Lifetime calculations in energy levels of Kr vii

Calculations of the lifetimes for all experimentally known energy levels of the spectrum of the six times-ionized krypton (Kr vii) are presented. The relativistic Hartree–Fock method including core-polarization effects were used. The energy matrix was calculated using energy parameters adjusted to the experimental energy levels. We also present a calculation based on a relativistic multiconfigurational Dirac–Fock approach. For some energy levels, a comparison of these results with the bibliography data was made.


Introduction
Six times ionized krypton (Kr VII) belongs to the Zn I isoelectronic sequence. The analysis of the atomic structure of this ion is important in astrophysical plasma research and recently Kr VI and Kr VII lines have been observed in the ultraviolet spectrum of the hot DO-type white dwarf RE 0503-289 [1], in the first detection of krypton in this kind of star. Reliable measurements and calculations of atomic data are a prerequisite for state-of-the-art nonlocal thermodynamic equilibrium stellar-atmosphere modeling (NLTE). Observed Kr V-VII line profiles in the UV spectrum of the white dwarf RE 0503−289 were simultaneously well reproduced with newly calculated oscillator strengths [2]. The energy levels measured by Raineri et al [3] were used to fit the parameters of 4s 2 , 4p 2 , 4s4d, 4s5d, 4s6d, 4s5s, 4s6s, 4p4f, 4s4p, 4s5p, 4s6p, 4s4f, 4s5f, 4s6f, 4p5s, and 4p4d configurations in Kr VII. In that work, the spectrum was recorded in the 300-4800 Å wavelength range, resulting in 115 new classified lines and extended the analysis to 38 new energy levels belonging to 4s5s, 4s6s, 4p4f, 4s6d and 4p4d, 4s5p, 4s4f, 4p5s, 4s5f, 4s6p, 4s6f even and odd configurations, respectively. The authors also mentioned several discrepancies between the values on Kr VII published in papers by Churilov [4], Raineri et al [5] and Cavalcanti et al [6]. In order to clarify these disagreements they presented a new revised and extended analysis for the Kr VII. A critical compilation of the energy levels and transitions of the Kr VII ion was reported by Saloman [7]. Liang et al [8] presented calculations of line strengths, oscillator strengths, radiative decay rates and fine structure collision strengths for 90 lines in Kr VII. In their calculations, using the AUTOSTRUCTURE code [9], they included nine configurations, i.e., 4s 2 , 4p 2 , 4s4d, 4s5s, 4s5d, and 4s4p, 4s4f,4p4d, 4s5p for the even and odd parities, respectively.
Lifetimes depend on the allowed and forbidden transition probabilities from the corresponding states. The only experimental results for lifetimes of energy levels for Kr VII was provided by Pinnington et al [10,11], who used the beam-foil method. They describe two techniques in their methods; from multiexponential curve fitting (d and f in in table 1) and from the arbitrarily normalized decay curve (ANDC) (e and g in table 1) where quoted errors are standard deviations. Empirical predictions are reported for the lifetimes of the 4s 2 -4s4p resonance and intercombination transitions in the Zn isoelectronic sequence [12]. Using configuration interaction wave functions, Hibbert et al [13] presented lifetime calculations of the 4s4p 3 P 0 1 energy level in Kr VII. This value is in excellent agreement with the experiment. They also included core polarization (CP) effects in the calculation.
In the present work, we calculated the lifetimes for all experimentally known energy levels of the Kr VII [3,7]. Four calculation methods were used to obtain the lifetimes of the energy levels. For the first three methods, Cowan's package [14] was used with corrections to the code made by Kramida [15], due to an error in Cowan's Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. atomic structure theory. The calculations were performed for different sets of configurations. We also included CP effects [16]. The fourth method is based on a multiconfigurational relativistic approach for the Dirac equation (MCDF), as described by Grant using the general relativistic atomic structure package (GRASP) [17]. For some levels we compared the lifetime values with experimental results [10,11].
2. Theory 2.1. The Hartree-Fock (HF) method In the codes rcn36/rcn2 of the Cowan program [14] the wavefunctions are calculated in a HF approximation with relativistic corrections (HFR). The wavefunctions are used to calculate a multiconfigurational energy matrix with the code rcg11. Both eigenvalues and eigenvectors of the matrix are functions of the Slater parameters, i.e., functions of the average configuration energy E av , electrostatic direct F k and exchange G k integrals, effective radial parameter α, configuration interaction integrals R k , and spin-orbit parameters ζ nl . We had used this method in several previous papers for example [18,19]. The values of these parameters were changed to fit the experimental values by means of a least-squares calculation. We also made the corrections proposed by Kramida [15] in the rcn2 code, in this way we obtained non-zero values for some configurationinteraction integrals of the Rydberg series, which were zero in the original version of rcn2.

HF plus CP
We included the CP effects (see, for example, Curtis [12] and Biémont et al [20]) just by replacing the dipole integral P r rP r r P r r r P r r r P r P r r Here α d is the electric dipole polarizability of the core, and r c is the cut-off radius, which defines the boundaries of the atomic core. This is the same modification used by Quinet et al [21] to correct transition matrix elements when including CP effects. In our case, the radial functions were obtained from the single configuration HF method with relativistic corrections, and no modification was done to include CP effects in the Hamiltonian.

Relativistic Dirac-Fock calculations
The GRASP package solves the Dirac equations within the framework of relativistic quantum theory [17,22]. This program offers energy levels, wavelength, dipole transition rates and lifetimes from a multiconfigurational relativistic approach. The configuration state functions are a linear combination of Slater determinants constructed from relativistic (Dirac) orbitals equation (4) of [18].

Results and discussions
The results of our four different lifetime calculations are presented in table 1. The experimental energy level values were taken from Raineri et al [3] and the experimental lifetimes from Pinnington et al [10,11], who used the beam-foil method. The HFR calculations were performed with different sets of configurations. In the first one (A), the following configurations were included: 4s 2 , 4p 2 , 4s4d, 4d 2 , 4s5s, 4s5d, 4p4f, 4f 2 , 4p5p, 4p5f, 4s6s, 4s6d, 3d 9 4s 2 4d and 4s4p, 4p4d, 4s5p, 4s4f, 4s5f, 4p5s, 4p5d, 4s6p, 4s6f, 4d4f, 3d 9 4s 2 4p for even and odd parity respectively, which is the same set used by Raineri et al [3]. The values of the adjusted parameters used in the present work are also similar to [3], where the details for the least-squares calculation are explained. The difference is that we considered the corrections made by Kramida [15] where we obtained non-zero values for some configuration-interaction integrals of Rydberg series (see point 2.1). These integrals were R 0 (4s5d,4s6d), R 0 (4s5p,4s6p) and R 0 (4s5f,4s6f) with values in 605, 819, 458 cm −1 , respectively. The change in the electrostatic integrals is expected to be equivalent to the inclusion of electronic correlation effects of higher order in the final values of the energy levels.
Percentage composition c Type of calculation Lifetime (ns) Experimental lifetime (ns)    [3]. b Calculated energy level values obtained using the fitted energy parameters [3]. c Percentages below 4% have been omitted [3]. d, e Experimental lifetimes reported by Pinnington [10]. f, g Experimental lifetimes reported by Pinnington [11]. A: Calculated HFR lifetimes values obtained using the fitted energy parameters obtained in [3]. The second calculation (B) is the same as the first one, except that in this case we did not consider core excited configuration, but instead, we took into account CP effects. This inclusion required the knowledge of the dipole polarizability of the ionic core, α d , and of the cutoff radius, r c . For the first parameter, we used the value computed by Fraga et al [23] for the Kr 8+ ion, i.e. α d =0.209 a 0 3 , while the cutoff radius, r c , was chosen equal to 0.56, which corresponds to the mean HFR 〈r〉 value of the outermost core orbital 3d 10 .
The third calculation (C) is the same as the second one including CP effects except that we considered the valence shell correlation [13] taking into account 5s 2 , 5p 2 and 5s5p configurations. The least-squares calculation results for the energy parameters are shown in tables 2 and 3 for the even and odd parities, respectively. For the even parity, all parameters (E av , F 2 , G 0 , G 2 , G 4 and spin-orbit) were left free for the known configurations. In order to reduce the standard deviation and to obtain parameter values in accordance with the scaled HF values, all the configuration-interaction integrals were scaled down at 85% of their HF values, except for the corresponding to 4s4d-4f 4p and 5s 2 -5p 2 that were held fixed at 75% of their HF values. For the odd parity, all parameters ( E av , F 2 , G 1 , G 3 and spin-orbit) were left free for the known configurations except for the G 3 (4s,6f) that was held fixed at 85% of the HF value. The 4s4p-4p 4d interaction integral was held fixed at 75% of their HF values and those corresponding to 4p4d-4s4f were let free to optimize their values and then fixed for the final calculation. The standard deviation for the energy adjustment was 155 cm −1 and 158 cm −1 for the even and odd parities, respectively.
The fourth calculation (D) was the fully relativistic MCDF approach. We used the GRASP [17]. Computations were carried out with the extended average level assuming a uniform charge distribution in the nucleus, with a krypton atomic weight of 83.80. We considered the same numbers and type of configurations as in the C calculation. The values presented in this work for lifetimes are in Babushkin gauge since this one, in the non-relativistic limits (length), has been found to be the most stable value in many situations, in the sense that it converges smoothly as more correlation is included and it is less sensitive to details of the computational method [24]. This method takes into account relativistic effects by means of a more complete approach than Cowan's package, but the HFR method takes into account correlation effects in a more complete way. In the HFR+CP method, not only are correlation effects considered more deeply, but CP effects are taken into account in the lifetime calculations. These concepts are reflected in  [13]. In some cases, where the calculated values are not in agreement with the experimental values as it is for the lifetime of the 4p 2 1 D 2 level and this could be due to the mixing of percentage composition of the levels involved in the calculation [3]. It is noteworthy that in all HFR calculations with and without CP, we considered optimized values of the energy parameters using least squares techniques where we adjusted the theoretical values to the experimental ones.

Conclusion
Four different calculations of lifetimes in Kr VII were carried out: A, considering the HFR approach [14] with the modifications very recently suggested by Kramida [12]; B and C, including CP effects HFR+CP [16] and D, using the GRASP code [17]. Three sets of configurations were taken into account in the calculation A, B and C. In the calculation D we considered the same set of configurations as in C. For some energy levels, we compared the lifetimes with the experimental values given in the bibliography [10,11]. In most cases the HFR+CP calculations are in better agreement with the experimental lifetimes, especially if we consider the maximum error in each case taking into account the ANDC technique, reported in the [10,11]. The values calculated with the GRASP code (D) are in better accordance with those calculated in the case A, where core excited configurations were included.