$\mu$-Atomic Hyperfine Structure in the $K$, $L$, and $M$ Lines of ${\mathrm{U}}^{238}$ and ${\mathrm{Th}}^{232}$

The $\mu$-atomic hyperfine structure in the $K$, $L$, and $M$ lines of ${\mathrm{U}}^{238}$ and ${\mathrm{Th}}^{232}$ was obtained using a large high-resolution Ge(Li) detector. The hyperfine structure is due to large deformations in these nuclei and arises from a dynamic electric quadrupole interaction between the muonic states and the lowest nuclear states. The measurements were interpreted using the Bohr-Mottelson rigid-rotator model with a symmetry axis for the nucleus and taking into account the first and second rotational states. The charge distribution chosen was a Fermi type whose constant-density shells are concentric ellipsoids of revolution of constant eccentricity. By numerically solving the Dirac equation, it was possible to determine three parameters of the size and shape of the nuclear charge distribution. The intrinsic quadrupole moments were found to be (11.47±0.13) × ${10}^{-24\mathrm{}}$ and (9.83±0.16) × ${10}^{-24\mathrm{}}$ ${\mathrm{cm}}^2$ for ${\mathrm{U}}^{238}$ and ${\mathrm{Th}}^{232}$, respectively, in good agreement with other measurements. The deformation parameter $\beta$ was found to be 0.244±0.002 and 0.222±0.003, respectively. The central density, found to be 0.153±0.002 and 0.153±0.003 nucleons per cubic fermi, respectively, differs only slightly from the value of 0.158 nucleons per cubic fermi that one finds for most spherical nuclei.