Planes

Direction and planes in a crystal

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= n₁a+n₂b+n₃c

Where n₁, n₂ and n₃ 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 [n₁ n₂ n₃]. 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 z respectively.

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, c/z

4. Find the least common multiple of the denominator, by which multiply the above three ratios. This operation reduces them to a set of 3 integers (h k l) called miller indices for the crystal plane.

For the plane given above,

1. x = 2a/3 y = 3b/2 z = 2c

2. (x/a y/b z/c) = (2/3 3/2 2)

3. Taking reciprocal, (3/2 2/3 1/2)

4. Multiplying throughout by the least common multiple 6 for the denominator, we have the miller indices, (9 4 3)

Expression for interplanar spacing

The interplanar spacing d_hkl is given by,

For a cubic lattice, a = b =c