br The following generalized empirical rigorous formula has

The following generalized empirical ‘rigorous’ formula has been suggested for full cross-sections of electron scattering by the atoms of all inert gases for energies higher than 100eV: where α0=0.11, Е0= 20keV, α is the static polarizability of a target. Vainshtein, Sobelman and Yukov [15] present the dependence of the excitation cross-section on the electron energy as where for optically allowed transitions the function Ф(u) has the form or, in Green\'s notation, and for optically forbidden transitions _
Here Е0, Е1 are the energies of the initial and the final atomic levels for the 0 → 1 transition; is the Q-factor depending only on the quantum numbers of the angular momenta. The dependence (18) that we are using is close to the Vainshtein approximation (19) and has been constructed based on its modification. Let us have the regression deviation (3) for the cross-sections
Let us examine the regression deviation for the logarithms of the quantities
It follows from its representation in the form that for small deviations of the value δ (see Eq. (20)), the logarithmic deviation δ′ (Formula (21)) approaches the relative deviation
In connection with this, we call the variance (6) of logarithmic deviations the relative variance (for brevity).
The results of hydrogen SAR 405 studies This section presents the data on electronic excitation in a hydrogen atom, obtained based on the above-described approach from the data in the existing information sources [9]. The problem of electron scattering by hydrogen atoms and determining the cross-sections of such a scattering, in particular, of the electron scattering of different levels of hydrogen atoms, is one of the oldest and most frequently studied problems in atomic collision physics due to the relative simplicity of the subject. However, after we reviewed the literary sources in detail, we discovered that this problem is far from completely solved, and the studies conducted by different methods yield dissimilar results. It is not difficult to assess from the below-listed data exactly how much they differ. Most of the studies on electron excitation of a hydrogen atom deal with the 1s→2s, 2p transitions. There has been substantially less data published on other transitions. To analyze the existing results, we employed the above-described approach. We used a four-parameter approximation of the relationship between the excitation cross-sections and the electron energies (18). The regression deviation (3) was constructed for logarithms of cross-section values, i.e., it had the form (21). We found the respective approximation parameters and the values of the aggregate sample relative variance (13). The information sources were taken with the same weight =1. Table 1 presents the obtained approximation parameters p0, p1 – p3, the value of the sample\'s relative variance D, the energy and the value of the cross-section in the maximum of the excitation function Emax, Qmax for the 1s→2s, 2p transitions in a hydrogen atom. Fig. 1 illustrates the initial data from various information sources and the approximation curves determined by the above-described method. A data spread in information sources can be clearly seen (all quantities are given in the atomic system of units). We tested the hypothesis (14) about exceeding the systematic error of a source of the aggregate sample variance (13) for partial information sources with the data most drastically differing from the regression curve. In all cases, the least probability value for incorrectly rejecting a true hypothesis (type I error) (16) was close to α=0.5.
Conclusion Our study has detailed an approach to representing aggregate information on the cross-sections of electron-atom scattering within the framework of mathematical statistics based on regression analysis. This approach allows to take into account the whole set of available cross-section values obtained independently by different authors using different theoretical and experimental methods in differing conditions. It is notable that this approach allows to combine the results of different methods, obtained in a limited narrow range of electron energies, in a single curve describing the Q – E relationship, and to extend it to the respective wide energy range. The data of the specific procedure cannot be extrapolated, within this wide range, to energy regions lying outside their determined value intervals. This conclusion is particularly important for plasma applications requiring knowing the rate constants in wide electron energy ranges.