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<p>Created page with &quot;&lt;!--\label{prob:cross_section}--&gt; Imagine a beam of neutrons arriving randomly over a surface of area \(A\) perpendicular to the beam, with an arrival rate of \(N\) neutrons p...&quot;</p> <p><b>New page</b></p><div>&lt;!--\label{prob:cross_section}--&gt;<br /> Imagine a beam of neutrons arriving randomly over a surface of area \(A\) perpendicular to the beam, with an arrival rate of \(N\) neutrons per second. In a semi-classical approximation, you can consider each neutron to be point shaped. Now, on the surface we place one nucleus with an effective radius of \(2b\). Assume that each neutron hitting the nucleus is scattered and all other neutrons are left unscattered.<br /> <br /> =====Question 1=====<br /> Calculate the neutron flux.<br /> <br /> {{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}<br /> The flux is \( \Psi = N / A \). <br /> {{hidden end}}<br /> <br /> =====Question 2=====<br /> Calculate the probability for one neutron to hit the nucleus.<br /> <br /> {{hidden begin|toggle=right|title=Hint|titlestyle=background:#ccccff}}<br /> Consider the area of the nucleus (\(a\)) versus the area of the beam (\(A\)).<br /> {{hidden end}}<br /> <br /> {{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}<br /> The area of the nucleus perpendicular to the beam is \( a = 4 \pi b^2 \).<br /> The probability for hitting is thus \(p = a/A = 4\pi b^2 / A\).<br /> {{hidden end}}<br /> <br /> =====Question 3=====<br /> Show that the scattering cross section of the nucleus is \(\sigma = 4 \pi b^2\).<br /> <br /> {{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}<br /> The number of neutrons scattered per second must (in average) be \(n_{\rm scatt} = N p\).<br /> Using the definition for cross section we reach<br /> \(\sigma = n_{\rm scatt} / \Psi = 4 \pi b^2 \).<br /> {{hidden end}}</div>

ucph>Tommy