U(3) → R(3) integrity-basis spectroscopy

The U(3) → R(3) algebra, widely used in nuclear spectroscopy studies, is revisited. The most general form of a U(3)-preserving interaction that is rotationally invariant and of given degree in the group generators is presented. Here the full purpose and beauty of the integrity-basis concept is realized. En route to the above it is shown that the structure of the U(3) → R(3) integrity basis can be deduced from a systematic counting of defining space matrix elements of p-shell, k-body scalar operators, k = 0, 1, 2?. The tensorial character of the so-called “missing label” operators and, more importantly, of the operators responsible for the splitting and inversion of rotational bands is obtained by relating integrity-basis multinomials through degree four in the U(3) generators to density operators of a standard U(3) → R(3) many-body spectroscopy. The results are used to show how K-band splitting as well as an L(L + 1)]² factor in the energy can be realized within a single representation of SU(3) by two-body interactions of the ds and higher nuclear shells. Parameter sets of model interactions associated with both normal and inverted K-band structures are given, as well as the results of a “best fit” theory for the ground and gamma bands of ²⁴Mg.