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The boundary region between the solar radiation zone and the convection zone ($ T\thicksim180$ eV, $ n_e\thicksim $$ 9\times10^{22}\;{\rm{cm}}^{-3}$) is a critical interface where energy transport in the solar interior transitions from radiation-dominated to convection-dominated regimes. This region also serves as a natural laboratory for studying hot dense plasma. The physical properties of this zone are essential for the reliability of stellar evolution models and the stability of energy transport mechanisms. One of major unresolved issue is how electron collision ionization affects the density of free electrons and radiation properties in this plasma, while accurately describing the impact of hot-dense environments on electron impact ionization (EII) (such as electron screening, ion correlation). To fill this gap, we systematically calculate EII cross sections for C, N, and O ions under realistic solar boundary conditions, with a focus on hot-dense environment impacts. We develop a novel computational framework that merges?hot-dense environment effects into atomic structure calculations: the Flexible Atomic Code (FAC) for atomic structure is combined with the Hyper-netted-Chain (HNC) approximation to capture electron-electron, electron-ion and ion-ion correlations, enabling self-consistent treatment of electron screening and ion correlation. Atomic wave functions are derived by solving the Dirac equation within the ion-sphere model, using a modified central potential that incorporates both free-electron screening and ion–ion interactions. EII cross sections are then computed via the Distorted-Wave (DW) approximation in FAC. The results demonstrate that hot-dense environment effects significantly enhance the electron-impact ionization cross sections of C, N, and O compared to those calculated under the free-atom model. Additionally, a notable reduction in the ionization threshold energy is observed. These effects are attributed to the overlap of atomic potentials due to strong ion coupling and the shift in bound-state energy levels caused by free-electron screening. For instance, under solar boundary conditions, the ionization cross section of C+ increased by up to 50%, with the ionization threshold decreasing from about 24 eV (isolated) to 18 eV (with screening). Similar enhancements were observed for nitrogen and oxygen ions across various charge states. By providing updated ionization cross sections for C, N, and O ions under realistic solar interior conditions, this work offers essential parameters for improving radiation transport models, ionization balance calculations, and equation-of-state models in stellar interiors. The results underscore the necessity of including hot-dense environment effects in atomic process calculations for hot dense plasmas, with implications for astrophysics and inertial confinement fusion research.
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Keywords:
- radiation/convection zone boundary /
- environment effect /
- electron impact ionization /
- hypernetted-chain approximation
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图 1 孤立情况下, 类铍离子电子碰撞电离截面随着入射电子能量变化的关系. (a) C2+的结果; (b) N3+的结果; (c) O4+的结果. 带有误差的黑色正方形是Fogle等人[5]实验测量的结果, 红色实线和绿色点线是Fogle等人分别采用组态平均扭曲波(CADW) 近似[56]与赝态R-matrix(RMPS) 近似[57]计算的理论结果, 蓝色虚线、橙色点-虚线和紫色点-点-虚线是本文分别采用DW近似、CBE近似和BED近似理论计算的结果
Figure 1. The electron impact ionization cross section of beryllium-like ions as a function of incident electron energy for the isolation ion. (a) The case of C2+; (b) The case of N3+; (c) The case of O4+. The black squares with error bars[5] represent the experimental measurements by Fogle et al. The red solid and green dotted lines represent the theoretical results calculated by Fogle et al. by using the CADW approximation[56] and the RMPS[57] approximation, respectively. The blue dashed, orange dotted-dashed, and violet dotted-dotted-dashed lines represent the theoretical results calculated in this paper using the DW, CBE, and BED approximation, respectively.
图 2 热稠密等离子体中C2+的$ 1 s^22 s^2\; ^{1}S_{0} $组态$ 2 s $束缚电子径向波函数. (a) 在温度为100 eV时, 径向波函数随等离子体密度的变化. 黑色实线为孤立情况下计算的结果; 蓝色点-虚线、绿色虚线、红色点线是T为100 eV, $ n_e $分别为$ 3\times10^{22} $ cm–3、$ 6\times10^{22} $ cm–3、$ 15\times10^{22} $ cm–3情况下计算的结果; (b) 在相同密度$ n_e $为$ 9\times10^{22} $ cm–3下, 径向波函数随等离子体温度的变化. 黑色实线为孤立情况下的计算结果; 青色点线、蓝色点-虚线、棕色点-点-虚线是T分别为50 eV、100 eV和180 eV情况下计算的结果.
Figure 2. The radial wave function of the $ 2 s $ bound electron in the $ 1 s^22 s^2\; ^{1}S_{0} $ configuration of C2+ in hot dense plasmas. (a) The black solid line represents the result calculated for the isolated case. The blue dotted-dashed, green dashed, and red dotted lines represent the results calculated for $ T = 100 eV $ and $ n_e $ values of $ 3\times10^{22} $ cm–3, $ 6\times10^{22} $ cm–3, and $ 15\times10^{22} $ cm–3, respectively; (b) The black solid line represents the result calculated for the isolated case. The cyan dotted, blue dotted-dashed, and brown dotted-dotted-dashed lines represent the results calculated for $ n_e = 9\times10^{22} $ cm–3 and T values of 50 eV, 100 eV, and 180 eV, respectively.
图 3 随着入射电子能量变化的C2+的$ 1 s^22 s^2\; ^{1}S_{0}\; \rightarrow 1 s^22 s^1 $的电子碰撞电离截面. (a) 温度T为100 eV、不同电子密度的碰撞电离截面计算结果. 黑色实线为孤立情况下计算的结果; 绿色点线、紫色点-点-虚线、红色虚线是电子密度$ n_e $分别为$ 3\times10^{22} $ cm–3、$ 6\times10^{22} $ cm–3、$ 15\times10^{22} $ cm–3情况下计算的结果. (b) 电子密度$ n_e $为$ 9\times10^{22} $ cm–3、不同温度的碰撞电离截面计算结果. 黑色实线为孤立情况下的计算结果; 橙色点-虚线、绿色点线、蓝色虚线是温度T分别为50 eV、100 eV和180 eV情况下计算的结果
Figure 3. The electron impact ionization cross section of $ 1 s^22 s^2\; ^{1}S_{0} \rightarrow 1 s^22 s^1 $ of C2+ as a function of incident electron energy. (a) The collision ionization cross sections with temperature T of 100 eV and different electron densities. The black solid line represents the result calculated for the isolated case. The green dotted, violet dotted-dotted-dashed, and red dashed lines represent the results calculated for $ n_e $ values of $ 3\times10^{22}cm^{-3} $, $ 6\times10^{22} $ cm–3, and $ 15\times10^{22} $ cm–3, respectively; (b) The collision ionization cross sections with electron density $ n_e = 9\times10^{22} $ cm–3 and different temperatures. The black solid line represents the result calculated for the isolated case. The orange dotted-dashed, green dotted, and blue dashed lines represent the results calculated for T values of 50 eV, 100 eV, and 180 eV, respectively.
图 4 碳元素在孤立情况下以及在$ T = 180 $ eV, $ n_e = 9\times10^{22} $ cm–3的太阳辐射/对流区域边界处条件下, 随着入射电子能量变化的电子碰撞电离截面. (a) 黑色、红色和绿色分别为C+、C2+和C3+的碰撞电离截面计算结果. (b) 蓝色和橙色分别为C4+和C5+的碰撞电离截面计算结果. 在计算中分别考虑了在孤立情况(isolated)、屏蔽效应(no_IPD)和屏蔽效应+电离能(IPD)下降情况下对碰撞电离截面的影响, 在图中分别采用实线、虚线和点线表示
Figure 4. The electron impact ionization cross sections of carbon under isolated conditions and at the solar radiation/convection zone boundary($ T = 180 $ eV, $ n_e = 9\times10^{22} $ cm–3) as functions of incident electron energy. (a) Black, red, and green lines represent the calculated results for C+, C2+, and C3+, respectively; (b) Blue and orange lines correspond to C4+and C5+, respectively. Here, the solid line represents the calculation results in the isolated case, the dashed line represents the calculation results considering the screening effect at the boundary of the solar radiation/convective region, and the dotted line represents the calculation results further considering the ionization potential depression on the basis of the dashed line.
图 5 氮、氧元素在$ T = 180 $ eV, $ n_e = 9\times10^{22} $ cm–3的太阳辐射/对流区域边界处条件下, 随着入射电子能量变化的电子碰撞电离截面. (a) 黑色实线、红色点线、绿色虚线和蓝色点-虚线分别为N+、N2+、N3+和N4+的计算结果; (b) 黑色实线、红色点线、绿色虚线、蓝色点-虚线和橙色点-点-虚线分别为O+、O2+、O3+、O4+和O5+ 的计算结果; (c) 紫色实线、青色点线、粉色虚线和棕色点-虚线分别为N5+、N6+、O6+和O7+的计算结果
Figure 5. The electron impact ionization cross sections for nitrogen and oxygen at the solar radiation/convection zone boundary($ T = 180 $ eV, $ n_e = 9\times10^{22} $ cm–3) as functions of incident electron energy. (a) Black solid, red dotted, green dashed, and blue dotted-dashed lines represent calculated results for N+, N2+, N3+, and N4+, respectively; (b) Black solid, red dotted, green dashed, blue dotted-dashed, and orange dotted-dotted-dashed lines correspond to O+, O2+, O3+, O4+, and O2+, respectively; (c) Purple solid, cyan dotted, pink dashed, and brown dotted-dashed lines show results for N5+, N6+, O6+, and O7+, respectively.
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