Fine-structure electric-dipole matrix elements of He-like ions for x-ray line-shape calculations

Dipole matrix elements have been calculated with different methods in the intermediate-coupling scheme to study their impact on the Stark effect of highly charged ions. Special emphasis has been devoted to the 1$s$3$l$ fine structure of He-like ions that is widely employed for Stark broadening analysis in dense plasmas. Apart from a wavelength shift in the total He${}_{\beta }$ group in the x-ray energy range, important differences in the Stark width, induced line dips, and relative intensities of maxima are demonstrated for different methods of calculation. We found that these differences were related to the precision of the fine-structure dipole matrix elements and related wavelengths and explored the particularities of atomic structure precision for line broadening purposes that was distinct from the current database approaches. Based on advanced multi-configuration-Dirac-Fock simulations, we propose a complete set of high-precision matrix elements and wavelengths. Detailed numerical results are presented for He-like aluminum. We also discuss the influence of relativistic approximations in atomic structure on the line broadening.

Figure 1

He${}_{\beta }$ Stark profile of aluminum showing different electric-field-induced transitions $k{T}_e$ = 100 eV and $n_e={10}^{20}$ cm${}^{-3}$.

Figure 2

He${}_{\beta }$ Stark profile of aluminum for different atomic data sets: black solid curve: mcdfgme; red dashed curve: hfr; blue dotted curve: fac; $k{T}_e$ = 100 eV; $n_e={10}^{20}$ cm${}^{-3}$. Simulations for different data sets have been shifted and have been normalized to the He${}_{\beta }$ peak mcdfgme.

Figure 3

(a) Stark profile simulations of aluminum He${}_{\beta }$ employing different atomic data $k{T}_e$ = 100 eV and $n_e={10}^{22}$ cm${}^{-3}$. (b) The same as Fig. 1a, however, line profiles have been shifted to the mcdfgme line center position and have been normalized to peak intensity. Large discrepancies between the different data sets are observed (indicated by the arrow).

Figure 4

(a) He${}_{\beta }$ Stark profile of aluminum for different mixed sets of atomic data: $k{T}_e$ = 100 eV and $n_e={10}^{22}$ cm${}^{-3}$. Black solid curve: mcdfgme data; red dashed curve: mcdfgme data, except for the matrix element $\langle 1s3d{}^1{D}_2\left|r\right|1s3p{}^1{P}_1\rangle$ that employs hfr data; and blue dotted curve: hfr data. (b) The same as Fig. 3a, however, the curves have been normalized and have been shifted to the peak intensity of mcdfgme.

Figure 5

(a) Variation in the dipole matrix element $\langle 1s3d{}^1{D}_2\left|r\right|1s3p{}^1{P}_1\rangle$and its influence on the He${}_{\beta }$ Stark profile of aluminum for $k{T}_e$ = 100 eV and $n_e$ = $3\times {10}^{22}$ cm${}^{-3}$. All data are mcdfgme data, black solid curve: ab initio data set; red dashed curve: only the matrix element $\langle 1s3d{}^1{D}_2\left|r\right|1s3p{}^1{P}_1\rangle$ is multiplied by a factor of 2; and blue dotted curve: the matrix element is multiplied by a factor of 0.5. (b) The same as (a), however, only a variation in the intercombination dipole matrix element $\langle 1s3d{}^3{D}_2\left|r\right|1s3p{}^1{P}_1\rangle$is involved. (c) The same as (a), however, only a variation in the dipole matrix element of the triplet system $\langle 1s3d{}^3{D}_2\left|r\right|1s3p{}^3{P}_1\rangle$is involved.

Figure 6

Variation in the 1$s$3$l$ fine-structure energies and their influence on the He${}_{\beta }$ Stark profile of aluminum for $k{T}_e$ = 100 eV and $n_e$ = $3\times {10}^{22}$ cm${}^{-3}$. All data are mcdfgme data, black solid curve: ab initio data set; red dashed curve: the energy difference between states 1$s$3$p$ ${}^1{P}_1$ and 1$s$3$d$ ${}^1{D}_2$ has been increased by a factor of 2; and blue dotted curve: the energy difference between states 1$s$3$p$ ${}^1{P}_1$ and 1$s$3$d$ ${}^1{D}_2$ has been decreased by a factor of 2.

Figure 7

Influence of relativistic effects on the He${}_{\beta }$ Stark profile of aluminum for different atomic data sets: black solid curve: mcdfgme; green dot-dashed curve: “mcdfgme nonrelativistic;” red dashed curve: hfr; and blue dotted curve: fac, $k{T}_e$ = 100 eV and $n_e={10}^{20}$ cm${}^{-3}$.

Figure 8

(a) Influence on the He${}_{\beta }$ Stark profile of aluminum for different atomic data sets: black solid curve: mcdfgme; green dot-dashed curve: mcdfgme nonrelativistic; red dashed curve: hfr; and blue dotted curve: fac; $k{T}_e$ = 100 eV; $n_e={10}^{22}$ cm${}^{-3}$. (b) The same as (a), however, all calculations have been shifted and have been normalized to mcdfgme data.