*ANDRÉ AUTHIER*

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198528920
- eISBN:
- 9780191713125
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528920.003.0012
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter describes the propagation of wavefields inside the crystal close to the Bragg angle. It shows that the direction of propagation of packets of wavefields as obtained by their group ...
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This chapter describes the propagation of wavefields inside the crystal close to the Bragg angle. It shows that the direction of propagation of packets of wavefields as obtained by their group velocity is identical to that of the Poynting vector. The geometrical properties of wavefields trajectories (ray tracing) within the Borrmann triangle are determined and the intensity distribution along the base of the Borrmann triangle is calculated. A detailed account of the experimental observation of the double refraction of the X-ray wavefields at the Bragg angle is given. The propagation of wavefields in finite crystals giving rise to partial reflections and interference effects is then described. The Bragg–Laue, Bragg–Bragg, and Laue–Bragg geometries are successively considered, and the formation of the Borrmann–Lehmann fringes in the latter case analyzed. In the last section, the coherence properties of X-ray sources are discussed.Less

This chapter describes the propagation of wavefields inside the crystal close to the Bragg angle. It shows that the direction of propagation of packets of wavefields as obtained by their group velocity is identical to that of the Poynting vector. The geometrical properties of wavefields trajectories (ray tracing) within the Borrmann triangle are determined and the intensity distribution along the base of the Borrmann triangle is calculated. A detailed account of the experimental observation of the double refraction of the X-ray wavefields at the Bragg angle is given. The propagation of wavefields in finite crystals giving rise to partial reflections and interference effects is then described. The Bragg–Laue, Bragg–Bragg, and Laue–Bragg geometries are successively considered, and the formation of the Borrmann–Lehmann fringes in the latter case analyzed. In the last section, the coherence properties of X-ray sources are discussed.

*ANDRÉ AUTHIER*

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198528920
- eISBN:
- 9780191713125
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528920.003.0010
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The ...
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This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The transmission and reflection geometries are handled separately. Two methods of integration are given — direct integration and integration by the stationary phase method. The amplitude and intensity distributions of the reflected and refracted waves on the exit surface are calculated. It is shown that equal-intensity fringes are formed within the Borrmann triangle (Pendellösung fringes) that can be interpreted as due to interferences between the waves associated with the two branches of the dispersion surface. The integrated intensity is calculated and the influence of the polarization of the incident wave discussed. The last section describes the diffraction of ultra-short pulses of plane-wave X-rays such as those emitted by a free-electron laser and which can be handled by considering their Fourier expansion in frequency space.Less

This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The transmission and reflection geometries are handled separately. Two methods of integration are given — direct integration and integration by the stationary phase method. The amplitude and intensity distributions of the reflected and refracted waves on the exit surface are calculated. It is shown that equal-intensity fringes are formed within the Borrmann triangle (*Pendellösung* fringes) that can be interpreted as due to interferences between the waves associated with the two branches of the dispersion surface. The integrated intensity is calculated and the influence of the polarization of the incident wave discussed. The last section describes the diffraction of ultra-short pulses of plane-wave X-rays such as those emitted by a free-electron laser and which can be handled by considering their Fourier expansion in frequency space.

*ANDRÉ AUTHIER*

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198528920
- eISBN:
- 9780191713125
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528920.003.0014
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are ...
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This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are extended so as to apply to highly deformed crystals. Their resolution is the discussed and the principle of their numerical integration in an inverted Borrmann triangle given. The ray concept is generalized to the case of strong deformations by noting that new wavefields are generated in the highly strained regions; this is known as the interbranch scattering effect. The last part of the chapter is devoted to an account of the statistical dynamical theories for highly imperfect crystals, with emphasis on Kato's statistical theories. Examples of experimental test of the dynamical theory are also given.Less

This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are extended so as to apply to highly deformed crystals. Their resolution is the discussed and the principle of their numerical integration in an inverted Borrmann triangle given. The ray concept is generalized to the case of strong deformations by noting that new wavefields are generated in the highly strained regions; this is known as the interbranch scattering effect. The last part of the chapter is devoted to an account of the statistical dynamical theories for highly imperfect crystals, with emphasis on Kato's statistical theories. Examples of experimental test of the dynamical theory are also given.

*Helmut Rauch and Samuel A. Werner*

- Published in print:
- 2015
- Published Online:
- March 2015
- ISBN:
- 9780198712510
- eISBN:
- 9780191780813
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198712510.003.0011
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The perfect crystal, LLL-geometry, neutron interferometer is geometrically analogous to the classical Mach–Zehnder interferometer. Its operation depends in exquisite detail on the dynamical theory of ...
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The perfect crystal, LLL-geometry, neutron interferometer is geometrically analogous to the classical Mach–Zehnder interferometer. Its operation depends in exquisite detail on the dynamical theory of diffraction in a perfect crystal. Although understanding the basic ideas of most of the experiments discussed in this book does not depend on these details, actually carrying out experiments does. This chapter is devoted to a dynamical diffraction calculation of the operation of a three-crystal LLL interferometer. A description of the importance of the Pendellösung interference fringes is discussed. The spatial profiles of the beams traversing and exiting the interferometer are calculated and graphically displayed. The original Pendellösung interference experiments of Shull are discussed. The multiple reflection process of neutrons within each crystal plate is discussed and calculated using the Takagi–Taupin equations, showing how the spatial width of the beams increases upon traversing each crystal blade.Less

The perfect crystal, LLL-geometry, neutron interferometer is geometrically analogous to the classical Mach–Zehnder interferometer. Its operation depends in exquisite detail on the dynamical theory of diffraction in a perfect crystal. Although understanding the basic ideas of most of the experiments discussed in this book does not depend on these details, actually carrying out experiments does. This chapter is devoted to a dynamical diffraction calculation of the operation of a three-crystal LLL interferometer. A description of the importance of the Pendellösung interference fringes is discussed. The spatial profiles of the beams traversing and exiting the interferometer are calculated and graphically displayed. The original Pendellösung interference experiments of Shull are discussed. The multiple reflection process of neutrons within each crystal plate is discussed and calculated using the Takagi–Taupin equations, showing how the spatial width of the beams increases upon traversing each crystal blade.