Inelastic nuclear neutron scattering

One of the early successes of neutron scattering was the study of dynamics of matter, in particular phonon dispersion relations. Here, the vibrational frequency (or phonon energy) is deduced from the change in neutron energy through the principle of energy conservation. Hence, for the study of dynamics we are dealing with inelastic neutron scattering.

This page naturally leads to the description on the Scattering from lattice vibrations page of neutron scattering from quantised lattice vibrations, or phonons. The related topic of inelastic neutron scattering from diffusion and molecular motion is not covered in this version of the notes. The most important part of the instrumentation for the general field of inelastic neutron scattering is described in Scattering from lattice vibrations.

We start by repeating the important definition of the inelastic (or partial differential) scattering cross section:

\begin{equation} \frac{d^2\sigma}{d\Omega dE_{\rm f}} = \frac{1}{\Psi} \frac{ \text{number of neutrons scattered per sec into } d \Omega \text{ with energies } [E_f;E_f + dE_f] } { d\Omega dE_{\rm f} } \, . \end{equation}

This defintion holds for all types of inelastic scattering, whether it is of nuclear or magnetic origin.

*Scattering from nuclear dynamics

The calculations the lead to the description of the inelastic nuclear scattering are of quantum mechanical nature and are specified in Scattering theory for nuclear dynamics. These calculations lead directly to the observable scattering cross section, which covers both elastic and inelastic nuclear scattering: \begin{align} \label{eq:inel_final_nuc} %\lefteqn{\frac{d^2\sigma}{d\Omega dE_{\rm f}}} &= %ToDo: Make this boxed %\boxed{ \frac{d^2\sigma}{d\Omega dE_{\rm f}} &= \frac{k_{\rm f}}{k_{\rm i}} \sum_{j,j'} \frac{b_j b_{j'}}{2 \pi \hbar} \\ &\quad \times \int_{-\infty}^{\infty} \big\langle \exp(-i {\bf q} \cdot {\bf R}_j(0)) \exp(i {\bf q} \cdot {\bf R}_{j'}(t)) \big\rangle \exp(-i\omega t) \, dt \nonumber . %} \, . \end{align} It should be noted that this is a rather general result, valid for any kind of nuclear motion. The result can (and will often be) interpreted semiclassically, in the sense that \({\bf R}_j(t)\) is viewed as the classical position of the atoms or molecules under study.

In Scattering from lattice vibrations, we will describe the scattering resulting from vibrations of nuclei around their equilibrium positions in a lattice. We can, however, already see now that since the nuclear position enter (\ref{eq:inel_final_nuc}), we will need to make a series expansion of the complex exponentials in order to describe the effects of the small movements.

*Scattering from magnetic dynamics

In the same way as for the inelastic nuclear scattering described above, the inelastic magnetic scattering requires a quantum mechanical treatment. This is detailed in Scattering theory for magnetic dynamics. These quantum mechanical calculations lead directly to the observable scattering cross section, which covers both elastic and inelastic magnetic scattering from magnetic moments on a lattice: \begin{align} \left(\frac{d^2\sigma}{d\Omega dE_{\rm f}}\right)_{\rm magn.} &= \left(\gamma r_0 \right)^2 \frac{k_{\rm f}}{k_{\rm i}} \left[ \frac{g}{2} F(q)\right]^2 \exp(-2W) \sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{\bf q}_\alpha\hat{\bf q}_\beta\right) \\ &\quad\times \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} \sum_{j,j'} \exp(i {\bf q} \cdot ({\bf r}_{j'}-{\bf r}_j)) %+\Deltabold_i)) \left\langle {\bf s}_{j}^\alpha(0) {\bf s}_{j'}^\beta(t) \right\rangle \exp(-i \omega t) dt. \nonumber \end{align} Due to the translational symmetry of the lattice, the index \(j\) can be chosen to be the Origin (\(j \rightarrow 0\)). We can then relabel \( ({\bf r}_{j'}-{\bf r}_j) \rightarrow {\bf r}_{j'}\), and thereby all terms in the sum over \(j\) becomes equal, giving a factor \(N\). The final result becomes: \begin{align} \label{eq:inel_final_mag} \left(\frac{d^2\sigma}{d\Omega dE_{\rm f}}\right)_{\rm magn.} &= \left(\gamma r_0 \right)^2 \frac{k_{\rm f}}{k_{\rm i}} \left[ \frac{g}{2} F(q)\right]^2 \exp(-2W) \sum_{\alpha \beta} \left( \delta_{\alpha\beta}-\hat{\bf q}_\alpha\hat{\bf q}_\beta\right) \\ &\quad\times \frac{N}{2\pi\hbar} \int_{-\infty}^{\infty} \sum_{j'} \exp(i {\bf q} \cdot {\bf r}_j) %+\Deltabold_i)) \left\langle {\bf s}_{0}^\alpha(0) {\bf s}_{j'}^\beta(t) \right\rangle \exp(-i \omega t) dt. \nonumber \end{align} In reality, we here describe contribution that is elastic in the nuclear positions (the phonon channel) and elastic or inelastic in the spins (the magnetic channel). To obtain this, we have multiplied with the Debye-Waller factor, \(\exp(-2W)\). In these note, we will not describe magnetic scattering that is inelastic in the phonon channel; this is discussed to some extend in Squires \cite{squires}.

The expression for the cross section should be understood as the space and time Fourier transform of the spin-spin correlation function, \(\left\langle {\bf s}_{0}^\alpha(0) {\bf s}_{j'}^\beta(t) \right\rangle\), and it is the starting point for most calculations of magnetic scattering cross sections. We note that since the spin operators do not appear as argument of the exponential function, there is no need to perform the series expansion of the complex exponential as needed for the lattice vibrations.

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