Quadrupole dominance in the light Sn and in the Cd isotopes

All nuclear species are equal, but some are more equal than others. The tin isotopes deserve pride of place, because $Z=50$ is the most resilient of the magic numbers, because they are very numerous, and many of them stable, starting at $A=112$. For these, accurate data have been available for a long time. As seen in Fig. 1, a parabola accounts very well for their $B(E2:2^{+}_{1}\to 0^{+}_{1})$ trend, except at ^112-114Sn.

That these early results (Jonsson et al. [2]) truly signaled a change of regime became evident through work on the unstable isomers, starting with the measurement in ¹⁰⁸Sn by Banu et al. [3]. A flurry of activity followed [4, 5, 6, 7, 8, 9, 10], from which a new trend emerged in which the parabola, characteristic of a seniority scheme, gives way to a platform, predicted by a pseudo-SU3 scheme (the squares). Here, I am going a bit fast, to follow the injunction of Montaigne: start by the last point (“Je veux qu’on comance par le dernier point” Essais II 10) [11, p. 296]. To slow down, I note that the idea to associate the plateau with pseudo-SU3, originated in a study of the cadmium isotopes, where quadrupole dominance is stronger and its consequences more clear-cut. Therefore, it is convenient to study the Cd and Sn families together. Section II provides the necessary tools.

The basic idea is inspired by Elliott’s SU3 scheme [12, 13] and consists in building intrinsic determinantal states that maximize $q_{0}$, the expectation value of the quadrupole operator $\hat{q}_{0}=2q_{20}$, i.e., $q_{0}$=$\langle 2q_{20}\rangle\,$ [14, 15, 16]. Fig. 2 implements the idea for ¹⁰⁴Cd ($Z=48,\,N=56$). The single-shell contribution (S) of the $g_{9/2}\equiv g$ proton orbit (S$g$) is given by Eq.(1) (with changed sign for hole states) with $m=9/2$ being the first unfilled orbit for prolate deformation. For the neutron orbits, the pseudo-SU3 scheme  [17, 18, 16] (P generically, P$r_{p}$ for specific cases) amounts to assimilating all the orbits of a major osillator shell of principal quantum number $p$, except the largest (the $r_{p}$ set) into orbits of the $p-1$ major shell. In our case the $sdg$ shell has $p=4$, and $r_{4}$ is assimilated into a $pf$ shell. As the $\hat{q_{0}}$ operator is diagonal in the oscillator quanta representation, maximum $\langle 2q_{20}\rangle\,$is obtained by orderly filling states $(n_{z}\,n_{y}\,n_{x})=(300),(210),(201)\ldots(012),(003)$, with $q_{0}=$$\langle 2q_{20}\rangle\,$=$2n_{z}-n_{y}-n_{z}=$ 6, 3, 3, 0, 0…-3, -3, as in Fig. 2. Using $q(n)$ for the cumulated $q_{0}$ value (e.g. 24 for ¹⁰⁴Cd in Fig. 2), the intrinsic quadrupole moment then follows as a sum of the single-shell (S) and pseudo-SU3 (P) contributions

where I have introduced effective charges and recovered dimensions through $Q=b^{2}q$ with $b^{2}\approx 41.467/\hbar\omega~{}{\rm fm}^{2}$ and $\hbar\omega=45A^{-1/3}-25A^{-2/3}$. To adapt Eq. (2) to Sn, simply drop the S part (i.e., the $8e_{\pi}$ term).

To qualify as a Bohr Mottelson rotor, $Q_{0}(SP)$ must coincide with the intrisic spectroscopic $Q_{0s}$ and transition $Q_{0t}$ quadrupole moments, defined through (as, e.g, in Ref. [16])

To speak of deformed nuclei two conditions must be met: $B(E2:4^{+}_{1}\to 2_{1}^{+})/B(E2:2_{1}^{+}\to 0_{1}^{+})$ =1.43 (the Alaga rule from Eq. (4)), and the “quadrupole quotient” rule, $Q/q$ which follows from Eqs.(3) and (4) by equating $Q_{0s}\approx Q_{0t}$ (for $K=0$):

Full verification demands calculations but Eq. (5) can be checked directly by inspecting Fig. 2 as done in Table 1, that will be analyzed once the shell-model results are in.

These results rely on diagonalizations in spaces defined by $(g^{X-u}{r}_{4}^{u})_{\pi}(g^{10-t}r_{4}^{n+t})_{\nu}$, $X=8$ for Cd and 10 for Sn. The proton ($u$) and neutron ($t$) excitations are restricted to have $u+t\leq M$. The calculations were done for $utM=000$ (the case in Fig. 2), 111, 101 and 202 using $V_{\text{low-}k}$ variants [21] of the precision interaction N3LO [22] (denoted as I in what follows) with oscillator parameter $\hbar\omega\,$= 8.4 MeV and cutoff $\lambda=2$ fm^-1. As a first step the monopole part of I is removed and replaced by single-particle energies for ¹⁰⁰Sn from Ref. [23] referred to as GEMO for general monopole: a successful description of particle and hole spectra on magic nuclei from O to Pb, in particular consistent with the analysis of Ref. [24] for ¹⁰⁰ Sn. Specifically $\epsilon_{j}=$ 0.0, 0.5, 0.8 and 1.6 MeV. for $j=$ 5/2, 7/2, 1/2, and 3/2, respectively.)

The I interaction is then renormalized by increasing the $\lambda\mu=20$ quadrupole and $JT=01$ pairing components by q$\times$10% and p$\times$10%, respectively, and subject to an overall 1.1 scaling to account for renormalizations of other origins. The resulting interactions are called I.q.p. According to Ref. [25], the quadrupole renormalization (due to 2$\hbar\omega\,$perturbative couplings) amounts to 30%, a theoretically sound result, empirically validated by the best phenomenological interactions in the $sd$ and $pf$ shells. By the same token the effective charges in 0$\hbar\omega\,$spaces are estimated as $(e_{\nu},e_{\pi})$ = (0.46, 1.31), as confirmed in Refs. [26, 27]. For the pairing component, perturbation theory is not a good guide, but comparison with the phenomenological interactions demands a 40% increase [25, 15]. It follows that I.3.4 and $(e_{\nu},e_{\pi})$ = (0.46, 1.31) should be taken as standard for full 0$\hbar\omega\,$spaces.

As I will be working in very truncated bases, which demand large effective charges, renormalizations should be implemented to account for polarization mechanisms that involve excitations to the $g$ shell. Proton jumps will contribute to $e_{\nu}$ and are expected to have greater impact than the corresponding neutron jumps, rapidly blocked by the $(r_{4}^{n+t})_{\nu}$ particles. As a consequence, I set $e_{\pi}=1.4$, a guess close to the standard value, and let $e_{\nu}$ vary, thus becoming the only adjustable parameter in the calculations, a choice validated later in Sec. V.

In Fig. 3 it is seen that $utM=000$ and 101 give the same results provided $e_{\nu}$ is properly chosen. There is little difference between $utM=111$ and $utM=101$ because as soon as neutrons are added they block the corresponding jumps, as mentioned above.

The calculation exhibits near perfect agreement with the Alaga rule:( $B(E2:4^{+}_{1}\to 2_{1}^{+})/B(E2:2_{1}^{+}\to 0_{1}^{+})$ $\approx 1.43$$B(E2:2^{+}_{1}\to 0^{+}_{1})$ ). In the figure it is shown for $utM=101$ but it holds as well for 000 and 111. The more stringent quadrupole quotient rule Eq. (6) yields an average $Q/q=0.96$ for ^106-110Cd, corroborating the existence of a deformed region. As announced immediately after Eq. (6), Eq. (5) can be checked directly by inspecting Fig. 2, as done in Table 1 describing the “back of an envelope” SP estimates. Note that the naive form of P used so far (in $q(n)_{n}$ and B20sp) is supplemented by the more accurate $q(n)_{f}$ and B20SP using fully diagonalized values of $\langle 2q_{20}\rangle\,$. The remarkable property of the $r_{4}^{n}$ space that produces four identical $q(n)_{s}$ values for $n=6-12$ has already been put to good use in Ref. [14] and Ref. [15, Fig. 38, TableVII]. In the present case, it is seen to do equally well.

It follows that the very simple estimates suggested by Fig. 2 are quantitatively reliable and can be associated with stable deformation in Cd. In Sn, the same estimates will remain reliable but they cannot be associated with stable deformation. A paradox examined in Sec. IV.1.

My interest in the Cd isotopes was stimulated by the work of Schmidt and collaborators [29], an extension of Ref. [30], where the $v3sb$ interaction (a renormalized version of the charge dependent Bonn potential [31]) was defined. The quadrupole coherence in these studies was clearly identified, but the $B(E2:2^{+}_{1}\to 0^{+}_{1})$ plateau was missed. I obtained the interaction from Nadya Smirnova [32], and reproduced their results. If $v3sb$ is made monopole free and the single particle spectrum replaced by the GEMO value for I.3.4, the $B(E2:2^{+}_{1}\to 0^{+}_{1})$  pattern becomes identical to that of Fig. 3.

The basic tenet of this paper is that quadrupole dominance is responsible for the $B(E2:2^{+}_{1}\to 0^{+}_{1})$ patterns in the light Cd and Sn isotopes, which means that they should exhibit a pseudo-SU3 symmetry. Hence, an intrinsic state should exist, implying the validity of the Alaga rule (Eq. (4)). The expectation is fulfilled in Cd (Fig. 3) but it fails in Sn, as seen in Fig. 4, where the $B(E2:2^{+}_{1}\to 0^{+}_{1})$ rates are consistent with pseudo-SU3 validity, and are immune to details, while the $B(E2:4^{+}_{1}\to 2^{+}_{1})$ rates are sensitive to the single-particle field and to the pairing strength.

To test how the $B(E2:4^{+}_{1}\to 2^{+}_{1})$ rates are influenced by the single particle field, the energy of the $s_{1/2}$ orbit in ¹⁰¹Sn was displaced by 0.0, 0.4 and 0.8 MeV with respect to the present GEMO choice [23], called DZ (Duflo Zuker) in Ref. [24, Fig. 3.2.1] where an extrapolated value (EX) is given as reference. The position of the $s_{1/2}$ orbit for DZ and EX differ by 800 keV. In the calculations reported in Fig. 4, I.3.4(0,0) and I.3.4(0.8) correspond to DZ and EX respectively. The $B(E2:4^{+}_{1}\to 2^{+}_{1})$ differences are significant. Thanks to the recent ¹⁰⁸Sn $B(E2:4^{+}_{1}\to 2^{+}_{1})$ measure of Siciliano et al. [33, Fig. 4b], the DZ choice is clearly favored.

Traditionally, the tin region was taken to be the prime example of pairing dominance, and calculations were based on seniority truncations which lead naturally to an overall parabolic trend, as shown schematically in Fig. 1.

The $B(E2:2^{+}_{1}\to 0^{+}_{1})$ trend in the upper panel of Fig. 4, is already well reproduced by the neutron-only ($utM=000$) case. It will be shown that the monopole field is responsible for this welcome result. Though Bäck and coworkers [37, Fig. 3] already suggested that in such spaces the $B(E2:2^{+}_{1}\to 0^{+}_{1})$ trend could be altered by changes in the single-particle behavior, their result was only indicative. The more complete calculations of Togashi et al. [38, Fig. 2] did better quantitatively, but $g$ proton excitations proved indispensable. (No $B(E2:4^{+}_{1}\to 2^{+}_{1})$ estimates were given in this reference).

The difficulty in reproducing the $B(E2:2^{+}_{1}\to 0^{+}_{1})$  enhancements for $A<114$ persists even when proton excitations are allowed in calculations  [3, 6, 37] using the CDB (charge dependent Bonn) potential [31], renormalized following Ref. [39] (B in what follows). This raises two questions: why the I.3.4 interaction succeeds where others fail? and why the neutron-only description is not only viable, but the correct basic model space? They can be answered simultaneously and I start by explaining how severely truncated spaces may represent the exact results, by comparing the largest calculation available with smaller ones. In Ref. [33, Table I], results are given for ^106-108Sn in $utM=444$ ($m$-dimensions $10^{10}$) with the B interaction used in Banu et al. [3] (but omitting the $h_{11/2}$ orbit). In Fig. 5 they are shown as B444 (circles) and compared with B202 (squares, the same interaction in our standard space). The agreement is very good for the two points in $B(E2:2^{+}_{1}\to 0^{+}_{1})$ and $B(E2:4^{+}_{1}\to 2^{+}_{1})$ . The result amounts to a splendid vindication of the shell model viewed as the possibility to describe in a small space the behavior of a large one. Although in general the reduction from large to small spaces demands renormalization of the operators involved, for our purpose, only the effective charges are affected, a non-trivial fact that invites further study.

Let me return now to the crucial role of the monopole field. In Fig. 5, the $B(E2:2^{+}_{1}\to 0^{+}_{1})$ discrepancies between I.3.4 and B in the upper panel disappear when B is replaced by BG: the interaction made monopole free and supplemented by the GEMO single-particle field used in the I.q.p forces. For the $B(E2:4^{+}_{1}\to 2^{+}_{1})$ pattern in the lower panel, the result is even more interesting: now BG is no longer close to I.3.4, but to I.3.0, which is not shown, but can be guessed by extrapolation in Fig. 4 (from I.3.4 to I.3.2) and from the type of multipole decomposition proposed in Ref. [25], revealing the same $q\cdot q$ content in I.3.4 and BG, and a much weaker pairing for the latter; so weak in fact, that the BG results move closer to the Alaga rule.

It follows that for I.3.0, say, the P$r_{4}$ symmetry will hold, at least partially. As the pairing force is switched on, the $J=0_{1},2_{1}$ states are not affected, while $J=4_{1}$ is. This points to an unusual form of interplay between the two coupling schemes, pairing and quadrupole, traditionally associated with collectivity. In single fluid species, such as Sn, the seniority scheme can operate fully. It breaks down in the presence of two kinds of particles, which turns out to be the condition for quadrupole to operate successfully, as indicated by the Cd isotopes. What is unusual is the presence of quadrupole coherence in the light tins. It is shaky and challenged by pairing and it is known that at about $A=120$ the seniority scheme will prevail.

For the transition nuclei ^116-118Sn, my original guess was that a mixing of spherical and weakly oblate states would take place. I also expected that neutron-only calculations, in which the monopole field would play a crucial role, would be likely to shed light on these matters. The guess was totally wrong, as explained in Sec. VI.3. I mention this anecdote to stress that these two nuclei were the hardest to understand in the region.

Throughout this study, energy spectra have been ignored in favor of electromagnetic rates which are less sensitive to details, so I close by showing spectra for the lowest $J=2$ and 4 levels. In Fig. 6, some pairing state dependence has been allowed, but I.3.4 is seen to be the right overall choice.

In Table 2, the naive P$r_{4}$ adimensional intrinsic quadrupole moments for prolate (q0p) and oblate (q0o) are compared. The former are the same as $q(n)_{n}$ in Table. 1. The latter are obtained by filling the platforms in reverse order (from the top). Up to $N=56$ prolate dominates. From $N=58$ to 62 there is oblate-prolate degeneracy. At $N=64$, oblate dominates. In the absence of strong quadrupole dominance, these intrinsic values only indicate a trend in sign, respected by the calculated spectroscopic moments that opt for “oblate” shapes for $A>108$. For ^112-114Sn, the shell-model results are close for the quadrupole moments, or agree for the magnetic moments, with the measured values. Note: The magnetic moments are very sensitive to the anomalous $g_{l\nu}$.

By suggesting a very different behavior for the Sn and Cd and families at N=64, that will be examined in what follows, Table 2 illustrates the heuristic value of relying on pseudo-SU3, even when the symmetry does not hold in the strict sense.

So far the $sdg$ space has proven sufficient, as the effects of the $h_{11/2}$ orbit ($h$ for short) remain perturbative. For Sn, it is known from classic $(p,d)$ work [42], that the $h$ occupancy, very small up to ¹¹⁰Sn, increases at ^112-114Sn, as borne out by calculations that indicate the need of a boost of some 10% in $B(E2:2^{+}_{1}\to 0^{+}_{1})$  [43] with respect to Fig. 4. Beyond $N=64$, the explicit inclusion of the $h$ orbit becomes imperative but the situation is different for the two families. In Sn, it is known that the traditional $hr_{4}$ space will eventually prove sufficient when the seniority scheme takes over at $N=70$. For ^116-118Sn, at this stage, nothing can be said. For Cd, the calculations give systematically prolate values in line with Stone’s tables of quadrupole moments [44], but in ¹¹²Cd (excluded from both Table 1 and Fig. 3) they yield severe underestimates whose correction necessitates the introduction of a quasi-SU3 mechanism (referred generically as Q in what follows, and Q$hfp$ for the case I introduced next). It is illustrated in Fig. 7, where it is seen that at $N=64$, promoting an extra particle to the $q_{0}=-3$ platform reduces prolate coherence in Cd, while filling the seven upper platforms makes it possible for ¹¹⁴Sn to stay oblate. To obtain realistic estimates demands calculating the quadrupole moment in the presence of a central field. An economic way of doing so is through Nilsson-SU3 self-consistency [16], which is explained next.

For a summary of what has been achieved in this paper, I refer to the “Results” item of the abstract, and concentrate here on the basic ingredients at the origin of the good results: the interaction and specifically, the monopole Hamltonian, and the SPQ interpretive framework.

The Interaction

The shell-model Hamiltonian can be separated into a monopole part, $H_{m}$, that contains all quadratic terms in number operators, and a multipole part, $H_{M}$, that contains all the rest, and is dominated by pairing and quadrupole components that demand a renormalization for use in restricted spaces. While $H_{M}$ depends very little on the realistic potential chosen [15, II D], and does well spectroscopically, $H_{m}$, responsible for saturation properties and single-particle behavior, usually necessitates ad hoc fits, adapted to the problem at hand. In this work, I have relied on the general monopole Hamiltonian GEMO [23], that reproduces, within a root mean square error of 200 keV, the particle and hole spectra on double-magic nuclei.

The monopole interaction

A key step in the implementation is to eliminate the monopole part $H_{m}$ and use the GEMO single-particle field, eventually modulated by the total number of active particles; not necessary for $N<64$ but essential beyond, when the $h_{11/2}$ orbit comes in. The GEMO choice has been found to ensure correct $B(E2:2^{+}_{1}\to 0^{+}_{1})$ and $B(E2:4^{+}_{1}\to 2^{+}_{1})$ patterns in Figs. 3, 4 and 5, and has lead to a prediction for the position of the $s_{1/2}$ orbit in ¹⁰¹Sn.

Conceptually, GEMO has made it possible to establish the neutron-only description in the tin isotopes as the natural starting model space.

The SPQ interpretive framework and self consistency

In doing calculations, the aim is not only to reproduce data, but also to understand underlying mechanisms. The SPQ representations, in addition to their predictive power, have served this purpose well. They are also the backbone of Nilsson-SU3 selfconsistent estimates that supplement existing calculations and open the way to future ones, as in Fig. 8.

In this paper, calculations have alternated between full rigor, heuristics and semi-quantitative estimates, regions have moved from well-developed deformed, to pairing-quadrupole coexistence, to the newly postulated q-vibrational states. Cadmium and tin come and go. “The world is but a perennial swing” (Le monde n’est qu’une branloire perenne. Essais III 2 [11, p. 582]).