Measurement of the B(E2↑) strengths of Ca36 and Ca

The $B(E2,\phantom{\rule{0.16em}{0ex}}{0}_{1}^{+}\ensuremath{\rightarrow}{2}_{1}^{+})$ strengths of $^{36}\mathrm{Ca}$ and $^{38}\mathrm{Ca}$ are measured to be 131(20) ${e}^{2}{\mathrm{fm}}^{4}$ and 101(11) ${e}^{2}{\mathrm{fm}}^{4}$, respectively. The $B(E2)$ value for $^{36}\mathrm{Ca}$ required a measurement of the $p$/$\ensuremath{\gamma}$ branching ratio because the ${2}^{+}$ state is proton unbound. This branching ratio is ${B}_{p}=0.087(8)$. These $B(E2)$ and branching-ratio values can be reproduced in the shell-model with the ZMB2 interaction, an interaction that predicts the $Z=20$ $sd$-shell closure is incomplete with large proton $pf$-shell occupancies in the ground state. These occupancies are at odds with other shell-model and energy-density-functional calculations of $^{36}\mathrm{Ca}$. New data are used to provide an update on constraints of the density dependence of the symmetry energy through mirror charge-radii differences as well as to help reduce uncertainties of the astrophysical important $^{35}\mathrm{K}(p,\ensuremath{\gamma}$) reaction.