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Contrib. Plasma Phys. 51, 615 (2011) Bennadji .pdf



Nom original: Contrib. Plasma Phys. 51, 615 (2011) Bennadji.pdf
Titre: Local Field Corrections Effect on Equation of State in Dense Hydrogen Plasma: Plasma Phase Transition
Auteur: Kamel Bennadji

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Contrib. Plasma Phys. 51, No. 7, 615 – 620 (2011) / DOI 10.1002/ctpp.201000073

Local Field Corrections Effect on Equation of State in Dense
Hydrogen Plasma: Plasma Phase Transition
K. Bennadji∗1,2
1
2

Laboratoire d’Electronique Quantique, Facult´e de Physique, Universit´e des Sciences et de la Technologie
Houari Boumediene (USTHB), El Alia BP 32 Bab Ezzouar 16111, Alger, Alg´erie
Laboratoire de Physique des Gaz et des Plasmas, Universit´e Paris-Sud, UMR8578, Orsay F-91405, France

Received 10 June 2010, accepted 27 October 2010
Published online 28 December 2010
Key words Plasma phase transition, local field corrections, equation of state, coupled electrons, thermodynamic instabilities.
We present calculations of Hydrogen plasma pressure including local field correction (LFC) effects modelled in
the Singwi Tosi Land and Sj¨olander model extended to arbitrary temperature (ATSTLS), neutrals contributions
are not included. We show that LFC induces thermodynamic instabilities (phase transition) in the vicinity of
the maximum electron coupling parameter which corresponds to maximum departure of the ATSTLS electron
polarizability from the Random Phase Approximation one. Comparison of our results to Monte-Carlo calculations induced us to think that free electron coupling plays a dominant role in plasma phase transition. Critical
temperature is estimated and compared to existing values.
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

1 Introduction
Dense hydrogen plasma appears in giant planets as Jupiter and Saturn and is made in laboratories by shock
waves. Different studies based on path integral Monte-Carlo [1-3], molecular dynamic [4] and density functional
theory [5] indicated the possibility of occurrence of plasma phase transition (PPT) and some experiments based
on shock waves [6, 7] revealed signatures of such transitions. PPT is explained as a possible result of competition between effective coulomb attraction and quantum repulsion [8-10]. It is also pointed out [11, 12] that
nonideality is a crucial factor for PPT. One possible way to involve free electron nonideality is to use LFC in
the electron dielectric function. In our earlier study of LFC by mean of arbitrary temperature Singwi Tosi Land
and Sj¨olander model (ATSTLS) [13], it is revealed attractive effective ion-ion potential in regions in which free
electron coupling parameter takes significant values. The departure of the ATSTLS screening from the Random
Phases Approximation (RPA) one is linked to the increase in the coupling parameter. It is pointed out in the
same study that the attractive behaviour of the effective ion-ion potential possibly causes PPT. It is the aim of
the present study to extend the ATSTLS calculations to the pressure of hydrogen plasma. For this purpose, both
ion-ion effective potential and ion-ion radial distribution function are needed. The first one is calculated within
this ATSTLS model and the second one is determined by solving an Hypernetted-Chain (HNC) equation for ions
interacting via the effective potential.
The contribution of neutrals to the pressure is not included. We attempted to justify this omission in the region
of interest by giving additional calculations of ionization energy lowering, including LFC effects. However, this
study is essentially devoted to a qualitative demonstration outlining the electron correlation responsibility in the
thermodynamic instabilities.

2 Effective ion-ion potential in the ATSTLS model
The system we consider here is constituted by ions screened by electrons (pseudo-ions or pseudo-atoms). The
screening is introduced via the static electronic dielectric function. The effective ion-ion potential in reciprocal


Corresponding author: E-mail: kbennadji@hotmail.com
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

616

K. Bennadji: Local field corrections effect on equation of state in dense Hydrogen plasma: Plasma phase transition

space is:
φ˜ii (k) =

v˜ (k)
ε (k, ω = 0)

(1)

!
v˜ (k) = 4πZ 2 e2 k 2 is the Fourier transform of the ion-ion Coulomb potential, Z = 1 in our case. The dielectric
function ε is calculated in the same manner as in the original paper of Singwi Tosi Land and Sj¨olander (STLS) [14]
but at arbitrary temperature [13]. The starting point of this model is the first equation of the Bogoliubov, Born,
Green, Kirkwood and Yvon hierarchy, this equation gives the one electron distribution function as a functional
of the two electrons distribution function. The main approximation made in STLS (ATSTLS) model is:
f (2) (%r1 , %r2 ; p%1 , p%2 ; t) = f (1) (%r1 , p%1 , t) f (1) (%r2 , p%2 , t) g(|%r1 − %r2 |),

(2)

where f (2) , f (1) and g stand for the two electrons distribution function, the one electron distribution function
and the equilibrium radial distribution function respectively. The right-hand side of relation (2) reproduces as
closely as possible electron correlations; we remember that RPA is equivalent to take g = 1 in relation (2),
thus, ATSTLS model reproduces more accurately electron correlations than RPA does [13]. Moreover, a closure
equation is added which is the fluctuation-dissipation theorem, written in our case at arbitrary temperature and in
quantum case [15]. The temperature is also introduced via the free electron polarizability. Exchange and quantum
effects are also considered in this model [13].

Fig. 1 Effective ion-ion potentials with ATSTLS screening at temperature of 10 000 K . (a) Densities from up to down: Log(ne
(cm−3 )) = 20, 20.2, 20.4, 20.6, 20.8, 21, 21.2, 21.4, 21.6, 21.8,
22. (b) Densities from left to right: Log(ne ) = 22.2, 22.4, 22.6,
22.8, 23, 23.2, 23.4, 23.6. (Colour figure: www.cpp-journal.org).

The effective ion-ion potential φii (r) is obtained from equation (1) by means of a Fourier transform. Figure 1
a and b shows ATSTLS potentials at fixed temperature of 10 000 K and various densities. For comparison, RPA
potentials are shown in figure 2 at the same temperature and densities as in figure 1. These densities are chosen
in the vicinity of the maximum electron coupling parameter; this one is shown in figure 2 of reference [13]. In
these figures, it is clearly seen that the departure of the ATSTLS potentials from the RPA ones follows exactly the
electron coupling parameter variations [13]; especially, a maximum departure corresponds to a maximum coupling parameter which about ne = 1022 cm−3 . This is not a strange feature because correlations are incorporated
systematically in the ATSTLS model via the electrons radial distribution function as mentioned above. We can
see (Fig. 1) that ATSTLS potentials make damped oscillations around zero [13,5,16,17] for some densities near
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

www.cpp-journal.org

Contrib. Plasma Phys. 51, No. 7 (2011) / www.cpp-journal.org

617

the maximum coupling, in contrast, no such oscillations are revealed for RPA potentials. The attractive behaviour
of these potentials can give rise to instabilities in the equation of state.
The peculiar behaviour of the ion-ion effective potential as well as the ion-ion structure factor in reference [13]
is demonstrated by the density functional calculations of Hong Xu and Hansen [5]. The authors signalled that the
oscillations in the ion-ion effective potential may be a signature of an incipient plasma-insulator transition.

Fig. 2 Effective ion-ion potentials with RPA screening at temperature of 10 000 K and densities from left to right: Log(ne ) = 20,
20.4, 20.8, 21.2, 21.6, 22, 22.4, 22.8, 23.2, 23.6. (Colour figure:
www.cpp-journal.org).

3 The equation of state
The PPT is related to a drastic increase in the electrical conductivity [6,7,22,23,1], this last one increases by 4 to 5
order of magnitude in a narrow density range [7, 22, 23]. This increasing is attributed to pressure ionization which
is enhanced by energy level shifts caused by free electron screening (Mott effect). A well justified hypothesis
is that PPT is caused by free electrons, it occurs when electron system reaches a high degree of correlation.
Thus, PPT is a consequence of the ionization. It is interesting to note from a figure (fig. 4) of reference [6], by
comparing experimental values of the pressure and experimental conductivity, that PPT occurs after the plasma
reaches a metallic state. It seems as if the onset of PPT occurs when free electron density reaches a critical value.
This observation suggests that PPT occurs as a consequence of the huge increasing in the conductivity and is
not this increasing itself. In these conditions, the hydrogen reaches a high degree of ionization but neutrals still
contribute to the pressure. The onset of PPT when the plasma reaches a metallic state is also revealed by some
theoretical studies [24,1].
In the framework of the hypothesis cited above, the equation of state is investigated by means of the ATSTLS
model. The pressure is calculated in conditions where the electron coupling parameter takes significant values,
electron correlations are added through the LFC. The main contributions to the pressure come from the electron
and ion kinetic energies and ion-ion interaction, this last one is an effective one due to electron screening. For
reason of commodity, the contribution of neutrals is not included. Indeed, exact evaluation of this one assumes a
narrow knowledge of the ionization process which is a very complex task (nearly bound and nearly free electrons,
tunnelling, overlap, etc.). On the other hand, the ionization in the region of interest must be in a high achievement.
To make an estimation of this process, we have calculated the ionization energy evolution of a hydrogen atom in
the plasma versus the electron density. Figure 3 shows this energy, calculated in both ATSTLS and RPA static
screening. At low electron densities (ne < 1018 cm−3 ), this energy coincides with that of isolated atom. As the
density increases, the energy level shifts to the continuum. The shift in the case of ATSTLS is more pronounced
than the one in RPA. The ionization potential reaches the order of magnitude of thermal energy at the density of
2 × 1022 cm−3 in the case of ATSTLS screening, and at the density of 7.2 × 1022 cm−3 in the case of RPA. We
can say that, at densities greater than 2 × 1022 cm−3 , the whole of hydrogen is in an ionized state.
To overcome the problem of the neutrals density, we have plotted the pressure versus the free electron density
instead of the total density of hydrogen. This is sufficient to identify instabilities. Indeed, Norman and Starostin
[9] demonstrated that instability with respect to the total density is equivalent to instability with respect to ions
density (∂P /∂nT < 0 ⇔ ∂P /∂ni < 0).
In light of these considerations, we suppose our system constituted by Ni positive hydrogen ions and Ne
electrons in a volume V (Ne = Ni ). This system is equivalent to a fictitious one which consists of pseudo-ions
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c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

618

K. Bennadji: Local field corrections effect on equation of state in dense Hydrogen plasma: Plasma phase transition

Fig. 3 Hydrogen ground state (1s) energy variations versus free
electron density. (Colour figure: www.cpp-journal.org).

interacting via screened potentials φii and free electrons constituting a neutralising background [17,25]. This
method of splitting the system was used for solid and liquid metals and was extended to metallic hydrogen [25].
Its applicability in the present study is evident when the density is above the solid one (ne ! 1022 cm-3 ), it is less
justified when the density is lower.
The partition function of this system is:


ZNe ,Ni

1
=
e 3Ni
Ne ! Ni ! λ3N
λi
e

"



Ni


&


dR1 .....dRNi exp −β
φii (|Rl − Rj |) .




l, j = 1
l<j

(3)

*
,
+
λe,i λe,i = 2π!/ 2πme,i kB T are the De Broglie wavelengths of electrons and ions respectively and β =
1/kB T . The Helmholtz free energy is:
(4)

F = −kB T ln (ZNe ,Ni )
The pressure of the system is:
∂F -P =−
∂V -T,Ni

(5)

After some calculus we obtain:
P =

kB T
4!3

(2me kB T )

3/2

I3/2 ( kµBeT ) + ni kB T

.∞ 2
2πn2 .
3
ii
ii
− 3 i r3 gii (r) ∂φ
r gii (r) ∂φ
∂r dr + 2πni
∂ni dr
0

(6)

0

here, kB is the Boltzmann constant, T , the temperature, me(i) , electron (ion) masse, ne(i) , electron (ion) density,
I3/2 , Fermi integral, µe , electron chemical potential and gii , the ion-ion radial distribution function.
The first term in the right-hand side of equation (6) is the electron kinetic energy contribution to the pressure,
the second term is the ion kinetic energy contribution and the third and the fourth terms are the ion interaction
energy contributions.
The partition function as written in (3) leads to a classical electron contribution to the pressure, that is ne kB T .
We have considered here densities and temperatures such as electrons are degenerate. The kinetic contribution
involves a quantum effect known as Fermi repulsion. We have thus replaced in expression (6) the classical term
by its quantum counterpart, that is the first term in the right-hand side of (6).
The radial distribution function gii is obtained by solving a combined HNC equation [18] and Ornstein-Zernike
relation [19], that is:
00
/
/
"
φii (r)
φii (r$ )
gii (r) = exp −
(7)
+ ni d3 r$ [gii (|%r −%r $ |) − 1] gii (r$ ) − 1 − ln(gii (r$ )) −
kB T
kB T
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

www.cpp-journal.org

Contrib. Plasma Phys. 51, No. 7 (2011) / www.cpp-journal.org

619

Given a potential φii , equation (7) is a self-consistent equation which is solved numerically. We point out that
gii is evaluated in the ATSTLS model, so, it involves LFC. It is different from the radial distribution function
obtained by RPA as it can be verified from a figure of reference [13].
Results of calculations in both HNC-ATSTLS and HNC-RPA are plotted in figure 4. This figure shows
isothermal pressures versus free electron density at the temperature of 10 000 K. It is obvious that instabilities (∂P /∂n < 0 or (and) P < 0) occur at high electron coupling parameter range. The system would avoid
these instabilities by undergoing a phase transition: It splits into two systems of different densities, n1 and n2
with n1 < n2 . It passes instantaneously from density n1 to density n2 without reaching any density in the range
[n1 , n2 ]; this is a forbidden density range at the temperature considered here. As the volume is lowered by compression, matter passes instantaneously from density n1 to density n2 by forming droplets of density n2 . This
droplet formation is established by Monte Carlo simulations of Filinov et al. [1].
In addition to the HNC-ATSTLS and HNC-RPA curves, it is shown in figure 4 isothermal pressures of :
ideal classical electron-ion gas, Holst and Redmer (HR) [20], Filinov, Levasnov, Bonitz and Fortov (FLBF) [2],
Debye-H¨uckel and Fortov et al. (experimental) [6]. The Debye-H¨uckel pressure is:
PDH = (ne + ni ) kB T −

kB T
24π

/

4πe2
kB T

Fig. 4 Pressure isotherms at the temperature of 10000 K.
The not filled area in FLBF curve corresponds to negative
pressures. (Colour figure: www.cpp-journal.org).

03/2

3/2

ni

.

(8)

Fig. 5 Pressure isotherms at temperatures from up to down:
12589 K, 10000 K, 8000 K and 5000 K. The not filled
areas correspond to negative pressures. (Colour figure:
www.cpp-journal.org).

We note that all calculated pressures join the classical ideal-gas one at low density; the system in this regime
is uncorrelated. Both HNC-ATSTLS and HNC-RPA curves join at low and high densities. The first curve is
always below the second one. The difference between them is correlated to the electron coupling parameter
[13]. The HNC-ATSTLS curve differs from the HR and experimental ones. These last curves are not correctly
reproduced in figure 4, the respective pressures have been given by their respective authors versus the total density
of hydrogen, atoms included. We have reported these pressures to figure 4 assuming a fully ionized plasma. To
correctly reproducing them, we need to know the exact free electron density in the respective studies. Thus, these
curves must shift to the left. In contrast, FLBF pressure is given by the authors versus the free electron density.
Although this pressure is calculated for a hydrogen-helium mixture, it reproduces instabilities in a density range
which is surprisingly close to the instability range of HNC-ATSTLS.
Isotherms for different temperatures are calculated in the HNC-ATSTLS model and shown in Figure 5. The
first remark is that the pressure increases as the temperature increases. All curves converge at high density
(ne > 1024 cm−3 ). The isotherms 5000 K, 8000 K and 10 000 K exhibit instabilities while no instability exists
for the isothermal 12589 K. We note that the width of the density range where the instability occurs decreases
with increasing temperature to vanish at some critical temperature. The latter is located between 10 000 K and
12 589 K and the critical density lies between ne = 1021 cm−3 and 1022 cm−3 . These results are in the same
www.cpp-journal.org

c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

620

K. Bennadji: Local field corrections effect on equation of state in dense Hydrogen plasma: Plasma phase transition

order of magnitude as many others found in the literature. For instance, Kremp, Schlanges and Kraeft [11] gave
a critical point at ncr = 6 × 1021 cm−3 and Tcr = 10 200 K. Norman and Starostin [8, 9] used two simple
models to obtain critical temperatures 2660 K and 10 640 K respectively. The second one is more justified for
correlated electrons. Saumon and Chabrier [17] predicted a critical point at Tcr = 15 300 K, Pcr = 0.614 Mbar
and ρcr = 0.35 g/cm3 . Path-integral Monte Carlo calculations of Magro et al. [21] give a critical temperature of
11 000 K.
As mentioned above, the instabilities reported in figure 5 define a forbidden area in the electron densitytemperature plane, the electron system never reaches this area and the system would avoid it by undergoing a
phase transition. It is worth to verify if the inconsistency of the STLS model revealed by some authors does
not lie into this area. Indeed, it is pointed out that the STLS model does not fulfil a sum rule [26, 27], that is
the isothermal compressibility deduced from the compressibility sum rule differs from the one deduced from
the equation of state; this fact is considered as a shortcoming of the model. On the other hand, the isothermal
compressibility, χT = n−1 (∂n/∂P )T , takes an infinite value at the critical point [28]. This fact is related to
density fluctuations and to droplets formation in the region of thermodynamic instability [28]. It is worth to
verify if eventual sum rule violation is related or not to thermodynamic instabilities.
This work is devoted to the pressure calculations of a proton-electron system including LFC effects. The LFC
evaluated here at arbitrary parameters of the plasma, incorporates electron exchange-correlations. These ones
are shown to be responsible for thermodynamic instabilities when the electron coupling parameter exceeds unity.
These calculations suffer of a shortcoming that is the neutrals contributions to the pressure are not considered.
Although effort is devoted to justify this point for densities exceeding the solid density, its application to lower
densities is not justified; thus, these calculations do not apply to real hydrogen. However, the method is of
methodological value showing the importance of electron correlations in the equation of state.
Acknowledgements The author is particularly grateful to Marie-Madeleine Gombert, Gilles Maynard, Claude Deutsch and
Abderrezeg Bendib for their criticisms and fruitful discussions.

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c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
!

www.cpp-journal.org


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