Crystalline State

 

            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:

 

  1. Lattice – A collection of mathematical points in a periodic arrangement.
                            - infinite in extent
                            - all points are identically  surrounded
                            - mathematically described by translation vector
  2. Basis – Set of atoms attached to each lattice point.  These are the actual physical things that make up the material.

 

 

 

 

Translation Vectors

 

Definition – A lattice can be mathematically described by a set of primative translation vectors denoted by a1, a2, and a3.  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+n1 a1+n2 a2+n3 a3       where n1, n2, n3  are integers

 

 

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

 

T=n1 a1+n2 a2+n3 a3                       where n1, n2, n3  are integers

 

From Kittel Fig 1.2
 

 

Primitive Lattice Cell      

 

      The parallelepiped defined by a1, a2, and a3 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.