Diffraction

1.Background

The electron microscope does not have good enough resolution for accurate direct determination of unknown crystal structures. Diffraction patterns provide the most accurate data about crystal structures.

A sample of the crystal being studied is bombarded with photons, electrons or neutrons with an associated wavelength comparable to the interatomic spacing. A single atom (theoretically) scatters the incident waves equally in all directions, but in a crystal cancellation due to destructive interference gives zero in most directions. In certain directions constructive interference gives maxima of intensity, producing a pattern characteristic of the crystal structure. The problem for crystallographers to solve is how the positions of the peaks observed can be converted into useful information about the crystal structure.

2. The Bragg Law

This law was derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence.

Derivation of the Bragg law; notice that reflection is specular

The Bragg law is based upon 2 assumptions:

1) Diffraction peaks are caused by constructive interference of waves reflected from parallel planes of atoms

2) The incident waves are reflected specularly, that is the angle of incidence is equal to the angle of reflection

For constructive interference, the path difference between waves reflected from the 2 planes must be an integer number of wavelengths. As can be seen in figure 11 above, the path difference is 2dsinq and so the Bragg law is

nl = 2dsinq (2)

There are many possible planes of atoms and so there are many peaks in the diffraction pattern. Only a few are shown on the diagram below, which shows a plane of atoms perpendicular to the c axis in an arbitrary crystal.

Some Miller planes which could cause diffraction

3. Experimental Methods

Peaks of intensity are hence obtained for an appropriate combination of l, d and q.

Since d is fixed, it is necessary to scan in either wavelength or angle.

In the Laue Method, a single crystal is stationary in a beam of x-ray or neutron radiation of continuous wavelength. Diffraction only occurs at the appropriate discrete values of l for which planes exist of spacing d and incidence angle q satisfying the Bragg law.

In the rotating crystal method, a single crystal is rotated about a fixed axis in a beam of monoenergetic x-rays or neutrons. The variation in angle brings different planes into the appropriate position for reflection.

In the powder method monochromatic radiation strikes a fine powder of the specimen. In this form the crystallites will be present in virtually every possible orientation. Diffracted rays are reflected from individual crystallites that happen to be in the appropriate orientation. The reflected beam is detected at an angle 2q to the original beam.