Introduction

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 89-90

Section 1.3.5.1. Introduction

A. Authier^a ^* and A. Zarembowitch^b

^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
Correspondence e-mail: aauthier@wanadoo.fr

1.3.5.1. Introduction

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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.

Figure 1.3.5.1 | top | pdf |

Temperature dependence of the elastic stiffnesses of an aluminium single crystal (after Landoldt-Börnstein, 1979).

Figure 1.3.5.2 | top | pdf |

Pressure dependence of the elastic stiffness $[c_{11} ]$ 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:

(i) From 0 K to about $[\Theta_{D}/5]$ , where $[\Theta_{D}]$ is the Debye temperature, the elastic stiffnesses decrease according to a $[\Theta^{4}]$ law. From $[\Theta_{D}/5]$ to the beginning of the instability domain, the dependence is linear with Θ. In addition, $[(\partial c_{ii}/\partial \Theta)_{p} = 0]$ at 0 K as predicted by the third principle of thermodynamics.
(ii) For stable crystals, the pressure dependence of the elastic stiffnesses is linear as long as the applied pressure is small compared to the elastic stiffnesses. As an example, a typical order of magnitude for $[c_{11}]$ in crystals is about 100 GPa and, within the experimental uncertainty, the pressure dependence of $[c_{11} ]$ does not depart from a linear behaviour up to at least 0.2 GPa.

These observations can be quantitatively justified on the basis of an equation of state of a solid: $[f(T_{ij},S_{ij},X,\Theta) = 0, ]$ where $[T_{ij}]$ represents the stress tensor, $[S_{ij}]$ 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:

(a) In the Einstein model, where the N atoms of a crystal are considered as 3N independent harmonic oscillators, the temperature increase affects only the amplitude of the oscillations and not the average positions of the atoms; consequently, this model can explain neither the thermal expansion nor the temperature dependence of the elastic constants . In addition, this theory ignores the difference between isothermal and adiabatic elastic constants . Similarly, if the oscillators are harmonic, the stiffness of the `springs' connecting atoms does not depend on the distances between atoms and the model cannot therefore explain the pressure dependence of the elastic constants , which requires anharmonic `springs' or, more accurately, anharmonic potentials.
(b) In the Debye model, the 3N oscillators are not independent but they are still harmonic. The result is that here again the elastic constants are pressure and temperature independent .
(c) In the Grüneisen model, the frequencies of the oscillators are volume-dependent; this so-called `quasiharmonic approximation' can justify the standard behaviour shown in Figs. 1.3.5.1 and 1.3.5.2.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 89-90