UBC Theses and Dissertations

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UBC Theses and Dissertations

Phase transitions in multiply-connected superconductors Fillmore, Keith Geddes 1971

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PHASE TRANSITIONS IN MULTIPLY-CONNECTED SUPERCONDUCTORS by  KEITH GEDDES FILLMORE B.Sc., Queen's U n i v e r s i t y , 1 9 5 5 M.A., Princeton University 1 9 5 7 s  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n t h e Department of PHYSICS  s  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 7 1  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  the U n i v e r s i t y  s h a l l make i t  f u r t h e r agree  in p a r t i a l  freely  f u l f i l m e n t o f the of B r i t i s h  available  for  requirements  Columbia, I agree  for  that  r e f e r e n c e and s t u d y .  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s of  this  representatives.  It  thesis for financial  i s understood that copying or p u b l i c a t i o n gain s h a l l  written permission.  Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  not be allowed without my  ABSTRACT  The  f i r s t two c h a p t e r s p r e s e n t a c r i t i c a l  of the d e r i v a t i o n of the Ginzburg-Landau  review  macroscopic  e q u a t i o n s and t h e i r a p p l i c a t i o n t o the d e t e r m i n a t i o n o f the c r i t i c a l  f i e l d s and temperatures  for  superconducting-  normal phase t r a n s i t i o n s i n s i m p l y connected Second o r d e r phase t r a n s i t i o n c r i t e r i a  bodies.  are obtained i n  the form o f volume i n t e g r a l s which do not r e q u i r e p r i o r s o l u t i o n o f the f i e l d  equations.  With the G-L  e f f e c t i v e wave f u n c t i o n (p  London gauge f o r d o u b l y - c o n n e c t e d  b o d i e s , we  i n the  obtain several  e q u i v a l e n t e x p r e s s i o n s f o r the e l e c t r o m a g n e t i c Free Energy which do not assume u n i f o r m to  /(^l  a s y s t e m a t i c method f o r expanding =  a  nd  One  F^  The  expansion  leads  i n powers of  o b t a i n e d i s used t o  second o r d e r t r a n s i t i o n c r i t e r i a  fluxoid states.  of t h e s e  t h i s method i s a p p l i e d to the l o n g h o l l o w  circular cylinder. determine  .  A closed expression f o r  c y l i n d e r which does not assume u n i f o r m /(pl  for a l lpossible F^  i n the  hollow  i s obtained.  A d e t a i l e d a n a l y s i s of the t h i n h o l l o w  cylinder  y i e l d s c u r v e s f o r the e q u i l i b r i u m v a l u e s of Free E n e r g y ,  ii  Super E l e c t r o n D e n s i t y , Magnetic Moment, and Super E l e c t r o n Momentum as f u n c t i o n s o f a g e n e r a l  field variable.  Critical  p o i n t s on t h e c u r v e s a r e i d e n t i f i e d and t h e l o c i o f c r i t i c a l p o i n t s t r a c e d as the g e o m e t r i c parameter v a r i e s . f u n c t i o n s a r e extended t o s u r f a c e s field  and t e m p e r a t u r e .  obtained  These  of the v a r i a b l e s of  Phase t r a n s i t i o n c r i t e r i a a r e  f o r i n c r e a s i n g temperature at constant  R e v e r s i b l e and i r r e v e r s i b l e c u r v e s a r e o b t a i n e d  field. describing  the v a r i a t i o n and d e s t r u c t i o n o f p e r s i s t e n t c u r r e n t s a t v a r y i n g t e m p e r a t u r e , and t h e r e s u l t s a r e compared w i t h experiment.  The f u l l range o f p o s s i b l e f l u x o i d  states  i s d e t e r m i n e d and i t i s shown t h a t the f a m i l i e s o f c u r v e s obtained  f o r low f l u x o i d s t a t e s may be a p p l i e d t o h i g h  f l u x o i d s t a t e s by s u i t a b l e s c a l i n g . undergo second o r d e r fluxoid.  A l l thin cylinders  t r a n s i t i o n s at s u f f i c i e n t l y  The c o r r e c t i o n due t o f i n i t e  of t h e o r d e r  high  coherence l e n g t h i s  o f the f o u r t h power o f t h e r a t i o o f t h e  cylinder wall thickness  t o t h e coherence l e n g t h .  Other t h i n d o u b l y connected s u p e r c o n d u c t o r s a r e described  by the same f u n c t i o n s a p p l i c a b l e t o c i r c u l a r  c y l i n d e r s by making t h e a p p r o p r i a t e  correspondence o f  parameters.  critical field  The e x a c t second o r d e r  torus i s c a l c u l a t e d .  iii  for a  Systems loops are of  two c o - a x i a l m a g n e t i c a l l y  a n a l y s e d and t h e  coupled  c r i t i c a l conditions  for  f o u n d t o d e p e n d i n a c o m p l i c a t e d way on t h e the  one, the  are  of  other.  the  a thin  behaviour  latter.  obtained  If  of  Second  for  systems  I n the results  obtained  viously  i n the  loop i s  the  former  o r d e r phase of  final  n  is  thesis  literature.  iv  coupled to a  thick  d o m i n a t e d by t h a t  loops of  loop  presence  transition criteria  co-axial  chapter  in this  closely  one  equal  of are  radius.  a c o m p a r i s o n i s made o f with those appearing  the  pre-  TABLE OF CONTENTS Page ABSTRACT  i  LIST OF FIGURES  i  v i i i'X  ACKNOWLEDGEMENTS Chapter 1  INTRODUCTION  2  SIMPLY-CONNECTED SUPERCONDUCTORS  3  MUI.TIPI.V-nONMEr.TF.M  4  THIN CYLINDERS  8l  P a r t 1.  Geometry  8l  P a r t 2.  Temperature and F i e l d  5  6  '  20  SUPERnOMnnr.TORS  -  $h  9^  THIN CYLINDERS CONTINUED  I l l  P a r t 1.  High F l u x o i d Number  P a r t 2.  F i n i t e Coherence Length  I l l 119  OTHER FORMS OF MULTIPLY-CONNECTED SUPERCONDUCTORS  7  1  '  130  COMPARISON OF THE LITERATURE WITH PRESENT RESULTS  156  BIBLIOGRAPHY  162  v  Page APPENDIX 1  SUMMARY OF FORMS OF THE G - L FREE ENERGY . . . .  APPENDIX 2  EXPANSION OF BOUNDARY FACTORS IN POWERS  169  OF^ APPENDIX 3  164  EXPANSION OF BOUNDARY FACTORS IN POWERS OF  (d/A )  171  vi  LIST OF FIGURES  A l l F i g u r e s a r e p l a c e d a t t h e end o f t h e i r r e s p e c t i v e c h a p t e r . i n d i c a t e d by the number p r e c e d i n g the hyphen. •Figure 2-1  . F u n c t i o n a l dependence on  2-2  Following o f Magnetic Free Energy  (//<J )  53  2  F u n c t i o n a l dependence  o f Magnetic F r e e Energy  on t h e r e l a t i v e super e l e c t r o n d e n s i t y 2-3  Reduced F r e e Energy  2-4  Reduced F r e e Energy  2-5  Second Order C r i t i c a l  2-6  S p l i t t i n g o f Second Order C r i t i c a l Three Branches a t t *  4-1 4-2  4-3  Page  f o r  F(*5^)  s m  all  53  body  f o r l a r g e body  Field i n H  - t  plane . . .  53 53 53  Curve i n t o 53  E q u i l i b r i u m v a l u e o f r e l a t i v e super e l e c t r o n density ^ Thin c y l i n d e r functions f o r v a r i a b l e f i e l d , geometry parameter (a) F r e e Energy F ( X ) (b) Super E l e c t r o n D e n s i t y y(X) (c) Magnetic Moment M(X) . < (d) Super E l e c t r o n Momentum P ( X ) Thin c y l i n d e r functions f o r v a r i a b l e f i e l d , t e m p e r a t u r e parameter (a) F r e e Energy f ( x ) (b) Super E l e c t r o n D e n s i t y ^ ( x ) (c) Magnetic Moment m(x) . (d) Super E l e c t r o n Momentum p ( x )  vii  110 110 110 110 110 110 110 110 110 110 110  Figure  Following  Page  4-4  Thin cylinder functions f o r v a r i a b l e t e m p e r a t u r e , f i e l d parameter (a) F r e e Energy f(t) (b) Super E l e c t r o n D e n s i t y 3 ( t ) . . . . . . . (c) Magnetic Moment m(t) . *\ (d) Super E l e c t r o n Momentum p ( t )  110 110 110 110 110  4-5  F r e e Energy o f n e i g h b o u r i n g f l u x o i d at 'onset o f s u p e r c o n d u c t i v i t y  110  4-6  S e p a r a t i o n of h-t plane i n t o r e g i o n s of n o r m a l , s t a b l e and m e t a s t a b l e s u p e r c o n d u c t i n g states  110  4-7  Reduced Magnetic Moment m ( t ) i n t h i n h o l l o w c y l i n d e r , showing m e t a s t a b l e l i m i t m^  110  4-8  Reduced Magnetic Moment m(t) i n t h i n h o l l o w c y l i n d e r , showing maxima o f m(x) l i n e m^ . . .  110  4- 9  M a g n i f i c a t i o n o f e q u i l i b r i u m c u r v e s near metastable l i m i t  1.10  5- 1  5-2  5- 3  6- 1  129  Dependence o f F r e e Energy F- on Y range o f p o s s i b l e f l u x o i d s t a t e s  129  Magnetically Magnetically Magnetically / *  6-4  for full  Dependence o f Super E l e c t r o n D e n s i t y on Y f o r f u l l range o f p o s s i b l e f l u x o i d s t a t e s . . . .  = 2  1  {  coupled loops w i t h :  coupled loops with  129  ,  /  155  ,££=.5 .  , 155  L  > ^ = 1 . 2  ,  °  Magnetically  yLC =10  M. =2.5  coupled loops with  ^ 2 = 4 . 0 6-3  , .  Change i n c r i t i c a l reduced f i e l d s a t l a r g e fluxoid  ^ 2 = 2 . 6 6-2  states  155  coupled loops with  y&S- = 4 . 0 , /  2  viii  155  ACKNOWLEDGEMENTS  I w i s h t o thank my s u p e r v i s o r Dr. R. E. Burgess f o r h i s a s s i s t a n c e d u r i n g the c o u r s e o f t h i s work and f o r the p r o v i s i o n o f summer r e s e a r c h g r a n t s .  I a l s o thank  the U n i v e r s i t y o f B r i t i s h Columbia f o r t h e award o f a Graduate F e l l o w s h i p from 1966 t o 1970.  ix  CHAPTER  1  INTRODUCTION  The dates p r o p e r l y (1933)  study of the thermodynamics of s u p e r c o n d u c t o r s from the d i s c o v e r y  by M e i s s n e r and  Ochsenfeld  t h a t the s t a t e of a pure s u p e r c o n d u c t o r , a t f i x e d  t e m p e r a t u r e and i n f i x e d e x t e r n a l magnetic  f i e l d , i s unique.  I n a s i m p l y - c o n n e c t e d s u p e r c o n d u c t o r i n low f i e l d p a r t i c u l a r c l a s s of m a t e r i a l s  now  known as Type I s u p e r -  f l u x i s e x c l u d e d from the m a t e r i a l . magnetic  f l u x i s now  Previously  (and f o r a  The e x c l u s i o n  of  r e f e r r e d t o as the " M e i s s n e r E f f e c t " . ,  i t had been assumed t h a t the s t a t e depended on  p a s t h i s t o r y , and t h a t phase t r a n s i t i o n s from normal t o super s t a t e s , and v i c e v e r s a , were i n g e n e r a l i r r e v e r s i b l e . An elementary thermodynamics of i d e a l M e i s s n e r i s g i v e n by London ( 1 9 5 0 ) .  Refinements  superconductors  o f t h i s thermo-  dynamics, c o n c e r n i n g i n p a r t i c u l a r the c r i t e r i a  f o r phase  change, and t h e i r dependence on the c o n n e c t i v i t y , s i z e and shape of the body, form the p r i n c i p a l s u b j e c t thesis.  1  of t h i s  2  The observed m a c r o s c o p i c  e x c l u s i o n o f the  field  from t h e body i s e v i d e n t l y due t o p e r s i s t e n t s u r f a c e c u r r e n t s which g e n e r a t e a magnetic field.  London was  field  opposite to the e x t e r n a l  the f i r s t t o e l u c i d a t e the a c t u a l  dis-  t r i b u t i o n o f the c u r r e n t d e n s i t y i n the " s u r f a c e " c u r r e n t s . In p a r t i c u l a r , London d e f i n e d the p e n e t r a t i o n depth or average of  t h i c k n e s s of the s u r f a c e c u r r e n t .  Except i n b o d i e s  z e r o d e m a g n e t i z a t i o n t h e s e c u r r e n t s a l s o a f f e c t the  o u t s i d e the body g i v i n g r i s e t o d i p o l e and h i g h e r moments which i n t e r a c t w i t h the e x t e r n a l f i e l d . to  field  magnetic In order  compare the f r e e e n e r g i e s o f s u p e r c o n d u c t i n g and  normal  p h a s e s , and hence t o determine the e n e r g e t i c a l l y f a v o u r a b l e phase, the k i n e t i c energy e x a c t magnetic  field  of the super c u r r e n t s and  energy  be t a k e n i n t o a c c o u n t .  the  o f the p e n e t r a t i n g f i e l d must  These become e x t r e m e l y  Important  i n b o d i e s h a v i n g at l e a s t one d i m e n s i o n comparable t o o r l e s s t h a n the p e n e t r a t i o n d e p t h . London's t h e o r y had one p a r t i c u l a r d e f e c t o f which he was w e l l aware. of  By s t r a i g h t f o r w a r d  application  t h i s t h e o r y i t i s e n e r g e t i c a l l y f a v o u r a b l e under c e r t a i n  c i r c u m s t a n c e s f o r a s u p e r c o n d u c t o r o f zero d e m a g n e t i z a t i o n to  split  i n t o very f i n e l a y e r s of superconducting  normal m a t e r i a l . does not o c c u r .  and  I n f a c t , i n Type I s u p e r c o n d u c t o r s ( T h i s s p l i t t i n g s h o u l d hot be  this  confused  3  w i t h t h e i n t e r m e d i a t e s t a t e which depends e s s e n t i a l l y on demagnetization.)  To c i r c u m v e n t  duced ad hoc a s u r f a c e energy f o r m a t i o n o f a normal-super  t h i s problem  t o be a s s o c i a t e d w i t h t h e interface.  G i n z b u r g and Landau ( 1 9 5 0 ) sented a macroscopic  London i n t r o -  ( h e r e a f t e r G-L) p r e -  t h e o r y o f s u p e r c o n d u c t i v i t y which  accounts f o r a wide range o f observed phenomena w i t h o u t t h e need f o r any a d d i t i o n a l h y p o t h e s i s on s u r f a c e energy. normal-super  " s u r f a c e " i s seen as a c o n t i n u o u s  i n super e l e c t r o n d e n s i t y and t h e energy  variation  associated with  the s u r f a c e c a n , i n p r i n c i p l e a t l e a s t , be c a l c u l a t e d the t h e o r y i t s e l f .  The  from  The G-L t h e o r y forms the b a s i s on which  the r e s u l t s o f t h i s t h e s i s a r e o b t a i n e d .  The G-L t h e o r y i s  c o n v e n i e n t l y p r e s e n t e d i n two s t a g e s , f i r s t t h e c o n d e n s a t i o n a s p e c t ;^hich l e a n s h e a v i l y on t h e g e n e r a l t h e o r y o f second o r d e r phase t r a n s i t i o n s o f Landau and L i f t s h i t z  ( 1 9 5 8 ) , and  secondly the electromagnetic aspect. Consider a superconductor  i n thermal contact with  a heat r e s e r v o i r and i n t h e absence o f any e x t e r n a l f i e l d . As i t s temperature  i s lowered below t h e c r i t i c a l  temperature  some o f t h e c o n d u c t i o n e l e c t r o n s condense i n t o a l o w e r energy  s t a t e c a l l e d t h e s u p e r c o n d u c t i n g s t a t e , o r more  s i m p l y , t h e super s t a t e .  The f r e e energy  d e n s i t y i n the  body i n t h e s u p e r c o n d u c t i n g s t a t e i s l e s s t h a n i n t h e normal  4  s t a t e where " f r e e " here means w i t h i n the c o n s t r a i n t o f constant temperature.  Throughout the t h e s i s we s h a l l t a k e  as z e r o r e f e r e n c e o f f r e e energy the f r e e energy of the normal s t a t e , so t h a t " t h e " f r e e energy o f the super, s t a t e i s synonymous w i t h the d i f f e r e n c e of f r e e e n e r g i e s o f super and normal s t a t e s .  C o n s e q u e n t l y , " t h e " f r e e energy  i s always z e r o when the super e l e c t r o n d e n s i t y use the symbol ( s c r i p t ) I? for  f r e e energy  density  i s zero.  f o r t o t a l f r e e e n e r g y , and  We /A  v  I?  density.  The f r e e energy d e n s i t y w i l l i n g e n e r a l be a f u n c t i o n of temperature density  n  g  .  T  and the super e l e c t r o n number  The t e m p e r a t u r e  c o n s t r a i n t whereas  T  i s an e x t e r n a l l y imposed  n  w i l l a d j u s t i t s e l f so as t o m i n i m i z e s the body i s n o r m a l , so ^St T) has a 0  <J~ .  At T > T  c  p h y s i c a l minimum at n s have a minimum a t some simplest  =  0  .  n > 0 s  At  T<T  ,  ~?(r? T)  must  S)  c a t which J~ ^ ®  .  The  f u n c t i o n s a t i s f y i n g t h i s r e q u i r e m e n t and  i s the p a r a b o l a where CK , j§  are d e f i n i t e p h y s i c a l f u n c t i o n s  on the m a t e r i a l , but as y e t unknown. make CX  of T  depending  The minus s i g n i s t o  t u r n out p o s i t i v e ( i n c o n t r a s t  t o the n o t a t i o n  G-L) w h i l e t h e 1 / 2 i n the l a s t term i s c o n v e n t i o n a l ( i n  of  5  agreement w i t h G-L). ^ Ic  <C 7"  0 The  ^3  must be p o s i t i v e i n the range  i n order  minimum o f  that  ~J~ have a minimum a t a l l .  o c c u r s when  CT) ^  where  means the e q u i l i b r i u m v a l u e o f  field. for  Prom the p r e c e d i n g  T < T  Near  0(/^ =  0  =  s  that i s , at ft&fT)  =  n /3(T)  +  T  , c  c  (T)  & < 0  for  3  we  know t h e n t h a t (T ~T).  0 0  The  c  n^fT)  (1>2  d i s c u s s i o n we T > T and c (X(T)  value  n  i n zero  see t h a t  (X ~  0  (T ~T) C  at  )  (X > 0 T=T  , and '  . c  of the f r e e energy d e n s i t y  In e q u i l i b r i u m i s ~z  of. 2  2(3 (1.3)  I n the absence of any  e x t e r n a l f i e l d the e q u i l i b r i u m f r e e  energy d e n s i t y i s u n i f o r m t h r o u g h o u t the body so t h a t f o r a body of volume t x o n s CK(T)  ,  , /3(T)  J~  w  may  =  v  "j^o)  Vj  .  The  func-  be deduced from measurements of  c e r t a i n r e l a t e d q u a n t i t i e s , one we  ( A  of which i s  3^  , which  proceed to d i s c u s s . When the body i s i n an e x t e r n a l magnetic  persistent surface  field  c u r r e n t s f l o w i n the sense t o e x c l u d e  6  the f i e l d  from t h e body.  I n t h i s c a s e , b o t h the i n e r t i a l  k i n e t i c energy o f the c u r r e n t s and the a s s o c i a t e d  magnetic  f i e l d energy must be i n c l u d e d i n t h e f r e e energy o f the superconductor.  S i n c e t h e k i n e t i c energy c o n t r i b u t i o n comes  from t h e s u r f a c e , whereas the f i e l d  c o n t r i b u t i o n comes from  the volume, f o r s u f f i c i e n t l y l a r g e volumes the l a t t e r dominates.  The f i e l d energy d e n s i t y i n G a u s s i a n u n i t s i s  /t/r  (H -H) e  pre-  where  absence o f t h e body and the body.  H  i s the magnetic f i e l d H  i n the  i s the f i e l d i n the p r e s e n c e o f  One might be tempted t o put ( i n c o r r e c t l y ) , as i n d e e d one sometimes f i n d s i n t h e  literature.  The c o r r e c t e x p r e s s i o n  may be d e r i v e d  from  f i r s t p r i n c i p l e s f o r example by c o n s i d e r i n g a loop o f zero r e s i s t a n c e s i t u a t e d i n t h e f i e l d o f a s o l e n o i d and carrying a current.  initially  The f r e e energy o f t h e loop i s the  work done by the loop when i t i s d i s p l a c e d s l o w l y a l o n g t h e s o l e n o i d a x i s t o the p o i n t where t h e l o o p c u r r e n t becomes z e r o , the s o l e n o i d c u r r e n t b e i n g h e l d c o n s t a n t .  I t i s found  t h a t t h i s f r e e energy can be e x p r e s s e d as a volume I n t e g r a l of t h e f i e l d energy d e n s i t y  (H ~ ft) / <Ff7 e  C o n s i d e r a l a r g e s u p e r c o n d u c t o r o f volume h a v i n g z e r o d e m a g n e t i z a t i o n c o e f f i c i e n t (a l o n g n e e d l e p a r a l l e l t o the f i e l d ) s i t u a t e d i n a uniform e x t e r n a l —»  H  e  .  Due t o s u r f a c e c u r r e n t s the f i e l d  field  i n s i d e the body  will  7  be z e r o , except i n a v e r y t h i n p e n e t r a t i o n s k i n t h e volume of w h i c h i s n e g l i g i b l e because o f body s i z e , w h i l e o u t s i d e the  body  H=H  complete.  e  .  That i s , t h e M e i s s n e r e f f e c t i s e s s e n t i a l l y  The magnetic f r e e energy  7^  is  (ffefc/V  =  d- *)  H£A  1  T h i s i s a p o s i t i v e c o n t r i b u t i o n t o t h e f r e e energy f o r a l l fields. I f the external f i e l d some f i e l d , say \ H l — HQ^  at  e  the  i s g r a d u a l l y i n c r e a s e d then  (cb s t a n d s f o r c r i t i c a l  magnetic energy e q u a l s t h e c o n d e n s a t i o n energy.  / fig} > r1 b  bulk), At  t h e super s t a t e i s no l o n g e r e n e r g e t i c a l l y  c  f a v o u r a b l e , hence a t r a n s i t i o n t a k e s p l a c e t o the normal state.  T h i s i s a c o n v e n i e n t way t o measure t h e c o n d e n s a t i o n •  energy as a f u n c t i o n o f t e m p e r a t u r e .  Equating the bulk  c r i t i c a l f i e l d f r e e energy d e n s i t y t o t h e n e g a t i v e o f t h e c o n d e n s a t i o n energy d e n s i t y we have  — Hk c  -  - oo  Z  f  3  i s a property of the superconducting m a t e r i a l quite  independent o f any magnetic f i e l d which may a c t u a l l y be present.  H j^ c  i s found by experiment t o have t h e approximate  8  t e m p e r a t u r e dependence H  ctl  where  *  He  t=T/T  c  -  (i  t)  and  extrapolated t o 0  o  1  i s the bulk c r i t i c a l  field  K .  U s i n g (1.2) and (1.1) we may r e w r i t e t h e g e n e r a l e x p r e s s i o n f o r {S^"3~ , e q u a t i o n ( 1 . 1 ) , i n t h e f o l l o w i n g form  The G-L t h e o r y i s f o r m u l a t e d i n quantum m e c h a n i c a l terms from the o u t s e t .  G-L i n t r o d u c e an " e f f e c t i v e wave  f u n c t i o n o f t h e super e l e c t r o n s "  1^ , which i s a k i n d o f  average over t h e super e l e c t r o n wave f u n c t i o n s . to n  s  I n analogy  quantum mechanics, IJfijf* r e p r e s e n t s t h e number d e n s i t y o f t h e super f l u i d , w h i l e t h e s p a t i a l l y dependent phase  of  c o n t a i n s i n f o r m a t i o n on t h e momentum o f t h e super  fluid.  ^  , and c o n s e q u e n t l y  —  ty*  , i s a continuous  s p a t i a l l y dependent f u n c t i o n i n s i d e t h e s u p e r c o n d u c t o r , and z e r o o u t s i d e . Let  IjJ^ be t h e e q u i l i b r i u m z e r o - f i e l d v a l u e o f 2  "ty  , t h a t i s / tp^l  -  ftco  .  Lpcc  , like  r] , i s M  a  p r o p e r t y o f t h e m a t e r i a l a t g i v e n t e m p e r a t u r e and i s n o t a s p a t i a l l y dependent q u a n t i t y .  C o n s e q u e n t l y t h e phase o f  9  can have no meaning and we t a k e  (poo  LjJgg t o be r e a l .  i n t h e G-L f o r m u l a t i o n , t h e f r e e . e n e r g y  Then  d e n s i t y i n zero  field i s  A  V  f_ IM .+ 2  ? „ = J ± £  2  Sn  \  Probably theory  9<»  m  j  ?  VS  z  f  t h e most p r o f o u n d i n n o v a t i o n i n t h e G-L  i s the f o r m u l a t i o n o f the k i n e t i c energy d e n s i t y  term w h i c h u l t i m a t e l y e n a b l e s G-L t o d e r i v e - a field  equation  equations  f o r the current d e n s i t y .  as p o s t u l a t e s .  London-like  London s t a t e d h i s .  He knew, o f c o u r s e ,  t h a t they  c o u l d n o t be d e r i v e d c l a s s i c a l l y , and supposed  correctly  t h a t they would one day be a consequence o f a m i c r o s c o p i c theory  of super-conductivity. A c l a s s i c a l charged p a r t i c l e , o f charge  a magnetic f i e l d momentum p  of vector p o t e n t i a l  — IT)*/"  +JL  A  ~K has c a n o n i c a l  and H a m i l t o n i a n  I n c l a s s i c a l mechanics s u b s t i t u t i o n o f p the u s u a l e x p r e s s i o n Newtonian e q u a t i o n s the H a m i l t o n i a n p  ~  into  { P ~ *t  ™ j  gives  for- k i n e t i c energy w h i c h g i v e s t h e of motion.  I n quantum m e c h a n i c s , however,  f o r m u l a t i o n has p r o f o u n d e f f e c t , f o r I t i s  and n o t tY)A/  ~-<i j? V  C^. , i n  ; we have  ;vhich i s r e p r e s e n t e d  by t h e o p e r a t o r  10  7*> The consequences o f t h i s form o f t h i s form i s found t o be c o r r e c t . G-L put f o r t h e k i n e t i c  are o b s e r v a b l e and I n s p i r e d by t h i s  form,  energy d e n s i t y o f t h e s u p e r f l u i d since the i n t e g r a l of t h i s  q u a n t i t y o v e r the body i s analogous t o the quantum m e c h a n i c a l expression  f o r the expectation  value  of k i n e t i c  energy.  T h i s form a l s o has the e s s e n t i a l f e a t u r e o f b e i n g gauge c o v a r i a n t w i t h t h e "wave f u n c t i o n " . Collecting  t e r m s , the t o t a l f r e e energy  is  g i v e n q u i t e g e n e r a l l y by  (1.8  Only t h e term i n (''"~ '^e) L  can c o n t r i b u t e o u t s i d e the volume  of the s u p e r c o n d u c t o r . F o l l o w i n g G-L, we i n t r o d u c e  the " r e l a t i v e  effective  wave f u n c t i o n " 9  =  (1.9  I n t h e c a s e oir z e r o e x t e r n a l f i e l d  and/or i n the  c a s e o f a b u l k s u p e r c o n d u c t o r , (p  has  1  I n t h e s u p e r s t a t e and The  by d e f i n i n g  0 i n the normal 7  expression for  "G-L  _  Imc  ratio  the  simplified "coherence  h  free  ==  of these c h a r a c t e r i s t i c  parameter"  /  d.io)  1  n  lengths i s the  so-called  K  I tmc \ I /Q  energy  $ r r  takes the  J/?ll fpece  (1 -11)  form  I  (1.12)  A~  K All  and  /3  X  The  i s considerably  the " p e n e t r a t i o n depth" A  f  The  state.  f  length" ^  the u n i f o r m v a l u e  the macroscopic  physical properties  J of the  superconductor  are c o n t a i n e d i n the temperature dependent q u a n t i t i e s H ^ As a l r e a d y m e n t i o n e d , and  i t s t e m p e r a t u r e dependence found.  a b l e and  c a n be A  ,  measured  i s also  measur-  i t s t e m p e r a t u r e d e p e n d e n c e i s f o u n d t o be a p p r o x i m a t e l y  12  k i t )  A  where  «  \ J l - t + )  i s the  0  calculated  from the  &  indicates  to  0  coherence length  F  K  a n d  absolute  •  is  The then  '°\i-f  cb  the  K  be  H tt)X(t)  obtain  i MQO  , ^3  the  extrapolated  above r e l a t i o n s to  /Tf S i m i l a r l y we  A  value of  t e m p e r a t u r e dependence of  (1.13)  f o l l o w i n g r e l a t i o n s f o r the  quantities  , where I n e v e r y case s u b s c r i p t  Q  zero.  v2  Various normalization quantities  i n the  integrand  i n o r d e r t o c o n c e n t r a t e on tion.  also  are the  A c o n f u s i n g v a r i e t y of  "dimensionless"  of  in this thesis.  These are  will  be  defined  the  different  i n p a r t i c u l a r cases the  equa-  "reduced," " r e l a t i v e , "  quantities  and  useful  and  f u n c t i o n a l form of  etc.  reference,  I?  appear i n the  l i t e r a t u r e and  summarized i n Appendix 1 i n the  text  as  needed.  for  13  We expect the body t o go i n t o the s t a t e d e f i n e d by (jj(fh)  and (\ (/i> J , f o r which the f r e e energy i s a minimum  under the c o n s t r a i n t s o f c o n t a c t w i t h a temperature and f i x e d e x t e r n a l magnetic f i e l d . e n t a i l equations  f o r (p  and A  differ-  by t a k i n g the f i r s t  w i t h r e s p e c t t o (p  tion of  G-L o b t a i n the  reservoir  and A  T a k i n g the v a r i a t i o n o f J"  varia-  respectively.  i n equation  (1.12)  •fc  w.r.t.  and p e r f o r m i n g a p a r t i a l i n t e g r a t i o n on the  terms c o n t a i n i n g  fzJ,  , we o b t a i n  /i h A c  K  *v<p  i  I  f A  y  Surface  S e t t i n g the c o e f f i c i e n t o f S<p i n the volume i n t e g r a l e q u a l t o zero g i v e s the f i r s t G-L e q u a t i o n : —9  v  2( 1 . 1 4 )  S i n c e the v a l u e o f <j) on the s u r f a c e i s not s p e c i f i e d as a boundary c o n d i t i o n , i d e n t i c a l l y zero.  i n the s u r f a c e i n t e g r a l i s not  To a s s u r e zero v a r i a t i o n i n "3" we must  14  r e q u i r e the integrand of the surface i n t e g r a l t o vanish, T h i s g i v e s t h e f o l l o w i n g boundary  where  IjJ  :  Jl = o  ^  K  c o n d i t i o n on  (ia5)  i s a normal t o t h e s u r f a c e . To o b t a i n t h e v a r i a t i o n w . r . t . —»  —0  s u b s t i t u t e M-V*A  and #  X , we  first  —o  —ja  = V X ,4  e  .  The  terms  i n 3^  —o  containing  A  are y  r,fV  '  2  where is  V,  i s the  a l l space  0-z  volume  external  i f  to  -  y  7  A  of V,  f ^  the .  superconductor, Taking  Hci> ,  the  ,  and  variation  , ,  15 Partial  As  3  L  i n t e g r a t i o n o f the second  ~*  j L H~~ ^ e J " " * 0  0 0  s u r f a c e i n t e g r a l i s 0. first  is  0 since  that  in  Y^  V X H —  3  external currents  V x A since  density is  (JJ — 0  V X  }  tf^  d  remain constant.  and t h e l a s t u e  term  to the constraint  Hence  F i n a l l y we n o t e t h a t  v"  2  contributes  the term  i n the i n t e g r a n d i s 0 t h r o u g h o u t t h e volume  &  this  quantity  i s p r o p o r t i o n a l t o the  producing the e x t e r n a l f i e l d  entirely  H  g  current  , and t h i s  e x t e r i o r to the superconductor.  current  Hence  fillip (fvf - (/J Vp*)+?M^ -hlVxVXA  ^J=±j Therefore  , hence the  I n the volume i n t e g r a t i o n o v e r  n o t h i n g t o the I n t e g r a l . V  a t l e a s t a s / /Ji  two t e r m s a r e 0 s i n c e  the  i n t e g r a l leads to  J  i s s t a t i o n a r y when t h e c o e f f i c i e n t o f dA  \-[A c/V is  0:  vx  A  ==-  -  - f - — — (yvy*  w h i c h i s t h e s e c o n d G-L e q u a t i o n .  -  -(1.16).  16 I t might be i n c o r r e c t l y c o n c l u d e d from t h e p r e ceding  c a l c u l a t i o n t h a t the v a r i a t i o n i n the magnetic energy  a s s o c i a t e d w i t h the f i e l d o u t s i d e t h e body i s z e r o . devided  ~3~  i n t o two i n t e g r a l s , over  I f we  arid .  respec-  t i v e l y , a t t h e o u t s e t , t h e n when t h e p a r t i a l i n t e g r a t i o n i s performed i n  t h e r e appears a non-zero s u r f a c e  over t h e s u r f a c e o f the body.  T h i s s u r f a c e i n t e g r a l con-  t a i n s a l l t h e magnetic f i e l d energy i n remaining  integral  i s 0.  volume i n t e g r a l over  , since the The i n t e g r a l over  when p a r t i a l l y i n t e g r a t e d a l s o c o n t a i n s a non-zero s u r f a c e i n t e g r a l which j u s t c a n c e l s t h e s u r f a c e from and  Vp . exterior  Integral  There i s thus a r e l a t i o n between the i n t e r i o r c o n t r i b u t i o n s which p e r m i t s  us t o w r i t e  J~  as an i n t e g r a l over the body o n l y . L e t us c o n s i d e r e q u a t i o n The  (1.16) more c l o s e l y .  LHS i s by Maxwell's e q u a t i o n p r o p o r t i o n a l t o t h e c u r r e n t — •>  density  .  The RHS i s , - w i t h i n a n o r m a l i z a t i o n  i d e n t i c a l w i t h the quantum m e c h a n i c a l e x p r e s s i o n  factor, f o r the  c u r r e n t d e n s i t y a s s o c i a t e d w i t h a s i n g l e quantum p a r t i c l e i n a magnetic f i e l d .  I t might be supposed t h a t (1.16) i s  an i d e n t i t y , o r a t most, a n o r m a l i z a t i o n c o n d i t i o n . i t s most p r o f o u n d s i g n i f i c a n c e  i s t h a t the G-L  wave f u n c t i o n " behaves under quantum o p e r a t o r s c o n s i s t e n t with r e a l Schrodinger  However,  "effective i n a fashion  wave f u n c t i o n s .  I t i s not  17 s u r p r i s i n g t h a t e q u a t i o n (1.16) i s o b t a i n e d by m i n i m i z a t i o n ; the S c h r o d i n g e r e q u a t i o n can a l s o be o b t a i n e d by a m i n i m i z a t i o n p r o c e d u r e , once the c o r r e s p o n d e n c e  of operators i s  asserted. When the e q u i l i b r i u m c o n d i t i o n  (1.16) i s f u l f i l l e d  t h e boundary c o n d i t i o n (1.15) and the e x p r e s s i o n , f o r f r e e energy  7  forms.  F i r s t we  where  (1.12) can be s i m p l i f i e d and t a k e on i n t e r e s t i n g  f = 1  , (7\(- )  ^[st)  v  rL  =  (  + /  2  Substituting in  separate  ^  e  i n t o i t s a m p l i t u d e and phase  U.  are r e a l .  *y  x2  (j) — *7 €  CA  Then  v<?s) e ,  /  "1 \<-  i n t o the second  '  G-L  (  1  equation gives  equilibrium  where  r e f e r s t o the super c u r r e n t d e n s i t y w i t h i n the  I s o l a t e d superconductor. density i s zero.  I n e q u i l i b r i u m the normal c u r r e n t  -  1  7  )  18 Ij) — ^ Q  Substituting tion  i n t o the boundary  (1.15), d i v i d i n g out the e x p o n e n t i a l , and  r e s u l t i n g r e a l and i m a g i n a r y  (jiH f\<n V^7  •  c?S  s e t t i n g the  p a r t s e q u a l t o 0 we  ify  ~  cb7  ~  condi-  obtain  I  0  0  -  J  I n view of (1.18) the f i r s t component of ^  equation  s t a t e s t h a t t h e r e i s no  normal t o the s u r f a c e w h i c h I s a n a t u r a l  boundary c o n d i t i o n .  The  the a m p l i t u d e of ^  s h a l l have no component of  second e q u a t i o n  which s t a t e s t h a t gradient  normal t o the s u r f a c e cannot be i n t u i t i v e l y j u s t i f i e d m a c r o s c o p i c arguments. microscopic  .  (1 19)  theory  De Gennes (1964) has  by  shown from the  t h a t t h i s boundary c o n d i t i o n h o l d s  at  the i n t e r f a c e between a s u p e r c o n d u c t o r and an i n s u l a t o r (or vacuum). S u b s t i t u t i n g i n the e x p r e s s i o n  (1.12) f o r  ~jf- g i v e s  quite generally  7 -  l-2ft  f  t 2 f'tlft  ) d-i  ^U^P^-th """•  )  I t w i l l be c o n v e n i e n t t o have names f o r each of t h e s e i n t e g r a l s ; we  c a l l the f i r s t  the " c o n d e n s a t i o n f r e e energy"  19  , and' the second the " e l e c t r o m a g n e t i c 3^  I n view o f ( 1 . 1 8 ) ,  .  f r e e energy"  i n e q u i l i b r i u m , "jE^ s i m p l i f i e s  to ZrrX 2  -?  z  j i/  I  _L_  T h i s e x p r e s s i o n f o r the e l e c t r o m a g n e t i c the c l a s s i c a l one, the term i n -g^r energy o f the super e l e c t r o n s .  -  (1.21)  f r e e energy I s j u s t  r e p r e s e n t i n g the k i n e t i c  However, I n the G-L  theory,  t h i s e x p r e s s i o n g i v e s the energy c o r r e c t l y o n l y i n e q u i l i b rium.  I n p a r t i c u l a r , the I n t e g r a l ( 1 . 2 1 )  not m i n i m i z e d  by the second G-L — 0  condition  i n t h i s form i s  e q u a t i o n , but r a t h e r by the  which i m p l i e s H ~M  58 &  G  .  Evidently  the minimum v a l u e of the i n t e g r a n d (and i n t e g r a l ) i s z e r o . On the o t h e r hand, A, — 0  does not i m p l y /9 ~ 0  when a n '  —»  "  e x t e r n a l magnetic f i e l d i s p r e s e n t . A  i n c r e a s e s so  as t o d e c r e a s e  H  i s increased.  The optimum compromise i s the second  equation.  , then  As -4s  i s decreased w h i l e  (H ~~ n ) e  G-L  E v i d e n t l y a London-type e q u a t i o n cannot be d e r i v e d  from the c l a s s i c a l e x p r e s s i o n f o r e l e c t r o m a g n e t i c  energy.  2  CHAPTER  SIMPLY-CONNECTED SUPERCONDUCTORS  S e c t i o n 1.  S i g n i f i c a n c e o f Simple C o n n e c t i v i t y  I n t h i s c h a p t e r we p r e s e n t a v e r y g e n e r a l d i s c u s s i o n o f phase changes i n s m a l l s i m p l y - c o n n e c t e d v i a t e d t o s-c) s u p e r c o n d u c t o r s .  (hereafter  abbre-  A s-c body i s one which  has no h o l e t h r o u g h I t , though i t may have t o t a l l y  enclosed  b u b b l e s w i t h i n i t - a s i m p l e t e s t i s t h a t i t cannot be hung up c n a s t r i n g .  A h o l l o w sphere o r s p h e r i c a l s h e l l I s s i m p l y -  c o n n e c t e d , w i t h one h o l e i n t h e w a l l i t i s s t i l l  s-c, but  w i t h two h o l e s • t h r o u g h t h e w a l l i t i s d o u b l y - c o n n e c t e d .  In  l a r g e s u p e r c o n d u c t o r s , t h e body may s p l i t i n t o r e g i o n s o f normal and super s t a t e .  I n t h a t case t h e c o n n e c t i v i t y  t o t h e super r e g i o n and not t h e whole body. t h i s c o m p l i c a t i o n does not o c c u r .  refers  I n s m a l l bodies  Some o f t h e r e s u l t s  p r e s e n t e d here have p r e v i o u s l y been d i s c u s s e d o n l y f o r p a r t i c u l a r g e o m e t r i e s , e.g. f o r t h i n f i l m s . s e r v e as a b a s i s o f comparison multiply-connected  geometries.  20  A l s o these r e s u l t s  f o r the d e t a i l e d a n a l y s i s o f  21  C o n s i d e r t h e second G-L i n which the a m p l i t u d e explicitly. ^ T T  Let  A" 2  fiT  *J  e q u a t i o n i n the form  and phase (71  p  of  R e a r r a n g i n g t h e terms t h i s may rr \f H , \7c7) -= + Ai f -_ /TAf//^ i  (1.18)  are w r i t t e n  be  written  £c  v  ^  us t a k e the f o l l o w i n g l i n e i n t e g r a l around a c l o s e d  contour  C  l y i n g w h o l l y w i t h i n the body.  t h i s q u a n t i t y the " f l u x o i d , " symbol  <fic  London has named  .  (2.1)  The f i r s t  term o f the i n t e g r a n d i n the f i r s t  meaning o u t s i d e the s u p e r c o n d u c t o r .  However  l i n e has A  no  i s defined  —?  I n a l l space, and t h e c o n t o u r i n t e g r a l o f  A  can be  trans-  formed t o a s u r f a c e i n t e g r a l over any s u r f a c e bounded by the c o n t o u r , whether or not i t l i e s w i t h i n the body.  p  \ 2.  —->  'c  where <y6 bounded by  i s t h e t o t a l magnetic C .  f l u x c r o s s i n g the s u r f a c e  S  I n case the c o n t o u r l i e s deep i n the i n t e r i o r  of a l a r g e body h a v i n g an almost complete  Meissner  effect,  22  the c u r r e n t i s n e g l i g i b l e the f l u x t h r o u g h Since the body, 0\  0\  and t h e f l u x o i d v e r y n e a r l y e q u a l s  C . <j)  and y ^  and ^7<T\  must be s i n g l e  s i n g u l a r i t y o f 0\  a r e everywhere c o n t i n u o u s i n  7  are also continuous.  I n a s-c body  valued, since the c o n t r a r y i m p l i e s a w i t h i n t h e body.  (In multiply-connected  b o d i e s t h i s i s n o t t r u e s i n c e (7) may i n c r e a s e by i n t e g r a l m u l t i p l e s o f 277* hole.)  a l o n g a c l o s e d p a t h g o i n g once around t h e  Consequently  i n a s-c body:  'C o n r l  cA  0  sss  ' c  fnr>  p>v£>r>v  n. n n  '  f: m  1  r> .  "  T h i s  -  T ' p s n ' l f.  i s  n p r f PP.tl v  -. - -  g e n e r a l as t o s i z e and shape o f the body, o r s p a t i a l t i o n of ^  , remembering t h a t ^  varia-  o c c u r s under t h e i n t e g r a l  sign i n the d e f i n i t i o n of  <p  So f a r n o t h i n g has been s a i d about t h e gauge o f A .  The gauge of. A ' and t h e phase ^  pendent.  London  (1950)  of W  are not i n d e -  has i n t r o d u c e d a s t a n d a r d gauge which  he shows i s u n i q u e l y d e f i n e d by  V  XA  —  H  Aj_ ~ 0 where  V'A  " 0  everywhere  on t h e s u r f a c e o f t h e s u p e r c o n d u c t o r Aj^  i s t h e component  of  A  normal t o t h e  23  surface.  This p a r t i c u l a r  gauge i s now r e f e r r e d t o as t h e  "London gauge." Let  A  be i n the London gauge.  Then t a k i n g  the d i v e r g e n c e o f b o t h s i d e s o f t h e second G-L e q u a t i o n ( 1 . 1 8 )  we o b t a i n  V  CD  0  —  From the boundary c o n d i t i o n  everywhere i n the body. ( 1 . 1 9 )  we have  everywhere on t h e s u r f a c e o f t h e body. it  follows that  t h a t 0.  0  By a s t a n d a r d p r o o f  everywhere i n t h e body, so  i s a t most a m e a n i n g l e s s c o n s t a n t which may be  set equal t o 0 . real.  7cn = 0  VcT> • c Z ? =  Hence  A  i n t h e London gauge i m p l i e s <p  I n t h e remainder o f t h i s c h a p t e r (and except f o r  the i n t r o d u c t o r y p a r a g r a p h s o f Chapter 3, i n the r e m a i n d e r of t h e t h e s i s ) ,  A  i s always i n t h e London gauge.  emphasize t h i s by u s i n g ^  instead of ^  We  and we s h a l l a l s o  remind t h e r e a d e r from time t o time when t h e gauge i s s i g nificant . I n the London gauge t h e two G-L e q u a t i o n s f o r a s-c body t a k e t h e form:  (2.2)  —>  (2.3)  I  24  S e c t i o n 2.  Formulations In  of t h e E l e c t r o m a g n e t i c  F r e e Energy  o r d e r t o c o n s i d e r c r i t e r i a f o r phase change i n  a s-c s u p e r c o n d u c t o r  one must f i r s t c a l c u l a t e t h e t o t a l f r e e  energy. . I n p r i n c i p l e , t h i s i s done by s o l v i n g the two equations  simultaneously f o r  s u b s t i t u t i n g i n equation  and  (1.20) f o r  and t h e n  .  The second term  i n t h i s e q u a t i o n which we c a l l t h e e l e c t r o m a g n e t i c energy  ~7H  where  G-L  free  is  i s the volume o f t h e body i t s e l f and  Is the  volume of- a l l space e x t e r n a l t o the body, i n c l u d i n g any h o l e s t o t a l l y e n c l o s e d by the body.  i s t h e super  c u r r e n t d e n s i t y i n t h e body o n l y , and does not i n c l u d e t h e currents producing  the e x t e r n a l f i e l d  H/ e  e  ,  We  consider  now some d i f f e r e n t f o r m u l a t i o n s of t h i s i n t e g r a l f o r  Jl  A l l t h e f o r m u l a t i o n s a r e v a l i d f o r a s-c body i n e q u i l i b r i u m . In  equation  i s 0 everywhere i n A  and  A  g  V  ( 2 . 4 ) , the second term i n the i n t e g r a n d p  ,  H ~  a r e everywhere c o n t i n u o u s  London gauge f o r the body. term :  We put  VXn  and  H  E  — V X A  and a r e b o t h i n t h e  P a r t i a l l y i n t e g r a t i n g the f i r s t  E  25  (2.5)  V,  S  The  surface  S  off  at least  as  the  first  i s at i n f i n i t y ,  2  integral, i n  V  , VK H  2  V_  JL  . V x l  -  "  since  e x t e r n a l to  (mil y  l  V  n  equation  third  H  .  The  %TT  (2.3)  Oi  -in  development d e v i a t e s have  3  a n  t h e sum  (For f u t u r e reference  not t r u e f o r m u l t i p l y - c o n n e c t e d at t h i s  simply  bodies,  point.)  integral.  zn>t^) fi  c*"?  —^-f-  .  e  entirelv  z  -  H  bv -  reduces to  +  f o r s-c b o d i e s ,  terms i s z e r o ,  We  to the  Kir V^H  on  the f i e l d  n - n n i i u c e d  integral  In  the  t o remain unchanged  ,/f _ inn  Using the Maxwell equation  is  , since  e  contributes nothing  =G  falls  i s 0.  integral  i n t r o d u c t i o n of the superconductor i n t o  currents  G-L  the i n t e g r a n d  ~ VX//  are c o n s t r a i n e d  Hence t h e v o l u m e Tn  and  so t h e s u r f a c e  external currents the  z  d  the  o/V j  second  of the f i r s t we and  note that the  and this  analogous  26  (2.6)  = - J- // 2c  In  obtaining this  Z F.-KUM Uv .  A  e  s i m p l e e x p r e s s i o n , we h a v e made no assump-  tions  or approximations  being  careful  on t h e s p a t i a l dependence o f ^  t o keep i t under t h e i n t e g r a l  sign.  Further-  m o r e , s o f a r we h a v e made n o a s s u m p t i o n s o n t h e s p a t i a l of  5  H  i s _ u n i f o r m , H^, d e p e n d s o n l y o n t h e f i r s t  g  moment o f t h e c u r r e n t . a constant.  triple  scalar  In this  The f i n a l .  c a s e we c a n p u t /A —  Then, u s i n g t h e c y c l i c p r o p e r t y  e  of the *  integral  =-^j^f.dv  i s just  the d e f i n i t i o n  -  (2 7)  o f m a g n e t i c moment  Hence  ; _  —  Since the currents are i n a diamagnetic relative  ff &sl/Z  product  I,-^f^(Z^)^ /9(  form  H , i t n e e d n o t be u n i f o r m , e If  H  ,  to  E  Q  and ^  i s positive.  (  2  .  8  )  s e n s e , 7P£ i s n e g a t i v e The g e n e r a l  validity  27 o f t h i s r e s u l t has not p r e v i o u s l y been made c l e a r .  This  f o r m , however, i s d e c e p t i v e l y s i m p l y - o n l y on t h e a p p r o x i mation of ^  u n i f o r m , and f o r s i m p l e geometry, can 0^ be  c a l c u l a t e d i n c l o s e d form. Another i n t e g r a l which t u r n s out t o be e q u a l t o is  ~Jtf  (H*-H*-) 7  Zrrk  2.  ^  *  t  \ ' )  (2.9  / s /  o  i  V  To p r o v e t h e e q u a l i t y , we adopt a p r o c e d u r e s i m i l a r t o t h a t f o l l o w i n g equation A , A  g  Jv,,v  (2.4). F a c t o r i n g  as b e f o r e , t h e I n t e g r a l become  i n  2  , andu s i n g  Jv,  c  f  /  Then, r e p e a t i n g t h e remarks f o l l o w i n g (2.5), we have  Jv,  I  g1T  C  Y  I  and removing terms w h i c h , by t h e second G-L e q u a t i o n  sum t o  zero, gives  which i s i d e n t i c a l t o (2.6), t h a t I s t o  3^ .  L e t us l o o k f o r t h e m o t i v a t i o n f o r w r i t i n g t h e particular integral  (2.9). C a l l t h e n e g a t i v e o f t h i s I n t e g r a  28  Q-  , where  Cr i m p l i e s t h i s p a r t i c u l a r form o f I n t e g r a n d ,  and not j u s t t h e t o t a l v a l u e .  Then Q- i s a q u a n t i t y of.  energy a s s o c i a t e d w i t h t h e body under t h e c o n d i t i o n t h a t the magnetic moment, t h e e x t e n s i v e magnetic v a r i a b l e , i s h e l d constant  r a t h e r t h a n t h e e x t e r n a l f i e l d which i s t h e  intensive variable.  Gr  i s the Legendre  thermodynamic p o t e n t i a l o f  transformed  > and d i f f e r s from i t by  the p r o d u c t o f t h e i n t e n s i v e and e x t e n s i v e v a r i a b l e s .  =  Gr  7  + H[- H  e  H  Therefore  ~ """  Gr  =  +j  ^  - -  • H  e  (2.4) and  J  =  H  ( 2 . 9 ) , we  j  Or, i f H ' e  iZ°(Z~H)  .  7y  d  C2.io)  V  i s uniform  depends on t h e p a r t i c u l a r  I f t h e d e m a g n e t i z a t i o n approaches 0 t h e l a s t  form i s s i m p l e , s i n c e i n t h a t l i m i t in  for  obtain  Which form we choose t o e v a l u a t e geometry.  H  , which i s j u s t ( 2 . 9 ) .  F i n a l l y , i f we average t h e two e x p r e s s i o n in  J  ' (H-PI )  I n many cases (2.6) o r ( 2 . 7 ) , i f H  approaches 0 e  i s uniform,  are s i m p l e s t , s i n c e they r e q u i r e i n t e g r a t i o n over t h e body only .  29 Section 3.  Order Parameter The  property  of s m a l l b o d i e s which we  shall  here i s the near u n i f o r m i t y of the o r d e r parameter The  exploit  >^  l e n g t h a g a i n s t which the word " s m a l l " i m p l i e s a comparison f  i s the coherence l e n g t h  .  Since  normal t o the s u r f a c e by e q u a t i o n  has  no component  V ^  (1.19) and s i n c e  2  ^2.  5  i s i n v e r s e l y p r o p o r t i o n a l to changes l i t t l e  A < f  , where  d  t h a n d<  A  f ^ X  i  a  , the body may represents  cant d i m e n s i o n of the body. superconductors,  by e q u a t i o n  X  In  be s m a l l  the g r e a t e s t  and signifi-  nd  smallness  i s a stronger  condition  .  the q u e s t i o n a r i s e s how  ^  ),  On the o t h e r hand, i n Type I I  Having e s t a b l i s h e d that  speaking,  (2.2  over d i s t a n c e s s m a l l compared t o  Type I s u p e r c o n d u c t o r s , s t i l l permit  (  i s nearly  t o f i n d i t s mean v a l u e .  the boundary c o n d i t i o n (1.19) s t i l l  , t h r o u g h the r e s i d u a l s p a t i a l v a r i a t i o n .  uniform, Strictly determines However,  t h i s c o n d i t i o n i s r a t h e r i n s e n s i t i v e and d i f f i c u l t I t i s e a s i e r t o assume ^  s t r i c t l y uniform  r e a l ) from t h e s t a r t i n e q u a t i o n g r a d i e n t t e r m drops out and  ^  to  apply.  (and of c o u r s e  (1.20) f o r jF f a c t o r s out of the  .  The integrals,  leaving  (2.11) $7T  J  Vj  X  30  Now we s i m p l y s e t t h e p a r t i a l d e r i v a t i v e o f 77 equal 0 ,  yielding  f A  w.r.t.  '  HSV,  A  J  i s also a f u n c t i o n of ^  2  d  Y  , t h r o u g h t h e second G-L equa-  t i o n , so t h i s i s an i m p l i c i t e x p r e s s i o n f o r ^  which  cannot i n g e n e r a l be s o l v e d e x a c t l y except i n c e r t a i n l i m i t i n g cases. I n p r a c t i c e i t i s u s u a l l y more c o n v e n i e n t t o retain  as a parameter u n t i l t h e c o m p u t a t i o n o f 7  ^  is  c o m p l e t e d , b y one o f the methods p r e v i o u s l y d i s c u s s e d . J~  i s d i f f e r e n t i a t e d w.r.t.  minimizes i t .  ^  to f i n d the  which  I n t h i s case we a r e t a k i n g t h e t o t a l  t i v e w.r.t.  , r a t h e r t h a n the p a r t i a l  However, s i n c e  A  Then  deriva-  derivative.  s a t i s f i e s t h e second G-L e q u a t i o n , w h i c h  i n t u r n makes t h e v a r i a t i o n o f /  w.r.t.  A  z e r o , the  t o t a l and p a r t i a l d e r i v a t i v e s must l e a d t o t h e same r e s u l t f o r *^^ by b o t h methods. e  S t r i c t l y s p e a k i n g a body i s s m a l l i f i t s g r e a t e s t d i m e n s i o n i s much l e s s t h a n  "f  .  Some b o d i e s may be con-  s i d e r e d s m a l l i f o n l y one o r two d i m e n s i o n s a r e s m a l l , p r o v i d e d t h e i r symmetry i s such as t o ensure u n i f o r m i t y o f ^7  i n the o t h e r d i r e c t i o n s .  Thus a p l a n e t h i n f i l m o f  i n f i n i t e extent i s small i f thickness d «  f .  A l ong  31  c y l i n d e r not  necessarily c i r c u l a r , placed p a r a l l e l  external f i e l d , sion  d  shell  of r a d i i  i s small i f i t s greatest transverse  satisfies  On  J l ) , Jl-^  the  , i s not  v a r i a t i o n from pole to the  external  dimen-  other hand, a s p h e r i c a l  will  to equator,  the  s m a l l , even though  since  parallel  to  have a  where the  significant,  polar axis i s  field.  Example It in  i s our p u r p o s e i n t h i s  chapter  t h e most g e n e r a l f o r m p o s s i b l e , and  and  w h e r e any  restrictions  to give  results  to note c a r e f u l l y i f  or approximations  a r e made.  seems, however, d e s i r a b l e t o c o n c r e t i z e the g e n e r a l by w o r k i n g  a specific  example.  We  to f a c i l i t a t e  comparison of d i f f e r e n t  procedures.  claimed  - only f o r the v a r i e t y  which the r e s u l t s  c a n be  Consider  No  a long t h i n J l , «  >  of r a d i u s J l , , placed  external field  H  ft)  9  in  the  }  Z-  equation  .  cylinderical $  , Z  The  the is  o f ways i n  solid  cylindrical  parallel  to a  uniform  c y l i n d e r i s c o - a x i a l to the  co-ordinate  d i r e c t i o n s and  becomes t h e  originality  a  obtained.  super-conductor  g  formulas  choose d e l i b e r a t e l y  s i m p l e w e l l - s t u d i e d geometry i n order  f o r the r e s u l t s  It  ^  system.  Assuming  uniform,  total differential  the  symmetry  second  equation  usual  G-L  32  with  solution 'A,  A  I  where t h e  n  are m o d i f i e d Bessel's  functions,  We p r o c e e d t o c a l c u l a t e t h e magnetic f r e e per u n i t l e n g t h  o f c y l i n d e r by t h r e e o f t h e p r e c e d i n g f o r m u l a s  From (2.4)  7.  H  energy  8TT  From e q u a t i o n (2.7)  33  He A  y  8*1  =  / /  A  0  e^/  x  J  2  rv^)  /o  far/*)  ?  Io(^y/A)  From e q u a t i o n  (2.10)  fir  H  L  e  1  /„ i _  ///.<2,*  Jo ^,?/A)/  J ; 1^/ ? /A )  The e q u a l i t y o f t h e v a r i o u s f o r m u l a t i o n s i s thus d e m o n s t r a t e d f o r t h i s example.  We s h a l l c a l l on t h i s same example  again  l a t e r i n the chapter.  Section 4 .  Order o f Phase  Transition  In zero e x t e r n a l f i e l d y~  ~  ~ tick V|/ ETt  and "J- —  >  0 .  . As  T-*T  i n equilibrium  , ™* ' H  c  5  0  s e e  cb  equation  A l t h o u g h the r e l a t i v e d e n s i t y ^  the d i m e n s i o n a l e l e c t r o n d e n s i t y  ^ ~. I  }  (1.6.)  remains 1,  z  ~ y  —  approaches  34  0 as  T —»T  c  .  Therefore, 3  as  T—»T c  s e c o n d o r d e r phase change o c c u r s . mentally  f o rlarge bodies  i n zero f i e l d ,  I t i s w e l l known e x p e r i -  i n an e x t e r n a l f i e l d  o r d e r p h a s e c h a n g e t a k e s p l a c e when \ H \ e  T  and t h e r e b y  H ^ ( T ) , see e q u a t i o n  F r o m t h e g e n e r a l f o r m o f ~J"  (1.6  also that there i s , i n p r i n c i p l e e  This  having  i s s o , and  a t l e a s t , an i n t e r v a l o f  interval  i s extremely  As we c o n s i d e r s m a l l e r a n d s m a l l e r  interval  increases u n t i l  Y  phase t r a n s i t i o n s  for a certain  a p p e a r s i n Jlj  the value  of ^  a r e second o r d e r . as w e l l as i n  f o r which  ~J"  when  small f o r bodies  critical  When  bodies size  .  —>  all  G-L  about 0 i n w h i c h a second o r d e r phase change o c c u r s  T. i s i n c r e a s e d .  this  ).  t o g e t h e r w i t h t h e second  e q u a t i o n , we c a n show i n a g e n e r a l way why t h i s  H  H ^ ,  o r by i n c r e a s i n g  e  lowering  that a f i r s t  exceeds  e i t h e r b y i n c r e a s i n g \H \  w h i c h may o c c u r  a  3  Hg v ^ 0  , and  ,  consequently  i s minimum i s n o t n e c e s s a r i l y  1.  Throughout t h e f o l l o w i n g d i s c u s s i o n , and i n d e e d  out  the remainder of t h i s  thesis  i n cases  i n which  through*^ i s 1  nearly uniform,  i ti s convenient  t o have a symbol f o r ^  We t h e r e f o r e d e f i n e  The  reader  nition,  i s c a u t i o n e d not t o pass l i g h t l y  f r o m h e r e on *\, w i l l  over  appear w i t h great  this  defi-  frequency.  35  We w i s h t o o b t a i n a q u a l i t a t i v e i d e a o f t h e dependence o f I? tions .  on ^  w i t h o u t a c t u a l l y s o l v i n g the G-L 3^  The form o f the c o n d e n s a t i o n energy  equa-  i s the  same f o r a l l b o d i e s  We t a k e t h e e x t e r n a l f i e l d u n i f o r m , and e v a l u a t e  3^y  by ( 2 . 6 )  of ^  (St )  or ( 2 . 7 ) , both o f which r e q u i r e a knowledge .  From the second G-L e q u a t i o n i n t h e form  (2.3)  fi  and  $  s  ~~~  A  enter only In the combination  A  . Then ( 2 . 1 5 )  p e n e t r a t i o n depth  £ c  For l a r g e ~f  —* 0  large  S  (2.15)  ^  +Tf  &  ,^  .  Put  i s j u s t t h e London e q u a t i o n w i t h  , that i s —o  A  (2.16)  becomes s m a l l .  and c o n s e q u e n t l y H  * /V-  t h e n , we can put t o f i r s t c  A f^  I n the l i m i t <3 ~" °° 9  and ,4  ?  /4.  then  For  approximation  A<  T h i s a p p r o x i m a t i o n i s good f o r cant d i m e n s i o n o f the body.  Ct  , the l a r g e s t  signifi-  U s i n g e q u a t i o n ( 2 . 6 ) we have  36  where  A  i s u n d e r s t o o d t o be i n t h e L o n d o n g a u g e .  g  i n t e g r a l depends o n l y  on t h e e x t e r n a l f i e l d  and geometry  of t h e body, and i s independent o f ^ and A Jtf  i s p r o p o r t i o n a l t o l/S J ^ , ^/A)  i.e.  starts H  portional to ^ As the 8 the  the current  from  $  when  i n a t h i n layer at the surface. l a y e r , but the t o t a l  l a y e r approaches a l i m i t i n g value  field  slope p r o -  2  flows  H  ,  e  the thinner  exclude  Therefore of W  finite  shown by L o n d o n q u i t e g e n e r a l l y  current  .  f o r small values  o f f from 0 w i t h  This  t-xie u o u y , x, «.  i n t h e body o p p o s i t e  to  H  since  0 ,  The  smaller  current i n  i t s job i sto  i t must g e n e r a t e a m a g n e t i c g  .  S —*• 0  As  approaches a s t a t e of i d e a l diamagnetism.  , t h e body  The i d e a l  dia—  —*  magnet r e p r e s e n t s  an upper l i m i t  for given  Consequently as  H  E  monotonically  .  and a s y m p t o t i c a l l y  w h i c h d e p e n d s on  H  E  summary, f o r s m a l l (//$ ) (//S ) z  with }  j£  slope  > < x >  )^>  )  ,  t o w a r d some u p p e r  $ ^  p r o p o r t i o n a l t o HQ  A typical  set of curves of  as  p a r a m e t e r i s shown i n F i g u r e  2-1.  limit, In  i s proportional to , while  becomes h o r i z o n t a l w i t h v a l u e  H  ~Jy  increases  and t h e s h a p e o f t h e b o d y .  to  E  o n f&( a n d h e n c e on  f o r large  proportional with  HQ  37  Having o b t a i n e d i t i s easy to o b t a i n parameter.  as a f u n c t i o n o f as a f u n c t i o n o f ^  ranges from 0 t o 1 ,  As^  -A  with  as  ranges a l o n g  curve of F i g u r e 2 - 1 from 0 t o  One  has  the  set o f curves o f F i g u r e 2 - 2 which have been n o r m a l i z e d a t "J, — I .  as t o g i v e the same A  limit  I n the  p o s i t i v e s l o p e whereas i n the l i m i t step f u n c t i o n .  A ~* 0  finite  i t becomes a  Of course n e i t h e r l i m i t e x i s t s  I t i s convenient  so  mathematical  the curve becomes a s t r a i g h t l i n e w i t h  0 0  the  physically.  t o d e f i n e the "Reduced  Free  Energy"  (2.17) —  pr  Then  F  c  c?77  =  ~z  Z.^. r y  independent o f  H  -2  with slope  rr  __  , a parabola passing through  and h a v i n g a minimum a t ^ — / ,  V/e s h a l l c o n s i d e r F(^-) 4  and  (b) \ s > cL  s  i n the two  and t h e n  A«  cl-  S  ee  Figure 2 - 3 .  the  ^  origin  F =-1 c  shown  2-4a. extreme  cases  (c) d e t e r m i n e the  e s s e n t i a l c r i t e r i o n s e p a r a t i n g t h e s e two Oase (a)  -~j  i s a simple u n i v e r s a l f u n c t i o n of  I n dashed l i n e i n F i g u r e s 2 - 3 a and  (a) X ^ - d ,  Sir  types of  behaviour.  I n F i g u r e 2 - 3 a the s e t  o f n e a r l y s t r a i g h t l i n e s i n the upper quadrant r e p r e s e n t  F^  38  as a f u n c t i o n o f shows t h e sum  2-3b  of  f o r i n c r e a s i n g parameter  F  w.r.t. ^  F = F +FJJ  .  For  F~ f o r J  > 0  minimizes ^  and  H  .  G  Figure  E v i d e n t l y t h e r e i s a minimum  t o the r i g h t of the o r i g i n u n t i l  i n c r e a s e s t o a value at which origin.  H  (^)  H  +2  has s l o p e  E  at the  g r e a t e r t h a n t h i s , t h e r e i s no minimum o f  G  .  I n e q u i l i b r i u m , ^ - t a k e s t h e v a l u e which  F*  .  As  F  approach z e r o c o n t i n u o u s l y .  H  i n c r e a s e s , the e q u i l i b r i u m values of  G  Thus, a t l e a s t  i n t h i s extreme c a s e , t h e r e i s a second o r d e r phase t r a n s i tion. Case (b) d ^ > A at  see F i g u r e 2 - 4 .  3  start  0 and r i s e r a p i d l y t o a. p l a t e a u , t h e h e i g h t o f which i s  p r o p o r t i o n a l t o Hg  .  F +F^ , t h e minimum o f c  of  The curves o f ^ ^  I n F i g u r e 2 - 4 b showing t h e sum F  c  i s superimposed on t h e p l a t e a u  F„ , g i v i n g a l o c a l minimum i n F  at J & I .  F  may  t a k e on n e g a t i v e o r p o s i t i v e v a l u e s a t t h e minimum, i n the l a t t e r case t h e body i s i n a m e t a s t a b l e  s t a t e b l o c k e d from  g o i n g I n t o t h e normal s t a t e by t h e p o t e n t i a l b a r r i e r .  The  p l a t e a u , i s not q u i t e f l a t ; i t s s l o p e as w e l l as i t s h e i g h t increases with  H  .  Eventually  e which t h e s l o p e o f minimum i n F  J  F^  H . reaches e  a value at  overcomes t h e d i p i n F  disappears.  c  , and t h e  There i s thus a c r i t i c a l e x t e r n a l  field at which F j u s t has a h o r i z o n t a l i n f l e c t i o n , and f o r H > H _ t h e r e i s no minimum o f F and no super e e3 s t a t e can e x i s t .  39  (c)  I t i s e a s i e r t o f i n d t h e d i v i d i n g l i n e between t h e s e  two extreme t y p e s o f b e h a v i o u r b e g i n n i n g w i t h case ( a ) , and n o t i n g t h e m o d i f i c a t i o n i n t h e p i c t u r e as A shown i n F i g u r e 2 - 2 , t h e l i n e s o f c u r v e d as t h e c o n d i t i o n  A  d , t h e r e i s some v a l u e o f  H  has s l o p e +2 a t t h e o r i g i n . at  the o r i g i n .  ceases t o be v a l i d . , say  G  At  E  F  q  c  F  has  , t h e r e w i l l be no  0 . But i f fjjj F  (  has zero s l o p e  Now i f i n t h i s case t h e curve o f  curvature at the o r i g i n than H > H  ,  E  F  F o r every  , f o r which  H =H  minimum f o r ^ e ^ ^ e j except at  As  (^) become more and more  l e s s curvature at the o r i g i n than  for  decreases.  l^)  has g r e a t e r  , t h e r e w i l l be a minimum .  . , and t h e type o f b e h a v i o u r o f (b) above w i l l .  o c c u r , a l t h o u g h t h e a c t i o n may take p l a c e much n e a r e r t h e o r i g i n t h a n appears  i n the f i g u r e .  F i n a l l y we summarize i n m a t h e m a t i c a l criteria  f o r phase t r a n s i t i o n s .  The uniqueness  o b t a i n e d by a p p l i c a t i o n o f t h e s e c r i t e r i a p o s s i b l e forms o f  r  ^  d i s c u s s e d above.  form t h e of the r e s u l t s  r e s t s on t h e These c r i t e r i a  been g i v e n by G i n z b u r g (1958) who o b t a i n e d c u r v e s to  F i g u r e 2 - 3 and 2-4 by d i r e c t c a l c u l a t i o n o f  have  similar  F for  p a r t i c u l a r b o d i e s f o r which t h e f i e l d e q u a t i o n can be s o l v e d In  c l o s e d form. I f a second o r d e r phase t r a n s i t i o n o c c u r s , i t  does so a t a f i e l d  H -, g i v e n by  40  A second i f , at the f i e l d  o r d e r phase t r a n s i t i o n does Indeed H  5  , el' n  An e q u i l i b r i u m f i r s t at  the f i e l d  H ^  which  o r d e r phase t r a n s i t i o n  '*-  r  Although the f i r s t F ,  F>0  e q u a t i o n may be s a t i s f i e d a t a  always a t such a maximum.  A metastable where  s t a t e may e x i s t f o r  i s g i v e n by the s i m u l t a n e o u s  • As  H -*>H ^ e  occurs  s a t i s f i e s simultaneously  (%)•> • '° • r° maximum o f  occur,  ^ ^  s o l u t i o n of  _ '""  &)-<> t h e maximum and minimum o f  e  ^ ^€3  f~ ( ^ )  erf \  c o a l e s c e , and • S e c t i o n 5-  n  a  Expansion In  s  a  of  F  i n powers o f  o r d e r t o I n v e s t i g a t e second  i t i s c o n v e n i e n t t o have powers o f ^  horizontal inflection point.  .  F  q  F  order  transitions  i n t h e form o f an e x p a n s i o n i n  i s a l r e a d y i n t h i s form,  -^2^, f"^.  4l  We have a l r e a d y noted t h a t 11  f o r a g i v e n body i s propor-  z  t i o n a l to h . and t h a t w r i t e the e x p a n s i o n of F^ e  as £ 0 . i n the form ^  A "/  where the  A*  2  Hc£{ A  '  J ? A %  -  /.  X  w i l l be j u s t i f i e d  shortly.  The  may  )  (2.22)  6  have the d i m e n s i o n s of l e n g t h and  s  depend on the geometry of the body.  i s given  A  Hence we  The  will  a l t e r n a t i n g signs  second order.' c r i t i c a l  field  by  - z ^ ^  = o  //cl, X The  c o n d i t i o n t h a t a second o r d e r t r a n s i t i o n o c c u r s i s  *  -  \il  e  He,  which t o g e t h e r w i t h ( 2 . 2 3 )  Once  F„ n  J?* i i s obtained  order c r i t e r i a If  g i v e s a c o n d i t i o n on  A  In the above s e r i e s f o r m , the  only : (2.24) second  can be o b t a i n e d by i n s p e c t i o n . F^  i s known i n c l o s e d f o r m , t h e n the s e r i e s  (2.22) can be o b t a i n e d by s t r a i g h t f o r w a r d e x p a n s i o n of i n a McLaurin s e r i e s .  F^  There i s , however, a method by which  the s e r i e s can be o b t a i n e d w i t h o u t  s o l v i n g the f i e l d  equations.  42  Furthermore  t h i s method can be a p p l i e d t o any body, a l t h o u g h  n u m e r i c a l i n t e g r a t i o n may be r e q u i r e d .  C o n s i d e r the e x p r e s -  —7  s i o n f o r Jr g i v e n i n e q u a t i o n ( 2 . 6 )  A  g  i s independent o f ^  series i n  ^  as a power  &  t h i s e x p r e s s i o n w i l l g i v e t h e r e q u i r e d ex-  3  pansion of  , so i f we e x p r e s s  F„ . ri  As ^  —  —S>  potential  p  0 , the supercurrent  —* 0 and t h e v e c t o r  —9  . We assume A and A e e London gauge f o r the body. We may w r i t e ^ A— A ?  expansions  I n powers o f y  are both i n the and  A  as s e r i e s  , w i t h terms o f the s e r i e s numbered  a  by s u b s c r i p t s i n p a r e n t h e s e s ,  where^^j  , A^j  are f u n c t i o n s o f p o s i t i o n . plane p a r a l l e l to H  For a plane f i l m i n the X ^ }  A Q  ~i^ \z  3J  £  .  Q  l^ly  ,  F o r a body h a v i n g a x i a l symmetry about  an a x i s p a r a l l e l t o t h e f i e l d , but o t h e r w i s e o f a r b i t r a r y —P  shape,  A  *.  i s i n the  g  c ' - d i r e c t i o n of a c y l i n d r i c a l co-ordinate  system c e n t r e d on t h e a x i s X  =  ~?  \^e\  9  (2.26)  43  F o r o t h e r shapes  A  g  can be found by a d d i n g g r a d i e n t s o f  s c a l a r s s a t i s f y i n g L a p l a c e ' s e q u a t i o n t o one o f the above e x p r e s s i o n s i n such a way as t o s a t i s f y the boundary c o n d i tion  A  dS  a  e  S u b s t i t u t i n g the above expansions  i n the second G-L  equation  i n the form  f*  ~^TT  =  h  and e q u a t i n g c o e f f i c i e n t s o f t h e same powers o f ^  on e i t h e r  side i>  r  fir  o>  _  — -JL_  )f  Ao>  etc.  T h i s g i v e s a l r e a d y the f i r s t term o f ^  .  From M a x w e l l ' s  e q u a t i o n we have  c Using ^  S7°A - 0  , s u b s t i t u t i n g the s e r i e s e x p a n s i o n s  and e q u a t i n g c o e f f i c i e n t s o f e q u a l powers o f ^ <  etc  for A ,  Since  i s a known f u n c t i o n , the components o f /4 f)>  e q u a t i o n f o r the  I n t h i s way  we  each s e r i e s . for  P  ri  TT  .  series for  may  obtain  Substitution  f i r s t l i n e i s Poisson' , with  solution  a l t e r n a t i v e l y the next term o f i n (2.25) t h e n g i v e s the  E x a m i n a t i o n of t h i s p r o c e d u r e shows t h a t  F^  series the  is alternating.  Example  C o n s i d e r a g a i n the  a x i a l symmetry  i s g i v e n by  T a k i n g advantage of 6  which has  the  l o n g t h i n c y l i n d e r of  and  Z  radius  (2.26). Then symmetry, we  have  s o l u t i o n s a t i s f y i n g the boundary  conditions  45  Consequently He  f  S u b s t i t u t i n g i n t o equation  I  (2.25)  for  3^ f o r u n i t l e n g t h  of c y l i n d e r we have c o r r e c t t o two terms  3SV-A*  6 ^ A*  ^  I t i s i n t e r e s t i n g t o note t h a t t h e f i r s t term i s p r o p o r t i o n a l t o t h e moment o f i n e r t i a o f t h e body. f r e e energy  r #  F o r t h e r e d u c e d magnetic  o J f y / /7c6 J*-i  z  from which we i d e n t i f y  L e t us compare t h i s w i t h t h e c l o s e d e x p r e s s i o n obtained  on page 3 2  f o r F^  4 6  •I  (i,llZ\)  Met  is  I  (Zr-k)l  k!  k =o  zk k!  k!  1  then  /6 A  2X  He, ' 7  1  2  and. t h e c o n d i t i o n t h a t a second o r d e r t r a n s i t i o n does occur i s  i n agreement w i t h G i n z b u r g  (1958).  We now r e t u r n t o the d i s c u s s i o n o f the e x p a n s i o n of  F  f o r a body of a r b i t r a r y shape.  only serves to evaluate the  F  i n the l i m i t ^-—^0  approximate b e h a v i o u r of  % ^ 0  •  For  °y  T h i s e x p a n s i o n not  sufficiently  F  , but d e s c r i b e s  i n the neighbourhood  of  s m a l l , we can approximate  by the f i r s t few terms o f the s e r i e s .  47  (  If  y  |—  6  V V ) z  [  * I <0  0^/4  and  ^ //  e/  , two terms w i l l do  which has a minimum a t  I  \Hd '»  / /  \  rich A  at which  F  -  -  S ~  W\  :  •—  /J.  H;I £ X  _  7/ T  l  H^ALAI.  S i n c e // _ \  {J*L£_7 f  far  » V^de-  / remains f i n i t e as /TL — * H^i >  2.  creases l i n e a r l y t o 0 w i t h  .  F  i s always n e g a t i v e ,  2  and  F e  q(^q)  behaviour  goes as ~ a s ^ - * ^ .  o f a second o r d e r phase I f (j —  approximate  ikJiJ*-)/Q  when  T h i s i s the t y p i c a l transition. H =H  , t h e n we must  by t h r e e terms o f t h e s e r i e s .  valid f o r situations  i n which t h e phase t r a n s i t i o n  f i r s t order, that i s , i t occurs at very s m a l l ^ t h r e e terms :  This i s i s barely .  To  48  F^-)  i s a cubic p a s s i n g through  the o r i g i n , having a maxi-  a t 'J-" max $ max and<f max <f / eVq  F  mum o f say  F  J  F max > Feq  /  r  variable  H  H  .  at  eq The dependence o f FMa' ^  / eq on t h e  has the g e n e r a l c h a r a c t e r o f F i g u r e 2-4b.  e  We can observe as  and a minimum o f  t h i s by n o t i n g the m i g r a t i o n s o f t h e extrema  i s increased.  For  , o n l y '•V ,  n  l  y  i s to the  r i g h t o f t h e o r i g i n and has p h y s i c a l s i g n i f i c a n c e .  F  < 0 eq  and r e p r e s e n t s a s t a b l e s u p e r c o n d u c t i n g At ^  i a i i  As  z s :  He  0  state.  H =H , ( d e f i n e d by e q u a t i o n ( 2 . 1 9 ) ) ,  , but  F  =0.,  < 0 s t i l l r e p r e s e n t s a unique minimum s t a t e  i n c r e a s e s t h e maximum moves up t o the r i•g h t flrnoL*. 'JiL ">D,'  F~rnax ^ ®  > representing a potential barrier.  t i n u e s t o r i s e v/ith  F  g a  con-  , and becomes zero a t a f i e l d e ' H e2„ • g i v e n by , t h e v a l u e o f He which s a t i s f i e s  H „ e2  H  i s the e q u i l i b r i u m f i r s t order t r a n s i t i o n ^  field.  If H > H „ e e2  metastable  then  super s t a t e .  F  eq  critical  > 0 , and the system i s i n a 3  J  I n p r i n c i p l e , i f we w a i t l o n g  enough, i t w i l l go over t o t h e normal s t a t e which has lower f r e e energy. F i n a l l y , at a f i e l d value of  H"  e  which  satisfies  H^ >H^ e  e  > g i v e n by t h e  19  _; WjWtf)  ' h i l l  3 .tfci'A  7( Hd A  1  z  6  /  I,  H / J X ]  (  /&fX* J  2  the two extrema c o a l e s c e , and t h e r e i s a h o r i z o n t a l t i o n i n the curve.  Section 6 .  ^e2  3  The the  , no super s t a t e can e x i s t .  equations f o r the v a r i o u s c r i t i c a l  ^e3  3  temperature  X  e  Temperature Dependence  The ^el  At H >  inflec-  Involve the f a c t o r s  fields  and X , whose  dependence i s w e l l d e s c r i b e d by  —  A  0  /(t  ~ t*)  where t=T/T  (2.27)  c  e q u a t i o n s f o r t h e c r i t i c a l f i e l d s a r e thus l i n e s i n H  e  - t plane.  When  H  ,t  are.caused  t o vary along  any p a t h whatever i n t h e p l a n e w h i c h c u t s one o f t h e s e l i n e s , some change o c c u r s i n t h e s t a b i l i t y o f one phase w i t h r e s p e c t t o the other. The  second o r d e r c r i t i c a l  field  H ^=H ^A/^/f o r w r i t i n g i n t h e temperature e  i s g i v e n by dependence e x p l i c -  itly H  e i  (  t ]  = Mxl*  ( 2  .  2 8 )  50  Its  slope i s  which i s 0 a t The  phase o r d e r - d e t e r m i n i n g c r i t e r i a , namely  ^ A IZ dent.  t = 0 and i n f i n i t e a t . t = l .  f  o  r  second o r d e r , i s a l s o temperature  I n g e n e r a l t h e r e i s a range o f  o r d e r changes o c c u r . t  t  depen-  f o r w h i c h second  The " o r d e r - d e t e r m i n i n g  temperature"  which i s t h e lower bound o f t h i s i n t e r v a l i s g i v e n by  If  t  ^ t  If  "t ^ i  I f \ -fl/ 0  a first  o r d e r change o c c u r s as  H"  a second o r d e r change o c c u r s as then  z  t * = 0 , and f o r  t i o n s a r e second o r d e r .  increases.  e  H"  increases.  e  a l l phase t r a n s i -  On t h e o t h e r hand as  t —*» 1 ,  A  i n c r e a s e s w i t h o u t l i m i t , so t h e r e I s always some range o f t  near  1  i n which second o r d e r t r a n s i t i o n s  Corresponding field  (t )  H  H* = ^H  If  e  If W ^ H e  e  to  t  , t h e r e i s an o r d e r - d e t e r m i n i n g  , w h i c h by e q u a t i o n  H u*) 0l  a first  occur.  (2.28)  i s found t o be  -  = HH ^-(,-e f 00  <2 30)  z  o r d e r t r a n s i t i o n o c c u r s as  a second o r d e r t r a n s i t i o n o c c u r s as  t  i s raised t  i s raised  51  For a l l b o d i e s h a v i n g t h e same shape, b u t d i f f e r ent s i z e , J^x^/ for  H  (t )  g  ^  s  a  Equation (2,30)  numerical constant.  i s a l i n e i n the H  - t  plane belonging t o  a p a r t i c u l a r shape o f body, which s e p a r a t e s t h e p l a n e two p a r t s . first  into  A l l t r a n s i t i o n s o c c u r r i n g i n s i d e t h i s curve a r e  o r d e r , and a l l t r a n s i t i o n s o u t s i d e t h e c u r v e a r e  second o r d e r .  T h i s c u r v e has f i n i t e n e g a t i v e s l o p e a t  t = 1 , so t h a t some p o r t i o n o f a l l t h e He l, 1  curves  1  to the r i g h t of t h i s d i v i d i n g l i n e . entire  H , el  For ^  curve l i e s t o the r i g h t . &  ^ X / fz  Curves o f  lies }  the  H , (t) el  are" shown i n F i g u r e 2 - 5 .  for various  H , ceases t o e x i s t as a second o r d e r phase el ^ t r a n s i t i o n boundary a t equation H ^(t)  £  g  o r  t  .  The c u r v e d e f i n e d by t h e same  t < t  has t h e meaning o f t h e m i n i -  mum f i e l d i n which a m e t a s t a b l e normal s t a t e c a n e x i s t . As  t  d e c r e a s e s below  curve s p l i t s  t  , t h e second o r d e r t r a n s i t i o n  i n t o three branches:  a t i o n of  the mathematical c o n t i n u -  which i s t h e " s u p e r - c o o l i n g " c u r v e , t h e e q u i l -  i b r i u m phase t r a n s i t i o n c u r v e h e a t i n g " curve  H (t) q  H^ ( t ) ,  which i s t h e maximum f i e l d i n which  the m e t a s t a b l e super s t a t e can e x i s t . . i n powers o f ^ of  t .  and t h e extreme "super-  Our e x p a n s i o n o f  i s v a l i d i n some neighbourhood  The e q u a t i o n s f o r H „ , H _ e2 ' e3  quadratics i n H ^ e  and  >  a n <  ^ ^  t o the l e f t  can be s o l v e d as  1  e  F  temperature  52  dependence w r i t t e n i n .  I t i s more i n f o r m a t i v e t o e s t a b l i s h •x  the f o l l o w i n g p r o p e r t i e s o f t h e s e curves near 1)  H  :  H _ ( V ) = H . (t ) e3 el  (t )  e 2  t  A l l curves are continuous at t 2)  The s l o p e s o f t h e c u r v e s a r e a l s o c o n t i n u o u s a c r o s s We s h a l l work w i t h  , the proof i s i d e n t i c a l f o r  D i f f e r e n t i a t i n g b o t h s i d e s o f eq. (2.26) f o r w.r.t.  t  H  g 2  t  Het. *&i  £j  J-i  f  Ah" A  dt  d  jMe-Z I HcL  2  -^2.  \*  The LHS and t h e f i r s t term on t h e r i g h t a r e 0 by t h e d e f i n i t i o n s of  dt  But  t  and  ( Hf  H  \  c  Z  Hence  1  also  d dt  I  -ii  Hei  \ Hcb  2  A  2  Expanding t h e l a s t two e q u a t i o n s and s e t t i n g them e q u a l y i e l d s c l H d t  t  M dt  (2.32)  53 3) We p r o c e e d t o c a l c u l a t e the second d e r i v a t i v e s a t t D i f f e r e n t i a t i n g b o t h s i d e s o f (2.. 31), the  terms which a r e z e r o at 2  Z  Hez  dt  _/_  obtains  ,2  (2.33)  A Z  Z  H ^ ( t ) we have  /I  (2.34)  0  Ue^A 1  d f  one  H-Hcb A ^  f Whereas f o r  Mei  •x  and l e a v i n g out  E x p a n d i n g the LHS of t h e s e l a s t two e q u a t i o n s , and usin< the  (2.32) above, a l l terms w i l l be e q u a l e x c e p t  result  those c o n t a i n i n g  c  ^ ie/ ' e  and  fez fx 2-  . S u b t r a c t i n g (2.34)  from (2.33) ~d rlz>~ Z  di  df  x  L  J  S i n c e t h e RHS  cfH&) df ^„ H e2  ei  Jz  t  if  (/-t*T  o f t h i s e q u a t i o n i s always p o s i t i v e , and  i s n e g a t i v e as seen i n P i g . 2-5, the c u r v e , o f  has l e s s convex c u r v a t u r e t h a n  H , and l i e s above i t . el n  5  By a s i m i l a r c a l c u l a t i o n , one shows t h a t above the  H  since  e2  to the l e f t of t  curves of  Hence we may  H^  sketch i n  H „ and H „ I n the neighbourhood of t  e2  e3  These are shown i n F i g u r e 2-6.  lies  Figure 2-1 F u n c t i o n a l dependence o f Magnetic Free Energy 7 on , where S i s t h e e f f e c t i v e penetration depth, f o r increasing values of uniform external f i e l d H . H  Q  F i g u r e 2-2 F u n c t i o n a l dependence o f Magnetic Free Energy on r e l a t i v e super e l e c t r o n d e n s i t y ^ , f o r values of A f r o m A « oL t o \»c£ .  F i g u r e 2-3 Reduced Free Energy f o r s m a l l body. (a) Separate curves of condensation energy F (dashed) and e l e c t r o m a g n e t i c f r e e energy FH (solid) at increasing external field K » (b) T o t a l f r e e e n e r g y . Dashed c u r v e shows l o c u s o f e q u i l i b r i u m s t a t e s l e a d i n g t o 2nd o r d e r t r a n s i t i o n . c  e  F i g u r e 2-kReduced Free Energy F ( ^-) f o r l a r g e body. (a) Separate curves of condensation energy F (dashed) and e l e c t r o m a g n e t i c f r e e energy F^ ( s o l i d ) a t increasing external field H . (b) T o t a l f r e e e n e r g y . Dashed c u r v e shows l o c u s o f e q u i l i b r i u m states leading to 1st order t r a n s i t i o n . c  e  He  0  0  { =  T/n  F i g u r e 2-5 P l o t s o f H ] _ / H -b a s f u n c t i o n s of t f o r different « Dashed l i n e i s l o c u s o f o r d e r - d e t e r m i n i n g p o i n t f o r all long cylinders. e  c  F i g u r e 2-6 S p l i t t i n g o f 2nd o r d e r t r a n i t i o n curve into three branches a t t * . Dashed l i n e as i n F i g u r e 2-5  CHAPTER . 3  MULTIPLY-CONNECTED SUPERCONDUCTORS  In t h i s c h a p t e r and c o n t i n u i n g t o t h e end o f t h e t h e s i s we d i r e c t  the d i s c u s s i o n p a r t i c u l a r l y t o m u l t i p l y -  connected s u p e r c o n d u c t o r s .  I n order that t h i s  discussion  may be s e l f c o n t a i n e d we s h a l l b r i e f l y r e p e a t from time t o time some o f t h e f o r m u l a s and d e f i n i t i o n s The  already given.  f i r s t part of t h i s chapter Is a g e n r a l d i s c u s s i o n of  multiply-connected superconductors without f u r t h e r r e s t r i c t i o n as t o t h e i r shape.  I n t h e l a t t e r p a r t we c o n s i d e r i n  more d e t a i l t h e s i m p l e s t g e o m e t r i c shape o f d o u b l y - c o n n e c t e d superconductor, the i n f i n i t e l y long r i g h t  circular  f o r w h i c h we o b t a i n f i e l d e q u a t i o n s f o r a r b i t r a r y outer r a d i i .  cylinder i n n e r and  F o r t h e most p a r t we r e f e r s p e c i f i c a l l y t o  d o u b l y - c o n n e c t e d b o d i e s , t h e g e n e r a l i z a t i o n t o h i g h e r conn e c t i v i t i e s w i l l u s u a l l y be o b v i o u s .  The a r c h e t y p a l d o u b l y -  connected body i s the t o r u s which may be kept i n mind f o r v i s u a l i z i n g the d i s c u s s i o n .  To a v o i d r e p e a t i n g t h e l o n g  phrase " d o u b l y - c o n n e c t e d body" we s h a l l use t h e word  54  "ring"  55  not i m p l y i n g t h e r e b y any p a r t i c u l a r shape except  t h a t the  body has one h o l e t h r o u g h i t . L e t us w r i t e as b e f o r e the G-L f u n c t i o n i n the f o r m  ij) = ^ Q  e f f e c t i v e wave and CT\  where ^  are  real  —*  continuous body.  f u n c t i o n s of /I  (jJ  d e f i n e d everywhere i n the  must be c o n t i n u o u s  (7\  and s i n g l e v a l u e d .  must  be c o n t i n u o u s  but need not n e c e s s a r i l y be s i n g l e v a l u e d  s i n c e t7*  and  <7)-f*2nTT g i v e i d e n t i c a l  integer.  Indeed <7\ a t a p o i n t  as  P  P  (JJ where  may  n  i s an .  increase continuously  moves a l o n g a c o n t o u r around the h o l e , so l o n g as  i t i n c r e a s e s by e x a c t l y position.  2nTf when i t r e t u r n s t o i t s o r i g i n a l  However, i n g o i n g around any c o n t o u r i n the  body not c i r c l i n g the h o l e , cn  must r e t u r n to i t s o r i g i n a l  v a l u e j u s t as i n a s i m p l y - c o n n e c t e d  body.  ments t a k e n t o g e t h e r show t h a t i f G> a l o n g one  i n c r e a s e s by  state2nff  c o n t o u r c i r c l i n g the h o l e i t must i n c r e a s e by  a l o n g every c o n t o u r c i r c l i n g the h o l e . f) — 0  valued then o f the G-L  These two  and v i c e v e r s a .  theory a superconducting  If_ G\  i s single  W i t h i n the framework r i n g may  be i n any  o f an i n f i n i t e number of d i s c r e t e s t a t e s l a b e l l e d by The  label  n  2nT7*  i s a p r o p e r t y of the h o l e .  one n .  I n a more h i g h l y  c o n n e c t e d body the s t a t e i s l a b e l l e d by a s e t o f  n^ ,  one f o r each h o l e . Let us i n v e s t i g a t e the s i g n i f i c a n c e of equation  n .  (2.1) f o r the f l u x o i d we have f o r a c o n t o u r  Prom C  56  g o i n g once around the h o l e :  ( 3 . 1 )  We  a r b i t r a r i l y choose one  Then the p o s i t i v e by  C  sense o f  sense o f the c o n t o u r as H  positive.  a c r o s s the s u r f a c e bounded  i s t h a t o f the magnetic f i e l d produced  c i r c u l a t i n g around the r i n g i n the p o s i t i v e  by c u r r e n t  sense.  If  t h e r e i s an e x t e r n a l f i e l d t h r e a d i n g the c o n t o u r , we choose the p o s i t i v e positive. positive  usually  sense so t h a t the e x t e r n a l f i e l d i s  With r e s p e c t t o t h i s c o n v e n t i o n , <p  may  c  be  or n e g a t i v e . (7\  If  I n c r e a s e s by  2n7t  i n g o i n g once around  the h o l e , the c o n t o u r i n t e g r a l i s j u s t quantity  ch/q  <fi . -fie  ^  0  n / c h )  .  The  i s c a l l e d the quantum o f f l u x o i d , symbol s  a  q u a n t i z e d q u a n t i t y h a v i n g the same v a l u e  H <p- f o r every c o n t o u r s u r r o u n d i n g the h o l e once. 0  q u e n t l y we may  p r o p e r l y speak of the f l u x o i d s t a t e o f the  r i n g as a whole. the v a l u e of electronic  Conse-  Prom e x p e r i m e n t ,  ch/2e  charge.  where The  e  Q  i s found t o have  i s the magnitude of the  effective  magnitude t o the charge of two  <j>  charge  electrons.  q  i s equal i n  57 The  c o n d i t i o n s d e f i n i n g t h e London gauge o f t h e  vector p o t e n t i a l  A  22  g i v e n on page  equally w e l l define  a unique gauge o f A  f o r a multiply-connected  s h a l l always t a k e  i n t h e London gauge.  A  body.  This  We  implies  c e r t a i n c o n d i t i o n s on <7> . T a k i n g t h e d i v e r g e n c e o f t h e second G-L e q u a t i o n  (1.18) y i e l d s V  1  <7)  0  —  everywhere i n  The boundary c o n d i t i o n (1.19) g i v e s  the body.  everywhere on t h e s u r f a c e o f the body.  V<7) *  I f n=0  and (7)  s i n g l e - v a l u e d one may show as f o r a s-c body t h a t everywhere and we may p u t (7) ~ 0 the c o n d i t i o n s on a constant  f o r a p a r t i c u l a r body.  u s u a l form; i f  OMSiJ  H ^  . For given  d e t e r m i n e a unique f u n c t i o n  <T\  The p r o o f  V<7> =  is  0  0 within  <J\(A)  follows the  and cni/t) a r e two f u n c t i o n s  satisfying  a l l t h e c o n d i t i o n s j t h e i r d i f f e r e n c e i s s i n g l e - v a l u e d and hence must be a c o n s t a n t .  The f i e l d  s o l e n o i d a l and c i r c l e the r i n g . cut a c r o s s  l i n e s o f V<T\  Surfaces  are  o f c o n s t a n t <J\  the r i n g meeting the surface o f the r i n g  normally.  C o n s i d e r now a r i n g i n t h e absence o f any e x t e r n a l field.  We s h a l l show t h a t t h e t o t a l l o o p c u r r e n t ^  c i r c u l a t i n g around t h e h o l e I s 0 J  C  ^  when n ^ 0  when  and t h a t  . Forming t h e s c a l a r p r o d u c t o f  b o t h s i d e s o f the second G-L e q u a t i o n rearranging  n=0  I c  t h e terms we have:  (1.18)  with ^  , and  58  T h i s e q u a t i o n i s v a l i d i n a l l space s i n c e ^ zero o u t s i d e t h e body. the M a x w e l l e q u a t i o n  i s identically  I n t e g r a t i n g over a l l s p a c e , u s i n g ^  — VX V XA  i n t h e second  term: •rrX  1  IM  l7  T  2  '  c  J  Z  7  r  h,  Space  On t h e LHS we p e r f o r m term.  a p a r t i a l i n t e g r a t i o n on t h e second  The s u r f a c e i n t e g r a l thus o b t a i n e d over a s u r f a c e  at i n f i n i t y i s zero.  P u t t i n g VxA = H , t h e LHS becomes:  Space  To i n t e g r a t e t h e RHS d i v i d e t h e volume i n t o t h i n s l i c e s bounded by s u r f a c e s  of the r i n g S  o f c o n s t a n t (7)  dV as d l ' ' olS  We may w r i t e t h e element o f volume  i s t h e t h i c k n e s s o f t h e s l i c e and ds  i s an element  o f t h e s u r f a c e o f the s l i c e , t h a t i s o f t h e s u r f a c e constant  (Ts . N o t i n g t h a t  where  S of  VO) i s p a r a l l e l t o dS , we  may w r i t e t h e RHS a s :  zrr The  £  q> V<7)>dl  „  c  i n t e g r a t i o n over  S  yields  I  the t o t a l c u r r e n t , c  w h i c h i n t h e steady c u t t i n g the r i n g . j u s t gives  2 n 77 .  s t a t e i s t h e same a c r o s s any s u r f a c e The i n t e g r a t i o n o f V^J) around t h e r i n g  59  Consequently:  Space  S i n c e the i n t e g r a n d on the LHS i s p o s i t i v e d e f i n i t e , if  n=0  ft =ft 0  = 0  we must have ^  (which i m p l i e s ^ ^ 0  everywhere.  (f>  have the same s i g n .  c  state I  n~=fcO  Conversely i f  a t l e a s t somewhere)  b o t h s i d e s a r e p o s i t i v e and  then  then  Furthermore  I  and  I n summary, i f a body i s i n a  then ( i ) t h e r e i s a p e r s i s t e n t l o o p c u r r e n t  i n z e r o e x t e r n a l f i e l d , and ( i i ) the f l u x o i d o f any  c o n t o u r around the h o l e ( o r " o f the h o l e " f o r s h o r t ) i s n  4  We have s t i l l exist physically.  t o demonstrate t h a t t h e s e s t a t e s  C o n s i d e r a r i n g a t temperature  T> T  i n an e x t e r n a l f i e l d which passes through t h e h o l e . the r i n g i s c o o l e d below  T  c  and the r i n g becomes  c  ,  As super-  c o n d u c t i n g , a l l quantum s t a t e s a r e a v a i l a b l e t o i t .  It  t h e r e f o r e chooses the s t a t e w i t h t h e l o w e s t f r e e energy. T h i s i s not i n g e n e r a l a s t a t e o f zero f l u x o i d , s i n c e t h e r e i s f l u x from  H  t h r o u g h the h o l e .  In general there  be a p e r s i s t e n t l o o p c u r r e n t i n the r i n g i n t h i s  state,  and a p e r s i s t e n t (though d i f f e r e n t ) l o o p c u r r e n t w i l l t i n u e t o c i r c u l a t e i f the e x t e r n a l f i e l d  will  i s removed.  con-  60 The f i n a l p r o o f i s always e x p e r i m e n t a l existence of persistent time and c o n s t i t u t e s  currents  The  has been known f o r some  one o f t h e most d r a m a t i c d e m o n s t r a t i o n s  of s u p e r c o n d u c t i v i t y .  Q u a n t i z a t i o n o f the f l u x o i d has been  o b s e r v e d d i r e c t l y i n t h e now c l a s s i c e x p e r i m e n t s o f Deaver and  F a i r b a n k (1961) and o f D o l l and Naubauer (1961). Having e x t a b l i s h e d  of t h e  n=0  t h e e x i s t e n c e and s i g n i f i c a n c e  s t a t e s , we now proceed t o c o n s i d e r t h e f r e e  energy o f t h e s e s t a t e s .  The i n t e g r a l f o r  can be put  i n a v a r i e t y o f I n t e r e s t i n g and u s e f u l forms when t h e body is i n equilibrium  i n t h e super s t a t e , and (1.18) h o l d s .  From e q u a t i o n (1.21)  w h i c h i s j u s t t h e c l a s s i c a l e x p r e s s i o n f o r t h e magnetic f r e e e n e r g y , c o m p r i s i n g t h e f i e l d energy and t h e k i n e t i c energy o f t h e c u r r e n t . P u t t i n g H ~ V XA A — 0  p  i n V„  ,H  e  ~ VX/Ag and n o t i n g  that  v/e have  2  P a r t i a l l y i n t e g r a t i n g t h e f i r s t t e r m , and d r o p p i n g t h e s u r f a c e i n t e g r a l over a s u r f a c e a t i n f i n i t y , s i n c e t h e  61  i n t e g r a n d drops o f f a t l e a s t as //^  ~*  V „ , V X //  In  2  —o  7" AP  R  5  c  '  3  , s i n c e the sources  e x t e r n a l t o t h e body a r e h e l d c o n s t a n t .  _»  of  H  and  Hence  H e  contri —<>  b u t e s n o t h i n g t o t h e volume i n t e g r a l . V X H  and  - *TTj£  In  V X <V = 0 e  /C 2.  •V,  3  2c  i  c?  1  Z  J  f  —<>  Combining t h e f i r s t  and t h i r d terms f a c t o r i n g out ^  u s i n g t h e second G-L e q u a t i o n  "  J  ZC  Vl  The f i r s t  (1.18)  irr  v  gives  »/  term i n t h e i n t e g r a n d i s p r o p o r t i o n a l t o t h e RHS*  of e q u a t i o n  ( 3 . 2 ) w h i c h we have j u s t i n t e g r a t e d over  .  U s i n g t h e r e s u l t o b t a i n e d t h e r e we have  j  = H  (»*>)lc Zc  _  f  I  J  • ~  Vl  d  (3A)  V  c  So f a r no assumptions have been made about t h e s p a t i a l —*>  p r o p e r t i e s of  H  can put  —p  .  If H  is_ u n i f o r m over t h e body, we S u b s t i t u t i n g this i n (3.4), using  the c y c l i c p r o p e r t y o f t h e t r i p l e s c a l a r p r o d u c t the c o n s t a n t  H  out o f t h e i n t e g r a l  gives  and f a c t o r i n g  62  ^  2c  2  The l a s t i n t e g r a l -7  2c  Jv,  i s j u s t t h e d e f i n i t i o n o f magnetic moment  _ (n<p )Ic 0  _  ±  Tj  fa  m  This form i s u s e f u l s i n c e i t r e q u i r e s i n t e g r a t i o n only  .(3  over  the body. We d e f i n e a n o t h e r q u a n t i t y  Q- , somewhat  arbitrarily  i t may a p p e a r , as  Jvi+\/  2  i  t  '  J  P r o c e e d i n g as b e f o r e , we a r r i v e a t  Comparing t h i s v/ith (3-4)  we have  where G i s t h e above i n t e g r a l  we o b t a i n  - f (j-  (3-6)  Averaging t h i s expression (3-3)  jr^ =  for  v/ith t h a t o f  (3-7)  63  a s i m p l e e x p r e s s i o n which does n o t c o n t a i n the c u r r e n t d e n s i t y under t h e i n t e g r a l .  The c h o i c e o f which form t o use depends  on the geometry o f the body. When S7 (T) - 0 Z  For  A  i s i n the London gauge, (A s a t i s f i e s  everywhere i n the body and V(7) dS O e  on  a body h a v i n g a x i a l symmetry about t h e z  c y l i n d r i c a l c o - o r d i n a t e system i t i s e a s i l y  the s u r f a c e .  =  axis ofa  shown t h a t t h e s e  c o n d i t i o n s a r e s a t i s f i e d by Shirr  and t h a t  c/0  2 rin-  =  ks a l r e a d y n o t e d , t h i s s o l u t i o n i s unique except f o r an arbitrary  meaningless .additive c o n s t a n t .  symmetric body, e q u a t i o n ( 3 - 3 . 1 ) f o r  (?)fio  7.H  Zc  \  ft  T  \  F o r an a x i a l l y becomes  /,/  (3-9)  Zrrst  The two G-L e q u a t i o n s t a k e t h e form I  ( J  ~  * O  C  \  2  U  ,2. ( ^  n<f>o  A* 2  ZTTSL  (3.10)  ¥-7T  Zrrsi,  where  1  i s r e a l and A  i s i n t h e London gauge.  (3.H)  64  In  some m u l t i p l y - c o n n e c t e d s u p e r c o n d u c t o r s  n e a r l y u n i f o r m t h r o u g h o u t , which the f i e l d equations.  is  s i m p l i f i e s the s o l u t i o n of  I f the body has a x i a l symmetry  (such  as a h o l l o w c i r c u l a r c y l i n d e r o r t o r u s ) about t h e e x t e r n a l f i e l d d i r e c t i o n , t h e n we assume ^  i s not a f u n c t i o n o f 0  At l e a s t we o b t a i n a s o l u t i o n o f t h e G-L e q u a t i o n s i n which this i s true. unique.  I t i s not so easy t o prove  I n an i n f i n i t e l y  does not depend on metry.  l o n g h o l l o w c y l i n d e r , we assume  Y  y  z , on t h e b a s i s o f t r a n s l a t i o n a l sym-  I f the w a l l t h i c k n e s s  length,  this solution i s  t h e coherence  w i l l be n e a r l y u n i f o r m i n t h e r a d i a l  direction,  i n view o f t h e f i r s t G-L e q u a t i o n and t h e boundary c o n d i t i o n on  Y  J  * °*^  ~ ^  •  Another  geometry i n which  *•/ i s  n e a r l y u n i f o r m i s a t h i n a x i a l l y symmetric t o r o i d whose c r o s s s e c t i o n need not be c i r c u l a r but whose maximum in cross-section i s « f c r o s s - s e c t i o n , cannot  .  dimension  T h i s geometry, even f o r c i r c u l a r  be s o l v e d I n c l o s e d form s i n c e London's  e q u a t i o n i s not s e p a r a b l e i n t o r o i d a l c o - o r d i n a t e s . The l o n g h o l l o w c y l i n d e r w i t h &  uniform merits  d e t a i l e d study f o r t h e complementary reasons t h a t t h e f i e l d e q u a t i o n can be r e a d i l y s o l v e d i n c l o s e d form, and t h a t t h i s . geometry i s amenable t o experiment (length) »  p r o v i d e d we can t a k e  ( d i a m e t e r ) as a p p r o x i m a t i n g " i n f i n i t e l y C o n s i d e r then an i n f i n i t e l y  c y l i n d e r o f i n n e r r a d i u s A.  f  long hollow  long." circular  and o u t e r r a d i u s Jl~, , p l a c e d  65  i n a u n i f o r m magnetic f i e l d restriction y  i s p l a c e d on A)  H  parallel  g  ,  chapter unless s p e c i f i c a l l y  the r e m a i n d e r o f  s t a t e d to the c o n t r a r y .  . The  vector potential  and  Z  components a t t h e s u r f a c e .  symmetry, A^  ,  AQ  have  9  Since  0  A%  , and  A  z  s  0  geometry of i n f i n i t e  obeys ( 3 . 1 1 ) .  may  a t the s u r f a c e , and  Prom ( 3 . 1 1 ) , If A  No  o t h e r t h a n t h a t i m p l i e d by  u n i f o r m , w h i c h i s assumed throughout  this  to i t s a x i s .  may  Because o f  be f u n c t i o n s o f Sl^  V'A  ~0  then  A  =  0  3  . I n  A^~  0  the  idealized  only.  everywhere.  length there i s mathematically  nothing  which p r e v e n t s an a x i a l f l o w o f c u r r e n t .  In a r e a l  c y l i n d e r t h e r e w i l l e v i d e n t l y be no a x i a l  current unless  we p a r t i c u l a r l y  cause i t t o f l o w w i t h e x t e r n a l c o n n e c t i o n s .  F o r p r e s e n t purposes we  U%  s i m p l y add /  condition.  The  finite  "  0  as a boundary A  3  v e c t o r p o t e n t i a l i s t h e n o f the form AJ-fti 9  p A  hence the f i e l d  i s H ^) z  This being understood we  Z  A  and the c u r r e n t i s  f o r c y l i n d e r s p a r a l l e l t o the  s h a l l drop the s u b s c r i p t s and  treat  A ,  H ,^  scalars. E q u a t i o n .(3.11) becomes the s i n g l e t o t a l JL(1 - 4 M ) = - £ / / ! e n t i a l equation: w h i c h has  the g e n e r a l  solution  ^L) $ field, as differ-  ;  66  where  CL ,  a r e c o n s t a n t s , and  Bessel's functions.  I  , K  are modified  Hence  and =  Since ^ &  4r  i s uniform i n &  d i r e c t i o n , the f i e l d  u n i f o r m and a x i a l . demagnetization,  (&)}  +  and H^  Z , and f l o w s o n l y i n t h e i n the c y l i n d r i c a l hole i s  S i n c e t h e i n f i n i t e c y l i n d e r has zero  the f i e l d  at the e x t e r n a l surface i s  u n i f o r m and a x i a l , and e a u a l t o t h e e x t e r n a l f i e l d  H . e  P u t t i n g ^"2/ , Ji-i. s u c c e s s i v e l y i n (3.14) we may s o l v e f o r Q,  , Jr  I n terms o f  E^  and  t h a t as boundary c o n d i t i o n s , quite d i f f e r e n t :  R^  H-^ . H  2  V/e p o i n t o u t , however,  and  are p h y s i c a l l y  on t h e e x t e r n a l . s u r f a c e i s an a p p l i e d  p h y s i c a l c o n s t r a i n t on t h e s y s t e m , whereas  H^  depends on  the c i r c u l a t i n g c u r r e n t i n t h e c y l i n d e r , and v a r i e s w i t h the s t a t e o f the body.  Determination  o f t h e p r o b l e m t o be s o l v e d . uniform Hg •  H^ , and we may w r i t e  of  i s thus p a r t  However, t h e r e i s some CL , <ir i n terms o f i t and  We might choose some o t h e r q u a n t i t y as t h e second  boundary c o n d i t i o n , such as  or  .  s i m p l y because i t l e n d s a c e r t a i n symmetry  We choose to the equations.  67 We  are c o m b i n a t i o n s of B e s s e l ' s and  > ^J-f-  d e f i n e the q u a n t i t i e s  outer surfaces.  >  functions evaluated  which at the  inner  These q u a n t i t i e s which o c c u r f r e q u e n t l y  i n the d i s c u s s i o n of the h o l l o w t o as the "Boundary F a c t o r s . " 1 or 2 r e f e r r i n g t o A.,  and  c y l i n d e r w i l l be r e f e r r e d ^  A  x  and ^  take on the  values  respectively!  (3.16)  Evidently r  From the boundary c o n d i t i o n s on  H , we  have  0  68  Since by  . i s uniform  A^) =  i n the hole,  A,  so t h a t  SlMi/Z  as w e l l p u t  A  i n the hole  == A / t y  for  .  / Z  i s given  We c o u l d  i n t h e boundary  just  condition  above. a t Slj  E v a l u a t i n g E q n . (3-13) f o r A above v a l u e s  o f a,b  Setting this  equal  using the  gives  to  , we c a n s o l v e f o r  J l i H i / Z  :  (3.17) in  equilibrium, this  quantities being  H  H-^  the  here.  The a u t h o r h a d t h i s  for.  conditions.  second boundary  One knows t h a t a p h y s i c a l  itself.  condition?  e  equation  depends on  H  e  situation requires  , what i s  I t i s not the f l u x o i d  explicitly  i n t o the d i f f e r e n t i a l  L e t us s e e how t h e b o u n d a r y c o n d i t i o n s a r e  applied physically. H  differential  If  q u a n t u m number w h i c h e n t e r s equation  feeling for  was t a k e n as a b o u n d a r y c o n d i t i o n a n d now  by a s e c o n d o r d e r  two b o u n d a r y  I t may a p p e a r t h a t a f r a u d i s  0  has been s o l v e d  described  r i . ' i n terms c f the f i x e d  and n<j/> .  perpetrated  some t i m e . it  g  gives  The f i r s t  condition, the external  field  , may be a p p l i e d by w i n d i n g a n i d e a l s o l e n o i d a r o u n d t h e  69  e x t e r i o r o f t h e c y l i n d e r and p a s s i n g some c u r r e n t , say desired f i e l d the  I  H '.  through the winding  , which i s c a l c u l a t e d t o produce t h e Now t h e u n i f o r m i t y  of  throughout  e n t i r e h o l e and the consequent r e l a t i o n A (A) =  depend upon t h e absence o f any e x t e r n a l c u r r e n t s hole.  hole. g  .  i n the  The second boundary c o n d i t i o n i s " a p p l i e d " by not  i n t r o d u c i n g any c u r r e n t s  H  H,A,/Z  from an e x t e r n a l source_ i n t o t h e  Under t h i s c o n d i t i o n Equation  (3-17)  o t h e r q u a n t i t y , namely and  ^  i s a d e f i n i t e f u n c t i o n of  contains, n  i n addition to  , which i s not f i x e d e x t e r n a l l y .  w i l l take values i n e q u i l i b r i u m which minimize  the f r e e e n e r g y , not i n d e p e n d e n t l y , but s u b j e c t (3-17)•  , one  The f l u c t u a t i o n s o f ^  a  n  d  H  ^  t o equation  about t h e e q u i l -  i b r i u m v a l u e s a r e not i n d e p e n d e n t . S i n c e the f l u x o i d i s c o n s t a n t d u r i n g  changes o f  t e m p e r a t u r e and e x t e r n a l f i e l d , i t i s c o n v e n i e n t t o e x p r e s s the f i e l d q u a n t i t i e s i n terms o f Hp  and  .  Hp  and n (f>  o  , rather  than  We note i n p a r t i c u l a r t h e f o l l o w i n g q u a n t i t i e s  at t h e i n n e r and o u t e r  surfaces:  70  TfJlj  f  #2  The t o t a l f l u x  A /,  K,  \  Z  >2  A  A  V/:  \_  frriJ-  7^2 i / J  2rrsii.Jl  e n c i r c l e d by a c o n t o u r a t t h e i n n e r  and o u t e r  s u r f a c e s c a n be found from  The magnetic moment fo[ p e r u n i t l e n g t h o f c y l i n d e r i s found by d i r e c t i n t e g r a t i o n w i t h (3.15)  ^ (Sl)  g i v e n by  and g i v e s .1  ^ =-A_  \H  Z  (A,  - l ) i-H,  U i  -/) (3.18)  Hz  A  The t o t a l c u r r e n t  I  cylinder i s  r  ^ 7 t  (Hz-H,)  around t h e h o l e , p e r u n i t l e n g t h o f  71  J  The e l e c t r o m a g n e t i c f r e e energy  may be found,  H  by d i r e c t s u b s t i t u t i o n o f t h e p r e c e e d i n g two q u a n t i t i e s i n the g e n e r a l f o r m u l a o f Eqn. (3-5) • to o b t a i n  directly  However, i t i s p o s s i b l e  from t h e f i e l d i n t e g r a l  (3-3) by  e x p l o i t i n g t h e c y l i n d r i c a l symmetry and the zero  demagnetiza-  t i o n , and w i t h o u t a c t u a l l y i n t e g r a t i n g any f u n c t i o n s . Furthermore  we do not r e q u i r e  o n l y t h a t t h e second ytsi}  m  a  to ^  e  constant a t a l l ,  G-L e q u a t i o n be s a t i s f i e d , whatever  y be.  Using equation (3-3), noting that i n the h o l e  H = H^  s/  Putting so t h a t  we o b t a i n  "7/  H  g  = H  a constant,  'JfVo/e  -?  n<fio  ,1  2  , and  72  i  SrrJv,  +  11  = —/  f  dV  +  (/7>(VxA')  +  H  J  '  c  +TT  (JthZJffl  (  V  K  ^  '  A  Jsj  t  f  '  }  ^  (VKA')dV  -  C o n v e r t i n g t o s u r f a c e i n t e g r a l s , where  i n c l u d e s both  the i n n e r and o u t e r s u r f a c e s o f t h e h o l l o w  irr is, •S,  + Since  H  T  8  7  _  "  cylinder  Jr. J, S  8  and  A  a r e u n i f o r m over t h e s u r f a c e s , they  out•of the i n t e g r a l s . simple  UV  factor  A f t e r some a l g e b r a , one o b r a i n s the  expression ^iJ!i  -  - ± ^ L -  A  ±  j  >  H  Hz(n<jo)  -  ( 3 . 1 9 )  T h i s e x p r e s s i o n i s exact i r r e s p e c t i v e o f t h e v a r i a t i o n o f *y i n t h e Jis and  d i r e c t i o n but i t c o n t a i n s t h e q u a n t i t i e s  , i n a d d i t i o n t o the c o n s t r a i n t s  Hp  Ap  and n<j/> . 0  To  73  express  "J^  c o m p l e t e l y i n terms of. the l a t t e r two  s t r a i n t s , we must f a l l back on the f i e l d have been s o l v e d f o r f Ot) c o n s t a n t .  We have made no r e s t r i c t i o n s on Sl, i m p l i e d by ^ H  constant.  equations  In t h i s  , Jl  conwhich  case  other than that  x  This expression i s a quadratic i n  andfoj4>),which i s u s u a l l y the most u s e f u l f o r m , s i n c e  2  these are the p h y s i c a l c o n s t r a i n t s . re-express  M a t h e m a t i c a l l y , we  i n terms of v a r i o u s o t h e r q u a n t i t i e s ,  any p a i r of  H"  2  ,  , A  H  2  , A  , 9?( , I  1  ,  Q  <f>  c  can  almost  w i l l do.  Using  the r e l a t i o n s a l r e a d y f o u n d , we w r i t e down two  such-expres-  sions f o r  terms.  which do not c o n t a i n i n t e r a c t i o n As a f u n c t i o n of e x t e r n a l f i e l d  current  I  H  and  0  total  . c  3  (3.21) As a f u n c t i o n of f l u x o i d 7-  We  _  ?»•  {  g  &  and magnetic moment  t j - J  - J  )  4-  rfJn  }  (  draw the f o l l o w i n g c o n c l u s i o n from e q u a t i o n ( 3 - 2 1 ) .  the c y l i n d e r i n i t i a l l y H  ^  <$  3  = 0 , that i s i f  I c  -  2  If  i n the -normal s t a t e , i n e x t e r n a l f i e l d  i s c o o l e d i n t o the s u p e r c o n d u c t i n g  be a minimum i f  3  s t a t e , then H ' = H 1  . e  We  3^  will  2  )  74 e x p e c t t h e c y l i n d e r t o go i n t o t h e f l u x o i d s t a t e f o r w h i c h as n e a r l y as p o s s i b l e e q u a l s of t h e d i s c r e t e f l u x o i d s .  H  Ginzburg  w i t h i n the l i m i t a t i o n s  g  has g i v e n  (1962)  this  r e s u l t f o r a t h i c k c y l i n d e r b u t i t i s here shown t o be t r u e for  a l l hollow c y l i n d e r s . The  energy  7  complete f r e e energy i n c l u d e s  as w e l l as t h e e l e c t r o m a g n e t i c  - 3 *  From e q u a t i o n  1  with  (1.20) H  _  f r e e energy  ,  H  - <±A  —7  the c o n d e n s a t i o n  (-Z  we have  V*£~0  + +)  Z  (3.23)  X  7  2 Chapter  equation  2,  (2.13).  ^  i s the " r e l a t i v e d e n s i t y  of super e l e c t r o n s , " t h a t I s r e l a t i v e t o t h e d e n s i t y i n zero f i e l d a t t h e same t e m p e r a t u r e . f r e e energy  ~Jjj  expansions of i.e.  The  electromagnetic  i s an even f u n c t i o n o f ^7  .  Consequently  w i l l i n v o l v e o n l y even powers o f *7  s  i n t e g r a l powers o f '5^- . The  defined  "Reduced F r e e Energy"  as i n e q u a t i o n  Hd?  V,  (2.17)  as  f  ?  F  (block l e t t e r ) i s  H e ?  V,  ^  75  I t s h o u l d be remembered f o r f u t u r e r e f e r e n c e t h a t t h e r e d u c t i o n f a c t o r from 7  to F  i s t e m p e r a t u r e dependent.  As we  use t h e term "Free Energy" w i t h s e v e r a l s l i g h t l y d i f f e r e n t meanings we adopt t h e p r a c t i c e o f appending t h e r e l e v a n t w e l l d e f i n e d symbol t o t h e words "Free Energy" i n t h e t e x t . The complete e x p r e s s i o n f o r F"  f o r a hollow  cylinder of w a l l thickness r e s t r i c t i o n s on Jl  , J^x. is<  f  +  tk H&>  vi 1th no o t h e r  +  J&L  neb  J  r  L  "  -y?,^ j-j l c  /(-, // ' A '  ?  ]  (3.24)  J  The e q u i l i b r i u m s t a t e s o f t h e system a r e t h o s e f o r w h i c h i s a minimum, w . r . t .  , t h a t i s when ^  has t h e p a r t i c u l a r  v a l u e which s a t i s f i e s  IE _ For  >o  ,  0  -  <3 25)  a g e n e r a l h o l l o w c y l i n d e r , e q u a t i o n s ( 3 . 2 4 ) and ( 3 . 2 5 )  taken together are very complicated, since ^ out t h e boundary  factors.  c y l i n d e r s f o r which  A  occurs through-  I n t h e next c h a p t e r we w i l l c o n s i d e r 3  i n which case ( 3 . 2 4 ) can be w e l l  a p p r o x i m a t e d by a much s i m p l e r e x p r e s s i o n .  76  F i n a l l y i n t h i s c h a p t e r we o b t a i n an e x p a n s i o n of  P  i n powers o f ^  w i t h o u t any r e s t r i c t i o n on Slj , A.^ .  I t i s p o s s i b l e t o expand. t~ a l l t h e bounadry algebra. are  i n powers o f  by expanding  f a c t o r s and s i m p l y w o r k i n g t h r o u g h t h e  E x p a n s i o n s o f t h e boundary  f a c t o r s i n powers o f  g i v e n i n Appendix 2. There i s however a much s i m p l e r and more d i r e c t .  method w h i c h does not even r e q u i r e t h e s o l u t i o n o f t h e f i e l d equations. for  The p r o c e d u r e i s analogous t o t h a t d e s c r i b e d  s i m p l y connected b o d i e s i n Chapter 2, page  42 . We  d i s c u s s f i r s t the procedure i n g e n e r a l , then apply i t t o hollow c i r c u l a r c y l i n d e r s . equation  Take  ~J"  H  i n t h e form g i v e n i n  (5«9)  (3.26)  3 The i n t e g r a t i o n i s over t h e body o n l y , n o t t h e h o l e . vector ^ p o t e n t i a l of the e x t e r n a l f i e l d  Ae  The  must be i n t h e  London gauge o f t h e body b e i n g c o n s i d e r e d , i . e . have no component normal t o i t s s u r f a c e . (3.26)  I n the integrand of  t h e e x p r e s s i o n i n b r a c k e t s i s independent o f  J  If ^  'v'* .  I s o b t a i n e d as an e x p a n s i o n i n  , the i n t e g r a l  gives d i r e c t l y the r e q u i r e d expansion of  . We proceed  to  obtain the c o e f f i c i e n t s  =  fJ  n)  *tfa  (^)  (JL)  i n the expansion  f  77  C o n s i d e r the second G-L form o b t a i n e d d i r e c t l y from  If  e q u a t i o n i n the f o l l o w i n g  (3-11)  , so t h a t ^ ^ O = 0  , then  .  are b o t h i n t h e London gauge, t h e n as ^ —*> 0  A , A , A —> A  /s A  A  i n powers o f ^  ( w . r . t . ^ - ) term  /\  , since e  i s t h e n due e n t i r e l y t o the e x t e r n a l c u r r e n t s .  pansion of  g  The  ex-  c o n t a i n s the c o n s t a n t  , i.e.  e  S u h s t i tut;] n<? t h e s e e x p a n s i o n i n (3.27) and e a u a t i n s c o e f f i c i e n t s of  0)  M  gives =  _  C  r  /  rc  A n  e  W i t h i n the s u p e r c o n d u c t o r , from Maxwell's  equation  S u b s t i t u t i n g the s e r i e s expansion f o r  and  equating c o e f f i c i e n t s of  A , and  , we have  —o where RHS  i s a known f u n c t i o n of  .  This i s e s s e n t i a l l y  78  Poison's equation  whose p a r t i c u l a r s o l u t i o n can be w r i t t e n  as a d e f i n i t e i n t e g r a l .  Continuing  i n t h i s way we may add  terms a l t e r n a t i v e l y t o t h e s e r i e s f o r ^  and  A .  For any body h a v i n g a x i a l symmetry the e x t e r n a l vector p o t e n t i a l f o r uniform co-ordinates  H  i s given i n c y l i n d r i c a l  i n the London gauge by  Then  7/;  We now r e s t r i c t t h e d i s c u s s i o n t o the " i n f i n i t e l y " lone: h o l l o w  c i r c u l a r c y l i n d e r and make use o f the B  symmetry t o c a l c u l a t e t h e next h i g h e r Keeping i n mind t h a t that  A ,^  H  Z  term i n the s e r i e s .  i s e n t i r e l y i n the  a r e e n t i r e l y i n the $  and  Z  direction,  d i r e c t i o n , and t h a t  a l l a r e f u n c t i o n s o f Jl- o n l y , we have  (V  X hi)  S o l v i n g f o r H^  =  ^-o)  $  w i t h boundary c o n d i t i o n  t)  I  f//  e  U Z - J l  1  - )  $  C  H  (l)  *  ~ 0 JI±  at  JL - Jl-  A  79  = £ JjLfafa) =  V * 4  Then  H M u>  S o l v i n g t h i s f o r Aj^  w i t h c o n d i t i o n s A^^°)  continuous  surface  A1  over i n n e r  [* 1  t  2  2.  Then e q u a t i n g  c o e f f i c i e n t s o f 'J-  ~0  1i n (3.27)  and /}  {))  gives  7(z)  U s i n g t h e s e two terms o f t h e e x p a n s i o n o f ^  , we may sub-  s t i t u t e i n t o t h e i n t e g r a n d o f (3.26) and o b t a i n "3^ c o r r e c t which i s s u f f i c i e n t t o d e t e r m i n e t h e second  5  order c r i t i c a l  field  and t h e c o n d i t i o n f o r second  phase t r a n s i t i o n s t o o c c u r . and  The i n t e g r a t i o n  order  i s elementary  gives  "  I  §2J?  [ Tl  r  Sir  •*  x  rr  "  z  r {  h  TT^  1  \  '  Zt,  '  1  Z  8tr  K  '  % }  '  1  "  -  x  j  zii  ~  iyt  '^l  (3.28)  f  Anyone who has o b t a i n e d t h i s r e s u l t a f t e r l o n ^ tedious expansion of Modified Bessel's functions  will  80  a p p r e c i a t e the s i m p l i c i t y o f t h i s method w h i c h appears not t o have been used b e f o r e .  I n a d d i t i o n , t h i s method can be  a p p l i e d t o g e o m e t r i e s f o r w h i c h a s o l u t i o n o f the f i e l d e q u a t i o n s cannot be o b t a i n e d .  i  CHAPTER  4  THIN CYLINDERS  In  t h i s c h a p t e r we c o n s i d e r I n f i n i t e l y l o n g t h i n -  walled hollow cylinders.  " T h i n - w a l l e d " ( o r more s i m p l y  and ctA/)  " t h i n " ) means  .  d i f f i c u l t t o s a t i s f y the f i r s t  I n p r a c t i c e i t would be  c o n d i t i o n without  satisfying so t h a t *f  the second.  I t i s a l s o understood  i s uniform.  For t h i n c y l i n d e r s the expression f o r the f r e e  CT*  I  C U C l ' g J  .. •  •  / -.  tvj u a u j . u a  \  ^  I. \  - - .-  <^ ci1.1  . c_ -r /  s h a l l study t h e b e h a v i o u r  of h  that  .  k> ^-  _..«.. -i ...  pi,x  ,r>_. .. .1  u x i i i | j x x i  J.^.^  i.r_  .  .»  ^  and some o t h e r p r o p e r t i e s  of t h e c y l i n d e r as t h e e x t e r n a l f i e l d varied.  cx u -x j  ~i  and temperature a r e  I n P a r t I o f t h e c h a p t e r we t a k e temperature  T  c o n s t a n t and c o n c e n t r a t e on t h e dependence o f t h e v a r i o u s q u a n t i t i e s on t h e geometry o f t h e c y l i n d e r , t h a t i s on and  d.  both  T  /l  f  I n P a r t 2 we choose a p a r t i c u l a r c y l i n d e r and l e t and  H e  vary, °  P a r t 1 - Geometry The to  first  s t e p i n a p p l y i n g e q u a t i o n (3.24) f o r F*  t h i n c y l i n d e r s i s to o b t a i n s u i t a b l e approximations of 81  82  the  boundary  by p u t t i n g J l ft/\ "' ) i  S  '  , hjj,  factors %  -A, m  a  ^  and J^^  .  These a r e o b t a i n e d  i n a l l B e s s e l f u n c t i o n s o f argument n  a  ^  ^ y^a  or  e x p a n s i o n about  ^  These e x p a n s i o n s a r e o b t a i n e d i n powers of (cZ/\) r e s u l t s f o r t h e boundary of  ' .  The  f a c t o r s and s e v e r a l c o m b i n a t i o n s  them a r e g i v e n i n Appendix  3.  T h i s method i s r e l a t i v e l y  easy f o r the p a r t i c u l a r c o m b i n a t i o n s o f B e s s e l f u n c t i o n s r e q u i r e d , and i s e a s i e r t h a n u s i n g the a s y m p t o t i c e x p a n s i o n s of  each B e s s e l f u n c t i o n s e p a r a t e l y s i n c e a r a t h e r  large  number of terms of the l a t t e r must be r e t a i n e d t o get s u f f i c i e n t accuracy.  These e x p a n s i o n s a r e v a l i d f o r a l l  except t h a t f o r y  0  the terms must be r e g r o u p e d t o form  v  • an e x p a n s i o n I n powers o f "V"  , and t h e c o e f f i c i e n t s o f  powers o f ^ a r e o n l y a p p r o x i m a t i o n s of the e x a c t c o e f f i c i e n t g i v e n i n Appendix  2.  We p r o c e e d t o o b t a i n l o w e s t o r d e r I n  approxi-  mations f o r v a r i o u s q u a n t i t i e s I n the c y l i n d e r f o r which e x a c t e x p r e s s i o n s were o b t a i n e d i n the p r e v i o u s c h a p t e r . I t I s c o n v e n i e n t t o use the q u a n t i t y "reduced, f l u x o i d "  <£  d e f i n e d by  The  "reduced f l u x o i d " has the d i m e n s i o n of magnetic  but i t i s not i d e n t i c a l w i t h the f i e l d i n the h o l e .  field The  83 dimensionless AC  ==  "geometry parameter" ^u.  £lA A  r  (4.2)  1  o c c u r s f r e q u e n t l y throughout The  field  :  the  analysis.  i n the h o l e i s  The d i f f e r e n c e between i n n e r and o u t e r f i e l d i s  The  i n n e r s u r f a c e and o u t e r s u r f a c e v e c t o r p o t e n t i a l s  (which  d i f f e r o n l y by terms o f h i g h e r o r d e r ) are  The  super c u r r e n t d e n s i t y ^  , u n i f o r m i n the w a l l t o w i t h i n  terms of second o r d e r , i s (4.5) Since  -4<  /  i s n e a r l v u n i f o r m i n the w a l l the t o t a l  around the h o l e per u n i t l e n g t h i s j u s t X  ^  c  t h i s v a l u e one quantity  11^ -  c o u l d c a l c u l a t e i n an elementary  • way  current Using the  g i v e n above.  L e t us now  w r i t e down the reduced  f r e e energy f~~ ,  c o r r e c t t o the l o w e s t o r d e r term I n each e x p a n s i o n , by  direct  84  s u b s t i t u t i o n , from Appendix  3  i n t o equation  This  (3.24).  gives  In t h i s e q u a t i o n ference  ( M p . ~ $ ^  approximation F*H  Hp  and  appear o n l y t h r o u g h t h e i r d i f -  This i s only t r u e f o r the lowest  -  of  .  More a c c u r a t e c a l c u l a t i o n shows t h a t  i s n o t independent o f quantum number ' n  n .  order  T h i s w i l l be d i s c u s s e d i n t h e next  for large  chapter.  We can a r r i v e a t ( 4 . 6 ) i n a d i f f e r e n t manner w h i c h g i v e s some p h y s i c a l i n s i g h t .  enerey  F~  i s the i n t e g r a l o f f i e l d  d e n s i t y o l u s k i n e t i c enersy d e n s i t y o f the c u r r e n t  as g i v e n i n e q u a t i o n  (2.4).  I n t h e t h i n c y l i n d e r the volume  of t h e w a l l  i s l e s s t h a n t h a t o f the h o l e by a f a c t o r o f  order  , hence we n e g l e c t the f i e l d  energy i n the w a l l .  U s i n g the above approximate e x p r e s s i o n s f o r . substituting performing  (Hp - H^) and  i n the i n t e g r a l i n e a u a t i o n  (2.4),  and  t h e i n t e g r a t i o n which i s t r i v i a l because of con-  s t a n t i n t e g r a n d s we a r r i v e p r e c i s e l y a t  (4.6).'  Of c o u r s e  we have j u s t done the same c a l c u l a t i o n i n two ways i n t r o d u c i n g the a p p r o x i m a t i o n agree.  a t d i f f e r e n t p o i n t s and the r e s u l t s must  The l a t t e r method however p r o v i d e s an approach which  may be used i n l e s s i d e a l g e o m e t r i e s  than the hollow  cylinder.  85  To s i m p l i f y t h e a l g e b r a and t o o b t a i n more u n i v e r s a l r e s u l t s we d e f i n e  ZJzX  A  H t, c  S i n c e we a r e c o n s i d e r i n g c o n s t a n t The b e h a v i o u r  T , yU> i s a c o n s t a n t .  o f . t h e f r e e energy and o t h e r q u a n t i t i e s i n t h e  c y l i n d e r depends s t r o n g l y on t h e g e o m e t r i c a l parameter y * ^ " equation.(4.2).  F  becomes 2. (1.8)  where (~2\t'i )  i s t h e reduced  2  the l a s t term i s the reduced  condensation  energy  e l e c t r o m a g n e t i c energy  F  c  and F^ .  This simple a l g e b r a i c expression m a n i f e s t l y e x h i b i t s the f o l l o w i n g behaviour  t y p i c a l of  F^  i n a more c o m p l i c a t e d e x p r e s s i o n . thatyU<&l  F^  3  portional to  which i s o f t e n For A  i s nearly linear i n  X^  v e r y l a r g e , so  , I t s slope i s pro-  ( l i k e t h e c u r v e s i n F i g . 2-3a) .  v e r y s m a l l , y U . » / , ^^"^  hidden  For A  r i s e s r a p i d l y t o a p l a t e a u and  s a t u r a t e s , the h e i g h t of the p l a t e a u b e i n g p r o p o r t i o n a l t o X ( l i k e t h e c u r v e s i n F i g . 2-4a). T y p i c a l c u r v e s f o r F = F„ + F, f o r t h e s e extreme cases a r e shown i n F i g s . 2-3b G n T  and  2-4b.  minimizes progress  Since ^ F of  i n e q u i l i b r i u m takes the value which  one can f o l l o w q u a l i t a t i v e l y i n t h e s e p l o t s t h e F  and  as  X  increases.  86 In e q u i l i b r i u m , ^  if  =  0  , that i s  which d e t e r m i n e s  ^  i m p l i c i t l y as a f u n c t i o n o f  f u n c t i o n c a n be w r i t t e n  /  '  3/"  ( o r d i n a t e ) vs ^  X .  This  explicitly  3 {  To v i s u a l i z e X  i s d e t e r m i n e d by t h e r e q u i r e m e n t  (yU +*) '  J  3 z  t h e dependence o f ^  .  on  X  curves o f  ( a b s c i s s a ) a r e p l o t t e d i n F i g u r e 4-1.  The p h y s i c a l l y m e a n i n g f u l p a r t s o f t h e c u r v e ( 0  - /  F  The dashed  a minimum w.r-.t. ^ ) a r e shown by s o l i d l i n e s .  lines  are simply a n a l y t i c continuation of equation (4.9).  A l l s o l i d curves c o n t a i n the p o i n t X  2  ,  has a maximum a t  ^  — j - ^ ~ ^ u )  =/ , X .  The symbol ^  used i n a n t i c i p a t i o n o f t h e d i s c u s s i o n on page o o r t i o n o f t h e curve i n t h e r e g i o n 0 ^ a maximum o f  F w.r.t. ^  .  As yM-  =0. F o r } is  89 . The  ^ 1^ c o r r e s p o n d s t o  becomes l a r g e , ^  and A 3 i n c r e a s e s w i t h o u t l i m i t l i k e '^ry  }  *3  ( u n t i l the approxi-  mation f o r t h i n n e s s b r e a k s down). F o r yd 6/ , a l l t h e s o l i d c u r v e s pass t h r o u g h ^~  0 ,  2 X  = 1, a t which p o i n t  F = 0 .  Consequently, i n a l l c y l i n d e r s  h a v i n g yU- ^ ' , a second o r d e r phase t r a n s i t i o n o c c u r s a t 2 2 X =1 since d e c r e a s e s c o n t i n u o u s l y from 1 t o 0 as X  87  i n c r e a s e s from  0  t o 1.  As yOt-*0  and more n e a r l y a s t r a i g h t decreases  l i n e a r l y with  , t h e curve becomes more  l i n e ; i n very t h i n  cylinders  X^ .  L e t us compare t h i s c o n c l u s i o n w i t h t h a t reached by a d i r e c t s e a r c h f o r second  order t r a n s i t i o n s c o n s i d e r i n g  the complete  i n powers o f 3^ .  expansion of  f i c i e n t s of i n Chapter  and ^  3>  F  have been c a l c u l a t e d i n c l o s e d form  i n equation  (3.28).  The second  order c r i t i c a l .  f i e l d i s found by s e t t i n g t h e c o e f f i c i e n t o f z e r o , and a second  The c o e f -  equal t o  o r d e r t r a n s i t i o n does o c c u r i f a t t h i s 2  f i e l d the c o e f f i c i e n t of ~ r of ^  and  positive.  i s  The c o e f f i c i e n t s  a r e q u i t e c o m p l i c a t e d f o r g e n e r a l Jt, JL-^ 3  but i f we approximate  f o r t h i n n e s s GC^Jij , and keep o n l y t h e  l o w e s t term i n t h e c o e f f i c i e n t s we o b t a i n , u s i n g the p r e s e n t notation /="«  - Z f + ^ X *  7  E v i d e n t l y tine c r i t i c a l - second call  1  order value of  X  which we  X-^ i s  and the c o n d i t i o n f o r second  o r d e r phase changes t o occur i s  S i n c e t h e s e r e s u l t s a r e i n agreement w i t h t h o s e o b t a i n e d by considering:  X  as a f u n c t i o n o f ^  use e q u a t i o n ( 4 . 8 ) f o r F  we may then  confidently  r i g h t down t o t h e l i m i t p, ~~ 0 9  .  88  We  consider  as a f u n c t i o n of  now  X .  the e q u i l i b r i u m f r e e energy  Throughout the r e m a i n d e r of  F  eq  the  t h e s i s we  d e a l almost e x c l u s i v e l y w i t h the e q u i l i b r i u m v a l u e s  of  , and  F ,^  other bodies. script has  e  ^  some o t h e r q u a n t i t i e s i n the c y l i n d e r For  s i m p l i c i t y of n o t a t i o n we  drop the  , i t being understood throughout.  Whenever  and sub^  the meaning of an independent v a r i a b l e we w r i t e i t i n  e x p l i c i t l y i n the argument of the f u n c t i o n . (4.9)  i n (4.8)  we  Substituting  have ' (4.11)  where ^  i s the f u n c t i o n of  d e r i v a t i v e s of  y  where a g a i n For  yi>L  F(X)  > /  F(X)  X  given  The  are  i s a f u n c t i o n of  there  i n (4.10).  X .  are f o u r s i g n i f i c a n t  p o i n t s on the  curve  :  P o i n t 0 : Minimum of  F  , at  X  0  ~ 0 , ^  0  -  1  >  •  0  and  i Point  2:  dx  Jo  E q u i l i b r i u m f i r s t o r d e r phase change d e f i n e d f-l — 0  which o c c u r s a t  by  89  P o i n t s 3 : Extreme l i m i t o f m e t a s t a b l e the maximum v a l u e o f  X^  s t a t e which o c c u r s a t  f o r which a minimum o f h (X  }  w.r. t . *y e x i s t s , namely (see F i g . 4 - 1 ) when ^ ~^ A n a l y t i c a l l y the c o n d i t i o n i s At t h i s  ^ l~ (Xj"^  Q  point  When—? /  from above, p o i n t s 2 and 3 c o a l e s c e  t o t h e second o r d e r t r a n s i t i o n p o i n t s 1 which have a l r e a d y been g i v e n except We d e s c r i b e  t o note  F ( X ) as b e i n g o f "second o r d e r  when i t t e r m i n a t e s h o r i z o n t a l l y a t  P o i n t 4:  Point of i n f l e c t i o n .  F = 0 .  A l l curves f o r a l l J A -  a p o i n t of i n f l e c t i o n at  type"  have  .  90  <r  ~')  =  F  The p o i n t k.  >  ( j j t  maximum v a l u e  o f magnetic moment o c c u r s a t  I n t h e l i m i t i n g case JA,-* 0  approaches a s i m p l e F M  —  —  limiting (/ - x ^ )  ^  =  the curve  P(X)  form: ^  z  —  —  >  w i t h the p o i n t of i n f l e c t i o n at  On t h e o t h e r hand, a s b e c o m e s  large,  P ( X ) becomes n e a r l y  p a r a b o l i c e x c e p t f o r a v e r y s h o r t s e c t i o n a t t h e upper e x t r e m i t y where i t i n e v i t a b l y t u r n s o v e r . and ^  M  f o r differentyCC  and  (4-2b).  1)  For  y  point For y U , point 2)  Families of  F M  a r e p l o t t e d i n F i g u r e s (4-2a)  We remark t h e f o l l o w i n g p r o p e r t i e s :  U I  3-  has i n f i n i t e s l o p e a t t h e t e r m i n a l  3.  < I j p~ fx)  has f i n i t e s l o p e a t t h e t e r m i n a l  1.  A l l the points of i n f l e c t i o n  of  F ( X ) l i e on t h e s t r a i g h t  .ine  1  J 3  91  3)  The extreme m e t a s t a b l e p o i n t s l i e on a c u r v e a s y m p t o t i c to t h i s  4)  l i n e f o r l a r g e yU* .  F o r large/CC  , Xj^/Ct  a l w a y s , and  and  ^ lyu- ,  °&/U- , whereas  - —/  Hence, f o r l a r g e j u u by f a r t h e  g r e a t e s t p a r t o f the c u r v e l i e s i n t h e m e t a s t a b l e r e g i o n .  5)  The dashed c u r v e (shown o n l y f o r yU.-iQ f o r c l a r i t y ) i n F(X)  i s a p l o t o f e q u a t i o n (4.8)  f o r v a l u e s o f ^ <p-  3  T h i s c u r v e c o r r e s p o n d s t o maxima o f  F  w.r.t.^  .'  and  not t o e q u i l i b r i u m states... • The. v e r t i c a l d i s t a n c e between the s o l i d and dashed c u r v e s i s t h e p o t e n t i a l b a r r i e r t o the normal  state.  The magnetic moment  i s of i n t e r e s t since i t  i s f r e q u e n t l y t h e q u a n t i t y w h i c h i s measured.  (3.18) and t h e e x p a n s i o n i n powers o f (d/Aj  Using equation t o lowest order  one o b t a i n s :  D e f i n e t h e "reduced magnetic  M  _  ^**-  moment"  = / c£F \  V\  Noting that UM  dX  _  z (3^.-2  ( 72\ / (ty-*)  -tZ/yU.)  o  n  e  h  a  s  i m m e  dlately  The  values  of  M  and  M  0  at the four points  =  Q  1&\  }  =  [otK/O  Point  2:  Mo  =  ;  2-)  I***] =  Point 3  —  3 Point 4:  discussed  } I :  above a r e , f o r Point 0 :  dM/dX  co  T h i s i s t h e maximum v a l u e o f magnetic moment  • &),-<> As  ~ * M,  f r o m above,  ForyU  ~*  0  and / s ^ j  "~* °°  I , p o i n t s 2 .and 3 a r e m e a n i n g l e s s , t h e e x p r e s s i o n s  at p o i n t s 0 and 4 r e m a i n v a l i d , w h i l e a t t h e second  order  c r i t i c a l point 1 :  "l  .  Curves o f  M(X)  (,/x/y are p l o t t e d i n Figure  (4-2c)  To t h e a c c u r a c y we a r e c o n s i d e r i n g the c u r r e n t density ^ ^  i n the w a l l of the c y l i n d e r i s uniform.  Hence  and s e v e r a l o t h e r q u a n t i t i e s a r e p r o p o r t i o n a l t o M  and have s i m i l a r b e h a v i o u r as f u n c t i o n s o f p o r t i o n a l i t y between ^ ference  , loop current  (H^ - H-^) , and magnetic moment  m a r i z e d as f o l l o w s :  I  X . q  The p r o -  , field  dif-  may be sum-  93  V Q>  Zc  °  ?c Zc  s  C  Froim t h e c l a s s i c a l p o i n t o f v i e w we c a l c u l a t e t h e average momentum  7^ where to.  n  -~  =  M A T  i s the number d e n s i t y o f super e l e c t r o n s , e q u a l  s  ." U s i n g e q u a t i o n s  ffgQ  ^  o f t h e super e l e c t r o n s  , and tine d e f i n i t i o n o f  X  i n terms o f the. f i e l d v a r i a b l e  = A  if  (4.7),  .  . . .  i s t h e coherence l e n g t h .  Momentum"  P == 'f'f'f  Second o r d e r the v a l u e  1,  f  we o b t a i n f o r  and parameter yCt* :  X  where ~f  -  ,' ( 4 . 5 ) f o r '  x  /9  P  for n^  (1.2)  ^  J  D e f i n i n g t h e "Reduced  and u s i n g e q u a t i o n  k  (4.9)  we have  -  t r a n s i t i o n s occur at ^  —0  a t which  P has  i . e . -when t h e de B r o g l i e w a v e l e n g t h o f t h e  electrons equals  / . T r i  .  For/U.'i  , t h e t e r m i n a l momentum  P-  i s somewhat s m a l l e r but n e v e r l e s s t h a n  of  P ( X ) a r e shown i n F i g u r e 4 - 2 d . In Figures  Curves  4 - 2 a , b , c, d we have t a k e n as a b s c i s s a  a variable proportional to  (Hp - $ ) •  A l t e r n a t i v e l y , we  may w i s h t o t a k e t h e a b s c i s s a p r o p o r t i o n a l ' t o  Hp  and d i s p l a y  94  the  c u r v e s f o r each f l u x o i d . n u m b e r s e p a r a t e l y .  <j5 to  , the curves f o r a l l  n  For small  a r e i d e n t i c a l but t r a n s l a t e d  t h e r i g h t o r l e f t by an amount p r o p o r t i o n a l t o  n .  The meaning o f " s m a l l " i n t h i s c o n t e x t w i l l be made p r e c i s e i n Chapter 5 .  I t i s o f i n t e r e s t t o have an i d e a o f t h e  s e p a r a t i o n o f n e i g h b o u r i n g c u r v e s , say A X  i n units of  X , w h i c h i s t h e n a t u r a l f i e l d u n i t o f t h e system.  AX i s  g i v e n by  / F A l rr^j  a  For  H t> c  a t i n c y l i n d e r o f d i a m e t e r 1 cm. ( s i m i l a r t o ones used  by Hunt and Mercereau  (1965) a t t = 0.9 , one f i n d s  - y-  £i A  rJC5  'O  .  The s u c c e s s i v e c u r v e s a r o e x t r e m e l y  c l o s e l y s p a c e d , q u i t e u n r e s o l v a b l e i n a p l o t such as F i g u r e  4-2. the  AX  i n c r e a s e s w i t h t e m p e r a t u r e but a t  t = 0 .999  -3  separation i s s t i l l  g i v e n by Douglass  small  ;  AX  ^ JO  .  The c u r v e s  (1963) g i v e a f a l s e i m p r e s s i o n o f t h e  s t a t e o f a f f a i r s i n t h i s c y l i n d e r s i n c e he shows s u c c e s s i v e c u r v e s s e p a r a t e d by amounts o f o r d e r A X P a r t 2 - Temperature In  /.  and F i e l d  t h i s p a r t we c o n c e n t r a t e on a p a r t i c u l a r  c h a r a c t e r i z e d by i t s yU  0  cylinder  , and study t h e dependence o f  f r e e e n e r g y , o r d e r p a r a m e t e r , magnetic moment, and s u p e r e l e c t r o n momentum on b o t h t e m p e r a t u r e and f i e l d .  In the  95  p r e v i o u s p a r t the d e f i n i t i o n s o f F t e m p e r a t u r e dependent. We  define:  *  ~ 2/1K  and X  are themselves  T h i s p r o c e d u r e i s not s u i t a b l e h e r e .  (/+t  //„,  %  l  A  •(4.13)  HjV,  3  (  and r e c a l l t h a t JUL' i s t e m p e r a t u r e • , _ .v / / _ -r) ¥  S~  f  "  *'  "  '  where  dependent: /y. =. Jl, c£/ A "  i s d e f i n e d so as t o be n o r m a l i z e d at  t = 0 j X = 0  a  f = - l .  f  t = 0 , that i s a t  has the same r e l a t i v e  t u r e dependence and as the d i m e n s i o n a l f r e e energy U s i n g the d e f i n i t i o n s f o r r e d u c e d f r e e energy  (4.13)  n  tempera•  and the e x p r e s s i o n  F , equation ( 4 . 8 )  we o b t a i n f o r  f (t x, %) 3  In equilibrium ditions  ^  i s a g a i n d e t e r m i n e d by the con-  96  The  first  c o n d i t i o n y i e l d s t h e same e q u a t i o n as  With  (4.9).  the temperafciure dependence w r i t t e n I n , t h i s e q u a t i o n becomes I-t-t  (4.15)  This gives  i m p l i c i t l y as a f u n c t i o n o f  Substituting  where ^*  in  (4.15)  (4.14)  x  and . t. .. •  gives  i s not independent but i s g i v e n by  have shown i n t h e p r e v i o u s p a r t t h a t t h e e x p a n s i o n i n powers ot(d^h)  use  , including  ^  —>  0  .  r  order)  We may then •  ( 4 . 1 4 ) t o study the o r d e r o f phase t r a n s i t i o n s . For a given t h i n c y l i n d e r equations  (4.15)  together give  f  and- ^  of the independent v a r i a b l e s  (4.16)  responding  m a g n e t i c moment  are also functions of  m x  and  r e s p e c t i v e l y as f u n c t i o n s  x  and  t , t h a t i s , as s u r -  f a c e s i n t h r e e d i m e n s i o n a l c o - o r d i n a t e systems.  p  or  (which l e a d s t o ( 4 . 1 4 ) . t o l o w e s t  is valid f o r a l l ^  We  (4.15).  The c o r -  and super e l e c t r o n momentum and  t .  To i l l u s t r a t e t h e  form o f t h e s e s u r f a c e s we s h a l l draw two s e t s o f c u r v e s : (a) curves o f  f ( x ) v/ith  t  c o n s t a n t , each c u r v e  labelled  97  w i t h t h e v a l u e o f t h e parameter f(t) of  with x .  x  c o n s t a n t , each curve l a b e l l e d by the v a l u e  Corresponding  constant  t  t , and (b) c u r v e s o f  curves o f ^ - ( x ) , m(x), p ( x ) at  and ^ - ( t ) , m(t)  are a l s o drawn.  The curves  as t h e p r o j e c t i o n s on the  and  p ( t ) at constant  (a) f o r f ( x ) may be i n t e r p r e t e d f - x  p l a n e o f t h e l i n e s formed  by the i n t e r s e c t i o n o f p l a n e s o f c o n s t a n t face  f(x,t)  m(x)  and  .  x  t  w i t h the s u r -  S i m i l a r remarks a p p l y t o t h e curves  ^(x),  p ( x ) , and a l s o t o a l l t h e curves' (b) which a r e  p r o j e c t i o n s on t h e  f - t  plane, e t c .  Curves (a) a r e r e a d i l y o b t a i n e d by an a d a p t a t i o n o f t h e c u r v e s o f p a r t 1. depends o n l y on f (x)  The shape o f  f(x) for fixed  i a d e f i n i t e Quantity f o r given  i s o b t a i n e d from  P(X)  t  t .  w i t h c o r r e s p o n d i n g /S-  by  2 (1 - t )  2  compressing the a b s c i s s a f(x) t  the ordinate X  P  by a f a c t o r o f  by a f a c t o r  i s i d e n t i c a l t o the curve  i n c r e a s e s , t h e curves  I'  /  P(X)  .  and  At. t = 0 ,  h a v i n g y^C ~y^o .  f ( x ) squeeze i n toward  As  the o r i g i n ,  more r a p i d l y i n o r d i n a t e t h a n a b s c i s s a , and a t the same time become more " s e c o n d - o r d e r - l i e k . "  At  t£t  where  t  is  g i v e n by  y.(i*)  » y j l -  i**)  - I  the c u r v e s become o f t h e second o r d e r type .  (1.17)  Similarly  curves  98  of  m(x)  toward  a r e o b t a i n e d from  M(X)  by s h r i n k i n g the l a t t e r  t h e oi°igin u n i f o r m l y on b o t h axes by a f a c t o r F o r purposes  o f i l l u s t r a t i n g t h e s e f u n c t i o n s we — 10 f o r two r e a s o n s :  choose a c y l i n d e r h a v i n g  first,  i t i s a convenient value to i l l u s t r a t e the i n t e r e s t i n g f e a t u r e s over a c o n v e n i e n t range o f t e m p e r a t u r e s ,  and  second,  from t h e d a t a g i v e n , i t seems to. c o r r e s p o n d .to the. c y l i n d e r s used by Hunt and M e r c e r e a u m(x)  and  p(x)  f o r s e v e r a l values of  plotted i n Figures 4-3a,b,c,d. c u r v e s a r e second  1  For  t  (constant) are  t£ t  = . 9 7 4 the  rn(x)  and  order type.  The c u r v e s o f  /M>  Curves o f . f ( x ) ,  (1964).  ±Xx)  ,  p.(jc)  s  a r e c h a r a c t e r i z e d by t h e same f o u r - s i g n i f i c a n t p o i n t s  as found i n p a r t 1 , w i t h s t r i c t l y analogous the meaningful p o i n t s are the the second  f  meanings.  For a p a r t i c u l a r value of  and t h e c o r r e s p o n d i n g  f , m ,^  f(x)  t , hence  the c o n d i t i o n d e f i n i n g each p o i n t d e t e r m i n e s x  For  minimum p o i n t 0 ,  o r d e r t r a n s i t i o n p o i n t 1 and t h e  t i o n p o i n t . 4.  of  p(x) f o r  unique  and  p .  inflec, values If t  v a r i e s from 0 t o 1 each s i g n i f i c a n t " p o i n t " t r a c e s out a l i n e o f f i n i t e l e n g t h on t h e a p p r o p r i a t e p l o t .  F o r example  i n the  X%0  f - x  p l a n e , ( o f which the h a l f p l a n e  shown i n F i g u r e s 4 - 3 and 4 - 4 )  f^  is  i s the s e c t i o n o f the  f  99  a x i s from - 1 t o 0 , w h i l e a x i s from 0 t o 2 . 5 3 . are shown. clarity.  fg  i s the s e c t i o n of the x  The l o c i o f t h e p o i n t s  which l i e s j u s t below A l l o f t h e a r e a bounded by  the l i n e s  f^  and f-^  f^  f^  and f ^  i s omitted f o r  f ( x ) at  represents possible  t = 0  and  superconducting  s t a t e s o f t h e system. In the three dimensional r e p r e s e n t a t i o n s ,  f(x,t)  f o r example, t h e l o c u s o f each s i g n i f i c a n t p o i n t s . , f ^ . e t c . i s a l i n e i n space.  The l i n e s shown on t h e  f - x  plane  are p r o j e c t i o n s o f t h e space l i n e s , onto t h i s p l a n e . f , 0  fg, f^  l i e on t h e s u r f a c e  f ( x , t ) , while  c o n s t i t u t e t h e upper edge o f t h i s s u r f a c e . j o i n s smoothly onto l i n e s has  t  f-^ .  with  X =  = 1 .  x  f^ ,f^  The l i n e  f^  The p o i n t s e p a r a t i n g t h e s e two  co-ordinate  ing f i e l d co-ordinate  Lines  t  g i v e n by  = x^(t- )  Solving this  (4,17)  and c o r r e s p o n d -  which i s g i v e n by  (4.J.'3)  explicitly ~~  (4.18)  In g e n e r a l , I f the s t a t e of the system.is by s l o w l y v a r y i n g b o t h m and p f(x,t)  x  and  t , the q u a n t i t i e s  f ,  w i l t r a c e out paths on t h e r e s p e c t i v e s u r f a c e s e t c . The p r o j e c t i o n o f t h e s e p a t h s on t h e  p l a n e may be shown on t h e p l o t s o f F i g u r e s 4 - ^ 3 . ticular,  changed  i f t h e temperature  t  f - x  I n par-  v a r i e s w h i l e the reduced  100  relative field  x  remains c o n s t a n t the s t a t e of the  p r o g r e s s e s a l o n g a v e r t i c a l l i n e i n F i g u r e s 4-3of  system  Examples  such p a t h s a r e shown by the v e r t i c a l l i n e s on w h i c h  arrows i n d i c a t e the d i r e c t i o n o f r i s i n g t e m p e r a t u r e .  If  the  t e m p e r a t u r e i s i n c r e a s e d s u f f i c i e n t l y , the p o i n t r e a c h e s  the  upper edge of the s t a b l e or m e t a s t a b l e r e g i o n and a  phase change o c c u r s . the  Paths t e r m i n a t i n g t o the l e f t , of  v e r t i c a l l i n e marked  x.  l e a d t o second order, t r a n s i -  t i o n s w h i l e p a t h s t o the r i g h t o f transitions. at  x;  lead to f i r s t order  The c o n d i t i o n f o r second o r d e r t r a n s i t i o n s  c o n s t a n t f i e l d and i n c r e a s i n g t e m p e r a t u r e i s  or w r i t t e n out  •  ( Zyt/o -  <  ,  explicitly  /  X  x£x  I -  The c o n d i t i o n yU- ^ /  -  ,  J ( y ^ o - l )  X  //£  )  cannot be a p p l i e d d i r e c t l y t o t h i s  case s i n c e t h e c o n d i t i o n i s t e m p e r a t u r e  dependent.  An a l t e r n a t i v e r e p r e s e n t a t i o n of the f u n c t i o n s f(x t) 3  the x  f - t  plane  l i n e s formed by the i n t e r s e c t i o n of p l a n e s of c o n s t a n t w i t h the  f(t), the  e t c . i s o b t a i n e d by p r o j e c t i n g on the  ^(t)>  f(x,t) (t)  m  parameter  surface.  This gives sets of curves  and p ( t ) j each c u r v e b e i n g l a b e l l e d by x .  These c o n t a i n e s s e n t i a l l y the same i n f o r m a t i o n as F i g u r e s 4 - 3 .  They a r e s i m p l y a remapping  of t h o s e p l a n e s  101  onto p l a n e s I  n which  t  i s measured l i n e a r l y as a b s c i s s a .  In practice„ t r a c i n g t h e curves o f  f ( t ) etc. i s rather  more tedious; ifchan t r a c i n g the c u r v e s o f  f ( x ) etc. since  i n t h e l a t t e a r case one e a s i l y o b t a i n s p a i r s o f v a l u e s o f f  and  x  uising ^  as a p a r a m e t e r , whereas i n t h e former  case t h e r e i s no way t o a v o i d u s i n g the e x p l i c i t for  ^  expression  i n e q i u a t i o n (4.10) p u t t i n g i n t h e t e m p e r a t u r e de-  pendence of' X F i g u r e s 4-4.  and yU-  . . T h i s gives, t h e .curves .shown i n  To f a c i l i t a t e  the' I n t e r p r e t a t i o n o f F i g u r e s  4-4 as a remapping o f F i g u r e s 4-3 t h e t h i c k l i n e s a g a i n c o r respond constant  to constant  t  and t h e t h i n l i n e s w i t h arrow's t o  x: ,. The " s i g n i f i c a n t p o i n t s " whose l o c i a r e  space l i n e s ; .are a l s o shown i n p r o j e c t i o n on t h e etc,  f - t  planes.. Tiihe curves  f ( t ) c o n t a i n no s u r p r i s e s .  s i b l e super-conducting by the l i n e s  f^  A l l pos-  s t a t e s l i e i n t h e r e g i o n bounded above  and  f^  and below by t h e l i m i t i n g  curve  2 2 x = 0 , f o r which t  at constant  f = (1-t )  x = x  .  The curve f o r I n c r e a s i n g  which separates the r e g i o n s of  f i r s t and second o r d e r t r a n s i t i o n s l i e s j u s t above t h e lower l i m i t i n g c u r v e . t h a t as  x — 0  I n t h e curves o f ^ ^  J  w  e  n  o  t  e  t h e curve hugs t h e a x i s ^ - } > and drops  abruptly t o 0 at  t = 1 .  T h i s i s t h e l i m i t i n g case o f a  phase t r a n s i t i o n i n zero f i e l d .  i  I t s h o u l d be remembered  102  that ^  i s the r e l a t i v e  c o n c e n t r a t i o n of s u p e r e l e c t r o n s .  The  absolute concentration  as  t  increases to The  ing  at f i r s t .  ^ ^ o o  decreases  1 .  curves  of  m(t)  For  x > x  a r e perhaps a l i t t l e  , as  t h e n drop down and t e r m i n a t e  on  surpris-  t i n c r e a s e s , the curves  c r o s s over t h e l i m i t i n g m e t a s t a b l e  forming  continuously  in^(t) ,  state line  m^  with i n f i n i t e slope  a s e r i e s o f bumps on t h e r i g h t  of  m .(£.)..  n i f i c a n c e i s attached to the cross-over p o i n t .  thus  No. s i g -  The curve  *  m(t) for  for x = x x < x  terminates v e r t i c a l l y at  , m(t)  tive slope.  As  terminates at  M(x)  with f i n i t e  x — » 0 , the t e r m i n a l value of  t e r m i n a l s l o p e — * 0. maxima o f  m = 0  m = 0 , while  Since  i n the  m^(t)  m - x  nega-  t —* 1  and  i s t h e l o c u s o f the a  plane  m,,(t)  must l i e t o  -r  the r i g h t the bumps.  of  m^Ct)  m^(t)  and i s i n f a c t t h e envelope c u r v e o f terminates h o r i z o n t a l l y at  t - 1 ,  m = 0 . The t  super e l e c t o n momentum  p ( t ) increases with  for- a l l x , s l o w l y at f i r s t and more r a p i d l y as  approaches the l i m i t i n g v a l u e . w h i l e f o r x <C x order t r a n s i t i o n  , p(t)  p ( t ) i s v e r t i c a l at  p^ ,  has f i n i t e s l o p e a t t h e second  temperature.  creases monotonically  t  The l i m i t i n g  curve  t o zero w i t h decreasing  x .  p^  de-  103  As b e f o r e , we may a b l e a q u a n t i t y , say  t a k e as independent f i e l d  h , p r o p o r t i o n a l to  x  which i s p r o p o r t i o n a l t o  we  put  (Hp - <j& ) .  Then i n the t h r e e d i m e n s i o n a l space  Hp  vari-  r a t h e r than  For d e f i n i t e n e s s  f , h, t  we o b t a i n a  v e r y l a r g e number o f s u r f a c e s l a b e l l e d by. the. f l u x o i d quantum number n  n. .  (large  A l l the s u r f a c e s have the same shape f o r s m a l l  n  i s d i s c u s s e d i n the next c h a p t e r ) and each i s  d i s p l a c e d f r o m i t s n e a r e s t neighbour  /\h =  <p l*f$Sl, Ao H 0  00  at c o n s t a n t  t  a l o n g the  h  i s not t e m p e r a t u r e  t  f(x)  h = n(4h).  Since  Ah  dependent the p o s i t i o n of these l i n e s does  not v a r y w i t h t e m p e r a t u r e .  At  On the  These c u r v e s are symmetric  about the s e t of v e r t i c a l l i n e s  axis.  axis.  p l o t s these s u r f a c e s appear as a s e t of  c l o s e l y spaced i d e n t i c a l c u r v e s .  s h r i n k s toward  by an amount  As  t  i n c r e a s e s each  the c o r r e s p o n d i n g p o i n t g i v e n by jJ.(i)~l  curve  n (A h)on the  h  the c u r v e s become second o r d e r  type f o r m i n g a s e t of o v e r l a p p i n g i n v e r t e d b e l l s , a  few  o f which a r e shown by the t h i n l i n e s i n F i g u r e 4-5.  As  c o n t i n u e s t o i n c r e a s e t h e r e i s some t e m p e r a t u r e , a t which the touch.  At  say  upper e x t r e m i t i e s of n e i g h b o u r i n g c u r v e s t > t^  t  t-^ , just  t h e r e i s a space between n e i g h b o u r i n g  104  c u r v e s and as on the  h  t—>1  the c u r v e s s h r i n k t o a s e t of p o i n t s  axis. On t h e  different  n  f ( t ) , constant  h  p l o t s the c u r v e s f o r  a r e a g a i n c l o s e l y spaced but they are not  merely d i s p l a c e m e n t s of t h e i r n e i g h b o u r s . a l r e a d y o b t a i n e d i n F i g u r e 4-.4 may  Let  n  }  h  but d i f f e r e n t  n = 0 .  Then f o r s l o w l y v a r y i n g  f^ = 0  and  which g i v e s  f o r yU> h  ± by  The upper edge o f f^  and f o r ytf « 1  as a f u n c t i o n o f  t .  In Figure  i s shown the p r o j e c t i o n of the upper edge of the  f(h,t) fl*  h  does not change and the s t a t e of the system i s  the s u r f a c e i s d e t e r m i n e d  4-6  n ,  (constant), x .  r e p r e s e n t e d b y a p o i n t on t h i s s u r f a c e .  by  as  us c o n c e n t r a t e on one p a r t i c u l a r s u r f a c e  f ( h , t ) , say f o r t  curves  be i n t e r p r e t e d  examples of c u r v e s h a v i n g the same and t h e r e f o r e d i f f e r e n t  The  s u r f a c e on the  ^ 1 , which l i e s  h - t  plane.  The l i n e  f^ = 0 ,  on t h e s u r f a c e i s a l s o shown.  I f the  p a t h of the p o i n t r e p r e s e n t i n g the s t a t e of the system s e c t s the  f = 0  3  jA. k l l i n e t h e n the c y l i n d e r l e a v e s  t h i s p a r t i c u l a r s u p e r c o n d u c t i n g f l u x o i d s t a t e by way  of a  second o r d e r phase t r a n s i t i o n .  I f the p a t h i n t e r s e c t s  line  of a f i r s t  f^  t h e n i t l e a v e s by way  transition.  inter-  I t cannot be d e t e r m i n e d  o r d e r phase  from t h i s  whether the c y l i n d e r w i l l go i n t o the normal  diagram  s t a t e or a  the  105  super s t a t e o f d i f f e r e n t f l u x o i d . of simply-connected  bodies  a "supercooling" state i s possible  whose l i m i t may be shown on t h e below  fg  I n s i m i l a r phase diagrams  h - t  p l o t as a l i n e  j o i n i n g smoothly onto t h e second o r d e r  line.  I n t h e case o f t h i n c y l i n d e r s such a l i n e i s not m e a n i n g f u l . Because o f t h e m u l t i p l i c i t y o f s t a t e s a v a i l a b l e s u p e r c o o l i n g does not o c c u r . One o f the most s t r i k i n g aspects, p f superconductivity  i s the existnece of persistent, currents i n m u l t i p l y As mentioned i n Chapter 3, a r i n g i n a  connected b o d i e s . super s t a t e w i t h  fl ^ 0  i n zero e x t e r n a l f i e l d .  barrier.  will  e x h i b i t a p e r s i s t e n t current  Such a s t a t e i s not t h a t o f l o w e s t  L e t us c o n s i d e r i n more d e t a i l t h e c r e a t i o n o f  persistent current states. The c y l i n d e r i n i t i a l l y normal a t to  t < 1  field  and becomes s u p e r c o n d u c t i n g  Hg ,  ambiguity  in  V/e ask two q u e s t i o n s : n?  second o r d e r ?  i s cooled  i n a uniform  external  1) I s t h e r e any  and 2) I s the phase t r a n s i t i o n f i r s t or  These two q u e s t i o n s  In Figure f(h)  t > 1  are i n t e r - r e l a t e d .  (4-5) i s shown i n t h e r e p r e s e n t a t i o n  t h e f i r s t few o f the f u l l s e t o f p o s s i b l e s u p e r c o n -  ducting states.  As a l r e a d y d i s c u s s e d , as  below t h i s s e t of curves on the  h  t —*» 1  from  shrinks to a set of p o i n t s  a x i s , that i s with  f = 0 .  n (A h)  This Figure a p p l i e s  106  e q u a l l y w e l l t o the case o f l o w e r i n g t h e t e m p e r a t u r e . t = 1,  the "curves"  t i o n e d , and as  t  f ( h ) a r e the s e t o f p o i n t s j u s t men-  drops below 1 the c u r v e s spread'out  those p o i n t s u n t i l a t some t e m p e r a t u r e , bouring curves j u s t touch. medium w e i g h t  say  t^  l i n e s i n F i g u r e 4-5.  type and t o u c h a t f = 0 ..  At - t ^  :  one f l u x o i d number f o r w h i c h  of  Hp .  the p o i n t s  If  Hp  t^  second-order-  t h e r e i s one and o n l y  f ^ 0 . f o r g i v e n .' Hp. .  say t h e t r a n s i t i o n t a k e s p l a c e a t between  two n e i g h -  In a n t i c i p a t i o n of a  sumably the c y l i n d e r i s i n t h i s s u p e r s t a t e .  t  t  1  i s such t h a t x  Pre-  T h i s i s not t o  , i t takes p l a c e at  and 1 depending on the e x a c t  n ( d h ) , then  from  T h i s s i t u a t i o n i s shown by the  r e s u l t t o be o b t a i n e d the c u r v e s a r e drawn as  some  At  value  h l i e s e x a c t l y on one o f  d e f i n e d i n e q u a t i o n (4.1-3)  e q u a l s z e r o and the t r a n s i t i o n o c c u r s a t  t = 1.  If  Hp  l i e s near t h e m i d p o i n t between two of. the p o i n t s on the a x i s then t h e t r a n s i t i o n t a k e s p l a c e at the maximum p o s s i b l e value of / x / "reduced"  , which i s lx.1 =<j>0/ z T T J i f  quantum o f f l u x o i d .  .  This i s h a l f a  I f t h i s v a l u e of \7t\  satisfies  the c r i t e r i o n o f e q u a t i o n (4.17) then t h e t r a n s i t i o n be second o r d e r .  For t h e example c y l i n d e r d e s c r i b e d on  9.4 we have X *» /O w h i l e from (4.17) Hence x < x * by a f a c t o r o f 10 5 . page  will  ?C &  /0  .  107  We  c o n c l u d e t h a t i n a l l t h i n c y l i n d e r s a second  order t r a n s i t i o n takes place f l u x o i d number.  i n t o a s t a t e o f unambiguous  I f the t r a n s i t i o n were not  i t might be p o s s i b l e t o s u p e r c o o l  second o r d e r  the normal body t o a  t e m p e r a t u r e at which more t h a n one  f l u x o i d s t a t e were  e n e r g e t i c a l l y a v a i l a b l e l e a v i n g some a m b i g u i t y as t o  the  quantum s t a t e a c t u a l l y r e a l i z e d . I f the magnetic f i e l d c o o l e d below  t = 1  i s s u b s e q u e n t l y removed the  i s i n a p e r s i s t e n t current to  n <P  Q  h = 0 .  since  i n which the. c y l i n d e r , was  q u a n t i t i e s i n Figures  s t a t e i n which  4 - 4 at d i f f e r e n t constant  example c y l i n d e r d e s c r i b e d The  i s proportional  I n t h i s case the c u r v e s of  curves f o r d i f f e r e n t f l u x o i d s t a t e s .  below each c u r v e .  x  on page  V a l u e s of 94  space " c u r v e s "  o f c l o s e l y spaced d i s c r e t e p o i n t s .  the r e l e v a n t p a r t s of F i g u r e n  .  Cur  4-4.  f^  experiment has but  are  n  for  the  brackets  discuss  4 - 7 and  4-8  s o l i d c u r v e s are  t h e o r y p r e d i c t s t h a t , as  the m a g n e t i c moment a c t u a l l y i n c r e a s e s .  x  e t c . are a s e t •  In order to  The  the  are shown i n  some changes of s t a t e we r e p r o d u c e i n F i g u r e s  constant  cylinder  The  been c a r r i e d out by Hunt and  t  at  i s lowered,  corresponding Mercereau  (1964),  t h e i r r e s u l t s have not p r e v i o u s l y been compared w i t h  a  complete a n a l y s i s of the s t a t e of a t h i n c y l i n d e r as a f u n c t i o n o f e x t e r n a l f i e l d and  temperature.  An example of  10  4-7.  t h e i r r e s u l t s i s shown to the r i g h t of F i g u r e We  suppose-in Figure  brought t o s t a t e field  A  by  then increased  to  t^  t h a t the  system has  c o o l i n g to temperature  /rf/l* which has  //-, —  4-7  t^  t h e n been removed.  been  In a If  t  is  the s t a t e moves r e v e r s i b l y a l o n g  the  curve  to point  the  s t a t e p o i n t moves around the bump t o the extreme meta-  stable point t£  t^//  <  /n l  t .  at  there  A  We  .  I f we  continue to i n c r e a s e  w h i c h by. . d e f i n i t i o n l i e s , on  the f l u x o i d s t a t e Inl  A'  M^.  f)^ becomes u n s t a b l e b u t . y  e x i s t s t a b l e or m e t a s t a b l e  make the h y p o t h e s e s t h a t at  A  Figure  4 - 7 we  state  I  see  line  The  .  On  From to  7^-/  curve  tooth path  at the apex of each t o o t h .  The  line  m^  ,  touching  teeth  on the p l o t of F i g u r e  A g r e a t l y m a g n i f i e d s e c t i o n of the p a t h i s shown i n The  are 4-7. Figure  system appears t o move i r r e v e r s i b l y down the .  As  t h e n drops i n t o  system f o l l o w s a saw  much too f i n e to be r e s o l v e d  4-9.  as  the o t h e r hand,  system f o l l o w s the  u n t i l t h i s s t a t e becomes u n s t a b l e ,  the  •  system  super e l e c t r o n momentum d e c r e a s e .  c o n t i n u e s t o r i s e the  so on.  states  the magnetic moment a c t u a l l y i n c r e a s e s ,  the f r e e energy and  and  for a l l  t h a t I n " d r o p p i n g " from s t a t e  does the r e l a t i v e o r d e r parameter ^  t  . . At .  the  " d r o p s " i n t o the n e i g h b o u r i n g f l u x o i d s t a t e 7^—/  t  109 In p r i n c i p l e , we may s t o p a t some p o i n t begin again lowering the temprature. moves up aro-und t h e bump t o p o i n t The p a t h  B"- B ' - B  i n c r e a s e d beyond  B  Then the s t a t e p o i n t and out t o B  ;  i sreversible.  Finally, i f t i s  t ^ we r e a c h a t e m p e r a t u r e  t  JD m = 0 , ^ = 0  and  f = 0 . Note t h a t  t ^ I c  . At t h i s p o i n t ,  e q u a l t o t h e second o r d e r c r i t i c a l Comparing  a t which  C  d e f i n e d by y U . ( t ) ~ / c  B " and  x  , but i s  has a v a l u e  field.  the t h e o r e t i c a l behaviour w i t h the  e x p e r i m e n t a l r e s u l t s we note t h a t no bumps a r e a p p a r e n t i n the  latter.  However, p o i n t s on the bump r e p r e s e n t a narrow  t e m p e r a t u r e i n t e r v a l and would be b o t h d i f f i c u l t t o o b t a i n and t o d i s t i n g u i s h from p o i n t s a l o n g  m^ .  I n the e x p e r i -  ment no a t t e m p t was made t o d i s t i n g u i s h t h e bump from  .  H. and M. show the i r r e v e r s i b l e c u r v e t e r m i n a t i n g a t t = 1 , whereas t h e c u r v e  m^(t) r e a c h e s z e r o a t  i s n o t c l e a r whether i n f a c t m = 0 for  H.  and  M.  as t h e e x p e r i m e n t a l d e f i n i t i o n o f  t = .975.  It  have not t a k e n t = 1.  Except  t h e s e two d e t a i l s the agreement o f t h e o r y and e x p e r i -  ment i s e x c e l l e n t which c o n f i r m s t h e h y p o t h e s i s t h a t the system d r o p s t h r o u g h s u c c e s s i v e f l u x o i d For  states.  c o m p l e t e n e s s , we c o n s i d e r a second p o s s i b l e  h y p o t h e s i s f o r comparison w i t h e x p e r i m e n t , namely t h a t t h e e q u i l i b r i u m s t a t e e x i s t s o n l y up t o m^(t) , t h e p o i n t a t  110  which  M  at f i x e d  t  i s a maximum w . r . t .  Hg .  V/e make  no argument why t h i s s h o u l d be so except t o note t h a t t h e e f f e c t o f f l u c t u a t i o n s w i l l be to' lower somewhat t h e c r i t i c a l temperature.  T h i s s i t u a t i o n i s . d e p i c t e d i n F i g u r e 4-8 i n  which the i n i t i a l s t a t e r e v e r s i b l e curve curve a t  F .  AA'  If t  A  i s reached  as b e f o r e .  The  j o i n s t a n g e n t i a l l y onto t h e m^(t) i s i n c r e a s e d beyond . t„  we assume  a g a i n t h a t t h e system drops, through s t a t e s , p f s u c c e s s i v e l y d e c r e a s i n g f l u x o i d number b u t here each s t a t e to to  m(x)  a maximum.  drop a l o n g  t = 1  3  corresponds  O b s e r v a t i o n a l l y , t h e system appears  m^(t).  T h i s curve t e r m i n a t e s a t  w i t h b e h a v i o u r (i — t )  i n the l i m i t  m = 0 ,  t —*» 1.  This  i s i n agreement w i t h a remark o f Mercereau and Hunt, not obviously a p p l i c a b l e t o the attached r e s u l t s . also gives a reasonable  f i t t o t h e observed  This  points.  hypothesis I t may  w e l l be t h a t t h e c o r r e c t curves l i e somewhere between F i g u r e 4.7 and F i g u r e 4.8.  Figure la) (b)  4-2  Thin cylinder  (above) (below)  functions  for variable  Free Energy F(X) Super E l e c t r o n D e n s i t y  Jr(X)  field,  F i g u r e 4~2 Thin cylinder functions f o r variable geometry parameter. "(c) ( a b o v e ) M a g n e t i c Moment M(X) (a) (below; s u p e r c - i e c x r o n Mo men cum P\AJ  field,  -1.0 F i g u r e 4-3 Thin cylinder functions temperature parameter. V,  a;  (b)  \a.uuve J (below)  rice Suoer  J L J I K S I .  g,,y  J.  Electron  \ - A .  for variable  /  Density  ^(x)  .5"  deduced  1.0  Field  X 2.0  field,  F i g u r e 4-3 Thin cylinder functions f o r variable temperature parameter, (c) (above) M a g n e t i c Moment m(x) (d) (belov/) S u p e r E l e c t r o n Momentum p(x)  field,  o  -V-  .b  .8  l.o  F i g u r e h-h Thin cylinder functions f o r variable tenmeratiire . field rameter(a) ( a b o v e ) Free Energy f ( t ) (b) (below) Super E l e c t r o n D e n s i t y ^"(t)  O  .X  .t  .6  .8  /-o  Relot'iVe F i g u r e h-kThin temperature, (c) (above) (d) ( b e l o w )  .9-  i  Temparotu.re  -8  T / Tc  '-°  cylinder functions f o r variable f i e l d parameter. M a g n e t i c Moment m ( t ) S u p e r E l e c t r o n Momentum p ( t )  in unit's  Figure  4-5  (above)  Free Energy of neighbouring f l u x o i d states at onset of s u p e r c o n d u c t i v i t y  Figure  4-6  (right)  Separation of h-t plane into regions of normal, stable superconducting, and m e t a s t a b l e superconducting states o  cf  (ftejiryi,  F i g u r e 4-? R e d u c e d M a g n e t i c Moment m(t) o f p e r s i s t e n t current i n t h i n hollow c y l i n d e r , showing metastable l i m i t xaj  F i g u r e 4-8 R e d u c e d M a g n e t i c Moment m(t) o f p e r s i s t e n t current i n t h i n h o l l o w c y l i n d e r , s h o w i n g maxima o f m ( x ) l i n e mlj,.  Figure  4-9  Magnification of equilibrium curves near metastable P e r s i s t e n t c u r r e n t drops* t h r o u g h s u c c e s s i v e f l u x o i d  limit. states.  Ill  CHAPTER  5  THIN CYLINDERS  Part In  1 - High  fluxoid  state  field say,  cannot there  would  (which  , H  side.  for  a thin  any  fluxoid. are  along  both  and  H  i n Figures  retain X  both  0  A s ws  F  axes  maximum  fluxoid.  i n this  section,,  otherwise  becomes  a n d we  small  important.  a high  field  the c r i t i c a l  consider  form  shall  X  super-  on  field  c a n be no s u p e r s t a t e a t  cease  consider  shall  of  F(X)  higher  b u t become  and f i n a l l y We  this  difference small)  numbers t h e c u r v e s  k—2.  separ-  i n which  with  reach  there  t h e same g e n e r a l and  film  there  Evidently  numcsr,  itself  a thin  d  For low f l u x o i d  the  particular  H„  ,  large but their  like  i sto  each  i s large but  the wall  of thickness  given  curves  from  That  FCHg)  fluxoid  When  both  2  behaves  .When  film  these  temperatures  wall  curves  to the external f i e l d  survive,  of the field  cylinder  either  might  reduced  calculations,  $o/TfSl^  f o runlimited high  implies  exclusion  ~  Z  other  on t h e e x t e r n a l only.  of the preceding  b y AH  and  i n a given  t o depend  set of identical  be no l i m i t  conductivity  cylinder  the difference  i t s neighbour  continue  the f r e e energy  have been found  an i n f i n i t e  from  chapter  i n a thin  w i t h i n the accuracy  ated  a  £  through  exists  The  F l u x o i d Number  the previous  dependent q u a n t i t i e s  CONTINUED  employ  fluxoids,  compressed  to exist only the  and  at  constant reduced  112  quantities  F We  and  occur  p o s e we in  f o r the  powers o f  closed The  and  the  equations  critical  fields  c o n d i t i o n s f o r second  .  first  two  f o r second  order  terms o f the  order  transitions For  expansion  T h e s e have b e e n o b t a i n e d  this  i n Chapter  second  order  of  ^  critical  equal  to  field  zero.  for arbitrary  i s found  This  by  Sl,  t  pur-  F(Mt ^)  of  t  3»  (3.28) i n w h i c h t h e c o e f f i c i e n t s o f powers o f  transcendental expressions  icient  (4.8), (4.7).  f u l l range of p o s s i b l e f l u x o i d s .  r e q u i r e the  equation  d e f i n e d by  obtain f i r s t  phase t r a n s i t i o n s to  X  are Sl  z  s e t t i n g the  0  coeff-  gives  (5.1) We  r e w r i t e t h e LHS  i n terms o f the  p o r t i o n s ! t o the d i f f e r e n c e solve  this  equation  for  quantities  (fi - 2 J J  x  X  . of  From t h e  definition  „ equation  #  2^XHUK  -  X  <£  This f a c i l i t a t e s  s o l u t i o n w i t h the r e s u l t s of  and  the p r e v i o u s  which i s pro-  itself„  ana  comparison of  5  we  (5.2)  A,  2  and by d i r e c t s u b s t i t u t i o n into the p r e c e d i n g e q u a t i o n we  1  This of  -*/7  A  equation determines X  the  have  6  +  then  chapter.  (4 7) 0  X  , at  ,  obtain  (\Hcbl (5.3)  the  second  order  critical  value  which t h e s y s t e m l e a v e s t h e p a r t i c u l a r f l u x o i d  state  U3 by  way  of a  second  restriction plies be  on  f  d«  this  , the  X  sidering  and  the  = A  X  +d  /  the  the  length.  Equation  , we  case  same o r d e r  coefficients  terms  important  r e c a l l t h a t the  +£•  i n powers o f  Z  We  i s that'  i t i s more  p a r t i c u l a r l y the  Retaining  with  but  J2  §  &/'McJ> "to b e  be  uniform,  instructive  expand d/X  the .  terms  to proceed  , we  are  do  (5«2)  as  X  „ and  then  of  now  not  con-  assume v/e  i n comparison  equation  imeasily  coefficients  o f magnitude as d$  which could  S i n c e we  of large $  like  (5«3)  only  retain X  to  .  becomes  solution  X  This  Xi  coherence  Putting  X<£  9  transition.  equation  solved e x p l i c i t l y  follows.  in  order  _  =  reduces  = - /  $  d  to  on  the  the  corresponding metrically ~  $  obtained  to  "low  case the  of  When t h e  between the  toward  the  origin  fields  become c l o s e r  previous  , which  chapter  i s the  mathe-  fluxoid", low  + and  fluxoid  the  two  critical  - signs i n equation  fluxoid two  Z  i n the 1  placed with respect to «  midpoint  result  d*~§  condition that  m a t l e a l meaning of In  I i  t  the  value  X  fields  (5<>4) a r e -  0  , i.e.  becomes s i g n i f i c a n t l y l a r g e  second  of the together.  order  axis  and  critical  fields is  a l s o the  F i n a l l y at  §  two  = /If  sym-  the shifted  critical  A// £/'d c  , the  Ilk  two  critical  exist.  ^ to  We  fields  g r e a t e r than  i s a p a r a m e t e r w h i c h compares a n y p a r t i c u l a r t h e maximum f l u x o i d  s t a t e which can e x i s t  Y  fluxoid  i n that  state  cylinder.  s p e a k i n a m a t h e m a t i c a l way o f t h e d e p e n d e n c e  c e r t a i n q u a n t i t i e s on Y  Evidently  t h i s can  define  We s h a l l of  merge, and no f l u x o i d s  , as though  Y  i s n o t a dynamic v a r i a b l e ,  were a v a r i a b l e .  f o r the f l u x o i d  served  during quasistatic  "as  Y  i n c r e a s e s " a r e s h o r t f o r "as we c o n s i d e r s t a t e s o f h i g h e r  and  higher  and  most o f t h e s t a t e m e n t s  fluxoid".  phrase l i k e  equilibrium processes.  i s con-  Furthermore,  Y  made h e r e  "to the nearest  , like should  allowable  $  Statements  like  , i s quantized,  be q u a l i f i e d  quantized  by a  value".  It is  omit i t . The  c o n d i t i o n t h a t a second o r d e r  p h a s e change d o e s . 2.  in  fact  occur  at  X^  i s found  by s e t t i n g t h e c o e f f i c i e n t  equal zero a t the reduced c r i t i c a l f i e l d X was o b t a i n e d i n e q u a t i o n (3.28) i n t e r m s o f  1  b e f o r e , we c o n v e r t  i t to an expression  We e x p a n d t h e c o e f f i c i e n t s the  fact  that  requirement  ^/^ci>  X  i n powers o f (d/X/  may be l a r g e .  f o r second o r d e r  2  *y  This coefficient and <f , As  and j  using  . keeping  m  (5»2) mind  This leads t o the f o l l o w i n g  transitions  2  //_  in  » H  of  t o occur?  2  +  11  )  4.  -A__  (5.6)  115  In t h i s  expression  Y  and  are taken  t o be o f o r d e r  we have r e t a i n e d a l l t e r m s up t o o r d e r would  appear  transitions  incorrectly  Y«  If term the  dominates other  \  ,  order d,  extreme,  c Y ^ f e X  .  fact,  , then  Putting i n  This  order  i n aY/Af I  as b e f o r e .  X^  (5«*0  from  order  of thin  f o r a l l the cylinders  At  term  gives the con-  t h e c o n d i t i o n f o r second  thin  films  of total  thickness ^ I  T h i s c o n d i t i o n i s much weaker t h a n  by our d e f i n i t i o n  order  the f i r s t  o n l y the second  i s precisely i n plane  second  fluxoids.  to lowest  i f Y ~ )  phase t r a n s i t i o n s  as e x p e c t e d .  that  then  high  undergo  and we have t h e c o n d i t i o n  appears on t h e l e f t . dition  Otherwise i t  that a l l cylinders  at sufficiently  1, and  cylinders,  o Y «  A  . I n  , we c a n s a y  v/e a r e c o n s i d e r i n g s e c o n d  order  phase  *  changes b e g i n extreme well  a t some  Y  l i m i t 't—*• / , t h e s e c o n a o r a e r p n a s e  expressed  g i v e n yl^  second order  order  We and  s i g n gives the value  transitions  in only  H  2  and  i n the  change c o n a i t i o n i s  begin.  At t h i s  of  value  Y  of  a t which ?  t h e second  valid  for a l l  f i e l d s are  t u r n now t o t h e t h i n v / a l l  One must r e t a i n  Appendix  Except  /  o b t a i n an e x p r e s s i o n f o r  value. in  *  the equal  critical  .  by  (/- Y^yU For  , t h a t i s a t some  F  valid  expansion f o r <jp  a l l terms t o o r d e r (  3°  A s b e f o r e , we t r a n s f o r m  §  t o one i n  X  terms o f h i g h e s t o r d e r ,  and where o r d e r  M  up t o i t s maximum Y i n the  the quadratic  expansions expression  Finally, i s determined  we  keep  by t h e  ll6 maximum v a l u e and  <£ / / / e i  one  obtains  t h e t e r m may  have.  (c£/X)  i s of order  Specifically  .  After  ' This  differs  from  were p r e v i o u s l y Define  equation  dropped  X  9  i s of order  considerable  algebra  *• '  (4.8)  1  •  (5.7)  by the terms  i n Y  , which  i n the assumption o f low  a s e t o f "prime"  quantities  as  follows:  (5.8)  Then i n t h e prime  A Iso  Jlfl-l  SZ Q  quantities  ^F  implies  r' - (fl/-i by  analogy  to equation The  exactly  take  j  __  over  quantities,  relations  o  that  f o r high  i n the prime quantities  fluxoid  a l l the r e s u l t s o f Chapter then  s  exactly  i n equilibrium  (4,11).  functional  the equations  p.  an equation o f  - sf)  t h e same a s f o r t h e u n p r i m e  interpret to  one o b t a i n s  interpret  these  quantities are a t low  ^  ,  To  number t h e p r o c e d u r e 4 applied  physically using  t o the prime the  inverse  it  transformations for  o f equations  a l l quantities at the four  89  hold  on  the functions  ing  f o r prime  paragraphs  which refer  As  Y  This  type,  found  which  X  (X)  This order  —  Q  0  critical X  ~  at  = H  F  occurs  and hence  XQ  — — /J Y  both  .  about t h i s  Hg  outer  surfaces  uated  at the inner  portional 0  to ^ ,  total  , so t h a t  loop  in  (3»1?)  ,  directions. to  i f H  , then  Using  one e a s i l y  than  ~ K  0  H^  phase  X ~  ,  88)at  F ( X ) and X ~ 0 ,  about  t h e second situated  I  o f 4^  C  0  Jl  ,^2,  f  When  i n the wall  a t t h e i n n e r and I f ^ plus <^>  i s eval-  a term Hj  and f l u x o i d  the explicit  shows t h a t  second  3 following  current  and^  The d i f f e r e n c e between f l u x ^)  order  f o rg e n e r a l  i ti s equal 1  "second-  0 ( s e epage  the distribution , ^  ,  F = 0  The c u r v e s  occurs  Y  i n powers o f ^  i n Chapter  F  must be i n o p p o s i t e  at large  at  t o be s y m m e t r i c a l l y  Hg =  surface  F  at point  As was n o t e d  When  significant equation  were found  are large  important.  becomes  ( 5 ° ^ )above where  equation  i s f o rzero  of  point rather  t h e minimum o f  , that  and  becomes  Y  , F(X)  ,  on  becomes more  horizontally  minimum o f  ~"  2  /  the expansion  fields  and hence  F(X)  above u s i n g  (3 = 2 1 ) ,  equation  depends on and  i n the follow-  10 .  =  t h e c o n d i t i o n f o r second  i s i n agreement w i t h  about  X  c y l i n d e r having  Is i t terminates  a r e symmetric  The e f f e c t o f  5 - 1 » 5 - 2 , and 5 - 3  i n Figures  F(X)  agrees with  .  summarized  s  = yU> (f ~ Y*) =•  that  result  ^  decreases,  At  The  ^  of  increases,yU.'  changes  and  and i l l u s t r a t e d  shape  order-like" . order  F(X)  the equations  p o i n t s , on pages 8 8 ,  significant  q u a n t i t i e s f o ra l l Y  t o a n example  The  In particular,  (5.8).  , and  becomes  expression ~~{3  pro-  Y  for  implies  H.^  118  Hg ~  for a l l For  lower is and  of  Xg  Xg  merge  the  with  as  , the f i e l d  at which  F = 0 .  , and for^a.'< I  into  and lower  Y  on  critical  5-1•  i n Figure  role  i s plotted  i s a more  ^[4. * /(J  as  discrete  is  plotted  the  whole  range  of  X  extremely vertical and  of  strate  Y  of  limits  with  vertical  Y  while  greatly  at  X  F  = =  The  for a l l "curves"  ,  i s  i n conformthe para-  o f one s t a t e  to scale to  horizontal  magnified  , i s  The maximum „  5  For  are  consist field  of Hg  appears  ,  F(X)  the  can exist i s as a  the horizontal $  p a r t i c u l a r curves  expanded  ~~ ^3 Y  On t h i s s c a l e  a particular fluxoid state  i s proportional  i s also  X^  i n v o l v i n g juJ  fluxoid states.  informative  number, f o u r  scale  , and  V/ithin the r e s o l u t i o n of t h i s diagram  which  i n pro-  i s s u c h as t o encompass  the v a r i a t i o n of the functions  fluxoid  Xg  as a b s c i s s a  of  5-2.  and t h e s c a l e  and the l i f e  line,  #  be-  h a v i n g y^-— 0  In p r i n c i p l e , the external  of possible  small  decreases  of the cylinder.  function  graph of Figure  over which  , X^  Q  , occurring  o f they^L  points.  i t seems more  terms  (d)  range  F  and lower  as a b s c i s s a ,  X^  same  , X^  the separation  cylinder  i s plotted  of  complicated  shown i n t h e upper  X  The  ordinate.  value  % the upper  s o m e 10  ,  and  '— I  At  as independent v a r i a b l e  independently F  X  increases.  fields  of the fields  The minimum  of  Y  decreases  i t s usual  Y  between t h e upper  F o r the example  dependence  meter  , the separation  to  illustrated ity  of the present calculation,,  fields  tween t h e upper portion  to the accuracy  > /  limiting  true  Y  In order  thin then  H%  axis i n to  i l l u -  f o r increasing  a r e shown i n ( a ) , ( b ) ,  scale.  by a f a c t o r  j>  I n ( c ) and of  25.  (d) the  (c),  119  5-3 i s a s i m i l a r  Figure ation  o f t h e maximum and minimum v a l u e s  maximum v a l u e  i s universal  " c u r v e " d e p e n d s onyU>'  vertical  scale  Finally ion  inder This  2 - Finite  a l l the r e s u l t s y  •=  lift  based  together with  0  this  o  energy  F  tative results  4 x 10^ Y  even  This implies  term  i n a hollow 0  „  0  used  uniform.  of very V  small  ~ Q  which  the boundary c o n d i t i o n ^  constant.  cylinder,  equation  For the t h i n  On t h e b a s i s  (3*24),  which  cY^  ,  cylinder  i n powers o f  to o b t a i n the primary  behaviour  o f the  i n the c y l i n d e r .  part 2 of this  estimate  as  cyl-  we have o b t a i n e d a c l o s e d e x p r e s s i o n f o r  for "f—*•  phase t r a n s i t i o n s In  negli-  f o r the t h i n hollow  c l o s e d e x p r e s s i o n has been expanded the f i r s t  on t h e d i s c u s s -  has b e e n t a k e n  the surface implies  n  approximation  becomes e x a c t  Y  on t h e assumption  t u r n when t a k e n  free  i n Figure 5-3.  Length  obtained  G - L equation.  this  scale.  t = , 9 9 9 » and, t o  Coherence  the second  the  and  at  The  s t a t e s as i n  horizontal  of  ,  whereas t h e lower  z  number i n F i g u r e 4 - 7 i s  ,00Z  Y —  i s an a p p r o x i m a t i o n  (-7y) ° o($ = of  to  Y  4, part 2 i s completely  i n Chapter  the order parameter  in  for a l l  /— y  that the e f f e c t  the v a r i -  temperature.  Part In  —  t h e same e x p a n d e d  The g r e a t e s t f l u x o i d  f o r lower  of ^  The same f o u r f l u x o i d  we n o t e  which corresponds less  "curve"  illustrating  i s t h e same f o r a l l g r a p h s  o f change o f s t a t e  gible.  in  ,  5 - 2 a r e shown w i t h  Figure The  diagram  c h a p t e r v/e w i s h  o f the e f f e c t  already obtained  8  of finite  to obtain a quanti-  coherence  l e n g t h on t h e  and t o d e f i n e more p r e c i s e l y  the con-  120  dition  under  Rather  than  with of  which  coherence  modifying  t h e two c o u p l e d  both  H  expansion term  to  approach. F  We  dependent  t o work  measured  co-ordinates  ables  we  i n script  has form  the f u l l  From  the outset  we  and employ  , and t o t h i s  coupled  ===  JX  of this  /  shall  start  direct the a  series  corrections  e n d we  make  such  chapter  letters  H  UJ  and d e f i n i n g  =  quantities.  we  « A  letters  reduced  G-L e q u a t i o n s The  i t i s con-  co-ordinates In  cylindrical  A  critical  write  dimensionless  , and d i m e n s i o n a l .  We  define with  field.  We  shall  field  field  vari-  G-L  also  use the  fluxoid"  i s a.lready d i m e n s i o n l e s s t h e form  we  and o b t a i n t h e d i s t r i b u t i o n s  to the penetration length.  i s the bulk  "dimensionless  IP  A"*" >0  define  i n block  where  on  relative  the remainder  variables  obtained  are looking f o r the f i r s t  i n dimensionless  /o For  .  infinite.  as are useful.  When u s i n g  are  already  G-L e q u a t i o n s ^ (st)  be assumed  to the thin-walled cylinder  approximations  venient  the results  and  calculation  l e n g t h may  as defined  where the quantity  ^  by equation  i s real.  Taking  (1.9) and i n this  121  where  n  i s the fluxoid  dimensionless  n u m b e r , t h e G-L  1  VxvxA  A'  = - y A  The  equations  f o r the quantities  the  equations  f o r ^  should  , A  i n the d  symmetry reduce and  t o two c o u p l e d  4  which  t o jO^  f  of  z  total  we  cylinder  will  i n a plane  tesian inder This  form wall  small. measured  cylindrical  t o assume p e r f e c t  T h e t w o G-L  differential  equations  d  by a plane  into We  ^1/  from  r  surface.  equations  *J f/o)  i n  thin  film  term  a 1  — A^-A,  the cylinder form  we  _  ^  form  subject to the  t o use the simple  car-  temporarily replace the cyl-  f o r the calculation introduce an e r r o r which  of  ^' (/O)  cartesian wall  ' ,  of order  i s already expected  of the cylinder  &±fr A.  of  o n one s u r f a c e and  as t h e d i m e n s i o n l e s s  the mid-"plane" m  d  In order  can only  the correction T  within  , the functional  H =  equations  procedure  define  of ^  of thickness  o f t h e G-L  unwinding  d/X  to the  i ti snot  b e n e a r l y t h e same a s t h e f u n c t i o n a l  film  on t h e o t h e r  0  as  superconductor,  directions.  same b o u n d a r y c o n d i t i o n s , n a m e l y H = H  t h e same f o r m  connected  continue  on the non-uniformity  For a thin jO  have  -  }  depends  wall. from  e  (  A*  In applying these  cylinder  and  ,  i s not a vector potential;  at the origin.  geometry o f t h e hollow  y  f o r a simply  that A  be n o t e d  continuous  become i n  quantities  J  It  equations  t o be  variable  The  G-L  equations f o r the plane are  The b o u n d a r i e s o f T  are defined  inner  surface  T  —  —A  outer  surface  T  —  -t" A  The b o u n d a r y  as  — jfi.  &i  that  series  =  1  =  T h e G-L in  terms  a  0  in  +  ^  solutions  i n T  &,r  +  f,  +  .  of  (°i  i n the t h i n  a  +  r  ^  ;  The  f o r large  £  f  }  ^  0  %r  }  ^  f o r /\  r ' f  z  the discussions  .  We  a  nd  *jr  ITI ^ A  I  i n the form  of  ^  i  (5.10)  •• •  i-  z  &  •  2  ;  ...  (5.ii) and  w h i c h must be f i x e d  coefficients  *L  i n the plane  oY^X  cylinder  Put  equations determine  conditions. cated  assume  expansions  fi  ~  conditions are  i n dimensionless quantities  T h e r e f o r e we  /°  at  In dimensional quantities so  at  note  (Z< here  < only  become  ^ by the rather  those which  ...  boundary compli-  are used  ?3  f  ^  =  r  Z  The in  2.  \lW{o  Oof,  +  exact expression f o r the  dimensional quantities  pression ities  c a n be  When t h e the  f o r the reduced  integrand  expansions  and  corresponding  the  F  +• F  i n  by  &  z  J-  energy A  i s given  similar  ex-  i n dimensionless quant7 *J  o n l y by are  carried  of the  each  out  a constant.  substituted  into  there results  a  form  -h  z  fields  • • •  X.^  critical  e t c , depend  field  may  also  on be  the expressed  A  the results F  and  F  series  approximating  for A  of  as  F  from  critical  form  All  for  free  differs  the  functional  energy  integration  + F) 6  Fo  Since  a power  the  series  =  F  free  (5oi4)  j  i n e q u a t i o n (1.20),  w r i t t e n which  series  T 2. . 1 3^  71 +  -  obtained previously  the f i r s t  present notation with  term  that  l~~  ,  of Chapter  are based  on  In order to k we  shall  correlate  first  ob-  r—  tain terms on *J  r  using of the  n 0  stant  since of  1  '  A *2  series 0  will  evidently  the boundary  2  expansions,  to s a t i s f y  not appear  i s required,  which  Applying  the present series  *j  in  was  satisfies conditions  t h e two H  ,  Only  the boundary A  we  need  boundary  previously  to  We  conditions  the f i r s t  assumed  obtain  term  t o be  conditions  t-d  three  on  con*J  i  oh.  X *-~r  so  that  e q u a t i o n (5«12)  from  we  have  Hz ~ Hi a  °  Assuming true  i  =  H g - H^  except  thickness A  ^  r  ( 5  i s o f t h e same o r d e r a s  i nfilms  i salready  so t h e second  that y U«  so t h i n  negligible)  term  4 <Z  less  i 5 )  (v/hich i s  1  , i nwhich  &/  then  i s A  i n f\  I  Hg + H  -  case  wall  by a f a c t o r o f  0  than the f i r s t ,  The  i  A  third  term  order  A  i salso ~  &  v a l u e o f CL For  two  parts  0  pendence  o n *J  terms  with  To free the  energy super  page  84,  terms  dependence  •  t o CL^ ' t o  find  on  A  f o r  F  into  free  and  with  e x p l i c i t de-  energy  H ,  free  includes  As b e f o r e ,  + i sjustified  i s equal t o the f i e l d  electron  kinetic  Both f i e l d to  energy  energy (Hg -  the present notation using  n/</^o  The " c o n d e n s a t i o n  i n the integrand  the accuracy which  proportional  -  0  condition  t h e magnetic  =  0  %  we s e p a r a t e t h e i n t e g r a l  only while  explicit  £.  ~  .  0  includes  c  A  To l o w e s t  i n the lowest approximation of  a s has been done - o r e v i o u s l y . F  and  <Z  convenience  energy"  In  only  to the f i r s t .  and  was n e c e s s a r y t o a p p l y t h e b o u n d a r y  the  are  compared  as g i v e n b y (5.15),  0  A l t h o u g h we r e t a i n it  of order / i  i nthewall  density )  energy  f o r  i n the hole as discussed  and k i n e t i c  a n d we  w  ©  can write  energy  plus on  density  have  t h e geometric parameter  e q u a t i o n (5.15)  t h e magnetic  yOC  ==• 2 /\jO^  125  f5 = 2*y(/*4^) C  Setting field terms  a s i t must be b e c a u s e t h e  i s u n i f o r m i n t h e h o l e we of  Hg  <|5  and  .  This  t o e q u a t i o n {3,17),  ing  for  Hg -  corresponds to the procedure  in lead-  In the present n o t a t i o n  7*>  =  H,-H,  can solve  (Hz -  $)  and  •>, (H  z  -  $)  (5.16)  2(, + The  first  which to  term  i n the f r e e  agrees w i t h equation  minimize  ,  c a n be w r i t t e n  <Z  Q  which  Z  -  i n view  free  F  ,  /£"  (4.8).  becomes  As b e f o r e ,  JLc° = Q  *J  gives  0  seeks a value  an e q u a t i o n w h i c h  i n the form  -t  )  of  Next we of  Setting  energy  f  = 0  2 0  (5»l6) i s e q u i v a l e n t t o l o o k a t t h e terms  Consider f i r s t  e n e r g y /~~  (5.17)  which  ff  (4,9),  and  the c o n t r i b u t i o n  Is given  exactly  by  i n the expansion  from t h e c o n d e n s a t i o n  126  where t h e l i m i t s the a  apply  t o t h e plane  series expression  region  terms  for  symmetric with  i nthe integrand  zero  terms  ^  .  film.  Since  respect  We s u b s t i t u t e i n t h i s  t h e i n t e g r a t i o n i s over of T  to theorigin  do n o t c o n t r i b u t e .  t h e odd  The f i r s t  two non-  give  (5.18)  The  first  before  and apparently  i s f  magnitude to  ^  or J  p so  that  Applying terms ^  2  w  e  »  of J  Using obtain  line.  r  this  /K  (5.9)  are symmetrical t h e boundary  ^  from  equation  terms ,  on f  the  conditions  (5*13)  t  Q  i s o f order  t h e same a r g u m e n t t o t h e s e c o n d  containing  o f ^ (r)  c o n d i t i o n , and t h e r e l a t i o n  0  T  Keeping  3  & a a, y  ^  has n o t appeared  L e tus determine the  conditions.  must s a t i s f y  for  - K  t h e term  fi  conditions  and even p a r t s  «  "the f i r s t  t h e boundary  +  t h e boundary  ^2  n  here which  we h a v e f o r t h e o d d p a r t  f,r  **  individually. for  i  i n ^  ?o»*  odd  ^ from  t  order  Since  t  l a r g e s t term  areat least  A  compared t o  line  o f order  o f (5.18), a l l  A  ( 5 . 1 4 ) i s i n v i e w o f (5.17)  .  The v a l u e o f  itself  o f order  127  at  A  least  bracket so  .  Hence  containing  ^  i-  2  ^  n  e  if-  i n the second  t h e second  correction  a l l t h e terms  line  line  contributes  t o fj*  A  (5»18) a r e o f o r d e r a t l e a s t  of  only  of order less  A  to order  than  A!~  .  There  ,  i s no  w.r.t. the f i r s t  term.  That i s  In ^  which  the  calculating  i sdefined  integration  cylinder. wind  i n the wall  i n the thin  To c a l c u l a t e  the plane back  energy  both  we a r e c o n c e r n e d only.  into  i n the wall  H e n c e we m a y c a r r y o u t  plane which  the magnetic a cylinder  r e p r e s e n t s t h e unwound  free  and i n t h e hole.  T  as a f u n c t i o n  ents  i n the expansions  *J  ,  Q  ^  which satisfies  Q  of  but  d o e s np_t r e q u i r e  culation.  of  A  +p ) z  ^  and  / 2.  .  F  .  ofthe  H  i n the  The  of  must  H  r  coeffici, H  ?  In particular,  V/e m a k e u s e o f e q u a t i o n  symmetry i n t h e $  Z  }  uniform.  and  (3.19)  directions  In dimensionless variables  reads s  outline  terms  (5.17).  assumes p e r f e c t  V/e w i s h t o e x p r e s s We  (^  , A  a r e expressed i n terms  equation  ~J-y  this  /O —  we  (5.10) and  Equation  are adjusted to minimize  for  which  =  Ffj  energy  and take account  (5»11) t h e n g i v e t h e d i s t r i b u t i o n s o f ^ wall  v/ith t h e q u a n t i t y  of  here  , H  i n terms  the procedure  of the constraints  b u t omit  the details  ( 5 . 9 ) a n d ( 5 . 1 0 ) we w r i t e  Using Hp  fj^  1  to the required  e v a l u a t e d a t JOJ  „ i . e .a t  A % ~  accuracy. T  —  ~™A  From  ,  Hg ,  ^  ,  of the calA ( A )  i n  the definition  and u s i n g t h e  128  relation  A j  ~ f>t ty j 2  Z  (i J  w h e r e RHS i s t h e s e r i e s be  , we p u t  s o l v e d f o r H^  cedure.  expansion  i n a power  This value  into  ( 5 . 1 9 ) a n d Fy  this  calculation  of  H  obtained  i s carried  di  w.r.t.  F , - 0 And  H  ,  2  ^  i s substituted  t o the required accuracy.  K  F^  F„ =0  like  t  less  That i s  J  z  When  dependent terms o f order  term.  pro-  F  H  3  = 0  consequently  We Since ion  the primary  >  H  of  This can  by an i t e r a t i o n  out i tturns out that  , does n o t c o n t a i n any than  i n A  series  i n terms  1  (5.10).  of equation  shall  n o t pursue  o u r v;hole p r o c e d u r e  term  i n A^  which  Ff.  when c a l c u l a t i n g  negligible.  i n the pursuit *  I  »  . the correct-  The c a l c u l a t i o n Several  of  In particular,  o f "unwinding"  to find  only f o r A  more c o m p l i c a t e d .  are justified  simplification  i svalid  i s indeed  becomes enormously  the calculation  of  F^.  approximations  a r e no l o n g e r  valid  we c a n n o t e m p l o y t h e  the cylinder  and working  i n cartes-  ian co-ordinates. Returning  to dimensional  quantities  the lowest  correction  terms w i l l  be o f o r d e r  conclude  that i n thin  order  2.  non-zero  effect  K  dependent  or  „_ff:.__  1"  of finite  ,  We  coherence  length w i l l  ~  cylinders the  be u n m e a s u r a b l y  s m a l l on  condition  that  j we Type  II  may  ^  » o C  -well have  superconductors.  .  This f  >^cd  does n o t imply even f o r  any c o n d i t i o n >2  t  i.e.  on  for  -r/.O  man  -  1  Reduce^/ Change at  fre/S in  large  I  0  X  critical fluxoid.  reduced  fields  Z  3  /•H  u3 V) Vj  . 4  H  Reduced  -z  k  i 1  \  \  i  Fie/c/  X  •i  I \  i  u  I  5-2 Dependence o f Free Energy of p o s s i b l e f l u x o i d s . C e n t r e and bottoms Examples of d i f f e r e n t values of Y .  Figure Top:  ~  -1.0  S2,  I-  F  on  Y  curves of  for full F(X)  at  range  -.01  l-o-  (dji/.i  Reo/acec/  f^ie./c/  X  S  — .  ^ " ,  ^  .1 .1 —  oFigure  _ i Z \  i  x  5-3 of Super E l e c t r o n  Density  y  Top:  Dependence  Centre  f o r f u l l range of p o s s i b l e fluxoids. and b o t t o m ; Examples of curves o f different values of / , "  on  ¥ at  13  6  CHAPTER  OTHER FORMS OF M U L T I P L Y - C O N N E C T E D  In the  thin  cuss  Chapters  hollow  some o t h e r  which r e s u l t s  circular  cylinder.  c a n be o b t a i n e d  without  equation  form,  <—•  volved outer to  here  expansion J2>  t  An exact  equation  and how  X  (4.8)  expression  (3°19),  Bessel*s  (3,20).  (4,1), above.  F  i n  this  were o b t a i n e d  were  f o r the c i r -  to thin  terms r e t a i n e d .  was  f o r certain which i n -  a t t h e i n n e r and  s u b s t i t u t e d i n (3-19)  cylinders,  f u n c t i o n s w a s made  o n p a g e 84  geometry.  use o f the c i r c u l a r  about  a  Taylor  t h e argument  •With t h e d e f i n i t i o n s  (4,2), (4.7), t h i s l e a d s t o t h e f o r m o f I t v/as n o t e d  form and  f o r the f r e e energy  , expressions  expressions  To a p p l y  order  was o b t a i n e d  functions evaluated  of the Bessel's  , and l o w e s t  of similar  i n the present  making s p e c i f i c  On t h e a s s u m p t i o n  s u r f a c e s , and these  obtain  circular  f r e e energy  i s t o o b t a i n an equation  the quantities  modified  method  /  q u a n t i t i e s o c c u r r i n g i n (3«19)  field  hollow  for  namely  cylinder.  symmetry.  resort to numerical  f o r the reduced  us r e c a p i t u l a t e b r i e f l y  obtained,  superconductors  to the analysis of the thin  identify  cular  v/e s h a l l d i s  Central  procedure  Let  chapter  detail  1 - Thin walled non-circular cylinders  I  to  In this  i n some  Section  normalized  The  have d i s c u s s e d  forms o f multiply-connected  w a s the  cylinder  4 a n d 5 we  SUPERCONDUCTORS  F  i n (4,8)  t h a t t h e same e x p r e s s i o n i s  obtained ^4cA  ( i )c o n d e n s a t i o n magnetic  energy  field  necessarily magnetic  its  field  H  parallel  g  .  and i s t h i n  state,  i s cooled  there w i l l density  This  induce  current w i l l  n  ^2T  =  ,  c  accuracy  where  $tf  1  0  as before  by e q u a t i o n  (2.3)  sur-  sense d  ^  of curvature  X of  l a r g e compared  to c / ,  t o be i n t h e circulating  i n a d i r e c t i o n normal to  a uniform  magnetic  H  field  field  i n the  i s also uniform,  i  and (6.1)  three  +(±kz±!AA  terms.  i s the r e l a t i v e  <^  c  J-  i s given to  first  V/e h a v e  +M£( ct)f  %rr  — 1^1  The f l u x o i d  plane  a normal to the  i n g e n e r a l be a  length of cylinder  /  electrons.  by  S  by t h e above  ff  hole  be t h e  and bounded  sufficiently  ^  5  a J  j ^ i ^ u - z i + f ) J  that the r a d i i  of nearly uniform  e  i n the double  not  external  of this  along  surfaces are everywhere  f r e e energy per u n i t  order  o i , measured  so t h a t t h e r e s u l t a n t  "  cylinder,  Let  to the cylinder  be t h e p e r i m e t e r  the cylinder  H» - H  thin walled  to the cylinder.  I t i s assumed  superconducting  hole,  three  kinetic  i n cross-section, i n a uniform  i n n e r and o u t e r  current  a long hollow  thickness  ,  If  The  now  i s uniform;  p d ^ < S  e  by  of super-electrons i n the w a l l ,  i n t h e h o l e , and ( i i i )  and l e t p  The w a l l  surface,  H  energy  surface normal  inner wall,  both  (justified  i n the free energy the f o l l o w i n g  energy  circular  o f a plane  face.  and  i n the wall  of super-electrons i n the wall. Consider  area  uniform  ) and r e t a i n i n g  terms (ii)  by t a k i n g ^  (6.2)  F  O f '  9  .  density of  associated with the hole  superi s given  132  Jp  Js which  i s readily  <r /7  integrated since  »/^/  t  a n c i  ^  a  r  e  uniform.  7 From in  t h e g e n e r a l argument g i v e n i n Chapter  units  ^  and  from  of  (j)  .  0  (6.2)  i s identical  Consequently, non-circular then  free  energy  t o (4.5)  thin  cylinders,  order  p"  <p  i s quantized  c  to eliminate  H  and  1  energy  STTJ -/  =  Hck> peY  1  p r o v i d e d v/e p u t .  a l l the results  a second  (6,3)  and  v/e o b t a i n f o r t h e f r e e  f o r the reduced  This  (6,1)  Using  3,  o f Chapter  4 may  In particular*  phase t r a n s i t i o n  occurs  be  i f  applied to  ZSd/.Ap  ^  I  }  at  2 For  a circular  (6,5)  reduces  cylinder  cylinder immediately  the simple  5 ~ TfA  l  t o (4.7)  relation  and p  (4,2).  and  between  ~ ZffSli  S  a  nd  p  9  equation  In the tends  circular to  obscure  the  way i n w h i c h  X  a n d yd-  see  t h a t yWp/Z  .  t h e s e two q u a n t i t i e s (6.1)  From  enter the definitions  and t h e l a s t  i s equal to the ratio  o f (6.2)  two terms of field  energy  of we  to kinetic  energy. If fluxoid  the external  field  due t o a p e r s i s t e n t  i s zero,  current,  but there i s a  e q u a t i o n (6,6)  non-zero  becomes  simply  =  <f>  a  VT  \pHc6  T h i s may b e i n t e r p r e t e d the  temperature  on  t h e maximum  condition is,  i s easily  n(  i n terms  dependence fluxoid  =  o f a c r i t i c a l temperature  of A  state  t>o and  which  Hb  can exist  T  .  D — p IJ  fluxoid  i s e q u a l t o t h e number o f c o h e r e n c e  condition  at given  transformed t o the form  t h e quantum number o f t h e g r e a t e s t  exist  ; o r as a  c  state  through  This  , that  which can  l e n g t h s around  the peri-  meter ,  Section So been studied.  2 - Torus  f a ronly There  and Loop  "infinitely  a r e two r e a s o n s  ing  two d i m e n s i o n a l geometry  and  secondly, the relevant  long  hollow cylinders  for  comparison with  end  effect  an a  theory.  unstable condition phase  transition.  connected  forthis,  simplifies  n o t always However,  length  increasing  hollow cylinders  been c a r r i e d  as long  superconductor which  analysis, out on  as one would  like  i s rendered negligible  F o r example,  a t t h e end o f t h e c y l i n d e r Evidently  the result-  i ti s not obvious that the  cylinder  i t s length,  first  have  the mathematical  experiments have  though  i n a finite  sufficiently  long"  i ti s desirable i s essentially  t h e r e may which  by  be  triggers  to study a doublyfinite,  Consider  134  then  the simplest  cross-section netic of  field  axial  uniform led  such,  radius H  parallel  e  throughout  —  Unfortunately ordinates.  from  f o r A  obtain  first  i s not separable  described  f o ra general o n page  The f i r s t  with  f\  replaced must  rotational  term  from  76  torus  by  f\  e  torus After  field f o r using the  this  ( o r "loop")  cz*i"fcics.j-  i n powers o f  v/e s h a l l  having  d  ^  R  this  co-ordinates  of  i s found series i n  by equation  (3=27)  potential of the external  be i n t h e L o n d o n Gauge,  i n the expansion  of the free  as a power  series i s given  the axis,  *js  i i s i - d i \ V G T~OC]_UH, IT S  (3»23) a n d  equation  , the vector  In cylindrical  term  .  i s expressed  of this  symmetry about  cylinder.  first  i s found  (3.26) w h e r e ^  .  A  i n toroidal co-  results.  o f the expansion  3^  equation  field,  where  u.s^33rnii!riG "til© s e c o n d , cx'cl G if  ,  J-  3 we a r e  #  some a p p r o x i m a t e  term  mag-  A  *L  A  procedure  and  On t h e b a s i s  as i n Chapter  c a n , however, c a l c u l a t e t h e c r i t i c a l  ^  The  equation Art  /?  external  of the torus.  Proceeding  the discussion to the thin  energy  radius  i s nearly  phase t r a n s i t i o n s  To the  to the axis  London's equation  We  order  iteration  and  of axial  , s i t u a t e d i n a uniform  the torus.  T-S>  A'  limit  torus  symmetry, and t a k i n g  —*>  a  ^6"  to a London-like  second  a circular  ~3~ i s  and s i n c e  the torus  i s t h e same g a u g e  has  as f o r  135  where and is  Vj  i s t h e volume o f t h e t o r u s .  the c r i t i c a l given  value  H  of  i s thin,  In  second  , this  t o the second  a straightforward of ^  order  }  which  We  to the consideration  density J  We  equations cular of  A  .  i s nearly  s  — -j^r n  loop.  the condition will  and ^  where prefer  From  /9 ~ T?".St o u s e /}  cross-section. terms: Energy  involve  examples,  thin  of  thinness.  s a t i s f y i n g both  loop  i s the cross-section 1Y<6-  involved  one i s l e d t o  condition  and t h e t o t a l  than  down t h e  the condition  some d e g r e e  the latter  rather  the  current  current area  of the  i n the following  the generalization to loops  The f r e e  i  Condensation  fora  Put the i n t e g r a l s  of a t h i n loop,  uniform  to facilitate  the three  field  t o determine  From p r e v i o u s  that  R  critical  manner, one c a n w r i t e  t r a n s i t i o n s to occur,  believe  4 «  reduces to  order  i s required  t o be e l e m e n t a r y .  turn  (6.?)  R  of radius  coefficient  *  1  t h e "reduced f l u x o i d "  i s identical  cylinder  cease  *  have used  the torus  for  order phase t r a n s i t i o n s  by  w h e r e we  which  f o r second  Q  I  If  The i n t e g r a l i s e l e m e n t a r y  e n e r g y c a n be w r i t t e n  of  non-cir-  a s t h e sum  116  Kinetic  Energy 2  Magnetic  /  \2-  r  Field  Energy  IL Z  Z where  L  given  by  Define and  cr i s the inductance  as b e f o r e  of 7  fluxoid  V  h  /'ftR  I  <j£>  c  i s  energy  we  ,  Adding  the three  o b t a i n f o r the reduced  * Hair  \  z  /  [  free  (6.8)  an e x p r e s s i o n i n standard  gHd'X  case  ^  —  and e l i m i n a t i n g  •  i s again  Again,  ^  free  The f l u x o i d  f  energy  This  the reduced  the reduced  terms  of the loop.  ' '  2  a l l the results  o f Chapter  form  V-TT^RA 4 may  2  f o r both  that  a second  thin  loop  the thin  order  phase  of circular  inductance i s  cylinder  and t h i n  change w i l l  ( 6  '  9 )  be a p p l i e d t o t h e p r e s e n t  using the appropriate quantity f o r y U .  quantity  where  occur  .  X  loop.  i s t h e same The c o n d i t i o n  i s y U ^ I ,  c r o s s - s e c t i o n an approximate  value  For a of the  137  In  this  case  "A / This  c o n d i t i o n depends  loop  radius.  loop  000*6-  t o compare  thickness  <?L  this  , which  on  f o r  and only  R ~ /0<tr , J-/A  , we h a v e  condition with i s  cd ^ A /R  we h a v e  < .36 that a  weakly  .  on t h e  *6-/k i  . 6 2  I t i s inter-  f o ra thin  cylinder of  The c o n d i t i o n o n t h e  i s much w e a k e r . Equation  trary  strongly  To i l l u s t r a t e ,  R ~  for  esting  becomes  *  1  while  & I  the condition  (6.8) a p p l i e s  cross-section,  than  o  evaluate  L  symmetry  i n order  Of c o u r s e ,  provided  equally  that  well  to loops  the greatest  of arbi-  dimension  f o r i t s a p p l i c a t i o n one must be a b l e  V/e d o h o w e v e r to justify  require taking  ^  the loop uniform  t o have around  i s less to  axial the loop.  138  not of  Section  3 - Two C o u p l e d .  In  s e c t i o n we c o n s i d e r  this  i n contact hut coupled t h e type  axially  described  with  We r e s t r i c t  Loops  magnetically.  i nthe previous  axes p a r a l l e l  the d i s c u s s i o n t o loops  remainder  loop  of this  and should  "significant  i nthis  section  areas  loops  have  chapter  n o t be c o n f u s e d  chapter. A  1  , A  inductance rng  I  0  in  ^  ,^  are positive t h e same s e n s e  metry sider  M  a n d k>  here  treated  field  and  A/ £  R  .  c  to a  H  .  e  , and  Throughout particular  subscripts referring to The l a t t e r  use i s e n t i r e l y  may h a v e d i f f e r e n t shape  L^ , L g  cases  sxaxe  cross-  (within reason).  respectively  H  i n v/hich  down t h e t h r e e  (ZnR)  g  .  The  and mutual  1  super  , Ig ,  inside  i nthis  positive.  changes  choosing  I  produce  Consequently  H  reiaxive  currents  they  are essentially  by a r b i t r a r i l y  Condensation  loop  when t h e f i e l d as  ny.ve  t h e .loops  and t o t a l  Following write  loops  v/here fc, i s t h e c o u p l i n g c o n s t a n t .  superconQuccxng  densities  4.  The l o o p s  self-inductances  A  same  of arbitrary  ?  circular  o f t h e same r a d i u s  with  rings  section are placed co-  subscripts refer  p o i n t s " i n Chapter  avoided  Two t h i n  t o t h e e x t e r n a l magnetic  m a d e o f t h e same m a t e r i a l , h e n c e the  two s u p e r c o n d u c t i n g  We  sign,  1  ' and  restricted shall  geo-  n o t con-  but these  of the previous  energy +  I  the loop i s  may b e  s e c t i o n we  t e r m s v/hich c o n t r i b u t e t o t h e f r e e  + f )  sxecx-ron  one a x i s as p o s i t i v e ,  the procedure  I A, (-ZJ-,  In  A  z  (-ty  +  energy.  139 Kinetic  energy  U,4 Magnetic  The  field  I,  -h  ft: 2. 2-/7  energy  respective fluxoids  a r e g i v e n by  .a.  +  c  v/hich a l s o f i x t h e s i g n c o n v e n t i o n equations  are easily  eliminated first  from  solved f o r  <pci  —  (j>  co  S  <^>£ .  t o loops  o f equal  are  produced  by cooling  e  loop w i l l  equal  radii  This R  Then  z  e  -  0 )ft, c  which  9  energy-.  We  have  field quantum  the loop  K  be  shall t h e same  +  i n  state) the flux  of  currents ar  ~  4*p  fluxoids  , the fluxoid  "2/2-  -M  These two  can then  » f o r i fthe non-zero  \  (n/? H  and 1  <£>ci «  i s consistent with the r e s t r i c t -  (to the nearest  C  c  ,  Cf  that the loops  i n a uniform  by the loop,  closed  1  restriction  ion  each  l  ^>  the expression f o rthe free  make t h e f u r t h e r  fluxoid  of  IzM  A  -  M  0 The A  two l o o p s a r e c h a r a c t e r i z e d , I>  „ Ag  1  the  form  1  of  ,  L  physically  , M , although only  2  7  I n the case  entered through the single  convenient  to define  the  coupling  the  yM-'s  and  two " U  However, all  o f each  "  A  1  A 2  affects  loop the geometric  p a r a m e t e r yCC-  ,  Here  i t i s  parameters: =  / M  \FtLx.  (6,10)  parameters  > ^ = y (i  -  2  V/e d e f i n e  i n e q u a t i o n (6,9)  F  • - /  quantities  loop  f o rreasons  five.  free  The  constant  =y,(i-k{jpi  v,  as  five  the ratio  of a single  properties  by t h e f i v e  o f symmetry  fe)  (  i n t h e e q u a t i o n s we s h a l l  the normalized f i e l d f o ra single  k.  loop,  variable  X  and d e f i n e  6  .  1  2  )  use  exactly  the reduced  energy  = reduced  $77 7/'Hct(Zrr*)(/>, free  energy  i s found  \  +4*) t c be  0  J  (6,13) (A,+A)\  ^  F  0  l4i  In and  in  ^  equilibrium,  ^2  »  F  i s a minimum w i t h r e s p e c t t o  both  consequently  A  ty .  Eliminating following  V/e s h a l l a  path  from  relation  refer  these  t  and  plane  equilibrium.  e q u a t i o n s we  equation  along which  The v a l u e  by s u b s t i t u t i o n  (6.It)  obtain the  ^  as the "path"  ,  c a n be f o u n d  simultaneous  between %  to this  i n the  quasistatic path  X  .  of  i n that i tdescribes  the system  X  travels i n  a t any p o i n t on the  i n either  of the preceding  equations. Although part  of the path  we  have n o t y e t c o n s i d e r e d whether  corresponds  (6.14) b e i n g n e c e s s a r y some p o s s i b l e within "unit All In the  paths.  the unit square"  paths  word  0  be u s e d  ^^)  /  path  ,  0  F  we  to that branch  first  meaningful £ Jr  2  £ 0  beginning at ^  {  (The  -  I  We ,  term  sense.)  , at which  branches.  discuss  p a t h must l i e  .  precise  , p..^ - )  (6,15) h a s s e v e r a l  (equations  shall  henceforth i n this  i n c l u d e t h e p o i n t p., - I  general, equation  of  but not sufficient) The o n l y p h y s i c a l l y  square  will  t o a minimum  a l l o r any  X = 0  restrict ~ I  >  ,  l42  lying within In  order  the unit  Y% (j-i)  '  V  3  0  t  h  e  is  l  e c  4-1 w i t h  u a  '  t i o n m  a  i n fact  y  replacing y  scripts  y  b  X  sect. and By  The  a t p., = ^  "path"  t  square.  the path  Three  Type  1  types 1  y  I I  :  jj  f  to  =*  /  ,  0,  I I I:  and  0  h  ~  t  ^  fte'(ps)  •  /  P  x  9  ' I  yl.^  then  i s  o f c u b i c s as i n parameter  l e t the subcase  lies  ®  ^  —  ^  below  where t h e y  decrease  from  c a l l the  function  e  fpi) ~  X .  1 to 0  from  the condition  inter-  y^.' f i  9  '  of the unit  occur: ,  p ,  a n d p2.  aecrease  t o 0, > j  , ^j  decreases  monotonically  monotonically to a  finite  value.  > ^ to  t  - Yz  and p  decreases  positive Type  S  terminates a t the boundary  i\  ^  H  t  by l e t t i n g  monotonically Type  R  Then y  on).  o f p a t h may ^  t  e  (the special  o b s e r v i n g t h e c o r r e s p o n d i n g p-^ definition  h  the discussion  ISj <  - I  i s found  d  as o r d i n a t e and t h e  To s i m p l i f y  discussed separately later « except  n  written  e  replacing  U ,  a  of the path  i t i s t h e same f a m i l y  be a s s i g n e d so t h a t  /z. ^ 1 )  a description  t h e f u n c t i o n ^i^>)  of cubics5  family  Figure  and c o r r e s p o n d i n g t o i n c r e a s i n g  to obtain graphically  LHS o f ( 6 . 1 5 )  a  square  • , p  passes  decreases monotonically  f  through  a positive  increases to a finite  minimum  value  when t h e p a t h t e r m i n a t e s . It  yU. -Z the a  f o r iZj  i s possible  Figure 4-1  f o rwhich  ^-^i^O  a r e concave  three types  physical  t o be s l i g h t l y  o f paths  situation  was d r a w n . upward  negative i n contrast to Curves  from p  ~0  t  described remain  not f a l l i n g  within  o f Y'^'} to p  valid,  F  - /  f  P  f  ~Z  O  R  , and implies  t h e bounds o f t h e p r e s e n t  143  discussion* It of  F  ,  the  path,  i sclear  We  turn  that  now t o t h e q u e s t i o n  i fany, a t which  condition  that  X = 0 ,^  at  F"fytj^-*)  condition  must have Since  a minimum c e a s e s  derivatives  mum  will  mum  reduced  found  by making  , equation  p ^  which  the point  to exist.  A  on  necessary  (6.16)  I^F  and and  ~  ~  ^  a n c i  s  i  n  c  i nthe unit  a metastable  straightforward  e  a  H  second  square,  (6.16) i s v i o l a t e d .  when  a t which  '  state  (6.16) a n e q u a l i t y , a n d s o l v i n g  3y  minimum •  0  J  are continuous  X  a  f o r a minimum must be p o s i t i v e .  for ^  t o e x i s t only  field  (6.14),  v/ith X  cease  /o ~  that  i s t h e case  i s  be a n extremum i s  t h e same s i g n  this  partial  implies  /  =  o f determining  /  This  ~^-x  t  this  mini-  The m a x i -  can e x i s t i s simultaneously  c a l c u l a t i o n , and by e l i m i n a t i n g  (6.16) r e d u c e s t o t h e f o l l o w i n g e q u a t i o n i n t h e  J.  ,  plane:  l l ,  C  ^  3 V~  Simultaneous point  I  s<i(1-^)1)if*  [}  beyond  with  which  the path  this  in  (6.15) w h i c h g i v e s  equation  f o r ^-y  ,  say f o r  The g r a p h i c a l  equation  superstate  explicitly  equation  W  s < x . ( i - ^ ) n  (6.17) c a n be s o l v e d the path  '>  (6,17)  solution of this  on t h e path  2(,  \)(,  ^2  a n c  cannot * "this  i ngeneral  solution  gives the exist. substituted  a sixth  i nthe ^  , ^  degree z  ,  ll4 JL-T plane of  i seasily  t h e phase  ular  transition.  hyperbola  refer  point  v/hose a s y m p t o t e s  The b a r r i e r  ( i f a n y ) where  path  curve,  k*  different this  ^  Point  P  Pivot  Point  P  2  the limit  as  i n the limit  > 0  k.  straint  of passing  detail  f o r yU  »  (  y^x,  shown t h a t a l l c u r v e s  the following  =  f  two " P i v o t P o i n t s "  ,  , the barrier  curve  ^  with of  inde-  P  of into  1  h-  to  0  where-  and t h e  0  c a n be v i s u a l i z e d  a straight  „ Pg „  a r e shown i n t h e upper correspond  f  line:  o f the corner through  -  becomes a c o r n e r  the asymptotes recede  f o ri n t e r m e d i a t e v a l u e s  Figures  curves  //-^)  k —*- /  bending  These  square.  I t i seasily  becomes t h e s t r a i g h t  barriers  meaning-  i s , f o r t h e same l o o p s  ^  ,  gradual  and  atthe  1  In  Curves  has no p h y s i c a l meaning e x c e p t  0 t o 1j t h a t  through  shall  J  Pivot  curve  We  (6.17) a s t h e " B a r r i e r  of  the family of barrier  separations.  of  identified.  i ti s i n t e r s e c t e d by t h e p h y s i c a l l y  v a r y i n g from  f a m i l y pass  pendent  are readily  hand b r a n c h  curve  the nature  (6.17) r e p r e s e n t s a r e c t a n g -  i.e. within the unit  Consider fixed,  and u s e f u l i n d e t e r m i n i n g Equation  t o the upper r i g h t  Curve".  ful  visualized  with  Several paths  part  to particular  line  of Figures  as a t h e con-  (with 6-1  to  arrows) 6-4,  cases  t o be d i s c u s s e d i n  types  o f phase  shortly, I n t h e two l o o p  system  three  transition  Lk5  0  are  possible:  l)  simultaneously 2)  mixed  at  first y,  p  > 0  v/hile order  , 3„  > O  •?*  1)  A  the  origin  second  origin.  The  k'  >o  one  transition  i f the  ^  lies  case  *  (  to  2.  v\  of  ^ /  (  order  and  order  same s u p e r s t a t e ,  system  becomes  i f the path  6  and  unstable  passes  .  independent}  through  through)the  2  either  other or -(b.iy;.  0  of  )  to-  ( 6 , 1 9 )  furthermore, i r  i s assured  ( 6 . 2 0 )  transition).  situation.  From  are  "*"¥'  for a l l .  origin i f  ( 6 . 1 2 ) ,  does not  and  (Mathematically i n which  s a t i s f i e s the  condition  correspond  In practice,  necessary  situation  (6.14)  the  equation ^  transition.  of which  through  i ^ /,  ( 6 . 2 0 )  a more c o m p l i c a t e d one  pass  .  realizable  ( 6 . 1 9 )  branches  a second  b e h i n d (or p a s s e s  » condition  path w i l l  definition  physically  second  field,  ,  ^  ditions  some  (6.19)  i m p l i e s the  "  The  assures  loops  when  /  (6.20)  From the  at  both  I  gether v/ith  possible  the  occurs  barrier  i s the  ,  ,  i n the  i n which  c o n d i t i o n s are not  z  transition  loop undergoes  other remains  These t h r e e  y&4f ^  order  i n which  .  ^  7  transition  a second  transition  latter  I  order  i n which  the  order  and  $ and  undergo  transition  transition 3)  second  to  for  l^,^^, ^ ~^ path  has  two  ~ 0  the  a gives stable  c o n d i t i o n s f o r a second  evaluated at ^  a  t h e r e f o r e , con-  sufficient  the  (6.19).  order  second  rise  order  transition It  F  Jl  search  i n powers , p2.  at  »  to o b t a i n these  , p^  t  i s  .  f o r second  o f p-  F  1  X =  i s interesting  straightforward of  occurs  given  ,  order  conditions using  transitions  Correct to  by  the  expansion  q u a d r a t i c terms  in  by  2.  The  critical  p^  equal  to  transition fulfilled  field  0 which  does  occur  and  (6.20),  2)  I n some c a s e s ,  in  the  we  apply  p,  the  which  X = 1  »  order  phase  at  pz.  > 0  for  v/hich the  so  „  The  and  •  This  that  the  two  loop.  To  w h i l e the  ?  transition  which  occurs  must l i e below t h e  ,  order  (6.16)  one  are  (6.19)  loop  other  remains  discussion  p a t h must t e r m i n a t e  c o n d i t i o n s are  second  i n which  keep the  £y[J^  A  o f pj.,  gives precisely  occurs  transition  state.  barrier  following  f o r each  transition  Then i f a mixed  changes phase,  1  coefficients  c o n d i t i o n s f o r a minimum  a mixed  =  the  i f the  subscripts .  setting  X  same s u p e r c o n d u c t i n g  1J2.  6  second  by  gives  ~px. ~ 0  at  undergoes a  i s found  succinct  implies i t i s loop  1  on  axis  the  path  necessary  3-,  terminus and  suffi-  cient : (i)  asymptote  pu, (ii)  path  (/-  lies k?)  intersects  intersection s  to the  left  of  the  unit  squares  above  the  barrier  </  the  p^  axis  curve  1  If  yi-f+^+W*) It  i s not necessary  by  itself  of  a second  value  undergo  of  loop  X  w h e r e p>~  loops.  second  order  transition  transition  this  3)  I f t h e above  order  Xwineet  >  i f X  may  which  i n the  transition.  6 o  2i)  would  presence  The  except  I  critical  of a single  This  loop  cannot  justifies  2 will  case  n o t undergo  occurs  at  undergo  the term  Since  f o r the t r i v i a l  i s further increased,  o f t h e same r a d i u s  for  i s , a loop  T h i s means t h a t l o o p  transition,  transitions.  ; that  (  transition i s  ,  order  order  order  )  i n t e r c e p t g i v e n b y LHS o f ( 6 , 2 1 ) .  second  loops  y^rl  a second  0  >  (, ^ /  f  undergo  f o r k  coupled  yU  a first  i s the path  uncoupled  2  that  f o r a mixed  1^2. ^y^x. of  \  Ii7  since  a  X = 1 .  successive  "mixed  a  Two second  transition"  case. conditions are not f u l f i l l e d  phase  transition  will  least  one l o o p - w i l l  occur  change  p  a t some  f  >Q  then ,  p^  a first  >0  order  .  At  state discontinuously. t  In  writing  involve  nothing  only with In the  general range  There system  down a n a l y t i c worse  c o n d i t i o n s f o r phase  than  quadratic  equations)  the terminal point of the physically t h e dependence  of  F  , 1^  o f s t a b l e and m e t a s t a b l e  a r e , however, reduces  three  special  t o "standard  form"  transitions we  are dealing  meaningful  , l g » pf  »  states i s quite cases  (which  i n which  on  path. X  over  complicated.  the two-loop  so t h a t t h e r e l a t i o n s  of  148  4 can  Chapter of  be applied  appropriate  correspondence  parameters.  Special  Case  1 t  Completely  Clearly by  the  the  analysis  two-loop  F  separates  of  b ymaking the  ^  as  »^ 2  j  of  the  arms a r e  into  sum  the  only.  e v e r p-^l type  Special  o f two  We c a n  type.  instability  strikes  in this  Equations  any  the  independently  preceding section. appears  be of the  loop acts  analysis  functions o f X .  v / h i c h may  the  each  The  barrier  2 s  Completely  coupled,  loops  are  not  the  too  iS  »  x  independently  f o r the  system whose  separately.  (or a na x i s ) f i r s t  of  the  n o t e how  i s a corner  ,  f  transition  If  >  curve  f o r p.  curve  us just  be t r e a t e d  functions respectively  draw a p a t h  barrier  limits  may  and  give  ( 6 , l 4 )  s t i l l  Let  and  limit.  terms,  of phase  Case  k - 0  uncoupled,  Which-  determines  system,  k, = /  different,  then  since  L  'de-  A  pends  logarithmically  circular \h>,\ l ^3,1 }  cross-section),  « i  .  Approximating y- J  9  7}  which p-l and  i s the  *  The  intersects  we may  path  lies  cross-section radius put  approximately//  very near  the  F  using the  F  with  ^  (for a ^-a  and  d i a g o n a l a n d Jf relations  • ^ O  L) ~ t  U<Aln  1  +f  standard  plane  the  free energy  the  2_  y  o n Jr  the  the  x  form  of  barrier  yU  becomes the  ~^M  straight  diagonal as expected a t  .  f  line  In  the  (6,18)  l>9  We n o t e to  i nparticular  a thicker  the  system  rather  a value  approximates  loop  &  t  2,  by loop  dependent the value  thin  ~* /  ^  loop .  takes  closely  The b e h a v i o u r  have  of  on the value o f  on the p r o p e r t i e s o f loop i t would  coupled  1  while  i n t h e absence o f  1.  Special  Case  3 *  Identical  Consider identical  loops.  //- ~p., ~p~2_ AT  of a very  one, i . e .  i sdominated  than  ^2  t h e case  t h e reduced  one f i n d s  , and t h a t a necessary  /f~~ 0  other  ~  , or p  f  equation which  » ^/ ~ ^z.  f r e e energy  equation  I f we m a k e a c h a n g e o f v a r i a b l e s  then  8  loops  o  that  F  This  non-physical branches  f o r which  into  *"p2. •>  i s an even f u n c t i o n o f F  i s  i s i n agreement w i t h t h e path  satisfied  SJ^.  &  c o n d i t i o n f o r a minimum o f  i scertainly  s t i t u t i n g p, - p^  (6.13) f o r  ~J/z this  the equation  » ^ u t " h i c h has  Is not true.  for  F  Sub-  and s i m p l i f y i n g  yields  F - -Z?  +  f  y-  +  ( which  i s just  I n t h e p.i barrier in  , p t ^ plane  curve  (6.17)  the standard  2 form  the path  of  intersects i t 3  i  w i t h y U . -ytf  F  f  (/  j-k)  i s t h e d i a g o n a l whereas t h e  i s symmetric w,r,t,  thebarrier  /  the diagonal.  Putting p  t h e diagonal as expected <  \  /<i(i+k))  ~J' 2  t  at  150  We  note  will  that two i d e n t i c a l  undergo a second  first  order  change  Let l o o p s yU.  2^- &ytf;(/-k)  f  f  o  r  this  A n example ,  pivot points are enclosed  nearly  ^  these  bilities  which  t othe functional  three  roughly  discussion  o f second  ( i ) /() < I  , the path  cases  meaningful  yU M < <  f /  x  order  /  , ( i i ) JS, £ /  When lies  The more &  i in-  close t o the the diagonal  a t t h e minimum o f ^ L ,  The f u n c t i o n a l loops  fall  remain  into  lies  approxi-  similar  I  a l a r g e number o f p o s s i loops  o f medium  We c o n s i d e r  , andc l a s s i f i e d as  , ( i i i ) /i,  loops.  o r near one  has a l r e a d y been covered  y  .  relations  a r e good  forphysically  phase t r a n s i t i o n s .  e x a m p l e s h a v i n g ytf^  i s shown i n  part o fthe path  identical  there  . The  intersects  categorized as "dis-similar  The case  three  t h e bend.  configurations will  special  „  a n dt h e phase  bends upward.  Theb a r r i e r  relations  ®  , /.0  follows the diagonal  , then  „  ,  arrows)  A - 0 , . *r  three paths  i s precisely  t omathematically  many  t o these  fora l l  (with  dots 5 thus  solid  ft *- '  /, ~  i s illustrated i n  The paths  the sharper  the physically  coupling".  finally  .  ' ^1)  t o make  3T (l " ''/^l)  While of  1  down t o t h e o r i g i n .  applicable  case  , the path  the loops  t othe diagonal  mations  A  ~ 3 (I ~  sufficiently  Consequently, close  with  and a  identical  We h a v e  i n small circles  corresponding  For small  identical  diagonal ^  ^ ^ ' ^  down t o ^  creases  at  p o i n t s a r e marked  6-lb.  closely  ^  .  z  a r e shown i n 6 - l a f o r c o u p l i n g s  The  Figure  y  o fthis  ^ =  harriers  function  , /S > I  >f  t  I  coupling.  d i s c u s s i o n t o twon e a r l y  t h e caseyX  - ZS  aT / ^ i ^ 1 j ^ ^y^z  p h a s e change when u n c o u p l e d ,  and  transition  having  a t some s u f f i c i e n t  .  6-1 f o r y U  Figure  order  us extend  ytfx  (  loops  »  I  .  They a r e  i nthe  151  illustrated part and  (a) gives part  k  At  ( i )  ~ 0  , at which  creased  line  ition  yU  .7^  at ^  occurs  ~ • $~  x  at  X  mixed  S^~*-  phase  decreases  For  I  ?  — . sT ^  f  ~  of which plane,  2  •  t r a n s i t i o n occurs  A-  k,  As  i s i n -  from  .55  a first  at  k. - * 7S" , ,  0 at  little  up t o a b o u t greater  ®  intercepting the  becoming  the barrier varies  .  ,^  1 becomes n o r m a l .  loop  square  i n the  monotonically  . A  intercept  the unit  respectively,  .  decreases  point  the  Within cal  ~  a n d 6-4  and b a r r i e r s  6-2,  , Figure  at  6-3»  X/S/)  , the path  axis X = 1  the paths  (b) gives  Example  6-2,  i n Figures  the v e r t i -  a mixed order  trans-  transition  occurs,  Example  ( i i )  k. ~ 0  At p  ,  axis  z  decreases the at  corner k. ~ 0  this  6-3,  , Figure  the path at  .97  -  decreases  0 when  a first  barrier.  order  As  k-  which tinues barrier occurs  ^  goes  within  the path  the unit  up t o k ~  1  intercepting the t h e p%  o f k/  '  the barrier a mixed  square,  until  swings  strikes  cut of the  t r a n s i t i o n occurs i n X > /  i tagain  and a f i r s t  i s  Consequently  when t h e p a t h  t o 0 a t some drops  intercept  T h e b a r r i e r a t It — 0  arm i s a t  i s increased  continuously  to increase  „  t r a n s i t i o n occurs  way a n d f o r a c e r t a i n r a n g e  --' /€>  i s increased  k. — . f  whose h o r i z o n t a l  yJ^  }  monotonically  , As c o u p l i n g  reaching  A 2  order  .  As  k*>  con-  s t r i k e s the transition  152  Example  At  k  (iii),  0  =  As  , a first  value  order  of  -  JO — ~5 (I~  at ^  occurs  lowers  but never  t o 0.  As  and t h e c r i t i c a l  f o r a l l k,  .  This  value  of  terminates  ^  f  order  the discussion  systems.  S e c t i o n 4 - Many C o u p l e d Consider placed  co-axially  field  H  N  circular  with  T h e W*^  .  e  contin-  A first  occurs  two l o o p  tfa')  s w i n g s down b u t n o t o u t , and t h e  to  transition  the  yrf^  J  transition  decreases  to increase the path  increases  of  -  6-4,  increases the barrier  critical ues  Figure  superconducting  Loops  loops  axes p a r a l l e l loop  t h e same r a d i u s  to the uniform  has c r o s s - s e c t i o n area  s t a t e has l o o p  super-electron density  ^2  i s Mj,j.  inductance  and the mutual  with  ,  current  Ij,  external  /fc  and  , and i n relative A.  The s e l f - i n d u c t a n c e o f of the z  R  a n d j.* * 4  loop loops  f is  M*±.  terms  ,  M:J  i s an  N  square  symmetric matrix.  c o n t r i b u t i n g t o the f r e e energy  Condensation neb  energy  i *P R) x.  Magnetic  ~i  field  energy  N  N  I  Super-electron Kinetic  . . •» .  M  Energy - 2.  7  are s  The  three  153  The  As  of  identical  reasonable  loop  s e c t i o n we fluxoids  as d i s c u s s e d  definitions  *4  S  of the  i n the previous  case  as  <p>  fluxoid  f o r reduced  i s  restrict  i n each  loop  previously. fluxoid  <|>  the d i s c u s s i o n to  We and  which use  is  physically  e x a c t l y the  reduced  the  field  same  variable  before:  ( H * - § ) R  The  reduced  r T h e yj.  f r e e energy  -  V- 7  parameter  ing matrix  s *p u  fo  ^MA^  ^  analogous  to the  yU-  It  i s convenient  Y_  / HcfR  J^A^  Mu  Note t h a t  coupling: constant  i s related  ^Qt  s  kjj  °f  a  -T  to  N  N -square  n  by  Aj,  i s generalized to the  elements  ==  F  loop  case  by  defin-  matrix  ~£-  symmetric.  's u s e d ~ kjZ  to define  i n the  The  diagonal  2 loop  between the  the matrix  with  elements  discussion, A*^  and  element  ^'"^  Ftc'^,  are The loop  X  T Cli. Lj-r  is The  not a parameter  determinant  of  i s  of thematrix  a l l the preceding  found  Rc^  i s \P\  , and t h e c o - f a c t o r  P't .  Then adding of  since i tcontains the variable  the three equations  i n 7  terms  the reduced  and talcing free  account  energy  F  i s  t o be  k--l We  limit  to  t h e problem  phase  the investigation  transitions  second  order  transition ZMIC<  h r  fa-vtiH  quadratic  The  of  exactly of  c i w n  X  terms  result.  I i r  W o  of  F  I n every  By t h e e x p r e s s i o n "second  T r n  go t o z e r o  T f i P  order  p v - n Q v i q  transitions  ^  equal  emphasizing  *i o  ^  71  order  continut'  n -n  h'  0.  and t h e second  ^  t o determine  ~ 0  „  This  *f- r\  order  X = l  generality X  has  critical  whether  X = 1 .  by  gives  considered  we r e q u i r e a l l t h e s e c o n d at  i s found  t h e extreme  g e o m e t r y v/e h a v e  I n order  evaluated  which  r o n n  o f each  I t i sworth  does o c c u r  and t h e c o n d i t i o n s under  which i s  f o r second  1.  order  we m e a n a l l ^  t h e same d e f i n i t i o n i s always  field  expression  f o r second  occur.  i nthe  field  loop.  transition tives  I t o w p A i i c <  the coefficient  every this  o f t h e system  transitions  rather complicated  the critical  o f t h e system"  critical  setting for  of finding  of this  value  a second  partial These  order  deriva-  ares  155  {/-/<#) >  7-' Then a second all  p^  in  terms  at Ji  order  =2^/<.^)  transition  —?> 0  »  of determinates  X = 1  .  following  i f  This  F  i s a minimum  c o n d i t i o n may  i n v o l v i n g the second  R e m o v i n g t h e c o m m o n f a c t o r s we the  occurs  state  be  partial  the condition  w.r.t,  expressed derivatives,  as a p p l i e d  to  matrix:  / The n e c e s s a r y is  that  and s u f f i c i e n t  every p r i n c i p a l  condition  minor  that  of the matrix  F  >  f o r one l o o p  and two l o o p s  minimum'  be p o s i t i v e ,  y~  The c o n d i t i o n s  be a  "v  >0  a  are  that i s  recognized,  0 F i g u r e 6-1 Above: Below:  .>  AO  A?  1.°  M a g n e t i c a l l y c o u p l e d l o o p s h a v i n g ^u, - Z r ^ -2.6 P a t h s ( w i t h a r r o w s ) and b a r r i e r s i n plane. Depression of super e l e c t r o n d e n s i t y w i t h reduced f i e l d X.  ,5*  Figure  6-2  Above! Below s  Magnetically  /.0 coupled  ^  A>~  loops  having M y  /  = o.€~  )  yj^  =  Paths ( w i t h arrows) and b a r r i e r s i n ^ - p l a n e . D e p r e s s i o n o f s u p e r e l e c t r o n d e n s i t y *•> v / i t h reduced f i e l d X . /'  .r k  AC  /-sr  io  3.o  z f  M a g n e t i c a l l y c o u p l e d l o o p s h a v i n g jit, = + /i = /O P a t h s ( w i t h a r r o w s ) a n d b a r r i e r s i n p,-fz plane. Depression of super e l e c t r o n d e n s i t y ^ w i t h reduced f i e l d X . ' }  x  156  7  CHAPTER  C O M P A R I S O N OF T H E L I T E R A T U R E W I T H P R E S E N T R E S U L T S  In  this  final  tained  and note  served  as p o i n t s o f departure  the  calculations  background original on  those  c h a p t e r we s u m m a r i z e  paper  this  HeIfand  local The  paper  adds  (1958).  little  some c o r r e c t i o n s  i s based  The l a t t e r bodies  on t h e  i s based  discusses  with uniform  fields  of the value  effects,  which  they  g i v e n here  calculate should  netic  energy  free  i n each  o f second  the  solution  for  the f i r s t  order  quantized.  ,  o f I'jsl  i s inadequate, Mauser and  fields  then  due t o non-  the microscopic certain  theory.  points, i n  The d i r e c t field  are being  procedure  f o rthe calcul-  criteria,  not requiring  equations, appears  t o be g i v e n  here  time,  London which  fy/  f i l m s and  what a p p r o x i m a t i o n s  critical  of the field  clarify  critical  o f v a r i o u s e x p r e s s i o n s f o r t h e mag-  and e x a c t l y  expression.  from  largely  i n thin  to that o f Ginzburg.  to the c r i t i c a l  the equivalence  ity  chapter  (1962) a f t e r a d i s c u s s i o n s i m i l a r t o t h e a b o v e  particular  ation  have  l and 2 a r e p r i m a r i l y  (1950) w h i l e t h e s e c o n d  but h i s treatment  treatment  made  Chapters  The f i r s t  of Ginzburg  which  ob-  o r as p o i n t s o f comparison f o r  (1963) d i s c u s s e s c r i t i c a l  filaments,  obtain  o f G-L  o f other workers  f o r small simply-connected  Kusnezov  and  made h e r e .  and r e v i e w ,  t h e work  fields  results  the results  (1950) r e c o g n i z e d t h e i m p o r t a n c e  he c a l l e d Deaver  the fluxoid and F a i r b a n k  and remarked  o f t h e quant-  t h a t i tshould  (1961) a n d D o l l a n d N a b a u e r  be  1.57 (1961) f i r s t in  micro-cylinders.  Byers a  observed  ring  t h e most p r o b a b l e  amounts o f e n c l o s e d further bining  Several theoretical  papers  followed.  (1961) s h o w e d t h a t f o r a c o n t o u r d e e p  and Yang  thick  experimentally the quantization of flux  flux,  stability  quantization with  having  o f these  states.  t h e London theory,  al  (1962) h a v e o b t a i n e d a n e x p r e s s i o n f o r t h e f l u x  by  a long circular  field, for  tion  a proof  The a u t h o r  a n d may t h u s  and e x t e r n a l  a circular  feels  that this  He g i v e s  demonstra-  t o o b t a i n thermodynamic  on t h e " l o w - f i e l d "  some a p p r o x i m a t e  cylinders.  should not  i s consistent with the microscopic  be u s e d  cylinder  equations  i n equilibrium to quanti-  (1962) h a s c a l c u l a t e d t h e m a g n e t i c f i e l d  Ginzburg  enclosed  of quantization, but rather a  t h a t t h e G-L t h e o r y  theory,  in  wave f u n c t i o n l e a d  of the fluxoid.  considered  i n terms o f f l u x o i d  Com-  Lipkin et  (1961) s h o w e d t h a t t h e G-L  and Zumino  the effective  zation be  Keller  cylinder  quantized  (1962) h a v e ,  Bohr and M o t t e l s o n  discussed the energy fluxoid  s t a t e s are those  inside  values  papers  distribution  assumption o f  of enclosed  A l l t h e above mentioned  results,  flux  f o r particular  have assumed  and  field-independent super-electron density throughout.  the  first  G-L e q u a t i o n  this  can never  be more t h a n  uniform From  an approxi-  mation. In postponing v/e a r e a b l e particular We  note  Chapter  3 we h a v e  any approximations t o make c e r t a i n case  t o proceed  the following  kept  the discussion very  as f a ras p o s s i b l e .  general  statements,  In this  way  and i n any  systematically to required  results s  general„  accuracy.  (i)  The v a r i o u s e x p r e s s i o n s  (3.9).  (3.8) do n o t r e q u i r e u n i f o r m  Equation  (3=5)  is  only  valid  (ii)  which  starting which  are  phase  (iii)  The m a g n e t i c  (3.19)  g i v e n by e q u a t i o n  error  (iv) is H  .  T i l l s form  given and I  e  then  i n a form  the f l u x o i d  state  (v)  The d e p e n d e n c e  for  without  over  which  cylinder i s  does n o t assume  uni-  f o r the determination of  the f i n i t e  coherence  length.  cylinder  definite i n I f  0  energy  length.  of a hollow  i s manifestly positive  of lowest  This  .  E  of a long hollow  i n a form  o f p.  t h e body o n l y . H  prior  H e  i s fixed,  i s that having the  current, f o ra l l cylinders.  the cylinder three  .  from  The c o e f f i c i e n t s  t h e IOOD c u r r e n t n e r u n i t  c  loop  only  energy  i s essential  which  G  i n powers o f ^  (3,21) t h e f r e e e n e r g y  lowest  of  ^  t o non-uniform  involved i n ignoring  In equation  H  i s the essential  are determined  integrals  also free  of  equations.  as definite  i s applicable  /(pl  nor uniform  .  criteria  o f t h e -G-L f i e l d  method  the  E  lip/  ( 3 . 9 ) , a p p a r e n t l y new h e r e ,  order  expressed  form  H  (3»3)»  i n equations  previously i n the l i t e r a t u r e  p o i n t .for t h e e x p a n s i o n  second  solution  has appeared  f o runiform  Equation  f o r  of various quantities  has been  "boundary  a l l i n powers  systematically  factors"  of  which  o r (<///[)  on t h e geometry  expressed  i n terms o f  may b e e x p a n d e d to give  once and  any r e q u i r e d degree  (/ of  accuracy. For  fluxoid, energy  a hollow  Lung-Tao  cylinder  and Zharkov  and u s i n g t h e phase  (1958) h a v e d e t e r m i n e d  i n the particular  state  o f zero  (1963) h a v e c a l c u l a t e d t h e f r e e  transition  the c r i t i c a l  criteria  fields.  of  Their  Ginzburg results,  1  although  less In  ders  4 we d i r e c t  Super-Electron  transition  Density ^ ^ )  Momentum  criteria.  approximating obtained  curves  calculations  In  is  equations  part  thin  second uses  equations  this  result  to discuss  appears  t o go i n t o  energy.  More r e c e n t  reported  by McLachlan  P  By  (1963)  and  f o rthe functions values  o fy U *  .  F(X),  The  t o f o l l o w as there a r e  Tinkham  the experiment  here  t  i s certainly curve.  of Little  the c r i t i c a l  t h e quantum  experiments  where  (1964) h a s shown f o r  t h e phase t r a n s i t i o n  near  the exten-  F(X,t)  on t h e c r i t i c a l  cylinder  and  points asthe  4 we h a v e d i s c u s s e d  ^  that very  M  o f phase  on t h e p l o t s .  F(X) etc. to surfaces  Xi  We h a v e  points  ours.  temperature.  that  here  M(X)  i sa generalized  f o r thinness, Douglass  aredifficult  ( 1 9 6 2 ) who f o u n d  ,  given  c y l i n d e r s i nwhich  order  Moment  e q u i v a l e n t t o (4.9) and (4.11),  2 o f Chapter  o f the curves  F(X),  and i n t e r p r e t e d i n terms  varies are given  o f Douglass  the relative  very  X  , where  f o r two p a r t i c u l a r  i n agreement w i t h  sion  > Magnetic  Loci of the significant  the field  M(X)  Energy  cylin-  "Significant  e q u i v a l e n t t o those and  are  P(X)  are identified  "geometry parameter"yU-  has  the discussion to thin  proportional to  t h e curves  ours.  o f t h e f u n c t i o n s Free  Super-Electron  variable on  Chapter  a r e i n agreement w i t h  and o b t a i n curves  Relative and  general,  and Parks  temperature  state o f lowest  on such  He  the free  c y l i n d e r s have  been'  (1969). obtained  curves  of the quantities  as f u n c t i o n s o f temperature.  We h a v e  F , given  160  the  criterion  field,  t  o r d e r phase  moment: -of a c y l i n d e r  external  = 0  second  increas&aag temperature.  magnetic zero  ftor  t — 1  to  versible  field .  The e x p e c t e d b e h a v i o u r o f t h e i n a persistent  The r e s u l t i n g  reversible  a set of closely  to t h e (experimental r e s u l t s  leading  t o the;- ( c o n c l u s i o n t h a t states  near  In  Chapter  5 "by u s i n g  the  analysis  aff t h i n  cylinders  the  effect  of Mgh  To d e t e r m i n e  fluxoid  t;lae c r i t e r i a  spaced  the cylinder  the metastable  i s extended  We  show t h a t  f o r second  F  o f Chapter  coherence  transition,  on  field  of  length.  i n p o w e r s o f p*  the c r i t i c a l  3  n  .  given  The  has been  first  found  (1968) b y e x p a n s i o n o f t h e B e s s e l ' s f u n c t i o n s .  a l lthin  A)  (ct«  at sufficiently  dary  F a c t o r s fco t h i r d  that  at high fluxoid  cylinders  high  n  order i n U  m  .  undergo  to those a t low  "second  order l i k e "  n  n  .  F(X)  compressed  order  o f t h e Boun-  have been used  , b u t become  at higher  second  Expansions  numbers t h e c u r v e s o f  similar  is  limit.  o r d e r phase  of the expansion of  and B a r k s  transitions  (1964),  to take account  e q u a t i o n (3-28) v/hich makes no r e s t r i c t i o n  Groff  irre-  through  the general r e s u l t s  by  by  from  points) are  drops  number and f i n i t e  determines  varying  o f Hunt and M e r c e r e a u  use two t e r m s  which  state i n  c u r v e s and  we  coefficient  current  f o r temperature  similar  fluxoid  at constant  i s calculated  "curve"' ( r e a l l y  successive  transitions  t o show  etc,  are  and more  The maximum p o s s i b l e  n  g i v e n b y e q u a t i o n (5»5)« In  effect  part  of finite  that  the f i r s t  tial  prerequisite  2 of this coherence  correction to this  chapter an i n v e s t i g a t i o n length  term  i s carried  i s of order  calculation  out,  of the  I t i s found An  essen-  i s e q u a t i o n (3°19) f o r  l6i ~3y  which  does  All functional to  other  of  loops loop  i s new  i n the  of  critical  curve.  second  at  thin  The  cal  c o n d i t i o n s i n the  the  transition i n either  obtained  f o r second  co-axial  loops.  critical  field  of  first,  the and  sense.  order  X =  1  .  may  of  analyzed  i  1  X  > 1  second may  by  types  making  loop  i n any  of  two  i t v/as  in  the  and  a  ( i . e . one  ( i i i ) both affects  criteria  the  ( i ) both  ( i i ) mixed and  the  adopted  and  curve  even change  Finally  order  he  graphically  three  X =  shown t h a t  Then systems  of a path  transitions  In a l l second i s  found  be  occurring at  presence  have  cylinder  were  intersection  order), occurring at  order.  We  superconductors  be  T r a n s i t i o n s may  order  here.  loops  p o i n t s may the  .  parameters.  magnetically coupled  second  first  6  correspondence  plane barrier  Chapter  thin multiply-connected  shown t h a t  ly>\  assume u n i f o r m  relationships  appropriate co-axial  not  the  the  criti-  order  have  number o f  t r a n s i t i o n s the  loops  of  been similar reduced  162  BIBLIOGRAPHY-  B o h r , A,, a n d M o t t e l s o n , Byers,  N,, a n d Y a n g ,  Christiansen, Deaver, Doll,  de  C.N.,  R.,  1961,  Phys, Rev, L e t t . 1968.  a n d S m i t h , H.,  and Nabauer, D.H.Jr.,  M.,  1963.  1964,  P.G.,  Benjamin, Ginzburg,  1961.  A.G.,  1961.  46. 445,  Phys, Rev, L e t t ,  Phys. Rev, L e t t ,  1966.  2»  P h y s , R e v . 121,  P h y s . R e v , 132,  H,J., and P r e s s o n ,  Gennes,  1_2J), 4 9 5 .  Phys, Rev,  B . S . J r , , a n d F a i r b a n k , W.M.,  Douglass, Fink,  P.V.,  1962.  2»  2»  51.  513.  1J5JL, 219.  Phys, Rev,  R e v . Mod, P h y s . 36,  1966, superconductivity New Y o r k ,  225.  o r ivietais  V . L . , 1958.  Soviet  Phys.  JETP  £»  1962,  Soviet  Phys.  JETP  1J5,  and .Alloys,  ?8. 207.  a n d L a n d a u , L . D . , 1950. Z h , e k s p . t e o r . F i z . 20, 1064 ( E n g l i s h t r a n s l a t i o n i n Men o f P h y s i c s : L.D. L a n d a u I e d . D, t e r H a a r , P e r g a m o n P r e s s , 1 9 6 5 ) . Groff,  R.P., a n d P a r k s ,  Hauser, Hunt,  J . J . , and Helfand,  T.K.,  Keller, Kuper,  and Mercereau,  1968.  Phys, Rev.  E . , 1962.  Phys.  J . E . , 1964.  J . B . , a n d Z u m i n o , B., 1961.  N.,  1963.  Phys. Rev.  567. 386,  R e v . 135A,  Phys, Rev, L e t t ,  1^0,  L . D . , a n d L i f s h i t z , E.M., Pergamon P r e s s , London,  I2i»  R e v . 12?,  Phys.  CG. An I n t r o d u c t i o n t o t h e Theory 1968, Clarenden Press, Oxford.  Kusnezov, Landau,  R.D.,  2»  944. 1o  ^»  of Superconductivity.  2253.  1959.  Statistical  Physics,  ^3»  163  K . J . , P e s h k i n , M., R e v . 126, 116.  Little,  W.A.,  £,  9.  London,  F , , 1950' S u p e r f l u i d s V o l . I : Macroscopic Theory S u p e r c o n d u c t i v i t y . W i l e y , New Y o r k .  of  Lung-Tao, H,,  and  Meissner,  W.,  21,  Shikin,  V.B.,  T i n k h a m , M,,  P a r k s , R ,D.,  and  M c L a c h l a n , D.S.,  Zharkov, 1969.  and  787.  1969. 1964,  and  L . J . , 1962.  Lipkin,  Tassie, 1962.  G.F.,  Phys,  Soviet  Phys.  R e v , Mod.  1963.  Rev.  O c h s e n f e l d , R.,  Phys.  Soviet  Lett, 1933»  JETP  Phys.  36,  Rev.  21*  Phys.  Lett.  Phys,  JETP  r?_,  1434.  Naturwissenschaften 29_, 268.  902.  1426,  APPENDIX 1 SUMMARY OF FORMS OF THE GINZBURG-LANDAU FREE ENERGY.  Notation:  Capital  i s used, f o r t h e d i m e n s i o n a l " e f f e c t i v e wave f u n c t i o n , " i  s m a l l (p i s used f o r t h e r e l a t i v e e f f e c t i v e wave f u n c t i o n . G-L w r i t e t h e f r e e energy d e n s i t y  initially  i n t h e form: (H~H )' e  Expanding t h e a b s o l u t e v a l u e b r a c k e t  S u b s t i t u t i n g f o r Igh — <jJ {jj^ , where (j) i s t h e r e l a t i v e wave f u n c t i o n , the zero f i e l d e q u i l i b r i u m super e l e c t r o n  z  2  m  density,  2mC  2r*c  x  (p^ —  ijJ^ —  Substituting  &I  t  (equation  (1.2))  Define  A  M u l t i p l y i n g through by  _* * 7r  L  tiff  +  8^' we have f o r  w^7<^/%  }  ±lA\ *.2iVf'f+  +7K>kfx(fvy-yvtp^-A  77A "  TT  AIL-  = 2HJ  f  +  1  r  fa  Zmo<  wlvf/f^  (H-H f e  Hence.  A  J  - ^  ["  I Hob  I  f/V  ^ /V/V  f W  V  &  A  fvVy  -~lf>V<f>*)'A  +  He)  J Space  Reduced Free Energy  3  where  = volume o f body  Vi JAII  Space  Reduced F r e e Energy w i t h a m p l i t u d e and phase o f (JJ e x p r e s s e d =  The*  e  Ivy/*'-  < W  }  (vyf  +  y (v<r>) l  }  x  (7) (sZ)  explicitly  teal  ' ((/j*vtjj - yj vy*)  ~  -2y Va~) z  J  ,1V  Space  u _ — , "Condensation  / Fr&e- trner^f  »  /  y  Et^ctro ao^eHc  Free-  m  Fc  F  Energy  H  S p e c i a l Cases:  1)  2)  (jJ i s r e a l , a p p l i c a b l e t o s i m p l y - c o n n e c t e d  uniform, a p p l i c a b l e to small bodies.  bodies w i t h  A  i n London gauge.  Define  Spa.ce  3)  Doubly connected  bodieSj  (p of t h e form i n c y l i n d r i c a l c o - o r d i n a t e s  ip =  ye  i—  1  —3  /e.  a)  ^  n  $  }  v<7)  = A  }  HcbfX  =  E=  $o  Spa.ce  D i m e n s i o n l e s s form.  Notation:  primes f o r " d i m e n s i o n l e s s "  A l l l e n g t h s measured r e l a t i v e t o A : f'-=-  f/\  -  '/K  quantities.  \7 ~ ^'  o/V~  A  d^  A  Define  / r '  Then  £p>cu.e'  s  / r / ^  „ '-  j'=  so  M a f  »' =  X A'  169  APPENDIX 2  EXPANSION OF BOUNDARY FACTORS IN POWERS OF y  (see page 67)  F o r h o l l o w c y l i n d e r s w i t h o u t r e s t r i c t i o n on A, , A?. .  L  I ht%  ^  _A_—.  \ i  (- ^L  +  ^f L h  I  A  Z  I  W  9  Y  %l  (  -  -h ^  +  .*)}'  ^' ^ ^ / i i ^  .1  <t  z «  z  V-  A  6  \i  \?'Z  z  (  t  J  A,  If  6  ?-z  A  )  e  *  h  z+  S  2" &  z  Z?  6  Z  e  j  ?<2  e  i*'*  6  Z  6  J  *i)  170  171  APPENDIX 3  EXPANSION. OF BOUNDARY FACTORS TN POWERS' OF ( d / A ) ( s e e page 6?) F o r h o l l o w c y l i n d e r s w i t h <s£«A Geometry parameter  /M*  s  . M-i  , valid fora l l  / A  .  

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