Relativistic Calculations for Be-like Iron

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1 Commun. Theor. Phys. (Beijing, China) 50 (2008) pp Chinese Physical Society Vol. 50, No. 2, August 15, 2008 Relativistic Calculations for Be-like Iron YANG Jian-Hui, 1 LI Ping, 2, ZHANG Jian-Ping, 1 and LI Hui-Li 2 1 Department of Physics, Leshan Teachers College, Leshan , China 2 Institute of Atomic and Molecular Physics, Sichuan University, Chengdu , China (Received September 5, 2007) Abstract Relativistic configuration interaction calculations for the states of 1s 2 2s 2, 1s 2 2s3l (l = s,p,d) and 1s 2 2p3l (l = s,p,d) configurations of iron are carried out using relativistic configuration interaction (RCI) and multi-configuration Dirac Fock (MCDF) method in the active interaction approach. In the present calculation, a large-scale configuration expansion was used in describing the target states. These results are extensively compared with other available calculative and experimental and observed values, the corresponding present results are in good agreement with experimental and observed values, and some differences are found with other available calculative values. Because more relativistic effects are considered than before, the present results should be more accurate and reliable. PACS numbers: Rj, z, Jv, f Key words: transition, Fe XXIII, multi-configuration Dirac Fock, relativistic configuration interaction 1 Introduction In the past few decades, the core-excited states of beryllium-like ions have attracted considerable attention to both theoretical [1 6] and experimental [7] researchers. It is now well established that energies and transition rates of these lines are of importance in various spectroscopic applications in astrophysics, fusion plasmas and also in study of atomic collisions. [5] In performing plasma diagnostics and tracing impurities, it is important to have a detailed knowledge of the spectra of the elements involved and Fe is one of these impurities and is frequently used for diagnostic purposes. [6] Proper identification of high-resolution measurements/observations requires precise calculations, therefore, a wide range of transition data for iron is needed. The relativistic and configuration-interaction effects play an important role in the correct assignment of different transitions and also in the accurate evaluation of atomic transition data of highly ionized atoms [6]. In this paper, we consider iron atoms of initial configurations 1s 2 2s 2, 1s 2 2s3l and 1s 2 2p3l (l = s,p,d). The energy levels, transition rates, and oscillator strengths have been computed in the framework of relativistic configuration interaction (RCI) formalism using multi-configuration Dirac Fock (MCDF) wavefunctions with the extended optimal level (EOL) scheme, The configuration interaction calculations include Breit interaction, quantum electrodynamics (QED) and nuclear mass corrections. The graspvu [8] package used in the present work to generate the MCDF wavefunctions and perform the structure calculations is the modified version of the grasp92 code. [9,10] 2 Method of Calculation The theoretical model employed in the present study has been described in detail in many papers. [8 13] Hence only the essential features are briefly repeated here. In the MCDF method the configuration state functions φ(γj P ) of a certain J and parity are formed by taking a linear combination of Slater determinants of the Dirac orbitals. A linear combination of these configuration state functions (CSFs) is then used in the construction of atomic state functions (ASFs) with the same J and parity: n csf ψ i (J P ) = c iα φ(γ α j P ), (1) α=i where c iα are the mixing coefficients for the state i, and n csf is the number of CSFs included in the evaluation of ASF. Γ α represents all the one-electron and intermediate quantum numbers needed to define the CSFs. The ASFs thus constructed were used in solving the Dirac Fock equation, and the Dirac Coulomb Hamiltonian is N N 1 N Ĥ DC = Ĥ D (i) + ˆr i ˆr j 1, (2) i=1 i=1 j=i+1 where the first term is the one-body contribution for an electron due to kinetic energy and interaction with the nucleus. The two-body Coulomb interactions between the electrons comprise the second term. The contributions from higher-order terms such as Breit interaction, vacuum polarization and finite nuclear mass effects are not included in Eq. (2) and are in general added as a first-order perturbation correction after self-consistency is obtained. The relativistic transition probabilities can be calculated either by using a Coulomb gauge or a Babushkin The project supported by the Leshan Teachers College under Grant No Corresponding author, lpscun@163.com

2 No. 2 Relativistic Calculations for Be-like Iron 469 gauge, i.e., Coulomb gauge, which in the nonrelativistic limit corresponds to the velocity form, and Babushkin gauge, which in the nonrelativistic limit corresponds to the length form. As the starting point, the CSF was generated from the referent configurations 1s 2 2s 2, 1s 2 2s2p, 1s 2 2p 2, 1s 2 2s3l, and 1s 2 2p3l (l = s,p,d) using the extended optimal level (EOL) scheme. In this method, the radial orbitals and the mixing coefficients are determined by optimizing the energy function, which is the weighted sum of energy values corresponding to a set of eigenstates. The zero-order eigenfunctions and energy eigenvalues were first calculated using this code. Then using the active space method, the multi-configuration expansion set was built. In this method, the CSFs of a certain J and parity were generated by substitution of electrons from the states in the reference configuration to an active set of orbitals. The active set in general consists of many virtual shells in addition to the unfilled shells in the reference configuration. The size of the active set was increased in a systematic way till the convergence of the expectation values of the orbitals and energies was obtained. In this paper, we consider the active space consisting of all the orbitals in the set with principal quantum number n = 1 6. We first carry out Dirac Fock (DF) calculations for the various reference configurations. Then we optimize the orbitals set {1s, 2s, 2p, 3s, 3p, 3d} for single and double excitation. Next, we included the n = 4 layer (an orbital set with the same principal quantum number is referred to as a layer) and optimized the orbitals. This procedure was carried out up to the n = 6 layer. The optimized orbitals are then used in the configuration interaction calculations. In this approach, the Breit interaction, vacuum polarization, nuclear motional and mass corrections are included in the Dirac Coulomb Hamiltonian and the mixing coefficients were recalculated without changing the radial functions. As the expansion set involves a large number of orbitals, the transverse interaction is limited by performing a first-order perturbation calculation, where only the diagonal part of the submatrix containing the Breit interaction is included. The calculation is performed in the low-frequency limit of the virtual photons. This code estimates the diagonal contributions to self-energy using screened hydrogenic wavefunctions. In this woke we use the blocked-structuring feature of the program, in which blocks corresponding to a given J and parity are grouped into that particular block. The advantage of this version is that the diagonalization of the Hamiltonian can be programmed for specific blocks of interest and this improves the efficiency of the diagonalization process. Once the RCI energies and wavefunctions are obtained, the mixing coefficient is transformed from the block format to non-bock format. Then the initial and final state orbital sets are transformed to become bi-orthonormal and the transition rates were calculated. 3 Results and Discussions In Table 1, we compare the energy levels in the n = 2 complex obtained by different theoretical methods, also various observed levels are given for comparison, which are obtained from solar flare spectra or laboratory measurement such as tokamak spectrum. The entries labeled Chen (A) and Chen (B) are obtained by Chen and P.P. Ong [14] using grasp2 code, the labeled Chen (A) with only the configurations in n = 2, 3 complex included, while the labeled Chen (B) with configurations in n = 2, 3, 4 complex is included. Table 1 Comparison of energy levels (in cm 1 ) between RCI results and other theoretical, observed values. The meaning for each entry is explained in the text. Configuration Level Present Chen (A) Chen (B) BM NF Edlén Expt. 1 Expt. 2 2s 2 1 S s2p 3 P s2p 3 P s2p 3 P s2p 1 P p 2 3 P p 2 3 P p 2 3 P p 2 1 D p 2 1 S The entries labeled BM are obtained by Bhatia and Mason [15] using the superstructure code, in which intermediate coupling is included. The entries labeled NF are obtained by Norrington and Grant [16] using the earlier version of the Grasp code with only the configurations in the n = 2 complex included. The entries labeled Expt. 1 are experimental

3 470 YANG Jian-Hui, LI Ping, ZHANG Jian-Ping, and LI Hui-Li Vol. 50 values taken from Ref. [17] and those labeled Expt. 2 are experimental or observed values from solar-flare spectra taken from Ref. [15]. The entries labeled Edlén were recommended by Edlén [17] obtained by polynomials fitting according to the experimental and/or observed results. Our computations are in good agreement with other theoretical values and experimental results. The discrepancies between our results and experimental or observed values are generally less than 6. This indicates that RCI effects (till n = 6) using multi-configuration Dirac Fock (MCDF) wavefunctions with the extended optimal level (EOL) scheme. The configuration interaction calculations including Berit interaction, quantum electrodynamics (QED) and nuclear mass corrections are all important for obtaining accurate energy levels. Our calculation considering more relativistic configurations than Chen (A) and Chen (B), so our results must be more believable. The weighted electric dipole oscillator strengths (gf) and the radiative transition probabilities (A) for the allowed transitions within the n = 2 complex are listed in Table 2. The entries labeled Zhang are the gf values calculated by Zhang and Sampson [18] using the MCDF method in Dirac Fock Slater (DFS) approximation but only configurations in the n = 2 complex are included in their procedure. The entries labeled NS are obtained in an elaborate CI calculation with a few tens of configuration bases in the nonrelativistic frame. [19] Oscillator strengths are gauge-dependent in the real calculation due to the approximate wave function obtained. It is well accepted that the inaccuracy in the velocity form for a given truncation scheme is larger than in the length form, [14] so in Table 2, we only list electric dipole oscillator strengths (gf), and transition probabilities (A in s 1 ) in Babushkin gauge form. In general, the gf or A values for transitions within the n = 2 complex are in good agreement with those obtained by various theoretical results with discrepancies less than a few percent. Table 2 Electric dipole oscillator strengths (gf) and transition probabilities (A in s 1 ) between levels n = 2 complex. The various data sources are explained in the text. gf A Transition Present Chen (A) Chen (B) BM Zhang Present Chen (A) Chen (B) BM NS 1s 2 2s 2-1s 2 2s 1 2p 1 1 S 0-3 P [ 3] 1.440[ 3] 1.510[ 3] 1.51[ 3] 1.500[ 3] 5.077[+7] 4.643[+7] 4.850[+7] 4.84[+7] 5.014[+7] 1 S 0-1 P [ 1] 1.562[ 1] 1.553[ 1] 1.56[ 1] 1.567[ 1] 1.948[+10] 2.053[+10] 2.000[+10] 2.00[+10] 1.974[+10] 1s 2 2s 1 2p 1-1s 2 2p 2 3 P 0-3 P [ 2] 6.420[ 2] 6.345[ 2] 6.39[ 2] 6.440[ 2] 6.510[+9] 6.614[ 9] 6.563[+9] 6.56[+9] 6.432[+9] 3 P 1-3 P [ 2] 5.537[ 2] 5.474[ 2] 5.50[ 2] 5.55[ 2] 1.216[+10] 1.240[+10] 1.231[+10] 1.22[+10] 1.225[+10] 3 P 1-3 P [ 2] 4.467[ 2] 4.414[ 2] 4.43[ 2] 4.47[ 2] 4.120[+9] 4.191[+9] 4.160[+9] 4.13[+9] 4.097[+9] 3 P 1-3 P [ 2] 8.370[ 2] 8.310[ 2] 8.36[ 2] 8.49[ 2] 5.308[+9] 5.400[+9] 5.382[+9] 5.39[+9] 5.206[+9] 3 P 1-1 D [ 3] 5.302[ 3] 5.035[ 3] 4.98[ 3] 4.800[ 3] 5.102[+8] 4.878[+8] 4.645[+8] 4.58[+8] 4.840[+8] 3 P 1-1 S [ 4] 2.424[ 4] 2.379[ 4] 2.32[ 4] 1.864[+8] 1.802[+8] 1.768[+8] 1.70[+8] 1.864[+8] 3 P 2-3 P [ 2] 6.469[ 2] 6.394[ 2] 6.49[ 2] 6.500[ 2] 4.392[+9] 4.469[+9] 4.439[+9] 4.46[+9] 4.454[+9] 3 P 2-3 P [ 1] 1.577[ 1] 1.555[ 1] 1.59[ 1] 1.580[ 1] 7.329[+9] 7.659[+9] 7.580[+9] 7.75[+9] 7.502[+9] 3 P 2-1 D [ 2] 6.675[ 2] 6.625[ 2] 6.51[ 2] 6.75[ 2] [+9] 4.856[+9] 4.830[+9] 4.73[+9] 4.773[+9] 1 P 1-3 P [ 4] 7.906[ 4] 7.658[ 4] 7.82[ 4] 9.000[ 4] 2.191[+7] 1.934[+7] 2.041[+7] 2.05[+7] 2.250[+7] 1 P 1-3 P [ 4] 4.183[ 4] 4.410[ 4] 4.46[ 4] 3.000[ 4] 7.841[+6] 6.361[+6] 7.149[+6] 7.17[+6] 7.574[+6] 1 P 1-3 P [ 2] 2.573[ 2] 2.633[ 2] 2.64[ 2] 2.670[ 2] 3.858[+8] 3.244[+8] 3.501[+8] 3.54[+8] 4.353[+8] 1 P 1-1 D [ 1] 1.615[ 1] 1.632[ 1] 1.67[ 1] 1.677[ 1] 4.443[+9] 4.220[+9] 4.419[+9] 4.54[+9] 4.353[+9] 1 P 1-1 S [ 1] 1.084[ 1] 1.048[ 1] 1.06[ 1] 1.080[ 1] 3.087[+10] 3.225[+10] 3.181[+10] 3.17[+10] 3.037[+10] Table 3 Present results of transition wavelength, electric dipole oscillator strengths (gf), and transition probabilities (A in s 1 ) in length gauge for L x-ray from Be-like iron. Transition Wavelength gf A Transition Wavelength gf A 1s 2 2s 2-1s 2 2s3p 1 S 0-3 D [ 3] 1.230[+11] 1 S 0-3 P [ 1] 4.654[+12] 1 S 0-3 P [ 4] 4.930[+9] 1 S 0-1 P [ 1] 7.644[+12] 1 S 0-1 P [ 2] 4.100[+11] 1s 2 2s 2-1s 2 2p3s 1s 2 2s2p-1s 2 2s3s 1 S 0-1 P [ 2] 2.474[+11] 3 P 0-3 S [ 2] 4.031[+11] 1 S 0-3 P [ 2] 4.257[+11] 3 P 1-3 S [ 2] 1.205[+12] 1s 2 2s 2-1s 2 2p3d 1 P 1-3 S [ 1] 2.104[+12]

4 No. 2 Relativistic Calculations for Be-like Iron 471 Continued Transition Wavelength gf A Transition Wavelength gf A 3 P 2-3 S [ 3] 3.004[+10] 3 P 0-1 P [ 4] 1.160[+10] 3 P 1-1 S [ 4] 9.081[+9] 1 D 2-1 P [ 5] 9.110[+8] 3 P 2-1 S [ 2] 1.335[+12] 1 D 2-3 P [ 3] 2.954[+10] 1s 2 2s2p-1s 2 2s3d 3 P 2-3 P [ 3] 3.953[+10] 3 P 0-3 D [ 1] 1.305[+13] 3 P 1-3 P [ 3] 8.345[+9] 3 P 1-3 D [ 1] 9.525[+12] 1s 2 2p 2-1s 2 2p3s 1 P 1-3 D [ 2] 6.265[+11] 1 D 2-3 P [ 2] 2.616[+12] 3 P 2-3 D [ 2] 1.926[+11] 3 P 0-1 P [ 2] 9.327[+11] 3 P 1-3 D [+0] 1.719[+13] 1 D 2-1 P [ 2] 5.177[+11] 1 P 1-3 D [ 1] 5.596[+12] 3 P 2-1 P [ 2] 1.334[+12] 3 P 2-3 D [ 2] 1.131[+11] 3 P 1-1 P [ 4] 1.708[+9] 1 P 1-3 D [+0] 2.241[+13] 1 S 0-1 P [ 3] 2.960[+10] 3 P 1-1 D [ 3] 9.941[+10] 1 D 2-3 P [ 1] 1.025[+12] 1 P 1-1 D [ 3] 2.390[+10] 3 P 2-3 P [ 1] 1.764[+12] 3 P 2-1 D [+0] 1.629[+13] 3 P 1-3 P [ 2] 5.122[+11] 1s 2 2s2p-1s 2 2p3p 3 P 0-3 P [ 3] 3.557[+10] 3 P 0-3 D [ 2] 1.021[+12] 1 D 2-3 P [ 3] 1.071[+11] 3 P 1-3 D [ 1] 2.052[+12] 3 P 2-3 P [ 3] 1.120[+11] 1 P 1-3 D [ 5] 3.471[+8] 3 P 1-3 P [ 1] 2.257[+12] 3 P 2-3 D [ 2] 1.425[+12] 1 S 0-3 P [ 2] 9.567[+11] 3 P 1-3 D [ 1] 4.089[+12] 1s 2 2p 2-1s 2 2p3d 1 P 1-3 D [ 2] 1.090[+11] 1 D 2-3 F [ 2] 2.266[+11] 3 P 2-3 D [ 2] 3.848[+11] 3 P 2-3 F [ 1] 1.631[+12] 3 P 0-1 P [ 1] 3.229[+12] 3 P 1-3 F [ 2] 7.753[+11] 3 P 1-1 P [ 2] 9.735[+11] 3 P 2-3 F [ 1] 6.861[+12] 1 P 1-1 P [ 3] 5.828[+10] 3 P 1-3 F [ 2] 2.104[+11] 3 P 2-1 P [ 2] 1.531[+12] 1 D 2-3 D [+0] 1.271[+13] 3 P 1-3 P [ 1] 6.241[+12] 3 P 2-3 D [ 1] 1.679[+12] 3 P 2-3 P [ 3] 2.426[+11] 3 P 1-3 D [ 1] 1.828[+12] 3 P 1-1 S [ 3] 2.086[+11] 3 P 0-3 D [+0] 2.137[+13] 3 P 2-1 S [ 1] 6.061[+12] 1 D 2-3 D [ 1] 4.498[+12] 3 P 0-3 P [ 2] 4.271[+11] 3 P 2-3 D [ 2] 2.010[+11] 3 P 1-3 P [ 2] 1.878[+12] 3 P 1-3 D [ 3] 3.554[+10] 1 P 1-3 P [ 2] 6.924[+11] 1 S 0-3 D [ 2] 3.122[+11] 3 P 2-3 P [ 1] 3.323[+12] 1 D 2-1 D [+0] 1.107[+13] 1 P 1-3 D [ 1] 4.377[+12] 3 P 2-1 D [ 1] 2.241[+12] 3 P 0-3 S [ 3] 2.141[+10] 3 P 1-1 D [ 1] 2.316[+12] 3 P 1-3 S [ 3] 1.376[+11] 1 D 2-3 P [ 1] 1.973[+12] 3 P 1-3 S [ 3] 1.375[+11] 3 P 2-3 P [+0] 1.105[+13] 3 P 2-3 S [ 2] 5.927[+11] 3 P 1-3 P [ 1] 7.615[+12] 3 P 1-3 P [ 2] 2.812[+11] 3 P 0-3 P [ 3] 3.469[+10] 1 P 1-3 P [ 1] 4.595[+12] 1 D 2-3 P [ 1] 1.179[+13] 3 P 2-3 P [ 1] 2.065[+12] 3 P 2-3 P [ 1] 5.584[+12] 3 P 1-1 D [ 2] 2.180[+11] 3 P 1-3 P [ 2] 1.616[+12] 1 P 1-1 D [ 1] 1.178[+12] 1 S 0-3 P [ 2] 1.713[+11] 3 P 2-1 D [ 1] 6.803[+12] 1 D 2-3 P [ 1] 1.615[+13] 1s 2 2p 2-1s 2 2s3p 3 P 2-1 F [ 1] 2.195[+12] 3 P 0-3 P [ 4] 7.963[+9] 3 P 1-1 F [+0] 3.435[+13] 1 D 2-3 P [ 4] 7.958[+9] 3 P 0-1 P [ 2] 4.131[+11] 3 P 2-3 P [ 3] 9.578[+10] 1 D 2-1 P [ 2] 2.513[+11] 3 P 1-3 P [ 3] 6.068[+10] 3 P 2-1 P [ 3] 2.269[+10] 1 S 0-3 P [ 4] 1.165[+10] 3 P 1-1 P [ 2] 1.048[+12] 1 D 2-3 P [ 4] 3.854[+10] 1 S 0-1 P [+0] 2.031[+13]

5 472 YANG Jian-Hui, LI Ping, ZHANG Jian-Ping, and LI Hui-Li Vol. 50 In Table 3, the data on transition wavelength, electric dipole oscillator strengths (gf), and transition probabilities (A) for Be-like iron are listed. Due to a large number of CSF having been generated and Breit interaction and QED corrections on the relativistic correlation having been included in our calculations, our calculation method and results are reliable. 4 Summary The RCI theory has been used to obtain the atomic structure for Be-like iron, the transition wavelength, absorption oscillator strengths, and transition probabilities of 1s 2 2s 2-1s 2 2s2p, 1s 2 2s2p-1s 2 2p 2 and L x-ray of correlative configurations. These accurate data are needed for the identifications of solar-flare spectra and the modeling of high-temperature plasmas such as those in x-ray laser and astrophysical research. Our results reveal some significant differences compared with those from previous nonrelativistic and relativistic calculations. Considering the present elaborate and accurate calculations, these new atomic data should enable a more objective reexamination of the Fe XXIII lines expected in laboratory experiments and solar-flare spectra. References [1] B. Saha and S. Fritzsche, Phys. Rev. E 73 (2006) [2] Bing-Cong Gou and Fei Wang, Phys. Rev. A 69 (2004) [3] X.W. Zhu and Kwong T. Chung, Phys. Rev. A 50 (1994) [4] Dae-Soung Kim, Phys. Rev. A 71 (2005) [5] Anuradha Natarajan and L Natarajan, J. Phys. B: At. Mol. Opt. Phys. 37 (2004) [6] Anders Ynnerman and Charlotte Froese Fischer, Phys. Rev. A 51 (1995) [7] E. Träbert, P. Beiersdorfer, G.V. Brown, et al., Phys. Rev. A 64 (2001) [8] L. Natarajan and Y.G. Mulye, J. Phys. B: At. Mol. Opt. Phys. 34 (2001) [9] P. Jonsson, X. He, and C. Froese Fischer, The graspvu Relativistic Atomic Structure Package, private communication (1998). [10] F.A. Parpia, C. Froese Fischer, and I.P. Grant, Comput. Phys. Commun. 94 (1996) 249. [11] K.G. Dyall, I.P. Gran, C.T. Johnson, et al., Comput. Phys. Commun. 55 (1989) 425. [12] CHENG Zhang, LI Ping, and DENG Xiao-Hui, Commun. Theor. Phys. (Beijing, China) 46 (2006) 723. [13] LI Hui-Li, LI Ping, CHENG Zhang, and MA Hai-Rong, Commun. Theor. Phys. (Beijing, China) 49 (2008) 217. [14] G.X. Chen and P.P. Ong, Phys. Rev. A 58 (1998) [15] A.K. Bhatia and H.E. Mason, Astron. Astrophys. 103 (1981) 324. [16] P.H. Norrington and I.P. Grant, J. Phys. B 20 (1987) [17] B. Edlén, Phys. Scr. 22 (1981) 593. [18] H.L. Zhang and D.H. Sampson, At. Data Nucl. Data Tables. 52 (1992) 143. [19] H. Nussbaumer and P.J. Stoey, J. Phys. B 12 (1979) 1647.

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