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

Ahmed Al-Asadi, Eugene Chudin, Oleg V. Tsodikov

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Pages (from-to)313-318
Number of pages6
JournalActa Crystallographica Section A: Foundations of Crystallography
Issue number3
StatePublished - May 2012


  • algebraic geometry
  • direct methods
  • phase problem
  • structure ambiguity

ASJC Scopus subject areas

  • Structural Biology


Dive into the research topics of 'Enumeration of one-dimensional crystal structures obtained from a minimum of diffraction intensities'. Together they form a unique fingerprint.

Cite this