Bravais Lattices
The 5 different types of planar lattices (i.e. the 2 dimensional lattices--oblique, rectangular, centered rectangular, diamond, hexagonal, and square) in combination with the symmetry constraints imposed by the 32 crystal classes (that is you can only have 1, 2, 3, 4, and 6 fold axes) give rise to the 14 space or Bravais lattices.
Bravais or space lattices: The 14 possible ways in which points or nodes within a crystal can be arranged periodically in three dimensions. These lattice structures define the way that unit cells can be repeated in three dimensions.
Restrictions on choices of unit cells.
1) The edges of the unit cell should coincide, if possible, with symmetry axes of the lattice.
2). The edges should be related to each other by the symmetry of the lattice.
3). The smallest possible unit cell should be chosed in agreement with 1 and 2.
The basic groups of space lattices are Triclinic, Monoclinic, Orthorhombic, Tetragonal, Rhombohedral, Hexagonal, and Isometric primitive lattices (nodes at the corners only).
The additional 7 types are generated by the presence of C, F, and I lattices in some of the systems (note that the rhombohedral and hexagonal primitive lattices are the only types present in the Hexagonal Crystal System.
Primitive (P): The lattice points occur only at its corners.
C-centered (C): Lattice points are found in the middle of the c-faces (or a or b) and at the corners.
Face centered (F): Lattice points are found in the center of all faces and at corners.
Body centered (I): Lattice points are found in the center of the cell and at corners.
THE SIX CRYSTAL SYSTEMS: Know these relationships!
CUBIC: a = b = c; alpha = beta = gamma = 90.
HEXAGONAL: a = b # c; alpha = beta = 120; gamma = 90.
TETRAGONAL: a = b not= c; alpha = beta = gamma = 90.
ORTHORHOMBIC: a # b # c; alpha = beta = gamma = 90.
MONOCLINIC: a # b # c; alpha = gamma = 90; beta > 90.
TRICLINIC: a # b # c; alpha and beta > 90; gamma.
The multiplicity of a unit cell is simply how many of the lattice nodes are shared with other cells. Each corner is shared by 8 adjacent cells, so only 1/8 of a specific corner belongs to a specific cell. Since each 3-D cell has 8 corners, the multiplicity of a primitive cell is 8 x 1/8 = 1. Centered faces are shared by only 2 cells, so add 1 (2x1/2) for each pair of centered faces. Body centered nodes (in the interior) are unique to each cell, so add 1 to each cell.
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The 5 different types of planar lattices (i.e. the 2 dimensional lattices--oblique, rectangular, centered rectangular, diamond, hexagonal, and square) in combination with the symmetry constraints imposed by the 32 crystal classes (that is you can only have 1, 2, 3, 4, and 6 fold axes) give rise to the 14 space or Bravais lattices.
Bravais or space lattices: The 14 possible ways in which points or nodes within a crystal can be arranged periodically in three dimensions. These lattice structures define the way that unit cells can be repeated in three dimensions.
Primitive (P): The lattice points occur only at its corners.
THE SIX CRYSTAL SYSTEMS: Know these relationships! The multiplicity of a unit cell is simply how many of the lattice nodes are shared with other cells. Each corner is shared by 8 adjacent cells, so only 1/8 of a specific corner belongs to a specific cell. Since each 3-D cell has 8 corners, the multiplicity of a primitive cell is 8 x 1/8 = 1. Centered faces are shared by only 2 cells, so add 1 (2x1/2) for each pair of centered faces. Body centered nodes (in the interior) are unique to each cell, so add 1 to each cell. |