
	Where results make sense

Topic: Hexagonal lattice

	Centered hexagonal number - Wikipedia, the free encyclopedia
		A centered hexagonal number, or hex number is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice.
		To find centered hexagonal numbers besides 1 that are also triangular numbers or squares, it is necessary to solve Diophantine equations.
		Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes.
	en.wikipedia.org /wiki/Centered_hexagonal_number (334 words)

	Bravais Lattices (Site not responding. Last check: 2007-10-10)
		For example, the hexagonal lattice is an example of a 2-dimensional Bravais lattice, but the honeycomb lattice is not since neighboring points are not equivalent (it is an example of a compound structure, which we will discuss next).
		For the bcc lattice, a point is added to the center of each cube; alternatively, this lattice can be viewed as two sets of sc sublattices, A and B, that inter-penetrate (however, note that there is no way to distinguish these two sets).
		This is completely obvious in 2-D, where the W-S cell is a square for a square lattice, a hexagon for a hexagonal lattice, and a rectangle for a rectangular lattice.
	carini.physics.indiana.edu /p615/lattices.html (773 words)

	Wallpaper Groups: lattices
		That's because in that case, the points in the lattice nearest any one point in the lattice are the vertices of a regular hexagon.
		Lattices are special kinds of patterns, and their symmetry groups of five of these 17 groups.
		Since a square is simultaneously a rectangle and a rhombus, square lattices have all the symmetries of both rectangular lattices and rhombic lattices.
	www.clarku.edu /~djoyce/wallpaper/lattices.html (864 words)

	VDIL
		The driven lattice gas (DLG) is a statistical mechanical model for the effects of an external field on a simple phase transition.
		The current overall project is a study of the driven lattice gas on triangular and hexagonal lattices, and especially the relationship of the field to the temperature at which the lattice gas freezes.
		Two field-lattice combinations are particularly interesting: the hexagonal lattice with the field aligned with a lattice vector, and the triangular lattice with the field bisecting lattice vectors.
	www.d.umn.edu /vdil/research/html_files/Dr_Siders_summer2004.htm (1131 words)

	[No title]
		Thus, we have identified 5 types of 2-D lattices: the general oblique lattice that does NOT have reflection or anything beyond 360o and 180o rotational symmetries; the square lattice, the hexagonal lattice, the rectangular lattice, and the centered rectangular lattice.
		For a hexagonal lattice, (a1(= (a2(, and that the angle, (, between a1 and a2, is 120o (or we could choose 60o) For a rectangular lattice, (a1(((a2(, and that the angle, (, between a1 and a2, is 90o.
		In this lattice, in addition to having lattice points at all the corners of a cube, there is a lattice point right in the middle of the volume of the cube.
	www.cbu.edu /~jholmes/P353/N100CrysStr.doc (2065 words)

	Three Dimensional Order
		Depending on the lattice you start with and how special or general your choice of a 3rd translation (in both direction (angle) and magnitude (distance)), you can generate the 14 3-D lattices that are known as the Bravais lattices.
		Hexagonal net: Stacking a hexagonal net along the (z) direction a random distance 'c' but constraining the angle to be 90 degrees produces a primitive hexagonal lattice.
		Because the translation distances in the lattice and the glide planes and the screw axes are on the order of a few angstroms they can not be seen by eye and so morphologically all 230 space groups appear to us as one of the 32 crystal classes.
	www.geology.wisc.edu /courses/g360/lect6.html (1648 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		describe a cladding structure based on an array of holes that are arranged at the vertices of a lattice defined by a regular hexagonal polygon and a regular triangle having a side length corresponding to that of the regular hexagonal polygon, and wherein hexagonal polygons exist, each side of which is shared with a triangle'.
		Kagome structure, the advantages of the hexagonal structure are maintained, and a further spacing of the high index areas defined in the hexagonals is provided via the triangles'.
		The principal elements of the fibre (the jacket, the Kagome lattice structure and the large air core) are visible in Fig.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=04/01461.031231&ELEMENT_SET=DECL (5611 words)

	Semiconductor Fundamentals
		The lattices in two dimensions are the square lattice, the rectangular lattice, the centered rectangular lattice, the hexagonal lattice and the oblique lattice as shown in Figure 2.2.2.It is customary to organize these lattices in groups, which have the same symmetry.
		This lattice can also be formed from two face-centered-cubic lattices, which are displaced along the body diagonal of the larger cube in Figure 2.2.5 by one quarter of that body diagonal.
		The cubic crystals are characterized by a single parameter, the lattice constant a, while the hexagonal structures are characterized in the hexagonal plane by a lattice constant a and by the distance between the hexagonal planes, c.
	ece-www.colorado.edu /~bart/book/book/chapter2/ch2_2.htm (1394 words)

	Direct X-Ray Observation of a Single Hexagonal Myofilament Lattice in Native Myofibrils of Striated Muscle -- Iwamoto ...
		The density distribution in the hexagonal myofilament lattice was reconstructed by performing Fourier synthesis by using the
		lattice, indicating that 40% of the x-ray path in the specimen
		FIGURE 4 Example of the electron densities of the hexagonal myofilament lattice reconstituted by the direct two-dimensional Fourier synthesis from the diffraction pattern from a single hexagonal lattice in Fig.
	www.biophysj.org /cgi/content/full/83/2/1074 (4597 words)

	Reference NASA Tech 1987 Watson & Ahumada (Site not responding. Last check: 2007-10-10)
		The seven basis functions occupy a point and a hexagon of six nearest neigbors on a hexagonal sample lattice.
		At each level, the output is subsampled in such a way as to yield a new hexagonal lattice with a spacing the square root of seven larger than the previous level, so that the number of coefficients is reduced by a factor of seven at each level.
		In the biological model, the input lattice is the retinal ganglion cell array.
	vision.arc.nasa.gov /publications/OrthogonalHexagonal.html (183 words)

	I
		Beginning with a single lattice point, all of the other lattice points in the array can be generated by shifting the lattice point by the translation vector.
		Note that a lattice point is not a physical object, it is simply a point in space upon which a real object may be placed.
		An alternative description would be to draw the unit cell as a rectangle, which contains a lattice point at each corner of the unit cell and one in the center of the unit cell (for a total of two lattice points per unit cell and an area twice that of the rhombus unit cell).
	www.chemistry.ohio-state.edu /~woodward/ch754/sym_2d.htm (1524 words)

	Biophysical Journal: Direct X-Ray observation of a single hexagonal myofilament lattice in native myofibrils of ... (Site not responding. Last check: 2007-10-10)
		Because the lattices are randomly oriented, the fiber gives rise to an equatorial x-ray diffraction pattern, which is essentially a rotary-averaged "powder diffraction," carrying only information about the distance between the lattice planes.
		We were able to record an x-ray diffraction pattern from a single myofilament lattice, very likely originating from a single myofibril from the flight muscle of a bumblebee, by orienting the incident x-ray microbeam along the myofibrillar axis (end-on diffraction).
		The pattern consisted of a number of hexagonally symmetrical diffraction spots whose originating lattice planes were readily identified.
	www.findarticles.com /p/articles/mi_qa3938/is_200208/ai_n9112152 (1287 words)

	Crystal Lattice, Hexagonal Closest Packing (Site not responding. Last check: 2007-10-10)
		Instead of stacking hexagonal closest packed planes directly above one another, they can be stacked such that atoms in successive planes nestle in the triangular "grooves" of the adjacent plane.
		If a third hexagonal closest packed plane is stacked above B but in the "A" orientation, and succeeding planes are stacked in the repeating pattern ABABA...
		For the stack of hexagonally closest packed spheres of equal radius (r) described above, the interplanar spacing between adjacent planes is proportional to r.
	www.chem.lsu.edu /htdocs/people/sfwatkins/MERLOT/lattice/07hcp.html (226 words)

	Promorphology of Crystals Preparation 3-D
		The only difference between these two lattices as they are furnished with atoms, is their mutual position : they are shifted with respect to each other along the longer diagonal of the rhombus, as indicated in the Figure, and in addition to that they are shifted half a vertical unit cell length along the c-axis.
		Indeed each lattice of our structure (A and B, which are identical) can be based on a rhombic prism of which the top and bottom faces are a rhombus which we already knew as a possible 2-dimensional unit cell for the 2-D hexagonal net.
		The hexagonal prism associated with the larger hexagon (thick fl lines) is a t o t a l m o t i f associated with a lattice point, and builts up the 3-D structure (where the glide and screw translations are eliminated) when it is repeated periodically.
	home.hetnet.nl /~turing/promorph_crystals_preparation_3dim.html (7265 words)

	Archimedes Plutonium (Site not responding. Last check: 2007-10-10)
		The face-centered-cubic lattice whose density is pi/3sqrt2 is the maximum density for the packing of identical spheres in 3-dimensional Euclidean space.
		PROOF that the hexagonal lattice is the densest packing in the Euclidean plane: Suppose there exists a denser pack of identical circles than the hexagonal lattice in the Euclidean plane.
		The hexagonal lattice pack in the Euclidean plane is the construction of regular 6-gon for each circle.
	www.iw.net /~a_plutonium/File112.html (948 words)

	ILL : ANNUAL REPORT 2000
		These lines are often ordered to form a lattice, the orientation and geometry of which reflects the anisotropy of the underlying interactions between the lines.
		In (II) the lattice was grown at 500mK in the A-phase.
		As the temperature at which the FLL was formed was increased, we observed a change from an hexagonal lattice aligned along crystal symmetry directions, to two equally populated domains of approximately hexagonal lattices aligned at +15° and -15° to the
	www.ill.fr /AR-00/p-22.htm (1072 words)

	CVM 1.1 (VW): Eigenfunctions with 3-Fold Rotations
		Armed with the lattice coordinates of the previous section, it is easy to find the eigenfunctions of the Laplacian that are periodic with respect to a special lattice we call the hexagonal lattice.
		The hexagons are formed from three lattice cells together.
		In this case, these would be the various symmetries in the five groups whose translational subgroup amounts to the hexagonal lattice.
	www.geom.uiuc.edu /~dpvc/CVM/1998/01/vw/article/constructing/ef3r.html (317 words)

	THEORY:
		The lattice structure of graphite was observed to be hexagonal layers, confirming that predicted by theory and experiment.
		The lattice layers of graphite are composed of hexagonal rings of carbon.
		Although we were not able to measure the vertical separation between the layers, the surface scan of the STM did reveal the size of the atomic radii of the carbon atoms.
	webphysics.davidson.edu /alumni/anselsb/lab/STM/STM.html (1096 words)

	[No title]
		As I explained in "week66", a lattice in the complex plane consists of all points that are integer linear combinations of two complex numbers, say omega_1 and omega_2.
		What I mean is, if you take the hexagonal lattice and pick a basis for it like this: omega_2 * * * * omega_1 * 0* * * * * * then in this basis the matrix ST represents a 60 degree rotation.
		And the point tau = exp(2 pi i/3) corresponding to the hexagonal lattice, is preserved by ST and all its powers.
	www.math.niu.edu /~rusin/known-math/98/PSL (2693 words)

	More complicated lattices (Site not responding. Last check: 2007-10-10)
		One choice for the primitive lattice vectors, a1 and a2, for the hexagonal lattice is for them to have the same length and to have a 60-degree angle between them.
		The highest symmetry operations for the Hexagonal lattice are rotations about a lattice point by 60 or 120 degrees.
		v1 at one of the vertices of a hexagon (0,0,0) and
	carini.physics.indiana.edu /p615/symmetries-compound.html (1208 words)

	CVM 1.1 (VW): The Hexagonal Lattice
		We have already done most of the work for the hexagonal lattice in an example in the first section
		In the table, we use these shorthands to list the terms, or combinations of terms, that achieve the desired pattern.
		Recall that parity considerations don't apply to this lattice type.
	www.maa.org /cvm/1998/01/vw/article/lattice/hexagonal.html (180 words)

	[No title]
		Ba, Ce, Li, K, molybdenum (less ductile metals) Face centered cubic- atom centered on each of the faces Atomic packing factor (APF) is 0.74 and represents the fraction of the unit cell occupied by the two atoms.
		Change to FCC if temperature is raised to 1670 F allotropes are also known as polymorphs Freezing point of material As the energy in a liquid system decreases, the forces that are grouping the atoms tend to form distinct patterns which become the characteristic lattice structure of the material.
		Lattice grows until it meets another energy block, e.g., lattice structure or container.
	www.csuchico.edu /~jpgreene/itec104/m104_ch03-02/m104_ch03-02.ppt (602 words)

	Crystals
		For a square lattice as shown on the left, the coordination number is 4 (the number of circles touching any individual).
		If we start with a hexagonal array of spheres (the blue "A" layer), that would be the most-packed arrangement - there is no way to pack more spheres into a given area.
		The resulting structure is a 3-D analog of the hexagonal packing in a plane - it is the most efficient way to pack spheres.
	www.chemistry.nmsu.edu /studntres/chem116/notes/crystals.html (1888 words)

	Cellular Automaton Fluids: Basic Theory (1986)
		Crystallographic lattices are classified not only by point symmetries, but also by the spatial arrangement of their sites.
		The lattices consist of ``unit cells'' containing a definite arrangement of sites, which can be repeated to form a regular tesselation.
		The hexagonal and face-centered cubic lattices, which have the largest point symmetry groups in two and three dimensions, respectively, are also the lattices that give the densest packings of circles and spheres (e.g., Ref. 42).
	www.stephenwolfram.com /publications/articles/ca/86-fluids/4/text.html (1983 words)

	Hexagonal to square lattice conversion in bilayer systems (Site not responding. Last check: 2007-10-10)
		The results clearly show that the character of the reconstructive hexagonal to square lattice conversion in bilayer colloid systems is potential dependent.
		In this case the hexagonal to square lattice transition is first order.
		from the square lattice by strong first order transitions, and from the hexagonal lattice by either weak first or second order transitions.
	mrsec.uchicago.edu /Publications/Rice/rice039.html (249 words)

	Lattice Model
		The lattice modeling uses the same lattice Green's function techniques we have used in the previous interface work[], and the general methodology is given in Thomson, etal[].
		It might be useful to note here that the use of the 2D lattice is not a simplification of something more general, but the 2D case is exactly the one pertaining to the physics of the problem.
		In the lattice case, we are at liberty to make the bonding between the layers different than that of either bulk, as in the real physical situation.
	www.ctcms.nist.gov /~robb/iI/section3_3.html (1357 words)

	Structure of BeB2
		There are three kinds of subunits, and near the vertices of the honeycomb lattice 4th subunit consisting of a chain of 4 atoms is placed.
		Unit 3 (a vertical column of 4 atoms -- see VRML) sits roughly at the midpoint of the triangle whose vertices are at the centers of the Unit 1, Unit 2a and Unit 2b objects, or equivalently near the vertices of the hexagonal lattice whose openings contain the 1 and 2 subunits.
		Be corresponds to Unit 2a's at half of the honeycomb lattice and Unit 2b's on the other half.
	www.phys.lsu.edu /faculty/browne/BeB2_old/xtal.html (437 words)

	Wallpaper classification
		In the glide reflection column, an asterisk * means that there is a glide mirror which is not a reflection mirror of the pattern.
		The lattice type of a pattern denotes the most general translation lattice which can occur for the pattern.
		A pattern with hexagonal lattice type can occur only with a hexagonal translation lattice.
	www.math.uga.edu /~clint/2005/5210/wallpaper.htm (210 words)

	Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals
		Beginning with the standard hexagonal lattice, the results of a symmetry analysis are used to determine a cavity geometry that produces a mode whose symmetry immediately leads to a reduction in vertical radiation loss from the PC slab.
		The Q is improved further by a tailoring of the defect geometry in Fourier space so as to limit coupling between the dominant Fourier components of the defect mode and those momentum components that radiate.
		We also consider defect cavities in a compressed hexagonal lattice, where the lattice compression is used to modify the in-plane bandstructure of the PC lattice, creating new (frequency) degeneracies and modifying the dominant Fourier components found in the defect modes.
	www.opticsexpress.org /abstract.cfm?URI=OPEX-11-6-579 (716 words)

Try your search on: Qwika (all wikis)