TY - JOUR

T1 - Enumeration of one-dimensional crystal structures obtained from a minimum of diffraction intensities

AU - Al-Asadi, Ahmed

AU - Chudin, Eugene

AU - Tsodikov, Oleg V.

PY - 2012/5

Y1 - 2012/5

N2 - A central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small (N 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N - 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (4 N /N 3/2 for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures.

AB - A central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small (N 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N - 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (4 N /N 3/2 for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures.

KW - algebraic geometry

KW - direct methods

KW - phase problem

KW - structure ambiguity

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U2 - 10.1107/S0108767312002231

DO - 10.1107/S0108767312002231

M3 - Article

C2 - 22514062

AN - SCOPUS:84860174170

SN - 0108-7673

VL - 68

SP - 313

EP - 318

JO - Acta Crystallographica Section A: Foundations of Crystallography

JF - Acta Crystallographica Section A: Foundations of Crystallography

IS - 3

ER -