Monte Carlo approach to calculate ionization dynamics of hot solid-density plasmas within particle-in-cell simulations

A physical model based on a Monte Carlo approach is proposed to calculate the ionization dynamics of hot-solid-density plasmas within particle-in-cell (PIC) simulations, and where the impact (collision) ionization (CI), electron-ion recombination (RE), and ionization potential depression (IPD) by surrounding plasmas are taken into consideration self-consistently. When compared with other models, which are applied in the literature for plasmas near thermal equilibrium, the temporal relaxation of ionization dynamics can also be simulated by the proposed model. Besides, this model is general and can be applied for both single elements and alloys with quite different compositions. The proposed model is implemented into a PIC code, with (final) ionization equilibriums sustained by competitions between CI and its inverse process (i.e., RE). Comparisons between the full model and model without IPD or RE are performed. Our results indicate that for bulk aluminium at temperature of 1 to 1000 eV, (i) the averaged ionization degree increases by including IPD; while (ii) the averaged ionization degree is significantly over estimated when the RE is neglected. A direct comparison from the PIC code is made with the existing models for the dependence of averaged ionization degree on thermal equilibrium temperatures and shows good agreements with that generated from Saha-Boltzmann model and/or FLYCHK code.

Figure 1

(a) The total plasma energy (A.U.), with the full model by summarizing over all free electrons within a computational cell, as a function of time, with initial plasma temperature $150\text{eV}$ and predefined charge state $4+$. (b) The same as shown in panel (a), but with the model excluding IPD. (c) The same as shown in panel (a), but with the model excluding RE. The inlets over panels (a), (b), and (c) are the corresponding final ionization distributions of aluminium after 3-ps relaxation. The red line covered on the inlets are the ionization distributions of aluminium calculated by Saha-Boltzmann equation with defined temperature (a) $T_e=74\text{eV}$ and (b) $T_e=77\text{eV}$ also excluding IPD.

Figure 2

The total plasma energy (A.U.), with the full model by summarizing over all free electrons within a computational cell, as a function of time, with the initial temperature of aluminium carbide $100\text{eV}$ and predefined charge states of $11+$ for aluminium and $6+$ for carbon. The inlet is the final ionization distributions of aluminium and carbon after 15-ps relaxation.

Figure 3

(a) The total plasma energy (A.U.), with the full model and merging particle technique by summarizing over all free electrons within a computational cell, as a function of time, with initial plasma temperature $150\text{eV}$ and predefined charge state $4+$. (b) The corresponding temporal fluctuation of the number of macroparticles.

Figure 4

The averaged ionization degree of bulk aluminium as a function of plasma temperature. (a) Blue, red, and green lines (square) are the results calculated by Saha-Boltzmann equation (FLYCHK code), with fixed electron density of ${10}^{20}{\text{cm}}^{-3}, {10}^{22}{\text{cm}}^{-3}$, and ${10}^{24}{\text{cm}}^{-3}$. (b) Red and green lines are the results calculated by Saha-Boltzmann equation with updated numerical scheme, including IPD and excluding IPD, with fixed aluminium density $2.7\text{g}/{\text{cm}}^3$. Furthermore, solid red line is with the SP model of IPD, while dashed red line is with EK model of IPD. Black square is picked up from the equilibrium states calculated by our PIC code with full model.