## 1. Introduction

Calculation of spectral line shapes is a most powerful tool for plasma diagnostic in both the star atmosphere and in laboratory plasmas. Perturbation of the radiating atom or ion by the surrounding particles leads to spectral line broadening (Stark broadening), while the coherent emission process is interrupted by collisions and influenced by plasma microfield. Several theoretical approaches have been applied to calculate Stark broadening, such as the well known semiclassical approximation the standard theory (ST) by Griem [

1], or the quantum statistical approach of many-particle theory [

2], where the motion of ion perturber is neglected during the inverse halfwidth of the line. Furthermore, the model microfield method (MMM) [

3,

4,

5,

6], the frequency fluctuation method (FFM) [

7] or computer simulations [

8,

9,

10,

11] are used for calculating the line broadening including ion-dynamics effects, which lead to further broadening of the line shapes.

In this work, we describe several theoretical approaches to study the He I 3889Å (

${2}^{3}S$–

${3}^{3}P$) transition line. Plasma parameters are determined by comparing our theoretical results with the measurement of Jovićević

et al. [

12]. In our quantum statistical approach, thermodynamic Green’s function is used for calculating emitted or absorbed radiation from bound–bound transition of charge particles by using two-particle polarization function [

13,

14,

15,

16], which is related to the Fourier transformation of the imaginary part of the dipole–dipole correlation function. In principle, this approach is able to describe the dynamic screening and strong collision by electrons in a systematic way. The quantum statistical approach can adequately treat electron collisions. To obtain full profiles, it has been supplemented by a calculation of ion broadening done by other methods and thereby has been successfully applied to calculate the line profile of hydrogen and H-like ions [

17]. The strong collisions of an electron-emitter pair are treated within a T-matrix approach by solving close-coupling equations to produce forward scattering amplitudes. Ions are treated in a quasistatic approximation via the microfield distribution function. Then the latest formulation of the FFM is applied to account for ion dynamics [

18]. On the other hand, computer simulations accounting simultaneously for the electronic and ionic fields on the same footing are used for comparison. Recently, the shift and width of this line was computed by Lorenzen

et al. [

19], comparison was made with various theoretical and measurement data in a wide range of electron density and temperature. In

Section 2 we describe briefly the experimental setup and result of Jovićević

et al. [

12]. Our theoretical approaches for calculating the line shapes are reviewed in

Section 3, followed by results and discussions

Section 4.

## 2. Experiment

In this paper, we analyze theoretically the He I 3889 Å Stark broadening, measured by Jovićević

et al. [

12] from a pulsed low-pressure capillary discharge. The plasma was a mixture of neon, helium and hydrogen, with the predominant H

${}^{+}$ perturber ions. Originally the experiment was set up for the measurement of Ne I spectral line profiles, while He was added to the gas mixture for plasma diagnostic purpose. The presence of hydrogen increases the electron density to the required value until a constant value of the Stark width was recorded. The optimum gas mixture was determined in a series of measurements starting from pure neon and then diluting until optically thin Ne I lines were recorded. In the experiment, a quartz discharge tube with 3 mm inner diameter and length of

$7.2$ cm was used, and aluminum electrodes with 3 mm diameter holes were located 200 mm apart. The measurements were performed in a premixed gas mixture of

$2.4\%$ Ne,

$5.6\%$ He and

$92\%$ H

${}_{2}$ by volume, at an initial pressure of 4 mbar. Spectroscopic plasma observations of Ne I and He I lines were performed, measuring the line shapes at the same time of plasma decay. The dilution of neon and helium with hydrogen was done in order to increase the electron density, since H has much lower ionization potential than Ne and He, and to generate a plasma with H

${}^{+}$ ions dominating He

${}^{+}$ and Ne

${}^{+}$. Special care was taken to keep the line profiles optically thin during measurements, in order to minimize self-absorption. This was achieved by using a continuous gas flow and diluting neon with helium and hydrogen in the operated arc. Moreover, plasma radiation was observed from the axial narrow discharge tube as well as from the radial expanded part simultaneously. However, the experimental design and procedure may have resulted in plasma inhomogeneity which may cause line distortion and line asymmetry. Electron density was determined from the measured He I line profiles, which were assumed to be optically thin under the same plasma conditions. The experimental profiles were fitted to a semiclassical calculation using two parameters, the electron impact width and the ion-broadening parameter

A of the quasistatic approximation [

1], which allow the authors to determine the electron density. The mean value of electron density

${n}_{e}$ was estimated to be

$4.8\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ with an estimated uncertainty of

$\pm 10\%$. By assuming local thermal equilibrium, the electron temperature

${T}_{e}={T}_{i}=3.3\times {10}^{4}$ K was determined from the Boltzmann plot of O II impurity lines with an uncertainty of ±

$12\%$, as the Stark broadening parameters weakly depend upon

${T}_{e}$. More details about the experimental set up and measurement are given in [

12].

## 4. Results and Discussions

In this study, we present several theoretical approaches for calculating He I line shapes.The plasma parameters are inferred by comparing our results with the experimental measurement of Jovićević

et al. [

12]. The measured profile is compared with the theoretical profiles at

$T=3.3\times {10}^{4}$ K for two values of electron densities

${n}_{e}=5.2\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and

${n}_{e}=4.58\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$, respectively. In [

12] the experimental He I 3889 Å line broadening was fitted by using the

${j}_{AR}(x)$ line profiles generated on the assumption of electron impact broadening combined with broadening by quasistatic ion field using the ST of Griem [

1]. The electron density of

${n}_{e}=4.8\times {10}^{22}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$ with an estimated uncertainty of

$\pm 10\%$ was determined from the line broadening, while the electron temperature of

$T=3.3\times {10}^{4}$ K with an uncertainty of

$\pm 12\%$ was estimated by using the relative intensities of O II impurity lines.

In our quantum statistical approach electrons and ions are treated separately. Green’s function is applied for calculating the electronic contribution from Equation (

3), while the dominant ionic contribution arises from the quadratic Stark effect and quadrupole interaction [

15,

16]. First, the ionic contribution is determined in the quasistatic approximation by using the calculated microfield distribution function in

Section 3.1.2, then the new FFM formulation [

18] is adopted to account for the fluctuation of the microfield on the spectral line shapes. The dynamic effects of the ionic motion are taken into account by substituting the area-normalized static profile into Equation (

15). The result is displayed in

Figure 1, which compares measured and theoretical profiles. Since the authors of the original experimental paper [

12] did not mention any measurement of absolute line shift, we displaced the calculated line profiles to the unperturbed wavelength

${\lambda}_{0}=3888.65$ Å in

Figure 1,

Figure 2 and

Figure 3 to determine the free electron density

${n}_{e}$ from the line width.

**Figure 1.**
The calculated He I (

${2}^{3}S$–

${3}^{3}P$) line profiles in dynamically screened Born approximation by including ion dynamics in the FFM . Comparison is made with the measured data by Jovićević

et al. [

12] at

$T=3.3\times {10}^{4}$ K. The profiles are plotted at the unperturbed transition wavelength

${\lambda}_{0}=3888.65$ Å, and the shift of profiles are shown.

**Figure 1.**
The calculated He I (

${2}^{3}S$–

${3}^{3}P$) line profiles in dynamically screened Born approximation by including ion dynamics in the FFM . Comparison is made with the measured data by Jovićević

et al. [

12] at

$T=3.3\times {10}^{4}$ K. The profiles are plotted at the unperturbed transition wavelength

${\lambda}_{0}=3888.65$ Å, and the shift of profiles are shown.

The electronic contribution is recalculated via the T-matrix by applying the CCC method in order to describe the strong electron-helium collisions within the Debye static screening model [

34]. Both real and imaginary part of forward scattering amplitudes are substituted in Equation (

4) to provide electronic shift and width [

19]. In contrast to the Born approximation, the contribution of each state is different for different values of the magnetic quantum number

$\left|m\right|=0,1$. Furthermore, an average over different spin-scattering channels

$S=0$ and

$S=1$ has been done. The Stark profiles are calculated in the quasistatic approximation from Equation (

1) by using the T-matrix approach for electron-emitter collisions, The dynamic profiles are evaluated from Equation (

15) and illustrated in

Figure 2. It may be seen that the electron density

${n}_{e}=5.2\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ gives better agreement with the experimental profile in

Figure 1 and

Figure 2.

Results from simulations are presented in

Figure 3. Dynamic profiles show very good agreement with the measured profile for the inferred plasma density

${n}_{e}=4.58\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and temperature

$T=3.3\times {10}^{4}$ K. In

Table 1, numerical results of the shift and FWHM from our theories are given. We compare the broadening parameters in the dynamically screened Born approximation, the T-matrix approach, and MD simulations for interacting particles.

**Figure 2.**
Comparison between the measured [

12] and calculated line shapes of He I (

${2}^{3}S$–

${3}^{3}P$) using the T-matrix approach with ion dynamics in the FFM at

$T=3.3\times {10}^{4}$ K. The profiles are given at the unperturbed transition wavelength

${\lambda}_{0}=3888.65$ Å, and the shift of profiles are shown.

**Figure 2.**
Comparison between the measured [

12] and calculated line shapes of He I (

${2}^{3}S$–

${3}^{3}P$) using the T-matrix approach with ion dynamics in the FFM at

$T=3.3\times {10}^{4}$ K. The profiles are given at the unperturbed transition wavelength

${\lambda}_{0}=3888.65$ Å, and the shift of profiles are shown.

**Figure 3.**
The calculated He I (

${2}^{3}S$–

${3}^{3}P$), line profiles from MD simulations compared with the experimental result of Jovićević

et al. [

12] at

$T=3.3\times {10}^{4}$ K. The profiles are given at the unperturbed transition wavelength

${\lambda}_{0}=3888.65$ Å, and the shift of profilesare shown.

**Figure 3.**
The calculated He I (

${2}^{3}S$–

${3}^{3}P$), line profiles from MD simulations compared with the experimental result of Jovićević

et al. [

12] at

$T=3.3\times {10}^{4}$ K. The profiles are given at the unperturbed transition wavelength

${\lambda}_{0}=3888.65$ Å, and the shift of profilesare shown.

**Figure 4.**
The shift of He I 3889 Å vs. electron density. Comparison between our calculated data and the experimental results are shown. The electron temperature is given for the corresponding T-matrix data points.

**Figure 4.**
The shift of He I 3889 Å vs. electron density. Comparison between our calculated data and the experimental results are shown. The electron temperature is given for the corresponding T-matrix data points.

Ion dynamics (FFM) are included in the Born approximation and the T-matrix approach, the dynamic effects lead to further shift. The shift is significantly affected by including strong collisions via the T-matrix and ion-dynamics, while the dynamic profiles appear slightly broadened compared to the static profile. The calculated line shift for different values of

${n}_{e}$ is illustrated in

Figure 1,

Figure 2 and

Figure 3 as well. Note, that the shift of this line is red, towards larger wavelengths with respect to the unperturbed transition line at

${\lambda}_{0}$. An example of the line measurement can be seen also in [

50].

**Table 1.**
Theoretical calculations of shift and FWHM of the He I 3889 Å line are illustrated in [Å], to analyze the experimental result of Jovićević

et al. [

12]. The calculated values correspond to dynamically screened Born approximation (quasistatic/FFM), T-matrix approach (quasistatic/FFM) and MD simulations at

$T=3.3\times {10}^{4}$ K.

**Table 1.**
Theoretical calculations of shift and FWHM of the He I 3889 Å line are illustrated in [Å], to analyze the experimental result of Jovićević et al. [12]. The calculated values correspond to dynamically screened Born approximation (quasistatic/FFM), T-matrix approach (quasistatic/FFM) and MD simulations at $T=3.3\times {10}^{4}$ K.
${n}_{e}$ ${10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ | Shift [Å] | FWHM [Å] |
---|

Born | T-Matrix | MD | Born | T-Matrix | MD |
---|

5.2 | 0.483/0.523 | 0.178/0.228 | 0.32 | 1.245/1.278 | 1.038/1.087 | 1.402 |

4.58 | 0.427/0.462 | 0.147/0.197 | 0.269 | 1.094/1.123 | 0.914/0.965 | 1.245 |

The illustrated approaches can provide the shift and full width at half maximum (FWHM) in a wide range of

${n}_{e}$ and

T. The comparison with different measurements for shift and FWHM as a function of electron density

${n}_{e}\approx ({10}^{22}-{10}^{24}){\mathrm{m}}^{-3}$ in temperature range

$T\approx (2-6)\times {10}^{4}$ K are shown in

Figure 4 and

Figure 5, respectively. The microfield distribution function of Hooper [

51] is used in both the Born approximation and the T-matrix method for evaluating the Stark parameters in

Figure 4 and

Figure 5 [

19]. However, outside the validity range of Hooper’s approach, we used the fit formula of Potekhin

et al. [

52] in our previous calculations in [

19,

53], which is based on Monte Carlo simulations and is appropriate for strongly coupled plasmas as well. The calculated FWHM in the T-matrix approach shows good agreement with the MD simulations data, especially at high densities. The ion dynamics slightly affects the width. However, the shift is rapidly reduced at low electron densities, and the discrepancy can be seen in contrast to MD simulations, but still the contribution of FFM is more pronounced in this region. The shift of this line is overestimated in the Born approximation compared to both the T-matrix method and MD simulations. The correlation between perturbers tends to decrease the line shift and width at high electron densities for interacting particles. In

Figure 4 and

Figure 5 the comparison is made with the following experiments: The measurement by Pérez

et al. [

54] was done in a low-pressure pulsed arc, in the plasma density range of

${n}_{e}=(1-6)\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and temperature interval of

$T=(0.8-3)\times {10}^{4}$ K with a mean value of

$2\times {10}^{4}$ K . The error bar of

${n}_{e}$ was

$\pm 10\%$, and uncertainty in the temperature evaluation was about

$20\%$. The experimental result of Kelleher [

55] was obtained in a helium plasma generated in a wall-stabilized arc, with

${n}_{e}=1.03\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and

${T}_{e}=2.09\times {10}^{4}$ K. Recently, the FWHM of the same transition line was measured by Gao

et al. [

56] from a helium arc for the density range

${n}_{e}=(0.5-4)\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ at

$T=2.3\times {10}^{4}$ K. The reported result by Kobilarov

et al. [

57] from a pulsed low-pressure arc at

${n}_{e}=(2-10)\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and

$T=(3.1-4.2)\times {10}^{4}$ K are included. Furthermore, values for the shift measured by Morris and Cooper [

58] within the density range

${n}_{e}=(0.6-2.3)\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and

$T=(1-1.6)\times {10}^{4}$ K are shown. The Stark parameters of this line were also measured by Berg

et al. [

59] at

${n}_{e}=1.5\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and

${T}_{e}=2.6\times {10}^{4}$ K. The measured widths by Milosavljević and Djeniže [

60] are included as well, by using a linear low-pressure pulsed arc. The measured electron density and temperature were in the ranges of

$(4.4-8.2)\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and

$(1.8-3.3)\times {10}^{4}$ K with error bars

$\pm 9\%$ and

$\pm 10\%$, respectively. Furthermore, a comparison is made with the measurement of Soltwisch and Kusch [

61], a wall-stabilized quasistationary pulsed discharge was used as a homogeneous plasma source. The spectra were recorded in a single shot, and the electron density range of

$(0.7-1.2)\times {10}^{23}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ was determined from the plasma reflectivity at two different wavelengths. The estimation of the temperature was about

$2\times {10}^{4}$ K.

As shown in

Figure 5, the T-matrix approach always gives a smaller FWHM than the Born approximation. This trend leads to a better agreement with the MD simulations results for interacting particles. For the data of Pérez

et al. [

54], the calculated width in the T-matrix approach agrees very well with the result of the Born approximation. However, the measurement may be influenced by self-absorption, as mentioned in [

54], and then the measured widths may be overestimated. At

${n}_{e}=9.8\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}$, the width in the Born approximation shows a very good agreement with both measurement and MD simulations for independent particle data. However, the results of T-matrix approach and MD simulations for interacting particles give slightly lower values than the experiment data. At the highest measured density, the FWHM of theories is below that of Berg

et al. [

59]. Good agreement is found between the MD simulations result and T-matrix approach. However, the shift of this line is overestimated in the Born approximation, where, the strong collisions with large momentum transfer are not treated appropriately, while the perturbation theory breaks down at small distances and the perturbative expansion has to be avoided [

20].

**Figure 5.**
The FWHM of He I 3889 Å as a function of electron density. Our theoretical approaches are compared with the experiments results.

**Figure 5.**
The FWHM of He I 3889 Å as a function of electron density. Our theoretical approaches are compared with the experiments results.

## 5. Conclusions

Spectral line shapes for He I are investigated theoretically in dense plasmas, the comparison is made with the measurements of Jovićević

et al. [

12] for diagnostic purposes. Free electron-emitter collisions are considered within both the Born approximation, and the T-matrix approach while ions are treated by the quasistatic approximation. The electronic contribution to the shift and width is computed by thermodynamic Green’s function by using Born approximation, which is the main contribution to the line broadening. Dynamic screening of the electron-atom interaction is included, which is a collective, many-particle effect. In contrast to ST, this effect modifies the broadening parameters with increasing free electron density, causing a non-linear behavior, where the plasma oscillations become relevant, see

Figure 4 and

Figure 5. A cut-off procedure for strong collisions is used according to Griem [

1], while the second-order Born approximation overestimated the electronic contribution and the strong collisions term are added [

23]. The Coulomb approximation is employed to evaluate the wavefunctions of helium. The accuracy of this approximation was approved for He-like ion by comparing our result with the Hartree-Fock wavefunctions, thus the radial part of transition matrix-element from various approaches were compared in [

62].

Then the electron-emitter interaction is investigated again by elastic scattering amplitudes in Debye plasmas from a two-particle T-matrix approach by using the close-coupling equations. The Debye–Hückel potential significantly affects the bound states and scattering processes of the metastable states. The result is improved by treating strong electron-helium collisions consistently within T-matrix approach and better agreement can be seen with the measurement than the Born approximation, especially for the line shift. The contribution of perturbing ions is taken into account in a quasistatic approximation, with both quadratic Stark effect and quadrupole interaction. The perturbing ionic microfield is considered as a static field during the radiation. The calculated spectral line shapes in the static limit are modified by applying the FFM for both the Born approximation and T-matrix approach, which provides the dynamic line shapes and leads to further broadening [

18]. In addition, MD simulations are used for comparison with the experiments and the analytically obtained profiles.

The shift of this line is over estimated in Born approximation even with a cut-off procedure and better agreement can be seen with the screened T-matrix approach and MD simulations. The FWHM of all theoretical approaches are in good agreement with the experimental result.