Pith. sign in

REVIEW 3 major objections 4 minor 45 references

Relaxation pathways in X-ray Free Electron Laser heated Iron

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Simulations show that the K-beta spectrum of iron, excited by an X-ray free-electron laser, is sensitive to non-thermal ionization: changing the assumed M-shell collisional ionization cross-section shifts K-beta satellite lines by 5-7 eV…

desk verdict Competent simulation study with a plausible spectral fingerprint for M-shell CI in iron, but the headline 5–7 eV shifts rest on an uncalibrated factor-10 cross-section reduction. read the letter →

arxiv 2501.17000 v1 pith:37L4FZSN submitted 2025-01-28 physics.plasm-ph

Lucas Ansia , Pedro Velarde , Marta Fajardo , Gareth Williams This is my paper · ORCID
classification physics.plasm-ph
keywords X-rayfree-electronlasernon-thermalelectrondistributionscollisionalradiativemodelK-betaspectroscopyionizationM-shellironplasmapotentialdepression
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper seeks to show that after an X-ray free-electron laser pulse hits solid-density iron, the electrons knocked out by photoionization do not relax through an instantly thermalized distribution; instead, the dominant relaxation channel is collisional ionization of the outer M-shell electrons. Because those M-shell cross-sections are poorly constrained, the authors look for an observable spectral fingerprint of this non-thermal pathway. They simulate the coupled evolution of atomic states and the electron distribution, and find that the K-beta satellite lines are sensitive to the assumed M-shell collisional ionization cross-section: reducing it by a factor of ten produces 5-7 eV line shifts at high pump intensities. This sensitivity, if confirmed, turns K-beta spectra of iron into a diagnostic of non-thermal ionization and a way to refine cross-section models that are hard to compute near the continuum.

What carries the argument

The central machinery is BigBarT, a collisional-radiative model that couples atomic rate equations for super-configurations (ionic states distinguished by K, L, and M shell occupations) to a degenerate, isotropic Fokker-Planck equation for the electron distribution. The coupling happens through source terms that deposit photoionized and Auger electrons and through inelastic collision terms that transfer energy from hot electrons to bound states via collisional ionization and excitation. M-shell collisional ionization rates use a Coulomb-Born-exchange fitting for the differential cross-section, with Pauli blocking included, and the ionization potential depression is fixed at its initial Ecker-Kröll value. This machinery is what lets the paper track how non-thermal electrons change ionization pathways and how those changes translate into K-beta satellite intensities and line positions.

What would settle it

Measure time-resolved K-$\beta$ spectra of solid-density iron at an XFEL with a 30 fs pulse at $10^{19}$-$10^{20}$ W/$cm^{2}$ and pump energies near 7.2, 7.31, and 8.1 keV, comparing the first four K-$\beta$ L-satellites to instantly-thermalized simulations. If the satellite lines do not show the predicted 5-7 eV blue shifts (or, for the reduced-cross-section case, the predicted enhancement and shift), the claim that K-$\beta$ spectra are sensitive to non-thermal M-shell ionization in this way is contradicted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that M-shell collisional ionization is the primary thermalization channel for non-thermal electrons in XFEL-heated solid iron, and that the K-$\beta$ satellite structure records this channel. In the first tens of femtoseconds the thermal bulk of the electron distribution sits below the M-shell ionization threshold, so only the non-thermal tail can ionize, producing an order-of-magnitude higher collisional ionization rate than an instantly thermalized distribution and creating plateaus in the electron spectrum. When the M-shell collisional ionization cross-section ($\sigma_{CI-M}$) is reduced by a factor of ten, non-thermal electrons accumulate, the plasma evolution slows, satellite line intensities change, and the K-$\beta$ lines shift by 5-7 eV, a shift the authors argue would be experimentally observable at current XFEL facilities. The final equilibrium state is the same in all cases because the inverse rates obey Fowler's relation; the spectral differences are purely dynamical.

Load-bearing premise

The load-bearing assumption is that the ionization potential depression stays fixed at its initial Ecker-Kröll value for the entire pulse; if the depression changes as the plasma ionizes, the non-thermal population fractions and the predicted satellite shifts could change substantially.

Editorial extensions

If this is right

  • At high XFEL intensities (10^19-10^20 W/cm^2), non-thermal electrons enhance K-beta satellite line strengths relative to instantly-thermalized models, and the enhancement becomes more pronounced when the M-shell collisional ionization cross-section is reduced by a factor of ten.
  • The predicted 5-7 eV shifts in the K-beta satellites are large enough to be resolved by existing X-ray spectrometers, so time-resolved K-beta spectra can serve as a direct probe of non-thermal relaxation in solid-density iron.
  • Because the final equilibrium states are identical for different cross-section assumptions, the spectral shifts are genuinely dynamical signatures: comparing time-resolved emission, rather than only the final state, is required to extract cross-section information.
  • The approach extends beyond iron to other 3d transition metals and higher-Z materials, where near-continuum collisional cross-sections are similarly uncertain and where non-thermal effects have not yet been studied with self-consistent models.
  • Non-thermal electrons dominate collisional ionization during the early femtoseconds of the pulse, which means that relaxation timescales and the initial heating of the thermal bulk are set by the inelastic M-shell channel rather than by elastic electron-electron collisions alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct experimental test would be to measure the K-beta satellite positions of solid-density iron at 10^19-10^20 W/cm^2 with pump energies around 7.2, 7.31, and 8.1 keV: if the predicted 5-7 eV shifts are observed, the measured spectra could be inverted to infer the M-shell collisional ionization cross-section without needing a full kinetic reconstruction.
  • The fixed ionization potential depression is the main modeling uncertainty; a natural extension would be to rerun the same simulations with a dynamically evolving IPD (e.g., from a Stewart-Pyatt or a density-functional-theory-based model) to see whether the 5-7 eV shifts survive or change size, which would tell experimenters how confident to be in interpreting the shifts as cross-section signature
  • The paper's delta parameter measures deviations mainly at low energy, so the early low-energy features it identifies (from M-shell three-body recombination and LM M Auger processes) could also appear in emitted electron spectra or XUV emission, offering an independent, complementary observable to the K-beta shifts.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript presents simulations of solid-density iron heated by an X-ray free-electron laser using the BigBarT model, which couples collisional-radiative atomic kinetics with a degenerate Fokker-Planck treatment of the isotropic electron distribution. The authors identify M-shell collisional ionization as the dominant inelastic relaxation channel, show that non-thermal electron populations enhance early-time ionization rates, and compare thermal and non-thermal Kβ spectra. The central claim is that reducing the M-shell collisional ionization cross section by a factor of 10 produces 5–7 eV shifts in Kβ satellite lines that would be experimentally observable, offering a route to refining M-shell cross sections.

Significance. If the sensitivity claim is quantitatively robust, the paper would give XFEL experiments a new spectroscopic handle on M-shell collisional ionization cross sections in transition metals, a regime where theoretical cross sections are uncertain. The model includes Pauli blocking and self-consistent continuum evolution, and the forward-modeling framework is appropriate for proposing such diagnostics. The paper is clearly written and the simulations are internally consistent, but the main experimental-observability claim currently rests on an uncalibrated cross-section perturbation and on idealized spectra. These issues are addressable with additional sensitivity analysis, so the work has clear potential after revision.

major comments (3)
  1. [Section IV.B, Figures 6 and 7] The 5–7 eV shifts are obtained by reducing σ_CI-M by an arbitrary factor of 10 from a single Fontes-based baseline. Section IV.A states that with the baseline cross sections there are no appreciable line shifts, so the headline signature appears only under this ad hoc perturbation. The authors should calibrate the perturbation against the actual spread among independent M-shell CI cross-section models (for example, FAC or HULLAC distorted-wave calculations or other published fits) and report the resulting range of shifts. Without this, the claim that Kβ spectra are sensitive to variations in σ_CI-M is a statement about one artificially chosen variation, not about physically realistic model uncertainty.
  2. [Section IV.B and Section V] The claim that the shifts would be experimentally observable is not supported by a spectral-resolution analysis. The spectra in Figures 4–7 are synthetic line intensities without convolution by an instrument response function or inclusion of Doppler, Stark, or natural broadening. For a quantitative comparison with XFEL experiments, the authors should estimate the expected line width and the relevant spectrometer resolving power (for example, E/ΔE for the instruments cited in Refs. [43,44]) and show that a 5–7 eV shift at 7–7.4 keV is resolvable above line widths and noise. This comparison is load-bearing for the proposed diagnostic.
  3. [Section II, fixed Ecker-Kröll IPD] The ionization potential depression is computed once with the Ecker-Kröll model from initial conditions and held constant, as stated in Section II. Because M-shell ionization thresholds are close to the continuum and IPD may shift by eV scales as the plasma heats and ionizes, the fixed-IPD approximation could shift M-shell satellite populations and either mimic or mask the cross-section signature. The authors should quantify the sensitivity of the predicted spectra to the IPD model and to a time-varying IPD, even with a simple estimate, or provide a bound on the resulting spectral shifts.
minor comments (4)
  1. [Section II, after Eq. (7)] The text refers to the "Rosseblunt potentials"; this should be "Rosenbluth potentials".
  2. [Figure 4 caption] The caption lists "L8 ∼ 7130, L7 ∼ 7240, L6 ∼ 7130, L5 ∼ 7420"; L6 and L8 appear at the same energy, which looks like a typographical error that should be corrected.
  3. [Section IV] The sentence "Spectra is focused in the first 4 Kβ L-satellites" should be reworded to "The spectra are focused on the first four Kβ L-satellites."
  4. [Section II and Eq. (9)] The definition of feq(ε) in the δ parameter is not fully explicit; the authors should state precisely how the temperature and chemical potential of the equivalent equilibrium distribution are obtained from the total energy and density.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral sensitivity result is a forward-model output, not an input restatement.

full rationale

The paper is a forward-modeling sensitivity study using the previously published BigBarT code. Its central claim, that Kβ spectra exhibit 5–7 eV shifts when the M-shell collisional ionization cross section is reduced by a factor of 10, is a computed consequence of the coupled Fokker-Planck and atomic-kinetics equations, not a quantity inserted into the model. No parameter is fitted to the target spectra, and the line shifts, population changes, and rate convergences are emergent outputs. Self-citations to BigBarT [16,17] and to related van den Berg et al. work [11] are code-development and experimental-context references; they do not supply a load-bearing uniqueness or existence argument, and the cited code has prior external use. The fixed Ecker-Kröll IPD and the uncalibrated 10x cross-section perturbation are modeling and validity limitations, not circular steps: they affect the credibility of the prediction but do not make the prediction equivalent to its input. Accordingly, no circular step meeting the quoted-evidence standard is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on a chain of modeling assumptions inherited from the BigBarT code and published atomic data. The paper introduces no new free parameters fit to data, but it also does not validate these assumptions for iron specifically, so the predicted spectral signatures are only as reliable as the underlying model choices.

assumptions (6)
  • domain assumption BigBarT's degenerate Fokker-Planck equation with Rosenbluth potentials correctly describes electron thermalization in solid-density iron.
    Invoked in Section II as the framework for evolving the electron distribution; if this kinetic description is inaccurate, all non-thermal predictions change.
  • domain assumption The Ecker-Kröll ionization potential depression model, evaluated at initial conditions and held constant, is adequate for this plasma.
    Section II states IPD is held constant because dynamic IPD is an open question; the sensitivity of results to this choice is not tested.
  • domain assumption The Fontes et al. Coulomb-Born exchange fitting gives reliable M-shell collisional ionization cross sections at solid density with IPD scaling.
    Section II states CI cross sections use this fitting; the entire sensitivity analysis varies these cross sections, so the baseline values matter.
  • domain assumption Grouping atomic states by super-configurations (KLM shells only) preserves the essential physics for K-beta satellite spectra.
    Section II describes the super-configuration approach; the coarse grouping may affect satellite intensities and shifts.
  • domain assumption The Shaffer-Starrett Coulomb logarithm, with a switch to the classical form for non-degenerate conditions, is appropriate.
    Section II adopts this model; the paper states this factor does not influence conclusions, but does not show the evidence.
  • domain assumption The initial configuration 288, obtained from the degenerate Saha equation, represents the unheated solid iron.
    Section II sets initial conditions; different initial charge states could affect the early time dynamics and satellite populations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Relaxation pathways in X-ray Free Electron Laser heated Iron." pith.science (2026). https://pith.science/paper/37L4FZSN

@misc{pith2026250117000,
  author       = {Pith},
  title        = {Pith review of: Relaxation pathways in X-ray Free Electron Laser heated Iron},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37L4FZSN}},
  note         = {Machine review of arXiv:2501.17000}
}
abstract

Non-thermal photo-ionized plasmas are now established in the laboratory, and require models that treat the atomic processes and electron distribution self-consistently. We investigate the effects of inelastic thermalization in iron under intense X-ray irradiation using the atomic model BigBarT, suited for the self-consistent evolution of the electron continuum, including degeneracy effects. Our study focuses particularly on collisional $M$-shell ionization, which we identify as the dominant relaxation process of the non-thermal electrons. We show that $M$-shell satellite intensities are sensitive to non-thermal ionization, providing a potential method to refine collisional cross sections that are otherwise difficult to compute due to their proximity to the continuum and the associated plasma screening effects.

Figures

Figures reproduced from arXiv: 2501.17000 by the authors.

Figure 1
Figure 1. FIG. 1. Electron distributions at the pulse peak (60 fs) for varying collisional ionization cross sections of the M-shell. The [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Percentage of electrons out of equilibrium ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Temporal evolution of the total collisional ioniza [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of thermal and non-thermal spectra [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of thermal and non-thermal spectra at [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of non-thermal simulation spectra with [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of the evolution of ionic states as a function of time for reduced and regular cross sections, with an [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 37 canonical work pages

  1. [1]

    E. D. Filippov, M. Khan, A. Tentori, P. Gajdos, A. S. Martynenko, R. Dudzak, P. Koester, G. Zeraouli, D. Mancelli, F. Baffigi, L. A. Gizzi, S. A. Pikuz, P. Nicola ¨ ı, N. C. Woolsey, R. Fedosejevs, M. Krus, L. Juha, D. Batani, O. Renner, and G. Cristoforetti, Matter and Radiation at Extremes 8, 10.1063/5.0157168 (2023)

    work page doi:10.1063/5.0157168 2023

  2. [2]

    S. H. Cao, D. Patel, A. Lees, C. Stoeckl, M. J. Rosen- berg, V. Gopalaswamy, H. Wen, H. Huang, A. Shvy- dky, R. Betti, and C. Ren, Physical Review E 106, 10.1103/physreve.106.055214 (2022)

    work page doi:10.1103/physreve.106.055214 2022

  3. [3]

    Cristoforetti, F

    G. Cristoforetti, F. Baffigi, D. Batani, R. Dudzak, R. Fe- dosejevs, E. D. Filippov, P. Gajdos, L. Juha, M. Khan, P. Koester, M. Krus, D. Mancelli, A. S. Martynenko, P. Nicolai, S. A. Pikuz, O. Renner, A. Tentori, L. Volpe, N. Woolsey, G. Zeraouli, and L. A. Gizzi, Scientific Re- ports 13, 10.1038/s41598-023-46189-7 (2023)

    work page doi:10.1038/s41598-023-46189-7 2023

  4. [4]

    L´ evy, P

    A. L´ evy, P. Audebert, R. Shepherd, J. Dunn, M. Cam- marata, O. Ciricosta, F. Deneuville, F. Dorchies, M. Fa- jardo, C. Fourment, D. Fritz, J. Fuchs, J. Gaudin, M. Gauthier, A. Graf, H. J. Lee, H. Lemke, B. Nagler, J. Park, O. Peyrusse, A. B. Steel, S. M. Vinko, J. S. Wark, G. O. Williams, D. Zhu, and R. W. Lee, Physics of Plasmas 22, 10.1063/1.4916103 (2015)

    doi:10.1063/1.4916103 2015

  5. [5]

    Cho, Applied Sciences 9, 4812 (2019)

    B. Cho, Applied Sciences 9, 4812 (2019)

    work page 2019

  6. [6]

    S. H. Glenzer, L. B. Fletcher, E. Galtier, B. Nagler, R. Alonso-Mori, B. Barbrel, S. B. Brown, D. A. Chap- man, Z. Chen, C. B. Curry, F. Fiuza, E. Gamboa, M. Gauthier, D. O. Gericke, A. Gleason, S. Goede, E. Granados, P. Heimann, J. Kim, D. Kraus, M. J. MacDonald, A. J. Mackinnon, R. Mishra, A. Ravasio, C. Roedel, P. Sperling, W. Schumaker, Y. Y. Tsui, J....

    work page 2016

  7. [7]

    Ranjan, K

    D. Ranjan, K. Ramakrishna, K. Voigt, O. S. Humphries, B. Heuser, M. G. Stevenson, J. L¨ utgert, Z. He, C. Qu, S. Schumacher, P. T. May, A. Amouretti, K. Appel, E. Brambrink, V. Cerantola, D. Chekrygina, L. B. Fletcher, S. G¨ ode, M. Harmand, N. J. Hartley, S. P. Hau- Riege, M. Makita, A. Pelka, A. K. Schuster, M. ˇSm ´ ıd, T. Toncian, M. Zhang, T. R. Pres...

    doi:10.1063/5.0146416 2023

  8. [8]

    Modern Methods in Collisional- Radiative Modeling of Plasmas

    Y. Ralchenko, ed., “Modern Methods in Collisional- Radiative Modeling of Plasmas” (Springer International Publishing, 2016)

    work page 2016

Show all 45 references
  1. [9]

    Chung, M

    H.-K. Chung, M. Chen, and R. Lee, High Energy Density Physics 3, 57 (2007)

    2007

  2. [10]

    Ciricosta, S

    O. Ciricosta, S. M. Vinko, H.-K. Chung, C. Jackson, R. W. Lee, T. R. Preston, D. S. Rackstraw, and J. S. Wark, Physics of Plasmas 23, 022707 (2016)

    2016

  3. [11]

    van den Berg, E

    Q. van den Berg, E. Fernandez-Tello, T. Burian, J. Chalupsk´ y, H.-K. Chung, O. Ciricosta, G. Dakovski, V. H´ ajkov´ a, P. Hollebon, L. Juha, J. Krzywinski, R. Lee, M. Minitti, T. Preston, A. de la Varga, V. Vozda, U. Zas- trau, J. Wark, P. Velarde, and S. Vinko, Physical Revi...

    2018 doi

  4. [12]

    Bretagne, J

    J. Bretagne, J. Godart, and V. Puech, Journal of Physics D: Applied Physics 15, 2205–2225 (1982)

    1982

  5. [13]

    Abdallah, G

    J. Abdallah, G. Csanak, Y. Fukuda, Y. Akahane, M. Aoyama, N. Inoue, H. Ueda, K. Yamakawa, A. Y. Faenov, A. I. Magunov, T. A. Pikuz, and I. Y. Skobelev, Physical Review A 68, 10.1103/phys- reva.68.063201 (2003)

    2003 doi

  6. [14]

    Abdallah and J

    J. Abdallah and J. Colgan, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 035701 (2012)

    2012

  7. [15]

    Abdallah, J

    J. Abdallah, J. Colgan, and N. Rohringer, Journal of Physics B: Atomic, Molecular and Optical Physics 46, 235004 (2013)

    2013

  8. [16]

    A. G. de la Varga, P. Velarde, F. de Gaufridy, D. Portillo, M. Cotelo, A. Barbas, A. Gonz´ alez, and P. Zeitoun, High Energy Density Physics 9, 542 (2013)

    2013

  9. [17]

    Garc ´ ıa de la Varga,Modelizaci´ on de las propiedades radiativas e interacci´ on de l´ aseres ultraintensos con la materia, Ph.D

    A. Garc ´ ıa de la Varga,Modelizaci´ on de las propiedades radiativas e interacci´ on de l´ aseres ultraintensos con la materia, Ph.D. thesis, Universidad Polit´ ecnica de Madrid - University Library (2014)

    2014

  10. [18]

    S. Ren, Y. Shi, Q. Y. van den Berg, M. F. Kasim, H.-K. Chung, E. V. Fernandez-Tello, P. Velarde, J. S. Wark, and S. M. Vinko, Communications Physics 6, 10.1038/s42005-023-01216-x (2023)

    2023 doi

  11. [19]

    Y. Shi, S. Ren, H. kyung Chung, J. S. Wark, and S. M. Vinko, Exploring relaxation dynamics in warm dense plasmas by tailoring non-thermal electron distribu- tions with a free electron laser (2024), arXiv:2405.04240 [physics.plasm-ph]

    2024 arXiv

  12. [20]

    C. Gao, Y. Li, F. Jin, J. Zeng, and J. Yuan, Physical Review E 110, 10.1103/physreve.110.015201 (2024)

    2024 doi

  13. [21]

    Ziaja, V

    B. Ziaja, V. Saxena, S.-K. Son, N. Medvedev, B. Barbrel, B. Woloncewicz, and M. Stransky, Physical Review E93, 10.1103/physreve.93.053210 (2016)

    2016 doi

  14. [22]

    Ziaja, J

    B. Ziaja, J. J. Bekx, M. Masek, N. Medvedev, V. Lipp, V. Saxena, and M. Stransky, Philosophical Transactions of the Royal Society A: Mathematical, Physical and En- gineering Sciences 381, 10.1098/rsta.2022.0216 (2023)

    2023

  15. [23]

    Ziaja, J

    B. Ziaja, J. J. Bekx, M. Masek, N. Medvedev, P. Piekarz, V. Saxena, M. Stransky, and S. Toleikis, The European Physical Journal D 75, 10.1140/epjd/s10053-021-00235-z (2021)

    2021 doi

  16. [24]

    Mercadier, A

    L. Mercadier, A. Benediktovitch, S. Krusic, J. J. Kas, J. Schlappa, M. Ag ˚ aker, R. Carley, G. Fazio, N. Gerasimova, Y. Y. Kim, L. Le Guyader, G. Mer- curio, S. Parchenko, J. J. Rehr, J.-E. Rubensson, S. Serkez, M. Stransky, M. Teichmann, Z. Yin, M. Zitnik, 10 A. Scherz, B. Z...

    2024

  17. [25]

    H. P. Le, M. Sherlock, and H. A. Scott, Physical Review E 100, 10.1103/physreve.100.013202 (2019)

    2019 doi

  18. [26]

    G. O. Williams, H.-K. Chung, S. K¨ unzel, V. Hilbert, U. Zastrau, H. Scott, S. Daboussi, B. Iwan, A. I. Gon- zalez, W. Boutu, H. J. Lee, B. Nagler, E. Granados, E. Galtier, P. Heimann, B. Barbrel, R. W. Lee, B. I. Cho, P. Renaudin, H. Merdji, P. Zeitoun, and M. Fa- jardo, Phys...

    2019 doi

  19. [27]

    Tallents, High Energy Density Physics 20, 9–16 (2016)

    G. Tallents, High Energy Density Physics 20, 9–16 (2016)

    2016

  20. [28]

    Gu, Canadian Journal of Physics 86, 675 (2008)

    M. Gu, Canadian Journal of Physics 86, 675 (2008)

    2008

  21. [29]

    Chung, R

    H.-K. Chung, R. Lee, and M. Chen, High Energy Density Physics 3, 342–352 (2007)

    2007

  22. [30]

    R. E. H. Clark, J. Abdallah, J., and J. B. Mann, The Astrophysical Journal 381, 597 (1991)

    1991

  23. [31]

    C. J. Fontes, D. H. Sampson, and H. L. Zhang, Physical Review A 48, 1975–1982 (1993)

    1993

  24. [32]

    Oxenius, Kinetic Theory of Particles and Photons (Springer Berlin Heidelberg, 1986)

    J. Oxenius, Kinetic Theory of Particles and Photons (Springer Berlin Heidelberg, 1986)

    1986

  25. [33]

    Shkarofsky, Johnston, and Bachynski, eds., The Particle Kinetics of Plasmas (Addison-Wesley, 1996)

    1996

  26. [34]

    Nicholson, ed., Introduction to plasma theory (Wiley, 1983)

    D. Nicholson, ed., Introduction to plasma theory (Wiley, 1983)

    1983

  27. [35]

    The numerical solution is carried out using the scheme presented by Bobylev et al

    associated with friction and diffusion, defined as: H(v, t) = 2 · ∇v Z dv′ f (v′, t) |v − v′| , (6) G(v, t) = ∇v∇v Z dv′|v − v′|f (v′, t), (7) here ∇v∇v and ∇v∇v : are the tensor generalization of the gradient and divergence, respectively [33]. The numerical solution is carrie...

  28. [36]

    M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd, Physical Review 107, 1–6 (1957)

    1957

  29. [37]

    Bobylev and V

    A. Bobylev and V. Chuyanov, USSR Computational Mathematics and Mathematical Physics 16, 121–130 (1976)

    1976

  30. [38]

    Tzoufras, A

    M. Tzoufras, A. Tableman, F. S. Tsung, W. B. Mori, and A. R. Bell, Physics of Plasmas 20, 056303 (2013)

    2013

  31. [39]

    S. R. Brown and M. G. Haines, Journal of Plasma Physics 58, 577–600 (1997)

    1997

  32. [40]

    D. J. Gardner, D. R. Reynolds, C. S. Woodward, and C. J. Balos, ACM Transactions on Mathematical Soft- ware (TOMS) 10.1145/3539801 (2022)

    2022 doi

  33. [41]

    A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, ACM Transactions on Mathematical Software (TOMS)31, 363 (2005)

    2005

  34. [42]

    Ecker and W

    G. Ecker and W. Kr¨ oll, Physics of Fluids6, 62 (1963)

    1963

  35. [43]

    N. R. Shaffer and C. E. Starrett, Physical Review E 101, 10.1103/physreve.101.013208 (2020)

    2020 doi

  36. [44]

    ˇSmıd, O

    M. ˇSmıd, O. Humphries, C. Baehtz, E. Brambrink, T. Burian, M. S. Cho, T. E. Cowan, L. Gaus, M. F. Gu, V. H´ ajkov´ a, L. Juha, Z. Konopkova, H. P. Le, M. Makita, X. Pan, T. Preston, A. Schropp, H. A. Scott, R. ˇStefanıkov´ a, J. Vorberger, W. Wang, U. Zastrau, and K. Falk, Pl...

    2024

  37. [45]

    H. J. Lee, S. Vinko, O. Humphries, et al. , Driving iron plasmas to stellar core conditions using extreme x-ray radiation (2023)

    2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.