Scattering: Unmasking Nature

Understanding the composition and structure of matter is perhaps the most ubiquitous objective across the physical sciences. The fundamental structures found within matter are typically on the nanometre scale and have thus proved unobservable for most of history. Following a number of developments during the 20th century, we are now capable of probing nature to such scales. Here, we discuss arguably the greatest tool at our disposal, analysis of the nature in which small particles are scattered from the substance of interest.

Crystal Structure

Most of the objects we interact with on a daily basis are solids. Unlike liquids and gases, particles within a solid are not free to move through space, but are fixed to a particular position. In many solids, particles organise themselves into orderly, repeating structures and are known as crystals. A crystal is an array of unit cells, small regions of space that, when stacked together, fill all of the space within the material.

The unit cells of crystals found in nature can be modelled by 1 of the 14 Bravais lattices, depending on the separation and orientation of its particles. A few of the Bravais lattices are shown below.

Real lattices contain deviations from these ideal structures due to the effects of temperature and other interactions with their surroundings. These are known as defects and result in structural distortions and, if on a large enough scale, changes in the macroscopic properties of the material. The typical size of interatomic distances (labelled a in the diagram above) is of the order 10-10m.

Scattering Theory

Small, energetic particles are deflected by the atoms within crystals, altering their direction and velocity. This phenomenon is known formally as scattering. The nature of the scattering is determined by the properties of the incident particles and the positions and nature of the atoms within the crystal. By measuring the angle at which a beam of particles is deflected (denoted θ in the diagram below) in a number of different directions, we can begin to build a model of the crystalline structure.

The structure of a crystal is periodic, meaning once you have successfully imaged the unit cell, you have a model of the entire material. The atoms within the unit cell create a potential that incident particles will experience if they travel close enough. The nature of the potential is expressed within the structure factor, a mathematical term that stores information on the nature of the unit cell.

A range of different particles are chosen for scattering experiments depending on the material we would like to probe. A couple of the main candidates, x-rays and neutrons, are discussed below.

X-ray Scattering

X-rays are a form of electromagnetic radiation with high energies and are scattered heavily by atomic electrons. The amount of scattering, known as the scattering intensity, will therefore be proportional to the number of electrons on each atom, or atomic number, Z, of the material. Crystals containing elements higher up on the periodic table can be well imaged via X-ray scattering whereas lighter elements will not be observable.

Neutron Scattering

Neutrons are uncharged and therefore do not experience the electromagnetic force. A beam of neutrons will interact only with the atomic nuclei within the crystal via nuclear forces and ignore nearby electrons. The strong force has an incredibly short range; the structure factor for unit cells during neutron experiments is proportional to the nuclear scattering length which is typically of the order 10-15m.

The main advantage of neutron scattering is that the scattering intensity doesn’t depend on the atomic number of the material at question. Neutrons are ideal for imaging structures containing large amounts of Hydrogen (Z = 1) and other elements lower down on the periodic table. Neutrons also possess a quantum mechanical property called spin, allowing the spin of particles within the crystal to be deduced.

Experimental Setup

There are a number of different practical approaches to scattering experiments. To generate a 3D image of the sample, either the beam or crystal must be regularly rotated so that the different planes of the unit cell can be observed. One way of getting around this is by using powder diffraction, a popular method that utilises a sample that is not single crystalline, but is powdered. The incoming beam of particles can be scattered by one of many small crystallites which may be orientated in any possible direction.

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