Date Published: September 01, 2018
Publisher: International Union of Crystallography
Author(s): Jack T. Fraser, Justin S. Wark.
Fewster [(2014), Acta Cryst. A70, 257–282] claimed that a new theory of X-ray diffraction is required, and that small crystallites will give rise to scattering at angles of exactly twice the Bragg angle, whatever their orientation. This article demonstrates that this theory is in error.
Despite the field of X-ray diffraction being more than a century old, in an article entitled A new theory for X-ray diffraction (Fewster, 2014 ▸), hereafter referred to as NTXRD, it is claimed that a new theory of diffraction is required to explain the intensities observed in powder diffraction and other diffraction geometries. Within NTXRD a theory of X-ray diffraction is proposed which predicts that ‘the scattering from a crystal or crystallite is distributed throughout space [which] leads to the effect that enhanced scatter can be observed at the ‘Bragg position’ even if the ‘Bragg condition’ is not satisfied’ and that ‘the scatter from a single crystal or crystallite, in any fixed orientation, has the fascinating property of contributing simultaneously to many ‘Bragg positions’’. If this new approach were correct it would certainly have significant implications for the whole field of X-ray diffraction, and given the prominence afforded to this new theory (it featured on the front cover of the published volume), its veracity or otherwise deserves appropriate scrutiny. However, we show here that the analysis presented within NTXRD is incorrect, and that the underlying concepts upon which the theory is based are not new but were known to the earliest pioneers of X-ray diffraction.
Consider the diffraction geometry shown in Fig. 1 ▸, adapted from Fig. 4(a) of NTXRD. Fewster derives the following formula [the square of the amplitude, , calculated in equation (5) of NTXRD] for the scattered intensity from a set of atoms, recorded by a detector placed at an angle to a beam of monochromatic radiation of wavelength λ which is incident at an angle Ω to the crystal plane: where is the length of the crystal, d is the plane spacing, n denotes the ‘order’ of planes from which the X-rays are diffracting and N is the number of planes in the stack contributing to the reflection.
Fewster’s analysis contains three errors – one minor and two major. Firstly, he states that the amplitude, , of X-rays diffracted from a single (the first) plane shown in Fig. 1 ▸ is given by This is clearly the scattering amplitude from a uniform plane. However, if instead we consider scattering from N discrete atoms (assumed here to be point-like, i.e. ignoring the atomic form factor) separated by a distance a, the scattered amplitude from a single plane of atoms is This is only a minor error since, in the small-angle limit, equations (2) and (3) are in very close agreement, but diverge for larger angles (we discuss further the relationship between the use of sinc functions to describe the diffraction and the ratio of two sine functions in §4).
In order to elucidate further errors described in NTXRD, in this section we note the well known result that equation (5) can, via the method of Poisson sums, be written in terms of the Fourier transform of the shape function of an orthorhombic shaped crystal (sinc functions) centred on the infinite reciprocal lattice [see equation (20) below]. By use of such shape functions we will, in §5, show results for diffraction from spherical crystals, which are also discussed erroneously in NTXRD.
A consideration of the geometry of the shape transform shown in Fig. 6 ▸ enables us to see why we observe a specular peak in intensity for this particular cubic shaped crystal, why this crystal also provides a peak in scattered intensity at an angle close to (but not exactly at) the Bragg angle when it is oriented away from the Bragg condition, and why in the general case NTXRD is incorrect.
Thus, contrary to the claims made within NTXRD, crystals with different shapes do not have a persistent peak at the Bragg condition when Ω differs from . Indeed, the effects discussed thus far were already well understood in the earliest days of X-ray diffraction, and the widths of the Bragg peaks have been (within the approximations of this simple model) understood for of the order of a century. The Scherrer equation (Scherrer, 1918 ▸; Patterson, 1939 ▸) relates the peak width (full width at half-maximum, FWHM), , to the crystallite dimension L for nano-scale particles ( µm): where K is the Scherrer constant, a function of crystal shape and which typically has a value of the order .
As well as calculating the diffracted intensity for crystallites rotated about an axis perpendicular to the plane containing the source and detector, results are also given within NTXRD for simultaneous rotations of the crystallites through angles χ about a second axis, perpendicular to the first – being parallel to the x axis and passing through the crystal, as shown in Fig. 1 ▸. We consider once more the cubic shaped crystal, initially set up for Bragg diffraction from (010). We calculate the intensity at any given scattering angle as a function of Ω and χ from equation (20).
The effects that the finite size of crystals has on X-ray diffraction have been discussed and considered since soon after the foundation of the field. Within NTXRD mistakes are made in summing the phases of scattered X-rays from a crystal with an orthorhombic shape, which lead to the incorrect conclusion that such crystals always have some peak in scattering at the Bragg condition. It is also claimed that this result holds for crystals of a general shape. As we have shown, these conclusions are in error, and the effects that the shape and finite size of crystals have on the diffraction pattern are well described by conventional diffraction theory.