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Progress of Theoretical Physics, Vol. 36, No. 2, August 1966

Residual Attenuation of Shear Waves in Aluminum

Mario YOKOTA, Hiroyuki KUSHIBE and Toshihiko TSUNETO:

Department of Applied Physics, Osaka City University, Osaka *Department of Physics, Faculty of Engineering Science

Osaka University, Toyonaka

(Received March 7, 1966)

237

The purpose of this work is to discuss the residual attenuation of shear waves in the superconducting state of pure metals. The attenuation constant at temperatures just below the transition point is calculated directly from the pseudo-potential theory in the case of aluminum and compared with the experimental data.

§I. Introduction

The study of ultrasonic attenuation in metals, especially in the presence of a static magnetic field, has been one of the most powerful means of finding the electronic structure. The theories of the attenuation developed by Pippard and other people1

) are quite adequate for the purpose of analysing the experimental data and extracting information pertinent, for example, to the determination of the Fermi surface. These theories are either based on the model of free electrons interacting with a sound wave via an electromagnetic field, or are to be called semiphenomenological because they assume a phenomenological deformation potential. There is, however, another interesting fact which calls for further theoretical consideration and may give us some information about the electronic structure and the electron-phonon interaction, namely the residual attenuation of shear waves in the superconducting state.2

)

In contrast to the case of longitudinal waves which 'is rather well understood on the basis of the BCS theory of superconductivity, the attenuation of transverse waves seems to involve a more complicated mechanism. Its temperature dependence as (T) in the superconducting state apparently consists of two parts, a sudden drop right below the transition temperature Tc followed by a more gradual decrease which obeys the simple BCS formula for the longitudinal case, as/ an= 2f(i1 (T)), where an is the attenuation constant in the normal state, L1 (T) the energy gap and f the Fermi function.~) The sudden drop at Tc is explained by the cutting out of the electromagnetic interaction between the transverse wave and electrons due to the Meissner effect. In an ideal superconductor with a spherical Fermi surface and with an infinite mean free path we would expect to have vanishing attenuation of the shear wave as long as

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238 M. Yokota, H. Kushibe and T. Tsuneto

the wave length· of the sound is much larger than the relevant screening length. The residual attenuation we observe in real metals, therefore, arises from interaction mechanisms other than the electromagnetic interaction and is now generally understood to be due to the collision drag effect, as pointed out by Morse and Claiborne,4

> and to the deformation potential tensor acting on electrons which is associated with the non-spherical Fermi surface.5

> Let us consider this problem in a little more detail.

Suppose that the electronic system interacting with the impre~sed sound wave is described by a hamiltonian

H=Ho+H', (1)

where H' is the interaction energy term. There is a concise formula for the time rate of the energy dissipation to second order in H' (for details see Appendix A):

t

Q=[·d<H)(t)] = -i[) dt'/[ oHJf_)_,. Hl(t')])] , . (2) dt Av \ 0[ I Av

-co

where H1 is the part of H' linear in the amplitude of the sound wave. Here <H) (t) is the total energy of the electronic system at time t, <A)= Tr (exp [- /3 (Ho- fJ.N)] A) /Tr exp [- /3 (Ho- fJ.N)], all the operators are in the interaction representation and finally the time average [ ] Av may be taken over a cycle of the sound wave. In general, the interaction energy H' consists of three parts,

H' = Helemag + Himp + Ha. (3)

The first term, the usual electromagnetic interaction between electrons and lattice wave, can be written as (1/mc)JA-j dx in terms of the vector potential A(x) describing the self-consistent (i.e. screened) electromagnetic field produced by the motion of the ions and the electronic cqrrent density j(x).. The second term describes the effect of the motion of impurities fixed in the lattice if they are present (see reference 6) and Appendix A). Finally, the third term represents the potential of the ions; it is in ·principle distinguishable from the first by the fact that it does not vanish in the limit c~oo, and which plays no role in the shear wave attenuation if one neglects the band structure.

If we take the free electron model and substitute the first two terms of (3), we can easily derive the express~on for an in the normal state obtained by Pippard and Cohen, Harrison and Harrison. In the superconducting state, since the transverse electromagnetic field is screened out, A= 0, we have only the second term in (3) . Hence, in this model the residual attenuation comes from the collision drag effect. It has been shown that in the limit ql~O, as/ an= 2f(iJ)

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Residual Attenuation of Shear ·Waves in Aluminum 239

and there is no sudden drop at Tc whereas in the opposite limit ql~1 the

residual attenuation vanishes as as/ an= (1/ ql) 2f(L1) (Appendix A) . In a real metal in the normal state we expect all three mechanisms to be

effective and the picture is rather complicated since, as one can see from (2),

they interfere with each other. The attenuation due to various interactions is

obviously not additive. In the superconducting state of a sufficiently pure metal

(ql~ 1), however, we have a very simple situation in that only Ha is effective.

We will confine our attention to this case in order to study this particular interaction.

The problem we are concerned with in the main part of this paper is

whether or not one can get reasonable agreement with the experimental data

if one calculates the residual attenuation in a pure superconductor directly from

the pseudo-potential theory of electrons in metals. All we need is the same

information as that which is necessary for calculating the Fermi surface. In

fact, once we choose the pseudo-potentials of ions and the Fermi energy, the

attenuation constant can be calculated unambiguously; the shape of the Fermi

surface, which is of course necessary for the calculation, is determined at the

same time. One should also note that, since as we will see later the tempera

ture dependence of the attenuation arising from Ha is the same as in the case

of the longitudinal wave, we just have to calculate the attenuation constant in

the normal state due to Ha only (H' = Ha); this should give the magnitude of

as right below Tc.

In § 2 we will give the theory of attenuation using the pseudo-potential and

in § 3 we will describe the results of our calculation in the case of aluminum.

Fortunately, the electronic structure of aluminum has been studied intensively by

a number of people such as Harrison, Segall, I-Ieine and Ashcrofe) and is the

best known of the polyvalent metals. Since the deviation from the free electron

model is known to be not too considerable in aluminum, this seems to be the

natural choice. We shall compare our results with the experimental data on

shear wave attenuation in aluminum obtained by David et al. 8) In particular

the dependence of the residual attenuation on the direction of propagation and

of the polarization of the wave will be discussed.

§ 2. Interaction of shear wave and electrons .

It is well known that, if the electronic mean free path is much larger than

the wave length of the sound wave, one can treat the interaction of the wave with electrons in the normal state by simple perturbation theory, regarding the displacement of the ions as· small. In this case we may readily quote the ex

pression for the attenuation constant

(4)

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where V 8 is the velocity of the sound, p the mass density of the ions, wq and q the frequency and the wave vector respectively, and finally Pq 1s given by

(5)

In this expression eq is the polarization vector of the wave and Ek the energy of the electron and f(k) denotes the Fermi function. If one adopts the rigid ion model and assumes that the potential at a point is the sum of the potentials of all the individual ions, the matrix element vector Ik, k+q is given by

lk,k+q= (k+qiPwik)N, (6)

where w is the effective potential of an ion, N is the number density of ions and the wave function lk) is the Bloch function with crystal momentum k.

Since the magnitude of q is much smaller than the Fermi momentum kF, we can rewrite ( 5) as

Pq= -vs~ilk,k+q·eqi 2- af o(n· PkEk -·vsJ, k aEk n I

(7)

where n is the unit vector in the direction of wave propagation, which we take as the z axis. Changing the variables (k:c, ky, kz) to Ek, Vz = (1/h) fJEkjf)kz and the azimuthal angle cp between k and the x axis, we get

[001]

Fig. 1. The first Brillouin zone of aluminum.

(8)

where the integral is to be taken along the curve Vz = V8 =0 on the Fermi surface.

In order to calculate the matrix element Ik,k+q, we use the pseudo-potential theory. For details of the problems involved in the use of the pseudo-potential in this connection we refer the reader to the references cited above.7), 9

) Let us denote the pseudo

wave function by ifJk· In the neighborhood of the symmetry point W 1 of the Brillouin zone of aluminum (see Fig. 1.), for example, it may be expanded 1n terms of the free electron wave functions cj;k :

(9)

where Km for example is the reciprocal lattice vector (n/ a) (1, 1, 1) with the lattice constant a. Following Ashcroft, we assume in our calculation that ifJk is· always given by the linear' combination of four free states as in (9). Obviously, if ifJk is a free electron state, the product (IkJHq · eq) vanishes for the shear wave.

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Residual Attenuation of Shear Waves in Aluminum 241

Now the matrix element of the effective potential V with respect to the free electron states depends on k as well as on the reciprocal lattice vectors K, because of the non-locality of the potential and because it includes effectively higher order perturbation terms. We shall neglect this k dependence of V and regard it as a function only of the K's. Then the coefficients C in (9) are determined by solving th~ following secular equation:

(Tk- A) Ck,k + v2 Ck,k-K200 + vl Ck,k-Km + vl Ck,k-Kt-11 = 0,

v2 Ck,k + (Tk-IC200- A) Ck,k-K200 + vlck,k-Km + vl Ck,k-ICl-11 = 0,

vlck,k+ vlck,k-IC200 + (Tk-Km -A)Ck,k-Kl.n + V2Ck,k-Kt-11 =0, (10)

vlck,k+ vl Ck,k-K200 + V2Ck,k-TCm + (Tk-Kl-1.1-A)Ck,k-Kl-1.1 =0 .

. . Tk=k2/2m and V1 and V2 are the Fourier components V(K111) and V(K200)

respectively. The energy eigenvalue A is given by the solution of the secular equation

=0.

(11)

From the normalization condition of (/Jk we have

1 +s =C k,k[1 + f~oo +fin+ fi.n], (12)

where

(13)

A constant s appears because (/Jk is not normalized to unity, and according to Harrison's calculation based on the orthogonalized plane wave theory it is equal to 0.076 in the case of aluminum.*)

Using the wave function (/Jk determined in this way we can write the explicit formula for the matrix element Ik,k+q (see Appendix B for the derivation):

*) Depending on the definition of the pseudo-potential, the expression for the true matrix element in terms of the pseudo-potential and the pseudo-wave function ({Jk may not be just (CfJki

V pseudo I (/Jk') but involve a correction term due to the non-orthonormality of ({Jk (see reference 9)). Since we use below the values for V1 and V 2 which are determined by Ashcroft to fit th~ experimental data, we shall ignore this problem altogether. We would expect that the error due to this is of the order of c.

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242 M~ Yokota, H. Kushibe and T. Tsuneto

(15)

where we have kept only the terms linear in q and dropped all the components which are parallel to the vector q. The velocity vz and its derivative 8vz/8kz can readily be calculated with the help of the following equations:

()2). -- ()2F I 8F 8k/ - 8k/ 8;:. (16)

Thus, once we know the values of V1 and V2, the attenuation constant can be calculated without introducing any other parameters.

Table I.

~I Ashcroft I Harrison (a) I Harrison (~) Heine Segall I Animal~-Heine I

V1

I

0.0179

I

0.026

i

0.039 0.030(W) 0.026(W)

I

0.020

V2 0.0562 0.071 0.088 0.055(W) 0.048(W) 0.057

The unit is Ry.

In Table I we tabulate the values of V1 and V2 given by both the theoretical and the experimental analysis. The fact that they are reasonably close to each other assures us of their reliability to a certain extent, enough to justify attempts such as ours. We will use the values determined by Ashcroft together with his value A.= 0.85605 Ry for the Fermi energy.

Finally we should like to note that the interaction whose matrix element is given by (15) is of the same type as a scalar potential interaction in the sense that in the superconducting state its matrix element has the property

where ~ is the spin index. Therefore, the temperature dependence of the attenuation due to this interaction is the same as that of the longitudinal wave.

§ 3. Discussion and results of calculation

Let us first give a qualitative discussion of the basis of our formula for the residual attenuation. We can compare our theoretical calculation with the experimental data obtained by David et al. who measured the residual attenuation (denoted by a 1) of the shear wave in aluminum with different frequencies and with various possible directions of the wave propagation and its polarization.

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In Fig. 2 we have plotted their data for the case ql I (110). Since our theory is valid in the limit of infinite mean free path l~oo, we should compare our calculation with the slope of a 1 in Fig. 2 for large values of the frequency. where a 1 becomes proportional to w.*)

Fig. 2. The experimental values of the residua) attenuation a 1 [q// (110)] obtained by David et al.8>

etc. From these factors and the

The salient features of the experimental data are, firstly, that the residual attenuation due to the mechanism we are discussing is apparently of the same order of magnitude as the attenuation due to the electromagnetic interaction in the normal state arid secondly that it is dependent on the direction of polarization in the case of ql I (110), as is evident in Fig. 2, whereas in the case of ql I (100) it is not.

Turning now to the· formulae (8), (14) and (15) we immediately notice the presence of the two factors (q · ah_lak) and ( eq • K.i), where i stands for (200), (111), condition Vz = 0 we can tell where on the

Fermi surface we should expect a nonvanishing contribution to a1. In the first place, the line Vz = 0 which lies on the plane of mirror symmetry

of the Fermi surface perpendicular to the z axis will not make any contribution because of the factor (q · ah_lak). This is rather obvious, although we should note that in our approximate formula (15) this symmetry is lost. Hence, for example, the equator of the Fermi surface of the second zone cannot contribute in the case of ql I (100).. A similar argument shows that in this case we have no contribution from the lines Vz = 0 on the planes perpendicular to the z axis and containing the symmetry point X in the third zone.

In general we would expect an appreciable~'contribution from the line vz=O

on the Fermi surface near the Brillouin zone boundaries which are neither parallel nor perpendicular to the z axis. If it is parallel to the z axis, then (aflak) z will be the derivative along the boundary and consequently small. If perpendicular, the relevant reciprocal vector K will be perpendicular to e so

· that its contribution vanishes. In Fig. 3-5 we plot schematically the lines vz = 0 on the Fermi surfaces of the second and the third zones along which the integral (15) is to be taken in the case of ql I (110).

From the above considerations we can understand why the attenuation a 1

depends on the direction of polarization in the case of ql I (110). We know

*> The fact that the tangent of the curve, for example, of eq= (001) for large values of w

intersects the vertical axis at a positive value may be explained' by the collision drag effect which is present if l is finite. In fact, in the limit ql)>l the contribution of the collision drag effect to a 1

is given by Nmv8 vo/(pv82 l).

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from the theoretical calculation and the experimental data that V1<V2 • In the case of e// (001) it is evident from (15) that only the terms with K 002 survive among the terms involving V2• But since the zone boundary containing the point X is parallel to q, its contribution is expected to be small. In the case of ej / (110), K2oo and Ko2o have a finite product with e and at the same time the relevant zone boundaries are inclined with respect to q. Therefore, we expect a large contribution to al due to the large potential v2.

Table II

Direction 3rd Zone of

Polarization

[1IO]

[001]

A I

The unit is db/em MHZ.

Fig. 3. A schematic plot of the Fermi surface in the second zone according to the free electron model. The bold line is the curve vz=O for q//(110) along which the integral (15) has to be evaluated.

B I

c

W,O,IJ

h-W'P (J,O,OJ ri,I,OJ

2nd Zone

0.281

0.0317

Total

0.891

0.0873

Experiment

=0.7

0.167

Fig. 4 and 5. Portions of the Fermi surface in the third zone, on which the curves vz=O for q//(110) are drawn (A, B, C).

The results of our calculation for the case of q/ / (110) are given in Table II where we tabulated separately the contributions from the curves vz = 0 on the third (A, B, C) and the second zone which are schematically depicted in Figs. 3-5. The agreement of the calculated and the experimental values of a 1

is rather satisfactory in the case of e/ / (1IO). Note that it is indeed comparable to the attenuation due to the electromagnetic interaction which is, if we take the free electron model, equal to 1.64 db/em MHz in the present case. The theoretical value for ej / (001) is considerably smaller than the experimental value. This may be partly due to our choice of V1 = 0.0179 Ry which is the smallest given in Table I. For the same reason the similar calculation for the

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case of q/ / (100) also yields a value much smaller than the experimental one. The fact that a 1 does not depend on the direction of e in this case is to be expected from the cubic symmetry.

Acknowledgment

Part of this work was carried out while one of the authors (T.T) was a visitor to Dept. of Physics, Rutgers University, New Brunswick. He would like to thank Professor E. Abrahams and Professor P. Weiss for their hospitality.

Appendix A*)

In this Appendix we derive the formula (2) for the attenuation and apply it to the shear wave attenuation in an impure superconductor with a spherical Fermi surface.

Suppose that the electronic system interacting with the sound wave IS described by the Hamiltonian (1) with

(A·1)

where H1 and H 2 are the interaction energy terms linear and quadratic in the amplitude of the sound wave respectively. The average total energy of the system at time t is given, in the interaction representation, by

<H)(t) =<S-1(t)H(t)S(t)), (A·2)

where

t

S (t) = T exp{- i~ dt' [H1 (t') + H2 (t')]}

and

<A>= Tr ce-f3Ho A) /Tr e-f3Ho.

Expanding this expression we get to the second order In the amplitude

<H) (t) = <Ho) + <H2 (t))

t

-i ~ dt' < [H1 (t), H1 (t') ])

t t'

- i ~ dt' ~ dt" < [ [Ho, H1 (t")], H 1 (t')]). (A·3) -oo -co

Taking the derivative of (A· 3) with respect to time and noting that

*> The content of this appendix was reported in Technical Report of Rutgers University (1964 unpublished).

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M. Yokota, H. Kush£be and T. Tsuneto

dH1 (t) I dt = fJH1 (t) jot+£ [Ho, H1 (t)],

we obtain

(A·4).

-co

where f) jfJt operates only on the explicit time dependence. What we really

want is the average rate of increase of the energy which is given by taking

the time average of the expression (A· 4) over a cycle of the wave. Then the

first term drops out and we get our expression (2), which is a general formula

and may be applied to other problems of energy dissipation.

When the sound wave is of frequency w0 and wave vector q (q = wo/vs), we

can write

H = ( dx H' (x) eiqx-iroot 1 J qroo •

In this case it IS easy to show that

Q =~ Re {iwoK (q, Wo)},

where K (q, w0) is the Fourier transform of

t

K(x, t) = -i ~ dt'~ dx'([H'__q-roo(x, t), H~"'0 (x', t')])e-iqx'-iroot'. -co

(A·5)

(A·6)

(A·7)

In an. impure metal the interaction energy to first order in the amplitude

of sound wave, consists of two terms :

(A·8)

The first term is the electromagnetic interaction via the self-consistent field A

induced by the ion motion,

He1=~-A·p me

(A·9)

while the second enables us to take into account the effect due to motion of

impurities,

1 Hi=-[q· (p+q/2)] [u· (p+q/2)] -u(p+q/2), (A·lO) mwo

where uexp (iqx- iwo t) is the velocity field of the sound wave (for details see

reference 6)) . In the case of the shear wave the calculation of the attenuation constant is

straightforward. If we assume isotropic scattering by impurities, it amounts

to evaluating quantities of the form

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(A ·11)

where G (p, (J)n) is the thermal Green's function of the electrons in the presence of (fixed) impurities, g(p) is a polynomial in p and (J)n= (2n+ 1)nT. For the normal state one can easily obtain the formula given by Cohen, Harrison and Harrison, 1>

(A·12)

Here

(A·13)

is the total electronic current, (J = (J (q, {)) 0) the conductivity and <v) = ie/ e. Note that the second term in (A ·12), to be interpreted as the collision drag effect, comes from < [Hi, He1 +I-Ii]) in the expression (A· 7) .

In the superconducting state the electromagnetic interaction becomes negligible because of the Meissner effect except in the region of temperature very close to Tc. One can show explicitly that all the terms in (A· 7) involving the self-consistent field A are indeed negligible compared to their values in the normal state. What is left is the effect of moving impurities described by · the <[Hi, Hi]) term in (A· 7), which is equal to the appropriate analytical continuation of

where

2n: 1

Kii= -iN(O) ~d~ ~ dtLlul 2 _g- (1-,LL2)tL o -1 (J)o

00

~ d~ XPo2 Vo ~-·nT~I,

2ni "' n

I= (qvo,LL-i(J)0)G(p)G(p+)- (qvo,LL+i(J)o)F(p)F(p+)

=G(p) -G(p+)

(A·14)

+ 1 ( iwn _ i(wn+(J)o) ) (G(p) G(p ) +F(p)F(p )] 2r V {)),/ + L12 V ((J)n + (J)o) 2 + J2 + +

~ (V(J)n21+iP - V((J)n+~o)2+J2) [G(p)F(p+) +F(p)G(p+)].

(A·15)

Here G (p) and F(p) are the Green's function of the superconducting state in

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.the presence of impurities*) and P+ = (p+ q, Wn + Wo), Po the Fermi momentum, §= (p2

- Po2) /2m and Vo= Po/m.

We shall evaluate this only in the two extreme cases:

1) -rJ, ql~1

)~~I~ ( ~-~) (1- w+w+ L12

) _L12(w+ + w) (_!_ __ 1 )

2nz e+ e e+e e+e e+ e

-2iar (1 w+w+L12) + 2r (e/-e2

) (w++w) ee+ (ee+) 2

(A·16)

2) r~oo

(A·17)

where W = Wn, (J)+ = Wn + Wo, a= qvofl. and e = v' Wn2 + A2• Substituting these into

(A ·14) we get

r~O, (A·18)

= i 2Nm qvo 2f(J) lul2, 3 rwo

(A·19)

Using Pippard's formula for the attenuation constant in the normal state, we :finally obtain

as!an=2f(J), r~O

= (3n I 4ql) 2f(J) , -r~ oo.

Our results confirm, in the two extreme limits, the formula

as =-3~ { (ql)2+ 1 tan-1 ql-1} 2f(J) an 2(ql) 2 ql

(A·20)

(A·21)

obtained by Claiborne and Morse with the help of a semi-classical theory.4) As

was suggested by them, one can interpret the residual attenuation as caused by moving impurities splashing thermally excited quasiparticles. It should be emphasized that the collision drag effect, which is negligible in normal metals except at very high frequency~this was the reason that one of the authors (T. T) incorrectly neglected it in former work6)~, becomes quite effective once the electromagnetic field is switched off by the Meissner effect.

*) The derivation of .(A·9) and the subsequent calculation can be carried out along the same lines as the calculation of the electromagnetic properties of impure superconductors; this is, for example, presented in reference 10) Chapt. 7, §§ 37 and 39.

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We have not carried out a calculation for intermediate values of ql since it is rather difficult and not so rewarding. For qli'"'V 1 it does not in general seem possible to separate out the factor depending on ql as in the two extreme cases discussed.

Appendix B

We proceed as follows to obtain the formula (15):

Ik,k' = )'/?_::, Vw '/?k dr

+···

+ ck',k'-IC200 Ck,k-Km) ¢k'-K200 Vw cfJk-Km dr

+ Ck,k-K200 ck',k'-Km) ¢k'-Km Vw cfJk-K200 dr

+···

+···

+···

+.:.

+···.

Rewriting the expressio· ns 1n the brackets ( ) 1·n terms of + etc J200, .,

(B·1)

(B·2)

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_ cz ( + 8hoo _ + 8fm ) - k,k J lllah J 2008k (B·3)

we readily obtain (15).

References

1) A. B. Pippard, Proc. Roy. Soc. A257 (1960), 165 ; Repor~s on Progress in Physics 23 (1960) ' 176. M. H. Cohen, M. ]. Harrison and W. A. Harrison, Phys. Rev. 117 (1960), 937.

2) R. W. Morse, Progress in Cryogenics I (1959), London, p. 221. 3) J. Bardeen and J. R. Schrieffer, Low Temperature Physics III, edited by C. J. Gorter,

(1961), Amsterdam. 4) L. T. Claiborne, Jr. and R. W. Morse, Phys. Rev. 136 (1964), A 893. 5) ]. R. Leibowitz, Phys. Rev. 133 (1964), A 84. 6) T. Tsuneto, Phys. Rev. 121 (1961), 402. 7) B. Segall, ·Phys. Rev. 124 (1961), 1797.

V. Heine, Proc. Roy. Soc. (London) A240 (1957), 340, 354 and 361. W. A. Harrison, a) Phys. Rev. 131 (1963), 2433; {1) Phys. Rev. 136 (1964), 1107. N. W. Ashcroft, Phil. Mag. 8 (1963), 2055. V. Heine and A. 0. E Animalu, Phil. Mag. 12 (1965), 1249.

8) R. David, H. R. Van der Laan and N. J. Poulis, Physica 29 (1963), 357. 9) B. J. Austin, V. Heine and L. J. Sham, Phys. Rev. 127 (1962), 276.

10) A. A. Abrikosov, L. P. Gor'kov and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, English Translation, Prentice-Hall, Inc., (1963).

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