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Nuclear Physics A 697 (2002) 655–688www.elsevier.com/locate/npe

Quasi-SU(3) truncation schemefor odd–even and odd–oddsd-shell nuclei

C.E. Vargasa,b,∗, J.G. Hirschb, J.P. Draayerca Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740

México 07000 DF, Mexicob Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543

México 04510 DF, Mexicoc Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Received 8 December 2000; revised 24 July 2001; accepted 1 August 2001

Abstract

The quasi-SU(3) symmetry, as found in shell-model calculations, refers to the dominance of thesingle-particle plus quadrupole–quadrupole terms in the Hamiltonian used to describe well-deformednuclei, and to the subspace relevant in its diagonalization. It provides a very efficient basis truncationscheme. It is shown that a small number ofSU(3) coupled irreps, those with the largestC2 valueswithin the direct product of the proton and neutronSU(3) irreps with spin 0 and 1 (for even numberof particles), and spin 1/2 and 3/2 for (for odd number of nucleons), are enough to describe the low-energy spectra,B(E2) transition strengths and band structure of21Ne,23Na and25Mg odd-mass and22Na,24Na,26Al and 28Al odd–odd nuclei. A simple but realistic Hamiltonian is employed. Resultscompare favorably both with experimental data and with full shell-model calculations. Limitationsand possible improvements of the schematic Hamiltonian are discussed. 2002 Elsevier ScienceB.V. All rights reserved.

PACS: 21.60.Fw; 21.60.Cs; 27.30.+t

1. Introduction

Since its introduction more than fifty years ago [1], the shell model has beena fundamental tool in the microscopic description of nuclear properties. Full shell-modelcalculations in thesd- [2] andfp-shells [3] provide very accurate predictions for energylevels, electromagnetic transition strengths and weak decay half lives. State of the art codesallow studies up toA = 50 [4]. Monte Carlo shell-model diagonalization is an original andpowerful tool which opens the possibility to study larger systems [5].

* Fellow of CONACyT. Corresponding author.E-mail address: [email protected] (C.E. Vargas).

0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9474(01)01261-1

656 C.E. Vargas et al. / Nuclear Physics A 697 (2002) 655–688

The system symmetries, although in general are not exact, provide a natural truncationscheme of the Hilbert space while keeping the predictive power of the theory, and supporta qualitative understanding of collective nuclear modes. Algebraic models are particularlywell suited to describe systems with symmetries, using either bosonic representations [6],fermionic representations [7,8], or combination of both [9].

In the present article we concentrate our attention in the fermionicSU(3) algebraicmodel developed by Elliott [7]. Based on the crucial role the quadrupole–quadrupoleinteraction plays in deformed systems, theSU(3) algebra is the natural language to describequadrupole excitations in a harmonic oscillator basis in light nuclei. In its present version,the nuclearSU(3) model is built using a truncation scheme that consists of the selection ofthe states withS = 0 and 1 for even number of protons or neutrons,S = 1/2 and 3/2 forodd number of protons or neutrons, and a very schematic but still realistic Hamiltonian.Such model has been called “quasi-SU(3) truncation scheme” [3,10,11], and it has thesame algebraic structure as the proposed by Elliott in 1958 [7]; the extra ingredients beingthe use of schematicSU(3) symmetry breaking Hamiltonian and the states withS = 0,1or 1/2,3/2.

In heavy deformed nuclei, a similar role is played by the pseudo-SU(3) model [8],a scheme built over the pseudo-spin symmetry. Used as an approximate symmetry, i.e.allowing the mixing of different irreps through the single-particle energies and the pairinginteraction, it provides a very good description of low-energy bands,B(E2) intra [12] andinterband [13] andB(M1) transition strengths in the rare earth region, both for even–even[14] and odd-A [15] deformed nuclei.

In the pseudo-SU(3) model the truncation of the Hilbert space usually excludes intruderorbits, which are known to provide an important contribution to the total quadrupolemoment. The quasi-SU(3) truncation scheme [3] offers a simple and consistent way toinclude intruder orbits in theSU(3) description of heavy deformed nuclei. In [10] it wasshown that including the leadingSU(3) irreps (those with the largestC2 values) and spin 0and 1, the interplay between the quadrupole–quadrupole interaction and the spin–orbitsplitting is well described. A detailed description of four even–even nuclei in thesd-shell,ranging from20Ne to 28Si was presented in [11]. The present article deals with the odd-mass and odd–odd nuclei in the same shell. The main goal is to prove that in well deformednuclei, even in the presence of a large spin–orbit splitting, it is possible to make a gooddescription of the low-energy spectra using a small number ofSU(3) irreps. This is themeaning of the “quasi-SU(3) truncation scheme”. The results presented here and in theprevious contribution [11] strongly support this conclusion. They open the possibility fora coherent description of nucleons in intruder levels in heavy deformed nuclei using theSU(3) formalism, as envisioned in [3].

In the present paper, the low-energy spectra andB(E2) transition strengths of21Ne,23Naand25Mg, and22Na, 24Na, 26Al and 28Al are studied using the quasi-SU(3) basis and aschematic Hamiltonian. The band structure is recovered and the wave functions along eachband are presented. Results are compared both with the experimental data and with fullshell-model calculations [2]. Section 2 reviews the essentials of theSU(3)model, includinga short description of theSU(3) basis and the Hamiltonian. In Sections 3 and 4, the energy

C.E. Vargas et al. / Nuclear Physics A 697 (2002) 655–688 657

spectra, band structure andB(E2) transition strengths of odd-mass and odd–odd nuclei,respectively, are presented together with the discussion of each case. Section 5 containsthe conclusions.

2. The SU(3) model

For a general review of theSU(3) model and the operator expansion in terms ofSU(3)tensors we refer the reader to Refs. [7,16–19], as well as to the previous article whereeven–even light nuclei were described using the same scheme [11]. In what follows weintroduce theSU(3) basis and the schematic Hamiltonian used in this work.

The basis states are written as

|βJM〉 = ∣∣{nπ [fπ ]απ(λπ ,µπ),nν[fν]αν(λν,µν)}ρ(λ,µ)KL{Sπ ,Sν}S;JM

〉, (1)

wherenπ is the number of valence protons in thesd-shell and[fπ ] is the irrep of theU(2)spin group for protons, which is associated with the spinSπ = (f 1π − f 2π )/2. TheSU(3)irrep for protons is(λπ ,µπ) with a multiplicity labelαπ associated with the reductionfrom U(6). Similar definitions hold for the neutrons, labeled withν. There are other twomultiplicity labels:ρ, which counts how many times the total irrep(λ,µ) occurs in thedirect product(λπ ,µπ )⊗ (λν,µν) andK, which classifies the different occurrences of theorbital angular momentumL in (λ,µ).

The vector states (1) span the complete shell-model space within only one active(harmonic oscillator) shell for each kind of nucleon. As an example we take21Ne. It hastwo protons (nπ = 2) in thesd-shell, which can be accommodated in three possible irreps:(λπ ,µπ) = (4,0), (2,1) and(0,2). The first and third irreps have spin zero, the secondone has spin 1. Each one occurs only once (απ = 1). For three neutrons in thesd-shellthere are five irreps:(λν,µν) = (4,1), (2,2), (3,0), (0,3) and(1,1). Three of them havespin 1/2, the other two have spin 3/2. TheSU(3) irreps are ordered by decreasing valuesof the expectation value of the second-order Casimir operator,C2:

〈(λ,µ)

∣∣C2∣∣(λ,µ)

〉 = (λ+µ+ 3) (λ+µ)− λµ. (2)Irreps for 2, 3, 4, 5, and 7 particles (proton or neutrons) in thesd-shell are listed in

Table 1 with their spin.Calculations performed using the full proton–neutron coupledSU(3) Hilbert space in

thesd-shell [10,11] show that the quasi-SU(3) [3] truncation scheme is quite efficient. TheSU(3) basis is built by taking the direct product of the proton and neutron irreps with thelargestC2 values andS = 0 and 1 (for even number of nucleons) or 1/2 and 3/2 (for oddnumber), and keeping from this list only those states with the largest totalC2 values [10,11,20]. The truncated list of their final couplings, which describes the Hilbert space usedfor each nuclei, can be seen in Tables 3, 5, 7, 9, 11, 12 and 15.

The basis states are labeled|JiM〉, whereJ is the total angular momentum,M itsprojection andi is an integer which enumerates the states with the sameJ , starting fromthe one with the lowest energy. They are built as linear combinations,


Table 1Irreps and spins for 2, 3, 4, 5 and 7 particles (protons or neutrons) in thesd-shell

# S Irreps

2 0 (4, 0), (0, 2)1 (2, 1)

3 1/2 (4, 1), (2, 2), (1, 1)3/2 (3, 0), (0, 3)

4 0 (4, 2), (0, 4), (3, 1), (2, 0)1 (5, 0), (2, 3), (3, 1), (1, 2), (0, 1)2 (1, 2)

5 1/2 (5, 1), (2, 4), (3, 2), (4, 0), (1, 3), (2, 1), (0, 2)3/2 (3, 2), (1, 3), (2, 1), (1, 0)5/2 (0, 2)

7 1/2 (1, 5), (4, 2), (2, 3), (0, 4), (3, 1), (1, 2), (2, 0)3/2 (2, 3), (3, 1), (1, 2), (0, 1)5/2 (2, 0)7/2 (0, 0)

The number of particles is shown in the first column, the spin in the second columnand theSU(3) irreps in the third.

|JiM〉 =∑

β

CJiMβ |βJM〉, (3)

of the strong coupled proton–neutron states|βJM〉 defined in Eq. (1) [12].The Hamiltonian used is

H = Hsp,π +Hsp,ν − 12χQ ·Q−GπHpair,π −GνHpair,ν + aK2J + bJ 2+AsymC2, (4)

whereHsp,α is the spherical Nilsson Hamiltonian forα = π or ν, and the quadrupole–quadrupole and pairing interaction strengthsχ , Gπ and Gν have been fixed fromsystematics as in previous works [11,15]:

χ = 17A5/3

, Gπ =Gν = 9.5A

. (5)

The parametersa, b, andAsym correspond to the three “rotor terms”. They are small, andprovide some freedom to perform a nuclei by nuclei best fit. The Hamiltonian parametersused in this work are listed in Table 2.

For the spherical Nilsson Hamiltonian

Hsp,α = h̄ωo{(η̂ + 32

) − 2κ�l · �s − κµ�l 2}, (6)the Ring and Schuck [21] parameterizationκ = 0.08 andµ= 0.00 was chosen. While boththe single-particle Hamiltonian and the pairing terms induce a mixing ofSU(3) irreps, forthe values used here the single-particle terms are by far and away the driving force behindthe mixing.


Table 2Hamiltonian parameters in MeV

Nucleus χ Gπ = Gν a b Asym21Ne 0.106 0.452 −0.200 −0.020 0.01023Na 0.091 0.413 0 −0.005 025Mg 0.079 0.380 −0.130 0.090 0.02422Na 0.098 0.432 −0.100 −0.025 024Na 0.085 0.396 0.100 −0.102 0.04026Al 0.074 0.365 0.300 −0.091 028Al 0.066 0.339 −0.080 0.051 0.025

Hamiltonian (4) proved to be very powerful in the description of the normal-parity bandsin heavy deformed odd-A nuclei [15], where protons and neutrons occupy different majorshells. In the description of light nuclei, the same Hamiltonian is missing the proton–neutron pairing term, and for this reason it is not isoscalar. Given that the main goal ofthe present work is to assert the validity of the quasi-SU(3) truncation scheme, we havekept this Hamiltonian, using its rotor terms to partially compensate for the missing terms.Further applications of this model to light- and medium-mass nuclei must use an isospin-invariant Hamiltonian if the model is intended to display its full predictive power.

The electric quadrupole operator is expressed as [22]

Qµ = eπQπ + eνQν, (7)whereeπ and eν are the effective charges for protons and neutrons, respectively. Theeffective charges used are indicated in each section (the current values used in this workare in all cases very close toeπ eff = 1.5e andeν eff = 0.5e, wheree is the charge of theproton).

In the following sections a detailed analysis of the odd-mass (Section 3) and odd–odd(Section 4) nuclei is presented. The energy spectra, band structure andB(E2) transitionstrengths are shown for each nuclei.

3. Odd-mass sd-shell nuclei

3.1. 21Ne

21Ne has two protons and three neutrons occupying thesd valence shell. As discussedin the previous section, they can be in any of the threeSU(3) proton irreps and fiveSU(3)neutron irreps listed in Table 1, which include proton states with spin 0 and 1, and neutronstates with spin 1/2 and 3/2. The truncated proton–neutron coupled basis is listed inTable 3. It includes the 17 coupled irreps with the largestC2 values.

The 21Ne energy spectra obtained with this basis, using the Hamiltonian parametersshown in the second row of Table 2, are presented in the left-hand side column of Fig. 1.


Table 3The 17 irreps used in description of21Ne

(λπ ,µπ )Sπ (λν,µν)Sν (λ,µ)S total

(4,0)0 (4,1)1/2 (8,1)1/2 (6,2)1/2 (7,0)1/2 (4,3)1/2(4,0)0 (2,2)1/2 (6,2)1/2 (4,3)1/2(2,1)1 (4,1)1/2 (6,2)1/2, 3/2 (7,0)1/2, 3/2 (4,3)1/2, 3/2(4,0)0 (3,0)3/2 (7,0)3/2(4,0)0 (0,3)3/2 (4,3)3/2(2,1)1 (2,2)1/2 (4,3)1/2, 3/2(0,2)0 (4,1)1/2 (4,3)1/2

In some couplings the total irrep can have spins of 1/2 and 3/2.

Fig. 1. Energy spectrum for21Ne. The first column displays the present results, the second showsexperimental data, and the third column displays full shell-model results [24] with a unifiedsdHamiltonian [2].


Fig. 2. Band structure in21Ne.

They are compared with the experimental results [23], shown in the second column, andwith the theoretical results obtained with full shell-model calculations using the unifiedsd-shell Hamiltonian [24] (third column). We also show the band structure in Fig. 2 andtheir wave function components in Fig. 3.

It can be seen from Fig. 1 that the low-lying energy spectra of21Ne is well describedusing the truncated quasi-SU(3) space, which includes only the 17 coupled irreps with thelargestC2 values, out of the 137 coupled irreps (some of them withρ multiplicity largerthan 1) which form the fullSU(3) basis.

Despite the very schematic form of Hamiltonian (4) and the missing proton–neutronterms mentioned above, the rotor terms allow a fine tuning of the spectra. The momentof inertia is slightly increased by selectingb = −0.02. It corrects the quadrupole momentof inertia I , where 1/(2I) = (3/2)χ ≈ 0.16. Some excited states are moved to higherenergies by usinga = −0.200. UsingAsym = 0.010 for the symmetry term enhances thecontributions of the irreps withλ andµ even relative to the others, because they belong todifferent symmetry types of the intrinsic VierergruppeD2 [25].

It is worth mentioning that, as it was the case in even–even nuclei [11], the main featuresof the spectra can be obtained by setting all the rotor parameters to zero. On the otherhand, if these rotor parameters are too large (in absolute value), the whole band structureis destroyed, and with it the agreement with the observedB(E2) values (see below).

The present model predicts the energies of the ground-state band, and the states1/21, 5/22, 5/23, 11/23, 9/23−5, 7/25, and 1/22 pretty close to their experimentalcounterparts. The ground-state band staggering effect forces the states to be clustered bypairs:(3/2, 5/2), (7/2,9/2), (11/2, 13/2) (see also Fig. 2). It is slightly exaggerated inthe last case. The levels 3/22, 7/22, 9/22, 7/24, and 11/22 are predicted at energies around2 MeV lower than observed. This is a clear limitation of the model, which can be related tothe limited and schematic nature of Hamiltonian. Further investigations including proton–neutron pairing would clarify this point. In general, the results reported here representa clear improvement from previousSU(3) based descriptions [26].

In the evaluation ofB(E2) transition strengths for the three odd-mass nuclei (21Ne,23Na and25Mg) the effective chargeseπ eff = 1.56e, eνeff = 0.56e were used in the electric


Fig. 3. Wave function components of states belonging to (a) the ground-state band, (b) theK = 1/2band and (c) theK = 7/2 band in21Ne. The percentage 100× |CJiMβ |2 each irrep contributes isshown as function of the angular momentum. The convention used is✸ for (8,1)1/2[(4,0)0 ⊗(4,1)1/2], + for (4,3)3/2[(2,1)1 ⊗ (2,2)1/2], ✷ for (7,0)3/2[(4,0)0 ⊗ (3,0)3/2], × for(6,2)1/2[(4,0)0 ⊗ (2,2)1/2], � for (6,2)3/2[(2,1)1 ⊗ (4,1)1/2] and + for (4,3)1/2[(2,1)1 ⊗(2,2)1/2].

quadrupole operator (see Eq. (7)).B(E2) values calculated with the present model, thoseobtained by the shell model [24] and experimental [23] ones for21Ne are shown in Table 4.The agreement with both the experimental and shell-model values is in general very good.All but the last transition connect states belonging to the ground-state band. The factor 5deviation found in the 5/2+1 → 7/2+4 transition seems to reflect effects of the truncation ofthe Hilbert space.

The21Ne band structure of this nuclei is presented in Fig. 2. Results for the ground-stateandK = 1/2 bands are very similar to those found in previous studies [24]. The staggeringin the ground-state band noted above is clearly seen.


Table 4B(E2) transition strengths for21Ne in [e2 b2 × 10−2]

B(E2;Ji → Jf) ↑ Expt. SU(3) SM [24]

3/2+1 →5/2+1 1.239± 0.155 1.215 1.1265/2+1 →7/2+1 0.505± 0.183 0.710 0.7387/2+1 →9/2+1 0.387± 0.215 0.348 0.3919/2+1 →11/2+1 0.248± 0.165 0.239 0.247

11/2+1 →13/2+1 0.187 0.13613/2+1 →15/2+1 0.129 0.07815/2+1 →17/2+1 0.077 0.07717/2+1 →19/2+1 0.040 0.0343/2+1 →7/2+1 0.675± 0.151 0.688 0.6335/2+1 →9/2+1 0.906± 0.097 0.771 0.7287/2+1 →11/2+1 0.618 0.7079/2+1 →13/2+1 0.540 0.564

11/2+1 →15/2+1 0.413 0.42213/2+1 →17/2+1 0.264 0.31815/2+1 →19/2+1 0.182 0.2155/2+1 →7/2+4 0.011 0.002

TheSU(3) wave function components of each band are shown in Fig. 3. The percentagedifferent SU(3) irreps contribute to each state (100× |CJiMβ |2, see Eq. (3)) is plotted asa function of the angular momentum of the members of the band. Different irreps arerepresented by the different lines and symbols listed in the figure caption. All irreps whichcontributes more than 2% are plotted. The bands are recognized by their largeB(E2)transition strengths. The slow (adiabatic) change in theirSU(3) content helps to confirmthe band assignment.

Fig. 3a shows the dominance of the irrep(8,1) with spin 1/2 in the ground-state bandup to J = 9/2, in agreement with previous studies [26,27]. AtJ = 11/2 there is a clearchange in the wave function, which forJ = 13/2, 15/2 is dominated by the(6,2) irrepwith spin 3/2. Irreps with spinS = 3/2 were not included in Ref. [26]. In the presentcontribution we are showing for the first time its relevance in the low-energy spectra. Thisis the main new feature in the present quasi-SU(3) truncation scheme. In Ref. [27] it wasfound that even in presence of strongSU(3) breaking interactions, like the one proposed byPreedom and Wildenthal, there is a clear dominance of the(8,1)1/2 irrep for the groundstate, in complete agreement with the present results. In the other two bands, insert (b) forK = 1/2 and (c) forK = 7/2 in Fig. 3, similar features can be found. While theK = 1/2band exhibits strong mixing between the(8,1)1/2 and(6,2)1/2 irreps, theK = 7/2 bandis dominated by the(6,2)3/2 band, underlining once again the crucial role played byspin 3/2 irreps.


3.2. 23Na

23Na has 11 protons and 12 neutrons. The valence space contains 3 protons and4 neutrons in thesd-shell, allowing for the five protonSU(3) irreps and ten neutronSU(3) irreps listed in Table 1. From them, only three proton irreps and five neutron irrepscontribute to the truncated basis, which include the 20 coupled irreps with spinS = 1/2,3/2 listed in Table 5. The complete space contains 670 coupled irreps plus their externalmultiplicities. Notice that most irreps can have both spins, due to the 1⊗ 1/2 coupling.There are also two(7,2)1/2 irreps, one coming from the(4,1)1/2⊗ (3,1)0 coupling, andthe other from the(4,1)1/2⊗ (3,1)1 coupling.

The left-hand side of Fig. 4 shows the results obtained for the low-energy spectra of23Na, calculated using the Hamiltonian parameters listed in the third row of Table 2,and the Hilbert space described above. This is compared with the experimental data [23],presented in the second column, and with full shell-model calculations performed by theGlasgow group [28] (third column). The ground state, as well as some excited states arewell described using the quasi-SU(3) truncation scheme. On the other hand, as can be seenin Fig. 4, the model fails to reproduce the energy of the states 7/22, 5/23, 3/23, 3/24, andothers, thereby exhibiting the limits of the model. The energy of these states is in generalunderestimated, probably due to the limited Hamiltonian used.

The band structure, reconstructed from the largestB(E2) calculated values, is shownin Fig. 5. Comparing Figs. 4 and 5 it is noticeable that the ground-state band is welldescribed, while the excitedK = 1/2 andK = 7/2 bands are displaced to lower energiesthan in the observed spectrum. Adjusting the Hamiltonian parameters did not prove to beuseful for correcting this feature, pointing again to the need of including other terms in theHamiltonian.

The negative parameterb = −0.005 provides a small correction to the moment of inertia,which is similar to what was found for21Ne. The other two rotor parameters were foundto be useless in this nuclei. Both tend to wash out the staggering in the ground-state band,and also destroy the band structure. For these reasons they were fixed at zero.

It is remarkable that many states are well described, and that in most cases the smalldeviations with respect to the experimental values show the same trend as the shell-model

Table 5The 20 irreps used in the description of23Na


(4,1)1/2 (4,2)0 (8,3)1/2 (9,1)1/2 (6,4)1/2 (7,2)1/2(4,1)1/2 (5,0)1 (9,1)1/2, 3/2 (7,2)1/2, 3/2(4,1)1/2 (2,3)1 (6,4)1/2, 3/2 (7,2)1/2, 3/2(2,2)1/2 (4,2)0 (6,4)1/2 (7,2)1/2(4,1)1/2 (3,1)0,1 (7,2)1/22, 3/2(2,2)1/2 (5,0)1 (7,2)1/2, 3/2(3,0)3/2 (4,2)0 (7,2)3/2


Fig. 4. Energy spectrum of23Na, with the same convention of Fig. 1. The SM data are from Glasgowgroup [28].

Fig. 5. Band structure in23Na.


Fig. 6. Wave function components of states belonging to (a) the ground-state band, (b) theK = 1/2band, (c) theK = 7/2 band and (d) theK = 5/2 band in23Na. The percentage 100× |CJiMβ |2associated with each irrep is shown as function of the angular momentum. The convention used is✷for (8,3)1/2[(4,1)1/2 ⊗ (4,2)0], + for (9,1)3/2[(4,1)1/2 ⊗ (5,0)1], ✸ for (9,1)1/2[(4,1)1/2 ⊗(4,2)0], × for (7,2)3/2[(4,1)1/2 ⊗ (3,1)1], � for (6,4)1/2[(2,2)1/2 ⊗ (4,2)0] and + for(6,4)3/2[(4,1)1/2 ⊗ (2,3)1].


results. Some small energy displacements from the observed data in the ground-state bandare similar to those found in previousSU(3) studies [29].

A study was also performed using the same Hamiltonian but increasing the basis for20 to 302 irreps, around 40% of the full space. The energy spectrum showed almost nochanges. This is a very strong argument in favor of the quasi-SU(3) truncation scheme. Themost sizable change happened for the 5/21 state, whose energy was reduced by 200 keV,and thereby approaching the experimental value.

Fig. 6 shows the wave function decomposition of the states belonging to (a) the ground-state band, (b) theK = 1/2 band, (c) theK = 7/2 band and (d) theK = 5/2 band in23Na. For each band the percentage (100×|CJiMβ |2) associated with differentSU(3) irrepsis shown as function of the angular momentum. The ground-state band is dominated bythe irrep (8,3)1/2 up to J = 13/2. For larger angular momentum the irrep(9,1)3/2has the largest component. TheK = 1/2 band has a similar structure, but for largeangular momentum there is competition between the(9,1)3/2 and (6,4)3/2 irreps toreplace the(8,3)1/2, resulting in a strong mixing. TheK = 7/2 band is dominatedby the (8,3)1/2 irrep, while theK = 5/2 band is by large made of the(9,1)3/2irrep.

It must be stressed again that the inclusion of the(9,1)3/2 and other spin 3/2 irreps isone of the most relevant features of the quasi-SU(3) truncation scheme, which accountsfor most of its present success, as compared with previousSU(3) studies [29].

B(E2) transition strengths are presented in Table 6. They are compared with theexperimental values, available for some transitions between states belonging to theground-state band and to the excitedK = 1/2 band. The agreement is in general very

Table 6B(E2) transitions for23Na in [e2 b2 × 10−2]

B(E2;Ji → Jf) ↑ Expt. SU(3)

3/2+1 →5/2+1 1.573± 0.233 1.6775/2+1 →7/2+1 0.777± 0.207 1.1235/2+1 →9/2+1 1.359± 0.194 1.0313/2+1 →7/2+1 0.987± 0.085 1.0151/2+1 →5/2+1 0.124± 0.023 0.0021/2+1 →5/2+2 2.681± 0.583 2.1037/2+1 →9/2+1 0.4477/2+1 →11/2+1 1.2169/2+1 →11/2+1 0.5129/2+1 →13/2+1 0.888

11/2+1 →13/2+1 0.15711/2+1 →15/2+1 1.05513/2+1 →15/2+1 0.32213/2+1 →17/2+1 0.67115/2+1 →19/2+1 0.800


good. The calculatedB(E2;1/21 → 5/22) value is within the experimental error, whileB(E2;1/21 → 5/21) is two orders of magnitude smaller than the experimental one.

3.3. 25Mg

25Mg has 4 protons and 5 neutrons in thesd-valence shell. The ten proton and 12 neutronirreps in which they can be accommodated are listed in Table 1. From the 1821 proton–neutron coupledSU(3) irreps we selected the 16 with the largestC2 values to build thetruncated Hilbert space. They are shown in Table 7, together with their spin. As in the

Table 7The 16 irreps used in description of25Mg


(4,2)0 (5,1)1/2 (9,3)1/2 (10,1)1/2 (7,4)1/2(4,2)0 (2,4)1/2 (6,6)1/2 (7,4)1/2 (4,7)1/2(5,0)1 (5,1)1/2 (10,1)1/2, 3/2(4,2)0 (3,2)1/2, 3/2 (7,4)1/2, 3/2(5,0)1 (2,4)1/2 (7,4)1/2, 3/2(2,3)1 (5,1)1/2 (7,4)1/2, 3/2(2,3)1 (2,4)1/2 (4,7)1/2, 3/2

Fig. 7. Energy spectra of25Mg. The first column shows theSU(3) results, the second theexperimental levels, and the third the results of full shell-model calculations [30].


Fig. 8. Band structure of25Mg.

previous cases, some coupled irreps can have both spin 1/2 and 3/2. Only the three protonSU(3) irreps and the four neutronSU(3) irreps with the largestC2 values were included inthe truncated basis.

The low-energy spectra of25Mg is depicted in Fig. 7. The first column shows thepredicted levels, in the second one the experimental values [23] and in third one thosefound in full shell-model calculations [30]. Its band structure is shown in Fig. 8 and thepercentage each irrep contributes to the wave functions of the ground-state,K = 1/2, 1/22,9/2, 13/2 and 11/2 bands is presented in Fig. 9. The Hamiltonian parameters used in thiscalculation are shown in the fourth row of Table 2.

The energy levels shown in Fig. 7 exhibit a complicated pattern. To identify the bandstructure depicted in Fig. 8, both theB(E2) values and the wave function decompositionwere employed. This band structure is fully consistent with the one reported in [31]. Whilegood agreement has been found for a number of states, others, like 9/21, 1/22, 11/21,and 13/21, show a deviation from the experimental and shell-model values, reflectinga limitation of the present model, probably due both to the Hamiltonian used and thetruncation of the Hilbert space.

The description of the25Mg energy levels and its wave functions in terms ofSU(3) irrepsreported here is very similar to the one described in the original work of Draayer [32],despite the schematic interaction and the smaller basis used in the present work. It stronglysupports the reliability of the quasi-SU(3) truncation scheme.

The general structure of the energy spectra in Fig. 7 reproduces some of the observedlevels. The limitations of the schematic Hamiltonian used can be gauged by the “bandshifts” [31] of three excited bands: those withK = 1/22, 9/2 and 13/2. The effect of otherterms in the Hamiltonian, like proton–neutron pairing, in shifting these bands in the rightdirection will be the subject of future research.

In view of the limitations of Hamiltonian (4), a very specific selection of rotor termswas needed to get a 5/2 ground state. The parameter set shown in Table 2 reflects a subtlebalance between theK2, J 2 and symmetry terms. The negativea value pushes bands with


Fig. 9. Wave function components of states belonging to (a) the ground-state band, (b) theK = 1/2 band, (c) theK = 1/22 band, (d) theK = 9/2 band, (e) theK = 13/2 band and (f)the K = 11/2 band in25Mg. The percentage 100× |CJiMβ |2 each irrep contributes is shown asfunction of the angular momentum. The convention used is✸ for (6,6)1/2[(4,2)0 ⊗ (2,4)1/2],+ for (4,7)3/2[(2,3)1⊗ (2,4)1/2], × for (9,3)1/2[(4,2)0⊗ (5,1)1/2], � for (10,1)3/2[(5,0)1⊗(5,1)1/2], + for (10,1)1/2[(4,2)0 ⊗ (5,1)1] and✷ for (7,4)3/2{[(2,3)1 ⊗ (5,1)1/2], [(5,0)1 ⊗(2,4)1/2], [(4,2)0 ⊗ (3,2)3/2]}.


Table 8B(E2) transitions for25Mg in [e2 b2 × 10−2]

B(E2;Ji → Jf) ↑ Exp. SU(3) [32]

1/2+1 →5/2+1 0.024± 0.001 0.398 0.5303/2+1 →5/2+1 0.033 0.276 0.1801/2+1 →3/2+1 0.868± 0.434 1.249 1.4405/2+1 →7/2+1 1.621± 0.289 1.968 1.8133/2+1 →5/2+2 0.202± 0.098 0.835 0.5101/2+1 →5/2+2 2.345± 1.042 2.323 2.3101/2+2 →5/2+1 0.156± 0.074 0.020 0.0443/2+1 →7/2+2 2.084± 0.261 1.488 1.5405/2+1 →7/2+2 0.010 0.001 0.0013/2+2 →5/2+1 0.074 0.002 0.0047/2+1 →9/2+1 0.814± 0.271 0.081∗ 1.3625/2+1 →9/2+1 0.550± 0.043 0.432 0.4007/2+1 →9/2+2 0.054± 0.038 1.420∗5/2+1 →9/2+2 0.097± 0.006 0.5455/2+2 →9/2+3 2.171± 0.362 2.0667/2+2 →11/2+1 0.847± 0.261 0.0017/2+1 →11/2+1 0.332± 0.078 0.3729/2+1 →13/2+1 0.128± 0.061 0.279

The second column shows the experimental values, the third column the present theoretical results,and fourth those obtained previously with theSU(3) model [32]. Stars denote transitions with unclearidentification in the theory.

largerK down in energy, while the positiveb decreases the moment of inertia, increasingthe level separation inside each band, and moving the band heads with largerJ to higherenergies. SettingAsym= 0.024 helps to make theJ = 5/21 state ground.

Table 8 lists theB(E2) transitions for25Mg, calculated with the same effective chargeseπ eff = 1.56e, eν eff = 0.56e as used in the previous cases. The second column showsthe experimental values, the third column the present theoretical results, and fourththose obtained previously using theSU(3) model with a renormalized Kuo–Browninteraction [32]. The agreement is in general quite good.

Stars were used to mark the calculated transition strengthsB(E2;7/21 → 9/21)= 0.081([e2 b2 × 10−2]) andB(E2;7/21 → 9/22) = 1.420. Due a shift of theK = 9/2 band, the9/21 and 9/22 are inverted in our level scheme. A more natural assignment would beB(E2;7/21 → 9/21) = 1.42 andB(E2;7/21 → 9/22) = 0.081, which are closer to theexperimental results than the later.

Calculated results for theB(E2) transition strengths are not very good. Only a few arewell described: 1/2+1 → 3/2+1 , 5/2+1 → 7/2+1 , 1/2+1 → 5/2+2 , 5/2+1 → 9/2+1 , 5/2+2 →9/2+3 , 7/2

+1 → 11/2+1 , and 9/2+1 → 13/2+1 . For the other reported transitions the


difference is one or more orders of magnitude, most being underestimated. These wrongpredictions show the limitations of the model. It is interesting to see that the present valuesand those reported in Ref. [32] are in general very close.

As mentioned above, by using theB(E2) transition strengths and the form of the wavefunction it was possible to identify the six rotational bands shown in Fig. 8, in agreementwith [31]. TheSU(3) content of these bands is shown in Fig. 9 as a function of the angularmomentum of the states belonging to the different bands. Beyond the regular structurefound in most of the bands, the most remarkable feature is that the first two bands aredominated by the(6,6) irrep. The “leading” irrep(9,3)1/2 is dominant in the secondexcitedK = 1/22 band. This is a remarkable result: the quadrupole–quadrupole interactionbuilds the ground-state band mostly from the leading irrep. As was pointed out in [32], italso implies a coexistence of prolate and triaxial shapes, associated with the(9,3) and(6,6) irreps, respectively. It also shows that, while developing clear rotational bands, the25Mg ground state is mostly triaxial. The yrast band, however, includes many differentband heads, which are either triaxial or prolate.

At variance from what was found in21Ne and23Na discussed above, the spin 3/2 irrepsplay only a marginal role in25Mg, except for the states with the largest angular momentumin theK = 9/2 band, insert (d), and theK = 11/2 band, insert (f), which is dominated bythe(7,4)3/2 irrep.

4. Odd-odd sd-shell nuclei

4.1. 22Na

22Na has 3 protons and 3 neutrons in thesd-shell, outside the16O inert core. In Table 1are shown the 5 possible irreps for 3 particles in thesd-shell. The list of the 20SU(3)irreps, which define the quasi-SU(3) Hilbert space is shown in Table 9; this is a small setof irreps selected from the 384 of the full Hilbert space (plus their external multiplicities).

The 22Na energies obtained with the Hamiltonian (4) diagonalized in this basis areshown in the first column (labeledSU(3)) of Fig. 10. They are compared with theexperimental energies [23] shown in the second column, and with those obtained withfull shell model [33], shown in the third column.

Table 9The 20 irreps used in description of22Na


(4,1)1/2 (4,1)1/2 (8,2)0,1 (9,0)0,1 (6,3)0,1 (7,1)0,1(4,1)1/2 (2,2)1/2 (6,3)0,1 (7,1)0,1(2,2)1/2 (4,1)1/2 (6,3)0,1 (7,1)0,1(4,1)1/2 (3,0)3/2 (7,1)1,2(3,0)3/2 (4,1)1/2 (7,1)1,2


Fig. 10. Energy spectra of22Na. For details see the text. The first column labeledSU(3) exhibits thepresent results, the second column shows the experimental values [23] and the third column displaysfull shell-model results [33].

It can be seen from Fig. 10 the good agreement between the predicted values of themodel and those experimentally measured. Below 5 MeV we have displayed all predictedand measured levels but 2+3 , 2

+4 , and 5

+3 states. 2

+3 and 2

+4 are underestimated by 1.5 MeV

and the 5+3 is overestimated by 1 MeV. It must be pointed out the good agreement betweenthe prediction of the first two 0+ levels and the experimental values. The rotor terms inHamiltonian (4) allow the fit of these excited band heads, which are not well reproducedin shell-model calculations.

In general, for the levels of Fig. 10, the agreement is as good as the one obtained withfull shell-model calculations [33], where the Hamiltonian included the realistic effectiveinteractions calculated from the Hamada–Johnston potential by Kuo [34] using 63 two-body matrix elements adjusted by rms best fit of 72 levels in theA = 18–22 region and the17O single-particle energies.

Bands can not be identified as easily as in even–even [11] or some odd-A nuclei. Theyare reconstructed from those states having the largest calculatedB(E2) transition strengthbetween them. In this way the 6 bands shown in Fig. 11 were found. They exhibit the usualrotor structure, while some of them only include states with even angular momentum.

The percentage each quasi-SU(3) irrep contributes to the wave function of states in theground-state band is shown in Fig. 12 as a function of the angular momentum of each state.The irrep(8,2) with spin 1 dominates each eigenstate of the band up toJ = 8, where the


Fig. 11. Band structure of22Na.

Fig. 12. Wave function components of states belonging to the ground-state band in22Na. The

percentage 100× |CJiMβ |2 each irrep contributes is shown as function of the angular momentum.

irrep (7,1) contributes with more than 30% to the wave function. ThisS = 2 irrep is builtwith the (3,0)3/2 proton or neutron irreps, underlining again the relevance of includingspin 3/2 states in theπ subspace (and not only theS = 1/2 states) for the description ofexcited states belonging to the low-lying bands.

The B(E2) transition strengths are shown in Table 10. The first column shows thetransition under consideration, the second column shows all experimental measured values,the third exhibits the values found in the present work witheπ eff = 1.4e andeν eff = 0.4eand the last column shows full shell-model values [33] witheπ eff = 1.5e andeν eff = 0.5e.It is interesting to observe the good agreement between experimental values and bothSU(3) and shell-model [33] predictions.

4.2. 24Na

The valence space of24Na contains 3 protons and 5 neutrons in thesd-shell. The fiveproton and twelve neutronSU(3) irreps are listed in Table 1.

The complete basis is built coupling all the proton and neutron irreps, it has 1040 irrepsplus their internal multiplicities. For this nuclei, we have performed calculations with the


Table 10B(E2) transition strengths for22Na in [e2 b2 × 10−2]

B(E2;Ji → Jf) ↑ Expt. Present SM [33]

11 → 31 (3.652± 0.932) × 10−4 9.835× 10−4 1.0× 10−311 → 32 1.577± 0.145 1.466 1.61021 → 33 0.381± 0.272 0.112 0.30831 → 41 1.284± 0.030 1.240 1.15731 → 43 0.005 1.118× 10−5 0.03031 → 51 0.312± 0.013 0.311 0.38532 → 52 0.855± 0.366 1.099 0.84941 → 51 1.049± 0.005 0.947 0.96541 → 61 0.483± 0.079 0.479 0.462

Experimental,SU(3) and full shell model [33] values are shown in columns 2, 3 and 4, respectively.In the present work the effective charges wereeπ eff = 1.4e andeν eff = 0.4e.

Table 11The 20 irreps used in description of24Na


(4,1)1/2 (5,1)1/2 (9,2)0,1 (10,0)0,1 (7,3)0,1(4,1)1/2 (2,4)1/2 (6,5)0,1 (7,3)0,1 (4,6)0,1(4,1)1/2 (3,2)1/2 (7,3)0,12,2(2,2)1/2 (5,1)1/2 (7,3)0,1(2,2)1/2 (2,4)1/2 (4,6)0,1

20 irreps with largestC2 values and spin 0, 1 and 2; they are listed in Table 11. Onlyproton and neutron irreps with spin 1/2 and 3/2 are present, irreps with spin 5/2 arealways coupled to states with smallerC2 value.

The left hand side of Fig. 13 shows the result obtained for the low-energy spectra of24Na, calculated using the Hamiltonian parameters listed in the fifth row of Table 2, and theHilbert space described above. This is compared with the whole set [23] of experimentalenergies measured below 2.5 MeV presented in the second column, and with calculationsperformed by the Glasgow group using the full shell model [35] (third column). They usedthe CWC interaction, one effective interaction developed by Chung and Wildenthal [36]together with an empirically-determined Coulomb interaction. CWC was an improvedversion of the interaction used forA = 22 nuclei [33]. It can be seen that the present schemereproduce the right ground state and several excited states. It is interesting to mention theset of states 1+1 , 2

+1 , 2

+2 , 2

+3 , 3

+2 , 3

+3 , and 3

+4 , because they were found very close to their

experimental energies. On other hand, for the states 3+1 and 5+1 fall very low in energy (the

5+1 is not shown in the figure). These states belong to theK = 1 band.The state 0+1 state is predicted at an energy 2 MeV lower than its observed value. It is

possible to select a different set of rotor parameters in order to fit the position of the 0+1and 0+2 . It is achieved usingb = −0.120 MeV instead ofb = −0.102 MeV (see Table 2).


Fig. 13. Energy spectra of24Na. The graph shows the energy levels for the present work in the firstcolumn, the experimental energies are displayed in the second column [23], and results of full shellmodel [35] are in the third column.

It increases the moment of inertia compressing the rotational bands. However, it spoils theglobal description of the energy spectra. A better option in order to rise the energy of the0+ states (and others) respect to 4+1 ground state, would be to modify the Hamiltonian (4),including proton–neutron pairing and quadrupole-pairing terms in the Hamiltonian.

If we set the parameterb to zero the state withJ = 2 becomes the ground state. Usingb = −0.102 is enough to lower the firstJ = 4 state. Settinga = 0.1 accommodates theK = 12 band-head.

In order to reconstruct the band structure we have analyzed the wave functions of thesestates and theirB(E2) transition strengths. It is shown in Fig. 15 with their wave functionspresented in Fig. 14. The ground-state band (g.s.b.), insert (a) in Fig. 14, shows a strongmixing of SU(3) representations. The leading contribution comes from theS = 1 state, asit was found in22Na, but only with∼ 25%. This state is built with the irreps(2,2)1/2 forπ and(2,4)1/2 for ν, coupled to(4,6)1. The band begins withJ = 4+ state, because wefound no 3+ state with strongB(E2) transition to 4+1 or with a similar wave function.

In the next two bands,K = 0, insert (b), andK = 1, insert (c), the largest contribution(70% approximately) comes from the[(4,1)1/2 ⊗ (5,1)1/2](10,0) irrep. In theK = 0band, the coupling is toS = 0 states (represented by triangles) and in theK = 1 it is toS = 1 (represented by circles). In insert (d), the largest contribution to the lower angularmomentum states comes from(10,0)1, nevertheless for higherJ the mixing grows and itis not possible to identify the dominant irrep. Given the first 0+ and the 1+1 and 1

+2 states


Fig. 14. Wave function components of states belonging to (a) the ground-state band, (b) theK = 0band, (c) theK = 1 band, and (d) theK = 12 band, in24Na. The percentage 100× |CJiMβ |2each irrep contributes is shown as function of the angular momentum. The convention used is� for (4,6)1[(2,2)1/2 ⊗ (2,4)1/2], + for (4,6)0[(2,2)1/2 ⊗ (2,4)1/2], ✷ for (4,6)0[(4,1)1/2 ⊗(2,4)1/2], × for (4,6)1[(4,1)1/2 ⊗ (2,4)1/2], � for (10,0)0[(4,1)1/2 ⊗ (5,1)1/2], + for(6,5)1[(4,1)1/2 ⊗ (2,4)1/2], ◦ for (6,5)0[(4,1)1/2 ⊗ (2,4)1/2], � for (9,2)1[(4,1)1/2 ⊗(5,1)1/2], © for (10,0)1[(4,1)1/2 ⊗ (5,1)1/2] and• for (7,3)2[(4,1)1/2 ⊗ (3,2)3/2].


Fig. 15. Band structure of24Na.

are all built by a(10,0) irrep, theB(M1;0+1 → 1+) transition strengths will be very strong,as it is expected in light odd–odd nuclei [37,38].

They are only few experimentally measuredB(E2) intensities. In the ground-state bandcalculated intrabandB(E2) strengths ranges from 0.5857 e2 b2 × 10−2 for 4+1 → 5+2transition to 0.1322e2 b2 × 10−2 for 7+2 → 8+4 . Interband transitions are one or moreorders of magnitude smaller. InK = 01 band, theB(E2) intraband transition strengths runfrom 2.8666 for 0+1 → 2+2 to 0.9393 for 4+3 → 6+3 . The effective charges used for thesetransitions wereeπ eff = 1.5e andeν eff = 0.5e.

In conclusion, the model predicts several states in the low-energy spectra which arein agreement with the experimental values and full shell-model results, and allows theidentification of four rotational bands.

4.3. 26Al

26Al has 5 protons and 5 neutrons in the activesd-shell.The 12 irreps for 5 particles (protons or neutrons) in thesd-shell are those listed in

Table 1. From the 1599 proton–neutron coupledSU(3) irreps we have selected the 28 withthe largestC2 values to build the truncated Hilbert space. They are listed in Table 12. The

Table 12The 28 irreps used in description of26Al


(5,1)1/2 (5,1)1/2 (10,2)0,1 (11,0)0,1 (8,3)0,1(2,4)1/2 (2,4)1/2 (4,8)0,1 (2,9)0,1 (0,10)0,1(5,1)1/2 (2,4)1/2 (7,5)0,1 (8,3)0,1(2,4)1/2 (5,1)1/2 (7,5)0,1 (8,3)0,1(5,1)1/2 (3,2)1/2 (8,3)0,12,2(3,2)1/2 (5,1)1/2 (8,3)0,12,2


Fig. 16. Energy spectra of26Al. The first column displays the results of the present article, the secondone the experimental data [23], and the third shows full shell-model results [39,40].

most important irreps for protons and neutrons are those with spinS = 1/2 and 3/2, whileproton–neutron coupledSU(3) irreps have spinS = 0, 1 and 2.

The values ofκ andµ in the Nilsson Hamiltonian for26Al were taken from Ring andSchuck [21], they produce better agreement with experimental energies than the Wildenthal[2] parameterization. The symmetry term in the Hamiltonian was set to zero.

The energy spectra is shown in Fig. 16; the present results, labeledSU(3), are in thefirst column, the experimental values [23] in the second column and those of the full shellmodel [39,40] in the third column. The gap observed in the experimental values from1.05 to 1.75 MeV is well reproduced by the present model (in the same way as the fullshell model predicts), and with the exception of the 2+1 and 1

+2 levels, the agreement is in

general good.No set of rotor parameters was able to avoid the depletion in energy of the levels 11, 31,

41, 42 and 52. As mentioned above, the description of the energy spectra could be improvedby using a more complete Hamiltonian.

26Al is one of the best studied odd–odd nuclei. It has a very complicated energy levelscheme, which Röpke and Endt [41] have been able to assign to rotational bands. Afteran exhaustive study 170 levels were grouped in 35 rotational bands. Maximum alignmentwas found in the yrast band. TheSU(3) model naturally describes these rotational bands;the first 8 are shown in Fig. 17. However, theSU(3) description of the ground-state banddiffers from the one given in [41]. We associate the states 3+1 and 1

+1 to the g.s.b., which


Fig. 17. Band structure of26Al.

Table 13Theπ = + rotational bands in26Al

Kπn Jπ E (band-head) (MeV)

Present Exp. [23] Estimated [41]

0+1 0+, 2+, . . . 0.50 0.23 0.200+2 0+, 2+, . . . 2.65 3.75 3.400+3 0+, 2+, . . . 4.60 5.19 4.901+3 1+, 2+, . . . 2.25 2.07 3.201+4 1+, 3+, . . . 2.93 2.74 1.20

has the sequence of angular momentaJ , (J + 2), (J + 4), . . . , (see Fig. 17), instead ofJ , (J + 1), (J + 2), . . . , as found in [41]. Given the experimentalB(E2) values discussedbelow, it seems to be a failure of our model.

In Table 13 we compare the band-head energies of five positive-parity bands.SU(3)energies are shown in the third column, labeled ‘present’. These excited bands wererecognized looking for theJ (J + 1) pattern and enhanced intrabandB(E2) strengthtransitions. They correlate well with the experimental energies. The last column, labeled‘estimated’, shows band-heads energies obtained by adding the single-particle energies ofthe valence nucleons given by the Nilsson model, neglecting the residual interaction [41].

In Fig. 18 the wave functions of the g.s.b.,K = 21,01,12 and 14 bands are plotted. Thewave function components of the g.s.b., with odd angular momentum states, insert (a),are very similar to those shown forK = 21 band, which has even angular momentumstates, insert (b): both have above 70% of the irrep(10,2)1. Nevertheless, they do notseem to form a single band, because theB(E2) values between members of these bandsare suppressed, and the g.s.b wave function includes(8,3)1 and(11,0)0 components whilethe 21 band has an important(8,3)2 component.


Fig. 18. Wave function components of states belonging to (a) the ground-state band, (b) theK = 21 band, (c) theK = 01 band, (d) theK = 12 band, and (e) theK = 14 band in26Al. The percentage 100× |CJiMβ |2 each irrep contributes is shown as function of the angularmomentum. The convention used is× for (10,2)1[(5,1)1/2 ⊗ (5,1)1/2], � for (8,3)1[(5,1)1/2 ⊗(3,2)1/2,3/2], ✷ for (11,0)0[(5,1)1/2 ⊗ (5,1)1/2], + for (8,3)2[(5,1)1/2 ⊗ (3,2)3/2], � for(10,2)0[(5,1)1/2 ⊗ (5,1)1/2], + for (11,0)1[(5,1)1/2 ⊗ (5,1)1/2], ◦ for (7,5)1[(5,1)1/2 ⊗(2,4)1/2], � for (2,9)0[(2,4)1/2 ⊗ (2,4)1/2], © for (4,8)1[(2,4)1/2 ⊗ (2,4)1/2] and • for(0,10)1[(2,4)1/2 ⊗ (2,4)1/2].


Table 14B(E2) transition strengths for26Al in [ e2 b2 × 10−2]

B(E2;Ji → Jf) ↑ Expt. Present

31 → 51 0.360± 0.010 1.48412 → 31 0.201± 0.037 0.27941 → 51 0.151± 0.027 0.07901 → 22 2.882± 0.137 4.18211 → 32 0.470± 0.096 0.12832 → 51 0.001± 0.000 0.00011 → 33 0.066± 0.017 0.00233 → 51 0.001± 0.000 0.00012 → 34 0.683± 0.117 0.12011 → 34 0.256± 0.043 0.14934 → 51 0.001± 0.000 0.00032 → 52 0.532± 0.122 1.81841 → 61 0.251± 0.073 1.42251 → 61 0.741± 0.135 0.24536 → 51 0.007± 0.002 0.00037 → 51 0.009± 0.014 0.00238 → 51 0.002± 0.001 0.00123 → 43 1.482± 0.329 1.38921 → 43 0.020± 0.004 0.06124 → 45 0.321± 0.049 0.783

Experimental and theoretical values are shown in column 2 and 3,respectively. The effective charges wereeπ eff = 1.3e andeν eff = 0.3e.

TheK = 14 band, insert (e), has between 40% and 60% of the irrep(4,8)1 (big cir-cles), except for theJ = 7 state which becomes an almost pureSU(3) state. The irreps(8,3)2 (+), (10,2)1 (×), and(7,5)1 (smaller open circle) are present in the other statesof the band.

The K = 01 and 12 bands seems very interesting, because very strong M1 transitionstrengths have been found inN =Z odd–odd nuclei [37,38]. Inserts (c) and (d) of Fig. 18show that the 0+1 and 1

+2 states are dominated by the same irrep(10,2), although for the

J = 0+1 state (insert (c)), the dominant irrep has spin 0 (� line), while for the 1+2 state(insert (d)), the coupling is toS = 1 (× line).

Table 14 lists theB(E2) transitions for26Al. While the general trend is reproduced,the 31 → 51, 32 → 52 and 41 → 61 are overestimated, exhibiting the limitations of themodel. As said above, the 31 state appears in our description at a very low energy. Its wavefunctions is so similar to the ground-state wave function that we were forced to considerit as a member of the ground-state band. For this reason theB(E2) transition strength islarger than the observed one. The other two transitions are overestimated in a similar way.

4.4. 28Al

28Al has 5 protons and 7 neutrons in the activesd-shell. The irreps for these particles intheη = 2 shell are listed in Table 1.


Table 15The 21 irreps used in description of28Al


(5,1)1/2 (4,2)1/2 (9,3)0,1 (10,1)0,1(2,4)1/2 (1,5)1/2 (3,9)0,1 (1,10)0,1 (7,4)0,1(5,1)1/2 (1,5)1/2 (6,6)0,1(2,4)1/2 (4,2)1/2 (6,6)0,1(3,2)1/2,3/2 (4,2)1/2 (7,4)0,1(5,1)1/2 (2,3)1/2,3/2 (7,4)0,1(3,2)1/2,3/2 (1,5)1/2 (7,4)0,1,2

Fig. 19. Energy spectra of28Al. The first column displays the results of the present article, the secondone the experimental data [23], and the third shows full shell-model results [42].

From the 2840 proton–neutron coupledSU(3) irreps we selected the 21 with the largestC2 values to build the truncated Hilbert space. They are shown in Table 15, together withtheir spin. The low-energy spectra of28Al is depicted in Fig. 19. The first column showsthe predicted levels, in the second one the experimental [23] values and in third one thosefound in full shell-model calculations [42]. TheJ = 3 ground state and itsJ = 2 partnerare properly described, as well as the gap between this pair of states and the others. Manyexcited states have their energies over- or under-predicted. Even though, the general pictureis better than the one obtained for26Al.


Fig. 20. Band structure of28Al.

To improve the description of the energy spectra we have included a term proportionalto the third-orderSU(3) Casimir operatorC3 in the Hamiltonian, with a coefficientc =−0.0017. It helps in spreading the energies of the first five states withJ = 2, puttingthem close to their observed energies. It makes the energy spectra of this odd–odd nucleito be closer to the experiment than the other odd-mass nuclei. This term has also shownits usefulness in the description of heavy deformed nuclei [14]. While we prefer to use thesmallest possible number of free parameters, the difficulties associated with the descriptionof 28Al forced us to its inclusion.

TheB(E2) transition strengths have not been measured for this nuclei; we have usedthem only to build the bands. In the Fig. 20 the28Al band structure is displayed. Atvariance with the band structure of26Al, the 28Al ground state (3+1 ) is the band-head ofthe ground-state band and there are no states withJ < 3 belonging to this band. The state2+1 , which is almost degenerated with the ground state (3

+1 ), is band-head of other rotational

band.TheSU(3) wave function components for each the band presented in Fig. 20 are shown

in Fig. 21. These bands exhibit a complicated structure. For the ground-state band shown ininsert (a), a strong mixture between the components with spin 0 and 1 of the irrep(9,3) isobserved. We found the ground state to be mainly prolate, despite the presence of the twooblate irreps(3,9)0,1 with the same value of the second-order Casimir (〈C2〉 = 153) as(9,3). For the bandK = 21 a strong mixture between irreps(9,3) and(10,1) with S = 1is observed. The bandK = 11 is mainly built by the irreps(9,3)0,1 for states withJ = 1and 2, whereas forJ > 2 other representations with spin 1 become important.

5. Discussion and outlook

The quasi-SU(3) truncation scheme was used in conjunction with Hamiltonian (4) todescribe the energy spectra andB(E2) transition strengths in21Ne, 22,23,24Na, 25Mg and26,28Al. Comparison was made with experimental data, with shell-model calculations and


Fig. 21. Wave function components of states belonging to (a) the ground-state band, (b) theK = 21band, (c) theK = 11 band, (d) theK = 01 band, (e) theK = 51 band, and (f) theK = 02 bandin 28Al. The percentage 100× |CJiMβ |2 each irrep contributes is shown as function of the angularmomentum. The convention used is� for (9,3)0[(5,1)1/2 ⊗ (4,2)1/2], + for (9,3)1[(5,1)1/2 ⊗(4,2)1/2], ✷ for (10,1)1[(5,1)1/2 ⊗ (4,2)1/2], × for (7,4)1[(5,1)3/2 ⊗ (2,3)1/2] and[(3,2)1/2⊗ (4,2)1/2], � for (6,6)1[(2,4)1/2⊗ (4,2)1/2], and◦ for (6,6)0[(2,4)1/2⊗ (4,2)1/2].


with previousSU(3) studies. The agreement was in general good, exhibiting the success

of the model.

The truncation recipe is quite simple. First select three or four proton and neutronSU(3)

irreps with largestC2 values, which in general will have spin 0 or 1 for even number of

nucleons and spin 1/2 and 3/2 for odd number of nucleons. Build from these the coupled

proton–neutronSU(3) irreps, and select again only a small number ofSU(3) irreps with

the largest totalC2 values. Usually 20 irreps are enough. Use this basis to diagonalize the

Hamiltonian. While the effect of the truncation in21Ne was small, forA ∼ 26 nucleusit implied a two orders of magnitude reduction in the basis size. The composition of the

calculated wave functions was found to be very similar to the ones reported in previous

SU(3) studies where larger basis and the Kuo interaction were used [32].

The Hilbert space built in this way is rich enough to include many excited rotational

bands which have a clear counterpart both in the experimental data and in full shell-

model calculations. The fact that most of these bands have one fairly dominantSU(3)

irrep underscores the strength of the model. This feature leads to a simple picture in terms

of the SU(3) decomposition of the wave functions, and exhibits the crossing and mixing

which occur within a band as a function of the angular momentum of the different band

members.B(E2) transition strengths were also found to be in close correspondence with

the experimental data, allowing a clean identification of band members. TheSU(3) content

of the ground-state band also showed that, while most nucleus (21Ne,22,23Na and26,28Al)

are definitively prolate,24Na and25Mg are mostly triaxial.

There were model limitations uncovered in the present study. The most striking are the

band shifts, i.e. the fact that some excited bands are predicted at energies 1 to 2 MeV

lower than they appear in the experiment. Given that the band structure and theB(E2)

values were in general well depicted, the band shifts seem to reflect on a limitation of the

Hamiltonian and not of the Hilbert space, since the same feature was found in full shell-

model calculations.

In addition to realistic single-particle energies, the Hamiltonian (4) includes quadrupole–

quadrupole and like particle-pairing terms with fixed interaction strengths taken from

systematics. It has also three rotor-like terms which allow for a fine tuning of the spectra.

In the case of25Mg, these terms played an important role in the reproducing of the energy

spectra, especially in pushing theJ = 5/2 band down to become the ground-state band ofthe system.

In these nuclei withN very close toZ, proton–neutron pairing, both in theT = 0 andT = 1 channels, plays a very important role [43]. Its inclusion in Hamiltonian (4) wouldmake it isoscalar. It could be relevant not only for improving the predicted energy spectra,

but also in the description of M1 excitations, which are known to be very challenging for

any theoretical model [44]. It would also allow the model to be tested in thefp-shell where

B(M1) transition strengths have been measured [37,38]. Future research on these subjects

is desirable.


Acknowledgements

The authors thank C. Johnson and S. Pittel for emphasizing the relevance of proton–neutron pairing in thesd-shell. The authors thank the Institute of Nuclear Theory of theUniversity of Washington for its hospitality and the Department of Energy for partialsupport during the completion of this work, which was also supported in part by Conacyt(México) and the National Science Foundation under Grant PHY-9970769 and CooperativeAgreement EPS-9720652 that includes matching from the Louisiana Board of RegentsSupport Fund.

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