Nuclear States in the RaE<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math>-Decay

Abstract

It is shown that the $\ensuremath{\beta}$-transforming neutron of ${\mathrm{Bi}}^{210}$ most probably has an ${i}_{\frac{11}{2}}$ character, despite a ${g}_{\frac{9}{2}}$ character of the ground-state neutron in ${\mathrm{Pb}}^{209}$. This makes a critical difference to the RaE spectrum parameter, $\ensuremath{\xi}=\frac{i〈\mathrm{r}〉}{〈\ensuremath{\sigma}\ifmmode\times\else\texttimes\fi{}\mathrm{r}〉}$, yielding $\ensuremath{\xi}\ensuremath{\approx}+1$ rather than $\ensuremath{\xi}\ensuremath{\approx}\ensuremath{-}\frac{1}{10}$. The effect of configuration mixing is also investigated but does not affect $\ensuremath{\xi}$ appreciably.To arrive at the above conclusions, it is necessary to show that the neutron-proton attraction in the ${({h}_{\frac{9}{2}}{i}_{\frac{11}{2}})}_{1}$ state is the large amount, 840 kev, greater than in the ${({h}_{\frac{9}{2}}{g}_{\frac{9}{2}})}_{0}$ state. The resulting shell-model problem has interest independent of the $\ensuremath{\beta}$-theory application which was the original objective of this work. True and Ford had found that two neutrons extra to the doubly-magic core, ${\mathrm{Pb}}^{208}$, as against nucleons deep in the core matter, interact with about the same strength and range of force as do two free nucleons. The problem here checks the extension of that important finding to neutron-proton and proton-proton pairs. The force strength is consequently not used as an adjustable parameter, as it has been in previous approaches to such problems.The True-Ford problem involved only singlet, central forces between like nucleons in an essential way. The RaE daughter, ${\mathrm{Po}}^{210}$, investigated here, has only Coulomb repulsion superposed. The resultant comparison with experiments is about as good as that obtained by True and Ford.The extension to the neutron proton pair of ${\mathrm{Bi}}^{210}$ is far more complex, since triplet forces, an exchange character, and non-central forces, may now come into play. We find that the finite-range, central forces alone cannot give substantially more attraction in the ${({h}_{\frac{9}{2}}{i}_{\frac{11}{2}})}_{1}$ state than in ${({h}_{\frac{9}{2}}{g}_{\frac{9}{2}})}_{0}$. Even using the strength as a parameter cannot help significantly. However, the tensor forces produce attraction in the former and repulsion in the latter state. Hence, the two-body forces must be imitated even in this detail in order to yield an explanation of the RaE level scheme. This is unfortunate for quantitative results, because the strength of the two-body tensor force seems never to have been determined unambiguously for potentials without cores.Our final conclusion is that the two-body neutron-proton force may be well represented by zero-range forces of the same volume energy as found experimentally. Tensor effects vanish identically in this limit and so an unambiguous representation of the strength can be obtained. The results for the relative positions of the $J=0 \mathrm{and} 1$ states, used as the test above, now turn out in almost perfect agreement with the observations. Configuration mixing plays a role in this result, and, in consequence, the work includes a generalization of de-Shalit's formulas, for the interaction energies with zero-range forces, to nondiagonal matrix elements.