Dynamical theory of diffraction
The dynamical theory of diffracton describes the interaction of wave fields with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice atomic crystal structures or nanometer scaled multi-layers or self arranged systems. In a wider sense, similar treatment is related to the interaction of light with optical band-gap materials or related wave problems in acoustics.
principle of theory
The dynamical theory of diffracton considers the wave field in the periodic potential of the crystal and takes into account all multiple scattering effects. Unlike the kinematic theory of diffraction which describes the approximate position of Bragg or Laue peaks in reciprocal space, dynamical theory corrects for refraction, shape and width of the peaks, extinction and interference effects. Graphical representations are described in dispersion surfaces around reciprocal lattice points which fulfill the boundary conditions at the crystal interface.
outcomes
- The crystal potential by itself leads to refraction and [specular reflection]] of the waves at the interface to the crystal and delivers the refractive index off the Bragg reflection. It also corrects for refraction at the Bragg condition and combined Bragg and specular reflection in grazing incidence geometries.
- A Bragg reflection is the splitting of the dispersion surface at the border of the Brillouin zone in reciprocal space. Regarding the quantum mechanical energy of the system, this leads to the band-gap structure which is commonly well known for electrons.
- Upon Laue diffraction, intensity is shuffled from the forward diffracted beam into the Bragg diffracted beam until extinction. The diffracted beam itself fulfills the Bragg condition and shuffles intensity back into the primary direction. This round-trip period is called the Pendellösung period.
- The extinction length is related to the Pendellösung period. Even if a crystal is infinitely thick, only the crystal volume within the extinction length contributes considerably to the diffraction in Bragg geometry.
- In Laue geometry, beam paths lie within the Borrmann triangle. Kato fringes are the intensity patterns due to Pendellösung effects at the exit surcace of the crystal.
applications
- neutron and X-ray interferometry.
- synchrotron crystal optics.
- X-ray standing waves.
- grazing incidence diffraction.
- neutron and X-ray topography.
- X-ray imaging.