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

International Tables for Crystallography (2006). Vol. D. ch. 1.4, pp. 103-104

## Section 1.4.4. Relation to crystal structure

H. Küppersa*

aInstitut für Geowissenshaften, Universität Kiel, Olshausenstrasse 40, D-24098 Kiel, Germany
Correspondence e-mail: kueppers@min.uni-kiel.de

### 1.4.4. Relation to crystal structure

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The anharmonicities of the interatomic potentials gain importance with increasing vibration amplitudes of the atoms. Since, at a given temperature, weakly bonded atoms oscillate with larger amplitudes, they contribute to a larger degree to thermal expansion in comparison with stronger bonds. This correlation follows also from the Grüneisen relation (1.4.2.14) because α (or β) is proportional to the compressibility, which, in turn, is a rough measure of the interatomic and intermolecular forces.

This simple consideration allows qualitative predictions of the thermal expansion behaviour of a crystal species if the structure is known:

 (1) Covalent bonds are associated with very small thermal expansions (diamond, graphite perpendicular to the c axis), whereas van der Waals bonds give rise to large thermal expansions (N2, graphite parallel to the c axis). In accordance with their relatively high elastic stiffness, hydrogen bonds, especially short hydrogen bonds, lead to comparably small thermal expansions. (2) In layer-like structures, the maximum thermal expansion occurs normal to the layers (mica, graphite, pentaerythritol). (3) Thermal expansion decreases when the density of weak bonds decreases: therefore, expansion is greater for crystals with small molecules (many van der Waals contacts per volume) than for their larger homologues (e.g. benzene–naphthalene–anthracene).

Buda et al. (1990) have calculated the thermal expansion of silicon by means of ab initio methods. It is to be expected that these methods, which are currently arduous, will be applicable to more complicated structures in the years to come and will gain increasing importance in this field (cf. Lazzeri & de Gironcoli, 1998).

It is observed rather frequently in anisotropic materials that an enhanced expansion occurs along one direction and a contraction (negative expansion) in directions perpendicular to that direction (e.g. in calcite). The volume expansion, i.e. the trace of , is usually positive in these cases, however. If the tensor of elastic constants is known, such negative expansions can mostly be explained by a lateral Poisson contraction caused by the large expansion (Küppers, 1974).

Only a few crystals show negative volume expansion and usually only over a narrow temperature range (e.g. Si and fused silica below about 120 K and quartz above 846 K) (White, 1993). Cubic ZrW2O8 was recently found to exhibit isotropic negative thermal expansion over the complete range of stability of this material (0.5–1050 K) (Mary et al., 1996). This behaviour is explained by the librational motion of practically rigid polyhedra and a shortening of Zr—O—W bonds by transverse vibration of the oxygen atom. By tailoring the chemical content (of TiO2 or LiAlSiO4) in a glassy matrix, an expansion coefficient can be achieved that is nearly zero over a desired temperature range.

A compilation of numerical values of the tensor components of more than 400 important crystals of different symmetry is given by Krishnan et al. (1979).

Phase transitions are accompanied and characterized by discontinuous changes of derivatives of the free energy. Since the thermal expansion β is a second-order derivative, discontinuities or changes of slope in the curve are used to detect and to describe phase transitions (cf. Chapter 3.1 ).

### References

Buda, F., Car, R. & Parrinello, M. (1990). Thermal expansion of c-Si via ab initio molecular dynamics. Phys. Rev. B, 41, 1680–1683.Google Scholar
Krishnan, R. S., Srinivasan, R. & Devanarayanan, S. (1979). Thermal expansion of solids. Oxford: Pergamon.Google Scholar
Küppers, H. (1974). Anisotropy of thermal expansion of ammonium and potassium oxalates. Z. Kristallogr. 140, 393–398.Google Scholar
Lazzeri, M. & de Gironcoli, S. (1998). Ab initio study of Be(001) surface thermal expansion. Phys. Rev. Lett. 81, 2096–2099.Google Scholar
Mary, T. A., Evans, J. S. O., Vogt, T. & Sleight, A. W. (1996). Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2O8. Science, 272, 90–92.Google Scholar
White, G. K. (1993). Solids: thermal expansion and contraction. Contemp. Phys. 34, 193–204.Google Scholar