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II. Crystal Structure
A. Lattice, Basis, and the Unit Cell B. Common Crystal Structures C. Miller Indices for Crystal Directions and Planes D. The Reciprocal Lattice
A. Lattice, Basis, and Unit Cell
An ideal crystalline solid is an infinite repetition of identical structural units in space. The repeated unit may be a single atom or a group of atoms.
An important concept: crystal structure = lattice + basis
=
+
lattice: a periodic array of points in space. The environment surrounding each lattice point is identical. basis: the atom or group of atoms “attached” to each lattice point in order generate the crystal structure.
The translational symmetry of a lattice is given by the base vectors or lattice vectors a , b , c . Usually these vectors are chosen either: 1. to be the shortest possible vectors, or 2. to correspond to a high symmetry unit cell
Example: a 2-D lattice
These two choices of lattice vectors illustrate two types of unit cells:
b
b
a
Conventional (crystallographic) unit cell: larger than primitive cell; chosen to display high symmetry unit cell
a
Primitive unit cell: has minimum volume and contains only one lattice point
A lattice translation vector connects two points in the lattice that have identical symmetry:
r n1a n2 b n3c
n1 n2 n3 integers
a b
In our 2-D lattice:
a 2b
b
a
B. Common Crystal Structures
2-D 3-D only 5 distinct point lattices that can fill all space only 14 distinct point lattices (Bravais lattices)
The 14 Bravais lattices can be subdivided into 7 different “crystal classes”, based on our choice of conventional unit cells (see text, handout). Attaching a basis of atoms to each lattice point introduces new types of symmetry (reflection, rotation, inversion, etc.) based on the arrangement of the basis atoms. When each of these “point groups” is combined with the 14 possible Bravais lattices, there are a total of 230
A. Lattice, Basis, and the Unit Cell B. Common Crystal Structures C. Miller Indices for Crystal Directions and Planes D. The Reciprocal Lattice
A. Lattice, Basis, and Unit Cell
An ideal crystalline solid is an infinite repetition of identical structural units in space. The repeated unit may be a single atom or a group of atoms.
An important concept: crystal structure = lattice + basis
=
+
lattice: a periodic array of points in space. The environment surrounding each lattice point is identical. basis: the atom or group of atoms “attached” to each lattice point in order generate the crystal structure.
The translational symmetry of a lattice is given by the base vectors or lattice vectors a , b , c . Usually these vectors are chosen either: 1. to be the shortest possible vectors, or 2. to correspond to a high symmetry unit cell
Example: a 2-D lattice
These two choices of lattice vectors illustrate two types of unit cells:
b
b
a
Conventional (crystallographic) unit cell: larger than primitive cell; chosen to display high symmetry unit cell
a
Primitive unit cell: has minimum volume and contains only one lattice point
A lattice translation vector connects two points in the lattice that have identical symmetry:
r n1a n2 b n3c
n1 n2 n3 integers
a b
In our 2-D lattice:
a 2b
b
a
B. Common Crystal Structures
2-D 3-D only 5 distinct point lattices that can fill all space only 14 distinct point lattices (Bravais lattices)
The 14 Bravais lattices can be subdivided into 7 different “crystal classes”, based on our choice of conventional unit cells (see text, handout). Attaching a basis of atoms to each lattice point introduces new types of symmetry (reflection, rotation, inversion, etc.) based on the arrangement of the basis atoms. When each of these “point groups” is combined with the 14 possible Bravais lattices, there are a total of 230