The document presents a theory of transport coefficients in metallic systems that accounts for the dynamics of the ion subsystem and its effect on electron scattering. Key points: - Transport coefficients are related to the dynamic structure factor of the ion subsystem, which describes its collective motions. - Integral equations are derived relating the electron Green's function to the dynamic structure factor. - For solid metals, known results are obtained, and for liquid metals, Ziman's formulas are accurately obtained relating transport coefficients to the static structure factor.

1. THEORY OF KINETIC PROCESSES IN METALLIC SYSTEMS
Yu. P. Krasnyi and V. M. Kostenko
The theory of transport coefficients in metals is developed with allowance for the effect that
the dynamics of the ion subsystem has on the scattering of electrons. A relation is found
between the transport coefficients and the dynamic structure factor of the ion subsystem.
For solid metals already known results follow from the relations that are obtained.
Ziman's formula is obtained with very high accuracy for liquid metals.
1. Introduction
The development of the method of pseudopotentials in the theory of metals has resulted in a number
of important advances [1, 2]. These include the derivation of the basic transport coefficients of liquid
metals: electrical conductivity, thermal conductivity, thermoelastic power. Such an approach leads to
Ziman's well-known formulas [3, 4], which relate the transport coefficients to the static structure factor,
S(x), of the ion subsystem. Numerical calculations based on these formulas give satisfactory agreement
between theory and experiment [5-9]. However, the impression has been gaining ground that the trans-
port coefficients of metals calculated with allowance for electron scattering by the ion system must depend
not only on the static disposition of the ions but also on the dynamics of the ion subsystem [10-13]. In such
a case it is natural to expect that the transport coefficients will be expressed in terms of the dynamic struc-
ture factor S(x; o3) (the van Hove function). In the present paper it is shown in the second order in the
pseudopotential that the transport coefficients can indeed be expressed in terms of the dynamic structure
factor. For solid metals we obtain results that are already known in the theory of solids. For liquid
metals, Ziman's formulas are obtained with a very high accuracy.
2. Derivation of the Basic Transport Equations
We consider a system of N ions with Z-fold charge and Ne = ZN electrons in a volume V. Applying
the diffraction model of a metal [1], in which the electron-electron interaction is replaced by screening
of the electron-ion interaction, we write the Hamiltonian of the system in the form
where
'~ jPJ l
n~l n;n ~
(n~an')
(2)
He- ~t.Ne= 2{[E(k)-- ~te]Skk'+p(k--k')w(k--k')}ak.+ak'.,
kk'~
(3)
N
t 2 e-lkn~" (4)
Here the subscripts e and i denote the electron and the ion subsystem; w(r) is a local screened pseu-
dopotential of the electron-ion interaction; V(R) is the potential of the direct ion-ion interaction [1]; P'e and
t~i are the chemical potentials of the electrons and ions and m and M are their masses, respectively;
Odessa State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 14,
No. 2, pp. 251-261, February, 1973. Original article submitted December 22, 1971.
9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
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187

2. ak+ and ak~ are the operators of creation and annihilation of electrons in a state with momentum 5k
and spin a; akk' is the Kronecker delta; and E(k) = ~2k2/2m.
In the temperature range with which we shall be concerned the ion subsystem can be regarded as one
that obeys the laws of classical statistical mechanics, while the electrons obey Fermi statistics. Therefore,
only the electron part is in fact operator in nature in (1). In addition, we shall for simplicity study homo-
geneous and isotropic systems. Then, assuming that the metal is a two-component system consisting of
ion and electrons, the electric and thermal fluxes, I and q, can be represented in the form [14]
VT
e -- L~---~, (5)
where
Lo =--~ +~'((L(t) ;Bo(O) }}dt,
+~
L, = -7 ~ <(A(t); Bl (0) >>dt,
L2= --ff ~ <(}~(t); Bl (0) >)dt, (6)
[~o(t)= eWA(t')dt ', /},(t)= ~e q~(t )dt, (7)
<<ffi(t) ;h(t') >>=T0(t -t') <[~l (t) ;B(t') ]>. (8)
In these formulas, the angular brackets denote quantum-mechanical and statistical averaging over
the ensemble with the Hamiltonian ~; Iv and qv are the projections of the operators of the charge flux
of the electron gas and the total energy flux along the coordinate axes: ~ = ~c-m#i/M.
Assuming that w, the pseudopotential of the electron-ion interaction, is a small quantity, we restrict
ourselves in the calculation of the coefficients Ln to the first nonvanishing order. In the operators Iv and
~tv it is therefore sufficient to retain only the terms of zeroth order in w. If one also ignores the terms of
order m/M in the expression for iv, the explicit form of these operators becomes
hk. +
I~=eE m ako ak~,
k,T
q% E [E(k)-- ~t~] hk~= ak~+ak~ (9)m
ka
It can be seen from 16) and (9) that the problem of determining the coefficients Ln reduces to finding
the Green's function
l+|
<(ak~+(t)ak,~(t) ;[~(O) >)~ <<ak~+ak,~;B)> = 2~ ~ e-~t<<ak~+ak'~ h))~d~.
To determine this, we write down the following equations [14]:
ih- 8 <(ako+ak,~;~>>= _ 5 (t) <[ak.+ak,o;B] >
at
-- [E(k) - E(k') ]<<ak.+ak,~;B>>-- E:w (ki -- k) <<p(kt -- k) ak,.+ak'o;B>>
h 1
+ Ew (k" -- k~)(<p(k' - k,) ak,,+ak~,,;B>>,
h a
(10)
188

3. ih-~ ((9(")(k, -- k) a~,.+ak.~;/~>>= - 6 (t) <[p(n)(k, -- k) ak,.+ak'~; B] )
--[E(k~)- E (k')] <<p(~)(k,- k)au,~+ak%;O))
- - Zw (k2 -- kt) ((p(")(k~ -- k) p (k~ -- k~)ak~+ak,o;B>)
k~
+ Z w (k' -- kz) ((9(") (k, - k) p (k' - k2)ak,r b~
k 2
+ ih<ipu~+t)(ki -- k) ar.,o+ak,.;B)), n = O,t, 2..... (Ii)
Here p(n)(k) is the n-th derivative of p(k) with respect to the time.
Going over in these equations to the Fourier representation, expressing the function ((pa+a; ]3)~o~by
means of equations (11) in terms of the functions ((pCn)pa+a; ]3))w and going to the limit k ~k', we find
(ho) + ie) ((aj,~+ako;B))~ = ( [a~o+ak~; Bl >
k" (/h)"< [p(") (k, - k) ak,~+a~,~;~B]>
k" (ih)"< [pc-) (k - k,) ak.+a,~,o;/3]>1
-- w ( k - O=[E(k)_E(k,)+~,~+;a].+,
/_.a~-I{ .... (,~)"<(p'"'(k,--'k)p(k2--k,)a,ja,.;B>>.
+Z w(k2--k,)w(,,--K) [Ei~,i .E (k) + h(o + ie] ~+] -
9 lka n~o
-t- w (k - k,) w (kt - k~). (ih) "<<9(~) (k -- kt) p (k, - k2) a ko+ak.; B))
[E(k)_ E (k,) + ~r + i~e],+, "
(ih) '~((p uo(k - k~),9(kz - k) ak~o+ak,~;B)),~
-- w(k- k,)w(k2-k) [E(k)_E(k,)+ho~4~e],,+ i........
-- w(k-- kz)w(k,- k) [E(kt)_E(k)+ho)+ie],+t ). (12)
This equation cannot be direetly solved because of the presence in it of a Green's ffmction of higher order.
To obtain a closed equation, we make the decoupling
([O(")(k-kOa%~ak,o; t?]> z <p(")(k- k,) ><[ak,,+ak,o; B]>, (13)
((p(,o(k - k~)p (k, -- k,) ako+aJ,~;t3))~0
<p(")(k-k,)p(k~- k2) >((ak~+ak~; B-)),~. (14)
The formal possibility of making the approximation (14) is due to its being made in the terms of Eq. (12)
that contain the square of the pseudopotential as a factor. With regard to the approximation {13), it too is
admissible, since it refers to terms that play the role of free terms in the equation for the electron's Green
function, and it is therefore sufficient to take terms of first and zeroth order in w in these terms.
Note that Fisher and Rice [1] make a similar decoupling, but they restrict themselves to terms with
only a first derivative on the right-hand side of (12). We shall show that all the series in (12) can be summed
if the approximations (13) and (14) are used.
Exploiting the homogeneity and isotropy of the system,
(9(")(k - kl)> = (p(") (0) )fk~.
<9(")(k -- k~)9(k,- k2)> = (p("' (k -- kt)9(k~ --k) >6k,k,
= (- ~) "<o (k - k,) o (")(k, - k) >~,,~,
189

4. we can reduce Eq. (12) to the form
(/~a)+~e) <<ako+ak.;B))~,= --< [ako+ako; B] ) +Z lw (k-k,)i2[ ((ak~+ak.;B))',~
k 1
-- <<ak,o+ak,o;b)>,~]v-~ (ih) n<p(,~)(k -- k~) p (k~ -- k) )
n~0
X [E (k) -- E(k,) +/ko + ie] ,~+V-- [E (k) -- ~ (k,) -- Boo-- is] "+' '
(15)
We transform the right-hand side of this equation by noting that the terms in the eurIy brackets allow the
integral representation
[E(k) - E (k') • a,,) • ,~l-
=, • +| 1~- iel}dt.
Then the series in (15) can be formally summed:
9 " -- E (kt) +/io) :t=ge ~-l (ih)"<p(") (k, k,)_P(k, - k) )"
n~9
= i _+.o > ho =t=gel} dt.exp {-~-- [E (k) -- E(k,) +_T_ _f0(• ) ~t
Allowing for these relations, we obtain
.(/~60+ ie) ((ako+akGB))z = -- ( [akz+akG B] )
~._~.L [w2~i r'~ (k __k,) 12{ Sk,.~' ( E(k)--E(ki)+ho+ie)ti
Ir 1
+ Sk(__2;( E(k)- E(k,)-- k(0 -- ie
h )} {((ak~176176 (16)
Equation (16) can serve as the basis for finding the frequency-dependent transport coefficients.
we shall content ourselves with the static case. Going to the limit w ~ 0 in (16) and noting that
S (~; ~o) = &(+) (o)) + &(-) ((o)
+co
1
exp {io3t}dt<Z exp (--i~n~, (t)} exp {i~R~, (0)}k,--
we obtain the equation
<[ak,,+ak,,;B])
2r~i ( E(k)-- E(k~) )
hN ZIw(k--k~)[2S k-k1; h
k I
Here
• {((ak~ - ((ak~o+ak,,;B))0}.
Thus, the integral equation satisfied by the function ((a.+ a. 9 B))0, is an equation of BoltzmanntypeKEr K0-'
[10]. The kernel of this equation depends on S(~; co), the dynamic structure factor. This factor reflects
all possible collective motions of the ion subsystem, in which, as one can show, the interaction between
the ions can be described by means of an effective potential [1].
To solve Eq. (17) it is necessary to determine the explicit form of its left-hand side. To do this we
differentiate the operator ak+q; ~ak_q; cr with respect to the time, multiply both sides of the resulting
expression by exp (et), integrate with respect to t from -~ to zero, and retain only the terms of zeroth
order in w:
) hZk___qqo
inai*+q.oak-q.~--ihe e~tak~q.,(t)ak-~.o(t)dt~ -- 2 Ie~tak+q.~(t)ak-q.o(t)dt.' ' ' ' m ' '
190

5. If this relation is differentiated with respect to q, which is then allowed to tend to zero, then to terms
of order 5 (~ ~ 0) we readily obtain
([a,+o+ako;ao]>= ie 0 <akr
Ok,
<[~o+~,o;h,]> = ~[E(k)- ~+la-%,<a,o+~o>.
Thus, Eqs. (17) after the transition to the thermodynamic limit N, V -- o~ (V/N = v0 = const) take the
form
On hk, vo ( E.~Et )
--e 0E~- (2~)~h~,~lw(k-k,)l~S k-k~;--__ .d~k~
X {((ako+ak~; Bo))o -- ((ak,~+ak,~; Bo))o},
--(E On hk, Vo (k-- k~;~-~)
-- ~)'-~- m ~- (2~)~ z~ [w(k--k,)[zS d~k,
X {((ak~+ak~;/~t))o-- ((ak,o+ak~o;~?t)>o},
(is)
(19)
where
r+ = (ak.+ak~) m [ 'i + exp E -- ~, ] -t
k~T J
3. Solution of the Transport Equations
The integral equations (18) and (19) can be solved by expanding the product I w 12S in a series in
Legendre polynomials and the functions ((ak ak~; B))0in a series in spherical functions:
Iw<k-L)PS( k-k~'E-E" h ]~=2 aZpz(c~
l
((a~o+ak.;Bo))o= e '~ f~,.(E) Y~,.(0, ~p),
lm
((ak++ak.;#~))o= ~ g,~(E) Y~..(O,q~).
lm
where 0 and ~pare the polar angles that determine the direction of the vector k; 7 is the angle between the
directions k and k'. Substituting these expansions into Eqs. (18) and (19) and applying the composition
theorem for Legendre polynomials, we obtain the following equations for fl0(E) =-f(E) and gl0(E) - g(E):
r~3 -'-~l -~ -- 2ah~ dk~k~ dysinTlw(k-k~)l~
0 O
(20)
(21)
/ E-Et
• S (k -- k,;---V--) It(E) --/(E,) cosy], (22)
V ! "4~ --~-E- m -2~h2 dk~ yIw(k-k,)l 2
0
• k,; ) (23)
Since the electron gas is strongly degenerate at the temperatures we are considering, only electrons with
energy E ~ E F' where E F is the Fermi energy, participate in the transport processes. The energy lost by
an electron when it interacts with the ions is E-E I. In metals this energy is of order kBT D (T D is the
Debye temperature). But E F >> k T D, and therefore at sufficiently high temperayures f(Et) and g(Et) can
be expanded in a Taylor series abound the point E:
S(E,)= V (E,--E)~ 0~ = s (e,--E): 0+g(e)
n! OE~ g(E~) n! OE'~
191

6. We substitute these expansions into Eqs. (22) and (23),
where
obtaining
V-~--( On l/ 2E .. + ~,A,,(E) o~E~'
V f(E)
n~t
4r~ __~On 9 ~
-5--
(24)
(25)
Vo ~dk, k~Sdysin7lw(k_kOl2S(k_kt E~E~)(~_cos7),~-'(E)= 9.h----T
0 D
(26)
k,; E-E~ (E,--E)"A~(E)= vo~(E)2nh~ ~ dk, k,2:~dysinvcosv[w(k_k0 i~S(k_ h ] n, .. (27)
0 0
If we go over in the integral of the expression (27) to the dimensionless variable of integration (EI-E)
/kBT D and in Eqs. (24) and (25) to the dimensionless variable E/EF, we see directly that the coefficients
A n are of order (kBTD/EF)n. For simple metals, the ratio kBTD/E F is very small (~ 10-3-5 9 10-3).
Therefore, it is meaningful to seek solutions of Eqs. (245 and (25) in the form of series:
[(E)=~/,,(E), g(E)=Lg~(E),
u~O n=O
assuming that fn and gn are quantities of order (kBTD/EF 5n. Substituting these expansions into /245 and
(255 and equating terms of the same order, we find that
(28)
go(E)= -7-- m '
a/o(E) @o(E)
]I(E)=-A,(E) OE ' g,(E)=-A,(E) O~ ..... (29)
Since the correction terms are very small, we shall restrict ourselves in what follows to terms of
zeroth order.
Substituting (9) and (21) into (6) and using the expressions (28) and (29), we obtain the following final
expressions for the coefficients Ln:
Lo=--a = 3:~ h~ dE E~h~(E) - ~ "~ yore '
0
e 2Y2m~ (0~) n2a(kBT)2[3 E dv]
Li-- h3 dEEV2(E- ee)x(E) - z +-- , (31)
3~ o 3eEF "~ dE E=%
L2
e
(32)
Here we have used the well-known approximate relations for integrals containing the Fermi function
and we have noted that ~e ~EF = (li2/2mS(z37r2/v053/2. Knowing the explicit form of the coefficients Ln,
we can readily show (see [1515 that the well-known Lorentz formulas with relaxation time ~- determined by
(26) are obtained for the resistivity, thermal conductivity, and thermoelectric power.
4. Discussion of the Results
Let us show that for solid metals the expression obtained for the relaxation time automatically yields
results already known in solid-state theory. For simplicity, we restrict the treatment to a crystal of cubic
symmetry. It is well known that for crystals [16-17]
192

7. ,,[-oo
S(~; co)= i ~'~exp {-tu(ni-n2)} Sdtexp {icot}<exp{- i~u~, (t) }exp{i~uo,(0) }>, (335
2~N nln~
where Un(t5 is the displacement of the ion from the equilibrium position and n is the radius vector character-
izing the ion position.
If Un(t5 is expanded in normal vibrations of the lattice ions, S(~t; co) can be represented in the form of
the series
S(u; co) = So(u; co) + Sl(u; ,co) +..., (34)
where S0(~; co) describes the elastic scattering of electrons by the crystal lattice and $1(~r co) the single-
quantum scattering of an electron in which the number of phonons in the scatterer changes by one, etc.
It can be shown that for temperatures that are greater then the Debye temperature [16]
nT~,'
kBT
S, (~; 0,) ~ ~ ~ (co)....
where s is the velocity of sound.
Substituting the expansion (34) into (265 and retaining only the first two terms (single-phonon approxi-
mation) we obtain the well-known expression for v-I [15]
2
x_~(EF)_ vo kBT mkF sin~d~f co 2kFsin-~-
2ah ~ h o
Retaining in (34) the following terms of the expansion, we can allow for the effect of many-phonon
processes on the transport coefficients.
We now turn to liquid metals. If we again restrict ourselves to the single-phonon scattering of elec-
trons, then to determine the dynamic structure factor it is sufficient to use the hydrodynamic approximation
(~; co ~ 05. If we also ignore the damping of spin waves, then [18]
kBT f
where Cp and cv are the specific heats at constant pressure and volume.
The first term in this expression is due to the disordered arrangement of the ions, the second de-
scribes the scattering of an electron on acoustic waves. For a crystalline metal, Cp = cv, and we auto-
matically arrive at the already known result (35). In liquid metals, Cp ~ cV, and therefore structure scat-
tering will make an important contribution to the transport coefficients.
Substituting ~36) into (26) and noting that E ~ EF>> s~, we obtain the expression already found in [19]:
O0
kBT~T mk[jsin'fd~l wt2kFsinf' " ~" (i--cos~f) (37)9-t--t(EF)=
2nh v0 h o
where fiT is the isothermal compressibility. This expression gives the correct qualitative result. For
example, it is readily seen from ~37) and ~35) that the resistivity, p = l/or, is discontinuous at the melting
point, the ratio of p at the melting point for the solid and the liquid phase being
oL = (13~)L (--~-, s" (38)
Ps
If one substitutes into (385 the well-known values for (~T)L [20] and (Ms2/v0)s [21] for Na, K, AI, and Pb,
the theoretical value of pL/PS for these metals is 1.38, I. 38, 2.61, and I. 63, while the experiment [22]
gives 1.49, 1.45, 2.85, and I. 9. However, the hydrodynamic approximation is very crude. We shall
show that Ziman's more exact formula [3] follows from the relation (26) for liquid metals.
To do this we exploit the fact that S(~; w) decreases fairly rapidly with increasing co at fixed ~ and
for almost all metals S04 ; E F/fi 5 = 0 [23-26].
193

8. In the expression (26) we reverse the limits of integration:
kF+U
IhF--M
In the integral over k I we make the change of variable co = (EF-E0/h, and we split the integral over
~4 into two: from 0 to 2k F and 2k F to infinity:
2k F
u0m
o
2m (2kP • co
• f S (x; co) 2kF +
h 2/~F
2"-~ (-2kF x+ •
n
2--m-(ZkF•215
[ •176176 9
n
(-~kF•
In the first term of this relation the limits of the integral over co can be allowed to tend to i~ [since
S(~t,; co) is in effect equal to zero at the upper and the lower limit], and the second term can be ignored
completely [since the integral is here over a region in which S(~t.; w) %-anishes]. In the integrand we then
expand with respect to ~co/E F (this a quantity of order or less than 5.10 -3 [23, 24]) and we use the known
sum rules [17]; then to terms of order m/M, we obtain Ziman's formula
ZhF
vora ! ,w(• 2•215 {S(• m ksT 3:4z-2kFzt_.." }. (40)
"~-'(E~) 4nh3kF3 M EF 4• ~'
Finally, we note that this theory of kinetic processes may be helpful in the study of the optical proper-
ties of liquid metals, in the investigation of the behavior of the transport coefficients of a metal near the
critical point, etc.
We are very grateful to Professor I. Z. Fisher and Professor D. N. Zubarev for stimulating discus-
sions and assistance and also to Yu. A. Tserkovinkov for interesting comments.
1.
2.
3.
4.
5.
6.
7.
8.
9.
I0.
Ii.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
W.
E.
J.
C.
L.
N.
N.
A.
S.
G.
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