducting devices instead of diodes to produce rectification have been developed by Olsen (76) and Buchhold (77).Laquer (78) has developed a similar device operating at a low cycling rate of about once per minute and using thermal switching with hard superconductors in the switch. Devices based on electromagnetic induction have been built by Volger and Admiraal (79) and Wipf (80). I n these devices, movement of a conductor in a magnetic field results in an induced voltage which causes the flow of current in a suitably coupled circuit and hence creates a magnetic field. Wherc superconducting circuits are involved, the magnetic field established is permanent since there is no power dissipation in the circuit. Instead of moving parts, moving patterns of conductivity and magnetic field may be used to create the induced voltage. Volger and Admiraal’s device is a superconducting homopolar generator and Wipf’s is more like an ordinary dc generator for which higher efficiency than the homopolar device is claimed. Losses occur in flux pumps although there is hope th a t these can be reduced. Buchhold’s experimental model was claimed to have a maximum power output of 3 W for an approximate loss of 0.2 W at 10 cps. The aggregate loss was due to transformer and reactor losses and to losses in the superconducting switches during reverse current operation. Thus, the device has an efficiency of about 93%. The losses in one of Wipf’s generators have been assessed and increase with the rotational speed of the exciting magnet. At 100 rpm the efficiency appeared to approach 100%; it was approximately 90% a t 200 rpm and 50%at700rpm. Typical voltage outputs from an experimental generator were about 60 pV a t 100 rpm and 200 pV a t 500 rpm. I n a later paper (81), Wipf gave a measured efficiency of 35% and felt that 50% was attainable. Thelossesin Volger’s device have not been measured, but a recent model was operated a t 12,800 Anip on a small load. The potential gain in using a flux pump may be much lower than

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expected unless losses can be kept low in the large models we would need. The argument is as follows: A large magnet with a stored energy of lo8 J which has to be charged in lo4 sec requires an average charging rate of lo4J/sec a t 10 kW. At 99% efficiency in the flux pump, 100 W would be dissipated in liquid helium during charge. At 95% efficiency, the situation would be completely intolerable since the loss rate would be too high unless a longer charge time can be accepted. Flux pumps are acceptable with smaller magnets. The ANL 67-kG, 7-in. magnet system has a stored energy of about 6 X lo6 J. A charging time of lo6 see is acceptable, corresponding to a n average power input of 6 W. At 90% efficiency, energy would be dissipated in the liquid helium a t a rate of 0.6 W during charge, and this is an acceptable rate. Superconducting dynamo and solenoid combinations operating a t high current level have been constructed (82, 83).Typical applications of the latest low power low voltage type pump are the testing of short samples to currents in excess of 12,000 Amp and the charging of a medium size 35-kG, 0.27-in. bore coil to 1400 Amp in 25 min. It has been suggested that while present flux pumps are inadequate for the charging of large coils, they may be used as steady state regulat,ors to replace the leakage of magnetic energy due to resistive contacts in a large coil. The writer suggests that the precision regulation and control of electron microscope lenses could be realized by the application of flux pumps.

C . Safety Considerations The problems with smaller magnets are trivial and have been solved. The ANL 67-kG, 7-in. system is well enough understood to operate without accidental discharges in normal operation. Accidental discharges of the system result in the loss of the liquid helium contained in the reservoir, a n acceptable transient pressure rise, and some inconvenience. It is now possible to construct such a magnet to operate completely without discharging when a normal region is developed. These stable magnets are safer and more convenient to operate since no loss of liquid helium or of magnetic field occurs when the normal region is reached. Magnets with stored energies of the order of lo8J or greater are components of expensive engineering installations where safety and reliability are paramount considerations. The windings can be designed with enough thermal capacity to result in a limited acceptable temperature rise in the unlikely event that complete and instantaneous loss of refrigeration occurs and a small fraction of the energy has to be dissipated in the coil. A current problem is that of ensuring that if a small portion of the winding is uncovered accidentally from liquid helium, the coil energy or some appreciable fraction of it will not be dissipated in a short length of

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conductor and thus result in its destruction and the resultant destruction of the system. One technique is to build in short circuits between layers or fractions of layers with an external shunt resistor connected across the whole coil to absorb most of the coil energy when emergency discharge is required. A disadvantage of this technique is that it may result in unduly high power dissipations where rapid charges of the system are required since energy is dissipated in the internal shunt. This may result in a prohibitively expensive refrigeration system. Another possibility is to incorporate a liquid helium blow-off system to ensure rapid elimination of the liquid helium environment under such emergency conditions. The problems are under consideration. In considering this aspect of large magnet design problems, it becomes obvious that emergency conditions may dictate the necessity for heavy current leads from the magnet to an external shunt resistor so that a large fraction of the coil energy can be dissipated outside the system if required. These leads constitute a substantial heat leak. If this course is adopted, the leads can be used equally well with an external power supply and the need for a flux pump becomes questionable in any case. Heavy current conductors for the magnet may therefore be used without extra complication because the heavy leads to room temperature, with their substantial heat leak, have to be installed in any case. These are current design problems under study. The writer suggests the use of low temperature discharge resistors at helium temperature in a separate enclosure to eliminate the heat leak problem, the removal of the power lead after charge, and the use of a flux pump to prevent field decay if the persistent circuit is resistive.

D . Persistent Switches Superconducting magnets can be operated as permanent magnets by switching superconducting shunts across them during operation. This can be accomplished by heating the superconducting shunt during the energizing operation so that it is resistive. When the coil is energized, the shunt heater is switched off and the shunt becomes superconducting. Theoretically, the input leads can now be disconnected if an external power supply is used and the heat leak to the liquid helium environment reduced to low levels. Flux pumps are automatically portions of the persistent circuit when they are no longer operating to energize or deenergize the circuit. The current in a persistent circuit will decay in a time which is proportional to L / R , the ratio of circuit inductance to series resistance. Where a number of series connections are made in the winding, the decay time is reduced. Typical joint resistances obtained so far with multi-strand

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stabilized cables are of the order of a. Thus, a coil of 10 H inductance will decay t o l / e of its initial field in 1Olo sec or approximately los days. Alternatively, we can say that the magnetic field is stable to 1 part in 106 per day. Work will continue on improving the performance and reliability of internal contacts in coils to reduce the internal resistance as much as possible and to maintain it at the initial low value. Techniques for making various types of superconducting and resistive contacts have been developed and can be improved.

E. Contacts Two types of contact are required in using supermagnets. One is the contact between the electrical power leads and the superconducting windings, and the other is the contact between adjacent lengths of superconductor in a winding. The electrical bond between superconductor and normal metal coatings in commercially produced coated superconductors is now so good that joint resistance of the order of with ordinary soldered contacts can be obtained between short lengths of copper input lead and the coated superconductor. The power dissipation a t normal current levels up to 1000 Amp is therefore extremely low and the heat dissipated is easily removed by the liquid helium bath without appreciable temperature rise. Contacts of this type are also used in most magnet windings. Such magnets are slightly resistive and consequently are not really supermagnets, but power dissipation rates are usually considerably less than 1 W and are acceptable. Resistive contacts of this type are acceptable in most applications requiring persistent mode operation since the decay in magnet field over the required period of operation is usually acceptable. In some few applications, zero resistance contacts are a necessity. Truly superconducting joints between adjacent lengths of NbsSn are yet to be developed. Superconducting systems such as NbZr and NbTi can be joined with zero resistance contacts. Typical techniques include the use of superconducting solders (usually low field, say 20 kG maximum), spot welded junctions, or mechanical contacts. Many contact types are field sensitive and it is therefore necessary to either bring the joints into a low-field or field free region or to ensure that the system to be chosen will operate satisfactorily in the external magnetic field in which it is t o be placed. We have found (84) th a t the current-carrying capacity of resistive and spot welded contacts can decrease with magnetic field. The welded contacts may degrade some of the superconducting material in the welded region, and consequently may have zero resistance over a wide range of field, but have a current-carrying capacity which

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decreases with applied field. They require mechanical support but have been used successfully in some commercial magnets. The use of knife-edge type contacts of niobium as a hard superconducting alloy with additional superconductor in the region of the clamped joint is usually adequate. Where large numbers of contacts are to be used in a magnet system, failure to maintain adequate quality control in the contact system can lead to disaster. The problems arising in the study of contacts are relevant to the study of modern stable magnets since the current may switch out of the superconductor a t any point along the length of a stable winding. The limitation on a contact is a thermal one and low resistivity alone does not ensure good performance. One can analyze the current distribution along a contact in a similar manner to the analysis used in the case of the propagation of current along a transmission line. A good contact has the following chararteristics (85): 1. High thermal and electrival conductivity in the normal conductor. 2. Low surface resistance. 3. A contact length longer than ( S / r ) ' l 2where r is the resistanre/ unit length of conductor and S is the surface resistance of unit length of contact. 4. Good cooling as far as exposing the surface to the bath, as well as good thermal contact between the superconductor and the normal conductor. 5 . High I , obtained by placing the contact in a low field region if possible.

Contacts should be oriented parallel to the magnetic field where possible since the critical currents for hard superconductors in longitudinal magnetic fields are enhanced as compared with their performance in transverse fields and can be as high as eight times greater a t some fields (86).

F. Refrigeration Requirements The amount of refrigeration required for a superconducting magnet is dictated by the temperature of the magnet environment, the heat leak through outside power leads or protective leads to the coil, and the energy dissipation in the low temperature environment. The latter parameter is influenced by the presence or absence of a flux pump, the energy dissillated in series cable connertions and power contacts, and the energy dissipated in protective internal interturn or interlayer shunts during charge and discharge where these are adopted. A typical ruleof thumb figure is to allow between 500 and 750 W of room tempera-

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ture refrigeration power for each watt of dissipation a t liquid helium temperature. The magnet environment may be a prime factor in determining whether or not it is economical to use a superconducting magnet. Magnets for use with liquid hydrogen bubble chambers have refrigeration provided to about 20°K in their external environment. The fractional refrigeration required t o operate at 4.2"K from the 20°K environment is small since the radiation losses can be made negligible and the internal heat leaks are small. Heat leaks due to internal power leads can be reduced to tolerable levels by clamping these leads a t liquid hydrogen and liquid nitrogen temperature if the boiled off helium is to be recirculated and liquefied a t low temperature. Alternatively, efficient power lead cooling systems can utilize the additional refrigeration capacity of the boiled off helium if the gas is not to be held a t low temperature. Such systems can operate a t about 1 W of heat input a t 4.2"K per 1000 Amp input power lead. Magnets with room temperature environments require refrigeration systems of much higher capacity since the temperature differential between room temperature and the magnet coil is 300°K. Intermediate nitrogen shields a t 77"K, superinsulation systems, or both are required, in association with spaces between the different temperature environments which are held a t high vacuum to prevent excessive heat leaks due to conduction and convection between the various containers. These additional refrigeration systems and vacuum systems represent further capital and operating expenditure where the magnetic fields are required a t room temperature. The greatest factor in determining the reliability of a large superconducting magnet is the reliability of the refrigeration system. The use of iron usually necessitates the employment of heavy structural members to contain the large loads which either exist during operation or occur during emergency conditions. The iron is held a t room temperature in large structures because of the prohibitive cooldown costs and other problems associated with cooling the iron. I n magnet designs of this type i t is necessary to contain the internal forces as much as possible by structural members a t 4.2"K, while a t the same time reducing the heat leak from the room temperature environment due to these members. Typical large magnets of this type include spark chamber magnets and cyclotron magnets. Spark chamber systems can be considerably reduced in size by taking advantage of the high fields now available with superconductors. Refrigeration costs for smaller magnets can also be deceptive. Superconducting magnets are often preferable for many research applications requiring a small volume of magnetic field a t between 10 and 50 kG; since

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the installations are small, power supplies are small, cheap and readily available, and closed cycle refrigeration systems are not necessary since liquid helium can be purchased and used when required. Above a certain percentage level of utilization closed cycle systems in the range of $10,000 to $20,000 may be desirable rind the magnet should be held a t low temperature continuously. A small ANL 8th-order corrected, high homogeneity 25-kG, 2-in. bore, 5-in. long coil weighing approximately 6 Ibs which was recently cooled down and used frequently over a &month period required about $10,000 for liquid helium. Thus, operating costs for small magnets are not often as small as they appear. Medium size superconducting magnets with no closed cycle refrigeration systems cost as much to operate as conventional magnets of similar performance but the cost of the power supply for the superconducting magnet is usually less by more than one order of magnitude. The problem has to be reexamined for each particular case, since where high rates of charge of magnetic field are required in each case, the power supply requirements and operating cost are similar for both conventional and superconducting coils. There is now a need for systems which will circulate liquid helium a t reasonable flow rates. Some of these systems must function below the X point, others a t 4.2"K1and some a t pressures up to about 50 atmospheres to utilize the heat transfer properties of supercritical helium.

G . Predictability of Magnet Performance It is possible to predict the performance characteristics of a large range of conductors in various coil types on the basis of experience (87). Since thermal instabilities are the cause of premature transitions, the thermal environment for the conductor and its mode of support will determine the transition current to a large extent. Windings containing uncoated 0.01-in. diameter NbZr superconductor will carry currents of up to 20 Amp in medium size coils (14, 17) and about 10 Amp in large coils (21). If the superconductor is heat treated to enhance its short sample performance ( S I ) , currents of 5 Amp in small coils may be difficult to attain. The situation can be improved by choosing coil designs which permit good cooling either by using cooling passages while maintaining good mechanical support of the individual turns or by using interleaving material of good thermal conductivity between layers. Transitions are now induced by a flux jump a t some point in the winding which dissipates enough energy to cause a thermal avalanche. The thermal environment for such system is indeterminate and it is not possible a t the moment to predict thermal conductivities, diff usivities, or heat transfer characteristics with any degree of accuracy.

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The situation changes with stable coils and it is possible to predict coil performance for such cases. Such coils have windings which are well cooled, well supported, and well shunted. The superconductor will now carry currents near the short sample characteristic a t which point some portion of the superconductor will become resistive and a portion of the supercurrent will be diverted into the shunt. The heat dissipated in the shunt is now removed by the helium bath in a manner which depends upon the heat transfer characteristic a t the liquid-to-metal interface, the properties of the liquid helium, and whether or not it is a t rest or in motion. The first attempts to analyze the processes involved are due to Stekly [see Kantrowitz and Stekly (2'7), Stekly and Zar (28), (@)I. The condition for thermal stability is that the joule heating in the normal region should be balanced by thermal conduction to the liquid helium bath. Thus,

h AT

=

i2R/S

where i is the shunt current (there may be some contribution, as yet indeterminate, from the superconductor), R the resistance per unit length, S the effective surface area per unit length, h the thermal boundary conductivity, AT the temperature differential between cable surface arid bath. It is conservative to take the shunt current, i, as the total conductor current. Maximum nucleate boiling heat transfer flux occurs a t AT, sz 03°K for most configurations in static liquid helium a t 4.2'K. Any further increase in AT decreases h by an order of magnitude. The current which flows in a composite current and a completely stable conductor is now defined as one in which the recovery current is equal to the transition current. Many stable systems in fact exist in which the normal region disappears when the current is reduced at some current below the critical value, but they are unstable in terms of the above definitions. The recovery current is this lower value. Stekly introduces a useful parameter a defined as: ff=

where I , is the critical current of the superconductor for the critical field under consideration, T , is the zero current critical temperature a t that magnetic field, P / A is the resistance per unit length of substation, h is the effective heat transfer coefficient from the surface of the conductor to the liquid, P is the cooled perimeter of conductor cross section. Thus, a is the ratio of the temperature rise that would result if the superconductor critical current flowed through the substrate to the zero current critical temperature of the superconductor in the presence of the

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externally applied magnetic field. On the basis of the analysis, (28) the recovery current, I,, is related to the critical current by the relation I, = I C / G . This analytical approach is a useful first step and progress toward more realistic models of the various situations is being made. Recent experimental work (33) indicates that the critical currents for fully stabilized cable are essentially independent of T, and are determined primarily by surface heat transfer characteristics. This is the more specific case for which recovery current levels are only slightly lower than short sample critical currents. I n such cases, a copper limited region exists which is completely independent of T,. Three experimentd techniques have been developed to predict the performance of stable coils from the results of measurements on short samples. These are due to Cornish (4.2), Stekly (@), and Purcell(88). Stekly measures the voltage-current characteristics of the conductor under these conditions. Cornish uses a small heater wrapped around the conductor and heats the conductor at various conductor currents in the desired magnetic field. The conductor current for which a small additional amount of heat results in the production of a normal region is the predicted current, and the subsequent behavior of the region can be studied. An objection to the Cornish technique is that the heat transfer characteristics of the conductor near 4.2"K vary rapidly with temperature. This experimental error can be reduced by breaking the superconductor a t some appropriate point and is the basis of Purcell's approach (88). The current has now to transfer from the superconductor to the normal conductor and then back into the other section of normal conductor. An artificial normal region of the worst kind (infinite superconductor resistance) has now been introduced since the total conductor current must be carried by the copper. The temperature of the system is not changed. We have found this method to be more realistic but less convenient to use than that introduced by Cornish. The error in the Cornish approach probably arises in increasing the temperature locally so that near the critical current the heat transfer characteristics are driven from those for nucleate boiling to those for film boiling by the heater. An even worse error due to this is introduced by the Stelrly technique. Combining these various techniques and calculations with an amount of empirical data, it can now be certain that the current-carrying capacity of most superconducting magnet designs can be predicted.

H . Accessibility An advantage which is claimed for superconducting magnets is that the high magnetic field region can be made more compact. This is generally true since the cross sectional area occupied by the windings can

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be made much less than in the case of copper coils. The space taken by the thermal insulation can be made small. Superconductors operate at current densities greater than lo6 Amp/cm2 up to about 100 kG; when combined with copper and wound so that cooling channels are distributed throughout the coil, average current densities of up t o 30,000 Amp/cm2 are attainable. High performance copper magnets have been operated at current densities of greater than 5000 Amp/cm2, but values less than 1000 Amp/cm2 are more customary.

I . Operation at Temperatures Above and Below 42°K The operating characteristics of the various materials vary with the nature of the material itself, the nature of any coating material, and the ratio of surface area tovolume. Many types of coil construction are used; some are well cooled, some are all metal, some are insulated, and some are completely potted in epoxy. Consequently, it is not surprising that there should be considerable variation in magnet operating characteristics over the operating temperature range. A coherent and complete picture is not yet available. Kunaler (13) found that his NbrSn coils had the enhanced currentcarrying capacity at temperatures below the X point (2.18"K) of liquid helium one might expect. Recently the same has been found to be true with tape wound RCA NbPSn ribbon (89),but the gain in current-carrying capacity is now much greater than had been ant.icipated. The gain is a function of coil design, being greatest for more unstable coils, and can be as much a factor of 2 at high fields. Coils using organically insulated, copper coated, NbZr wire usually operate at currents well below the short sample characteristic when wound into coils. Consequently, they can be regarded as unstable. It has been shown that such coils can be operated to higher currents a t elevated temperatures such as 5 or 6°K (90, 91) for the following reason. The material has been desensitized in the sense that its current-carrying capacity at a given magnetic field has been reduced because of the increase in temperature. The flux junips associated with field penetration become less severe as the temperature is increased and eventually at some temperature they cease to occur. Thus, although the short sample characteristic is depressed, the current-carrying capacity of the coil is enhanced because the material is now more stable. Similarly, a t temperatures below 4.2"K the short sample characteristic is enhanced because of the lowered temperature, but the current-carrying capacity of the coil is degraded because of the extra energy liberated in the flux jumps. Coils which operate on the short sample characteristic at 4.2"K degrade in current-carrying capacity at higher temperatures; thus, their current-

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carrying capacit,y will be enhanced a t lower temperatures if they have been,designed with a sufficient margin of stability for 4.2"K ( 1 7 ) .Unstable coils which are well cooled can have reasonable performance with minimum degradation by attention to these aspects of the design (92). We have recently observed that even a well shunted nylon insulated NbZr cable in a large coil containing cooling passages every layer had a degraded performance below the X point.

J . Homogeneous Fields There are many applications in which fields of high homogeneity, high stability in time, and high intensity are required. Superconductors offer the possibility of making coils of this type. Helniholtz coils are the classical arrangement for achieving such fields. When the axially symmetric fields are developed in spherical harmonics, it is possible to nullify the coefficients of various orders of the Legendre polynomials by suitable shaping of the coil cross section, by suitable variation of the current density in the cross section, or both. Thus the use of compensating windings with a split coil system can result in sixth or eighth-order correction and in fact even higher order harmonic corrections can be made. Such techniques enable the calculation of coils of high homogeneity over a given volume with a considerable reduction in the number of ampere turns required as the order of the correction is increased. Using Garrett's technique (93), sixth order systems of high homogeneity have been produced (94, 95). We produced an eighth-order corrected system as a joint project with Argonne and Northwestern University in 1963 (96). Sauzade and Grivet obtained a homogeneity better than A H / H = over a volume of 150 mm3. Polarized proton target systems do not need such high homogeneity. DesPortes (9'7) of Saclay, France, has constructed and operated a high homogeneous field superconducting magnet for a polarized proton target with an inner bore of 8 in., a central field of 21 kG, a homogeneity of A H / H = low4over a cylindrical volume of length 0.8 in. and diameter of 1 in. The magnet has greater flexibility than a conventional magnet because of the wide angle radial and axial accessibility. The problem of measuring the estimated degree of homogeneity is difficult a t the highest levels. Mechanical tolerances have to be held to very close limits in the conductor, the coil form, and the insulation. Mechanical changes in the system due to the high loads can also cause the highly homogeneous volume to change in shape with current. The amount of diamagnetism exhibited by the superconductor varies with field and consequently varies throughout the volume of the winding. Distortions in the field due to this effect can be calculated approximately

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but i t is necessary to measure the degree of homogeneity achieved and correct for this by adjusting the turns or currents in parts of the coil. As a consequence of these effects, the residual field in a solenoid varies along its length and the central field is not a linear function of current. The effect is worse in NbsSn than in NbZr and, in the former, fields as high as 6 kG have been trapped on return to zero current (98). Foss (99) has pointed out t hat certain errors are associated with coil winding. An example would be the pitch of the helix in layer winding, for instance, so that axial components of current are present as well as the cylindrically symnietrical current sheet. It is possible to correct for many of these, but they should be taken into account when highest accuracy is required in any system. The advantage of ferromagnetic materials in the lower field regions where they can be used is that they screen out such winding errors.

K . Radiation Eflects It is the object of this section to draw attention to some radiation effects but not to present an over-all view of this topic. The possibility of using superconducting magnets for bending high energy particle beams or ejecting beams from circular machines is under consideration. T h e energy content of the beams of the great accelerators is large enough to raise the temperature of objects in their paths by a large amount and this would, of course, destroy the superconductivity. Consequently, the windings of superconducting coils should be screened from any beam which may accidentally strike it. Even a 5-MeV electron beam can cause sufficient local heating to cause trouble if the eiiergy is absorbed by a small local section of the winding either directly or by thermal conduction from a container wall. Since hard superconductors are improved if controlled defects are introduced, the use of radiation to produce better quality superconductors is a possibility. Many effects of proton or neutron bombardment can be annealed out so materials treated in this way may be easily degraded. The main questions to be answered are whether or not the lifetime of a winding can be reduced by continual exposure to radiation, even if only intermittent operation is required, and whether or not normal operation is interfered with by certain types of radiation. Since the effects can be annealed out by raising the superconductor temperature above T,, or above room temperature even for some types of damage, it is necessary to consider tests in which the coil is maintained a t the low temperature throughout the experiment. The choice of insulation system is important. Little work of this type has been carried out but there is obviously a great deal of interest in the results. In some recent preliminary work at Argonne in which an NbZr coil was maintained in liquid helium in a

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unique facility while under bombardment by fission neutrons, the coil was not superconducting during bombardment although the computed temperature rise was about 3°K in the coil, and when tested while still cold a t the end of one week there was evidence of a n approximate 20% reduction in current-carrying capacity which annealed out after the coil had been raised to room temperature. This exposure was computed to be roughly equivalent to a lifetime for a cyclotron magnet. More results are required but the experience points out the need for caution in considering the use of supermagnets under certain conditions of radiation and the need for further studies of this nature. VI. MAGNETECONOMICS AND LARGEMAGNET STUDIES

A . General Remarks The question as to whether or not a superconducting magnet is more economical than a conventional magnet can only be resolved for particular cases. Some excellent studies have been made on the subject of superconductor economics and, in the case of older material, merely require the use of new cost figures to take into account present superconductor and magnet performance (100-102). A number of typical examples are given here to show what factors are involved in choosing between the two systems on a basis of cost. I n an earlier comparison of the Argonne 67-kG system with an existing conventional magnet system of almost identical size and performance, we found that the costs of supermagnet systems were less than half those of a conventional system. The cost of fabricated conventional coils was about $15,000 out of a total of $305,000 as compared with $100,000 for the superconducting coil and $140,000 for the whole system. A comparison of the cost of a superconducting magnet facility for plasma physics research with that of a suitable hypothetical conventional magnet system (103) revealed that the cost of this system ($66,000) was considerably less than that estimated for the conventional system ($150,000). The operating costs for the superconducting magnet system were estimated to be higher in this case because the intermittent use requires 100 liters/week of liquid helium to cool the coils to 4.2'K about once per week. This superconducting system was claimed to be superior to the conventional system since it provided more instrumental and visual access to the working volume than the conventional system, could be operated without electrical power since it i s fitted with persistent switches, and provided a better working vacuum in the experimental system because of the cryo-pumping action of the cold surface. I n the c:ise of a proposed high energy experiment a t 75,000 ft (104)

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which requires a 12-kG1 1-meter bore, 2-meter long magnet, the comparison between conventional and superconducting magnets takes a different form. A conventional magnet system is now out of the question since the cost of lifting the experimental equipment is approximately $500/lb averaged over the total cost of the experiment. Permanent magnets would be too heavy and bulky, and have too low a field. Copper coils and a portable high current generator would also be out of the question. The environment a t 75,000 f t favors the low temperature system since the heat losses can be made low. A comparison is now almost ridiculous since only a superconducting coil system can be used. The possibility of placing superconducting coils in a large spark chamber magnet with a 70-ton iron yoke and a useful volume of 50 in. by 60 in. by 30 in. with a field of 17 kG was considered. I n this case, the 11-MW power supply was available and the capital cost of either type of coil would have been the same. The cost of the necessary closed cycle refrigeration facilities would have been of the order of $200,000 since it would have been necessary to provide cooling from room temperature to 4.2"K. This is equivalent to the cost of one year's continuous operation of the spark chamber. It would have been a reasonable economic proposition to install a superconducting winding if continuous operation over a period of years had been anticipated. Intermittent operation with a low duty cycle was expected so it was decided that it would be unwise to use a superconducting coil. I n fact, the wrong question was really studied. The conventional system actually operated a t 12 kG in this case and the large scale of the equipment is necessary because of this working field. I n fact, a split coil system of 60 or 80 kG is feasible in about 1-ft bore and the whole apparatus could have been reduced in size by a factor of 5 or 6. Naturally, this course has only recently become available because of the rapid advances in supermagnet technology, but the reader's attention is drawn to the fact that in replacing large conventional systems with superconducting coil systems, the possibilities in size reduction by changing to higher field intensities should be considered.

B . Large Bubble Chamber Magnet Studies A number of large hydrogen bubble chambers are being considered seriously at the moment. Funds for the construction of the ANL chambers have been allocated and it has been decided that a superconducting magnet with iron will be used. The arrangement of the pancake-type windings in bubble chamber system is shown in Fig. 31. Two current design studies are under way in the United States, one a t Brookhaven National Laboratory (105) and one a t Argonne National Laboratory

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(106). A 5-meter diameter chamber is under consideration in CERN. Serious consideration is being given to the possibility of using superconducting coils with both systems, and, in the Brookhaven case, consideration is being given to an air core design. Bubble chambers are used in high energy physics to detect and photograph the tracks of charged particles in a magnetic field. In this way the charge of the particles can be detected and the dynamics of an event involving interactions between various particles can be reconstructed. Hydrogen bubble chambers are

12-FT HYDROGEN BUBBLE CHAMBER

FIG. 31. The projected Argonne National Laboratory 12-ft hydrogen bubble chamber with superconductingmagnet (Courtesy of E. G . Pewitt, Argonne Natl. Lab.).

particularly attractive for superconducting magnets because the low temperature environment is already in existence and it is only necessary to refrigerate from 20"K, the boiling point of liquid hydrogen a t 760 mmHg, to 4.2"K. Some details of the ANL and BNL systems are given to illustrate some of the design problems and to compare estimated costs of large conventional coils with those of superconducting coils. The final design characteristics of the Argonne superconducting magnet are listed in Table 11. The coil appears as a section of an infinite solenoid because of the presence of the iron which acts as a screen for the large stray magnetic field which would otherwise surround the coil. This iron also reduoes the magnetomotive force necessary to attain the required 20-kG central field. The number of ampere turns needed to attain the

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field with the iron is 5 x loe. The winding for a conventional coil requires 90 tons of copper, water cooling, and a 10-MW power supply (107). Estimates of the costs of both conventional and superconducting magnets are given in Table 111. I n the case of the superconducting magnet, I have based my estimate of the cost of the coil on the published prices for NbZr cable with a n allowance for copper. The coil requires 6,000,000 Amp turns a t 50 ft/ turn. The length of heat treated 0.010-in. NbZr conductor cable involved (if indeed this were to be the most acceptable material at the time a decision had to be made) a t 1000 Amp/strand would be 300,000 ft. TABLE I1 DESIGNCHARACTERISTICS OF THE SUPERCONDUCTING MAGNET FOR THE ANL 12 BT. HYDROGEN BUBBLECHAMBER^ Field Ampere turns Current Inductance of winding Energy stored in field Wt. of copper in windings Wt. of superconductor Hoop stress on winding Coil axial compressive force Wt. of iron Inside diameter of coil Length of coil Power supply voltage Charging time

18.6 kG x 106 2000 Amp 40 H 80 X lo6 J 100,000 lbs 700 to 1000 lbs 6000 psi 1.5 X loe lbs 1600 tons 16 ft 10 ft 10 v 2$ hrs 5

See (100-106).

With a 20% increase in price for copper coatings to stabilize the conductor, the cost of the conductor would be about $300,000. The cost of coil form, Dewar, and installation would be about $630,000 since about 200,000 tons of iron have to be used as in the case of the conventional coil. Thus, the estimated cost for the NbZr magnet would be as seen in Table 111. The table indicates that a superconducting magnet would save about $4,000,000 over 10 years and cost somewhat less than a conventional magnet. Only basic costs have been given in each case to give some idea of what is involved. Engineering, labor, substation cost for the conventional magnet, and contingencies have been omitted. The estimated cost of the chamber is $17,000,000 of which $2,000,000 is for building and $10,000,000 for the chamber. The cost of the superconductor is thus a small fraction of the cost of the system. I n fact, ANL has chosen a more

46 1

SUPERCONDUCTING MAGNET TECHNOLOGY

TABLE I11 COMPARISON O F THE ESTIMATED COSTS O F 20-90 CONVENTIONAL SUPERCONDUCTING COILSWITH IRON YOKESFOR THE ANL 12-FT I.D. HYDROGEN BUBBLECHAMBERa ANL estimate for a standard 18 kG magnet (as in proposal) Yoke (1900 tons) Coils (90 tons) copper Installation Power supply Bus bars Controls and misc. Cooling tower (12 MW)

AND

ANL estimate for standard NbZr magnet

-

-

$1,000,000 $1,000,000 Yoke 225,000 540,000 Coils 180,000 Allow 20% safety factor (i.e., increase in Amp-turns) 45,000 200,000 300,000 Dewar, coil form 50,000 70,000 Installation 25,000 150,000 Power supply 200,000 150,000 Helium refrigerator

Total capital costs $1,740,000 Total capital costs $2,380,000 Power for 10 yr (1 MW for Power for 10 yr of 50% operrefrigerator) 400,000 ation at $.008 kWh 4,000,000 Total 10-year cost

$6,400,000 ~~

Total 10-year cost

$2,140,000

~

These are preliminary estimates; the design study is completed and more realistic figures will shortly be published.

TABLE I V MAGNET P A R A M E T E R S FOR 20-KG CONVENTIONAL IRON MAGNET FOR A 1 4 - I.D. ~ ~ BNL HYDROGEN BUBBLECHAMBER

PROPOSED

Magnetic field intensity Weight of coil (copper) Weight of yoke (iron) Cross section of copper conductor Number of turns Length of copper conductor Radius of water hole Cooling water flow (30°C rise) Pump power output Coil voltage Coil current Coil power Inductance Resistance Time constant Stored energy

20 kG 141 T 2865 T 2.52 X 2.88 sq in. 270 15,400 ft 0.90 in. 1400 gal/min 82 hp 560 v 19,600 A 11 MW 0.59 H 28.6 m0 21 sec 1.1 x lo8J

~

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CHARLES LAVERICK

conservative conductor design with considerably more copper at about three or four times the price per foot of conductor. This would increase the supermagnet cost by $600,000 to $900,000 and would still favor a superconducting coil. The first BNL proposal contained comparisons of a 20-kG conventional coil and superconducting coil with iron. The latest proposal is now for an air core superconducting coil of the same diameter (108). The TABLE V EQUIPMENT AND MANPOWER COSTS FOR 20-KG COMPARISON O F ESTIMATED CONVENTIONAL SUPBRCONDUCTING COILSIN IRON YOKEFOR THE PROPOSED BNL 14-FT DIAMETERHYDROGEN BUBBLECHAMBER Conventional coil system Equipment $2,970,000 $915.000 Magnet coil Raw material, extrusion of copper, winding, insulation, and installation (242,400 lbs at $3.50/lb) $850,000 Coil connections, water manifolds, and hoses 50,000 Bus connections to power supply 15,000 Magnet yoke 1,480,000 Raw material, machined, and assembled, 5,728,000 lbs at $0.25/lb or 28,654 tons at $500/ton 1,430,000 Platforms and external structures 50,000 Power supply 460,000 Install 13.8 kV ac to 560 V regulated dc Instrumentation and misc. 115,000 Current, voltage, temperature, pressure, and flow monitoring, magnet regulation, 15,000 and safety interlocks 100,000 Misc. Manpower Scientific and professional 14 man-years Design/drafting 8 man-years All other 22 man-years Engineering, design, and inspection 176,000 manpower costs 330,000 330,000 Construction manpower costs $506,000 Total manpower costs $3,300,000 Total construction costs

-

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TABLE V (Continued) Superconducting Coil Systcni Equipment $2,992 ,000 (a) Magnet coil $650.000 Superconducting material $300,000 Normal conductor (includes processing to produce cable) 150,000 Coil structure 75,000 Support structure 125,000 (b) Magnet yoke 1 ,482,000 Raw material, machined and assembled, (5,728,000 lb at $0.25/lb or 28,654 tons at $500/ton) 1,432,000 Platforms and external structure 50,000 (c) Dewars and plumbing 150,000 (d) Radiation shield 20 ,000 (e) Power supply and controls 50 ,000 (f) Helium refrigerator 450,000 (g) Addition t o other parts 90,000 Multiple layer insulation 20,000 Vacuum tank 70,000 (h) Misc. 100,000 Manpower Scientific and professional 14 man-years Design/drafting 8 man-years All other 22 man-years Engineering, design, and inspection manpower 176,000 costs Construction manpower costs 330,000 330.000 Total manpower costs $506,000 Total construction costs $3 ,322 ,000

conventional magnet parameters are given in Table IV. Estimated costs for the conventional and superconducting coils are given in Table V. In this case, the costs are approximately the same for both systems and the saving to be achieved is the power cost for the operation of the conventional coil. Both ANL and BNL estimates are given t o show the spread in estimate for two separate studies of this type and to outline the more important considerations in each type of design as seen by two separate groups. A safe figure for the average current density in each case would be 3000

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TABLE VI ESTIMATED COSTSFOR A ~O-KG,16-FT I.D. AIR CORESPLITSUPERCONDUCTING SYSTEM FOR THE PROPOSED 14-FTI.D. BNL HYDROGEN SOLENOID BUBBLECHAMBER $3,200,000 Magnet $2,860,000 1. Equipment $2,350,000 a. Magnet Coil Superconducting material processed into stabilized cable $980,000 High purity normal conductor processed into stabilized 700 ,000 cable Metal stiffener processed into stabilized cable 456,000 Coil structure 134,000 Coil spacers 80 ,000 b. Dewar and Support Structures 400,000 Dewar, Dewar support platform support cylinder, and brackets (stainless steel 89,400 lbs at $3.30/lb) 295,000 Coil bridge, (stainless steel, 25,800 lbs at $3.30/lb) 85,000 Assembly Fixtures 20 ,000 60 ,000 c. Power Supply and Leads It 40,000 Power supply, 15 V, IO,OOO Bus bar and-connectors 20; 000 50 ,000 d. Instrumentation 340,000 2. Manpower

Amp/cm2, but ANL is considering 1000 Amp/cm2 which is even safer and more conservative. The latest BNL proposal envisages a 30-kG magnet with no iron a t very little difference in cost from the 20-kG coil with iron. Each magnet coil is 6 f t high, has an inner diameter of 1 s t f t and outside diameter of 20f f t . The two coil assemblies are separated by 21 in. to allow for beam entry into the bubble chamber and are attracted to one another by a force of the order of 10,000 tons. A conductor current of lo4 Amp is envisaged. The estimated costs for this magnet are given in Table VI.

VII. TESTFACILITIES AND INSTRUMENTATION Test facilities are required for investigating the properties of superconducting materials and magnets. These are elaborate for larger systems ; but since it is possible to predict the properties of many large magnets

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with reasonable certainty by investigating the performance of short lengths of conductor, a great deal of useful design and development work can be carried out with a small test facility. Short sample tests on a few feet of conductor in which the current-carrying capacity and performance are determined as functions of magnet field can be carried out in a small Dewar system of about 4 to 6 in. diameter. I n earlier work we used lengths of between 400 and 1000 f t of cable to determine the properties of the conductor and wrapped these on coil forms of between 1 and 6 in. diameter. This type of testing develops the workshop facilities since reasonable lengths of conductor have to be fabricated. A typical useful test system requires two vacuum insulated Dewars placed one inside the other. This is necessary since liquid helium (4°K) has to be well insulated from the room temperature environment (300°K) if it is not to be evaporated quickly. A heat leak of 1 W for 1 hour evaporates about 1 liter of liquid helium at 4.2"K. The liquid nitrogen, which is much cheaper and has a greater thermal capacity, is placed in the outer Dewar to provide an intermediate temperature environment (77°K) between the helium of the inner Dewar and the outside world. Some means of providing current is necessary and batteries with rheostats or transistor controlled supplies are necessary. Currents of up to 200 Amp are adequate for most work with smaller coils but lo3, lo4,or even lo6 Amp may be used in future systems. Voltages of 1 V or less at the higher currents are satisfactory. A minimum of two power supply systems is usually required, one for the external magnetic field and one for the specimen. Voltages across terminals, coils, or cable specimens are usually monitored during a test and a useful measuring range is from 1 pV to 100 V for the lower current systems. Test magnets to provide the magnetic field are usually superconducting coils because of their over-all convenience, compactness, and comparatively low price in small bore coils. Magnetic fields of between 60 and 100 kG in bores of 1 or 2 in. are desirable. The magnetic field can be measured most simply by using a small coil of known area and number of turns and pulling it out of the magnetic field. The magnetic field difference between the two positions is then measured on a standard fluxmeter. More convenient for continual recording of the magnetic field are devices which use the Hall Effect (Hall Plates), and magneto-resistance probes which are usually made from lengths of pure copper or bismuth wire. Rate of change of field can be recorded using a built-in pickup coil of the required sensitivity in conjunction with a recording millivoltmeter. Transients are not common in most stable, insulated coils and no special recording equipment is necessary. We find it convenient to use a 12-

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channel Visicorder in our work since this enables us to record simultaneously on a number of channels at a variety of writing speeds when desired. Large test facilities require ample head room, a crane of some sort, facilities for continuous transfer of liquid nitrogen and perhaps even helium. It should be possible to operate them with vacuum equipment or at internal pressures of one or two atmospheres. We have found it cheaper in cooldown costs to cool larger coils first with liquid nitrogen to 77'K, with liquid hydrogen to 20"K, and then with liquid helium. The installation becomes more complicated and the use of liquid hydrogen involves special care, but it is economically justifiable. More complicated test installations involve the collection, compression, and subsequent liquefying of the boiled off helium or completed closed cycle helium refrigeration systems. We have not yet used such systems, but they are essential for long-term experiments or continuously operating magnets.

FACILITIES VIII. COILFABRICATION A set of simple experimental facilities for coil development have been built up at Argonne and are mentioned here to indicate what types of equipment are desirable in this type of work. They have been developed gradually as a necessary part of the conductor and coil development program. They are as simple as possible since once a given technique is established, it can be accepted by industry if required. Existing manufactured superconductors are used in the experiments and are evaluated and modified as required to promote advances in magnet design. Commercially available winding machines have been installed for the smaller coils but a winding machine for coils up to 4 ft in diameter and 16 in. long has been designed and constructed at ANL. Heavy cables can be wound a t high tension and the tension monitored using a spring balance. Cable making machines are expensive and accepted cable designs can be made best by commercial cable makers. Superconducting wire is too valuable to be wasted in cabling experiments and it is necessary to keep every part of the process under control in an experimental cable making and development program. We solved the problem by modifying a small workshop lathe and mounting a rotating plate to hold the spools of copper or nylon which were to be braided around the central conductor. The finished braid can be cleaned or impregnated with a pure metal by passing it through the appropria.te bath immediately following the braiding operation before winding on the finished spool. The machine is also used to unwrap strands from a cable when required. The fabrication of coils from strip and the manufacture of sandwich

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type strip conductor also requires a special machine, which is under construction at ANL, since strip is not so flexible as cable and it is necessary to move the final wind-up spool axially a t the correct pitch during coil winding to spool the final strip without damage. We have found it convenient to make simple cleaning facilities in which wires are passed through cleaning solutions. Facilities for heat treatment are also desirable. Simple facilities of the type described are useful in developing and maintaining a flexible experimental program. Each step in the development of a new cable or coil system takes too long if the work is carried out with complete reliance on outside facilities.

IX. FUTURE DEVELOPMENTS It is hazardous to speculate in a field where so much has happened so quickly. Plans for larger superconducting magnet installations are being considered in many parts of the world. I n addition to the spectacular large bubble chambers and balloon borne magnet systems (104), serious thought is being given to the practicability of larger superconducting magnets for the direct conversion of electrical power from flames or plasmas (Magneto-Hydrodynamic Power Generation). M H D systems may result in a dra.stic reduction in cost of electrical power throughout the world in the years to come. The British Electricity Authority is considering the possibility of systems with an energy storage of 1O'O J. (109). An 80-kG, 5-ft bore coil is being considered by one set of experimenters in a European High Energy Physics program. Stanford Linear Accelerator Center is now planning a 70-kG, 12-in. bore split coil system. Superconducting beam transport solenoids in lengths up to 60 f t have been proposed (110) and are the subject of design studies. Superconducting magnetic shields for space vehicles are being studied and proposed (76) ; feasibility studies have already been carried out. The possibility of using superconducting magnets to prevent communications blackout on reentry of a vehicle from space has been discussed. Superconducting bending magnets and focusing magnets have been operated and will further be developed. Superconducting magnetic shields have been operated and systems using this principle are being considered in beam extraction schemes that may cause considerable reductions in the price of variable energy cyclotrons. Consideration is being given to the use of high magnetic fields in medicine to induce anesthesia and this would involve superconducting coils. The possibility of ultrahigh energy accelerators with pulsed superconducting coils is being considered although obviously many problems have to be solved.

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The proposed 200-GeV accelerator has experimental facilities which are 2 miles long and require a very large number of magnets of every type, including bending, focusing, and detector coils, with consequent consumption of large quantities of electrical power. Superconducting coils may be used extensively in this facility to reduce power costs. The cost of electrical power is becoming of less importance than its availability in many large laboratories, and even in some countries. Operation of the MIT Magnet Laboratory's 255-kG., 04411. bore coil, for instance, requires about of the town's electrical power generating capacity. The superconducting magnet will obviously exert a strong influence on the cost and development of many programs. One aspect of pulsed magnet development to which strong programs are directed is that of energy storage. Superconducting coils offer the most economical form of energy storage where large energies are involved. Some programs require the discharge of this energy from the coil in very short periods of time. Thus, the problems of dealing with high currents, achieving high current densities in small volumes, pulsing superconducting coils, and removing large percentages of stored energy are being studied from this viewpoint, A dramatic possibility which is under study in Japan is that of using such energy storage systems to balance the load of large power stations so as to eliminate the variations in daily demand. Such a system may involve the storage of up to 1OI2 J and its transfer to the supply line at a rate of 10" J/hour. While other methods of storage may be cheaper after due consideration, the fact that such systems are under consideration is indicative of the interest being aroused in this new technology. Conductor developments are continuing a t a rapid pace. The group directed by D. C. Freeman of the Union Carbide Corporation has developed a plasma-spraying technique that permits the coating of unusual shapes with NbrSn and offers a promise of great savings in cost. Small solid sintered cylinders of NbaSn have been made and offer the possibility of providing larger magnets with high stability permanent fields that can be developed inside the sintered cylinders by the use of an auxiliary coil. The cylinders can be removed from the charging system together with the trapped magnetic field which will remain in the cylinder as long as the system is cold. The potential current density in suitably designed solid cylinders of NbBn is in excess of 100,000 Amp/cm2 at 100 kG. Materials research is also continuing on an intensive scale in the hope of increasing critical temperatures and fields so that magnets can be operated without liquid helium but with cheaper, higher temperature cryogenic fluids, increasing upper critical fields and current densities. The development of thin film conductors is an important avenue which may permit the realizition of some of these goals.

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The scale and rate of development OI these magnets and their application to practical problems are continuing to increase as their properties and potential become known to a n ever widening circle of scientists and technologists in many countries. Note: Superconducting magnet developments and applications have continued a t a rapid pace since the preparation of this article. Funding for this purpose has increased considerably and consequently, larger groups are now involved in applying them in various fields of research in many countries. A final article in preparation for the succeeding volume will discuss more recent developments, changes in design philosophy, and possibilities for the future.

GENERALREFERENCES AND REVIEWS 1. J. Bardeen, Theory of superconductivity, “Encyclopedia of Physics,” Vol. 15, Springer, Berlin, 1956. 8. T. G. Berlincourt, High magnetic fields by means of superconductors, Brit. J . Appl. Phys. 14, 9 (1963). 3. P. G. deGennes, “Superconductivity of Metals and Alloys,” Benjamin, New York, 1966. 4. B. B. Goodman, Rev. Mod. Phys. 36, 12 (1964). 6. B. B. Goodman, Type I1 superconductors, Report in Progress in Physics (1966). 6. H. Kolm, B. Lax, F. Bitter, and K. Mills, (eds.), “High Magnetic Fields,” M.I.T. Press, Cambridge, Massachusetts and Wiley, New York, 1962. 7. F. London, Superfluids 1 (New York, 1950). 8. C. Laverick, Cryogenics 6, 152-158 (1965). 9. J. Lowell, Cryogenics 6, 185 (1965). 10. E. A. Lynton, “Superconductivity,” 2nd ed. Methuen, London, 1966. 11. B. Montgomery, Generation of intense magnetic fields, J. Appl. Phys. 36, 893 (1965). 18. Proc. Intern. Conf. Sci. Superconductivity, published in Rev. Mod. Phys. 36, 1-331 (1964). 13. Proc. Intern. Symp. Magnet Technol., Slanford, 1966. AEC Conf. 650922, January 1966. 14. J. R. Schrieffer, “Theory of Superconductivity,” Benjamin; New Yo&, 1964.

16. B. Serin, “Superconductivity Experimental P&rt,” Encyclopedia of Physics, vol. XV, Springer-Verlag, 1956. 16. D. Schoenberg, “Superconductivity,” 2nd ed. Cambridge Univ. Press, London and New York, 1952.

REFERENCES 1. See General References and Reviews above for comprehensive bibliography and extensive theoretical and historical treatment. 8. B. T. Matthias, Superconductivity: Recent results, Talk presented at the uniU. of Chicago Colloq. Chicago, Illinois, April 7, 1966. 3. C. P. Bean, Phys. Rev. Letters 8, 250 (1962). 4 . C. P. Bean and M. V. Doyle, J. Appl. Phys. 33, 3334 (1962). 6. K. Yasucochi, T. Ogasawara, N. Usui, H. Kobayashi, and 8. Ushio, Effectof external currents on the magnetization of nonideal type I1 SUperconductors, J. Phys. Soc. Japan 21, 89 (1966). 6. J. D. Livingston and H. W. Schadler, G. E. Res. Rept. No. 6 4 - R L - 3 7 6 5 ~1964. ,

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Colloq. on Intense Magnetic Fields, their production, their application, Centre Nationale de la Recherche Scientifique, Grenoble, September, 1966. 30. J. E. Kunzler, i n “High Magnetic Fields,” p. 574. M.I.T. Press, Cambridge, Massachusetts, and Wiley, New York, 1962. 31. J. B. Vetrano and R. W. Boom, J . Appl. Phys. 36, 1179-1181 (1965). 32. C. N. Whetstone, G. G. Chase, J. W. Raymond, J. B. Vetrano, and R. W. Boom, Thermal stability for Ti 22% Nb superconducting cables and solenoids, Proc. Intern. Magnet Conf., Stuttgart, April, 1966. 33. R. G. Treuting, J. H. Wernick, and F. S. L. HSU,in “High Magnetic Fields,” p. 597. M.I.T. Press, Cambridge, Massachusetts, and Wiley, New York, 1962. 34. R. D. Cummings and W. N. Latham, An experimental measurement of dependence of the H-I curve of NbZr wire in temperature, J . Appl. Phys. 36,2971 (1965). 36. Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Reu. Letters 9, 306 (1962), Phys. Rev. 129, 528 (1963). 36. S. L. Wipf and M. S. Lubell, Phys. Letters 16, No. 2, p. 103 (1965). 37. C. Laverick, Experiments with superconducting cable, Proc. Intern. Symp. Magnet Technol. Stanford, 1966, p. 560. AEC Conf. 650922, 1966. 38. C. E. Taylor and C. Laverick, A superconducting ‘Minimum B’ magnet, Proc. Intern. Symp. Magnet Technol., Stanford, 1966, p. 594. AEC Conf. 650922. 39. S. H. Minnich, Dynamic protection of superconductive coils, Intern. Aduan. Cryog. Eng. 11, 675 (1966). 40. H. T. Coffey, J. K. Hulm, W.T. Reynolds, D. K. Fox, and R. E. Span, A protected 100 kG superconducting magnet, J . Appl. Phys. 36, 128-136 (1965). 41. J. R. Purcell and H. Brechna, Meeting on the design of large superconducting bubble chamber, (C. Laverick, ed.), ANL Rept. 7192, November 18, 1965. 42. D. N. Cornish, Superconducting coils using stranded cables, UKAEA, Culham Lab. Rept. CLM-P83, United Kingdom Atomic Energy Authority, 1965. 43. Z. J. J. Stekly, Thermal characteristics of short samples of superconductors and their effect on coil behavior, Proc. Intern. Symp. Magnet Technol., Stanford, 1966, p. 550. AEC Conf. 650922. 44. C. Laverick, G. M. Lobell, J. R. Purcell, and J. M. Brooks, Use of aluminum in superconducting cables, Rev. Sci. Instr. 37, 806-807 (1966). 46. Proc. Intern. Symp. Magnet Technol., Stanford, 1966. AEC Conf. 650922. 46. H. Kolm, B. Lax, F. Bitter, and R. Mills, (eds.). “High Magnetic Fields.” M.I.T. Press, Cambridge, Massachusetts, and Wiley, New York, 1962. 47. J. Hord, Stress analysis of tape wound magnet coils, J . Res. Natl. Bur. Std. C69, p. 287 (1965). 48. V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i Teor. Fiz. 20, 1064 (1950). 49. T. H. Fields, High magnetic fields, Nucl. Znstr. Methods 20, 465-476 (1963). 60. P. F. Smith, Rutherford Lab. Private communication. 1966. 61. D. A. Gandolfo and C. M. Harper, Study of properties of high field superconductors-AC field induced flux jumps, Tech. Sum. Rept., Contract NAS 8-11272. NASA, Washington, I). C., 1966. 62. H. A. Ullmaier, AC measurements on hard superconductors, Oak Ridge Natl. Lab., Oak Ridge, Tennessee, 1966. 63. R. Domchick, Airborne Instruments, Private communication. 64. R. B. Flippen, Superconductive transitions in fast-pulsed magnetic fields, Phys. Rev., 137, No. 6 8 , (1965). 66. P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39 (1964).

472

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66. L. Donadieu, Compagnie Generale d’Electricite, France, Private communica-

tion, 1966. 67. H. T. Coffey, J. K. Hulm, W. T. Reynolds, D. K. Fox, and R. E. Span “A protected 100 kGsuperconducting magnet, J . Appl. Phys. 56, No. 1, p. 128 (1965). 68. D. L. Martin, M. G. Benz, C. A. Brach, and C. H. Rosner, Cryogenics S, 114 (1963). 69. Z. J. J. Stekly et al., A large experimental superconducting magnet for MHD power generation, Meeting Intern. Inst. Refrig., Bodder, Colorado, June, 1966. 60. J. C. Laurence and W. D. Coles, A superconducting magnetic bottle, Adban. Cryog. Eng. 11, 643 (1966). 61. J. C. Laurence and W. D. Coles, Design, construction and performance of cryogenically cooled and superconducting electromagnets, Proc. Intern. Symp. Magnet Technol., Stanford, 1966, p. 574. AEC Conf. 650922. 62. J. Reese Roth, D. C. Freeman, and D. A. Haid, Proc. 6th Symp. Engineering Aspects of MHD, Pittsburgh, April, 1966. Rept. No. Conf. 650436, Am. Inst. of Aeronautics and Astronautics. 69. C. H. Rosner and M. G. Bena, Stranded niobium-tin wire solenoids design and fabrication techniques, Proc. Intern. Conf. Magnet Technol., Stanford, p. 597, 1965. AEC Conf. 650922. 64. H. Schindler, Private communication, 1966. 66. W. B. Sampson, 112 kG superconducting magnet, Rev. Sci. Instr. 56, 565 (1965). 66. W. B. Sampson, A high field superconducting solenoid for magnetic moment measurements on short lived particles, Proc. Intern. Symp. Magnet Technol., Stanford, 1966, p. 530. AEC Conf. 650922. 67. New superconducting magnetic lens tested successfully at the AEC, Brookhaven Natl. Lab., AEC News Release No. H-203, Sept. 1965. 68. P. Gerald Kruger, W. B. Sampson, and R. B. Britton, A superconducting quadrupole lens, BNL Rept. No. AADD-104, March 30, 1966.

J. Snyder, and W. B. Sampson, Digital calculations of magnetic fields owing to currents in NbsSn ribbon in superconducting quadrupole lens, BNL Rept. No. June, 1966. 70. H. Fernandez-Moran, Proc. Nat. Acad. Sci. U.S., 65, 445451 (1965). 71. H. Fernandez-Moran, Science and Technology, in “1966 McGraw-Hill Year Book.” McGraw-Hill, New York, 1966. 72. Proc. ANL-AMU Electron Microscope Workshop. ANL Rept. 7275, p. 51. Argonne Nat. Lab., January, 1967. 79. H. Fernandea-Moran, Low temperature electron microscope with high field superconducting lenses, 6th Intern. Conf. Eleclron Microscopy, Kyoto, Japan,

69. P. Gerald Kruger,

.

1966. 74. L. Donadieu and J. Royet, A 4500 Gauss superconducting magnet for corpuscular ionography experiments used a t CERN, Proc. Zniern. Conf. Magnet Technol. Stanford, 1965, p. 604. AEC Conf. 650922. 76. S. R. Hawkins, A six-foot superconducting magnet system for magnetic orbital satellite shielding, Intern. Advan. Cryog. Eng. 10, 124 (1965). 76. J. L. Olsen, Superconducting rectifier and amplifier, Rev. Sci. Instr. 29, 537 (1958). 77. T. A. Buchhold, Superconductive power supply and its application for electrical flux pumping, Cryogenics 4, 212 (1964). 78. H. L. Laquer, An electrical flux pump for powering superconducting magnetic coils, Cryogenics, 5, 27 (1963).

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79. J. Volger and P. S. Admiraal, A dynamo for generating a persistent current in a superconducting circuit, Philips Tech. Rev. 26, 16 (1963-1964). 80. S. L. Wipf, A nuperconducting direct current generator, Advan. Cryog. Eng. 9, 342 (1964). 81. 8. L. Wipf, Flux pumps as power supplies for supcrconducting coils, Proc. Intern. Conf. Magnet Technol., Stanford, 1965, p. 615. AEC Conf. 650922, 1966. 82. D. van Houwelingen, A. L. Luiten, and J. Volger, A superconducting dynamo and solenoid combination operation a t high current level, Advan. Cryog. Eng. 12 (1967). 83. T. A. Buchhold, Superconductive rectification and its use for flux pumping, Meeting Intern. Inst. Refrig., Boulder, Colorado, June, 1966. 84. C. Laverick and E. G. Pewitt, Superconducting joint studies, ANLAD Rept. No. 78, June, 1964, Argonne Natl. Lab. 86. E. J. Lucas, 2. J. J. Stekly, C. Laverick, and E. G. Pewitt, Current transfer in contacts involving superconductors, Intern. Advan. Cryog. Eng. 10, 113 (1965). 86. S. T. Sekula, R . W. Boom, and C. J. Eergeron, Longitudinal critical current in cold-drawn superconducting alloys, Appl. Phys. Letters 6, 102-104 (1963). 87. C. Laverick, The performance of large superconducting coils, Intern. Advan. Cryog. Eng. 11, 659 (1966). 88. J. R. Purcell, Private communication. To be published ANL. 89. W. B. Sampson, M. Strongin, A. Paskin, and G. M. Thompson, Appl. Phys. Letlers 8, 191 (1966). 90. R. W.Meyerhoff and B. H. Heise, Dependence of current degradation in superconducting solenoids on flux pinning energy, Conf. Phys. Type I I Superconductivity, Western Reserve Univ., Cleveland, Ohio, paper 32, August, 1964. 91. D. N. Cornish and J. E. C. Williams, Eight-inch aperture superconducting coils, Cryogenics 6, 132-135 (1965). 92. It. E. Hintz and C. Laverick, An inorganically insulated superconducting solenoid for operation at 1"K, Intern. Symp. Magnet Technol., Stanford, 1966,p. 568. AEC Conf. 650922, January. 93. M. W. Garrett, J . Appl. Phys. 22, 1091 (1951). 94. H. L. Marshall and H. E. Weaver, J . Appl. Phys., 94, 3175 (1963). 96. P. Grivet and M. Sauzade, A high homogeneity-high field superconducting magnet, Proc. Intern. Conf. Magnet Technol., Stanford, 1966. AEC Conf. 650922. 96. C. Huang and C. Laverick, Unpublished work. 97. H. desPortes and B. Tsai, A homogeneous field superconducting magnet for a polarized proton target, Proc. Intern. Conf. Magnet Technol., Stanford, 1965. AEC Conf. 650922. 98. W. B. Sampson, High field superconducting magnets, Invited paper presented Intern. Magnet Conf. Stuttgart, Germany, April, 1966. 99. M. H. Foss, Proc. ANL-AMU Electron Microscope Workshop June, 1966. To be published. 100. G. P. Haskell and P. F. Smith, The economics of superconducting magnets in high energy physics, Natl. Inst. Res. Nucl. Sci. NIRL/R/63 (Gt. Brit.) (1963). 101. Roland W. Schmitt and W. A. Morisen, Economic aspects of superconductivity, G. E. Schenectady, New York, July, 1966. 102. J. Reece Roth, Optimization of adiabatic magnetic mirror fields for controlled fusion research, N A S A Tech. Memo. TMX-1261,(1966). 103. J. Reece Roth, D. C. Freeman, and D. A. Haid, Superconducting magnet facility for plasma physics research, Rev. Sci. Znstr. 36, 1481-1485 (1965).

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W.Alvarez, Proposal for high altitude particle physics experiment, Space Sci. Lab., Univ. of California, Berkeley, UC. BSSL No. 9. 106. Proposal for construction of 14-foot diameter liquid hydrogen bubble chamber for use at the Brookhaven alternating gradient synchrotron, Phys. Dept. Brookhaven Natl. Lab., BNL-8266, June, 1964 (revised June, 1965). 106. Proposal for the construction of a 12-foot hydrogen bubble chamber, High Energy Phys. Div. Argonne Natl. Lab. June, 1964. 107. T. H. Fields, Some estimates of magnet cost, Argonne Natl. Lab. Memo., ANLBBC-31, September, 1964. Internal unpublished. 108. 14-Foot cryogenic bubble chamber project, Final Pre-Title I Rept., Phys. Dept., Brookhaven Natl. Lab., December, 1965. 109. P. F. Chester, Superconducting magnets for MHD generators, Meeting, Royal Society M H D CERL, Leatherhead, Surrey, England, November, 1965. London. 110. M. H. Foss and R. Sutton, Carnegie Inst. of Technol., Private communication. 104. L.

Author Index Numbers in parentheses are reference numbers and indicate that a n author's work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed.

A

Baum, W. A., 348(5), 352, 353, 357(25), 358, 360(5), 382, S83 Abragam, A., 41, 64(75, 80), 66, 72(5), Bean, C. P., 31, 37, 400, 469 Beaty, E. C., 93(128), 160 146, 148 Becking, A. G. T., 307, 309(10), 346 Abrahams, E., 33, 37 Bell, W. E., 42,48(34), 86(9), 94(140), 99, Abrikosov, A. A., 13, 14, 19, 36 146,147,160 Adler, R. B., 312(22), 344, 346, 346 Bellier, M., 357, 386 Admiraal, P. S., 445, 473 Bender, P. L., 41,93,94,96(147), 98, 103, Agouridis, D., 326, 346 104, 107(160), 146, 160 Aitken, M. J., 68, 148 Bene, G. J., 50(39), 147 Aliev, Yu. M., 229, 232, 301, 298 Benoit, H., 69, 71(96), 72(88), 75, 149 Alldredge, L. R., 45(26), 49(38), 147 Benroff, H., 48(32), 147 Aller, L. H., 350, 357, 382 Benz, M. G., 430(58), 439, 476 Alley, C. O., 90, 96(151), 160 Bergcron, C. J., 449(86), 47s Alvarea, L. W., 457(104), 467(104), 474 Anderson, P. W., 25, 32, 33, 36, 37, 428, Berlincourt, T. G., 401 (96), 469, 470 Bernheim, It. A., 77(102b), 96, 149, 160 4 71 Bernstein, H. J., 62(73), 66(73), 148 Angell, J. B., 312(22), 346 Besson, R., 66(84a), 148 Arditi, M., 84(107), 94, 98, 149, 160 Birdsall, C. K., 246, 251, 299 Arnold, A. J. P. T., 144(187), 261 Bitter, F., 401(9), 469, 470 Arnold, J. T. I., 48, 147 Blaquiere, A., 58, 60, 61, 74(58), 148 Aron, P. Q., 402(14), 451(14), 470 Bloemberger, N., 71, 149 Arsac, J., 357(18), 382 Bloom, A. L., 42, 48(4), 86, 93, 94(140), Arsenev, Yu. I., 251, 300, 298, 300 95, 99, 102, 104, 128, 132, 138(131), Askaryan, G. A., 155, 165, 222, 266, 268, 146, 147,160 270, 698, 300 Bogolyubov, N. N., 159,239 Augarde, R., 355(13), 386 Bon Mardion, G., 21, 36 Autler, S. H., 401, 470 Bonnet, G., 58(58), 60(58), 61(58), 73, 74(58), 148 B Boom, R. W., 402(18), 405, 406(32), 449(86), 451(3i), 4ro,471,4r3 Baelde, A., 308, 309(15), 346 Boot, H. A. H., 155, 156, 164, 208, 260, Bakos, G. A., 369, 383 899, 301 Bardeen, J., 3, 6, 28, 29, 30, 36, S6, 469 Borcherds, P. H., 30, 36 Bardet, R., 282, 283, 298, 699 Borghini, M., 64(80), 148 Barker, W. A., 64(78), 148 Bouchiat, M. A., 96(153), 112(153), 160 Barrat, J. P., 94, 99, 160, 161 Bowen, I. S., 359, 383 Basov, N. G., 71(95), 149 Boyd, R. L. F., 47, 147 475

476

AUTHOR INDEX

Brach, C. A., 430(58), 472 Bravo-Zhivotovsky, D. M., 242, 246,899 Brechna, H., 417(41), 424(41), 471 Breckinridge, J. B., 355, 357(12, 27), 389, 383 Brewer, G. R., 94(141), 160 Britton, R. B., 440(68), 478 Brooks, J. M., 420(44), 471 Brossel, J., 86, 95(145), 112(145, 177), 149, 160,161 Brown, L. S., 188, 899 Brown, R. J. S., 62, 64(74), 148 Brown, S. C., 260, $99,SO8 Bruce, P. H., 302 Bruncke, W. C., 314(23, 27), 322(37), 323, 326, 327(27), 346,346 Buchhold, T. A.,445, 446(83), 479,473 Buchsbaum, S. J., 190, 194, 301 Buehler, E., 401(12), 470 Burrows, K. J., 44(22), 147 Butler, H. S., 250, 899 Butler, J. W., 221, 253, 899,302 Butslov, M. M., 363, 383

C Cackenberghe, J., 45(24), 147 Cadart, M., 899 Cagnac, B., 94, 99,160 Cahill, L. J., Jr., 48(33a), 53, 147, 148 Cain, J. C., 53(46), 53(47), 148 Campbell, W. J., 48(33), 147 Canobbio, E., 184, 899 Capellaro, D. F., 377(59), 884 Cardona, M., 21, 23, 27, 36 Carbon, D. E., 25, 36 Caroli, C., 16, 27, 36 Carver, T. R., 84(107), 86(114), 89(114), 94, 95, 108,149,160 Causse, J. R., 368(40), 383 Champliu, K. S., 320, 346 Chandrasekhar, B. S., 402(19), 416(19), 470 Chapman, S., 47, 147 Chase, G. G.,406(32), 471 Chenette, E. R., 309(19, 20), 322(19, 37, 38), 323, 326(20), 329, 330(46), 333 (47), 334(46), 346,346 Cheremisin, F. G., 265, 899

Chester, P. F., 467(109), 474 Cheverev, N. S., 251(898), 300(898),898 Chi, A. R., 93(128), 160 Chuvaev, K. K., 363(33), 383 Clark, G. L., 245, 308 Clark, W. L., 106(170), 161 Clauser, M. U.,208, 302, $99 Clogson, A. M., 248, 299 Coffey, H. T., 415(40), 430(57), 471,478 Cohen-Tannoudji, C., 77(102a), 86, 93 (132), 94,96(148), 99, 100, 112(177), 135(121), 149,160,161 Colegrove, F. D., 84(106), 94(137), 104, 106, 108, 110(106), 149, 160,161 Coles, W.D., 439(60), 439(61), 47.9 Collins, L. F., 339, 346 Combrisson, J., 41(5), 66(5), 66(86), 69 (86), 70(89), 71, 72(5), 72(86), 146, 148, 149 Comentez, G., 302 Consoli, T., 158, 181, 194, 220, 221, 253, 256, 259, 278, 279, 282(898), 283 (898),293, $98, 899 Cooke, H. F., 325, 346 Cooper, L. N., 3(3), 6(3), 36 Cope, A. D., 369(42), 383 Corenzwit, E., 401 (7), 470 Cornish, D. N., 418(42), 453, 454(91), 471,473 Cribier, D., 17, 36 Cummhgs, R. D., 409(34), 410(34), 471 Cushing, V. S., 202, 899

D Davidovits, P., 84(107a), 149 Dayem, A. H., 32, 37 DeBlois, R. W., 32, 37 de Bruyn-Ouboter, R., 144(187), 161 de Gennes, P. G., 4, 15, 16(14), 18, 21, 23, 26(39), 27(14), 28, 30, 31, 32, 36,36, 37, 469 Dehmelt, H. G., 42, 80(8), 85, 86, 94 (138), 96, 99, 108, 112(146, 176), 149,160,161 Delone, G. A., 287, 300 De Sorbo, W., 31, 32, 37 des Portes, H., 455, 473 de Winter, T. A., 403(21), 415(21), 451 (211, 470

477

AUTHOR INDEX

DeWitt, D., 330(46), 334(46), 346 de Zafra, R. L., 86(115), 149 Dicke, R. H., 96,160 Dietz, R. D., 372, 384 Dingle, R. B., 13, 36 Dolginov, S. S., 44(15), 147 Domchick, R., 427(53), 471 Donadieu, L.,428(56), 443(74), 472 DOW,D. G., 302 Doyle, M. U., 400(4), 469 Driscoll, R. L., 41, 98(160b), 146, 160 Dubeck, L., 15, 37 Duchesne, M., 355, 357(19), 382 Dunlap, J. R., 370, 383 Dupas, L., 899 Duprd, M. F., 355, 357(18), 382

E Early, J. M., 312(22), 346 Ebers, J. J., 309, 346 Eder, G., 235, 300 Eliseev, G. A., 251(298), 300(298), 298 Emberson, D. L., 368(39), 383 Emslie, A. G., 86(112), 90, 95(112), 149 Ensberg, E. S., 96(149), 112(176), 160, 161 Epstein, M., 44, 147 Erbeia, A., 41(4), 48(4), 62(70), 146, 148 Erickson, E. F., 374(57), 384 Ericson, M., 302 Esipov, V. F., 371(51), 383

F Faber, T. E., 31, 37 Faini, G., 58(59), 60, 61, 148 Fairbank, WZM., 240(301), 301 Fakan, J. C., 402(15), 470 Farnoux, B., 17(16), 36 Fassett, J. R., 304(4), 346 Fawcett, E., 30(47), 36 Fellgett, P., 348(3), 353, 368(3), 382 Fernandez-Moran, H., 442, 472 Fessenden, T., 169, 173, 300 Feynman, R. P., 144,161 Field, G. B., 108(173), 161 Fields, T. H., 427(49), 460(107), 471, 474 Fink, H. J., 32, 37

Firor, J., 371, 384 Fisher, R. R., 372(55), 384 Fitch, J, L., 51(45a), 52(45a), 147 Flippen, R. B., 428(54), 471 Flory, L. E., 369(42), 383 Folkes, J. R., 361, 384 Fonger, W. H., 328, 346 Ford, W. K., Jr., 363(32), 383 FOES, M. H., 456, 467(110), 473, 474 Fowler, T . K., 229, 300 Fox, D. K., 415(40), 430(57), 471, 472 Francis, P. D., 51(56), 61, 148 Franken, P. A., 106,161 Franz, F. A., 94(143), 160 Franzen, W., 86, 90, 95, 96(150), 149, 160 Fraser, M. J., 401(9a), 470 Fredrick, L. W., 371, 383 Freeman, D. C., 439(62), 457(103), 472, 473 Frenkel, V. Ya., 181, 184, 187, 278, 301 Friedel, J., 26, 36 Friis, H. T., 344, 346 Fromm, W. E., 44(18), 147 Fukui, H., 325, 346 Fuortes, A., 58(59), 60(59), 148

G Gandolfo, D. A,, 427(51), 471 Ganson, A., 357(24, 25), 382 Gapgnov, A. V., 164, 165, 266, 300 Garrett, M. W., 455, 473 Gauthier, R. V., 368(40), 383 Geballe, T. H., 401(7), 470 Gekker, I. R., 283, 285, 300 Geller, R., 282(898), 283(898), 898, 899 Geller, S., 401(7), 470 Gendrin, R., 60(65), 148 Gerard, V. B., 94(142), 160 Geyger, W. A., 44(23), 147 Giacoletto, L. J., 307, 313, 330(11), 346 Gibbons, J. F., 334, 346 Gildenburg, V. B., 222,266,267,270,272, 300 Ginzburg, V. L., 6, 36,427(48), 471 Gittleman, J., 21(19), 35, 36 Glagolev, V. M., 251(898, S o l ) , 298, 300, 301

478

AUTHOR INDEX

Goldenberg, H. M., $001 Good, M. L., 154, SO0 Goodman, B. B., 9, 21(21), S6,469 Goodrich, G. W., 376, S84 Gorbunov, L., 229, 233, SOO, 301 Gordeev, G. V., SO2 Gordon, G. D., 302 Gordon, J. P., 65, 148 Gor'kov, L. P., 6, 21, 25, 32, 33, S6 Gough, C. E., 30(48), 96 Gourber, J. P., 94(144), 127, 160 Grant, G. R., 374(57), S84 Gray, P. E., 330(46), 334(46), ,946 Greenhow, R. C., 111(174a), 161 Greenstein, L. S., 44(17), 147 Grivet, P., 44, 51(44), 52(44), 58(16, 58), 60(58, 63), 61(58), 71(91), 74(58), 75(16), 84(91), 99(16), 146,147,148, 455(95),47s Grosse, A., 355, S82 Grossetete, F., 96(154), 160 Gubanov, A. I., 302 Guggenbuehl, W., 320, 346 Guinau, T., 144(187),161 Gunter, W. D., Jr., 374, 384 Gurevich, A. V., 274, 300, 301

H Haaland, C. M., SO2 Haenssler, F., 31, S7 Hahn, E. L., 55(55), 59(60), 148 Haid, D. A., 439(62), 457(103), 472, 473 Hake, R. R., 401,470 Hall, J. S., 348(5), 352(5), 353(5), 360(5), 382 Hall, R. B., 181, 278, 279, 293,299, SO2 Halladay, H. E., 314(27), 327(27), ,946 Hannaford, W. L. W., 44(21), 147 Hansen, 0. Kofoed, 235(301), 301 Hanson, G. H., 322, 323(36), S46 Hanson, R. J., 51(45), 52(45), 147 Harper, C. M., 427(51), 471 Harris, E. G., 300 Harris, F. K., 50(38a), 147 Harrison, E. R., 53, 148 Haskell, G. P., 457(100), 460(100), 47s Hatch, A. J., 221 (299),253, 299, SO2 Haus, H. A., 304(1), 344, S46,348 Hawkins, S. R., 443(75), 467(75), 472

Hawkins, W. B., 80(104), 112(178), 149, 161 Hays, D. A,, 31, 37 Heffner, H., 248, 299 Heims, S . P., 54, 148 Heine, V., 34, 37 Heise, B. H., 454(90), 475 Hempstead, C. F., 25(37), 26(37), 36,409 (351, 471 Hennequin, J., 75, 149 Heppner, J. P., 44(14), 45(27), 93(49), 105, 107(49), 146, 147, 148, 161 Hernandez, H. P., 402(14), 451(14), 470 Hielscher, F. H., 314(54), 327(54), S46 Hiltner, W. A., 348(2), 357, 881, 382 Hinder, G. W., 364(35), ,983 Hindmarsh, W. R., 52(45b), 147 Hintz, R. E., 455(92), 47s Hipple, J. A., 41, 146 Hitchcock, H. C., 402(14), 451(14), 470 Hochstrasser, G., 41(34), 48(4), 62(70), 146,148 Holtz, E. A., 283(300), 300 Honig, A., 70(89), 149 Honig, R. E., 103 Hopkins, R. E., 377(60), S84 Hord, J., 424(47), 471 HSU,F. S . L., 401(12), 406(33), 417(33), 453(33), 470, 471 Huang, C., 455(96), 473 Hulm, J. K., 401, 402(19), 415(40), 416 (19), 430(57), 470, 471, 472 Hyman, J. T., 241, SO1 Hynek, J. A., 370, 98s

I Ichtchenko, G., 299 Iredale, P., 364, 983 Iriarte, B., 359(31), 383 Ishizuka, H., SO2 Izyumova, T. G., 86(118), 149

1 Jacquot, B., 299 Jacrot, B., 17(16), 56 Jakleavic, R. C., 143, 161 Jaynes, E. T., 54, 148

479

AUTHOR INDEX

JelTries, C. D., 64(7Y), 148 Jensen, T. Hrsselberg, 235(301), 301 Johnson, E. J., 31, 37 Johnson, H., 316, 346 Johnson, H. L., 359, 383 Johnson, K. H., 309(20), 326, 345 Johnston, T. W., 157, 222, 300 Jones, R. C., 353, 368(10), 382 Jordan, A. G., 316, 346 Jordan, N. A., 316, $46 Josephson, B. D., 27, 32, 36, 37 Jovnovich, M. L., 155(298), 298 Judge, D. L., 44(13), 146 Jung, J., 45(24), 147

K

Kulinsky, S., 176, 300 Kunalcr, J. E., 401,402(13), 405(30), 454, 470,471

Kuprevich, N. F., 372, 384 Kurochkin, S., 74, 149 Kurwitr, L., 45(25), 147 Kushnir, A. J., 31, 37

L Lacaze, A., 21(21), 36 Lallemand, A., 347, 355, 357, 371, 381, 382, 383

Lambe, J., 143(182), 161 Landau, L. D., 6, 36, 188, 199, 270, 300, 427(48), 471

Kadomtsev, B. B., 253, 301 Kantrowitr, A. R., 404(27), 417(27), 452, 470

Kapany, N. S., 377(59), 384 Karplus, R., 116(180), 161 Kastler, A., 42, 77, 85, 86, 95, 108, 112 (145), 146, 145, 160

Kelly, J. C. R., 302 Keonjian, E., 328(41), 346 KhiliI, V. V., 251(301), 300(301), 301 Khogali, A., 357(24, 25), 382 Kibble, T. W. B., 188, 299 Kim, Y. B., 22, 25, 26, 30(47), $6, 409 (35), 428, 471

King, E. R., 49, 147 Kino, G. S., 250, 299 Kinsel, T., 21, 36 Kittel, C., 66, 68, 148 Klose, G., 58(57), 61, 148 Knechtli, R. C., 302 Knox, F. B., 154, 155, 221, 223, 227,300 Kobayashi, H., 400(5), 469 Kolm, H., 401(9), 469,47O Kolondra, F., 403(23), 470 Kondratev, I. G., 270, 300 Kononov, B. P., 283, 300 Konstantinova, T. G., 283(300), 285 (300),300 Kopylov, I. M., 363(33), 383 Kron, G. E., 355(12), 357, 382, 383 Kruger, P. Gerald, 440(68), 441(69), 472 Krustalev, 0. A., 100(164), 161 Kuhn, H. G., 88

Landesman, A. J., 68(87), 148 Langmuir, R. V., 302 Lapierre, A. Blanc, 60(62), 148 Laquer, H. L., 445, 472 Latham, W. N., 409(34), 410(34), 471 Laverick, C., 402(16, 17), 403(21, 22, 25, 26), 404(29), 405(16), 411(16), 413 (16), 414(17, 37, 38), 415(21), 416 (17), 417(22, 25, 37), 418(37), 420 (44), 421(25, 26), 422(37), 430(22, 25), 435(22), 437(38), 448(84), 449 (85), 451 (17, 21, 87), 455(17, 92, 96), 469,470, 471,473 Lawrence, J. C., 439(60, 61), 472 Lax, B., 401(9), 469, 470 Leaton, B. R., 46(28a), 147 le Borgne, E., 116(181), 136(181), 161 Le Gardeur, R., 253, 256(299), 299 Leighton, R. B., 144(185), 151 Lemoine, H., 66(84a), 148 le Mouel, J., 116(181), 136(181), 161 Leroy, L., 299 Leslie, D. H., 401(9a), 470 Levin, M. L., 155, 268, 298, 300 Lichtenberg, A. J., 251, 299 Lifshitr, E. M., 6(7), 36, 188, 199, 270,

300 Lindenfeld, P., 15(66), 37 Litvak, A. G., 228, 243, 300 Livingston, J. D., 31, 37, 400, 469 Livingston, R. S., 402(18), 470 Livingston, W. C., 366, 369, 379, 383, 384

480

AUTHOR INDEX

Lobell, G. M., 403(22), 417(22), 420(44), 430(22), 435(22), 470, 471 London, F., 3, 4, 36, 469 Longmire, C. L., 229, 301 Lorber, H. W., 302 Lory, A., 51 (44), 52(44), 147 Losche, A., 59(61), 148 Lowell, J., 469 Lowes, F. J., 52(45b), 147 Lubell, M. S., 409(36), 471 Lucas, E. J., 403(21), 415(21), 449(85), 470, 473 Ludwig, G. H., 53(48), 54, 148 Luedicke, E., 369, 383 Luiten, A. L., 446(82), 473 Lukyanchikov, G. S., 283(300), 285(300), 300 Luttinger, J. M., 28, 33(63), 36, 37 Lynch, J., 94(140), 160 Lynds, C. R., 366, 383 Lynton, E. A,, 1, 15(66), 21(20), 36, 36, 37,469

M McConnell, A. J., 207, 300 McConville, T., 22, 36 McDonald, A. D., 260, 299 McGee, J. D., 348(4), 357, 382 McLeod, M. G., 44(13), 146 Maki, K., 21, 25, 36 Malin, S. R. C., 46(28a), 147 Malnar, L., 115, 117(179), 161 Mansir, D., 62, 148 Marchant, J. C., 349, 350, 382 Marjerie, J., 95(145), 112(145), 160 Marshall, H. L., 455(94), 473 Martin, D. L., 430(58), 472 Marton, L. L., 348(5), 352(5), 353(5), 360(5), 382 Marzetta, L. A., 51(40), 147 Massey, H. S., 47, 147 Matricon, J., 16(14), 26(39), 27(14), 32, 36, 37 Matthks, B. T., 388(2), 401, 469, 470 Maxfield, B. W., 31, 37 Maxwell, A. E., 49(36), 147 Maxwell, D. E., 59(60), 148 Mayall, N. U., 348(6), 382

MQallet,M., 355, 382 Melton, B. S., 44, 147 Mercereau, J. E., 143, 161 Meyerhoff, 13.. W., 454(90), 473 Miller, M. A., 155, 163, 164, 165, 166, 167, 169, 173, 222, 228(300), 242 (299), 243(300), 246(299), 266, 267, 270, 272, 299, 300, 302 Millikan, A. G., 349, 382 Minnich, S. H., 415(39), 471 Mitropolski, Yu. A., 159(299), 199 Miyamoto, G., 301 Mochel, J. M., 23(28), 36 Mochenev, S. B., 242(299), 246(299), $99 Mockler, R. C., 78(103), 88, 149 Moll, J. L., 309, 346 Montgomery, B., 469 Morisen, W. A., 457(101), 460(101), 473 Morton, G. A., 358, 383 Mota, H., 202, 235, 300, 301 Mourier, G., 194, 220(199), 256(299), 259 (199), 299 Myint, T., 84(108), 110(108), 149

N Narayans, S., 62(69), 148 Nelson, J., 45(25), 147 Ness, N. F., 44(14, 20), 53, 55(20), 105 (167), 107(20), 146, 147, 161 Neville, R. F., 359(31), 383 Nickonov, V. B., 363 (33), 383 Nielsen, E. C., 330, 346 Niessen, A. K., 30, 36 Nirtbet, A., 216, 301 Nordseth, M. P., 370(49), 376,379, 383 Novikov, L. N., 87(123), 160 Noyce, R. N., 305, 306(6), 346 NoaiBres, P., 28, 30, 31, 36, 37 Nyquist, H., 341, 346

0 Ochs, S. A., 370(47), 383 Ogasawara, T.,400(5), 469 Okress, E. C., 302 Olsen, J. L., 445, 472 Omar, M. H., 144(187), 161 Osovets, S. M., 234, 253, 301

48 1

AUTHOR INDEX

P Packard, M., 55,148 Pake, G. E., 64(76), 71(76), 148 Panofsky, W. K. H., 216, 220, 239, 301 Papiashvili, I. I., 355(12), 357(12, 26, 27), 382, 383 Parks, R. D., 23, 36 Parlange, F., ,998, 299 Parsons, L. W., 92, 160 Paskin, A., 454(89), 473 Patton, J., 51 (45a), 52(45a), 147 Paul, W., 302 Petritz, R. L., 308, 346 Pewitt, E. G., 448(84), 449(85), 473 Phillips, M., 216, 220, 239, 301 Picinbono, B., 60(62), 148 Pierce, A. K., 366(37, 38), 383 Pierce, J. M., 240(301), SO1 Pierce, J. R., 244, 301 Pilia, A. D., 181, 184, 187, 278, 301 Pipkin, M. F., 51 (45), 52(45), 147 Pippard, A. B., 2, 9, 10, 36,36 Plumb, J. L., 329, 346 Podgoretskij, M. I., loo(164), 161 Pohl, H. A., 302 Poincare, H., 301 Polder, D., 308, 309(15), 346 Policky, G. J., 325, 346 Popesco, I. M., 87(123), 160 Pople, J. A., 62, 66(73), 148 Post, R. F., 235, 301 Potter, R. J., 377(60), 384 Pound, R. V., 71,149 Pritchard, R. L., 312(22), 346 Prokhorov, A. M., 71(95), 149 Purcell, E. M., 108(173), 161 Purcell, J. R., 417(41), 420(44), 424(41), 453, 471, 473 Purgathofer, A., 363(32), 383 Puskov, N. V., 44(15), 147

R Rabinovich, M. A., 165, 298 R,abinovich, M. S., 155(298, SOO), 268 (300), 298, 300 Raether, M., 302 Raff, A. D., 49,147

Ramsey, N. F., 92(127),112(175),160,lbl Rao, L. Madhav, 17(16), 36 Rassat, A., 66(84a), 148 Rayfield, G. W., 246, 299 Raymond, J. W., 406(32), 471 Rebod, J. J., 302 Reboul, T. T., 302 Reed, W. A., 30, 36 Reinhard, H. P., 302 Reynolds, W. T., 415(40), 430(57), 471, 472 Riemersma, H., 401(9a), 402(19), 416 (19), 470 Rikitake, T., 62, 148 Rinderer, L., 31, 37 Robbins, C. D., 370(48), 383 Roberts, C. S., 190, 194, 30 Roberts, L. D., 402(18), 470 Roberts, P. H., 52(45b), 147 Robinson, H. G., 84(108), 96(149), 110 (108), 112(176), 149, 160, 161 Rohrer, H., 15(66), 37 Rome, M., 368, 383 Rosch, J., 355, 357(18), 382 Rosenblum, B., 21(19), 23, 25, 27, 36 Rosenbluth, M. N., 229, 301 Rosner, C. H., 430(58), 439, 472 Roth, J. Reece, 439(62), 457(102, 103), 460(102), 472,473 Rothwarf, A., 20, 24, 36 Row, R., 220(299), 256(299), 259(899), 299 Royet, J., 443(74), 472 Ruchadze, A. A., 238, 301 Rudakov, L. I., 251 (Sol), 300(301), 301 Ruddock, K. A., 105,162 Ruggles, P. C., 363(34), 383 Runcorn, S. K., 46, 47(28), 52(45b), 147 Ryden, D. J., 364(35), 383 Rymer, K. R., 370, 383

S Sachs, H. M., 44(17), 147 Sagdeev, R. Z., 202, 227, 251(301), 300 (Sol), 301 Sah, C. T., 305, 306(6), 314(26, 54), 327 (26, 541, 346, 346 Saint-James, D., 23, 24, 36 Salle, F., 51(43), 147

482

AUTHOR INDEX

Salvi, A., 73, 149 Sampson, W. B., 440, 441(69), 454(89), 456(98), 472, 473 Sandiford, D. J., 24, 36 Sands, M., 144(185), 161 Sarkasyan, K. A., 283(300), SO0 Sarles, L. R., 48(34), 147 Sator, A., 66(84a), 148 Sauzade, M., 44(11), 51(43, 44), 52(44), 146, 147, 455(95), 473 Savchenko, M. M., 287, 300 Scearce, C. S., 44(14, 20), 53(20), 55(20), 105(167), 107(20), 146, 147, 161 Schadler, H. W., 400, 469 Schaffner, J. S., 328(41), 346 Schemer, L. D., 51, 84(106), 94(136,137), 104(106), 107(171), 108(106, 172), 110(106), 147, 149, 160, 161 Scheglov, P. V., 371 (51), 383 Schindler, H., 439, 472 Schmelzer, C., 74, 149 Schmitt, Roland, W., 457(101), 460(101), 473 Schneider, B., 307, 309(7, 18), 312(18), 320, 326, 346 Schneider, W. G., 62(73), 66(73), 148 Schoenberg, D., 469 Schottky, W., 341, 346 Schrader, E. R., 402(15), 403(23), 470 Schrieffer, J. R., 3(3), 6(3), S6, 469 Schweitzer, D. G., 24, 56 Schwettmann, H. A., 240, 301 Scull, W. E., 54, 148 Seachman, N. J., 351, 373, 374, 379, 384 Seek, J. B., 44(20), 53(20), 55(20), 107 (201, 147 Sekula, S. T., 449(86), 473 Self, S. A., 155(299), 156(299), 202, 208 (.299), 260, 299, 301 Sergeichev, K. F., 283(300), 285(300), 300 Serin, B., 17, 21(20), 22, 36,469 Serson, P. H., 44(21), 147 Servos-Gavin, P., 66(84a), 148 Severnyi, A. B., 363(33), 383 Shapiro, A. R., 302 Shapiro, I. R., 45(27), 53(46), 147 Shelton, H., 308 Shersby Harvie, R. S. R., 155(299), 156 (299), 164, 208(699), 299, 302

Shimoda, M., 65,148 Shockley, W., 305, 306(6), 316, 346, 346 Shoji, M., 315, 327, 346 Sholokhov, N. V., 228(300), 243(300), 300 Siegman, A. E., 84(109), 90, 149 Silin, S. P., 274, 301, 300 Silin, V. A., 229, 283(300), SO0 Silk, V. P., 225, 229(298), 232, 233, 238, 301(298), 298, 300, 301 Sillesen, A. H., 235(301), 301 Silver, A. H., 143(182), 161 Sims, A. R., 44(13), 146 Singer, J. R., 65, 70(81), 148 Skalinski, T., 86(116), 88, 149, 160 Skillman, T. L., 44(14), 53(47), 98(160), 105(167), 107(160), 146, 148, 160, 161 Skrotskii, G. V., 86(118), 149 Slama, L., 220(299), 256(299), 259(299), 299 Slark, N. A., 363(34), 383 Slichter, C. P., 99(161), 160 Smith, P. F., 427(50), 457(100), 460(100), 471, 473 Snyder, J., 441 (69), 472 Sodha, M. S., 202, 299 Solomon, I., 41(5), 66(5, 85), 68(85), 69 (85, 93), 71, 72(5, 85), 146, 148 Solomon, K. A., 402(14), 451(14), 470 Solow, M., 308, 346 Sonnett, C. P., 44(12), 146 Span, R. E., 415(40), 430(57), 471, 47.2 Spiess, F. N., 49(36), 147 Staas, F. A., 30, 36 Stefant, R., 44, 48(10), 60(65), 146, 148 Stekly, Z. J. J., 402,403(21), 404(27, 28), 415(21), 417(27, 28), 418(43), 421 (28), 431,449(85), 451(21), 452,453, 470,472, 473 Stephen, J. M., 28, 29, 30, 36 Stolarik, J., 45(27), 53(46), 147 Strand, A. R., 25(37), 26(37), 56 Stratton, J. A., 216, 268, 269, 301 Strnad, A. R., 22, 36, 409(35), 471 Strongin, M., 454(89), 473 Strutt, M. J. O., 307, 309(7, 18), 312(18), 320, 346, 346 Sturrock, P. A., 251, 301 Surgent, L. V., Jr., 23(28), S6

483

AUTHOR INDEX

Sutton, R., 467(110), 474 Suzuki, Y., 30% Svelto, O., 58(59), 60(5Y, 64), 61, 148 Swartz, G. A., 302

Veksler, U. I., 154, 268, 272, 283, 286, 301 Venturino, A. J., 401(9a), 470 Vetrano, J. B., 405, 406(32), 451(31), 471

Vial, P., 220(299), 256(299), 259(299),

T Taconis, K. W., 144(187), 161 Tanoka, I., 62, 148 Taylor, C. E., 414(38), 437, 471 Thomas, H. A., 41(1), 146 Thompson, D. D., 62, 64(74), 148 Thompson, G. M., 454(89), 473 Thompson, W. B., 235, SO1 Thornton, R. D., 330(46), 334(46), 346 Tien, P. K., 248, 301 Tinkham, M., 23, 24,36 Tite, M. S., 68, 148 Todkill, A., 368(39), 383 Townes, C. H., 65(82, 83), 70, 148, 149 Treuting, R. G., 406(33), 417(33), 453 (331, 471 Tsai, B., 455(97), 473

Tsopp, A. E., SO0 Tsopp, L. E., 283(300), 300 Tsuneto, T., 33, 37 Tuve, M. A., 348(5), 352(5), 353(5), 360 (51, 382

U Uebersfeld, J., 64(77), 148 Uhlir, A., 307, 346 UIImaier, 1%.A., 427(52), 471 Ulrich, A. J., 221(298), 253, 299, 502 Ushio, S., 400(5), 469 Usui, N., 400(5), 469

V Vachaspati, 190, 301 van der Ziel, A., 304(3), 305, 307, 308, 309, 314(23, 24, 25, 281, 315, 319(3, 33), 322, 323, 326, 327(28), 346, 346 van Houwelingen, D., 446(82), 473 van Vliet, K. M., 304(2, 4), 346 van Zahn, U., 302 Varian, R., 55, 148 Vedenov, A. A., 251, 300, SO1

299

Viner, W. F., 30, 36 Vladimirsky, R. V., 71, 149 Volger, J., 445, 446(82), 473 Volkov, I. V., 371, 383 Volkov, T. F., 202, 203, 251(301), 253, 300 (301) , 301

von Hippel, A. R., 87(122), 108(122), 160

W Walker, G. B., 241, 301 Walker, M. F., 350, 354, 355, 357, 58% Walters, G. K., 84(106), 94(137), 104 (106), 106(172), 108(106), 110(106), 149,160,161

Wandel, C. F., 235, 301 Wang, T. C., 65(83), 148 Ward, C. S., 302 Ward, S. A., 370(48), 383 Warren, A. C., 30(48), 36 Waters, G. S., 57(56), 61, 148 Watson, C. J. H., 229, 232(298), 301 (298), 298, 301

Watson, W. K. R., 302 Weaver; H. E., 455(94), 473 Webster, W. M., 312(22), 346 Weibel, E. S., 155, 156, 164,202,208,,227, 234, 245, 299, 302

Weill, M., 298, 299 Weimer, R. K., 369, 383 Weisskopf, V. F., 307, 346 Wernick, J. H., 406(33), 417(33), 453(33), 471

Werthamer, N. R., 33(63), 37 Whetstone, C. N., 406, 471 Whipple, R. T. P., 227, 30% Wiatr, Z. M., 92, 160 Wiehl, C., 401(12), 470 Wien, R. E., 401(9a), 470 Wilcock, W. L., 368(39), 383 Wiley, W. C., 376, 384 Williams, H. R., 302 Williams, J. E. C., 454(91), 473

484

AUTHOR INDEX

Wilson, O.,363,383 Wilson, P.B.,240(301),3'01 Winckler, J., 48(35),147 Winter, J., 100, 103(166),161 Wipf, S. L.,409(36), 445,471, 473 Wittke, J. P., 96(156),160 WltSrick, G.,355,357(18),388 Wolff, N.,51, 147 Wolgast, R.C., 402(14),451(14),470 Wood, R.W., 30.9 Woolard, G.,62(69), 148 Wroughton, D.M., 30.9 Wu, S. Y., 314(54),327(54),346 Wuercker, R.F., 30.9

Y Yankov, V. V., 227, 30.9 Yasucochi, K.,400,469

Yeliseyev, G. A., 251(301), 300(301),

so1

Yeremin, B. G.,242(899),246(.999),299, 30.9 Yntema, G.B.,401, 470

Z Zagryadsky, E. Y.,242(.999), 246(.999), 899 Zar, J. L.,408(28),417(28),421(28),431, 451(21),452,453(28),470 Zeiger, H.J., 65(82),148 Zharkov, G.F., 144(186),161 Zhugov, L. N., 44(15), 147 Zietz, I., 49(38), 147 Zimmerman, J. E., 143(183, 184), 144 (183,la), 161 Zirin, H.,371, 372,384

Subject Index A Absorption, optical, of resonance bulb, 85-86 Acceleration radiofrequency, of plasma, 153-302 experiments, 283-298 theory, 264-283 resonant conservative, 276 nonconservative, 276-277, 291-298 traveling wave, 281-282 Alkali atoms in optical pumping, 86-91 Amplifiers bipolar transistor, 33(t334 comparison of bipolar and FET, 337338 field effect transistor, 334-337 practical low-noise, 329-340 tuned, 338-340 ANL 67-kG, 7 inch bore coil system, 431-437 Astronomy, image intensifiers in, 347-384 AVCO model M H D magnet, 435-437 stabilized magnet, voltage-current characteristic, 430

B Benoit’s liquid flow method, 75 Bloch’s equations, 98-101 Breit-Rabi formula, 133 Bubble chamber magnet studies, 458-464 Buffer gas, 95

C Cables, superconducting configurations, 417-422 types, 432 Cesium energy levels, 117 vapor magnetometer, 111-143

Clocks, optical, 98 Coherence length, 8-10, 390, 395-396 Coil system, ANL 67-kG, 7 inch bore, 431-435 Conductance, noise, 342-343, 344 Conductors heavy section, 412-415 multiple, configurations, 417-422 tapetype,440-441 working stress, 424-426 Confinement combined rf and magnetic mirror, 253259 radiofrequency, 241-264 direct evidence, 250-260 electron beam focusing, 246-250 single particle, 242-246 Contacts in superconducting magnets, 448-449 Cooper pairs, 392 Current-carrying capacities of conimercial wires, 405 ff Current equivalent noise, 322-324 generator, noise, 307, 312, 316-317 recombination, 306 saturated diode, 343 transitions in early super magnets, 415 Cyclotron resonance, 173-187, 190-194

D Density matrix theory, 98-101 Depth, penetration, 5, 8, 394 Diamagnetism, superconductor, 2 Diffusion theory, 261-262 Diode equation, 306 junction, noise in, 30&309, 340 collective approach, 308-309 corpuscular approach, 305-308 Dipole moment, point, 270 Discharge, transition, in superconducting coils, 402-403 485

486

SUBJECT INDEX

E Earth’s field, 46 diurnal variations, 47 Ebers-Moll equations, 309 ff Electron motions, collective, 267 ff nuclear resonance, optical detection, 76-1 11 Electronography, 354-360 Energy, free per unit volume, 19-20 of superconductor, 7-8, 33-35 Energy-momentum tensor, 198-200 Equilibria in cylindrical cavities, 216-217 infinitely cylindrically symmetric, 207-215 cutoff cylindrical TEo mode, 214215 energy-momentum tensor, 207-208 TMomode, 208-214 one-dimensional, 200-207 in spherical cavities, 220-221 truncated, 205-206 Eschenhagen oscillations, 48

Figure of merit for amagnetometer,. 119.. 120 Flicker noise, 328-329 Flux escape, time dependent, 259 flow, steady-state, 25-28 gate method of measuring magnetic fields, 44-45 jumping, 400 jump intensity increase in supercoils, 414-415 pumps, 444-446 quantization, 5-6 Focusing, electron beam, by rf confinement, 246-250 Force components, electromagnetic, 424 exerted by plane traveling wave, 267 on lossfree plasmoid, 271 Frequency closed-loop cut-off, 120-121 measurement in precession magnetometer, 58-61 modulation, detection by, 114-115 transition, in alkali atoms, 92

F

G

Field(s) equations, self-consistent, 194-198 geomagnetic, 45-52 compensating natural and manmade fluctuations, 50-52 exceptional variations, 48 exploration with mobile magnetometers, 48-49 geophysical observatory, 49-50 magnetic storms, 48 normal conditions, 45-48 quiet days, 45-48 spatial va,riations, 45-47 time variations, 47-48 interplanetary, 41-42, 52-55 minimum detectable, 119 natural magnetic, order of magnitude, 45-55 superconductor critical magnetic, >3 temperature dependence, 393-394 weak magnetic, measured by magnetic resonance, 39-151

Gain, magnetometer open loop, 121-122 Ginsburg-Landau equations, 6-10 order parameter, 391-392 GLAG theory, 393 Granularity film, RMS, 358-360 photographic, dependence on spatial sample size, 358-360 Guiding center motion, 271, 273 theory, 176 Gyromagnetic ratio, 40

H Helium magnetometer, 42-43, 105-111

I Image dissection, 379 intensifiers

SUBJECT INDEX

in astronomy, 347-384 description, 354-372 orthicon, 368-370 as finding and pointing aid, 370 tubes cascade, 36G363 comparison with photographic plates, 349-352 fiber optic faceplates for, 377 geometrical stability in, 352 image movement in, 352 with infrared photocathodes, 371372 microstructure elements for, 376 mosaic target, 370 potential use for detection of faint stars, 358-360 preformed photocathodes for, 377 pulse counting with, 368 quantitative evaluation, 352-354 Induction method of measuring magnetic fields, 44 Inform ation passband, 124 ratio, 353-354

J Josephson effect in thin film bridges, 32-33 Junction noise theory of Polder and Baelde, 30&309

K Kappa parameter, 393 Kulinsky theory, 176 ff

L Lallemand tube, 354-360 Length, coherence, 8-10, 395-396 Lenses, superconducting magnet, 441442 Levels, optical and magnetic, 76-79 Line broadening, by Doppler effect, 96-98 width, 94-98 London equations, 4-5 Lorentz force, 25-27

487 M

Magnet (8) coil, first 18 inch i.d., 431 economics, 457-464 Livermore minimum B, 437 MHD, AVCO model, 435-437 RCA 150 kG, 6 inch bore, 439 superconduc ting accessibility in, 453-454 coil fabrication facilities, 466-467 condition for thermal stability, 452 contacts in, 448-449 design problems, 443-457 early problems, 414-416 general development, 401-404 homogeneous fields with, 455-456 operation above and below 4.2"K, 454-455 performance predictability, 451-453 radiation effects on, 456-457 recent, 429-443 refrigeration requirements, 449-451 safety considerations, 446-447 technology, 385-474 test facilities, 464-466 for thermonuclear fusion studies, 437-439 synchrotron, 428-429 Magnetometer cesium vapor, 111-143 principles, 112-113 controlled, 113-127 control error, 120 dynamic range, 120, 130 electronic problems, 120-124 limitations in use, 129-143 practical design, 125-127 dynamic range, 120, 130 helium, 42-43, 105-11 1 mobile, 129-136 precession, 58-61 transition in earth's field, 98 turnover effects, 132-136 Maser action in optical pumping, 84 oscillations, 68-72 Microoscilla.tions, 162-163 Mixing of sublevel populations, 94-96 Modulation, transverse, 98-102

488

SUBJECT INDEX

Motion charged particle, in periodic electrostatic field, 248-249 collective electron, 267 ff nonresonance, in nonuniform magnetic field, 181-187 particle, in traveling electromagnetic wave, 187-190 of plasma confined and accelerated by nearly standing rf waves, 273-275 single particle, 159-194 in field of plane electromagnetic wave, 159-163

N NbsSn vapor deposited ribbon, 406, 408 NbZr effect of copper coating on, 416-417 embrittlement, 406-409 Noise in bipolar transistors, 309-314 current generator, drain-source, 316318 in diodes, 340 electromotive, base region, 312 factor (figure), 344 in field effect transistors, 314-319 generator frequency, 122-1 24 in junction diodes, 305-309 in linear two-ports, 343-345 modulation, 305 representation, 341-345 in semiconductor devices, 303-346 experimental verification, 319-329 sources, 304-305 shot, 304 temperature, 344 transition, 304 transport, 304-305 of two-terminal networks, 342-343 Nyquist’s theorem, 341

0 Order parameter of Ginsburg and Landau, 391-392 Ordering mechanism in superconductors, 3-5 Orientation in pumped state, 83-86

Oscillator hydrogen maser, 109-110 one-beam optical spin, 101-102 self-, magnetometer system, 127-129 Overhauser effect, 64 ff polarization, 64-75

P Passband limits, magnetometer, 137-143 Penetration depth, magnetic field, 5, 8, 394 Photocathode, reflective, 372-376 Plasma acceleration combined rf and magnetic fields, 275-298 fast traveling waves, 266-276 purely rf, 264-275, 283-285 slow traveling waves, 266 standing waves, 265-266 combined rf and magnetostatic confinement, 223-227 low pressure plasma distributions, 226-227 one-dimensional equilibria, 225-226 self-consistent field equations, 223225 heating due to rf field penetration, 237-239 Maxwellian, 200-205 fusion power generated by, 235 pressure, 235-236 radiofrequency confinement and acceleration, 153-223 stability theory, 227-234 theory, 194-223 Polarization of fluorescent light, 85 Overhauser, 64-75 Power supply, superconducting magnet, 443-444 Precession, free nuclear, 55-64 Prepolarization, 55-64 Pulling effect, 72-73 Pumping, optical, 76-91 alkali atoms in, 86-91 Dehmelt, 82-83 in hydrogen, 87

489

SUBJECT INDEX

Kastler, 82 light for, 78-79 maser action in, 84 mechanism 80-83

Q Quantum efficiency detective, 352-353 responsive, of photoelectric surface, 349 Quasi-potential concept, indirect support from breakdown measurement, 260264

R Radiofrequency cavity, superconducting, 240 confinement experiments related to, 241-264 of plasma, direct evidence, 250-260 field(s) arbitrary, guiding center theory, 163-165 penetration, heating of plasma due to, 237-239 plus nonuniform field, 168-173 plus uniform magnetostatic field, 166-168 losses in cavity wall, 239-241 pulse length a s function of gas pressure, 259-260 Reactors fusion, 234-241 rf thermonuclear, 234-241 Relaxation, 94-98 transitions, 82 Resettability, magnetometer, 136-137 Resistance, noise, 342, 344 Resonance cyclotron, 173-187, 190-194 double, 84 electron, “optical detection” of, 42, 43, 76-111 line asymmetry, 133-135 magnetic, measurement of weak magnetic fields, 39-151 nuclear, 55-75

S Schottky’s theorem, 341 Secondary electron tube, transmission, 363-368 for spectroscopy, 366367 Semiconductor devices, noise in, 303-346 experimental verification, 319-329 theory, 304-319 Sensitivity limit, of controlled magnetometer, 115120 magnetometer, orientation dependence, 130-132 Short sample characteristic, 405 Sodium, level structure, 87-91 Spectrograph, “double-resonance,” 65-66 Spin@) coupled, in very low fields, 62-64 exchange, 108-109 oscillator, 73-75 SQUID, 144-146 Stability, magnetometer, 136-137 Static properties, mixed state, 13-23 Stekly Q parameter, 452-453 Storage, open-loop, 124 Superconductor(8) a.c. characteristics, 426-429 braided copper coated, 417 classes, 388 interferometers as magnetometers, 143-146 intermediate state, 391, 396-397 magnet characteristics, 404-414 materials, 401-429 magnetization, 397-398 loss in, 426-427 pulsed, 426-429 specific heat, temperature dependence, 395 strip type, 420-422 type 11, 1-37 dynamic effects, 25-33 electrical resistance in, 27 magnetic behavior, 16-19 mixed state, 392-393 Superconductivity concepts, 388-401 surface, 23-25

490

SUBJECT INDEX

Superfluid, 389 Surface barrier, vortex, 31-32 energy between superconducting and normal laminar planes, 389 interpha,se, 10-13 superconductivity, 23-25 Switches, persistent, 447-448

T Ti-22 atomic percent N b superconductor, 405-406 Transistor bipolar noise in, 309-314, 320-326 noise factor, 330-332 field effect noise in, 314-319, 326-328 noise figure, 335, 336 minimum noise resistance, 338 theoretical noise generators, 311-313

v Vidicon, 370 Vortex lines, 398-399

motion, 28-30 in thin films, 32-33 surface barrier, 31-32 waves, 30-31 Vortices, current in type I superconductors, 23 in type I1 superconductors, 13-16

W Wave(s) of arbitrary polarization, 206 equation, nonlinear solutions, 200 ff fast traveling, 266-276 slow traveling, 266 standing, 173-187, 190-194, 265-266 vortex, 30-31 Wire, superconducting choice of bonding and shunting materials, 422-424 copper coated, 403-405 degradation due to heating, 409-412

Z Zeeman excitation, 91-102 resonance frequency, 91-94 transitions, 77-78