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

The scattered amplitude $f\left(\vartheta \right)$ 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 $V\left(r\right)\mathrm{}$ and without tensor forces. The result for the symmetrical theory is: ${\sigma }_{p-n}=0.96\mathrm{}\times {10}^{-25\mathrm{}}$ ${\mathrm{cm}}^2$, ${\sigma }_{p-p}=0.22\mathrm{}\times {10}^{-25\mathrm{}}$ ${\mathrm{cm}}^2$, as compared with the first approximation values 1.40×${10}^{-25\mathrm{}}$ and 0.28×${10}^{-25\mathrm{}}$ ${\mathrm{cm}}^2$, respectively. These give for ${\sigma }_{p-n}+{\sigma }_{p-p}$ the value 1.18, in much closer agreement with the observed value 1.17 for ${\sigma }_{p-d}$ than the other two forms of interaction potential.