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Функция релаксации. В общем, ширина релаксационной функции измерена для половины ширины пика потерь и составляет от 4 до 6 декад, но она симметрична. С увеличением температуры ширина P-пика уменьшается. Предполагается, что форма функции релаксации для случая Р-релаксации связана с распределением как по энергии активации, так и по предэкспоненциальному множителю, т. е. связана с молекулярным окружением релаксирующих диполей. Обычно сложно получить информацию о преобладающем механизме подвижности, исходя из анализа таких широких релаксационных пиков.
СПИСОК ЛИТЕРАТУРЫ
1. Белоусов А. М., Пазников Е. А., Орлова Н. А. Исследование процессов структурирования тетра-золсодержащих полимеров различными системами отверждения // Известия высших учебн. завед.: Сер. «Хим. и хим. технология». 2006. Т. 49. № 8. C. 3—9.
2. Kremer K., Schonhals A. (Eds.). Broadband dielectric spectroscopy, Springer, Berlin Heidelberg, 2003. 729 с.
REFERENCES
1. Belousov A. M., Paznikov E. A., Orlova N. A. Issledovanie protsessov strukturirovanija tetrazolso-derzhashchih polimerov razlichnymi sistemami otverzhdenija // Izv. vyssh. uchebn. zaved.: Ser. “Him. i him. Tehnologija” 2006. T. 49. № 8. S. 3-9.
2. Kremer K., Schonhals A. (Eds.). Broadband dielectric spectroscopy, Springer, Berlin Heidelberg, 2003. 729 с.
D. Rodionov, A. Belyaev, M. Guitou,
A. Spielfiedel, N. Feautrier, P. Barklem
EXTENDED STUDY OF LOW-ENERGY INELASTIC MAGNESIUM-HYDROGEN COLLISIONS
Quantum calculations of cross sections for inelastic processes in Mg + H and Mg+ + H collisions are improved. It is shown that the largest cross section among the endothermic processes, with a value of approximately 100 A2, corresponds to the process of ion-pair formation: Mg(3s4s S + H ^ Mg + FT. The mechanism of the process is based on nonadiabatic transitions between the MgH(2Z+) molecular states, which provide the main mechanism for inelastic processes in Mg + H collisions. On the other hand, nonadiabatic transitions between MgH(2n) states affect some cross sections rather significantly. For example, transitions between the MgH(2n) states increase the cross section for the excitation process Mg(3s3p 1P) + H ^ Mg(3s3d 1D) + H by almost an order of magnitude as compared to the cross section obtained within the MgH(2Z+) symmetry.
Keywords: atomic collisions, nonadiabatic transitions, inelastic cross sections.
Д. С. Родионов, А. К. Беляев, М. Гиту,
А. Спилфидель, Н. Фотриер, П. С. Барклем
РАСШИРЕННОЕ ИЗУЧЕНИЕ НЕУПРУГИХ НИЗКОЭНЕРГЕТИЧЕСКИХ MG + H СТОЛКНОВЕНИЙ
Улучшены квантовые расчёты сечений неупругих процессов Mg + Hu Mg+ + HT. Показано, что наибольшее сечение среди эндотермических процессов со значением примерно 100 A2 соответствует процессу формирования ионной пары:
Mg(3s4s 1S) + H ^ Mg + НГ. Доминирующим механизмом неупругих процессов, рассмотренных в данной работе, являются переходы между MgH(2Z+) молекулярными состояниями. С другой стороны, неадиабатические переходы между MgH(2n) молекулярными состояниями могут значительно повлиять на некоторые сечения. Например, переходы между MgH(2n) состояниями увеличивают сечение процесса возбуждения Mg(3s3p 1P) + Н ^ Mg(3s3d 1D) + H почти на порядок, по сравнению с сечением, полученным с учётом только MgH(2Z+) симметрии.
Ключевые слова: атомные столкновения, неадиабатические переходы, неупругие сечения.
1. Introduction
The measurement of abundances of chemical elements in stellar atmospheres, as interpreted from stellar spectra, is of fundamental importance in modern astrophysics, see, e.g., [16, 2, 20]. Inelastic processes in collisions of different atoms with hydrogen atoms are important for the non-local thermodynamic equilibrium modeling of stellar spectra which is the main tool for relative and absolute chemical abundances, see, e.g., [2, 3]. Magnesium is an element of significant astrophysical importance, see [1, 19] and references therein. Thus, the need for investigation of inelastic collisions of hydrogen atoms with magnesium atoms is well justified.
In the earlier works, Ref. [14, 9] quantum dynamical calculations have been performed for three MgH(2E+) plus two MgH(2n) molecular states, as well as for the eight lowest-lying MgH(2E+) states, respectively, based on accurate ab initio quantum chemical data [15]. In Ref. [12] it was shown that among the low-lying states the transitions between 2E+ molecular states dominate over transitions involving states of other symmetries. For this reason, the inelastic cross sections were calculated in Ref. [9] by taking into account only the transitions between the eight lowest-lying MgH(2E+) states. In the present paper, we extend the earlier work in Ref. [9] through a calculation including nine 2E+ states up to and including the ionic channel, and the five lowest 2П states based on improved ab initio quantum-chemical data.
2. MgH interaction potentials, nonadiabatic couplings and cross sections
2.1. Ab initio quantum-chemical calculations
The adiabatic potentials and nonadiabatic couplings were calculated using large active spaces and basis sets since avoided crossings occur due to the Mg+ + H- ionic configuration [15].
2 + 2 2 4 + 4
All E , П, A, E and П electronic molecular states arising from Mg + H for energies up to about 6 eV above the lowest atomic asymptote Mg(1S) + H(2Sg) were calculated at the MRCI
level, as well as the couplings between these states. The details are described in Refs. [14, 15]. These new potentials (and couplings) are improved thanks to of an extension of the basis set, comparing to ones used in Refs. [14, 15]. The presently used basis set is larger and will be published elsewhere.
Table 1
The MgH molecular channels, the corresponding asymptotic atomic states and the calculated and experimental (NIST [20] weighted average values) asymptotic energies with respect to the ground state.
Some calculated asymptotic values were adjusted to fit the experimental data
j Molecular states Atomic asymptotic states Asymptotic energies (eV)
Calculation Experiment
1 1 2I+ Mg(3s2 'S) + H 0.0 0.0
2 2 2I+ Mg(3s3p 3P) + H 2.7142 2.7142
3 3 2I+ Mg(3s3p JP) + H 4.3894 4.3458
4 4 2I+ Mg(3s4s 3S) + H 5.1342 5.1078
5 5 2I+ Mg(3s4s JS) + H 5.4237 5.3937
6 6 2I+ Mg(3s3d JD) + H 5.7532 5.7532
7 7 2I+ Mg(3s4p 3P) + H 5.9321 5.9321
8 8 2I+ Mg(3s3d 3D) + H 5.9460 5.9459
9 9 2I+ Mg+(3s 2S) + H- 6.8916 6.8916
10 1 2n Mg(3s3p 3P) + H 2.7142 2.7142
11 2 2n Mg(3s3p JP) + H 4.3894 4.3458
12 3 2n Mg(3s3d JD) + H 5.7532 5.7532
13 4 2n Mg(3s4p 3P) + H 5.9321 5.9321
14 5 2n Mg(3s3d 3D) + H 5.9460 5.9459
The nine lowest MgH(2E+) and the five lowest MgH(2n) molecular states treated in the present work are collected in table 1, including their asymptotic atomic limits. The corresponding potential energy curves (PECs) are represented in figs. 1 as a function of the internuclear distance.
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\ ' \ f уЛ 8T* Mg(3s3d’D)+H Mg(3s2 'S)+H
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. i . . .
10 100 Mg-H intemuclear distance R, (a.u.)
Fig. 1. Potential energy curves for the lowest nine MgH(2E+) states
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Fig. 2. Potential energy curves for the lowest
five MgH(2E+) states
Mg-H intemuclear distance R, (a.u.)
A series of avoided crossings for the 2E+ states is clearly seen. The background of these avoided crossings is an interaction of covalent states with the Mg+ + H- ionic state. For the high-lying molecular states the avoided crossings occur at large distances with small adiabatic splittings. These avoided crossings provide the main mechanism for inelastic processes in low-energy Mg + H collisions, that is, for the processes of excitation, de-excitation, ion-pair production, and mutual neutralization. The nonadiabatic couplings needed for performing quantum dynamical calculations are depicted in figs. 3 and 4. They clearly confirm the presence of the nonadiabatic regions, where nonadiabatic transitions take place.
Fig. 3. Nonadiabatic radial coupling matrix elements ( j---------к j plus offsets,
dR
between the nine MgH(2Е+) states used in the present study. The molecular state label j is indicated in each panel. The keys for the labels к are given in the right panel and are common to all panels. Offsets are used for better representation of the radial couplings
Fig. 4. Nonadiabatic radial coupling matrix elements
dR
к ) plus offsets,
between the five MgH(2n) states used in the present study. The molecular state label j is indicated in each panel. The keys for the labels k are given in the right panel and are common to all panels. Offsets are used for better representation of the radial couplings
It is worth emphasizing that the five lowest MgH^^) molecular states have practically the same adiabatic potentials and nonadiabatic couplings as in the previous calculations [14, 9, 1З]. The improvement mainly concerns the higher-lying states, starting from j = б. Thus, inelastic cross sections obtained with the more accurate quantum-chemical data presented here may be expected to deviate more substantially for processes involving high-lying states.
2.2. Nonadiabatic nuclear dynamics
The nonadiabatic nuclear dynamical calculations are carried out within the standard adiabatic Born-Oppenheimer approach by means of the multi-electron reprojection method [б] (see also references therein) to account for the so-called electron translation problem. The method takes into account non-vanishing asymptotic nonadiabatic matrix elements, provides the correct incoming and outgoing asymptotic total wave functions, and removes nonadiabatic transitions between atomic-state channels in the asymptotic region. First, the nuclear dynamics was performed for the three low-lying 2Е+ and the first two 2П states [14]. It was concluded that the main mechanism for nonadiabatic transitions between molecular states at low energies is due to the radial couplings associated with the avoided ionic crossings in the 2Е+ symmetry. The same mechanism, though in the 1Е+ symmetry, was found to be dominant in calculations for Li + H and Na + H low-energy collisions [10, 11, 12, 13, 7, S]. Based on this conclusion, the nonadiabatic dynamics is then studied for the eight lowest molecular states in the 2Е+ symmetry including the ionic channel [З], as the astrophysical applications [19, 4, 17, 1S] show the importance of ion-pair production processes.
In the present study, the inelastic cross sections for Mg + H and Mg+ + H- collisions are calculated for all transitions between the nine lowest 2Е+ symmetry states, as well as for all transitions between the five lowest 2П symmetry states, based on the improved quantum-chemical data. The cross sections calculated within the 2Е+ symmetry for the endothermic processes are presented in fig. З for each particular initial channel. The collision energy E is the kinetic energy in a given initial channel. The comparison of the present cross sections with those obtained in Ref. [9] shows that the calculations are stable, though some cross sections deviate due to the new and more accurate quantum-chemical data. It is seen that the ion-pair formation process, Mg(3s4s 1S) + H ^ Mg+ + H-, has the largest cross section, of approximately 107 А2, among the endothermic processes.
The previously calculated cross section for the same process had a value of 100 А2 [9], that is, the deviation is within a few percent. This is within the expected accuracy of 20 % [З].
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-к = 2 -к = 3 -к = 4 -к = 5 к = 6 -к = 7 к = 8 - к = 9
Collision energy (eV)
Fig. З. The inelastic cross sections ojk(E) for transitions j ^ к (к > j) in low-energy Mg + H collisions.
The label j of the initial state from which transitions occur is indicated in each panel.
The key for the final states к is given in the bottom right panel
Nonadiabatic transitions between MgH(^) states may affect the excitation (but not ion-pair formation nor mutual neutralization) cross sections due to transitions between highly excited states of magnesium atoms colliding with hydrogen atoms. The inelastic cross sections calculated within the 2П symmetry are shown in Fig. [б]. The labels for the processes within the MgH(^) are presented in table 1. One should keep in mind that not all atomic channels collected in table 1 can produce MgH(^) molecular channels. Thus, the inclusion of the additional channels of the MgH(^) molecular symmetry increases inelastic cross sections for those transitions that can occur within both the MgH(^) and the MgH(^+) symmetries.
For example, in the case of the excitation process Mg(3s3p 1P)+H ^ Mg(3s3d 1D) + H, accounting for transitions between the MgH(^) states increases the cross section by almost an order of magnitude as compared with the cross section obtained within the MgH(^+) states.
The importance of 2П channels should be confirmed by taking into account the rotational couplings that mix Е and П states. This work is in progress.
-------к = 3
к = 6
-------к = 7
к = 8 — к = 11 к = 12 к = 13 — к = 14
Collision energy (eV)
Fig. 6. The inelastic cross sections ojk(E) for transitions j ^ k (k > j) in low-energy Mg + H collisions calculated within the 2X+ symmetry (solid lines) and within the 2n symmetry (dashed lines). Note that only the transitions corresponding to the same initial and the same final atomic states of Mg but occurring in different molecular symmetries (2X+ and 2n) are presented in the figure. For example, the transitions j = 3 ^ k = 6 and j=11 ^ k = 12 correspond to the excitation Mg(3s3p 1P ^ 3s3d 1D) + H in the 2X+ and 2n symmetries, respectively
The physics for the inelastic processes in the treated collisions was carefully studied in Ref. [9] and several mechanisms were found. The present calculations confirm the previous conclusions. It is shown that some of the mechanisms are determined by interactions between adjacent molecular states due to interactions between ionic and covalent configurations. However, mechanisms at short distances, not restricted to adjacent states, are found to be important for some processes. These mechanisms explain the relatively large cross sections for excitation of highly excited states.
3. Concluding remarks
In the present study, the cross sections for the inelastic processes in Mg + H and Mg+ + H-collisions are calculated based on improved quantum-chemical data and on estimates of accounting for nonadiabatic transitions not only in the MgH(2E+) symmetry, but also in the MgH(2n) symmetry. It is shown that the process of ion pair formation: Mg(3s4s 1S) + H ^ Mg+ + H- has the largest cross section, approximately 100 A2, among the endothermic processes. This process occurs due to the nonadiabatic transitions between the 2E+ states. On the other hand, nonadiabatic transitions between MgH(2n) states significantly affect some excitation cross sections due to transitions between highly excited states of magnesium atoms colliding with hydrogen. As an example, accounting for transitions between the MgH(2n) states increases the cross section for the excitation process Mg(3s3p 1P) + H ^ Mg(3s3d 1D) + H by almost an order of magnitude as compared with the cross section obtained within the MgH(2E+) symmetry.
4. Acknowledgements
The work partly was supported by the Russian Foundation for Basic Research (Grant No. 13-03-00163-a), by the Wenner-Gren Foundation (Sweden), by the French CNRS-PNPS (Programme National de Physique Stellaire) and by the GAIA program of Paris Observatory.
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