Extinction in X-ray and neutron diffraction

Abstract

Accurate neutron-diffraction measurements from crystals suffering from severe extinction have been used to test the recent general theory of extinction of Zachariasen (Acta Cryst. (1967). 23, 558). Analysis of these measurements indicates that certain of the approximations made in the theory are not generally valid and result in systematic deviations between theory and experiment, namely a marked angle-dependent effect and an inadequacy of the theory for strong extinction. The original theory is therefore extended to take these factors into account and to give agreement with the observed data. IntroductionTheoretical formulae for the Bragg intensities of diffracted X-rays or neutrons have been derived rigorously only in the limiting cases of an ideally perfect crystal (the dynamical theory) and an ideally imperfect crystal (the kinematical theory). In general a given crystal will lie somewhere between these two extreme cases and modification of the kinematical theory is necessary to take into account the degree of perfection of the crystal. The treatment is normally based on the mosaic model for which the crystal is assumed to consist of a number of small perfect crystal domains, each slightly misoriented with respect to its neighbours. Zachariasen (1967) has recently described a general theory of X-ray diffraction in crystals, based on an approximate treatment of the coupling between incident and diffracted beams. In this theory he derives a general formula for the intensity diffracted by a finite perfect crystal and hence the intensity diffracted by a finite mosaic crystal. Because of the complexity of the problem a number of approximations are introduced and the ultimate test of this theory is therefore, as Zachariasen states, a test of its agreement with experiment. Zachariasen himself has carried out a number of experimental tests using X-ray diffraction data (see Zachariasen, 1968aZachariasen, , b, c, 1969 and has obtained excellent agreement for the crystals studied: lithium fluoride, quartz, phenakite, hambergite and calcium fluoride. In addition, other experimental X-ray diffraction tests have provided good agreement with the theory (see e.g. Chandrasekhar, Ramaseshan & Singh, 1969).The Zachariasen formulae have also been applied to accurate neutron diffraction measurements on several crystals. These include barium fluoride (Cooper, Rouse & Willis, 1968) and strontium fluoride and calcium fluoride (Cooper & Rouse, 1970). For barium fluoride only limited strongly extinguished data were collected but the theory held up to a level of extinction of about 16 per cent in intensity (at 2= 1-038 A.). For calcium fluoride complete two-dimensional hkk data were collected initially at a wavelength of 0-877 A and a preliminary analysis gave a value of r* [see equations (16)] comparable to that obtained by Zachariasen (1968b)" However, once again the theory did not correct the strongest intensities adequately and so a more extensive study was undertaken. Complete two-dimensional neu-M. J. COOPER AND K. D. ROUSE ...

Inhomogeneous mosaicity in extinction correction

“…By giving appropriate expressions to /3, this formulation is compatible with those of Darwin (1922), Zachariasen (1963Zachariasen ( , 1967 type I and type II crystals as well as with the corrected expression for type I crystals given by Becker & Coppens (1974). It is, however, not compatible with the more elaborate formulae of Cooper & Rouse (1970) or Becker & Coppens (1974) which contain a more complex expression in the denominator.…”

Extinction correction in white X-ray and neutron diffraction

“…The extinction theories given by Zachariasen (1967) and Cooper & Rouse (1970) are applicable without any modification in white-beam diffraction, since the expression of x as a function of F z, 2, sin 20 and other quantities is invariant for the two diffraction methods.…”

Extinction, polarization and crystal monochromators

“…Bonnet et al (1976) discuss this matter within the framework of the Becker & Coppens theory, but we believe this discussion to be erroneous as will be detailed below. Cooper & Rouse (1970 (C&R) make a good fit to various moderately extinguished neutron diffraction data and would be expected to apply to graphite in an appropriate range. The formula of Olekhnovich et al (1975) was not meant to apply near y± _~ 1, and the graph is only indicative, as discussed in the text.…”

“…Cooper & Rouse, 1970Sakata, Cooper, Rouse & Willis, 1978). These workers have developed expressions which meet the requirements of (29) The right-hand and upper horizontal hatched region shows that the deviation from ideally mosaic behavior is even more pronounced for transmission geometry with respect to primary and/or secondary extinction.…”