Matter at extremely high temperatures and densities has attracted considerable interest from researchers in recent years. Experimental study of warm dense matter (WDM) at such laser facilities as the National Ignition Facility (NIF),1 the European XFEL,2 and OMEGA3 opens a path to reveal the internal structure of planets down to their cores, both in our solar system and beyond.4 Such high-pressure experiments will also allow us to test theories of states of matter under extreme conditions of density and temperature.5–10 This attention is prompted not only by the significance of the physical properties of dense plasmas for fields such as astrophysics,11 inertial confinement fusion,4 and laser technology,12–14 but also by other potential practical applications such as the synthesis of novel materials.15,16

Under experimental conditions, plasma and heated dense matter are created by rapid energy release processes and subsequent relaxation. As a rule, plasmas are created with an initial state that is out of equilibrium, and therefore understanding temperature relaxation is critical for modeling the evolution of multitemperature high-energy-density plasmas.17–20 In this regard, direct and accurate measurement of relaxation rates is challenging because of the ultrashort time scales and complex initial conditions. Therefore, a theoretical description of the relaxation and relevant transport properties is necessary to interpret these experiments. Temperature relaxation has been theoretically studied extensively for electron–ion systems.21–24 These theories have been compared with molecular dynamics (MD) simulations with varying degrees of success.25,26 However, explicit electron–ion MD simulations often rely on quantum statistical potentials,27,28 which may only be valid in thermodynamic equilibrium. This complicates comparisons of MD simulations with theory, because disagreements could be attributed to uncertainties in the interaction potentials rather than to the theoretical models. Carefully designed and accurately diagnosed laboratory experiments are required to test the reliability of these theoretical approaches. One such experiment was presented in Ref. 29, where ion–ion temperature relaxation in a binary mixture was studied by examining a dual-species ultracold neutral plasma and measured relaxation rates were compared with those from atomistic simulations and a range of popular theories.

First-principles computational modeling methods, such as density functional theory (DFT)30–33 and the quantum Monte Carlo method (QMC),34–36 are indispensable for reliable calculation of WDM properties. For example, QMC results have been used as input for various approximations.37 For WDM applications, Groth et al.38 provided accurate ab initio QMC results for the exchange-correlation free energy for various densities, temperatures, and spin polarization degrees.

In addition to ab initio methods, various approximate models are still relevant for describing the transport properties of WDM. This is because of the high computational costs of ab initio methods, such as DFT at high temperatures. For example, the Boltzmann equation with the Landau or Lenard–Balescu collision integral can be used to calculate the transport coefficients. One of the methods for solving the kinetic equation is Chapman–Enskog theory,39 based on expansion in orthogonal Sonin polynomials. In the case of a fully ionized plasma, by including electron–electron collisions in the Chapman–Enskog method, Spitzer and Härm40 were the first to provide an adequate analytical result for the plasma conductivity. Often, plasma properties are evaluated using the Debye–Hückel potential for the classical plasma and the Thomas–Fermi potential for degenerate ideal electrons.41–44 There are several modifications of these potentials taking into account various quantum and nonideality effects.45–55 These potentials have been widely used to study various characteristics of nonideal plasmas,56–61 and good agreement with experimental data has been obtained in the temperature and density ranges studied. There have been many studies of electron–ion scattering in the dense plasma regime, taking account of charge screening in partially and weakly degenerate cases.62–65

A key approach for computing the transport properties of many-particle systems is linear response theory. Linear response theory can be used to analyze how a plasma responds to small changes in external conditions or fields. For example, using Kubo linear response theory, Holst et al.66 computed electrical conductivity and thermal conductivity in a strongly correlated electronic system within the framework of ab initio MD simulation. Arkhipov et al.61 obtained an approximate expression for the generalized Coulomb logarithm based on the Ornstein–Zernike integral equation in the hyperchain approximation with different interaction potentials. They considered the electron–ion plasma as a classical system of pseudoparticles interacting through effective pair potentials that mimicked the effects of quantum diffraction at short distances. In Refs. 64, 67, and 68 an average atom model based on DFT was used to compute the potential of average force, which was determined by solving the Ornstein–Zernike equations and took into account the correlations between ions and valence electrons.69 This approach provided good agreement with experimental electrical conductivity measurements. In Refs. 70–72 expressions were obtained for the electrical conductivity and absorption coefficients of a high-temperature plasma on the basis of linear response theory. We note that within the framework of the virial expansion of the electrical conductivity of a fully ionized plasma, which takes into account many-particle effects, various limiting cases have been considered, and the corresponding interpolation formulas for the electrical conductivity coefficient have been given73 and compared with experimental values.

For a satisfactory description of WDM and dense plasmas, it is critical to take account of exchange-correlation effects. In an earlier series of studies by Ichimaru et al.74–76 it was shown that in a dense plasma, exchange-correlation effects between charged particles become crucially important. To go beyond the random phase approximation (RPA), the local field correction within linear response formalism is used. In Refs. 77–80, by utilizing local field correction, various analytical formulas were obtained for screened potentials using the long-wave approximation approach and taking into account exchange-correlation effects.

In the present work, we adopt an analytical parameterization of the QMC results for the free energy density of a uniform electron gas in WDM regime38 in combination with a simple screened Thomas–Fermi-type potential to compute the electron–ion scattering cross-sections and generalized Coulomb logarithm of dense hydrogen plasmas. In addition to the electronic contribution, the approach presented here takes into account the screening due to ions using a simple recipe suggested by Stanton and Murillo.44 We show that our simple and fast computational scheme allows us to achieve remarkably good agreement with first-principles DFT results.

To study the dynamic properties of a WDM plasma, we use a Yukawa-type potential and the classical Chapman–Enskog method to calculate the Coulomb logarithm based on the transport scattering cross-section. Particle scattering cross-sections are determined from the Calogero equation for scattering phases.

The main goal of this work is to quantify the influence of electron degeneracy and the effect of electron exchange correlations on the temperature relaxation process and the transport properties of a WDM plasma in comparison with the approach adopted in standard plasma physics. Similar analyses can be carried out for other properties, such as kinetic and optical properties.

The screened electron–ion potential that we adopt isΦY=Zerexp(−ksr),(1)with the inverse screening parameter in the form54,81,82ks2=ke2(Te,ne)+ki2(Ti,ni),(2)where k_e and k_i are the contributions of electrons and ions to the screening. We consider a general case where the temperatures of electrons T_e and ions T_i are not equal. As well as the temperatures, the screening parameter also depends on the densities of electrons n_e and ions n_i. The potential (1) follows from the use of the long-wavelength limit of the density response function of the uniform electron gas (UEG).41,42,54,77

The quality of the calculation results using the potential (1) depends on the plasma screening model used to calculate the screening parameter k_s. Moldabekov et al.41 have shown that the electron screening parameter of the UEG has the following form:ke−2(Te,ne)=kid−2(Te,ne)−kF−2(ne)γ(Te,ne).(3)Here, k_F(n₀) is the Fermi wavenumber and k_id(T_e, n_e) denotes the screening parameter of the ideal UEG:kid2(Te,ne)=kTF2(ne)θ1/2I−1/2(η)/2,(4)where kTF2(ne)=3ωp/vF is the Thomas–Fermi screening parameter, θ = k_BT_e/E_F is the degeneracy parameter of electrons, and I_−1/2(η) is the Fermi integral of order −1/2 at the chemical potential of electrons η = μ/k_BT.83

The dimensionless coefficient γ(T_e, n_e) in Eq. (3) takes into account the correction to the electronic screening due to electronic exchange correlations (nonideality)41,55,84 and represents the contribution to the electronic density response function due to the static local field correction in the long-wavelength limit:85G(k)≃γ(k2/kF2)+⋯,(5)which also defines the exchange-correlation kernel of the electrons.86

The value of γ(T_e, n_e) is computed using the compressibility sum rule:41γ(Te,ne)=−kF24πe2∂2[nfxc(Te,ne)]∂n2,(6)where k_F = v_Fm/ℏ, and f_xc is the exchange-correlation free energy per electron of the UEG from the quantum Monte Carlo simulation-based parameterization presented in Ref. 38.

Although a simple form, the electron screening model (3) has a simple form, Moldabekov et al.41 demonstrated that it provides an accurate description of test charge screening in plasmas compared with a model using the exact static density response function of the UEG. They also showed this by performing MD simulations of the static and dynamic structure factors of ions using various screening models.78,87

The next ingredient needed to complete the plasma screening model is the contribution of ions k_i(T_i, n_i) in Eq. (2). Stanton and Murillo44 performed a detailed analysis of the screening due to ions in different ionic nonideality regimes. They showed that the following simple model provides an accurate description in a wide range of parameters:44ki2=1λi2+ai2,(7)where λi=(kBTi/4πnie2Z2)1/2 is the ideal ion screening length and a=(3Z/4πni)1/3 is the ion sphere radius (the mean distance between ions). In the case of multicomponent plasmas, the density of ions is computed as n_i = ∑_αZ_αn_α, with α denoting the type of ions in the plasma. In this case, the screening parameter in Eq. (2) is defined as44ki2=∑α1λα2+aα2,(8)whereλα=kBTα4πnαe2Zα21/2,aα=3Zα4πni1/3.(9)

Equations (2)–(8) define our plasma screening model for dense plasmas across plasma degeneracy and coupling regimes. As it is based on the results of the analyses performed by Moldabekov et al.41,42,78,87 and by Stanton and Murillo,44 we refer to this model as the nonideal plasma screening (NPS) model. We use the NPS model to compute the generalized Coulomb logarithm in dense plasmas,88 which allows us to calculate the temperature relaxation rate and conductivity.

Following recent work by Rightley and Baalrud,88 the generalized Coulomb logarithm is computed using the transport cross-section Q defined by electron–ion collisions:Ξ=12∫0∞dgG(g)QT(g)σ0,(10)where σ0=ℏ2/(mevTe)2, g = u/v_Te, u is the relative velocity of the scattering particles, and v_Te is thermal velocity of electrons. In Eq. (10), the function G(g) takes into account the Fermi–Dirac statistics of electrons and represents the relative availability of states that contribute to the scattering:88G(g)=η⁡exp(−g2)g5[−Li3/2(−η)][η⁡exp(−g2)+1]2,(11)where−Li3/2(−η)=43πθ−3/2,(12)η = exp(μ/k_BT_e), and μ is the electron chemical potential.88,89

The transport cross-section Q is computed according to two-particle quantum scattering theory:90QT(k)=4πk2∑l(l+1)sin2[δl(k)−δl+1(k)],(13)where the phase shift δ_l(k) ≡ δ_l(k, r → ∞) is calculated by solving the Calogero equation:91–93dδl(kr)dr=−1kΦ(r)[cosδl(kr)jl(k,r)−sinδl(k,r)nl(k,r)]2,(14)with the condition δ_l(k, 0) = 0. In Eq. (14), k = mu/ℏ is the wavenumber, l is the orbital quantum number, j_l and n_l are Riccati–Bessel functions, and Φ(Y) is the pair interaction potential between colliding particles.

The relaxation rates of the electron and ion temperatures are determined by the collision rates (frequencies) between electrons and ions and the difference of electron and ion temperatures:25dTedt=Ti−Teτei,dTidt=Te−Tiτie,(15)where the collision frequencies areνei=1τei=82πniZ2e4Ξ3memikBTeme+kBTimi−3/2,(16)νie=1τie=Zνei.(17)

The electrical conductivity is another important transport coefficient that we examine in this work. We consider a dense plasma regime where the resistivity is dominated by the scattering of electrons on ions. Electrical conductivity is defined in terms of the electron–ion collision frequency as88Ω=e2nemeνei.(18)

We compute the generalized Coulomb logarithm, the temperature relaxation rates, and electrical conductivity using the NPS model. In addition, we compare the NPS model results with those of the ideal plasma model with γ = 0 and with the case where ionic nonideality is neglected by setting ki=λi−1, as well as with the data from DFT simulations.

The approach adopted in this work describes electron–ion collisions and can be directly applied to compute electron–ion temperature relaxation. In addition, the model for the electron–ion collision frequency presented here can be used to describe processes in plasmas where electron–ion collisions are dominant over electron collisions. As discussed in Refs. 94–96, in the case of plasmas with a low level of electron degeneracy, a plasma model requires the inclusion of electron–electron collisions. For hot plasmas, one can use, for example, the model for electron–electron collisions derived by Potekhin et al.97 in the case of arbitrary electron degeneracy. We note that under plasma conditions defined within a standard WDM regime with T ∼ 10 eV, electron–electron collisions are usually negligible (see Fig. 8 and related discussions).

For multicomponent plasmas, the plasma screening model considered here can be used to compute the collision frequency ν_eα for each type of ion α separately. This also provides the total collision frequency ν_ei = ∑_αν_eα.

Figure 1 shows the effective potential of electron–ion interaction given by Eq. (1) with the NPS model for screening. We compare the NPS results (γ ≠ 0, dashed lines) with those of calculations performed setting γ = 0 (solid lines). Results are shown for degeneracy parameters θ = 0.1 and 1 at density parameters r_s = 0.5 [Fig. 1(a)], 1 [Fig. 1(b)], and 3 [Fig. 1(c)]. As can be seen, the correction to electron screening due to electron exchange correlations noticeably affects the effective potential when the degeneracy parameter is small. Here, we note that the importance of the difference in the considered screened potentials depends on the energy of collision during electron–ion scattering. Therefore, it is not possible to gauge from Fig. 1 the degree to which the electronic exchange-correlation effects are essential for plasma transport properties. We can, however, conclude from Fig. 1 that these effects do lead to stronger screening of the electron–ion interaction.41

In Fig. 2, the dimensionless coefficient γ appearing in the NPS model is shown as a function of the degeneracy parameter θ at different densities. We can conclude from Fig. 2 that with increasing density and degeneracy parameters, the effect on the screening properties of plasma electrons becomes weaker.

Figures 3(a) and 3(b) show the inverse screening lengths k_e and k_s, respectively, in atomic units as functions of the degeneracy parameter θ for different values of the density parameter r_s with and without account being taken of exchange-correlation effects. It can be seen that electron exchange correlations have an impact on the screening length for r_s ≳ 1.

Next, we analyze the effect of the correction to electron screening due to electron exchange correlations by considering the transport cross-section for electron–ion scattering. The corresponding results are presented in Figs. 4(a)–4(c), which show the transport cross-sections for electron scattering on a proton at degeneracy parameters θ = 0.5, 1, and 3, respectively. The calculations were carried out by solving the Calogero Eq. (14). It can be seen that taking account of electronic nonideality leads to an increase in the scattering cross-sections at low densities with r_s ≳ 1 and low collision velocities with v < v_th. In addition, we see that the effect of electronic nonideality on the transport scattering cross-section becomes weaker with increasing degeneracy parameter. According to Fig. 4, at the densities considered, the ideal electron approximation for screening becomes accurate at θ = 3.

Figure 5 shows the generalized Coulomb logarithm (10) as a function of θ. It can be seen that the electronic exchange correlations can be safely neglected at r_s = 0.5, but the effect of electronic nonideality starts to become noticeable at r_s ≥ 1. We can also see that taking into account the electronic local field correction (i.e., γ) leads to smaller values of the Coulomb logarithm in the transition region from a nondegenerate to a degenerate plasma at θ < 3.

We now investigate the effect of electronic local field correction on temperature relaxation in a hydrogen plasma with electrons hotter than the ions. For this, we solve Eqs. (15) and (16). Based on the calculation of the Coulomb logarithm Ξ, the temperature relaxation times are computed for different densities with the initial temperature of electrons T_e = 10 eV and that of ions T_e = 1 eV. The results for the temperature relaxation are presented in Fig. 6, where we compare the NPS model results with those obtained by setting γ = 0. With increasing density, the frequency of collisions between electrons and ions increases, leading to more rapid equalization of the electron and ion temperatures. From Fig. 6, it is clear that the lower the density, the more time it takes for the system to reach thermodynamic equilibrium. The effect of electron change correlations leads to longer relaxation times. This can be explained by recalling that the electronic nonideality leads to a stronger screening of the electron–ion pair potential, which reduces the collision rates between electrons and ions.

As another example of the application of the NPS model, in Fig. 7, we show the relaxation rate (i.e., the inverse collision frequency) computed for warm dense aluminum at solid density. We compare the NPS model results with the data obtained using the IPS model, with the results of the well-known Landau–Spitzer model,98 and with the results of advanced state-of-the-art methods, including quantum Landau–Fokker–Planck (QLFP) model results from Daligault99,100 and the model developed recently by Rightley and Baalrud88 on the basis of the average-atom model.

As one might expect, Fig. 7 demonstrates that all the models give similar values at high temperatures owing to the weakening of correlation effects. By contrast, at low temperatures, the models that include strong correlation effects and those designed for weakly coupled plasmas (or ideal plasmas) start to differ from each other with decreasing temperature. In a nondegenerate plasma with T > 100 eV, the relaxation time can be calculated using the ideal plasma model for screening. At high and partial degeneracy with T ≲ 10 eV, both quantum effects and plasma nonideality must be taken into account. Therefore, as shown in Fig. 7, the consideration of exchange-correlation effects is crucial for the WDM regime. We find that the NPS model agrees with the results of Rightley and Baalrud.88

As a third example of the application of the NSP model, in Fig. 8, we show the electrical conductivity of aluminum at a density ρ = 2.7 g·cm⁻³ and different temperatures. In addition to the aforementioned models, here we present a comparison with the results from the models of Shaffer and Starrett (SS)101 and of Lee and More (LM),102 as well as the quantum molecular dynamics (QMD) results of Witte et al.103 obtained using the exchange-correlation functionals of Perdew–Burke–Ernzerhof (pbe) and Heyd–Scuseria–Ernzerhof (hse). It can be seen from Fig. 8 that the NPS model is in good agreement with the results of the more elaborate calculations by Witte et al.,103 Shaffer and Starrett,101 and Rightley and Baalrud.88 The IPS model overestimates the conductivity at T ≲ 10 eV.

Using screening theory in quantum dense plasmas, we have proposed a simple model for calculating the relaxation and transport properties of dense plasmas. The model uses quantum Monte Carlo results for the exchange-correlation free energy density and the ionic nonideality in the screening length of plasmas. This screening length is used in Yukawa-type pair interaction potential of particles and combined with a generalized Coulomb logarithm. Comparison of the results with those from state-of-the-art methods shows that the proposed model can be used to evaluate the transport and relaxation properties of warm dense matter. We should note, however, that the model is limited to systems with a high degree of ionization. The calculations are carried out using a plasma physics approach, initially derived from classical kinetic equations. The quantum scattering cross-section considered in this work can also be used within the framework of quantum kinetic plasma theory.104 Such a more advanced approach could provide additional insights, particularly in regimes where nonequilibrium effects or more complex plasma behaviors are significant.

Acknowledgment. This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan Grant No. AP19678033 “The study of the transport and optical properties of hydrogen at high pressure.”