Electron-ion Temperature Equilibration Research Articles

Calculation of electron-ion temperature equilibration rates and friction coefficients in plasmas and liquid metals using quantum molecular dynamics.

We discuss a method to calculate with quantum molecular dynamics simulations the rate of energy exchanges between electrons and ions in two-temperature plasmas, liquid metals, and hot solids. Promising results from this method were recently reported for various materials and physical conditions [Simoni and Daligault, Phys. Rev. Lett. 122, 205001 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.205001]. Like other ab initio calculations, the approach offers a very useful comparison with the experimental measurements and permits an extension into conditions not covered by the experiments. The energy relaxation rate is related to the friction coefficients felt by individual ions due to their nonadiabatic interactions with electrons. Each coefficient satisfies a Kubo relation given by the time integral of the autocorrelation function of the interaction force between an ion and the electrons. These Kubo relations are evaluated using the output of quantum molecular dynamics calculations in which electrons are treated in the framework of finite-temperature density functional theory. The calculation presents difficulties that are unlike those encountered with the Kubo formulas for the electrical and thermal conductivities. In particular, the widely used Kubo-Greenwood approximation is inapplicable here. Indeed, the friction coefficients and the energy relaxation rate diverge in this approximation since it does not properly account for the electronic screening of electron-ion interactions. The inclusion of screening effects considerably complicates the calculations. We discuss the physically motivated approximations we applied to deal with these complications in order to investigate a widest range of materials and physical conditions. Unlike the standard method used for the electronic conductivities, the Kubo formulas are evaluated directly in the time domain and not in the energy domain, which spares one from needing to introduce an extraneous undetermined numerical parameter to account for the discrete character of the numerical density of states. We highlight interesting properties of the energy relaxation rate not shared by other electronic properties, in particular its self-averaging character. We then present a detailed parametric and convergence study with the numerical parameters, including the system size, the number of bands and k points, and the physical approximations for the dielectric function and the exchange-correlation energy.

An ALMA+ACA measurement of the shock in the Bullet Cluster

Context. The thermal Sunyaev-Zeldovich (SZ) effect presents a relatively new tool for characterizing galaxy cluster merger shocks, traditionally studied through X-ray observations. Widely regarded as the “textbook example” of a cluster merger bow shock, the western, most-prominent shock front in the Bullet Cluster (1E0657-56) represents the ideal test case for such an SZ study. Aims. We aim to characterize the shock properties using deep, high-resolution interferometric SZ effect observations in combination with priors from an independent X-ray analysis. Methods. Our analysis technique relies on the reconstruction of a parametric model for the SZ signal by directly and jointly fitting data from the Atacama Large Millimeter/submillimeter Array (ALMA) and Atacama Compact Array (ACA) in Fourier space. Results. The ALMA+ACA data are primarily sensitive to the electron pressure difference across the shock front. To estimate the shock Mach number ℳ, this difference can be combined with the value for the upstream electron pressure derived from an independent Chandra X-ray analysis. In the case of instantaneous electron-ion temperature equilibration, we find ℳ = 2.08−0.12+0.12, in ≈ 2.4σ tension with the independent constraint from Chandra, MX = 2.74 ± 0.25. The assumption of purely adiabatic electron temperature change across the shock leads to ℳ = 2.53−0.25+0.33, in better agreement with the X-ray estimate ℳX = 2.57 ± 0.23 derived for the same heating scenario. Conclusion. We have demonstrated that interferometric observations of the thermal SZ effect provide constraints on the properties of the shock in the Bullet Cluster that are highly complementary to X-ray observations. The combination of X-ray and SZ data yields a powerful probe of the shock properties, capable of measuring ℳ and addressing the question of electron-ion equilibration in cluster shocks. Our analysis is however limited by systematics related to the overall cluster geometry and the complexity of the post-shock gas distribution. To overcome these limitations, a simultaneous, joint-likelihood analysis of SZ and X-ray data is needed.

Analytic expressions for electron-ion temperature equilibration rates from the Lenard-Balescu equation.

In this work, we elucidate the mathematical structure of the integral that arises when computing the electron-ion temperature equilibration time for a homogeneous weakly coupled plasma from the Lenard-Balescu equation. With some minor approximations, we derive an analytic formula, requiring no input Coulomb logarithm, for the equilibration rate that is valid for moderate electron-ion temperature ratios and arbitrary electron degeneracy. For large temperature ratios, we derive the necessary correction to account for the coupled-mode effect, which can be evaluated very efficiently using ordinary Gaussian quadrature.

Molecular dynamics studies of electron-ion temperature equilibration in hydrogen plasmas within the coupled-mode regime.

We use classical molecular dynamics (MD) to study electron-ion temperature equilibration in two-component plasmas in regimes for which the presence of coupled collective modes has been predicted to substantively reduce the equilibration rate. Guided by previous kinetic theory work, we examine hydrogen plasmas at a density of n=10^{26}cm^{-3},T_{i}=10^{5}K, and 10^{7}K<T_{e}<10^{9}K. The nonequilibrium classical MD simulations are performed with interparticle interactions modeled by quantum statistical potentials (QSPs). Our MD results indicate (i) a large effect from time-varying potential energy, which we quantify by appealing to an adiabatic two-temperature equation of state, and (ii) a notable deviation in the energy equilibration rate when compared to calculations from classical Lenard-Balescu theory including the QSPs. In particular, it is shown that the energy equilibration rates from MD are more similar to those of the theory when coupled modes are neglected. We suggest possible reasons for this surprising result and propose directions of further research along these lines.

Ion potential in warm dense matter: wake effects due to streaming degenerate electrons.

The effective dynamically screened potential of a classical ion in a stationary flowing quantum plasma at finite temperature is investigated. This is a key quantity for thermodynamics and transport of dense plasmas in the warm-dense-matter regime. This potential has been studied before within hydrodynamic approaches or based on the zero temperature Lindhard dielectric function. Here we extend the kinetic analysis by including the effects of finite temperature and of collisions based on the Mermin dielectric function. The resulting ion potential exhibits an oscillatory structure with attractive minima (wakes) and, thus, strongly deviates from the static Yukawa potential of equilibrium plasmas. This potential is analyzed in detail for high-density plasmas with values of the Brueckner parameter in the range 0.1≤r(s)≤1 for a broad range of plasma temperature and electron streaming velocity. It is shown that wake effects become weaker with increasing temperature of the electrons. Finally, we obtain the minimal electron streaming velocity for which attraction between ions occurs. This velocity turns out to be less than the electron Fermi velocity. Our results allow for reliable predictions of the strength of wake effects in nonequilibrium quantum plasmas with fast streaming electrons showing that these effects are crucial for transport under warm-dense-matter conditions, in particular for laser-matter interaction, electron-ion temperature equilibration, and stopping power.

Electron-ion temperature equilibration in warm dense tantalum

We present measurements of electron-ion temperature equilibration in proton-heated tantalum, under warm dense matter conditions. Our results agree with theoretical predictions for metals calculated using input data from ab initio simulations. However, the fast relaxation observed in the experiment contrasts with much longer equilibration times found in proton heated carbon, indicating that the energy flow pathways in warm dense matter are far from being fully understood.

ELECTRON-ION EQUILIBRIUM AND SHOCK PRECURSORS IN THE NORTHEAST LIMB OF THE CYGNUS LOOP

We present an observational study using high-resolution echelle spectroscopy of collisionless shocks in the Cygnus Loop supernova remnant. Measured Hα line profiles constrain pre-shock heating processes, shock speeds, and electron-ion equilibration (T{sub e} /T{sub i} ). The shocks produce faint Hα emission line profiles, which are characterized by narrow and broad components. The narrow component is representative of the pre-shock conditions, while the broad component is produced after charge transfer between neutrals entering the shock and protons in the post-shock gas, thus reflecting the properties of the post-shock gas. We observe a diffuse Hα region extending about 2.'5 ahead of the shock with line width ∼29 km s{sup –1}, while the Hα profile of the shock itself consists of broader than expected narrow (36 km s{sup –1}) and broad (250 km s{sup –1}) components. The observed diffuse emission arises in a photoionization precursor heated to about 18,000 K by He I and He II emission from the shock, with additional narrow component broadening originating from a thin cosmic-ray precursor. Broad to narrow component intensity ratios of ∼1.0 imply full electron-ion temperature equilibration T{sub e} ≅ T{sub i} in the post-shock region. Broad component line widths indicate shock velocities of about more » 400 km s{sup –1}. Combining the shock velocities with proper motions suggests that the distance to the Cygnus Loop is ∼890 pc, significantly greater than the generally accepted upper limit of 637 pc. « less

Molecular dynamics simulations and generalized Lenard-Balescu calculations of electron-ion temperature equilibration in plasmas

We study the problem of electron-ion temperature equilibration in plasmas. We consider pure H at various densities and temperatures and Ar-doped H at temperatures high enough so that the Ar is fully ionized. Two theoretical approaches are used: classical molecular dynamics (MD) with statistical two-body potentials and a generalized Lenard-Balescu (GLB) theory capable of treating multicomponent weakly coupled plasmas. The GLB is used in two modes: (1) with the quantum dielectric response in the random-phase approximation (RPA) together with the pure Coulomb interaction and (2) with the classical (ℏ→0) dielectric response (both with and without local-field corrections) together with the statistical potentials. We find that the MD results are described very well by classical GLB including the statistical potentials and without local-field corrections (RPA only); worse agreement is found when static local-field effects are included, in contradiction to the classical pure-Coulomb case with like charges. The results of the various approaches are all in excellent agreement with pure-Coulomb quantum GLB when the temperature is high enough. In addition, we show that classical calculations with statistical potentials derived from the exact quantum two-body density matrix produce results in far better agreement with pure-Coulomb quantum GLB than classical calculations performed with older existing statistical potentials.

Effects of electron-ion temperature equilibration on inertial confinement fusion implosions

The electron-ion temperature relaxation essentially affects both the laser absorption in coronal plasmas and the hot-spot formation in inertial confinement fusion (ICF). It has recently been reexamined for plasma conditions closely relevant to ICF implosions using either classical molecular-dynamics simulations or analytical methods. To explore the electron-ion temperature equilibration effects on ICF implosion performance, we have examined two Coulomb logarithm models by implementing them into our hydrocodes, and we have carried out hydrosimulations for ICF implosions. Compared to the Lee-More model that is currently used in our standard hydrocodes, the two models predict substantial differences in laser absorption, coronal temperatures, and neutron yields for ICF implosions at the OMEGA Laser Facility [Boehly et al. Opt. Commun. 133, 495 (1997)]. Such effects on the triple-picket direct-drive design at the National Ignition Facility (NIF) have also been explored. Based on the validity of the two models, we have proposed a combined model of the electron-ion temperature-relaxation rate for the overall ICF plasma conditions. The hydrosimulations using the combined model for OMEGA implosions have shown ∼6% more laser absorption, ∼6%-15% higher coronal temperatures, and ∼10% more neutron yield, when compared to the Lee-More model prediction. It is also noticed that the gain for the NIF direct-drive design can be varied by ∼10% among the different electron-ion temperature-relaxation models.

Balmer-Dominated Shocks: A Concise Review

AbstractA concise and critical review of Balmer-dominated shocks (BDSs) is presented, summarizing the state of theory and observations, including models with/without shock precursors and their synergy with atomic physics. Observations of BDSs in supernova remnants are reviewed on an object-by-object basis. The relevance of BDSs towards understanding the acceleration of cosmic rays in shocks is emphasized. Probable and possible detections of BDSs in astrophysical objects other than supernova remnants, including pulsar wind nebulae and high-redshift galaxies, are described. The case for the continued future of studying BDSs in astrophysics is made, including their relevance towards understanding electron–ion temperature equilibration in collisionless shocks.

Temperature equilibration in a fully ionized plasma: Electron-ion mass ratio effects

Brown, Preston, and Singleton (BPS) produced an analytic calculation for energy exchange processes for a weakly to moderately coupled plasma: the electron-ion temperature equilibration rate and the charged particle stopping power. These precise calculations are accurate to leading and next-to-leading order in the plasma coupling parameter and to all orders for two-body quantum scattering within the plasma. Classical molecular dynamics can provide another approach that can be rigorously implemented. It is therefore useful to compare the predictions from these two methods, particularly since the former is theoretically based and the latter numerically. An agreement would provide both confidence in our theoretical machinery and in the reliability of the computer simulations. The comparisons can be made cleanly in the purely classical regime, thereby avoiding the arbitrariness associated with constructing effective potentials to mock up quantum effects. We present here the classical limit of the general result for the temperature equilibration rate presented in BPS. In particular, we examine the validity of the m(electron)/m(ion)-->0 limit used in BPS to obtain a very simple analytic evaluation of the long-distance collective effects in the background plasma.

Molecular Dynamics Simulations of Electron-Ion Temperature Equilibration in anSF6Plasma

We use classical molecular dynamics to investigate electron-ion temperature equilibration in a two-temperature SF6 plasma. We choose a density of 1.0 x 10;{19}SF_{6} molecules per cm;{3} and initial temperatures of T_{e} = 100 eV and T_{S} = T_{F} = 15 eV, in accordance with experiments currently underway at Los Alamos National Laboratory. Our computed relaxation time lies between two oft-used variants of the Landau-Spitzer relaxation formula which invoke static screening. Discrepancies are also found when comparing to the predictions made by more recent theoretical approaches. These differences should be large enough to be measured in the upcoming experiments.

Temperature equilibration rate with Fermi-Dirac statistics

We calculate analytically the electron-ion temperature equilibration rate in a fully ionized, weakly to moderately coupled plasma, using an exact treatment of the Fermi-Dirac electrons. The temperature is sufficiently high so that the quantum-mechanical Born approximation to the scattering is valid. It should be emphasized that we do not build a model of the energy exchange mechanism, but rather, we perform a systematic first principles calculation of the energy exchange. At the heart of this calculation lies the method of dimensional continuation, a technique that we borrow from quantum field theory and use in a different fashion to regulate the kinetic equations in a consistent manner. We can then perform a systematic perturbation expansion and thereby obtain a finite first-principles result to leading and next-to-leading order. Unlike model building, this systematic calculation yields an estimate of its own error and thus prescribes its domain of applicability. The calculational error is small for a weakly to moderately coupled plasma, for which our result is nearly exact. It should also be emphasized that our calculation becomes unreliable for a strongly coupled plasma, where the perturbative expansion that we employ breaks down, and one must then utilize model building and computer simulations. Besides providing different and potentially useful results, we use this calculation as an opportunity to explain the method of dimensional continuation in a pedagogical fashion. Interestingly, in the regime of relevance for many inertial confinement fusion experiments, the degeneracy corrections are comparable in size to the subleading quantum correction below the Born approximation. For consistency, we therefore present this subleading quantum-to-classical transition correction in addition to the degeneracy correction.

Prospects for the diagnosis of electron–ion temperature equilibration rates of warm dense matter by ultra-short pulse hard X-ray diffraction with an X-ray free electron laser

Ultra-short pulse kiloelectronvolt X-ray diffraction with an X-ray free electron laser and its potential for the diagnosis of electron–ion equilibration rates of warm dense matter are evaluated. A simple experimental configuration is suggested for the generation and subsequent probing of warm dense aluminium with the TESLA X-ray free electron laser. Differential scattering cross-sections are computed in an approximate manner with Thomas–Fermi form factors and tabular ion–ion static structure factors of one-component plasmas, inclusive of electron screening and degeneracy. This requires simulation of the sample, for which we use a hydrodynamic code featuring the Sesame equation of state, Thomas–Fermi ionisation and cold solid opacities (for the calculation of energy deposition). The effect of electron–ion equilibration rate on the evolution of the diffraction pattern on a picosecond time-scale is investigated. Finally, the signal level expected from experiment is estimated, indicating that measurements with good angular-resolution are possible.

Ablation of metals with picosecond laser pulses: Evidence of long-lived nonequilibrium conditions at the surface

We report here experimental results on laser ablation of metals in air and in vacuum in similar irradiation conditions. The experiments revealed that the ablation thresholds in air are less than half those measured in vacuum. Our analysis shows that this difference is caused by the existence of a long-lived transient nonequilibrium surface state at the solid-vacuum interface. The energy distribution of atoms at the surface is Maxwellian-like but with its high-energy tail truncated at the binding energy. We find that in vacuum the time needed for energy transfer from the bulk to the surface layer to build the high-energy tail, exceeds other characteristic timescales such as the electron-ion temperature equilibration time and surface cooling time. This prohibits thermal evaporation in vacuum for which the high-energy tail is essential. In air, however, collisions between the gas atoms and the surface markedly reduce the lifetime of this nonequilibrium surface state allowing thermal evaporation to proceed before the surface cools. We find, therefore, that the threshold in vacuum corresponds to nonequilibrium ablation during the pulse, while thermal evaporation after the pulse is responsible for the lower ablation threshold observed in air. This paper provides direct experimental evidence of how the transient surface effects may strongly affect the onset and rate of a solid-gas phase transition.

Electron–ion temperature equilibration at collisionless shocks in supernova remnants

The topic of this review is the current state of our knowledge about the degree of initial equilibration between electrons, protons and ions at supernova remnant (SNR) shocks. Specifically, the question has been raised as to whether there is an inverse relationship between the shock velocity and the equilibration. This review aims to compile every method that has been used to measure the equilibration and every SNR on which they have been tested. I review each method, its problems and uncertainties and how those would effect the degree of equilibration (or velocity) inferred. The final compilation of observed electron to proton temperature ratios as a function of shock velocity gives an accurate, conservative picture of the state of our knowledge and the avenues we need to pursue to make progress in our understanding of the relation between the velocity of a shock and the degree of equilibration.

The Physics of Supernova Blast Waves. I. Kinematics of DEM L71 in the Large Magellanic Cloud

We present the results from Fabry-Perot imaging spectroscopy of the Balmer-dominated supernova remnant DEM L71 (0505-67.9) in the LMC. Spectra extracted from the entire circumference of the blast wave reveal the broad- and narrow-component Hα line emission characteristic of nonradiative shocks in partially neutral gas. The new spectra of DEM L71 include portions of the rim that have not been previously observed. We find that the broad-component width varies azimuthally along the edge of DEM L71, ranging from 450 ± 60 km s-1 along the eastern edge to values as high as 985 km s-1 along the faint western edge. The latter width is nearly 60% larger than the value determined by earlier spectroscopy of the brightest Balmer-dominated filaments. In part of the faint northern rim the broad component is not detected, possibly indicating a lower density in these regions and/or a broad-component width in excess of 1000 km s-1. Between the limits of zero and full electron-ion temperature equilibration at the shock front, the allowed range of shock velocities is 430-560 km s-1 along the east rim and 700-1250 km s-1 along other parts of the blast wave. The Hα broad-to-narrow flux ratios vary considerably around the remnant, ranging from 0.4 to 0.8. These ratios lie below the values predicted by our shock models. We find that narrow-component Hα emission from a cosmic-ray precursor may be the cause of the discrepancy. The least decelerated portions of the blast wave (i.e., regions excluding the brightest filaments) are well characterized by Sedov models with a kinetic energy E51 = (0.37 ± 0.06)D, where D50 is the LMC distance in units of 50 kpc. The corresponding age for DEM L71 is (4360 ± 290)D50 yr. This is the first time that velocity information from the entire blast wave has been utilized to study the global kinematics of a nonradiative supernova remnants at a known distance.

The Physics of Supernova Remnant Blast Waves. II. Electron‐Ion Equilibration in DEM L71 in the Large Magellanic Cloud

We present analysis and modeling of X-ray spectra from the blast wave shock of DEM L71 in the Large Magellanic Cloud. This remnant exhibits widespread Balmer-dominated emission characteristic of nonradiative shocks in partially neutral gas. We have used Chandra ACIS-S data and optical Fabry-Perot spectra of the blast wave to measure the electron and proton temperatures, respectively. In principle, when combined, these measurements can determine the degree of electron-ion temperature equilibration (g0 ≡ Te/Tp) immediately behind the shock front. In our X-ray analysis we fit Chandra spectra of three nested regions behind the blast wave under three different scenarios: (1) a planar, initially unequilibrated shock (g0 = me/mp), where the downstream electron and proton temperatures equilibrate through Coulomb collisions, (2) a planar, immediately equilibrated shock (g0 = 1), and (3) a spherical, equilibrated shock under Sedov evolution. Using independent measurements of Te and Tp, we find that the X-ray spectra from the fastest blast wave locations (Vs ~ 700-1000 km s-1) are consistent with little or no equilibration at the shock front and are inconsistent with full equilibration. In contrast, spectra from regions showing slower blast wave speeds (Vs ~ 400-600 km s-1) allow full equilibration but exclude zero equilibration. In order to directly constrain the initial equilibration, we incorporated our knowledge of the proton temperatures into our X-ray models to build planar shock models that allow for a variable g0. This model confirmed and strengthened the above results. Specifically, we found that X-ray spectra from an intermediate-velocity shock (Vs ~ 800 km s-1) were consistent with intermediate equilibration, excluding both g0 = me/mp and g0 = 1 at greater than 1 σ. Overall, our results support the picture of decreasing electron-ion equilibration with increasing shock speed found from previous studies of optical spectra in other Balmer-dominated supernova remnants.

The X‐Ray Spectrum of Supernova Remnant 1987A

We discuss the X-ray emission observed from Supernova Remnant 1987A with the Chandra X-ray Observatory. We analyze a high resolution spectrum obtained in 1999 October with the high energy transmission grating (HETG). From this spectrum we measure the strengths and an average profile of the observed X-ray lines. We also analyze a high signal-to-noise ratio CCD spectrum obtained in 2000 December. The good statistics (~ 9250 counts) of this spectrum and the high spatial resolution provided by the telescope allow us to perform spectroscopic analyses of different regions of the remnant. We discuss the relevant shock physics that can explain the observed X-ray emission. The X-ray spectra are well fit by plane parallel shock models with post-shock electron temperatures of ~ 2.6 keV and ionization ages of ~ 6 x 10^10 cm^3/s. The combined X-ray line profile has a FWHM of ~ 5000 km/s, indicating a blast wave speed of ~ 3500 km/s. At this speed, plasma with a mean post-shock temperature of ~ 17 keV is produced. This is direct evidence for incomplete electron-ion temperature equilibration behind the shock. Assuming this shock temperature, we constrain the amount of collisionless electron heating at the shock front at T_e0 / T_s = 0.11 (+0.02,-0.01). We find that the plasma has low metallicity (abundances are ~ 0.1 -- 0.4 solar) and is nitrogen enriched (N/O ~ 0.8 by number), similar to that found for the equatorial ring. Analysis of the spectra from different regions of the remnant reveals slight differences in the parameters of the emitting plasma. The plasma is cooler near the optical Spot 1 (at position angle ~ 30 degrees) and in the eastern half of the remnant, where the bright optical spots are found, than in the western half, consistent with the presence of slower (~ 500 km/s) shocks entering denser ring material.