Beam attenuation due to scattering and absorption

The cross sections for scattering and absorption are additive due to the rule of addition of probabilities (see Problem: Attenuation of the neutron beam), giving the total volume specific cross section: \begin{equation} \Sigma_{\rm t} = \Sigma_{\rm s} + \Sigma_{\rm a}. \end{equation}

Since the number of neutrons scattered or absorbed is necessarily limited by the number of incoming neutrons, the total cross section cannot be truly proportional to the number of nuclei, at least not for large, strongly scattering/absorbing systems. Hence, (\ref{eq:Sigma_def}) should be understood only as what is called the em thin sample approximation or the Born approximation. This equation is valid only when the total scattering cross section of a given sample is much smaller than its area perpendicular to the beam. Note that the total cross section of a sample cannot exceed its area. This would lead to the number of scattered neutrons exceeding the number of incoming neutrons, which is not possible.

For a thick sample, we must consider successive thin slices of thickness \(dz\), each attenuating the incident beam (which we take to travel in the positive \(z\) direction):

\begin{equation} {\rm \; no.\; of\; neutrons\; scattered\; or\; absorbed\; per\; sec.\; from\;} dz = \Psi(z) \Sigma_{\rm t} A dz , \end{equation} where \(A\) is the area of a sample slice perpendicular to the beam. We assume that \(A\) and \(\Sigma_{\rm t}\) are constants and that the scattering and absorption cross section is uniform within the sample. The flux of the incident beam in the neutron flight direction is then attenuated inside the sample according to

\begin{equation} \label{eq:attenuation} \Psi(z) = \Psi(0) \exp(-\mu_{\rm t} z) \, , \end{equation}

where we have defined the total attenuation coefficient \begin{equation} \mu = \mu_{\rm t} = \Sigma_{\rm t} . \end{equation}

The derivation is simple and is left as an exercise to the reader, see Problem: Attenuation of the neutron beam.

When the attenuation coefficient varies along the neutron path, (\ref{eq:attenuation}) is generalized to \begin{equation} \label{eq:attenuation2} \Psi(z) = \Psi(0) \exp\left(-\int_0^z \mu(z') dz' \right) \, . \end{equation}

This equation is essential in the use of neutron transmission for real-space imaging of samples, in analogy to medical X-ray images. This application of neutrons will be elaborated more in Imaging.