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Cauchy distribution, intensity statistics and phases of reflections from crystal planes

Cauchy distribution, intensity statistics and phases of reflections from crystal planes Recently near-Gaussian distributions have been of much interest in the field of crystallographic statistics. In the present work, expressions for a truncated Cauchy distribution corresponding to acentric and centric cases have been derived. Expressions for P+, the probability of sign relations for centric crystals, and for P, the probability of the tangent relationship for acentric crystals, have been derived on the basis of the Cauchy distribution of structure factor components. Theoretical N(Z) curves for centric and acentric Cauchy distributions have been compared with those for acentric, centric and bicentric Gaussian distributions. The N(Z) curve for the Cauchy acentric distribution follows closely that for the Gaussian acentric up to Z = 0.5. It then takes an upward turn and surpasses the Gaussian bicentric curve at high Z values. A similar trend is shown by the N(Z) curve for the Cauchy centric distribution after being approximately intermediate between the Gaussian centric and bicentric cases up to Z = 0.5. The results of P+ and P have been compared with some known crystal data and the agreement is quite satisfactory for the cases studied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Foundations of Crystallography International Union of Crystallography

Cauchy distribution, intensity statistics and phases of reflections from crystal planes

Cauchy distribution, intensity statistics and phases of reflections from crystal planes


Abstract

Recently near-Gaussian distributions have been of much interest in the field of crystallographic statistics. In the present work, expressions for a truncated Cauchy distribution corresponding to acentric and centric cases have been derived. Expressions for P+, the probability of sign relations for centric crystals, and for P, the probability of the tangent relationship for acentric crystals, have been derived on the basis of the Cauchy distribution of structure factor components. Theoretical N(Z) curves for centric and acentric Cauchy distributions have been compared with those for acentric, centric and bicentric Gaussian distributions. The N(Z) curve for the Cauchy acentric distribution follows closely that for the Gaussian acentric up to Z = 0.5. It then takes an upward turn and surpasses the Gaussian bicentric curve at high Z values. A similar trend is shown by the N(Z) curve for the Cauchy centric distribution after being approximately intermediate between the Gaussian centric and bicentric cases up to Z = 0.5. The results of P+ and P have been compared with some known crystal data and the agreement is quite satisfactory for the cases studied.

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References (4)

Publisher
International Union of Crystallography
Copyright
Copyright (c) 1989 International Union of Crystallography
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S0108767388013194
Publisher site
See Article on Publisher Site

Abstract

Recently near-Gaussian distributions have been of much interest in the field of crystallographic statistics. In the present work, expressions for a truncated Cauchy distribution corresponding to acentric and centric cases have been derived. Expressions for P+, the probability of sign relations for centric crystals, and for P, the probability of the tangent relationship for acentric crystals, have been derived on the basis of the Cauchy distribution of structure factor components. Theoretical N(Z) curves for centric and acentric Cauchy distributions have been compared with those for acentric, centric and bicentric Gaussian distributions. The N(Z) curve for the Cauchy acentric distribution follows closely that for the Gaussian acentric up to Z = 0.5. It then takes an upward turn and surpasses the Gaussian bicentric curve at high Z values. A similar trend is shown by the N(Z) curve for the Cauchy centric distribution after being approximately intermediate between the Gaussian centric and bicentric cases up to Z = 0.5. The results of P+ and P have been compared with some known crystal data and the agreement is quite satisfactory for the cases studied.

Journal

Acta Crystallographica Section A: Foundations of CrystallographyInternational Union of Crystallography

Published: May 1, 1989

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