the strong polarization interaction with neutral atoms, by the quantity ∆I1 = na 2√πq/2m. Here, q is the total electron–atom scattering cross–section and na is the atomic concentration (see Eq. 1.12). Second, the ion energy in a gas is also reduced by the quantity ∆I2 = 2πnaαe2/ra due to the polarization of the neighboring atoms (see Eq.1.10). The decrease of ionization potential leads to an exponential growth of the electron concentration with density (in this section, we take the absolute value of ∆I). Saha’s equation gives

Let us assume that the electron mobility has a weak dependence on density. Then the electrical conductivity, following variations of ne, grows exponentially with density,

Here na is the density value starting from which this growth becomes substantial. These formulas yield the following expression for na:

(na)−1 = (2mkT 2/√πq)−1 . (4.10)

The calculated na (4.10) is in good agreement with the experimental value, na 1.7 · 1020 cm−3 (α = 400 a30, ra = 5a0, q = 2 · 10−13 cm2). Smirnov (1970) and Popielawski and Gryko (1977) independently drew the same conclusion.

It should be noted that the mean field approximation used by these authors to calculate ∆I ignores the correlation of atoms in the neighborhood of charged particles. This is valid if the depth of the charge–neutral interaction potential, g, is not too large. However, the value of g/kT is in fact much larger than unity. For estimation, one can use D2+ (the bond energy of Cs+2 or Hg+2 ions) or the polarization well depth αe2/(2ra4). It is easy to see that the correlation is substantial.

4.2.3Charged clusters

Above, we have already discussed the necessity to take into account the correlation of atoms around charged particles. Indeed, in a dense medium the state of charged particles experiences radical changes. Because of attractive forces acting between the charged and neutral particles, charged clusters are formed. The clusters are stabilized due to the presence of charge. The structure of clusters depends strongly on the degree of nonideality. Let us follow the variation of the cluster structure in a vapor upon isobaric heating. On the saturation line (T = Ts) one observes charged droplets. For temperatures somewhat higher than Ts, these are less dense formations which can no longer be regarded as liquid but which are still in dynamic equilibrium with the vapor. At substantially higher temperatures, these are cluster ions containing several atoms and are bound by polarization forces. Finally, in a weakly nonideal plasma the cluster ions dissociate and there remain ordinary molecular ions. The existence of clusters might strongly a ect the properties of matter. For instance, in saturated cesium vapors

the thermal emission of electrons from droplets raises the degree of ionization and the electrical conductivity of the plasma by several orders of magnitude. This is responsible for the anomalously high electrical conductivity of high–temperature metallic vapors near the saturation line.

The clusters and similar particles were observed in di erent temperature ranges in numerous gases and liquids, see review by Khrapak and Yakubov (1979, 1981) and Khomkin et al. (1995). Based on the review by Lagar’kov and Yakubov (1980), we shall discuss below the ionization equilibrium in a metal vapor plasma.

We shall start with the simplest case. Let us consider a plasma consisting of electrons, atoms A, ordinary ions A+, and molecular ions A+2 . The electron density is defined by Saha’s equation

In Eqs. (4.11)–(4.13), n+, ne, na, n+2 stand for the concentration of ordinary ions, electrons, atoms and molecular ions, respectively, λe and λa are the thermal wavelengths for the electron and atom, and D2+ is the bond energy of the atom in the ion A+2 . Let us assume that the degree of ionization is low. Then, by excluding n+2 and n+ from (4.13) with Eqs. (4.11) and (4.12), we obtain

where k1 is the constant of ionization equilibrium in A A++ e and k2 the constant of dissociation equilibrium in A+2 A+ + A. For cesium, k2−1 = 1.45 · 1021T −1/2 cm−3 where T is expressed in eV and D2+ 0.7 eV. It follows from Eq. (4.14) that the inclusion of the molecular ions led to a shift of ionization equilibrium towards increased electron concentration. The higher the molecular ion dissociation energy, the larger is the shift. Under conditions when T = 1400 K and na = 1020 cm3 (near the saturation point for cesium vapors at p = 2 MPa), the inclusion of Cs+2 ions gives an approximately fivefold increase of ne. The electrical conductivity increases accordingly. The important role played by molecular ions in alkali plasmas was confirmed by measurements of the electrical conductivity of sodium vapors near the saturation line by Morrow and Craggs (1973).

The inclusion of Cs+2 ions is not su cient at high pressures and low temperatures, because under these conditions we have nak2 exp(D2+β) 1. This means that n+2 n+. One can naturally expect that a substantial number of Cs+3 ions is present in a plasma. Similarly to Eq. (4.13), for the A+3 A+2 + A equilibrium

we have the equation n+3 = n+2 nak3 exp(βD3+). Taking these ions into account, we obtain for n2e instead of Eq. (4.14),

Evidently, as the formation of more and more complex ions (referred to as cluster ions) is taken into account, subsequent terms will appear in the square brackets in Eq. (4.8). Assuming that D3+ = D2+ and k3 = k2/2, the right side of Eq. (4.15) can be regarded as the beginning of a series whose sum is expressed by the exponent. Such summation yields (Likal’ter 1978)

One of the first papers containing the discussion of heavy cluster ions and the summation similar to that performed above, was the paper by Magee and Funabashi (1959).

The procedure of summation in Eq. (4.15) is heuristic. Indeed, even the third term in the series, calculated with due regard for the real values of D3+ and k3, does not correspond to the third term of the exponential series. Nevertheless, Eq. (4.16) represents the major features of the phenomenon. It describes the presence on the isobar of a sharp temperature minimum of free electron concentration and, thereby, of conductivity. Indeed, if σ ne, then

where σ0 is some constant. The position of the minimum of electrical conductivity on the isobar approximately corresponds to the value of

Tmin D2+/k ln(nak2D2+/I) .

If p = 2 MPa, then Tmin = 1500 K for cesium, which is in good agreement with experiment. The emergence of the minimum of electrical conductivity is due to the competition between the dissociation of cluster ions and the thermal ionization of atoms.

4.2.4Thermodynamics of multiparticle clusters

As the nonideality becomes stronger, the number of particles in the cluster increases. Therefore, we shall approach the formation of ion clusters from the standpoint of macroscopic thermodynamics. Let an ion be placed in a gas with a mean atomic concentration na and temperature T . Then, if the density of particles attracted by the ion reaches a value corresponding to the loss of thermodynamic stability, the vapor–liquid phase transition occurs in the neighborhood of the ion, in the r < R region. Let us now estimate the value of R. In a steady state, the gradient of the vapor pressure in the neighborhood of the ion must be

balanced by the force of attraction of an atom by the ion located at the origin of coordinates,

where V (r) is the potential of the ion–atom interaction and M the atomic mass. Given the equation of state for the vapor, one can integrate Eq. (4.18) to find the density distribution na(r) in the neighborhood of the ion,

In an ideal gas, Eq. (4.19) gives the obvious result,

na(r) = na exp[−βV (r)].

Near the saturation line the interatomic attraction is substantial, p = nakT −an2a, where a is the parameter characterizing the attraction in van der Waals’ equation. In this case, Eq. (4.19) gives, after the integration,

For na = 1020 cm−3 and T = 1400 K at a distance from the ion equal to R = 17a0, the density na(R) exceeds the mean density by a factor of two. The number of particles in a drop of the calculated radius is close to 20. As the number of atoms in a complex increases, the valence electrons will be located on molecular orbitals around the entire complex and will form a band whose width increases as yet another atom is added.

Properties of an ion cluster containing a few dozens of particles are close to metallic. On the other hand, of course, it might still be far from having all the inner properties of a large metal sample (Montano et al. 1986). If equal numbers of positive and negative ion clusters were formed, no anomalous growth of free electron concentration would be observed. In fact, the electron–atom interaction is weaker than the ion–atom interaction (although their polarization asymptotes are identical). This is reflected even by the fact that the bond energy of a positive ion D2+ is substantially higher than that of a negative ion E. For cesium, D2+ 0.7 eV and E = 0.47 eV, and for mercury the bond energy of the ion is D2+ 1 eV while the mercury atom does not bind an electron at all. Therefore, it turns out that the number of positive ion clusters in a plasma is much higher than the number of negative clusters. Consequently, a large number of free electrons is present in the plasma. The electronic transport coe cients are proportional to the free electron concentration and, therefore, they are very sensitive to the concentration ratio between the positive and negative clusters.

Let us now consider how the emergence of ion clusters shifts the ionization equilibrium. We follow a simple model described by Khrapak and Yakubov (1981). This model takes into account the ion–atom interactions and ignores the

interatomic and electron–atom interactions. Assume that F0 is the free energy of a mixture of ideal gases of electrons, atoms, and ions. The free energy of the system, F (with due regard for interaction), is lower than F0 by the quantity kT n+g, where g is the number of ions in a cluster. This quantity is close to the excess number of atoms around the ion,

By implementing for F a conventional procedure of variation with respect to the concentrations of particles involved in the ionization reactions and taking into account the quasineutrality, we obtain the equation of ionization equilibrium,

n2e = nak1 exp{−βI + na dr[exp(−βV ) − 1]}. (4.21)

Equation (4.21) includes the exponential density dependence of electron concentration, which describes the increase of ne upon compression. Of considerable importance is the presence in the exponent of strong temperature dependence,

∆I = nakT dr[exp(−βV ) − 1]. The potential V (r) can be characterized by two parameters, namely, by the (spherical) volume of the force action between the ion and atom, v0, and the potential well depth, q. At low temperatures, the principal relations are reproduced by a simple formula analogous to Eq. (4.17),

At low pressures, the densities na are low and therefore the conductivity grows monotonically with heating. At pressures of the order of p pc the nonideality leads to the emergence of a minimum of conductivity on the isobar. This occurs at temperature

As in the case discussed above, let us assume q = 0.7 eV for cesium. The radius of the ion–atom forces is assumed to be equal to 10a0 (which is close both to the polarization radius (αe2β/2)1/4 and the internuclear distance of the Cs+2 ion). Then, Eq. (4.23) gives Tmin = 1700 K for the 2 MPa isobar, and Tmin = 2500 K for the 15 MPa isobar.

4.2.5“Vapor–liquid” phase transition and metallization

It follows from diagrams showing regions of the existence of nonideal plasma of alkali metals and mercury (see Fig. 1.4) that these regions adjoin the high– temperature part of the two–phase region.

Yakubov 1981). Measurement results: ◦ — (Hensel and Franck 1968); • — (Kikoin and Senchenkov 1967); 1 — level of electrical conductivity for metal; 2 — electrical conductivity for ideal plasma.

If the density increases to approach the value of 3/(4πα) (where α is the polarizability in the gaseous state) the dielectric permeability increases indefinitely, which is typical for metals. Case na > 3/(4πα) corresponds to spontaneous ionization, i.e., the transition of electrons to the free or, rather, collectivized state. In other words, if na 3/(4πα), the material is a metal. The value of 3/(4πα) equals approximately 4 · 1021 cm−3 for cesium and 4 · 1022 cm−3 for mercury. These values coincide to within the order of magnitude with the observed transition values.

High–temperature vapors of some metals in the vicinity of the saturation line are nonideal because of the charge–neutral and charge-charge interactions. However, the results of measurements of p–ρ–T -dependencies for cesium and mercury, given in Figs. 2.20 and 2.21, failed to point explicitly to any peculiarity in the equation of state that would be due to the plasma e ects. Nevertheless, it was suggested by Khrapak and Yakubov (1970), and Starostin (1971) that the vapor condensation in the high–temperature region is largely due to plasma e ects. The critical parameters of cesium shown in Table 4.1 were estimated by using the methods of the nonideality description developed by Kraeft et al. (1985).

Apparently, these assumptions about the nature of phase transition in metals are valid. Then one can hardly expect that, at higher temperatures, plasma interactions will once again cause the phase separation. The interesting possibility of a new “plasma” phase transition was discussed by Norman and Starostin (1970) (see Chapter 5 below).