Quantum mechanics of scattering

We will now go through the principles of neutron scattering from nuclei in a way, which is more strictly quantum mechanical than that of in the section Wave description of nuclear scattering. This section does not contain new results, but may be more satisfactory for readers with a physics background. Furthermore, the formalism developed here carries on to the detailed treatment of inelastic scattering of phonons and magnetic scattering in subsequent chapters.

This section is strongly inspired by the treatments in the textbooks by Marshall and Lovesey and Squires.

* The initial and final states

We define the state of the incoming wave as

\begin{equation}\label{dummy1729998104} | \psi_{\rm i} \rangle = \dfrac{1}{\sqrt{Y}} \exp(i {\mathbf k}_{\rm i} \cdot {\mathbf r}) , \end{equation}

where \(Y=L^3\) can now be identified as the (large) normalization volume for the state which is assumed enclosed in a cubic box with a side length \(L\). The incoming neutron flux is given as equation \eqref{eq:flux_plane}

\begin{equation}\label{dummy634398538} \Psi_{\rm i} = |\psi_{\rm i}|^2 v = \dfrac{1}{Y} \dfrac{\hbar k_{\rm i}}{m_{\rm n}} . \end{equation}

In contrast to the spherical outgoing wave from the section Wave description of nuclear scattering, we express the final state as a (superposition of) plane wave(s)

\begin{equation}\label{dummy411059591} | \psi_{\rm f} \rangle = \dfrac{1}{\sqrt{Y}} \exp(i {\mathbf k}_{\rm f} \cdot {\mathbf r}) . \end{equation}

We here ignore the spin state of the neutron, which will be discussed in the later chapter on neutron polarization. For the spinless states, we calculate the number density in \({\bf k}\)-space:

* Density of states

For the spinless states, we calculate the number density in \(\bf k\)-space:

\begin{equation}\label{dummy138597603} \dfrac{dn}{dV_k} = \left(\dfrac{2\pi}{L}\right)^{-3} = \dfrac{Y}{(2\pi)^3}. \end{equation}

We now consider a spherical shell in \({\bf k}\)-space to calculate the (energy) density of states,

\begin{equation}\label{eq:DOS1} \dfrac{dn}{dE_{\rm f}} = \dfrac{dn}{dV_k} \dfrac{dV_k}{dk_{\rm f}} \left(\dfrac{dE_{\rm f}}{dk_{\rm f}}\right)^{-1} = \dfrac{Y}{(2 \pi)^3} 4\pi k_{\rm f}^2 \dfrac{m_{\rm n}}{k_{\rm f} \hbar^2} = \dfrac{Y k_{\rm f} m_{\rm n}}{2 \pi^2 \hbar^2} . \end{equation}

In order to describe the differential scattering cross sections, we would like to describe the fraction of the wavefunction which is emitted into directions of \({\bf k}_{\rm f}\), corresponding to a solid angle \(d\Omega\). Here, the densities are given by

\begin{equation}\label{dummy1062809500} \dfrac{dn}{dV_k} \biggr|_{d\Omega} = \dfrac{dn}{dV_k} \dfrac{d\Omega}{4\pi} = \dfrac{Y}{(2\pi)^3}\dfrac{d\Omega}{4\pi} . \end{equation}

Following the calculations leading to equation \eqref{eq:DOS1}, we can now calculate the density of states within the scattering direction \(d\Omega\):

\begin{equation}\label{eq:DOS} \dfrac{dn}{dE} \biggr|_{d\Omega} = \dfrac{Y k_{\rm f} m_{\rm n}}{(2 \pi)^3 \hbar^2} d\Omega . \end{equation}

We will need this expression in the further calculations.

* The master equation for scattering

We describe the interaction responsible for the scattering by an operator denoted \(\hat{V}\). The scattering process itself is described by the Fermi Golden Rule . This gives the rate of change between the neutron in the single incoming state, \(|\psi_{\rm i}\rangle\) and a final state, \(|\psi_{\rm f}\rangle\), where \(|\psi_{\rm f}\rangle\) resides in a continuum of possible states.

\begin{equation}\label{eq:goldenrule} W_{{\rm i} \rightarrow {\rm f}} = \dfrac{2\pi}{\hbar} \dfrac{dn}{dE_{\rm f}} \big| \big\langle \psi_{\rm i} \big| \hat{V} \big| \psi_{\rm f}\big\rangle \big|^2 . \end{equation}

We wish to consider only neutrons scattered into the solid angle \(d\Omega\). Using equation \eqref{eq:DOS} and \eqref{eq:goldenrule}, we reach

\begin{equation}\label{dummy1031223580} W_{{\rm i} \rightarrow {\rm f},d\Omega} = \dfrac{Y k_{\rm f} m_{\rm n}}{(2 \pi)^2 \hbar^3} d\Omega \big| \big\langle \psi_{\rm i} \big| \hat{V} \big| \psi_{\rm f}\big\rangle \big|^2 . \end{equation}

\(W_{{\rm i} \rightarrow {\rm f},d\Omega}\) is the number of neutrons scattered into \(d\Omega\) per second. We now only need the expression for the incoming flux, equation \eqref{eq:flux_plane}, to reach the result for the differential scattering cross section in equation \eqref{eq:dscs}

\begin{equation}\label{eq:master_scatt} \dfrac{d\sigma}{d\Omega} = \dfrac{1}{\Psi} \dfrac{W_{{\rm i} \rightarrow {\rm f},d\Omega}}{d\Omega} = Y^2 \dfrac{k_{\rm f}}{k_{\rm i}} \left(\dfrac{m_{\rm n}}{2\pi \hbar^2}\right)^2 \big| \big\langle \psi_{\rm i} \big|\hat{V}\big| \psi_{\rm f}\big\rangle \big|^2 . \end{equation}

In this expression, the normalization volume, \(Y\), will eventually vanish due to the factor \(1/\sqrt{Y}\) in the states \(|k_{\rm i}\rangle\) and \(|k_{\rm f}\rangle\), since the interaction, \(\hat{V}\), is independent of \(Y\). We will thus from now on neglect the \(Y\) dependence in the states and in the cross sections.

The factor \(k_{\rm f}/k_{\rm i}\) in equation \eqref{eq:master_scatt} is of importance only for inelastic neutron scattering, where it always appears in the final expressions. For elastic scattering, \(k_{\rm f} / k_{\rm i} = 1\) and is thus removed from the expression.

* Elastic scattering from one and two nuclei

The interaction between the neutron and the nuclei is expressed by the Fermi pseudopotential

\begin{equation}\label{dummy770422586} \hat{V}_j({\mathbf r}) = \dfrac{2 \pi \hbar^2}{m_{\rm n}} b_j \delta({\mathbf r}-{\mathbf r}_j) . \end{equation}

Here, \(b_j\) has the unit of length and is of the order fm. It is usually denoted the scattering length. The spatial delta function represents the short range of the strong nuclear forces and is a sufficient description for the scattering of thermal neutrons.

It should here be noted that a strongly absorbing nucleus will have a significant imaginary contribution to the scattering length. We will, however, not deal with this complication here.

For a single nucleus, we can now calculate the scattering cross section. We start by calculating the matrix element

\begin{equation} \label{eq:matrixelem} \big\langle \psi_{\rm f} \big|\hat{V}_j \big| \psi_{\rm i}\big\rangle = \dfrac{2\pi \hbar^2}{m_{\rm n}} \,b_j \displaystyle\int \exp(-i {\mathbf k}_{\rm f} \cdot {\mathbf r}) \delta({\mathbf r}-{\mathbf r}_j) \exp(i {\mathbf k}_{\rm i} \cdot {\mathbf r}) d^3{\mathbf r} = \dfrac{2\pi\hbar^2}{m_{\rm n}} \,b_j \exp(i \mathbf{q} \cdot {\mathbf r}_j) , \end{equation}

where we have defined the very central concept of neutron scattering, the scattering vector, as

\begin{equation}\label{dummy195610253} \mathbf q = {\mathbf k}_{\rm i} - {\mathbf k}_{\rm f} . \end{equation}

Inserting \eqref{eq:matrixelem} into equation \eqref{eq:master_scatt}, we reassuringly reach the same result as in equation \eqref{eq:nuclear_diff_cross}:

\begin{equation}\label{dummy1725912318} \dfrac{d\sigma}{d\Omega} = b_j^2. \end{equation}

For a system of two nuclei, we obtain interference between the scattered waves. We can write the scattering potential as a sum \(\hat{V}=\hat{V}_j+\hat{V}_{j'}\). In this case, the matrix element becomes

\begin{equation}\label{eq:scatter_matrix_2} \big\langle \psi_{\rm f} \big|\hat{V} \big| \psi_{\rm i}\big\rangle = \dfrac{1}{Y}\dfrac{2\pi\hbar^2}{m_{\rm n}} \left( b_j \exp(i \mathbf{q} \cdot {\mathbf r}_j) + b_{j'} \exp(i \mathbf{q} \cdot {\mathbf r}_{j'}) \right) . \end{equation}

Inserting into equation \eqref{eq:master_scatt}, we reach the same result, \eqref{eq:interference2}, as found by the simpler approach in the Wave description of nuclear scattering section.

* Formalism for inelastic scattering

When describing the quantum mechanics of the inelastic scattering process, it is important to keep track of the quantum state of the scattering system (the sample), since it changes during the scattering process (for \(\hbar \omega \neq 0\)). The initial and final sample states are denoted \(|\lambda_{\rm i}\rangle\) and \(|\lambda_{\rm f}\rangle\), respectively. The partial differential cross section for scattering from \(|\lambda_{\rm i},{\bf k}_{\rm i}\rangle\) to \(|\lambda_{\rm f},{\bf k}_{\rm f}\rangle\) is given in analogy with equation \eqref{eq:master_scatt} by

\begin{equation}\label{eq:master_scatt_inel} \dfrac{d^2\sigma}{d\Omega dE_{\rm f}} \biggr|_{\lambda_{\rm i}\rightarrow \lambda_{\rm f}} = \dfrac{k_{\rm f}}{k_{\rm i}} \left(\dfrac{m_{\rm n}}{2\pi\hbar^2}\right)^2 \left| \left\langle \lambda_{\rm i}\psi_{\rm i} \left|\hat{V}\right| \psi_{\rm f} \lambda_{\rm f}\right\rangle \right|^2 \delta(E_{\lambda_{\rm i}}-E_{\lambda_{\rm f}}+\hbar\omega) , \end{equation}

where the \(\delta\)-function expresses explicit energy conservation and the normalization factor \(Y^2\) is omitted.

This expression will be our starting point in the chapters on inelastic scattering from lattice vibrations and magnetic excitations.