Energy levels, oscillator strengths, radiative transition probabilities, level lifetimes and electron-impact excitation rate coefficients for Ne-like Mo XXXIII

Abstract

The energy levels, oscillator strengths, radiative decay rates, lifetimes, collision strengths, direct and resonance electron-impact excitation rate coefficients have been computed for the 257 fine-structure levels arising from $1s^{2}2s^{2}2p^{5}nl$ and $1s^{2}2s2p^6{n}^{\prime }{l}^{\prime }$ configurations belonging to the $M{o}^{32+}$ ion with $n\le 7$, $l\le 4$ and $n^{\prime }\le 5$, $l^{\prime }\le 4$. The model-potential approach is used for the target ion structure calculations. Additionally, we employ the multiconfiguration Dirac–Hartree–Fock method to further assess the energy levels and transition probabilities. The collision strengths for the electron-impact direct excitation are computed within the relativistic distorted-wave approximation at 34, 135, 680, 1700, 5436, and 12,740 eV scattered electron energy values. We also perform collision calculations at 85, 175, and 450 keV electron energies using the plane-wave approximation and interpolate the reduced cross section within Fano plots, thus accounting for relativistic effects at asymptotic energies. This assures the convergence of Maxwellian integration for effective collision strengths calculation at electron temperatures up to $80$ keV. The resonance contribution to the excitation rates is accounted for within the independent process isolated resonance approximation by including Na-like $M{o}^{31+}$ doubly excited autoionization states arising from the $1s^2{2}^{7}nl{n}^{\prime }{l}^{\prime }$ configurations with $n\le 7,l\le 4,n^{\prime }\le 20,l^{\prime }\le 8$. Contributions from high $n^{\prime }\ge 21$ Rydberg states to the excitation rate coefficients are included by employing the ${n{}^{\prime }}^{-3}$ extrapolation law for radiative and autoionization decay rates. Radiative decay of resonances to lower autoionization states followed by autoionization cascade as well as radiative damping via core transitions are included in our model. The highest rate coefficients corresponding to ground state excitations are the $2p$–$3d$ allowed and $2p$–$3p$ electric monopole transitions, respectively. Intercombination and generally higher-order electric multipole transitions amount to lower excitation rate coefficients but are not to be neglected. Magnetic transitions are only relevant for low electron temperatures since both their direct and resonance contributions to excitation significantly decrease with increasing electron energy. Present results compare well with existing data from literature. The resonance contributions play an important role in the accuracy of rate coefficients, especially for weak forbidden transitions at low electron temperatures. These results may be useful in fusion related plasmas, astrophysics and fundamental physics.

Introduction

In the past decades the X-ray emission from molybdenum ions has been measured in tokamaks [1], [2], [3], [4], vacuum sparks[5], [6] and Z-pinch plasmas [7]. An increased interest for accurate determination of position and transition probability of recorded lines for plasma diagnostic purposes lead to further measurements accompanied by numerous theoretical studies. X-ray lines in the spectral range of $4.3$–$5.3$ Å arising from the $2p$–$3d$, $2s$–$3p$, and $2p$–$3s$ transitions between $M{o}^{28+}$ and $M{o}^{32+}$ ions detected in the Alcator-C tokamak [8] have been investigated by E. Kallne et al. [9]. The authors stated that the $2p$–$3d$ transitions gave the highest line intensity and originated from the $M{o}^{32+}$ ion at plasma temperature $1$–$1.5\text{keV}$, despite the fact that the maximum abundance of this ion is situated in the $2.5$–$3\text{keV}$ temperature range. T. Nakano et al. [4] measured two $M{o}^{32+}$ lines together with four tungsten lines, belonging to the $W^{45+}$ and $W^{46+}$ ions, in the emission spectra of ITER-like wall tokamak at JET [10]. This is, in part, due to the fact that the neon-like ionization stage of molybdenum is dominant for a rather large temperature range.

Strong radiative transitions belonging to neon-like ions from $Z=$ 36–92, including $M{o}^{32+}$, are measured in vacuum sparks and compiled along with available theoretical results by E. V. Aglitskii et al. [11]. Results contain wavelengths for the dipole transitions from the $2p^{-1}3s\left({}^{1,3}{P}_1^o\right)$, $2p^{-1}3d\left({}^{1,3}{P}_1^o{,}^3{D}_1^o\right)$, and $2s^{-1}3p\left({}^{1,3}{P}_1^o\right)$ levels to the ground state $1s^{2}2s^{2}2p^6\left({}^1{S}_0\right)$ of neon-like ions. J. Nielsen and J. H. Scofield [12] computed the wavelengths and laser gains on $3s$–$3p$ transitions belonging to the $S^{6+}$ to $X{e}^{44+}$ neon-like ions using the relativistic multiconfiguration Hartree–Fock (MCHF) method. X-ray spectroscopy measurements by B. A. Bryunetkin et al. [13] in laser-produced plasma reveal wavelengths of transitions between $n=3$ complex levels to the ground state for neon-like $A{l}^{3+}$, $G{e}^{22+}$, $M{o}^{32+}$ and $T{a}^{63+}$ ions.

Radiative transitions belonging to $n\ge 4$ complexes have been studied by Burkhalter et al. [14] and Burkhalter et al. [15] by performing experiments with exploding molybdenum wires. The $2l-n{l}^{\prime }$ X-ray transitions have been measured in the Alcator C and Alcator C-Mod plasma and studied thoroughly for $n\le 19$ and $l\le 1,l^{\prime }\le 2$ in $M{o}^{30+}$ to $M{o}^{33+}$ ions by J. E. Rice et al. [16] and in $M{o}^{23+}$ to $M{o}^{33+}$ ions by J. E. Rice et al. [17], for $n\le 12$ and $l\le 1,l^{\prime }\le 2$ in near neon-like $Z{r}^{30+}$, $N{b}^{31+}$, $M{o}^{32+}$ and $P{d}^{36+}$ ions by J. E. Rice et al. [18], and for $n\le 9$ and $l\le 1,l^{\prime }\le 2$ in near neon-like $K{r}^{26+}$, $M{o}^{32+}$, $N{b}^{31+}$ and $Z{r}^{30+}$ ions by J. E. Rice at al. [19]. These results contain measured and calculated wavelengths along with oscillator strength values for the mentioned transitions. The structure calculations were performed via an optimized model-potential approach. It is worth mentioning that relatively high intensity lines belonging to the $M{o}^{32+}$ ion also appear in the $3000$–$3070$ mÅ and $3730$–$3770$ mÅ wavelength ranges of the measured spectra of Alcator-C tokamak plasma [20], [21] along with lines belonging to $A{r}^{16+}$ and $A{r}^{17+}$ ions, respectively.

Forbidden transitions in molybdenum ions are a useful tool for plasma diagnostics. B. K. F. Young et al. [22] have shown that the line intensities corresponding to electric quadrupole transitions in neon-like $M{o}^{32+}$ and $A{g}^{37+}$ ions are sensitive to electron densities variations. Electric quadrupole transitions of type $2s$–$3d$ and $2p$–$3p$ along with magnetic quadrupole transitions of type $2p$–$3s$ have relatively high line intensities in tokamak plasma for the electron density in the range of $10^{13}$–$10^{15}{\mathrm{cm}}^{-3}$ [19]. Theoretical forbidden transitions, such as $M1$, $E2$, and $M2$, belonging to neon-like ions have been studied by U. I Safronova et al. [23] and P. Beiersdorfer et al. [24] using the relativistic many-body perturbation-theory (RMBPT) approach.

The possibility to diagnose the magnetic field strength in a tokamak plasma by measuring magnetic field induced transitions [25] such as $2s^{2}2p^{5}3s\left({}^3{P}_{0,2}^o\right)-2s^{2}2p^6\left({}^1{S}_0\right)$ belonging to neon-like ions has been studied by J. Li et al. [26]. The magnetic field treated as perturbation induces coupling of states with the same angular momentum projection or magnetic quantum number $M$ and thus mixes states with different $J$ values. For example, the forbidden $2p^{5}3s\left({}^3{P}_{0,2}^o\right)-2p^6\left({}^1{S}_0\right)$ transitions may become magnetic induced electric dipole-allowed due to strong mixing of the excited states with the $2p^{5}3s\left({}^{1,3}{P}_1^o\right)$ perturbing states of the same $M\Pi$ symmetry.

The electric dipole and magnetic quadrupole $2s^{2}2p^{5}3s\left({}^3{P}_{1,2}^o\right)-2s^{2}2p^6\left({}^1{S}_0\right)$ transitions belonging to the $M{o}^{32+}$ ion have been measured at JET among other lines belonging to $W^{46+}$ and $W^{45+}$ ions in the $5200$–$5220$ mÅ wavelength range by Nakano et al. [4]. The divertor is made out of tungsten material which may result in possible contamination of the fusion plasma with tungsten ions. Despite the fact that the molybdenum source is unknown the line intensities from these two materials are comparable. The concentrations of 10⁻⁵ for tungsten and $7\times 10^{-7}$ for molybdenum ions in the tokamak plasma with electron density $\approx 6\times 10^{13}{\mathrm{cm}}^{-3}$ are determined from the measured spectrum. These authors also estimated the abundance ratio for the $W^{45+}$, $W^{46+}$, and $M{o}^{32}$ ions at $4\text{keV}$ electron temperature obtaining 1.0:0.3:0.7. This suggests that the $M{o}^{32+}$ ionization stage is robust to changes in plasma parameters. Further collisional-radiative modeling and diagnosis of fusion plasma are required in order to determine the molybdenum source material.

H. Zhang et al. [27] used the Hartree–Fock approach treating the relativistic corrections perturbatively within the framework of COWAN code [28] to compute the atomic data for the 88 fine-structure levels arising from $2^{7}nl$ with $n=3,4$ and $l\le n-1$ for all neon-like ions between $18\le Z\le 74$. Part of this study is devoted to labeling the fine-structure levels among the Ne-like isoelectronic series showing different crossing of states for increasing Z elements. Their results include energy levels, oscillator strengths, and collision strengths between all available levels computed at 9 electron-impact energy values within the distorted-wave approximation (DW).

M. J. Seaton [29] highlighted the importance of resonance structures in the collision strengths for electron-impact excitation processes and their effects on the overall rate coefficients. R. D. Cowan [30] computed the contribution of resonance excitations (RE) to the rate coefficients belonging to the $O^{4+}$ ion by employing the independent process isolated resonance distorted wave (IPIRDW) approximation. This author treated the RE as a two step process, the dielectronic capture followed by autoionization, similar to the treatment of dielectronic recombination which consists of dielectronic capture followed by emission of radiation. A series of papers accounting for RE via the IPIRDW approach have emerged for highly-charged medium to heavy ions, such as H-like $A{l}^{12+}$ to $M{o}^{41+}$ [31] and $F{e}^{25+}$ [32], B-like $F{e}^{21+}$ [33], Mg-like $A{r}^{6+}$, $T{i}^{10+}$, $F{e}^{14+}$ and $S{e}^{22+}$ [34], P-like $C{u}^{14+}$ [35] ,Ni-like $G{a}^{36+}$ [36], $T{a}^{45+}$ [37] , $W^{46+}$ [38] and $A{u}^{51+}$ [39], and Cu-like $G{a}^{35+}$ [40] and $A{u}^{50+}$ [41] ions.

M. H. Chen and K. J. Reed [42] computed the direct and resonance contribution to the electron-impact excitation rate coefficients for transitions between states arising from $2s^{2}2p^{5}3l$ with $l\le 2$ configurations and the ground state in the case of $P^{5+}$, $A{r}^{7+}$, $F{e}^{16+}$, $S{e}^{24+}$, $M{o}^{32+}$, and $X{e}^{44+}$ neon-like ions for an electron temperature in the $50\le T_E\le 4000\text{eV}$ range. These authors accounted for the contribution of resonance excitation via IPIRDW by including resonances belonging to the corresponding Na-like ions which arise from $2s^{2}2p^{5}nl{n}^{\prime }{l}^{\prime }$ configurations with $n\le 4,l\le 3$ and $n^{\prime }\le 5,l^{\prime }\le 4$. Their electron-impact excitation rate coefficients have been used in literature for solving the rate equation in the collisional-radiative modeling of tokamak plasmas, including molybdenum plasma [43], [44]. In another work [45], M. H. Chen and K. J. Reed improved their previous calculations of excitation rate coefficients for transitions between fine-structure levels arising from $1s^2{2}^{7}3l$ $(l\le 2)$ configurations in neon-like $S{e}^{24+}$. In order to account for the resonance contribution to the excitation rate coefficients they included resonances arising from $2^{7}3l{n}^{\prime }{l}^{\prime }$ $(n^{\prime }\le 15,l^{\prime }\le 4,l\le 2)$, $2s^{2}2p^{5}4l{n}^{\prime }{l}^{\prime }$ $(n^{\prime }\le 5,l,l^{\prime }\le 3)$, and $2s^{2}2p^{4}3l3l^{\prime }3l^{\prime \prime }$ $(l,l^{\prime },l^{\prime \prime }\le 2)$ configurations belonging to Na-like $S{e}^{23+}$. The $S{e}^{24+}$ excitation rate coefficients for transitions among the first 241 fine-structure levels of arising from the $1s^2{2}^{7}nl$ $(n^{\prime }\le 6,l^{\prime }\le n^{\prime }-1)$ configurations, have been studied by K. Wang et al. [46]. Significant improvements over previous calculations have been made by including relevant resonance states belonging to Na-like $S^{23+}$ arising from $1s^2{2}^{7}nl{n}^{\prime }{l}^{\prime }$ configurations with $3\le n\le 7,l\le n-1$ and $n\le n^{\prime }\le 50,l^{\prime }\le 8$. However, no attempts have been made to improve and further extend the calculations of excitation rate coefficients for transitions belonging to $M{o}^{32+}$, such as $2p$–$3d$ which dominate from low to high electron temperatures.

The purpose of this article is to provide accurate structure data and electron–ion collision calculations for transitions between the 257 fine-structure levels arising from the $1s^{2}2s^{2}2p^{5}nl$ and $1s^{2}2s^{1}2p^6{n}^{\prime }{l}^{\prime }$ configurations, with $n\le 7$, $n^{\prime }\le 5$, $l\le 4$ and $l^{\prime }\le 4$. This study is motivated by the importance of atomic data in spectroscopic diagnostics of fusion and astrophysical plasmas, as well as in fundamental physics. Up to the authors knowledge there are no complete data sets for levels, radiative transition probabilities, levels lifetime, collision strengths and rate coefficients belonging to Ne-like $M{o}^{32+}$ ion in the literature and our work provides such results assessed with existing measured and theoretical data. The paper is structured as follows. Section 2 provides a brief description of the relativistic methods used for computing the structure for the Ne-like $M{o}^{32+}$ and contains results such as energy levels, wavelengths, transition probabilities and lifetime of levels. Section 3 presents the approaches for computing the direct and resonance excitation rate coefficients and comparison is provided with other papers from literature. Section 4 summarizes the conclusions.