Theory of Scattering of Protons by Protons

Abstract

The experiments of Tuve, Heydenburg and Hafstad and those of White are discussed by means of the standard theory of scattering in central fields. The theoretical formulas are presented in a form convenient for numerical computation and are supplemented by tables. These are arranged so as to enable an experimentalist to compute the effect of phase shifts due to angular momenta $L=0,\ensuremath{\hbar},2\ensuremath{\hbar}$, and to infer these phase shifts from the experimental material (Tables I, II, III, IV, V, VI, VII, VIII, IX). Tables of necessary Coulomb wave functions are also given for zero angular momentum. By means of these the interaction energy can be computed from the experimental material (Tables X, XI, XII, XIII).Statistical fluctuations make conclusions drawn from White's data somewhat uncertain. The experiments of Tuve, Heydenburg and Hafstad are comparatively free of statistical effects and their comparison with theory shows that (a) There is an unmistakable difference between the observed scattering and that to be expected according to Mott's formula which uses the inverse square law. (b) This difference can be explained by using practically entirely effects of the phase shift in the partial wave having $L=0$ (head on collisions; $s$ wave distortions). The distortion of $p$ and $d$ waves ($L=\ensuremath{\hbar},2\ensuremath{\hbar}$) is secondary and the experimental accuracy does not yet suffice to enable their quantitative determination. (c) The variation of the scattering anomaly with proton energy is in approximate agreement with that to be expected from an interaction potential independent of the energy. (d) For a given range of nuclear forces the interaction potential is accurately determined by the data. The values obtained are in good agreement with those found by Feenberg and Knipp and by Bethe from the mass defects of ${\mathrm{H}}^{2}$, ${\mathrm{H}}^{3}$, ${\mathrm{He}}^{4}$ provided the mass defect calculations are made on the basis of a proton-neutron interaction which depends on the relative orientation of the spins of proton and neutron in accordance with Wigner's explanation of the large scattering of slow neutrons by hydrogen. Mass defect calculations based on a proton-neutron interaction indicated by the binding energy of ${\mathrm{H}}^{2}$ without dependence on the spin orientation give a much lower value for the interaction between like particles than that obtained from the proton-proton scattering experiments. The "like-particle" interaction for a Gauss error potential is $39m{c}^{2}{e}^{\ensuremath{-}17{r}^{2}}$ with 8.97\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm as the unit of length and the interaction energy is 11.1 mev for a potential which is constant (except for its Coulombian part) within a distance $\frac{{e}^{2}}{m{c}^{2}}=2.82\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm. (e) The interaction between protons as derived from the scattering experiments is found to be very nearly equal to that between a proton and a neutron in the corresponding condition of relative spin orientation and angular momentum ($^{1}S$ state). The proton-neutron values which come closest to being equal to the proton-proton values are those obtained by Fermi and Amaldi from the scattering and absorption of slow neutrons.The close agreement between the empirical values of the proton-proton and proton-neutron interactions in $^{1}S$ states suggests that aside from Coulombian and spin effects the interactions between heavy particles are independent of their charge and that the apparent preference for equal numbers of protons and neutrons in the building up of nuclei is conditioned more by the operation of the exclusion principle than by the greater values of proton-neutron forces.