Sum rules for B (M1, 01+ → 1+) strength in IBM-3 and IBM-4

Sum rules for B (M1, 01+ → 1i+) strength are derived for even-even nuclei in the isospin-invariant forms of the IBM, IBM-3 and IBM-4, in the cases where the respective natural internal symmetries, isospin U (3) and U (6) ⊃ SU (4), are conserved. Subsequently, the total strength is resolved into its component partial sums to the allowed isospins (and SU (4) representations in IBM-4). In cases where the usual IBM dynamical symmetries are also valid, a complete description of all B (M1, 01+ → 1i+) follows. In contrast to IBM-2, there is fragmentation of the strength even in the dynamical symmetry cases, for T ≠ 0, over two states in IBM-3, and over three states in IBM-4. An application to the first half of the sd-shell reveals differences in strength between IBM-4 and the shell model which may also indicate areas for further study in IBM-2 analyses of heavier nuclei.