Large-amplitude pairing fluctuations in atomic nuclei

Pairing fluctuations are self-consistently incorporated on the same footing as the quadrupole deformations in present state-of-the-art calculations including particle-number and angular-momentum conservation as well as configuration mixing. The approach is complemented by the use of the finite-range density-dependent Gogny force which, with a unique source for the particle-hole and particle-particle interactions, guarantees a self-consistent interplay in both channels. We have applied our formalism to study the role of the pairing degree of freedom in the description of the most relevant observables like spectra, transition probabilities, separation energies, etc. We find that the inclusion of pairing fluctuations mostly affects the description of excited states, depending on the excitation energy and the angular momentum. $E0$ transition probabilities experience rather big changes while $E2$'s are less affected. Genuine pairing vibrations are thoroughly studied with the conclusion that deformations strongly inhibits their existence. These studies have been performed for a selection of nuclei: spherical, deformed, and with different degrees of collectivity.

Figure 1

Energies of the lowest $0^{+}$ and $2^{+}$ states as a function of the number of $\delta$ points (3, 5, 10, or 19) used in the calculations in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ approach for ${}^{54}$Cr. The lines connecting the points are to guide the eye.

Figure 2

Energies of the four lowest $0^{+}$ states of the nucleus ${}^{24}$Mg. Left (right) panels correspond to the 2D (1D) calculation both of them have been performed with the three different approaches.

Figure 3

The same plot as Fig. 2, but for the ${}^{32}$Mg.

Figure 4

(Left panels) One-dimensional potential energy curves for the nuclei ${}^{52}$Ti, ${}^{24}$Mg, and ${}^{32}$Mg in the HFB and in several approaches for $I=0^{+}$. (Right panels) Proton and neutron pairing energies of the intrinsic w.f.'s in the HFB and PN-VAP for the same nuclei.

Figure 5

Potential energy contour plots for ${}^{52}$Ti in the $(\delta ,q)$ plane in different approaches. The dashed lines show equipotential lines from 0 to 3 MeV in steps of 1 MeV. The continuous lines are contours from 4 to 10 MeV in steps of 2 MeV. In each panel the energy origin has been chosen independently and the energy minimum has been set to zero. The bullets in each panel represent the $\delta$ values of the self-consistent solution (HFB or PN-VAP) extracted from the 1D ($q$-constrained) approach and are displayed as a discussion guide. Because all HFB based approaches have the same intrinsic w.f., all of them have the same bullet pattern. The same applies to all PN-VAP-based approaches.

Figure 6

(Left panels) Potential energy surfaces for the nuclei ${}^{24}$Mg and ${}^{32}$Mg in the HFB approach. (Right panels) Potential energy surfaces for the nuclei ${}^{24}$Mg and ${}^{32}$Mg in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ approach for $I=0$. In both cases the contours follow the same interval as in Fig. 5.

Figure 7

(Left panels) Potential energy surfaces for the nuclei ${}^{50}$Ca, ${}^{52}$Ca, and ${}^{54}$Ca in the PN-VAP approach. (Right panels) Potential energy surfaces for the same nuclei in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ approach for $I=0$. In both cases the contours follow the same interval as in Fig. 5.

Figure 8

Renormalized norm overlaps of the ${}^{52}$Ti nucleus. The original matrix elements have been divided by a factor to have in each panel the maximum value equal to unity. (Top panels) The $\mathrm{HFB}+\mathrm{AMP}$ approach, for $I=0$ ($\mathrm{factor}=1.0$), $I=2$ ($\mathrm{factor}=0.484$), and 4 ($\mathrm{factor}=0.316$). (Bottom panels) The $\mathrm{HFB}+\mathrm{PNAMP}$, for $I=0$ ($\mathrm{factor}=0.191$), $I=2$ ($\mathrm{factor}=0.218$), and 4 ($\mathrm{factor}=0.202$). The continuous lines represent contours from 0.2 to 0.8 in steps of 0.2 and in dashed lines we depict the 0.1 contour and the 0.9 one.

Figure 9

Norm matrix elements for ${}^{52}$Ti for different $q$ and $\delta$ values (see main text) in the $\mathrm{HFB}+\mathrm{PNAMP}$ (blue circles), $\mathrm{HFB}+\mathrm{AMP}$ (magenta squares), and HFB (green triangles and only for $I=0\hslash$). Because the HFB w.f. are normalized to unity these norms are multiplied by $0.5$ to make the pictures more legible.

Figure 10

Evolution of the total energies of the ground-state energy $0_1^{+}$ (blue squares) and excited energies $0_2^{+}$ (magenta circles) and $0_3^{+}$ (green diamonds) as a function of the different approximations for ${}^{52}$Ti. Dashed lines correspond to 1D calculations and continuous lines are for 2D calculations.

Figure 11

Spectra of ${}^{52}$Ti in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ (left), $\mathrm{HFB}+\mathrm{PNAMP}$ (middle), and $\mathrm{HFB}+\mathrm{AMP}$ (right) approaches. The four lowest states for spin $0^{+},2^{+},4^{+}$, and $6^{+}$ are represented in 1D (dashed lines) and 2D (continuous lines).

Figure 12

Spectra of ${}^{32}$Mg in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ approach (left) and the $\mathrm{HFB}+\mathrm{AMP}$ approach (right).

Figure 13

Spectra of ${}^{50}$Ca, ${}^{52}$Ca, and ${}^{54}$Ca in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ approach.

Figure 14

The w.f.'s of the lowest $0^{+}$ collective states for the nucleus ${}^{52}$Ti in various approaches: (left panel) in the $\mathrm{PN}-\mathrm{VAP}+\mathrm{PNAMP}$ approach, (middle panel) in the $\mathrm{HFB}+\mathrm{PNAMP}$ approach, and (right panel) in the $\mathrm{HFB}+\mathrm{AMP}$ approach in the 1D calculations. $0_1^{+}$ continuous blue line, $0_2^{+}$ dashed magenta line, and $0_3^{+}$ dotted green line. The w.f. of the $0_2^{+}$ state in the $\mathrm{HFB}+\mathrm{AMP}$ approach has been multiplied by a factor 0.5.

Figure 15

Contour lines of the w.f.'s of the three lowest $0^{+}$ states of ${}^{52}$Ti in different approaches in the 2D calculations. The step size is 0.02 for the states. The thick dashed lines correspond to the zeros of the w.f. To get larger resolution the $x$ axis runs from $-$180 fm${}^2$ up to 240 fm${}^2$ at variance with former figures.

Figure 16

Potential well and w.f.'s of the two lowest states in 1D calculations for the nucleus ${}^{52}$Ti. (Top) For fixed $q$ value and with $\delta$ as a generator coordinate, (left panel) $q=-80$ fm${}^2$ and (right panel) $q=+80$ fm${}^2$. (Bottom) For fixed $\delta$ value and with $q$ as a generator coordinate, (left panel) $\delta =1.5$ and (right panel) $\delta =3.5$.

Figure 17

Wave functions of the lowest $0^{+}$ states of ${}^{52}$Ti of combined 1D generator coordinate calculations; see main text for further explanations. (Left) Calculations for a fixed $q$ value taking $\delta$ as generator coordinate. (Right) Calculations for a fixed $\delta$ value taking $q$ as generator coordinate.

Figure 18

Particle-number distribution in the ground and lowest excited states for ${}^{24}$Mg states in 1D in the $\mathrm{HFB}+\mathrm{AMP}$ approach.

Figure 19

Particle-number distribution in the ground and lowest excited states with $I=0^{+}\hslash$ for ${}^{24}$Mg states in 2D for the $\mathrm{HFB}+\mathrm{AMP}$ approach.

Figure 20

Two neutron separation energies ($S_{2n}$) in MeV for the magnesium isotopes. The experimental data are taken from Ref. [47].

Figure 21

Excitation energies for the $0_2^{+}$ states and $E0$ strength ${\rho }^2($E0$,0_2^{+}\to 0_1^{+})$ for the Mg isotopes (top panels), the Si isotopes (middle panels) and S isotopes (bottom panels). The experimental values are taken from [47, 48, 50, 51].

Figure 22

$E2$ transition probabilities for the Mg and Ca. Experimental data are taken from Refs. [47, 50, 53, 54].

Figure 23

(a) Potential energy surfaces in the PN-VAP approach for different number of integration points. (b) Potential energy surfaces in the PN-VAP approach and in the PNAMP approach for $I=0$, 2, and 4 for different number of integration points.