IGNOU MPHE 26 2026 - Show that the average cosine of scattering angle in the LAB system ()

Question

Show that the average cosine of scattering angle in the LAB system ( $\bar{\mu}_L$ ) is given by $\bar{\mu}_L = \frac{2}{3A}$ , where A is the mass number of nucleus. Is this relation applies to hydrogen?

19 Jan 2026

Answer :

Word Count : 432

Let’s solve this step by step. We are asked to show that the average cosine of the scattering angle in the LAB system, (\bar{\mu}_L = \langle \cos\theta_L \rangle), is: [ \bar{\mu}_L = \frac{2}{3A} ] where (A) is the mass number of the nucleus. --- Step 1: Relation between CM and LAB angles For elastic scattering of a neutron on a nucleus of mass number (A), the relation between the scattering angle in the centre of mass (CM) frame (\theta_{CM}) and the LAB frame (\theta_L) is: [ \cos \theta_L = \frac{\cos \theta_{CM} + \alpha}{1 + \alpha \cos \theta_{CM}} ] where (\alpha = \frac{1}{A}) (since for neutron scattering, the neutron mass (m \approx 1), nucleus mass (M \approx A)). --- Step 2: Average cosine in LAB frame The average cosine is defined as: [ \bar{\mu}*L = \langle \cos \theta_L \rangle = \frac{1}{4\pi} \int \cos \theta_L , d\Omega*{CM} ] For isotropic scattering in the CM frame: [ \langle \cos \theta_L \rangle = \frac{1}{2}

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