research papers

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*) =^{-1}exp{-*I*/}, and 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*) = (2)^{½}exp{-*F*^{2}/2}, 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 parameter*k*, 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 , a fact that can be used to put relative intensities on an absolute basis. The mean values of |*F*| or*I*^{2}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.