Derive Bragg's equation .

Explanation

When a beam of X - rays strikes a crystal, the constituent particles of the crystal scatter or diflect some of the X - rays from their original path. The scattered X - rays can be detected on a photographic plate. From the diffraction pattern, the distance between constituent particles in a crystal can be studied.
In the figure X - rays with wave - length `lambda` strike a crystal face at an angle `theta` and then reflect at the same angle. The rays that strike an atom in the second layer are diffracted at the saem angle `theta`, but because the second layer of atoms away from the X - ray source, the distance that the X - rays has to travel is more to reach the second layer. The path difference is indicated by AB in the figure. AB is equal to the distance between atomic layers d (=zB) times the sine of the angle `theta`.

`sin theta = (AB)/(d) , " So " AB = d sin theta`
The X - rays have to travel a total extra distance AB + BC.
AB + BC = ed `sin theta` `(therefore AB = BC)`.
The different X - rays striking the two layers of atoms are in - phase initially and also they will be in - phase after reflection only the extra distance AB + BC is equal to a whole number of wavelenghts `n lambda` where n is an integer (1, 2, 3, 4......)
`therefore AB + BC = 2d sin theta = n lambda`
This is known as Bragg.s equation.
By knowing the value of `phi, lambda` the value of `theta` can be measured, and the value of n is a small integer, usually 1. Thus the distance d between layers of atoms in a crystal can be calculated.