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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., P r i n c e t o n U n i v e r s i t y s 1 9 5 7 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 7 1 In present ing th i s thes is in pa r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying o f th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on of th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The f i r s t two chapters present a c r i t i c a l review of the d e r i v a t i o n of the Ginzburg-Landau macroscopic equations and t h e i r a p p l i c a t i o n to the determination of the c r i t i c a l f i e l d s and temperatures f o r superconducting-normal phase t r a n s i t i o n s i n simply connected bodies. Second order phase t r a n s i t i o n c r i t e r i a are obtained i n the form of 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 of 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 i n the London gauge f o r doubly-connected bodies, we o b t a i n s e v e r a l e q u i v a l e n t expressions f o r the electromagnetic Free Energy which do not assume uniform /(^l . One of these leads to a systematic method f o r expanding F^ i n powers of = and t h i s method i s a p p l i e d to the long hollow c i r c u l a r c y l i n d e r . The expansion obtained i s used to determine second order t r a n s i t i o n c r i t e r i a f o r a l l p o s s i b l e f l u x o i d s t a t e s . A c l o s e d expression f o r F^ i n the hollow c y l i n d e r which does not assume uniform /(pl 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 hollow c y l i n d e r y i e l d s curves f o r the e q u i l i b r i u m values of Free Energy, i i Super E l e c t r o n Density, Magnetic Moment, and Super E l e c t r o n Momentum as f u n c t i o n s of a general f i e l d v a r i a b l e . C r i t i c a l p o i n t s on the curves are i d e n t i f i e d and the l o c i of c r i t i c a l p o i n t s traced as the geometric parameter v a r i e s . These f u n c t i o n s are extended to surfaces of the v a r i a b l e s of f i e l d and temperature. Phase t r a n s i t i o n c r i t e r i a are obtained f o r i n c r e a s i n g temperature at constant f i e l d . R e v e r s i b l e and i r r e v e r s i b l e curves are obtained d e s c r i b i n g the v a r i a t i o n and d e s t r u c t i o n of p e r s i s t e n t currents at va r y i n g temperature, and the r e s u l t s are compared with experiment. The f u l l range of p o s s i b l e f l u x o i d s t a t e s i s determined and i t i s shown that the f a m i l i e s of curves 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 to high 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 . A l l t h i n c y l i n d e r s undergo second order t r a n s i t i o n s at s u f f i c i e n t l y high f l u x o i d . The c o r r e c t i o n due to f i n i t e coherence len g t h i s of the order of the f o u r t h power of the r a t i o of the c y l i n d e r w a l l t h i c k n e s s to the coherence l e n g t h . Other t h i n doubly connected superconductors are desc r i b e d by the same f u n c t i o n s a p p l i c a b l e to c i r c u l a r c y l i n d e r s by making the appropriate correspondence of parameters. The exact second order c r i t i c a l f i e l d f o r a torus i s c a l c u l a t e d . i i i Systems o f two c o - a x i a l m a g n e t i c a l l y c o u p l e d l o o p s a r e a n a l y s e d and the c r i t i c a l c o n d i t i o n s f o r one l o o p a r e f o u n d t o depend i n a c o m p l i c a t e d way on the p r e s e n c e o f t h e o t h e r . I f a t h i n l o o p i s c l o s e l y c o u p l e d t o a t h i c k o n e , t h e b e h a v i o u r o f t h e f o r m e r i s d o m i n a t e d by t h a t o f t h e l a t t e r . 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 a r e o b t a i n e d f o r s y s t e m s o f n c o - a x i a l l o o p s o f e q u a l r a d i u s . I n t h e f i n a l c h a p t e r a c o m p a r i s o n i s made o f t h e r e s u l t s o b t a i n e d i n t h i s t h e s i s w i t h t h o s e a p p e a r i n g p r e -v i o u s l y i n t h e l i t e r a t u r e . i v TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES v i i ACKNOWLEDGEMENTS i ' X Chapter 1 INTRODUCTION ' 1 2 SIMPLY-CONNECTED SUPERCONDUCTORS 20 3 MUI.TIPI.V-nONMEr.TF.M SUPERnOMnnr.TORS - $h 4 THIN CYLINDERS 8 l Part 1. Geometry 8 l Part 2. Temperature and F i e l d 9^  5 THIN CYLINDERS CONTINUED I l l Part 1. High F l u x o i d Number I l l Part 2. F i n i t e Coherence Length 119 6 OTHER FORMS OF MULTIPLY-CONNECTED SUPERCONDUCTORS ' 130 7 COMPARISON OF THE LITERATURE WITH PRESENT RESULTS 156 BIBLIOGRAPHY 162 v Page APPENDIX 1 SUMMARY OF FORMS OF THE G-L FREE ENERGY . . . . 164 APPENDIX 2 EXPANSION OF BOUNDARY FACTORS IN POWERS O F ^ 169 APPENDIX 3 EXPANSION OF BOUNDARY FACTORS IN POWERS OF (d/A ) 171 v i LIST OF FIGURES A l l F igures are placed at the end of t h e i r r e s p e c t i v e chapter. i n d i c a t e d by the number preceding the hyphen. •Figure . F o l l o w i n g Page 2-1 F u n c t i o n a l dependence of Magnetic Free Energy on ( / / < J 2 ) 53 2-2 F u n c t i o n a l dependence of Magnetic Free Energy on the r e l a t i v e super e l e c t r o n d e n s i t y 53 2-3 Reduced Free Energy f o r s m a l l body 53 2-4 Reduced Free Energy F(*5^) f o r la r g e body 53 2-5 Second Order C r i t i c a l F i e l d i n H - t plane . . . 53 2-6 S p l i t t i n g of Second Order C r i t i c a l Curve i n t o Three Branches at t * 53 4-1 E q u i l i b r i u m value of r e l a t i v e super e l e c t r o n d e n s i t y ^ 110 4-2 Thin c y l i n d e r f u n c t i o n s f o r v a r i a b l e f i e l d , geometry parameter 110 (a) Free Energy F(X) 110 (b) Super E l e c t r o n Density y(X) 110 (c) Magnetic Moment M(X) . < 110 (d) Super E l e c t r o n Momentum P(X) 110 4-3 Thin c y l i n d e r f u n c t i o n s f o r v a r i a b l e f i e l d , temperature parameter 110 (a) Free Energy f ( x ) 110 (b) Super E l e c t r o n Density ^ ( x ) 110 (c) Magnetic Moment m(x) . 110 (d) Super E l e c t r o n Momentum p(x) 110 v i i F i g u r e 4 - 4 4 - 5 4 - 6 4 - 7 4 - 8 4 - 9 5 - 1 5 - 2 5 - 3 6 - 1 6 - 2 6 - 3 6 - 4 F o l l o w i n g Page Thin c y l i n d e r f u n c t i o n s f o r v a r i a b l e temperature, f i e l d parameter 1 1 0 (a) Free Energy f ( t ) 1 1 0 (b) Super E l e c t r o n Density 3 ( t ) . . . . . . . 1 1 0 (c) Magnetic Moment m(t) . *\ 1 1 0 (d) Super E l e c t r o n Momentum p ( t ) 1 1 0 Free Energy of neighbouring f l u x o i d s t a t e s at 'onset of s u p e r c o n d u c t i v i t y 1 1 0 Separation of h-t plane i n t o regions of normal, s t a b l e and metastable superconducting s t a t e s 1 1 0 Reduced Magnetic Moment m(t) i n t h i n hollow c y l i n d e r , showing metastable l i m i t m^  1 1 0 Reduced Magnetic Moment m(t) i n t h i n hollow c y l i n d e r , showing maxima of m(x) l i n e m^  . . . 1 1 0 M a g n i f i c a t i o n of e q u i l i b r i u m curves near metastable l i m i t , . 1.10 Change i n c r i t i c a l reduced f i e l d s at l a r g e f l u x o i d 1 2 9 Dependence of Free Energy F- on Y f o r f u l l range of p o s s i b l e f l u x o i d s t a t e s 1 2 9 Dependence of Super E l e c t r o n Density on Y f o r f u l l range of p o s s i b l e f l u x o i d s t a t e s . . . . 1 2 9 M a g n e t i c a l l y coupled loops w i t h M. =2.5 , ^ 2 = 2 . 6 { / 1 5 5 M a g n e t i c a l l y coupled loops w i t h , £ £ = . 5 , ^ 2 = 4 . 0 : . L 1 5 5 M a g n e t i c a l l y coupled loops w i t h > ^ = 1 . 2 , / * 2 = 1 ° 1 5 5 M a g n e t i c a l l y coupled loops w i t h y&S- = 4 . 0 , yLC2=10 / 1 5 5 v i i i ACKNOWLEDGEMENTS I wish to thank my sup 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 course of t h i s work and f o r the p r o v i s i o n of summer research grants. I a l s o thank the U n i v e r s i t y of B r i t i s h Columbia f o r the award of a Graduate F e l l o w s h i p from 1966 to 1970. i x CHAPTER 1 INTRODUCTION The study of the thermodynamics of superconductors dates p r o p e r l y from the di s c o v e r y by Meissner and Ochsenfeld (1933) that the s t a t e of a pure superconductor, at f i x e d temperature 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. In a simply-connected superconductor i n low f i e l d (and f o r a 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 super-f l u x i s excluded from the m a t e r i a l . The e x c l u s i o n of magnetic f l u x i s now r e f e r r e d to as the "Meissner E f f e c t " . , P r e v i o u s l y i t had been assumed that the s t a t e depended on past h i s t o r y , and that phase t r a n s i t i o n s from normal to super s t a t e s , and v i c e v e r s a , were i n general i r r e v e r s i b l e . An elementary thermodynamics of i d e a l Meissner superconductors i s given by London (1950). Refinements of t h i s thermo-dynamics, concerning 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 subject of t h i s t h e s i s . 1 2 The observed macroscopic e x c l u s i o n of the f i e l d from the body i s e v i d e n t l y due to p e r s i s t e n t surface currents which generate a magnetic f i e l d opposite to the e x t e r n a l f i e l d . London was the f i r s t to e l u c i d a t e the a c t u a l d i s -t r i b u t i o n of the current 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 defined the p e n e t r a t i o n depth or average t h i c k n e s s of the surface c u r r e n t . Except i n bodies of zero demagnetization these currents a l s o a f f e c t the f i e l d o utside the body g i v i n g r i s e to d i p o l e and higher magnetic moments which i n t e r a c t with the e x t e r n a l f i e l d . In order to compare the f r e e energies of superconducting and normal phases, and hence to determine the e n e r g e t i c a l l y favourable phase, the k i n e t i c energy of the super currents and the exact magnetic f i e l d energy of the p e n e t r a t i n g f i e l d must be taken i n t o account. These become extremely Important i n bodies having at l e a s t one dimension comparable to or l e s s than the p e n e t r a t i o n depth. London's theory had one p a r t i c u l a r defect of which he was w e l l aware. By s t r a i g h t f o r w a r d a p p l i c a t i o n of t h i s theory i t i s e n e r g e t i c a l l y favourable under c e r t a i n circumstances f o r a superconductor of zero demagnetization to s p l i t i n t o very f i n e l a y e r s of superconducting and normal m a t e r i a l . In f a c t , i n Type I superconductors t h i s does not occur. (This s p l i t t i n g should hot be confused 3 w i t h the intermediate s t a t e which depends e s s e n t i a l l y on demagnetization.) To circumvent t h i s problem London i n t r o -duced ad hoc a surface energy to be a s s o c i a t e d w i t h the formation of a normal-super i n t e r f a c e . Ginzburg and Landau (1950) ( h e r e a f t e r G-L) pre-sented a macroscopic theory of 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 of observed phenomena without the need f o r any a d d i t i o n a l hypothesis on surface energy. The normal-super " s u r f a c e " i s seen as a continuous v a r i a t i o n i n super e l e c t r o n d e n s i t y and the energy a s s o c i a t e d w i t h the surface can, i n p r i n c i p l e at l e a s t , be c a l c u l a t e d from the theory i t s e l f . The G-L theory forms the b a s i s on which the r e s u l t s of t h i s t h e s i s are obtained. The G-L theory i s conveniently presented i n two stages, f i r s t the condensation aspect ;^hich leans h e a v i l y on the general theory of second order phase t r a n s i t i o n s of 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 w i t h a heat r e s e r v o i r and i n the absence of any e x t e r n a l f i e l d . As i t s temperature i s lowered below the c r i t i c a l temperature some of the conduction e l e c t r o n s condense i n t o a lower energy s t a t e c a l l e d the superconducting s t a t e , or more simply, the 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 the superconducting s t a t e i s l e s s than i n the 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 of constant temperature. Throughout the t h e s i s we s h a l l take as zero reference of f r e e energy the f r e e energy of the normal s t a t e , so that "the" f r e e energy of 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 energies of super and normal s t a t e s . Consequently, "the" f r e e energy d e n s i t y i s always zero when the super e l e c t r o n d e n s i t y i s zero. We use the symbol ( s c r i p t ) I? f o r t o t a l f r e e energy, and /A v I? f o r f r e e energy d e n s i t y . The f r e e energy d e n s i t y w i l l i n general be a f u n c t i o n of temperature T and the super e l e c t r o n number de n s i t y n g . The temperature T i s an e x t e r n a l l y imposed c o n s t r a i n t whereas n w i l l adjust i t s e l f so as to minimize s 0 <J~ . At T > T c the body i s normal, so ^St T) has a p h y s i c a l minimum at n = 0 . At T<T , ~?(r?S)T) must s c have a minimum at some n > 0 at which J~ ^ ® . The s simplest f u n c t i o n s a t i s f y i n g t h i s requirement and i s the parabola 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 of T depending on the m a t e r i a l , but as yet unknown. The minus s i g n i s to make CX t u r n out p o s i t i v e ( i n co n t r a s t to the n o t a t i o n of G-L) w h i l e the 1 / 2 i n the l a s t term i s conventional ( i n 5 agreement w i t h G-L). ^3 must be p o s i t i v e i n the range 0 <C 7" ^ Ic i n order that ~J~ have a minimum at a l l . The minimum of occurs when CT) + ns/3(T) = 0 that i s , at ^ = 0 ( / ^ = n^fT) ( 1 > 2 ) where ft&fT) means the e q u i l i b r i u m value of n i n zero f i e l d . Prom the preceding d i s c u s s i o n we see that (X > 0 f o r T < T , & < 0 f o r T > T and (X ~ 0 at T=T . c 3 c c Near T we know then that (X(T) (TC~T) , and c ' (T) 0 0 (Tc ~T). The value 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) In 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 uniform throughout the body so that f o r a body of volume , J~w = ( A v "j^o) Vj . The func-txons CK(T) , /3(T) may 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 of which i s 3 ^ , which we 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 f i e l d p e r s i s t e n t surface currents flow i n the sense to exclude 6 the f i e l d from the body. In t h i s case, both the i n e r t i a l k i n e t i c energy of the currents 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 the f r e e energy of the superconductor. Since the k i n e t i c energy c o n t r i b u t i o n comes from the 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 pre-dominates. The f i e l d energy d e n s i t y i n Gaussian u n i t s i s (He-H) /t/r where H i s the magnetic f i e l d i n the absence of the body and H i s the f i e l d i n the presence of the body. One might be tempted to put ( i n c o r r e c t l y ) , as indeed one sometimes f i n d s i n the l i t e r a t u r e . The c o r r e c t expression may be de 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 of zero r e s i s t a n c e s i t u a t e d i n the f i e l d of a s o l e n o i d and i n i t i a l l y c a r r y i n g a c u r r e n t . The f r e e energy of the loop i s the work done by the loop when i t i s d i s p l a c e d slowly along the s o l e n o i d a x i s to the point where the loop current becomes zero, the s o l e n o i d current being held constant. I t i s found that t h i s f r e e energy can be expressed as a volume I n t e g r a l of the f i e l d energy d e n s i t y (He~ ft) / <Ff7 Consider a l a r g e superconductor of volume having zero demagnetization c o e f f i c i e n t (a long needle p a r a l l e l to 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 f i e l d —» H . Due to surface currents the f i e l d i n s i d e the body w i l l e 7 be zero, except i n a very t h i n p e n e t r a t i o n s k i n the volume of which i s n e g l i g i b l e because of body s i z e , w h i l e outside the body H=He . That i s , the Meissner e f f e c t i s e s s e n t i a l l y complete. The magnetic f r e e energy 7^ i s (ffefc/V = H£A d - 1 * ) This i s a p o s i t i v e c o n t r i b u t i o n to the f r e e energy f o r a l l f i e l d s . I f the e x t e r n a l f i e l d i s g r a d u a l l y i n c r e a s e d then at some f i e l d , say \ Hel — HQ^ (cb stands f o r c r i t i c a l b u l k ) , the magnetic energy equals the condensation energy. At / fig} > r1cb the super s t a t e i s no longer e n e r g e t i c a l l y f a v o u r a b l e , hence a t r a n s i t i o n takes place to the normal s t a t e . This i s a convenient way to measure the condensation • energy as a f u n c t i o n of temperature. 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 to the negative of the condensation energy d e n s i t y we have — - - oo Z f 3 Hck i s a property of the superconducting m a t e r i a l q u i t e independent of any magnetic f i e l d which may a c t u a l l y be present. Hcj^ i s found by experiment to have the approximate 8 temperature dependence Hctl * He ( i - t1) o where t=T/T and i s the bulk c r i t i c a l f i e l d c e x t r a p o l a t e d to 0 K . Using (1.2) and (1.1) we may r e w r i t e the general expression f o r {S^"3~ , equation (1.1), i n the f o l l o w i n g form The G-L theory i s formulated i n quantum mechanical terms from the out s e t . G-L introduce an " e f f e c t i v e wave f u n c t i o n of the super e l e c t r o n s " 1^ , which i s a kin d of average over the super e l e c t r o n wave f u n c t i o n s . In analogy to quantum mechanics, IJfijf* represents the number d e n s i t y n of the super f l u i d , w h i l e the s p a t i a l l y dependent phase s of contains i n f o r m a t i o n on the momentum of the super f l u i d . ^ , and consequently — 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 the superconductor, and zero o u t s i d e . Let IjJ^ be the e q u i l i b r i u m z e r o - f i e l d value of 2 "ty , that i s / tp^l - ftco . Lpcc , l i k e r]M , i s a property of the m a t e r i a l at given temperature and i s not a s p a t i a l l y dependent q u a n t i t y . Consequently the phase of 9 (poo can have no meaning and we take LjJgg to be r e a l . Then i n the G-L f o r m u l a t i o n , the free.energy d e n s i t y i n zero f i e l d i s A V ? „ = J ± £ f_2IM2.+ m ? j Sn \ 9<»z VS  f Probably the most profound i n n o v a t i o n i n the G-L theory i s the f o r m u l a t i o n of the k i n e t i c energy d e n s i t y term which u l t i m a t e l y enables G-L to d e r i v e - a London-like f i e l d equation f o r the current d e n s i t y . London s t a t e d h i s . equations as p o s t u l a t e s . He knew, of course, that they could not be d e r i v e d c l a s s i c a l l y , and supposed c o r r e c t l y that they would one day be a consequence of a microscopic theory of s u p e r - c o n d u c t i v i t y . A c l a s s i c a l charged p a r t i c l e , of charge C^. , i n a magnetic f i e l d of vec t o r p o t e n t i a l ~K has c a n o n i c a l momentum p — IT)*/" +JL A and Hamiltonian ~ { P ~ *t ™ j In c l a s s i c a l mechanics s u b s t i t u t i o n of p i n t o gives the u s u a l expression for- k i n e t i c energy which gives the Newtonian equations of motion. In quantum mechanics, however, the Hamiltonian f o r m u l a t i o n has profound e f f e c t , f o r I t i s p and not tY)A/ ;vhich i s represented by the operator ~-<i j? V ; we have 10 7*> The consequences of t h i s form of are observable and t h i s form i s found to be c o r r e c t . I n s p i r e d by t h i s form, G-L put f o r the k i n e t i c energy d e n s i t y of the s u p e r f l u i d q u a n t i t y over the body i s analogous to the quantum mechanical expression f o r the exp e c t a t i o n value of k i n e t i c energy. This form a l s o has the e s s e n t i a l f e a t u r e of being gauge co-v a r i a n t w i t h the "wave f u n c t i o n " . C o l l e c t i n g terms, the t o t a l f r e e energy i s given q u i t e g e n e r a l l y by of the superconductor. F o l l o w i n g G-L, we introduce the " 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 " s i n c e the i n t e g r a l of t h i s ( 1 . 8 Only the term i n ('L'"~ '^e) can c o n t r i b u t e outside the volume 9 = ( 1 . 9 I n t h e case oir z e r o e x t e r n a l f i e l d and/or i n t h e case of a b u l k s u p e r c o n d u c t o r , (p has the u n i f o r m v a l u e 1 I n t h e s u p e r s t a t e and 0 i n the normal s t a t e . The e x p r e s s i o n f o r 7 i s c o n s i d e r a b l y s i m p l i f i e d by d e f i n i n g the " p e n e t r a t i o n d e p t h " A and t h e "coherence l e n g t h " f h f == / n 1 d . i o ) ^ _ Imc /3 The r a t i o o f t h e s e c h a r a c t e r i s t i c l e n g t h s i s t h e s o - c a l l e d "G-L parameter" K X I tmc \ I /Q (1 -11) The f r e e energy t a k e s the form $ r r J/?ll fpece I (1.12) K A~ J A l l the m a c r o s c o p i c p h y s i c a l p r o p e r t i e s o f the s u p e r c o n d u c t o r a r e c o n t a i n e d i n the t e m p e r a t u r e 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 , can be measured and i t s t e m p e r a t u r e dependence f o u n d . A i s a l s o measur-a b l e and i t s t e m p e r a t u r e dependence i s found t o be a p p r o x i m a t e l y 12 k i t ) « \ J l - t + ) (1.13) where A 0 i s t h e v a l u e o f A e x t r a p o l a t e d t o 0 K • The t e m p e r a t u r e dependence of the coherence l e n g t h F i s t h e n c a l c u l a t e d from the above r e l a t i o n s t o be /Tf Hcbtt)X(t) '°\i-f S i m i l a r l y we o b t a i n the f o l l o w i n g r e l a t i o n s f o r t h e q u a n t i t i e s & , ^ 3 i MQO a n d K , where I n e v e r y case s u b s c r i p t Q i n d i c a t e s a b s o l u t e z e r o . v2 V a r i o u s n o r m a l i z a t i o n o f I? and the d i f f e r e n t q u a n t i t i e s i n the i n t e g r a n d a r e u s e f u l i n p a r t i c u l a r c a ses i n o r d e r t o c o n c e n t r a t e on the f u n c t i o n a l form o f the equa-t i o n . A c o n f u s i n g v a r i e t y of " r e d u c e d , " " r e l a t i v e , " " d i m e n s i o n l e s s " e t c . q u a n t i t i e s appear i n the l i t e r a t u r e and a l s o i n t h i s t h e s i s . These a r e summarized i n Appendix 1 f o r r e f e r e n c e , and w i l l be d e f i n e d i n t h e t e x t as needed. 13 We expect the body to 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 of contact w i t h a temperature r e s e r v o i r and f i x e d e x t e r n a l magnetic f i e l d . G-L o b t a i n the d i f f e r -e n t a i l equations f o r (p and A by t a k i n g the f i r s t v a r i a -t i o n of wit h respect to (p and A r e s p e c t i v e l y . Taking the v a r i a t i o n of J" i n equation (1.12) •fc w.r.t. and performing 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 , we o b t a i n f z J , /ich A *v<p i I f A K y Surface S e t t i n g the c o e f f i c i e n t of S<p i n the volume i n t e g r a l equal to zero gives the f i r s t G-L equation: —9 v 2-Since the value of <j) on the surface 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 n the surface i n t e g r a l i s not i d e n t i c a l l y zero. To assure zero v a r i a t i o n i n "3" we must ( 1 . 1 4 ) 1 4 r e q u i r e the integrand of the surface i n t e g r a l to vanish, This gives the f o l l o w i n g boundary c o n d i t i o n on IjJ : ^ J l = o  (ia5) K where i s a normal to the s u r f a c e . To obta i n the v a r i a t i o n w.r.t. X , we f i r s t — 0 —» — j a —o s u b s t i t u t e M-V*A and # = V X , 4 e . The terms i n 3^ —o c o n t a i n i n g A are y - y 7 r,fV2 '  A where V, i s t h e v o l u m e o f t h e s u p e r c o n d u c t o r , and i s a l l s p a c e e x t e r n a l t o V, . T a k i n g t h e v a r i a t i o n 0 - z i f f ^ Hci> , , , , 15 P a r t i a l i n t e g r a t i o n o f the second i n t e g r a l l e a d s t o As 3L ~* 0 0 j L H~~ ^ e J " " * 0 a t l e a s t as //Ji , hence t h e s u r f a c e i n t e g r a l i s 0. I n the volume i n t e g r a t i o n o v e r the f i r s t two terms a r e 0 s i n c e (JJ — 0 } and the l a s t term i s 0 s i n c e i n Y 3^ V X H — V X tf^ d u e t o t h e c o n s t r a i n t t h a t e x t e r n a l c u r r e n t s r e m a i n c o n s t a n t . Hence v"2 c o n t r i b u t e s n o t h i n g t o the I n t e g r a l . F i n a l l y we note t h a t t h e term V V x A& 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 the volume s i n c e t h i s q u a n t i t y i s p r o p o r t i o n a l t o the c u r r e n t d e n s i t y p r o d u c i n g the e x t e r n a l f i e l d H g , and t h i s c u r r e n t i s e n t i r e l y e x t e r i o r t o the s u p e r c o n d u c t o r . Hence ^J=±j fillip (fvf - (/J Vp*)+?M^ -hlVxVXA \-[A c/V T h e r e f o r e J i s s t a t i o n a r y when the c o e f f i c i e n t o f dA i s 0 : v x A ==- - - f - — — (yvy* - -(1.16). w h i c h i s the second G-L e q u a t i o n . 16 I t might be i n c o r r e c t l y concluded from the pre-ceding c a l c u l a t i o n that 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 the body i s zero. I f we devided ~3~ i n t o two i n t e g r a l s , over arid . respec-t i v e l y , at the o u t s e t , then when the p a r t i a l i n t e g r a t i o n i s performed i n there appears a non-zero surface i n t e g r a l over the surface of the body. This surface i n t e g r a l con-t a i n s a l l the magnetic f i e l d energy i n , sin c e the remaining volume i n t e g r a l over i s 0. 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 contains a non-zero surface i n t e g r a l which j u s t cancels the surface I n t e g r a l from Vp . There i s thus a r e l a t i o n between the i n t e r i o r and e x t e r i o r c o n t r i b u t i o n s which permits us to w r i t e J~ as an i n t e g r a l over the body only. Let us consider equation (1.16) more c l o s e l y . The LHS i s by Maxwell's equation p r o p o r t i o n a l to the current —•> d e n s i t y . The RHS i s , - w i t h i n a n o r m a l i z a t i o n f a c t o r , i d e n t i c a l w i t h the quantum mechanical expression f o r the current 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 that (1.16) i s an i d e n t i t y , or at most, a n o r m a l i z a t i o n c o n d i t i o n . However, i t s most profound s i g n i f i c a n c e i s that the G-L " e f f e c t i v e wave f u n c t i o n " behaves under quantum operators i n a f a s h i o n c o n s i s t e n t w i t h r e a l Schrodinger 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 that equation (1.16) i s obtained by m i n i m i z a t i o n ; the Schrodinger equation can a l s o be obtained by a minimiza-t i o n procedure, once the correspondence of operators i s a s s e r t e d . 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 the boundary c o n d i t i o n (1.15) and the expression, f o r f r e e energy 7 (1.12) can be s i m p l i f i e d and take on i n t e r e s t i n g forms. F i r s t we separate ^ i n t o i t s amplitude and phase U. CA f = 1 e where ^[st) , (7\(- rL) are r e a l . Then v = ( + *y v<?s) e "1 2 / x2 , / \<- ' ( 1 - 1 7 ) S u b s t i t u t i n g (j) — *7 € i n t o the second G-L equation gives i n e q u i l i b r i u m where r e f e r s to the super current d e n s i t y w i t h i n the I s o l a t e d superconductor. In e q u i l i b r i u m the normal current d e n s i t y i s zero. 18 S u b s t i t u t i n g Ij) — ^ Q i n t o the boundary c o n d i -t i o n (1.15), d i v i d i n g out the e x p o n e n t i a l , and s e t t i n g the r e s u l t i n g r e a l and imaginary p a r t s equal to 0 we o b t a i n (jiHcb7f\<n ~ ify ~ 0 I (1.19) V^7 • c?S - 0 J In view of (1.18) the f i r s t equation s t a t e s that there i s no component of ^ normal to the surface which Is a n a t u r a l boundary c o n d i t i o n . The second equation which s t a t e s that the amplitude of ^ s h a l l have no component of gradient normal to the surface cannot be i n t u i t i v e l y j u s t i f i e d by macroscopic arguments. De Gennes (1964) has shown from the microscopic theory that t h i s boundary c o n d i t i o n holds at the i n t e r f a c e between a superconductor 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 expression (1.12) f o r ~jf- gives q u i t e g e n e r a l l y 7 - l-2ft f t 2 f'tlft ) d-i ^U^P^-th """•) I t w i l l be convenient to have names f o r each of these i n t e -g r a l s ; we c a l l the f i r s t the "condensation f r e e energy" 19 , and' the second the "electromagnetic f r e e energy" 3^ . In view of ( 1 . 1 8 ) , 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 ZrrX2- -?z _L_ I j i / - (1.21) This expression f o r the electromagnetic f r e e energy Is j u s t the c l a s s i c a l one, the term i n -g^ r r e p r e s e n t i n g the k i n e t i c energy of the super e l e c t r o n s . However, In the G-L theory, t h i s expression gives the energy c o r r e c t l y only i n e q u i l i b -rium. In p a r t i c u l a r , the I n t e g r a l ( 1 . 2 1 ) i n t h i s form i s not minimized by the second G-L equation, but r a t h e r by the c o n d i t i o n — 0 which i m p l i e s H ~M& 58 G . E v i d e n t l y the minimum value of the integrand (and i n t e g r a l ) i s zero. On the other hand, A, — 0 does not imply /9 ~ 0 when an' " —» e x t e r n a l magnetic f i e l d i s present. As -4s i n c r e a s e s so as to decrease H , then A i s decreased wh i l e (H ~~ ne) i s i ncreased. The optimum compromise i s the second G-L equation. E v i d e n t l y a London-type equation cannot be d e r i v e d from the c l a s s i c a l expression f o r electromagnetic energy. CHAPTER 2 SIMPLY-CONNECTED SUPERCONDUCTORS Se c t i o n 1. S i g n i f i c a n c e of Simple C o n n e c t i v i t y In t h i s chapter we present a very general d i s c u s s i o n of phase changes i n sm a l l simply-connected ( h e r e a f t e r abbre-v i a t e d to s-c) superconductors. A s-c body i s one which has no hole through I t , though i t may have t o t a l l y enclosed bubbles w i t h i n i t - a simple t e s t i s that i t cannot be hung up cn a s t r i n g . A hollow sphere or s p h e r i c a l s h e l l Is simply-connected, with one hole i n the w a l l i t i s s t i l l s-c, but wi t h two holes•through the w a l l i t i s doubly-connected. In la r g e superconductors, the body may s p l i t i n t o regions of normal and super s t a t e . In that case the c o n n e c t i v i t y r e f e r s to the super r e g i o n and not the whole body. In sm a l l bodies t h i s c o m p l i c a t i o n does not occur. Some of the r e s u l t s presented here have p r e v i o u s l y been discussed only f o r par-t i c u l a r geometries, e.g. f o r t h i n f i l m s . A l s o these r e s u l t s serve as a b a s i s of comparison f o r the d e t a i l e d a n a l y s i s of mult i p l y - c o n n e c t e d geometries. 20 21 Consider the second G-L equation i n the form (1.18) i n which the amplitude *J and phase (71 of p are w r i t t e n e x p l i c i t l y . Rearranging the terms t h i s may be w r i t t e n ^ T T A2" T i i f _ rr \f H , - £c v ^ fi + A - /TAf//^ \7c7) = Let us take 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 wholly w i t h i n the body. London has named t h i s q u a n t i t y the " f l u x o i d , " symbol <fic . (2.1) The f i r s t term of the integrand i n the f i r s t l i n e has no meaning outside the superconductor. However A i s de f i n e d — ? In a l l space, and the contour i n t e g r a l of A can be t r a n s -formed to a surface i n t e g r a l over any surface bounded by the contour, whether or not i t l i e s w i t h i n the body. p \ 2. —-> 'c where <y6 i s the t o t a l magnetic f l u x c r o s s i n g the surface S bounded by C . In case the contour l i e s deep i n the i n t e r i o r of a l a r g e body having an almost complete Meissner e f f e c t , 22 the current i s n e g l i g i b l e and the f l u x o i d very n e a r l y equals the f l u x through C . Since <j) and y 7 ^ are everywhere continuous i n the body, 0\ and ^7<T\ are a l s o continuous. In a s-c body 0\ must be s i n g l e v a lued, since the cont r a r y i m p l i e s a s i n g u l a r i t y of 0\ w i t h i n the body. (In mu l t i p l y - c o n n e c t e d bodies t h i s i s not true s i n c e (7) may increase by i n t e g r a l m u l t i p l e s of 277* along a cl o s e d path going once around the hole.) Consequently i n a s-c body: 'C o n r l cA sss 0 f n r> 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 P P . t l v ' c ' " - - . - -general as to s i z e and shape of the body, or s p a t i a l v a r i a -t i o n of ^  , remembering that ^ occurs under the i n t e g r a l s i g n i n the d e f i n i t i o n of <p So f a r nothing has been s a i d about the gauge of A . The gauge of. A ' and the phase ^ of W are not inde-pendent. London ( 1 9 5 0 ) has introduced a standard gauge which he shows i s uniquely d e f i n e d by V X A — H V'A " 0 everywhere Aj_ ~ 0 on the surface of the superconductor where Aj^  i s the component of A normal to the 23 s u r f a c e . This p a r t i c u l a r gauge i s now r e f e r r e d to as the "London gauge." Let A be i n the London gauge. Then t a k i n g the divergence of both sides of the second G-L equation everywhere on the surface of the body. By a standard proof that 0. i s at most a meaningless constant which may be set equal to 0 . Hence A i n the London gauge i m p l i e s <p r e a l . In the remainder of t h i s chapter (and except f o r the i n t r o d u c t o r y paragraphs of Chapter 3, i n the remainder of the t h e s i s ) , A i s always i n the London gauge. We emphasize t h i s by using ^ i n s t e a d of ^  and we s h a l l a l s o remind the reader from time to time when the gauge i s s i g -n i f i c a n t . In the London gauge the two G-L equations f o r a s-c body take the form: ( 1 . 1 8 ) we o b t a i n V CD — 0 everywhere i n the body. ( 1 . 1 9 ) we have VcT> • c Z ? = 0 From the boundary c o n d i t i o n i t f o l l o w s that 7cn = 0 everywhere i n the body, so ( 2 . 2 ) — > ( 2 . 3 ) I 24 Se c t i o n 2. Formulations of the Electromagnetic Free Energy In order to consider c r i t e r i a f o r phase change i n a s-c superconductor one must f i r s t c a l c u l a t e the t o t a l f r e e energy. . In 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 G-L equations simultaneously f o r and and then s u b s t i t u t i n g i n equation (1.20) f o r . The second term i n t h i s equation which we c a l l the electromagnetic f r e e energy ~7H i s where i s the volume of the body i t s e l f and Is the volume of- a l l space e x t e r n a l to the body, i n c l u d i n g any holes t o t a l l y enclosed by the body. i s the super current d e n s i t y i n the body o n l y , and does not i n c l u d e the / e A l l the formulations are 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 (2.4), the second term i n the integrand i s 0 everywhere i n V p , We put H ~ V X n and HE — V X AE A and A g are everywhere continuous and are both i n the London gauge f o r the body. 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 term : currents producing the e x t e r n a l f i e l d He , We consider now some d i f f e r e n t formulations of t h i s i n t e g r a l f o r Jl 25 ( 2 . 5 ) V, Sz The s u r f a c e S 2 i s a t i n f i n i t y , and the i n t e g r a n d f a l l s o f f a t l e a s t as so t h e s u r f a c e i n t e g r a l i s 0. I n t h e f i r s t i n t e g r a l , i n V 2 , VK H ~ V X / / e , s i n c e the e x t e r n a l c u r r e n t s are c o n s t r a i n e d t o r e m a i n unchanged on the i n t r o d u c t i o n o f the s u p e r c o n d u c t o r i n t o the f i e l d He . Hence th e volume c o n t r i b u t e s n o t h i n g t o the i n t e g r a l . Tn V_ . V x l =G s i n c e H - i n n - n n i i u c e d e n t i r e l v bv JL - " Oi -c u r r e n t s e x t e r n a l t o V n . The i n t e g r a l r e d u c e s t o (mil , /f _ inn + zn>t^) o/V y l %TT Kir c*"?z fi j U s i n g t h e M a x w e l l e q u a t i o n V^H - —^-f- 3 a n d the second G-L e q u a t i o n ( 2 . 3 ) f o r s-c b o d i e s , the sum o f the f i r s t and t h i r d terms i s z e r o , ( F o r f u t u r e r e f e r e n c e we note t h a t t h i s i s 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 b o d i e s , and the analogous development d e v i a t e s a t t h i s p o i n t . ) We have s i m p l y 26 (2.6) / F.-KUM 2c = - J-  Z . A e vI n o b t a i n i n g t h i s s i m p l e e x p r e s s i o n , we have made no assump-t i o n s or a p p r o x i m a t i o n s on the s p a t i a l dependence o f ^  , b e i n g c a r e f u l t o keep i t under the i n t e g r a l s i g n . F u r t h e r -more, so f a r we have made no assum p t i o n s on the s p a t i a l form o f H , i t need not be u n i f o r m , e 5 I f H g i s _ u n i f o r m , H^, depends o n l y on the f i r s t moment o f the c u r r e n t . I n t h i s case we can put /A — ffe&sl/Z H a c o n s t a n t . Then, u s i n g t h e 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 I,-^f^(Z^)^ =-^j^f.dv  (2- 7) The f i n a l i n t e g r a l i s j u s t t h e d e f i n i t i o n o f magnetic moment /9( . Hence ; _ — ( 2 . 8 ) S i n c e the c u r r e n t s a r e i n a d i a m a g n e t i c s e n s e , 7P£ i s n e g a t i v e r e l a t i v e t o EQ and ^  i s p o s i t i v e . The g e n e r a l v a l i d i t y 27 of 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 form, however, i s d e c e p t i v e l y simply - only on the a p p r o x i -mation of ^  uniform, and f o r simple geometry, can 0^ be c a l c u l a t e d i n closed form. Another i n t e g r a l which turns out to be equal to ~Jtf i s 2. (H*-H*-) Zrrk ^ \ / s / (2.9 7 * t ' ) o i V To prove the e q u a l i t y , we adopt a procedure s i m i l a r to that f o l l o w i n g equation (2.4). F a c t o r i n g , and using A , A g as be f o r e , the I n t e g r a l become Jv,,v2 i n Jv,c f / Then, r e p e a t i n g the remarks f o l l o w i n g (2.5), we have Jv, I g1T C Y I and removing terms which, by the second G-L equation sum to zero, gives which i s i d e n t i c a l to (2.6), that Is to 3^  . Let us look f o r the m o t i v a t i o n f o r w r i t i n g the p a r t i c u l a r i n t e g r a l (2.9). C a l l the negative of t h i s I ntegra 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 of Integrand, and not j u s t the t o t a l value. 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 the body under the c o n d i t i o n that the magnetic moment, the extensive magnetic v a r i a b l e , i s hel d constant r a t h e r than the e x t e r n a l f i e l d which i s the i n t e n s i v e v a r i a b l e . Gr i s the Legendre transformed thermodynamic p o t e n t i a l of > and d i f f e r s from i t by the product of the i n t e n s i v e and extensive v a r i a b l e s . Gr = 7H + H[- He = +j ^ • He - - JH Therefore ~ """ Gr , which i s j u s t (2.9). F i n a l l y , i f we average the two expression f o r 7 y i n (2.4) and (2.9), we o b t a i n J H = j iZ°(Z~H) d V C 2 . i o ) Or, i f H i s uniform ' e Which form we choose to evaluate depends on the p a r t i c u l a r geometry. I f the demagnetization approaches 0 the l a s t form i s simple, since i n that l i m i t ' (H-PI ) approaches 0 i n . In many cases (2.6) or (2.7), i f H 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 the body only . 29 S e c t i o n 3 . Order Parameter The property of small bodies which we s h a l l e x p l o i t here i s the near u n i f o r m i t y of the order parameter >^  The 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 i s the coherence l e n g t h f . Since has no component normal to the surface by equation (1.19) and s i n c e V 2 ^ ^2. ( i s i n v e r s e l y p r o p o r t i o n a l to 5 by equation (2.2 ), changes l i t t l e over distances s m a l l compared to X In Type I superconductors, A < f , the body may be small and s t i l l permit , where d represents the g r e a t e s t s i g n i f i -cant dimension of the body. On the other hand, i n Type I I superconductors, f ^ X i a n d smallness i s a stronger c o n d i t i o n than d< A . Having e s t a b l i s h e d that i s n e a r l y uniform, the q u e s t i o n a r i s e s how to f i n d i t s mean va l u e . S t r i c t l y speaking, the boundary c o n d i t i o n (1.19) s t i l l determines ^ , through 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 . 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 to apply. I t i s e a s i e r to assume ^ s t r i c t l y uniform (and of course r e a l ) from the s t a r t i n equation (1.20) f o r jF . The gradient term drops out and ^ f a c t o r s out of the i n t e g r a l s , l e a v i n g $7T JVj X (2.11) 30 Now we simply set the p a r t i a l d e r i v a t i v e of 77 w.r.t. equal 0 , y i e l d i n g f ' HSV, J A 2 d Y A i s a l s o a f u n c t i o n of ^ , through the second G-L equa-t i o n , so t h i s i s an i m p l i c i t expression f o r ^ which cannot i n general be solved 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. In p r a c t i c e i t i s u s u a l l y more convenient to r e t a i n ^ as a parameter u n t i l the computation of 7 i s completed, by one of the methods p r e v i o u s l y d i s c u s s e d . Then J~ i s d i f f e r e n t i a t e d w.r.t. ^ to f i n d the which minimizes i t . In t h i s case we are t a k i n g the t o t a l d e r i v a -t i v e w.r.t. , r a t h e r than the p a r t i a l d e r i v a t i v e . However, s i n c e A s a t i s f i e s the second G-L equation, which i n t u r n makes the v a r i a t i o n of / w.r.t. A zero, 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 lead to the same r e s u l t f o r *^ e^  by both methods. S t r i c t l y speaking a body i s small i f i t s g r e a t e s t dimension i s much l e s s than "f . Some bodies may be con-s i d e r e d s m a l l i f only one or two dimensions are s m a l l , provided t h e i r symmetry i s such as to ensure u n i f o r m i t y of 7^ i n the other d i r e c t i o n s . Thus a plane t h i n f i l m of i n f i n i t e extent i s small i f t h i c k n e s s d « f . A l ong 31 c y l i n d e r not n e c e s s a r i l y c i r c u l a r , p l a c e d p a r a l l e l t o t h e e x t e r n a l f i e l d , i s s m a l l i f i t s g r e a t e s t t r a n s v e r s e dimen-s i o n d s a t i s f i e s On t h e o t h e r hand, a s p h e r i c a l s h e l l o f r a d i i J l ) , Jl-^ , i s not s m a l l , even though s i n c e w i l l have a s i g n i f i c a n t , v a r i a t i o n from p o l e t o e q u a t o r , where the p o l a r a x i s i s p a r a l l e l t o the e x t e r n a l f i e l d . Example I t i s our purpose i n t h i s c h a p t e r t o g i v e r e s u l t s i n t h e most g e n e r a l form p o s s i b l e , and t o n ote c a r e f u l l y i f and where any r e s t r i c t i o n s o r a p p r o x i m a t i o n s are made. I t 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 f o r m u l a s by w o r k i n g a s p e c i f i c example. We choose d e l i b e r a t e l y a s i m p l e w e l l - s t u d i e d geometry i n o r d e r t o f a c i l i t a t e t h e c o m p a r i s o n of d i f f e r e n t p r o c e d u r e s . No o r i g i n a l i t y i s c l a i m e d f o r t h e r e s u l t s - o n l y f o r t h e v a r i e t y o f ways i n w h i c h the r e s u l t s can be o b t a i n e d . C o n s i d e r a l o n g t h i n J l , « > s o l i d c y l i n d r i c a l s u p e r - c o n d u c t o r o f r a d i u s J l , , p l a c e d p a r a l l e l t o a u n i f o r m e x t e r n a l f i e l d H g . The c y l i n d e r i s c o - a x i a l t o t h e u s u a l ft) 9} Z- c y l i n d e r i c a l c o - o r d i n a t e system. Assuming symmetry i n the $ , Z d i r e c t i o n s and ^ u n i f o r m , the second G-L e q u a t i o n becomes the t o t a l d i f f e r e n t i a l e q u a t i o n 32 with s o l u t i o n 'A, A where the I are modified Bessel's f u n c t i o n s , n We proceed to c a l c u l a t e the magnetic f r e e energy per u n i t l e n g t h of c y l i n d e r by three of the preceding formulas From (2.4) 7. H 8TT From equation (2.7) 3 3 He A y0 A /o r v ^ ) 8 * 1 = / / e ^ / x J 2 far/*) ? Io(^y/A) From equation (2.10) H fir L e 1 /„ i Jo ^,?/A)/ _ ///.<2,* J ; 1^/ ? /A ) The e q u a l i t y of the various formulations i s thus demonstrated 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. S e c t i o n 4 . Order of Phase T r a n s i t i o n In zero e x t e r n a l f i e l d i n e q u i l i b r i u m ^ ~. I} y~ ~ ~ tick V|/ETt . As T-*Tc , H c b™* 5' 0 s e e equation (1.6.) and "J- — > 0 . Although the r e l a t i v e d e n s i t y ^ remains 1, z the dimensional e l e c t r o n d e n s i t y ~ y — approaches 34 0 as T —»T . T h e r e f o r e , as T — » T i n zer o f i e l d , a c 3 c 3 second o r d e r phase change o c c u r s . I t i s w e l l known e x p e r i -m e n t a l l y f o r l a r g e b o d i e s i n an e x t e r n a l f i e l d t h a t a f i r s t o r d e r phase change t a k e s p l a c e when \ He\ exceeds H ^ , whic h may o c c u r e i t h e r by i n c r e a s i n g \He\ o r by i n c r e a s i n g T and t h e r e b y l o w e r i n g H ^(T) , see e q u a t i o n (1.6 ). From t h e g e n e r a l form o f ~J" t o g e t h e r w i t h t h e second G-L e q u a t i o n , we can show i n a g e n e r a l way why t h i s i s s o , and a l s o t h a t t h e r e i s , i n p r i n c i p l e a t l e a s t , an i n t e r v a l o f H e about 0 i n which a second o r d e r phase change o c c u r s when T. i s i n c r e a s e d . T h i s i n t e r v a l i s e x t r e m e l y s m a l l f o r b o d i e s h a v i n g As we c o n s i d e r s m a l l e r and s m a l l e r b o d i e s t h i s i n t e r v a l i n c r e a s e s u n t i l f o r a c e r t a i n c r i t i c a l s i z e —> . a l l phase t r a n s i t i o n s a r e second o r d e r . When Hg v^ 0 , Y appears i n Jlj as w e l l as i n , and c o n s e q u e n t l y t h e v a l u e o f ^ f o r which ~J" i s minimum i s not n e c e s s a r i l y 1. Throughout the 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 t h r o u g h -out t h e r e m a i n d e r of t h i s t h e s i s i n cases i n whi c h *^  i s 1 n e a r l y u n i f o r m , i t i s c o n v e n i e n t 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 r e a d e r i s c a u t i o n e d not t o pass l i g h t l y o v e r t h i s d e f i -n i t i o n , from here on *\, w i l l appear w i t h g r e a t f r e q u e n c y . 35 We wish to o b t a i n a q u a l i t a t i v e i d e a of the depen-dence of I? on ^ without a c t u a l l y s o l v i n g the G-L equa-t i o n s . The form of the condensation energy 3^  i s the same f o r a l l bodies We take the e x t e r n a l f i e l d uniform, and evaluate 3^ y by ( 2 . 6 ) or ( 2 . 7 ) , both of which r e q u i r e a knowledge of ^ (St ) . From the second G-L equation i n the form ( 2 . 3 ) fi ~~~ +Tf ^ ( 2 . 1 5 ) and A enter only In the combination A f^ . Put $ s A . Then ( 2 . 1 5 ) i s j u s t the London equation w i t h p e n e t r a t i o n depth £ , that i s — o c A ( 2 . 1 6 ) For l a r g e & , ^  becomes s m a l l . In the l i m i t <3 ~"9 °° then ~f —* 0 and consequently H * /V- and ,4 ? / 4 . For l a r g e S then, we can put to f i r s t approximation c A< This approximation i s good f o r Ct , the l a r g e s t s i g n i f i -cant dimension of the body. Using equation ( 2 . 6 ) we have 36 where A g i s u n d e r s t o o d t o be i n the London gauge. T h i s 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 the body, and i s independent o f ^ and A . T h e r e f o r e Jtf i s p r o p o r t i o n a l t o l/S f o r s m a l l v a l u e s o f W , i . e . J ^ , ^/A) s t a r t s o f f from 0 w i t h f i n i t e s l o p e p r o -2 p o r t i o n a l t o H ^ e As shown by London q u i t e g e n e r a l l y when $ 0 , the c u r r e n t f l o w s i n a t h i n l a y e r a t t h e s u r f a c e . The s m a l l e r 8 the t h i n n e r the c u r r e n t l a y e r , but t h e t o t a l c u r r e n t i n the l a y e r approaches a l i m i t i n g v a l u e s i n c e i t s j o b i s t o e x c l u d e H f r o m t-xie u o u y , x, «. i t must g e n e r a t e a magnetic f i e l d i n the body o p p o s i t e t o H g . As S —*• 0 , the body approaches a s t a t e o f i d e a l d iamagnetism. The i d e a l d i a — — * magnet r e p r e s e n t s an upper l i m i t on f&( and hence on ~Jy f o r g i v e n HE . C o n s e q u e n t l y as > < x > ) ^ > i n c r e a s e s m o n o t o n i c a l l y and a s y m p t o t i c a l l y toward some upper l i m i t , w h ich depends on HE and the shape o f the body. I n summary, f o r s m a l l ) , $ ^ i s p r o p o r t i o n a l t o (//$ ) w i t h s l o p e p r o p o r t i o n a l t o HQ , w h i l e f o r l a r g e (//Sz) } j£ becomes h o r i z o n t a l w i t h v a l u e p r o p o r t i o n a l t o HE A t y p i c a l s e t of c u r v e s o f w i t h HQ as parameter i s shown i n F i g u r e 2-1. 37 Having obtained as a f u n c t i o n of i t i s easy to o b t a i n as a f u n c t i o n o f ^ w i t h -A as parameter. A s ^ ranges from 0 to 1 , ranges along the curve of Figure 2 - 1 from 0 to One has the set of curves of Figure 2 - 2 which have been normalized so as to give the same at "J, — I . In the mathematical l i m i t A 0 0 the curve becomes a s t r a i g h t l i n e w i t h f i n i t e p o s i t i v e slope whereas i n the l i m i t A ~* 0 i t becomes a step f u n c t i o n . Of course n e i t h e r l i m i t e x i s t s p h y s i c a l l y . I t i s convenient to d e f i n e the "Reduced Free Energy" ( 2 . 1 7 ) pr — c?77 ~z rr __ Sir -~j Then F = c Z.^. r y i s a simple u n i v e r s a l f u n c t i o n of ^ independent of H , a parabola passing through the o r i g i n w i t h slope - 2 and having a minimum a t ^ — / , F c = - 1 shown In dashed l i n e i n Figures 2 - 3 a and 2 - 4 a . V/e s h a l l consider F(^-) i n the two extreme cases (a) X 4 ^ - d , and (b) \ s > cL s and then (c) determine 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 these two types of behaviour. Oase (a) A « cl- See Figure 2 - 3 . In F i g u r e 2 - 3 a the set of n e a r l y s t r a i g h t l i n e s i n the upper quadrant represent F^ 38 as a f u n c t i o n of f o r i n c r e a s i n g parameter H G . Figu r e 2 - 3 b shows the sum F=F +FJJ . E v i d e n t l y there i s a minimum of F w.r.t. ^ to the r i g h t of the o r i g i n u n t i l H E i n c r e a s e s to a value at which (^) has slope + 2 at the o r i g i n . For H G greater than t h i s , there i s no minimum of F~ f o r J > 0 . In e q u i l i b r i u m , ^ - takes the value which minimizes F* . As H G i n c r e a s e s , the e q u i l i b r i u m values of ^ and F approach zero c o n t i n u o u s l y . Thus, at l e a s t i n t h i s extreme case, there i s a second order phase t r a n s i -t i o n . Case (b) d ^ > A 3 see Figu r e 2 - 4 . The curves of ^ ^ s t a r t at 0 and r i s e r a p i d l y to a. p l a t e a u , the height of which i s p r o p o r t i o n a l to Hg . In Figu r e 2 - 4 b showing the sum F c+F^ , the minimum of F c i s superimposed on the p l a t e a u of F„ , g i v i n g a l o c a l minimum i n F at J & I . F may take on negative or p o s i t i v e values at the minimum, i n the l a t t e r case the body i s i n a metastable s t a t e blocked from going Into the normal s t a t e by the p o t e n t i a l b a r r i e r . The plateau, i s not q u i t e f l a t ; i t s slope as w e l l as i t s height incr e a s e s w i t h H . E v e n t u a l l y H . reaches a value at e J e which the slope of F^ overcomes the d i p i n F c , and the minimum i n F disappears. There i s thus a c r i t i c a l e x t e r n a l f i e l d 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 _ there i s no minimum of F and no super e e 3 s t a t e can e x i s t . 39 (c) I t i s e a s i e r to f i n d the d i v i d i n g l i n e between these two extreme types of behaviour beginning w i t h case ( a ) , and no t i n g the m o d i f i c a t i o n i n the p i c t u r e as A decreases. As shown i n Figure 2 - 2 , the l i n e s of (^ ) become more and more curved as the c o n d i t i o n A ceases to be v a l i d . For every d , there i s some value of H G , say , f o r which ( has slope +2 at the o r i g i n . At H E = H , F has zero slope at the o r i g i n . Now i f i n t h i s case the curve of F has l e s s curvature at the o r i g i n than F q , there w i l l be no minimum f o r ^ e ^ ^ e j except at 0 . But i f fjjj l^) has gr e a t e r curvature at the o r i g i n than F c , there w i l l be a minimum . f o r H E > H . , and the type of behaviour of (b) above w i l l . occur, although the a c t i o n may take place much nearer the o r i g i n than appears i n the f i g u r e . F i n a l l y we summarize i n mathematical form the c r i t e r i a f o r phase t r a n s i t i o n s . The uniqueness of the r e s u l t s obtained by a p p l i c a t i o n of these c r i t e r i a r e s t s on the p o s s i b l e forms of r ^ discussed above. These c r i t e r i a have been given by Ginzburg (1958) who obtained curves s i m i l a r to F igure 2-3 and 2-4 by d i r e c t c a l c u l a t i o n of F f o r p a r t i c u l a r bodies f o r which the f i e l d equation can be solved In c l o sed form. I f a second order phase t r a n s i t i o n occurs, i t does so at a f i e l d H -, given by 4 0 A second order phase t r a n s i t i o n does Indeed occur, i f , at the f i e l d H n , 5 e l ' An e q u i l i b r i u m f i r s t order phase t r a n s i t i o n occurs at the f i e l d H ^ which s a t i s f i e s simultaneously (%)•> • r'° • r° '*-Although the f i r s t equation may be s a t i s f i e d at a maximum of F , F > 0 always at such a maximum. A metastable s t a t e may e x i s t f o r ^ ^ ^ ^€3 where i s given by the simultaneous s o l u t i o n of • &)-<> _ '"" As He-*>He^ the maximum and minimum of f~ ( ^ ) erf \ coalesce, and • n a s a h o r i z o n t a l i n f l e c t i o n p o i n t . S e c t i o n 5- Expansion of F i n powers of In order to I n v e s t i g a t e second order t r a n s i t i o n s i t i s convenient to have F i n the form of an expansion i n powers of ^ . F q i s already i n t h i s form, -^2^, f"^. 4 l We have already noted that f o r a given body i s propor-11 z t i o n a l to he . and that as £ 0 . Hence we may w r i t e the expansion of F^ i n the form ^ J ? A % X ) (2.22) Hc£{ A 2"/ A* / . - A 6 where the A ' s have the dimensions of length and w i l l depend on the geometry of the body. The a l t e r n a t i n g signs w i l l be j u s t i f i e d s h o r t l y . The second order.' c r i t i c a l f i e l d i s given by - z ^ ^ = o //cl, X The c o n d i t i o n that a second order t r a n s i t i o n occurs i s * \ile - He, which together w i t h ( 2 . 2 3 ) gives a c o n d i t i o n on A only : J?* i (2.24) Once F„ i s obtained In the above s e r i e s form, the second n order c r i t e r i a can be obtained by i n s p e c t i o n . I f F^ i s known i n closed form, then the s e r i e s (2.22) can be obtained by s t r a i g h t f o r w a r d expansion of F^ i n a McLaurin s e r i e s . There i s , however, a method by which the s e r i e s can be obtained without s o l v i n g the f i e l d equations. 4 2 Furthermore t h i s method can be a p p l i e d to any body, although numerical i n t e g r a t i o n may be r e q u i r e d . Consider the expres-—7 s i o n f o r Jr given i n equation ( 2 . 6 ) A g i s independent of ^ , so i f we express ^ & as a power s e r i e s i n 3 t h i s expression w i l l g ive the r e q u i r e d ex-pansion of F„ . ri As ^ — p 0 , the supercurrent —* 0 and the ve c t o r —S> — 9 p o t e n t i a l A — ? A . We assume A and A are both i n the e e London gauge f o r the body. We may w r i t e ^ and A as s e r i e s expansions I n powers of y , w i t h terms of the s e r i e s numbered a by s u b s c r i p t s i n parentheses, w h e r e ^ ^ j , A^j are f u n c t i o n s of p o s i t i o n . For a plane f i l m i n the X} ^ plane p a r a l l e l to HQ l ^ l y , A Q ~i^£\z 3J . For a body having a x i a l symmetry about an a x i s p a r a l l e l to the f i e l d , but otherwise of a r b i t r a r y —P *. shape, A g i s i n the 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 centred on the a x i s X = ~? \^e\ 9 ( 2 . 2 6 ) 4 3 For other shapes A g can be found by adding g r a d i e n t s of s c a l a r s s a t i s f y i n g Laplace's equation to one of the above expressions i n such a way as to s a t i s f y the boundary con d i -t i o n Aea dS 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 equating c o e f f i c i e n t s of the same powers of ^  on e i t h e r s ide i> ro> fir )f _ — -JL_ Ao> e t c . This gives already the f i r s t term of ^ . From Maxwell's equation 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 expansions f o r A , ^ and equating c o e f f i c i e n t s of equal powers of ^< etc Since i s a known f u n c t i o n , the f i r s t l i n e i s Poisson' equation f o r the components of /4 f ) > , w i t h s o l u t i o n In t h i s way we may o b t a i n a l t e r n a t i v e l y the next term of each s e r i e s . S u b s t i t u t i o n i n (2.25) then gives the s e r i e s f o r PTT . Examination of t h i s procedure shows that the ri s e r i e s f o r F^ i s a l t e r n a t i n g . Example Consider again the long t h i n c y l i n d e r of r a d i u s a x i a l symmetry i s given by (2.26). Then Taking advantage of 6 and Z symmetry, we have which has the s o l u t i o n s a t i s f y i n g the boundary c o n d i t i o n s 4 5 Consequently He f I S u b s t i t u t i n g i n t o equation ( 2 . 2 5 ) f o r 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 to two terms 6 ^ A* 3SV-A* ^ I t i s i n t e r e s t i n g to note that the f i r s t term i s p r o p o r t i o n a l to the moment of i n e r t i a of the body. For the reduced magnetic f r e e energy r # o J f y / /7c6 J*-i z from which we i d e n t i f y Let us compare t h i s with the cl o s e d expression f o r F^ obtained on page 3 2 4 6 (i,llZ\) •I k! k! zk k! (Zr-k)l I k = o Met1 i s then He,7' 2X1 /6 A 2 and. the c o n d i t i o n that a second order t r a n s i t i o n does occur i s i n agreement with Ginzburg (1958). We now r e t u r n to the d i s c u s s i o n of the expansion of F f o r a body of a r b i t r a r y shape. This expansion not only serves to evaluate F i n the l i m i t ^-—^0 , but describes the approximate behaviour of F i n the neighbourhood of % ^ 0 • For °y s u f f i c i e n t l y s m a l l , we can approximate by the f i r s t few terms of the s e r i e s . 4 7 ( y V V z ) I f | — 6 [ * I <0 and 0^/4 ^ //e/ , two terms w i l l do which has a minimum at I \Hd '» / / \ rich A at which F - - {J*L£_ 7 f /J. _ H ; I X £ S ~ W\ l 7 / T far Since // _ H^ALAI. / remains f i n i t e as /TL — * H^i » V^de-\ : •— > 2. creases l i n e a r l y to 0 w i t h . F i s always ne g a t i v e , 2 and F e q ( ^ q ) goes as ~ a s ^ - * ^ . This i s the t y p i c a l behaviour of a second order phase t r a n s i t i o n . I f (j — i k J i J * - ) / Q when H =H , then we must approximate by three terms of the s e r i e s . This i s v a l i d f o r s i t u a t i o n s i n which the phase t r a n s i t i o n i s b a r e l y f i r s t order, that i s , i t occurs at very s m a l l ^ . To three terms : 48 F^ -) i s a cubic passing through the o r i g i n , having a maxi-mum of say F at 'J-" and a minimum of F at J max $ max eq / eq F > F and <f /V . The dependence of FM on the max r eq <f max /eq ^ a' v a r i a b l e H e has the general character of Figure 2-4b. We can observe t h i s by no t i n g the migrations of the extrema as H i s i n c r e a s e d . For , only '•V i s to the , n l y r i g h t of the 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 represents a s t a b l e superconducting s t a t e . At H =H , (defined by equation (2.19)), F =0., ^ i a i i z s : 0 , but < 0 s t i l l r epresents a unique minimum s t a t e As H increases the maximum moves up to the r i g h t 'JiL ">D, e • flrnoL*. ' F~rnax ^  ® > r e p r e s e n t i n g a p o t e n t i a l b a r r i e r . F g a con-t i n u e s to r i s e v/ith H , and becomes zero at a f i e l d e ' H „ given by the value of H which s a t i s f i e s e2 • , e 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 c r i t i c a l e2 ^ f i e l d . I f H > H „ then F > 0 , and the system i s i n a e e2 eq 3 J metastable super s t a t e . In p r i n c i p l e , i f we wait long enough, i t w i l l go over to the normal s t a t e which has lower free energy. F i n a l l y , at a f i e l d He^ > He^ > given by the value of H"e which s a t i s f i e s 19 3 ' h i l l _; WjWtf) I, H / J X ] 2 .tfci'A1 7( Hdz A6 / ( /&fX* J the two extrema c o a l e s c e , and there i s a h o r i z o n t a l i n f l e c -t i o n i n the curve. At He > , no super s t a t e can e x i s t . S e c t i o n 6 . Temperature Dependence The equations f o r the various c r i t i c a l f i e l d s ^ e l 3 ^e2 3 ^e3 Involve the f a c t o r s and X , whose temperature dependence i s w e l l described by X — A 0 / ( t ~ t*) where t=T/T (2.27) c The equations f o r the c r i t i c a l f i e l d s are thus l i n e s i n the H e - t plane. When H , t are.caused to vary along any path whatever i n the plane which cuts one of these l i n e s , some change occurs i n the s t a b i l i t y of one phase wit h respect to the other. The second order c r i t i c a l f i e l d i s given by He^=H ^A/^/f or w r i t i n g i n the temperature dependence e x p l i c -i t l y H e i ( t ] = Mxl* ( 2 . 2 8 ) 5 0 I t s slope i s which i s 0 at t = 0 and i n f i n i t e at . t = l . The phase order-determining c r i t e r i a , namely ^ A IZ f o r second order, i s a l s o temperature depen-dent. In general there i s a range of t f o r which second order changes occur. The "order-determining temperature" t which i s the lower bound of t h i s i n t e r v a l i s given by I f t ^ t a f i r s t order change occurs as H"e i n c r e a s e s . I f "t ^  i a second order change occurs as H"e i n c r e a s e s . I f \0-fl/z then t * = 0 , and f o r a l l phase t r a n s i -t i o n s are second order. On the other hand as t —*» 1 , A increases without l i m i t , so there Is always some range of t near 1 i n which second order t r a n s i t i o n s occur. Corresponding to t , there i s an order-determining f i e l d H ( t ) , which by equation ( 2 . 2 8 ) i s found to be H* = H0lu*) = HH00^-(,-ezf <2-30) I f ^ He a f i r s t order t r a n s i t i o n occurs as t i s r a i s e d I f We ^ He a second order t r a n s i t i o n occurs as t i s r a i s e d 51 For a l l bodies having the same shape, but d i f f e r -ent s i z e , J^x^/ ^ s a numerical constant. Equation ( 2 , 3 0 ) f o r H g (t ) i s a l i n e i n the H - t plane belonging to a p a r t i c u l a r shape of body, which separates the plane i n t o two p a r t s . 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 are f i r s t order, and a l l t r a n s i t i o n s outside the curve are second order. This curve has f i n i t e negative slope at t = 1 , so that some p o r t i o n of a l l the H , curves l i e s 1 1 e l to the r i g h t of t h i s d i v i d i n g l i n e . For ^ ^ X / fz } the e n t i r e H , curve l i e s to the r i g h t . Curves of H , ( t ) e l & e l f o r v a rious are" shown i n Figure 2-5 . H , ceases to e x i s t as a second order phase el ^ t r a n s i t i o n boundary at t . The curve d e f i n e d by the same equation H g^(t) £ o r t < t has the meaning of the m i n i -mum f i e l d i n which a metastable normal s t a t e can e x i s t . As t decreases below t , the second order t r a n s i t i o n curve s p l i t s i n t o three branches: the mathematical c o n t i n u -a t i o n of which i s the "s u p e r - c o o l i n g " curve, the e q u i l -i b r i u m phase t r a n s i t i o n curve H ^ ( t ) , and the extreme "super-h e a t i n g " curve H q ( t ) which i s the maximum f i e l d i n which the metastable super s t a t e can exist.. Our expansion of F i n powers of ^  i s v a l i d i n some neighbourhood to the l e f t of t . The equations f o r H „ , H _ can be solved as 1 e2 ' e3 q u a d r a t i c s i n He^ and > a n <^ ^ e 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 to 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 of these curves near t : 1) H e 2 (t ) H _ ( V ) = H . ( t ) e3 e l A l l curves are continuous at t 2) The slopes of the curves are a l s o continuous across t We s h a l l work wi 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 both sides of eq. (2.26) f o r H g 2 w.r.t. t fHet. *&i J-i £ j d jMe-Z -^2. Ah" A dt I HcL2 \* The LHS and the f i r s t term on the r i g h t are 0 by the d e f i n i -t i o n s of t and H Hence dt ( Hcf \ Z 1 But a l s o d I Hei -ii dt \ Hcb2 A 2 Expanding the l a s t two equations and s e t t i n g them equal y i e l d s c l H t d t M d t (2.32) 53 3) We proceed to c a l c u l a t e the second d e r i v a t i v e s at t D i f f e r e n t i a t i n g both sides of (2.. 31), and l e a v i n g out •x the terms which are zero at one obtains 2 Hez dt Z ,2 _/_ M e i A f H-HcbZAZ^ (2.33) Whereas f o r H ^ ( t ) we have /I d f Ue^A 1 0 (2.34) Expanding the LHS of these l a s t two equations, and usin< the r e s u l t (2.32) above, a l l terms w i l l be equal except c ^ e ' and . Su b t r a c t i n g (2.34) those c o n t a i n i n g from (2.33) ie/ fez fx 2-~dZrlz>~ dix L J df Jz t ei if (/-t*T Since the RHS of t h i s equation i s always p o s i t i v e , and since cf H&) df i s negative as seen i n P i g . 2-5, the curve,of H ^ „ has l e s s convex curvature than H n , and l i e s above i t . e2 e l 5 By a s i m i l a r c a l c u l a t i o n , one shows that H ^ l i e s above H e2 to the l e f t of t Hence we may sketch i n the curves of H „ and H „ In the neighbourhood of t e2 e3 These are shown i n Fi g u r e 2-6. F i g u r e 2 - 1 F u n c t i o n a l d e p e n d e n c e o f M a g n e t i c F r e e E n e r g y 7H o n , where S i s t h e e f f e c t i v e p e n e t r a t i o n d e p t h , f o r i n c r e a s i n g v a l u e s o f u n i f o r m e x t e r n a l f i e l d HQ . F i g u r e 2-2 F u n c t i o n a l d e p e n d e n c e o f M a g n e t i c F r e e E n e r g y o n r e l a t i v e s u p e r e l e c t r o n d e n s i t y ^ , f o r v a l u e s o f A f r o m A « oL t o \»c£ . F i g u r e 2-3 R e d u c e d F r e e E n e r g y f o r s m a l l b o d y . ( a ) S e p a r a t e c u r v e s o f c o n d e n s a t i o n e n e r g y F c ( d a s h e d ) a n d e l e c t r o m a g n e t i c f r e e e n e r g y F H ( s o l i d ) a t i n c r e a s i n g e x t e r n a l f i e l d K e » ( b ) T o t a l f r e e e n e r g y . D a s h e d 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 2 n d o r d e r t r a n s i t i o n . F i g u r e 2-k- R e d u c e d F r e e E n e r g y F( ^-) f o r l a r g e b o d y . ( a ) S e p a r a t e c u r v e s o f c o n d e n s a t i o n e n e r g y F c ( d a s h e d ) a n d e l e c t r o m a g n e t i c f r e e e n e r g y F^ ( s o l i d ) a t i n c r e a s i n g e x t e r n a l f i e l d H e . ( b ) T o t a l f r e e e n e r g y . D a s h e d 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 1 s t o r d e r t r a n s i t i o n . He 0 0 { = T/n F i g u r e 2-5 P l o t s o f H e ] _ / Hc-b a s f u n c t i o n s o f t f o r d i f f e r e n t « D a s h e d 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 l o n g c y l i n d e r s . 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 c u r v e i n t o t h r e e b r a n c h e s a t t * . D a s h e d l i n e a s i n F i g u r e 2-5 CHAPTER . 3 MULTIPLY-CONNECTED SUPERCONDUCTORS In t h i s chapter and c o n t i n u i n g to the end of the 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 to m u l t i p l y -connected superconductors. In order that t h i s d i s c u s s i o n may be s e l f contained we s h a l l b r i e f l y repeat from time to time some of the formulas and d e f i n i t i o n s already g i v e n . The 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 to t h e i r shape. In the l a t t e r p a r t we co n s i d e r i n more d e t a i l the simplest geometric shape of doubly-connected superconductor, the i n f i n i t e l y long r i g h t c i r c u l a r c y l i n d e r f o r which we o b t a i n f i e l d equations f o r a r b i t r a r y i n ner and outer r a d i i . For the most part we r e f e r s p e c i f i c a l l y to doubly-connected bodies, the g e n e r a l i z a t i o n to higher con-n e c t i v i t i e s w i l l u s u a l l y be obvious. The a r c h e t y p a l doubly-connected body i s the torus 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 avoid r e p e a t i n g the long phrase "doubly-connected body" we s h a l l use the word " r i n g " 54 55 not i m p l y i n g thereby any p a r t i c u l a r shape except that the body has one hole through i t . Let us w r i t e as before the G-L e f f e c t i v e wave f u n c t i o n i n the form ij) = ^ Q where ^ and CT\ are r e a l — * continuous f u n c t i o n s of /I d e f i n e d everywhere i n the body. (jJ must be continuous and s i n g l e valued. (7\ must be continuous but need not n e c e s s a r i l y be s i n g l e valued s i n c e t7* and <7)-f*2nTT give i d e n t i c a l (JJ where n i s a n . i n t e g e r . Indeed <7\ at a p o i n t P may i n c r e a s e c o n t i n u o u s l y as P moves along a contour around the h o l e , so long as i t i n c r e a s e s by e x a c t l y 2nTf when i t r e t u r n s to i t s o r i g i n a l p o s i t i o n . However, i n going around any contour 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 value j u s t as i n a simply-connected body. These two s t a t e -ments taken together show that i f G> i n c r e a s e s by 2nff along one contour c i r c l i n g the hole i t must i n c r e a s e by 2nT7* along every contour c i r c l i n g the h o l e . If_ G\ i s s i n g l e valued then f) — 0 and v i c e v e r s a . W i t h i n the framework of the G-L theory a superconducting r i n g may be i n any one of 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 n . The l a b e l n i s a property of the hole. In a more h i g h l y connected body the s t a t e i s l a b e l l e d by a set of 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 n . Prom equation (2.1) f o r the f l u x o i d we have f o r a contour C 56 going once around the h o l e : ( 3 . 1 ) We a r b i t r a r i l y choose one sense of the contour as p o s i t i v e . Then the p o s i t i v e sense of H across the surface bounded by C i s that of the magnetic f i e l d produced by current 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 sense. I f there i s an e x t e r n a l f i e l d t h reading the contour, we u s u a l l y choose the p o s i t i v e sense so that the e x t e r n a l f i e l d i s p o s i t i v e . With respect to t h i s convention, <pc may be p o s i t i v e or negative. I f (7\ Increases by 2n7t i n going once around the h o l e , the contour i n t e g r a l i s j u s t n / c h ) . The q u a n t i t y ch/q i s c a l l e d the quantum of f l u x o i d , symbol <fi0 . -fie ^ s a quantized q u a n t i t y having the same value H <p-0 f o r every contour surrounding the hole once. Conse-quently 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 of the r i n g as a whole. Prom experiment, <j>Q i s found to have the value of ch / 2 e where e i s the magnitude of the e l e c t r o n i c charge. The e f f e c t i v e charge q i s equal i n magnitude to the charge of two e l e c t r o n s . 57 The c o n d i t i o n s d e f i n i n g the London gauge of the vect o r p o t e n t i a l A given on page 22 e q u a l l y w e l l d e f i n e a unique gauge of A f o r a multiply-connected body. We  s h a l l always take A i n the London gauge. This i m p l i e s c e r t a i n c o n d i t i o n s on <7> . Taking the divergence of the second G-L equation (1.18) y i e l d s V1 <7) — 0 everywhere i n the body. The boundary c o n d i t i o n (1.19) gives V<7) * everywhere on the surface of the body. I f n=0 and (7) i s si n g l e - v a l u e d one may show as f o r a s-c body that V<7> = 0 everywhere and we may put (7) ~ 0 . For given H ^ 0 the c o n d i t i o n s on <T\ determine a unique f u n c t i o n <J\(A) w i t h i n a constant f o r a p a r t i c u l a r body. The proof f o l l o w s the usual form; i f OMSiJ and cni/t) are two f u n c t i o n s s a t i s f y i n g a l l the 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 constant. The f i e l d l i n e s of V<T\ are s o l e n o i d a l and c i r c l e the r i n g . Surfaces of constant <J\ cut across the r i n g meeting the surface of the r i n g normally. Consider now a r i n g i n the absence of any e x t e r n a l f i e l d . We s h a l l show that the t o t a l loop current I ^ c c i r c u l a t i n g around the hole Is 0 when n=0 and that J C ^ when n ^ 0 . Forming the s c a l a r product of both side s of the second G-L equation ( 1 . 1 8 ) w i t h ^ , and rea r r a n g i n g the terms we have: 58 This equation i s v a l i d i n a l l space since ^ i s i d e n t i c a l l y zero outside the body. I n t e g r a t i n g over a l l space, u s i n g the Maxwell equation ^ — V X V X A i n the second term: •rrX1 T2 IM lc7 ' J Z 7 r h , Space On the LHS we perform a p a r t i a l i n t e g r a t i o n on the second term. The surface i n t e g r a l thus obtained over a surface at i n f i n i t y i s zero. P u t t i n g VxA = H , the LHS becomes: Space To i n t e g r a t e the RHS d i v i d e the volume of the r i n g i n t o t h i n s l i c e s bounded by surfaces S of constant (7) We may w r i t e the element of volume dV as d l ' ' olS where i s the t h i c k n e s s of the s l i c e and ds i s an element of the surface of the s l i c e , that i s of the surface S of constant (Ts . Noting that VO) i s p a r a l l e l to dS , we may w r i t e the RHS as: £ q> V<7)>dl zrr „c The i n t e g r a t i o n over S y i e l d s I the t o t a l c u r r e n t , c which i n the steady s t a t e i s the same across any surface c u t t i n g the r i n g . The i n t e g r a t i o n of V^J) around the r i n g j u s t gives 2 n 77 . 59 Consequently: Space Since the integrand on the LHS i s p o s i t i v e d e f i n i t e , then i f n=0 we must have ^ = 0 everywhere. Conversely i f ft =ft 0 (which i m p l i e s ^ ^ 0 at l e a s t somewhere) then both s i d e s are p o s i t i v e and Furthermore I and (f>c have the same s i g n . In summary, i f a body i s i n a s t a t e n~=fcO then ( i ) there i s a p e r s i s t e n t loop current I i n zero e x t e r n a l f i e l d , and ( i i ) the f l u x o i d of any contour around the hole (or "of the ho l e " f o r short) i s n 4 We have s t i l l to demonstrate that these s t a t e s e x i s t p h y s i c a l l y . Consider a r i n g at temperature T> T c , i n an e x t e r n a l f i e l d which passes through the h o l e . As the r i n g i s cooled below T c and the r i n g becomes super-conducting, a l l quantum s t a t e s are a v a i l a b l e to i t . I t t h e r e f o r e chooses the s t a t e w i t h the lowest f r e e energy. This i s not i n general a s t a t e of zero f l u x o i d , s ince there i s f l u x from H through the h o l e . In general there w i l l be a p e r s i s t e n t loop current i n the r i n g i n t h i s s t a t e , and a p e r s i s t e n t (though d i f f e r e n t ) loop current w i l l con-t i n u e to 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 i s removed. 60 The f i n a l proof i s always experimental The exi s t e n c e of p e r s i s t e n t currents has been known f o r some time and c o n s t i t u t e s one of the most dramatic demonstrations 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 of the f l u x o i d has been observed d i r e c t l y i n the now c l a s s i c experiments of Deaver and Fairbank (1961) and of D o l l and Naubauer (1961). Having e x t a b l i s h e d the exist e n c e and s i g n i f i c a n c e of the n=0 s t a t e s , we now proceed to consider the f r e e energy of these 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 of I n t e r e s t i n g and u s e f u l forms when the body i s i n e q u i l i b r i u m i n the super s t a t e , and (1.18) holds. From equation (1.21) which i s j u s t the c l a s s i c a l expression f o r the magnetic f r e e energy, comprising the f i e l d energy and the k i n e t i c energy of the cu r r e n t . P u t t i n g H ~ V XA , He ~ VX /Ag and no t i n g that A — 0 i n V„ v/e have p 2 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 term, and dropping the surface i n t e g r a l over a surface at i n f i n i t y , s ince the 61 i n t e g r a n d drops o f f at l e a s t as //^ 3 ~* —o _» In V „ , V X // R 7" AP , sin c e the sources of H and H 2 5 c ' e e x t e r n a l to the body are held constant. Hence c o n t r i —<> butes nothing to the volume i n t e g r a l . In 3 V X <Ve = 0 and V X H - *TTj£ /C •V, i 2c c? 1 f J —<> Combining the f i r s t and t h i r d terms f a c t o r i n g out ^ using the second G-L equation (1.18) gives 2. Z " JVl ZC v irr »/ The f i r s t term i n the integrand i s p r o p o r t i o n a l to the RHS* of equation ( 3 . 2 ) which we have j u s t i n t e g r a t e d over . Using the r e s u l t obtained there we have j = (»*>)lc _ f I • d V (3A) H Zc JVl ~ c So f a r no assumptions have been made about the s p a t i a l —*> —p p r o p e r t i e s of H . I f H is_ uniform over the body, we can put S u b s t i t u t i n g t h i s i n (3.4), using the c y c l i c property of the t r i p l e s c a l a r product and f a c t o r i n g the constant H out of the i n t e g r a l g ives 62 ^ 2c 2 Jv, 2c The l a s t i n t e g r a l i s j u s t the d e f i n i t i o n of magnetic moment - 7 _ (n<p0)Ic _ ± Tj m fa . ( 3 . 5 ) 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 over the body. We d e f i n e another q u a n t i t y Q- , somewhat a r b i t r a r i l y i t may appear, as Jvi+\/2 i t ' J Proceeding as be f o r e , we a r r i v e at Comparing t h i s v/ith (3-4) we have jr^ = -f (j- (3-7) where G i s the above i n t e g r a l (3-6) Averaging t h i s expression f o r v/ith that of (3-3) we o b t a i n 63 a simple expression which does not co n t a i n the current d e n s i t y under the i n t e g r a l . The choice of which form to use depends on the geometry of the body. When A i s i n the London gauge, (A s a t i s f i e s S7 Z(T) - 0 everywhere i n the body and V(7)e dS=Oon the s u r f a c e . For a body having a x i a l symmetry about the z a x i s of a c y l i n d r i c a l co-ordinate system i t i s e a s i l y shown that these c o n d i t i o n s are s a t i s f i e d by and that Sh-irr c/0 = 2 rin-ks already noted, t h i s s o l u t i o n i s unique except f o r an a r b i t r a r y meaningless .additive constant. For an a x i a l l y symmetric body, equation ( 3 - 3 . 1 ) f o r becomes (?)fio ft T \ / , / ( 3 -9 ) 7. H Zc \ Zrrst The two G-L equations take the form I ( J ~ * O C \ 2 U ,2. ( ^  n<f>o ZTTSL A2* ( 3 . 1 0 ) Zrrsi, ¥-7T ( 3 . H ) where 1 i s r e a l and A i s i n the London gauge. 64 In some multi p l y - c o n n e c t e d superconductors i s ne a r l y uniform throughout, which s i m p l i f i e s the s o l u t i o n of the f i e l d equations. I f the body has a x i a l symmetry (such as a hollow c i r c u l a r c y l i n d e r or to r u s ) about the e x t e r n a l f i e l d d i r e c t i o n , then we assume ^ i s not a f u n c t i o n of 0 At l e a s t we o b t a i n a s o l u t i o n of the G-L equations i n which t h i s i s t r u e . I t i s not so easy to prove t h i s s o l u t i o n i s unique. In an i n f i n i t e l y long hollow c y l i n d e r , we assume y does not depend on z , on the b a s i s of t r a n s l a t i o n a l sym-metry. I f the w a l l t h i c k n e s s the coherence l e n g t h , Y w i l l be n e a r l y uniform i n the r a d i a l d i r e c t i o n , i n view of the f i r s t G-L equation and the boundary c o n d i t i o n on Y J * °*^ ~ ^ • Another geometry i n which *•/ i s ne a r l y uniform 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 dimension i n c r o s s - s e c t i o n i s « f . This geometry, even f o r c i r c u l a r c r o s s - s e c t i o n , cannot be solved In c l o s e d form s i n c e London's equation i s not separable i n t o r o i d a l c o - o r d i n a t e s . The long hollow 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 the complementary reasons that the f i e l d equation can be r e a d i l y solved i n closed form, and that t h i s . geometry i s amenable to experiment provided we can take (length) » (diameter) as approximating " i n f i n i t e l y l ong." Consider then an i n f i n i t e l y long hollow c i r c u l a r c y l i n d e r of inner r a d i u s A.f and outer r a d i u s Jl~, , placed 65 i n a uniform magnetic f i e l d H g p a r a l l e l to i t s a x i s . No r e s t r i c t i o n i s placed on A) , other than that i m p l i e d by y uniform, which i s assumed throughout the remainder of t h i s chapter unless s p e c i f i c a l l y s t a t e d to the c o n t r a r y . . The v e c t o r p o t e n t i a l A obeys (3.11). A may have 9 and Z components at the s u r f a c e . Because of symmetry, A^ , AQ , and A% may be f u n c t i o n s of Sl^ o n l y . Since 0 at the s u r f a c e , and V'A  = 0 3 A^~ 0 everywhere. Prom (3.11), If A z s 0 then ~0 . I n the i d e a l i z e d geometry of i n f i n i t e l e n g t h there i s mathematically nothing which prevents an a x i a l flow of c u r r e n t . In a r e a l f i n i t e c y l i n d e r there 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 to flow w i t h e x t e r n a l connections. For present purposes we simply add U% " 0 as a boundary / 3 A c o n d i t i o n . The vector p o t e n t i a l i s then of the form AJ-fti 9 ; p A A hence the f i e l d i s Hz^) Z and the current i s ^ L ) $ This being understood f o r c y l i n d e r s p a r a l l e l to the f i e l d , we s h a l l drop the s u b s c r i p t s and t r e a t A , H , ^  as s c a l a r s . Equation .(3.11) becomes the s i n g l e t o t a l d i f f e r -e n t i a l equation: JL(1 - 4 M ) = - £ / / ! -which has the general s o l u t i o n 66 where CL , are cons t a n t s , and I , K are modified Bessel's f u n c t i o n s . Hence and = 4 r + ( & ) } Since ^ i s uniform i n & and Z , and flows only i n the & d i r e c t i o n , the f i e l d H^ i n the c y l i n d r i c a l hole i s uniform and a x i a l . Since the i n f i n i t e c y l i n d e r has zero demagnetization, the f i e l d at the e x t e r n a l surface i s uniform and a x i a l , and eaual to the 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 solve f o r Q, , Jr In terms of E^ and H-^  . V/e point out, however, that as boundary c o n d i t i o n s , H 2 and are p h y s i c a l l y q u i t e d i f f e r e n t : R^ on the 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 the system, whereas H^ depends on the c i r c u l a t i n g current i n the 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 of the body. Determination of i s thus part of the problem to be solved. However, there i s some uniform H^ , and we may w r i t e CL , <ir i n terms of i t and Hg • We might choose some other q u a n t i t y as the second boundary c o n d i t i o n , such as or . We choose simply because i t lends a c e r t a i n symmetry to the equations. 67 We define the q u a n t i t i e s > ^J-f- > which are combinations of Bessel's f u n c t i o n s evaluated at the inner and outer s u r f a c e s . These q u a n t i t i e s which occur 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 hollow c y l i n d e r w i l l be r e f e r r e d to as the "Boundary F a c t o r s . " ^ and ^ take on the values 1 or 2 r e f e r r i n g to A., and Ax r e s p e c t i v e l y ! (3.16) E v i d e n t l y r 0 From the boundary c o n d i t i o n s on H , we have 6 8 S i n c e . i s u n i f o r m i n the h o l e , A i n the h o l e i s g i v e n by A^) = SlMi/Z so t h a t A, = = A / t y / Z . We c o u l d j u s t as w e l l put f o r i n t h e boundary c o n d i t i o n above. E v a l u a t i n g Eqn. (3-13) f o r A a t Slj u s i n g the above v a l u e s o f a,b g i v e s S e t t i n g t h i s e q u a l t o J l i H i / Z , we can s o l v e f o r : (3.17) i n e q u i l i b r i u m , t h i s g i v e s r i . ' i n terms c f the f i x e d q u a n t i t i e s H g and n<j/>0 . I t may appear t h a t a f r a u d i s b e i n g p e r p e t r a t e d h e r e . The a u t h o r had t h i s f e e l i n g f o r some t i m e . H-^  was t a k e n as a boundary c o n d i t i o n and now i t has been s o l v e d f o r . One knows t h a t a p h y s i c a l s i t u a t i o n d e s c r i b e d by a second o r d e r d i f f e r e n t i a l e q u a t i o n r e q u i r e s two boundary c o n d i t i o n s . I f depends on H e , what i s the second boundary c o n d i t i o n ? I t i s not t h e f l u x o i d quantum number wh i c h e n t e r s e x p l i c i t l y i n t o the d i f f e r e n t i a l e q u a t i o n i t s e l f . L e t us see how t h e boundary c o n d i t i o n s a r e a p p l i e d p h y s i c a l l y . The f i r s t c o n d i t i o n , t h e e x t e r n a l f i e l d H e , may be a p p l i e d by w i n d i n g an i d e a l s o l e n o i d around the 69 e x t e r i o r of the c y l i n d e r and passing through the winding some c u r r e n t , say I , which i s c a l c u l a t e d to produce the d e s i r e d f i e l d H ' . Now the u n i f o r m i t y of throughout the e n t i r e hole and the consequent r e l a t i o n A (A) = H,A,/Z depend upon the absence of any e x t e r n a l c u r r e n t s i n the hol e . 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 currents from an e x t e r n a l source_ i n t o the hol e . Under t h i s c o n d i t i o n i s a d e f i n i t e f u n c t i o n of H g . Equation ( 3 - 1 7 ) c o n t a i n s , i n a d d i t i o n to , one other q u a n t i t y , namely n , which i s not f i x e d e x t e r n a l l y . and ^ 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 energy, not independently, but subject to equation ( 3 - 1 7 ) • The f l u c t u a t i o n s of ^ a n d H ^ about the e q u i l -i b r i u m values are not independent. Since the f l u x o i d i s constant dur i n g changes of temperature and e x t e r n a l f i e l d , i t i s convenient to express the f i e l d q u a n t i t i e s i n terms of Hp and n (f>o , r a t h e r than Hp and . We note i n p a r t i c u l a r the f o l l o w i n g q u a n t i t i e s at the inner and outer s u r f a c e s : 7 0 TfJlj \ Z A V / : >2 f #2 A /, K, A \ _ frriJ-7^ 2 i/J 2rrsii.Jl The t o t a l f l u x e n c i r c l e d by a contour at the inner and outer surfaces can be found from The magnetic moment fo[ per u n i t length of 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 ^ (Sl) given by ( 3 . 1 5 ) and giv e s .1 ^ =-A_ \HZ (A, - l ) i-H, U i -/) Hz A ( 3 . 1 8 ) The t o t a l c u r r e n t I around the h o l e , per u n i t l e n g t h of c y l i n d e r i s r ^ 7 t (Hz-H,) 71 The electromagnetic f r e e energy JH may be found, by d i r e c t s u b s t i t u t i o n of the preceeding two q u a n t i t i e s i n the general formula of Eqn. (3-5) • However, i t i s p o s s i b l e to o b t a i n d i r e c t l y from the 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 the c y l i n d r i c a l symmetry and the zero demagnetiza-t i o n , and without 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 to ^ e constant at a l l , only that the second G-L equation be s a t i s f i e d , whatever ytsi} m a y be. Using equation (3-3), n o t i n g that H g = H 2 , and i n the hole H = H^ a constant, s/'JfVo/e P u t t i n g "7/ -? n<fio so that ,1 we o b t a i n 72 SrrJv, i c ' J +TT Jsjt H UV + 11 f dV + (JthZJffl f = — / (/7>(VxA') + ( V K ^ ' A ' } ^ - (VKA')dV Converting to surf a c e i n t e g r a l s , where i n c l u d e s both the inner and outer surfaces of the hollow c y l i n d e r irr is, Jr. •S,  T " JS, + 8 8 Since H and A are uniform over the s u r f a c e s , they f a c t o r out•of the i n t e g r a l s . A f t e r some a l g e b r a , one obrains the simple e x p r e s s i o n 7 _ ^ i J ! i - - ± ^ L - A ± j H> - Hz(n<jo) ( 3 . 1 9 ) This expression i s exact i r r e s p e c t i v e of the v a r i a t i o n of *y i n the Jis d i r e c t i o n but i t contains the q u a n t i t i e s Ap and , i n a d d i t i o n to the c o n s t r a i n t s Hp and n<j/>0 . To 73 express "J^ completely i n terms of. the l a t t e r two con-s t r a i n t s , we must f a l l back on the f i e l d equations which have been s o l v e d f o r f Ot) constant. In t h i s case We have made no r e s t r i c t i o n s on Sl, , Jlx other than that i m p l i e d by ^ constant. This expression i s a quadratic i n H 2 and foj4>), which i s u s u a l l y the most u s e f u l form, s i n c e these are the p h y s i c a l c o n s t r a i n t s . M a t h e m a t i c a l l y , we can re-express i n terms of various other q u a n t i t i e s , almost any p a i r of H " 2 , H , A 2 , A 1 , 9?( , I Q , <f>c w i l l do. Using the r e l a t i o n s already found, we w r i t e down two such-expres-sions f o r which do not c o n t a i n i n t e r a c t i o n terms. As a f u n c t i o n of e x t e r n a l f i e l d H 0 and t o t a l (3.21) current I . c 3 As a f u n c t i o n of f l u x o i d <$& and magnetic moment 7- _ ?»• { ^ t j - J - J ) 4- rfJn } ( 3 - 2 2 ) We draw the f o l l o w i n g c o n c l u s i o n from equation (3-21). I f the c y l i n d e r i n i t i a l l y i n the -normal s t a t e , i n e x t e r n a l f i e l d H g 3 i s cooled i n t o the superconducting s t a t e , then 3^  w i l l be a minimum i f I = 0 , that i s i f H ' = H . We c 1 e 74 expect the c y l i n d e r to go i n t o the f l u x o i d s t a t e f o r which as n e a r l y as p o s s i b l e equals H g w i t h i n the l i m i t a t i o n s of the d i s c r e t e f l u x o i d s . Ginzburg ( 1 9 6 2 ) has given t h i s r e s u l t f o r a t h i c k c y l i n d e r but i t i s here shown to be true f o r a l l hollow c y l i n d e r s . The complete f r e e energy i n c l u d e s the condensation energy as w e l l as the electromagnetic f r e e energy , 7 - 3 * 1H From equation ( 1 . 2 0 ) w i t h V*£~0 we have -  H<±A (-Z7 Z + X+) (3.23) —7 _ 2 Chapter 2 , equation ( 2 . 1 3 ) . ^ 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 , " that Is r e l a t i v e to the d e n s i t y i n zero f i e l d at the same temperature. The electromagnetic f r e e energy ~Jjj i s an even f u n c t i o n of 7^ . Consequently expansions of w i l l i n v o l v e only even powers of *7 s i . e . i n t e g r a l powers of '5^ - . The "Reduced Free Energy" F (block l e t t e r ) i s de f i n e d as i n equation ( 2 . 1 7 ) as Hd? V, f ? H e ? V, ^ 75 I t should be remembered f o r f u t u r e r eference that the reduc-t i o n f a c t o r from 7 t o F i s temperature dependent. As we use the 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 the p r a c t i c e of appending the r e l e v a n t w e l l d e f i n e d symbol to the words "Free Energy" i n the t e x t . The complete expression f o r F" f o r a hollow c y l i n d e r of w a l l t h i c k n e s s vi 1th no other r e s t r i c t i o n s on Jlf , J^x. is< + t k J&L + -y?,^? ] ( 3 . 2 4 ) H&> neb  J r L " j-jcl /(-, // ' A ' J The e q u i l i b r i u m s t a t e s of the system are those f o r which i s a minimum, w.r.t. , that i s when ^ has the p a r t i c u l a r value which s a t i s f i e s IE _ 0 , > o  <3- 25) For a general hollow c y l i n d e r , equations (3.24) and (3.25) taken together are very complicated, s i n c e ^ occurs through-out the boundary f a c t o r s . In the next chapter we w i l l consider c y l i n d e r s f o r which A 3 i n which case (3.24) can be w e l l approximated by a much simpler expression. 76 F i n a l l y i n t h i s chapter we o b t a i n an expansion of P i n powers of ^  without 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 to expand. t~ i n powers of by expanding a l l the bounadry f a c t o r s and simply working through the alge b r a . Expansions of the boundary f a c t o r s i n powers of are given i n Appendix 2. There i s however a much simpler and more d i r e c t . method which does not even r e q u i r e the s o l u t i o n of the f i e l d equations. The procedure i s analogous to that d e s c r i b e d f o r simply connected bodies i n Chapter 2, page 42 . We dis c u s s f i r s t the procedure i n g e n e r a l , then apply i t to hollow c i r c u l a r c y l i n d e r s . Take ~J"H i n the form given i n e q u a t i o n (5«9) 3 ( 3 . 2 6 ) The i n t e g r a t i o n i s over the body o n l y , not the hole. The vect o r p o t e n t i a l of the e x t e r n a l f i e l d A must be i n the ^ e London gauge of the body being considered, i . e . have no component normal to i t s s u r f a c e . In the integrand of J I f ^  Is obtained as an expansion 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 o b t a i n the c o e f f i c i e n t s (^) i n the expansion ( 3 . 2 6 ) the expression i n brackets i s independent of 'v'* . = fJn) *tfa(JL) f 7 7 Consider the second G-L equation i n the f o l l o w i n g form obtained d i r e c t l y from (3-11) I f , then , so t h a t ^ ^ O = 0 . A , A g are both i n the London gauge, then as ^ —*> 0 , A —> A , since /s e A i s then due e n t i r e l y to the e x t e r n a l c u r r e n t s . The ex-pansion of A i n powers of ^ contains the constant ( w . r . t . ^ - ) term /\ e , i . e . Suhsti tut;] n<? these expansion i n (3.27) and eauatins c o e f f i c -i e n t s of g i v e s M = _ C / A rc r0) n e W i t h i n the superconductor, 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 A , and equating c o e f f i c i e n t s of , 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 to the s e r i e s f o r ^  and A . For any body having 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 H i s given i n c y l i n d r i c a l co-ordinates i n the London gauge by Then 7 / ; We now r e s t r i c t the d i s c u s s i o n to the " i n f i n i t e l y " lone: hollow c i r c u l a r c y l i n d e r and make use of the B and Z symmetry to c a l c u l a t e the next higher term i n the s e r i e s . Keeping i n mind that H i s e n t i r e l y i n the Z d i r e c t i o n , that A , ^  are e n t i r e l y i n the $ d i r e c t i o n , and that a l l are f u n c t i o n s of Jl- o n l y , we have (V X hi) = ^-o) $ S o l v i n g f o r H^t) w i t h boundary c o n d i t i o n H(l) ~ 0 at JL - Jl-A I f//e U Z - J l 1 - ) $C * JI± 79 Then V * 4 = £ JjLfafa) = Hu>M 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^^°) ~ 0 and /} { ) ) continuous over inner surface A 1 [ * 1 2 t  1 2. Then equating c o e f f i c i e n t s of 'J- i n (3.27) gives 7(z) Using these two terms of the expansion of ^  , we may sub-s t i t u t e i n t o the integrand of (3.26) and o b t a i n "3^ c o r r e c t 5 which i s s u f f i c i e n t to determine the second order c r i t i c a l f i e l d and the c o n d i t i o n f o r second order phase t r a n s i t i o n s to occur. The i n t e g r a t i o n i s elementary and gives " I r Sir x •* " rrz zii j §2J? [ Tl { h 1 ' ' 1 8tr K ' ' 1 " r - x ~ iyt'^l TT^ \ Zt, Z % } f (3.28) Anyone who has obtained t h i s r e s u l t a f t e r lon^ tedious expansion of Modif i e d Bessel's f u n c t i o n s w i l l 8 0 appreciate the s i m p l i c i t y of t h i s method which appears not to have been used before. In a d d i t i o n , t h i s method can be a p p l i e d t o geometries f o r which a s o l u t i o n of the f i e l d equations cannot be obtained. i CHAPTER 4 THIN CYLINDERS In t h i s chapter we consider I n f i n i t e l y long t h i n -w a l led hollow c y l i n d e r s . "Thin-walled" (or more simply " t h i n " ) means and ctA/) . In p r a c t i c e i t would be d i f f i c u l t t o s a t i s f y the f i r s t c o n d i t i o n without s a t i s f y i n g the second. I t i s a l s o understood that so that *f 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. \ - - .- . _..«.. -i ... ~i ,r>_. .. .1 i . r _ C U C l ' g J I t v j u a u j . u a \ . c_ -r / <^  ci.11 k> ^ - pi,x cx u -x j u x i i i | j x x i J.^ .^  . .» ^ s h a l l study the behaviour of h and some other p r o p e r t i e s of the c y l i n d e r as the e x t e r n a l f i e l d and temperature are v a r i e d . In Part I of the chapter we take temperature T constant and concentrate on the dependence of the v a r i o u s q u a n t i t i e s on the geometry of the c y l i n d e r , t h a t i s on /lf and d. In Part 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 both T and H vary, e ° Part 1 - Geometry The f i r s t step i n app l y i n g equation (3.24) f o r F* to 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 8 1 82 the boundary f a c t o r s , hjj, and J^^ . These are obtained by p u t t i n g J l % - A , i n a l l B e s s e l f u n c t i o n s of argument ft/\i"'S) ' m a ^ n ^ a ^ay^-or expansion about ^ ' These expansions are obtained i n powers of (cZ/\) . The r e s u l t s f o r the boundary f a c t o r s and s e v e r a l combinations of them are g i v e n i n Appendix 3. This 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 combinations of Bessel 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 than using the asymptotic expansions 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 since a r a t h e r l a r g e number of terms of the l a t t e r must be r e t a i n e d to get suf-f i c i e n t accuracy. These expansions are 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 regrouped to form v • an expansion In powers of "V" , and the c o e f f i c i e n t s of powers o f ^ are only approximations of the exact c o e f f i c i e n t given i n Appendix 2 . We proceed to o b t a i n lowest order In a p p r o x i -mations f o r v a r i o u s q u a n t i t i e s In the c y l i n d e r f o r which exact expressions were obtained i n the previous chapter. I t Is convenient to use the q u a n t i t y "reduced, f l u x o i d " <£ defined by The "reduced f l u x o i d " has the dimension of magnetic f i e l d 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 hole. The 83 dimensionless "geometry parameter" ^u. : AC == £lA ( 4 . 2 ) r A1 occurs f r e q u e n t l y throughout the a n a l y s i s . The f i e l d i n the hole i s The d i f f e r e n c e between inner and outer f i e l d i s The i n n e r s u r f a c e and outer surface v e c t o r p o t e n t i a l s (which d i f f e r only by terms of higher order) are The super c u r r e n t d e n s i t y ^ , uniform i n the w a l l to w i t h i n terms of second order, i s (4.5) Since -4< i s n e a r l v uniform i n the w a l l the t o t a l current / around the hole per u n i t length i s j u s t Xc ^ • Using t h i s value one could c a l c u l a t e i n an elementary way the q u a n t i t y 11^ - given above. Let us now w r i t e down the reduced f r e e energy f~~ , c o r r e c t to the lowest order term In each expansion, by d i r e c t 84 s u b s t i t u t i o n , from Appendix 3 i n t o equation ( 3 . 2 4 ) . This gives In t h i s equation Hp and appear only through t h e i r d i f -ference ( M p . ~ $ ^ - This i s only t r u e f o r the lowest order approximation of . More accurate c a l c u l a t i o n shows that F*H i s not independent of quantum number ' n f o r l a r g e n . This w i l l be discussed i n the next chapter. We can a r r i v e at ( 4 . 6 ) i n a d i f f e r e n t manner which gives some p h y s i c a l i n s i g h t . F~ i s the i n t e g r a l of f i e l d enerey d e n s i t y olus k i n e t i c enersy d e n s i t y of the current as given i n equation ( 2 . 4 ) . In the t h i n c y l i n d e r the volume of the w a l l i s l e s s than that of the hole by a f a c t o r of order , hence we neglect the f i e l d energy i n the w a l l . Using the above approximate expressions f o r (Hp - H^) and . s u b s t i t u t i n g i n the i n t e g r a l i n eauation ( 2 . 4 ) , and performing the i n t e g r a t i o n which i s t r i v i a l because of con-stant integrands we a r r i v e p r e c i s e l y at ( 4 . 6 ) . ' Of course 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 approximation at 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 agree. The l a t t e r method however provides an approach which may be used i n l e s s i d e a l geometries than the hollow c y l i n d e r . 8 5 To s i m p l i f y the algebra and to 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 A ZJzX Hct, Since we are c o n s i d e r i n g constant T , yU> i s a constant. The behaviour of.the f r e e energy and other q u a n t i t i e s i n the c y l i n d e r depends s t r o n g l y on the geome t r i c a l parameter y * ^ " equation.(4.2). F becomes 2. ( 1 . 8 ) where (~2\t'i2) i s the reduced condensation energy F and c the l a s t term i s the reduced electromagnetic energy 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^ which i s o f t e n hidden i n a more complicated expression. For A very l a r g e , so thatyU<&l 3 F^ i s ne a r l y l i n e a r i n , I t s slope i s pro-p o r t i o n a l to X^ ( l i k e the curves i n F i g . 2-3a) . For A very small,yU.» / , ^^"^ r i s e s r a p i d l y to a p l a t e a u and s a t u r a t e s , the height of the pla t e a u being p r o p o r t i o n a l to X ( l i k e the curves i n F i g . 2-4a). T y p i c a l curves f o r F = F„ + F,T f o r these extreme cases are shown i n F i g s . 2-3b G n and 2-4b. Since ^ i n e q u i l i b r i u m takes the value which minimizes F one can f o l l o w q u a l i t a t i v e l y i n these p l o t s the progress of F and as X i n c r e a s e s . 86 In e q u i l i b r i u m , ^ i s determined by the requirement i f = 0 , that i s which determines ^ i m p l i c i t l y as a f u n c t i o n of X . This f u n c t i o n can be w r i t t e n e x p l i c i t l y / ' 3/" 3 { (yU +*)3'z J . To v i s u a l i z e the dependence of ^  on X curves of X (o r d i n a t e ) vs ^  (abscissa) are p l o t t e d i n Figu r e 4-1. The p h y s i c a l l y meaningful parts of the curve ( 0 - / , F a minimum w.r-.t. ^ ) are shown by s o l i d l i n e s . The dashed l i n e s are simply a n a l y t i c c o n t i n u a t i o n of equation (4.9). A l l s o l i d curves contain the poin t = / , X =0. F o r } } X 2 has a maximum at ^ — j - ^ ~ ^ u ) . The symbol ^ i s used i n a n t i c i p a t i o n of the d i s c u s s i o n on page 89 . The o o r t i o n of the curve i n the r e g i o n 0 ^  ^ 1^ corresponds to a maximum of F w.r.t. ^ . As yM- becomes l a r g e , ^ * 3 and A 3 i n c r e a s e s without l i m i t l i k e '^ry ( u n t i l the a p p r o x i -mation f o r thinness breaks down). For yd 6/ , a l l the s o l i d curves pass through ^~ 0 , 2 X = 1, at which p o i n t F = 0 . Consequently, i n a l l c y l i n d e r s having yU- ^ ' , a second order phase t r a n s i t i o n occurs at 2 2 X = 1 s i n c e decreases continuously from 1 to 0 as X 8 7 increases from 0 to 1 . As yOt-*0 , the curve becomes more and more n e a r l y a s t r a i g h t l i n e ; i n very t h i n c y l i n d e r s decreases l i n e a r l y w ith X^ . Let us compare t h i s c o n c l u s i o n w i t h that reached by a d i r e c t search 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 expansion of F i n powers of 3^ . The coef-f i c i e n t s of and ^ have been c a l c u l a t e d i n closed form i n Chapter 3> i n equation ( 3 . 2 8 ) . 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 the c o e f f i c i e n t of equal to zero, and a second order t r a n s i t i o n does occur i f at t h i s 2 f i e l d the c o e f f i c i e n t of ~ r i s p o s i t i v e . The c o e f f i c i e n t s of ^ and are q u i t e complicated f o r general Jt, 3 JL-^ but i f we approximate f o r thinness GC^Jij , and keep only the lowest term i n the c o e f f i c i e n t s we o b t a i n , using the present n o t a t i o n /="« - Z f + ^ X * 7 1 E v i d e n t l y tine c r i t i c a l - second order value of X which we c a l l X-^  i s and the c o n d i t i o n f o r second order phase changes to occur i s Since these r e s u l t s are i n agreement w i t h those obtained by considering: X as a f u n c t i o n of ^ we may then c o n f i d e n t l y use equation ( 4 . 8 ) f o r F r i g h t down to the l i m i t p, ~~9 0 . 88 We consider now the e q u i l i b r i u m f r e e energy F eq as a f u n c t i o n of X . Throughout the remainder of 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 values of F , ^  , and some other q u a n t i t i e s i n the c y l i n d e r and other bodies. For s i m p l i c i t y of n o t a t i o n we drop the sub-s c r i p t e ^ , i t being understood throughout. Whenever ^ has 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 . S u b s t i t u t i n g (4.9) i n (4.8) we have ' (4.11) where ^ i s the f u n c t i o n of X given i n (4.10). The d e r i v a t i v e s of F(X) are where again y i s a f u n c t i o n of X . For yi>L > / there are four s i g n i f i c a n t p o i n t s on the curve F(X) : P o i n t 0 : Minimum of F , at X 0 ~ 0 , ^ 0 - 1 > • 0 and i dx Jo Point 2 : E q u i l i b r i u m f i r s t order phase change de f i n e d by f-l — 0 which occurs at 89 P o i n t s 3: Extreme l i m i t of metastable s t a t e which occurs at the maximum value of X^ f o r which a minimum of 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 ^ l~ (Xj"^ Q At t h i s p o i n t W h e n — ? / from above, p o i n t s 2 and 3 coalesce t o the second order t r a n s i t i o n p o i n t s 1 which have already been given except to note We de s c r i b e F(X) as being of "second order type" when i t terminates h o r i z o n t a l l y at F = 0 . P o i n t 4: P o i n t of i n f l e c t i o n . A l l curves f o r a l l J A - have a p o i n t of i n f l e c t i o n at 90 F<r = ~') > ( j j t = ^ The maximum value of magnetic moment occurs at point k. In the l i m i t i n g case JA,-* 0 the curve P(X) approaches a simple l i m i t i n g form: F M — — (/ - x ^ ) z ^ — — > w i t h the po i n t of i n f l e c t i o n at On the other hand, a s b e c o m e s l a r g e , P(X) becomes n e a r l y p a r a b o l i c except f o r a very short s e c t i o n at the upper extremity where i t i n e v i t a b l y turns over. F a m i l i e s of F M and ^ M f o r d i f f e r e n t y C C are p l o t t e d i n Figures (4-2a) and (4-2b). We remark the f o l l o w i n g p r o p e r t i e s : 1) For y U I 3- has i n f i n i t e slope at the t e r m i n a l p o i n t 3. For y U , < I j p~ fx) has f i n i t e slope at the t e r m i n a l p o i n t 1. 2) A l l the p o i n t s of i n f l e c t i o n of F(X) l i e on the s t r a i g h t . i n e 1 J 3 91 3) The extreme metastable p o i n t s l i e on a curve asymptotic to t h i s l i n e f o r l a r g e yU* . 4) For large/CC , Xj^/Ct and °&/U- , whereas - —/ always, and ^ lyu- , Hence, f o r largejuu by f a r the gre a t e s t part of the curve l i e s i n the metastable r e g i o n . 5) The dashed curve (shown only 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 of equation (4.8) f o r values of ^ <p-3 .' This curve corresponds to maxima of F w . r . t . ^ and not to 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 curves i s the p o t e n t i a l b a r r i e r to the normal s t a t e . 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 the qu a n t i t y which i s measured. Using equation (3.18) and the expansion i n powers of (d/Aj to lowest order one o b t a i n s : Define the "reduced magnetic moment" M _ ^**- = / c£F \ Noting that V\ ( 72\ / o n e h a s i m m e d l a t e l y UM _ z (ty-*) dX (3^.-2 -tZ/yU.) The values of M and dM/dX at the four p o i n t s discussed above are, f o r } I : P o i n t 0 : M0 = Q } 1&\ = [otK/O 2-) Point 2 : Mo = ; I***] = P o i n t 3 — co 3 Point 4: This i s the maximum value of magnetic moment • &),-<> As from above, ~ * M, ~ * 0 and / s ^ j "~* °° F o r y U I , p o i n t s 2 .and 3 are meaningless, the expressions at p o i n t s 0 and 4 remain v a l i d , while at the second order c r i t i c a l p o i n t 1 : " l . (,/x/y Curves of M(X) are p l o t t e d i n Figure ( 4 - 2 c ) To the accuracy we are c o n s i d e r i n g the current d e n s i t y ^ 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 other q u a n t i t i e s are p r o p o r t i o n a l to M and have s i m i l a r behaviour as f u n c t i o n s of X . The pro-p o r t i o n a l i t y between ^ , loop current I q , f i e l d d i f -ference (H^ - H-^ ) , and magnetic moment may be sum-marized as f o l l o w s : 9 3 Vs ?c ° C Zc Q> Z  Froim the c l a s s i c a l p o i n t of view we c a l c u l a t e the average momentum of the super e l e c t r o n s 7^ -~ M A T = where n i s the number d e n s i t y of super electrons, equal s to. ffgQ ." Using equations ( 1 . 2 ) f o r n ^  ,' ( 4 . 5 ) for' ^ , and tine d e f i n i t i o n of X ( 4 . 7 ) , we o b t a i n f o r i n terms of the. f i e l d v a r i a b l e X and parameteryCt* : = A  x if /9 . . . . where ~f i s the coherence l e n g t h . D e f i n i n g the "Reduced Momentum" P == 'f'f'f k and using equation ( 4 . 9 ) we have P - f ^ J -Second order t r a n s i t i o n s occur at ^ — 0 at which P has the value 1, i . e . -when the de B r o g l i e wavelength of the e l e c t r o n s equals / . T r i . F o r / U . ' i , the t e r m i n a l momentum P- i s somewhat smaller but never l e s s than Curves of P(X) are shown i n Figure 4 - 2 d . I n Figures 4 - 2 a , b , c, d we have taken as a b s c i s s a a v a r i a b l e p r o p o r t i o n a l to (Hp - $ ) • A l t e r n a t i v e l y , we may wish to take the 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 curves f o r each fluxoid.number s e p a r a t e l y . For sm a l l <j5 , the curves f o r a l l n are i d e n t i c a l but t r a n s l a t e d to the 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 to n . The meaning o f " s m a l l " i n t h i s context w i l l be made p r e c i s e i n Chapter 5 . I t i s of i n t e r e s t to have an i d e a of the sep a r a t i o n of neighbouring curves, say A X i n u n i t s of X , which i s the n a t u r a l f i e l d u n i t of the system. A X i s given by a /FA l rr^j Hct> For a t i n c y l i n d e r of diameter 1 cm. ( s i m i l a r to ones used by Hunt and Mercereau (1965) at t = 0.9 , one f i n d s - y-£i A r J C 5 'O . The successive curves aro extremely c l o s e l y spaced, q u i t e u n r e s olvable i n a p l o t such as Figure 4-2. AX i n c r e a s e s w i t h temperature but at t = 0.999 -3 the s e p a r a t i o n i s s t i l l s m a l l ; A X ^ JO . The curves given by Douglass (1963) give a f a l s e impression of the st a t e of 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 successive curves separated by amounts of order AX / . Part 2 - Temperature and F i e l d In t h i s part we concentrate on a p a r t i c u l a r c y l i n d e r c h a r a c t e r i z e d by i t s yU0 , and study the dependence of fr e e energy, order parameter, magnetic moment, and super-e l e c t r o n momentum on both temperature and f i e l d . In the 95 • ( 4 . 1 3 ) previous part the d e f i n i t i o n s of F and X are themselves temperature dependent. This procedure i s not s u i t a b l e here. We d e f i n e : * ~ 2/1K //„, ( / + t % l  A 3 HjV, ( and r e c a l l t h a t JUL' i s temperature dependent: • , _ .v / / _ -r ¥) where /y. =. Jl, c£/ A n S~ " * ' " ' " f i s d e f i n e d so as to be normalized at t = 0 , that i s at t = 0 j X = 0 a f = - l . f has the same r e l a t i v e tempera-ture dependence and as the dimensional f r e e energy • Using the d e f i n i t i o n s ( 4 . 1 3 ) and the expression f o r reduced f r e e energy F , equation ( 4 . 8 ) we ob t a i n f o r f ( t 3 x , % ) In e q u i l i b r i u m ^ i s again determined by the con-d i t i o n s 96 The f i r s t c o n d i t i o n y i e l d s the same equation as ( 4 . 9 ) . With the temperafciure dependence w r i t t e n I n , t h i s equation becomes I-t-t ( 4 . 1 5 ) This gives i m p l i c i t l y as a f u n c t i o n of x and . t. .. • S u b s t i t u t i n g ( 4 . 1 5 ) i n ( 4 . 1 4 ) gives where ^* i s not independent but i s given by ( 4 . 1 5 ) . We have shown i n the previous part that the expansion or r i n powers ot(d^h) (which leads to ( 4.14).to lowest order) i s v a l i d f o r a l l ^  , i n c l u d i n g ^ —> 0 . We may then • use (4.14) t o study the order of 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 . 1 6 ) and ( 4 . 1 5 ) together give f and- ^  r e s p e c t i v e l y as f u n c t i o n s of the independent v a r i a b l e s x and t , that i s , as sur-faces i n three dimensional co-ordinate systems. The cor-responding magnetic moment m and super e l e c t r o n momentum p are a l s o f u n c t i o n s of x and t . To i l l u s t r a t e the form of these surfaces we s h a l l draw two sets of curves: (a) curves of f ( x ) v/ith t constant, each curve l a b e l l e d 97 w i t h the value of the parameter t , and (b) curves of f ( t ) w i t h x constant, each curve l a b e l l e d by the value of x . Corresponding curves of ^ - ( x ) , m(x), p(x) at constant t and ^-(t) , m(t) and p ( t ) at constant x are a l s o drawn. The curves (a) f o r f ( x ) may be i n t e r p r e t e d as the p r o j e c t i o n s on the f - x plane of the l i n e s formed by the i n t e r s e c t i o n of planes of constant t w i t h the sur-face f ( x , t ) . S i m i l a r remarks apply to the curves ^ ( x ) , m(x) and p(x) , and a l s o t o a l l the curves' (b) which are p r o j e c t i o n s on the f - t plane, e t c . Curves (a) are r e a d i l y obtained by an adaptation of the curves of part 1. The shape of f ( x ) f o r f i x e d t depends only on i a d e f i n i t e Quantity f o r given t . f (x) i s obtained from P(X) w i t h corresponding /S- by 2 2 compressing the ordinate P by a f a c t o r of (1 - t ) and the a b s c i s s a X by a f a c t o r / I ' . At. t = 0 , f ( x ) i s i d e n t i c a l to the curve P(X) having y^C ~y^o . As t i n c r e a s e s , the curves f ( x ) squeeze i n toward the o r i g i n , more r a p i d l y i n ordinate than a b s c i s s a , and at the same time become more "second-order-liek." At t £ t where t i s given by (1.17) y.(i*) » y j l - i**) - I the curves become of the second order type . S i m i l a r l y curves 98 of m(x) are obtained from M(X) by s h r i n k i n g the l a t t e r toward the oi°igin unif o r m l y on both axes by a f a c t o r For purposes of i l l u s t r a t i n g these f u n c t i o n s we choose a c y l i n d e r having — 10 f o r two reasons: f i r s t , 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 convenient range of temperatures, and second, from the data g i v e n , i t seems to. correspond .to the. c y l i n d e r s used by Hunt and Mercereau ( 1 9 6 4 ) . Curves of . f ( x ) , m(x) and p(x) f o r s e v e r a l values of t (constant) are p l o t t e d i n F i g u r e s 4 - 3 a , b , c , d . For t£ t = . 9 7 4 the curves are second order type. The curves of ±Xx) , p.(jc) s rn(x) and p(x) f o r /M> 1 are c h a r a c t e r i z e d by the 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 meanings. For the meaningful p o i n t s are the f minimum p o i n t 0 , the second order t r a n s i t i o n p o i n t 1 and the f ( x ) i n f l e c -t i o n point. 4. For a p a r t i c u l a r value of t , hence , the c o n d i t i o n d e f i n i n g each poi n t determines unique values of x and the corresponding f , m , ^  and p . I f t v a r i e s from 0 to 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 of f i n i t e l e n g t h on the appropriate p l o t . For example i n the f - x plane, (of which the h a l f plane X%0 i s shown i n F i g u r e s 4 - 3 and 4 - 4 ) f ^ i s the s e c t i o n of the f 99 a x i s from - 1 to 0 , while f g i s the s e c t i o n of the x a x i s from 0 t o 2 . 5 3 . The l o c i of the p o i n t s f ^ and f ^ are shown. which l i e s j u s t below f ^ i s omitted f o r c l a r i t y . A l l of the area bounded by f ( x ) at t = 0 and the l i n e s f ^ and f-^ represents p o s s i b l e superconducting s t a t e s of the system. I n 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, the locus of each s i g n i f i c a n t points. , f ^ . e t c . i s a l i n e i n space. The l i n e s shown on the f - x plane are p r o j e c t i o n s of the space l i n e s , onto t h i s plane. Lines f 0 , f g , f ^ l i e on the surface f ( x , t ) , w h i l e f ^ , f ^ c o n s t i t u t e the upper edge of t h i s s u r f a c e . The l i n e f ^ j o i n s smoothly onto f-^ . The po i n t s e p a r a t i n g these two l i n e s has t co-ordinate t given by ( 4 , 1 7 ) and correspond-i n g f i e l d c o-ordinate x = x^(t- ) which i s given by ( 4 . J . ' 3 ) w i t h X = = 1 . S o l v i n g t h i s e x p l i c i t l y ~~ ( 4 . 1 8 ) In g e n e r a l , I f the s t a t e of the system.is changed by slowly v a r y i n g both x and t , the q u a n t i t i e s f , m and p w i l t r a c e out paths on the r e s p e c t i v e surfaces f ( x , t ) e t c . The p r o j e c t i o n of these paths on the f - x plane may be shown on the p l o t s of Figures 4 - ^ 3 . In par-t i c u l a r , i f the temperature t v a r i e s w h i l e the reduced 100 r e l a t i v e f i e l d x remains constant the s t a t e of the system progresses along a v e r t i c a l l i n e i n Figures 4-3- Examples of such paths are shown by the v e r t i c a l l i n e s on which arrows i n d i c a t e the d i r e c t i o n of r i s i n g temperature. I f the temperature i s increased s u f f i c i e n t l y , the p o i n t reaches the upper edge of the s t a b l e or metastable r e g i o n and a phase change occurs. Paths t e r m i n a t i n g to the l e f t , of the v e r t i c a l l i n e marked x. lead to second order, t r a n s i -t i o n s w h i l e paths to the r i g h t of x; lead to f i r s t order t r a n s i t i o n s . The c o n d i t i o n f o r second order t r a n s i t i o n s at constant f i e l d and i n c r e a s i n g temperature i s x £ x , or w r i t t e n out e x p l i c i t l y / • - , X / / £ X < ( Zyt/o - I - J ( y ^ o - l ) ) The c o n d i t i o n yU- ^ / cannot be a p p l i e d d i r e c t l y to t h i s case s i n c e the c o n d i t i o n i s temperature 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 3 t ) e t c . i s obtained by p r o j e c t i n g on the f - t plane the l i n e s formed by the i n t e r s e c t i o n of planes of constant x w i t h the f ( x , t ) s u r f a c e . This gives sets of curves f ( t ) , ^ ( t ) > m(t) and p ( t ) j each curve being l a b e l l e d by the parameter 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 Figures 4-3. They are simply a remapping of those planes 101 onto planes 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 the curves of f ( t ) e t c . i s r a t h e r more tedious; ifchan t r a c i n g the curves of f ( x ) e t c . s i n c e i n the lattear case one e a s i l y obtains p a i r s of values of f and x uising ^ as a parameter, whereas i n the former case there i s no way to avoid using the e x p l i c i t expression f o r ^ i n eqiuation (4.10) p u t t i n g i n the temperature de-pendence of' X and yU- . . This gives, the .curves .shown i n Figures 4-4. 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 of Figures 4-4 as a remapping of Figures 4-3 the t h i c k l i n e s again cor-respond to constant t and the t h i n l i n e s with arrow's to constant x: ,. The " s i g n i f i c a n t p o i n t s " whose l o c i are space li n e s ; .are a l s o shown i n p r o j e c t i o n on the f - t et c , planes.. Tiihe curves f ( t ) c o n t a i n no s u r p r i s e s . A l l pos-s i b l e super-conducting s t a t e s l i e i n the r e g i o n bounded above by the l i n e s f ^ and f ^ and below by the l i m i t i n g curve 2 2 x = 0 , f o r which f = (1-t ) . The curve f o r I n c r e a s i n g t at constant x = x which separates the regions of f i r s t and second order t r a n s i t i o n s l i e s j u s t above the lower l i m i t i n g curve. In the curves of ^ ^ J w e n o t e that as x — 0 the curve hugs the a x i s ^ - } > and drops a b r u p t l y t o 0 at t = 1 . This i s the l i m i t i n g case of a phase t r a n s i t i o n i n zero f i e l d . I t should be remembered i 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 c o n c e n t r a t i o n ^ ^ o o decreases continuously as t in c r e a s e s to 1 . The curves of m(t) are perhaps a l i t t l e s u r p r i s -i n g at f i r s t . For x > x , as t i n c r e a s e s , the curves cross over the l i m i t i n g metastable s t a t e l i n e in^(t) , then drop down and terminate on m^  with i n f i n i t e slope thus forming a s e r i e s of bumps on the r i g h t of m .(£.).. No. s i g -n i f i c a n c e i s attached to the cross-over p o i n t . The curve * m(t) f o r x = x terminates v e r t i c a l l y at m = 0 , whi l e f o r x < x , m(t) terminates at m = 0 wi t h f i n i t e nega-t i v e s l o p e . As x — » 0 , the t e r m i n a l value of t —* 1 and t e r m i n a l slope — * 0. Since m^(t) i s the locus of the a maxima of M(x) i n the m - x plane m,,(t) must l i e to -r the r i g h t of m^ Ct) and i s i n f a c t the envelope curve of the bumps. m^(t) terminates h o r i z o n t a l l y at t - 1 , m = 0 . The super e l e c t o n momentum p ( t ) increas e s w i t h t for- a l l x , slowly at f i r s t and more r a p i d l y as t approaches the l i m i t i n g value. p ( t ) i s v e r t i c a l at p^ , while f o r x <C x , p ( t ) has f i n i t e slope at the second order t r a n s i t i o n temperature. The l i m i t i n g curve p^ de-creases monotonically to zero w i t h decreasing x . 103 As b e f o r e , we may take as independent f i e l d v a r i -able a q u a n t i t y , say h , p r o p o r t i o n a l to Hp r a t h e r than x which i s p r o p o r t i o n a l to (Hp - <j& ) . For d e f i n i t e n e s s we put Then i n the three dimensional space f , h, t we o b t a i n a very l a r g e number of surfaces l a b e l l e d by. the. f l u x o i d quantum number n. . A l l the surfaces have the same shape f o r small n ( l a r g e n i s discussed i n the next chapter) and each i s d i s p l a c e d from i t s nearest neighbour by an amount /\h = <p0 l*f$Sl, Ao H00 along the h a x i s . On the f ( x ) at constant t p l o t s these surfaces appear as a set of c l o s e l y spaced i d e n t i c a l curves. These curves are symmetric about the set of v e r t i c a l l i n e s h = n ( 4 h ) . Since A h i s not temperature dependent the p o s i t i o n of these l i n e s does not vary w i t h temperature. As t i n c r e a s e s each curve s h r i n k s toward the corresponding p o i n t n (A h)on the h a x i s . At t given by jJ.(i)~l the curves become second order type forming a set of overlapping i n v e r t e d b e l l s , a few of which are shown by the t h i n l i n e s i n Figure 4-5. As t continues to i n c r e a s e there i s some temperature, say t-^ , at which the upper e x t r e m i t i e s of neighbouring curves j u s t touch. At t > t ^ there i s a space between neighbouring 104 curves and as t — > 1 the curves s h r i n k to a set of p o i n t s on the h a x i s . On the f ( t ) , constant h p l o t s the curves f o r d i f f e r e n t n are again c l o s e l y spaced but they are not merely displacements of t h e i r neighbours. The curves already obtained i n Figure 4-.4 may be i n t e r p r e t e d as examples of curves having the same h but d i f f e r e n t n , and t h e r e f o r e d i f f e r e n t (constant), x . Let us concentrate on one p a r t i c u l a r surface f ( h , t ) , say f o r n = 0 . Then f o r s l o w l y v a r y i n g h and t } n does not change and the s t a t e of the system i s represented by a p o i n t on t h i s s u r f a c e . The upper edge of the surface i s determined f o r yU> ± by f ^ and f o r ytf « 1 by f ^ = 0 which gives h as a f u n c t i o n of t . In Figure 4-6 i s shown the p r o j e c t i o n of the upper edge of the f ( h , t ) s u r f a c e on the h - t plane. The l i n e f ^ = 0 , fl* ^ 1 , which l i e s on the surface i s a l s o shown. I f the path 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 i n t e r -sects the f = 0 3 jA. k l l i n e then the c y l i n d e r leaves t h i s p a r t i c u l a r superconducting f l u x o i d s t a t e by way of a second order phase t r a n s i t i o n . I f the path i n t e r s e c t s the l i n e f ^ then i t leaves by way of a f i r s t order phase t r a n s i t i o n . I t cannot be determined from t h i s diagram whether the c y l i n d e r w i l l go i n t o the normal s t a t e or a 105 super s t a t e of d i f f e r e n t f l u x o i d . In s i m i l a r phase diagrams of simply-connected bodies a " s u p e r c o o l i n g " s t a t e i s p o s s i b l e whose l i m i t may be shown on the h - t p l o t as a l i n e below fg j o i n i n g smoothly onto the second order l i n e . In the case of t h i n c y l i n d e r s such a l i n e i s not meaningful. Because of the m u l t i p l i c i t y of 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 occur. One of the most s t r i k i n g aspects, pf superconduc-t i v i t y i s the existnece of p e r s i s t e n t , c u r r e n t s i n m u l t i p l y -connected bodies. As mentioned i n Chapter 3, a r i n g i n a super s t a t e w i t h fl ^  0 w i l l e x h i b i t a p e r s i s t e n t current i n zero e x t e r n a l f i e l d . Such a s t a t e i s not that of lowest b a r r i e r . Let us consider i n more d e t a i l the c r e a t i o n of p e r s i s t e n t c u r r e n t s t a t e s . The c y l i n d e r i n i t i a l l y normal at t > 1 i s cooled to t < 1 and becomes superconducting i n a uniform e x t e r n a l f i e l d Hg , V/e ask two questions: 1) Is there any ambiguity i n n? and 2) Is the phase t r a n s i t i o n f i r s t or second order? These two questions are i n t e r - r e l a t e d . I n Figure (4-5) i s shown i n the r e p r e s e n t a t i o n f(h) the f i r s t few of the f u l l set of p o s s i b l e supercon-du c t i n g s t a t e s . As already d i s c u s s e d , as t —*» 1 from below t h i s set of curves s h r i n k s to a set of p o i n t s n (A h) on the h a x i s , that i s w i t h f = 0 . 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 of lowering the temperature. At t = 1, the "curves" f ( h ) are the set of p o i n t s j u s t men-t i o n e d , and as t drops below 1 the curves spread'out from those p o i n t s u n t i l at some temperature, say t ^ two neigh-bouring curves j u s t touch. This s i t u a t i o n i s shown by the medium weight l i n e s i n Figure 4-5. In a n t i c i p a t i o n of a r e s u l t to be obtained the curves are drawn as second-order-type and touch at : f = 0 .. At - t ^ there i s one and only one f l u x o i d number f o r which f ^ 0 . f o r given .' Hp. . Pre-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 . This i s not to say the t r a n s i t i o n takes place at t 1 , i t takes place at some t between t ^ and 1 depending on the exact value of Hp . I f Hp i s such that h l i e s e x a c t l y on one of the p o i n t s n ( d h ) , then x d e f i n e d i n equation (4.1-3) equals zero and the t r a n s i t i o n occurs at t = 1. I f Hp l i e s near the midpoint between two of. the p o i n t s on the a x i s then the t r a n s i t i o n takes place at the maximum p o s s i b l e value of /x/ , which i s lx.1 = <j>0 / z T T J i f . This i s h a l f a "reduced" quantum of f l u x o i d . I f t h i s value of \7t\ s a t i s f i e s the c r i t e r i o n of equation (4.17) then the t r a n s i t i o n w i l l be second order. For the example c y l i n d e r described on page 9.4 we have X *» /O while from (4.17) ?C & /0 . * 5 Hence x < x by a f a c t o r of 10 . 1 0 7 We conclude that 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 i n t o a s t a t e of unambiguous f l u x o i d number. I f the t r a n s i t i o n were not second order i t might be p o s s i b l e to supercool the normal body to a temperature at which more than 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 ambiguity as to 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 i n which the. c y l i n d e r , was cooled below t = 1 i s subsequently removed the c y l i n d e r i s i n a p e r s i s t e n t current s t a t e i n which x i s p r o p o r t i o n a l to n <PQ s i n c e h = 0 . In t h i s case the curves of the q u a n t i t i e s i n Figures 4 - 4 at d i f f e r e n t constant x are 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 . Values of n f o r the example c y l i n d e r d e s c r i b e d on page 94 are shown i n brackets below each curve. The space "curves" f ^ e t c . are a set • of c l o s e l y spaced d i s c r e t e p o i n t s . In order to d i s c u s s some changes of s t a t e we reproduce i n Figures 4 - 7 and 4 - 8 the r e l e v a n t p a r t s of Figure 4 - 4 . The s o l i d curves are at constant n . Cur theory p r e d i c t s t h a t , as t i s lowered, the magnetic moment a c t u a l l y i n c r e a s e s . The corresponding experiment has been c a r r i e d out by Hunt and Mercereau ( 1 9 6 4 ) , but 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 of e x t e r n a l f i e l d and temperature. An example of 10 t h e i r r e s u l t s i s shown to the r i g h t of Figure 4 - 7 . We suppose-in Figure 4 - 7 that the system has been brought to s t a t e A by c o o l i n g to temperature t ^ In a f i e l d //-, — /rf/l* which has then been removed. I f t i s then i n c r e a s e d to t ^ the s t a t e moves r e v e r s i b l y along the curve to point A' . I f we continue to i n c r e a s e t the s t a t e p o i n t moves around the bump to the extreme meta-s t a b l e p o i n t t^// which by. . d e f i n i t i o n l i e s , on M^ . . . At . t£ the f l u x o i d s t a t e f)^ becomes unstable ybut. f o r a l l Inl < /nAl there e x i s t s t a b l e or metastable s t a t e s at t . We make the hypotheses that at A the system "drops" i n t o the neighbouring f l u x o i d s t a t e 7^—/ • From Figure 4 - 7 we see that In "dropping" from s t a t e to s t a t e I the magnetic moment a c t u a l l y i n c r e a s e s , as does the r e l a t i v e order parameter ^ . On the other hand, the f r e e energy and super e l e c t r o n momentum decrease. As t continues to r i s e the system f o l l o w s the 7^-/ curve u n t i l t h i s s t a t e becomes un s t a b l e , then drops i n t o , and so on. The system f o l l o w s a saw t o o t h path touching the l i n e at the apex of each t o o t h . The t e e t h are much too f i n e to be r e s o l v e d on the p l o t of Figure 4 - 7 . A g r e a t l y magnified s e c t i o n of the path i s shown i n Figure 4 - 9 . The system appears to move i r r e v e r s i b l y down the l i n e m^  . 109 In p r i n c i p l e , we may stop at some point B " and begin again l o w e r i n g the temprature. Then the s t a t e p o i n t moves up aro-und the bump to point B ; and out to B The path B " - B ' - B i s r e v e r s i b l e . F i n a l l y , i f t i s increas e d beyond t ^ we reach a temperature t at which JD C m = 0 , ^ = 0 and f = 0 . Note that t c ^  I , but i s defi n e d by y U . ( t c ) ~ / . At t h i s p o i n t , x has a value equal to the second order c r i t i c a l f i e l d . Comparing the t h e o r e t i c a l behaviour w i t h the experimental r e s u l t s we note that no bumps are apparent i n the l a t t e r . However, p o i n t s on the bump represent a narrow temperature i n t e r v a l and would be both d i f f i c u l t to o b t a i n and to d i s t i n g u i s h from p o i n t s along m^  . In the e x p e r i -ment no attempt was made to d i s t i n g u i s h the bump from . H. and M. show the i r r e v e r s i b l e curve t e r m i n a t i n g at t = 1 , whereas the curve m^(t) reaches zero at t = .975. I t i s not c l e a r whether i n f a c t H. and M. have not taken m = 0 as the experimental d e f i n i t i o n of t = 1. Except f o r these two d e t a i l s the agreement of theory and e x p e r i -ment i s e x c e l l e n t which confirms the hypothesis that the system drops through successive f l u x o i d s t a t e s . For completeness, we consider a second p o s s i b l e hypothesis f o r comparison w i t h experiment, namely that the e q u i l i b r i u m s t a t e e x i s t s only up to m^(t) , the po i n t at 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 should be so except to note that the e f f e c t of f l u c t u a t i o n s w i l l be to' lower somewhat the c r i t i c a l temperature. This s i t u a t i o n i s . d e p i c t e d i n Figure 4-8 i n which the i n i t i a l s t a t e A i s reached as before. The r e v e r s i b l e curve AA' j o i n s t a n g e n t i a l l y onto the m^(t) curve at F . I f t i s increa s e d beyond . t„ we assume again that the system drops, through states, pf s u c c e s s i v e l y decreasing f l u x o i d number but here each s t a t e corresponds to m(x) a maximum. O b s e r v a t i o n a l l y , the system appears to drop along m^(t). This curve terminates at m = 0 , t = 1 3 w i t h behaviour (i — t ) i n the l i m i t t —*» 1. This i s i n agreement with a remark of Mercereau and Hunt, not obviously a p p l i c a b l e to the attached r e s u l t s . This hypothesis a l s o gives a reasonable f i t to the observed p o i n t s . I t may w e l l be that the c o r r e c t curves l i e somewhere between Figure 4.7 and Figure 4.8. F i g u r e 4-2 T h i n c y l i n d e r f u n c t i o n s f o r v a r i a b l e f i e l d , l a ) ( a b o v e ) F r e e E n e r g y F ( X ) ( b ) ( b e l o w ) S u p e r E l e c t r o n D e n s i t y J r ( X ) F i g u r e 4~2 T h i n c y l i n d e r f u n c t i o n s f o r v a r i a b l e f i e l d , geometry p a r a m e t e r . "(c) (above) M a g n e t i c Moment M(X) (a) (below; s u p e r c-iecxron Mo men cum P\AJ -1.0 F i g u r e 4-3 T h i n c y l i n d e r f u n c t i o n s 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 p a r a m e t e r . V, a ; \a.uuve J r i c e J L J I K S I . g,,y J. \ - A . / ( b ) ( b e l o w ) S u o e r E l e c t r o n D e n s i t y ^ ( x ) .5" deduced Field X 1.0 2.0 F i g u r e 4-3 T h i n c y l i n d e r f u n c t i o n s 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 p a r a m e t e r , ( c ) ( a b o v e ) 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 ) o -V- .b .8 l.o F i g u r e h-h T h i n c y l i n d e r f u n c t i o n s f o r v a r i a b l e t e n m e r a t i i r e . f i e l d r a m e t e r -(a) ( a b o v e ) F r e e E n e r g y f ( t ) ( b ) ( b e l o w ) S u p e r E l e c t r o n D e n s i t y ^ " ( t ) O .X .t .6 .8 /-o .9- i -8 '-° Relot'iVe Temparotu.re T / Tc F i g u r e h-k- T h i n c y l i n d e r f u n c t i o n s 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 p a r a m e t e r . ( c ) ( a b o v e ) M a g n e t i c Moment m ( t ) (d) ( b e l o w ) S u p e r E l e c t r o n Momentum p ( t ) in unit's cf (ftejiryi, F i g u r e 4 - 5 ( a b o v e ) F r e e E n e r g y o f n e i g h b o u r i n g f l u x o i d s t a t e s a t o n s e t o f s u p e r c o n d u c t i v i t y F i g u r e 4-6 ( r i g h t ) S e p a r a t i o n o f h - t p l a n e i n t o r e g i o n s o f n o r m a l , s t a b l e s u p e r c o n d u c t i n g , 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 s t a t e s o 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 c u r r e n t 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 m e t a s t a b l e 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 c u r r e n t 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,. F i g u r e 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 n e a r m e t a s t a b l e l i m i t . 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 s t a t e s . I l l CHAPTER 5 THIN CYLINDERS CONTINUED P a r t 1 - H i g h F l u x o i d Number I n t h e p r e v i o u s c h a p t e r t h e f r e e e n e r g y and o t h e r d e p e n d e n t q u a n t i t i e s i n a t h i n c y l i n d e r i n a g i v e n r e d u c e d f l u x o i d s t a t e £ h a v e b e e n f o u n d t o d e p e n d o n t h e e x t e r n a l f i e l d t h r o u g h t h e d i f f e r e n c e o n l y . T h a t i s t o s a y , w i t h i n t h e a c c u r a c y o f t h e p r e c e d i n g c a l c u l a t i o n s , t h e r e e x i s t s a n i n f i n i t e s e t o f i d e n t i c a l c u r v e s FCHg) e a c h s e p a r -a t e d f r o m i t s n e i g h b o u r b y AHZ ~ $o/TfSl^ , E v i d e n t l y t h i s c a n n o t c o n t i n u e f o r u n l i m i t e d h i g h f l u x o i d n u m c s r , o t h e r w i s e t h e r e w o u l d be no l i m i t t o t h e e x t e r n a l f i e l d i n w h i c h s u p e r -c o n d u c t i v i t y m i g h t s u r v i v e , When i s l a r g e b u t X s m a l l ( w h i c h i m p l i e s , H 2 b o t h l a r g e b u t t h e i r d i f f e r e n c e s m a l l ) e x c l u s i o n o f t h e f i e l d f r o m t h e w a l l i t s e l f becomes i m p o r t a n t . The c y l i n d e r w a l l b e h a v e s l i k e a t h i n f i l m w i t h a h i g h f i e l d o n e i t h e r s i d e . .When H„ and H 0 b o t h r e a c h t h e c r i t i c a l f i e l d f o r a t h i n f i l m o f t h i c k n e s s d t h e r e c a n be no s u p e r s t a t e a t a n y f l u x o i d . F o r l o w f l u x o i d n umbers t h e c u r v e s o f F(X) and a r e g i v e n i n F i g u r e s k—2. As ws c o n s i d e r h i g h e r f l u x o i d s , t h e s e c u r v e s r e t a i n t h e same g e n e r a l f o r m b u t become c o m p r e s s e d a l o n g b o t h t h e X and F a x e s and f i n a l l y c e a s e t o e x i s t a t a p a r t i c u l a r maximum f l u x o i d . We s h a l l c o n s i d e r o n l y c o n s t a n t t e m p e r a t u r e s i n t h i s s e c t i o n , , a n d we s h a l l e m p l o y t h e r e d u c e d 112 q u a n t i t i e s F and X d e f i n e d by equations (4.8), (4.7). We o b t a i n f i r s t the c r i t i c a l f i e l d s f o r second order phase t r a n s i t i o n s and c o n d i t i o n s f o r second order t r a n s i t i o n s to occur f o r the f u l l range of p o s s i b l e f l u x o i d s . For t h i s pur-pose we r e q u i r e the f i r s t two terms of the expansion o f F(Mtt^) i n powers o f . These have been o b t a i n e d i n Chapter 3» e q u a t i o n (3.28) i n which the c o e f f i c i e n t s o f powers o f are c l o s e d t r a n s c e n d e n t a l e x p r e s s i o n s f o r a r b i t r a r y Sl, t Slz 0 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 the c o e f f -i c i e n t o f ^ equal to z e r o . T h i s g i v e s (5.1) We r e w r i t e the LHS i n terms of the q u a n t i t i e s X which i s p r o -p o r t i o n s ! t o the d i f f e r e n c e (fix - 2J J and <£ itself„ ana then s o l v e t h i s e q u a t i o n f o r X . T h i s f a c i l i t a t e s comparison o f the s o l u t i o n w i t h the r e s u l t s of the p r e v i o u s c h a p t e r . From the d e f i n i t i o n of X „ e q u a t i o n (4 07) 5 we have # - 2^XHUK + 6 (5.2) 2 A, and by d i r e c t substitution into the p r e c e d i n g e q u a t i o n we o b t a i n 1 -*/7 A , (\Hcbl (5.3) T h i s e q u a t i o n determines the second order c r i t i c a l value o f X , a t which the system l e a v e s the p a r t i c u l a r f l u x o i d s t a t e U 3 b y way o f a s e c o n d o r d e r t r a n s i t i o n . We r e c a l l t h a t t h e o n l y r e s t r i c t i o n o n t h i s e q u a t i o n i s t h a t ' +£• be u n i f o r m , w h i c h i m -p l i e s d« f , t h e c o h e r e n c e l e n g t h . E q u a t i o n (5«3) c o u l d e a s i l y be s o l v e d e x p l i c i t l y b u t i t i s more i n s t r u c t i v e t o p r o c e e d a s f o l l o w s . P u t t i n g J2X = A/ +d , we e x p a n d t h e c o e f f i c i e n t s o f X 9 X<£ a n d § Z i n p o w e r s o f d/X . S i n c e we a r e now c o n -s i d e r i n g p a r t i c u l a r l y t h e c a s e o f l a r g e $ , we do n o t assume &/'McJ> "to be t h e same o r d e r o f m a g n i t u d e a s X „ a n d v/e r e t a i n i n t h e c o e f f i c i e n t s t e r m s l i k e d$ i n c o m p a r i s o n t o X . R e t a i n i n g t h e i m p o r t a n t t e r m s e q u a t i o n (5«2) t h e n becomes w i t h s o l u t i o n X = _ d $ t I i d*~§Z T h i s r e d u c e s t o t h e r e s u l t o b t a i n e d i n t h e p r e v i o u s c h a p t e r Xi = - / o n c o n d i t i o n t h a t 1 , w h i c h i s t h e m a t h e -m a t l e a l m e a n i n g o f " l o w f l u x o i d " , I n t h e c a s e o f l o w f l u x o i d t h e two c r i t i c a l f i e l d s c o r r e s p o n d i n g t o t h e + and - s i g n s i n e q u a t i o n (5<>4) a r e sym-m e t r i c a l l y p l a c e d w i t h r e s p e c t t o t h e v a l u e X - 0 , i . e . ~ $ « When t h e f l u x o i d becomes s i g n i f i c a n t l y l a r g e t h e m i d p o i n t b e t w e e n t h e two s e c o n d o r d e r c r i t i c a l f i e l d s i s s h i f t e d t o w a r d t h e o r i g i n o f t h e a x i s a n d a l s o t h e two c r i t i c a l f i e l d s become c l o s e r t o g e t h e r . F i n a l l y a t § = /If A//c£/'d , t h e Ilk two c r i t i c a l f i e l d s merge, and no f l u x o i d s g r e a t e r than t h i s can e x i s t . We d e f i n e ^ i s a parameter which compares any p a r t i c u l a r f l u x o i d s t a t e to the maximum f l u x o i d s t a t e which can e x i s t i n t h a t c y l i n d e r . We s h a l l speak i n a mathematical way o f the dependence o f c e r t a i n q u a n t i t i e s on Y , as though Y were a v a r i a b l e . E v i d e n t l y Y i s not a dynamic v a r i a b l e , f o r the f l u x o i d i s con-se r v e d d u r i n g q u a s i s t a t i c e q u i l i b r i u m p r o c e s s e s . Statements l i k e "as Y i n c r e a s e s " are 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 h i g h e r f l u x o i d " . Furthermore, Y , l i k e $ , i s quan t i z e d , and most of the statements made here should be q u a l i f i e d by a phrase l i k e "to the n e a r e s t a l l o w a b l e q u a n t i z e d v a l u e " . I t i s omit i t . The c o n d i t i o n t h a t a second order phase change does . 2. i n f a c t occur a t X^ i s found by s e t t i n g the c o e f f i c i e n t o f *y equal zero a t the reduced c r i t i c a l f i e l d X 1 » T h i s c o e f f i c i e n t was o b t a i n e d i n e q u a t i o n (3.28) i n terms o f H 2 and <f , As b e f o r e , we c o n v e r t i t to an e x p r e s s i o n i n X and j u s i n g (5»2) We expand the c o e f f i c i e n t s i n powers o f (d/X/ . keeping m mind the f a c t t h a t ^/^ci> may be l a r g e . T h i s l e a d s t o the f o l l o w i n g requirement f o r second order t r a n s i t i o n s to occur? 2 2 //_ + 11 ) 4. -A__ ( 5 . 6 ) 115 In t h i s e x p r e s s i o n Y and are taken to be o f order 1, and we have r e t a i n e d a l l terms up to order Otherwise i t would appear i n c o r r e c t l y t h a t a l l c y l i n d e r s undergo second order t r a n s i t i o n s a t s u f f i c i e n t l y h i g h f l u x o i d s . I f Y« \ , then to lowest order i n aY/Af the f i r s t term dominates and we have the c o n d i t i o n I as b e f o r e . At the other extreme, i f Y ~ ) , then o n l y the second order term appears on the l e f t . P u t t i n g i n X^ from (5«*0 g i v e s the con-d i t i o n c Y ^ f e X . T h i s i s p r e c i s e l y the c o n d i t i o n f o r second order phase t r a n s i t i o n s i n plane t h i n f i l m s o f t o t a l t h i c k n e s s d, as expected. T h i s c o n d i t i o n i s much weaker than ^ I . I n f a c t , by our d e f i n i t i o n o f t h i n c y l i n d e r s , o Y « A , we can say t h a t f o r a l l the c y l i n d e r s v/e are c o n s i d e r i n g second order phase * changes b e g i n at some Y , t h a t i s a t some . Except i n the extreme l i m i t 't—*• / , the secona o r a e r pnase change c o n a i t i o n i s w e l l expressed by (/- Y^yU * / For g i v e n yl^ the equal s i g n g i v e s the value o f Y at which second order t r a n s i t i o n s b e g i n . At t h i s v alue o f ? the second order c r i t i c a l f i e l d s are We t u r n now to the t h i n v / a l l expansion v a l i d f o r a l l and o b t a i n an e x p r e s s i o n f o r F v a l i d f o r <jp up to i t s maximum v a l u e . One must r e t a i n a l l terms to order ( M Y i n the expansions i n Appendix 3° As b e f o r e , we tr a n s f o r m the q u a d r a t i c e x p r e s s i o n i n H 2 and § to one i n X and F i n a l l y , we keep o n l y terms o f h i g h e s t o r d e r , where order i s determined by the l l6 maximum v a l u e t h e t e r m may h a v e . S p e c i f i c a l l y 9 X i s o f o r d e r 1 and <£ / / / e i i s o f o r d e r (c£/X) . A f t e r c o n s i d e r a b l e a l g e b r a • one o b t a i n s ' *• ' ( 5 . 7 ) T h i s d i f f e r s f r o m e q u a t i o n ( 4 . 8 ) b y t h e t e r m s i n Y , w h i c h w e r e p r e v i o u s l y d r o p p e d i n t h e a s s u m p t i o n o f l o w D e f i n e a s e t o f " p r i m e " q u a n t i t i e s a s f o l l o w s : ( 5 . 8 ) T h e n i n t h e p r i m e q u a n t i t i e s one o b t a i n s a n e q u a t i o n o f e x a c t l y Iso Jlfl-l SZ Q i m p l i e s ^F j __ p. s o t h a t i n e q u i l i b r i u m A r' - (fl/-i - sf) b y a n a l o g y t o e q u a t i o n ( 4 , 1 1 ) . The f u n c t i o n a l r e l a t i o n s i n t h e p r i m e q u a n t i t i e s a r e e x a c t l y t h e same a s f o r t h e u n p r i m e q u a n t i t i e s a t l o w ^ , To i n t e r p r e t t h e e q u a t i o n s f o r h i g h f l u x o i d number t h e p r o c e d u r e it t o t a k e o v e r a l l t h e r e s u l t s o f C h a p t e r 4 a p p l i e d t o t h e p r i m e q u a n t i t i e s , t h e n i n t e r p r e t t h e s e p h y s i c a l l y u s i n g t h e i n v e r s e t r a n s f o r m a t i o n s o f e q u a t i o n s ( 5 . 8 ) . I n p a r t i c u l a r , t h e e q u a t i o n s f o r a l l q u a n t i t i e s a t t h e f o u r s i g n i f i c a n t p o i n t s , o n p a g e s 8 8 , 8 9 h o l d f o r p r i m e q u a n t i t i e s f o r a l l Y . The e f f e c t o f o n t h e f u n c t i o n s F ( X ) and ^ s s u m m a r i z e d i n t h e f o l l o w -i n g p a r a g r a p h s a n d i l l u s t r a t e d i n F i g u r e s 5 - 1 » 5 - 2 , a n d 5 - 3 w h i c h r e f e r t o a n e x a m p l e c y l i n d e r h a v i n g = 10 . The s h a p e o f F(X) d e p e n d s o n and h e n c e o n Y , A s Y i n c r e a s e s , y U . ' d e c r e a s e s , a n d F ( X ) becomes more " s e c o n d -o r d e r - l i k e " . A t = yU> (f ~ Y*) =• / , F ( X ) becomes s e c o n d o r d e r t y p e , t h a t I s 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 F = 0 T h i s r e s u l t a g r e e s w i t h t h e c o n d i t i o n f o r s e c o n d o r d e r p h a s e c h a n g e s f o u n d a b o v e u s i n g t h e e x p a n s i o n o f F i n p o w e r s o f ^ , The minimum o f F o c c u r s a t p o i n t 0 ( s e e p a g e 8 8 ) a t w h i c h XQ — 0 a n d h e n c e XQ — — /J Y , The c u r v e s F ( X ) a n d ^ ( X ) a r e s y m m e t r i c a b o u t t h i s p o i n t r a t h e r t h a n a b o u t X ~ 0 , T h i s i s i n a g r e e m e n t w i t h e q u a t i o n ( 5 ° ^ ) a b o v e w h e r e t h e s e c o n d o r d e r c r i t i c a l f i e l d s w e r e f o u n d t o be s y m m e t r i c a l l y s i t u a t e d a b o u t X ~ ~" Y . A s was n o t e d i n C h a p t e r 3 f o l l o w i n g e q u a t i o n ( 3 = 2 1 ) , t h e minimum o f F o c c u r s f o r g e n e r a l Jlf , ^ 2 , a t = H 2 , t h a t i s f o r z e r o t o t a l l o o p c u r r e n t I C 0 When b o t h and Hg a r e l a r g e t h e d i s t r i b u t i o n o f 4^ i n t h e w a l l becomes i m p o r t a n t . When Hg = , ^ a n d ^  a t t h e i n n e r and o u t e r s u r f a c e s m u s t be i n o p p o s i t e d i r e c t i o n s . I f ^ i s e v a l -u a t e d a t t h e i n n e r s u r f a c e i t i s e q u a l t o H^ p l u s a t e r m p r o -p o r t i o n a l t o ^  , so t h a t i f H 1 ~ K0 , t h e n <^> Hj , a n d X 0 , The d i f f e r e n c e b e t w e e n f l u x a n d f l u x o i d becomes s i g n i f i c a n t a t l a r g e ^) , U s i n g t h e e x p l i c i t e x p r e s s i o n f o r H.^  i n e q u a t i o n (3»1?) one e a s i l y shows t h a t X ~ ~ ~ { 3 Y i m p l i e s 118 Hg ~ f o r a l l Y t o t h e a c c u r a c y o f t h e p r e s e n t c a l c u l a t i o n , , F o r > / , t h e s e p a r a t i o n b e t w e e n t h e u p p e r a n d l o w e r l i m i t i n g f i e l d s d e c r e a s e s a s Y i n c r e a s e s . The same i s t r u e o f Xg , t h e f i e l d a t w h i c h F = 0 . A t ' — I , X^ an d X g merge i n t o , a n d f o r ^ a . ' < I , t h e s e p a r a t i o n b e -t w e e n t h e u p p e r a n d l o w e r c r i t i c a l f i e l d s X^ d e c r e a s e s i n p r o -p o r t i o n t o F o r t h e e x a m p l e c y l i n d e r h a v i n g y ^ - — 0 , t h e d e p e n d e n c e o n Y o f t h e f i e l d s X Q , X^ # Xg , a n d X^ i s i l l u s t r a t e d i n F i g u r e 5-1• X i s p l o t t e d a s a b s c i s s a i n c o n f o r m -i t y w i t h i t s u s u a l r o l e a s i n d e p e n d e n t v a r i a b l e w h i l e t h e p a r a -m e t e r Y i s p l o t t e d a s o r d i n a t e . The minimum v a l u e o f F , o c c u r r i n g a t X = = ~~ ^ 3 Y , i s i n d e p e n d e n t l y o f t h e y ^ L o f t h e c y l i n d e r . The maximum o f F i s a more c o m p l i c a t e d f u n c t i o n o f Y i n v o l v i n g juJ „ F o r ^[4. * /(J % t h e u p p e r a n d l o w e r l i m i t s o f F f o r a l l 5 a r e shown i n t h e u p p e r g r a p h o f F i g u r e 5 - 2 . The " c u r v e s " c o n s i s t o f some 10 d i s c r e t e p o i n t s . I n p r i n c i p l e , t h e e x t e r n a l f i e l d Hg i s p l o t t e d a s a b s c i s s a , a n d t h e s c a l e i s s u c h a s t o encompass t h e w h o l e r a n g e o f p o s s i b l e f l u x o i d s t a t e s . On t h i s s c a l e t h e r a n g e o f X o v e r w h i c h a p a r t i c u l a r f l u x o i d s t a t e c a n e x i s t i s e x t r e m e l y s m a l l a n d t h e l i f e o f one s t a t e a p p e a r s a s a t h i n v e r t i c a l l i n e , V / i t h i n t h e r e s o l u t i o n o f t h i s d i a g r a m t h e n H% j> a n d i t seems more i n f o r m a t i v e t o s c a l e t h e h o r i z o n t a l a x i s i n t e r m s o f Y w h i c h i s p r o p o r t i o n a l t o $ , I n o r d e r t o i l l u -s t r a t e t h e v a r i a t i o n o f t h e f u n c t i o n s F ( X ) f o r i n c r e a s i n g f l u x o i d number, f o u r p a r t i c u l a r c u r v e s a r e shown i n ( a ) , ( b ) , ( c ) , ( d ) w i t h g r e a t l y e x p a n d e d h o r i z o n t a l s c a l e . I n ( c ) and ( d ) t h e v e r t i c a l s c a l e i s a l s o m a g n i f i e d b y a f a c t o r o f 25. 1 1 9 F i g u r e 5-3 i s a s i m i l a r diagram i l l u s t r a t i n g the v a r i -a t i o n o f the maximum and minimum v a l u e s o f ^  f o r a l l Y , The maximum value i s u n i v e r s a l "curve" — / — yz whereas the lower "curve" depends onyU>' , The same f o u r f l u x o i d s t a t e s as i n F i g u r e 5-2 are shown w i t h the same expanded h o r i z o n t a l s c a l e . The v e r t i c a l s c a l e i s the same f o r a l l graphs i n F i g u r e 5 - 3 . F i n a l l y we note t h a t the e f f e c t o f Y on the d i s c u s s -i o n o f change o f s t a t e i n Chapter 4 , p a r t 2 i s completely n e g l i -g i b l e . The g r e a t e s t f l u x o i d number i n F i g u r e 4 - 7 i s 4 x 10^ which corresponds to Y — ,00Z a t t = ,999 » and, to Y even l e s s f o r lower temperature. P a r t 2 - F i n i t e Coherence Length In a l l the r e s u l t s o b t a i n e d f o r the t h i n hollow c y l -i n d e r the o r d e r parameter y •= lift has been taken as uniform. T h i s i s an approximation based on the assumption o f v e r y s m a l l i n the second G - L e q u a t i o n . T h i s i m p l i e s V ~ Q which i n t u r n when taken t o g e t h e r w i t h the boundary c o n d i t i o n (-7y) ° o($ = 0 o n the s u r f a c e i m p l i e s ^ c o n s t a n t . On the b a s i s of t h i s approximation 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 the f r e e energy F i n a hollow c y l i n d e r , e q u a t i o n ( 3*24 ) , which becomes exact f o r " f — * • 0 0 „ For the t h i n c y l i n d e r cY^ , t h i s c l o s e d e x p r e s s i o n has been expanded i n powers o f and the f i r s t term used to o b t a i n the primary behaviour o f the phase t r a n s i t i o n s i n the c y l i n d e r . In p a r t 2 o f t h i s chapter v/e wish to o b t a i n a q u a n t i -t a t i v e estimate o f the e f f e c t o f f i n i t e coherence l e n g t h on the r e s u l t s a l r e a d y o b t a i n e d 8 and to d e f i n e more p r e c i s e l y the con-1 2 0 d i t i o n u n d e r w h i c h c o h e r e n c e l e n g t h may be assumed i n f i n i t e . R a t h e r t h a n m o d i f y i n g t h e r e s u l t s a l r e a d y o b t a i n e d we s h a l l s t a r t w i t h t h e two c o u p l e d G-L e q u a t i o n s and o b t a i n t h e d i s t r i b u t i o n s o f b o t h H and ^ (st) . From t h e o u t s e t we d i r e c t t h e c a l c u l a t i o n t o t h e t h i n - w a l l e d c y l i n d e r and employ a s e r i e s e x p a n s i o n a p p r o a c h . We a r e l o o k i n g f o r t h e f i r s t c o r r e c t i o n s t e r m t o F d e p e n d e n t on A"*" >0 , and t o t h i s end we make s u c h a p p r o x i m a t i o n s a s a r e u s e f u l . When u s i n g t h e f u l l c o u p l e d G-L e q u a t i o n s i t i s c o n -v e n i e n t t o work i n d i m e n s i o n l e s s q u a n t i t i e s . The c o - o r d i n a t e s a r e m e a s u r e d r e l a t i v e t o t h e p e n e t r a t i o n l e n g t h . I n c y l i n d r i c a l c o - o r d i n a t e s we d e f i n e /o == JX / A F o r t h e r e m a i n d e r o f t h i s c h a p t e r we w r i t e d i m e n s i o n l e s s f i e l d v a r i a b l e s i n b l o c k l e t t e r s H « A , and d i m e n s i o n a l f i e l d v a r i -a b l e s i n s c r i p t l e t t e r s . We d e f i n e w i t h G-L where i s t h e b u l k c r i t i c a l f i e l d . We s h a l l a l s o u s e t h e " d i m e n s i o n l e s s r e d u c e d f l u x o i d " IP i s a.l r e a d y d i m e n s i o n l e s s a s d e f i n e d b y e q u a t i o n ( 1 . 9 ) and has t h e f o r m UJ = where ^ i s r e a l . T a k i n g i n t h i s f o r m and d e f i n i n g t h e q u a n t i t y 121 w h e r e n i s t h e f l u x o i d number, t h e G-L e q u a t i o n s become i n d i m e n s i o n l e s s q u a n t i t i e s J  1 A' V x v x A = - y A The e q u a t i o n s f o r t h e q u a n t i t i e s y , A* h a v e t h e same f o r m a s t h e e q u a t i o n s f o r ^ , A f o r 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 t s h o u l d be n o t e d t h a t A i s n o t a v e c t o r p o t e n t i a l ; i t i s n o t c o n t i n u o u s a t t h e o r i g i n . I n a p p l y i n g t h e s e t o t h e c y l i n d r i c a l g e o m e t r y o f t h e h o l l o w c y l i n d e r we c o n t i n u e t o assume p e r f e c t s y m m e t r y i n t h e d and z d i r e c t i o n s . The two G-L e q u a t i o n s r e d u c e t o two c o u p l e d t o t a l d i f f e r e n t i a l e q u a t i o n s i n *J f/o) 4 ( e } -and w h i c h d e p e n d s o n t h e n o n - u n i f o r m i t y o f ^ w i t h i n t h e c y l i n d e r w a l l . F o r a t h i n c y l i n d e r d ^1/ , t h e f u n c t i o n a l f o r m o f ^ f r o m jOf t o jO^ w i l l be 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 f o r m o f i n a p l a n e f i l m o f t h i c k n e s s d — A^-A, s u b j e c t t o t h e 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 = o n one s u r f a c e a n d H = H 0 o n t h e o t h e r s u r f a c e . I n o r d e r t o u s e t h e s i m p l e c a r -t e s i a n f o r m o f t h e G-L e q u a t i o n s we t e m p o r a r i l y r e p l a c e t h e c y l -i n d e r w a l l b y a p l a n e t h i n f i l m f o r t h e c a l c u l a t i o n o f ^' (/O) ' , T h i s u n w i n d i n g p r o c e d u r e c a n o n l y i n t r o d u c e a n e r r o r o f o r d e r d/X i n t o t h e c o r r e c t i o n t e r m w h i c h i s a l r e a d y e x p e c t e d t o be s m a l l . We d e f i n e T a s t h e d i m e n s i o n l e s s c a r t e s i a n v a r i a b l e m e a s u r e d f r o m t h e m i d - " p l a n e " o f t h e c y l i n d e r w a l l r m a _ &±fr 1 A . The G-L e q u a t i o n s f o r t h e p l a n e a r e The b o u n d a r i e s o f T a r e d e f i n e d a s i n n e r s u r f a c e T — —A — jfi. &i a t / ° ~ (°i o u t e r s u r f a c e T — -t" A a t The b o u n d a r y c o n d i t i o n s a r e I n d i m e n s i o n a l q u a n t i t i e s i n t h e t h i n c y l i n d e r oY^X s o t h a t i n d i m e n s i o n l e s s q u a n t i t i e s i n t h e p l a n e ITI ^  A I T h e r e f o r e we assume s o l u t i o n s f o r /\ a n d *jr i n t h e f o r m o f s e r i e s e x p a n s i o n s i n T . P u t fi = a0 + &,r + az r ' f • • • (5.10) 1 = ^ + f, r + %rz i- • ( 5 . i i ) The G-L e q u a t i o n s d e t e r m i n e ^ i & 2 ; . . . and ^ ... i n t e r m s o f ^ ; £ f } ^ 0 } ^ w h i c h must be f i x e d b y t h e b o u n d a r y c o n d i t i o n s . The c o e f f i c i e n t s (Z< < become r a t h e r c o m p l i -c a t e d f o r l a r g e *L . We n o t e h e r e o n l y t h o s e w h i c h a r e u s e d i n t h e d i s c u s s i o n s ^ Z r 2. T 2. . 1 (5oi4) ?3 = f \lW{o + Oof, - 71 + 3 ^ j The e x a c t e x p r e s s i o n f o r t h e f r e e e n e r g y J- i s g i v e n i n d i m e n s i o n a l q u a n t i t i e s i n e q u a t i o n (1.20), A s i m i l a r e x -p r e s s i o n f o r t h e r e d u c e d f r e e e n e r g y F i n d i m e n s i o n l e s s q u a n t -i t i e s c a n be w r i t t e n w h i c h d i f f e r s f r o m 7 o n l y b y a c o n s t a n t . When t h e s e r i e s e x p a n s i o n s f o r A and *J a r e s u b s t i t u t e d i n t o t h e i n t e g r a n d a n d t h e i n t e g r a t i o n c a r r i e d o u t t h e r e r e s u l t s a c o r r e s p o n d i n g s e r i e s f o r F o f t h e f o r m F = Fo + F) 6 +• Fz &z -h • • • S i n c e t h e c r i t i c a l f i e l d s X.^  e t c , d e p e n d on t h e f u n c t i o n a l f o r m o f F e a c h c r i t i c a l f i e l d may a l s o be e x p r e s s e d a s a power s e r i e s i n A A l l t h e r e s u l t s o b t a i n e d p r e v i o u s l y a r e b a s e d o n a p p r o x i m a t i n g F b y t h e f i r s t t e r m l~~ , I n o r d e r t o c o r r e l a t e t h e p r e s e n t n o t a t i o n w i t h t h a t o f C h a p t e r k we s h a l l f i r s t o b -r— t a i n r u s i n g t h e p r e s e n t s e r i e s e x p a n s i o n s , We n e e d t h r e e t e r m s o f t h e A s e r i e s t o s a t i s f y t h e two b o u n d a r y c o n d i t i o n s o n n s i n c e *20 w i l l n o t a p p e a r i n H , O n l y t h e f i r s t t e r m *J0 o f 1 i s r e q u i r e d , *j was p r e v i o u s l y a s sumed t o be c o n -s t a n t w h i c h e v i d e n t l y s a t i s f i e s t h e b o u n d a r y c o n d i t i o n s o n *J A p p l y i n g t h e b o u n d a r y c o n d i t i o n s t o A we o b t a i n ' 2 t-d i oh. X *-~r s o t h a t f r o m e q u a t i o n ( 5 « 1 2 ) we have Hz ~ Hi a° = i ^ r ( 5 - i 5 ) A s s u m i n g Hg - H^ i s o f t h e same o r d e r a s Hg + H 1 ( v / h i c h i s t r u e e x c e p t i n f i l m s s o t h i n t h a t y U « I , i n w h i c h c a s e w a l l t h i c k n e s s i s a l r e a d y n e g l i g i b l e ) t h e n &/ 4 <Z0 b y a f a c t o r o f A s o t h e s e c o n d t e r m i n f\ i s A l e s s t h a n t h e f i r s t , The Ai t h i r d t e r m i s a l s o o f o r d e r / i c o m p a r e d t o t h e f i r s t . To l o w e s t o r d e r A ~ & 0 a s g i v e n b y ( 5 . 1 5 ) , a n d A ~ % 0 - n/</^o • A l t h o u g h we r e t a i n o n l y <Z0 i n t h e l o w e s t a p p r o x i m a t i o n o f i t 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 c o n d i t i o n t o CL^ ' t o f i n d t h e v a l u e o f CL0 . F o r c o n v e n i e n c e we s e p a r a t e t h e i n t e g r a l f o r F i n t o t wo p a r t s a s h a s b e e n done - o r e v i o u s l y . The " c o n d e n s a t i o n f r e e e n e r g y " F c i n c l u d e s t e r m s i n t h e i n t e g r a n d w i t h e x p l i c i t d e -p e n d e n c e o n *J o n l y w h i l e t h e m a g n e t i c f r e e e n e r g y i n c l u d e s t e r m s w i t h e x p l i c i t d e p e n d e n c e o n A and H , As b e f o r e , £. = + To t h e a c c u r a c y w h i c h i s j u s t i f i e d f o r t h e m a g n e t i c f r e e e n e r g y 0 i s e q u a l t o t h e f i e l d e n e r g y i n t h e h o l e p l u s t h e s u p e r e l e c t r o n k i n e t i c e n e r g y i n t h e w a l l a s d i s c u s s e d o n page 8 4 , B o t h f i e l d e n e r g y d e n s i t y a n d k i n e t i c e n e r g y d e n s i t y a r e p r o p o r t i o n a l t o (Hg - ) and we h a v e I n t h e p r e s e n t n o t a t i o n t h e g e o m e t r i c p a r a m e t e r yOC ==• 2 /\jO^ a n d u s i n g e q u a t i o n (5.15) w © c a n w r i t e 1 2 5 f5C = 2*y(/*4^ ) S e t t i n g as i t must be because the f i e l d i s uniform i n the hole we can s o l v e f o r Hg - i n terms o f Hg and <|5 . T h i s corresponds to the procedure lead-i n g to e q u a t i o n {3,17), In the pres e n t n o t a t i o n = 7*> (Hz - $) H,-H, and •>, (Hz - $) (5.16) 2(, + The f i r s t term i n the f r e e energy /£" becomes which agrees w i t h e q u a t i o n ( 4 . 8 ) . As b e f o r e , *J0 seeks a value to minimize , S e t t i n g JLc° = Q g i v e s an e q u a t i o n which can be w r i t t e n i n the form <ZQZ - ) -t f 0 2 = 0 (5.17) which i n view o f (5»l6) i s e q u i v a l e n t to ( 4 , 9 ) , Next we look a t the terms ff and i n the expansion of F , Consider f i r s t the c o n t r i b u t i o n from the condensation f r e e energy /~~ which Is g i v e n e x a c t l y by 126 w h e r e t h e l i m i t s a p p l y t o t h e p l a n e f i l m . We s u b s t i t u t e i n t h i s t h e s e r i e s e x p r e s s i o n f o r ^ . S i n c e t h e i n t e g r a t i o n i s o v e r a r e g i o n s y m m e t r i c w i t h r e s p e c t t o t h e o r i g i n o f T t h e odd t e r m s i n t h e i n t e g r a n d do n o t c o n t r i b u t e . The f i r s t two n o n -z e r o t e r m s g i v e (5.18) The f i r s t a n d a p p a r e n t l y l a r g e s t t e r m h e r e w h i c h h a s n o t a p p e a r e d b e f o r e i s f ^ t ^ i n "the f i r s t l i n e . L e t u s d e t e r m i n e t h e m a g n i t u d e o r Jt f r o m t h e b o u n d a r y c o n d i t i o n s . K e e p i n g t e r m s t o o r d e r i n ^ we ha v e f o r t h e odd p a r t o f ^ (r) , ?o»* ** f,r + fi r 3 S i n c e t h e b o u n d a r y c o n d i t i o n s ( 5 . 9 ) a r e s y m m e t r i c a l o n f t h e odd a n d e v e n p a r t s o f J m u s t s a t i s f y t h e b o u n d a r y c o n d i t i o n s i n d i v i d u a l l y . U s i n g t h i s c o n d i t i o n , a n d t h e r e l a t i o n (5*13) f o r ^2 « w e o b t a i n f o r ^ t p » - K & a0a, yQ s o t h a t t h e t e r m ^T /K i s o f o r d e r A c o m p a r e d t o A p p l y i n g t h e same a r g u m e n t t o t h e s e c o n d l i n e o f ( 5 . 1 8 ) , a l l t e r m s c o n t a i n i n g a r e a t l e a s t o f o r d e r A . The v a l u e o f ^ 2 f r o m e q u a t i o n ( 5 . 1 4 ) i s i n v i e w o f (5.17) i t s e l f o f o r d e r 127 a t l e a s t A . Hence a l l t h e t e r m s c o n t a i n i n g ^ 2 i - n ^ e if-b r a c k e t i n t h e s e c o n d l i n e o f (5»18) a r e o f o r d e r a t l e a s t A , s o t h e s e c o n d l i n e c o n t r i b u t e s o n l y t o o r d e r A . T h e r e i s no c o r r e c t i o n t o fj* o f o r d e r l e s s t h a n A!~ w . r . t . t h e f i r s t t e r m . T h a t i s I n c a l c u l a t i n g we a r e c o n c e r n e d v / i t h t h e q u a n t i t y ^ w h i c h i s d e f i n e d i n t h e w a l l o n l y . Hence we may c a r r y o u t t h e i n t e g r a t i o n i n t h e t h i n p l a n e w h i c h r e p r e s e n t s t h e unwound c y l i n d e r . To c a l c u l a t e t h e m a g n e t i c f r e e e n e r g y Ffj we must w i n d t h e p l a n e b a c k i n t o a c y l i n d e r a n d t a k e a c c o u n t o f t h e e n e r g y b o t h i n t h e w a l l a n d i n t h e h o l e . E q u a t i o n (5 .10) and (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 ^ , A and H i n t h e w a l l a s a f u n c t i o n o f T = /O — (^ +pz) / 2. . The c o e f f i c i -e n t s i n t h e e x p a n s i o n s a r e e x p r e s s e d i n t e r m s o f H r , H ? and *JQ , w h i c h a r e a d j u s t e d t o m i n i m i z e F . I n p a r t i c u l a r , ^ Q s a t i s f i e s e q u a t i o n ( 5 . 1 7 ) . V/e make u s e o f e q u a t i o n (3 .19) f o r ~J-y w h i c h a s sumes p e r f e c t s y m m e t r y i n t h e $} Z d i r e c t i o n s b u t d o e s np_t r e q u i r e ^ u n i f o r m . I n d i m e n s i o n l e s s v a r i a b l e s t h i s r e a d s s V/e w i s h t o e x p r e s s fj^ i n t e r m s o f t h e c o n s t r a i n t s Hg , ^ , We o u t l i n e h e r e t h e p r o c e d u r e b u t o m i t t h e d e t a i l s o f t h e c a l -c u l a t i o n . U s i n g (5 .9) and (5 .10) we w r i t e A % ~ A ( A ) i n t e r m s o f Hp , H 1 t o t h e r e q u i r e d a c c u r a c y . From t h e d e f i n i t i o n o f A e v a l u a t e d a t JOJ „ i . e . a t T — ~™A , and u s i n g t h e 128 r e l a t i o n A j ~ f>t ty j 2 , we p u t Z (Ji w h e r e RHS i s t h e s e r i e s e x p a n s i o n o f e q u a t i o n ( 5 . 1 0 ) . T h i s c a n be s o l v e d f o r H^ i n a powe r s e r i e s i n A b y a n i t e r a t i o n p r o -c e d u r e . T h i s v a l u e o f H 1 i n t e r m s o f H 2 , ^ i s s u b s t i t u t e d i n t o (5 .19) and Fy o b t a i n e d t o t h e r e q u i r e d a c c u r a c y . When t h i s c a l c u l a t i o n i s c a r r i e d o u t i t t u r n s o u t t h a t F^ t l i k e , d o e s n o t c o n t a i n a n y K d e p e n d e n t t e r m s o f o r d e r l e s s t h a n di w . r . t . t h e p r i m a r y t e r m . T h a t i s F H , - 0 > F„z=0 J F H 3 = 0 And c o n s e q u e n t l y We s h a l l n o t p u r s u e t h e c a l c u l a t i o n t o f i n d » S i n c e o u r v;hole p r o c e d u r e i s v a l i d o n l y f o r A I . t h e c o r r e c t -i o n t e r m i n A^ i s i n d e e d n e g l i g i b l e . The c a l c u l a t i o n o f F^. becomes e n o r m o u s l y more c o m p l i c a t e d . S e v e r a l a p p r o x i m a t i o n s w h i c h a r e j u s t i f i e d i n t h e p u r s u i t o f a r e no l o n g e r v a l i d when c a l c u l a t i n g Ff. * I n p a r t i c u l a r , we c a n n o t e m p l o y t h e s i m p l i f i c a t i o n o f " u n w i n d i n g " t h e c y l i n d e r a n d w o r k i n g i n c a r t e s -i a n c o - o r d i n a t e s . R e t u r n i n g t o d i m e n s i o n a l q u a n t i t i e s t h e l o w e s t o r d e r 2. n o n - z e r o K d e p e n d e n t c o r r e c t i o n t e r m s w i l l be o f o r d e r o r „_ff:.__ , We c o n c l u d e t h a t i n t h i n c y l i n d e r s t h e 1" ~ e f f e c t o f f i n i t e c o h e r e n c e l e n g t h w i l l be u n m e a s u r a b l y s m a l l o n c o n d i t i o n t h a t ^ » o C . T h i s d o e s n o t i m p l y a n y c o n d i t i o n o n j we may - w e l l h ave f >^cd e v e n f o r >2 t i . e . f o r Type I I s u p e r c o n d u c t o r s . -r/.O man - 1 0 I Z 3 Reduce^/ fre/S X Change i n c r i t i c a l r e d u c e d f i e l d s a t l a r g e f l u x o i d . /•H u3 V) Vj . 4 H k Reduced Fie/c/ X ~ S2, -1.0 -z 1 i i I \ • i i \ I u \ I- -.01 F i g u r e 5 - 2 Top: D e p e n d e n c e o f F r e e E n e r g y F o n Y f o r f u l l r a n g e o f p o s s i b l e f l u x o i d s . C e n t r e a n d b o t t o m s E x a m p l e s o f c u r v e s o f F ( X ) a t d i f f e r e n t v a l u e s o f Y . l-o-(dji/.i .1 -.1 — o-Reo/acec/ f^ie./c/ X S — . ^ " , ^ _ i Z \ i x F i g u r e 5-3 Top: D e p e n d e n c e o f S u p e r E l e c t r o n D e n s i t y y o n ¥ f o r f u l l r a n g e o f p o s s i b l e f l u x o i d s . C e n t r e and b o t t o m ; E x a m p l e s o f c u r v e s o f a t d i f f e r e n t v a l u e s o f / , " 13 CHAPTER 6 OTHER FORMS OF MULTIPLY-CONNECTED SUPERCONDUCTORS I n C h a p t e r s 4 and 5 we have d i s c u s s e d i n some d e t a i l t h e t h i n h o l l o w c i r c u l a r c y l i n d e r . I n t h i s c h a p t e r v/e s h a l l d i s c u s s some o t h e r f o r m s o f 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 f o r w h i c h r e s u l t s c a n be o b t a i n e d w i t h o u t r e s o r t t o n u m e r i c a l m e t h o d S e c t i o n 1 - T h i n w a l l e d n o n - c i r c u l a r c y l i n d e r s C e n t r a l t o t h e a n a l y s i s o f t h e t h i n h o l l o w c i r c u l a r c y l i n d e r was the e q u a t i o n f o r t h e r e d u c e d f r e e e n e r g y F i n n o r m a l i z e d f o r m , n a m e l y I <—• / The p r o c e d u r e h e r e i s t o o b t a i n a n e q u a t i o n o f s i m i l a r f o r m a nd t o i d e n t i f y t h e q u a n t i t i e s a n d X i n t h e p r e s e n t g e o m e t r y . L e t us r e c a p i t u l a t e b r i e f l y how (4.8) was o b t a i n e d f o r t h e c i r -c u l a r c y l i n d e r . A n e x a c t e x p r e s s i o n f o r t h e f r e e e n e r g y was o b t a i n e d , e q u a t i o n ( 3 ° 1 9 ) , m a k i n g s p e c i f i c u s e o f t h e c i r c u l a r s y m m e t r y . On t h e a s s u m p t i o n , e x p r e s s i o n s f o r c e r t a i n f i e l d q u a n t i t i e s o c c u r r i n g i n ( 3 « 1 9 ) were o b t a i n e d w h i c h i n -v o l v e d m o d i f i e d B e s s e l * s f u n c t i o n s e v a l u a t e d a t t h e i n n e r a n d o u t e r s u r f a c e s , a n d t h e s e e x p r e s s i o n s w e re s u b s t i t u t e d i n ( 3 - 1 9 ) t o o b t a i n ( 3 , 2 0 ) . To a p p l y t h i s t o t h i n c y l i n d e r s , a T a y l o r 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 was made a b o u t t h e a r g u m e n t J2>t , and l o w e s t o r d e r t e r m s r e t a i n e d . •With t h e d e f i n i t i o n s ( 4 , 1 ) , (4,2), (4.7), t h i s l e a d s t o t h e f o r m o f F i n (4,8) a b o v e . I t v/as n o t e d o n page 84 t h a t t h e same e x p r e s s i o n i s o b t a i n e d b y t a k i n g ^ u n i f o r m i n t h e w a l l ( j u s t i f i e d b y ^4cA ) and r e t a i n i n g i n t h e f r e e e n e r g y t h e f o l l o w i n g t h r e e t e r m s ( i ) c o n d e n s a t i o n e n e r g y o f s u p e r - e l e c t r o n s i n t h e w a l l , ( i i ) m a g n e t i c f i e l d e n e r g y i n t h e h o l e , a n d ( i i i ) k i n e t i c e n e r g y o f s u p e r - e l e c t r o n s i n t h e w a l l . C o n s i d e r now a l o n g h o l l o w t h i n w a l l e d c y l i n d e r , n o t n e c e s s a r i l y c i r c u l a r i n c r o s s - s e c t i o n , i n a u n i f o r m e x t e r n a l m a g n e t i c f i e l d H g p a r a l l e l t o t h e c y l i n d e r . L e t 5 be t h e a r e a o f a p l a n e s u r f a c e n o r m a l t o t h e c y l i n d e r a n d b o u n d e d b y i t s i n n e r w a l l , a n d l e t p be t h e p e r i m e t e r o f t h i s p l a n e s u r -f a c e . The w a l l t h i c k n e s s o i , m e a s u r e d a l o n g a n o r m a l t o t h e s u r f a c e , i s u n i f o r m ; a n d i s t h i n i n t h e d o u b l e s e n s e d ^ X and p d ^ < S , I t i s assumed t h a t t h e r a d i i o f c u r v a t u r e o f b o t h i n n e r a n d o u t e r s u r f a c e s a r e e v e r y w h e r e l a r g e c o m p a r e d t o c / , I f t h e c y l i n d e r i s c o o l e d s u f f i c i e n t l y t o be 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 , t h e r e w i l l i n g e n e r a l be a c i r c u l a t i n g c u r r e n t o f n e a r l y u n i f o r m d e n s i t y ^ i n a d i r e c t i o n n o r m a l t o He . T h i s c u r r e n t w i l l i n d u c e a u n i f o r m m a g n e t i c f i e l d i n t h e h o l e , s o t h a t t h e r e s u l t a n t h o l e f i e l d Hi i s a l s o u n i f o r m , a n d H» - H = ^2T a J ( 6.1) " e n , c 0S The f r e e e n e r g y p e r u n i t l e n g t h o f c y l i n d e r J- i s g i v e n t o f i r s t o r d e r a c c u r a c y b y t h e a b o v e t h r e e t e r m s . V/e h a v e j ^ i ^ u - z i + f ) +(±kz±!AA +M£(Fct)f ( 6 . 2 ) J $tf 1 ff / %rr O f ' 9 . w h e r e a s b e f o r e — 1^1 i s t h e r e l a t i v e d e n s i t y o f s u p e r -e l e c t r o n s . The f l u x o i d < c^ a s s o c i a t e d w i t h t h e h o l e i s g i v e n b y e q u a t i o n ( 2 . 3 ) 132 Js Jp <r w h i c h i s r e a d i l y i n t e g r a t e d s i n c e /7t » / ^ / a n c i ^ a r e u n i f o r m . 7 From t h e g e n e r a l a r g u m e n t g i v e n i n C h a p t e r 3, <pc i s q u a n t i z e d i n u n i t s o f (j)0 . U s i n g (6,1) and (6,3) t o e l i m i n a t e H 1 and ^ f r o m (6.2) v/e o b t a i n f o r t h e f r e e e n e r g y a n d f o r t h e r e d u c e d f r e e e n e r g y p" = STTJ1-/ Hck> peY T h i s i s i d e n t i c a l t o (4.5) p r o v i d e d v/e p u t . C o n s e q u e n t l y , a l l t h e r e s u l t s o f C h a p t e r 4 may be a p p l i e d t o n o n - c i r c u l a r t h i n c y l i n d e r s , I n p a r t i c u l a r * i f Z S d / . A p ^ I} t h e n a s e c o n d o r d e r p h a s e t r a n s i t i o n o c c u r s a t 2 F o r a c i r c u l a r c y l i n d e r 5 ~ TfAl and p ~ Z f f S l i 9 e q u a t i o n (6,5) r e d u c e s i m m e d i a t e l y t o (4 . 7 ) and (4,2). I n t h e c i r c u l a r c y l i n d e r t h e s i m p l e r e l a t i o n b e t w e e n S a n d p t e n d s t o o b s c u r e t h e way i n w h i c h t h e s e two q u a n t i t i e s e n t e r t h e d e f i n i t i o n s o f X and yd- . From (6.1) and t h e l a s t two t e r m s o f (6.2) we s e e t h a t yWp/Z i s e q u a l t o t h e r a t i o o f f i e l d e n e r g y t o k i n e t i c e n e r g y . I f t h e e x t e r n a l f i e l d i s z e r o , b u t t h e r e i s a n o n - z e r o f l u x o i d due t o a p e r s i s t e n t c u r r e n t , e q u a t i o n (6,6) becomes s i m p l y <f>a = V T \pHc6 = n(t>o T h i s may be i n t e r p r e t e d i n t e r m s o f a c r i t i c a l t e m p e r a t u r e t h r o u g h t h e t e m p e r a t u r e d e p e n d e n c e o f A and Hcb ; o r a s a c o n d i t i o n o n t h e maximum f l u x o i d s t a t e w h i c h c a n e x i s t a t g i v e n T . T h i s c o n d i t i o n i s e a s i l y t r a n s f o r m e d t o t h e f o r m D — p I J , t h a t i s , t h e quantum number o f t h e g r e a t e s t f l u x o i d s t a t e w h i c h c a n e x i s t i s e q u a l t o t h e number o f c o h e r e n c e l e n g t h s a r o u n d t h e p e r i -m e t e r , S e c t i o n 2 - T o r u s a n d L o o p So f a r o n l y " i n f i n i t e l y l o n g " h o l l o w c y l i n d e r s h a v e b e e n s t u d i e d . T h e r e a r e two r e a s o n s f o r t h i s , f i r s t t h e r e s u l t -i n g two d i m e n s i o n a l g e o m e t r y s i m p l i f i e s t h e m a t h e m a t i c a l a n a l y s i s , a n d s e c o n d l y , t h e r e l e v a n t e x p e r i m e n t s h a v e b e e n c a r r i e d o u t o n l o n g h o l l o w c y l i n d e r s t h o u g h n o t a l w a y s a s l o n g a s one w o u l d l i k e f o r c o m p a r i s o n w i t h t h e o r y . H o w e v e r , i t i s n o t o b v i o u s t h a t t h e end e f f e c t i n a f i n i t e l e n g t h c y l i n d e r i s r e n d e r e d n e g l i g i b l e b y s u f f i c i e n t l y i n c r e a s i n g i t s l e n g t h , F o r e x a m p l e , t h e r e may be a n u n s t a b l e c o n d i t i o n a t t h e end o f t h e c y l i n d e r w h i c h t r i g g e r s a p h a s e t r a n s i t i o n . E v i d e n t l y i t i s d e s i r a b l e t o s t u d y a d o u b 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 w h i c h i s e s s e n t i a l l y f i n i t e , C o n s i d e r 134 t h e n t h e s i m p l e s t s u c h , a c i r c u l a r t o r u s o f a x i a l r a d i u s /? and c r o s s - s e c t i o n r a d i u s ^6" , s i t u a t e d i n a u n i f o r m e x t e r n a l mag-n e t i c f i e l d He p a r a l l e l t o t h e a x i s o f t h e t o r u s . On t h e b a s i s o f a x i a l s y m m e t r y , a nd t a k i n g i s n e a r l y u n i f o r m t h r o u g h o u t t h e t o r u s . P r o c e e d i n g a s i n C h a p t e r 3 we a r e l e d t o a L o n d o n - l i k e e q u a t i o n f o r A w h e r e —*> T-S> Art *L A A ' — A # U n f o r t u n a t e l y L o n d o n ' s e q u a t i o n i s n o t 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 . We c a n , h o w e v e r , c a l c u l a t e t h e c r i t i c a l f i e l d f o r s e c o n d o r d e r p h a s e t r a n s i t i o n s f o r a g e n e r a l t o r u s u s i n g t h e i t e r a t i o n p r o c e d u r e d e s c r i b e d o n page 76 . A f t e r t h i s v/e s h a l l l i m i t t h e d i s c u s s i o n t o t h e t h i n t o r u s ( o r " l o o p " ) h a v i n g d ^ R and o b t a i n some a p p r o x i m a t e r e s u l t s . To u.s^33rnii!riG "til© second, cx'cl G if cz*i"fcics.j- i i s i - d i \VG T~OC]_UH, IT S t h e f i r s t t e r m o f t h e e x p a n s i o n i n p o w e r s o f *js o f t h e f r e e e n e r g y J- , 3^ i s f o u n d f r o m e q u a t i o n (3»23) a n d i s f o u n d f r o m e q u a t i o n (3.26) w h e r e ^ i s e x p r e s s e d a s a p o w e r s e r i e s i n ^ . The f i r s t t e r m o f t h i s s e r i e s i s g i v e n b y e q u a t i o n (3=27) w i t h f\ r e p l a c e d b y f\e , t h e v e c t o r p o t e n t i a l o f t h e e x t e r n a l f i e l d , A must be i n t h e L o n d o n Gauge, a n d s i n c e t h e t o r u s h a s r o t a t i o n a l s y m m e t r y a b o u t t h e a x i s , t h i s i s t h e same gauge a s f o r a c y l i n d e r . I n c y l i n d r i c a l c o - o r d i n a t e s The f i r s t t e r m i n t h e e x p a n s i o n o f ~3~ i s 135 w h e r e Vj i s t h e v o l u m e o f t h e t o r u s . The i n t e g r a l i s e l e m e n t a r y and t h e c r i t i c a l v a l u e o f HQ f o r s e c o n d o r d e r p h a s e t r a n s i t i o n s i s g i v e n b y I * 1 * (6.?) w h e r e we h a v e u s e d t h e " r e d u c e d f l u x o i d " I f t h e t o r u s i s t h i n , , t h i s r e d u c e s t o w h i c h i s i d e n t i c a l t o t h e s e c o n d o r d e r c r i t i c a l f i e l d f o r a t h i n c y l i n d e r o f r a d i u s R I n a s t r a i g h t f o r w a r d m anner, one c a n w r i t e down t h e c o e f f i c i e n t o f ^ } w h i c h i s r e q u i r e d t o d e t e r m i n e t h e c o n d i t i o n f o r s e c o n d o r d e r t r a n s i t i o n s t o o c c u r , P u t t h e i n t e g r a l s i n v o l v e d c e a s e t o be e l e m e n t a r y . From p r e v i o u s e x a m p l e s , one i s l e d t o b e l i e v e t h a t t h e c o n d i t i o n w i l l i n v o l v e some d e g r e e o f t h i n n e s s . We t u r n t o t h e c o n s i d e r a t i o n o f a t h i n l o o p , s a t i s f y i n g b o t h 4 « R and A . From t h e l a t t e r c o n d i t i o n t h e c u r r e n t d e n s i t y ^ s i s n e a r l y u n i f o r m and t h e t o t a l l o o p c u r r e n t J — -j^r n w h e r e /9 ~ T?".S- i s t h e c r o s s - s e c t i o n a r e a o f t h e l o o p . We p r e f e r t o u s e /} r a t h e r t h a n 1Y<6- i n t h e f o l l o w i n g e q u a t i o n s t o f a c i l i t a t e t h e g e n e r a l i z a t i o n t o l o o p s o f n o n - c i r -c u l a r c r o s s - s e c t i o n . The f r e e e n e r g y c a n be w r i t t e n a s t h e sum o f t h e t h r e e t e r m s : i C o n d e n s a t i o n E n e r g y 116 K i n e t i c E n e r g y 2 / r \2-M a g n e t i c F i e l d E n e r g y IZL Z cr w h e r e L i s t h e i n d u c t a n c e o f t h e l o o p . The f l u x o i d <j£>c i s g i v e n b y D e f i n e a s b e f o r e t h e r e d u c e d f r e e e n e r g y a n d t h e r e d u c e d f l u x o i d ^ — ^ /'ftR , A d d i n g t h e t h r e e t e r m s o f 7 a n d e l i m i n a t i n g I we o b t a i n f o r t h e r e d u c e d f r e e e n e r g y f ( 6 . 8 ) h V • * Hair \ z / [ T h i s i s a g a i n a n e x p r e s s i o n i n s t a n d a r d f o r m w h e r e gHd'X2 ' ' V-TT^RA2 ( 6 ' 9 ) A g a i n , a l l t h e r e s u l t s o f C h a p t e r 4 may be a p p l i e d t o t h e p r e s e n t c a s e u s i n g t h e a p p r o p r i a t e q u a n t i t y f o r y U . . X i s t h e same q u a n t i t y f o r b o t h t h e t h i n c y l i n d e r a n d t h i n l o o p . The c o n d i t i o n t h a t a s e c o n d o r d e r p h a s e c h a n g e w i l l o c c u r i s y U ^ I , F o r a t h i n l o o p o f c i r c u l a r c r o s s - s e c t i o n a n a p p r o x i m a t e v a l u e o f t h e i n d u c t a n c e i s 137 I n t h i s c a s e t h e c o n d i t i o n & I becomes "A1 / * T h i s c o n d i t i o n d e p e n d s s t r o n g l y o n and o n l y w e a k l y o n t h e l o o p r a d i u s . To i l l u s t r a t e , f o r R ~ /0<tr , we h a v e *6-/k i . 6 2 w h i l e f o r R ~ 000*6- , we h a v e J-/A < .36 . I t i s i n t e r -e s t i n g t o compare t h i s c o n d i t i o n w i t h t h a t f o r a t h i n c y l i n d e r o f t h i c k n e s s <?L , w h i c h i s cd ^ A /R a The c o n d i t i o n o n t h e l o o p i s much w e a k e r . E q u a t i o n (6.8) a p p l i e s e q u a l l y w e l l t o l o o p s o f a r b i -t r a r y c r o s s - s e c t i o n , p r o v i d e d t h a t t h e g r e a t e s t d i m e n s i o n i s l e s s t h a n o Of c o u r s e f o r i t s a p p l i c a t i o n one must be a b l e t o e v a l u a t e L , V/e do h o w e v e r r e q u i r e t h e l o o p t o h a v e a x i a l s y m m e t r y i n o r d e r t o j u s t i f y t a k i n g ^ u n i f o r m a r o u n d t h e l o o p . 138 S e c t i o n 3 - Two Coupled. L o o p s I n t h i s s e c t i o n we c o n s i d e r two s u p e r c o n d u c t i n g r i n g s n o t i n c o n t a c t h u t c o u p l e d m a g n e t i c a l l y . Two t h i n c i r c u l a r l o o p s o f t h e t y p e d e s c r i b e d i n t h e p r e v i o u s s e c t i o n a r e p l a c e d c o -a x i a l l y w i t h a x e s p a r a l l e l t o t h e e x t e r n a l m a g n e t i c f i e l d H e . We r e s t r i c t t h e d i s c u s s i o n t o l o o p s o f t h e same r a d i u s R , and made o f t h e same m a t e r i a l , h e n c e same A and A/c£ . T h r o u g h o u t t h e r e m a i n d e r o f t h i s c h a p t e r s u b s c r i p t s r e f e r t o a p a r t i c u l a r l o o p a n d s h o u l d n o t be c o n f u s e d w i t h s u b s c r i p t s r e f e r r i n g t o " s i g n i f i c a n t p o i n t s " i n C h a p t e r 4. The l a t t e r u s e i s e n t i r e l y a v o i d e d i n t h i s c h a p t e r . The l o o p s may have d i f f e r e n t c r o s s -s e c t i o n a r e a s A 1 , A ? o f a r b i t r a r y s h a p e ( w i t h i n r e a s o n ) . The l o o p s h a v e s e l f - i n d u c t a n c e s L^ , L g r e s p e c t i v e l y a n d m u t u a l i n d u c t a n c e v/here fc, i s t h e c o u p l i n g c o n s t a n t . I n rng s u p e r c o n Q u c c x n g s x a x e t h e .loops n y . v e r e i a x i v e s u p e r s x e c x - r o n d e n s i t i e s ^ , ^  and t o t a l l o o p c u r r e n t s I 1 , I g , I 1 ' and I 0 a r e p o s i t i v e when t h e f i e l d t h e y p r o d u c e i n s i d e t h e l o o p i s i n t h e same s e n s e a s H g . C o n s e q u e n t l y i n t h i s r e s t r i c t e d g e o -m e t r y M and k> a r e e s s e n t i a l l y p o s i t i v e . We s h a l l n o t c o n -s i d e r h e r e c a s e s i n v / h i c h H c h a n g e s s i g n , b u t t h e s e may be t r e a t e d b y a r b i t r a r i l y c h o o s i n g one a x i s a s p o s i t i v e , F o l l o w i n g t h e p r o c e d u r e o f t h e p r e v i o u s s e c t i o n we w r i t e down t h e t h r e e t e r m s v / h i c h c o n t r i b u t e t o t h e f r e e e n e r g y . C o n d e n s a t i o n e n e r g y (ZnR) I A, (-ZJ-, + f ) + A z ( - t y + 139 K i n e t i c e n e r g y U ,4 -h I, M a g n e t i c f i e l d e n e r g y ft: 2-/7 2. The r e s p e c t i v e f l u x o i d s a r e g i v e n b y .a. c + IzM v / h i c h a l s o f i x t h e s i g n c o n v e n t i o n o f ^>Cf , <£>ci « T h e s e two e q u a t i o n s a r e e a s i l y s o l v e d f o r l 1 and 1 9 w h i c h c a n t h e n be e l i m i n a t e d f r o m t h e e x p r e s s i o n f o r t h e f r e e energy-. We s h a l l f i r s t make t h e f u r t h e r r e s t r i c t i o n t h a t t h e l o o p s h a v e t h e same  f l u x o i d <pci — (j>co S <^ >£ . T h i s i s c o n s i s t e n t w i t h t h e r e s t r i c t -i o n t o l o o p s o f e q u a l r a d i i R » f o r i f t h e n o n - z e r o f l u x o i d s a r e p r o d u c e d b y c o o l i n g i n a u n i f o r m f i e l d K , t h e f l u x o i d i n e a c h l o o p w i l l e q u a l ( t o t h e n e a r e s t quantum s t a t e ) t h e f l u x o f e c l o s e d b y t h e l o o p , T h e n t h e l o o p c u r r e n t s a r C \ "2/2-c (n/?zHe - 0c)ft, -M + ~ 4*p A - M 0 The two l o o p s a r e c h a r a c t e r i z e d p h y s i c a l l y b y t h e f i v e q u a n t i t i e s A 1 „ Ag , I>1 , L 2 , M , a l t h o u g h o n l y t h e r a t i o • A- 1/ A 2 a f f e c t s t h e f o r m o f 7 I n t h e c a s e o f a s i n g l e l o o p t h e g e o m e t r i c p r o p e r t i e s e n t e r e d t h r o u g h t h e s i n g l e p a r a m e t e r yCC- , H e r e i t i s c o n v e n i e n t t o d e f i n e f i v e p a r a m e t e r s : t h e c o u p l i n g c o n s t a n t = \FtLx. / M (6 ,10) t h e yM-'s o f e a c h l o o p and two " U " p a r a m e t e r s v, =y,(i-k{jpi > ^ = y2(i - k. fe) ( 6 . 1 2 ) H o w e v e r , f o r r e a s o n s o f s y m m e t r y i n t h e e q u a t i o n s we s h a l l u s e a l l f i v e . V/e d e f i n e t h e n o r m a l i z e d f i e l d v a r i a b l e X e x a c t l y a s i n e q u a t i o n ( 6 , 9 ) f o r a s i n g l e l o o p , and d e f i n e t h e r e d u c e d f r e e e n e r g y F = $77 7/'Hct(Zrr*)(/>, +4*) The r e d u c e d f r e e e n e r g y i s f o u n d t c be \ 0 J (6,13) (A,+A)\ ^ F 0 l 4 i I n e q u i l i b r i u m , F i s a minimum w i t h r e s p e c t t o b o t h a n d ^ 2 » c o n s e q u e n t l y in ^ A ty . . ( 6 . I t ) E l i m i n a t i n g X f r o m t h e s e s i m u l t a n e o u s e q u a t i o n s we o b t a i n t h e f o l l o w i n g r e l a t i o n b e t w e e n % t a n d ^ V/e s h a l l r e f e r t o t h i s a s t h e " p a t h " e q u a t i o n i n t h a t i t d e s c r i b e s a p a t h i n t h e , p l a n e a l o n g w h i c h t h e s y s t e m t r a v e l s i n q u a s i s t a t i c e q u i l i b r i u m . The v a l u e o f X a t a n y p o i n t o n t h e p a t h c a n be f o u n d b y s u b s t i t u t i o n i n e i t h e r o f t h e p r e c e d i n g e q u a t i o n s . A l t h o u g h we h a v e n o t y e t c o n s i d e r e d w h e t h e r a l l o r a n y p a r t o f t h e p a t h c o r r e s p o n d s t o a minimum o f F ( e q u a t i o n s (6.14) b e i n g n e c e s s a r y b u t n o t s u f f i c i e n t ) we s h a l l f i r s t d i s c u s s some p o s s i b l e p a t h s . The o n l y p h y s i c a l l y m e a n i n g f u l p a t h must l i e w i t h i n t h e u n i t s q u a r e 0 ^^) / , 0 £ Jr2 £ 0 . (The t e r m " u n i t s q u a r e " w i l l be u s e d h e n c e f o r t h i n t h i s p r e c i s e s e n s e . ) A l l p a t h s i n c l u d e t h e p o i n t p., - I , p..^ - ) , a t w h i c h X = 0 , I n g e n e r a l , e q u a t i o n (6,15) h a s s e v e r a l b r a n c h e s . We r e s t r i c t t h e w o r d p a t h t o t h a t b r a n c h b e g i n n i n g a t ^ { - I , ~ I > l 4 2 l y i n g w i t h i n t h e u n i t s q u a r e a nd 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 X . I n o r d e r t o o b t a i n g r a p h i c a l l y a d e s c r i p t i o n o f t h e p a t h c a l l t h e LHS o f ( 6 . 1 5 ) t h e f u n c t i o n ^i^>) a n d t h e R H S t h e f u n c t i o n Y% (j-i) ' 3 0 t h e e c l u a ' t i o n m a y b e w r i t t e n - Yz • fte'(ps) i s a f a m i l y o f c u b i c s 5 i n f a c t i t i s t h e same f a m i l y o f c u b i c s a s i n F i g u r e 4 - 1 w i t h y r e p l a c i n g X a s o r d i n a t e a n d t h e p a r a m e t e r V r e p l a c i n g y U , To s i m p l i f y t h e d i s c u s s i o n l e t t h e s u b -s c r i p t s be a s s i g n e d s o t h a t ISj < ( t h e s p e c i a l c a s e ^ — ^ i s d i s c u s s e d s e p a r a t e l y l a t e r o n ) . T h e n yt fpi) l i e s b e l o w /z. ^ 1 ) « e x c e p t a t p., = ^  - I and pt ~ ~ ® w h e r e t h e y i n t e r -s e c t . The " p a t h " i s f o u n d b y l e t t i n g ^ d e c r e a s e f r o m 1 t o 0 9 and 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-^ f r o m t h e c o n d i t i o n y^.' f i ' By d e f i n i t i o n t h e p a t h t e r m i n a t e s a t t h e b o u n d a r y o f t h e u n i t s q u a r e . T h r e e t y p e s o f p a t h may o c c u r : T ype 1 1 yt ^ i\ =* / , p , and p2. a e c r e a s e m o n o t o n i c a l l y t o 0 , Type I I : jjf ^ / , Px > j , ^j d e c r e a s e s m o n o t o n i c a l l y t o 0 , d e c r e a s e s m o n o t o n i c a l l y t o a f i n i t e p o s i t i v e v a l u e . Type I I I : > ^ ' I • , p f d e c r e a s e s m o n o t o n i c a l l y t o 0 9 yl.^ p a s s e s t h r o u g h a p o s i t i v e minimum and t h e n i n c r e a s e s t o a f i n i t e v a l u e when t h e p a t h t e r m i n a t e s . I t i s p o s s i b l e f o r iZj t o be s l i g h t l y n e g a t i v e i n c o n t r a s t t o y U . f o r w h i c h F i g u r e 4 - 1 was d r a w n . C u r v e s o f Y'^'} F O R -Z ^-^i^O a r e c o n c a v e u p w a r d f r o m pt ~ 0 t o pf - / , and t h e t h r e e t y p e s o f p a t h s d e s c r i b e d r e m a i n v a l i d , Pf ~Z i m p l i e s a p h y s i c a l s i t u a t i o n n o t f a l l i n g w i t h i n t h e b o u n d s o f t h e p r e s e n t 1 4 3 d i s c u s s i o n * I t i s c l e a r t h a t a t X = 0 , ^ t ~^-x = / i s a minimum • o f F , We t u r n now t o t h e q u e s t i o n o f d e t e r m i n i n g t h e p o i n t o n t h e p a t h , i f a n y , a t w h i c h a minimum c e a s e s t o e x i s t . A n e c e s s a r y c o n d i t i o n t h a t F"fytj^-*) be a n extremum i s / 0 (6 .16) a n d I^F T h i s c o n d i t i o n i m p l i e s t h a t / o ~ J a n d m u s t h a v e t h e same s i g n w h i c h f o r a minimum mu s t be p o s i t i v e . S i n c e t h i s i s t h e c a s e f o r ^ ~ ~ ^ ' a n c i s i n c e a H s e c o n d p a r t i a l d e r i v a t i v e s a r e c o n t i n u o u s i n t h e u n i t s q u a r e , t h i s m i n i -mum w i l l c e a s e t o e x i s t o n l y when (6 .16) i s v i o l a t e d . The m a x i -mum r e d u c e d f i e l d X a t w h i c h a m e t a s t a b l e s t a t e can e x i s t i s f o u n d b y m a k i n g (6 .16) a n e q u a l i t y , a n d s o l v i n g s i m u l t a n e o u s l y v / i t h ( 6 . 1 4 ) , 3y s t r a i g h t f o r w a r d c a l c u l a t i o n , a n d b y e l i m i n a t i n g X , e q u a t i o n (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. , , p ^ p l a n e : C l l , I \)(, 2(, '> W ^ 3 V~ [} s<i(1-^)1)if* s < x . ( i - ^ ) n (6,17) S i m u l t a n e o u s s o l u t i o n o f t h i s w i t h t h e p a t h e q u a t i o n g i v e s t h e p o i n t o n t h e p a t h b e y o n d w h i c h t h i s s u p e r s t a t e c a n n o t e x i s t . (6.17) c a n be s o l v e d e x p l i c i t l y s a y f o r ^2 a n c * " t h i s s u b s t i t u t e d i n t h e p a t h e q u a t i o n (6 .15) w h i c h g i v e s i n g e n e r a l a s i x t h d e g r e e e q u a t i o n f o r ^ -y , The g r a p h i c a l s o l u t i o n i n t h e ^ , ^ z ll JL-T 4 p l a n e i s e a s i l y v i s u a l i z e d and u s e f u l i n d e t e r m i n i n g t h e n a t u r e o f t h e p h a s e t r a n s i t i o n . E q u a t i o n (6.17) r e p r e s e n t s a r e c t a n g -u l a r h y p e r b o l a v/hose a s y m p t o t e s a r e r e a d i l y i d e n t i f i e d . We s h a l l r e f e r t o t h e u p p e r r i g h t h and b r a n c h o f (6.17) a s t h e " B a r r i e r C u r v e " . The b a r r i e r c u r v e h a s no p h y s i c a l m e a n i n g e x c e p t a t t h e p o i n t ( i f a n y ) w h e r e i t i s i n t e r s e c t e d b y t h e p h y s i c a l l y m e a n i n g -f u l p a t h c u r v e , i . e . w i t h i n t h e u n i t s q u a r e . C o n s i d e r t h e f a m i l y o f b a r r i e r c u r v e s f o r yU( » y^x, f i x e d , k* v a r y i n g f r o m 0 t o 1j t h a t i s , f o r t h e same l o o p s w i t h d i f f e r e n t s e p a r a t i o n s . I t i s e a s i l y shown t h a t a l l c u r v e s o f t h i s f a m i l y p a s s t h r o u g h t h e f o l l o w i n g two " P i v o t P o i n t s " i n d e -p e n d e n t o f ^ J P i v o t P o i n t P 1 P i v o t P o i n t P 2 , ^ = f //-^) , ^ - f I n t h e l i m i t k. > 0 , t h e b a r r i e r c u r v e becomes a c o r n e r w h e r e -a s i n t h e l i m i t k —*- / t h e a s y m p t o t e s r e c e d e t o 0 0 and t h e c u r v e becomes t h e s t r a i g h t l i n e : C u r v e s f o r i n t e r m e d i a t e v a l u e s o f h- c a n be v i s u a l i z e d a s a g r a d u a l b e n d i n g o f t h e c o r n e r i n t o a s t r a i g h t l i n e w i t h t h e c o n -s t r a i n t o f p a s s i n g t h r o u g h P 1 „ Pg „ S e v e r a l 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 a r e shown i n t h e u p p e r p a r t o f F i g u r e s 6-1 t o 6 - 4 , T h e s e F i g u r e s c o r r e s p o n d t o p a r t i c u l a r c a s e s t o be d i s c u s s e d i n d e t a i l s h o r t l y , I n t h e two l o o p s y s t e m t h r e e t y p e s o f p h a s e t r a n s i t i o n 0 Lk5 a r e p o s s i b l e : l ) s e c o n d o r d e r t r a n s i t i o n i n w h i c h b o t h l o o p s s i m u l t a n e o u s l y u n d e r g o a s e c o n d o r d e r t r a n s i t i o n a t some f i e l d , 2) m i x e d t r a n s i t i o n i n w h i c h one l o o p u n d e r g o e s a s e c o n d o r d e r t r a n s i t i o n v / h i l e t h e o t h e r r e m a i n s i n t h e same s u p e r s t a t e , and 3) f i r s t o r d e r t r a n s i t i o n i n w h i c h t h e s y s t e m becomes u n s t a b l e a t y, > O , 3„ > o . p > 0 •?* 1) A s e c o n d o r d e r t r a n s i t i o n o c c u r s i f t h e p a t h p a s s e s t h r o u g h t h e o r i g i n and i f t h e b a r r i e r l i e s b e h i n d (or p a s s e s t h r o u g h ) t h e o r i g i n . The l a t t e r i s t h e c a s e when $ I , ^ * I ( 6.19) and k7' ^ ( / ( 6 . 2 0 ) T h e s e t h r e e c o n d i t i o n s a r e n o t i n d e p e n d e n t } e i t h e r o f ( 6 , 1 9 ) t o -g e t h e r v / i t h ( 6 . 2 0 ) i m p l i e s t h e o t h e r o r - ( b . i y ; . f u r t h e r m o r e , i r y&4f ^ z , " 2. » c o n d i t i o n ( 6 . 2 0 ) i s a s s u r e d f o r a l l . p o s s i b l e , The p a t h w i l l p a s s t h r o u g h t h e o r i g i n i f From t h e d e f i n i t i o n o f i ^ / , e q u a t i o n ( 6 . 1 2 ) , c o n d i t i o n ( 6 . 1 9 ) . a s s u r e s ^ v\ ^ / . ^ "*"¥' d o e s n o t c o r r e s p o n d t o a p h y s i c a l l y r e a l i z a b l e s i t u a t i o n . I n p r a c t i c e , t h e r e f o r e , c o n -d i t i o n s ( 6 . 1 9 ) and ( 6 . 2 0 ) a r e n e c e s s a r y and s u f f i c i e n t f o r a s e c o n d o r d e r t r a n s i t i o n . ( M a t h e m a t i c a l l y l^,^^, ^ ~^ g i v e s r i s e t o a more c o m p l i c a t e d s i t u a t i o n i n w h i c h t h e p a t h h a s two s t a b l e b r a n c h e s one o f w h i c h s a t i s f i e s t h e c o n d i t i o n s f o r a s e c o n d o r d e r t r a n s i t i o n ) . From ( 6 . 1 4 ) e v a l u a t e d a t ^ ~ 0 t h e s e c o n d o r d e r t r a n s i t i o n o c c u r s a t X = 1 . I t i s i n t e r e s t i n g t o o b t a i n t h e s e c o n d i t i o n s u s i n g t h e s t r a i g h t f o r w a r d s e a r c h f o r s e c o n d o r d e r t r a n s i t i o n s b y e x p a n s i o n o f F i n p o w e r s o f p-t , p^ , C o r r e c t t o q u a d r a t i c t e r m s i n J l , p2. » F i s g i v e n by 2. The c r i t i c a l f i e l d i s f o u n d b y s e t t i n g t h e c o e f f i c i e n t s o f pj., , p^ e q u a l t o 0 w h i c h g i v e s X = 1 f o r e a c h l o o p . A s e c o n d o r d e r t r a n s i t i o n d o e s o c c u r i f t h e c o n d i t i o n s f o r a minimum (6.16) a r e f u l f i l l e d a t ~px. ~ 0 » X = 1 • T h i s g i v e s p r e c i s e l y (6 .19) a n d ( 6 . 2 0 ) , 2) I n some c a s e s , a m i x e d t r a n s i t i o n o c c u r s i n w h i c h one l o o p u n d e r g o e s a s e c o n d o r d e r p h a s e t r a n s i t i o n w h i l e t h e o t h e r r e m a i n s i n t h e same s u p e r c o n d u c t i n g s t a t e . To k e e p t h e d i s c u s s i o n s u c c i n c t we a p p l y t h e s u b s c r i p t s s o t h a t £y[J^ ? w h i c h i m p l i e s p, 6 1J2. . T h e n i f a m i x e d t r a n s i t i o n o c c u r s i t i s l o o p 1 w h i c h c h a n g e s p h a s e , and t h e p a t h m u s t t e r m i n a t e o n t h e 3-, a x i s a t pz. > 0 „ The b a r r i e r must l i e b e l o w t h e p a t h t e r m i n u s f o r v / h i c h t h e f o l l o w i n g two c o n d i t i o n s a r e n e c e s s a r y a n d s u f f i -c i e n t : ( i ) a s y m p t o t e l i e s t o t h e l e f t o f t h e u n i t s q u a r e s pu, (/- k?) </ ( i i ) p a t h i n t e r s e c t s t h e p^ a x i s a b o v e t h e b a r r i e r c u r v e i n t e r s e c t i o n s 1 I i 7 If yi-f+^+W*) \ 2 (, y^rl ) ( 6 o 2 i ) I t i s n o t n e c e s s a r y t h a t yUf ^ / ; t h a t i s , a l o o p w h i c h w o u l d b y i t s e l f u n d e r g o a f i r s t o r d e r t r a n s i t i o n may i n t h e p r e s e n c e o f a s e c o n d l o o p u n d e r g o a s e c o n d o r d e r t r a n s i t i o n . The c r i t i c a l v a l u e o f X f o r a m i x e d t r a n s i t i o n i s w h e r e p>~ i s t h e p a t h i n t e r c e p t g i v e n b y LHS o f ( 6 , 2 1 ) . S i n c e 1^2. ^y^x. f o r k > 0 , Xwineet  > I e x c e p t f o r t h e t r i v i a l c a s e o f u n c o u p l e d l o o p s . T h i s means t h a t l o o p 2 w i l l n o t u n d e r g o a s e c o n d o r d e r t r a n s i t i o n , i f X i s f u r t h e r i n c r e a s e d , s i n c e a s e c o n d o r d e r t r a n s i t i o n o f a s i n g l e l o o p o c c u r s a t X = 1 . Two c o u p l e d l o o p s o f t h e same r a d i u s c a n n o t u n d e r g o s u c c e s s i v e s e c o n d o r d e r t r a n s i t i o n s . T h i s j u s t i f i e s t h e t e r m " m i x e d t r a n s i t i o n " f o r t h i s c a s e . 3 ) I f t h e a b o v e c o n d i t i o n s a r e n o t f u l f i l l e d t h e n a f i r s t o r d e r  p h a s e t r a n s i t i o n w i l l o c c u r a t some p f >Q , p ^ >0 . A t l e a s t one l o o p - w i l l c h a n g e s t a t e d i s c o n t i n u o u s l y . t I n w r i t i n g down a n a l y t i c c o n d i t i o n s f o r p h a s e t r a n s i t i o n s ( w h i c h i n v o l v e n o t h i n g w o r s e t h a n q u a d r a t i c e q u a t i o n s ) we a r e d e a l i n g o n l y w i t h t h e t e r m i n a l p o i n t o f t h e p h y s i c a l l y m e a n i n g f u l p a t h . I n g e n e r a l t h e d e p e n d e n c e o f F , 1^ , l g » pf » o n X o v e r t h e r a n g e o f s t a b l e a n d m e t a s t a b l e s t a t e s i s q u i t e c o m p l i c a t e d . T h e r e a r e , h o w e v e r , t h r e e s p e c i a l c a s e s i n w h i c h t h e t w o - l o o p s y s t e m r e d u c e s t o " s t a n d a r d f o r m " so t h a t t h e r e l a t i o n s o f 1 4 8 C h a p t e r 4 c a n be a p p l i e d b y m a k i n g t h e a p p r o p r i a t e c o r r e s p o n d e n c e o f p a r a m e t e r s . S p e c i a l C a s e 1 t C o m p l e t e l y u n c o u p l e d , k - 0 C l e a r l y e a c h l o o p a c t s i n d e p e n d e n t l y and may be t r e a t e d b y t h e a n a l y s i s o f t h e p r e c e d i n g s e c t i o n . L e t us j u s t n o t e how t h e t w o - l o o p a n a l y s i s a p p e a r s i n t h i s l i m i t . a n d iSx » F s e p a r a t e s i n t o t h e sum o f two t e r m s , f u n c t i o n s r e s p e c t i v e l y o f ^ j » ^ 2 o n l y . E q u a t i o n s ( 6 , l 4 ) g i v e > i n d e p e n d e n t l y a s f u n c t i o n s o f X . We c a n s t i l l d r a w a p a t h f o r t h e s y s t e m v / h i c h may be o f a n y t y p e . The b a r r i e r c u r v e i s a c o r n e r whose arms a r e t h e i n s t a b i l i t y l i m i t s f o r p.f , s e p a r a t e l y . W h i c h -e v e r p-^l s t r i k e s t h e b a r r i e r c u r v e ( o r a n a x i s ) f i r s t d e t e r m i n e s t h e t y p e o f p h a s e t r a n s i t i o n o f t h e s y s t e m , S p e c i a l C a s e 2 s C o m p l e t e l y c o u p l e d , k, = / I f t h e l o o p s a r e n o t t o o d i f f e r e n t , t h e n s i n c e L 'de-A p e n d s l o g a r i t h m i c a l l y o n Jr t h e c r o s s - s e c t i o n r a d i u s ( f o r a c i r c u l a r c r o s s - s e c t i o n ) , we may p u t a p p r o x i m a t e l y / / ^ ^-a a n d \h>,\} l ^ 3,1 « i . The p a t h l i e s v e r y n e a r t h e d i a g o n a l and Jf • A p p r o x i m a t i n g t h e f r e e e n e r g y F u s i n g t h e r e l a t i o n s L)t ~ ^ O y- J y 2_ 9 1 U<Aln 7} +f x w h i c h i s t h e s t a n d a r d f o r m o f F w i t h yU ~^Mf . I n t h e p-l * p l a n e t h e b a r r i e r becomes t h e s t r a i g h t l i n e (6,18) a n d i n t e r s e c t s t h e d i a g o n a l a s e x p e c t e d a t l>9 We n o t e i n p a r t i c u l a r t h e c a s e o f a v e r y t h i n l o o p c l o s e l y c o u p l e d t o a t h i c k e r o n e , i . e . t & ~* / . The b e h a v i o u r o f t h e s y s t e m i s d o m i n a t e d b y l o o p 2 , ^ t a k e s o n t h e v a l u e o f r a t h e r t h a n a v a l u e d e p e n d e n t o n t h e p r o p e r t i e s o f l o o p 1 w h i l e ^2 a p p r o x i m a t e s t h e v a l u e i t w o u l d h a v e i n t h e a b s e n c e o f l o o p 1 . S p e c i a l C a s e 3 * I d e n t i c a l l o o p s » ^/ ~ ^z. C o n s i d e r t h e r e d u c e d f r e e e n e r g y e q u a t i o n ( 6 . 1 3 ) f o r i d e n t i c a l l o o p s . I f we make a chan g e o f v a r i a b l e s & *"p2. •> //- ~p., ~p~2_ 8 t h e n one f i n d s t h a t F i s a n e v e n f u n c t i o n o f AT , and t h a t a n e c e s s a r y c o n d i t i o n f o r a m i n i m u m o f F i s /f~~ 0 , o r pf ~ o T h i s i s i n a g r e e m e n t w i t h t h e p a t h e q u a t i o n w h i c h i s c e r t a i n l y s a t i s f i e d ~J/z » ^ u t " h i c h h a s o t h e r n o n - p h y s i c a l b r a n c h e s f o r w h i c h t h i s I s n o t t r u e . S u b -s t i t u t i n g p, - p^ S J ^ . i n t o t h e e q u a t i o n f o r F and s i m p l i f y i n g y i e l d s F - -Z? + f + y -( 2 / w h i c h i s j u s t t h e s t a n d a r d f o r m o f F w i t h y U . -ytff (/ j-k) I n t h e p.i , p t ^ p l a n e t h e p a t h i s t h e d i a g o n a l w h e r e a s t h e b a r r i e r c u r v e i s s y m m e t r i c w , r , t , t h e d i a g o n a l . P u t t i n g p t ~J'2-i n ( 6 . 1 7 ) t h e b a r r i e r i n t e r s e c t s t h e d i a g o n a l a s e x p e c t e d a t i t < \ 3 i /<i(i+k)) 150 We n o t e t h a t t wo i d e n t i c a l l o o p s h a v i n g aT / ^ i ^  1 j ^ ^y^z I w i l l u n d e r g o a s e c o n d o r d e r p h a s e c h a n g e when u n c o u p l e d , a n d a f i r s t o r d e r c h a n g e a t some s u f f i c i e n t c o u p l i n g . L e t u s e x t e n d t h i s d i s c u s s i o n t o two n e a r l y i d e n t i c a l l o o p s yU.( ytfx f o r t h e caseyX t >f ,y/Sz> I . We ha v e /, ~ , 2^ - &ytf;(/-k) . An e x a m p l e o f t h i s c a s e i s i l l u s t r a t e d i n F i g u r e 6-1 f o r y U f - ZS , ^ = ^ . The p a t h s ( w i t h a r r o w s ) and h a r r i e r s a r e shown i n 6 - l a f o r c o u p l i n g s A - 0 , . *r , /.0 „ The p i v o t p o i n t s a r e e n c l o s e d i n s m a l l c i r c l e s ® a n d t h e p h a s e t r a n s i t i o n p o i n t s a r e m a r k e d w i t h s o l i d d o t s 5 t h u s . The f u n c t i o n ^ ^ ' ^ c o r r e s p o n d i n g t o t h e s e t h r e e p a t h s i s shown i n F i g u r e 6 - l b . F o r s m a l l A , t h e p a t h f o l l o w s t h e d i a g o n a l c l o s e l y down t o ^  ~ 3 (I ~ 1 ' ^1) , t h e n b e n d s u p w a r d . The more n e a r l y i d e n t i c a l t h e l o o p s t h e s h a r p e r t h e b e n d . When & i i n -c r e a s e s s u f f i c i e n t l y t o make ft *- ' , t h e p a t h l i e s c l o s e t o t h e d i a g o n a l down t o t h e o r i g i n . The b a r r i e r i n t e r s e c t s t h e d i a g o n a l a t ^ ^ 3T (l " ''/^l) w h i c h i s p r e c i s e l y a t t h e minimum o f ^ L , . C o n s e q u e n t l y , t h e p h y s i c a l l y m e a n i n g f u l p a r t o f t h e p a t h l i e s c l o s e t o t h e d i a g o n a l f o r a l l „ The f u n c t i o n a l r e l a t i o n s a p p l i c a b l e t o m a t h e m a t i c a l l y i d e n t i c a l l o o p s a r e good a p p r o x i -m a t i o n s t o t h e f u n c t i o n a l r e l a t i o n s f o r p h y s i c a l l y s i m i l a r l o o p s . W h i l e many c o n f i g u r a t i o n s w i l l f a l l i n t o o r n e a r one o f t h e s e t h r e e s p e c i a l c a s e s t h e r e r e m a i n a l a r g e number o f p o s s i -b i l i t i e s r o u g h l y c a t e g o r i z e d a s " d i s - s i m i l a r l o o p s o f medium c o u p l i n g " . The c a s e yUf </Mx< / has a l r e a d y b e e n c o v e r e d i n t h e d i s c u s s i o n o f s e c o n d o r d e r p h a s e t r a n s i t i o n s . We c o n s i d e r f i n a l l y t h r e e e x a m p l e s h a v i n g ytf^ y I , and c l a s s i f i e d a s ( i ) /() < I , ( i i ) JS, £ / , ( i i i ) / i , » I . The y a r e 1 5 1 i l l u s t r a t e d i n F i g u r e s 6-2, 6-3» and 6-4 r e s p e c t i v e l y , o f w h i c h p a r t ( a ) g i v e s t h e p a t h s a nd b a r r i e r s i n t h e , ^ 2 p l a n e , and p a r t ( b ) g i v e s X/S/) . E x a m p l e ( i ) , F i g u r e 6-2, yUf — . sT ^ S^~*- ~ ® A t k ~ 0 , t h e p a t h d e c r e a s e s m o n o t o n i c a l l y i n t e r c e p t i n g t h e a x i s a t ~ .7^ . A m i x e d p h a s e t r a n s i t i o n o c c u r s a t X = 1 , a t w h i c h p o i n t l o o p 1 becomes n o r m a l . A s k, i s i n -c r e a s e d t h e i n t e r c e p t d e c r e a s e s b e c o m i n g 0 a t k. - * 7S" , , W i t h i n t h e u n i t s q u a r e t h e b a r r i e r v a r i e s l i t t l e f r o m t h e v e r t i -c a l l i n e a t ^ x ~• $~ . F o r up t o a b o u t . 5 5 a m i x e d t r a n s -i t i o n o c c u r s a t X ? I • g r e a t e r A- a f i r s t o r d e r t r a n s i t i o n o c c u r s , E x a m p l e ( i i ) , F i g u r e 6-3, - A 2 } yJ^ --' /€> A t k. ~ 0 , t h e p a t h d e c r e a s e s m o n o t o n i c a l l y i n t e r c e p t i n g t h e pz a x i s a t .97 , As c o u p l i n g i s i n c r e a s e d t h e p% i n t e r c e p t d e c r e a s e s r e a c h i n g 0 when k. — . f „ The b a r r i e r a t It — 0 i s t h e c o r n e r whose h o r i z o n t a l arm i s a t ' C o n s e q u e n t l y a t k. ~ 0 a f i r s t o r d e r t r a n s i t i o n o c c u r s when t h e p a t h s t r i k e s t h i s b a r r i e r . A s k- i s i n c r e a s e d t h e b a r r i e r s w i n g s c u t o f t h e way a n d f o r a c e r t a i n r a n g e o f k/ a m i x e d t r a n s i t i o n o c c u r s i n w h i c h ^ g o e s c o n t i n u o u s l y t o 0 a t some X > / . A s k*> c o n -t i n u e s t o i n c r e a s e t h e p a t h d r o p s u n t i l i t a g a i n s t r i k e s t h e b a r r i e r w i t h i n t h e u n i t s q u a r e , a n d a f i r s t o r d e r t r a n s i t i o n o c c u r s up t o k ~ 1 152 E x a m p l e ( i i i ) , F i g u r e 6-4, - J yrf^ - JO A t k = 0 , a f i r s t o r d e r t r a n s i t i o n o c c u r s a t ^ — ~5 (I~ tfa') As i n c r e a s e s t h e b a r r i e r s w i n g s down b u t n o t o u t , and t h e c r i t i c a l v a l u e o f d e c r e a s e s b u t n e v e r t o 0. A s c o n t i n -u e s t o i n c r e a s e t h e p a t h l o w e r s and t h e c r i t i c a l v a l u e o f ^ f i n c r e a s e s t o A f i r s t o r d e r t r a n s i t i o n o c c u r s f o r a l l k, . T h i s t e r m i n a t e s t h e d i s c u s s i o n o f two l o o p s y s t e m s . S e c t i o n 4 - Many C o u p l e d L o o p s C o n s i d e r N c i r c u l a r l o o p s w i t h t h e same r a d i u s R p l a c e d c o - a x i a l l y w i t h a x e s p a r a l l e l t o t h e u n i f o r m e x t e r n a l f i e l d H e . The W*^ l o o p h a s c r o s s - s e c t i o n a r e a /fc , and 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 h a s l o o p c u r r e n t Ij, and r e l a t i v e s u p e r - e l e c t r o n d e n s i t y ^ 2 , The s e l f - i n d u c t a n c e o f A. l o o p i s Mj,j. a n d t h e m u t u a l i n d u c t a n c e o f t h e z a n d j.*4* l o o p s f i s M*±. , M:J i s a n N s q u a r e s y m m e t r i c m a t r i x . The t h r e e t e r m s c o n t r i b u t i n g t o t h e f r e e e n e r g y 7 a r e s C o n d e n s a t i o n e n e r g y neb i *P R) x. ~ i M a g n e t i c f i e l d e n e r g y N N I S u p e r - e l e c t r o n K i n e t i c E n e r g y . . •» . M - 2. The f l u x o i d <p> o f t h e S l o o p i s 1 5 3 *4 A s i n t h e p r e v i o u s s e c t i o n we r e s t r i c t t h e d i s c u s s i o n t o t h e c a s e o f i d e n t i c a l f l u x o i d s i n e a c h l o o p w h i c h i s p h y s i c a l l y r e a s o n a b l e a s d i s c u s s e d p r e v i o u s l y . We u s e e x a c t l y t h e same d e f i n i t i o n s f o r r e d u c e d f l u x o i d <|> a n d r e d u c e d f i e l d v a r i a b l e X a s b e f o r e : ( H * - § ) R s u*p == ~£-The r e d u c e d f r e e e n e r g y F i s r e l a t e d t o -T by r - V- 7 / HcfR Y_ Aj, The yj. p a r a m e t e r i s g e n e r a l i z e d t o t h e N l o o p c a s e by d e f i n -i n g m a t r i x e l e m e n t s J^A^ °f a n N - s q u a r e m a t r i x fo Mu N o t e t h a t ^MA^ ^ s ^Qt s y m m e t r i c . The d i a g o n a l e l e m e n t s a r e a n a l o g o u s t o t h e yU- 's u s e d i n t h e 2 l o o p d i s c u s s i o n , The c o u p l i n g : c o n s t a n t kjj ~ kjZ b e t w e e n t h e A*^ and ^'"^ l o o p I t i s c o n v e n i e n t t o d e f i n e t h e m a t r i x w i t h e l e m e n t Ftc'^, T Cli. Lj-r i s n o t a p a r a m e t e r s i n c e i t c o n t a i n s t h e v a r i a b l e The d e t e r m i n a n t o f t h e m a t r i x Rc^ i s \P\ , a n d t h e c o - f a c t o r o f i s P't . T h e n a d d i n g t h e t h r e e t e r m s i n 7 and t a l c i n g a c c o u n t o f a l l t h e p r e c e d i n g e q u a t i o n s t h e r e d u c e d f r e e e n e r g y F i s f o u n d t o be k--l We l i m i t t h e i n v e s t i g a t i o n o f t h i s r a t h e r c o m p l i c a t e d e x p r e s s i o n t o t h e p r o b l e m o f f i n d i n g t h e c r i t i c a l f i e l d f o r s e c o n d o r d e r p h a s e t r a n s i t i o n s o f t h e s y s t e m and t h e c o n d i t i o n s u n d e r w h i c h s e c o n d o r d e r t r a n s i t i o n s o c c u r . By t h e e x p r e s s i o n " s e c o n d o r d e r t r a n s i t i o n o f t h e s y s t e m " we mean a l l ^  go t o z e r o c o n t i n u -ZMIC< h r f a - v t i H c i w n I t o w p A i i c < I i r W o r o n n T r n T f i P p v - n Q v i q *i o71 ^ t' h' n -n *f- r\ q u a d r a t i c t e r m s i n t h e w h i c h i s The c r i t i c a l f i e l d f o r s e c o n d o r d e r t r a n s i t i o n s i s f o u n d b y s e t t i n g t h e c o e f f i c i e n t o f e a c h ^ e q u a l 0 . T h i s g i v e s X = l f o r e v e r y l o o p . I t i s w o r t h e m p h a s i z i n g t h e e x t r e m e g e n e r a l i t y o f t h i s r e s u l t . I n e v e r y g e o m e t r y v/e ha v e c o n s i d e r e d X h a s e x a c t l y t h e same d e f i n i t i o n a n d t h e s e c o n d o r d e r c r i t i c a l v a l u e o f X i s a l w a y s 1. I n o r d e r t o d e t e r m i n e w h e t h e r a s e c o n d o r d e r t r a n s i t i o n d o e s o c c u r we r e q u i r e a l l t h e s e c o n d p a r t i a l d e r i v a -t i v e s o f F e v a l u a t e d a t ^ ~ 0 „ X = 1 . T h e s e a r e s 155 {/-/<#) > = 2 ^ / < . ^ ) 7-' T h e n a s e c o n d o r d e r t r a n s i t i o n o c c u r s i f F i s a minimum w . r . t , a l l p^ a t J i — ? > 0 » X = 1 . T h i s c o n d i t i o n may be e x p r e s s e d i n t e r m s o f d e t e r m i n a t e s i n v o l v i n g t h e s e c o n d p a r t i a l d e r i v a t i v e s , R e m o v i n g t h e common f a c t o r s we s t a t e t h e c o n d i t i o n a s a p p l i e d t o t h e f o l l o w i n g m a t r i x : / The n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t F be a minimum' i s t h a t e v e r y p r i n c i p a l m i n o r o f t h e m a t r i x be p o s i t i v e , t h a t i s y~ " v > 0 > a The c o n d i t i o n s f o r one l o o p and two l o o p s a r e r e c o g n i z e d , 0 .> AO A? 1.° F i g u r e 6-1 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 Above: P a t h s ( w i t h arrows) and b a r r i e r s i n p l a n e . Below: D e p r e s s i o n o f super e l e c t r o n d e n s i t y w i t h r e d u c e d f i e l d X. F i g u r e 6 - 2 Above! Below s , 5 * /.0 A>~ ^ 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 yM/ = o.€~) yj^ = 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 ^ - 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 r e d u c e d f i e l d X . / ' . r AC /-sr i o z f 3.o k 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, = +} /ix = /O 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 p,-fz 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 ^ w i t h r e d u c e d f i e l d X . ' 156 CHAPTER 7 COMPARISON OF THE LITERATURE WITH PRESENT RESULTS I n t h i s f i n a l c h a p t e r we s u m m a r i z e t h e r e s u l t s o b -t a i n e d a n d n o t e t h o s e r e s u l t s o f o t h e r w o r k e r s w h i c h have s e r v e d a s p o i n t s o f d e p a r t u r e o r a s p o i n t s o f c o m p a r i s o n f o r t h e c a l c u l a t i o n s made h e r e . C h a p t e r s l and 2 a r e p r i m a r i l y b a c k g r o u n d a n d r e v i e w , The f i r s t c h a p t e r i s b a s e d o n t h e o r i g i n a l p a p e r o f G-L (1950) w h i l e t h e s e c o n d i s b a s e d l a r g e l y o n t h e w o r k o f G i n z b u r g ( 1958) . The l a t t e r d i s c u s s e s c r i t i c a l f i e l d s f o r s m a l l s i m p l y - c o n n e c t e d b o d i e s w i t h u n i f o r m fy/ , K u s n e z o v (1963) d i s c u s s e s c r i t i c a l f i e l d s i n t h i n f i l m s a n d f i l a m e n t s , b u t h i s t r e a t m e n t o f t h e v a l u e o f I'jsl i s i n a d e q u a t e , and t h i s p a p e r a d d s l i t t l e t o t h a t o f G i n z b u r g . M a u s e r a n d H e I f a n d (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 t h e n o b t a i n some c o r r e c t i o n s t o t h e c r i t i c a l f i e l d s due t o n o n -l o c a l e f f e c t s , w h i c h t h e y c a l c u l a t e f r o m t h e m i c r o s c o p i c t h e o r y . The t r e a t m e n t g i v e n h e r e s h o u l d c l a r i f y c e r t a i n p o i n t s , i n p a r t i c u l a r t h e e q u i v a l e n c e 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-n e t i c f r e e e n e r g y and e x a c t l y w h a t a p p r o x i m a t i o n s a r e b e i n g made i n e a c h e x p r e s s i o n . The d i r e c t p r o c e d u r e f o r t h e c a l c u l -a t i o n o f s e c o n d o r d e r c r i t i c a l f i e l d c r i t e r i a , n o t r e q u i r i n g t h e s o l u t i o n o f t h e f i e l d e q u a t i o n s , a p p e a r s t o be g i v e n h e r e f o r t h e f i r s t t i m e , L o n d o n (1950) r e c o g n i z e d t h e i m p o r t a n c e o f t h e q u a n t -i t y w h i c h he c a l l e d t h e f l u x o i d a n d r e m a r k e d t h a t i t s h o u l d be q u a n t i z e d . D e a v e r a nd F a i r b a n k (1961) and D o l l a n d N a b a u e r 1.57 (1961) f i r s t o b s e r v e d e x p e r i m e n t a l l y t h e q u a n t i z a t i o n o f f l u x i n m i c r o - c y l i n d e r s . S e v e r a l t h e o r e t i c a l p a p e r s f o l l o w e d . B y e r s a n d Y a n g (1961) showed t h a t f o r a c o n t o u r deep i n s i d e a t h i c k r i n g t h e most p r o b a b l e s t a t e s a r e t h o s e h a v i n g q u a n t i z e d a m o u n t s o f e n c l o s e d f l u x , B o h r a n d M o t t e l s o n (1962) h a v e , f u r t h e r d i s c u s s e d t h e e n e r g y s t a b i l i t y o f t h e s e s t a t e s . Com-b i n i n g f l u x o i d q u a n t i z a t i o n w i t h t h e L o n d o n t h e o r y , L i p k i n e t a l (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 e n c l o s e d b y a l o n g c i r c u l a r c y l i n d e r i n t e r m s o f f l u x o i d a n d e x t e r n a l f i e l d , K e l l e r a n d Zumino (1961) showed t h a t t h e G-L e q u a t i o n s f o r t h e e f f e c t i v e wave f u n c t i o n l e a d i n e q u i l i b r i u m t o q u a n t i -z a t i o n o f t h e f l u x o i d . The a u t h o r f e e l s t h a t t h i s s h o u l d n o t be c o n s i d e r e d a p r o o f o f q u a n t i z a t i o n , b u t r a t h e r a d e m o n s t r a -t i o n t h a t t h e G-L t h e o r y i s c o n s i s t e n t w i t h t h e m i c r o s c o p i c t h e o r y , a n d may t h u s be u s e d t o o b t a i n t h e r m o d y n a m i c r e s u l t s , G i n z b u r g (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 d i s t r i b u t i o n i n a c i r c u l a r c y l i n d e r o n t h e " l o w - f i e l d " a s s u m p t i o n o f He g i v e s some a p p r o x i m a t e v a l u e s o f e n c l o s e d f l u x f o r p a r t i c u l a r c y l i n d e r s . A l l t h e a b o v e m e n t i o n e d p a p e r s h a v e assumed u n i f o r m and f i e l d - i n d e p e n d e n t s u p e r - e l e c t r o n d e n s i t y t h r o u g h o u t . From t h e f i r s t G-L e q u a t i o n t h i s c a n n e v e r be more t h a n a n a p p r o x i -m a t i o n . I n C h a p t e r 3 we ha v e k e p t t h e d i s c u s s i o n v e r y general„ p o s t p o n i n g a n y a p p r o x i m a t i o n s a s f a r a s p o s s i b l e . I n t h i s way v/e a r e a b l e t o make c e r t a i n g e n e r a l s t a t e m e n t s , a n d i n a n y p a r t i c u l a r c a s e t o p r o c e e d s y s t e m a t i c a l l y t o r e q u i r e d a c c u r a c y . We n o t e t h e f o l l o w i n g r e s u l t s s ( i ) The v a r i o u s e x p r e s s i o n s f o r i n e q u a t i o n s (3»3)» ( 3 . 9 ) . ( 3 . 8 ) d o n o t r e q u i r e u n i f o r m lip/ n o r u n i f o r m H G . E q u a t i o n (3=5) w h i c h h a s a p p e a r e d p r e v i o u s l y i n t h e l i t e r a t u r e i s v a l i d o n l y f o r u n i f o r m H E . ( i i ) E q u a t i o n ( 3 . 9 ) , a p p a r e n t l y new h e r e , i s t h e e s s e n t i a l s t a r t i n g p o i n t . f o r t h e e x p a n s i o n o f ^ i n p o w e r s o f ^  f r o m w h i c h s e c o n d o r d e r p h a s e c r i t e r i a a r e d e t e r m i n e d w i t h o u t p r i o r s o l u t i o n o f t h e -G-L f i e l d e q u a t i o n s . The c o e f f i c i e n t s o f p. a r e e x p r e s s e d a s d e f i n i t e i n t e g r a l s o v e r t h e b o d y o n l y . T h i s m e t h o d i s a p p l i c a b l e a l s o t o n o n - u n i f o r m H E . ( i i i ) The m a g n e t i c f r e e e n e r g y o f a l o n g h o l l o w c y l i n d e r i s g i v e n b y e q u a t i o n (3 .19) i n a f o r m w h i c h d o e s n o t assume u n i -f o r m /(pl . T i l l s f o r m i s e s s e n t i a l f o r t h e d e t e r m i n a t i o n o f t h e e r r o r i n v o l v e d i n i g n o r i n g t h e f i n i t e c o h e r e n c e l e n g t h . ( i v ) I n e q u a t i o n (3 ,21) t h e f r e e e n e r g y o f a h o l l o w c y l i n d e r i s g i v e n i n a f o r m w h i c h i s m a n i f e s t l y p o s i t i v e d e f i n i t e i n H and I t h e IOOD c u r r e n t n e r u n i t l e n g t h . I f H i s f i x e d , e c 0 e t h e n t h e f l u x o i d s t a t e o f l o w e s t e n e r g y i s t h a t h a v i n g t h e l o w e s t l o o p c u r r e n t , f o r a l l c y l i n d e r s . ( v ) The d e p e n d e n c e o f v a r i o u s q u a n t i t i e s o n t h e g e o m e t r y o f t h e c y l i n d e r h a s b e e n s y s t e m a t i c a l l y e x p r e s s e d i n t e r m s o f o n l y t h r e e " b o u n d a r y f a c t o r s " w h i c h may be e x p a n d e d o n c e a nd f o r a l l i n p o w e r s o f o r (<///[) t o g i v e a n y r e q u i r e d d e g r e e (/ o f a c c u r a c y . F o r a h o l l o w c y l i n d e r i n t h e p a r t i c u l a r s t a t e o f z e r o f l u x o i d , L u n g - T a o and Z h a r k o v (1963) h a v e c a l c u l a t e d t h e f r e e e n e r g y and u s i n g t h e p h a s e t r a n s i t i o n c r i t e r i a o f G i n z b u r g (1958) have d e t e r m i n e d t h e c r i t i c a l f i e l d s . T h e i r r e s u l t s , 1 a l t h o u g h l e s s g e n e r a l , a r e i n a g r e e m e n t w i t h o u r s . I n C h a p t e r 4 we d i r e c t t h e d i s c u s s i o n t o t h i n c y l i n -d e r s a n d o b t a i n c u r v e s o f t h e f u n c t i o n s F r e e E n e r g y F ( X ) , R e l a t i v e S u p e r - E l e c t r o n D e n s i t y ^ ^ ) > M a g n e t i c Moment M(X) and S u p e r - E l e c t r o n Momentum P(X) , wh e r e X i s a g e n e r a l i z e d v a r i a b l e p r o p o r t i o n a l t o " S i g n i f i c a n t p o i n t s o n t h e c u r v e s a r e i d e n t i f i e d and i n t e r p r e t e d i n t e r m s o f p h a s e t r a n s i t i o n c r i t e r i a . L o c i o f t h e s i g n i f i c a n t p o i n t s a s t h e " g e o m e t r y p a r a m e t e r " y U - v a r i e s a r e g i v e n o n t h e p l o t s . By a p p r o x i m a t i n g t h e f i e l d e q u a t i o n s f o r t h i n n e s s , D o u g l a s s (1963) h a s o b t a i n e d e q u a t i o n s e q u i v a l e n t t o (4.9) a n d (4 . 1 1 ) , and c u r v e s e q u i v a l e n t t o t h o s e g i v e n h e r e f o r t h e f u n c t i o n s F ( X ) , and M ( X ) f o r two p a r t i c u l a r v a l u e s o f y U * . The c a l c u l a t i o n s o f D o u g l a s s a r e d i f f i c u l t t o f o l l o w a s t h e r e a r e a r e i n a g r e e m e n t w i t h o u r s . I n p a r t 2 o f C h a p t e r 4 we have d i s c u s s e d t h e e x t e n -s i o n o f t h e c u r v e s F ( X ) e t c . t o s u r f a c e s F ( X , t ) w h e r e t i s t h e r e l a t i v e t e m p e r a t u r e . T i n k h a m (1964) h a s shown f o r v e r y t h i n c y l i n d e r s i n w h i c h t h e p h a s e t r a n s i t i o n i s c e r t a i n l y s e c o n d o r d e r t h a t Xi ^ o n t h e c r i t i c a l c u r v e . He u s e s t h i s r e s u l t t o d i s c u s s t h e e x p e r i m e n t o f L i t t l e and P a r k s (1962) who f o u n d t h a t v e r y n e a r t h e c r i t i c a l t e m p e r a t u r e t h e c y l i n d e r a p p e a r s t o go i n t o t h e quantum s t a t e o f l o w e s t f r e e e n e r g y . More r e c e n t e x p e r i m e n t s o n s u c h c y l i n d e r s h a v e b e e n ' r e p o r t e d b y M c L a c h l a n (1969). We h a v e h e r e o b t a i n e d c u r v e s o f t h e q u a n t i t i e s F , , M and P a s f u n c t i o n s o f t e m p e r a t u r e . We h a v e g i v e n 160 t h e c r i t e r i o n f t o r s e c o n d o r d e r p h a s e t r a n s i t i o n s a t c o n s t a n t f i e l d , i n c r e a s & a a g t e m p e r a t u r e . The e x p e c t e d b e h a v i o u r o f t h e m a g n e t i c moment: -of a c y l i n d e r i n a p e r s i s t e n t c u r r e n t s t a t e i n z e r o e x t e r n a l f i e l d i s c a l c u l a t e d f o r t e m p e r a t u r e v a r y i n g f r o m t = 0 t o t — 1 . The r e s u l t i n g r e v e r s i b l e c u r v e s a nd i r r e -v e r s i b l e " c u r v e " ' ( r e a l l y a s e t o f c l o s e l y s p a c e d p o i n t s ) a r e s i m i l a r t o t h e ( e x p e r i m e n t a l r e s u l t s o f H u n t and M e r c e r e a u (1964), l e a d i n g t o the;- ( c o n c l u s i o n t h a t t h e c y l i n d e r 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 s t a t e s n e a r t h e m e t a s t a b l e l i m i t . I n C h a p t e r 5 "by u s i n g t h e g e n e r a l r e s u l t s o f C h a p t e r 3 t h e a n a l y s i s aff t h i n c y l i n d e r s i s e x t e n d e d t o t a k e a c c o u n t o f t h e e f f e c t o f M g h f l u x o i d number and f i n i t e c o h e r e n c e l e n g t h . To d e t e r m i n e t;lae c r i t e r i a f o r s e c o n d o r d e r p h a s e t r a n s i t i o n , we u s e two t e r m s o f t h e e x p a n s i o n o f F i n p o w e r s o f p* g i v e n by e q u a t i o n (3-28) v / h i c h makes no r e s t r i c t i o n o n n . The f i r s t c o e f f i c i e n t w h i c h d e t e r m i n e s t h e c r i t i c a l f i e l d h a s b e e n f o u n d b y G r o f f a n d B a r k s (1968) by 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 . We show t h a t a l l t h i n (ct« A ) c y l i n d e r s u n d e r g o s e c o n d o r d e r t r a n s i t i o n s a t s u f f i c i e n t l y h i g h n . E x p a n s i o n s o f t h e B o u n -d a r y F a c t o r s fco t h i r d o r d e r i n U m h a v e b e e n u s e d t o show t h a t a t h i g h f l u x o i d numbers t h e c u r v e s o f F ( X ) e t c , a r e s i m i l a r t o t h o s e a t l o w n , b u t become c o m p r e s s e d a n d more " s e c o n d o r d e r l i k e " a t h i g h e r n . The maximum p o s s i b l e n i s g i v e n b y e q u a t i o n (5»5)« I n p a r t 2 o f t h i s c h a p t e r a n i n v e s t i g a t i o n o f t h e e f f e c t o f f i n i t e c o h e r e n c e l e n g t h i s c a r r i e d o u t , I t i s f o u n d t h a t t h e f i r s t c o r r e c t i o n t e r m i s o f o r d e r A n e s s e n -t i a l p r e r e q u i s i t e t o t h i s c a l c u l a t i o n i s e q u a t i o n (3°19) f o r l 6 i ~3y w h i c h d o e s n o t assume u n i f o r m ly>\ . A l l o f C h a p t e r 6 i s new h e r e . We have shown t h a t t h e f u n c t i o n a l r e l a t i o n s h i p s i n t h e t h i n c y l i n d e r may he a d o p t e d t o o t h e r t h i n 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 b y m a k i n g t h e a p p r o p r i a t e c o r r e s p o n d e n c e o f p a r a m e t e r s . T h e n s y s t e m s o f two c o - a x i a l m a g n e t i c a l l y c o u p l e d l o o p s w e r e a n a l y z e d and i t v/as shown t h a t c r i t i c a l p o i n t s may be f o u n d g r a p h i c a l l y i n t h e b a r r i e r c u r v e . T r a n s i t i o n s may be o f t h r e e t y p e s ( i ) b o t h l o o p s s e c o n d o r d e r o c c u r r i n g a t X = 1 i ( i i ) m i x e d ( i . e . one l o o p s e c o n d o r d e r ) , o c c u r r i n g a t X > 1 and ( i i i ) b o t h l o o p s f i r s t o r d e r . The p r e s e n c e o f t h e s e c o n d l o o p a f f e c t s t h e c r i t i -c a l c o n d i t i o n s i n t h e f i r s t , a n d may e v e n c h a n g e t h e o r d e r o f t h e t r a n s i t i o n i n e i t h e r s e n s e . F i n a l l y c r i t e r i a h a v e b e e n o b t a i n e d f o r s e c o n d o r d e r t r a n s i t i o n s i n a n y number o f s i m i l a r c o - a x i a l l o o p s . I n a l l s e c o n d o r d e r t r a n s i t i o n s t h e r e d u c e d c r i t i c a l f i e l d i s X = 1 . p l a n e a t t h e i n t e r s e c t i o n o f a p a t h c u r v e a n d a 162 BIBLIOGRAPHY-B o h r , A,, and M o t t e l s o n , 1962. P h y s , R e v , 1_2J), 495. B y e r s , N,, and Y a n g , C.N., 1961, P h y s , R e v , L e t t . 2» 46 . C h r i s t i a n s e n , P.V., and S m i t h , H., 1968. P h y s , R e v . 121, 445, D e a v e r , B . S . J r , , a nd F a i r b a n k , W.M., 1961. P h y s , R e v , L e t t , 2» ^3» D o l l , R., and N a b a u e r , M., 1961. P h y s . R e v , L e t t , 2» 51 . D o u g l a s s , D . H . J r . , 1963. P h y s . R e v , 132, 513. F i n k , H , J . , and P r e s s o n , A.G., 1966. P h y s , R e v , 1J5JL, 219. de G e n n e s , P.G., 1964, R e v . Mod, P h y s . 36 , 225. 1 9 6 6 , s u p e r c o n d u c t i v i t y o r i v i e t a i s a n d . A l l o y s , B e n j a m i n , New Y o r k , G i n z b u r g , V.L., 1958. S o v i e t P h y s . J E T P £» ?8. 1962, S o v i e t P h y s . J E T P 1J5, 207. and 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 , 1965) . G r o f f , R.P., a n d P a r k s , R.D., 1968. P h y s , R e v . I2i» 567. H a u s e r , J . J . , a n d H e l f a n d , E., 1962. P h y s . R e v . 12?, 386, H u n t , T.K., a n d M e r c e r e a u , J . E . , 1964. P h y s . Rev. 135A, 944. K e l l e r , J . B . , a n d Z u m i n o , B., 1961. P h y s , R e v , L e t t , 2» 1 o ^» K u p e r , C G . A n I n t r o d u c t i o n t o t h 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 . 1968, C l a r e n d e n P r e s s , O x f o r d . K u s n e z o v , N., 1963. P h y s . R e v . 1^0, 2253. L a n d a u , L.D., and L i f s h i t z , E.M., 1959. S t a t i s t i c a l P h y s i c s , P e r g a m o n P r e s s , L o n d o n , 163 L i p k i n , K . J . , P e s h k i n , M., and T a s s i e , L . J . , 1962. P h y s . R e v . 126, 116. L i t t l e , W.A., and P a r k s , R ,D., 1962. P h y s . R e v . L e t t . £, 9 . L o n d o n , F,, 1950' S u p e r f l u i d s V o l . I : M a c r o s c o p i c 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 . W i l e y , New Y o r k . L u n g - T a o , H , , and Z h a r k o v , G.F., 1963. S o v i e t P h y s , J E T P r?_, 1426, M c L a c h l a n , D.S., 1969. P h y s , R e v . L e t t , 21* 1434. M e i s s n e r , W., and O c h s e n f e l d , R., 1933» N a t u r w i s s e n s c h a f t e n 21, 787. S h i k i n , V.B., 1969. S o v i e t P h y s . J E T P 29_, 902. T i n k h a m , M,, 1964, R e v , Mod. P h y s . 3 6 , 268. APPENDIX 1 SUMMARY OF FORMS OF THE GINZBURG-LANDAU FREE ENERGY. Notati o n : C a p i t a l i s used, f o r the dimensional " e f f e c t i v e wave f u n c t i o n , " i small (p i s used f o r the 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 the free energy density i n i t i a l l y i n the form: (H~He)' Expanding the absolute value bracket S u b s t i t u t i n g f o r Igh — <jJ {jj^ , where (j) i s the r e l a t i v e wave f u n c t i o n , (p^ — 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 d e n s i t y , z 2m 2mC 2r*cx S u b s t i t u t i n g ijJ^ — & I t (equation ( 1 . 2 ) ) Define A M u l t i p l y i n g through by 8^' } we have f o r _ *7r*L tiff + w^7<^/% ±lA\ *.2iVf'f+ +7K>kfx(fvy-yvtp^-A + w l v f / f ^ (H-Hef T T 77A1" r A I L - fa Zmo< = 2HJ f Hence. A A J - ^ [ " I Hob I f / V ^ / V / V f W V & fvVy -~lf>V<f>*)'A + He) J ,1V Space J Reduced Free Energy 3 where = volume of body Vi JAII Space Reduced Free Energy with amplitude and phase of (JJ expressed e x p l i c i t l y = e } < W (7) (sZ) teal The* Ivy/*'- (vyf + yl(v<r>) } x' ((/j*vtjj - yj vy*) ~ -2yzVa~) Space u _ — , / » y / "Condensation Fr&e- trner^f Et^ctromao^eHc Free- Energy Fc F H S p e c i a l Cases: 1) (jJ i s r e a l , a p p l i c a b l e to simply-connected bodies w i t h A i n London gauge. Spa.ce 2) uniform, a p p l i c a b l e to small bodies. Define 3) Doubly connected bodieSj (p of the form i n c y l i n d r i c a l co-ordinates ip = y e i—1 —3 /e. a) ^ n $ } v<7) = A } HcbfX = E= $o Spa.ce Dimensionless form. Notation: primes f o r "dimensionless" q u a n t i t i e s . A A l l lengths measured r e l a t i v e to A : f'-=- f/\ - '/K \7 ~ ^' o/V~ A d^ Define / r ' s / r / ^ „ ' - j'= so M a f » ' = X A' T h e n £p>cu.e' 169 APPENDIX 2 EXPANSION OF BOUNDARY FACTORS IN POWERS OF y (see page 67) For hollow c y l i n d e r s without r e s t r i c t i o n on A, , A?. . I L %l Y 9 ( .*)}' ht% ^ _A_—. \ i + (- ^L -h ^  - ^ ' ^ ^ / i i ^ ) ^fhL AZ I + <t .1 A, J W I z « z 6 I f ? - z e A z ( t * V- \ih z + z6 Z? j A 6 \?'Z S 2&" Z e ?<2e i*'*6 Z 6 J *i) 1 7 0 1 7 1 APPENDIX 3 EXPANSION. OF BOUNDARY FACTORS TN POWERS' OF (d / A ) (see page 6?) For hollow c y l i n d e r s w i t h <s£«A , v a l i d f o r a l l . Geometry parameter /M* s . M-i / A 

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