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. 1.4, p. 102
Section 1.4.3.2. Diffraction^{a}Institut für Geowissenshaften, Universität Kiel, Olshausenstrasse 40, D-24098 Kiel, Germany |
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 . Differentiation of the Bragg equation , giving , yields the thermal expansions in directions normal to lattice planes (hkl) (i.e. along ) and, if h has direction cosines with respect to the chosen Cartesian coordinate system,The coefficient permits a tremendous increase of sensitivity and accuracy if . That means, if possible, high-angle reflections should be used for measurement because, for a given Δd, the changes of Bragg angles to be measured increase with .
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 ( 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), Part 5 . 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.
References
International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, 3rd ed., edited by E. Prince. Dordrecht: Kluwer Academic Publishers.