Brillouin zone - Revision history

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-09T16:53:08Z

Tidied translations and added German and Spanish (U. Mueller)

← Older revision		Revision as of 16:53, 9 November 2017
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	<~~Font~~ color=blue>~~Zones~~ de Brillouin </Font>(''Fr''). <~~Font~~ color=red>Brillouin ~~Zonen~~ </Font>(''Ge''). <~~Font~~ color=~~green~~> ~~Zonas de~~ Brillouin </~~Font~~>(''Sp''). <~~Font~~ color=~~"black"~~>~~Zone di Brillouin~~ </~~Font~~>(''It''). <~~Font~~ color=~~purple~~> ~~ブリュアンゾーン~~ </~~Font~~>(''Ja'').		<font color=blue>Zone de Brillouin</Font> (''Fr''). <font color=red>Brillouin-Zone</Font> (''Ge''). <font color="black">Zona di Brillouin</font> (''It''). <font color=purple>ブリュアンゾーン</font> (''Ja''). <font color=green>Zona de Brillouin</font> (''Sp'').

	== Definition ==		== Definition ==

BrianMcMahon: moved Brillouin zones to Brillouin zone: Adopting singular for all pages

2017-11-09T16:50:36Z

moved Brillouin zones to Brillouin zone: Adopting singular for all pages

← Older revision		Revision as of 16:50, 9 November 2017
(No difference)

BrianMcMahon at 09:49, 13 May 2017

2017-05-13T09:49:21Z

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	*[[Domain of influence]]		*[[Domain of influence]]
	*Sayre, D. (1952). ''Acta Cryst.'' '''5''', 60-65. ''The squaring method: a new method for phase determination.''		*Sayre, D. (1952). ''Acta Cryst.'' '''5''', 60-65. ''The squaring method: a new method for phase determination.''
	*Chapter 2.2 of ''International Tables of Crystallography, Volume D''<br>		*Chapter 2.2 of ''International Tables for Crystallography, Volume D''<br>

	[[Category:Fundamental crystallography]]<br>		[[Category:Fundamental crystallography]]<br>

BrianMcMahon: Edits to align with printed edition

2017-05-13T09:48:58Z

Edits to align with printed edition

← Older revision		Revision as of 09:48, 13 May 2017
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	<Font color=blue> Zones de Brillouin </Font>(''Fr''). <Font color=red> Brillouin Zonen </Font>(''Ge''). <Font color=green> Zonas de Brillouin </Font>(''Sp'').<Font color="black"> Zone di Brillouin </Font>(''It''). <Font color=purple> ブリュアンゾーン </Font>(''Ja'')		<Font color=blue>Zones de Brillouin </Font>(''Fr''). <Font color=red>Brillouin Zonen </Font>(''Ge''). <Font color=green> Zonas de Brillouin </Font>(''Sp''). <Font color="black">Zone di Brillouin </Font>(''It''). <Font color=purple> ブリュアンゾーン </Font>(''Ja'').

	== Definition ==		== Definition ==

	A ''Brillouin ~~Zone~~'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[Wigner-Seitz cell]] (also called ''Dirichlet'' or ''Voronoi ~~Domain~~'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (~~See~~ figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap		A ''Brillouin zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[Wigner-Seitz cell]] (also called ''Dirichlet'' or ''Voronoi domain of influence'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (see figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap with copies of such a Brillouin zone.
	with copies of such a Brillouin ~~Zone~~.

	For high-symmetry lattices one introduces sometimes the notion of ''n-th Brillouin Zone''. This is the set of points one reaches with a straight line from the origin and passing through ''n''-1 Bragg ~~Planes~~. In this terminology, the Brillouin ~~Zone~~ defined above is the first Brillouin ~~Zone~~. The ''n''-th Brillouin ~~Zone~~ is a shell around lower Brillouin ~~Zones~~ and		For high-symmetry lattices one introduces sometimes the notion of ''n-th Brillouin Zone''. This is the set of points one reaches with a straight line from the origin and passing through ''n''-1 Bragg planes. In this terminology, the Brillouin zone defined above is the first Brillouin zone. The ''n''-th Brillouin zone is a shell around lower Brillouin zones and its shape becomes for higher values of ''n'' rapidly rather complicated (see figure).
	its shape becomes for higher values of ''n'' rapidly rather complicated (~~See~~ figure).

	~~Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonon or electron states. States are non-equivalent if~~
	~~they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin Zone. Especially, for low-symmetry systems Brillouin Zones~~
	~~are sometimes difficult to visualize, and another choice of unit cell may be useful, e.g. the parallelepiped spanned by the basis vectors.~~

	~~The Brillouin Zone boundary consists of pieces of Bragg planes. Since two points on the boundary may differ by a reciprocal lattice vector, only a part of the boundary~~
	~~may be used for characterization of states. There is no simple rule for that choice. Dispersion curves at the zone boundary have either a maximum or minimum there~~
	~~or they merge with other curves giving rise to a change of degeneracy~~.

			Vectors in the Brillouin zone or on its boundary characterize states in a system with lattice periodicity, ''e.g.'' phonon or electron states. States are non-equivalent if they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin zone. Especially, for low-symmetry systems Brillouin zones are sometimes difficult to visualize, and another choice of unit cell may be useful, ''e.g.'' the parallelepiped spanned by the basis vectors.

			The Brillouin zone boundary consists of pieces of Bragg planes. Since two points on the boundary may differ by a reciprocal lattice vector, only a part of the boundary may be used for characterization of states. There is no simple rule for that choice. Dispersion curves at the zone boundary have either a maximum or minimum there or they merge with other curves giving rise to a change of degeneracy.

	[[Image:BraggPl.gif\|frame\|Bragg plane perpendicular to and going through the midpoint of the line connecting the origin and ''K''. Particles in the crystal potential are elastically scattered from wave vector ''k'' to wave vector ''k'''=''k''-''K''. The tips are in the Bragg plane.]]		[[Image:BraggPl.gif\|frame\|Bragg plane perpendicular to and going through the midpoint of the line connecting the origin and ''K''. Particles in the crystal potential are elastically scattered from wave vector ''k'' to wave vector ''k'''=''k''-''K''. The tips are in the Bragg plane.]]
	[[Image:NBZ2.001.gif\|frame\|Up to the 3rd BZ for a 2D square lattice; red lines are Bragg planes]]		[[Image:NBZ2.001.gif\|frame\|Up to the 3rd Brillouin zone for a two-dimensional square lattice; red lines are Bragg planes.]]

	== See also ==		== See also ==

	~~Section~~ 2.2 of ''International Tables of Crystallography, Volume D''<br>		*[[Domain of influence]]
			*Sayre, D. (1952). ''Acta Cryst.'' '''5''', 60-65. ''The squaring method: a new method for phase determination.''
			*Chapter 2.2 of ''International Tables of Crystallography, Volume D''<br>

	[[Category:Fundamental crystallography]]<br>		[[Category:Fundamental crystallography]]<br>

MassimoNespolo: /* Definition */ double parentehses removed

2015-03-17T13:06:22Z

Definition: double parentehses removed

← Older revision		Revision as of 13:06, 17 March 2015
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	== Definition ==		== Definition ==

	A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[[[Wigner-Seitz cell]]]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (See figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap		A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[Wigner-Seitz cell]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (See figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap
	with copies of such a Brillouin Zone.		with copies of such a Brillouin Zone.

BrianMcMahon at 17:08, 3 February 2012

2012-02-03T17:08:44Z

← Older revision		Revision as of 17:08, 3 February 2012
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	<Font color=blue> Zones de Brillouin </Font>(''Fr''). <Font color=red> Brillouin Zonen </Font>(''Ge''). <Font color=green> Zonas de Brillouin </Font>(''Sp'').<Font color="black"> Zone di Brillouin </Font>(''It''). <Font color=purple> ブリュアンゾーン </Font>(''Ja'')		<Font color=blue> Zones de Brillouin </Font>(''Fr''). <Font color=red> Brillouin Zonen </Font>(''Ge''). <Font color=green> Zonas de Brillouin </Font>(''Sp'').<Font color="black"> Zone di Brillouin </Font>(''It''). <Font color=purple> ブリュアンゾーン </Font>(''Ja'')


	== Definition ==		== Definition ==


	A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[[[Wigner-Seitz cell]]]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (See figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap		A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[[[Wigner-Seitz cell]]]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (See figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap
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	Section 2.2 of ''International Tables of Crystallography, Volume D''<br>		Section 2.2 of ''International Tables of Crystallography, Volume D''<br>

	~~----~~

	[[Category:Fundamental crystallography]]<br>		[[Category:Fundamental crystallography]]<br>

TedJanssen: /* Definition */

2011-07-27T11:46:47Z

Definition

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	[[Image:BraggPl.gif\|frame\|Bragg plane perpendicular to and going through the midpoint of the line connecting the origin and K]]		[[Image:BraggPl.gif\|frame\|Bragg plane perpendicular to and going through the midpoint of the line connecting the origin and ''K''. Particles in the crystal potential are elastically scattered from wave vector ''k'' to wave vector ''k'''=''k''-''K''. The tips are in the Bragg plane.]]
	[[Image:NBZ2.001.gif\|frame\|up to the 3rd BZ for a 2D square lattice; red lines are Bragg planes]]		[[Image:NBZ2.001.gif\|frame\|Up to the 3rd BZ for a 2D square lattice; red lines are Bragg planes]]

	== See also ==		== See also ==

TedJanssen: /* Definition */

2011-07-27T10:49:50Z

Definition

← Older revision		Revision as of 10:49, 27 July 2011
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	A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[[[Wigner-Seitz cell]]]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint. Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap		A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[[[Wigner-Seitz cell]]]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint (See figure). Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap
	with copies of such a Brillouin Zone.		with copies of such a Brillouin Zone.

	For high-symmetry lattices one introduces sometimes the notion of ''n-th Brillouin Zone''. This is the set of points one reaches with a straight line from the origin and passing through ''n''-1 Bragg Planes. In this terminology, the Brillouin Zone defined above is the first Brillouin Zone. The ''n''-th Brillouin Zone is a shell around lower Brillouin Zones and		For high-symmetry lattices one introduces sometimes the notion of ''n-th Brillouin Zone''. This is the set of points one reaches with a straight line from the origin and passing through ''n''-1 Bragg Planes. In this terminology, the Brillouin Zone defined above is the first Brillouin Zone. The ''n''-th Brillouin Zone is a shell around lower Brillouin Zones and
	its shape becomes for higher values of ''n'' rapidly rather complicated.		its shape becomes for higher values of ''n'' rapidly rather complicated (See figure).

	Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonon or electron states. States are non-equivalent if		Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonon or electron states. States are non-equivalent if
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	~~<center>~~
			[[Image:BraggPl.gif\|frame\|Bragg plane perpendicular to and going through the midpoint of the line connecting the origin and K]]
	[[Image:NBZ2.001.gif\|frame\|up to the 3rd BZ for a 2D square lattice; red lines are Bragg planes]]		[[Image:NBZ2.001.gif\|frame\|up to the 3rd BZ for a 2D square lattice; red lines are Bragg planes]]
	~~</center>~~

	== See also ==		== See also ==

TedJanssen: /* Definition */

2011-07-27T10:00:45Z

Definition

← Older revision		Revision as of 10:00, 27 July 2011
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	A Brillouin Zone is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[Wigner-Seitz cell]] (also called Dirichlet or Voronoi Domain) of the reciprocal lattice. It is constructed as the set of points enclosed by the Bragg planes, the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint. Alternatively, it is the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap		A ''Brillouin Zone'' is a particular choice of the unit cell of the reciprocal lattice. It is defined as the [[[[Wigner-Seitz cell]]]] (also called ''Dirichlet'' or ''Voronoi Domain'') of the reciprocal lattice. It is constructed as the set of points enclosed by the ''Bragg planes'', the planes perpendicular to a connection line from the origin to each lattice point and passing through the midpoint. Alternatively, it is defined as the set of points closer to the origin than to any other reciprocal lattice point. The whole reciprocal space may be covered without overlap
	with copies of such a Brillouin Zone.		with copies of such a Brillouin Zone.

	For high-symmetry lattices one introduces sometimes the notion of n-th Brillouin Zone. This is the set of points one reaches with a straight line from the origin and passing through n-1 Bragg Planes. In this terminology, the Brillouin Zone defined above is the first Brillouin Zone. The n-th Brillouin Zone is a shell around lower Brillouin Zones and		For high-symmetry lattices one introduces sometimes the notion of ''n-th Brillouin Zone''. This is the set of points one reaches with a straight line from the origin and passing through ''n''-1 Bragg Planes. In this terminology, the Brillouin Zone defined above is the first Brillouin Zone. The ''n''-th Brillouin Zone is a shell around lower Brillouin Zones and
	~~their~~ shape becomes ~~fast~~ rather complicated.		its shape becomes for higher values of ''n'' rapidly rather complicated.

	Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. ~~phonons~~ or ~~electrons~~. States are non-equivalent if		Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonon or electron states. States are non-equivalent if
	they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin Zone. Especially, for low-symmetry systems Brillouin Zones		they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin Zone. Especially, for low-symmetry systems Brillouin Zones
	are sometimes difficult to visualize, and another choice of unit cell may be useful, e.g. the parallelepiped spanned by the basis vectors.		are sometimes difficult to visualize, and another choice of unit cell may be useful, e.g. the parallelepiped spanned by the basis vectors.

			The Brillouin Zone boundary consists of pieces of Bragg planes. Since two points on the boundary may differ by a reciprocal lattice vector, only a part of the boundary
			may be used for characterization of states. There is no simple rule for that choice. Dispersion curves at the zone boundary have either a maximum or minimum there
			or they merge with other curves giving rise to a change of degeneracy.


	<center>		<center>
			[[Image:NBZ2.001.gif\|frame\|up to the 3rd BZ for a 2D square lattice; red lines are Bragg planes]]
	</center>		</center>

TedJanssen: /* Definition */

2011-07-27T09:30:14Z

Definition

← Older revision		Revision as of 09:30, 27 July 2011
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	Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonons or electrons. States are non-equivalent if		Vectors in the Brillouin Zone or on its boundary characterize states in a system with lattice periodicity, e.g. phonons or electrons. States are non-equivalent if
	they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin Zone. Especially, for low-symmetry systems Brillouin Zones		they belong to different vectors in a unit cell of the reciprocal lattice. This is not necessarily the Brillouin Zone. Especially, for low-symmetry systems Brillouin Zones
	are sometimes difficult to visualize, and another choice of unit cell may be useful, e.g. the parallelepiped spanned b the basis vectors.		are sometimes difficult to visualize, and another choice of unit cell may be useful, e.g. the parallelepiped spanned by the basis vectors.