Isospin-symmetry breaking corrections for the description of triplet energy differences

The charge-independence breaking of the nuclear interaction is analyzed by means of energy differences among analog states in $T=1$ isobaric multiplets. Data on triplet energy differences in the $sd, pf$, and $pfg$ shells, i.e., $18\le A\le 66$, are reproduced with very good accuracy by large-scale shell-model calculations taking into account, aside from the Coulomb interaction, a single isotensor schematic interaction of monopole-pairing type. It is shown that the effect on the triplet energy differences of this isospin-breaking interaction is of the same magnitude as the Coulomb one. Moreover, its strength is the same for every single-particle orbital of the considered model space.

Figure 1

Data points: The isotensor INC matrix elements, $V_B^{\left(2\right)}\left(J\right)$, for $f_{7/2}$ pairs, extracted from the fits across the whole $f_{7/2}$ shell—see final row of Table 1. Dashed line (blue) shows results for the same parameters extracted from Brown and Sherr [12] who fit the Coulomb displacement energies to a single-$f_{7/2}$-shell-model calculation.

Figure 2

Comparison of experimental triplet energy differences (TED) in the $sd$ shell to the predictions of the shell model using the USD interaction—see text for details. Dots (black) show the experimental data. Solid line (black) shows the total TED calculation from the shell model. Dashed line (blue) shows the contribution to the calculated TED from the Coulomb two-body interaction alone. Dot-dashed line (red) shows the contribution to the calculated TED from the INC ($V_B^{\left(2\right)}$) interaction alone. The black line is the sum of the blue (dashed) and the red (dot-dashed) lines.

Figure 3

Comparison of experimental triplet energy differences (TED) in the $f_{7/2}$ shell to the predictions of the shell model using the KB3G interaction—see caption of Fig. 2 for further details.

Figure 4

Comparison of experimental triplet energy differences (TED) in the upper $pf$ shell to the predictions of the shell model using the GXPF1A interaction—see caption of Fig. 2 for further details.

Figure 5

Comparison of experimental $A=66$ triplet energy differences (TED) to the predictions of the shell model using the JUN45 interaction—see caption of Fig. 2 for further details.

Figure 6

The contribution to the calculated TED from the INC ($V_B^{\left(2\right)}$) interaction, determined for each orbital individually for selected examples. For each panel the sum of the single-orbital contributions (full thick black line) corresponds to the dot-dashed (red) curve in Figs. 2 to 5.