The Intensity of Reflected X-Rays and the Distribution of Electrons in Crystals

Abstract

From the intensities of x-rays reflected in different orders, atomic structurefactor (or $F$) curves may be obtained from Darwin's formula. Duane and Compton, using different arguments, have shown that the method of Fourier analysis may be applied to a set of $F$ values to obtain radial electron-distribution (or $U$) curves for the atoms of a crystal. However, Williams (Phil. Mag. 2, 657) and Jauncey (Phys. Rev. 29, 757) have suggested that the intensity of reflected x-rays is less than that on the classical theory because of the modified scattering of the Compton effect occurring at the expense of regular reflection. According to Williams and Jauncey, the number of electrons scattering coherently in the reflection process is a function of the angle of scattering and hence the method of Fourier analysis is invalid because this method depends on the assumption that the number of diffracting centers per atom of the crystal is independent of the angle of reflection. Havighurst (Phys. Rev. 31, 16) takes issue with Williams and Jauncey on the following grounds: (1) A Fourier analysis of an experimental $F$ curve gives a $U$ curve which is reasonable (2) A Fourier analysis of the unmodified $F$ values calculated for a model atom gives a $U$ curve which agrees with the model and which roughly agrees with the experimental $U$ curve (3) A Fourier analysis of the modified $F$ values (i.e. $F$ values modified by the Compton effect) gives a $U$ curve which is unreasonable and does not at all agree with the experimental $U$ curve.