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There can be more than one way to choose a primitive cell for a given crystal and each choice will have a different primitive cell shape, but the primitive cell volume is the same for every choice and each choice will have the property that a one-to-one correspondence can be established between primitive unit cells and discrete lattice points ...
A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by n cells are counted as 1 / n of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain 1 / 8 of each of them. [3]
The smallest group of particles in material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects symmetry and structure of the entire crystal, which is built up by repetitive translation of unit cell along its principal axes. The translation vectors define the nodes of Bravais lattice.
For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3, or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.
In either case, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes. [4] The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice). This is a unit cell with parameters a = b = c; α = β = γ ...
In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group. These are the Bravais lattices in three dimensions:
Unit cell For the base-centered monoclinic lattice, the primitive cell has the shape of an oblique rhombic prism; [ 1 ] it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes.
The letters A, B and C were formerly used instead of S. When the centred face cuts the X axis, the Bravais lattice is called A-centred. In analogy, when the centred face cuts the Y or Z axis, we have B- or C-centring respectively. [5] The fourteen possible Bravais lattices are identified by the first two letters: