^{†}Corresponding author. Email: sunny.kmc87@gmail.com
Energy levels, radiative rates, and lifetimes are calculated for all levels of 3s^{2}3p, 3s^{2}3d, 3s3p^{2}, 3s3d^{2}, 3s3p3d, 3p^{2}3d, 3s3d^{2}, 3p3d^{2}, 3p^{3}, and 3d^{3} configurations of Allike tungsten ion (W XLVII). Multiconfigurational Dirac–Fock (MCDF) method is adopted for calculating energy levels and radiative rates. Oscillator strengths, radiative rates, and line strengths are reported for some E1 transitions from the ground level. Comparisons are made with the available data in the literature and good agreement has been found which confirms the reliability of our results.
Spectroscopic parameters of tungsten ions are essential for exploring the physical conditions in tokamak plasmas such as ITER in which tungsten is currently considered to be a promising candidate for the plasmafacing material in the divertor region.^{[1]} As sputtering at the surfaces with plasma contact cannot be avoided completely, tungsten will be an intrinsic impurity in these plasmas. The electron temperatures in ITER will range from below 0.1 keV at the edge up to about 30 keV in the core of the plasma. The measured spectra often contain many spectral lines from several ionization stages forming spectral features. To understand its influence as a plasma impurity, reliable atomic data are needed for many ionic stages of tungsten.
Many investigations have been carried out for the Allike tungsten in the last two decades both theoretically as well as experimentally. For instance, on the theoretical side, Feng et al.^{[2]} have presented a comprehensive study of atomic characteristics of ten tungsten ions including W XLVII in a broad range of wavelengths, energy levels and transition probabilities. They have also presented results of multiconfiguration Dirac– Fock and relativistic configuration interaction calculations including the Breit interaction. Safronova and Safronova^{[3]} have calculated the excitation energies for Allike W^{61+ } of the [2p^{6}]3s^{2}3p, 3s3p^{2}, 3d^{2}3d, 3s3p3d, and 3p^{3} levels and radiative rates of the E1 transitions between them. Further, Huang^{[4]} has tabulated transition rates and oscillator strengths for Allike ions, calculated by the multiconfiguration Dirac– Fock (MCDF) method. Relativistic manybody calculations of electric dipole lifetimes, transition rates and oscillator strengths for n = 3 states in Allike ions were presented by Safronova et al.^{[5]} They calculated the transition rates, oscillator strengths, and line strengths for electricdipole (E1) transitions between evenparity 3s3p^{2}, 3s^{2}3d, 3p^{2}3d, 3d^{2}3s, and 3d^{3} states and oddparity 3s^{2}3p, 3p^{3}, 3s3p3d, and 3d^{2}3p states in Allike ions with the nuclear charges ranging from Z = 15 to 100.^{[5]} Further, Lavin et al.^{[6]} have reported oscillator strengths corresponding to dipoleallowed transitions for a group of Allike ions (Z = 13– 103).
On the experimental side, Ralchenko et al.^{[7]} reported the measurements of extreme ultraviolet (EUV) spectra (4– 20 nm) of highlycharged tungsten ions W^{54+ } to W^{63+ }obtained with an electron beam ion trap (EBIT) facility at the National Institute of Standards and Technology (NIST). The three levels ^{2}P_{1/2}, ^{2}P_{3/2} of 3s^{2}3p, and ^{4}P_{1/2} of 3s3p^{2} of Allike tungsten were found using EBIT at NIST.
The aim of this work is to upgrade the database of energy levels, oscillator strengths, transition probabilities, and lifetimes of Allike tungsten ion. The investigations of highly charged Allike tungsten ion considered in the present work require the simultaneous consideration of electronic correlation and relativistic effects. For this purpose, we use MCDF technology which was originally developed by Grant et al.^{[8]} and revised by Norrington.^{[9]} It includes the most important configuration effects as well as quantum electrodynamics (QED) correction and has been applied successfully to calculate atomic data for highly charged ions.^{[10– 18]} In our calculations, all the orbitals are simultaneously optimized on the average energy of all configurations by using the option of EAL (extended average level).
The rest of this paper is organized as follows. In Section 2, we present the method of calculations, then the energy levels and wavefunction compositions are presented in Section 3. In Section 4, we report the radiative rates for electric dipole (E1) and magnetic quadrupole (M2) transition. In Section 5, we present the lifetimes of excited states. Finally, the conclusion is given in Section 6.
We obtain results using the multiconfiguration Dirac– Fock method employed in the GRASP code of Grant et al., ^{[8]} revised by Norrington^{[9]} and recently used by various authors.^{[10– 18]} In the MCDF approach, the Hamiltonian for an Nelectron atom or ion Dirac– Coulomb Hamiltonian is given by
where
is the single particle Hamiltonian consisting of kinetic energy and its interaction with the nucleus. In Eq. (1), α and β represent 4× 4 Dirac matrices where c is the speed of light. The Nelectron wavefunction is constructed from centralfield Dirac orbitals given by
where k is the Dirac angular quantum number, k = ± (j + 1/2) for l = j ± 1/2, so j = k − 1/2, m is the projection of the angular momentum j, and P_{nk} and Q_{nk} are radial functions. The spin angular momentum χ _{km}(θ , ϕ ) is a 2 component function defined by
An atomic state function (ASF) can now be formed for an Nelectron system with the given total angular momentum J, M and parity P is approximated by a linear combination of n_{c} electronic configuration state functions (CSFs)
where n_{c} is the number of CSFs included in the expansion, C_{i}(α ) are the expansion mixing coefficients, and α represents all information such as orbital occupation numbers, coupling, etc. The CSFs are usually chosen to be orthonormal, so that
For expectation of the Dirac– Hamiltonian, we get the energy of the Nelectron system as
The Hamiltonian matrix, H^{DC} has elements
Using the normalization condition
where I is the n_{c} × n_{c} unit matrix. Thus the predicted atomic energy level
We have performed calculations on transitions energies for the 148 fine structural levels belong to 3s^{2}3p, 3s^{2}3d, 3s3p^{2}, 3s3d^{2}, 3s3p3d, 3p^{2}3d, 3s3d^{2}, 3p3d^{2}, 3p^{3}, and 3d^{3} configurations of Allike tungsten. The Nelike (1s^{2}2s^{2}2p^{6}) core was treated as an inactive core. Energy levels of all the fine structural levels generated by above configurations have been calculated by the fully relativistic Multiconfigurational Dirac– Fock method. The flexible atomic code (FAC), which is also fully relativistic and based on the j jcoupling scheme has been used to generate all relevant atomic data.
The energy levels with their labels obtained using MCDF and FAC lifetimes of excited levels for Allike tungsten are presented in Table 1. The calculation includes ([1s^{2}2s^{2}2p^{6}]3s^{2}3p, 3p^{3}, 3p3d^{2}, 3p4s^{2}, 3p4f^{2}, 3p4d^{2}, 3s^{2}4p, 3s^{2}4f, 3p^{2}4p, 3p^{2}4f, 3s3p^{2}, 3s^{2}3d, 3p^{2}3d, 3s3d^{2}, 3d^{3}, 3s^{2}4s, 3d4s^{2}, 3d4p^{2}, 3d4d^{2}, 3d4f^{2}, 3s ^{2}4d, 3p^{2}4s, 3s3p3d, 3s3p4s, 3s3p4p, 3s3p4d, 3s3p4f, 3s3d4s, 3s3d4p, 3s3d4d, 3s3d4f, 3p3d4s, 3p3d4p, 3p3d4d, and 3p3d4f) configurations, which generate 894 energy levels, of which 148 are listed in Table 1. As QED corrections are important for transition energy calculations, therefore we present the QED corrections in our results. Further, we show the contributions to the excitation energies arising from different parts of the Hamiltonian and compare our level energies calculated by MCDF (included Breit and QED effects) with the FAC.
In Table 1, we report many new spectral lines which have not been reported either experimentally or theoretically. Energy values for 13 fine structure levels were compiled by NIST. We present energy values for the fine structural levels belonging to 3s3d^{2}, 3p3d^{2}, and 3d^{3} configurations, which are reported for the first time. We also present new results for many levels belonging to 3p^{3}, 3p^{2}3d, and 3s3p3d configurations.
In Table 2, we compare our calculated results for energy values from MCDF and FAC with the energy levels compiled by NIST, which are based on the data listed by Kramida.^{[19]} The experimental energy values compiled by NIST were measured by some authors.^{[3, 7, 20– 22]} In Table 2, we also present the calculated values of Safronova.^{[3]} For the levels belonging to 3s^{2}3p, 3p^{2}3d, and 3s^{2}3d configurations, our calculated energy values with MCDF are in good agreement with the NIST values and other theoretical results.^{[3]} For the level 3s3p^{2}, our energy values calculated with MCDF agree well (error 0.2– 1.1%) with NIST values. The disagreement between our calculated MCDF energies and FAC energies of the 3s3p^{2} level is about (0.06– 0.23%) except for the level 3s3p^{2} (^{4}P_{1/2}) (level 2) for which both calculated results differ by 1%. Further, the calculated energies for the levels 3p^{3} and 3s3p3d by the MCDF method agree well with those calculated by FAC and Safronova.^{[3]} The maximum disagreement between MCDF and FAC results for these levels is 0.36% while between MCDF and results of Safronova^{[3]} is 0.49%. Further, the ordering of Safronova and our calculated results is the same. In our MCDF calculation, we include those configurations which can provide reliable energy levels and whose energy levels interact with other energy levels. Therefore, we conclude that our MCDF results agree well with other available experimental and theoretical results.
In Table 3, we present the transition wavelengths (λ in Å ), radiative rates (A_{ji} in s^{− 1}, oscillator strengths (f_{i j}, dimensionless), and line strengths (S in a.u. (atomic units)), in length form only, for electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions from the ground state (3s^{2}3p ^{2}P_{3/2}) of Allike W. To show the reliability of our results, in Table 4, we report the transition wavelengths for some levels and compare them with the available experimental and theoretical work. One can see from Table 4 that our transition wavelengths calculated with MCDF are in reasonable agreement with other theoretical and experimental results. For 3s^{2}3p ^{2}P_{1/2}− 3s3p^{2}^{4}P_{1/2} transition, our results differ 1.1% from the experimental transition wavelength, while the theoretical results of Pü tterich et al.^{[23]} differ by 1.8% from experimental results. For all other transitions, our transition wavelength agrees within an error range of 0.02− 0.4%. In addition, the mean ratio λ _{This work}/λ _{exp.} is found to be 0.9943. This is a clear indication of the accuracy of our calculated results.
In Table 5, we compare our calculated transition probabilities for some transitions with other theoretical results. For 3s^{2}3p ^{2}P_{1/2}− 3s3p^{2}^{4}P_{1/2} transition, the results of Pü tterich et al.^{[23]} differ by an order of magnitude from the results of Ralchenko et al.^{[7]} as well as other theoretical results. Our calculated transition probability agrees well (within 1%) with Ralchenko et al.^{[7]} for the 3s^{2}3p ^{2}P_{1/2}− 3s3p^{2}^{4}P_{1/2} transition. The Avalue calculated by Quinet^{[10]} differs by an order of magnitude with all available theoretical results. From Table 5, it can be seen that our calculated value of transition probabilities are in good agreement with the other available results for all transitions.
Our calculated oscillator strength using MCDF for some transitions with other theoretical results are compared in Table 6. For the transition 3s^{2}3p ^{2}P_{1/2}− 3s^{2}3d ^{2}D_{3/2}, we compare our results with the results of Lavin et al.^{[6]} who calculated fvalues using three different approaches (QDO, RQDO, and MCDFEAL). From Table 6, it can be seen that our MCDF fvalues for the 3s^{2}3p ^{2}P_{1/2}− 3s^{2}3d ^{2}D_{3/2} transition is in close agreement with their MCDFEAL results.^{[6]} For the 3s^{2}3p ^{2}P_{1/2}− 3s^{2}3p ^{2}P_{3/2} transition, the maximum disagreement between our calculated fvalue and the other available fvalue is about 5%. It shows that our results are quite reliable.
The lifetime (τ ) for a level j is defined as
Calculations for transition probabilities have become advanced and require relativistic effects to be included. All possible E1 (electric dipole), E2 (electric quadrupole), M1 (magnetic dipole), and M2 (magnetic quadrupole) transitions are computed from which the lifetimes of levels have been determined. To the best of our knowledge, no calculations or measurements are available in the literature for lifetimes in Allike tungsten. We hope that our calculated values of lifetimes will be beneficial to experimentalists in their future work.
In this paper, we have presented the energies of 3s^{2}3p, 3s^{2}3d, 3s3p^{2}, 3s3d^{2}, 3s3p3d, 3p^{2}3d, 3s3d^{2}, 3p3d^{2}, 3p^{3}, and 3d^{3} configurations of Allike tungsten ion (W XLVII) levels and E1, E2, M1, and M2 transitions from the ground state by taking into account the various checks for accuracy of wavefunctions. We have also estimated the lifetimes of all excited levels. Our results may be useful for plasma physicists for interpretation of spectra emitted by plasmas produced in fusion reactors.
M. M. is thankful to D. S. T. (INDIA), U. G. C. (INDIA) and the University of Delhi under R & D research programme for financial support.
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