Influence of pairing and deformation on charge exchange transitions

Carranza M., A.; Pittel, S.; Hirsch, Jorge G.; Carranza M., A.; Pittel, S.; Hirsch, Jorge G.

doi:10.31349/revmexfis.66.710

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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.66 n.6 México Nov./Dec. 2020 Epub Jan 31, 2022

https://doi.org/10.31349/revmexfis.66.710

Research

Reviews

Influence of pairing and deformation on charge exchange transitions

A. Carranza M.^a

S. Pittel^b

Jorge G. Hirsch^a

^{^a}Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510, México CDMX, México.

^{^b}Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA.

Abstract

We describe the importance of charge-exchange reactions, and in particular Gamow-Teller transitions, first to astrophysical processes and double beta decay, and then to the understanding of nuclear structure. In our review of their role in nuclear structure we first provide an overview of some of the key steps in the emergence of our current understanding of the structure of nuclei, including the central role played by the isovector pairing and the quadrupole-quadrupole channels in the description of energy spectra and in the manifestation of collective modes, some associated with deformation of the nuclear shape. We then turned our focus to Gamow-Teller (GT) transitions in relatively light nuclei, especially in the 2p1f shell, where isoscalar pairing may be playing a role in competition with the isovector pairing that dominates in heavier regions. Following a summary of the progress made in recent years on this subject, we report a systematic shell model study aimed at providing further clarification as to how these pairing modes compete. In this study, we use a schematic Hamiltonian that contains a quadrupole-quadrupole interaction as well as both isoscalar and isovector pairing interactions. We first find an optimal set of Hamiltonian parameters for the model, to provide a starting point from which to vary the relevant pairing strengths and thus assess how this impacts the behavior of GT transitions and the corresponding energy spectra and rotational properties of the various nuclei involved in the decays. The analysis includes a comparison with experimental data as an important theme. The need to suppress the isoscalar pairing mode when treating nuclei with a neutron excess to avoid producing spurious results for the ground state spin and parity with the simplified Hamiltonian is highlighted. Varying the strength parameters for the two pairing modes is found to exhibit different but systematic effects on GT transition properties and on the corresponding energy spectra, which are detailed.

Keywords: Charge-exchange; Gamow-Teller; isovector and isoscalar pairing; quadrupole-quadrupole interaction; shell model; fp shell

PACS: 21.60.Cs; 23.20.Lv; 23.40.-s; 25.40.Kv; 27.40.+z

1.Introduction

Gamow-Teller (GT) transitions provide an important and useful tool in the exploration of nuclei ¹^-⁴. They play a key role in the β decay and electron capture processes that arise in stellar evolution ⁵^,⁶, in the double beta decay process ⁷^,⁹ and in neutrino nucleosynthesis ⁹^-¹¹. Furthermore, they are very useful for the testing of nuclear models.

There are two types of GT transitions; GT₊ in which a proton is changed into a neutron, and GT_ in which a neutron is changed into a proton. The transition strength B(GT) can be obtained from 𝛽 decay studies, but with excitation energies limited by the Q values of the decays. On the other hand, with charge-exchange (CE) reactions, such as (p,n), (n,p), (d.²He), or (³He,t), it is possible to access GT transitions for large values in energy without the Q value restriction. Experimental measurements for such reactions at angles close to 0^O and with an incident energy above 100 MeV/nucleon provide valuable information about GT transitions.

In this work, we describe the relevance of charge-exchange reactions, and in particular Gamow-Teller transitions, in the various areas noted above in which it is known to play an important role. We begin with a description of its role in nucleosynthesis in astrophysical environments and in double beta decay, a rare process whose neutrinoless mode, if observed, would shed light on the understanding of neutrino properties. While some experimental results are mentioned, the emphasis here is on the theoretical developments in the last few decades.

We then turn to the important role of Gamow Teller transitions in nuclear structure. The dominance of the quadrupole-quadrupole interaction in the particle-hole channel and isovector (J = 0) pairing in the particle-particle channel serves as the microscopic foundation for the use of the pairing plus quadrupole Hamiltonian, whose development is briefly reviewed.

For a meaningful description of relatively light nuclei with N ≈ Z, however, it is important to also consider the possible importance of proton-neutron (pn) pairing. Proton-neutron pairing can arise in two channels, isoscalar (T = 0) and isovector (T = 1), in both of which the neutron and proton can have net zero orbital angular momentum and thus exploit the short-range nuclear force. Its role in relatively light nuclei has been studied in recent years with a variety of formalisms. Some of the earliest work considered the extension of mean field techniques like BCS or Hartree-Fock-Bogolyubov (HFB) ¹²^,¹³ to include the proton-neutron pairing channel. Its possible importance in N ≈ Z nuclei has also been studied recently in the context of the nuclear shell model in several recent works (14),(15). Of particular interest is the isoscalar pairing channel as it is expected to be especially important for nuclei with N = Z and N ≈ Z ¹⁶^,¹⁷.

We focus our attention on GT transition intensities in the 2p1f shell and particularly nuclei in the A = 40 - 48 region. For these nuclei, the transitions GT_ and the reactions (p,n) are the primary experimental tools used to obtain the distribution of intensities, but the reaction (³He,t) has served as an alternative experimental tool. This review includes many of the most cited published works, but we apologize in advance for any omissions.

Though as noted earlier, mean field techniques have often been used to treat the interplay of the various pairing modes in nuclei near N ≈ Z, it is known that they can lead to serious errors because of their violation of symmetries ¹⁸. [Note: Recent efforts ¹⁹ to build a symmetry-restored mean field approach have proven promising.] For this reason it is especially useful to study how the various modes of pairing compete in nuclear systems near N = Z in the context of the nuclear shell model, whereby it is possible to treat all pairing modes on an equal footing, to preserve all symmetries, and also to naturally incorporate the effects of deformation which are critical in the regions in which these pairing modes are thought to be important.

For the above reasons, we have chosen to use the shell model to assess how the various pairing modes affect Gamow-Teller intensities in light N ≈ Z nuclei. Our approach is inspired by earlier work ¹⁴ where a simple parametrized shell model Hamiltonian was used in order to have a a tool for systematically isolating the effects of the two pairing channels. The model includes a quadrupole-quadrupole interaction as well as both isoscalar and isovector pairing interactions. By leaving out other components of realistic nuclear Hamiltonians, we can focus directly on the effects of the different pairing modes, albeit at the cost of missing some important features of the structure of the nuclei we consider.

We consider in this analysis even-mass nuclei from A = 42 - 48. We focus on the fragmentation of GT transition strengths, but also consider the energy spectra of both the parent and daughter nuclei involved.

A summary of the paper is as follows. In Sec. 2 we review the role of Gamow-Teller transitions in astrophysics and particle physics, briefly describe the pairing and quadrupole-quadrupole interactions and then summarize earlier shell model investigations of Gamow-Teller transitions in the 2p1f shell. In Sec. 3 we briefly describe the model we use and in Sec. 4 describe selected results of our analysis for GT transition strengths. We leave a discussion of the corresponding results for energy properties to the Appendix, where we also show results of a related Hamiltonian ¹⁴ differing only in the choice of single-particle energies.

2.The relevance of Gamow-Teller transitions

2.1.Charge exchange reactions and the Gamow-Teller strength function

Charge exchange reactions employing nuclear projectiles and targets play a fundamental role in the study of the isospin- and spin-isospin dependent response of nuclei. They reveal important aspects of nuclear dynamics, connecting strong and weak interactions ²⁰. One of them is the Gamow-Teller strength distribution, which is usually inaccessible to beta decays, but can be obtained with high resolution using reactions such as (p, n) (i.e., isospin-lowering) and (n, p) (i.e., isospin-raising), performed at intermediate energies and at nearly zero momentum transfer ²¹^,²². There is a proportionality between single charge-exchange reaction cross sections in the forward direction, e.g. (p, n) and (³He,t), and the Gamow-Teller (GT) strength into the same final nuclear states, which also holds or the (n, p)-type of charge-exchange reactions, such as (⁷Li,⁷Be) ²³. In nuclei with a neutron excess, the main contribution to the GT strengths comes from the removal of a neutron from an occupied single-particle state and the placement of a proton into an unoccupied state having either the same quantum numbers or those of the spin-orbit partner. However, the opposite channel, explored with (n, p and (d,²He) reactions, is Pauli forbidden in medium-heavy nuclei and can only be effective if the Fermi surface is smeared out, which can introduce a radial dependence that is usually not included in the analysis ²⁴.

The smearing of the Fermi surface is in many cases obtained introducing like-particle isovector (J= 0) pairing correlations at mean field level through the BCS approach, while more sophisticated deformed selfconsistent Hartree-Fock calculations with density-dependent Skyrme forces are also employed ²⁵^,²⁶. Residual proton-neutron interactions in the particle-hole and particle-particle channels are introduced employing the proton-neutron Quasiparticle Random Phase Approximation (pnQRPA) ²⁷^-³⁰. This interaction gives rise to collective spin modes such as the giant Gamow-Teller resonance ³¹.

It has been observed that the measured total Gamow-Teller transition strength in the resonance region is much less than a model-independent sum rule predicts, the so called “quenching” phenomena ³¹^-³³. It has been partially explained introducing coupling with Δ-hole excitations ³⁴^-³⁶, associated with nuclear correlations, which cause many of the resulting peaks to be weak enough to become unobservable in full shell model calculations ¹^,³⁷^,³⁸, to a Schiff part of the GT strength to higher energies due to 2-particle-2-hole excitations ³⁹, tensor correlations ⁴⁰, finite momentum transfer in relativistic QRPA descriptions ⁴¹, and with the presence of two- and three-body weak currents ³⁸^,⁴².

The theoretical estimate of GT nuclear matrix elements of beta-decay and electron-capture processes in heavy nuclei, which are usually deformed, has seen important developments in the last decades. One of the latest involves employing the projected shell model with many multi-quasiparticle configurations included in the basis, which shows that the B(GT) distributions can have a strong dependence on the detailed microscopic structure of the relevant states of both the parent and daughter nuclei ⁴³.

2.2.Charge-exchange reactions in astrophysics

Many relevant astrophysical processes like stellar burning, neutrino nucleosynthesis, explosive hydrogen burning and core-collapse supernovae involve nuclear weak interactions ⁴⁴. Their description requires state of the art nuclear models associated with experimental data coming from radioactive ion-beam facilities ⁵^,⁴⁵. Supernova explosions are associated with the collapse of the core of a massive star. The dynamics of this process involves electron capture on thousands of nuclei, whose rates must be estimated employing microscopic models ⁴⁶.

Neutrinos generated in supernova explosions induce nuclear reactions which play an important role in the nucleosynthesis of heavy elements ⁴⁷^,⁴⁸. Recent advances in their description have benefited from new experimental data on allowed Gamow-Teller strength distributions, which are accurately reproduced employing improved nuclear models and computer hardware capabilities, including shell model diagonalization and the proton-neutron Quasiparticle Random Phase Approximation ⁶. As most nuclei relevant to astrophysical processes have a large neutron excess, with proton and neutrons occupying different nuclear major shells, partial occupation numbers must be estimated. In the stellar high density environments, finite temperature dependent occupations are estimated employing the shell model Monte Carlo approach, later combined with the RPA to unblock the Gamow-Teller transitions at all temperatures relevant to core-collapse supernovae ⁴⁹.

2.3.The double beta decay

One of the nuclear processes associated with the weak interaction that continues to attract much experimental and theoretical attention is double beta decay ⁵⁰ ⁵¹ ⁵³^-⁶¹. Two-neutrino double-beta decay has the longest radioactive half-lives ever observed, an outstanding experimental achievement that continues with new developments ⁷. Its neutrinoless mode is a forbidden, lepton-number-violating nuclear transition whose observation would have fundamental implications for neutrino physics and cosmology. A wide range of experiments has been performed and are in execution or planned to discover this decay, unseen up to now ⁸. The theoretical estimate of the decay rates for the two-neutrino and neutrinoless modes are defined as second-order perturbative expressions starting from an effective electroweak Lagrangian ⁶¹. The neutrinoless mode can proceed through mechanisms involving light Majorana neutrinos, heavy Majorana neutrinos, sterile neutrinos and Majorons.

Theoretical calculations involve the evaluation of nuclear transition matrix elements, which have been estimated using a variety of nuclear models. Shell model calculations have been used to estimate the double-beta decay nuclear matrix elements for ⁴⁸Ca ⁶²^-⁶⁶ and in heavier nuclei ⁶⁷^,⁶⁸. Large-scale shell-model calculations for ⁴⁸Ca, including two harmonic oscillator shells (2s1d and 2p1f) found that the neutrinoless double-beta decay nuclear matrix element is enhanced by about 30% compared to 2p1f -shell calculations ⁶⁹. With this formalism a very good linear correlation between the double Gamow-Teller transition to the ground state of the final nucleus and the neutrinoless double beta decay matrix element has been observed ⁷⁰. Shell model Monte Carlo methods also provide valuable results ⁷¹. The Quasiparticle Random Phase Approximation (QRPA) provides a consistent treatment of both particle-hole and particle-particle interactions in calculations for the nuclear matrix elements governing two-neutrino and neutrinoless double-beta decay ⁷². It opened the way to resolve the discrepancy between experimental and calculated two-neutrino decay rates, employing both schematic ⁷³^-⁷⁵ and realistic interactions ⁷⁶^-⁷⁸. The suppression was also found employing a generalized-seniority-based truncation scheme ⁷⁹, and in QRPA calculations including particle number projection ⁸⁰. Extensions of these techniques have been widely applied to estimate the neutrinoless nuclear matrix elements ⁸¹^-⁸⁴, the decay to excited states ⁸⁵^-⁸⁹, and the influence of proton-neutron pairing ⁹⁰. To avoid divergences found in the calculations, a renormalized QRPA version was introduced ⁹¹^-⁹³ which had its own difficulties ⁹⁴^-⁹⁹.

The influence of deformation on double beta decay rates was studied with the deformed QRPA formalism ¹⁰⁰^-¹⁰³ and the Pseudo SU(3) scheme ¹⁰⁴ ¹⁰⁵. The projected-Hartree-Fock-Bogolyubov (PHFB) approach, employing a pairing plus multipole type of effective two-body interaction, shows that deformation plays a crucial role in the nuclear structure aspects of the decays ¹⁰⁶^-¹¹⁰.

The NUMEN project employs an innovative technique to access the nuclear matrix elements entering the expression of the lifetime of the double beta decay by cross section measurements of heavy-ion induced Double Charge Exchange (DCE) reactions. It has reached the experimental resolution and sensitivity required for an accurate measurement of the DCE cross sections at forward angles. However, the tiny values of such cross sections and the resolution requirements demand beam intensities much larger than those manageable with the present facility ¹¹¹.

2.4.The quadrupole-quadrupole and pairing Hamiltonians

As needed background for our discussion of the role of Gamow Teller transitions in nuclear structure physics, we first give a brief reminder of some important concepts and early developments in nuclear structure theory.

Isovector pairing interactions, those coupling like particles to zero total angular momentum ¹¹², play a fundamental role in low energy nuclear structure, and are particularly relevant in the understanding of mass differences between even and odd-mass nuclei. Extending the Bardeen, Cooper and Schrieffer (BCS) description of superconductivity ¹¹³ as a number-nonconserving state of coherent pairs, Bohr, Mottelson and Pines ¹¹⁴ proposed a similar physics mechanism to explain the large gaps seen in the spectra of even-even atomic nuclei, later corrected for finite size effects employing particle number projection ¹¹⁵^-¹¹⁷. The exact solution of the isovector pairing Hamiltonian was presented by Richardson and Sherman in 1963 ¹¹⁸. In the last decades it was extended to families of exactly-solvable models, called generically Richardson-Gaudin (RG) models ¹¹⁹^,¹²⁰, which found application in different areas of quantum many-body physics including mesoscopic systems, condensed matter, quantum optics, cold atomic gases, quantum dots and nuclear structure ¹²¹.

Neutron-proton (np) pairing has long been expected to play an important role in N = Z nuclei. both through its isovector and isoscalar character. While the relevance of isovector (J = 0) np-pairing is well established, the role of isoscalar np-pairing continues is still being debated ¹²². The first exact solution of the proton-neutron isoscalar-isovector (T = 0, 1) pairing Hamiltonian with nondegenerate single-particle orbits and equal pairing strengths was presented in 2007, as a particular case of a family of integrable SO(8) Richardson-Gaudin models ¹⁶. There is clear evidence for an isovector np condensate as expected from isospin invariance. However, and contrary to early expectations, a condensate of deuteron-like pairs appears quite elusive ¹²³.

Particle-number-conserving formalisms have been explored for the treatment of isovector-isoscalar pairing in nuclei, but the agreement with the exact solution is less satisfactory than in the case of the SU(2) Richardson model for pairing between like particles ¹⁷. This leads to the important conclusion that the isoscalar and the isovector proton-neutron pairing correlations cannot be treated accurately by models based on a proton-neutron pair condensate ¹²⁴.

A major step in the microscopic description of the low-energy spectrum of nuclei was made in the early 60s by Baranger ¹²⁵ in terms of quasiparticle fermions with residual two-body interactions, the most important of which is the quadrupole-quadrupole interaction, associated with quadrupole deformation of the nuclear shape and the existence of rotational bands ¹²⁶. Soon thereafter this led to birth of the pairing-plus-quadruple-model ¹²⁷. Mean-field quadrupole-quadrupole correlations allow for shell corrections in the single-particle structure of spherical and deformed nuclei ¹²⁸. Collective properties of vibrational and rotational nuclei have been described employing boson expansion techniques with this Hamiltonian ¹²⁹^-¹³³.

A detailed tensorial analysis of realistic shell model Hamiltonians has conclusively shown that isovector pairing is the most important component in the particle-particle channel, while the quadrupole-quadrupole interaction is the most important in the particle-hole channel ¹³⁴. Extensive shell model calculations confirm the dominance of the quadrupole-quadrupole component of the interaction ¹³⁵^,¹³⁶.

The quadrupole-quadrupole interaction can be associated with the second order Casimir operator of the SU(3) algebra, allowing for an algebraic description of nuclear dynamics ¹³⁷^,¹³⁸ in the rotational regime, generalized for heavy nuclei in the pseudo SU(3) model ¹³⁹^,¹⁴⁰, with the pseudo-spin symmetry that enters in this model now known to have a relativistic origin ¹⁴¹^-¹⁴⁴. It has been found that the SU(3) symmetry of the quadrupole term is broken mainly by the one-body spin-orbit term, but that the energies depend strongly on pairing ¹⁴⁵. The inclusion of a quadrupole-pairing channel allowed for a very detailed description of rotational bands in heavy deformed nuclei with many quasiparticle excitations in the Projected Shell Model ¹⁴⁶^,¹⁴⁷. While these methods were generalized to include realistic effective interactions in Hartree-Fock-Bogolyubov calculations ¹⁴⁸, and in extensive analysis of shape coexistence in nuclei ¹⁴⁹, in this review we will focus on the pairing-plus-quadrupole Hamiltonian, but with the inclusion of isoscalar pairing as well for the reasons noted in the Introduction.

2.5.Shell model description of Gamow-Teller strengths in 2p1f -shell nuclei

The growth in computational power, the development of sophisticated shell model codes and the use of realistic potentials, consistent with two-nucleon data, has opened in the last decades new avenues for a detailed microscopic description of the dynamics of medium mass nuclei, both its single-particle and collective character. In particular, it enabled a quantitative description of rotational motion and Gamow-Teller transitions ¹³⁶. Full shell model calculations in the 2p1f shell, including the orbitals f7/2,p3/2,p1/2 and f5/2 successfully reproduce the spectra, binding energies, quadrupole moments and transition rates ¹⁵⁰^,¹⁵¹, providing a microscopic description of the onset of rotational motion ¹³⁵^,¹⁵²^,¹⁵³ and the quenching of Gamow-Teller transitions ¹^,³⁷^,¹⁵⁴. More recently, novel realistic interactions, like the slightly monopole-corrected version of the well-known KB3 interaction, denoted as KB3G, were able to reproduce the measured Gamow-Teller strength distributions and spectra of 2p1f -shell nuclei in the mass range A = 45 - 65 ¹⁵⁵, while a new shell model interaction, GXPF1J, has been employed to describe the electron capture reaction rates, and the strengths and energies of the Gamow-Teller transitions in the even isotopes of Ni from 56-64 ² ¹⁵⁶. These interactions have been successfully employed ³^,⁴ to describe the most recent experimental results ¹⁵⁷^,¹⁶⁰, and have enabled a study of the evolution of the GT strength distribution from stable nuclei to very neutron-rich nuclei ¹⁶¹. In the 2p1f shell the estimated quenching factor is q = 0.744, slightly smaller but statistically compatible with the 2𝑠1d-shell value ¹⁶².

An anticorrelation between the total Gamow-Teller strength and the transition rate of the collective quadrupole excitation has been observed ¹⁶³^,¹⁶⁴, which can be simulated with artificial changes of the spin-orbit splitting ¹⁶⁵.

Full shell model Monte Carlo calculations for N= Z 2p1f -shell nuclei ¹⁶⁶ with a schematic Hamiltonian containing isovector pairing and quadrupole-quadrupole interactions found a transition with increasing temperature from a phase of isovector pairing dominance to one where isoscalar pairing correlations dominate ¹⁶⁷. The appearance of T = 1 ground states in N = Z odd-odd nuclei has been connected to the combined effect of the isoscalar and isovector 𝐿=0 pairing components of the effective nucleon-nucleon interaction ¹⁶⁸. While in general isovector pairing dominates in the ground states, the isoscalar pair correlations depend strongly on the spin-orbit splitting ¹⁶⁹. The isoscalar (T =0, S = 1) neutron-proton pairing interaction plays a decisive role for the concentration of GT strengths at the first-excited 1+ state in ⁴²Sc, but this effect is suppressed in heavier N = Z nuclei by the spin-orbit force supplemented by the quadrupole-quadrupole interaction ¹⁷⁰. The isoscalar pairing interaction enhances the GT strength of lower energy excitations in N = Z nuclei (¹⁷¹). The competition between T = 1 and T = 0 pairing correlations was also studied using self-consistent Hartree-Fock-Bogolyubov (HFB) plus quasiparticle random-phase approximation calculations, showing that it can cause the inversion of the J^π =0⁺ and J = 1⁺ states near the ground state ¹⁷². An analysis of Gamow-Teller transitions and neutron-proton-pair transfer reactions reveals that the SU(4) -symmetry limit is not realized in ⁴²Sc ¹⁷³ and it is strongly broken by the spin-orbit interaction and by increasing neutron excess ¹⁵.

3.Systematic model calculations of Gamow Teller transitions in the 2p1f major shell

In this section and in the remainder of this review we describe model calculations through which we can systematically appraise the role of the various key components of the nuclear Hamiltonian introduced in Subsection 2.4 on properties of Gamow-Teller transition rates (and to a lesser extent the associated nuclear spectra) for nuclei in the vicinity of N=Z. For such nuclei, we can readily focus on the interplay between all of the key interactions discussed and especially between isovector and isoscalar pairing correlations.

3.1.Model and Optimal Hamiltonian

We focus our attention on even-mass nuclei in the region from A = 42 - 48, in which the valence neutrons and protons reside outside an assumed doubly-magic ⁴⁰Ca core and are restricted to the orbitals of the 2p1f major shell. The Hamiltonian we use is

Hˆ2=∑iεinˆi+χ:Qˆ⋅Qˆ:+aPˆ†⋅Pˆ+bSˆ†⋅Sˆ. (1)

Here Qˆ=Qˆn+Qˆp is the quadrupole mass operator and :Qˆ⋅Qˆ: is the two-body part of the quadrupole-quadrupole operator. Also Pˆ† is the operator that creates a correlated pair with L = 0, S = 1 , J = 1, T = 0, whereas Sˆ† is the operator that creates a correlated pair with L = 0, S = 0, J = 0, T = 1. Finally, the first term is the contribution of single-particle energies, which are taken from the realistic interaction 𝐾𝐵3 (see ¹⁵⁰):ϵ7/2=0.0 MeV, ϵ3/2=2.0 MeV, ϵ1/2=4.0 MeV, and ϵ5/2=6.5 MeV.

Note that our Hamiltonian indeed contains the key quadrupole-quadrupole interaction emphasized in Subsection 2.4 and also contains both isovector and isoscalar pairing, whose relative importance, as we noted, will be a key focus of our interest. It of course also includes an underlying single-particle field, which as also noted in Subsect. 2.4 can play a critical role.

It is interesting to note here that much the same Hamiltonian was used in Ref. ¹⁴, which also systematically explored some features of the same nuclear region. However, in that work a different set of single-particle energies was used, involving a sum of the one-body parts of the quadrupole-quadrople interaction and the spin-orbit interaction. We will briefly show the corresponding results for such a choice of Hamiltonian in Appendix A.

We will focus here on the effect of varying the strength parameters for the isovector and isoscalar pairing terms, leaving the strength of the quadrupole-quadrupole interaction and the single-particle energies unchanged. As noted above, we restrict the analysis to even-mass nuclei near the beginning of the 2p1f shell, namely with A 42 - 48. All calculations reported here have been carried out using the ANTOINE shell-model code ¹³⁶^,¹⁷⁴^,¹⁷⁵.

As our goal is to leave the quadrupole-quadrupole strength and the single-particle energies unchanged as we search for the effect of varying the two pairing strengths, we need to first choose an optimal set of parameters. We can then vary the pairing strength parameters away from their optimal values, while keeping the others fixed, to see how these changes impact the description of the properties of interest. Our approach for choosing the optimal Hamiltonian parameters is based on two primary criteria:

Good reproduction of the low-energy spectra of the nuclei of interest (especially the 1⁺ states of odd-odd nuclei),
Good description of GT properties and especially their fragmentation.

Our analysis, described in some detail in Appendix B, suggests:

For N = Z nuclei the optimal set of parameters for even-mass nuclei with A = 42 - 48 are: χ=-0.065MeV, a = b = 6 .
When N≠Z, as likewise discussed in Appendix B, the ground states of odd-odd nuclei with such a Hamiltonian and attractive isoscalar pairing typically have spin and parity1⁺, at variance with the experimental data. Thus, for N≠Z nuclei, we simply turn off the isoscalar pairing, i.e. set a = 0. This leads to a good overall description of most features exhibited by the nuclei we consider, thereby providing the starting part for our analysis of the impact of changes to the pairing strengths.

In what follows we first very briefly discuss the energy spectra emerging from this optimal Hamiltonian and then focus on the role of the various pairing modes on GT transitions.

A further discussion of the role of the pairing modes on energy spectra is reserved for Appendix B.

4.Results

4.1.Energy spectra

The energy spectra that emerge from our optimal Hamiltonian for the eight nuclei involved in the GT transitions we study are shown in Fig. 1. All experimental data was obtained from ¹⁷⁶. In all cases the ground state angular momentum agrees with the observed values. Furthermore, in all odd-odd daughter nuclei the energies of the first 1⁺ are accurately reproduced by the model. Lastly, the energy of the first 2⁺ state in all of the nuclei we consider is fairly well reproduced.

Figure 1 Energy spectra of the daughters (a, c, e and g) and parents (b, d, f and h) nuclei, comparing the results obtained with the optimal Hamiltonian to experimental data ¹⁷⁶.

In the case of ⁴⁸V the model predicts a near degeneracy between the lowest three states, those with J^π = 0⁺, 1⁺ and2⁺, somewhat closer than in experiment, but nonetheless in reasonable agreement.

On the other hand, the calculated energies of states in even-even nuclei with angular momenta higher than 2⁺ are somewhat more compressed than the experimental energies, thereby losing the rotational properties that were better described in Ref. ¹⁴. In Appendix B, we show how increasing the pairing strengths expands these energy spectra, however at the cost of losing the fragmentation of GT intensities that emerges nicely for the optimal Hamiltonian (see the following subsection).

4.2.GT Transitions

Below are the results obtained for GT transition intensities, and their dependence on the pairing strengths. The values of B(GT) are multiplied by the usual quenching factor (0.74)²³⁸^,¹⁶².

4.2.1.A = 42

The GT transition strengths for ⁴²Ca → ⁴²Sc are shown in Fig. 2, as functions of the isovector pairing strength b, which applies to both the parent and daughter nucleus, and the isoscalar pairing strength a, which only contributes to the properties of the ⁴²Sc daughter nucleus.

Figure 2.Comparison of the experimental ¹⁷⁷ and theoretical results for B(GT) transition strengths for ⁴²Ca→⁴²Sc as a function of the isovector pairing strength b and the isoscalar pairing strength a, which only acts in the daughter nucleus.

When a = b = 6 (the optimal pairing strengths) there is a single strong peak at almost the exact experimental energy and strength. There are small satellite peaks at higher energy with somewhat more strength than seen experimentally.

As 𝑎 is increased the strength to the lowest 1⁺ state increases, albeit slowly, while the energy of that state goes down in energy and eventually for a = 12 becomes the ground state. As b is increased, for a given a, the main peak moves up in energy, but with no noticeable change in its strength. The presence of a single dominant peak is an indication that these nuclei have good SU(4) symmetry.

4.2.2.A = 44

Figure 3 depicts the GT intensities for ⁴⁴Ca → ⁴⁴Sc. As both nuclei have a neutron excess, we set a = 0 for both and vary the isovector pairing strength b only.

Figure 3 Comparison of the experimental ¹⁷⁸ and theoretical results for B(GT) transition strengths for ⁴⁴Ca→⁴⁴Sc as a function of the isovector pairing strength b. Since both nuclei involved have neutron excesses, the isoscalar pairing strength is set to a = 0.

In this case, and in contrast to A = 42, the experimental data shows several satellite peaks at fairly low energies. For the optimal b = 6 isovector strength, the lowest excitation is roughly five times more strongly populated than the next few, in contrast with the experimental data where the relative enhancement is roughly two. The lowest peak moves up in energy as b is increased and becomes progressively more dominant.

Though not part of the A =44 GT decay, it is worth commenting briefly here on the N = Z nucleus ⁴⁴Ti. The spectrum of ⁴⁴Ti is shown in Fig. 11 of Appendix B.2 for several choices of the isovector and isocalar pairing strengths (since it has N = Z both pairing modes are relevant). The first point to note is that like ⁴⁴Sc and ⁴⁴Ca the lowest states are well described by the optimal Hamiltonian. In contrast to them, however, the higher angular momentum states are too compressed, so that the resulting spectrum is even more removed from that of an SU(3) rotor. We can partially restore the SU(3) pattern and get a better overall description description of the energy spectrum by increasing the isovector pairing strength to b = 12. On the other hand, as we see from Fig. 3, an increase in the isovector pairing strength would lead to a loss of fragmentation in the corresponding mass-44 GT pattern, in worse agreement with the experimental data. Thus there is a competition between the fragmentation produced by the spin-orbit operator ¹⁵ and by isovector pairing.

4.2.3.A = 46

Next, we analyze the GT results obtained for the nuclei with mass A = 46. While the parent nucleus involved in the GT decay ⁴⁶Ti → ⁴⁶V has a neutron excess, the daughter nucleus does not. Thus we assume a = 0 for the parent nucleus and present the results as a function of the a value used in describing the daughter. In addition, the results are shown as a function of the isovector pairing strength b used for both nuclei. These results are shown in Fig. 4.

Figure 4 Comparison of the experimental ¹⁵⁸ and theoretical results for B(GT) transition strengths for ⁴⁶Ti →⁴⁶V as a function of the isovector pairing strength b and the isoscalar pairing strength a, which only acts in the daughter nucleus.

The results are interesting. Here there is strong fragmentation of the strength for the optimal a = b = 6 parameters, though the lowest state has slightly more strength than the next few. However, the overall strength to these states is in reasonable accord with what is seen in experiment.

Other features of the results worth noting are that (1) even though we produce an appropriate fragmentation pattern the individual strengths for a = b = 6are substantially larger than in the data, (2) the effect of increasing a is, as for the other masses studied, to focus increasing strength in the lowest 1⁺ state while lowering its energy and (3) the effect of increasing b is, again as for other masses, to likewise enhance population of the lowest 1 + state but now while lifting its energy.

As in our mass-44 discussion, we also comment here on the rotational energy pattern for the even-even nucleus ⁴⁶Ti. As can be seen from Fig. 1f), the higher angular momentum states are compressed with respect to experiment for the optimal Hamiltonian and would require an increase in isovector pairing (see Fig. 13 in Appendix B.3) to improve the overall description of those states and in doing so partially restore the rotational SU(3) symmetry. But as just noted, such an increase of the isovector pairing strength has the effect of focussing the GT strength in the lowest peak, thereby also serving to partially restore SU(4) symmetry.

Though isoscalar pairing is only of importance to the daughter nucleus ⁴⁶V, its increase likewise has the effect of restoring SU(4) symmetry, as is evident from Fig. 4, and to worsen the reproduction of the experimental results.

4.2.4.A = 48

Finally, we treat the GT transitions in 𝐴=48. In this case the relevant decay is ⁴⁸Ti → ⁴⁸V, for which both the parent and daughter nuclei have a neutron excess. Thus, in Fig. 5, where we compare the experimental and calculated transition rates, the theoretical analysis is only shown as a function of the isovector pairing strength b.

Figure 5 Comparison of the experimental ¹⁷⁹ and theoretical results for B(GT) transition strengths for ⁴⁸Ti → ⁴⁸V as a function of the isovector pairing strength b. Since both nuclei involved have neutron excesses, the isoscalar pairing strength is set to a = 0 for both.

The model with the optimal b = 6 isovector strength produces the lowest energy 1⁺ excitation in close agreement with experiment. However, the level of fragmentation in the data is not well reproduced. The strength is concentrated in a single peak at 2 MeV, which is where the strongest state lies in the data, but it is several times more strongly populated than any other. As b is increased, the overall effect is to increase the level of fragmentation across an increasingly wider range of states, in worse the agreement with experiment.

In this case, the optimal parameters give acceptable results for the energy spectra of the daughter and parent nuclei (Fig. 1g and 1h), neither of which extends to particularly high angular angular momentum. In the case of ⁴⁸Cr, where the data extends to high angular momentum and whose results are shown in Fig. 16 in Appendix B.4, the optimal Hamiltonian gives a good description of the spectrum up to J^π = 16⁺. The fact that the rotational pattern of this nucleus is fairly well described without having to dramatically modify the pairing strengths reflects the fact that in a system with a large product of the number of valence neutrons and protons N_p x N_n, the effects of the quadrupole-quadrupole interaction are increasingly more dominant and changes in the pairing strengths have a less important effect.

5.Summary and Conclusions

In this work, we first reviewed the role played by Gamow-Teller transitions in stellar nucleosynthesis and in double beta decay, and summarized theoretical results obtained employing shell model and mean field techniques over the last half century. The pairing plus quadrupole Hamiltonian was introduced and the role of the isoscalar pairing interaction was discussed.

In the second part of this review we explored the effects of proton-neutron pairing on even-mass nuclei in the beginning of 2p1f shell. The analysis was done in the framework of the nuclear shell model using a parametrized Hamiltonian that contains not only isoscalar and isovector pairing but also a quadrupole-quadrupole interaction that produces background deformation.

The first step in this analysis was the choice of the optimal Hamiltonian parameters within our model. This was done by focussing on the parameters that provided an optimal description of the energies of the lowest 1⁺ states in the nuclei of interest, a good description of the other low-lying states of these nuclei, and an optimal GT fragmentation pattern. A significant outcome of this part of the analysis was the realization that for a restricted model Hamiltonian of the type we used we must turn off isoscalar pairing when dealing with nuclei having a neutron excess to avoid producing the incorrect ground state spin and parity for many such nuclei. Our optimal Hamiltonian is able to achieve good overall fits to experimental data for both energy spectra and GT decay properties, albeit with the limitations inherent in the relatively simple parametrization we use. Highlights are an accurate reproduction of the properties of the lowest 1⁺ states, and a reasonable description of the GT fragmentation pattern.

We then varied the isoscalar and isovector pairing strengths from their optimal values to systematically study how the two pairing modes affect first the GT properties of these nuclei and subsequently, in the Appendix, the energy systematics. Our analysis extends from A = 42 to A = 48.

Our analysis of the effect on GT transition properties showed that an increase in the isoscalar pairing strength in those systems in which it is active (N = Z nuclei) focuses GT strength on the lowest 1⁺ state while lowering its energy. Increasing the isovector pairing strength, which is always active, also focuses GT strength on the lowest 1⁺ state but raises its energy.

Our analysis of the impact of the two pairing modes on energy spectra showed that the isoscalar and isovector pairing modes focus primarily on the odd-J states and the even-J states, respectively. Enhancing the isoscalar strength systematically lowers the first 1⁺ state and expands the set of odd-J states. In contrast, enhancing the isovector pairing strength expands the even-J part of the spectrum.

Acknowledgements

We acknowledge helpful advice of Alfredo Poves on the use of the code ANTOINE and Lei Yang for sharing valuable information. This work received partial economic support from DGAPA-UNAM projects IN109417 and IN104020.

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Appendix

A. The original model

As mentioned earlier, the work describe in Sec. 3 took inspiration from the model introduced in ¹⁴, for which the Hamiltonian is

Hˆ1=χ(Qˆ⋅Qˆ+aPˆ†⋅Pˆ+bSˆ†⋅Sˆ+α∑il⃗iˆ⋅s⃗iˆ). (1)

For the sake of comparison, we show here the energy spectra that emerged for that Hamiltonian. The energy spectra of the daughters (a, c, e and g) and parents (b, d, f and h) nuclei, compared with the experimental data, are shown in Fig. 6. They are calculated with the same optimal parameters as for the model discussed in Sec. 3, namely χ=-0.065 MeV, a = b = 6, but with a = 20 as was used in . Here too in the case of N≠Z we must the isoscalar pairing, (i.e. set a = 0), to produce the correct energy of the lowest 1⁺ states in those nuclei.

Figure 6 Energy spectra of the daughters (a, c, e and g) and parents (b, d, f and h) nuclei, comparing experimental data and results obtained with the Hamiltonian (1).

Having the full quadrupole-quadrupole interaction, and not just its two-body part as in Eq. (1), provides a better description of the rotational properties of these nuclei and in particular the spreading of the higher part of their spectra. On the other hand, no combination of parameters reproduces the 4⁺ ground state in ⁴⁸V. This is the main reason we have chosen to use the single-particle energies of the KB3 interaction in the results presented in Sec. 3 of the main body of this paper.

B. Energy Spectra and rotational properties

In this part of the Appendix, we return to the Hamiltonian described in Sec. 3 of the main body of this review and show the effect of varying the strengths of the isoscalar (where appropriate) and isovector terms in the Hamiltonian (1) on the energy spectra of the nuclei under discussion. Some of this material has been referred to already in Sec. 3 and in such cases we do not repeat the discussion.

B.1 A = 42

In Fig. 7, results for the energy spectra for the daughter nucleus ⁴²Sc are shown.

Two points should be noted:

The best overall reproduction of the experimental spectrum occurs for equal values of a and b, and
the optimal choice is a = b = 6.

Figure 7 Energy spectra of 42Sc as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results are shown for a; b = 0; 6, and 12.

With these values the lowest 1⁺ state has essentially the correct energy and there is a reasonable density of low-lying

states. The results, though optimal for this Hamiltonian parametrization, still show clear limitations of the model. In particular, it is not possible to produce a very low-lying 7⁺ state, as is present in the data. It would be necessary to include other components of the two-body interaction to reproduce this.

Another point worthy of note in Fig. 7 concerns the effect of the isovector pairing in the presence of isoscalar pairing. When the isovector pairing strength 𝑏 is weak the ground state is a 1⁺. As the isovector strength is ramped up the 0⁺ state emerges as the ground state. Thus in the presence of non-degenerate single-particle levels it is the interplay of isovector and isoscalar pairing that produces the ground state spin in N = Z nuclei, a conclusion that in fact carries through to heavier N = Z nuclei as we will see when we present those results.

Next we consider the corresponding results for ⁴²Ca, the parent nucleus for GT decay. These results are shown in Fig. 8. We only present the levels of the ground state rotational band in these figures.

Figure 8 Rotational band of ⁴²Ca as a function of the strength of the isovector pairing interaction b in comparison with the experimental data.

Since there are only valence neutrons in ⁴²Ca isoscalar pairing is omitted from the figure. Note that the spectrum does not exhibit a rotational pattern even for b = 0. Clearly the presence of the single-particle energies erases the SU(3) rotational behavior of the purely quadrupole interaction, and in the presence of isovector pairing the rotational pattern is not recovered. In all cases, the2⁺, 4⁺ and 6⁺ tend to group together, in marked contrast to the experimental data where only the 4⁺ and 6⁺ states are grouped. Nevertheless the optimal results seem to arise when we choose b = 6, in accord with the conclusion from the ⁴²Sc analysis.

B.2 A = 44

The calculated spectra of the daughter ⁴⁴Sc are shown in Fig. 9 with the experimental data.

Figure 9 Energy spectra of ⁴⁴Sc as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results are shown for a; b = 0; 6, and 12.

We include isoscalar pairing in the figure as it enables us to demonstrate that isoscalar pairing must be removed when N≠Z. Notice that only in the case a = 0 is the ground state spin 2⁺ reproduced. Using a = 6 it is nearly degenerate with the 1⁺ state, while for a = 12 the ground state spin and parity are1⁺.

The results for the parent nucleus ⁴⁴Ca are shown in Fig. 10. The features observed as a function of the isovector strength parameter are similar to those seen in ⁴²Ca: the optimal energy spread arises for b = 6, the spectrum is compressed as the isovector strength is decreased with the gap between the ground state and the first excited 2⁺ state becoming very small, and the first excited 0⁺ state remains too high in energy for all isovector strengths.

Figure 10 Energy spectra of ⁴⁴Ca as a function of the isovector pairing strength b in comparison with the experimental data.

Next we consider ⁴⁴Ti with 2 valence neutrons and 2 valence protons, for which the results are shown in Fig. 11. Though not a part of the GT decay, it is worth looking at it nevertheless to study the effect of the different pairing modes on its rotational properties. Without pairing and single-particle effects the quadrupole-quadrupole interaction would produce a pure rotor with SU(3) symmetry. Single-particle effects by themselves perturb this pattern, as can be seen from the a = b = 0 results in the figure. Both isovector and isocalar pairing spread the spectrum, in better agreement with experiment, but do not restore the pure rotational character.

Figure 11 Rotational band of ⁴⁴Ti as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results are shown for a; b = 0; 6, and 12.

B.3 A = 46

Now we consider the energy spectra of the A = 46 nuclei involved in the GT decay ⁴⁶Ti → ⁴⁶V.

The results for ⁴⁶V are shown in Fig. 12. The same basic features as for lighter N = Z nuclei are seen here. When the intensity of both pairings decrease simultaneously, the spectrum energies are reduced. When we reduce the isoscalar pairing alone (note this is an N = Z nucleus so that isoscalar pairing is relevant), there is no appreciable effect on the states with even angular momentum, whereas those with odd angular momentum gradually compress. Finally, when decreasing the isovector strength alone, a compression of the overall spectrum occurs and the ground state gradually changes from ⁰⁺ to ¹⁺.

Figure 12 Energy spectra of ⁴⁶V as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results are shown for a; b = 0; 6, and 12.

Next we turn our discussion to ⁴⁶Ti for which the relevant results for the rotational band are shown in Fig. 13. As a nucleus with a neutron excess we only show the results as a function of b. The main features of these results were already described in Subsec. 4.2.3.

Figure 13 Energy spectra of ⁴⁶V as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results are shown for a; b = 0; 6, and 12.

B.4 A = 48

Lastly we treat nuclei with A = 48. We first discuss those that participate in the GT decay ⁴⁸Ti → ⁴⁸V and afterwards briefly comment on ⁴⁸Cr.

The results for ⁴⁸V are shown in Fig. 14. We include 𝑎≠0 values even though this nucleus has a relatively large neutron excess, for reasons that will be made clearer soon.

Figure 14 Rotational band of ⁴⁶Ti as a function of the isovector b pairing strength in comparison with the experimental data. Results are shown for a = 0; b = 0; 6, and 12.

The first point to note is that the experimental ground state of this nucleus has spin and parity 4⁺. The model is able to reproduce the 4⁺ ground state for the optimal set of parameters a = 0 and b = 6. If a is increased, however, it no longer produces the correct ground state spin and parity, confirming again that it is critical to set a = 0 to suppress erroneous features in the low-energy spectrum when treating nuclei with a neutron excess.

Results for the energy spectra of ⁴⁸Ti are exhibited in Fig. 15 as a function of the parameter b. While the model with the optimal value of b = 6 reproduces the experimental spectrum reasonably well, the agreement is rapidly lost when we increase or decrease b.

Figure 15 Energy spectra of ⁴⁸V as a function of the isovector b and isoscalar a pairing strengths in comparison with experimental data. Results are shown for a; b = 0; 6, and 12.

Finally we show results in Fig. 16 for the states of the ground state rotational band in the N = Z nucleus ⁴⁸Cr. These results were already discussed in Subsec. 4.2.4 in the main body of the paper.

Figure 16 Energy spectra of ⁴⁸Ti as a function of the isovector b pairing strength in comparison with the experimental data. % Results are shown for a = 0; b = 0; 6, and 12.

Received: June 11, 2020; Accepted: July 13, 2020

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