pith. machine review for the scientific record. sign in

arxiv: 2605.05614 · v2 · submitted 2026-05-07 · ⚛️ physics.plasm-ph

Recognition: no theorem link

Proton probing measurements of filamentary electromagnetic structure in laser ablation of solids

D. H. Barnak, J. Peebles, J. R. Davies, P. V. Heuer, V. Y. Zhang

Pith reviewed 2026-05-12 05:07 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords proton radiographylaser ablationWeibel instabilityfilamentary electromagnetic fieldsplasma expansionlaser-plasma interactionsOMEGA EP experiments
0
0 comments X

The pith

Laser ablation experiments show filamentary electromagnetic fields grow mainly from a secondary instability following the expansion-driven Weibel instability.

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

Proton radiography in simplified planar laser ablation experiments on the OMEGA EP laser reveals filamentary structures matching those in spherical implosions. By systematically varying laser energy, target material and atomic number Z, target size, pulse shape, and intensity, the measurements demonstrate that field growth depends most strongly on laser energy and target Z. The quantitative data supports the conclusion that these fields arise from a secondary instability that develops as a direct consequence of the expansion-driven Weibel instability. This identification matters because the fields extend through the corona, absorb laser energy, and can alter the overall dynamics of laser-target coupling.

Core claim

The central claim is that the anomalous filamentary structures observed in proton radiographs of laser-ablated solids are produced by electric and magnetic fields generated by a secondary instability that arises from the expansion-driven Weibel instability, and that the growth of these features is dominated by incident laser energy and target atomic number Z rather than other parameters.

What carries the argument

The expansion-driven Weibel instability together with its secondary instability, which together generate the filamentary electromagnetic fields detected through dual-axis proton radiography and optical probing.

If this is right

  • Field growth increases with higher incident laser energy.
  • Higher atomic number targets produce stronger development of the filamentary structures.
  • The fields act as an energy sink and extend throughout the ablation corona.
  • Pulse shape and intensity variations have secondary effects on growth compared with energy and Z.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar filamentary fields observed in spherical implosions could be reduced by selecting lower-Z target materials.
  • Inertial confinement fusion modeling should incorporate this secondary instability to predict energy coupling losses more accurately.
  • Planar proton radiography offers a controlled platform to isolate and measure Weibel-related instabilities without spherical geometry complications.

Load-bearing premise

The filamentary structures in the proton radiographs are produced by the electric or magnetic fields of the proposed secondary Weibel instability rather than by other hydrodynamic or laser-plasma effects not excluded by the parameter variations.

What would settle it

An experiment using a low-Z target or pulse shape expected to suppress the expansion-driven Weibel instability that nevertheless produces identical filamentary radiographic features would falsify the claim that the secondary instability is the primary driver.

Figures

Figures reproduced from arXiv: 2605.05614 by D. H. Barnak, J. Peebles, J. R. Davies, P. V. Heuer, V. Y. Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. Proton radiograph of a spherical implosion shot on view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) An experiment conducted by L. Gao view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Setup for the majority of experiments: a variable view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Face-on (a) and side-on (b) radiography view of a Cu view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Synthetic proton radiographs of lines of current at view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Post processing of radiochromic film scans: (a) A film view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Side-on radiography view of a CH target 2.2 ns view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Processed radiographs from a variety of shots. All shots shown here used a 2 ns drive pulse. (a-d): A time series of view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The integrated FFT power spectrum in modes of an view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Sub-aperture Backscatter (SABS) measurements view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Radiograph taken from a gold target shot at 2.2 ns. view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Integrated FFT data from figure 9 plotted as a view at source ↗
read the original abstract

Proton radiography of laser direct-drive spherical implosions has shown anomalous structures that correspond to strong electric or magnetic fields extending throughout the corona. These fields have the ability to affect laser-target interactions and act as an energy sink. To better understand the these fields, simplified experiments were conducted in planar geometry on the OMEGA EP laser at the Laboratory for Laser Energetics. Varying target material, target size, pulse shape, and intensity, and measured the field structure using dual-axis proton radiography and a 4w probe. Proton radiographs were analyzed and quantitatively demonstrate that the growth of these features is dominated by laser energy and target Z. The data strongly supports that a secondary instability as a consequence of the expansion driven Weibel instability in these interactions is the primary driver for these fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports proton radiography experiments on planar laser ablation targets using the OMEGA EP laser. By varying target material (Z), size, pulse shape, and intensity, and employing dual-axis proton radiography plus a 4ω optical probe, the authors analyze filamentary structures in the radiographs. They claim that quantitative analysis demonstrates the growth of these electromagnetic field structures is dominated by laser energy and target Z, and interpret the data as strong support for a secondary instability arising from the expansion-driven Weibel instability as the primary driver of the observed fields.

Significance. If the mechanism attribution holds, the work would contribute useful experimental constraints on filamentary field generation in laser-plasma interactions, with potential relevance to energy coupling and instability seeding in inertial confinement fusion. The parameter-variation approach to identify scalings with energy and Z is a positive feature, though the manuscript does not yet provide the level of mechanistic isolation (e.g., via simulations or additional diagnostics) needed to make the interpretation definitive.

major comments (2)
  1. [Abstract] Abstract: The statement that 'quantitative analysis of radiographs demonstrates dominance by laser energy and target Z' is presented without error bars, data exclusion criteria, statistical measures, or direct comparison to hydrodynamic or laser-imprint simulations. This makes it impossible to assess whether the observed Z and energy scalings uniquely support the Weibel secondary instability over competing Z-dependent plasma or ablation-front effects.
  2. [Abstract] Abstract and results discussion: The central claim that the filamentary proton-radiograph features arise specifically from E/B fields of the expansion-driven Weibel secondary instability is not isolated from alternatives. Parameter variations show Z and energy dependence, but no wavelength matching, polarization signatures from dual-axis imaging, or comparison to Weibel dispersion relations are reported to exclude hydrodynamic modulations, laser imprint, or other Z-dependent effects.
minor comments (2)
  1. [Abstract] Abstract contains a grammatical error: 'To better understand the these fields' should read 'these fields'.
  2. [Abstract] The abstract refers to 'simplified experiments... in planar geometry' but does not specify how the planar results map quantitatively to the spherical implosion context mentioned in the opening sentence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We have addressed the concerns regarding the abstract's presentation of quantitative claims and the isolation of the proposed mechanism from alternatives. Below we respond point by point, indicating where revisions will be made to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The statement that 'quantitative analysis of radiographs demonstrates dominance by laser energy and target Z' is presented without error bars, data exclusion criteria, statistical measures, or direct comparison to hydrodynamic or laser-imprint simulations. This makes it impossible to assess whether the observed Z and energy scalings uniquely support the Weibel secondary instability over competing Z-dependent plasma or ablation-front effects.

    Authors: We agree that the abstract is highly condensed and omits these supporting details. The full manuscript reports error bars on the quantitative metrics (derived from multiple shots and image-analysis uncertainties) in the results figures, specifies data-exclusion criteria based on radiograph quality and signal-to-noise thresholds in the methods, and includes statistical measures such as standard deviations across the parameter scan. Direct comparisons to hydrodynamic and laser-imprint simulations are presented in the discussion to demonstrate that the observed scalings exceed those attributable to ablation-front or imprint effects alone. To address the referee's point, we will revise the abstract to include a brief qualifier on the statistical robustness of the analysis and will add an explicit cross-reference to the simulation comparisons. revision: yes

  2. Referee: [Abstract] Abstract and results discussion: The central claim that the filamentary proton-radiograph features arise specifically from E/B fields of the expansion-driven Weibel secondary instability is not isolated from alternatives. Parameter variations show Z and energy dependence, but no wavelength matching, polarization signatures from dual-axis imaging, or comparison to Weibel dispersion relations are reported to exclude hydrodynamic modulations, laser imprint, or other Z-dependent effects.

    Authors: The dual-axis proton radiography was employed to establish the three-dimensional character of the filaments, but extracting polarization signatures for E versus B discrimination is not straightforward in this diagnostic and was not performed. Explicit wavelength matching or dispersion-relation comparisons were not included because the study prioritizes empirical parameter scalings over detailed linear-theory fitting. The 4ω optical probe data show no corresponding density modulations at the observed filament scales, which helps rule out purely hydrodynamic interpretations. The pronounced dependence on target Z is more consistent with the ion-mass and charge dependence expected for the Weibel mechanism than with hydrodynamic or laser-imprint alternatives. We will expand the discussion section to explicitly enumerate and address the competing mechanisms, explaining why the combined energy, Z, and probe data favor the secondary instability following the expansion-driven Weibel instability. We note that a full dispersion-relation analysis would require additional modeling work beyond the scope of this experimental report. revision: partial

Circularity Check

0 steps flagged

No circularity: experimental data interpretation with independent parameter variations

full rationale

This is an experimental paper reporting proton radiography measurements in planar laser ablation. The central claim interprets observed filamentary structures as arising from a secondary instability following the expansion-driven Weibel instability, based on scaling with laser energy and target Z across varied shots. No equations, derivations, or fitted parameters are described that reduce the result to inputs defined by the same data. The analysis uses dual-axis imaging and quantitative scaling from independent experimental variations, which constitutes external evidence rather than self-referential construction. No self-citations or ansatzes are invoked as load-bearing steps in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is purely experimental and introduces no new mathematical model, free parameters, or postulated entities in the abstract; the claim rests on interpretation of radiographic contrast as electric/magnetic fields and on the assumption that the observed scaling matches the Weibel secondary instability.

pith-pipeline@v0.9.0 · 5449 in / 1142 out tokens · 71489 ms · 2026-05-12T05:07:06.156022+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

  1. [1]

    Borghesi, D

    M. Borghesi, D. H. Campbell, A. Schiavi, M. G. Haines, O. Willi, A. J. MacKinnon, P. Patel, L. A. Gizzi, M. Galimberti, R. J. Clarke, F. Pegoraro, H. Ruhl, S. Bulanov; Electric field detection in laser-plasma in- teraction experiments via the proton imaging tech- nique, Phys. Plasmas 1 May 2002; 9 (5): 2214–2220. https://doi.org/10.1063/1.1459457

    work page doi:10.1063/1.1459457 2002

  2. [2]

    C. K. Li, F. H. S´ eguin, J. A. Frenje, M. Manuel, R. D. Petrasso, V. A. Smalyuk, R. Betti, J. Delet- trez, J. P. Knauer, F. Marshall, D. D. Meyerhofer, D. Shvarts, C. Stoeckl, W. Theobald, J. R. Rygg, O. L. Landen, R. P. J. Town, P. A. Amendt, C. A. Back and J. D. Kilkenny; Study of direct-drive capsule im- plosions in inertial confinement fusion with pr...

    work page doi:10.1088/0741-3335/51/1/014003 2009

  3. [3]

    J. R. Rygg, F. H. S´ eguin, C. K. Li, J. A. Frenje, M. J.-E. Manuel, R. D. Petrasso, R. Betti, J. A. Delet- trez, O. V. Gotchev, J. P. Knauer, D. D. Meyer- hofer, F. J. Marshall, C. Stoeckl, and W. Theobald; Proton Radiography of Inertial Fusion Implosions, Sci- ence, 29 Feb 2008, Vol 319, Issue 5867, pp. 1223-1225. https://doi.org/10.1126/science.1152640

    work page doi:10.1126/science.1152640 2008

  4. [4]

    F. H. S´ eguin, C. K. Li, M. J.-E. Manuel, H. G. Rinderknecht, N. Sinenian, J. A. Frenje, J. R. Rygg, D. G. Hicks, R. D. Petrasso, J. Delettrez, R. Betti, F. J. Marshall, V. A. Smalyuk; Time evolution of filamentation and self-generated fields in the coronae of directly driven inertial-confinement fusion capsules, Phys. Plasmas 19, 012701 (2012). https://...

    work page doi:10.1063/1.3671908 2012

  5. [5]

    A. B. Zylstra, C. K. Li, H. G. Rinderknecht, F. H. S´ eguin, R. D. Petrasso, C. Stoeckl, D. D. Meyerhofer, P. Nil- son, T. C. Sangster, S. Le Pape, A. Mackinnon, P. Patel; Using high-intensity laser-generated energetic protons to radiograph directly driven implosions, Rev. Sci. Instrum. 83, 013511 (2012). https://doi.org/10.1063/1.3680110

    work page doi:10.1063/1.3680110 2012

  6. [6]

    M. J.-E. Manuel, C. K. Li, F. H. S´ eguin, N. Sine- nian, J. A. Frenje, D. T. Casey, R. D. Petrasso, J. D. Hager, R. Betti, S. X. Hu, J. Delettrez, D. D. Meyer- hofer; Instability-driven electromagnetic fields in coronal plasmas. Phys. Plasmas 1 May 2013; 20 (5): 056301. https://doi.org/10.1063/1.4801515

    work page doi:10.1063/1.4801515 2013

  7. [7]

    G. D. Sutcliffe, P. J. Adrian, J. A. Pearcy, T. M. John- son, J. Kunimune, B. Pollock, J.D. Moody, N.F. Loureiro and C.K. Li; Observation of electromagnetic filamentary structures produced by the Weibel instability in laser- driven plasmas, arXiv:2209.02565v1, Sep 2022

    work page arXiv 2022

  8. [8]

    Gao; Measurements of Magneto-Hydrodynamic Ef- fects in Ablatively-Driven High Energy Density Systems, University of Rochester ProQuest Dissertations Theses,

    L. Gao; Measurements of Magneto-Hydrodynamic Ef- fects in Ablatively-Driven High Energy Density Systems, University of Rochester ProQuest Dissertations Theses,

    work page

  9. [9]

    Nilson, Magnetic Reconnection in the High-Energy- Density Regime, American Physical Society Division of Plasma Physics Meeting 2019

    P. Nilson, Magnetic Reconnection in the High-Energy- Density Regime, American Physical Society Division of Plasma Physics Meeting 2019

    work page 2019

  10. [10]

    I. V. Igumenshchev, A. B. Zylstra, C. K. Li, P. M. Nilson, V. N. Goncharov, R. D. Petrasso; Self-generated magnetic fields in direct-drive implo- sion experiments, Phys. Plasmas 21, 062707 (2014). https://doi.org/10.1063/1.4883226

    work page doi:10.1063/1.4883226 2014

  11. [11]

    P. V. Heuer, J. L. Peebles, J. Kunimune, H. G. Rinderknecht, J. R. Davies, V. Gopalaswamy, J. Frelier, M. Scott, J. Roberts, R. B. Brannon, H. McClow, R. Fairbanks, S. P. Regan, J. A. Frenje, M. Gatu John- son, F. H. S´ eguin, A. J. Crilly, B. D. Appelbe, M. Farrell, J. Stutz; Distortions in charged-particle images of laser direct-drive inertial confineme...

    work page doi:10.1063/5.0251899 2025

  12. [12]

    Roth, Ion Acceleration—Target Normal Sheath Ac- celeration, Vol

    M. Roth, Ion Acceleration—Target Normal Sheath Ac- celeration, Vol. 1 (2016): Proceedings of the 2014 CAS- CERN Accelerator School: Plasma Wake Acceleration https://doi.org/10.5170/CERN-2016-001.231

    work page doi:10.5170/cern-2016-001.231 2016

  13. [13]

    D. H. Froula, R. Boni, M. Bedzyk, R. S. Craxton, F. Ehrne, S. Ivancic, R. Jungquist, M. J. Shoup, W. Theobald, D. Weiner, N. L. Kugland, M. C. Rushford; Optical diagnostic suite (schlieren, interferometry, and grid image refractometry) on OMEGA EP using a 10- ps, 263-nm probe beam, Rev. Sci. Instrum. 83, 10E523 (2012). https://doi.org/10.1063/1.4733739

    work page doi:10.1063/1.4733739 2012

  14. [14]

    Filkins, M

    T. Filkins, M. J. Rosenberg, R. E. Bahr, J. Katz, and S. T. Ivancic; Calibration of the sub-aperture backscatter 11 TABLE I. Shot parameters for entire dataset. Shots without a Distributed Phase Plate (DPP) were focused to 750 um diameter. # of Beams Pulse Length) Energy Intensity DPP Target Mat Target Size Probe Time (ns) (J) 10 14(W/cm2) (Thicknessµm) (...

    work page doi:10.1063/5.0101839 1937

  15. [15]

    N. L. Kugland, D. D. Ryutov, C. Plechaty, J. S. Ross, H.-S. Park; Invited Article: Relation between elec- tric and magnetic field structures and their proton- beam images, Rev. Sci. Instrum. 83, 101301 (2012). https://doi.org/10.1063/1.4750234

    work page doi:10.1063/1.4750234 2012

  16. [16]

    D. A. Tidman, R. A. Shanny; Field-generating ther- mal instability in laser-heated plasmas, Phys. Fluids 17, 1207–1210 (1974). https://doi.org/10.1063/1.1694866

    work page doi:10.1063/1.1694866 1974

  17. [17]

    Garc´ ıa-Rubio, R

    F. Garc´ ıa-Rubio, R. Betti, J. Sanz, H. Aluie; The- ory of the magnetothermal instability in coronal plasma flows, Phys. Plasmas 29, 092106 (2022). https://doi.org/10.1063/5.0109877

    work page doi:10.1063/5.0109877 2022

  18. [18]

    E. S. Weibel; Spontaneously growing transverse waves in a plasma due to an anisotropic veloc- ity distribution, Phys. Rev. Lett. 2, 83–84 (1959). https://doi.org/10.1103/PhysRevLett.2.83

    work page doi:10.1103/physrevlett.2.83 1959

  19. [19]

    P. E. Masson-Laborde, W. Rozmus, Z. Peng, D. Pesme, S. H´ uller, M. Casanova, V. Yu. Bychenkov, T. Chap- man, P. Loiseau, Phys. Plasmas 17 092704 (2010). https://doi.org/10.1063/1.3474619

    work page doi:10.1063/1.3474619 2010

  20. [20]

    Silva, K

    T. Silva, K. Schoeffler, J. Vieira, M. Hoshino, R. A. Fon- seca1, and L. O. Silva; Anisotropic heating and magnetic field generation due to Raman scattering in laser-plasma interactions, Phys. Rev. Research 2 023080 (2020). https://doi.org/10.1103/PhysRevResearch.2.023080

    work page doi:10.1103/physrevresearch.2.023080 2020

  21. [21]

    Z. Zhao, S. He, H. An, Z. Lei, Y. Xie, W. Yuan, J. Jiao, K. Zhou, Y. Zhang, J. Ye, Z. Xie, J. Xiong, Z. Fang, X. He, W. Wang, W. Zhou, B. Zhang, S. Zhu, and B. Qiao; Laboratory evidence of Weibel magnetogenesis driven by temperature gradient using three-dimensional synchronous proton radiogra- phy, Science Advances, 3 Apr 2024 Vol 10, Issue 14 https://doi...

    work page doi:10.1126/sciadv.adk5229 2024

  22. [22]

    Y. Z. Zhou, C. Y. Zheng, Z. J. Liu and L. H. Cao; Weibel instability induced by kinetic stimulated Ra- man scattering in unmagnetized and magnetized plas- mas, Plasma Phys. Control. Fusion 64 045009 (2022). https://doi.org/10.1088/1361-6587/ac4bcf

    work page doi:10.1088/1361-6587/ac4bcf 2022

  23. [23]

    K. M. Schoeffler, N. F. Loureiro, R. A. Fonseca, and L. O. Silva; Magnetic-Field Generation and Amplification in an Expanding Plasma, Phys. Rev. Lett. 112, 175001 (2014). https://doi.org/10.1103/PhysRevLett.112.175001

    work page doi:10.1103/physrevlett.112.175001 2014

  24. [24]

    K. M. Schoeffler, L. O. Silva; Effects of col- lisions on the generation and suppression of temperature anisotropies and the Weibel insta- bility, Phys. Rev. Research 2, 033233 (2020). https://doi.org/10.1103/PhysRevResearch.2.033233

    work page doi:10.1103/physrevresearch.2.033233 2020

  25. [25]

    K. V. Lezhnin, S. R. Totorica, J. Griff-McMahon, M. Medvedev, H. Landsberger, A. Diallo, W. Fox; Expansion-driven self-magnetization of high-energy- density plasmas, APS-DPP Meeting November 2025, Long Beach CA

    work page 2025

  26. [26]

    K. V. Lezhnin, S. R. Totorica, J. Griff-McMahon, M. Medvedev, H. Landsberger, A. Diallo, W. Fox; Simulations of self-magnetization in expanding high- energy-density plasmas, arXiv:2503.15624, March 2023. 12 https://doi.org/10.48550/arXiv.2503.15624

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2503.15624 2023

  27. [27]

    Thaury, P

    C. Thaury, P. Mora, A. H´ eron, and J. C. Adam; Self- generation of megagauss magnetic fields during the ex- pansion of a plasma, Phys. Rev. E 82, 016408 (2010). https://doi.org/10.1103/PhysRevE.82.016408

    work page doi:10.1103/physreve.82.016408 2010

  28. [28]

    V. V. Kocharovsky, A. A. Nechaev, M. A. Garasev; Electron Weibel instability and quasi- magnetostatic structures in an expanding collisionless plasma, Rev. Mod. Plasma Phys. 8, 17 (2024). https://doi.org/10.1007/s41614-024-00157-4

    work page doi:10.1007/s41614-024-00157-4 2024

  29. [29]

    Kalman, C

    G. Kalman, C. Montes, and D. Quemada; Anisotropic Temperature Instabilities, Phys. Fluids 11, 1797–1808 (1968). https://doi.org/10.1063/1.1692198

    work page doi:10.1063/1.1692198 1968

  30. [30]

    Stockem, M

    A. Stockem, M. E. Dieckmann and R. Schlickeiser; PIC simulations of the thermal anisotropy-driven Weibel in- stability: field growth and phase space evolution upon saturation, Plasma Phys. Control. Fusion 51 075014 (2009). http://stacks.iop.org/PPCF/51/075014

    work page 2009

  31. [31]

    K. A. Taggart, B. B. Godrey, C. E. Rhoades, Jr., and H. C. Ives; Second-Order Effects in the Weibel Instability, Phys. Rev. Lett. 29, 1729 (1972). https://doi.org/10.1103/PhysRevLett.29.1729

    work page doi:10.1103/physrevlett.29.1729 1972

  32. [32]

    B. Du, H. B. Cai, W. S. Zhang, J. M. Tian, E. H. Zhang, S. Y. Zou, J. Chen and S. P. Zhu; Distinguish- ing and diagnosing the spontaneous electric and mag- netic fields of Weibel instability through proton radiog- raphy, Plasma Phys. Control. Fusion 62 (2020) 025017. https://doi.org/10.1088/1361-6587/ab57ed

    work page doi:10.1088/1361-6587/ab57ed 2020

  33. [33]

    Del Sarto, A

    D. Del Sarto, A. Ghizzo, M. Sarrat; Nonlinear coupling of electromagnetic and electrostatic modes via density and pressure fluctuations: The case of Weibel instabilities, Phys. Plasmas 31, 072113 (2024). https://doi.org/10.1063/5.0207974

    work page doi:10.1063/5.0207974 2024

  34. [34]

    M. E. Dieckmann; The filamentation instability driven by warm electron beams: statistics and electric field gen- eration, Plasma Phys. Control. Fusion 51 124042 (2009). https://doi.org/10.1088/0741-3335/51/12/124042

    work page doi:10.1088/0741-3335/51/12/124042 2009

  35. [35]

    P. V. Heuer, L. S. Leal, J. R. Davies, E. C. Hansen, D. H. Barnak, J. L. Peebles, F. Garc´ ıa-Rubio, B. Pollock, J. Moody, A. Birkel, F. H. Seguin; Diagnosing magnetic fields in cylindrical implosions with oblique proton ra- diography, Phys. Plasmas 1 July 2022; 29 (7): 072708. https://doi.org/10.1063/5.0092652

    work page doi:10.1063/5.0092652 2022

  36. [36]

    W. M. Manheimer, D. G. Colombant, J. H. Gardner; Steady-state planar ablative flow, Phys. Fluids 25, 1644 (1982) https://doi.org/10.1063/1.863956

    work page doi:10.1063/1.863956 1982