# A geometrical-approach to solving crystal structures

A geometrical-approach to solving crystal structures The Karle & Hauptman and Goedkoop matrices are regarded as metric tensors in Hilbert space. This enables an E value to be expressed in terms of a finite number of other E's. It is shown that the knowledge of a finite number of E's is theoretically sufficient to determine all the atomic coordinates. This number is smaller than 8(N + 3) divided by the order of the point group, counting a complex E twice (N is the number of atoms in the unit cell). The phase problem is analyzed from a geometrical point of view and it is shown how probability becomes certainty with a finite number of data. The 2 relationship is obtained as a particular approximation of an exact relationship between E's. This theory enables a criterion to be established which is equivalent to Tsoucaris's maximum determinant rule but more restrictive. The theory is valid for structures with both equal and unequal atoms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography International Union of Crystallography

# A geometrical-approach to solving crystal structures

, Volume 35 (2): 266 – Mar 1, 1979

## A geometrical-approach to solving crystal structures

, Volume 35 (2): 266 – Mar 1, 1979

### Abstract

The Karle & Hauptman and Goedkoop matrices are regarded as metric tensors in Hilbert space. This enables an E value to be expressed in terms of a finite number of other E's. It is shown that the knowledge of a finite number of E's is theoretically sufficient to determine all the atomic coordinates. This number is smaller than 8(N + 3) divided by the order of the point group, counting a complex E twice (N is the number of atoms in the unit cell). The phase problem is analyzed from a geometrical point of view and it is shown how probability becomes certainty with a finite number of data. The 2 relationship is obtained as a particular approximation of an exact relationship between E's. This theory enables a criterion to be established which is equivalent to Tsoucaris's maximum determinant rule but more restrictive. The theory is valid for structures with both equal and unequal atoms.

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# References (4)

Publisher
International Union of Crystallography
Copyright (c) 1979 International Union of Crystallography
ISSN
0567-7394
DOI
10.1107/S0567739479000577
Publisher site
See Article on Publisher Site

### Abstract

The Karle & Hauptman and Goedkoop matrices are regarded as metric tensors in Hilbert space. This enables an E value to be expressed in terms of a finite number of other E's. It is shown that the knowledge of a finite number of E's is theoretically sufficient to determine all the atomic coordinates. This number is smaller than 8(N + 3) divided by the order of the point group, counting a complex E twice (N is the number of atoms in the unit cell). The phase problem is analyzed from a geometrical point of view and it is shown how probability becomes certainty with a finite number of data. The 2 relationship is obtained as a particular approximation of an exact relationship between E's. This theory enables a criterion to be established which is equivalent to Tsoucaris's maximum determinant rule but more restrictive. The theory is valid for structures with both equal and unequal atoms.

### Journal

Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General CrystallographyInternational Union of Crystallography

Published: Mar 1, 1979