Fig. 1. Stopping power of monoenergetic 5 MeV protons in fully ionized aluminum at solid density and 1 keV as a function of penetration depth: comparison between the BPS and classical stopping models.

Fig. 2. Penetration depth of monoenergetic protons in solid aluminum as a function of plasma temperature for different stopping power models: SCAALP,^35,36 the model described by Kim et al.,²³ and the model used here based on combining the BPS model for free electrons¹⁶ and the BC model for bound electrons.^30,33 The results obtained by the combined classical stopping and BC models are also shown for comparison.

Fig. 3. Penetration depth vs plasma temperature of monoenergetic 5 MeV protons in different materials as obtained with the BPS and BC model and the combined classical and BC models.

Fig. 4. Resistivities of different materials as functions of the plasma temperature obtained with the Eidmann–Chimier model.

Fig. 5. (a) Penetration depth vs current density of monoenergetic and perfectly collimated 5 MeV proton beams in a solid aluminum target at 20 ps obtained using the classical and BPS stopping power models. (b) and (c) Energy density of a proton beam with a current density of 10¹¹ A/cm² impinging on the same aluminum target at 20 ps using the classical and BPS stopping models, respectively. The bound electron stopping is obtained using the BC model.³⁰

Fig. 6. B-field distributions of a perfectly collimated 5 MeV proton beam impinging on solid aluminum at 20 ps with the current density as a parameter: (a) 10¹¹ A/cm²; (b) 10¹² A/cm²; (c) 10¹³ A/cm².

Fig. 7. Electron temperature distribution at 20 ps for the case of Fig. 6. The current densities are as follows: (a) and (b) 10¹⁰ A/cm²; (c) and (d) 10¹¹ A/cm²; (e) and (f) 10¹² A/cm²; (g) and (h) 10¹³ A/cm². The left panels show the results with the resistive fields artificially suppressed and the right panels those with the resistive fields on.

Fig. 8. Electron temperature distribution at 20 ps for different proton divergence half-angles (HWHM): (a) and (b) 0°; (c) and (d) 5°; (e) and (f) 10°; (g) and (h) 15°. The beam current density is 10¹¹ A/cm². The left panels show the results with the resistive fields artificially suppressed and the right panels those with the resistive fields on. Other parameters are the same as in Fig. 7.

Fig. 9. Ohmic heating fraction as a function of divergence half-angle (HWHM) for different current densities. Other parameters are the same as in Fig. 8.

Fig. 10. B-field distribution and electron temperature of a 5 MeV perfectly collimated proton beam with a current density of 10¹² A/cm² impinging on different materials at solid density: (a) and (b) aluminum; (c) and (d) copper; (e) and (f) gold. The left panels show the azimuthal B-field, and the right panels show the electron temperature. Coulomb collisions and resistive fields are turned on. The simulation time is 20 ps.

Fig. 11. Ohmic heating fraction as a function of beam current density for different materials.

Fig. 12. Electron temperature map for a 5 MeV perfectly collimated proton beam with current density 10¹² A/cm² impinging on an aluminum target at 20 ps. (a)–(d) Transverse cuts at z = 2, 60, 140, and 200 μm, respectively. (e) Comparison of 3D simulation results with 2D simulation results: the blue stars indicate the electron temperatures on the four different cross-sectional distributions at the central axis for the 3D case, and the red curve shows to the electron temperature distribution at the central axis of Fig. 10(b) for the 2D case. Other parameters are the same as in Fig. 10(b).