190 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

ring approximation in the grand canonical ensemble of statistical mechanics is convenient (Gryaznov and Iosilevskii 1976).

In the ring approximation, the expression for the thermodynamic potential of a plasma in the grand canonical ensemble takes the form (Gryaznov and Iosilevskii 1976)

where ξ0k = λ−k 3 exp(µk β) is the activity and Zk is the charge of the kth ion, and f = e2β is the Coulomb scattering amplitude. Using the relation

one can readily obtain in the case of single ionization,

where N0 is the neutral number density and α(2/Γ) is the positive root of the equation

The correction to the enthalpy, β∆H, and the reduction of the plasma ionization potential, ∆I, in this approximation take the form

The ring approximation in the limit Γ → 0 has the correct asymptotes, and for high Γ it possesses acceptable extrapolation properties which permit the description of shock–wave compression experiments up to Γ 2.5. This is the reason why model (5.50)–(5.55) was used in the engineering calculations of the thermophysical properties of working media in a gas–phase nuclear reactor (Gryaznov and Iosilevskii 1976).

5.3Pseudopotential models. Monte Carlo calculations

Perturbation theory methods, as discussed in Section 5.2, are based on the expansion over small parameters and therefore are valid, strictly speaking, in the limit of a weakly nonideal plasma. In order to describe the thermodynamics of substantially nonideal plasmas, the Monte Carlo technique (Zamalin et al. 1977) is e ective: it does not involve expansion over small parameters and, hence, is especially attractive in the case of dense gases and liquids as well as OCP (see

4

3

Fig. 5.8. Coulomb and exchange corrections in hydrogen plasma for kT = 50 eV (a) and kT = 200 eV (b): p3L – ladder correction (5.46); pring – ring correction (5.43); p1ex – exchange correction (5.44); p4 – total value of Coulomb corrections.

Section 5.1.2), where the form of the interparticle interaction is defined. Based on the first principles of statistical physics, this method enables one to perform direct computer calculations of mean thermodynamic values (see Section 5.1.1). It is important that, within this method, the interparticle interaction potential is assumed to be pre–assigned and, in most concrete calculations, two–particle. Therefore, all physical hypotheses in this approach refer to a particular form of interaction potential, and then the Monte Carlo technique enables one to carry out all thermodynamic calculations through to the end.

However, the application of this general method to multicomponent plasmas runs into specific di culties of taking into account – within the classical Monte Carlo formalism – quantum e ects in the electron–ion interactions at short distances. Quantum e ects play a decisive role in real plasmas, because they ensure the stability and lead to the emergence of bound states (Zamalin et al. 1977; Zelener et al. 1981; Filinov 2000; Filinov and Norman 2000).

The inclusion of the specific quantum–mechanical features within the classical Monte Carlo formalism is provided by a pseudopotential model, where the electron–ion interaction is described via the e ective potential Φ(r, T ) which differs from the initial potential Φ(r) only at small distances r λe. This di erence

192 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

.

.

.

.

.

.

Fig. 5.9. Pseudopotentials Φep and Φee (Zelener et al. 1981): 1 – T = 103 K; 2 – T = 104 K; 3 – T = 105 K. The dashed line represents the Coulomb law.

is governed by the quantum spatial uncertainty of the electron within the thermal de Broglie wavelength and by exchange e ects (for like particles), as well as by the possibility of bound state formation at low (kT Ry) temperatures. For the pseudopotential approach, the quantum partition function reduces to an expression that is classical in form, thus enabling one to employ the classical Monte Carlo calculation technique. In this case, it is possible to take into account rigorously the so–called pair quantum e ects governing the deviation of the two–particle pseudopotential from the initial potential.

5.3.1Choice of pseudopotential

Let us consider two particles with the interaction potential Φ(r). The probability density of finding these particles at a distance r in classical statistics is proportional to exp{−βΦ(r)}, while in quantum statistics it is expressed in terms of the Slater sum,

α

where ψα and Eα are the wavefunctions and the respective eigenvalues of the energy of two particles, respectively, and the summation in Eq. (5.56) is performed over all states of discrete and continuous spectra.

˜

By defining the pseudopotential Φ(r, T ) as a potential giving in the classical case the same particle distribution in space as the potential Φ(r) gives in the quantum case, we obtain

194 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

hydrogen, the following expression is valid (Zamalin et al. 1977) for the depth of the pseudopotential Φep at r = 0:

βΦep(0, T ) = − ln{π1/2ξ3[ξ(3) + βRyξ(5)] + 2π1/2ξ},

∞

ξ = 2(βRy)1/2; ξ(k) = n−k ; ξ(3) = 1.202; ξ(5) = 1.0369.

n=1

Pseudopotentials Φei constructed in this manner were calculated for a number of chemical elements and temperatures (Zamalin et al. 1977). Because of the weak dependence of Φei on the temperature and the type of chemical element, Zamalin et al. (1977) proposed a simple approximation which forms the basis of the plasma pseudopotential model of “zero” approximation (the dependence of the pseudopotential on the density was not studied),

βΦ0ee(x) = βΦ0ii(x) = x−1,

where the numerical parameter of the model, ε = 2, is selected based on experimental data. Because of the simple form of Φ0 in Eq. (5.61), this model satisfies the similarity relations, which allows one to present the results in compact form (Figs. 5.10 and 5.11) and greatly reduces the calculations. This fact simplifies implementation of the model for concrete thermodynamic calculations, where it is convenient to use approximation of Zamalin et al. (1977) with ε = 4,

Deviations between the real pseudopotential (5.60) and the zero approximation model (5.61) can then be taken into account with the thermodynamic perturbation theory developed by Zamalin et al. (1977) for arbitrary local potential.

As a whole, the range of validity of the pseudopotential Monte Carlo model proves limited because of the fact that the multiparticle interactions, degeneracy e ects, and charge–neutral and neutral–neutral interactions are ignored. Moreover, one should know the discrete energy spectrum (5.51) and (5.57), which might be distorted in a dense plasma as a result of strong interaction and, generally speaking, is not known in advance (see Section 5.4).

Iosilevskii (1980) has proposed, for describing the thermodynamics of a partially ionized degenerate plasma, a model in which the interparticle interaction takes the form

Monte Carlo calculations of the equation of state for a plasma with the pseudopotential approximation (5.61) (Zamalin et al. 1977), γ3 = Z2(Z + 1)e6neβ3: 1 – ε = 2 in (5.61); 2 – ε = 4; 3 – Debye dependence; 4 – ideal plasma

.

.

Fig. 5.11. Dimensionless plasma energy (Iosilevskii 1980) (a) and plasma pseudopotential (b). 1 – Debye approximation; 2 (points) – energy calculated using the Monte Carlo method for pseudopotential (5.61) with the depth βΦei(0) = −3 (Zamalin et al. 1977); 3

– linearized approximation (5.63)–(5.71) for di erent depths of pseudopotential (5.63); 4 – nonlinearized approximation with the depth βΦei(0) = −6 of pseudopotential (5.63); 5 – pseudopotential of hydrogen at T = 105 K (Zamalin et al. 1977); 6 – Coulomb potential; 7 – pseudopotential (5.61); 8 – pseudopotential (5.63).

196 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

Following Gryaznov and Iosilevskii (1973), we shall write the binary correlation functions in the following form:

as derived within the self–consistent ring approximation for potential (5.63) in the limit Γ → 0. The amplitude of the screening cloud, ψ0, and the screening radii, ν−1 and 1/ω, are found from the screening condition and the approximate relation between ψ0 and potential depth Φei(0) (Iosilevskii 1985),

Corrections to the potential energy, ∆Π , and internal energy, ∆E , as well as to the pressure and chemical potential, have the following form (Iosilevskii 1980):

Relation (5.70) follows from the virial theorem, whereas Eq. (5.71) highlights the condition that the correlation occurring upon inclusion of the additional charge is proportional to its magnitude. With this model, calculations are reduced to the solution of algebraic equations. In the weak nonideality limit, the obtained results tend to the Debye corrections. At higher densities the corrections are smaller than the Debye values, and become positive at σ/rD 1. Figure 5.11 shows the dimensionless energy of a free charge system (Iosilevskii 1980). One can see that for like potentials the results of simple calculations using model (5.63)–(5.71) are in adequate agreement with the Monte Carlo results. This is apparently because the general conditions of local electroneutrality, which proved very important (as in the case of the OCP model), are satisfied in model (5.63)– (5.71).

Selection of the single numerical parameter of the model (5.63)–(5.71) – the potential depth Φei(0) – should be based on experimental data, as was done

198 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

.

.

.

.

.

D

.

.

.

Fig. 5.12. Deformation of the hydrogen energy spectrum (Savukinas and Chizhunas 1974): I – Debye–H¨uckel potential; II – cut-o Coulomb potential; III – boundary condition for wavefunction f (r0) = 0; IV – boundary condition ∂f /∂r|r=r0 = 0 (condition of periodicity). rD is the Debye screening distance and r0 is the boundary atomic radius.

phenomena which cannot be described with perturbation theory and require the solution of the complete quantum–mechanical problem, with all forms of interaction taken into account.

The simplest problem is to calculate the bound states of a single electron in potentials of di erent structures modeling the plasma environment. The results of such calculations by Savukinas and Chizhunas (1974) are given in Fig. 5.12. It shows a sharp variation of the energy structure in a relatively narrow range of atomic compression.

The thermodynamic consequences of such deformation of the energy spectrum for model (5.74) have been analyzed by Graboske et al. (1969) and illustrated in Fig. 5.13, where the relative contributions of various thermodynamic corrections are shown. One can see that in the range of increased plasma densities, even for hydrogen one should expect substantial changes in the plasma compressibility due to deformation of bound states. Unfortunately, this simplest model of a hydrogen plasma cannot be compared with experimental data which are only available for multielectron atoms.

In order to describe the thermodynamic properties of strongly compressed plasmas, one has to calculate the internal structure of atoms and ions as well as to include the e ects of compression on the position of energy spectra of bound electrons. Such a problem can only be solved by employing numerical methods, with the leading one being the Hartree–Fock method which is successfully used for calculations of atomic structures.

This method deals with a system of N electrons in the field of the atomic

. Pa

g cm

Fig. 5.13. Equation of state for the hydrogen plasma, kT = 2 eV (Graboske et al. 1969): p1 is the ideal gas ion pressure; p2 is the pressure where the deformation of the discrete spectrum is taken into account (Section 5.4); p3 is the electron pressure (5.41); p4 is the Coulomb correction; p5 takes into account the atom size is taken into account; pΣ is total pressure.

nucleus with a charge Z. In the atomic unit system (me = 1, = 1, e = 1), the Hamiltonian takes the form

where U (ri) is the e ective potential which is equal to Z/ri for an isolated atom. Based on the variational principle of quantum mechanics, the multielectron

wavefunction is determined from the condition of minimum of the functional

N ˆ N N

E = ψ (q )Hψ(q )dq , (5.75)

where qN is a combination of {r1ξ1, r2ξ2, . . . , rN ξN } coordinates and spin variables of electrons, and the integration over dqN includes, along with integration over coordinates, the summation over spin variables.

Selected as the approximate atomic wavefunction in the Hartree–Fock method is a determinant consisting of one–electron wavefunctions. Assuming that each electron is in the central field of the nucleus and of the remaining electrons, and

200 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

also noting that the Hamiltonian (5.74) does not depend on spin variables, the one–electron wave function can be represented as

Hence, in this approximation the motion of each electron is characterized by the following quantum numbers: the principal number, n, the orbital, l the magnetic, m, and the spin number, S. For given n and l, there exist 2(2l + 1) electron states corresponding to di erent m and S values. It is assumed that all electrons with preassigned n and l have identical radial parts fnl(r). In the absence of spin–orbit interaction, the atomic state is characterized by the complete orbital moment L and spin moment S which are equal to zero when, for any n and l, all 2(2l + 1) states are occupied, i.e., all the nl–shells are filled. In the case when the atom has unfilled shells, in order to ensure that the atomic wavefunction is the eigenfunction of complete orbital and spin moment operators (i.e., the atom is characterized by given L and S), it is necessary to seek that function in the form of a linear combination of determinants. The coe cients of this combination are found from the condition that the complete orbital and spin moments are equal to the preassigned values.

The application of the variational principle yields the following set of integrodi erential equations (Hartree–Fock equations):

nj n=i

where Uni li (r) is the Coulomb potential of the electron interaction with the nucleus and with each other, Gnili is the nonlocal part of the potential, or exchange term, and the quantum numbers ni and li vary in accordance with the chosen atomic configuration. The eigenvalues εni li and nondiagonal Lagrangian multipliers εni li nj lj , introduced to satisfy the condition of orthogonality of the radial functions fni li (r) and fnj lj (r) at ni =nj , are found from the condition of orthonormality,

and from the boundary conditions. One of these conditions follows from the finiteness of the wavefunction at the origin of coordinates,

fni li (0) = 0,

and the other one is selected based on the particular statement of the problem. For instance, in the approximation of an isolated atom, the electron wavefunction

should decrease exponentially at large distances. In the “confined” atom approximation (Gryaznov et al. 1980, 1989), which is one of the methods that include the influence of external e ects on the bound states, the following condition is imposed on the radial wavefunction:

Let us consider now a three–component plasma consisting of atoms, singlecharged ions and electrons. We assume that the atoms are spheres of variable radius rc, whereas the sizes of electrons and ions are ignored for simplicity. We shall treat the subsystem of finite–size atoms as a set of hard spheres which do not interact when the distance between them exceeds 2rc. The free energy of such a model can be written as

F (Na, Ni, Ne, V, T ) = Fid + Fhs + ∆Fcoul.

The first term is the free energy of the ideal plasma, with the only di erence that now the atomic partition function depends, in accordance with the boundary condition (5.80), on the radius. The second term is the contribution of the hard sphere repulsion which also depends on rc via the dimensionless parameter ν = na(4πrc3/3). In order to include this contribution, the thermodynamic results for the hard-sphere systems are used, as obtained from molecular dynamics calculations and described with the Pade approximation (Carnahan and Starling 1969),

Thus, the free energy in this model depends on rc due to the compression of atoms, on the one hand, and because of their interaction as hard spheres, on the other. An equilibrium value of atomic radius can be determined from the condition of the minimum of the free energy,

The dependence of the atomic partition function on rc is found from the solution of the Hartree–Fock equations for the ground and excited atomic states,

202 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

Fig. 5.14. Quantum–mechanical calculation of the energy spectrum of compressed cesium

(a) and argon (b) based on the bound atom model (5.77)–(5.83). The dashed regions correspond to experiments (Bushman et al. 1975; Bespalov et al. 1975; Fortov 1982), rc is the atomic cell radius in units of the Bohr radius and hν = 2.14 eV is the energy of the detected light radiation. The dot–dashed line shows the continuous spectrum boundary.

with boundary condition (5.79) at di erent rc (see Fig. 5.14). The solution of Eq. (5.83) gives an equilibrium value of rc(V, T ), which makes the model thermodynamically closed.

In contrast to the solid–state cell models (Bushman and Fortov 1983), this approximation is constructed in the framework of the quasichemical method of description, with explicit inclusion of the translational degrees of freedom for individual particles. Yet the electrons are separated into two types and are located inside and outside the cell, with the volume (3/4)πrc3 being just a part of the average volume per particle.

The thermodynamic calculations performed for model (5.77)–(5.83) demonstrate that the e ective repulsion and deformation of the discrete atomic spectrum in the selected potential have a significant influence both on the thermal and caloric plasma equations of state. Given the pressure and temperature, the density calculated from this model has a smaller value than that obtained in the ring Debye approximation; the same is true for the enthalpy at fixed pressure and volume. The latter is important because it is one of the principal qualitative results obtained so far experimentally in nonideal plasmas (see Section 5.6).

The e ect of energy level deformation has significant influence on the optical plasma properties, since atomic photoionization by visible radiation occurs from highly excited energy levels that are distorted even at relatively low compressions. In Chapter 7 we shall discuss this e ect in detail.