Monte Carlo approach to calculate proton stopping in warm dense matter within particle-in-cell simulations

A Monte Carlo approach to proton stopping in warm dense matter is implemented into an existing particle-in-cell code. This approach is based on multiple electron-electron, electron-ion, and ion-ion binary collision and accounts for both the free and the bound electrons in the plasmas. This approach enables one to calculate the stopping of particles in a more natural manner than existing theoretical treatment. In the low-temperature limit, when “all” electrons are bound to the nucleus, the stopping power coincides with the predictions from the Bethe-Bloch formula and is consistent with the data from the National Institute of Standard and Technology database. At higher temperatures, some of the bound electrons are ionized, and this increases the stopping power in the plasmas, as demonstrated by A. B. Zylstra et al. [Phys. Rev. Lett. 114, 215002 (2015)]. At even higher temperatures, the degree of ionization reaches a maximum and thus decreases the stopping power due to the suppression of collision frequency between projected proton beam and hot plasmas in the target.

Figure 1

(a) Configuration to account for proton stopping calculation in our PIC simulations. A proton beam of energy $1\text{MeV}$ irradiates the aluminum bulk. Two diagnostic planes separated by a distance of $0.039\mu \text{m}$ are placed to record the energy spectra of protons. (b) The black-square line represents the spectra recorded by the first diagnostic plane, while the red-square line represents that by the second one.

Figure 2

Stopping power as a function of projected proton energy. Results from our PIC simulations are compared with those from the NIST database. (a) Results of the PIC simulations: The red-square line is the one provided the average ionization degree is $Z=0.3$ and the blue-square line is the one provided the average ionization degree is $Z=3.0$. Data from the NIST database are plotted by the black solid line. (b) Results of PIC simulations: The red-square line is the one provided $T_e=100\text{eV}$ with average ionization degree $Z=8.0$, the blue-square line is the one provided $T_e=500\text{eV}$ with average ionization degree $Z=12.0$, and the green-square line is the one provided $T_e=1500\text{eV}$ with average ionization degree $Z=13$. Data from the NIST database are plotted by the black solid line. Note the average ionization degree under different temperature is calculated based on either the model we proposed in Ref. [37] or the external equation of state (EOS) tables [38, 39].

Figure 3

(a) Particle kinetic energies as functions of time in a typical PIC simulation. A proton beam, with energy of $5\text{MeV}$, density of ${10}^{18}{\text{cm}}^{-3}$ and pulse duration $1\text{ps}$, irradiates the bulk aluminum, with initial temperature $T_e=1\text{eV}$ and average ionization degree $Z=0.3$. The black line represents the energy of the injected protons, the blue line is the kinetic energy of ions (excluding the projected protons) in the simulation box, and the red line is the total electron energy in the simulation box. (b) Proton density (black line) distribution and electron background temperature (red line) at the final time of simulations.

Figure 4

Variation of proton beam range with the increase of bulk temperatures. A proton beam, with energy of $5\text{MeV}$ and/or $3\text{MeV}$, density of ${10}^{18}{\text{cm}}^{-3}$, and pulse duration $1\text{ps}$, irradiates the bulk aluminum with different initial temperatures. (a) Temperature $T_e=1\text{eV}$ with average ionization degree $Z=0.3$, (b) temperature $T_e=30$ eV with $Z=3.5$, (c) temperature $T_e=100$ eV with $Z=8.0$, (d) temperature $T_e=500\text{eV}$ with $Z=12$, (e) temperature $T_e=1000\text{eV}$ with $Z=13$.