The Bragg law
We study crystal structure through the diffraction of photons, neutrons,
and electrons (Fig. 1).
The diffraction depends on the crystal structure and on the wavelength. At optical wavelengths such as 5000 Å, the superposition of the waves scattered elastically by the individual atoms of a crystal results in ordinary optical refraction.
When the wavelength of the radiation is comparable with or smaller than the lattice constant, we may find diffracted beams in
directions quite different from the incident direction.
Fig. 1 –
W. L. Bragg presented a simple explanation of the diffracted beams from a crystal.
The Bragg derivation is simple but is convincing only because it reproduces the correct result.
Suppose that the incident waves are reflected specularly from parallel planes of atoms in the crystal, with each plane reflecting only a very small fraction of the radiation, like a lightly silvered mirror.
In specular (mirror like) reflection the angle of incidence is equal to the angle of
The diffracted beams are found when the reflections from parallel planes of atoms interfere constructively, as in Fig. 2.
Fig. 2 –
We treat elastic scattering, in which the energy of the x-ray is not changed on reflection.
Consider parallel lattice planes spaced d apart. The radiation is incident in
the plane of the paper.
The path difference for rays reflected from adjacent planes is 2d sin θ , where is measured from the plane.
Constructive interference of the radiation from successive planes occurs when the path difference is an integral number n of wavelengths , so that
2d sin θ = n λ. (1)
This is the Bragg law, which can be satisfied only for wavelength λ ≤ 2d
Although the reflection from each plane is specular, for only certain values
of will the reflections from all periodic parallel planes add up in phase to give
a strong reflected beam.
If each plane were perfectly reflecting, only the first plane of a parallel set would see the radiation, and any wavelength would be reflected.
But each plane reflects 10-3 to 10-5 of the incident radiation, so that
10-3 to 10-5 planes may contribute to the formation of the Bragg-reflected beam in
a perfect crystal.
Reflection by a single plane of atoms is treated on surface physics.
The Bragg law is a consequence of the periodicity of the lattice. Notice
that the law does not refer to the composition of the basis of atom associate with every lattice point.
We shall see, however, that the composition of the basis determines the relative intensity of the various orders of diffraction
(denoted by n above) from a given set of parallel planes.
Bragg reflection from a single Crystal shown in fig.3 and a powder in fig. 4