# primitive lattice

## primitive lattice

(crystallography)
A crystal lattice in which there are lattice points only at its corners. Also known as simple lattice.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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For simplification, we assume that [a.sub.[mu]], [mu] = 1, 2, 3, 4, are mutually perpendicular primitive lattice vectors.
 Wenguang Zhai, On primitive lattice points in planar domains.
Furthermore, as the cubic primitive lattice is common for inorganic materials, the above type of ambiguity can often present difficulties in common practice.
The above discussion has focused on the cubic crystals characterized by a primitive lattice. However, for cubic crystals, mathematical ambiguities in indexing are not confined to crystals characterized by a cubic primitive lattice.
Image data are usually given on homogeneous point lattices, e.g., the cubic primitive lattice [L.sup.n] = a[Z.sup.n], a > 0, where Z denotes the set of integers and a is the lattice distance.
In the case of a cubic primitive lattice which is highly symmetric, the pixel configurations [[xi].sub.l] [subset or equal to] [F.sup.0] (C) are commonly grouped in equivalence classes.
For simplicity we restrict ourselves to the special case where X is observed on a cubic primitive lattice. Nevertheless, this is not a substantial restriction and the following considerations can be extended to arbitrary homogeneous lattices, too.
For the special case of a cubic primitive lattice [L.sup.3] = a[Z.sup.3], a > 0, we have [[gamma].sub.i.sup.(1)] = [[gamma].sub.i.sup.(2)] for all i, and the numerical values are [[gamma].sub.i.sup.(k)] = 0.091556 for i = 1, 2, 3; [[gamma].sub.i.sup.(k)] = 0.073 961 for i = 4, ...
Again, for the sake of simplicity we concentrate on samplings on cubic primitive lattices.
Table 4 shows that most organic compounds crystallize in a triclinic, monoclinic, or orthorhombic Bravais lattice with the primitive lattice (87.1 %) by far the most common.
We remark that the method of Ohser and Mucklich (2000) is designed for the general case of cuboidal lattices, while the restriction on the particular case of cubic primitive lattices allows to exploy symmetry properties (Ohser and Schladitz, 2009), and to present the weights in a very condensed form (depending on representatives of the 22 equivalence classes, as in Table 1).
First, the method can simply be extended to arbitrary homogeneous lattices so that we are no longer restricted to cubic primitive lattices (Ohseretal., 2009; Ohser and Schladitz, 2009).

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