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 40 Å
What is the range of wavelength for X-rays? Answer: Ranges from 0.5 to 10.0 Å
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, 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 has vectoral properties with the ray defined a the direction of 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 (they are not all in the same place) some destructive interference occurs.
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.
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 (&theta), 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, &theta= &theta' and AC =DCsin &theta.
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 = n&lambda), but it does not.
Note that the distance AC is the interplanar d-spacing.
Under "reflecting" conditions then &theta = &theta' and sin &theta = AC / DC and sin &theta' = AE / CE
Let: d = AC and n&lambda = DC + CE
then: 2AC sin &theta = DC + CE or n&lambda = 2dsin &theta