Electron correlation and Breit interaction effects on the one- and two-electron one-photon transitions in double K hole states of He-like ions(10\!\leq\!Z\!\leq\!47)

Electron correlation and Breit interaction effects on the one- and two-electron one-photon transitions in double K hole states of He-like ions()

Xiaobin Ding dingxb@nwnu.edu.cn Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering,Northwest Normal University, Lanzhou 730070, China    Mingxin Cao Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering,Northwest Normal University, Lanzhou 730070, China    Denghong Zhang Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering,Northwest Normal University, Lanzhou 730070, China    Mingwu Zhang Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China    Yingli Xue Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China    Deyang Yu Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China    Chenzhong Dong Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering,Northwest Normal University, Lanzhou 730070, China
August 2, 2019
Abstract

The x-ray energies and transition rates of single and double electron radiative transitions from double K hole state to and configuration of 11 He-like ions () are evaluated using fully relativistic multi-configuration Dirac-Fock method. An appropriate electron correlation model was constructed with the aid of active space method, with which the electron correlation effects can be studies efficiently. The contributions from electron correlation, Breit interaction to the transition properties have been analyzed in detail. It is found that the two-electron one-photon (TEOP) transition is correlation sensitive. The Breit interaction and electron correlation both have significant contribution to the radiative transition properties of double K hole state of He-like ions. A good agreement between the present calculation and the previous work was achieved. The calculated data will be helpful for the future investigations on double K hole decay process of He-like ions.

pacs:
31.25.-v,31.15.vj

I Introduction

The energy level structures and radiative decay processes of the inner-shell hole state are one of the important issues of atomic physics Hoogkamer et al. (1976); Briand (1976); Nagel et al. (1976); Åberg et al. (1976); Stoller et al. (1976); Safronova and Senashenko (1977). The inner-shell hole state is a special excited state which refers to the inner-shell orbital of an atom or ion is unoccupied, while the outer shell orbital is occupied by electrons. The inner shell hole state have been observed in high-energy ion-atom collisions process Kumar et al. (2006); Hutton et al. (1991); Zou et al. (2003), synchrotron radiation Diamant et al. (2000, 2000), laser-produced plasmas Boiko et al. (1977), ion beam-foil spectroscopy Andriamonje et al. (1991), tokamak Bitter et al. (1984), and solar flares Phillips (2004). Moreover, it can also be produced by the electron excitation or ionization of the inner-shell of atom or ions López-Urrutia et al. (2005), as well as the inner-shell photoionization or photoexcitation process with high-energy photons Fennane et al. (2009). These exotic atoms are extremely unstable and mainly decay through non-radiative Auger process Chen (1991); Natarajan and Natarajan (2008); Inhester et al. (2012) and radiative process. The former is usually faster than the later. With the development of X-ray spectroscopy, the technology of detection on the weak signals helps scientists to understand such processes from the photon perspective.

It is also possible to create an ion where innermost shell were empty, forming a double K shell hole state Hoszowska et al. (2009). Generally, the radiative de-excitation of an atom with an initially empty K shell may take place either through the more probable one-electron one-photon (OEOP) transition or through the competing weak two-electron one-photon (TEOP) transition. For the initially double K hole state in He-like ion system, it can decay through either OEOP transition to single excited states in which a electron transition to with a spectator electron, or TEOP transition in which both two electrons in and orbitals transition to orbital simultaneously to ground states due to the electron correlation effects. The TEOP process was first predicted theoretically by Heisenberg in 1925 Heisenberg (1925) and was observed from the ion-atom collision experiments between Ni-Ni, Ni-Fe, Fe-Ni and Fe-Fe by Wölfli et al. in 1975 Wölfli et al. (1975). Since then, the TEOP transitions have been widely studied both theoretically and experimentally Porquet et al. (2010); Decaux et al. (1997); Natarajan and Natarajan (2007); Natarajan (2013, 2016); Lin et al. (1977); Kadrekar and Natarajan (2011); Natarajan and Kadrekar (2013); Natarajan (2014); Elton et al. (2000); Trabert et al. (1982); Schäffer et al. (1999); Tawara and Richard (2002).

TEOP process is strictly forbidden in independent particle approximation of atom. Investigations on this process are helpful to explain the electron correlation effects, relativistic effects and quantum electro-dynamics (QED) effects on the energy level structure and radiative transition of these exotic atoms. It is also helpful to reveal the situation of the electron coupling in the complex atom system. For the astrophysics and laboratory plasma, some important diagnostics information, such as the composition, temperature and density were also provided by these basic atomic physics processes Porquet et al. (2010); Decaux et al. (1997).

There are many works related to the energy levels and transition properties of inner-shell hole state in the past several decades Natarajan and Natarajan (2007); Natarajan (2013, 2016); Lin et al. (1977); Kadrekar and Natarajan (2011); Natarajan and Kadrekar (2013); Natarajan (2014); Elton et al. (2000); Trabert et al. (1982); Schäffer et al. (1999); Tawara and Richard (2002); Kadrekar and Natarajan (2010); Li et al. (2010), where only a few studies focused on He-like ions Lin et al. (1977); Kadrekar and Natarajan (2011); Natarajan and Kadrekar (2013); Natarajan (2014); Tawara and Richard (2002); Trabert et al. (1982); Schäffer et al. (1999); Elton et al. (2000). The He-like ion is a two-electron system with simple structure and electron correlation effect and it can be used as a good candidate to study the TEOP process. Kadrekar and Natarajan calculated transition property and the branching ratios between OEOP and TEOP transition in He-like ions with configuration using multi-configuration Dirac-Fock (MCDF) method Kadrekar and Natarajan (2011) and they find that the contribution from TEOP transition is considerable for low-Z ions. The influence of configuration interaction on single-electron allowed E1 transitions is negligible. They also calculated both OEOP and TEOP transition from and of He-like Ni, including electric dipole transition (E1)and magnetic quadrupole transition (M2) Natarajan and Kadrekar (2013) and they find that higher order corrections are more important for the than for the transitions of He-like Ni. After that, Natarajan conducted a research on the orthogonality of the basis. The biorthogonal and common basis sets give almost the same transition rates for light and medium heavy elements while the differences are substantial for heavy elements Natarajan (2014). The contributions from correlation and higher-order corrections consist of Breit and QED effects to the energies and transition rates were analyzed. Experimentally, the transitions from - in He-like Si has been observed in laser-produced plasma experiments at the TRIDENT facility by Elton et al. Elton et al. (2000). Tawara and Richard et al. observe the Ar K X-rays under 60 keV/u Ar-Ar collisions from the KSU EBIS Tawara and Richard (2002).

Previous theoretical and experimental investigation on the OEOP and TEOP transition mostly focused on the low-Z atoms, while only few work on the high-Z ions Kadrekar and Natarajan (2010); Li et al. (2010). The present work reports a MCDF calculation of OEOP and TEOP transition from double K hole configuration in 11 selected He-like ions (). The electron correlation effects were taken into account by choosing appropriate electron correlation model using active space method. The Breit interaction and QED effects were included perturbatively in relativistic configuration interaction (RCI) calculation. The finite nuclear size effects were described by a two-parameter Fermi distribution model. The purpose of present calculations is to explore the variation trend of electron correlation effect and Breit interaction on the transition energies and rates of OEOP and TEOP transitions with increase of Z. The present results will be helpful for the future theoretical and experimental work on the radiative decay processes of the double K hole states. The calculation has been performed by using Grasp2K code Jönsson et al. (2007).

Ii Theoretical method

The multi-configuration Dirac-Fock (MCDF) method is widely used to investigate the relativistic, electron correlation, Breit interaction and quantum electrodynamics effects (QED) effects on the structure and transition of complex atoms or ions, which based on a relativistic atomic theory Ding et al. (2011, 2017); Aggarwal and Keenan (2016). The method was expounded in Grant’s monograph Grant (2007) and had been implemented by Grasp family code Grant et al. (1980); Desclaux (1984); Dyall et al. (1989); Parpia et al. (1996); Jönsson et al. (2007). Here, only a brief introduction about MCDF method was given.

In the MCDF method, the atomic state wave function (ASFs) () for a given state with certain parity P, total angular momentum J, and its z component is represented by a linear combination of configuration state functions (CSFs) () with the same P, J, , which can be expressed as:

(1)

where is the number of CSFs and denotes all the other quantum numbers necessary to define the configuration, is the mixing coefficient. The CSFs are the linear combinations of Slater determinants of the many-particle system consisting of single electron orbital wave functions. The extended optimal level (EOL) mode is used in the self-consistent field (SCF) calculation to optimize the radial wave functions. The mixing coefficients of CSFs are determined variationally by optimizing the energy expectation value of the Dirac-Coulomb Hamiltonian, which is defined as in the following equation:

(2)

The RCI calculations have been done by including higher-order interactions in the Hamiltonian. The transverse photon interaction plays a dominant role in the calculations, especially for high-Z ions, which can be expressed as follows:

(3)

The Breit interaction is the low-frequency limitation of eq. (3). QED effect comprise of vacuum polarization and self-energy are also taken into account in the present calculation perturbatively.

Iii Electron correlation model and calculation strategy

The electron correlation effects was taken into account by choosing appropriate electron correlation model. The correlation model used in the present calculation is similar to the model used by Kadrekar and Natarajan Kadrekar and Natarajan (2011). The major electron correlation effects can be captured by including the CSFs that can be formed from configurations obtained by allowing single and double (SD) excitations from the interested reference configurations to some virtual orbitals space. The configuration space was extended by increasing the active orbital set layer by layer to study the correlation contributions. Generally, the zero-order Dirac-Fock (DF) wave functions were firstly generated from the reference configurations of He-like ions in EOL mode for initial and the final state. In the EOL method, the radial wave functions and the mixing coefficients are determined by optimizing the energy functional which is the weighted sum of the selected eigenstates. For double K hole state, the minimum basis (MB) was generated by considered limited expansion allowing SD substitutions of electrons from the reference configurations. Since this procedure result in better optimized wave functions than the DF functions, all the study on the correlation effects were carried out with respect to MB. Then, the active space was expanded to the first layer, i.e. ({n3l2}) virtual orbitals and all the newly added orbital functions were optimized while , and orbitals were kept fixed from MB. Repeat these step by step and increase the virtual orbitals to make sure that eigenenergy and wave function were converged. To ensure the stability of the numeric data and reduce the calculation time, only new added layer was optimized and the previous calculated orbits were all kept frozen. As the virtual orbitals increasing, the number of CSFs increased rapidly. In order to maintain calculation manageable, the principle quantum number of the virtual orbitals was limited to .

Iv Result and discussion

He-like Ne
Active sets 2s2p 1s2s
P P P P S S
DF -645.71 -629.80 -645.30 -645.58 -1653.06 -1643.25
MB -645.49 -629.38 -645.08 -645.36 -1652.95 -1641.73
n3l2 -645.68 -630.16 -645.26 -645.55 -1653.00 -1642.07
n4l3 -645.72 -630.36 -645.31 -645.59 -1653.01 -1642.16
n5l3 -645.74 -630.74 -645.38 -645.62 -1653.03 -1642.23
n6l3 -645.80 -630.92 -645.39 -645.67 -1653.03 -1642.26
E 1911.70 1926.58 1912.11 1911.83 904.47 915.24
Ref 1911.48 1926.13 1911.89 1911.60 904.41 914.82
NIST 1912.26 1926.63 1912.83 1911.97 905.08 915.34
He-like Ag
Active sets 2s2p 1s2s
P P P P S S
DF -15437.72 -15146.59 -15203.39 -15417.21 -38550.42 -38489.71
MB -15437.46 -15146.22 -15203.17 -15416.90 -38550.33 -38487.92
n3l2 -15437.65 -15146.84 -15203.36 -15417.21 -38550.38 -38488.33
n4l3 -15437.70 -15147.04 -15203.41 -15417.29 -38550.39 -38488.42
n5l3 -15437.72 -15147.42 -15203.48 -15417.35 -38550.41 -38488.51
n6l3 -15437.78 -15147.53 -15203.50 -15417.48 -38550.41 -38488.54
  • ReferenceKadrekar and Natarajan (2011)

Table 1: Energies (in eV) of initial and final states of He-like Ne and Ag ions in various active space sets. The notation DF denotes Dirac-Fock calculation, MB the minimum basis, {ab} the active set consists of all orbitals from =a to =b, the energy is the energy relative to the ground state , see details in text of Sec. III.

One of the main purposes of this work is to study the effect of electron correlation effects on the transition properties of TEOP and OEOP transition process of He-like ions. The energy levels and transition properties of He-like Ne, Si, Ar, Ca, Fe, Ni, Cu, Zn, Kr, Nb, and Ag ions were calculated by using MCDF method with active space method. The eigenvalues (in eV) of the energy levels of double excited configuration and the single excited configuration of He-like Ne and Ag were presented in Table 1 to show the convergence. From the table the correlation model of MB led to better optimized wave functions than the DF functions, so all our investigations on the effect of correlation and higher-order corrections were carried out with respect to MB. It can be speculated from the table that with the increase of the active space, the eigenenergy tend to be converged both for low-Z and high-Z ions. The energy of relative to the ground state of He-like Ne was provided with available theoretical results. An excellent agreement with relative error between the present calculation and the previous work, which also use the MCDF method, was achieved. Therefore the present calculation was restricted to the correlation models.

Energy
Z P-S P-S P- S P-S P-S P-S
10 996.38 1007.09 1007.22 1007.50 1011.14 1021.97
Ref 996.79 1007.07 1007.20 1007.48 1011.32 1021.73
Expt. 1007.86
Theory 996.92 1007.0 1007.2 1007.5 1011.2 1021.4
14 1968.59 1983.91 1984.41 1985.54 1990.22 2006.04
Ref 1968.97 1983.88 1984.39 1985.52 1990.39 2005.80
Expt. 1985.8 1991.7
Theory 1969.1 1983.9 1984.4 1985.5 1990.3 2005.5
18 3272.06 3291.73 3293.05 3296.28 3301.40 3322.40
Ref 3272.44 3291.69 3293.02 3296.26 3301.56 3322.15
Theory 3272.6 3291.7 3293.0 3296.2 3301.5 3321.9
20 4048.97 4070.68 4072.63 4077.67 4082.73 4106.39
Ref 4049.34 4070.64 4072.60 4077.66 4082.89 4106.15
Theory 4049.5 4070.6 4072.6 4077.6 4082.8 4105.9
26 6886.05 6913.88 6918.15 6933.78 6937.35 6969.45
Ref 6886.41 6913.32 6918.10 6933.77 6937.52 6969.20
Expt. 6910 6942
Theory 6886.7 6913.4 6918.1 6933.8 6937.6 6969.0
28 8002.38 8031.47 8037.45 8059.10 8061.75 8096.83
29 8592.88 8622.86 8629.48 8654.75 8656.88 8693.48
30 9205.07 9235.93 9243.22 9272.57 9274.13 9312.28
36 13338.82 13375.35 13386.87 13452.79 13450.21 13498.25
41 17399.28 17441.38 17456.47 17573.86 17567.01 17624.2
47 23036.86 23086.18 23105.36 23319.99 23307.07 23376.57
  • ReferenceKadrekar and Natarajan (2011).

  • ReferenceGoryaev et al. (2017).

  • ReferenceMosnier et al. (1986).

  • ReferenceNandi (2008).

  • ReferencePhillips (2004)

Table 2: Transition energies (in eV) of the one-electron radiative transitions from states of configuration in He-like ions. ’*’ denotes the spin-forbidden transition

The transition energy in eV of OEOP transition from configuration to configuration of He-like ions () were presented in Table 2. The results for He-like ions agree well with the available experimental data and other theoretical calculation results. The average relative error of the current calculation compare to the experimental observation is about -. The ions with were also calculated in the present work. To the best of the authors’ knowledge, the corresponding experimental and theoretical data was unavailable. Therefore, it will be helpful for the future experimental and theoretical investigation.

Rate
Z P-S P-S P- S P-S P-S P-S
10 1.042(9) 5.667(12) 5.661(12) 5.650(12) 5.661(12) 8.316(8)
Ref 1.246(9) 5.755(12) 5.751(12) 5.744(12) 5.946(12) 1.046(9)
Theory 1.17(9) 5.79(12) 5.80(12) 5.80(12) 6.02(12) 1.22(9)
14 3.090(10) 2.245(13) 2.239(13) 2.236(13) 2.237(13) 2.736(10)
Ref 3.375(10) 2.268(13) 2.262(13) 2.259(13) 2.318(13) 2.938(10)
Ref 3.19(10) 2.29(13) 2.28(13) 2.29(13) 2.35(13) 3.31(11)
18 3.611(11) 6.240(13) 6.179(13) 6.198(13) 6.178(13) 3.318(11)
Ref 3.818(11) 6.285(13) 6.238(13) 6.246(13) 6.346(13) 3.465(11)
Theory 3.63(11) 6.36(13) 6.33(13) 6.38(13) 6.47(13) 3.75(11)
20 9.8669(11) 9.568(13) 9.432(13) 9.489(13) 9.419(13) 9.153(11)
Ref 1.034(12) 9.628(13) 9.509(13) 9.554(13) 9.646(13) 9.472(11)
Theory 9.86(11) 9.76(13) 9.67(13) 9.81(13) 9.87(13) 1.01(12)
26 1.067(13) 2.768(14) 2.655(14) 2.730(14) 2.636(14) 1.005(13)
Ref 1.100(13) 2.780(14) 2.666(14) 2.744(14) 2.683(14) 1.025(13)
Theory 1.06(13) 2.84(14) 2.74(14) 2.87(14) 2.79(14) 1.08(13)
28 1.992(13) 3.738(14) 3.576(14) 3.676(14) 3.496(14) 1.881(13)
29 2.652(13) 4.304(14) 4.029(14) 4.232(14) 3.990(14) 2.505(13)
30 3.472(13) 4.936(14) 4.578(14) 4.847(14) 4.529(14) 3.283(13)
36 1.318(14) 1.031(15) 8.977(14) 1.005(15) 8.817(14) 1.245(14)
41 3.013(14) 1.744(15) 1.440(15) 1.686(15) 1.405(15) 2.884(14)
47 6.548(14) 3.029(15) 2.371(15) 2.897(15) 2.283(15) 6.141(14)
  • ReferenceKadrekar and Natarajan (2011).

  • ReferenceGoryaev et al. (2017).

Table 3: Transition rates (in sec ) of the one-electron radiative transitions from states of configuration in He-like ions, ’*’ denotes the spin-forbidden transition.

In the calculation of the transition properties in relativistic atomic theory, Babushkin (B) and Coulomb (C) gauge were often used, which corresponding to the length and velocity gauge in non-relativistic quantum mechanics, respectively. Both are equivalent when exact wave functions are used, but they usually give rather different results when approximate wave functions are used. The consistency of the transition rate from different gauge indicates the accuracy of the wave function in some extent. The ratio respect to the transition rate from Babushkin and Coulomb gauge often has been adopted to be a criteria for the accuracy of wave function and the calculation results. It was found in the present calculation, the ratio of the transition rate from different correlation model will tend to 1.00 with the increase of active space. It indicate that the wave function used in the present calculation was good and the most important correlation effects were included in the present calculation.

The transition rate of OEOP transition from to of He-like ions () were presented in Table 3. For brevity, only transition rate in Babushkin gauge are given in the table. The current calculated transition rates is in good agreement with the numeric data calculated by Kadrekar and Natarajan using MCDF method Kadrekar and Natarajan (2011) and by Goryaev et al. using Z-expansion method Goryaev et al. (2017). The Z-expansion method based on the perturbation theory and hydrogen-like basis, while MCDF could include the electron correlation effectively. Four allowed transitions and two dipole forbidden transitions are listed in the table. For transitions from the same initial state P to different final state S and S, the ratio of the two transitions rate are approximately when Z=10, while the ratio increase to when Z=47. It is indicates that the intensity of these dipole forbidden transition increases sharply with the increasing of Z, which provide a good candidate that could be used to observe the E1 forbidden transitions in high-Z ions. For the high temperature plasma, some important diagnostics information are provided by these transitions.

P-S P-S
Z Energy Rate Energy Rate
10 1926.00 6.030(9) 1911.25 1.648(6)
Theory 1926.027 4.813(9) 1911.507 1.343(6)
Theory 1928.844 1.269(10)
14 3844.42 1.232(10) 3822.79 1.955(7)
Theory 3843.901 9.391(9) 3822.746 1.715(7)
Theory 3835 9.125(9)
18 6425.41 2.077(10) 6396.07 1.307(8)
Theory 6424.385 1.568(10) 6396.059 1.138(8)
Theory 6399 1.534(10)
Exp. 6390
20 7966.43 2.587(10) 7932.67 2.860(8)
Theory 7967.522 1.945(10) 7933.026 2.491(8)
Theory 7978.211 5.56(10)
26 13604.75 4.346(10) 13553.46 1.828(9)
Theory 13604.082 3.342(10) 13553.011 1.599(9)
28 15827.11 5.001(10) 15767.73 2.955(9)
29 17003.44 5.338(10) 16939.45 3.676(9)
30 18223.52 5.682(10) 18154.46 4.512(9)
36 26475.88 7.897(10) 26364.5 1.220(10)
41 34603.61 1.002(11) 34435.88 2.232(10)
47 45918.8 1.313(11) 45647.6 3.917(10)
  • ReferenceKadrekar and Natarajan (2011).

  • ReferenceSafronova and Senashenko (1977).

  • ReferenceTrabert et al. (1982)

  • ReferenceTawara and Richard (2002).

Table 4: Transition energies (in eV) and rates (in sec) in length gauge of two-electron one-photon transitions from states of to configuration in He-like ions. The numbers in the parentheses are powers of ten.

The transition energy and the rate of TEOP transitions from the initial state to the final state configuration are listed in Table 4. The ratio respect to the transition rate from Babushkin and Coulomb gauge is about -. The TEOP transition energy is approximately twice of the corresponding OEOP transition energy as expected. The magnitude of the transition rate indicates that the intensity of the transition line. In general, a good agreement between the present rate and the length gauge rate of Kadrekaret al. Kadrekar and Natarajan (2011) can be obtained.

Figure 1: Electron correlation effect on the transition energies and rates for OEOP and TEOP transitions in He-like ions. (a)Correlation contribution to energies of the OEOP transitions from states of configuration of He-like ion. (b)Correlation contribution to energies of the TEOP transitions from states of configuration of He-like ions. (c)Percentage correlation contribution to the OEOP transition energies from transitions of He-like ions. (d)Percentage contribution to the TEOP energies from transitions of He-like ions. (e)Percentage contribution to the OEOP transition rate in length gauge from transitions of He-like ions. (f)Percentage contribution to the TEOP transition rate in length gauge from transitions of He-like ions
Figure 2: Breit interaction effect on the transition energies and rates for OEOP and TEOP transitions in He-like ions. (a)Breit contribution to the OEOP transition energies from configuration of He-like ion. (b)Breit contribution to TEOP transition energies from configuration of He-like ions. (c)Percentage Breit contribution to the OEOP transition energies from transition of He-like ions. (d)Percentage Breit contribution to the TEOP transition energies from transition of He-like ions. (e)Breit contribution to the OEOP transition rates in length gauge from transition of He-like ions. (f)Breit contribution to the TEOP transition rates in length gauge from transition of He-like ions

The electron correlation effect on the OEOP and TEOP transition energies and rates are shown on Fig. 1. The correlation contribution to the transition energy from the are presented in Fig. 1(a) and (b) respect to that of MB. From Fig. 1(a), the correlation contribution to the transition energies from P upper level are decrease smoothly, while others are increases with Z increase. The correlation effect contribute to the transition energy of the allowed transitions by 0.2eV to 1.0eV, while it is 0.2 eV to 1.5 eV for dipole forbidden transition. However, for the TEOP transition in Fig. 1(b), the contribute to the transitions energy is 0.2 eV to 1.5 eV.

The percentage of the correlation contribution to the OEOP and TEOP transition energies from with respect to MB are given in Figs. 1(c) and (d). The contribution from the electron correlation on the transition energy increases with increasing of Z for TEOP transitions and P-S OEOP transition while others are decrease.

The percentage contribution of electron correlation to the transition rates in Babushkin Gauge of OEOP and TEOP transition from with respect to MB are shown on Fig. 1(e) and (f). The general trend of the correlation contribution to the transition rate of OEOP process become smaller while that to TEOP transition become larger with increasing of Z. However, for the low-Z elements especially , irregularities in the E1 forbidden transition rate was observed. This might caused by the fact that the Coulomb interaction between the nucleus and electrons competes with the electron correlation for low-Z elements. For high-Z ions, the Coulomb interaction between the nucleus and electrons dominated and the correlation contribution becomes smaller.

Fig. 2 shows the contribution from the Breit interaction to the transition energy and rate of OEOP and TEOP transitions. In Fig. 2(a)-(d), it is found that the Breit interaction decrease the P-S and P-S transition energies of both OEOP and TEOP transition, while it slightly increase the transition energy of the transition to S state in OEOP transitions. This is due to the Breit interaction reduces the binding energy of each state of 2s2p configuration and also reduces the S of 1s2s configuration and slightly increase the binding energy S state of 1s2s configuration, which makes the transition energy of P-S and P-S become smaller and the energy of other transition to S state slightly increase. It can be inferred from Fig. 1 and 2 that the contribution from electron correlation larger than that from Breit interaction for low-Z element, while the latter becomes significant for high-Z ions.

In Fig. 2(e) and (f), the contributions from Breit interaction to the transition rate in length gauge of OEOP and TEOP was given. Unlike correlation contribution, the Breit interaction reduces the rates of P-S and P-S, and slightly increase the transition rate of other transitions to S state of OEOP process. For TEOP transitions, the Breit interaction increase the transition rate of P-S and P-S with the increase of Z. As seen clearly from the figs, the Breit interaction contribution to P-S and P-S OEOP transition rates is about and at Z = 10, respectively. It decreases with the increase of atomic number Z reaching about at Z = 47 for both transitions. However, for the TEOP transition, the Breit interaction contribution to the transition rate is about - and - for P-S and P-S transitions, respectively. Since TEOP is a multi-electron process, the electron correlation effect plays a essential role in this transition, and Breit interaction becomes more and more significant with the increase of Z, which can be inferred from the Fig. 1(f) and 2(f).

The mixing of the CSFs lead to the feasibility of TEOP transition which was strictly forbidden according to the selection rule. The main contribution for P and P of the CSFs change from 67% for Ne to 98% for Ag which indicate the change of coupling scheme from to with the change of the nucleus and the interaction in these ions. The mixing contribution from P and P was tiny (less than ), even though it contribute the main parts for the TEOP transitions. Because the resonance transition was strong, the TEOP transition matrix elements were obtained by this tiny mixing. Besides the mixing of the in the excited states , there are also some tiny mixing of the ground states, such as , have contribution to the S. Therefore, the and transition matrix elements could also contribute to the TEOP transition by mixing.

V Conclusion

The energy levels, transition energies and rates for the one- and two-electron radiative transitions from double K hole states to and configurations of He-like ions were calculated using multi-configuration Dirac-Fock method. An active space method was employed and reasonable electron correlation model was constructed to study the electron correlation effects. The Breit interaction and QED effects were taken into account efficiently. The convergence of the energy levels were obtained with the increase of active space. The transition energy and rates show a good agreement between the present calculation and the previous work. It is emphasized in the present work that the TEOP transition was essentially caused by the electron correlation effects. It is also found in the present work that the electron correlation effect and Breit interaction contribution to the transition energies of both OEOP and TEOP transition will decrease with the increase of Z. The competition between the nucleus-electron Coulomb interaction and electron correlation was obviously found for lower Z ions. The former will be dominated in high Z ions. The calculated data will be helpful for the future investigations on OEOP and TEOP transition of He-like ions.

Acknowledgements.
This work was supported by National Key Research and Development Program of China, Grant No:2017YFA0402300, National Nature Science Foundation of China, Grant No: U1832126, 11874051, 11264035.

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