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

International Tables for Crystallography (2006). Vol. D, ch. 1.4, p. 102

Section Diffraction

H. Küppersa*

aInstitut für Geowissenshaften, Universität Kiel, Olshausenstrasse 40, D-24098 Kiel, Germany
Correspondence e-mail: Diffraction

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Thermal expansion expresses itself, on a microscopic scale, by a change of the interplanar spacings of lattice planes. These can be measured by use of diffraction methods from changes of Bragg angles [\theta]. Differentiation of the Bragg equation [2d \sin \theta = \lambda], giving [\Delta d/d = - \cot \theta \Delta \theta], yields the thermal expansions [\alpha'_{11}] in directions normal to lattice planes (hkl) (i.e. along [{\bf h} = h{\bf a}^{*} + k{\bf b}^{*} + l{\bf c}^{*}]) and, if h has direction cosines [a_{1j}^{(hkl)}] with respect to the chosen Cartesian coordinate system,[\alpha_{11}'^{(hkl)} = a_{1i}^{(hkl)} a_{1j}^{(hkl)} \alpha_{ij} = {1 \over d^{(hkl)}} {\partial d^{(hkl)} \over \partial T} = - \cot \theta {\partial \theta \over \partial T}.]The coefficient [\cot\theta] permits a tremendous increase of sensitivity and accuracy if [\theta \rightarrow 90^{\circ}]. That means, if possible, high-angle [(\theta \;\gt\; 70^{\circ})] reflections should be used for measurement because, for a given Δd, the changes of Bragg angles [|\Delta \theta|] to be measured increase with [(\cot \theta)^{-1} = \tan \theta].

The most important diffraction techniques (X-radiation is preferentially used) are: the rotating-crystal method, the Weissenberg method and diffractometers with counter recording. If small single crystals ([\gt] approximately 50 µm) are not available, powder methods (using a Debye–Scherrer film camera or powder diffractometer) must be used, although the advantage of the highly accurate back-reflections, in general, cannot be used.

Experimental aspects of measuring absolute d-values are discussed in detail in Volume C of International Tables for Crystallography (2004)[link], Part 5[link] . Since only relative displacements are to be measured in the present case, many complications connected with the determination of absolute values do not apply for thermal expansion measurements, such as zero-point correction, eccentricity of the mounted sample, refraction, absorption and diffraction profile.


International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, 3rd ed., edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

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