Microscopic Analysis of the (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:math>) and (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>′) Reactions on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mi/><mml:mi>p</mml:mi></mml:math>-Shell Nuclei

The ($^{3}\mathrm{He},t$) and ($^{3}\mathrm{He}$,$^{3}\mathrm{He}$\ensuremath{'}) reactions on $^{9}\mathrm{Be}$, $^{12}\mathrm{C}$, $^{13}\mathrm{C}$, $^{14}\mathrm{C}$ [($^{3}\mathrm{He},t$) reaction only], $^{14}\mathrm{N}$, and $^{15}\mathrm{N}$ have been investigated at $^{3}\mathrm{He}$ energies varying between 40 and 50 MeV. Angular distributions were obtained for all prominent states up to excitations of 20 MeV. A microscopic analysis of these data has been carried out, using a local two-body interaction with an arbitrary spin-isospin exchange mixture. Spectroscopic factors were calculated, using intermediate-coupling wave functions for $p$-shell states, while simple $j\ensuremath{-}j$ configurations were assumed for the levels which were populated by promoting a $p$ nucleon to the $s\ensuremath{-}d$ shell. A Yukawa interaction with a range of 1.2 F was found to give the best results. The strength of the effective nucleon-nucleon interaction required to fit these data is in good agreement with recent analyses of the ($p,{p}^{\ensuremath{'}}$) and ($p,n$) reactions on light nuclei. In particular, dominant $L=0$ transitions observed in the ($^{3}\mathrm{He},t$) reaction give values for the isospin-dependent (${V}_{\mathrm{ST}}={V}_{01}$) and spin-isospin-dependent (${V}_{11}$) terms (converted to an effective nucleon-nucleon interaction at a range of 1.0 F) of 20.6 and 16.5 MeV, respectively, while the strengths required to fit ($^{3}\mathrm{He},t$) $L=2,3$ transitions were generally enhanced. For inelastic transitions, the average strengths obtained for ${V}_{00}$, assuming a Serber exchange mixture, varied from 47.2 to 67.3 MeV, depending upon the $L$ transfer involved. A comparison of the ($^{3}\mathrm{He},t$) and ($^{3}\mathrm{He}$,$^{3}\mathrm{He}$\ensuremath{'}) reactions populating analog final states (where ${T}_{f}={T}_{i}+1$) is also presented. In general, these transitions were weak; however, it was possible to observe the lowest $T=\frac{3}{2}$ levels in the mirror nuclei $^{9}\mathrm{B}$-$^{9}\mathrm{Be}$ and $^{13}\mathrm{N}$-$^{13}\mathrm{C}$, and several $T=1$ levels in $^{12}\mathrm{N}$-$^{12}\mathrm{C}$.