# Neutron Diffraction

Neutron diffraction can conceptualized in terms of three parts enumerated below:

1) Incident Neutron Beam Momentum, ${\mathbf{k}}_{o}$
2) Scattered Neutron Beam Momentum, ${\mathbf{k}}_{s}$
3) Change in Momentum due to Scattering, ${\mathbf{Q}}_{}$

When a collimated or focused neutron beam with and an incident momentum described by the vector ${\mathbf{k}}_{o}$ impinges some crystal medium and scatters from it, the constructive scattering will emerge from the crystal traveling in a new direction described by the vector ${\mathbf{k}}_{s}$. For elastic scattering the magnitude of ${\mathbf{k}}_{o}$ and the magnitude of ${\mathbf{k}}_{s}$ are equal, i.e.,

Elastic scattering Condition
$|{k}_{o}|=|{k}_{s}|$

The incident neutron beam can be described as a plane wave equation and the scattered neutron beam can be described as spherical wave:

$\psi =exp\left(i{\mathbf{k}}_{o}\cdot \mathbf{y}\right)+F\left(\mathbf{r}\right)exp\left(i\mathbf{k}\cdot \mathbf{r}\right)$

The quantum current density is then the expectation value of the continuity equation, i.e.,

$〈\mathbf{J}〉=-i\genfrac{}{}{0.1ex}{}{\hslash }{2m}\left({\Psi }^{*}\mathbf{\nabla }\Psi -\Psi \mathbf{\nabla }{\Psi }^{*}\right)$

And the scattering cross section is then,

$\left(\genfrac{}{}{0.1ex}{}{d\sigma }{d\Omega }\right)=\genfrac{}{}{0.1ex}{}{〈{J}_{s}〉}{〈{J}_{o}〉}{r}^{2}=|F{|}^{2}$

Where ${\mathbf{J}}_{s}$ is the scattered current density and ${\mathbf{J}}_{o}$ is the incident current density. Since the scattered wave function must account for scattering throughout the scattering medium, the scattered scoss section becomes dependent on the lattice structure of that medium, i.e.,

$F\left({\mathbf{k}}_{s}-{\mathbf{k}}_{o}\right)=\underset{𝕍}{\int }\rho \left(\mathbf{R}\right)exp\left[i\left({\mathbf{k}}_{s}-{\mathbf{k}}_{o}\right)\cdot \mathbf{R}\right]{d}^{3}\mathbf{R}$

Here we can define the scattering vector $\mathbf{Q}={\mathbf{k}}_{s}-{\mathbf{k}}_{o}$ such that constructive interference, which satisfies Bragg's law, only occurs at values of ${\mathbf{Q}}_{HKL}$ for which $|F\left(\mathbf{Q}\right)|\ne 0$. The set of all ${\mathbf{Q}}_{HKL}$ forms the reciprocal lattice of the crystal where $\mathbf{Q}$ defines the scattering geometry.

The reciprocal lattice represents the frequency structure of the crystal in the same way a Fourier transformation of a periodic signal describes the frequency composition of that signal.

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