Nuclear structure of the first $2^+$ state in radioactive $^{68}$Ge based on $g$ factor and lifetime measurements

The $g$ factor of the $2_1^+$ state of radioactive $^{68}$Ge (T_{1/2}=270 d) has been measured for the first time. The technique used is based on $\alpha$ transfer from a $^{12}C target to energetic $^{64}$Zn projectiles that incorporates the favorable conditions of inverse kinematics as in projectile Coulomb excitation. It also includes features of the transient field technique applied to nuclear spin precessions. Because the reaction cross section is large the method is a significant alternative to Coulomb excitation of low-intensity radioactive ion beams. In addition, we have remeasured the lifetimes of several excited states using the Doppler-shift-attenuation method. In these measurements, the inherent focusing nature of the reaction in the forward direction was optimally exploited for the resulting fast-moving nuclei. The g factor value obtained, g(2_1^+)=+0.55(14), is in good agreement with the collective value, g=Z/A=0.47, and is also consistent with the precise data of the stable even-A Ge isotopes. The newly determined lifetimes partially agree with those quoted in the literature and are of comparable accuracy. The deduced B(E2) values and the new measured $g$ factor are well reproduced by some $fp$ shell model calculations in which excitations from the $f_{7/2}$ orbital play an important role