Large-scale second random-phase approximation calculations with finite-range interactions

Second RPA (SRPA) calculations of nuclear response are performed and analyzed. Unlike in most other SRPA applications, the ground state, approximated by the Hartree-Fock (HF) ground state, and the residual couplings are described by the same Hamiltonian and no arbitrary truncations are imposed on the model space. Finite-range interactions are used and thus divergence problems are not present. We employ a realistic interaction, derived from the Argonne V18 potential using the unitary correlation operator method (UCOM), as well as the simple Brink-Boeker interaction. Representative results are discussed, mainly on giant resonances and low-lying collective states. The focus of the present work is not on the comparison with data, but rather on technical and physical aspects of the method. We present how the large-scale eigenvalue problem that SRPA entails can be treated, and demonstrate how the method operates in producing self-energy corrections and fragmentation. The so-called diagonal approximation is conditionally validated. Stability problems are traced back to missing ground-state correlations.

Figure 1

Isoscalar monopole response of ${}^{16}\mathrm{O}$, calculated within a single-particle basis of seven oscillator shells. Strength function is calculated within SRPA and SRPA0 [diagonal approximation; Eq. (4)], as well as RPA, in (a) linear and (b) logarithmic scale.

Figure 2

Quality of the diagonal approximation. (a) IS quadrupole response of ${}^{48}\mathrm{Ca}$ in a single-particle basis with ${\epsilon }_{max}=8$ and using $V_{\mathrm{UCOM}}$. (b) IV dipole response of ${}^{16}\mathrm{O}$ in a single-particle basis with ${\epsilon }_{max}=12$ and ${\ell }_{max}=8$ and using $V_{\mathrm{BB}}$. Results are shown obtained with RPA, solving the full SRPA problem and using the diagonal approximation (SRPA0). (In all cases, $\Gamma =0.5$ MeV.)

Figure 3

$0^{+}$ component of the double dipole resonance of ${}^{16}\mathrm{O}$, calculated within a single-particle basis of seven oscillator shells. The strength function ($\Gamma =0.5$ MeV) is calculated within SRPA and SRPA0 [diagonal approximation; Eq. (4)], as well as using the unperturbed (HF) ph and 2p2h states.

Figure 4

For the $0^{+}$ response of ${}^{16}\mathrm{O}$, calculated within a single-particle basis of seven oscillator shells. Number of SRPA and SRPA0 [diagonal approximation; Eq. (4)] eigenstates per 5-MeV excitation energy, in logarithmic scale. The corresponding density of unperturbed (HF) ph and 2p2h states is also shown.

Figure 5

Fragmentation and shift of ph states—$0^{+}$ response of ${}^{16}\mathrm{O}$ in a space of seven shells. Thin dark bars show how the spectroscopic strength $S_{\mathrm{ph}}(E)$, Eq. (15), of the ph configurations $|(\nu {\mathrm{p}}_{3/2})(\mathrm{\nu }0{\mathrm{p}}_{3/2}){}^{-1};0^{+}\rangle$ (contributing to the monopole strength) is distributed within (a) SRPA, (b) RPA, and (c) HF. Thicker bars denote the distribution of $|(\nu 1{\mathrm{p}}_{3/2})(\mathrm{\nu }0{\mathrm{p}}_{3/2}){}^{-1};0^{+}\rangle$ (one particle shell only).

Figure 6

Same as Fig. 5 without HF, both axes in logarithmic scale.

Figure 7

For the $0^{+}$ response of ${}^{16}\mathrm{O}$, calculated within a single-particle basis of seven oscillator shells, but imposing an energy cutoff $E_{2\mathrm{p}2\mathrm{h},\max }$ on the 2p2h configurations taken into account: energies of the four lowest eigenstates (axis on the left) and percentage of basis states taken into account (axis on the right), versus $E_{2\mathrm{p}2\mathrm{h},\max }$. The arrow indicates the point at which all available states are used (maximal 2p2h energy).

Figure 8

Spurious admixtures in the SRPA dipole spectrum. Dipole strength distributions of ${}^{16}\mathrm{O}$ using $V_{\mathrm{UCOM}}$ and a basis with $e_{max}=12$, ${\ell }_{max}=10$. The scale on the left corresponds to the IS strength and the scale on the right to the folded IV strength. The IS strength is shown for the usual IS dipole transition operator and for its uncorrected form (see text).

Figure 9

Exploring the influence of ground-state correlations: Isoscalar $3^{-}$ response of ${}^{16}\mathrm{O}$ (${\epsilon }_{max}=14$, ${\ell }_{max}=10$) within RPA or SRPA and “renormalized” RPA or SRPA, (RRPA, RSRPA; see text), as well as TDA and STDA (SRPA with $B=0$). (a) Evolution of the three lowest (closest to zero) eigenvalues as we increase the number of 2p2h configurations considered, starting from zero and up to all 2p2h states available in the single-particle space, $N_2$. The numbers assigned to the various points indicate the corresponding 2p2h-energy cutoff imposed, $E_{2\mathrm{p}2\mathrm{h},max}$, in MeV. The energy of the lowest eigenstate becomes imaginary for larger $N_2$ in (R)SRPA; then its amplitude is indicated. Note that the STDA and SRPA results for the second state almost coincide. (b) Strength function for $E_{2\mathrm{p}2\mathrm{h},max}=150$ MeV in SRPA, RSRPA, and STDA ($\Gamma =2$ MeV). (c) Strength function in RPA, RRPA, and TDA.