International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 5.1, p. 548   | 1 | 2 |

Section 5.1.8.1. Introduction

A. Authiera*

aLaboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.8.1. Introduction

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The preceding sections have dealt with the diffraction of a plane wave by a semi-infinite perfect crystal. This situation is actually never encountered in practice, although with various devices, in particular using synchrotron radiation, it is possible to produce highly collimated monochromated waves which behave like pseudo plane waves. The wave from an X-ray tube is best represented by a spherical wave. The first experimental proof of this fact is due to Kato & Lang (1959)[link] in the transmission case. Kato extended the dynamical theory to spherical waves for non-absorbing (Kato, 1961a[link],b[link]) and absorbing crystals (Kato, 1968a[link],b[link]). He expanded the incident spherical wave into plane waves by a Fourier transform, applied plane-wave dynamical theory to each of these components and took the Fourier transform of the result again in order to obtain the final solution. Another method for treating the problem was used by Takagi (1962[link], 1969[link]), who solved the propagation equation in a medium where the lateral extension of the incident wave is limited and where the wave amplitudes depend on the lateral coordinates. He showed that in this case the set of fundamental linear equations (5.1.2.20)[link] should be replaced by a set of partial differential equations. This treatment can be applied equally well to a perfect or to an imperfect crystal. In the case of a perfect crystal, Takagi showed that these equations have an analytical solution that is identical with Kato's result. Uragami (1969[link], 1970[link]) observed the spherical wave in the Bragg (reflection) case, interpreting the observed intensity distribution using Takagi's theory. Saka et al. (1973[link]) subsequently extended Kato's theory to the Bragg case.

Without using any mathematical treatment, it is possible to make some elementary remarks by considering the fact that the divergence of the incident beam falling on the crystal from the source is much larger than the angular width of the reflection domain. Fig. 5.1.8.1(a)[link] shows a spatially collimated beam falling on a crystal in the transmission case and Fig. 5.1.8.1(b)[link] represents the corresponding situation in reciprocal space. Since the divergence of the incident beam is larger than the angular width of the dispersion surface, the plane waves of its Fourier expansion will excite every point of both branches of the dispersion surface. The propagation directions of the corresponding wavefields will cover the angular range between those of the incident and reflected beams (Fig. 5.1.8.1a[link]) and fill what is called the Borrmann triangle. The intensity distribution within this triangle has interesting properties, as described in the next two sections.

[Figure 5.1.8.1]

Figure 5.1.8.1 | top | pdf |

Borrmann triangle. When the incident beam is divergent, the whole dispersion surface is excited and the wavefields excited inside the crystal propagate within a triangle filling all the space between the incident direction, AC, and the reflected direction, AB. Along any direction Ap within this triangle two wavefields propagate, having as tie points two conjugate points, P and P′, at the extremities of a diameter of the dispersion surface. (a) Direct space; (b) reciprocal space.

References

Kato, N. (1961a). A theoretical study of Pendellösung fringes. Part I. General considerations. Acta Cryst. 14, 526–532.
Kato, N. (1961b). A theoretical study of Pendellösung fringes. Part 2. Detailed discussion based upon a spherical wave theory. Acta Cryst. 14, 627–636.
Kato, N. (1968a). Spherical-wave theory of dynamical X-ray diffraction for absorbing perfect crystals. I. The crystal wave fields. J. Appl. Phys. 39, 2225–2230.
Kato, N. (1968b). Spherical-wave theory of dynamical X-ray diffraction for absorbing perfect crystals. II. Integrated reflection power. J. Appl. Phys. 39, 2231–2237.
Kato, N. & Lang, A. R. (1959). A study of Pendellösung fringes in X-ray diffraction. Acta Cryst. 12, 787–794.
Saka, T., Katagawa, T. & Kato, N. (1973). The theory of X-ray crystal diffraction for finite polyhedral crystals. III. The Bragg–(Bragg)m cases. Acta Cryst. A29, 192–200.
Takagi, S. (1962). Dynamical theory of diffraction applicable to crystals with any kind of small distortion. Acta Cryst. 15, 1311–1312.
Takagi, S. (1969). A dynamical theory of diffraction for a distorted crystal. J. Phys. Soc. Jpn, 26, 1239–1253.
Uragami, T. (1969). Pendellösung fringes of X-rays in Bragg case. J. Phys. Soc. Jpn, 27, 147–154.
Uragami, T. (1970). Pendellösung fringes in a crystal of finite thickness. J. Phys. Soc. Jpn, 28, 1508–1527.








































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