# Direction And Planes In A Crystal

**Description:**

Many physical properties of crystalline solids are dependent on the direction of measurement or the planes across which the properties are studied. In order to specify directions in a lattice, we make use of lattice basis vectors **a**, **b **and **c**.

In general, any directional vector can be expressed as

**R**= n_{1}**a** n_{2}**b** n_{3}**c **

where n_{1}, n_{2} and n_{3} are integers. The direction of the vector **R **is determined by these integers. If these numbers have common factors, they are removed and the direction of R is denoted as [n_{1} n_{2} n_{3}]. A similar set of three integers enclosed in a round bracket is used to designate planes in a crystal.

**Lattice planes and Miller indices: **

The crystal lattice may be regarded as made up of a set of parallel, equidistant planes passing through the lattice points. These planes are known as lattice planes and may be represented by a set of three smallest possible integers. These numbers are called ‘Miller indices’ named after the crystallographer W.H.Miller.

**Determination of Miller indices: **

Consider a crystal plane intersecting the crystal axes as shown:

The procedure adopted to find the miller indices for the plane is as follows:

1. Find the intercepts of the plane with the crystal axes along the basis vectors **a**, **b**and **c**. Let the intercepts be x, y and zrespectively.

2. Express x, y and z as fractional multiples of the respective basis vectors. Then we obtain the fractions,

x/a, y/b, z/c .

3. Take the reciprocal of the three fractions to obtain 𝐚/𝐱, 𝐛/𝐲, /𝐳.