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Extracting single-particle information for the superheavy mass region by studying excited structure in transfermium nuclei
Nuclear Physics A 834 (2010) 41c–44c www.elsevier.com/locate/nuclphysa
Extracting single-particle information for the superheavy mass region by studying excited structure in transfermium nuclei Yang Sunab∗ a
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, PR China
b
Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, PR China
Recent experimental advances have made it possible to study spectroscopy in very heavy nuclei. We show that from the excited high-spin structure of transfermium isotopes, one may gain useful information on single-particle states for the superheavy mass region, which is the key to locating the anticipated ‘island of stability’. In this work, we employ the Projected Shell Model for Cf, Fm, and No isotopes to study rotation alignment of the particles that occupy particular high-j intruder orbitals. 1. Introduction It has been suggested that an “island of stability” [1] exists in the superheavy mass region possibly with new magic numbers. The occurrence of superheavy elements (SHE) is attributed to the nuclear shell effect because the macroscopic liquid-drop model would predict that such heavy elements can not exist due to large Coulomb repulsion. The singleparticle (SP) states as a consequence of the shell effect have thus become the discussion focus in the SHE problem. The precise location of the predicted island beyond the known magic number 126 for neutrons and 82 for protons depends sensitively on the SP structure. While some SHE have been successfully synthesized in laboratory, little or nothing about their structure has been known. The heaviest nuclei for which detailed spectroscopy measurement can currently be performed lie in the transfermium mass region [2]. These nuclei, typically with Z ≈ 100 and N ≈ 150 − 160, are not really SHE. However, they are at the gateway to the SHE region, and furthermore, they are well deformed. With deformation, the Fermi surfaces are surrounded by some orbitals [3] originating from the subshells near the anticipated new magic numbers. Thus, the study of these deformed transfermium nuclei may provide an indirect way to access unknown SP states of the closed spherical shells, which are of direct relevance to the location of the predicted island. Not all the structure information of the transfermium nuclei are equally useful. Inbeam data for yrast γ-ray spectroscopy of even-even transfermium nuclei reveal that at low-spins near their ground state, they all exhibit very similar collective behavior with regular rotational level sequence [2]. This tells us that near the ground state, these well∗ The author would like to thank F. Al-Khudair and G.-L. Long for their contribution. Work supported by the National Natural Science Foundation of China under Contract 10875077 and by the Chinese Major State Basic Research Development Program through Grant 2007CB815005.
0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.01.013
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Y. Sun / Nuclear Physics A 834 (2010) 41c–44c
deformed nuclei behave just like a heavy, rigid rotor. They show a strong collectivity, diluting any individual role of single-particles. Therefore, nothing can be learnt from these low-spin rotor states of even-even nuclei. Useful information may be gained from quasiparticle (QP) excitations, such as high spin structure along the yrast line, structure of K-isomers, and multi-QP configurations in odd-mass or odd-odd nuclei. We study one of the QP excitations, namely, the structure change along the yrast line in even-even transfermium nuclei. It is understood [4] that when a nucleus is rotating, the Coriolis antipairing force acts on the QP pairs, which is proportional to the size of angular momentum j of nucleons under consideration. One thus expects those pairs with the highest j value to break first and align their rotation along the rotational axis. Rotation-alignment lowers the energy of the high-j configurations, and at a certain spin, the states with aligning nucleons can become so low energetically that the aligning bands cross the ground state band (g-band), constituting a situation of so-called band-crossing [4]. The characteristic feature of rotation-alignment in transfermium nuclei is that the aligning pairs come mainly from the two high-j intruder orbitals: proton pairs from the i13/2 orbital and neutron pairs from the j15/2 orbital. The presence of these two highj orbitals near the Fermi level and their response to rotation can lead to interesting observations. Therefore, from the study of rotation-alignment in transfermium nuclei one may gain information on the high-j orbitals and the spin-orbit interaction of SP states. 2. Projected Shell Model study on rotation alignment
7
4-qp 2-qp, π i13/2 2-qp, ν j15/2 0-qp after mixing
6
Enegy (MeV)
5 4 3 2
252
1 0
0
4
8
12
16
20
254
Cf
24
28
0
4
8
12
16
20
256
Fm
24
28
0
4
8
12
16
20
No
24
28
32
Spin
Figure 1. Band diagrams for 252 Cf, 254 Fm, and 256 No. Important configurations are shown: 0-qp (solid curves), neutron 2-qp (dashed curves) and proton 2-qp (dotted curves), and 4-qp (dotted-dashed curves). Filled diamonds denote the yrast states obtained after configuration mixing.
We extend the applicability of the Projected Shell Model (PSM) [5] to the transfermium region. For the model parameters in this mass region, we refer to Ref. [6]. In a so-called
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Y. Sun / Nuclear Physics A 834 (2010) 41c–44c
J (1)
band diagram [5], each angular-momentum-projected state of the PSM basis produces ˆ Pˆ I |Φκ /Φκ | Pˆ I |Φκ . The first one, a rotational band κ given by Eκ (I) = Φκ | H KK KK E0 (I), represents the g-band in which all the particles are paired. The remaining states represent bands built upon multi-quasiparticle states [5]. In a band diagram, one may observe several band-crossings at various spins and study their structure. For the transfermium nuclei studied in this work, we have found interesting bandcrossings. We show in Fig. 1 band diagrams for three N = 154 isotones: 252 Cf, 254 Fm, and 256 No. As the neutron number is unchanged in an isotonic chain but the proton Fermi level varies with the shell filling, the relative position of proton and neutron Fermi levels, and therefore the SP states in the vicinity of the Fermi levels, differ in the three. In Fig. 1, those curves starting from 2 – 2.5 MeV are the bands with 2-qp high-j configurations. At low-spins, the g-band is low, and is the dominant component in the yrast wave function. However, it is seen that in all the three diagrams, several 2-qp bands cross the g-band around spin I = 24 and become energetically lower after the band-crossing. With a careful inspection, very delicate differences in the three cases can be found: After the band-crossing, the lowest band in 252 Cf is a 2-qp band of i13/2 protons (dotted curve) and in 256 No a 2-qp band of j15/2 neutrons (dashed curve), whereas in 254 Fm, the lowest dotted and the dashed curves are nearly degenerate after the band-crossing.
252
254
Figure 2. Top row shows calculated static moments of inertia (open symbols) and available data (filled symbols). Middle and bottom rows are the predicted B(E2) (in e2 b2 ) and g factor values, respectively.
Figure 3. Calculated dynamic moments of inertia J (2) (open symbols) and experimental data (filled symbols) for 252,254 No.
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Y. Sun / Nuclear Physics A 834 (2010) 41c–44c
The above observation suggests a picture that at the band-crossing region, proton and neutron 2-qp high-j configurations compete strongly in the yrast states. After the band-crossing, the proton (neutron) configuration dominates the yrast structure in 252 Cf (256 No). In 254 Fm, the strongest competition between the proton and neutron configurations is predicted. Consequences of the competition can lead to observable effects. The top row of Fig. 2 are the calculated energy levels presented in terms of static moment of inertia. Although they show some differences, the evolution against rotation looks very similar in all the three nuclei; the curves are nearly identical for low spins. The dynamic moment of inertia shown in Fig. 3 is more sensitive in emphasizing the changes, but cannot distinguish proton and neutron contributions. B(E2) values in the middle row show a clear signal of the structure change around I = 24, but cannot extarct proton and neutron contributions either because one cannot experimentally separate the proton and neutron contributions to B(E2). A good probe in this regard is the magnetic moments through the study of g factors because they show rather clearly which kind of particle is aligning. In the bottom row of Fig. 2, a drastic rise in g factor around I = 24 in 252 Cf indicates a proton contribution to the wave functions. This is the consequence of a proton i13/2 2-qp alignment in this nucleus. In a sharp contrast, 256 No shows a sudden drop in g factor around the same spin because of the neutron j15/2 2-qp alignment. The balance in competition between the high-j protons and neutrons leads to near-constant g factors in 254 Fm in the spin region where rotation-alignment takes place. We thus expect that the future g factor measurements will provide a direct test for the present prediction. 3. Discussion It is important to comment on the employed SP states in the present calculation. Strictly speaking, SP states in deformed nuclei are not directly measurable. SP states in our shell-model basis are constructed by using the deformed Nilsson potential, and the above results and discussions depend on the Nilsson SP states. The adopted Nilsson parameters are the 1985 parameterisation of Bengtsson and Ragnarsson [7]. For the mass region of the present interest, this parameterisation gives a SP distribution very similar to another popular set of SP states produced by the Woods-Saxon potential of Chasman et al. [3]. Thus in a sense, experimental confirmation or repudiation of the present PSM results is a test of the SP states in the standard Nilsson model, and will provide a reference for us to improve the SP states for the superheavy mass region. REFERENCES 1. 2. 3. 4.
M. A. Stoyer, Nature 442 (2006) 876. R.-D. Herzberg and P. T. Greenlees, Prog. Part. Nucl. Phys. 61 (2008) 674. R. R. Chasman et al., Rev. Mod. Phys. 49 (1977) 833. P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, New York, 1980). 5. K. Hara and Y. Sun, Int. J. Mod. Phys. E 4 (1995) 637. 6. Y. Sun, F. Al-Khudair, G.-L. Long, and J. A. Sheikh, Phys. Rev. C 77 (2008) 044307. 7. T. Bengtsson and I. Ragnarsson, Nucl. Phys. A 436 (1985) 14.
Extracting single-particle information for the superheavy mass region by studying excited structure in transfermium nuclei Yang Sunab∗ a
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, PR China
b
Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, PR China
Recent experimental advances have made it possible to study spectroscopy in very heavy nuclei. We show that from the excited high-spin structure of transfermium isotopes, one may gain useful information on single-particle states for the superheavy mass region, which is the key to locating the anticipated ‘island of stability’. In this work, we employ the Projected Shell Model for Cf, Fm, and No isotopes to study rotation alignment of the particles that occupy particular high-j intruder orbitals. 1. Introduction It has been suggested that an “island of stability” [1] exists in the superheavy mass region possibly with new magic numbers. The occurrence of superheavy elements (SHE) is attributed to the nuclear shell effect because the macroscopic liquid-drop model would predict that such heavy elements can not exist due to large Coulomb repulsion. The singleparticle (SP) states as a consequence of the shell effect have thus become the discussion focus in the SHE problem. The precise location of the predicted island beyond the known magic number 126 for neutrons and 82 for protons depends sensitively on the SP structure. While some SHE have been successfully synthesized in laboratory, little or nothing about their structure has been known. The heaviest nuclei for which detailed spectroscopy measurement can currently be performed lie in the transfermium mass region [2]. These nuclei, typically with Z ≈ 100 and N ≈ 150 − 160, are not really SHE. However, they are at the gateway to the SHE region, and furthermore, they are well deformed. With deformation, the Fermi surfaces are surrounded by some orbitals [3] originating from the subshells near the anticipated new magic numbers. Thus, the study of these deformed transfermium nuclei may provide an indirect way to access unknown SP states of the closed spherical shells, which are of direct relevance to the location of the predicted island. Not all the structure information of the transfermium nuclei are equally useful. Inbeam data for yrast γ-ray spectroscopy of even-even transfermium nuclei reveal that at low-spins near their ground state, they all exhibit very similar collective behavior with regular rotational level sequence [2]. This tells us that near the ground state, these well∗ The author would like to thank F. Al-Khudair and G.-L. Long for their contribution. Work supported by the National Natural Science Foundation of China under Contract 10875077 and by the Chinese Major State Basic Research Development Program through Grant 2007CB815005.
0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.01.013
42c
Y. Sun / Nuclear Physics A 834 (2010) 41c–44c
deformed nuclei behave just like a heavy, rigid rotor. They show a strong collectivity, diluting any individual role of single-particles. Therefore, nothing can be learnt from these low-spin rotor states of even-even nuclei. Useful information may be gained from quasiparticle (QP) excitations, such as high spin structure along the yrast line, structure of K-isomers, and multi-QP configurations in odd-mass or odd-odd nuclei. We study one of the QP excitations, namely, the structure change along the yrast line in even-even transfermium nuclei. It is understood [4] that when a nucleus is rotating, the Coriolis antipairing force acts on the QP pairs, which is proportional to the size of angular momentum j of nucleons under consideration. One thus expects those pairs with the highest j value to break first and align their rotation along the rotational axis. Rotation-alignment lowers the energy of the high-j configurations, and at a certain spin, the states with aligning nucleons can become so low energetically that the aligning bands cross the ground state band (g-band), constituting a situation of so-called band-crossing [4]. The characteristic feature of rotation-alignment in transfermium nuclei is that the aligning pairs come mainly from the two high-j intruder orbitals: proton pairs from the i13/2 orbital and neutron pairs from the j15/2 orbital. The presence of these two highj orbitals near the Fermi level and their response to rotation can lead to interesting observations. Therefore, from the study of rotation-alignment in transfermium nuclei one may gain information on the high-j orbitals and the spin-orbit interaction of SP states. 2. Projected Shell Model study on rotation alignment
7
4-qp 2-qp, π i13/2 2-qp, ν j15/2 0-qp after mixing
6
Enegy (MeV)
5 4 3 2
252
1 0
0
4
8
12
16
20
254
Cf
24
28
0
4
8
12
16
20
256
Fm
24
28
0
4
8
12
16
20
No
24
28
32
Spin
Figure 1. Band diagrams for 252 Cf, 254 Fm, and 256 No. Important configurations are shown: 0-qp (solid curves), neutron 2-qp (dashed curves) and proton 2-qp (dotted curves), and 4-qp (dotted-dashed curves). Filled diamonds denote the yrast states obtained after configuration mixing.
We extend the applicability of the Projected Shell Model (PSM) [5] to the transfermium region. For the model parameters in this mass region, we refer to Ref. [6]. In a so-called
43c
Y. Sun / Nuclear Physics A 834 (2010) 41c–44c
J (1)
band diagram [5], each angular-momentum-projected state of the PSM basis produces ˆ Pˆ I |Φκ /Φκ | Pˆ I |Φκ . The first one, a rotational band κ given by Eκ (I) = Φκ | H KK KK E0 (I), represents the g-band in which all the particles are paired. The remaining states represent bands built upon multi-quasiparticle states [5]. In a band diagram, one may observe several band-crossings at various spins and study their structure. For the transfermium nuclei studied in this work, we have found interesting bandcrossings. We show in Fig. 1 band diagrams for three N = 154 isotones: 252 Cf, 254 Fm, and 256 No. As the neutron number is unchanged in an isotonic chain but the proton Fermi level varies with the shell filling, the relative position of proton and neutron Fermi levels, and therefore the SP states in the vicinity of the Fermi levels, differ in the three. In Fig. 1, those curves starting from 2 – 2.5 MeV are the bands with 2-qp high-j configurations. At low-spins, the g-band is low, and is the dominant component in the yrast wave function. However, it is seen that in all the three diagrams, several 2-qp bands cross the g-band around spin I = 24 and become energetically lower after the band-crossing. With a careful inspection, very delicate differences in the three cases can be found: After the band-crossing, the lowest band in 252 Cf is a 2-qp band of i13/2 protons (dotted curve) and in 256 No a 2-qp band of j15/2 neutrons (dashed curve), whereas in 254 Fm, the lowest dotted and the dashed curves are nearly degenerate after the band-crossing.
252
254
Figure 2. Top row shows calculated static moments of inertia (open symbols) and available data (filled symbols). Middle and bottom rows are the predicted B(E2) (in e2 b2 ) and g factor values, respectively.
Figure 3. Calculated dynamic moments of inertia J (2) (open symbols) and experimental data (filled symbols) for 252,254 No.
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Y. Sun / Nuclear Physics A 834 (2010) 41c–44c
The above observation suggests a picture that at the band-crossing region, proton and neutron 2-qp high-j configurations compete strongly in the yrast states. After the band-crossing, the proton (neutron) configuration dominates the yrast structure in 252 Cf (256 No). In 254 Fm, the strongest competition between the proton and neutron configurations is predicted. Consequences of the competition can lead to observable effects. The top row of Fig. 2 are the calculated energy levels presented in terms of static moment of inertia. Although they show some differences, the evolution against rotation looks very similar in all the three nuclei; the curves are nearly identical for low spins. The dynamic moment of inertia shown in Fig. 3 is more sensitive in emphasizing the changes, but cannot distinguish proton and neutron contributions. B(E2) values in the middle row show a clear signal of the structure change around I = 24, but cannot extarct proton and neutron contributions either because one cannot experimentally separate the proton and neutron contributions to B(E2). A good probe in this regard is the magnetic moments through the study of g factors because they show rather clearly which kind of particle is aligning. In the bottom row of Fig. 2, a drastic rise in g factor around I = 24 in 252 Cf indicates a proton contribution to the wave functions. This is the consequence of a proton i13/2 2-qp alignment in this nucleus. In a sharp contrast, 256 No shows a sudden drop in g factor around the same spin because of the neutron j15/2 2-qp alignment. The balance in competition between the high-j protons and neutrons leads to near-constant g factors in 254 Fm in the spin region where rotation-alignment takes place. We thus expect that the future g factor measurements will provide a direct test for the present prediction. 3. Discussion It is important to comment on the employed SP states in the present calculation. Strictly speaking, SP states in deformed nuclei are not directly measurable. SP states in our shell-model basis are constructed by using the deformed Nilsson potential, and the above results and discussions depend on the Nilsson SP states. The adopted Nilsson parameters are the 1985 parameterisation of Bengtsson and Ragnarsson [7]. For the mass region of the present interest, this parameterisation gives a SP distribution very similar to another popular set of SP states produced by the Woods-Saxon potential of Chasman et al. [3]. Thus in a sense, experimental confirmation or repudiation of the present PSM results is a test of the SP states in the standard Nilsson model, and will provide a reference for us to improve the SP states for the superheavy mass region. REFERENCES 1. 2. 3. 4.
M. A. Stoyer, Nature 442 (2006) 876. R.-D. Herzberg and P. T. Greenlees, Prog. Part. Nucl. Phys. 61 (2008) 674. R. R. Chasman et al., Rev. Mod. Phys. 49 (1977) 833. P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, New York, 1980). 5. K. Hara and Y. Sun, Int. J. Mod. Phys. E 4 (1995) 637. 6. Y. Sun, F. Al-Khudair, G.-L. Long, and J. A. Sheikh, Phys. Rev. C 77 (2008) 044307. 7. T. Bengtsson and I. Ragnarsson, Nucl. Phys. A 436 (1985) 14.