Derive Bragg's equation .

Explanation

Derivation of Bragg's equation : When X - rays are incident on the crystal or plane , they are diffracted from the lattice points (lattice points may be atoms or ions or molecules) . In the crystal the lattice points are arranged in regular pattern . When the waves are diffracted from these points , the waves may be constructive or destructive interference .
The `1^(st)` and `2^(nd)` waves reach the crystal surface. They undergo constructive interference . Then from the figure `1^(st)` and `2^(nd)` rays are parallel waves . So they travel the same distance till the wave form AD . The second ray travels more than the first by an extra distance (DB + BC) after crossing the grating for it to interfere with the first ray in a constructive manner .
Then only they can be in the same phase with one another . If the two waves are to be in phase , the path difference between the two ways must be equal to the wavelength `(lambda)` or integral multiple of it (n `lambda` where n = 1 , 2 , 3 .... )
(i.e.,) `n lambda = (DB + BC) " " ` [where n = order of diffraction ]
`DB = BC= d sin theta " " [ theta ` = angle of incident beam,]
`(DB + BC) = 2 d sin theta " " ` d= distance between the planes
`n lambda = 2 d sin theta ` This relation is known as Bragg's equation .