crystallograpy-SS2011

crystallograpy-SS2011 - Crystal Structures A crystal is a...

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Crystal Structures A crystal is a solid whose atoms are arranged in a periodic pattern in three dimensions. A point lattice can be constructed by taking three non-coplanar sets of parallel, equally spaced planes, whose intersections form a set of lines which further intersect to form a set of points. When this is done such that, for a given crystal system, each of the points has identical physical surroundings, a point lattice is formed. A lattice point does not, however, need to correspond to a particular atom location, and many systems are constructed with lattices in which each lattice point is associated with two or more atoms. The lattice construction also gives a set of identical unit cells , which are parallelepipeds. Any crystal may be described with a number of different point lattices, but the one commonly used generally gives the smallest and most symmetrical unit cell. Both the unit cell and the point lattice may be constructed by successive application of a set of three lattice vectors (or ‘basis’ vectors) a , b and c , which define the edges of the unit cell when drawn from a common origin. These vectors may be considered as unit vectors in a crystallographic coordinate system (which need not be orthogonal). A point anywhere in the lattice may then be specified in terms of the unit vectors as the crystal coordinates , (h a , k b , l c ). The magnitudes of a , b , and c , together with the angles between the vectors, α , β and γ are called lattice parameters (or lattice constants). 7 crystal systems and 14 Bravais lattices Cubic systems have a = b = c and α = β = γ = 90 ο . (Simple cubic; body-centered; face-centered) Tetragonal systems have a = b c and α = β = γ = 90 ο . (Simple; body-centered) Orthorhombic systems have a b c and α = β = γ = 90 ο . (Simple; base-centered; body-centered; face-centered) Rhombohedral (or ‘trigonal’) systems have a = b = c and α = β = γ 90 ο . Hexagonal systems have a = b c and α = β = 90 ο and γ = 60 ο . Monoclinic systems have a b c and α = β = 90 ο γ . (Simple, base-centered) Triclinic systems have a b c and α β γ 90 o .
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The Simple Cubic Lattice For simple cubic lattices, there is one lattice point per unit cell. The lattice may be described as a three dimensional array of lattice points by either beginning with a lattice point at the position (0,0,0), and adding additional lattice points by translating indefinitely in all three principal directions by the lattice parameter, a , or alternatively, by allotting 1/8 of a lattice point at each of the eight corners of the unit cell and stacking up unit cells in all three directions to build up the lattice. (In reality, there is one lattice point at each cube corner, with each contributing 1/8 of itself to each unit cell.) The physical surroundings of each lattice point in the unit cell of any crystal lattice are identical. The number of atoms contained within the unit cell is the number of lattice points per unit cell
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This document was uploaded on 10/28/2011 for the course MSE 250 at Michigan State University.

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crystallograpy-SS2011 - Crystal Structures A crystal is a...

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