Medium-mass nuclei with $\Delta$ excitations under compression

The ground state properties of ${}^{90}\mathrm{Zr},$ ${}^{100}\mathrm{Sn},$ and ${}^{132}\mathrm{Sn}$ at equilibrium and at large amplitude compression are investigated. We use a realistic effective baryon-baryon Hamiltonian that includes $N-N,$ $N-\Delta ,$ and $\Delta -\Delta$ interactions. We perform the calculations in no-core model spaces within the framework of the constrained spherical Hartree-Fock approximation. We specifically investigate the sensitivity to the sizes of the nucleon and $\Delta$ model spaces. At equilibrium, we find no case of mixing between nucleons and $\Delta ’\mathrm{s}$ in our largest model space of eight major nucleon shells plus 16 $\Delta$ orbitals. On the contrary, there is mixing in ${}^{90}\mathrm{Zr},$ and ${}^{132}\mathrm{Sn}$ in the smaller model space of seven major nucleon shells plus eight $\Delta$ orbitals. Expanding the nucleon model space has a larger effect on reducing the static compression modulus and softening the nuclear equation of state than increasing the number of $\Delta$ states. Most of the excitation energy delivered to the system during compression is employed by two nuclei with a neutron excess (i.e., ${}^{90}\mathrm{Zr},$ ${}^{132}\mathrm{Sn})$ to create massive $\Delta$ resonances. On the other hand, in the ${}^{100}\mathrm{Sn}$ nucleus most of the excitation energy goes to a simple reduction in the binding, suggesting a suppressed role for the $\Delta$ states. Under extreme compression, at a density 2–3 times the normal nuclear density, the excitation of nucleons to $\Delta ’\mathrm{s}$ increases sharply up to 10% of the total number of constituents. At fixed excitation energy under compression, the number of $\Delta$ excitations is not dependent on the number of $\Delta$ states over the range studied. The $\Delta$-excitation results are consistent with heavy-ion collision data, and suggest an important mean field mechanism for subthreshold pion production in particle-nucleus and nucleus-nucleus collisions.