Dynamical theory of diffraction

The dynamical theory of diffraction describes the interaction of wave

Wave

In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s with a regular lattice. The wave fields traditionally described are X-rays, neutron

Neutron radiation

Neutron radiation is a kind of ionizing radiation which consists of free neutrons. A result of nuclear fission or nuclear fusion, it consists of the release of free neutrons from atoms, and these free neutrons react with nuclei of other atoms to form new isotopes, which, in turn, may produce...

s 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 diffraction 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

Diffraction formalism

- Quantitative description and analysis :Because diffraction is the result of addition of all waves along all unobstructed paths, the usual procedure is to consider the contribution of an infinitesimally small neighborhood around a certain path and then integrate over all paths from the source...

 which describes the approximate position of Bragg or Laue diffraction

X-ray crystallography

X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

 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
  Refraction
  Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
  
   and specular reflection
  Specular reflection
  Specular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected into a single outgoing direction...
  
   of the waves at the interface to the crystal and delivers the refractive index
  Refractive index
  In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
  
   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
  Brillouin zone
  In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
  
   in reciprocal space. There is a gap between the dispersion surfaces in which no travelling waves are allowed. For a non-absorbing crystal, the reflection curve shows a range of total reflection, the so-called Darwin plateau. Regarding the quantum mechanical energy
  Energy
  In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
  
   of the system, this leads to the band gap
  Band gap
  In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
  
   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 surface of the crystal.

• Anomalous absorption effects take place due to a standing wave
  Standing wave
  In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
  
   patterns of two wave fields. Absorption is stronger if the standing wave has its anti-nodes on the lattice planes, i.e. where the absorbing atoms are, and weaker, if the anti-nodes are shifted between the planes. The standing wave shifts from one condition to the other on each side of the Darwin plateau which gives the latter an asymmetric shape.

Applications

• X-ray diffraction
• Neutron diffraction
  Neutron diffraction
  Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material: A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of...
  
  
• Electron diffraction
  Electron diffraction
  Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern...
  
   and transmission electron microscopy
  Transmission electron microscopy
  Transmission electron microscopy is a microscopy technique whereby a beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through...
  
  
• Structure determination in Crystallography
  Crystallography
  Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
  
  
• grazing incidence diffraction
  Grazing incidence diffraction
  Grazing incidence X-ray and neutron diffraction , typically from a crystalline structure uses small incident angles for the incoming X-ray or neutron beam, so that diffraction can be made surface sensitive. It is used to study surfaces and layers because wave penetration is limited. Distances are...
  
  
• X-ray standing waves
  X-ray standing waves
  - The X-ray standing wave technique :The X-ray standing wave technique can be used to study the structure of surfaces and interfaces with high spatial resolution and chemical selectivity. Pioneered by B.W...
  
  
• neutron and X-ray interferometry.
• synchrotron crystal optics
• neutron and X-ray diffraction topography
  Diffraction topography
  Diffraction topography is an X-ray imaging technique based on Bragg diffraction.Diffraction topographic images record the intensity profile of a beam of X-rays diffracted by a crystal....
  
  
• X-ray imaging
  Radiography
  Radiography is the use of X-rays to view a non-uniformly composed material such as the human body. By using the physical properties of the ray an image can be developed which displays areas of different density and composition....
  
  
• Crystal monochromator
  Crystal monochromator
  A crystal monochromator is a device in neutron and X-ray optics to select a defined wavelength of the radiation for further purpose on a dedicated instrument or beamline. It operates through the diffraction process according to Bragg's law....
  
  s
• Electronic band structure
  Electronic band structure
  In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
  
  s

Further reading

• J. Als-Nielsen, D. McMorrow: Elements of Modern X-ray physics. Wiley, 2001 (chapter 5: diffraction by perfect crystals).
• André Authier: Dynamical theory of X-ray diffraction. IUCr monographs on crystallography, no. 11. Oxford University Press (1st edition 2001/ 2nd edition 2003). ISBN 0-19-852892-2.
• R. W. James: The Optical Principles of the Diffraction of X-rays. Bell., 1948.
• M. von Laue: Röntgenstrahlinterferenzen. Akademische Verlagsanstalt, 1960 (German).
• Z. G. Pinsker: Dynamical Scattering of X-Rays in Crystals. Springer, 1978.
• B. E. Warren: X-ray diffraction. Addison-Wesley, 1969 (chapter 14: perfect crystal theory).
• W. H. Zachariasen: Theory of X-ray Diffraction in Crystals. Wiley, 1945.
• Boris W. Batterman, Henderson Cole: Dynamical Diffraction of X Rays by Perfect Crystals. Reviews of Modern Physics, Vol. 36, No. 3, 681-717, July 1964. (PDF 7.7 MB)
• H. Rauch, D. Petrascheck, “Grundlagen für ein Laue-Neutroneninterferometer Teil 1: Dynamische Beugung”. , AIAU 74405b, Atominstitut der Österreichischen Universitäten, (1976)
• H. Rauch, D. Petrascheck, “Dynamical neutron diffraction and its application” in “Neutron Diffraction”, H. Dachs, Editor. (1978), Springer-Verlag: Berlin Heidelberg New York. p. 303.
• K.-D. Liss: "Strukturelle Charakterisierung und Optimierung der Beugungseigenschaften von Si(1-x)Ge(x) Gradientenkristallen, die aus der Gasphase gezogen wurden", Dissertation, Rheinisch Westfälische Technische Hochschule Aachen, (27 October 1994), urn:nbn:de:hbz:82-opus-2227