The partial wave theory of low energy s wave n-p scattering

The partial wave theory of low-energy s-wave neutron-proton (n-p) scattering is a theoretical framework used to describe the interactions between neutrons and protons when they are scattered at low energies. This theory is based on the assumption that the n-p interaction can be approximated by a central force, which means that it depends only on the distance between the particles.

In the partial wave theory, the n-p scattering amplitude is decomposed into a sum of partial waves, where each partial wave describes the scattering of neutrons and protons with a specific angular momentum quantum number, denoted by l. The s-wave corresponds to the case where the angular momentum is zero (l=0).

The partial wave analysis of the n-p scattering amplitude yields a set of scattering parameters, which describe the properties of the n-p interaction at low energies. These parameters include the scattering length, which describes the strength and sign of the n-p interaction, and the effective range, which describes the range over which the n-p interaction is effective.

At low energies below 10 MeV, the scattering is essentially low due to neutrons with angular momentum l=0. Only 9% scattering is due to neutron with l =1. Therefore, angular diffraction of scattered neutrons is isotropic in centre of mass system. More explicity, for neutron energy < 10 MeV, S wave overlap with nuclear potential with central potential V(r). The SWE for two body system (n-p) system is

where M = Mass of proton or neutron = 2𝜇

E = Kinetic energy of incident proton.

Let us consider a low energy beam is incident on a proton along z axis. Then at large distance from centre of scattering complete wavefunction is

e^ikz represents a plane wave describing a beam of particles moving in z direction towards the scattering centre, f(𝜃) is scattered amplitude in direction of 𝜃 and is evaluated in terms of k. f(𝜃) doesn't depend on azimuthal angle Φ.

Schrodinger wave equation is

This can be expanded in terms of spherical harmonic functions as

Rr(r) is radial part of equation 3

is spherical harmonics in absence of azimuthal coordinates and 'l' is integer representing the number of partial waves. The equation 6 has two solutions which is finite at origin can be represented by Bessel function as

The average value of quantity in bracket in overall direction in space is zero. The first term corresponds to spherically symmetry potentially wave. In presence of potential v(r) for s waves scattering only first term is affected and can be written as 𝜓sc . We can write as u(r)/r. Outside the range of scattering potential, the amplitude of outgoing wave is unchanged. The only possible change in wave is change in phase.

Thus at r-> ∞, the solution u(r) assumes the form Csin(kr + δo) where C is arbitrary and δo is phase shift. Hence complete wavefunction outside scattering potential is

Since the scattered wave contain no incoming wave, we have

Thus the differential scattering cross section

𝜎 sc will be maximum if 𝛿o = (n + 1/2)\𝜋. This is total scattering cross section for l =0. If all the contribution due to higher 'l' are taken into account then we could have,

which clearly gives result 11 for l=0 and total scattering cross section becomes maximum for 𝛿l = (n + 1/2 )𝜋

In this method, a proton captures a neutron to become deuteron and a gamma ray emerges out. The gamma ray has an energy almost equal to deuteron Binding energy.

The partial wave theory of low-energy s-wave n-p scattering has been used to study a wide range of phenomena in nuclear physics, including the properties of the deuteron, the simplest nucleus composed of a neutron and a proton, and the nucleon-nucleon scattering phase shifts. It is a fundamental tool for understanding the interactions between nucleons and the properties of atomic nuclei.

This note is a part of the Physics Repository.