Derive Bragg's equation .

Explanation

When a beam of X-rays strikes a crystal, the constituent particles of the crystal scatter or deflect some of the X-rays from their original path. The scattered X- ray.s can be detected on a photographic.plate. From the diffraction pattern, the distance between constituent particles in a crystal can be studied.

In the above figure X -rays with wavelength `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 same 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=2 d sin theta( :. 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 wavelengths `n lambda` where n is an integer (1, 2, 3, 4....)
`:. AB+B=2 d 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 distanced between layers of atoms in a crystal can be calculated