The derivation of force constant information from experimental data

Abstract

It is shown that a direct least-squares fitting procedure leads to inaccurate force constants, even when there is sufficient knowledge to determine them. The inaccuracies occur because of the neglect of more distant interactions which are especially important for the limiting velocities and hence the elastic constants. Mathematically the inaccuracies are apparent because of the requirements of translational invariance and of fitting the elastic constants. The mathematical behaviour of the long-range forces is analysed. A new procedure for deriving force constants is suggested: the dispersion curve is fitted to an appropriate parametric form, the force constants then being obtained by an analytic Fourier transform which automatically incorporates the contribution from the long-range interactions. This procedure is first developed for one-dimensional systems and then extended to three-dimensional crystals with one ion per unit cell. The amount of information available from the experimental data is discussed and it is argued that our procedure, in contrast with other procedures, does not try to extract more information than is available. The difficulties of extracting information in more complicated situations is also discussed.