Theory of the Effective Range in Nuclear Scattering

The scattering of neutrons up to about 10 or 20 Mev by protons can be described by two parameters, the scattering length at zero energy, a, and the effective range, r₀. A formula (16), expressing the phase shift in terms of a and r₀ is derived; it is identical with one previously derived by Schwinger but the derivation is very much simpler. Reasons are given why the deviations from the simple formula are very small, as shown by the explicit calculations by Blatt and Jackson. The theory is then applied to proton-proton scattering, with a similarly simple result. Moreover, a method is developed to compare proton-proton and proton-neutron scattering without explicit calculation of a nuclear potential. The most recent experimental results are evaluated on the basis of the theory, and accurate values for the effective ranges are obtained for the triplet scattering of neutrons, and for proton-proton scattering. The nuclear force between two protons is found to differ by a slight amount, but beyond doubt, from that between neutron and proton in the singlet state. All actual results agree with those obtained by Breit and collaborators, and by Blatt and Jackson.