Three-body model calculations for N=Z odd-odd nuclei with T=0 and T=1 pairing correlations

We study the interplay between the isoscalar (T=0) and isovector (T=1) pairing correlations in N=Z odd-odd nuclei from 14N to 58Cu by using three-body model calculations. The strong spin-triplet T=0 pairing correlation dominates in the ground state of 14N, 18F, 30P, and 58Cu with the spin-parity J^{\pi}=1+, which can be well reproduced by the present calculations. The magnetic dipole and Gamow-Teller transitions are found to be strong in 18F and 42Sc as a manifestation of SU(4) symmetry in the spin-isospin space. We also discuss the spin-quadrupole transitions in these nuclei.


I. INTRODUCTION
The pairing correlation is one of the most remarkable effects in nuclear physics. It appears in many properties of nuclei, including odd-even mass staggering, as well as the large energy gap between the first excited and the ground states in even-even nuclei compered to oddeven nuclei. In literature, the spin-singlet T = 1 pairing has been mainly discussed in nuclear physics, since the large spin-orbit splitting prevents to couple a spintriplet (T = 0, S = 1) pair in the ground state [1,2]. Another reason for this is that the large neutron-excess along the stability line of the nuclear chart suppresses the proton-neutron pairing. A recent availability of radioactive beams has opened up an opportunity to measure structure properties of unstable nuclei along the N = Z line, strongly enhancing a possibility to measure new properties of nuclei such as pairing correlations related with the spin-triplet T = 0 pairing. It is thus quite interesting and important to study the competition between the spin-singlet T = 1 and the spin-triplet T = 0 pairing interactions in odd-odd N = Z nuclei and seek an experimental evidence for the competition in the spins of low-lying states. In this paper, we focus our study in sd-and pf -shell nuclei, in which the ground state spins and spin-isospin transitions are observed. In order to study the ground state and the low-lying excited states in odd-odd N = Z nuclei in these mass regions, we apply a three-body model with a density-dependent contact interaction between the valence neutron and proton.
The paper is organized as follows. In Sec. II, we explain the three-body model employed in the present study. In Sec. III, we present the results of the calculations and discuss the ground state properties of oddodd N = Z nuclei. We also discuss the magnetic moments, the magnetic dipole transitions, the isovector spin-quadrupole transitions, and the Gamow-Teller transitions in these nuclei. We summarize the paper in Sec. IV.

II. MODEL
We first describe the model Hamiltonian for N = Z nuclei, assuming the core+ p+n structure [3]. This model is based on the three-body model for describing the properties of Borromean nuclei such as 11 Li and 6 He [4][5][6].
In the rest frame of the three-body system, the model Hamiltonian is given by where m is the nucleon mass and A C is the mass number of the core nucleus. V pC and V nC are the mean field potentials for the valence proton and neutron, respectively, generated by the core nucleus. These are given as where V (N ) and V (C) are the nuclear and the Coulomb parts, respectively. In Eq. (1), V pn is the pairing interaction between the two valence nucleons. For simplicity, we neglect in this paper the recoil kinetic energy of the core nucleus, that is, the last term in Eq. (1). The nuclear part of the core-valence particle interaction, Eq. (2), is taken to be where f (r) is a Fermi function defined by f (r) = 1/(1 + exp[(r − R)/a]). For 18 F nucleus, as in Ref. [3], we set v 0 = −49.21 MeV and v ls = 21.6 MeV·fm 2 . For the other nuclei, we adjust v 0 so as to reproduce the neutron separation energies, while v ls is kept constant for all the nuclei considered in this paper. The radius and the diffuseness parameters are set to be R = 1.27A 1/3 C fm and a = 0.67 fm, respectively. The Coulomb potential V (C) in the proton mean field potential is generated by a uniformly charged sphere of radius R and charge Z C e, where Z C is the atomic number of the core nucleus. We use a contact interaction between the valence neutron and proton, V np , given as [3], whereP s andP t are the projectors onto the spin-singlet and spin-triplet channels, respectively: In each channel in Eq. (4), the first term corresponds to the interaction in vacuum while the second term takes into account the medium effect through the density dependence. Here, the core density is assumed to be a Fermi distribution of the same radius and diffuseness as in the core-valence particle interaction, Eq. (3). The strength parameters, v s and v t , are determined from the proton-neutron scattering length as [5] v where a (s) pn = −23.749 fm and a (t) pn = 5.424 fm [7] are the empirical p-n scattering lengths in the spin-singlet and spin-triplet channels, respectively. k cut is the momentum cut-off introduced in treating the delta function, which is related with the cutoff energy as E cut = 2 k 2 cut /m. The strengths v s and v t determined from the scattering lengths depend on the cutoff energy, E cut , as will be discussed in Sec. III. The three parameters x s , x t , and α in the density-dependent terms in Eq. (4) are determined so as to reproduce energies of the ground (J π = 1 + ), the first excited (J π = 3 + ), and the second excited (J π = 0 + ) states in 18 F with respect to the three-body threashold (See also Ref. [3]). The density ρ(r)/ρ 0 is replaced by a Fermi function f (r) hereafter.
The Hamiltonian (1) is diagonalized in the valence twoparticle model space. The basis states for this are given by a product of proton and neutron single particle states with the single particle energy ǫ (τ ) , which are obtained with the single-particle potential V τ C in Eq. (1) (τ = p or n). To this end, the single-particle continuum states are discreized in a large box. We include only those states satisfying ǫ β ≤ E cut . We use the proton-neutron formalism without antisymmetrization in order to take into account the breaking of the isospin symmetry due to the Coulomb interaction. !"

III. RESULTS
The spin-orbit potential in the mean-field potentials plays a crucial role in determining the properties of T = 0 pairing as discussed in Refs. [1,8,9]. In Fig. 1, we plot the energy differences between the first 0 + and 1 + states and between the first 3 + and 1 + states in 18 F as a function of the spin-orbit coupling strength v ls . We use the cutoff energy of E cut = 20 MeV. It is clearly seen in Fig.  1 that the T = 0 pairing correlations decreases as the spin-orbit interaction increases. That is, the energy difference E 0 + − E 1 + decreases and eventually the spectrum is reversed so that the 0 + state becomes the ground state, where the T = 1 pairing overcomes the T = 0 pairing.
The calculated spectra for 14 N, 18 F, 30 P, 34 Cl, 42 Sc, and 58 Cu nuclei are shown in Fig. 2 together with the experimental data. The spin-parity for the ground state of the nuclei in Fig. 2 are J π = 1 + except for 34 Cl and 42 Sc. This feature is entirely due to the interplay between the isoscalar spin-triplet and the isovector spinsinglet pairing interactions in these N = Z nuclei. In the present calculations, the ratio between the isoscalar and the isovector pairing interactions is v t /v s = 1.9 for the energy cutoff of the model space, E cut = 20 MeV. This ratio is somewhat larger than the value ∼1.6 obtained in Ref. [9] from the shell model matrix elements in p-and sd-shell nuclei. For a larger model space with E cut = 30 MeV, the ratio becomes 1.6, but the agreement between the experimental data and the calculations somewhat worsens quantitatively even though the general feature remains the same. It is remarkable that the energy differences ∆E = E(0 + 1 ) − E(1 + 1 ) are well reproduced in 34 Cl and 42 Sc both qualitatively (the inversion of the 1 + and 0 + states in the ground state) and quantitatively (the absolute value of the energy difference). The model description is somewhat poor in 14 N and 30 P because the cores of these two nuclei are deformed, although the ordering of the two lowest levels are correctly reproduced.
The probability of the total spin S = 0 and S = 1 components for the 0 + and the 1 + states, respectively, are listed in Table I. The total spin S = 0 and S = 1 components in two particle configurations can be calculated with a formula with the 9j symbol and a factorL ≡ √ 2L + 1. For a j π = j ν = j = l + 1/2 configuration, the S = 0 and S = 1 components are given by the factors (j + 1/2)/2j and (j − 1/2)/2j, respectively, for J=0. For a j π = j ν = j = l − 1/2 configuration, on the other hand, they are (j + 1/2)/(2j + 2) and (j + 3/2)/(2j + 2) for S = 0 and S = 1, respectively. Notice that s 2 1/2 configuration has only S = 0 component if J = 0. Otherwise, all the two particle states have a large mixture of the S = 0 and S = 1 components. In general, the S = 1 and S = 0 components are thus largely mixed in the wave functions of both the ground and the excited states. An exception is 30 P. In this nucleus, the dominant configuration in the 0 + state is (2s π 1/2 ⊗ 2s ν 1/2 ), which can couple only to the total spin S = 0. On the other hand, in the 1 + state, the dominant configuration is (2s 1/2 ⊗ 1d 3/2 ) T = 0 which states, ∆E = E(0 + 1 ) − E(1 + 1 ), in N = Z nuclei. The probabilities of the S = 0 component P (S = 0) in the wave functions for the 0 + 1 state are shown in the fourth line. The fifth line shows the probability of the S = 1 component in the 1 + state. The probabilities P (j π ⊗ j ν ) for the dominant valence shell proton-neutron configuration are also given for the 0 + 1 and 1 + 1 states in the 7th and 8th lines, respectively. The experimental data is taken from Ref. [10]. 14  can couple only to the total spin S = 1 with the total angular momentum L = 2. We next discuss the magnetic moment for the 1 + state, and the magnetic dipole transition strength B(M 1) ↓ and the isovector spin-quadrupole transition strength B(IV SQ) ↑ between 0 + 1 and 1 + 1 states. The symbol ↓ (↑) means the transition from the excited (ground) to the ground (excited) states. The magnetic operator is defined as where g s (i) and g l (i) are the spin and the orbital g factors, respectively. The reduced magnetic dipole transition probability is given by where the double bar means the reduced matrix element in the spin space. We take the bare g factors g s (π) = 5.58, g s (ν) = −3.82, g l (π) = 1, and g l (ν) = 0 for the magnetic moment and the magnetic dipole transitions in the unit of the nuclear magneton µ N = e /2mc. The spin-quadrupole transition is defined by The calculated magnetic moments and the magnetic dipole transitions are listed in Table II together with the spin quadrupole transitions. The calculated magnetic moment in 14 N reproduces well the observed one, II: The magnetic moments µ, the magnetic dipole transitions, and the isovector spin quadrupole transitions in the N = Z nuclei. The experimental data of B(M 1) values are taken from Ref. [10], while the data for the magnetic moment are taken from Ref. [11]. The symbol ↓ (↑) means the transition from the excited (ground) to the ground (excited) states.
14 N 18 F 30 P 34 Cl 42 Sc 58 Cu while the agreement is worse in 58 Cu. This is due to the fact that the core of 56 Ni might be largely broken and the f 7/2 −hole configuration is mixed in the ground state of 58 Cu [12]. The values for B(M 1) are also shown in Fig. 2. Very strong B(M 1) values are found both experimentally and theoretically in two of the N = Z nuclei in Table II, that is, in 18 F and 42 Sc. The B(M 1) value from 0 + to 1 + in 18 F is the largest one so far observed in the entire region of nuclear chart. We notice that our three-body calculations provide remarkable agreements not only for these strong transitions in 18 F and 42 Sc but also quenched transitions in the other N = Z nuclei such as in 14 N and 34 Cl.
In the case of 18 F, the 0 + and 1 + states are largely dominated by the S = 0 and S = 1 spin components, respectively, with the orbital angular momentum l = 2 (see Table I). Therefore, the two states can be considered as members of SU(4) multiplet in the spin-isospin space. This is the main reason why the B(M 1) value is so large in this nucleus, since the spin-isospin operator g IV s sτ z connects between two states in the same SU(4) multiplet, that is, the transition is allowed, and the isovector g− factor is the dominant term in Eq.  [13][14][15]. In nuclei 30 P and 58 Cu, the 1 + state is dominated by 1d 3/2 2s 1/2 and 2p 3/2 1f 5/2 configurations, respectively, while the 0 + state is governed by the 2s 2 1/2 and 2p 2 3/2 configurations, respectively. Therefore the isovector spin-quadrupole transi- tions are largely enhanced in the two nuclei even though the B(M 1) is much quenched.
We also calculate the Gamow-Teller (GT) strength where g A is the axial-vector strength, and summarize the results in Table III. One can again see the strong GT transition between the lowest 0 + and 1 + states in A = 18 and 42 systems, which exhaust a large portion of the GT sum rule value. This can also be interpreted as a manifestation of SU(4) symmetry in the wave functions of these nuclei. We note here again that the result obtained in Ref. [13] by an analysis of GT transition also implies a good SU(4) symmetry in the A = 18 system. On the other hand, for 58 Cu, the GT strength is largely fragmented and no strong state in B(GT ) is seen near the ground state. The experimental data are consistent with the calcuted results as can be seen in Table III.

IV. SUMMARY
We have studied the properties of the lowest 0 + and 1 + states in the odd-odd N = Z nuclei in the sd-and pf -shell region with the three-body model with valence proton and neutron and a core. The ratio between the spin-triplet isoscalar and the spin-triplet isovector pairing interactions, v s /v t , is determined to be 1.9 based on the neutron-proton scattering lengths and the energy cut-off of the model space. It was pointed out that the energy ordering of the 0 + and 1 + is very sensitive to the strength of spin-orbit coupling, i.e., the spin-orbit splitting prevents the strong spin-triplet pairing interactions and makes the ground states of 34 Cl and 42 Sc to have J π = 0 + . The energy differences between the lowest 0 + and 1 + states are well reproduced by our model qualitatively (that is, the inversion of the level ordering between the two states) and quantitatively (that is, the excitation energy). It was shown that the calculated wave functions of the lowest 0 + and 1 + states in 18 F and 42 Sc have typical features of the SU(4) multiplets in the spin-isospin space and give the strong magnetic dipole transitions strength between the 0 + and 1 + states. The GT transitions from the neighboring even-even T = 1, T z = 1 nuclei 18 O and 42 Ca with the J π = 0 + to the 1 + states in the odd-odd T = 0 nuclei 18 F and 42 Sc are also shown to be very strong, exhausting a substantial amount of the GT sum rule. The calculated transitions give quantitatively good accounts of the observed strong B(M 1) and B(GT ) values in the two nuclei. In the other N = Z nuclei, B(M 1) transitions are rather hindered, while the spin-quadrupole transitions are found to be rather strong.