ISSN:
1600-5724
Source:
Crystallography Journals Online : IUCR Backfile Archive 1948-2001
Topics:
Chemistry and Pharmacology
,
Geosciences
,
Physics
Notes:
Recently the application of the maximum-entropy method to direct methods has been initiated for a priori uniformly and independently distributed atoms, introducing non-uniformity in direct space by putting constraints on the expected values of the distribution [Bricogne (1984). Acta Cryst. A40, 410–445; (1988). Acta Cryst. A44, 517–545]. In this paper a start is made in using the maximum-entropy principle for deriving exponential joint probability distributions of structure factors for a chemically more realistic model of a priori non-uniformly and non-independently distributed atoms. The maximum-entropy equations are obtained by treating the atomic positions as well as the reciprocal vectors as random variables and applying constraints on the maximum of the distribution. The interdependence of the Lagrange multipliers leads to inequalities which may be compared with the Karle–Hauptman inequalities. The radial interatomic correlations such as minimal interatomic distances lead to integrals whose evaluation via the cluster integral mechanism is shown to be equivalent to those of the classical hard-sphere gas in an external field [Van Kampen (1961). Physica (Utrecht), 27, 783–792]. The Debye scattering equation results from these calculations. The exponential multiplet terms are expressed as cluster integrals. From the distribution of the single structure factor the influence of the interatomic correlations on the normalization procedure is assessed. The exponential triplet distribution up to order N−3/2 is derived and is shown to be in agreement with the exponential Edgeworth result [Karle & Gilardi (1973). Acta Cryst. A29, 401–407]. The effect of the radial interatomic correlations on the triplet distribution is discussed. The exponential quartet distribution up to order N−1 is derived, and found to be equal to the well known result [Hauptman (1975). Acta Cryst. A31, 617–679, 680–687] except for some normalization terms resulting from the interatomic correlations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1107/S0108767391004634
