Constraints on the neutron equation of state using the difference in 54Ni-54Fe mirror pair charge radii

Just like the ideal gas law is used to characterize a `perfect' gas, equations of state can be used to describe nuclear matter. Two such equations of state include the symmetric matter equation of state and the neutron matter equation of state. While symmetric matter is well known, neutron matter is not, especially when extrapolating to higher densities. The nuclear matter equation of state is of interest because a greater understanding of it is required to predict properties of both super-heavy nuclei and neutron stars. There is ongoing debate about whether the neutron equation of state is `soft' or `stiff', where a `stiff' equation of state implies the pressure in the nucleus increases rapidly with increasing density and consequently implies a larger neutron star radius. A way to constrain this neutron equation of state is through the slope of the symmetry energy (the $L$ parameter), where the symmetry energy is the difference between the symmetric and neutron matter equations of state. Conceptually, $L$ is proportional to the pressure of pure neutron matter at a specific density, and can also be thought of as a restoration force (`spring constant') between protons and neutrons when they are dislocated in the nucleus.The complication is that $L$ is not a physical observable and cannot be directly measured in the laboratory. However, it has been shown that the neutron skin thickness ($\Delta R_\mathrm{np}$) of neutron-rich nuclei are correlated to $L$. By measuring the neutron skin we can therefor place constraints on the $L$ parameter and ultimately the neutron equation of state. Many experimental and theoretical techniques have been used to constrain $L$. It is noted that all of them are model-dependent in some way. Even though many of these analyses agree within $1\sigma$, they each have tendencies toward either the `soft' or the `stiff' nuclear equation of state. Results from the PREX and CREX experiments, highly regarded benchmarks for the neutron skin value, also show tension between their results, highlighting that discretion is needed when addressing the model-dependent components in these analyses. This model-dependence brings about the need for increased systematic measurements of the $L$ parameter to add to the discussion on constraints on $L$.The difference in charge radii ($\Delta R_\mathrm{ch}$) is a new, purely electromagnetic probe to deduce the neutron skin and constrain $L$. Assuming perfect charge symmetry, the distribution of the protons is equal to the distribution of the neutrons in the mirror nucleus. By taking the difference in charge radii, the neutron skin can be obtained. In reality, however, the charge symmetry is broken by the Coulomb interaction that pushes protons out relative to neutrons, leading to a weaker correlation between $\Delta R_\mathrm{np}$ and $\Delta R_\mathrm{ch}$. However, even with this Coulomb disruption, $\Delta R_\mathrm{ch}$ shows tighter correlation to $L$ than that of $\Delta R_\mathrm{np}$. It was also shown that $\Delta R_\mathrm{ch}$ is correlated to $|N-Z| \times L$. Due to this correlation, ideally a mirror pair would be chosen with a high $|N-Z|$ to provide a tighter constraint, where the maximum possible is $|N-Z|=6$ for the $^{22}$Si-$^{22}$O pair. The present $^{54}\mathrm{Ni}$-$^{54}\mathrm{Fe}$ mirror pair has a rather low $|N-Z|=2$, and therefor requires a highly sensitive technique to be able to provide good enough results to place constraints on $L$, otherwise too large of an uncertainty would void any meaningful discussion.Isotope shift measurements using bunched beam collinear laser spectroscopy of $^{54}$Ni ($I^\pi=0^+,\,\,t_{1/2}=114\,\mathrm{ms}$) and other nickel isotopes were performed at the BEam COoling and LAser spectroscopy (BECOLA) facility at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. These precise measurements were used to extract the charge radius for $^{54}$Ni for the first time to be $R(^{54}\mathrm{Ni})=3.737 \pm 0.003$\,fm. Using the already known $^{54}$Fe charge radius from literature, the difference in charge radii between the mirror pairs was taken in order to obtain $\Delta R_\mathrm{ch}=0.049 \pm 0.004$\,fm. Based on the correlation between $L$ and $\Delta R_\mathrm{ch}$ calculated by density functional theory using the Skyrme energy density functional, the present $\Delta R_\mathrm{ch}(A=54)$ set a constraint on the $L$ parameter as $21 \le L \le 88$\,MeV. The model takes into account corrections for the quadrupole deformation, which was evaluated through the $\beta_2$ deformation parameter obtained by the reduced $E2$ transition probability $B(E2,\uparrow)$ for $^{54}$Ni.These constraints on $L$ from BECOLA are in good agreement with the GW170817 neutron star merger, whose results also favor the `soft' neutron equation of state, providing an link between a new terrestrial-based experimental method and an astrophysical observation. These results are different from the PREX results, which favor the `stiff' EOS. To add to the systematics, the same method using parity violating electron scattering to measure the electroweak form factor was used with the CREX experiment.Using the $\Delta R_\mathrm{ch}$ method has also enabled constraints on the neutron skin for $^{48}$Ca which agree with the CREX results. The tension between CREX and PREX stems from the model-dependent step during the analyses demonstrated by a reevaluation of the PREX results which resulted in a smaller neutron skin and $L$ value consistent with GW170817, BECOLA, and CREX.A global trend analysis evaluated the relationship between $\Delta R_\mathrm{ch}$ and $L$. The results concluded that while there was correlation between the observable and the $L$ parameter, it could not place stringent constrains on the neutron equation of state within the model, hearkening back to model-dependence playing a critical role in the determination of $L$.