Isobaric Analog States and Coulomb Displacement Energies in the (1d52) Shell

The ($p,t$) and ($p$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$) reactions have been used to locate three previously unknown $T=\frac{3}{2}$ isobaric analog states in ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$, ${}_{}{}^{19}{}_{}{}^{}\mathrm{Ne}$, and ${}_{}{}^{23}{}_{}{}^{}\mathrm{Mg}$, in addition to significantly improving the precision on the energies of the $T=\frac{3}{2}$ state in ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ and the $T=2\mathrm{}$ states in ${}_{}{}^{20}{}_{}{}^{}\mathrm{F}$ and ${}_{}{}^{24}{}_{}{}^{}\mathrm{Na}$. These multiplets all lie within the "($1d_{\frac{5}{2}}$) shell" in that they have fewer than $n=12\mathrm{}$ nucleons outside ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, and $T\le \mathrm{minimum} (\frac{1}{2}n,6-\frac{1}{2}n)$. Including these data, twenty-eight displacement energies are now known throughout this shell for all possible multiplets with $T\le 2$ (except $T=2\mathrm{}$, mass-22). The experimental displacement energies were compared in detail with calculations which used Hecht's Coulomb-energy equations; the excellent agreement obtained appeared to be relatively insensitive to the assumed nuclear wave functions since both the low-seniority $j-j$ coupling limit and the Wigner supermultiplet scheme produced similar results. Four parameters related to the two-body Coulomb-energy matrix elements were treated as adjustable in fitting the data but their final values are in reasonable agreement with matrix elements calculated using harmonic-oscillator wave functions. A fifth adjustable parameter took account of the $Z$ and $N$ dependence of the charge radius. Using the experimental parameters, the unmeasured masses of ${}_{}{}^{19}{}_{}{}^{}\mathrm{Na}$, ${}_{}{}^{20}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{21}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{22}{}_{}{}^{}\mathrm{Al}$,