Session BM10: Mini-Conference: Proton Transport in HED I: Stopping Power

Introduction

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A Review of Stopping Power Theory and Experimental Tests

Stopping power is of direct relevance to the understanding of energy deposition of charged particles products in a fusion burning plasma and charged particle heating and diagnostics as well as being a probe of fundamental collision physics in a many-body medium. The different modeling schemes, from linear response theory and binary scattering theory to molecular dynamics and density functional theory, will be discussed and compared in the context of recent experiments. Several unitless parameters will be used to emphasize regions of validity of different models and point to regimes where experiments can most contribute.   

A status update on MIT's studies of ion-stopping power in Warm Dense Matter (WDM) plasmas

A platform has been developed and extensively used to accurately measure ion-stopping power in Warm Dense Matter (WDM) plasmas at conditions characterized by x-ray Thomson scattering at the OMEGA. A cylindrical geometry has been used to allow charged-particle be transported along the symmetry axis of the WDM plasma. Either a solid-density beryllium, boron or carbon cylinder was isochorically heated by L-shell x-ray emission generated on the outside of the cylinder to temperatures up to about 30 eV, corresponding to moderately-coupled (Γ ~ 0.3) and moderately-degenerate (θ ~ 2) WDM conditions. The results from these experiments illustrate an increase energy loss in WDM relative to cold matter, consistent with a reduced mean ionization potential, which is well-described by ion-stopping-power models based on an ad-hoc treatment of free and bound electrons, as well as the average-atom local-density approximation. With this experimental platform, the insignificance of electromagnetic fields around the target was demonstrated. Going forward, the plan is to build on these results and use lower-velocity ions for studies of WDM ion-stopping power closer to the Bragg peak.   

Measurements of Proton Stopping Power in Warm Dense Matter approaching the Bragg Peak

The transport of protons through plasma is a highly active field of research due to its myriad applications in high energy density science. A crucial accounting of proton transport is the energy loss rate dε/dx, or stopping power, that comes with propagation in some medium. In the gray space between condensed matter and plasma, warm dense matter (WDM) has proven to be one such perplexing medium, evading accurate modeling from both classical and quantum treatments. Models of proton stopping power in WDM generally agree for fast projectiles (v_p » v_th,e), but when compounded with the Bragg peak regime (v_p ≈ v_th,e) in which protons lose most of their energy quickly, model predictions vary by as much as 30-40%. The first experimental measurements of proton stopping power in WDM approaching the Bragg peak (v_p / v_th,e ~ 3-10) were made on the VEGA-II laser [1]. Here, we report on platform advancements made on the CSU ALEPH laser, where 500±10 keV protons with short bunch duration (<200 ps) were generated [2] and directed toward a laser-heated WDM sample. We present a preliminary analysis of the stopping power measurements along with WDM characterization through streaked optical pyrometry.

[1] S. Malko et al., Nature Communications 13.1 (2022): 2893.

[2] J. I. Apiñaniz et al., Scientific Reports 11.1 (2021): 6881.    

Benchmark calculations of electronic stopping power in warm dense matter using time-dependent density functional theory

Scarcity of experimental data in the warm dense regime makes first-principles calculations particularly valuable for constraining tabulated data and characterizing limitations of more efficient models. Real-time time-dependent density functional theory (TDDFT), which can capture mean-field excited electron dynamics in extended systems, ranks among the most accurate but computationally intensive methods of predicting electronic stopping powers. However, TDDFT calculations still involve various choices and approximations contributing to uncertainties in computed values. For the case of alpha particle stopping in warm dense hydrogen, we cross-benchmark different TDDFT codes and methodologies, demonstrate reproducibility, and scrutinize best practices. We then consider effects that may challenge more approximate stopping power models, including nonlinear behavior manifesting as deviations from Z² scaling between proton and alpha stopping in carbon and aluminum and the validity of additivity laws for proton stopping in carbon-deuterium and carbon-hydrogen mixtures. This work takes important steps toward reducing uncertainties in first-principles stopping power predictions and improving the accuracy of more efficient treatments.   

Calculation of Proton Stopping Powers in Warm Dense Matter using the Mean Force Kinetic Theory

The averaging done to move from a kinetic to hydrodynamic description of a plasma can lead to loss of information. One key energy transfer and heating mechanism lost in the averaging is the stopping power. This refers to the drag force and resulting energy transfer experienced by a high energy charged particle as it interacts with a surrounding plasma. This can be especially important for fusion and other high energy density physics experiments where energy balance in the system is important. We present a method based on the Mean Force Kinetic Theory (MFKT) [S. D. Baalrud and J. Daligault, Phys. Plasmas 26, 082106 (2019)], to calculate the free-free contribution to proton stopping powers in a Warm Dense Matter system. The MFKT is based on an expansion about equilibrium instead of about the strength of correlations and extends plasma theory into the strongly coupled regime. In order to extend the MFKT to Warm Dense Matter, degenerate electrons must be treated separately. Their degenerate screening is modeled through the potential of mean force calculated using an Average Atom and Quantum Hyper-Netted Chain model [C. E. Starrett and Saumon, HEDP 10, 35-42 (2014)]. Additionally, collisions involving the degenerate electrons are treated quantum mechanically to calculate the correct scattering cross-sections.   

Energy Deposition in Proton Fast Ignition and HEDP Plasmas Using an eRPA-LDA Electronic Stopping Model, Including Inflight and Secondary Fusion Reactions

We report on a new wide range electronic stopping power model that extends the random phase approximation (RPA) dielectric response formalism to include a strong collision correction based on the binary collision theory for k>kmax, a static local field correction, an electron binding energy correction, and the Barkas effect. This eRPA model, when used with the local density approximation (LDA) calculated in an average atom model using the Flexible Atomic Code (FAC), produces proton stopping powers in cold targets that are in close agreement with experiments across the periodic table (PSTAR database), including high-Z elements, such as gold. We also report on validation of ion stopping powers in warm dense plasmas compared with published data. Our model is valid for classical and degenerate plasmas, and we report on its use for proton and ion transport in HEDP plasmas. We also describe calculations of energy deposition for proton fast ignition that include the impact of inflight beam fusion reactions and secondary reactions from fuel ions elastically scattered by beam proton (both Rutherford and nuclear elastic). This effect is particularly significant for proton fast ignition of p-¹¹B fuels.