An Analysis of the Energy Levels of the Mirror Nuclei, ${\mathrm{C}}^{13}$ and ${\mathrm{N}}^{13}$

An analysis employing the recent nuclear reaction theories of Wigner and others is given of the experimental data on the low energy interactions of $s$, $p$, $d$ orbital neutrons and protons with ${\mathrm{C}}^{12}$ and $s$ neutrons and protons with ${\mathrm{O}}^{16}$. Assuming the equality of $\mathrm{nn}$ and $\mathrm{pp}$ nuclear interactions, it is possible to account for the data on the $s$ interactions if the level spacing is considered in addition to the customary two resonance parameters: reduced width and level position; in particular, the displacement of conjugate levels can be attributed to the difference of the external wave functions for the odd particle, although with an uncertainty of about 25 percent which is due primarily to the lack of precise knowledge of the internal Coulomb energy of the excited states. The large magnitudes of the reduced width and level spacing indicate that two-body potential interactions exist between the odd particle and the ${\mathrm{C}}^{12}$ and ${\mathrm{O}}^{16}$ cores, and the values of the respective logarithmic derivatives indicate that these interactions are of about equal strengths. The energy dependence of the radiative capture cross section of $s$ neutrons and protons with ${\mathrm{C}}^{12}$ can be understood if an additional quantity, the final-state reduced width, is included in the theory to take into account the energy-dependent external contribution to the transition moment. The experimental data are only sufficient to treat the $p$ and $d$ interactions in the one-level approximation; a reasonable explanation can be given of the observed displacements of conjugate levels in terms of the differences of the electromagnetic properties of the odd particle such as: external wave functions, spin-orbit interactions, and variations of the internal Coulomb energy. There is some indication from the data on radiative transitions that the independent-particle model also prevails in the $p$ states; on the other hand, the small reduced widths of these states suggest a many-body description. Derivations based on the recent theories are given of the one-channel formulas and of the general one-level formulas which include the negative-energy alternatives. The radial dependences of the resonance parameters are discussed.