Lattice Point/Motif/Basis
- Practical Electron Microscopy and Database -
- An Online Book -
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A lattice point is known as a motif or basis. We can obtain a crystal structure by combining the lattice with the motif (i.e., crystal structure = lattice + motif). Figure 3076a shows a periodic pattern consisting of a two-dimensional (2-D) net and a motif. The motif is arranged symmetrically and is repeated at each point of the 2-D net to create the periodic pattern, and thus the lattice structure is also symmetric.

 (a) (b) (c)
Figure 3076a. (a) A 2-D net, (b) A motif, and (c) Formed periodic pattern (2-D lattice structure).

Note that all lattice points must be identical. The unique arrangements of lattice points are so-called Bravais lattice, named after Auguste Bravais.

The number of the lattice points per unit cell in 2-D lattices can be given by,
---------------------------- [3076a]
where,
NInterior and NCorner - The numbers of the lattice points inside the unit cell and at the corners, respectively, as shown in Figure 3076b.

Figure 3076b. Lattice points inside the unit cell and at the corners in 2-D lattices.

On the other hand, the number of the lattice points per unit cell in 3-D lattices can be given by,
---------------------------- [3076b]
where,
NFace - The number of the lattice points at the faces as shown in Figure 3076c.

Figure 3076c. Lattice points inside the unit cell and at the corners in 3-D lattices.

Table 3076. Typical lattices and their basis.
 Crystal Basis CsCl, TlBr, TlI, RbCl, CsI, CsBr, CuZn, AgMg, LiHg, TlCl, AlNi, BeCu SC (simple cubic) lattice with a two-atom basis. E.g. for CsCl, Cl at 0 0 0 and Cs at 0.5 0.5 0.5 Diamond structure: C, Si, Ge (single type of atom) FCC lattice with a two-atom basis. E.g, for Si, Si at (0, 0, 0) and Si at 1/4, 1/4, 1/4. NaCl, KCl, LiF, AgCl, MgO, CaO, GaAs, GaP, InP, ZnSe FCC lattice with a two-atom basis. E.g. for GaAs, Ga at (0, 0, 0) and As at 1/4, 1/4, 1/4. CaF2, FeS2 FCC lattice with a three-atom basis. E.g. for CaF2, space group Fm-3m, Ca at 0, 0, 0 and F at 1/4, 1/4, 1/4. AlN HCP (hexagonal close-packed) lattice with a two-atom basis. E.g. Al at 1/3, 2/3, 0.0 and N at 1/3, 2/3, 0.38. Cu2O More complex but still cubic. E.g. for Cu2O, Cu at 0, 0, 0 and O at 1/4, 1/4, 1/4. TiO2, CuO Much more complex. E.g. for TiO2 (a = 0.4593 nm and c = 0.2959 nm): Ti at 0, 0, 0 and O at 0.3053, 0.3053, 0. Al2O3, CdI2 "HCP" anions but not HCP structure. E.g. for Al2O3, Al at 0, 0, 0.355 and O at 0.303, 0, 1/4. MoS2 Layered material. For MoSi2, Mo at 0, 0, 0 and Si at 0, 0, 1/3.

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