The probability distribution of X-ray intensities

Abstract

In a crystal without symmetry elements and containing a sufficiently large number of atoms the probability of the hkl reflexion having an intensity between I and I + dI is P(I) dI, where P(I) = Σ−1 exp{−I/Σ}, and Σ is the sum of the squares of the scattering factors of the atoms. In a centrosymmetric crystal the probability of the structure amplitude of the hkl reflexion lying between F and F + dF is P(F) dF, where P(F) = (2πΣ)½ exp{−F2/2Σ}, a result noticed empirically. In a centred crystal (k−1)/k of the reflexions are zero, and the remaining 1/k of them are distributed like those of an uncentred crystal with parameterkΣ, where k is 4 for face-centring and 2 for end- or body-centring. Other symmetry elements do not produce important effects on the general reflexions, but may make a zone or line of intensities behave as if centred or centrosymmetric. The mean value of I is Σ, a fact that can be used to put relative intensities on an absolute basis. The mean values of |F| or I2 can also be used, but the mean value of I is the only one independent of the symmetry. The difference between the ratios of 〈|F|〉2 or 〈I〉 for centrosymmetric and noncentrosymmetric crystals may serve for the purely X-ray determination of a centre of symmetry.