View source for Coherent and incoherent scattering

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Very often, the neutron scattering length varies from nucleus to nucleus in a sample. This can be caused by the variation of the nuclear spin direction with time, or by variations between isotopes of the same element - or between different elements. <!--The latter is a static disorder that for a macroscopic sample can be treated in the same way as a time-dependent disorder, since we can assume the sample is large enough to essentially represent an ensemble average.-->We here describe how this affects the scattering cross sections.

==The coherent and incoherent cross sections==

We will here consider what happens to the scattering cross section if the nuclear scattering lengths vary randomly within the sample. 

===Variation in scattering lengths===

We have to types of variation in nuclear scattering length: Element or isotope disorder, which is a static effect, and variations in the value of \(b_j\) themselves due to variations in the nuclear spin directions, which is a dynamic effect. However, for a macroscopic sample the calculations can be treated in the same way, since we can assume that 1) the sample is large enough to essentially represent an ensemble average and 2) we observe the system over times much longer than nuclear fluctuation times, meaning that the time average equals an ensemble average.

Let us for simplicity assume that the scattering length at site \(j\) has the stochastic value 

\begin{equation} \label{eq:b_av}
  b_j=\langle b_j \rangle + \delta b_j ,  
\end{equation}

where \(\langle b_j \rangle\) denotes the (time and ensemble) average. \(\delta b_j\) is the local deviation from the average, defined so that \(\langle \delta b_j \rangle = 0\). We assume that the deviations are uncorrelated between sites, \(\langle \delta b_j \, \delta b_{j'} \rangle = 0\). 

To derive the mean differential scattering cross section, it is sufficient to consider the two-atom problem.

\begin{equation} \label{eq:sigma_av}
  \left\langle \dfrac{d\sigma}{d\Omega}\right\rangle =
  \left\langle|b_j \exp(i \mathbf{q}\cdot {\mathbf r}_j)
             + b_{j'}\exp(i \mathbf{q}\cdot {\mathbf r}_{j'})|^2 \right\rangle
  = \left\langle|b^2_j + b^2_{j'} + 2b_j b_{j'} cos({\bf q} \cdot ({\bf r}_j - {\bf r}_{j'})) \right\rangle
\end{equation}

Using (\ref{eq:b_av}), we now see that the "square terms" in \eqref{eq:sigma_av} give \(\langle b_j^2 \rangle = \langle b_j \rangle ^2 + \langle \delta b_j^2 \rangle\), because \( \langle \langle b_j \rangle \delta b_j \rangle = \langle b_j \rangle \langle \delta b_j \rangle \). In a similar way, the "interference terms" are seen to give \(\langle b_j b_{j'} \rangle = \langle b_j\rangle \langle b_{j'}\rangle\). We thus reach

\begin{equation} \label{eq:sigma_av2}
  \left\langle \dfrac{d\sigma}{d\Omega}\right\rangle =
  \langle (\delta b_j)^2 \rangle + \langle (\delta b_{j'})^2 \rangle + 
  \langle b_j \rangle^2 + \langle b_{j'} \rangle ^2 + 2 \langle b_j \rangle \langle b_{j'} \rangle \cos ({\bf q} \cdot ({\bf r}_j - {\bf r}_{j'})).
\end{equation}

We introduce the ''incoherent scattering cross section

\begin{equation} \label{eq:sigma_av3}
  \sigma_{inc,j} = 4 \pi \langle ( \delta b_j )^2 \rangle
\end{equation}

and rewrite \eqref{eq:sigma_av2} to give

\begin{equation}\label{eq:nuclear_incoherent1}
  \left\langle \dfrac{d\sigma}{d\Omega}\right\rangle = 
  \dfrac{\sigma_{{\rm inc},j} + \sigma_{{\rm inc},j'}}{4 \pi}
  + \left|\langle b_j\rangle \exp(i \mathbf{q}\cdot {\mathbf r}_j)
             + \langle b_{j'}\rangle \exp(i \mathbf{q}\cdot {\mathbf r}_{j'})\right|^2 .
\end{equation}

Here we notice that the incoherent cross section, \(\sigma_\text{inc}\), represents a constant scattering of neutrons, ''i.e.'' in all directions, without interference between scattering from neighbour atoms. The average value \(\langle b_j\rangle\) represents the strength of the interfering scattering and is denoted the ''coherent scattering length''. In general, the coherent scattering depends on the scattering vector \(\bf q\), and hence on the scattering angle. One defines ''coherent scattering cross section'' for a single nucleus \(j\) as \(\sigma_{\rm{coh},j}= 4 \pi \langle b_j\rangle^2 \).

Usually, the explicit average notation \(\langle b \rangle\) is dropped, and the symbol \(b\) almost exclusively means the average scattering length of a certain isotope or element. This is also the notation used in <xr id="tab:cross">Table %i</xr><!--Table~\ref{tab:cross}-->.

==Incoherent nuclear scattering from randomness==
There are several sources of the incoherent scattering, described in general terms above. One source is the spin-dependent term, which is described in detail in <ref name="marshall">W. Marshall and S.W. Lovesey. ''Theory of Thermal Neutron Scattering''. (Oxford, 1971)</ref><!--\cite{marshall}-->, and which is the one given by the isotope tables. Below, we will deal with incoherent scattering caused by variations in the scattering length due to isotopic mixture or chemical randomness. From the point of view of unpolarized neutron scattering, all these mechanisms are very similar, as described above. The values of the incoherent scattering cross sections for the elements, found in <xr id="tab:cross">Table %i</xr><!--Table~\ref{tab:cross}-->, deal with the combined effect from spin and isotopic mixture. 

For a simple example, assume that a material consists of two isotopes with the abundances \(a_c=a\), and \(a_d=1-a\), the scattering lengths \(b_c\) and \(b_d\), respectively, and no nuclear spin. The average scattering length is 

\begin{equation}\label{dummy104856507}
  \langle b \rangle = a b_c + (1-a) b_d ,\, 
\end{equation}

and the average incoherent cross section can be calculated by an average over the isotope abundances:

\begin{equation} \label{eq:sigma_mix}
  \dfrac{\sigma_{\rm inc}}{4\pi} = \langle (\delta b)^2 \rangle  
= a (b_c-\langle b \rangle)^2 + (1-a) (b_d-\langle b \rangle)^2  
= a (1-a) (b_c - b_d)^2 . 
\end{equation}

This means that we see an incoherent scattering due to the isotope mixture, strongest at 50%-50% mixing ratio.

With a little effort, \eqref{eq:sigma_mix} can be generalized to more than two isotopes.

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