Background: Mixed-symmetry $2^{+}$ states in vibrational nuclei are characterized by a sign change between dominant proton and neutron valence-shell components with respect to the fully symmetric $2^{+}$ state. The sign can be measured by a decomposition of proton and neutron transition radii with a combination of inelastic electron and hadron scattering [C. Walz et al., Phys. Rev. Lett. 106, 062501 (2011)]. For the case of ${}^{92}$Zr, a difference could be experimentally established for the neutron components, while about equal proton transition radii were indicated by the data.

Purpose: Determination of the ground-state (g.s.) transition strength of the mixed-symmetry $2_2^{+}$ state and verification of the expected vanishing of the proton transition radii difference between the one-phonon $2^{+}$ states in ${}^{92}$Zr.

Method: Differential cross sections for the excitation of one-phonon $2^{+}$ and $3^{-}$ states in ${}^{92}$Zr have been measured with the ($e,e^{\prime }$) reaction at the S-DALINAC in a momentum transfer range $q\simeq 0.3–0.6$ fm${}^{-1}$.

Results: Transition strengths $B(E2;2_1^{+}\to 0_1^{+})=6.18\left(23\right)$, $B(E2;2_2^{+}\to 0_1^{+})=3.31\left(10\right)$, and $B(E3;3_1^{-}\to 0_1^{+})=18.4\left(1.1\right)$ Weisskopf units are determined from a comparison of the experimental cross sections to quasiparticle-phonon model (QPM) calculations. It is shown that a model-independent plane wave Born approximation (PWBA) analysis can fix the ratio of $B\left(E2\right)$ transition strengths to the $2_{1,2}^{+}$ states with a precision of about 1$\%$. The method furthermore allows to extract their proton transition radii difference. With the present data $\Delta R=-0.12\left(51\right)$ fm is obtained.

Conclusions: Electron scattering at low momentum transfers can provide information on transition radii differences of one-phonon $2^{+}$ states even in heavy nuclei. Proton transition radii for the $2_{1,2}^{+}$ states in ${}^{92}$Zr are found to be identical within uncertainties. The g.s. transition probability for the mixed-symmetry state can be determined with high precision limited only by the available experimental information on the $B(E2;2_1^{+}\to 0_1^{+}$) value.

• Received 27 November 2012

DOI:https://doi.org/10.1103/PhysRevC.87.014337