Generalized intensity statistics: the subcentric distribution and effects of dispersion

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.