From WolfWikis

Diffraction

Diffraction is Mother Nature's way of decomposing scattering power in terms of translational irreps. In the case of X-ray diffraction (XRD) the scattering power is provided by the electron density ρ_e. This concept is quite familiar from quantum mechanics from ρ_e=Ψ*Ψ.

Under appropriate conditions XRD can be described very well as a Fourier transform of ρ_e=Ψ*Ψ, albeit that Ψ represents the wave function of the entire sample. One way of looking at what an XRD experiment does is that it makes (part of the) reciprocal space of the sample visible.

If the sample is a crystal with absolute ideal translation symmetry the scattering is absolutely restricted to the totally symmetric representation Γ₀. That means we can only excite those points of reciprocal space that differ from the origin by whole Q vectors.

Bragg reflections

These points are known as Bragg reflections and the space in between them will remain dark.

If we ignore the dark space and simply collect the set of Bragg reflections, what we get is very is very similar to the simple discrete DFT decomposition of the function ρ inside one unit cell. We just have more of them and our problem is three dimensional rather than 1D

The Bragg intensities map out how the electron density fluctuates inside one unit cell. (The dark space between them is related to fluctuations that involve more than one unit cell.)

This means that if we put them back together (inverse DFT) we should get the structure. There is a problem though. We measure the intensity as a real number, not as a complex amplitude. In other words we measure A*A, not A for each Bragg reflection and lose its phase information. This is why solving the structure is not always a straightforward exercise.

Bragg reflections are typically labeled with their Miller indices (hkl). A reflection (100) represents the first harmonic over the fluctuating electron density inside one unit cell in the x-direction, (030) is the third harmonic in the y-direction and so on.

Again it is useful to think in terms of resolution here. Each harmonic has its own repeat unit, known as the distance d. For a third harmonic the distance d₀₃₀ = d₀₁₀/3, which means you are observing the fluctuation of ρ_e inside the unit cell with three times larger resolution.

Bragg's law

The angle θ under which we observe the scattering of a Bragg reflection is given by:

2d.sinθ= λ

The angle between the incident beam and the scattered one is always taken as 2θ in X-ray and neutron scattering.

Obviously this angle will vary if we vary the wavelength λ. Nowadays with synchrotrons varying the wavelength is commonplace.
Therefore it is best to thing in terms of the scattering vector q:

q = 4π.sinθ/λ = 2π/d

Again we can look at this vector in a variety of ways:

1. it is derived from the scattering angle
2. it is the variable of reciprocal space (k-space)
3. it is a resolution
4. it is a yardstick we probe our system with
5. it is a momentum transfer vector

The latter is best understood by considering the experiment as an elastic collision of a photon with the electron cloud of the sample. The photon bounces off in different direction without losing energy. Thus its energy remains the same but its momentum changes vectorially (in direction, not in magnitude)

There is a natural limit to the highest resolution we can get out of our diffraction pattern, because we cannot observe scattering for angles higher than θ= 90^o. That means that the smallest d we can observe is equal to λ/2. If we want to see atoms we better use something like X-rays or neutrons with a wavelength in the order of Angstroms.

Non-ideal cases

The Bragg spots contain all the information of what happens inside a unit cell. The space in between reflects what happens from cell to cell. Ideally, i.e. if translation symmetry is strictly observed that space remarks dark.

Of course no crystal has true translation symmetry, if only because it is finite, i.e. it has no closure. On top of that there are always defects, like mozaic structure and at finite temperatures the atoms do not form a strict translation lattice because they vibrate.

The finite size and mozaic structure tends to broaden the Bragg points to spots of finite width and that is actually helpful for us to be able to see them better.

Vibrations are best described as phonons (we'll come back to that) and they are not limited to totally symmetric modes. They are waves that can involve many, many unit cells and cause the crystal to temporarily deviate from strict translation symmetry. They do contribute to a tiny bit of scattering in the empty space between the Bragg spots

If the crystal contains other forms of deviations from true symmetry, e.g. there is a molecule in it that can sit in more than one orientation, say up and down, this tends to lead to much more intense diffuse scatter between the Bragg points. The Bragg spots now tell the tell of where the photons encounter electron density (read: your atoms) on the average in the cell. So if a molecule can sit either right side up or upside down and does so 50:50 the Bragg peaks would reveal the superposition of the molecule with itself.

Information on how neighboring molecules relate to each other e.g. will end up as diffuse intensity in what otherwise is the empty space between the Bragg points.