Stopping power modeling in warm and hot dense matter

A model is presented to calculate the stopping power of ions propagating in dense matter. Comparisons with experiment in the cold dense regime are presented and discussed. Further, we present results from the warm dense matter regime and the field of high energy density physics.     

Effective ion–ion potentials in warm dense matter

The effective ion–ion potential is extracted from first principle simulations and from experimental structure factors for several warm dense matter systems. Results of three different methods are compared for simple elements like hydrogen or beryllium as well as for composite materials like lithium-hydride and hydrogen–helium plasmas. It is shown that iterative techniques based on the pair distribution function are not unique in their solution and direct force-matching from first principle simulations is subject to finite size effects. Moreover, both methods are not able to provide potentials for small distances. These disadvantages can be avoided by using the static structure factor as input, although higher order correlations are only accounted for within the hypernetted chain approximation in this case. Furthermore, we discuss possibilities to use the extracted effective potentials to investigate the dielectric function beyond linear response.     

The quantum hypernetted chain model of warm dense matter

Modeling warm dense matter, where a combination of partial ionization, partial electron degeneracy, and strong ion–ion and ion–electron coupling occur, is a frontier of equation of state research. We present the quantum hypernetted chain model which can be applied to studies of liquid metals, warm dense matter, and plasmas. This is an all-electron model that considers a mixture of a classical fluid of ions (with bound electrons) and a quantum electron fluid. The model describes self-consistently the structure of the ion fluid as well as the bound states of the ions and the non-linear response of the electron fluid. We present our initial results and compare them with experimental and ab initio results for liquid metals and low-temperature plasmas.     

Analysis of semi-classical potentials for molecular dynamics and Monte Carlo simulations of warm dense matter

Effective, semi-classical potentials may present a powerful tool for the determination of properties of warm dense matter, systems characterized by both moderate coupling and moderate degeneracy. However, this requires the use of these potentials in a regime where the approximations employed in their derivation begin to break down. This work presents a careful analysis of the methodology and approximations used to derive semi-classical potentials for Coulomb systems. Particular attention is paid to the appearance of many-body effects and the techniques that may be used to model them. Analytical arguments and simple examples indicate that the role of many-body effects cannot be ignored in the warm dense matter regime, and those semi-classical Coulomb potentials that focus on the pair interaction do not adequately treat many-body effects.     

Thermodynamic conditions of shock Hugoniot curves in hot dense matter

We use a self-consistent model combining the Quotidian Equation Of State and a nonrelativistic screened-hydrogenic model with ?-splitting to study the thermodynamic conditions encountered along shock Hugoniot curves. We focus on the maximum compression ratio reached on shock Hugoniot curves for simple elements that are solid at normal temperature and pressure. Electron shell effect and pressure ionization can have a strong impact on the results. Numerical calculations are presented and discussed. Some of the physical points addressed in this work could be tested experimentally when the National Ignition Facility laser in the USA attains full power or when the Laser Mégajoule in France comes online.     

Equation of state calculations for hot,dense matter at arbitrary densities and temperatures

Within the approximations of spherical lattice cell, central-field, and relativistic Fermi statistics, an algorithm with average atom model is presented to calculate the electronic energy levels and equation of state for hot and dense matter at arbitrary densities and temperatures. Choosing Zink's analytical potential as initial potential, we have solved the Dirac-Slater equation which satisfies the Weigner-Seitz boundary condition. The electronic energy bands are not taken into account. Taking energy level degeneracy as a continuous function of density, we have considered the pressure ionization effects for highly dense matter. Results for¹³Al atom are shown.     

Viscosity estimates of liquid metals and warm dense matter using the Yukawa reference system

A practical method of computing the viscosity of liquid metals and warm dense matter over wide ranges in parameters is proposed. The method is based on mapping the system of interest onto the Yukawa model, for which the viscosity is well known and can be written in a quasiuniversal form. Comparisons are made with quantum molecular dynamics results for compressed iron relevant to the Earth's core, experimental data for many liquid metals, and simulation results for dense deuterium relevant to inertial confinement fusion experiments. Finally, the dispersion and damping of ion-acoustic waves in warm dense matter are considered in this context.     

Resonant bound-free contributions to Thomson scattering of X-rays by warm dense matter

Recent calculations [Nilsen et al. arXiv:1212.5972] predict that contributions to the scattered photon spectrum from 3s and 3p bound states in chromium (Z = 24) at metallic density and T = 12 eV resonate below the respective bound-state thresholds. These resonances are shown to be closely related to continuum lowering, where 3d bound states in the free atom dissolve into a resonant l = 2 partial wave in the continuum. The resulting d-state resonance dominates contributions to the bound-free dynamic structure function, leading to the predicted resonances in the scattered X-ray spectrum. Similar resonant features are shown to occur in all elements in the periodic table between Ca and Mn (20 ≤ Z ≤ 25).     

Tungsten L transition line shapes and energy shifts resulting from ionization in warm dense matter

Spectra of the W L transitions in the energy range 8–12 keV from warm dense plasmas generated by the Naval Research Laboratory's Gamble II pulsed power machine were recorded by a newly developed high-resolution transmission-crystal X-ray spectrometer with ±2 eV accuracy. The discharges have up to 2 MV voltage, 0.5 MA current, and produce up to 2.4 MJ/cm^?3 energy density. The plasma-filled rod pinch (PFRP) diode produces a plasma with N_e ≈ 10²² cm^?3 and T_e ≈ 50 eV during the time of maximum X-ray emission. By analyzing the line shapes, it was determined that the Lβ₂ inner-shell transition from the 4d_5/2 level was shifted to higher energy by up to 23 eV relative to nearby Lβ transitions from n = 3 levels. In addition, the Lβ₂ transition was significantly broader and asymmetric compared to the n = 3 transitions. The energy shift of the Lβ₂ transition results from the ionization of electrons outside the 4d shell that perturbs the transition energies in the ions to higher values. The increased line width and asymmetry result from unresolved transitions from a range of ionization states up to +28. The ionization distribution was determined by comparison of the measured energy shifts and widths to calculated transition energies in W ions, and the ionization was correlated with Gamble discharge parameters such as the anode type and the high voltage delay time. This work demonstrates a new hard X-ray spectroscopic diagnostic technique for the direct measurement of the ionization distribution in warm dense plasmas of the heavy elements W through U that is independent of the other plasma parameters and does not require interpretation by hydrodynamic, atomic kinetics, and radiative simulation codes.     

Measurements of opacity and temperature of warm dense matter heated by focused soft X-ray laser irradiation

The transmission of plasma-based soft X-ray lasers through thin targets can be used to measure the target opacity. Measurements of warm dense matter transmission obtained using a focused 59 eV photon energy laser irradiation on thin targets of polyimide (C₂₂H₁₀N₂O₅) and aluminum are shown to produce simultaneous heating and probing enabling opacity and temperature measurements of warm dense matter. It is shown that the opacity of the warm dense matter considered in the experiments follows closely tabulated cold ‘room temperature’ opacities at temperatures below ～10 eV. Transmission measurements of thin iron targets which are highly opaque to the X-ray laser radiation are also presented.     

Equation of state studies of warm dense matter samples heated by laser produced proton beams

Heating of matter by proton beams produced by short pulse, laser-solid target interaction has been demonstrated over the last ten years by a number of workers. In the work described in this paper heating by a pulse of laser produced protons has been combined with high-resolution soft x-ray radiography to record the expansion of thin wire targets. Analysis of the radiographs yields material properties in the warm dense matter regime. These measurements imply initial temperatures in the experimental samples over a range from 14 eV up to 40 eV; the sample densities varied from solid to a tenth solid density. Assuming an adiabatic expansion after the initial proton heating phase isentropes of the aluminium sample material were inferred and compared to tabulated data from the SESAME equation of state library. The proton spectrum was also measured using calibrated magnetic spectrometers and radiochromic film. The accuracy of the technique used to infer material data is discussed along with possible future development.     

X-ray scattering from warm dense iron

We have carried out X-ray scattering experiments on iron foil samples that have been compressed and heated using laser-driven shocks created with the VULCAN laser system at the Rutherford-Appleton Laboratory. This is the highest Z element studied in such experiments so far and the first time scattering from warm dense iron has been reported. Because of the importance of iron in telluric planets, the work is relevant to studies of warm dense matter in planetary interiors. We report scattering results as well as shock breakout results that, in conjunction with hydrodynamic simulations, suggest the target has been compressed to a molten state at several 100 GPa pressure. Initial comparison with modelling suggests more work is needed to understand the structure factor of warm dense iron.     

Electrical conductivity of warm dense tungsten

The electrical conductivity of warm dense tungsten plasma has been investigated successfully by a linear mixture rule considering various interactions of electrons with electrons, atoms, and ions. The plasma composition is calculated by the nonideal Saha equation. The interesting regime for tungsten plasma spans from weakly coupled and nondegenerate regime to strongly coupled and partial degenerate state. The electrical conductivity calculated is in reasonable agreement with the exploding wire experiments and other theoretical models. The present result demonstrates that the theoretical model is valid for the electrical conductivity of tungsten plasma in the warm dense matter regime.     

Prospects of warm dense matter research at HiRadMat facility at CERN using 440 MeV SPS proton beam

In this paper we present numerical simulations of heating of a solid copper cylinder by the 440 GeV proton beam delivered by the Super Proton Synchrotron (SPS) at CERN. The beam is made of 288 proton bunches while each bunch comprises of 1.15·10¹¹ so that the total number of protons in the beam is about 1.3·10¹³. The bunch length is 0.5 ns while two neighboring bunches are separated by 25 ns so that the beam duration is 7.2 μs. Particle intensity distribution in the transverse direction is a Gaussian and the beam can be focused to a spot size with σ = 0.1 mm–1.0 mm. In this paper we present results using two values of σ, namely 0.2 mm and 0.5 mm, respectively. The target length is 1.5 m with a radius = 5 cm and is facially irradiated by the beam. The energy deposition code FLUKA and the two-dimensional hydrodynamic code BIG2 are employed using a suitable iteration time to simulate the hydrodynamic and the thermodynamic response of the target. The primary purpose of this work was to design fixed target experiments for the machine protection studies at the HiRadMat (High Radiation Materials) facility at CERN. However this work has shown that large samples of High Energy Density (HED) matter will be generated in such experiments which suggests an additional application of this facility. In the present paper we emphasize the possibility of doing HED physics experiments at the HiRadMat in the future.     

A fast method to generate collisional excitation cross-sections of highly charged ions in a hot dense matter

A method to compute collisional excitation cross-sections in jj-averaged configuration sets is presented in the framework of plane-wave Born approximation using Dirac–Hartree–Slater wave functions with appropriate low-energy corrections. When averaged into the ls configuration or hydrogenic superconfiguration sets, the results are found to compare with distorted wave calculations well within 30% on average. The cross-sections are averaged into hydrogenic cross-sections and fitted using the Gaunt factor formalism. We present analytic fit coefficients of Gaunt factors for 12 atoms of Z between 5 and 79 for hydrogenic transitions.     

Radiative shock solutions with grey nonequilibrium diffusion

This study describes a semi-analytic solution of planar radiative shock waves with a grey nonequilibrium diffusion radiation model. The solution may be used to verify radiation-hydrodynamics codes. Comparisons are made with the equilibrium diffusion solutions of Lowrie and Rauenzahn (Shock Waves 16(6):445–453, 2007). The solution also gives additional insight into the structure of radiative shocks. Previous work has assumed that the material temperature reaches its maximum at the post-shock state of the embedded hydrodynamic shock (Zel’dovich spike). We show that in many cases, the temperature may continue to increase after the hydrodynamic shock and reaches its maximum at the isothermal sonic point. Also, a temperature spike may exist even in the absence of an embedded hydrodynamic shock. We also derive an improved estimate for the maximum temperature.      

Electron scattering in hot/warm plasmas

Electrical and thermal conductivities are presented for aluminum, iron and copper plasmas at various temperatures, and for gold between 15,000 and 30,000 K. The calculations are based on the continuum wave functions computed in the potential of the temperature and density dependent self-consistent ‘average atom’ (AA) model of the plasma. The cross-sections are calculated by using the phase shifts of the continuum electron wave functions and also in the Born approximation. We show the combined effect of the thermal and radiative transport on the effective Rosseland mean opacities at temperatures from 1 to 1000 eV. Comparisons with low temperature experimental data are also presented.     

Free–free opacity in warm dense aluminum

We present calculations of the free–free opacity of warm, solid-density aluminum at photon energies between the plasma frequency at 15 eV and the L-edge at 73 eV, using both density functional theory combined with molecular dynamics and a semi-analytical model in the RPA framework which includes exciton contributions. As both the ion and electron temperature is increased from room temperature to 10 eV, we see a marked increase in the opacity. The effect is less pronounced if only the electron temperature is allowed to increase, while the lattice remains at room temperature. The physical significance of these increases is discussed in terms of intense light-matter interactions on both femtosecond and picosecond time scales.     

Radiative shock solutions in the equilibrium diffusion limit

A semi-analytic solution is described for planar radiative shock waves in the equilibrium diffusion (1−T) limit. The solution requires finding numerically the root of a polynomial and integrating a nonlinear ordinary differential equation. This solution may be used as a test problem to verify computer codes that use the equilibrium–diffusion radiation model, or for more advanced radiation models in the optically-thick limit. The structure of the shock profiles is also discussed, including new accurate estimates on the conditions for continuous solutions. We also discuss how the Zel’dovich spike may be estimated from the equilibrium diffusion solution. Finally, results from a computer code are shown to compare well with a semi-analytic solution.