ONE–COMPONENT PLASMA

175

lattice in the limit of strong interaction, making use of the short–range e ect of the interparticle interaction, which is characteristic for cell models.

In order to clarify the role of this short–range e ect and to describe the second ( γ1/4) term in Eq. (5.9), the hard–sphere model popular in the theory of liquids was employed by DeWitt and Rosenfeld (1979). By treating the hard spheres as “zero” approximation models (H0, F0) in describing OCP (H, F ), using the Gibbs–Bogoliubov inequality,

F F0 + H − H0 ,

and employing solutions of the Percus–Yevick equations, we derive after the variational procedure (see Section 5.4.3 for details):

uex

9

8γ

1/4

1

7 18

1/4

= −

γ +

−

+

− . . .

nikT

10

9

2

80

γ

In fact, this relation reproduces the first term in the ion–sphere model (5.13) and explains the appearance of the second term in (5.9), which can be interpreted as a manifestation of the short–range e ects in highly compressed plasmas.

5.1.4Wigner crystallization

The crystallization was first considered by Wigner (1934) for a degenerate plasma, where it was shown that at fairly low concentrations the electron gas with the compensating background charge should become ordered. Indeed, as the plasma expands the Coulomb energy that stabilizes the lattice, VC e2n1e/3 e2/rs, decreases slower than the kinetic energy that destroys the lattice, the scale of the latter being the Fermi energy εkin εF 2n2e/3/2m. Therefore, at su ciently low density the kinetic energy εkin n2e/3 becomes smaller than the potential energy VC n1e/3, and is incapable destroying of the ordered electron structure formed due to the repulsion.

The problem of determining the conditions of the Wigner crystallization is treated in numerous studies, which are analyzed by Tsidil’kovskii (1987). We will briefly describe the results of this analysis. A simple comparison of the potential and kinetic energy of electrons gives the limits of crystallization,

rsc (1.1 − 3.7)a0, n1c/3a0 0.56 − 0.17,

where the lower estimate for rsc was derived by neglecting the exchange and correlation e ects, and the upper estimate by taking them into account in the Hartree–Fock approximation. Based on the Lindemann melting criterion, one derives the estimate

rsc 20a0, n1c/3a0 3, 1 · 10−2,

which is very sensitive to the details of this empirical criterion.

176 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

Assuming that the melting corresponds to the disappearance of bound states of the electron, one can derive the stability limit,

50 rsc/a0 100, 6.2 · 10−3 n1c/3a0 1.2 · 10−2.

Detailed calculations of the ground state yield

rsc (5–6)a0, n1c/3a0 0.1–0,12.

The Monte Carlo calculations for paramagnetic and ferromagnetic Fermi liquids result in the critical values

rsc (75 ± 5)a0, n1c/3a0 (8.3 ± 0.5) · 10−3, rsc (100 ± 20)a0, n1c/3a0 (6.2 ± 1.3) · 10−3.

It is believed that the latter values of rs are the most reliable ones (Tsidil’kovskii, 1987).

For a nondegenerate plasma, the estimates of the melting parameters are obtained by comparing the free energy of the gas and the crystal phases. From Eq. (5.12), the expression for the free energy density (Slattery et al. 1980) is derived,

f (γ) = −0.895 929 + 9γ/2 − 1.8856 − 1490/γ2.

(5.14)

The free energy curves for gaseous (5.10) and solid (5.14) phases intersect at γm = 165 (Slattery et al. 1980). The Wigner crystallization in OCP occurs for this value of nonideality parameter, which is very sensitive to the details of the calculation procedure. Di erent authors give, in their opinion, the most probable values of γm, which lie in the range between 155 and 171.

Apparently, Wigner crystallization was experimentally observed in a two– dimensional system of electrons localized at the surface of liquid helium by Grimes and Adams (1979). A classical electron crystal with triangular lattice occurred at γm 140, with γ = (Ze)2(πne)1/2/kT . This value is close to the calculated value, γ = 125 ± 15 (Gann et al. 1979).

The quasiperiodic structure of an ion cloud with ni 108 cm−3 and T 102 K in a magnetic field was also interpreted by Gilbert et al. (1988) as a manifestation of the Wigner crystallization. Oscillations of the plasma binary function, which are typical for ordered structures, were measured by Hall et al. (1988) in laser-driven shock waves and were treated from the same perspective. It is possible that at γ > γm, supercooled OCP is in an amorphous glassy state (Ichimaru 1977), and the phase transition is accompanied by pronounced hysteresis phenomena. This state is a disordered system of monocrystals with a size of several lattice spacings. It is interesting to note that this might be the state of matter in the interior of white dwarfs because, according to some estimates, the parameter γ there may even exceed γm. Properties of Wigner crystals are discussed in detail in Chapter 10.

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177

5.1.5Integral equations

In the theory of liquids, integral equations relate the binary correlation function to the two–particle interaction potential V (r) (Rushbrooke 1968; Kovalenko and Fisher 1973). The Percus–Yevick equation, which had shown itself to be advantageous in describing the interaction of the system properties with the short–range potential (Rosenfeld and Ashcroft 1979), turned out to be hardly suitable for the OCP (Springer et al. 1973). For plasmas, the hypernetted chain (HMC) approximation is, apparently, the most appropriate one. One can conclude this from the qualitative analysis of the HMC equations as well as from the good agreement between the results of the Monte Carlo simulations and the results of numerical integration of hypernetted chain equations (Ng 1974; Springer et al. 1973). These equations have the following form:

g(r) = 1 + h(r) = exp{−V (r)/kT + h(r) − c(r)},		(5.15)
h(r) = c(r) + N	dr c(|r − r |)h(r ),	(5.16)

where h(r) and c(r) are the complete and direct correlation functions, respectively, and V (r) = Z2e2/r. The solution procedure is as follows: Starting from some initial h(r), one calculates its Fourier transform h(k) and obtains c(r),

c(r) =	dk		h(k)		(5.17)
		eikr		.
	(2π)3		1 + h(k)

Upon substituting Eq. (5.17) in Eq. (5.16), one derives the next interpretation of h(r), and so on.

Springer et al. (1973) found g(r) and uex(r) in the 0.05 γ 50 range. Ng (1974) did that with high accuracy for 20 γ 7000. The binary correlation functions in the HMC approximation describe well the values obtained with the Monte Carlo method (see, e.g., Fig. 5.4).

Numerical HMC results were approximated by DeWitt (1976) with the following expression:

uex/(nikT ) = aγ + bγ1/2 + c ln γ + d,

(5.18)

where a = −0.900 470, b = 0.268 8263, c = 0.071 9925, and d = 0.053 7919. Because of the predominance of the static term, Eqs. (5.9) and (5.18) coincide with an accuracy better than 1%. As for the thermal terms in uex, they di er both qualitatively [scaling γ1/2 in (5.18) instead of γ1/4 in (5.9)] and numerically (45 % at γ = 150).

At the highest values of γ, the strict constraint (5.13) is not satisfied. In spite of the considerable advantages of the HMC approximation in general, it proves unsatisfactory in situations when the thermal part plays a major role. This is the case, for instance, in calculating the heat capacity,

cV /ni − 1 = −γ2d(uex/nikT γ)/dγ.

(5.19)

178 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

Here, the error is as high as 20%. The shortcomings of the HMC approximation arise because the contribution of the ladder diagrams is neglected in calculating uex. The semiempirical inclusion of this contribution proved successful by DeWitt (1976). It was performed in such a manner as to satisfy the sum rule for compressibility violated by the HMC approximation. As a result, the thermodynamic quantities and correlation functions were found to be very close to those obtained using the Monte Carlo method.

5.1.6Polarization of compensating background

The degenerate electronic gas, which provides the compensating background, is polarized due to the nonuniformity of the ion charge distribution. An electron cloud is formed around each ion, which modifies the ion–ion potential. In the Fourier representation,

V (q) = 4π(Ze)2q−2εe−1(q).

(5.20)

The static dielectric permeability εe(q) is known in various approximations (Gorobchenko and Maksimov 1980) and has the form

εe(q) = 1 −	4π(Ze)2χ0(q)q−2	(5.21)
	1 + 4π(Ze)2G(q)χ0(q)q−2 ,

where χ0(q) is the static dielectric permeability of the free electronic gas (Lindhard function), and G(q) is the local field correction due to the exchange and correlation e ects (DeWitt and Rosenfeld 1979). In the limit of high electron density (rs → 0), the exchange and correlation can be neglected and εe(q) is given by a random phase approximation (RPA),

εe(q) = 1

−

4π(Ze)2χ0(q)

q2

1

1

qF

1

q2

q + 2qF

.

= 1

+

+

−

ln

(5.22)

(qλTF)2

2

2q

4qF2

q

−

2qF

1/3

Here λTF = rs(π/12Z)

is the Thomas–Fermi screening distance and qF is the

Fermi momentum, qF−1 = rs(4/(9πZ))1/3. Actually, Eq. (5.22) works fairly well as long as rs a0.

To the rough Thomas–Fermi approximation, the interion potential is shielded

as follows:
V (r) =	(Ze)2	exp −	r	.	(5.23)
	r		λTF

If λTF r, screening is not important, that is, the background polarization can be neglected. In the opposite case, the results of the calculation depend on two, rather than one, parameters, that is, γ and rs (Galam and Hansen 1976).