Analysis of Debye-Waller-Factor and Mössbauer-Thermal-Shift Measurements. I. General Theory

We give a simple theoretical analysis of the dynamics of an arbitrary atom in a general harmonic solid. The atom under discussion may be an impurity. Several general results are found which limit the possible temperature dependences of the mean-square displacement and mean-square velocity in a way which is described. These results are expected to be most useful in analyzing experiments involving Debye-Waller-factor and Mössbauer-thermal-shift measurements. As an illustration the allowed range of mean-square displacements at $T=0\mathrm{}$ and $T=80\mathrm{}°$K corresponding to a measured value at $T=298\mathrm{}°$K is given. These results also provide strong consistency relations that experimental data or numerical calculations should satisfy. One especially interesting result indicates a possible method for determining a simple sum over atomic force constants from Debye-Waller-factor measurements. This sum, which in general is not obtainable from any other type of measurements, could be used as a convenient check on atomic-force-constant models. The dependence of the mean-square displacement and mean-square velocity on the various masses and force constants in the lattice is described. The relation between our results derived in the harmonic approximation and experimentally measurable quantities is discussed. Finally, several experiments which appear to be interesting are mentioned.