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  • 1
    Electronic Resource
    Electronic Resource
    On the libration of 9,10-anthraquinone at five temperatures (1974)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 30 (1974), S. 768-771 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Potential energy profiles corresponding to libration of 9,10-anthraquinone about its molecular axes were constructed for the five structures derived from data which were collected at - 170, - 112, -72, - 12.5 and 20.5°C [Lonsdale, Milledge & El Sayed (1966). Acta Cryst. 20, 1-13]. These profiles were represented by fourth-degree least-squares polynomials, whereafter r.m.s. libration amplitudes and rigid rotator frequencies of 9,10-anthraquinone were evaluated in the quadratic approximation. The temperature dependence of the calculated quantities is in most cases close to that of the observed ones thus reproducing, by comparison with observed Raman frequencies, the pseudoharmonic behaviour of 9,10-anthraquinone. Calculated r.m.s. libration amplitudes are only qualitatively comparable to the experimental ones and appear to be somewhat too low. The present representation of energy profiles makes it possible to estimate conveniently the contribution of anharmonicity to the profile shape.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739474001859
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  • 2
    Electronic Resource
    Electronic Resource
    A study of generalized intensity statistics: extension of the theory and practical examples (1982)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 38 (1982), S. 362-371 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Generalized probability density functions, cumulative distribution functions and moments of the normalized structure amplitude |E|, depending on space-group symmetry of the crystal and on the composition of the asymmetric unit, were extended to include the tenth moment of |E| and five-term expansions. The formalism was also simplified and is presented in a concise and unified form. The equations linking the formalism to practical problems, the composition and space-group terms, are discussed from a practical point of view and a convenient implementation of the above statistics in a computer program is indicated. The generalized cumulative distributions of |E| and of the normalized intensity z = |E|2 are compared with corresponding distributions based on five published structures, each containing one outstandingly heavy atom (Pt, Rh and Br) and about twenty light ones in the asymmetric unit, excluding hydrogens. These examples indicate that the above formalism is a valuable tool for resolving space-group ambiguities which cannot be treated by conventional methods because of effects of atomic heterogeneity. N(z) distributions for a structure belonging to the space group Fddd show that the theoretical expressions correctly predict the existence of different intensity distributions in reflection subsets with hkl all even and hkl all odd for this space group.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739482000758
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  • 3
    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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  • 4
    Electronic Resource
    Electronic Resource
    Symmetry- and composition-dependent cumulative distributions of the normalized structure amplitude for use in intensity statistics (1979)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 35 (1979), S. 282-286 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Centric and acentric cumulative distribution functions of the normalized structure amplitude, which explicitly account for the presence of outstandingly heavy atoms in crystals of any symmetry, have been derived. These cumulative distributions can now be readily evaluated for all triclinic, monoclinic and orthorhombic space groups, with the exceptions of Fdd2 and Fddd, and thus constitute an extension of the commonly employed cumulative distributions based on the Wilson statistics. Expected discrepancies between the distributions derived in this work and the corresponding Wilson-type distributions are illustrated, and their symmetry and composition dependence is discussed in view of relevant applications to intensity statistics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739479000590
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  • 5
    Electronic Resource
    Electronic Resource
    Generalized intensity statistics: the subcentric distribution and effects of dispersion (1983)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 39 (1983), S. 225-233 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The well-known acentric and centric distributions apply, asymptotically in the number of atoms in the unit cell, when there is no crystallographic symmetry or centrosymmetry only. Series expansions, involving Laguerre or Hermite polynomials, can be obtained, which take into account paucity of (heterogeneous) atoms and higher space-group symmetries. The asymptotic as well as the generalized distributions are further modified if (i) the crystal exhibits partial (non-space group) symmetry, and (ii) if some atoms exhibit appreciable dispersion. This article deals with the generalization of the asymptotic 'subcentric' distribution of the normalized intensity P(z) dz = (α2 - β2)1/2 exp(-αz) I0(βz) dz which accommodates both partial (non-crystallographic) centrosymmetry and effects of dispersion. A four-term Gram-Charlier expansion with appropriate orthogonal polynomials has been derived for the subcentric distribution and detailed expressions for the required moments of z have been obtained for the case of dispersion. This generalization, i.e. the orthogonal polynomials, the moments of z and the asymptotic subcentric distribution, incorporates the generalized acentric and dispersionless centric expansions as limiting cases. The above derivation has been brought to completion using computer-algebraic techniques, which permit the use of well-established but rarely used mathematical methods in an ab-initio generalization of a given asymptotic distribution of intensity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767383000501
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  • 6
    Electronic Resource
    Electronic Resource
    Moments of the trigonometric structure factor (1983)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 39 (1983), S. 615-621 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The eighth moment of the magnitude of the trigonometric structure factor has been computed for all the space groups and all the reflection subsets giving rise to different functional forms of this quantity. This extension of previously published computations of lower moments permits the construction of four-term generalized distributions of normalized intensity, which are necessary in treating problems arising from highly heterogeneous atomic compositions in various space-group symmetries. The related problem of odd-even mixed partial moments of the trigonometric structure factor has also been investigated, and these mixed moments were found to vanish for all the three-dimensional space groups, confirming the correctness of the hitherto published theoretical statistics. Similar computations for the plane groups showed that non-zero values of the mixed partial odd-even moments are obtained for p3, p31m, p3ml, p6 and p6m. This result calls for some modifications of the statistical formalism to be applied to two-dimensional sets of intensity data. The modifications required for the centrosymmetric case are indicated in some detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767383001282
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  • 7
    Electronic Resource
    Electronic Resource
    Space-group algorithms. I. The space group and its symmetry elements (1984)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 40 (1984), S. 559-567 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A well-defined and efficient algorithm, which enables the derivation of any crystallographic space group and full characterization of its symmetry operations and elements, is described and illustrated. The algorithm is based on a representation of crystallographic point groups in terms of cyclic groups, and on isomorphism relations between the point groups and the corresponding factor-group representations of the space groups. The characterization of the symmetry operations and the corresponding symmetry elements is also presented in an algorithmic manner, with particular emphasis on the orientation of the axes of rotation and their location vectors. The above algorithms have been implemented in a computer program, an application of which to the space group Pa\bar 3 is shown and some relevant programming considerations are given. The input to this general program can be fully adapted to the space-group tables in Vol. A of International Tables for Crystallography [(1983). Dordrecht: Reidel].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0108767384001161
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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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  • 9
    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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  • 10
    Electronic Resource
    Electronic Resource
    Covariances of thermal parameters and their effect on rigid-body calculations (1972)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 28 (1972), S. 648-652 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: An isotropic approximation is developed for the covariance matrix of atomic vibration components in an arbitrary coordinate system. The effect of alternative treatments of these covariances on derived rigid-body vibration parameters is illustrated by a numerical comparison. Serious errors may result from neglect of the correlations between different components of the vibration tensors U if these are referred to non-orthogonal axes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739472001597
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