Date Published: September 01, 2018
Publisher: International Union of Crystallography
Author(s): Paul F. Fewster.
In response to the comments by Fraser & Wark [(2018), Acta Cryst. A74, 447–456], experimental evidence and an explanation of the new theory in the context of a modified Ewald sphere construction are presented.
The new theory of X-ray diffraction arose from trying to account for inexplicable experimental observations. Neither the conventional dynamical nor kinematical theories could explain the measurements. The microstructure would have to be fantastical to account for some of these observations. Several experimental examples are included in this article that support the theoretical interpretation. My questioning of conventional theory started in the 1990s when using the near-perfect diffraction space probe (Fewster, 1989 ▸) to study polycrystalline materials and perfect semiconductors, with work on a different description beginning in the mid-2000s. It was clear that the observed features could no longer be dismissed as artefacts of the instrument, requiring an alternative explanation of experimental data.
The whole basis of the new theory is that a strong scattering feature, e.g. a Bragg peak, can still be observed as the crystal is rotated away from its position on the Ewald sphere. This applies to all the diffraction features, e.g. thickness fringes and crystal truncation rods, but will be weak. The distance of a diffraction feature from this ‘conventional’ Ewald sphere surface is given by the length of the arc of a vector (for the feature of interest) rotated about 000 (Fig. 1 ▸). The length of the vectors in the figure corresponds to 1/dhkl. The arcs touch the Ewald sphere at 2θhkl with a residual amplitude given by equation (4) of Fewster (2014 ▸). The next section explains why there is intensity at this position. Thus, a considerable proportion of the full diffraction pattern should be observed if there is sufficient intensity. This is exactly what we would expect from optical diffraction. Rotating the crystal just increases or decreases the intensity of the features in the diffraction pattern, e.g. Bragg peaks, thickness fringes, crystal truncation rods, fringes from spherical crystals etc., and when they coincide with the surface of the ‘conventional’ Ewald sphere the intensity for that feature reaches its maximum value. The ‘conventional’ Ewald sphere just represents the specular condition and has no width. The new theory just expresses that there is a residual specular contribution that does not go to zero as soon as the feature giving rise to it is rotated away from the optimum position on the sphere surface.
A series of diagrams (Fig. 3 ▸) is given that explains the thinking behind the new theory and the reasoning of Fraser & Wark to make it clear where their misunderstanding has occurred.
(i) The first example was an early test of my theory. The sample is a large, perfect crystal wafer of 111-oriented silicon. The incident beam (Cu K) is collimated to give an angular divergence of 0.03° and the crystal is set to several incident angles, Ω, either side of the 111 Bragg angle (θB). The scattering is captured by scanning in 2θ (Fig. 5 ▸a). Peaks are observed that correspond to the intersection of the crystal truncation rod at 2θ = 2Ω and further peaks at 2θ = 2θB for both the Cu Kα and Cu Kβ wavelengths for the d111 crystal planes. The 2θ = 2θB peaks are observed for incident angles up to 6° away from the Bragg condition.
The crystal shape will modify the intensity close to the Bragg peak, which was recognized by Fewster (2014 ▸) p. 262: ‘Hence a powder sample that has a distribution of orientations will create fringes associated with its size and surface shape and an enhancement at 2θB for each crystallite plane’. The main thrust of this theory is to concentrate on the persistent intensity at 2θB, whereas all shape effects will modify the intensity around the Bragg condition peak and will not form intensity at 2θB unless by chance. Equation (5) in Fewster (2014 ▸) can be considered as the formula for a crystal wafer with crystal planes parallel to the surface. For other crystal shapes, the full shape transform can be included, but the position of the Bragg condition is unchanged. To include the shape transform for a parallelepiped, as in the work of James (1962 ▸) and Authier (2001 ▸), for a small crystal, would involve extra terms in equation (5), i.e. of the form and . Since so few crystals conform to this shape I refer to my original statement above, i.e. any shape can be included but the persistent intensity at 2θB still exists.
Requiring crystals to be mosaic to suppress dynamical effects (Darwin, 1922 ▸) for the kinematical approximation to be applied in structure determination puts a big onus on all crystals. Is that reasonable? The number of crystals required to form a reliable polycrystalline diffraction pattern is greater than in a typical sample, in which case microdiffraction will not work; but it does, so what is going on? This did not go unnoticed by Alexander et al. (1948 ▸) who suggested crystals in a powder diffraction sample must be mosaic; but how small are they? De Wolff (1958 ▸) suggested that slack gearing in diffractometers may be the cause, but high-quality diffractometers of today would rule that out. Smith (1999 ▸) concluded that the data cannot be reliable even with the numbers of crystals used in Bragg–Brentano geometry. More recently, the data from XFELs show that there are reflections simultaneously observed in a snapshot from a single crystal, which should be a very rare event but is very common. This has led to a plethora of complex explanations to account for the data, e.g. Wojtas et al. (2017 ▸).
The new theory explains the experimental results. There is, as far as I know, no alternative explanation within the confines of conventional theory. Those who can understand my description as well as the conventional theory should be able to compare these two approaches and make a judgement on which best describes their data. The new theory could be considered as defining a thickness profile for the Ewald sphere surface. In conventional theory this surface has no thickness, placing all the experimental interpretation on changing the shape of the reciprocal-lattice point, e.g. mosaic crystals. Shape effects cannot explain the results described above and therefore the conventional theory can only be an approximation. I consider my theory to be a better description of X-ray diffraction. The criticisms of my theory by Fraser & Wark are therefore based on an invalid argument.