Geometrical Factors Affecting the Contours of X-Ray Spectrometer Maxima. II. Factors Causing Broadening

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Abstract
By applying Stokes' Fourier transform method for the analysis of diffraction maxima it is shown that the pure diffraction contour generated by a crystallite size distribution is apt to be approximated rather closely by the function 1/(1+k2φ2). In the case of the x-ray spectrometer this pure diffraction contour is broadened significantly by the action of the following five geometrical factors: (I) the x-ray source width, (II) flat rather than curved sample surface, (III) vertical divergence of the x-ray beam, (IV) penetration of the sample by the beam, and (V) the receiving slit width. The broadening of the pure diffraction contour due to the action of each of the five factors and the breadth of the final contour generated by the instrument can be deduced by employing the convolution approach suggested by Spencer. The effect of each instrumental factor is expressed by a convolution equation of the form fi(φ)= ∫ −∞+∞Wi(ζ)fi−1(φ−ζ)dζ,in which φ is the angular displacement from twice the ideal Bragg angle, 2θ, fi−1 is the contour before the action of the ith geometrical factor, Wi is the form of the ith geometrical factor, and fi is the contour after the action of Wi on fi−1. Starting with a pure diffraction contour of the form 1/(1+k2φ2), generalized broadening curves are derived for the effect of each of the five geometrical factors. Using these curves it is possible to predict the breadth of the final diffraction contour generated by the spectrometer from an initial contour of any breadth.