A lattice symmetry has the following basic characteristics:
i) Each lattice point must have identical surroundings, i.e., have the same environment.
ii) In a given direction, all lattice points must be separated by an identical distance, so-called lattice parameter.
As shown in Figure 4545a, for an one-dimensional (1-D) lattice, a translation of l from one lattice point to another will present an “identical” lattice:
n -- An integer.
l and a -- Vectors.
Figure 4545a. One-dimensional (1-D) lattice.
As shown in Figure 4545b, a two-dimensional (2-D) lattice has the following characteristics:
i) It has two non-collinear basis vectors (a and b). The interaxial angle γ determines the relationship between the two basis vectors.
ii) A translation of na + pb (e.g. 1a + 3b in Figure 4545b) from one lattice point to another must form an “identical” lattice structure. Here, n and p are integers.
Figure 4545b. Two-dimensional (2-D) lattice.
As shown in Figure 4545c, a three-dimensional (3-D) lattice has the following characteristics:
i) It has 3 non-collinear basis vectors (a, b, and c) and 3 interaxial angles (α, β, and γ).
ii) All points can be determined by a series of vectors:
Figure 4545c. Three-dimensional (3-D) lattice.
Note that crystal lattices must exhibit a specific minimal amount of symmetry, and each crystal system has a certain symmetry. Table 4545 lists the relationship between three-dimensional crystal families, crystal systems, and lattice systems.
Table 4545. Relationship between three-dimensional crystal families, crystal systems, and lattice systems.