Odd-even mass differences of well and rigidly deformed nuclei in the rare earth region: A test of a newly proposed fit of average pairing matrix elements

In our approach, the sp canonical basis states are obtained within self-consistent HF+BCS calculations. Let us recall that throughout this paper the BCS correlations are determined within the seniority force simple approximation. As an alternative presentation for the values of the the average matrix elements $V_{q}$ we provide $G_{q}$ parameters introduced in Ref. [15] to the effect of removing approximately the dependence of these matrix elements from $N_{q}$ values

The particle-hole interaction in use is of the standard Skyrme type. Calculations are performed with the SIII parametrisation [16] which has been shown in many instances to provide a rather good description of the spectroscopic properties of well and rigidly deformed nuclei (see, e.g., for a recent account Ref. [17]). The averaged pairing matrix elements $V_{q}$ are given for each nucleus according to the method summarized in Section II for even-even nuclei. For odd-$Z$ (odd-$N$ respectively) nuclei the retained values of the matrix elements are interpolated between the values found for the neighbouring isotones (isotopes respectively).

The axial and intrinsic parity symmetries have been imposed to the solutions. The nuclei are thus specified by their projection $K$ of their angular momentum on the symmetry axis and their parity $\pi$. To solve the Schrödinger equation determining the canonical basis states, we have expanded these states on eigenstates of an axially symmetrical harmonic oscillator Hamiltonian. It is defined by a basis size parameter $N_{0}=16$ defining a deformation-dependent selection of relevant basis states corresponding at sphericity to the consideration of 17 shells. This basis is truncated according to (see Ref. [18])

where $n_{z}$ and $n_{\bot}$ are the number of oscillator quanta in the symmetry-axis direction ($z$-axis) and in the perpendicular direction and $N_{0}+1$ corresponds to the number of spherical shells.

The inverse-length $b$ and deformation $q$ parameters are defined, $m$ being the average nucleonic mass, such that [18]

These two parameters are related to the angular frequencies on the $(x,y)$ plane, $\omega_{\bot}$, and $z$-axis, $\omega_{z}$, while $\omega_{0}^{3}=\omega_{\bot}^{2}\omega_{z}$ is the angular frequency at sphericity. The two parameters $b$ and $q$ have been optimized for even-even nuclei to give the lowest energy for each equilibrium solution, whereas for odd-$A$ nuclei they have been interpolated from those obtained for even-even nuclei. Integrals involving the local densities are performed using the Gauss-Hermite and Gauss-Laguerre approximate integration methods with 50 points along the symmetry axis and 16 points in the perpendicular direction, respectively.

The data on odd-even mass differences are extracted through a 3-point formula. As discussed in Refs. [19, 20] such differences $\delta_{q}^{(3)}$ centered on an odd-neutron or odd-proton nucleus are good markers on the degree of pairing correlations. They are, to a large extent, free from single-particle filling effects and are given, e.g., for an isotopic series by

where $N$ is odd and $S_{n}(N,Z)$ is the neutron separation energy of a nucleus composed of $N$ neutrons and $Z$ protons whose total energies are denoted as $E(N,Z)$. Similar expressions are easily deduced from the above for odd-proton nuclei.

These energies are compared with those extracted directly from calculated binding energies within the HF+BCS approach. In the case of odd-$A$ nuclei, we have performed selfconsistent blocking calculations (i.e. placing a nucleon in the relevant sp orbit specified by the $K^{\pi}$ quantum numbers). The breaking of the time-reversal symmetry implies the presence of new (time-odd) local densities in the expression of the Hamiltonian density resulting in new terms in the corresponding Hartree-Fock potential. As explicitely detailed in Ref. [21], when using the SIII interaction we have considered a restricted choice of time-odd potential fields, yet preserving the Galilean invariance (namely the vector spin field $\mathbb{S}(\mathbb{r})$ and the vector current field $\mathbb{A}(\mathbb{r})$, with usual notation). This choice is dubbed as the minimal scheme in Ref. [21].

In the spirit of the Bohr-Mottelson unified model, well suited for these deformed nuclei, we assimilate the nuclear angular momentum and parity quantum numbers $I^{\pi}$ to those $K^{\pi}$ of the blocked blocked-nucleon sp state. This is of course not free for any perturbation of the low-energy nuclear spectra from possible Coriolis coupling which will be ignored here. Particular cases concern solutions where $K=1/2$ with a decoupling parameter $a$ outside the range $-1\leq a\leq 4$. They will be specifically discussed in Section IV and shown to be unable to perturb the natural rotational band ordering of states. The decoupling parameter $a$ is defined, as well known [22], through the relation

where $\widehat{J}_{+}$ is the usual angular momentum ladder operator (sum of orbital and spin angular momenta) in $\hbar$ unit and $|\widetilde{i}\rangle$ is the single-particle state canonically conjugate of the blocked state $|i\rangle$. These two states are such that $\widehat{J}_{z}|i\rangle=\Omega_{i}|i\rangle$ and $\widehat{J}_{z}|\widetilde{i}\rangle=-\Omega_{i}|\widetilde{i}\rangle$. In our HFBCS code with selfconsistent blocking, the time-reversal symmetry is broken in the one-body sector and the definition of the two sp states forming the equivalent of Cooper pairs in our BCS wavefunction is provided in Appendix A of [23].

We have systematically calculated solutions for odd-$A$ nuclei corresponding to the ground state $I^{\pi}$ experimental values [24] as well as those where the calculated energies lie below the former or in some cases (see the discussions of Section IV) above generally up to a couple of 100 keV.

We have chosen odd-$A$ nuclei belonging to the rare-earth region plus Hafnium isotopes (loosely called below rare-earth nuclei), i.e. some odd-proton isotopes from Europium to Lutetium and odd-neutron isotopes from Samarium to Hafnium. They are chosen to be well and rigidly deformed and, moreover, far enough from a region of transition between deformed and soft nuclei. The first criterion is retained to be able to reduce approximately the collective dynamics to a pure rigid rotation within the Bohr-Mottelson unified model, allowing in particular to limit ourselves to a single BCS state, i.e. ignoring quantal shape fluctuations. The second criterion is retained to avoid wide shape variations (and therefore large sp spectrum changes) between the three isotopes (isotones respectively) entering the calculation of the energy differences $\delta_{n}^{(3)}$ ($\delta_{p}^{(3)}$ respectively).

Table 1 displays the ratio $R_{42}$ of the excitation energies of the first $4^{+}$ and $2^{+}$ states [24] of the 22 even-even nuclei bracketing the odd-$A$ nuclei whose energy differences $\delta_{q}^{(3)}$ are evaluated in our calculations. It appears that they satisfy reasonably well the first criterion since for all of them one has $R_{42}\geq 3.29$.

On this table also, the intrinsic axial charge quadrupole moments $Q^{\rm int}_{20}$ obtained in our calculations are compared, when available, with the corresponding experimental values deduced either from reduced B(E2) data [25] or those deduced from the spectroscopic moments $Q^{\rm spect}_{20}$ of the first $2^{+}$ state [26], upon using the unified model relation for $I=2,K=0$, namely $Q^{\rm int}_{20}=-3.5$ $Q^{\rm spect}_{20}$. One notices a rather good reproduction of the $Q^{\rm int}_{20}$ data with the interaction SIII in use, as shown long ago [27] (somewhat less good however for the ^156,158Sm isotopes).

We compare here the lowest-energy configurations as obtained in our calculations with the experimental values of the angular momentum and parity quantum numbers $I^{\pi}$ given in the current version of the NUDAT compilation [24].

As seen on Table 2, for 8 nuclei out of 13 (¹⁶¹Tb, ¹⁶³Tb, ¹⁶⁷Ho, ¹⁶⁹Ho, ¹⁶⁹Tm, ¹⁷¹Tm, ¹⁷⁷Lu, ¹⁷⁹Lu), our theoretical assignments agree with the data.

We confirm the suggested assignments for 3 nuclei, namely: ¹⁶⁵Tb and ¹⁶⁷Tb as $3/2^{+}$ and ¹⁷³Tm as $1/2^{+}$. No assignment is proposed in Ref. [24] for the ¹⁶¹Eu nucleus. We suggest a $5/2^{-}$ configuration noting however that we have obtained a $5/2^{+}$ solution $179$ keV above the $5/2^{-}$ configuration. In one case (¹⁵⁹Eu) the experimental lowest configuration $5/2^{+}$ is obtained at 145 keV above a $5/2^{-}$ state. Finally, we remark that the decoupling constant values corresponding to the $1/2^{+}$ state considered in the three calculated isotopes of Thulium belong to the $[-0.64,-0.60]$ interval ensuring that the band head spin assignment as $1/2$ is correct.

As seen on Table 3, for 9 nuclei out of 16 (¹⁵⁹Sm, ¹⁶¹Gd, ¹⁶³Dy, ¹⁷¹Er, ¹⁷³Yb, ¹⁷⁵Yb, ¹⁷⁷Yb, ¹⁷⁹Hf, ¹⁸¹Hf), our theoretical assignments agree with the data.

We confirm the suggested assignment for one nucleus, namely: ¹⁵⁷Sm as $3/2^{-}$. No assignment is proposed in Ref. [24] for the ¹⁶⁵Gd nucleus. We suggest a $7/2^{+}$ configuration. In ¹⁶⁷Dy the suggested ground state configuration $1/2^{-}$ suggested in Ref. [24] is calculated only 58 keV above a $7/2^{+}$ state. In one nucleus (¹⁶⁹Er) the experimental lowest configuration $1/2^{-}$ is obtained only 72 keV above a $7/2^{+}$ state.

Three cases deserve a particular attention. In Ref. [24] two assignments ($5/2^{-},7/2^{+}$) are suggested for ¹⁶³Gd. We found the latter ($7/2^{+}$) as the ground state with the former ($5/2^{-}$) lying 579 keV higher with a $1/2^{-}$ configuration located in between at 59 keV. In ¹⁶⁵Dy one has found experimentally a $7/2^{+}$ ground state with a $1/2^{-}$ isomeric state with an excitation energy of 108 keV. We found both states as the lowest ones but with an inversion, the $1/2^{-}$ being found 48 keV lower in energy. A similar situation is also present in ¹⁷¹Yb with an experimental $1/2^{-}$ ground state and a $7/2^{+}$ isomeric state 95 keV above. Here, too, we found these states as the lowest ones but with an inversion whereby the $1/2$ being found 73 keV lower.

In our calculations for both odd-$Z$ and odd-$N$ nuclei, we thus found that we are in agreement in 21 out 29 cases where an assignment of spin and parity has been reported or suggested in Ref. [24]. We have proposed an assignment for two nuclei. In two instances where an isomeric state has been experimentally found at excitation energies in the 100-150 keV range, we have obtained them, yet with an inversion in their respective ordering, corresponding to an error of 50-70 keV. From all these results implying a sample of 29 odd-$A$ nuclei, yet limiting ourselves to the consideration of intrinsic states within the unified model, we can reasonably hint that our estimates of the low lying bandhead spectra provides a good reproduction of relative energies within not much more than about 150 keV.

Finally we discuss the decoupling constant values corresponding to the $1/2$ states considered in our calculations of these odd-$N$ nuclei. It is found to be equal to $-0.61$ for the $1/2^{-}$ state of ¹⁶³Gd, $-0.64$ and $-0.65$ for the $1/2^{-}$ states of ¹⁶⁵Dy and ¹⁶⁷Dy, $-0.73$ for the $1/2^{-}$ state of ¹⁶⁹Er, $-0.76$ for the $1/2^{-}$ state of ¹⁷¹Yb and $+0.24$ for the $1/2^{-}$ state of ¹⁸¹Hf. In all cases, such values warrants that the bandhead spin assignment as $1/2$ is correct.

Tables 2 and 3 display the three-points odd-even mass differences $\delta_{q}^{(3)}$ for protons and neutrons. The rms differences between experimental (as deduced from the separation energies given in Ref. [24]) and calculated in all cases for the experimental $I^{\pi}$ assignments are 165 keV for protons and 141 keV for neutrons. We can compare these values with what has been obtained in Ref. [6] where direct fits of the pairing matrix elements $V_{n}$ and $V_{p}$ have been performed on the odd-even mass differences $\delta_{n}^{(3)}$ and $\delta_{p}^{(3)}$ for a similar sampling of nuclei. There, the rms deviations were calculated as 182 keV for protons and 78 keV for neutrons. Given the expected accuracy on energies of our approach, we can deem the quality of our current approach to be comparable to those of Ref. [24].

To conclude this Subsection, it is instructive to assess numerically on two test cases, the effect of the two improvements with respect to the standard fitting procedure which have been considered in this paper (the Möller-Nix prescription and the correction for the deficiency of the Slater approximation)

We display in Table 4 the average matrix elements $V_{q}$ and the resulting odd-even mass-differences with and without taking the $R_{p}$ corrective factor for two odd-mass nuclei and consider two types of empirical formula, namely that of Jensen and that of Möller-Nix.

We first compare the results obtained using the Jensen and Möller-Nix formulas without Slater correction. Using the Jensen pairing gaps, the absolute values of the estimated neutron and proton pairing matrix elements are always lower than in the Möller-Nix case. This decrease of about 10 keV in both neutron and proton pairing matrix elements is translated into a decrease of the odd-even mass differences of about 200 keV.

For the odd-neutron ¹⁷⁷Yb nucleus and the neighbouring even-even nuclei, using the Möller-Nix gaps, inclusion of the $R_{p}$ corrective factor decreases the absolute value of the proton pairing matrix elements (i.e. less pairing) by about 10 keV, having no significant effect, as expected, on the neutron odd-even mass difference around the ¹⁷⁷Yb nucleus.

For the odd-proton ¹⁷⁷Lu nucleus, including the Slater correction yields also a decrease of about 10 keV of the average pairing matrix element resulting in a decrease of the calculated $\delta_{p}^{3}$ of about 250 keV.