Ionic interaction and conductivity of metallic hydrogen
We calculate the electroresistivity of metallic hydrogen within the framework of perturbation theory in electronproton interaction. To this end we employ the Kubo linear response theory while using the two-time retarded Green functions method to calculate the relaxation time. The expressions for t...
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Інститут фізики конденсованих систем НАН України
2006
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| Цитувати: | Ionic interaction and conductivity of metallic hydrogen / V.T. Shvets, S.V. Savenko, Ye.K. Malynovski // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 127–133. — Бібліогр.: 15 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859845708307234816 |
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| author | Shvets, V.T. Savenko, S.V. Malynovski, Ye.K. |
| author_facet | Shvets, V.T. Savenko, S.V. Malynovski, Ye.K. |
| citation_txt | Ionic interaction and conductivity of metallic hydrogen / V.T. Shvets, S.V. Savenko, Ye.K. Malynovski // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 127–133. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | We calculate the electroresistivity of metallic hydrogen within the framework of perturbation theory in electronproton
interaction. To this end we employ the Kubo linear response theory while using the two-time retarded
Green functions method to calculate the relaxation time. The expressions for the second and third order contributions
are given. To describe the electron subsystem, the random phase approximation is used, allowing
for the exchange interactions and correlations in a local field approximation. Thermodynamics of the proton
subsystem is assumed to be given by the Percus-Yevick equation. At a given density and temperature the
only parameter of the theory is the hard sphere diameter, which is calculated from effective pair ionic interaction.
For a completely degenerated electron gas, the latter is determined by the density of the system. The
dependence of the second and the third order contributions on the parameters of the theory is investigated.
For all densities and temperatures examined here the third order contribution constitutes more than half of the
second order term. The corresponding magnitude of resistivity is about 100 ∼ 250µΩ cm.
|
| first_indexed | 2025-12-07T15:39:10Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 127–133
Ionic interaction and conductivity of metallic hydrogen
V.T.Shvets1, S.V.Savenko1,2, Ye.K.Malynovski1
1 Odesa State Academy of Refrigeration
13 Dvoryanska str., 65026, Odesa, Ukraine
2 Debye Institute, Soft Condensed Matter, Utrecht University,
Princetonplein 5, 3584 CC Utrecht, The Netherlands
Received August 31, 2005
We calculate the electroresistivity of metallic hydrogen within the framework of perturbation theory in electron-
proton interaction. To this end we employ the Kubo linear response theory while using the two-time retarded
Green functions method to calculate the relaxation time. The expressions for the second and third order con-
tributions are given. To describe the electron subsystem, the random phase approximation is used, allowing
for the exchange interactions and correlations in a local field approximation. Thermodynamics of the proton
subsystem is assumed to be given by the Percus-Yevick equation. At a given density and temperature the
only parameter of the theory is the hard sphere diameter, which is calculated from effective pair ionic inter-
action. For a completely degenerated electron gas, the latter is determined by the density of the system. The
dependence of the second and the third order contributions on the parameters of the theory is investigated.
For all densities and temperatures examined here the third order contribution constitutes more than half of the
second order term. The corresponding magnitude of resistivity is about 100 ∼ 250µΩ cm.
Key words: metallic hydrogen, electrical conductivity
PACS: 71.10.+x, 72.10.Bg, 72.15.Cz
1. Introduction
Hydrogen is the most widespread element in the universe. The major portion of the mass of the
planets in the solar system is contributed by the hydrogen in metallic state. From the theoretical
point of view hydrogen is a unique object for which the electon-ion interaction is known exactly. It
allows us to perform calculations of various properties involving the minimal number of simplifying
assumptions, and therefore, with maximal plausibility.
The possibility for the metallic phase of hydrogen to exist was predicted in 1935 [12]. It was
suggested that at high densities hydrogen should exhibit the transition from diatomic molecular
state, with insulator properties, to an atomic state with metallic conductivity. The actual dis-
covery of metallic hydrogen and the detailed investigation of conductivity as a function of the
pressure and the temperature followed at 1996 [11]. In this study the molecular hydrogen in a
liquid state was subjected to shock compression up to 1.80 Mbar at the temperatures in the
2200 − 4400 K range. The dielectric-metal transition was observed at the pressure of 1.4 Mbar
while temperature 3000 K, while the resistivity of metallic phase measured 500µΩ cm. In the
pressure and the temperature range 1.4 − 1.8 Mbar and 3000 − 4400 K the electroresistivity
remained virtually unchanged. As a matter of fact, the examined transition was of a metal-
semiconductor type, as the band-gap did not vanish completely, but decreased from 15 eV to
0.3 eV, which is almost equal to the temperature of a sample. It should be noted that the ear-
lier experimental and theoretical studies of the conjectured metallic state exist. In the work [6]
the electric conductivity of molecular hydrogen was measured at substantially lower pressure of
0.1 − 0.2 Mbar, the results followed the exponential law with the rate specific to semiconductors
with the 12 eV band-gap. The discovery of metallic hydrogen was first reported in [3] in 1978. The
authors reported the discovery of metallic hydrogen at the pressure 2 Mbar with the resistivity of
1000µΩ cm.
c© V.T.Shvets, S.V.Savenko, Ye.K.Malynovski 127
V.T.Shvets, S.V.Savenko, Ye.K.Malynovski
When the new substances are discovered, the examination of the static properties usually pre-
cede the studies of transport phenomena. At present the research on the static properties of metallic
hydrogen is carried out intensively [1,4,5,7–9]. Yet, there are virtually no theoretical investigations
of the electron transport phenomena in metallic hydrogen. The distinctive feature of the treatises
on static properties of metallic hydrogen is the adoption of the nearly free electron model. Here
we are going to employ this model to calculate the electroresistivity of metallic hydrogen. Herein
we assume that the hydrogen is in the true metallic state with no band-gap at all, and not in
the experimentally observed semiconductor state with the band-gap of 0.3 eV. This state can be
realized either at a higher pressure, or at a higher temperature. We should like to note that the
Jupiter core with the radius equal to half the radius of the planet, consists of a hydrogen at the
pressure of 3 − 40 Mbar and the temperature 10000− 20000 K.
2. Hamiltonian
We assume the hydrogen to be in a disordered atomic state with all the electrons collectivized.
In this case the Hamiltonian of the electron subsystem of metallic hydrogen can be taken in the
form similar to that of the liquid simple metals [15]
H = H0 + Hie. (1)
Again, just like in the case of the liquid simple metals, the electron gas is considered degener-
ated, and the ionic subsystem to be a disordered and static one. Using the nearly free electrons
approximation, the Hamiltonian of noninteracting electron gas can be written as
H0 =
∑
k
εka+
k ak , (2)
where a+
k , ak are the operators of nucleation and annihilation of electrons with the wave vector k,
εk = ~
2k2/2m is the free electron energy and m is its mass.
The Hamiltonian of the electron-proton interaction can be taken in line with the diffraction
model of a metal, which incorporates the electron-electron interaction via screening of the electron-
ion interaction
Hie = V −1
∑
q
W (q)ρi(q)ρe(−q). (3)
Here V is the volume of the system, W (q) = −V (q)/ε(q) is the screened potential energy of
electron-proton interaction, V (q) = 4πe2/q2 is the Fourier transform of the Coulomb interaction,
e is the electron charge, ε(q) = 1 + [V (q) + U(q)]π0(q) is the dielectric permittivity of the electron
gas in the random phase approximation, U(q) = −2πe2/(q2 + λk2
F) is the potential energy of the
exchange interaction and the correlations of the electron gas with λ ≈ 2 [2], kF is the Fermi wave
vector,
π0(q) =
mkF
π2~2
(
1
2
+
4k2
F − q2
8kFq
ln
∣
∣
∣
∣
2kF + q
2kF − q
∣
∣
∣
∣
)
(4)
is the polarization function of a free electron gas, ρe(q) =
∑
k a+
k ak+q is the Fourier transform
of the density operator of the electron gas and ρi(q) =
∑
n exp(−iqRn) with Rn denoting the
radius-vector of the n-th ion.
We should like to stress that the metallic hydrogen is the only system with the exactly known
unscreened potential of the electron-ion interaction, which is the Coulomb law. This fact signifi-
cantly simplifies the calculation of various properties of metallic hydrogen, as long as the problem of
modelling the electron-ion interaction, leading to the introduction of additional fitting parameters,
is avoided.
Just like in the theory of many of the disordered simple metals, the ratio of the potential energy
of the electron-ion interaction to the Fermi energy is the small parameter of the theory not for all
values of the wave vector. This situation is illustrated in figure 1. The actual small parameter of
the theory is the product of the above mentioned ratio on the structure factor of the ion subsystem.
128
Ionic interaction and conductivity of metallic hydrogen
-0.8
-0.6
-0.4
-0.2
0.0
0 1 2 3 4 5
ω
(q
)/
ε F
qσ/2π
kF 2kF -0.6
-0.4
-0.2
0.0
0 1 2 3 4 5
ω
(q
)S
(q
)/
ε F
qσ/2π
kF 2kF
Figure 1. The form factor of the screened
electron-proton pseudopotential. Two vertical
lines denote the values of the wave vector
equal to kF and 2kF correspondingly (scaled
with σ/2π). These values correspond to σ =
4au and η = 0.45.
Figure 2. The small parameter of the theo-
ry. Two vertical lines denote the values of the
wave vector equal to kF and 2kF correspondi-
ngly (scaled with σ/2π). These values corre-
spond to σ = 4au and η = 0.45.
From figure 2 it can be seen that again this parameter cannot be considered small for every value
of the wave vector. Inasmuch as these values of the wave vector provide substantial contribution to
the corresponding integrals, one can expect the perturbation series in this parameter of the theory
to converge slowly. This circumstance does not allow us to restrict ourselves to the first term of
the perturbation series when calculating the resistivity.
3. Resistivity
The electroresistivity of well conducting disordered simple metals can be found using the well
known Drude formula
R =
m
ne2
τ−1. (5)
Here n is the electron gas density and τ is the relaxation time for the electroconductivity process.
We are going to exploit this formula to calculate the resistivity of metallic hydrogen as well. Here
we will consider only the electron contribution to the resistivity.
Within the framework of the linear response theory of Kubo and the method of two-time
retarded Green functions [2,10] the inverse relaxation time can be written in the form of the series
in electron-proton interaction
τ−1 =
∞
∑
n=2
τ−1
n . (6)
The general term of this expansion is of the form
τ−1
n =
N
V n
∑
q1,...,qn
W (q1) . . . W (qn)S(q1, . . . ,qn)Γ(q1, . . . ,qn). (7)
Here S(q1, . . . ,qn) is the n-particle structure factor of the ion subsystem, N is the number of ions
in the system, Γ(q1, . . . ,qn) is the electron multipole for the electric conductivity process.
The second order contribution to the inverse relaxation time for the disordered simple metals,
first obtained by Ziman [13], is well explored and reads
τ−1
2 =
m
4π~3νk3
F
∫ 2kF
0
dxx3W 2(x)S(x), (8)
129
V.T.Shvets, S.V.Savenko, Ye.K.Malynovski
where ν is the inverse proton number density. We now examine the third order contribution in
more detail. It is of the form
τ−1
3 =
N
V 3
∑
q1,q2,q3
W (q1)W (q2)W (q3)S(q1,q2,q3)Γ(q1,q2,q3). (9)
For the noninteracting electron gas, the electron multipole, obtained by using the kinetic equation
and characterizing the electric conductivity, is of the form
Γ(k1 − k2,k2 − k3,k3 − k1) =
βπ~
3mN
(k1 − k2)
2n(k1)[1 − n(k1)]
δ(εk2
− εk1
)
(εk2
− εk1
)
, (10)
where β is the inverse temperature and n(k) is the Fermi-Dirac distribution function. With some
manipulation [9,11] the third order contribution can be expressed as a principal value integral
τ−1
3 =
m2
24π5~5k2
F
∫ ∞
0
dk
f(k)
kF − k
. (11)
As long as the electron-electron interaction is exactly known, the major approximation we use is the
superposition approximation for the 3-particle structure factor S(q1,q2,q3) = S(q1)S(q2)S(q3).
As a consequence of this approximation we obtain the following expression for the function f(k)
f(k) =
1
kF + k
∞
∑
n=0
(2n + 1)AnB2
n(k), (12)
with
An =
∫ 2kF
0
dq q3W (q)S(q)Pn
(
2k2
F − q2
2k2
F
)
,
Bn(k) =
∫ k+kF
|k−kF|
dq qW (q)S(q)Pn
(
k2 + k2
F − q2
2kkF
)
, (13)
where Pn(x) is the Legendre polynomial of order n.
4. The effective pair proton interaction
The effective pair proton interaction is an essential component in estimating the conductivity
of metallic hydrogen. It is a particularly valuable feature since the only parameter on which it
depends is the number density of the system. Since the effective pair interaction makes it possible
to calculate the effective hard sphere diameter for a given temperature, it enables us to determine
the hard sphere diameter as a function of density and temperature.
The expression for the effective pair potential is known from the second order perturbation
theory in electron-ion interaction and reads
Veff(R) =
e2
R
−
ν
π2
∫
dqq2 sin(qR)
qR
Q(q) , (14)
where
Q(q) =
ν
2
V 2(q)
π(q)
1 + V (q)π(q)
. (15)
The polarization function of electron gas with exchange interaction and correlations taken into
account in a local field approximation can be written as
π(q) =
π0(q)
1 + V (q)π0(q)
. (16)
130
Ionic interaction and conductivity of metallic hydrogen
2 3 4 5 6 7
10
10
20
30
40
50
Interionic potential in metal hydrogen
R (a.u.)
V
(R
)
(
10
00
K
)
Figure 3. The effective pair interaction, n = 0.001 gm/cm3.
This expression does not contain any fitting parameters characterizing the ion subsystem and
depends only on the Fermi wave vector, i.e. on the number density of the system. The only ap-
proximation it involves is the random phase approximation for the electron subsystem taking into
account the exchange interaction and electron correlations in a local field approximation.
Figure 3 provides the plot of the effective pair ion interaction for a number density n =
0.001 gm/cm3. Now we can introduce the effective hard sphere diameter as the minimum ion
approach distance for a given temperature. This distance can be found from equating the total
energy of protons to zero at the closest approach
Veff(σ) = 3kT/2. (17)
Here the reference point for potential energy is the lowest value of the potential well. For instance
at n = 1mole/cm3 and T = 103 K σ = 2.77, while raising the temperature to T = 104 K at the
same density we obtain σ = 1.64. In the first case the packing fraction is η = 0.45, which roughly
corresponds to the value for the liquid alkali metals at melting point, in the second case η = 0.08.
0
1
2
3
0 1 2 3
S(
q)
qσ/2π
kF 2kF
Figure 4. The pair structure factor of the ion subsystem. Two vertical lines denote the values of
the wave vector equal to kF and 2kF correspondingly (scaled with σ/2π). These values correspond
to σ = 4au and η = 0.45.
5. Results and discussion
If for the structure factor of the ion subsystem we use the Percus-Yevick equation [14] (see
figure 4), then the resistivity depends on three parameters. Namely these are the hard sphere
diameter, the packing fraction and the Fermi wave vector. If all the electrons are considered col-
lectivized, then the Fermi wave vector can be related to the number density through ν = 3π2/k3
F,
131
V.T.Shvets, S.V.Savenko, Ye.K.Malynovski
while on the other hand νη = πσ3/6. It is convenient to choose the hard sphere diameter as the
independent parameter. If the temperature is known, this parameter can be uniqely defined as well.
Table 1. The coefficients of the series in (12).
n 1 2 3 4 5 6 7 8
An –2.031 1.210 –0.487 0.168 –0.050 0.013 –0.002943 0.0005285
As we see from the table 1 the coefficients An, Bn(q) quickly decay with the increase of n. This
means that we need to allow for only a few terms of the corresponding series. As long as liquid
hydrogen exists in nature in a wide pressure and temperature range it is interesting to investigate
the dependence of resistivity on parameters σ and η. From figures (5), (6) we can see that the
0 3 6 9 12
0
22
44
66
88
110
ρ
µΩ
cm
T×103 K
0.014 0.018 0.022 0.026 0.030 0.034
0
30
60
90
120
150
ρ
µΩ
cm
ρ au-3
Figure 5. The dependence of the resistivity
on the temperature. Solid line: 2nd order con-
tribution; dotted line: 3rd order contribution.
n = 0.025 au−3.
Figure 6. The dependence of the resistivity on
the number density. Solid line: 2nd order con-
tribution; dotted line: 3rd order contribution.
T = 5000 K.
electron gas is strongly degenerated at all densities and temperatures considered. The second order
contribution to electoresistivity is of the same order as for majority of the liquid metals that are well
described using the nearly free electrons approximation. On the other hand, the third order contri-
bution constitutes more than 50% of the second one, which is typical of liquid metals with relatively
high resistivity. Consequently, we can expect a slow convergence of the pertutabation theory series
for electric conductivity. This is caused by a relatively large magnitude of the above mentioned
small parameter of the theory. If we approximate the perturbation series by an infinite geometric
progression with the ratio R3/R2 = 0.43, then the electric resistivity measures R ≈ 250µΩ cm at
T = 5000 K, n = 0.016 au and R ≈ 195µΩ cm at T = 5000 K, n = 0.032 au. As the density
increases at a fixed temperature, the magnitudes of both second and third contributions decay,
while the relative magnitude of the third order contribution increases. The estimated resistivity
here decays as well, which is typical of metallic conductivity. If we keep the density fixed and in-
crease the temperature, then both contributions grow together with the estimated resistivity which
is again characteristic of metallic conductivity. This can be illustrated with the following numbers:
R ≈ 135µΩ cm at T = 1000 K and n = 0.025 au, R ≈ 243µΩ cm at T = 10000 K and n = 0.025 au.
The experimentally observed resitivity of R ≈ 500µΩ cm was not the intent of this work. However,
the values obtained are in reasonable agreement with this value. To provide a better estimate of the
experimentally observed resisitivity, the theory should allow for the band gap in the energy spec-
trum of electrons, and for the presence of the hydrogen in molecular state in experimental setups.
132
Ionic interaction and conductivity of metallic hydrogen
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133
134
|
| id | nasplib_isofts_kiev_ua-123456789-121306 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T15:39:10Z |
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| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Shvets, V.T. Savenko, S.V. Malynovski, Ye.K. 2017-06-14T04:38:01Z 2017-06-14T04:38:01Z 2006 Ionic interaction and conductivity of metallic hydrogen / V.T. Shvets, S.V. Savenko, Ye.K. Malynovski // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 127–133. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 71.10.+x, 72.10.Bg, 72.15.Cz DOI:10.5488/CMP.9.1.127 https://nasplib.isofts.kiev.ua/handle/123456789/121306 We calculate the electroresistivity of metallic hydrogen within the framework of perturbation theory in electronproton interaction. To this end we employ the Kubo linear response theory while using the two-time retarded Green functions method to calculate the relaxation time. The expressions for the second and third order contributions are given. To describe the electron subsystem, the random phase approximation is used, allowing for the exchange interactions and correlations in a local field approximation. Thermodynamics of the proton subsystem is assumed to be given by the Percus-Yevick equation. At a given density and temperature the only parameter of the theory is the hard sphere diameter, which is calculated from effective pair ionic interaction. For a completely degenerated electron gas, the latter is determined by the density of the system. The dependence of the second and the third order contributions on the parameters of the theory is investigated. For all densities and temperatures examined here the third order contribution constitutes more than half of the second order term. The corresponding magnitude of resistivity is about 100 ∼ 250µΩ cm. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Ionic interaction and conductivity of metallic hydrogen Міжіонна взаємодія і провідність металевого водню Article published earlier |
| spellingShingle | Ionic interaction and conductivity of metallic hydrogen Shvets, V.T. Savenko, S.V. Malynovski, Ye.K. |
| title | Ionic interaction and conductivity of metallic hydrogen |
| title_alt | Міжіонна взаємодія і провідність металевого водню |
| title_full | Ionic interaction and conductivity of metallic hydrogen |
| title_fullStr | Ionic interaction and conductivity of metallic hydrogen |
| title_full_unstemmed | Ionic interaction and conductivity of metallic hydrogen |
| title_short | Ionic interaction and conductivity of metallic hydrogen |
| title_sort | ionic interaction and conductivity of metallic hydrogen |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/121306 |
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