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Field theoretical description of the superconducting state Pugh, Robert Edward 1955

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FIELD THEORETICAL DESCRIPTION OF THE SUPERCONDUCTING STATE ROBERT EDWARD PUGH A Thesis Submitted i n Partial Fulfilment of the Requirements for the Degree of MASTER OF ARTS In the Department of Physics We accept this thesis as conforming to the standard required for the degree of MASTER OF ARTS Members of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA April, 1955 ABSTRACT The superconducting s t a t e i s described i n terms of the e l e c t r o n i c plasma i n i n t e r a c t i o n w i t h the electromagnetic p o t e n t i a l s . The i n t e r a c t i o n of the plasma w i t h i t s e l f and w i t h the l a t i c e , Is represented by a p o t e n t i a l energy deter-mined such that the equations of motion are s e l f - c o n s i s t e n t . The r e s u l t i n g n o n - l i n e a r f i e l d equations are solved i n the l i n e a r approximation. There e x i s t f o u r independent modes of v i b r a t i o n of the f i e l d v a r i a b l e s corresponding t o the plasma o s c i l l a t i o n s , and one l o n g i t u d i n a l and two transv e r s e o s c i l l a t i o n s of the electromagnetic p o t e n t i a l s . The c o n t r i -b u t i o n of these modes to the s p e c i f i c heat i s discussed. ACKNOWLEDGEMENT I wish t o thank P r o f e s s o r F. A. Kaempffer f o r suggesting the research problem and f o r h i s advice and encouragement throughout the performance of t h i s work. I am g r a t e f u l t o the N a t i o n a l Research C o u n c i l of Canada f o r the donation of a Bursary (1954-55) i n support of the research. TABLE OF CONTENTS I INTRODUCTION 1 A. The Experimental Facts 1 B. The Present State of the Theory of Sup e r c o n d u c t i v i t y 4 I I THE LAGRANGIAN FORMULATION OF THE SUPERCONDUCTING STATE AND THE RESULTING EQUATIONS OF MOTION 7 I I I SPECIFIC HEAT 14 IV CONCLUSIONS 21 APPENDIX A 22 REFERENCES 26 LIST OF FIGURES FIGURE 1. Molar Heat C a p a c i t i e s of Niobium 3 FIGURE 2. V a r i a t i o n of the S p e c i f i c Heat from a T^ Law 18 I INTRODUCTION: I n t h i s work an attempt i s made to describe the phenomenon of s u p e r c o n d u c t i v i t y by repr e s e n t i n g the s o - c a l l e d "superconducting e l e c t r o n s " as a c l a s s i c a l charged f l u i d . I t w i l l be shown that the hydrodynamle equations d e s c r i b i n g the f l u i d are c o n s i s t e n t w i t h London's phenomenological equations. Since as yet the behaviour of the s p e c i f i c heat of a superconducting metal below i t s t r a n s i t i o n temperature has not been s a t i s f a c t o r i l y e x p l a i n e d , we s h a l l t r y to des-c r i b e t h i s phenomenon i n terms of the energies of the wave motions of the e l e c t r i c and magnetic f i e l d p o t e n t i a l s . We s h a l l begin our d i s c u s s i o n by reviewing b r i e f l y some of the experimental f a c t s associated w i t h the phenomenon of super-c o n d u c t i v i t y . A The Experimental F a c t s . I n 1911» Kamerlingh Onnes discovered that at a c e r t a i n very low temperature, T , the r e s i s t a n c e of c e r t a i n conductors dropped s h a r p l y to zero, i n d i c a t i n g that no e l e c -t r i c f i e l d could be maintained i n s i d e t h i s 'superconductor'. Kamerlingh Onnes f u r t h e r discovered t h a t t h i s phenomenon of su p e r c o n d u c t i v i t y was destroyed by the a p p l i c a t i o n of a s u f f i c i e n t l y strong magnetic f i e l d , H C ( T ) . Considering the superconductor to be a p e r f e c t conductor ( i n f i n i t e c o n d u c t i v i t y ) , then i t f o l l o w s from Maxwell's equations that the magnetic f i e l d i n s i d e a super-conductor i s constant (B = 0 ) . One would then expect that i f a superconducting metal were placed i n a magnetic f i e l d and then cooled below i t s t r a n s i t i o n temperature, t h a t the magnetic f i e l d would be 'frozen i n 1 . I t was found, however, by Meissner i n 1933 that when the metal became superconducting the f i e l d i n the neighborhood of the specimen i n c r e a s e d , thus showing that the f i e l d i n s i d e the superconductor was pushed out, l e a v i n g no magnetic f i e l d i n s i d e . I t was also shown that when a superconductor i n i t i a l l y i n zero f i e l d i s placed i n a magnetic f i e l d , the f i e l d i n s i d e the superconductor remains zero. Hence one concludes that t h i s process i s a r e v e r s i b l e one and that the magnetic f i e l d may t h e r e f o r e be t r e a t e d as a thermodynamic v a r i a b l e . T r e a t i n g H as a thermodynamic v a r i a b l e of the s t a t e of the superconductor, one can e a s i l y d e rive from the f i r s t and second laws of thermodynamics that the d i f f e r e n c e between the s p e c i f i c heat i n the superconducting s t a t e and the spec-i f i c heat i n the normal s t a t e i s c - r = i A l ^5 u r v 9T J y i - 3 -The s p e c i f i c heat at constant volume i n the normal state at low temperatures can be represented by C n = « T +• UWfef . The T^ term, which i s h e r e a f t e r denoted by Cj,, i s a t t r i b u t e d to the l a t t i c e v i b r a t i o n s according to the w e l l known Debye theory of s p e c i f i c heats. The l i n e a r term has been shown (3) to r e s u l t from the a p p l i c a t i o n of Fermi-Dirac s t a t i s t i c s t o a f r e e e l e c t r o n gas, and i s hence u s u a l l y r e f e r r e d t o as the e l normal e l e c t r o n i c s p e c i f i c heat and i s designated by C n . I t i s g e n e r a l l y found t h a t at the t r a n s i t i o n temper-ature T c, where the change from normal to superconducting s t a t e s takes p l a c e , the s p e c i f i c heat as a f u n c t i o n of temperature i s discontinuous (see F i g u r e 1 ) . I t i s seen from Figure 1 that the s p e c i f i c heat i n the superconducting s t a t e , C s, i s greater than the l a t t i c e c o n t r i b u t i o n , C L , at a l l p o i n t s , but not always greater than the normal e l e c t r o n i c s p e c i f i c heat. Hence i t i s g e n e r a l l y assumed that the l a t t i c e v i b r a t i o n s s t i l l c o n t r i b u t e to the s p e c i f i c heat i n the superconducting s t a t e , whereas the normally conducting e l e c t r o n s do not c o n t r i b u t e . Since the d i f f e r e n c e C g -e l cannot be accounted f o r by C n , t h i s d i f f e r e n c e i s c a l l e d the superconducting s p e c i f i c heat, and i s designated by C®3-. e l el Experimental measurements of C g g e n e r a l l y i n d i c a t e that C g obeys a T^ law. Recent work on the s p e c i f i c heat of super-e l conducting Niobium (1) shows that i n t h i s case C g r i s e s much f a s t e r than any T^ dependence. We s h a l l l a t e r d i s c u s s the c o n t r i b u t i o n of var i o u s modes of v i b r a t i o n to the supercon-ducting s p e c i f i c heat, and w i l l give a p o s s i b l e reason f o r t h i s d e v i a t i o n from a T^ dependence. B The Present State of the Theory of Su p e r c o n d u c t i v i t y . There e x i s t s at present no a t o m i s t i c theory of su p e r c o n d u c t i v i t y . One can, however, s p e c i f y some of the co n d i t i o n s that a s u c c e s s f u l theory of s u p e r c o n d u c t i v i t y must f u l f i l l . I t must ( i ) agree w i t h the phenomenological theory (London's equ a t i o n s ) , ( i i ) give the t r a n s i t i o n temperature and the c r i t i c a l magnetic f i e l d , and ( i i i ) give the isotope e f f e c t (M^ T. = c o n s t . ) . Taking the usual view that s u p e r c o n d u c t i v i t y i s caused by the conduction e l e c t r o n s , and i s not a property of the l a t t i c e , one can say that a s u c c e s s f u l approach towards the problem has been made i f one can e x p l a i n the o r i g i n of a la r g e gap I n the energy l e v e l s of the conduction e l e c t r o n s near the surface of the Fermi sphere. I n f a c t , i f e l e c t r o n s were Bose p a r t i c l e s , one could construct a f a i r l y reasonable model of the superconductor, i n which the existence of the c r i t i c a l magnetic f i e l d i s explained i n terms of the Zeeman e f f e c t f i e l d which would tend to c l o s e the gap between the ground s t a t e and the f i r s t e x c i t e d s t a t e of the e l e c t r o n s i n a superconductor of reasonable dimensions. This e f f e c t , - 5 -comes out i n the r i g h t order of magnitude i n the approximation i n which the i n t e r a c t i o n between the e l e c t r o n s themselves and between the e l e c t r o n s and the l a t t i c e i s neglected. Such a model i s u n f o r t u n a t e l y of l i t t l e use i n the case of a Fermi gas of e l e c t r o n s , since the l e v e l d e n s i t y near the surface of the Fermi sphere i s so l a r g e that the magnetic f i e l d needed to make neighboring energy l e v e l s c o i n c i d e by Zeeman e f f e c t i s extremely small (10~^ o e r s t e d s ) . I t can th e r e f o r e be only by t a k i n g i n t o account the i n t e r a c t i o n between e l e c t r o n s them-s e l v e s , and between the e l e c t r o n s and l a t t i c e v i b r a t i o n s , that one can hope to o b t a i n a s u c c e s s f u l s t a r t towards an a t o m i s t i c theory of s u p e r c o n d u c t i v i t y . The very existence of the isotope e f f e c t i n d i c a t e s that the width of the energy gap at the surface of the Fermi sphere depends e s s e n t i a l l y on the i n t e r a c t i o n s between the e l e c t r o n s and the l a t t i c e v i b r a t i o n s . I t i s beyond the scope of the present work t o t r y to describe the mechanism which sets up the superconducting s t a t e , which i s presumably due to the i n t e r a c t i o n of e l e c t r o n s among themselves and w i t h the l a t t i c e i o n s . I t w i l l be assumed tha t t h i s i n t e r a c t i o n leads to a r i g i d ground s t a t e f o r the e l e c t r o n s , i . e . , most of the e l e c t r o n s must be i n t h e i r lowest s t a t e s , and the p r o b a b i l i t y that they can r i s e to s t a t e s of higher energy must be s m a l l . I n the f o l l o w i n g , the e l e c t r o n i c d e s c r i p t i o n of the superconductor w i l l be c h a r a c t e r i z e d by a system of continuous - 6 -e l e c t r i c charges which i n t e r a c t w i t h the electromagnetic f i e l d which they themselves produce. In other words, the e l e c t r o n -i c behaviour i s c h a r a c t e r i z e d by a system of simultaneous d i f f e r e n t i a l equations c o n s i s t i n g of Maxwell's equations and the c l a s s i c a l hydrodynamic equations f o r a continuous f l u i d . - 7 -I I THE LAGRANGIAN FORMULATION OF THE SUPERCONDUCTING STATE AND THE RESULTING EQUATIONS OF MOTION: We s h a l l t r y the f o l l o w i n g Lagrangian d e n s i t y : * where H i s the magnetic f i e l d , E the e l e c t r i c f i e l d , (f) and A the e l e c t r i c and magnetic v e c t o r p o t e n t i a l s r e s p e c t i v e l y , i the d i e l e c t r i c constant, and v (defined by equation [2] below) i s the v e l o c i t y of the f l u i d , (p i s the v e l o c i t y p o t e n t i a l , and j o i s the mass d e n s i t y of the f l u i d . The f i r s t two terms are the us u a l c l a s s i c a l hydrodynamic terms f o r a f r e e c o n t i n -uous f l u i d of d e n s i t y ^ ; by themselves, they give upon independent v a r i a t i o n w i t h respect to j o and <p , the equation of c o n t i n u i t y and the B e r n o u i l l i equation f o r a continuous f l u i d . The l a s t three terms are the electro-magnetic terms which g i v e , by themselves, upon v a r i a t i o n w i t h respect to A and , Maxwell's equations. There i s , however, one import-ant d i f f e r e n c e : the us u a l term ~f>^ has been replaced by THJ? w n e r e jf3 Hf a r e the d e v i a t i o n s from the average d e n s i t y j o and the average p o t e n t i a l (jj> r e s p e c t i v e l y . This means tha t the background of p o s i t i v e l a t t i c e charge * This Lagrangian i s chosen f o r s i m p l i c i t y ; one can t r y other Lagrangians which are derived from analogy w i t h the c l a s s i c a l Schroedinger f i e l d , but i t turns out tha t t h i s more complicated Lagrangian leads to the same s o l u t i o n s i f one demands that London's f i r s t equation be s a t i s f i e d . The treatment of t h i s Lagrangian i s out-l i n e d i n Appendix A. - 8 -e f f e c t i v e l y cancels out the Coulomb i n t e r a c t i o n * of the f l u i d l e a v i n g only the i n t e r a c t i o n between the net e l e c t r i c charge densityJ/O, and the net e l e c t r i c f i e l d p o t e n t i a l (|) . j o and (|) are imagined to be w r i t t e n a s j o = i j 3 + j i and ^ = + I n a d d i t i o n we define the f o l l o w i n g : [2] = VOP -' Tne " [3] tf= CJUUXjtrT' [4] ff = -V$ - ±{T . The choice [2] has been made to s a t i s f y i d e n t i c a l l y London's equation [2a] OUUtlj\.J = - ± ht , where ytj= ?pj]L . while [3] and [4] are the usual expressions f o r the e l e c t r i c and magnetic f i e l d s i n terms of the p o t e n t i a l s (|) and A. We w i l l now assume that the d e n s i t y of the f l u i d i s almost constant and that v a r i a t i o n s i n d e n s i t y are s m a l l , i . e . , we assume that i n the expansion jo ~jo +jo , the d e v i a t i o n i s much smaller thanj> , and that JD i s constant. This assumption w i t h the previous assumption t h a t the p o s i t i v e l a t t i c e charge e f f e c t i v e l y cancels out the e l e c t r i c f i e l d due to the e l e c t r o n s , makes i t reasonable to assume that <p , A, and , are s m a l l , so t h a t second order terms i n them can be neglected. * This i s shown to be true i n Appendix A. - 9 -By independent v a r i a t i o n of the Lagrangian d e n s i t y w i t h respect to cp ,jc> , ^ , and A , we get the f o l l o w i n g equations of motion; [ 5 ] JO +J?^<P - ~ 0 [7] rfi-tt+sijeftnp-jkfii'o [8] &$-0+!jgf.o , n e g l e c t i n g second order terms. Equation [5] i s the usual equation of c o n t i n u i t y expressing the law of conservation of matter. Equation [6] i s B e r n o u i l l i ' s equation f o r the conser-v a t i o n of energy i n the case that the k i n e t i c energy of the f l u i d i s n e g l i g i b l e . Equations [7] and [8] are the usual electr|o„ f i e l d equations. London's f i r s t equation f o l l o w s d i r e c t l y from equation [6]. We have which i s London's f i r s t equation. The equations [5] to [8] may be solved f o r JO as f o l l o w s : d i f f e r e n t i a t e [5] w i t h respect t o time and s u b s t i -t u t e <p from [6]. Making use of the Lorentz c o n d i t i o n 7 equation [5] becomes: - 10 -Hence j o = ^ e ? O J ° r where [10] GJ$ = jfclSfx, and f ( r ) I s , as f a r as equation [9] i s concerned, an a r b i t -r a r y f u n c t i o n of the co-ordinates. However, i n order to solve the r e s t of the equations, we must know the e x p l i c i t form of f ( r ) . I t turns out tha t i f f ( r ) i s taken to be a constant, the Lorentz c o n d i t i o n cannot be s a t i s f i e d . I n Appendix A, where a more general Lagrangian i s considered, f ( r ) i s shown to be of the form exp ( i K ^ ' r ) where K 0 i s a vect o r whose magnitude depends on the c o e f f i c i e n t of a term which does not appear i n the Lagrangian used above. I t i s there f o r e convenient to take [11] jf(jt) = const, where f o r t h i s d i s c u s s i o n K Q can be taken to be any a r b i t r a r y non-zero v e c t o r . Hence we have: i [12] JD = e o N » . e l ' * , * * i a * r We may now solve equation [8] f o r ^  . We get: [13] $ = co* s r . e i 7 ? , x > " I < O T + R P where [14] K 4= - 11 -For b r e v i t y we put £>' =«~ s,e 7 ' * and < = : ^ ^ _ z c o ^ — ' Equations [5] and [6] y i e l d the f o l l o w i n g s o l u t i o n f o r <p : [15] <p = QJ± p + where #?'is a general s o l u t i o n of £?=o and (7^'=o . F i n a l l y equation [7] y i e l d s : [16] t g c 7 ^ + + A^^/Ce* ' * * "^* where / } 0 i s a constant v e c t o r , and [17] co/- = <^< ^ co-l t i s seen from [13] and [16] that the Lorentz c o n d i t i o n i s s a t i s f i e d provided that AQ'K-^O, i . e . provided that the v e c t o r p o t e n t i a l [18] # = A l e ^ ^ + ? c o < t i s a transverse wave. I t should be pointed out t h a t the magnetic v e c t o r p o t e n t i a l A given by [16] has a very unusual f e a t u r e : i t has  a l o n g i t u d i n a l component, ^  \7(§f . We see from the above, that the s o l u t i o n s are con-s t r u c t e d from four independent modes of o s c i l l a t i o n : j o ,. , and the two components of A ' which are orthogonal to i t s propa-g a t i o n v e c t o r K-^ . jP has a f i x e d frequency, aja y given by [10] and an a r b i t r a r y propagation v e c t o r , K^. (§> has a frequency, - 12 -u) , and a propagation v e c t o r , K, which are r e l a t e d by i A' has a frequency cJ,,and a propagation v e c t o r K-p r e l a t e d by oJ?~ = + c o -I t i s shown i n S e c t i o n I I I , t h a t , f o r thermodynamic consider-a t i o n s , i s n e g l i g i b l e compared w i t h <^J<<Z . The c o n t r i b u t i o n of these f o u r modes of o s c i l l a t i o n to the s p e c i f i c heat of the system i s considered below. Equation [10] Is the clue to a p o s s i b l e e x p l a n a t i o n of the high s p e c i f i c heats of superconductors. I t s t a t e s simply that the frequency of the c l a s s i c a l charged f l u i d r e p r e s e n t i n g the superconducting e l e c t r o n s , i . e . , the frequency of the s o - c a l l e d plasma o s c i l l a t i o n s , i s a constant. We could then imagine the f o l l o w i n g model. I n the normal s t a t e , the conducting e l e c t r o n s are e s s e n t i a l l y f r e e , as e l shown by the existence of C n . At some c r i t i c a l temperature the e l e c t r o n s , under the i n f l u e n c e of the l a t t i c e i o n s , v i b r a t e w i t h some f i x e d frequency which i s a sub-harmonic of the maximum l a t t i c e frequency, i . e . a sub-harmonic of the upper l i m i t of the Debye spectrum of l a t t i c e f r e q u e n c i e s . Let us assume that t h i s plasma frequency i s of order of magnitude kT c rJ 10"*"°to 10^". This frequency r e s u l t s i n an extremely "~ET" l a r g e d i e l e c t r i c constant as was pointed out by Ginsburg (2). For example, i f we take the number of superconducting - 13 -el e c t r o n s per u n i t volume to be 5 x 10 , then from equation [10] we have i - ~ /Oero /0'° This l a r g e d i e l e c t r i c constant plays a very important r o l e i n the behaviour of the superconductor. Such a l a r g e £ means a very small v e l o c i t y of l i g h t ( ^  ~ to^m/^. Now the smaller the v e l o c i t y of l i g h t , the l a r g e r i s the momentum £ associated w i t h b l a c k body r a d i a t i o n of given energy E. Hence, s i n c e the number of photons which can have energy between E and E+dE i s p r o p o r t i o n a l to p 2dp, there are more photons w i t h energy E when £ i s l a r g e than when £.is s m a l l . I n f a c t , the number of photons of given energy i s p r o p o r t i o n a l t o £ V l". Because of the g r e a t l y increased number of photons present, a much greater expenditure of energy w i l l be re q u i r e d to r a i s e the energy of the superconductor. This means a l a r g e r s p e c i f i c heat i n the superconducting sta t e than i n the normal s t a t e . This subject i s t r e a t e d q u a n t i t a t i v e l y i n the next s e c t i o n . - 14 -I I I SPECIFIC HEAT From the s o l u t i o n s f o r the equations of motion given i n S e c t i o n I I , we see t h a t there are f o u r independent modes of v i b r a t i o n i n the superconductor. They are: (1) the plasma o s c i l l a t i o n , (2) the e l e c t r i c f i e l d p o t e n t i a l , (3) and (4) the two independent modes of v i b r a t i o n due to A'. There are two modes because A 1 i s a t r a n s -verse wave and hence there are two independent mutually orthogonal o s c i l l a t i o n s each perpendicular to the d i r e c t i o n of propagation. In thermal e q u i l i b r i u m , each of these modes w i l l c o n t r i b u t e s e p a r a t e l y to the t o t a l energy of the system. We w i l l f i r s t show that the c o n t r i b u t i o n of the plasma o s c i l l a t i o n s to the s p e c i f i c heat of the system i s n e g l i g i b l e and then proceed to c a l c u l a t e the c o n t r i b u t i o n of the other three modes of v i b r a t i o n . (1) Plasma C o n t r i b u t i o n I t was shown i n S e c t i o n I I that the d e n s i t y of the continuous charged f l u i d r e p resenting the superconducting e l e c t r o n s was of the form y -Ji +J> = ^ + co"sr- c We define the energy of such an o s c i l l a t i o n as E Q = tuco 0 and the momentum as£=fc-C • Then the t o t a l energy per u n i t - 1 5 -volume due to the plasma o s c i l l a t i o n i s where p o m a x i s the maximum momentum associated w i t h the plasma o s c i l l a t i o n s , k i s Boltzmann's constant and T i s the temperature. Because E Q Is independent of P q, the s i z e of E J t o t w i l l be determined e n t i r e l y by E Q and P0max. Since we know nothing about the s i z e of Pomax' ^he best we can do i s to assume some upper l i m i t f o r i t . We w i l l assume that Pomax = ^-E0 •= T ( 'r^^y^ 5 presumably i t would be much sma l l e r than t h i s . We t h e r e f o r e have: We w i l l now assume f o r the sake of the f o l l o w i n g order of magnitude c a l c u l a t i o n , t h a t p o m a x and E 0 are temperature independent. I n general, of course, they w i l l not be inde-pendent of temperature, but t a k i n g t h i s i n t o account would not change the r e s u l t s obtained below by more than a s m a l l f a c t o r . The c o n t r i b u t i o n of the plasma o s c i l l a t i o n s to the s p e c i f i c heat i s then where ?c = -§s . The expression y*" *L. i s a monotonously fe T (e > c- ») decreasing f u n c t i o n , having i t s l a r g e s t value ( f o r p o s i t i v e x) at x = 0, at which point i t i s u n i t y . I t f o l l o w s then that - 16 -C v has i t s l a r g e s t value when E Q « k T ; i . e . , Taking the number of superconducting e l e c t r o n s per u n i t volume, J , to be about 5 x 10 , we get: Since measured values (1) of the s p e c i f i c heat i n Niobium and 4 other superconducting metals are about 10 ergs/cc, we con-clude t h a t the c o n t r i b u t i o n from the plasma o s c i l l a t i o n s t o the s p e c i f i c heat i s n e g l i g i b l e . (2) E l e c t r i c F i e l d P o t e n t i a l C o n t r i b u t i o n . We saw i n S e c t i o n I I that the e l e c t r i c f i e l d poten-t i a l <|> could be expressed as = <|> - f - ^ =• $ o *•. const ClK'A + oCJj> w i t h Kl= • The c o n t r i b u t i o n of the o s c i l l a t i o n p to the energy of the system has been taken i n t o account i n I I I ( l ) . We w i l l now consider the c o n t r i b u t i o n of the mode of v i b r a t i o n represented by . We define the energy and momentum of t h i s o s c i l l a t i o n i n the usual f a s h i o n by E =JUo and £_ = fc.K, respect-i v e l y , and we define the t o t a l energy per u n i t volume of such o s c i l l a t i o n s by f l 9 l F* - JUL. I EWptJp W s S ^ T (3) C o n t r i b u t i o n of the Magnetic P o t e n t i a l . From equation [16] we see that the magnetic v e c t o r - 17 -p o t e n t i a l A can be expressed as the sum of the waves grad^», grad , and the complimentary f u n c t i o n A' = AQel*'':**to*T ((p'i being independent of time, i s not a wave motion.) The wave represented by A 1 can be t r e a t e d i n the same way as that represented by <§>t . We define the energy and momentum of the A' wave as E,= tiu>, a n d ^ = ^ ^ r e s p e c t i v e l y . Equation [17] gives where £0~KC^0% Hence the t o t a l energy of the A ' wave i s given by The f a c t o r of two, appearing i n f r o n t of the i n t e g r a l i n equation [20] comes from the f a c t that A ' has two p o s s i b l e p o l a r i z a t i o n s , i . e . , two independent d i r e c t i o n s f o r AQ perpen-d i c u l a r to the propagation v e c t o r K^. Equation [20] i s s i m p l i f i e d a l i t t l e by p u t t i n g u = ^ and u Q = , and by denoting by I the i n t e g r a l Then 'TOT ( T_ 77 tic)* I t w i l l be shown below that the s p e c i f i c heat due to the energy of the <3?, wave as given by equation [19] i s of - 18 -the same order o f magnitude as e x p e r i m e n t a l l y measured v a l u e s i f the d i e l e c t r i c constant i s taken to be about 10^. With t h i s B and w i t h the number of superconducting e l e c t r o n s 21 ~ taken to be 5 x 10 / c c , we get u Q = 0.05. Hence a v e r y good approximation i s made to I by t a k i n g u 0 = 0.00, i n which case I = . We t h e r e f o r e have f o r the t o t a l energy o f the and A ' waves [21] £ +- E ~ / a y f # C o n s i d e r i n g the d i e l e c t r i c constant t o be a f u n c t i o n of temperature, we get f o r the s p e c i f i c heat c o n t r i b u t i o n from these waves: [22] C „ = sUasr =. aT\^ + ±T d T dT J e l 1> F i g . 2 shows a p l o t o f C g / a T J v e r s u s T c a l c u l a t e d from the experimental d a t a f o r Niobium ( i ) . Since we cannot expect to f i n d the f u n c t i o n a l dependence of the d i e l e c t r i c constant on the temperature from t h i s present theory, we must content o u r s e l v e s w i t h f i n d i n g what f u n c t i o n w i l l give the c l o s e s t f i t between the experimen-t a l and t h e o r e t i c a l s p e c i f i c heat c u r v e s . By a process o f t r i a l and e r r o r , a curve o f £ ver s u s T was found which c o u l d be used t o c o n s t r u c t the curve i n F i g u r e 2 by the r e l a t i o n - 19 -I t was found that t h i s e m p i r i c a l curve of £Viversus T could be described quite a c c u r a t e l y by the f u n c t i o n [23] £ v - = 4*- - £ « p f $ >4. 15 where £0 = 16.4 x 10 f o r Niobium. Hence we can w r i t e = /•4,ZS-x/oiT2-e Y?+^=9 f o r Niobium. e l Table I shows a comparison of C s c a l c u l a t e d from [2#j w i t h e l the experimental values ( l ) of C_ . I t should be noted that the experimental values l i s t e d f o r T<2.5°K are e x t r a p o l a t i o n s and not measurements. Hence the l a r g e d i f f e r e n c e s i n the f i r s t f o u r e n t r i e s are not s i g n i f i c a n t . - 20 -TABLE I T°K r e l ergs u s °-K cc. experimental r e l ergs s °K cc. T h e o r e t i c a l % d i f f e r e n c e .5 20 .00030 1.0 70 3.23 1.5 230 97.4 2.0 700 616 2.5 1900 2021 +5.9 3.0 4.56 x 103 4.74 x 103 +3.8 3.5 9.25 x 103 9.12 x 103 -1.4 4.0 1.601 x 104 1.528 x 104 -4.8 4.5 2.458 x 104 2.350 x 104 -4.4 5.0 3.48 11 " 3.38 » » -2.9 5.5 4.69 M " 4.64 " " -1.1 6.0 6.12 11 •» 6.10 " " -0.3 6.5 7.75 " » 7.78 « « +0.4 7.0 9.59 " " 9.74 » " +1.5 7.5 11.7 " 11 11.9 " " +1.7 8.0 14.2 " " 14.3 " " +0.7 8.5 17.0 " " 17.0 " « 0.0 8.7 18.1 » » 18 .1 " 0.0 - 21 -IV CONCLUSIONS Demanding that London's phenomenologieal equations be c o n s i s t e n t w i t h the hydrodynamic d e s c r i p t i o n of a c l a s s i c a l charged f l u i d has l e d us to f i n d that s o l u t i o n s of the equations of motion c o n t a i n four independent modes of o s c i l l a -t i o n . One of these, the plasma o s c i l l a t i o n , does not co n t r i b u t e appreciably to the s p e c i f i c heat of the system. The other three o s c i l l a t i o n s , the one e l e c t r i c p o t e n t i a l o s c i l l a t i o n and the two magnetic v e c t o r p o t e n t i a l o s c i l l a t i o n s , have been shown to c o n t r i b u t e a p p r e c i a b l y to the s p e c i f i c heat provided that the d i e l e c t r i c constant i s , as was suggested by Ginsburg, of the order of 1 0 1 0 . Whether or not t h i s i s the c o r r e c t i n t e r p r e t a -t i o n of the high s p e c i f i c heat i n superconducting metals can only be ascertained by an experiment determining the d i e l e c t r i c constant i n superconductors. * Ginsburg (2) mentions that an experiment has been per-formed by G a l k i n and Besugly which i n d i c a t e s that the d i e l e c -t r i c constant of t i n i s about 5 x l o " . The author, however, has been unable to f i n d the paper r e f e r r e d to by Ginsburg. - 22 -APPENDIX A In analogy w i t h the c l a s s i c a l Schroedinger f i e l d we could s t a r t w i t h the Lagrangian d e n s i t y : 111 L =-J>9 - ^ ( S f / ' - ^ i r ^ & f c ^ where U C/*,^?) i s some energy d e n s i t y which i s to be deter-mined. By independent v a r i a t i o n w i t h respect to <p A, and <p , we get the equations of motion: Making the expansions: - 23— and assuming that A and (f are small such that second order terms i n them may be neglected, we f i n d that i n order to s a t i s f y the zero order equations we must have fit = B( 1 1 O . Hence the f i r s t order equations of motion become: [5"] j*j;r<p -Zj&t-o [6" ] V-g^tt-*^**.*, [8"] = m $ , ^ x > C7"] We can determine A 2 and C 1 as f o l l o w s : take the divergence of [7" ] and demand that the Lorentz condition;*7./T=-J.<jf , holds, We get £7"a] Vlfy - J L £ " UZZJ. ^cp ~ $ = o S u b s t i t u t i n g vz<p from [5M ] we get D i f f e r e n t i a t i n g equation [8"] w i t h respect to t , we get [8-b] v%-lM^tr(aA,*cJ) Comparison of equation [7"b] and [8"b] shows that A 2 = 0 and C - - e Therefore = -€.<?'•<--•. higher order terms and ~ - ~ £•($*- B P * higher order terms Hence U - U0- % J > § . + ^ y ?V-higher order terms - 24 -The absence of any terms i n U l i k e J=jJ>c£ or l^jgq^ means that there i s no Coulomb i n t e r a c t i o n between the e l e c t r o n s on the average. This means e s s e n t i a l l y that the i n t e r a c t i o n w i t h the p o s i t i v e l a t t i c e charges cancels the i n t e r a c t i o n w i t h the average negative e l e c t r o n charges represented b y ^ and c£. Using these c o e f f i c i e n t s , the equations of motion become-. [5b] j+j'W-e-^v.fco-[7b] rt-tf+zg, X? ff [8b] v*$-t$ We w i l l now demand that London's f i r s t equation i s s a t i s f i e d . Taking the gradient of a l l terms i n equation [6b] we have: Hence V ^ ^ ^ = o The s o l u t i o n of the above equation i s : [12a] p _ qMe^K ***** *^ Hj££^ I f we demand that the s p a t i a l average of J2 i s zero, then i t - 25 -f o l l o w s immediately from [12a] that the constant C, must be zero. Hence we have: [12b] j o _ ^(t)eiJ^'^ Proceeding as before we can now construct the s o l u t i o n s to the equations of motion. - 26 -REFERENCES (1) Brown, Zemansky, and Boorse, Phys. Rev., %2, 57, 1953• (2) Ginsburg, W.L., F o r t s c h r . d. Phys., Band 1, Heft 3/4, 101, 1953-(3) Sommerfeld, A., Z. f. Physik, 4£, 1, 1928. OTHER REFERENCES USED BUT NOT REFERRED TO IN THE TEXT Becker, R., H e l l e r , G., and Sauter, F. Z e i t . f . P h y s i k , Band 85, 772, 1933-Ginsburg, W. L., Zh. Eksper. Teor. F i z . , 21, 979, 1951. Horowitz, S i l v i d i , Malaker, and Daunt, Phys. Rev., 88, 1182, 1952. Kaempffer, F. A. Z e i t . f . Physik, Band 7, 487, 1949* Keesom, W.H., and van Laer, P.H., P h y s i c a , 4, 487, 1937« Keesom, W.H., and van Laer, P. H., P h y s i c a , 193, 1938. F. and H. London, Proc. Roy. Soc. A, 142, 71, 1935* F. London, Rev. Mod. Phys., 310, 1945. F. London, S u p e r f l u i d s , V o l . I . 

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