Effect of Angular Localization on Direct Nuclear Reactions

Some interesting effects of the angular localization of reaction sites caused by the damping of the incident and exit particle waves by the nuclear optical potential are described. Using the reactions ${\mathrm{Li}}^7\mathrm{}(p,2p)\mathrm{}{\mathrm{He}}^6$ and ${\mathrm{Be}}^9\mathrm{}(p,2p)\mathrm{}{\mathrm{Li}}^8$, as examples, we show that, due to the angular localization, the total reaction cross section is different for aligned than for unaligned target nuclei. For the ${\mathrm{Li}}^7$ target in $j-j$ or $L-S$ coupling the fractional change, $f$, in the cross section for an incident energy of 6 BeV was given by $f=0.165\mathrm{}(2x-1)\mathrm{}$, where $x$ is the fractional population of ${\mathrm{Li}}^7$ in the $M=\left|\frac{3}{2}\right|$ substates. At 140 MeV the value of the constant in the above expression was slightly but not significantly smaller. For the reaction ${\mathrm{Be}}^9\mathrm{}(p,2p)\mathrm{}{\mathrm{Li}}^8$ the fractional change in the cross section is one half of and of the opposite sign of the ${\mathrm{Li}}^7$ value for the same amount of target alignment. Another result of angular localization of reaction sites in the nucleus is that $J\ne 0$ product nuclear states will be oriented with respect to the incident beam. Using ($p,pn$) and ($p,2p$) reactions on ${\mathrm{O}}^{16}$ and ${\mathrm{Ni}}^{58}$ as examples, the amount of product nuclear state alignment was computed for the $1p_{\frac{3}{2}}$ and $1f_{\frac{7}{2}}$ excited hole states of ${\mathrm{O}}^{15}$, ${\mathrm{N}}^{15}$, and ${\mathrm{Ni}}^{57}$. The population ratios obtained for the substates in ${\mathrm{O}}^{15}$ or ${\mathrm{N}}^{15}$ were $\left|\frac{3}{2}\mathrm{}\right|:\left|\mathrm{}\frac{1}{2}\right|=0.58:0.42\mathrm{}$ and for ${\mathrm{Ni}}^{57}$, $\left|\frac{7}{2}\mathrm{}\right|:\left|\mathrm{}\frac{5}{2}\mathrm{}\right|:\left|\mathrm{}\frac{3}{2}\mathrm{}\right|:\left|\mathrm{}\frac{1}{2}\right|=0.34:0.25:0.21:0.20\mathrm{}$. The angular anisotropies, $\frac{\left[W\right(\frac{\pi }{2})-W(0\left)\right]}{W\left(\frac{\pi }{2}\right)]}$, expected from gamma decay of these states were computed to be -0.125 assuming an $M1\mathrm{}$ transition for the ${\mathrm{O}}^{15}$ or ${\mathrm{N}}^{15}$ $\frac{3}{2}$-states and 0.162 assuming an $E2\mathrm{}$ transition for the ${\mathrm{Ni}}^{57}$ $\frac{7}{2}$-state.