TY - JOUR

T1 - Probability density functions of the average and difference intensities of Friedel opposites

AU - Shmueli, U.

AU - Flack, H. D.

PY - 2010/11

Y1 - 2010/11

N2 - Trigonometric series for the average (A) and difference (D) intensities of Friedel opposites were carefully rederived and were normalized to minimize their dependence on sinν/λ. Probability density functions (hereafter p.d.f.s) of these series were then derived by the Fourier method [Shmueli, Weiss, Kiefer & Wilson (1984). Acta Cryst. A40, 651-660] and their expressions, which admit any chemical composition of the unit-cell contents, were obtained for the space group P1. Histograms of A and D were then calculated for an assumed random-structure model and for 3135 Friedel pairs of a published solved crystal structure, and were compared with the p.d.f.s after the latter were scaled up to the histograms. Good agreement was obtained for the random-structure model and a qualitative one for the published solved structure. The results indicate that the residual discrepancy is mainly due to the presumed statistical independence of the p.d.f.s characteristic function on the contributions of the interatomic vectors.

AB - Trigonometric series for the average (A) and difference (D) intensities of Friedel opposites were carefully rederived and were normalized to minimize their dependence on sinν/λ. Probability density functions (hereafter p.d.f.s) of these series were then derived by the Fourier method [Shmueli, Weiss, Kiefer & Wilson (1984). Acta Cryst. A40, 651-660] and their expressions, which admit any chemical composition of the unit-cell contents, were obtained for the space group P1. Histograms of A and D were then calculated for an assumed random-structure model and for 3135 Friedel pairs of a published solved crystal structure, and were compared with the p.d.f.s after the latter were scaled up to the histograms. Good agreement was obtained for the random-structure model and a qualitative one for the published solved structure. The results indicate that the residual discrepancy is mainly due to the presumed statistical independence of the p.d.f.s characteristic function on the contributions of the interatomic vectors.

KW - Friedel opposites

KW - interatomic vectors

KW - probability density functions

KW - trigonometric series

UR - http://www.scopus.com/inward/record.url?scp=77958599931&partnerID=8YFLogxK

U2 - 10.1107/S0108767310035920

DO - 10.1107/S0108767310035920

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AN - SCOPUS:77958599931

SN - 0108-7673

VL - 66

SP - 669

EP - 675

JO - Acta Crystallographica Section A: Foundations of Crystallography

JF - Acta Crystallographica Section A: Foundations of Crystallography

IS - 6

ER -