Practical Electron Microscopy and Database

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:

         ----------------------------- [4545]

where,
        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.

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:
                 ---------------------------------- [3073]

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.