Theory of the Lithium Two-Alpha-Reactions. I. Angular Distribution ofLi7(p,α)α

Abstract

The reactions ${\mathrm{Li}}^7(p,\alpha )\alpha$ and ${\mathrm{Li}}^6(d,\alpha )\alpha$ have a special interest because the Bose statistics and consequent even parity of the final pair of alphas simplifies the analysis. It is assumed on reasonable grounds that ${\mathrm{Li}}^7$ has odd parity and ${\mathrm{Li}}^6$ even. This means that only $p,f\cdots$ protons entering ${\mathrm{Li}}^7$ are relevant to this reaction and only entering $s,d\cdots$ deuterons of ${\mathrm{Li}}^6$. An expression for the energy variation of the angular distribution in the ${\mathrm{Li}}^7$ reaction is derived from the dispersion formula, with the specific assumption that only two levels of the compound nucleus, having angular momentum 0 and 2, are important in the range of energies employed in the experiments. (An alternative assumption is investigated but is more involved and it is considered less plausible.) The angular factor is $1+A\left(E\right)\mathrm{}{cos}^2\theta +B\left(E\right)\mathrm{}{cos}^4\theta$. A formula for $A\left(E\right)\mathrm{}$ as far as it results from entering $p$ waves has been derived previously by Critchfield and Teller, but it may not be adjusted to fit the data in the energy range up to 3 Mev even within the rather large experimental uncertainty. The effect of entering $f$ waves is also included in the present analysis, and this contribution is not only essential to the existence of $B\left(E\right)\mathrm{}$ but also makes it possible to obtain agreement with the recent measurements of $A\left(E\right)\mathrm{}$. Certain matrix elements are treated as arbitrary parameters in obtaining this fit. In addition to the low-energy node apparent in the early data, a second node of $A\left(E\right)\mathrm{}$ is required at higher energy. The formula derived for $B\left(E\right)\mathrm{}$ contains the same resonance denominator as does $A\left(E\right)\mathrm{}$ and is compatible with the present experimental results. The contrasting facts, first that the easy entrance of $s$ waves makes the ${\mathrm{Li}}^6(d,\alpha )\alpha$ reaction symmetric at very low energies and second that the $p$ waves responsible for the ${\mathrm{Li}}^7$ reaction at low energies may introduce asymmetry, are associated in a simple way with the qualitative experimental observation that both $A\left(E\right)\mathrm{}$ and $B\left(E\right)\mathrm{}$ rise more quickly in the ${\mathrm{Li}}^7$ reaction than in the ${\mathrm{Li}}^6$ reaction as the bombarding energy is increased from zero. Presentation of a detailed formulation of the ${\mathrm{Li}}^6$ reaction is deferred until a later paper.