The basics of crystallography and diffraction (3rd Edition)

196 X-ray diffraction: Bragg’s law
8.3 Bragg’s analysis of X-ray diffraction: Bragg’s law
Laue’s analysis is in effect an extension of the idea of a diffraction grating to three
dimensions. It suffers from the severe practical disadvantage that in order to calculate
the directions of the diffracted beams, a total of six angles α n , α 0 , β n , β 0 and γ n , γ 0 ,
three lattice spacings a, b and c, and three integers n x , n y and n z need to be determined.
As discussed in Section 5.5, W. L. Bragg envisaged diffraction in terms of reflections
from crystal planes giving rise to the simple relationship (Bragg’s law, derived below):
nλ = 2d hkl sin θ.
It can be seen immediately, by comparing the Laue equations with Bragg’s law, that
the number of variables needed to calculate the directions of the diffracted beams are
much reduced.
Bragg’s law may be derived with reference to Fig. 8.3(a) which shows (as for the
derivation of the Laue equations) a simple crystal with one atom at each lattice point.
The path difference between the waves scattered by atoms from adjacent (hkl) lattice
planes of spacings d hkl is given by
(AB + BC) = (d hkl sin θ + d hkl sin θ) = 2d hkl sin θ.
Hence for constructive interference:
nλ = 2d hkl sin θ,
where n is an integer (the order of reflection or diffraction).
As explained in Section 5.5, n is normally incorporated into the lattice plane
symbol, i.e.
( )
dhkl
λ = 2 sin θ = 2d nh nk nl sin θ
n
where nh nk nl are the Laue indices for the reflecting planes of spacing d hkl /n. In other
words, to repeat the important point made in Section 5.5, n is not written separately but
(a)
u
u
A
B
C
u
u
u u
d hkl
u
C
A
B
(b)
u
Fig. 8.3. (a) Bragg’s law for the case of a rectangular grid, i.e. AB = BC = d hkl sin θ; the path
difference (AB + BC) = 2d hkl sin θ. (b) Bragg’s law for the general case in which AB ̸= BC. Again,
the path difference (AB + BC) = d hkl sin θ.