266 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

interatomic correlation and ignoring the latter. The scatterers were assumed to interact as hard spheres of radius δ. Interatomic repulsion limits the ρ(E) tail.

6.3.2Electron mobility and electrical conductivity

The linear response theory permits writing exact initial expressions for the kinetic coe cients. In particular, as long as the linear relationship between current and field is valid (Ohm’s law), the electrical conductivity is expressed in terms of a correlation function of currents and, hence, velocities. Neglecting quantum e ects we have

σµν (ω) = (nee2β) vν (0)vµ(t) exp(−iωt − δt)dt, δ = +0. (6.29)

Here, ne is the concentration of electrons, ω the external field frequency, andvν (0)vµ(t) is the time autocorrelator of electron velocities. The averaging is performed over the electron velocity distribution and over the position of scatterers,

. . . = A	dv exp	− 2			dR1 . . . dRN Ω−N exp (−βVa) × (. . .),
			βmv2

where A is the normalization constant of averaging. The expression for static electrical conductivity σ ≡ σxx(0) can be rewritten as

∞
σ = (e2/m) −∞ ne(E)τ (E)dE.	(6.30)

Here, τ (E) is the autocorrelation time of an electron with energy E (in a medium of massive scatterers the electron energy can be regarded as constant). The value of τ (E) is given by the following expression:

∞	ϕE (t) = vx(0)vx(t) / vx(0)2	,	(6.31)
τ (E) = −∞ ϕE (t)dt,
	0	1

where ϕE (t) is the autocorrelation function of the electron velocity, and vx is the velocity component in the direction of electric field.

In a rarefied gas, τ (E) is inversely proportional to the free path between two successive collisions of the electron with atoms. The integration range is restricted by positive energies. Therefore, Eq. (6.30) reduces to Lorentz’s formula. It is only for very fast electrons that an ideal–gas approximation of the free time of flight is valid. In a dense system, electrons with di erent energies make substantially di erent contributions to the electrical conductivity. Let us discuss

these contributions and follow Hensel and Franck (1968).

√

If naW0/kT 1 most of the conduction electrons are not free even if their energy is positive. They are constantly in the field of scatterers. This field is an alternation of “wells” and “hills” with the most probable depths (heights) close

CONDUCTIVITY OF NONIDEAL WEAKLY IONIZED PLASMA

267

√

to naW0 and space dimensions of the order of l. The value of l corresponds to

the correlation length of the field of scatterers; it is close to the e ective range

√

of the potential V (r). Therefore, slow electrons with E < naW0 travel with velocities v (naW0)1/4/√m which exceed thermal velocities. The electrons are scattered from density fluctuations. Apparently, τ is close to the time of flight of the field correlation length, i.e., close to l/v.

The electrons of the ρ(E) tail make no contribution to static conductivity

because they correspond to low–mobility clusters. However, electrons with nega-

√

tive energies of lower absolute value, |E| < naW0, may be conduction electrons. The point is that field fluctuations lead to the emergence of “percolation channels”. The possibility of formation of percolation channels permeating the entire macroscopic volume of the medium is a sharp, practically step, function of energy (Shklovskii and Efros 1984). The minimum energy, starting from which the electrons contribute to conductivity, is referred to as the percolation energy EP. Let us now determine EP for a classical particle in an arbitrary pattern of potential energy U (r). It is equal to such a minimum value of energy at which it is

still possible to find a region in space in which U (r) < EP and which permeates

√

the entire plasma volume. Numerical modelling shows that EP −0.3 naW0.

Consequently, along with electrons of positive energies, electrons with negative

√

energies (E > EP) are conduction electrons. If naW0/kT 1, it is the electrons in the percolation energy range which make the major contribution to

σ.

The most detailed information on the electron dynamics is contained in the correlation function ϕE (t). It is only for fast electrons that ϕE (t) = exp(−t/τ ) where τ = ν−1 and ν is the frequency of successive electron–atom collisions. The correlator ϕE (t) in the case of low positive energies decreases nonexponentially, but under the Gaussian law ϕE (t) = exp(−t2/τ 2), where τ = l/v. Finally, the correlator of trapped electrons oscillates, ϕE (t) = cos ω0t, where ω0 is the natural frequency of electron oscillation in the potential well of the cluster. This gives the autocorrelation time τ close to zero.

The peculiarities of the behavior of the velocity autocorrelation function for di erent values of relative energy ε (calculated from the mean field level and normalized to the potential depth g) are well defined in Fig. 6.8, taken from Lagar’kov and Sarychev (1975). These results were obtained from molecular dynamics simulation of the electron dynamics. At positive energies, the electron is free and ϕε(t ) decays monotonically (t is dimensionless time). With an energy decrease, the function ϕε(t ) acquires a minimum: The electron is as though trapped until it finds a passage (percolation channel) in a field of complex shape. In these conditions the autocorrelation time τ (ε) is still substantially di erent from zero. With further decrease of energy, the electron becomes localized.

Figure 6.9 illustrates the di usion coe cient of electrons in the same medium, D = 0∞ vx(0)vx(t) ε dt. This quantity goes to zero at the energy correponding to the percolation threshold.

Based on ρ(E) and D(E), Lagar’kov and Sarychev (1979) calculated from

268 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

.

.

Fig. 6.8. Autocorrelation function of electron velocity ϕ vs. dimensionless times t for several values of relative energy ε (Lagar’kov and Sarychev 1975) and na = 4.8 · 1021 cm−3.

Eq. (6.30) the conductivity of mercury plasma in a wide range of temperatures and densities (Fig. 6.10). Good agreement is observed both with experiment and with qualitative assumptions on the initial stage of the “metal–dielectric” transition.

6.4The thermoelectric coe cient

In the presence of a temperature gradient, Ohm’s law assumes the form

F + (1/e) µ = (1/σ)j + S T.

(6.32)

The left–hand part includes µ, the gradient of the chemical potential of electrons. The thermoelectric coe cient S appears in the right–hand part.

In a Lorentz gas, the thermoelectric coe cient is calculated analogously to the coe cient of electrical conductivity (see, for example, Section 6.1 and Lifshitz and Pitaevskii 1981). The electron distribution function is f = f0 + δf , f0 exp [(µ − E)/kT ], where E = mv2/2 is the electron energy and δf is a small correction which is linear on the field and gradients of the chemical potential and temperature. Proceeding from the kinetic equation

eF ∂f0

∂f0

−

+ v

= −ν(v)δf,

(6.33)

m

∂v

∂r

we get for δf

δf =

f0

(eF + µ)v + f

µ − E

v T.

−kT ν(v)

0 kT 2ν(v)

The thermoelectric coe cient is calculated from the coe cient in the equation j = −sσ T at eF + µ = 0. The current is

THE THERMOELECTRIC COEFFICIENT

269

Fig. 6.9. Energy–dependent electron di usion coe cient D(ε) (Lagar’kov and Sarychev

1975). Designations are the same as in Fig. 6.8; na = 4.8 · 1021 cm−3.
		j = −ene					vδf dv.
Finally, we get for the thermoelectric coe cient
S = eT µ −			0∞		E3/2	/ν(E)		exp(−βE)dE .	(6.34)
1			∞	+	E5/2	/ν(E) exp( βE)dE
		0					,	−
				+			,

In a weakly ionized plasma under conditions when I kT , the thermoelectric coe cient is mainly defined by the temperature dependence of the degree of ionization. Because µ ≈ kT ln(neλ3), then, in view of Saha’s equation (4.5), we have

S	≈	+−	a		3/2		−	const.T	, ≈ −	I/2eT.	(6.35)
		(1/eT )	I/2 + (kT /2) ln(n	λ3)
In a strongly ionized plasma, ν E−						, and equation (6.34) yields
			S = (1/eT )(µ − 4kT ).								(6.36)

This expression is valid only if Z >> 1. For small values of Z, electron–electron collisions a ect the thermoelectric electrical conductivity coe cients. A discussion of this problem may be found in Kraeft et al. (1985).

Figure 6.11 gives the results of calculation by Redmer et al. (1990) of the thermoelectric coe cient of cesium on isobars. The calculation procedure is close to

270 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

ohm cm

.

Fig. 6.10. Isotherms of electrical conductivity of mercury plasma (Lagar’kov and Sarychev 1975). Dots indicate the measurements (Hensel and Franck 1968), and curves the calculation at T = 1823 K (1), 2200 K (2), 2600 K (3), 3000 K (4), 4000 K (5), 5000 K (6), 6000 K (7), 7000 K (8), 10 000 K (9), 15 000 K (10). The dashed line defines the region with the degree of ionization is smaller than 0.1.

that developed by researchers from the Rostock group to calculate the coe cient of electrical conductivity (see Section 7.3). Also shown in Fig. 6.11 is the behavior of the asymptotes for fully ionized (6.36) and weakly ionized (6.35) plasmas. These asymptotes correlate well with the numerical calculation results. Note that the calculation covers the T ≤ 3000 K range, in which the e ect of nonideality is still fairly small. At lower temperature (at about 2000 K), a sharp increase of the absolute magnitude of S occurs, due to the dielectric–metal transition (Fig. 4.2).

We will now turn to the discussion of the thermoelectric coe cient of a nonideal plasma. In a weakly nonideal plasma, one should use the equation of ionization equilibrium, which contains the quantity (I − ∆I) instead of the ionization potential I and allows for the presence of complex ions in the plasma composition. Then, we assume, for example, that the ionization equilibrium on the subcritical isobars of metal vapors is mainly defined by the A+3 and A− ions, and derive

S = [−I − (E3 + E5 − E2)]/(2eT ),

(6.37)

where the designations of Table 4.9 are used. In Figure 4.2 results following from relations (6.36) and (6.37) are shown.

As was discussed in Section 4.2 on supercritical isobars the e ect of nonideality shows up as a decrease of the ionization potential rather than as a variation

THE THERMOELECTRIC COEFFICIENT			271
	T K
.
.	50 MPa
.
.
.
.	.
.
	mV K−1

Fig. 6.11. Thermoelectric coe cient of cesium on isobars (Redmer et al. 1990). Solid curves indicate the isobars of a partly ionized plasma; dashed curves, asymptotes (6.36); dot–dashed curves, asymptote (6.35).

of the composition of charged components of plasma. Then, we use expression (4.9) to derive

S =	−	(I	−	∆I)/2eT = [	I + kT (n	/n )]/(2eT ).	(6.38)
				−	a	a

Hence it follows that the quantity S(T ) must have a minimum on the isobars. Indeed, at low temperature |S| increases with heating, because ∆I decreases. At high temperature, however, |S| decreases inversely proportional to T . The existence of a minimum of S was predicted by Alekseev et al. (1970),

|S|min ≈ 1 mV/K.

Expression (6.35) may be transcribed to

S = −∆E/(2eT ),

(6.39)

where ∆E = 2kT 2(d ln σ/dT ) is the temperature coe cient of conductivity. The quantity ∆E is sometimes referred to as the “transport energy gap”. One can use formula (6.39) to obtain S if σ is known. In Figure 6.12, this is done for cesium. At low temperature, the thermoelectric coe cient of liquid cesium is proportional to temperature, S T /eEF, because d ln σ/d ln EF is almost independent of temperature (Lifshitz and Pitaevskii 1981). At high temperature, in accordance with (6.35), the thermoelectric coe cient of a weakly ionized plasma is

272 ELECTRICAL CONDUCTIVITY OF PARTIALLY IONIZED PLASMA

µV K−1

MPa

.

.

.

103 K

Fig. 6.12. Isobars of the thermoelectric coe cient for cesium, calculated by Alekseev et

al. (1970).

inversely proportional to T . The intermediate region, in which the thermoelectric coe cient varies sharply, corresponds to the metal–dielectric transition.

The pattern described above corresponds to the measurements results shown in Fig. 4.2. The subcritical isobars (2 and 6 MPa) are shown in this drawing in accordance with the theory of the metal–dielectric transition discussed in Section 4.5 (Iakubov and Likalter 1987).

Lagar’kov and Sarychev (1979) studied the thermoelectric coe cient using molecular dynamics simulation. The general expression

S = (1/eT ) µ −	∞ ρ(E)τ (E)E exp(−βE)dE
	−∞∞	ρ(E)τ (E) exp(	−	βE)dE
	−∞

allows one to calculate S provided ρ(E) and τ (E) are known. Figure 6.13 shows supercritical isobars of thermoelectric coe cient for mercury calculated in this way. Good agreement with experimental data (Alekseev et al. 1976; Schmutzler and Hensel 1973) is observed.

It is known that the thermoelectric coe cient is very sensitive to slight variations of the state of substances. Therefore, the value of the thermoelectric co- e cient may su er drastic variations in the neighborhood of the critical point (at ρ ρc). Neale and Cusack (1979) registered such variations on the 170 MPa isobar for mercury.

Note that many of the interesting problems associated with the behavior of the thermoelectric coe cient in the region of the metal–dielectric transition