Note on Second Born Approximation and Proton-Neutron and Proton-Proton Scattering

The scattered amplitude f(ϑ) is regarded as a series in powers of the interaction potential V, and the second Born approximation consists in calculating the scattered intensity up to the fourth power of V. The calculation can be carried out analytically for a Gaussian potential, leading to a simple way of calculating the second approximation for the phases. The result is applied to the proton-neutron and proton-proton scattering at 100 Mev, on the basis of the three forms of nucleon interaction suggested by Rarita and Schwinger, with a Gaussian $V(r)$ and without tensor forces. The result for the symmetrical theory is: σ_p-n=0.96×10^-25 cm², σ_p-p=0.22×10^-25 cm², as compared with the first approximation values 1.40×{}10^-25 and 0.28×{}10^-25 cm², respectively. These give for σ_p-n+σ_p-p the value 1.18, in much closer agreement with the observed value 1.17 for σ_p-d than the other two forms of interaction potential.