Directions and Planes in a Crystal - Civil Engineering (CE) PDF Download

Directions and Planes in a crystal Directions:
 In crystals there exists directions and planes in which contain concentration of atoms. It is necessary to locate these directions and planes for crystal analysis. Arrows in two dimensions show directions.The directions are described by giving the coordinates of the first whole numbered point ((x, y) in two dimension,(x,y,z) in three dimension) through which each of the direction passes. Directions are enclosed within square brackets.

Planes:
The crystal may be regarded as made up of an aggregate of a set of parallel equidistant planes, passing through the lattice points, which are known as lattice planes. These lattice planes can be chosen in different ways in a crystal. The problem in designating these planes was solved by Miller who evolved a method to designate a set of parallel planes in a crystal by three numbers (h k l) called Miller Indices.

Miller indices 
Miller indices may be defined as the reciprocals of the intercepts made by the plane on the crystallographic axes when reduced to smallest numbers.

Steps to determine Miller Indices of given set of parallel planes:
Consider a plane ABC which is one of the planes belonging to the set of parallel planes with miller indices (h k l). Let x, y and z be the intercepts made by the plane along the Three crystallographic axes X, Y and Z respectively.

(1) Determine the coordinates of the intercepts made by the plane along the three crystallographic axes.
(2) Express the intercepts as multiples of the unit cell dimensions, or lattice parameters along the axes.
(3) Determine the reciprocals of these numbers.
(4) Reduce them into the smallest set of integral numbers and enclose them in simple brackets. (No commas to be placed between indices)

Example: Lets determine the Miller indices for the plane shown in the fig (7.7)
(1) The intercepts x = 2a , y = 2b and z = 5c. In general, x = pa , y = qb , z = rc

(2) The multiples of lattice parameters are

Taking the reciprocals

(3) Reducing the reciprocals to smallest set of integral number by taking LCM.

Note:
(a) All parallel equidistant planes have the same Miller indices.
(b) If the Miller indices have the same ratio, then the planes are parallel.
(c) If the plane is parallel to any of the axes, then the corresponding intercepts is taken to be ∞.

Expression for Interplanar spacing in terms of Miller Indices
Consider a Lattice plane ABC, which is one of the planes belonging to the set of planes with Miller indices (h k l). Let x, y and z be the intercepts made by the plane along the Three crystallographic axes X, Y and Z respectively.

Let OP be the perpendicular drawn form the origin to the plane. Let  and be the angles made by OP with the crystallographic axes X, Y and Z respectively. Let another consecutive plane parallel to ABC pass through the origin. Let a, b and c be the lattice parameters. OP is called interplanar spacing and is denoted by dhkl. From right angled triangle OCP

From right angled triangle OBP

From right angled triangle OAP

but we know that

Therefore

for the rectangle Cartesian coordinate system we have

is the expression for interplanar spacing.
For a cubic lattice a=b=c,.Therefore