Fig. 1. (a) The total plasma energy within a computational cell as a function of time, with initial plasma temperature and pre-defined charge state . The red line covered on the inlets is the ionization distributions of Al calculated by the Saha–Boltzmann equation with defined temperature, . (b) The averaged ionization degree as a function of temperature, where red and green lines are the results calculated by the Saha–Boltzmann equation, including IPD and excluding IPD, with fixed Al density . The solid red line is with the SP^[49] model of IPD, while the dashed red line is with EK^[50] model of IPD. The black square line is picked up from the equilibrium states calculated by our PIC code.

Fig. 2. Schematic of charged particle collision. For neutral atoms, when all electrons are bounded at the nuclei with the radius on the order of Bohr unit , only projectile that could penetrate through the electron can be deflected by the Coulomb force of the nuclei. When temperature is high, some of the bound electrons are ionized and form plasmas around the nuclei, projectile with a collision distance smaller than (usually is much larger than ) can also be deflected by the Coulomb force.

Fig. 3. For given Al solid of density , (a) the electron–electron (black square line) and electron–ion (black diamond line) collision frequency given by our PIC code as functions of temperatures; (b) the resistivity given by our PIC code versus experimental values^[55]. In these calculations, the variation of averaged ionization degree with temperature is also taken into account. The values given by the PIC code are averaged over 1000 particle pairs.

Fig. 4. Stopping power of different materials, (a) for Al and (b) for Cu, as a function of projected electron kinetic energy. Results from our PIC simulations, at low-temperature limit, are compared with that from the NIST database. The solid black line is the collisional stopping power (), the solid blue line is the radiation stopping power () and the solid red line is the total stopping power, with from the NIST database. The black square line is the collisional stopping power calculated by PIC code, and the red square line is the total stopping power calculated by PIC code, where dashed lines represent the one excluding density effect . (c) The stopping power of Al as a function of projected electron kinetic energy at different temperatures.

Fig. 5. Comparison of PIC simulations when including and excluding Bremsstrahlung radiation correction. Initially, a mono-energetic electron beam of is launched into a bulk Al. The final energy spectrum after is shown in (a), where the red line is the case including Bremsstrahlung and the black line is the one excluding Bremsstrahlung. (b) is the angular distribution of emitted photons due to Bremsstrahlung radiation. See text for the explanation of coordinate setup. (c) is the frequency spectra of emitted photons due to Bremsstrahlung radiation, where we have plotted as a function of cut-off frequency . Note  eV, corresponding to the energy of a photon with wavelength .

Fig. 6. The value of self-heating as a function of simulation time. Plasma is of density ( is the corresponding critical density for electromagnetic wave of wavelength ), plasma temperature is , the simulation grid size is and 100 electrons are filled into a computational cell. Different coloured lines represent different combinations of numerical schemes, fourth/second order and with/without current smoothing.

Fig. 7. The schematic of ‘layered density’ method. Here ‘layered density’ means electrons are divided into two groups, i.e., electron-0 and electron-1. During the PIC simulation, electron-0 updates following the ionization dynamics. For the calculation of electromagnetic fields, only electron-1 is involved. For collisions, both electron-0 and electron-1 are involved.

Fig. 8. Thermal equilibrium benchmark of the ‘layered density’ (LD) method. Electron and ion kinetic energy as a function of time. Initial plasma density is set to be , initial electron temperature is and initial proton temperature is . For the LD method, electrons are divided into two groups, and the density of each group is . In PIC simulations, these two groups of electrons are treated as different species.

Fig. 9. (a) and (c) The current density distribution, , of forward-propagating fast electrons (red line) and returning background electrons (black line), when a fast electron beam of 1 MeV with density is launched into uniform plasmas. (b) and (d) The resistive electric fields, normalized by , generated by the launched electron beam. Here background plasma density in (a) and (b) is of ( is the corresponding critical density for electromagnetic wave of wavelength and ) and temperature is of . In (c) and (d), plasma density of is used. Thick lines are the results calculated by ‘layered density’ methods, and thin lines are the ones obtained from full-PIC method.

Fig. 10. (a) The initial parameter setup, with pre-plasma scale length , initial density () and temperature . In the ‘layered density’ method, density of ele-0 is and ele-1 is . Here is the corresponding critical density of electromagnetic wave with wavelength . (b) Electron density and temperature at the end of simulations.

Fig. 11. (a) The final energy spectra of electrons, with the black line representing the reference case without considering atomic processes, and the red line representing the one including both ionization and collision with Bremsstrahlung radiation corrections. (b) The angular distribution of emitted photons. (c) The frequency spectra of emitted photons, where we have plotted as a function of cut-off frequency . Note  eV, corresponding to the energy of a photon with wavelength .

Fig. 12. The – phase-space plot of electrons, with the same simulation parameters as shown in Figure 10. (a) The one without considering ionization, collision and Bremsstrahlung radiation correction. (b) The one turning on both ionization and collision with Bremsstrahlung radiation corrections. Different columns represent values at different times, here  ps for (1),  ps for (2) and  ps for (3). The red curves covered on the phase-space plots are the electrostatic potential curves (), normalized by . The blue lines are the , normalized by , components of the superposition of incoming and reflected laser pulses.

Fig. 13. Dynamics of an electron calculated with single particle simulations. The gained energy from laser beam as a function of propagation length. The total simulation time is . (a) An electron with initial momentum , a single laser pulse of amplitude . (b) An electron with initial momentum , a single laser pulse of amplitude , and a constant external electric field of . (c) An electron with initial momentum , a single laser pulse of amplitude , a constant external electric field of and initial collision frequency of . (d) An electron with initial momentum , a single laser pulse of amplitude , a constant external electric field of and initial collision frequency of .

Fig. 14. Results of 2D PIC simulations. The plasma density perturbations in front of the target at the end of simulation.

Fig. 15. Results of 2D PIC simulations. (a) The plasma density in the inner part of the target at the end of simulation time. (b) The plasma temperature in the inner part of the target at the end of simulation time. (c1) The magnetic fields, normalized by , generated by the forward-propagating fast electrons. (c2) The resistive magnetic fields, normalized by , generated by the Ohmic return current.

Fig. 16. Results of 2D PIC simulations. Figure shows the angular distribution of emitted photons and the frequency spectra of emitted photons, where we have plotted as a function of cut-off frequency . Note  eV, corresponding to the energy of a photon with wavelength .