nakT

INTERPARTICLE INTERACTIONS. CRITERIA OF NONIDEALITY

7

where Ψ(r) is the electron wavefunction. If the interaction is weak, one may neglect the electron–atom correlations and, using Eq. (1.11), obtain

U = 2π 2Lna/m.

(1.12)

The criterion of plasma ideality relative to the electron–atom interaction takes the form

γae = 2π|L| 2naβ/m 1.

(1.13)

The approximation (1.11) is valid if λe |L|, i.e., at low temperatures.

Due to their high intensity and long–range character, the plasma interactions (Coulomb and polarization) can be strong under conditions when the interaction between neutral particles is still minor. Simple inequalities can be written using the van der Waals type equation of state

(p + n2aa) = 1 − nab .

Then, the ideality criteria take the form

nab 1, naaβ 1,

(1.14)

where a and b are the parameters in the van der Waals equation, which can be expressed in terms of the critical temperature and density

Tc = 8a/27b,

nc = 1/3b.

(1.15)

Interatomic interactions are substantial at densities close to and higher than the critical values and, in vapors, in the range of loss of thermodynamic stability.

A discussion of plasma nonideality criteria can also be found in Vedenov (1965); Kikoin et al. (1966); Kudrin (1974); Khrapak and Iakubov (1981); Iosilevskii (2000); and Iosilevskii et al. (2000).

1.1.4Compound particles in plasma

Consider first how atoms are formed in a plasma. We shall proceed from a two– component classical plasma consisting of electrons and singly charged ions only. The inequality (1.3) can be rewritten as

βRy 1,

Ry = me4/2 2.

(1.16)

It follows from this inequality that the energy of the bound state of an electron and ion, i.e., the bond energy of the atom, considerably exceeds the energy of thermal motion of free particles, thus pointing to the thermodynamic e ciency of the presence of atoms in the plasma. Such partly ionized plasma is often referred to as a three–component plasma, in view of the atoms and of the electrons and ions that remain unbound.

8 NONIDEAL PLASMA. BASIC CONCEPTS

The concentrations of atoms, electrons, and ions are related by Saha equation

	neni		2Σ+		m		3/2
		=					exp(−βI),	(1.17)
	na		Σ		2π 2β

where I is the ionization energy of the atom, and Σ+ and Σ are the internal statistical sums of ion and atom, respectively. The quantity x = ne(na + ni)−1 is referred to as the degree of ionization.

It is important that the electron and ion bound in the atom are separated from each other by a distance of the order of the Bohr radius a0, which is another characteristic dimension in the system. Therefore, the electron and ion making up the atom are screened by each other such that, in a first approximation, the interaction of such pairs with the surrounding particles may be generally ignored. Indeed, note that the bond energy of the hydrogen atom, H = p + e, is equal to Ry = 13.6 eV whereas the bond energy of the ion, H− = p + 2e, is only 0.8 eV, while the possibility of the stability of H−2 = p + 3e is not confirmed and only discussed in the literature.

The residual quantum e ects in a classical plasma can be accounted for by corrections. While proportional to λe/rD, they are small if

λe/rD γ3/2(βRy)−1/2 1.

Given a high density or su ciently low temperatures, a plasma may become multicomponent. The A2 molecules, A+2 molecular ions, and A− negative ions are formed in the plasma. These are the result of interatomic interaction and charged–neutral particle interactions. The concentrations of these composite particles are calculated using equations of chemical equilibrium, Saha’s equation being one of those. For instance, for the reaction of negative ion formation, A− A + e, we obtain the equation

n−		Σ−		2π 2	β		3/2
	=						exp(βE),	(1.18)
nena		2Σ		m

where n− is the density of negative ions, Σ− and Σ are the internal statistical sums of the negative ion and atom, respectively, and E is the energy of the electron–atomic a nity.

1.2The range of existence of nonideal plasma. The classification of states

1.2.1Two–component plasma

If a plasma is fully ionized, it is two–component, with the electron concentration ne being equal to the ion concentration ni (for singly charged ions) due to quasineutrality. Such a plasma is characterized by two independent parameters, namely, the degeneracy parameter ξ (Eq. 1.4) and the Coulomb coupling (nonideality) parameter γ (Eq. 1.1) if ξ 1 and γq (Eq. 1.5) if ξ 1.

THE RANGE OF EXISTENCE OF NONIDEAL PLASMA

9

log(ne [cm-3])

log(T [K])

Fig. 1.2. Regions of parameters of a two–component plasma

It is convenient to show on the (T, ne) plane in logarithmic coordinates the lines corresponding to the following conditions: γ = 1, γq = 1, and ξ = 1. These lines will divide the (T, ne) plane into a number of characteristic regions. This is shown in Fig. 1.2 for the case of hydrogen plasma (Klyuchnikov and Triger 1967). The degeneracy parameter for electrons is ξe 2n2e/3β/m, for ions ξi 2n2i /3β/M , where M is the ion mass. The quantum coupling parameter for electrons is γqe = rs/a0, where a0 = 2/me2, and for ions γqi = (rs/a0)(M/m).

Let us now analyze the parameter regions emerging in Fig. 1.2:

Region I: ξe < 1, ξi < 1, γ < 1: a classical plasma with weak interaction of the electrons and ions.

Region II: ξe < 1, ξi < 1, γ > 1: a classical plasma with strong interaction of the electrons and ions.

Region III: ξe > 1, ξi < 1, γqe > 1, γ > 1: the electrons form a degenerate system with strong interaction while the ions form a classical system with strong interaction.

Region IV: ξe > 1, ξi > 1, γqe > 1, γqi > 1: a quantum plasma with strong interaction of the electrons and ions.

Region V: ξe > 1, ξi < 1, γqe < 1, γ > 1: the electrons form a degenerate system with weak interaction while the ions form a classical system with strong interaction.

Region VI: ξe > 1, ξi > 1, γqe < 1, γqi > 1: the electrons are degenerate and interact weakly, the ions are degenerate and interact strongly.

10	NONIDEAL PLASMA. BASIC CONCEPTS

Region VII: ξe > 1, ξi < 1, γqe < 1, γ < 1: an electron/ion plasma with weak interactions, in which the electron component is degenerate.

Region VIII: ξe < 1, ξi < 1, γ < 1: a classical plasma with weak interaction of the electrons and ions.

It is seen from this analysis that regions I, VII, and VIII represent gaseous plasmas at various temperatures and densities; regions V and VI correspond to a solid in which the electrons form a degenerate gas with weak interaction; and in regions III and IV, the states corresponding to moderate values of γ for electrons (a typical value of γ in metals is in the range 2–5) and rather high values of γ for ions are also condensed.

Region I represents a weakly nonideal low–temperature plasma well known in the physics of gas discharge and realized in numerous natural phenomena. Region VIII corresponds to high–temperature almost ideal plasma. The plasma parameters observed in the various systems are shown schematically in Fig. 1.3; see also Ebeling et al. (1976) and Smirnov (1982).

In the regions of γ < 1, perturbation theory can be employed to calculate the thermodynamic quantities. In those regions the interparticle interactions are weak and the system may be regarded as a mixture of almost ideal gases. In the regions where γ > 1, perturbation theory is inapplicable; the particle interaction is appreciable and the system is similar to a liquid. Such is region III where the liquid–metal states are located. The region of classical strongly nonideal plasma, region II, is restricted from below with respect to charge concentrations by the condition of strong interaction, γ = 1, and from above by the electron degeneracy. Naturally, the regions of nonideality are restricted from above with respect to temperature as well, because at high temperatures the kinetic energy prevails over the interaction energy.

The information provided by the (T, ne) diagram is incomplete. The regions of real existence of a stable two–component plasma are not indicated in Fig. 1.2. With a temperature decrease electrons and ions recombine, the plasma becomes partly ionized, may become molecular, and, finally, the matter condenses and crystallizes.

1.2.2Metal plasma

Figure 1.4 shows in what range of temperature and density one can experimentally realize the nonideal plasma of mercury and cesium, which have the lowest values of critical temperatures for metals.

Curve 5 corresponds to the degree of ionization of x = 0.5. This curve conventionally separates the region of two–component fully ionized plasma from the region where the ionization is only partial. A rise in temperature causes an increase of the configurational weight of free states of electrons and ions. As a result, thermal ionization occurs. In accordance with this, the high–temperature branch of curve 5 is constructed using Saha equation (1.17). It is known, however, that, with a strong interparticle interaction, the conventional Saha equation does not describe real ionization equilibrium. One of the most important e ects

THE RANGE OF EXISTENCE OF NONIDEAL PLASMA

11

log(Te [K])
log(n	e	[cm-3	])

Fig. 1.3. Plasma parameters realized in nature and in various technical devices (Ebeling et al. 1976): 1, solar corona; 2, tokamak; 3, laser–induced fusion; 4, core of Sun; 5, Z–pinch; 6, stellarator; 7, gas lasers; 8, plasmotron; 9, chromosphere of Sun ; 10, plasma of hydrocarbon fuel combustion products; 11, electric arcs; 12, cathode spot; 13, spark; 14, MHD generator utilizing nonideal plasma; 15, semiconductor plasma; 16, metal–ammonia solutions; 18, metals.

is that, in becoming strong, the interaction leads to a decrease in the bond energy of atoms or, in other words, reduces the ionization potential. Given a very strong interaction, the bound states of the electron and ion completely disappear. It is sometimes said that pressure ionization takes place. These situations correspond to the high–density branch of curve 5.

The region occupied by a nonideal electron plasma on the low–temperature side is limited by the curves of the existence of vapor and liquid. On the high– density side, the region of classical nonideality is limited by the electron degeneracy, neλ3e = 1. At ρ > ρc, the metallization of the plasma occurs. The states above curve 4 can be regarded as liquid–metal ones. It should be noted, however, that the direct measurements in mercury by Kikoin et al. (1966) have shown that a truly metal state exists only at ρ = 11 g cm−3, i.e., at na = 3.3 · 1022 cm−3.

On the low–density side, the region of nonideality is defined by the condition of equality between the interaction energy and thermal energy. The charge– neutral interaction becomes strong at na 1020 cm−3 in cesium and at na 1021

12

NONIDEAL PLASMA. BASIC CONCEPTS

cm

cm

a

b

Fig. 1.4. Density–temperature diagram for cesium (a) and mercury (b). The numbered curves are calculated from the following conditions: Γ = 1 (1); γ = 1 (2 or dashed line if ne is determined according to Saha equation); γai = 1 (3); electron degeneracy (4); x = 0.5

(5). The regions of liquid–vapor coexistence are shaded.

cm−3 in mercury (the large di erence in these values of na is mainly due to the di erence in polarizability). In the plasma of metal vapors it is this interaction which becomes pronounced with an increase of na at low temperatures. This interaction is responsible for a whole number of qualitatively new e ects observed in metal vapors at temperatures in the neighborhood of the critical temperature.

The charge–neutral interaction shifts the ionization equilibrium toward the increase of ne. At high densities the degree of ionization does not drop down with an increase of na according to Eq. (1.17), but increases. As a result, the ionization observed in the neighborhood of the critical point of cesium is almost full while in the neighborhood of the critical point of mercury the degree of ionization x amounts to several tens of percent.

In view of this, in Fig. 1.4 are plotted the curves Γ = 1 and γ = 1 indicative of nonideality in interaction between charges. In order to illustrate the extent of the discrepancy between the ionization equilibrium in nonideal plasma and ideal plasma equlibrium (Eq. 1.17), further shown in Fig. 1.4(b) with a dashed line is a γ = 1 curve calculated using Saha equation (1.17).

It is evident from Fig. 1.4 that the region occupied by a non–Debye plasma is very wide. In the last few years it was subjected to very intensive studies. Much less data are available on a strongly nonideal plasma in which γ 1, i.e., Γ 5.