Statistical dynamical direct methods. I. The effective kinematical approximation

Abstract

While it is known that the kinematical approximation works poorly if at all for transmission electron diffraction, substantial success has been achieved over the last few years in applying it via direct methods to determine atomic structures. This raises an interesting quandary; is the established theory of electron diffraction wrong, or are the apparent successes mirages? The intention of this note is to look more deeply into this question and it is found that the correct answer is neither of the above. Beyond vanishingly thin samples when the kinematical approximation holds rigorously, the distribution of phases can remain effectively kinematical; the summation operator(0) distribution given by the sum of the phase of +g and -g reflections remains peaked, albeit not at zero phase, and has a relatively narrow distribution. This fact is shown via both exploiting prior works on including anomalous-scattering effects into direct methods, and numerical calculations. Provided that the summation operator(0) distribution remains narrow, direct methods and indeed structural refinements have some validity. Even larger unit-cell structures with close to statistically random atomic positions do not approach a kinematical limit but instead an effective statistical kinematical approximation. While there are similarities to what there is in conventional (kinematical) direct methods, there remain major differences; for instance, positivity is no longer a valid constraint and the scattering need not be dominated by heavy atoms.