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4Physics of Complex Systems, 2023, vol. 4, no. 3 _www.physcomsys.ru
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UDC 539.1 EDN QEBNZO
https://www.doi.org/10.33910/2687-153X-2023-4-3-124-130
Atomic data on inelastic collisional process for YH
V. A. Vasileva1, S. A. Yakovleva™
1 Herzen State Pedagogical University of Russia, 48 Moika Emb., Saint Petersburg 191186, Russia
Theoretical physics. Physics of atoms and molecules
Authors
Vera A. Vasileva, ORCID: 0009-0009-9622-9128, e-mail: veravasilieva2000@gmail.ru Svetlana A. Yakovleva, ORCID: 0000-0002-8889-7283, e-mail: cvetaja@gmail.com
For citation: Vasileva, V. A., Yakovleva, S. A. (2023) Atomic data on inelastic collisional process for YH. Physics of Complex Systems, 4 (3), 124-130. https://www.doi.org/10.33910/2687-153X-2023-4-3-124-130 EDN QEBNZO Received 9 June 2023; reviewed 5 July 2023; accepted 5 July 2023.
Funding: This study was supported by the Russian Science Foundation (the Russian Federation), Project No. 22-23-01181. Copyright: © V. A. Vasileva, S. A. Yakovleva (2023) Published by Herzen State Pedagogical University of Russia. Open access under CC BY-NC License 4.0.
Abstract. The cross sections and rate coefficients for inelastic processes in low-energy collisions of yttrium and hydrogen atoms and ions are calculated. Three ionic states of Y+ + H- are considered. Calculations of non-adiabatic nuclear dynamics in all molecular symmetries of each ionic state are performed. Inelastic processes due to non-adiabatic transitions between 65 different states of the YH quasimolecule are considered. In total, 1,796 inelastic processes are treated and their cross sections for collision energy range from 0.001 to 100 eV and rate coefficients for temperatures from 1,000 to 10,000 K are calculated and analyzed. Inelastic processes with large rate coefficients are important for treating Non-LTE effects in astrophysical modeling.
Keywords: atomic data, inelastic processes, non-adiabatic transitions, yttrium, hydrogen
Introduction
Chemical elements that are formed in neutron-capture processes (n-processes) such as Sr, Ba, Y, La, Zr are of great interest to astrophysical studies. The information on the abundances of these elements in stars and star clusters is important in order to understand n-processes as a source of gaseous environment enrichment (Busso et al. 2001), chemical evolution of galactic disks (Chiappini, Gratton 1997; Serminato et al. 2009) and the production of star clusters (Brewer, Carney 2006).
Neutron capture involves two main processes that depend on the density of the neutron flux: a slow process (s-process) and a rapid process (r-process). The main sources of elements produced in neutron capture (primarily, yttrium and barium) are stars with small and intermediate masses in the asymptotic giant branch. They enrich the interstellar medium with these chemical elements that enter stars during their formation (Travaglio et al. 1999).
In astrophysical modeling of various gaseous environments, a model atom for each chemical element is constructed. It includes energy levels and the data on radiative and non-radiative inelastic processes due to collisions with electrons, atoms, and molecules. Collisions of different atoms and ions with hydrogen atoms and negative ions are one of the greatest uncertainties in astrophysical modeling as hydrogen is the most abundant element in the universe. For this reason, a theoretical calculation of molecular data on inelastic hydrogen collisions is important for astrophysics.
Brief theory
Inelastic processes in collisions of yttrium atoms and positive ions and hydrogen atoms and negative ions are studied within the Born-Oppenheimer formalism using asymptotic model approach (Belyaev 2013).
This approach allows to model non-adiabatic regions due to ionic-covalent interaction via constructing the electronic Hamiltonian matrix in the diabatic representation. The diagonal matrix elements are the diabatic potential energies determined by the Coulomb potential in case of the ionic term and by the asymptotic energy of the scattering channel in case of the covalent term; off-diagonal matrix elements that represent ionic-covalent interaction are calculated using a semi-empirical formula from (Olson et al. 1971).
Non-adiabatic nuclear dynamics is treated within the Landau-Zener multichannel model. Transition probabilities in one non-adiabatic region are calculated using an adiabatic potential-based formula (Belyaev, Lebedev 2011), while state-to-state transition probability p.f (J, E) for a particular collision energy E and total angular momentum quantum number J is calculated using analytical expressions that take all non-adiabatic regions into account (Yakovleva et al. 2016). Inelastic cross sections a.f (E) and rate coefficients K.f for exothermic processes (i >f are then calculated using the following equations:
4-2 „ slat Jmax
•rlfl^Efrl^^t').
J=0
Kif=
8
¡<rif{E)Eexp
kBT
dE,
where p is the reduced nuclear mass, pf'is the statistical probability of population of the initial channel i, kB is the Boltzmann constant. For endothermic processes (k < j) cross sections and rate coefficients are calculated using detailed balance equations:
PT E+AE*j ps'at E
stat
Kkj(T) = Kjk(T)^exp Pj
'AV
V kBT J
where AEk. is the energy defect between the asymptotic energies Ek and E..
YH calculations
The present study of inelastic processes in ytterbium-hydrogen collisions is performed for three sets of molecular states. As the asymptotic model approach (Belyaev 2013) allows taking into account only ionic-covalent interaction, we include only those covalent molecular states that have the same molecular symmetries. Each set contains one of the ionic molecular terms Y++H- and the covalent molecular states that have the same core electrons configuration. All the molecular states, corresponding scattering channels and asymptotic energies for these tree sets are summarized in Tables 1, 2 and 3:
Table 1: 5 covalent Y(5s2 2L) + H(1s 2S) molecular states and ionic state Y+(5s2 1S) + H-(1s2 1S) with 1E+ molecular symmetry;
Table 2: 35 covalent Y(4d5s 24L) + H(1s 2S) molecular states and ionic state Y+(4d5s 3D) + H-(1s2 1S) with 3Z+, 3n, 3A molecular symmetries;
Table 3: 22 covalent Y(4d5s 13L) + H(1s2S) molecular states and ionic state Y+(4d5s 1D) + H-(1s2 1S) with 1E+, 1n, 1A , molecular symmetries.
Table 1. The Y(2L) + H(1s 2S) and Y+(5s2 1S) + H (1s2 1S) molecular channels, the corresponding asymptotic atomic states, and the asymptotic energies with respect to the ground state
Asymptotic atomic states Molecular symmetries Asymptotic energies (eV)
1 Y(4d5s2 2D) + H(1s 2S) 1Z+ 0.03885
2 Y(5s2 5p 2P) + H(1s 2S) 1Z+ 1.37165
3 Y(5s2 6s 2S) + H(1s 2S) 1Z+ 3.92677
Table 1. Completion
4 Y(5s2 (2D)5d 2D) + H(1s 2S) 4.24606
5 Y(5s2 (S )6p 2P) + H(1s 2S) 4.61472
ionic Y+ (5s2 1S) + H- (1s2 S 5.46326
Table 2. The Y(2,4L) + H(1s 2S) and Y+(4d5s 3D) + H (1s2 1S) molecular channels, the corresponding asymptotic atomic states, and the asymptotic energies with respect to the ground state
Asymptotic atomic states Molecular symmetries Asymptotic energies (eV)
1 Y(4d5s2 2D) + H(1s 2S) 3Z+ 3n 3A 0.03885
2 Y(4d2(3i)5s 4F + H(1s 2S) - 3n 3A 1.39762
3 Y(4d2 (3P)5s 4P) + H(1s 2S) - 3n - 1.90689
4 Y(4d2 (3F)5s 2F) + H(1s 2S) - 3n 3A 1.93805
5 Y(4d5s(3D)5p 4F) + H(1s 2S) 3Z+ 3n 3A 1.94401
6 Y(4d2 (1D)5s 2D) + H(1s 2S) 3Z+ 3n 3A 1.99508
7 Y(4d5s(3D)5p 2D) + H(1s 2S) - 3n 3A 1.99599
8 Y(4d5s(3D)5p 4D) + H(1s 2S) - 3n 3A 2.08841
9 Y(4d2(1G)5s 2G) + H(1s 2S) 3Z+ 3n 3A 2.29434
10 Y(4d5s(3D)5p 4P) + H(1s 2S) 3Z+ 3n - 2.36513
11 Y(4d2 (3P)5s 2P) + H(1s 2S) - 3n - 2.39859
12 Y(4d5s(3D)5p 2F) + H(1s 2S) 3Z+ 3n 3A 2.69638
13 Y(4d2(1S)5s 2S) + H(1s 2S) 3Z+ - - 2.90930
14 Y(4d5s(3D)5p 2P) + H(1s 2S) 3Z+ 3n - 3.04487
15 Y(4d5s(1D)5p 2F) + H(1s 2S) 3Z+ 3n 3A 3.06669
16 Y(4d5s(1D)5p 2D) + H(1s 2S) - 3n 3A 3.06818
17 Y(4d5s(1D)5p 2P) + H(1s 2S) 3Z+ 3n - 3.47492
18 Y(4d5s(3D)6s 4D) + H(1s 2S) 3Z+ 3n 3A 4.15074
19 Y(4d5s(3D)6s 2D) + H(1s 2S) 3Z+ 3n 3A 4.51635
20 Y(4d5s(1D)6s 2D) + H(1s 2S) 3Z+ 3n 3A 4.58576
21 Y(4d5s(3D)5d 4D) + H(1s 2S) 3Z+ 3n 3A 4.79801
22 Y(4d5s(3D)6p 4D) + H(1s 2S) - 3n 3A 4.81911
23 Y(4d5s(3D)6p 4F) + H(1s 2S) 3Z+ 3n 3A 4.82798
24 Y(4d5s(3D)5d 4G) + H(1s 2S) 3Z+ 3n 3A 4.82840
25 Y(4d5s(3D)6p 2F) + H(1s 2S) 3Z+ 3n 3A 4.83337
26 Y(4d5s(3D)5d 4S) + H(1s 2S) 3Z+ - - 4.83556
27 Y(4d5s(3D)5d 2P) + H(1s 2S) - 3n - 4.85594
28 Y(4d5s(3D)6p 2D) + H(1s 2S) - 3n 3A 4.86874
29 Y(4d5s(3D)6p 4P) + H(1s 2S) 3Z+ 3n - 4.90231
30 Y(4d5s(3D)5d 4F) + H(1s 2S) - 3n 3A 4.92456
31 Y(4d5s(3D)5d 2F) + H(1s 2S) - 3n 3A 4.92681
32 Y(4d5s(3D)5d 4P) + H(1s 2S) - 3n - 5.01564
Table 2. Completion
33 Y(4d5s(3D)5d 2D) + H(1s 2S) - 3n 3A 5.14169
34 Y(4d5s(D)5d 2D) + H(1s 2S) 3X+ 3n 3A 5.16591
35 Y(4d5s(D)5d 2P) + H(1s 2S) - 3n - 5.29103
ionic Y+(4d5s 3D) + H-(1s2 1S) 3X+ 3n 3A 5.61116
Table 3. The Y(1,3L) + H(1s 2S) and Y+(4d5s 1D) + H (1s2 1S) molecular channels, the corresponding asymptotic atomic states, and the asymptotic energies with respect to the ground state
Asymptotic atomic states Molecular symmetries Asymptotic energies (eV)
1 Y(4d5s2 2D) + H(1s 2S) 0.03885
2 Y(4d2(3F)5s 2F) + H(1s 2S) - 1.93805
3 Y(4d2(D)5s 2D) + H(1s 2S) 1.99508
4 Y(4d5s(3D)5p 2D) + H(1s 2S) - 1.99599
5 Y(4d2(1G)5s 2G) + H(1s 2S) 2.29434
6 Y(4d2(3P)5s 2P) + H(1s 2S) - - 2.39859
7 Y(4d5s(3D)5p 2F) + H(1s 2S) - 2.69638
8 Y(4d2(1S)5s 2S) + H(1s 2S) - - 2.90930
9 Y(4d5s(3D)5p 2P) + H(1s 2S) - 3.04487
10 Y(4d5s(1D)5p 2F) + H(1s 2S) 3.06669
11 Y(4d5s(1D)5p 2D) + H(1s 2S) - 3.06818
12 Y(4d5s(1D)5p 2P) + H(1s 2S) - 3.47492
13 Y(4d5s(3D)6s 2D) + H(1s 2S) 4.51635
14 Y(4d5s(1D)6s 2D) + H(1s 2S) 4.58576
15 Y(4d5s(3D)6p 2F) + H(1s 2S) 4.83337
16 Y(4d5s(3D)5p 2P) + H(1s 2S) - - 4.85594
17 Y(4d5s(3D)6p 2D) + H(1s 2S) - 4.86874
18 Y(4d5s(3D)5d 2F) + H(1s 2S) - 4.92681
19 Y(4d5s(3D)5d 2D) + H(1s 2S) 5.14169
20 Y(4d5s(1D)5d 2D) + H(1s 2S) 5.16591
21 Y(4d5s(1D)5d 2P) + H(1s 2S) - - 5.29103
22 Y(4d5s(1D)5d 2F) + H(1s 2S) - 5.46192
ionic Y+(4d5s D) + H-(1s2 1S) 5.87193
The data for the atomic levels of yttrium atom and ion and their energy values are taken from NIST (Kramida et al. 2022). Asymptotic energies are averaged over the total angular momentum quantum number J. It is worth noting that only one-electron transitions are included in the present investigation as two-electron transitions generally have negligible probabilities, cross sections and rate coefficients because of the narrow non-adiabatic regions (Belyaev et al. 2016; 2019; Vlasov et al. 2018).
For all the transitions between the states in the three sets of molecular states, calculations of partial cross sections and rate coefficients are performed within each molecular symmetry separately and then summed over initial and final states of the process. Cross sections are calculated for the collision energy range from 0.01 eV to 100 eV and rate coefficients — for the temperature range from 1,000 K to 10,000 K.
Results and analysis
Graphical representation of rate coefficients for the temperature T = 6,000K for all the treated processes is shown in Figs 1, 2 and 3 for the three sets of molecular states from Tables 1, 2 and 3, respectively. The rate coefficient order of magnitude is shown with color from blue to red (see the respective legend). The initial and final states labels in graphical representations correspond to those in Tables 1, 2 and 3. White color denotes either the elastic processes that are not studied in the present work (diagonal squares) or transitions between molecular states that do not have the same symmetries.
Final state f 1 2 3 4 5 6
Fig. 1. Graphical representation of the inelastic processes rate coefficients in Y(2L) + H(1s 2S) and Y+(5s2 S) + H-(1s2 S) collisions. The labels for the initial and final states are given in Table 1.
Final state f
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Fig. 2. Graphical representation of the inelastic processes rate coefficients in Y(2,4L) + H(1s 2S) and Y+(4d5s 3D) + H-(1s2 S) collisions. The labels for the initial and final states are given in Table 2.
Final state f
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Fig. 3. Graphical representation of the inelastic processes rate coefficients in Y(1,3L) + H(1s 2S) and Y+(4d5s 1D) + H-(1s21S) collisions. The labels for the initial and final states are given in Table 3.
For the first group of molecular states (see Table 1) the largest rate coefficients at T = 6,000 K correspond the neutralization processes Y+(5s2 1S ) + H- — Y(5s2 nl 2L) + H (j = 6 to j = 3, 4, 5) with values:
K , = 3.59 x 10-8 cm3 s-1,
6—'3 '
K , = 5.86 x 10-8 cm3 s-1,
6— 4
K K = 1.24 x 10-8 cm3 s-1.
6— 5
In the second molecular states group (Table 2) the largest rate coefficients at T = 6,000 K correspond the neutralization processes Y+(4d5s 3D) + H- — Y(4d5s nl 2 4L) + H (j = 36 to j = 18, 19, 20) with values:
K„ „ = 4.72 x 10-8 cm3 s-1,
36— 18
K„ 10 = 3.67 x 10-8 cm3 s-1,
36— 19
K. ,n = 3.13 x 10-8 cm3 s-1.
36— 20
In case of the third group (Table 3) the largest rate coefficients at T = 6,000 K correspond the neutralization processes Y+(4d5s 1D) + H-—Y(4d5s nl 13L) + H (j = 23 to j = 13 - 18) with values:
K23—13 = 3.01 x 10-8 cm3 s-K„—= 4.12 x 10-8 cm3 s-
23—»14
K23—15 = 2.93 x 10-8 cm3 s-K„—,5 = 1.05 x 10-8 cm3 s-
23—16
K23—17 = 2.33 x 10-8 cm3 s-IC3—= 1.75 x 10-8 cm3 s-
23 18
Conclusions
The reported study of inelastic yttrium and hydrogen collisions is performed for the three sets of ionic molecular states of Y+ + H-. For each set, the study of non-adiabatic nuclear dynamics is carried out within all the molecular symmetries of a particular ionic state. Inelastic processes due to nonadiabatic transitions between 65 different states of the YH quasimolecule are considered. In total, 1,796 inelastic processes are studied, cross sections are calculated for the collision energy range from 0.001 eV to 100 eV and the rate coefficients are calculated for temperatures from 1,000 K to 10,000 K. Inelastic processes with large values of the rate coefficients may find application in non-LTE astrophysical modeling.
Conflict of Interest
The authors declare that there is no conflict of interest, either existing or potential.
Author Contributions
All the authors discussed the final work and took an equal part in writing the article.
Acknowledgements Authors gratefully acknowledge discussions with Prof. Andrey K. Belyaev.
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