Generalized seniority with realistic interactions in open-shell nuclei
Generalized seniority provides a truncation scheme for the nuclear shell model, based on pairing correlations, which offers the possibility of dramatically reducing the dimensionality of the nuclear shell-model problem. Systematic comparisons against results obtained in the full shell-model space are required to assess the viability of this scheme. Here, we extend recent generalized seniority calculations for semimagic nuclei, the isotopes, to open-shell nuclei, with both valence protons and valence neutrons. The even-mass and isotopes are treated in a full major shell and with realistic interactions, in the generalized seniority scheme with one broken proton pair and one broken neutron pair. Results for level energies, orbital occupations, and electromagnetic observables are compared with those obtained in the full shell-model space. We demonstrate that, even for the isotopes, significant benefit would be obtained in going beyond the approximation of one broken pair of each type, while the isotopes require further broken pairs to provide even qualitative accuracy.
The generalized seniority scheme [1, 2], or broken pair approximation [3, 4], has long been proposed as a truncation scheme for the nuclear shell model, based on pairing correlations, with the potential to dramatically reduce the dimensionality of the nuclear shell-model problem. The generalized seniority scheme has also been proposed as a microscopic foundation for the phenomenologically successful interacting boson model (IBM) , through the Otsuka-Arima-Iachello mapping [6, 7]. The underlying premise of the generalized seniority scheme is that the ground state of an even-even nucleus can be well approximated by a condensate built from collective pairs. These are defined as a specific linear combination of pairs of nucleons in the different valence orbitals, each pair coupled to angular momentum zero. A shell-model calculation for the ground state and low-lying states can then be carried out in a truncated space, consisting of states built from a condensate of collective pairs together with a small number (the generalized seniority) of nucleons not forming part of an pair.
Although the generalized seniority approach has been applied in various contexts [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], only recently has it been benchmarked against calculations carried out in the full shell-model space, with realistic interactions. Comparisons have so far focused on semimagic nuclei [21, 22, 23]. Systematic comparisons for both even-mass and odd-mass isotopes, across the full shell (), with the FPD6  and GXPF1  interactions, are presented in . These benchmark calculations for semimagic nuclei are based on the assumption of at most one broken pair, i.e., for even-mass isotopes, which is found to provide a quantitatively successful reproduction of many of the full-space results (energies, occupations, and electromagnetic observables) for the lowest-lying (and, in particular, yrast) states. In the interior of the shell, where both valence protons and neutrons are present, the obvious challenge to the generalized seniority truncation is the seniority-nonconserving, or pair-breaking, nature of the proton-neutron quadrupole-quadrupole interaction . As this interaction induces deformation, it also suppresses pairing correlations.
The purpose of the present work is to investigate the viability of a highly-truncated generalized seniority description as one moves beyond semimagic nuclei, introducing valence nucleons of both types, and to map the breakdown of this description. The generalized seniority scheme, as a simple representation of the full nuclear shell-model problem, is foremost of conceptual interest. The generalized seniority model space of generalized seniority is equivalent to the space spanned by BCS states with at most quasiparticles, projected onto definite particle number. Therefore, the principal question being addressed by the present calculations is to what extent the shell-model wave functions and predictions quantitatively reflect a simple structure based on BCS-like pairing, in a form which nonetheless fully accounts for particle number conservation. However, the generalized seniority scheme may also be of computational relevance, as a practical truncation scheme. In the shell, the dimensions of the full shell-model space for semimagic and near-semimagic nuclei are computationally tractable. However, truncation is still necessary if single-particle spaces significantly larger than the shell are considered. A generalized seniority truncation is more likely to be advantageous for weakly-deformed nuclei near closed-shell, in large single-particle spaces, than for strongly-deformed nuclei with large numbers of both protons and neutrons in the valence shell.
We extend the investigations of the isotopes ()  into the interior of the shell, to the even-even () and () isotopes, establishing a benchmark comparison of results obtained in the generalized seniority truncation against results obtained in the full shell-model space. We consider calculations truncated to one broken proton pair plus one broken neutron pair, i.e., . These calculations should be viewed as a baseline, in that they are based on the most restricted generalized seniority truncation for both protons and neutron. Calculations involving more broken pairs could be expected to provide a description of the dynamics which extends further into the interior of the shell. The present study is therefore also intended to provide an indication of the dependences upon valence proton and neutron numbers, interactions, and observables likely to influence successful treatment of nuclei in such spaces of higher generalized seniority.
The generalized seniority basis, in a proton-neutron scheme, and computational method are summarized in section 2. Then, calculations for the and isotopes in the generalized seniority scheme are compared with full shell-model calculations, for level energies (section 3.1), orbital occupations (section 3.2), and electromagnetic observables (section 3.3). Preliminary results were presented in .
2 Generalized seniority scheme
We briefly review the construction of the generalized seniority basis for like particles [10, 28, 4, 23] and its extension to a proton-neutron scheme. Let be the creation operator for a particle in the shell-model orbital , with angular momentum projection quantum number . Then the collective pair of the generalized seniority scheme is defined by , where runs over the active orbitals, and . This operator creates a linear combination of pairs in different orbitals , with amplitudes .
A basis state within the generalized seniority scheme then consists of a “condensate” of collective pairs, together with additional nucleons not forming part of a collective pair. The number is termed the generalized seniority. Thus, if we consider semimagic nuclei, so only like valence particles (all neutrons or all protons) are present, the condensate state is constructed as , with angular momentum . States with and angular momentum are then obtained as , states with as , etc., where, if is the number of valence nucleons, is the total number of pairs.
To construct a generalized seniority basis, the parameters must first be determined. This is usually accomplished variationally, so as to minimize the energy functional [3, 29], subject to the conventional normalization . The -condensate state has the same form as a number-projected BCS ground state, and these coefficients are related to BCS occupancy parameters and by . The generalized seniority states as defined above are in general unnormalized and nonorthogonal (for , they also form an overcomplete set). However, a suitable basis is obtained by a Gram-Schmidt procedure, which yields orthonormal basis states as linear combinations of the original states, e.g., for ,
where is simply a counting index for the orthogonalized states. Several approaches [28, 13, 4, 30, 31] have been developed for evaluating matrix elements of one-body and two-body operators in the generalized seniority basis. The present calculations make use of recurrence relations derived in , where the notation and methods used in the present work are also established in detail.
A generalized seniority basis can be defined for nuclei with valence particles of both types via a proton-neutron scheme, that is, by taking all possible products of proton and neutron generalized seniority states, with generalized seniorities and [4, 19]. Here we consider states with one broken pair of each type  and thus have basis states
The coefficients appearing in the proton collective pair operator  and the coefficients appearing in the neutron collective pair operator  are distinct, and these are chosen independently, by minimizing and , respectively. The values of the amplitudes and obtained in this manner for the isotopes are shown in figure 1, for both the FPD6 and GXPF1 interactions. The variational expression for the proton pair has no manifest dependence upon neutron number and, similarly, the variational expression for the neutron pair has no manifest dependence upon proton number. However, an implicit dependence is induced, in each case, by the -dependent definitions of the interaction two-body matrix elements, which are proportional to for FPD6  or for GXPF1 . A simple multiplicative scaling of the Hamiltonian would not affect the collective pair structure, but note that here rather the -dependent two-body contribution varies in strength relative to the -independent single-particle energies. Consequently, the values of in the shell interior vary slightly () between successive isotopic chains, and the values of vary slightly along each isotopic chain [figure 1 (bottom)].
For semimagic nuclei, the size of the generalized seniority basis is the same as for the shell-model problem with only particles in the same set of orbitals, regardless of the number of pairs. Thus, the generalized seniority model space with one broken pair ( for any even isotope taken in the shell has the same dimension as the shell-model space for — dimension for , for , for , etc. . Similarly, if both valence protons and neutrons are present, the basis size is the same as for the shell-model problem with only protons and nucleons. Thus, the generalized seniority model space with one broken proton pair and one broken neutron pair  for any even-even nucleus taken in the shell has the same dimension as the shell-model space for — dimension for , for , for , etc. For comparison, the dimensions of the full shell-model spaces in the shell are shown, in the scheme, in figure 2.
The reduction in model space dimension from the full shell-model space to the generalized seniority truncated space considered in this work exceeds two orders of magnitude for the isotopes midshell and approaches four orders of magnitude for the isotopes midshell (e.g., factors of and in the spaces for and , respectively).
Here we consider some basic energy observables, for the even-even () and () isotopes, in the generalized seniority space. Calculations are shown for , carried out in the full shell (, , , and orbitals) for both protons and neutrons, with the FPD6  and GXPF1  interactions. The generalized seniority results are benchmarked against those obtained in the full shell-model space, calculated using the code NuShellX .
We begin by considering the energy eigenvalue for the ground state. The calculated values are summarized in figure 3 (left). The ground state energy is shown as obtained both for the proton-neutron -pair condensate and in the generalized seniority model space, as well as in the full shell-model space.
To provide a baseline for comparison, it is helpful to review the analogous results for the semimagic isotopes in the space, from , shown at top in figure 3. It should first be noted (see appendix of ) that the ground state obtained in the space is simply the -pair condensate, provided the coefficients have been chosen variationally as in section 2. Therefore, the ground-state energies obtained in and spaces are identical. [In contrast, for nuclei in the interior of the shell, the ground state obtained in the space is not in general the -pair condensate, and these states are considered separately in figure 3.] The ground state energies [figure 3(a,b)] obtained for in the generalized seniority truncation are essentially indistinguishable from those obtained in the full shell-model space, when viewed on the scale of the valence shell interaction energies. A more useful measure of the level of agreement is provided by the residual energy difference , shown in figure 3 (right), obtained by subtracting the full space result from the generalized seniority result. This difference may be thought of as the missing correlation energy, not accounted for by the generalized seniority description of the ground state. For the isotopes [figure 3(g,h)], the maximum difference occurs approximately midshell, peaking at for the FPD6 interaction or for the GXPF1 interaction. It is worth noting at this point that the deviations between the generalized seniority results and the full-space results were found to be consistently smaller for the GXPF1 interaction than for the FPD6 interaction in the study of isotopes — not just for the ground state energy, but for excitation energies, orbital occupations (section 3.2), and electromagnetic observables (section 3.3), as well.
Considering now the ground-state eigenvalues for in the generalized seniority scheme [figure 3(c,d)], a calculation, which simply takes the expectation value of the Hamiltonian in the proton-neutron pair condensate, misses – in binding energy. However, the model space accounts for most of the missing correlation energy, leaving a difference of between the generalized seniority and full-space results [figure 3(i,j)]. Since has two valence protons in the shell, a calculation encompasses the full space of proton configurations. Moreover, for () and (), there are only two valence neutrons or neutron holes, so the space is in fact identical to the full shell-model space, and the generalized seniority and full shell-model results are strictly identical. The generalized seniority results lie furthest from the full-space results in the middle of the subshell (), with a maximum deviation of for () with the FPD6 interaction or with the GXPF1 interaction. The generalized seniority results more closely match the full-space results in the upper shell (), where the average deviations are for the FPD6 interaction or for the GXPF1 interaction.
For the evolution of ground-state eigenvalues across the shell [figure 3(e,f)], similar qualitative observations apply as to the isotopes. However, the energy differences between the generalized seniority and full shell-model results are much larger, up to for the calculation. Breaking one pair of each type yields a less dramatic improvement than observed for the isotopes. The generalized seniority results again lie furthest from the full-space results in the middle of the subshell, missing of binding energy for () with FPD6 or with GXPF1. The generalized seniority results again more closely match the full-space results in the upper shell, where the average deviations are for the FPD6 interaction or for the GXPF1 interaction. It has been hypothesized by Monnoye et al.  that the generalized seniority description should improve at subshell closures. The residual energy difference does indeed have a sharp local minimum at , for both interactions, in both the -condensate  and broken-pair  calculations.
FPD6 GXPF1 0.31 0.48 0.62 0.88 0.13 0.25 0.34 0.63 0.65 1.10 1.14 2.68 0.45 0.76 0.82 1.66 2.48 4.54 5.07 5.74 1.62 3.04 3.55 4.02
In considering excitation energies, we focus on the low-lying states, for which the generalized seniority description was found to be most successful in the semimagic nuclei, especially the first and states. The succussful reproduction of the properties of the first excited state requires a greater number of broken pairs, especially in the subshell . For the energy eigenvalues of the ground state and other low-lying states, the average (root-mean-square) deviations of the generalized seniority model space results from the full shell-model results across the shell are summarized in table 1, along with the corresponding results for .
The excitation energies of the first and states, calculated relative to the ground state, are shown in figure 4. We again begin by reviewing the results for the isotopes, at top in figure 4. The broad features of the evolution of excitation energies across the shell are reproduced within the model space. For instance, for the state [figure 4(a,b)], spikes in excitation energy are obtained at the subshell closure () and subshell closure (–). The excitation energy calculated for the state deviates from that calculated in the full model space by at most for FPD6 or for GXPF1. The deviations in excitation energy for the state [figure 4(g,h)] are only modestly larger, at most for FPD6 or for GXPF1.
For the excitation energies of the [figure 4(c,d)] and [figure 4(i,j)] states in the isotopes, the values obtained in the generalized seniority truncated space and the full shell-model space necessarily agree at and , where, as noted above, the space is identical to the full space. However, otherwise, substantial differences arise. It is interesting to note the cases in which these calculations at low generalized seniority are most successful. Differences are largest in the subshell. Then, at the subshell closure (), the excitation energies calculated in the generalized seniority truncated space closely reproduce those in the full space — moderately closely for the energy, to for FPD6 and for GXPF1, and much more closely for the energy, to (or ) for FPD6 and (or ) for GXPF1. In the upper shell, specifically for the GXPF1 interaction, the excitation energies of the [figure 4(d)] and states [figure 4(j)] are reproduced with a quantitative accuracy comparable to that observed in the semimagic nuclei. Deviations from the full shell-model results average only for the state or for the state. The excitation energies obtained in the generalized seniority truncated model space are systematically higher than those in the full model space, i.e., more correlation energy is missed in the excited state eigenvalue than in the ground state eigenvalue.
For the isotopes, it is seen that considering only one broken pair of each type is markedly inadequate for description of the excitation energies, for both the [figure 4(e,f)] and [figure 4(k,l)] states. The actual excitation energies calculated in the full space are comparable to those for the isotopes, but the excitation energies obtained in the generalized seniority truncation are about twice as high.
The occupations of orbitals provide a direct measure of the structure of an eigenstate, here as obtained either in a generalized seniority calculation or the full shell-model space. Unlike conventional shell-model basis states, the generalized seniority basis states do not have definite occupation for each orbital, rather, involving a BCS-like distribution of occupations. Therefore, the occupation of an orbital in an eigenstate represented in this basis cannot be obtained as a simple average over the contributing basis states, but rather must be evaluated as the expectation value of a one-body operator, the number operator for the orbital , by the process described in [32, 23]. Occupations of each of the -shell orbitals in the ground state are shown in figure 5, for the , , and isotopes, both for neutron orbitals [figure 5 (left)] and proton orbitals [figure 5 (right)].
We again begin by reviewing the situation for the isotopes [figure 5(a,b)]. Recall that the ground state obtained in the space, for semimagic nuclei, is simply the -pair condensate (section 3.1). For , the neutrons are almost exclusively found in the orbital. Therefore, the generalized seniority scheme essentially reduces to conventional seniority in a single -shell. However, in the upper shell, for , neutrons fill multiple orbitals simultaneously, and the generalized seniority scheme prescribes correlations among these -shells beyond the conventional seniority correlations within each -shell. The generalized seniority calculations closely reproduce the occupations obtained in the full shell-model space, to within nucleon across the entire shell, and with deviations averaging only nucleon for FPD6 or for GXPF1. The occupations obtained in the first and states follow very similar patterns to those for the ground state. For the first excited state, the occupations are only well-described in the upper shell (see figure 5 of ) — for , the single-particle energies strongly favor an excitation involving promotion of a single particle from the orbital to the orbital, which is not supported in the highly-restricted (dimension ) , space.
Let us consider now the occupations of these same neutron orbitals for the ground state of the isotopes [figure 5(c,d)], as obtained in the generalized seniority space. The occupations are reproduced to within nucleon throughout the shell, with deviations averaging nucleon for FPD6 or for GXPF1 — greater than for the semimagic nuclei, but still leaving the generalized seniority and full-space curves in figure 5(c,d) largely indistinguishable. The deviations which do arise are localized in and are seen to follow a systematic pattern. Over most of the shell, but especially around the subshell closure at , the generalized seniority calculation overestimates the occupation of the neutron orbital. Similarly, at the nominal subshell closure at , the generalized seniority calculations overestimate the occupation relative to the occupation. In this limited set of examples, the generalized seniority calculations would appear to favor a filling order more strictly reflecting the ordering of single-particle energies (, , , ) than is actually found in the solutions in the full shell-model space. It is worth noting that the quantitative accuracy obtained for the occupations in figure 5(c,d) is obtained only through the breaking of collective pairs. For comparison, occupations calculated in the proton-neutron -pair condensate for the isotopes are shown in figure 6 (a,b). These do reflect the overall evolution of the neutron occupations with , but with substantially larger deviations, averaging nucleon for FPD6 or nucleon for GXPF1.
For the isotopes [figure 5(e,f)], the accuracy with which the neutron orbital occupations for the ground state are reproduced within the generalized seniority truncation deteriorates, averaging nucleon for both FPD6 and GXPF1. The systematic deviations continue to follow the trend described above for the isotopes — in which the generalized seniority results follow nominal filling order more strictly than the full shell-model results — but now increased in magnitude, extending over larger ranges of neutron number, and observed in an overestimate of the orbital occupation relative to as well.
FPD6 GXPF1 0.03 0.09 0.11 0.29 0.015 0.03 0.08 0.26 0.10 0.18 0.17 0.37 0.08 0.13 0.09 0.30 0.30 0.34 0.36 0.27 0.30 0.35 0.42 0.24
In both the and isotopes, occupations obtained in the generalized seniority scheme for the first and states follow very similar patterns to those for the ground state, and with a comparable (or marginally lower) level of accuracy, summarized in table 2. Occupations for the first excited state are obtained with comparable accuracy to that obtained in the semimagic nuclei, and, in , comparable to that for the lower-lying states as well (table 2). This is perhaps surprising, given the much poorer description of energies for the excited state in the generalized seniority scheme (table 1).
The occupations of the proton orbitals along the and isotopic chains, by the two valence protons for [figure 5(g,h)] or four for [figure 5(i,j)], depend on the filling of the neutron orbitals. In the full shell-model space, these protons have a significant probability of occupying orbitals other than the orbital, especially for . As noted in section 2, the coefficients in the proton pair are nearly constant along an isotopic chain [figure 1 (bottom)]. Therefore, the proton occupations obtained in a simple -pair condensate, shown in figure 6 (right), are likewise nearly constant along the chain, varying by only – for the and isotopes. However, for the isotopes, the generalized seniority space [figure 5(g,h)] largely accounts for the actual variation in the proton orbital occupation, to within nucleon throughout the shell. The agreement is significantly better for , to within nucleon, with differences from the full-space results averaging only nucleons for FPD6 or nucleons for GXPF1. The nature of the deviation is, as for the neutron orbitals, to overestimate the occupation of the orbital. For the isotopes [figure 5(i,j)], the deviations follow the same pattern, but with significantly larger magnitudes.
3.3 Electromagnetic observables
Electromagnetic moments and transition strengths probe correlations in the eigenstates which are not simply apparent from the occupations of section 3.2. Of particular interest is the extent to which electric quadrupole correlations are reproduced in a space truncated to low generalized seniority, as one moves into the shell interior, where the proton-neutron quadrupole-quadrupole interaction is expected to play an increasingly significant role. The electric quadrupole reduced transition probability and quadrupole moment are shown for the , , and isotopes in figure 7. Electromagnetic transition matrix elements are obtained from the one-body densities, which are evaluated in the generalized seniority scheme as described in [32, 23].
To review the situation for the isotopes , the evolution of strengths [figure 7(a,b)] across the shell is reproduced in the generalized seniority space in its qualitative features (minima and maxima) and with a quantitative accuracy averaging for FPD6 or for GXPF1. The magnitude of [figure 7(g,h)] is markedly attenuated in the generalized seniority calculations relative to the full-space results, although the generalized seniority results demonstrate the same qualitative features as the results in the full space, notably exhibiting the same alternations in sign as a function of .
For the isotopes, the
strengths [figure 7(c,d)] are an order of magnitude larger than for the
isotopes [note the change of scale from figure 7(a,b) to
figure 7(c,d)]. The variation with neutron number is
relatively flat, but downsloping with increasing .
Both these broad characteristics are well-reproduced in the generalized seniority
space, although the generalized seniority calculations do not reproduce the detailed
fluctuations with neutron number. The deviations under FPD6 are
[averaging , on