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 -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 given 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), sin theta`
The X-rays have to travel a total.extra distance AB + BC:
`AB+BC=2d 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 na where n is an integer (1, 2, 3, 4.....)
`:. 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.