WEAKLY BOUND NEUTRON RICH C ISOTOPES WITHIN RMF+BCS APPROACH

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1 NUCLEAR PHYSICS WEAKLY BOUND NEUTRON RICH C ISOTOPES WITHIN RMF+BCS APPROACH G. SAXENA 1,2, D. SINGH 2, M. KAUSHIK 3 1 Department of Physics, Govt. Women Engineering College, Ajmer India, gauravphy@gmail.com 2 Department of Physics, University of Rajasthan, Jaipur , India, dsingh5@gmail.com 3 Physics Department, Shankara Institute of Technology, Kukas, Jaipur , India, mkaushik007@gmail.com Received March 29, 2013 In the present investigations we have employed relativistic mean-field plus BCS (RMF + BCS) approach to carry out a systematic study for the ground state properties of eveneven C Isotopes. One of the prime reasons of this study has been to look into the role of low lying states in neutron rich region near neutron drip line. It is found that the occupancy of weakly bound neutron single particle states having zero orbital angular momentum (l = 0) with a well spread wave function due to the absence of centrifugal barrier, helps to cause the occurrence of nuclei with widely extended neutron density. Such nuclei are found to have characteristically very small two-neutron separation energy and large neutron rms radius akin to that observed in weakly bound systems. Key words: Relativistic Mean Field Theory, C isotopes, Weakly Bound Structure, Resonant States, Single Particle Levels. 1. INTRODUCTION From the last few decades experiments with radioactive nuclear beams continue to yield invaluable results for very short-lived nuclei, especially for the neutron rich nuclei with unusually large isospin [1-3]. For several neutron rich nuclei, it is found that the neutron density distribution shows a much extended tail with a diffused neutron skin while the Fermi level lies close to the single particle continuum [1, 2]. In few cases, it may lead to weakly bound system or the phenomenon of neutron halo, as observed in the case of light nuclei [1, 2, 4], made of several neutrons outside a core with separation energy of the order of 100 kev or less. A theoretical consideration on the possibility of occurrence of such loosely bound structures has been made by Migdal [5] already in early 70's. So far, because of experimental advancements there are ample examples of neutron halos in the light mass region of the periodic chart [1, 4]. Most of these light halo nuclei have been described within the framework of few-body models [4, 6]. Recently, 22 C has Rom. Journ. Phys., Vol. 59, Nos. 1 2, P , Bucharest, 2014

2 2 Weakly bound neutron rich C isotopes within RMF+BCS approach 87 been of special interest as a halo nucleus as has been observed experimentally by K. Tanaka et at. [3]. Neutron rich C isotopes have been studied by various theoretical and experimental methods in recent past in various ref. [7-10]. For such exotic nuclei, due to the weak binding and large spatial dimension of the outermost nucleons, the role of continuum states and their coupling to the bound states become exceedingly important, especially for the pairing energy contribution to the total binding energy of the system. It has been found that the occupancy of weakly bound neutron single particle states having zero orbital angular momentum, (l = 0), with a well spread wave function due to the absence of centrifugal barrier, helps to cause the occurrence of nuclei with widely extended neutron density and very small two-neutron separation energy. The role of pairing correlations is found to be consistent with the conclusions of non-relativistic HFB studies of neutron rich weakly bound nuclei discussed recently in Ref. [11] and that of spherical RMF+BCS calculations in Ref. [12, 13]. Encouraged by the success of RMF+BCS approach [12-15], and the impetus provided by the recent experimental developments [3], in this communication we present the results of our calculations for the C isotopes carried out within the framework of relativistic mean-field (RMF) theory to study the role of continuum states lying very close to the Fermi level. 2. RELATIVISTIC MEAN-FIELD THEORY Theoretical descriptions of drip line nuclei in terms of mean field theories, both non-relativistic as well as relativistic mean field (RMF), have been well received [16-19]. The advantage of the RMF approach is that it provides the spinorbit interaction in the entire mass region, which is consistent indeed has been found to be very important for the study of unstable nuclei near the drip line. Recently it has been shown [12, 13] that the relativistic mean-field (RMF) plus BCS approach wherein the continuum has been replaced by the discrete single particle states for the calculation of the pairing energy provides an alternative fast approach to the relativistic Hartree-Fock-Bogoliubov (RHB) description of the drip-line nuclei. In particular, it has been demonstrated [12] that the resonant states which lie close to zero energy in continuum and gradually come down to become bound with increasing neutron number, play the crucial role in producing extremely neutron rich loosely bound nuclei due to coupling of these resonant states with the bound states near the Fermi level through the pairing interaction. Results of our studies for the even-even Carbon isotopes reveal a similar physical situation as is described below. Our RMF calculations have been carried out using the model Lagrangian density with nonlinear terms both for the σ and ω mesons (shown below) along with the TMA parameterization as described in detail in Refs. [12] and [17].

3 88 G. Saxena, D. Singh 3 where the field tensors H, G and F for the vector fields are defined by and other symbols have their usual meaning. State dependent BCS calculations employing a delta function interaction with the same interaction strength throughout have been performed for the pairing correlation energy. More details of calculations can be found in Ref. [12-16]. 3. RESULTS AND DISCUSSIONS For simplicity and transparency we have not considered the deformation degree of freedom. Indeed, by using such an approach we intend to utilize the advantage one has in the analysis of results in terms of spherical single particle wave functions. This is especially true for the behavior of the single particle states near the Fermi surface which in turn plays an important role in producing weakly bound systems and halo structures. Similarly, within such a framework contributions of neutron and proton single particle states to the density profiles, pairing gaps, total pairing energy etc. which are also equally important in the study of shell closures and halo structures can be demonstrated with clarity. These aspects of the spherical framework indeed make this approach very useful especially for the study of poorly understood exotic nuclei. Nevertheless, we eventually compare our results with those of the RMF calculations including deformation [16], to check if these nuclei persist to be spherical. Our calculations for the isotopes C show that the proton number Z = 6 remains a magic number throughout for the entire chain of isotopes due to a large energy gap between the proton 1p 3/2 and 1p 1/2 sp states. Similarly the neutron numbers N = 6 and 16 are also found to be magic numbers apart from the traditional magic numbers 2 and 8. Moreover, calculations show that the isotope

4 4 Weakly bound neutron rich C isotopes within RMF+BCS approach C lies at the two-neutron drip-line which is a doubly magic nucleus. These findings along with the calculated two neutron separation energy for the C isotopes are found to be in agreement with the available measurements [18]. It is gratifying to note that the relativistic Hartree-Bogoliubov (RHB) approach which provides a unified description of mean-field and pairing correlations, yields for the isotopes C results [19] which are very close to those obtained in the present RMF+BCS calculations. These results of spherical RMF+BCS approach [12, 13] are compared with deformed RMF+BCS approach [14, 15, 16] and it is found that Z = 6 persist as a magic number for all isotopes of C as proton pairing energy is always found zero. N = 6 and 16 provide zero neutron pairing energy for deformed RMF calculations which are in agreement with spherical RMF calculations. Moreover, 10,12,14,22 C isotopes are indeed spherical with zero quadrupole deformation parameter and motivate us to discuss loosely bound structure of C isotopes in detail with the use of spherical RMF+BCS approach. Neutron density S. P. Energy S. P. Energy and Occupancy RMF(TMA) C RMF(TMA) Neutron Neutorn Density Neutron Single Particle Levels C Z=6 N= RMF(TMA) 8 N= Radius [fm] C S.P.Energy 1d5/2 Occupancy 2s1/ Neutron Number N 1d5/2 2s1/2 1d3/2 3s1/2 2p3/2 2p1/2 1d5/2 2s1/2 Fig. 1 Lower panel: Variation of the neutron single particle energies obtained with the TMA force for the Carbon isotopes C with increasing neutron number. The neutron Fermi energy has been shown by filled circles connected by solid line to guide the eyes. Middle panel: Variation of the position and occupancy (no. of neutrons occupying the levels) of the neutron 1d 5/2 and 2s 1/2 single particle states in the C isotopes with increasing neutron number. Upper Panel: The variation in the radial dependence of the neutron density for the nuclei with increasing neutron number N in C isotopes. These results have been obtained in the spherical RMF+BCS calculations with the TMA force parameters. 1p1/2 1p3/2

5 90 G. Saxena, D. Singh 5 A detailed analysis of the calculated results of spherical approach for the C isotopes shows that the neutron rich C isotopes represent examples of somewhat loosely bound system due to simultaneous and gradual filling in of the neutron sp 2s 1/2 and resonant 1d 5/2 states in these isotopes. In order to elucidate this point, we have displayed in the lower panel of Fig. 1 the variation in neutron sp spectrum with increasing neutron number N. It also shows the position of Fermi level. The middle panel in Fig. 1 explicitly shows the variation in position of neutron sp 2s 1/2 and 1d 5/2 states along with their occupancy in terms of number of neutrons. It is found that the neutron sp 1d 5/2 state which lies just above the continuum threshold in C is a resonant state having large pairing gap (1d 5/2 ). With increasing number of neutrons it moves down and becomes bound as can be seen in the middle panel of Fig. 1. Characteristically, the wave function of the resonant 1d 5/2 state is always confined within the potential region and thus has appreciable overlap with the other lower bound states. This results in increased pairing gap and an enhanced contribution to the pairing energy. Due to this the neutron 1d 5/2 state which always lies above the 2s 1/2 state begins to be partially occupied already in 16 C, even before the 2s 1/2 state is totally filled in. This simultaneous filling in of the two states continues until the drip-line isotope 22 C is reached when both the 2s 1/2 and 1d 5/2 are totally occupied. The simultaneous filling in of sp states helps in accommodating more neutrons with comparatively little increase in total binding energy and consequently we have somewhat loosely bound neutron rich isotopes C. The neutron 2s 1/2 state which begins to be occupied in the isotope 16 C essentially gives rise to an extended neutron density distribution. Consequently, an extended tail of density is obtained for 16 C in comparison to that for 14 C as is evidently seen from upper panel of Fig. 1. Correspondingly, an enhancement in the rms radius r n for the neutron distribution is also found from 14 C to 16 C. The matter density of 16 C extracted from cross section measurements [1, 7, 20] also supports an extended neutron tail in the isotope 16 C. With further addition of neutrons to 16 C the density remains almost unchanged for the isotopes C as can be seen in upper panel of Fig. 1. For the drip-line nucleus 22 C the Fermi level moves just above the continuum threshold at ε f = 0.43 MeV as can be seen in the lower panel of Fig 1. The neutron sp spectrum of the isotope 22 C shows that the sp state next to fully occupied 1d 5/2 is higher in energy by about 3.2 MeV in the continuum. Due to such a large gap in the sp energy the neutron number N=16 shows magicity. Thus 22 C represents an example of a spherical dripline nucleus for which the neutrons are somewhat loosely bound. It is gratifying to note that this result of 22 C is in the agreement with the recent experiment by Tanaka et al. [3].

6 6 Weakly bound neutron rich C isotopes within RMF+BCS approach 91 Fig. 2 Lower panel: The RMF potential energy (sum of the scalar and vector potentials), for the nucleus 22 C shown by the solid line as a function of radius r. Lower panel: Radial wave functions of a few representative neutron single particle states with energy close to the Fermi surface for the nucleus 22 C. The single particle spectrum in Fig. 1 also suggests that the odd-neutron isotopes C with neutron number N = 9, 11, 13 and 15 should be even more weakly bound as compared to even-neutron isotopes (nonmagic) since the binding energy of the even neutron isotopes has a sizable contribution additionally from the pairing energy of the last two neutrons besides the single particle energy contributions. This is especially interesting for the isotope 15 C since the neutron sp spectrum in Fig. 1 evidently provides support to a halo like formation for this isotope due to the fact that the odd neutron falls in the 2s 1/2 state while it lies close to the zero energy as well as the Fermi level. It is evident from the above discussion that the ground state properties of the even-even C isotopes are to an extent influenced by the gradual and simultaneous filling in of the close lying neutron 2s 1/2 and 1d 5/2 sp states. These isotopes are somewhat weakly bound due to the wide spatial spread in the wave function of the 2s 1/2 state. As an example in Fig. 2, we have plotted the calculated RMF potential, a sum of scalar and vector potential, along with the wave function of the neutron single particle states for drip line doubly magic nucleus 22 C. In upper panel of Fig.2 it is seen that even though 2s 1/2 state with s.p. energy MeV lie below 1d 5/2 state of s.p. energy MeV state in the spectrum but due to lack of centrifugal barrier (2s state l=0) in 2s 1/2 state its wave function is spread over large

7 92 G. Saxena, D. Singh 7 spatial extension as can be seen from upper panel of Fig. 2 and contributes substantially to the neutron density distribution at large distances. It results well spread density distribution for 22 C as discussed earlier in upper panel of Fig. 1. Thus the filling in of the 2s 1/2 single particle state with increasing neutron number in the C isotopes causes the formation of weakly bound structure in neutron rich C isotopes. Fig. 3 The present RMF results for binding energy (upper panel), two neutron separation energy (middle panel) and neuron rms radii (lower panel) for C isotopes obtained with the TMA (circle), NL3 (triangle) and TM2 (diamond) force parameters are compared with the deformed RMF (hexagon) calculations using TMA force parameters and available experimental data filled square) [1, 18]. In order to check the possible dependence and sensitivity of results on the force parameterization, we have also carried out these calculations employing other popular RMF parameterizations, the TM2 and NL3 as given in detail in Refs. [17], and [21] respectively. With this in view, we have plotted in Fig. 3. Binding energy, two neutron separation energy and neutron radii calculated by spherical RMF approach using TMA, NL3, TM2 parameters and by deformed RMF approach using TMA parameters along with available experimental data. A comparison of the results for the C isotopes shows that these forces (the TMA, NL3 and TM2 parameterizations) essentially yield similar results. Indeed the results for the separation energy, neutron radii, binding energy and drip lines etc. are almost identical for all the C isotopes and are in good agreement with experimental data.

8 8 Weakly bound neutron rich C isotopes within RMF+BCS approach SUMMARY Inspired by the recent experiments [3] indicating doubly magic halo nucleus 22 C and encouraged by the success of our relativistic mean-field (RMF) plus state dependent BCS approach for the description of the ground state properties of the drip-line nuclei [12-15], we have employed this approach to explore loosely bound structures in neutron rich C isotopes. The Lagrangian density with nonlinear terms for the σ and ω mesons along with the TMA force parameters [17] has been employed for the purpose. The calculated results comprising wave function, single particle energy, two-neutron separation energies, neutron radii, and the neutron density distribution profiles have been analyzed to examine the occurrence of weakly bound nuclei. Our calculations in the case of neutron rich nuclei addition to filling in of single particle s state lying close to the continuum threshold, gives rise to a sudden growth in the rms radius leading to loosely bound system. The neutron rich carbon isotopes C are found to represent weakly bound system due to interplay of the 2s 1/2 and resonant 1d 5/2 states lying close to the continuum threshold. These results are found to be consistent with the experimental data [18], calculations with other force parameters and calculations with deformation degree of freedom. Acknowledgements. Authors would like to express deep and sincere gratitude to Prof. H. L. Yadav, Banaras Hindu University,Varanasi, India for his guidance and constructive comments. The authors are indebted to Dr. L. S. Geng, Beihang University, China, for valuable correspondence. REFERENCES 1. I. Tanihata, J. Phys. G 22,157 (1996); I. Tanihata et al., Phys. Lett. B 512, 261 (2001). 2. A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida and I. Tanihata, Phys. Rev. Lett. 84, 5493 (2000); A. Ozawa et al., Nucl. Phys. A 709, 60 (2002). 3. K. Tanaka et al., Phys. Rev. Lett.104, (2010). 4. A. Jensen, K. Riisagar, D. V. Fadarov and E. Garrido, Rev.Mod. Phys.76, 215 (2004) and reference therein; A. S. Jensen and K. Riisager, Phys. Lett. B 480, 39 (2000); P. G. Hansen and A. S. Jensen, Annu. Rev. Nucl. Part. Sci. 45, 591(1995). 5. A. B. Migdal, Jour. Nucl. Phys. 16, 238 (1973). 6. S. Dasgupta, I. Mazumdar and V.S. Bhasin, Phys. Rev. C 50, 551(R) (1994). 7. T. Yamaguchi et al., Nucl. Phys. A 734, E73 (2004). 8. W. Horiuchi and Y. Suzuki, Phys. Rev. C 74, (2006). 9. H.G. Bohle et al., Nucl. Phys. A 734, 345 (2004). 10. J.S. Wang et al., Nucl. Phys A 691, 618 (2001). 11. J. Dobaczewski et al., Phys. Rev. C 53, 2809 (1996); K. Bennaceur, J. Dobaczewski and M. Ploszajczak, Phys. Lett. B 496, 154 (2000). 12. H. L. Yadav, M. Kaushik and H. Toki, Int. Jour. Mod. Phys. E 13, 647 (2004). 13. G. Saxena, D. Singh, H. L. Yadav, A. Haga and H. Toki, Mod. Phys. Lett. A 23, 2589 (2008).

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