A model for the electron thermal conductivity of a two-component, strongly coupled plasma

Volume 112A, number 9 PHYSICS LETTERS 18 November 1985

A MODEL FOR THE ELECTRON THERMAL CONDUCTIVITY OF A TWO-COMPONENT, STRONGLY COUPLED PLASMA

W. ROZMUS Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1

and

R. CAUBLE Berkeley Research Associates, P.O. Box 852, Springfield, VA 22150, USA

Received 5 June 1985; accepted for publication 12 September 1985

A n expression for the electron thermal conductivity in the two-component strongly coupled plasmas is derived. The new formulation is examined in the context of laser fusion plasmas, using static correlations derived from the bypernetted chain equations. The model yields results in good agreement with molecular dynamics simulations.

The correct model for heat conduction is especially important in formulating a proper hydrodynamic descrip- tion of laser fusion experiments, where dense, laser-compressed plasmas become strongly coupled. In this letter the kinetic calculation of the electron thermal conductivity for two-component strongly coupled plasmas is pre- sented.

Assume a plasma being in thermal equilibrium at the temperature T = T e = T i, consisting of ions (qi = +Ze, mi) and electrons (qe = -e , me).

The plasma will be considered as weakly degenerate two-component fluid, i.e. for such densities n and T that the electron de Broglie wavelength 7~ = h/(2nmek B T) 1/2 is less than the interparticle spacing a = (3/4,nni)1/3. Such a semiclassical plamaa can be described by classical statistical mechanics with the assumption that electrons and ions interact via effective pair potentials which take approximate account of quantum diffraction effects at short distances. Here we adopt the following form [ 1 ] of the effective potential,

V a1~2 (r) = (q~lq~2/r) [1 - exp(-r/kal,~2) ] , (1)

where a l , t~2 = e, i, Xa, ~ = h/(2~rl~a.~kBT)ll2 is the thermal de Broglie wavelength for the pair t~l, a 2 with the reduced mass/a s a'2_'The effects ~lu~ to the Pauli exclusion principle can be included in eq. (1) by adding an extra term to the ef'f'ective electron-electron potential [2]. This'has been done in hypernetted chain calculations discussed below.

The calculations of the electron thermal conductivity will be based on the kinetic theory derived by Mazenko [3,4]. The starting point in his fully renormalized kinetic theory is an exact kinetic equation for the two-point phase space density correlation function,

/ _Nctl

¢(1,2;t)= 8(p l -P i ( t ) )8 ( r l - r i ( t ) ) -n~ ,~ '~ l ) )

/=1 8 ( P 2 - p/(O))8(r 2 - r / ( 0 ) ) - nc~ 2 f/~l 2 ~ 2 ) ) ) ,

440 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


0 0--7 Ca ~ a2(k 'P l 'P2 ; t ) - ~ dt 1 f d3p3~a~aa(k,Pl ,P3;t l ) C ~ a a 2 ( k , P 3 , P 2 ; t - t l ) = O , (2)

a a 0

where 1 ~ (rl , P l , °tl),f/~l~ is the maxwellian distribution function, and

Cala2 (k, Pl ,P2; t) = fd3(r 1 - r 2 ) exp[ - ik . ( r 1 - r 2 ) ]C(1 , 2; t ) .

The memory function ~b~ 1"2 (k, P l , P2 ; t) consists of a static contribution ~(s) ~1~2 = i(k 'Pl/m~l ) ~(Pl - P 2 ) ~ala2 - i (k 'Pl /mal) nt~l- JM"al"ala2t" D (k)

(Cg 1 a3 is the direct correlation function) and a collisional part ~b(~ c)_ The approximate form of the collisional x~ t~lCX 2 •

memory function is derived by using the renormalization procedure discussed by Wallenbom and Bans [5] (cf. also ref. [6]) and by assuming the disconnected form [3] for a four-point contracted correlation function. Final- ly the non-local and non-markovian ~(c) reads

i~b(e) 2 (k, p 1, P2; t) ha2 f/~t 2 (P2)

=-½kar f d3p3d3p4 f d3t ~aa4 87r-- ~ C~)la'(l) l" ~Pl ap2 Cala2 (k - l ,p 1 ,p2 ; t)Ca4aa(l , P4,P3 , t)

+ Va2+a(k-O'ap2C~laa(k-k ! - I , Pl,P3;t)Ca4"~2(1,P4,P2 ;t)] +k 'P l fMI (p l ) ~ala2(k,P2 't)' +(1 ~ 2 ) , (3)

where ~k a 1 a~ is a complicated operator which involves three-point static correlation functions. However, because it is multiplied by (k.p 1), ~ 1 a2 does not contribute to the thermal conductivity. The time dependence of C a 1 ~ 2 in eq. (3) will be approximated by free particle propagators.

In 6rder to extract the hydrodynamical limit (k ~ 0, t ~ co) from eq. (2), particularly the expressions for trans- port coefficients, one may follow the method originally introduced by Forster and Martin [7]. Their analysis showed that as a remit of the conservation and symmetry properties of the system the hydrodynamical modes can be calculated from the space-time correlation functions of conserved densities. This method in which the correlations of conserved quantities are found by projecting eq. (2) onto the hydrodynamic basis states was adopted to the two-component Coulomb system by Bans [8]. The electron thermal conductivity r e can be identified with the following expression [8]

r e = C V ni(1 + Z ) D , (4)

where C V is the specific heat and D is a thermal diffusion coefficient

O =(1 + Z ) - I (ZD ee + v ~ D ei + v ~ D ie +Dii) .

The coefficients D a 1 a2 above are related to the elements of the projected memory function,

Dala2 = lira lim ik-2I(Heallep(k,z)ln~2><n~ll~b(k,z)Q(z - Qdp(k,z)Q)-lQCp(k,z)lH~2)] (5) z~O k-*O

where ~(k, z) is an operator whose matrix elements are ~,~ l a 2 (k, p 1, P2; z) the Laplace-Fourier transformed memory function. The hydrodynamical state

H~ 1 = (2/3n~1)l/2(p21/2mal kBT - 3/2)

corresponds to the excess kinetic energy of species ot 1 . It is one of ten multi-fluid hydrodynamical states [8,9].

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And Q is a projection operator which projects on the entire non-hydrodynamical subspace. We will assume Q in one-polynomial approximation [9],

Q..~ IH~)(H~] , (6)

where

H~ =(1/lOne)ll2[k.p/(mekBT)ll2](p2/mekBT - 5), k =k/k .

The main result of this work, the electron thermal conductivity r e, eq. (4) with m i ~ ~, reads

r e = -C v (A + 257r3/2n e [(k B T)3/2/ml/2] (1 - B)2/E) , (7)

where the direct part A has the following form

(; A = [n2/(mekBT)l/2](l/72r¢3/2) dlRee(/)[~ lC~e(l) - 12ac~e/al- 13a2c~e(l)/al 2]

+ O; dllC~)i(l)Rei 8xlr~Z )"

The second term in the brackets, an indirect part, contains coefficients B and E,

ni(, ; B ~r ~ SkB------ T Z dll2C~e(l)Ree(l) "t" 1 4

0

+ ~ ; dl l2[Rei(l) - Rie(l)] la~(l)fi)l ) , 0

; dll2C~i(l)Rei(l) -~ f dll2C~)i(l)Rie(l) 0 0

13 J d113Cl~i(/) Rei(/) . E=ne 0 f dl l3C~e(l)Ree(l) + 4--~ni O

The potential energy contributions to re(7) are included through terms A and B and in the expression for C v (eq. (8) below). The thermal conductivity re(7) is defined in terms of integrals from the direct and static correlation functions which enter eq. (7) through the coefficients

Ralaa = ~ . V~1~3 S ~2~3 , Ot 3 = e r, 1

where Saza3 is the usual static formfactor, V~la: is the Fourier transformed effective potential (1). The specific heat (per particle), Cv, can also be evaluated [3] from the memory function ~alaz • It has the following form

3 { 2___2__[ Z (' fdll2C~e(l)Ree(l) .ni;dll2C~i(l)Rei(l)) Cv=~kB 1 - 3 k B T 27r2~ +Z) ne0 0

These expressions were evaluated using direct correlation functions and static formfactors derived from the solu- tion of the hypernetted chain (HNC) equations. It is known [10] that the HNC equations describe the equilibrium properties of charged systems quite well.

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Table 1 The numerical values for normalized electron thermal eonducUvities. The K~4N¢ ~ axe evaluated from eq. (7) and with HNC corre-

• • * - e - 2 1 / 2 - " - ~ 1 / 2 * - e l a t i on f u n c t i o n s KHN C - K /kBCOpekDe [kDe - (4~re ne/kBT) , tope - (47re he~me) , KDH - KDH/kBtOeekDe (el. eq. (9 ) ] , • " * e KBR = ~BR/kBtOpekDe (ef . eq. (9) w i t h t he B r y s k [10] f o r m for A) , ~ D E U = KDEU/kBt°pekDe [ef. eq. ( 1 0 ) ] .

n i (cm -a) * . . • I" r s KDH ~HNC KBR ~DEU

T e = I00 eV 1020 0.01 25.2 1.6 X 10 4 1.6 X 10 4 1.6 X 104 1.6 X 10 4 1021 0.02 11.7 2500 2400 2000 2400 1022 0.05 5.4 350 310 250 320 1023 0.1 2.5 56 48 35 49 1024 0.23 1.2 14 10 5.1 10 1025 0.5 0.54 1.1 3.2 0.7 3.5

T e = 50 eV 1020 0.02 25.2 3100 2700 2300 2700 1021 0.04 11.7 420 340 290 350 1022 0.1 5.4 66 47 38 51 1023 0.22 2.5 15 8.8 5.4 9.2 1024 0.46 1.2 i0 2.6 0.7 2.6

T e = 10 eV 1020 0.11 25.2 57 27 26 33 1021 0.23 11.7 14 3.9 3.3 5.3 1022 0.5 5.4 1.1 1.1 0.44 1.2 1023 1.07 2.5 69 0.6 0.06 0.6

T e = 5 eV 102o 0.22 25.2 15 5.5 3.7 5.5 1021 0.46 11.7 10 0.8 0.5 1.1

The numerical results indicate, first o f all, that the term A is usually very small for plasmas which are only weakly degenerate (P < rs; F = e2/kBTa , r s = a/h2mee2 ). The term B however, which also describes potential energy contributions, may have significant values and account for changes up to 60% in Ke(7) (for P ~ 1). The range of parameters which were tested corresponds to the conditions generated at the heat front of laser-fusion plasmas (of. table 1). The analysis relevant for these plasmas was reported in refs. [ 1 1,12]. However the thermal conductivit~¢ used there was basically Spitzer's formula [ 13 ] with a collision logarithm of the form

In(1 + k~ax/k~) , where kma x = m i n ( b e 1 , h~ 1), b c = e2/k a T, k2D = 41re2n e (1 + Z)/k B T. The electron conductivity analogous to Spitzer's result can be obtained from eq. (7) if the Debye-Hi ickel (DH) expressions are used for the direct correlation functions and interparticle potentials, C V = 3/2kB, A = B = 0, and S a 1"2 = 8 a 1~2" With these assumptions expression (7) has the following form

K~)H -~-s X / ~ k B (k B/')5/2 (roll2 Z e 4 ) - I ( ~ V ~ / Z + ~s ) - 1 A - 1 , (9)

where A is identical to Spitzer's Coulomb logarithm [13]

A = / ~ d x x 3(1 +x2) - 2 = ~ - 1 n ( 1 + ~ ) - ~ / 2 ( 1 + ~ ) , ~=(kmax/kD) 2 . 0

The numerical coefficient in eq. (9) is different from the one obtained by Spitzer [13] or Braginskii [14]. This is a consequence of our one-polynomial approximation (6), which was important for obtaining general expression (7).

* is eq. (7) with HNC static correlations. K~) H is the Spitzer result (9). It is obvious that at In the tables,/~HNC high densities and/or low temperatures, the latter form can be orders of magnitude off. Eq. (9) with the Brysk [11] formulation for A, K~R provides improvement over much of the range considered, but still can predict a thermal conductivity too a'aall by factors of five to ten.

Returning to the full expression, eq. (7), employing DH static correlations and the Fourier transform of eq. (1)

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Table 2 Comparison with molecular dynamics data by Bernu and Hansen [18 ], g BH = K/kB tOPe kDe, where K is the thermal conductivity defined in ref. [181 (el. eq. (8) of ref. [18]).

n i (cm -3) F r s KDH KHN C KBR rDE U KBH

T = 54 eV 1.6 X 1024 0.5 1.0 10.56 2.42 0.87 2.51 2.14 ± 0.6 T = 27 eV 1.6 X 1024 1.0 1.0 54.1 0.91 0.22 1.17 0.77 -+ 0.35

for Val ~2, the integrals can be evaluated. In the tables, this is K~)EU. This form provides a good fit to the HNC data. The result is similar to eq. (9), except for a small contribution from the non-ideal part o f C v , i f A is re- placed by (cf. refs. [ 1 5 - 1 7 ] )

~(~ - 1 - In ~1)/(1 - rt)2(1 - B ) 2 , (10)

where

r? -- k D2 hie,2 B = - [k3/ne(1 + Z)3/2 ] (0.0050 + 0.0119Z)

The direct part can be ignored. We do compare our results with molecular dynamics (MD) calculations by Bernu and Hansen [ 18]. As one can see from table 2 the coefficient K~INC (and also K~EU) approximate the MD data, K~H [18], reasonably well. The formula (7) and HNC results suggest that one may expect very different values for r e at the heat front o f laser produced plasmas, compared to the results obtained by simple interpolation o f Spitzer's theory [ 11,12]. The practical conclusion of this work is, that one can replace usual Coulomb logarithm by expression (10) in order to describe the heat conductivity in the region of cold and dense plasmas,

R.C. would like to acknowledge discussions with J. Davis and the support o f the Naval Research Laboratory. W.R. would like to acknowledge the generous support o f the Natural Sciences and Engineering Research Council of Canada.

References

[1] C. Deutsch, Phys. Lett. 60A (1977) 317. [2] H. Minoo, M.M. Gombert and C. Deutsch, Phys. Rev. A23 (1981) 924. [3] G.F. Mazenko, Phys. Rev. A9 (1974) 360. [4] G.F. Mazenko and S. Yip, in: Statistical mechanics, part B, ed. B.J. Berne (Plenum, New York, 1977). [5] J. Wallenborn and M. Bans, Phys. Rev. A18 (1978) 1737. [6] D.B. Boerekez, Phys. Rev. A23 (1981) 1969. [7] D. Forster and P.C. Martin, Phys. Rev. A2 (1970) 1515. [8] M. Baus, Physica 88A (1977) 319, 336. [9] W. Rozmus, J. Plasma Phys. 24 (1980) 265.

[ 10] M. Baus and J.P. Hansen, Phys. Rep. 59 (1980) 1. [ 11 ] H. Brysk, P.M. Campbell and P. Hammerling, Plasma Phys. 17 (1975) 473. [ 12] Y.T. Lee and RaM. More, Phys. Fluids 27 (1984) 1273. [13 ] L. Spitzer Jr., Physics of fully ionized gases (Interscience, New York, 1956). [ 14 ] S.I. Braginskii, in: Review of plasma physics, ed. M.A. Leontovich (Consultants Bureau, New York, 1965). [15 ] M. Baus, J.P. Hanson and L. Sjogxen, Phys. Lett. 82A (1981) 180. [ 16] L. Sjogren, J.P. Hanson and E.L Pollock, Phys. Roy. A24 (1981) 1544. [ 17] W. Rozmus and A.A. Offenberger, Phys. Rev. A (1985), to be published. [ 18] B. Bernu and J.P. Hanson, Phys. Rev. LetL 48 (1982) 1375.

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A model for the electron thermal conductivity of a two-component, strongly coupled plasma