Transition sum rules in the shell model

An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the non-energy-weighted sum rule (NEWSR), or total strength, and the energy-weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, which in the case of the EWSR is a double commutator. While most prior applications of the double commutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We apply this simple tool to a number of nuclides and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole ($E1$) sum rule against the famous Thomas-Reiche-Kuhn version. We also find surprising systematic behaviors for ground-state electric quadrupole ($E2$) centroids in the $sd$ shell.

Figure 1

Energy-weighted sum rules (EWSR) and transition strength function centroids as a function of initial energy $E_i$. Results are put into 2-MeV bins with the average and root-mean-square flucutation shown; the fluctuations are not sensitive to the size of the bins. (a) EWSRs for isoscalar $E2$ for ${}^{34}\mathrm{Cl}$ in the $sd$ shell. The (red) solid line is the secular behavior predicted by spectral distribution theory, as described in Ref. [7]. (b) Centroids for $M1$ transitions in ${}^{21}\mathrm{Ne}$ in the $sd$ shell. (c) EWSR for $E1$ transitions in ${}^{10}\mathrm{B}$ in $0p\text{−}1s\text{−}0d_{5/2}$ space. (d) Centroids for GT transitions, sum of $\beta \pm$, for ${}^{27}\mathrm{Ne}$ in the $sd$ shell.

Figure 2

Ground-state $E1$ energy-weighted sum rule (EWSR) for $Z=N$ nuclides computed in the $0p\text{−}1s\text{−}0d_{5/2}$ shell model space (SM), normalized by the Thomas-Reiche-Kuhn (TRK) EWSR.

Figure 3

In the $sd$-shell using the Brown-Richter USDB interaction. (a) Centroids of $E2$ transitions from the ground state as a function of neutron number $N$. Includes nuclides with both even and odd proton numbers, with symbols in the boxes on both panels. (b) Excitation energies of the first $2^{+}$ state for even-even nuclides only.