The Ewald and Darwin solutions for perfect crystals

Weiss, R. J.

doi:10.1107/s0365110x65001858

Cited by 26 publications

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“…The reason for this apparent discontinuity in is th at for zero absorption the precise treatm ent of the back face (even for a very thick crystal) becomes crucial. In particular, the assumption of a finite-thickness para llel-sided slab and zero absorption implies th a t the back face is always ' seen ' and leads to the Ewald solution, while the standard thick-crystal result, involving the assumption of finite but negligibly small absorption (implying th a t the back face is not 'seen' for sufficiently thick crystals), leads to the Darwin result (see also Zachariasen 1945 andWeiss 1964). The difference in behaviour of the reflectivity curves for the two cases is illustrated in figure 3, where it can be seen th a t the reflectivity curves for finite A 0 always possess Pendellosung fringes arising from interference between wave fields excited at the front and back face, while the 'thick-crystal' result does not (since it relates more correctly to a semi-infinite crystal).…”

Section: -Ray Bragg Reflexion From 'Perfect Crystals 581mentioning

confidence: 99%

X-ray Bragg reflexion from perfect crystals – dependence on geometrical factors with particular regard to the zero-level-of-interaction limit

Wilkins

1978

Proc. R. Soc. Lond. A

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The variation of X-ray Bragg reflexion properties of centro-symmetric perfect crystals (both absorbing and non-absorbing) with thickness and degree of asymmetry of the reflexion is explored systematically by direct numerical evaluation of the dynamical theory. In particular, it is shown that well-defined universal limits exist where the integrated reflectivity of a perfect crystal (i. e. dynamical theory) approaches asymptotically that for an ideally imperfect crystal (i. e. kinematical approximation) of the same material under the same diffraction conditions. That is, in these limits the level of extinction goes to zero with zero gradient when plotted against an appropriate parameter. The first case occurs when the crystal thickness tends to zero, while the second case occurs when the degree of asymmetry tends toward the asymmetric limits. In each case it is shown that the level of interaction , as indicated for example by the maximum reflectivity, also goes to zero. If absorption is small it is found that the integrated reflectivity for a finite crystal can exceed that for the corresponding semi-infinite crystal, a particular example being the difference between the Ewald and Darwin results which occur for zero and negligible absorption respectively. If absorption and anomalous dispersion are both large, then for some range of parameter values the integrated reflectivity for a perfect crystal can exceed that for an ideally mosaic crystal leading to the phenomenon of negative extinction . A cautionary message arising from the present investigations relates to the dubious practical value of theoretical results for variation of diffraction properties with asymmetry, when these are based on the assumption of zero absorption.

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Section: -Ray Bragg Reflexion From 'Perfect Crystals 581mentioning

confidence: 99%

X-ray Bragg reflexion from perfect crystals – dependence on geometrical factors with particular regard to the zero-level-of-interaction limit

Wilkins

1978

Proc. R. Soc. Lond. A

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Measurement of the Integrated X-Ray Intensities of Ge and Si

Jennings¹

1969

Journal of Applied Physics

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The integrated Bragg intensities of the (111) reflections of perfect crystal Ge and Si have been measured with a higher accuracy than heretofore. Ge was measured using x-ray wavelengths of 0.56 and 1.54 Å, and Si using 1.54 Å. Comparison of the scattering factors so obtained with those measured previously by the same method or by the Pendellösung fringe method leads us to the conclusion that there is not a systematic difference between the two methods, as has been suggested. We find that the reflectivity in the wings of the Bragg peak is much lower than the calculated value, and this effect would have to be taken into account if accuracies of the order of 0.1% are envisaged. We also measured Si (222) and found results in agreement with other reflection measurements, but not with recent transmission measurements.

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Dynamical Theory of X-Ray Diffraction

Authier

2003

392

658

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X-ray diffraction is a major tool for the study of crystal structures and the characterization of crystal perfection. Since the discovery of X-ray diffraction by von Laue, Friedrich, and Knipping in 1912 two basic theories have been used to describe this diffraction. One is the approximate geometrical, or kinematical theory, applicable to small or highly imperfect crystals; it is used for the determination of crystal structures and the study of powders and polycrystalline materials. The other one is the rigorous dynamical theory, applicable to perfect or nearly perfect crystals and, for that reason, is the one used for the assessment of the structural properties of high technology materials. It has witnessed exciting developments since the advent of synchrotron radiation. This book provides an account of the dynamical theory of diffraction and of its applications. The first part serves as an introduction to the subject, presenting early developments, Ewald's theory of dispersion and the basic results of Laue's dynamical theory. This is followed in the second part by a detailed development of the diffraction and propagation properties of X-rays in perfect crystals, including the study of anomalous absorption, Pendellösung, grazing incidence diffraction (GID) and n-beam or multiple-beam diffraction. The third part constitutes an extension of the theory to the case of slightly and highly deformed crystals. The last part gives three applications of the theory: X-ray optics for synchrotron radiation, location of atoms at surfaces and interfaces and X-ray diffraction topography.

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The Ewald and Darwin solutions for perfect crystals

Cited by 26 publications

References 1 publication

X-ray Bragg reflexion from perfect crystals – dependence on geometrical factors with particular regard to the zero-level-of-interaction limit

X-ray Bragg reflexion from perfect crystals – dependence on geometrical factors with particular regard to the zero-level-of-interaction limit

Measurement of the Integrated X-Ray Intensities of Ge and Si

Dynamical Theory of X-Ray Diffraction

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