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NONIDEAL PLASMA. BASIC CONCEPTS

1.1Interparticle interactions. Criteria of nonideality

1.1.1Interparticle interactions

At low densities, a low–temperature, partly ionized plasma can be regarded as a mixture of ideal gases of electrons, atoms, and ions. The particles move with thermal velocities mainly along straight lines, and collide with each other only occasionally. In other words, the free path times are much greater than those of interparticle interaction. With an increase in density, the mean distances between the particles decrease and the particles start spending more time interacting with each other.

Under these conditions, the mean energy of interparticle interaction increases. When this energy gets to be comparable with the mean kinetic energy of thermal motion, the plasma becomes nonideal. The properties of such a plasma become very unusual, and cannot be described by simple relationships of the theory of ideal gases.

If the plasma is fully ionized, its state is defined by Coulomb interactions, whose specific feature is their long–range character. Therefore, in a rarefied plasma the particles move in weak but self–consistent fields developed by all of the particles. The interaction energy increases with compression, however, the contribution from strong short–range interactions becomes more and more important. Finally, under conditions of strong nonideality, the role of these interactions becomes predominant.

Discussed below are the peculiarities of interparticle interactions; parameters of nonidealliy are introduced and estimated. This enables one to perform the classification of states of a nonideal plasma. The conditions of strong nonideality correspond to a high energy density in the plasma. Under natural conditions the plasma nonideality plays a substantial role in a variety of phenomena which have always attracted researchers’ attention. In recent years, investigations of nonideal plasmas assumed a practical significance in technological applications. Therefore, the last section of this chapter deals with nonideal plasma in natural phenomena and in technology.

1.1.2Coulomb interaction. Nonideality parameter

The criterion of ideality of a plasma with respect to the interaction between charged particles may be provided by the smallness of the ratio between the average potential energy of Coulomb interaction and the mean thermal energy characterized by the temperature T . For nondegenerate singly ionized plasma this condition has the form

1

2	NONIDEAL PLASMA. BASIC CONCEPTS
	γ = e2/kT rs e2ne1/3β 1,	(1.1)

where ne is the electron number density, β = 1/kT , k is the Boltzmann constant, and rs is the mean interparticle distance. The value of rs is usually defined as the radius of the Wigner–Seitz cell and is related to the plasma density by the simple relation

(4π/3)ner3	= 1.	(1.2)
s

Given high temperatures, multiple ionization can be attained. In this case the degrees of correlation between di erent species are di erent and, hence, di erent nonideality parameters for ion–ion, ion–electron, and electron–electron interactions should be used. For example, in a fully ionized plasma with ions having charge number Z we have

γZZ = Z5/3e2β/rs = Z5/3γee,

γZe = Ze2β/rs = Z2/3γee,

γee = e2β/rs.

Note that γ is the parameter of nonideality of a classical Coulomb system. The electrons and ions in a plasma form a classical system if they are sel-

dom found to be at distances comparable with the thermal electron wavelength

√

λe = / 2mkT . Since the characteristic radius of the ion–electron interaction is Ze2/kT , the condition of classicality can be written as

λe Ze2/kT.

(1.3)

Therefore, a plasma is classical at relatively low temperatures, kT Ry = e4m/ 2. In a more heated plasma one must allow for interference quantum e ects due to the uncertainty principle.

Another manifestation of quantum e ects is degeneracy and, primarily, the degeneracy of electrons having the greatest thermal wavelength. The possibility of employing classical statistics is defined by the smallness of the number of electrons in an elementary volume with a linear size of λe. In other words, the inequality neλ3e 1 should be met. The condition of applicability of classical statistics corresponds to the smallness of Fermi energy εF as compared to temperature:

ξ = ε	F/T2		1,				(1.4)
			2/3		2	/2m,
εF = (3π ne)

where ξ is the degeneration parameter.

Therefore, the isothermal compression of low–temperature plasma (T Ry/k = 1.58 · 105 K) leads to an increase of the Coulomb interaction energy which, after the parameter γ becomes larger than unity, exceeds the kinetic energy of particle motion. This complicates considerably the theoretical description of nonideal plasma while making impossible the use of perturbation theory and forcing one to employ qualitative physical models. Further compression of the

INTERPARTICLE INTERACTIONS. CRITERIA OF NONIDEALITY

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plasma causes an increase of nonideality, although to a certain limit only. The point is that with increasing density degeneracy of electrons occurs at neλ3e ≈ 1. For example, in metals ne 1023 cm−3, and electrons are degenerate at T 105 K, i.e., almost always. The Fermi energy, which increases with the plasma density, becomes the kinetic energy scale. The quantum criterion of ideality has the form

γq = e2ne1/3/εF 1.

(1.5)

Since εF n2e/3, the parameter γq decreases with increasing electron density. Therefore, a degenerate electron plasma becomes even more ideal with compression.

It should be emphasized that, at higher densities, only electrons represent an ideal Fermi gas. The ion component is nonideal. Depending on the degree of its nonideality, one may talk about ionic liquids, cellular or crystalline structures, and other model representations of the ion subsystem.

It is well known that even in a rarefied plasma, when γ 1, one cannot directly employ the formulas of ideal gas theory for determining the thermodynamic and transport properties of the plasma. Quantities such as the second virial coe cient or the mean free path are diverging. This is due to the specific character of the Coulomb potential, i.e., its slow decrease at large distances and infinite increase at small distances. The divergence at small distances is eliminated when quantum e ects are taken into account, while the e ect of charge screening by the surrounding plasma eliminates divergences of the mean free path and the second virial coe cient at large distances.

Let us consider charge screening in plasmas. As a result of the long–range character of the potential Ze2/r, multiparticle interactions at large distances r rs prove to be substantial. Consider now the charge density distribution in the neighborhood of an arbitrary test particle with charge number Z. Such a particle repels like charges and attracts charges of the opposite sign. The resulting self–consistent potential created by the selected test particle and its plasma environment is known as the Debye–H¨uckel potential:

ϕ = (Ze/r) exp(−r/rD),

(1.6)

where r is the distance from the test particle, and rD is the Debye screening distance or Debye radius

−1/2

rD = 4πe2β Zk2nk	,	(1.7)

k

where subscripts k correspond to di erent charged plasma species. Therefore, the Coulomb field of the test particle is screened over a distance of the order of rD. This, in fact, leads to the convergence of the basic quantities such as the mean free path and virial coe cient. The presence in the Debye sphere of a

4	NONIDEAL PLASMA. BASIC CONCEPTS

su ciently large number of charged particles is the essential condition of validity of expressions (1.6) and (1.7).

Let us now estimate the intensity of interparticle interaction in a Debye plasma. Note that Eq. (1.6) represents a superposition of the Coulomb potential created by the test particle and the potential created by all of the remaining particles in the plasma. Deducting the test particle potential Ze/r from Eq. (1.6) and assuming r → 0 we obtain the potential created by charged particles of the screening cloud at the location of the test particle, ϕ = Ze/rD. Then, the criterion of ideality for singly charged plasma can be written as

Γ = e2β/rD 1.

(1.8)

The parameter Γ is referred to as the plasma parameter or the Debye parameter of nonideality.

It is readily seen that the criterion (1.8) can be expressed in terms of ND = (4π/3)nerD3 , i.e., in terms of the number of electrons in the Debye sphere. We have Γ = (2γ3)1/2 = (3ND)−1 and, therefore, the plasma ideality criterion, i.e., the smallness of the energy of the Coulomb interaction as compared with kinetic energy, coincides with the condition of applicability of the Debye approximation: In both cases the number of charged particles in the Debye sphere must be large, ND 1.

Under conditions when the electron component is degenerate, neλ3e 1, the screening distance by degenerate electrons is defined by the Thomas–Fermi length,

rTF = (π/3ne) 2/4me2.

In a two–component electron–ion system, in which the electrons are degenerate while ions remain classical, the screening distance of the test charge is defined by the expression

rscr−2 = (rTFe )−2 + (rDi )−2 ≈ (rDi )−2.

Hence the screening is mostly due to the ion component.

As the density increases, the Debye screening distance decreases and may become smaller than the interparticle distance rs. Under these conditions, rD loses the meaning of screening distance. For example, such is the situation in a liquid metal, where the charge is screened over distances of the order of the radius of the Wigner–Seitz cell.

Another characteristic singularity of the Coulomb potential is its behavior at small distances which makes a purely classical Coulomb system unstable. However, it is at small distances (of the order of the wavelength λe) that quantum e ects leading to the formation of atoms and molecules become appreciable. One can speak of short–range quantum repulsion which is connected to the e ect of the uncertainty principle and does not allow close approach of two particles with a preset relative momentum. This eliminates the divergencies and leads to quantum corrections at low temperatures (at kT Ry in a plasma with singly

INTERPARTICLE INTERACTIONS. CRITERIA OF NONIDEALITY

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cm
Thomas	Fermi
	Debye
	Ry

Fig. 1.1. Ranges of existence of nonideal classical and degenerate low–temperature plasma.

ionized ions). Such a plasma is referred to as a “classical plasma”. In the opposite case the plasma is called a “quantum plasma”.

The ranges of existence of nonideal classical and degenerate plasma are given in Fig. 1.1. They are defined by the conditions γ = 1, neλ3e = 1, and γq = 1. Also shown in the diagram are the ranges of applicability of limiting approximations describing the states of a weakly nonideal plasma, namely, the Debye and Thomas–Fermi approximations.

As we can see, only the states with extremely high pressures and temperatures located at the periphery of the phase diagram of matter are accessible to consistent theoretical analysis. The range of existence of nonideal plasma appears in Fig. 1.1 to be located within the triangle defined by the condition εF = e2n1e/3 and γ = 1, with the upper portion of this range relating to a degenerate plasma and the lower portion to a Boltzmann plasma, while the maximum possible values of the nonideality parameter are finite and do not exceed several units. At higher densities the plasma remains strongly nonideal due to ion–ion interaction as long as the ions are not degenerate.

1.1.3Electron–atom and ion–atom interactions

Given the low ionization fraction in a low–temperature plasma, the interactions between charged particles can be ignored. Here, the interactions between charged and neutral particles can be dominant in view of the high density of the neutrals. In this case, the electron and ion components of a weakly ionized

6	NONIDEAL PLASMA. BASIC CONCEPTS

plasma cannot be described in an ideal–gas approximation. The nonideality due to charge–neutral interaction is appreciable primarily for the properties caused by the presence of charged particles, such as the electrical conductivity, thermal conductivity, and thermoelectromotive force.

In order to estimate the importance of charge–neutral interaction, let us calculate the potential produced by atoms (molecules) at the ion (electron) location. Contributions to the ion–atom (electron–atom) interaction are made by the exchange, electric, and polarization forces. Because of their long–range character, it is primarily the polarization force that is appreciable at low densities. This is especially true for the ions. An atom polarized by an ion sets up at the ion location a potential ϕ(r) = −αe/2r4, where r is the distance between them and α is the atomic polarizability. It is also assumed that r > ra, where ra is the atomic radius. The total potential set up by all atoms at the ion location is

∞

ϕ = 4π ϕ(r)na(r)r2 dr, (1.9)

ra

where na(r) is the atomic concentration which depends on the distance from the ion. If the interaction is not yet strong, this dependence (ion–atom correlations) can be neglected. Then,

ϕ = −2πenaα/ra.
The ideality criterion can be written as
γai = 2παe2naβ/ra 1,	(1.10)

where ra is the radius of the polarization interaction “cut-o ”, which is close to the atomic size. The nonideality caused by charge–neutral interaction can occur in highly polarizable gases such as metal vapors. For cesium, α = 400a30, and ra = 4a0, where a0 is the Bohr radius. In a plasma of cesium vapors at T = 2000 K, γai 0.1 as long as na 1019 cm−3.

The electron–atom interaction potential has the same polarization asymptote ϕ(r); however, this potential may not always be determined unambiguously at small distances. Full information on the electron–atom interaction is contained in the scattering phases. A set of those phases enables one to calculate the mean interaction energy using the Beth–Uhlenbeck expressions (Landau and Lifshitz 1980). This leads to rather bulky expressions. However, the electron– atom interaction at low temperatures can be described by a single parameter, namely, the electron–atom scattering length L (as long as na|L|3 1). Then, in solving a number of problems, the real potential V (r) can be replaced with a

δ–like potential
V (r) = 2π 2Lδ(r)/m.	(1.11)

In this approximation, one can readily calculate the electron–atom interaction energy:

U = na(r )V (r − r )|Ψ(r)|2drdr ,