International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 89-90
Section 1.3.5.1. Introduction
a
Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France |
In a solid, the elastic constants are temperature and pressure dependent. As examples, the temperature dependence of the elastic stiffnesses of an aluminium single crystal within its stability domain (the melting point is 933 K) and the pressure dependence of the elastic stiffnesses of the ternary compound KZnF3 within its stability domain (the crystal becomes unstable for a hydrostatic pressure of about 20 GPa) are shown in Figs. 1.3.5.1 and 1.3.5.2, respectively.
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Temperature dependence of the elastic stiffnesses of an aluminium single crystal (after Landoldt-Börnstein, 1979). |
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Pressure dependence of the elastic stiffness of a KZnF3 crystal. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980). Copyright (1980) IEEE. |
We can observe the following trends, which are general for stable crystals:
These observations can be quantitatively justified on the basis of an equation of state of a solid: where represents the stress tensor, the strain tensor, X the position of the elementary elements of the solid and Θ the temperature.
Different equations of state of solids have been proposed. They correspond to different degrees of approximation that can only be discussed and understood in a microscopic theory of lattice dynamics. The different steps in the development of lattice dynamics, the Einstein model, the Debye model and the Grüneisen model, will be presented in Section 2.1.2.7 . Concerning the temperature and the pressure dependences of the elastic constants, we may notice that rather sophisticated models are needed to describe correctly the general trends mentioned above: