Physical Properties of Dense Low Temperature Plasmas.pdf

PHYSICA ERTIES OF DENSE, LOW-TEMPERATURE PLASMAS

LAUSANNE - NEW YORK - OXFORD - SHANNON - TOKYO

PHYSICS REPORTS

ELSEVIER Physics Reports 282 (1997) 35-157

Physical properties of dense, low-temperature plasmas Ronald Redmer

Universitiit Restock, Fachbereich Physik, Universitiitsplatz 3, D-18051 Restock, Germany

Received June 1996; editor: R.N. Sudan

Contents

1. Introduction 1.1. Plasma parameters 1.2. Dense plasmas

2. Plasma properties and many-particle effects 2.1. Thermodynamic relations and equation

of state 2.2. Dyson equation and elements of the

diagram technique 2.3. Cluster decomposition of the self-energy 2.4. Generalized laws of mass action

3. Thermodynamic properties of dense plasmas 3.1.

3.2.

3.3.

3.4.

3.5.

Thermodynamic functions and plasma

composition Critical phenomena and phase transitions in dense plasmas Equation of state for expanded alkali-atom fluids Thermodynamic properties of dense alkali-atom plasmas Equation of state and coexistence curve in expanded fluid Hg

4. Transport properties of dense plasmas 4. I. Nonequilibrium statistical operator 4.2. Linear response theory 4.3. Evaluation of the correlation functions 4.4. Dynamic screening

38 38

39 42

42

44 45 49 55

55

59

66

70

72 78 78 81 83 86

4.5. Fully ionized plasmas: improved Spitzer results

4.6. Transport coefficients in partially ionized plasmas

4.7. Expanded fluid metals: improved Ziman theory

5. Magnetic susceptibility of expanded fluid metals 5.1. Theoretical model for the magnetic

susceptibility 5.2. Contributions to the total volume

susceptibility

5.3. Results for the magnetic properties 6. Structure factor for dense plasmas

6.1. Structure factor and correlation functions 6.2. Polarization function and effective ion-ion

potentials 6.3. Results for the static structure factor of

expanded fluid metals 7. Spectral line shape

7.1. Introduction 7.2. Green’s function approach to the spectral

line shape 7.3. Results

8. Conclusions References

90

96

112 118

118

123 126

130 130

133

135 137 137

139 141 144 146

Abstract

Plasmas occur in a wide range of the density-temperature plane. The physical quantities can be expressed by Green’s functions which are evaluated by means of standard quantum statistical methods. The influences of many-particle effects such as dynamic screening and self-energy, structure factor and local-field corrections, formation and decay of bound

0370-1573/97/$32.00 0 1997 Elsevier Science B.V. All rights reserved PII SO370-1573(96)00033-6

R. Redmer IPhysics Reports 282 (1997) 35-157 31

states, degeneracy and Pauli exclusion principle are studied. As a basic concept for partially ionized plasmas, a cluster decomposition is performed for the self-energy as well as for the polarization function. The general model of a partially ionized plasma interpolates between low-density, nonmetallic systems such as atomic vapors and high-density, conducting systems such as metals or fully ionized plasmas.

The equations of state, including the location of the critical point and the shape of the coexistence curve, are determined for expanded alkali-atom and mercury fluids. The occurrence of a metal-nonmetal transition near the critical point of the liquid-vapor phase transition leads in these materials to characteristic deviations from the behavior of nonconducting fluids such as the inert gases. Therefore, a unified approach is needed to describe the drastic changes of the electronic properties as well as the variation of the physical properties with the density. Similar results are obtained for the hypothetical plasma

phase transition in hydrogen plasma. The transport coefficients (electrical and thermal conductivity, thermopower) are studied within linear response theory

given here in the formulation of Zubarev which is valid for arbitrary degeneracy and yields the transport coefficients for the limiting cases of nondegenerate, weakly coupled plasmas (Spitzer theory) as well as degenerate, strongly coupled plasmas (Ziman theory). This linear response method is applied to partially ionized systems such as dense, low-temperature plasmas. Here, the conductivity changes from nonmetallic values up to those typical for plasmas in a narrow density range above 10” cme3 for alkali-atom plasmas and 10” cm-3 for hydrogen plasma, respectively. Furthermore, the thermopower can change its sign in the same region which indicates that a nonmetal-to-metal transition occurs also in dense, low-temperature plasmas.

The magnetic susceptibility and the Korringa relation are calculated for expanded fluid metals along the coexistence line within the partially ionized plasma model. The various contributions to the total susceptibility are derived from an extended RPA which takes into account local-field corrections as well as the influence of localized electron states. The metal-nonmetal transition indicated by an enhancement of the electronic paramagnetic volume susceptibility is strongly connected with the occurrence of charged clusters.

Static structure factors are determined for expanded cesium and mercury within the MHNC scheme via effective ion- ion potentials which are derived from the polarization function within an extended RPA. The optical properties of dense plasmas, the shift and broadening of spectral lines, can also be derived within the Green’s function technique. Some new results for the spectral line shape are discussed.

PACS: 52.20.-j; 52.25.-b

38

1. Introduction

1. f _ Pravda ~~~~~e ters

The physical properties of plasmas are mainly influenced by free charge carriers. Typical nonrela- as are shown in the de~s~~-t~m~era~r~ plane in Fig. 1. Besides low-density plasmas,

high-density plasmas are of special interest for, e.g., astrophysics and future technical applications such as laser fusion.

We consider here neutral plasmas consisting of electrons (mass m, = m, charge qe = -e) and ions (mi = M, qi = +Ze) so that ZN, = Ni. The following parameters are ~~~oduced to characterize the plasma state: the Coulomb coupling constant r far the ion system and the degeneracy parameter 0 for the electron system,

f = (2~)‘/4~~~~~~~i 3 0 =kBTJEF . U)

EF denotes the Fermi energy, a,, is the mean distance between particles of species c = e, i, and a0 is the Bohr radius,

EF = fi2/2nz(3n2n,)*“, I& = (3/47cn,)“3 ) a0 = 47r&()fi2/%?” 0 (2)

We usually distinguish the cases of weak and strong coupling for the ions (r Q 1: nearly ideal, I’ < 1: weakly nonideal, r > 1: s~o~g~y coupled plasmas)~ The electron system belongs to the

4 8 12 16 20 24 28 32 log n[cm -Q

Fig, I. Density-temperature plane for the nonrelativistic case. Characteristic values for the plasma parameters r, 0, T20, and YS are indicated by lines, Examples for typical weakly and strongly coupled plasmas are given. The location of the critical points for Cs, Hg, H, and Xe is indicated by asterisks (*),

R. Redmerl Physics Reports 282 (1997) 35-157 39

nondegenerate (0 > 1) or degenerate (0 < 1) domain. For the nondegenerate domain, the classical (Boltzmamr) statistics apply, whereas for the degenerate region quantum corrections are important. The heavy ions (n/r % m) obey in nearly all cases the classical (Boltzmann) statistics [niA: + 1, nf = 2di2/(m,kBT): thermal wave length of particles of species c]. The parameter rs = a,/~ denotes the density parameter for the electron system (I es > 1: low densities, rs < 1: high densities). The

lines in Fig. 1 (r = 0.1, 1,lO; 0 = 0.1, 10; rs = 1) divide the n-T plane into the above-mentioned regions. The parameter

(3)

depends only on temperature and defines the classical (r20 % 1) and the high-temperature domain (T*O 4 1) for which the Born approximation holds. Further plasma parameters are the coupling parameter ~1, the Landau length (, and the Debye screening length RD:

e= e*

4nEokBT ’ R;* = C n,.q~/(ksT&O) .

c

This paper is devoted to dense, low-temperature plasmas, and, especially, to alkali-atom plasmas. There is a close connection to the domain of expanded fluids at subcritical conditions where a metal-nonmetal transition occurs near the critical point of the liquid-vapor phase transition. Precise experimental data are available for expanded fluids from the melting point up to the critical point so that the model of a partially ionized system originally developed for low-temperature plasmas is also applied in that region.

1.2. Dense plasmas

Besides hydrogen as the simplest and most abundant element in the universe and the inert gases, both expanded fluid metals and metal plasmas (alkali-atom metals, mercury, and also aluminum and copper, etc.) are studied. These materials have a fluid range up to high temperatures (see Table 1) and great heats of evaporation. Furthermore, they have both high electrical and thermal conductivities so that they pass for optimum high-temperature working fluids in technical equipments for the production, storage and conversion of energy (see the review of Ohse [OHS85]). Therefore, reliable data for the thermodynamic (ionization degree, equation of state, compressibility), optical (index of refraction, coefficient of absorption and emission) as well as the transport properties (thermoelectric power, electrical and thermal conductivity, Hall coefficient) of plasmas and expanded fluid metals are needed in a wide region of the n-T plane.

Experimental methods to produce high-pressure plasmas as, for instance, shock-wave techniques, wire explosions, capillary discharges, ballistic compressors, or high-intensity laser pulses on target material have been reviewed [GRY80, KHR81, BIB87, YON82, EBE83, ALE83a, GUE84, OHS85, GAT86, BUS88, FOR90, EBE91].

Theoretical approaches to the properties of dense plasmas as many-particle systems of charged particles are based upon quantum statistical methods (for reviews, see [KLI75, EBE76, BAU80, ICH82, ICH86, KRA86]). The powerful Green’s function method is often utilized to treat quantum many-particle systems (for details, see [KAD62]). In general, the singularities of the propagators in momentum space determine the spectrum and lifetime of excitations in the system.

40 R. Redmer I Physics Reports 282 (1997) 35-157

Table 1 Values for the melting (m), boiling (b), and critical point (c) of various elements and alloys

Element/

alloy T”,

(K)

Tb

(K) [ZIPa) PC (g/cm’)

Reference for

critical point

4He 1.764 4.214

HZ 13.34 20.26 Ne 24.544 27.15

N2 63.18 77.36 co 68.10 81.66 Ar 83.81 87.29

CH4 90.68 111.66

Kr 115.78 119.93 Xe 161.36 165.03

NH3 195.4 239.73

Hg 234 630 Fr 300 947 cs 302 955 Rb 327 967 K 336 1031

Na 371 1156 Li 453 1597 Cd 594 1040

Al

Ag AlI cu Pt V

MO W

932 2720 5730 182 0.42 [GAT86] 1234 2436 5130 114 2.89 [MAR831 1336 3081 6520 129 4.69 [MAR831 1356 2839 5890 169 2.19 [MAR831 2036 2573 9300 950 4.7 [GAT86] 2175 3682 6400 920 1.63 [GAT79] 2890 5100 14300 570 2.9 [SEY78] 3680 5890 13400 337 4.28 [FUC79]

5.21 0.229 0.06945 [DAN671 33.24 1.296 0.0301 [DAN671 44.44 2.654 0.483519 [DAN671

126.3 3.383 0.311 [DAN671 133.0 3.499 0.301 [DAN671 150.72 4.864 0.53 I [DAN671 191.1 4.629 0.138 [DAN671 209.39 5.492 0.908 [DAN671 289.75 5.904 1.105 [DAN671 405.5 1 11.30 0.235 [DAN671

1751 167.3 5.8 [GOE88]

1924 9.25 0.38 [JUE85] 2017 12.45 0.29 [JUE85] 2178 14.8 0.18 [HEN911 2485 24.8 0.30 [BIN841 3223 68.9 0.105 [FOR751 2610 150 2.77 [MAR831

Problems such as the derivation of the effective interaction between the constituents of the plasma including dynamic screening, exchange and vertex corrections, the formation and the decay of bound states, the renormalization of the particle energies due to interaction with the surrounding particles (self-energy), degeneracy and phase-space occupation (Pauli exclusion principle) can be treated in a consistent way. Explicit results were given for the whole spectrum of thermodynamic, optical, and transport properties in a wide range of the n-T plane [EBE83, KRA86, ZIMWJ.

This review is organized as follows. Physical properties of dense plasmas are expressed in terms of thermodynamic Green’s functions in Section 2. The determination of the self-energy is the main problem for which a cluster decomposition is employed. For the evaluation of the Green’s functions, standard diagram techniques are utilized (see [KAD62, KRA86]).

New results for the equilibrium properties (composition, equation of state, critical point, coexistence curve) of hydrogen, alkali-atom, and mercury plasmas are given in Section 3 (see also [EBE91]). The parameter domain studied here extends from weakly nonideal, nondegenerate plasmas

R. Redmer I Physics Reports 282 (1997) 35-157 41

[T = (3-10) x lo3 K, n = (10’5-10’9) cm-3] up to the critical point [T = (2-3) x lo3 K, IZ =

( 1021-1022) cm- 3 for alkali-atom metals], for which strong coupling effects and degeneracy are

characteristic. The transport properties of dense plasmas are treated in Section 4 within linear response theory,

given here in the formulation of Zubarev [ZUB74]. We compare with other standard methods to determine transport coefficients and with available experimental results. The general expressions for the transport coefficients are evaluated for limiting cases. For fully ionized plasmas, correlation functions are derived on the level of the Landau, Boltzmann and Lenard-Balescu collision integrals [RED90]. Various effects were studied for the nondegenerate case: improvements of the Born approximation utilizing the T matrix [MEI82, SIG88], dynamic screening [MOR89, RED90], electron-electron scattering and structure factor [ROESla, ME182, RED92], or transport due to the heavy particles (ions, atoms) [REI89].

For the degenerate domain, the Ziman and Mott formulae for the electrical conductivity and thermoelectric power are recovered. Especially, the influence of structure factor and screening including local-field corrections is studied. Interpolation formulae are presented for the transport coefficients that reproduce the experimental results in a wide range of density and temperature within the experimental error bars [ROE88, ROE89a, REI92]. These Pade formulas are based upon analytical expressions for the transport coefficients for the classical limit r*O B 1, the Born limit r*O Q 1, as well as for the degenerate case 0 4 1 and interpolate the behavior in the intermediate region (cf. also [ICH85b, RIN85]).

The transport properties of partially ionized plasmas change drastically when neutral bound states (atoms, molecules) occur. For instance, isotherms of the electrical conductivity can show a minimum as function of the pressure, and the thermoelectric power can change its sign. These effects are connected with the Mott transition [EBE83, HOE84, SCH84, SCH85, BLU86] and are treated here especially for alkali-atom and hydrogen plasmas [REI89, BIA89, ARN90, REI95, STA96].

The magnetic properties (susceptibility, Korringa relation) are calculated in Section 5 for expanded fluid metals along the coexistence line within the partially ionized plasma model. The different contributions to the total susceptibility are derived from an extended random phase approximation (RPA) which takes into account local-field corrections as well as the influence of neutral bound states [RED93].

In Section 6, the static structure factor for dense plasmas and expanded fluid metals is determined. Effective ion-ion potentials are also derived from an extended RPA for the polarization function including local-field corrections. Modified hypemetted chain (MHNC) calculations are performed for the static structure factor in H, Cs, and Hg.

The broadening and shift of spectral lines in dense plasmas is treated in Section 7 (see [GR174, SOBSl]. In contrast to the natural line width and Doppler broadening of spectral lines due to the thermal motion of the radiating atoms, pressure broadening is caused by interactions with the surrounding plasma. The Green’s function approach to the optical properties of partially ionized plasmas is again based upon a cluster decomposition of the dielectric function and yields general expressions for the shift and broadening of spectral lines [ROE8lb, HIT86, HIT88, ROE89b, GUE89, ROE89, GUE9 1, GUE95]. Many-particle effects such as dynamic screening, vertex and self-energy corrections can be studied in a systematic way. Furthermore, the influence of strong collisions and of the dynamic microfield distribution of the ions on the spectral line shape have to be considered. Concluding remarks are given in Section 8.

42 R. Redmer I Physics Reports 282 (I 997) 35-157

2. Plasma properties and many-particle effects

2. I. Thermodynamic relations and equation of‘ state

The thermodynamic ical potentials p, and canonical ensemble,

properties of electron-ion plasmas with a temperature p = l/(knT) and chem- pl, respectively, can be gained from the statistical operator for the grand-

, (5)

The Hamilton operator Hs is given in second quantization by creation and anihilation operators a:( 1 ), a,( 1) with the cumulative variables 1 3 {k, ,st },

Hs = CE,.( 1 )a:( l)ac( 1) + i C I’,(!( 12,1’2’)aJ( l)aJ(2)ad(2’)ac( 1’) . (6) c, I <d.12,1’2’

E,( 1) = h2k:/(2m,.) is the kinetic energy of particles of species c with wave number k, . The matrix elements of the Coulomb interaction are symmetric with respect to spin and conserve the momentum so that &(12,1’2’) = &(k) = qcqd/(SZo&ok2). Q. is the volume of the system and k = k, - k2 the transfer momentum. J&(k) is the Fourier transform of the Coulomb potential V,(r) = qcqd/(4z0r).

Identifying the quantity S = -k,Tr {p. In po} with the thermodynamic entropy S = U/T+ pOo/T - ,uN/T, the thermodynamic potential J = -pOo is given by

J(Qo,p,T) = -kBTlnZo(Qo,iu,T) . (7)

The thermodynamic functions can be derived from this equation of state (EOS) because the sum of states Z. contains all microscopic information about the system via the Hamilton operator Hs and the particle number operators NC. For instance, the particle number density IZ, = (N,)/sZo is obtained by partial differentiation of J with respect to the chemical potential Pi, where the expectation value of arbitrary operators (A) is given by the trace over po, (A) = Tr{poA}.

For low-temperature plasmas with Z = 1, we have n, = n, = n. The particle number n and

the temperature T are then more convenient variables for which the free energy density f(n, T) = F(n, T)/Go is the relevant thermodynamic potential,

J’(n, T) = n Mn, T) + ~,(n, T)l - PC/L /A, T) . (8)

The chemical potential ,u,(n, T) follows from the functional n&L,, T) by inversion or, alternatively, by partial differentiation of the free energy f with respect to the particle number density n,

a.f(n, T)lan = p,(n, T) + pi(n, T) E P(K T) , (9)

so that simple relations for the thermodynamic functions can be derived:

f(n, T) = f(n = 0, T) + I’

’ dn’p(n’, T) , . 0 P/i (10)

P(PC~, T) = .I_, d,u’ n(p’, T >

R, Red~~r~~~.~~~~s Repports 282 (f997j 35-157 43

Tn the following, the relation between the particle number density n&, T) and the the~od~amic Green’s function G,(k,z) is utilized to derive the thermodynamic to Eq. (10) [KAD62, KRA86, ZIM88],

s

d(k) - Im G,(k, w - i0’) fc(fiw) .

n:

_fm) = {expW-I4)1+11- ’ is the Fermi distribution function. the the~odynami~ average of time ordered products of creation

a,( l’, t, i ) according to

The Green’s function is defined as and anihilation operators ~5 ( 1, tl >,

G&J’) = (~/ifi)(T,{~~(l,t,)a,(l’,t~~)t) cw

functions of the plasma according

(11)

and describes the propagation of disturbances in the system which are due to the adding and removing of a particle at (1, tl ) and (l’, tlP), respectively. Because of the symmetry properties of the statistical operator (5) and the Hamilton operator (6), the correlation functions (12) are only dependent on the time difference T = tl - tlf and, furthermore, diagonal in the wave number k (k, = klj ) and the spin variable s’ (of = of, )_ Therefore, we can introduce G,(k, z) = G,( 1,l’ ), and the Fourier transform GJk, W) in the equation of state (11) follows from anaI~ica~ contin~t~on of the Matsubara Green’s function,

(13)

Alternatively, introducing a dynamic properties can be relation,

Pb4 T) = PO04 T) -

coupling constant ,I in the Hamilton operator, Hs = T + ;.V, the thermo- derived after partial differentiation of (7) with respect to ,I from the

p. is the pressure of the ideal system (A = 0). The integration over the coupling parameter ;1 at constant chemical potentials p, (c = e, i) yields in general an integration over nonneutral states with n, # ni. The average (V} can also be expressed by Green’s functions, in this case by the two-particle Green’s function Gcd( 12,1’2’; 3L),

G&l2,1’2’:;) = (l/(i~)2)~T~(a,(l,tl)a~(2,~~)a~(2’,~~~)a~(l’,tl~)))~. . WI

The dependence of the expectation value {Y} from the coupling constant ;1 becomes rather complex when evaluating the two-particle Green’s function [EBE76,KRE84]. Considering the free energy density f(n, 7’) instead of the pressure p(p, T), a similar expression as (24) can be derived which avoids an integration over nonneutral states. However, both expressions contain the contributions of the ideal system which are, in principle, of no importance for an interacting many-particle system. Therefore, Eq. ( 11) seems to be more appropriate for the calculation of the thermodynamic properties of dense electron-ion plasmas as well as electron-hole plasmas [ZIM88f and hot nuclear matter [SCH90].

44 R. Redmerl Physics Reports 282 (1997) 35-157

2.2. Dyson equation and elements of the diagram technique

Utilizing Eq. (11) for the determination of the thermodynamic functions, physical relevant approximations have to be derived for the Green’s function G,(k, w). The hierarchy of equations of motion for the Green’s functions is formally decoupled by introducing the self-energy C,(k,z,). Within the Matsubara representation and employing the Kubo-Martin-Schwinger relation, the Dyson equation can be derived,

G,(k,z,)-’ = h, - E,(k) - C,(k,z,.) . (16)

Introducing the propagator of free particles according to G,O(k,z,j)p’ = hz, -E,(k), and representing it by a simple arrow, the Dyson equation ( 16) for the full or dressed one-particle propagator, characterized by a double arrow, can be rewritten as an integral equation,

G,(l, 1’) = G,o(l, 1’) + s

d2d2’G,0(1,2)&.(2,2’)G,(2’, 1’) , (17)

G, G'j ---tt-=-+

yx)L I I’ I I’ I 2 2’ I'

The self-energy is given by the sum of all irreducible diagrams. The problem of solving an infinite hierarchy of equations of motion for the Green’s function was transformed to the summation of an infinite number of Feynman diagrams within a perturbative expansion for the self-energy. Usually, only the lowest-order terms are treated explicitly, or partial summations of certain types of diagrams are performed. However, the appropriate choice of relevant diagrams that describe the physical effects is an open problem. For dense electron-ion plasmas, those diagrams are taken into account which describe dynamic screening between the charge carriers and the formation of bound states between them. Both effects can be represented by a partial summation of a certain type of Feynman diagrams.

2.2.1. Dynamic screening Perturbative expansions with respect to the Coulomb potential diverge due to its long-range char-

acter. However, the polarizability of the surrounding medium which is caused by the mobility of the charges in a plasma leads to the effect of dynamic screening that is described by the polarization function I7(q,u) according to:

v,d(q) v,s,~q,ql) = ~ =

v,d(q)

Et% qI 1 1 - Ca,b Vob(q)flba(q,WP) ’

Of = 7cp/-ij? , p = 0, f2, f4,. . . ,

d d d :

(18)


The bare Coulomb potential f&(q) is denoted by a broken line. The integral equation (18) for the screened potential l$(q, o), given by a wavy line, is solved by summing all orders of bubble-like diagrams. This partial summation leads to the dielectric function E(q, o) that describes the polarizability of the medium and, thus, the screening of the bare Coulomb potential. The standard method for the treatment of the polarization function II&q, w) is the random phase approximation (RPA). For dense systems, higher-order terms of the perturbative expansion, i.e. correlations, have to be considered.

2.2.2. Bound stutes The formation of bound states between the elementary charge carriers such as atoms in electron-

ion plasmas or excitons in electron-hole plasmas gives rise for drastic changes of the physical properties of the system and should, therefore, be included in the theoretical approach. Two-particle bound states are described by the ladder approximation for the two-particle Green’s function which can be expressed in terms of Feynman diagrams:

This Bethe-Salpeter equation is solved within a bilinear expansion with respect to the eigenfunctions Y$ and the eigenvalues E$ of the isolated two-particle bound state (atom),

G;f( 12,1’2’, L’;,) = C Y$( 12)Y$,“( 1’2’)

%P hL’; - E$ ’ fi2; = z + ,& + ,& ,

-lb /I = 0, f2, f4,. . .)

(19)

[ha;. - E,(l) - E&)l[EfP - &Cl’) - &@‘)I .

P = pI + p2 is the total momentum and c( denotes a complete set of internal quantum numbers, for instance c( = {nZms”}. For dense systems, the ladder approximation ( 19) has to be improved by considering screening, self-energy or Pauli blocking terms in the effective wave equation (see [ZIM78b, ROE82, ROE83, ROE84, RED8.5, KRA90]. Notice, that the sum over LX runs over the discrete bound as well as continuous scattering states.

2.3. Cluster decomposition of the self-energy

The EOS (11) is given by the imaginary part of the one-particle Green’s function G,(k,z) after analytic continuation of z, into the complex z-plane, known as spectral function,

A,(ko) = G(k o-iO+)-G,(k,o+iO’), (20)

e occupation of energies E = &w by one-particle states with a lifetime z. It obeys the sum rule J(du/2@4,(k,~0) = 1 and has to be calculated from the Dyson equation ( 16). This requires an appropriate treatment of the self-energy which can be expressed in the general form Z,(k, CO) = &(k, OJ) + ic,(k, CO). Usually, the following cases are considered:

1. The self-energy vanishes for an ideal system so that the spectral function is given by _4z(S co) = 27rS[tiw - E,(K)). The density p2,(& ~4) = (l/L&)Ck fJ&(k)] is that of an ideal fermion gas.

2. First-order perturbation theory (Hartree-Fock approximation) yields for the self-energy ZFF (k, CO) = dyF(k) and f,!” = 0. From the spectral function AyF(k, 8) = 27c@&co - E,(k) - rlFF(k)], we get for the density ~~F(~,~~) = ( l/QO)Ck &[&(il) + dtf”(h-)].

3. The spectral ~n~t~o~ for the general case is given by

ZT,.(k, r-11)

AcckT u, = [iko - E,(k) - A,(k, co>]2 -t [r&t, a)]” *

Sharp one-particle energies in the ideal system (1) are accompanied by an infinite lifetime of the respective states. The Hartree-Fock approximation (2) yields shifted one-particle energies. The damping of one-particle states yields a finite lifetime and is only obtained within an approximation for the self-energy according to (3). Therefore, a systematic improvement of the perturbative expansion beyond the Hartree-Fock level is of minor significance than the treatment of correlations, dynam~~ screening, or cluster formation. This can be done within the frame of a cluster decomposition for the self-energy, consider~~g not only the two-pa~i~le bound states (atoms)~ but also higb~r-order clusters as, for instance, three- e bound states ~molec~lar ions as A, A,‘, or dimers

A& This co~es~onds to a the at treats composite particles on the same leve elementary ones [EBE76]. The self-energy is then given by [ OE83, RED85, 4X487]:

=

e quantities rk denote the ladder g~nera~i~ed ~etbe-Sa~peter equation:

0 T.k + + . . . G3 I 1’

T matrixes for tile ~-~a~ic~e cluster which follow from a

lN is the symmetrical N-particle interaction, whereas I$ contains all interactions between the N particles,

1, = c Kj 7 &,I = tl: &j . .i t<j

The e@zctive two-panicle interaction J, - I - cd’( 3 2,1’2’ ) contains besides the dynamical screened potential yzI further terms, which describe the influence of bound states on this potential and which

47

are of the same order in density as the s~lf-en~r~~ cu~ections in (22):

v:‘$; (12,1’2’)= l I’

2,

-t 4” + 1 t . +

2 2’ 2 2’ 2 2”

The vertices in the last diagram describe the interaction between a two-particle cluster tic’tes [RQE79]. For the ~o~ar~~at~on function ~(~~~~~~ a similar cluster d~c~l~~osjt~un

+

The first term in (25) describes the ~~~ar~~abi~i~y of a free electron gas within the WA whereas the additional diagrams represent the polarizabilities of two-particle, three-particle, and hither-order clusters. This extended .RPA scheme was derived by R6pke and Der [ROE791 beginning with ~~(~~ CD) and ~~~~~~~d for the dete~inat~on of the equilibrium ~ro~e~i~s of dense systems fROE82, ROE83, RQE84, RED85, REDS8, RED89].

The system of equations (X2)--(25) is evaluated to dateline the th~~ud~namic pr~~~~~es. First, within the frame of the chemical picture, we consider only the bound state part of the Ti matrices and neglect the sca~er~n~ state co~~ib~t~ons because the latter belong to a higher density order and are pa~ial~~ compensated by diagrams that have to be sub~act~d in order to avoid double courting.

Second, the ~-~a~~cl~ bound states follow from the solution of the corresponding N-particle Bethe-Salpeter equation with an effective two-particle interaction F$ that takes into account


many-particle effects. Already the ladder approximation with respect to the dynamically screened potential v,S,(q, o) is rather involved. The dependence of three Matsubara frequencies has been treated up to now only within the so-called Shindo approximation that yields an one-frequency expression for the effective two-particle interaction (see [ZIM88]). Therefore, the usual ladder approximation is performed with respect to a statically screened potential as, for instance, the Debye potential. Those terms in the expression for the effective interaction (24) and the polarization function (25) that go beyond the RPA can be interpreted as self-energy contributions characterizing the interaction between free particles and N-particle bound states within the Born approximation. From these diagrams, effective potentials for the respective scattering processes can be derived such as the polarization potential VJT) N Y -’ for the interaction between charges and neutrals, and the van der Waals potential V,,,(v) N Y -’ for the interaction between neutrals (see [RED87, NAG92]).

Third, the dressed one-particle propagators (double arrow) in the Feynman diagrams contain a self-energy. These contributions are of higher density order and, thus, neglected. However, a self- consistent determination of the Green’s function and the polarization function as, e.g., in the so- called V” approximation [KRA86] or the GW approximation known from semiconductor band theory [HYB85] should be the aim of further efforts. An alternative treatment avoids the quasiparticle picture at all and determines Green’s function and polarization function self-consistently in terms of the spectral function (2 1) [ALM95].

The EOS (11) is determined when inserting the cluster decomposition for the self-energy (22) into the Dyson equation (I 6). Stolz and Zimmermann [ST079a] utilized the two-particle T matrix Tk in the cluster decomposition which allows for a genuine description of the formation of atoms out of the elementary constituents electrons and ions. Kremp et al. [KRE84] improved this approach by a systematic expansion of the spectral function A,.(/& (0) with respect to the imaginary part of the self-energy T,.(k,w) which yields an EOS for quasiparticles. The extended cluster decomposition introduced here was employed by Ropke et al. [ROE82, ROE83, ROE841 as the so-called ladder Hartree-Fock approximation for hot nuclear matter and, later, also for dense electron-ion plasmas [RED85, RED88, RED89]. As a result, the total density of species c is given by

n,(B, p) = !?;.” + n:,*) + 2?$3’ + 2nj4’ + 3nb.5’ + 3n;” + . . (26)

The partial densities II, cnr) of m-particle clusters are given by respective partition functions,

(27)

where fC is Fermi distribution function of one-particle states and

gm(E) = {exp[P(E - PI - ~2 - . . . - ,h)l - C-1 Y-’

denotes the distribution function of m-particle clusters (m odd/even: Fermi/Bose statistics). The arguments of these distribution functions are the one-particle energies I, determined by the Dyson equation ( 16), and the energy spectrum E,~ (m) of m-particle clusters following from the Bethe-Salpeter

equation (22), respectively,

I?$‘, @$” are the eigenvalues and eigenfunctions of isolated m-particle clusters, whereas d,!$’ contains the corrections due to interactions with the surrounding particles. The one-particle energies Ed are those of free q~asipa~ic~es taking into a~~o~~t the interaction ~o~e~t~ons via quasipa~i~le shifts d,(k). These effects yield a lowering of the ionization energy (M&t ~ausitio~). A solution of the self-consistent system of equations (22)-(25) is rather complex. However, an approximate solution can be given within the chemical picture.

Reactions in how-temperature plasmas can be described by respective mass action laws (MAL). We have, for instance, (1) ionization equilibrium e + AS G$ A’: !?A” = N~~~+K~o,

(2) dissociation equilibrium mA” Z$ AZ: nA;U, = FI$&;:,, (3) positively charged molecular ions A’ t ~A~ + Az,r: tzAz+, = ~~*~~~~~~~~~,

(4) negatively charged molecular ions e + pnAO = A;: nA, = n~“n~JCA,,

The quantities K characterize the chemical equilibrium between the reacting species according to (27) [RED89],

The partition functions of the species involved in the reaction are dependent on the density and temperate via (28). The quantities K reduce to t e weal-mown constants of mass action of respective Saha equations only in the low-density limit where interaetion corrections can be neglected. Supposing charge neutrality, a generalized MAL can be derived [ALE83a],

Eqs. (29) and (30) are the starting point for all further calculations. Similar approaches have already been used by Alekseev and Iakubov [ALE83a] and ~e~and~z [HER841 for

50 R. Redtner I Physics Reports 282 ( 1997) 35-157

alkali-atom plasmas and fluids who also found a strong influence of higher clusters on the thermodynamic as well as transport properties.

The evaluation of (30) requires an approximative scheme due to the nontrivial density dependence of the K’s. The ionization equilibrium describes the formation of atoms as the simplest bound state and KAo influences the generalized MAL (30) strongly. The formation of charged and neutral clusters A; and Ai is also closely connected with the ionization equilibrium, see Eqs. (29). Therefore, the terms K*I on the r.h.s. of Eq. (30) are considered to be (small) corrections which lead to an II/ increase or decrease of the number of free electrons, respectively. We will take into account all interaction contributions to the energy spectrum of single-particle states as well as of atoms according to Eq. (28). For the higher-order clusters, the usual separation of the partition function with respect to the internal degrees of freedom is performed. The interactions between these clusters are considered via the respective second virial coefficients.

The ionization equilibrium between free particles and atoms is treated. Zimmermann and Stolz [ZIM85] derived from (11) an EOS which is the sum of the densities of free quasiparticles, nF,and correlated two-particle states, n~.‘~cor’). Replacing the slightly k-dependent self-energy shift in Eq. (28) by a constant quantity A,. which is fixed by the quasiparticle density,

A, = &T .f~[~,,(k)] Re C,.(k,E,.(k) + i0) I/{

and utilizing the optical theorem, a generalized Beth-Uhlenbeck formula the correlated density,

(31)

[BET361 was derived for

(2.am) _ n, -$c (2/$1) l- Oy.d./

( q&l) {~~~~~&,,,+~+A~,+A~,~

fi2k2 - a&,/

+ ?vq2 2~, + A,. + &

1 d&(k) 2 . 7c dk 2sm b,(k) .

‘ d (32)

MCd and j&d denote the total and reduced mass of a pair of particles (cd), respectively, and Y&E) =

{exp[P(E - i4 - k)l - 1 >F’ is the Bose function for two-particle states. The correlated density consists of the sum over the discrete bound state energies E,/ and the integral over the continuous scattering states, characterized by their scattering phase shifts 6,(k). Zimmermann and Stolz [ZIM85] pointed out that the difference to the standard Beth-Uhlenbeck formula, the additional factor 2 sin2 6,(k) in the scattering state part, is a result of putting as much correlation as possible into the definition of the quasiparticle density n,” via the full self-energy shift A,.. They demonstrated the well-known compensation of discontinuities in the bound state part which occur whenever a bound state disappears due to self-energy and screening effects. The respective contributions are taken over by the scattering state part and the whole partition function (32) remains a smooth function.


For the nondegenerate case, Bose and Fermi functions are replaced by Boltzmann factors. The

ionization degree aion is defined according to (32) via

%m = 4

(2,corr) ' n: + a2

where

4 = 2K3 exp[-PC4 - ~11 , n;2.c0rr) = [n;]2A;*+Ze ) z, = Ze*+ + 23’2z,, )

/. (k)

(33)

(34)

The use of higher-order Levinson theorems [BOL79] projects the first expansion terms of the bound state part with respect to DE,/ into the scattering state part. In this way, the finite Planck-Larkin partition function for the bound state part,

ZJk+ = C (2& + 1 )C [exp(-BE,,) - 1 + &I , (35) / n

can be derived in a natural way [ROG86, KRE93a] and has not to be introduced ad hoc in order to avoid its divergence for low densities. The bound state part dominates the partition function of alkali and mercury atoms for low temperatures considered here so that the scattering state part can be neglected.

For hydrogen and inert gas plasmas such as xenon, the situation is more involved. For comparison, we have calculated the two-particle partition function Z, = Z,, + 23i2Zee, see (34) for hydrogen plasma. The bound state energies E,/ and scattering phase shifts 6/ were determined by solving the Schrodinger equation for the attractive and repulsive Debye potential numerically. The respective values were checked against earlier results [ROG71, ZIM88, SIG88]. The results for the Planck- Larkin, the standard Beth-Uhlenbeck, and the modified Zimmermann-Stolz partition functions are compared in Table 2 for given temperatures and inverse screening lengths IC = l/Rn.

All three partition functions are smooth functions with respect to the density so that no unphysical jumps in the thermodynamic functions occur whenever a bound state vanishes. The bound state part dominates the partition function for low temperatures, i.e. T 5 2 x lo4 K, and inverse screening lengths k’ao 2 0.5. The scattering state contribution is not negligible for larger screening parameters lcaO where only a small number of bound states remains and compensation effects between bound and scattering contributions are considerable. For higher temperatures and low densities, the number of scattering states increases strongly so that their contribution to the two-particle partition function becomes dominant.

To study the influence of two-particle scattering states on the thermodynamic functions in hydrogen plasma, the Planck-Larkin and the Zimmermann-Stolz partition function are inserted into the MAL e + p H H, Eqs. (29) and (34). The Planck-Larkin convention may serve as the simplest version of the chemical picture, while the quasiparticle picture of Zimmermann and Stolz contains

Table 2 Two-particle partition function for the Debye potential for various inverse screening lengths K and temperatures T [R 2:: is the Planck-Larkin partition function ( 15 ). 2,“” the standard Beth-Uhie~beGk result (without the sin-term), and Z,“” the Z~~e~~~-Stol~ version ( 14). The no~tio~ 0.1352 f 2 stands for 0.3352 x 10”’

7 = 10 x 10’ K

2.aa 1.25

1.00 0.67

0.50 0.40 0.25 0.10

0.01

T = 20 x IO3 K

2.00 1.25

1 .oo 0.67 0.50

0.40 0.25 0.10

0.01

T = 30 x IO3 K 2.00 1.25 1.00 0.67 0.50 0.40

0.25 0.10 0.01

0.00 0.6602 -I 0.3386 -2

0.00 0.3398 +o 0.2314 +O 0.5896 - 1 0.1009 fl 0.1127 +1 0.99~8 i-1 0.1336 +2 0.1352 +2 0.1018 +3 0.1073 $3 0.1075 f3 0.5182 +3 0.5255 +3 0.5255 t3 0.9756 +4 0.9767 14 0.9768 f4

0.3823 +6 0.3824 +6 0.3824 f6

0.5260 +7 0.5260 f7 0.5260 -57

0.00 0.8192 -1 0.6689 -2 0.00 0.3313 +O 0.2326 -+O 0.1393 -1 0.7237 +O 0.8111 +o 0.1374 +1 0.3386 +I 0.3546 +1

0.7028 +I 0.1021 i-2 0.1033 +2 0.1879 tt 0.2301 12 0.2306 f2 0.9323 +2 0.1001 +T 0.1006 +3

0.6124 +3 0.6295 +3 0.6319 i-3

0.2302 +4 0,2502 +4 0.2656 t4

0.00 0.9041 - I 0.9287 -2 6.00 0.3209 +O 0.2261 +o 0.6078 -2 0.6174 +o 0.6832 t-0 0.5139 +O 0.2044 +1 0.2191 fl 0,2195 +1 0.4575 +l 0.4698 + 1

0.4982 fl 0.8129 $1 0.8211 +I

0.1731 f2 0.2255 +2 0.2368 +2 0.6789 t2 0.8144 +2 0.8395 +2 0.1741 +3 0.3552 -1-3 0.5493 +3

at least the correct two~pa~i~le ~a~itio~ Knutson for the ~on~e~enerate case. The shifts A, can be decomposed into the Hartree-Fock (HF) and Montroll-Ward (MW) contributions, characterizing the self-energy of charged particle interactions in second order with respect to the Coulomb potential, and a polarization contribution (PP) which is due to interactions between the charged particles and the neutral bound states,

We have used the Pad6 approximations of Ebeling et al. [EBE82, EBE85, EBE88a, EBE91J for the self-energy shift of charged particles, Agas = d,HF + dyw, which interpolate between the limiting cases of nondegeneracy (Debye-Htickel theory), the strong coupling limit for electrons ((Sell-Mann

R. Redmer I Physics Reports 282 (I 997) 35-157 53

Table 3 Polarization potential VPP(r = 0), dipole polarizabilities go, and quadrupole polarizabilities %Q for the alkali atoms

H Li Na K Rb CS

VPP(r = 0) in Ryd (a) 1.0 1.8224 1.4175 1.0304 0.9960 0.9512

aD(&a 4.5 162.9 158.0 280.5 306.0 382.4

aD(a;)b 164 159.3 293 319 402

aQ(d )” 15 1442 1912 5398 6845 10049

xQ(‘d 1’ 1383 1799 4597 5979 9478

a Calculated within the QDT [RED87]. b Experimental values [MIL77]. ’ More detailed calculations [MAE79].

and Brueckner result [GEL57]), and the case of strongly correlated ions (see [DEW76]). These formulas cover the entire density region from the low-density to the high-density plasma within an estimated error of about 20% [EBE88b]. Numerical results for these contributions were given earlier

by [ZIM78a, ST079b, KRA79, KRA84]. Alternatively, Tanaka et al. [TAN85a, TAN85b] derived interpolation formulas for the thermody-

namic functions (from MHNC calculations using the local density formalism) which are valid from the strongly coupled, degenerate domain to the weakly nonideal, nondegenerate region. Computer simulation data [BAU80] are reproduced within 5%.

The polarization contribution A, ” for the interaction of electrons with alkali atoms was calculated via the quantum defect theory (QDT) for arbitrary densities [RED87]. The results can be given in a parametrized form as linearized virial coefficients B,“A , with respect to a local polarization potential VPP(r) [RED85],

(37)

The dipole polarizability an and the cutoff radius r. can be determined within the QDT [RED87, RED891 and are compared in Table 3 with more refined calculations and the experimental values available. A reasonable agreement within 5% can be stated.

The shift of the energy spectrum of atoms, AE$, is due to the interaction with free quasiparticles (PP) as well as with other atoms and clusters (vdW, HC), and can be derived from the different in-medium corrections to the Bethe-Salpeter equation [RED85, RED89],

L~E’~’ LZZ AEPP + AE”dW t1.P n,p n,~ + AE,$ . (38)

The polarization contribution is given by AE$, = TZ~~+‘*)B$, similar to (37). The second term describes the long-range van der Waals attraction between atoms and can be calculated again by means of the QDT, AE,,, vdw = TZ(~,~“~)B~~. The last contribution is due to the short-range, hard-core repulsion between clusters. Thfs contribution was treated utilizing the modified Carnahan-Starling expression derived by Mansoori et al. [MAN711 for the chemical potential of a mixture of hard spheres, AEz$ = ,ufcA.

54 R. Redmer I Physics Repports 282 (1997) 35-157

Table 4 Spectroscopic data for alkali atoms A, dimers AZ, and molecular ions Ai which are necessary for the calculation of the

mass action laws (29)

Li Na K Rb cs

RA+ (ao )” 1.29 1.83 2.51 2.78 3.16 RA(ao)” 2.95 3.61 4.46 4.78 5.18

dA2 (ao )b 5.05 5.82 7.41 7.90 8.79

dA;(ao)b 5.88 6.80 8.31 9.04 9.88

rota0 1’ 3.075 3.249 4.062 4.187 4.478

rn(a:)c 162.9 158.0 280.5 306.0 382.4

&,(eYd 5.39 5.14 4.34 4.18 3.89

D.+(eV)’ 1.05 0.72 0.52 0.49 0.39

BA2(cmp’ 1’ 0.6726 0.1547 0.0567 0.0226 0.0127

WA2 (cm-’ )’ 351.43 159.12 92.02 1 57.75 42.019

DA+(eVY 1.30 0.97 0.82 0.72 0.66

BA:(cm-‘)e 0.496 0.113 0.042 0.018 0.0092

’ QA+ (cm- )” 262.2 120.8 73.7 45.0 34.0

a Ion and atom radius. b Equilibrium distances.

’ Cut-off parameter ro and dipole polarizability c(n for the polarization potential [RED87]. d Ionization energy.

e Spectroscopic constants [HUB79].

For the evaluation of the mass action laws for dimers and molecular ions in (29) expressions for the partition functions I& and KAz- are needed which usually separate with respect to the translational, spin, electronic, rotational, and vibrational degrees of freedom [HUB79],

(39)

a:: = [l - exp(-~,+/j?hc)]- , ~2, = exp(bDA,. + AE,, ) .

BA, and oA\ are the characteristic rotational and vibrational constants. DA,, is the dissociation energy of the cluster AN and c is the speed of light.

We have taken into account interaction corrections to these mass action laws via the quantity A_!?*,. Comparing the dipole polarizabilities of alkali atoms and dimers [MIL77], f = CXA~/CIA, one finds f z 1.5. Applying the London relation for the van der Waals constants, CA,, M ~cc,M,AE,AE,/(AE, + AEB), and considering the respective resonance energy of the atom and the dissociation energy of the dimer for AEA and AE,, the following estimates can be given for the virial coefficients:


Effective hard-core radii for the dimers and molecular ions can be derived from the known atomic and ionic radii and the equilibrium distances in these clusters. The parameters needed for the evaluation of the mass action laws (29) are given in Table 4 [RED89, REI95].

3. Thermodynamic properties of dense plasmas

3.1. Thermodynamic functions and plasma composition

In this section, we give explicit results for the thermodynamic properties of the alkali-atom metals, mercury, and hydrogen in the expanded fluid and plasma states (see [RED85, RED88, RED89, ROE93, NAG94, EBE91, EBE92, VAN93]).

The standard model for the calculation of the EOS for dense plasmas considers the interaction between charged particles on the Montroll-Ward level. The interaction between charged and neutral particles is treated via the second virial coefficient with respect to a screened polarization potential. For the interaction between heavy particles, extended fluid perturbation theory applies [BAR67, WEE71]. The short-range, repulsive terms can be described by a hard-sphere reference system, whereas the attractive, long-range part is treated by the second virial coefficients again.

The thermodynamic properties of dense plasmas are treated alternatively within (a) activity or fugacity expansions for the EOS [ROG73, ROG74, ROG81, GAU81], (b) the one-component plasma (OCP) model [BAU80], (c) modified hypernetted chain (MHNC) calculations for the pair correlation functions [ROG84, TAN85a, IYE86, BER91], (d) computer simulation techniques such as the Monte Carlo and Molecular Dynamic method [HAN81, CEP91, PTE94, COL95], or (e) integral equation methods [ROG84].

Hydrogen or hydrogen-helium plasmas are of special relevance for astrophysics. Utilizing the data of shock wave experiments for the parametrization of the interaction potentials and considering self-energy, screening, and local-field effects in the way described above, new results for the thermodynamic functions and their stability behavior can be gained [SAU89, CHA90, VAN90, TAN90, SAU91, SAU92]. Experimental data for the thermodynamic properties of dense alkali-atom plasmas are given in a number of papers [FOR82, ALE83a, ALE83b, BUS83, DIK85, FOR90].

3.1. I. Composition for alkali-a tom plasmas The composition of dense plasmas is determined from the general MAL (30). Before presenting

explicit results, the influence of various nonideality corrections on the ionization equilibrium K*o (32) is shown for Na in Fig. 2

Neglecting nonideality corrections due to interactions with and between neutral particles, the MAL (34) reduces to the usual Saha equation valid for nondegenerate, weakly nonideal plasmas with densities yt 5 lO’*~rn-~ where the concept of lowering of the ionization energy, given by the Debye- Hiickel value --Ice*, applies. For higher densities, the ionization degree aion becomes ambigious which leads to instabilities in the thermodynamic functions. For instance, the critical temperatures obtained from this simple model are usually too high by a factor of about 2-5 compared with the experimental values for the alkali-atoms.

The nonideality corrections due to neutral particles A, , ” A!& and AAA have to be considered in partially ionized plasmas at elevated densities n > lOI cm- 3_ They lead to a drastic increase of

Fig. 2. Degree of ionization of Na plasma as function of the total electron density n, within different approximations [R~~~~~: ttl (dotted line) - neg~~ct~n~ all co~~ct~o~s due to ~~te~ct~~ns with or between neutral atoms, et2 (dashed tine) - taking into account these interactions, c13 (solid line) - additional consideration of dimers.

the ionizat~o~ degree after pass~n~ through a minimum at about n 2 1 Ozo cm-3_ For densities n > 5 x 102r cm-3p the plasma becomes fully ionized again which refers to the Mott effect or ~r~ssure ionization. The pronounced minimum behavior of the ionization degree affects physical quantities such as the electrical or thermal conductivity strongly.

In addition, higher clusters A,, have to be ~o~~~de~~d evaluating tbe general ML (30) for dense plasmas with n > 102@ cm-3 and low temperatures T < SC@0 K. At least dimers Aa and molecular ions Al reach remarkable concentrations of about 22% and 37%, respectively, near the critical points of the alkali-atom metals Na-Cs, i.e. around T~ZOOO K [RED89, D93]. For temperatures T > 5000 K, these clusters are negligible. The composition of Cs plas a as function of the total ion density is shown in Fig. 3 for 7” = 2000 K which demonstrates the variation of the partial densities as discussed above [RED89, RED93].

Other model ~al~ul~t~ons for the ~om~osition of ~a~ially ionized plasmas have also found a considerable amount of neutral and charged clusters in that region. In particular, Iermohin et al. [IER’I l] and Gogoleva et al. [COG841 considered clusters up to mmax = 3 for the calculation of the electrical conductivity in partially ionized Cs plasma. Hemandez EHER84, HER85, HER86f treated the expanded fluid near the critical point of Cs, Rb, and Hg as a generate, pa~ially ionized system with clusters up to mmax = 6 and calculated the EOS as well as the electrical conductivity. Taking into account interaction corrections via the Debye-Hiickel theory for the interaction between charged particles, the excluded volume due to the heavy particles, and local-field effects of clusters via the Clausius-~oso~i relation, reasonable results were obtained. Especially, positively charged cluster ions as Al reach considerable concentrations because of their relatively large binding energy of about IeV.

The composition of hydrogen plasma was determined similarly to that of the alkali-atom plasmas, In some models, the Hl molecules are dissociated into atomic hydrogen which is then pressure

R. Redmer I Physics Reports 282 (1997) 354.57 57

Eogfn cm31

0.8

0.6

18 20 22 24

tog f ne cm3 )

Fig. 3. Composition of Cs plasma at T = 2000 K as function of the total ion density n [FtED89]. The fraction of free

electrons e, atoms Cs, dimers CSZ, and molecular ions Cs,f is shown.

Fig. 4. Composition of H plasma at T = 15000 K as function of the total electron density n [REI95]. The fraction of free

electrons e, atoms H, dimers H2, and molecular ions Ht is shown.

ionized wi~~i~ a narrow d~nsi~ range [ROBSJ, EBE85, HAR87, HUM88], Saumon and Chabrier [SAIJ89, SAU91, SAU927 have found a discreet behavior of fluid by~ogen at high density from their model calculations for the chemical equilibrium between a lower-de~si~, insulating phase and a higher-density, metallic phase which are supposed to consist both of a mixture of Hz, H, H’“‘, and

electrons e. Their results indicate a very gradual increase of the ionization degree in the metallic phase, whereas the metal-nonmetal transition can be located in the lower-density phase by the drastic and discontinuous rise of ionization at about p = 0.35 g cm-j.

This so-called plasma phase transition (PPT) occurs between a weakly ionized phase (CIi,, < 0.01) and a partially ionized one (aion < 0.5). Following Girardeau [GIR90], the system exhibits such a plasma ionization transiton if there is a transition line in the p-T plane separating two different fluid phases I and 11 with different degrees of ionization and a coexistence curve in the pn plane separating a two-phase region from a single-phase region The equation of state would then be non-

ical across the transition line, and the ~a~sition is a phase tr~sition in the the~od~~rn~~

Dimers Hz can reach ~on~en~tions of more than 70% for t~mpera~res less than T < I O4 K and of about 20% for a temperature of T m 2 x lo4 K at densities around y1= 1 023 cmp3. The most striking feature of the model of Saumon and Chabrier is that molecular dissociation and pressure ionization occur almost at the same density for any temperature, in contrast to the papers quoted above, Such a gradual transition was also found recently for H-He mixtures [SCH95].

In Fig. 4, the composition of H plasma as derived from the T= 15 x lo3 K [REI%]. We an clearly distinguish between three plasma is weakly ionized w an ionization degree of less than 5% It consists mainly of atoms H and dimers Hz, the latter roaching

nt approach is displayed for nt regions. The low-density tends up to about 1 023em-3.

~on~e~t~tio~ of about

9~% for T = 104 K and 70% for 7’ = 15 x 103 K at about this density_ Dimers are not ~e~~e~ibl~ in the region above 1 0i9 cmM3.

The second region from 10z3 cmm3 up to 2 x I 02' cmm3 is characterized by partial ionization from 5% to 80%. Here, all species have strongly varying concentrations which are a result of the nonideality corrections to the mass action laws. Dimers vanish at 2 x 1O24 cmm3, where molecular ions H,f reach their maximum concentration of about 35%. Then, the ionization degree is sharply rising with increasing densities due to pressure ionization.

1t would be interesting to study the effect of trimers H3 and molecular ions HT on the equation of state. The rno~e~~lar ions are formed via the reaction HI -t- Hz --+ I-!;” -!- H, which becomes operative ifb Hl and H2 are present in higher ~oncen~at~ons. This is not the case for the pararn~t~~ domain

con red, see Fig. 4. Since bydro~en trimers are formed in a charge transfer reaction of the type

H; + M + J-J; + M+, where M can be an atom H or a molecule H2 (see [FIG84]), this argument applies also here. Therefore, these species have been neglected.

Hummer and Mihalas [HUM881 found a maximum concentration for molecular ions Hz’ of only 6.2 x lo-” in nearly the same density-temperature region within a free energy minimization scheme for the coupled chemical equilibria. This discrepancy is possibly caused by the use of other parameters for the internal partition functions and the simpler neutral-neutral interaction for the evaluation of the occupation probabilities. The latter forces pressure dissociation of H2 for Q 2 0.1 g crnm3 at

arbi~a~ temperatures. The third region above 2 x 102” c -’ is nearly fully ionized and completely described by the

e~enerat~ electron gas im~~erse~ in the positive background protons and molecular ions. For hush-temperature plasmas with T 2 5 x 104K, the fraction of neut and charged clusters is decreasing,

we have simply a three-component system of electrons, ons, and hydrogen atoms that is reasonably well described within an extended Debye-Htickel theory [BER91]. The PPT in hydrogen and the inert gases is discussed in the next section.

The chemical picture can be generalized by introducing excited states or higher ionized atoms (2 - Zj > 1) as new species which increases the number of coupled nonlinear equations (mass action laws) to be solved. Excited states are globally j~cluded in the (Pla~~k-harking pa~itio~ function. They have a larger polari~ab~l~~ and a greater volume than the gro state and cause consider-

ably higher nonidea~~ty ~o~ectio~s. However, the oGcupa~io~ of excit states is less than 1*/o for

T < 2~0~~ in alkali-atom plasmas and T 5 15 x 1Oj K in hydrogen and inert gas plasmas so that they can be neglected. Although the occupation number increases for higher temperatures, the ionization degree itself tends to unity more quickly due to the chemical equilibrium with the free particles so that excited states are of no importance at all (see also [EBE87, ROG90]),

Higher ionized atoms with Z -Zj > 1 are relevant for high-temperature plasmas where the thermal energy is comparable to the second ionization energy. Typical values for I!$,‘, are, for instance, 11,9eV (Ca), 15.0eV (Mg), 18.8eV (Hg), 21.2eV (Xe), 25.1 eV (Cs), 27.5eV (Rb), 54.4eV (He) [DIK85]. These conditions are relevant especially for as~ophysical plasmas such as the interior of the sun ~EBE9Ob~ TUR93], the Brown Dwarfs and the giant planets Jupiter and Saturn [SAU87, HUB87], or White Dwarfs [FON87]. Higher ionized atoms affect the tb~~od~arni~ prop~~igs via their ~~nidea~ity ~o~ectio~s which are propo~ional to (2 - Zi) for electron-ion interaction, and {(Z - Zi 1”) for ion-ion interaction. Subsequently, the ionization energies of bound states are lowered more


rapidly so that complete ionization is reached at lower densities compared to models where only single charged ions are considered. In that way, higher ionization states may be excited in higher fractions as expected from the thermal energy of the system [EBE89, EBE90a, EBE91, FOE92a,

FOE93, EBE95]. However, the influence of heavy ions on the energy levels of bound states is more likely caused by

their microfield distribution than by the respective Debye shift as we know from the study of the shift and broadening of spectral lines in dense plasmas, see Section 8. The subsequent Stark ionization is therefore a more reliable process for the region of high densities than the self-amplification

mechanism mentioned above.

3.2. Critical phenomena and phase transitions in dense plasmas

Besides the composition of dense plasmas, the chemical potential p, the free energy density .f, and the pressure p were calculated via Eqs. (11) dependent on density II and temperature T. Especially, the stability behavior (i3p/&z), > 0 and (ap/aV)7. < 0 was studied.

3.2.1. Critical phenomena and metal-nonmetal transition Classical Coulomb systems such as ionic fluids (dielectric constant E, hard-core radius R = R, =

R_, particle density n = n+ = n_ ) already exhibit phase separation and criticality (for a review, see [FIS94]). Monte Carlo simulations yield a critical point at T* = 2kBT&R/e’ = (0.057 f 0.001) and Q* = 16nR3 =(0.030 f 0.008) [PAN92]. Based on the simple Debye-Htickel theory but allowing for association into dipolar pairs and considering the dipole-ionic-fluid coupling in addition, already a reasonable agreement has been achieved, T* =0.057 and Q* =0.028 [FIS93]. Therefore, the effect of association (bound states) and dipole-ionic-fluid coupling (polarization) should be taken into account also in the equation of state for quantum plasmas.

Considering nonconducting fluids such as hydrogen or the inert gases, the conventional liquid- vapor phase transition occurs at temperatures T < 200 K with a critical point at (T,, pl, pl ), see Table 1. The most striking feature in the critical behavior of conducting fluids such as mercury and the alkali-atoms Cs-Na is the simultaneous appearance of a metal-nonmetal transition near the critical point which is located at temperatures of about 2000 K, see Table 1. For temperatures T > 1 O4 K and at high pressures in the range 10 GPa I_ p 5 100 GPa, the transition from a dielectric state with a low conductivity to a fully ionized state with a plasma-like conductivity is expected to occur in hydrogen and the inert gas plasmas. The question whether or not this transition is a first-order phase transition with an instability region and a corresponding second critical point ( T2, p2, y2) has been studied intensively. We will shortly review the various theoretical approaches to this so-called plasma phase transition (PPT) (see also [EBE91, SAU92, REI951).

Dependent on the location of this PPT in the density-temperature plane, different coexistence lines and supplemental phase transitions are possible, for instance between a dielectric and a metallic solid, or between a dielectric and a metallic fluid. A similar behavior is also expected to occur in highly excited semiconductors such as Si [SMI86]. The corresponding multicritical behavior is illustrated in proposed phase diagrams for H, Xe, and Si in Fig. 5.

The treatment of this up to now hypothetical PPT is of high relevance for some astrophysical objects. For instance, a verification of that transition would imply the possibility for metallization within the giant planets Jupiter and Saturn. Further interesting critical phenomena which might


H PLASMA

“-3 0 3 6 9 12

(a) log (p/borl

0 3

4 ‘a2 0)

1

0

-1

-2

-3

-L

-5 2 3 L 5

(b) lg IT/K)

8 ?! 3

5 $ Q 20 E

F

.

i : ‘3

EHL : b

0

10” 10” 10’” 10” 10’” 10”

Density km-3) (c)

Fig. 5. Schematic phase diagrams for various plasmas: (a) Hydrogen plasma [ROB83]. a - degree of ionization, fi - degree of dissociation, dash-dotted curves - estimated profiles through Jupiter and the Sun, short-broken curve - Monte Carlo results. (b) Xenon plasma [EBE88a]. Solid lines indicate first-order phase transitions. Cr - critical point of the liquid-gas phase transition, Trr - triple point between the solid (s), liquid (l), and gaseous (g) phase. For T > lo4 K, the plasma state (p) is indicated by the degree of ionization CY. Dashed lines show possible coexistence curves. Tr2 ~ triple point between the dielectric solid, the dielectric fluid, and the metallic fluid (mf) phase. Trx - triple point between the metallic solid (ms), the metallic fluid, and the dielectric solid phase. Dotted lines indicate the domain of shock-wave experiments: N - Nellis et al. [NEL82], F ~ Zaporozhets et al. [ZAP84]. (c) Excitonic matter in Si [SMI86]. EHL - electron-hole liquid with a (first) critical point at rr” z 24K. CP - condensed plasma phase with a (second) critical point at Ti2’ % 45 K. A triple point where the gas, liquid and condensed phase coexist is expected to occur at r, % 18.5 K

[see (a) and (b)].

occur in astrophysical objects are crystallization or a glass transition inside of white dwarfs or in the crusts of neutron stars (for a review, see [VAN90]). Treating these systems as one-component plasmas (GCP) consisting of positively charged ions immerzed in a neutralizing background of electrons, ~~s~llization to a bee 1 iee takes place at r = 178 i 1 [SLA#)]. However, this classical approximation is si~i~~a~tly mod d by q~a~~rn effects and e freezing process can be unde~tood as a ~ansfo~ation of a quantum liquid to a quantum solid, where the freezing temperature may be reduced from the classical value [CHA92,DEW93]. These new considerations may have an important effect on the cooling rate of white dwarfs, and thereby on their inferred evolution and ages. In case of a rapid cooling process, ~~stallization may be prevented for 17 1 < r < 2 10 and an amorphous state, the so-called Coxcomb glass, is assumed [ICH83].

A possible phase separation between Fe and H in typical stars, or between H and He within the giant planets may cause an increased gravitational energy which has to be considered in the energy balance of these systems. A Fe-H phase separation within the sun could explain the solar neutrino problem, i.e. the variation of the theoretical value for the solar neutrino flux from the experimental

axially only 25% of the theoretical Iyetomi and ~~hirna~ [IYE%] predicted ase separation to occur for 7* < 5.5 x and p> 10 GPa for which the sun would

be too hot. Other potential candidates for such a process are low-mass main sequence stars or brown dwarfs. A H-He phase separation within the giant planets could be the source of that excess energy by which the energy emitted these planets exceeds the energy received from the sun [STE77].

Advanced high-pressure ex ents such as shock wave compression of solid metals can realize pressures of up to lo3 GPa, tempera~res of up to 5 x I O5 K, and densities of twu-to-four times the crystal density (for a recent review, see Appendix I in [FOR90]). Applying shock wave techniques to gases like xenon, the maximum pressures reached are near 1 OGPa. However, up to now there has been no evidence for the existence of the PPT at elevated temperatures of about 1 O4 K in all gases studied so far (Xe: ~NELg2,ZAF$4]~ H2: [NEL83, ROS83, RQS85, ROS87]; NZ: [NEL80, NEL84, RAD%]).

The nonmetal-metal transition may also occur in solid insulators exposed to high pressures uf p > 10 GPa due to band gap closure, This so-called ~~~~~~~z~~jQ~ of hydrogen was first predicted to occur at pressures of 25 GPa by Wigner and Huntington [WIG351 in their classical study. Such a strongly compressed quantum solid could possess a number of exotic properties such as high- tempera~re superconductivity with a critical temperature of about 100-200 K (see, e.g., [ASH68a]).

New band stickle tribulations for hy~oge~ have been ~er~~~ed by Friedl~ and Ash~ro~ [FRI77]. They predicted a critica density of 1 .I x 1 023 cm- 3 for the transition from insulating to metallic hydrogen which corresponds to five times the melting point density, or a compression of about 200 GPa. Recent calculations predict metallization pressures between 200 and 400 CPa [BAR89]. The unc retical estimates are due to the unknown high-pressure seethes of both molec drogen, and the role of both vibrational and rotational zero-point motion,

The domain of ultrahigh, static pressures above 250 GPa is now accessible with the diamond- anvil cell technique, and the behavior of hydrogen at very high densities has been studied with infrared absorption measurements, X-ray and neutron diffraction, Brillouin scattering measurements, and synchrotron infrared spectroscopy (for a review, see [MA094]$. Aro~d 150 GPa, the Raman- active vibron mode drops in frequency by IOOcm- which is accompanied by an increased reflectivity in the infrared at higher pressures. First interpreted as a precursor of metallization due to band overlap, this behavior may be caused by a transition to a lower-symmetry structure [MAQ90].

62 R. Redmrr I Physics Rrpovts 28.2 (1997) 35-157

Herzfeld [HER271 proposed an empirical criterion for that metallization based upon the lowering of the ionization energy due to the increase of dielectric pe~eab~~~~ with density according to

he pe~eab~li~ of the medium is given in his model by the Clausius-Moso~i relation,

E = 1 -I- 4nna/( 1 - 47ma/3 ), so that at N,, = 3/(4nra) spontaneous ionization occurs which refers to

free conduction electrons as characteristic of metals. Far densities n < ncr, a dielectric (nonmetallic) behavior is obtained. This model was applied to metal plasmas by Alekseev and Vedenov [ALE considering the lowering of the ionization energy by ~~~~~o~-at~rn and ion-atom s~a~~~~g via polarization potential.

The interesting problem whether this metallization is directly accompanied by pressure dissociation of molecular to atomic solid hydrogen or separated from that structural phase transition has not been finally solved. Reliable results for that problem would also give some hints for the similar question discussed above for the PPT at elevated temperatures. Recent experimental and theoretical results indicate a nonmetal-metal transition at p M ( 170 + 20) GPa ~MIN86, AOSB], and a subsequent structural phase transition from molecular to atomic hydrogen around y 4f 1) x lO”GPa [MIN86],

Metallization was clearly verified experimentally for a variety of materials. For instance, the

transition pressures are p = (15 f 2) GPa for Si [ZAP87], p = (132 r4r: 5) GPa for Xe [GOE89], and p = (110 rt: 10) GPa for CsI (which is isoelectronic with Xe; for a review, see [ROS87, ROS95]). The behavior of rno~e~ul~r iodine Tz under pressure is of special interest because of the s~rn~~ar~t~

to molecular hydrogen H-2. Solid iodine undergoes a transition from semiconducting to metallic

conduction at 16 GPa due to band gap closure [BALG I]. At even higher pressures around 2 1 GPa, a structural transition from molecular to atomic iodine occurs [SHT78], Recent high-pressure Raman spectroscopy studies have indicated a more subtle way of dissociation via a quasi-ore-dimensional rnole~~~a~ phase at inte~ediate pressures ~~L~94].

The electrical ~ondu~tivi~ of ~~~~ hydrogen and deuteri~m has been measured in single-s experiments up to 20 GPa and 4600 K, indicating a semiconducting behavior with a band-gap of 11.7 eV at 7.5 cm3/mol [NEL92]. These measurements have been extended recently up to the region of 93-l 80 GPa utilizing a double-shock technique, covering a density and temperature domain of 0.28-0.36 mol~cm’ and 2200--4400 K, respe~ti~~ely [WEI96]. The resistivi~ decreases from about 1 52 cm at 93 GPa to 5 x lo-’ ft cm at 140 GPa, and remains constant at this value for higher pressures. The corresponding conductivity is 2 x lo3 (52 cm)-‘, as typical for expanded fluid metals such as Cs and Rb (see Section 4.7). Thus, at about 140 GPa and 3000 K, fluid hydrogen becomes metallic via a continuous transition from a semiconducting to a metallic fluid due to band overlap. The dissociation fraction of molecular hydrogen at this point was estimated to be only 5%. The authors predict that the hypothetical PPT at about 10 x I O3 K and 100 GPa probably does not occur because the transition to metallic hydrogen occurs already within the molecular phase at lower temperatures, as verified for the first time by their conductivity data. At higher (plasma) temperatures T 2 lo4 K also ionization becomes effective which may be even enhanced by pressure ionization so that the full mechanism of the transition is more involved than in the dense fluid phase. Ex- ~a~olatin~ their data to the solid state at T = 0 K, a ~~eta~~~zat~on pressure of about 300 GPa is predicted.

Nevertheless, supposing the existence of the PPT in hydrogen and the inert gas plasmas for temperatures T > 1 O4 K, the coexistence lines for the liquid and solid dielectric phases, and for the solid dielectric and solid metallic phases at low temperatures T < 1 O3 K, have to be extended into the plasma domain. Dependent on the slope of these curves in the density-tempe~ture plane, several


triple points and a second critical point are possible (cf. Fig. 5 and [EBE85, HAR87, HUB87, EBE95] for H, [EBE88a, FOE92a] for Xe, and [FOE92b, FOE931 for He).

3.2.2. Theoretical results jbr the PPT

Many theoretical studies of the PPT were performed within the chemical picture which considers the various species in the plasma (e,A+,A,At,A,, . . .) on the same footing. As long as these species are the well-defined elements of statistical physics, the density- and temperature-dependent corrections to the thermodynamic functions due to the interactions in the system can be derived by standard methods such as virial expansions.

However, the transition from a dielectric to a metallic state at high densities and elevated temperatures (Mott transition) is forced by pressure ionization (and pressure dissociation for H), so that the concentrations of all species vary strongly in a narrow density range. The most striking feature is a sharp rise of the ionization degree and a subsequent increase of the electrical conductivity. A leap of the reflectivity in that region should clearly indicate the transition from dielectric to metallic behavior. The plasma phase instability and the related critical point derived from those approaches should not be produced by discontinuities in the partition functions of the species involved in these reactions. Furthermore, the theoretical approach should yield the correct results for different regions of the phase diagram.

A rigorous approach to that problem has not yet been developed. The quasiparticle picture of Zimmermann and Stolz [ZIM85] is capable of treating at least the two-particle partition function for arbitrary densities by a generalized Beth-Uhlenbeck formula which considers bound as well as scattering states. Similar expressions have to be derived also for the other reactions in the plasma such as dimerization or molecular ion formation with their respective internal degrees of freedom.

A review of the former approaches to hydrogen was given by Saumon and Chabrier [SAU92]. They studied dissociation and ionization dependent on pressure and temperature within a Helmholtz free-energy model for a nonideal mixture of HZ, H, H+, and e; molecular ions such as Hi and H- were neglected. The nonideality corrections were calculated by means of extended fluid perturbation theory using realistic interatomic potentials. They found thermodynamic instability in that region where pressure ionization occurs and determined the critical point for this PPT. These data are given in Table 5 together with the results of other approaches to H and Xe plasma and reveal two interesting features.

First, the inclusion of the charged-neutral interaction in partially ionized plasma models (PIP) reduces the critical temperatures by about 30% compared to fully ionized plasma models (FIP). The critical temperatures T, are located in the narrow range between ( 15 and 16.5) x lo3 K and are almost insensitive with respect to a variation of the parameters for the nonideality corrections.

Second, the critical pressures show a strong variation. Including higher clusters such as dimers or molecular ions into the models results in a considerable lowering of the values. Furthermore, the density dependence of the nonideality corrections is of significant influence on the stability behavior at high densities. Therefore, different models for the treatment of the short-range interactions between the particles yield also deviating critical pressures and especially, a strong dependence on the respective parameters for, e.g., the hard-sphere reference system can be stated. As long as rigorous results are not available, the critical pressure pc are located in the broad range between 20 and 100 GPa.

64 R. RedrnrrlPhysics Rqwrts 2&? (1997) 3.5-157

Table 5

Theoretical results for the critical point of the hypothetical plasma phase transition (FPT) in hydrogen and the inert gases

Element TC (IO3 K)

PC (g/cm” 1

Method Reference

H 12.6 12.6 6-9

19.0 16.5 16.5

15.8

15.0 15.3

14.9 16.5

95 3.6

0.95

24 22.8 95

194

64.6 61.4

72.3 57

1

0.14 0.13 0.427

I .4& 0.36 0.35

0.29 0.42

PIP [EBE73] PIP [EBE76] Ising [ FRA80] PIP [ROB831 PIP [EBE&5] PIP [ HAR87] FIT [HES87] PIP{ I) [SAU91] PIP(2) [SAU92] PIP @RE93a, SCH95] PIP [REI95]

He 35 660 2.2 PIP cp 1 [FOE92b] 120 10x io3 8.6 PIP cp2 [FOE92b]

19 500 2.1 PIP & MSA cp 1 [FOE931

61 20x loJ 16 PIP & MSA cp2 [FOE931 17 722 2.5 PIP [KI%E93a, SCH95]

Xe 12.0 I.81 1.51 FIP [DIE803 14.1 2.1 2.45 FIT [HESS71 16.0 2.0 1.75 PIP /KAH88] 12.6 9.9 3.7 PIP [EBE88a] 8.7 25.5 4.6 PIP & MSA [FOE92a]

Methods: FIP - fully ionized plasma model, PIP - partially ionized plasma model, FIT - estimates based on empirical rules, Ising - transverse Ising model, MSA -- mean spherical approximation for charged hard spheres. For He, two subsequent PPT have been predicted [FOE92b, FOE93J, one for each ionization stage, where the respective critical points

are denoted by cpl and cp2.

3.2.3. Chemical picture nrzd PPT The chemical potential of hydrogen plasma is given in the present approach via

In Fig. 6, we display this quantity for T =( 10,15,20) x 1 O3 K within three different models. The ideal part of the chemical potential is always calculated for arbitrary degeneracy via the Fermi integral. The electran and proton gas contributions are taken in the parametrized form of Ebeling and Richert tEBE82, EBESS].

The simplest model (a) (dash~otted line) considers only electrons, p~otons~ and h~~~~e~ atoms, the fraction of which was determined by means of the Flick-Larkin partition functian (35). Dimers and molecular ions were neglected. Furthermore, the hard-core contributions which might be the source of substantial uncertainties at high densities are also neglected,

Molecular ions and dimers as well as the respective hard-core contributions have been included in model (b) (dashed line). Zn case (c) (solid line), the Pluck-L~kin p~~itio~ action was replaced


-1.c

PWY)

-1.2

-1.5

-1.7:

-2.0

-2.2,

,

5-

j-

/

lwo[ 4cmw3) 1

Fig. 6. Chemical potential p of hydrogen plasma for T =( 10,15,20) x lo3 K within different models [REI95]. Dash-dotted line (a) - using the Planck-Larkin partition function and neglecting clusters; dashed line (b) - considering clusters Hz, Hz+ in addition; solid line (c) - using the correct two-particle partition function and considering clusters.

by the Zimmermann-Stolz two-particle partition function so that the influence of scattering states is included. All other contributions are the same as in case (b).

The resulting curves for the chemical potential indicate that a thermodynamic instability occurs in all three models. The instability is inherent already in the simplest model (a) and, therefore, produced by the electron and proton gas contributions to the chemical potential which were taken from Ebeling and Richer-t [EBE82,EBE85]. Not surprisingly, the results for the critical point of the plasma phase transition within this model coincide very well with their estimates. We find T, = 15 x lo3 K,

66 R. Redmer I’ Ph_wics Reports 282 (I 997) 3S-1Si’

pc=o.2 gem-3, and pc ==26 GPa compared with their data, r, = 16.5 x I O3 K, pc = 0. I3 g cmm3, and pc = 23 GPa.

The consideration of molecular ions and dimers as well as of eflective int~~ole~u~ar interactions in model (b) changes the results only slightly between log,,(n x cm3) = 22.5-23.5. The instability region is now smaller and shifted towards higher densities, indicating that the consideration of further neutrals (dimers) tends to stuhilize the system. This is a result of the polarization contributions &ii

and &,i~ in the mass action laws which force the ionization degree towards the value @ion =0.5. In the region of the Mott effect, where Qn inc.reases sharply, these terms yield a smoother transition. Furthermore, the small discrepancies between the results of models (a) and (b) for high densities indicate, that the influence of the effective intermolecular interactions on the location of the PPT is surprisingly small

The strong influence of two-panicle s~atte~~g states on the tbe~modynamic prope~ies at hig densities is to be seen from curve (c). Here, instead of the Pla~~k-~arki~ pa~ition ~u~~tio~ f IS), the correlated two-particle density YI:T) ( 14) was take n into account. The consideration of two-particle scattering states tends to enlarge the instability region compared with model (b). This behavior is an effect of the smoother decrease of Z: at high densities compared with .Z$, see Table 2. As a result, the ~oncen~atio~ of atoms H, dimers HZ, as well as of rtloiecular ions H2+ decreases gradually at high densities (see Fig. 4) so that these species con~b~t~ to the chemical potential up to higher densities As a further consequence, the strong increase of the chemical potential in the hig -density limit occurs only at higher densities where the effective intermolecular interactions already become important.

The critical point of the PPT within model (c) is located at T, = 16.S x 1 O3 K, pc = 0.42 g cme3,

and pc = 57 GPa which is in good agreement with the data given by Saumon and Chabrier [SAU92], T, = 15.3 x 10’ K, 0, =0.35 gem-“, and pc = 61.4 GPa, see Table 5.

Hydrogen and the inert gases have relatively high ionization energies Eion > 12 eV, so that the conventional liquid-vapor phase transition (T, 5 300 K, p! < 6 MPa, see Table I ) Es well separated from the hypothetical PPT ( T2 > lo4 I(, pz => 1 GPa, see Table 1). The alkali atoms Cs-Li have considerable lower ionization energies between 3.89 eV < Eion < 5.14 eV so that the metal-nonmetal transition occurs near the critical point of the liquid-vapor phase transition which is located in the temperature range I920 K I_< T, <= 3500 K and at pressures of 9 MPa 5 pc < 70 MPa (see Table 1). Ad- vanced static high-temperature, high-pressure experiments ave been ~erfo~ed for the alkali-atom elements with the lowest ionization energies Cs-K up to the critical point (for a review, see [WIN89, HEN89, HEN91]). The complex behavior of these materials during the expansion along the liquid- vapor coexistence curve from the melting point (liquid metal) up to the critical point (expanded fluid) is apparent in the EOS data, the transport, structure, optical, and magnetic properties.

The most striking feature of the the~odynamic properties of the alkali-atom fluids is the extreme asymmetry of their coexistence curve, see Fig. 7. Cailletet and Mathias [CAB61 advanced a linear decrease of the averaged density of the saturated liquid and its vapor with temperature, which is known as the empirical law of rectilinear diameter Pd = i(p: + &)=A -BT. This law is satisfied for nonconducting fluids with high accuracy, whereas expanded fluid metals show pronounced deviations

CL6 07 0.8 a9 IO 1.1

T/T,

Fig. 7. Reduced densities of the coexisting vapor and iiquid of the inert gases Ar and Xe ~~rn~ar~d with those of the metals K, Rb, Cs, and Hg [HENPI].

near the critical point. A weak singularity in slope in that region has been predicted theoretically ~~~~~Q]. Jiingst et al. ~J~~$5] found from the analysis of their data for Cs and Rb an expansion

of the form

(42)

and were able to extract the prefactors 91 and the exponents 01 and di. The exponents 01 are very e tbeoreti~al value 0.11 of the re~o~ali~ation group theory [NKXl] (0.13 -;t: 0.03 for 4 f 0.03 for Rb) and dominate the scaling behavior for 10s3 < r < IO-‘. These re-

sults strongly support the suggestion of Goldstein and Ashcroft [GOLM] that the state dependence of the interparticle interaction and, especially, their changes in the course of the metal-nonmetal transition lead to very large amp~i~des of the 1 - IX anomaly in the diameter liquid-vapor coexistence curve. He~and~z et al. [HER93] have recently obliged as~rne~i~ stence curves already for the homogeneous electron fluid treated in Hartree-Fock approximation, similar to earlier results of Iasilevski for the OCP [IOSSS]. These results indicate the significant influence of the charged component on the special features of the EOS for expanded metal fluids.

Furthermore, the empirical principle of corresponding states (PCS) which was successfully applied to describe the EOS of simple fluids such as the inert gases or molecular fluids is apparently not valid for liquid metals. The PCS applies to any EOS containing two parameters in addition to the gas constant _22. For instance, a universal function of the reduced pressure, temperature, and volume f( pr, T,, I$) = 0 exists already for the van der Waals EOS which is, thus valid for all substances belonging to that particular EOS. The high accuracy of ple fluids obtained with the PCS is a consequence of the pai ise additivity of t tentials, the vapidly of

e same potential for the liquid and vapor phase, and that this potential depends only on two parameters. Therefore, the EOS for this particular group of substances scales also with respect to the potential parameters as given, e.g., by the Lennard-Jones potential.

68 R. Retirnerl~~ysics Reports 262 (1997) 3.5-151

Table 6

Comparison of theoretical and experimental values for the critical point of Cs

z 61 &Pa,

PC

(g/cm’) Method Ref.

1924 9.25 0.38 EXP [JUE85]

2040 I I.5 0,404 PCS [~~L~7]

2~~~ 14.2 0.350 PCS [~~R44] 1867 7.00 ~ PCS [HOH77]

2045 1 1.62 Q.422 Estimates [MAG85]

9500 5600

4500

2600 2200 2000

180 22

46

1942 23.3

2325 61.5

2350 6.0

0.22 0.02

0.10

0.665 0.22

0.55

0.534

I .32 0.47

FIP [DIE801 FIP [ZIM77] FIP [EBE79]

PIP [RIC84] PIP [RED851 PIP [RED891

VDW & HCM [YOU7 1 f

VDW ~d~mers~ [RUS94] MC [CHAoSJ

Methods: EXP - experiment, PCS - principle of corresponding states, FIP - fully ionized plasma model, PIP - partially

ionized plasma model, VDW - van der Waals equation of state, HCM - hard-core model, MC - Monte Carlo simulation.

These assumptions are at bes iled for the dilute metal vapor, whereas the behavior of arid alkali-atom metals (and Hg) is involved. There, the ions interact via a dynamieally screened Coulomb potential where the dielectric function of the degenerate electron gas includes local-field corrections” The degeneracy is weakened during the thermal expansion from the melting point up to the critical point along the coexistence line and th temperature becomes of irn~o~nce. This strong state dependence of the interaction implies that ere is no rigorous PCS for nonmetallic and metallic fluids and even not for the alkali-atom metals as a group. However, the alkali metals K, Rb, and Cs behave very similarly [HENQI] and relatively simple parametrizations for the EOS such as a van der Waals equation combined with the hard-sphere model [YOU71] or a two-state van der Waals equation with a denser-dependent cohesive energy [ROSM] yield reliable results.

3.3.2. Results for the critical point

The present approach for the thermodynamic functions p(p, T) and ~(n, T) via the EOS for quasiparticles (1 I ) is valid for arbitrary degeneracy, i.e. for dense plasmas as well as for expanded fluids. Earlier ca~eulations have dealt exclusively with overcritical temperatures, i.e. the plasma state (see next section). As a special ~bara~teristi~ of the EOS, the critical points have been de alkali-atom metals and mercury from the stability Gond~ti~~s (~~/~~)~ 1 0 and (~p/~~)~ 5 0, respectively. They are compared with other theoretical results and the experimental values in Tables 6 (Cs) and 7 (Li-Rb).

Although the PCS is of limited validity as discussed above, it yields reliable results with deviations of about 10% for the critical temperatures and up to 20% for the critical pressures which are usually

R. Redmer I Physits Reports 282 (1997) 35-157 69

Table I Comparison of theoretical and experimental values (if available) for the critical point of the alkali-atom metals Li-Rb (see table 6 for Cs)

Element PC (S/cm3 )

Method Reference

Rb 2017 12.45 0.29 EXP [JUE85] 2085 14.27 0.352 Estimates [MAG85] 2061 30.8 0.421 VDW & HC [YOU711 2200 65 0.45 PIP [RED891 2475 7.3 G.35 MC [CHA95]

K 2178 14.8 0.18 EXP [HEN911 2239 15.44 0.192 Estimates [MAG85] 2185 39.6 0.237 VDW & HC [YOU711 2350 69 0.21 PIP [RED891 2550 7.0 0.22 MC [CHA95]

Na 2485 24.8 0.30 EXP [BIN841 2507 26.16 0.210 Estimates [MAG85] 2635 92.1 0.269 VDW & HC [YOU711 2400 140 0.27 PIP [RED891 2970 12.8 0.22 MC [CHA95]

Li 3223 68.9 3503 38.42 4176 84.0 3831 242.2 3500 71

0.105 0.110

0.147

0.025

EXP (dyn.) [FOR751 Estimates [MAC%51 Estimates [HES89] VDW & HCM [YOU7 1] PIP [RED891

Methods: EXP - experiment, FIP - fully ionized plasma model, PIP - partially ionized plasma model, VDW - van der Waals equation of state, HCM - hard-core model, MC - Monte Carlo simulation.

estimated by extrapolating the rectilinear diameter to the value of T, where the slight singularity in the slope of that curve occurs. The best fits [MAG85] were performed for each critical parameter independently.

The simplest model for the EOS of dense alkali-atom plasmas, the fully ionized plasma (FIP), holds for high temperatures T > 5000 K and yields, not surprisingly, critical temperatures which are too high by a factor of at least two. The critical temperatures are decreased considerably by taking into account nonideality corrections due to neutral particles in the more general model of a partially ionized plasma (PIP). Furthermore, also the critical pressures are decreasing when including not only atoms A [RIC84], but also dimers A2 and molecular ions as Al [ALE83a, RED85, HER85, RED89]. However, the critical pressures and densities are still too large by a factor of 2-5. A better agreement with the experimental values is expected when treating the short-range repulsive forces between the heavy particles (ions, atoms, dimers) in a more sensitive way than the hard-sphere reference model, for instance, utilizing fluid variational theory [ROS87].

In the density region near the critical point, the ion-ion structure factor plays a crucial role for the metal-nonmetal transition. The ion contributions to the EOS, so far estimated via the OCP reference system, have to be calculated with standard methods of liquid state theory, for instance within the pseudopotential method [OSM87, SIN911 or the MHNC scheme [MAT90, MAT91].

70 R. RedmeriPh.ysics Reports 282 (1997) 35-W

Recently, a lattice gas ode1 has been proposed [CHA93, TAR93, TAR951 which allows an inte- grated study of spatial, the~od~amic and electronic progenies of expanded alkali metals. Grand canonical Monte Carlo simulations have shown that some features of the simultaneous liquid-vapor and metal-nonmetal transition near the critical point can be described within this model. Especially, nonadditive interactions due to valence electron localization are taken into account. The critical data and the shape of the coexistence curve for Cs, Rb, K, and Na are in good agreement with the ex~e~rn~nta~ obse~ations [CHA95], see Tables 6 and 7,

Likalter ~LIK9~] supposed a virtual atomic s~c~~re in a strongly coupled metallic plasma. The metal-nonmetal transition was studied within the percolation model for overlapping classically accessible spheres of valence electrons where mutual screening leads to a mixing of bound and free electron shells [LIK92]. A three-parameter equation of state was fitted by means of the critical data for Cs. The critical data of other metallic fluids such as Li and Al are estimated by scaling the ionization energy and the valency through the Cs data. Qnly a flat part of the coexistence curve near the critical point is reproduced, but the critical exponents [see Eq. (42)] are close to the theoretical values [G0L85]. The coexistence curve shows a larger asymmetry than obtained experimentally. Most interestingly, this percolation model explains condensation of gaseous metals as a result of exchange interaction in some atomic st~~ture Contras to the ~onde~~~tion of nonmetallic, atomic

gases due to a~active van der Waals forces.

3.4. Thermodynamic properties of’ dense alkali-atom phmus

The the~od~amic functions p(p, T) and ~(YE, T) of dense alkali-atom plasmas are also calculated via the EOS for quasi~a~i~les (I 11, solving the system of coupled mass action laws (29) and (30). Earlier calculations have dealt with high-temperature, fully ionized plasmas [ZIM77, EBE79, DIE80]. Then, the thermodynamic properties of partially ionized alkali-atom plasmas have been studied [ALE83a, RIC84, RED85, HER& HER%, RED& RED89], The concentration of neutral particles (atoms, dimers) and the strength of their interaction (polarization and van der Waals potentials depends on density and temp~ra~re ich is of papillar signi~~ance near the critical point where the bound states disappear due to pressure ionization and dissociation and a subsequent

Table 8 Degree of ionization for alkali-atom plasmas at T = 4000 K [STA96]

Li Na K Rb CS

16.0 0.13 16.5 0.76-I

17.0 0.45-l 17.5 0.25-I 18.0 0.25-1 18.5 0.88-2 19.0 0.54-2 19.5 0.38-2

20.0 0.35-2 20.5 0.72-2 21.0 0.47- 1

0.18 0.11

0.65-l 0.38-l 0.22-I 0.13-I 0.82-2 0.55 -2

0.50-2 0. I l-l 0.60-I

0.48

0.31 0.20 0.12 0.73-l 0.45-l 0.30-I 0.24-l

0.32-I 0.88-I 0.2 1

0.56

0.38 0.25 0.15 0.94-I 0.60-l 0.40-l 0.33-l

0.45-l 0.10 0.24


Table 9 Degree of ionization for alkali-atom plasmas at T = 6000 K [STA96]

log(ni[cmP3]) Li Na K Rb CS

16.0 0.88 0.92 0.98 0.98 0.99

16.5 0.74 0.82 0.94 0.96 0.97

17.0 0.56 0.65 0.86 0.89 0.92

17.5 0.39 0.48 0.73 0.77 0.83

18.0 0.26 0.32 0.57 0.62 0.70

18.5 0.16 0.22 0.42 0.48 0.55

19.0 0.11 0.14 0.32 0.37 0.44

19.5 0.78-l 0.1 I 0.27 0.31 0.36

20.0 0.70-I 0.10 0.27 0.29 0.33

20.5 0.94-l 0.14 0.30 0.31 0.34

21.0 0.16 0.21 0.37 0.38 0.41

Table IO Degree of ionization for alkali-atom plasmas at T = 10000 K [STA96]

Li Na K Rb cs

16.0 1.0 1.0 1.0 1.0 1.0

16.5 1.0 1.0 1.0 1.0 1.0 17.0 0.99 0.99 1.0 1.0 1.0 17.5 0.96 0.97 0.99 0.99 0.99 18.0 0.91 0.93 0.96 0.97 0.97 18.5 0.81 0.86 0.92 0.93 0.94 19.0 0.69 0.75 0.84 0.86 0.87 19.5 0.56 0.64 0.76 0.77 0.79 20.0 0.46 0.55 0.67 0.67 0.68 20.5 0.40 0.48 0.58 0.57 0.58 21.0 0.38 0.45 0.53 0.52 0.53

transition from nonmetallic to metallic behavior occurs as has been shown for the electrical conductivity [RED92]. Molecular ions such as Cst may be of importance in the course of that transition as it can be concluded from the magnetic properties [RED93].

The final results for the degree of ionization of all alkali-atom plasmas are shown in Tables 8-l 0 as a function of the total number density for various temperatures [STA96]. The parameters necessary for the evaluation of the given formulas are summarized in Table 4. The following general behavior can be stated for the composition of alkali-atom plasmas. For isotherms T 2 IO4 K, the plasma is nearly fully ionized with C(ion > 0.5. For lower temperatures, partial ionization has to be taken into account for densities 1 016 cmP3 2 y1 5 3 x lo*’ cmb3. There, the different nonideality corrections become of importance. Especially, the charged-neutral interaction yields significant shifts because of the large polarizability of neutral alkali atoms. These contributions shift the ionization degree towards the value 0.5 compared with the ideal Saha equation and, thus, tend to stabilize the plasma.

For the low-density region, n < lOI cm-3, the Debye-Htickel theory is applicable, whereas for the high-density domain, yt 1 3 x 102’ cmP3, the plasma reaches full ionization by pressure


ionization, The hard-core repulsion terms dominate the behavior in that region and a strsng dependence upon the chosen bard-core radii occurs. We have taken here the ~~s~llo~aphi~ radii because simple alkali ions A’ have closed electron shells w ich should not be a eted very much by the medium.

Higher clusters have significant concentrations only for relatively high densities and for temperatures T < 3000 K. Dimers AZ, for instance, reach maximum concentration of about 22% at a density

wher~~~ molec~~ar ions AZ may have ~on~en~ations of up to 38% at n M (2- ?) x IO*’ ernp3 [RED89, RED93]. The ionization degree is sharply rising with ~n~~eas~~g densities which is a result of pressure ionization (Mott effect) as discussed above and, finally, all the clusters disappear.

Mercury has the lowest critical temperature (175 1 K) of any fluid metal so that it is the most extensively investigated substance for the study of the interreIations of the metal-nonmetal transition and the liquid-vapor phase transition [HEN89, HEN90]. Precise measurements of the thermoelectric [GOE88] and optical [UCH88] properties including the exact location of the critical point (r, = 175 1 K, pc = I673 bar, and ec = 5.8 g ernm3 ) are available for fluid rner~~~ over the whole liquid- vapor coexistence range, see Fig. 7.

The data clearly demonstrate that radical changes in the electronic states occur from phase to

phase similar to the alkali-atom fluids. For example, far below the critical point, the liquid phase is cunduct~ng but the coexisting vapor p ase is not. Nearer to the liquid-vapor critical point, where the distinctions between the coe~~st~~~ phases van&h, the data for the e~e~~~a~ conductivity the the~oelec~ic power, the Knight shift [WAR82], and optical properties show that nonmetallic behavior is present in both phases. The metal-nonmetal transition appears to occur at about 9 gcme3, i.e. above the critical density.

The existence of the electronic transition makes the problem of describing theoretically the liquid- vapor transitions more di~cult for mere than for simple nonmetallic fluids. The two limiting cases of the dense liquid metal and low-density vapor phase are, in general, reasonably well understood. However, the connection between these limits through the critical region of the phase diagram has been described only within simplified models [ALE83a, HERSS]. A complete and satisfactory solution of the problem requires the simultaneous calculation of the electronic structure and the phase behavior over wide ranges of pressure and tempera~re starting from realistic atomic properties of rner~~. A unified treatment has to include both ~~rn~t~~g cases, van der Waals and metallic bonding, and their interplay as function of density. This is strongly sup-

ported by experimental results for the size dependence of the ionization potential for Hg clusters produced by nozzle beams [RAD87]. For cluster sizes of IZ > 14, the electronic states are expected to be delocalized and the van der Waals bonding is ~dually converted into a metallic behavior.

Such a unified, quantum statistical treatment of the electronic and thermodynamic phase transition was introduced above for the alkali-atom metals [RED89, RED93, CHA93]. Simple models such as an ideal mixture of quantum gases or a fermion liquid of quasiparticles are not sufficient to describe strongly correlated systems where effects such as, e.g., the Mott transition from localized, bound electrons to itinerant, free electrons are of ~mpo~ance.


3.5.1. Partially ionized mercury Starting point for study of the EOS of mercury is again the relation n = n(jI, p) ( 11). Stability

against phase separation requires (ap/&z)r > 0. If this stability condition is violated, a Maxwell construction can be used to determine the phase transition region.

We consider mercury along the liquid-vapor coexistence curve as a charge neutral system consisting of electrons, density 2n, and mercury ions Hg 2+ density II [ROE93, NAG94, NAG96]. Potentials ,

are assumed between these constituents which, in principle, can be deduced from a more elementary description of the system. The cluster decomposition for the self-energy (22) allows to account for the formation of bound states such as neutral Hg atoms and Hg+ ions. The total density is decomposed into a contribution of free particles and the contributions of bound and scattering states.

With respect to the variety of bound states, we restrict ourselves to the ground states of Hg+ and atomic Hg, neglecting excited states or further possible constituents such as Hg-, Hg2, or Hg:. The Green’s function approach allows us to introduce formally new chemical potentials pC corresponding to the different components in the system, {c} = {e, Hg2+, Hg+, Hg} (chemical picture). The relation

iUHg = pHg+ + /& = fiHg’+ + be = /l (43)

expresses the chemical equilibrium of the respective chemical reaction, Hg Z$ Hg+ + e, and Hg+ L$ Hg*” + e. The total density is decomposed into the partial densities IZ, of the respective constituents in this chemical picture, n = nug + It&.+ + nHgL+, 2n = n, + ring+ + hHg, where the partial densities are again given by Eqs. (27).

For further evaluation, it is useful to separate an ideal part py of the chemical potential according to

nHg = /‘$ exp[h$g - Ekg)l , &g+ = 24: exp[B(&g+ - E&+ >I ,

nHg:+ = A;tg’ exp[8&2+], ne = 243Q2[Pc1:dl 9 (44)

where Eig+ = -18.8 eV and EEg = -29.2 eV are the energies to bind one (Hg+) or two electrons (Hg) to the Hg2+ ion, respectively. LI, is again the thermal wave length, and F,,* denotes the Fermi integral; spin multiplicities are given explicitly. The excess parts ApC = pC - ,uLd are determined by the in-medium shifts of the energies E,(p). Before specifying A,uL, as function of the partial densities n, due to the interactions with the various constituents, we give the solution of the EOS n(fl,p). Effective binding energies which are density- and temperature-dependent are introduced via A,uL,:

E Hg+ = Eig+ + A PHI+ - APHID+ - A,uL, , EH~ = Ei, + ApHg - A/.LH~~+ - 2Ape . (45)

Utilizing the condition of chemical equilibrium (43) and eliminating the ideal parts of the chemical potentials, the partial densities are given by

nngz+ = 3 { 1 + exp@(yy - EHg+)]}-’ , nHg = nHgl+ exp[fi(2$ - EHg)] . (46)

From the condition of charge neutrality follows the relation &g+ = ~1, - 2&g’+. In this way, the partial densities nugZ+, nHg+, nHg are given in terms Of j,LLd or, solving the Fermi integral in (44), by n,. The EOS is then obtained in parametric form according to n(n,, T) and, furthermore, we have

p(?ze, T) = k,T h(nHg2+/iLg) + A,LLH~Z+ + 2~: + 2Apu, . (47)

74 R. Redmer / Physics Reports 282 (1997) 35-157

From the effective binding energies EHg+ ,EHL!, Eqs. (45) and (46), the ionization energies Zng+ = E Hg+ -EHg and IHg’+=E,+ -EHg (with EHg:+=ApHg:+ ) are obtained which are necessary to separate one or two electrons, respectively, from neutral Hg atoms. Due to the interaction parts of the chemical potentials Apu,., these ionization energies are density- and temperature-dependent. We will discuss these effective ionization energies in relation with the metal-nonmetal transition.

With increasing density, the binding energies EHg and EHg+ go to zero. As a result, the corresponding bound states vanish - they are dissolved in the continuum of scattering states (Mott effect). The corresponding partial density no longer contributes to the total density above the Mott density. However, the physical properties change smoothly at the Mott density as proven by the Levinson theorem [BOL79]. The jump in the bound state contribution is compensated by a corresponding jump in that of the scattering states [ZIM85, ROG86] as already shown for hydrogen. These discontinuities are avoided using a simplified version of the Planck-Larkin partition function (??) for the

low-temperatures relevant for mercury fluid,

Z PL,eK = @(I,.) [exp(-b1(. - 1 + /UC] ,

where O(x) is the step function.

(48)

The interaction parts of the chemical potentials were determined as function of the partial densities similar to the case of hydrogen and the alkali-atom plasmas and we have

APU,. = C AILI (49)

Aped denotes the contribution of species d to the shift of species c. For the long-range Coulomb interactions between charged particles, efficient interpolation formulas

[ERE88a] are utilized, for Hg in the generalized form valid for the case of multiply charged ions. It has been shown already for H and the alkali-atom metals that the long-range Coulomb interaction leads to a PPT from a phase of low ionization degree to a phase of high ionization degree (Mott transition). However, in order to obtain relevant results for mercury, we have to take into account short-range interactions as well which are of special importance for ions and neutrals. In principle, these short-range interactions can be derived from an analysis of the cluster expansion of the self-

energy [RED87, NAG92]. The van der Waals interaction between neutrals (Ap Hg,Ht:) has been derived from the experimental

second virial coefficient using a Lennard-Jones potential. Comparing with measurements in the low- density region [GOE88], a parameter set c = 0.00652 Ryd and B = 5.07~ has been used [ROE93]. The hard-core contributions to the chemical potentials were treated by a Carnahan-Starling formula [CAR69]. For the hard-core diameters we have adopted the values dHg = 5.1a0 in correspon- dence with the Lennard-Jones parameters, dHg+ = 4.5~ according to the reduction of the atomic

wave functions in Hg+, and dHg:+ = 2.04~ from the measured compressibility in the liquid state; dc,d = (dc + dcl)/2. The results given below are not very sensitive with respect to the parameters dHg+, dtlg’ f because of the low concentration of the respective species.

The electron-ion interaction is modified at short distances by the repulsion of the bound electrons due to the Pauli principle. In a simplified treatment, this contribution is taken into account by an excluded volume which is also determined by the hard-core radius.

The treatment of the interaction between charged particles and neutrals is very important as already shown for the alkali-atom metals. The screened polarization potential (37) has been used to

0 2 4 6 8 10 12 14

total mass d&&y p [g/cnG]

Fig. 8. Ionization energies of mercury atoms as a function of the mass density at temperatures TI = I260 K and TZ = 1750 K

[NAG94]. Experimental data (solid circles) are taken from [UCHSS]. The theoretical Mot? density at 13 gem-’ is too high compared with the experimental value of 9 g cm-’ I

100

(1

7

8

6

r

partick density [cm’3]

Fig. 9. Composition of mercury vapor at T = 1840 K as function of the particle number density [NAG96].

determine corresponding virial coefficients. The cutoff parameter rO = 1.4~ was chosen in accordance to [GOL64]. The screening length should also reflect the contribution of neutrals, especially for the very weakly ionized mercu~ vapor. This can be ac ieved within the extended RPA given by Rijpke and Der [ROE79]. In the simplest approximation, the inverse screening length is defined by a Clausius-Mosotti-like relation,

76

-2.34 2 4 6 8

tstal mass densjty p [gbn”]

Fig. 10. Chemical potential of mercury for various temperatures as function of the total mass density [NAG96].

Fig. 11. Theoretical ~~ex~ste~ce curve of mer~u~ (broken line [NAG96]) ~~rn~~d with that of a pure ~en~ard-JQ~es reference system (dotted fine) and the expe~me~tal data (solid curve [GOE88j).

where E is the dielectric function, and an = 34.4~~: is the dipole polarizability of mercury atoms. Using these contributions to Jpc, self-consistent solutions for the partial densities can be found. The results for the relevant quantities are shown in Figs. 8-11.

In Fig. 8, the ionization energy I”(n, T) is shown for different temperatures as function of the total density n. Starting with the atomic ionization energy of 10.44 eV in the low-density limit, it decreases with density due to the pol~i~abi~i~ of the rn~d~~ as described by the polarization potential. Passing trough zero indicates the Mott density nM,,(T) where the bound state disappears, and a transition to the double ionized state occurs. We have neglected the two excited states of neutral Hg at energies which are almost not populated at the relatively low temperatures considered here.

The density dependence of the calculated ionization energies can be contrasted with the experimental behavior of the optical properties of fluid mercury [UCH88]. Dense liquid mercury has a behavior cfose to a metal showing Dude-bike optical conductivi~ G(W) deriving from the free electrons. This gradually changes with density until by 9 gem-’ O(CU) is characteristic of materials with energy gaps. At even lower densities, down to about 3 gcmm3 in the vapor, optical absorption edges are observed from which effective optical gaps can be determined (solid points in Fig, 8). For still lower densities~ this is no longer possible because here the expe~menta~ly observed abso~tion edge is dominated by transitions to the broadened singlet and triplet states of the neutral mercury atom [UCH81]. The latter hampers the observation of the transitions to the continuum states. It should be emphasized here that the 1~Gatio~ af the tt density at 13 gcmm3 as ob~ined from this calculation (see Fig. 8) is sensitive with respect to e reagent of screening in the medium, and a more refined calculation should account for dynamical screening.

The composition of mercury, shown in Fig. 9, is mainly determined by the lowering of the ionization energy discusse above. Up to the Mott density, the composition is gove~ed by the contribution of neutral atoms. The contribution of the single charged mercury ions is small, and the partial density of Hg*+ is negligible. Therefore, mercury vapor can be considered as a no~metall~~ dielectric system up to the Mott density. The reason for this behavior is found from

Table 11

~~~~a~i~~~ of theoretical and e~per~menta~ values for the critical point of mercu~

7-C (K) &Pa, PC

(s/cm3 1

Method Reference

1751 167.3 5.8 EXP [GOES83

43 000 3 x lo4 20 FIP [ROE931

2Q74 178 3.9 VDW+SSM [YOU71] 1563 92 4.0 VDW+HC~ [YOU711 1740 6.2 FIP+HC~ ~AG94~ 1840 173 4.5 PIP+HCM WAG961

Methods: EXP - experiment, FIP - fully ionized plasma model, PI - partially ionized plasma model, SSM - soft-sphere

model, VDW - van der Waals equation of state, HCM - hard-core model.


the ionization energy the value of which is large compared with the temperature. Above the Mott density, mercury becomes fully ionized.

Fig. 10 shows the chemical potential ,u for different temperatures as a function of the density. The instability region (all/&z), < 0 should be replaced by a Maxwell construction. It can be shown that the lowering of the ionization energy with increasing density destabilizes the system so that the liquid-vapor phase transition which is already present in the atomic system is shifted to higher temperatures. Considering different isotherms, the critical parameters T, = 1740 K and ec = 6.2 g cm-3 are found which are in good agreement with the experimental values, see Table 11.

The coexistence curve is obtained from the Maxwell construction and shown in Fig. 11. The theoretical values are in reasonable agreement with the experimental results. However, the theoretical Mott density at 13 g cm-3 is higher than the experimental value of 9 g cme3. These discrepancies in the electronic properties near the critical point may be the reason for the deviations of the theoretical coexistence curve from the experimental one which are still to be seen. Furthermore, the treatment of the ion contributions to the self-energy has to go beyond the OCP model, then accounting for the finite volume of the heavy particles and the structure in the system.

4. Transport properties of dense plasmas

4.1. Nonequilibriunz statistical operator

The nonequilibrium properties of dense plasmas and expanded fluids have been studied both experimentally and theoretically. The response of the system to external fields (electric, magnetic, radiation) and/or gradients of macroscopic parameters (temperature, concentration) is an irreversible process. Following the definitions of Kubo [KUB57], mechanical perturbations due to external fields can be described by respective terms in the Hamilton operator. Nonmechanical perturbations such as heat or particle exchange with a reservoir cannot be treated in that way.

4.1.1. Approaches to transport properties The microscopic equations of motion such as the Schrodinger or Liouville equation are invariant

with respect to time reversion and, thus, reversible. A central problem of statistical physics is the derivation of respective irreversible equations from general principles. The description of transport processes is based on balance equations such as the Boltzmann equation which can be solved by means of various methods of kinetic theory [CHA52, GRA58, SHK66, KL175, KL182, BAL63, LIB79]. Assuming the principle of weakening of initial correlations leads to a decoupling of the hierarchy for the N-particle distribution function. Although this assumption is fulfilled rigorously only for the low-density limit, more general principles were derived so that also dense systems with nonideality effects, chemical reactions, and the formation and decay of bound states can be described [KRE89].

An equivalent method determines the expectation values of operators for physical variables via a nonequilibrium statistical operator (NESO) similar to the equilibrium case. The main problem is the construction of that NESO. Kubo [KUB57] has considered the linear response of the system with respect to an external electric field E and utilized the equilibrium statistical operator (ESO) p. for the calculation of correlation functions.

Van Hove and Prigogine [VAN55, PRf59] developed a perturbation theory with respect to a diago~ali~~d Hamilton operator and derived kinetic equations. Zwanzig [ZWA60] and Mori [MOR65] reformulated this approach by introducing a projection operator technique in Hilbert space of Gibbsian ensemble densities. Within that technique, the ensemble densities are separated into a relevant part

is needed for the ~a~eulation of mean values of specified observables, and the remanding irrelevant part. The relevant part satisfies a kinetic equation which is a gen~ra~~~at~o~ of Van Hove’s master equation to general order.

Robertson [ROB%] introduced a generalized density operator which is in any case a functional of the present values of the the~odynami~ coordinates, no matter how far the system is from equilibrium. The entropy of the system is obtained by using that operator which does not fulfill the Liouville equation but a modified one. The thermodynamic coordinates are then given by a set of coupled, no~~~ne~r, integr~di~erent~al equat~u~s of motion.

4.12. Zuharevs method to comtrucrt a NESO Zubarev ~Z~~74~ has derived a general method for the ~eatrn~~t of rne~ha~i~a~ and n#~rne~hani~al

perturbations in an open system based on Robertson’s approach. The NESO is constructed by means of a set of relevant observables. The principle of weakening of initial correlations for t -+ --cc

as assumed in kinetic theory is utilized for an in~nites~mal variation of the equation of motion lim i+ff that breaks the time reversibility Both in~iples are connected via Abel’s theorem ~ZU~7~]. Zubarev’s approach to nonequilibrium statistic mechanics is capable of treating stationary and non- stationary processes, homogeneous and inhomogeneous systems, and the linear or nonlinear response to rn~~bani~a~ and ~o~rn~~hani~a~ ~~~~bations (for a review, see [CHR85, ROE87]).

The Zubarev method contains the Kubo formalism [KUB57] as a special case, when replacing the relevant statistical operator Q,.~~ by the equilibrium statistical operator eo. A close connection exists also to the projection operator method of Mori [MOR65] for the construction of a NESO when expressing the so-called memos function by correlation unctions (see ~R~~87, DER87]). However, the projection operator changes the dynamics of the correlation functions so that the conventional methods of perturbation theory are not applicable for their calculation.

The Zubarev method is applied to derive expressions for the inverse transpo~ ~oe~~~~~~s by means of correlation functions. An open system in th~~odynami~ equilibrium is described by the Hamilton operator (6), and characterized by the ES0 (5). Under the influence of an external per- turbatio~ H,,., the system changes its state which is now represented by a NESU Q, Consi small pe~rbatjo~s of the equilibrium state, the system is supposed to respond linearly to the external fields.

The nonequilibrium properties can be gained from the time-dependent expectation values of respective opemtors A, averaged with the NESQ a(r) via

(A)’ = Tr(p(t)A} . (51)

The evolution of the system is completely defined by the equation of motion for the NESO and appropriate initial conditions. However, the quantum statistical Liouville equation has to be modified in order to account for the time irreversibility of the evolution. Especially, the relevant part erel(f> is needed for the &al~u~at~o~ of mean values (A)‘.


Usually, Bogolyubov’s principle of weakening of initial correlations provides the appropriate initial condition for the construction of a NESO:

lim eiU-h) 10 - - cc

edto) = e(t) 2

L : Liouville operator , i&t) = -iLQ(t) .

The entropy reaches a maximum with given mean values (B,)f, i.e. S so that

(&)’ = Tr{e(tPy} = Tr{erel(t)B,,} = (Be):,, .

Utilizing Abel’s theorem,

(52)

-h Tr(e,,l ln erel 1 -+ max,

!ic k 1-O .f(T)dt = Iii E/O e”“f( t’) dt’ , T --x

a formal solution of the equation of motion can be found by partial integration,

(53)

(54)

t’j e(t) = erel(t) - /’ dt'e"(f'p') w t’){(ilh)[e,,l(t’),Hl + (aiat’)ere,(t’)}u+(t, I , ,

--‘x

in(a/at)u(t, t’) = H(t)u(t, t’) , u(t, t) = 1 . (55)

U(t, t’) is the time evolution operator. e(t) depends on the mean values B, at former times t’ which can be interpreted as a memory effect. We can derive a modified, irreversible Liouville-von Neumann equation with a so-called source term from (55) and (52),

(56) lim u(t,t0){e(t0) - e,dt0)}u+(t,t0) = 0.

l”--CC

Now, we have to specify the relevant part of the NESO which can be done within the projection operator technique [ZWA60, MOR65]. Robertson [ROB661 introduced a generalized Gibbs ensemble where the Lagrange parameters 6, are fixed by the relation (By)f = (I?,):,,:

Zdt) = Tr {exp [-I$F;,(t)&]} . (57)

For the case of thermodynamic equilibrium, the following set of relevant observables applies: {B,,} =

J&N; {F,} = B,BP. We have no general criterion to distinguish relevant observables from irrelevant ones so that we

even do not know the precise number of such relevant observables which fix the nonequilibrium state. The statement of Robertson [ROB661 is still valid in this context, that “. . . In general, the number of coordinates used will be quite small compared with the immense number of microscopic

coordinates of the system. Usually, it will be immediately apparent which coordinates should be used in the theory”.

For instance, using the set of relevant observables {B,) = (Hs,Nnk) leads to a kinetic description of the system where ylk = a$~ are mean occupation numbers for particles in the state /rC>. utilizing an aIte~at~v~ set of relevant obse~ab~es (By) = (Hs,sV, P,>, the force-force co~elation function method follows where P, = k fik@& ya;u, are generalized momenta. A hydrodynamic description of the system follows when using the energy density h = H/V’, the concentration n = N/Y, and the local mean velocity (U(Y)} as a set of relevant observables, {B,,} = {h, ~1, (U(Y)>}.

Mean values of arbitrary operators are given by (5 1). The time derivatives of relevant observables are the starting point for the derivatiun of quantum kinetic equations:

$B,.)’ = Tr { p(f)i [H(t),&]} = (a,)‘. (58)

Within the frame of linear response theory, the deviations from equilibrium are considered to be proportional to the external perturbations. The operators in the time evolution operator and in the NESO have to be expanded according to the Kubo formula,

Assuming stationary perturbations (for the dynamical case, see [CHR85]), the mean values of the relevant obs~~ables are also s~t~o~a~,

-&!I,.) 1-i Tr { Q(I)i[N,B,,]} = 0. (60)

If we choose, r instance, fiBi = j3&, %;B;z = BE, JQ%, e&, lVZ3 = F,B, jnZ3, the relevant statistical o~~mtor can be writes as

where the field term HF has been added for simplicity JY = Hs +- HF_ For equilib~um sit~tions, the Lagrange multipliers vanish, i.e. F% = 0, because there are no additional observables that have to be fixed except fl and /3,u_

For nonequilibrium situations, only the deviations AI;;, = x;l, - $” are relevant. Inserting the general solution for the NESO (54) into the stationarity conditions (60) and expanding the exponentials with respect to the response parameters and e field con~butio

82 R. Redmer/Plzysics Reports 282 (‘1997) 35-157

response equations for the parameters Fti are derived that can be interpreted as generalized, linearized

Boltzma~~ equations [RQE83, HOE84, CHR85],

(A ) A,) - {k(e) I&) = 0 *

The brackets in (63) denote correlation functions defined via

(63)

(A 1 B) = ,l” dzTr{e&-i&iz)B} , {A(E) 1 B) = lim /“’ db et@(t) 1 B) , 1.--o

A(t) = exp(iH#)A(O)exp( -iHt/fi,) . (64)

For a pra~tiGa1 solution, a finite set of relevant obse~ables (B,,) has to be chosen for which the corresponding response parameters LIF;, have to be determined from (63) by means of Cramer’s rule for an inhomogeneous system of equations. We will choose here generalized momenta as relevant

observables,

The lowest-order terms can be interpreted as the electrical current density (j,,) and the electronic energy current density (jo) according to (h-enthalpy per particle):

It is interesting to note that standard methods of kinetic theory for solving the Boltzmann equation can be reproduced within the present correlation function method by choosing the corresponding ~ol~omials as relevant observables {B,?). In this way, Sonine polynomials lead to the Chapman- Enskog method [C A52], and ~ermitian polynomials to the Grad met

Introducing a local description in the Hamilton operator, gradients of the chemical potential V{L (diffusion current) and of the temperature VT (thermal current) are also tractable [HOE84], A modified Boltzmann equation is then obtained from (63) which yields the complete set of thermoelectric despot Goeffi~ie~ts:

&&Tl - T{k, 7X&,,, p_ hQo,} = & AK&L, . (67)

otential. The correlation factions in (67) are defmed

R. Redmer / Physics Reports 282 (1997) 35-157 83

L~~~~r~zing the expressions (661 for the cutest densities with respect to the response parameters yields:

The response parameters AF, can be gained from (67) by Cramer’s rule. Utilizing the Onsager relation between the generalized forces and the currents in the system via the Onsager coefhcients Lik [DEG62],

the th~~o~l~ctric mansion coefficients can be expressed as

The transport coefficients electrical conductivity 0, thermopower a, and thermal conductivity A are given by the Onsager coefficients Lliik for which a general expression can be found from (69) and (70) [HOE84],

N,, = (72)

4.3. J. Polarization approximation The correlation functions N,,, can be evaluated for arbitrary degeneracy which gives generalized

particle numbers

(73)

N is the electron particle ~urnb~~~ and &(x) denutes Fermi integrals of order r.


The force--force correlation Eunctions .&, are evaluated by means of the Hamilton operator (6).

For an e~~e~o~-~on system we have snappy fls = If” -t- H”’ + Hei + H” so at the Co~elat~Q~

tinctions can be separated according to I&, = @i,,$ -t- II& -t DE. Usually, the ion-ion cont~b~tion

I& is neglected. Furthermore, the relaxation terms (r), 1 pm) vanish in the Born approximation so that we obtain:

Different approximations for the correlation functions D,, can be derived which are fully equivalent

to respective expressions for the collision integrals I,, in standard kinetic theory. For this, we apply the relation between the correlation functions (64) and thermodynamic Green’s functions according

to [ROE83]:

F(k,p,q; k’,p’,q/;t - ifir) = - dcu exp(iw@) 7 2~1 exp(@0) - 1

[Gd((-u + ir) - Gd(w - ir)].

fW

The quantity F is given by the four-particle Green’s function C&(w) and consists of four summands

with respect to the product K, x K,, which can be represented by diagrams of the following type:

k+q k

P-Y

P

k'

k”+q’

P'

P’--q’

(76)

e diagram technique allows for partial summations in evaluating C.&(W) which yield different approximations for the collision term of the generalized Boltzmann equation (63), known as Landau (L), Boltzmann (B), or Lenard-Balescu (LB) collision integrals [RED90].

The simplest approximation for G4 is given by the product of fo free-particle Green’s functions Gp which yields the Landau collusion ~~te~~~, re~rese~ti~~ the Born for the ~o~elation functions ZIn, :

(771

R. Redmer I Physics Reports 282 (1997) 35-1.57 85

This expression is of the order V2 and applies to a weak, short-range potential. For stronger interactions, we can extract from all higher-order diagrams those of importance in the low-density limit. Performing an expansion with respect to the density, the lowest density order (WV) characterizes the Landau collision integral (77) for which all ladder-type diagrams with respect to a short-range potential (broken line) have to be summed up,

cl G, = T,L is the two-particle ladder T matrix (23) so that the Boltzmann collision integral is derived:

x fk( 1 - fk+q >f,< 1 - f& KG, qYG@ + 4, -4) + “ex” . (79)

As it is well known, the Coulomb potential cannot be treated in this way because of its long-range character. As a consequence, the expansion of the transport coefficients with respect to the density shows another behavior. The Born approximation II:* is diverging for q -+ 0 (long-wavelength limit), which is removed by considering the dynamic screening of the interaction potential, i.e. by summing up self-energy-type contributions for the Coulomb propagator, which are given by the polarization function I7(q,z), see (25). In more detail, the Born approximation is improved considering the diagrams

where the particle-hole propagator Lz(q,z) is related to the dynamically screened potential Y’(q,z)

according to

V(qPz(q,z)V(q) = @AZ> - V(q) = Vq)s_“__ f’mc;l’“$+) . The correlation functions D,, (75) are given by

X fk(l - fk+$G(k,q)Km(k + 4, -q)6(cd + &ty - -&I 7 (82)

86 R. Redvtzer / Physics Reports 282 (1997) 35-1.57

where ~(~) = [ex#k~)- I]-’ denotes Bose s~~butio~ ~~ct~o~s. Eq. (82) represents the scattering of free particles at a system of scatterers characterized by the dielectric function ~(q, co) in first Born approximation [KL175,ROE83]. For the further evaluation of the correlation functions (821, we have to treat the dielectric function c(g, ~3) which is related to the polarization function 17(q, w) via (18) and we have in RPA:

FRPA(q,o.2) = 1 + V(q)X -6 ~ fp-y , (83)

This approximation describes the polarization effects of a system of free scatterers {electrons and ions). Generalizations have been proposed which account for neutral bound states [ROE791 or co~e~ations ~~CH86~. The LB collision integral is obtained from (82) and (83) using the relation

Im E-‘(g,~) = V(g) Im LZRPA(q, Cti)/ICKPA(qV W)i” , (84)

which leads to the final epression:

As a consequence, the ladder T matrix T. for the Bohzrnann collision integral (79) should also e constructed with respect tu the dynamically screened interaction potential (18).

a solution of the ladder T matrix equation is not within reach at present, and we have to discuss approximations. For instances one can adopt the experiences from the~od~a~ics that only the lowest orders of a ladder T matrix with respect to the bare Coulomb potential should be replaced by a dynamically screened potential in order to obtain convergent results, and that dynamic screening of h~gber-order terms becomes operative only in higher orders of de sity. fn this way, the convergent collision integral

has already been derived by Hubbard [HUB611 and Gould and DeWitt [GOU67], and was solved by Williams and Dewitt [WIL69] within the frame of the Chapman-E~skog method. Further improvements of the convergent collision integral (X6) are expected by a successive replacement of higher-order Born approximations with respect to the bare Co~~ornb potential Y(q) by the respective dynamically screened one.

We want to study in pa~i~ular the influence of d~amic screening on the ~a~spo~ coe~ci~nts of fully ionized plasmas starting from the convergent collision integral (86) with the LB collision term (85). The dielectric function c(q,co) is treated in RPA, We will compare with the usual cases of static screening (CO = 0) and the long-w~velen~b limit (4 + O), where analytical results for the LB collision term are available.

R. RedmerlPhysics Reports 282 (1997) 35-157 87

4.4. I. Lenard-Balescu collision integrals We write the ei- and ee-collision terms explicitly, and neglect exchange contributions:

x$(1 -fk’,,)f;(l -1;:_,)6(m+E,:_,-E~)d(u,-E,e+,+E;),

x f;( 1 - .fj;+q >f;( 1 - _f& )6(w + E;_q - E;)S(o - E;,, + E;) ,

CVvz) = WE;)” - (k + q)(W:+,)” +zGE;)” - (P - WE;-,Y .

Introducing the abbreviations

h9 Q = @&J’)‘!2 ’

“$2,1, 4 m’12e4nN

mi ’ = $(2n)“2 (k,J;“2(4nCo)~ ’

(87)

(88)

(89)

the LB collision terms (87) and (88) can be expressed for the nondegenerate case by a double integral which contains the full momentum and frequency dependence of the dielectric function [RED90]:

Q2 3v2 R$=l, R;;=Rf, = I+--+-,

Q2 Q4 RF, =2+q+64+

s

m dQ ev(-Q2/4)

(90)

RE=O for n=0,1,2 ,.,., L, Rf; =2+4v2 ,... (91)


The RPA dielectric function for the nondegenerate case is given by [KLI74],

ERPA(Q,V) = 1 + $ (45 [D(x:+)) - D(xp)] + fi [D(xI”) - D(x;-I)]

-i Ji{exp(-xi+‘:) -exp(-x:-12) +-$ [exp(-x4f’*) -exp(-xk-“>]}},

(92)

where XT = -&( v k Q/2), and x: = $=(v f yQ/2). D(x) = exp( -x2) Jl dt exp(t2) is the Dawson

integral. Before presenting numerical results for the LB collision integrals (90) and (9 1) utilizing the RPA

dielectric function (92), usual limiting cases for the treatment of screening effects are considered which secure the convergence of the Q integral at the lower limit Q + 0. The convergence in the large Q domain is produced by quantum effects with an effective cutoff at the inverse thermal wavelength A,.

Analytical expressions for the LB collision integrals can be derived for the cases of static screening cWA(Q, v = 0) and for the long-wavelength limit sRPA(Q + 0, v + 0), which are then compared with

the numerical results accounting for dynamic screening.

4.4.2. Static screening The static case aRPA(Q, v = 0) is the simplest way to treat screening effects. The numerical

results for the collision integrals DkE@ with c = e, i indicate that quantum effects play no role

in the dielectric function for the nondegenerate case chosen here (cf. also [I&483]). The results are already produced in the long-wavelength limit Q ---f 0, which corresponds to the simple Debye model for the screening function via cRPA (Q + 0, v = 0) = sD(Q, 0) = 1 + K’/Q’ (denoted by the superscript “D”). Neglecting terms of the order of the mass ratio 741, the first correlation functions 0:; are given by

&I ei,D = (d/2)[- 1 - ( 1 + x)e”Ei( -x)] ,

DIN ei3D = (d/2)[-x + (1 - x - x2)eXEi(-x)] ,

$” = (d/2)[1 - x2-(2-2x+x2+x3)eXEi(-x)],..., D (93)

D 77” = (d/2)[-1 - (1 + 2x)e2”Ei(-2x)],...,

where x = K/841 and E(x) = /“, dt exp(t)/t denotes the exponential integral. Furthermore, performing the low-density limit T<l, the following asymptotic expansions are ob-

tained for the correlation functions:

D yiD = d(n + m)![i ln(O/r) + $4(n + m) - c+~],

2” = dfib,,,,[f ln(O/I) + i&,, - ~~‘~~1, (94)

D

where A(m) = 1 + $ + f + . . . + k, A(0) = 0. The prefactors are given by born = bmo = 0, b,, = 1, b12 = b2, = 2, b2* = y,. . The values for fnm are f,, = 0, fi2 = f2, = y, f2? = $$,,.. . The parameters c”~*~ are characteristic for the chosen treatment of screening effects and are given in Table 12.


Table 12

Effective parameters ccc of the low-density expansion (94) for the correlation functions DzA, c = e,i, within different treatments of screening effects [RED901

D(e+i) D(e) LW GD RPA BCI cc1

yi 0.3395 -0.0071 0.1895 0.1860 0.1895 0.2680 0.1180

c= 0.6860 0.3394 -0.0805 -0.0810 -0.2320 -0.9990

Note: D: Static (Debye) screening due to electrons and ions (e+i) and only due to electrons (e). LW: Long-wavelength limit (95). GD: Modified LW limit of Gould and Dewitt [GOU67]. RPA: Numerical solution with the full RPA dielectric function (92).

BCI: Numerical solution of the Boltzmann collision integral (79) with respect to a statically screened potential and performing the low-density expansion (94).

CCI: Final result according to (86) i.e. CCI=BCI+RPA-D(e+i).

4.4.3. Long-wavelength limit Usually, the first step done in order to account for dynamic screening is to expand the RPA

dielectric function (92) with respect to small values for Q as well as for v which yields (indicated by the superscript “LW”)

cLW(Q, v) = &RpA(Q --$ 0 ,v + 0) = 1 + (K/Q’>[A(v) + iG(v)] ,

A(v) = 1 - wJzP(v/Jz) - (Nmxwm , (95)

G(v) = &{(v/h) exp(-v2/2) + (vi&) exp( -v2/2y)} .

The numerical results for the ei- and ee-collision integrals (90) and (91) inserting the RPA dielectric function in the long-wavelength limit (95) indicate an asymptotic expansion for the low- density limit r+l similar to (94). The parameters cec,LW are given in Table 12.

Gould and Dewitt [GOU67] have applied a modified version of the LW limit (95), which represents complete static screening of electrons and dynamic screening only of ions. An analytical expression for the ei-collision integral DtdGD (indicated by the superscript “GD”) can be derived. The respective numerical results agree very well (deviations less than 2%) with those obtained from (95). The corresponding parameter cei,GD for a similar low-density expansion as given by (94) is also shown in Table 12.

4.4.4. RPA results The full RPA expression was utilized for the calculation of the LB collision integrals (90) and

(91) for densities and temperatures that cover the low-density limit (r41, 09 1 ), as well as weakly nonideal (r < 1) and slightly degenerate (0 2 1) conditions. The numerical results indicate an asymptotic behavior for r$l and 0 & 1 similar to the expansion (94) where the parameters cec,RPA are again given in Table 12. These asymptotic expressions are applicable in a wide density and temperature domain although they were derived for the low-density limit, and fit the exact numerical results for O/r > 80 within an estimated error of 1%. For typical plasma temperatures of T = ( 1 - 5) x lo4 K, this refers to weakly nonideal conditions up to densities of 12 I ( 1 - 10) x 1 019 cm-3.

90 R. Redmerl Physics Reports 282 (19971 35-157

The different methods in treating screening effects are characterized by the corresponding parameters tee and Pi, see Table 12. One can conclude that for the nondegenerate case dynamic screening is completely described by the LW limit (95) and that quantum effects play no role in the dielectric function. Furthermore, static screening due to electrons and ions [D(e+i) in Table 121 overestimates screening effects considerably. Therefore, static screening is often treated in adiabatic approximation, ‘J + 0, where only electrons contribute to screening [D(e) in Table 121.

The parameters cc’ that account for dynamic screening (LW, GD, RF’A) of electron-ion scattering processes are given by values between these two limiting cases of static screening. This indicates that the potential strength of dynamic screening in ei-collisions corresponds to static screening given by the total electron density and about one-half of the ion density. Comparing the parameters tee for ee-collisions, static screening of only electrons already overestimates screening effects considerably. Dynamic screening corresponds in this case to a potential strength of static screening by only about one-half of the electron density.

The above statement reflects the long discussion and controversy over the appropriate choice of a statically screened potential. For instance, there are different positions as to whether or not ions should be included in the determination of the Debye screening length RD. The rigorous treatment of screening as a dynamical effect, at least in Born approximation, gives the possibility of choosing a static Debye potential so that the Born approximation coincides with the results for the dynamic screening. Introducing effective screening lengths Rz of statically screened (Debye) potentials

[ROE89a],

P(r) = E exp( -r/R”,“), c = e,i,

the open parameters R”,’ are usually fixed by performing the static limit in the dielectric function, i.e. c(q, o = 0). Determining these parameters by the condition that the corresponding collision integrals with effectively screened static interactions coincide with the respective LB results (90) and (91), the following values are obtained:

(RE)P2 = 1.4713ne2fi/c0, (Re,“)-2 = 0.4313 ne2b/.s0 . (97)

The numerical prefactor of static screening by electrons and ions would be 2.0 and that of static screening only by electrons 1.0, which illustrates the above statement quantitatively.

4.5. Fully ionized plasmas: improved Spitzer results

The transport coefficients of fully ionized hydrogen (and helium) plasmas are of relevance for the computation of the physical properties of the interior of astrophysical objects such as the giant planets or White Dwarfs [MIN76, FL076, IT083]. We consider first nondegenerate, fully ionized plasmas at high temperature and/or low densities. The correlation functions N,,, (73) are then simply multiples of the particle number, i.e. No0 = N, No, = N10 = ;N, NI1 = No2 = N20 = YN, . . .

4.5.1. Correlation jimctions and transport cross sections Based on the convergent collision integral (86) we will give here explicit results for the transport

coefficients by evaluating the correlation functions D,,. The Boltzmann collision term D,“, (79) is

related to the transport cross sections Q?(k) for electron-ion (c = i) and electron-electron scattering

(c = e) via:

D B, ee nm O” dk k’R,,,,,(/3fi’k”/m,=_) exp( -/Pi2k2/m,)Q~(k>,

R&c) = R&) = ; +x2, RT2(x) = y + 7x2 +x4,. . . (98)

The transport cross sections QT(k) can be represented by integrals over the scattering angle x or the transfer rnome~~m 4 = 2k sin(~/2),

(9%

where da/diZ denotes the differential cross section for the respective scattering process (1 = 1 for ei- and E = 2 for ee-scattering). For the Coulomb interaction, t e behavior of the integrand in the region of small my values leads to the bell-God div~rge~~~s, and we have to treat the efFects of dynamos screening. However, at present a rigorous treatment of a dynamically screened T matrix, where the ladder sum with respect to the dynamically screened Coulomb interaction is considered, is not in reach. As shown in the previous section, the regularization of the Coulomb divergences for small

les x is already obtained from the Born approximation - the Lenard-Balescu equation. We have introduces effective static scree~~~~ lengths Rg (97) such that the collusion terms in Born approximation coincide with the respective Lenard-Balesc~ results. Therefore, for small s~a~eri~g angles (weak scattering limit), the correct Lenard-Balescu behavior is reproduced, whereas for large scattering angles (strong scattering limit) the differential cross section of the Coulomb potential in Born approximation comes out.

Considering the convergent collision integral (86) under these ass~mptions~ the Landau and ~enard-Balesc~ collision integrals just cancel and the ~ol~rna~n collision ~~te~a~ with respect to the effective statically screened potentials (96) seems to be a reasonable approximation for the dynamically screened T matrix. Then, the transport cross sections QT(k) can be evaluated from a phase-shift calculation according to

sin2(d;F(k) - d?+*(k)) .

Extensive nurne~~a~ ~a~~ulat~ons have been performed to dete~i~e the ~a~s~o~ cross sections in a large k domain for screening lengths of Ro =( l--10”& [SK%8, ARN!?O]. The scattering phase shifts (SPS) 6?(k), c = e, i, were calculated numerically by solving the radial Schriidinger equation for the respective scattering process by means of the Numerov and the amplitude-phase methods [PEA67,

92

b P%

Fig. 12. (a) S~a~e~ng phase shifts for the attractive Debye (dashed line) and Thomas-Fermi potential (solid line} for various screening lengths RD and quantum numbers / as function of the wave number. (b) Same as (a), but for the repulsive Debye potential [SIC%%].

-\ \---

200 / -‘r

%q.

‘. ‘* 4

_5+,--- - +, --t---- Ro’a,

. . 1 1

- 50 T

. . t ----

0.2 0.4 0.6 1.2 P-a,

Fig. 13. Transport cross sections for the Debye (DB) and Thomas-~e~j (TF) potential for electron-ion s~a~~~~g, and for the Debye potential for electron-electron scattering (E-E) [SIG88].

SIG88]. These values have been checked against earlier calculations [ROG71, ROG713. For illus- tration, the scattering phase shifts for the Debye and the comas-Fe~i-Luke potential of Green et al. [GRE69] are shown in Fig. 12 as function of the wave number for various screening lengths. The corresponding transport cross sections are displayed in Fig. 13 and show a very systematic behavior


Table 13 Parameters xi in the interpolation formula (102) for the transport cross section of ei- and ee-collisions

ei 0.50 0.5389 1.9804 3.4035 2.0298 0.6278 4.50 ee 0.25 0.3662 I .4644 102.69 3.4166 0.2500 2.25

with respect to the screening parameter. These results for the Boltzmann collision integrals can be parametrized in a low-density expansion according to

D$(T<l) = d(n + m)! 0.0905 0.0196

-i ln(6r3) + A(n + m) - ce’,B - m - - 1 r4@2 ’ (101) D:~~(r<l) =d&b,, + fnm - cee%B - E - s , * 1

where the parameters ceisB and cee,B can be taken from Table 12. Eq. (101) includes the first and second-order WKB corrections [HAH’Il]. The classical asymptotes (101) are valid up to weakly nonideal conditions r < 0.25 for which the deviations from the numerical results do not exceed 10%. For typical plasma temperatures of T = (l-5) x lo4 K, this refers to densities of about n 5 (8-40) x lOI cme3.

The following interpolation formula can be given for the transport cross sections in terms of the classical parameter I: = k2aORD/2 and the quantum parameter K = k2ai/n2 [ROE89a],

ze;‘(k) = a0 In { 1 + e2 cI I

+1UTK’iKa3K2 [1 + CI~ ln( 1 + x~/F)]‘} . (102)

The parameters LX~ are given in Table 13 for ei- and ee-collisions. The values for CI~ are taken from the correct classical asymptote [HAH71]. The coefficients CY~, a3, and ~1~ are chosen to reproduce the Born approximation for ~3 1, and the first and second WKB approximation for small quantum corrections ~41. The parameters ~1~ and &, are introduced to fit the behavior of &(k) in the quasiclassical limit for small values of E. In contrast to the interpolation formulas given in [HAH71, KIH63], Eq. (102) yields a strictly positive behavior for &(k).

4.5.2. Fit _formula for the electrical conductivity Similar to Eq. (102), interpolation formulas can also be derived for the correlation functions D,,

which reproduce the correct limiting cases in the quasiclassical and in the Born approximation. However, we will give here immediately the results for the inverse conductivity for which a low- density expansion can be found [ROE88],

a-‘(n,T)=A(T)lnn+B(T)+C(T)filnn+... (103)

The virial coefficient A(T) is given by the Spitzer result [SPI53], whereas the evaluation of B(T) includes the treatment of dynamic screening and ladder T matrix approaches. C(T) is determined by self-energy effects and nonequilibrium two-particle correlations [ROE88, ROE89a, RED90].


Table 14 Prefactors f, u, and L in Eqs. (105) within different approaches [RE189]. The results for the Lorentz plasma include only electron-ion interactions (ei). The influence of electron-electron scattering processes is also shown (ei+ee). The results

of common methods for solving the Boltzmann equation are given in addition

0 0.2992 0.2992

O,I 0.9724 0.5781

0,1,2 1.0145 0.5834

0.1,2,3 1.0157 0.5875

0, 1,2,X4 1.0158 0.5892

1.1538 0.8040

1.5207 0.7110 1.5017 0.7139 I .5003 0.7093

Spitzer theory”

relaxation time approximationb

1.0159

1.0159 0.5908 1.5

1.5

0.7033

Grad’s 13-moment methodC 0.972 0.578 1.154 0.804

Appel’s I. approximationd 0.972 0.578 1.15 0.81

Appel’s 2. approximationd 1.014 0.580 1.52 0.72

Chapman-Cowling 1 st’ Chapman-Cowling 2nd’ Chapman-Cowling 3rd’ Chapman-Cowling 4th’

0.2945 0.5693 0.5743 0.5777

0.5971 0.6936

3.6781 1.6215 3.9889 1.6114 4.0000 I .6695

4.0 1.5966 4.0

3.678 1.621

3.84 1.64

a Spitzer and Harm [SPI53]. ’ Brooks and Blatt [BR051]. ’ Grad [GRA58]. d Appel [APP61]. ’ Landshoff [LAN5 I].

The electrical conductivity of a (hydrogenic) plasma is a universal function of the parameters P

and 0,

(104)

In the low-density limit 0% 1, the asymptotic expansions (94) can be used to show explicitly the correct reproduction of the Spitzer results within the present linear response approach. The convergence of this method is studied with respect to a systematic extension of the set {P,} of momenta used to fix the NESO [HOE84, REI89]. The reduced electrical conductivity g*, as well as the thermopower x and thermal conductivity A are then fixed by prefactors according to

g*( r, 0) = f/ln( rp3:*) ,

(105)

which are shown in Table 14 for various sets of {Pn}. The present method converges rapidly to the Spitzer results valid for the low-density limit. For practical applications, three momenta P,, i.e. PO, P, , P2, are sufficient for a reasonable description of the transport coefficients in the nondegenerate case. Notice that the present figures coincide with those obtained from the Chapman-Enskog method in solving the Boltzmann equation within every level of approximation [SIG88].


Table 15 Electrical conductivity (T [in l/( Qcm)] of dense plasmas at given temperature T (in IO3 K) and electron density n, (in 1 019 cm-l )

Gas T 4% r 0 o,xp ORRR' (T,Tf f&g (rEh

Ar”

Xe”

Nea

Air”

CZH~~ z,,,, = I .oo Z,“” = 0.28”

Arc

XeC

Hd

22.2 20.2 19.3 19.0 17.8

30.1 27.5 27.0 26.1 25.1 24.6 22.7

19.8 19.6

11.0

116

2.8 0.368 56.9 190 226 200

5.5 0.505 33.2 155 232 203

8.1 0.604 24.4 170 241 209

14 0.736 16.7 255 272 234

17 0.838 13.7 245 274 232

25 0.564 17.9 450 458 442

59 0.822 9.24 680 544 506 79 0.922 7.47 740 598 546

140 1.150 4.93 690 767 657

160 1.260 4.34 780 822 660 200 1.380 3.66 1040 945 728

200 1.500 3.38 930 950 694

1.1 1.9

0.13

0.303 94.6 0.367 65.0

130 165

60

174 148 187 160

0.267 218 67.4 53.1

6000 400 0.909 I .789 9750 9370 1890 0.595 4.180 5000 5680 530

16.4 0.059 0.128 551 83 96.1 79.0 0.101 0.165 385 79 102 84.3 0.131 0.180 324 76 106 87.1 0.154 0.190 291 64 108 88.9

12.4

12.6

0.062 0.185 403 46.4 70.1 55.8 0.112 0.226 272 43.8 75.6 60.1 0.125 0.234 252 41.1 76.7 61.0 0.072 0.192 371 48.0 72.8 58.0 0.126 0.230 255 46.3 78.2 62.3 0.140 0.239 238 43.5 79.2 63.1

15.4 0.10 0.175 364 62.5 95.4 77.8 18.7 0.15 0.165 337 91.3 125 104 21.5 0.25 0.170 276 114.3 156 133

192 216 204 217 209 222

246 244 241 243

223 441 241 491

262 520 368 607

371 622 481 661 476 630

150 165 168 179

- Note: Experiments: “[IVA76], b[SHE88], ‘[POP90], d[RAD76]. Theory: ‘[ROE89a, REI92], f[ICH85b], g[RIN85],

h[EBE91, p. 2221.

Furthermore, other common methods for the determination of the transport coefficients such as the Grad method [GRASS] or the Kohler variational principle [KOH48, APP61] can be related to different levels of the present scheme. The inclusion of electron-electron scattering leads to a characteristic lowering of the transport coefficients.

Applying the units rnp3 for the number density, K for the temperature T, and l/(0 m) for the electrical conductivity, the following interpolation formula is proposed within a four-momentum

96

approximation [ROE89a],

R. Redmer I Physics Reports 2&? (1997) ~-157

(1 +&J [In(l SASB)-C- h2$J,

A = r-3 1 - a4/(r20)

1 - a2/(r20) - a3/(r4G2) [a, + cl ln( 1 + c?r312)]2 ,

B=b 1 +c304'5

2ro(i+c30)'

C = c4[ln( 1 + rp3) + c5r20]-I . (106)

The parameters a, are fixed by the low-density limit, whereas the values for the bj are given by the strong degenerate limit. Only the parameters ci were fitted to the numerical data for the correlation functions. The whole set of parameters reads: a0 = 0.03064, al = 1.1590, a2 = 3.136, a3 = 3.0369, a4 = 3.6590, h, = 0.3831, b2 = 3.274, cl = 1.5, c2 = 6.2, c3 = 0.3, c4 = 0.6, c5 = 0.1.

Comparison between Eq. (106) and experimental values for the electrical conductivity of strongly coupled plasmas is performed in Table 15 [ROE89a, REI92]. Taking into account an experimental error of about 30%, theory and experiment are not in contradiction. However, the experimental values are systematically smaller for the low-density plasmas with r N 0.2-0.7. For high-density plasmas, they are larger than the theoretical ones.

In addition, we have presented in Table 15 the results of Ichimaru and Tanaka [ICH85b] and Rinker [RIN85], who derived similar interpolation formulas based on the Ziman theory for liquid metals. They included especially local-field corrections and the ion-ion structure factor which are of relevance in the high-density domain. Ebeling et al. [EBE91] tabulated conductivity data for dense metal plasmas based on the relaxation time approximation, following the scheme given by Lee and More [LEE84]. These authors found also an overall agreement with the experimental values within the error bars of about 30%. A semi-empirical interpolation formula for the electrical conductivity as function of r was given in [BAI96].

Referring to the virial expansion

A(T) = -32.8013 TP3’* ,

B(T) = 98.4040 Tp3:2(1n T +

( 103 ), the coefficients A(T) and B(T) are given by

10.6177). (107)

4.6. Transport cot$fkients in partially ionized plasmas

4.6.1. Electron-utom scattering in hydrogen plasma The force-force correlation functions d,,, are related to the four-particle Green’s functions via

Eq. (75). The polarization approximation (82) describes the interaction of free particles with a system of scatterers in first Born approximation that is characterized by the dielectric function E(q,co) (for details, see [ROE83]). The dielectric function relates the dynamically screened potential V’(q,o) to the polarization function II(q, CO) via (18) for which a cluster decomposition is employed similar to that for the self-energy, i.e. Ll(q,o) = L7,(q, o) + L’z(q,o) + . . . II, represents the RPA for free-particle states, whereas II2 describes the contribution of two-particle states to the polarization


function [ROE79]. The correlation functions d,,, can then be separated with respect to electron-ion, electron-electron and electron-atom scattering, i.e. d,,,,, = DfM + DT,“, + Dp,. The electron-electron and electron-ion contributions have the structure of Lenard-Balescu collision terms with respect to the relevant interaction potentials, see Eqs. (87) and (88).

The electron-atom correlation function stemming from I7, is treated within second Born approximation so that besides the atomic form factor (first Born approximation), the polarization contributions (second Born approximation) are also included. The influence of exchange processes on the cross sections and the conductivity has been studied for alkali-atom plasmas [STA96]. In the static case, we have

0% = nfifi x- pT,, 1 V”‘(knP,k + qn’P - q) + V’2’(knP, k + qn’P - q)12 3 , 1

x We(k) + E~P - Ec(k + q) - Env-qlfeWl - fe(k + 411

x QnP[l + Qn’P-ql K7(kqYcn(~ + 4, -4). (108)

gnp = [exp(jE,p - pe - pi) - 11-l is the Bose distribution function for atoms in the internal state n with a total momentum P. The Born approximations for the effective electron-atom potential read:

V”‘(knP, k + qn’P - q) = v,;(q)M,,,,(q) ,

~‘2’(knm + Cln’P - 4) = c 4x-4’vxd - 4) Pf,o,(d)12

4’ Enp + E (k) _ Enp_q, _ E (k + q,)

e e

+C Yi&JmcL(-q’)

nl’#n Enp + E,(k) - E,,,~P+,J - Ee_(k + q’) .

The vertex function M,,,(q) describes the coupling between free charges and bound states,

KNz) = J

*Y;(k){ZeY$ (k-ffq) -e’& (k+zq)}. (*71)j

(109)

(110)

Yn are the wave functions for the bound state between the electron (charge -e, mass m,) and the ion (charge Ze, mass mi). Usually, we have m,<m, and 2 = 1.

The transport cross sections for electron-ion and electron-electron scattering have already been determined according to Eq. ( 100) in the previous section for the Debye potential, see Figs. 12 and 13. For electron-atom scattering, the scattering phase shifts were calculated numerically with respect to the effective potential (109). Utilizing the optical potential method for electron-neutral scattering, the second-order potential contains the atomic form factor (first Born approximation) as well as the polarization potential (second Born approximation). The respective results for hydrogen are displayed in Fig. 14.

The transport cross sections within the partial wave expansion (100) with respect to the polarization potential V (2) = VPP(r) (37) alone are compared for R. = (5,20, co) x a0 with those where the atomic form factor [MOT651

p’(r) = _g ( > 1 + 1 exp(-2r/ao) :o a0 Y (111)

98

5.1

k * aB

Fig. 14. Tran~~~~ cross sections for electron scattering at hydrogen atoms as function of the wave number [REI95]:

(a) ~ti~~~ng only the ionization potential Y j2) for the unscreened case (o), RD = 20ao (i-1, and Rn = 5 erg (a);

(b) considering the atomic form factor Y (I) in addition, i.e. P’(l) + VC2), for the unscreened case (x ), RD = %?a~ (A),

and RD = 5a~ (*>. For comparison, the ordinary Born approximation with respect to the unscreened polarization poten-

tial (dotted line) and the results of Meyer and Bartoli [MEYSI] within a T ma@ix calculation (dash-dotted line) are

shown.

has been inc~udcd, i-e, V(r) + Vg2) as an elective potential. In addition, the ordinal Born a~~r~xima- tion for the transport cross section with respect to the unscreened (R, = co) polarization patential (dotted line), and the T matrix results of Meyer and Bartoli [MEY81] (dashed-dotted line) which obey the Friedel sum rule are shown.

In Born approximation, the e~~c~on-atom ~a~sp~~ cross section is considerably underestimated. The partial wave expansion with respect to the polarization potential alone gives results which are too high for low transfer momenta k. This has a pronounced influence on the transport coefficients in regions with low degree of ionization, i.e. the pa~~ally ionized plasma region. A very good overall agreement with the consistent Meyer-Bartoli curve is found within the partial wave expansion with respect to the atomic form factor plus the polarization potential (see also [BIA89] for alkali-atom plasmas). One can conclude, that the atomic form factor is important for large transfer momenta k, and that there is no si~ifi~ant in uence of screening on the erector-ato ~ra~sp~~ cross section. Therefore, the Meyer-Bartoli formula can be utilized for the further evaluation of the correlation functions DE for hydrogen.


-.-- partially icrnized ptasma

- fully ionized plasma

--- genera-alized Ziman fo~ul~

- - - - Spitzer result

T=lSOOOK

I $7 18 19 20 21 22 23 24 25

log,~[n~cm-3)l

Fig. 15. Electrical conductivity a of hydrogen plasma for T = I5 x 10” K within different models [REI35]: Spitzer curve (dash-dotted line), generalized Ziman formula (dashed line), fully ionized plasma within the T matrix approximation (dotted line), partially ionized plasma within the T matrix approximation (fill line) including (a) and neglecting (b) the

structure factor.

The electrical con al ~ondu~tivi~ R, and the opower c1 of racially ionized hydrogen plasma were caI~~~ated for II” = ( 11 O4 - 105) K an I 0’6-1 025 ) mr3 [REI95]. The composition was dete~ined ~o~sid~~~g the ~onideali~ corrections to the respective laws of mass action (see Section 3.1) so that also the region of partial ionization at intermediate densities and low temperatures has been covered,

In the high-density limit, the consideration of higher momenta P, corresponds to a generalization of the Ziman (and Mott) formula which has been employed, e.g., in [BOE82, ICHUb]. We have included structure factor effects as well as local-field corrections when calculating the correlation

functions 0:; in that region. For high temperatures, structure factor effects can be neglected. For xao > 1, i.e. in the high-density limit, we find a pronounced lowering of the electron-ion correlation function by up to one order of ma~itude because the small k values are damped out when integrating over k.

The numerical results obatined within the present linear response approa~b inte~olate between the Spitzer theory for nondegenerate, weakly nonideal plasmas and the Ziman theory for degenerate, strongly coupled plasmas. This is demonstrated in Figs. 15-17 where the transport coefficients are shown for a given temperature of T = 15 x IO3 K as function of the number density. A three- momentum approximation has been applied, i.e. utilizing P&P,, and P2, which yields an accuracy of 1% compared with the values obtained within an infinite series expansion of the distribution function,

The correlation functions were calculated on T matrix level using ( 100) for electron-electron and electron-proton scattering, and, for simplicity, the fit formula of Meyer and Bartoli [MEY81] for electron-atom scattering, see Figs. 13 and 14. The scattering of electrons at dimers was supposed to follow a potential strength similar to that of atoms so that the ~nt~~~tion of eIe~tro~s with all ~eu~als is described by this facula,

100

-60

-80

-100

~ partially ionized plasma -.“.“...-“’ fully ionized plasma - - - generalized hAott formula

a

If 18 19 20 21 22 23 24 25

log,,[n@m~%

Fig, 16. ~e~o~ow~r r of hydrogen plasma for T= I5 x E03 I( within thereat models [REI95]: gene~~i~ed Mott formula for the Lore&z plasma (only electron-~0~ interaction, dashed line), fully ionized plasma within the T matrix approximation (dotted line), partially ionized plasma within the T matrix approximation (full line) including (a) and neglecting (b) the

structure factor.

T=150OOK

. ..--

Spiker result Wiedemann-Franz law partially ionized plasma fully ionized plasma

I.5 1 I t t I ! L

17 18 19 20 21 22 23 24 25

~og,,b-Wf3)1

Fig. 17. Lorentz number L of hydrogen plasma for T= 15 x lo3 K as a function of the particle number density n within di~ere~t models [REI95]: Spitzes result (d~h~o~ed line), ~iede~~-~~anz law (dashed line), f&y ionized plasma within the T matrix approximation (dotted line) including (a) and neglecting (b) the structure factor, partially ionized plasma within the T matrix approximation (full line).


a - T(lO’K)

7 -

3 -

2 - I

17 16 19 20 21 22 23 24 25

loMWO1

Fig. 18. Electrical conductivity (r for partially ionized hydrogen plasma for various temperatures as a function of the particle number density n [REI95].

The results for the electrical conductivity and thermopower indicated with generalized Ziman and Mott formula have been deduced from the first Born approximation for the electron-ion correlation function, see Eq. (115) within a one- and two-momentum approximation, respectively. Electron- electron and electron-neutral interactions have been neglected in this Lorentz plasma model. How- ever, the effects of structure factor and of a smeared-out Fermi surface (arbitrary degeneracy) are included.

The electrical conductivity in Fig. 15 is given by the Spitzer formula in the low-density limit up to 10” cme3 and has typical values of c M lo4 (0 m))’ . The Spitzer curve diverges for higher densities. The T matrix results for the fully ionized plasma merge into the Ziman formula above 102* cmp3. This is a result of the strong screening of the Coulomb interaction so that the Born approximation becomes valid. Furthermore, the electron distribution function is determined already by one momentum PO in the degenerate domain, whereas twothree momenta P,, are needed in the nondegenerate region. The Ziman formula yields conductivities in that region which are too low by a factor of about two.

Partial ionization of the plasma, i.e. the formation of neutral and charged clusters, leads to a strong decrease and a typical minimum behavior of the isotherms for the electrical conductivity. The decrease is a result of the diminishing fraction of free electrons and their reduced mobility due to scattering at neutrals. The minima occur at about lO22 cmp3 and run to values less than G w lo2 (0 m)- ’ dependent on temperature, see Fig. 18. The conductivity shows then a subsequent steep increase due to the lowering of the ionization energy which leads to a vanishing of neutral bound states, and the plasma becomes fully ionized again. Furthermore, electron-electron and electron- neutral scattering is no longer of importance so that the free electron mobility is also increasing. Ion-ion correlations are described by the structure factor in the electron-ion correlation function (115) and lead to a sharper increase of the conductivity in the high-density (liquid) branch compared with the case of S(k) = 1.

102 R. Redmer I Physics Reports 282 (I 997) 35-1.57

This typical behavior at low temperatures can be interpreted as a nonmetal-to-metal transition. Utilizing the Mott criterion for the minimum metallic conductivity of CJ cz lo4 (0 m))’ also for T # 0 in order to locate this transition, a critical density of about 0.16 g cmP3 can be found which is close to the critical density of the proposed thermodynamic instability at 0.42g cmP3, see Section 3.2.

The general behavior of the thermopower is shown in Fig. 16. Again, the low-density asymptote within the present T matrix approximation for the fully ionized plasma coincides with the Spitzer result, a = -60.60 pV K-‘. The high-density limit is given by the Mott formula. However, already the T matrix results for the fully ionized case show an interesting behavior between 102’ cmP3 and the critical density of 1 023 cm- 3. In this region, a resonance-like behavior of the scattering phase shift sum which enters into the expression for the electron-ion correlation function (98) is obtained when the lowest bound states (1 s, 2s, 2p) disappear. For the thermopower, two Onsager coefficients & have to be determined according to Eq. (72) which average this resonance-like behavior in different ways so that we obtain a “wiggle”. This distinct behavior of the thermopower has already been found for hydrogen-like systems such as cesium [ALE75]. Furthermore, the density region between the local maximum and minimum of the thermopower is connected with the decreasing slope of the curve for the electrical conductivity in Fig. 15.

The Mott formula (Born approximation) yields values for the thermopower which are too small by a factor of about two in the low-density limit, In addition, the influence of electron-electron scattering which is important in that region is not accounted for in this Lorentz plasma model.

Inclusion of partial ionization leads to drastic effects. At about 1Or8 cmP3, a strong increase up to slightly negative values is found. However, positive values for the thermopower as found earlier in that domain [HOE841 could not be verified. This might be an inherent feature of the Born

approximation for the collision integrals D,, and/or result from the simpler evaluation of the EOS within the Debye-Hiickel approximation utilized in that paper.

Taking into account ion-ion correlations again, the sharp increase of the thermopower becomes more pronounced in the high-density low-temperature region, i.e. for T 5 15 x lo3 K and it 2 1O24 cmM3. Furthermore, small positive values and, thus, a change of sign have been obtained. A similar behavior was recently found for liquid cesium [REI93] as a result of structure factor effects and local-field corrections.

It would be of high interest to calculate the thermopower along the liquid-vapor coexistence curve of mercury within this improved model where such strong positive values for the thermopower have been measured [GOE88]. Then, general conditions for the coincidence of the density for the zero- point transition of the thermopower and the critical density of the liquid-vapor phase transition as found for mercury can be verified.

The thermal conductivity behaves very similar to the electrical conductivity so that we have plotted in Fig. 17 the Lorentz number L according to Eq. (105). For the nondegenerate case, the Lorentz number is 4.0 for a Lorentz plasma (only electron-ion interaction), or 1.5966 considering also electron-electron scattering, see Table 14. For the degenerate domain, the Wiedemann-Franz relation is fulfilled which gives L = 7c2/3. In the intermediate region, the numerical results for the Lorentz number interpolate between these limiting cases dependent on temperature. Again, the oscillatory behavior at densities between 1O23 cmP3 and 1 024 cme3 stems from the resonance-like structures in the scattering phase shift sum as already discussed for the thermopower. Ion-ion correlations have little influence on the Lorentz number in the partially ionized domain and yield only a small reduction of the values.


-80 ' 17 18 19 20 21 22 23 24 25

W,lNcm~3)1

Fig. 19. Thermopower c( for partially ionized hydrogen plasma for various temperatures as a function of the particle number density n [REI95].

5

4

3 =: E

gf 2

x a $ 1

0

-1

17 18 19 20 21 22 23 24 25

log,,Wm-3)1

Fig. 20. Thermal conductivity i for partially ionized hydrogen plasma for various temperatures as a function of the particle number density n [REI95].

4.6.3. Metal-nonmetal transition and comparison In Figs. 18-20, the transport coefficients are shown for various temperatures as function of the

number density. The deep minima for the electrical (Fig. 18) and thermal conductivity (Fig. 20) vanish above T = 30 x lo3 K because the plasma remains nearly fully ionized over the entire density domain. For low temperatures, a steep increase of the conductivities over four orders of magnitude in the narrow density region around 1O23 cmp3 can be seen which clearly demonstrates the occurrence of a smooth nonmetal-to-metal transition near the critical point of the hypothetical PPT.

104 R. RedmerlPhysics Reports 282 (1997) 35-157

Comparing with the conductivities of 2 x 105(flm))’ as measured recently by Weir et al. [WE1961 in dense fluid hydrogen at 3000 K and 40.31 mol/cm3, the present calculation yields values of only 2 x lo4 (R m))’ at the lowest (plasma) temperature of lo4 K considered, see Fig. 18 [REI95]. However, the systematic of the conductivity curves with the temperature is just inverted at the relevant densities above 1O23 cm-3 so that at lower temperatures the conductivity may be higher. Furthermore, the electron-ion pseudopotential becomes essentially weaker when the density increases up to the fluid phase which is not well described by a simple (Debye) screening of the Coulomb potential. Thus, the conductivity will be enhanced at lower temperatures.

Another indication for the nonmetal-to-metal transition is the strong increase of the thermopower towards positive values and the subsequent steep slope in that density region, see Fig. 19. This behavior is similar to that observed experimentally for mercury along the coexistence line [GOE88], although high positive values for the thermopower could not be verified for hydrogen plasma contrary to the mercury results.

There are few experimental data for the transport coefficients of dense plasmas with coupling parameters up to r _< 2, see [EBE83, GUE84, FOR90, lAK93]. We will compare with the famous shock wave experiments of Ivanov et al. [IVA76] for inert gases. Gunther and coworkers performed conductivity measurements for weakly nonideal hydrogen [RAD76] and inert gas plasmas [GUE83] in a pulsed high-pressure arc. Further experiments were performed with flashtubes [BAK70, AND75, POP90]. Data for C2H3Cl are also shown [OGU74].

These experimental data for the reduced electrical conductivity cr* (105) are given in Fig. 21 as function of the coupling parameter 0.1 < f < 2 (compare Table 15, and Fig. 6 in [MEI82]). The values derived from shock wave experiments, high-pressure arcs, and flashtubes lie between the Spitzer curve as an upper bound for the conductivity of fully ionized, weakly nonideal plasma, and the Born approximation - the Ziman curve gives the lower bound for the conductivity. The results for hydrogen, argon, xenon, and air plasma in the range r < 0.2 with 9 x lo3 K 5 T <

21.5 x lo3 K show a strong dispersion. Partial ionization for low temperatures, occurrence of ionization stages Z > 1 for higher temperatures, and non-Coulombic contributions to the effective electron-ion potential lead to the scattering of the data which were derived under the assumption Z = 1 and show the standard T’ 2 dependence of the plasma conductivity. Another source of un- certainty are the relatively large experimental errors of about 30% which are indicated for two values.

In the range 0 .2 < r 5 1.5, the conductivities extracted from the shock wave experiments of Ivanov et al. [IVA76] for neon, argon, and xenon plasma with 20 x lo3 K 5 T 5 30 x lo3 K approach the T matrix results systematically. The classical result for the Coulomb cross section becomes invalid and the Spitzer curve for the conductivity diverges so that the correct quantum transport cross sections (100) have to be used.

Furthermore, ion-ion coupling becomes more pronounced in that region which can be accounted for by the ion-ion structure factor Sii(k). A reasonable agreement with these experimental data has already been found by Boercker et al. [BOE82] within the Chapman-Enskog scheme for solving the Gould-Dewitt kinetic equation [GOU67]. They calculated the correct quantum, Debye screened cross sections for electrons scattering from finite sized ions, neutrals, and other electrons. This procedure coincides completely with the present approach in the nondegenerate case.

The< correlation function method utilized by Ichimaru and Tanaka [ICH85b] is based on the Ziman formula for Z = 1 and yields conductivities between the T matrix results and the generalized Ziman

105

Fig. 21. Reduced electrical conductivity 6* (104) within the Z’ matrix approach and the generalized Ziman formula as function of the coupling parameter r I 2, see also [MEI82]. Comparison with other theoretical models (Spitzer curve [SPI53], Ic~jrn~ and Tanaka rIC~~5~], r>jur% et al. [DJWl]), and -with ~xpe~rn~ta~ results is feinted: Radtke and G&her [RAD%f for H (+); G.&her, Lang and Radtke [GIJE83j for Ar (v); Bakeev and R~vi~skii [BAK7QJ for tilr (A) and Xe (A); Andreev and Gavrilova [ANB’B] for air ( q ); Ivanov et al. [IVA76] for Ar (m). Xe co), Ne (a >, and air (0); PopoviC et al, [POP901 for Ar (I) and Xe (Cl); Ogurzowa, Podmozhenskii, and Smirnova [OGU74] for C2H3CI (X ).

fo~ula~ Were, a pref~~tor was adopted to give the right Spitter values in the Io~-d~~si~ limit. In the hi~h~densi~ limit, structure factor effects and local-field co~eGtio~s have been included self- consistently. Another self-consistent field model of DjuriC et al. [DJU91] is based on au extended Lorentz fo~~la for fully ionized plasmas and yields slightly smaller results than those of Ich~a~ and Tanaka (for the transport coefficients in fully ionized plasmas with and without ma~eti~ fiefd, see aIso [ADA94

New methods have been developed to reach the strongly coupled plasma domain experimentally. For instance, Shepherd, Kania, and Jones [SIIE88] reposed conductivities for polyurethane using a capiII~ ~s~harge. il~bbe~~ et al, ~~IL~~] meas~ed the ~esistivi~ of solid-densi~ metal from room temperate to IO6 K by focussing uhrashort, high-ener~ laser pulses on a planar aluminum target. Intense laser pulses were also utilized by ~osto~~h et al. ~~~~91] to study the opacity (Coulomb logarithm) of strongly coupled aluminum plasma. RecentIy, Benage et al. [BEN941 reported new data for polyurethane derived from capillary discharges apin. Althuugh the plasma densities reached in these experiments are near the solid-sate defied ~o~~li~g parameters of only f 2 2 have been produced due to relatively high temperat~~s- ReSilva and Kunze ~D~~9~] employed papillae discharges to measure ~onductivities in dense copper plasmas at relativefy tow temperatures 8 x IO3 K 5 T 2 30 x lo3 K with ionization stages in the range 2 M l-25, Coupling

106 R. RedtneriPhysics Reports 282 (1997) 35-157

6.0 I Milchbera e’_ai.

:1oq ’ .. Benage et al. (4%-i-( ,* Al

5.5 PU (25-30) (65)

-1-I

, &$ ‘b

_ / ‘e a0 /,e

/ /

5.0 7. , , ojd$!“’ / /

/ / Ebeling et al.

z I PU(10; /” $. 4.5 Shepherd

’ 3T P et al. , /;v:

B ’ / i :

Ichim&

c

l . i

,

_~ ’ ’ 4.0 and Tanaka L’

I (El)

DeSilva and Kunze

3.5

!

/ Rinker

-0.5 0.0 0.5 1 .o 1.5 2.0

log,J-

Fig. 22. Electrical conductivity 0 of strongly coupled metal plasmas. The experimental results of DeSilva and Kunze [DES941 for Cu plasma (full dots) are compared with the theories of Lee and More [LEE84], Rinker [RIN85], Ichimaru and Tanaka [ICH85b], Djuric et al. [DJU91]), and Ebeling et al. [EBE91] for T=20 x lo3 K. In addition, the experimental results of Milchberg et al. [MIL88] for Al plasma, and of Shepherd et al. [SHE881 and Benage et al. [BEN941 for polyurethane (PU) are shown. The numbers in brackets give the estimated temperature in eV.

parameters up to r < 70 have been realized. DeSilva and Kunze found that the conductivity becomes a function of only the coupling parameter for r > 10.

The conductivities for these high-temperature plasmas are shown in Fig. 22. Despite the problems of determining the plasma parameters correctly, these results indicate that for plasmas at high temperatures and solid-state densities new transport mechanisms have to be studied. Furthermore, stronger electron-electron correlations for plasmas with elevated ionization stages Z > 1 lead to an effective decrease of the conductivity compared with the inert gas plasmas at lower temperatures.

DeSilva and Kunze compared their experimental values for copper with various theories for the conductivity (see Figs. 5-7 in [DES94]). The Rinker theory [RIN85] yields a lower bound for the conductivity whereas the conductivity model of Lee and More [LEE841 gives an upper bound. Both models are valid for Z 2 1 and arbitrary degeneracy. This behavior indicates that electron-electron correlations neglected in the relaxation time approximation of Lee and More yield a pronounced lowering of the conductivity compared with the Lorentz gas results (cf. Table 14).

The transport theories of Ichimaru and Tanaka [ICH85b], Djuric et al. [DJU91], and Ebeling et al. [EBE91] have been derived for Z = 1 and show, surprisingly, a better agreement with the copper data of DeSilva and Kunze, especially for the large-r domain. Strong ion-ion correlations and screening effects of the degenerate electron gas seem to fix the conductivity in that region alone. Furthermore, only one moment (PO) is needed to determine the electron distribution function, and the Born approximation with respect to the electron-ion interaction is sufficient for the calculation of the cross section. Therefore, the Ziman formula which is recovered within these theories and the


present linear response approach in a more general form should give the right high-density limit for

the electrical conductivity.

4.6.5. Transport cross sections for the alkali metals The calculation of the transport cross sections for alkali-atom plasmas is, compared with the case

of hydrogen, more difficult because of the complicated inner shell electron structure of the ions and atoms. A detailed analysis of the respective cross sections has been performed starting from quantum kinetic equations for the electron distribution function in three-particle collision approximation [BIA89, STA96] which can be derived by means of nonequilibrium Green’s functions [KRE89]. Especially, the three-particle collision integral Iubc contains all possible scattering processes between free and bound particles which are described by transition T matrices for the respective channel states. The Chapman-Enskog method [CHA52] was utilized to determine the anisotropic part f,’ of the electron distribution function for elastic scattering processes, and the electrical conductivity is given again by determinants with collision integrals 1”” as elements [instead of correlation functions in linear response theory, see Eq. (72)] which can be split with respect to the various scattering processes, i.e. Z”fl = &’ + 4:‘” + 4:. These quantities are integrals over the respective transport cross sections (99).

We restrict ourselves to elastic scattering from alkali ions and atoms in the ground state. Expanding the full wave function in terms of atomic states and employing the adiabatic exchange approximation, the following close coupling equation is derived [TAY75]:

(HP - E + & + ~~ti’<~l) + vr’<ri))Fo(rl)

(112)

E. and HP define the energy eigenvalue of the valence electron and the incident electron, respectively. vLti’ represents the static interactions with the nucleus (charge z), the inner shell electrons, and the valence electron. vr’ is the polarization potential. The electron exchange contribution is given on the r.h.s. where the upper/ lower sign refer to the singlet /triplet state. F. is the wave function for elastic electron-atom scattering. The static interaction, vr(rI ) = petit + P$‘i(r, ), can be given by an effective, two-particle potential [GRE69]:

~,S~~(r) = -2 [ eCKr +

z-l

0 1 H,(e@ - 1)+ 1 . (113)

Static Debye screening applies for the long-range part. The parameters Hi and di were fitted to the energy spectrum of the valence electron. The static electron-electron potential, J$:2Stat(r, ), was determined numerically. The total static potential, yitit(r, ), was then parametrized by means of the second term in (113) replacing (z - 1) by z. The respective parameters can be found in Table 16 for all alkali metals, see [BIA89, STA96].

The polarization potential ~~‘(~i ) is determined using the dipole approximation and assuming static screening. Introducing again the atomic dipole polarizability @o and a radius r. which yields the behavior for small distances, we find the form (37) where the parameters are given in Table 3.

Performing a partial wave expansion, an integro-differential equation [ST0631 follows from (112) which can be solved with the method derived by Marriott [MAR58]. The electron-electron transport

108 R. Redrner I Physics Reports 282 (I 997) 35-157

Table 16 Parameters for the effective electron-ion and electron-atom potentials of the alkali metals [STA96]

IOn

H d (ao)

Atom

H d (ao)

Li 0.85 0.4335 2.62 1.635 Na 1.337 0.493 2.33 0.849 K 1.85 0.667 2.62 0.95 Rb 3.5 0.8 4.65 1.066 CS 5.121 1.0 6.54 1.29

lo*

IO'

106

“P IO 10'

‘0 10’

lo*

10’

0.0 0.2 0.4 0.6 0.8 1.0

h

10’

lo6

lo5

-aa 4 lo4

‘0 10’

lo2

10’

I I

0.0 0.2 0.4 0.6 0.8 1 .o

kas

Fig. 23. Transport cross sections for e-Li+ scattering for different screening parameters [STA96]: (1) tiao = 0.001, (2)

Kao=O.l, (3) Kao=O.6.

Fig. 24. Transport cross sections for e-Cs’ scattering for different screening parameters [STA96]: (1) /cao = 0.001, (2)

tiao=O.l, (3) tiao=0.6.

cross section was determined via the two-particle Schrodinger equation for a Debye potential. For example, the transport cross sections for e-L? and e-C$ scattering are shown in Figs. 23 and 24 as function of the wave number for various screening parameters &, = rcao, see also Fig. 13. The transport cross sections for elastic scattering of electrons on Li and Cs atoms in the ground state are displayed in Figs. 25 and 26 (for details, see [STA96]).

The cross sections are lowered with increasing plasma screening. Especially, resonance states appear which lead to a typical resonance structure in the low-energy region. For higher &values, this behavior disappears, but a characteristic minimum is formed as it is observed in the curves for An = 0.6. A comparison of the results for different alkali ions shows the expected differences in the high-energy region due to the influence of the ion core electrons on the effective electron-ion interaction potential. Here, the cross section is larger for elements with higher nuclear charge z which results from the more complex ionic core.

The transport cross sections for elastic scattering of electrons on Li and Cs atoms in the ground state show a decrease at low energies kuo < 0.1. In the region of intermediate energies, the cross

lo5

lo4

n_ I?

c QIOT

10:


e- -Li

1)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1 .o

kaB La,

Fig. 25. Transport cross sections for elastic e-Li scattering for different screening parameters [STA96]: (I (2) K&1=0.1, (3) tiao=0.6.

) KU0 = 0.001,

Fig. 26. Transport cross sections for elastic e-Cs scattering for different screening parameters [STA96]: (I ) tiuo = 0.001, (2) icao=0.1, (3) Kao=0.6.

sections for higher &values can reach larger values in comparison to the case of low screening (Au = 0.001). Similar to electron-ion scattering, a resonance-like behavior is observed in the low- energy region. The resonance peaks disappear for higher energies and the cross section decreases with increasing screening parameters.

It is interesting to show the effect of the different electron-atom interaction contributions on the results for the transport cross sections. First, all contributions were included, i.e. the effective Schrodinger equation (112) was solved taking into account the static and polarization potential as well as exchange. Then, we have neglected the exchange term on the right-hand side of Eq. (112) to determine the region where electron exchange is of importance for the numerical results. In a third approximation, the exchange term and the static potential were neglected so that the Schrodinger equation with respect to the screened polarization potential has been solved. This simplest treatment of electron-atom collisions has frequently been utilized to determine the effect of partial ionization on the mobility of free electrons [REI89, ARN90, RED92].

The respective transport cross sections for electron scattering on Li atoms are given in Fig. 27 for a screening parameter of lo = 0.001. Th e numerical results show a significant effect of exchange in the low-energy region. At higher energies ku O > 0.5, the transport cross section including all contributions merges into that which accounts only for static and polarization interaction without exchange. If only the polarization potential is taken into account, considerable deviations in the low- as well as in the high-energy region occur. A similar behavior has been found for other values of the screening parameter 1&u.

4.6.6. Electrical conductivity of alkali-atom plasmas For example, the electrical conductivity of Li plasma is shown in Fig. 28 as function of the total ion

density for different temperatures. At high temperatures, the typical behavior of the conductivity for a fully ionized plasma is observed. It increases monotonically with the density. This behavior changes

110 R. Redmu I Physics Reports 282 (1997) 35-157

lo5

,- IO4 QF

“a 103

lo*

0.0 0.2 0.4 0.6 0.8 1.0 16 17 18 19 20 21

kaB log “i [cni31

Fig. 27. The elastic e-Li transport cross section for WQ =O.l within different approximations: static and polarization potential including exchange (solid curve), static and polarization potential without exchange (dotted curve), only polarization potential (dashed curve) [STh96].

Fig. 28. ElectricaE conductivity of Li pfasma as a tin&ion of the total ion density ni for the tern~e~~s T =(4,6,10) x i03 K fSTA96]. The d~~~re~t curves ~~~esp~nd to the ~atm~nt of elastic ~I~~tro~-at~m scattering as shown in Fig. 27, i.e. static and polarization potential including exchange {solid curve), static and polarization potential without exchange

(dotted curve), only polarization potential (dashed curve).

drastically at lower temperatures T < 5000K. The degree of ionization is lowered (see Section 4.3.) so that the ~~rnber fraction of atoms becomes ~rn~o~ant which ~o~~spo~ds to the ~~~iaIly ionized

plasma state. As a result, the electrical conductivi~ decreases because of the reduction of of free electrons and their mobility in the plasma.

Let us consider the effect of different approximations for the transport cross sections of electron- atom scattering on the results for the electrical conductivity. The solid curves are calculated from those cross sections which account for static and polarization interaction as well as exchange. The ~o~es~ondi~g results without exchange are presented by the dotted curves. We find an impo~a~t influence of exchange contributions on the results for the conductivity in the region of the partially ionized plasma. The inclusion of exchange leads to a lowering of the conductivity which is more developed for lower temperatures. If the cross sections only with respect to the polarization potential are utilized (dashed curves), lower values for the co~du~~vi~ are obtained at higher dgnsities~

For densities YES > 5 x lOi cmW3, the conductivity is strongly increasing which indicates the occurrence of a Mott transition in the plasma. That transition can be interpreted as a nonmetal-to- metal transition between a state with low degree of ionization (partially ionized plasma) and another one with high degree of ionization (fully ionized plasma) because of pressure ionization of the atoms in the dense plasma [ISRA86].

The same general behavior is obtained for the other alkali-atom plasmas, As in Fig. 28, the curves increase at high temperatures which is typical far the fully ionized plasma state whereas at low temperatures the characteristic minimum behavior for the partially ionized plasma state is observed. In order to provide a complete set of conductivity data for all alkali-atom plasmas which is based upon the same theoretical level and respective approximations for the numerical evaluation,


Table 17

Electrical conductivity log (cr[ l/( .Q m)]) f or alkali-atom plasmas at T = 4000 K [STA96]

Li Na K Rb cs

16.0 2.927 2.968 3.058 3.068 3.080

16.5 2.913 2.937 3.095 3.112 3.137

17.0 2.874 2.887 3.125 3.149 3.178

17.5 2.787 2.812 3.137 3.171 3.209

18.0 2.676 2.700 3.124 3.170 3.214

18.5 2.560 2.583 3.092 3.147 3.215

19.0 2.439 2.462 3.055 3.130 3.212 19.5 2.338 2.380 3.060 3.141 3.264 20.0 2.358 2.425 3.242 3.353 3.485 20.5 2.770 2.816 3.591 3.638 3.861 21.0 3.660 3.656 4.148 4.203 4.173

Table 18 Electrical conductivity log( a[( l/( Q m)]) for alkali-atom plasmas at T = 6000 K [STA96]

Li Na K Rb cs

16.0 3.282 3.290 3.292 3.291 3.286 16.5 3.341 3.358 3.365 3.365 3.361 17.0 3.395 3.421 3.442 3.444 3.442 17.5 3.430 3.467 3.518 3.524 3.526 18.0 3.441 3.489 3.590 3.601 3.610 18.5 3.436 3.480 3.658 3.675 3.694 19.0 3.416 3.462 3.728 3.752 3.783 19.5 3.420 3.462 3.831 3.858 3.898 20.0 3.498 3.540 3.947 3.976 4.064 20.5 3.749 3.838 3.121 3.100 4.207 21.0 4.082 4.153 4.490 4.467 4.334

Table 19 Electrical conductivity log (c[ l/(Q m)]) f or alkali-atom plasmas at T = 10000 K [STA96]

log(~l[cm-31) Li Na K Rb cs

16.0 3.533 3.538 3.529 3.530 3.521 16.5 3.600 3.610 3.593 3.594 3.587 17.0 3.673 3.689 3.663 3.665 3.660 17.5 3.754 3.778 3.742 3.744 3.743 18.0 3.837 3.873 3.829 3.832 3.833 18.5 3.920 3.970 3.925 3.930 3.932 19.0 3.994 4.064 4.030 4.036 4.040 19.5 4.066 4.157 4.151 4.147 4.162 20.0 4.148 4.272 4.281 4.262 4.306 20.5 4.257 4.418 4.424 4.326 4.398 21.0 4.383 4.607 4.652 4.571 4.400

112 R. Redmer / Physics Reports 282 f 1997) 35-157

Fig. 29. Electrical conductivity of Cs plasma as a function of the total ion density Pli for different temperatures. The theoretical results of Starzynski et al. [STA96] are compared with the experimental data by Lomakin and Lopatin [LOM83] (x), Borzhiyevskii et al. [BOR88] (i-), Isakov et al. [ISA841 (o), and with the calculations of Gogoleva et al.

[GOG84] (0).

explicit results for Li, Na, K, Rb, and Cs are presented in Tables 17-19 for various screening

parameters J.n = Icao. A comparison with other available data for the eIectriea1 conductivity of Cs plasma is performed

in Fig. 29. Our results (soXid curves) show the same qualitative behaviour as already discussed for H (see Fig. 15) and Li plasma (see Fig. 28). ~ogoleva et al [GOG843 have eal6ulated the conductivity of par-Gaily ionized alkali-atom plasmas within the reIaxatio~ time appruximatio~ for a semi-~1assicaI Boltzmann equation. They used empirical data for the transport cross sections and considered also clusters in the EOS, however, without treating nonideality effects. This approach leads to smaller values for the electrical conductivity (open circles) in the low-density region compared with the

present results. There are few experimental data available for the electoral ~o~du~tivi~ of dense Cs plasma

[LOM83, ISA84, BOR88]. These data show a characteristic scattering and, due to the complexity of the experiments, rather large error bars, However, the experimental values are systematically smaller than the present results. Unfortunately, there are no data available up to now for the high-density plasma region where the Mott effect leads to the sharp increase of the ~ondu~tivi~. Here, only data for the low-tempera~re~ high-densi~ Cs vapor are available [HEN9 l].

4.7. Expanded fluid metals: improved Ziman theory

The present linear response fo~alism is valid for arbitrary degeneracy. The limiting case of complete degeneracy @ Q I is relevant for h~~~-d~nsi~y plas as as, for i~s~~~~, in astrophysical objects, see Fig. 1 /t For lower temperatures, liquid metals are usually considered as degenerate electron-ion plasmas [CHI78]. Ion-ion correlations are described by the structure factor Sij(q) where the interactions between the charges are screened by the electron dielectric function which is given in RPA by the Lindhard function [LIN54]. The electron-ion interaction in metals is weak and can be described by a pseudopot~ntial such as the As croR emp~-core potential.

Measurements of the liquid-vapor coexistence curve, the electrical conductivity and thermopower, the static magnetic susceptibility and the Knight shift of liquid alkali metals expanded by heating towards their liquid-vapor critical points have indicated that a metal-nonmetal transition (MNT) occurs in the critical region, see [HENSI), HENBO]. This imbibes that the interatomic forces must exhibit drastic changes as the density of the fluid is decreased and, simultaneously, the temperature

is increased towards their critical values. These precise experimental data for expanded fluid metals allow the study of the variation of

the physical properties in a large density--temperature domain: from the degenerate electron gas in the liquid metal near the milting point up to the no~degenerate, partially ionized vapor through the region of the critical point, see Fig. 1, where the metal-nonmetal ~ansitio~ occurs.

There is a ctose connection to the behavior of dense and low-temperature plasmas which are continuously following the density-temperature domain of expanded fluids in the supercritical region. The relation between the ordinary liquid-vapor phase transition and the electronic transition is not yet clear. Aspects of s~ct~re and disorder, of lucalizatio~ and bound state formation, and of electron co~e~ation have been studied witb~~ the Anderso Mott modet (see, e.g., ~~~T7~, EDW85, MOT87, LOG91, LOG95, EDW951).

A general theoretical treatment of fluid metals at low densities is still lacking. For instance, the percolation theory has been utilized for the description of the thermodynamic, electric, and optical properties of yaseous rnetcrls near the critical point [LIKSEI, LIK92j.

The present linear response fo~aIism has been applied to calculate the electrical cond~ctivi~ of

Cs along the coexistence curve [RED92]. Measurements of the electrical conductivi~ that the metallic properties of the fluid phase are lost at conditions close to the critic that the metal-nonmetal transition probably coincides with the liquid-vapor critical point [NOL88]. By correlating the structure and electrical conductivity data with the known EOS data, further ’

fo~ation about the electrical transport in expanded fl d metals can be gained. In compa~so~ with the standard theoretical approach to electrical prosodies of liqu Ziman theory [ZlM41], exhibits the limiting validity of the nearly free electron (NFE) model. Sub- sequent calculations have been performed also for the thermopower in expanded fluid Cs and the electrical conductivity in dense Cs vapor [RE193],

For a degenerate electron gas as in liquid metals near the elting point, the general expression (72) for the electrical conductivity is evaluated within a one-momentum approximation (i.e. considering only PO) which leads to the well-known Ziman formula [ZIM61],

Xii(q) is the static ion-ion structure factor, g:(q) the static electronic dielectric function, and f(q) the Fermi distribution function. The usual Ziman formula describes the resistivity of a liquid metal within the NFE model. It depends on the appropriate choice for the tweaks eIectron-ion pse~d~~ote~t~al

e approximation for the dielectric function E(q). Winter et al. [WIN87, WIN891 observed deviations of the electrical conductivity from the Ziman

formula ( 114) at densities below 1.3 g cm-j when applying the measured data for the static structure


factor &i(q) for Cs. These deviations occur in that range where the onset of the magnetic suscep- t~bi~i~ ~n~a~~erne~t [F: 79, ~A~$9~ and the increase of the effective electron mass ~K~~9~~ is obtained. This is usually ~ute~reted as an effect of strong electron-electron correlations.

The present approach (72) allows for different improvements of the usual Ziman formula which can be considered for the explanation of the deviations observed:

1. allowance for arbitrary degeneracy, 2. ~onsidg~tion of e~ec~on-e~eG~o~ sca~er~ng processes, 3. intrusion of higher momenta P, in dete~i~~ng the electron d~s~butio~ function, 4. treatment of an ionization equilibrium which accounts for localized electrons and, thus, leads

to a decrease of the number of free (conduction) electrons. Furthermore, the influence of various pseudopotentials as well as of different screening functions

g(q) [Lindhard function ~~~~q) plus local-meld corrections G(q)] on the numerical results can be

studied. The correlation functions N,.,, and D,,, which determine the transport coefficients (70) have to

be evaluated for arbitrary degeneracy. The generalized particle numbers N,, are given by (73). The force-force correlation functions D,,, can be decomposed with respect to the relevant scattering ~o~~buti~~s of free (c#~du~tion) electrons, i.e. electron-ion and ~l~~tron~lectron sca~er~ng. On the level of the Landau collision integral (77) relevant for a weak electron-ion pseud~~otent~al and

The electron-ion contribution was determined by means of the Ashcroft empty-core potential yt [ASH68b], the Heine-Abarenkov potential pi HA [HEI64], and a new local empty-core potential

cy proposed recently by Hasegawa et al. [HAS90]:

Y?(q) = 471ze2

- - cost ql?,. ) , 4”

y;“(q) = - 4nZe2 - si~(q~~) , $I?,.

4KZe2 CH(q) = - yzco”(q&)

C I + aq”

q2 f b2 exp( -bR,.) 1 + $ tan(qR,)

L 0

.

(116)

The respective cut-off- radii were chosen to match the electrical ~~nd~~tiv~~ near the melting point [COOSZ]. The values for LI = 22 and b = 1.2/ao were determined by Matsuda et al. [MAT911 so as to fit the form of the pseudopotential to the electron--ion potential that was calculated in local density approximation. The cut-off radius of R, = 3.2%~ was chosen ta reproduce theoretically the position of the first peak and the low-q behavior of the observed structure factor near the triple point.

The electronic dielectric ~Gt~on (83) is modified by loyal-meld ~o~e~tions G(q) and can be represented as

E(9) = 1 + wlvM~)c1 - GWl ‘i (117)

where no(q) denotes the Lindhard function. The influence of local-field corrections G(q) is studied utilizing the expressions derived by Sham [SHA65], Shaw [SHA70], and Icbimaru and


Utsumi [ICHSl]. A detailed discussion of the various local-field corrections can be found, e.g., in [KUS76, IWA84, GOL93, MOR95].

The static structure factor which enters into the Coulomb logarithm L(k) in (115) was taken from the experiment [WIN87]. The electron-electron contribution to the correlation function D,$ tends to lower the values for the electrical conductivity compared with the Lorentz gas (only electron- ion scattering; see [ROE81a]). However, these corrections are small for the conditions along the liquid-vapor coexistence line of Cs.

The following effects have been studied for an explanation of the deviations between the experimental values for the electrical conductivity and the Ziman formula.

4.7.2. Arbitrary degeneracy and injkence of higher momenta P,, The replacement of the sharp Fermi surface (integration s,‘” dq . . .) for complete degeneracy by

the Fermi distribution function (integration Sow dq f(q/2). . .) f or arbitrary degeneracy has only little

effect on the results, because the electron system is still in the degenerate domain above the critical density of Cs. However, for lower densities characteristic of the vapor phase, these effects become important. For instance, the electrical conductivity is increased by about 8% for the lowest density given here.

The Zubarev method for calculating the electrical conductivity (72) is rapidly converging with respect to a systematic extension of the set of relevant observables {P,}, see Table 14. For instance, the Spitzer result valid for the low-density limit is already obtained within a four-momentum approximation. With increasing density, the influence of higher momenta becomes less important and, in the limit of complete degeneracy, a one-momentum approximation yields already the Ziman formula (114). The electron distribution function is in this case a step function with a sharp cutoff at the Fermi momentum.

For nearly critical conditions, the degeneracy of the electron system is not complete and the consideration of higher momenta P, for calculating the electrical conductivity (72) leads to increased values. However, this increase amounts to less than 1% for the lowest density considered here.

4.7.3. Electron-electron scattering When including higher momenta, we have to consider, in addition to the electron-ion correlation

function D,$,, the electron-electron correlation functions 0:: (0:: = 0). Electron-electron scattering processes lead to a lowering of the electrical conductivity. In the low-density limit, the prefactor J’ for the co n uctivity is therefore 0.591 instead of 1.015 for a Lorentz gas (see Table 14). The d influence of electron-electron scattering decreases with increasing density [ROE8la] and vanishes in the limit of complete degeneracy where the final states for an electron-electron scattering process are occupied. The electrons can only respond to charge fluctuations which is described by the dielectric function E(q). For the conditions considered here, electron-electron scattering decreases the electrical conductivity by up to 10% for the lowest density of about 1 x 102’ cme3.

4.7.4. Pseudopotential and screening function The electrical conductivity (114) is calculated utilizing the Ashcroft, Heine-Abarenkov and

Hasegawa pseudopotentials ( 116) and different expressions for the local-field corrections G(q) in

1 I6

Table 20

Electrical conductivity (114) of fluid Cs in l/(12 cm) using various pseudopotentials V(q) and local-field corrections G(q) in the dielectric function z(q) for different temperatures T and densities p. The cut-off radii R, were fitted to match the rn~~~~~~~ electrical co~ductivi~ [CD0 821 for T= 373 K and p = 1.8 g cm-’

T (K) p &cm’-‘“) Ashcroft Heine-Abarenkov Hasegawa

Sh%N Shaw IU Sham Shaw IU Sham Shaw II..?

373 I .800 22 160 22 140

773 1.567 10240 9196

973

1.452 8237 7095

1173 1.332 6123 4880

1373 1.209 5140 3830

1673 0.956 2934 1916

1923 0.590 1488 815

22 180 22 190 22 210 22 190 22 240 22010 22 190

8890 10670 9244 8999 9992 10470 10810

668 1 X669 7189 6845 7978 8224 8329

4560 6502 5030 4767 5894 5618 5618

3567 548 1 3998 3783 4936 4342 4297

1731 3151 2040 1877 2808 2119 2003

701 1592 875 763 1430 868 767

Rc (au) 2.214 2.516 2.502 4.125 4.685 4.483 2.576 2.85 1 2.855

the dielectric function (117) which were given by Sham, Shaw, and Ichimaru and Utsumi (IU). The results are shown in Table 20.

The choice of the electron-ion ~se~dopotentia~ affects the results for the electrical conductivity only weakly. The use of the Hasegawa empty-core potential gives rise to small improvements as compared with the Ashcroft potential. However, local-field corrections have a strong influence on the results and yield a lowering of the electrical conductivity of about 50% in comparison to the values observed with the Sham dielectric fuIl~t~on for the lowest density considered here. Both the Ashcroft and the Heine-Abarenkov potentials yield conductivities lower than the experimental values for densities

greater than about 1.3 g cm- 3. The last row in Table 20, the Hasegawa empty-core potential with local-field corrections in the ~~hirna~-Wtsum~ form [ICH81], shows the best overall agreement.

In Table 21 we compare these values (B) with the experimental data of Cook [C0C%2] and Nofl et al. [NOL88]. For comparison, we present the results of the simplest approach using the ordinary Ziman formula (sharp Fermi surface) with the Ashcroft potential and the Sham dielectric function. ~u~~e~ore, we give the Co~duGt~vit~es of Hoshino et al. ~H~S~O] (A) which were obtained by means of the Ziman formula (114) with the Hasegawa potential and the Ichimaru-Utsumi dielectric function. The cut-off radius R,. was fitted to observed structure data [MATgIl.

The deviation of the current numerical results (B) from the experimental data is less than 10% for densities from the melting point up to 1.3 g cm- 3. For lower densities, the theoretical results are still up to 50% too high. For conditions less than about three times the critical density, another mechanism obviously becomes operative which leads to a further lowering of the electrical conductivity.


Table 21 Electrical conductivity (114) of fluid Cs in l/(G cm) compared to the experimental values of Cook” [COO 821 and No11 et aLb [NOL88]

T (K) Ashcroft

P (gcmP3) and Sham

Hasegawa and IU

(GJ”) A B C Experiment

373 1.800

773

1.567

973 1.452

1173

1.332

1373

1.209

1673

0.956

1923

0.590

Rc (au)

22 200 16600

10 100 14 100

8060 12900

5940 10 000

4940 7080

2760 3090

1370 490

2.214 3.25

22 000

10 800

8330

5620

4300

2000

770

2.855

(1.00) 22 000 22 196”

( 1 .OO)

10 800 10200b

(1.00)

8330 7800’

(0.95)

5040 5070b

(0.94)

3840 3550b

(0.90)

1670 1 770b

(0.70)

460 500b

Ashcroft and Sham: Ashcroft empty-core potential and Sham dielectric function; Hasegawa and IU: Hasegawa empty- core potential and Ichimaru-Utsumi dielectric function. (A) results of Hoshino et al. [HOS90] with R, fitted to observed structure data; (B) present calculation with R, fitted to the conductivity at the melting point; (C) consideration of the ionization equilibrium (ionization degree is given in brackets), see [RED92].

4.7.5. Ionization equilibrium The analysis of EOS data for dense, partially ionized alkali-atom plasmas (see Sections 3.3.

and 3.4.) indicates that the ionization degree is strongly reduced near the critical point due to the formation of bound states (localized electrons). Besides atoms Cs, also dimers Csz and molecular ions C$ are formed. They can reach maximum concentrations of about 20% and 40%, respectively, near the critical point (see Fig. 3). This is in good agreement with the results obtained by measurements of the magnetic susceptibility in expanded fluid Cs [FRE79].

The present theoretical approach for the transport coefficients (72) has been applied to the more general case of partial ionization [RED92]. The ionization equilibrium in Cs becomes effective for densities below 1.3 g cm- 3. The number of free charge carriers, and thus also the electrical conductivity, decreases on approaching the critical point. The respective values for 0 are indicated in Table 21 by the label C and show a better agreement with the experimental values for nearly critical conditions. The ionization degree resulting from the EOS within the partially ionized plasma model, Eqs. (29) and (30), is given in the brackets. Fig. 30 shows the electrical conductivity within model C in comparison with the experimental values and the nearly free electron model. The main effect in the drastic lowering of the conductivity below 1.3 g cmP3 as it was experimentally observed, and could not be described by any systematic improvement of the Ziman formula, seems to be the


5

4

F s 0

3

2 c

0 NFE

l exp.

+ part. ion. plasma model

!2 3 P

I .2

g _;._r:::

VAPOR / c:P LIQUI cl 0.4 -

0.2 -,$

3 ,,,I, I, olv I / I I , I I I I I 0.4 0.8 1.2 1.6 2.0 0 02 04 06 0.8 1.0 1.2 1.4 1.6 1.8 2.0

P (c3 Cm-3) DENSITY (g Cm-3)

z i m -

I.41 I I I I/ I I I I I 1

Fig. 30. Electrical conductivity of expanded fluid Cs [in l/(12 cm)] along the coexistence curve [RED92]: experiment [NOLSS] (o), NFE model (o), partially ionized plasma model (+). The critical density ec is indicated.

Fig. 31. Spin susceptibility per unit volume versus density along the liquid-gas coexistence curve of Cs extracted from Freyland’s measurements [FRE79] by Warren et al. [WAR89]. Dashed line indicates the Curie law for one electron per

atom.

reduction of the number of free charge carriers due to the formation of bound states. This leads simultaneously to a possible explanation of the enhancement of the magnetic susceptibility in that region. Unpaired localized spins in Cs atoms and molecular ions Csl give rise to strong Curie-like contributions to the electronic paramagnetic susceptibility near the critical point, whereas the spin- paired electrons in dimers Cs2 lead to the diamagnetic deviations in the dense vapor as obtained by Freyland [FRE79]. A detailed discussion of these effects is given in the next section.

The electrical conductivity of dense Cs vapor has been measured for subcritical temperatures [HEN911 and shows a strongly pressure-dependent, nonmetallic behavior which can be explained within the present partially ionized plasma model [REI93] by the strongly density-dependent ionization equilibrium. The same general behavior was also derived by Zhukhovistkii [ZHU89] within the Frenkel drop model by fitting two parameters so as to match the experimentally observed compressibility factor at the saturation line.

5. Magnetic susceptibility of expanded fluid metals

5.1. Theoretical model jbr the magnetic susceptibility

5.1.1. Experimental results and theoretical models The metal-nonmetal transition (MNT) in expanded fluid metals occurs near the critical point of the

liquid-vapor phase instability and leads to some peculiarities in the behavior of physical quantities compared with that of nonconducting fluids. As shown above, the metallic conductivity shows a sharp decrease in a narrow density range near the critical point to values less than 104/(R m),


see Fig. 30. This value, the minimum metallic conductivity as estimated from the Mott criterion, is utilized to locate the MNT in disordered systems also at finite temperatures. In the vapor, the conductivity decreases further when leaving the coexistence line and behaves like that of a weakly

ionized gas [HEN9 11. The coexistence curve of Hg and the alkali-atom metals in reduced units (p/pc versus T/T,) shows

a strong asymmetry relative to that of nonconducting fluids such as the inert gases, see Fig. 7 [JUE85, HEN9 11. There is clearly no common law of corresponding states for both the liquid metals and the

inert gases, and even not for the liquid metals as one group. These experimental data give no complete insight into the nature of the MNT. The magnetic

properties such as the susceptibility x(4,0) of Cs and Rb [FRE79, FRE80], of Na [BOT83] and Li [VAN881 (see also [WAR84]), or the Knight shift K [ELH83, WAR891 as function of the density p and temperature T are more sensitive with respect to the interactions in these systems. Generally, many-particle theories are needed to explain the behavior of these quantities [PIN89].

The volume electronic susceptibility extracted from Freyland’s measurements [FRE79, FRE80] of the mass susceptibility along the coexistence curve of Cs and Rb exhibits an interesting behavior. We have displayed the Cs data in Fig. 31 (see [WAR89]). At the melting point, the electronic susceptibility is enhanced by a factor of 1.6-2.2 compared to the Pauli spin susceptibility of a free electron gas which is typical for metals. The enhancement of the spin susceptibility derives from the local-field corrections to the dynamic dielectric function, e(q, co), and from the effective electron mass m* which is described by the conventional Stoner model. These quantities are not directly measurable and there is still a great variation in the theoretical results for the susceptibility.

This situation becomes much more complex for expanded fluid metals, since we have to deal with coupling strengths of 5 5 r, 5 15, a gradual transition to nondegenerate conditions, and thermal excitations near the critical point. When the density of fluid Cs is lowered by thermal expansion, the electronic susceptibility first decreases and reaches a minimum value at three times the critical density where the enhancement factor is about unity. The electronic susceptibility is enhanced again for lower densities and reaches its maximum value at about twice the critical density, where the enhancement factor is nearly two. For densities below this point, a strong decrease is obtained. However, the data provide a systematic deviation from the Curie law even at vapor densities.

This behavior together with that of the nuclear spin-lattice relaxation rate l/T1 and of the charge density at the nucleus (I Y(0) 12), w ic are derived from both the susceptibility and Knight shift h’ h measurements [WAR89], are not explainable by an extension of the conventional Stoner model to these conditions.

Various aspects of this MNT have been studied on the basis of existing theories [MOT71, MOT74, EDW85, MOT87, LOG91, EDW95]. The Hubbard model treats electron correlation effects in crystalline materials. Assuming that a band structure concept is applicable up to the critical region of fluid metals, the MNT can be explained by a splitting of the conduction band into two Hubbard bands caused by the intra-atomic electron-electron interaction at reduced densities. Thus, the transport should be thermally activated like in semiconductors.

A correlation enhancement of both the spin susceptibility and the electronic specific heat is expected from the calculation of Brinkman and Rice [BRI70]. The possibility of a ferromagnetic ground state for expanded fluid metals as inherent in the Stoner model is avoided due to the derivation of a reduced degeneracy temperature so that the susceptibility saturates at the Curie value for elevated temperatures.

120 R. Redmer I Ph~vics Reports 282 (I 997) 35-157

The liquid state is disordered and has only locally an ordered structure. We know from neutron scattering experiments [WIN87, WIN891 that the thermal expansion of Cs and Rb leads to an almost linear decrease of the coordination number rather than to an increased near-neighbor distance, see next section. This feature has been used as a basis for electronic structure calculations for expanded fluid metals within the one-electron (band) theory. One-electron theory has been highly successful in explaining the electronic properties of simple metals at solid state densities. The fluid metal at lower densities is modeled by assuming various crystal structures with different coordination numbers but fixed near-neighbor distance [WAR84, KEL86]. In addition, the effects of changing the interatomic separation for a given structure can be studied [KEL86]. In these calculations, antiferromagnetic ordering seems to be the favored ground state for the alkali-atom metals rather than a ferromagnetic one, similar to the behavior of H [SAN81, MINS6]. These one-electron band structure calculations fail completely in explaining the behavior of the Korringa relation and of the charge density at the nucleus.

Focussing on the effects of disorder, one has to deal with the phenomenon of Anderson localization [AND581 and a subsequent disorder-induced MNT. In fluid metals, a pseudogap might develop in the density of states at the Fermi level during the expansion and the states around EF become localized [MOT74]. We have seen that the one-electron picture is not sufficient for the description of the changes of the electron structure during the MNT and that electron correlation effects play an important role.

Logan [LOG91,LOG94] studied the effects of both electron correlation (on Hartree-Fock level) and of disorder (given by a random distribution of sites and characterized by a classical pair potential) in a disordered Hubbard model. The introduction of localized magnetic moments as a first effect of electron correlation and a simple scheme for averaging over all distributions of sites leads to the dis- tinction of three density domains: a nonmagnetic metallic state at high densities, a metallic state with local magnetic moments at intermediate densities, and a nonmetallic state with magnetic moments at low densities. A qualitative agreement with the experimentally observed behavior of the magnetic susceptibility and of the electrical conductivity was achieved for expanded fluid alkali-atom metals.

5.1.2. Susceptibility within the partially ionized plasma model

An alternative approach to such systems does not utilize methods of solid state physics which are clearly applicable for the high-density metal, but starts with the low-density vapor. For supercritical temperatures T B T,, a dense plasma state is reached which is characterized by partial ionization. The concentration of neutral and charged clusters can be determined dependent on density and temperature. Correlations lead to the lowering of the ionization energy and, at densities high enough to let even the deepest energy level merge into the continuum of scattering states, the plasma becomes fully ionized, and a corresponding nonmetal-to-metal-like transition takes place in a relatively narrow density region.

Contrary to hydrogen and the inert gases, where this MNT takes place at plasma conditions well separated from the ordinary liquid-vapor phase transition, in the expanded fluid alkali-atom metals and in Hg both transitions are superposed due to the relatively low ionization potentials in these materials. The model of a partially ionized plasma has been successfully applied to calculate the equation of state [RED88, RED891 (see Section 3) and the transport properties [RED92, REI93] (see Section 4) for the density-temperature region near the critical point of alkali-atom plasmas and Hg.

R. Redmrrl Physics Reports 282 (1997) 35-157 121

We will utilize this approach to calculate the magnetic susceptibility of Cs and Rb along the liquid-vapor coexistence line and compare with the experimental results of Freyland [FRE79,FRE80]. These experiments indicated not only susceptibility enhancement at expanded liquid densities, but also strong deviations from the Curie law at vapor densities. First, we will focus our attention on the possible mechanism which causes this special behavior, the formation of spin-paired dimers. The concept of particle clustering is then generalized to allow for higher-order clusters such as molecular ions Csl and Rbl so that even higher densities can be treated, especially the region up to two or three times the critical density. For higher densities, the fully ionized state is approached and the conventional Stoner model for the magnetic susceptibility yields the limiting behavior for this metallic state.

The dielectric function .z(q, co) is given by the retarded density fluctuation correlation function and describes the response of the charged particle system to the external longitudinal field as well as the induced density fluctuations. It is more convenient to relate the dielectric function to the associated time ordered correlation function, the polarization function Il(q, co), via (18).

Besides the self-energy C,.(k,z), also the polarization function ZI(q,o) has to be decomposed with respect to the contributions of N-particle clusters in systems with bound states, i.e. U(q, co) =

n, (4, (0) + fl,(q, w) + . . .. We will first discuss the contributions of one-particle states, n7, (q, co), to the polarization function. In the simplest case, this part is given by the RPA (83), which neglects correlation effects as well as density fluctuations due to the Coulomb interaction and is valid for the high-density limit, i.e. rs Q 1. Considering only the electronic contribution, the Lindhard expression is derived [LIN54].

For metallic (2 < rs I 5.5) and lower densities, correlation effects are important especially at short distances where they prevent the electrons from feeling the full consequences of the induced density fluctuations. These correlations are usually taken into account by introducing the concept of static local-field corrections G(q), which have been calculated for a uniform electron gas in various approximations (see, e.g., [KUS76, IWA84, ICH86, MOR951). The polarization function is then

given by

U,(q, (0) = noccl> WI

1 - G(q)V(q)no(q,o) (118)

In contrast with the spin-symmetric dielectric response to an external field coupling to the (charge) density fluctuations, the response of the electron system to a magnetic field is spin-antisymmetric. The magnetic susceptibility x(q,w) is therefore calculated from the spin-density fluctuation correlation function. The treatment of correlation effects analogous to the dielectric function leads to the following expression:

x,(q,w) = - &flO(!z> 0)

1 - Gs(q)~(q)~o(q> 0) ’ (119)

where ,& is the Bohr magneton. For the case of a noninteracting [G,,(q) = 0] and degenerate electron gas, the Pauli spin susceptibility follows from Eqs. (83) and (119) in the static and long-wavelength

limit,

x0 = lim {

limX,(q,o = 0) = 2E , I

3&r, kc, ++s q-o F

(120)


where EF is the Fermi energy and ~1, the free electron density. Band structure effects in solid and also liquid metals are usually described by introducing an effective electron mass m* in (120) which leads to a modified expression for the Pauli spin susceptibility in accordance with the Landau theory of Fermi liquids, xp = Xom*/me, see [PIN89].

The usual Stoner model for the explanation of the enhancement of the magnetic susceptibility in metals is derived from Eq. ( 119) by taking into account exchange and correlation effects in form of static local-field corrections G,,(q), and considering the effective mass of the electrons m*. The Stoner parameter @r = lim,_o { V(q)G(q)~o(q)m*/m,} is about 0.2-0.6 for simple metals at the melting point which yields typical enhancement factors of about 1.25-2.5. As mentioned above, the local-field corrections are not directly measurable and the effective electron mass can be measured only under limited conditions. Thus, the results for the Stoner parameter depend on the theoretical model applied for their calculation. A summary of a large number of theoretical results was given by Kushida et al. [KUS76] together with the experimental data available for the alkali metals near the melting point.

An additional quantity which can give insight into the behavior of susceptibility is the Korringa relation. Expressing the nuclear relaxation Knight shift K one finds for the case of noninteracting electrons,

ye and Y,, are the gyromagnetic ratios of the electrons and the nuclei, respectively. Utilizing the representation of l/T by the integral over the imaginary part of the susceptibility, and of K by the static uniform susceptibility, a more general relation can be derived [WAR89, MOR63],

the q-dependent magnetic rates l/q in terms of the

(121)

which expresses the relaxation rate relative to the Korringa value for arbitrary nuclear resonance frequency. Within the Stoner model, this ratio becomes

(122)

densities. w. is the

(123)

where the q-dependent Stoner parameter is defined by a(q) = V(q)G(q)Xo(q)m*/m,. For alkali metals at the melting point, qsr has typical values of 0.5-0.7. The agreement of both the enhancement factor (1 - xsT)-’ and of the ratio qsr would be a hint for a consistent model for the susceptibility.

From the behavior of the real and imaginary part of x(q,co) within the Stoner model one can conclude that q has to be a decreasing function when expanding the metal along the liquid- vapor coexistence curve. Warren et al. [WAR891 found the opposite behavior, a strong increase up to values greater than 1, for Cs from their measurements of the Knight shift K and the nuclear relaxation rates l/q. This indicates once more the limited validity of the Stoner model.

The next terms in the cluster expansion for the polarization function describe the contributions of N-particle bound states to the polarization function lI(q,w). Ropke and Der [ROE791 derived an expression for the dielectric function considering two-particle bound states in addition to the


free-particle states, ~~(4, w) = 1 + V(q)Ii’,(q, co)+ I7*(q, co). In the static and long-wavelength limit, this dielectric function can be written as

liiOcRD(q, w = 0) = c/q2 + D + 0(q2) . (124)

C stems from II, and is connected with the free-particle density via the inverse screening length K. D is derived from I772 and shows a Clausius-Mosotti-like behavior via the two-particle density PZ~“‘~) and their dipole polarizability c(u according to

(2,corr)

c = K20, D= 1+ ItA aD

1 - (4n/3)n~co%u . (125)

We adopt this approach in order to include the bound state contributions to the magnetic susceptibility. Evaluating the respective diagrams, one finds a Curie-like behavior for the localized spins of two-particle bound states (atoms A) in the static and long-wavelength limit,

xc = lim 1

lim x2(4, co = 0) /Q&‘-X 9-o I

= ff$ . B

(126)

Generalizing this result for the higher clusters with N 2 3, molecular ions Al with one localized spin yield a Curie-like contribution according to (126), proportional to their partial density nA;. Dimers A2 have paired spins and thus no Curie-like paramagnetic contribution.

All bound states have diamagnetic contributions due to orbital magnetization. The respective molar susceptibilities are given by the expectation value (Y’) of their wave functions,

(127)

where L is Avogadro’s number. Considering the relations between the volume susceptibility xv, the mass susceptibility xs, and the molar susceptibility xmol,

Ilv = PXg 3 xg = Xmol/M,ol > (128)

where p is the density and Mm01 the molar mass, the total volume susceptibility is given by,

x:“’ = Xv”++&+&+,;t +$ . (129)

5.2. Contributions to the total volume susceptibility

Within the frame of the given model, the expanded fluid metal consists of free electrons e and ions A+, of neutral atoms A and dimers AZ, and of molecular ions At with strongly varying concentrations along the liquid-vapor coexistence curve. All these species contribute to the magnetic susceptibility of the system so that we need the respective density and temperature-dependent expressions.

5.2.1. Susceptibility of the ion cores The first contribution in Eq. ( 129) is the diamagnetic susceptibility of the alkali ion cores A+

which have closed electron shells. Therefore, this part is considered to be almost constant, whether or not the ions are free or bound in atoms, dimers, or molecular ions. Furthermore, it is assumed to be


independent of density and temperature. Then, this contribution is simply given by xc’ = ~x~~,/M,~~,;. The densities p for a given temperature T were taken from the data of Jiingst et al. [JUE85] for the coexistence curves of Cs and Rb. Using Hat-tree-Fock wave functions [JOH83] in Eq. (127) the molar susceptibilities are given by -40.6 x lop6 cm3 mol-’ for Cs and -24.9 x lop6 cm3 mol-’ for Rb.

5.2.2. Free electron susceptibility The contribution of free electrons consists of both paramagnetic and diamagnetic parts, xG =

x:par + x:dia. The paramagnetic susceptibility is given by the Stoner model, Eq. (119). The full Lindhard function (83) is calculated for densities and temperatures along the coexistence curve of Cs and Rb so that we accounted for the gradual transition to nondegenerate conditions and thermal excitations when approaching the critical point. We have considered the local-field corrections in the form given by Ichimaru and Utsumi [ICH81] which are valid for rs 5 15. The use of this parametrization has already lead to improved data for the conductivity of Cs from the melting point up to twice the critical density (see Section 4.7. and [RED92]). The resulting enhancement factors and Korringa ratios for Cs and Rb are compared in Table 22 with the experimental findings and a satisfactory agreement can be pointed out.

The diamagnetic part was calculated by Vignale et al. [VIG88] in RPA. They found that the many-particle corrections have a considerably smaller effect on x;d’a than on x;par. Furthermore, the diamagnetic susceptibility is decreased by the interactions and not enhanced as the paramagnetic part. The numerical results indicate an almost linear decrease with respect to rs. The high-density expansion of Kanazawa and Matsudawa [RAN601 which was often used to extract the paramagnetic susceptibility from the measured mass susceptibility, shows an opposite behavior and may be the source of (small) systematic errors in the earlier curves for the spin susceptibility as in Fig. 3 1.

5.2.3. Susceptibility oj’ localized states The unpaired electrons bound in atoms A and molecular ions Al yield a paramagnetic susceptibility

according to the Curie law (126) with the respective partial densities instead of n,. Furthermore, they show also orbital magnetization with a corresponding diamagnetic susceptibility according to Eq. (37). For the calculation of the expectation value (Y’), we need the wave functions of the atoms, molecular ions, and also of the spin-paired dimers dependent on density.

These wave functions are tabulated in certain approximations such as the Hartree-Fock-Roothan scheme [CLE74] for the isolated atoms of almost all elements. Using the quantum defect theory for the determination of (r’),., for the alkali atoms [as earlier for the calculation of the quasiparticle shifts AI’, Eq. (37)] yields reasonable results compared with more detailed calculations [MUE84] with deviations of less than 10%. The respective values for the valence electron contributions are for Cs -25 x 10e6 cm3 mol-’ and for Rb -22 x 10e6 cm3 mol-’

Considerable less information is available for the magnetic susceptibilities of isolated molecular ions Al and dimers A2 [CAD81]. Though powerful methods such as the molecular orbital method or ingenious variational schemes have been developed for the construction of their wave functions, most of the results refer to the molecular ion and the molecule of hydrogen, to those of the lighter elements such as Li and Na, and to some simple molecules like NH3 and CH4. If there are experimental data from molecular beam studies available, typical deviations from the calculated values are less than 10%.

Table 22

Calculated en~ncement factors of the Pauli spin susce~tib~li~ and Ko~n~ ratios for Cs and Rb at the meting point using the local-field corrections of Ichima~ and Utsumi [ICH81] and the cyclotron resonance mass given by Grimes and Kip [GR163] compared to experimental data and more detailed theoretical calculations, see [RED931

E~erne~t

~nh~cement Korringa factor ratio

CS 2.336 1.76f0.06 2.24~~.~6

2.44 2.14Zto.01

2.20

0.412

0.578

0.6110.02 0.590

Redmer and Warren [RED931 Knecht,B dHvA Knecht,a dHvA

Dupree and Seymourb Springford, Templeton, and Coleridge,” dHvA

Vosko, Perdew, and MacDonald: theory (DFT) Narath and Weavere

EI-Hanany, Brennert, and Warren,’ NMR Shaw and Warren,” theory (XC)

Rb 0.627

0.617 0.628

i .703 I .724&0.008

1.55*0. IO

1.93

1.78

Redmer and Wades [RED931 Knecht,” dHvA

Dunifer, Pinkel, and Schulz,h spin waves.

Dupree and Seymou? Vosko, Perdew, and Mac~o~ald,~ theory (DFT) Narath and Weaver”

Shaw and Warrenp theory (XC)

a B. Knecht, J. Low Temp. Phys. 21 (1975) 679; de Haas-van Alphen effect (dHvA). b R. Dupree and E.F.W. Seymour, Phys. Kondens. Mater. 12 (1970) 97; see also R. Dupree and D.J.W. Cefdart, Solid

State Commun. 9 (1971) 145. ’ M. Springford, I.M. Templeton, and P.T. Coleridge, J. Low Temp. Phys. 53 (1983) 563; de Haas-van Alphen effect

(dHvA). ’ Vosko, Perdew, and MacDonald [VQS75]; density-~nctional theory (DFT). ’ A. Narath and H.T. Weaver, Phys. Rev, 175 (1968) 373. ’ El-Hanany, Brennert, and Warren [ELH83]; nuclear magnetic resonance (NMR). g R.W. Shaw, Jr., and W.W. Warren, Jr., Phys. Rev. B 3 11971) 1562; exchange-co~elatio~ (XC) potential taken from

Shaw [SHA 701. h G.L. Dunifer. D. Pinkel, and S. Schulz, Phys. Rev. B 10 (1974) 3159.

Aside from the fact that there are to our knowledge no data for the diamagnetic susceptibilities of the molecular ions and dimers of Rb and Cs up to now, those data would apply only for the dilute vapor, but not for the dense vapor near the critical point where these clusters are relevant. There, the wave captions have to be s~~~t~~~s of r~s~e~tiv~ ~-panicle eth~-Sa~~eter equations which take into account density effects. Noting the complexity of calculating atomic wave functions and energy levels [ZIM78b, KRA90] even for the case of N = 2, and the extensive numerical ~al~~~ations ~~~essa~ for the dete~ination of the electronic s~G~re of isolated higher clusters, a solution seems to be out of reach,

Therefore, we make a simple estimate for these contributions neglecting their density dependence. The wave function of the valence electron in the molecular ion AZ is constructed within a variational method utilizing the atomic wave functions already known from the quantum defect theory_ This

method will lead to smaller binding energies and to larger equilibrium distances compared with more detailed theories as we know from the study of the hydrogen molecular ion [BRA83], The resulting diama~etic s~scep?ibilities according to Eq. (127) should therefore be upper limits, The values for Csl and Rbl are -47 x IO-” cm3 mol-’ and -38.5 x lo-” cm3 mol-‘, respectively.

We adopt these values also for the diamagnetic susceptibilities of the neutral dimers. Though there is orbital ma~~tization by two valence electrons compared to one in molecular ions, these are more strongly bonded and also the eq~ili~rium distance is shorter in dimers. The two effects tend to compensate.

The total diamagnetic susceptibility of the dimers, the sum of the valence electron and the ionic core co~~ibutions, amounts to about - 128 x IQ-’ cm3 mol- ’ for Csz and -88 x 10v6 cm3 mol- ’

for Rbz within the given model. These values seem to be reasonable in comparison, for instance, with that of the I2 molecule, -90 x 10h6 cm3 mol-' . Applying the current estimate to Li2, a value of --23 x fO_” cm3 mol-’ is obtained which is also in reasonable agreement with that of -33 x 1 O-’ cm3 mol-” given in fKAR63].

A further mechanism which might contribute to the total susceptibility of molecules is Van Vleck paramagnetism [VAN321 which is due to virtual transitions to excited states similar to the electric pola~~abili~. Co~sidcr~ng the rough approximations made for the d~te~ina~on of the diamagnetic contributions and the relatively s all Van Vleck susceptibility of H2 (less than 5%~ of the diamagnetic susceptibility [TIL57]), we neglect these processes. However, the latter argument has to be checked when the matrix elements for these transitions are available for the alkali dimers, because the respective ~~sitio~ energies are much smaller than in H.

Ross et al. [ROS94] have recently performed total energy calculations for expanded solid Cs in the local density approximation and found that the diatomic form is more stable than the monoatomic form. F~~hc~ore~ cn~rgetica~ly a#racti~~~ low-lying electron excited states were predicted for CsZ-Csz tetramers from a molecular orbital configuration interaction calculation. These states would be thermally populated in the expanded metal region and might contribute to the paramagnetic susceptibility dependent on their symme~,

The stabili~ of an electron gas to fo~atio~ of bound states around a pair of ions was tested earlier by Ferraz et al. [FER84] considering Thomas-Fermi screening between the charges within the Heitler-London method for HZ. They argued that this mechanism may be responsible for the NNT in dense H and gave also reasonably estimates for the co~es~o~di~g critical rs values for the alkali-metal elements by a scaling procedure.

5.3.1. Maynetic susceptibility The calculated magnetic mass and volume susceptibilities along the coexistence curves of Cs and

Rb are shown in Figs. 32 and 33, respectively. Within the present approach, we are able to reproduce the experimental behavior for the mass susceptibility xs over the whole density range from the low- density, nonmetallic vapor up to the high-density metal at the melting point. The deviations from Freyland’s data do not exceed 25% for densities up to twice the critical density which is a reasonable result considering the approximations made for the solution of the EOS and the calculation of the respective quasiparticle shifts and partition functions, the uncertainties in determining some of the density-dependent contributions to the total susceptibility, and the error bars for the experimental

R. Redn-ter I Pfqyrics Reports 282 {1997/ 3S-157 127

-I I 1 , ’ I * i - I

Extracted from exp. -I Present calculation

o-0 0.2 0,4 0,6 0,8 I,6 1,2 1.4 4.6 I.8 DENSITY [ g cmd )

9,4

I,2

z,o

‘V

w

034

w

0.0

Fig. 32. Electronic paramagnetic volume susceptibility of Cs along the coexistence line: (solid line) experimental data [FRE79]; {ahoy-broken line) ~~~~a~ly ionized plasma model [RED93]; (~bo~-d~~ed line) Curie law for one electron per

atom. Taking into account the effective mass M* at the melting point [GRf63] yields values indicated by an arrow which are in reasonable agreement with the measured susceptibilities.

Fig. 33. Elec~o~ic ~~ramagnctic volume ~u~ceptjbil~~ of Rb along the coexistence line: (solid line) experimental data [FREW]; (short-broken line) partially ionized plasma model [RED93]; (short-dotted line) Curie law for one electron per atom. Taking into account the effective mass rn* at the melting point [GRI63] yields values indicated by an arrow which are in reasonable agreement with the measured susceptibilities.

values, especially in the vapor, and the absence of experimental results for the region near the critical point.

For densities between two and three times the critical d~nsi~~ the present values are systematically too small indicating that the chemical picture of ~ndividua~ species as applied here becomes more and more invalid with increasing density. The behavior of the mass susceptibility in that region can also not be derived from the one-electron theories developed for solid state densities. An extended cluster model which considers large and ~uct~ating cEusters in addition seems to be more appropriate for this region. The results for the magnetic properties of small metallic clusters in Li, Na, and K available so far [JEN87, MAR851 indicate stable localized spins in the A3 and A7 clusters.

Approaching the melting point, the enhancement of the susceptibility is described by the conventional Stoner model for the electron gas in metals (see Table 8). Considering the cyclotron resonance mass as given by Grimes and Kip [GRI63] for the effective electron mass together with the local-field corrections of Ichimaru and Utsumi [JCH81], the arrows in Figs. 32 and 33 show

ag~ee~~~~ with the experimental values for the mass and vo~~rn~ sus~~~t~b~l~ty. However, such an agreement is not unique and has also been achieved with other combinations of these parameters (see [KUS76]). The density-functional formalism developed by Vosko et al. [VOS75] yields the best ~~~an~ern~~t factors at the melting point for all alkali metals compared to the expe~n~enta~ values.

Furthermore, we have displayed the electronic paramagnetic susceptibility per volume x;pa’. We compare with the previous results which were extracted from the experimental data by subtracting the diama~eti~ ~on~ibut~o~s of the simple ions and the ~o~du~t~on electrons (see Fig. 31) The

128 R. Redmu I Physics Reports 282 (I 997) 35-I 57

Curie law for one electron per atom is considered to yield an upper limit for the paramagnetic susceptibility.

The experimentally observed deviations from the Curie law at vapor densities are clearly a result of the formation of spin-paired dimers in that region as already pointed out by Freyland [FRE79,FRE80]. He extracted dimer fractions for Cs and Rb by applying simple mass action laws to his data without considering nonideality corrections similar to our values in the dilute vapor. Analogous estimates can be given for Cs, Rb, and K based on new experimental data for the EOS [JUE85, HEN91]. The deformation of the Curie curve can thus serve as a measure for the amount of spin-paired species in the system.

The enhancement of the electronic paramagnetic susceptibility at about twice the critical density can be explained in the present model by the formation of molecular ions Ai which give a Curie-like contribution due to their localized electrons. This peak in the susceptibility-density plot for Cs was interpreted by Chapman and March [CHA88] as the limitation of enhanced Pauli paramagnetism and a consequent crossover to a Curie-like regime. They treated a correlation-induced MNT in expanded fluid alkali metals phenomenologically by a finite temperature extension of the theory of Brinkman and Rice [BRI70]. At this special point, p % 0.8 gcmp3 and T z 1780 K, they found a ratio kBT/EF M 0.18 from the renormalization of the electron degeneracy temperature due to the effects of correlation. Taking the fraction of free electrons from our composition data, see Fig. 3, we find a very similar value of 0.25. As the molecular ions vanish with increasing density, a susceptibility minimum occurs at about three times the critical density.

The results for the paramagnetic susceptibility between two and three times the critical density may change when higher clusters are taken into account. Although chemically stable higher clusters such as A3 or At are of less importance within our model, fluctuating charged and neutral clusters may develop in that region at elevated temperatures. This idea is supported by results of Likalter [LIK88,LIK92,LlK96] who utilized the model of percolation clusters in order to explain the behavior of gaseous metals in the vicinity of the critical point where the MNT occurs.

The decrease of the electronic paramagnetic susceptibility in the liquid metal range from the melting point down to three times the critical density seems to be a consequence of a reduced effective electron mass. This assumption is supported by the one-electron band structure calculations mentioned earlier [WAR84, KEL86], where a reduced density of states at the Fermi level was found for various crystal structures simulating the decreasing coordination number during the expansion of the liquid.

5.3.2. Korrinya relation Though the consideration of bound states leads to a reasonable overall agreement between the

calculated and measured susceptibilities, the experimentally observed behavior of the Korringa ratio q cannot be explained. For the normal liquid metal domain, the Korringa ratio rsr is recovered from the one-particle contribution xl to the total susceptibility and a reasonable agreement with the experimental value can be pointed out, see Table 22.

In the expanded metal domain, a decrease is obtained for the Korringa ratio from the Stoner model which is in contrast to the experimental findings [WAR89]. This failure is not removed by the two-particle contribution x2. In fact, the decrease becomes more pronounced. Because the scaling of the nuclear relaxation rate with the Knight shift as expressed by the Korringa relation does not apply for localized spins, this result is not surprising.

Warren et al. [WAR891 have d~v~l~~ed a s~~iquantitativ~ d~s~r~~tiQ~ of the Korringa ratio behavior in the expanded metal domain in terms of enhancement of the d~~rni~, ~onunifo~ s~sceptibili~.

rom this point of view, the observed increase in the Ko~~~a ratio at low density is seen to reflect a change in the character of electron spin fluc~ations from ferromagnetic enhancement in the normal metal to antiferromagnetic enhancement in the expanded metal. In terms of the present description, the development of antiferromagnetic spin fluctuation character corresponds to formation of Csz dimers and other spin-paired clusters. A similar phenomenon in low-dimensional solids is the well-known Peierls distortion. The physics may be the same in both cases, i.e. a lowering of the electronic energy by formation of the spin-paired bound state which more than compensates an increase in the ion-ion repulsion (elastic energy in the solid).

The charge density at ~~Cl~~S, (1 !Y(0)[2)F, averaged at the Fermi IeveE, can be extracted from the NMR and susc ility measurements. Compa~ng the Cs results with the atomic value

of /Y(O>li = 2.58 x t025 cmV3 [KUS49], Warren et al. [WAR891 found a ratio of < M i for the normal liquid metal at high densities. Below three times the critical density, the ratio is decreasing markedly while, after reaching a minimum value of about 0.28 at twice the critical density, it starts to increase. The value 1 has to be approached in the low-density vapor. They have pointed out that this behavior cannot be due to spin pairing in dimers because these particles do not contribute to the paramagnetic susceptibility.

ESR studies of neutral Li, Na, and K clusters [GAR83] have shown a dim~ishing s character with increasing cluster size. The ratio c decreases monotonically fro the atomic value 1 towards the respective value in bulk metal and shows no sign of a rn~~~rn~ ehavior. The band sickle catculations for the expanded fluid [WAR~4,KEL~6] also yield a simple monotonic increase towards the atomic value and fail to account a decrease of <.

A displacement of charge density away from the ion cores does occur in molecular ions in the expanded metal. The charge density in Hl, for instance, reaches about 40% of the value in atomic

H. ‘Using the same wave functions as for the determination of the diamagnetic susceptibilities, the

respective values are 47% for Cs,’ and 46% for Rb: which yield a decrease in the ratio <, but

not the minimum of < M 0.28 as found experimentally. A further decrease would appear if excited s of charged and neutraE clusters reach considerable concentrations, and if fluctuating

at elevated t~rn~e~at~~es.

5.3.4. ~~~a~-~~~~~ tal t~a~~~t~~~ The deviations of the paramagnetic susceptibility from the Curie law at vapor densities are clearly

due to the formation of spin-paired dimers A 2, as pointed out already by Freyland [FRE79, FRE80J in his experimental papers. The maximum of the paramagnetic susceptibility at about two times the critical density is explainable in the present model by the drastic increase of the fraction of molecular ions AZ. Both features support the applicability of the concept of cluster formation for the alkali-metal elements when increasing the density from the vapor to the expanded fluid domain. This model has already lead to reasonable results for the thermodynamic (Section 3) and transport properties (Section 4) of alkali-atom plasmas and fluids (see also [ALE83a, HER853 ).

130 R. Redmrr I Physics Reports 282 (1997) 35-1.57

A modified cluster model which considers large and fluctuating clusters in addition to the present model seems to be more appropriate for the region between two and three times the critical density, where our values are too small systematically. This seems to be an ideal subject for the concepts of cluster physics which are considered to bridge the gap between the molecular domain with its well-defined individual properties and the condensed matter domain where the collective behavior dominates the physical properties.

The MNT occurs in that domain where neutral and charged clusters, and eventually also higher fluctuating clusters, are formed. Such a picture coincides with the reverse Monte Carlo calculation of Nield et al. [NIE91], who treated the MNT as a bond percolation problem. They found that some finite atomic clusters are present close to the critical point as well as weak links within infinite clusters.

These ideas are strongly supported by the results of Ross et al. [ROS94] for the MNT in Cs. The formation of excited neutral dimers Cs; at elevated temperatures in the dense vapor is very likely to occur as one can conclude from their total energy calculations for the Csz-Csz cluster. The respective extra paramagnetic contributions of the excited states and the Van Vleck term of the ground state would diminish the present diamagnetic deviations from the experimental susceptibilities in the vapor and the expanded fluid.

The consideration of excited higher clusters or fluctuating clusters would also lead to a decrease of the charge density at the nucleus in this density region as it was deduced from NMR measurements [ELH83, WAR89]. The corresponding excitation mechanisms should be reflected in the imaginary part of x(q, co) so that the simultaneous increase of the Korringa ratio may be derived. Neither feature is explainable by considering only the ground states of the smallest clusters as in the present model.

6. Structure factor for dense plasmas

6.1. Structure jkctor and correlation jimctions

6.1.1. Pair distribution Jim&ion Besides the thermodynamic, transport, and magnetic properties, the study of the structural changes

in many-particle systems with respect to the density and temperature can give more insight into the relevant physical mechanisms for, e.g., the MNT. Therefore, considerable efforts have been put into both the experimental and theoretical investigation of the structural properties (for reviews, see [HAN86, LOV87, LEE88, EGE92]).

Physical quantities can be calculated by means of correlation functions as (12) or, alternatively, by means of distribution functions. The expectation values of dynamical variables A(v,, . . . , rN;pl,. . . ,pN)

in a system of N identical particles with the Hamiltonian H N = TN + UN are expressed by the N- particle distribution function &(v,,. . . , rN;pI , . . . ,pN; t) within the canonical ensemble. Supposing that A depends only on the coordinates as, e.g., the potential energy, and decomposing A into s-particle contributions according to

&J?,...,h) = a0 + 5aI(ri)+ i C a2(ri, vi) + f . . , (130) !=I i./=li#/

R.

the expectation value (A}, can

W‘I, * - * f G 0,

~~~~~r i Physics Rt?ports 282 jl997) 35-157 I31

be expressed by means of reduced s-particle distribution functions

(131)

The one-particle distribution function F1 (Q j t) denotes the probability for finding a particle at Y! at the time t and is normalized to the particle number N(t) = j’ dvlFl(ul, t). The stationary and homogeueo~s case yields F1 (Q, t) t= n. The two-pa~cl~ dist~b~t~on f~~t~o~ F*(q) rzr t) defines the probability for finding simultaneously two particles at rl and ~2 at a time t and is connected with

l;l (Q , t > by the relation

The bather-order s-particle distribution f~~~ti~~s are defined analogously. supposing radial symme~

of the interaction, i.e.

(133)

and the stationary case, the pair correlation function can be defined, g2(q2) = F2(q2 )/n*. Employ- ing the relation between the free energy an e config~~atio~a~ part of the canonical sum of states F = -kBr In QN analogous to (7), the the~odynamie and st~~~ral progenies of the ~-pa~i~~e system can be gained from this pair correlation function. For the internal energy U, tile pressure p, the isothermal compressibility K T, and the static structure factor S(q), the following relations hold (see [EGE92] ):

Especially, for the long-wavelength limit one obtains S(O) = nkBTq-. The pair correlation function g(r) can be determined from the Ornstein-Zemike equation for the total correlation function

h(r) = g~(~~ - I,

’ h(v) = c(r) + yt .I

dr’ c(r - r’) h(u’) , WV

where, besides the direct part C(T), the binary interactions with all other particles of the system are contained. The integral equation ( I 35) is usually solved replacing c(r) by h(r) and U(T). These clo- sures are known as hypemetted chain (HNC), Percus-Yevick (PY), or mean spherical approximation (MSA):

GE+,(~) = -/W,r> for r > 42, y?(r) = 0 for r < Rlz .

The PY method and the MSA are valid for systems with short-range, repulsive interactions such as nonconducting gases and fluids and yield identical results in the limiting case of bard-sphere interactions with a contact radius R12. The HNC method is suitable for systems with long-range interactions such as plasmas, fluid metals or electrolytes. For ail three methods, improved versions have been developed, For instance, the modi~ed HNC method (MHNC~ utilizes on the r.h.s. of Eq. (136) the bridge fu~~t.io~ B(r) of a hard-sphere reference system which leads to reasonable a~eement with molecular dynamic simulations or experimental results available for strongly coupled plasmas [ICH85a] and expanded fluid metals [MAT90].

6./,2. Fluctuation-dissipation theorem

Another possibility in determining the structure factor arises from the fluctuation-dissipation theorem which can be fo~ulated in terms of the densi~-density correlation faction L(g, ~1) according

to [ICH86]

S&o>) = 1 1

7I exp(~~~~ - I

The density-density correlation

ImL(q,w - iot). WV

function L is related to the polarization function Kl (25) via [KRA86]

The advantage of Eq. ( 137) for the determination of the structural properties in a many-particle system is that dynamic effects are taken into account in a genuine way. Therefore, the study of the dynamic structure factor is possible evaIuating the dynamic polarization Knutson. For instance, the specific behavior of the dynamic structure factor of expanded fluid Rb as found from neutron diffraction experiments near the critical point [WIN91, WIN941 should be studied within this model.

However, considering only the static severe factor

R. Redmerl Physics Reports 282 (1997) 3~5-157 133

the s~ndard approach is to solve the Epstein-Zemik~ integral equation (135) n~meri~alIy [LAB853 with respect to an approp~ate potenial. Having in mind the drastic changes of the int~~a~icle ~o~elations along the coexistence curve of expanded fluid metals which rest& in a MNT near the critical point, this effective potential has to include density (and tempera~re) effects which can be derived from the polarization function (see next section).

Considering the RPA for the one-particle contribution n,, applying a Kramers-Kronig relation,

and performing the classical as well as long-wavelength limit, the Debye-Htickel structure factor

and the related pair correlation function follow:

PH(q) = q2/(q2 + K2) ) @“(p”) = 1 - (&/r)eXp(-Kr). (140)

62. ~~~~~~~~~~u~ .~~~~~~~n and ~~~~~~~~ ~~~-~~n ~~~~n~~~~~

The ion-ion s~&tur~ factor i&(q) has been rn~as~r~d for expanded fluid metals such as Rb [~~~O~ WIN91], Cs [WIN87,WIN94], and Hg [TA~93,TA~94] along the liquid-vapor coexistence curve. The neutron and X-ray diffraction experiments indicate that the expansion of the fluid metal derives from a reduced coordination number and that the next-neighbor distance remains almost constant as can be seen from the pair distribution function gii(r) displayed in Figs. 34 and 35.

The static structure factor Sii(k) of expanded fluid metals is calculated for effective, density and temperature dependent ion-ion potentials which can be derived from the polarization function n(q, u), Eq. (138). Rewriting the relation (138) between the correlation functions L(q, w) and 17(q, w> by making the contributions of electrons and ions to the p~lari~atio~ function explicit, we find for the ion-ion density~ens~ty ~o~e~atio~ function [KOV903:

(141)

x [nii(q3 w)neeCqcr, a) - n,‘i(q, m)] .

For pure Coulomb interactions between electrons and ions we ave l&(q) = e2/(Eoq2), V,i(q) =

-Ze”J(E&), and Q(q) = ~Ze)2/(&~q2)~ so that the last term for Eq. ( 141) vanishes. For

metals, the NFE model is utilized so that flei = 0. Eq. (141) ca

Y;Tff(q9 w, = &i(q) + I/,:(q) aA,& (4

&hz, 4

5 (142)

Eq. (142) may serve as a starting point for the direct dete~ination of the dynamic structure factor via (137), For this, an ap~r~~~iate approximation has to be made for the ion-ion densi~y~e~sity ~o~elatio~ function Lii which, up to now, has not yet been derived witb~n the Green’s ~~~tio~

134 R. Redmrrl Physics Reports 282 (1997) 35-157

10

i a _

6 f

ORb ( I

0 0.5 1.0

wnp

f

d (glcm3)

Fig. 34. Next-neighbor distance RI and coordination number NI for expanded fluid Cs and Rb as extracted from neutron diffraction experiments [WIN87].

Fig. 35. Next-neighbour distance RI and coordination number NI for expanded fluid Hg as extracted from X-ray diffraction experiments [TAM93]. Three different methods to obtain NI are shown above the figure.

technique. For example, we know from the study of the thermodynamic and optical properties, that a RPA for the ion contributions to the polarization function gives poor results and that the ions rather produce a micro-field distribution than react like a polarizable background as in the case of electrons.

Another possibility for the determination of the static ion-ion structure factor Sii(q) is to utilize the Fourier transform of the effective ion-ion potential c:‘(q) defined via (142) for a solution of the Ornstein-Zernike equation (135) within the HNC or MHNC approximation, for which fast computer codes are available [LAB85]. The resulting pair correlation function gii(r) can be transformed via (134) to get the static ion-ion structure factor &(q). This method combines pseudopotential perturbation theory for simple metals (derivation of effective potentials) with standard classical liquid state theory (Omstein-Zernike equation).

Within this method, density- and temperature-dependent interionic potentials can be derived from the electron polarization function Uec which is treated in RPA (n,) including local-field corrections G,(q) [ICH81] via Eq. (118). We evaluate the full RPA expression (83) valid for arbitrary degeneracy of the electron system numerically in order to account for the thermal expansion of the fluid along the liquid-vapor coexistence curve. For the weak electron-ion interaction I’ei(q) in the fluid metal, we take an Ashcroft empty-core potential where the cut-off radius R, is fitted to the observed structure data at the melting point.


Fig. 36. Effective interparticle potentials in Hg for T = 1673 K and @ = 9.24 g cme3 [NAG96]: (solid line) Lennard-Jones potential valid for the atomic vapor, dotted line: NFE potential as derived from the polarization function. A hard-sphere

potential is shown in addition.

To check our calculation, the resulting structure factors at r = 1 and 0 = 0.1 were compared with self-consistent results of Mitake et al. [MIT851 for S&q) and G,(q) for strongly coupled hydrogen plasma, and a very good agreement can be stated.

With decreasing density, the electron-ion interaction becomes stronger so that electrons are localized at ions as bound states. For instance, the fraction of atoms is already dominant near the critical point of Hg and in Cs vapor (see Sections 3.3. and 3.5). For such a partially ionized system, the partial structure factors have to be considered. For instance, a Lennard-Jones-type potential which can be derived from the two-particle contribution Zi’ 2, see Eq. (25) [ROE79], should be applied to calculate the atom-atom structure factor.

6.3. Results for the static structure factor of expandedjuid metals

6.3.1. Comparison with experimental results The potentials relevant for these different regions of the density-temperature plane are compared

for Hg in Fig. 36. The repulsive branches of the effective ion-ion and atom-atom (Lennard-Jones) potential are almost identical and, thus, independent of the density. The attractive branches change from a screened Coulomb interaction in the liquid metal to an induced dipole-dipole interaction as characteristic for the atomic vapor. For comparison, a hard-core potential is also shown.

Utilizing standard integral equation techniques, we calculate the static structure factor Sii(k) in MHNC approximation [ROS79] for the effective interaction potentials as displayed in Fig. 36 along the coexistence curve of Hg and Cs. The resulting structure factors are compared in Fig. 37 with those derived from X-ray diffraction experiments for expanded fluid Hg [TAM93], and in Fig. 38 with the neutron diffraction experiments for Cs [WIN87].

The oscillatory behavior in the large k-domain for Hg is described within the Lennard-Jones as well as the NFE potential model because of their nearly identical short-range repulsive branches which can be replaced by an effective hard core potential. However, the asymmetry of the first peak as observed experimentally could not be reproduced within these calculations. Furthermore, the height of the first peak is overestimated within all three models in the liquid metal domain. For low k-values, the deviations from the experimental data become more pronounced with decreasing density.


0.0

0.0

E 3i

0.0

I 1400°C. 9.25 g/cm3

! 1200°C. 10.26 g/cm’ I

20”C, 13.55 g/cm’

r-t\

.! G J .c..7

./f 1 0.0 2.0 4.0

k [ft.‘]

i;,;T ._. - ,__- _.~ ___ _ _i 0.0 L

4

0.0

L.? z

0.0

0.0

2.0

1.5

1.0

0.5

i

1 1400°C. 0.96 #cm3 I

2.0 3.0 4.0 5.0 k [A-']

Fig. 37. Static structure factor of expanded fluid Hg calculated within the MHNC approximation utilizing different model potentials [NAG96], see Fig. 36: Lennard-Jones potential (solid line), NFE model potential (dotted line), hard-sphere potential (broken line), experimental data of Tamura and Hosokawa [TAM931 (dots).

Fig. 38. Static structure factor of expanded fluid Cs calculated within the MHNC approximation for the NFE model potential (dotted line) compared with the experimental data of Winter et al. [WIN871 (solid line) and the hard-sphere model (broken line), see [NAG96].

The small differences in the behavior of the structure factors derived from an effective ion-ion potential for the liquid metal and a Lennard-Jones potential for the insulating vapor, as well as the pronounced deviations from the experimental results indicate, that further effects have to be included in order to get a more detailed insight into the structural changes along the liquid-vapor coexistence curve of Hg. Jank and Hafner [JAN901 included relativistic as well as d-band effects when describing the ion core of heavy divalent metals. The interionic potential at the melting point


is then purely repulsive around the nearest-neighbor distance, and Friedel oscillations are almost damped out. Furthermore, partial structure factors Sii, $,, and S,, for the partially ionized system of ions (i=Hg 2+ Hg+) and atoms (a=Hg) have to be determined from respective effective potentials ,

which can be derived via the polarization function 17,d (see, e.g., [NAG92]). The structure factor for expanded fluid Cs along the coexistence curve is shown in Fig. 38 for

various densities [NAG961 in comparison with the experimental data [WIN871 and the hard-sphere

model. With decreasing density and increasing temperature from the melting point up to nearly critical conditions, the first maximum of Sii(k) is strongly damped and the higher maxima even vanish, contrary to the behavior of Hg in Fig. 37. The position of the first maximum is almost constant. There are pronounced changes in the behavior for small k-values due to the strong changes of the interionic potential. For high densities near the melting point, the compressibility of the system is low, and Sii(q -+ 0) is nearly zero. With decreasing density, the long-wavelength limit for the structure factor increases strongly. This behavior is mainly due to the attractive part of the interionic potential which becomes more pronounced in that region. There, the influence of two-particle bound-states via Z12, of dynamic screening, and of density fluctuations has to be studied in order to give possible explanations for the deviations observed. The present data for the structure factor of Cs coincide with the MHNC results of Hoshino et al. [HOS90], who have utilized a modified electron-ion pseudopotential and the Lindhard screening function for T = 0 when deriving the effective pair potentials along the coexistence curve.

6.3.2. Other theoretical results The structural properties of expanded alkali-atom fluids have been studied by means of computer

simulations [MOU78, TAN801 utilizing effective pair potentials calculated also within pseudopotential perturbation theory. The standard approach is to use an Ashcroft empty-core potential for the weak electron-ion interaction and exchange-correlation corrections to the dielectric function in the Ichimaru-Utsumi form [ICH81]. It has been shown that the results found for the static structure factor within the molecular dynamics simulation fully agree with those obtained within the MHNC approximation using the same effective pair potentials [MOR90, MAT91].

The dynamic properties of liquid alkali metals have also been studied within molecular dynamics simulations (see [KAH94]). A very good quantitative agreement with the data derived from inelastic neutron scattering experiments is found for the dynamic structure factor Sii(q, co) of Cs near the melting point [KAM92] and of Rb expanded along the coexistence curve up to 1700 K [KAH94]. Discrepancies occur at conditions near the critical point where the MNT occurs due to the drastic changes of the effective pair potentials with the density.

7. Spectral line shape

7.1. Introduction

The study of the optical properties of dense plasmas gives new insight into the behavior of partially ionized multi-component plasmas, For instance, reflectivity and absorption of the plasma are determined by the dielectric function and are, thus, closely connected with the optical conductivity, see [BER92].


Due to the rapid development of laser technology, high-intensity ultrashort pulses are now available so that plasmas at solid densities can be generated [MIL88, TEU93, SAU94]. The various absorption mechanisms (electron collisions, inverse bremsstrahlung, anomalous skin effect, X-ray emission, etc.) depend on the laser intensity and the plasma parameters so that reliable models for the light-matter interaction are required, see [ROZ90].

For astrophysical problems such as the age of globular cluster stars (sets a lower limit on the age of the universe), the luminosity of Cepheid variable stars (utilized to fit distance scales), or the density-temperature profile within a star (input for the helioseismology), the opacity plays an important role. The plasma opacity calculations are based on many-particle theory for the equation of state and standard methods for atomic structure calculations (for a review, see [RIC95]). As examples for more than twenty codes which have been developed are mentioned here the Los Alamos code (LAAOL) [HUE77], the Livermore code (OPAL) [IGL87, ROG92], and that of the opacity project (OP) [PRA93].

The broadening and shift of spectral lines emitted from plasmas are of special interest because spectroscopic methods have direct access to these quantities so that they are widely used for plasma diagnostics [LOC68, GR174, AUC89, HUT87]. In contrast to the natural line width and the Doppler broadening of spectral lines due to the thermal motion of the radiating atoms or ions, pressure broadening is caused by interactions with the surrounding plasma.

A quantum statistical approach to the optical properties has, therefore, to consider the microscopic processes in dense plasmas and was derived within the scope of kinetic theory as well as linear response theory [BAR58, BAR62, VOS69, VlD70]. An alternative approach employs the relation between the optical properties and the dielectric function that can be given by means of Green’s functions [KLE69, ROS66,ZAI68, KUD74, DHA80].

The influence of electron collisions on the shape of the spectral line profile was mainly studied within the semiclassical impact approximation [BAR62, GR174, SOB8 11. The radiating atom (or ion) is treated quantum mechanically, while the colliding electron is characterized by the classical impact parameter. Furthermore, it is supposed that the collision time is less than the time relevant for the radiation process. Due to the perturbative treatment of the electron-radiator interaction, the integrals for the shift and broadening of spectral lines diverge for small impact parameters. To treat also strong collisions, the integrand is cutoff at a minimum impact parameter that is usually determined within the WeiDkopf theory [WEl32]. The Coulomb divergence of the integrals for large impact parameters is removed by utilizing a screened potential, or simply by performing another cutoff at the Debye length. The impact approximation yields good results for the line center.

Strong electron-radiator collisions as well as collision times longer than the time necessary for the radiation process can be treated within the un$ied theory [SMI67, SM169, VOS69, GRE82a]. Neglecting virtual transitions between states with different main quantum numbers n, i.e. An = 0 (no-quenching approximation), explicit results can be given for the lines of hydrogen and hydrogen- like ions [LIS72, VID73, GRE75, ROS75] and a good agreement for the whole line profile can be stated. However, the shift of the lines derives mainly from virtual transitions with An # 0 so that the no-quenching approximation has to be avoided.

Within quantum kinetic theory [BAR58], the shift and width of spectral lines are connected with the scattering amplitudes for elastic scattering. Inelastic scattering contributes to the width via the electron cross sections at the radiator in the initial and final state. Utilizing the close coupling method for the calculation of scattering phase shifts for electron scattering at hydrogen or helium atoms,


the shifts of respective lines were obtained [TSU70, UNN90] which are, in general, too small. The numerical evaluation of the close coupling equations is expensive because a rather large number of states has to be included.

The influence of the ionic microfield distribution on the spectral line shape is studied within quasistatic and dynamic approaches [SEI81]. Compared with the simple Holtsmark distribution of static, independent ions [HOL19], the screening of the electric microfield in plasmas [ECK57] and pair correlations between the charged particles [BAR59, HO0661 lead to a higher probability of low field strengths in the classical limit. Furthermore, quantum corrections to the microfield distribution [BOE81, IGL82] tend to lower the probability of high field strengths compared with the classical distribution.

The influence of ion dynamics on the spectral line shape was shown experimentally by Wiese et al. [WIE72] by systematically increasing the reduced mass of the system radiator-perturbing-ion. Within the model microfield method (MMM), a stochastic process is utilized to derive the relevant contribution to the Hamilton operator which describes the dynamic correlation between the ions [BRI7 11.

Compared to static treatments of the ions, a better agreement with the experimental H, and HLj line profiles has been achieved [SEI77, MAZ78]. Similar results were obtained within the relaxation theory for ionic radiators such as He+ [OZASS] and Cs+ [OZA86].

Utilizing computer simulations for the time evolution of the ionic microfield distribution, the experimental results for the width of the first hydrogen lines were reproduced within the error bars [STA79, GIG86]. However, shift and asymmetry of the lines cannot be described well within these simulations due to the utilization of the no-quenching approximation and the restriction to the dipole approximation for the interaction between the perturbing ion and the radiator.

7.2. Green’s function approach to the spectral line shape

A special Green’s function approach to the optical properties of partially ionized plasmas was derived in a series of papers [ROE8lb, HIT86, HIT88, ROE89b, GUE89, ROE89, GUE91, GUE95]. Starting point is the relation between the transversal part of the dielectric function Etr(q, w) and the complex refractive index,

n(c0) + (ic/2<0) ~((0) = [Et,(q, w)]li2 . (143)

Considering the long-wavelength limit q -+ 0 which is relevant for emission and absorption of visible light, the transversal and longitudinal parts of the dielectric function are identic, and the absorption coefficient derives from the imaginary part of the dielectric function,

a(o) = [44o)l Im 44, ~1,

n(o) = (l/A) {ReE(q,o) + [(ReE(q,W))* + (ImE(q,o))2]“2}“2 .

Again, the cluster decomposition (25) for the dielectric function is applied. The one-particle contribution n71 describes transitions between free particle states and, therefore, the continuous spectrum. Transitions between the discrete energy levels of two-particle states are described by n2 from which the required line profiles can be derived. Taking into account the propagator of an isolated two- particle state Gi (19), the resulting line profile is only a b-function. A realistic line profile is

140 R. Redmer I Ph,vsics Reports 282 (1997) 35-157

obtained by taking into account self-energy corrections to the one-particle states ( 17) and dynamical screening of the Coulomb potential (18), the effective two-particle interaction I$ (24), as well as vertex corrections to the effective two-particle interaction [GUE91, GUE94]. The following final expression can be derived for f12, see Eq. (6.27) in [GUE95]:

x exp(-~~~,.p)(n;l(nIl[L(do) + ir”l-‘l4)l~2) . (145)

The vertex function AI:,,,,?(k) is given by (110). dw = (I) - Etl, + Et’, is the deviation from the line center. The other quantities are defined by

P.k k2

(146)

s = dti X - Im EC’(4, 6 + iO+)[ 1 + nu(f%)]S(o) .

--x 71

The evaluation of (145) requires approximations for the two-particle self-energy C, and the dynamic dielectric function E(q, co). The electron and ion contributions are separated as usual. The electronic, two-particle self-energy is treated in Born approximation with respect to the dynamical screened potential similar to Eqs. (17) and (IS), replacing the one-particle propagator by the two-particle Green’s function and utilizing the RPA (83) for the polarization function.

Strong electron collisions with the radiator can also be treated by performing the partial summation of ladder-type diagrams with respect to a statically screened potential [GUE93a]. The ion contributions are usually treated via the microfield distribution of a quasistatic reference system. However, ion dynamics are essential for the center of spectral lines so that the results of the MMM or of computer simulations for the dynamic microfield distribution have to be taken into account.

Averaging over all static microfield distributions W(b), the line profile for a transition i -+ j’ can be derived from the expression (see [GUE91]):

L(h) N C 4’;-“(W J’ & exp II’./ I’

(-&) ~=hWMW-l

1 -I

Re[C,(dw, p) - C,]+i Im[Ci(du,p)+Cf]+iT’ If’W) * (147)

I:;“(dw), the intensity of all transitions contributing to the spectral line, contains the trivial asymmetry via the frequency dependence of the dipole radiation and the occupation probability. Evaluating Eqs. (146) and (147) within simplest approximations reproduces the results of the uni$ed theory and of the semi-classical impact approximation for the shift and width of spectral lines [GUE95]. The advantage of Eqs. (145) and (146) is the genuine inclusion of many-particle effects such as


80

60

50

z 40

E 6 30

20

IO

0

.

.

.’ 8 I * I 8 . I. I I ., .,

0 1 2 3 4 5 6 7 8 9 10 Electron density n, [lop1 m-3]

Fig. 39. Shift of the maximum of the hydrogen H, line dependent on the particle number density n within static (0) and

dynamic screening (W) compared with the experimental results (0) for temperatures between T = (6 - 12) x 10’ K, see [BOD93, GUE95].

self-energy and dynamic screening, the quantum mechanical treatment of the perturbing particles, as well as the avoidance of the no-quenching approximation. Therefore, the shift and the asymmetry of spectral lines can be calculated within the present theory, to my knowledge, for the first time without any arbitrary parameters.

7.3. Results

The hydrogen lines (H, P, and L series) have been calculated up to IZ = 5 [GUE96]. The agreement of the shift and the asymmetry with the experimental results is reasonable for all lines. For the width of the lines, the ion dynamics is relevant which was treated in the MMM so that a good agreement with the experimental data can be stated also for that quantity.

7.3.1. Injuence of dynamic screening The shijt of the hydrogen H, and P, line have been calculated taking into account dynamic

screening via the full RPA dielectric function (84) [GUE93b, BOE93, DOE94, HIT95]. In Fig. 39, the shift of the H, line is shown as function of the electron density in hydrogen plasma [BOE93]. The experimental results for the shift dependent on density show a significant nonlinear behavior which cannot be reproduced within static treatments of screening in the plasma. The agreement between a dynamic theory and the experimental results is excellent for densities n, < 4 x 10” cmem3. For higher densities, the experimental error bars are large because the shift of the center of a strongly broadened line is not well defined.

142 R. Rrdmerl Physics Reports 282 (IYY7) 35-157

The same behavior has been found for the hydrogen P, line [DOE94]. Here, the energy differences between the levels are smaller so that the onset of dynamic screening effects, for which the plasma frequency can serve as a measure, are significant already at lower densities. The agreement between theory and experiment is excellent up to N, < 3 x 10” cm-‘.

The same behavior is also expected for other systems with comparable energy differences between the levels such as Xe or Cs. For instance, the plateau in the shift of the XeI line (transition 3ps + 1 sg with 467.1 nm) at about 2 x 10’” cme3 as found experimentally [KET85, KET92] can be explained by dynamic screening because the plasma frequency equals at that density the energy of the virtual 3p, + 4d4 transition which gives the main contribution to the shift [HIT95].

An interesting behavior is also found for the CsI line (transition 8d512 + 6pji2 with 621.3 nm) [KET92]. First, a blue shift is obtained which derives mainly from the 9~~;~ state slightly below the 8dS.2 level. At about 6 x lOI cmp3, the plasma frequency equals the energy difference between the 8d5.2 level and the 9~~:~ perturbing state so that another perturbing state, the 6f state slightly above the 8d5!2 level, becomes operative which yields a red shift. This inversion of the shift could again be verified theoretically when considering dynamical screening [HIT95].

7.3.2. ShiJt and asymmetry of the lines The shift of spectral lines in dense plasmas can be defined in various ways. For instance, the

determination of the maximum of a strongly broadened line is not simple. Another possibility is to study the shift of the center of a line. However, the cutoffs in the wings of that line which are due to the limited wavelength domain which is experimentally accessables produce some uncertainties. The shift of asymmetric spectral lines is even more complex. Therefore, the definition of the corresponding shift has to be taken into account when comparing between theoretical and experimental results, and even between different theories.

The main contributions to the asymmetry of a line, besides the trivial asymmetry due to the frequency dependence of the dipole radiation and the occupation probability, derive from the inhomogeneous ionic microfield and the quadratic Stark effect [SH069, HAL90]. Furthermore, fine-structure splitting is relevant for low densities or ionic radiators.

For instance, Fig. 40 shows the L, line for the 1s + 2s transition in hydrogen plasma [SEI94]. The experimental results, obtained from a Doppler-free two-photon polarization spectroscopy, can only be explained when treating the dynamic microfield of the ions and considering the fine-structure splitting of the line, see also [KOE94].

7.3.3. Line projiles of hydrogen-like ions Besides neutral radiators, also charged radiators (ions) are of importance for, e.g., astrophysical or

laser produced plasmas. Considering ionic radiators, the vertex function (110) yields also monopole contributions for all virtual transitions II --f r. The corresponding shift is the same for all levels so that the spectral line itself is not affected. Besides the dipole interaction between electrons and neutral radiators, also the quadrupole interaction becomes of importance for ionic radiators [GRI74,ST096]. The natural line width and the fine-structure splitting are proportional to Z4 so that these effects have significant influence on the line profile with increasing charge of the radiator.

For instance, Fig. 41 shows the experimental HeII H, line [PIE841 and the influence of fine- structure splitting, ion dynamics, and Doppler effect on the theoretical results [ST0961 which are in good agreement with the measured line profile only when considering all three effects. However, the

R. RedmerlPhysics Reports 28.2 (1997) 35-157 143

Fig. 40. Line profile of the hydrogen L, line in arbitrary units as derived from a Doppler-free two-photon polarization spectroscopy [SE1941 (0) for ne =3.7 x 102’ rnv3 and T=8000 K. The static treatment of the ions {broken line) yields poor results for the line center, while a d~~~~~ treatment (solid line) gives a good overall agreement, see [SE194: KOE94],

n, I 0 exrmiment PIE-484 f: --- static ions

I’ - static ions ancf fine structure

i i ZZ dZ%Z~~rbroadenin~ I! I- with Doppler and fine structure

i ‘,

Fig., 41. The expe~m~~tal He11 H, line profile in arbitra~ units FOR nB = 5.5 x 1O22 mA3 and T = 44 x 103 K [PIE841 compared with the theoretical results which account for various effects [ST096]: (broken line) static ions; (thin solid line) static ions and fine-structure splitting; (dash-dotted line) dynamic ions; (solid line) dynamic ions and Doppler broadening; (bold solid line) d~n~~~ ions and Doppler broadening and ~~e-s~~~re s~li~in~~ see fGUE95].

width of the H, (and P,z) line is too small even when cons~de~ng fink-secure s~~~tt~~g~ as it has already been found in other theories where this effect was neglected [STE94, GRE82b].

The resulting shift of the H, line is too small while that of the P, line is in excellent agreement with the experiment ~~R~94]. The theory af ~r~ern [GRI&8] yields t e opposite behaviors the shift

of the H, line is in good agreement, while that of the P, line is too high.

The spectral lines of further hy~og~~-~~ke ions such as t ose of C5+ or AI-“+ can also be calculated within the Green’s function approach [GUE95]. However, the dynamic microfield of the strongly coupled ions has to be determined rather from, e.g., the HNC pair correlation function similar to Section 6 than from Monte Carlo simulations [IGL83]. Considering ion dynamics, Doppler broadening, and fine-structure splitting, a reasonable agreement can also be stated for the H, and L,

lines.

statistical treatment of the physical properties of dense plasmas and expanded flui metals can be given in terms of Green’s functions. Especially, the partially ionized plasma model was utilized to calculate the EQS, transport coefficients, structure factor, magnetic susceptibility, and the spectral line shape of hydrogen, alkali-atom, and mercury plasmas as well as dense metallic vapors, Reasonable agreement of the theoretical results with the experimental findings can be stated.

The occurrence of a me&l-to-nortnzetul transition is a special feature in dense, low-temperature plasmas. In hydrogen and the inert gas plasmas, this MNT is well separated from the ordinary liquid-vapor phase transition and probably connected with a ~~~~s~~~ phase t~~~sitiu~ with a second critical point. This PPT has not yet been verified experimentally up to now, Recent co~du~t~vi~ measurements in dense hydrogen fluid ~W~I96] indicate that ~~e~~~Z~~~~~~~ already occurs in the molecular Auid due to band overlap at low temperatures, say 3000 K and 14OGPa, so that a first- order phase transition between a weakly dissociated and a (nearly) fully dissociated fluid phase at about lo4 K and 100 GPa probably does not occur. However, ionization may have a significant influence on this electronic transition at higher (plasma) temperatures,

Static high-pressure experiments have reached the domain of 250 GPa at T = 0 K where metallization and structural transitions may occur (see [MA094]). The study of the phase diagram of matter at temperatures between lo3 and 106 K and solid density ~0, say in the range e/e0 ==: 0.1 _I 10, is of high interest for the understanding of such a. While in dynamic high-pressure experiments driven by explosives [FOR821 or gas guns ure plasmas at solid densities have been produced, the potential of ~~~~=~u~~~~~ gion systematically has found litt ~~t~rest so far (see, e.g., rL~E94]). Aoki an [AQK94] have proposed rn~~t~p shock compression of solid hydrogen by high-power lasers in order to reach the phase instability region. The use of heavy ion beams is another way to produce the desired high energy density in matter [MEYBO]. These efforts are closely connected with the projects for inertial confinement fusicn (ICF).

Ultrashort (femtosecond) high-intensity laser pulses on solid targets make it possible to study matter at intensities above 10’” W cm-’ (see, for instance, ]TEU93, SAU94]). The behavior of matter under such conditions is described by plasma physics for strong nonequilibrium situations, i.e. the laser field exceeds the inner atomic field strengths many times. Future applications are connected with an efficient conversion of the laser radiation in the solid target into X-rays, leading to new s~~~~es for vacuum ~~~av~o~et ~VUV~ and extreme u~t~violet ~X~V) light,

~e~~ab~e data for the equation of state, the ~a~spo~ ~oe~~~e~ts, and the optical pro~e~~~s of dense plasmas are also needed in astrophysics, for instance, when calculating opacities [ROG92] or


modeling the Sun by means of helioseismology [TUR93]. The determination of nuclear fusion rates in dense plasmas has also attracted much attention in this context (for a recent review, see [ICH93]). The possible enhancement of these reaction rates which are given by the short-range behavior of the pair correlation function is due to many-particle correlations and of relevance for stellar interiors, interiors of giant planets or brown dwarfs, metal hydrides such as PdH and TiD2, or ICF experiments.

In expanded fluid metals such as cesium or mercury, the MNT occurs near the critical point of the liquid-vapor phase transition. Unified treatments of the thermodynamic, electronic, and structural transition along the liquid-vapor coexistence curve from the melting up to the critical point are needed to explain the special behavior of the thermophysical properties as found experimentally.

Besides the present accurate experimental data for the equation of state and the transport coefficients of expanded metal fluids (see [HEN89,HEN90]), future experiments on the dynamic structure factor [WIN91, WIN94], the magnetic susceptibility [FRE79], and the Knight shift [WAR891 will give new insight into the mechanism of the combined transitions.

The theoretical description of such systems requires the treatment of many-particle correlations as well as of disorder and electron localization. Approaches as the present partially ionized plasma model are based on the chemical picture where the density- and temperature-dependent interaction potentials between the various species have to be determined. This model is appropriate for the dense (atomic or molecular) vapor through the critical region and treats the (metallic) fluid state as a fully ionized plasma (see [RED95]). Alternatively, the model of percolation clusters is applicable in the critical region [LIK96].

Combined Mot&Hubbard-Anderson models are a promising tool to study such disordered interacting systems within condensed matter theory [LOG91, LOG95]. Although even the limiting cases _ the disordered but noninteracting Anderson model, and the interacting but nondisordered Hubbard model - are far from being fully understood, certain features of the MNT such as the formation of local magnetic moments in a system of itinerant electrons or the variation of the electric properties with disorder have been explained.

Due to the rapid development of computer capacity, ab initio calculations of the statistical properties of Coulomb systems will be performed on a larger scale. For instance, restricted path integral Monte Carlo simulations [PIE94], quantum molecular dynamics simulations [COL95], or wave-packet molecular dynamics simulations [KLA94] have been performed for strongly coupled hydrogen plasma. Grand canonical Monte Carlo simulations have also been carried out for a unified, self-consistent study of the structural, thermodynamic, and electronic properties of alkali fluids [CHA95].

There are, of course, many other interesting topics in the physics of dense plasmas which have not been treated in this review. For instance, the stopping power - describing the energy loss per time - is of central importance for, e.g., the deposition of energy by heavy ion beams in the outer shell of a fusion pellet in the ICF scenario (see [YAN85, KRA88, ZWI95]). The study of bremsstrahlung,

rtlfle&ivity, and absorption is of special relevance for fully ionized plasmas produced by high- intensity laser radiation, where the light interaction with matter has to be described (see [BER92]). The kinetics of low-temperature, chemically reacting, and partially ionized plasmas is treated via rate coefJic.ients which play a fundamental role in, e.g., ICF plasmas or for the plasma chemistry [BIB82, KL187, KRE89, KRE93b, SCH93]. These reaction rates as well as the full time evolution of nonequilibrium states can be derived by means of the nonequilibrium Green’s function technique which is outlined elsewhere [KAD62, DAN84, KRE89].

f would fike TV thank W. Ebe~~~~~ A. Fiirster, V.E. Fortov, J-P, Hernandez, LT. Iakubov, 6. Kahl, Y.W. Kim, A.A. Likalter, BE. Logan, J. Meyer-ter-Vehn, R/I+ Ross, R, Sauerbrey, M. Schmidt, S.A. Trigger, W.W. Warren, and IX, Witty for many stimulating discussions which have been very helpful in preparing and writing this review. I am very grateful to F. Hensel, W.D. Kraefi, D, Krcmp, S. Nagel, H. Reinholz, G. R6pke, and M. Schlanges for the pleasant and ~uit~t cooperation and support. Special thanks to S. Gun&r for providing me with the material for the spectral line shape in dense plasmas and helpful comments. I thank S. Nagel, A. Wierling and M. Reigrotzki for the help in preparing sume of the

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