*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/9780198528920.003.0011
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations ...
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This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations are generalized as a set of partial differential equations (Takagi's equations). Their solutions for an incident spherical wave are first obtained by the method of integral equations for both the transmission and reflection geometries. The hyperbolic nature of Takagi's equations is shown and their solution derived using the method of Riemann functions for a point source located on the entrance surface or away from the incident surface. An appendix describes the properties of hyperbolic partial differential equations.Less

This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations are generalized as a set of partial differential equations (Takagi's equations). Their solutions for an incident spherical wave are first obtained by the method of integral equations for both the transmission and reflection geometries. The hyperbolic nature of Takagi's equations is shown and their solution derived using the method of Riemann functions for a point source located on the entrance surface or away from the incident surface. An appendix describes the properties of hyperbolic partial differential equations.

*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This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid.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.