The diffraction condition

Consider Fig.21, the change ?? in wave vector ? is in the direction perpendicular to the (hkl) planes.
Provided that scattering is elastic (I? ?I =I?I=2?/?)
I??I= 2sin? IKI
? I??I = (4? sin??)/?
??= (4? sin??)/? n ?
where n ?=Ghkl / IGhklI unit vector along Ghkl
???= (4? sin??)/? G/(IGI ) =(4? sin??)/(?(2?/dhkl )) Ghkl

???= (2dhkl sin??)/?.Ghkl

(2dhkl sin??)/?=1 Bragg’s law for n=1
???=Ghkl (Bragg condition)
? ? = Ghkl + ?

Consider Fig.21, the change ?? in wave vector ? is in the direction perpendicular to the (hkl) planes.
Provided that scattering is elastic (I? ?I =I?I=2?/?)
I??I= 2sin? IKI
? I??I = (4? sin??)/?
??= (4? sin??)/? n ?
where n ?=Ghkl / IGhklI unit vector along Ghkl
???= (4? sin??)/? G/(IGI ) =(4? sin??)/(?(2?/dhkl )) Ghkl

???= (2dhkl sin??)/?.Ghkl

(2dhkl sin??)/?=1 Bragg’s law for n=1
???=Ghkl (Bragg condition)
? ? = Ghkl + ?

When each side of this equation is squared and the quantity ? ? = ? , the Bragg condition appears in the form:
IGI2+2k.G=0
This is the central result of the theory of elastic scattering of waves in a periodic lattice.
If G is a reciprocal lattice vector, so is - G , then
2k.G= IGI2
This Eq. is another statement of the Bragg condition or called Bragg equation in reciprocal lattice