MODEL STUDY

Recently, two computer codes capable of solving the self-consistent equations for the isospin-invariant pnEDFs with the p-n mixing, have been developed in parallel. The recent study [27] describes the scheme based on the code HFODD [28,29], which can treat symmetry-unrestricted nuclear shapes. In this work, we present the implementation based on the code HFBTHO, which assumes axial and time-reversal symmetries. The two codes complement each other in that HFODD is more general whereas HFBTHO is much faster, and thus they have different scopes and application ranges. While HFBTHO can employ the transformed oscillator basis that is particularly useful for weakly bound nuclei, the focus of the present work is to benchmark HFBTHO with HFODD; hence, we shall use the standard harmonic oscillator basis.

As in Ref. [27], we diagonalize the s.p. Routhian,

$$\hat{h}' = \hat{h} - \vec{\lambda}\circ\hat{\vec{t}},$$ (16)

where $\hat{h}$ is given by Eq. (11) and contains kinetic, Skyrme-pnEDF, and Coulomb-energy terms. The isocranking term [32,33], $-\vec{\lambda}\circ\hat{\vec{t}}$, depends on the isocranking frequency (isovector Fermi energy) $\vec{\lambda}$ and the s.p. isospin operator $\hat{\vec{t}}=\hat{\vec{\tau}}/2$.

For systems obeying the time-reversal symmetry, $\langle\,\hat{t}_y\,\rangle$ vanishes [17] and the rotation in isospace is described by a two-dimensional isocranking, that is,

$$-\vec{\lambda}\circ\hat{\vec{t}} = - \lambda_x \hat{t}_x - \lambda_z \hat{t}_z.$$ (17)

The isocranking frequencies, $\lambda_x$ and $\lambda_z$, can be varied to control the isospin of the system. Following the methodology developed in Ref. [27], they are parametrized as

$$\lambda_z = \lambda'\cos\theta' + \lambda_{\text{off}}, \quad \lambda_x = \lambda' \sin\theta',$$ (18)

and the isocranking tilting angle $\theta '$ is varied between $0^\circ$ and $180^\circ$, that is, the isovector Fermi energy on the $\lambda_x$-$\lambda_z$ plane moves along a shifted semicircle. In this work, numerical calculations were performed for the $A=78$ IASs with $T\simeq11$. We used the Skyrme EDF parametrization SkM* [34] and the s.p. basis space consisting of $N_{\rm sh}=16$ spherical harmonic-oscillator (HO) shells. Fore more details on the parameters employed, see Sec. 4.

In the absence of the Coulomb interaction, choosing $\lambda' = 21$MeV, $\lambda_{\text{off}} = 0$MeV, and varying $\theta '$ from $0^\circ$ to $180^\circ$, generates all the $A=78$ and $T=11$ IASs. The angles $\theta '=0^\circ$, $90^\circ$, and $180^\circ$ correspond to the HF solutions for $^{78}$Ni ($T_z=11$), $^{78}$Y ($T_z=0$ in the odd-odd system), and $^{78}$Sn ($T_z=-11$), respectively. Our example involves very exotic nuclei, including those beyond the proton dripline. We find this case interesting because the nuclei at both ends of the isobaric chain are heavy and doubly magic, thus spherical.

As discussed in Ref. [27], with the Coulomb term off, the value of $\lambda'$ is roughly equal to the absolute value of difference between the proton and neutron Fermi energies in $^{78}$Ni or $^{78}$Sn, $\vert\lambda_n - \lambda_p\vert = 21.18$MeV. Then, the isocranking term makes the Fermi energies of neutrons and protons almost equal. In the presence of the Coulomb interaction, however, a large asymmetry between $\vert\lambda_n - \lambda_p\vert$ develops between $^{78}$Ni (12.31MeV) and $^{78}$Sn (33.62MeV). Therefore, to offset the difference of Fermi energies at $\theta '=0^\circ$ and $180^\circ$ with Coulomb interaction present, we set the values:

Using these expressions, $\lambda' = 22.94$MeV and $\lambda_{\text{off}} = -10.92$MeV.

Figure 1: (Color online) Total HF energy of the $A=78$, $T\approx 11$ IASs as a function of $\theta '$ with (solid line) and without (dashed line) the isospin-symmetry-breaking Coulomb term.

Figure 1 shows that in the absence of the Coulomb interaction, the total energy is independent of $\theta '$. This should be the case, as the pnEDF is isospin-invariant and thus the energy must be independent of the direction of the isospin vector. This also turned out to be an important test on the derived expressions and numerical code, as different terms of pnEDF become active for different values of $\theta '$. For $\theta '=0^\circ$ and $180^{\circ}$, solutions are unmixed and the densities are block-diagonal in neutron and proton subspaces. At intermediate values of $\theta '$, the solutions are p-n mixed. For the special case of $\theta'=90^{\circ}$, proton and neutron densities are equally mixed. When the Coulomb interaction is turned on, the total energy increases with $\theta '$ (Fig. 1), because more and more protons replace neutrons and the Coulomb repulsion grows.

Figure: (Color online) Similar as in Fig. 1, but for the expectation values of $\langle\,\hat{T}_x\,\rangle$ (a) and $\langle\, \hat{T}_z\,\rangle$ (b).

The degree of p-n mixing can be directly inferred from the expectation values of $\langle\,\hat{T}_x\,\rangle$ plotted in Fig. 2(a). As expected, the p-n mixing increases with $\theta '$ and reaches its maximum value for $\theta'=90^{\circ}$, and then drops again. In Fig. 2(b), we show $\langle\, \hat{T}_z\,\rangle$ and it is seen that the values of $\theta '=0^\circ$, $\sim$$90^\circ$, and $180^\circ$ do correspond to $^{78}$Ni, $^{78}$Y, and $^{78}$Sn, respectively. The behavior of $\langle\, \hat{T}_z\,\rangle$ and $\langle\,\hat{T}_x\,\rangle$ weakly depends on whether the Coulomb term is included or not. This is entirely due to our choice of the shifted semicircle (18), whereupon the linear constraint $\lambda_{\text{off}} \hat{t}_z$ absorbs the major part of the isovector component of the Coulomb interaction.

Figure: (Color online) Similar as in Fig. 1, but for $\langle\,\hat{T}^2 \,\rangle$.

The Coulomb interaction breaks isospin and thus induces the isospin mixing in the HF wave function. To illustrate this, Fig. 3 shows the average value of $\langle\,\hat{T}^2 \,\rangle$ for the converged HF solutions. For the considered case of the $T=11$ systems, $\langle\,\hat{T}^2 \,\rangle$ should be exactly equal to $11\times 12=132$ in the absence of isospin mixing. However, as shown in Fig. 3, even in the absence of the Coulomb interaction, $\langle\,\hat{T}^2 \,\rangle$ slightly deviates from this value. At the origin of this effect is the spurious isospin mixing [35,36,37]. Indeed, within the mean-field approximation, the isospin symmetry is broken spontaneously as the HF wave function is not an isospin eigenstate. However, since the Skyrme EDF is isospin covariant [38,15], the HF solutions corresponding to different orientations in the isospin space are degenerate in energy. While the neutron-proton mixing changes with the angle $\theta$, $\langle\,\hat{T}^2 \,\rangle$ must remain the same in the absence of the Coulomb interaction. In the presence of the Coulomb term, the isospin mixing is very small in the isospin-stretched $\vert T_z\vert=11$ configurations (for $\theta'=0^{\circ}$ and $180^{\circ}$) and reaches its maximum around $\theta'=90^{\circ}$ for $T_z=0$ [37,39].

Figure 4: (Color online) Single-particle Routhians as functions of $\theta '$ for $T\approx 11$ configurations in the $A=78$ systems. The Coulomb interaction is not included. Points are colored according to the s.p. expectation values of $\langle\,\hat{\tau}_z\,\rangle$. At $\theta '=0^\circ$, neutron and proton states are plotted up to 1$g_{7/2}$ and 2$p_{1/2}$, respectively.

Figure 5: (Color online) Similar to Fig. 4, but with the Coulomb interaction included.

The s.p. Routhians as functions of $\theta '$ are shown in Figs. 4 (without Coulomb) and 5 (with Coulomb). Eleven spherical neutron levels and seven proton levels are occupied at $\theta '=0^\circ$, and the neutron and proton Fermi energies are shifted in such a way that the gaps in the s.p. spectra appear at $A=78$ around $-15$MeV (Fig. 4) and $-10$MeV (Fig. 5). Our choice of $\vec{\lambda}$ guarantees that, in the presence of the Coulomb interaction, the s.p. Routhians near the Fermi surface do not cross as functions of $\theta '$; this would have caused a drastic structural changes of the mean-field and made the adiabatic tracing of the $T\simeq11$ IAS as a function of $\theta '$ extremely difficult. At $\theta '=0^\circ$ and $180^{\circ}$, the s.p. states have pure values of $\langle\,\hat{t}_z\,\rangle =\pm\tfrac{1}{2}$. At $\theta' \sim 90^{\circ}$, most of the s.p. Routhians have $\langle\,\hat{t}_z\,\rangle$ close to zero, that is, they are fully p-n mixed.

Figure 6: (Color online) Similar to Fig. 4 but for the s.p. energies with the Coulomb interaction included. Only the occupied states are plotted, that is, at $\theta '=0^\circ$, neutron and proton states are plotted up to 1$g_{9/2}$ and 1$f_{7/2}$ shells, respectively.

Figure 6 displays s.p. HF energies, that is, s.p. Routhians with the isocranking term removed. Note that these are not eigenvalues but the average values of the HF Hamiltonian, calculated for states that are eigenstates of the Routhian (16). With increasing $\theta '$, owing to the increasing Coulomb field, s.p. states that increase proton (neutron) component gradually increase (decrease) in energy.

Figure 7: (Color online) Evolution of s.p. energies of occupied HF states (with Coulomb) originating from proton (top) and neutron (bottom) shells at $\theta '=180$ with respect to the energy of the 1$f_{7/2}$ shell, and relative to the corresponding values at $\theta '=0^\circ$. That is, the energies plotted are $[\varepsilon _i(\theta ') - \varepsilon _{f_{7/2}}(\theta ')] - [\varepsilon _i(0^\circ ) - \varepsilon _{f_{7/2}}(0^\circ )]$.

To better visualize the relative shifts of s.p. levels with $\theta '$, in Fig. 7 we show s.p. energies relative to the energy of the 1$f_{7/2}$ shell. The figure nicely illustrates the effect of the Coulomb interaction on the proton components of the s.p. orbits: the relative level shifts correlate with their binding energies and $\ell$ values [40,41]. Indeed, the deeply (loosely) bound levels, which have smaller (larger) rms radii and thus experience stronger (weaker) Coulomb repulsion, are shifted up (down) in energy relative to the high-$\ell$ 1$f_{7/2}$ shell.

Some of the calculated $A=78$, $T\simeq11$ IASs are predicted to appear beyond the proton drip line. As seen in Fig. 6, energy of the 1g$_{9/2}$ level (which is neutron at $\theta '=0^\circ$ and proton at $\theta'=180^\circ$) becomes positive at around $\theta'=100^\circ$, where $\langle\hat{T}_z\rangle \approx -1.5$. At $\theta'=180^\circ$, energies of the 1$g_{9/2}$, 1$f_{5/2}$, 2$p_{1/2}$, and 2$p_{3/2}$ shells are positive. However, all these states are well localized by the Coulomb barrier, and thus correspond to narrow resonances, whose energies can be reasonably well described within the HO basis expansion [41].

Figure 8: (Color online) The total HF energy (with Coulomb) for $A=78$, $T\approx 11$ calculated with $N_{\rm sh}=10, 12,$ and $14$ HO shells relative to that with $N_{\rm sh}=16$ shells as a function of $\theta '$.

Figure 9: (Color online) Neutron and proton rms radii as functions of $\theta '$.

In Fig. 8, we show the convergence of the total HF energy with respect to the total number of HO shells $N_{\rm sh}$. Calculations using $N_{\rm sh}=14$ are not yet completely converged, with the energy difference between $N_{\rm sh}=16$ and $N_{\rm sh}=14$ varying around 73keV at $\theta '=0^\circ$ and 135keV at $\theta'=180^\circ$. Although it is expected that by increasing $N_{\rm sh}$ one may still lower the energy, the change is expected to be less than 100keV, and the results presented in this study are not expected to change significantly. A delayed convergence beyond $\theta'=140^\circ$ shows that the higher HO shells are more important at $\theta'=180^\circ$ than at $\theta '=0^\circ$; that is, a larger model space is required for the description of the unbound proton resonances [42]. Nevertheless, as seen in Fig. 9, even near $\theta'=180^\circ$ no sudden increase of proton rms radii is obtained.

Figure 10: (Color online) Lines: proton effective HF potential (with Coulomb and centrifugal terms included), calculated for the $A=78$, $T\simeq \vert T_z\vert$ isobars for $\ell =4$ (a) and $\ell =1$ (b). Dots: rms radii and s.p. energies of the proton 1$g_{9/2}$ (a) and 1$p_{1/2}$ states.

To investigate properties of the unbound proton orbits, for the ground states of $A=78$, $T\simeq \vert T_z\vert$ nuclei, we performed the HFBRAD [43] calculations (without p-n mixing). In Fig. 10, we show results obtained for the 1$g_{9/2}$ and 2$p_{1/2}$ proton states. For each $A=78$ isobar, a dot is placed at the values of s.p. energies and radii, and lines show standard total effective HF proton potentials. The total effective HF proton potential consists of the standard central, spin-orbit, centrifugal, and Coulomb terms. The proton 1$g_{9/2}$ orbit in $^{78}$Zr is bound, and in $^{78}$Mo it becomes slightly unbound. This result is consistent with the experimental observation, whereby the last bound nucleus of the $A=78$ isobaric chain, which is experimentally known, is $^{78}$Zr $(T_z=-1)$. The rms radii of the proton 1$g_{9/2}$ orbits are about 5fm, and the s.p. wave functions are still localized, even if the orbits become unbound. This is because the 1$g_{9/2}$ and 2$p_{1/2}$ protons occupy states well below the potential barrier, which pushes the proton continuum up in energy, thus effectively extending the range of nuclear landscape into the proton-unstable region [44,21,22].

It is worth noting that the 2$p_{1/2}$ orbit, which has a small centrifugal barrier, is bound up to around ( $\theta'=125^\circ, \langle\hat{T}_z\rangle\approx-5.7$) in Fig. 6. This is consistent with Fig. 10 that shows that the s.p. energy of 2$p_{1/2}$ is unbound in $^{78}$Ru $(\langle\hat{T}_z\rangle=-5)$. In the presence of p-n mixing, the proton components of the s.p. states are smaller than those in the pure proton states, and this effectively reduces the repulsive Coulomb energies of the 1$g_{9/2}$ orbits.