# Systematic study of even-even Mg isotopes

###### Abstract

A systematic study for 2 and 4 energies for even-even Mg by means of large-scale shell model calculations using the effective interaction USDB and USDBPN with SD and SDPN model space respectively. The reduced transition probability B(E2;) were also calculated for the chain of Mg isotopes.Very good agreement were obtained by comparing the first 2 and 4 levels for all isotopes with the recently available experimental data and with the previous theoretical work using 3DAMP+GCM model, but studying the transition strengths B(E2; 02) for Mg isotopes using constant proton-neutron effective charges prove the limitations of the present large-scale calculations to reproduce the experiment in detail.

###### pacs:

23.20.Lv, 21.60.Cs## I Introduction

The low energy structure of magnesium nuclei has attracted considerable interests in the last decade, both experimental and theoretical. In particular, the sequence of isotopes Mg encompasses three spherical magic shell numbers : N=8, 20 and 28 and, therefore presents an excellent case for studies of the evolution of shell structure with neutron number, weakening of spherical shell closures, disappearance of magic numbers, and the occurrence of islands of inversion OM08 ().

Extensive experimental studies of the low-energy structure of Mg isotopes have been carried out at the Institute of Physical and Chemical Research, Japan (RIKEN) H01 (); S09 (), Michigan State University (MSU) BV99 (); JM06 (); AL07 (); AG07 () , the Grand Accélérateur National d’Ions Lourds, France (GANIL) VC01 () and CERN ON05 (); WS09 ().

In addition to numerous theoretical studies based on large-scale shell-model calculations EC98 (); YU99 (); TO01 (); TT01 (); EC05 (); FM05 (), the self-consistent mean-field framework, including the nonrelativistic Hartree-Fock-Bogolibov (HFB) model with Skyrme JT97 () and Gogny forces RJ02 () and the relativistic mean-field (RMF) model SK91 (); ZZ96 () as well as the macroscopic-microscopic model based on a modified Nilsson potential QJ06 (), have been used to analyze the ground-state properties (binding energies, charge radii, and deformations) and low-lying excitation spectra of magnesium isotopes.

The purpose of present work is to study the ground state 2 and 4 excitation energies and the reduced transition probabilities B(E2; 02) (efm) of the even-even Mg isotopes using the new version of Nushell@MSU for windows BW07 () and compare these calculations with the most recent experimental and theoretical work.

## Ii Shell Model Calculations

The calculations were carried out in the SD and SDPN model spaces with the USDB and USDBPN effective interactions BW06 () using the shell model code Nushell@MSU for windows BW07 ().

The core was taken as O with 4 valence protons and 4,6,8,10,12,14,16 valence nucleons for Mg, Mg, Mg, Mg, Mg, Mg and Mg respectively distributed over 1d , 1d and 2s.

The effective interaction USDB with model space SD where used in the calculation of the Mg isotopes, while USDBPN in pn formalism where employed with SDPN model space for Mg nucleus.

## Iii Results and Discussion

The test of success of large-scale shell model calculations is the predication of the low-lying 2 and 4 and the transition rates B(E2; 02) using the optimized effective interactions for the description of sd-shell nuclei.

Figure 1 presents the comparison of the calculated E(2) energies from the present work (P.W.) with the experiment NN12 (), the work of J. M. Yao et al.JM11 () using 3DAMP+GCM model with the relativistic density functional PC-F1.

The comparison shows that our calculation are in better agreement with the experiment than the work of Ref.JM11 ()

Figure 2 shows the comparison of the calculated low-lying E(4) excitation energies from present work (P.W.)with the experiment NN12 (), the work of J. M. Yao et al.JM11 () using 3DAMP+GCM model with the relativistic density functional PC-F1. The comparison shows very clear that our prediction for the E(4) are in better agreement with the experiment.

Figure 3 presents the comparison of the calculated B(E2; 02) (efm) from present work (P.W.) with the experimental data taken from the Institute of Physical and Chemical Research, Japan (RIKEN) H01 (); S09 (), the Grand Accélérateur National d’Ions Lourds, France (GANIL) VC01 () and CERN ON05 (); WS09 (), the previous theoretical work of J. M. Yao et al.JM11 () using 3DAMP+GCM model and with the work of R. Rodríguez-Guzmán et al.RJ02 () using HFB-Gogny force. The effective charges were taken to be e=1.25e for proton and e=0.8e for neutron. With these effective charges our prediction for the reduced transition probability B(E2; 02) are more closer to the experimental values than the previous work of Refs. JM11 (); RJ02 ().

## Iv Summary

Unrestricted large scale-shell model calculations were performed using the effective interactions USDB and USDBPN in pn formalism with the model space SD and SDPN to study the low lying 2 and 4 energies for even-even Mg isotopes and the transition strengths B(E2; 02) for the mass region A=20-32. Good agreement were obtained in comparing our theoretical work with the recent available experimental data and with the most recent theoretical work of Ref.JM11 () using 3DAMP+GCM model with the relativistic density functional PC-F1 for both excitation energies and transition strengths.

## References

- (1) O. Sorlin and M.-G. Porquet, Prog. Part. Nucl. Phys. 61, 602 (2008).
- (2) H. Iwasaki et al., Phys. Lett. B 522, 227 (2001).
- (3) S. Takeuchi et al., Phys. Rev. C 79, 054319 (2009).
- (4) B. V. Pritychenko et al., Phys. Lett. B 461, 322 (1999).
- (5) J. M. Cook, T. Glasmacher, and A. Gade, Phys. Rev. C 73,024315 (2006).
- (6) A. Gade et al., Phys. Rev. Lett. 99, 072502 (2007).
- (7) A. Gade et al., Phys. Rev. C 76, 024317 (2007).
- (8) V. Chisé et al., Phys. Lett. B 514, 233 (2001).
- (9) O. Niedermaier et al., Phys. Rev. Lett. 94, 172501 (2005).
- (10) W. Schwerdtfeger et al., Phys. Rev. Lett. 103, 012501 (2009).
- (11) E. Caurier, F. Nowacki, A. Poves, and J. Retamosa, Phys. Rev. C 58, 2033 (1998).
- (12) Y. Utsuno, T. Otsuka, T. Mizusaki, and M. Honma, Phys. Rev. C 60, 054315 (1999).
- (13) T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki, Phys. Rev. Lett. 87, 082502 (2001).
- (14) T. Otsuka, Y. Utsuno, T. Mizusaki, and M. Honma, Nucl. Phys. A 685, 100 (2001).
- (15) E. Caurier, G. Martiez-Pinedo, F. Nowacki, A. Poves, and A.P. Zuker, Rev. Mod. Phys. 77, 427 (2005).
- (16) F. Maréchal et al., Phys. Rev. C 72, 044314 (2005).
- (17) J. Terasaki, H. Flocarda, P. H. Heenen, and P. Bonche, Nucl. Phys. A 621, 706 (1997).
- (18) R. Rodríguez-Guzmán, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 709, 201 (2002).
- (19) S. K. Patra and C. R. Praharaj, Phys. Lett. B 273, 13 (1991).
- (20) Z. Z. Ren, Z. Y. Zhu, Y. H. Cai, and G. Xu, Phys. Lett. B 380, 241 (1996).
- (21) Q. J. Zhi and Z. Z. Ren, Phys. Lett. B 638, 166 (2006).
- (22) NuShell@MSU, B. A. Brown and W. D. M.Rae, MSU-NSCL report (2007).
- (23) B. Alex Brown,. and W. A. Richter, Phys. Rev. C 74, 03431 (2006).
- (24) National Nuclear Data Center (ENSDF). http://www.nndc.bnl.gov
- (25) J. M. Yao, H. Mei, H. Chen, J. Meng, P. Ring, and D. Vretenar, Phys. Rev. C 83, 014308 (2011).