MAICRM: A general model for rapid simulation of hot dense plasma spectra

To accurately model the properties of matter under non-local thermodynamic equilibrium (NLTE) conditions is critical to simulate the evolution of plasmas produced in inertial confinement fusion (ICF), magnetic fusion and other laboratory experiments as well as to diagnose the states of these systems by simulating the spectra. For most NLTE simulations the collisional-radiative (CR) modelBates1962 is used. BookRalchenkobook reviewed the history and the recent developments of modern methods employed in CR modeling. In general, NLTE atomic models can be classified into two kinds: one is to characterize the plasma properties by experientially choosing the important atomic levels, such as the detailed configuration accounting (DCA), superconfiguration accounting (SCA) or hybrid modelsSOSA1991 ; STA1989 ; the other one is average-atom (AA) model which characterizes the plasma simply by the orbital occupations of one ion, which is the average of all the atomic levels. Although much effort has gone into developing the NLTE simulations, such as extended or reduced DCA modelsRDCA2008 ; RDCA2009 ; Scott2010 , the questions of which (or how many) states and transitions to include in an atomic model and what degree of averaging to use as well as the computational cost are still critical and unsolved completely for an on-line detailed atomic calculation incorporated in radiation-hydrodynamic simulations.

In our workMAICRM-1 we reported a general atomic model MAICRM (Multi-Average Ion Collisional-Radiative Model) to rapidly simulate the average ionization $\langle Z\rangle$ and the charge state distribution (CSD) of hot dense plasmas (labeled as paper I in the following). The main thought of MAICRM is using an average ion to characterize the plasma properties of all the configurations at one charge state. The orbital occupation of an average ion is the average of the orbital occupations of all the configurations at one charge state and its population is the sum of the configuration populations. The idea of using multi-average ions to simulate plasma has been presented beforeKiyokawa2014 ; Itoh1987 but the procedures to get the orbital occupations and the populations of the average ions are different. Ref.Kiyokawa2014 proposed a multi-average ion model for LTE plasma, in which a Fermi distribution for the orbital occupation numbers of each average ion is assumed and the populations of the ions are calculated self-consistently through minimizing the free energy of the whole system established by the finite temperature density functional theory. Ref.Itoh1987 reported a hybrid-atom model, which estimated the excited level populations approximately by quasisteady equations of bound-bound transitions in an ion of charge $Z$. The normalized populations of excited levels constructed from the most populated ion are scaled and used for other ions at different charge states. Then the rate coefficients for each average ion are constructed by the summation over all excited states weighted by the level populations and the populations of the ions are obtained by solving the rate equations only including ionization/recombination processes.

Differing from the former work, our model calculates the orbital occupation numbers by solving the rate equations involving bound-bound (excitation/de-excitation) transitions, and the populations by solving the rate equations involving ionization/recombination processes with fixed occupation numbers. These rate equations derive from the rate equations of detailed configurations. During the transformation from the detailed configuration rate equations into two sets of rate equations for average ions, two assumptions are adopted: one is the single orbital rate coefficients (without the occupation and hole numbers) are only the functions of the charge state; the other one is the coupled rate equations of occupation numbers and populations of average ions are separated.

Paper I has shown the calculated $\langle Z\rangle$ and CSD of Fe, Xe and Au plasmas under various plasma conditions by MAICRM codes agree with the experimental and DCA/SCA results. This work will continue the development of the method to calculate the plasma spectral properties of hot dense plasmas, including the formulas to calculate the position and width of transition array for average ions. Using the developed codes, our calculated single orbital transition spectra agree with those of super transition arrays (STA) model with a single supershell for each charge stateSTA1989 . The present calculated emissivity spectra of Xe and Au plasmas agree well with the detailed SCRAMScott2010 and DCA resultsRDCA2009 . Since for each ionic stage only one configuration is calculated in MAICRM, the computational cost is lower than SCA/DCA by magnitudes due to their hundreds or thousands superconfigurations/configurations at each charge state.

In this work, 65 single orbital bases $nlj(n\leq 5)$ and $nl(5<n\leq 10)$ are used (listed in paper I).

Fig.1 shows the comparisons of the one-electron transition emission spectra for Fe plasma at $T$=200eV, $N_{i}=8.5\times 10^{22}$ cm^-3 and Pr plasma at $T$=250eV, $N_{i}=10^{20}$ cm^-3. The emissive intensity of Ref.STA1989 is in arbitrary units. As shown in Fig.1, compared with the STA curves, which are of a single supershell for each charge state, the widths of MAICRM curves are in good agreement, while the line positions show about $2\%$ red shifts, namely about 11eV for Fe and about 0.2${\mathbf{\AA}}$ for Pr. The shifts of the positions should result from the difference of the ionization distribution and the atomic data including the orbital energies and transition matrix elements. The widths of the AA curves are obviously broader than MAICRM and STA. From the statistic angle, the spectra of MAICRM and STA with a single supershell are both the function of the average orbital occupations of the configurations at one charge stage while the average orbital occupations are determined by different ways, namely by solving the excitation/de-excitation rate equations for MAICRM and by assuming a Boltzmann distribution for STA.

Fig.2 shows the comparison of the emissivity of Xe plasma at $T_{e}$=4keV and $N_{e}=3.6\times 10^{20}$ cm^-3. The detailed gray curve is calculated by SCRAM codes which uses hybrid atomic data, including levels, configurations and superconfigurations, constructed from a combination of fine structure and unresolved transition array (UTA) data from FAC, supplemented with the extended screened-hydrogenic model (including satellite shifts)SCRAM2007 . The red curve is calculated by extended ’DCA’ codes which construct atomic levels as superconfigurations or Layzer complexes by shells described by principal quantum numbers with two extensions: one is to split each photoexcitation transition between principal quantum numbers into multiple term-to-term transitions with individual energies and oscillator strengths; the other one is to assign an additional width to each transition, representing additional unresolved fine structure multiplets, modeling each transition as an UTAScott2010 . Here SCRAM and ’DCA’ use a set of similar superconfigurations for each ionic stage, thus their calculated average ionizations $<Z>_{\mathbf{SCRAM}}=42.9$ and $<Z>_{{}^{\prime}\mathbf{DCA}^{\prime}}=43.0$ agree well. $<Z>_{\mathbf{MAICRM}}=43.4$ is higher by about 0.5.

In Fig.2 the mean features of the three curves are in general agreement although MAICRM and extended ’DCA’ have no the fine structures of SCRAM. Compared to the detailed gray spectrum of SCRAM, MAICRM curve agrees better than ’DCA’ curve at some places such as the peaks at 310eV and the relative intensities of three peaks at 155eV, 190eV and 220eV. The peak of MAICRM curve at 105eV is closer to the maximum position of gray curve than the red curve. The continue background of MAICRM at energies larger than 1keV is lower than the grey and red curves which should result from the difference of $<Z>$.

Fig.3 shows the comparison of the emissivity of Au plasma at $T_{e}$=1keV, $T_{r}$=400eV and $\rho$=$0.0155$g$/$cm³. Here DCA is the traditional detailed configuration accounting model, RDCA is a reduced DCA model and XSN is a AA modelScott2010 . RDCA model makes an average of the energies and cross-sections of the detailed states falling within a spectral energy grid divided according to the DCA energy level structure of an ionScott2010 . Compared with $<Z>_{\mathbf{DCA}}$=51.8, $<Z>_{\mathbf{MAICRM}}$=52.6 is much closer than $<Z>_{\mathbf{XSN}}$=56.5. For clarity, the blue curve of MAICRM is added into the three panels. Panel (a) shows the MAICRM curve agrees well with the DCA curve especially at energies larger than 1keV. In the energy range of 80eV-120eV, the widths of MAICRM curve are larger than DCA. Panel (b) shows the RDCA curve differs from the MAICRM curve in many places. In panel (c) the XSN curve shows simpler structure and broader peaks than MAICRM since in AA model only one average ion is considered.

We have developed a general model to simulate the plasma properties of hot dense plasma. The average orbital occupations and total population for the configurations within one charge state are characterized by an average ion. The orbital occupation $\{\mathbf{\Omega}^{\mathbf{\Lambda_{n_{e}}}}_{n}\}$ and population $\{P_{\mathbf{\Lambda_{n_{e}}}}\}$ of the average ion $\mathbf{\Lambda_{n_{e}}}$ are obtained by solving two sets of rate equations sequentially and iteratively. Our calculated CSDs as well as the spectra of hot dense plasmas agree with DCA/SCA models as shown in paper I and this work. On the other hand, since MAICRM only considers one configuration at each charge state the computational cost reduces magnitudes compared to DCA/SCA models due to their hundreds or thousands atomic levels at one charge state, which means the possibility to be coupled into radiation-transport hydrodynamic simulations in the future.