MULTICOMPONENT PLASMA

185

If we use the concept of the mean ion charge, Eq. (5.36) suggests the following law of charge averaging (Salpeter 1954):

2				¯	1/3
	= Z		5/3			.	(5.37)
Ze				(Z)

Note that this averaging technique di ers substantially from the Debye– H¨uckel one, where Ze2 = Z2. Since the Madelung term predominates in uex, it is Eq. (5.37) that corresponds to strong nonideality.

DeWitt and Hubbard (1976) and Hansen et al. (1977) have performed the Monte Carlo calculations of the thermodynamic properties of mixtures for Z2/Z1 = 2 and 3, and have obtained the solution to the hypernetted chain equations. It turned out that the excess free energy can be presented with very good accuracy as a linear superposition of free OCP energies, which are homogeneous with respect to charge (Slattery et al. 1980),

Fex/(nkT ) = f = x1f0(Z15/3γ ) + (1 − x1)f0(Z25/3γ ).

(5.38)

A binary ionic mixture can be separated into homogeneous phases (Stevenson 1975), such as the H+ and He++ mixtures in the interior of giant planets. This occurs at temperatures below some pressure–dependent critical value Tcr. Hansen et al. 1977 found that in a hydrogen-helium OCP (xHe = 0.28) at pressure of 6 TPa the critical temperature is Tc = 6300 K. Under these conditions, rs = 0.85a0. Essential is that if the linear superposition (5.38) is quite accurate, the phase separation is impossible. Of considerable importance in describing the separation e ect is the inclusion of the electron background correlation. As shown by Pollock and Alder 1977, the separation can occur because the local quasineutrality is more e ective in homogeneous phases than in the mixture.

5.2Multicomponent plasma. Results of the perturbation theory

The main disadvantage of the one–component plasma model resides in the extremely simplified inclusion of the opposite sign charge, which is treated as an unstructured compensating background. More sophisticated plasma models provide an explicit inclusion of the structure and interaction of charges of all signs, with the necessary description of quantum e ects for the Coulomb interaction. The point is that if the quantum–mechanical features of the problem are neglected, this can lead to major di culties with the classical description of the particle motion at small distances ( λe), because of the divergence of the coordinate part of the Gibbs probability for the oppositely charged particles (Norman and Starostin 1970). In quantum theory, such a divergence is naturally eliminated by the growth of the mean momentum and kinetic energy upon the localization of charges, which ensures the stability of the entire system. Yet the quantum effects lead to the formation of bound states (that is, molecules, atoms, and ions) which, in turn, a ect the interaction of free charges. Numerous attempts have been made to maintain the classical formalism, by introducing a cuto in the Coulomb potential at small distances with the elimination of configurations with

186 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

closely spaced charges. In this case, the cuto parameter is included in the final answer and the models become thermodynamically unstable at γ 1, when the cuto radius becomes comparable with the interparticle distance.

A consistent quantum–mechanical treatment of the problem is based on the Hamiltonian description of the full interaction between all charges. This corresponds to the “physical” model of a multicomponent plasma, where the contribution of discrete spectrum is finite and occurs simultaneously with the free charge contribution (Ebeling et al. 1976). The “physical” model is the most general and consistent one to describe real plasmas. However, in practice the calculations with this model are very laborious and have not yet been implemented widely. The problem is that, being applied to partially ionized plasmas of multielectron elements, the physical model requires the quantum–mechanical calculations of the internal structure of bound states, which is analogous to the calculations of atoms and ions with the Hartree–Fock method.

In a rarefied plasma, where configurations with closely spaced particles are quite unlikely, considerable simplifications are possible. The principal simplification consists of the separate description of the discrete and continuous spectra states. The first spectrum corresponds to the internal structure of atoms and ions, and the second one to an electrically charged component. This is the essential approximation of the so–called “chemical” model (Ebeling et al. 1991), which currently is the most popular one in plasma physics. In this model, the number of particles of a di erent type, {Ni}, is governed by the conditions of

	V, T, N )
chemical ionization equilibrium, µi		= 0:
	∂F (	i	V,T	= 0.	(5.39)
	∂Nk

Here, all hypotheses concerning the structure of particles and their interactions are contained in the expression for the free energy, F (V, T, {Ni}),

F (V, T, N ) = F k + F c + F b + F l.

(5.40)

The contribution of the discrete spectrum, F b, in this model is calculated independently of the continuous spectrum contribution, which is represented by the kinetic part, F k, and various corrections for the interparticle interaction, F c.

If radiation is in local thermodynamic equilibrium with matter, the contribution of the free energy of the photon gas becomes significant upon extremely high heating or strong rarefaction, F l = −(4σ/3c)V T 4, where σ is the Stefan– Boltzmann constant and c is the speed of light (Landau and Lifshitz 1980). For instance, for tungsten of solid state density the contribution of thermal radiation becomes noticeable at T 107 K (p 103 TPa). The free energy of an ideal gas is

F k = kT k	Nk	µk − F1/2		(µk ) , Fp(µk ) =	Γ(p + 1)
			F3/2	(µk )	1

∞	pdt
0	t	,	(5.41)
	et−µk − 1

MULTICOMPONENT PLASMA

187

where the reduced chemical potential µk (in units of kT ) is a measure of the degree of plasma degeneracy and is determined by the relationship

F1/2(µk ) =	Nk	λk3 nk λk3 .
	gkV

For heavy particles we have nk λ3k 1 and µk 1, which provides the transition to the classical limit of Boltzmann statistics,

F k = kT	k	Nk ln	gk V λk3			− 1 .	(5.42)
				Nk

The inclusion of the electron component degeneracy reduces the degree of plasma ionization while the kinetic part of the electron pressure is increased.

The chemical model usually involves relations that are derived for a fully ionized plasma with the methods of modern perturbation theory (Ebeling et al. 1976). The perturbation theory is developed up to an arbitrary order with respect to the expansion parameter and is equipped with a diagram technique, which facilitates classification and identification of terms in the corresponding series. The major correction for plasmas is calculated by the summation of the so– called “ring” diagrams (Graboske et al. 1971; Ebeling et al. 1976). For degenerate plasmas, it corresponds to the Gell–Mann and Brueckner (1957) model, whereas in the Boltzmann limit it leads to the Debye–H¨uckel model,

βF2c = − i	1	ΓP (γ).	(5.43)
	Ni 3

Here, the parameter of nonideality of a multicomponent plasma is

		Zi2θiNi/N	3/2
Γ = 2π1/2e3β3/2(N/V )1/2			,	(5.44)
		i

where N = Ni and the parameter θi = F−1/2(µi)/F1/2(µi) counts the de-

i

generacy of the i species. The additional quantum–mechanical correction in Eq. (5.43) takes into account the e ects caused by the uncertainty principle in a high–temperature plasma (that is, when βe2 Z2 < λi, see Fig. 5.7 ) and corresponds to the e ective repulsion of charges at small distances (DeWitt 1962; Graboske et al. 1969, 1971):

P (γ) = 1 − (3/16)π1/2[(Ze4Ne2γee + 2Ze2Zi2NeNiγei

+Zi2Zj2NiNjγij)/N 2 Z2 2]

+(1/4)γe2{[Ze4Ne2 + Ze2Zi2NeNi(1 + me/mi)

+Zi4Ni2(me/mi)]/N 2 Z2 2}, (5.45)

188 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

.

.

.

.

. . .

Fig. 5.7. E ect of the uncertainty principle on the equations of state (5.43) and (5.45)

(Graboske et al. 1971).

γi =	λi	; γei =	λei	=	/(2µeikT )1/2	; µei =	memi	,
	rD		rD		rD		me + mi
where rD = [4πβe2(N/V ) j			Zj2Nj θj ]−1/2 is the Debye screening distance. The

next (after Eq. 5.43) expansion term is referred to as the ladder term; it describes the binary interactions of charges in terms of the dynamically screened Coulomb potential. The general expression for the ladder correction is given by Ebeling et al. (1976) and includes multidimensional integrals of complex structure. In the high–temperature approximation, γii < Γ < γei and γee < 1 (for a hydrogen plasma, 50 kT 2000 eV), the ladder correction takes the form (Ebeling et al. 1976)

βF3c = −		1				Γ2		Ze6Ne2(ln γee + 0.887)
	i	Ni
			12		N 2	Z2	2

+2Ze3Ne Zi3Ni(ln γei + 0.887)

+ Zi3Zj3NiNj [ln ZiZj βe2/rD + 0.420] . (5.46)

ij

The first term in this expression describes electron–electron interactions, the second one describes electron–ion interactions and the third one describes ion– ion interactions. In order to describe quantum and di raction e ects in this and subsequent terms of perturbation theory, a number of approximations of a much more complex structure have been proposed (Graboske et al. 1971).

Along with the Coulomb interaction, perturbation theory methods allow one to calculate the corrections for exchange interaction between free charges of identical spins. The first–order correction for arbitrary statistics takes the form

MULTICOMPONENT PLASMA

189

exch1

−

2

Zi2Ni

µe

dµ F 2

(µ )

e

F12/2

(µe)

βF c

=

Ne

1

γ2

Ne

e −1/2

e

.

(5.47)

i

−∞

The exchange integrals for the cases of a Boltzmann and degenerate plasma are calculated analytically, thus bringing about the following asymptotes:

lim

βF = Ne

1

γ2

Ne

,

(5.48)

4

µe−1

e i

Zi2Ni

1

1

4

µe2

−

2

π ln µe + c3

Ne

lim

γ2

π

3

βF = −Ne 2

Z2N 2

2 .

(5.49)

µe 1

e

i

i

3

√

µe

π

i

4

3/2

One can see that the classical asymptote (5.48) is applicable for −∞ µe 1.5. Second–order exchange corrections (DeWitt 1962) can only be calculated for the Boltzmann case (for µe 4),

βFexchc

2 = Ne 27/2 ln

2 2S + 1

Z2N

1/2

−

π3/2

λFγe

Ze2Ne

e

i

i

i

2

−

12π

12λ

F

λ

3

N

i

.

F

i

Ne

1

βe2

Ne

The general expression for the third–order exchange correction, as obtained by Ebeling et al. (1976) in the case of nondegenerate plasma, reduces to

2

βFexchc

3

=

−

12π

12λ

λ

3

N

i

.

F

F

i

Ne

1

βe2

Ne

The relative contribution of the corrections to the pressure of hydrogen plasma is illustrated in Fig. 5.8.

These models obtained with perturbation theory are asymptotic and hence applicable, strictly speaking, only for small values of the expansion parameters, Γ 1 and γ 1. According to Graboske et al. (1971), the range of validity of the ring model (5.43) is estimated as Γ 0.5, whereas in Ebeling et al. (1976) the estimate is more moderate, Γ 0.1. According to Graboske et al. (1971), the interference e ects for hydrogen are adequately described by Eq. (5.45) at γ < 1.1. In the region of increased nonideality, Γ 1 and γ 1, the results of perturbation theory are inapplicable, and here one must either employ nonparametric methods or use extrapolations. For the latter case, the