Normalized Structure Factors Research Articles

Interface response function for a model of sodium: A molecular dynamics study

Molecular dynamics studies of the liquid epitaxial crystallization and melting of a model of sodium have been carried out to determine the (001) velocity vs interface temperature and to investigate the microscopic growth and melting mechanisms. The growth involves the correlated growth of at least seven planes. We discuss two independent ways of determining the velocity of growth and melting from the simulation data. Details of how the interface temperature is determined and the variation of the system energy with time in both crystallization and melting are included. The maximum growth velocity is 150 m/s and occurs at approximately one-half the melting temperature. The system continues to crystallize to the lowest temperature investigated (81 K) and the growth is not diffusion limited. If conventional transition state theory is used to describe the simulation data then there is a slope discontinuity at the melting temperature with the melting velocities being much higher than predicted. The average structure factors of planes buried in the crystal vs temperature show similar discontinuities near the melting point and suggests that the higher melting velocities are associated with anharmonic softening.

Extension of molecular replacement: a new search strategy based on Patterson correlation refinement

A new search strategy is presented to obtain initial phases for single-crystal diffraction data by molecular replacement. It consists of carrying out 'Patterson refinements' of a large number of the highest peaks of a rotation function. The target function for Patterson refinement is proportional to the negative correlation coefficient between the squared amplitudes of the observed and the calculated normalized structure factors. If the root-mean-square difference between the search model and the crystal structure is within the radius of convergence of the minimization procedure employed, the correct orientation can be identified by having the lowest value of the target function after refinement. Similar to conventional crystallographic R-factor refinement, the target function for Patterson refinement may be combined with an empirical energy function describing geometric and non-bonded interactions. Patterson refinement of individual atomic coordinates or of rigid-group parameters may be carried out. Search models of crambin and of myoglobin with 1.6-2.0 Å backbone atomic r.m.s, differences from the target crystal structures show that the Patterson refinement strategy can solve crystal structures that cannot be solved by conventional molecular replacement or even by full six-dimensional searches.

Explicit Fourier representations of non-ideal hypercentric P.D.F.'s of magnitude of E.

Explicit non-ideal hypercentric distributions of the magnitude of the normalized structure factor have been derived and investigated for the space group P1. One of the distribution types investigated arises when the asymmetric unit of P1 consists of several identical centrosymmetric motifs, interrelated by additional non-crystallographic centres of symmetry. The ideal version of such distributions was studied by Rogers & Wilson [Acta Cryst. (1953), 6, 439-449]. The other distribution type studied originates from atomic arrangements in which the asymmetric unit is composed of several unrelated centrosymmetric fragments that may differ in their sizes and chemical compositions. Explicit non-ideal probability density functions (p.d.f.'s) of the magnitude of the normalized structure factor were formulated for both above types of distribution as Fourier series, were evaluated numerically and were compared with appropriately simulated distributions of magnitude of E. The computations were carried out for a range of atomic compositions, and numbers of unrelated centrosymmetric fragments that comprise the asymmetric unit; both of these factors have a significant influence on the deviation of the hypercentric p.d.f.'s from the values predicted on the basis of the central-limit theorem approximation.

TWO-WAVELENGTH NEIGHBORHOOD PRINCIPLE OF TWO-PHASE STRUCTURE INVARIANTS

An approximate result of first neighborhood is given in ref [5] . In consideration of the importance of strict treatment about this problem, a new formula is derived. According to the computational comparision between the two formulas, it can be concluded that the approximate method in ref [5] is very simple, convenitnt and reliable. It is also concluded by adding random errors to the magnitudes of normalized structure factors that Hauptman's conclusion, namely, the two-phase structure invariants are only dependent on chemical composition and independent of the structure factors, is still correct in the two-wavelength case. Considering this fact, we suggest a fast computational method of the two-wavelength struc-ture invariants which may be useful in practical macrostructure crystallography.

Exact random-walk models in crystallographic statistics. IV. P.d.f.'s of [E] allowing for atoms in special positions.

The effect of scatterers, located in variable special positions, on the probability density function of the magnitude of the normalized structure factor has been investigated. Exact characteristic functions have been obtained for all the statistically different variable special positions in triclinic, monoclinic and ortho-rhombic space groups except in Fdd2 and in the space groups based on the point group 222, and the probability density functions have been evaluated from their Fourier or Fourier-Bessel series expansions. It is seen that the effect of heavy scatterers, located in the special positions investigated, is very marked and should be accounted for in cases of space-group ambiguities.

On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula

The Sayre-equation tangent formula (SETF) develops sets of phases tending to satisfy Sayre's equations for both large and small normalized structure factors. There are two components in the SETF, corresponding to contributions from phase triplets and quartets respectively. The development of objective algorithms for properly weighting these components and for gradually building up the quartet contribution has enabled the SETF, within the procedure SAYTAN, to be incorporated into MULTAN87, the latest version of the package. Examples of tests of MULTAN87 and its use in solving unknown structures are given.

The role of crystallographic symmetry in the direct methods of X-ray crystallography

The number of equations relating the intensities of X-ray diffraction maxima with the crystal structure is in general far greater than the number of parameters needed to define the structure. Since the elucidation of the crystal structure requires also a knowledge of the complex numbers E H = | E H|exp( iφ H), the normalized structure factors, of which only the magnitudes | E H| are obtainable from the measured intensities while the needed phases φ H are lost in the diffraction experiment, it is necessary first to determine the phases. Owing to the redundancy of observed magnitudes | E H|, relationships exist among the complex numbers E H means of which the desired phases φ may be expressed in terms of the known magnitudes | E|. Thus, the lost phase information is to be found among the observed intensities. The direct methods are those which exploit relationships among the normalized structure factors E H in order to extract the values of the individual phases φ H from the measured magnitudes | E H|. When elements of crystallographic symmetry are present then there exist corresponding, space group dependent, relationships among the structure factors E H which may also be employed. The major goal of the present article is to show how these additional relationships, arising from the space group symmetries, may be effectively exploited.

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.

Exact random-walk models in crystallographic statistics. III. Distributions of |E| for space groups of low symmetry

Exact univariate probability density functions (p.d.f.'s) of the magnitude of the normalized structure factor, taking into account space-group symmetry and the chemical composition of the asymmetric unit, have been investigated. The p.d.f.'s that represent distributions for centrosymmetric space groups are given by single Fourier series, while those for the non-centrosymmetric ones can be obtained from double Fourier series or, more conveniently, from single Fourier-Bessel expansions. Analytical expressions for the expansion coefficients are given for all triclinic, monoclinic and orthorhombic space groups, except Fdd2 and Fddd. These results are applied to a comparison of simulated distributions, based on a C9U asymmetric unit and the space groups P1, P1, P2 (or Pm), P2/m, P222, Pmm2 and Pmmm, with the theoretical p.d.f.'s derived here and with approximate generalized p.d.f.'s given by previously published five-term Hermite and Laguerre expansions. The performance of the Fourier and Fourier-Bessel p.d.f.'s is very good throughout the range of symmetries investigated, while that of the approximate ones is rather poor for the lowest symmetries and improves - albeit not uniformly - for the higher ones. Pertinent programming considerations, which suffice for the implementation of the new results in appropriate software, are presented.

Estimation of quartet phase sums from a new joint probability distribution of normalized structure factors

A new joint probability distribution of normalized structure factors is derived for equal-atom structures in space group P1. From this general distribution, a series expansion, the conditional joint probability distribution of the quartet phase sum is obtained, when restrictive conditions among the reciprocal vectors are imposed. The main difference from existing quartet distributions is the possibility of enclosing higher-order terms to any given order of N, although an approximation employed in the derivation limits the number of them considerably. The higher-order terms present are easily employed in the series since the determination of their explicit appearance has been automated: a computer program derives the terms up to a desired order and stores them in a coded form. In general, the incorporation of selective terms up to order N-3 appears to yield sufficient convergence. Only high |E| values or a low N value may necessitate the use of higher-order terms. Test results show that, in contrast to results from the quartet distributions of Hauptman and Giacovazzo, systematic estimation errors are hardly present, while absolute estimation errors are reduced as well.

Exact joint distribution of E h , E k and E h+k , and the probability for the positive sign of the triple product in the space group P\overline{1}

A recently formulated method of deriving exact Fourier-series representations of joint probability density functions (p.d.f.'s) of several normalized structure factors is applied to the derivation of an exact expression for the conditional probability that the sign of the triple product E h E k E h + k is positive. The relevant joint and conditional probabilities are derived for the space group P\bar 1. The Fourier coefficients of the p.d.f. are given by rapidly convergent series of Bessel functions, and the convergence properties of the Fourier summations are also found to be favourable. The exact conditional probability is compared with the currently employed approximate one, well known as the hyperbolic tangent formula, for several hypothetical structures. The examples illustrate the effects of the number of atoms in the unit cell, the magnitude of the E values and the atomic composition on the exact and approximate probabilities. It is found, in agreement with previous studies, that the hyperbolic tangent formula may indeed significantly underestimate the probability when the number of equal atoms is small and the E values are only moderately large, and when the structure contains outstandingly heavy atoms. The opposite behaviour, i.e. the approximate probability overestimating the exact one, was not observed in the present calculations. For large values of the triple product in equal-atom and heterogeneous models, the agreement between the approximate and exact probabilities is usually good.

A systematic procedure for estimating the orientation distribution for nonspherical Rayleigh particles

The inverse problem of estimating the orientation distribution of monodisperse nonspherical Rayleigh particles via electromagnetic scattering is systematically analyzed. Using all the possible angular and polarization information available in Rayleigh scattering, at most 15 averages or moments of the orientation distribution function can be obtained. For monodisperse axisymmetric Rayleigh particles with a plane of symmetry normal to the axis (spheroids, finite cylinders, etc.), the orientation distribution function is expanded in terms of the complete set of surface harmonics. The first 15 coefficients of the expansion can be determined unambiguously without any assumption about the form of the distribution. No additional coefficients can be determined with all conceivable experiments. For distributions which are azimuthally symmetric around the direction of incident light, only three coefficients are nonzero. Average orientation and structure factors are calculated in terms of these coefficients. Computer simulation of experimental errors reveals certain instrument precision requirements for successful measurements. The procedure is applied to a simulated example of spheroids whose orientation distribution is determined by rotational Brownian diffusion.

Exact joint probability distribution for centrosymmetric structure factors. Derivation and application to the Φ1relationship in the space groupPgif">\overline{1}

Recent applications of exact random-walk techniques to crystallographic structure-factor statistics have now been extended to multivariate joint probability density functions of several structure factors. The technique of deriving such multivariate exact density functions is introduced, and is applied to the study of the simplest sign relationship: Σ1, in the space group P\bar 1. An exact expression is obtained for the probability that the sign of the normalized structure factor E(2h 2k 2l) is positive, given the magnitudes of E(2h 2k 2l) and E(h k l), and an extensive numerical examination of this new expression - as compared with the conventional asymptotic formula for this probability - is presented. It is shown that the asymptotic formula usually underestimates the probability that E(2h 2k 2l) is positive, the discrepancies between the exact and asymptotic results being rather serious when the atomic composition of the asymmetric unit is heterogeneous (even moderately so), and when the number of atoms is small; a paucity of atoms leads to significant discrepancies even in the equal-atom case. On the other hand, for large asymmetric units of low heterogeneity and for high E values, the exact and asymptotic expressions agree very well in their predictions. The qualitative behaviour of the new exact expression is consistent with the known features of the Σ1 relationship and its statistical interpretations.

Estimating the three-phase structure invariantsviatheir second neighborhoods

The quintet extension of the three-phase structure invariant is used to define the second neighborhood of the invariant, which in turn serves to define a new kind of probability distribution. The latter leads to novel estimates of the three-phase structure invariants and the magnitudes of the normalized structure factors for the 'squared' structure.

On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula

The application of the condition that a true set of phases for a substantial subset of normalized structure factors should satisfy Sayre's equation leads to a phase-refining equation called the Sayre tangent formula. Phases refined by this formula tend to satisfy Sayre's equation for a subset of E's containing some of large magnitude and some of small (ideally zero) magnitude. Trials indicate that the new formula, incorporated into a computer program SAYTAN, is more effective than MULTAN80, especially for symmorphic structures.

Effect of heavy atoms on the two-phase structure seminvariant φh+ φkinP\overline{1}

A crystal structure in P\bar 1 consisting of P heavy atoms and Q light atoms in the unit cell is considered, and the random variable vectors h and k, subject to well defined restrictions, are uniformly and independently distributed in reciprocal space. The linear combination of two phases ψ = ϕh+ ϕk is a structure seminvariant if and only if h + k = 0 (mod ω) where ω = (2, 2, 2). Eh and Ek are the two normalized structure factors associated with phases ϕh and ϕk and EPh and EPk are the contributions of the heavy atoms respectively. The first neighborhood of ψ consists of the four magnitudes |Eh|, |Ek|, |E(h + k)/2|, |E(h - k)/2|- The conditional probability distribution of ψ, given the four magnitudes |Eh|, |Ek|, |E(h + k)/2|, |E(h - k)/2| and the known contributions EPh, EPk from the heavy atoms, is derived to the approximation of 1/N. In favorable cases, i.e. when phase indications from the four-magnitude neighborhood of the seminvariant and the heavy-atom contributions coincide, a reliable estimate for ψ is obtained. The result obtained is applied to the structure of a derivative of β-lactam and its usefulness is demonstrated.

THE TRIAL AND ERROR PROCEDURES IN THE DIRECT METHOD (Ⅰ)

Some trial and error procedures are often used in the proccess of structure determination (calculating normalized structure factor, determining phase and builting molecular model) by the direct method. The present paper provides a procedure to calculate a trial molecular formula (TMF) on the basis of the diffraction data. Five unknown structures have been solved succesfully by means of the TMF. This method is helptul in extending the application of the direct method.

Exact random-walk models in crystallographic statistics. II. The bicentric distribution for the space groupP\overline{1}

An exact probability density function for the magnitude of the normalized structure factor |E| has been derived for the space group P\bar 1, taking account of the presence of one non-crystallographic center of symmetry. The function is based on the exact solution of the corresponding random-walk model and its expansion into a Fourier series. The above result is compared with simulated semi-cumulative distributions based on hypothetical structures and very good agreement is obtained for the equal-atom case, as well as for a heterogeneous asymmetric subunit containing fourteen carbon atoms and one uranium atom. The new exact bicentric probability density functions of |E|, for the space group P\bar 1, reduce to the well known asymptotic expressions that are valid for equal-atom structures and a large number of atoms in the asymmetric unit of the space group.

Exact random-walk models in crystallographic statistics. I. Space groupsP\overline{1} andP1

Probability density functions that are exact solutions to classical random-walk problems have been adapted to represent distributions of the magnitude of the normalized structure factor, for the space groups P\bar and P1. The functions are given by readily summable Fourier and Fourier-Bessel series, and account explicitly for the atomic composition of the asymmetric unit. These new probability density functions have been extensively tested by comparison with simulated histograms of |E|, for a wide range of atomic compositions. The most heterogeneous compositions examined are C14U and C29U, for P\bar 1 and P1, respectively. Very good agreement between the simulated and theoretical distributions has been obtained in all these tests, over the entire (useful) range 0 < |E| < 3. A distribution of |E| values, recalculated from published data on a triclinic platinum complex with chloroorganic ligands, has also been compared with the new probability functions and excellent agreement with the (expected) P\bar 1 theoretical distribution has been obtained. The discrepancy between the recalculated distribution and the P1 theoretical rules out the latter space group both by visual comparison and quantitative discrepancy criteria. It is concluded that probability density functions are definitely preferable to moments in attempting to resolve a space-group ambiguity. Measures of discrepancy to be used in such statistical tests are proposed and discussed in some detail.

Structural-order effects in low-energy electron transmission spectra of condensed Ar, Kr, Xe,N2, CO, andO2

Low-energy electron transmission spectroscopy is sensitive to crystal order and electronic band structure. The latter is responsible for minima in transmitted current whenever the incident electron energy coincides with that of a band gap. We propose a quasi-free-electron model to describe the experimental results for Ar, Kr, Xe, ${\mathrm{N}}_{2}$, CO, and ${\mathrm{O}}_{2}$ films deposited on Pt. It contains two parameters: the energy of the bottom of the conduction band ${V}_{0}$ and an average electron effective mass ${m}^{*}$. The onset of inelastic processes is extremely valuable in determining ${V}_{0}$. Further knowledge of the reciprocal-lattice vectors allows a reliable fit to ${m}^{*}$ and ${V}_{0}$. The results for Xe join smoothly to those of higher-energy low-energy electron diffraction experiments. We are also able to correlate the average atomic structure factor to the minima in the elastic mean free path of Xe.