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	The angular dependence of the scattered neutrons is a most important aspect of all neutron scattering. To describe this dependence, we define the ''differential scattering cross section'' as the intensity of neutrons scattered into a particular solid angle, \(d\Omega\). We denote the wave vector of the incoming neutron by \(k_i\) and that of the outgoing neutron by \(k_f\), where the indices "i" and "f" denote ''initial'' and ''final'', respectively. This is illustrated in <xr id="fig:Fig2dot1v3.png">Figure %i</xr> and is quantified as:		The angular dependence of the scattered neutrons is a most important aspect of all neutron scattering. To describe this dependence, we define the ''differential scattering cross section'' as the intensity of neutrons scattered into a particular solid angle, \(d\Omega\). We denote the wave vector of the incoming neutron by \(k_i\) and that of the outgoing neutron by \(k_f\), where the indices "i" and "f" denote ''initial'' and ''final'', respectively. This is illustrated in <xr id="fig:Fig2dot1v3.png">Figure %i</xr> and is quantified as:

	<figure id="fig:Fig2dot1v3.png">[[File:Fig2dot1v3.png \| thumb \| ~~400px~~ \| <caption>An illustration of the scattering geometry. The incoming neutrons, labeled \(k_i\), scatter at the center of the sphere with the scattering angle \(2\theta\) and azumital angle \(\phi\). The neutrons scattered into the solid angle element \(d\Omega\) (or detector area \(dA\) ar labeled with \(k_f\).</caption>]]</figure>		<figure id="fig:Fig2dot1v3.png">[[File:Fig2dot1v3.png \| thumb \| 450px \| <caption>An illustration of the scattering geometry. The incoming neutrons, labeled \(k_i\), scatter at the center of the sphere with the scattering angle \(2\theta\) and azumital angle \(\phi\). The neutrons scattered into the solid angle element \(d\Omega\) (or detector area \(dA\) ar labeled with \(k_f\).</caption>]]</figure>

	\begin{equation}\label{eq:dscs}		\begin{equation}\label{eq:dscs}

ucph>Tommy: /* The differential scattering cross section */

2019-08-27T13:58:09Z

The differential scattering cross section

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We fist introduce the terms by which we describe the scattering of a neutron beam. In particular, we describe the interaction of a neutron beam with materials by introducing the central concept of cross sections.

== Neutron flux ==
We define the flux of a neutron beam as the neutron rate per area

\begin{equation}\label{dummy378179674}
\Psi = \dfrac{ {\rm number\; of\; neutrons\; impinging\; on\; a\; surface\; per\; second}}
{ {\rm surface\; area\; perpendicular\; to\; the\; neutron\; beam\; direction}} ,
\end{equation}

usually given in the unit n/(cm\(^2\)s).

=====Experimental considerations=====
At most neutron scattering experiments, the incoming flux at the sample position lies in the range \(10^4\) to \(10^9\) n/(cm\(^2\)s). At ESS, however, some instruments will reach a flux of \(10^{10}\) n/(cm\(^2\)s).

==The scattering cross section==
The ''neutron scattering cross section'', \(\sigma\), of a system is defined by the system's ability to scatter neutrons:

\begin{equation}\label{eq:sigma_def}
\sigma = \dfrac{1}{\Psi} \cdot { {\rm number\; of\; neutrons\; scattered\; per\; second}} ,
\end{equation}

which has units of area. The scattering intensity is divided by the neutron flux to ensure that \(\sigma\) is an intrinsic property, independent on the neutron flux at the particular experimental set-up.

For a single nucleus, \(\sigma\) can now be seen as the effective area of the nucleus perpendicular to the neutron beam, as will be elaborated in the problem [[Exercises in Basics of neutron scattering #The cross section|the cross section]]. The scattering cross section used here is the total cross section, which depends on the system (sample) volume, \(V\). For thin samples, \(\sigma\) can be described by the volume specific cross section, \(\Sigma\), through

\begin{equation}\label{eq:Sigma_def}
\sigma = V \Sigma .\,
\end{equation}

For thicker samples, beam attenuation must be taken into account; see the section [[Basics of neutron scattering#Beam attenuation due to scattering|Beam attenuation due to scattering]].

==The differential scattering cross section==
The angular dependence of the scattered neutrons is a most important aspect of all neutron scattering. To describe this dependence, we define the ''differential scattering cross section'' as the intensity of neutrons scattered into a particular solid angle, \(d\Omega\). We denote the wave vector of the incoming neutron by \(k_i\) and that of the outgoing neutron by \(k_f\), where the indices "i" and "f" denote ''initial'' and ''final'', respectively. This is illustrated in <xr id="fig:Fig2dot1v3.png">Figure %i</xr> and is quantified as:

<figure id="fig:Fig2dot1v3.png">[[File:Fig2dot1v3.png | thumb | 400px | <caption>An illustration of the scattering geometry. The incoming neutrons, labeled \(k_i\), scatter at the center of the sphere with the scattering angle \(2\theta\) and azumital angle \(\phi\). The neutrons scattered into the solid angle element \(d\Omega\) (or detector area \(dA\) ar labeled with \(k_f\).</caption>]]</figure>

\begin{equation}\label{eq:dscs}
\frac{d\sigma}{d\Omega} = \frac{1}{\Psi} \frac{{\rm \; number\; of\; neutrons\; scattered \; into\; solid\; angle\;} d\Omega{\rm \; per\; second}}{d\Omega} .
\end{equation}

The total number of scattered neutrons is the integral of neutrons per solid angle over all of the \(4\pi\) solid angle, hence

\begin{equation}\label{eq:dscs_int}
\sigma = \displaystyle\int \dfrac{d\sigma}{d\Omega} d\Omega .
\end{equation}

==The partial differential scattering cross section==
In some scattering processes, the neutron delivers energy to or absorbs energy from the scattering system. This type of scattering we denote ''inelastic scattering''. We define the neutron energy transfer by

\begin{equation}\label{eq:inelastic_basic}
\hbar \omega = E_{\rm i} - E_{\rm f},
\end{equation}

which can also be written as

\begin{equation}\label{eq:inelastic_basic2}
\hbar \omega = \dfrac{\hbar^2(k_{\rm i}^2-k_{\rm f}^2)}{2 m_{\rm n}}.
\end{equation}

Note that the energy change is defined with the opposite sign of most definitions of changing properties, so that neutron energy loss gives a positive value of \(\hbar \omega\), while ''neutron energy gain'' gives negative values.

In inelastic scattering processes, energy is transfered to - or taken from - the sample. Energy conservation gives that the energy change, \(\Delta E\), of the sample is given by

\begin{equation}\label{dummy1737456334}
\Delta{}E = \hbar \omega . \,
\end{equation}

For describing inelastic scattering, one needs to take into account the energy dependence of the scattered neutrons. This is described by the ''partial differential scattering cross section'':

\begin{equation}\label{eq:pdscs}
\dfrac{d^2\sigma}{d\Omega dE_{\rm f}} = \dfrac{1}{\Psi}
\dfrac{ { {\rm number\; of\; neutrons\; scattered\; per\; sec.\; into\;}} d\Omega\; { {\rm with\; energies\;}} [ E_{\rm f};E_{\rm f}+dE_{\rm f} ] }
{d\Omega dE_{\rm f}} .
\end{equation}

Integrating over the all final energies, \(E_{\rm f}\), we reach the differential cross section described earlier:

\begin{equation}\label{dummy222151533}
\dfrac{d\sigma}{d\Omega} = \displaystyle\int \dfrac{d^2\sigma}{d\Omega dE_{\rm f}} dE_{\rm f}.
\end{equation}

Following equation \eqref{eq:dscs_int}, the total cross section is found by a double integration:

\begin{equation}\label{dummy350986584}
\sigma = \displaystyle\iint \dfrac{d^2\sigma}{d\Omega dE_{\rm f}} d\Omega dE_{\rm f}.
\end{equation}

For a closer description of inelastic scattering, a quantum mechanical treatment of the scattering process is required, as initiated in the section [[Basics of neutron scattering#* Quantum mechanics of scattering|Quantum mechanics of scattering]] and described in detail in the part on dynamics of materials.

==The absorption cross section==




Neutron absorption takes place as a result of neutron-induced nuclear processes, which destroy the neutrons, emitting secondary radiation (\(\alpha\), \(\beta\), or \(\gamma\)) as a result. In most cases, the absorption cross section, \(\sigma_{\rm a}\), of thermal neutrons is inversely proportional to the neutron velocity. In other words, the absorption is proportional to the neutron wavelength: \(\sigma_{\rm a} \propto \lambda\).

The neutron absorption cross sections are measured and tabelized for all but the rarest isotopes, see e.g. the Neutron Data Booklet<ref name="ILLblue">A.-J. Dianoux and G. Lander, ''ILL Neutron Data Booklet'' (ILL, 2003)</ref>, or the [http://www.nist.gov/ NIST home page]. Traditionally, the absorption cross section of isotope \(j\) is given as \(\sigma_{\text{a,j,th}}\) per nucleus in units of "barns" (1 barn \(=10^{-28}\) m\(^{2}\)) and is listed at the standard "thermal" velocity \(v_\text{th} = 2200\) ms\(^{-1}\) (\(\lambda_\text{th} = 1.7982\) Å, see also the [[Introduction to neutron scattering#label-tab:conversion|neutron unit conversion table]] on the [[Introduction to neutron scattering]] page). The actual absorption cross section is then given by

<figtable id="tab:cross">[[File:Tabel2.1.png | frame | <caption>Neutron cross sections and scattering lengths for the first 32 elements (isotopic average using the natural abundancies). In addition, data for selected isotopes and for some heavier elements are presented. Data taken from <ref name="ILLblue"></ref>. <nowiki>*</nowiki> It should be noted that the important incoherent cross section for H varies with neutron wavelength. The value listed is valid for \(\lambda > 1.798\) Å<ref name="howard87">J. Howard, ''J. Appl. Cryst.'', vol. 20, p. 120-122 (1987)</ref><ref name="henry11">P. Henry, ''in preparation'' (2011)</ref></caption>]]</figtable>

\begin{equation}\label{dummy2114890051}
\sigma_{\rm a,j} = \sigma_{\text{a,j,th}} \dfrac{v_\text{th}}{v}
= \sigma_{\text{a,j,th}} \dfrac{\lambda}{\lambda_\text{th}}.
\end{equation}

Also for absorption, the volume specific cross section is defined as

\begin{equation}\label{dummy21148900512}
\sigma_{\rm a,j} = V \Sigma_{\rm a,j}.
\end{equation}

In a general material, many nuclei contribute to the total absorption. As in equation \eqref{eq:attenuation}, the resulting attenuation coefficient is

\begin{equation}\label{dummy74943618}
\Sigma_{\rm a} = \sum_j \dfrac{N_j \sigma_{ {\rm a},j}}{ V}
= \sum_j n_j \sigma_{ {\rm a},j}.
\end{equation}

In this sum, \(N_j\) represents the number of nuclei of isotope, \(j\) in the sample volume \(V\), and \(n_j=N_j/V\) is their respective atomic density.

An abbreviated list of absorption and scattering cross sections for selected isotopes/elements is given in <xr id="tab:cross">Table %i</xr>.

<references />

The neutron cross sections - Revision history

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