Question

Calculate in the Born approximation the differential scattering cross section for neutron-neutron scattering, assuming that the interaction potential responsible for scattering vanishes for the triplet spin state and is equal to V(r) = \(V_0\frac{e^{-\mu r}}{r}\) for the singlet spin state. [Evaluate the cross section for an unpolarized (random spin orientation) initial state.]

Answer 1

The Born approximation gives

where k is the wave vector of the relative motion of the neutrons, m = m_n/2 is the reduced mass.

As the spin wave function of the spin singlet state is antisymmetric, its spatial wave function must be symmetric. Thus

Because the neutrons are initially unpolarized, the scattering cross section is