# Benchmark Angle-Differential Cross-Section Ratios for Excitation of the 4p5s Configuration in Krypton

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Cross-Section Ratios

## 3. Theory

`GRASP2K`relativistic atomic-structure package [33]. After that, the valence orbitals were generated in a frozen-core calculation for Kr${}^{+}$ by using the average-term approximation. All these states of Kr${}^{+}$ were then used as target states in B-spline bound-state close-coupling calculations in order to generate the low-lying states of atomic Kr (with $N=36$ electrons) employing non-orthogonal, term-dependent orbitals for each Kr state.

## 4. Experiment

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## Abbreviations

DCS | Angle-differential cross section; |

DBSR | Dirac B-Spline R-matrix. |

## References

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**Figure 1.**Electron energy loss spectrum of Kr taken at ${E}_{0}=13.5$ eV and $\theta ={130}^{\circ}$. The data are the red dots, and the line is a linear least squares fit using a multi-Gaussian instrumental line profile centered at the empirical ${E}_{L}$ values given by Moore [10] and also listed in Table 2. The features m = 1–4 are discussed in the text following Equations (1)–(4).

**Figure 2.**Ratio r for various ${E}_{0}$ values as a function of $\theta $; ( ) present experimental work; ( ) past experimental work of Guo et al. [18]; ( ), DSBR-31 calculation.

**Table 1.**Intermediate-coupling coefficients for the 4p${}^{6}$ and 4p${}^{5}$5s states of Kr in a multi-configuration expansion, produced with the Cowan code [8] by Fontes [9]. ${E}_{L}$ values from the Cowan code and from Moore [10] are also listed. The notation ${\left[K\right]}_{J}$ (see also below) indicates the total electronic angular momentum of the ${\left(4{\mathrm{p}}^{5}\right)}^{2}{\mathrm{P}}_{\mathrm{K}}$ ionic core. After coupling to the outer orbital (here 5s), this results in the total electronic angular momentum J of the state. Refer to the text for discussions.

Level # | Configuration | Intermediate Coupling | ${\mathit{E}}_{\mathit{L}}$ (eV) | |
---|---|---|---|---|

Cowan | Moore | |||

1 | $4{\mathrm{p}}^{6}$ | $0.998\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{6}\right)}^{1}{\mathrm{S}}_{0}$ | 0.00 | 0.00 |

2 | $4{\mathrm{p}}^{5}5\mathrm{s}{[3/2]}_{2}$ | $0.998\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{5}5\mathrm{s}\right)}^{3}{\mathrm{P}}_{2}$ | 9.88 | 9.92 |

3 | $4{\mathrm{p}}^{5}5\mathrm{s}{[3/2]}_{1}$ | $0.67\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{5}5\mathrm{s}\right)}^{3}{\mathrm{P}}_{1}+0.74\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{5}5\mathrm{s}\right)}^{1}{\mathrm{P}}_{1}$ | 9.97 | 10.03 |

4 | $4{\mathrm{p}}^{5}5\mathrm{s}{[1/2]}_{0}$ | $0.996\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{5}5\mathrm{s}\right)}^{3}{\mathrm{P}}_{0}$ | 10.50 | 10.56 |

5 | $4{\mathrm{p}}^{5}5\mathrm{s}{[1/2]}_{1}$ | $0.74\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{5}5\mathrm{s}\right)}^{3}{\mathrm{P}}_{1}-0.67\phantom{\rule{0.166667em}{0ex}}{\left(4{\mathrm{p}}^{5}5\mathrm{s}\right)}^{1}{\mathrm{P}}_{1}$ | 10.56 | 10.64 |

**Table 2.**Transmission parameters that affect the ratios r, ${r}^{\prime}$, ${r}^{\u2033}$, and ${r}^{\u2034}$. The ${E}_{L}$ values, listed in decreasing order for the $5\mathrm{s}{[3/2]}_{2}$, $5\mathrm{s}{[3/2]}_{1}$, $5{\mathrm{s}}^{\prime}{[1/2]}_{0}$, and $5{\mathrm{s}}^{\prime}{[1/2]}_{1}$ states, are taken from the recommended values of Moore [10]. See text for a detailed discussion.

${\mathit{E}}_{0}$ (eV) → | 11.5 | 12.0 | 13.5 | 15.0 | 20.0 |
---|---|---|---|---|---|

${E}_{L}$ (eV) ↓ | ${E}_{R}$ (eV) ↓ | ||||

9.876 | 1.624 | 2.124 | 3.624 | 5.124 | 10.124 |

9.969 | 1.531 | 2.031 | 3.531 | 5.031 | 10.031 |

10.495 | 1.005 | 1.505 | 3.005 | 4.505 | 9.505 |

10.566 | 0.934 | 1.434 | 2.934 | 4.434 | 9.434 |

$\Delta $${E}_{R}$/${E}_{\overline{R}}$↓ | |||||

r | 0.470901 | 0.341141 | 0.186755 | 0.12857 | 0.06307 |

r${}^{\prime}$ | 0.484381 | 0.344589 | 0.184687 | 0.126149 | 0.061341 |

r${}^{\u2033}$ | 0.014739 | 0.011191 | 0.006499 | 0.004579 | 0.002307 |

r${}^{\u2034}$ | 0.018308 | 0.012079 | 0.005977 | 0.003971 | 0.001874 |

**Table 3.**Ratio ${r}^{\u2033}$ for Kr taken at ${E}_{0}=11.5$ eV, with one standard deviation uncertainty. See text for discussion.

$\mathit{\theta}{(}^{\circ})$ | ${\mathit{r}}^{\u2033}$ | Err |
---|---|---|

20 | 0.238 | 0.018 |

25 | 0.242 | 0.019 |

30 | 0.195 | 0.015 |

35 | 0.174 | 0.014 |

40 | 0.255 | 0.020 |

45 | 0.290 | 0.023 |

50 | 0.359 | 0.028 |

60 | 0.679 | 0.053 |

70 | 1.25 | 0.10 |

80 | 2.37 | 0.19 |

90 | 2.66 | 0.22 |

100 | 1.70 | 0.10 |

110 | 0.970 | 0.055 |

120 | 0.520 | 0.030 |

130 | 0.370 | 0.019 |

**Table 4.**Ratios r, ${r}^{\prime}$, ${r}^{\u2033}$, and ${r}^{\u2034}$ for Kr taken at ${E}_{0}=12.0$ eV with one standard deviation error. See text for discussion.

$\mathit{\theta}{(}^{\circ})$ | r | Err | ${\mathit{r}}^{\prime}$ | Err | ${\mathit{r}}^{\u2033}$ | Err | ${\mathit{r}}^{\u2034}$ | Err |
---|---|---|---|---|---|---|---|---|

20 | 10.3 | 1.3 | 2.22 | 0.22 | 0.383 | 0.039 | 0.082 | 0.009 |

25 | 10.8 | 1.5 | 2.18 | 0.22 | 0.492 | 0.050 | 0.099 | 0.012 |

30 | 11.3 | 1.4 | 2.12 | 0.21 | 0.507 | 0.051 | 0.096 | 0.010 |

35 | 10.9 | 1.4 | 2.00 | 0.20 | 0.485 | 0.049 | 0.088 | 0.010 |

40 | 10.8 | 1.4 | 1.67 | 0.17 | 0.592 | 0.060 | 0.091 | 0.010 |

45 | 9.56 | 1.20 | 1.43 | 0.14 | 0.645 | 0.065 | 0.096 | 0.010 |

50 | 7.59 | 0.95 | 1.36 | 0.14 | 0.665 | 0.067 | 0.119 | 0.013 |

60 | 5.89 | 0.74 | 1.22 | 0.12 | 0.782 | 0.080 | 0.162 | 0.017 |

70 | 3.62 | 0.45 | 1.04 | 0.11 | 0.973 | 0.100 | 0.281 | 0.030 |

80 | 2.46 | 0.31 | 0.994 | 0.103 | 1.13 | 0.12 | 0.458 | 0.048 |

90 | 2.73 | 0.34 | 1.06 | 0.11 | 1.08 | 0.11 | 0.420 | 0.044 |

100 | 3.46 | 0.43 | 1.17 | 0.12 | 0.93 | 0.09 | 0.314 | 0.033 |

110 | 4.47 | 0.56 | 1.31 | 0.13 | 0.76 | 0.08 | 0.224 | 0.023 |

120 | 6.75 | 0.85 | 1.33 | 0.13 | 0.65 | 0.07 | 0.129 | 0.014 |

130 | 12.7 | 1.7 | 1.58 | 0.16 | 0.54 | 0.06 | 0.068 | 0.008 |

$\mathit{\theta}{(}^{\circ})$ | r | Err | ${\mathit{r}}^{\prime}$ | Err | ${\mathit{r}}^{\u2033}$ | Err | ${\mathit{r}}^{\u2034}$ | Err |
---|---|---|---|---|---|---|---|---|

15 | 5.49 | 0.99 | 1.69 | 0.22 | 0.111 | 0.012 | 0.034 | 0.004 |

20 | 4.95 | 0.80 | 1.41 | 0.18 | 0.172 | 0.017 | 0.049 | 0.005 |

25 | 5.65 | 0.93 | 1.35 | 0.18 | 0.253 | 0.026 | 0.061 | 0.006 |

30 | 5.56 | 0.91 | 1.26 | 0.17 | 0.339 | 0.034 | 0.078 | 0.008 |

35 | 6.14 | 1.01 | 1.22 | 0.16 | 0.418 | 0.043 | 0.083 | 0.009 |

40 | 6.14 | 1.00 | 1.26 | 0.17 | 0.478 | 0.049 | 0.099 | 0.010 |

45 | 7.80 | 1.30 | 1.32 | 0.17 | 0.513 | 0.052 | 0.087 | 0.009 |

50 | 7.50 | 1.24 | 1.41 | 0.19 | 0.514 | 0.052 | 0.097 | 0.011 |

60 | 8.28 | 1.38 | 1.48 | 0.20 | 0.591 | 0.060 | 0.107 | 0.012 |

70 | 8.15 | 1.43 | 1.649 | 0.223 | 0.66 | 0.07 | 0.134 | 0.016 |

80 | 7.54 | 1.29 | 1.67 | 0.22 | 0.70 | 0.07 | 0.156 | 0.018 |

90 | 6.04 | 1.02 | 1.54 | 0.21 | 0.69 | 0.07 | 0.176 | 0.020 |

100 | 4.98 | 0.83 | 1.42 | 0.19 | 0.57 | 0.06 | 0.164 | 0.018 |

110 | 4.84 | 0.80 | 1.36 | 0.18 | 0.53 | 0.05 | 0.149 | 0.016 |

120 | 5.87 | 0.97 | 1.38 | 0.18 | 0.54 | 0.06 | 0.127 | 0.014 |

130 | 6.8 | 1.1 | 1.44 | 0.19 | 0.53 | 0.05 | 0.111 | 0.012 |

$\mathit{\theta}{(}^{\circ})$ | r | Err | ${\mathit{r}}^{\prime}$ | Err | ${\mathit{r}}^{\u2033}$ | Err | ${\mathit{r}}^{\u2034}$ | Err |
---|---|---|---|---|---|---|---|---|

15 | 5.40 | 0.78 | 1.81 | 0.09 | 0.058 | 0.006 | 0.019 | 0.002 |

20 | 5.92 | 0.87 | 1.76 | 0.09 | 0.097 | 0.010 | 0.029 | 0.004 |

25 | 4.98 | 0.71 | 1.69 | 0.09 | 0.160 | 0.017 | 0.054 | 0.007 |

35 | 6.35 | 0.95 | 1.51 | 0.09 | 0.347 | 0.037 | 0.083 | 0.011 |

40 | 5.50 | 0.78 | 1.48 | 0.09 | 0.404 | 0.043 | 0.109 | 0.014 |

45 | 6.01 | 0.87 | 1.44 | 0.08 | 0.404 | 0.043 | 0.097 | 0.012 |

50 | 6.42 | 0.92 | 1.50 | 0.09 | 0.446 | 0.047 | 0.105 | 0.013 |

60 | 5.84 | 0.82 | 1.45 | 0.09 | 0.510 | 0.054 | 0.130 | 0.016 |

70 | 5.60 | 0.78 | 1.50 | 0.09 | 0.577 | 0.061 | 0.154 | 0.019 |

80 | 5.68 | 0.82 | 1.45 | 0.09 | 0.659 | 0.071 | 0.169 | 0.022 |

90 | 5.24 | 0.79 | 1.26 | 0.09 | 0.703 | 0.078 | 0.170 | 0.023 |

100 | 4.93 | 0.74 | 1.29 | 0.08 | 0.581 | 0.064 | 0.155 | 0.020 |

110 | 4.40 | 0.62 | 1.23 | 0.08 | 0.553 | 0.060 | 0.155 | 0.019 |

120 | 4.88 | 0.67 | 1.23 | 0.07 | 0.539 | 0.057 | 0.136 | 0.016 |

130 | 5.22 | 0.70 | 1.21 | 0.07 | 0.620 | 0.065 | 0.142 | 0.016 |

$\mathit{\theta}{(}^{\circ})$ | r | Err | ${\mathit{r}}^{\prime}$ | Err | ${\mathit{r}}^{\u2033}$ | Err | ${\mathit{r}}^{\u2034}$ | Err |
---|---|---|---|---|---|---|---|---|

10 | 5.51 | 0.73 | 1.49 | 0.08 | 0.086 | 0.009 | 0.023 | 0.003 |

15 | 5.69 | 0.75 | 1.47 | 0.08 | 0.146 | 0.015 | 0.038 | 0.004 |

20 | 5.10 | 0.67 | 1.42 | 0.07 | 0.225 | 0.023 | 0.062 | 0.007 |

25 | 5.98 | 0.76 | 1.32 | 0.07 | 0.278 | 0.028 | 0.061 | 0.007 |

30 | 5.61 | 0.73 | 1.27 | 0.07 | 0.290 | 0.030 | 0.066 | 0.007 |

35 | 6.12 | 0.82 | 1.27 | 0.07 | 0.264 | 0.027 | 0.055 | 0.006 |

40 | 4.97 | 0.75 | 1.30 | 0.07 | 0.237 | 0.026 | 0.062 | 0.008 |

45 | 5.29 | 0.75 | 1.24 | 0.07 | 0.194 | 0.020 | 0.046 | 0.006 |

50 | 4.87 | 0.80 | 1.20 | 0.07 | 0.196 | 0.022 | 0.050 | 0.007 |

60 | 4.35 | 0.70 | 1.23 | 0.07 | 0.201 | 0.023 | 0.057 | 0.008 |

70 | 4.68 | 0.69 | 1.19 | 0.07 | 0.259 | 0.028 | 0.066 | 0.009 |

80 | 5.03 | 0.75 | 1.04 | 0.06 | 0.387 | 0.042 | 0.080 | 0.011 |

90 | 5.28 | 0.81 | 1.13 | 0.07 | 0.452 | 0.050 | 0.097 | 0.013 |

100 | 5.25 | 0.85 | 1.11 | 0.07 | 0.471 | 0.054 | 0.102 | 0.015 |

110 | 5.58 | 0.94 | 1.09 | 0.08 | 0.513 | 0.059 | 0.100 | 0.015 |

120 | 4.85 | 0.69 | 1.01 | 0.06 | 0.581 | 0.062 | 0.121 | 0.015 |

130 | 4.74 | 0.65 | 0.964 | 0.056 | 0.624 | 0.066 | 0.128 | 0.015 |

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Sakaamini, A.; Faure, J.-B.; Khakoo, M.A.; Zatsarinny, O.I.; Bartschat, K.
Benchmark Angle-Differential Cross-Section Ratios for Excitation of the 4p5s Configuration in Krypton. *Atoms* **2021**, *9*, 61.
https://doi.org/10.3390/atoms9030061

**AMA Style**

Sakaamini A, Faure J-B, Khakoo MA, Zatsarinny OI, Bartschat K.
Benchmark Angle-Differential Cross-Section Ratios for Excitation of the 4p5s Configuration in Krypton. *Atoms*. 2021; 9(3):61.
https://doi.org/10.3390/atoms9030061

**Chicago/Turabian Style**

Sakaamini, Ahmad, Jean-Baptiste Faure, Murtadha A. Khakoo, Oleg I. Zatsarinny, and Klaus Bartschat.
2021. "Benchmark Angle-Differential Cross-Section Ratios for Excitation of the 4p5s Configuration in Krypton" *Atoms* 9, no. 3: 61.
https://doi.org/10.3390/atoms9030061