Ambiguity of structure determination from a minimum of diffraction intensities

Abstract

Although the ambiguity of the crystal structures determined directly from diffraction intensities has been historically recognized, it is not well understood in quantitative terms. Bernstein's theorem has recently been used to obtain the number of one-dimensional crystal structures of equal point atoms, given a minimum set of diffraction intensities. By a similar approach, the number of two- and three-dimensional crystal structures that can be determined from a minimum intensity data set is estimated herein. The ambiguity of structure determination from the algebraic minimum of data increases at least exponentially fast with the increasing structure size. Substituting lower-resolution intensities by higher-resolution ones in the minimum data set has little or no effect on this ambiguity if the number of such substitutions is relatively small.

Determination of small crystal structures from a minimum set of diffraction intensities by homotopy continuation

“…To our knowledge, we are the first to apply one of such methods, polynomial homotopy continuation, to this problem. The exponentially increasing ambiguity of crystal structure determination directly from the minimum of diffraction intensities (Al-Asadi et al, 2012, 2014 precludes brute-force root computation even for relatively small structures. Experimental uncertainty exacerbates this problem further.…”

“…1a). In these cases Bernstein's bound, four unique structures (Al-Asadi et al, 2014), is exact, and all of the solutions are physically meaningful. For the other 38 systems, one or more paths diverged or ended on a physically meaningless solution.…”

“…a crystal structure. As N increases, the size and the degree of system (2) rapidly increase, resulting in the exponentially increasing number of its solutions (Al-Asadi et al, 2012, 2014. To resolve this ambiguity, higher-order intensity data are needed, overdetermining the system.…”

“…It was realized early on that the problem of determining an N-atom structure from the algebraic minimum of 3(N À 1) error-free intensities has multiple solutions (Hauptman & Karle, 1951). Recently, we rigorously enumerated this ambiguity and demonstrated that the number of crystal structures that yield the same minimum set of intensities increases exponentially with increasing N (Al- Asadi et al, 2012Asadi et al, , 2014. The solutions themselves can be determined by methods of elementary algebra only for theoretically relevant one-dimensional crystals of small numbers of atoms (N < 5) (Shkel et al, 2011), whereas such methods do not work for larger structures or for higher dimensionality.…”