International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 190-209   | 1 | 2 |
https://doi.org/10.1107/97809553602060000554

Chapter 2.1. Statistical properties of the weighted reciprocal lattice

U. Shmuelia* and A. J. C. Wilsonb

a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

References

First citation Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover.Google Scholar
First citation Barakat, R. (1974). First-order statistics of combined random sinusoidal waves with application to laser speckle patterns. Opt. Acta, 21, 903–921.Google Scholar
First citation Bernstein, S. (1922). Sur la théorème limite du calcul des probabilités. Math. Ann. 85, 237–241.Google Scholar
First citation Bernstein, S. (1927). Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Ann. 97, 1–59.Google Scholar
First citation Cramér, H. (1951). Mathematical methods of statistics. Princeton University Press.Google Scholar
First citation Faggiani, R., Lippert, B. & Lock, C. J. L. (1980). Heavy transition metal complexes of biologically important molecules. 4. Crystal and molecular structure of pentahydroxonium chloro(uracilato-N(1))(ethylenediamine)platinum(II)chloride (H5O2)[PtCl(NH2CH2CH2NH2)(C4H5N2O2)]Cl, and chloro(thyminato-N(1))(ethylenediamine)platinum(II), PtCl(NH2CH2CH2NH2)(C5H5N2O2). Inorg. Chem. 19, 295–300.Google Scholar
First citation French, S. & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Cryst. A34, 517–525.Google Scholar
First citation Gerhard, O. E. (1993). Line-of-sight velocity profiles in spherical galaxies: breaking the degeneracy between anisotropy and mass. Mon. Not. R. Astron. Soc. 265, 213–230.Google Scholar
First citation Giacovazzo, C. (1977). On different probabilistic approaches to quartet theory. Acta Cryst. A33, 50–54.Google Scholar
First citation Giacovazzo, C. (1980). Direct methods in crystallography. London: Academic Press.Google Scholar
First citation Harker, D. (1953). The meaning of the average of [|F|^{2}] for large values of interplanar spacing. Acta Cryst. 6, 731–736.Google Scholar
First citation Hauptman, H. & Karle, J. (1953). Solution of the phase problem. I. The centrosymmetric crystal. Am. Crystallogr. Assoc. Monograph No. 3. Dayton, Ohio: Polycrystal Book Service.Google Scholar
First citation Howells, E. R., Phillips, D. C. & Rogers, D. (1950). The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centers of symmetry. Acta Cryst. 3, 210–214.Google Scholar
First citation International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.Google Scholar
First citation International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar
First citation Kendall, M. & Stuart, A. (1977). The advanced theory of statistics, Vol. 1, 4th ed. London: Griffin.Google Scholar
First citation Lipson, H. & Woolfson, M. M. (1952). An extension of the use of intensity statistics. Acta Cryst. 5, 680–682.Google Scholar
First citation Lomer, T. R. & Wilson, A. J. C. (1975). Scaling of intensities. Acta Cryst. B31, 646–647.Google Scholar
First citation Marel, R. P. van der & Franx, M. (1993). A new method for the identification of non-Gaussian line profiles in elliptical galaxies. Astrophys. J. 407, 525–539.Google Scholar
First citation Myller-Lebedeff, W. (1907). Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen. Math. Ann. 64, 388–416.Google Scholar
First citation Nigam, G. D. (1972). On the compensation of X-ray intensity. Indian J. Pure Appl. Phys. 10, 655–656.Google Scholar
First citation Nigam, G. D. & Wilson, A. J. C. (1980). Compensation of excess intensity in space group P2. Acta Cryst. A36, 832–833.Google Scholar
First citation Rabinovich, D. & Shakked, Z. (1984). A new approach to structure determination of large molecules by multi-dimensional search methods. Acta Cryst. A40, 195–200.Google Scholar
First citation Rabinovich, S., Shmueli, U., Stein, Z., Shashua, R. & Weiss, G. H. (1991a). Exact random-walk models in crystallographic statistics. VI. P.d.f.'s of [|E|] for all plane groups and most space groups. Acta Cryst. A47, 328–335.Google Scholar
First citation Rabinovich, S., Shmueli, U., Stein, Z., Shashua, R. & Weiss, G. H. (1991b). 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|]. Acta Cryst. A47, 336–340.Google Scholar
First citation Rayleigh, Lord (1879). Investigations in optics with special reference to the spectroscope. Philos. Mag. 8, 261–274.Google Scholar
First citation Rogers, D. (1950). The probability distribution of X-ray intensities. IV. New methods of determining crystal classes and space groups. Acta Cryst. 3, 455–464.Google Scholar
First citation Rogers, D. & Wilson, A. J. C. (1953). The probability distribution of X-ray intensities. V. A note on some hypersymmetric distributions. Acta Cryst. 6, 439–449.Google Scholar
First citation Shmueli, U. (1979). Symmetry- and composition-dependent cumulative distributions of the normalized structure amplitude for use in intensity statistics. Acta Cryst. A35, 282–286.Google Scholar
First citation Shmueli, U. (1982). A study of generalized intensity statistics: extension of the theory and practical examples. Acta Cryst. A38, 362–371.Google Scholar
First citation Shmueli, U. & Kaldor, U. (1981). Calculation of even moments of the trigonometric structure factor. Methods and results. Acta Cryst. A37, 76–80.Google Scholar
First citation Shmueli, U. & Kaldor, U. (1983). Moments of the trigonometric structure factor. Acta Cryst. A39, 615–621.Google Scholar
First citation Shmueli, U., Rabinovich, S. & Weiss, G. H. (1989). Exact conditional distribution of a three-phase invariant in the space group P1. I. Derivation and simplification of the Fourier series. Acta Cryst. A45, 361–367.Google Scholar
First citation Shmueli, U., Rabinovich, S. & Weiss, G. H. (1990). Exact random-walk models in crystallographic statistics. V. Non-symmetrically bounded distributions of structure-factor magnitudes. Acta Cryst. A46, 241–246.Google Scholar
First citation Shmueli, U. & Weiss, G. H. (1985a). Centric, bicentric and partially bicentric intensity statistics. Structure and statistics in crystallography, edited by A. J. C. Wilson, pp. 53–66. Guilderland: Adenine Press.Google Scholar
First citation Shmueli, U. & Weiss, G. H. (1985b). Exact joint probability distributions for centrosymmetric structure factors. Derivation and application to the Σ1 relationship in the space group [P\bar{1}]. Acta Cryst. A41, 401–408.Google Scholar
First citation Shmueli, U. & Weiss, G. H. (1986). 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\bar{1}]. Acta Cryst. A42, 240–246.Google Scholar
First citation Shmueli, U. & Weiss, G. H. (1987). Exact random-walk models in crystallographic statistics. III. Distributions of [|E|] for space groups of low symmetry. Acta Cryst. A43, 93–98.Google Scholar
First citation Shmueli, U. & Weiss, G. H. (1988). Exact random-walk models in crystallographic statistics. IV. P.d.f.'s of [|E|] allowing for atoms in special positions. Acta Cryst. A44, 413–417.Google Scholar
First citation Shmueli, U., Weiss, G. H. & Kiefer, J. E. (1985). Exact random-walk models in crystallographic statistics. II. The bicentric distribution in the space group [P\bar{1}]. Acta Cryst. A41, 55–59.Google Scholar
First citation Shmueli, U., Weiss, G. H., Kiefer, J. E. & Wilson, A. J. C. (1984). Exact random-walk models in crystallographic statistics. I. Space groups [P\bar{1}] and [P1]. Acta Cryst. A40, 651–660.Google Scholar
First citation Shmueli, U., Weiss, G. H. & Wilson, A. J. C. (1989). Explicit Fourier representations of non-ideal hypercentric p.d.f.'s of [|E|]. Acta Cryst. A45, 213–217.Google Scholar
First citation Shmueli, U. & Wilson, A. J. C. (1981). Effects of space-group symmetry and atomic heterogeneity on intensity statistics. Acta Cryst. A37, 342–353.Google Scholar
First citation Shmueli, U. & Wilson, A. J. C. (1982). Intensity statistics: non-ideal distributions in theory and practice. In Crystallographic statistics: progress and problems, edited by S. Ramaseshan, M. F. Richardson & A. J. C. Wilson, pp. 83–97. Bangalore: Indian Academy of Sciences.Google Scholar
First citation Shmueli, U. & Wilson, A. J. C. (1983). Generalized intensity statistics: the subcentric distribution and effects of dispersion. Acta Cryst. A39, 225–233.Google Scholar
First citation Spiegel, M. R. (1974). Theory and problems of Fourier analysis. Schaum's Outline Series. New York: McGraw-Hill.Google Scholar
First citation Srinivasan, R. & Parthasarathy, S. (1976). Some statistical applications in X-ray crystallography. Oxford: Pergamon Press. Google Scholar
First citation Stuart, A. & Ord, K. (1994). Kendall's advanced theory of statistics. Vol. 1. Distribution theory, 6th ed. London: Edward Arnold.Google Scholar
First citation Weiss, G. H. & Kiefer, J. E. (1983). The Pearson random walk with unequal step sizes. J. Phys. A, 16, 489–495.Google Scholar
First citation Wilson, A. J. C. (1942). Determination of absolute from relative intensity data. Nature (London), 150, 151–152.Google Scholar
First citation Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–320.Google Scholar
First citation Wilson, A. J. C. (1950). The probability distribution of X-ray intensities. III. Effects of symmetry elements on zones and rows. Acta Cryst. 3, 258–261.Google Scholar
First citation Wilson, A. J. C. (1952). Hypercentric and hyperparallel distributions of X-ray intensities. Research (London), 5, 588–589.Google Scholar
First citation Wilson, A. J. C. (1956). The probability distribution of X-ray intensities. VII. Some sesquicentric distributions. Acta Cryst. 9, 143–144.Google Scholar
First citation Wilson, A. J. C. (1964). The probability distribution of X-ray intensities. VIII. A note on compensation for excess average intensity. Acta Cryst. 17, 1591–1592.Google Scholar
First citation Wilson, A. J. C. (1975). Effect of neglect of dispersion on apparent scale and temperature factors. In Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 325–332. Copenhagen: Munksgaard.Google Scholar
First citation Wilson, A. J. C. (1976). Statistical bias in least-squares refinement. Acta Cryst. A32, 994–996.Google Scholar
First citation Wilson, A. J. C. (1978a). On the probability of measuring the intensity of a reflection as negative. Acta Cryst. A34, 474–475.Google Scholar
First citation Wilson, A. J. C. (1978b). Variance of X-ray intensities: effect of dispersion and higher symmetries. Acta Cryst. A34, 986–994.Google Scholar
First citation Wilson, A. J. C. (1978c). Statistical bias in scaling factors: Erratum. Acta Cryst. B34, 1749.Google Scholar
First citation Wilson, A. J. C. (1979). Problems of resolution and bias in the experimental determination of the electron density and other densities in crystals. Acta Cryst. A35, 122–130.Google Scholar
First citation Wilson, A. J. C. (1980a). Relationship between `observed' and `true' intensity: effects of various counting modes. Acta Cryst. A36, 929–936.Google Scholar
First citation Wilson, A. J. C. (1980b). Effect of dispersion on the probability distribution of X-ray reflections. Acta Cryst. A36, 945–946.Google Scholar
First citation Wilson, A. J. C. (1981). Can intensity statistics accommodate stereochemistry? Acta Cryst. A37, 808–810.Google Scholar
First citation Wilson, A. J. C. (1986a). Distributions of sums and ratios of sums of intensities. Acta Cryst. A42, 334–339.Google Scholar
First citation Wilson, A. J. C. (1986b). Fourier versus Hermite representations of probability distributions. Acta Cryst. A42, 81–83.Google Scholar
First citation Wilson, A. J. C. (1987a). Treatment of enhanced zones and rows in normalizing intensities. Acta Cryst. A43, 250–252.Google Scholar
First citation Wilson, A. J. C. (1987b). Functional form of the ideal hypersymmetric distributions of structure factors. Acta Cryst. A43, 554–556.Google Scholar
First citation Wilson, A. J. C. (1993). Space groups rare for organic structures. III. Symmorphism and inherent symmetry. Acta Cryst. A49, 795–806.Google Scholar