Dynamical theory of diffraction

The dynamical theory of diffraction describes the interaction of wave

Wave

A wave is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. There also exist waves capable of traveling through a vacuum,...

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

Neutron radiation

Neutron radiation is a kind of non-ionizing radiation which consists of free neutrons.-Sources:Neutrons may be emitted during either spontaneous or induced nuclear fission, nuclear fusion processes, very high energy reactions such as in the Spallation Neutron Source and in cosmic ray interactions,...

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.



The dynamical theory of diffraction considers the wave field in the periodic potential of the crystal and takes into account all multiple scattering effects.

The dynamical theory of diffraction describes the interaction of wave

Wave

A wave is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. There also exist waves capable of traveling through a vacuum,...

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

Neutron radiation

Neutron radiation is a kind of non-ionizing radiation which consists of free neutrons.-Sources:Neutrons may be emitted during either spontaneous or induced nuclear fission, nuclear fusion processes, very high energy reactions such as in the Spallation Neutron Source and in cosmic ray interactions,...

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

Bragg diffraction

Bragg diffraction was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays...

 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 diffracts into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture...

 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 velocity. This is most commonly observed when a wave passes from one medium to another...
  
   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
  The refractive index of a medium is a measure of how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index close to 1.5, which means that in glass, light travels at 1 / 1.5 = 2/3 the speed of light in a vacuum...
  
   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 of 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 a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
  
   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 exist...
  
   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.

• Anomaleous absorption effects take place due to a standing wave
  Standing wave
  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 in opposite...
  
   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 is a method for the determination of the atomic and/or magnetic structure of a material. It can be equally well applied to study crystalline solids , gasses, liquids or amorphous materials. Neutron diffraction is a form of elastic scattering where the neutrons exiting the...
  
  
• Electron diffraction
  Electron diffraction
  Electron diffraction is a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern. This phenomenon occurs due to wave-particle duality, which states that a particle of matter can be described as a wave. For this reason, an electron can be...
  
   and transmission electron microscopy
• Structure determination in Crystallography
  Crystallography
  Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals...
  
  
• 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 the property of X-rays to cross materials to view inside objects. The impact on society of this technique has also been immense : application fields are medical, non-destructive testing, food inspection, security, archeology, ......
  
  
• 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 ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice...
  
  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