Quartet structure of $N=Z$ nuclei in a boson formalism: the case of $^{28}$Si

The structure of the $N=Z$ nucleus $^{28}$Si is studied by resorting to an IBM-type formalism with $s$ and $d$ bosons representing isospin $T=0$ and angular momentum $J=0$ and $J=2$ quartets, respectively. $T=0$ quartets are four-body correlated structures formed by two protons and two neutrons. The microscopic nature of the quartet bosons, meant as images of the fermionic quartets, is investigated by making use of a mapping procedure and is supported by the close resemblance between the phenomenological and microscopically derived Hamiltonians. The ground state band and two low-lying side bands, a $\beta$ and a $\gamma$ band, together with all known $E2$ transitions and quadrupole moments associated with these states are well reproduced by the model. An analysis of the potential energy surface places $^{28}$Si, only known case so far, at the critical point of the U(5)-$\overline{\rm SU(3)}$ transition of the IBM structural diagram.


Introduction
The important role played by quartets in N = Z nuclei has been known for a long time [1][2][3][4][5][6]. By quartets we denote here alpha-like four-body correlated structures formed by two protons and two neutrons coupled to total isospin T = 0. Recently, microscopic quartet models have been successfully employed to describe the proton-neutron pairing [7][8][9][10][11][12] as well as general two-body interactions [13][14][15][16] in N = Z nuclei. As a basic outcome, the J = 0 quartet has been found to play a leading role but other low-J quartets have also been found essential to describe the spectra of N = Z nuclei.
The difficulties associated with a microscopic treatment of N = Z nuclei in a formalism of quartets rapidly grow with increasing the number of active nucleons. To make the application of this formalism possible also for large systems, in the present work we propose an approach where elementary bosons replace quartets. Based upon the above fermionic studies, we search for a description of N = Z nuclei in terms of only two building blocks, the T = 0, J = 0 and T = 0, J = 2 quartets. These quartets are therefore represented as elementary s and d bosons, respectively. This bosonic architecture clearly coincides with that of the Interacting Boson Model (IBM) in its simplest version [17]. The application of this model, in terms of quartets, to N = Z nuclei is, however, without precedent. We remark that in the standard IBM framework a proper treatment of even-even N = Z nuclei would imply the use of the much more elaborate IBM-4 version of the model [18] which carries 10 different types of pair bosons. We also notice that an IBM-type approach based on quartet bosons was applied long ago [19], on a purely phenomenological basis, to nuclei with protons and neutrons occupying different major shells, i.e. nuclei which are commonly described by IBM-2 [17].
The manuscript is structured as follows. In Section 2, we illustrate our formalism. In Section 3, this formalism is applied to a description of 28 Si. In Section 4, we discuss the geometric structure of this nucleus. Finally, in Section 5, we give the conclusions.

The formalism
We start by setting the general quartet boson formalism for the treatment of N = Z nuclei. We describe these nuclei in terms of collective T = 0 (J = 0 and 2) quartets that we represent as elementary sd bosons. By denoting the corresponding boson creation operators as b † 0 = s † and b † 2µ = d † µ (µ being the angular momentum projection), the most general one-body plus two-body Hamiltonian takes the standard IBM form To evaluate to what extent the quartet bosons can be associated to microscopic quartets as well as to have an initial guess for the parameters of this Hamiltonian we shall resort to a mapping procedure. Mapping procedures allow to establish a link between spaces of composite and elementary objects and have been largely employed in a microscopic analysis of the IBM [20]. In this work we will follow the general lines of the procedure of Ref. [21] adapted for the quartet case.
We begin by introducing the most general quartet with isospin T = 0 and angular momentum (projection) J(M ) With N such quartets we construct the fermionic quartet space where Q † i ≡ Q † JiMi . To the quartet operator Q † i we associate the boson b † i and, in correspondence with the fermion space F (N ) , we define the boson space where N i1i2...iN is a normalization factor. There is a one-to-one correspondence between the states of F (N ) and B (N ) , the basic difference being that the boson states are orthonormal while the fermion ones are not. In correspondence with a fermion Hamiltonian H F , we define a boson hamiltonian H B such that where |N, i and |N, i) are generic states of F (N ) and B (N ) , respectively, and

The spectrum of 28 Si
We apply the formalism just described to the nucleus 28 Si. 28 Si has 6 protons and 6 neutrons outside the 16 O core. Thus we describe this nucleus in terms of three collective quartets that we represent as elementary sd bosons. The corresponding theoretical spectrum has only 10 states. The angular momenta of the states are such that these can be arranged into a ground state band and two side bands, a β and a γ band. Correspondingly, as experimental spectrum of 28 Si we consider only the ground state band and two low-lying β and γ bands. These β and γ bands have their band heads at 4.98 MeV and 7.42 MeV, respectively. According to Ref. [24], these ground, β and γ bands share a common intrinsic structure, all being classified as "oblate". These experimental bands are shown on the left side of Fig. 1. Some uncertainties are present for the J = 4 state of the β band due to the lack of experimental information.
The state which has been tentatively inserted in Fig. 1 is the J = 4 state at E = 10.67 MeV. It is worth mentioning that the experimental spectrum shown in Fig. 1 is only a part of the complex spectrum of 28 Si, which contains many other bands [24]. .746, (7)=-9.316. Some differences can be seen between microscopically derived and phenomenologically fitted parameters (particularly at point (3)). These differences, which have significant effects on the final spectrum, are expected to reflect a renormalization of the boson parameters which takes into account the lack of J > 2 quartets (whose role has been previously pointed out [13]) as well as the lack of three-body terms in H B . The overall agreement between the two set of parameters of Fig. 2 is, however, such to support the microscopic interpretation of the bosons as images of T = 0 quartets.
The theoretical spectrum of the Hamiltonian (1) with the parameters fitted as discussed above is shown Fig. 1 To evaluate the E2 transitions we have adopted the standard IBM operator   Any IBM-type Hamiltonian has associated with it an intrinsic geometric structure [27,28]. This is defined by means of the coherent state where The variables β and γ identify the intrinsic shape of the nucleus. γ = 0 o corresponds to a prolate deformation while γ = 60 o to an oblate one. The equilibrium shape of the nucleus is defined by the values β 0 , γ 0 which minimize the potential energy surface E(β, γ) = N ; β, γ|H B |N ; β, γ . In Fig. 3 we show E(β, γ) for 28 Si,  Previous studies of the potential energy surface have evidenced that the presence of two shallow coexisting minima tipically occurs at the critical point of a first order phase transition [29]. To illustrate this point we introduce the schematic Hamiltonian [29,30] H (T ) where (2) and χ = + √ 7 2 . In this Hamiltonian the control parameter η allows to move continuously from the spherical U(5) limit to the oblate SU(3) limit. In between these two limits the system undergoes a first order phase transition at η cr = 0.129. This criticality emerges from a discontinuity at η cr in the first derivative of the minimum of the potential energy surface E(β, γ) similarly to what found in the U(5)-SU(3) case [17,31]. In Fig. 4 we show the potential energy surface associated with H In what follows we will explore whether precursors of a phase transition can be observed in a system of only N = 3 bosons, such as 28 Si. To do so it is appropriate to look at observables that are particularly sensitive to the control parameter in the critical region. In Fig. 6 we show two such observables. The first one, shown in the main panel, is the ratio B(E2; 0 + 2 − 2 + 1 )/B(E2; 2 + 1 − 0 + 1 ). The experimental value of this ratio, 0.72(0.05), corresponds to a value of η close to η cr . From Fig. 6 one can notice that this B(E2) ratio strongly depends on the control parameter η and this dependence further grows with increasing N (inset (a)). For N = 10 one observes a precipitous drop in this ratio which marks the occurrence of a phase transition. As it can also be inferred from a glance at the inset (a), η cr significantly decreases with increasing N . However, such  (7) is rescaled by a factor 4N , as often done in the literature [29].
In Fig. 6 we also show (inset (b)) the ratio R 42 = E(4 + 1 )/E(2 + 1 ). For 28 Si the experimental value for this ratio is equal to 2.6 and this too corresponds to a value η close to η cr . As seen in Fig. 6

Conclusions
In this paper we have proposed a quartet description of N = Z nuclei in a formalism of elementary bosons. As an application we have studied 28 Si.
The microscopic foundation of the quartet bosons has been supported by the outcomes of a mapping procedure. An analysis of the potential energy surface has evidenced the peculiar nature of 28 Si by placing it, only known case so far, at the critical point of the U(5)-SU(3) transition of the IBM structural diagram.
Oblate nuclei are rather rare in nature [31] and no sequence of nuclei exhibiting a spherical-oblate transition has ever been observed to our knowledge. This holds true in particular for the small sd shell which hosts 28 Si just in the middle and where, in particular, nuclei differing by one boson quartet from this nucleus exhibit a deformed prolate spectrum on one side ( 24 Mg) and a vibrational-like spectrum on the other side ( 32 S). The case of 28 Si thus appears different from those of other well-established critical nuclei such as 150 Nd, 152 Sm or 154 Gd which lie along isotopic chains exhibiting a spherical-prolate transition [33][34][35]. Although the present analysis has been limited to 28 Si we expect that the quartet boson model will represent an useful tool to describe heavier unstable N = Z nuclei, for which new experimental data are going to be provided by the radioactive beam facilities.        Comparison of spectra generated with the Hamiltonians (7) (left) and (1) (right).