4 - Lecture notes for Clay Mineralogy


Readings:


X-ray Diffraction Principles


The essential feature for all diffraction phenomena is that the wavelength of the wave is about the same as the distance between the scattering points through which the waves are traveling.

What are typical interplanar distances in a clay mineral? Answer: Can range from 0.1 to 20 Å (0.01 to 2 nm)

What is the range of wavelength for X-rays? Answer: Ranges from 0.5 to 20.0 Å (0.05 to 2 nm)

So it is a coincidence that we use X-rays for diffraction studies of crystalline material?

 
There are two theoretical approachs to the study of X-ray diffraction. Kinematic theory considers scattering from each atom is independent of all other atoms and once scattered and the X-rays pass beyond without further scattering. Dynamical theory takes into account all the wave interactions within a crystal. In other words, the total electromagnetic field is considered in dynamic theory because the incident and diffracted beams swap energy back and forth. Dynamic theory must be employed if a large single crystals involved, because the scattered beam may be rescattered to recombine again with the primary beam. As it turns out for fine crystal powders such as clay minerals, the underlying assumptions for kinematic theory can explain most of the observed phenomenon. We will therefore focus of our kinematic theory for our discussions.

Coherent Scattering

When X-rays encounter electrons they are scattered....

Electromagnetic radiation (EM) has vectoral properties with a ray path defined as the direction of EM propagation.

As with all electromagnetic radiation there is an electric E component vibrating perpendicular to the ray and a magnetic H vector perpendicular to the electric vector.

The electric component interacts with the electrons of the atoms, vibrating in resonance, essentially absorbing and re-emitting the same frequency radiation in all directions.

The electrons around the nucleus do the scattering, therefore the scattering power of an atom increases with the number of electrons bound to the atom.

The scattering power is not exactly proportional to the number of electrons, because as the number of electrons increase some destructive interference occurs. In other words, electrons are not all in the same place and there is a phase shift.

 

Interference

A diffracted beam is a beam that results from a great number of constructively interfered wave fronts. The two waves in the diagram above have path-length-difference of 1/4 of the wavelength.

Click here to download a simple Excel spreadsheet that allows you to add two waves of equal wavelength together.

Click here to see a movie of spherical wave fronts coming from a single scattering point in a row of waves.

What condition allows for numerous wave fronts to come together in a constructive way?

Scattering from a row of atoms.

Regularly spaced scattering centers (i.e., atoms) result in the constructive interference at specific points in space and destructive interference in all other points in space.

Note the points of constructive interference in the above figure

Wave fronts of constructive interference result.

These fronts form cones of constructive interference.

Where cones intersect in 3-dimensions, further constructive interference occurs.


Braggs's Law

Scattering from a three dimensional crystal structure

Until now we have only considered a single row of atoms. We know that atoms in a crystal lattice are arranged in an orderly three dimension array. For each linear set of scattering atoms (row) there is a set of diffraction cones that emanate from atom centers. The places where the cones coincide (i.e., constructively interfere) occur under unique geometric conditions.

If we consider an incident beam approaching scattering centers at some angle (θ), it can be shown that the only place the scattered beam will be in phase is at the same or "reflected" angle that leaves the scattering points.

Geometrically, the conditions of constructive interference are met only when DC = CE therefore, θ = θ' and AC = DC sinθ.

Under these conditions there is zero path length difference between rays 1 and 2.

Unlike light, which can be reflected at all angles, X-rays are "reflected" only at specific angles.

The wave fronts that pass through a crystal must have path-length-differences exactly 1,2,3...n integers away or they will destructively interfere.

Compare Rays 1 and 3.

Note path length difference into the plane of atoms is the distance FB + BG.

This additional distance must be equal to some integer distance (i.e., FB + BG = nλ), but it does not.

Note that the distance AC is the interplanar d-spacing.

Under "reflecting" conditions then θ = θ' and sin θ = AC / DC and sin θ' = AE / CE

Let: d = AC and nλ = DC + CE

then: 2AC sin θ = DC + CE or     nλ = 2d sinθ