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Theoretical Calculation for the Energy and Transition Energy of O6+ Ion in Confined Environment
DOI: 10.12677/MP.2023.134010, PDF, HTML, XML, 下载: 272  浏览: 412  科研立项经费支持

Abstract: Based on the variational principle, an analytical method is presented to deal with the nonrelativistic energies of atoms (ions) and relativistic corrections in Debye and quantum plasma environments. This method can not only deal with single-electron atoms, but also be easily extended to multi-electron atomic systems. The energy level and transition energy of O6+ ion in ground state 1s2(1S) and excited state 1s2p(1P) and 1s2p(3P) are calculated, and the influence of plasma shielding parameters on the energy level and transition energy is analyzed.

1. 引言

2. 理论模型

2.1. 哈密顿量

${H}_{NR}=-\frac{1}{2}{\nabla }_{1}^{2}-\frac{1}{2}{\nabla }_{2}^{2}+\frac{Z}{{r}_{1}}+\frac{Z}{{r}_{2}}+\frac{1}{{r}_{12}}$(1)

$V\left(r\right)=\frac{Z}{r}{\text{e}}^{-\mu r}$ (2)

$V\left(r\right)=\frac{Z}{r}{\text{e}}^{-\mu r}\mathrm{cos}\left(\mu r\right)$ (3)

2.2. 变分法

$E=〈\gamma LS{M}_{L}{M}_{S}|{H}_{NR}|\gamma LS{M}_{L}{M}_{S}〉$(4)

$|{}^{1}\text{S},0,0〉=‖{\psi }_{1s0+}\left({\stackrel{⇀}{x}}_{1}\right){\psi }_{1s0-}\left({\stackrel{⇀}{x}}_{2}\right)‖$ (5)

$|{}^{1}\text{P},1,0〉=\frac{1}{\sqrt{2}}\left[‖{\psi }_{1s0+}\left({\stackrel{⇀}{x}}_{1}\right){\psi }_{2p1-}\left({\stackrel{⇀}{x}}_{2}\right)‖-‖{\psi }_{1s0-}\left({\stackrel{⇀}{x}}_{1}\right){\psi }_{2p1+}\left({\stackrel{⇀}{x}}_{2}\right)‖\right]$ (6)

$|{}^{3}\text{P},1,1〉=‖{\psi }_{1s0+}\left({\stackrel{⇀}{x}}_{1}\right){\psi }_{2p1+}\left({\stackrel{⇀}{x}}_{2}\right)‖$ (7)

${\psi }_{{n}_{i}{\mathcal{l}}_{i}{m}_{\mathcal{l}i}{m}_{si}}\left({\stackrel{⇀}{x}}_{i}\right)={R}_{{n}_{i}{\mathcal{l}}_{i}}\left({r}_{i}\right){Y}_{{\mathcal{l}}_{i}{m}_{\mathcal{l}i}}\left({\theta }_{i}{\varphi }_{i}\right){\chi }_{{m}_{si}}\left({s}_{i}\right)$(8)

${R}_{n\mathcal{l}}\left(r\right)={N}_{n\mathcal{l}}\mathrm{exp}\left(-\frac{{\alpha }_{n\mathcal{l}}r}{n}\right)\underset{\nu =0}{\overset{n-1}{\sum }}\frac{n!}{\nu !\left(n-\nu -1\right)!\left(\nu +1\right)!}{\left(-\frac{2{\alpha }_{n\mathcal{l}}r}{n}\right)}^{\nu }$(9)

$\underset{0}{\overset{\infty }{\int }}{R}_{n\mathcal{l}}^{\ast }\left(r\right){R}_{n\mathcal{l}}\left(r\right){r}^{2}\text{d}r=1$ . (10)

$I\left(n\mathcal{l}\right)=\frac{1}{2}{\int }_{0}^{\infty }r{R}_{n\mathcal{l}}\left(r\right)\left[-\frac{{\text{d}}^{2}}{\text{d}{r}^{2}}+\frac{\mathcal{l}\left(\mathcal{l}+1\right)}{{r}^{2}}-2V\left(r\right)\right]r{R}_{n\mathcal{l}}\left(r\right)\text{d}r$(11)

${R}^{\left(k\right)}\left({n}_{i}{\mathcal{l}}_{i}{n}_{j}{\mathcal{l}}_{j},{n}_{i\text{'}}{\mathcal{l}}_{i\text{'}}{n}_{j\text{'}}{\mathcal{l}}_{j\text{'}}\right)={\int }_{0}^{\infty }{\int }_{0}^{\infty }{R}_{{n}_{i}{\mathcal{l}}_{i}}\left({r}_{1}\right){R}_{{n}_{i}{\mathcal{l}}_{i}}\left({r}_{1}\right){R}_{{n}_{i\text{'}}{\mathcal{l}}_{i\text{'}}}\left({r}_{2}\right){R}_{{n}_{j\text{'}}{\mathcal{l}}_{j\text{'}}}\left({r}_{2}\right)\frac{{r}_{<}^{k}}{{r}_{>}^{k+1}}{r}_{1}^{2}{r}_{2}^{2}\text{d}{r}_{1}\text{d}{r}_{2}$ . (12)

2.3. 能级表达式

${E}_{\text{SCP/ESCP}}\left({\text{1s}}^{\text{2}}{}^{\text{1}}\text{S}\right)=2I\left(\text{1s}\right)+{R}^{0}\left(\text{1s1s},\text{1s1s}\right)$ (13)

${E}_{\text{SCP/ESCP}}\left(\text{1s2p}{}^{\text{1}}\text{P}\right)=I\left(\text{1s}\right)+I\left(\text{2p}\right)+{R}^{0}\left(\text{1s2p},\text{1s2p}\right)-\frac{1}{6}{R}^{1}\left(\text{1s2p},\text{2p1s}\right)$ (14)

${E}_{\text{SCP/ESCP}}\left(\text{1s2p}{}^{\text{3}}\text{P}\right)=I\left(\text{1s}\right)+I\left(\text{2p}\right)+{R}^{0}\left(\text{1s2p},\text{1s2p}\right)-\frac{1}{3}{R}^{1}\left(\text{1s2p},\text{2p1s}\right)$ (15)

${E}_{\text{SCP}}\left({\text{1s}}^{\text{2}}{}^{\text{1}}\text{S}\right)=\frac{5a}{8}+{a}^{2}-\frac{8{a}^{3}z}{{\left(2a+u\right)}^{2}}$ (16)

$\begin{array}{c}{E}_{\text{SCP}}\left(\text{1s2p}{}^{\text{1}}\text{P}\right)=\frac{{a}^{2}}{2}+\frac{{c}^{2}}{8}-\frac{56{a}^{3}{c}^{5}}{3{\left(2a+c\right)}^{7}}+\frac{8{a}^{5}c}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{4}{c}^{2}}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{3}{c}^{3}}{{\left(2a+c\right)}^{5}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{10{a}^{2}{c}^{4}}{{\left(2a+c\right)}^{5}}+\frac{a{c}^{5}}{{\left(2a+c\right)}^{5}}-\frac{4{a}^{3}z}{{\left(2a+c\right)}^{2}}-\frac{{c}^{5}z}{4{\left(c+u\right)}^{4}}\end{array}$ (17)

$\begin{array}{c}{E}_{\text{SCP}}\left(\text{1s2p}{}^{\text{3}}\text{P}\right)=\frac{{a}^{2}}{2}+\frac{{c}^{2}}{8}-\frac{112{a}^{3}{c}^{5}}{3{\left(2a+c\right)}^{7}}+\frac{8{a}^{5}}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{4}{c}^{2}}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{3}{c}^{3}}{{\left(2a+c\right)}^{5}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{10{a}^{2}{c}^{4}}{{\left(2a+c\right)}^{5}}+\frac{a{c}^{5}}{{\left(2a+c\right)}^{5}}-\frac{4{a}^{3}z}{{\left(2a+u\right)}^{2}}-\frac{{c}^{5}z}{4{\left(c+u\right)}^{4}}\end{array}$ (18)

${E}_{\text{ESCP}}\left({\text{1s}}^{\text{2}}{}^{\text{1}}\text{S}\right)=\frac{5a}{8}+{a}^{2}-\frac{8{a}^{5}z}{{\left(2{a}^{2}+2au+{u}^{2}\right)}^{2}}-\frac{8{a}^{4}uz}{{\left(2{a}^{2}+2au+{u}^{2}\right)}^{2}}$ (19)

$\begin{array}{c}{E}_{\text{ESCP}}\left(\text{1s2p}{}^{\text{1}}\text{P}\right)=\frac{{a}^{2}}{2}+\frac{{c}^{2}}{8}-\text{}\frac{56{a}^{3}{c}^{5}}{3{\left(2a+c\right)}^{7}}+\frac{8{a}^{5}c}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{4}{c}^{2}}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{3}{c}^{3}}{{\left(2a+c\right)}^{5}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{10{a}^{2}{c}^{4}}{{\left(2a+c\right)}^{5}}+\frac{a{c}^{5}}{{\left(2a+c\right)}^{5}}-\frac{4{a}^{5}z}{{\left(2{a}^{2}+2au+{u}^{2}\right)}^{2}}-\frac{4{a}^{4}uz}{{\left(2{a}^{2}+2au+{u}^{2}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{c}^{9}z}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}-\frac{{c}^{8}uz}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}+\frac{2{c}^{6}{u}^{3}z}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}+\frac{{c}^{5}{u}^{4}z}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}\end{array}$ (19)

$\begin{array}{c}{E}_{\text{ESCP}}\left(\text{1s2p}{}^{\text{3}}\text{P}\right)=\frac{{a}^{2}}{2}+\frac{{c}^{2}}{8}-\frac{112{a}^{3}{c}^{5}}{3{\left(2a+c\right)}^{7}}+\frac{8{a}^{5}c}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{4}{c}^{2}}{{\left(2a+c\right)}^{5}}+\frac{20{a}^{3}{c}^{3}}{{\left(2a+c\right)}^{5}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{10{a}^{2}{c}^{4}}{{\left(2a+c\right)}^{5}}+\frac{a{c}^{5}}{{\left(2a+c\right)}^{5}}-\frac{4{a}^{5}z}{{\left(2{a}^{2}+2au+{u}^{2}\right)}^{2}}-\frac{4{a}^{4}uz}{{\left(2{a}^{2}+2au+{u}^{2}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{c}^{9}z}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}-\frac{{c}^{8}uz}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}+\frac{2{c}^{6}{u}^{3}z}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}+\frac{{c}^{5}{u}^{4}z}{{\left[{u}^{2}+{\left(c+u\right)}^{2}\right]}^{4}}\end{array}$ (20)

3. 结果和讨论

3.1. 变分参数和非相对论能级的计算

Table 1. Energy level of the ground state 1s21S of O6+ ion in plasma environment (Unit: a.u.)

Table 2. Energy level of the 1s2p1P state O6+ ion in plasma environment (Unit: a.u.)

Table 3. Energy level of the 1s2p3P state O6+ ion in plasma environment (Unit: a.u.)

3.2. 跃迁能计算结果

Figure 1. Transition energies as function of shied parameters

4. 总结

NOTES

*通讯作者。

 [1] Debye, P. and Hückel, E. (1923) Debye-Hückel-Type Relaxation Processes in Solid Ionic Conductors: The Model. Physikalische Zeitschrift, 24, 185-190. [2] Shukla, P.K. and Eliasson, B. (2008) Screening and Wake Potentials of a Test Charge in Quantum Plasmas. Physics Letters A, 372, 2897-2899. https://doi.org/10.1016/j.physleta.2007.12.067 [3] Ray, D. (2000) Influence of a Dense Plasma on the Fi-ne-Structure Levels of a Hydrogenic Ion. Physical Review E, 62, 4126-4130. https://doi.org/10.1103/PhysRevE.62.4126 [4] Saha, J.K., Bhattacharyya, S., Mukherjee, T.K. and Mukherjee, P.K. (2010) 1,3Do and 1,3Pe States of Two Electron Atoms under Debye Plasma Screening. Journal of Quantitative Spec-troscopy and Radiative Transfer, 111, 675-688. https://doi.org/10.1016/j.jqsrt.2009.11.026 [5] Fang, T.K., Wu, C.S., Gao, X. and Chang, T.N. (2017) Redshift of the Heα Emission Line of He-Like Ions under a Plasma Environment. Sical Review A, 96, Article ID: 052502. https://doi.org/10.1103/PhysRevA.96.069906 [6] Kar, S. and Ho, Y.K. (2005) Doubly-Excited 2s21Se Resonance State of Helium Embedded in Debye Plasmas. Chemical Physics Letters, 402, 544-548. https://doi.org/10.1016/j.cplett.2004.12.099 [7] Xie, L.Y., Wang, J.G., Janev, R.K., Qu, Y.Z. and Dong, C.Z. (2012) Energy Levels and Multipole Transition Properties of C4+ Ion in Debye Plasmas. The European Physical Journal D, 66, Article No. 125. https://doi.org/10.1140/epjd/e2012-20594-6 [8] Chen, Z.B. (2017) Circular Polarization of X-Ray Radiation Emitted by Longitudinally Polarized Electron Impact Excitation: Under a Screened Coulomb Interaction. Physics of Plasmas, 24, Article ID: 122119. https://doi.org/10.1063/1.5005550 [9] Chen, Z.B., Ma, K., Hu, H.W. and Wang, K. (2018) Relativistic Effects on the Energy Levels and Radiative propeRties of He-Like Ions Immersed in Debye Plasmas. Physics of Plasmas, 25, Arti-cle ID: 072120. https://doi.org/10.1063/1.5040806 [10] Chaudhuri, S.K., Mukherjee, P.K. and Fricke, B. (2017) Atomic Structure under External Confinement: Effect of Plasma on the Spin Orbit Splitting, Relativistic Mass Correction and Darwin Term for Hydrogen-Like Ions. The European Physical Journal D, 71, Article No. 71. https://doi.org/10.1140/epjd/e2017-70511-6 [11] Hu, H.W., Chen, Z.B. and Chen, W.C. (2016) Radiative Transi-tion of Hydrogen-Like Ions in Quantum Plasma. Radiation Effects and Defects in Solids, 171, 890-903. https://doi.org/10.1080/10420150.2016.1257618