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Topic: Brillouin zone

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	Fermi Surfaces and Brillouin Zones
		Brillouin zones are an important characteristic of crystal structures.
		The construction and illustration of Brillouin zones for a three dimensional lattice are somewhat difficult to follow.
		The first Brillouin zone for a point in a lattice is the set of points that are closer to the point than the Bragg "plane" of any point.
	www.sjsu.edu /faculty/watkins/brillouin.htm (411 words)

	Brillouin zone (Site not responding. Last check: 2007-10-08)
		Its importance is that every momentum allowed for the electron s in a crystal is included in a Brillouin zone.
		In general, the n -th Brillouin zone consist of the set of points that can be reached from the origin by crossing n'' − 1 Bragg planes.
		Zone 3 Hobart, Tasmania The host for the 2003 Nationals Tournament.
	www.serebella.com /encyclopedia/article-Brillouin_zone.html (397 words)

	DefaultSample.nb
		The most important fact for high order Brillouin zones is that their shape approaches that of a thin spherical shell and their volume is a constant.
		Figure 7 shows the (outside of) 15th Brillouin zone for the simple cubic, Figure 8 shows the 18th for the face-centered cubic and Figure 9 shows the 10th for the body-centered cubic lattice.
		Figure 10 shows the area of the Brillouin zones normalized to the area of a sphere of the same volume.
	www.wolfram.com /products/publicon/samples/default/DefaultSample.html (1535 words)

	Brillouin zone -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Taking the surfaces at the same distance from one element of the lattice and its neighbours, the (The magnitude of sound (usually in a specified direction)) volume included is the first Brillouin zone.
		Its importance is that every momentum allowed for the (An elementary particle with negative charge) electrons in a crystal is included in a Brillouin zone.
		The concept of a Brillouin zone was developed by (Click link for more info and facts about Leon Brillouin) Leon Brillouin, a French physicist.
	www.absoluteastronomy.com /encyclopedia/b/br/brillouin_zone.htm (212 words)

	Density of States dsn (Site not responding. Last check: 2007-10-08)
		The bcc structure has a zone bounded by planes perpendicular to (1 1 0) vectors, whereas the fcc structure has a zone bounded by planes perpendicular to both (1 1 1) and (2 0 0) vectors.
		Brillouin zones containing one million cubes are defined for the sc, fcc, and bcc structures, and are scanned to generate a histogram with one hundred energy divisions.
		One can "see" where an expanding sphere first touches the zone boundary (at the expected 1/3 of the maximum energy for sc), and the fact that the sphere touches at a lower energy for fcc than for bcc is used to explain phase transitions in alloys.
	www.kw.igs.net /~jackord/bp/ds.html (370 words)

	MATTER Glossary (Site not responding. Last check: 2007-10-08)
		A Brillouin zone is a property of a crystal.
		Wave vectors lying within the zone are in the same energy band: there is a jump in energy before the state given by the shortest vector in the next Brillouin zone.
		The Fermi sphere is usually considered to have the same origin as the zones and is thought of as filling the zone.
	www.matter.org.uk /glossary/detail.asp?dbid=47 (89 words)

	Bloch wave - Wikipedia, the free encyclopedia
		The plane wave wavevector (or Bloch wavevector) k (multiplied by Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone.
		The band structure is the collection of energy eigenstates within the first Brillouin zone.
		All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation.
	en.wikipedia.org /wiki/Bloch_wave (568 words)

	The Wigner-Seitz Method - Electronic States (Site not responding. Last check: 2007-10-08)
		And because of this the Wigner-Seitz cell in reciprocal space is the Brillouin zone of the crystal represented by this particular reciprocal lattice.
		The number of quantum (wave) states in the Brillouin zone is equal to the number of unit cells in the crystal.
		The nth Brillouin zone is the zone bounded by the set of points that can be reached from the origin by crossing n-1 bisecting planes.
	www.chembio.uoguelph.ca /educmat/chm729/wscells/electronic.htm (405 words)

	Choice of the k-point mesh
		This essentially means that the sampling k-points are distributed homogeneously in the Brillouin zone, with rows or columns of k-points running parallel to the reciprocal lattice vectors that span the Brillouin zone.
		The entire Brillouin zone is tiled by small polyhedra of the same shape as the Brillouin zone itself.
		To perform the integrals in the Brillouin zone , it is sufficient to sample the contribution from a subset of non-symmetry-equivalent k-points only.
	www.fhi-berlin.mpg.de /th/fhi98md/doc/main/node15.html (1547 words)

	ILL : ANNUAL REPORT 2000
		Figure 2: Schematic visualisation of the folding of the Brillouin zone in a superlattice on the spin wave dispersion curves.
		In bulk the Brillouin zone is determined by the periodicity of the lattice, i.e.
		This superlattice period L, which is the sum of the individual Dy and Y layer thicknesses, is expected to induce a folding of the Brillouin zone like in the case of phonons in superlattices [10] or graphite intercalated compounds [11].
	www.ill.fr /AR-00/p-26.htm (1310 words)

	7 The Brillouin Condition
		This result is called the Brillouin condition and shows that diffraction occurs about planes which are perpendicular bisectors of reciprocal lattice points.
		The smallest such volume is the 'first Brillouin zone', the second smallest is the 'second Brillouin zone' and so on.
		Figure 17: The first Brillouin zone for a simple cubic lattice lattice in 2 dimensions; in 3 dimensions the first Brillouin zone is a cube.
	www.chm.bris.ac.uk /webprojects2003/cook/brillouincondition.htm (194 words)

	Fermi Surfaces
		It is inconvenient to have sections of the Fermi surface that belong to the same Brillouin zone appear detached one from another.
		the fl square shown is the first Brillouin zone, the blue circle is the surface of constant energy for free electrons, and the shaded area represents occupied electron states.
		the fl square shown is the first Brillouin zone, the blue lines are the Fermi surfaces for free electrons on the fourth zone, and the shaded area represents occupied electron states.
	phycomp.technion.ac.il /~nika/fermi_surfaces.html (536 words)

	Energy band formation
		The band structure in figure 5 is plotted in several Brillouin zones.
		We can transform figure 5 into a first Brillouin zone plot by adding reciprocal lattice vectors until we get the proper format as in figure 6.
		The warped sphere is located at the center of the first Brillouin zone.
	www.mtmi.vu.lt /pfk/funkc_dariniai/quant_mech/bands.htm (1339 words)

	1. Crystal structures
		A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice.
		The allowed energy levels within the first Brillouin zone is directly related to the electrical properties of the material.
		The band structure in figure 1.6.5 is plotted in several Brillouin zones.
	people.deas.harvard.edu /~jones/ap216/lectures/ls_2/ls2_u7/sse_tut_1/solid1.html (1687 words)

	Encyclopedia: Bloch wave
		In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction.
		The plane wave wavevector k (multiplied by Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the Brillouin zone.
		It can be shown that the wavefunction of a particle in a periodic potential must have this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian.
	www.nationmaster.com /encyclopedia/Bloch-wave (607 words)

	A Lattice Model for Computing the Transmissivity of the Cornea and Sclera -- Ameen et al. 75 (5): 2520 -- Biophysical ...
		The first Brillouin zone is the hexagon shown and is formed by the perpendicular bisectors of the shortest reciprocal lattice vectors.
		The small dots outline the triangular path around the irreducible segment of the first Brillouin zone that we use to illustrate the photonic band structure in subsequent figures.
		The shaded regions are believed to be two-dimensional band gaps extending throughout the Brillouin zone.
	www.biophysj.org /cgi/content/full/75/5/2520 (5926 words)

	Brillouin Zones
		The first Brillouin zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice, or it could be defined as the set of points in k space that can be reached from the origin without crossing any Bragg plane.
		The second Brillouin zone is the set of points that can be reached from the first zone by crossing only one Bragg plane.
		Brillouin zone is the set of points not in the
	phycomp.technion.ac.il /~nika/brillouin_zones.html (309 words)

	Invariants Help & Info
		The coordinates for the points in the first Brillouin zone are given in terms of "conventional" vectors of the reciprocal lattice, as defined by Miller and Love.
		The directions of order parameters for the case of points of symmetry in the first Brillouin zone are obtained from a data base.
		For other points in the first Brillouin zone, these directions are obtained by calculating the isotropy subgroups associated with the irreducible represenation chosen.
	www.ccp14.ac.uk /ccp/web-mirrors/isotropy/~stokesh/invariantshelp.html (1190 words)

	Semiconductor Crystals Section (Site not responding. Last check: 2007-10-08)
		The Brillouin zone is defined in the reciprocal lattice as the volume enclosed within a Wigner-Seitz cell.
		At the boundaries of the Brillouin zone, the Bragg diffraction condition in the reciprocal lattice must be satisfied.
		The transformation of the WS cell of bcc lattice in real space transforms to a Brillouin zone in a fcc lattice in reciprocal space while the WS cell of a fcc lattice transforms to a Brillouin zone of a bcc lattice in reciprocal space.
	britneyspears.ac /physics/crystals/wcrystals.htm (1642 words)

	Brillouin zone - Information
		The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.
		, Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used more rarely.
		A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice.
	www.logicjungle.com /wiki/Brillouin_zone (318 words)

	2.1.5 Band Structures and Standard Representations
		The lower part (the "cup") is contained in the 1st Brillouin zone, the upper part (the "top") comes from the second BZ, but is now folded back into the first one.
		The top pictures show the elementary cell of the diamond lattice or of the ZnS type lattice; the lower left picture the Bravais lattice of the fcc type and the primitive (non-cubic) lattice which is an equally valid, if less symmetric, representation of the fcc lattice..
		The first Brillouin zone with the proper indexing of the relevant points and some dispersion parabola along prominent directions are shown.
	www.tf.uni-kiel.de /matwis/amat/semi_en/kap_2/backbone/r2_1_5.html (880 words)

	Re: Q: Dual lattices/spaces
		But I should have written: "Aliasing an electron's _dispersion relation_ back onto the first Brillouin zone gives the band structure." Many solid-state texts introduce band structure this way: treat the electrons as nearly free, as a first approximation.
		This is the "aliasing" I meant -- "mod out by the reciprocal lattice." Turning on interactions (with ions and other electrons) tends to distort things at the edges of the first B.Z., separating the aliased portions into bands.
		Defining unit (reciprocal) volumes by "second-nearest neighbors" (and higher) gives the second and higher Brillouin zones; I'd have to look in a solid-state book for the precise definition.
	www.lns.cornell.edu /spr/1999-04/msg0015743.html (660 words)

	Bass Lab Facilities (Site not responding. Last check: 2007-10-08)
		This picture shows the Brillouin Scattering apparatus that we use to measure sound velocities (comparable to seismic wave velocities) on small crystals at high pressures and temperatures.
		A high-temperature furnace is on the left (with tubes attached for controlling the atmosphere in the furnace), and a Eularian cradle on the right (for single crystal work and diamond cell work).
		Below is a Brillouin experiment using a high temperature furnace (the silver furnace at top center).
	www.geology.uiuc.edu /~jaybass/lab.html (166 words)

	Léon Brillouin - Wikipedia, the free encyclopedia
		Léon N. Brillouin (August 7, 1889-1969) was a French physicist.
		His father Marcel Brillouin was a physicist as well.
		There, he taught at the University of Wisconsin and Harvard University, and later Columbia University.
	en.wikipedia.org /wiki/Leon_Brillouin (96 words)

	TETPACK Library
		There are no tetrahedra, C is ignored, the k-points are defined in the 1/12th (+NH) or 1/6th part (-NH) of the 1st Brillouin zone for C6v or C6 symmetry at gamma depending on whether one enters +NH or -NH as first argument.
		The sampling volume is 1/8 irreductible wedge of the first Brillouin zone.
		The sampling volume is 1/16 irreductible wedge of the first Brillouin zone.
	www.physique.fundp.ac.be /administration/tetpack/setk.html (1373 words)

	Solid State Simulations - "peierls" (Site not responding. Last check: 2007-10-08)
		The blue square is the boundary of the Brillouin zone and the red line the contour in k-space of the Fermi energy (the Fermi surface).
		There is a wide variety of orbits, some (near the center of the zone) which are free electron like, but others near the Fermi energy which are quite complex.
		Second, we take advantage of the periodicity of E(k) surfaces in reciprocal space to displace three of the quadrants of the Brillouin zone to share a common corner.
	www.physics.cornell.edu /sss/peierls/peierls.html (426 words)

	Band Structure p.5
		Thus for any given k in the Brillouin zone there are infinitely many eigenenergies and wavefunctions, each one "generated" from a reciprocal lattice vector G.
		I've coded the central Brillouin zone in red, the six nearest neighboring cells in green, the second nearest neighboring cells in blue, etc. Consider the red k vector moving out from k=0, along the x axis, toward the zone boundary.
		When k reaches the zone boundary the first-zone eigenenergy (red k vector) will be degenerate with a pair of nearest neighbor eigenenergies.
	www.physics.csbsju.edu /QM/band.05.html (685 words)

	Re: Brillouin zone special points (Site not responding. Last check: 2007-10-08)
		It's origin is: Bouckaert, Smoluchowski, and Wigner, "Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals," Phys.
		These labels may make some sense in the context of group theory (or perhaps in Hungarian), but they are commonly regarded as an arbitrary but common convention.
		By the way, these are simply known as "symmetry points" in the Brillouin zone.
	www.lns.cornell.edu /spr/2004-03/msg0059719.html (263 words)

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