6

ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED

PLASMA

6.1Electrical conductivity of ideal partially ionized plasma

6.1.1Electrical conductivity of weakly ionized plasma

The electrophysical properties of matter are defined primarily by the state of the electron component. During the transition of metal from the liquid to weakly conducting gas phase, the state of charged particles varies from the state of almost free electrons of liquid metal to that of electrons strongly bound in atoms. In so doing, the value of the coe cient of electrical conductivity varies by many orders of magnitude.

The electrical conductivity σ is defined by the concentration of electrons ne and their mobility ,

σ = eµne.

(6.1)

In an ideal plasma, these quantities are defined by expressions well known from kinetic theory. The concentrations of electrons, ne, and ions, ni, are related to each other by Saha’s formula (4.5). Complex ions are absent in an ideal plasma, therefore, ni = ne. At low temperatures the degree of ionization is low, ne na, and

ne = n0e = K1na. (6.2)

In a weakly ionized plasma, one can ignore electron–ion and electron–electron interactions and consider only interactions of electrons with atoms (molecules) of a neutral gas. The latter can be regarded as a sequence of independent binary collisions. Such a system is well described by the Lorentz gas model (see, for example, Lifshitz and Pitaevskii 1981).

Under steady–state and spatially homogeneous conditions, Boltzmann’s equation for the distribution function of electrons f (v) in an electric field F takes the form

−(eF/m)∂f (v)/∂v = Ic(f ).

The left–hand side of this equation describes the field e ect and the right–hand side the variation of the number of electrons in an element of phase volume due to collisions, Ic being the collision integral. In the following we use the fact that the electron mass is much smaller than the atomic mass (m M ) and assume

248

ELECTRICAL CONDUCTIVITY OF IDEAL PARTIALLY IONIZED PLASMA 249

small deviations of f from equilibrium. Then, f (v) should be close to spherically symmetric and linear on the electric field. This enables us to represent it as

f = f0 + δf,

where δf (v) = cos θf1(v) and θ is the angle between the directions of the velocity and electric field. The symmetric part of the distribution function, f0(v), under conditions of thermodynamic equilibrium is Maxwellian. The nonsymmetric part f1(v) is essential for calculating the electron mobility and hence plasma conductivity.

On substituting the expansion for f (v) in the initial kinetic equation, we obtain the equation in f0(v):

−(eF/m)∂f0/∂v = Ic(δf ).

The collision integral Ic(δf ) in these conditions can be written as (Lifshitz and Pitaevskii 1981)

Ic(δf ) = −ν(v)δf (v),

ν(v) = navq(v),

where q(v) is the transport cross–section of electron–atom scattering and ν(v) the corresponding (velocity dependent) electron–atom collision frequency.

In the electric field F the electrons, which are mainly in chaotic thermal motion, on average drift in the direction opposite to the field. The drift velocity (mean electron velocity over the time exceeding greatly the time between individual collisions) is

u = vδf (v) dv = v cos θf1(v) dv,

since f0 makes no contribution to u. The mobility is determined from the equation u = −µF. Taking the projections on the direction of the electric field and using the fact that ∂f0/∂v = −f0v/vT2e , so that

δf (v) = −	eF v cos θf0(v)	,	f1(v) = −	eF vf0(v)	,
	mvT2e ν(v)			mvT2e ν(v)

we obtain for the mobility (in the Lorentz plasma approximation)

µ =	e		v2f0(v)	cos2 θdv.
	T		ν(v)

Integrating over the angles and substituting the Maxwellian distribution for f0(v) we finally get

	1					e		∞ v4
µ =			/	2
	3			π mvT5e				0 ν(v) exp(−v2/2vT2e )dv,

where vTe = kT /m is the thermal velocity of the electrons.

250 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

This expression is valid when certain conditions are met. These include the following: The neutral gas must be su ciently rarefied for the binary collision approximation to be valid, naq3/2 1. The temperature must be su ciently high and the thermal wavelength of the electron λ su ciently small, so that one could ignore the interference of the electron on neighboring atoms (naqλ 1). The potential energy of the Coulomb interaction between electrons must be much smaller than their kinetic energy, e2n1e/3/kT 1. The plasma must be nondegenerate, 2n2e/3/mkT 1. The interatomic correlation can be neglected, nab 1, naa/kT 1, where a and b are the coe cients of the van der Waals equation of state for the neutral gas.

It is convenient to integrate over the electron energy, E = mv2/2, instead of the velocity. Then, we have for the mobility

4

e

0

∞ E3/2

µ =

3√

exp(−E/T ) dE,

(6.3)

m(kT )5/2

ν(E)

π

ν and

where ν(E) = 2E/mnaq(E). Introducing the mean (e ective) collision fre-

quency

cross–section

, one can write

q

µ = e/mν,

= 3√

(6.4)

2π/4 na

(T )vTe ,

ν

q

where

1

1

∞

E

=

0

exp(−E/kT ) dE.

( T )

(kT )2

q(E)

q

In the simplest case when the electron–neutral collision can be approximated as a scattering on a hard sphere of a diameter d, the transport cross–section is independent of energy, q(E) = πd2/4. In this case averaging over energies yields

Table 6.1 Averaged transport cross–sections of scattering of electrons from atoms of alkali metals, q(T ) in units of 102a20 (Gogoleva et al. 1984).

T , 103 K	Li	Na	K	Cs	T , 103 K	Li	Na	K	Cs
1.0	16.5	15.0	15.3	14.1	2.6	6.88	7.31	7.00	8.98
1.2	14.4	14.0	13.6	12.8	2.8	6.41	6.73	6.52	7.63
1.4	12.6	12.9	12.1	11.7	3.0	5.99	6.23	6.10	7.32
1.6	11.1	11.7	10.9	10.8	3.2	5.63	5.79	5.73	7.04
1.8	9.91	10.6	9.84	10.1	3.4	5.32	5.41	5.41	6.79
2.0	8.93	8.73	8.96	9.42	3.6	5.05	5.07	5.12	6.57
2.2	8.11	8.73	8.20	8.89	3.8	4.80	4.77	4.87	6.37
2.4	7.46	7.97	7.56	8.41	4.0	4.59	4.51	4.64	6.20

ELECTRICAL CONDUCTIVITY OF IDEAL PARTIALLY IONIZED PLASMA 251

the same value, q = πd2/4, and thus ν = (3π3/2/27/2)nad2vTe . In a more realistic situation the transport cross–section is a function of energy. If the dependence q(E) is known, the mean cross–sections q(T ) can also be readily calculated. A large amount of reference data on electron–atom and electron–molecule scattering cross–sections is available (e.g., Bederson and Kie er 1971; Kie er 1971; Gilardini 1972). Table 6.1 gives the values of averaged cross–sections for atoms of alkali metals.

One can readily perform numerical estimation of the coe cient of electrical conductivity using the following expression:

σ = 3.8 · 106	ne				1		ohm−1 cm−1 ,	(6.5)
					√
	n
		a		q		T

where q is the average cross–section in units of 10−16 cm2, and T is the temperature in K.

6.1.2Electrical conductivity of strongly ionized plasma

Equation (6.4) describes well the electron mobility until the electron–atom collision frequency ν becomes comparable with the electron–ion collision frequency νi. Let us now dwell on the peculiarities associated with the determination of νi (Spitzer 1962).

The cross–section of thermal electron scattering by an ion with charge number Z is defined by the square of the Coulomb (or Landau) length, RC = Ze2/kT . However, the Coulomb cross–section scale 4πRC2 only accounts for the collisions which lead to large–angle scattering. The specific character of the long–range Coulomb interaction resides in the prevalence of small-angle scattering. As a result, to get the transport cross–section one should multiply 4πRC2 by the so–called “Coulomb logarithm” ln Λ. The quantity Λ in a classical plasma is often defined as the ratio between the Debye radius and the Coulomb length. The energy– dependent momentum transfer cross–section in the case of Coulomb scattering

is

q(E) = π(Ze2/E)2 ln Λ.

The expression for the electrical conductivity of a fully ionized plasma (ions with a charge Z and electrons) can be readily written assuming that the ions are immobile uncorrelated scatterers while the electrons do not interact with each other. Such a model corresponds with the Lorentz gas model used above to describe the conductivity of weakly ionized plasmas. Averaging over energies with the assumption that ln Λ does not depend on energy yields for the e ective collision frequency

			π3/2			Ze2	2
		i =					nivTe ln Λ.	(6.6)
	ν		4√
						kT
				2

Using Eqs. (6.1) and (6.4) and the neutrality condition ne = Zni one can readily obtain

252 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

	√			3/2	2	1	1
		2/π						.	(6.7)
	σ = (4				)(Ze	/kT )−	vTe (ln Λ)−

The value of the Coulomb logarithm is uncertain to some extent, because variation of Λ with energy was neglected. However, the e ect of such variations is no larger than that of other terms neglected in the evaluation of the transport cross–section. Thus, one may simply write Λ rD/RC, where rD ≈ [4πe2ne(1 + Z)/kT ]−1/2 is the Debye screening radius and RC Ze2/kT . Using the plasma parameter Γ = Ze2/rDkT we may write ln Λ ≈ ln(1/Γ). The approach described here is only applicable when Λ 1, i.e., for ideal plasmas with Γ 1. In this case the numerical factor, which is sometimes present under the logarithm, e.g., Spitzer form Λ = 3/Γ (Cohen et al. 1950; Spitzer 1962), is of minor importance.

The specific nature of the Coulomb long–range interaction shows up in the strong e ect produced by electron–electron correlations on the electrical conductivity even in the range of very small values of Γ. In order to account for these correlations, the integral of electron–electron collisions must be added to the integral of electron–ion collisions in the right–hand side of the kinetic equation. Such an equation was solved numerically by Spitzer and H¨arm (1953) and corrections were calculated to the Lorentz formula (6.7). These corrections are given by the factor γE dependent on the ion charge Z:

Ion charge, Z	1	2	4	16	∞
Factor γE	0.582	0.683	0.785	0.923	1.000

The electron–electron interactions cause a decrease of the coe cient of electrical conductivity. When the electric field is applied, the velocity distribution function of electrons, which was initially spherically symmetric, stretches along the field. In opposing this, the electron–electron interactions somewhat symmetrize the electron distribution in the velocity space, thus leading to a decrease of the transport coe cients. In the range of high values of Z, the factor γE tends to unity, because the importance of electron–electron collisions decreases.

The resulting expression for the electrical conductivity of fully ionized plasma, usually referred to as Spitzer’s formula, is

	√			3/2	2	1	1
		2/π						.	(6.8)
	σSp = (4				)(Ze	/kT )−	vTe γE(Z)(ln Λ)−

For singly charged ions (Z = 1) we have γE(1) = 0.582. Then, numerically,

	T 3/2
σSp = 1.53 · 10−4	ln Λ ohm−1 cm−1 ,	(6.9)

where the temperature is expressed in K. In an ideal classical plasma, Eq. (6.9) is asymptotically exact in the limit of very large Coulomb logarithm, ln Λ 1.

At high temperatures, the Coulomb length, RC = Ze2/kT , becomes comparable with the proper radius of a complex ion. In such a case the scattering

ELECTRICAL CONDUCTIVITY OF IDEAL PARTIALLY IONIZED PLASMA 253

process becomes non–Coulomb. If an ion is fairly heavy (for example, an ion of xenon), its radius is large and, for T ≈ Ry, non–Coulomb corrections are significant (Podlubbnyi and Rostovskii 1980).

A plasma may be regarded as strongly ionized if the electrons collide more frequently with the ions rather than with the atoms. Equating the frequencies of these collisions, we have for singly charged ions

na		(T ) (e4/k2T 2)ne ln Λ.	(6.10)
	q

For example, if T 1 eV, q(T ) 10−16 cm2, and ln Λ 5, then electron–ion collisions dominate at an ionization fraction greater than 10−3, i.e., already at relatively small ionization fraction. This is because the Coulomb cross–section for electron–ion collisions is appreciably larger than that for electron–neutral collisions.

In Fig. 6.1 the ρ − T diagram for cesium is divided into a number of regions, with every one of those regions having a di erent charge transfer mechanism. Equation (6.10) describes the curve separating the regions VII and VI – the regions of strongly and weakly ionized plasma, respectively.

The electrical conductivity of the plasma in the region of intermediate degrees of ionization can be calculated using the Chapman–Enskog method of successive approximations. In the first approximation, the frequency ν(E) is averaged in the expression for mobility, instead of ν−1(E). In this way the expression for the characteristic frequency of momentum transfer of the electron gas in collisions

with ions		√				2	2
	4		2π		Ze
ν˜i =							nivTe ln Λ	(6.11)
		3			kT

appears instead of the Lorentz frequency νi given by Eq. (6.6).

The Chapman–Enskog method is characterized by poor convergence if the q(E) cross–section depends nonmonotonically on E. In view of this, a number

	of interpolation formulas were proposed of the type of the Frost formula:
		4			e2ne	∞	E3/2	exp −	E	dE.
	σ =	3√				0					(6.12)
					m(kT )5/2		ν(E) + νi(E)/γE		kT
			π

Let us briefly explain the structure of the interpolation formula (6.12). It is based on the Lorentz approximation rather than on the Chapman–Enskog series. If one could ignore the electron–electron correlations, then, in order to determine the e ective frequency of electron collisions, it would be su cient to sum the frequencies of collisions with atoms and ions, ν(E) + νi(E). The inclusion of the factor γE in expression (6.12) renders the latter exact in the weakly ionized and fully ionized limits and gives adequate interpolation to the intermediate region. Note that it follows from Eq. (6.12) that in ideal plasmas Spitzer’s values of electrical conductivity at a fixed temperature limit the conductivity from above,

σ(T ) ≤ σSp(T ).

Fig. 6.1.

254 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

cm

ρ −T diagram for cesium (Alekseev and Iakubov 1986): Region I represents liquid metal; II – metal–dielectric transition; III – nonideal fully ionized plasma; IV – nonideal plasma of metal vapors (weakly ionized plasma); V – weakly nonideal strongly ionized plasma; VI – weakly nonideal plasma of metal vapors; VII – ideal strongly ionized plasma.

Figure 6.2 plots the electrical conductivity of the plasma as a function of temperature. At high temperatures, plasmas of various compositions lose their individuality, and all σ(T ) curves reach Spitzer values. They remain close to each other until the temperatures are reached for which the second ionization becomes important. At low temperatures, in a weakly ionized plasma the parameter I/T , which determines the degree of ionization, is of principal importance. This is the reason for the high values of σ for cesium plasma and a sharp increase in the value of σ for hydrogen plasma when the latter is seeded with cesium.