Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Stochastic processes and thermal fluctuations in superconductors Leung, Man-Chiu 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1970_A1 L48.pdf [ 8.04MB ]
Metadata
JSON: 831-1.0084800.json
JSON-LD: 831-1.0084800-ld.json
RDF/XML (Pretty): 831-1.0084800-rdf.xml
RDF/JSON: 831-1.0084800-rdf.json
Turtle: 831-1.0084800-turtle.txt
N-Triples: 831-1.0084800-rdf-ntriples.txt
Original Record: 831-1.0084800-source.json
Full Text
831-1.0084800-fulltext.txt
Citation
831-1.0084800.ris

Full Text

STOCHASTIC PROCESSES AND THERMAL FLUCTUATIONS IN SUPERCONDUCTORS . oy Man-Chiu Leung B. S c , U n i v e r s i t y of Hong Kong, 1961. M.Sc, Lou i s i a n a State U n i v e r s i t y , 1965.. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS' FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics. We accept t h i s t h e s i s as conforming to the required standard. -THE UNIVERSITY OF BRITISH COLUMBIA 1970 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t f r e e l y available for reference and study. I further agree tha permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or p u b l i c a t i o n o f t h i s thesis f o r f i n a n c i a l gain shall not be allowed without my written permission. Department of Physics The University of B r i t i s h Columbia Vancouver 8, Canada Date 1 6 O c t o b e r , 1Q70 i i • ABSTRACT • The g e n e r a l i z e d L a n g e v i n e q u a t i o n , which i s d e r i v e d b y H. Mori,as a g e n e r a l , m i c r o s c o p i c f o r m u l a t i o n of the s t o c h a s t i c t h e o r y of Brownian motions,- i s a p p l i e d .to e v a l u a t e the l o n g i t u d i n a l a t t e n u a t i o n r a t e of sound i n weak-coupling, pure s u p e r c o n d u c t o r s . In or d e r t o q u a l i f y the u l t r a s o n i c a b s o r p t i o n as s t o c h a s t i c p r o c e s s e s , the r e l a x a t i o n time of the random i n t e r a c t i o n s on the phonon must be much s h o r t e r than the r e l a x a t i o n time of the phonon. We show.that t h i s i s i n d e e d t h e case b y showing t h a t the former i s of t h e o r d e r of the r e l a x a t i o n time of the d i s -t r i b u t i o n f u n c t i o n of the e x c i t a t i o n s from the BGS conden-s a t e . As compared, w i t h the e a r l i e r r e s u l t d e r i v e d by f i r s t o r d e r p e r t u r b a t i o n t h e o r y , the i n c l u s i o n of the l i f e - t i m e e f f e c t has improved, the agreement with e x p e r i m e n t a l d a t a . When T A Y w , i t i s shown t h a t Cooper p a i r s p a r t i c i p a t e i n the a b s o r p t i o n p r o c e s s e s even when to<2A , A sum r u l e f o r the l o n g i t u d i n a l u l t r a s o n i c a b s o r p t i o n r a t e i s a l s o d e r i v e d . We a l s o employ the Lamrevin approach to e v a l u a t e the s p e c t r a , v a r i a n c e s and c o v a r i a n c e s of the magnetic f l u x , the c u r r e n t s and the m a g n e t i s a t i o n i n a lohec, t h i n , h o l l o w s u p e r c o n d u c t i n g c y l i n d e r . When the w a l l t h i c k n e s s of a c y l i n d e r i s a r b i t r a r y , the l a n g e v i n approach becomes i n c o n v e n i e n t . M o d i f y i n g s l i g h t l y Kubo's f o r m u l a t i o n of p e r t u r b a t i o n t h e o r y to take care of the f a c t t h a t g e n e r a l -i z e d s u s c e p t i b i l i t y may. not be zero a t ' i n f i n i t e f r e q u e n c y , we e s t a b l i s h i n the c l a s s i c a l l i m i t , a g e n e r a l r e l a t i o n between g e n e r a l i z e d s u s c e p t i b i l i t y a-g^(w) and the c r o s s -spectrum of two q u a n t i t i e s A and B, where co i s the f r e c u e n c y . P u t t i n g 0) t o z e r o , a p a r t i c u l a r l y simple r e l a t i o n between I l l c o v a r i a h c e and g e n e r a l i z e d s u s c e p t i b i l i t y i s o b t a i n e d . T h i s r e l a t i o n and f l u c t u a t i o n d i s s i p a t i o n theorem are then a p p l i e d to e v a l u a t e the s p e c t r a , v a r i a n c e s and c o v a r i a n c e s of magnetic f l u x , c u r r e n t s and m a g n e t i s a t i o n f o r the. case of a l o n g , hallow, s u p e r c o n d u c t i n g c y l i n d e r of a r b i t r a r y t h i c k n e s s . S e t t i n g the i n n e r r a d i u s of the c y l i n d e r to be z e r o , we o b t a i n the p a r t i c u l a r case of a s o l i d c y l i n d e r and our t h e o r y is•compared w i t h an experiment by Vant-Hull e t . a l . , which measures the f l u c t u a t i o n s of the magnetic f l u x i n s o l i d , m e tal c y l i n d e r s . A c c o r d i n g to our t h e o r y , the t h e r m a l magnetic n o i s e does not d i s a p p e a r when the metal becomes s u p e r c o n d u c t i n g . With the•dimensions of the t i n r o d used i n the experiment, we expect t h e o r e t i c a l l y , however, the s t a n d a r d d e v i a t i o n of the magnetic f l u x to drop more than an o r d e r of magnitude from i t s normal s t a t e v a l u e when - 5 becomes g r e a t e r than 1 0 . Experiment agrees with our t h e o r e t i c a l e x p e c t a t i o n . More a c c u r a t e d a t a , however, are needed to d e c i d e whether our t h e o r y i s q u a n t i t a t i v e l y adequate. In the above two approaches, we have assumed the f l u c t u a t i o n of the e l e c t r o n d e n s i t y n i n the BCS-con-densate t o he n e g l i g i b l e . To take c a r e of the f l u c t u a t i o n of p a i r d e n s i t y , we make use of the f a c t t h a t the probability' d e n s i t y of n s and § i s p r o p o r t i o n a l to exp ^ ~ 3 ^ s ( n s » § ) ] where $. i s the magnetic f l u x and G i s the Gibbs f r e e energy. To avoid, mathematical c o m p l i c a t i o n s , we keep to the case of a l o n g , t h i n , h o l l o w c y l i n d e r i n the Landau-Ginzburg temperature domain. We a l s o d i s c u s s the responses of magnetic f l u x and c u r r e n t s to an e x t e r n a l magnetic f i e l d which changes stepwise at time t = 0 . TABLE OF CONTENTS Abstract Table of Contents L i s t of I l l u s t r a t i o n s Acknowledgments INTRODUCTION' PAGE i i i v v i i ix. 1 CHAPTER I SOME THEORIES OP IRREVERSIBLE PROCESSES 1.1 Generalized Langevin Equation 36 1.2 C o r r e l a t i o n and Generalized S u s c e p t i b i l i t y 4-3 1.3 Stepwise Change of External Force 50 CHAPTER II. LONGITUDINAL ULTRASONIC ABSORPTION RATE IN PURE SUPEP.CONDUCTORS : LANGEVIN APPROACH Ji- .1 Introduction 5.4 l l .2 L o n g i t u d i n a l u l t r a s o n i c Attenuation Rate 56 I I .3 Damping of the Random Force 78 I I .4 Sum Rule and I n t e r r e l a t i o n s 81 CHAPTER IE FLUCTUATIONS IN SUPERCONDQCTIKG CYLINDERS: LANGEVIN AND FREE-ENERGY APPROACHES 111.1 Introduction 89 111.2 Two-Fluid Model 94 111.3 Long, Thin, Hollow, Superconducting Cylinder: Langevin Approach 96 111.4 F l u c t u a t i o n of Superconducting E l e c t r o n P A G E CHAPTER I I I (Continued) Density and Magnetic Flux i n Long, Thin, Hollow, Superconducting Cylinder no CHAPTER IV . FLUCTUATIONS IN SUPERCONDUCTING CYLINDERS: FLUCTUATION-DISSIPATION'APPROACH . IV.1 Introduction 121. IV.2 L o c a l i z e d Electromagnetic Response 123 IV.3 Generalized S u s c e p t i b i l i t i e s f o r Superconducting Cylinders 131 IV.4 Steady State 135 IV.5 F l u c t u a t i o n of Magnetisation 138 IV.6 R e l a t i o n between Magnetisation and Magnetic Flux .146. • IV. 7 F l u c t u a t i o n of Magnetic Flux 148 CHAPTER V RESPONSE TO STEPWISE CHANGE OF EXTERNAL  FIELD V. l Introduction 162 V.2 Response of Currents 163 V.3 Response of Magnetic Flux 168 V.4 Response of Magnetic Moments 170 V.5 Responses i n Long, Thin, Hollow, r Superconducting Cylinders 172 V.6 .... Energy and D i s s i p a t i o n 175 References Appendix A: Appendix B: F A G E 180 P h y s i c a l i n t e r p r e t a t i o n of :ReLftiu^i| Metastable Values of Cooper ._; P a i r Density i n A Long, Thin Hollow, Superconducting Cylinder .186"" INDEX OF SYMBOLS AND ABBREVIATIONS ' 194 v i i LIST OF ILLUSTRATIONS P A G E Pig. 1 . A p l o t of $ i versus T/T f o r the attenuation of sound i n pure t i n . 6 P i g . 2 . The r a t i o of A(T)/A(o)as a function of temperature for weak-coupling super-conductors, based on BCS theory 8 Pig. 3 . A r a t i o of attenuations of sound i n superconducting and normal states as a function of frequency f o r d i f f e r e n t temperatures. ' 9 P i g . 4. Longitudinal attenuation of a 33.5 MHz sound wave i n pure t i n . 11 Pig. 5. Longitudinal attenuations of sound i n normal metals and superconductors as a function of frequency. 13 Pig . 1 . 1 . Response to stepwise change of F ( t ) . 52 P i g . 2 . 1 . Coordinate system with q_ as the polar a x i s . 68 P i g . 2 . 2 . <X-i/oLn as a function of temperature 74a Pig . 3 . 1 . Var(§)/Var(?) i n t h i n , superconducting c y l i n d e r s as a function of temperature. 103 Pi g . 3 . 2 . A cross - s e c t i o n of a superconducting, t h i n c y l i n d e r of a r b i t r a r y c r o s s - s e c t i o n a l shape. 107 Pig. 3 . 3 . Sketches of Gibbs Free Energy [ < ^ n s , $ ) - ^ Y O J as a func t i o n of magnetic f l u x . 1 1 8 F i g . 3.4. Sketches of Gibbs Free Energy fo(< 6ri(<ni>3 v i i i PAGE as a fun c t i o n of (n > . 119 s Pig.3.5. Sketches of 1^-$^ as a fun c t i o n of xn s> 119 Pig.4.1. Var(M)/Var(M) as a fun c t i o n of temperature i n a hollow c y l i n d e r . 141 Fig.4.2. Magnetic noise i n s o l i d c y l i n d e r s . 150 Fig.4.3. % t ^ ^ /S§^o) as a function of frequency i n a superconducting, s o l i d c y l i n d e r . 154. Fig.4.4. ^ i t ^ A S g f o " ) as a fun c t i o n of temperature i n a superconducting, s o l i d c y l i n d e r . 155 Fig.4.5. The r a t i o of covariances of EL and i n the superconducting and the normal phases as .a function of temperature i n hollow c y l i n d e r s . l r58: Fig.5.1. Responses i n a t h i n c y l i n d e r to a. step-wise change of .external f i e l d H ( t ), -as a fun c t i o n of time t . 173 ) i x ACKN OWLEDG-MENTS This t h e s i s was prepared under the d i r e c t i o n of Professor R.E. Burgess. I t i s a pleasure to thank him f o r h i s help during the course of the work and the preparation of the manuscript. I would also l i k e to thank Dr. N. Kumar f o r many valuable discussions. I t i s also g r a t e f u l l y acknowledged that t h i s research has been f i n a n c i a l l y supported by the UBC Fellowship and the Defence Research Board (Canada-).. 1 INTRODUCTION 0)(2).' In the past decade, an enormous amount of e f f o r t has been spent i n the search, f o r some general and convenient microscopic formulations of a stochastic theory of Brownian motion. One of the most f l e x i b l e and general formulation of t h i s problem i s the generalized Langevin equation" -derived by H. Mori. As we are going to apply only Mori's formalism, we s h a l l not spend our time i n comparing merits of d i f f e r e n t formulations. We s h a l l go d i r e c t l y to bring out the e s s e n t i a l contents of the generalized Langevin equation derived by H. Mori. The equation of motion f o r a quantity A i n a c e r t a i n p h y s i c a l system can be written as dt v J F ( t ) i s the t o t a l i n t e r a c t i o n , on the p h y s i c a l quantity A, at any i n s t a n t , a r i s i n g from other degrees of freedom i n the system. These other degrees of freedom are very^ large and s h a l l be c a l l e d the r e s e r v o i r . In the ordinary Langevin equation,'F(t) i s written as F ( t ) = - p A ( t ) + f (t) where FA(t) represents the average action of the r e s e r v o i r and f ( t ) takes account of the extremely r a p i d l y varying part of the i n t e r a c t i o n s . V i s the damping rate of A ( t ) , or e q u i v a l e n t l y , f ~ ^ i s the time A(t) needs to r e l a x to i t s equilibrium value. The r e l a x a t i o n time of f ( t ) i s much shorter than T . In f a c t , i n the ordinary Langevin equation, i t i s assumed to be zero; i . e . , the auto c o r r e l a t i o n 2 function'of f (t) , <(f ( t ) f (t+r)}, i s prop o r t i o n a l to a d e l t a f u n c t i o n of t. In the theory of a Brownianian p a r t i c l e i n a f l u i d environment, A(t) i s the momentum of the p a r t i c l e . Due to c o l l i s i o n of t h i s Brownian p a r t i c l e with other p a r t i c l e s i n the f l u i d , the average e f f e c t gives r i s e to the damping rate and the s o - c a l l e d random force f (t ) represents the r a p i d l y varying part of s c a t t e r i n g . While the ordinary Langevin equation i s phenomenologically devised to provide an explanation f o r Brownian motion, H. Mori has shown that the generalized Langevin equation"*" (1.1.5) (GLE) r + _ ^AU) -itt>0A(t) +• u t - o A t o d f =m flt J o can be derived from the Heisenberg equation of motion. This equation (GLE) can deal both with quantal and c l a s s i c a l q u a n t i t i e s . In the s p e c i a l case, wQ = 0, V(t-t')= 2 T S(t-t') and f (t) a r e a l f u n c t i o n , we recover the ordinary/ Langevin equation (1.1.1a). The s c a l a r product (f ( t),f (t+r)) i s a quantum analogue of ^ f ( t ) f ( t + r ) ^ f o r the ordinary Langevin equation. The d e f i n i t i o n s of a s c a l a r product, w0 » 700 a n d fCt) are given by (1.1.10), and (1.1.13)-(1.1.15). Kt) i s d i r e c t l y r e l a t e d to ( f ( t ) , f * ) by f(t) = ( A , A * ) _ 1 ( f ( t ) , f * ) . This i s the s o - c a l l e d ^ s e c o n d f l u c t u a t i o n - d i s s i p a t i o n theorem. In the ordinary Langevin equation with constant damping r a t e , ( f ( t ) , f * ) must be prop o r t i o n a l to.a d e l t a f u n c t i o n of t i n order to be s e l f -c o n s i s t e n t . The p h y s i c a l s i g n i f i c a n c e of toQ and Kt) i n (1.1.5) i s demonstrated i n the r e l a t i o n , obtained by applying the The numbering r e f e r s to numbering of equations i n the main text; e.g., (1.1.5) r e f e r s to the 5th eq. i n ^ 1 of chapter I. 3 Laplace t r a n s f o r m t o G-LF, ; ( W W " ^ % ) t ( ^ A t = •/ ( A / f r - * F H i L ' ' i < * > - W . - I » C 7 ( W g j + l?e[rli-51 o r . s i n c e ( Att),A*) - CA, A*<t)), ( e ~ i a ) t ( A ( t ) , A ) dt h a s 2 the p h y s i c a l s i g n i f i c a n c e of the spectrum of A. co0 i s thus the energy of the mode A ( t ) when there i s no i n t e r a c t i o n between A ( t ) and other degrees of freedom^. Im[?(io))} i s the s h i f t of energy due t o i n t e r -a c t i o n of A ( t ) w i t h other degrees of freedom of the system. Re(jf(ito)] i s the damping r a t e of A ( t ) . When the l i n e shape i s sharp, we can w r i t e r- -Coot * ZRetUtcA)] \ e ( Aft).A )dt = —. *r~5 _ R ( W . N . u i — where co = (OQ + Im[jf(ia))] • This i s the L o r e n t z i a n l i n e shape w i t h the peak at w =• <S. As an example, l e t us take the case where ( f ( t ) , f ( t + T ) ) i s p r o p o r t i o n a l t o e 1 fc,/i.c# The damping r a t e Rejjf(iu>)} i s then p r o p o r t i o n a l t o ^c"^ 2 ' -2 (I) + T C 2 To be p r e c i s ! 5 In the c l a s s i c a l l i m i t [ e ~ i w t ( A ( t ) , A * ) d t i s equal t o j"e~ i a ) t(A(t)A> dt which is the spectrum of A. V^w)is the Laplace t r a n s f o r m o f [\tjl 4-As another example l e t us take where ^ k i s a func t i o n depending on k, the damping rate i s of the form s -\ 4- X»» The damping rate (or attenuation r a t e ) f o r the l o n g i t u d i n a l sound wave i n superconductors i s of such a form. To evaluate the u l t r a s o n i c attenuation by the generalized Langevin formalism, we equate A(t) i n ( 1 . 1 . 5 ) to where b^  + i s the phonon cr e a t i o n operator with momentum q*. From~ the generalized Langevin equation, we obtain 3 For example, take a system of free phonons,the Hamiltonian of which i s ^ w^b^b^. The Heisenberg equation of motion f o r b + i s then q fL b^ CO -t t£jb^ (t) = o V/e can consider t h i s equation a s p e c i a l case of GLE; i . e . , a GLE with 0 = f ( t ) . u)Q i s thus equal to w^. However, we would l i k e to point out here the GLE i s not appropriate f o r such system of independent p a r t i c l e s . a s the d e f i n i t i o n of the s c a l a r product (A , B ) becomes meaningless., i n such a system. 5 w i t h Wo = co - I m U ( i c o ) Or, ,-and. co .=w='"energy of free phonon, 4 t c (b iC tWt ' ) ) = (br.bi ) • r — ^ : — I f one adds,a phonor with momentum 5* at time t ' to a system, what i s the p r o b a b i l i t y that one can f i n d the phonon' with the same momentum at a l a t e r time t?. According to quantum mechanics, the p r o b a b i l i t y i s (t) > 1 . . With the Lorentzian form f o r the Fourier transform of ( b-(t ) , t-* +(t»)), i t i s straightforward to show that the p r o b a b i l i t y i s (Appendix A). The average time during which ' phonon can stay i n the state ^ i s then given by l / 2 R e ^^(ioo^)] or the attenuation rate of the phonon^ i s In sections I I . 2 and I I . 3 , we apply- the GLE to derive an e x p l i c i t expression f o r Re£^(ito)j f o r pure, weak-coupling superconductors, while the mathematical d e t a i l s 4 Let us, f o r s i m p l i c i t y , make the assumption that if ( t ) . = rS("k-°)" Then, the equation of motion f o r b ^ + i s I t i s straightforward to show that, since v^( t ) ,b | ) - 0 6 P i g . 1. A p l o t of <*sld* versus T/T f o r "the a t t e n u a t i o n of sound i n t i n (Ref .21 ). The s o l i d curve i s from BCS model.. (Ref. 2 .). ( P « ^ C O ) 7 can be found i n these two sections, we s h a l l devote the next s i x paragraphs to discussing some phy s i c a l aspects of the problem. Since .Mori 1s formulation i s very general and "formal, i t i s o f i n t e r e s t to. apply i t . to .some p a r t i c u l a r s case. ^ In Chapter I I , we apply bis' general formalism to evaluate the l o n g i t u d i n a l u l t r a s o n i c absorption rate in pure, weak-coupling superconductors. Longitudinal u l t r a s o n i c absorption rate i n pure weak-coupling suner-conductors has been evaluated hv BCS^ - } f o r freouencv smaller than twice the energy gap- and by P r i v o r . o t s k i i v y / f o r frequency higher than twice the energy gap. B o b e t i c ^ ^ has numerically evaluated the r a t i o of l o n g i t u d i n a l u l t r a -sonic-absorption rate i n superconducting and normal phase They have used f i r s t order perturbation theory based; on BCS theory of superconductivity. There have also been large number of experiments dealing with the l o n g i t u d i n a l u l t r a s o n i c absorption i n pure ;weak-coupling superconductors. Most o f the experiments have been c a r r i e d out with u l t r a -8 TO sonic frequency i n the range 1 0 Hz to 1 0 Hz;. The v a r i a t i o n of oCg/c^ as a func t i o n of temperature f o r the values of- w much l e s s than k^T i s i l l u s t r a t e d i n q- r> F i g . 1. ( a f t e r Ref. ( 1 2 ) ) . oc0 denotes the u l t r a s o n i c absorp-s t i o n rate of sound waves i n metal i n superconducting phase and denotes the u l t r a s o n i c absorption rate of sound waves i n the normal phase. A ( 0 ) i s the energy gap at (bq »kq) h a s Physical s i g n i f i c a n c e of number of phonons with momentum c£. . Therefore, the decaying rate of the number of t>bononswith momentum . q i s 2 f . The assumption o f 4 d e l t a f u n c t i o n f o r / ( t ) i s i n f a c t o v e r s i m p l i f i e d and would lead to incorredt results. However the example serves the purpose of i l l u s t r a t i o n . 8 F i g . ^ The r a t i o of ZlT^ 0)as a fu n c t i o n of temperature f o r weak-coupling supercon duetors, based on BCS theory. 9 10 at 0°K. The s o l i d l i n e follows from BCS one-parameter model, with A(0)/k BT = 3.52, and, was f i r s t derived by BCS i n Ref .'(.8) employing f i r s t o r d e r perturbation theory. The s o l i d l i n e ' i s a n a l y t i c a l l y represented by the o:s/ccn = 2 f(A) =. 2/[l+exp(pA)] A i s a s i m p l i f i e d svmbol f o r A(T) which denotes the energy gap at temperature T°K. According to the BCS theory, there e x i s t s ^ an energy gap between the excited states and the BCS condensate (the ground s t a t e ) . The energy gap decreases as temperature increases. It becomes zero at the c r i t i c a l temperature T. when the metal becomes normal. In F i g . 2?, the r a t i o A(T)/ A ( 0 ) i s pl o t t e d as a function of temper-a t u r e f o r pure., weak-coupling superconductors. F i g . 3-is' plotted., according to Bobetic's numerical c a l c u l a t i o n s , wi t h a /o^ as a function of frequency at d i f f e r e n t temper-a t u r e s . When the energy of the excited state i s p e r f e c t l y d e f i n e d , i . e . the l i f e - t i m e of the state i s i n f i n i t e , the .' minimum energy t o excite two q u a s i - p a r t i c l e s by destroying one Cooper p a i r i s equal to twice the energy gap. When the f r e q u e n c y of sound (h = l ) i s l e s s than 2 A attenuation y F o r superconductors w i t h paramagnetic i m p u r i t i e s , the energy gap can vanish-while the metal i s s t i l l i n the superconducting -chase. They are the s o - c a l l e d gapless s u p e r c o n d u c t o r s . However we are only i n t e r e s t e d i n pure, weak-coupling superconductors. In these superconductors, a l t h o u g h the order parameter i s i n general complex, i t can be represented as r e a l when there i s no external f i e l d s , and, i t can then be i d e n t i f i e d as the energy gap. 11 P i g . 4.. L o n g i t u d i n a l a t t e n u a t i o n of a 3 3 . 5 sound wave i n pure t i n ( a f t e r R e f . ( 1 2 ) ) . 12 i s due c o r m l c t e l ^ to s c a t t e r i n g of q u a s i - p a r t i c l e s from one er-ci-ed state to another. ';/hen the freouency i s great-er than or enual to twice the energy gap, absorption due te destroying Cooper p a i r s can occur. Therefore i n F i g . 3 , whenever the freruencv i s twice the energy gap, there i s ••' ,'\vr.r> In tho absorption. In F i g . 1 of Ref. (10), there i s , however, a mistake or a misprint: the s c a l e ^ f o r w/2A(0) should be halved. In F i g . 4 , an experimental curve ( a f t e r Ref. (12)) i s p l o t t e d f o r sound attenuation i n pure t i n at freouency 33.5 Mc/sec. There i s a sharp drop of attenuation a f t e r the metal becomes superconduct-ing. The unit used i n F i g . 4 f o r attenuation i s decibel/cm. In case the attenuation i s equal to one decibel/cm., the u l t r a s o n i c absorption rate i s given by 0 .1 u_ log_10/cm. where u i s the v e l o c i t y of sound. In the experiments of s . Refs. (11) and (12) f o r pure metals, q *o i s much greater than one where jf0 i s the e l e c t r o n free path. In t h i s l i m i t , i t can be shown t h a t ^ ^ a ^ oc_ i s proportional to frequency (12a ) co^  i f we assume^ 'the phonon-electron coupling constant va = S g i s a coupling constant depending on the l a t t i c e dynamics of each metal but independent of q. Under t h i s assumption, o- can be written as 2 2 ft i s taken as 1. 14 or oo where 4 = 24(0) m 2 [ g l 2 /ir In .Fig. 5 , r a t i o s f o r ae/s{ and oc^/d are p l o t t e d as a f u n c t i o n of freouency at temperature T=.7.6T^... The r a t i o c o:.s/fli can be obtained from F i g . 3.. by simpl y n o t i n g t h a t cc a z .• s n ^ - ~n * In chapter I ' l , we s h a l l use q u a s i - p a r t i c l e operators and T$ . When acts on the system, i t w i l l 'create an e x c i t a t i o n w i t h momentum k. The operator «j+ , when a c t i n g on the system 1, s h a l l .destroy an e x c i t a t i o n w i t h momentum k. We s h a l l c a l l the' e x c i t a t i o n a q u a s i -p a r t i c l e . The number of q u a s i - p a r t i c l e s i s not conserved. In the o u a s i - n a r t i c i e . r e p r e s e n t a t i o n , the mechanisms of phonon a b s o r p t i o n and emission are presented i n the E a m i l - , t o n i a n by terms i n v o l v i n g (a) fCvf ifj? , (b) t>^  fn+f ^ , (c) b^it^Ti? and t h e i r complex conjugate. b c i s the phonon a n n i h i l a t i o n o p e r a t o r . The process (a) i s e q u i v a l e n t to s c a t t e r i n g of q u a s i - p a r t i c l e from k to k+q by absorbing a p-onon of v e c t o r a. (b) i s e q u i v a l e n t to the c r e a t i o n of a p a i r of q u a s i - p a r t i c l e s by absorbing a phonon. When the energy of the e x c i t e d s t a t e s i s p e r f e c t l y d e f i n e d , the minimum of energy needed f o r s u c h • c r e a t i o n s i s equal to twice the energy gap and, t h e r e f o r e , t h i s process i s im p o s s i b l e u n l e s s the phonon energy 10n i s g r e a t e r than 2A» 15 However, t h e l i f e t i m e -r o f t h e state i s i n f a c t f i n i t e . (15) q Tewprdt v 'has shown that,due to phonon-scattering alone, -a i t i s of t h e order 10' y sec. f o r t h e temperature range of experimental i n t e r e s t . Therefore the indeterminacy of the e n e r g y o f a state' i s priven by T J ^ ' . When T ^ - 1 i s comparable to A. , we have some sort of gapless s i t u a t i o n . In t h i s case, a p a i r o f q u a s i - p a r t i c l e s can be created even though co < 2& . (c) i s corresponding to the case of absorbing a phonon and destroying a p a i r o f o u a s i - p a r t i c l e s simultaneously. When the energy of the q u a s i - p a r t i c l e state i s p e r f e c t l y defined, such a process can never conserve energy. However, as the f i n i t e n e s s o f l i f e t i m e i s taken i n t o consideration, t h i s process i s also possible near the c r i t i c a l temperature. Mathematically the e f f e c t of f i n i t e l i f e t i m e i s represented by r e p l a c i n g the d e l t a functions -^fc) by Lorentzian f a c t o r s "^ '^/[u'lti^ Vr^u*] i n the expression f o r sound attenuation. I t s h a l l be shown i n $11.4- that the r e l a x a t i o n time T of the random term f ( t ) i s of the same order as the c l i f e t i m e T ^ u of the q u a s i - p a r t i c l e density (or the l i f e t i m e of the q u a s i - p a r t i c l e energy state.) As the r e l a x a t i o n time f o r the phonon i s of the order 10 ^  sec. or longer the re l a x a t i o n time T of f ( t ) i s much shorter than the r e l a x -c ation time of phonon. We have a well-defined case where the Langevin equation i s ap p l i c a b l e . Besides the anharmonic terms i n the phonon Hamil-tonian, which gives r i s e to phonon-phonon i n t e r a c t i o n , a phonon can also i n t e r a c t with another phonon i n d i r e c t l y through i t s i n t e r a c t i o n with the o u a s i - p a r t i c l e s . When we neglect the phonon-phonon i n t e r a c t i o n s which are of higher order, we have the p h y s i c a l p i c t u r e of a phonon i n a re s e r -v o i r of q u a s i - p a r t i c l e s . Absorption of sound i s due to i n -t e r a c t i o n of phonons with the r e s e r v o i r . As the r e l a x a t i o n time of the r e s e r v o i r i s the r e l a x a t i o n time of the random 16 term f ( t ) , i t i s much shorter than, that of the phonon and we can consider the r e s e r v o i r as always i n equilibrium. CD Using Mori's formalism, we are a b l e v y t o obtain an expression f o r the l o n g i t u d i n a l attenuation rate of sound wave i n \yeak-coupling, pure superconductor. In t h i s formal-ism, i t i s e s s e n t i a l that the r e l a x a t i o n time of f ( t ) i s f i n i t e . I t should be, i n f a c t , much shorter than the re l a x a -t i o n time of phonon. The i n c l u s i o n of a f i n i t e r e l a x a t i o n time of f ( t ) has the e f f e c t of replacing the de l t a functions i n the expressions f o r sound attenuations a i n Refs. (8) and (9) by Lore n t z i a n ' f a c t o r s . Corresponding to the processes ( a ) , (b) and ( c ) , we replace (see Eq. (2.2.28)) V/hen A » T ~ 1 and co<2A,..the p r o b a b i l i t y of the process (b) and (c) (creation and destruction of quasi-p a r t i c l e s ) i s extremely small. This i s equivalent to putting It i s not a bad approximation to replace '/[(*»-fei.'-Wj'J+rc2] by T T S ( t ^ - C y . However, instead of making such approxima-t i o n , we evaluate numerically the r a t i o of l o n g i t u d i n a l u l t r a -sonic absorption rate i n superconducting and normal phase. Our r e s u l t s ( F i g . 2.2) agree much better with the experimental d a t a v 'than the e a r l i e r r e s u l t s derived i n Refs. (8) and ( 9 ) . V/hen co >> x and co > 2 A . replacement of a l l the Lorentzian O C ' Q functions by d e l t a functions gives us the expression f o r a s derived i n Ref. ( 9 ) . ° ^ . V/hen co - jA, the Lorentzian functions cannot c ' 17 be r e p l a c e d by d e l t a f u n c t i o n s . Process (b) and ( c ) are n o t i m p o s s i b l e even when co<2A. That i s , even when to< 2A, a p a i r o f o u a s i - p a r t i c l e s " c a n be c r e a t e d by absorbing a phonon. What i s more: . a phonon can he absorbed w i t h a . p a i r o f q u a s i - p a r t i c l e s destroyed at the same time. The p h y s i c a l r e a s o n s u n d e r l y i n g such p o s s i b i l i t i e s have been d i s c u s s e d e a r l i e r . Such e f f e c t can he f e l t o n l y when th e t e m p e r a t u r e i s v e r v near the c r i t i c a l temperature T . _c C I n f a c t , i t can he f e l t o n l y when IrT/T < 10 y . I n ^ I I . 4 , .we d e r i v e a sum r u l e f o r the l o n g i -t u d i n a l u l t r a s o n i c a b s o r p t i o n r a t e . I t i s s t r a i g h t f o r w a r d (19) •to a r r i v e at t h i s sum r u l e v 'once we e s t a b l i s h e d c onnection between t h e u l t r a s o n i c a b s o r p t i o n r a t e and the s t r u c t u r e f u n c t i o n ( f o r d e f i n i t i o n , see (2.4.4)). The sum r u l e f o r t h e s t r u c t u r e f u n c t i o n has been d e r i v e d ; e.g. i n Ref. (14), by a i d o f t h e p r i n c i p l e of c o n s e r v a t i o n of matter. We a l s o e s t a b l i s h e d t h e r e l a t i o n s between the l o n g i t u d i n a l u l t r a -s o n i c a b s o r p t i o n r a t e , t h e l o n g i t u d i n a l c o n d u c t i v i t y and d i e l e c t r i c c o n s t a n t , (2.4.12) and (2.4.11). We a l s o e s t a - • b l i s h t h e i r r a t i o s i n normal phase and superconducting phase. I n t h e d e r i v a t i o n of t h e sum r u l e s , we would l i k e to s t r e s s In t h e d e r i v a t i o n of sum r u l e s f o r a (q,co) there i s no d i s p e r s i o n r e l a t i o n s h i p between ' co. and q at a l l , l e t alone s u c h simple r e l a t i o n s h i p as co = v q. i n the j e l l i u m model s o f n a t u r a l propagation o f sound or co = cq as f o r l i e r h t pro-pagating i n npn-absorbin? media (v and c are r e s p e c t i v e l y v e l o c i t i e s o f sound and l i f f h t ) . The experiments i n Ref.(11) (12) are c o n d u c t e d f o r n a t u r a l . p r o p a g a t i o n of sound where a d e f i n i t e d i s p e r s i o n r e l a t i o n between frequency and wave v e c t o r e x i s t s . The r e l a t i o n between u l t r a s o n i c a b s o r p t i o n and f r e q u e n c y d e r i v e d f r om t h e s e experiments i s , t h e r e f o r e , n o t a p p l i c a b l e as f a r as t h e sum r u l e i s concerned. . The c u r v e s i n F i g . 5. a r e not obeyed by a„(q,co) i n the i n t e g r a n d o f (.2.4.9) where q i s kept constant as to v a r i e s . 18 th? t the v.rave vector q end the freouencv to are two indepen-dent q u a n t i t i e s . There i s no dis p e r s i o n r e l a t i o n between co and "r as i n the case of natural wave propagation. These remarks are true whether we are t a l k i n r of a sum ru l e f o r l o n g i t u d i n a l c o n d u c t i v i t y or d i e l e c t r i c constant or long-i t u d i n a l u l t r a s o n i c absorption. In Chapter I I , we apply the Generalized Langevin Equation to evaluate u l t r a s o n i c attenuation where quantum mechanical treatments are e s s e n t i a l . In $111.3, we s h a l l use the Langevin approach to i n v e s t i g a t e , i n the c l a s s i c a l l i m i t , the f l u c t u a t i o n s of qua n t i t i e s l i k e magnetic f l u x and currents i n a l o n f , t h i n , hollow superconducting c y l i n d e r . The c y l i n d e r i s of length JL , radius P. and thickness d with the condition that £ » R » d . In f a c t , there are more str i n g e n t conditions to be imposed on the thickness d, which we s h a l l discuss i n more d e t a i l l a t e r . There may e x i s t an external magnetic f i e l d p a r a l l e l to the axis of the c y l i n d e r . In t h i s lonf? c v l i n d e r , the magnetic f i e l d F in the hole can. be considered as s p a t i a l l y constant and the magnetic f l u x $ i s equal to $=1TRLH . In f i l l . 3 , we evaluate the aut o c o r r e l a t i o n f u n c t i o n , the variance and the spectrum of the magnetic f l u x ? . V7e are also i n -terested i n the variances of the currents and the c o v a r i -ances of the magnetic f l u x and the currents. However, as the magnetic f l u x and the currents are l i n e a r l y r e l a t e d , t h e v can he evaluated without d i f f i c u l t y once our evalua-tions on the magnetic f l u x are completed. The autocorrelation function of a nuantity, say magnetic f l u x , i s defined by [ <§(t)f(£+'0> - . Pron the autocorrelation we know how the quantity relaxes to i t s average value. T^e variance of a quantity $ i s \/ar4$W$a>-fe)* 19 It i s equal to tbe autocorrelation function with r = 0. The variance gives an estimate of how much a quantity f l u c t u a t e about i t s average value. The spectrum i s defined as the Fourier transform of the autocorrelation function, i t i s r e l a t e d to the absorption of.energy at a given frequency. The covariance of two n u a n t i t i e s , say the superconducting component of the current I s and magnetic f l u x § :.is defined as c o v 2 <r$£> -<*><9 I t t e l l s , on the average, how I v a r i e s with respect to • s In the case of t h i n superconducting cylinder,, we are t r e a t i n g , Cov (I e,<?) i s shown to be negative. From t h i s , s we expect that as increases, there i s greater p r o b a b i l i t y f o r I to decrease than to increase. s ' At temperature above 0 K., there are e x c i t a t i o n s from the B C S condensate. The dominant source of the f l u c t u a t i o n l i e s i n the random s c a t t e r i n g of these e x c i t a t i o n s by impurities i n , or the l a t t i c e s of, the superconductor. ' Mathematically, the random' s c a t t e r i n g of the e x c i t a t i o n s (or normal electrons) can be represented by.the.Langevin . equation (3.3.2) f o r the d r i f t v e l o c i t y v of the normal -» n e l e c t r o n s . The average of \j i s aero. However, due to the random a c c e l e r a t i o n f^(t) i n ( 3 . 3 . 2 ' ) , j at any instant may not be zero. I t f l u c t u a t e s about i t s average zero value. f f t(t) represents the r a p i d l y varying part of the s c a t t e r i n g of the normal e l e c t r o n s . 20 In chapter I I I we are t r e a t i n g the case of t h i n c y l i n d e r s . By "t h i n c y l i n d e r " , we mean we can assume' s p a t i a l constancy f o r the superconducting e l e c t r o n density n , the superconducting component i of current density (the part of current c a r r i e d by Cooper p a i r s ) and the normal component of current density (the part of current c a r r i e d by the e x c i t a t i o n s ) . We s h a l l discuss l a t e r i n more d e t a i l the conditions under which such s p a t i a l constancy can be assumed. In f i l l . 3 where the Langevin approach i s used, we also neglect the f l u c t u a t i o n s of the superconducting e l e c -tron density n s and the t o t a l e l e c t r o n density n. That i s , n and n are assumed to be always at t h e i r average values. For t h i n c y l i n d e r , the t o t a l current I (I = j<£d) and magnetic f l u x § i s simply r e l a t e d by$ = Ll+§ 6 where L' i s the s elf-inductance. As the normal component I of I f l u c t u a t e d because of random s c a t t e r i n g by impurities and l a t t i c e s , ^ fluctuates.. However and I are connected through the f l u x o i d ^ 1 5 ^ 1 6 ^ c o n d i t i o n (3.1.1). Thus a l l the q u a n t i t i e s , <& , I g and I f l u c t u a t e i n such a way that (3.1.1),(3.3.2),(3.3.8), and (3.3.9a) are obeyed s e l f -c o n s i s t e n t l y . Using (3.3.1),(3.3.2),(3.3.8), and (3.3.9a), we obtain a l i n e a r r e l a t i o n (3.3.11) between 4 and f ( t ) . Once the spectrum (3.3.6) of ^ n ( t ) i s obtained, we can immediately evaluate the spectrum of $ , (3.3.14). The a u t o - c o r r e l a t i o n (3.3.15) and the variance (3.3.16) then . fo l l o w . Most of the r e s u l t s i n §3.3 have been reported by (IT) Prof. R. E. Burgess to the Symposiumv 1 ; o n Physics of Super-conducting Devices (1967). (2 8 ) There are two important c h a r a c t e r i s t i c lengths^' J i n superconductivity. One i s the penetration depth at temperature T, A. (T), and the other i s the coherence length 21 at temperature T, £(T). The penetration depth measures how a magnetic f i e l d can e f f e c t i v e l y penetrate into a superconduc It also i n d i c a t e s how the magnetic f i e l d v a r i e s s p a t i a l l y . The coherence length at temperature T, ^ ,(T) i s a measure of how the order parameter & v a r i e s s p a t i a l l y . That i s , i f the dstance between two points i s small compared v/ith ^ ( T ) j we-can reasonably assume that the order parameter : A at these two points i s the same. The superconducting (g) e l e c t r o n density n i s i n general a f u n c t i o n of the order parameter, S p e c i f i c a l l y , near the c r i t i c a l temperature, n^ , i s prop o r t i o n a l to lAj . (we are e s s e n t i a l l y i n t e r e s t e d s. i i n the region near ^ c)» ^ (T) i s thus a measure of how' n v a r i e s s p a t i a l l y . Near the c r i t i c a l temperature Tc ZJT^YTI z _ T / r c where hS0) i s the London's penetration depth at • - 2 ifi-zero temperature, equal to mc I , £ i s the coherence L^ -JTe n J length (at zero temperature) equal to , T c i s the c r i t i c a l temperature of the superconductor, A.(0) i s ^^e order parameter at zero temperature. In the t h e s i s , A. (T) s h a l l be simply denoted bv A. and A T (0) by A. \ i s defined by | >n c.2 / itire*^"] z , When t>^ e thickness of the c y l i n d e r i s smaller than £(T) and Ag(T) , n s and 3* can be* ' assumed to be s p a t i a l l y constant. Such assumptions have been made by T i n k h a m ^ ^ i n de r i v i n g the v a r i a t i o n of c r i t i c a l current i n a t h i n c y l i n d e r with temperature and there were good agreement between h i s theory and L i t t l e - P a r k ^ 2 0 ^ experiment. .... - - ; - -'When d < A(T), the magnetic f i e l d can he assumed to • vary l i n e a r l y with the'distance from the surface. Since the'" current density i s proportional to the c u r l of the f i e l d , i t can be assumed to be constant. 22 In general, the current density at a point r i s influenced by the vector p o t e n t i a l at another point r'. Mathematically, t h i s non-local e f f e c t can be represented by A d.VS(r,r')Atf')_ The kernal matrix S i s proportional to ^ j pL - ' "^^  ' ^ *U i v;here ? i s the coherence length eaual t o <-£-and JLthe free electron p a t h . The s p a t i a l v a r i a t i o n of the vector p o t e n t i a l i s measured by 'A, , (the London penetration depth at temperature T ) . If X$ » (^r ' the r e l a t i o n between the current density and vector p o t e n t i a l can be written as *" ' r ' jo?> = ' m V s i ? . r ' ) ] A ( 7 > In such a case, the electromagnetic response i s l o c a l i z e d ; that i s , the vector p o t e n t i a l a t another point r has no influence on the current density at A superconductor with l o c a l i z e d electromagnetic response i s c a l l e d London superconductor. For impure superconductors''where f0 >P0 , they are London superconductors i f A . s > J [ 0 . Pure super-conductors are London superconductors i f A. > f . Pure (257 S * 0 Type II superconductors are those superconductors where the London's penetration depth at zero temperature, A., i s greater than f • Since A. , beine: i d e n t i c a l to A at 0° K . , ^ o s 7 increases with temperature, Type I I superconductors are . . always London superconductors. For Type I superconductor's,' ' on the other hand, i s greater than A.. However, when the temperature i s near enough to the c r i t i c a l temperature T . A. can be greater than and the Type I superconductors c' s ' o then become London's superconductors. Near T , A and Al u c b are r e l a t e d by \-\JL(\~T/T<.J\ z . The temperature range where a Type I superconductor becomes London superconductor i s g i v e n by • > J z ( , _ T / T c ) 23 The two-f luid model used i n chapter III , chapter IV and chapter V and the Landau Ginzburg's for of Gibbs free energy used i n §3.4- are only v a l i d for London superconductors. In §3.3, a regular thin cylinder was discussed. The arguments for the regular thin cylinder are extended i n a straightforward fashion to a cylinder with i r regular cross-sectional shape. B. Free Energy Approach In ^111.3, we have assumed that the order para-meter does not varv with time; that i s , i t always remains at i t s average value. In f i l l . 4 - , we relax such r e s t r i c t i o n . We investigate the contribution of the f luctuat ion of super-conducting electron density n g with respect to time by "free-energy" approach as the Langevin approach turns out to be inconvenient. The superconducting electron density i s s t i l l s p a t i a l l y constant by reauiring d<A . The t o t a l electron density, as before, i s constant both s p a t i a l l y and with res -pect to time. Due to this restraint and the restraints im-posed by Eos. (3.1.1), (3.3.8), and (3.3.9a), there are only two independent variables i n the system at a given tempera-ture. We choose these two independent variables to be the superconducting electron density n and magnetic f l u x $ . The Gibbs free energy G g can therefore be expressed as a function of n_ and $ . The probabil i ty that n_ and $ may s s assume values between n s , n g + dn g and $, $ + d§ i s given by ~£(r,(ils,$) . 24 We expand i n T a y l o r ' s - . s e r i e s t h e Gibb s f r e e e n e r g y and s t o p a t t h e q u a d r a t i c t e r m . where n, = n s - <«f> and . <$>and<nA a r e t h e 3 average v a l u e s o f n and $ . They a r e g i v e n by t h e . s o l u t i o n s o f The s o l u t i o n o f t h e s e two e q u a t i o n s a r e d i s c u s s e d i n d e t a i l i n A p p e n d i x B. When t h e r e i s no r e a l , p o s i t i v e s o l u t i o n s ofi£n \ t h e sys t e m i s i n t h e n o r m a l phase. W i t h p, q d e n o t i n g i n t e g e r s . 0 , 1, 2, t h e c e n t r a l moments o f $ and n a r e d e f i n e d by s - 0 0 W i t h q u a d r a t i c a p p r o x i m a t i o n f o r G ( n g , $ ) , g ( n ,§) i s a b i v a r i a t e G a u s s i a n d i s t r i b u t i o n o f n a n d $ . By s t r a i g h t -f o r w a r d i n t e g r a t i o n , i t f o l l o w s f r o m t h e d e f i n i t i o n s of t h e moments.that ( l - ) ^ C**>M> 25 The c o r r e l a t i o n c o e f f i c i e n t i n the quadratic approximation i s given, by .'. <?>,<"s> In t h e Landau Ginzburg temperature domain (near T , so that A i s s m a l l ) , t h e Gibbs free energy possesses a rather simple f o r m ( 3 . 4 . 1 ) . From t h i s form of Gibbs free energy, i!!£U , 7 n ? ) (r._ I can he expressed as some rather simple functions of <n s^ and ^ §), while ^ n ^ and & ^  themselves can be expressed as functions of § 0 and ^  by means of the eouilibrium conditions 4^r=0 and -f^=o . The c o r r e l a t i o n c o e f f i c i e n t , f , i s a measure of coupling between n and 2 . When r = o » the f l u c t u a t i o n S . <v» of n does not influence the f l u c t u a t i o n of $ • Under such s i t u a t i o n , we need not consider the f l u c t u a t i o n of n s at a l l , i f we are only. i n t e r e s t e d i n the f l u c t u a t i o n of $ . When the average current ( I ) i s equal to zero ( i . e , the magnetic f l u x §. i s eoual to §. ) f i s zero. C. F l u c t u a t i o n - D i s s i p a t i o n Approach Due to the presence of an external time-dependent perturbation F ( t ) , there e x i s t s a perturbation term yti'sAFU) i n the Hamiltonian of the system, A being an.operator conjugate to F ( t ) . When we are only i n t e r e s t e d i n l i n e a r response, " 26 an operator can, i n general, be written as B = B + X F(t) where B and % are operators that do not involve F ( t ) e x p l i -c i t l y . Due to the external perturbation, there i s a response A B(t) at time t defined, by AB(t) = B(t) - B where B(t) i s the eouilibrium average of the operator B i n the presence of. an external time-dependent perturbation F ( t ) and B i s the equilibrium average of B v/hen there i s no time-dependent perturbation, B(t) i s a function of time while B i s time-independent. When F ( t ) = F ^ e 1 ^ , the generalized s u s c e p t i b i l i t y , ct B A(w) v. .is defined by • AB(t) = Re[a B A(co) F w e i u ) t } Due to the presence of the t e r m ^ F C t ) , the l i m i t of c.-g^ Cco), when co goes to i n f i n i t y , i s non-vanishing.- In f a c t , i t i s shown i n §1.2 that lim o:B4(co) = <<X> where ^ i s the eouilibrium average of the operator . Modifying s l i g h t l y Kubo's formalism^^to take care of the (27 ) f a c t that lim o'.(co) ma:/ not be zero, we o b t a i n v , i n the c l a s s i c a l l i m i t , (1,2.5), and, when' co = 0, (1.2.8), o i ^ o ) -U6A{.oo) = C o v ( A , B ) where the covariahce of two quantity A and B, Cov ( A,B), i s defined by CovCA,B) s <AB> - < A > < B > i 27 and the cross spectrum S-p^(to) i s the Fourier transform of Cov(A,B(t)). The d e t a i l of the d e r i v a t i o n i s presented i n §1.2. (22) " Landau and L i f t s h i f t v 'have pointed out that when the r e l a x a t i o n times T. and T-n of A and B s a t i s f i e s Q , . A B the i n e q u a l i t i e s ' (n = 1 ) we can t r e a t A and B as c l a s s i c a l q u a n t i t i e s . Refs. (22), (23). and (24) have mentioned the p a r t i c u l a r case of (1.2.8) when A = B. The Kramer-Kronig r e l a t i o n has been employed to derived t h i s p a r t i c u l a r case. By aid of ( 1 . 2 . 8 ) , we can immediately obtain the covariance of A and B once t h e i r s u s c e p t i b i l i t y Cy-g(co) at frequencies co = 0 and co =oo .is . known. '..'hen F ( t ) i s changing stepwise at time t = 0, the response to such change i s denoted b v £ B ( t ) . (The sub-z a • s c r i p t a i s a reminder of the abrupt change of F ( t ) at t = 0 ) . There i s an i n i t i a l or immediate response lim A B ( t ) at time t = 0 and a f i n a l response l i m AB ( t ) at t much + t -*°* a greater than the r e l a x a t i o n time of the Quantity B. The , i n i t i a l response, limAB(t), i s equal to the generalized s u s c e p t i b i l i t y . . a B^(co) when the frequency goes to i n f i n i t y : ( 1 . 3 . 6 ) . The f i n a l response, l i ^ ^ K t ) , i s equal to the generalized s u s c e p t i b i l i t y a-g^(co) at zero frequency; (1 . 3 . 5 ) • Thus, we see that we can only have a non-vanishing, immediate, abrupt response when non-vanishing. We s h a l l employ these r e l a t i o n s i n chapter V to inv e s t i g a t e the i n i t i a l and f i n a l magnetic responses of cy l i n d e r s of a r b i t r a r y thickness. J The condition T « k^T i s equivalent to the quantum uncertainty At i n energy being much smaller than k-oT which i s a measure of thermal f l u c t u a t i o n . The quantum e f f e c t , therefore, becomes unimportant when "t"'«I^ T i s true. 28 R . K U D O " has d e r i v e d t h e p a r t i c u l a r c a s e o f (1.2.8) where 0.^.(00) i s v a n i s h i n g , by arguments somewhat s i m i l a r t o t h o s e p r e s e n t e d i n §1.3. H. T a k a h a s i J , a f t e r some p h y s i c a l arguments, has come t o t h e c o n c l u s i o n t h a t Cov(5(t)>/\)=-^J[^5A((.o)-A^(t)] When t=0, we have Cov(B,A") =^L^Bdl00)-AB<lCo)]. T h i s r e l a t i o n and (1.2.8) a r e t h e n r e l a t e d by (1.3.5) and (1.3.6). I t i s o n l y a t t i m e s t=0 and t=<o- t h a t t h e r e s p o n s e A B _ ( t ) i s more s i m p l y a r e l a t e d t o t h e g e n e r a l i z e d s u s c e p t i b i l i t y . I n g e n e r a l , we have t h e r e l a t i o n (1.2.5). We have a r r i v e d a t (1.2.5) and (1.2.8) i n §1.2 by p o i n t i n g o u t t h a t t h e r e a s o n o f <*-g^ (°°) b e i n g n o n - z e r o i s due to, t h e e x i s t e n c e o f a " d i a m a g n e t i c " t e r m 1CF(t; i n t h e o p e r a t o r f o r B. I n f l . 3 , we show t h a t we c a n r e - d e r i v e T a k a h a s i * s r e s u l t more r i g o r o u s l y u s i n g Kubo's l i n e a r r e s p o n s e t h e o r y . (21) I n c h a p t e r IV,. we i n v e s t i g a t e v 'the m a g n e t i c f l u c t u a t i o n s o f r e g u l a r c y l i n d e r s o f a r b i t r a r y t h i c k n e s s . To be p r e c i s e , we c o n s i d e r a c y l i n d e r o f i n n e r r a d i u s a, o u t e r r a d i u s b and length<£. The t h i c k n e s s d i s e q u a l t o b-a. The r e s t r i c t i o n oh t h e t h i c k n e s s i n f i l l . 3 i s removed. F o r t h e same r e a s o n s as we d i s c u s s e d e a r l i e r (pp. 18-19), we a r e i n t e r e s t e d i n know i n g t h e v a r i a n c e s and s p e c t r a o f ma g n e t i c f l u x , m a g n e t i s a t i o n , and t h e c o v a r i a n c e between c u r r e n t s and ma g n e t i c f l u x . The most c o n v e n i e n t system o f c o - o r d i n a t e s i n t h i s c a s e i s o b v i o u s l y t h e c y l i n d r i c a l c o - o r d i n a t e ( r , 0 , z ) w i t h t h e a x i s o f t h e c y l i n d e r f o r m i n g t h e z - a x i s . W h i l e we s t i l l assume c y l i n d r i c a l symmetry ( i . e . i n d e p e n d e n t o f&) the c u r r e n t d e n s i t i e s c a n no l o n g e r be assumed s p a t i a l l y c o n s t a n t as i n £.111.3. I n g e n e r a l , t h e y have d i f f e r e n t v a l u e s a t d i f f e r e n t v a l u e s o f r a d i u s r . We c o n t i n u e , how-e v e r , t o assume as i n §111.3 t h a t t h e s u p e r c o n d u c t i n g e l e c t r o n d e n s i t y .and t h e t o t a l e l e c t r o n d e n s i t y a r e c o n s t a n t b o t h s p a t i a l l y and w i t h r e s p e c t t o t i m e . The s p a t i a l h o m o geneity o f t h e s u p e r c o n d u c t i n g e l e c t r o n d e n s i t y i s a r e a s o n a b l e a s s u m p t i o n f o r Type I 29 superconductor. For Type I superconductors, the coherence length ^(T) i s always greater than the penetration depth A . s # As mentioned before i n connection with chapter I I I , ^(T) i s a measure of s p a t i a l v a r i a t i o n of the electron superconduct-ing density and X i s a measure of the penetration of a magnetic f i e l d . Due to Messner e f f e c t , only points within the penetration depth A. contribute s i g n i f i c a n t l y to the generalized s u s c e p t i b i l i t y . When ^ ( T ) ^ ^ s * * n e supercon-ducting e l e c t r o n density can be considered constant within the penetration depth and thus can be considered as constant f o r our purpose. For pure Type I I superconductors, homo-geneity of electron d e n s i t i e s can be assumed i f the cohe-rence length £(T) i s greater than the thickness d ( i f hollow) or the radius of the c y l i n d e r ( i f s o l i d ) . Due to the r a d i a l dependence, the Langevin ap-proach i n $ 3 . 3 or the free energy approach i n §3.4- are no longer convenient. Instead, we employ the r e l a t i o n (1.2.8) between covariance and generalized s u s c e p t i b i l i t y and the f l u c t u a t i o n - d i s s i p a t i o n theorem (FDT)^ 2^^ to evaluate the variances, covariances, and spectra. The c e n t r a l point i s then to derive an e x p l i c i t expression f o r the generalized s u s c e p t i b i l i t i e s f o r the magnetic f l u x , currents, and mag-n e t i s a t i o n . Once these generalized s u s c e p t i b i l i t i e s are known, the properties of magnetic f l u c t u a t i o n s of the system s h a l l be obtained by a i d of (1.2.8) and FDT. To evaluate the s u s c e p t i b i l i t i e s , we assume that there i s a time-dependent magnetic f i e l d " i f l ( t ) e H e"*"w* 30 p a r a l l e l to the z-axis. Due to t h i s external f i e l d , the perturbation term i n the Hamiltonian i s where **| i s an operator denoting magnetisation,, Let M(t) denote t h e average of ^  at time t i n the presence of H^t).!. M denotes the average of *| when there i s no external time-dependent magnetic f i e l d . The response of magnetisation v i s given by AM(t) = M(t) - M. Let us denote the Fourier transform of AM(t) by M^. The s u s c e p t i b i l i t y a M M(to) i s then defined by An e x p l i c i t expression f o r M , on the other hand, can be obtained' from the Maxwell equations: can be f i n a l l y . wri t t e n as (4 . 5.2a): where g^jM(co) i s an e x p l i c i t expression i n v o l v i n g modified Bessel functions (see ( 4 . 5 . 2 a ) ) . Comparing these t w o - i L d expressions, we obtain ( 4 . 5 . 3 ) : By a i d of (1.2.8) and FDT,,the variance and low frequency spectrum of magnetisation can be obtained. Due to the perturbation term H" =JV|H(t), there i s a current response &I(t). We denote the Fourie r transform of &I(t) by•I . The generalized s u s c e p t i b i l i t y aJVjj(co) i s defined 0<„Aoo) - I*. Mtv ' fi 31 On the other hand, an e x p l i c i t expression can be obtained by means of Maxwell equation. I can be f i n a l l y expressed as (4.5.12): Comparing these two expressions, the s u s c e p t i b i l i t y cx^j(co) i s therefore obtained as an e x p l i c i t expression i n v o l v i n g Bessel f u n c t i o n s . From (1.2.8), we a r r i v e at an expression f o r the covariance, Cov(M,l), of magnetisation and t o t a l current. .. Due to an external time-dependent current i n a surrounding solenoid c o a x i a l with the superconducting c y l i n d e r , we have a perturbation term a, = c~ cptItt) i n the Hamiltonian. f t i s an operator f o r the t o t a l magnetic f l u x i n the cyl i n d e r . . ( § t i s equal to the magnetic f l u x i n the hole, 5 . ( = Ua 2H.), plus the magnetic f l u x i n the metal of. the c y l i n d e r ) . S i m i l a r agruments as f o r the case of magnetisations are then applied to evaluate the variance and spectrum of t o t a l magnetic f l u x ^ e c o*" variances of § t and I4t" and the covariances of <$t and currents. Using the two- f l u i d model, which i s v a l i d f o r London superconductors, the r e l a t i o n between the Fourier transforms of current density response £j(rt)-j(r) and vector p o t e n t i a l response j i t ( r t ) - A ( r ) ] i s where ^(co) i s defined by (4.2.6). In case one i s only i n t e r e s t e d i n evaluating the covariances, we need only know /*(co) at two pa r t i c u l a r - values of frequency, namely, co = 0 and co = t o . ^(co) at these two values of frequency can be obtained by perturbation method (^4.2B) without appealing to the two-fluid model. 32 Due to the c y l i n d r i c a l symmetry that we mentioned i such that before, A (r) depends on radius r and we can choose a gauge With t h i s gauge f o r A ( r ) , the Maxwell equation s h a l l give us a Bessel d i f f e r e n t i a l equation (4 .3.1) f o r A ^ ( r ) . The s o l u t i o n f o r A (r) i n (4 .3.1) are modified Bessel functions *~ co I-^(^r) and K- (^yi*r). Using the Boundary conditions (4 .3 .3 ) we obtain an e x p l i c i t s o l u t i o n f o r A (r) i n terms of modified /^CO Bessel functions (see (4 . 3.10)). Once an e x p l i c i t expression f o r A (r) i s obtained, e x p l i c i t expressions f o r M , I , CO CO CO *cco and I can be obtained through t h e i r r e l a t i o n s , (4.5.2), SCO (4.5.11), (4.7.2a) and (4 . 7 . I f ) with ^ ^ ( r ) . This method to obtain e x p l i c i t s o l u t i o n s f o r vector p o t e n t i a l , e t c . , has been employed i n Refs. (28) and (29 a-c) f o r the s t a t i c case ( i . e . when there i s no time-dependent.magnetic f i e l d ) . We have generalized t h e i r c a l c u l a t i o n s to dynamic s i t u a t i o n . Ref. (31) has obtained an expression f o r the generalized s u s c e p t i b i l i t y f o r the magnetisation i n a s o l i d c y l i n d e r . By putting the inner radius a = 0 i n our general consideration, we can obtain • r e s u l t s f o r the p a r t i c u l a r case of a s o l i d c y l i n d e r . By require ing the innder radius Ct to be much greater,than the penetration depth at temperature T ( i . e . A ) and the thickness d much smaller than A, , we obtain the r e s u l t s of the t h i n c y l i n d e r considered i n §111.3. By r e q u i r i n g A. to go to i n f i n i t y , we 3 3 obtain r e s u l t s f o r normal c y ^ ^ d e r s . L. Vant-Hull et. a l . have measured the noise o f the magnetic f l u x i n a s o l i d c y l i n d e r of t i n (T = 3 , 7 2 K). The v a r i a t i o n of the standard deviation of magnetic f l u x (r.m.s. value of the magnetic noise). with temperature i s shown i n F i g . The s o l i d curve i s p l o t t e d according to the t h e o r e c t i c a l expression(4>7.4cT) i n ^4-.J.. According to t h i s expression, the thermal magnetic noise does not disappear when the metal becomes superconducting. With the dimensions (10 cm. x 0 , 4 7 cm, d i a . ) of the t i n rod used i n the experiment, we expect the standard de v i a t i o n " to drop more than an order of magnitude from i t s normal T - T state value as the order of c becomes greater than -ST 10 . While there i s no c o n t r a d i c t i o n between our theory and experiment, more accurate data are needed to decide . whether our theory i s an adequate one to describe the magnetic f l u c t u a t i o n s i n superconducting c y l i n d e r s . We hope that the present work can stimulate more e f f o r t s in t h i s d i r e c t i o n . As pointed out by L, Vant-Hull et al^ fHhat the existence of thermal magnetic noise has serious lm i m p l i c a t i o n s i n the design of magnetic s h i e l d i n g and magnetometry i n the sub-microgauss regime, better under-standing about the thermal magnetic f l u c t u a t i o n i s important. When an e x t e r n a l - f i e l d changes stepwise-at a . c e r t a i n time; say,, at time t = 0, from H a to H^ + AH^ ,; how s h a l l the c y l i n d e r respond to i t ? In chapter V, we discuss t h i s question,^ Before time t' = 0, the system i s at equil i b r i u m and the averages of the system'' are independent of time. A f t e r t = 0, the system i s disturbed. I t s h a l l r e l a x , i n the end, to a new equi l i b r i u m value consistent with the f i e l d H'" + AH.. The i n i t i a l response of a quantity, s a y , t h e t o t a l current, i s denoted by A I ( 0 + ) , I t i s equal 34-t o the d i f f e r e n c e between the va l u e at time t = 0 and the value f o r time before t £ 0 . AT(o+l = I (00 - I ( t « o ) This i n i t i a l v a l u e , as p o i n t e d out i n ^ 1 . 2 , i s equal t o the g e n e r a l i z e d s u s c e p t i b i l i t y O^IMC^AE^ at i n f i n i t e frequency. The f i n a l response o f , f o r example, I i s denoted by AI(°°)» I t i s equal t o the d i f f e r e n c e between the c u r r e n t at time t much g r e a t e r than the r e l a x a t i o n time of the c u r r e n t and the e q u i l i b r i u m v a l u e of the c u r r e n t before time t = 0: A I M - T(«) - 1 (t s o ) I(*°) i s the average v a l u e of the c u r r e n t when the system has r e t u r n e d t o an e q u i l i b r i u m under the new e x t e r n a l f i e l d H & + A H A . *-s equal t o the g e n e r a l i z e d s u s c e p t i b i l i t y oc-]-M(to) at z e r o a f r e q u e n c y . With the g e n e r a l i z e d s u s c e p t i b i -l i t i e s d e r i v e d i n chapter IV, the i n i t i a l and f i n a l responses of the t o t a l magnetic f l u x the normal c u r r e n t I n , - the superconducting current, I g , and the mag n e t i s a t i o n M can be r e a d i l y o b t a i n e d . I n the case of a t h i n c y l i n d e r , we can f o l l o w i n more d e t a i l how these q u a n t i t i e s of system changes w i t h time. The changes are i l l u s t r a t e d i n P i g . 5 « 1 . In t h i s f i n a l paragraph i n the I n t r o d u c t i o n , we are going t o summarize the c o n t r i b u t i o n s of the ca n d i d a t e . I n ^1.2 , we have d e r i v e d , i n the c l a s s i c a l l i m i t , a r e l a t i o n between g e n e r a l i z e d s u s c e p t i b i l i t y and cross-spectrum (1.2.5) and a r e l a t i o n between g e n e r a l i z e d s u s c e p t i b i l i t y and covar-iance ( 1 .2 .8 ) . The m a t e r i a l i n t h i s s e c t i o n s h a l l be p u b l i s h -e d ( 2 7 ) i n t h e J o u r n a l o f s t a t i s t i c a l p h y s i c s , V o l . 2 (1970), along w i t h most of the m a t e r i a l i n chapter IV. In$ 1 . 3 , we have 35 re-derived Takahasi's r e s u l t ^ ;(1.3.9), using Kubo's l i n e a r response theory. In $3.2 and ^ 3.3, our merits c o n s i s t s of applying a general formalism (Mori's generalized Langevin e q u a t i o n ^ ^ ) to re-derive the l o n g i t u d i n a l attenuation rate of sound wave i n pure, weak-coupling superconductors. The f i n i t e n e s s of the l i f e - t i m e of the random i n t e r a c t i o n i s e s s e n t i a l i n t h i s formalism. The i n c l u s i o n of t h e v l i f e - t i m e e f f e c t gives b e t t e r agreement with experimental data than those e a r l i e r expressions derived i n Ref. (8) and ( 9 ) . Our r e s u l t s also d i f f e r i n some aspects from the e a r l i e r r e s u l t s when the temperature becomes nearly c r i t i c a l . Most of the materials i n §2.2 and §2.3 s h a l l be published^"1"-^in Physica (19T0) 568. A sum r u l e f o r the l o n g i t u d i n a l attenuation rate of sound i n superconductors i s de r i v e d i n §2.4-. In §3.3, we have independently derived the r e s u l t s f o r the regular, t h i n c y l i n d e r s . We also extend the arguments to t h i n c y l i n d e r s of i r r e g u l a r c r o s s - s e c t i o n a l shapes. In § 3.4, we use the free energy approach to t r e a t the f l u c t u a t i o n s of the superconducting e l e c t r o n density and the magnetic f l u x . In sections 4.5 - we a p p l y ( l . 2 . 8 ) and the f l u c t u a t i o n -d i f f i p a t i o n theorem to long, superconducting c y l i n d e r s of a r b i t r a r y thickness. Our theor^ i s compared with an experi-(32) ment by L. Vant-Hull e t. a l . J . In Chapter V, we discuss the responses of magnetic f l u x , etc., i n a superconducting c y l i n d e r of a r b i t r a r y thickness, to a stepwise change of an external magnetic f i e l d . I. SOME THEORIES OF IRREVERSIBLE PROCESSES  1.1 Generalized Langevin Equation Consider the equation of motion of a p h y s i c a l quantity A(t) i Mt) = FU) d . i .n dit P(t) i s a force due to i n t e r a c t i o n of A(t) with other degrees of freedom. In the theory of Brownian Motion, P(t) i s assumed to be able to.be separated i n t o two p a r t s : - -P 1(A(s),t> /.s^t 0) and f ( t ) . P 1 i s a f u n c t i o n a l of A(s) depending on the past h i s t r y of A ( t ) . I t represents average a c t i o n of the other degrees of freedom. These other degrees of freedom can be thought of as some sort -of r e s e r v o i r i n which thermal:equilibrium can be considered as always present with respect to A ( t ) . The random term f ( t ) takes account of the r a p i d l y varying part of the i n t e r a c t i o n s -In case F-^ can be approximated as l i n e a r i n A and the r e l a x a t i o n tims of A" can be considered as constant, we have This i s the phenomenological Langevin equation which appear^in the theory of Brownian Motion v . f ( t ) i s assumed^^to have the following c o r r e l a t i o n function'. 37 In thermal equilibrium, the ensemble average of a quantity-i s given by .  s. • .. -where |3^£gT) and 0<i i s the Hamiltonian of the system. In case we are dealing with, f o r example, monochromatic sound wave or spin wave with frequency 00 o , the Langevin equation can be . generalized to £ AttWw.ACt)+rMt)=-*lt) , where A(t) denotes the normal coordinate. We can fur t h e r r e l i n q u i s h the d e l t a f u n c t i o n assumption f o r The generalized version^"^ of the Langevin equation then becomes - . ,4M»-«A.A&>*C rct-t-j A(ndf -jio. o-'.o . at to H. Mori has shown that (1.1.5) can be d e r i v e d ^ ) from the equation of motion; dt 38 where L i s the L o u i l l e operator defined by V C 5SIA a* lA) / . .... i v Od i s the Hamiltonian of the system which i s assumed to a closed system without time-depandent external f o r c e . (1.1.5:) i s thus formally exact. Whether f ( t ) possesses the charac-t e r i s t i c of a random force is^determined by the nature of the system under consideration. H. Mori has defined the sc a l a r product of two q u a n t i t i e s B.and'C as Ct)d & -Ct ) i BOO = -e B « He also defined a p r o j e c t i o n operator P as P 13 - (B, A* )•(£,** )~'A where B stands f o r any p y s i c a l quantity or any operator. I t can then be shown that-and that the terms Ct3 0 /f(t ) <x*dl 4^  ) i n (1.1.5) are' 39 g i v e n by I c0 0 ( A , A * ) - ( A, A*')"' 0.1.13). Yto = ( -HO , f* ) (A, A*)" 1 , t > o ( i . i . i 4 ) K = (\- p .) A ( i . i . i f e ) A (1) Remembering t h a t v y ( f ( t ) , A ) = 0, t h e e q u a t i o n o f m o t i o n f o r ( A ( t ) , A ) can be w r i t t e n as ± ( A U ) , A*) - ioOoUUVA* )+ ^ t 7 ( t - - t / ) (AU ' ) y A*)a t , = o The L a p l a c e t r a n s f o r m o f t h i s e q u a t i o n g i v e s \ ( A ( t ) , A ) e - — r - — — ^ . where Jf(iw) i s t h e L a p l a c e t r a n s f o r m o f J ( t ) : r ( iu3) = U « f at e V(t )=(A / )G-Jate S - * o J o Wo J 0 S i n c e ( A , B * ) * = (B,A*), t o e F o u r i e r t r a n s f o r m o f ( A ( t ) , A * ) i s g i v e n by r ^ * Ci.i.iT) 5 \\A^ A^ ) has t h e p h y s i c a l s i g n i f i c a n c e o f s p e c t r u m . The l o g a r i t h m i c r a t e o f d i s s i p a t i o n i s t h u s g i v e n by 2Re [_ & (ico)J . 40 Although the Generalized Langevin equation can be formally derived from the Heisenberg's equation of motion, there i s some difference between these two equations. While the Generalized Langevin equation can only be applied mean-i n g f u l l y to a many-body system i n thermal equilibrium, the Heisenberg eauation can formally handle any number of par-t i c l e s which are not neccessarily i n thermal equilibrium. The properties of thermal equilibrium and many-body system are introduced through the def ini t ions of the scalar product (1.1.10) and the projection operator (1.1.11). When we are dealing with a system consisting of , say, one p a r t i c l e , the d e f i n i t i o n of the scalar product, and therefore that of the projection operator, become meaningless. The evolution of the random term f ( t ) i s rather pecul iar . It i s defined by the equation of motion with l(o) = k = . ( I - P ) A The underlying reason for th is peculiar evolution i s that f ( t ) i s not the total force but only the random part of the force . Defining K(t) by K(t) = e ' K 41 •^(tt a ( K t t ) , « * ) U . ^ r 1 ^ ( t ) and )f(t) have somewhat d i f f e r e n t p h y s i c a l connotations. While Kt) includes only' the r e l a x a t i o n ^ "^of the f a s t random process, the J^(t) includes both the random process and the slow process of A. However, i n case T c R e ^ ( i t o ) J « 1,where T i s the r e l a x a t i o n time of the f a s t process and i s the r e l a x a t i o n time of the slow process ( A ( t ) , A * ) , the f a s t and slow processes are well separated. In such a case, the r e l a x a t i o n time of A may he considered to be given bv the r e a l part of Jo where T' i s a time i n t e r v a l s a t i s f y i n g the i n e q u a l i t i e s -1 When f ( t ) i s neglected, (1.1.5) w i l l describe a smoothed-out, secular motion;- The random term f ( t ) describes the a c t u a l deviations from such a secular motion due to f l u c t u a t i o n s . Suppose we are only i n t e r e s t e d i n the most probable value of A ( t ) subject to the c o n d i t i o n that the average values of the extensive constants of motion and the v a r i a b l e s A have prescribed values at the i n i t i a l time. The i n i t i a l ensemble which w i l l give ^ ^ t h i s most probable value i s the e q u i l i b r i u m ensemble subject to the corresponding 4 2 r e s t r a i n t s and i s determined by minimizing the Gibbs H-fun c t i o n . The r e s u l t i s f =. Z *ex|>[-(?>(Td6-A*B)1 where (ouantal) I J exp^-p(X6-A*B")} dt»d<J ( c l a s s i c a l ) where A denotes the deviations from the i n v a r i a n t part and B i s the cononical conjugate. In the l i n e a r approximations, f0 {V e * * A * e ^ B ] The value of the random force along the most probable path i s thus given h y ^ ^ •Un - [f dU <^ A*,Ci&)>B' = O c t ) , A*)-B> In the Stochastic Theory, t h i s average value i s required to be zero. When the projector P i s defined by (1.1.11), i t can be shown k 'that ( f ( t ) , A ) = 0. Therefore, by d e f i n i n g a s c a l a r product according to (1.1.10), we ensure f ( t ) = 0. In other words the s c a l a r product has been defined i n such a way that, i f one neglects f ( t ) , then (1.1.5) describes the most probable path of A(t) i n the l i n e a r approximation. 43 1,2 C o r r e l a t i o n and Generalized S u s c e p t i b i l i t y . ( 5 ) R. Kubo v 'has introduced a very convenient formal-ism connecting time c o r r e l a t i o n function and generalized sus-c e p t i b i l i t y a B A ( c o ) , In the case ^ " ^ a ^ C w ) * °> n i s formal-ism, however, needs to be s l i g h t l y modified, V/e keep to l i n e a r approximation. An operator f o r a p h y s i c a l quantity £ i n the presence of an external force F ( t ) can then be, i n general, w r i t t e n i n the f o r m ^ where B and \ are operators which do not involve P(t) e x p l i -c i t l y . Suppose the perturbation energy can be written as 9Al(t) = - A F ( t ) . The l i n e a r response i s observed through the change AB(t) which can be expressed a s ^ ^ A B l t ) - ' 4> ( t - f ) P(t') dt ' (1.2.2-) j fin ^ For example: the current density i n an i n f i n i t e medium i n the presence of an external f i e l d i s where f = ^ [ ^ V ^ - C V H ^ ) ^ ] i s the current density i n the absence of external f i e l d , , In t h i s case, V 44 where the a f t e r - e f f e c t f u n c t i o n 0 B A ( t - t . ' ) d i f f e r s from Kubo's (1.2.3) ^•' B A(t) i s K U D O ' S a f t e r - e f f e c t f u n c t i o n ^ ^ d e f i n e d by s < L A , B i t ) ] ) 1A d^ i. J (classical) - l [ A , B ( t ) ] _ a - i [ A B l t ) - B ( t ) A ] ( q u a n t a l ) Without l o s s of g e n e r a l i t y , we put <A) = 0 = (B)„ The g e n e r a l i z e d s u s c e p t i b i l i t y a B A ( t o ) i s g i v e n by CO where ODACCO) i s d e f i n e d by CO ^ - ^ \ Wo Jo S A ( t ) e dt S i n c e 0 ^ A ( t ) p o s s e s s e s no d e l t a f u n c t i o n a t t=0 'BA1 and i s almost p i e c e w i s e c o n t i n u o u s ^ ° ^ f o r t> 0 , l i m a>>A(co)=0. 45 In such c a s e s ^ \ In the c l a s s i c a l l i m i t w / % ^ *-jl<*B(t>> where As <AB(t)> i s expected to be a continuous function, ^g^Ct) does not possess any delta function charac ter is t ic . We s h a l l keep to the c l a s s i c a l l i m i t . (1.2.4) can be written i n the form 00 where 7J(t) i s the unit step function {0 . f < 0 For example: for systems with loca l ized electromagnetic response, _^ where -Q(to) i s the generalized s u s c e p t i b i l i t y . ,3 I ivy, Q0») J I £ _ = ^ w»c* ' (See $4.2) 46 iojt P u t t i n g i n an i n t e g r a l f o r m , Y|(t) = - i [{rn J ^ ^ and c h a n g i n g o r d e r of i n t e g r a t i o n , we have OO to' where S^^(to) i s the cross-spectrum of A and B defined by (6 ) The second e q u a l i t y i n ( 1 . 2 . 5 ) follows from the r e l a t i o n v J This i s the general, hut rather involved, r e l a t i o n s h i p bet-ween s u s c e p t i b i l i t y and the cross-spectrum of two quantities In case S-g^(to) i s r e a l , we h a v e ^ ^ " ^ ) $B|H = - k 8T 2 0.2.?/) We have used the f a c t that Tm(a^to)j i s an odd function of frequency and therefore Im Jjo^to)| vanishes at to = When A = B, Sg^(to) i s always r e a l and we have the f a m i l i a r f l u c t -u a t i o n - d i s s i p a t i o n theorem. (1.2.Y) i s i n general much simpler than ( 1 . 2 . 5 ) as i t does not involve the p r i n c i p a l value of an i n t e g r a l of the cross-spectrum. Ih general, how-12 ever, S B A(to) i s not r e a l and ( 1 . 2 . 7 ) i s not true. Never-th e l e s s , some simple and general r e l a t i o n s h i p can be esta-b l i s h e d between generalized s u s c e p t i b i l i t y and covariance. 1 2 For example: i n the case • SBA^ W^ i s P u r e imaginary. 47 From (1.2.5), p u t t i n g to = 0, we obtain o o _ CO This follows from the f a c t that Sg A(co) has no pole at co = 0. Therefore we.obtain Cov C A,B) = i B T [ ° < B A ( 0 ) - ^ & A ( o o ) ] (1.2.8) In the p a r t i c u l a r case A = B, Cov(A,B) becomes the variance of A < 2 2 The general form of f l u c t u a t i o n - d i s s i p a t i o n theorem SA A(W) = - Coru^I^{SAft(co^ / V -where oc-n^  denotes the generalized s u s c e p t i b i l i t y v a l i d f o r a l l frequencies. Remembering that Var(A) = 2 j SA^(co)dco, we can obtain an expression f o r Var(A) by simply performing the i n t e g r a t i o n , i f we can derive an expression f o r the generalized s u s c e p t i b i l i t y v a l i d f o r a l l frequencies. However, such d e r i v a t i o n i s u s u a l l y not po s s i b l e . Some approximations have then to be introduced. V/hen Xn ^< I^ T where i s re l a x a t i o n time of A, we can consider the quantity A as c l a s s i c a l . In such a case, we expect the spectrum S^(co) vhth co y k-gT does not contribute s i g n i f i c a n t l y to the i n t e g r a l . 48 The, variance of A can then be written as \!xr A = 2 S A ACu)")du) When — <-&eT , where s i s a constant of order 10. Only those frequencies smaller than -| contribute s i g n i f i c a n t l y . For the frequencies smaller than ~-, S^Cco) can u s u a l l y A. approximated by a simpler form where ocAA(co) i s the generalized s i m p l i f i e d s u s c e p t i b i l i t y which i s only v a l i d for"frequencies l e s s than ~-; i . e . A °t*hW - ^ A f t ^ , ' " 4-rv CO < JE or Usually, we have s* frfl J ^ i f ^ i d * » f O to IT dlu) 3- to Then, we can write ( 1 . 2 , 8 a ) a s Using Kramer-Kronig r e l a t i o n , we have 49 which i s a p a r t i c u l a r case of (1.2.8). The l a s t step can be considered as a mathematical t r i c k to evaluate (1.2.8a). To a r r i v e at (1.2.8), we have i m p l i c i t l y assumed lim ^AB(t)) =<A>(B> . As pointed out by K u b o ^ , t h i s needs t->o° not be the case unless the degrees of freedom associated with the observed q u a n t i t i e s A and B are much smaller than those of the t o t a l system. In chapter IV, where Eq. (1.2.8) i s going to be applied, we can assume that a heat r e s e r v o i r . has also been included i n the unperturbed system. In t h i s way, we ensure the v a l i d i t y of the r e l a t i o n l i m ^ AB(t)^ ={A)^B^. t-»*> In order to obtain (1.2.8), we only made use of the co n t i n u i t y of ^AB(t)} . However, we can mention i n passing that should ^AB(t)^ possess a delta f u n c t i o n at t = 0 ( i . e . <(AB(t)) = & (t.) + ^(t)r :w]aere g ( t ) - i s - a smooth.function of t.:), i t produces ho a d d i t i o n a l mathematical d i f f i c u l t y . (1.2.8) i s simply modified to °<to) - ^ C o o ) = ^ < A B ( 0 + ) > - < A > < B > In t h i s hypothetical case AgA(°°) i s equal to (ty.j> + <X>) instead of £>C> • " 50 1.5. Stepwise Change of F ( t ) In case F ( t ) changes discontinuously at t = 0, such that Rt) = \ (1.5-0 1 o .. , ir <o the response AB (t) i s given by The subscript a of B (t) reminds us of the abrupt change of F ( t ) at t = 0. From Eq. (1.3.1), A 3 a l t / / AF = <*8ft(t) (1.3.3) where Therefore, o i u ^ ) = U» fo~ABa(t)' < T U i ° + S ) t d t / A F (1-3.4) W<> In the l i m i t co approaches zero * b ^ ) = Km f o°°AB a(t)' e ~ S t d t / A F = ^ ' ^ ^ A ( t ; / A F In the l i m i t co approaches i n f i n i t y JU*» <*6A(u>) = ( AB a(t)'dt/AF = i i w AB a(t)/AF J o t->0+ ( 1 . 3 . G) 51 The f i r s t e q u a l i t y f o l l o w s f r o m t h e fact^°^that u * » f d t A B a W e =0 The s e c o n d e q u a l i t y f o l l o w s f r o m A B _ ( t ^ O ) = 0 . The s o - c a l l e d r e l a x a t i o n f u n c t i o n ^ , ( t ) , d e f i n e d i n . R e f . (5) as c» St' o« o- -St' i s c o n n e c t e d t o AB ( t ) by S i n c e * - 5 ) $ (t) = - ( < A ( - i A ) B <t )> ax , BA We have P I B A " > =• L < A C - i M B ( t ) > d X . We assume h e r e t h a t Li*n <A5(t)> = <A><8> I n t h e c l a s s i c a l l i m i t $ B A l t > = . • [ A B A ( « ) - 4 B f t ( f ) ] / A F ( 1 . 3 . 7 ) | B A l t ) = £ < A B t t )> (1.3.5) S i n c e (AB(t))> i s a c o n t i n u o u s f u n c t i o n of t , 0B.4^'O p o s s e s s e s no d e l t a f u n c t i o n a t t = 0 . 52 s I I > t P i g . 1.1. Response t o stepwise c h a n g e K e x t e r n a l " f o r c e " , P ( t ) . 53 U s i n g ( 1 . 3.7) a n d ( 1 . 3 . 8 ) , we h a v e ^ 5 ^ -<AB(t)> - <A><B> = i B T [ A B 4 ( ^ ) - AB a ( t ) ] /AF (1.3.9) T h e r e f o r e , Cov/(A ;B) = Vr CA B A ( ° ° ) -AB * ( c O j/dF The second, e q u a l i t y f o l l o w s from ( 1 . 3 . 5 ) and ( 1 . 3 . 6 ) . . . 5 4 II LONGITUDINAL ULTRASONIC ABSORPTION RATE IN PURE, WSAK-COUPLING SUPERCONDUCTORS I I . 1 Introduction The generalized form of Langevin Equation of motion ( 1 . 1 . 5 ) i s a very general formalism dealing with st o c h a s t i c processes. I t i s of i n t e r e s t to apply t h i s general s t o c h a s t i c approach to some problems i n Supercon-d u c t i v i t y . We are going to apply i t to evaluate the long-i t u d i n a l u l t r a s o n i c attenuation rate i n Superconductors by aid of t h i s method. The r e l a x a t i o n time-of phonon i n normal metal i s of an order 10 ^  sec. The r e l a x a t i o n time of phonon i n the superconducting phase ^ ^ ^ ^ i s greater than or of the same order as that i n the normal phase. It s h a l l be shown that the r e l a x a t i o n time ( t ) of the random "force" i s of the same order as the q u a s i - p a r t i c l e energy r e l a x a -t i o n time (T ). The order of the l a t t e r i s 10~^ s e c . ^ 1 ^ . Therefore the generalized Langevin approach i s an appro-p r i a t e ^ ^one f o r the present problem. The problem of l o n g i t u d i n a l u l t r a s o n i c absorption f o r pure Type I superconductor has been i n v e s t i g a t e d t h e o r e t i c a l l y by B C S ^ a n d I.A. P r i v o r o t s k i i ^ u s i n g the perturbation theory. ' . The generalized Langevin approach, however, requ.ires 55 the r e l a x a t i o n time of the random "force" to he f i n i t e . We are able to show that the i n c l u s i o n of the l i f e - t i m e e f f e c t , improves the agreement with experiments. V/hen A (order parameter) < T \ due to the l i f e -time e f f e c t , the superconducting.pairs begin to p a r t i c i p a t e i n the attenuation process even though co i s s t i l l smaller than 2 A. . With terms l i k e neglected, the general-i z e d Langevin approach gives the f o l l o w i n g p h y s i c a l p i c t u r e . The phonon i s placed i n a r e s e r v o i r of q u a s i - p a r t i c l e s or e x c i t a t i o n s from the BCS condensate. The absorption rate of sound represents the average i n t e r a c t i o n s of these e x c i t a t i o n s on the phonon. 5 6 .11.2 Longitudinal U l t r a s o n i c Attenuation Rate C540 V a l a t i n v 'and Bogolyubov^ 'have shown that s i n g l e p a r t i c l e e xcited states can be obtained from the ground state by operating i t with q u a s i - p a r t i c l e operators v/hich obey Fermi-Dirac commutation r e l a t i o n . This q u a s i - p a r t i c l e opera-tors are defined as (-2.2.' I) where -le k E k = 2m" " E F ( 2 . 2 - 4 - ) A. i s the order parameter. The s u i t a b i l i t y of these d e f i n i t i o n s i s c l e a r . since the operator defined i n t h i s way s a t i s f i e s : ( 1 ) the fermion commutation r u l e s + + • where § g c s = T[ V k Q g ^ ) i s the ground state wave 57 f u n c t i o n of BCS with no e x c i t a t i o n . The d e f i n i t i o n s (2.2.1) show that an excited state can be created e i t h e r by adding an e l e c t r o n (it/**) to the condensed state or by removing an electron^k^G') from the p a i r ( " V " 5 " ^ i n the condensed s t a t e . U,^  can be phys i c a l l y , i n t e r p r e t e d as the p r o b a b i l i t y that a p a i r state i s not occupied and Vk i s the p r o b a b i l i t y that a p a i r state i s occupied. • ^ i s the energy neccessary to excite a q u a s i - p a r t i c l e with momentum k from the BCS condensate The minimum e x c i t a t i o n energy (energy gap) i s therefore equal to |A| • In the absence of f i e l d s , A can be taken as r e a l ; i . e . [A1=A . ( 3 5 ) The i n t e r a c t i o n s 'between the electrons and phonons contribute the fo l l o w i n g i n the Hamiltonian: + -t H-j = V%Y> Z*+l<r Ci?<rb£ •+ C C ( 2 . 2 . 6 ) " where V i s the electron-phonon coupling constant, b-» and C£ are r e s p e c t i v e l y phonon and el e c t r o n operators. Since the sound attentuation process involves s c a t t e r i n g , c r e a t i o n and des t r u c t i o n of q u a s i - p a r t i c l e s , i t i s more convenient to work with q u a s i - p a r t i c l e operators. In terms of these operators, we have, s u b s t i t u t i n g (2.2.1) 58 i n t o ( 2 . 2 . 6 ) , where (2.2.?) The k i n e t i c t e r m i n t h e H a m i l t o n i a n f o r q u a s i - . p a r t i c l e s i and phonons a r e g i v e n by 2 ^^hf a n d ^ ^ ^ % where c5 i s t h e f r e q u e n c y o f a f r e e phonon w i t h wave v e c t o r q. I d e n t i f y i n g A ( t ) i n ( 1 . 1 . 5 ) w i t h t h e phonon c r e a t i o n o p e r a t o r h^* +, t h e g e n e r a l i z e d L a n g e v i n e q u a t i o n , f o r phonon i s i = iu>0 b j c t i - \l»(t-t<> b?V)dt' + Act) 59 where •mo =  (^ >H ] (t.a.iQ (bf,b|) and f ( t ) I s t h e random p a r t o f t h e i n t e r a c t i o n o f q u a s i -p a r t i c l e s on a phonon. b'v i s a phonon o p e r a t o r t h a t w o u l d c r e a t e a p h o n o n , w i t h momentum q% 4- * + The s c a l a r p r o d u c t (b^ a n d C"b^ i^q*) a r e d e ~ f i n e d as i n (1.1.10). The k e r n e l j f ( t ) i s r e l a t e d ^ t o t h e t i m e - c o r r e l a t i o n o f f ( t ) by t h e e q u a t i o n ? ( t ) = i i i ^ -W i t h t h e p r o j e c t i o n o p e r a t o r P d e f i n e d as i n (1.1.11), t h e p r o j e c t i o n o f a Quantity G ( t ) on t o t h e a x i s b-»+ i s p ( T ( t ) =, ( G r ^ ) , k | ) - U ^ b ^ r 1 ^ (2.2.13) The.random f o r c e f ( t ) i s d e f i n e d by ^ C t ) - e u - 1 M k ( o ) (2.Z.14-) 60 where K(o) = ( l - P ) -b| ( J 2 . 2 . 1 5 - ) L i s t h e L i o u v i l l e . o p e r a t o r d e f i n e d by (1.1.9a) From t h e arguments i n A p p e n d i x A, t h e u l t r a s o n i c a b s o r p t i o n r a t e i s g i v e n ' b y cxltO^) =- 2-Re f ( i ^ ) (2.2.16 V where r0^|) = L i m \ 0 e lr(t)dt (2.2.17) The e q u a t i o n o f m o t i o n f o r b->+ i s 4 j . (2.2.18) A c c o r d i n g t o (2.2.11) - Z ^ 4 [ c . *?> ^ / ^ r S t , b ? ) ] } 61 Using (2.2.13), ' where co i s sriven bv (2.2 .19) o In Ref. ( 8 ) , ( 9 ) , when the matrix elements are c a l c u l a t e d , f r e e q u a s i - p a r t i c l e and phonon approximation has been taken. Using the same kind of approximation, qu a n t i t i e s l i k e ( If^ ) i n (2.2 .19) can be neglected. Therefore, co = w and according to (1.1.16) 0 q £ V (Z.7..7.0) (2.2.20) shows that'the random force i s due to the s c a t t e r i n g of q u a s i - p a r t i c l e s . Therefore the approximation of neglecting terms l i k e ( f £ t , k | ) i s equivalent to considering a phonon (renormalized) i n a r e s e r v o i r of q u a s i - p a r t i c l e s and d i s s i p a -t i o n i s due to i n t e r a c t i o n between the phonon and the r e s e r v o i r . This p i c t u r e i s s i m i l a r to that proposed^ ' i n the problem of l a s e r noise. I f instead of assuming terms l i k e (^£+1^?+?^) to be zero, we can take the f i r s t order approximation of them, which 62 not zero. K w i l l then he a mixture of b ^ and fl£ and . Since these higher approximation w i l l introduce unnecessary complication, we w i l l l i m i t ourselves to putting terms l i k e (lf|?+|*^ ct> ^ to zero. Since a s i s of order 10^ s e c . ^ " ^ or smaller and the r e l a x a t i o n time of tne random forceCr,) i s of the order 10~9 sec. (see § 2 . 3 ) , o£ 4< 1. The f a s t process ( f ( t ) , f + ) and the slow process ( h ^ + , b^) are then well separated. As remarked i n p. 41, we can write ("^ . Y ( t ) = ( K t t ) , k + ) . ( b f > b ? ) M (2-2.21) From (2.2.20), i t follows that r + + (2.2.2$) I f we use the free q u a s i - p a r t i c l e approximation, the Hamiltonian )d" ^  f o r the q u a s i - p a r t i c l e s can be represented as 63 -v The equation of motion for frgv^Cff" i s J i t f W f c r ) fori t ( - 6 k ) TUT According (1.1.10), where f k i s the d i s t r i b u t i o n function of the quasi -part ic les * = —i " 4 1 The three other scalar products i n (2:23$)can be expressed i n a s imilar way: 64 -I (2.2.24-I)) 4- - 1 (j2.2..Ucl ") 65 To include damping, we f o l l o w , f o r the time being, the method used by Kawasaki and Mori and T a n i ^ 6 ' * by m u l t i p l y i n g bhe s c a l a r products (2.2.24a—d) by a pbenomenological damping f a c t o r exp (- tA c ) . In ^2.3, we s h a l l j u s t i f y t h i s f a c t o r and show that T i s of the order of the r e l a x a t i o n time t of the q u a s i - p a r t i c l e density. Thus, s u b s t i t u t i n g (2.2,24a—d) (with the damp-ing f a c t o r ) into (2.2.23), we have V ± f ^^X-A* i(€k-6k4t)t j f ^ - f ^ 1 D e f i n i t i o n s i n -.(2... 2.8.)_,_f or L m ^ and n ^ .have- been used. In the second summation over k, we transform k to -k-q and we obtain 0«),K+ ) = p-e-^ -IVJ* Z h ^ - ^ f ) «i(*'-Wt--t-V + i Q - I ^ ) > - f , - V r ^ ^ ^ ^ ^ e , ) t j l 66; According to (2.2.21), Using free phonon approximation, • the-JJamiltaa-. f o r phonon can he written as:, cy where &)^is the renormalized frequency, K ^ ) i s the Laplace transform of fr(t.)' : v(ccov=. \ ^ rctwt The equation of motion f o r the phonon i s or 67 . + A c c o r d i n g t o ( 1 . 1 . 1 0 ) , the s c a l a r product (b^ ,b-0 can be w r i t t e n as 4 r ^ « = C^O"' (2.2.^a) As discussed i n Appendix A, the attenuation rate f o r phonon i s -1 A . ! **** '"IT^T bt-t^+TC* (2.2.-2.6) D e f i n i t i o n s (2.2.8) f o r m k k and n f e k have been used. Since q i s small compared to k^, we need only to include normal —• — » _ > processes; i . e . k'-k = q. 68 69 The summation c a n he r e p l a c e d by i n t e g r a l sip~n y \ £± L > 1 (air) 1 . -» -* where x i s t h e a n g l e between q and k. Cos x = 0sa+4* -o r i e ' d V T h e r e f o r e , I—I d E \ d e ' (2.2.27) where m i s t h e e f f e c t i v e mass o f t h e q u a s i - p a r t i c l e . 7 0 .Replacing the -summation s i g n i n (2.2.26) by an i n t e g r a l sign according to (2.2.2 7), oi, • » ' - * ' ) - I + »6 6 6 ' "g'-e (e'-t-«j)a+r;a Tt -I ~l (2.2.28) q2/2m has been n e g l e c t e d i n the l i m i t s of i n t e g r a t i o n s i n c e k r , » q . In the normal phase, the a t t e n u a t i o n i s due t o s c a t t e r i n g of e l e c t r o n s from one s t a t e t o another by a phonon. The a t t e n u a t i o n r a t e i n the normal phase (A=o) i s ( 2 . 2 . 2 9) where £ = (1+ e P E )"' and i ' = («+c ^  . V/e have used the f a c t t h a t when A = 0 , €={E| and 6'»lf.|a.nd I £-»^ c. 71 F o r most experiments, kgT >>o)n. We can approximate The a t t e n u a t i o n r a t e i n normal phase t h e n becomes - I We can b r e a k up the i n t e g r a l i n the f o l l o w i n g f a s h i o n : 4-o f dE o E + E - *1M& 1 r ~ 1 E i s always n e g a t i v e i n the f i r s t i n t e g r a l , and, t h e r e f o r e , E i s p o s i t i v e i n the second and the t h i r d i n t e g r a l and 1 •+ — -2 1 IP I X F i n a l l y , we have dB' + 4E r 4* ' x L 2 L \ — — ^ £ — -72 The second i n t e g r a l i n (2.2.30) can then "be written as 2 W ( ^ v F r c ) • d E ( - | | - ) ( 2 . 2 . 3 0 O The f i r s t i n t e g r a l i n (2.2.30) can he written as If co ' i s much greater than ^ Q " 1 * the i n t e g r a l ( 2 . 2 . 3 0 b ) i s given by Otherv;ise i t can be approximated by where W i s of the order ^ . Since k^T i s much greater _ h c than r , i t involves very l i t t l e e rror i f we replace the lower l i m i t w i n the i n t e g r a l i n (2.2.30c) by zero. 73 Adding (2.2.30a) .and (2.2.30c), the attenuation rate i n < normal phase then becomes <Crt)lv= ^f<H 2 w W ) ( V U ) I f we assume^ 1 2 a^ V = gq^ where g i s a constant depending • Q. on l a t t i c e dynamics of the metal but independent of q, we have . When n v P T C » 1, R Taking co'a as .335x10 Hz as used i n the experiment of Morse and Bohm^12^, q i s of the order lO^cm" 1. v^q i s then of the order 1 0 1 0 sec."" 1 . We s h a l l show i n s e c t i o n 2.3 that T i s of the same order as the r e l a x a t i o n time of quasi-c (13) p a r t i c l e density. The l a t t e r has been shownv ^'to be of the order 10~^ secj* Therefore,OV^T i s of the order 10. (2.2.30d) i s then not a bad . approximation f o r o^Oo ). 7 4 T o e s t i m a t e ( 2 . 2 . 2 8 ) , w e c o n s i d e r t h e f o l l o w i n g c a s e s : ( 1 ) A o r co » T ~ 1 T a k e t h e c a s e w h e r e t h e e n e r g y g a p i s m u c h g r e a t e r t h a n T q ^ . W h e n t h e p h o n o n e n e r g y i s s m a l l e r t h a n 2 A , t h e . p r o b a b i l i t y f o r c r e a t i n g t w o q u a s i - p a r t i c l e s b y a p h o n o n i s e x t r e m e l y s m a l l . T h i s i s e q u i v a l e n t t o s a y i n g t h a t T c - t (€'-<- e%C0 t) 24-T c-* ^ ^ ^ ^ =° y f o r C0t<2A I n n e a r l y a l l t h e e x p e r i m e n t s , t h e f r e q u e n c y o f s o u n d u s e d i s m u c h s m a l l e r t h a n . - I n s u c h c a s e s , t h e u l t r a s o n i c a b s o r p t i o n f o r m o s t o f t h e t e m p e r a t u r e b e l o w T q i s d u e t o s c a t t e r i n g o f q u a s i - p a r t i c l e s f r o m o n e e x c i t e d s t a t e t o a n o t h e r e x c i t e d s t a t e . ( 2 . 2 . 2 8 ) c a n t h e n b e w r i t t e n a s I t i s n o t a b a d a p p r o x i m a t i o n t o r e p l a c e TT -CA^) . I f w e f o l l o w R . S c h r i e f f e r ^ ^ b y r e p l a c i n g 1 3 4 t h e l i m i t s o f i n t e g r a t i o n o f E ' ' b y ( - * > , » » ) » w e o b t a i n t h e e a r l i e r r e s u l t s . i n R e f . ( 8 ) a n d ( 9 ) ; I t i s r a t h e r d u b i o u s w h e t h e r s u c h r e p l a c e m e n t ; JSJ v a l i d u n l e s s q V j , » k - g T w h i c h ' r e q u i r e s f r e q u e n c y fa£ »~lcP.Kz. 74a F i g . 2 . 2 . a / a . as a f u n c t i o n of temperature . ( 2 A ( 0 ) / k R T =3 .52) . ^ 3 2*1, E x p e r i m e n t a l d a t a are taken a f t e r Ref. ( 2 1 ) . 75 where I ( A ) = ( l + e ^ A ) 1 and f»(x) = 5- f ( x ) . Remembering t h a t A i s a f u n c t i o n of temperature, cc^/o^ i n (2 .2.31a) i s a f u n c t i o n o f temperature, co f*(A) i s s m a l l compared to 2f(&).- (2.2.31a) i s represented i n F i g . ' 2 . 2 by the dash-dot l i n e ( • ) . We have ev a l u a t e d ( 2 .2.31) n u m e r i c a l l y without making t h e two assumptions mentioned i n the preceding paragraph. We s h a l l ' s h o w i n §11.3 t h a t T i s approximately G ( 1 3 ) e q u a l t o (2T ). Tewordt^ 'has shown t h a t the r e l a x a t i o n t i m e f o r t h e q u a s i - p a r t i c l e s t a t e with momentum equal to k^, due to phonon s c a t t e r i n g , i s g i v e n by .oo 3 \ I . * 3 1 T V "I J 0 r OQ ? 4- \ dx ( 2 . 2 . 5 1 b ) 1 _ J I I The normal s t a t e v a l u e of the r e l a x a t i o n time of energy due to phonon s c a t t e r i n g i s denoted by n ^ q U which i s g i v e n by ^ n^u 2 It* <JOpt w p l ^ e r e ^e 3 1 0' 1" 6 5 "khe plasma frequency which i s g i v e n by topi - ^ n e * Z / M 76 where e i s the charge on ar} electron, Ze i s the charge on an ion, n i s the electron density and M i s the mass of an ion. c If we assume that the energy r e l a x a t i o n i s pre-dominantly contributed by nhonon s c a t t e r i n g , t (and there -' qu fore T ) at a p a r t i c u l a r temperature can be evaluated numerically bT- a i d of (2.2.31 b). S u b s t i t u t i n g i n ( 2 .2.31) the value of T £ thus obtained, we can obtain numerically the value of ^£ at a p a r t i c u l a r temperature. . We denote the dimensionless quantity (at temperature T), by F(T). In F i g . 2 . 2 , . the s o l i d l i n e represents the case where P(T ) - . 0 0 4 while the dash l i n e represents the case c where P(T ) = . 0 0 2 . In both cases, qV-n,/A(0) i s taken to be . 0 2 . The numerical integrations have been c a r r i e d out. by computer basing on Gauss-Legendre quadrature schemes. Impurities in the metal can also contribute to the r e l a x a -t i o n of energy (see p. 3Z8 of Ref. (14-)). The damping due to impurities i s independent of. temperature. Numerical c a l c u l or-ations show that i s v i r t u a l l y represented again by the s o l i d l i n e i f , at T = T , the phonon c o n t r i b u t i o n to P(T ) c c i s four times as great as that of i m p u r i t i e s , with P(T ) taken as .004-. The s o l i d l i n e shows much better agreement with 77 e x p e r i m e n t a l data than the e a r l i e r r e s u l t ( 2 . 2 . 3 1 a ) . Numerical c a l c u l a t i o n s show t h a t the dash-dot l i n e ( •-) a l s o r e p r e s e n t s the case where ^ T J J = »02 and i — c (53) - 7 7 T T = 10 y . I t has been c o n j e c t u r e d w 'that the d i s c r e -pancy between (2.2.31a) and experimental data can be e x p l a i n -ed away by i n c l u d i n g l i f e - t i m e e f f e c t and. a n i s o t r o p y of the energy gap.. While we have not i n c l u d e d the e f f e c t of a n i s o -t r o p y , we show t h a t the l i f e - t i m e e f f e e t . i s ' q u i t e important. when co > 2 A . , the Cooper p a i r s a l s o p a r t i c i p a t e i n the processes of a t t e n u a t i o n s . R e p l a c i n g the L o r e n t z i a n f u n c t i o n s i n (2.2.28) by d e l t a f u n c t i o n s and l i m i t s of i n -t e g r a t i o n of S by ( - 0 0 , " 0 ) , we have 2. €6 (2.2.32) as a f u n c t i o n of frequency has been n u m e r i c a l l y computed by B o b e t i c ( 1 0 ^ a s i s i l l u s t r a t e d i n F i g . 4-. When = 2^, there i s a jump i n a b s o r p t i o n which i s gi v e n i n Ref. (9)'as 2 78 (2) X c ' >, COg *«A A ; The L o r e n t z i a n f a c t o r i n t h i s case cannot be r e p l a c e d . Even when w^<2(\ ,. the second term-in (2.2.8) i s no lo n g e r z e r o . This means t h a t the e f f e c t of. f l u c t u a t i o n becomes impor t a n t . The Cooper p a i r can break up to absorb a phonon even though to^ i s s m a l l e r than 2A . What i s more: a superconducting p a i r can be formed and s i m u l t a n e o u s l y _q -p -p-a phonon i s em i t t e d . I f T q i s taken as 10 s e c . , — - £_ -5 i s of order 10 , where T Q i s the t r a n s i t i o n temperature and T^ i s the temperature at which the e f f e c t of f l u c t u a t i o n s begins to be f e l t . To make the above e s t i m a t e , T i s taken ' c to be 1°K. and the r e l a t i o n s h i p ^ " ^ L = S.I & B T C ( 1-T/TC) 1/2 i s assumed. I I . 3 Damping of the Random Force Using the r e l a t i o n s h i p ^ ^ and 79 we have fill Assuming t h a t t h e r e l a x a t i o n t i m e o f q u a s i -p a r t i c l e e n e r g y s t a t e i s i n d e p e n d e n t o f f r e o u e n c y . a n d -t h a t .the r e l a x a t i o n time, o f s t a t e ^ d i f f e r s v e r y l i t t l e f r o m t h a t o f s t a t e £ k , we have T d e n o t e s r e l a x a t i o n t i m e o f s t a t e € b ( a t t e m p e r a t u r e T ) . S u b s t i t u t i n g t h e s e two e x p r e s s i o n s i n t o (2.3.1) and u s i n g t h e r e l a t i o n ( e ^ ' " * ^ - I ) -fCuJ,)[l - f Cola)] = -fCcO^-fW.) we have, a f t e r s i m p l i f i c a t i o n , O b t a i n i n g t h e o t h e r three- s c a l a r : p r o d u c t s i n a 80 s i m i l a r f a s h i o n and s u b s t i t u t i n g them, into (2.2.23), the u l t r a s o n i c a b s o r p t i o n r a t e i s then eoual to, using (2.2.8) f o r m k q and , ot(u,) = ariM ^ i ^ i . y 5£ h» This expression, though i n a s l i g h t l y d i f f e r e n t form, c a r r i e s the same meaning as expression (2.2.26). Comparing with (2.2.26), we note that T C ^ S 2 T Q U . 81 II.4- Sum Rule and I n t e r r e l a t i o n s To obtain the sum r u l e , i t i s more convenient to express K i n terms of el e c t r o n operators + , C^ .. Using • (2.2.1)\ K i n (2.2.20) can be written as where fa i s the s p a t i a l F o u r i e r transform of the density operator where T^r-t) and Y ( r t ) are s i n g l e p a r t i c l e c r e a t i o n and a n n i h i l a t i o n operators. = 2 | V ? l a ( f - ( t ) , f _ ? ) (2.4.2) The f i r s t e q u a l i t y follows from and the conservation of momentum which gives 82 The second e q u a l i t y i n (2.4.2) follows from t r a n s l a t i o n a l and r o t a t i o n a l invariance of the system.. The F o u r i e r transform of (K ( t ) , K + ) i s f a t ( W + ) e - i w t = 2 I V , I 2 [ > e" C M t ( f ?tt), r_|) — OO 2 J^dt e- iat <[fj(t),f.|]> £ Ob ' • To obtain the second e q u a l i t y , we have used^'the r e l a t i o n s h i p < C A , B l t > ] > = - p ( A , B ( t ) ) The s o - c a l l e d s t r u c t u r e f u n c t i o n ^ 1 ^ S(q,co) i s defined by S (> ) = - f <[ f(?t), f h > e ^ ? _ C < ° T d 3 f d t (5.4.4) —00 ' where [A,B]^5 AB - BA = i [A,BJ . S(q,to) i s a r e a l , odd func t i o n of co. From ( 2 . 2 . I f ) and (2.2.21), we obtain T ( ^ =(fcf,b}) H U [ " d t e- C l" w + s ) t ( K ( t ) , K+) d t N - t [ °° —C tco-v- S) tr where ^ ( t ) i s a u n i t step f u n c t i o n defined by F 1 -fa v t > o Its i n t e g r a l representation can be written as v\(t ^  = — i iCiAA 1 _ e 83 U s i n g t h i s i n t e g r a l r e p r e s e n t a t i o n of ^ ( t ) , The l o n g i t u d i n a l u l t r a s o n i c a b s o r p t i o n r a t e c a n t h e n be e x p r e s s e d as ^ H , » ) ^ f a [ ? w ) ] - * |y t i a S(t ,»») ( • H (i>l*t 7) The law o f c o n s e r v a t i o n o f m a t t e r r e q u i r e s t h a t " n + v . 9 ( ? , r t = = where ^ ( r t ) i s t h e f l u x d e n s i t y o p e r a t o r d e f i n e d by [ f C O , P (^ ] _ = : - ? , [ £ ( . ) , f ( z ) ] _ T h e r e f o r e U s i n g t h e commutation r e l a t i o n w h i c h g o v e r n s t h e f e r m i o n o p e r a t o r s ¥(rt) and Y + ( r t ) : + . + 84 we obtain M i < t f < r t ) , f L > = ^ V V ) t - > 0 v/here n i s average t o t a l e l e c t r o n density: n = <_o) . From the d e f i n i t i o n (2.4.4) of S(q,oo), 'we obtain •oo '-CO du) 2vr As S(q,to).=. -S(q,co), 50 w S ( ^ = i f ( « . * . « ) From (2.4.6) and (2.4.8), .we obtain the sum r u l e f o r the l o n g i t u d i n a l u l t r a s o n i c absorption r a t e Using the fr e e phonon approximation (2.2.25a), the r e l a t i o n 85 ( 2 . 4 . 9 ) becomes M c a ' c ^ t y o ) = n l V t ' ' " ' 1 i (2.4.9,) I f we assume that or = u a and v„ = gq where u i s thet q s q ° ' s v e l o c i t y of sound, the sum r u l e becomes f d(0 r\* , / , "-III M* )V1 Also, " ^ <M$,<0^ = 2 l V t ! 2 S ( t ^ ) (2.4.10) I t must be stressed here that co i n the i n t e g r a l i n (2.4.9a) and co^  are very d i f f e r e n t , co and q are two -independent q u a n t i t i e s , not r e l a t e d by d i s p e r s i o n r e l a t i o n -ship of any kind, co^  i s , however, a fu n c t i o n of q. In the j e l l i u m model to = u a. q s -Due to an external charge r (q ,co)e^"a)*, the i n t e r -a c t i o n between the e l e c t r o n system and the charge can be written i n the form i ' A-ire* / v ' t o t 86 A c c o r d i n g t o (1.2.2) the F o u r i e r t r a n s f o r m of the l i n e a r r esponse of the d e n s i t y of the system i s The l o n g i t u d i n a l d i e l e c t r i c c o n s t a n t ^ o f the system i s g i v e n by • • J P t - . i > v . « i ' > « " f a " « oo ^ 1 l i t -co' €^ * T h e r e f o r e , the r e l a t i o n between the s t r u c t u r e f u n c t i o n and the i m g i n a r y p a r t of the d i e l e c t r i c c o n s t a n t i s g i v e n by The r e l a t i o n between the l o n g i t u d i n a l u l t r a s o n i c a b s o r p t i o n r a t e and t h e l o n g i t u d i n a l d i e l e c t r i c c o n s t a n t i s 87 As an i l l u s t r a t i o n , we t a k e t h e c a s e o f p u r e w e a k - c o u p l i n g 1 ' s u p e r c o n d u c t o r and e x p r e s s t h e d e n s i t y o p e r a t o r i n • term's 'of. q u a s i - p a r t i c l e o p e r a t o r s . M a king t h e a p p r o x i m a t i o n s l i k e and ioo air 1 (to- 6 K ) S +(arJ-a where t h e r e l a x a t i o n t i m e T ^ i s t a k e n as constant^, i t i s easy t o show t h a t SUM — ^ • ^ -Comparing w i t h Eq. (2.3.2) c o n f i r m s t h e r e l a t i o n (2.4.10). I t i s a l s o s t r a i g h t f o r w a r d t o s h o w t h a t or 88 where C[ i s the r e a l part of the l o n g i t u d i n a l c o n d u c t i v i t y . Therefore, from (2.4.6), we have Using free phonon approximation (2.2.2&>), 2 GO, - 2 e.1 co: Since the random force given by (2.4.1) i s the same f o r superconductor as f o r normal metal, (2.4.6), (2.4.11) and (2.4.12) are true f o r superconductors as well as f o r normal metal. We therefore obtain ($,W) where the subfixes s.,n denotes.respectively superconducting and normal phase of the metal. 89 I I I . ' FLUCTUATIONS IN SUPERCONDUCTORS I I I . l Introduction . For a multiply-connected superconductor, there e x i s t s a non-fluctuating'quantity c a l l e d f l u x o i d . The f l u x o i d $| i s defined ^15^15 a^s where m and e are r e s p e c t i v e l y mass and charge of an ele c t r o n ; C i s a curve l y i n g i n s i d e the superconducting material /%/ enclosing a non-superconducting area; . -*• v i s the v e l o c i t y of a superconducting p a i r s —f (k + mv t , -kV); <3> I s ^ t t e magnetic f l u x enclosed by curve £; 5_ i s an in t e g e r . The subscript £ of i$j denotes how many "quanta" (^) the system contains.., ( 3 . 1 . 1 ) i s a consequence of the single-valuedness of the wave -> f u n c t i o n of the condensate, v i s i n general not the same s f o r a l l the p a i r s . At temperatures above zero, a superconductor con-s i s t s of both condensate.(superconducting pairs)and 90 excited states ( q u a s i - p a r t i c l e s ) . The q u a s i - p a r t i c l e s have properties s i m i l a r to electrons i n normal metals. They are scattered at random by-ithe l a t t i c e s and i m p u r i t i e s . The s c a t t e r i n g provides the source of momentum f l u c t u a t i o n . As a consequence, although the f l u x o i d i s conserved and quantized as long as the material remains superconducting, the magnetic f l u x and current do f l u c t u a t e . However the f l u c t u a t i o n s behaves i n such a way that the f l u x o i d i s kept constant In a superconducting r i n g or c y l i n d e r where a pe r s i s t e n t current e x i s t s i n the absence of an external magnetic f i e l d ( i . e . 1^0), d i s s i p a t i o n and r a d i a t i o n due to f l u c t u a t i o n of current do not lead to decay of p e r s i s t e n t current. The energy loss i s provided by the l a t t i c e and impurities which act- l i k e r e s e r v o i r of energy. When a superconducting current i s disturbed i t relaxes very r a p i d l y to i t s metastable value. This r e l a x a t i o n time i s very d i f f e r e n t . f r o m the decay time of the p e r s i s t e n t current. In the absence of an external f i e l d , the l a t t e r i s the average length of time needed f o r the system i n a metastable state (i %0) to r e l a x to i t s e q u i l i b r i u m .> state ( £ and the p e r s i s t e n t current i s equal to zero). In s e c t i o n 3 . 3 , we adopt the two - f l u i d model (38)-0 and Langevin equation f o r the q u a s i - p a r t i c l e v e l o c i t y , 91 various c o r r e l a t i o n s and variances are c a l c u l a t e d f o r a long, t h i n , h o l l o w superconductor. The d e n s i t i e s of the .'. . superconducting electrons and normal electrons are assumed to he constant. The t w o - f l u i d model i s only v a l i d f o r London superconductors where the electromagnetic response i s l o c a l i z e d . A superconductor i s a London superconductor w h e n^ 1 8\ (T) i s greater than the smaller of ^ 6 W J 0, where X (T) i s the penetration depth at temperature T, f Q i s the • s coherence length (at zero temperature) and 1 0is the el e c t r o n f r e e path. The reason underlying t h i s c r i t e r i o n i s that the current density can be, i n general,written where R = r - r and A ( r ) i s the vector p o t e n t i a l . <j(R,T) can be approximately represented by exp(-R./jk) over the e n t i r e range of R and T where Vp/n"A(o). Therefore when X (T), xvhich i s a measure of the s p a t i a l v a r i a t i o n of A ( r ) , i s greater than the smaller of £ 6 and fl© , we can . fr take A ( r ' ) out of the i n t e g r a l and the response becomes 92 l o c a l i z e d . For the type I superconductor, t h i s c o n d i t i o n i s s a t i s f i e d f o r temperature near c r i t i c a l temperature such that i . e , A(oy T c > T > 0 - ^ f - ) ^ where T Q i s the c r i t i c a l temperature and A . ( o ) i s the London penetration depth at zero temperature given by SrM - (4-1TTie2'T1^. , , ' / U u J - " L w c x J . For the type II superconductor, A.(T) i s always greater than However, f o r magnetic f i e l d H greater than H ^ vertex states are formed i n the superconductors. The metal i s normal at the centre ofeach vertex and superconducting i n the space surrounding i t . The order parameter or superconducting electron-:' density can' no longer be assumed homogeous. Therefore we s h a l l l i m i t ourselves f o r the s i t u a t i o n E« H c l Near the c r i t i c a l temperature, the equations f o r two - f l u i d model can be.dbtaine by g e n e r a l i z i n g Gorkov s method of d e r i v i n g Landau-Ginzburg equations. Such d e r i v a t i o n i s p ossible only when A. (T) i s ;v ... greater than the smaller of £ oand t0 . The microscopic d e r i v a t i o n for the t w o - f l u i d model f o r London superconductors at a l l temperatures has been 93 discussed i n d e t a i l i n Refs.(40) . In section 3.4-, the assumption that the densit ies of superconducting electrons and normal electrons are temporarily constant i s relaxed. ' Using the Landau-Ginzburg form of free energy, we obtain expressions for the variances of the magnetic f lux and the superconducting electron density and t h e i r c o r r e l a t i o n . 94-UL2 Two-fluid M o d e l^ 8>^)^0 ) ( 4 1 ) Within the two-fluid model, the equations of motion can be written as d ^ _ _ 6_ £ (3.2,-0 dt " m dvn , ? _ e g dt + Th ~ *\ where v g i s the veloci ty of the superconducting p a i r s , v n i s the d r i f t ve loc i ty of the quasi -par t ic les , T n i s the transport relaxation time i n normal metal, e i s the charge of an electron, m i s the electron mass and E i s the e lec t r i c f i e l d . (3.2.1) i s the London equation, which i s v a l i d for London superconductors as pointed out i n the \ 3.1 The t o t a l current density i s given D y (?8K39) (4-0) (4-1) J = e (-nsv£ + ^ v n ) where n„ i s the quasi-part ic le . density( 8)(59)(4-0) n I n_ i s the superconsucting electron density: 5 and n i s the t o t a l electron density which i s assumed to n_ = n — n s n be constant. 95 f^ i s the Fermi d i s t r i b u t i o n f u n c t i o n : i £ k i s the o u a s i - p a r t i c l e energy. As the temperature approaches the c r i t i c a l tem-p e r a t u r e , we c a n expand (3.2.4) i n a s c e n d i n g power o f (440 and the s u p e r c o n d u c t i n g e l e c t r o n d e n s i t y becomes v J To i n c l u d e f l u c t u a t i o n due t o random s c a t t e r i n g by i m p u r i t i e s , a random a c c e l e r a t i o n ^ n ( t ) i s added to (17) (22)(24) ( 3 # 2 . 2 ) which now becomes % + = - — E - f . f t j (3.2.1) remains as i t i s s i n c e t h e r e i s no damp-i n g f o r the superconducting, c u r r e n t . 96 III.3 Long, Thin, Hollow, Superconducting Cyli n d e r s : ' Langevin Approach . Consider a long, t h i n , superconducting c y l i n d e r of radius R, thickness d, length<£in the presence of a s t a t i c , c o a x i a l , external f i e l d H . We use the t w o - f l u i d model and o assume that the average v e l o c i t i e s of the superconducting and normal electrons are constant s p a t i a l l y . We also assume the e l e c t r o n d e n s i t i e s to be constant s p a t i a l l y and tempor-a r i l y . The transport equation of the s u p e r f l u i d component i s given by (3.2.1) dvs(t) _ e d$(t) d t 2TTR.MC d t (.5.3.1) (3.2.1) i s a r e s u l t of absence of damping f o r superconducting p a i r s . The Langevin equation of the normal component i s given by (3.2.6) : . d v>ft) X L(t)___e d§Ct) + IJ. ^ dt T « 2nR.^ c d t h In (3.3.1-2), we have assumed that only the ^-components 12a of the currents are non-zero. (3.3.2) can be w r i t t e n as where I"N = N N E V N 12a The axis of the c y l i n d e r i s the z-axis of the c y l i n d r i c a l co-ordinates ( r , 6 , z ) . 97 We n o t i c e t h a t t h e t e r m I = ^ i i H 0,3.3) has t h e d i m e n s i o n o f s e l f - i n d u e t a n c e . ^ ( t ) I s Wie random e.m.f. V..»^pH<W ( 5 . 5 , 4 ) . We assume t h a t t h e s p e c t r u m f o r V N i n t h e s u p e r c o n d u c t i n g p h a s e i i s t h e same as i n t h e n o r m a l p h a s e . F o r n o r m a l me'tal, we have dl(t) j ^ . VM where I i s t h e t o t a l c u r r e n t i n t h e ^ - d i r e c t i o n : where R n i s t h e r e s i s t a n c e o f t h e n o r m a l c y l i n d e r ; i . e . & a - ••^•1T&^ Her e , we have assumed t h a t a c o n s t a n t momentum r e l a x a t i o n t i m e T e x i s t s . A c c o r d i n g t o t h e se c o n d f l u c t u a t i o n -n d i s s i p a t i o n t h e o rem (1.1.14), s u c h an a s s u m p t i o n i s e q u i v a -l e n t t o assuming a d e l t a f u n c t i o n c o r r e l a t i o n f o r t h e random e l e c t r o m o t i v e f o r c e V . U s i n g (1.1.14) and t h e e q u i p a r t i t i o n p r i n c i p l e w h i c h g i v e s we have 98 or the spectrum where the spectrum of a quantity i s defined by 5A(co) = ( °° dx e " ^ < A(t)A(t-nO> The spectrum of f i s then given by (3.3.4) where N n = 2irR£dn i s the t o t a l number of q u a s i - p a r t i c l e s . (3.3.1) i s equivalent to conservation of f l u x o i d where f l u x o i d §j i s defined by (3.1.1). Since V& i s s p a t i a l l y constant i n the model, (3.1.1) becomes = M | C j n ^ + § (3.5.T) — Js + 1 where the t o t a l magnetic f l u x $ i s the sum of the f l u x due 2 to the e x t e r n a l l y applied f i e l d (<J ' =TTR H Q) and the magnetic f l u x induced by current. For a long, t h i n c y l i n d e r , we have. $ = LI + f 0 (5.3.?) where L i s the self-inductance of the c y l i n d e r 1 _ (5.5.^) and I i s the t o t a l current I = Is+ In. The penetration depth \ i s t r a d i t i o n a l l y defined as . »3 4-TT n ^ e * & 3.10a.). 99 We define a corresponding X as xn = [ - m - c - - i ' A (3.5.10b) We define another quantity X by * = [^v^Y2 (5.5.10c") where n = n s + n n i s the t o t a l density of el e c t r o n s . X i s the same as the London penetration depth at zero temperature. I t i s c l e a r from ( 3 . 3 . 1 0 ) that A r 2 = A;1 Using ( 3 . 3 . 2 ) , (3.3.7) and ( 3 . 3 . 8 ) , we obtain ( l + R d / 2 \ 2 ) ^ § ( t ) + ' ± (l+ R d / 2 A f ) ( $ ( t ) - < I » = NnWt) where " ( 3 . S . M ) <£> = fe-#5j/(l+|f) ( 5 . 3 . . * ) From ( 3 . 3 . 1 1 ) we see'that (§)is the quasi-equilibrium.value of the magnetic f l u x ; i . e . we would obtain (?)as the magnetic f l u x i f we neglect f l u c t u a t i o n . The r e l a x a t i o n time i s now given by t : <-S - T,—T " (5-5.13) Since A. w e have T > T , and the e q u a l i t y can only occur at temperature T = 0 . However, when the temperature i s near to 0°K, the arguments leading to ( 3 . 3 . 1 3 ) i s to be doubted. ( 3 . 3 . 2 ) i s b a s i c a l l y a d e s c r i p t i o n of many-body e f f e c t . But near 0°K, there are very few normal electrons and the v a l i d i t y of ( 3 . 3 . 2 ) , therefore, becomes questionable. 100 Above the c r i t i c a l temperature the r e l a x a t i o n time becomes This i s the r e l a x a t i o n time f o r th& magnetic f l u x i n a normal Rd metal c y l i n d e r . Usually — ^ i s much greater than one and x' 2kd n can then be written i n the f a m i l i a r form where R n i s re s i s t a n c e and L i s the self-inductance. Consider a d i r t y f i l m where T n i s , say, of the order -13 1 10 sec. ~- i s then greater than k-JT. However, i n most n li Rd r e a l i s t i c cases, — ~ i s much greater than one and we can f i n d a c e r t a i n range of temperature near T.Q such that , Rd 1-- + T n > O T , Rd " B 1 + *j 2 ; V s 2 i s v a l i d . For example, when l»kgTr v> ,•the temperature range where the c l a s s i c a l l i m i t (e.g. f o r the f l u x ) - i s v a l i d , i s given by J ^Wc > i ;- T / Tc 101 In order that the method of evaluating variances and covariances, discussed i n t h i s chapter, to be v a l i d , we must s a t i s f y the con d i t i o n s : (1) the relevant q u a n t i t i e s such as magnetic f l u x can be considered as c l a s s i c a l , (2) the electromagnetic response must be l o c a l i z e d , and (3) the superconducting e l e c t r o n density must be s p a t i a l l y constant. To s a t i s f y the f i r s t c o n d i t i o n , the temperature of a Type I superconductor must be near the c r i t i c a l temperature. The Type I I superconductors can s a t i s f y c o n d i t i o n (2) at a l l temperatures. However, i n order to be able to assume s p a t i a l constancy f o r n g , the thickness of the c y l i n d e r must be small-er than the coherence length. For temperature not near T., i t means that the thickness d must be smaller than £ 0which i s of the order v ^'10 ^cm. The r e l a x a t i o n time f o r the -13 normal electrons i s then of the order 10 sec. As discussed on p.100, with such a value of T , the c o n d i t i o n (1) ( i . e . •'• r k-r,T)can on.lv be met at temperatures near T . Thus our s B / '- c treatment i n t h i s chapter i s v a l i d only when the temperature i s near T c. (4-5") For pure, very t h i n f i l m v there are a few electrons whose v e l o c i t y i s just i n the plane of the f i l m and which thus s u f f e r no c o l l i s i o n s . For s i m p l i c i t y , we assume, f o r very t h i n f i l m , there e x i s t also volume impur i t i e s and thus a bulk mean free path L 0^^o (but" JU >dt )« Under such condition there i s no s i g n i f i c a n t c o n t r i b u t i o n - Jt \ from these "anomalous el e c t r o n s " . 102 The spectrum of the magnetic f l u x i s ^7) c(\+RA/aA*) « ) 2 4 - r f - a 2-c L^T £d/2A2 04f5a/2A 2) 2(cO 24-T s " Z ) (5.3.14) / 1 V/e have made the s i m p l i f y i n g assumption that the magnetic f i e l d i s s p a t i a l l y constant i n the hole of the superconduct-ing c y l i n d e r . For t h i s assumption to he v a l i d , the frequency to must be le s s that c/R. That i s , the l a r g e r the average radius R , the smaller i s the range of frequency i n which (3.3.14) i s v a l i d . This c r i t e r i a f o r s p a t i a l constancy of the magnetic f i e l d i s discussed i n some d e t a i l on p. 134". • The a u t o c o r r e l a t i o n of H i s given by 2TT i oo Putting t = 0, we obtain the variance of magnetic f l u x ( i+ Rd/aAj )( I + fcd/2A2) f - . - . 104 Since the magnetic f i e l d i n the hole has been assumed constant, (3.3.IS) i s only v a l i d when the radius Rj.is much smaller than C T . This c r i t e r i a i s s u f f i c i e n t because only the part of the spectrum with frequency l e s s than or of the same order as T ""^  contribute s i g n i f i c a n t l y to the f l u c t u a t i o n of $ . s As a matter of f a c t , the expression i n (3.3.15") i s only an approximation f o r Cov GSfc),!) • There should be a sum £ % i ^  added to the right-hand side of (3.3.15"), where T ^ ' S are r e l a x a t i o n times such that C T ^ < R and k-^T. In our model, i t i s assumed that the expression on the right-hand side of (3.3.16 ) i s much greater than ^*X{ .. • . When Rd/2A. >> 1 and X ' i s much greater than A., the variance of the magnetic f l u x can be writ t e n as V4*(S} = cLfc BT/( 1+W*A.?) ( 3 . 5.Ua) At T = T , the c y l i n d e r becomes normal and A. becomes i n f i n i t e c' s and A. = X . The variance of the magnetic f l u x i s then given by Var($) n = c U s T / C \+ 2 \ 7 f c d ) p When Rd/2A. i s much greater than one, we obtain the f a m i l i a r expression Wr(£)„ = c U B T The r a t i o of variance of magnetic f l u x i n superconducting phase and normal phase i s p l o t t e d i n F i g . 3.1 as a fu n c t i o n of, _ 2 temperature f o r d i f f e r e n t values of Z q = Rd/2A. . I t i s noted 105 that as the T decreases below T , the variance of magnetic f l u x drops r a p i d l y , e s p e c i a l l y when Z Q i s lar g e . .. The superconducting current i s (17) Some c o r r e l a t i o n s between d i f f e r e n t p h y s i c a l q u a n t i t i e s are given below Rd Rd 1 + 2Aa CovCls,*) = - 7»V ?\ . „ > ^ c { ^ ( 3 . 3 . ^ 3 ) (i-t-Rd/2Af )(\+Ra/aAa) The negative sign i n (3.3.21) and (3.3.23) show tha t , on the average, as I increases,.I" and? decrease. The p o s i t i v e signs i n (3.3.22) and (3.3.24-) show th a t , on the average, as I increases,§. and I increase. In f a c t , from the r e l a t i o n s (3.1.1) and (3.3.8): We expect negative signs i n (3.3.21) and (3.3.23) even before we go through a l l the mathematics. 2 When Rd/2\ i s much greater than one and A.„ i s much greater than A, we have 106 2.rrte.wc As mentioned b e f o r e , ^ . a — has the s i g n i f i c a n c e of s e l f - i n d u c t a n c e . ,We c a l l t h i s term L . We d e f i n e s i m i l a r l y L* - yr-:  " (5.5. 2 2 The r a t i o s , Rd/2\ and Rd/2\ , which o c c u r v e r y o f t e n above, have the p h y s i c a l s i g n i f i c a n c e of the r a t i o s between 107 P i g . 3.2. A c r o s s - s e c t i o n (P) of a s u p e r c o n d u c t i n g >thin c y l i n d e r o f a r b i t r a r y c r o s s - s e c t i o n a l shape. The p l a n e of P i s p e r p e n d i c u l a r to the a x i s . 108 s e l f - i n d u c t a n c e s j i . e . (.i.S.Zk) The r e s u l t s of t h i s section can e a s i l y extend to the case of a c y l i n d e r of a r b i t r a r y c r o s s - s e c t i o n a l shape and v a r i a b l e , but small wall thickness ( F i g . 3.2), i f we can s t i l l main-t a i n the assumptions that the e l e c t r o n d e n s i t i e s of the condensate and the q u a s i - p a r t i c l e are s p a t i a l l y constant. We need only to replace the expressions L and L i n (3.3.3) (3.3.25) by more general expressions C^ i s a curve l y i n g i n a superconducting c r o s s - s e c t i o n P perpendicular to the a x i s . The tangent of C would give the d i r e c t i o n of the current density. dr„ i s an i n f i n i t e s i m a l s e c t i o n of curve C^ between r (j and r„ + dr|( . <£ r^ i s the c r o s s - s e c t i o n a l area perpendicular to the flow of the current where oL i s the length of the c y l i n d e r . To be accurate,rj_ 109 should be given by the i n t e g r a l where Cp i s the curve that,, l y i n g i n the c r o s s - s e c t i o n P, i n t e r s e c t s C^ at r„ and i s perpendicular to the current density everywhere. The self-inductance is.no longer given by (3.3.9(). As we have not been able to' f i n d a general a n a l y t i c expres-sion f o r a cylinder, of i r r e g u l a r c r o s s - s e c t i o n , we s h a l l denote the self-inductance by L which i s to be eventually . Rd L Rd determined by experiment. I f we replace ~ by — , * 2 V L n 2 V by , and by L(i--+ —) i n (3.3.13) - (3.3.24-) with 2 ^ h L s s n L and.I .. given by (3.3.28)and(3.3.29), we obtain the spectrum, c o r r e l a t i o n functions, variances and covariances of d i f f e r e n t relevant p h y s i c a l q u a n t i t i e s i n a long t h i n c y l i n d e r with a r b i t r a r y c r o s s - s e c t i o n a l shape. 110 III.4 F l u c t u a t i o n s of Superconducting Electron D e n s i t y and Magnetic Flux i n Long Thin Hollow Superconducting Cylinders In s e c t i o n 3.2, we have assumed the e l e c t r o n d e n s i t i e s , both superconducting and normal, to be non-fluctuating. Here we r e l a x such assumption.- V/e again consider a long, t h i n superconducting c y l i n d e r i n the presence, of an external c o a x i a l s t a t i c f i e l d H Q but we use a d i f f e r e n t approach to. . . evaluate the variances. 13 In the Landau-Ginzburg temperature domain , the (28) (43) (44) Gibbs free energy of a system i s given by v y ' K J 0-,= . € r H - d ' l V l * + . ( d l 5 r ^ r ? ) ( 7 - € A ) ^ ( ? ) ^ ( H - H . ) V f where (3.4-. I ) r' = T r ( 3 ) E p / l 2 - F T 2 ( W 8 T c ) 2 = a e T is- the c r i t i c a l temperature and n is., e l e c t r o n ...density, The Landau-Ginzburg domain^ ^ i s very near c r i t i c a l temperature such that A. (T) i s l a r g e r than the smaller of and X 0 and the order parameter i s small so that we can. expand the f r e e energy i n Taylor's s e r i e s and stop-at the quadratic term. J> I l l G n i s the free energy f o r a f i e l d - f r e e normal metal, the second and t h i r d term due to existence of energy gap,-the fourth term gives the k i n e t i c energy of the superconductor, '.: and the f i f t h : term gives the magnetic energy, i n the space i n c l u d i n g the e n c i r c l e d non-superconducting area. 1^ 1 has the p h y s i c a l s i g n i f i c a n c e of Cooper p a i r density. I t i s r e l a t e d to. the superconducting e l e c t r o n density n b y It i s to be noted that a' and B' In the expression of the Gdbb's free energy (3.4.1) are defined d i f f e r e n t l y from those (a,B) defined i n references (28), (43), and (44), where The volume (ZllRiidl ) i s placed i n a' and B' f o r convenience. The f a c t o r s }£ i n a', and % i n p* are due to the choice of m"* instead of ra i n the renormalization of the order parameter A.(r) With such a choice, the k i n e t i c energy density can be written as 2. 2- » 112 To be c o n s i s t e n t , t h e d e f i n i t i o n s o f p e n e t r a t i o n d e p t h A, (T) a n d t h e c o h e r e n c e d e p t h £(T) a r e now g i v e n b y (3.3.10a) » A Jl 5 • . t. 1 4 . T [ e " 2 | i p l 2 4"Tf Q2 y\j and instead of (6-17) and (6-21) i n Ref. (18). With = *R.aHo and r e p l a c i n g l ^ h y ng/2" which i s the p a i r -density, (3.4.1.) can be written as * :L- (3.4.7) Using (3.3. J ) and (3.3.8) we can express V and Vn i n terms o f $ 2C S 2cLC 2Af/|^l ) where A. and A. are given by (3.3.10a) and (3.3.10b) <' The p r o b a b i l i t y that n g and $ have values between n^, n g+dn c and $ , $+d3> i s given by (5.4,5) I 113 F o r t e m p e r a t u r e n o t t o o n e a r T , we c a n expand t h e f r e e e n e r g y about t h e s t e a d y s t a t e v a l u e ^n g>, and up t o t h e q u a d r a t i c t e r m s . The G i b b s f r e e e n e r g y t h e n becomes ' $-<5> ~ C r t f ^ , * ) l C S - ^ > ) ( H , - < n F > ) 4 . i r ^ 1 C r ( n , , i ) l (.V-Terms h i g h e r t h a n t h e q u a d r a t i c have been d i s c a r d e d . From (3.3.7) (3.3.8), i t i s s t r a i g h t f o r w a r d t o d e t e r m i n e t h e d e r i v a t i v e s ..in G Q w i t h r e s p e c t t o n s and §5 and we have (3. l.ir) where ( n g ^ and a r e d e t e r m i n e d by t h e e q u a t i o n s 2 K\< = o - o i = <€> A l s o n i s assumed c o n s t a n t so t h a t n = n - n , 114 These c o n d i t i o n s g i v e s and - * •''+ £'<v> +'2r(?ia *i <^>2 = o (3.4.8) U s i n g t h e f a c t t h a t <r > = <if> =i Id n< e <v«> (3.4.8) can he w r i t t e n as (see A p p e n d i x B) where v - (T,<i>) _ (ns> 8 TT £  zd.' The e x i s t e n c e of a r e a l p o s i t i v e s o l u t i o n o f t h e c u b i c equat (3.4.9) i s r e q u i r e d f o r t h e c y l i n d e r t o be s u p e r c o n d u c t i n g ^ 115 s h a l l d i s c u s s (3.4 .9) i n d e t a i l i n Appendix B. Within the quadratic a p p r o x i m a t i o n , the c o r r e l a t i o n c o e f f i c i e n t i s g i v e n by J 3 A* i& (3.4,10) i s a n e g a t i v e q u a n t i t y , f i s p o s i t i v e . As S u b s t i t u t i n g e x p r e s s i o n s of \ r , and i ^ L at n Q = <n V M I i$ a an* s * and into (3.4.10), we ;have 1 + ^a/j2<A $ 2> l a. (3.4. I l ) f i s exactly zero when n = 2a'/3'« From (3.4.9), we observe that mQ = 2oc'/(3' i f ? 0 = § j . This i s equivalent to ^1 > = 0. Therefore, i n order that the f l u c t u a t i o n s of the superconducting p a i r density and of the magnetic f l u x • are uncorrelated, the steady state current has to be zero. This s i t u a t i o n can only be r e a l i z e d when the external f l u x € c i s equal to 5. which i s an i n t e g r a l number of quanta (hc/2e). i 116 L i m i t i n g ourselves here'only to the case where f i s smaller than one, the variance of the magnetic f l u x i s given by V«c($) = V*7 (3.4 .U) (3.4-.l<Z^ i s the same as (3.3.16.) where the f l u c t u a t i o n of superconducting p a i r density i s neglected. The variance o f superconducting e l e c t r o n density i s given by k8T ( i -5 =<§> 2 L a. p V 2 * ' ; J U 4 - . I 3 ) 117 When 1 = 0 , j.= 0, t h e n we have ' As t h e m a g n e t i c f i e l d has been assumed c o n s t a n t i n t h e d e r i v a t i o n s o f (3.4.11), (3.4.12) and ( 3.4 . 13 ) , t h e v a l i d i t y o f t h e s e e x p r e s s i o n s i s l i m i t e d t o c y l i n d e r o f av e r a g e r a d i u s R much l e s s t h a n C T where T i s t h e r e l a x a t i o n t i m e o f t h e s s m a g n e t i c f l u x i n t h e c y l i n d e r ( s ee p. 13$). V a r ( I ) , V a r ( I n ) , V a r ( l g ) , Cov(I,n g) and C o v ( l s , n s ) can he o b t a i n e d f r o m (3.4.12) and (3.4.13) by t h e r e l a t i o n s T - ^ ~ ^° I* = ( l + Kd/z\\ ) ( l - < i » K e e p i n g n c o n s t a n t , 0„(n_,5) - 0 ( n ) i s s k e t c h e d as a f u n c t i o n o f m a g n e t i c f l u x & i n -Fig. 3.3, where G (n_) d e n o t e s The c u r v e s i n F i g . 3.3 a r e p a r a b o l a s w i t h minimum a t ^ _ From (3.4.9), we c a n a l s o o b t a i n t h e s k e t c h e s of (<"*>/§>)-fs(<>\>] Jf 118 F i g . 3.3. Sketch of G- ( n - 1 )-G (n ) as a f u n c t i o n of s s s s magnetic f l u x J i . §>0is the magnetic f l u x due to an external f i e l d , n i s kept constant;. s 119 I (a) \20^d w Ao >,U P i g . 3.4-. Sketch of [^>/*>)-^l<T,i>)-]astfunction of<n^. A 0 & [Wf fane^ ' V 4 i (a) A 2 ^ R d • - o <b> X o > R d ( f i r s t order t r a n s i t i o n ) (second order t r a n s i t i o n ) F i g . 3.5. Sketch of [ € t - $ 6 ( as a function of (n s). A 6 = [_™c*(&7<P/re2^']^ ; 7\sc i s the c r i t i c a l value of<n s>at which £ns>jumps from n„„ to zero, sc 120 as a f u n c t i o n o f t h e s u p e r c o n d u c t i n g e l e c t r o n d e n s i t y (n \ i n F i g . 3.4- and l l ^ - ^ l as a f u n c t i o n o f (n g> i n F i g . 3.5. X i s t h e p e n e t r a t i o n d e p t h a t t e m p e r a t u r e T v/hen t h e a v e r a g e c u r r e n t ( l ) ( = ( l g ^ ) i s z e r o ( i . e . ? 0 = 3fy ) : > - T i ^ f ' 1 / z V/hen A. * Rd, t h e phase t r a n s i t i o n i s o f t h e f i r s t o r d e r , o 7 The c r i t i c a l s u p e r c o n d u c t i n g d e n s i t y i s and t h e c o r r e s p o n d i n g v a l u e f o r 1 i s 4 2 When X Rd, t h e phase t r a n s i t i o n i s o f t h e s e c o n d o r d e r , The c r i t i c a l s u p e r c o n d u c t i n g d e n s i t y i s e q u a l t o z e r o , a t w h i c h t h e v a l u e of |54-5„l i s 121 IV. FLUCTUATIONS IN SUPERCONDUCTOR' T I IV,1 Introduction In s e c t i o n 3.3, we evaluated the spectrum and variances of p h y s i c a l q u a n t i t i e s f o r a long, t h i n super-conducting c y l i n d e r . .In t h i s .section, we are going to consider f l u c t u a t i o n i n . c y l i n d e r s of a r b i t r a r y wall t h i c k -ness. The Langevin Equation approach i n se c t i o n 3.3 or the free energy approach of s e c t i o n 3.4 become inconvenient. In t h i s s e c t i o n , we are going to employ the f l u c t u a t i o n - d i s -(22) s i p a t i o n theorem v yand the r e l a t i o n between the generalized s u s c e p t i b i l i t y and covariance we have derived i n §1.2. One of the approximation we use to s i m p l i f y the mathematics i s to assume the electromagnetic response to be (21) l o c a l i z e d . This i s a reasonable approximation^ yas long as A.(T) i s greater than the smaller of ^ 0 and {^where ^ Q i s the coherence length at zero temperature and j j 0 i s the el e c t r o n free path. We have discussed t h i s c r i t e r i o n i n some d e t a i l i n §3.1. While Type I I superconductors always s a t i s f y t h i s c r i t e r i o n , Type I superconductors can also s a t i s f y i t as the temperature approaches the c r i t i c a l temperature. For Type II superconductors, however, we s h a l l keep to the s i t u a t i o n s where the magnetic f i e l d i s much smaller than the c r i t i c a l (M-G) magnetic f i e l d H , v 'to avoid inhomogeneity of the order C X parameter. To keep the order parameter s p a t i a l l y constant, J 1 2 2 t h e t h i c k n e s s o f c y l i n d e r w a l l must he s m a l l e r t h a n t h e c o h e r e n c e l e n g t h £(T). F o r solid;,."' c y l i n d e r , t h e i n n e r r a d i u s a must be s m a l l e r t h a n £(T). We have t o a b i d e by the. r e s t r i c t i o n s m e n t i o n e d i n t h e p r e c e d i n g p a r a g r a p h when we a r e i n t e r e s t e d i n t h e s p e c t r u m o f a q u a n t i t y , s a y , m a g n e t i c f l u x , f o r f r e q u e n c y l e s s t h a n k^T and c/a where a i s t h e i n n e r r a d i u s o f t h e c y l i n d e r and.c i s v e l o c i t y o f l i g h t . I n c a s e we a r e I n -t e r e s t e d i n . e v a l u a t i n g v a r i a n c e s and c o v a r i a n c e s o f a quan-t i t y , s a y , m a g n e t i c f l u x , we must impose more r e s t r i c t i o n s ; n a m e l y , U ) t h e q u a n t i t y can be t r e a t e d as c l a s s i c a l ( i i ) cit > a where T i s t h e r e l a x a t i o n t i m e o f t h i s q u a n t i t y . (22) I t i s a good a p p r o x i m a t i o n v / t o use c l a s s i c a l r e l a t i o n s f o r a q u a n t i t y s a t i s f y i n g k-^ T > i where T i s t h e t i m e f o r t h a t q u a n t i t y , s a y , m a g n e t i c f l u x , t o r e l a x t o i t s s t e a d y s t a t e v a l u e . F o r T = 3°K, t h i s c o n d i t i o n i s s a t i s f i e -12 by T>3x10 s e c . F o r v e r y p u r e m e t a l s , t h e momentum r e l a x a t i o n t i m e T i s o f o r d e r lO -"*"^ s e c . and t h e t i m e f o r t h e n m a g n e t i c f l u x , c u r r e n t s , e t c . t o . r e l a x t o t h e i r m e t a s t a b l e v a l u e s a r e u s u a l l y l o n g e r t h a n T . T h e r e f o r e , f o r most s i t u a t i o n s o f p r a c t i c a l i n t e r e s t , we can. use t h e c l a s s i c a l r e l a t i o n Eq. (1.2.8). Even f o r t h e c a s e s o f d i r t y o r v e r y t h i n c y l i n d e r s where T i s a few o r d e r s s m a l l e r t h a n 10 "^se n t h e c o n d i t i o n k-gT > i can s t i l l , be k e p t u n d e r t h e s i t u a t i o n s d i s c u s s e d i n §3.3. •as/ 123 IV.2 L o c a l i z e d Electromagnetic Response A. Two f l u i d model •Prom (3.2.1) with E = ~;r^Mt) , we ohtain where 'A(rt) i s the vector p o t e n t i a l , The l a s t term on the r i g h t i s due to f l u x o i d conservation and quantization. The conservation follows from i n t e g r a t i n g (3.3 . 1 ) . The quantization i s obtained by comparing with (3 . 1 * 1). C y l i n d r i c a l symmetry of j i s also assumed i n order to obtain the l a s t term on the r i g h t s i d e . When there i s no time-dependent f i e l d present, the steady-state value of current density i s given by ^ ) = 2ki![_ h)^) (4.2.* where A(r) i s the steady-state value of the vector p o t e n t i a l at the point r when there i s no time-dependent f i e l d present. (4- / L . l ) 124 Denoting the Fourier transforms of A( r t ) - A ( r ) J r e s p e c t i v e l y - b y ^-( .r ) and kj^) , we have ^ ( 4 - 2 3 ) From (3.2.2,), the Fouri e r transform of the current c a r r i e d by q u a s i - p a r t i c l e s (the normal current) i s J (?) =. - -.J*n*± _±ob_ £ { t ) (4,2 .4-) In the absence of a ' time-dependent f i e l d , the steady-state value of the normal current vanishes. Following the d e f i n i t i o n s (3.3.10) f o r ^ and / l ^ » we obtain _„ _> r» r - ; ; - y / = ' u - r ) and i s the complex c o n d u c t i v i t y defined 0(0.= ( ^ ^ E t o • I f we are i n t e r e s t e d i n evaluating the variances and , covariances, we use (1.2.8.). In t h i s case V/e only need to know the values of o r Q(°0 at. "(ii = 0 and w = ©6 , 125 From Eq. ( 4 . 2 . 6 ) .>CPV = ^ <$<©) = -L = A l L i ke i . j P e r t u r b a t i o n . method .. To e v a l u a t e Q(G) and 0(°°)j we need n o t use t h e t w o - f l u i d model a t a l l . We c a n a r r i v e a t e x p r e s s i o n s f o r Q(co) a t a) = 0 and co = °o d i r e c t l y f r o m p e r t u r b a t i o n t h e o r y . We a r e 5 as m e n t i o n e d i n t h e i n t r o d u c t i o n , l i m i t i n g o u r s e l v e s t o London s u p e r c o n d u c t o r s . From p e r t u r b a t i o n t h e o r y ' t h e c u r r e n t d e n s i t y i s g i v e n hy Where t h e e l e c t r o m a g n e t i c r e s p o n s e k e r n e l i s d e f i n e d by J 126 ^(t-t'j i s a u n i t step fu n c t i o n 1 , t > t ' ^Ct - t ' ) = f O t < t When we are only i n t e r e s t e d i n London l i m i t at to = 0, i t has been shown that^®^ ^ . \ . ; 5 x ?w C ~-where^n i s the superconducting e l e c t r o n density defined by (3.2.4). Comparing with (4.2.5), we see t h a t 4TT _ \ -2. c Q<°) = A, For the case co =co, we proceedC^ 8 H^9) a s f 0 n 0 W S : Since we are only i n t e r e s t e d i n London superconductors, we can write < L j > ? t > > J ^ ? t ) ] _ > = S ^ S ^ - f ' J K t - V ) U2.1i) The f a c t o r Su^ i s due to isotropy of the medium. ( 4 , 2 . 1 1 ) can be written as (4 .2 . d) 1 2 7 wrier e coated Due to c o n s e r v a t i o n o f charge - v - <[ j ( T t ) , 5 (? ' f ) ] ,> = e <[ ^ 1 , f^'t')l> ( 4 . 2 . 1 2 ) . i m p l i e s t h a t 0 0 'icD(t-t') e <r(o)} I T 0» A p p l y i n g the commutation r u l e , we o b t a i n T h e r e f o r e we o b t a i n the sum r u l e : -00 TT (T(ui) - ne (4.2.13) By a i d of (4.2.12), the e l e c t r o m a g n e t i c k e r n e l , ( 4 . 2 . 1 0 ) 128 can be w r i t t e n as (4,2 .14-) The s e c o n d e q u a l i t y f o l l o w s from the sum r u l e (4.2.13). S u b s t i t u t i n g (4.2.14) i n t o (4.2 .9) and i n t e g r a t i n g by p a r t s , we o b t a i n F o u r i e r t r a n s f o r m g i v e s where Q(co) i s gi v e n by <j(-) = ^ f a t e ^ ^ i o f ^ - e i o t « r t 3 > ««.2..<) 129 Using the i n t e g r a l r e p r e s e n t a t i o n of the step f u n c t i o n .00 ioit r ~ J 2TT U)-«S ( 4 . 2 . 1 6 ) becomes - 00 C TT Oi-ui c The complex c o n d u c t i v i t y OJI(u>) i s d e f i n e d by =. c Q(c*)) /i*co As Im^Q(o))]' i s an odd f u n c t i o n of co, i t i s equal to zero at co = <*> • Therefore Gf(co) goes t o zero more q u i c k l y than co - 1 as co goes t o i n f i n i t y . T h e r e f o r e C ^ 8 a ) ( ^ 9 ) Q«o) = l . ^ *efo«4 = i [^4^ <r(&) 1 3 0 o r Q(oo) = w h i c h i s ( 4 . 2 . 8 ) * I n t h e d e r i v a t i o n of l i m Q(to), we need n o t m a k e a n y d i s t i n c t i o n between superconductor or normal m e t a l . 1 3 1 .py.3 G e n e r a l i z e d S u s c e p t i b i l i t i e s f o r Superconducting  C y l i n d e r s C o n s i d e r a hollow c y l i n d e r w i t h w a l l t h i c k n e s s d, outer r a d i u s b and i n n e r r a d i u s a, and l e n g t h X . There i s no assumption about the dimension; of the t h i c k n e s s d except t h a t we can assume the magnitude of the. order parameter and hence the superconducting e l e c t r o n " d e n s i t y t o be s p a t i a l l y c o n s t a n t . . This assumption i s reasonable as long as £(T) i s g r e a t e r than the s m a l l e r of A(T) or the width d. ^ ( T ) ^ 1 8 ^ i s the coherence l e n g t h at temperature T. I t i s r e l a t e d to T -T the coherence l e n g t h at zero temperature by £ Q 12C—fjr—)] • C / - i o \ W i t h i n the coherence l e n g t h , vie can assume.the d e n s i t y ^ 'to be c o n s t a n t . The p e n e t r a t i o n depth at temperature T i s r e l a t e d to p e n e t r a t i o n depth at zero temperature \(0) by T —T —}£ \(T) = \(0)\?(-^—)] . For Type I s u p e r c o n d u c t o r s ^ 2 5 ^ , £ o > ^ - ( 0 ) . Therefore i t i s always t r u e t h a t f (T) > A ( T ) . For Type I I s u p e r c o n d u c t o r s ^ 2 - ^ , ^ ( T ) i s always g r e a t e r than ^(T). However, when the t h i c k n e s s d i s s m a l l e r than the co-herence l e n g t h £(T), we can assume the order parameter to be s p a t i a l l y c o n s t a n t . A homogeneous e x t e r n a l magnetic f i e l d i s p l a c e d p a r a l l e l to the a x i s of the c y l i n d e r . We assume t h a t the e x t e r n a l f i e l d c o n s i s t s of two p a r t s , a s t a t i c f i e l d H Q •". and a time dependent p a r t . H e '^ •(0* . With the a x i s of symmetry as the z - a x i s , we i n t r o d u c e the c y l i n d r i c a l c o o r d i n a t e ( r , 8 , z ) . We use the approximation t h a t the 132 c u r r e n t o n l y f l o w s i n t h e ^ - d i r e c t i o n so t h a t t he c u r r e n t , v e c t o r p o t e n t i a l and o t h e r r e l e v a n t p h y s i c a l q u a n t i t i e s depend o n l y on r . We c a n t h e n choose a gauge t o make t h e v e c t o r p o t e n t i a l A(r*t) have t h e f o r m ( 0 , A ( r t ) , 0 ) . We u s e t h e G a u s s i a n u n i t s * , Remembering t h a t A«(r) i s t h e o n l y n o n - v a n i s h i n g component o f A(rt) and u s i n g t h e e x p r e s s i o n .2..$") .» t h e M a x w e l l e q u a t i o n c a n be w r i t t e n a s T r ^ ' + T t o - ^ V T ) A J r V : = 0 - : . . .;. . r (4.5.1). The s o l u t i o n s o f t h i s e q u a t i o n a r e t h e m o d i f i e d B e s s e l f u n c t i o n s I^ (j*r) and K^jvc). We have and a r e t h e c o e f f i c i e n t s w h i c h c a n be d e t e r m i n e d f r o m t h e boundary c o n d i t i o n s . The boundary c o n d i t i o n s • (as) ( 1 ) L^ xAC -^t)]* -H© e + on t h e o u t e r s u r f a c e ' (2) ff^A ( r t ) ] =. J L - A ^ t ) on t h e i n n e r s u r f a c e (4.3.1) 133 The second b o u n d a r y c o n d i t i o n comes f r o m t h e a s s u m p t i o n t h a t t h e f i e l d a c r o s s t h e b o u n d a r y i s c o n t i n u o u s and t h e f i e l d i n t h e h o l e i s s p a t i a l l y c o n s t a n t . I y and K y obey t h e r e l a t i o n s •'••(HlVw*)] =>'-H W.v). U.'-KO w i t h K = 1, P= 1 i n (4.3.4), c o n d i t i o n (2) becomes C , I 0 ( ^ b ) - C 2 k o ( / ^ b ) - M I P =-0 (4.5.*0 The s e c o n d c o n d i t i o n g i v e s Q I ^ O . - C 2 M / i a ) = O ( 4 . 3 . 4 ) . S o l v i n g (4.3.5) and (4.3.6) f o r and C g , we have c z = i i ^ l q (4.*.s) z p>(co) where Ota) = I 0 ( > « b ) K 2 ( ^ a ) - I29ua)Kft(jub) ( 4 . 3 . < 3 ) S u b s t i t u t i n g (4.3.T) and (4.3 .8 ) i n t o (.4.3.2), we o b t a i n ~w(r) " T^ ool [^*)r,(^o 4.1^)^(^0] (4.5,10) From (4.3.10) g e n e r a l i z e d s u s c e p t i b i l i t i e s o f m a g n e t i c f l u x , e t c . c an be o b t a i n e d i n a s t r a i g h t f o r w a r d manner. ^ '• - F o r t h e empty space i n s i d e a h o l l o w . c y l i n d e r , t h e J 134-Maxwell equation c a t c a n be w r i t t e n as o r , i n c y l i n d r i c a l c o - o r d i n a t e s (r,0,z)j.as C»VU + i 4. / toz _ J_ ) A = 0 -» where A # t d e n o t e s t h e (3-component o f A ( r ) i n s i d e t h e h o l e , o co 17 ^co Knowing t h e v e c t o r p o t e n t i a l t o be f i n i t e a t r = 0, t h e s o l u t i o n i s , ( d e n o t i n g t h e B e s s e l f u n c t i o n o f o r d e r ^byj^), „ A J - > ' = C 0 T , ( f r ) . . : The m a g n e t i c i n d u c t i o n i n s i d e t h e h o l e i s I n t h e bo u n d a r y c o n d i t i o n ( 4 . 3 . 3 ) , we have made t h e s i m p l i f y -i n g c o n d i t i o n t h a t t h e m a g n e t i c i n d u c t i o n i s s p a t i a l l y c o n s t a n t i n s i d e t h e h o l e o f t h e c y l i n d e r . F o r t h i s a s s u m p t i o n t o be v a l i d , ^ a has t o be much s m a l l e r t h a n 1. V/hen t h e r e l a x a t i o n ' t i m e o f a q u a n t i t y , s a y , m a g n e t i c f l u x > i s t , o n l y t h e p a r t o f t h e s p e c t r u m w i t h f r e q u e n c y o f t h e o r d e r l7*' c o n t r i b u t e s i g n i -f i c a n t l y t o f l u c t u a t i o n . I t i s r e a s o n a b l e t o assume s p a t i a l c o n s t a n c y f o r t h e m a g n e t i c i n d u c t i o n i n s i d e t h e h o l e i f — i s C T m u c h ' s m a l l e r t h a n 1 ( i . e . i f t h e i n n e r r a d i u s a i s much s m a l l e r t h a n c r . ) 135 IV.4 Steady States Putting H = 0, the steady state s o l u t i o n A(r) can now be wri t t e n as \Z (29c) The s o l u t i o n of t h i s equation i s then A ( r ) ' = _ l L _ + C.LCrAs) + C 2 k ,Cr/A0' 0 .4 - . I ) 2>irr 'Obtaining C-^  and Cg from the boundary conditions (4.3.3), we f i n a l l y o b t a i n v ' ' (4.4.2 where Ds = K 2 U / M l 0 ( b / A $ ) - L>(a/A<) K 0 (b/Aj ) 0-4.3) The current c a r r i e d by the q u a s i - p a r t i c l e s i s zero: J_ = 0 v v v The t o t a l f l u x * - 2 9 0 ^ i s It = 2H«b) = ( l - | ^ _ ) 5, + ^ [ K i ( ^ I , ( i ) - l , ( f ^ , ( ^ ] H „ ^ (4.4.4) To obtain the second e q u a l i t y i n (4.4.4), the I'/ronskian e has been used. 136 . (29c) The f l u x i n the hole of the c y l i n d e r i s V l i , = 2Tva^A C O • = - | L [ K 6(a/ A s)I e (VA0-rc(a/ A slKc(b/AO] + ^ H o When a>?Aj and eU<A s We recognise that t h i s i s equal to the quasi-equilihrium value of the magnetic f l u x f o r a long , t h i n c y l i n d e r , ( 3 . 3 . 1 ^ . The t o t a l current I i s given by 1 r i = * i . W>* Using the r e l a t i o n s f o r modified Bessel functions, i t = w and . = - k , ( x ) we nave 1 ~ HIT \ V f c  J ° -nft'Ds J It i s noted that $t and I are r e l a t e d by 137 .The c u r r e n t I i s c a r r i e d by s u p e r c o n d u c t i n g p a i r s . . The m a g n e t i s a t i o n i s g i v e n ^ ^ ^ y v _ 1U f M - — lr2 J ( r ) d r azD< (.4.4.7) 138 IV.5 F l u c t u a t i o n of Magnetisation The perturbation term i n the Hamiltonian due to the external f i e l d can. be written as l i f t ) = - f h a i o ' H C O d V =-MH(t) 0.5.1) where m(r) i s the magnetic moment of the c y l i n d e r at r and M i s . t h e t o t a l magnetisation of the system. In the presence • of the external magnetic f i e l d H(t) = H e 1 , a magnetic moment M ( ( ) e i c D ' b i s established• where 'M i s the F o u r i e r transform of - M ] :.. Since i L x ' K . W ] = - x \ u ) The generalized s u s c e p t i b i l i t y a M(co) of the magnetic moment 13.9 i s d e f i n e d by Hence, W = ~£x-> V / * * (*.«). The s p e c t r u m o f M i s g i v e n by f l u c t u a t i o n - d i s s i p a t i o n t heorem U.2 .7 ) .-. To evaluate., t h e v a r i a n c e o f m a g n e t i s a t i o n , we have o n l y t o c a l c u l a t e a M ( 0 ) and a M(°°). A t co = 0 , = , ^ M l 0 ) = I ^ ^ k a ( < » / ^ - I t ( * / A « ) K 1 ( b / A , i ] where D i s d e f i n e d by (4.4.3) s A t a) = rt , ^ =L \ - ' , where • ' D x = s k 2 U / M I 0 ( b / M ~ r 2 ( a M ) K o ( t / A ) ( 4 * . 4a) 14-0 We notice that a^(<*>) has the same value whether we are de a l -ing with superconducting or normal c y l i n d e r s . The reason f o r t h i s i s that f o r .co»A , the energy gap has l i t t l e e f f e c t i n i n f l u e n c i n g electromagnetic absorption. Therefore, super-conductors behave l i k e normal conductors at high frequency. When i t i s also true that coT n»l, a normal conductor behaves l i k e free e l e c t r o n gas. The.same remarks apply to the generalized s u s c e p t i b i l i t y of other q u a n t i t i e s we consider l a t e r . For superconducting c y l i n d e r s , the variance of the magnetic moment i s then given by (1.2.8) with A = M = B, -In the boundary c o n d i t i o n (4.3.3), the s i m p l i f y i n g assumption that the magnetic f i e l d i s s p a t i a l l y constant i n the hole of the c y l i n d e r has been made. To maintain the v a l i d i t y of t h i s assumption, the inner radius a must be smaller than C T ^ where i s the r e l a x a t i o n time of magneti-sati o n M. v - . . F o r normal metal, A s becomes i n f i n i t e and we have VartM)„ =ihiLr r (b/A) Ma/A) - I2U/A} K2(WA)1 V T 4 u \ L (4-^,4) J 142 Var(M)/Var(M) n as a function of temperature.is '"; plotted' i n F i g . 4.1 f o r a/b = 0.5 and b A = 100. These r a t i o s are chosen just to be s p e c i f i c . In f a c t , numerical c a l c u l a t i o n s have shown that the behaviour of Var(M)/Var(M) n i s well represented f o r values of a/b ranging from 0.1 to O.T. V/hen a/b = 0.8, the behaviour of Var(M)/Var(M) deviates s l i g h t l y above the curve i n F i g . 4.1 when the temperature i s jus t below T . In case that the c y l i n d e r i s not h o l l o w ( J l ) ^ w e . obtain, by pu t t i n g the inner radius a = 0 The variance of the magnetic moment of the superconducting (31) s o l i d c y l i n d e r i s given by v y Var(M) = [ I a (b/A )/ lo(b/M - I a CI>/As)/lo(bA $ )] We are again going to t r e a t some l i m i t i n g cases. When a » \jx\ ' , we have, f o r superconducting hollow c y l i n d e r , V/hen \ j * \ d « l , (4.5.8) becomes o(^<H») = - i-v i /^Ra ] M (4.*.<H where ^--(a^) ss. a z> b ( s i n c e a » [^\" l»d ) For frequency l e s s than the smaller of k-o'T and c/R, JD " the spectrum has the form: r i t u ) = £ L M _^ ' (4.t.|0) 143 where The variance of magnetisation of the t h i n c y l i n d e r i s given by According to (4.2.6) .^•v-.14? A* i ui + _t i s a monotonic fun c t i o n of to, i n c r e a s i n g from A to A"""*" s as to increases from zero to i n f i n i t y . As pointed out on ;p.l'00, the temperature of the superconductor must be near T i n order to s a t i s f y the c o n d i t i o n s : (1) homogeneous order c parameter, (2) l o c a l i z e d electromagnetic response and ( 3 ) , " the relevant-.quantities ;•>::•<? c l a s s i c a l . Near T , A g » A.^ I n tW« case 'When the thickness d i s much l e s s than A.,^  \jU>\d « I i s automatically s a t i s f i e d since A^" 1. When the thickness d i s l a r g e r or equal to A, Rd/2A i s much greater than one since., to a r r i v e at ( 4 . 5 . 8 ) , we have to assume <* » 2 —1 For Rd/2A greater or of the same order as one, r i s , s s according to (4.5.10a), of the order of ("Xf^w ) • The most important contributions to the i n t e g r a l i n (4.5.10b) comes from the spectrum with frequency l e s s or of the same order -1 as T . For these frequencies, 144 Since, near T , A. i s much, greater than A., we have ' . . C o Therefore, f o r a l l the frequencies-that contribute s i g n i f i -c a n t l y to f l u c t u a t i o n s , lyul >?>.can".be\-Ta:;k&n-i.as? of the same . order as A. . Thus f o r our purposes of estimating f l u c t u a t i o n s , s the c o n d i t i o n jjUl <A.<< t . can be replaced by a weaker one: d < Ks 2 1 5 When the temperature i s very near T , Rd/2A. can be much . / c s smaller than one. In t h i s case, or, Since R i s much greater than d, [^ 1 *is greater than d f o r a l l those frequencies ( £ — ) that contribute s i g n i f i c a n t l y to - .. T s . Rd f l u c t u a t i o n -when T i s near enough to T Q such that ^\ 2 1* s (£<£_>>!). Therefore, i n order to obtain (4-.5.10c), the 2\d c o n d i t i o n , A. > d, i s quite s u f f i c i e n t . 1 5 R > A. and Rd/2\ c 2 «l imply that \ » d. 145 The F o u r i e r component o f t h e t o t a l r e s p o n s e c u r r e n t [ l ( r t ) - I ( r ) j t o H i s g i v e n by l a ) ~ 4-IT. U.4r. I I ) 4-TT I D(00) 4 T 2. Using (1.2.8), with A=M and B=I, we have f o r t h e r m a l f l u c t u a t i o n s F o r t h i n c y l i n d e r s , s u c h t h a t ,we have C o v ( M , l ) = 1TR." F o r v e r y t h i n c y l i n d e r s s u c h t h a t &»X$ and el«AJ, we have 146 IV.6 Relation between Magnetisation and Magnetic Flux From the d e f i n i t i o n (4.5.2) and the Maxwell equation We have M s f b co 7~ 4 r 2 (v* B W ) A d r Since i n the c y l i n d r i c a l co-ordinate ( r , 0 , z ) , = (0,0,B ( r ) ) , we have The second e q u a l i t y follows from p a r t i a l i n t e g r a t i o n and the r e l a t i o n Using the boundary conditions (4.3.5), the f i r s t term i n (4.6.1) becomes The second term i n (4.6.1) i s S u b s t i t u t i n g these expressions back to (4.6.1) and s i m p l i f y i n g , we obtain 147 'where i s the F o u r i e r t r a n s f o r m of the t o t a l magnetic f l u x response [ ^ ( t ) - . i s the magnetic f l u x i n the. h o l e p l u s the magnetic f l u x i n the m e t a l . T h i s i s j u s t a nother form of the well-known r e l a t i o n v/here B i s the magnetic f l u x d e n s i t y , H i s the magnetic f i e l d and m i s the m a g n e t i s a t i o n per u n i t volume. 148 IV.7 -Fluctuation of Magnetic Flux Consider an external f i e l d produced by a long solenoid c a r r y i n g a current I ( t ) . A long, hollow super-conducting c y l i n d e r i s placed i n s i d e the solenoid. The axis of the c y l i n d e r i s p a r a l l e l to the f i e l d H(t) induced by the current l ( t ) . From (4.5.1) and (4.6.2), we obtain the pertur-bation term i n the Hamiltonian where 3>(t) H ( t ) . The t o t a l magnetic f l u x i s the sum of the magnetic f l u x i n the hole plus the magnetic f l u x i n the metal of the c y l i n d e r . Denoting the Fourier transform of : the t o t a l magnetic f l u x response £? tCrt) —3>(r)J by §Et(£, we have • fctoa * 2.1Tb A^Cb) (4>7i*<0 The generalized s u s c e p t i b i l i t y i s then given by I d e n t i f y i n g A and B i n (1.2.8) as A = § = B, we have 149 ( 4 . 7 - 4 ) In the boundary condition (4.3.3), we have made the s i m p l i f y -ing assumption that the magnetic f i e l d in the hole of the cy l i n d e r i s s p a t i a l l y constant. Therefore, f o r (4.7.4) to he v a l i d , the inner radius a should he much smaller than C T , where xx i s the r e l a x a t i o n time of the t o t a l magnetic * t * t f l u x «f , . When T ^ T , the c y l i n d e r becomes normal. Therefore A =oo. V/e have then, from Eq. (4.7.4) We can see that the variance changes continuously when the metal passes from the superconducting to the normal phase. For a s o l i d superconducting c y l i n d e r , we put a = 0 i n (4."f';4) and the variance of the magnetic f l u x i n the c y l i n d e r becomes *sI,(i>/AO _ A I, U A ) I^M I 6G>M) (4.7.4- a.) The v a r i a t i o n of Var($^) with temperature i s pl o t t e d i n F i g . 4 .2 according to (4.7.4a),' with = 10 cm., b = 0.235 cm., and A = 3.55 x 10 cm. With these dimensions, 150 151 b1 i s much, greater than A. and consequently the magnetic noise i s the same f o r a l l the s o l i d metal rods i n normal phase. J Var ($^.) f o r normal metal rods i s represented i n Fie:. 4 . 2 by a dash-line when the temnerature i s smaller than 3 . " [ 2 °K . and by a s o l i d - l i n e when the temperature i s greater than 3 . J 2 0K. The s o l i d - l i n e represents the t h e o r e t i c a l value of the magnetic noise i n the t i n rod which undergoes phase t r a n s i t i o n at 3.72°K. Near the c r i t i c a l temperature, X/\ i s approximated by [2(1-T/T . According to (4.7.4a), the thermal f l u c t u a t i o n s of magnetic f l u x does not disappear x^hen the rod becomes superconducting. However, with the given dimensions, Var($^.) drops nearly two orders of magnitude when (l-T/T ) i s only 10"^. L. Vant-Hull et a l . ^ 2 ^ c have measured the thermal magnetic noise of metal rods With dimensions given above. While there i s no c o n t r a d i c t i o n between our theory and the experimental, data, more accurate measurements are needed to decide whether or not our theory i s adequate to describe q u a n t i t a t i v e l y the thermal magnetic noise i n c v l i n d e r s . There are some other l i m i t i n g cases which i s of i n t e r e s t . For a hollow c y l i n d e r , s a t i s f y i n g a»A. g, we would have hjtla >> 1. Then we have 152 where d(=b-a) i s the wall thickness. < i i ^ ( d A H -ir-Ur+4rO c o s U d M V 1 * Near the c r i t i c a l temperature, X beco^ec: much greater than X, Let us consider the case Xs » d » \ The expression f o r the variance of magnetic f l u x i n ( 4 . 7 . 8 ) can then be wri t t e n as Vflr(?l = C V L / ( » 4* W / 2 A J ) (4,7. ) where R i s the average r a d i u s : Since R and d are bot(a much greater than \, Rd/2A i s much greater than one. We have therefore confirmed (3.3.16 a i n Chapter III. For very t h i n c y l i n d e r such that A » d , ( 4 . 7 . 8 ) can be.-written as VarCl) - c t 6 T L . M / 2 A ' fZ,-7m\ ( l ^ R d / 2 / t f ) ( ( 4 - R . a / 2 A * ) L T } which confirms (3.3.16). 153 The low f r e q u e n c y s p e c t r u m o f t h e m a g n e t i c f l u x i s s;iven by (1.2.7) and ( 4 . 7 . 3 ) S ? t(.) = - i ^ - ± _ [ ^ V / b ) + tyoKOW . (4.7.11 ) where The a u t o c o r r e l a t i o n f u n c t i o n i s g i v e n by C o v ( %t).*M) = j ^ e ' ^ j ^ -oo To go f u r t h e r , we have t o c o n s i d e r some l i m i t i n g c a s e s . I n t h e case o f t h i n - w a l l c y l i n d e r , l^uld << I a n d l y U l a > > 1 , we have s5 w = - 2 C ^ T L R D / 2 A " ( C O V T - ) - ' 4T\* where L i s t h e s e l f - i n d u c t a n c e L — , w h i c h c o n f i r m s ( 3 . 3 . 1 4 ) . The a u t o c o r r e l a t i o n f u n c t i o n i s t h e n g i v e n by C o v ( § t < t ) , $ t C o ) ) =- c fcBT L e l t l / r i where 1+ U/zf ^ ^ — ^ T* 154 S $(o) £ = 1 0 0 A - . 8 -.6 -.4 -.2 0 .2 ,4 . 8 P i g . 4.3. .6 r-- ..... • • • . . U>Vn S $t(co)/ S $ t ( 0 ) - a s . aUfiaric-tion of frequency i n a super-conducting, s o l i d c y l i n d e r . S § (to) i s the spectrum of of t o t a l magnetic f l u x ^ i n the c y l i n d e r . 155 10 -1 10 -2 ^ 10 J to \< o ^—'*j to 10 -4 - £ ^ 1 0 0 A. 1 0" 10" . 8 • 9 T/T C F i g . 4 . 4 , s I t C ° ) / n s $ (°-) a s . a f u n c t i o n of tenper^We i n a s u p e r c o n d u c t i n g , s o l i d c y l i n d e r . ". 156 In the case of s o l i d c y l i n d e r , the spectrum of magnetic f l u x s i m p l i f i e s to o f . oo The peak value of (co) at co>=0 i s equal' to At T = T , the peak values becomes The v a r i a t i o n of (ftO/f.S^(0) with t o T n i s p l o t t e d i n Pig.4.5, f o r b/A = 100, T./T c= 0.8, 0.9, and 0.98. The r e l a t i o n \ A S = [2(1-T/0? C)J ^  has been used. From the l i n e shape of the spectrum, we notice that the r e l a x a t i o n time r g of §^ i n a s o l i d superconducting c y l i n d e r increases with tempera-ture. V/hen T/T = .9,T_ i s about an order higher than T_. c * s n When T/T = .98, T_ i s of two orders higher. The v a r i a t i o n of ( O ) A S , (0) with T/T i s p l o t t e d i n F i g . 4.4 f o r -**t n 2^ c b A = 100. V/hen the thickness d i s much greater than A.s, (4.7.8) can he written as 15 7 The F o u r i e r component of the t o t a l response current [ l ( r t ) - I ( r ) ] i s g i v e n by - C 4-TT J a ~ - \ T h e r e f o r e , the covariance i s given by Eq. (1.2.8)' with A =|t and B = I G>v(i ,$ t i = — — — a 2 I D< For very t h i n c y l i n d e r s such that a»A. and d>>\, we have Rd /2Ai For t h i n c y l i n d e r such that a » X »d»A., (4 .7.13) can be w r i t t e n as ; (4.T. 14a) 158 T/T c F i g . 4 . 5 . The r a t i o of c o v a r i a n c e ^ o f H. and £ t i n t h e s u p e r -c o n d u c t i n g and the"'^rmal";ph"ss.e as a f u n c t i o n of temperature i n h o l l o w . c y l i n d e r s . .'. -159 Denote magnetic f i e l d i n the hollow of the c y l i n d e r by H^. For a < C T , , H. can be.considered s p a t i a l l y constant. The 4 t 1 • * • y j • Fourier transform of H^(t) - H'\ i s given by Therefore, using Eq. (1.2.8) with A = § t and B = B^, . ? i r f A s (4.1.10 This expression can be, i n f a c t , obtained from (5.7.13) hy aid of the r e l a t i o n When the c y l i n d e r i s i n the normal phase, the covariance/ becomes The v a r i a t i o n of Cov(H i,§ t)/Cov(H i,f 1_) n with temperature i s i l l u s t r a t e d i n F i g . 4.5 f o r cases where b/A = 100, a/b = 0 .1 , 0 .5 , and 0 . 9 . I t i s not due to any s p e c i a l s i g n i f i c a n c e that the p a r t i c u l a r r a t i o of b/A i s chosen. It i s chosen simply to be s p e c i f i c . For t h i n c y l i n d e r s , C o v ( $ i f $ t ) * V a r ( $ t ) f c V a r ( $ i ) where ^ = TTa H^. 160 . The Fourier transform of the current response carried r i by the Cooper pairs, |J[$(rt) "* -^rM -*-s S i v e n by 1 S M J .= -JLJL 4ir A (4.7.17) - \ I aa*D(">) Using (1.2.8) with A= J T and B=I_, a* D, A " D A . ) -X 3 A,* (4.7-1?) For thin cylinders such that a »A. >? d>>\, we have 1 + R d In this case, Rd/2A. is much greater than one, i t confirms (3.3.23a). When a » d and we have 2 A 2 ) w h i c h c o n f i r m s (3.3.23). The n e g a t i v e s i g n o f Cov(I ,$+.) i s e x p e c t e d f r o m t h e f l u x o i d c o n d i t i o n (3.1.1). S i n c e Cov(I ,1^) i s e q u a l t o t h e d i f f e r e n c e b e ^ tween Cov(l,$ t) and Cov(I we o b t a i n f r o m (4.7.13) and (4.7.18) and e x p r e s s i o n for t h e c o v a r i a n c e of t h e 161 n o r m a l component o f t h e c u r r e n t and t h e t o t a l m a g n e t i c f l u x i s From (4.7.14a> and (4.7 . 19 ) , we o b t a i n , f o r a » \ » d » \ , Cov U«,l?t) = c trBT w h i c h c o n f i r m s ( 3 . 3 . 2 2 a ) . 162 V RESPONSE TO STEPWISE CHANGE OF EXTERNAL FIELD V . l Introduction In s e c t i o n 1.3, we have shown that, i n case of an external f i e l d changing stepwise at time t = 0, we can ex-press the i n i t i a l and f i n a l response of the system i n terms of the generalized s u s c e p t i b i l i t y by a i d of (1.3.5) and (1.3.6). We have also shown i n se c t i o n 1.4 how these i n i t i a l and f i n a l responses of a quantity are connected with thermal equilibrium variance of that quantity. In the fol l o w i n g sec-t i o n s , we assume that there i s an external magnetic f i e l d along the axis of a long c y l i n d e r . This f i e l d changes abrupt-l y at t = 0, and, t h i s abrupt change can be i d e a l i z e d as a stepwise change at t = 0. This external f i e l d can be ex-pressed e x p l i c i t l y i n the form We are going to give some e x p l i c i t expressions f o r the i n i -t i a l and f i n a l responses f o r p h y s i c a l q u a n t i t i e s such as currents, magnetic f l u x , and magnetic moment. t >o (£.1.1) t < o .163 V .2 Responses of-Currents The F o u r i e r component of the t o t a l current response \ I ( t ) - l J i s , according to ( 4 . 5 . 1 2 ) , = _.cJL ^ 1 H where i s the F o u r i e r transform of H ( t ) . Corresponding to the stepwise change of H ( t ) , the i n i t i a l change of the current i s , according to ( 1 . 3 . 6 ) , Urn A I M = # ( ! £ ' • - ' ) A H * ( " • ^ w h e r e • i s defined hv ( 4 . 5 . 4 a ) . In the case of normal c y l i n d e r s , the i n i t i a l change of the current due to H(t) defined by ( 5 . 1 . 1 ) i s s t i l l given by ( 5 . 2 . 2 ) . There i s no d i f f e r e n c e i n the i n i t i a l change of current, whether the c y l i n d e r i s superconducting or normal. The same remark would apply to a l l other q u a n t i t i e s such as magnetic f l u x , magnetic moments, etc. The reason f o r such i d e n t i c a l behaviours i s that the impulse i s received, during an i n f i n i t e s m a l period of time that the s c a t t e r i n g cannot be brought into a c t i o n . Therefore the normal metal and superconductor behave i n i t i a l l y i n the same way: they both behave l i k e free e l e c t r o n gas. . 164 When t - » o o , 'the f i n a l change i s g i v e n by B e f o r e t = 0, t h e s y s t e m i s i n a s t e a d y s t a t e and 1 ( 0 0 i s g i v e n by (4.4.6) w i t h H = H ( t S " 0 ) = H . The c u r r e n t a t ^~J"° i s e q u a l t o I(oo) = [ K0(a/Aj).'.Io(b/A») - I 0 ( a / A ) M b / A . ) ] * ! where D i s d e f i n e d by (4.4.3). T h i s i s t h e s t e a d y s t a t e v a l u e o f c u r r e n t u n d e r a new e x t e r n a l s t a t i c f i e l d (H +AJI ). a a As t->«o , we know t h a t t h e s y s t e m r e t u r n s t o s t e a d y s t a t e . . F o r s o l i d s u p e r c o n d u c t i n g c y l i n d e r s , we p u t t h e i n n e r r a d i u s a t o z e r o . i n (5.2.2) and (5.2.3). The i n i t i a l r e s p o n s e o f t h e t o t a l c u r r e n t i s t h e n g i v e n The f i n a l change i s 165 F o r t h e c a s e o f h o l l o w c y l i n d e r s w i t h i n n e r r a d i u s much g r e a t e r t h a n \Q, (5.2.2) and (5.2.3) become and AI(co) = 4rT -1 A H a I f , i n a d d i t i o n , t h e t h i c k n e s s i s s m a l l e r t h a n A , we have A 1 ( 0 + ) = -L- I+2AVM. and AI(*>) = r~ where ^-TrR^H . I f , on t h e o t h e r hand, t h e w a l l t h i c k n e s s i s much g r e a t e r t h a n A 4 we have o s 1 A I(o, and A I M = 4TT 4-TT ^ a. 4 ) AH< 166 V/hen t h e c y l i n d e r s become n o r m a l , AI(«) i s a l ways z e r o s i n c e t h e r e i s no s t e a d y s t a t e c u r r e n t i n a s t a t i c f i e l d , The' i n i t i a l r e s p o n s e A 1(0+) i n a n o r m a l c y l i n d e r i s the. same as t h a t o f t h e s u p e r c o n d u c t i n g c a s e , as m e n t i o n e d b e f o r e . B e f o r e t h e s u p e r c o n d u c t i n g s y s t e m has r e a c h e d t h e s t e a d y s t a t e , p a r t o f t h e c u r r e n t i s c a r r i e d by t h e q u a s i -p a r t i c l e s . C a l l i n g t h i s - component I , 'we have — • 0^ ATT A. i ( A J . T _ > I b 4-TT A » {CO TTvT1 o-, c x - * _ i ^ _ _ l \ f f w (4,25) i co T " A*;** iio+V The i n i t i a l change of I i s , a c c o r d i n g t o (1.3.6), A * n The r e l a t i o n -Va —TT1- has been u s e d . The sec o n d e c u a l i t y o c c u r s b e c a u s e t h e q u a s i - p a r t i c l e and t h e s u p e r c o n d u c t i n g ; 167 p a i r r e s p o n d i d e n t i c a l l y a t t = 0 : v £ ( t ) = V;U) The f i n a l change i s , a c c o r d i n g t o (1.3.5), T h i s i s a g a i n e x p e c t e d ; as a t , t h e sys t e m has r e t u r n e d t o i t s s t e a d y s t a t e , and, i n a s t e a d y s t a t e , t h e c u r r e n t i s c a r r i e d by s u p e r c o n d u c t i n g e l e c t r o n a l o n e ( f l u c t u a t i o n , o f c o u r s e , i s n e g l e c t e d h e r e ) . The p a r t o f t h e c u r r e n t c a r r i e d by t h e s u p e r c o n -d u c t i n g e l e c t r o n s i s The i n i t i a l change of I i s s AljCOt) A I ( 0 + ) The f i n a l change o f I ' i s s Als(°°) = AU*) o r The above r e l a t i o n i n f o r m s us t h a t , n e g l e c t i n g f l u c t u a t i o n , t h e s t e a d y s t a t e c u r r e n t i s c a r r i e d c o m p l e t e l y by s u p e r -c o n d u c t i n g p a i r s . 168 V.5 Response o f M a g n e t i c F l u x . U s i n g ( 4 . 7 . 2 ) , t h e F o u r i e r t r a n s f o r m o f t h e t o t a l . m a g n e t i c f l u x r e s p o n s e i s The i n i t i a l change o f t h e t o t a l m a g n e t i c f l u x i s t h e n g i v e n by \ A § t ( 0 + ) = i l k * [ I|C-fe-) + k ((£)] A H * C « . 2 ) The f i n a l change o f t h e t o t a l m a g n e t i c f l u x i s O b t a i n i n g §t(o_) f r o m (4.4.4) T h i s i s t h e s t e a d y s t a t e v a l u e o f f l u x u n d e r t h e new e x t e r n a l m a g n e t i c f i e l d H a + A H a . When a»Aj , we have Aj.tO+l = 2irbA " . " a b — *H«. 169 For t h i n c y l i n d e r d ^ < A * l + Rd/2V V/hen d » A. s A§t(o+) = 2 TT bA AH a -fke ' . . . For a s o l i d c y l i n d e r , we put innerrradius to zero i n - ' ( 5 . 3 . 2 - 3 ) b A I t ( 0 + ) = 2.TTbA A I A L AU For normal cylinder,1A$(o+) i s the same as that of super-conducting c y l i n d e r s hut the f i n a l change i s A § t ( * o ) = i r b 2 A H o . As the normal c y l i n d e r i n a s t a t i c f i e l d can support no steady state current, the f i n a l change i n magnetic f l u x i s completely due to the change of the external f i e l d . 170 V.4 Response o f M a g n e t i c Moments : U s i n g (4.5.2a) t h e r e l a t i o n between t h e F o u r i e r component o f m a g n e t i c moment M^ and e x t e r n a l m a g n e t i c f i e l d -i s g i v e n by T h e r e f o r e t h e i n i t i a l change i s AM(o+) = - -f^I^b/A) k zU/A)- I i(aA)K i(b/A)]AH a and f i n a l change i s . AM(oo) = - - i ^ ' T I ^ b / A ^ a / M -yaM,) k2(fe/A$j| AH*. 4D S L J When a » A , we have A M , F o r t h i n c y l i n d e r s u c h t h a t d « A , we have A M ( o + ) = - ^ 1 — - — A H . 171; F o r s o l i d c y l i n d e r ^ A M ( o + ) = -~ L A H 4 Io(b/A) A \ ^ ^ ( b / A s V A „ F o r normal c y l i n d e r s , M(«*») = 0. 172 V.5 Responses of Thin C y l i n d e r To b r i n g out the p h y s i c a l p i c t u r e more clearly,; we consider the t h i n c y l i n d e r (a >> A. d<< \) i n greater d e t a i l . With f w = TTR* .',-we .obtain, from (5.3.1), (5.2 . 1 ) , '.(5.2.5), (5.2.6), f o r . t h i s p a r t i c u l a r case, the. f o l l o w i n g approximations i - L 1 The Fo u r i e r transform of [H(t) - H(t£0)] i s Su b s t i t u t i n g (5.5.5) int o (5.5.1), where ^ = ^ A H * The change of f l u x at time t i s 173 H(t)/AH a Rd/2A?=100 T / T C = .99'-.3334 P i g . 5.1 Responses i n a t h i n c y l i n d e r to a step-wise change of external f i e l d H ( t ) , as- a func t i o n of time t . 174 F o r t > 0 we have t o take the c o n t o u r i n the upper to-plane S i m i l a r l y , we o b t a i n from ( 5 . 5 . 2 ) - ( 5 . 5 . 4 ) Aftt) = ~ V 2 ^ A I s ( t ) = - — L — [ i - R d / ^ a e 41* (r.s.9] L l-HW/zA? ( 5 . 5 . 6 ) - ( 5 . 5 . 9 ) are p l o t t e d i n F i g . 5.1. From ( 5 . 5 . 6 ) and ( 5 . 5 . 9 ) . , we o b t a i n the r e l a t i o n .. -.• -.-. L AIs(0 4 A§Ct) = O which i s j u s t a statement of c o n s e r v a t i o n of f l u x o i d . 175' V.6 Energy and D i s s i p a t i o n Due to the stepwise change of external magnetic f i e l d , the system i s perturbed from the steady s t a t e . Because of t h i s , part of the current i s c a r r i e d by the q u a s i - p a r t i c l e s which i s scattered by the l a t t i c e and i m p u r i t i e s . D i s s i p a t i o n therefore occurs and the d i s s i p a t i o n i s eoual to •OO •> n T , _ .2. 'o+ Z 2c L (l+ ^ /2A|)(i+Ra/2A2) The k i n e t i c and magnetic energy of the system i s The f i r s t term i s the k i n e t i c energy due to current c a r r i e d by Cooper p a i r s . The second term i s due to current c a r r i e d by normal e l e c t r o n s . The t h i r d term denotes magnetic energy, Using the f l u x o i d conservation r e l a t i o n (3.1.1) and remembering that $ = L I * I ( t ) where 1 U( t ) = ±- ( i ^ ) ( i + [ l (t) - i i z l f c L l * 176 F o r t< 0 , <£(t) - : ICt<oV = - i § t - ^ , U(t<o) .= 2cL( 1+ 2A$/e<0 F o r t> 0 , we have 1>(t) = " § a + A 5 a . S i n c e we have t-»» 2cL(i-+£X*/fc<*) T h e r e f o r e , f o r t > 0 , •UU) = ^^U+W/2A.tU\+^/2A*)(AI(t)-AIH^E(^) 2c Rd The l o s s o f e n e r g y i s t h e n g i v e n by 2cL (_H ad/2At)(\-tRcH/2A.x) Jo 177; The l a s t e q u a l i t y f o l l o w s f r o m c o m p a r i n g • w i t h ( 5 . 6 . 1 ) . The t o t a l d e c r e a s e o f e n e r g y of. t h e system i s e q u a l t o t h e energy d i s s i p a t e d t h r o u g h t h e s c a t t e r i n g o f n o r m a l e l e c t r o n s . -We have d e r i v e d ( 5 . 6 . 3 ) f o r t h e p a r t i c u l a r c a s e o f t h i n c y l i n d e r s . However, a more g e n e r a l r e l a t i o n between d i s s i p a t i o n and t h e d e c r e a s e o f e n e r g y o f a s y s t e m can a l s o be d e r i v e d . From t h e e q u a t i o n s o f m o t i o n ( 3 . 2 . 2 ) and ( 3 . 2 . 1 ) i n t w o - f l u i d model -. • , ^ - ^ , f t e . = ^ 4 _ j . ( f r t We o b t a i n , a f t e r some s i m p l e m a n i p u l a t i o n s , We t h e n i n t e g r a t e ( 5 . 6 . 4 ) o v e r t h e V o l ^ where V o l ^ d e n o t e s volume ( V 0 I 2 ) o f t h e c y l i n d e r p l u s t h e h o l e . The d i s s i p a t i o n o f t h e s y s t e m i s g i v e n by Vol 2 The i n t e g r a t i o n i s o n l y o v e r . t h e volume o f t h e c y l i n d e r as 178 n n i s zero o u t s i d e the c y l i n d e r . From ( 5 . 6 . 4 ) , we o b t a i n Uo = [ t U 0 ^ -lU<->] - L° *J dV jtn) -ten) Vol, where the k i n e t i c energy of the system i s g i v e n by Vola fd'r(-y.£) r e p r e s e n t s c o v e r s i o n of m e c h a n i c a l or t h e r m a l energy V o U V (50) i n t o e l e c t r o m a g n e t i c e n e r g y ^ • . U s i n g the Maxwell, equation. -•» *"* 4-TV "* the v e c t o r i d e n t i t y V U * S ) = B-(WE) - E (VxH ) and Faraday's law we o b t a i n ^6 at Vol t V ( 179 We have used magnetic f l u x density B i n s t e a d of H i n (5.6.5). T h i s i s because J ( r t ) i s the t o t a l c u r r e n t at the p o i n t r . D i s p l a c e m e n t c u r r e n t ' i s a l s o n e g l e c t e d i n (5.6.5). 4- J - d*r B- dB i , . J J3(cV> ~ ~ If the medium i s l i n e a r i n e l e c t r i c and magnetic properties as i t i s i n the case f o r the superconductors we consider, we have For the case of t h i n c y l i n d e r , S and B are s p a t i a l l y constant, The l a s t i n t e g r a l then-vanishes and we.have-(5.6.3). 180 References. H. Mori, Prog. Theor. Phys. 33 (1965), 423. R. Zwanzig, Lectrues i n Uheorectical Physics, Ed. W.E. B r i t t i n , B. W. Downs and J. Downs (Interscience Publishers, Inc. New York), Vol I I I . Prigogine, Non-Equilibrium S t a t i s t i c a l Mechanics (Int-erscience Publishers,,Inc, N.Y., 1962). I. R. Senitzky, Phys. Rev. 119 (I960), 67/0; 124, (1961) 642. J. P. Gordon, L.R. Walker and W, H a L o u i s e l l , Phys. Rev. 130 (1963),806. W. E. Lamb,. J r . , Phys. Rev. 134 (1964) A1924. M. S c u l l y , W. E. Lamb, J r . , and M. J. Stephen, Fhysics o f Quantum E l e c t r o n i c s , P. L. K e l l y , B. Lax and P. E. Tannenwald,. Eds., New York, McGraw-Hill, (1966), pp . 759 . W. H. L o u s i e l l and L. R. Walker, Phys. Rev. 13 (1965) B204. M. Lax, Phys. Rev. 129 (1963) 2342; J. Phys. Chem. Sol i d s 25 (1964), 4S7; Phys. Rev. 145 (1966), 110.' 181 „ (3) S . Chandrasekhar, Rev. Mod. Phys. 15(194-3), 1. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys*.I7 (1945), 398. (4) R. Ktibo, Tokyo Summer Lectures i n Theor. Phys. Part I, Ed. R. Kubo (Syokabo and W. A. Benjamin 1965), pp. I-../:: , (5) R, Kubo, J. Phys. Soc. Japan 12 (1957) , 570. .-(6> R. Kubo, Reports of Prog. Phys. V o l . 29 (part 1) (1966) ,255. (6a) J . R. S c h r i e f f e r , Theory of Superconductivity, (W.A. • Benjamin 1964), £5-6. " ! " " ' I . L. P. Kadanoff and G. Baym, Quantum S t a t i s t i c a l - ~ ~ Mechanics (W.. A. Benjamin 1962) pp. 28. (7) M. C. Leung, ' Physica -: (1970) , 568, • (to be published). (8) J. Bardeen, L.N. Cooper and J. R. S c h r i e f f e r , Phys. Rev. 108 (1957) , 1 : L T 5 . (9) I. A. P r i v o r o t s k i i , . Soviet Phys.-JETP 16 (1963), 945. (1'0.) V.:.M. Bofeetic, Phys. Rev. 136, (1964) A1535. (10a) A. Pippard, P h i l , Mag. 46(1965), 388. . (11) V/. P. Mason and H. E. Bommel, J. Acoustic Soc 0 Am. 28 (19*Sf), 930. E. A. Pagan and M. P. Garfunkel, PhysV^Rev. L e t t . 18 (1967), 897-(12) R. W. Morse and H. V. Bohm, Phys. Rev. 108 (1957) , 1094. (3.2a) R. Rickayzen, Theory of Superconductivity, (Inter*-' sdience Publishers 1965)", ^  3.13. (13) L.~ Tewordt. Phys. Rev. 128 (1958), 12 . (14) A. A. Abrikosov, L. P. Gorkov and I. Ye D z y a l o s h i n s k i i , Quantum F i e l d T h e o rectical Methods i n S t a t i s t i c a l Mechanics (Pergamon Press 1965), pg . 150 . 182 (15) P. London, . Superfluids, V o l . 1, (Dover 1961), p. 151. (16) N. Byers and C. N. Yang, Phys. Rev. L e t t . 7, 46 (1961); L. Onsager, Phys. Rev. L e t t . 7, 50 (1961); W. Brenig, Phys. Rev. L e t t . 7, 337 (1967). (17) R. E. Burgess, Proc. Symposium on.Physics of Superconducting Devices, C h a r l o t t e s v i l l e (1967), pp. H.1-H.17. (18) P. G. De Gennes, Superconductivity of Metals and A l l o y s (W. A. Benjamin 1966), 6-4. (19) R.SD. P u f f and N. S . G i l l i s , Ann. of Phys. (N.Y.) 46 (1968), 364.. (20). M. Tinkham, Phys. Rev. 129 (1963), 2413. (20a) W. A. L i t t l e and R. D. Parks, Phys. Rev. L e t t . 9 (1962), 9. • (21) A. B Pippard, Proc, Roy. Soc. A216 (1955) , 5^7. (22) L. D. Landau and L i f t s h i t z , S t a t i s t i c a l Physics (Pergamon 1958), Chap. 12. (23) A. G. Sitenko, Electrodynamics F l u c t u a t i o n i n Plasma (Academic Press 1967), p. 15; p..31. (24) R.'E. Burgess, Proceedings of the Asilomar Conference, Fluctuations i n Superconductors ( 1968) , pp. 4-7-79. (25) See Ref. (18), 1-2=; ' 1-4. (26) See Ref. (18), 6-1. (27) M. C. Leung, J . S t a t i s t i c a l Phys., V o l . 2 (1970), ("to he published). (28) D. H. Douglass, J r . , Phys. Rev. 132 (1963), 513. 183 (29cD J . J . Hauser and E. Helfand, Phys. Rev..127 (1962), 389. (Z9b) T. HSU and G. P . Kharkov, Soviet Phys.-JETP' IT (1963), 1426. (2.9c) H. J . L i p k i n , M. Peshkin and L. J . Tassie, Phys. Eev. 126 (1962), 116. (30) See, f o r example, W.R. Le Page, Complex Variables and Laplace transform f o r engineers (McGraw-Hill 1961), . pp. 306. (31) R. E. Burgess, Canadian J . Phys. 47 (1969), 2583.. (32) L. Vant-Hull, R.-A. Simpkins and J . T. Harding, •• Phys. Rev. L e t t . 24A, (1967), 736. (33) J . G. V a l a t i n , Nuovo Cimento 7 (1958), 843. (34) N. N. Bogoliubov, Huovo Cimento 7 (1958), 794. (35) P. Rhodes, Proc. #oy. Soc. (London) A204 (1950), 396. • . . t. H. F r o h l i c h , Proc. Roy. Soc. (london) A215 (1952), 291. (36) K. Kawasaki and H. Mori, Prog. Theor. Phys. 28 (1962), 784. K. Tani, Prog. Theor. Phys. 41 (1969) , 891. (37) R. Rickayzen, Theory of Superconductivity ( I n t e r -science Publishers 1965). p.178. (38) See Ref.(15) | 3 3 184 (39) J . Bar de en, Phys. Rev. L e t t . 1, 399 (1958). (40) M. J . Stephen, Phys. Rev. 139 (1965) A197. A. V. S v i d z i n s k i i and V. A. Slyusarev, Soviet Phys.-Doklady 12 (1967), 59. (41) A. Schmidt, Physik Kondensierten Materie 5, 302 (1966). (42) See Ref. (14) 38. (43) V. L. Ginzburg, Soviet Phys.—JETP 7 (1962), 78. . (44) P. G. Be Gennes, Superconductivity of Metals and A l l o y s (V/. A. Benjamin 1966), 6-4. (45) See (18), p. 226; K. Maki, Progr. Theor. Phys. 29 (1963); 31, (1964), 731. (46) See Ref. (18), 3-1. (48) C P . Martin and J . Schwinger, Phys. Rev. 115 (1959), 1342. (48a) L. P. Kadanoff and C. P. Martin, Phys. Rev. 115 (1959). (49) A. W. B. Taylor, Proc. Phys. Soc. 78 (1962), 1372. (50) J . D. Jackson, C l a s s i c a l Electrodynamics, (John Wiley and Sons 1963), p. 189. (51) See (15), 9. (52) H. Takahasi, J . Phys. Soc. Japan (1952), 439. (53) G. Rickayzen, Theory of Superconductivity ( I n t e r -science Publishers 1965), p. 194. (54) J . R. S c h r i e f f e r , Theory of Superconductivity (W. A. Benjamin 1964), 3-2. 185 APPENDIX A: Physical I n t e r p r e t a t i o n ^ 6 a ^ o f Y('\^) {h^(T)b^ +^, with T >(), possesses the ph y s i c a l s i g n i f i c a n c e of gi v i n g the average p r o b a b i l i t y amplitude f o r f i n d i n g the system i n the same state at time T > 0 as i t was i n at T = 0. This s i g n i f i c a n c e can be shown as f o l -lows. Phases are chosen so that the Heisenberg and Schro-dinger p i c t u r e s are i d e n t i c a l at t = 0. For the moment we speak i n the Schrodinger language. Let us create at t = 0 a phonon i n the bare state a. The wave function f o r the system i s then ^mr.(o)> assuming the system i n state At a l a t e r time T the wave func t i o n has evolved into -4 1 I f the phonon were created i n state q at t = T rather than t = 0, the wave.function would be by e' 1^  Thus, the p r o b a b i l i t y amplitude f o r f i n d i n g the system at t = x >0 i n the same state as i t was i n at t = 0 i s the sc a l a r product of the two Schrodinger states at time T: 18.6. which i n the Heisenberg p i c t u r e becomes At temperature T*0, the p r o b a b i l i t y that the system i s i n state ^ i s where energy of .-the state | i>. Therefore the average p r o b a b i l i t y amplitude f o r f i n d i n g a system undisturbed i n the time i n t e r v a l Tj-t^O i s i or the average p r o b a b i l i t y of f i n d i n g the system undisturbed i s I f we add at t=0 and remove at t=r>0 a phonon with the same momentum, we should come back to the same state only i f i n the intervening time the phonon has not i n t e r a c t v/ith any other p a r t i c l e s i n the system. There, we should expect that the p r o b a b i l i t y (Al) should decay as e " S where a i s the c o l l i s i o n or damping rate of phonon. s 1 8 7 We s h a l l employ the f o l l o w i n g r e l a t i o n s - ft*0 ~ i -r oo • oo and ( I . I . I 7 ) 0 0 -1 tot J - J » • * (co- CA^)a + , \K*C rCv w^j [?(ico)] and coo i s t h e energy o f a f r e e phonon w i t h momentum c|. U s i n g t h e s e r e l a t i o n s , -00 where co = co - lm q 0 V/hen the di s p e r s i o n i n energy 2 Re£j(i<j>Q)j i s much less than |3, |<b|U)b|Co)>l = l i ^ ( b , f , ^ | 4 5 ^ , J We can t h u s i d e n t i f y t h e a b s o r p t i o n r a t e o f phonon w i t h 2R« [ ?0 ' a > s ) ] S i n c e Bco^ i s much s m a l l e r t h a n one f o r most e x p e r i m e n t a l s i t u a t i o n s , we can x v r i t e l < b | l t H | ( o ) > | 3 = (^,^fexb^-lRjf(ia3 s)]tlr • From experiment, a g [=2 Re JJ i s o f t h e o r d e r 10 s e c . o r s m a l l e r . (3 R e ^ « 1 f o r p r a c t i c a l i n t e r e s t s . 186-: APPENDIX B : Metastable Values of Cooper P a i r Density i n A Long, Thin, Hollow C y l i n d e r . V/ith the Gibbs fr e e energy given by ( 3 . 4 . 1 ) , the metastable values of superconducting e l e c t r o n density n and s magnetic f l u x are given by the simultaneous solutions of the equations 7 i r ° and g i v i n g and where ^ •* X o ( B O 1 - . . I -( x - n c ^ x ) 2 = ^ M r ( J l - i ) - ^ I 5. i s the number of quanta §futb.e c y l i n d e r contains. £(T) i s the coherence length: £(TJ* = S ^ f ^ T " ^ 2 1 1 " ^ ^ ) and \ i s the penetration depth i n the absence of a pe r s i s t e n t current ( i . e . when <j>0-= 5T ): A 0 = £-v*.ca^/£TreV]^ The s o l u t i o n s of (Bl) and (B2) would give the values of OU> at which the extrema of the Gibbs free energy occurs. We consider two .-cases (1) $ < Z Of A 6 and (2) ? >2. or Af >K4 p Noting that the maximum of y = (l-x)(z+x) occurs at x = (2-z)/3 and y i s zero at x = 1, the sketches of y are given i n F i g . B l . A and B give the extrema of G . From the form of (3.4.1), i t i s c l e a r that the value of G s corresponding to y = A i s a minimum o f a n d the value of G„ * • s corresponding to y = B i s 'a- maximum of G . The correspond-s ing sketches f o r the Gibbs f r e e energy are given by F i g . (3 .4- ) -Case (1) : z < 2*. The curves i n to have maximum and minimum f o r x > 0 when 191 or wh ere ^(T) i s the coherence length at temperature T (see (3 . 4.1a)). The maximum of y i s y^= -|j(B-J-) which occurs at X c 3 ^ In t h i s case, the t r a n s i t i o n i s of f i r s t order. The c r i t i c a l value of the superconducting density i s non-zero and i s given by The corresponding value of ! 5 o ~ " 5 j i l i s 15-$,! = § (i+M-.fz-A±. f3L Without loss of g e n e r a l i t y , we can assume ? 0>§^ . Suppose we increase €Q u n t i l |€0~§jif i s greater than H^-Til^. Then the c y l i n d e r becomes normal. The f l u x o i d i s no longer conserved. .However with the Meissner current becoming zero, the c y l i n d e r becomes superconducting again. It s h a l l become superconducting again with .a new f l u x o i d <§£/, such that I'>l , 192 While the minimum value of l ? 0 - ~ $ t ' | v/ould give smallest Gibbs fr e e energy, i t i s not c l e a r with what p r o b a b i l i t y s h a l l the c y l i n d e r pick up a c e r t a i n value Jl'when i t becomes super-conducting. However due to supercooling e f f e c t , we would expect "§jto increase to a value such that (B 3) I s true; i . e . I ^ o - ^ ' l < ^j$u . . . . . When $ 0 =• 0 , the maximum number J8C of quanta 5^  the c y l i n d e r can contain i s given by the maximum value of H such that . . .' £tr) A z Case (2) : z>2 (or A 6 < Rd). In t h i s case, there i s no supercooling or super-heating as i s c l e a r from F i g . 3.4b, where t h e r e . i s only one extremurn (minimum) of the curve Q s f o r p o s i t i v e values of x. The phase t r a n s i t i o n i s of the second order. At the t r a n s i -t i o n point, the value . of l<S>0-3>j[ - i s given by. / When $ 0 = 0 , the maximum possible f l u x o i d number i s given by the greatest i n t e g r a l part of 193~ ' "The v a r i a t i o n of (n ^  with 1 i s sketched i n F i g 3 . $ ~ . Hers, we use the c r i t e r i o n that, when there e x i s t s no r e a l s o l u t i o n f o r (62), the t r a n s i t i o n from superconduct-ing to normal phase would occur. This c r i t e r i o n i s equivalent the c o n d i t i o n s ^ -where ¥ i s defined i n (3.4.1), f o r the f i r s t order t r a n s i t i o n ; and, * = ° ^ t l - ° f o r the second order t r a n s i t i o n . As c l e a r from the disc u s s i o n above, the s o - c a l l e d c r i t i c a l d i m e n s i o n ^ = (r"KcT)_ i s given by L = \ . When L WSd o • o c o c ^ ( i . e . A o£jRd), i t i s second order t r a n s i t i o n , V/hen I ^ R d ( i . e . XQ <JRd), i t i s f i r s t order t r a n s i t i o n . Hauser and. (45) (Q&) Helfand^ v y a n d Douglass J r . ^  0 yhave assumed that the con d i t i o n f o r phase t r a n s i t i o n from superconducting to normal phase i s given by G - G = 0. While t h i s i s true f o r the case of second order phase t r a n s i t i o n , i t i s not a well-founddcriterio.v f o r the case of f i r s t order t r a n s i t i o n , 194 SYMBOLS AND ABBREVIATION A v A a = inner radius of a hollow cylinder A = a physical ouantity or an operator for a physical Q u a n t i t y A = vector potential 6 - component of T i n the c y l i n d r i c a l coordinates ( r , d , z ) = steady-state value of the vector potential A A w ( r ) = Fourier transform of (A(rt) - A(r)) bg, b ^ + = phonon operators of wave number q* b = outer radius of a hollow cylinder B = a physical quantity- or an operator for a physical quantity B ss magnetic f lux density c = ve loc i ty of l i g h t C^, Cj* ss electron operators of wave number k d ss thickness of a hollow cylinder D(o>) = Eq. (4 .3.9) D G = EQ. (4 .4 .3) D^ ss Eo. (4.5.4a) = energy of an electron with wave number k E p s= Fermi energy f =  e l e c t r i c f i e l d e = the charge of an electron e* = 2e f (A ) = (1 + e P * ) " 1 „ p 6 f k = Fermi function: f f e • (1 + e k ) - 1 f (t) ss a random force f n ( t ) ss random force due to random scattering of electrons by impurit ies . F( t ) ss an external force 195 G = Gibbs free energy h = Planck's constant ( f i i s taken as 1 for convenience) I i = magnetic f i e l d H(t) = time-dependent, external magnetic f i e l d H = Fourier component of H(t) = He*' 0* H Q = s tat ic external magnetic f i e l d 7 i »> Hamiltonian I = current ( total) I = current carr ied by Cooper pairs (condensate) I = current carried by quasi-part icles (the excitations) steady-s current the Fou] current response modified Bessel J 5 = f lux density operator T(t) = external current: l"(t) = ' i e l a > * J *= current density or current density operator k-g = Boltzmann constant ic, k = wave number kp = wave number at Fermi Surface K, K( t ) : d e f i n i t i o n on p. 40 and Eq. (2.2.20) = modified Bessel function of order V U = electron free path ^ = number of quanta (hc/2e) i n the f luxoid L = self-inductance L = L o u i v i l l e operator ot = length of a cylinder v v v I , I s , I n = steady-state value of the corresponding l , . i I~.*= urier transform of the corresponding co' sco' nco ° Iy = function of order y L = c r i t i c a l dimension c m = mass of an electron m a mass of a quasi -part ic le m* = 2m 196 m = magnetic moment density M = t o t a l magnetisation of a system n = t o t a l electron density n_ = superconducting electron density s n n = auasi -part ic le density P = projection operator Q(u>) = electromagnetic response function q t q = wave number r = space coordinate i n (r,e,z) R = radius of a thin cyl inder : b*Rfca R = resistance due to normal electrons n SA(co) = spectrum of A SA-D (W ) = cross-spectrum of A and B AD S(q,w) = structure function t = time T = temperature T Q = c r i t i c a l temperature of a superconductor U ss energy u k : d e f i n i t i o n : (2.2.2) v ss d r i f t ve loc i ty of ouasi-part ic les n v k : d e f i n i t i o n : (2.2.3) v = ve loc i ty of the condensate (Cooper pairs) s 2\ 2 / R d o <xAB(co) * generalized s u s c e p t i b i l i t y a_((o) = longitudinal ultrasonic absorption rate i n s superconductors 0^(0)) = longitudinal ultrasonic absorption rate i n normal conductors a* = coeff i c ient i n Landau-Ginzburg eauation:(3.4.1) ^ = ( k g T ) " * 1 197 B' = c o e f f i c i e n t i n Landau-Ginzburg equation ( 3 . 4 - . I ) = q u a s i - p a r t i c l e energy Y[ (t) = a,unit step f u n c c t i o n aX 0°k 2 2 2 Aj = \. = London penetration deptly X-^ = mc /(4-Jfne ) Xs * X2 = mo.2/ ( 4 T n s e 2 ) V : \>*'' = mc 2 / ( g f r | . ' ^ ) A n ; X2 = mc 2 / (4 im ne 2) &o = coherence length (at zero temperature) = coherence length at temperature T = c o r r e l a t i o n c o e f f i c i e n t j 5 = density operator ? = magnetic fluxe ; = t o t a l magnetic f l u x §C. = magnetic f l u x i n the hole of a hollow c y l i n d e r t^<o,5(&= F o u r i e r transform of the corresponding magnetic f l u x responses = hc/2e, a auantnm i n the f l u x o i d \ = * ' t h e f l w x o i d 5i 6 f l = r e l a x a t i o n f u n c t i o n 198 «£0 = magnetic f l u x due to an external f i e l d &oo . = c r i t i c a l magnetic flux, due to. an external f i e l d . 51^ , #|L s i n g l e p a r t i c l e Fermi operator ^ = wave fu n c t i o n f o r BGS condensate % ' f = ( K ( t ) , f ) , see ^ .A-l <y* = co n d u c t i v i t y Tc - complex condu c t i v i t y T $ = r e l a x a t i o n time of magnetic flux,, etc. i n t h i n " superconducting c y l i n d e r s . = momentum r e l a x a t i o n time f o r normal- electrons T c = r e l a x a t i o n time of the random i n t e r a c t i o n on a phonon. = r e l a x a t i o n time of the q u a s i - p a r t i c l e density <*>fr = frequency (renormalized) of sound wave with wave number q = frequency of free phonon with momentum q. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084800/manifest

Comment

Related Items