A solution and solid state study of niobium complexes University of ...

3.5.1 Bragg’s law
Chapter 3
Bragg’s law 10 defines the conditions under which diffraction occurs and gives the
position of a diffracted beam without any reference to its intensity. X-rays are
diffracted by “lattice planes” within a crystal. All X-rays reflected from a given plane
are in phase after reflection. Neighboring plane X-rays travel different path lengths
and are out of phase after reflection. Bragg’s law is used to correct phase difference
for diffracted rays by two adjacent planes:
nλ = 2dhklsin θ (3.6)
where n is an integer, λ is the wavelength of the radiation, dhkl is the interplanar
spacing of the (hkl) planes and θ is the diffraction angle (Bragg angle).
X-ray beam
dhkl
Figure 3.3: Diffraction of X-rays as proposed by Bragg. 11
There are an infinite number of sets of atom planes within a crystal, and Bragg’s law
applies to all these. Subsequently, if a crystal is rotated in an X-ray beam, each set
of planes will in turn diffract the radiation when the value of sin θ becomes
appropriate. The part of the X-ray that is not reflected off the upper atomic plane with
angle θ can pass on to the following plane and reflect at the same angle. The
spacing between these lattice planes is represented as d and all X-rays reflected of a
given plane are in phase afterwards. The arrangement of the diffracted beams gives
10 D. Sherwood, J. Cooper, Crystals, X-ray’s and Proteins: Comprehensive Protein Crystallography, Oxford
University Press, New York, 291, 2011.
θ
48
2θ
Diffracted
beam
Atom planes
(hkl)
Undiffracted
beam