# Relativistic B-Spline R-Matrix Calculations for Electron Scattering from Thallium Atoms

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{A}is a measure of the spin dependence of the differential cross-section (DCS) in collisions of spin-polarized electrons with a given target. It is a very sensitive parameter to test the quality of a theoretical approach to electron–atom collisions [2,3]. This is particularly true for excitation processes, and the difficulties increase significantly with the complexity of the target. For open-shell heavy targets, significant contributions can be attributed to both electron exchange via the so-called “fine-structure effect” [4] and the spin–orbit interaction of the continuum electron in the field of the target atom (Mott scattering). The competition between them makes S

_{A}extremely sensitive to the details of the theoretical approach.

^{2}6p with nonvanishing orbital angular momentum as its ground-state configuration. Benchmark experimental data for e-Tl collisions at low energies were produced by the Münster group [5,6,7]. Kaussen et al. [5] investigated the spin polarization in elastic scattering of unpolarized electrons from several heavy atoms, including thallium, for selected incident-electron energies between 6 and 180 eV. Geesmann et al. [6] used a source of spin-polarized electrons, which were scattered elastically and inelastically from an ensemble of unpolarized thallium atoms, to study the spin asymmetry function for incident energies ranging from 3 to 14 eV and scattering angles in the range of 35° to 125°. Later, the measurements were extended to energies below 2 eV by Dümmler et al. [7].

_{A}, the angle-differential cross-sections (DSCs) are also presented below.

_{A}in Section 3.3. We finish with some conclusions in Section 4.

## 2. Computational Method

^{2}nl (l = 3 for n = 5, l = 1–3 for n = 6, l = 0–3 for n = 7 and 8, l = 0–2 for n = 9 and 10) and 6s6p

^{2}, respectively. The number of physical states that we can generate in this method depends upon the size of the R-matrix box. The R-matrix radius was set to 60 a

_{0}(a

_{0}= 0.529 × 10

^{−10}m, Bohr radius) and the target Hamiltonian was diagonalized in this box. This choice allowed us to obtain a sufficiently good description for the low-lying bound states of Tl up to 6s

^{2}8f (see Table 1 in Section 3.1). All Dirac–Fock calculations for the core and the correlated orbitals were performed with the relativistic atomic structure packages GRASP2K [19] and DBSR_HF [20]. Different sets of correlated nl orbitals were optimized separately for the lowest state of each symmetry with total target electronic angular momentum and parity (nl)J

^{π}.

_{A}were determined by using the formulas given, for example, in Refs. [3,23].

_{A}is expressed as

_{A}[12], which can be obtained as

_{0}and J

_{1}denote the total angular momenta of the target in its initial and final states, respectively, during the transitions. In the non-relativistic limit, electron exchange still has contributions to S

_{A}, while Mott scattering does not. Therefore, the pure fine-structure effect is considered as a non-relativistic approximation, in which the spin–orbit interaction is neglected. Then, the energy levels of a pair of fine-structure states are equal. As a result, the average values of spin asymmetry function S

_{A}will vanish. A detailed discussion of the fine-structure effect can be found in the review by Hanne [4].

## 3. Results and Discussion

#### 3.1. Energy Levels and Oscillator Strengths

^{2 4}P

_{1/2}, 6s

^{2}10d

^{2}D

_{3/2,5/2}, and 6s

^{2}8f

^{2}F°

_{1/2,3/2}states have somewhat larger discrepancies above 0.2 eV. However, the deviations do not exceed 5%.

^{2}6p

^{2}P°

_{1/2}and 6s

^{2}6p

^{2}P°

_{3/2}. Specifically, we show our theoretical values obtained in the velocity (V) and length (L) forms of the electric dipole operator. The experimental values are from the data recommended by NIST [25], and references [26] (obtained with the optical-double-resonance technique, ODRT) and [27] (employing beam-foil technique, BFT). The present results are in good to moderate agreement with the experimental data. In some cases, deviations of up to 40% are seen, with the results obtained in the length gauge generally being larger than those obtained in the velocity gauge. The most likely reason for the deviations between the two sets as well as the recommended data is the computational necessity to limit the number of configurations in the target description in a way that makes the subsequent collision calculation possible.

#### 3.2. Differentical Cross-Sections

^{2}6p

^{2}P°

_{1/2}for incident electron energies between 1 and 24 eV. No experimental data are available for comparison. The magnitudes of the DCS for all incident energies are close at scattering angles smaller than 5 degrees. A systematic move of the DCS minimum is found going from around 110 degrees at 1 eV to approximately 90 degrees at 11 eV. The DCS curves become broader and flatten out in their bottom region as the energy increases, and more minima appear when the energy is higher than 15 eV. Three significant minima appear at 20 and 24 eV. Their angular positions are around 40, 90, and 130 degrees, respectively. The curves for energies higher than 11 eV intersect near 145 degrees.

**Table 1.**Energy levels of Tl (in eV) relative to the ground state obtained in this work, compared with those in the NIST [25] tables (J denotes the total angular momentum quantum number).

No. | State | Term | J | NIST [25] | This Work | Difference |
---|---|---|---|---|---|---|

1 | 6s^{2}6p | ^{2}P° | ^{1}/_{2} | 0.00000 | 0.00000 | 0.00000 |

2 | 6s^{2}6p | ^{2}P° | ^{3}/_{2} | 0.96617 | 0.93535 | −0.03082 |

3 | 6s^{2}7s | ^{2}S | ^{1}/_{2} | 3.28279 | 3.35298 | 0.07019 |

4 | 6s^{2}7p | ^{2}P° | ^{1}/_{2} | 4.23529 | 4.26149 | 0.02620 |

5 | 6s^{2}7p | ^{2}P° | ^{3}/_{2} | 4.35942 | 4.30594 | −0.05348 |

6 | 6s^{2}6d | ^{2}D | ^{3}/_{2} | 4.47805 | 4.47796 | −0.00009 |

7 | 6s^{2}6d | ^{2}D | ^{5}/_{2} | 4.48822 | 4.48293 | −0.00529 |

8 | 6s^{2}8s | ^{2}S | ^{1}/_{2} | 4.80388 | 4.81059 | 0.00671 |

9 | 6s^{2}8p | ^{2}P° | ^{1}/_{2} | 5.12899 | 5.12293 | −0.00606 |

10 | 6s^{2}8p | ^{2}P° | ^{3}/_{2} | 5.1752 | 5.15013 | −0.02507 |

11 | 6s^{2}7d | ^{2}D | ^{3}/_{2} | 5.20875 | 5.18402 | −0.02473 |

12 | 6s^{2}7d | ^{2}D | ^{5}/_{2} | 5.21341 | 5.17956 | −0.03385 |

13 | 6s^{2}5f | ^{2}F° | ^{5}/_{2} | 5.24681 | 5.22070 | −0.02611 |

14 | 6s^{2}5f | ^{2}F° | ^{7}/_{2} | 5.24681 | 5.22151 | −0.02530 |

15 | 6s^{2}9s | ^{2}S | ^{1}/_{2} | 5.35193 | 5.34096 | −0.01097 |

16 | 6s^{2}9p | ^{2}P° | ^{1}/_{2} | 5.50253 | 5.48915 | −0.01338 |

17 | 6s^{2}9p | ^{2}P° | ^{3}/_{2} | 5.52505 | 5.50476 | −0.02029 |

18 | 6s^{2}8d | ^{2}D | ^{3}/_{2} | 5.5387 | 5.51819 | −0.02051 |

19 | 6s^{2}8d | ^{2}D | ^{5}/_{2} | 5.54119 | 5.51857 | −0.02262 |

20 | 6s^{2}6f | ^{2}F° | ^{5}/_{2} | 5.55741 | 5.53341 | −0.02400 |

21 | 6s^{2}6f | ^{2}F° | ^{7}/_{2} | 5.55741 | 5.53366 | −0.02375 |

22 | 6s6p^{2} | ^{4}P | ^{1}/_{2} | 5.607 | 5.32811 | −0.27889 |

23 | 6s^{2}10s | ^{2}S | ^{1}/_{2} | 5.61609 | 5.61945 | 0.00336 |

24 | 6s^{2}10p | ^{2}P° | ^{1}/_{2} | 5.69575 | 5.74420 | 0.04845 |

25 | 6s^{2}10p | ^{2}P° | ^{3}/_{2} | 5.70868 | 5.76181 | 0.05313 |

26 | 6s^{2}9d | ^{2}D | ^{3}/_{2} | 5.71549 | 5.75791 | 0.04242 |

27 | 6s^{2}9d | ^{2}D | ^{5}/_{2} | 5.71695 | 5.75781 | 0.04086 |

28 | 6s^{2}7f | ^{2}F° | ^{5}/_{2} | 5.72625 | 5.75303 | 0.02678 |

29 | 6s^{2}7f | ^{2}F° | ^{7}/_{2} | 5.72625 | 5.75313 | 0.02688 |

30 | 6s^{2}10d | ^{2}D | ^{3}/_{2} | 5.82105 | 6.05811 | 0.23706 |

31 | 6s^{2}10d | ^{2}D | ^{5}/_{2} | 5.82205 | 6.06582 | 0.24377 |

32 | 6s^{2}8f | ^{2}F° | ^{5}/_{2} | 5.82783 | 6.03291 | 0.20508 |

33 | 6s^{2}8f | ^{2}F° | ^{7}/_{2} | 5.82783 | 6.03290 | 0.20507 |

**Table 2.**Selected oscillator strengths for excitation of Tl from 6s

^{2}6p to upper levels, as obtained in the velocity (V) and length (L) forms of the electric dipole operator. (The subscript number denotes the total electronic angular momentum quantum number).

Lower | Upper | This Work | ||||
---|---|---|---|---|---|---|

Level | Level | NIST [25] | ODRT [26] | BFT [27] | V | L |

6s^{2}6p_{1/2} | 6s^{2}6d_{3/2} | 0.29 | 0.29 ± 0.022 | 0.34 | 0.2271 | 0.3307 |

6s^{2}7s_{1/2} | 0.134 | 0.133 ± 0.007 | 0.13 | 0.1276 | 0.1399 | |

6s^{2}7d_{3/2} | 0.075 | 0.074 ± 0.009 | 0.09 | 0.0555 | 0.0814 | |

6s^{2}8s_{1/2} | 0.018 | 0.0176 ± 0.0016 | 0.0169 | 0.0199 | ||

6s^{2}8d_{3/2} | 0.028 ± 0.004 | 0.0237 | 0.0331 | |||

6s^{2}9s_{1/2} | 0.0062 ± 0.0008 | 0.0050 | 0.0050 | |||

6s^{2}6p_{3/2} | 6s^{2}6d_{3/2} | 0.0411 | 0.04 ± 0.004 | 0.0349 | 0.0499 | |

6s^{2}6d_{5/2} | 0.346 | 0.346 ± 0.035 | 0.37 | 0.3100 | 0.4273 | |

6s^{2}7s_{1/2} | 0.151 | 0.151 ± 0.007 | 0.14 | 0.1295 | 0.1870 | |

6s^{2}7d_{3/2} | 0.0091 ± 0.0009 | 0.0079 | 0.0109 | |||

6s^{2}7d_{5/2} | 0.08 | 0.081 ± 0.009 | 0.08 | 0.0619 | 0.0858 | |

6s^{2}8s_{1/2} | 0.0135 | 0.0136 ± 0.0014 | 0.0118 | 0.0172 | ||

6s^{2}8d_{3/2} | 0.004 ± 0.0004 | 0.0031 | 0.0043 | |||

6s^{2}8d_{5/2} | 0.028 ± 0.003 | 0.03 | 0.0233 | 0.0316 | ||

6s^{2}9s_{1/2} | 0.0048 ± 0.0005 | 0.0027 | 0.0043 | |||

6s^{2}9d_{3/2} | 0.002 ± 0.0002 | 0.0025 | 0.0035 | |||

6s^{2}10s_{1/2} | 0.003 ± 0.0003 | 0.0023 | 0.0035 |

^{2}6p

^{2}P°

_{3/2}state from the ground state. It is not surprising that the DCSs for the inelastic transitions are generally much smaller than for the elastic case. Most of the curves exhibit peaks at scattering angles of around 7 degrees, where the values of the DCSs at different impact energies are very similar. Unlike for the elastic case, the minima in these curves show a systematic movement towards the direction of larger angles, and turn into a single deep valley near 90 degrees for incident energies higher than 15 eV.

^{2}7s

^{2}S

_{1/2}state from the ground state exhibits a different dependence in the curves of DCSs compared to the former two cases, which is displayed in Figure 3. The DCSs decrease rapidly from 0 to 15 degrees, especially for the higher impact energies. Most curves intersect at scattering angles of around 15 degrees. There is still a systematic movement of the DCS minima towards smaller angles. The DCSs for excitation from the ground state to 6s

^{2}7s

^{2}S

_{1/2}have generally a smaller magnitude than those for excitation from the ground state to 6s

^{2}6p

^{2}P°

_{3/2}. However, this is not the case near the forward direction at scattering angles smaller than 15 degrees. The deep valleys also emerge in Figure 3 near 90 and 130 degrees for incident energies higher than 17 eV.

#### 3.3. Spin Asymmetries

_{A}for elastic electron collisions from Tl atoms in the ground state 6s

^{2}6p

^{2}P°

_{1/2}for incident electron energies between 1 and 24 eV. The present DBSR predictions are compared with the experimental data of the Münster group [5,6,7], along with the theoretical results of BPRM method by Bartschat [11] and Goerss et al. [12], GDF calculations by Haberland and Fritsche [13], as well as the RCCC predictions by Bostock et al. [15]. The present DBSR prediction results slightly overestimate the values of S

_{A}at scattering angles near 90 degrees for the cases of low impact energies displayed in Figure 4. However, it is clear that the present results are still in overall good agreement with the experimental data, especially for the cases at higher energies such as at 17 eV and 24 eV. One should note that there are three narrow peaks in the S

_{A}curve for 24 eV at scattering angles near 40, 90, and 135 degrees. These positions coincide with the three significant minima in the corresponding DCS curve for 24 eV, displayed in Figure 1. This is a typical characteristic of Mott scattering, where high values of |S

_{A}| are correlated with minima in the DCS. This suggests that the dominant contribution to the polarization mechanism is the spin–orbit interaction in this case. In contrast, such characteristic correlations between the DCS and S

_{A}are not so evident for the low incident energies below 6 eV shown in Figure 4. This indicates that electron exchange processes may also play an important role for the spin polarization in elastic e-Tl collisions at low energies.

_{A}for electron impact excitation of Tl atoms from their ground state 6s

^{2}6p

^{2}P°

_{1/2}to the 6s

^{2}6p

^{2}P°

_{3/2}and 6s

^{2}7s

^{2}S

_{1/2}states, respectively. We also show the experimental measurements of Geesmann et al. [6], along with RCCC calculations by Bostock et al. [15] and BPRM calculations by Goerss et al. [12]. The present calculations represent a clear improvement over the semi-relativistic BPRM results. This indicates that relativistic effects should be fully considered when studying electron scattering from heavy atoms such as thallium, especially for excitation. The agreement between the present DBSR results and the experimental measurements is generally good.

_{A}were calculated according to Equation (2) for the transitions between the fine-structure levels of the ground-state configuration 6s

^{2}6p

^{2}P° and the excited-state configurations 6s

^{2}7s

^{2}S, 6s

^{2}7p

^{2}P°, and 6s

^{2}6d

^{2}D, respectively. Figure 7 depicts the present DBSR predictions for incident electron energies of 5, 7.5, 10, 12.5, and 15 eV, along with the theoretical results of the BPRM method by Goerss et al. [12]. We also show S

_{A}for individual fine-structure transitions from the 6s

^{2}6p

^{2}P°

_{1/2}and 6s

^{2}6p

^{2}P°

_{3/2}states, respectively, to the 6s

^{2}7s

^{2}S

_{1/2}state at 5 eV. The average S

_{A}values for transitions from 6s

^{2}6p

^{2}P° to 6s

^{2}7s

^{2}S are significantly smaller than the values for the individual fine-structure transitions. The situation is similar for the other two sets of transitions, in agreement with the findings of Goerss et al. [12]. This indicates that the fine-structure effect cannot be ignored for these cases. Especially for the 6s

^{2}6p

^{2}P° to 6s

^{2}6d

^{2}D transitions, the angular dependence of the average S

_{A}is almost flat, while significant contributions to the spin asymmetry by the fine-structure effect are possible in the fine-structure-resolved transitions. On the other hand, the average S

_{A}values for each set of transitions show a stronger angular dependence as the impact energy increases. This suggests that the spin–orbit interaction plays a more important role at higher energies.

## 4. Conclusions

^{2}6p

^{2}P°

_{3/2}and 6s

^{2}7s

^{2}S

_{1/2}states in thallium atoms. Our ab initio calculations are based on the fully relativistic Dirac B-spline R-matrix (DBSR) approach that Oleg Zatsarinny developed for years. The excellent agreement between theoretical predictions and experimental measurements indicates, once again, the reliability of the DBSR method to treat the process of electron–atom collisions. We also conclude that, at low energies, even for the open-shell heavy target thallium, the spin polarization in the electron scattering process can still be attributed to a significant extent to electron exchange effects, in addition to the spin–orbit interaction.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**DCS for elastic electron collisions from Tl atoms in their 6s

^{2}6p

^{2}P°

_{1/2}ground state for a variety of projectile energies.

**Figure 2.**DCS for electron impact excitation of the 6s

^{2}6p

^{2}P°

_{1/2}→ 6s

^{2}6p

^{2}P°

_{3/2}transition in Tl for a variety of projectile energies.

**Figure 3.**DCS for electron impact excitation of the 6s

^{2}6p

^{2}P°

_{1/2}→ 6s

^{2}7s

^{2}S

_{1/2}transition in Tl for a variety of projectile energies.

**Figure 4.**Spin asymmetry function S

_{A}for elastic electron collisions from Tl atoms in the ground state 6s

^{2}6p

^{2}P°

_{1/2}for incident electron energies between 1 and 24 eV. The DBSR results are compared with the experimental data of Kaussen et al. [5], Geesmann et al. [6], and Dümmler et al. [7], along with the theoretical results of Bartschat [11], Goerss et al. [12], Haberland and Fritsche [13], and Bostock et al. [15].

**Figure 5.**Spin asymmetry function S

_{A}for electron impact excitation of the 6s

^{2}6p

^{2}P°

_{1/2}→ 6s

^{2}6p

^{2}P°

_{3/2}transition in Tl for incident energies between 3 and 14 eV. The DBSR results are compared with the experimental data of Geesmann et al. [6], along with the theoretical results of Goerss et al. [12] and Bostock et al. [15].

**Figure 6.**Spin asymmetry function S

_{A}for electron impact excitation of the 6s

^{2}6p

^{2}P°

_{1/2}→ 6s

^{2}7s

^{2}S

_{1/2}transition in Tl for incident energies between 4 and 14 eV. The DBSR results are compared with the experimental data of Geesmann et al. [6], along with the theoretical results of Goerss et al. [12] and Bostock et al. [15].

**Figure 7.**Average spin asymmetry function S

_{A}for electron impact excitation of the 6s

^{2}6p

^{2}P° → 6s

^{2}7s

^{2}S, 6s

^{2}6p

^{2}P° → 6s

^{2}7p

^{2}P° and 6s

^{2}6p

^{2}P° → 6s

^{2}6d

^{2}D transitions in Tl for incident energies between 5 and 15 eV. The DBSR results are compared with the theoretical results of Goerss et al. [12]. As one example for comparison, fine-structure-resolved predictions are shown in the top left panel.

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**MDPI and ACS Style**

Wang, Y.; Du, H.-L.; Zhu, X.-M.; Zatsarinny, O.; Bartschat, K.
Relativistic B-Spline R-Matrix Calculations for Electron Scattering from Thallium Atoms. *Atoms* **2021**, *9*, 94.
https://doi.org/10.3390/atoms9040094

**AMA Style**

Wang Y, Du H-L, Zhu X-M, Zatsarinny O, Bartschat K.
Relativistic B-Spline R-Matrix Calculations for Electron Scattering from Thallium Atoms. *Atoms*. 2021; 9(4):94.
https://doi.org/10.3390/atoms9040094

**Chicago/Turabian Style**

Wang, Yang, Hai-Liang Du, Xi-Ming Zhu, Oleg Zatsarinny, and Klaus Bartschat.
2021. "Relativistic B-Spline R-Matrix Calculations for Electron Scattering from Thallium Atoms" *Atoms* 9, no. 4: 94.
https://doi.org/10.3390/atoms9040094