Perturbation theory for electrical resistivity of liquid transition metals
Nearly free electron model is used for the s-electron subsystem of liquid transition metals. d-electrons are considered to be bound and serve as an additional scattering factor. Model Hamiltonian of the electron subsystem contains two small parameters: pseudopotential of s-electron-ion interactio...
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| Опубліковано в: : | Condensed Matter Physics |
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| Дата: | 2002 |
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Інститут фізики конденсованих систем НАН України
2002
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| Цитувати: | Perturbation theory for electrical resistivity of liquid transition metals / V.T. Shvets, S. Savenko, S. Datsko // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 511-522. — Бібліогр.: 13 назв. — англ. |
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Shvets, V.T. Savenko, S. Datsko, S. 2017-06-12T16:28:29Z 2017-06-12T16:28:29Z 2002 Perturbation theory for electrical resistivity of liquid transition metals / V.T. Shvets, S. Savenko, S. Datsko // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 511-522. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 72.14.E, 72.15.C DOI:10.5488/CMP.5.3.511 https://nasplib.isofts.kiev.ua/handle/123456789/120673 Nearly free electron model is used for the s-electron subsystem of liquid transition metals. d-electrons are considered to be bound and serve as an additional scattering factor. Model Hamiltonian of the electron subsystem contains two small parameters: pseudopotential of s-electron-ion interaction, and s-d hybridization potential. The linear response theory of Kubo and the method of two-time retarded Green functions are used to investigate the conductivity of the system. The perturbation series by two small system parameters are first derived for the electrical resistivity of liquid transition metals. To introduce the electron-electron interaction, a random phase approach is employed. The analysis of the second and the third terms of perturbation series is given. Their contribution to the electrical resistivity of all liquid transition metals is estimated. Для підсистеми s-електронів перехідних рідких металів використане наближення майже вільних електронів, d-електрони вважаються зв’язаними і грають роль додаткових факторів розсіювання. Модельний гамільтоніан електронної підсистеми містить два малих параметри: псевдопотенціал взаємодії s-електронів з іонами і потенціал гібридизації s- і d-електронів. Для знаходження коефіцієнта електропровідності використовується теорія лінійної реакції Кубо і метод двочасових загаяних функцій Гріна. Для електроопору перехідних рідких металів вперше отримано ряд теорії збурень за двома малими параметрами теорії. Електрон-електронна взаємодія врахована у наближенні випадкових фаз. Детально проаналізовані члени другого і третього порядків теорії збурень. Оцінена їх роль у формуванні електроопору перехідних рідких металів. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Perturbation theory for electrical resistivity of liquid transition metals Теорія збурень для електроопору перехідних рідких металів Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Perturbation theory for electrical resistivity of liquid transition metals |
| spellingShingle |
Perturbation theory for electrical resistivity of liquid transition metals Shvets, V.T. Savenko, S. Datsko, S. |
| title_short |
Perturbation theory for electrical resistivity of liquid transition metals |
| title_full |
Perturbation theory for electrical resistivity of liquid transition metals |
| title_fullStr |
Perturbation theory for electrical resistivity of liquid transition metals |
| title_full_unstemmed |
Perturbation theory for electrical resistivity of liquid transition metals |
| title_sort |
perturbation theory for electrical resistivity of liquid transition metals |
| author |
Shvets, V.T. Savenko, S. Datsko, S. |
| author_facet |
Shvets, V.T. Savenko, S. Datsko, S. |
| publishDate |
2002 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Теорія збурень для електроопору перехідних рідких металів |
| description |
Nearly free electron model is used for the s-electron subsystem of liquid
transition metals. d-electrons are considered to be bound and serve as an
additional scattering factor. Model Hamiltonian of the electron subsystem
contains two small parameters: pseudopotential of s-electron-ion interaction,
and s-d hybridization potential. The linear response theory of Kubo
and the method of two-time retarded Green functions are used to investigate
the conductivity of the system. The perturbation series by two small
system parameters are first derived for the electrical resistivity of liquid
transition metals. To introduce the electron-electron interaction, a random
phase approach is employed. The analysis of the second and the third
terms of perturbation series is given. Their contribution to the electrical resistivity
of all liquid transition metals is estimated.
Для підсистеми s-електронів перехідних рідких металів використане наближення майже вільних електронів, d-електрони вважаються зв’язаними і грають роль додаткових факторів розсіювання. Модельний гамільтоніан електронної підсистеми містить два малих параметри: псевдопотенціал взаємодії s-електронів з іонами і потенціал гібридизації s- і d-електронів. Для знаходження коефіцієнта електропровідності використовується теорія лінійної реакції Кубо і метод двочасових загаяних функцій Гріна. Для електроопору перехідних рідких металів вперше отримано ряд теорії збурень за двома малими параметрами теорії. Електрон-електронна взаємодія врахована у наближенні випадкових фаз. Детально проаналізовані члени другого і третього порядків теорії збурень. Оцінена їх роль у формуванні електроопору перехідних рідких металів.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120673 |
| citation_txt |
Perturbation theory for electrical resistivity of liquid transition metals / V.T. Shvets, S. Savenko, S. Datsko // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 511-522. — Бібліогр.: 13 назв. — англ. |
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| first_indexed |
2025-11-24T06:16:05Z |
| last_indexed |
2025-11-24T06:16:05Z |
| _version_ |
1850844174712045568 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 3(31), pp. 511–522
Perturbation theory for electrical
resistivity of liquid transition metals
V.T.Shvets, S.Savenko, S.Datsko
Odesa State Academy of Refrigeration,
1/3 Dvorianskaia Str., 65026 Odesa, Ukraine
Received April 10, 2002
Nearly free electron model is used for the s-electron subsystem of liquid
transition metals. d-electrons are considered to be bound and serve as an
additional scattering factor. Model Hamiltonian of the electron subsystem
contains two small parameters: pseudopotential of s-electron-ion interac-
tion, and s-d hybridization potential. The linear response theory of Kubo
and the method of two-time retarded Green functions are used to investi-
gate the conductivity of the system. The perturbation series by two small
system parameters are first derived for the electrical resistivity of liquid
transition metals. To introduce the electron-electron interaction, a random
phase approach is employed. The analysis of the second and the third
terms of perturbation series is given. Their contribution to the electrical re-
sistivity of all liquid transition metals is estimated.
Key words: electroresistivity, metal
PACS: 72.14.E, 72.15.C
1. Introduction
The theory of electron transport phenomena in the liquid simple metals dates
back from the Ziman’s work, where the inverse relaxation time was calculated in
the second order by pseudopotential of electron-ion interaction. In succeeding years
the perturbation theory was developed and the numerical calculations of the terms
up to the third order were accomplished (the detailed list of the corresponding pa-
pers is available in the work [2]). At present, the theory of the electron transport
phenomena in the liquid simple metals has taken its complete shape. The investiga-
tion of transition metals started much later and much of the work done is confined
within two various trends. The first one goes back to the work [3] and represents
the generalization of the pseudopotential theory to the case of transition metals.
From the formal point of view it comes to the substitution of the single-particle
T -matrix for the formfactor of pseudopotential. Further on, the resistivity is calcu-
lated in the second order by T -matrix [4,5]. Another trend is connected with Mott
c© V.T.Shvets, S.Savenko, S.Datsko 511
V.T.Shvets, S.Savenko, S.Datsko
[6,9,10], and is based on the idea that the basic factor causing the scattering of con-
duction electrons is hybridization of s- and d- states of the electron subsystem of a
metal. At that d-electrons are considered to be localized and the effect is described
using a hybridization potential. The formula proposed suits the second order of
the perturbation theory by hybridization potential. Subsequently, no attempts were
made neither of combining these two approaches, nor of constructing the perturba-
tion series in the framework of each one. Unexpectedly, the progress was made in
quite different direction. The starting point of these investigations was the tight-
binding approximation [7–10], unlike all the previous cases where the nearly free
electron model was employed. It was found that the development of perturbation
series is more convenient for the metals with relatively high resistance (more than
200 µΩ · cm) are put under consideration. Since nearly half of the transition metals
do not meet these conditions, the problem remains rather acute for transition metals
with a relatively small resistivity. The purpose of the current paper is to build the
perturbation series for resistivity of liquid transition metals using two small theory
parameters: pseudopotential and the hybridization potential, with the subsequent
analysis of the third order term.
2. Hamiltonian
Let us choose the model Hamiltonian of the electron subsystem of transition
liquid metal in the form:
H = H0 + His + Hsd , (1)
where
H0 =
∑
k
εka
+
k ak +
∑
n
εnb+
n bn (2)
is the Hamiltonian of non-interacting subsystems of s- and d- electrons;
εk =
h̄2k2
2m
(3)
is the energy of a free s-electron, εd is the energy of a bound d-electron, a+
k and ak are
the operators of creation and annihilation of s-electrons in the state with quantum
number k, b+
n and bn are the operators of creation and annihilation of d-electrons
in the state with the set of quantum numbers n, which includes the ion number as
well;
His = V −1
∑
q
w(q)ρi(q)ρe(−q) (4)
is the Hamiltonian of s-electron-ion interaction. It’s assumed here that this inter-
action can be described using pseudopotential, just like in the case of a simple
metal. In the expression above w(q) is the screened formfactor of the local model
pseudopotential,
ρi(q) =
∑
n
exp(−iqRn) (5)
512
Electrical resistivity of liquid transition metals
is the Fourier transform of the ion density, Rn is the radius-vector of the n-th ion
and
ρe(q) =
∑
k
a+
k ak+q (6)
is the Fourier transform of s-electron density operator. The last term of expres-
sion (1)
Hsd = V −1/2
∑
k,n
[
∆kna+
k bn + ∆nkb
+
n ak
]
(7)
is the Hamiltonian of s-d hybridization, where V is the volume of the system and
∆kn = ∆k exp (−ikRn), (8)
∆nk = ∆k exp (ikRn) (9)
are the matrix elements of hybridization potential, ∆k is the formfactor.
The present Hamiltonian doesn’t include the electron-electron interaction explic-
itly. However, it is accounted implicitly via the screening of electron-ion interaction.
Here we also neglected the exchange of d-electrons between ions, considering that
this effect gains importance only in the metals with relatively high resistivity.
3. Conductance coefficient
According to the Kubo’s linear response theory we can write the conductivity
coefficient σ in the form
σ = −
1
3V kBT
Im 〈〈I(t)I(0)〉〉0 . (10)
In the right part of this equation there is a zero component of the two-time retarded
Green function, constructed from the operators of electric current
〈〈I(t)I(0)〉〉 =
1
ih̄
〈I(t)I(0)〉, (11)
the angular brackets have the meaning of the grand canonical ensemble average.
Assuming that the electric current is fully determined by s-electrons, we can write
the current operator as
I(t) =
eh̄
m
∑
k
ka+
k (t)ak(t). (12)
Thus, to obtain the conductivity coefficient we should perform calculations of the
following Green function: 〈〈a+
k (t)ak(t)I(0)〉〉. Henceforth, to save place, we will use
the shorthand 〈〈a+
k ak〉〉 for this expression.
513
V.T.Shvets, S.Savenko, S.Datsko
4. The motion equations
In contrast to amorphous alloys at a low temperature, the influence of the dy-
namics of ion subsystem is inessential for the liquid metals. So, it is possible to
regard the ion subsystem as a static one. We will consider it to be static as well that
allows us to use only the Hamiltonian of the electron subsystem while constructing
the motion equations. The Fourier transform of the first equation in the infinite set
of dependent equations will look as follows
h̄ω〈〈a+
kak〉〉ω = 〈a+
k ak〉 +
∑
q
[
〈〈a+
k W (q)ak−q〉〉ω − 〈〈a+
k+qW (q)ak〉〉ω
]
+
∑
k,n
[
〈〈a+
k ∆knbn〉〉ω − 〈〈b+
n ∆nkak〉〉ω
]
. (13)
Here we have denoted
W (q) = V −1w(q)ρi(q). (14)
For each of the new Green functions, the motion equation should be constructed in
turn. These four equations will take the form
(εk − εk−q + h̄ω)〈〈a+
kW (q)ak−q〉〉ω =
= 〈a+
k ak−q〉 +
∑
q′
[
〈〈a+
kW (q)W (q′)ak−q−q′〉〉ω − 〈〈a+
k+q′W (q′)W (q)ak−q〉〉ω
]
+
∑
n
[
〈〈a+
kW (q)∆k−q,nbn〉〉
ω
− 〈〈b+
n ∆nkW (q)ak−q〉〉ω
]
, (15)
(εk+q − εk + h̄ω)〈〈a+
k+qW (q)ak〉〉ω =
= 〈a+
k+qak〉 +
∑
q′
[
〈〈a+
k+qW (q)W (q′)ak−q〉〉
ω
− 〈〈a+
k+q+q′W (q′)W (q)ak〉〉ω
]
+
∑
n
[
〈〈a+
k+qW (q)∆knbn〉〉
ω
− 〈〈b+
n ∆n,k+qW (q)ak〉〉ω
]
, (16)
(εk − εn + h̄ω)〈〈a+
k ∆knbn〉〉ω =
= 〈a+
k ∆kn〉 −
∑
q
〈〈a+
k+qW (q)∆knbn〉〉ω+
∑
k′
〈〈a+
k ∆kn∆nk′ak′〉〉ω
−
∑
n′
〈〈b+
n′∆n′k∆knbn〉〉ω, (17)
(εn − εk + h̄ω)〈〈b+
n ak∆nk〉〉ω =
= 〈b+
n ∆nk〉 +
∑
q
〈〈b+
n ∆nkW (q)ak−q〉〉ω+
∑
n′
〈〈b+
n ∆nk∆kn′bn′〉〉ω
−
∑
k′
〈〈a+
k′∆k′n∆nkak〉〉ω. (18)
5. Second order perturbation theory
Certain sums in the right parts of the last four equations contain an item of the
value comparable to the contribution of the rest summands. These items are the
Green functions containing weakly coupled averages. If we take into account only
514
Electrical resistivity of liquid transition metals
these and neglect all the other summands we will find the conductivity in the lowest
(quadratic) order of perturbation theory by hybridization and by pseudo-potential.
Considering only the weakly coupled averages we can write down the system:
(εk − εk−q + h̄ω)〈〈a+
kW (q)ak−q〉〉ω =
= 〈〈a+
k W (q)W (q′)ak〉〉ω − 〈〈a+
k−qW (−q)W (q)ak−q〉〉ω , (19)
(εk+q − εk + h̄ω)〈〈a+
k+qW (q)ak〉〉ω =
= 〈〈a+
k+qW (q)W (q′)ak+q〉〉ω − 〈〈a+
k W (−q)W (q)ak〉〉ω , (20)
(εk − εn + h̄ω)〈〈a+
k ∆knbn〉〉ω = 〈〈a+
k ∆kn∆nkak〉〉ω , (21)
(εn − εk + h̄ω)〈〈b+
n ∆nkak〉〉ω = −〈〈a+
k ∆kn∆nkak〉〉ω . (22)
At that, in the lowest order of perturbation theory we have neglected the absolute
terms of equations. From the formal point of view this is plausible in the calculations
of the terms up to the 4-th order.
To derive the closed set of equations for the needed Green function we must
uncouple all the functions that differ from the sought one. Our approach consists in
the assumption that only the functions containing weakly coupled averages should
be uncoupled, namely
〈〈a+
k W (q)W (−q)ak〉〉ω = 〈W (q)W (−q)〉〈〈a+
k ak〉〉ω , (23)
〈〈a+
k ∆kn∆nkak〉〉ω = 〈∆kn∆nk〉〈〈a
+
kak〉〉ω . (24)
Uncoupling of this type is widely used in various fields of nowadays theoretical
physics and is usually referred to as the mean field approximation. In the particular
case of electron transport phenomena it means that we consider the scattering of the
electrons on the averaged potential produced by the whole ion subsystem, instead
of examining the scattering on the specific ion. The precision of this approximation
increases as the mean free path of the conductance electron grows. Finally, we can
rewrite the first five equations of the set in the form of a single equation for the
wanted Green function. In the static case this equation looks as follows:
〈a+
k (0)ak(0)〉 = i
2πN
V 2
∑
q
w2(q)S(q)δ(εk − εk+q)
×
[
〈〈a+
k (t)ak(t)〉〉0 − 〈〈a+
k+q(t)ak+q(t)〉〉0
]
+ i
2π
V
∑
n
∆2
kδ(εk − εn)〈〈a+
k (t)ak(t)〉〉0 . (25)
Here we have used the well-known operator relation
1
x + iδ
= P
1
x
− iπδ(x) (26)
and have introduced the static structure factor of ion subsystem
S(q) =
1
N
〈ρi(q)ρi(−q)〉. (27)
515
V.T.Shvets, S.Savenko, S.Datsko
The kinetic equation derived differs from the one for the simple metals by the
last term only. As well as its exact solution is quite analogous and based on the
result precise to a second order of perturbation theory
〈〈a+
k+q(t)ak+q(t)〉〉0 =
k + q
|k + q|
〈〈a+
k (t)ak(t)〉〉0 (28)
(see e.g. [1]).
As a result, the solution takes the form
〈〈a+
k (t)ak(t)〉〉0 = −
i
h̄
〈a+
k (0)ak(0)〉 τ(k), (29)
where
τ−1(k) =
2πN
h̄V 2
∑
q
w2(q)S(q)δ(εk − εk+q) [1 − cos (k∧k + q)]
+
2π
h̄V
∑
n
∆2
kδ(εk − εn). (30)
is the inverse relaxation time for the conductivity of transition metals. If in the last
expression we put ∆k → 0, we are left with a well-known Ziman formula for simple
metals [1]. After some simplification it can be brought to the form
τ−1(k) =
Nm
4πV h̄3k3
2k
∫
0
dx x3w2(x)S(x). (31)
On the other hand, if we tend w → 0, our result comes to Mott formula [6]. After the
transformation consisting in switching from summation over the quantum numbers
to the integration over the energy we will derive
τ−1(k) =
2π
h̄
∆2
kG(εk), (32)
where G(ε) is the density of d-states.
6. Third order of perturbation theory
To do calculations of the terms of the third order we should supply the previously
derived set with the following equations:
(εk − εk−q−q′ + h̄ω)〈〈a+
k W (q)W (q′)ak−q−q′〉〉ω =
= 〈〈a+
kW (q)W (q′)W (−q − q′)ak〉〉ω
− 〈〈a+
k−q−q′W (q)W (q′)W (−q − q′)ak−q−q′〉〉ω , (33)
(εk+q′ − εk−q + h̄ω)〈〈a+
k+q′W (q′)W (q)ak−q〉〉ω =
= 〈〈a+
k+q′W (q′)W (q)W (−q − q′)ak+q′〉〉ω
516
Electrical resistivity of liquid transition metals
− 〈〈a+
k−qW (−q − q′)W (q′)W (q)ak−q〉〉ω , (34)
(εk − εn + h̄ω)〈〈a+
kW (q)∆k−q,nbn〉〉ω =
= 〈〈a+
k W (q)∆k−q,n∆nkak〉〉ω , (35)
(εn − εk−q + h̄ω)〈〈b+
n ∆nkW (q)ak−q〉〉ω =
= −〈〈a+
k−q∆k−q,n∆nkW (q)ak−q〉〉ω , (36)
(εk+q − εk−q′ + h̄ω)〈〈a+
k+qW (q)W (q′)ak−q′〉〉ω =
= 〈〈a+
k+qW (q)W (q′)W (−q − q′)ak+q〉〉ω
− 〈〈a+
k−q′W (−q − q′)W (q)W (q′)ak−q′〉〉ω , (37)
(εk+q+q′ − εk + h̄ω)〈〈a+
k+q+q′W (q′)W (q)ak〉〉ω =
= 〈〈a+
k+q+q′W (q′)W (q)W (−q − q′)ak+q+q′〉〉ω
− 〈〈a+
kW (−q − q′)W (q′)W (q)ak〉〉ω , (38)
(εk+q − εn + h̄ω)〈〈a+
k+qW (q)∆knbn〉〉ω =
= 〈〈a+
k+qW (q)∆kn∆n,k+qak+q〉〉ω , (39)
(εn − εk + h̄ω)〈〈b+
n ∆n,k+qW (q)ak〉〉ω =
= −〈〈a+
k ∆kn∆n,k+qW (q)ak〉〉ω , (40)
(εk+q − εn + h̄ω)〈〈a+
k+qW (q)∆knbn〉〉ω =
= 〈〈a+
k+qW (q)∆kn∆n,k+qak+q〉〉ω , (41)
(εk − εk−q + h̄ω)〈〈a+
k ∆kn∆n,k−qak−q〉〉ω =
= 〈〈a+
k ∆kn∆nkW (−q)ak〉〉ω
− 〈〈a+
k−qW (−q)∆kn∆n,k−qak−q〉〉ω , (42)
(εn − εk−q + h̄ω)〈〈b+
n ∆nkW (q)ak−q〉〉ω =
= −〈〈a+
k−q∆k−q,n∆nkW (q)ak−q〉〉ω , (43)
(εk+q − εk + h̄ω)〈〈a+
k+q∆k+q,n∆nkak〉〉ω =
= 〈〈a+
k+q∆k+q,n∆nkW (−q)ak+q〉〉ω
− 〈〈a+
kW (−q)∆k+q,n∆nkak〉〉ω . (44)
Examining the structure of these equations one can see that the kinetic equation
will be widened with terms of the third order by pseudopotential as well as with
the terms of the second order by hybridization potential together with the first one
by pseudopotential. Here we will discuss the latter only as far as the former is well
described in the works devoted to simple metals [12,13]. In view of the preceding,
the kinetic equation will take the form
h̄ω〈〈a+
kak〉〉ω = 〈a+
k ak〉 + T̂ww〈〈a
+
k ak〉〉ω + T̂∆∆〈〈a
+
k ak〉〉ω
+ T̂www〈〈a
+
k ak〉〉ω + T̂∆∆w〈〈a
+
kak〉〉ω. (45)
After the weakly coupled averages are uncoupled, we derive
T̂∆∆w〈〈a
+
k ak〉〉ω =
∑
q,n
〈W (q)∆k+q,n∆nk〉×
517
V.T.Shvets, S.Savenko, S.Datsko
×
{
2
[
1
εk − εk+q + h̄ω
·
1
εk − εn + h̄ω
−
1
εk+q − εk + h̄ω
·
1
εn − εk + h̄ω
]
× 〈〈a+
k ak〉〉ω
+
[
1
εk − εk+q + h̄ω
·
1
εn − εk + h̄ω
−
1
εk+q − εk + h̄ω
·
1
εk+q − εn + h̄ω
+
1
εn − εk + h̄ω
·
1
εn − εk+q + h̄ω
−
1
εk − εn + h̄ω
·
1
εk+q − εn + h̄ω
+
1
εn − εk + h̄ω
·
1
εk+q − εk + h̄ω
−
1
εk − εn + h̄ω
·
1
εk − εk+q + h̄ω
]
× 〈〈a+
k+qak+q〉〉ω
}
. (46)
Using the well-known operator identity in a static case ω = 0 we can write the term
under consideration as
T̂∆∆w〈〈a
+
k ak〉〉0 = −4πi
∑
q,n
〈W (−q)∆k+q,n∆nk〉
×
{
δ(εk − εk+q)
εk − εn
[
〈〈a+
k ak〉〉0 − 〈〈a+
k+qak+q〉〉0
]
+
δ(εk − εn)
εk − εk+q
〈〈a+
k ak〉〉0
}
. (47)
Let us do a thorough analysis of the average under summation
〈W (q)∆k+q,n∆nk〉 =
1
V 2
w(q)∆k∆|k+q|〈ρ
i(−q) exp(−iqRn)〉. (48)
It is helpful to divide the summation over the “n” into two sums. The first of them
corresponds to the summation over the set of quantum numbers, determining the
electron energy. The second one corresponds to the summation over the ion positions.
Taking the assumption that the electron energy εn doesn’t depend on the position
of a specific ion we can do a significant simplification. The potential of hybridization
depends in turn on the ion positions only. In that case we can easily carry out the
summation and express the result
∑
n
〈W (q)∆k+q,n∆nk〉 =
N
V 2
w(q)∆k∆|k+q|S(q) (49)
via the static structure factor of the ion subsystem. There is only a summation over
the electron states in the ion left in the kinetic equation. As far as the formfactor
of hybridization potential depends purely on the absolute value of wave vector, the
only multiplier in the integral equation kernel that depends on the angle between
k and k + q is delta function. On the whole, the structure of the third order term
under consideration is equivalent to the structure of the second order term as the
angle dependence is regarded. So, we can put the third order term in a similar way
T̂∆∆w〈〈a
+
k ak〉〉0 = −ih̄τ−1
∆∆w(k)〈〈a+
kak〉〉0 , (50)
518
Electrical resistivity of liquid transition metals
where the contribution of the third order term to the inverse relaxation time has
the form
T̂∆∆w(k) = −
4πN
h̄V 2
∑
q,n
w(q)∆k∆|k+q|S(q)
×
{
δ(εk − εn)
εk − εk+q
+
δ(εk − εk+q)
εk − εn
[1 − cos (k∧k + q)]
}
. (51)
Usually, when the summation over the electron energy states in all the ions is per-
formed, one can allow for a disorder of the ion system introducing the energy dis-
tribution function, namely
∑
n
→
∫
dε G(ε). (52)
It is assumed that the function G(ε) is equal to electron density states up to nor-
malization factor. In our case it is d-states. So that
τ−1
∆∆w(k) =
4πN
h̄V 2
∑
k′
w(|k − k′|)∆k∆k′S(|k− k′|)
×
G(εk)
εk − εk′
+ [1 − cos (k∧k′)] δ(εk − εk′)
∞
∫
−∞
G(ε)
εk − ε
dε
. (53)
To derive the final result we should replace the summation over the wave vector by
integration. In a traditional approach
τ−1
∆∆w(kF) =
1
6π3z
∆2
kF
2kF
∫
0
w(x)S(x)x3dx
∞
∫
−∞
G(ε)
εF − ε
dε
+
2k2
F
3π3z
∆kF
G(εF)
∞
∫
0
dk k
∆k
k2
F − k2
kF+k
∫
|kF−k|
w(x)S(x)xdx, (54)
where z is the valence of ion. The electrical resistance can be found using a well-
known formula of Lorenz at that.
7. Discussion
If within the framework of Mott model we suppose that the only considerable
potential determining the electron scattering is the hybridization potential, then the
third order term in the expansion of the inverse relaxation time will disappear. It
is easy to see if we tend pseudopotential to zero. The term following the quadratic
will be of the fourth order in this approach.
First of all, we should notice that the derived result generalizes the Mott formula
yet in the second order of perturbation theory. The importance of the terms of
expansion by pseudopotential is clarified by the following estimation. Let us, for
example, put the disordered 3d-metals under consideration. This row in periodic
519
V.T.Shvets, S.Savenko, S.Datsko
table is closed by the Cu, that has completed 3d-states. An accurate calculation of
its resistance is possible only while taking into account the pseudopotential, and
results approximately in 20 µΩ · cm. For all the other metals in the row, it varies
from 80 µΩ · cm to 400 µΩ · cm. Considering the pseudopotential contribution to be
equal for all 3d-metals it amounts from 5 to 25 percent from the total value. The
estimation of the third order term can be done in the same approach. To do this, we
need to multiply the second order hybridization term by a square root of the second
order pseudopotential term. The resulting range is 20–50%.
There is a qualitative conclusion that can be drawn from the result obtained. It
is known that the electrical resistance decreases as the number of d-electrons grows.
Our correction contains two components. The sign of the first one doesn’t depend
on mutual disposal of the Fermi level and the middle of d-zone. It is completely
determined by the behaviour of hybridization potential formfactor in the entire
region of integration. On the contrary, the sign of the second component depends
on the mutual disposal of the energy levels. As long as
∞
∫
−∞
G(ε)
εF − ε
dε > 0, εF > ε0
and
∞
∫
−∞
G(ε)
εF − ε
dε < 0, εF < ε0 .
Usually we have
2kF
∫
0
W (x)S(x)x3dx < 0,
so, the second component will be positive for the metals with a few d-electrons (less
than 5) and negative for those with a large number of d-electrons (more than 5).
Correspondingly, the electrical resistance is higher in the first case.
References
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521
V.T.Shvets, S.Savenko, S.Datsko
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