Intrinsic Electric Quadrupole Moment of the $K^\pi=8^-$ Isomeric State in $^{178}$Hf

The lifetime of the 9$^-$ state in the rotational band based on the 4.0 s, $K^\pi =8^-$, isomeric state ($^{178}$Hf$^{m_{1}}$) from the decay of the 31-yr isomer ($^{178}$Hf$^{m_{2}}$) was determined to be 99(2) ps by means of the fast-timing technique using two LaBr$_3$(Ce) scintillators. The $\delta(E2/M1)$ mixing ratios of the $\Delta I =1$ $\gamma$ rays depopulating levels in this band were deduced from $\gamma$-$\gamma$ angular correlations by using a $^{178}$Hf$^{m_{2}}$ radioactive source located at the center of the Gammasphere HPGe detector array. The new results, together with previous spectroscopic information, provide a different way to extract the intrinsic quadrupole moment of $Q_0=6.45(14)~eb$ for the $^{178}$Hf$^{m_{1}}$ band. A possible explanation for the reduction of the $^{178}$Hf$^{m_1}$ nuclear charge radius is presented.