Effects of deformation on the coexistence between neutron-proton and particle-like pairing in $N=Z$ medium-mass nuclei

A model combining self-consistent mean-field and shell-model techniques is used to study the competition between particle-like and proton-neutron pairing correlations in $fp$-shell even-even self-conjugate nuclei. Results obtained using constant two-body pairing interactions as well as more sophisticated interactions are presented and discussed. The standard BCS calculations are systematically compared with more refined approaches including correlation effects beyond the independent quasiparticle approach. The competition between proton-neutron correlations in the isoscalar and isovector channels is also analyzed, as well as their dependence on the deformation properties. Besides the expected role of the spin-orbit interaction and particle number conservation, it is shown that deformation leads to a reduction of the pairing correlations. This reduction originates from the change of the single-particle spectrum and from a quenching of the residual pairing matrix elements. The competition between isoscalar and isovector pairing in the deuteron transfer is finally addressed. Although a strong dependence the isovector pairing correlations with respect to nuclear deformation is observed, they always dominate over the isoscalar ones.

Figure 1

Binding energy as a function of the quadrupole deformation parameter obtained in the $\mathrm{HF}+\mathrm{BCS}$ calculations for different $Z=N$ nuclei (see text for details).

Figure 2

Mean proton (blue dashed line) and neutron (red solid line) gaps as a function of the average quadrupole deformation parameter $\beta$. For each nucleus, the black solid circle indicates the equilibrium configuration that minimizes the EDF.

Figure 3

Evolution of the neutron single-particle energies as a function of the deformation $\beta$ obtained in the mean-field calculations for ${}^{52}\mathrm{Fe}$. The calculations are performed using EV8 with only particle-like pairing. Negative and positive parity states are plotted with solid and dashed lines, respectively. The dotted solid green line indicates the Fermi energy. Note that here, the single-particle quantum numbers have been assigned by continuity with the spherical case.

Figure 4

Same as Fig. 3 for ${}^{56}\mathrm{Ni}$.

Figure 5

Correlation energy obtained assuming only pairing between like particles. The BCS (black circles) and $|T_z|=1$ SM results (blue squares) are shown for different $N=Zf$-shell nuclei. Spherical symmetry is imposed for all the nuclei and a constant pairing interaction is used (see text).

Figure 6

Correlation energy obtained in SM calculation including all channels in $T=1$ (red solid circles) compared to the case where only $|T_z|=1$ are included (blue solid squares). In the former case, the red open circles indicates the correlation energy associated to one $T_z$ component. Note that owing to the assumed symmetry between protons and neutrons the three components leads to the same contribution equal to $1/3$ of the total energy. The blue open squares correspond to the correlation energy associated to $n\text{−}n$ pairs for the $|T_z|=1$ calculation. Again, owing to the isospin symmetry, this correlation energy is half of the total energy and equals the one associated to the $p\text{−}p$ contribution.

Figure 7

Correlation energy in the $T=1$ case with (red solid circles) and without (blue solid squares) spin-orbit interaction in the mean-field calculations. The results are obtained assuming a spherical symmetry for all nuclei and a constant pairing interaction.

Figure 8

(a) Correlation energy in the $(S=0,T=1)$ obtained with SM calculation in spherical (black solid circles) and deformed (red solid triangles) $N=Z$ nuclei. In the latter case, the equilibrium deformation value ${\beta }_0$ obtained with EV8 was used. Open symbols are obtained by considering states in a window of $\pm 5$ MeV around the Fermi energy allowing thus for the possible mixing of the $f$ and $p$ shells.

Figure 9

Averages of the pairing matrix elements in the $T=1$ (black solid lines) and $T=0$ (red dashed lines) channels as functions of the deformation for ${}^{56}\mathrm{Ni}$ (top panel) and ${}^{52}\mathrm{Fe}$ (bottom panel) are plotted. The diagonal part of the interaction is not considered in the calculations and not shown here (see the text for more details). The matrix elements are computed here using the same strength in the two channels, i.e., $v_1=v_0$ $\left(x=1\right)$ parameter in Eq. (1).

Figure 10

Pairing matrix elements in the $T=1$ (black points) and $T=0$ (red points) channels for eight single-particle states (four hole and four particle states) around the Fermi energy as a function of the deformation for ${}^{56}\mathrm{Ni}$ (top panel) and ${}^{52}\mathrm{Fe}$ (bottom panel) are plotted. The index $I(i,j)=1,...,28$ enumerates the pair indices of the matrix elements $V_{ij}$, with $i=1,...,8$ and $j>i$. The wave functions used correspond to the $\beta$ equilibrium value. The matrix elements are computed here for $v_1=v_0$ $\left(x=1\right)$.

Figure 11

Correlation energies obtained in $T=1$ SM calculations with residual pairing interaction computed directly from EV8. For deformed nuclei, see panel (b); the results obtained using the spherical mean-field EV8 solution (black solid circles) and the deformed one (red solid triangles) are shown. The results obtained by using a constant pairing interaction and same valence space (open symbols) are also shown for comparison.

Figure 12

The total energy (9) as a function of the deformation parameter calculated in the mean-field (MF) approach (red solid line) is compared with different kind of SM calculations: $|T_z=1|$ results (red dotted line), where only particle-like pairing is considered, and $T=1$ (black dashed line), where all the components of the isovector pairing are included. See the text and Table 1 for more details.

Figure 13

Total energy (9) as a function of the deformation obtained using only $p\text{−}n$ pairing in the $T=1$ channel (black dashed line, $T_z[x=0]$), including also the isoscalar one with equal strength (red thin solid line, $T_z[x=1]$) and using the more realistic value $x=1.6$ (blue dot-dashed line, $T_z[x=1.6]$). The results obtained in the mean-field (MF) calculations are plotted for comparison.

Figure 14

The isovector (red dashed line) and isoscalar (blue dotted line) energy contributions, defined by Eqs. (10) and (11), respectively, and the sum of them (black solid line) are plotted as a function of the deformation and corresponding to a $\left(|T_z|=0\right)$ calculation with $x=1.6$.

Figure 15

Isovector (red dashed lines) and isoscalar (blue dotted lines) total deuteron transfer probability obtained for $x=1.6$, corresponding, respectively, to the $Q$ and $R$ quantities defined in the text.