Hartree-Fock calculations with the T=0 n-p pairing configuration mixing

Apart from the 4²4² configuration discussed above, in ⁶⁰Zn we also calculated 6 other configurations, namely those that correspond to exciting the 4² proton and neutron simultaneously to the negative-parity orbitals $f_\pm$ and $p_\pm$. In principle, there are 16 such excitations possible, however, the lowest ones are obtained by putting the neutron and the proton in the same orbitals. This gives 4 configurations denoted by 4¹f₊4¹f₊, 4¹f_-4¹f_-, 4¹p₊4¹p₊, and 4¹p_-4¹p_-. In addition, we also study 2 other configurations obtained by putting the neutron and the proton into the [303]7/2 orbital with different signatures, i.e., those denoted by 4¹f₊4¹f_- and 4¹f_-4¹f₊.

In Fig. 7(b) the energies of the seven configurations selected above are shown with respect to a common rigid-rotor reference energy of 0.025$\times$I(I+1)MeV. Similarly, Figs. 7(a) and (c) show the analogous configurations in ⁵⁸Cu and ⁶²Ga. Because orbitals $f_\pm$ and $p_\pm$ are very close in energy (cf. Fig. 1), they strongly interact and mix, which very often precludes the convergence of the HF procedure, see discussion in Ref. [36]. Apart from that, the bands of Fig. 7 are shown up to the so-called termination points, i.e., up to the point where the angular-momentum contents of the involved orbitals does not allow for a further angular momentum build up, see Ref. [37], without a significant rearrangement of the nucleons.

By considering the available projections of the total angular momentum I_y for oblate shapes with the y axis as the symmetry axis, one can easily determine the values of the termination-point angular momenta I_t, see Table 2. The bands obtained in the HF calculations do not always terminate at the oblate axis and can usually be continued beyond I_t. However, at angular momenta I_t there always occur significant changes in the structure of bands. Below we discuss and present results only up to the termination points I_t.

  

Figure 7: Energies of selected configurations in ⁵⁸Cu (a), ⁶⁰Zn (b), and ⁶²Ga (c), calculated within the HF method with the Skyrme SLy4 interaction, and plotted with respect to a rigid-rotor reference energy. The configurations shown in the legend correspond to the numbers of occupied N₀=4 intruder orbitals, n and p, that are indicated in each panel.

A conspicuous feature of the HF energies presented in Fig. 7 is the significant energy separation between the n-p paired configurations 4ⁿf₊4^pf₊ and 4ⁿf_-4^pf_- on one side, and the broken-pair configurations 4ⁿf₊4^pf_- and 4ⁿf_-4^pf₊ on the other side. The former and latter configurations have opposite total signatures, i.e., in the even-even nucleus ⁶⁰Zn, configurations 4¹f₊4¹f₊ and 4¹f_-4¹f_-( 4¹f₊4¹f_- and 4¹f_-4¹f₊) correspond to r=+1 (r=-1), while in the odd-odd nuclei ⁵⁸Cu and ⁶²Ga the analogous configurations correspond to r=-1 (r=+1). Such a signature-separation effect has been for the first time discussed for the SD bands in ³²S [38]. Here it is obtained in the heavier SD region of the A$\simeq$60 nuclei, as a mutatis mutandis identical effect occurring for all the orbitals promoted to the next HO shell.

Configuration		Table 2: Values of the termination-point angular momenta It(in $\hbar$) for the seven selected configurations in 58Cu, 60Zn, and 62Ga. For convenience, the second column gives the configurations shown in the convention of Refs. [37,17], that, however, does not allow for distinguishing between the signatures of the occupied orbitals. 58Cu n=p=0	60Zn n=p=1	62Ga n=p=2
4n+14p+1	[2(p+1),2(n+1)]	29	36	41
4nf+4pf+	[1p,1n]	15	24	31
4nf-4pf-	[1p,1n]	13	22	29
4np+4pp+	[2p,2n]	23	32	39
4np-4pp-	[2p,2n]	21	30	37
4nf+4pf-	[1p,1n]	14	23	30
4nf-4pf+	[1p,1n]	14	23	30

In Ref. [38] the signature-separation effect was interpreted as a result of the strong n-p attraction transmitted through the time-odd mean fields. Such an attraction is typical for any realistic effective interaction, and it has its origin in the spin-spin components of the interaction. (The signature separation vanishes when in the Skyrme energy functional [39] the coupling constants corresponding to terms $\mbox{{\boldmath {$s$ }}}$$\cdot$ $\mbox{{\boldmath {$s$ }}}$and $\mbox{{\boldmath {$s$ }}}$$\cdot$ $\Delta\mbox{{\boldmath {$s$ }}}$ are set equal to zero.) When averaged within the mean-field approximation, the spin-spin components lead naturally to the time-odd mean fields [39]. Within the phenomenological mean fields, like those given by the Woods-Saxon or Nilsson potentials [40], the time-odd mean fields vanish, and therefore all the four configurations $4^nf_{\pm}4^pf_{\pm}$ are nearly degenerate, i.e., the signature-separation effect occurs only for self-consistent mean fields generated from the spin-spin interactions.

One should note that the four configurations $4^nf_{\pm}4^pf_{\pm}$ have purely independent-particle character (Slater-determinant wave functions), i.e., no collective pair correlations are built into the wave functions. Nevertheless, configurations 4ⁿf₊4^pf₊ and 4ⁿf_-4^pf_- contain one more T=0 n-p pair as compared to the 4ⁿf₊4^pf_- and 4ⁿf_-4^pf₊ configurations, and therefore are sensitive to the n-p pairing component of the effective interaction that is attractive. As a result, the paired configurations 4ⁿf₊4^pf₊ and 4ⁿf_-4^pf_- cross the magic configurations 4ⁿ⁺¹4^p+1 at I=11, 18, and 27$\hbar$ in ⁵⁸Cu, ⁶⁰Zn, and ⁶²Ga, respectively.

The n-p pairing correlations should be, in principle, studied by using methods beyond the mean-field approximation, i.e., by taking into account the configuration-mixing effects for configurations that differ by the n-p pair occupations. The generator-coordinate method (GCM) [40] is the approach of choice for including such effects. It allows for a consistent improvement of wave functions, while staying in the framework of the variational approach. Therefore, the same interaction can be/should be used in the HF method and in the mixing of the HF configurations via the GCM method.

At present, the GCM approach in the rotating frame has not yet been implemented, and in the present study we discuss the same physics problem by introducing a model T=0 n-p pair-interaction Hamiltonian in the form of

$$\!\!\! \hat{H}_{\mbox{\rm\scriptsize {n-p}}} = \hat{H}_0 + \... ...{\alpha\beta} \hat{P}^\dagger_{\alpha{r}}\hat{P}_{\beta{r}} ,$$ (1)

where the particle-number ( $\hat{N}_{\tau\alpha{r}}$) and T=0 n-p pair-creation ( $\hat{P}^\dagger_{\alpha{r}}$) operators read

  

$$\hat{N}_{\tau\alpha{r}} a^\dagger_{\tau\alpha{r}}a_{\tau\alpha{r}} ,$$ = (2)

$$\hat{P}^\dagger_{\alpha{r}} a^\dagger_{\nu\alpha{r}}a^\dagger_{\pi\alpha{r}} ,$$ = (3)

$\tau$ denotes neutrons ($\nu$) or protons ($\pi$), and $\alpha$ and $\beta$ denote the Nilsson labels without the signature quantum number r that is shown explicitly.

Hamiltonian (1) is meant to replace the usual effective-interaction (Skyrme) Hamiltonian when studying the n-p correlation aspects of the nuclear wave functions, and not to be added on top of it. Therefore, the effective single-particle energies $\epsilon_{\tau\alpha{r}}$ and the coupling constants $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\beta}$= $G_{\mbox{\rm\scriptsize {n-p}}}^{\beta\alpha}$ have to be angular-momentum and configuration dependent, and Hamiltonian (1) should be understood as a phenomenological interaction operator between configurations that differ by the n-p pair occupations.

The diagonal pairing term can be transformed as

$$\hat{P}^\dagger_{\alpha{r}}\hat{P}_{\alpha{r}} \equiv \hat{N}_{\nu\alpha{r}}\hat{N}_{\pi\alpha{r}} ,$$ (4)

i.e., it gives a non-zero contribution only if both a neutron and a proton occupy the given $\{\alpha{r}\}$ orbital. Therefore, the diagonal matrix elements of the n-p pairing Hamiltonian (1) in any given configuration,

$$E(\mbox{\rm\scriptsize {conf.}})=\langle\mbox{\rm\scriptsize... ...m\scriptsize {n-p}}}\vert\mbox{\rm\scriptsize {conf.}}\rangle,$$ (5)

can be immediately calculated for each Slater determinant. In particular, since in all the four configurations $4^nf_{\pm}4^pf_{\pm}$the effective single-particle energies are identical (cf. the Routhian diagram in Fig. 1), the differences of the total energies in the signature-separated configurations read

   

E(4ⁿf₊4^pf_-)-E(4ⁿf_-4^pf₊) = 0 , (6)

E(4ⁿf₊4^pf₊)-E(4ⁿf_-4^pf_-) = 0 , (7)

$$G_{\mbox{\rm\scriptsize {n-p}}}\mbox{[303]7/2${}$ } ,$$ E(4ⁿf₊4^pf_-)-E(4ⁿf₊4^pf₊) = (8)

where $G_{\mbox{\rm\scriptsize {n-p}}}$[303]7/2 stands for the diagonal matrix element $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}$ for $\alpha$=[303]7/2.

By subtracting the total HF energies of configurations in Eq. (8), see Fig. 7, one thus obtains an estimate of the n-p pairing diagonal matrix element $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}$. Such relative energies (8) in ⁵⁸Cu, ⁶⁰Zn, and ⁶²Ga are plotted in Fig. 8. One can see that the effective matrix elements depend strongly on the angular momentum, and decrease from $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}(I$=0)$\simeq$1.6 (⁵⁸Cu) or 1.9MeV (⁶⁰Zn and ⁶²Ga), reaching zero at the termination angular momentum I_t. This dependence can be very well parameterized by a simple cubic expression,

$$G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}(I)= G_{\mbo... ...\mbox{=}0)\times \left(1-\left(\frac{I}{I_t}\right)^3\right),$$ (9)

shown by dashed lines in Fig. 8.

  

Figure 8: Energies of configurations 4ⁿf₊4^pf_- (closed circles) and 4ⁿf_-4^pf₊ (open circles) in ⁵⁸Cu, ⁶⁰Zn, and ⁶²Ga, relative to the corresponding 4ⁿf₊4^pf₊ configurations. Dashed lines show the simple cubic approximations of Eq. (9). The relative energies can be identified with the angular-momentum-dependent T=0 n-p pairing matrix elements $G_{\mbox{\rm\scriptsize {n-p}}}$ in the [303]7/2 orbital (see text).

A large standard signature splitting of the other single-particle orbitals, which have lower values of the K quantum numbers, does not allow us to determine the other diagonal matrix elements $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}$ directly from the HF results, as in Eq. (8). Of course, such a determination of the non-diagonal matrix elements is not possible either. However, we may use the I-dependence of Eq. (9) to postulate a simple separable approximation for the n-p pairing interaction matrix $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\beta}$ in the form

$$G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\beta}(I)= \sqrt{G_... ...pha\alpha}(I)G_{\mbox{\rm\scriptsize {n-p}}}^{\beta\beta}(I)}.$$ (10)

Such a postulate is motivated by the fact that the pairing matrix elements of short-range interactions are given primarily by overlaps between the space wave functions, or more precisely, by the integrals of products of squares of the wave functions. Then, Eq. (10) stems from approximating the integral of products by the product of integrals.

Within the separable approximation (10), the T=0 n-p pairing interaction $\hat{V}_{\mbox{\rm\scriptsize {n-p}}}$ in Hamiltonian (1) takes the simple form of

$$\hat{V}_{\mbox{\rm\scriptsize {n-p}}}=-G(I)\left(\hat{P}^\dagger_{+i}\hat{P}_{+i} +\hat{P}^\dagger_{-i}\hat{P}_{-i}\right),$$ (11)

where the I-dependent collective n-p pair operators read,

$$\hat{P}^\dagger_{r} = \sum_\alpha x_\alpha(I)\hat{P}^\dagger_{\alpha{r}},$$ (12)

for

  

$$x_\alpha^2(I) G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}(I)/G(I) ,$$ = (13)

$$\sum_\alpha G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}(I) .$$ G(I) = (14)

Even then, however, the problem is defined by one parameter per orbital, $G_{\mbox{\rm\scriptsize {n-p}}}^{\alpha\alpha}(I$=0), i.e., it cannot be defined without explicit microscopic configuration-mixing calculations.

  

Figure 9: Same as in Fig. 7(b) but for configurations 4²4² and 4¹f₊4¹f₊ interacting through the T=0 n-p pairing interaction (11) with $G_{\mbox{\rm\scriptsize {n-p}}}$[303]7/2(I=0)=1.9MeV and $G_{\mbox{\rm\scriptsize {n-p}}}$[440]1/2(I=0)=0.65MeV. Crosses show the experimental data in the absolute energy scale. The symbols used in the figure indicate which are the pure configurations of Fig. 7(b) that dominate in the given mixed configurations.

  

Figure 10: Experimental (full squares) and calculated (open squares) dynamic moments of inertia ${\cal J}^{(2)}$ in the SD band of ⁶⁰Zn. Calculations correspond to the lowest band shown in Fig. 9.

Before these become available, in the present study we perform the simplest two-level mixing calculation, in which the two configurations that cross in ⁶⁰Zn, 4²4² and 4¹f₊4¹f₊, see Fig. 7(b), are allowed to interact through the T=0 n-p pairing interaction (11). With the diagonal matrix elements of Hamiltonian (1) taken from the HF calculations, and the interaction matrix element defined by the value of $G_{\mbox{\rm\scriptsize {n-p}}}$[303]7/2(I=0)=1.9MeV, also taken from the HF calculations, we are left with one free parameter, i.e., with the value of $G_{\mbox{\rm\scriptsize {n-p}}}$[440]1/2(I=0).

By fixing this parameter at $G_{\mbox{\rm\scriptsize {n-p}}}$[440]1/2(I=0) = 0.65MeV, we obtain at the crossing point of I=18$\hbar$ the effective interaction strength of 0.79MeV. With the I-dependent matrix elements given by Eqs. (9) and (10), we obtain the energies and dynamic moments of inertia shown in Figs. 9 and 10, respectively. It is clear that the mixing and interaction of the 4²4² and 4¹f₊4¹f₊ configurations correctly reproduces the magnitude of the bump in the ${\cal J}^{(2)}$ of ⁶⁰Zn.

The position of the crossing point is obtained at frequency or spin that are too large by 0.2MeV or 4$\hbar$, respectively, as compared to experiment. As seen in Fig. 7(b), this position is dictated by the diagonal matrix element $G_{\mbox{\rm\scriptsize {n-p}}}$[303]7/2 that shifts down configuration 4¹f₊4¹f₊ with respect to the broken-pair degenerate configurations 4¹f₊4¹f_-and 4¹f_-4¹f₊. As discussed above, such a shift is a direct consequence of the time-odd mean fields resulting from the Skyrme energy density. In the present work we have used the time-odd terms as directly given by the SLy4 Skyrme functional, see Ref.[39], i.e., those that result from fitting the time-even, and not time-odd properties of nuclei. It is clear that a modification of these time-odd terms, that is permitted in the local density approximation, may move the crossing frequency from its current position in Fig. 10. In fact, it is obvious that by decreasing this intensity one may easily decrease the crossing frequency. We do not attempt such a fit here, because the problem of finding good physical values of the time-odd coupling constants is much more general, and it would not make too much sense to make such an adjustment based solely on the specific effect discussed in the present study. We only note in passing that an analogous readjustment of the isovector time-odd coupling constants[41] has led to values that are quite different from those resulting directly from the Skyrme functional.