Inverse-kinematics proton scattering on ${}^{50}\mathrm{Ca}$: Determining effective charges using complementary probes

We have performed measurements of the $0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+}$ excitations in the neutron-rich isotopes ${}^{48,50}\mathrm{Ca}$ via inelastic proton scattering on a liquid hydrogen target, using the GRETINA $\gamma$-ray tracking array. A comparison of the present results with those from previous measurements of the lifetimes of the $2_1^{+}$ states provides the ratio of the neutron and proton matrix elements for the $0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+}$ transitions. These results allow the determination of the ratio of the proton and neutron effective charges to be used in shell model calculations of neutron-rich isotopes in the vicinity of ${}^{48}\mathrm{Ca}$.

Figure 1

Doppler-corrected spectra of $\gamma$ rays measured in coincidence with incoming and outgoing (a) ${}^{48}\mathrm{Ca}$ and (b) ${}^{50}\mathrm{Ca}$ particles. The solid curves are the geant4 fits described in the text. The shaded region is the background, consisting of nonprompt (dark gray) and prompt (light gray) components.

Figure 2

Partial level schemes of (a) ${}^{48}\mathrm{Ca}$ from Ref. [12] and (b) ${}^{50}\mathrm{Ca}$ from Ref. [13], showing levels populated in the present work. Arrow widths are proportional to the measured $\gamma$-ray intensities.

Figure 3

The ratio of neutron to proton transition matrix elements $M_n/M_p$ in the neutron-rich calcium isotopes. The open symbols are taken from Ref. [16] and the filled circles are from the present work.