Ambiguity of structure determination from a minimum of diffraction intensities

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.