⇦Chinese Physics Letters, 2019, Vol. 36, No. 8, Article code 083401 Influence of Debye Plasma on the KLL Dielectronic Recombination of H-Like Helium Ions * Deng-Hong Zhang (张登红)**, Lu-You Xie (颉录有), Jun Jiang (蒋军), Chen-Zhong Dong (董晨钟) Affiliations Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070 Received 27 March 2019, online 22 July 2019 *Supported by the National Key Research and Development Program of China under Grant No 2017YFA0402300, and the National Natural Science Foundation of China under Grant Nos 11864036, 11564036 and 11774292.
**Corresponding author. Email: zhangdh@nwnu.edu.cn Citation Text: Zhang D H, Jie L Y, Jiang J and Dong C Z 2019 Chin. Phys. Lett. 36 083401    Abstract Using the Debye shielding model, the effects of plasma shielding on the dielectronic recombination processes of the H-like helium ions are investigated. It is found that plasma shielding causes a remarkable change in the Auger decay rate of the doubly excited $2p^2$ $^3P_2$ state. As a result, the dielectronic recombination cross sections from the doubly excited $2p^2$ $^3P_2$ state increases with the decreasing Debye shielding length. DOI:10.1088/0256-307X/36/8/083401 PACS:34.80.Lx, 34.80.Dp, 52.20.Fs © 2019 Chinese Physics Society Article Text With the development of many new applications for x-ray radiation from hot plasma, research on atomic structure and impact processes in hot and dense plasma environment has undergone rapid progress in recent years.[1–3] In contrast to free ions in a vacuum, the atomic/ionic potential in the hot plasma is shielded by the surrounding ions and electrons. Such shielding alters the electron wave function of ions embedded in the plasma, and therefore the energy levels and decay property will be different from those of the free ions or atoms.[4] Dielectronic recombination (DR) can be regarded as a resonance radiative recombination process, in which an electron undergoes radiationless capture into an autoionizing doubly excited state, subsequently the doubly excited state stabilizes by radiative decay. DR process will play important roles in the determination of the ionization balance and formation of an excited state population in high temperature plasmas.[5] The DR cross sections are needed for the theoretical modeling of laboratory and astrophysical plasma. As is known, the DR processes can also lead to the emission of satellite lines, which can be observed in many high temperature and high density plasma such as tokamak and inertial-confinement fusion (ICF) plasma. These satellite lines are extremely useful for plasma diagnostics. Recently, many investigations have shown that the hot and dense plasma environment can affect the energy levels and decay property of the doubly excited states.[3,6–13] Therefore, studies of the plasma shielding effect on the DR processes are very important for the hot and dense plasma diagnostic. In the present work, the influence of the plasma shielding on the KLL DR processes of H-like helium ions is investigated in the framework of the multi-configuration Dirac–Fock (MCDF) theory. The KLL DR processes of H-like helium ions can be written as $$\begin{alignat}{1} \!\!\!\!\!\!\!\!e^-(\varepsilon l)+{\rm He}^+(1\,s)\rightarrow {\rm He}({2l2l}')\rightarrow {\rm He}({1s2l}'')+h\nu.~~ \tag {1} \end{alignat} $$ To model a weakly coupled hot plasma, a Debye–Hückel potential has been used. In this work, the self-consistent multi-configuration Dirac–Fock (MCDF) method and the corresponding GRASP92 program[14] are used. The MCDF method has been described in detail in literature (see, e.g.,[14,15] and reference therein). Only a brief outline of the theory is given in the following. In the MCDF model, an atomic state wavefunction (ASF) is represented as a linear combination of configuration state functions (CSF) with the same parity $P$ and angular momentum $(J, M)$, $$\begin{align} \psi_{\alpha}(PJM)=\sum_{r=1}^{n_{\rm c}} c_{r}(\alpha)|{\gamma_rPJM}\rangle,~~ \tag {2} \end{align} $$ where $n_{\rm c}$ is the number of CSFs, and $c_{r}(\alpha) $ is the configuration mixing coefficient. The CSFs are antisymmetrized products of a common set of orthonormal orbitals, which are optimized on the basis of the relativistic Dirac–Coulomb Hamiltonian. To describe the plasma shielding effect on the electronic property, the Dirac–Coulomb Hamiltonian is modified in the following form $$\begin{align} H_{\rm DC}=\sum_{i} (c \alpha \cdot {\boldsymbol p}+ \beta c^{2})+V^{\rm DH},~~ \tag {3} \end{align} $$ where $V^{\rm DH}$ is the Debye–Hückel potential. In this work, we have used Whitten's[16] and Pindzola's[17] approximation model, where they used a simple but effective model to describe interaction between charged particles, $$\begin{align} V^{\rm DH}=\Big(\sum_{i}V(r_i)+\sum_{i > j} \frac{1}{r_{ij}}\Big)e^{-\frac{r_i}{\lambda}}.~~ \tag {4} \end{align} $$ In this model, a spherical shielding is assumed. Here $\lambda$ is called the Debye shielding length and it is a function of the temperature and density of the plasma, given by $$\begin{align} \lambda=\Big[\frac{kT_{\rm e}} {4\pi n_{\rm e}}\Big]^{1/2},~~ \tag {5} \end{align} $$ where $T_{\rm e}$ and $n_{\rm e}$ are the plasma electron temperature and density, respectively, and $k$ is the Boltzmann constant. The effect of plasma screening can be described by the coupling strength ${\it \Gamma}$ and plasma non-ideality parameter $\gamma$. The coupling strength is defined as the ratio of average Coulomb potential energy between pairs of particles and their kinetic energy. The plasma non-ideality parameter is defined as $\gamma=e^2/(ak_{\rm B}T_{\rm e})$, where $a=[3/(4\pi n_{\rm e})]^{1/3}$ is the average interparticle distance. If the Coulomb coupling parameter ${\it \Gamma}\leq 1$ and plasma non-ideality parameter $\gamma \ll 1$, the representation of charged particle interaction in a plasma by the potential Eq. (4) is valid. This means that the present calculations are valid for the plasma with the high temperature and low density or the mean thermal energies of the particles are much greater than the magnitude of their mean electrostatic interaction energy. The DR strength $S_{ij}$ is given by[18] $$\begin{align} S_{ij}(\varepsilon_j)=\frac{\pi^2\hbar^3}{m_e\varepsilon_j} \frac{g_j}{2g_i}{A^{\rm a}_{ji}B_{j}},~~ \tag {6} \end{align} $$ where $\varepsilon_j$ is the free electron energy, $A^{\rm a}_{ji}$ is the auger rate from the resonant state $j$ to the non-resonant electron-continuum state $i$, while $B_{j}$ is the radiative branching ratio given by $$\begin{align} B_{j}=\frac{\sum_{k}A^r_{jk}}{\sum_{k'}A^r_{jk'} +\sum_{i'}A^{\rm a}_{ji'}},~~ \tag {7} \end{align} $$ with $A^r_{jk}$ the Einstein coefficient for spontaneous radiative decay from the excited state $j$ to the final state $k$. The convoluted cross section is given by[19] $$\begin{align} \sigma_t{^{\rm DR}}{(\varepsilon)}=\sum_{j}\frac{S_{ij} (\varepsilon_j)}{\sqrt{2\pi}{\it \Gamma}} \exp\Big[-\frac{{(\varepsilon -\varepsilon_j)}^2}{{2{\it \Gamma}}^2}\Big],~~ \tag {8} \end{align} $$ where ${\it \Gamma}$ is an energy resolution of the Gaussian distribution in an experimental resolution.
Table 1. The eigenenergies (in units of a.u.) and widths (in units of a.u.) of doubly excited state $2p^2$ ${^1}D_2$ and $2p^2$ $^3P_2$ of helium atom in plasma for different Debye shielding lengths $\lambda$.
$2p^2$ $^1D_2$ $2p^2$ $^3P_2$
Eigenenergies Widths Eigenenergies
$\lambda$ ($a_0$) This work Ref.  [8] This work Ref.  [8] This work Ref.  [9]
$\infty$ $-$0.700 $-$0.702 2.60$\times$10$^{-3}$ 2.35$\times$10$^{-3}$ $-$0.70955 $-$0.7105
2.24$\pm$0.4$\times$10$^{-3}$(Ref.  [20]) $-$0.7105(Ref.  [21])
$-$0.70999(Ref.  [22])
100 $-$0.667 $-$0.672 2.57$\times$10$^{-3}$ 2.35$\times$10$^{-3}$ $-$0.67706 $-$0.68081
50 $-$0.635 $-$0.643 2.53$\times$10$^{-3}$ 2.34$\times$10$^{-3}$ $-$0.64557 $-$0.65173
30 $-$0.594 $-$0.605 2.47$\times$10$^{-3}$ 2.32$\times$10$^{-3}$ $-$0.60506 $-$0.61390
20 $-$0.546 $-$0.560 2.38$\times$10$^{-3}$ 2.29$\times$10$^{-3}$ $-$0.55671 $-$0.56810
cpl-36-8-083401-fig1.png
Fig. 1. KLL Dielectronic recombination cross sections of the H-like helium ions versus the incident electron energy for different Debye shielding lengths $\lambda$.
To demonstrate the reliability of our calculation, as an example, we list the eigenenergies and widths of the doubly excited $2p^2$ $^1D_{2}$ and $2p^2$ $^3P_2$ states for different Debye shielding lengths $\lambda$ in Table 1. For the purpose of comparison, other available results[8,9,20–22] are also presented. It is found that with the decrease of the Debye shielding length $\lambda$, the eigenenergies increase while the widths decrease. Very good agreement can be found between the present calculations and other calculations for both eigenenergies and widths.[8,9] The KLL DR cross sections of the H-like helium ions for different Debye shielding lengths $\lambda$ are given in Fig. 1. It can be seen that the DR spectroscopy shows three different separate peaks in the case of no plasma shielding. With the decrease of the Debye shielding length, all these peaks shift to high energy. It can also be found that the cross sections for $2s_{1/2}2p_{1/2}$ $^1P_1$ and $2p^2_{1/2}$ $^1S_0$ are nearly unchanged. However, the left shoulder of the middle peak, that is, the cross sections for $2p^2_{3/2}$ $^3P_2$, rises gradually when the Debye shielding length decreases. To explain the above phenomenon, Fig. 2 shows the eigenenergies of the doubly excited $2p{^2}$ ${^1D_2}$ and $2p{^2}$ ${^3P_2}$ states of helium atom versus the inverse of the Debye shielding length $\lambda$. It can be seen that the eigenenergies of the both states will shift toward to higher energy with the decrease of the Debye shielding length $\lambda$. This shifting to the higher energy will also lead to a change of configuration interaction. The $2p^2$ $^3P_2$ state is mainly written as the linear combination of $(2p^2_{3/2})_2$ and $(2p_{1/2}2p_{3/2})_2$, that is, $$\begin{align} | 2p^2, ^3P_2 \rangle= c_1 |(2p^2_{3/2})_2\rangle+c_2 |(2p_{1/2}2p_{3/2})_2\rangle+\ldots,~~ \tag {9} \end{align} $$ where $c_1$ and $c_2$ are the mix coefficients of $(2p^2_{3/2})_2$ and $(2p_{1/2}2p_{3/2})_2$ configuration states, respectively. Figure 3 shows the square of configuration mixing coefficients $c_1$ and $c_2$ versus the inverse of the Debye shielding length $\lambda$. It can be seen that the mixing coefficient of $(2p^2_{3/2})_2$ decreases gradually and the mixing coefficient of $(2p_{1/2}2p_{3/2})_2$ increases with the decrease of the Debye shielding length $\lambda$.
cpl-36-8-083401-fig2.png
Fig. 2. Eigenenergies as a function of the inverse of the Debye shielding length $\lambda$ for doubly excited $2p^2$ $^1D_2$ and $2p^2$ $^3P_2$ states of the helium atom.
cpl-36-8-083401-fig3.png
Fig. 3. Square of configuration mixing coefficients of the $2p^2$ $^3P_2$ state versus the inverse of the Debye shielding length $\lambda$ for the helium atom.
Figure 4 shows the Auger rates versus the inverse of the Debye shielding length $\lambda$ for the doubly excited $2p^2$ $^1D_2$ and $2p^2$ $^3P_2$ states, respectively. It can be found that the excited $2p^2$ $^1D_2$ state has very large Auger decay rate and the order of magnitude reaches $10^{14}$. The Auger rate of the doubly excited $2p^2$ $^1D_2$ state decreases with the Debye shielding length $\lambda$ and it is relatively small. However, Auger rate of the doubly excited $2p^2$ $^3P_2$ state increases with the decrease of the Debye shielding length $\lambda$. The variety is about three hundred times, from $\lambda$=infinity (8.18$\times10^7$ $s^{-1}$) to $\lambda=20a_0$ (2.56$\times10^{10}$ $s^{-1}$). From this fact, we can deduce that the Auger matrix element $\langle1s\varepsilon l|\frac{1}{r_{12}}|(2p_{1/2}2p_{3/2})_2\rangle$ is much larger than $\langle1s\varepsilon l|\frac{1}{r_{12}}|(2p^2_{3/2})_2\rangle$. With the decrease of the Debye shielding length $\lambda$, the component of the $(2p_{1/2}2p_{3/2})_2$ in the excited $2p^2$ $^3P_2$ state increases gradually (see Fig. 3). Thus the increase in the mixing of $(2p_{1/2}2p_{3/2})_2$ is just the reason for the increase of the Auger rate for the excited $2p^2$ $^3P_2$ state when the Debye shielding length $\lambda$ decreases.
cpl-36-8-083401-fig4.png
Fig. 4. Auger rates as a function of the inverse of the Debye shielding length $\lambda$ for the doubly excited $2p^2$ $^1D_2$ state of helium atom.
cpl-36-8-083401-fig5.png
Fig. 5. Radiative branching ratio for doubly excited $2p^2$ $^3P_2$ and $2p^2$ $^1D_2$ states of the helium atom versus the inverse of the Debye shielding length $\lambda$.
In Fig. 5, we plot radiative branching ratio of the doubly excited $2p^2$ $^3P_2$ and $2p^2$ $^1D_2$ states of helium atom versus the inverse of Debye shielding length $\lambda$. We can find that the radiative branching ratio of the $2p^2$ $^1D_2$ state is very small and increases with the decrease of the Debye shielding length. The variety range is also very small. However, the radiative branching ratio of the $2p^2$ $^3P_2$ state decreases with the Debye shielding length $\lambda$. In comparison of the case of no plasma shielding and the Debye shielding length $\lambda=20$, the radiative branching ratio of the doubly excited $2p^2$ $^3P_2$ state reduces by about one order of magnitude. Since the auger rate increases by approximately three hundred times (as mentioned in Fig. 4), we can conclude that the DR strength of the doubly excited $2p^2$ $^3P_2$ state increases with the decrease of the Debye shielding length $\lambda$. Consequently, it results in the rise of the left shoulder, as shown in Fig. 1. In summary, we have investigated the KLL DR processes of the H-like helium ions for different Debye lengths within the framework of the MCDF approximation. In contrast to other doubly excited $2l2l'$ states, the influence of the Debye plasma on the excited $2p^2$ $^3P_2$ state is very strong and can increase its Auger decay rate by several orders of magnitude. Thus the shape of DR cross sections of the H-like helium ions changes significantly. References Dense plasmas, screened interactions, and atomic ionizationDoubly excited 2 s 2 p P o 1 , 3 resonance states of helium in dense plasmas Be I isoelectronic ions embedded in hot plasmaThe Ionization Equilibrium for Iron in the Solar CoronaIntroductionDoubly-excited 2s21Se resonance state of helium embedded in Debye plasmasThe doubly-excited 2p 2 1 D e resonance state of the helium atom in hot-dense plasmasDoubly excited 1,3 P e meta-stable bound states and resonance states of helium in weakly coupled plasmasDynamics of He 2 + + H ( 1 s ) excitation and electron-capture processes in Debye plasmasDynamics of O 8 + + H electron capture in Debye plasmasCrossover of Feshbach Resonances to Shape-Type Resonances in Electron-Hydrogen Atom Excitation with a Screened Coulomb InteractionAtomic excitation in dense plasmasGRASP92: A package for large-scale relativistic atomic structure calculationsThe transverse electron-electron interaction in atomic structure calculationsPlasma-screening effects on electron-impact excitation of hydrogenic ions in dense plasmasElectron-impact ionization of atoms in high-temperature dense plasmasTheoretical study of the KLL dielectronic recombination for highly charged iodine ionsDoubly excited states in some light atomsEigenvalues of the 2 p 3 p P 3 and 2 p 3 d D 1 , 3 Bound States of the Helium Isoelectronic SequenceTwo-electron excited states of helium
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