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.The present paper is a reply to the criticisms of Havighurst. By trial a reasonable Bohr model of the chlorine ion has been found which gives modified $F$ values at all angles in good agreement with the experimental values. Furthermore, the theoretical modified $F$ curve has a kink at the same angle as that at which a kink occurs in the experimental $F$ curves. This kink occurs at $sin\ensuremath{\theta}=0.45$ (for Mo $K\ensuremath{\alpha}$ x-rays) and is connected with the result on Jauncey's theory that at this angle the $M$ electrons entirely cease to act as diffracting centers. Havighurst's $F$ curves do not show this kink because he draws smooth curves. From the theoretical modified $F$ values a $U$ curve is plotted for a grating space of $D=3.06\mathrm{A}$. This $U$ curve is similar to Havighurst's experimental $U$ curve for the chlorine ion and has an area of 16.64 electrons in good agreement with 16.74 electrons for the experimental curve. Yet the theoretical $U$ curve carries no particular information concerning the model from which it is derived. This is because the method of Fourier analysis is invalid.The accuracy of experimental $F$ values and $U$ curves is discussed. The experimental $F$ values for chlorine seem to be subject to an absolute error of 0.3, while the area under a $U$ curve is easily subject to an error of 0.5 electron. Objection is raised to the practice by some authors of obtaining $U$ curves for fictitious grating spaces. It is shown that the area under a $U$ curve is a function of $D$ and has no precise meaning even if crystal reflection is unaffected by modified scattering.
Published Version
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