NORMAL MODE ANALYSIS FROM ELASTIC DIFFRACTION DATA IS POSSIBLE! WHY AND HOW? H. B. Bürgi, Labor für Kristallographie, Freiestr. 3, Universität, CH-3012, Bern, Switzerland

It is generally believed that atomic mean square displacement parameters (ADP) obtained from an elastic diffraction experiment provide information on the motion of individual atoms, but cannot say anything on the cooperative displacements of groups of atoms unless assumptions of questionable validity are made. This opinion is correct with respect to ADP's pertaining to a single temperature; it is untenable, however, when ADP's at several temperatures are available. Why is this so?

ADP's can be considered as a superposition of the contributions from many normal modes. The absolute atomic displacements of a mode depend on its energy ([[nu]]), its effective or reduced mass (u), on temperature (T) and on relative displacement (A) according to A*h/(8*[[pi]]²*u*[[nu]])*coth [h*[[nu]]/(2*k*T/]. At sufficiently low temperatures this expression can be approximated by A*h/(8*[[pi]]²*u*[[nu]]). At sufficiently high temperatures the corresponding approximation is A*k*T/(4*[[pi]]²*u*[[nu]]²).

Now, suppose that a mean squared displacement U is the result of a superposition of two normal modes only and that it has been measured at a low and at a high temperature; then

U(lowT) =A1*h/(8*[[pi]]²*u1*[[nu]]1) + A2*h/(8*[[pi]]²*u2*[[nu]]2)

U(high T) = A1*k*T/(4*[[pi]]²*u1*[[nu]]1²)+A2*k*T/(4*[[pi]]²²).

From the two independent observations the two normal frequencies can be determined, provided A1 and A2 are known. This simple argument can be generalized to more than two frequencies and to the corresponding relative displacements if the coth-dependence and the three-dimensional information contained in the ADP's are taken into account. In practice, ADP's need to be measured at a sufficient number of appropriate temperatures.