# On the Stark Broadening of Be II Spectral Lines

^{1}

^{2}

^{3}

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## Abstract

**:**

^{11}cm

^{−3}and 10

^{13}cm

^{−3}. The influence of the temperature and the role of the perturbers (electrons, protons and He

^{+}ions) on the Stark width and shift have been discussed. Results could be useful for plasma diagnostics in astrophysics, laboratory, and industrial plasmas.

**Dataset:**Supplementary File.

**Dataset License:**CC-BY 4.0

## 1. Introduction

^{6}K for densities similar to those in the Sun. In solar-type stars, these temperatures are reached not far below the convection zone and well outside the core, and circulation and destruction of the light elements can result in observable abundance changes. Observations of these changes can provide an invaluable probe of stellar structure and mixing. Be and B have been observed in extreme Population II with low Z, in a number of low metallicity halo dwarf stars [5]. Some papers suggest that observed Be and B were generated by cosmic-ray spallation in the early Galaxy, and the standard model of primordial nucleosynthesis is unable to produce significant yields of both light elements [5]. According to [3], there are no theoretical explanations for the reduction in the abundances, a trend of decreasing with effective temperature and a dip at Teff ~6600 K in F, G, and K-dwarfs that have been found in the Hyades and other old clusters. Interactions between emitting atoms and surrounding electrons and ions result in Stark broadening of spectral lines. This broadening mechanism of line profiles is usually a principle one in the case of white dwarfs, and is of interest for main sequence stars from A type and late B type, and sometimes dominant ones [8,9]. Stark broadening parameters of Be II lines could serve effectively for the adequate modelling of stellar objects, opacity calculations, and diagnostics of astrophysical objects, laboratory and technological plasmas.

## 2. Data Description and Method of Research

_{kk’}(v), k = i,f, can be expressed by an integration of the transition probability P

_{kk’}(ρ,v):

_{el}is the elastic cross section, while φ

_{p}and φ

_{q}are phase shifts due to the polarization and quadrupolar potential (see Section 3 of Chapter 2 in [10]). For the cut-offs R

_{1}, R

_{2}, and R

_{D}, see Section 1 of Chapter 3 in [11]. The quantity σ

_{r}denotes the contribution of Feshbach resonances (see [14] and references therein), which concerns only electron-impact widths.

_{if}= E

_{i}− E

_{f}/ħ and E

_{i,}E

_{f}are the energies of the initial and final state. Therefore, if we know the Stark broadening parameters, width W and shift d, we can determine the spectral line profile.

^{2}4d–1s

^{2}np; 1s

^{2}4d–1s

^{2}nf and 1s

^{2}4f–1s

^{2}nd, where n = 6–8. The temperature varied from 2500 K to 50,000 K and the perturber density was 10

^{11}cm

^{−3}and 10

^{13}cm

^{−3}. The values of energy levels were taken from [15], and the oscillator strengths were calculated using [16]. For atoms such as beryllium an error around 20% was expected [17]. The dataset (Table S1) contains full Stark widths at half intensity maximum and shifts of Be II spectral lines due to collisions with electrons, protons and ionized helium, the main constituents of stellar atmospheres.

## 3. Results and Discussion

#### 3.1. Series 1s^{2}4d–1s^{2}np

#### 3.1.1. Temperature Dependence

^{2}4d–1s

^{2}5p. In order to complete the data, using available energy values from [15], results for transitions from higher levels have been obtained. With the increase in the principal quantum number of the upper atomic energy level within a spectral series, the maximal perturber density for which the impact approximation is valid, decreases. For a density of 10

^{13}cm

^{−3}, it is valid for all electron-impact broadening parameters within the considered data set, but not for all other widths and shifts in the case of collisions with heavier particles, protons and helium ions. Therefore, we performed calculations for 10

^{11}cm

^{−3}too, where impact approximation was valid for all perturbers within the considered data set. Moreover, these electron densities are typical for stellar atmospheres and for lower ones the linear extrapolation could be applied. In Figure 1a,b the temperature (T) dependence of Stark width and shift due to collisions with electrons, protons and ionized helium for 1s

^{2}4d–1s

^{2}6p transition have been illustrated.

^{+}ions were almost the same and they had the same trend, very slow increases with T. These values were notably lower (within an order) than the electron impact width. We observed the same trends with temperature for impact shifts, but the values for the three perturbers were much closer except for temperatures below 10,000 K, where the electron-impact shift started to dominate. For 20,000 K, the shifts due to electron- and He

^{+}ion-impacts were the same. For higher temperatures, the He

^{+}shifts were highest.

^{+}width. Comparing the shifts for the two transitions, we observed analogy in the behavior and differences in the values for different components. The impact component from He

^{+}ions was dominant for practically all temperatures. For temperatures above 15,000 K, proton shift also overcame the electron shift.

#### 3.1.2. Dependence on Principal Quantum Number

^{2}4d

^{2}D–1s

^{2}np

^{2}P

^{o}, for n = 5–8 have been presented for electron-impact broadening. Stark broadening parameters, widths, and shifts have been expressed in angular frequency units and as decimal logarithms. The values for n = 5, have been extrapolated linearly to the electron density of 10

^{11}cm

^{−3}. Namely, from references [12,13], where Stark broadening parameters for perturber densities from 10

^{13}cm

^{−3}to 10

^{19}cm

^{−3}are provided, it follows that the dependence on the electron density is linear towards the lower densities. Their increasing with principal quantum number is very regular.

#### 3.2. Series 1s^{2}4d–1s^{2}nf

#### 3.3. Series 1s^{2}4f–1s^{2}nd

^{2}4d–1s

^{2}nf and 1s

^{2}4f–1s

^{2}nd we can see that the behavior of widths due to collisions with electrons is very similar and that for all considered temperatures they increased with the increasing of the principal quantum number in a similar manner, and decreased uniformly with the increase in temperature. On the other hand, the shifts within these two series had different behavior, especially at a low temperature limit. However, in the case of the 1s

^{2}4f–1s

^{2}nd series, shifts uniformly decreased with the increase in temperature; in the 1s

^{2}4d–1s

^{2}nf series they considerably increased from T = 2500 K to T = 5000 K, and then uniformly decreased. A common characteristic for both series was a shift convergence with temperature. The width values did not converge.

## 4. Conclusions

^{11}and 10

^{13}cm

^{−3}. The dependence of Stark broadening parameters with temperature and the role of different perturbers (electrons, protons and He

^{+}ions) on the Stark width and shift have been discussed. Additionally, the regularity of behavior of Stark broadening parameters within the three considered spectral series was confirmed, and it was found that such regularities can be used for the interpolation and extrapolation of missing values and for a confirmation of experimental and theoretical results.

## Supplementary Materials

^{11}cm

^{−3}and 10

^{13}cm

^{−3}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Stark broadening width (

**a**) and shift (

**b**) for multiplet 1s

^{2}4d

^{2}D–1s

^{2}6p

^{2}P

^{o}(6638.3 Å) versus temperature from different types of perturbers: electrons—circle; protons—rhombus; ionized helium ions—triangle. Perturber density was 1 × 10

^{13}cm

^{−3}.

**Figure 2.**Stark broadening width (

**a**) and shift (

**b**) for multiplet 1s24d2D–1s28p2Po (4874.1 Å) versus temperature from different types of perturbers: electrons—circle; protons—rhombus; ionized helium ions—triangle. Perturber density was 1 × 1013 cm

^{−3}.

**Figure 3.**Decimal logarithm of full Stark width (

**a**) and electron-impact shift (

**b**) for spectral lines within the 2s

^{2}4d–2s

^{2}np (n = 5–8) spectral series versus principal quantum number. Electron density was 1 × 10

^{11}cm

^{−3}and temperature was 1 × 10

^{4}K. The Stark full width at half intensity maximum (FWHM) values for n = 5 have been taken from [13].

**Figure 4.**Stark broadening parameters for electron-impacts: width (

**a**) and shift (

**b**) versus temperature for the 2s

^{2}4d–2s

^{2}nf series: n = 6—circle; n = 7—rhombus; n = 8—triangle. Perturber density was 1 × 10

^{13}cm

^{−3}.

**Figure 5.**Electron-impact width (

**a**) and shift (

**b**) versus temperature for the 2s

^{2}4f–2s

^{2}nd series: n = 6—circle; n = 7—rhombus; n = 8—triangle. Electron density is 1 × 10

^{13}cm

^{−3}.

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**MDPI and ACS Style**

Dimitrijević, M.S.; Christova, M.; Sahal-Bréchot, S. On the Stark Broadening of Be II Spectral Lines. *Data* **2020**, *5*, 106.
https://doi.org/10.3390/data5040106

**AMA Style**

Dimitrijević MS, Christova M, Sahal-Bréchot S. On the Stark Broadening of Be II Spectral Lines. *Data*. 2020; 5(4):106.
https://doi.org/10.3390/data5040106

**Chicago/Turabian Style**

Dimitrijević, Milan S., Magdalena Christova, and Sylvie Sahal-Bréchot. 2020. "On the Stark Broadening of Be II Spectral Lines" *Data* 5, no. 4: 106.
https://doi.org/10.3390/data5040106