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In [1], spontaneous radiation arising from the channeling of weakly relativistic positrons along the c-axes in hexagonal crystals of Be, Mg, Sc, Ti, Co, Zn, Se, Y, Zr, Tc, Ru, Cd, Te, La was investigated. In [2] the dechanneling lengths were calculated for the listed crystals and such positron beams. In the present work, we consider cases of orientational motion along inclined directions marked in Fig. a by numbers 1, 2, 3 (as well as along directions equivalent to them). In the basis $\vec{a}_1 = a(1, 0, 0), \quad \vec{a}_2 = a(-1/2, \sqrt{3}/2, 0), \quad \vec{a}_3 = a(-1/2, -\sqrt{3}/2, 0), \quad \vec{c} = c(0, 0, 1)$
(see, for example, [3]) the indicated directions are collinear, with the vectors $
\vec{d}_1 = \vec{a}_1 + \vec{a}_3 - \vec{c}, \quad \vec{d}_2 = \vec{a}_1 - \vec{a}_2 - \vec{c}, \quad \vec{d}_3 = 2\vec{a}_1 - \vec{c}
$
respectively. As calculations show, the depths of 3D-potential wells in these directions for a number of hexagonal crystals attain sufficiently large values, which is important for practical applications. In Fig. b, the level lines are shown for the Co crystal, with a contour spacing of 1 eV (figure also depicts, in relative units $\zeta = \rho / a$, the axial potential $V(\zeta)/V_0$ averaged over thermal vibrations, where $V_0 = 54e/d$, as well as a dotted approximation curve superimposed on it).
As a result of numerically solving the Schrödinger equation in the Cartesian coordinate system, the transverse energy levels and wave functions were calculated for positron beams channeled at zero angles relative to the directions 1, 2, 3 at given Lorentz-factors $\gamma = 30 \div 150$ (the spontaneous radiation distributions $G(\omega)$ were analyzed for various angular divergences $\theta_{0\xi}, \theta_{0\eta}$). In particular, Fig. c shows these dependencies for a Co crystal at $\theta_{0\xi, 0\eta} = (0.5, 0)$ mrad and $\theta_{0\xi, 0\eta} = (0.0, 5)$ mrad. It can be seen that largest deviation arises in direction 2. These spectra can be used for quality control of hexagonal crystals (e.g., as indicators of defects, dislocations, etc. [3]).
References
[1] N.V. Maksyuta, V.I. Vysotskii, D.N. Maksyuta, S.V. Efimenko. Nuclear Instruments and Methods, A, 1058 (2024) 168891.
[2] N.V. Maksyuta, V.I. Vysotskii, D.N. Maksyuta, S.V. Efimenko. Nuclear Instruments and Methods, A, 1075 (2025) 170372.
[3] Anthony Kelly and Kevin M. Knowles, Crystallography and Crystal Defects, 2012 Jhon Wiley & Sons, Ltd.