\documentclass{utns} % \usepackage{graphicx} \usepackage{txfonts} \begin{document} \title{STARK WIDTHS OF Sc IV SPECTRAL LINES WITHIN 4s-4p TRANSITION ARRAY} \author{MILAN S. DIMITRIJEVI\'C \inst{1,2}\fnmsep\thanks{Corresponding author: {mdimitrijevic@aob.rs}} } \institute{Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia \and LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universities, UPMC Univ. Paris 06, 5 Place Jules Janssen, 92195 Meudon Cedex, France} \abstract{Stark full widths at half maximum of six 4s - 4p Sc IV spectral lines, broadened by collisions with electrons, have been calculated for electron density of 10$^{17}$ cm$^{-3}$ by using the modified semiempirical method. The results are provided for temperatures from 10 000 K to 160 000 K. They are used also to discuss the similarities of Sc IV spectral lines within multiplet, supermultiplet and transition array. Obtained results will be included in the STARK-B database which is also a part of Virtual atomic and molecular data center (VAMDC).} \keywords{Stark broadening, atomic data, atomic processes, line profiles.} \maketitle % %________________________________________________________________ \section*{INTRODUCTION} Broadening of spectral lines by fluctuating electric microfield produced by charged particles moving near an emitter/absorber, known as Stark broadening, is important for many topics in physics and astronomy. Data on Stark broadening are needed for laboratory plasmas diagnostic and investigation \citep{Ko99, To06, Pe09}, laser produced plasma analysis, modelling and diagnostics \citep{Go99, So00, Ri70, Se01} and inertial fusion plasma \citep{Ga83, Ke90, Gr92}. Stark broadening data are also useful in technology, as for example for laser welding and piercing of various metals \citep{Ho05, Di14}, design, investigation and improvement of effectivity of various light sources based on plasma \citep{Se01, Wi97}, and the design and developing of lasers \citep{Cs04, Di14}. In many astrophysical plasmas Stark broadening of spectral lines is very important or at least non negligible and should be taken into account \citep{Be97, Po01b, Di03, Di14}. Stark broadening data are particularly important for stellar plasma research, and for analysis and synthesis of stellar spectra \citep{Be97, Di14}. In atmospheres of white dwarfs, pre white dwarf stars, and post AGB (Asymptotic Giant Branch) stars this is usually the main broadening mechanism \citep{Ta03,Mi04,Si06,Dufour11}. It can not be neglected and in some atmospheric layers of A and late B stars \citep{Si05a, Si05b, Simic09}. As an example, \citet{Po01b} have shown that, for A-type star atmospheres, the inclusion of Stark broadening can change the equivalent widths by 10-45\%. Consequently, for determination of abundances, the importance of Stark broadening of analysed spectral lines should be checked. For some astrophysical problems, like modelling of stellar atmospheres, radiative transfer or derivation of accurate atmospheric parameters, we need a very large number of various atomic data, including Stark broadening \citep{Dufour11}. We note as well that \citet{Rauch07} emphasized the crucial importance of an accurate and as much as possible complete Stark broadening data set for large number of atoms and ions, ``for sophisticated analysis of stellar spectra by means of NLTE model atmospheres''. The development of satellite-born astronomy, enabling to obtain stellar spectra with earlier not possible resolution, as well as development of computers enabling very sophisticated NLTE synthesis and modelling of stellar spectra, made that earlier astrophysically insignificant data on trace elements now are important. Scandium is present and observed in stellar spectra \citep{Ad00, Ad01, Ka17} so that its Stark broadening parameters are needed. However, there is no at all experimental data for Stark broadening of scandium lines. For Sc IV there is no neither theoretical data. Results of Stark broadening parameter calculations exist only for Sc II \citep{Po96, Po97}, Sc III \citep{Di92}, Sc X \citep{El11, Di98a, Di98b} and Sc XI \citep{Di98a, Di98b}. In order to provide the Stark broadening data, needed first of all for stellar astrophysics, for Sc IV spectral lines, which are absent completely in the literature, Stark Full Widths at Half intensity Maximum (FWHM) $W$ for six transitions have been calculated here, by using the modified semiempirical method (MSE, \cite{Di80, Di86, Di01}) for collisions of Sc IV ions with electrons. The obtained results will be used for the discussion of similarities of Stark widths within a supermultiplet. Also, the obtained results will be implemented in the STARK-B database \citep{starkb, Sah15}, a repository for spectral line broadening and shifts due to collisions with charged particles, and also a node of Virtual Atomic and Molecular Data Center - VAMDC \citep{vamdc, Rixon11, Dub16}. %__________________________________________________________________ \section*{THE MODIFIED SEMIEMPIRICAL METHOD} Within the frame of the modified semiempirical (MSE) approach \citep{Di80, Di86, Po01b} the electron impact full width at half intensity maximum (FHWM) of an isolated non hydrogenic ion line is given as $$W_{MSE}=N{8\pi\over 3}{\hbar^2\over m^2}\bigl({2m\over{\pi kT}}\bigr)^{1/2}{\pi\over\sqrt{3}}{{\lambda}^2\over 2{\pi}c}\cdot \{\sum_{\ell_i\pm1}\sum_{A_{i'}J_{i'}}\vec{\Re}^2\times$$ $$\times[n_i\ell_iA_iJ_i,n_i(\ell_i\pm 1)A_{i'}J_{i'}]\widetilde{g}(x_{\ell_i,\ell_i\pm 1})+$$ $$+\sum_{\ell_f\pm1}\sum_{A_{f'}J_{f'}}\vec{\Re}^2[n_f\ell_fA_fJ_f,n_f(\ell_f\pm 1)A_{f'}J_{f'}]\widetilde{g}(x_{\ell_f,\ell_f\pm 1}) +$$ $$+\bigr(\sum_{{i'}}\vec{\Re}_{i{i'}}^2\bigl)_{\Delta n\ne 0}g(x_{n_i,n_i+1})+\bigr(\sum_{{f'}}\vec{\Re}_{f{f'}}^2\bigl)_{\Delta n\ne 0}g(x_{n_f,n_f+1})\},\eqno(1) $$ \noindent where the initial level is denoted as $i$ and the final one as $f$. The square of the matrix element $\{\vec{\Re}^2[n_k\ell_kA_kJ_k,(\ell_k\pm 1)A_{k'}J_{k'}], \ \ \ k=i, f\}$ is given by the expression $$\vec{\Re}^2[n_k\ell_kA_kJ_k,n_k(\ell_k\pm 1)A_k'J_k']=$$ $$={\ell_>\over{2J_k+1}}Q[\ell_k A_k,(\ell_k\pm 1)A_k']Q(J_k,J_k')[R^{n_k^*(\ell_{k}\pm 1)}_{n_k^*\ell_k}]^2.\eqno(2).$$ \noindent Here, $\ell_>={\rm max}(\ell_k,\ell_{k}\pm 1)$ and $$\bigl(\sum_{k'}\vec{\Re}^2_{kk'}\bigr)_{\Delta n\neq 0}= ({3n_k^*\over{2Z}})^2{1\over 9}(n{^*_k}^2+3\ell_k^2+3\ell_k+11)\eqno(3)$$ In Eq. (1) $$x_{\ell_k,\ell_{k'}}={E\over{\Delta E_{\ell_k,\ell_{k'}}}},\hskip0.3cm k=i, f$$ \noindent $E={3\over{2}}kT$ is the electron kinetic energy and $\Delta E_{\ell_k,\ell_{k'}}=|E_{\ell_k}-E_{\ell_{k'}}|$ is the energy difference between levels $\ell_k$ and $\ell_k\pm$1 ($k=i, f$), $$x_{n_k,n_k+1}\approx {E\over{\Delta E_{n_k,n_k+1}}},$$ \noindent where for $\Delta n\neq 0$, the energy difference between energy levels with $n_k$ \hskip1mm and \hskip1mm $n_k$+1, $\Delta E_{n_k,n_k+1}$ is approximated as $$\Delta E_{n_k,n_k+1} = 2Z^2E_H/n^{*3}_k, \eqno(4)$$ \noindent $n^*_k=[E_HZ^2/(E_{ion}-E_ k)]^{1/2}$ is the effective principal quantum number, $Z$ is the residual ionic charge (charge ``seen'' by optical electron; for example $Z$=1 for neutrals) and $E_{ion}$ is the appropriate spectral series limit. $N$ and $T$ are electron density and temperature, respectively and $Q(\ell A,\ell'A')$, $Q(J,J') $ multiplet and line factors. The value of $A$ depends on the adopted coupling scheme (see e.g. \citet{So79}). For the $LS$ coupling scheme, which is applied in this work, $A=L$, for the $jK$, $A=K$ and for the $jj$ approximation $A=j$. With $g(x)$ \citep{Griem68, Griem74} and $\widetilde{g}(x)$ \citep{Di80} are denoted the corresponding Gaunt factors. The needed radial integrals $[R^{n_k^*\ell_{k}\pm 1}_{n_{k}^*\ell_{k}}]$ have been calculated here within the Coulomb approximation by using the method of \citet{b10} and the tables of \citet{Oer68}. We note that if for the higher principal quantum numbers there is no the corresponding data in \citet{Oer68}, the radial integrals may be calculated with the help of the article of \citet{Van79}. \begin{table*} \caption{This table gives electron-impact broadening (Stark broadening) Full Widths at Half Intensity Maximum (W) for Sc IV spectral lines, for a perturber density of 10$^{17}$ cm$^{-3}$ and temperatures from 10 000 to 160 000 K. Also are given multiplet numbers and wavelengths from NIST database \citep{Kr17} and $3kT/2\Delta E$ for $T$ = 20 000 K, where $\Delta E$ is the energy difference between closest perturbing level and the closer of initial and final levels.} \label{tab:2} % Give a unique label % For LaTeX tables use \centering \begin{tabular}{crrrr} \hline Transition & T(K) & W[\AA]&W[10$^{12}$ s$^{-1}$]\\ \hline Sc IV 3s$^2$3p$^5$4s$^1$P$^o$-3s$^2$3p$^5$4p$^1$P & 10000.& 0.0735& 0.294\\ (Mult. 100) & 20000.& 0.0519& 0.208\\ 2171.2 \AA & 40000.& 0.0367& 0.147\\ 3kT/2$\Delta$E= 0.541 & 80000.& 0.0260& 0.104\\ & 160000.& 0.0194& 0.0776\\ \hline Sc IV 3s$^2$3p$^5$4s$^1$P$^o$-3s$^2$3p$^5$4p$^1$D & 10000.& 0.0802& 0.337\\ (Mult. 101) & 20000.& 0.0621& 0.260\\ 2119.0 \AA & 40000.& 0.0479& 0.201\\ 3kT/2$\Delta$E= 0.526 & 80000.& 0.0399& 0.167\\ & 160000.& 0.0344& 0.144\\ \hline Sc IV 3s$^2$3p$^5$4s$^1$P$^o$-3s$^2$3p$^5$4p$^1$S & 10000.& 0.0472& 0.321\\ (Mult. 102) & 20000.& 0.0334& 0.227\\ 1665.9 \AA & 40000.& 0.0240& 0.163\\ 3kT/2$\Delta$E= 0.453 & 80000.& 0.0180& 0.122\\ & 160000.& 0.0146& 0.0989\\ \hline Sc IV 3s$^2$3p$^5$4s$^3$P$^o_2$-3s$^2$3p$^5$4p$^3$S$_1$ & 10000.& 0.0964& 0.271\\ (Mult. 85) & 20000.& 0.0682& 0.192\\ 2586.9 \AA & 40000.& 0.0482& 0.136\\ 3kT/2$\Delta$E= 0.606 & 80000.& 0.0344& 0.0969\\ & 160000.& 0.0272& 0.0765\\ \hline Sc IV 3s$^2$3p$^5$4s$^3$P$^o_1$-3s$^2$3p$^5$4p$^3$S$_1$ & 10000.& 0.104 & 0.274\\ (Mult. 85) & 20000.& 0.0737& 0.194\\ 2678.0 \AA & 40000.& 0.0521& 0.137\\ 3kT/2$\Delta$E= 0.606 & 80000.& 0.0373& 0.0980\\ & 160000.& 0.0295& 0.0774\\ \hline Sc IV 3s$^2$3p$^5$4s$^3$P$^o_0$-3s$^2$3p$^5$4p$^3$S$_1$ & 10000.& 0.125 & 0.279\\ (Mult. 85) & 20000.& 0.0886& 0.197\\ 2906.5 \AA & 40000.& 0.0626& 0.140\\ 3kT/2$\Delta$E= 0.606 & 80000.& 0.0451& 0.101\\ & 160000.& 0.0357& 0.0797\\ \hline \end{tabular} \end{table*} \section*{RESULTS AND DISCUSSION} Stark widths (FWHM) of six Sc IV spectral lines, broadened by collisions with electrons, have been calculated using Eqs. (1-4) wihin the frame of MSE method \citep{Di80, Po01b}. Energy levels and ionization energy needed for their calculations have been taken from \citet{Su85}. In Table 1, the results of our MSE calculations of Stark widths (FWHM) for six spectral lines broadened by electron-impacts, for a perturber density of 10$^{17}$ cm$^{-3}$ and for a set of temperatures from 10 000 K to 160 000 K, are shown. The chosen temperature range is of interest in astrophysics, laboratory plasma, fusion research, technology and for lasers and laser produced plasma. If one needs perturber densities lower than 10$^{17}$ cm$^{-3}$ the extrapolation is linear. For higher perturber densities extrapolation is linear if the influence of Debye screening is negligible or reasonably small. In Table 1 are also given the observed wavelength and the multiplet number from the NIST database \citep{Kr17}. It is shown as well the quantity 3$kT/2\Delta E$, representing the ratio of the average energy of free electrons, $E = 3kT/2$, and the energy difference of initial or final and the closest perturbing level, $\Delta E$. It is calculated for $T$=20 000 K in the following way: $$\Delta E = {\rm Max}[E/\Delta E_{i,i'}, E/\Delta E_{f,f'}, \Delta E_{n_i,n_i+1}, \Delta E_{n_f,n_f+1}] \eqno(5)$$ This ratio shows which collisions are dominant. $3kT/2\Delta E = 1$ is the threshold for the corresponding inelastic transition. If it is lower than one, elastic collisions are dominant and it is so called low temperature limit. If it is large, say larger than 50, high temperature limit approximation can be applied. The calculated triplets belong to the same 4s$^3$P$^o$-4p$^3$S multiplet, singlets to the same 4s$^1$P$^o$-4p$^1$L (L=S, P, D) supermultiplet and all calculated transitions to the same, 4s-4p transition array. \citet{Wi82} concluded after examination of regularities and similarities in plasma broadened spectral line widths that line widths in angular frequency units in multiplets usually agree within a few per cent, in supermultiple within about 30 per cent and within a transition array within about 40 per cent. The obtained here results give us an opportunity to check the similarities of Stark line widths within multiplet, supermultiplet and transition array for Sc IV in order to see if they could be used for derrivation of missing values. The transformation of the Stark widths expressed in \AA-units to the widths in angular frequency units may be performed by the following formula: $$W(\mathring{A})=\frac{\lambda ^{2}}{2\pi c}W(s^{-1}), \eqno(6)$$ \noindent where $c$ is the speed of light. From Table 1 we obtained that within the Sc IV 4s$^3$P$^o$-4p$^3$S multiplet the smalest Stark width value is 2.87\% smaller from largest at $T$=10 000 K and 4.02\% at $T$=160 000 K. Within Sc IV 4s$^1$P$^o$-4p$^1$L (L=S, P, D) supermultiplet, the smallest W is 12.76\% smaller from the largest one at $T$=10 000 K and 46.11\% at $T$=160 000 K. For 4s-4p transition array these values are 19.58\% and 46.88\%. We can conclude that in average, these values are in agreement with \citet{Wi82} conclusions and that disagreement increases with the increase of temperature. If we have Stark width value for one member of multiplet, supermultiplet or transition array, the needed width for another member, can be obtained with the help of the expression: $$W_{1}=\left( \frac{\lambda _{1}}{\lambda }\right) ^{2}W. \eqno(7)$$ \noindent Here, with $W$$_{1}$ is denoted the corrected width, for the spectral line with the wavelength $\lambda$$_{1}$, and $\lambda$ is the wavelength of the line for which we have the Stark width value $W$. The Stark FWHM for Sc IV spectral lines obtained within the modified semiempirical method and shown in Table 1, will be also implemented in the \textsc{STARK-B} database \citep{starkb, Sah15}, created first of all for the investigations, modelling and diagnostics of the plasma of stellar atmospheres,but useful also for the diagnostics of laboratory plasmas, investigation of laser produced and inertial fusion plasma as well as for plasma technologies. We want to draw attention that \textsc{STARK-B} database is one of the databases which enter also in the Virtual Atomic and Molecular Data Center - VAMDC \citep{vamdc,Rixon11, Dub16}, created in order to enable an efficacious search and mining of atomic and molecular data. scattered in different databases and to make more convenient their adequate use. VAMDC portal with 30 databases with atomic and molecular data, including STARK-B, is on the web site: http://portal.vamdc.org/. \section*{CONCLUSION} Within the frame of MSE method we have calculated Stark widths for six Sc IV spectral lines broadened by collisions with electrons. The obtained data are used to check similarities of Stark widths within a multiplet, supermultiplet and transition array. Also the Stark widths for six Sc IV spectral lines will be implemented in STARK-B database. Other theoretical as well as experimental data for Stark broadening of Sc IV spectral lines do not exist, so that we hope that the obtained results will be of interest, for various problems, especially in stellar physics and laboratory plasma diagnostics. \section*{ACKNOWLEDGEMENTS} This work is a part of the project 176002 "Influence of collisional processes on astrophysical plasma line shapes" supported by the Ministry of Education, Science and Technological Development of Serbia. \begin{thebibliography}{43} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \bibitem[{Adelman \& Pintaldo (2000)}]{Ad00} Adelman, S.~J., \& Pintaldo, O. 2000, Elemental abundance analyses with Complejo Astronomico El Leoncito REOSC echelle spectrograms. IV. 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