Thermodynamical theory of galvanomagnetic and thermomagnetic phenomena. I

Fieschi, R. De Groot, S . R . Mazur, P. 1954

Physica XX 67-76

THERMODYNAMICAL THEORY OF GALVANOMAGNETIC AND THERMOMAGNETIC PHENOMENA. I RECIPROCAL

RELATIONS

IN ANISOTROPIC METALS

b y R. F I E S C H I *), S. R. D E GROOT and P. MAZUR Instituut voor theoretische natuurkunde, Universiteit, Leiden, Nederland

Synopsis T h e t h e r m o d y n a m i c t h e o r y of g a l v a n o m a g n e t i c a n d t h e r m o m a g n e t i c p h e n o m e n a in a n i s o t r o p i c m e t a l s is d e v e l o p e d . W i t h t h e m e t h o d of D e G r o o t and M a z u r, w h i c h allows to t r e a t v e c t o r i a l p h e n o m e n a , t h e following reciprocal relations are d e r i v e d f r o m microscopic r e v e r s i b i l i t y Lss(B) = Lsts(-

B), Lu(B) = L~e(-- B), Les(B) = L~e( - B),

w h e r e Lss is t h e h e a t c o n d u c t i o n tensor, Lte t h e electrical c o n d u c t i o n tensor, Lt. a n d Lse t h e tensors describing t h e cross-effects, a n d w h e r e t h e sign t i n d i c a t e s t r a n s p o s e m a t r i x . T h e r e l a t i o n b e t w e e n t h e cross-effects is a n o n - t r i v i a l e x a m p l e of a reciprocal r e l a t i o n d e r i v e d f r o m t h e f o r m u l a t i o n of microscopic r e v e r s i b i l i t y w i t h variables, w h i c h are e v e n a n d odd f u n c t i o n s of p a r t i c l e velocities (Casimir's a- a n d fl-variables).

§ 1. Introduction. The galvanomagnetic and thermomagnetic phenomena have been treated b y C a l l e n l ) and b y M a z u r and P r o g o g i n e 3 ) using the methods of thermodynamics of irreversible processes 3) 4). These authors write down reciprocal relations between corresponding irreversible phenomena. As is usually done these reciprocal relations are supposed to be examples of the relations, which 0 n s a g e r 6) derived from the property of microscopic reversibility. However, as C a s i m i r pointed out, Onsager's proof is really valid for scalar phenomena. The reason for this is that O n s a g e r assumes that the irreversible fluxes can be considered as time derivatives of thermodynamic state variables. For scalar processes, such as chemical reactions and relaxation phenomena, this is correct. But it is not true for vectorial processes (such as heat conduction, diffusion and electrical conduction) and tensorial processes (such as viscous flow) and therefore an extension of the theory is necessary. Two alternative methods have been proposed to solve the problem of finding reciprocal relations for vectorial and tensorial processes. In fhe first method the problem under investigation is reformulated in such a w a y that it is possible to apply the original Onsager formalism, To *) On leave from the "Istituto di Seienze Fisiche dell' Universith" Milan, Italy. m

67

- -

68

R. FIESCHI, S. R. DE GROOT AND P. MAZUR

achieve this it is necessary to introduce appropriate auxiliary phenomenological equations and coefficients, which are only indirectly related to the ordinary macroscopic laws and coefficients. C a s i m i r 6) has used this method for heat conduction in anisotropic crystals without a magnetic field. Two of us ~) have also considered systems with magnetic fields, and derived reciprocal relations for the heat and electric conduction tensors. The second method is in a certain sense the opposite of the first. Now the problem is not rewritten in a form appropriate for the application of the original Onsager theory, but the formalism is generalized in such a way that it can be applied also for fluxes which describe vectorial or tensorial irreversible phenomena, or, in other words, for fluxes which are not necessarily time derivatives of state variables. In this theory, which has been proposed b y two of us 8) 9), the ordinary macroscopic laws can be used as phenomenological equations, and thus the complications inherent in the approach of the first method are avoided. With the help of a generalization of the fluctuation theory, used by 0 n s a g e r, reciprocal relations for vectorial and tensorial processes can be derived directly from the property of microscopic reversibility. The method described above has been applied to heat conduction, diffusion, viscosity and cross-effects in anisotropic mixtures s), and also to electric conduction in anisotropic metals with magnetic fields 9). In this paper we shall derive, again with the help of the second method, reciprocal relations for heat conduction, electric conduction and cross-effects in anisotropic metals, in the presence of a magnetic field. These effects are usually called galvanomagnetic and thermomagnetic processes 10). In subsequent papers the observable relations between these effects will be derived and the more general case of mixtures will be treated, which, in addition to heat conduction, diffusion (including electric conduction) and cross-effects, show also viscous flow.

§ 2. Entropy production in local [orm. Let us consider a system, consisting of a rigid ion lattice and of electrons, in an electromagnetic field and with a non-uniform temperature. In order to set up the phenomenological equations it is necessary to know the entropy production in local form, expressed as a sum of products of phenomenological "fluxes" and "forces". This can be calculated from the fundamental laws of macroscopic physics. The law of conservation of charge reads Oee/Ot ------ - div i,

(1)

where O/at is the local time derivative, ~oethe electrical charge density due to both electrons and ions, and i the electric current. Conservation of energy can be written as {u~ + -~(E2 + B2)}/~t = - - div (Jq + c E A

B).

(2)

GALVANOMETIC

AND THERMOMAGNETIC

PHENOMENA.

I

69

The term on the left is the change in time of the density of internal energy % and of electromagnetic energy ½(E2 + B2). (Here E is the electric and B the magnetic field. We consider systems without electric or magnetic polarization). The right hand side gives the (negative) divergence of the energy flow (heat flow dq and Poynting vector). From (2) and Poynting's theorem, one obtains the expression for the change of internal energy, Ou,,/Ot = - - div Jq + i.E.

(3)

The entropy equation of G i b b s is Tdsv/dt = d u d d t - - Xk t*~ dek/dt,

(4)

where T is the temperature, s, the density of entropy, t*k the chemical potential per unit mass, and 0k the density of component k (ions or electrons). The differentials are substantial derivatives with respect to the centre of mass motion. Taking the velocity of the ion lattice zero, the centre of mass motion can be neglected, because the ions are heavy compared to the electrons. When also the fact that the density of the ions is constant is taken into account, one Can write instead of (4) T OsJOt = Outlet - - (~,/e) OcelOt,

(5)

where e is the specific charge of the electrons and [, the chemical potential per unit mass of the electrons. From (1), (3) and (5) the entropy balance equation can be derived : Os,,/Ot = - - div J, + ~,

(6)

with the entropy flow Js

=

(Jq --

i t, le)lr,

(7)

and the entropy source g given by Ta =

--

Jq.

(grad T ) / T + i. {E - - T grad ( / , / e T ) } > O.

(8)

Using the entropy flow (7) instead of the heat flow, (8) becomes Ta = - - Js.grad T + i.{E - - grad (/,/e)} > 0.

(9)

Expressing E in the electromagnetic potentials 9 and A, E

=

- -

grad 9 - - c- I OA/Ot,

( 1O)

one has finally from (9) the required expression for the local entropy production Ta = - - d,.grad r - - i.{grad (~/e) + c - ' OA/Ot} > 0, (11) where ~ = ~, + e9 is called the electrochemical potential of the electrons. § 3. E n t r o p y production in terms o/[luctualions.

In order to apply the fluc-

70

R. FIESCHI, S. R. DE GROOT AND P. MAZUR

tuation theory one should know dS/dt, the change of entropy per unit time, of the whole energetically insulated system, in terms of the fluctuations of the parameters which determine the thermodynamical state of the system, and of the variables conjugated to these parameters. Integration of (11) over the volume of the system gives the internal entropy production d~S/dt" d,S/dt = f o a d V = - - f v T - t [J,.grad T + i.{grad (~/e) + c -1 ~A/at}] dV. (12)

Let us put A T = T - - T o, A ~ = ~ - - ~ o , AA=A--A o and A i = i - - i o (To, ~o, Ao and i o being the equilibrium values) and remember that the equilibrium conditions require the constancy in time of all parameters, the uniformity of T o, ~0, Ao and the vanishing of i o. Then (12) can be written d~S/dT = - - f o T - I {d,.grad A T + Ai.grad (A~/e) + A i . c -~ aAA/at} dV. (13)

On integrating by parts the first and second terms on the fight hand side, using equations (I) and (6), and applying the condition that the system is insulated for heat and electrical current, (13) yields d,S/dt = - - T o ~fo (AT OAsdOt + A~ e =1 OAqe/at + A i . c -1 OAA/Ot) dV.

(14)

In the derivation third order terms have been neglected, since we need the entropy production up to second order terms only. We have introduced As o ~ - s o --(so) o and Ape = q e - (Qe)0, where the suffixes 0 indicate again equilibrium values. Integration of (6) over the whole volume and application of Gauss' theorem to the first term on the right hand side yields f~ (asd~t) d V = - - fa J,.df~ + f ~ a dV,

(15)

which can be written as dS/dt = d,S/dt + d,S/dt,

(16)

deS being the entropy supplied to the system by its surroundings. The condition of insulation of the system for heat and electrical current gives, from (16), (15), (14) and (7), dS/dt = d~S/dt = = --Tolfo(AT

~Asd~t + A~ e -~ ~Ao~/~t + A i . c -~ ~AA/~t)dV. (17)

One should remark that the condition of energetical insulation of a system in which electromagnetic phenomena occur requires special attention. It has been shown, however, that the expression (17) is left unmodified by this condition 7) 9). We have thus obtained the required form for the change of entropy of an energetically insulated system expressed in terms of fluctuations of infinite number of variables, which are continuous functions of space coordinates and time. The state variables Aso(r ) and A~,(r) are even functions of the indi-

GALVANOMAGNETIC AND THERMOMAGNETIC PHENOMENA. I

71

vidual particle velocities (so-called a-type variables 6) 8)), whereas the components of AA(r) are odd functions of the individual particle velocities (r-type variables); the variables AT(r), A~(r) and the components of Ai(r) are the corresponding conjugated quantities. R e m a r k. It should be noted that the local parameters are not completely independent, as is required in order to apply the fluctuation theory. The conservation laws for the total electric charge and for the total energy actually give two integral relations among the parameters themselves. This does not affect our conclusions, however, as can easily be shown.

§ 4. The phenomenological equations. With the flows and forces which occur in (l l) we can now establish the phenomenological equations as the linear relations i

=

- -

Lee.{grad (~/e) + c-1 aA/at} - - Le,. grad T, and

J , = - - L,e. {grad (~/e) + c-1 aA/at} - - L,,. grad T.

(18) (19)

Here Le~ is the electrical conductivity tensor, L,, is the heat conductivity tensor, and L~, and L,~ describe cross-effects. All these coefficients are, of course, functions of position and of the magnetic field. For our purpose it is necessary to transform these phenomenological equations in such a way that they give time derivatives of the state variables appearing in (17) in terms of the other state quantities in(17). We first write (18) in the form

c- 1 aA/at = - - grad (~/e) - - L2 ~. i - - L2 t . (L~,. grad T).

(20)

Using the electrical resistivity tensor R = L21 and the fluctuations o f the variables just as in the end of last section, (20) can be written

c -I aAA/at = - - grad (A~/e) - - R . A i - - R. (L~,.grad AT),

(21)

which we shall use below. Inserting (21) into (19) and again writing fluctuations of the variables, it follows that J, = L , ¢ . ( R . A i ) - L,,.grad AT + L,~,{R.(Le,.grad AT)};

(22)

Using (6) now and neglecting higher order terms, one obtains also for the second phenomenological equation the required form

~Asv/~t = - - div [L,~. (R.Ai) - - L,,.grad A T + L,~.{R. (L~,.grad AT)}]. (23) The formulae (21 ) and (23) give time derivatives of fluctuations, occurring in the expression (17), as a function of the other fluctuations, which appear in (17).

§ 5. Fluctuation theory. The fluctuation theory can now be used in order to calculate the following averages, which we shall need later. The form in which it will be applied here 8) is an extension of Onsager's method, and will provid~

72

R. F I E S C H I , S. R. DE GROOT AND P. MAZUR

the possibility of a straightforward derivation of the reciprocal relations (§ 7) amongst phenomenological coefficients from microscopic reversibility. Ape(r) .(2(r') A~(r')/e = k TI2(r') 6 ( r -

r'),

Ape(r) 12(r') Aii(r' ) ---- 0,

(24) (i = 1, 2, 3)

Ape(r ) $2(r') AT(r') = 0,

(25) (26)

c - ' AAi(r ) 12(r') Aii(r') = k Tbii f2(r') ~(r - - r'),

(i, j = 1, 2, 3)

(27)

c -l AAi(r ) 12(r') AT(r') = 0,

(i = 1, 2, 3)

(28)

c -1 AA,(r) g2(r') A~z(r')/e = 0,

(i = 1, 2, 3)

(29)

Asv(r) f2(r') AT(r') = kTg2(r') 6(r - - r'),

(30)

Asp (r) 12(r') A~(r')/e = O,

(31)

As,(r) f2(r') Aii(r' ) = 0.

(/' = 1, 2, 3)

(32)

Indices i and i indicate Cartesian components, r and r' indicate different points of the system, Oii is the Kronecker symbol and 6(r - - r') the Heaviside-Dirac-6-function. K2(r) is a differential operator of the general form g'2(r) = Xp,q,, am,,(r ) o~+q+'/ax~ ax~ a~,

(33)

where the coefficients ap,q,, are independent of the state variables, and xl, x 2, x 3 denote Cartesian coordinates.

§ 6. Microscopic reversibility. The fact that, on the average, the future behaviour of an aged system is identical with its past behaviour, will now be expressed with the help of correlation functions. Since a magnetic field B is present, it is necessary to reverse its direction in every point in order to have the particles retrace their paths. Minus signs arise (see (35) and (39)) when an a-type variable (Aso, Ape) is connected with a fl-type variable (components AA) 6). So the expressions for the microscopic reversibility, as they are used to derive reciprocal relations, read here

c-lAAi(r)(a/at) c-lAAi(r'){B. B'}=c-lAAi(r')(a/at)c-tAAi(r){--B,--B')}, C- 1

(34)

AAi(r) (a/at) Asv(r') {B, B'} = --Asv(r') (a/at) c-' AAi(r ) {-- B, - - B'}, (35)

Aso(r) (a/at) As~(r') {B, B'}

Aso(r') (a/at) Asv(r) {-- B , -

B'},

(36)

(a/at) Aoe(r ) { - - B, --B'}, Ape(r ) (alat) As~(r') {B, B'} = Asv(r') (a/at) Ap,(r) {-- B, - - B'},

(37)

=

Ape(r ) (a/at) Ape(r' ) {B, B'} = Ape(r' )

Ape(r) (a/0t) c -1AA,(r') {B, B'} = - - c -1AA,(r') (a/Ot) Ape(r)

(38)

{-- B , - B'}. (39)

B and B' indicate the magnetic field strengths at the positions r and r', where t h e averages are performed. The time derivatives which appear in

G A L V A N O M A G N E T I C AND T H E R M O M A G N E T I C P H E N O M E N A . I

73

these averages have to be considered as the average decay of the fluctiations, which is described b y difference quotients 5) 4). § 7. Derivation o/the reciprocal relations. With the preceding results, and the usual assumption that the average decay of the fluctuations follows the phenomenological macroscopic laws, the derivation of the reciprocal relations is straightforward• One should insert the phenomenological laws into the equations which express the macroscopic reversibility, and then use the results of the fluctuation theory. Introducing (21) into (34) one obtains c - ' A A i ( r ) [ - - grad {A~(r')/e} - - R(r', B').Ai(r') - - R(r', B') • • {L,s(r', B'). grad A T(r')}]i = c -l AAi(r' ) [ - - grad {A~,(r)/e} - - - R(r, - - B).Ai(r) - - R(r, - - B).{Le, (r, - - B ) . g r a d AT(r)}]/. (i, i = 1, 2, 3) Using (28) and (29) we are left with

(40)

c -l AAi(r ) [R(r', B').Ai(r')]i = c -1 AAi(r' ) [R(r, - - B).Ai(r)] i, (i, j ---- 1, 2, 3) or, writing explicitly in components,

(41)

c-' A A i ( r ) E,. Rim(r', B') Ai,.(r') = c - ' AA/(r') X,. Rim(r, - - B) Ai,,,{r ).

(42)

With (27) this becomes E., 6i., Rm(r', B') 6(r - - r') = E,,, 6/., Rim(r, - - B) 6(r - - r'),

(43)

or, eliminating the Kronecker 6's R/i(r', B') O ( r - r') = R i / ( r , - - B ) 6 ( r -

(44)

r').

The d-functions can be eliminated by multiplying both members with an arbitrary f u n c t i o n / ( r ' ) and integrating over r'. This gives Rii(r , B ) / ( r ) = Rii(r, - - B ) / ( r ) .

(45)

Since the f u n c t i o n / ( r ) is arbitrary, we find, equating its coefficients, R#(B) = Ri, ( - B), (i, j : 1, 2, 3) (46) which holds in every point of the system. In matrix notation (46) reads

R(B) = Rt(-- B),

(47)

where the sign* indicate the transpose matrix. This is the desired result for the s y m m e t r y properties of the electrical resistivity tensor• From (35), inserting (21) and (23), and using (28), (29), (31) and (32), one has, in components, C- 1 A A , t r ) [ - - Z i (O/Ox~) Zm. Lse,i,,, (r', B') R,..(r', B') Ai.(r')] = -~ - - Asv(r') [ - - Zmi {-- Rim( r, - - B) L ..... i(r, - - B) (O/Oxi) AT(r)}]. (i = I, 2, 3)

(48)

74

R. F I E S C H I , S. R. D E GROOT A N D P. M A Z U R

Using then (27) and (30) this gives Zim . O,.(O/Ox;) {L.,,i.Cr', B') Rm.(r', B') 0(r - - r')} =

= -- Y'-i R,.(r, -- B) L,,..i(r. -- B) (O/Oxi) 0(r - - r').

(49)

Elimination of the Kronecker 0 and of the &functions yields

y.;. L.,j.(r,B) R.,(r,B) (O/axi)/(r)=Z~iRi~(r,--B) L,,.~i(r,--B) (O/Oxi)[(r). (50) Since [(r) is an arbitrary function, we obtain, equating the coefficients of the same derivative, (for every point of the system)

Y'm Ls.,i.(B)R.,(B)

= X. R , . ( - - B) L .... i(-- B),

(51)

or, using (46),

Z. {L..,i,.(B ) -- L ..... i(-- B)} Rm,(B) = 0.

(i, m = l, 2, 3)

(52)

Multiplication of this equation by the invers tensor (R-1)i~ and summation over i gives L,.,i.,(B) = L.,,~i(-- B), (i, m = 1, 2, 3) (53) or, in matrix notation L..(B) = L.. t (-- B).

(54)

This is the s y m m e t r y relation for the coefficients of the cross-effects between heat and electrical conduction. From (36), inserting (23), and using the formulae (30)-(32) of fluctuation theory, we have

Z,i., . (O/Ox~) [{L,,,ii(r', B') - - L~.,#.(r', B') R,..(r', B') L~.,., (r', B')} (O/Ox~) 6(r - - r')] = = Zii.. (O/Ox,) [{L.~,~j(r, - - B) - - L..,,.(r, - - B ) R..,(r, - - B) L,s,mi(r, -- B)} (~/Oxi) 0 ( r - - r')].

(55)

On eliminating the 0-functions and writing in comPonents, we have, for every point of the system;

Z,i.. (OlOx,) [{L..,j,(B) --L,.,jm(B ) R...(B) L..,,,,(B)} (~lOxi)/(r)] = Y~,jm,, (a/ax,)

=

[{L.A--B )-L.,,.(-B) R.A--B) L..,~i(-- B)} (a/ax;)/(r)]. (56)

We equate now the coefficients of the same first order derivatives of/(r).

x;.,. (a/ax,) {L.,,(B) --L.,;AB) R.dB) L,,,.,CB)} = = X,,.. (Olex,) {L=,,;s (-- B) - - L,,,,.(-- B) R . . (-- B) L .... s(--

B)},

(57)

(i----- 1 , 2 , 3 ) or, w i t h (46) and (53),

x, (a/~x,) {t,s,j,(B) --L,,,,i(--B)}

= 0.

(j =

1, 2, 3)

(58)

GALVANOMAGNETICAND THERMOMAGNETICPHENOMENA. I

75

Moreover, equating in (56) the coefficients of the same second order derivatives of [(r), and applying again (46) and (53), we have, for the symmetric part L~ I of the heat conductivity tensor, L~I(B) = L~:I(-

B).

(59)

From (58), as we accept the independency of L,, of the shape of the solid, and assume that the tensor is zero in the e m p t y space e), it can be concluded that L,,.is(B ) = L,,.ii(-- B). (i, j = 1, 2, 3) (60) which reads, in matrix notation, L,,(B) = LL(-- B).

(61)

This is the desired relation for the heat conductivity tensor. Analogous calculations for (37), (38) and (39) will easily show that these expressions for the microscopic reversibility give no results. When (1) is introduced into (37), it is seen that both members vanish identically, since no correlations exist between de~ and Ai i ((25)). From (38), introducing (1) and (23), and using (25) and (26), one has again the vanishing of both members of (38). From (39), using (21), (1), and (24)-(27), one has -- (O/Ox~) O ( r - - r ' ) = Z i ~o.(O/Oxi) 6 ( r - - r ' ) ,

which is an identity. Hence all three relations (37), (38) and (39) are identically satisfied, which means .that microscopic reversibility can give no new results from these formulae. § 8. Remarks on correlations between a- and r-variables. It m a y be worthwhile to dwell on the various consequences of the expressions (34)-(39) for microscopic reversibility. The fact that results of a different kind are obtained is related to the character of the variables, which appear in the entropy production (17). The quantities Asv and A0~ are a-type variables with the corresponding conjugated a-type quantities AT and A~, whereas the components of AA are r-type variables with as the corresponding r-type quantities the components of Ai. The a-variables Asv and AO, are not exactly of the same character, because A~,/Ot is connected with the r-variable Ai through equation (1), whereas we have not a relation of such a kind for As~. For convenience we shall adopt the nomenclature a*-variables for variables, such as Ae~, which are connected with r-variables. We can now discuss the results obtained from formulae (34)-(39). The relation (34) states the equality of two correlations between r-variables, and gives as a result the reciprocal relations (47) foF the electrical conduction. The relation (35) is perhaps the most interesting; it states that a correlation between an a-variable and a (non-related).r-variable is equal to (but with oppo-

76

GALVANOMAGNETIC AND T H E R M O M A G N E T I C P H E N O M E N A . I

site sign) the correlation between this/5- and a. This is a (non trivial) example of the Casimir's formulation of microscopic reversibility with a- and/5variables combined, and in our case it leads to the reciprocal relations (54) for the cross-effects between heat and electrical conduction. The relation (36) is the usual case with correlations between a-type variables. Here it gives us the reciprocal relations for the heat conduction. The relation (37) yields no results, since both members vanish. Such a behaviour for correlations between two a*-type variables was first noted b y M a c h 1 u p and O n s a g e r 11). (38) also leads to the result of b o t h members becoming zero. This means t h a t no correlation between an a* and an a exist, nor is there a correlation between an a and an a*. Finally we found t h a t (3) gives an identity. This means t h a t a correlation between an a*.and a related/5 is identically equal (but with opposite sign) to the correlation between this/3 and a*. It is thus clear t h a t the complete result is obtained from the relations (34), (35) and (36) alone. If, however, we h a d started out with a t h e o r y in which we had not taken the "inertia t e r m s " Ai and OAA/atin (17) into account, this would not have been the case. In the first place, of course, (34), (35) and (39) would not have existed. F u r t h e r m o r e (3 i ) and its results would have remained unaltered. However (37) and (38) would now have given, as can easily be checked, the results for the electrical conduction and the cross-effects respectively. (Of course, in accordance with the form of the phenomenological equations, we find these results for the reciprocal relations in first instance preceded by the tensorial divergence operator, b u t it disappears when the limiting conditions explained in § 7 are adopted for all phenomenological coefficients. Thus exactly the same results as before are found). The authors are indebted to Mr J. V 1 i e g e r for useful comments. One of us (R.F.) is indebted to the Netherlands G o v e r n m e n t for a grant which enabled him to do research at the University of Leiden. Received 5-1-54. REFERENCES I) C a l l e n , H. B., Phys. Rev. 7:l (1948) 1349; 85 (1952) 16. 2) M a z u r , P. and P r i g o g i n e , I . , J . Phys. et Radium 12 (1951) 616. 3) P r i g o g i n e, I., l~tucle thermodynamique des phdnom~nes irr('versibles, Editions Deso('r, Liege, 1947. 4) G r o o t, S. R. d e, Thermodynan{ics of irreversible processes, North-Holland Publishing Company, Amsterdam and Interscience Publishers, New York, 1951. 5) O n s a g e r , L., Phys. Rev. 37 (1931) 405; 3B (1931) 2265. 6) C a s i m i r , H. B. G., Rev. rood. Phys. 17 (1945) 343. 7) M a z u r , P. and G r o o t , S. R. d e , Physica 19 (1953) 961. 8) G r o o t , S. R. d e and M a z u r , P., to be published. 9) M a z u r, P. and G r o o t , S. R. d e , to be published. 10) G e r l a c h , W., Handbueh der Physik, Vol. 13, Chapter 6, Julius Springer, Berlin, 1928; "M e i s s n e r, W., Handbuch der Experimentalphysik, Vol. 11, Akademische Verlagsgesellschaft M. B. H., Leipzig, 1935. 11) M a c h l u p , S. and O n S a g e r , L., Phys. Rev. 91 (1953) 1512.