Quasiparticle-rotor model description of carbon isotopes

Quasiparticle-rotor model description of carbon isotopes

T. Tarutina^I; A.R. Samana^II; F. Krmpotic^{I, III, IV}; M.S. Hussein^I

^IDepartamento de Física Matemática, Instituto de Física da Universidade de São Paulo, Caixa Postal 66318, CEP 05315-970, São Paulo, SP,Brazil

^IICentro Brasileiro de Pesquisas Físicas, CEP 22290-180, Rio de Janeiro, Brazil

^IIIInstituto de Física La Plata, CONICET, 1900 La Plata, Argentina

^IVFacultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, 1900 La Plata, Argentina

ABSTRACT

In this work we perform quasiparticle-rotor coupling model calculations within the usual BCS and the projected BCS for the carbon isotopes ¹⁵C, ¹⁷C and ¹⁹C using ¹³C as the building block. Owing to the pairing correlation, we find that ¹³C as well as the cores of the other isotopes, namely ¹⁴C, ¹⁶C and ¹⁸C acquire strong and varied deformations. The deformation parameter is large and negative for ¹²C, very small (or zero) for ¹⁴C and large and positive for ¹⁶C and ¹⁸C. This finding casts a doubt about the purity of the supposed simple one-neutron halo nature of ¹⁹C.

Keywords:¹³C, ¹⁵C, ¹⁷C, ¹⁹C nuclei; Energy spectra; Particle projected BCS model; Core excitation; Quasiparticle-rotor model

I. INTRODUCTION

Many experimental and theoretical investigations for over two decades were concentrated upon study of light nuclei near neutron drip line (see references in recent reviews [1, 2]). Pairing correlations play an important role in the structure of these nuclei, because of the proximity of the Fermi surface to the single-particle continuum. This gives rise to many interesting phenomena and creates a challenge for conventional nuclear structure models. The experimental evidence for the N = 8 shell melting [3] and the appearance of the 1s_1/2 intruder state in¹¹ Be are famous indications of the complicated structure of light nuclei. In Refs.[4, 5] it was shown that the increase in pairing correlations and the shallow single-particle potentials for nuclei close to the driplines may result in a more uniformly spaced spectrum of single particle states.

To mention a few recent works on this subject, a series of interesting articles appeared in the literature which study the pair correlation in spherical and deformed nuclei near the drip line using a simplified HFB model in coordinate representation with the correct asymptotic boundary conditions. In Ref.[6] the effects of continuum coupling have been studied, it was shown that for small binding energies, the occupation probability decreases considerably for neutrons with low orbital momentum. In Ref.[7] the weakly bound neutrons in s_1/2 state have been studied. The effective pair gap was found to be much reduced compared with that of neutrons with larger orbital momentum. In the presence of pair correlations, the large rms radius was obtained for neutrons close to the Fermi level, thus favouring the halo formation.

Nowadays it is generally accepted that nucleus tends to form a halo when the valence particles are loosely bound and the relative angular momentum is small. In this case, the last nucleons and the core are to a large extent separable and therefore these nuclei can be approximated as an inert core plus the halo formed by the valence particles. This supports the use of the cluster models for such systems. But for real nuclei, the admixtures between the core and valence particles have to be taken into account. One of the simple models that include core degrees of freedom is a core + particle cluster model for one-neutron halo systems including core excitation via deformation assuming a rotational model for the core structure [8, 9]. Although this model allowed successful description of the 1s_1/2 intruder state in ¹¹Be, it has a serious drawback as it does not treat Pauli Principle correctly.

In this work we incorporate the residual interaction between nucleons moving in an overall deformed potential and perform a Bogoljubov-Valatin canonical transformations from particles to quasiparticles, thus modifying the band-head energies and the non-diagonal particle core matrix elements. The resulting model is known as the quasiparticle-rotor coupling model (QRCM). The inclusion of pairing effects allows a correct treatment of Pauli Principle. This model is then applied to the series of odd-mass carbon isotopes.

Heavy carbon isotopes have recently been studied extensively, and ¹⁷C and especially ¹⁹C are suggested to be candidates to possess neutron halo. To calculate the quasiparticle energies for ¹³C and heavy carbon isotopes the BCS calculation with particle projection procedure was used (PBCS). It is known that in BCS formalism the number of particles is not conserved, which becomes a serious drawback for the case of light nuclei. It was shown in Refs.[12, 15], that the projection procedure is very important in such nuclei. The Fig. 1 shows the BCS and PBCS calculations confronted with experimental data, using the results obtained in [12], where we evaluate the ground states energies for the remaining odd-mass carbon isotopes, by only modifying the numbers of neutrons: N = 8,10 and 12 for ¹⁵C, ¹⁷C and ¹⁹C, respectively. One immediately sees that the pairing interactions account for the main nuclear structure features in these nuclei.

II. THE QRCM MODEL

Most applications of the particle-rotor coupling model (PRCM) use the simplest picture of the last odd nucleon moving outside the even-even core, and any kind of residual interaction between nucleons is neglected (see, for example, Refs.[9, 13, 14]). The core is assumed to be an axially symmetric rigid rotor. The Hamiltonian of the core-neutron system can be written as

where H_sp = T + U₀ is single-particle (shell-model) Hamiltonian, that is the sum of central Woods-Saxon potential and standard spin orbit interaction, whose eigenvalues e_j are specified by the particle angular momentum j. We consider here the quadrupole deformation only, specified by the deformation parameter b. In this case, the core-neutron coupling hamiltonian

where k(r) is the radial part of the interaction.

The total angular momentum is I = j + R (in this work we consider only I = 0,2) and the coupling between j and R is evidenced in the weak-coupling representation of the total wave function, which reads

where |RM_Rñ is the rotor wave function. The matrix elements of the Hamiltonian (1) are:

The resulting eigenfunctions are of the form:

with energies , and where the are constants that result from the solution of the coupled channel system of equations obtained by substituting the total wave function into the Schöedinger equation and projecting out the basis wave functions.

We now incorporate the residual interaction, V_res, between nucleons moving in an overall deformed potential. The total Hamiltonian becomes H_qp-rot = H_p-rot + V_res. After carrying out a Bogoljubov-Valatin canonical transformation from particle to quasiparticle operators, one gets that the matrix elements of H_qp-rot for odd-mass systems continue being of the form (4), except for (i) the band-head energies for a particle state are modified as e_j ® E_j in BCS, and e_j ® in PBCS, where E_j is the quasiparticle BCS energy, and = , where quantities R^K and I^K, for K = Z,N, are defined in Ref.[15]; and (ii) the non-diagonal matrix elements are renormalised by the following overlap factors

where u_j and v_j are the occupational numbers for the state j. I^K(j¢j), I^K-2(j¢j)) and I^K(j) are the PBCS number projection integrals (see Refs.[12, 16]).

The energies to be confronted with the experimental data are

where l is the BCS chemical potential. Of course, in the resulting quasiparticle-rotor coupling model (QRCM) the state |jmñ is now a quasiparticle state. The QRCM differs in several important aspects from the PRCM. First, the BCS (PBCS) energies are quite different from the single-particle energies e_j. Second, the factors F_jj¢ correctly take into account the Pauli principle. In addition, the particle-like states do not couple to the hole-like states. As a brief test, in Fig. 2 we show a comparison of the energy levels for ¹³C and ¹¹Be in the Pure Rotor Model with those in the QRCM: Rotor + BCS and Rotor + PBCS as a function of the parameter of deformation b for the single particles taken from the Vinh-Mau's work [17] for ¹³C and ¹¹Be. From a careful scrutinising of the energy levels in Fig. 2, one can easily convince oneself that, none of our particle-rotor coupling calculations is able to reproduce the experimentally observed spin sequence 1/2⁺-1/2^--5/2⁺ in ¹¹Be, for such a value of b neither positive nor negative. The constant of pairing for ¹³C is v_s = 37.80 in both BCS and PBCS, while that in ¹¹Be they are v_s = 21.02 in BCS and v_s = 24.25 in PBCS.

III. RESULTS AND CONCLUSIONS

In our procedure we use the nucleus ¹³C as a building block of the calculation, assuming core ¹²C deformation b = -0.6. In our calculations we adjust the single particle state energies e_j (for 1p_3/2, 1p_1/2, 1d_5/2 and 2s_1/2) so that the resulting quasiparticle energies including the core excitation give correct binding energies of four bound states in ¹³C. The same set of single particle energies are then used in PBCS calculation to obtain quasiparticle energies of ¹⁵C, ¹⁷C and ¹⁹C, occupation numbers for the states in these nuclei and the reduction factors of the matrix elements. Subsequently, the quasiparticle energies and reduction factors then enter the coupled channel system to include core excitation and give the binding energies of ¹⁵C, ¹⁷C and ¹⁹C. For ¹⁵C, ¹⁷C and ¹⁹C we adjust the deformation parameter b to obtain a correct one-neutron separation energy in each of the isotopes. Using this deformation parameter we calculate the energy of the first excited state. For simplicity, in the present calculation we use fixed values for averaged radial matrix elements between different states, that is, áj¢|k(r)|jñ = 25, 30 and 35 MeV. These values provide a reasonable estimation of non-diagonal part of the interaction and correctly reproduce the spin-orbit splitting between initial single-particle 1p_3/2 and 1p_1/2 states. The energies obtained within the Rotor + PBCS model on different carbon isotopes for ák(r)ñ = 25 MeV (all energies are in MeVs) are presented in the Table 1.

The results for the odd-mass carbon isotopes can be summarised as:

¹⁵C: In our calculation we obtain a zero deformation for the core ¹⁴C in case of ák(r)ñ = 25 MeV and a binding energy of 1.25 MeV. The calculation with larger average matrix element 30 and 35 MeV, gives an oblate shape with deformation parameter equal to -0.23 and -0.32 respectively. (In this calculation we introduced a lower 2⁺ rotational energy in ¹⁴C e_2⁺ = 3 MeV). This confirms the idea that the core nucleus ¹⁴C is nearly spherical and supports the possibility of halo formation for ¹⁵C. Finally, our calculation also predicts a first excited state ⁺ with the binding energy -936 keV, -991 keV and -867 keV for ák(r)ñ equal to 25 MeV, 30 MeV and 35 MeV, respectively. The binding of the first excited state is overestimated in our model.

¹⁷C: Small separation energy of ¹⁷C 0.728 MeV suggests a possible halo structure of this isotope. There is an uncertainty about ground state spin assignment for this nucleus. In our model we obtain ⁺ ground state for ¹⁷C, although the experimental value is ⁺ [11]. A simple explanation for this experimental result could be found in the so called J = j-1 anomaly discussed by Bohr and Mottelson [18]. Core-particle calculations without pairing similar to those in Ref.[13] used deformation parameter b = 0.55. We obtain the b equal to 0.505, 0.583 and 0.609 for average radial matrix element ák(r)ñ equal to 25, 30 and 35 MeV, respectively. We see that the deformation changes slowly with the value of the matrix element and is rather large. The first excited state of ¹⁷C in our model is ⁺ state. The binding energy of this state is as follows: -0.616 for ák(r)ñ = 25 MeV, -0.527 MeV for ák(r)ñ = 30 MeV and -0.430 for ák(r)ñ = 35 MeV. These energies are close to the experimental separation energy of -0.433 MeV.

¹⁹C: Ground state with spin-parity ⁺ and small one-neutron separation energy favours the formation of the halo in ¹⁹C. In our analysis we adjust the deformation parameter in order to obtain the adopted separation energy of 0.58 MeV. Core-particle calculations without pairing used deformation parameter b = 0.5. For ák(r)ñ = 25,30 and 35 MeV we need, respectively, b = 1.27,1.51 and 1.82. That is, we obtain very a large prolate deformation for the core ¹⁸C. The first excited state we obtain to be ⁺ state with the binding energy -0.21MeV (ák(r)ñ = 25 MeV).

The Fig. 3 summarises the above results and compares them with the results of other models and experimental data.

Finally, in Fig. 4 we show the dependence of the deformation parameter on the number of neutrons in the core N for three values of ák(r)ñ. It is seen that in each case, the deformation changes from negative values (for cores ¹²C and ¹⁴C) to large positive values (for cores ¹⁶C and ¹⁸C). The dependence is nearly linear and crosses the zero deformation in the region of the core ¹⁴C.

To summarise, in this work we analyse the structure of heavy carbon isotopes ¹⁵C, ¹⁷C and ¹⁹C in the quasiparticle-rotor model using the ¹³C as the building block of the calculation. The quasiparticle energies and the reduction factors in the matrix elements are calculated in the projected BCS model. As it was mentioned in the introduction, we first explore the results of the pure BCS and PBCS models, that use the single-particle energies and pairing strengths of Ref. [12] fixed so that the experimental binding energies of ¹¹C and ¹³C, together with the ¹³C energy spectra, are reproduced by the calculations. With these parameters we evaluate next the low-lying states in ¹⁵C, ¹⁷C and ¹⁹C, by augmenting correspondingly the number of neutrons only. We found that proceeding in this way both models are capable to explain fairly well the decrease of the binding energies in going from ¹³C to ¹⁹C. Second, the QRCM was implemented, and in this approximation we obtain ¹⁵C, ¹⁷C and ¹⁹C with ⁺ ground state and ⁺ first excited state. For ¹⁵C we obtain zero or small oblate deformation. For ¹⁷C large prolate deformation is given. Finally ¹⁹C has a very large prolate deformation, indicating that the nature of this nucleus is more complicated than the simple one-neutron halo picture of this nucleus would suggest.

The further analysis of this model and the details of its application to describe the low-lying spectra of light nuclei model will be presented in our next work [19].

Acknowledgments

F.K. acknowledges the support of the CONICET, Argentina. A.R.S. appreciates support received from Conselho Nacional de Ciência e Tecnologia (CNPq) and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ). T.T. and M.S.H. would like to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and CNPq.

[19] A.R. Samana, T. Tarutina, F. Krmpotic, and M.S. Hussein, in preparation.

Received on 18 March, 2006

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Publication Dates

• Publication in this collection
  13 Apr 2007
• Date of issue
  Dec 2006

History

• Received
  18 Mar 2006