Quasicharacteristic radiation of relativistic electrons at orientation motion in lithium halides crystals along charged planes and axes

The paper deals with the investigation of the orientation motion of relativistic electrons in charged (111) planes and charged [110] axes of lithium halides ionic crystals of LiF, LiCl, LiBr and LiI. On the basis of these investigations the spectra of quasicharacteristic radiation for the electron beams with various Lorentz-factors both in planar and axial cases have been calculated numerically.


Introduction
The given paper is a continuation of some papers dedicated to the investigation of the processes of the channeling and quasicharacteristic radiation (QCR) of relativistic electrons in charged planes and axes of LiM crystals having NaCl-type structure. Thus, in [1] the interaction potential of relativistic electrons with the charged (111) planes of LiH crystal has been calculated. More detailed research of similar potentials at M = H, D is given in [2] where a comparative analysis of analytically calculated QCR spectra with experimentally measured in [3] has been also done. The paper [4] gives analogous calculations for the case of M = F and the comparison of analytically calculated QCR spectra with experimental data given in [5] for three values of Lorentz-factor have been also made.
The given paper investigates the channeling of relativistic electrons in main charged (111) planes and main charged [110] axes of lithium halides of (M = F, Cl, Br, I). For such regime of motion some corresponding wave functions, energy levels of the channeling motion and QCR spectra were calculated by numerical solutions of Schrödinger equations.

The calculation of interaction potentials of electrons with the charged (111) planes of lithium halides crystals
Describing two 1s-electrons by a wave function of the form  One-particle potentials for negatively charged ions of F -, Cl -, Brand Imay be presented by Thomas -Fermi approximation [6] where Z is the nucleus charge of M  ion, Note that the first two conditions in (3) follow from the requirement that the potentials should be pure Coulomb near nuclei when their charges are not screened by electrons [8].
The potentials of electro-neutral stems of   0 Vx  of the charged planes of A + -(111) and B --(111) (in the formula (1) the first and the second components and in formula (2) the first component are used) are calculated in a traditional way [6].
The calculation of the potential created by long-range Coulomb components based on the method given in [9]. The sense of this method is the following. At first the Coulomb potentials of ionic crystals (the last components in (1), (2) formulas) are substituted by the screened Coulomb potentials, namely, Then after these potentials averaging along the planes, we come to the following expression for the summary potential of the full ensemble of the charged planes of (111) of the ionic A + Bcrystal: where 3 da  is the distance between the same charged crystallographic of (111) planes, 2 43 ga  is the density of atoms in (111) planes, а is the lattice constant of the crystal. If first in the expression (4) pass to the limit there arises a conditionally convergent series. The methods summations of similar series for the ionic crystals of finite dimensions have been proposed in [10]. This paper deals with another approach which may be more correct from a mathematical point of view. First fulfill a decomposition of the potential (4) by one-dimensional vectors At last, we confirm that the expression (5) is the expansion into a Fourier series of the following periodic function: where xd . Further the potential (6) must be averaged over thermal oscillations (with using of thermal oscillations amplitudes of u  and u  correspondingly in the planes of А + and B -). For example, figure 1 (а) presents the graph of interaction potential of   Figure 1а presents by a solid curve the graph of the interaction potential of electrons with electro-neutral stems of the charged planes in LiF and also dash-dot curve shows a graph of the interaction potential with Coulomb components of the same planes. All the graphs are calculated at T = 300 K.
Note that in the calculation of the considered interaction potentials with Bplanes other approximations may be used. For example, figure 1 (а) presents by a dot curve the potential   U  in the case of LiF crystal calculated by means of Moliere approximation [1] and a dash curve by means of a Barrette approximation correspondingly [7]. However, as further analysis shows, QCR spectra for a LiF crystal coincide with the ones got experimentally (see [5]) only in the case of the channeling of relativistic electrons in potential wells calculated using Firsov approximation. This is just the basis for the choice of this approximation for all other lithium halides crystals.

The spectra of quasicharacteristic radiation for the channeling electrons in (111) planes of lithium halides crystals
The method of calculation of QCR was the following. The one-dimensional Schrödinger equation with a relativistic mass (see [6]) reduced to the Sturm -Liouville problem. Then this problem was solved by means of corresponding program from IMSL library of Fortran 90 in the result of which transverse energies of ε k and their corresponding wave functions for all lithium halides crystals at the same Lorentz-factor values which was chosen in [5] for the calculation of QCR spectra in LiF crystal was found. Numerical values of energies ε k are shown in table 1.   From the graphics of wave functions in figure 2 (b) is seen that spontaneous dipole transitions of 1  0, 3  1 and 3  2 are connected with the electrons localized in potential wells of F --planes. The transition of 4  2 is realized on the same electrons which localized mainly in Li + -planes. At last, the transition of 4  1 relates to so-called non-direct transitions, since a channeling electron in the result of the loss of the transverse energy changes a region of localization on a spontaneous QCR.
Further, on the basis of the table 1 and using their corresponding wave functions spectral distributions of electron QCR energy were calculated numerically in a dipole approximation on a unit of the path length in correspondence with the following formula:     where       ). All the graphs present identification between the arising spectral peaks of QCR and displayed dipole radiation transitions.
As it is seen from figure 3 channeling electrons energy increases and also the transition from a crystal LiF to a crystal LiI causes a rise of spectral QCR peaks and their intensities (after a small falling in the case of LiCl crystal) for LiBr and LiI crystals increase approximately in one and a half, two times comparing with the intensities in LiF crystal.

The spectra of quasicharacteristic radiation at the channeling of electrons along the charged [110] axes of lithium halides crystals
The interaction potentials (quickly falling with a distance) of   0 , U   of relativistic electrons with electro-neutral stems of the charged [110] axes of ionic lithium halides crystals are calculated in a traditional way [6]. The interaction potential of   , U   of the channeling electrons with a Coulomb components of all the ensemble of opposite charged axes is found analogically as in a planar case, namely, it is calculated as the result of the interaction potentials summation (with axes grouping by electro-neutral shells shown in figure 4 (а) (more detailed argumentation for the axial case is shown in [12]). As illustration in figure 4 (b) the 3D-graph is built on one two-dimensional period of 1    As it is seen from figure 5 (b), even for a small value of Lorentz-factor the following tendency is evident: at the transition from LiF crystal to LiI one a quantity of radiation transitions increases abruptly and spectral QCR peaks shifting in high-frequency region rise by the intensity. Note that this results correlate with experimental data of orientation motion of electrons with longitudinal energies up to 5 MeV along different axial channels of crystal silicon (see [6,13]).

Conclusions
By using the methods of the calculation of interaction potentials of the charged particles to the charged planes and axes in ionic crystals of NaCl-type there were got analytically interaction potentials of electrons with (111) planes and [110] axes in lithium halides of LiF, LiCl, LiBr and LiI. Wherein the interaction potentials calculation in crystallographic planes built of negative halides ions was made in the framework of Firsov approximation (as it follows from experimental data it is more optimal for LiF crystal).
On the basis of the numerical solutions of Schrödinger equations both in planar and axial cases the systems of energy levels and their corresponding wave functions arising at the channeling of relativistic electrons with various Lorentz-factors were calculated (they coincided with those numerical values which were chosen in [5] and [13]).
Proceeding from these data the QCR spectra in dipole approximation were got numerically both in planar and axial cases. At planar channeling with the transition from LiF crystal to LiI one, QCR energy increases but it is concentrated in the same spectral regions (it is observed for all Lorentzfactors). For an axial case this fact does not take place.