Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential
A theory of transport coefficients of fully ionized strongly coupled plasmas,
 based on the self-consistent field concept and having no adjustable parameters, is presented. The proposed approach is not connected directly with
 the kinetic equation. A comparison of the obtained results...
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| Zitieren: | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential / V.M. Adamyan, I.M. Tkachenko // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 641-656. — Бібліогр.: 29 назв. — англ. |
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| citation_txt | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential / V.M. Adamyan, I.M. Tkachenko // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 641-656. — Бібліогр.: 29 назв. — англ. |
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| description | A theory of transport coefficients of fully ionized strongly coupled plasmas,
based on the self-consistent field concept and having no adjustable parameters, is presented. The proposed approach is not connected directly with
the kinetic equation. A comparison of the obtained results with the computer simulation data on the classical hydrogen-like plasmas macroscopic
properties shows that using the Yukhnovsky-Kelbg-Deutsch effective pair
potentials instead of the genuine Coulomb potentials can compensate, to
a certain extent, inaccuracies related to the standard approximations of the
quantum theory of Coulomb systems.
На основі уявлень про самоузгоджене внутрішньоплазмове поле
пропонується методика обчислення коефіцієнтів переносу у повністю іонізованій густій плазмі. Ця методика прямо не використовує ані
кінетичне рівняння, ані параметри підгонки. Порівняння отриманих
результатів з даними комп’ютерного моделювання параметрів густої
майже класичної воднево-подібної плазми показує, що використання з самого початку ефективного парного потенціалу Юхновського-Кельбга-Дойча замість кулонівського може до певної межі компенсувати похибки простіших стандартних наближень квантової теорії багатьох заряджених частинок.
|
| first_indexed | 2025-12-07T17:08:34Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 641–656
Linear static response of model
plasmas and Yukhnovsky-Kelbg-
Deutsch effective potential
V.M.Adamyan 1 ∗, I.M.Tkachenko 2 †
1 Department of Theoretical Physics, University of Odesa,
65026 Odesa, Ukraine
2 Department of Applied Mathematics, Polytechnic University of Valencia,
46071 Valencia, Spain
Received June 7, 2000
A theory of transport coefficients of fully ionized strongly coupled plasmas,
based on the self-consistent field concept and having no adjustable param-
eters, is presented. The proposed approach is not connected directly with
the kinetic equation. A comparison of the obtained results with the com-
puter simulation data on the classical hydrogen-like plasmas macroscopic
properties shows that using the Yukhnovsky-Kelbg-Deutsch effective pair
potentials instead of the genuine Coulomb potentials can compensate, to
a certain extent, inaccuracies related to the standard approximations of the
quantum theory of Coulomb systems.
Key words: hydrogen plasmas, transport coefficients, pair potentials
PACS: 52.25.Fi, 52.25.Ub, 52.65.-y
Dedicated to academician Igor Yukhnovsky on the occasion of his 75th birthday.
1. Introduction
The aim of this work is to bring together some results of the quantum statisti-
cal theory of the Coulomb systems transport properties with the results related to
the completely ionized nearly classical hydrogen plasmas(HP). The “experimental”
basis of this work is the computer simulation of the HP microscopic characteristics
obtained in [1,2]. The molecular dynamics (MD) method used there was borrowed
from the theory of classical liquids.
In order to avoid the Coulomb collapse while treating HP in the framework of the
classical statistical mechanics, instead of bare Coulomb potentials the method of MD
∗E-mail: vma@dtp.odessa.ua
†E-mail: imtk@mat.upv.es
c© V.M.Adamyan, I.M.Tkachenko 641
V.M.Adamyan, I.M.Tkachenko
employs effective pair potentials, which account, to a certain extent, for diffraction
and symmetry effects.
Let Ψn and En be the eigenfunctions and eigenvalues of the complete Hamiltonian
of the N electrons and N protons of HP, while {~ra} denotes the set of position
vectors of all plasma particles, rab = |~ra − ~rb|, and let V (rab) be the effective (self-
consistent) pair potential between particles a and b. Effective pair potentials can
be in principle derived by transforming based on a reasonable approximation of the
quantum-mechanical Slater sum
W ({~ra}) =
∑
n
Ψ∗
n exp (−βEn) Ψn (1)
as
W ≃ exp
(
−β
∑
a<b
V (rab)
)
. (2)
reminding the classical Boltzmann factor [3]. In the classical (high-temperature)
limit, V (rab) should reduce to the bare Coulomb potential between the two particles,
i.e.,
lim
T→∞
V (rab) =
ZaZbe
2
rab
, (3)
where Za = +1 (ions, a = i), or −1 (electrons, a = e). An effective potential
determined this way is generally dependent both on temperature and density of the
particles. Effective pair potentials appropriate to the low-density limit were first
derived for the hydrogen plasma by Yukhnovsky [4], Kelbg, and Deutsch [5] and
collaborators from an exact numerical computation of the two-particle Slater sum
for electron-electron and electron-proton pairs. At sufficiently high temperatures
(kBT & Ry), the contribution of bound states to the electron-proton Slater sum
may be ignored. If, moreover, the scattering states are limited to s waves, and if the
effects of permutative symmetry are neglected even for electron-electron pairs, the
following very simple effective potentials can be derived:
V (rab) =
ZaZbe
2
rab
[
1− exp
(−rab
λab
)]
, (4)
where
λab =
[
~β
2π
(
m−1
a +m−1
b
)]1/2
. (5)
The fact that the potentials V (rab) remain finite as rab → 0 is a consequence
of the uncertainty principle, which prevents the aforementioned Coulomb collapse.
At the temperatures of interest, λii is much smaller than the Wiegner-Seitz radius
d = (3/4πn)1/3, where n = ne = ni is the number density of the system particles.
Thus the effective ion-ion interaction is virtually identical to the bare Coulomb
potential at all separations.
An important approximation made in the MD simulations [2] is the neglect
of the density dependence of the effective pair potentials, even at densities of the
642
Linear static response of model plasmas
order of 1024 cm−3, which is reasonable as long as λab/d ≪ 1, i.e., for a sufficiently
nondegenerate plasma [3]. It was also implicitly assumed in [2] that the effective
potentials of clusters of three or more particles are pairwise additive. Despite all
these limitations, the results of [2] have not been surpassed in quality and precision:
the character of collective properties of the plasma is dictated primarily by the long
range part of the potential and therefore it is insensitive to the details of the short
range behaviour. For the same reason, the numerical simulations were carried out
in [2] without including the symmetry part of the electron-electron potential.
In this paper we study the electric conductivity of HP in the same near-classical
region as in [2].
In the three sections that follow we outline the general theoretical approach to
determining the strongly-coupled plasmas transport properties [6], based on the idea
of independent particles walking randomly in the fluctuating internal self-consistent
field. Thus first we obtain the expression for the plasma static conductivity using
the two-particles Green functions calculated in the random–phase approximation
(RPA). Then we improve this approximation taking into account the local field
corrections (LFCs), which at the same time may be treated as those resulting from
the substitution of the bare Coulomb potentials by the effective potentials according
to Yukhnovsky-Deutsch. To compare our outcome with the simulation results the
calculations [7] are carried out with the model effective potential of [5] employed in
[2]. The corresponding data is summarized and discussed in section 5. It follows from
this data that the substitution of the screened Coulomb potential in the quantum
statistical expressions for the transport coefficients obtained in the LFC-corrected
RPA by the screened Yukhnovsky-Deutsch potential brings results substantially
closer to the experimental data. This fact deserves a further investigation.
The approach possesses no adjustable parameters, except for the Monte-Carlo
one-component plasma equation of state [8], interpolated with the low-density ex-
pansion by Abe [9], see appendix 1.
From now on we use the following notations. The thermodynamic state of the
system is characterized by the standard dimensionless parameters:
Γ = βe2(4πn/3)1/3 which measures the rate of Coulomb coupling in the system,
D = θ−1 = βEF = β~2k2F/2m, measuring the plasma degeneracy rate, and
the Brückner parameter rs = Γθ/0.543.
As usual, kF = (3π2n)1/3 is the Fermi wavevector, and EF is the Fermi energy.
2. The Lorentz formula
The starting-point of the approach developed in [6] is the idea of self-consistent
field: each charge carrier of HP moves in a self-consistent field generated by all
charges of the system. The finite values of the transport coefficients then result from
the electron’s scattering on the self-consistent field fluctuations. This approach was
643
V.M.Adamyan, I.M.Tkachenko
prompted by the paper [10], which connects the Lorentz-model expression for the
electrical conductivity with the Green’s-function formalism for electrons in crystals
with randomly distributed impurities as well as in metallic alloys and in conducting
liquids.
We begin the calculation of the HP static conductivity with the general expres-
sion
σ =
∂
∂F
jx(~r, t)|F=0, (6)
where ~j(~r, t) is the averaged current density generated in the system by an external
electric field
~F (~r, t) = (F, 0, 0) exp(δ t), δ > 0, δ −→ 0+. (7)
The specific time dependence of the field is introduced to avoid coherent currents
induced at the switch-on moment t = −∞. In the general quantum-mechanical
expression
jx(~r, t) = Tr ρ̂(t){− i~e
2m
[Ψ+(~r,−∞)
∂
∂x
Ψ(~r,−∞)− ∂
∂x
Ψ+(~r,−∞)Ψ(~r,−∞)]}, (8)
where (−e) and m are the electronic charge and mass; only the density operator
ρ̂(t) depends on the external field (7) through the interaction contribution
−eF e−δt
∫
Ω
xΨ+(~r, s)Ψ(~r, s)d~r
to the system Hamiltonian H(t). Therefore
[
∂ρ̂(t)
∂F
]
F=0
=
ie
~
∫ ∞
0
e−δsds
∫
Ω
x[Ψ+(~r, s)Ψ(~r, s), ρ̂(−∞)]d~r, (9)
where
ρ̂(−∞) = exp {−β[H(−∞)− φ]} ,
exp (−βφ) = Tr exp [−βH(−∞)], (10)
φ being the system free energy of Helmholtz and Ω the system volume, β−1 = kBT .
Hence
σ =
e2
2m
ℜe
∫ ∞
0
e−δsds
∫
Ω
d~r ′x′ Tr ρ̂(−∞)
×
[
Ψ+(~r,−∞)
∂
∂x
Ψ(~r,−∞)− ∂
∂x
Ψ+(~r,−∞)Ψ(~r,−∞),Ψ+(~r ′, s)Ψ(~r ′, s)
]
.(11)
The second-quantized wave function Ψ(~r, t) in (11) is expressible in terms of the
one-electron normalized eigenfunctions ψν(~r) of the one-electron free Hamiltonian
ĥ0,
ĥ0ψν(~r) = ενψν(~r), (12)
Ψ(~r, s) =
∑
ν
aν exp
(
− i
~
ενs
)
ψν(~r), (13)
644
Linear static response of model plasmas
aν being the corresponding annihilation operator.
For the averaged commutator [a+µ′aµ, a
+
ν′aν ] we have
Tr ρ̂(−∞)[a+µ′aµ, a
+
ν′aν ] = δµν′δµ′ν [wν − wµ], (14)
where
wν = w(εν) = {exp [β(εν − µ)] + 1}−1
is the Fermi-Dirac distribution, µ being the electronic subsystem chemical potential.
Using (14) in (11) yields
σ =
e2
2m
ℜe
∫ ∞
0
∫ ∞
0
dε1dε2
∫
Ω
d~r ′x′
〈
G(~r, ~r ′; ε1)
∂
∂x
G(~r, ~r ′; ε2)
−G(~r, ~r ′; ε2)
∂
∂x
G(~r, ~r ′; ε1)
〉
~[w(ε1)− w(ε2)]
i(ε1 − ε2 − i~δ)
, (15)
where
G(~r, ~r ′; ε) =
∑
ν
ψ+
ν (~r )ψν(~r
′)δ(εν − ε)
is the Green’s function for the Schrödinger equation involving the self-consistent
field V (~r ):
− ~
2
2m
∆G + eV (~r )G = εG+ δ(~r − ~r ),
G(~r, ~r ′; ε) = G(~r ′, ~r; ε). (16)
The symmetry properties of the Green’s function lead to
σ =
πe2~3
m2
ℜe
∫ ∞
0
dε
dω(ε)
dε
∫
Ω
d~r ′
〈
∂G(~r ′, ~r; ε)
∂x′
· ∂G(~r, ~r
′; ε)
∂x
〉
. (17)
Averaging in equation (15) is to be carried out over the self-consistent field fluctu-
ations.
An important advantage of formula (17) for σ is that the diagrammatic technique
of perturbation theory can be directly applied to calculate its right-hand side. In
addition, the present problem lacks the divergence difficulties characteristic of quan-
tum electrodynamics and the expression (17) permits one to carry out the averaging
over the fluctuations of the self-consistent field before the coordinate integration.
Applying a diagrammatic technique analogous to that of the quantum field theory
to the calculation of the electron conductivity in crystals with randomly distributed
defects, Edwards [10] established that the averaging of the product of the Green
functions in (17) and subsequent coordinate integration yields the Lorentz formula
σ = −4e2
3m
∫ ∞
0
EdE
dw(E)
dE
ρ(E)τ(E), (18)
where ρ(E) = (2m3E)1/2/2π2
~
3 is the density of one-electron states in the energy
space and τ(E) is the mean relaxation time.
645
V.M.Adamyan, I.M.Tkachenko
Under certain approximations and in the case of finite conductivity conditioned
by the scattering on point defects, the mean relaxation time τ(E) in (18) can be
expressed through the exact scattering cross-section of electron scattering on a single
defect and the pair correlation functions of defects [10]. Using arguments similar to
those of [10], it was shown in [6] that formula (18) is also true for strongly correlated
plasmas of high density and, in particular, for dense HP. However, the finite values of
the mean relaxation time for electrons in plasmas are due to the electron scattering
on thermal fluctuations of the self-consistent electric field formed by all charged
particles of a plasma.
3. Calculation of conductivity
According to the results obtained in [6] as a modification of the approach devel-
oped in [10], the main contribution to the expressions for the averaged one-electron
Green’s function 〈G〉 and the averaged product 〈GG〉 in dense fully ionized plasmas
produces an infinite sum of a certain subclass of diagrams of the formal perturbation
series in the powers of the fluctuations of the self-consistent field potential. These
diagrams are actually those appearing in the H.Brückner method. Their summation
leads, in particular, to the following expression for the inverse mean relaxation time
in terms of the self-consistent field correlation function
τ−1(E) =
me2
4π(2mE)3/2
∫ Q
0
q3dq
∫ ∞
−∞
〈
|V̂s(~q, ω)|2
〉
dω. (19)
Here Q = (8mE/~2)1/2, the momentum ~Q being the maximum possible variation
of the electronic momentum as a result of the scattering process, and
V̂s(~q, ω) =
4πe
q2ε(q, ω)
∑
a=e,i
Za ρ̂a(~q, ω) (20)
is the screened field potential operator complete Fourier transform, ρ̂a(~q, ω) being the
a-species density operator in the (~q, ω)-space, and ε−1(q, ω) – the plasma dynamic
screening function. The field potential correlation function thus equals
〈|V̂ (~q, ω)|2〉 =
(
4πe
q2ε(q, ω)
)2 ∑
a,b=e,i
ZaZbSab(~q, ω). (21)
The dynamic structure factor of the species a and b,
Sab(~q, ω) = 〈ρ̂∗a(~q, ω)ρ̂b(~q, ω)〉 , (22)
is related, by the fluctuation dissipation theorem
Sab(~q, ω) =
~
2π
coth(β~ω/2)ℑmXab(~q, ω), (23)
646
Linear static response of model plasmas
to the partial density-response (Green’s) function
Xab(~q, ω) = Πa(q, ω)δab − Πa(q, ω)Πb(q, ω)J
ab(~q, ω), (24)
Jab(~q, ω) =
4πe2
q2
ZaZb
ε(q, ω)
(25)
being the full vertex part and Πa(q, ω) the a-species polarization operators, which
also determine the dielectric function in equation (20) and
ε(q, ω) = 1 +
4πe2
q2
∑
a=e,i
Z2
aΠa(q, ω). (26)
Substitution of equations (21)–(25) into equation (19) and integration [11], yields
τ−1(E) =
4πme4
β(2mE)3/2
∫ Q
0
dq
q
∑
a,l
Z2
aΠa(q, l)
ε3(q, l)
. (27)
Here the l-summation is spread over the poles
Ωl = 2πl/β~ (l = 0,±1,±2, . . .) (28)
of coth(β~z/2) on the imaginary z-axis, i.e., over the Matsubara frequencies [12]
and Πa(q, l) are the real parts of the Πa(q, ω) operators at ω = iΩl. Equation (27),
together with equation (18), forms a general algorithm of conductivity calculation, as
soon as specific approximate expressions are used for the density-response functions
and the polarization operators, see appendix 1.
Observe that the expression obtained for the relaxation time coincides with that
calculated in the Born approximation for an extraneous particle having the electron
mass and charge and scattering on the fluctuating internal field of HP.
4. Limiting cases
Despite the approximations made to obtain equation (27) expression for the
plasma conductivity, it possesses correct limiting forms corresponding to the cases
of dilute gas plasma and metal-density Coulomb systems. In particular, if we omit
the electronic contribution in equation (27) and neglect the screening effects (i.e. set
ε (q, l) = 1) and the momentum dependence of the ionic form factors, the sum on
the right-hand side of equation (27) becomes a constant βn. If we further presume
E to be equal to the mean kinetic energy of an electron, we retrieve from equa-
tion (27) the Coulomb logarithm, and equation (18) with w(E) substituted by the
Boltzmann distribution takes the form of the Spitzer formula without corrections
due to electron-electron interactions [13].
On the other hand, if we consider the low-temperature limiting case (β−1 → 0),
the Fermi-Dirac distribution derivative in equation (18) turns into −δ(E −EF) and
647
V.M.Adamyan, I.M.Tkachenko
Table 1. σ∗
L are the results of the extrapolation procedure according to equa-
tion (31) and σ∗
D were calculated in terms of the diffusion coefficients as explained.
σ∗
Cp are the results of the present work computed using the Coulomb potential,
and σ∗
ep represent our results; σ∗
BRD stands for the results of [15]
ne × 10−24(cm−3) T × 10−5(K) Γ rs σ∗
L σ∗
D σ∗
Cp σ∗
ep σ∗
BRD
1.611 1.579 2.0 1.00 1.1 0.60 0.59 1.20 3.72
1.610 6.315 0.5 1.00 2.15 0.86 1.00 1.40 2.13
25.170 15.79 0.5 0.40 3.6 1.47 1.80 2.70 4.13
Q becomes equal to 2kF, so that we immediately regain the Ziman specific resistance
formula [14].
Notice that no special effort was done ab initio to guarantee the correct limiting
behaviour of our model. Nevertheless, further studies of the limiting behaviour of our
model and a comparison with other general expressions for the collision frequency
(e.g., [15]) or for the conductivity itself (see [16]) are to be carried out.
In general, the difference between our expression and that of Ziman (widely used
lately to calculate conductivity [17–19]) is that we include the energy-dependent
relaxation time [equation (27)], and the Ziman formula takes it at E = EF. In addi-
tion, we have the electron-electron interaction included explicitly via the structure
factor See(q, ω).
As it is well known, the σ(T ) dependence at a constant density is character-
ized by a minimum corresponding to a transition from the low-temperature regime
with decreasing (with growing T ) conductivity characteristic of metals and liquid
metals (Ziman regime) to that of increasing conductivity at higher temperatures,
characteristic of dilute plasmas (Spitzer regime) with σSp(T ) ∝ T 3/2.
5. Results and conclusions
Here we report our results on the conductivity computation obtained for the
conditions corresponding to model Coulomb plasmas [2]. Other results, including
those for the thermal conductivity (see appendix 2), can be found elsewhere [20,7].
The dynamical results of [2] were successfully considered in [21].
A reasonable agreement with the conductivity results of [2] was obtained in
[15] (see table 1, where the results are presented for the dimensionless conductivity
σ∗ = σ/ωp , ωp = (4πne2/m)
1/2
being the electronic plasma frequency).
The static conductivity of model plasmas was obtained in [2] based on the Nernst-
Einstein law in terms of electronic and ionic diffusion coefficients directly estimated
by MD simulations:
σD = βnee
2(Di +De). (29)
In addition, σ was determined, at least for Γ = 2 and rs = 1, through the electric
current autocorrelation function in the relaxation time approximation [2].
648
Linear static response of model plasmas
Notice that the simulation data for Γ = 2 and rs = 1 were obtained in [2] by MD
calculations; in this case the value σ∗
D was calculated as
σ∗
D =
3Γ
4π
(m
M
)
D∗
i +D∗
e , (30)
whereM is the proton mass,D∗
e andD
∗
i being the dimensionless diffusion coefficients
determined in [2]. Other results were found in [2] by extrapolation. In these cases
D∗
i was set equal to zero (not determined in [2]).
The value of σ∗
L was obtained in [2] by a limiting procedure over the dynamic
conductivity σ(k, ω),
σL = lim
ω→0
lim
k→0
ℜe σ(k, ω) , (31)
and thus related via the fluctuation-dissipation theorem to the dynamic charge-
charge structure factor. The limiting value of equation (31) could be found in [2]
only by extrapolation of long-wavelength MD data (see table IV of [2]). The point
with Γ = 2.0 was the only point really simulated in [2]. The other two points were
obtained in this work using an extrapolation procedure, its precision is not known
to us. We would rather not consider σ∗
L = σL/ωpe (characterized in [2] as the true
value) to be much more reliable than σ∗
D.
We computed the conductivity of strongly coupled hydrogen plasma for all three
cases considered in [2]. The calculations were carried out for both the Coulomb
interaction and the effective potential of [5], and employed in [2]. In the case of
Coulomb interactions the relaxation time was calculated according to equation (27)
with Z2
a = 1, a = e, i; see the σ∗
Cp data in table 1.
The effective potential of [5] is determined by charge numbers of the interacting
particles and by their reduced masses. The species form factors cannot be introduced,
so that equation (21) should be modified:
〈
|V̂ (~q, ω)|2
〉
=
(
4πe
q2ε(q, ω)
)2∑
a,b
Yab(q)Sab(~q, ω), (32)
where
Yab(q) = Yba(q) = ZaZb
[
1 + (qλab)
2]−1
, (33)
ε(q, ω) = 1 + ϕ (q)
∑
a
Y 2
aa(q)Πa(q, ω); (34)
ϕ (q) =
4πe2
q2
. (35)
The screened interaction energy of pseudoparticles is equal to 4πe2Yab/q
2ε(q, ω),
and the relaxation time expression of equation (27) becomes more complicated:
τ−1
pp (E) =
4πme4
β(2mE)3/2
∫ Q
0
dq
q
∑
l
Y 2
eeΠe + Y 2
ii Πi + 2ϕ (q) (YeeYii − Y 2
ei) ΠeΠi
ε3 (q, l)
. (36)
649
V.M.Adamyan, I.M.Tkachenko
The results of our computations with all these changes included, labelled σ ∗
ep,
are also provided in table 1. We cannot overestimate the fact that σ∗
ep virtually
coincides with the true conductivity value σ∗
L at Γ = 2.0. More simulation results
on both transport and dynamic plasma properties are needed to decide whether,
and to what extent, the behaviour of the classical pseudoparticles with the effective
potential of [4,5] imitates the behaviour of the true quantum system. We conclude
that overall satisfactory agreement with the plasma-simulation data available is
achieved.
It is worth mentioning that all Yab appearing in (36) on the other hand can
also be treated as factors improving the LFC-corrected RPA expressions for the
polarization operators Πa(q, ω) and the vertex operator.
In conclusion, notice that the proposed self-consistent field theory of plasma
transport properties is not based on the solution of kinetic equations. In particu-
lar, we do not have to introduce into our expression the order of 2 correction [15]
that takes into account the higher-order Sonine polynomials contributions to the
solution of the kinetic equation. The theory is applicable to multiple-component
(non-hydrogen-like) plasmas with variable ionization states, and is shown to possess
correct low-density (Spitzer) and metal density (Ziman) limiting forms.
6. Acknowledgements
The authors are grateful to W.Ebeling, J.Ortner, and A.A.Mihajlov for valuable
discussions, and to P.Fernández de Córdoba and J.Alcober for their aid in numerical
calculations.
A. Static structural characteristics
A natural approach to the investigation of static correlations in strongly coupled
plasmas, is based on the separation of electronic and ionic components of the system,
so that the interionic interactions are assumed to be screened by the electronic static
dielectric function εe(k). In dense systems the latter should be treated beyond the
RPA, i.e., the calculation of εe(k) involves the electronic static local field correction
(LFC) Ge(k).
There exists a number of various approaches to the computation of the LFC
Ge(k) (see, e.g. [22]), but mostly they are applicable in specific realms of the system
phase diagram. In our calculations we used a simple alternative model [23] which is
to also serve as a basis for future studies of various properties of strongly coupled
systems.
The interpolating formula for the electronic LFC suggested and tested in [23],
Ge(z) =
(
b+ a/4z2
)−1
z = k/2kF, (37)
incorporates both long and short wavelength asymptotic values of Ge(k). In partic-
ular,
b−1 = lim
k→∞
Ge(k). (38)
650
Linear static response of model plasmas
The short-range behaviour ofGe(k) in the low-temperature limit has been studied
in the papers of Shaw [24] and Kimball [25]. Namely, it has been shown that if T → 0
in hydrogen-like systems,
b−1 = 1− ge(0), (39)
ge(r) being the usual electronic radial distribution function. This result is based on
the “cusp” condition which can be obtained from the s-solution of the two-particle
Schrödinger equation at r=0 (see, e.g.[25]).
On the other hand, since Ge(k → ∞) involves only the short-range properties of
the system, one expects the asymptotic value of equation (38) to be finite and the
relation (39) to hold at arbitrary values of temperature T .
One can further notice that the long-wavelength behaviour of Ge(k → 0) ≈
a−1(k/kF)
2 is responsible for the screening of a static impurity in the plasma.
On the other hand, the parameter a is determined by the system thermodynamic
properties via the compressibility sum rule:
a = − (12/π)2/3Γ
f (Γ) + Γf ′ (Γ) /3
. (40)
Since the best Monte-Carlo (MC) fit [8] for the dimensionless excess interaction
energy f (Γ) in equation (40),
fMC (Γ) = AΓ +B + CΓ−1/3 +DΓ1/3, (41)
where
A = −0.8993749, B = −0.2244699,
C = −0.0178747, D = 0.5175753, (42)
is valid only for Γ > 1, we sought a smooth interpolation between equation (41) and
the Abe low-Γ expansion [9]
fAbe (Γ) = −
√
3
2
Γ3/2 − 3Γ3
[
3
8
ln (3Γ) +
γ
2
− 1
3
]
, (43)
γ = 0.577215665 being the Euler’s constant, applicable for Γ < 0.1. We used a
four-parameter interpolation
F (Γ) =
s+ Γσ
t+ Γτ
, (44)
to satisfy the conditions of continuity and smoothness between the functions equa-
tions (41) and (43) at the points Γ = 0.1 and Γ = 1. The resulting values of the
parameters are
s = −1.800549291, t = 0.2826351523,
σ = 0.02932213806, τ = −1.528433658. (45)
651
V.M.Adamyan, I.M.Tkachenko
Thus in our computations we used
f (Γ) =
fAbe (Γ) , Γ < 0.1;
F (Γ) , 0.1 6 Γ 6 1;
fMC (Γ) , Γ > 1.
(46)
It remains to be seen how our results would be modified if instead of equation (46)
other interpolations [26] were used.
No quantum effects are included in the EOS (5) and, hence, there is a discrepancy
between (40) (and, thus, equation (37) as well) and our desire to apply it to electron
liquids under “quantum” thermodynamic conditions.
To diminish the influence of this inconsistency, electronic radial distribution func-
tion ge(r) and its zero separation value ge(0) (and b of equation (37)) were deter-
mined by a self-consistent procedure. In effect, the value of ge(r) was computed via
a simultaneous solution of two integral equations,
Se(z) =
l1∑
l=−l1
Pe(z, l)
εe(z, l)
, (47)
ge(r) = 1 +
6
rkF
∫ ∞
0
z sin(2kFrz) (Se(z)− 1) dz. (48)
In equation (47) the summation is over the Matsubara frequencies vl=(πθl)/(2z),
and
εe(z, l) = 1 +
Γ
(12π2)1/3
Pe(z, l)
z2
; (49)
the l1-parameter was determined by the numerical precision. Pe(z, l) in equation (47)
is the dimensionless polarization operator Πe(k, ω) with the LFC, included,
Pe(z, l) = P 0
e (z, l)
(
1− Γ
(12π2)
1
3
Ge(z)P
0
e (z, l)
z2
)−1
, (50)
The RPA dimensionless polarization operator P 0
e (z, l) can be calculated (for each
value of density and temperature, z and l) by simple integration [27],
P 0
e (z, l) =
3θ
4z
∫ ∞
0
ydy
ey2/θ−η + 1
ln
∣∣∣∣
z + y + ivl
z − y + ivl
∣∣∣∣ , (51)
while the dimensionless chemical potential η = βµ is determined by the normaliza-
tion condition ∫ ∞
0
t1/2dt
e(t−η) + 1
=
2
3
θ−3/2 . (52)
Thus we used in equation (27)
Πe(q, l) = neβP
0
e (z, l) (53)
652
Linear static response of model plasmas
Table 2. The excess Coulomb interaction energy density (normalized to tempera-
ture) f(Γ), and the results of the electronic liquid MC-simulation, equation (41).
ne(10
24cm−3) T (105K) Γ θ f(Γ) fMC(Γ)
0.258 1.715 1.00 1.00 –0.54 –0.62
1.610 6.315 0.5 1.09 –0.24 –0.29
1.611 1.579 2.0 0.27 –1.11 –1.38
2.579 ·105 1.715 ·103 0.10 0.10 –0.044 –0.11
for the electronic polarization operator and [22]
Πi(z, l) = βniδl,0
(
1− Γ
(12π2)1/3
Gi(z)
z2
)−1
(54)
with
Gi(z) =
{
b [εe(z, l)] + a/(2z)2
}−1
, (55)
for the ionic one (δl,m is the Kronecker delta symbol) and thus obtained a closed
expression for the conductivity coefficient. Notice that the influence of the value of
ge(0) proved to be quite small.
Using our data for the radial distribution function we calculated the excess
Coulomb interaction energy density normalized to β, f(Γ):
f(Γ) =
1
8
k2D
k2F
∫ ∞
0
(ge(r)− 1) r dr (56)
with k2D = 4πne2β. The results are given in table 2, where we also provided the
corresponding values of fMC(Γ) of equation (41). Both estimates are closer in less
degenerate electronic liquids.
Further studies of ge(0) should be carried out to include low-temperature and
dynamic effects.
In addition, to improve the physical self-consistency of our approach, one needs
the quantal EOS, either theoretical or numerical (obtained, e.g. within a quantum-
statistical variant of the MC method), see, e.g. [28].
B. Other transport coefficients
If the initial state of plasma is not far from that of thermodynamic equilibrium,
the generalized transport equations for the mean current density ~J and for the
thermal flux ~Q can be written as [29]
~J = e2K0
~F + T−1eK1(−~∇T ), (57)
~Q = eK1
~F + T−1K2(−~∇T ). (58)
653
V.M.Adamyan, I.M.Tkachenko
T is the plasma temperature and no magnetic effects are taken into account. The
transport coefficients Ki (i = 0, 1, 2) in equations (57) and (58) satisfy the Onsager
relations [29] and within the same approximation instead of equation (18) we have
Ki = − 4
3m
∫ ∞
0
Eρ(E)τ(E)
dw(E)
dE
(E − µ)idE , (59)
where µ is the electronic subsystem chemical potential and τ(E) is the same relax-
ation time defined by equation (27). The transport coefficient K0 determines the
static conductivity
σ = e2K0 , (60)
while the static thermal conductivity
κ =
1
T
(K2 −K2
1/K0), (61)
and the thermal electromotive potential
α = K1(eTK0)
−1. (62)
In the case of complete degeneracy of the electronic subsystem the conductivities
κ and σ are related by the Wiedemann-Franz law
κ
σ
=
π2
3
(
kB
e
)2
T. (63)
If the degeneracy is incomplete, like in our case, there appear temperature-
dependent corrections to equation (63). Nevertheless, these corrections under the
conditions we consider are quite small [20].
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655
V.M.Adamyan, I.M.Tkachenko
Лінійний статичний відгук модельної плазми та
ефективний потенціал Юхновського-Кельбга-Дойча
В.М.Адамян 1 , І.М.Ткаченко 2
1 Одеський державний університет, кафедра теоретичної фізики,
65026 Одеса
2 Валенсійський політехнічний університет,
кафедра прикладної математики,
46071 Валенсія, Іспанія
Отримано 7 червня 2000 р.
На основі уявлень про самоузгоджене внутрішньоплазмове поле
пропонується методика обчислення коефіцієнтів переносу у повніс-
тю іонізованій густій плазмі. Ця методика прямо не використовує ані
кінетичне рівняння, ані параметри підгонки. Порівняння отриманих
результатів з даними комп’ютерного моделювання параметрів густої
майже класичної воднево-подібної плазми показує, що використан-
ня з самого початку ефективного парного потенціалу Юхновського-
Кельбга-Дойча замість кулонівського може до певної межі компенсу-
вати похибки простіших стандартних наближень квантової теорії ба-
гатьох заряджених частинок.
Ключові слова: воднева плазма, транспортні коефіцієнти, парні
потенціали
PACS: 52.25.Fi, 52.25.Ub, 52.65.-y
656
|
| id | nasplib_isofts_kiev_ua-123456789-121003 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T17:08:34Z |
| publishDate | 2000 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Adamyan, V.M. Tkachenko, I.M. 2017-06-13T12:27:21Z 2017-06-13T12:27:21Z 2000 Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential / V.M. Adamyan, I.M. Tkachenko // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 641-656. — Бібліогр.: 29 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.641 PACS: 52.25.Fi, 52.25.Ub, 52.65.-y https://nasplib.isofts.kiev.ua/handle/123456789/121003 A theory of transport coefficients of fully ionized strongly coupled plasmas,
 based on the self-consistent field concept and having no adjustable parameters, is presented. The proposed approach is not connected directly with
 the kinetic equation. A comparison of the obtained results with the computer simulation data on the classical hydrogen-like plasmas macroscopic
 properties shows that using the Yukhnovsky-Kelbg-Deutsch effective pair
 potentials instead of the genuine Coulomb potentials can compensate, to
 a certain extent, inaccuracies related to the standard approximations of the
 quantum theory of Coulomb systems. На основі уявлень про самоузгоджене внутрішньоплазмове поле
 пропонується методика обчислення коефіцієнтів переносу у повністю іонізованій густій плазмі. Ця методика прямо не використовує ані
 кінетичне рівняння, ані параметри підгонки. Порівняння отриманих
 результатів з даними комп’ютерного моделювання параметрів густої
 майже класичної воднево-подібної плазми показує, що використання з самого початку ефективного парного потенціалу Юхновського-Кельбга-Дойча замість кулонівського може до певної межі компенсувати похибки простіших стандартних наближень квантової теорії багатьох заряджених частинок. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential Лінійний статичний відгук модельної плазми та ефективний потенціал Юхновського-Кельбга-Дойча Article published earlier |
| spellingShingle | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential Adamyan, V.M. Tkachenko, I.M. |
| title | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential |
| title_alt | Лінійний статичний відгук модельної плазми та ефективний потенціал Юхновського-Кельбга-Дойча |
| title_full | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential |
| title_fullStr | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential |
| title_full_unstemmed | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential |
| title_short | Linear static response of model plasmas and Yukhnovsky-Kelbg-Deutsch effective potential |
| title_sort | linear static response of model plasmas and yukhnovsky-kelbg-deutsch effective potential |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/121003 |
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