156	WEAKLY IONIZED PLASMAS

ne , cm−3

1017 Cs

Rb

K

1015

Na

1013

1011

109

1.0 1.2

1.4

1.6 T, 103 K

Fig. 4.12. Calculations of ne(T ) for saturated vapors of alkali metals by Pogosov and Khrapak (1988). The broken line shows nes(T ) for cesium.

n+g / n+g*

1.0

0.5

T= 1400 K

1800

0

10

20

30

40

50

60

g

Fig. 4.13. Distribution of Cs+g clusters over the number of bound atoms g (Pogosov and Khrapak 1988).

4.5Metallization of plasma

4.5.1Mott’s transition

A continuous decrease in the density of metals during their heating causes, in the long run, a sharp decrease in electrical conductivity. Reference is usually made to the metal–dielectric transition or to Mott’s transition (Mott and Devis 1971). Mott and Hubbard have demonstrated that the metal–dielectric transition is caused by the electron–electron interaction, which leads to band splitting. As a

METALLIZATION OF PLASMA

157

σ, ohm−1 cm−1

102

102

102

10−1

10−2

10−3

10−4

10−5

1.4	1.6 T, 103 K

Fig. 4.14. Specific electrical conductivity of saturated cesium vapors (Pogosov and Khrapak 1988). The solid curve represents the droplet model, broken curve is for ideal plasma, points are the experimental data.

result, the empty and filled bands become separated by an energy gap.

The initial Mott estimation of the parameters of the dielectric–metal transition is very simple: The spectrum of bound states of the shielded potential V (r) = −(e2/r) exp(−r/r0) is investigated, and it is demonstrated that the bound states disappear completely when r0 decreases to a value close to

rM = 0.84a		.	(4.51)
0	0

Depending on the absence or presence of electron degeneracy, r0 is the Debye– H¨uckel or Thomas–Fermi screening radius.

Unlike a phase transition, Mott’s transition in a metal occurs at high temperature. Therefore, the transition is smeared out and has no critical point. Upon the expansion of a metal, the electrical conductivity falls o sharply but continuously. Mott has introduced the concept of the minimum electrical conductivity of a metal, which is estimated from the condition that the mean free path of the electron is equal to its wavelength. Now, it is commonly assumed that this value is approximately 200 ohm−1cm−1.

Let us discuss the transition from a di erent point of view, namely, as the metallization of the plasma due to the compression. The increase of the electrical conductivity upon compression is due to a decrease in the ionization potential. The rate of this decrease rises during compression. It is often suggested that nonthermal ionization or ionization by pressure occurs. These assumptions, however, remain valid as long as one can speak about the individual atoms as well as bound and free electrons. At higher values of density, the shells of the ground atomic

158	WEAKLY IONIZED PLASMAS

states overlap. The description of such a system requires a di erent approach developed by Likal’ter (1982, 2000); see, also, Vorob’ev and Likal’ter (1988).

4.5.2Quasi–atomic model

Atoms of a metal come so close together that the classically accessible regions of motion of valence electrons start overlapping (the radius of such a region is e2/I). Moreover, the classic transition of electrons from one atom to another becomes possible. The quasi–atomic state is formed when the density reaches a value corresponding to the percolation threshold of

ζ0 = (4π/3)(e2/I)3ni 1/3.

At this point, electrons can cross the entire metal sample, passing from one ion to another. Each quasi–atom occupies a spherical cell with a volume of n−i 1. A valence electron spreads out over its own cell and, to some extent, over the neighboring cells. Each ion is mainly shielded by its own electron and, partly, by other electrons as well. This constitutes the essential di erence from the liquid– metal state, when the entire plasma volume is accessible to free electron motion. Roughly speaking, this is realized when the close packing of accessible regions is achieved, that is, at ζ0 = 0.74. Therefore, the discussed region is narrow with respect to density (approximately from one–half to two times the value of the critical metal density nc), but important changes occurring there govern the critical state of matter.

Because βI 1, it follows from the foregoing that the range of validity of the theory corresponds to the condition of strong nonideality, γ 1. These are the conditions in the neighborhood of the critical points of a number of

metals. In cesium, for example, the total ionization is achieved, ne = nc, so that

γc = e2n1c/3β 10.

The plasma energy is the sum of the interaction energy, ∆E, and the kinetic energy of ions and electrons,

E = ∆E + cv nkT = −(1/2)nI − αe2n4/3 + (3/2)nkT,

(4.52)

where n = ne +ni. The term −nI/2 in ∆E is peculiar to the quasi–atomic model. This suggests that the renormalization of the Madelung constant α is required. The latter is usually calculated for a system of ions against the background of uniformly smeared electrons and is equal to 0.57. In our case, the value of α should be smaller. It takes into account the electrostatic interaction between an ion and the periphery of the cell containing the electric charge brought about by “extraneous” electrons. According to the estimate by Vorob’ev and Likal’ter (1988), the Madelung constant in (4.52) is α 1/4.

Based on the model discussed above, one can write the expression for free energy and derive the equation of state,

p = nkT − αe2n4/3/3.

(4.53)

METALLIZATION OF PLASMA

159

This equation reveals thermodynamic instability. The finite compressibility of the liquid phase is included in (4.53) in Van der Waals’ approximation,

p = nkT (1 − nb)−1 − αe2n4/3/3.

(4.54)

The parameter b corresponds to the minimum volume per particle and is defined by the quantity I, via b (e2/I)3.

4.5.3Phase transition in metals

The degrees of ionization of a number of metals in the neighborhood of the critical points are so high that plasma interactions are predominant. Therefore, a number of researchers suggested that the phase transition in its high–temperature region might be due to plasma interactions, i.e., be a “plasma” phase transition. More generally, the issue of plasma phase transition is discussed in Section 5.8.

Equation (4.54) provides a qualitative description of the phase transition in metals and helps to estimate the critical parameters,

nc = (1/7)b, Tc = (48/77/3)(a/b1/3), pc = 7−7/3(a/b4/3),

(4.55)

where a = αe2/3. Parameter a, which describes attraction, is universal for all monovalent metals. It is only parameter b that is specific. This is an important peculiarity of the equation of state (4.54), which distinguishes it from the Van der Waals’ equation. The similarity relations for the critical parameters (Likal’ter 2000; Vorob’ev and Likal’ter 1988) follow from Eq. (4.55),

nc I3, Tc I, pc I4.

The theory was generalized by Likal’ter (1985) to metals with Z valence electrons, which yielded more general similarity relations,

nc (Z + 1)I3, Tc	Z2	(4.56)
	Z + 1 I, pc Z2I4.

For alkali metals, Eq. (4.55) yields fairly good estimates. For instance, for cesium with α = 0.21 (Likal’ter 2000) the deviation of the calculated critical temperature from the experimental value (J¨ungst et al. 1985) is about 15%, and the deviation of pressure is about 5%. One can see from the experimental data shown in Table 4.12, that the similar correspondence holds for other alkali metals (with the only exception being the sodium density, which has an unreliable experimental estimate). One can conclude that the Van der Waals’ equation provides a rather good description of the major properties of “good” metals in the vicinity of the vapor–liquid phase transition.

The similarity relations (4.56) make it possible to express the critical parameters of a number of metals (which maintain the metallic state in the critical point) in terms of the parameters measured for one of them. It is natural to use the critical parameters of cesium for this purpose. By taking into account that

160	WEAKLY IONIZED PLASMAS

cesium is monovalent, and using a bar to indicate the parameters related to the analogous parameters of cesium, we have

¯	Z2	¯	2	¯4
Tc =	Z + 1	I,	p¯c = Z	I	.	(4.57)

The compressibility at the critical point is (Likal’ter 2000)

7

zc = pc/(nckTc) = 48 (Z + 1).

The value of this factor is approximately equal to 0.29, which is about 45% larger than the experimental value 0.2 for alkali metals (J¨ungst et al. 1985). Likal’ter (1996) proposed to use a three–parameter modification of the Van der Waals’ equation, which allowed him to approximate critical values of three thermodynamic variables and make direct comparison of theoretical estimates of the hard–sphere radius and the Madelung constant with the experimental values. By replacing the Van der Waals’ denominator with the fourfold excluded volume, b = 4η, with the Carrnahan–Starling (1969) function,

	F (η) =	1 + η + η2 − η3	,	(4.58)
		(1 − η)3

and taking into account the next after the Madelung term in the expansion of the interaction energy, the following equation of state was derived:

π = 10ν[τ F (η) − A0η1/3 − B0η−1/3],

(4.59)

where A0 = 2.854, B0 = 0.03643, and π = p/pc, ν = ni/nic and τ = T /Tc are the reduced pressure, number density, and temperature.

Critical parameters determined from the equation of state (4.59) can be expressed in terms of the functions I and Z used for the Van der Waals’s equation, but with prefactors normalized by the parameters of cesium at the critical point (Likal’ter 2000),

Tc	0.085 Z2I	, pc 0.405	· 105Z2I4,
	k Z + 1			(4.60)

nic 2.92 · 1019I3, Zc = 0, 1(Z + 1),

where I and T are measured in eV, pc is in Pa, and nic is in cm−3. Similarity relations (4.60) allow us to estimate critical parameters without performing detailed calculations for a particular metal.

The range of validity of the theory is discussed by Vorob’ev and Likal’ter (1988). As mentioned above, this range is not wide. Nevertheless, it has been verified that a number of metals satisfy the condition of the metallic state at the critical point. These include all alkali metals, copper and silver of the noble

	METALLIZATION OF PLASMA			161
	π
102		Ni
		Cr B	Ni
	Cu
	Al	Cu U
		U

101

Li

Na

K

1

10

τ

Fig. 4.15. Relative critical pressure of metals π versus relative critical temperature τ : points indicate the data from a number of works (see references in the work by Likal’ter 1985) normalized by the critical parameters of cesium.

metals, beryllium of the elements of Group II, and all elements of the main subgroup of Group III, including aluminum. The transition metals include all elements found in the iron group, some elements in the palladium and platinum groups, including molybdenum, and uranium. However, there is no sharp dividing line between the di erent types of states. In a number of metals, which are not on this list, the interaction at the critical point is not of a pure Coulomb type, but makes a transition to the Van der Waals interaction due to the mutual polarization of atoms. This transition is terminated in the case of uncommon metals, such as mercury, arsenic, bismuth, and tellurium.

The critical temperature and pressure of some simple and transition metals, as calculated with formulas (4.57) and (4.60) on the basis of the critical parameters of cesium, are given in Table 4.12. Figure 4.15 shows the relative critical pressure as a function of the relative critical temperature. These quantities correspond to formula (4.56) if the values of Tc and pc are normalized by the corresponding values known for cesium. The estimates of the critical parameters, available from the literature and made on the basis of experimental data and semiempirical relations, correlate well with the dependencies given above. One can conclude that metallic states of this type with Coulomb interaction at the critical point are quite typical for metals.

162

			Table 4.12 Critical parameters of metallic liquids (Likal’ter 2000)
	Metal	Tc, K	pc, bar	ρc, g cm−3	Method	Reference
		1600	87	0.3	Van der Waals’ equation	Likal’ter and Schneidenbach 2000
	Cs (6s)	2000	460	0.55	Partially ionized gas	Redmer 1997
		2350	60	0.47	MC simulations	Chacon et al. 1995
		1924	92.5	0.38	Experiment	J¨ungst et al. 1985	WEAKLY
	Rb (5s)	2475	73	0.35	MC simulations	Chacon et al. 1995
		1720	115	0.24	Van der Waals’ equation	Likal’ter and Schneidenbach 2000
		2200	650	0.45	Partially ionized gas	Redmer 1997	IONIZED
		2060	123	0.3	Scaling with cesium normalization	Likal’ter 1996
		2017	124	0.29	Experiment	J¨ungst et al. 1985
		1790	134	0.12	Van der Waals’ equation	Likal’ter and Schneidenbach 2000
		2350	690	0.21	Partially ionized gas	Redmer 1997	PLASMAS
	K (4s)	2550	70	0.22	MC simulations	Chacon et al. 1995
		2140	144	0.16	Scaling with cesium normalization	Likal’ter 1996
		2178	150	0.17	Experiment	Hensel et al. 1991
		2115	263	0.12	Van der Waals’ equation	Likal’ter and Schneidenbach 2000
		2400	1400	0.27	Partially ionized gas	Redmer 1997
	Na (3s)	2970	128	0.22	MC simulations	Chacon et al. 1995
		2535	282	0.15	Scaling with cesium normalization	Likal’ter 1996
		2485	248	0.30	Experiment	Hensel et al. 1991

						Table 4.12 (continued )
	Metal		Tc, K	pc, bar	ρc, g cm−3	Method	Reference
	Al (3s2 3p)		3700	530	0.042	Van der Waals’ equation	Likal’ter 2000	METALLIZATION
			3500	770	0.025	Partially ionized gas	Redmer 1997
	Li (2s)
			2660	342	0.053	Scaling with cesium normalization	Likal’ter 1996
			3225	690	0.1	Extrapolation	Fortov et al. 1975
	Cu (4s2)		7620	5770	1.4	Scaling with cesium normalization	Likal’ter 1996
			8390	7460	2.4	Extrapolation	Fortov et al. 1975
			8860	4680	0.28	Scaling with cesium normalization	Likal’ter 1996
			8000	4470	0.64	Extrapolation	Fortov et al. 1975	OF
	Be (2s2)		9200	12200	0.35	Scaling with cesium normalization	Likal’ter 1996
								PLASMA
			8100	11700	0.55	Extrapolation	Fortov et al. 1975
	U (7s2	7p)	9000	5000	2.6	Scaling with cesium normalization	Likal’ter 1997
			11600	6100	5.3	Extrapolation	Fortov et al. 1975
	La (6s2 6p)		8250	3500	1.2	Scaling with cesium normalization	Likal’ter 1997
			11000	3500	1.8	Extrapolation	Fortov et al. 1975
	Y (5s2	5p)	9500	6000	1.1	Scaling with cesium normalization	Likal’ter 1997
			10800	3700	1.3	Extrapolation	Fortov et al. 1975

163