Tài liệu The thermal conductivity of an arbitrarily dense plasma

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LA-10202-MS UC-34 Issued: Ohol~er 1984 The Thermal Conductivity of an ArbitrarilyDense Plasma .. . - , . , .—.— - LOSES anrilos LosAlamosNationalLaboratory LosAlamos,NewMexico87545 THE THERMRL CONDUCTIVITY OF RN RRBXTRRRILY DENSE PLRSHR by George Rinker RBSTRRCT reports This is the second in a series of concerning the transport properties of dense plasmas. In this aork. we use the formalism of Lampe to extend our previous calculations of electrical conductivity to thermal conductivity and the calculation of thermoelectric coefficient. Quantitative result are temperatures r nging from M -2 to gi~en for iron at -4 to 10sglcm3. 10 eV and for densities from 3x1O Lampe for [19681 has calculated a weakly-coupled Chapman-Enskog Sonine carried is Ct31CUliltiOtl plasma with any by so[ving out method, using electrical and thermal degree of the Fermi-statistical transport electron degeneracy. His equation by the Lenard-Ba\escu generalizations coefficients of the first two polynomials. The physical electron-electron These are calculated for structure electron approximation approximate fluctuations. coupling at high density. report [Rinker of the Ziman and are scheme Our valid to The validity formula for much stronger of this for section a method structure avoid for the Born effects of complote using and couplings. but of resistivity, arbitrary the shortcomings electron-ion is unknown. use and neg\ect factors, mu(tip(e-scattering scheme potentials. his the electrical thus logarithms. Coulomb Debye-shielded cross and electron-ion from we described realistic compensate with by principally calculations for the expressed scattering 19841. iOl)iC potentials, degeneracy. ultimately arises e(ectron-ion analysis for incorporates Born approximation the In a previous self-consistent are using of weak approximation partial-wave he interactions The requirement lattice models of Born Ue use an and density we have obtained good 1 for agreement liquid generally do as well only those by reproduce metals near as theoretical et al. [19821 equivalent in the weak-coupling results combining approach, we logarithm with electrical Simply a numerical in that case consistent for set conduct,v,ty, these calculat~ons than prev~ous L&mpe’s all to have a ●,der the present highly are surpassed parametrized ●e for to expect the of val,d, ,on,z~t,on ty the formal stateO calculated for are J = electr~c current e = electron charge ~ = applied electric field def,ned ●,th respect to the transport the to be val~d an ,nternally electrical coeff,c,ent. ,n temper~ture ow Coulomb expression se obta,n th,s In our Born approx~mathon Thus are explo,t wtth reproduce &nd thermoelectric range we the electron-,on h,s the equat,on Ue expect and dens,ty calculations. coeff,c,ents to for coeff4cAents. modify cond, t,ons. ty. work. transport not choices calculations ad,us[ed for conduct,v, where 2 IS as calculations thermal and Lenard-Balescu expression do Iogarlthm, virtually of our calculations appropriate the thermal that Ue Coulomb are con~t,v,ty Lampe”s value conductivity. electron-electron In improved replace m,th and limit. electrical of Lampe to obtain that formula Ziman our models that calculations have shosn factor. by In fact. data. structure fact the point. pseudopotent,al pseudopotential known conduction-band Boercker the melting equ~t,ons 2 p = ~ ‘ee ‘e ~ ‘ ‘T = electron ‘e ‘ pressure number density T = temperature heat (j= f{ux ~ = mean kinetic The electrical constraint conductivity S,12=S21 Wi(( Lampe’s reproduced generalized electron. Uand thermal expressions here. In the present for Chebyshev of extreme potential). The combinations of conductivity c (*ith to the the conventional Fulierton Each integral logarithms. for the have not has because with has an asymptotic as where S they involve M expansion previously. efficient effective is the expressions numerically factors of of be statistics. highly are become as many not been available Lampe’s ultimately will electron obtained (WkT>lk14, fail terms and These approximations that include lengthy Coulomb [19821 degeneracy functions coefflc~ent. rather account integrals approximations the are which these electron thermoelectric Sij approximations. combinations be evaluated. for addition application, in cases the integrals. expressions 10-decimal be catted explicit Fermi-llirac Flccurate These per J=ldl are The quantity For energy except chemical conta,n unstable. the integrals to the form m Ik(Z) &Zk ~diZ-i o i=O where z=u/kT i= the degeneracy parameter. The instabilities arise through 3 cancellation of the leading terms in z. manipulations can be done analytically we have chosen and equivalent the easier approximations, combining numerically before f!s an computation for of the for 10-2 the ionization accommodate all arise ~S to 10sg/cm3. bound strongly range states are as kl 1 shows 2 shows the ionization state formed to increases apparent. and Surface analysis. partial-wave to these computed the readily the coupled various Figure and Fig. ionization is in at Temperatures surface, thermal structure iron grid. 3x10-4 increases p of the results previous 1-8 show numerically sufficient of we consider our IOU densities~ 3 and 4 show the electrical peak near and 3d states metal. In other temperature Here regions no and point of compar our mode The states iC)niz~tiOn in the probably not probable errors point. scattering, of and p physical of at comparisons value for arises from which kl, the conductivity which the S-l. The of the 4s ionization iron makes behavior data our model electrical the of of 138.6. a exist is strictly resistivity the surprisingly a its transition electrical good the The inapplicability solid of of of 2 phase the model of with other the model. in the the agreement as experience factor in regions applicable. good resutt This y significant, least U in the units normal. son is gives kl d-nave for density conductivity at smatl experimental the experimental additional density more nearly is Virtually point. normal and the strong conductivity melting coefficients from Figures from very The onset are asymptotic model. sharp With because coefficients Figures clear kl-kl shell Instead. Fullerton’s procedure. on a logarithmic range fit from terms. evaluating conductivity 19841. plot. ionization irregularities 4 as this as a three-dimensional electrons. pressure transport [Rinker Zi as a contour of coefficients and densities state the and series in. electrical and densities zero determining application and the surviving method of simply transport state to 104eV. approaches present the the remaining temperatures same data sets and densities from of of ic)nizatit)n temperatures values instability example to extract and them, the expansions In principle, [iqUid The only at 118 ti(l~cm. is of melting compared satisfying elements indicates Table at room temperature for the solid phase but I gives and at is the readily The apparent. experimentally, calculated the resistivity This decreased. considered decreases presumably arises S and 6 show the transition-metal peak magnitude of because does not in dramatically from electrical conductivity, is negligible. as the additional with temperature at whereas the are whereas temperature the is calculated values available for the cm-is-i. scattering, Table conductivity. good The decrease Table “ Comparison Of and experimental Phase calculated the the melting point but values transport monotonical decrease processes lye Experimental T (K) — bUi(son — I \ resistiv,ty for ~c (~(’)ocm) solid and liquid ,d? p (g/cm3) nc become phase. electrical resistivity gives with experimental as additional II % phase. near The in relative electron-electron is liquid units is smatler the solid lowered. then rise in for agreement but first of electrical measurements the c --------------------------------------------------------------------------------------------------------------------------------------------------------------- ‘%east is processes not transport conductivity process temperature not conductivity electrical to experimental as the measurements the contribute with deteriorates thermal the additional comparisons active, dependence here. Figures which temperature n8 ~cdl iron ~d\ ~dl — Solid 7.86 293 13s 9.7a Solid 7.36 1810 126 127.Sb 1.lXM-S Liquid 7.0S 1810 118 138.6b l.Ox10-s 1.3X1O-S 6.SX10-3 ““0 2.4x10-4 [1983al. [196S1. -------------------------------------------------------------------------------________________________________________________________________________________— 5 --------------------------------------------------------------------------------------------------------------------------------------------------------------- Table Comparison of calculated conductivity Lx for solid conductivity and experimental aUeast thermal II cc(erg~K-ls-lcm-ll iron at zero pressure p (g/cm3) T (K] 7.86 300 0.ssx106 8.03x106 7,76 600 ~.f0xf06 S.47x106 7.66 900 1.68x106 3.80x106 7.S6 1200 2.26x106 2.82x106 7.46 1s00 2.86x106 3.18x106 C1983bl. -------------------------------------------------------------------------------------------------------------------------------------------------------------- Figures 7 and 8 show the Figures 9 and 10 show the conductive related simply to the thermal thermoelectric opacity conductivity Le in coefficient cm2g-1 . Sj2 This by where ~sb/k It 6 is = 4.106g6XIfj11 included here for S-l cm-2 convenient K-3 o comparison with radiative Opacities. in cm-1s-1 quantity . is Figures 11 electrical result and and 2 are where assumed. to 10-2, The ratit) temperature arises in both 1/2 low density. from errors with cases the of in and Born of the Hubbard our work. Plotted it)niz~tion states temperature and Lampe do not between and comparison he regions In Hubbard varies a calculations conductivity to theirs, is set ~}S•P…»€• 2 show 2 their throughout The fact approximation it and the ratio of this where is not the 1 ratio to be valid. region uniformly differences Of our in Figs. displayed calculation most that is and density consider and Lampe [19691 in high of 1 apparently the structure factor. Figures 13-18 electron-electron give compwisons contributions with the Lorentz to thermal conduction gas model. and which yields neglects the simple relationships and Figures 13 and 14 show the ratAo of result. Deviation ratio from electron COlliSiOIIS, density. Figures In ~ddition regions 18 show Large ratio deviations change the above degeneracy the Figures from low temperature These region (low 8S12/3L0 occur for high 19 and 20 display a are thermal important nondegenerate is readi of at the the is to the Lorentz importance intermediate importance. temperature which conductivity indicates 1S and 16 show the same for to of high 1 in this the of very and high identically large electrcm- temperature thermoelectric and low coefficient. deviations occur Figures density). gas 1 in the Lorentz 17 in and gas model. degeneracy. the degeneracy system parameter to a degenerate system u/kT. at high The dramatic density and ly apparent. 7 The ion-ion is plotted in Figs. CrYStalliZatiOfl va[ue Values at given is at are phase the and r=170 One-component 22. for any material our model at solid and experimental On the scale hardly of the figures, The heavy visible. I,ne . plasma R rough calculations semiemp,rical the zero-pressure Ilquld on F,g. check melting densttles differences indicate are between 22 at logr=2.2S well inside on this temperature. r=204 these and 190, values indicates of this transition. Figures 23 and 24 show the plasma [h:]2, where 21 constant by evaluating respectively. r ing coupl 3thc12 kTR{HNc2 t$J iS the nuclear Figures r , mass. 2S and 26 shorn the Rd —= [3r]-1/2 frequency ,on,c Debye radius . ‘t This region 8 parameter . becomes too small to be meaningful the solid-phase In Figs. 27 and 28. *e interpret classical mean free path A, defined by The quantity to equal the structure as

is a suitable are wou(d Fermi momentum dramatic at at pF the values for high the electron the in Figs. given momentum degeneracy. near low temperature becomes completely and The effects ,s of shell metal-insulator and dens,ty 1-4 a transition. (insulating phase), this per unit meaningless. 29 is a contour Figure value low temperature Rt be expected. parameter average from plot of the free electron number dens,ty volume. Ue expect the (see Fig. 221. region of They should feasible be to be retiable particularly useful well 12 and 22: Figs. throughout beyond because the see also the region Itoh they extend limits imposed et 1983. al. r<200 the by Born Flitake et 19841. Difficulties (ow results calculation [compare approximation al. present are temperature. shell altering the potentials the long-range systematic Ue have is clear as temperature. of a great difficulties and it the investigated not arise [see crystal Some efforts way shell [Hubbard purely lab~es from structure great of can be adjusted with,n for ~200. more of In temperature in this results limits by to the and density the region that our approach tightly bound for Lee and tlore empirical adjustment at absence of sotution. have been made to account and Lampe 1969. our be attached of freedom to accurate I and 11) becomes cannot region many degrees amenable transition phase the sensitivity significance An this problem and is so far the metal-insulator arise but the interaction order inadequate used, reality. Further occurs, This structure. In involves near These difficulties to ionic results. encountered schemes crystallization quick(y becomes ,n relation these effects 1983. Itoh to force et al. to the In a 19841. our results to 9 fit solid-state experimental data. It fits, but whether extrapolation accurate unavailable regions can be expected is not particularly these of to make sense difficult fits remains into to produce experimentally to be seen. RCKNOULEDGMENT I would routines like to thank L. U. Fullerton ate the generalized to Ca[CU for inventing Fermi-Dirac and providing me with integrals. REFERENCES Boercker. D. B.. F. J. Rogers. and H. E. DeUitt. Techniques NH]. Fullerton, L. U., 1982. Numerical Inte~rals [Coyote Press. Velarde. Hubbard, Itoh. Itoh, N., and M. Lampe. S. Hitake, 1969. H. Iyetomi, Rstrophys. and S. H., Y. 1., tli take, for J. Rev. f12S. — Generalized Suppl. Ichimaru. Ser. 1983. S,, 1968. Phys. Rev. and R. H. More, S, Ichimeru, ~. ~, 1623. Ferm,-Dirac 297. Flstrophys. J. 273, 774. 276. 1983, Phys. and N. Itoh, FluIds 1984, 27. — Ueast, R. C., cd., 19B3a. CRC Handbook BociI RatonO Florida)O~F-12S. Ueast- R. C., cd., 1983bo C&Handbook Boca Raton. Florida]. p. E-9. Wilson. J. R., 196S. Hetall. Rev. ~. 1273. l%trophys. “The Electrical Conduct,v, Rinker* G. R., 1984, Plasma.” Los Rlamos Nat,onal Laboratory report 10 Phys. and Thermal N., Y. Kohyama, N. Hatsumoto. and H. Seki. 1984. “Electrical Conductive ties of Dense Hatter in the Crystalline Lattice Phase.” (to be published in Flstrophys. J.) Lampe, Lee, U. B.. 1982. of Chem,stry — J. 37S. ty of an flrb, trar,ly Dense Lfl-9872-HS (January 19841. a- of — Chemhstry 381. 277. Phys,cs. and Physics. (CRC Press. [CRC Press. 1 I 891!J @“b I So’@) u’ I / I / { n @ s’ . -, . u) k J $ &fi”E G lg. 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