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 above figure X-rays with wavelength `lambda` strike a crystal fạce at an angle `theta` and then reflect at the same angle. The rays that strike an atom in the second layer are diffracted at 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 angel ` theta `
the same 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 na where n is an integer (1,2,3,4....)
` :. AB + BC = 2 d sin theta =n lambda ` This is known as Bragg.s equation.
By knowing the values 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.