#### Abstract

Recently it has become increasingly evident that some assumptions in the nuclear model used for the Monte Carlo calculations yield cross section values which are not in accord with experiment. In particular, calculations of ($p,\mathrm{pn}$)-reaction cross sections in the Bev energy range give values which are low by factors of two to nine when compared to experimental values. The calculated cross sections also show a smooth variation with the target atomic weight whereas the experimental values show quite an erratic variation. Reasons which have been advanced to account for this lack of agreement are the lack of a nuclear surface and failure to account for shell effects in the nuclear model used.

In this work a theory is developed to take account of surface and shell effects and thereby describe the observed magnitude and variation of the cross sections for simple nuclear reactions as exemplified by the ($p,\mathrm{pn}$) reaction. At multi-Bev energies to which this treatment is restricted, the main contribution to the ($p,\mathrm{pn}$)-reaction cross section comes from inelastic collisions between the incident protons and target neutrons, with all the $p-n$ collision products escaping without further interaction. Approximations and assumptions used include the impulse approximation, 0° lab scattering angle for the inelastic $p-n$ collision products, classical trajectories for the incident and scattered particles, and a quantum-mechanical treatment for the target nucleons. The multi-Bev, $n-p$, cloud-chamber data was used to determine the average total exit cross section for the inelastically scattered particles. The only neutron shells in the target nucleus contributing to the ($p,\mathrm{pn}$) reaction are those for which the instantaneous knocking out of a neutron creates a product-neutron hole state stable to particle emission. The combination of these assumptions gives integral expressions which, when evaluated on the IBM-701 computer for the independent particle harmonic-oscillator shell model, give the ($p,\mathrm{pn}$) reaction cross sections as a function of the nuclear density distribution and the number of available shells.

For the low $Z$ nuclei where the available shells can be unambiguously determined, the results give a half-central-density radius parameter, ${r}_{0}$, (${r}_{0}=\frac{{R}_{\frac{1}{2}}}{{A}^{\frac{1}{3}}}$), of about 1.2 fermis compared to 1.03 fermis for the charge half radius from the electron-scattering work. Use of reasonable limits on the value of ${r}_{0}$ allows one to set the minimum number of shells available for some targets. For example, the ${\mathrm{Zn}}^{64}$, ${\mathrm{Cu}}^{65}$, and ${\mathrm{Cu}}^{63}$ ($p,\mathrm{pn}$) cross sections require that a large part or all the $1{f}_{\frac{7}{2}}$ neutrons be available, or, equivalently, that a $1{f}_{\frac{7}{2}}$ neutron hole state (across a major shell) in the product nucleus have less than 8- to 9-Mev excitation energy. The results also show that the energy associated with nuclear rearrangement to particle-stable product states must be less than 8 to 9 Mev. In several cases, the upper limit can be lowered considerably (to 1.5 Mev and 0 Mev in the cases of ${\mathrm{O}}^{16}$ and ${\mathrm{N}}^{14}$, respectively).

DOI: http://dx.doi.org/10.1103/PhysRev.119.324

- Received 18 December 1959
- Revised 9 March 1960
- Published in the issue dated July 1960

© 1960 The American Physical Society