Residual Interaction in Finite Nuclei

Equation (14) gives the most general form of the residual interaction in finite nuclei. Only a few terms contribute to the 1⁺ isovector excitations of the even-even nuclei we are interested in. First of all, only the isovector densities contribute. Next, the conditions $\Delta J=1$ and $\Delta \pi = 0$ between ground state and excited states imply that the only terms in the energy functional that can contribute are quadratic in local tensor or vector parity-even densities/currents. As can be seen from Table 2 in [20], all possible contributions are time-odd. One finally obtains

$${ v_{\rm res} (\vec{r}_1, \vec{r}_2) }$$

$$\frac{\delta^2 {\cal E}} {\delta \vec{s}_{1 t} (\vec{r}_1) \, \de... ...a$}}}_2) \, (\mbox{{\boldmath {$\tau$}}}_1 \cdot \mbox{{\boldmath {$\tau$}}}_2)$$ =

$$\Big[ 2 C_1^{s} [\rho_{00}] \; \delta (\vec{r}_1 - \vec{r}_2)$$ =

$$+ {\textstyle\frac{{1}}{{2}}} ( C_1^{T} - 4 C_1^{\Delta s} ) \; (... ...a (\vec{r}_1 - \vec{r}_2) + \delta (\vec{r}_1 - \vec{r}_2)\hat{\vec{k}}{}^{2} )$$

$$+ ( 3 C_1^{T} + 4 C_1^{\Delta s} ) \; \hat{\vec{k}}{}' \cdot \del... ...; \hat{\mbox{{\boldmath {$\tau$}}}}_1 \cdot \hat{\mbox{{\boldmath {$\tau$}}}}_2$$

$$- 2 \mbox{\rm\scriptsize {i}}C_1^{\nabla J} \; \hat{\mbox{{\boldm... ...t \hat{\vec{k}}{}^\prime \times \delta (\vec{r}_1 - \vec{r}_2) \, \hat{\vec{k}}$$

where $\hat{\vec{k}}$ and $\hat{\vec{k}}{}^\prime$ are defined in Appendix 8. Since the coupling constants depend only on the scalar isoscalar density $\rho_{00}$, there are no rearrangement terms in the spin-isospin channel of the residual interaction. Unsymmetrized proton-neutron matrix elements of this interaction are to be inserted into the QRPA equations as outlined in Ref. [11].