Measurement and analysis of muonic x rays ofHf176,177,178,179,180

Abstract

Monopole and quadrupole charge distributions of $^{176}\mathrm{Hf}$, $^{177}\mathrm{Hf}$, $^{178}\mathrm{Hf}$, $^{179}\mathrm{Hf}$, and $^{180}\mathrm{Hf}$ were investigated by muonic atom $K$ and $L$ x-ray measurements. The model-independent Barrett charge radii ${R}_{k}$ and the isotope shifts $\ensuremath{\Delta}{R}_{k}$ were measured, and values of $〈{r}^{2}〉$ and $\ensuremath{\Delta}〈{r}^{2}〉$ were deduced. A weak odd-even staggering of the nuclear charge radii was observed for the series $^{176\ensuremath{-}178}\mathrm{Hf}$ and $^{178\ensuremath{-}180}\mathrm{Hf}$. A large negative isomer shift was observed in the ${2}^{+}$ state of the $^{176}\mathrm{Hf}$ nucleus, a fact that existing theories do not explain. The quadrupole moments of the first excited states of the hafnium nuclei were determined to be ${Q}^{176}({2}^{+})=\ensuremath{-}2.10(2) e$ b, ${Q}^{177}({\frac{9}{2}}^{\ensuremath{-}})=1.30(2) e$ b, ${Q}^{178}({2}^{+})=\ensuremath{-}2.02(2) e$ b, ${Q}^{179}({\frac{11}{2}}^{+})=1.88(3) e$ b, and ${Q}^{180}({2}^{+})=\ensuremath{-}2.00(2) e$ b. These quadrupole moments and the simultaneously determined $B(E2)$ values for the respective nuclei are in satisfactory agreement with the predictions of the axially symmetric rotor model.