Influence of pairing and deformation on charge exchange transitions
IInfluence of pairing and deformation on charge exchangetransitionsA. Carranza M. (1) , S. Pittel (2) , Jorge G. Hirsch (1) . (1) Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma deM´exico, 04510, M´exico CDMX, M´exico.(2) 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 particularGamow-Teller transitions, first to astrophysical processes and double betadecay, and then to the understanding of nuclear structure. In our review oftheir role in nuclear structure we first provide an overview of some of thekey steps in the emergence of our current understanding of the structureof nuclei, including the central role played by the isovector pairing and thequadrupole-quadrupole channels in the description of energy spectra and inthe manifestation of collective modes, some associated with deformation ofthe nuclear shape. We then turned our focus to Gamow-Teller (GT) tran-sitions in relatively light nuclei, especially in the 2 p f shell, where isoscalarpairing may be playing a role in competition with the isovector pairing thatdominates in heavier regions. Following a summary of the progress made inrecent years on this subject, we report a systematic shell model study aimedat 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 interac-tions. We first find an optimal set of Hamiltonian parameters for the model,to provide a starting point from which to vary the relevant pairing strengthsand thus assess how this impacts the behavior of GT transitions and thecorresponding energy spectra and rotational properties of the various nucleiinvolved in the decays. The analysis includes as an important theme a com-parison with experimental data. The need to suppress the isoscalar pairingmode when treating nuclei with a neutron excess to avoid producing spuriousresults for the ground state spin and parity with the simplified Hamiltonianis highlighted. Varying the strength parameters for the two pairing modes isfound to exhibit different but systematic effects on GT transition properties1 a r X i v : . [ nu c l - t h ] J un nd on the corresponding energy spectra, which are detailed. Gamow-Teller (GT) transitions provide an important and useful tool in theexploration of nuclei [1–4]. They play a key role in the β decay and electroncapture processes that arise in stellar evolution [5,6], in the double beta decayprocess [7, 8] and in neutrino nucleosynthesis [9–11]. Furthermore, they arevery useful for the testing of nuclear models.There are two types of GT transitions; GT + in which a proton is changedinto a neutron, and GT − in which a neutron is changed into a proton. Thetransition strength B(GT) can be obtained from β decay studies, but withexcitation energies limited by the Q values of the decays. On the otherhand, with charge-exchange (CE) reactions, such as ( p, n ), ( n, p ), ( d, He ), or( He, t ), it is possible to access GT transitions for large values in energy with-out the Q value restriction. Experimental measurements for such reactionsat angles close to 0 ◦ and with an incident energy above 100 M eV /nucleonprovide valuable information about GT transitions.In this work, we describe the relevance of charge-exchange reactions, andin particular Gamow-Teller transitions, in the various areas noted above inwhich it is known to play an important role. We begin with a descriptionof its role in nucleosynthesis in astrophysical environments and in doublebeta decay, a rare process whose neutrinoless mode, if observed, would shedlight on the understanding of neutrino properties. While some experimentalresults are mentioned, the emphasis here is on the theoretical developmentsin the last few decades.We then turn to the important role of Gamow Teller transitions in nu-clear structure. The dominance of the quadrupole-quadrupole interaction inthe particle-hole channel and isovector (J=0) pairing in the particle-particlechannel serves as the microscopic foundation for the use of the pairing plusquadrupole Hamiltonian, whose development is briefly reviewed.For a meaningful description of relatively light nuclei with N ≈ Z , how-ever, 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 andproton can have net zero orbital angular momentum and thus exploit theshort-range nuclear force. Its role in relatively light nuclei has been stud-2ed in recent years with a variety of formalisms. Some of the earliest workconsidered the extension of mean field techniques like BCS or Hartree-Fock-Bogolyubov (HFB) [12,13] to include the proton-neutron pairing channel. Itspossible importance in N ≈ Z nuclei has also been studied recently in thecontext of the nuclear shell model in several recent works [14, 15]. Of partic-ular interest is the isoscalar pairing channel as it is expected to be especiallyimportant for nuclei with N = Z and N ≈ Z [16, 17].We focus our attention on GT transition intensities in the 2 p f shelland particularly nuclei in the A = 40 −
48 region. For these nuclei, thetransitions GT − and the reactions ( p, n ) are the primary experimental toolsused to obtain the distribution of intensities, but the reaction ( He, t ) hasserved as an alternative experimental tool. This review includes many of themost cited published works, but we apologize in advance for any omissions.Though as noted earlier, mean field techniques have often been used totreat the interplay of the various pairing modes in nuclei near N = Z , itis known that they can lead to serious errors because of their violation ofsymmetries [18]. [Note: Recent efforts [19] to build a symmetry-restoredmean field approach have proven promising.] For this reason it is especiallyuseful to study how the various modes of pairing compete in nuclear systemsnear N = Z in the context of the nuclear shell model, whereby it is possibleto treat all pairing modes on an equal footing, to preserve all symmetries,and also to naturally incorporate the effects of deformation which are criticalin 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 as-sess how the various pairing modes affect Gamow-Teller intensities in light N ≈ Z nuclei. Our approach is inspired by earlier work [14] where a simpleparametrized shell model Hamiltonian was used in order to have a a tool forsystematically isolating the effects of the two pairing channels. The modelincludes a quadrupole-quadrupole interaction as well as both isoscalar andisovector pairing interactions. By leaving out other components of realisticnuclear Hamiltonians, we can focus directly on the effects of the differentpairing modes, albeit at the cost of missing some important features of thestructure of the nuclei we consider.We consider in this analysis even-mass nuclei from A=42-48. We focus onthe fragmentation of GT transition strengths, but also consider the energyspectra of both the parent and daughter nuclei involved.A summary of the paper is as follows. In Sec. II we review the roleof Gamow-Teller transitions in astrophysics and particle physics, briefly de-3cribe the pairing and quadrupole-quadrupole interactions and then sum-marize earlier shell model investigations of Gamow-Teller transitions in the2 p f shell. In Sec. III we briefly describe the model we use and in Sec.IV describe selected results of our analysis for GT transition strengths. Weleave a discussion of the corresponding results for energy properties to theAppendix, where we also show results of a related Hamiltonian [14] differingonly in the choice of single-particle energies. Charge exchange reactions employing nuclear projectiles and targets playa fundamental role in the study of the isospin- and spin-isospin dependentresponse of nuclei. They reveal important aspects of nuclear dynamics, con-necting strong and weak interactions [20]. One of them is the Gamow-Teller(GT) strength distribution, which is usually inaccessible to beta decays, butcan be obtained with high resolution using reactions such as (p, n) (i.e.,isospin-lowering) and (n, p) (i.e., isospin-raising), performed at intermediateenergies and at nearly zero momentum transfer [21,22]. There is a proportion-ality between single charge-exchange reaction cross sections in the forwarddirection, e.g. (p, n) and ( He,t), and the Gamow-Teller (GT) strength intothe same final nuclear states. In nuclei with a neutron excess, the main con-tribution to the GT strengths comes from the removal of a neutron from anoccupied single-particle state and the placement of a proton into an unoccu-pied state having either the same quantum numbers or those of the spin-orbitpartner. However, the opposite channel, explored with (n, p) and (d, He) re-actions, is Pauli forbidden in medium-heavy nuclei and can only be effectiveif the Fermi surface is smeared out, which can introduce a radial dependencethat is usually not included in the analysis [23].The smearing of the Fermi surface is in many cases obtained introducinglike-particle isovector (J=0) pairing correlations at mean field level throughthe BCS approach, while more sophisticated deformed selfconsistent Hartree-Fock calculations with density-dependent Skyrme forces are also employed[24,25]. Residual proton-neutron interactions in the particle-hole and particle-particle channels are introduced employing the proton-neutron Quasiparticle4andom Phase Approximation (pnQRPA) [26–29]. This interaction givesrise to collective spin modes such as the giant Gamow-Teller resonance [30].It has been observed that the measured total Gamow-Teller transitionstrength in the resonance region is much less than a model-independent sumrule predicts, the so called ”quenching” phenomena [31–33]. It has beenpartially explained introducing coupling with ∆-hole excitations [35, 36, 75],associated with nuclear correlations, which cause many of the resulting peaksto 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 to2-particle-2-hole excitations [39], tensor correlations [40], finite momentumtransfer 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 andelectron-capture processes in heavy nuclei, which are usually deformed, hasseen important developments in the last decades. One of the latest involvesemploying the projected shell model with many multi-quasiparticle config-urations included in the basis, which shows that the B(GT) distributionscan have a strong dependence on the detailed microscopic structure of therelevant states of both the parent and daughter nuclei [43].
Many relevant astrophysical processes like stellar burning, neutrino nucle-osynthesis, explosive hydrogen burning and core-collapse supernovae involvenuclear weak interactions [44]. Their description requires state of the artnuclear models associated with experimental data coming from radioactiveion-beam facilities [5, 45]. Supernova explosions are associated with the col-lapse of the core of a massive star. The dynamics of this process involveselectron capture on thousands of nuclei, whose rates must be estimated em-ploying microscopic models [46].Neutrinos generated in supernova explosions induce nuclear reactionswhich play an important role in the nucleosynthesis of heavy elements [47,48]. Recent advances in their description have benefited from new exper-imental data on allowed Gamow-Teller strength distributions, which areaccurately reproduced employing improved nuclear models and computerhardware capabilities, including shell model diagonalization and the proton-neutron Quasiparticle Random Phase Approximation (pnQRPA) [6]. Asmost nuclei relevant to astrophysical processes have a large neutron excess,5ith proton and neutrons occupying different nuclear major shells, partialoccupation numbers must be estimated. In the stellar high density environ-ments, finite temperature dependent occupations are estimated employingthe shell model Monte Carlo approach, later combined with the RPA tounblock the Gamow-Teller transitions at all temperatures relevant to core-collapse supernovae [49].
One of the nuclear processes associated with the weak interaction that con-tinues to attract much experimental and theoretical attention is double betadecay [50, 51, 53–61]. Two-neutrino double-beta decay has the longest ra-dioactive half-lives ever observed, an outstanding experimental achievementthat continues with new developments [7]. Its neutrinoless mode is a for-bidden, lepton-number-violating nuclear transition whose observation wouldhave fundamental implications for neutrino physics and cosmology. A widerange of experiments has been performed and are in execution or plannedto discover this decay, unseen up to now [8]. The theoretical estimate ofthe decay rates for the two-neutrino and neutrinoless modes are defined assecond-order perturbative expressions starting from an effective electroweakLagrangian [61]. The neutrinoless mode can proceed through mechanismsinvolving light Majorana neutrinos, heavy Majorana neutrinos, sterile neu-trinos and Majorons.Theoretical calculations involve the evaluation of nuclear transition ma-trix elements, which have been estimated using a variety of nuclear mod-els.
Shell model calculations have been used to estimate the double-betadecay nuclear matrix elements for Ca [62–66] and in heavier nuclei [67, 68].Large-scale shell-model calculations for Ca, including two harmonic oscil-lator shells (2 s d and 2 p f ) found that the neutrinoless double-beta decaynuclear matrix element is enhanced by about 30% compared to 2 p f -shell cal-culations [69]. With this formalism a very good linear correlation between thedouble Gamow-Teller transition to the ground state of the final nucleus andthe 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 consistenttreatment of both particle-hole and particle-particle interactions in calcula-tions for the nuclear matrix elements governing two-neutrino and neutrino-less double-beta decay [72]. It opened the way to resolve the discrepancy be-6ween experimental and calculated two-neutrino decay rates, employing bothschematic [73–75] and realistic interactions [76–78]. The suppression was alsofound employing a generalized-seniority-based truncation scheme [79], and inQRPA calculations including particle number projection [80]. Extensions ofthese techniques have been widely applied to estimate the neutrinoless nu-clear matrix elements [81–84], the decay to excited states [85–89], and theinfluence of proton-neutron pairing [90]. To avoid divergences found in thecalculations, a renormalized QRPA version was introduced [91–93] which hadits own difficulties [94–99].The influence of deformation on double beta decay rates was studied withthe deformed QRPA formalism [100–103] and the Pseudo SU(3) scheme [104,105]. The projected-Hartree-Fock-Bogoliubov (PHFB) approach, employinga pairing plus multipole type of effective two-body interaction, shows thatdeformation plays a crucial role in the nuclear structure aspects of the decays[106–110].The NUMEN project employs an innovative technique to access the nu-clear matrix elements entering the expression of the lifetime of the dou-ble beta decay by cross section measurements of heavy-ion induced DoubleCharge Exchange (DCE) reactions. It has reached the experimental resolu-tion and sensitivity required for an accurate measurement of the DCE crosssections at forward angles. However, the tiny values of such cross sectionsand the resolution requirements demand beam intensities much larger thanthose manageable with the present facility [111].
As needed background for our discussion of the role of Gamow Teller tran-sitions in nuclear structure physics, we first give a brief reminder of someimportant concepts and early developments in nuclear structure theory.Isovector pairing interactions, those coupling like particles to zero to-tal angular momentum [112], play a fundamental role in low energy nuclearstructure, and are particularly relevant in the understanding of mass differ-ences between even and odd-mass nuclei. Extending the Bardeen, Cooperand Schrieffer (BCS) description of superconductivity [113] as a number-nonconserving state of coherent pairs, Bohr, Mottelson and Pines [114] pro-posed a similar physics mechanism to explain the large gaps seen in the7pectra of even-even atomic nuclei, later corrected for finite size effects em-ploying particle number projection [115–117]. The exact solution of theisovector pairing Hamiltonian was presented by Richardson and Sherman in1963 [118]. In the last decades it was extended to families of exactly-solvablemodels, called generically Richardson-Gaudin (RG) models [119, 120], whichfound application in different areas of quantum many-body physics includingmesoscopic 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 impor-tant 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 roleof isoscalar np-pairing continues is still being debated [122]. The first exactsolution of the proton-neutron isoscalar-isovector (T=0,1) pairing Hamilto-nian with nondegenerate single-particle orbits and equal pairing strengthswas 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 npcondensate as expected from isospin invariance. However, and contrary toearly expectations, a condensate of deuteron-like pairs appears quite elu-sive [123].Particle-number-conserving formalisms have been explored for the treat-ment of isovector-isoscalar pairing in nuclei, but the agreement with theexact solution is less satisfactory than in the case of the SU(2) Richardsonmodel for pairing between like particles [17]. This leads to the importantconclusion that the isoscalar and the isovector proton-neutron pairing corre-lations cannot be treated accurately by models based on a proton-neutronpair condensate [124].A major step in the microscopic description of the low-energy spectrumof nuclei was made in the early 60s by Baranger [125] in terms of quasi-particle fermions with residual two-body interactions, the most important ofwhich is the quadrupole-quadrupole interaction, associated with quadrupoledeformation 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 inthe single-particle structure of spherical and deformed nuclei [128]. Collectiveproperties of vibrational and rotational nuclei have been described employingboson expansion techniques with this Hamiltonian [129–133].A detailed tensorial analysis of realistic shell model Hamiltonians hasconclusively shown that isovector pairing is the most important component in8he particle-particle channel, while the quadrupole-quadrupole interaction isthe most important in the particle-hole channel [134]. Extensive shell modelcalculations confirm the dominance of the quadrupole-quadrupole componentof the interaction [135, 176].The quadrupole-quadrupole interaction can be associated with the secondorder Casimir operator of the SU(3) algebra, allowing for an algebraic de-scription of nuclear dynamics [137, 138] in the rotational regime, generalizedfor heavy nuclei in the pseudo SU(3)model [139, 140], with the pseudo-spinsymmetry that enters in this model now known to have a relativistic ori-gin [141–144]. It has been found that the SU(3) symmetry of the quadrupoleterm is broken mainly by the one-body spin-orbit term, but that the ener-gies depend strongly on pairing [145]. The inclusion of a quadrupole-pairingchannel allowed for a very detailed description of rotational bands in heavydeformed nuclei with many quasiparticle excitations in the Projected ShellModel [146, 147]. While these methods were generalized to include realisticeffective interactions in Hartree-Fock-Bogolyubov calculations [148], and inextensive analysis of shape coexistence in nuclei [149], in this review we willfocus on the pairing-plus-quadrupole Hamiltonian, but with the inclusion ofisoscalar pairing as well for the reasons noted in the Introduction. p f -shell nuclei The growth in computational power, the development of sophisticated shellmodel codes and the use of realistic potentials, consistent with two-nucleondata, has opened in the last decades new avenues for a detailed microscopicdescription of the dynamics of medium mass nuclei, both its single-particleand collective character. In particular, it enabled a quantitative descriptionof rotational motion and Gamow-Teller transitions [176]. Full shell model cal-culations in the 2 p f shell, including the orbitals f / , p / , p / f / p f -shell nuclei in the mass range A = 45-95 [155], while a new shell model interaction, GXPF1J, has been employedto describe the electron capture reaction rates, and the strengths and energiesof 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 mostrecent experimental results [157–160], and have enabled a study of the evolu-tion of the GT strength distribution from stable nuclei to very neutron-richnuclei [161]. In the 2 p f shell the estimated quenching factor is q = 0 . s d -shell value [162].An anticorrelation between between the total Gamow-Teller strength andthe transition rate of the collective quadrupole excitation has been observed[163, 164], which can be simulated with artificial changes of the spin-orbitsplitting [165].Full shell model Monte Carlo calculations for N=Z 2 p f -shell nuclei [166]with a schematic Hamiltonian containing isovector pairing and quadrupole-quadrupole interactions found a transition with increasing temperature froma phase of isovector pairing dominance to one where isoscalar pairing cor-relations dominate [167]. The appearance of T=1 ground states in N=Zodd-odd nuclei has been connected to the combined effect of the isoscalarand isovector L=0 pairing components of the effective nucleon-nucleon in-teraction [168]. While in general isovector pairing dominates in the groundstates, the isoscalar pair correlations depend strongly on the spin-orbit split-ting [169]. The isoscalar (T = 0, S = 1) neutron-proton pairing interac-tion plays a decisive role for the concentration of GT strengths at the first-excited 1+ state in Sc, but this effect is suppressed in heavier N=Z nucleiby the spin-orbit force supplemented by the quadrupole-quadrupole inter-action [170]. The isoscalar pairing interaction enhances the GT strength oflower energy excitations in N=Z nuclei [171]. The competition between T=1and T=0 pairing correlations was also studied using self-consistent Hartree-Fock-Bogoliubov (HFB) plus quasiparticle random-phase approximation cal-culations, 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 transitionsand neutron-proton-pair transfer reactions reveals that the SU(4)-symmetrylimit is not realized in Sc [173] and it is strongly broken by the spin-orbitinteraction and by increasing neutron excess [15].10
Systematic model calculations of GamowTeller transitions in the p f ma jor shell In this section and in the remainder of this review we describe model calcu-lations through which we can systematically appraise the role of the variouskey components of the nuclear Hamiltonian introduced in Subsection II.Con properties of Gamow-Teller transition rates (and to a lesser extent theassociated nuclear spectra) for nuclei in the vicinity of N=Z. For such nu-clei, we can readily focus on the interplay between all of the key interactionsdiscussed and especially between isovector and isoscalar pairing correlations.
We focus our attention on even-mass nuclei in the region from A=42-48, inwhich the valence neutrons and protons reside outside an assumed doubly-magic Ca core and are restricted to the orbitals of the 2 p f major shell.The Hamiltonian we use is (cid:98) H = (cid:88) i ε i (cid:98) n i + χ (cid:16) : (cid:98) Q · (cid:98) Q : + a (cid:98) P † · (cid:98) P + b (cid:98) S † · (cid:98) S (cid:17) . (1)Here (cid:98) Q = (cid:98) Q n + (cid:98) Q p is the quadrupole mass operator and : (cid:98) Q · (cid:98) Q : is the two-body part of the quadrupole-quadrupole operator. Also (cid:98) P † is the operatorthat creates a correlated pair with L = 0, S = 1, J = 1, T = 0, whereas (cid:98) S † isthe 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, whichare taken from the realistic interaction KB ε / = 0 . M eV , ε / = 2 . M eV , ε / = 4 . M eV and ε / = 6 . M eV .Note that our Hamiltonian indeed contains the key quadrupole-quadrupoleinteraction emphasized in Subsection II.3 and also contains both isovectorand isoscalar pairing, whose relative importance, as we noted, will be a keyfocus of our interest. It of course also includes an underlying single-particlefield, which as also noted in Subsect. II.3 can play a critical role.It is interesting to note here that much the same Hamiltonian was usedin Ref. [14], which also systematically explored some features of the samenuclear region. However, in that work a different set of single-particle en-ergies was used, involving a sum of the one-body parts of the quadrupole-11uadrople interaction and the spin-orbit interaction. We will briefly showthe corresponding results for such a choice of Hamiltonian in Appendix A.We will focus here on the effect of varying the strength parameters for theisovector and isoscalar pairing terms, leaving the strength of the quadrupole-quadrupole interaction and the single-particle energies unchanged. As notedabove, we restrict the analysis to even-mass nuclei near the beginning of the2 p f shell, namely with A = 42 −
48. All calculations reported here havebeen carried out using the ANTOINE shell-model code [174–176].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 pair-ing strengths, we need to first choose an optimal set of parameters. We canthen vary the pairing strength parameters away from their optimal values,while keeping the others fixed, to see how these changes impact the descrip-tion of the properties of interest. Our approach for choosing the optimal
Hamiltonian parameters is based on two primary criteria:1. Good reproduction of the low-energy spectra of the nuclei of interest(especially the 1 + states of odd-odd nuclei),2. 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 nucleiwith A = 42 −
48 are: χ = − . M eV , a = b = 6. • When N (cid:54) = Z , as likewise discussed in Appendix B, the ground states ofodd-odd nuclei with such a Hamiltonian and attractive isoscalar pairingtypically have spin and parity 1 + , at variance with the experimentaldata. Thus, for N (cid:54) = Z nuclei, we simply turn off the isoscalar pairing, i.e. set a = 0. This leads to a good overall description of most featuresexhibited by the nuclei we consider, thereby providing the starting partfor our analysis of the impact of changes to the pairing strengths.In what follows we first very briefly discuss the energy spectra emergingfrom this optimal Hamiltonian and then focus on the role of the variouspairing modes on GT transitions. A further discussion of the role of thepairing modes on energy spectra is reserved for Appendix B.12 Results
The energy spectra that emerge from our optimal Hamiltonian for the eightnuclei involved in the GT transitions we study are shown in Fig. 1. Allexperimental data was obtained from [177]. In all cases the ground stateangular momentum agrees with the observed values. Furthermore, in all odd-odd daughter nuclei the energies of the first 1 + are accurately reproduced bythe model. Lastly, the energy of the first 2 + state in all of the nuclei weconsider is fairly well reproduced.In the case of V the model predicts a near degeneracy between thelowest three states, those with J π =0 + , 1 + and 2 + , somewhat closer than inexperiment, but nonetheless in reasonable agreement.On the other hand, the calculated energies of states in even-even nucleiwith angular momenta higher than 2 + are somewhat more compressed thanthe experimental energies, thereby losing the rotational properties that werebetter described in Ref. [14]. In Appendix B, we show how increasing thepairing strengths expands these energy spectra, however at the cost of los-ing the fragmentation of GT intensities that emerges nicely for the optimalHamiltonian (see the following subsection). Below are the results obtained for GT transition intensities, and their depen-dence on the pairing strengths. The values of B(GT) are multiplied by theusual quenching factor (0 . [38, 162]. The GT transition strengths for Ca → Sc are shown in Fig. 2, as functionsof the isovector pairing strength b , which applies to both the parent anddaughter nucleus, and the isoscalar pairing strength a , which only contributesto the properties of the Sc daughter nucleus.When a = b = 6 (the optimal pairing strengths) there is a single strongpeak at almost the exact experimental energy and strength. There are smallsatellite peaks at higher energy with somewhat more strength than seenexperimentally. 13igure 1: Energy spectra of the daughters (a, c, e and g) and parents (b, d, fand h) nuclei, comparing the results obtained with the optimal Hamiltonianto experimental data [177]. 14igure 2: Comparison of the experimental [178] and theoretical results forB(GT) transition strengths for Ca → Sc as a function of the isovectorpairing strength b and the isoscalar pairing strength a , which only acts inthe daughter nucleus.As a is increased the strength to the lowest 1 + state increases, albeitslowly, 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 mainpeak moves up in energy, but with no noticeable change in its strength. Thepresence of a single dominant peak is an indication that these nuclei havegood SU(4) symmetry. Fig. 3 depicts the GT intensities for Ca → Sc . As both nuclei have aneutron excess, we set a = 0 for both and vary the isovector pairing strength b only.In this case, and in contrast to A = 42, the experimental data showsseveral satellite peaks at fairly low energies. For the optimal b = 6 isovectorstrength, the lowest excitation is roughly five times more strongly populatedthan the next few, in contrast with the experimental data where the relativeenhancement is roughly two. The lowest peak moves up in energy as b isincreased and becomes progressively more dominant.Though not part of the A=44 GT decay, it is worth commenting brieflyhere on the N=Z nucleus T i . The spectrum of T i is shown in Figure11 of Appendix B.2 for several choices of the isovector and isocalar pair-15igure 3: Comparison of the experimental [179] and theoretical results forB(GT) transition strengths for Ca → Sc as a function of the isovectorpairing strength b . Since both nuclei involved have neutron excesses, theisoscalar pairing strength is set to a = 0.ing strengths (since it has N=Z both pairing modes are relevant). The firstpoint to note is that like Sc and Ca the lowest states are well describedby the optimal Hamiltonian. In contrast to them, however, the higher an-gular momentum states are too compressed, so that the resulting spectrumis even more removed from that of an SU(3) rotor. We can partially restorethe SU(3) pattern and get a better overall description description of the en-ergy spectrum by increasing the isovector pairing strength to b = 12. Onthe other hand, as we see from Fig. 3, an increase in the isovector pairingstrength would lead to a loss of fragmentation in the corresponding mass-44GT pattern, in worse agreement with the experimental data. Thus there isa competition between the fragmentation produced by the spin-orbit opera-tor [15] and by isovector pairing. Next, we analyze the GT results obtained for the nuclei with mass A = 46.While the parent nucleus involved in the GT decay T i → V has a neutronexcess, the daughter nucleus does not. Thus we assume a = 0 for the parentnucleus and present the results as a function of the a value used in describingthe daughter. In addition, the results are shown as a function of the isovectorpairing strength b used for both nuclei. These results are shown in Fig. 4.The results are interesting. Here there is strong fragmentation of thestrength for the optimal a = b = 6 parameters, though the lowest state hasslightly more strength than the next few. However, the overall strength tothese states is in reasonable accord with what is seen in experiment.Other features of the results worth noting are that (1) even though weproduce an appropriate fragmentation pattern the individual strengths for16igure 4: Comparison of the experimental [158] and theoretical results forB(GT) transition strengths for T i → V as a function of the isovectorpairing strength b and the isoscalar pairing strength a , which only acts inthe daughter nucleus. a = b = 6 are 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 lowest1 + state while lowering its energy and (3) the effect of increasing b is, againas for other masses, to likewise enhance population of the lowest 1 + statebut now while lifting its energy.As in our mass-44 discussion, we also comment here on the rotationalenergy pattern for the even-even nucleus T i . As can be seen from Fig. 1 f,the higher angular momentum states are compressed with respect to experi-ment for the optimal Hamiltonian and would require an increase in isovectorpairing (see Fig. 13 in Appendix B.3) to improve the overall description ofthose 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 theeffect of focussing the GT strength in the lowest peak, thereby also servingto partially restore SU(4) symmetry.Though isoscalar pairing is only of importance to the daughter nucleus V , its increase likewise has the effect of restoring SU(4) symmetry, as isevident from Fig. 4, and to worsen the reproduction of the experimentalresults. 17 .2.4 A=48 Finally, we treat the GT transitions in A = 48. In this case the relevant decayis T i → V , for which both the parent and daughter nuclei have a neutronexcess. Thus, in Fig. 5, where we compare the experimental and calculatedtransition rates, the theoretical analysis is only shown as a function of theisovector pairing strength b .Figure 5: Comparison of the experimental [180] and theoretical results forB(GT) transition strengths for T i → V as a function of the isovectorpairing strength b . Since both nuclei involved have neutron excesses, theisoscalar pairing strength is set to a = 0 for both.The model with the optimal b = 6 isovector strength produces the lowestenergy 1 + excitation in close agreement with experiment. However, the levelof fragmentation in the data is not well reproduced. The strength is con-centrated in a single peak at 2 M eV , which is where the strongest state liesin 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 fragmentationacross an increasingly wider range of states, in worse the agreement withexperiment.In this case, the optimal parameters give acceptable results for the energyspectra of the daughter and parent nuclei (Fig 1g and 1h), neither of whichextends to particularly high angular angular momentum. In the case of Cr ,where the data extends to high angular momentum and whose results areshown in Figure 16 in Appendix B.4, the optimal Hamiltonian gives a gooddescription of the spectrum up to J π = 16 + . The fact that the rotationalpattern of this nucleus is fairly well described without having to dramaticallymodify the pairing strengths reflects the fact that in a system with a largeproduct of the number of valence neutrons and protons N p × N n , the effects ofthe quadrupole-quadrupole interaction are increasingly more dominant andchanges in the pairing strengths have a less important effect.18 Summary and Conclusions
In this work, we first reviewed the role played by Gamow-Teller transitions instellar nucleosynthesis and in double beta decay, and summarized theoreticalresults obtained employing shell model and mean field techniques over thelast half century. The pairing plus quadrupole Hamiltonian was introducedand the role of the isoscalar pairing interaction was discussed.In the second part of this review we explored the effects of proton-neutronpairing on even-mass nuclei in the beginning of 2 p f shell. The analysiswas done in the framework of the nuclear shell model using a parametrizedHamiltonian that contains not only isoscalar and isovector pairing but also aquadrupole-quadrupole interaction that produces background deformation.The first step in this analysis was the choice of the optimal Hamiltonianparameters within our model. This was done by focussing on the parametersthat provided an optimal description of the energies of the lowest 1 + states inthe nuclei of interest, a good description of the other low-lying states of thesenuclei, and an optimal GT fragmentation pattern. A significant outcome ofthis part of the analysis was the realization that for a restricted model Hamil-tonian of the type we used we must turn off isoscalar pairing when dealingwith nuclei having a neutron excess to avoid producing the incorrect groundstate spin and parity for many such nuclei. Our optimal Hamiltonian is ableto achieve good overall fits to experimental data for both energy spectra andGT decay properties, albeit with the limitations inherent in the relativelysimple parametrization we use. Highlights are an accurate reproduction ofthe properties of the lowest 1 + states, and a reasonable description of theGT fragmentation pattern.We then varied the isoscalar and isovector pairing strengths from theiroptimal values to systematically study how the two pairing modes affect firstthe GT properties of these nuclei and subsequently, in the Appendix, theenergy systematics. Our analysis extends from A=42 to A=48.Our analysis of the effect on GT transition properties showed that anincrease 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 loweringits 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 spectrashowed that the isoscalar and isovector pairing modes focus primarily onthe odd-J states and the even-J states, respectively. Enhancing the isoscalar19trength systematically lowers the first 1 + state and expands the set of odd-J states. In contrast, enhancing the isovector pairing strength expands theeven-J part of the spectrum. Acknowledgements
We acknowledge helpful advice of Alfredo Poves on the use of the codeANTOINE and Lei Yang for sharing valuable information. This work re-ceived partial economic support from DGAPA- UNAM projects IN109417and IN104020.
A The original model
As mentioned earlier, the work describe in Sect. 3 took inspiration from themodel introduced in [14], for which the Hamiltonian is (cid:98) H = χ (cid:32) (cid:98) Q · (cid:98) Q + a (cid:98) P † · (cid:98) P + b (cid:98) S † · (cid:98) S + α (cid:88) i (cid:99) −→ l i · (cid:99) −→ s i (cid:33) . (2)For the sake of comparison, we show here the energy spectra that emergedfor 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, areshown in Fig. 6. They are calculated with the same optimal parameters asfor the model discussed in Sect, 3, namely χ = − . M eV , a = b = 6, butwith α = 20 as was used in [14]. Here too in the case of N (cid:54) = Z we must turn off the isoscalar pairing, (ı.e. set a = 0), to produce the correct energyof the lowest 1 + states in those nuclei.Having the full quadrupole-quadrupole interaction, and not just its two-body part as in Eq. (1), provides a better description of the rotationalproperties of these nuclei and in particular the spreading of the higher partof their spectra. On the other hand, no combination of parameters reproducesthe 4 + ground state in V . This is the main reason we have chosen to usethe single-particle energies of the KB3 interaction in the results presented inSect. 3 of the main body of this paper.20igure 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 theHamiltonian (2). 21 Energy Spectra and rotational properties
In this part of the Appendix, we return to the Hamiltonian described in Sect.3 of the main body of this review and show the effect of varying the strengthsof the isoscalar (where appropriate) and isovector terms in the Hamiltonian(1) on the energy spectra of the nuclei under discussion. Some of this materialhas been referred to already in Sect. 3 and in such cases we do not repeatthe discussion.
B.1 A=42
In Fig. 7, results for the energy spectra for the daughter nucleus Sc areshown.Two points should be noted: (1) The best overall reproduction of theexperimental spectrum occurs for equal values of a and b , and(2) the optimal choice is a = b = 6. With these values the lowest 1 + state hasessentially the correct energy and there is a reasonable density of low-lyingstates. The results, though optimal for this Hamiltonian parametrization,still show clear limitations of the model. In particular, it is not possibleto produce a very low-lying 7 + state, as is present in the data. It wouldbe necessary to include other components of the two-body interaction toreproduce this.Another point worthy of note in Fig. 7 concerns the results in the presenceof pure isovector pairing. When the isovector pairing strength b is weakthe 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-degeneratesingle-particle levels it is the interplay of isovector and isoscalar pairing thatproduces the ground state spin in N = Z nuclei, a conclusion that in factcarries through to heavier N = Z nuclei as we will see when we present thoseresults.Next we consider the corresponding results for Ca , the parent nucleusfor GT decay. These results are shown in Fig. 8. We only present the levelsof the ground state rotational band in these figures.Since there are only valence neutrons in Ca isoscalar pairing is omittedfrom the figure. Note that the spectrum does not exhibit a rotational patterneven for b = 0. Clearly the presence of the single-particle energies erases theSU(3) rotational behavior of the purely quadrupole interaction, and in thepresence of isovector pairing the rotational pattern is not recovered. In all22igure 7: Energy spectra of Sc as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results areshown for a, b = 0 ,
6, and 12. 23igure 8: Rotational band of Ca as a function of the strength of the isovec-tor pairing interaction b in comparison with the experimental data.cases, the 2 + , 4 + and 6 + tend to group together, in marked contrast to theexperimental data where only the 4 + and 6 + states are grouped. Neverthelessthe optimal results seem to arise when we choose b = 6, in accord with theconclusion from the Sc analysis. B.2 A=44
The calculated spectra of the daughter Sc are shown in Fig. 9 with theexperimental data..We include isoscalar pairing in the figure as it enables us to demonstrate that isoscalar pairing must be removed when N (cid:54) = Z . Notice that only in thecase a = 0 is the ground state spin 2 + reproduced. Using a = 6 or a = 12the ground state spin and parity are 1 + .The results for the parent nucleus Ca are shown in Fig. 10. The featuresobserved as a function of the isovector strength parameter are similar to thoseseen in Ca : the optimal energy spread arises for b = 6, the spectrum iscompressed as the isovector strength is decreased with the gap between theground state and the first excited 2 + state becoming very small, and the firstexcited 0 + state remains too high in energy for all isovector strengths.Next we consider T i with 2 valence neutrons and 2 valence protons, forwhich the results are shown in Fig. 11. Though not a part of the GT decay, itis worth looking at it nevertheless to study the effect of the different pairingmodes on its rotational properties. Without pairing and single-particle ef-fects the quadrupole-quadrupole interaction would produce a pure rotor withSU(3) symmetry. Single-particle effects by themselves perturb this pattern,24igure 9: Energy spectra of Sc as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results areshown for a, b = 0 ,
6, and 12. 25igure 10: Energy spectra of Ca as a function of the isovector pairingstrength b in comparison with the experimental data.as can be seen from the a = b = 0 results in the figure. Both isovector andisocalar pairing spread the spectrum, in better agreement with experiment,but do not restore the pure rotational character. B.3 A=46
Now we consider the energy spectra of the A = 46 nuclei involved in the GTdecay T i → V .The results for V are shown in Fig. 12. The same basic features asfor lighter N = Z nuclei are seen here. When the intensity of both pairingsdecrease simultaneously, the spectrum energies are reduced. When we reducethe isoscalar pairing alone (note this is an N = Z nucleus so that isoscalarpairing is relevant), there is no appreciable effect on the states with evenangular momentum, whereas those with odd angular momentum graduallycompress. Finally, when decreasing the isovector strength alone, a compres-sion of the overall spectrum occurs and the ground state gradually changesfrom 0 + to 1 + .Next we turn our discussion to T i for which the relevant results for therotational band are shown in Fig. 13. As a nucleus with a neutron excess weonly show the results as a function of b . The main features of these resultswere already described in Section 3.2.3.26igure 11: Rotational band of T i as a function of the isovector b andisoscalar a pairing strengths in comparison with the experimental data. Re-sults are shown for a, b = 0 ,
6, and 12.27igure 12: Energy spectra of V as a function of the isovector b and isoscalar a pairing strengths in comparison with the experimental data. Results areshown for a, b = 0 ,
6, and 12. 28igure 13: Rotational band of T i as a function of the isovector b pairingstrength in comparison with the experimental data. Results are shown for a = 0 , b = 0 ,
6, and 12.
B.4 A=48
Lastly we treat nuclei with A = 48. We first discuss those that participatein the GT decay T i → V and afterwards briefly comment on Cr .The results for V are shown in Fig. 14. We include a (cid:54) = 0 values eventhough this nucleus has a relatively large neutron excess, for reasons thatwill be made clearer soon.The first point to note is that the experimental ground state of this nu-cleus has spin and parity 4 + . The model is able to reproduce the 4 + groundstate 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, con-firming again that it is critical to set a = 0 to suppress erroneous features inthe low-energy spectrum when treating nuclei with a neutron excess.Results for the energy spectra of T i are exhibited in Fig. 15 as a functionof the parameter b . While the model with the optimal value of b = 6 repro-duces the experimental spectrum reasonably well, the agreement is rapidlylost when we increase or decrease b .Finally we show results in Fig. 16 for the states of the ground staterotational band in the N = Z nucleus Cr . These results were alreadydiscussed in Section 3.2.4 in the main body of the paper.29igure 14: Energy spectra of V as a function of the isovector b and isoscalar a pairing strengths in comparison with experimental data. Results are shownfor a, b = 0 ,
6, and 12. 30igure 15: Energy spectra of T i as a function of the isovector b pairingstrength in comparison with the experimental data. Results are shown for a = 0 , b = 0 ,
6, and 12.
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