Solids are classified according to regularity and structure of their building blocks, typically atoms and can be the following:

1. Amorphous – No periodic structure at all. All constituent atoms are disordered. An example of this category is glass.

2. Crystal – Periodical arrangement of constituent species. Perfect array of atoms. Examples are diamonds, metals and salt.

3. Polycrystalline – Many small regions of single crystals with various orientations connected by grain boundaries.

To describe crystals, we use the concepts of lattice and basis:

- Lattice
– A collection of mathematical points in a periodic arrangement.

- infinite in extent

- all points are identically surrounded

- mathematically described by*translation vector* - Basis – Set of atoms attached to each lattice point. These are the actual physical things that make up the material.

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Definition** – **A lattice can
be mathematically described by a set of** primative translation vectors**
denoted by **a**_{1}, **a**_{2}, and **a**_{3}. These are vectors such that the atomic
arrangement looks the same in every respect when viewed from one point **r**
as when viewed from the point:

**r’**=**r**+n_{1}**
a**_{1}+n_{2}** a**_{2}+n_{3}** a**_{3}
where n_{1}, n_{2},
n_{3} are integers

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A lattice translation operation is defined as the displacement of a crystal by a crystal translation vector

**T**=n_{1}**
a**_{1}+n_{2}** a**_{2}+n_{3}** a**_{3} where n_{1}, n_{2},
n_{3} are integers

From Kittel Fig 1.2

The
parallelepiped defined by **a**_{1}, **a**_{2}, and **a**_{3}
is called a primitive lattice cell and is a minimum volume cell and has only
one lattice point per unit cell. Other
unit cells may be chosen that have more than one lattice point per unit cell.

There is not a unique primitive cell because the primitive lattice vectors themselves are not unique. One commonly used method to define the primitive cell is by constructing a Wigner-Seitz cell. This is done in a three-step procedure:

1. Draw lines from a lattice point to all nearby lattice points.

2. Draw perpendicular lines and the midpoint of the lines drawn in step 1.

3. The smallest volume enclosed by the lines drawn in step 2 is a valid

primitive cell

Primitive unit cells will have only one lattice point per cell as illustrated below. Other commonly used unit cells have more than one lattice point per cell as will be shown later.

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