Solids are classified as either amorphous or crystalline on the basis of their three-dimensional internal structure.

Amorphous solids like fused silica glass lack an ordered internal arrangement of their constituent particles, whereas crystalline solids like quartz have their constituent particles arranged in a repeating three-dimensional pattern throughout the solid. 

The structure of a crystalline solid is represented by a unit cell, which is the smallest repeating unit of the crystalline structure that retains the symmetry of the structure.

The overall three-dimensional pattern is known as a crystal lattice, which is composed of lattice points and lattice vectors. The lattice vectors delineate the edges of the unit cell, and the lattice points may be at the corners, on the faces, or at the center of the unit cell.

Lattice systems are defined by the dimensions of the unit cell. There are 7 types of lattice systems: cubic, tetragonal, orthorhombic, rhombohedral, monoclinic, triclinic, and hexagonal.

The positions of the atoms in a unit cell are not necessarily the same as those of the lattice points. The pattern of atoms in the unit cell, or motif, is often defined in terms of the locations of the atoms relative to a given lattice point.

The number of atoms in a unit cell reflects the packing efficiency of the solid, or the amount of its volume occupied by atoms rather than the space between them. A higher number of atoms in the unit cell generally corresponds to more efficient packing.

Atoms assigned to a unit cell may not be wholly contained within the cell. One way to count these partial atoms is to consider each atom on a corner as one-eighth of an atom and each atom on a face as one-half of an atom.

Alternatively, if a unit cell has an atom on each corner, one is assigned to the unit cell and the other seven are ignored. If a unit cell has an atom on each of two faces, one is assigned to the unit cell and the other is ignored.