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Systematic analysis of neutron-rich carbon isotopes
Nuclear
ELSEVIER
Physics
A7 19 (2003)
3 12~3
15~ www.elsevier.comllocate/npe
Systematic
analysis
G. Thiamovaa’ “Department
, N’. Itagaki”, of Physics,
bThe Institute Japan
of neutron-rich T. Otsuka”,b
University
of Physical
carbon
and K. Ikedab
of Tokyo,
and Chemical
isotopes
Hongo,
Research
Tokyo
(RIKEN),
113-0033,
Japan
Wako, Saitama,
351-0198,
The low-lying structure of the C isotopes is investigated using the Antisymmetrized Molecular Dynamics (AMD) Multi-Slater Determinant model. The calculated 2+ energies of the even-even isotopes indicate the change from spherical to deformed structure at N = 8. This is consistent with the isotope change of the single particle energy deduced from recent experiments. Binding energies and r.m.s. radii are also calculated and compared with the experimental data. 1. INTRODUCTION Structure of light neutron-rich nuclei around the C region is extensively studied using radioactive isotopes beams. Newly discovered magic number of N=16 corresponds to the driplines of C; N, 0 isotopes [1,2]. The dripline nucleus of the C isotopes is 22C. igC is known to have the halo structure due to the valence neutron in the s-orbit. However, Therefore, the the ground state of 15C is l/2+ with the valence neutron in the s-orbit. s-orbit is already occupied in i5C . So, how the large r.m.s. radius and sharp momentum distribution in igC, suggesting the s-orbit, can be explained? One possible explanation is a structure change in C isotopes. Even if l/2+ state (s-orbit) is already occupied in 15C, the order of orbits in energy would change in neutron-rich C isotopes, if the system had different shape with increasing neutron number. To study the structure changes we perform systematic analysis of the carbon isotopes using the improved version of the AMD method [3] and compare the results with the known experimental data. 2. MULTI-SLATER
DETERMINANT
AMD
In this paper we apply the improved version of the AMD method [3] and re-analyze the systematics of the C isotopes. In this approach the value of the r.m.s. radius is constrained during the cooling process and afterwards a lot of Slater determinants with different intrinsic structure (corresponding to different constrained r.m.s. radii) are superposed. The mixing amplitudes of these Slater determinants are determined after the angular momentum projection by the diagonalization of the Hamiltonian. The approximation of *Permanent
address:
Nuclear
Physics
0375-9474/03/$ - see front matter doi: 10.101 S/SO3759474(03)00939-4
Institute,
0 2003 Elsevier
Czech
Academy
Science
B.V.
of Sciences, All rights
Prague-Rez,
reserved.
Czech
Republic
G. Thiamoua et al. /Nuclear Physics A719 (2003) 312c-315~
313c
variation before projection (VBP) is drastically improved. Furthermore, when we solve the cooling equation, different initial sets of parameters are prepared to overcome the local minimum problem and after obtaining the cooled states we superpose these Slater determinants. The r.m.s. radii, the excitation energies of the 2+ states for the even-even C isotopes and the binding energies are calculated and compared with the experimental data. The behaviour of these quantities suggests the appearance of the N=16 magic number for the C isotopes. The detailed explanation of the method can be found in [4]. The Hamiltonian and the effective nucleon-nucleon interaction used is the same as in Ref.[5], and the Majorana parameter M of the Volkov No. 2 interaction and the strength of the G3RS spin-orbit interaction are determined by the Q-C): and o-n scattering phase shift analysis. 3. RESULTS We show summarized basis states value of the first results
the results of the C isotopes. The number of basis states employed are in Table 1. To overcome the problem of variation before (J) projection, 15 calculated from different initial parameter sets are prepared for each constraint r.m.s. radius in case of the even-even isotopes. For the even-odd isotopes our are presented where one basis function is employed for each r.m.s. constraint.
Table 1 The number of the employed basis states for the C isotopes r.m.s. radius (fm). constrained r.m.s. (fm) r2C 13C 14C r5C 16C 17C 2.3 15 1 2.4 15 1 15 1 2.5 15 1 15 1 15 15 1 15 1 2.6 2.7 15 1 2.8 1 2.9 3.0 3.1 3.2
as a function ‘sC
rgC
15 15 15
of constrained “C
15 15 15
‘ICI
22C
1 1 1
15 15 15
1 1 1
The calculated binding energies; the excitation energies of the 2+ states and the r.m.s. radii of the even-even isotopes are listed in Table 2 and 3. Comparison of the calculated and experimental binding energies for the even-odd isotopes is presented in Table 4. The experimental data are taken from [6-lo]. The binding energies of the even-even isotopes are well reproduced except for “C. The r2C! nucleus has been known to have both components of the shell and cluster structure, and the AMD method is still too simple for this nucleus. However, other binding energies and the 2+ excitation energies describe the experimental tendency. Both the calculation and the experimental data show very large 2+ excitation energy of 14C (7~8 MeV) due to
314c
G. Thianzoua et al. /Nuclear Physics A719 (2003) 312c-315~
Table 2 The calculated binding energies (B.E.) and the excitation energies of the 2+ states (& (2’))of the even-even isotopes. The values in the last row are the experimental data. All units are in 41eV. 12C 14C 16C 18C Z°C 22C B.E. 85.8 107.7 110.3 114.0 123.2 122.0 (92.2) (105.3) (110.8) (115.7) (119.2) (120.3) & @+I 3.6 8.5 2.5 1.8 3.4 5.7 exp. 4.4 7.0 1.8 1.6
Table 3 The calculated r.m.s. radii of the even-even C isotopes (Cal.), which are compared with the experimental values (exp.) deduced from the interaction cross section using the Glauber model [lo]. All units are in fm. 12C 14C 1% 18C Z°C Z2C 2.55 2.65 2.65 2.79 cal. 2.40 2.39 exe. 2.351t0.2 2.30f0.07 2.7OztO.03 2.821tO.04 2.98zkO.05
the closed shell effect (N = 8), and it suddenly decreases beyond 14C. This may suggest the structure change from spherical to deformed shape. Although the calculated values of the r.m.s.radii in Table 3 are relatively smaller than the experimental ones, both show drastic increase at “C. This also suggest the structure change. The spins of the ground states of the even-odd isotopes are well reproduced except for the “C nucleus which is known to have J”=1/2+ in the ground state. It seems it is necessary to solve the tail effect of the s-wave for the l/2+ state to explain the experimental data. The agreement of the calculated and experimental binding energies is reasonable. However, we expect that the agreement will be improved by a few MeV when more basis functions are employed in our future calculations. In summary; the calculated binding energies, 2+ excitation energies and r.m.s. radii of the even-even isotopes are in reasonable agreement with the experimental values. Very for the large 2+ excitation energy of 14C and its sudden decrease with the minimum midshell 18C isotope and the behavior of the r.m.s radii suggest the structure change from
Table 4 The calculated binding energies (B.E.) of the even-odd isotopes. The spin and parity of the lowest state which come out from the calculations is also indicated. The values in the last row are the experimental data. All units are in MeV. 13C 15C 17C 19C 21C J” l/25/2+ 3/2+ l/2+ l/2+ B.E. 94.2 104.2 104.7 98.6 116.2 exp. 97.2 106.5 111.5 115.8 118.8
G. Thiamoua et al. /Nuclear
Physics A719 (2003) 312c-315~
31%
spherical to deformed shape. We have also presented our first results for the even-odd isotopes. The experimental data are well reproduced except for the r5C nucleus. Further improvements are in progress.
Acknowledgments The authors thank members of the RI beam science laboratory in RIKEN and Nuclear Theory Group at the University of Tokyo for discussions and encouragements. One of the authors (NJ) thanks Prof. H. Horiuchi, and Dr. Y. Kanada-En’yo for fruitful discussions. This work is supported in part by Grant-in-Aid for Scientific Research (13740145) from the Ministry of Education, Science and Culture. REFERENCES 1,
A Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, Phys. Rev. Lett. 84 (2000) 5493. 2. T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, &I. Honma, and T. Mizusaki, Phys. Rev. Lett. 87 (2001) 082502. 3. Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C 54 (1996) R468. 4. N. Itagaki and S. Aoyama, Phys. Rev. C 61 (2000) 024303. 5. N. Itagaki and S. Okabe, Phys. Rev. C 61 (2000) 044306. 6. F. Ajzenberg-Selove, Nucl. Phys. A 506 (1990) 1.; 523 (1991) 1.; 475 (1987) 1. 7. D. R. Tilley, H. R. Weller and C. IVI. Cheves, Nucl. Phys. A 564 (1993) 1. 8. P. M. Endt, Nucl. Phys. A 521 (1990) 1. 9. G. Audi and A. H. Wapstra, Nucl. Phys. A 595 (1995) 409. IO. ,4. Ozawa, T. Suzuki, and I. Tanihata, Nucl. Phys. A 693 (2001) 33.
ELSEVIER
Physics
A7 19 (2003)
3 12~3
15~ www.elsevier.comllocate/npe
Systematic
analysis
G. Thiamovaa’ “Department
, N’. Itagaki”, of Physics,
bThe Institute Japan
of neutron-rich T. Otsuka”,b
University
of Physical
carbon
and K. Ikedab
of Tokyo,
and Chemical
isotopes
Hongo,
Research
Tokyo
(RIKEN),
113-0033,
Japan
Wako, Saitama,
351-0198,
The low-lying structure of the C isotopes is investigated using the Antisymmetrized Molecular Dynamics (AMD) Multi-Slater Determinant model. The calculated 2+ energies of the even-even isotopes indicate the change from spherical to deformed structure at N = 8. This is consistent with the isotope change of the single particle energy deduced from recent experiments. Binding energies and r.m.s. radii are also calculated and compared with the experimental data. 1. INTRODUCTION Structure of light neutron-rich nuclei around the C region is extensively studied using radioactive isotopes beams. Newly discovered magic number of N=16 corresponds to the driplines of C; N, 0 isotopes [1,2]. The dripline nucleus of the C isotopes is 22C. igC is known to have the halo structure due to the valence neutron in the s-orbit. However, Therefore, the the ground state of 15C is l/2+ with the valence neutron in the s-orbit. s-orbit is already occupied in i5C . So, how the large r.m.s. radius and sharp momentum distribution in igC, suggesting the s-orbit, can be explained? One possible explanation is a structure change in C isotopes. Even if l/2+ state (s-orbit) is already occupied in 15C, the order of orbits in energy would change in neutron-rich C isotopes, if the system had different shape with increasing neutron number. To study the structure changes we perform systematic analysis of the carbon isotopes using the improved version of the AMD method [3] and compare the results with the known experimental data. 2. MULTI-SLATER
DETERMINANT
AMD
In this paper we apply the improved version of the AMD method [3] and re-analyze the systematics of the C isotopes. In this approach the value of the r.m.s. radius is constrained during the cooling process and afterwards a lot of Slater determinants with different intrinsic structure (corresponding to different constrained r.m.s. radii) are superposed. The mixing amplitudes of these Slater determinants are determined after the angular momentum projection by the diagonalization of the Hamiltonian. The approximation of *Permanent
address:
Nuclear
Physics
0375-9474/03/$ - see front matter doi: 10.101 S/SO3759474(03)00939-4
Institute,
0 2003 Elsevier
Czech
Academy
Science
B.V.
of Sciences, All rights
Prague-Rez,
reserved.
Czech
Republic
G. Thiamoua et al. /Nuclear Physics A719 (2003) 312c-315~
313c
variation before projection (VBP) is drastically improved. Furthermore, when we solve the cooling equation, different initial sets of parameters are prepared to overcome the local minimum problem and after obtaining the cooled states we superpose these Slater determinants. The r.m.s. radii, the excitation energies of the 2+ states for the even-even C isotopes and the binding energies are calculated and compared with the experimental data. The behaviour of these quantities suggests the appearance of the N=16 magic number for the C isotopes. The detailed explanation of the method can be found in [4]. The Hamiltonian and the effective nucleon-nucleon interaction used is the same as in Ref.[5], and the Majorana parameter M of the Volkov No. 2 interaction and the strength of the G3RS spin-orbit interaction are determined by the Q-C): and o-n scattering phase shift analysis. 3. RESULTS We show summarized basis states value of the first results
the results of the C isotopes. The number of basis states employed are in Table 1. To overcome the problem of variation before (J) projection, 15 calculated from different initial parameter sets are prepared for each constraint r.m.s. radius in case of the even-even isotopes. For the even-odd isotopes our are presented where one basis function is employed for each r.m.s. constraint.
Table 1 The number of the employed basis states for the C isotopes r.m.s. radius (fm). constrained r.m.s. (fm) r2C 13C 14C r5C 16C 17C 2.3 15 1 2.4 15 1 15 1 2.5 15 1 15 1 15 15 1 15 1 2.6 2.7 15 1 2.8 1 2.9 3.0 3.1 3.2
as a function ‘sC
rgC
15 15 15
of constrained “C
15 15 15
‘ICI
22C
1 1 1
15 15 15
1 1 1
The calculated binding energies; the excitation energies of the 2+ states and the r.m.s. radii of the even-even isotopes are listed in Table 2 and 3. Comparison of the calculated and experimental binding energies for the even-odd isotopes is presented in Table 4. The experimental data are taken from [6-lo]. The binding energies of the even-even isotopes are well reproduced except for “C. The r2C! nucleus has been known to have both components of the shell and cluster structure, and the AMD method is still too simple for this nucleus. However, other binding energies and the 2+ excitation energies describe the experimental tendency. Both the calculation and the experimental data show very large 2+ excitation energy of 14C (7~8 MeV) due to
314c
G. Thianzoua et al. /Nuclear Physics A719 (2003) 312c-315~
Table 2 The calculated binding energies (B.E.) and the excitation energies of the 2+ states (& (2’))of the even-even isotopes. The values in the last row are the experimental data. All units are in 41eV. 12C 14C 16C 18C Z°C 22C B.E. 85.8 107.7 110.3 114.0 123.2 122.0 (92.2) (105.3) (110.8) (115.7) (119.2) (120.3) & @+I 3.6 8.5 2.5 1.8 3.4 5.7 exp. 4.4 7.0 1.8 1.6
Table 3 The calculated r.m.s. radii of the even-even C isotopes (Cal.), which are compared with the experimental values (exp.) deduced from the interaction cross section using the Glauber model [lo]. All units are in fm. 12C 14C 1% 18C Z°C Z2C 2.55 2.65 2.65 2.79 cal. 2.40 2.39 exe. 2.351t0.2 2.30f0.07 2.7OztO.03 2.821tO.04 2.98zkO.05
the closed shell effect (N = 8), and it suddenly decreases beyond 14C. This may suggest the structure change from spherical to deformed shape. Although the calculated values of the r.m.s.radii in Table 3 are relatively smaller than the experimental ones, both show drastic increase at “C. This also suggest the structure change. The spins of the ground states of the even-odd isotopes are well reproduced except for the “C nucleus which is known to have J”=1/2+ in the ground state. It seems it is necessary to solve the tail effect of the s-wave for the l/2+ state to explain the experimental data. The agreement of the calculated and experimental binding energies is reasonable. However, we expect that the agreement will be improved by a few MeV when more basis functions are employed in our future calculations. In summary; the calculated binding energies, 2+ excitation energies and r.m.s. radii of the even-even isotopes are in reasonable agreement with the experimental values. Very for the large 2+ excitation energy of 14C and its sudden decrease with the minimum midshell 18C isotope and the behavior of the r.m.s radii suggest the structure change from
Table 4 The calculated binding energies (B.E.) of the even-odd isotopes. The spin and parity of the lowest state which come out from the calculations is also indicated. The values in the last row are the experimental data. All units are in MeV. 13C 15C 17C 19C 21C J” l/25/2+ 3/2+ l/2+ l/2+ B.E. 94.2 104.2 104.7 98.6 116.2 exp. 97.2 106.5 111.5 115.8 118.8
G. Thiamoua et al. /Nuclear
Physics A719 (2003) 312c-315~
31%
spherical to deformed shape. We have also presented our first results for the even-odd isotopes. The experimental data are well reproduced except for the r5C nucleus. Further improvements are in progress.
Acknowledgments The authors thank members of the RI beam science laboratory in RIKEN and Nuclear Theory Group at the University of Tokyo for discussions and encouragements. One of the authors (NJ) thanks Prof. H. Horiuchi, and Dr. Y. Kanada-En’yo for fruitful discussions. This work is supported in part by Grant-in-Aid for Scientific Research (13740145) from the Ministry of Education, Science and Culture. REFERENCES 1,
A Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, Phys. Rev. Lett. 84 (2000) 5493. 2. T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, &I. Honma, and T. Mizusaki, Phys. Rev. Lett. 87 (2001) 082502. 3. Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C 54 (1996) R468. 4. N. Itagaki and S. Aoyama, Phys. Rev. C 61 (2000) 024303. 5. N. Itagaki and S. Okabe, Phys. Rev. C 61 (2000) 044306. 6. F. Ajzenberg-Selove, Nucl. Phys. A 506 (1990) 1.; 523 (1991) 1.; 475 (1987) 1. 7. D. R. Tilley, H. R. Weller and C. IVI. Cheves, Nucl. Phys. A 564 (1993) 1. 8. P. M. Endt, Nucl. Phys. A 521 (1990) 1. 9. G. Audi and A. H. Wapstra, Nucl. Phys. A 595 (1995) 409. IO. ,4. Ozawa, T. Suzuki, and I. Tanihata, Nucl. Phys. A 693 (2001) 33.