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  • 1990-1994  (7)
  • 1985-1989  (8)
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  • 1
    Electronic Resource
    Electronic Resource
    Exact random-walk models in crystallographic statistics. II. The bicentric distribution for the space group P\overline{1} (1985)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 41 (1985), S. 55-59 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767385000095
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Exact joint probability distribution for centrosymmetric structure factors. Derivation and application to the Φ1 relationship in the space group Pgif"〉\overline{1} (1985)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 41 (1985), S. 401-408 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767385000861
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Exact random-walk models in crystallographic statistics. IV. P.d.f.'s of E allowing for atoms in special positions (1988)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 44 (1988), S. 413-417 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: 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 orthorhombic 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767387009255
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Exact conditional distribution of a three-phase invariant in the space group P1. III. Construction of an improved Cochran-like approximation (1993)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 49 (1993), S. 260-265 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: An exact representation of the accurately computable conditional probability density function (c.p.d.f.) of the three-phase invariant for the space group P1 was developed in paper I of this series [Shmueli, Rabinovich & Weiss (1989). Acta Cryst. A45, 361–367]. The computation of this function is too time consuming for it to be of practical value. It is therefore desirable to find simple approximations based on the exact result that may be more accurate than the familiar Cochran approximation or its extensions. One such approximation, presented here, has the same functional form as the Cochran approximation but with a modified parameter in place of that appearing in Cochran's distribution. Some of the numerical procedures used in the estimation of this modified parameter are also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767392005506
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Exact conditional distribution of a three-phase invariant in space group P1. IV. Further improvements of Cochran-like approximations (1994)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 50 (1994), S. 633-636 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A new approximate method of computing the conditional probability density function (c.p.d.f.) of a three-phase invariant is investigated, the results being compared to accurate calculations [Shmueli, Rabinovich & Weiss (1989). Acta Cryst. A45, 361–367]. A direct dependence on N, the number of atoms in the unit cell, is incorporated into the new approximation and its performance in the equal-atom case appears to be excellent over a large range of N values. The polynomial approximation of the recently published method [Posner, Shmueli & Weiss (1993). Acta Cryst. A49, 260–265] has been improved in both accuracy and range.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767394002382
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Effects of noncrystallographic symmetry on the Σ1 relationship. I. Bicentric arrangements in Pl (1994)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 50 (1994), S. 605-608 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: An exact Fourier representation of the probability for the Σ1 relationship is derived, which takes into account the presence of noncrystallographic centrosymmetry in the asymmetic unit of the space group P{\bar 1}. Illustrative examples show that the main effect of noncrystallographic symmetry is to decrease the probability that E2h is positive, as compared to the corresponding probability in the absence of such symmetry. The effect appears to be more pronounced in the equal-atom case than it is for a structure having a heterogeneous atomic composition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767394002400
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Exact random-walk models in crystallographic statistics. VII. An all-space-group study of the effects of atomic heterogeneity on the P.d.f.'s of E (1991)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 47 (1991), S. 336-340 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Exact expressions have been found for the probability density functions (p.d.f.'s) of the magnitude of the normalized structure factor for all the two-dimensional and most three-dimensional space groups [Part VI: Rabinovich, Shmueli, Stein, Shashua & Weiss (1991). Acta Cryst. A47, 328–335]. The results of that investigation are used in the present article to examine some effects of atomic heterogeneity, in the various space-group symmetries, on the p.d.f.'s. Some typical comparisons are made between p.d.f.'s based on the central limit theorem and p.d.f.'s computed from exact formulae. In addition, the exact results are compared to histograms of simulated values of |E|. It is found that the p.d.f.'s for some space groups are influenced rather strongly by the presence of outstandingly heavy scatterers, but they are quite insensitive to the presence of such scatterers in other space groups. The often made general statement `The presence of outstandingly heavy scatterers may invalidate the indications of Wilson's statistics' is made more precise here, insofar as it depends on the particular space group.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767390013319
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    Exact joint distribution of Eh, Ek and Eh+k, and the probability for the positive sign of the triple product in the space group P\overline{1} (1986)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 42 (1986), S. 240-246 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: 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 EhEkEh + 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767386099361
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    Exact random-walk models in crystallographic statistics. III. Distributions of |E| for space groups of low symmetry (1987)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 43 (1987), S. 93-98 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767387099811
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Explicit Fourier representations of non–ideal hypercentric P.D.F.'s of E (1989)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 45 (1989), S. 213-217 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Explicit non-ideal hypercentric distributions of the magnitude of the normalized structure factor have been derived and investigated for the space group P\bar 1. One of the distribution types investigated arises when the asymmetric unit of P\bar 1 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 |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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767388011043
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    Paper (German National Licenses)
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