A Kinetic Model for Electron-Ion Transport in Warm Dense Matter
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b A Kinetic Model for Electron-Ion Transport in Warm Dense Matter
Shane Rightley ∗ Department of Physics and AstronomyUniversity of IowaIowa City, IA 52242 USA
Scott D. Baalrud † Department of Nuclear Engineering and Radiological SciencesUniversity of MichiganAnn Arbor, MI 48109 USA (˙Dated: February 11, 2021)We present a model for electron-ion transport in Warm Dense Matter that incorporates Coulombcoupling effects into the quantum Boltzmann equation of Uehling and Uhlenbeck through the useof a statistical potential of mean force. Although this model has been derived rigorously in theclassical limit [S.D. Baalrud and J. Daligault, Physics of Plasmas
8, 082106 (2019)], its quantumgeneralization is complicated by the uncertainty principle. Here we apply an existing model for thepotential of mean force based on the quantum Ornstein-Zernike equation coupled with an average-atom model [C. E. Starrett, High Energy Density Phys. , 8 (2017)]. This potential containscorrelations due to both Coulomb coupling and exchange, and the collision kernel of the kinetictheory enforces Pauli blocking while allowing for electron diffraction and large-angle collisions. Bysolving the Uehling-Uhlenbeck equation for electron-ion relaxation rates, we predict the momentumand temperature relaxation time and electrical conductivity of solid density aluminum plasma basedon electron-ion collisions. We present results for density and temperature conditions that span thetransition from classical weakly-coupled plasma to degenerate moderately-coupled plasma. Ourfindings agree well with recent quantum molecular dynamics simulations. I. INTRODUCTION
The microscopic physics of WDM is subject to a mul-titude of physical effects, including electron degener-acy, partial ionization, large-angle scattering, diffraction,and moderate Coulomb coupling leading to correlations.Such conditions are present in experiments involving ex-treme compression of materials [1–3], in astrophysics[4, 5], and along the compression path in inertial con-finement fusion (ICF) experiments [6]. As a result of thedemanding conditions for theoretical modeling, the de-scription of WDM has been highly reliant on computa-tional techniques. However, ab initio computation provestoo expensive for many problems, whereas faster meth-ods often involve uncontrolled approximations or haveuncertain applicability. In order to support computa-tional efforts, explore larger regions of parameter space,and expediently provide data tables for hydrodynamicsimulations, reliable and fast tools for the computationof transport coefficients in WDM remain desirable.In this work, we introduce a model for electron-iontransport based on the quantum Boltzmann equationof Uehling-Uhlenbeck [7], but with a modification mo-tivated by the classical mean force kinetic theory [8] inwhich aspects of many-body interactions are modeled bytreating binary collisions as occurring via the potentialof mean force. The model accounts for at least some de- ∗ [email protected] † [email protected] gree of partial ionization, electron degeneracy, moderateCoulomb coupling, diffraction, and large-angle collisions.The approximate regimes in which these different phys-ical processes are important can be roughly understoodin terms of the degeneracy parameter Θ ≡ T e /T F andCoulomb coupling parameter Γ = h U i / h K i with the sta-tistical averages taken using a Maxwell-Boltzmann distri-bution for ions and a Fermi-Dirac distribution for elec-trons. T e is the electron temperature, T F ≡ E F /k B theFermi temperature, h U i the average interaction energyand h K i the average kinetic energy of a particle. The av-erage speed of electrons shifts from the thermal speed tothe Fermi speed as degeneracy increases, a phenomenonthat causes electrons to become increasingly weakly cou-pled at high density. The Coulomb couplings Γ ii and Γ ie for ion-ion and electron-ion interactions, respectively, canbe expressed Γ ii = Z e /ak B T , (1)and Γ ie = Ze /ak B T Li / [ − ξ ]Li / [ − ξ ] , (2)where a = (3 / πn ) / is the Wigner-Seitz radius, Li isthe polylogarithm function (closely related to the Fermiintegral) and ξ ≡ exp( µ/k B T ) where µ is the electronchemical potential related to Θ through the normaliza-tion of the Fermi-Dirac distribution [9]: − Li / [ − ξ ] = 43 √ π Θ − / . (3)The conditions Γ = 1 and
Θ = 1 divide the density-temperature parameter space into multiple regions, asseen in figure 1. The regimes can be broken down into (1)classical weakly coupled, (2) classical strongly coupled,(3) quantum weakly coupled, and (4) classical stronglycoupled ions with degenerate weak or strongly coupledelectrons. WDM exists at the intersection of all of theseregions marked by the blue region, where no small ex-pansion parameter is available. Transport in region (1)is well-understood in terms of the Landau-Spitzer the-ory [10], and region (3) has been successfully modeledthrough quantum weak-coupling theories such as thequantum Landau-Fokker-Planck equation [11]. Progresshas recently been made extending classical plasma trans-port theory into region (2) for Γ . through use ofmean force kinetic theory (MFKT) [8, 12, 13], whichhas also been successfully applied in region (4) for WDMin the case of ion transport [14]. Other existing kineticmethods for predicting transport in WDM typically fallinto the categories of binary collision theories [11, 15–18], linear response theories [19–21], and non-equilibriumGreen’s functions and field-theoretic methods [22–25].The model presented in this work is physically intu-itive, contains much of the relevant physics, and can beevaluated relatively quickly. It is based on the Uehling-Uhlenbeck equation (named BUU equation from thispoint on, with the letter B referencing Boltzmann), whichaccounts for degeneracy and diffraction [7]. Correlationsin a moderate Coulomb coupling regime are modeledthrough the assertion that the binary scattering is me-diated by the equilibrium statistical potential of meanforce. This mean force is computed using a recent com-bined Average-Atom + Two-Component-Plasma model[18, 26]. The result has the advantage of retaining thedominant aspects of the relevant physics, while remainingrelatively fast to evaluate in comparison to fully dynam-ical calculations. In the classical limit the model can berigorously derived [8], but while this derivation cannot beeasily extended to the quantum domain due to the un-certainty principle, it is reasonable to apply the potentialof mean force to the BUU equation.Explicit results are computed for momentum and en-ergy relaxation rates of aluminum at conditions spanningthe WDM regime. The results for energy relaxation arefound to be equivalent to a recent model by Daligaultand Simoni [27] if interactions are assumed to occur viathe potential of mean force in that theory. An unan-ticipated result is observed for momentum relaxation,whereby degeneracy influences the relaxation rate in adifferent manner than for energy relaxation. This effectis not predicted by previous reduced kinetic theories, butappears to lead to better agreement with quantum molec-ular dynamics simulations of electrical conductivity atWDM conditions [28].We begin by detailing the model in section II. We in-troduce the potential of mean force into the BUU equa-tion and discuss what the concept means in the contextof a degenerate plasma. In section III we apply this to electron-ion momentum and temperature relaxation,where we obtain degeneracy- and correlation-dependent“Coulomb integrals” that replace the traditional Coulomblogarithm. In section IV, we evaluate the model forthe solid-density aluminum and compare to common andsimple alternatives and discuss the relative importanceof the effects of correlation, large-angle scattering, Pauliblocking, and diffraction. We conclude and summarizein section V. Figure 1. Parameter regimes of fully ionized hydrogen plasma.The solid black line is the boundary between weak and strongelectron coupling Γ e = 1 and turns over due to the electrondegeneracy; the dotted lined is the separation between weakand strong ion coupling Γ i = 1 ; and the dashed line is theseparation between classical and degenerate electrons Θ = 1 .The darker blue oval denotes the sector of WDM. Region 1(yellow) is classical weakly coupled plasma; region 2 (lightblue) is characterized by classical strong coupling; region 3(pink) by quantum weak coupling, and region 4 (green) byboth quantum electrons and strongly-coupled ions. We expectthe theory presented here to apply to each region 1-4. The redline demarcates the region of validity of plasma-type transporttheories; beyond this is the regime of condensed matter.
II. A KINETIC EQUATION FOR TRANSPORTIN WDMA. The Uehling-Uhlenbeck Collision Operator
We consider the collision integral from the right handside of the BUU equation [7], C ss ′ q = Z d v ′ d Ω dσd Ω u (cid:20) ˆ f s ˆ f s ′ (1 + θ s f s ) (1 + θ s ′ f s ′ ) − f s f s ′ (cid:16) θ s ˆ f s (cid:17) (cid:16) θ s ˆ f s ′ (cid:17)(cid:21) (4)where the “hatted” quantities ˆ f s are evaluated atthe post-collision velocity ˆ v = v + ∆ v and θ s =( ± /g s )( h/m s ) where g s is an integer accounting forparticle statistics with g s = g e = 2 for electrons, the + sign corresponds with Bosons and the − sign withFermions. Calculation of dσ/d Ω is carried out via a par-tial wave expansion in terms of the phase shifts δ l ( η ) . Thedetermination of the phase shifts from the Schrödingerequation is discussed in Appendix A.The BUU equation describes the evolution of theWigner quasi-probability distribution function f s . It wasoriginally proposed as an extension of the Boltzmannequation to account for degeneracy [7], but a consistentderivation of the equation was not accomplished for sometime. Early methods involved applying the BBGKY hier-archy to the kinetic equation for the Wigner function andoften fell short of fully obtaining the BUU equation, i.e.to include the θ s terms [29, 30]. Ultimately, a derivationwas carried out using the BBGKY hierarchy in the den-sity operator formalism [31]. This required a modifica-tion of the typical weak-correlation assumption in deriva-tions of the Boltzmann equation. Instead of neglectingthree-body correlations entirely, Boercker and Dufty in-cluded the quantum correlations of two scattering parti-cles with a third spectator particle to preserve Fermionanti-symmetry, without including correlations due to theinteraction. By this method they self-consistently de-rived the BUU equation with the statistical θ s factors,but came to the conclusion that the degeneracy mustsimultaneously affect the scattering cross section in ad-dition to the statistical availability of scattering statesencapsulated in the θ s terms.The BUU equation as originally formulated is appli-cable to moderately dense gases in which degeneracy ispresent but the amount of correlation is small. In the caseof WDM, the equation has several deficiencies. First, itdepends on the degree of degeneracy, which in turn de-pends on the electron number density and therefore theaverage ionization state of the system, which must be pro-vided as an input. Second, in a plasma it is well knownthat transport rates predicted by equation (4) diverge ifthe cross section is computed using the Coulomb poten-tial because the Coulomb force is of an infinite range.This is typically resolved in an ad hoc manner by en-forcing a large distance limit on the impact parameter.Third, the derivation of the BUU equation, while includ-ing correlations due to Fermi statistics, does not allow forcorrelations due to the interaction and thus applies onlyin the limit of weak coupling. The remainder of this sec-tion describes how all three deficiencies can be addressedin a consistent fashion in the WDM regime.For a tenuous and hot (read classical and weakly cou-pled) plasma the equilibrium ionization state is deter-mined by the Saha equation [32]. The divergence in theCoulomb logarithm is related to the neglect of correla-tion: in plasmas the collective affect of the surroundingplasma introduces Debye screening that limits the rangeof the interaction. A recent approach called “mean force kinetic theory” has derived a self-consistent approachto plasmas through a new expansion parameter of theBBGKY hierarchy [8]. In standard derivations of theBoltzmann equation, the BBGKY hierarchy is truncatedvia neglecting correlations involving three or more parti-cles and making certain assumptions about two-particlecorrelations. In mean-force kinetic theory the BBGKYhierarchy is re-arranged in terms of an expansion pa-rameter that is the difference between the exact non-equilibrium distribution function and the its equilibriumlimit. The hierarchy is then truncated by assuming thisdifference is negligible for reduced distribution functionsin three or more particle coordinates; i.e. that the highorder correlations take their equilibrium values. The re-sult is a collision integral identical in form to that of theBoltzmann equation, but in which the scattering parti-cles interact through the potential of mean force. In ad-dition, there is a term on the left hand side of the kineticequation that enforces the non-ideality of the equilibriumlimit in the equation of state. The result is capable ofdescribing transport in weak to moderately coupled plas-mas ( Γ . ) based on the equilibrium structural prop-erties of the plasma. B. The Quantum Potential of Mean Force
Extending mean-force kinetic theory to include quan-tum effects is complicated by two issues: the exclusionprinciple complicates the mathematics of the necessarystatistical averaging, and more significantly the uncer-tainty principle muddles the very meaning of a potentialof mean force. Classically, the mean force is the forceexperienced between two particles at rest with a givenseparation, with a statistical averaging over all of theremaining particles in the plasma at equilibrium. In thequantum case, knowing particles are “at rest with a givenseparation” is impossible according to the uncertaintyprinciple. Mathematically, this prevents factoring of thekinetic and potential (configuration) terms in the equi-librium density matrix, and ultimately prevents a generalderivation of the potential of mean force by extension ofknown classical means.Despite this complication, the potential of mean-forcemust have some meaning in at least a semi-classical sense.An electron-ion pair will still induce well-defined corre-lations in the plasma, and these correlations can in turninfluence the force felt by the interacting pair at least overthe average of many scattering events at many velocities.This is reflected in the the screened potential U sc ( r ) = φ ( r ) k B T e − r/λ sc (5)with degeneracy-dependent screening length (as per [9]) λ = λ s Li / ( − ξ )Li / ( − ξ ) (6)which can be seen as a weak-correlation limit of the po-tential of mean force both for classical and quantum plas-mas. The essential challenge of applying the mean forceconcept to WDM is how to encapsulate this effect in abinary potential when the coupling is no longer weak. Ithas long been known that weak correlations influence thepotential in the form of plasma screening in both the clas-sical (Debye-Huckel) and quantum (Thomas-Fermi lim-its). One other classical derivation of the potential ofmean force is via the Ornstein-Zernike equation, whichdefines the direct correlation function [33]. Fortunately, aquantum analog of the Ornstein-Zernike equation exists,and this equation has been used successfully to calcu-late the equilibrium pair correlation function in WDM[18, 26, 34]. Furthermore, it has been used to define aquantum potential of mean force for electron-ion inter-actions, and this potential has been used to predict elec-trical conductivities with good agreement with quantummolecular dynamics simulations [35].We postulate that a quantum potential of mean forcemust arise naturally from the quantum mean force ki-netic theory, and that this potential is that which isderived from the quantum Ornstein-Zernike equation.We turn to such a potential obtained with the quan-tum hypernetted-chain-approximation, coupled with anaverage-atom model that accounts for the structure andionization state of the ions [18, 26, 34, 36] (subsequentlyreferred to as the AA-TCP model for “Average-AtomTwo-Component-Plasma”. This potential can be ex-pressed as V MF ( r ) = − Zr + Z d r ′ n ion e ( r ′ ) | r − r ′ | + V xc [ n ion e ( r )] + n i Z d r ′ C ie ( | r − r ′ | ) − β h ii ( r ′ )+¯ n e Z d r ′ C ee ( | r − r ′ | ) − β h ie ( r ′ ) , (7) Figure 2. Electron-ion potential of mean force for solid-density ( . · cm − ) warm dense aluminum at , and eV (thick curves). In the high-temperature limit the poten-tial approaches the screened Coulomb potential of a classicalplasma (thin curves); as the temperature decreases it is al-tered by both degeneracy and correlations leading to differentscale lengths of the potential in addition to the non-monotonicbehavior. where C ie and C ee are the electron-ion and electron-electron direct correlation functions respectively, h ie and h ee are the electron-ion and electron-electron pair cor-relation functions respectively, n ion e ( r ) is the density ofbound electrons, β = k B T , and V xc is the exchangecorrelation functional (in the case of [26] it is the zero-temperature Dirac exchange functional [37]). Calculationof the potential requires closure, which in this case is pro-vided by the quantum hypernetted-chain-approximationfor the ion-ion correlations and through coupling toan Average-Atom model for the electron-ion correla-tions. Such methods can be substantially faster thanfull dynamical calculations such as molecular dynamics,wherein lies the primary benefit of the theory proposedin this work. In figure 2 we show example electron-ionscattering potentials from the AA-TCP model for warmdense aluminum at conditions that span the weakly cou-pled classical to moderately coupled degenerate regimes;see regions (1) and (4) of figure 1. The figure demon-strates the convergence of the potential of mean forcewith a screened Coulomb potential in the weakly-coupledlimit, and the importance of correlations in the calcula-tion of the potential in the region of moderate coupling. III. TRANSPORT RATES
Comprehensive methods to derive hydrodynamic equa-tions, such as that of Chapman and Enskog have beendeveloped for the Boltzmann equation [38], but their ex-tension to the BUU equation faces considerable math-ematical challenges and has not been accomplished toour knowledge. To demonstrate predictions for macro-scopic transport rates, we focus on electron-ion relax-ation in which the respective electron and ion distribu-tion functions are known but the species are not in equi-librium with each other. We consider both temperaturerelaxation and momentum relaxation, which is relatedto the electrical conductivity. A restriction imposed byconsidering only electron-ion relaxation is that it pro-vides only one contribution to processes such as elec-trical conductivity that are also influenced by electron-electron interactions. Although models such as the quan-tum Landau-Fokker-Planck equation have been solvedusing a Chapman-Enskog technique to address both con-tributions in a comprehensive hydrodynamic theory [15],they do not address strong coupling. A recent modifi-cation has been proposed to incorporate strong couplingvia a modified Coulomb logarithm computed using thepotential of mean force, and finds that in the strongly de-generate regime and for high-Z systems the electron-ioncollisions are dominant [35]. However, the Fokker-Planckform of the collision operator itself is only expected toapply when momentum transfer during collisions is small(i.e., weak coupling). For instance, it can be derived froma small momentum transfer expansion of the BUU equa-tion. Here, we focus on the electron-ion relaxation usingthe full BUU equation in order to isolate the influence oflarge momentum transfer in the collision operator.Concentrating on the electron-ion contribution also al-lows for a commensurable comparison with quantum MDsimulations of electrical conductivity [28]. Since electronsare often treated using the Born-Oppenhiemer approxi-mation in simulations, they are also limited to treat onlythe electron-ion contribution to transport processes. Al-though electron-electron interactions are expected to con-tribute to the total conductivity, it is only recently be-coming possible to simulate dynamic electrons in WDMfollowing advancements in wave-packet MD [39], mixedquantum-classical MD [40, 41], Bohmian quantum meth-ods [42], Kohn-Sham DFT [43] and quantum MonteCarlo [44]. Addressing contributions from both electronand ion dynamics will be the next step in both the theoryand simulation development.
A. General Formalism
A binary mixture of two species s and s ′ out of equilib-rium will relax towards equilibrium through s − s , s − s ′ and s ′ − s ′ collisions, which are modeled by moments ofthe collision operator (4), h χ i s − s ′ = Z d v χ ( v ) C s − s ′ qB , (8)where χ ( v ) is some polynomial function of the veloc-ity. To simplify, we utilize the following properties: d Ω dσd Ω is invariant under reversal of the collision, i.e. ( v , v ′ ) ↔ (cid:0) ˆ v , ˆ v ′ (cid:1) where v and v ′ are the pre-collision ve-locities of particles one and two respectively, the “hat” ˆ indicates a post-collision quantity, and the phase-spacevolume element is invariant, i.e. R d v d v ′ = R d ˆ v d ˆ v ′ . Wethus obtain h χ i s − s ′ = Z d v Z d Ω dσd Ω u Z d v ′ [ χ (ˆ v ) − χ ( v )] × f s f s ′ (cid:16) θ s ˆ f s (cid:17) (cid:16) θ s ˆ f s ′ (cid:17) . (9)Relevant χ ( v ) include χ ( v ) = → [ χ (ˆ v ) − χ ( v )] = 0 m s v → [ χ (ˆ v ) − χ ( v )] = m s ∆ vm s v → [ χ (ˆ v ) − χ ( v )] = m s ∆ v (10)where ∆ v = ˆ v − v . Substituting variables v = v ′ + u , defining m ss ′ = m s m s ′ / ( m s + m s ′ ) , and utilizingthe following relations obtained from the collision kine-matics: m s ∆ v = m ss ′ ∆ u , ∆ u · ∆ u = − u · ∆ u and (cid:0) v · ∆ v + ∆ v (cid:1) = ( m ss ′ /m s )∆ u · [ v ′ + ( m ss ′ /m s ) u ] , shows that (see [45]), χ ( u ) = → [ χ (ˆ v ) − χ ( v )] = 0 m s v → [ χ (ˆ v ) − χ ( v )] = m ss ′ ∆ u m s v → [ χ (ˆ v ) − χ ( v )] = m ss ′ (cid:16) v ′ − V s + m ss ′ m s ′ u (cid:17) · ∆ u (11)where ∆ u = u (cid:18) sin θ cos φ ˆ x + sin θ sin φ ˆ y − θ u (cid:19) . (12)The preceding discussion and the collision operator(4) are in principle applicable to transport in any semi-classical system. As it pertains to WDM, ion-ion scat-tering is contained within this formalism as ion dynamicsare classical and electron degeneracy effects enter onlyvia the potential of mean force. Application of the the-ory to ion-ion scattering was validated in [14]. The caseof the electron-electron terms requires further work dueto the subtleties associated with defining the potentialof mean force that are discussed in section II and willbe investigated in another work. However, the model atthe level to which we have developed it has immediateapplicability to the case of electron-ion scattering. B. The Relaxation Problem
We restrict our analysis to the class of problems inwhich electrons and ions in the plasma are in respectiveequilibrium with themselves with different fluid quanti-ties T e , T i , V e and V i , respectively. In such a system,the electron and ion fluid variables will equilibrate ona timescale long compared to the respective electron-electron and ion-ion collision times. The ions have aclassical Maxwellian velocity distribution f i ( v ′ ) = n i v T i e − ( v ′ − V i ) /v Ti π / (13)and the electrons have a Fermi-Dirac velocity distribution f e ( v ) = n e " v T e (cid:16) − π / Li ( − ξ ) (cid:17) ( v − V e ) /v Te ξ ! − (14)where v T s = p k B T s /m s and ξ = exp ( µ/k B T ) , the ionvelocity is v ′ and electron velocity is v . We can write f e f i (cid:16) θ e ˆ f e (cid:17) = n i v T i e − ( v ′ − V i ) /v Ti π / n e × " v T e (cid:16) − π / Li ( − ξ ) (cid:17) v ′ + u − V e ) /v Te ξ ! − × − v ′ + u +( m ei /m e )∆ u − V e ) /v Te ξ ! − , (15)from which the relation of the factor (cid:16) θ e ˆ f e (cid:17) to Pauliblocking can be seen in terms of the Fermi-Dirac occu-pation number: the contribution to the collision inte-gral from collisions to or from occupied states is zero.This simplification occurs from the combination of θ e =( − / h/m s ) with the prefactor n e v T e / Li ( − ξ ) in theFermi Dirac distribution through the relation (3).Electron-ion temperature and momentum relaxationrates depend on the energy exchange density Q s − s ′ andfriction force density R s − s ′ , respectively. These can inturn be written in terms of the moments (9), assuming auniform plasma, as Q ei = (cid:28) m e ( v − V e ) (cid:29) e − i = 3 n e dT e dt (16)(where in the last equality we have taken V e = V i = 0) and R ei = h m e v i e − i = m e d V e dt (17)which, in the respective limits of ∆ T ≪ T and ∆ V ≪ V yield simple relaxation rates dT e /dt = ν ( ǫ ) ei ∆ T and d V e /dt = ν ( p ) ei ∆ V .The integration over the ion velocity can be simplifiedsignificantly in the limit that the ion velocities are muchsmaller than the electron velocities: m e T i ≪ m i T e , which(due to the small electron-to-ion mass ratio) is true whentemperature differences are not extreme, coinciding withour expansion about the equilibrium state. Note that wealso make the simplifying replacement m ei ≈ m e . Byexpanding equation (15) in the limit that the electrondistribution is approximately constant over the range ofaccessible ion velocities, the integral over the ion veloc-ities can be carried out analytically. The evaluation ofthis integral differs for the calculation of Q ei versus R ei .Therefore we examine each case separately.
1. Temperature Relaxation
The energy-exchange density (16) in this case becomes Q ei = Z d u Z d Ω dσd Ω u ∆ u · Z d v ′ m ei (cid:18) v ′ + m ei m i u (cid:19) f i f e (cid:16) − | θ e | ˆ f e (cid:17) . Inserting equation (15), applying the expansion | v ′ | ≪| u | , assuming zero drift velocities and | T e − T i | ≪ T e , T i we perform the integral over v ′ and write ∆ u · Z d v ′ m ei (cid:18) v ′ + m ei m i u (cid:19) f i f e (cid:16) − | θ e | ˆ f e (cid:17) ≈ n e n i ηξe − η sin (cid:0) θ (cid:1) π v T e Li ( − ξ ) (cid:0) ξe − η + 1 (cid:1) where η ≡ u/v T e . The result is written to facilitate com-parison with the classical limit, Q ei = − m e m i n e ν ( ǫ ) ei ( T e − T i ) . (18)in terms of a collision frequency ν ( ǫ ) ei = ν Ξ ( ǫ ) ei , (19)where ν ≡ √ πn i Z e √ m e ( k B T e ) / = 2 . × − Zn i [m − ]( T e [eV]) / (20)and a generalized Coulomb integral Ξ ( ǫ ) ei . Effects ofdegeneracy and strong coupling are contained in theCoulomb integral, Ξ ( ǫ ) ei = 12 Z ∞ dηI ǫ ( η ) (21) I ǫ ( η ) ≡ G ( η ) σ (1)1 ( η, Γ) σ where σ (1)1 ( η, Γ) = 4 π Z π dθ sin θ θ dσd Ω (22)is the momentum transfer cross section, which can bewritten in terms of the phase shifts δ l as σ (1)1 = 4 πη ∞ X l =0 ( l + 1) sin ( δ l +1 − δ l ) . (23)The function G ( η ) ≡ ξ e − η η h − Li ( − ξ ) i (cid:0) ξe − η + 1 (cid:1) (24)determines the relative availability of states that con-tribute to the scattering. This is plotted in figure 3 forseveral values of the degeneracy parameter Θ , where itis shown that in the classical limit scattering is domi-nated by energy transfers around the thermal energy, andas degeneracy increases the envelope of relevant energy-transfers narrows about the Fermi energy. It should benoted that the relaxation rate obtained in equation (19)is identical to that obtained in equation (71) of reference[27] by very different means.
2. Momentum Relaxation
Momentum relaxation occurs through collisions be-tween electron and ion populations with different averagevelocities. The force density (17) associated with thesecollisions is (cid:1) ( Collision speed ) G s ( W e i gh t i ng F un c t i on s ) (cid:1) ( Collision speed ) G s ( W e i gh t i ng F un c t i on s ) (cid:1) ( Collision speed ) G s ( W e i gh t i ng F un c t i on s ) Figure 3. Statistical contributions to the integrands for tem-perature and momentum relaxation, G (solid), G (dashed)and G (dotted), for three conditions: Θ = 12 . and ξ = 0 . (top, weak degeneracy), Θ = 0 . and ξ = 1 . (middle, mod-erate degeneracy), Θ = 0 . and ξ = 1135 (bottom, strongdegeneracy). The relevant collision velocities for both momen-tum and temperature relaxation become narrowly centeredaround the Fermi velocity at strong degeneracy. The relativeimportance of the two different functions that contribute tomomentum relaxation is degeneracy dependent. R ei = Z d u Z d Ω dσd Ω u Z d v ′ m ei ∆ u f e f i (cid:16) θ e ˆ f e (cid:17) . (25)Inserting equation (15), applying the expansion | v ′ | ≪| u | , and assuming | T e − T i | ≪ T e , T i , V i ≪ v T i and V e ≪ v T e , the integral over v ′ can be performed analytically, Z d v ′ m ei f e f i (cid:16) θ e ˆ f e (cid:17) ≈ m e n e ξn i e − η h u · ∆ V − ξe − η ( ∆ u + u ) · ∆ V i π / v T e h − Li ( − ξ ) i (cid:0) ξe − η + 1 (cid:1) . We follow the classical example and write R ei = − n e m e ν ( p ) ei ( V e − V i ) (26)where the frequency ν ( p ) ei = ν Ξ ( p ) ei (27)involves a Coulomb integral Ξ ( p ) ei = 12 Z ∞ dηI p ( η, Γ , ξ ) I p ( η, Γ , ξ ) ≡ G ( η, ξ ) σ (1)1 ( η, Γ) σ − G ( η, ξ ) σ (1)2 ( η, Γ) σ (28) = G ( η, ξ ) σ p ( η, Γ , ξ ) σ (29)which is different from that involved in the energy-exchange density. Here, σ (1)1 is defined in equation (23), σ (1)2 ( η, Γ) = 4 π Z π dθ sin θ θ dσd Ω cos θ, (30)and σ p ( η, Γ , ξ ) = σ (1)1 ( η, Γ) − ξ e − η σ (1)2 ( η, Γ) . (31)It is interesting to note that the cross section arising inthe energy relaxation rate from equation (23) differs fromthat associated with momentum relaxation from equation(31). This is a purely quantum mechanical effect, as thecross section definitions are the same in the classical limit[45]. It is also an effect that is predicted by the full BUUequation, but not the Landau-Fokker-Plank limit asso-ciated with small momentum transfer interactions [35].The weighting functions G ( η, ξ ) = ξ e − η η h − Li ( − ξ ) i (cid:0) ξe − η + 1 (cid:1) , (32) G ( η, ξ ) = ξ e − η η h − Li ( − ξ ) i (cid:0) ξe − η + 1 (cid:1) , (33)are shown in figure 3 and compared with the statisticalweighting factors in the case of temperature relaxation.The presence of the differing angular integrals betweenthe energy and momentum relaxation cases warrants fur- ther discussion.Through the use of the Wigner-3j function, σ (1)2 can beexpanded in the phase shifts (see appendix B) as σ (1)2 = 4 πη ∞ X l =0 sin δ l l ( l + 1) − × (cid:8) ( l + 1)(2 l −
1) [( l + 2) sin( δ l − δ l +2 ) − (2 l + 3) sin( δ l − δ l +1 )] − l (2 l + 3) sin δ l (cid:9) . (34)While it is tempting to interpret the quantity σ (1)2 as a cross-section, σ (1)2 can become negative and therefore has nosuch interpretation. We will show in the next section that it is only in the combination defined in equation (31) thatthis interpretation is justified. We thus refer to σ p as an effective transport cross section. We note that this secondterm arises due to degeneracy, and has no analog in the classical relaxation problem.
3. Electrical Conductivity
The electrical conductivity is an important transportcoefficient that depends largely on the electron-ion colli-sional momentum relaxation rate. Considering a Fermi-Dirac electron population flowing through a stationaryMaxwellian ion population due to an applied electricfield, the frictional force balances the electric force R ei = − en e E , (35)which in the form of equation (26) is connected to thecurrent through Ohm’s law J = σ E (36)where J = − en e V e . Using the electron-ion collisionalfriction [equation (26)], the resulting electrical conduc-tivity is σ = e n e m e ν ( p ) ei , (37)where ν ( p ) ei is defined in equation (27). The assumption ofa Fermi-Dirac electron distribution means that electron-electron (e-e) collisions do not contribute to the relax-ation; distortions in the electron distribution away fromequilibrium amount to a higher order approximation thatcould be explored e.g. through the Chapman-Enskog ex-pansion. The e-e collisions do not contribute substan-tially in the degenerate regimes due to Pauli blocking,and at high temperatures the e-e contribution is well un-derstood via the Landau-Spitzer theory. The interme-diate regime where both degeneracy and e-e collisionsare important is discussed by Shaffer and Starrett [35]in the context of the quantum Fokker-Planck equation.The application of the BUU equation to this regime torelax the assumption of small-momentum-transfer colli-sions will require a Chapman-Enskog expansion of theBUU equation and will be addressed in further studies. IV. RESULTS AND DISCUSSION
To illustrate the application of the model, we now turnto evaluating it, with input potentials provided by theAA-TCP model ([26, 34]) for aluminum at a density of . · cm − , over a range of temperatures spanning fromthe degenerate moderately coupled to classical weakly-coupled regimes. However, firstly we demonstrate thebehaviors of the two functions σ (1)2 and σ (1)1 in figure 4 attwo example temperature-density points. The combinedinfluence of the negative values of σ (1)2 and the precedingnegative sign in equation (28) leads to interesting behav-ior in the integrand for the Coulomb integral. The fullintegrand of equation (28) is shown in figure 5 where itis seen that the resulting integrals are positive, as re-quired. Note that the integrands are peaked functions;broad and peaked near the thermal velocity v T s in theclassical limit, and narrow with peak near the Fermi ve-locity in the degenerate limit. Also note that I p and I ǫ are identical in the classical limit, but differ substantiallyin the degenerate case due to the presence of the σ (1)2 fac-tor. A. Relaxation Rates in Solid Density AluminumPlasma
Figure 6 shows a comparison of the momentum andenergy relaxation rates. Each model is compared withthe well-established Landau-Spitzer result [10], which inthe limit m e T i ≪ m i T e reduces to ν LS ei ≈ ν ln Λ LS (38)which has been verified in the classical limit given suffi-ciently weak coupling [46, 47]. The relaxation rate pre-dicted by the LFP model is(see equations 14-17 of [11]), ν LFP ei = ν (cid:18) ln Λ LFP ξ ξ √ π Θ / (cid:19) . (39) (cid:1) ( ) (cid:1) ( ) - - (cid:2) ( v / v Te ) T r an s po r tf un c t i on (cid:1) j ( ) ( a B ) - Collision speed (cid:1) ( v / v Te ) T r an s po r tf un c t i on (cid:2) j ( ) ( a B ) Figure 4. Functions σ (1)1 and σ (1)2 calculated using the PMFfor solid density (2 . · cm − ) aluminum at (top) and (bottom). While σ (1)2 plays the role of a cross section, it isevident from its negative value at many velocities that it is notone. Interestingly, it behaves (only approximately) inverselyto the momentum transfer cross section σ (1)1 . While σ (1)2 isnon-zero at high temperatures, its influence is negligible dueto the suppression of the term that contains it when T ≫ E F . We further note that our expression for the tempera-ture relaxation rate (given by equations (18)-(23)) is thesame as that recently obtained by a substantially differ-ent by Daligault and Simoni (see equations (71)-(75) of[27]) if the potential of mean force is used for calculatingthe transport cross section there. This equivalency canbe seen through use of the relation n e ( h/ √ πm e v T e ) = − / ( − ξ ) from the normalization of the Fermi-Diracdistribution.At a given density, as temperature decreases theCoulomb logarithm will eventually reach zero due to ne-glect of strong coupling physics. The resulting divergenceof the Landau-Spitzer result due to the presence of the(inverse) Coulomb logarithm ln Λ LS = ln b max b min . (40)The maximum impact parameter is modeled as the largerof the screening length λ sc [equation (6)] or the Wigner-Seitz radius a = (3 / πn i ) / , and the minimum is (cid:1) ( v / v Te ) F u ll I n t eg r and I s Collision speed (cid:1) ( v / v Te ) F u ll I n t eg r and I s (cid:1) ( v / v Te ) F u ll I n t eg r and I s Figure 5. Integrands ( I ǫ ( η ) solid and I p ( η ) dashed) appearingin equations (21) and (28) for aluminum at . · cm − atthree different temperatures:
100 eV ( Θ ≫ ) (top),
10 eV ( Θ ∼ ) (middle), ( Θ ≪ ) (bottom). the larger of the classical distance of closest approach r L = e /k B T or the thermal de Broglie wavelength λ dB = ~ / ( m e k B T e ) / [17]. In WDM, the vanishingCoulomb logarithm is often resolved through the mod-ification (see e.g. [17]) ln Λ LFP = 12 ln (cid:18) b b (cid:19) , (41)0 Figure 6. Electron-ion collisional relaxation times ( τ = ν − ei, )as a function of temperature in solid density ( . · cm − )aluminum. which we apply in our evaluation of the LFP model.This is often further altered, as is done in the Lee-Mooreconductivity model [17], by enforcing that the minimumvalue of the Coulomb logarithm be 2: ln Λ fix = max (cid:20) ,
12 ln (cid:18) b b (cid:19)(cid:21) . (42)The approximations inherent in this approach are two-fold: small-angle collisions must be assumed to obtain theLFP equation, and the choice of maximum and minimumimpact parameters represents an uncontrolled expansionin the strongly coupled regime. The convergent kineticequation in our approach avoids these limitations.Figure 6 confirms the expectation that all expres-sions agree at high temperatures associated with the theweakly-coupled classical regime, while at low tempera-ture the models differ as a result of the different levels ofinclusion of the physics of strong coupling and degener-acy. In each case there is a minimum in the relaxationtime. In all cases except the Landau-Spitzer result, thisminimum can be attributed to a combination of both de-generacy and strong coupling: strong coupling increasesthe collisionality of the system while the onset of degen-eracy reduces the collisionality through Pauli blocking.The decreased level of ionization at lower temperaturesalso reduces the collisionality. If the density is less than cm − as temperature is reduced the plasma will firstbecome strongly coupled and then degenerate, and if thedensity is greater than cm − the electrons will be de-generate when the transition to strong coupling occurs.The quantum mean force model predicts that the re-laxation time τ for energy and momentum relaxation donot have the same behavior with temperature. The ratesare equal in the classical limit as expected, but differfor lower temperatures when degeneracy arises. Gen-erally, the rates are smaller for energy relaxation, withthe maximum difference being a factor of ∼ . Experi-mental validation of this phenomenon will require accu-rate measurements of both momentum and temperature relaxation rates in WDM, a matter of considerable dif-ficulty. However, further consideration of the physicalbasis for the difference between momentum and temper-ature relaxation is called for, and perhaps computationalmethods will prove to be effective to this end. This effectis not present in the LS or QLFP theories and is a resultof allowing for strong quantum collisions, which thesetheories do not account for. The LS theory ignores bothdegeneracy and large-angle scattering. The QLFP theoryextends further into the degenerate regime and has fixedthe vanishing Coulomb log, but does not account for ei-ther correlations or large-angle scattering when there isstrong Coulomb coupling. The divergence between theQLFP results using the two different prescriptions forthe Coulomb log illustrates the lack of strong-couplingphysics in the method. B. Electrical Conductivity of Solid-DensityAluminum Plasma
We proceed to evaluate the electrical conductivity ac-cording to equation (37) for aluminum at . · cm − , asa demonstration of the model in a regime marked by par-tial ionization and a simultaneous transition from weak tostrong coupling and classical to degenerate statistics. Forcomparison we select the Lee-Moore model, the model ofShaffer and Starrett [35] and the QMD simulations ofWitte et al [28]. The electrical conductivity coefficientpredicted by the LM model [17] is σ e = ne m (cid:26) √ m ( kT ) / √ πZ n i e ln Λ fix (cid:27) R ∞ t dt t − µ/kT ) R ∞ t / dt t − µ/kT ) (43)which we relate to the friction force density R and thusthe scattering rate: ν ( p ) ei = e n e /σm e giving ν LM ei = ν (cid:20) ln Λ fix Li / ( − ξ )Li ( − ξ ) (cid:21) . (44)The Starrett and Shaffer model similarly uses the quan-tum potential of mean force to mediate scattering, but inthe context of the QLFP equation. In order to introducethe effect of large-angle collisions into the model they in-troduce a Coulomb logarithm defined via the relaxation-time approximation (RTA) which we will refer to as lnΛ SS . For a commensurate comparison with our method(where we assume a Fermi distribution for the electrons)and the QMD simulations, we neglect the higher orderChapman-Enskog corrections associated with electron-electron interactions that can be obtained in the SSmodel. The electron-ion contribution corresponds withthe first order of the Chapman-Enskog expansion, σ , qLFP = 3(4 πǫ ) ( k B T ) / √ πm e Ze ln Λ ss Li / ( − ξ )Li ( − ξ ) Li ( − ξ ) = − ξ ξ and (from 39) ln Λ ξ ξ √ π Θ / → ln Λ , and equation (3) it can be seen this is equivalent in formto equation (37) with the difference being the Coulomblogarithm. Figure 7. Electron-ion contribution to the electrical con-ductivity of solid density aluminum ( . · cm − ) as derivedthrough the current work (solid line) the Starrett and Shaffermodel evaluated at first order in the Chapman-Enskog expan-sion (dashed line), the Lee-More model (dotted line), and theLS conductivity (dot-dashed line), along with QMD resultsof Witte et al [28] using the Perdew–Burke-Ernzerhof andHeyd–Scuseria–Ernzerhof exchange-correlation functionals. The resulting predictions for the conductivity areshown in figure 7. Similarly to the relaxation times,there is a minimum in the conductivity near the Fermitemperature. This again can be attributed to both corre-lations and Pauli blocking [35]. Also as in the case of therelaxation times, the LS theory fails to accurately pre-dict the conductivity when degeneracy and correlationsare important, as expected. Furthermore, the commonlyused Lee-Moore theory performs poorly as a result of thecorrelations. Interestingly, the Lee-Moore theory can bereproduced by replacing the Coulomb logarithm in thestandard QLFP formulation with the fixed version pre-scribed in the Lee-Moore theory. Although we focus onthe electron-ion contribution, it is known that electron-electron interactions cause a contribution of compara-ble magnitude in the classical weakly coupled limit (theSpitzer correction) [10]. However, it can be expected thate-e collisions will be greatly suppressed below the Fermitemperature due to Pauli blocking and therefore the cor-rections due to a higher-order Chapman Enskog expan- sion will be diminished at lower temperatures. Indeed,this is seen for the QLFP equation [15].More interesting are the comparisons of the presenttheory with the Shaffer-Starrett formulation of the QLFPtheory [35], and with the QMD simulations of Witte etal. [28]. The QLFP equation being the limit of the BUUequation with only small-angle scattering, it may be ex-pected that these formulations should agree in the limitof weak coupling. However, the present theory and the SStheory differ in their treatment of the potential of meanforce, and the curves appear to not yet have reached thislimiting behavior at . The QMD simulations alsomake an interesting direct comparison. QMD simulationsdo not directly include e-e collisions because they use theBorn-Oppenhiemer approximation, but account for somelevel of the electronic interactions through the mean field.[48]. Thus, it seems most appropriate to compare theQMD simulations with theories evaluated to treat onlythe electron-ion interactions, as is done in figure 7. In-deed, the agreement with these simulations is remarkablefor most of the range of available data, from
15 eV downto approximately . At the lowest temperatures thepredictions begin to diverge, but it is unclear at thesevery low temperatures whether the BUU equation canbe expected to be valid as higher-order quantum correla-tions come into play. To fully address the contribution ofe-e collisions will require solutions of the BUU equationat higher orders of the Chapman-Enskog expansion.The good agreement between QMD and the BUU pre-dictions provides evidence that large momentum transfercollisions, and the associated second (quantum) contri-bution to the momentum scattering rate [equation (30)]are real and significant effects influencing the electricalconductivity. This points to important physics beyondwhat is captured by the QLFP theory, or its modifi-cations, as is shown by comparing with the first-orderChapman-Enskog solution of the Shaffer-Starrett modelfrom [35] (the first order of this method is equivalent tothe electron-ion relaxation model described in the previ-ous section, and therefore provides a commensurate com-parison). At the same time, it is also important to notethat electron-electron contributions may influence the to-tal conductivity at these conditions. Shaffer and Starrettpredict these to make order-unity contributions over therange of conditions plotted in figure 7 [35]. Further de-velopment will be required to evaluate this contributionfrom the BUU equation, as well as to provide a conclusivetest using QMD.
V. CONCLUSIONS
We have presented a model for transport in plasmaswith weak to moderate Coulomb coupling and weak tomoderate electron degeneracy. The model is based on thequantum Boltzmann equation of Uehling and Uhlenbeck,in which the two-body scattering is mediated by the equi-librium potential of mean force. This incorporates cor-2relations in the equilibrium limit while maintaining thesimplicity of binary collisions in the dynamical equation.This is relevant to electron-ion collisions in WDM. Asinput into the model, we utilized an existing model forthe potential of mean force derived from the quantumOrnstein-Zernike equations and an average-atom quan-tum hypernetted-chain-approximation model [18, 26, 34].The model was used to compute momentum and en-ergy relaxation rates. The transport coefficients werewritten analogously to the classical Landau-Spitzer (LS)result in terms of a “Coulomb integral” that takesthe place of the traditional Coulomb logarithm. TheCoulomb integral depends on the level of degeneracy, andCoulomb coupling enters through the calculation of themomentum-transfer cross section solving the Schrödingerequation with the PMF as the scattering potential. Themomentum relaxation rate was found to differ from tem-perature relaxation in that it depends on a differenttransport cross section, which includes a term that issolely associated with degeneracy, and has no analog inthe classical limit. The dependence of the integrandsof the Coulomb integrals on the level of degeneracy wascompared for the temperature and momentum relaxationcases.We concluded by calculating the temperature and mo-mentum relaxation rates and electrical conductivity insolid density aluminum plasma over a range of temper-atures that covered the transitions between weak andmoderate coupling and weak and moderate degeneracy.Predictions were compared with other leading models. Itwas found that all models behave as expected in the clas-sical weak-coupling limit, and diverge widely in the limitof a degenerate moderately-coupled plasma. We assessedthe relative importance of the different relevant physi-cal processes that complicate the problem as degeneracyand coupling simultaneously increase: diffraction, Pauliblocking, correlations, and large-angle scattering. Inter-estingly, in the degenerate regime there is a quantitativedifference in the predicted relaxation rates for momen-tum versus energy. Ultimately, current and near-futureexperimental measurements [1, 49, 50] and ab-initio sim-ulations [39–44] will be able to shed light on the applica-bility of the different models of transport for WDM.This work can be improved through inclusion ofelectron-electron collisions and higher-order terms of aChapman-Enskog expansion. Additionally, further workwill be required to obtain a rigorously derived conver-gent kinetic equation with the appropriate potential ofmean force. Finally, recent and upcoming experimentalmeasurements of electrical conductivity and temperaturerelaxation [49, 50] may soon open the door for discrimi-nation between the validity of the various models of re-laxation in WDM. This will enhance our understandingof the basic physics of WDM, and allow increased fidelityin the rapid calculation of transport coefficients for usein hydrodynamic simulations of naturally and experimen-tally occurring WDM.
ACKNOWLEDGMENTS
The authors wish to acknowledge Charles Starrett andNathaniel Shaffer for the provision of input data at equi-librium for the potential of mean force and for their valu-able comments on this work. This material is based uponwork supported by the U.S. Department of Energy, Officeof Science, Office of Fusion Energy Sciences under AwardNumber DE-SC0016159.
Appendix A: Determination of Phase Shifts
Solution of the scattering problem comes down to so-lution of the radial Schrödinger equation [51] d u l dr + (cid:20) k − l ( l + 1) r − m ei ~ W (2) ( r ) (cid:21) u l = 0 , For each angular quantum number l there is a phase shift δ l that can be extracted from the asymptotic behavior ofthe wavefunction u l beyond the range of the potential atpoint R (defined as a point beyond with the influence ofthe potential on the wavefunction is negligible) throughthe relation: tan δ l = kRj ′ l ( kR ) − β l j l ( kR ) kRy ′ l ( kR ) − β l y l ( kR ) with β l = 1 u l /r d ( u l /r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = R where j l ( y l ) are the spherical Bessel (Neumann) func-tions. For l > it is faster and still accurate to use theWKB phase shifts δ (WKB) l = − Z ∞ ( l +1 / /k r k − ( l + 1 / r dr + Z ∞ r C r k − ( l + 1 / r − m e ~ U ( r ) dr (A1) Appendix B: Simplification of Cross Sections
Cross sections are calculated in the partial wave ex-pansion dσd
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