∞

where J1/2(y) =

QUASICLASSICAL APPROXIMATION

203

5.5Quasiclassical approximation

As the pressure increases, molecular and ion–molecular structures in the plasma become destroyed, the outer valence electrons leave atoms, the electron shells of atoms and ions reorganize themselves and, at extremely high pressures, disintegrate, which provides a quasi–uniform charge distribution in the atomic cell. In this case, it is possible to use a quasiclassical description expressed via the mean electron density distribution, ne(x), instead of a quantum–mechanical formalism in terms of the wavefunctions and discrete energy spectrum. The corresponding approximation is known as the Thomas–Fermi model and represents a quasiclassical limit ( → 0) with respect to the Hartree self–consistent field equations (5.77). A comprehensive analysis of this method has been given in the review by Kirzhnits et al. (1975).

In this approximation, the quasi–uniform degenerate electronic gas relations are used for the mean electron density (Kirzhnits et al. 1975; Shpatakovskaya

2000),

√

2

(x)

2

ne(x) =

2

T

3/2

J1/2

pF

T

→

pF(x)

,

(5.84)

π2

2T

0

3π2

→

τ y1/2

exp(τ − y) + 1 dτ is the Fermi–Dirac function (in this section

0

we keep using the atomic units). The Fermi energy p2F(x)/2 is related to the chemical potential via

		p2
	µ =		F	+ U (r).	(5.85)
		2

In the thermodynamic Thomas–Fermi model the matter is broken down into electroneutral spherical Wigner–Seitz cells containing a nucleus and Z electrons surrounding the latter. The electrons and nucleus create a self–consistent potential which satisfies the Poisson equation,

− 2U = 4πne,				(5.86)
with the boundary conditions,
1		;	U (r) → (R − r)2,
U (r) →				(5.87)
	r
r→0			r→R

where the cell radius R is given by the condition of electroneutrality, ne(r)dr = Z. Numerical integration of (5.84)–(5.87) enables one to determine the intraatomic density ne(x) from which all thermodynamic functions of the electronic gas of atomic cell are recovered in accordance with the expression for the free energy,

√

T

dr

2

2

−

3 J3/2

2

+

U (r)ne(r)dr. (5.88)

F = π2

5/2

2T J1/2

2T

2T

2

pF

pF

2

pF

204 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

In order to obtain the total thermodynamic characteristics of the model, the contribution due to the motion of nuclei is added to the electron terms, as described either in the ideal–gas (5.42) or quasi–harmonic approximations (Kopyshev 1978), or by the OCP model (Kopyshev 1978) (see Section 5.1). At kT 1 keV, one should take into account also the pressure and energy of equilibrium radiation.

The Thomas–Fermi model is characterized by self–similarity with respect to the nuclear charge Z, i.e., the results obtained for one particular element can be used for an arbitrary substance. The temperature, volume, chemical potential, pressure, and energy in the model have the following scalings on Z (Shpatakovskaya 2000):

T (Z) = T (1)Z4/3, V (Z) = V (1)/Z, µ(Z) = µ(1)Z4/3,

p(Z) = p(1)Z10/3, E(Z) = E(1)Z7/3.

Modifications of the Thomas–Fermi model are associated with the more detailed inclusion of correlation and quantum e ects. The correlation corrections are caused by the di erence between the Hartree self–consistent field and the true field within the atomic cell. The corrections that result from the antisymmetry of the electron wavefunctions are interpreted as exchange and correlation e ects. In addition, force correlation e ects occur because of inaccuracy of the independent particle model. The quantum–mechanical corrections are due to the quasiclassical formalism and are divided into a regular, with respect to 2, part (referred to as quantum) reflecting the presence of nonlocal correlation between ne(x) and the potential U (x) due to the uncertainty principle, and an irregular correction, which reflects nonmonotonic physical quantities due to the discrete energy spectrum. It is important that the most modern modifications of the Thomas–Fermi model (Kirzhnits et al. 1975) are associated with the introduction of the oscillation correction, whereas the inclusion of exchange, correlation, and quantum corrections (Kalitkin 1960, 1989) are traditional for this model.

The relative magnitude of the correlation and exchange e ects is controlled by

dimensionless parameters (Kirzhnits et al. 1975) δcorr (n1e/3/p2F)ν and δexch δquant ne/p4F. In the degenerate region (for n2e/3 T , pF n1e/3, δ0 n−e 1/3,

ν = 2) these are δcorr n−e 1/3 and δexch n−e 2/3, whereas in the classical region

(n2e/3 T , pF T 1/2, δ0 n−e 1/3/T , ν = 3/2) we have δcorr n1e/2/T 3/2 and

δexch ne/T 2. The dimensionless parameter δ0 characterizes the nonideality and equals the ratio of the mean energy of the pair Coulomb interaction to the kinetic energy of the electrons. The range of validity of the quasiclassical method of describing the atomic cell electrons, estimated (Kirzhnits et al. 1975) on the basis of these criteria, is defined at T = 0 by the condition ne 1 and in hot matter by the condition ne T 3, which corresponds to the conditions of smallness of the binary interaction energy of electrons as compared with their kinetic energy.

QUASICLASSICAL APPROXIMATION

205

The physical conditions of the validity of the quasiclassical model correspond to extremely high pressures p pa (where pa = e2/a40 30 TPa is the atomic unit of pressure) and temperatures T 105 K, which are realized in the interior of superdense stars and in other astrophysical objects, but are still inaccessible in experiments under terrestrial conditions because they require extremely high local concentrations of energy (Fortov 1982). At present, record-breaking pressures and temperatures are attained by dynamic methods using the powerful shock wave technique. The data obtained in these experiments correspond to pressures from a few to a few tens of TPa, which is much below the range of formal validity of quasiclassical models. Therefore, the existing estimates of the lower range of validity of the Thomas–Fermi model (Altshuler et al. 1977) are based on the extrapolation of these experimental results. It is noted that the inclusion of quantum, exchange and correlation corrections (oscillation corrections were not included) substantially improves the extrapolation. According to Altshuler et al. (1977), in this case the extrapolation is possible for a cold (T = 0) plasma up to pressures of 30 TPa and, at T 10 keV, up to about 5 TPa.

Recently, however, such estimates were questioned. The point is that the commonly used version of the quasiclassical model (Latter 1955, 1956; Kalitkin 1960, 1989; Kalitkin and Kuzmina 1975, 1976; Altshuler et al. 1977) describes “averaged” characteristics of matter and ignores the so–called shell e ects caused by individual peculiarities of the population of atomic energy levels and bands. Elegant studies performed by Kirzhnits et al. (1975) and Kirzhnits and Shpatakovskaya (1995) have shown that the shell e ects can be qualitatively described within the quasiclassical approximation by including an irregular (with respect to 2) correction corresponding to the oscillating part of the electron density which was previously erroneously discarded. The inclusion of shell e ects changes noticeably the equation of state for superdense plasma and causes the emergence of sharp nonmonotonicities of thermodynamic functions in the high– pressure region (where this approximation is justified) (Kirzhnits et al. 1975). In addition, discontinuities emerge in the equations of state, associated with the first–order electron phase transitions caused by the “forcing out” of energy levels from the discrete spectrum to the continuous one. This, in turn, causes peculiarities in other physical quantities. Note that the substantial contribution by shell e ects at ultra–high pressures follows from the results of quantum–mechanical calculations using the more accurate method of attached plane waves (Bushman and Fortov 1983). These e ects are predicted in a wide range of parameters and must disappear only at ne Z4 in the uniform region, when all atomic energy levels pass over to the continuous spectrum (Bushman and Fortov 1983).

Therefore, the question concerning the lower range of validity of the quantum– static model for superdense plasma is currently open, whereas the behavior of matter in the region of p > 30 TPa appears more diversified than previously assumed on the basis of simplified representations (Latter 1955, 1956; Kalitkin 1960, 1989; Kalitkin and Kuzmina 1975, 1976; Altshuler et al. 1977). At the same time, the simplicity and universality of the Thomas–Fermi model (5.88) make

206 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

Fig. 5.15. Thermal equation of state for lithium plasma (Iosilevskii and Gryaznov 1981) at p = 0.1 MPa: 1 – calculation from Saha’s chemical model; 2 – Thomas–Fermi theory; 3 – Thomas–Fermi theory with corrections (Kalitkin 1960; Kalitkin and Kuzmina 1975, 1976).

it very attractive in the case of an “averaged” description of plasma properties at ultrahigh pressures, when the use of more complex approaches requires laborious quantum–mechanical calculations. Based on various modifications of the quasiclassical model, extensive tables of thermodynamic functions have now been published (Latter 1955, 1956; Kalitkin 1960, 1989; Kalitkin and Kuzmina 1975, 1976, 2000; Altshuler et al. 1977; Kirzhnits and Shpatakovskaya 1995) which cover the range of parameters far beyond the range of validity of this approximation. In particular, Kalitkin and Kuzmina (1975, 1976) propose to use their data for low–density “gas” plasmas, where the “chemical” model of a plasma (see Section 5.2) is traditionally used. Detailed analysis of the validity of the Thomas–Fermi model for this range of parameters is given by Iosilevskii and Gryaznov (1981).

Of special importance in using the quasiclassical method for the description of plasmas of reduced density (as compared with the density of solids) are specific errors introduced by the cell model itself. In this model, all correlations are automatically restricted by the atomic cell size and, therefore, cannot exceed the mean distance between the nuclei. This fact, in particular, sets limits on the applicability of the Thomas–Fermi model for the description of plasmas under conditions typical for this state of matter – when the Debye sphere contains a large number of charges, Γ 1. Therefore, the limit of weakly nonideal (Debye) plasma cannot be retrieved from the Thomas–Fermi theory.

Figures 5.15 and 5.16 show the comparison between the thermal and caloric equations of state for the lithium plasma calculated from the “chemical” plasma model, with the ionization from Saha’s equation (see Section 5.2) and in the Thomas–Fermi approximation with quantum and exchange corrections (Kalitkin