In this section we discuss the arrangements of atoms in various solids. We shall distinguish between single crystals and other forms of materials and then investigate the periodicity of crystal lattices.

**Periodic Structures:**

- A crystalline solid is distinguished by the fact that the atoms making up the crystal are arranged in a periodic fashion. That is, there is some basic arrangement of atoms that is repeated throughout the entire solid.
- Thus the crystal appears exactly the same at one point as it does at a series of other equivalent points, once the basic periodicity is discovered.
- However, not all solids are crystals (figure below), some have no periodic structure at all (amorphous solids), and others are composed of many small regions of single-crystal material (polycrystalline solids).

- The periodicity in a crystal is defined in terms of a symmetric array of points in space called the lattice.
- We can add atoms at each lattice point in an arrangement called a basis, which can be one atom or a group of atoms having the same spatial arrangement, to get a crystal.
- In every case, the lattice contains a volume or cell that represents the entire lattice and is regularly repeated throughout the crystal.
- As an example of such a lattice, Fig. given below shows a two-dimensional arrangement of atoms called a rhombic lattice, with a primitive cell ODEF, which is the smallest such cell.
- Notice that we can define vectors a and b such that if the primitive cell is translated by integral multiples of these vectors, a new primitive cell identical to the original is found (e.g., O’D’E’F’).
- These vectors, a and b (and c if the lattice is three dimensional), are called the primitive vectors for the lattice. Points within the lattice are in distinguishable if the vector between the points is

where p, q, and s are integers.

- A primitive cell has lattice points only at the corners of the cell. It is not unique, but the convention is to choose the smallest primitive vectors.
- Note that, in a primitive cell, the lattice points at the corners are shared with adjacent cells; thus, the effective number of lattice points belonging to the primitive cell is always unity.
- Since there are many different ways of placing atoms in a volume, the distances and orientation between atoms can take many forms, but it is the symmetry that determines the lattice, not the magnitudes of the distances between the lattice points.