High-Resolution Fourier Transform Spectra of Atomic Sulfur: Testing of Modified Quantum Defect Theory

Abstract

QDT (quantum defect theory) is an effective technique for calculating processes involving highly excited (Rydberg) states of atoms, ions, and molecules with one valence electron outside filled shells, whose spectrum generally resembles a hydrogen-like atom’s spectrum. At the expense of some modification of QDT, in this paper, we extend its applicability to describe low- and intermediate-excited levels of atoms with more complex spectra (on the example of atomic sulfur S I). Transitions between just such states are responsible for the infrared (IR) spectra of atoms. While the quantum defects (QDs) of the highly excited Rydberg levels are determined by the energies of individual levels near the ionization threshold, the radial wave functions of low excited electronic states, in the framework of our modification of QDT, include the QD dependence on energy over a wide energy range; this dependence is determined from the whole spectral series. We show that, outside the atomic core domain, the electron radial functions calculated using modified semi-phenomenological QDT agree well with ab initio calculations. As another assessment of QDT accuracy, we show satisfactory agreement of the probabilities of dipole transitions in S I, taken from the NIST Atomic Spectra Database, with our QDT calculations. We perform an indirect experimental verification of QDT on the basis of spectra of S I in gas-discharge plasma measured by time-resolved high-resolution Fourier transfer spectroscopy (FTS). The Boltzmann plot built from our measured spectra demonstrates that QDT provides a satisfactory approximation for calculating the experimental lines’ intensities.

1. Introduction

One of the fundamental tasks of astrophysics is to determine the detailed chemical composition of as many objects (stars, planets, interstellar clouds) as possible. In recent decades, astronomical observations in the infrared (IR) range have become increasingly important. The instrumental capabilities of infrared astronomy are constantly evolving and now include many observatories—ground, airborne, and space [1].

To meet the growing demands of infrared (IR) astrophysics, we need large amounts of spectroscopic data (such as level energies, line wavenumbers, probabilities of radiative transitions or oscillator strengths [2], electron scattering cross sections [3], atomic collisions [4], photoionization and recombination [5,6], Stark line broadening parameters [7], etc.) for a large number of atoms, including transitions involving excited (Rydberg) levels [8].

It should be noted that ab initio calculations of atomic structures for Rydberg states encounter significant difficulties [9], and the accuracy of the energy level calculations is much lower than the experimental accuracy of their spectroscopic measurement. Therefore, of particular relevance are methods that can combine the possibility of analytical representation of the results (and, therefore, computational simplicity) with the possibility of calculating electron transition matrix elements through experimental energy levels.

QDT (quantum defect theory, [10]) meets the above requirements. QDT assumes that the processes of radiation–atom interaction involve one “optical” Rydberg, which moves at a considerable distance from the atomic (molecular) core. The energies of this electron, $E_{nl}$, in the core potential form a spectral series similar to those of a hydrogen-like atom; these spectral series are given by the following well-known expression:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E_{nl}=E_{\mathrm{IP}}-{\displaystyle \frac{Z^2}{2{(n-{\mu }_l)}^2}}=E_{\mathrm{IP}}-{\displaystyle \frac{Z^2}{2{\nu }_{nl}^2}}.\end{array}\end{array}$$

(1)

According to the Rydberg Formula (1), each Rydberg series of levels with the orbital quantum number l consists of infinitely many levels, numbered by the integer principal quantum number n, that converge (at $n\to \infty$) to the ionization threshold, or ionization potential (IP), $E_{\mathrm{IP}}$. The energies of this level series, $E_{nl}$, are expressed in terms of quantum defect (QD)—${\mu }_l=n-{\nu }_{nl}$, where ${\nu }_{nl}$ is the effective (non-integer) quantum number $\nu$—and the integer principal quantum number. For neutral atoms, the core charge $Z=1$.

QDT can be useful for many problems in atomic and plasma physics [11], in particular, for describing processes involving Rydberg states, such as photoionization [12,13,14], collisions of excited atoms and molecules [15,16,17,18] (including those accompanied by the development of dynamic chaos [19]), etc.

The simplicity of Rydberg Formula (1) is due to the main approximation of QDT: the potential of the atomic core can be considered Coulombic at large distances, i.e., $r>r_{\mathrm{c}}$, where $r_{\mathrm{c}}$ is an effective dimension of the core. The shape of the potential in the intra-core domain ($r<r_{\mathrm{c}}$) is unknown, but a whole spectral Rydberg $l$-series (with high n corresponding to sufficiently highly excited levels) can be determined from one quantum defect, ${\mu }_l$, the latter being computed from any (sufficiently high) level energy of the experimental l-series. The quantum defects ${\mu }_l$ play in QDT a role similar to $\left\{l_{\mathrm{eff}}\right\}$ values appearing in the method of model potential [20,21,22]. Here, the curly brackets surround the fractional part, with $\left\{l_{\mathrm{eff}}\right\}$ being the effective orbital number, which is not an integer parameter of the model potential technique.

The semi-phenomenonlogical character of QDT (and the model potential method) consists in the fact that the quantum defects ${\mu }_l$ (or, in other words, the effective principal quantum numbers ${\nu }_{nl}$) are determined from experimental data, and the wave functions of the Rydberg electron in the region $r>r_{\mathrm{c}}$ (outside the ionic core) are calculated analytically. Using these wave functions, one can calculate amplitudes of atom–photon interaction processes. In particular, to calculate multiphoton processes in atoms with one valence electron over closed shells in the QDT approximation, an analog of the Coulomb Green’s function (QDT-GF) [23,24] has been constructed. From QDT-GF, the radial electron wave functions can also be calculated. These functions, like the QDT-GF, depend on the quantum defects ${\mu }_l$, which contain the experimental information about the spectrum of the atom.

As mentioned above, in numerous applications, QDT is used for atoms with a single electron outside the filled shells only. Here, we use wave functions obtained using a modified QDT-GF method to analyze spectra of more complex atoms that have several electrons over the filled shells. As an application, we compute an array of transition probabilities in neutral sulfur atom S I to analyze its high-resolution Fourier spectra in a gas discharge plasma.

Sulfur is a vital element in chemical, biological, and environmental processes, making it a major focus for researchers [25]. It is one of the most abundant elements in the universe, closely following hydrogen, carbon, nitrogen, and oxygen in the solar system. Sulfur lines have been observed in the spectra of the Sun and other astrophysical objects. The abundance and distribution of sulfur provide valuable insights into the chemical evolution of galaxies, stars, and planets. For instance, understanding the distribution of sulfur helps us comprehend the connection between the composition of planets and the formation processes in protoplanetary disks [26]. Sulfur is a crucial element in studies of the atomic composition of planetary nebulas, as it unveils the chemical evolution occurring in specific regions of the galaxy [27].

Let us describe the general outline of this work. Section 3.1 presents explicit formulae for the electron radial wave functions in QDT approximation together with general formulae for dipole transition matrix elements in $LS$ coupling. Section 3.2 briefly describes the design of the experiment in which the spectra of sulfur-containing plasma discharge were recorded. Section 2 compares the radial wave functions computed by QDT with those calculated using an ab initio (Hartree–Fock) method. This comparison proves the suitability of QDT for low excited terms of S I, not only for highly excited Rydbergs. We compare the probabilities of transitions involving S I ${(}^{4}S)nl$ levels taken from the NIST ASD (Atomic Spectra Database) [28] with those calculated within the QDT framework. From the measured IR spectra of S I, we construct a Boltzmann plot that demonstrates the reasonable accuracy of the QDT calculations of transition probabilities to describe the observed line intensities in discharge plasma spectra.

2. Results and Discussion

The strongest S I spectral lines in the 7000–11,000 ${\mathrm{cm}}^{-1}$ region corresponding to triplet–triplet and quintet–quintet $4s$–$4p$ transitions, as well as quintet–quintet $4p$–$5s$ transitions, are shown in Figure 1.

In Figure 2, we compare two calculations for radial wave functions of some of the above-mentioned terms. Solid lines correspond to QDT calculations (4), and the dots represent ab initio (using Robert D. Cowan’s code described in the book [29]; see also [30]). Figure 2 shows quantitative agreement between QDT and ab initio calculations even for lowest levels in s- and p-series over a wide range of the radial variable r except for a small intra-core region, $r\lesssim r_{\mathrm{c}}$. In QDT approximation, the radial wave function $P_{nl}\left(r\right)$ has $n_r=\mathrm{Int}\left({\nu }_{nl}\right)-l-1$ nodes (Int denotes the nearest integer number) outside the core ($r\gtrsim r_{\mathrm{c}}$). The radial quantum number $n_r$ is assigned in the model potential theory [20,21,22,31,32] for levels in a $nl$-series in the order of increasing energy, starting from $n_r=0$ for the lowest-excited level of each $nl$-series. The effective principal quantum number, $n_{\mathrm{eff}}={\nu }_{nl}=n_r+l_{\mathrm{eff}}+1$, in the model potential technique is calculated through the so-called effective orbital number $l_{\mathrm{eff}}={\nu }_{nl}-n_r-1$, which is non-integer.

QDs ${\mu }_l$ and principal quantum numbers ${\nu }_{nl}$ of some terms of S I (including those indicated in Figure 1 and Figure 2) are presented in

Table 1. QDs ${\mu }_l$, integer principal (n) and orbital (l) quantum numbers, and non-integer effective principal (${\nu }_{nl}$) and orbital ($l_{\mathrm{eff}}$) quantum numbers of the levels appearing in Figure 1 and Figure 2.

Term ${(}^4\mathit{S})\mathit{nl}{(}^{2\mathit{S}+1}\mathit{L})$	n	l	${\mathit{\nu }}_{\mathit{nl}}$	${\mathit{\mu }}_{\mathit{l}}\left({\mathit{\nu }}_{\mathit{nl}}\right)$	${\mathit{n}}_{\mathit{r}}$	${\mathit{l}}_{\mathbf{eff}}$
${(}^4\mathrm{S})4\mathrm{s}{(}^3\mathrm{S})$	4	0	1.972	2.028	1	−0.02835
${(}^4\mathrm{S})4\mathrm{s}{(}^5\mathrm{S})$	4	0	1.883	2.117	1	−0.1166
${(}^4\mathrm{S})4\mathrm{p}{(}^3\mathrm{P})$	4	1	2.425	1.575	0	1.425
${(}^4\mathrm{S})4\mathrm{p}{(}^5\mathrm{P})$	4	1	2.337	1.663	0	1.337
${(}^4\mathrm{S})4\mathrm{d}{(}^3\mathrm{D})$	4	2	3.799	0.2014	1	1.799
${(}^4\mathrm{S})4\mathrm{d}{(}^5\mathrm{D})$	4	2	3.575	0.4247	1	1.575
${(}^4\mathrm{S})5\mathrm{s}{(}^3\mathrm{S})$	5	0	2.998	2.002	2	−0.001832
${(}^4\mathrm{S})5\mathrm{s}{(}^5\mathrm{S})$	5	0	2.922	2.078	2	−0.07844
${(}^4\mathrm{S})5\mathrm{p}{(}^3\mathrm{P})$	5	1	3.437	1.563	1	1.437
${(}^4\mathrm{S})5\mathrm{p}{(}^5\mathrm{P})$	5	1	3.374	1.626	1	1.374
${(}^4\mathrm{S})6\mathrm{s}{(}^3\mathrm{S})$	6	0	4.006	1.994	3	0.005860
${(}^4\mathrm{S})6\mathrm{s}{(}^5\mathrm{S})$	6	0	3.933	2.067	3	−0.06727
${(}^4\mathrm{S})6\mathrm{p}{(}^3\mathrm{P})$	6	1	4.403	1.597	2	1.403
${(}^4\mathrm{S})6\mathrm{p}{(}^5\mathrm{P})$	6	1	4.386	1.614	2	1.386

. As one can see from this table, the indicated values of $n_r$ coincide with the number of nodes (outside the core) of the corresponding wave functions shown in Figure 2. Note that some negative $l_{\mathrm{eff}}$ appear in Table 1. The radial wave function in the model potential framework has a behavior $P_{nl}\left(r\right)\propto r^{1+l_{\mathrm{eff}}}$ at $r\to 0$, so small negative $l_{\mathrm{eff}}$ do not influence on the convergence of radial integral at small r. They are given in Table 1 for illustrative purposes only; our calculations do not use model potential wave functions.

The QDT-calculated wave functions in Figure 2 are in satisfactory agreement with the ab initio wave functions because we took into account the dependence of QDs $\mu \left(E\right)$ on energy in a wide range, not only in near-threshold region $E\to -0$, as well as the function $\Xi$ (5) in the normalization. This means that the $nl$-level wave function is not constructed from the experimental energy of one $nl$-level but includes information about the entire $l$-series (several values of n). We interpret this as some account of the multi-particle effects that distinguish the low excited states from the highly excited (Rydberg) ones. In high Rydberg states, the optical electron interacts quite weakly with the core electrons, whereas the low excited optical electron’s wave functions overlap essentially with the core electron wave functions.

Note that the radius of the core $r_{\mathrm{c}}$ may be different for states with different orbital quantum numbers l, e.g., because of the centrifugal contribution $l(l+1)/2r^2$ to the intra-core potential. It follows from Table 1 that the radial $4p$ functions should have no nodes outside the core; therefore, we can estimate $r_{\mathrm{c}}\simeq 2.5$ a. u. for p-electrons. Note that QDT wave functions demonstrate non-physical behavior inside the core because of the singularity of the Whittaker function at $r\to 0$. However, the radial integral (9) is affected only insignificantly by this small domain ($0<r\lesssim r_{\mathrm{c}}$) inside the core. Indeed, this integral contains the core-polarization correction

$$\kappa (r,r_{\mathrm{c}})={\displaystyle \frac{\alpha r}{{(r^2+r_{\mathrm{c}}^2)}^2}},$$

(2)

where $\alpha$ is the static dipole polarizability of the core. This correction accounts for some multi-particle effects that are due to the polarization of the atomic core [33,34,35,36]. At the $r\to 0$ limit, $\kappa (r,r_{\mathrm{c}})$ tends to zero, which reduces (together with the r factor) the contribution of the core domain ($0<r\lesssim r_{\mathrm{c}}$) into the radial integral (9). For large $r\gg r_{\mathrm{c}}$, the polarization correction $\kappa (r,r_{\mathrm{c}})\sim \alpha /r^3$ vanishes as well so that $r(1-\kappa (r,r_{\mathrm{c}}))\sim r$, and therefore, it is the large r domain that forms the main contribution to the integral (9). This is clearly seen for high-n Rydberg levels, for which the mean electron radius is of the order of $\sim n^2$. Given this fact, it is clear why both the QDT and the model potential theory give similar results for the matrix elements between high Rydberg states [31,32]. What is nontrivial is the fact that QDT, considered previously to be applicable for high n levels only, describes quantitatively even the wave functions of levels with low $n\gtrsim 4$ and even for the lowest of the ${(}^{4}S)nl$ terms (i.e., for $n=4$ and $s=0$); hence, the same can be expected for the transition matrix elements (9).

We compare the transition probabilities (A-values) between levels of ${(}^{4}S)nl$ terms (total 135) taken from the NIST Atomic Spectra Database [28] with our QDT-calculated A-values (see Formulas (7)–(9)). The comparison results are shown in Figure 3, from which it can be seen that most of the A-values calculated using QDT differ by no more than 12% from the values listed in the NIST ASD. The A-values compared in Figure 3 range from ∼$5\times {10}^4$ to ∼$2\times {10}^7$, which correspond to relatively strong transitions. For weaker transitions, the difference from the NIST is somewhat greater; some examples are given in

Table 2. Comparison of QDT-calculated probabilities for some weak transitions with the corresponding NIST values.

Wavenumber $\mathit{\sigma }$ (${\mathbf{cm}}^{\mathbf{-}1}$)	Lower Level a	Upper Level b	${\mathit{A}}_{\mathit{b}\leftarrow \mathit{a}}$ (${\mathbf{s}}^{\mathbf{-}1}$)
			This Work	NIST
3567.46	${(}^4\mathrm{S})5\mathrm{d}{(}^5{\mathrm{D}}_4)$	${(}^4\mathrm{S})8\mathrm{f}{(}^5{\mathrm{F}}_4)$	$1.12\times {10}^4$	$1.28\times {10}^4$
3567.46	${(}^4\mathrm{S})5\mathrm{d}{(}^5{\mathrm{D}}_4)$	${(}^4\mathrm{S})8\mathrm{f}{(}^5{\mathrm{F}}_3)$	$9.59\times {10}^2$	$1.10\times {10}^3$
3567.12	${(}^4\mathrm{S})5\mathrm{d}{(}^5{\mathrm{D}}_3)$	${(}^4\mathrm{S})8\mathrm{f}{(}^5{\mathrm{F}}_3)$	$2.01\times {10}^4$	$2.31\times {10}^4$
3567.12	${(}^4\mathrm{S})5\mathrm{d}{(}^5{\mathrm{D}}_3)$	${(}^4\mathrm{S})8\mathrm{f}{(}^5{\mathrm{F}}_2)$	$2.68\times {10}^3$	$3.07\times {10}^3$
3566.79	${(}^4\mathrm{S})5\mathrm{d}{(}^5{\mathrm{D}}_2)$	${(}^4\mathrm{S})8\mathrm{f}{(}^5{\mathrm{F}}_1)$	$4.47\times {10}^3$	$5.13\times {10}^3$
3566.79	${(}^4\mathrm{S})5\mathrm{d}{(}^5{\mathrm{D}}_2)$	${(}^4\mathrm{S})8\mathrm{f}{(}^5{\mathrm{F}}_2)$	$2.68\times {10}^4$	$3.08\times {10}^4$
3527.40	${(}^4\mathrm{S})4\mathrm{f}{(}^3{\mathrm{F}}_2)$	${(}^4\mathrm{S})6\mathrm{d}{(}^3{\mathrm{D}}_2)$	$1.24\times {10}^4$	$1.12\times {10}^4$
3525.89	${(}^4\mathrm{S})4\mathrm{f}{(}^3{\mathrm{F}}_3)$	${(}^4\mathrm{S})6\mathrm{d}{(}^3{\mathrm{D}}_3)$	$8.95\times {10}^3$	$8.00\times {10}^3$
3525.87	${(}^4\mathrm{S})4\mathrm{f}{(}^3{\mathrm{F}}_2)$	${(}^4\mathrm{S})6\mathrm{d}{(}^3{\mathrm{D}}_3)$	$2.56\times {10}^2$	$2.29\times {10}^2$

.

Another way to test the capabilities of the modified QDT for calculating the radial wave functions of electrons is to compare the transition probabilities with experimental results. The simplest (albeit indirect) way to check these values experimentally is to construct Boltzmann plots, which are valuable tools in spectral diagnostics of plasmas.

For a line corresponding to emission the upper state $|b\rangle$ to the lower state $|a\rangle$, its intensity $I_{b\to a}^{\mathrm{calc}}$ is proportional to the transition probability $A_{b\leftarrow a}$ (in optically thin LTE plasma characterized by electron temperature T):

$$I_{b\to a}^{\mathrm{calc}}\sim g_b{A}_{b\leftarrow a}{\sigma }_{ab}{\mathrm{e}}^{-E_b/T},$$

(3)

where $E_b$ and $g_b$ are the upper level’s excitation energy and statistical weight, respectively, and the transition wavenumber is ${\sigma }_{ab}={\displaystyle \frac{1}{2\pi \hslash }}(E_b-E_a)$.

Using the measured intensities, $I_{b\to a}^{\mathrm{obs}}$, and the calculated ones, $I_{b\to a}^{\mathrm{calc}}$ (3), we plotted the so-called Boltzmann plot, i.e., the logarithm of the $I^{\mathrm{obs}}/I^{\mathrm{calc}}$ ratio as a function of the upper level energy $E_b$. After fitting these data to the linear relationship of $\mathrm{const}-{\displaystyle \frac{E_b}{T}}$, the effective plasma temperature can be estimated together with the so-called instrumental correction function $H\left(\sigma \right)$. The latter accounts for the spectral dependence of the recorded signal. Both the instrumental H function and temperature T are computed simultaneously with the iterating procedure described in Ref. [37].

Figure 4 shows a Boltzmann plot constructed on the base of our measured spectra. As can be seen from this figure, the points on the Boltzmann plot show a reasonable scatter from the approximating straight line, and the electronic temperature can be extracted from this plot with satisfactory (about 14%) accuracy. In the course of building the plot in Figure 4, we also calculated the instrumental correction function $H\left(\sigma \right)={\mathrm{e}}^{-0.005447+0.00005015\sigma }$ in the $790<\sigma <\mathrm{11,000}{\mathrm{cm}}^{-1}$ spectral domain, where this function varies monotonically from 1.4 to 1.7, which corresponds to the minimal correction introduced to the experimental points. This fact is additional evidence in favor of the fact that QDT gives a satisfactory approximation for calculating the intensities of the lines measured in experimental spectra. The error bars in Figure 4 show the uncertainties (on the level of one standard deviation) of the $log(I^{\mathrm{obs}}/I^{\mathrm{calc}})$ values. These uncertainties were calculated by adding in quadrature of (i) uncertainty of QDT-calculated A-values (which was estimated as 14%) and (ii) uncertainty of the measured $I^{\mathrm{obs}}$ values. The latter arises because of the uncertainty of fitting the measured spectral peaks to a Lorentzian or Gaussian profile.

3. Methods

3.1. Theory

Knowing the exact analytic expression for the GF of an atom allows one to easily derive the wave functions of both continuum and bound electronic states. It also enables the calculation of the properties due to atom–field interactions [38,39,40]. In QDT approximation, the analytical formulae for the electron radial wave functions were first derived using the QDT-GF [23] technique through hypergeometric functions, utilizing a relationship between the continuum states’ scattering phases and the bound states’ QDs [24,41]. The above-mentioned radial electron wave functions are constructed by calculating residues of the QDT-GF in its poles (which are determined by experimental spectroscopic data). The QDT-GF technique allows the calculation of two-photon matrix elements: frequency-dependent polarizability of alkali metal atoms [24] and the molecular hydrogen ion (see Ref. [42], where the two-photon ionization of ${\mathrm{H}}_2^{+}$ was also calculated). For single-photon transitions involving very high atomic and molecular Rydberg states, a WKB–QDT was developed in Refs. [43,44].

All the above calculations are performed for systems with a single electron outside the filled shells. Ref. [45] proposed a modification of QDT-GF, called reduced-added GF (RAGF). The RAGF technique essentially uses the QD dependence on the electron energy, ${\mu }_l\left(E\right)$. It means that the QDs, ${\mu }_l$, are to be determined not from the experimental energy of one level with a given $l$ but rather from experimental energies of the entire $l$-series.

RAGF was used in Ref. [45] to calculate two-photon transitions (namely dynamical polarizabilities) of many atoms having several valence (including Rydberg) electrons outside the filled shells. In this paper, we use the results of the RAGF technique to calculate the single-photon transition probabilities in atomic sulfur S I. Specifically, we use an analytical expression [45] for the radial wave functions in terms of Whittaker functions [46]:

$$\begin{array}{c}\hfill P_{nl}\left(r\right)={\displaystyle \frac{Z^{1/2}}{{\nu }_{nl}}}{W}_{{\nu }_{nl},l+1/2}\left({\displaystyle \frac{2Zr}{{\nu }_{nl}}}\right){\left[{\displaystyle \frac{{\Xi }_l\left(E_{nl}\right)}{{\Pi }_l\left({\nu }_{nl}\right)\Gamma (l+1+{\nu }_{nl})\Gamma ({\nu }_{nl}-l)\left(1+\frac{\partial {\mu }_l\left({\nu }_{nl}\right)}{\partial \nu }\right)}}\right]}^{1/2}.\end{array}$$

(4)

The normalization factor in Equation (4) is different from that traditionally appearing in QDT wave functions [10]. This factor contains an entire analytic function of energy, $\Xi \left(E\right)$, which can be fully determined by its values on a countable set of points, as an approximation to which we can take the experimental spectrum points, $E_{nl}$:

$$\begin{array}{l}{\Xi }_l\left(E_{nl}\right)={\Pi }_l\left(n\right),n\ge l+1,\hfill \end{array}$$

(5)

$$\begin{array}{c}{\Pi }_l\left(\nu \right)={\displaystyle \frac{{\nu }^{2l}}{{\displaystyle \prod _{k=0}^{l-1}}(k+{\mu }_l+\nu )(k+1-{\mu }_l-\nu )}}.\hfill \end{array}$$

(6)

The RAGF theory introduces $\Xi$ and $\Pi$ functions to eliminate “waste” pure-Coulombic QDT-GF poles; such poles arise if the QD ${\mu }_l$ is obtained from only one experimental energy $E_{nl}$ of a high-n level. However, this is a satisfactory approximation for highly excited Rydberg levels only, i.e., in the limit $n\to \infty$ or, what is the same, $E\to -0$. In fact, the functions (5) and (6) bring an additional energy dependence into the radial wave function (4) together with the derivative $\partial {\mu }_l\left(\nu \right)/\partial \nu$ of the quantum defect ${\mu }_l$ $(E=-1/2{\nu }^2)$ over a wide energy range (not just near the $E\to -0$ threshold). The practical calculation of this derivative is performed with the help of fitting of ${\mu }_l\left(E\right)$ from a discrete set of its values, ${\mu }_l\left(E_{nl}\right)$, at the experimental levels of l-series. The fitting can be performed, for example, according to Ritz expansion; see [47] (Equation (12)) or [48] (Equations (10.10) and (11.52)). Obviously, one needs as many experimental energies in a Rydberg l-series as possible to provide a reasonable accuracy of the $\partial {\mu }_l\left(\nu \right)/\partial \nu$ calculation procedure.

The probability $A_{b\leftarrow a}$ of radiative transition from the upper $|a\rangle$ state to the lower $|b\rangle$ state is expressed through the line strength $S_{a\to b}$ or oscillator strength $f_{a\to b}$ [47,49]:

$$\begin{array}{ccc}\hfill S_{a\to b}& =& \sum _{M_a,M_b}{\left|\langle {\gamma }_a{J}_a{M}_a\left|\mathcal{D}\right|{\gamma }_b{J}_b{M}_b\rangle \right|}^2,\hfill \\ \hfill -f_{a\to b}& =& {\displaystyle \frac{2m}{3\hslash e^2}}{\displaystyle \frac{\omega }{g_a}}{S}_{a\to b},\hfill \\ \hfill A_{b\leftarrow a}& =& {\displaystyle \frac{4{\omega }^3}{3\hslash c^3}}{\displaystyle \frac{1}{g_b}}{S}_{a\to b}={\displaystyle \frac{2{\omega }^2{e}^2}{m{c}^3}}\left|f_{a\to b}\right|,\hfill \end{array}$$

(7)

where $\omega ={\omega }_{a\to b}=(E_a-E_b)/\hslash$ is the transition frequency and $g_{b,a}=\left[J_{b,a}\right]:=2J_{b,a}+1$ are the statistical weights of the corresponding levels. The line strength $S_{a\to b}$ is the square of the dipole transition matrix element summed by projections $M_{a,b}$ of the total orbital moments $J_{a,b}$ of the initial and final term. The additional quantum numbers, ${\gamma }_{a,b}$, are determined, for example, by the momentum coupling scheme. For the $LS$ coupling scheme, which is applicable to the neutral sulfur states, ${\gamma }_{a,b}$ include the total orbital momentum $L_{a,b}$ and spin $S_{a,b}$ of the initial and final terms, and the line strength can be expressed through the $6j$-symbols using the angular momentum [49] technique:

$$\begin{array}{cc}{S}_{a\to b}=\hfill & \sum _{M_a,M_b}{\left|\langle S_a{L}_a{J}_a{M}_a\left|\mathcal{D}\right|S_b{L}_b{J}_b{M}_b\rangle \right|}^2\hfill \\ & =\left[J_a\right]\left[J_b\right]\left[L_a\right]\left[L_b\right]{\left\{\begin{array}{ccc}{L}_a& J_a& S_a\\ J_b& L_b& 1\end{array}\right\}}^2{\left\{\begin{array}{ccc}{l}_a& L_a& L_a^{\mathrm{c}}\\ L_b& l_b& 1\end{array}\right\}}^2{\left|\langle n_a{l}_a\parallel \mathcal{D}\parallel n_b{l}_b\rangle \right|}^2,\hfill \end{array}$$

(8)

where $L^{\mathrm{c}}$ is the core orbital momentum. Most of the terms that form prominent lines in our S I spectra are characterized by the $3s^{2}3p^{3 4}S$ core term, i.e., $L^{\mathrm{c}}=0$; in this paper, we restrict ourselves to terms with the $3s^{2}3p^3{(}^{4}S)$ core.

The following radial integral gives the reduced dipole matrix element (8) (where e is the electron charge):

$$\langle n_a{l}_a\parallel \mathcal{D}\parallel n_b{l}_b\rangle =emax\{l_a,l_b\}{\int }_{r_{\mathrm{c}}}^{\infty }r(1-\kappa (r,r_{\mathrm{c}}))P_{n_a{l}_a}\left(r\right)P_{n_b{l}_b}\left(r\right)\mathrm{d}r,$$

(9)

where $\kappa (r,r_{\mathrm{c}})$ accounts for (multi-particle) core polarization effects (see Equation (2) and the discussion in Section 2).

3.2. Experiment

A microwave discharge in a buffer gas of helium was used to generate sulfur emission lines. The measurements were carried out in a quartz tube equipped with windows (KBr and CaF₂) at its ends. This allowed us to obtain spectra in two spectral domains: 760–4000 ${\mathrm{cm}}^{-1}$ and 2000–6500 ${\mathrm{cm}}^{-1}$. Two detectors, HgCdTe (MCT) and InSb, were used for these domains, respectively. Each detector had a different sensitivity and was cooled by liquid nitrogen. To cover the spectral range from 2000 ${\mathrm{cm}}^{-1}$ to 7000 ${\mathrm{cm}}^{-1}$, we used several Northumbria Optical Coatings Ltd. (Boldon, UK) bandpass interference filters. The spectral regions with wavelength numbers below 2000 ${\mathrm{cm}}^{-1}$ and above 7000 ${\mathrm{cm}}^{-1}$ were measured without interference filters.

We conducted a plasma discharge in a 99.996% helium stream at a pressure of 93–200 Pa (0.7–1.5 Torr). For a microwave source, we used a Microtron 200 W instrument operating at 2.5 GHz and 70 W power. At the start of the microwave discharge, it was necessary to optimize the helium pressure to ensure easy ignition and stabilize the microwave power at the optimum level. Sulfur powder with a purity of 99.99% was used as a sample. During the measurement, the discharge conditions varied in space and time as the sulfur diffused into the colder edges. It was observed that the plasma spectra show traces of impurities, such as O, N, and Ar, in the discharge.

The Fourier transfer spectroscopy (FTS) was performed on a Bruker IFS 120 spectrometer (Bruker Corporation, Leipzig, Germany). The spectral resolution was 0.04 ${\mathrm{cm}}^{-1}$ in the 790–2000 ${\mathrm{cm}}^{-1}$ domain and 0.05 ${\mathrm{cm}}^{-1}$ in the 2000–110,000 ${\mathrm{cm}}^{-1}$ domain. During prolonged measurements in the He plasma discharge, the windows were covered with sulfur layers, resulting in a decrease in the intensity of the signal. Several series of S I spectra recordings were performed to maintain stable results, and in each case, the sulfur samples were replaced together with the KBr and CaF₂ optical windows. To increase SNR, the spectrometer performed about a hundred scans.

Water absorption lines were used to calibrate the measured spectra; we took the high-precision wavenumbers of these lines from HITRAN [50]. For a more detailed description of the calibration, as well as the time-resolved FTS, see Refs. [51,52].

4. Concluding Remarks

QDT (quantum defect theory) is a semi-phenomenological theory that is successfully used for analytic calculations of a number of data needed in astrophysics, low-temperature plasma, and other applications of atomic and molecular physics, such as radiative transition probabilities, polarizabilities, photoionization cross-sections, etc. It was believed that QDT is applicable only for highly excited (Rydberg) electrons in a field of a closed-shell atomic core that essentially takes place in atoms with hydrogen-like spectrum (such as alkali metal atoms).

In the present paper, we use a modification of QDT that allows us to calculate transition probabilities for an atom with a more complex spectrum, sulfur S I. This modification is due to taking into account the energy dependence of quantum defects. We show that the modified QDT provides a good description for radial wave functions even of low excited S I terms as well as for radiation transitions involving such terms (not only the high-Rydberg terms).

Note that in QDT, because of its semi-phenomenological nature, the level energies (and, therefore, the line wavelengths) are not calculated, but rather they are considered as external input parameters from which the quantum defects $\mu \left(E\right)$ are calculated. In turn, once $\mu \left(E\right)$ are known, the radial electron wave functions can be calculated according to Equations (4) and (7). Using these wave functions, the transition probabilities (8) are calculated through the squares of one-photon matrix elements (9).

To check our QDT calculation of dipole transition probabilities (A-values), we took 135 relatively strong ($5\times {10}^4\lesssim A\lesssim 2\times {10}^7$) transitions between levels of ${(}^{4}S)nl$ terms from the NIST Atomic Spectra Database [28]; see Figure 3. This list includes transitions ${(}^4\mathrm{S}){\mathit{n}}_1\mathrm{s}$–${(}^4\mathrm{S}){\mathit{n}}_2\mathrm{p}$ with $n_1=4,5$ and $n_2=4–7$; ${(}^{4}S)4p$–${(}^{4}S)n_{2}s$ with $n_2=5–9$; ${(}^{4}S)4p$–${(}^{4}S)n_{2}d$ with $n_2=3–8$; ${(}^{4}S)3d$–${(}^{4}S)n_{2}f$ with $n_2=4–9$. Most of these A-values calculated using QDT differ by no more than 12% from the values listed in the NIST ASD. For weaker transitions, the difference from the NIST is about 15%; some examples are given in Table 2 for quintet ${(}^4\mathrm{S})5\mathrm{d}$–${(}^4\mathrm{S})8\mathrm{f}$ transitions and triplet–quintet ${(}^4\mathrm{S})4\mathrm{f}$–${(}^4\mathrm{S})6\mathrm{d}$ transitions.

We also made a comparison of QDT-calculated A-values with experimental results. High-resolution Fourier transfer spectroscopy was used to record S I spectra in the infrared domain from 790 to 11,000 ${\mathrm{cm}}^{-1}$ in a gas discharge plasma. By making Boltzmann plots for these spectra, we show that QDT gives reasonable estimates for the intensities of the spectral peaks in our experimental spectra.