Elastic scattering on crystals ↩

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aliases: [Elastic wave scattering on crystals, Elastic X-ray scattering on crystals]
shorthands: {}
created: 2021-10-31 20:53:23
modified: 2022-01-20 18:29:37

A common practice is to use scattering experiments to examine the structure of crystals. In these experiments we analyze the scattering pattern made by the deflected waves or particles. In elastic scattering, the energy of the scattering particle/wave remains unchanged, only its direction of propagation changes.

In these experiments we describe the scattering amplitude like this:

Where denotes the lattice vectors of the crystal lattice, is the change of wavenumber between the incoming and the scattered wave and are the position of atoms inside the unit cell. For example in the case of X-ray scattering, is the charge density of electrons for the atom.

The atomic scattering factor is defined as: ^{^37701d}

It describes the scattering on individual

atoms.
Similarly we define the structure factor like this:

Now this describes how the scattering happens on the whole unit cell.
We define the lattice sum as:

With these definitions, the scattering amplitude is given as:

The intensity of a wave scattered with is:

Lattice sum

Let's consider a finite size crystal with lattice vectors where , and . The number of lattice vectors is then .

With a now omitted derivation we get that in the thermodynamic limit, the lattice sum is:

(Where are the reciprocal lattice vectors)
We can see that the lattice sum gives us a strong indication for the possible wavenumbers for which a scattering particle can be deflected.
Based on this, in the scattering pattern, only the Bragg-peaks show up corresponding to the reciprocal lattice vectors. The waves destructively interfere in the other directions.

Atomic form factor for spherically symmetric charge distributions

Let's consider a spherically symmetric charge distribution: . The atomic form factor is then:

Atomic form factor for 1s orbits

The wavefuction of a 1s orbit (see Hydrogen atom in quantum mechanics):

The charge distribution based on this:

Where is the Bohr radius.

From these, the atomic form factor is: