Crystal: Graz 2012
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 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 [...]. 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 a wide range of crystalline structures are given, based on the original works of the authors and literature analysis. In particular...
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For HCP metals moment interaction proved to be efficient even if deviations in geometrical proportions of real metal's lattice are not taken into account [..]. ...does not exceed the difference in experimental data. Meanwhile, the choice of the interaction depends on the type of the metal electron shell; e.g. d-elements can be described with sufficient accuracy by central force models.
Three models for describing the elastic characteristics of metals with HCP lattice are constructed: (i) reference particle is connected with its closest neighbours by identical linear springs, (ii) reference particle is connected with its closest neighbours by two types of linear springs which reflect the geometry of the lattice; (iii) reference particle is connected with its closest neighbours by identical rods with transversal and longitudinal stiffnesses (model of moment interaction). It is concluded that in the majority of cases, a correct choice of the interaction is more important than taking account of the geometric characteristics of a specific lattice and the choice of the interaction depends on the type of the metal electronic subshell; in particular, d-elements can be described with sufficient accuracy by purely force models.
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.
