Bond force constants and thermal vibrations

The dynamical behaviour of the atoms in a crystal is described by the phonon (sound) spectrum which can be measured by inelastic neutron spectroscopy, though in practice this is only possible for relatively simple materials. Infrared and Raman spectra provide images of the phonon spectrum in the long wavelength limit but, because they contain relatively few lines, these spectra can only be used to fit a force model that is too simple to reproduce the full phonon spectrum of the crystal. Nevertheless a useful description of the bond dynamics can be obtained from such force constants using the methods described by Turrell (1972). 

A selection of bond stretching force constants, G, taken from the literature is plotted in Fig. 9.2 as a function of the bond valence, S. What is surprising is not that there is some scatter (this is expected from the crudeness of the Urey-Bradley force field) but that an expression such as eqn (9.5) gives such a good fit  

The amplitude. A, of the thermal motion of a bond is a property directly related to the stretching force constant, G, since the vibrational energy of a harmonic bond is given by eqn (9.6)  

In the classical approximation, which is sufficiently accurate at room temperature and above, the mean square amplitude of vibration of a bond, A ), at 

Uncorrelated motion is most likely to be found when the bonds are weak, for example, when the cation is a large alkali metal. In this case the mean square amplitude, A, is given by the sum of the components of the atomic displacement parameters, U, of the two atoms along the bond direction, that is. 

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