Crystal: Graz 2012

A.M. Krivtsov, O.S. Loboda, E.A. Podolskaya

Elastic properties of ideal crystals: from macro to micro

Recent advance in nanotechnologies has increased the interest to determination of mechanical properties of crystalline structures at nanolevel. Mechanical description of nanostructures is impossible without thorough knowledge about elastic characteristics of interatomic bonds. Molecular dynamics simulation of solids also requires parameterization of interatomic potentials to fit the known elastic properties. Although the existing potentials give acceptable description of the physical characteristics of solids, still there are certain problems in precise description of mechanical properties, and in particular the elastic properties of crystals when all components of the stiffness tensor of crystals are required [M. Arroyo et al., 2004; I.E. Berinskiy et al., 2009]. The quantum mechanical analysis can give additional information for the interatomic potentials, however, up to now this cannot solve all the problems for description of the anisotropic elastic properties of crystalline solids. An attractive way for obtaining the necessary information is to use connection between macroscopic elastic properties of ideal crystals and elastic properties of interatomic bonds, which can be acquired analytically on the basis of the long-wave approximation or elastic energy correlation. The attempts to obtain such analytical connections have been made for decades, starting with works by M. Born et al. [Born M.- Ann. Phys.1914, Bd. 44, S. 605], and in some cases they gave quite a good correspondence [Martin R. M. – Phys. Pev. B,1970, v.1, p.4005 ] [Keating P. N. Phys. Rev., 1966, v. 145, p. 637][Krivtsov 2010][Kuzkin 2011] In the lecture a review of the models, connecting parameters of macroscopic stiffness tensor of ideal crystals and parameters of interatomic bonds is presented. For description of elastic properties of the atomic bonds three models are considered and compared: central force interaction, multibody interaction, moment interaction. For these models formulae giving explicit connection between macro and micro parameters for a wide range of crystalline structures are given, based on the original works of the authors and literature analysis.

A set of HCP metals with different degree of geometric imperfection (Be, Hf, Cd, Co, Mg, Re, Ti, Zn, Zr) is considered. It is shown that using the moment model leads to more accurate or similar (for d-elements) description of the elastic properties than taking into account the deviations in geometrical proportions of real metal's lattice [Krivtsov_2010]. The difference between calculated elastic modulae and experimental data does not exceed the divergence in experimental data from various sourses. Thus, moment interaction is proved to be more universal for HCP structure.

A number of crystals with covalent preferred bonds are considered. They include diamond type crystals of the carbon group: C, Si, Ge, Sn, and crystals of sphalerite type, such as ZnS (sphalerite), BN, SiC, GaAs, and more then twenty other items. It is shown that moment model of atomic interaction describes with approximately equal precision both diamond and sphalerite type of crystal structures, while the other existing models are mainly orientated to one type of the structure and provide bigger errors or not acceptable for another type.

Summarizing the above it can be stated that the moment interaction gives the best description for the properties of interatomic bonds when the elastic properties of a wide range of crystalline structures should be described in a uniform manner.

Ivanova E.A., Krivtsov A.M., Morozov N.F. Macroscopic relations of elasticity for complex crystal latices using moment interaction at microscale // Applied mathematics and mechanics. 2007. Т. 71. N. 4. С. 595-615.

• [Krivtsov 2010] A.M. Krivtsov, E.A. Podol’skaya. Modeling of Elastic Properties of Crystals with Hexagonal Close-Packed Lattice. Mechanics of Solids, 2010, Vol. 45, No. 3, pp. 370–378.
• [Kuzkin 2011] Kuzkin V.A, Krivtsov A.M. Description of mechanical properties of graphene using par-ticles with rotational degrees of freedom. Doklady Physics, 2011, Vol. 56, No. 10. pp. 527-530