# Neutron

Neutrons are subatomic particles with no electric charge. Along with protons, they form the nuclei of atoms. Like all particles, neutrons behave as waves; thus beams of neutrons can be used for scattering experiments (e.g. SANS). In fact, there are neutron analogues of most light or x-ray methods (including diffraction, reflectivity, inelastic scattering, etc.). The unique properties of neutrons also make possible some methods not possible with other probes (e.g. neutron spin echo).

X-rays interact predominantly through a material's electrons (one can imagine the oscillating E-field of the incident x-ray photons driving oscillations in the electron clouds). Thus, x-ray scattering probes the realspace electron-density distribution. Neutrons, on the other hand, do not interact with charges. Instead, neutrons interact with atomic nuclei (with the potential-well defined by the nuclear energy states). The neutron scattering power of each element is given by the neutron scattering lengths. The scattering power for a given material is then given by the Scattering Length Density (SLD), which accounts for the atoms present, as well as their volume-density.

Neutron scattering lengths as a function of element.

Because of the complexity of the neutron-nucleus interaction, the neutron scattering lengths vary in a non-intuitive way as a function of atomic number. In fact, different isotopes of the same element can have drastically different neutron scattering lengths. This makes neutron scattering a powerful probe, since we can vary the scattering power independently of the chemistry (through isotopic substitution). The fact that neutron beams probe a different contrast-condition from x-rays makes neutron and x-ray scattering highly complementary. For instance, while x-ray scattering excels at resolving heavy atoms, neutrons are able to more easily probe hydrogen-rich materials, including hydrocarbons. One can also measure a sample using both techniques, and use the different contrast conditions to resolve structural ambiguities.

## Generating Neutron Beams

There are two main methods used to generate neutron beams for research purposes:

• Reactor source: A nuclear reactor can be used as a neutron source. Small ports ('holes') in the reactor's shield-wall allow neutrons to escape, which are then directed, using neutron-shielded guides, towards different experimental endstations. The wavelengths of the exiting neutrons are dictated by their velocities; i.e. their temperature. The thermal spectrum can be tuned by using a 'moderator' with which the neutrons collide, bringing them to a desired temperature. A monochromator can be used to select a particular neutron wavelength (e.g. a diffraction crystal, or a velocity-selector). One thus obtains a neutron-beam that can be used for various experiments. Research reactors need not be particularly large (<100 MW); they are much smaller than power-plant reactors (~1,000 MW).
• Spallation source: A particle accelerator can also be used to generate neutrons. For instance, at SNS, a ~1 GeV proton beam is fired at a liquid mercury target (60 Hz). The proton-nuclear collisions release neutrons across a broad range of energies. Instead of using a monochromator, spallation sources typically exploit the fact that the neutrons arrive in pulses (owing to the original pulsed proton source). In particular, they use 'time-of-flight' detection, where the wavelength of any detected neutron is inferred based on its arrival time. Thus the wavelength spread is used to provide an inherent and measurable spread in the data (e.g. in a TOF SANS instrument, the scattering from a single pulse yields data across a range of q-values).

## Wavelength

The wavelength of a neutron is given by the de Broglie relation:

$\lambda = \frac{h}{p} = \frac{h}{m v}$

Where h is Planck's constant (6.62606957(29)×10−34 Js), and p is momentum (given by velocity, v, multiplied by mass, m = 1.674927351(74)×10−27 kg). Neutrons are often characterized by their temperature, where kinetic energy allows one to convert between velocity and temperature (T):

$\mathrm{KE} = \frac{1}{2} m v^2 = \frac{3}{2} kT$

where k is Boltzmann's constant (1.3806488×10−23 m2 kg s−2 K−1)

A neutron source generally produces a broad range of energies, and thus a distribution of wavelengths. A 'hot' thermal reactor source will generate neutrons of ~1 Å, while a cryogenic 'cold' moderator source will generate neutrons of ~6 Å. A spallation source is typically also moderated to control the output wavelengths (e.g. SNS beamlines may generate neutrons of 0.5 Å to 4 Å).