Phenomenological Many-Body Exchange Forces

Phenomenological many-body exchange potentials are defined in group-theoretical analogy with the usual 2-body exchange potentials of Heisenberg, Bartlett, and Majorana. The uniform model of Wigner is then extended to include many-body Wigner and Majorana forces. The theory predicts a symmetry energy between neighboring isobars of even-odd nuclei which depends upon whether the isobar of highest $T_{\zeta }$ is $\mathrm{OE}$ or $\mathrm{EO}$, where $\mathrm{OE}$ signifies that the odd number of particles is greater than the even number, and $\mathrm{EO}$ signifies the reverse. A comparison is made with such energy differences in $T_{\zeta }=\frac{3}{2} \mathrm{and} \frac{1}{2}$ isobars, taken from $\beta$-decay data in the range $15\leqq A\leqq 59$. This leads to a rough estimate of the magnitude and sign of the contribution from 3-body potentials. On this basis, the contribution from 3-body potentials is not negligible in comparison with that from 2-body potentials.