Conditional ambiguity of one-dimensional crystal structures determined from a minimum of diffraction intensity data

Abstract

When the number of intensities greatly exceeds the number of unknown atomic coordinates, the problem of obtaining a crystal structure from the intensities is overdetermined and, for a sufficiently small structure, a chemically meaningful solution can be found by direct methods. A difficulty in determining a structure has been historically attributed to the non-uniqueness of such a structure owing to multiple, or homometric, structures that yield the same set of intensities. The number of homometric structures has not been rigorously analyzed owing to the complexity of this problem. By using the method of elementary symmetric polynomials with a new origin definition, one-dimensional crystal structures of a small number of identical atoms (N < 5), determined from a minimum (N - 1) of the lowest-resolution intensities, are enumerated. It is demonstrated that such a structure is unique for N ≤ 3. Interestingly, for N = 4, the structure can be determined either uniquely or twofold ambiguously, depending on the intensity values. These results suggest that, even for larger structures, a minimum set of (or not many more) accurately measured intensities can yield a unique structure.

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

“…Previously we demonstrated for small one-dimensional crystal structures that the number of physically meaningful solutions of the crystal structure determination problem given by system (2) depends on the structure itself (Shkel et al, 2011). Here, we find that for three-dimensional crystals far fewer than the maximum number of solutions are typically physically meaningful; therefore, system (2) need not (and should not) be solved exhaustively.…”

“…As we demonstrated previously for small one-dimensional crystals, ambiguity of crystal structure determination from an algebraic minimum of intensities generally varies depending on a given set of intensities (Shkel et al, 2011). Even though the maximum ambiguity increases exponentially with increasing N (AlAsadi et al, 2012, 2014), how likely is it to achieve such ambiguity, given a set of intensities?…”

“…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. Cervellino and Ciccariello elegantly demonstrated that, in principle, the structure ambiguity can be reduced by using chemical information such as bond lengths, allowing one to determine onedimensional crystal structures of up to 20 atoms in some cases (Cervellino & Ciccariello, 1999).…”

Enumeration of one-dimensional crystal structures obtained from a minimum of diffraction intensities

“…It is known that even the idealized complete set of perfectly measured intensities cannot yield a unique structure due to enantiomeric and homometric ambiguities (Patterson, 1939(Patterson, , 1944. It was recently rigorously proven that at most two homometric onedimensional crystal structures of N = 4 equal atoms can be obtained (Shkel et al, 2011). Only lower bounds on the number of such structures are known for N = 5 and N = 6 (Bullough, 1963) and no information about crystal structure ambiguity is available, to our knowledge, for larger N or for any N in the cases of two-and three-dimensional crystals of equal atoms, whereas an example of three-dimensional homometric structures has been known for 80 years (Pauling & Shappell, 1930).…”

“…The number of crystal structures that yield (or can be obtained from) a realistic, incomplete, set of intensities is larger, because it may include non-homometric structures. Recently, we investigated the ambiguity of structure determination for small (N 4) one-dimensional crystal structures of equal atoms given the minimum set of lowestresolution intensities, by applying the method of elementary symmetric polynomials with a new origin definition (Shkel et al, 2011). The ambiguity has not been investigated for larger N. Here we report the analysis of the number of onedimensional crystal structures of any number of equal atoms that can be obtained from the minimum of diffraction intensity data, by using approaches of modern computational algebraic geometry.…”