Experimental Study of the $2I+1$ Rule Using the ($d,\alpha$) Reaction on ${\mathrm{F}}^{19}$, ${\mathrm{Na}}^{23}$, ${\mathrm{Al}}^{27}$, and ${\mathrm{P}}^{31}$

The angular distributions corresponding to the ($d,\alpha$) production cross sections ($E_d\approx 9.3$ MeV) for the ground states and some of the low-lying states of ${\mathrm{O}}^{17}$, ${\mathrm{Ne}}^{21}$, ${\mathrm{Mg}}^{25}$, and ${\mathrm{Si}}^{29}$ have been measured using silicon surface-barrier detectors. The discussion of the $2I+1$ rule within the context of these experimental observations is based on the premise that both the direct interaction (DI) and compound nucleus (CN) mechanisms contribute incoherently to the angular distributions, and the approximate validity of an ad hoc method used for the decomposition of these differential cross sections into DI and CN components. The integrated experimental differential cross sections, ${\sigma }_T({10}^{\circ }-{170}^{\circ })$ and ${\sigma }_B({90}^{\circ }-{170}^{\circ })$, and the integrated CN component, ${\sigma }_{\mathrm{CN}}({10}^{\circ }-{170}^{\circ })$, are analyzed in terms of the $2I+1$ rule. The ${\sigma }_{\mathrm{CN}}$ values are found to be more nearly proportional to $2I+1$ than either the ${\sigma }_T or {\sigma }_B$ values. This result is interpreted as indicating that the $2I+1$ dependence of the ($d,\alpha$) cross sections is a characteristic associated with the CN mechanism rather than the DI mechanism. Furthermore, it is concluded that spin assignments based solely on the assumption of a $2I+1$ dependence of either the ${\sigma }_T or {\sigma }_B$ values derived from ($d,\alpha$) reactions would not as a rule be very reliable.