Probability Distribution Of Structure Factors Research Articles

The probability distribution of structure factors with non-integral indices. II. The P1; case.

The probability distribution of the structure factors with non-integral indices is derived in P1;. For integral values of at least one of the indices, the intensity distribution coincides with that provided by Wilson's statistics, but may strongly differ when the indices are (or are close to) half-integers. The deviations are stronger when the integral part of the indices is small, and increase with the size of the structure. In favourable circumstances, moduli and phases of the reflections may be accurately estimated.

The Probability Distribution of Structure Factors with Non-Integral Indices. I. TheP1 Case

The probability distribution of structure factors with non-integral indices is derived. The distributions are first studied in the one-dimensional case, to understand their main features, then the three-dimensional case is treated. Only the P1 group is taken into consideration. For integral values of the indices, the distributions coincide with those provided by Wilson statistics but may strongly differ from them when the indices are (or are close to) half-integrals and are sufficiently small. In these cases, the moduli and phases of the reflections may be accurately estimated in the absence of any structural information. Conditional distributions are also derived which are able to estimate moduli or phases by exploiting the prior information on the specific crystal structure.

The effect of atomic heterogeneity on the exact probability distribution of structure factors – crystals of space groupP[unk

AbstractAn attempt has been made to study the effect of atomic heterogeneity of various degrees on the probability distribution of the normalised structure factors. This treatment provides a method of studying separately the effect of two factors affecting heterogeneity, namelyp=NH/NLandg=fH/fLwhereNHandNLare numbers per asymmetric unit of heavy and light atoms, respectively andfHandfLare atomic scattering factors of the heavy and light atoms, respectively. A number of real examples of variouspandgvalues indicating different degrees of heterogeneity have been studied. We have shown that in cases of atomic heterogeneity a modified structure factorE′ may be defined which is Gauss distributed irrespective of whetherEis Gauss distributed or not. Depending on the parameterspandg, the agreement between the two distributionsP(E′) vsE′ andP(E) vsEvary from being poor to excellent. In cases where the agreement is excellent it should be possible, in principle, to obtain the heavy atom position.

Probability distribution of structure factors and the role of atomic heterogeneity

Probability distribution of structure factors and the role of atomic heterogeneity

Joint probability distributions of structure factors in case of incommensurately displacive modulations

Joint probability distributions of structure factors in case of incommensurately displacive modulations

The joint probability distribution of any set of phases given any set of diffraction magnitudes. IV. The active use of psi-zero triplets

In some recent papers [Giacovazzo, Burla & Cascarano (1992). Acta Cryst. A48, 901-906; Burla, Cascarano & Giacovazzo (1992). Acta Cryst. A48, 906-912; Cascarano, Giacovazzo, Moliterni & Polidori (1994). Acta Cryst. A50, 22-27], the method of the joint probability distribution of structure factors has been used to define a function that is frequently a maximum for the correct structure. Such a function was the basis for a modified tangent formula using P10 (negative and positive) triplet estimates and negative quartet estimates, which proved more efficient than the classical tangent formula of Karle & Hauptman [Acta Cryst. (1956), 9, 635-651]. The method is here combined with a recent formulation [Giacovazzo (1993). Z. Kristallogr. 206, 161-171], which suggests the supplementary active use in the phasing process of psi-zero triplets. Experimental tests prove the higher efficiency of the method and justify the default active use of psi-zero relationships in SIR92.

Fourier series probability distribution of structure factors and its relation with other distributions

Fourier series probability distribution of structure factors and its relation with other distributions

Derivation of exponential joint probability distributions of structure factors in P1 with maximum entropy and irreducible cluster integrals

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.

The joint probability distribution of structure factors incorporating anomalous-scattering and isomorphous-replacement data

A method to derive joint probability distributions of structure factors is presented which incorporates anomalous-scattering and isomorphous-replacement data in a unified procedure. The structure factors FH and F−H, whose magnitudes are different due to anomalous scattering, are shown to be isomorphously related. This leads to a definition of isomorphism by means of which isomorphous-replacement and anomalous-scattering data can be handled simultaneously. The definition and calculation of the general term of the joint probability distribution for isomorphous structure factors turns out to be crucial. Its analytical form leads to an algorithm by means of which any particular joint probability distribution of structure factors can be constructed. The calculation of the general term is discussed for the case of four isomorphous structure factors in P1, assuming the atoms to be independently and uniformly distributed. A main result is the construction of the probability distribution of the 64 triplet phase sums present in space group P1 amongst four isomorphous structure factors FH, four isomorphous FK and four isomorphous F−H−K. The procedure is readily generalized in the case where an arbitrary number of isomorphous structure factors are available for FH, FK and F−H−K.

New methods for deriving joint probability distributions of structure factors. II. Strengthening the triplet relationship inP1

Two methods are discussed in detail. In the first method the triplet relationship is treated using the first neighborhood, and the quartet relationship using its second neighborhood. For the triplet relationship it is found that the reliability ϕh +ϕk - ϕh+k ≃ 0 is enhanced when Rh ≃ Rk ≃ Rh+k and large. This conclusion is drawn from formula (16) giving the conditional probability of ϕh + ϕk - ϕh+k using an asymptotic development up to and including terms of order N-1/2. For the quartet relationship it is found that the reliability that ϕh + ϕk + ϕl - ϕh+k+l ≃ π given Rh+k ≃ Rh+l ≃ Rk+l ≃ 0 is diminished when Rh ≃ Rk ≃ Rl ≃ Rh+k+l and large. This conclusion is drawn from formula (19) using similar calculations for the triplet relationship. A heuristic theoretical discussion of this last result trying to explain this difference with the usual theories is given. In the second method the triplet relationship is treated using its first neighborhood. These calculations have been done using a 'normal' asymptotic development up to and including terms of order N-1/2. As a result a formula (28) is obtained that is (at least theoretically) able to predict negative cosine values. A third method that is proposed where one uses the ideas of Patterson superposition will be discussed in detail in a forthcoming paper.

A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors

In this first of three papers on a full Bayesian theory of crystal structure determination, it is shown that all currently used sources of phase information can be represented and combined through a universal expression for the joint probability distribution of structure factors. Particular attention is given to situations arising in macromolecular crystallography, where the proper treatment of non-uniform distributions of atoms is absolutely essential. A procedure is presented, in stages of gradually increasing complexity, for constructing the joint probability distribution of an arbitrary collection of structure factors. These structure factors may be gathered from one or several crystal forms of an unknown molecule, each comprising one or several isomorphous structures related by substitution operations, possibly containing solvent regions and known fragments, and/or obeying a set of non-crystallographic symmetries. This universal joint probability distribution can be effectively approximated by the saddlepoint method, using maximum-entropy distributions of atoms [Bricogne (1984). Acta Cryst. A40, 410-445] and a generalization of structure-factor algebra. Atomic scattering factors may assume arbitrary complex values, so that this formalism applies to neutron as well as to X-ray diffraction methods. This unified procedure will later be extended by the construction of conditional distributions allowing phase extension, and of likelihood functions capable of detecting and characterizing all potential sources of phase information considered so far, thus completing the formulation of a full Bayesian inference scheme for crystal structure determination.

Computer-aided derivation of theoretical joint probability distributions of normalized structure factors

A general theoretical and practical procedure is presented for deriving joint probability distributions of any number of structure factors in any space group. The distributions include all higher-order terms up to a present order of N and thus may be used at any approximation. The procedure combines and extends the two different methods introduced by Naya, Nitta & Oda [Acta Cryst. (1964). 17, 421-433; Acta Cryst. (1965). 19, 734-747] for deriving joint probability distributions of phase-restricted and not-phase-restricted normalized structure factors respectively. The general algorithm for deriving joint probability distributions of structure factors has been implemented in a computer program, thus resulting in the possibility of computer-aided derivations of probabilistic relations for any set of structure factors. Optionally, the program transforms the resulting series-expansion form of the joint probability distribution into an exponential expression. In low-order approximation these exponential expressions usually turn out to be identical to expressions known from the literature.

FourierversusHermite representations of probability distributions

Finite Fourier approximations fitted by use of orthogonality properties are identical with Fourier approximations fitted by least squares, but finite Gram-Charlier or Edgeworth (Hermite) approximations fitted by the two methods differ. This may give a partial explanation of the better performance of Fourier expressions for the probability distributions of structure factors. An alternative Hermite (Myller-Lebedeff) expansion exists, for which the orthogonality and the least-squares coefficients are identical, but they are not readily evaluated.

New methods for deriving joint probability distributions of structure factors. I

With new probabilities, based on the Patterson function, for the 'atomic' random variables x1,..., xN in P\bar 1, it is shown that an improved estimate can be obtained for the sign of the seminvariant E2h in P\bar 1. Two probability measures are considered. A method is also given for the case of a known Patterson vector of the form 2r1, giving an estimate for the sign of any structure factor Eh by using its first neighborhood.

The influence of rational dependence on the probability distribution of structure factors

An index subgroup of strong main reflexions and cosets of weak reflexions are typical features of crystal structures with systematic rational dependence of the atom coordinates exhibiting a pseudotranslational symmetry. The mean squares of these normalized structure-factor sets which deviate significantly from unity are interpreted in terms of correlation coefficients of the atom coordinates. An asymptotic form of the von Mises distribution of a structure factor phase is derived which allows for rational dependence and makes explicit use of the |\bar E|2 values of the different structure-factor sets. The formula provides a basis for the use of phase relationships of the type 'weak-strong-weak' proposed in the recent literature. The limits of the method are estimated. In particular, symmetry and homometry problems in superstructures are more complex than in usual cases and their careful consideration is essential for the success of procedures intending an automatic solution.

Estimating invariant phase sums from a new joint probability distribution of structure factors

Estimating invariant phase sums from a new joint probability distribution of structure factors

Conditional phase probability distributions of structure factors in a Karle–Hauptman matrix

From the joint probability distribution of all structure factors in a Karle-Hauptman matrix new phase probability distributions are obtained. These calculations lead to a reformulation of the maximum-determinant rule for phase determination. In addition a new function is derived whose maximum corresponds to the most probable values for the phases of an arbitrary subset of the structure factors in a Karle-Hauptman matrix. This function accounts for the interaction among phases in a Karle-Hauptman matrix through triple products and quartets simultaneously.

The derivation of joint probability distributions of structure factors for space groupP\overline{1} from the corresponding distributions for space groupP1

A general procedure is given to obtain joint probability distributions of structure factors for structures in space group P{\bar 1} from the corresponding distributions for structures in space group P1. Some examples are gaven.

The joint probability distribution of structure factors: the first-order term

The first-order term of the joint probability distribution of Eh1 + k, . . . , Ehm + k, for h1, . . . , hm fixed and k variable, is derived for both space groups P{\bar 1} and P1. It appears that the first-order term affects the most probable values for the moduli of the structure factors, but that it has no influence on the most probable values for the phases.

Joint probability distribution of structure factors and the phase problem

Joint probability distribution of structure factors and the phase problem