Gravity-Induced Electric Fields in Metals1

M-C. LEUNG, G. PAPINI, AND R. G. RYSTEPHANICK Department oJPhysics, University of Saskatchewan, Regina Campus, Regina, Saskatchewan

Received June 25, 1971

Gravitational perturbations induce electric fields in metals by causing the lattice to deviate system- atically from its periodic structure. In the microscopic approach used in this paper, the induced electric fields inside the metals are calculated from the periodicity deviations of the lattice. The results hold for weak stationary or quasi-stationary gravitational fields of permanent or nonpermanent nature. Explicit calculations are carried out for centrifugal and Lense-Thirring fields and for the gravitational field of the earth. In the latter case, size and direction of the electric field agree with those determined by Dessler et al. and lend support to the experimental results of Beams. The formalism used can also be applied to superconductors. It is found that the gravitational field introduces some anisotropic terms in the electron- phonon interaction.

Les perturbations gravitationnelles induisent des champs dlectriques dans les mktaux en deviant systkmatiquement le rdseau de sa structure pkriodique. Dans la thkorie microscopique utiliske dans cet article, les champs Clectriques induits a I'intkrieur des mktaux sont calculds a partir des dkfauts de pkrio- dicitd du rkseau. Les rksultats obtenus sont valables pour des champs gravitationnels faibles, stationnaires ou quasi stationnaires, de nature permanente ou non permanente. Des calculs explicites sont prdsentks pour des champs centrifuges, des champs de Lense-Thimng et pour le champ gravitationnel terrestre. Dans ce dernier a s , I'intensitk et la direction du champ klectrique correspondent avec I'intensitk et la direction du cham^ determinks Dar Dessler et concordent avec les rdsultats ex~6imentaux de Beams. Le formalisme employ6 peut aussi Etre applique aux supraconducteurs. On a trouvk que le champ gravi- tationnel introduit quelque terrnes anisotropiques dans I'interaction klectron-phonon.

Canadian Journal of Physics, 49. 2754 (1971)

1. Introduction where M is the ion mass. Rieger (1970) has at-

Some recent experimental work by Witteborn tempted a microsco~ic approach to evaluate the

and ~ a i ~ b ~ ~ k (1967) and witteborn and pallesen electric field induced inside the metal. In all these (1967) aimed at determining whether positrons works, only the static gravitational field of the

rise or fall when placed in the E ~ ~ ~ Y ~ gravitational Earth is considered. Leung (1971) has indicated a field has motivated a number of authors to study simple way of calculating the effect of the Earth's

the effect of gravity on metals. since in the experi- field as well as the centrifugal and Lense-Thirring

ments of Witteborn and Fairbank, electrons and fields.

positrons are exposed to the gravitational field of From the experimental point of view, the situa-

the earth in metal pipes in which a vacuum has tion is not too clear. The results of Witteborn and

been created, the effect of the gravitational field Fairbank seem to agree with the conclusions of

on the pipe, if not properly taken into account, Schiff and Barnhill- These experiments have,

may lead to an incorrect interpretation of the however, been criticized (Tannhouser 1968;

data. Theoretical discussions have centered on the Michel 1968). Stress-induced have been

actual magnitude and direction of the electric field measured semiquantitatively Craig (l969). . E induced both inside and outside a metal. Schiff This work suPPortstheviewsex~ressedb~ Dessler

and Barnhi11 (1966) have, in fact, estimated E to et a[. Still other apparent ~ ~ d l r m a t i o n of the

be equal in magnitude to rngje and directed down- results of Dessler et a[. comes from Beams (1968)

ward, where rn and are respectively the mass and experiments on rotation-induced electric fields.

the charge of the electron and g is the accelera- data are not presented in this paper,

tion due to gravity. ~~~~l~~ et (1968) and H ~ ~ - Beams claims that his results are in reasonable

ring(1968) take into account the lattice compressi- agreement with the conclusions of Dessler et a[-

bility and in their calculation the electric field is research, particularly on the behavior of

directed and of order of magnitude Mgje superconductors in gravitational fields, has in the meantime widened interest in the topic mainly

'Work supported by the National Research Cou'ncil of because of the possibility, at least in principle, of Canada. using superconductors to measure gravitational

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRIC FIELDS IN METALS 2755

effects (DeWitt 1966; Papini 1966, 1967). From this point of view, the problem acquires a new perspective and should be treated as part of the new research field of gravitational effects in solids recently founded by the experimental work of Weber and collaborators (Weber 1969, 1970; Sinsky and Weber 1967). To our knowledge, no attempt has yet been made to measure the gravity- induced electric field in superconductors.

Our approach to the problem differs from those of other authors in several respects. First of all, the gravity field is treated within the framework of general relativity and in so far as this theory

describes gravity accurately, our results are more unified and general since they apply to any type of stationary or quasi-stationary gravity fields. In addition, our formalism applies to normal metals as well as to superconductors, although our approach differs from that of Rieger. To see this, let us examine Rieger's approach more closely.

In Rieger's paper, the gravity-induced electric field has been calculated through the electron- phonon interaction. Using the Born-Oppen- heimer approximation, a solid can be described by a wave equation for the electrons

a wave equation for the lattice

and an electron-~honon interaction

where r denotes the position of an electron; R, symbolizing (R,, ... R, ...), stands for the position of the ions, and hoo denotes the gravitational field which shall be further discussed in Section 2. The standard procedure (Ziman 1960) is to obtain normal modes that shall diagonalize the lattice Hamiltonian:

Rieger represents the normal mode as

Fq(R) = exp [i(qy Y + q,Z)] sin q,X where

qy,z=2nmy,zlLy,z m y , z = 0 , * 1 , f 2 , . . .

q, = (2p + 1)n/2Lx p = 0, 1, 2, ... and L,,y,, are the lengths of the sides of the metal sample and q, is defined so as to satisfy the boundary conditions of a free upper surface and a fixed lower one. The Fq given above, however, cannot

- . diagoglize the lattice Hamiltonian even in the absence of the gravitational field. We would like to remark at this point that although Fq does not appear to be the proper normal mode of the lattice for the more rigorous Born-Oppenheimer approach, it still can be used in the less demanding Debye model (Dekker 1 960).

In Sections 3 and 4, we calculate the gravity-induced electric field through the evaluation of the gravity-induced deviations of the lattice from periodicity. This does not require any knowledge of the normal mode and therefore the problem of diagonalization does not come into the picture. In the case of the Earth's gravitational field, our results agree with those of Dessler et al.

Another problem is to decide whether the gravitational field has any effect on the electron-phonon interaction (which is responsible for superconductivity). We show in Section 5 that the gravitational field causes an anisotropy in the electron-phonon interaction. While properties of superconductors such as the critical temperature would most probably not be affected appreciably by the existence of a

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2756 CANADIAN JOURNAL O F PHYSICS. VOL. 49, 1971

gravitational field, it is possible that the gradient of the order parameter may be nonzero along the direction of the field. As a consequence, the electric field induced inside a superconductor may be different from that induced inside a normal metal. Rieger has, however, come to the conclusion that the gravitational field has no effect on the electron-phonon interaction.

Since He, as given in [3] is unwieldy, it is usually replaced (Ziman 1960) by

or, in the jellium model (Abrikosov et al. 1965 ; Rieger 1970)

where p,, p , denote, respectively, the electron density and the ion density, Z is the ionic valency, and h is the screening wave vector. The scalar part, which Rieger calls H2 (i.e. the part which is independent of phonon operators) of He, is interpreted by Rieger as the one-body potential on the electrons. Due to the approximation [4] used for He,, it is not immediately clear whether Rieger intends to include H2 in the electron-phonon interaction He, or whether he considers H2 remaining as a perturbation in [I]. The placement of H2, however, has some important physical significance. From Rieger's descrip- tion, the form of his electron Hamiltonian, H2, appears to be considered as part of He,. In this case, aside from the small perturbation

m 2 C hoo(ri)

which can be neglected, there is no gravity-induced perturbation in [I]. The electron wave function is then the same as in the absence ofgravity. It then follows that the electron-phonon interaction remains unchanged, which is the conclusion of Rieger. However, with no gravity-induced perturbation term in [I], the electron density is uniform along the direction of the gravitational field. A current would then result, which is unphysical. In fact, as shown by Leung, the gravity-induced electric field can be deduced very easily from the knowledge that there is no current flowing.

The correct place for the gravity-induced perturbation is in the electron wave equation [I]. With this gravitational perturbation in [I], the electron wave function is no longer periodic. The gradient of the electron density along the direction of the gravitational field then does not vanish. In Sections 5,6 we show that the electron-phonon interaction is modified as a result of the electron wave function being perturbed from its gravity-free solutions.

2. The Gravitational Field In general relativity, gravity is introduced by means of the components of the fundamental tensor

g,,, with p, v = 1, 2, 3, 0. When the gravitational field is weak, g,, can be written as

. . "- - . gpv = qrv + hpv

where qpv = (1, 1, 1, - 1) is the Minkowski metric and h,, are small quantities of first order. By neglecting second- and higher-order terms, Einstein field equations can be linearized in the form

where T,, is the energy momentum tensor and G the Newtonian gravitational constant. Once the distribution of momentum and energy of the source is specified, T,, is known and the gravitational field may be calculated by means of eq. 5. The equations of motion of a particle of mass m in a gravita- tional field follow from the action principle

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRlC FlELDS 1N METALS 2757

where L is the Lagrangian of the system. From L it is possible to obtain, in linearized form in the limit of small velocities, the Hamiltonian

If we neglect effects proportional topihoi, then the gravitational field in H is simply represented by hoo. In the case of the Earth, hoo and the Newtonian potential 4 are related by (Landau and Lifshitz 1965)

24 hoo = - c

so that

where ME and R are mass and radius of the Earth respectively, and z the coordinate of the field point with respect to an axis whose origin is on the surface of the Earth. In particular, if R >> z we have

2GM, hoo - z = 2gz

3. Gravity-Induced Deviations from Periodicity in the Lattice

We consider a conductor under the influence of a gravitational field. For simplicity, we assume a cubic lattice and zero Poisson ratio for the conductor. The gravitational force Fl acting on the Ith ion is directed along the x axis. One end of the conductor is fixed and the origin of the coordinates lies in the plane of the fixed end.

We denote the instantaneous position of the Ith ion by

where 2,9, and 2 are unit vectors along the x, y, and z axes respectively. In the absence of any external field, the equilibrium position of the lth ion is denoted by

[8 I 1 = a(v2 + 1,9 + I,?) where v, I,, and I, are integers and a is the distance between nearest neighbor ions in equilibrium without external perturbation. With the gravitational field acting along the x direction, the new equilibrium position of the Ith ion is denoted by

[91 1, = X,,g2 + 1,ap + [,a2

an&(Xl;,/a) is no loxger an integer. Its value is now determined by the equation

where U[= U(R,, R2 ...)I is the potential energy in the lattice Hamiltonian given by [2] and 6 X l defined as

6 X = X,,, - va

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2758 CANADIAN JOURNAL OF PHYSICS. VOL. 49, 1971

is the gravity-induced deviation of the Ith ion from its periodic position. The force F, is equal to

where hoo is the gravitational field defined in Section 2. We assume that interactions exist only among the nearest neighbors (Kittel 1953). The potential

U(Rl, R, ...) in the lattice Hamiltonian can then be written as

While R, denotes the position of the Ith ion, Rl+ denotes the position of the ion at the site (v + 1, I,, I,) or (v, I, + 1, I,) or (v, I,, I, + 1). We substitute [ l l ] into [lo] and assume

to be a constant independent of 1 where s = Rl - R,+ and s, the x component of s. We also denote the length of the conductor along the x axis by L, and let

Fv denotes the field acting on the ion at the site where the x component of 1 is equal to va. We then obtain from [lo]

SXn - SXn-, = Fn/@

-(6Xv+1 -6Xv) + (6Xv - 6XV-,) = FV/a for 1 < v < n - 1

Adding up all these previous equations from v = v to v = n, we obtain

from which we get by summing v = 1 to v = v

In our model, the amount of deviation of an ion from its periodic position is independent ofthe values of I, and I, but depends only on v. This can be expressed by writing

[I31 SX, = SX,

As 6Xv is a function of v, we denote it by - .

‘*"[I~I sxV = s(va>

4. Gravity-Induced Electric Fields in Metals Since we expect the gravity-induced deviations from periodicity in the lattice to be small, the

electron Harniltonian for metals can be written in the form

The third term is the potential energy due to gravity-induced deviations of the lattice from periodicity. It is obtained by expanding u(v, R,) in a Taylor series about the gravity-free equilibrium position 1 and by neglecting all terms higher than the first one. It can be rewritten as

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRIC FIELDS IN METALS

S d3r (- epe)d(x)

where pe is the local electron density at u and

1 C d U " I +'(" = (-el , ax, .=, 6Xl

is the gravity-induced potential on the electrons, e being the charge on an electron. As the screening length is small, we assume that ua(u, R,) can be approximated by (Abrikosov et al.

1965; see also Appendix)

C161 47L

ua(r, R,) = -Ze2 - 6(r - R,) h2

where Z is the ionic valency and h the screening wave vector (Ziman 1965). We then have

Replacing the discrete sums over ions by an integral in [15] and using [16], the gravity-induced potential is

~ 1 7 1 4nZe a

+e(x) = -- - C~~(x)s (x) l h2 ax where p ,(x) is the "coarse-grain" local ion density. S(x) is defined by [14] and will be computed below for three different situations. The gravity-induced electric field is given by

This expression applies to any type of stationary or quasi-stationary weak gravitational field in the limit of small velocities. To illustrate our results, we calculate [18] for the following three cases.

(i) Earth's Gravitational Field The Earth's gravitational field is acting along the x axis of the conductor. The origin lies in the

conductor's bottom plane which is held fixed. In this case, from the expression for hoo given in Section 2 we obtain

F, = -Mg

where M is the mass of an ion. Substituting this expression for F, into [12], the gravity-induced deviation of the lattice from its periodic structure is given by

When elasticity is taken into account, the gradient of ion density along the x direction is (Musk- helishvili 1963; Kittel 1953)

where N, is the total number of ions, V is the volume of the metal, and Ye is the Young's modulus. To be self-consistent with the assumptions made earlier, the Poisson ratio is assumed here to be zero.

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2760 CANADIAN JOURNAL O F PHYSICS. VOL. 49, 1971

Integrating [20], the ion density is

Substituting [19] and [21] into [18], the electric field induced by the Earth's gravitational field is given by

Ex = --- - - h2 V C ,

where C , denotes the longitudinal velocity of sound. The relations

122 1 C12 = aa2/M

and C12 = Y,V/MNI

have been employed. After using the relation (Ziman 1965)

123 1 h2 = ~ K ~ ~ Z N ~ I V E ~ the gravity-induced electric field becomes

where E, is the Fermi energy of the electrons and the gravity-induced electric field Ex is, therefore, proportional to (M/m)g and is the same as Rieger's result.

(ii) Thirring's Field Consider a metal rod inside a shell of mass M , and radius R. The rod lies horizontally along a

diameter of the shell. This diameter is taken as the x axis and the center of the shell the origin. The rotation of the shell at an angular velocity o about its vertical diameter produces, according to Thirring (191 8), a force on the lth ion directed along the x axis:

125 I F, = - BMCO' x,,, where

C261

and M , is the mass of the shell. With the condition

1 >> B M w ~ / c I i.e.

.- ". 0 - . 127 I 1 >> B(wa/C,)'

which is satisfied in all practical situations, the force acting on the lth ion can be approximated by

Substituting this value into [12], we obtain the deviations from periodicity as

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRIC FIELDS IN METALS

The gradient of the ion density along x is

After integration, we obtain

Substituting the values of S(x) and p ,(x) into [18], the horizontal electric field induced by the Thirring field is then given by

Since the Earth's gravitational field is also present, there are in fact a vertical electric field and a horizontal electric field induced in the metal rod.

(iii) Rotation A metal rod lying in a horizontal plane rotates about a fixed vertical axis with an angular velocity o.

The distance along the rod from the center of rotation is denoted by x. This case is similar to the experiment performed by Beams.

An expression of Ex can be obtained in this case directly from eqs. 18 and 29 by taking

In several cosmological models, this is the condition relating mass and radius of the universe. The argument is frequently used in general relativity as evidence in support of Mach's principle. The result is

The electric field induced by the Earth's gravitational field is constant along the direction of the field, but the electric fields induced by the Thirring's field and by rotation are proportional to the distance from the center of rotation.

5. Electron-Phonon Interaction

In the absence of gravity-induced perturbation, the electron wave equation is given by

1321 h2

where ,

is the self-consistent periodic potential seen by each electron. Ek is the energy of the k state. The elec- tron wave function (unperturbed by gravity) satisfies Bloch's theorem, i.e.

[331 +k(r + I ) = exp [ik . I] +,(r)

where

When the gravitational field is present, the Hamiltonian for an electron becomes

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2762 CANADIAN JOURNAL O F PHYSICS. VOL. 49. 1971

where B is the gravity-induced potential given by

The electron-phonon interaction is represented by

where 1, is defined in [9],

with

[36 I qy,z=2nmy,zlLy,z m Y , , = 0 , * 1 , f 2 ... q x = ( 2 p + + ) x / L x p = 0 , 1 , 2 , ...

This is similar to the normal mode used by Rieger (1970). As pointed out in the introduction, this does not diagonalize the gravity-free lattice Hamiltonian, but can satisfy the less demanding Debye model. In our derivation of gravity-induced electric fields, we need not make use of the phonon mode.

The probability amplitude for the scattering of an electron from state k to state k t is given by

According to perturbation theory

C381

with

Substituting [35] into [39] with 6Xgiven by [I21 and [13], we obtain

Pk'k = 8ky -ky~,~ ,~k , -~ .~ .~ .bk~k

where K is the reciprocal lattice vector and

bk'k = [2NII1l2 C (') k'k C Uq,(O)r(kx - k,', 4,) n Ek - Ek, qx

T(k, - k,', q,) = C exp [i(k, - k,')va] sin q,v v

The longitudinal wave is denoted by h = 1. Ckjk, is independent of 1 and is defined by

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRIC FIELDS IN METALS

C k . k = S d 3 r $ v * ( r ) s l ax1 R z = I +k(r)

Substituting [38] into [37], we obtain

a~ ( r - R I ) C411 (klHeplk0 = J d3r C*(r ) x ' a R l

I 1 qI$k'(.) R= I

- R' I ql+k.(r) + x ~k,k* S d 3 r $ k ~ * ( ~ ) 5: a R l kl R l = I

au'(r - Rl) + x ~ k ~ k ' s d3r 4k*(r) x a R I k l 1

I q l $ k ~ ( ~ ) R1= I

6Xlql$kj(r) + higher terms ' I R l = I

Expressed in terms of the phonon operator, A, and A,+, 1411 becomes

where T(k, - k,,, q,) is defined by [40] and

ckk,(') and c,,,(~) are independent of I .

Q(k, k ' , q ) = 2 Uqx.'O' 1 exp [i(k' - k ) . la] exp [iq . I, sin q,x] 4x' I

= x, Uqxr 1 - exp [i(q, + q,')nl 1 - exp [i(qx + q,')nl

q x 1 - exp [i(k,' - k , + q , + q,')] + 1 - exp [i(k,' - k , - q , - q,')]

The evaluation of Ckk.( ') , Ckk,(2) , and Ckk, depends on our choice of a particular model for u,(r - Rl).

The first term in the square bracket corresponds to the gravity-free electron-phonon interaction, with boundary conditions Fq(x = 0) = 0 and Fq(x = L,) = 1. The other three terms are of higher order, somewhat similar to the matrix elements of two phonon processes. They differ, however, from the two phonon process in their anisotropic characters. This anisotropy is introduced by gravity. Since E is small in the metals, the gravity-induced anisotropy in the electron-phonon interaction may cause the induced electric fields in the superconductors to be different from those induced in the normal metal.

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2764 CANADIAN JOURNAL OF PHYSICS. VOL. 49, 1971

6. Electron-Phonon Interaction-Jellium Model

In this section, v,(v - R,) is approximated by [16]. We also represent $,(r) as

[42 1 $k(r) = exp [ik-v/V1I2]

where k = kxP + ky$ + kzZ

Using [16] for v, and [42] for in [41], the matrix element of the phonon4ectron interaction is given by

where qxx fX(kx - k,', qJ = )qX{-%SoLL. dx exp [i(k,' - kJx] sin - a

+ iP -SoLx dx exp [i(k,' - k,)x] cos - a a

and 41 = (qy9 + qz@

a W,(k, kl, q) = P , ~ S ~ ~ ~ dx exp [i(k,' - k,)x] + i $ + %) [ S(x) a

+ i S(L) + (k,' - k,) P(k - k,') + 4-L dx exp [i(kl - k)x]S(x) sin - a a I SoL a

As an example, we calculate explicitly the electron-phonon interaction for a metal subject to the Earth's gravity field.

In this case, from [17] and [19] we obtain

using relation [22] and the approximation p , = Nl/V. Using this expression for 6 and [42] for $,, we obtain

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRIC FIELDS IN METALS

kB T 1 bkkr =

P i ( J Z k - Ek.)(k,' - k,) and

with

The matrix element of the phonon-electron interaction is then equal to

f, is the electron-phonon interaction in the absence of gravitational effects. W,, N , , and N2 are gravity-induced and their anisotropy is due to the fact that the gravitational fields act only along the x direction. Similar results can be obtained in a straightforward fashion for the other two gravitational fields. The expressions Nl(k,, k,', q) and N2(k,, k,', q) are defined and evaluated as follows:

We shall change the summation into contour integration. The contour C is indicated in Fig. 1.

[44] Nl(k , , k,', q) = - ig 1 1

4 i - 1 + 4 X 4 x p - k , ' - 2 p - k , + - a a

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2766 CANADIAN JOURNAL OF PHYSICS. VOL. 49, 1971

FIG. 1. Integration contour for eq. 44.

1 4 [k," - (k, + %)I2(kx - k,' + ' a

Appendix

Instead of the seemingly drastic approximation [16], the potential u,(r - R) can be quite rea- sonably approximated by

- Ze2 exp [-hlr - RIJ

lr - RI .* "-- - -

The gravity-induced electric field in the metal (see [I51 and [17]) is

Replacing the sum by an integral and retaining only the zeroth order of the ion density, we have

For a point r not too near the surface, the integral can be written as

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LEUNG ET AL.: GRAVITY-INDUCED ELECTRIC FIELDS IN METALS 2767

= "5"-' I - x d X S(X + x) exp [-hlXll

(i) Earth's Gravitational Field Using [19], we have

where A,(x) and T,(x) are polynomials of second degree. For a point not near the surface, we have, after replacing I' by [23]

which is [24].

(ii) Thirring Field and Rotation Using [29], we obtain

4nZe NI Bw2 a2 X? E x = - - - - h2 v zcI2 ax2 x - + exp [-I(Lx - x)]A2(x) + exp [-Ixrz(x)]

where A2(x) and r2(x) are polynomials of third LANDAU, L. D. and LIFSHITZ, E. M. 1965. The classical

degree. For a point not near the surface, we theory of fields (Pergamon Press, New York),

obtain Chap XI. LEUNG, M. C. 1971. Phys. Lett. T o be published. MICHEL, F. C. 1968. Phys. Rev. Lett. 21, 104.

E = ---- E ~ 2 BW'X MUSKHELISHVILI, N. I. 1963. Some problems of the " 3 eC, mathematical theory of elasticity (Nordhoff, New

York). which is [381 when B = 4GM,/1 5c2R, and is [311 PAPINI, G. 1966. ~ h y s . ~ e t t . 23,418. when B -- 1. 1967. Phys. Lett. A, 24, 32.

Approximation [16] appears, therefore, to be R~EGER, T. J. 19%. Phys. R e v . . ~ , 2, 825-

quite reasonable. SCHIFF, L. I., and BARNHILL, M. V. 1966. Phys. Rev. 151, 1067.

ABRIKOSOV, A. A., GORKOV, L. P., and DZYALOSHINSKII, I. Y. 1965. Quantum field theoretical methods in statistical physics (Pergamon Press, New York), p. 76.

BEAMS, J. W. 1968. Phys. Rev. Lett. 21, 1093. CRAIG, P. P. 1969. Phys. Rev. Lett. 22, 700. DEKKER, A. J. 1960. Solid state physics (Prentice-Hall,

New York), Chap 11.

. . DESSLER, A. J., MICHEL~ F. C., RORSCHACH, H. E., and TRAMMEL, G. T. 1968. Phys. Rev. 168, 737.

DEWITT, B. S. 1966. Phys. Rev. Lett. 16, 1092. HERRING, C. 1968. Phys. Rev. 171, 1361. KITTEL, C. 1953. Introduction to solid state physics

(J. Wiley and Sons, New York), Chap. 111.

SINSKY, J. and WEBER, J. 1967. Phys. Rev. Lett. 18, 795. THIRRING, H. 1918. Phys. Zeits. 19, 33. TANNHAUSER, D. S. 1968. Phys. Rev. Lett. 20, 1123. WEBER, J. 1969. Phys. Rev. Lett. 22, 1320.

1970. Phys. Rev. 24, 276. WITTEBORN, F. C . and FAIRBANK, W. M. 1967. Phys. Rev.

Lett. 19, 1049. WITTEBORN, F. C. and PALLESEN, M. R. 1967. Phys. Rev.

Lett. 19, 1123. ZIMAN, J. M. 1960. Electrons and phonons (Clarendon

Press. Oxford), Chaps. I, V. 1965. Principles of the theory of solids (Cam-

bridge Press, London), Chap. V.

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