X-ray diffraction by random layers: ideal line profiles and determination of sructure amplitudes from observed line profiles

Abstract

Warren's calculation of the line profile for diffraction from random layers depends on the use of an approximation function. This can be. avoided, and expressions found for (a) the ideal line profile for slow variation of the structure amplitude, and (b) the variation of the intensity of diffraction as a function of position along the `rod' of high intensity in reciprocal space, in terms of the layer shape and the observed line profile. The ideal line profile is, within a trigonometrical factor, {I(\sigma) = FF{^*}S{_0^{1 \over 2}} \int_{-\infty}^{\infty} A(t)\,\, |t|{^{1 \over 2}} (\cos 2\pi\sigma t + \sin 2\pi\sigma\,\, |t|)\,\, dt,} where S0(= 2 sin θ0/λ) is the perpendicular distance from the origin of the reciprocal lattice to the centre line of the `rod', σ = (sin2 θ − sin2 θ0)/λ sin θ0, F is the structure amplitude, and A(t) is the area common to a layer and its `ghost' shifted a distance t parallel to S0. This expression, evaluated for various layer shapes, gives intensities of diffraction slightly lower than those found by Warren and depending somewhat on the layer shape. In particular, the intensity for large negative volumes of σ is proportional to |σ|−3/2 multiplied by the maximum breadth of the layer. If w is measured along the `rod' from the foot of S0, F(w) F*(w) + F(−w) F*(−w) can be obtained by `unfolding' the observed I(σ) by means of a double Fourier transformation. Diffraction by random layers can thus give more information than diffraction by a perfect crystal, as the latter gives FF* only for integral values of the indices.