Cauchy distribution, intensity statistics and phases of reflections from crystal planes

Recently near-Gaussian distributions have been of much interest in the field of crystallographic statistics. In the present work, expressions for a truncated Cauchy distribution corresponding to acentric and centric cases have been derived. Expressions for P₊, the probability of sign relations for centric crystals, and for P_φ, the probability of the tangent relationship for acentric crystals, have been derived on the basis of the Cauchy distribution of structure factor components. Theoretical N(Z) curves for centric and acentric Cauchy distributions have been compared with those for acentric, centric and bicentric Gaussian distributions. The N(Z) curve for the Cauchy acentric distribution follows closely that for the Gaussian acentric up to Z = 0.5. It then takes an upward turn and surpasses the Gaussian bicentric curve at high Z values. A similar trend is shown by the N(Z) curve for the Cauchy centric distribution after being approximately intermediate between the Gaussian centric and bicentric cases up to Z = 0.5. The results of P₊ and P_φ have been compared with some known crystal data and the agreement is quite satisfactory for the cases studied.