Temperature relaxation and generalized Coulomb logarithm in two-temperature dense plasmas relevant to inertial confinement fusion implosions

Detailed knowledge of energy exchange between electrons and ions is of fundamental importance for the description of temperature relaxation and also other nonequilibrium physics in Inertial Confinement Fusion (ICF). We present a theoretical model for the temperature relaxation rate and the related generalized Coulomb logarithm based on the Quantum Lenard–Balescu (QLB) kinetic equation, where no special cutoffs are needed to be introduced. To describe the collective modes characterizing the ionic acoustic waves, a single-pole approximation is introduced for the ionic dielectric response. The proposed model for the generalized Coulomb logarithm is examined over a wide range of plasma conditions for electron temperatures between 102 and 105 eV and electron densities between 1022 and 1026cm−3 . The values of the generalized Coulomb logarithm are demonstrated to be in excellent agreement with the ones evaluated using the original QLB kinetic approach but with much less computational cost. Comparisons with molecular dynamics simulations and other theoretical approaches are presented. For further applications of our model, we present results for the recently measured experimental Coulomb logarithm. Compared to other widely applied models such as the Landau–Spitzer Coulomb logarithm, our model provides more consistent descriptions for the results of molecular dynamics simulations and also for the experimental outcomes. Our model for the generalized Coulomb logarithm is easy to calculate and can benefit efficient and reliable simulations for the ICF implosions.


Introduction
Accurate knowledge of electron-ion temperature equilibration is of essential importance in nonequilibrium plasmas with wide applications such as in shock waves [1,2], in warm dense matters [3][4][5][6], and in Inertial Confinement Fusion (ICF) [7][8][9][10][11][12]. For ICF simulations, the electron-ion temperature relaxation rate is one of the central ingredients for modelling the shock propagation through the shell and the hot-spot formation in ICF implosions. Moreover, the fusion burn rate depends very sensitively on the ion temperature [8,13], so that uncertainty in the temperature relaxation rate will significantly affect the equilibration of ICF plasmas and hence on the burn rates. Additionally, the neutron yield and target gain predicted in the ICF simulations depend sensitively on the electron-ion temperature relaxation model utilized [9]. Therefore, the precise knowledge of the electron-ion temperature relaxation rate is of fundamental concern in ICF modelling and has an important impact on experimental design and analysis of experimental outcomes.
The theoretical and experimental study on relaxation rates in two-temperature electron-ion systems is a long-standing problem in plasma physics and in fusion physics. The ICF plasmas span over many orders of magnitude in both density and temperature including the warm dense matter and the high energy density physics regimes. Hence, an accurate description of the relaxation rates in such interactive manybody environment is extremely complicated, because different physical effects such as collective excitations, strong ionic correlations, strong collisions, dynamic screening, and quantum degeneracy have to be taken into account consistently. The earliest theories on relaxation problem in plasmas were developed by Landau [14] and Spitzer [15] (LS) for classical plasmas with weak collisions. In LS theory, some ad hoc cutoffs have to be introduced to avoid divergence of the collision integral over Coulomb collisions, where the Coulomb logarithm is defined. However, the value of LS Coulomb logarithm becomes negative under certain plasma conditions, which leads to unphysical temperature relaxation rates. Additionally, the LS relaxation rate generally leads to an underestimation of the peak ion temperature in ICF experiments [16,17]. To overcome the shortcomings of the LS formula, various improvements for the LS theory have been proposed by introducing modified Coulomb logarithm models [18][19][20][21][22][23].
The existence of ad hoc cutoffs and other inconsistencies within the LS-related theories encourage researchers to pursue more advanced theoretical approaches for the relaxation rates and the Coulomb logarithm. These theoretical investigations include the Molecular Dynamics (MD) simulations [13,17,[24][25][26][27][28][29][30][31] and the quantum statistical methods including the many-body Green's function approach [32], the quantum kinetic theories [32][33][34][35][36][37][38][39][40][41][42][43][44][45], the stochastic Langevinlike equation [46][47][48][49], and the method based on average-atom model [11,50]. Besides these theoretical developments, the ion-electron Coulomb logarithm was recently measured from the low-velocity energy loss experiments [12]. The numerical computation of the relaxation rates based on quantum statistical methods and the MD simulations are very complicate and time-consuming, which seriously hampers the efficiency of ICF simulations. It is desirable to have analytic representation for the relaxation rates [41] (or the Coulomb logarithm) as accomplished in the LS theory. If this aim cannot be achieved, developing models containing only single integral with smooth integrand is a viable compromise [40], which is the purpose of the present study.
In the current work we develop reduced descriptions for the Quantum Lenard-Balescu (QLB) kinetic equation by means of introducing single-pole approximation for ionic dielectric response. The resulting generalized Coulomb logarithm can be evaluated quickly and efficiently. The remaining part of this work is organized as follows: in section 2.1 some relevant physical parameters are defined. The general theory for the electron-ion energy and temperature relaxation in two-temperature two-component plasma systems is presented in section 2.2, where the ion-electron Coulomb logarithm is defined from the energy relaxation rate in section 2.2.1. Subsequently, the model for the generalized Coulomb logarithm is described in detail in section 2.2.2. The only approximation within our approach is the single-pole approximation. The Coupled Modes (CM) effects are effectively described by the ionic acoustic modes. To examine the validity of the proposed model, the Coulomb logarithms for hydrogen plasmas in a wide range of densities and temperatures are calculated in section 3.1, where comparisons with other theoretical models and the MD simulations [13,24,26,31] are carried out. As a further application of our approach, we study the Coulomb logarithm for deuterium-tritium plasmas relevant to ICF in section 3.2. The recent experimental measurements on the ion-electron Coulomb logarithm performed at the OMEGA laser facility [12] are also analyzed in this subsection. Finally, conclusions are drawn in section 4.

Relevant physical parameters
The two-temperature two-component systems under study consist of electrons with mass m e and number density n e at temperature T e and ions with nuclear charge number z i , mass m i and number density n i at temperature T i . Such systems can be fully characterized by following five parameters: the charge neutrality z i = n e /n i , the temperature ratio α 1 = T i /T e , the mass ratio α 2 = m i /m e , the Brueckner parameter r S , and the electron degeneracy parameter θ e defined as with the Bohr radius a B and the averaged interparticle distance of species c defined as a cc = 3 3/(4π n c ). The Fermi energy of electrons is given by E F =h 2 k 2 F /(2m e ) with the Fermi wavevector k F = 3 3π 2 n e . Other plasma parameters can be derived from the aforementioned five parameters. For example, the plasma coupling parameter introduced in the BPS (Brown, Preston and Singleton) theory is given as [17,51] (2) with α 0 = (9π/4) −1/3 . Here r L = ze 2 /(4πϵ 0 k B T e ) is the Landau length and κ De = e 2 n e /(ϵ 0 k B T e ) denotes the inverse Debye length of electrons. For degenerate plasmas, the electronic screening length is different from the Debye length. In the current work, following definition for inverse screening parameter is employed where T cc denotes the effective temperature of species c. In the current work, following parametrization T ee = T e 1 + θ e 1.3251 − 0.1779 √ r S −2 is utilized for the effective electron temperature [52], which accounts for the quantum degeneracy effect on electronic screening. In the case of high temperatures, the inverse Debye length κ De is recovered. For the plasma conditions considered in this work, the effective ion temperature T ii is simply the ionic kinetic temperature, i.e. T ii = T i .

General theory for electron-ion energy relaxation.
In two-temperature two-component plasma systems, the energy flow characterizing the equilibration evolution between the electronic and ionic subsystem is described by the energy density transfer rate [25,39] where E i (E e ) is the energy density of the ionic (electronic) subsystem and R i ↔e is the corresponding electron-ion coupling parameter. The energy conversation of the total system is guaranteed by the relation dE e /dt = −dE i /dt. Generally, the electron-ion coupling parameter R i ↔e can be expressed as [25,39] where L is an universal factor describing the collision strength between electronic and ionic components, which is conventionally termed as the Coulomb logarithm.
If only the conservation of total kinetic energy of the twotemperature two-component systems is taken into account, then the energy density is given by E c = 3 2 n c k B T c . This relation leads to the definition of collision frequency (or relaxation time) such as [25] with the ion-electron collision frequencies defined as ν ie = 2R i ↔e /(3n i k B ). For the real two-temperature two-component systems, the relation E c = 3 2 n c k B T c does not hold any more and the interaction energy has to be accounted for in the temperature relaxation through the equation of state [28,38]. However, this procedure is extremely complicate, because the apportion of the electron-ion interaction energy to the electronic and ionic subsystems is ambiguous. Moreover, for plasma conditions relevant to ICF implosions, the contribution of potential energy arising from intraspecies and interspecies interaction plays a secondary role in the temperature relaxation. As handled in [10,11,50] for ICF plasmas, the contribution of interaction energy is neglected in this work and only the conservation of the total kinetic energy is considered.
Theoretically, one can always recast the temperature relaxation model into the LS form given by equation (6), so that different many-body effects such as dynamical screening, electron-ion coupling and degeneracy effects can be incorporated into the generalized Coulomb logarithm. Consequently, the essential task for studying the temperature and energy relaxation is to determine the form of the Coulomb logarithm for the two-temperature two-component plasma systems, which is still an active and extensive research topic [53][54][55]. Traditionally, efforts have been made to develop models with simple analytic forms to incorporate as many physical effects as possible. A brief review of different expressions for the Coulomb logarithm is given in appendix. In the remaining context of this subsection, we propose a generalized Coulomb logarithm based on the QLB kinetic equation.

Generalized Coulomb logarithm based on the Lenard-Balescu kinetic equation.
For weakly coupled twotemperature two-component plasma systems relevant to the ICF implosions, the QLB kinetic equation is believed to be a very good approximation for determining the energy density transfer rate and has been widely applied. Within this theory, the flow of energy density from ionic subsystem to electronic component is described by [38] where k andhω are the momentum and energy transfer during the collisions. n c is the dielectric function of the species c, and ϵ(k, ω) = ϵ ee (k, ω) + ϵ ii (k, ω) − 1 is the total dielectric function of the two-temperature two-component plasma systems. Inserting the rate (7) into equation (4a) in combination with the definition of electronion coupling parameter (5) results in the following form for the generalized Coulomb logarithm e /4 and α 3 = α 1 /α 2 . The auxiliary function I QLB (k) is given by with C ei (k, ω)= Imϵ ii (k,ω) Imϵ(k,ω) . Here the electron-ion coupling term D ei (k, ω) (10c) describes the CM effects in the QLB kinetic equation. The function Im ϵ −1 ii (k, ω) = Im ϵ(k,ω) |ϵ(k,ω)| 2 |ϵ ee (k, ω)| 2 denotes the effective ionic response in plasmas, which takes into account the modification of dielectric response within the pure ionic subsystem due to the presence of screening cloud from the slowing-moving free electrons. If the electronic and ionic subsystems are regarded as uncoupled twocomponent system, i.e. the CM effects are neglected, the dielectric function of the two-component system can be fac- [32,38,41]. This approximation is the well-known Fermi's golden rule (FGR) approximation for energy relaxation. In this approximation, the function D ei (k, ω) = Im ϵ −1 ii (k, ω) is simply the dielectric response of the pure ionic subsystem.
The direct numerical calculation of the generalized Coulomb logarithm L QLB (8) is computationally complicate, since it contains an ω-integration over sharply peaked functions involved in the function W(k, ω) as shown in the upper panel of figure 1. The resulting integrand for the kintegral given in equation (8) is oscillatory as depicted in the lower panel of figure 1 (green curve). Such oscillation causes further numerical difficulty for the evaluation of the generalized Coulomb logarithm L QLB (8). Therefore, some special numerical techniques [13,28,56] or approximate treatments have to be introduced for the double integral associated in equation (8). Generally, the dielectric fluctuations Im ϵ −1 cc (k, ω) is strongly peaked for small momenta and decreases very rapidly to zero when the frequency ω exceeds the plasma frequency ω pl,c = z 2 c e 2 n c /(ϵ 0 m c ) [33,37,57,58]. In the case of large momenta, the dielectric response function Im ϵ −1 cc (k, ω) also drops to zero very fast when the phase velocity ω/k of the particle-hole excitation exceeds the corresponding thermal velocity (m c β c ) −1/2 [33,57,58]. Because of the large mass difference of electrons and ions, the effective region for the ω-integral in equation (9) is strongly limited by the ionic spectral function and is restricted to the domain of small frequency values [38,42].
Actually, it can be seen from the upper panel of figure 1 that, for small momenta, the integrand of the ω-integral is strongly peaked. For the plasma conditions reported in figure 1, the height of the collective acoustic mode (single particle excitation) with CM effects is 1.9 (0.0051). For the FGR approach, the ratio of the heights of the collective plasmon mode to the single particle excitation is 1093.73 to 0.0030. Therefore, the main contribution to the ω-integral in equation (9) comes from the strongly peaked domain of the ionic dielectric function as shown in the upper panel of figure 1, which can be excellently approximated by a Dirac-delta distribution function. This approximate description can be generalized to large momenta and is widely employed in the research field of electronic structure of many-body systems [59,60]. This approximation is the so-called single-pole approximation for the inverse dielectric function [58]. In the present work, this approximation is utilized for the effective ionic response function Im ϵ −1 ii (k, ω). In the two-component plasmas, the electron-ion coupling effects make the peak positions of collective ionic modes shifting from the plasmon modes to the ion acoustic modes [57,58,61,62]. Consequently, the imaginary part of the effective ionic dielectric function is primarily characterized by the collective modes representing ion acoustic waves. Hence, the single-pole approximation for the effective ionic dielectric response can be adopted and is described by where the ionic acoustic dispersion (IAD) relation ω iad (k) is described by the ionic static structure factor [61,62] For weakly coupled ions, the Debye ionic structure factor in two-component plasmas S TCP ii (k) = k 2 +κ 2 e k 2 +κ 2 e +κ 2 i can be applied [57]. Consequently, the CM effects are encoded in the IAD relation ω iad (k) and in the C ei (k, ω) function. Substituting the effective ionic response function (11) in combination with equation (10c) into the integrand (9) yields following Coulomb logarithm with the corresponding function I IAD (k) given by This is the central result of the current work. The ionic acoustic modes exist only for T i < c z i T ee with a certain constant c, which can be determined from the condition Re ϵ(k, ω) = 0. Vorberger and Gericke [37] estimated the constant to be c ≈ 0.27 with the use of classic ionic dielectric function and electronic dielectric function in static long wavelength limit. A precise value c of stating ionic acoustic modes deserves further investigation, since it depends sensitively on the dielectric function applied in the calculation. This point was also discussed in [41]. When T i ≫ T e , the ionic acoustic modes disappear, so that the approximate expression of the ionic dielectric response (11) becomes inapplicable for α 1 ≫ 1. For the plasma conditions relevant to ICF physics studied in this work, the approximation (11) can provide a reasonable description for the temperature relaxation and the related generalized Coulomb logarithm as manifested in section 3. A similar approximation for the effective ionic dielectric response proposed by Gregori and Gericke [63] should be mentioned here. Their ansatz is equivalent to adopt for the effective ionic response [57,58]. However, this ansatz can only give an reasonable description for ionic response near the pole positions of ionic acoustic modes. With the approximate formula (11), the validity of the pole approximation can be extended [57,58], which is crucial for determining the proper mode structure of CM effects. In other words, the damping of the ion plasmon wave onto an IAD relation is captured by the approximation (11). If the ionic and electronic subsystems are assumed to be decoupled, the single-pole approximation for the FGR Coulomb logarithm is obtained by replacing the function I IAD (k) in equation (13) by with being the dispersion relation for ionic plasmon modes (IPMs). For weakly coupled ions, the structure factor S OCP ii (k) = k 2 /(k 2 + κ 2 i ) is used. For further evaluations, the dielectric functions involving in D ee (k, ω) (10b) and C ei (k, ω) = Im ϵ ii (k,ω) Im ϵ(k,ω) with Im ϵ(k, ω) = Im ϵ ee (k, ω) + Im ϵ ii (k, ω) have to be specified. In the current work, the dielectric function in random phase approximation (RPA) is applied for the electronic response [64] Imϵ ee (k, ω) = α 0 r S θ e 8q 3 ln with q = k/(2k F ), u = m e ω/(hk F k) and the function For fast and efficient computation of this integration, the parametrization proposed in [65] can be utilized. For ionic response, the classic dielectric function is used, whose imaginary part is given by [57,58] Im with the argument x defined as x = m i β i w 2 /k 2 . In comparison to the Coulomb logarithm (8) based on the QLB kinetic equation, the model (13) is numerically more convenient to evaluate, because it contains only a single integral over smooth functions. This fact is manifested in the lower panel of figure 1, where the integrand of Coulomb logarithm (8) is oscillatory in the region of 0.3 ≲ k/k F ≲ 0.55. For the integration over this region, more refine grid is required to achieve the numerical convergence. The final integral for the values of the corresponding Coulomb logarithms differs with each other within 0.75% for green (CM) and red (IAD) curves under the plasma conditions given in figure 1. More strongly oscillatory integrand for other plasma conditions was reported in [28]. Furthermore, the CM effects on the energy transfer rate and also the resulting Coulomb logarithm is demonstrated by the difference of the integrand (14) and (16) as shown in figure 1. Additionally, the only approximation in the Coulomb logarithm (13), i.e. L IAD , is the implementation of single-pole approximation in the ionic dielectric function. The influence of other physical effects inherent in the QLB kinetic theory is preserved. For example, the ionic correlation effect can be accounted for by using more advanced ionic structure factor.  [13]. The green diamonds and magenta squares are results derived from the MD simulations by Gao et al [31]. The black circles represent the MD simulation data of Glosli et al [24]. For comparison, the models of BPS [45], GMS [22], LM [20], and DD [17] as well as our models with CM effects (equation (13)) and without CM effects (equation (15)) are utilized to evaluate the Coulomb logarithm.

Comparison with other approaches and MD simulations for fully ionized hydrogen plasmas
In this subsection, we present results for the values of Coulomb logarithm in fully ionized hydrogen plasmas. To compare with data from MD simulations, the models of LS (A1), LM (Lee and More [20], No. 1 of table 2 in appendix), GMS (Gericke et al [22], No. 5 of table 2 in appendix), and BPS formula [45] as well as the fit formula proposed by Dimonte and Daligault [17] are used in addition to our model of Coulomb logarithm, i.e. IAD model L IAD (13). For comparison, the predictions using L IPM (15) are also shown.
We first compare the values of Coulomb logarithm for hydrogen plasmas with the number density n e = 10 22 cm −3 at various temperatures as shown in figure 2. The ion-electron temperature ratio is fixed to be α 1 = 2 in the calculation. The MD simulation results of Glosli et al [24] and Benedict et al [13] are the relaxation time. We use the relation (6) in combination with equation (5) to calculate the Coulomb logarithm from the corresponding MD data. The MD simulation results of Gao et al [31] are the electron-ion coupling parameter R i ↔e , so that the equation (5) is applied to obtain the corresponding Coulomb logarithm. All simulations make use of electron-ion pseudopotential to mimic the quantum behavior of electrons as adopted by Hansen and McDonald [66]. It should be emphasized that the MD data from Benedict et al lie much above other MD simulation results. However, it will becomes consistent with other simulations if their results are halved [67]. Note that throughout this work, the plasma systems are assumed to be fully ionized. For low temperatures shown in figure 2, the hydrogen plasmas are indeed in partial ionization. In this case the neutral hydrogen atoms come into play and will affect the energy and temperature relaxation processes. This fact was extensively discussed by Dharmawardana [68] and recently studied in [55]. For quantitative comparisons with other relaxation models such as the LS and BPS results, which treat only the fully ionized states, the influence of neutrals is ignored in the following discussions.
It can be seen from figure 2 that all approaches except for the BPS theory predict the increase of the Coulomb logarithm values with the growth of the electron temperatures (corresponding to the decrease of the plasma coupling parameters g BPS .) The behavior of upwards bending of the BPS Coulomb logarithm is also reported in [9]. It manifests the break down of the BPS theory for moderately and strongly coupled twotemperature two-component systems, so that it fails to give a reasonable Coulomb logarithm at the electron temperature T e = 10 eV. Compared to the MD-based Coulomb logarithm values at T e = 10, 30, 100 eV, the values of Coulomb logarithm from the GMS and DD formulas locate well within the error bar of MD simulation of Glosli et al The values of LM Coulomb logarithm are slightly larger than all other models and also the MD data in this region. At high temperature region, the BPS, LM, GMS models lead to Coulomb logarithm in consistent with the prediction of L IPM (15). The model L IAD (13) (DD) predicts smaller (larger) Coulomb logarithm than other approaches. In the case of lower temperatures, the Coulomb logarithm obtained from the model L IAD (13) and L IPM (15) are comparable with each other, which are larger than the ones of DD, GMS, and LM theories. Furthermore, both our models (L IAD (13) and L IPM (15)) display an overall excellent agreement with different simulation results in a wide temperature range. In contrast to the model L IPM (15), the model L IAD (13) matches better with the halved data of Benedict et al [67]. Figure 3 highlights the comparison between the Coulomb logarithm derived from the MD simulations of Murillo and Dharma-wardana [26] and theoretic predictions calculated from different models. The number density and ion temperature of the hydrogen plasmas is fixed at r S = 0.5 and T i = 10 eV, respectively. For high density plasmas, the quantum effects such as the quantum exchange effect, the collective effect and the dynamical screening effect play a crucial role in the temperature and energy relaxation process. In this case, the Coulomb logarithms predicted by these theoretical approaches differ strongly as illustrated in figure 3. The difference between the model L IPM (15) and the model L IAD (13) at high temperatures demonstrates the influence of CM effects on the nonequilibrium evolution. In this high temperature regime, the models of BPS, GMS, and L IPM (15) lead to Coulomb logarithm with merely slight difference. Remarkably, the Coulomb logarithm predicted by the model L IAD (13) agrees outstandingly with the MD data for all electron temperatures.  [26]. The values of Coulomb logarithm from the BPS [45], GMS [22], LM [20], and DD [17] as well as our approaches (equations (13) and (15)) are shown for comparisons.

Coulomb logarithm under plasma conditions relevant to the ICF implosions
In this subsection, we study the Coulomb logarithm in deuterium-tritium plasmas relevant to the ICF implosions. The recent experimental measurements on the ion-electron Coulomb logarithm performed at the OMEGA laser facility [12] will also be discussed.
The difference between our models (L IAD (13) and L IPM (15)) and other approaches is shown in detail in figure 4 for typical ICF plasma conditions. The ablator shell in the ICF implosions is substantially compressed to extremely high density with relative low temperatures as given in the upper panel of figure 4. As discussed in the previous subsection, the BPS Coulomb logarithm blends upwards from a certain plasma coupling parameter g BPS . For the temperature conditions in upper panel (a) of figure 4, the critical value of g BPS is about 0.7, which corresponds to a compressed dense shell with ρ DT ∼ 500 g cm −3 . The related plasma system is moderately correlated and partially degenerated, which might be out of the validity of the BPS model. For the same reason, the LS Coulomb logarithm becomes negative in this regime. The results from the GMS, L IPM (15), and the SSBEG parametrization proposed by Scullard et al [41] are close to each other in low density region, but derivations appear near the critical coupling parameter g BPS ≈ 0.7. Under hot-spot conditions (lower panel (b) of figure 4), the values of Coulomb logarithm calculated from different approaches are all greater than 1, since the hot-spot plasmas are weakly coupled. In this case, the BPS results match excellently with the results from the L IPM (15) and SSBEG models. The outcomes  [45], SSBEG [41], GMS [22], LM [20], and LS (A1) as well as our models (equations (13) and (15)) are utilized to evaluate the Coulomb logarithm.
of GMS model are in consistent with the predictions of LS, which are larger than the BPS results. The model of L IAD (13) gives smaller values of Coulomb logarithm compared to other approaches.
By analyzing the validity of different models under the plasma conditions relevant to the ICF implosions, Xu and Hu have suggested a combined model that incorporates the BPS formula for the high-density shell with g BPS ⩾ 0.1 and the MD fitting formula for the corona and hot-spot regimes with g BPS < 0.1 [9]. However, the BPS model is inappropriate for highly compressed fuel shell as shown in the upper panel of figure 4. The Coulomb logarithm based on more advanced quantum statistical theory is indispensable for such moderately to strongly coupled and partially to extremely degenerate plasmas. Now we discuss the recent experimental measurements performed by Adrian et al where the low-velocity ion stopping powers were measured and the ion-electron Coulomb logarithm is extracted from the stopping power data. The experimental results are detailed in table 1, where comparisons with different predictions are also performed. Relative differences Table 1. Plasma conditions and the inferred experimental Coulomb logarithm for different shots. For comparisons, models of LS (A1), LM [20], GMS [22], SSBEG [41], BPS [45] and DD [17] as well as our models (equations (13) and (15) [12] for different shots. Here the subscript 'X' implies the corresponding theoretical models. The gray areas mark the relative error of the experimental measurements.
of the theoretical Coulomb logarithm with respect to the experimental data for different approaches, i.e. (L X − L expt )/L expt , are highlighted in figure 5. Although the experimental errors are relatively large, the validity of different theoretical models can still be assessed. The inferred experimental Coulomb logarithm for different shots can be consistently and systematically explained by the SSBEG [41], BPS [45], and our models (L IAD (13) and L IPM (15)) as displayed in  (15)), significant quantitative differences between different theoretical models exist. Therefore, more precise experiments are necessary in order to benchmark the validity of different models for the Coulomb logarithm.

Conclusion
In the present work, we have proposed a theoretical model for the ion-electron Coulomb logarithm based on the QLB kinetic theory, where the only approximation employed is the singlepole approximation for the ionic dielectric function involved in the corresponding kinetic equation. The resulting expression for the Coulomb logarithm, i.e. L IAD (13), can be easily computed by integrating over some smooth integrand. By comparing with the MD simulations and also with the experimental measurements on the Coulomb logarithm, the proposed model L IAD (13) is demonstrated to be applicable for plasmas with electron degeneracy θ e ≳ 5 and ion coupling parameter Γ ii ≲ 1. Additionally, a reduced description, i.e. L IPM (15), was also derived from the energy transfer rate in FGR approximation, if the CM effects between the ionic and electronic subsystems are neglected. The MD simulation results of the Coulomb logarithm in fully ionized hydrogen plasmas with wide density and temperature conditions can be excellently explained by the model of L IAD (13). The CM effects are shown to have a significant influence on the prediction of Coulomb logarithm at high temperatures, which is consistently and effectively accounted for in the model of L IAD (13). In two-temperature two-component plasma systems, the motion of ions is always accompanied by the electron screening clouds, which shifts the ionic plasmon characters to ionic acoustic modes. By performing the single-pole approximation for the ionic response with the consideration of electron screening, the CM effects are approximately described by the IAD relation. The CM effects are not taken into account in the model of L IPM (15) and also other approaches such as LS and GMS models. Central to the discussion in the present investigation is that these different models differ strongly in the evaluation of the Coulomb logarithm. This fact is more visible by comparing the predicted Coulomb logarithm from diverse theoretical approaches to the inferred experimental results. The accuracy of the Coulomb logarithm has a nonnegligible influence on the neutron yield in the ICF implosion, so that the discrepancy of different models manifests the necessity of more advanced quantum statistical theory for the description of the Coulomb logarithm and the related energy relaxation rate. A conclusion that can be drawn from the analysis of MD simulations in section 3.1 and experimental measurements in section 3.2 is that the model of L IAD (13) works best for weakly to moderately coupled and nondegenerate to partially degenerate plasmas. Additionally, the model of L IAD (13) is also suitable for fast and reliable computations within a larger numerical simulation such as in the ICF modeling.
In this work, the plasmas are assumed to be in fully ionized state and only elastic scattering process is taken into account. However, the plasma system may be partially ionized, so that inelastic collisions, such as electron-neutral scattering for hydrogen plasmas [55,68,69], can have a remarkable contribution to the energy and temperature relaxation. This problem requires further study. Furthermore, the influence of equation of state (or the potential energy) [38] on the energy and temperature relaxation is totally neglected in the present investigation. Additionally, the RPA dielectric response for electrons is adopted in the quantum kinetic equation. These approximate treatments are assumed to be suitable in weakly coupled plasmas. However, as recently reported in [28], the RPA response might be inadequate to construct model for the energy and temperature relaxation in two-temperature two-component plasma systems, even in the case of weakly coupled plasmas. More advanced dielectric response such as response function with the dynamical local field corrections should be applied to describe the interparticle interactions. We will discuss the influence of the potential energy and the dynamical response beyond RPA level on the energy and temperature relaxation in a forthcoming work. Contract No. 24243/R0.

Appendix. Models for Coulomb logarithm
Within the framework of LS theory for the relaxation rate [14,15], the Coulomb logarithm is obtained after introducing ad hoc impact parameter cutoffs to eliminate the divergence in the collision integral Landau argued that the maximal impact parameter b max is determined by the total Debye screening length λ D,tot from both electrons and ions [14], whereas Spitzer proposed that only the screening length of electrons λ D,e contributes to the largest impact parameter [15]. For classical collisions, the minimal impact parameter b min is selected to be the closest-collision distance r ⊥ = z i e 2 /(4π ϵ 0 µv 2 rel ). Here µ = m e m i / (m e + m i ) and v rel are the reduced mass and the relative velocity of electron-ion scattering pair, respectively. In the case of high temperatures, the quantum diffraction effects have to be accounted for and a natural choice for b min is the quantum mechanical wavelength r qd =h/(2µv rel ). Hence we have b min = max(r ⊥ , r qd ) in L LS (A1). Note that the detailed expressions for r ⊥ and r qd are ambiguous due to different definitions for the averaged relative velocity v rel in the literature. For example, µv 2 rel = k B T e [14,15,22], 2k B T e [23,72] and 3k B T e [9,20,24,53,55] are all widely utilized. In this work, the definition µv 2 rel = k B T e is adopted, which results in Under certain plasma conditions, the values of the LS Coulomb logarithm (A1) become negative. To avoid such unphysical values for the Coulomb logarithm, extension for the definition of Coulomb logarithm was suggested. In deriving the Fokker-Planck equation from the chain of the Bogoliubov equations, Temko obtained following expression for the Fokker-Planck-Landau (FPL) Coulomb logarithm [18] where b max = λ D,tot appeared in a natural way, while the Landau length was used as a cutoff for the smallest impact parameter b min = r L = r ⊥ . The expression (A3) can be also derived by assuming the hyperbolic trajectories for Coulomb collisions [22]. The form of FPL Coulomb logarithm (A3) is widely applied in plasma physics for the determination of diverse physical properties. However, for quantum plasmas, more advanced expressions for the cutoffs b min and b max have to be utilized to account for the quantum diffraction, quantum degeneracy, finite ion-size effects, collective excitations and other many-body effects. We do not discuss the details of different suggestions for the cutoffs from different authors, but give some examples in table 2.
Another two analytic expressions without cutoffs for the Coulomb logarithm, which should be mentioned here, are developed by Brown et al (BPS formula) [45] and by Scullard et al (SSBEG parametrization) [41]. Both models are derived from the quantum kinetic theory with the inclusion of strong scattering in the presence of quantum effects such as the dynamical screening and the collective behavior. For the elaborate derivations and the corresponding analytic expressions, we refer readers to the more detailed discussion in [41,45]. In contrast to analytical studies, numerical simulations have also been broadly used to investigate the electron-ion energy relaxation problem. For example, Dimonte and Daligault (DD) [17] performed classical MD simulations for classic electron-ion Coulomb plasmas with like charges and have suggested the formula ln (1 + 0.7/g BPS ) for the Coulomb logarithm by fitting their MD results.