Quartetting in odd-odd self-conjugate nuclei

We provide a description of odd-odd self-conjugate nuclei in the sd shell in a formalism of collective quartets and pairs. Quartets are four-body structures carrying isospin T=0 while pairs can have either T=0 or T=1. Both quartets and pairs are labeled by the angular momentum J and they are chosen so as to describe the lowest states of 20Ne (quartets) and the lowest T=0 and T=1 states of 18F (pairs). We carry out configuration interaction calculations in spaces built by one quartet and one pair for 22Na and by two quartets and one pair for 26Al. The spectra that are generated are in good agreement with the shell model and experimental ones. These calculations confirm the relevance of quartetting in the structure of N=Z nuclei that had already emerged in previous studies of the even-even systems and highlight the role of J>0 quartets in the composition of the odd-odd spectra.


Introduction
A distinctive feature of self-conjugate nuclei is that of carrying an equal number of protons and neutrons distributed over the same single particle levels.
In these nuclei, owing to the charge independence of the nuclear interaction, the isovector proton-neutron (pn) pairing is expected to come into play on equal footing as the like-particle proton-proton and neutron-neutron pairing of the more common N > Z nuclei. In addition, pn pairing is also expected to occur in an isoscalar form. The competition between these two types of pn pairing in N = Z nuclei has been matter of great debate in recent years (for a recent analysis on the subject see Ref. [1]).
In the above context, particular attention has been addressed to odd-odd N = Z nuclei [2,3,4,5]. These nuclei exhibit a peculiar coexistence of isospin T = 0 and T = 1 states at very low excitation energies. This feature is clearly visible in Fig. 1 which shows the experimental positive-parity spectra of the lightest odd-odd N = Z nuclei in the sd shell, namely 18 F, 22 Na and 26 Al.
By focusing on the low-lying states within the circles, one may notice that: a), there is always a T = 1, J = 0 state which coexists with some T = 0 states; b), the energy of this T = 1, J = 0 state decreases with increasing the mass of the nucleus up to becoming almost degenerate with the T = 0 ground state in 26 Al; c), the low-lying T = 0 states form a group of three states with angular momenta J = 1, 3, 5 (with the only exception of 22 Na which exhibits an "intruder" J = 4 state) which one after the other become the ground state of the system. The nuclear shell model represents a powerful tool to study sd shell nuclei (see, for instance, the exhaustive comparison of experimental spectra and theoretical spectra obtained with the USD/A/B interactions in Ref. [6]) and, as it will be seen more in detail in the following, accounts well for all the above features. However, the complexity of the shell model wave function is such not to allow a simple description of these features. Providing such a simple description represents the basic goal of the present paper.
We have recently carried out an analysis of the pn pairing in even-even N = Z nuclei both in the isovector and in the isoscalar channels [7,8,9,10,11,12]. This analysis has evidenced, on the one hand, that a description of the ground state correlations induced by this interaction in terms of a condensate of collective pairs (of various form [12]) is not satisfactory and, on the other hand, that these correlations can be accounted for to a high degree of precision by approximating the ground state as a product of identical T = 0 quartets. T = 0 quartets are four-body correlated structures formed by two protons and two neutrons and, in the case of a spherical mean field, they are also characterized by a total angular momentum J = 0. One needs to remark that quartets have a long history in nuclear structure [13,14,15,16,17,18,19,20,21] but their complexity has undoubtedly represented a hindrance to the development of quartet models.
We have also explored a more sophisticated approximation which consists in letting the quartets to be all distinct and we have verified that it leads to basically exact results in the case of the pn isovector pairing in deformed systems [10]. In all cases the quartets have been constructed variationally for each nucleus.
The pn pairing is a key ingredient of the nuclear force for N = Z nuclei but it is nonetheless only a part of it. In the presence of a full Hamiltonian, other quartets are reasonably expected to come into play besides the T = 0, J = 0 ones emerging from the analysis of the pn pairing. This has been verified in a recent analysis of even-even nuclei in the sd shell which has evidenced a significant role of T = 0, J = 2, 4 quartets in the low-lying states of 24 Mg and 28 Si [22]. The quartets employed in this analysis have not been constructed variationally, as in the quoted works on pn pairing [7,8,9,10,11,12], owing to the difficulty in applying this procedure in the presence of quartets of various nature. T = 0 quartets have been instead simply assumed to represent the lowest states of 20 Ne (two protons and two neutrons outside the 16 O core). Once fixed, these quartets have been no longer modified throughout the calculations. A similar criterion has been adopted also for the T = 1 and T = 2 quartets which have been employed in an analysis of the whole isobaric chain of A = 24 nuclei [22].
These quartets have been associated with the lowest levels of 20 F (T = 1) and 20 O (T = 2). No need for T = 0 quartets in the structure of the even-even N = Z nuclei 24 Mg and 28 Si has been found due to the large gap in energy existing between these and the T = 0 quartets.
The analysis of odd-odd N = Z nuclei in the sd shell that we are going to illustrate is fully inspired to our previous work on even-even N = Z nuclei in the same shell. We assume that an odd-odd nucleus can be described by resorting to two distinct families of building blocks, one formed by collective T = 0 quartets and the other by collective pairs. These latter can have either T = 0 or T = 1 depending on the isospin of the state that we want to represent.
More precisely, we assume that any state with isospin T of an odd-odd nucleus can be represented as a superposition of products of one or more T = 0 quartets and one extra pair with isospin T . Therefore, within this scheme, the total isospin of the state coincides with that of the pair. Based on the conclusions of Ref. [22], we involve in the calculations only three T = 0 quartets, namely the J = 0, 2, 4 quartets describing the lowest three states of 20 Ne. In analogy, as collective pairs with isospin T , we assume those describing the lowest three states with that isospin in 18 F (one proton and one neutron outside the 16 O core). These states are characterized by angular momenta J = 1, 3, 5 for T = 0 (see Fig. 1) and J = 0, 2, 4 for T = 1. Once fixed, these quartets and pairs are no longer modified.
The manuscript is structured as follows. In Section 2, we describe the formalism. In Section 3, we present the results. Finally, in Section 4, we give the conclusions.

The formalism
We work in a spherically symmetric mean field and label the single-particle states by i ≡ {n i , l i , j i }, where the standard notation for the orbital quantum numbers is used. The T = 0 quartet creation operator is defined as where a + i creates a fermion in the single particle state i. No restrictions on the intermediate couplings J 1 T ′ and J 2 T ′ are introduced. Similarly, the pair creation operator is defined as In the above expressions M (T z ) stands for the projection of J(T ). The coefficients q i1j1J1,i2j2J2,T ′ and p ij are fixed by carrying out shell model calculations for 20 Ne (quartets) and for 18 F (pairs). The interaction that is used throughout this paper is the USDB interaction [23]. Once the quartets and the pairs have been fixed, we perform configuration interaction calculations in the space (we adopt the m-scheme) for 22 Na and in the space for 26 Al, with |0 representing the reference vacuum. 2) generates the wrong order of T = 0 states (as compared to the experimental spectrum of Fig. 1 and to the shell model result that will be discussed below); b), adding the J = 2 quartet (the case Q(J=0,2)) results in the appearance of the "intruder" J = 4 state mentioned above but leaves a wrong ordering of the T = 0 states; c), further including the J = 4 quartet (case Q(J=0,2,4)) the spectrum acquires the right structure. In passing from Q(J=0) to Q(J=0,2,4) one can also observe a significant increase of the ground state correlation energy.

Results
This quantity is defined as the difference between the ground state energies that are calculated with and without interaction and it is indicated by the number below each ground state. The number in parenthesis gives the relative error of the correlation energy with respect to the corresponding shell model value.

Conclusions
In this paper we have provided a description of the odd-odd N = Z nu- nuclei that had already been evidenced in our previous studies of the even-even systems [7,8,9,10,11,12,22]. With respect to these previous works, it appears even more clearly the role of T = 0, J > 0 quartets. Their presence has not only significant effects on the ground state correlation energy (as already remarked in Ref. [22]) but it also turns out crucial to generate the proper spectrum of the odd-odd nucleus. An even more direct confirmation of the role played by these quartets could arise from a study of odd N = Z + 1 systems that we leave for future work.