The Determination of Parameters in Crystal Structures by means of Fourier Series

The Fourier representation of the results of X-ray analysis was first suggested by W. H. Bragg. It was developed independently by Duane, and used by Havighurst and Compton to give striking representations of the distribution of scattering matter in crystals. Duane and Havighurst were first in applying the method to the much more accurate X-ray measurements available in 1925, and in showing how useful it could be. Duane used a formula derived by Epstein and Ehrenfest. They showed that the intensity of an X-ray reflexion from a plane ( h k l ) of a crystal is proportional to the square of the coefficient of a term in the Fourier series, representing the density ρ ( x, y, z ) of the diffracting material in the crystal as a function of x, y, z . The general term may be written A ( h k l ) sin (2π hx/a — δ h ) sin (2π k/b — δ k ) sin (2π lz/c — δ i Duane reversed the line of thought, and showed that it is possible to deduce the density in the crystal from the measured intensities of X-ray reflexion. The X-ray measurements give the values of (A ( h k l ) )2 but not the phase angles δ h , δ k , δ i . This difficulty, Duane showed, could be surmounted in certain simple cases.