In crystalline materials, Umklapp scattering (also U-process or Umklapp process) is a scattering process that results in a

wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength) ...

(usually written ''k'') which falls outside the first

Brillouin zone In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry whic ...

. If a material is periodic, it has a Brillouin zone, and any point outside the first Brillouin zone can also be expressed as a point inside the zone. So, the wave vector is then mathematically transformed to a point inside the first Brillouin zone. This transformation allows for scattering processes which would otherwise violate the

conservation of momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

: two wave vectors pointing to the right can combine to create a wave vector that points to the left. This non-conservation is why

crystal momentum In solid-state physics, crystal momentum or quasimomentum is a Momentum#Momentum in quantum mechanics, momentum-like Vector (geometric), vector associated with electrons in a Crystal structure, crystal lattice. It is defined by the associated Rec ...

is not a true momentum. Examples include electron-lattice potential scattering or an anharmonic

phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...

-phonon (or

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

-phonon)

scattering In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...

process, reflecting an electronic state or creating a phonon with a momentum ''k''-vector outside the first

. Umklapp scattering is one process limiting the

thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...

in crystalline materials, the others being phonon scattering on crystal defects and at the surface of the sample. The left panel of Figure 1 schematically shows the possible scattering processes of two incoming phonons with wave-vectors (''k''-vectors) ''k''₁ and ''k''₂ (red) creating one outgoing phonon with a wave vector ''k''₃ (blue). As long as the sum of ''k''₁ and ''k''₂ stay inside the first Brillouin zone (grey squares), ''k''₃ is the sum of the former two, thus conserving phonon momentum. This process is called normal scattering (N-process). With increasing phonon momentum and thus larger wave vectors ''k''₁ and ''k''₂, their sum might point outside the first Brillouin zone (''k'''₃). As shown in the right panel of Figure 1, ''k''-vectors outside the first Brillouin zone are physically equivalent to vectors inside it and can be mathematically transformed into each other by the addition of a reciprocal lattice vector ''G''. These processes are called Umklapp scattering and change the total phonon momentum. Umklapp scattering is the dominant process for

electrical resistivity Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...

at low temperatures for low defect crystals (as opposed to phonon-electron scattering, which dominates at high temperatures, and high-defect lattices which lead to scattering at any temperature.) Umklapp scattering is the dominant process for thermal resistivity at high temperatures for low defect crystals. The thermal conductivity for an insulating crystal where the U-processes are dominant has 1/T dependence.

History

The name derives from the German word ''umklappen'' (to turn over).

Rudolf Peierls Sir Rudolf Ernst Peierls, (; ; 5 June 1907 – 19 September 1995) was a German-born British physicist who played a major role in Tube Alloys, Britain's nuclear weapon programme, as well as the subsequent Manhattan Project, the combined Allied ...

, in his autobiography ''Bird of Passage'' states he was the originator of this phrase and coined it during his 1929 crystal lattice studies under the tutelage of

Wolfgang Pauli Wolfgang Ernst Pauli ( ; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...

. Peierls wrote, "…I used the German term ''Umklapp'' (flip-over) and this rather ugly word has remained in use…". The term Umklapp appears in the 1920 paper of Wilhelm Lenz's seed paper of the

Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...

References

{{reflist Scattering

History

See also

References