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Topic: Reciprocal lattice

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Bravais lattice

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	Reference.com/Encyclopedia/Reciprocal lattice
		The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.
		The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction.
		The Brillouin zone is a primitive unit cell of the reciprocal lattice.
	www.reference.com /browse/wiki/Reciprocal_space (676 words)

	Reciprocal lattice - Wikipedia, the free encyclopedia
		The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.
		The direction of the reciprocal lattice vector corresponds to the normal to the real space planes, and the magnitude of the reciprocal lattice vector is equal to the reciprocal of the interplanar spacing of the real space planes.
		The Brillouin zone is a primitive unit cell of the reciprocal lattice.
	en.wikipedia.org /wiki/Reciprocal_lattice (639 words)

	Reciprocal lattice -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		In (The branch of science that studies the formation and structure of crystals) crystallography, the reciprocal lattice of a (A 3-dimensional geometric arrangement of the atoms or molecules or ions composing a crystal) Bravais lattice is the set of all (A variable quantity that can be resolved into components) vectors K such that
		We find that the reciprocal simple cubic Bravais lattice, with cubic primitive cell of side, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side (in the crystallographer's definition).
		The first, which generalises directly the reciprocal lattice construction, uses (Analysis of a periodic function into a sum of simple sinusoidal components) Fourier analysis.
	www.absoluteastronomy.com /encyclopedia/r/re/reciprocal_lattice.htm (619 words)

	Reciprocal Lattice
		The concept of the reciprocal lattice was devised to tabulate two important properties of crystal planes: their slopes and their interplanar distances.
		The relationship between the period and frequency is similar to that of the reciprocal and the direct lattice.
		Because the rods are infinitely dense lattice points, the diffraction from a surface occurs continuously with changes in the direction and the magnitude of the incident wavevector as long as the wave is short enough to be diffracted.
	www.chembio.uoguelph.ca /educmat/chm729/recip/vlad.htm (1065 words)

	Advanced diffraction 2
		This again relates to the idea of reciprocal space: the planes (and corresponding diffraction spots) with larger indices correspond to closer spacings; the (3 0 0) planes are one-third as far apart as the (1 0 0) planes.
		Its length is the reciprocal of the distance between one bc plane and the next, and it is perpendicular to the bc plane.
		Reciprocal axes are defined as having a direction perpendicular to the other two real space axes and a length equal to the reciprocal of the spacing between the planes defined by the other two real space axes.
	www-structmed.cimr.cam.ac.uk /Course/Adv_diff2/Diffraction2.html (2847 words)

	Miller indices (Site not responding. Last check: 2007-11-05)
		Each vector on the reciprocal lattice is normal to a plane on the direct lattice, so to specify a direct-lattice plane it is sufficient to specify its reciprocal-lattice vector.
		Any lattice vector is expressible as a sum of integer multiples of three primitive vectors, so three integers are sufficient to specify the reciprocal vector, and hence the direct-lattice plane.
		The reciprocal of the three integers in the Miller index are the intercepts of the plane with the a, b, and c axes, respectively.
	www.ccr.buffalo.edu /etomica/app/modules/sites/CrystalViewer/Background5.html (476 words)

	Lattice Planes (Site not responding. Last check: 2007-11-05)
		In general, the direction perpendicular to a family of planes is not necessarily in the direction of one of the lattice vectors.
		This definition emphasizes the symmetric nature of the relationship between K and R: We say that {K} is the reciprocal lattice of {R}, but it is equally valid to say that {R} is the reciprocal lattice of {K}.
		Lattices in the oblique and centered rectangular families also have reciprocal lattices in the same families.
	carini.physics.indiana.edu /p615/lattice-planes.html (1707 words)

	Reciprocal Lattice Exercises
		In a reciprocal lattice, the reciprocal lattice points (relpoints) are labeled (or indexed) the same, since they represent the planes in the real lattice.
		Drag the white labels from the yellow panel to index the reciprocal lattice points for the unit cell of the reciprocal lattice shown.
		A infinite reciprocal lattice has a specified origin and all the other points are relative to this point.
	www.matter.org.uk /diffraction/geometry/reciprocal_lattice_exercises.htm (98 words)

	BritenySpears.ac: Semiconductor Crystal Structure
		Lattice vectors are the shortest distances to the nearest neighbouring points on the lattice and are conventionally denoted by a, b and c.
		The reciprocal lattices is the collection of points that represent allowed values of wavevectors for Fourier series and Fourier transforms with the periodicity of the lattice.
		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 transform to a Brillouin zone of a bcc lattice in reciprocal space.
	britneyspears.ac /physics/crystals/wcrystals.htm (1587 words)

	MIJ-NSR Volume 3, Article 46
		A relatively high lattice mismatch between GaN and AlN (2.5%) induces a high degree of strain in GaN/GaAlN heterostructures which, depending on the substrate choice and whether or not strain relaxation has occurred, can be present as well in the wells as in the barriers.
		N buffer and cap layer, and the middle one is the zero-order peak of the eleven period superlattice.
		Using Vegard’s law for the lattice parameter and a linear interpolation between the available Poisson ratios for GaN [8] and AlN [9] we find an aluminium content of 17% in our alloy, which is in fair agreement with the nominal value.
	nsr.mij.mrs.org /3/46/complete.utf.html (1429 words)

	Reciprocal lattice - Definition, explanation
		In X-ray diffraction, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector.
		There are actually two versions in mathematics of the abstract dual lattice concept, for a given lattice L in a real vector space V, of finite dimension.
		The other aspect is seen in the presence of a quadratic form Q on V; if it is non-degenerate it allows an identification of the dual space V* of V with V.
	www.calsky.com /lexikon/en/txt/r/re/reciprocal_lattice.php (540 words)

	Crystals and Lattices
		The method used here to derive the reciprocal lattice is somewhat different than is normally used by either crystallographers or solid-state physicists, and rests more on the general mathematical properties of dual or reciprocal spaces, as the mention of contravariant and covariant components indicates.
		In terms of the reciprocal lattice, let n be a unit vector in the direction of the incident radiation, and n' a unit vector in the direction of the diffracted beam.
		That of a face-centered cubic lattice is a truncated octahedron.
	www.du.edu /~jcalvert/phys/lattice.htm (6338 words)

	Problem set (Site not responding. Last check: 2007-11-05)
		The cubic lattice constant of Fe is 0.29 nm.
		Make a drawing of the surface (2D) lattice for the (0001) plane of Be (hcp structure) and draw the corresponding reciprocal lattice.
		The hexagonal lattice constants of Be are a=0.23 nm and c=0.36 nm.
	users.mrl.uiuc.edu /cahill/405/ps8_04.html (166 words)

	X-Ray Crystallography Notes (Site not responding. Last check: 2007-11-05)
		Thus there is a reciprocal relationship between the unit cell of the crystal and the lattice pattern of the diffracted spots.
		The reciprocal lattice is a property of the crystal.
		The reciprocal lattice is then drawn on the same scale as the sphere with its origin located 1/λ from the center of the circle on the opposite side of the incident beam.
	xrayweb.msg.ku.edu /notes/crystallography.html (5053 words)

	Background (Site not responding. Last check: 2007-11-05)
		The modulus of a* is equal to the ratio of the area of the face OBCG opposite to a to the volume of the cell built on the three vectors a, b, c.
		The basic vectors of the reciprocal lattice possess therefore the properties that we were looking for in the introduction.
		We shall see in the next section that with each family of lattice planes of the direct lattice a reciprocal lattice vector may be thus associated.
	shuttle.ps.uci.edu /haihui/background.html (1099 words)

	math lessons - Reciprocal lattice
		The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.
		The Brillouin zone is a primitive unit cell of the reciprocal lattice.
		The cubic lattice is therefore said to be dual, having its reciprocal lattice being identical (up to a numerical factor).
	www.mathdaily.com /lessons/Reciprocal_lattice (542 words)

	DoITPoMS TLP - X-ray Diffraction - Reciprocal lattice 1 (Site not responding. Last check: 2007-11-05)
		The use of the reciprocal lattice to understand diffraction was demonstrated in 1913 by P. Ewald.
		The direct lattice determines (through defined relationships) the reciprocal lattice vectors, the lattice point spacings and the associated reciprocal directions.
		The reciprocal lattice consists of an array of points which can be labelled as shown.
	www.doitpoms.ac.uk /tlplib/xray-diffraction/reciprocal1.php (263 words)

	25 - Lecture notes for Clay Mineralogy
		When a beam is diffracted by a crystal structure, the positions of the diffracted beam provide a map of the reciprocal lattice of the crystal.
		The reciprocal lattice is difficult to comprehend from a physical standpoint.
		The properties of a reciprocal lattice are such that : a.a = b.b = c.c = 1, and α + α= 180°, β* + β= 180° and γ* + γ= 180° and ρ
	www.gly.uga.edu /schroeder/geol6550/CM25.html (425 words)

	Biology 750: Diffraction/Fourier Transforms
		Thus, the continuous Fourier transform of the motif is sampled at the points of the Reciprocal lattice; the Fourier transform of the crystal is only non-zero at the points of the Reciprocal lattice.
		Compare this "Reciprocal Lattice" with the original crystal lattice outlined in the filtered image of the crystal shown at the bottom.
		in the real lattice the longest spacing is along the vertical axis (y-axis) while in the Reciprocal lattice the longest spacing is in the horizontal axis (the h-axis which corresponds to the x-axis in the crystal).
	www.sci.sdsu.edu /TFrey/Bio750/FourierTransforms.html (1418 words)

	Reciprocal Lattice
		A reciprocal lattice of a crystal structure is a set of vectors denoted by K that satisfies the constructive interference condition, i.e.
		are the primitive lattice vectors describing the structure of the crystal.
		Which are the primitive basis vectors of the body-centered cubic lattice with a lattice constant 4p/a.
	ees2cy.engr.ccny.cuny.edu /www/web/crouse/EE339/Lectures/Reciprocal_Lattice.htm (196 words)

	No Title (Site not responding. Last check: 2007-11-05)
		The Reciprocal Lattice of a Reciprocal Lattice is the original Bravais Lattice.
		The reciprocal lattice is an important mathematical tool, needed to describe all properties of periodic structures for electrons, phonons, magnons, etc.
		As a result we see that each family of lattice planes can be associated with a reciprocal lattice vector, and that each reciprocal lattice vector defines to a family of lattice planes.
	www.physics.orst.edu /~henri/Ph671_old/lect06/lecture_06.html (593 words)

	26 - Lecture notes for Clay Mineralogy
		In reciprocal space, the size of the reflection sphere or Ewald sphere is related to the wave length of radiation.
		Thus, to a first approximation, the first plane section through the reciprocal lattice is nearly normal to the incident wave vector.
		The vector R is therefore, a direct measure of the reciprocal lattice vector (ρ).
	www.gly.uga.edu /schroeder/geol6550/CM26.html (498 words)

	[No title]
		Show that the area of the reciprocal lattice unit cell is the inverse of the area of the real space unit cell.
		Having illustrated how the real space and reciprocal space lattices are related, we’ll return to a discussion of the assertion at the start of this section: a LEED pattern is a direct representation of the surface reciprocal lattice.
		The presence of reciprocal lattice rods as opposed to points means that, unlike the bulk X-ray diffraction case, a LEED pattern will be observed over a very wide range of electron energies and geometries and thus we do not need to use techniques such as the Laue or Debye-Scherrer methods to produce a diffraction pattern.
	www.nottingham.ac.uk /~ppzpjm/sect2_6.htm (1725 words)

	reciprocal lattice tutor
		In the figure below, the direct lattice is on the left while the corresponding reciprocal lattice (frequency-space transform) is on the right.
		That bright central spot in the reciprocal space model is oft referred to as the "DC peak" or "unscattered beam".
		Reciprocal lattice spots show up in diffraction experiments when they intersect the Ewald sphere.
	newton.umsl.edu /~run/nano/reltutor2.html (322 words)

	Reciprocal lattice (Site not responding. Last check: 2007-11-05)
		As the name implies, the reciprocal lattice is indeed a lattice, and it can be specified by its own set of primitive vectors.
		The significance of the reciprocal lattice is that each of the vectors it defines (by joining any site on on the lattice to another on it) corresponds to a plane in the original, or direct lattice---each reciprocal-lattice vector is a normal to a direct-lattice plane.
		This one-to-one correspondence between a lattice (which defines the sites) and its reciprocal (which describes its planes) indicates that the lattice planes are no less elementary a description of the lattice than are the lattice sites themselves.
	www.ccr.buffalo.edu /etomica/app/modules/sites/CrystalViewer/Background4.html (255 words)

	CREDUC : Check cell choice
		The essential objective of cell reduction is the recognition of the metric (lattice) symmetry of the input unit cell.
		In the normal case, the maximum values of the indices of these direct and reciprocal lattice rows may be limited to a value of two by working from a primitive Burger reduced cell generated from the input cell.
		This process places the lattice in a conventional orientation and the choice of rows to be selected as edges of the conventional cell is guided by the coincidence of lattice rows with pre-determined symmetry axes.
	xtal.sourceforge.net /man/creduc-desc.html (1006 words)

	MOD2635
		Lattice vibrations affect the motion of electrons in the lattice.
		It may be explained by the interaction of electrons with lattice waves that couple the stationary atomic spins in a parallel direction.
		It may be explained by the interaction of electrons with lattice waves that couple both the spins and the momenta associated with the conduction electrons.
	www.sjsu.edu /faculty/Acrivos/mod2635.html (1950 words)

	Fundamental law of the reciprocal lattice
		Let M be a reciprocal lattice point whose coordinates h, k, l have no common divider (M is the first node on the reciprocal lattice row OM), and P a point in direct space.
		This is the fundamental relation of the reciprocal lattice which shows that with any node M of the reciprocal lattice whose numerical coordinates have no common divider we may associate a set of direct lattice planes normal to OM.
		In other words, with the reciprocal lattice node M may be associated a set of fictitious planes in direct space whose spacing is n times smaller than the real lattice spacing.
	www.cartage.org.lb /en/themes/Sciences/Physics/SolidStatePhysics/AtomicBonding/ReciprocalLattice/Fundamental/Fundamental.htm (769 words)

	Physics 7440 Solutions 5
		Aluminum: The fcc lattice with conventional cubic cell of side, a, has a bcc reciprocal lattice of conventional cubic cell (in k-space) of side 4p/a.
		Therefore, the shortest reciprocal lattice vector could be either in the triangular plane, or perpendicular to the triangular plane, depending upon the actual values of a and c for the direct lattice.
		In other words, the direction of the second reciprocal lattice vector is 90 degrees to the x-axis and 30 degrees beyond the second direct lattice vector.
	www.colorado.edu /physics/phys7440/phys7440_sp03/HOMEWORK/Homework/S5.htm (2219 words)

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