The problems of theoretical calculation of the crystals forms are of great fundamental and practical importance. A number of publications are available now devoted to faceting of real crystals of corundum. Faceting sapphire single crystal rods grown by Stepanov method is described in deteil [1], giving the ratio of the size of faces on the side surface of cylindrical rod c {0001}> r{10-11}> a{11-20} as 8:4:1. There are, in eddition, reports calculat concerning on of theoretical faceting. Thus, [2] gives the result of quantum calculation, i.e. c {0001}> m{10-10}> R{10-12}> a{11-20}> r{10-11}. As in seen, the result of theoretical calculation does not agree with the data of the experiment.

M.D. Lubalinsuggested the mathematical tool for the theory of Periodic Bond Chain (PBC) to be applied in practice [3]. Here he introduces the parameters of bond unsaturation (H) and of atomic surface roughness (X). In the present work the parameters H and X for crystal lattice of α-Al₂O₃ are calculated. It is shown that energetically the most advantageous are c, r and a. The quantitative ratio of bond unsaturation (H) for the main faces is close to c> r> a as 4:3:2. Also shown is the difference in parameter H for negative and positive rhombohedrons. Thus, the result of calculation of corundum lattice according to the PBC theory describes to a good rather well the sequence of face display, although the quantitative values differ significantly.

It should be pointed out that on corundum crystals grown by flux method as well as on natural crystals there is observed quite different sequence of face sizes. This gives grounds to say that faceting of corundum crystals is influenced by the impurities of other atoms and the medium of crystal formation. On the other hand, the results on calculation by the PBC method best of all faceting of crystals of pure α-Al₂O₃ grown form reflect the melts in neutral atmosphere.

[1] Ю.Г. Носов, С.И. Бахолдин, В.М. Крымов Журнал технической физики, 2009, Т.79 №2, с. 76-82.

[2] I. Manassidis, M.J. Gillan. Journal American Ceramic Society, 1994, 77[2], p. 335-338

[3] М.Д. Любалин. Рост кристаллов в расплаве. Кристаллографический анализ и эксперимент. СПб.: Наука, 2008. 390с.

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