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The cross section is measured by measuring the intensity of beam, scattered from a thin foil, in the forward direction as a function of thickness of the foil.
To measure the total cross section, we measure the intensity of the beam in forward direction as a function of thickness of the foil. Assume that a the scattering take place from a foil containing $N$ scattering centres per unit volume and having a thickness $x$. We will show that the intensity of the beam \(I(x)\) in the forward direction after passing through the foil of thickness \(x\) is
\begin{equation}\label{EQ01}
I(x)=I_0 \exp(-\sigma N x)
\end{equation}
When the scattering takes place from one scattering centre, The total cross section is defined as the ratio of number of particles removed from the beam per second divided by the intensity of the incident beam. The number of particles removed from the beam per second by a target having \(n\) scattering centers is, therefore, given by \(n\sigma\). This expression is valid under the assumptions that
- the scattering takes place from each scattering centre independently;
- there is no interference between the scattering from different scattering centers;
- there is no multiple scattering.
To derive \eqref{EQ01}, consider the change in intensity when the thickness of the foil is changed from $x$ and $x+dx$ with $dI$ denoting the corresponding change in intensity.
\begin{equation} A dI = - \sigma I (A N dx) \Longrightarrow \frac{dI}{dx}=-\sigma N I
\label{EQ02} \end{equation}
where $I$ is the intensity of the beam after it has travelled a thickness $x$. Integrating \EqRef{EQ02} gives
the desired result.
\begin{equation}
I=I_0 \exp(-\sigma N x)
\end{equation}
where $I_0$ is the intensity of the incident beam.
If the cross section from different centres does not add up in a simple fashion,nwe will still have
\begin{equation}
I=I_0 \exp(-\lambda x)
\end{equation}
only the relation of $\lambda$ to the total cross section will be complicated.
