Crystal: Graz 2012 — различия между версиями

Версия 14:12, 30 ноября 2011

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

Elastic properties of ideal crystals: from macro to micro

Recent advances of nanotechnologies have increased 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, there is still existing 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 needed [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 obtained analytically on the basis of the long-wave approximation or elastic energy correlation. The attempts to obtain such analytical connections where 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 [...]. In the lecture a review of the models, connecting parameters of macroscopic stiffness tensor of ideal crystals and parameters of interatomic bonds are 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 wide number of crystalline structures are given, based on the original works of the authors and literature analysis. In particular...

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Moment interaction.

The particles in the lattice have translational and rotational degrees of freedom, and they interact by means of forces and moments. Crystal’s macroscopic characteristics depend on longitudinal C_А and transversal C_D stiffnesses of interatomic bond:

C_11=√3/12a(C_A+2C_D) ; C_12=√3/12a(C_A-C_D) ; C_44=(3√3)/8a (C_A C_D)/((C_A+2C_D)).

The discrete mechanical model of the complex crystal lattice is proposed in: 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. In moment’s model three modulus of elasticity are connected as

2C_44 C_11=(C_11+2C_12) (C_11-C_12);

The moment interaction gives the very good description for the elastic properties for elements with diamond lattice (C, Si, Ge) and sphalerits lattice (ZnS, BN, Si, GaAs et al).

As a result, it is shown that the moment interaction gives the best description for the elastic properties of interatomic bonds when the elastic properties of a wide number of crystalline structures should be described in a uniform manner.