ANNALS OF PHYSICS: 46, 76-113 (1968)

Inequalities, Instabilities, and Renormalization in Metals and Other Fermi Liquids*

A. J. LEGGET?

Department of Physics, University of Illinois, Urbana, Illinois 61801 and

Department of Physics, Harvard University, Cambridge, Massachusetts 02138

This paper discusses various aspects of microscopic Fermi-liquid theory, with particular emphasis on its application to metals. It is shown that under certain conditions we have the inequalities

The appropriate formulation of the theory is presented for the case that the fermions interact with an arbitrary background (e.g. 3He-4He mixtures). A microscopic proof of the stability conditions is given. The effects of the Coulomb and electron-phonon interaction in metals are treated by completely parallel techniques, and explicit expres- sions are given for the electronic correlation functions in the limits w 3 wn and w < on . Results for the static properties, etc., are in agreement with the recent work of Prange and Sachs. It is shown that the electron-phonon interaction can only enhance the “dynamic effective mass” m*/(l + +F,), and that the enhancement tends to zero for forward electron-phonon scattering. It is also shown explicitly that phonon effects on the Landau parameters can affect the static properties in the superconducting state even though they cannot in the normal state.

The inequalities obtained are compared as far as possible with experiment, and it is shown that the rough value of &/12(=&) recently obtained for Na from CESR experiments is almost certainly not compatible with the optical data. Tentative application is also made to the recently observed Knight shift in superconducting Al.

I. INTRODUCTION

It is by now clear that Landau’s semiphenomenological theory of Fermi liquids (I) can be generalized so as to provide a convenient framework for the discussion not only of liquid 3He but also of the electron “liquid” in metals (2)

* Work partially supported by DA-31-124-ARO(D)-114, U. S. Army Research Office (Durham) and NSF-GP-5321.

+ On leave of absence from Magdalen College, Oxford, England. Present address: School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, Sussex, England.

76

METALS AND OTHER FERMI LIQUIDS 77

and of the low-energy properties of medium and heavy nuclei (3). A further extension of the theory is the combination of Fermi-liquid theory with quantum hydrodynamics to give an exact semiphenomenological description of the low- temperature properties of sHe-4He solutions (4). Apart from the quasi-particle effective mass m*, the fundamental semiphenomenological parameters of the theory are the coefficients in the expansion of the quasi-particle interaction parameter f(pp’, aa’) for quasiparticles at the Fermi surface. For an isotropic system of fermions we have

f(PP’, 4,s,=,P’,‘* = (2)-l c {Fz + Zp - CT’} P,(cos f?).

Here (dn/&) is the total density of states (of both spins) at the Fermi surface, cos 6 = p * p’jp Fa, Pz is a Legendre polynomial and Q, Q’ are spin (not Pauli) matrices.l In principle, a knowledge of the quantities Fl and ZI (plus m*) fixes all the low-temperature, low-frequency, and long-wavelength static and dynamic properties of the system. Determination of the transport coefficients requires, in addition, a knowledge of real scattering processes, which is not incorporated in the Fl .

The aims of this paper are two. First, we shall derive some inequalities for the Landau parameters, which apparently have not appeared previously in the literature. These inequalities will apply to pure 3He, 3He-4He mixtures and (under certain rather general assumptions) to metals, and we shall discuss how far the experimental data presently available on the alkali metals agrees with the theory. Secondly, we shall attempt to give a concise account of the Landau theory for metals, taking into account both the Coulomb interaction and the electron-phonon interaction. While the general form of this theory is well-known, and it has recently been worked out in detail by Heine, Nozieres, and Wilkins (5) and by Prange and Sachs (6), there are certain questions which are not answered explicitly in these papers. For instance, while it is well known that phonon renormalization effects on the static and kinetic properties in the normal state always cancel, it is neither obvious nor even (as we shall show) true that they cancel in the superconducting state. It is also not entirely obvious whether the phonon-induced interaction between quasiparticles can lead to instability of the system even though it was stable in the presence of the Coulomb interactions alone, and we shall discuss this question.

In section II we discuss the case of a system of fermions alone or in an environment with which they do not form bound states (such as dilute solutions

1 The situation in the literature regarding the normalization conventions for the Fz and Z, is chaotic. Here we follow the “Russian” wnventionl; thus our Z, , for instance, is 4Fal in the language of Ref. (2) and 4(2f + 1) & in that of Ref. (8).

78 LEGGETT

of SHe in 4He). We shall show that subject to certain very general conditions we have the relations

m* l+)F, Zm’ (24

Pb)

In the process we shall also be able to give a microscopic proof of the well-known stability conditions (7), which for a pure system read

l+2z+l ’ Fz >o &I4

l+21+1>O.

As a byproduct we shall have a microscopic theory of mixed systems which will be applicable, e.g., to SHe4He mixtures.

In Section III we give a formulation of the Landau theory for metals which is suitable for our purposes. We derive the explicit form of the electronic correlation functions for frequencies low and high compared to the Debye frequency and show how the parameters entering the two expressions are related. We discuss how the electron-phonon interaction affects static properties such as the spin susceptibility in the superconducting state. We further show that we always have the relation

m* 1 +iFl >mo&

where moat is the “optical effective mass” which appears in the infrared dielectric constant, and argue further that it is very plausible that

m*

l+&Z1 2 m0pt . (5)

These results depend only on the nature of the electron-phonon interaction and are independent of how we initially set up the problem (for instance, whether we describe the conduction electrons by a momentum-dependent pseudopotential).

In Section IV we discuss briefly the conditions under which we can prove the further inequalities (2a) and (2b) for metals, put together all the inequalities we have obtained and compare them as far as possible with experimental results [optical and CESR (conduction electron spin resonance) data] for the alkali metals. It will be shown that the rough values of the Landau parameter Z,/12( =B,), recently obtained from CESR data (8), (9), are almost certainly incompatible with the optical data. We also discuss briefly phonon effects in the supercon- ducting state, with particular reference to the Knight shift in Al.

METALS AND OTHER FERMI LIQUIDS 79

Section V is a brief conclusion; we review the results obtained and discuss the physical interpretation of our inequalities. Since the ratio of formalism to new results in this paper is (regrettably but, apparently, unavoidably!) rather high, readers interested only in the results may find it convenient to read only Sections IV and V, which we have tried to make as self-contained as possible.

II. INEQUALITIES FOR “SIMPLE” SYSTEMS

In this section we shall derive the inequalities (2) for a system of N fermions interacting with an arbitrary set of “background” particles via isotropic spin- and velocity-independent two-body potentials, and satisfying the following relation between the Fermi momentum pp and the total number of particles N,

N = pp3/3r2, (6)

where we have set A = 1 and taken the volume of the system equal to unity. The condition (6) excludes, prima facie, both liquid and solid metals and, as we shall see in Section IV, some extra assumptions are needed in order to apply the results of this section directly to these cases. We make no assumption in what follows about the presence or absence of any kind of long-range order (e.g., Bose condensation or lattice periodicity) in the background particles. However, it is of course essential that the Fermi component can be described by Fermi-liquid theory, in the sense that the irreducible one-particle self-energy and two-particle scattering amplitude are sufficiently well-behaved near the Fermi surface.

For the special case of an isolated Fermi liquid (or one interacting with a static background), the inequalities (2) were essentially derived, without being stated explicitly, in Ref. (20). The technique we shall use here is a more or less trivial generalization of the one used in that reference. Before embarking on the mathe- matical details, let us sketch the principle of the method.

The crucial physical assumption is that at sufficiently low temperatures and low excitation energies, the behavior of the fermion component (and possibly, but not necessarily, of the background also) can be characterized in terms of the motion of quasiparticles, which are described in terms of a few semiphenomeno- logical parameters. This is equivalent to the assumption that Fermi-liquid theory holds for the fermion component. To put it more quantitatively, we require that the inverse lifetime 7-l of a typical fermion quasiparticle should be small compared to a characteristic energy of the fermion system, which we take to be of the order of the Fermi energy 6F ; only if this is so is the quasiparticle concept a simply- defined one. Then let us consider, say, the quantity

(7)

80 LEGGETT

where JI, is the Fourier component of the fermion current operator. The principle of our argument consists in writing this quantity in the form

K,&JJ) = @, + J~&J), (8)

where @?, is a constant and J,(kw) a function of k and w which tends to zero in the limit k/o + 0. J,(ko) is then identified with the contribution to the response function (7) from the excitation of quasi-particles, and 0, with the contribution from highly excited states which can be regarded as involving the breakup of a quasi-particle (or, perhaps, as two- or multi- quasi-particle states). In the limit o/k + 0, the function J&O) can be written down explicitly in terms of the Fermi- liquid parameters, while under the assumptions made above we have a sum rule for the whole quantity K,(k, 0). Thus we get an explicit expression for the “non quasi-particle” contribution QD . On the other hand, it follows from (7) that QP must be negative (or zero), which gives us the required inequalities.

We now fill in the mathematical details. For simplicity we shall first consider the case that the “background” particles are themselves fermions which obey a Fermi-liquid theory, as this enables the derivation to be put in a particularly compact form. It is then easy to see how to generalize it to the case of arbitrary background behaviour. We shall continue to refer to the system of primary interest to us as the “fermions” and the rest as “background.”

As in Ref. (IO), we consider the autocorrelation function (retarded Greens function)

K,(kw) = <<& : &&(w) = -i 1” e(t) < [6(t), L(O)] > eiWLdt, -co (9)

Here a+, is a fermion creation operator and &p~) some function of p and a; while we can consider general functions, the formalism is considerably simplified if we restrict ourselves to functions &cr) which form irreducible representations of the rotation and spin-rotation groups2 (in the general case of an anisotropic Hamiltonian, we should restrict ourselves to the irreducible representations of the corresponding symmetry groups of the Hamiltonian). Examples of &pa) which we shall want to consider are 1, u, pr , ppa; the corresponding expressions Kl , K, , K, , KD,, are respectively the density, spin-density, current and spin-current autocorrelation functions3

* This point was not mentioned in Ref. (IO). However, all functions [[PO) considered there do in fact fall into this class.

8 For notational simplicity we quantize the spin along the z axis and deal only with operators &IO) which are diagonal with respect to spin in this representation [cf. ref. (IO)].

METALS AND OTHER FERMI LIQUIDS 81

In the standard manner we can write

(10

If we were dealing with a one-component system, then we could write in the usual manner [see, e.g., ref. (lo)]

K&w) = ((GG + GGrGG) 8,

and the Dyson-like equation

(12)

r = r(l) + P’GGT, (13)

where r is the complete two-particle scattering amplitude and F(l) the part irreducible with respect to an intermediate particle-hole pair of momentum- energy (k, CO). we define the exact meaning of the implied matrix multiplications in (12) and (13) below.] In our case (a two-component system), Eq. (12) is unchanged but Eq. (13) must be changed to allow for propagation of an intermediate background particl*hole pair; i.e., we must have

r,, = r,,(l) + ~,,YGG>J-‘,, + r,,‘“‘(GG)J,, (14)

and a similar equation for r,, ; here (only) the subscripts f and b refer to fermion and background particles, respectively, and rtdl’, for instance, is the part of the total fermion-background scattering amplitude which is irreducible with respect to both fermion and background particle-hole pairs of energy-momentum (k, w). The simplest way to make the appropriate generalization is to treat (12) and (13) as matrix equations in a 2 x 2 “space” corresponding to fermions and background.4 We therefore define matrix multiplication and the trace operation as follows:

whfl (PP’, cm’, EC’) E c c / dSplde”A,Jpp”, ud, EE”) B&p”p’, u”u’, E”C’), (15) 0. Y

Tr A = c c 1 dgpdeA,(pp, uu, GE). (16)

Then we can write the generalized forms of (12) and (13) as

(17)

(18)

4 This 2 x 2 space should, of course, not be confused with either the 4 x 4 “arrow” space used in Ref. (IO) to treat pairing correlations, or the i-j space used there to deal with the effects of finite temperature.

82 LEGGE’lT

where g s g(ku) is the diagonal matrix

g&pp’, cm’, EE’: kw) = 8,6,d(p - p’) 8(e - E’)

x I &- Ga@ + k/2, E + 49 GAP - k/&e - 44

= S,S,4@ - p’) 8(~ - E’) g&e : kw)

and ,$ the matrix

cw

Here for notational simplicity we have assumed that &p, 0) is diagonal in the a,-representation (cf. Footnote 3); we have taken 01 = 1 to correspond to the fermions in the 2 x 2 “space.” Notice that (k, w) are parameters in the matrix equations (17) and (18); we shall hereafter not usually write them explicitly. Finally, we note that P)(ko) = P)&.BP’, uu’, EE’ : km) is the graph shown in Fig. 1. To tist order in the interparticle potential V, , the graph reduces to

V,(k) - S,&,V,,(p - p’), the sign being guaranteed by the factor --i in the definition of g [Eq. (19)]. The formula given above for g is appropriate to the zero-temperature case; at finite temperatures a slight generalization of the technique is needed [see ref. (lo)], but the results are very similar.

We are now in a position to generalize the techniques of Ref. (20) in a straight- forward way. In g&km) = g&w) we separate out the part gn coming from the product of the pole parts of G, in (19) from the rest, which is insensitive to the ratio k/w :

or

We then have explicitly (II)

where a, , v,(O) and Up, are, respectively, the Greens function renormalization constant, density of states at the Fermi surface, and Fermi velocity for the oath set

METALS AND OTHER Fl3RMl LIQUIDS 83

of particles. Delming now a “quasi-particle-irreducible” (matrix) amplitude by the equation

PJ = P + P’g,P, (23)

and using the same matrix-algebraic trick as in Ref. (IO), we can finally write for Kt

K,(kw) = Qe + Tr I [R+ c 1 -g$gI;kw) ‘tgly (24)

where CD, is a constant

Q-S = Tr 5kf + d%~ 5 (25)

and R, is a renormalization factor [“effective charge” in the language of Migdal(3)], which in our case is, for given 6, a constant matrix in the 2 x 2 space; explicitly, (R&, = 6,,(R& is the quantity (1 + Pg,) 4 evaluated at I p I = pp. , 6 = eF, . Since we restricted 5 to belong to an irreducible representation of the rotation and spin-rotation groups, the quantity (R,& has the same symmetry properties as 5, so that (R,), is just a number. It is a trivial generalization of the standard technique [ref. (Il),p. 1581 to show that if the quantity &,, &u)u+&,,, is conserved, i.e., if the physical quantity in question is conserved for the “fermions” separately, then

Qc = 0, (R&b = Las-l. (26)

In the case of interest to us the “fermion” density and spin density are conserved, but not the current or spin current; the renormalization factor R, for the latter will then be a nondiagonal matrix [cf. Ref. (3)].

Let us write (24) in the abbreviated form

K&o) = Qp, + J&w). (27)

From the explicit form of g&o), [Eq. (22)] we see that J&J) tends to zero in the limit k/w -+ 0. Now we compare (27) with the spectral expansion

(28)

We see that J,(kw) is to be identified with the contribution to the sum in (28) from states for which o,,, tends to zero with k, i.e., the “quasi-particle” states (including possible collective oscillations of the quasi-particles), while 0, is the contribution to the sum from states with w,, independent of k in the limit k + 0, which we may regard as involving the breakup of quasi-particles. Taking the limit w + 0, and noting that if our ground state is stable we must have w, > 0, we therefore draw

84 LEGGETT

the conclusion, which is fundamental to our proof, that both Qi, and J&k, 0) separately must be negative or zero:

Let us pause at this point to examine the case when the quasi-particles are not perfectly well-defined (say at finite temperatures). Qualitatively, the effect is to replace the quantity v l k/( w - v * k) in (22) by (v l k - i/~)/(w - v * k + i/~), where T is a characteristic quasi-particle lifetime. (There are actually also effects on the quantities P [cf. Ref. (IO)], but these do not affect the nature of the results.) Thus we can still use the above arguments provided there exists a region of k and w such that l/~<(w, vk)<c,, where E, is some characteristic energy at which we can no longer write K(kw) in the simple form (27). We may take this to be of the order + . Thus provided EAT > 1 the argument should go through (actually, if this condition does not hold the simple Landau theory is meaningless anyway). We shall show below, however, that this condition need only be fulfilled for the fermions and can be relaxed for the background (though in all cases of practical interest it is in fact fulfilled).

The next stage in our proof consists in noting that if K1 is the density auto- correlation function and K,, the longitudinal current autocorrelation function, i.e., K$ = <Jk : J-& where by definition

w&A, = k(J,bno 9 (30)

then we have directly from (30) and the spectral expansion (28)

(31)

where

C = 2k-a C w,,~ l@dno la = --k-?[pk, [pk, H]l> (32) R

is a quantity for which we generally have a sum rule. In particular, for the case assumed (velocity-independent potentials) we have

C = N/m. (33)

We now substitute (27) and (32) in the right-hand side of (31), and take the limit k/w + 0. Then since J&kw) + 0 in this limit and Qe < 0, we get

(34)

METALS AND OTHl3R FERMI LIQvrDS 85

On the other hand, the quantity on the left-hand side of (34) can be evaluated directly in terms of the Fermi-liquid parameters. Using (26), we have

K,(kw) = al-* Tr I E (35)

Since according to (22) g&w) tends to zero as k/w + 0, we can expand this expression in powers of g, . Writing out the indices 1 and 2 corresponding respectively to “fermion” and “background” particles explicitly, we get from (22) up to order gn2 :

+ O(k4/co4). (37)

We see that the properties of the “background” particles do not enter to this order. Thus, writing as usual (Z2)

(38)

we get

lim (02/k2) K,(kw) = &u~~v~(O)(~ + )Fd, kltLl+0

(39)

and thus from (34)

h2dO)(1 + UT3 < N/m. (40)

But according to (6) and the definitions u relation

F = (dc/dp)% = pp/m*, we have the

-&2vl(0) = N/m* (41)

and thus we finally get the desired result

m*/(l + 44) > m. (42)

We can go through a precisely similar argument as regards the spin density. However, to get Eq. (33) in this case it is essential that the Hamiltonian be spin- independent. If this is so, we get the inequality (2b)

m*/(l + $2&) 2 m. (43)

86 LEGGETT

In the case of a single system (no “background”), total particle current is conserved; hence @I = 0 according to (26) and in inequality (42) becomes an equality, the well-known Landau effective mass relation (I). Since total spin current is never conserved for a nontrivial spin-independent Hamiltonian (13), we should expect that (43) is never an equality. In the case of a one-component system such as SHe we should, therefore, conclude

2114 < 4 ; (44

i.e., the first harmonic of the scattering amplitude of particles of opposite spin is always repulsive. This conclusion is in agreement with the explicit calculations of Rice for an electron gas at metallic densities [see Fig. 5.15 of Ref. (2); while &,(cos 0) is negative, it is less negative for forward angles].

Indeed, it is extremely plausible that the inequality (44) should generally hold even for a multicomponent system. We see from the above argument that it will hold if

While it does not seem possible to prove this rigorously in general, the fact that all processes which conserve spin current conserve particle current, while the converse is not true, makes it extremely probable that (45) will hold in most normal cases. We shall argue below that it is likely to hold even for metals.

We notice in passing that the second inequality of (29), J&k, 0) < 0, will also give some inequalities for the Landau parameters. For a one-component system J,(k, 0) is simply evaluated for any 5 of the form (o)piP,(cos (I), and the resulting inequalities are

FZ 1+21+1 20, l+21+1 1

2,/4>() ’

which were written down long ago by Pomeranchuk (7) on the basis of the semi- classical Landau theory. [To the author’s knowledge no microscopic proof has previously appeared in the literature (14).]

We now turn to the question of how the above derivation needs to be modified for the case of a general background. We first consider the case where the fermion one-particle Greens function remains diagonal in the momentum representation (liquid metals or 8He-4He solutions). We do not wish to presuppose any particular information about the background, so it is not convenient to select for special treatment background particle-hole propagation as we did above. Instead, we proceed as follows. (In what follows the subscript 1 refers to the fermions and 2 to the background, as above.) We separate out the sum of all graphs which begin and end with two background lines (in the case of Bose condensation, one of these

METALS AND OTHBR FERMI LIQUIDS 87

may be a line corresponding to a zero-momentum particle) and are irreducible with respect to fermion particle-hole pairs of energy-momentum (It, o). This sum of graphs we call &(k, o)(=&(p, E : k, 0). We can then write the following symbolic equations (which are of course still matrix equations with respect to p and E):

4&J) = g1 + gJ,,g, (47)

r11 = ~(l$l + ~“‘llgl~l, + W2&~21 , (48)

r,, = r%l + ~'1'21gJ,, , (49)

where PJaB is, as above, the irreducible scattering amplitude which converts an 01 particle-hole pair into a /3 one. (Again, a background “particle-hole pair” may contain a zero-momentum line.) We notice that P)az and T’,, need not be defined; this was the point of introducing & instead of g, . We now consider a range of (k, w) such that 7-l < (w, vk) < up where T is a characteristic fermion lifetime, and consider the structure of & in this region. In most cases of interest we know the (k, o) dependence of & from general phenomenological considerations (for instance, it can be argued (4) that in the case of 3He-4He mixtures at T = 0 we should have

where N4 is the number of 4He atoms and M*, a boson effective mass). Even if this is not so, however, we can always fix a value of o, say w,, , in the region 7-l < (w, ok) < .+ and write

g”dk ~0) = P&J~) + P,@, wok lim zn,(k, wO) = 0. (50) k/q,+0

While everything that follows will go through even in the case that g’fi(wo) is a function of o. , the “Landau theory” for the fermions will be rather complicated in this case, since the effective I;l’s and ZI)s will depend on the frequencies of the disturbance considered (it seems that this will probably be the case for liquid metals in the region w - kT < t~rp);~ we shall, therefore, assume for simplicity that ~~2(wo) can be taken to be a constant in the region of interest. This should be so if Tb-1 5 +, Eb 2 ep , where 71, and Eb are the characteristic lifetime and energy associated with the background. This criterion is certainly satisfied at sufficiently low temperatures for liquid 8He-4He mixtures and for metals.

5 A similar situation, of course, holds in solid metals when the electron-phonon interaction is taken into account; however, the reason there is rather different (see next section).

88 LEGGETT

After making the separation (50) and the usual separation (21) for the fermions, we can define quantities rW, by

PII = m1 + W&jlr”11 + r’l’lzgfir”21 , ‘(50

J-%1 = r(l)21 + r’l’2*gflr”ll , (52)

l-y, = r(l)12 + Pngf*r”la . (53)

Now if we compare Eqs. (47)-(49) and (51)-(53) with the explicit forms of the matrix equations (17), (la), and (23), we see that they are identical in form except that now & has replaced g, everywhere (and similarly for gfi , etc.) and r(1)22 , PS, are identically zero. Thus we can immediately write down formula (24), simply making the same replacements in it; of course, we do not now in general have a simple formula for gnZ(kw), so that (24) is in fact a complicated integral equation. However, all the essential properties we need for the subsequent derivation are preserved; in particular R, is still the unit matrix for any conserved fermion quantity 6, and, by definition gn2(k, UJ) + 0 as k/w --t 0. Hence Eq. (37) is still valid [as is the statement that J&w) (Eq. (27) tends to zero for k/w + 0)] and our results follow.

In any particular case, of course, we may often be able to use apriori information on the structure of in2 (see above) to simplify Eq. (24) considerably. However, we shall not discuss this further here.

Finally, we should consider the possible effects of lattice periodicity in the background. Apart from the effect of this on the fermion one-particle propagators, it can be treated, or rather ignored, in the same way that Bose condensation was above. Actually, however, for the only physically interesting case (solid metals) we can dispose of this complication and at the same time of the difficulties due to the long range of the Coulomb interaction (which would invalidate the above derivation, since it was implicitly assumed that P was independent of k for small enough k), by the following simple device. We assume that we will be interested only in values of k less than some cutoff value kc , where kc < kP . We then simply drop from the Hamiltonian all potential terms with k < kc which diverge as k + 0; the resultant Hamiltonian then still has all the properties (e.g., Hermiticity) which we require in the proof used above, and the effect on the Landau parameters, etc., as usually defined for a charged system (2) will be at most of order kc/k= . Of course, we will still have processes in which background particl+hole pairs (i.e., phonons of a hypothetical neutral lattice) of energy-momentum (k, w) propagate and are coupled to the motion of the electron system, but the characteristic velocity of these pairs is of order c, the speed of sound, so that provided only that we consider values of w, k such that w > ck (we can still have o << vk, since c/v < 1) we can treat the lattice as essentially static and the electron system as effectively an isolated one-component system. This procedure is to all intents and purposes

METALS AND OTHER FERMI LIQUIDS 89

equivalent to considering the screened electron response function (which, of course, includes the effects of exchange of virtual phonons and photons; cf. next section). There remains the problem that if the background is periodic the one-fermion Greens function is not diagonal in the momentum indices. This, however, is not a serious difficulty; as has been shown by Jones and McClure (15), the Landau theory is easily generalized to this case, and all the basic formulas which we have used remain valid, with the important exception of Eq. (6). (We notice, incidentally, that in this case Qp, is generally nonzero even in the limit of no interelectron interaction, since it now includes the contribution of interband transitions.) Thus prima facie we can conclude, for metals, only (40) and not (42). We shall return to this point in Section IV.

To conclude, we point out that there appears no reason why the theory developed in this section should not apply to the electrons in liquid metals at frequencies such that kT < w < cF , (which would correspond in some cases to the frequencies used in infrared optical experiments). At such frequencies we can neglect real propagation of the “background” (the ions), which behaves classically and, therefore, has a characteristic energy of order kT, indeed, we should expect that in this region gi [Eq. (48)] is given by Nk2/Mw2, where M is the same as the ion mass to order m/M. The effect of virtual propagation of the background on the effective mass and Landau parameters (i.e., effects analogous to that of the electron-phonon interaction in solid metals) may presumably also be ignored at such frequencies, by an argument analogous to the one used in the next section to show that the electron-phonon interaction is ineffective when o > we . However, it is by no means true that the ions therefore affect the dynamic effective mass only to order l/prI, (where I is the electron mean free path), so that we certainly cannot conclude that (2a) is an equality to that order as is argued by Ginzburg et al. (16). Indeed, prima facie, if the argument of these authors were valid we could apply it equally well to 3He4He mixtures, and show that the effective mass of an isolated 3He atom in 4He is the same as that of a free atom. It cannot be overemphasized that the existence of mass renormalization of an electron in a solid has basically nothing to do with the periodicity of the lattice (though, of course, its magnitude may depend strongly on this).

III. EFFECTS OF THE ELECTRON-PHONON INTERACTION IN METALS

In this section we shall review the application of Fermi-liquid theory to metals in the presence of Coulomb and electron-phonon interactions. This subject has recently been treated in some detail by Heine, Nozieres, and Wilkins (5) and by Prange and Sachs (6) (hereafter referred to as PS); we shall try not to duplicate more of the work of these papers than necessary. Three results not given explicitly

90 LEGGE’IT

in these papers are proved: (a) the electron-phonon interaction can only enhance the “dynamic effective mass” m*/I + *F1 ; the enhancement tends to zero in the limit of forward electron-phonon scattering; (b) if the electron liquid is stable in the presence of the Coulomb interaction, it is stable when the phonon interaction is turned on as well; (c) the electron-phonon interaction, while it cannot affect the static properties in the normal state, can afIect them quite appreciably in the superconducting state (quite apart, of course, from providing a mechanism for superconductivity). While none of these conclusions is very surprising and (a) and (b) at least could have been proved directly from the results of PS, we believe that for some purposes the formulation we shall give below is more clear-cut. In particular, it enables us to see very simply under what circumstances the electronic correlation functions can be parametrized by a few constants and how these are related, and also to see the close analogy between renormalization effects in the usual Fermi-liquid theory and in the electron-phonon interaction case, which is perhaps somewhat obscured by the kinetic-equation approach. A further advantage of our formalism is that it is trivially transposed to the very similar problem of fermion-“paramagnon” coupling in nearly ferromagnetic Fermi liquids (I 7).

In accordance with the remarks made at the end of the last section, we shall consider only the “screened” electronic correlation functions (2), i.e. we shall ignore any graph contributing to the electronic correlation functions K&w) which is reducible with respect to the polarization propagator as defined in PS. We shall also follow PS in not treating explicitly the presence of different electronic bands, different types of phonons, and Umklapp processes; as remarked by them, this does not represent an approximation. We also assume isotropy for notational simplicity; the results are trivially extended to the anisotropic case. As far as possible in this section our notation will conform to theirs.

The analysis of the graphs contributing to the effective interelectron interaction is by now standard, and we refer to PS for the details. The result of incorporating all self-energy corrections in the internal photon and phonon lines is that the electrons interact through the complete polarization propagator 9(kw), which may be separated into a screened Coulomb part V&o) and a part due to the exchange of dressed phonons D(h):

Wkw) = V&w) + D@), (54) V&o) = 4myc%(kw), (55)

(56)

Here r&w) is the electronic dielectric constant, (dn/&) is the density of states at the Fermi surface in the free-electron gas, wk is the physical phonon energy

METALS AND OTHER FERMI LIQUIDS 91

and 01~ is a dimensionless constant of order 1; in the presence of short-range electron-ion coupling in the original Hamiltonian ak: may also depend on the momentum p of the electron emitting the phonon, but this is of no importance for what follows. When calculating the unscreened electronic correlation functions, we should treat LS(kw) in the same way as we should normally treat a short-range potential [remembering of course that &o) in (55) itself depends on the quantities we are trying to calculate, so that the problem should be treated self-consistently]; it is convenient to treat V&o) and D&J) as independent “interactions” and denote them by a broken and wiggly line respectively. The fundamental simplifying circumstance in the problem (18), (19) is that while the maximum phonon wave vector kD is quite comparable to the electronic Fermi wave vector kF , the maximum phonon energy wn is much smaller than the characteristic electron energy cp (or, equivalently, the speed of sound c is much less that the Fermi velocity 21~). All results subsequently quoted will be valid to zeroth order in Wn/Er .

The analysis of PS shows that the contributions of Vc and of D to the one- fermion self-energy Z(p, G) are additive, in the sense that Z can be separated into a part L’c which contains only screened Coulomb lines (Vc) and a part which is of the form shown in Fig. 2, where the blobs are electron-phonon vertices completely renormalized by the Coulomb interaction.

p-p’, C-6’

~+(P,E)= a pP -’

FIG. 2

In algebraic form, this reads

where .T(pp’, EE’, kw) is the renormalized electron-phonon vertex. In general F will have a singularity corresponding to the zero-sound pole of the electron correlation functions, but since the phonon propagator D is very small in this region, the effect of this pole is negligible (cf. ref. (Is)]. We can, therefore, treat Y as a smooth function of its variables, of order (dn/d+ll* [cf. Eq. (56)].

From Eq. (57) we easily verify the following well-known properties of Z+ :

92 LEGGETI-

(where E is measured from the Fermi energy);

(3) aLpeD - (o~/Q) aqjaa -g a2qae; (4) for E < wn, a.zejae N -b,

where b is a positive constant of order unity. It follows, then, that the electron- phonon interaction distorts the one-particle Greens functions only for E 5 we ; in this region the Coulomb contribution Zc(p, l ) is slowly varying. We can then choose some energy E,, such that wn < E,, < l p and assert that, for E, Ed < E,, , the Greens function can be represented in the form

ac ac WP, 4 = ~ - ~ E - ED@) - acZ+(p, E) = E - ED(C) - ,&&I, E) ’ (58)

where ac , E,(c), and 2&, are defined by:

Eptc) - E?, - &(p, E,(O)) = 0, (59)

( a& -1 ac= l-- 1 ae 2

4414 4 = eJ&h 4. (61)

We now consider an electronic correlation function K&k, OJ) assuming k < kp , w < EF but making for the moment no aSSUUIptiOnS about the ratios Vpk/wD , w/wD . Just as in the simple case, we can write the matrix equations (see Eq. (15) above-now, of course, we have lost the $Ldegree of freedom):

K = TrMb-4 + g&4 W-4 g@419 (62)

r = F(l) + P’gF, (63)

where T(l) is the part of the total electron scattering amplitude which is irreducible with respect to a single pair of lines g(ko). An analysis similar to that made by PS for the one-particle self-energy shows that the contributions of Vc and D to r(l) are additive: T(l) = T&l) + T,(l) , where r,+(l) is shown in Fig. 3.

p+k/2,o, l +W2 NN

p-k/2, Q,E-u/2 NN p’-k/2, u, E’- w/2

FIG. 3

METAJS AND OTHER FERMI LIQUIDS 93

The algebraic form of P,(l) is

~~‘YPP’, ug’, ~6: kw) = -I ~(PP’, 412 (e _ + y2;;;-+& + i8)z . 6,/ , (64)

the minus sign entering because from the point of view of the electron correlation function K,(ko) the graph of Fig. 3 is an “exchange” 1F;raph.O (We recall that the corresponding “direct” graph is omitted because we are considering the screened correlation functions.)

As a first step in evaluating (62) we proceed as in Section II and divide the regions of integration over energy and momentum (or equivalently over E,‘c)) into the regions “far from” the Fermi surface (1 E ] or 1 ED@)] > c,-, where c0 was defined above) and regions “near” the Fermi surface (1 E (, ] Ep(c)I < ~~0). Note that the “near” region includes I E 1, I Ep)@)/ - wD . We divide the product g(ko) = G@ + k/2, E + w/2) G@ - k/2, E - w/2) into parts gt and gn corre- sponding respectively to the integration over the “far” and “near” regions; then gn is just the product of terms like (58). The quantity gr is insensitive to the electron-phonon interaction, according to the property (2) of ,Z4 listed above. Moreover, the quantity r,(l) is appreciable only for I E - z’ I 2 wD ; hence in any integration in which E - E’ varies over a region of order .+ , its contribution relative to that of P,(l) is of order w n /C r , i.e., negligible. We therefore renormalize PJ by defining

r(2) = r(l) + r'ugJ'2,;

and we can then write, in analogy with Eq. (24) above,

(65)

K,(kw) = @t’G) + Tr I .?jRt(C)t

I = @Je) + RC(C)*l~(k~), (66)

where

(Pp@) = Tr(gr(1 + P)gr}, (67)

R,(%t = (1 + p'2)gr) 4 I,,mW,r=o , (68)

Z,(kw) = Tr [ 1 I

[By arguments similar to those used above Re(C) is practically constant over the region E, Epfc) - u,, and hence can be evaluated at p = pp , E = 01. According

6 Whether or not we attach a minus sign to this graph depends, of course, on our detinitions of g and of matrix multiplication in J&s. (62), (63). With the definitions (15), (16), and (19), the minus sign should be added. Equation (41) in Ps appears to have a sign wrong.

94 LEGGErr

to what was said above about r+(l), we can replace P2) in the expressions for #tc), Rt(c) by the amplitude I’cw defined by:

(69)

Thus the quantities I$~ (c), Retc) depend only on the Coulomb interaction. Just as for the case of a simple Fermi liquid, we can show that for a quantity which is conserved by the Coulomb interaction we have

&c) = 0, R$C’ = ~-1. (70)

In the term in round brackets in (66) (i.e., It(kw) we cannot, of course, replace P2) by I’,o. We can, in fact, write in general (again to zeroth order in U&P>

r(2) = r,w + r(l)+ . (71)

However, this still does not simplify the term in question very much, since both gn and r,(l) are fast-varying functions of their arguments E, E‘, according to (58) and (64). Thus this term still involves a complicated integral equation.

To simplify the expression I&k@) we fist consider a more general expression of the form

J&J) = $4 II‘ @P de’% + k/2, E + w/2) G@ - k/2, E - w/2)f(p, E) E gnf, (72)

where G@, e) is given by Eq. (58) and f(p, ) E is an arbitrary function obeying the conditions (1) its dependence on 1 p 1 is negligible compared with its dependence on ~(1 aj-1 &, I < 1 af/lae I) (2) f(c > wn) = f(e < -OJD) = const, An example of such a function is r,(l)(p - p’, E - E’) for p’ --m, , I E’ 1 2 uD . We shall show that in the limiting cases w > wn and w < wo , the integral (72) can be put in a simple form. In what follows, we use the notation

vc = aE(c’D/i$ ,

N, = p2d+u, ,

mc = p& .

From (58) we have for small k by a simple identity

G@ + k/2, E + 42) G@ - k/2, E - 42)

(73)

(74)

(75)

= w - vc l k - [.&(E 742) - a?$(~ - o/2)]

x [G@ - k/Z E - 42) - G(p + k/2, z + w/2)3, (76)

METALS AND OTHER FERMI LIQUIDS 95

where we took into account that the p-dependmx of Z,, can be neglected. Making the substitution (27r)-3 J d$Nc j dC?/4n. J dEDCc), we therefore have

i dE [a-iG(p - k/2, e - 42) - dG@ + k/&e + w/2)] f’ic),

OJ - vc l k - [,C,(E + 42) - a&,(~ - w/2)] 07) where we have not written explicitly the dependence off on direction on the Fermi surface. Actually, this double integral is not absolutely convergent, and its value depends on whether we integrate first over dEsfc) and then over dc or vice versa. For our purposes the second way is the physically correct one (as can be seen by taking the limit 2, --+ 0), but it is mathematically more convenient to integrate first over dEsfc) and add a term J,, which gives the correction arising from interchange of the limits of integration. Since for E > On , we have a,-lG(p, E) -+ (E - Eptc) + ia) and for E < -mu , a-iG(p, l ) -+ (E - EP(c) - ia), we easily find

(78)

where f( co) means f(l E [ > on). Thus we can write

where

A(k, w, c) = zf; j dEDce)[e + 42 - Es(c) - vc * k/2 - &,(E + w/2)]4

- [E - 42 - E,,@) + v, l k/2 - &(E - w/2)]-’ (80)

1 - x - (E - w/2 - &(E - 42)) * (811

Using the fact that Im C,(E) changes sign at E = 0, we find

A(k, w, e) = S(E + 42) - 8(c - w/2), (82)

where 19(x) is the usual step function. Thus we get

fiky w) = %Jac j $f I::,, w - vc * k - [,&(i-‘;’ $2) - ,&(E - w/2)] + Jo . (83:)

96 LEGGEm

This expression is exact (to order u&). However, the integral can be evaluated in a simple closed form only in the limiting cases o > q, and u Q we . We consider these two cases separately:

(a) u > on . In this case we have &,(e + w/2) - .&,(E - o/2) g 0 for most of the region of integration, and also f(c) = f(co). If f(c) has poles, then since the characteristic range of variation off(E) is of order wn , the residue at a pole will be of order q,J thus the relative contribution of the poles will be at most of order We/O and can be neglected. We therefore get from (83)

(84)

(b) w < wn . In this case we can put

and so, denoting f(l E I < oa) by f(O), we have

Now, in the limit E << wn the one-particle Green function (58) is again of simple pole form: Gb, 4 = a

E - E,, + i8 sgn E, ' (86)

where ED is defined by

E,, - Epfc) - &,(p, Ep)(c)) = 0, (87) and

a = a,(1 - EED/i3e)-1co . m

Defining v = aE,/ap, N* = pF2/dv (the fully renormalized density of states which enters the expression for the specific heat) and m* = pF/v, and remembering that the p-dependence of &,(p, l ) can be neglected, we easily find (6)

( l-L$-)-l 3c12+A, C=O ac vc

and hence (85) can be written

(89)

JOG, = N*a2 I g o -“, . k f(O) - Ncazc j gf(a) = dQ I I G N*a2 w “:. k f(O) + W*aY(O) - NcaBdXm)}/.

(90)

(91)

METALS AND OTHJ3R FERMI LIQUIDS 97

From Eq. (90) we see that J(lc, 0) = J(k, w > wn); this will lead to the well-known conclusion that normal-state static properties are never affected by the electron- phonon interaction (apart from the specific heat). Equation (91) has a useful interpretation. Suppose that for o < WD we had chosen to calculate the integral (72) by the following alternative method, which is analogous to the one used in getting to equations of the form (66). We choose some energy Ed such that w < c1 << wn and divide the integral (72) into the region ] E ] < <I (but arbitrary p), where the Greens functions have the form (86), and the region 1 E ] > Ed . Then from (86) we easily find that the contribution from the first region is just the first term in (91) so that we can identify the term in curly brackets with the contribution of the region 1 E ) > Ed . It follows then that if the Greens functions are distorted for some reason over an energy region E 5 Ed [as happens in the formation of the superconducting phase, for instance, provided d < wn (weak-coupling case)], the term in curly brackets in (91) will be unaffected and the only effect will be on the first term.

A finite-temperature generalization of the above arguments is more or less trivial; the appropriate expression for (GG) is given, for instance, in the appendix to Ref. (IO), and it turns out that the principal effect is to replace the theta-function in (82) by the expression

k (tanh (6 + ?)/2T - tanh (e - $-)/2T).

Thus if T > wn , expression (84) for J(kw) is valid even if o < Wn . If T < wD , we can still argue as above that all corrections to J(kw) due to the onset of (weak-coupling) superconductivity should affect only the first term of (91).

We finally notice that in the case where we wish to take into account the finite quasiparticle lifetimes the imaginary parts of both .Zc and &, must be kept in the Green functions. The result is to replace the denominator in the first term of (83) by the expression

w - vc * k - [Re L’,(e + w/2) - Re C,(E - w/2)] - a,[Im Z(C + w/2) - Im Z(e - w/2)], (92)

where C is the full self-energy, (including possibly small contributions from impurity scattering which are important only here). If T = 0 and Im Z(E) -- const. sgn E (as would be the case for impurity scattering), then we can simply replace the (w - v * k) in the denominator of the first term of (90) by (o - v * k + i/~) where

I/T = 2a I Im Z(E)I.-~. (93)

595/46/I-7

98 LEGGEIT

The fact that it is the full renormalization constant a rather than ac which occurs in T leads to the result, proved by PS, that there are no explicit renormalization factors in the kinetic coefficients in the normal state due to the electron-phonon interaction. This, of course, can also be proved easily in the present formalism, but we shall not give the details here.

We now proceed to apply Eqs. (84) and (90) [or (91)] to the evaluation of 1&ku), Eq. (68a). Consider first the limit w > UD . Eq. (68) can be expanded in the form

I, = Tr{5(gn + gPa)g, + gP%P)g~ + --> 61, (94)

assuming for the moment that the series converges. According to Eq. (84) this expression is equal to

X (

vc l k w-vc*k + ” l k

w-vc*k N&2c s g F2)(cc, cc: I I , I I ’ ) w : ; ; , k + - ) 8

(95)

where we have taken into account that r(2)(pp’, EE’ : kw) can depend on the relative directions (specified by n, n’) of the vectors p and p’. We easily verify that r(*)(oo, co) is to be taken to mean r(2)(] c - E’ I> wn), which according to (71) and (64) is just r,o. The case where the series (94) does not converge can be treated similarly by constructing an integral equation for the matrix (1 - rt2) g&l. Thus, finally, defining a dimensionless quantity which is a matrix only with respect to spin and direction on the Fermi surface

P&I, n’ : uu’) = N&Jwc(n, n’ : cd),

we can put It into the form [cf. ref. (IO)]

(96)

(97)

and hence we finally get

KAWw>, = tDc(c) + [RC(C)]* ac2Nc vc l k o - vc * k(1 + PC) 4

(98)

in complete analogy to the expression for K,(ko) in a simple Fermi liquid (10). Thus for w > OJP the phonon interaction has no effect whatever on the correlation functions.

The case w < Wb is rather more complicated. It is convenient and perhaps illuminating to deal with it by a technique completely parallel to that used in Section II to divide the correlation function into “quasiparticle” and

METALS AND OTHER FERMI LIQUIDS 99

“nonquasiparticle” contributions, and above to get to Eq. (66). That is, we shall show that for o < wn , we can Write

I&ko) = @$*) + [lIE(*)I8 Z’#w), c-9

where Z’&Isw) is an integral involving only excitation of fully dressed quasi-particles (dressed by the electron-phonon interaction as well as the Coulomb one), Z$(d) is a renormalization factor arising from the dressing process, and (PC(d) is a constant expressing the contribution to Z[ of the “nonquasiparticle” states with w - WD . mus, @p,(+) is parallel to Qi, of Section II, or the @‘e’c) of Eq. (66).]

To carry out this program, we go back to Eq. (91) and note that in our matrix notation the definition of .Z(kw), Eq. (72), is just gnf It is convenient to split up gn formally in such a way that one part, say g(l), gives the first term in (91) and the other, say gc2), the term in curly brackets. As we have seen above, g(l) is precisely the contribution from the region E < an where the fully dressed quasi-particles are well defined, i.e., it is the “quasi-particle” contribution. With an eye to an application to the superconducting case, we shall consider the general case where the “quasiparticle” Green functions are distorted over a region E < WD by, say, the onset of Cooper pairing. Then the form (91) is not valid, but we can still write

or in other words

P!f = 1 f$ ~*@f(O) Q(kw), (101)

g’2,f = lv*ay-(0) - iv&2f(co), W)

where Q(kw) is some function which in general will depend on the nature of the distortion of the one-particle Green functions and on the temperature. For the superconducting case in the limit ku, w < d the form of Q(kw) can be found, for instance, from the formulas of Ref. (20) [in general Q, like f(O), is also a function of n [cf. Eq. (91)], but we shall not write this explicitly so as not to complicate the notation).’ It is obvious that the separation into g(l) and gc2) is precisely analogous to the separation in Section II into gn and gr .

’ Here, of course, we have assumed that the definition of J(lr~) has been appropriately general- ized to include terms in FF as well as GG. To do this problem properly one would have to adopt some technique like that of Refs. (IO) and (20) and examine also integrals containing GF, etc. This brings in considerable notational complication, and we shall not go into the details here. Clearly, the end result will always be an expression of the form (96), with the form of Q(kw) the same as that for a simple superfluid Fermi liquid. The basic physical point we are making is that we can always make a separation of the type (93), (94), whatever is going on very near the Fermi surface.

100 LEGGETT

Guided by this analogy, we def?ne the quantity rU(n, n’ : CJ, CJ’ : E, E’) by

p = F(2) + jwg(2,p.J. (103)

After the usual matrix-algebraic manipulations, we can then write in analogy to (24):

4&4,+, = (Pp + [Rp’y I’,(b), VW where

@,((+J = Tr(&gt2) + g(2)Pg(2)) t}, (105)

RP’5 = (1 + rwg’2’) 5 Ir<oD , (106)

(107)

In contrast to the case discussed in section 2, we now have explicit expressions for the constants Qc(+) and RC(+), whether or not the quantity f is conserved. In fact, we get directly from (102)

@(W = i g MN *a2 - N(N*u~)~ P(0, 0) + (Ndc)” P(m, co)

-N*a2 Ncac2 PC=), 0) + rT4 m)>l 0, (108)

R$"' = 1 + N*a2P(0, 0) - Nc~~~cr”(0, co). (109)

Here P(0, co) for instance, means P(l E 1 < wn), ( E’ 1 > wn), and so on. In the expression (107) for Il&lcw), the quantity P should be taken to be P(0, 0), according to Eq. (101).

All that remains is to find the quantities P(0, 0), P(0, co), etc., in terms of Pc and F(l)+ . We can do this by writing out (95) explicitly for the various possible arguments:

P(0, 0) = F2)(0, 0) + rc2’(0, 0) N*u2P(0, 0) - r(2)(0, co) Ncu2,P(m, 0), (110)

etc. We recall that F2)(0, 00) = P2)(co, 0) = P2)(co, co) = P’c , while P)(O, 0) = rcU + FJl)(O, 0), (where r,“)(O, 0) = r,Cl)(~ - E’ = 0)). (Hereafter we omit the arguments on r,(l).) Solving the simultaneous equations (1 lo), we find

rv4 0) = r,(l) + P 1 + N*a2P(0, 0) 1+ NCQ~CFC ’

(111)

(1 + N*a2P(o,O)), (112)

(113)

METALS AND OTHER FERMI LIQUIDS 101

It is convenient to express the final results for @c(9), R,‘+), and I’,(kw) in terms of the dimensionless quantities

F = N*aaP(O, 0), (114)

Then substituting (11 l)-( 113) into (108), (109) and remembering that

N*aalNGc2 = m,lm*,

we finally get

Qe(4) = N*a2

(117)

Substituting (116)-(118) into (104) and this into (66) gives the complete expression for the correlation functions; their general form is

&(kwL, uD = @$o) + [Rt(c)]2 {@$+) + [Re(*)la I',(kw)}, (119)

showing the successive effects of Coulomb and phonon renormalization. In particular, for a quantity which is conserved by the Coulomb interaction, we can write explicitly [cf. Eq. (70)]

(where the F’s of course denote the appropriate harmonics). Eqs. (116)-(118) are the final results of this section, along with the explicit

expression (111) for P(0, 0), i.e., for F. We shall now apply them to investigate the inequalities which hold between the quantities flc and F, possible phonon- induced instabilities, and the effect of the electron-phonon interaction in the superconducting phase.

According to the same sort of considerations on the meaning of Gc(@ as used for @[ in Section II, we can conclude that Qbp(@) < 0, the equality holding for a quantity which is conserved by the electronphonon interaction. Since the condition

102 LEGGm

1 + flc > 0 is necessary to ensure stability of the system (cf. below), we immediately get

l+F, = 1-t&/4 m* 1+&o 1 + &/4 = K’

m* 1 + NW + 11 a 1 + *c:;21+ 1)

1 # 0, (122) m*

1 + &G/(21 + 1) a 1 + zc;;21+ 1)

(we put back in the factors of 21+ 1, which were omitted in the matrix notation used above). These relations could of course have been proved directly. Indeed, from (111) we find that (121) is equivalent to

mc -= m*

1 - N*ua l g r+(l), (123)

which could have been obtained from (57) by differentiation with respect to E, integration by parts and a little matrix algebra. Similarly, (122) together with (121) and (111) turns out to be simply the statement that

I dQ F~Vz(cos @clz,,) < I

dQ F+(l), (124)

which is obvious, since T,(l) is everywhere positive. An important example of (122) is the inequality

m* 1+ 64 ’ 1 Z&l ’

(125a)

which states that the “dynamic effective mass” can only be increased by the electron-phonon interaction. We notice that (125a) becomes an equality as (124) tends to an equality for I = 1, which is just the condition for the vertex F-(pp’, k) to become a delta-function of (p l p’ - pp’), i.e., for the emission of virtual phonons to be entirely in the forward direction with respect to the electron momentum. Thus, in this hypothetical case, the electron-phonon interaction would increase the thermodynamic effective mass m* while leaving the dynamic effective mass unchanged (like the fermion-fermion interaction in a translationally invariant system).

We note that if we accept the inequality (44) for the “Coulomb” quantities, then we,also get the relation

m* , m, 1+ +$1 ’ 1+ UGl

(125b)

METALS AND OTHER FERMI LLQvID.5 103

which is actually easier to compare with experiment (see next section). Whether (44) is in fact valid for metals is briefly discussed at the end of this section.

We next enquire whether the electron-phonon interaction can ever render unstable an electron system which was stable in its absence. One obvious way in which this apparently might happen is if the expression on the right-hand side of (123) should tend to zero, since a negative thermodynamic effective mass is meaningless. It was shown by Migdal(Z8) that this instability never occurs for the electron-phonon system without Coulomb interaction, but it appears rather more difficult to show this when the Coulomb interaction is taken into account, since the latter renormalizes the electron-phonon vertices. Leaving aside this possibility, we inquire whether there can be “Pomeranchuk” type (7) instabilities in the fully renormalized Fermi liquid if there are none in the presence of the Coulomb interactions only. Such instabilities are signalled (cf. Section II) by a positive value of I’,(k, 0) [Eq. (118)]. Since in the normal state Q(k, 0) = - 1, an instability would require

I-l-F-CO. (126)

We see from Eqs. (ill), (121), and (124) that this can never be fulfilled unless 1 + Fc < 0, in which case the electron liquid would already be unstable in the absence of the electron-phonon interaction.

We might, at fist sight, think that the converse situation could occur, i.e., that the electron-phonon interaction could rescue the system from an instability which would occur if the Coulomb forces alone were operative. However, a little thought shows that the electron-phonon interaction, which affects only a small region near the Fermi surface, can do nothing to prevent the system going to a lower energy state by large distortions of the Fermi surface (at most it could make the “normal” state metastable). In particular, the electron-phonon interaction can do nothing to inhibit ferromagnetism.

Finally, we investigate the effect of the electron-phonon interaction on the static properties. According to Eqs. (66), (104), and (116)-(118), we have for the normal state, since then Q(k0) = -1,

(127)

104 LEGGETT

i.e., the normal-state static properties are not affected by the electron-phonon interaction. In the superconducting state, however, the situation is quite different, since then Q(k0) # -1. Consider, for example, the spin susceptibility; we then have limk+OQ(k, 0) = --f(T), where f(T) is the “relative density of states near the Fermi surface” [cf. Ref. (IO)]. Since the spin density is conserved by both Coulomb and electron-phonon interactions, we have in accordance with Ref. (IO),

f(T) x= -$3"liiKo(k,0)=~/32N*-- 1 + tzlfvl ' (128)

where Z, includes the effect of the electron-phonon interaction (fi is the gyromagnetic ratio). A possible application will be discussed briefly at the end of the next section.8

We should finally emphasize that the relation (44), and hence (125b), is very likely to be valid for metals. If we neglect the electron-phonon interaction, then the electrons undergo collisions only with the static lattice and with one another. The first type of collisions will clearly give equal contributions to 1 Q9 1 and 1 @DO 1 (Section II). As to electron-electron interactions, we can argue as in Section II that, since any collision which conserves spin current conserves particle current, while the converse is not true, the contribution to ] @So ( is likely to be larger than that to I QD I. Hence we should conclude that .&/4 < fllc , and hence (125b). (We can also argue, from (111) and the fact that r,(l) is proportional to &s-even if Umklapp terms are included-that Z,/4 < Fl , but it does not seem possible at present to compare this result with experiment.) A point worth noting is that &c/4 = pl:c in the Hartree-Fock approximation; any difference between &c/4 and &C , and hence between Z,/4 and Fl , is therefore at least of second order in the electron-electron interaction, whatever the strengths of interaction with the static lattice and with phonons. (The same applies of course to harmonics ZrandFrforI> 1.)

IV. COMPARISON WITH EXPERIMENT FOR METALS

In this section we shall collect the results we obtained and compare them with the (not very copious) available experimental data. For pure liquid 3He the relation (2a) is an equality, while there seems to be no way at present of checking (26) since Z, is not experimentally accessible. For dilute 3He-4He mixtures (2a) is apparently well fultiled,9 but this is not very surprising or interesting

s Similar remarks clearly apply to the kinetic coefficients in the superconducting state. e The only rigorous way of obtaining m*/l + +F, experimentally is from the density of the

normal component at T = 0. However, it is obvious that at low concentrations this quantity cannot be very different from the effective mass of a single *He atom in ‘He, to which it must reduce in the zero-concentration limit.

METALS AND OTHER FERMI LIQUIDS 105

since Fl (and presumably also 2,) is very small at attainable concentrations (21). Consequently, we concentrate on the case of metals.

We begin with a discussion of the various “effective masses” occurring in metals and the extent to which they can be determined from experiment (see also ref. (6)). As we saw in the last section, Fermi-liquid theory in its simple form describes the response of a metal for T < TD , the Debye temperature, to a space- and time-varying external field in two separate regions of frequency (but for arbitrary wave vector k < kr): The “low-frequency”region w < or, and the “high frequency” region w,, <UJ < +, where wn is the Debye frequency. The Fermi-liquid parameters appropriate to the two regions are different. It can also be shown that in the temperature region T > TD the “high-frequency” form of the theory should be used for arbitrary frequencies of the external field. The parameters appropriate to the “high-frequency” theory are determined by the interaction with the static lattice and the screened Coulomb interaction only (and have been denoted by the subscript C), while the “low-frequency” parameters are determined also by the electron-phonon interaction (and are written without subscript).

Altogether in the theory as we have developed it there are five “masses”r0:

(129)

plus two quantities obtained by replacing the spin-independent Landau parameter Fl by its spin-dependent partner Z,/4. The first quantity in (129) is just the free-electron mass. The second, the “thermodynamic effective mass” is defined in terms of the (fully renormalized) density of states near the Fermi surface and is measured by the electronic specific heat at very low temperatures (T < TD) and also [cf. Ref. (22)] from cyclotron resonance in the Azbel’Xaner geometry. The third quantity has a clear theoretical significance; it is just the ratio of the pseudomomentum of a “fully dressed” electron quasi-particle (dressed, that is, by all interactions including these with phonons) to the current carried by it [cf. ref. (2)]. However, its experimental determination is rather more problematical. In addition to the possibility of cyclotron-resonance type experiments discussed at the end of ref. (6) and the references cited there, this quantity is in principle accessible from the behavior of the penetration depth with temperature in the rare case that the metal becomes a London super- conductor (10). At the time of writing there seems to be no data available on this quantity (23). The analogous quantity m*/l + &Zl , however, is obtainable

lo We assume that the Fermi surface is effectively spherical. I f not, the “effective masses” measured in various types of experiment will differ even in the absence of many-body effects [see M. H. Cohen, Phil. Mug. 3, 762 (1958)].

106 LEGGETT

from CESR experiments (9), which measure Z, directly (8) (note that these experiments were conducted in the frequency region w << wu). The fourth quantity in (129), mc , is defined theoretically in terms of the density of states near the Fermi surface as renormalized by the static-lattice and Coulomb interactions only; it does not seem likely to become experimentally accessible in the near future. Finally, the quantity me/l + #rc is the ratio of pseudomomentum to current for a quasi-particle dressed only by the static-lattice and Coulomb interactions. This is just the “optical effective mass” mopt as conventionally defined from the infrared dielectric constant (note that the infrared region usually corresponds to w > w,,) by the formula

1 - ~(0) = ~condez/moptwa (130)

when Ncond is the number of conduction electrons per unit volume. The corresponding quantity involving & is apparently not experimentally accessible.

Finally we may be interested in the “band mass,” a quantity not previously mentioned in this paper, which may be defined as the effective mass calculated from static-lattice effects alone (note that in this calculation the lattice should not be screened by the other conduction electrons). This quantity, which is of course not experimentally accessible, is of interest mainly because it has been calculated for a number of individual metals from a first-principles approach (24). We may argue that if (and only if) the effective lattice pseudopotential is weak (so that the band mass is almost the same as the free-electron mass, as is the case for Na and to a less extent for K) then also the band mass should be almost the same as the optical mass (in this case the Bloch states are almost plane-wave states, so that the Landau (I) argument would be expected to be “nearly” valid). However, it should be emphasized that this is the case only if the pseudopotential is weak for UN conduction electron states, not only for those near the Fermi surface. In so far as this is true the above criterion may be used to give a rough check on the credibility of the experimental data (or the theoretical calculations of ??la , according to taste).

We turn to the results we have derived. We have to deal with two types of inequalities: these relating phonon-renormahzed (low frequency) quantities Cm*, 4, etc.) and phonon-unrenormalized ones (mc , r;(,c , etc.) (Section III), and those relating the phonon-unrenormalized quantities to the bare electron mass (Section II). The first set are not problematical but the second are. We saw at the end of Section II that if we start out from the full Hamiltonian of the N nuclei and ZN electrons, then we can only prove the result

(131)

METALS AND OTHER FERMI LIQUIDS 107

(we could, of course, write the phonon-renormalized quantities on the left-hand side, but (131) as it stands gives the “best” inequality). Since the density of states iVc is related to Ncond, the number of electrons in the conduction band, by

(132)

(131) only gives us the inequality

(133)

which, while rigorous, is hardly very useful in practice, since Ncond/N is generally a small quantity.

If, however, we were prepared to start by assuming a Hamiltonian of the form

- rd -I- f&d%, Pm) + C U(ri - IQ, tm

(134)

where R, are the coordinates of the ions (nuclei plus core electrons), and the sums over i, j run over the conduction electrons only (Hc describing the core), then we would be in a position to apply the arguments of Section II in total and derive the results (2):

(135a,b)

Such an assumption would be equivalent to neglecting the effects of the identity of the core and conduction electrons. This model is, of course, a very popular one and has in fact usually been used as a starting-point for the consideration of many-body effects in metals; however, strictly speaking, one should try to take into account the necessity of antisymmetrizing the total wavefunction, as can be done to a certain extent by using a full pseudopotential approach (25). In so far as this leads to an effective Hamiltonian of the form (134), our results are justified. We shall use Eq. (134) with appropriate reservations, taking the view that the experimental confirmation of the relations (135) may be regarded to some extent as (negative) evidence for its validity.

We showed in Section III that independently of our starting Hamiltonian we have (among other things)

(136a)

(136b)

108 LEGGETT

where mopt is the optical effective mass as defined from the infrared dielectric constant [we recall that the relevant experiments are always in the region w > or, , so that according to the results of Section III electron-phonon effects do not enter (6)]. We also showed in Section II that it is very plausible that also

&c > &l4, (137)

the equality holding to first order in the Coulomb interaction. Finally we have suggested that if mb E m, then also mopt s m to the same order of accuracy. The quantities currently experimentally accessible are m*Z, [in CESR experiments (8), (9)], mopt and, of course, m. Fr is in principle obtainable from the temperature- dependent penetration depth in the case of a London superconductor (IO), but at present there appears to be no data on metals of sufficiently simple band structure which are London superconductors. Thus the predictions which can be compared directly with experiment are

m*/(l + ii-G) > mopt 2 m. (138)

In table I we show the relevant experimental quantities for the alkalis Na and K, for which the effects of anisotropy are very small (~2 %) (26). The quoted value of 2, is a rough estimate which should be capable of considerable improvement (9). Other quantities are subject to an error of order 2 %. We also tabulate for reference some calculated VdUeS of mb/m, r;(,c and (m*/mc).

TABLE I

QUASI-PARTICLE PARAMETERS IN Na AM) K

Parameter Na K Source (Ref. no.)

Cm*/4

~mo,tlm)

z,/12 = Bl

m*/m(l + ZJ12)

(m/m) f;‘lcl3 m*lmc

(cwt.)

(theor.)

1.24 1.21 (26)

I 1.26 1.01 (27) 1.08 (28) O-O.4 (9

1.24X1.88 1.00 1.02” (24)

0.06-0.1 (-0.1)” (30) 1.18 (30)

a “Optical” band mass. Cyclotron band masses for some orbits are as large as 1.09. b Extrapolated value. The approximations of Ref. (30) are not expected to be very good for K.

METALS AND OTHER FERMI LIQUIDS 109

We see that in the case of K all predictions which can be tested are fulfilled and further that from the relation (138) we should expect that CESR experiments will give B, < 0.2. For Na the situation is a good deal more disturbing. First, although our inequality mopt 3 m is fulfilled, it is in a sense too well fulfilled; while the calculated band mass is exactly equal to the free-electron mass, the experimental optical mass is different by 25 % according to the most recent data. This would imply that band effects are in reality quite important in Na, the equality of the band and free-electron masses being essentially an accident. While such a conclusion is not necessarily unexpected within the framework of the pseudo- potential approach, the magnitude of the discrepancy remains disturbing. Secondly, we see that the first inequality of (138) (which, as we have seen, does not depend on detailed assumptions about the nature of the effective potential felt by the conduction electrons) is not fulfilled if we use the optical data of Mayer et al; we should require Bl < -0.02. Since the quoted value of Bl is a very rough estimate, it is perhaps too early to become seriously worried by this; it will be interesting to see if more precise measurement confirms a positive value. If so, we should have to conclude either that the optical data of ref. (27) are in error or that not only is .& greater than f;ilc-in itself a highly improbable state of affairs-but that the difference is too great to be cancelled by the phonon enhancement of the quantity m*/(l + &Zl), which according to the last line of the table should be of order of magnitude lo-20 %. It is much to be hoped that more accurate data will shortly become available for both alkali metals.

We shall not discuss the comparison with optical experiments for liquid metals [see ref. (16)], because at the moment the experimental uncertainty is large, and in addition, it is not clear whether we always are in a frequency region where we should expect a simple Landau theory to apply.

While the results of this section are somewhat meagre, it should not be difficult to generalize the considerations advanced here so as to apply to metals with appreciable anisotropy, so that it should be possible to compare theory with experiment for a large number of metals when data become available.

We conclude this section with a note on a somewhat disconnected point. We saw at the end of Section III that the spin susceptibility of a superconductor should be given by the formula

1 f(T) x(T) = 4 ,BN* 1 + &&f(T) ’ (139)

where Z,, includes the effect of Coulomb and electron-phonon interactions, according to Eq. (11 l), andf(T) is the usual BCS-Yosida curve (31). The contribu- tion of the Coulomb interaction to Z,, is usually negative, but that of the electron- phonon interaction is always repulsive, and if it outweighs the Coulomb interaction, the spin susceptibility will, therefore, fall off less steeply below Tc than predicted

110 LEGGETT

by the BCS theory, (in contrast to the suggestion of ref. (IO). It is possible that this behavior has been seen in recent Knight-shift measurements on Al [Ref. (32)). According to Eqs. (111) and (114)-(116), we have

z,/4 = (m*/mc)(l + &c/4) - 1, (140)

where .& is the value of Z, given by taking into account only the Coulomb interaction, and (m*/mc) is the renormalization of the effective mass due to the electron-phonon interaction. The calculations of Rice (29) give [cf. Ref. (2)] &cl4 - -0.2 for Al, while the recent work of Eytte (33) gives m*/mc - 1.55- 1.60, so that Z,/4 would be about +0.3 according to these calculations. Such a value of 5, while it could not by itself account for the very strong departure of the Knight-shift curves from the curve f(T), would tend to enhance any other mechanisms, such as the nonlinear effects suggested in ref. (32), which tend to make the curve more convex [cf. the arguments of Ref. (IO)]. It would be extremely interesting to obtain data on the Knight shift in lower external magnetic fields, where nonlinearity could be neglected, and also to try to obtain Z,, directly from measurements of the normal-state spin susceptibility.

V. CONCLUSION

Let us first review the quantitative results of this paper. We have shown that for an arbitrary Fermi liquid whose particles interacts with one another and with their environment via spin- and velocity-independent forces, and which satisfies the relation dnlde = 3N/m*v 2 r , we have the rigorous inequalities

m* 1 + aFl ’ my

m* >m 1+&G’ -

(141a,b)

We have shown that this result should be applicable to metals in so far as we neglect the effects of identity of the core and conduction electrons. In addition, we have argued that most nonpathological systems will satisfy the relation

r;, 2 Z,/4. (142)

This inequality is rigorous in the case that total fermion momentum is conserved. With regard to the electron-phonon interaction in metals, we have shown that the following relations are rigorous for arbitrary 1.

m* , mc m* b

mc Fz ’

l+21+1 P ’

l+21;1 ZZ

’ + 4(2Z+ 1) 2 ;

’ + 4(21: 1) (143a,b)

METALS AND OTHER FERMI LIQUIDS 111

the inequality being replaced by an equality for 1 = 0. In addition, we have shown various subsidiary results which will be reviewed below.

The above inequalities are applicable to aHe, 3He-4He mixtures and, with reservations, to the electrons in liquid and solid metals (in the first three cases only the first set of inequalities is applicable). They are, unfortunately, not applicable to nuclear matter, since the potential between nucleons, in so far as it can be defined consistently at all, is strongly dependent on both spin and isotopic spin. We can, of course, derive inequalities for such cases, but they always contain an unknown expectation value which cannot be simply expressed in terms of the semiphenomenological parameters of the theory (though, of course, it may be of interest to insert calculated values). A similar difficulty prevents us from getting a useful inequality for F, in the case where fermion momentum is conserved.

The inequality (141a) actually has a very simple physical interpretation. Consider, for instance, a set of NaHe atoms moving in *He. Suppose we start with the N SHe quasi-particles filling the Fermi sphere centered at p = 0, then shift the whole distribution so that it is moving with a small velocity v while keeping the background at rest. It is easy to show that the change in energy of the system in iNma*+, where m*d E m*/(l + )FI) is the “dynamic effective mass”. Thus the fermion system behaves, for this purpose, like a set of independent particles with effective mass ma*, and Eq. (135a) simply says that this “effective mass” must be greater than (or equal to) the real aHe mass. This is clearly intuitively plausible since we can picture the 8He atom as dragging along a cloud of *He atoms with it. However, to prove relation (143a) rigorously in the macroscopic theory does not seem a completely trivial problem (even for the case of a single isolated SHe atom, i.e., F1 = 0).

In the case where total fermion momentum is conserved, relation (141b) also has a simple physical interpretation. If we write down the kinetic equation for the quasiparticles, multiply by u and sum, we can verify that the spin current carried by a quasiparticle is just qs(1 + &/12)/m*. (This could also, of course, be obtained by direct evaluation of R,). The particle current is just p/m. Hence, (14lb) says that the spin current carried by a quasiparticle cannot be greater that the product of the spin and the current carried by it. Again, this statement, although intuitively plausible, does not seem trivial to prove directly.

In addition to the above inequalities, certain other potentially useful results have been obtained. We have shown how to formulate a microscopic Landau-type theory for a Fermi liquid interacting with an arbitrary background, and have given a microscopic proof of the stability conditions (3). We have also proved explicitly the important result that effects of the electron-phonon interaction on the Landau parameters do in principle show up in the superconducting state even though they do not in the normal state. As we have seen, experimental data

112 LEGGETT

for comparison with the theory is at present rather scarce, but it is to be hoped that a good deal more will become available in the near future.

ACKNOWLEDGMENTS

I am very grateful to Professor Paul Martin and the Department of Physics at Harvard University for their hospitality during February and March 1967, when this work was begun, and to Professor Martin also for helpful discussions. I am also grateful to Professor Leo Kadanoff, Professor Gordon Baym, Dr. C. J. Pethick, and Professor David Pines for stimulating conver- sations. I would also like to acknowledge the hospitality of the Physics Division of the Aspen Institute, where the last stages of this work were completed.

RECEIVED: September 6, 1967

REFERENCES

I. L. D. LANDAU, Zh. Eksperim. i Teor. Fiz. 30, 1058 (1956 @%glish transl.: Soviet Phys.-JETP

3, 920 (1957)]. 2. See, for example, D. PINES AND P. NOZIERES, “Theory of Quantum Liquids,” Vol. I. Benjamin,

New York, 1966. 3. A comprehensive and up-to-date account of the semiphenomenological theory of nuclei

due to Migdal and co-workers is to be found in A. B. MIGDAL, “Metod Vsaimodeist- vuyushykh Kvazichastits v Teorii Yadra” (“The Quasiparticle Method in Nuclear Theory”). Moscow, 1966. (translation to be published by Benjamin, New York). See also A. B. MIGDAL, Nucl. Phys. 75, 441 (1964).

4. Work along these lines has been done by G. BAYM and by the author (unpublished). 5. V. HEINE, P. NOZIERES, AND D. WILKINS, Phil. Mdg. 13,741 (1966). 6. R. E. PRANGE AND M. SACHS, Technical Report No. 638. University of Maryland, College

Park (November 1966). 7. I. YA POMERANCHUK, Zh. Eskperim. i Teor. Fiz. 35,524 (1958) [English transl.: Soviet Phys.-

JETP 8, 361 (1959)]. 8. P. M. PLATZMAN AND P. A. WOLFF, Phys. Rev. Letters 18,280 (1967). 9. S. SCHULTZ AND G. DUNIFER, Phys. Rev. Letters 18,283 (1967).

10. A. J. LEGGED, Phys. Rev. 140A, 1869 (1965). 11. A. A. ABRKOSOV, L. P. GOR’KOV, AND I. E. DZYALOSHINSKII, “Quantum Field-Theoretic

Methods in Statistical Physics,” Pergamon, New York, 1965. 12. L. D. LANDAU, Zh. Eksperim. i Teor. Fir. 35,95 (1958) [English transl.: Soviet Phys.-JETP 8,

70 (1959)l. 13. Despite some assertions to the contrary in the literature. 14. N. D. MERMIN, Phys. Rev. 159, 161 (1967). 15. B. L. JONES AND J. W. MCCLURE, Phys. Rev. 143,133 (1966). 16. V. L. GINZBCJRG, G. P. M~TULEVICH AND L. P. Prr~~vs~n, Dokl. Akad. Nauk SSSR 163,

1352 (1966) [English transl.: Soviet Phys. Doklady 10,765 (1966)]. These authors claim that the only dimensionless parameter characterizing the breakdown of Galilean invariance in a liquid metals is ?i/p~I; however, the parameter ?i/ppa (u = interionic mean spacing) is also relevant.

METALS AND OTIIER FERMi LIQUIDS 113

17. S. DONIACH AND S. ENGELSBERG, Phys. Rev. Letters 17,750 (1966). 18. A. B. MIGDAL, Zh. Eksperim. i Teor. Fiz. 34, 1438 (1958) [English trand.: Soviet Phys.-

JETP 7, 996 (1958)]. 19. R. E. PRANGE AND L. P. KWANOFP, Phys. Rev. 134A 566 (1964). 20. A. J. LEGGElT, Phys. Rev. 147, 1, 119 (1966). 21. G. BAYM, J. BARDEEN, AND D. PINES, Phys. Rev. 156,207 (1967). 22. M. YA. AZBEL, Zh. Eksperim i Teor. Fiz. 34, 766 (1958) [English transl.: Soviet Phys.-JETP

7, 527 (1958)]. 23. The values quoted for various metals by V. P. SKIN, Zh. Eksperim. i Teor. Ez. 33, 1282

(1957) [English transl.: Soviet Phys.-JETP 6, 985 (1958)] are in fact values of mopt , that is they are obtained by neglecting the electron-phonon interaction completely.

24. For the alkali metals such calculations have been performed by F. S. Ham, Phys. Rev. 128, 82 (1962); ibid. 2524 (1962).

25. W. A. HARRISON, “Pseudopotentials in the Theory of Metals.” Benjamin, New York, 1966. 26. C. C. GRIMJZS AND A. F. KIP, Phys. Rev. 132, 1991 (1963). 27. M. EL NABY, Z. Physik. 174, 269 (1963); H. MAYER AND M. EL NABY, ibid. 280 (1963); B.

HIETEL, unpublished doctoral dissertation, Bergakademie, Ciausthal.; H. MAVER and B. HIETEL, in “Optical Properties and Electronic Structure of Metals and Alloys” (F. Abeles Ed.), p. 47. North-Holland, Amsterdam, 1966.

28. J. N. HODGSON, J. Phys. Chem. Solids 24,1213 (1963). 29. T. M. RICE, Ann. Phys. (N.Y.) 31, 100 (1965). 30. N. W. ASHCROFT AND J. N. WILKINS, Phys. Letters 14, 285 (1965). 31. K. YOSJDA, Phys. Rev. 110, 769 (1958). 32. R. H. HAMMOND AND G. M. KELLY, Phys. Rev. Letters 18,156 (1967). 33. E. Pvrrn, J. Phys. Chem. Solids 28, 93 (1967).

595/46/1-8