(1)Bragg's law is given by,
(1)
Multiply both sides by an integer m such that md=t, the thickness of the crystal. This leads to,
(2)
Eqn. (2) can also be interpreted as the mth order reflection from a set of planes with a interplanar distance t.
Differentiate both sides of eqn. (2) remembering ml is a constant. This gives,
(3)
Remembering that as D q can be positive or negative (we are only interested in absolute values) leads to,
(4)
Since the smallest increment in t is d, using D t=d, and substituting l /2 for dsinq (from Bragg's law) we get,
(4)
Substituting B for 2D q , the angular width, we get,
(5)
which is essentially Scherrer equation.
A more sophisticated analysis of the problem only adds a prefactor of 0.9 to the right hand side of eqn. (5) and leads to the correct Scherrer equation.
(6)