Experimental methods for determining crystal structure

Bragg's Law

A visual depiction of the relation of wavelength, atomic spacing, and scattering angle.

The core principle behind using waves scattered off atomic centers to obtain information about the internal structure of a crystal is that of Bragg's Law. Bragg's Law gives a relation between wavelength of the incident waves which are being used to scatter $\lambda$, the distance between the scattering centers, in this case, the atomic nuclei $d$, and the angle between the waves and the plane formed by the atoms in the lattice $\theta$.

When the incident waves reflect of the plane formed by the atomic centers, they reflect and can interfere either constructively or destructively. Where they constructively interfere, a film placed in the path of the waves will record a dot, and places with destructive interference record nothing. Doing this sort of scattering process from many different angles will give a 3-D pattern representing the location of atoms in the lattice.

X-ray scattering

X-rays are scattered off the periodically arranged atoms, the particular ways the X-rays scatter give information about the location of atoms in the crystal lattice.

Electron scattering

Because electrons do have a wave nature, they can also be scattered off atoms in a crystal lattice, this process is similar to X-ray scattering. Some differences are that

1. Electrons used in this way have a smaller wavelength than X-rays, this allows for greater resolution.
2. The thickness of the sample must be very small to be transparent with respect to the electrons, on the order of ~100nm.

Neutron diffraction

Similar to electron and X-ray diffraction, neutrons with wavelength ~0.1nm are used to scatter off atomic centers in a crystal to create a pattern from which its atomic structure can be derived.