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Super deformation to maximum triaxiality in A=100–112 ; superdeformation, chiral bands and wobbling motion
Nuclear Physics A 834 (2010) 28c–31c www.elsevier.com/locate/nuclphysa
Super deformation to maximum triaxiality in A = 100 − 112; superdeformation, chiral bands and wobbling motion J.H.Hamilton a , S.J.Zhu b , Y.X.Luo a c , A.V.Ramayya a , S.Frauendorf d , J.O.Rasmussen , J.K.Hwang a , S.H.Liu a , G.M.Ter-Akopian e , A.V.Daniel e , Y.Oganessian e
c
a
Physics Department, Vanderbilt University, Nashville, TN 37235, USA
b
Physics Department, Tsinghua University, Beijing 100084, People’s Republic of China
c
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
d e
Physics Department, University of Notre Dame, Notre Dame, IN 46556, USA
Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Russia
The neutron-rich nuclei in the A = 100− 112 regions have revealed a surprising richness of different collective motions. In this paper we present three examples of the different motions as these nuclei go from superdeformed ground states around A = 100 to maximum triaxial shapes around A = 112. These studies were carried out with our high statistics data set of 5.7 × 1011 γ − γ − γ prompt coincidences following the spontaneous fission of 252 Cf taken with 102 Ge detectors in Gammasphere. It has long been known that the sudden shift from spherical to superdeformed ground state around A = 100 arises from the occurrence of shell gaps at Z = 38 for protons and N = 60, 62 at the same deformation: β2 ≈ 0.45, reinforcing each other [1,2]. These reinforcing shell gaps at the same deformation quickly disappear as Z or N change by 4 particles. The shift from spherical to superdeformed shapes occurs between N = 58 and 60 in even-even neutron-rich Sr and Zr nuclei. The question is what happens at N = 59. Previously the ground states of the N = 59 isotones of 97 Sr, 98 Y, and 99 Zr were found to be spherical but with bands with well deformed shapes observed above 500 keV. In N = 59, 100 Nb only near spherical states were known previously, with a first excited state, (5+ ), 3 s isomer at 313 keV (see Ref. [3] for references). The levels in 100 Nb were investigated by studying the γ-rays emitted by the Nb-La spontaneous fission partners [3]. From double gates on gamma rays in La, La + Nb and Nb, a K π = 1+ , ΔI = 1 band built on the 1+ ground state of 100 Nb was found as shown in Fig. 1 [3]. The level energies are remarkably similar to the K π = 1+ strongly deformed band in N = 61 102 Nb up to spin 4+ as shown in Fig. 1 and in 100 39 Y61 . The J1 moments of inertia in 100 Y and 102 Nb are essentially identical and constant with increasing h ¯ ω at values just below the rigid body values. In 100 Nb, J1 exhibits a sharp increase and then drop above 4+ that indicates a band crossing. With its strongly deformed 1+ ground state band in contrast to other N = 59 nuclei and oscillating moment of inertia, 100 Nb is very interesting case for theoretical analysis. 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.01.010
29c
J.H. Hamilton et al. / Nuclear Physics A 834 (2010) 28c–31c
Figure 1. Ground band levels in
4428.3 (13-)
4032.5 (12-)
.0
2.5 102
8.3 135
7.5 124
4.4
0.2 135
101
(5-)
0
768
.1
764
109
8.8
.1
.6
84
2147.9
15
112
7(5-)
9.7
Figure 2. Two sets of chiral doublet bands in
(9-)
58
13
(11-)
1.
1 .9 4 99 1 0. 7.2 66 116 .9 2
.6
.1
.8 3420.8 (10-) 347 692.0 7 3379.8 (10-) 34 4. (9-) 85.6550.6 3 3076.5 649.0 2 26 8 4.8 2829.2 (8-) 2771.7 (8-) 304. 587.3 34 .1 28 0. 7- 255(495.1) 2.5 2489.2 0 541.5 (24 0.0 2334.2 (6 ) 426 . 2230.2 (6 ) 59 0486.0 .3 ) 33 22 2 1. .1 0 1995.1 235 (4-) 6.9 2003.2 33
44
0.7
83
.5
54
1 3768.6
.0 37 0 12 32. 9 0.7 .6 99 12 1 9. 91 40.6 0
12
0.
93
04
793.1
43
1.4
10
778.0
34
10+ (g b) 8+ (γ b) 7+ (γ b) 8+ (g b) 6+ (γ b) 5+ (γ b) + 6 + (g b) 4 + (γ b) 3 (γ b) 4+ (g b) (4+) ( two-phonon γ b)
0)
733
519.8 2574.3
4198.8 (12-)
. (11-) (321 652.7
11
11
48
.8
617.4 3094.1
4561.7 (13-)
874.1
(737.1)
716.8
(6)
(7) 5072.9 (14-)
4769.6 (14-)
799.5
3711.5
Nb.
(5)
(4)
5227.8 (15-)
112 44 Ru 68
100,102
Ru.
The calculations of M¨oller et al. [4] indicate that the deepest minimum for triaxial, asymmetric shapes should be found around 108 Ru. We have found that the odd Z nuclei in this region go from superdeformed 99 Y to maximum triaxiality (γ ∼ −30◦ ) in 111,113 Rh. In well deformed triaxial nuclei, two quite different collective motions can occur in such nuclei in addition to rotational motion, namely chiral bands and wobbling motion. Frauendorf and coworkers [5] proposed that when there are significant angular momentum along each of the three axes of a deformed, triaxial nucleus, chiral bands can occur. The classic case of chiral structures occurs in odd-odd nuclei where high j particles and holes align their angular momentum along the short and long axis and rotational angular momentum aligns along the intermediate axis. Then, two sets of ΔI = 1 bands with the same spins and parities occur where states of the same spin should have nearly identical energies. In addition, the two sets should have S(I) = [E(I) − E(I − 1)]/2I equal and constant as a function of h ¯ ω and similar B(E2)/B(M 1) ratios. We have discovered in 108,110,112 Ru two sets of ΔI = 1, doublet bands with the same spins and parities [6]. These bands show an evolution of chiral structure from γ-soft 108 Ru
30c
J.H. Hamilton et al. / Nuclear Physics A 834 (2010) 28c–31c
Figure 3. Energy level difference and S(I) values.
to maximum triaxiality in 112 Ru. Our high statistics data set allowed us to observe these weakly populated bands as shown for 112 Ru in Fig. 2. The data also were divided into 64 different angles and angular correlations used to determine the spins and depopulating transition multipolarities of at least one level of each of the new bands in 108,110,112 Ru. In Fig. 3, one sees that the energy differences between levels of the same spins are smaller than and the S(I) values comparable to those in 104,106 Rh which were earlier proposed as the best examples of chiral doublet bands in this region (see Ref. [6] for reference). The B(E2)/B(M 1) ratios also were found to be similar for our two sets of bands. The levels in the non-yrast set of doublet bands in 108 Ru exhibit an odd-even spin staggering not seen in any of the other doublets. Such staggering was seen in the nonyrast set of doublets in 106 Ag and was interpreted as a perturbation of the chirality by the γ-softness in 106 Ag [7]. In the theoretical analysis of the levels of 108,110,112 Ru, the levels of 108 Ru indicated it too is γ-soft while 110,112 Ru are more rigid triaxial [8]. Thus we interpreted the staggering in 108 Ru as likewise arising from its γ-softness Tilted axis cranking (TAC) and random phase approximation (RPA) calculations were carried out for different two-quasiparticle neutron configurations in 110,112 Ru [6]. The TAC and RPA calculations show that these doublets in 110,112 Ru do not arise from the accidental degeneracy of bands built on different neutron configurations [6]. The bands are interpreted as two quasi-neutron excitations involving the h11/2 and mixed d5/2 − g7/2 levels. They support the interpretation that these bands arise as chiral vibrations. These soft chiral vibrations cannot be reduced to the simple picture for odd-odd nuclei. The chirality comes from the interplay of all the neutrons in the open shell [6]. In addition to the chiral doublet bands, triaxial nuclei can exhibit wobbling motion of the angular momentum. Wobbling motion is described as a deviation of the axial collective motion away from the axis with largest moment of inertia as illustrated in Fig. 4. This gives rise to a series of wobbling bands with quantum numbers nw = 0, 1, 2, . . .. A classic example of stable triaxial wobbling is found in H2 O where the nw = 0, 1, 2, . . . bands are found. Recently, wobbling bands have been discovered at high spin in 163,165,167 Lu [9]. In O(6) nuclei, the odd spin members of the γ-vibrational band occur at higher energies
J.H. Hamilton et al. / Nuclear Physics A 834 (2010) 28c–31c
31c
50
2
E(I)-0.017I [MeV]
2
E-1.2*I [meV]
2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
H2O H 2O
40
First e-e-wobbler
30 20 10 0
0
2
4
6
8
10
I
odd I low 112
0
2
Ru
4
6
8
nwobble
2
nwobble
1
nwobble
0
Wobbling
Rotation
10 12 14 16 18 20 22
I
Figure 4. Wobbling bands in
112
Ru and H2 O and illustration of wobbling.
than the even spin member with near equal spacings between its even spin neighbors. Such a behavior is seen in the γ bands in nuclei like 140 Gd and the γ-band in 108 Ru. We have identified the γ band to 17+ in 112 Ru and its behavior is strikingly different. The odd spin members come increasingly closer to the lower even-spin levels. The levels of the ground and odd and even spin members of the γ band are proposed as the nw = 0, 1, 2 wobbling bands (Fig. 4). The behaviors of these three bands in 112 Ru as a function of spin look very much like the nw = 0, 1, 2 wobbling bands in H2 O. The data for 108,110,112 Ru indicate a transition from γ-unstable-O(6) structure in 108 Ru to stable triaxial wobbling in 112 Ru. This is the first example of wobbling motion in an even-even nucleus. Recall the calculation of M¨oller et al. [4] predicted that 108 Ru would be the center of a stable triaxial, ground-state region. Our data indicate that the center is 112 Ru not 108 Ru. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
J.H. Hamilton et al., J. Phys. Lett. G10, (1984) L87. J.H. Hamilton, In Treatise on Heavy Ion Science (Allan Bromley, ed.), Vol. 8, p. 2. Y. X. Luo et al., Nucl. Phys. A285, (2009) 1. P. M¨oller, et al., PRL 97, (2006) 162502. S. Frauendorf, J. Meng. Nucl. Phys. A617, (1997) 131; S. Frauendorf, Rev. Mod. Phys. 73, (2001) 463. Y.X. Luo et al., Phys. Lett. B670, (2009) 307. P. Joshi et al., Phys. Rev. Lett. 98, (2007) 102501. I. Stefanescu et al., Nucl. Phys. A789, (2007) 125. H. Amro et al., Phys. Lett. B553, (2003) 197: G. Sch¨onwasser et al., Phys. Lett. B552, (2003) 9.
Super deformation to maximum triaxiality in A = 100 − 112; superdeformation, chiral bands and wobbling motion J.H.Hamilton a , S.J.Zhu b , Y.X.Luo a c , A.V.Ramayya a , S.Frauendorf d , J.O.Rasmussen , J.K.Hwang a , S.H.Liu a , G.M.Ter-Akopian e , A.V.Daniel e , Y.Oganessian e
c
a
Physics Department, Vanderbilt University, Nashville, TN 37235, USA
b
Physics Department, Tsinghua University, Beijing 100084, People’s Republic of China
c
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
d e
Physics Department, University of Notre Dame, Notre Dame, IN 46556, USA
Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Russia
The neutron-rich nuclei in the A = 100− 112 regions have revealed a surprising richness of different collective motions. In this paper we present three examples of the different motions as these nuclei go from superdeformed ground states around A = 100 to maximum triaxial shapes around A = 112. These studies were carried out with our high statistics data set of 5.7 × 1011 γ − γ − γ prompt coincidences following the spontaneous fission of 252 Cf taken with 102 Ge detectors in Gammasphere. It has long been known that the sudden shift from spherical to superdeformed ground state around A = 100 arises from the occurrence of shell gaps at Z = 38 for protons and N = 60, 62 at the same deformation: β2 ≈ 0.45, reinforcing each other [1,2]. These reinforcing shell gaps at the same deformation quickly disappear as Z or N change by 4 particles. The shift from spherical to superdeformed shapes occurs between N = 58 and 60 in even-even neutron-rich Sr and Zr nuclei. The question is what happens at N = 59. Previously the ground states of the N = 59 isotones of 97 Sr, 98 Y, and 99 Zr were found to be spherical but with bands with well deformed shapes observed above 500 keV. In N = 59, 100 Nb only near spherical states were known previously, with a first excited state, (5+ ), 3 s isomer at 313 keV (see Ref. [3] for references). The levels in 100 Nb were investigated by studying the γ-rays emitted by the Nb-La spontaneous fission partners [3]. From double gates on gamma rays in La, La + Nb and Nb, a K π = 1+ , ΔI = 1 band built on the 1+ ground state of 100 Nb was found as shown in Fig. 1 [3]. The level energies are remarkably similar to the K π = 1+ strongly deformed band in N = 61 102 Nb up to spin 4+ as shown in Fig. 1 and in 100 39 Y61 . The J1 moments of inertia in 100 Y and 102 Nb are essentially identical and constant with increasing h ¯ ω at values just below the rigid body values. In 100 Nb, J1 exhibits a sharp increase and then drop above 4+ that indicates a band crossing. With its strongly deformed 1+ ground state band in contrast to other N = 59 nuclei and oscillating moment of inertia, 100 Nb is very interesting case for theoretical analysis. 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.01.010
29c
J.H. Hamilton et al. / Nuclear Physics A 834 (2010) 28c–31c
Figure 1. Ground band levels in
4428.3 (13-)
4032.5 (12-)
.0
2.5 102
8.3 135
7.5 124
4.4
0.2 135
101
(5-)
0
768
.1
764
109
8.8
.1
.6
84
2147.9
15
112
7(5-)
9.7
Figure 2. Two sets of chiral doublet bands in
(9-)
58
13
(11-)
1.
1 .9 4 99 1 0. 7.2 66 116 .9 2
.6
.1
.8 3420.8 (10-) 347 692.0 7 3379.8 (10-) 34 4. (9-) 85.6550.6 3 3076.5 649.0 2 26 8 4.8 2829.2 (8-) 2771.7 (8-) 304. 587.3 34 .1 28 0. 7- 255(495.1) 2.5 2489.2 0 541.5 (24 0.0 2334.2 (6 ) 426 . 2230.2 (6 ) 59 0486.0 .3 ) 33 22 2 1. .1 0 1995.1 235 (4-) 6.9 2003.2 33
44
0.7
83
.5
54
1 3768.6
.0 37 0 12 32. 9 0.7 .6 99 12 1 9. 91 40.6 0
12
0.
93
04
793.1
43
1.4
10
778.0
34
10+ (g b) 8+ (γ b) 7+ (γ b) 8+ (g b) 6+ (γ b) 5+ (γ b) + 6 + (g b) 4 + (γ b) 3 (γ b) 4+ (g b) (4+) ( two-phonon γ b)
0)
733
519.8 2574.3
4198.8 (12-)
. (11-) (321 652.7
11
11
48
.8
617.4 3094.1
4561.7 (13-)
874.1
(737.1)
716.8
(6)
(7) 5072.9 (14-)
4769.6 (14-)
799.5
3711.5
Nb.
(5)
(4)
5227.8 (15-)
112 44 Ru 68
100,102
Ru.
The calculations of M¨oller et al. [4] indicate that the deepest minimum for triaxial, asymmetric shapes should be found around 108 Ru. We have found that the odd Z nuclei in this region go from superdeformed 99 Y to maximum triaxiality (γ ∼ −30◦ ) in 111,113 Rh. In well deformed triaxial nuclei, two quite different collective motions can occur in such nuclei in addition to rotational motion, namely chiral bands and wobbling motion. Frauendorf and coworkers [5] proposed that when there are significant angular momentum along each of the three axes of a deformed, triaxial nucleus, chiral bands can occur. The classic case of chiral structures occurs in odd-odd nuclei where high j particles and holes align their angular momentum along the short and long axis and rotational angular momentum aligns along the intermediate axis. Then, two sets of ΔI = 1 bands with the same spins and parities occur where states of the same spin should have nearly identical energies. In addition, the two sets should have S(I) = [E(I) − E(I − 1)]/2I equal and constant as a function of h ¯ ω and similar B(E2)/B(M 1) ratios. We have discovered in 108,110,112 Ru two sets of ΔI = 1, doublet bands with the same spins and parities [6]. These bands show an evolution of chiral structure from γ-soft 108 Ru
30c
J.H. Hamilton et al. / Nuclear Physics A 834 (2010) 28c–31c
Figure 3. Energy level difference and S(I) values.
to maximum triaxiality in 112 Ru. Our high statistics data set allowed us to observe these weakly populated bands as shown for 112 Ru in Fig. 2. The data also were divided into 64 different angles and angular correlations used to determine the spins and depopulating transition multipolarities of at least one level of each of the new bands in 108,110,112 Ru. In Fig. 3, one sees that the energy differences between levels of the same spins are smaller than and the S(I) values comparable to those in 104,106 Rh which were earlier proposed as the best examples of chiral doublet bands in this region (see Ref. [6] for reference). The B(E2)/B(M 1) ratios also were found to be similar for our two sets of bands. The levels in the non-yrast set of doublet bands in 108 Ru exhibit an odd-even spin staggering not seen in any of the other doublets. Such staggering was seen in the nonyrast set of doublets in 106 Ag and was interpreted as a perturbation of the chirality by the γ-softness in 106 Ag [7]. In the theoretical analysis of the levels of 108,110,112 Ru, the levels of 108 Ru indicated it too is γ-soft while 110,112 Ru are more rigid triaxial [8]. Thus we interpreted the staggering in 108 Ru as likewise arising from its γ-softness Tilted axis cranking (TAC) and random phase approximation (RPA) calculations were carried out for different two-quasiparticle neutron configurations in 110,112 Ru [6]. The TAC and RPA calculations show that these doublets in 110,112 Ru do not arise from the accidental degeneracy of bands built on different neutron configurations [6]. The bands are interpreted as two quasi-neutron excitations involving the h11/2 and mixed d5/2 − g7/2 levels. They support the interpretation that these bands arise as chiral vibrations. These soft chiral vibrations cannot be reduced to the simple picture for odd-odd nuclei. The chirality comes from the interplay of all the neutrons in the open shell [6]. In addition to the chiral doublet bands, triaxial nuclei can exhibit wobbling motion of the angular momentum. Wobbling motion is described as a deviation of the axial collective motion away from the axis with largest moment of inertia as illustrated in Fig. 4. This gives rise to a series of wobbling bands with quantum numbers nw = 0, 1, 2, . . .. A classic example of stable triaxial wobbling is found in H2 O where the nw = 0, 1, 2, . . . bands are found. Recently, wobbling bands have been discovered at high spin in 163,165,167 Lu [9]. In O(6) nuclei, the odd spin members of the γ-vibrational band occur at higher energies
J.H. Hamilton et al. / Nuclear Physics A 834 (2010) 28c–31c
31c
50
2
E(I)-0.017I [MeV]
2
E-1.2*I [meV]
2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
H2O H 2O
40
First e-e-wobbler
30 20 10 0
0
2
4
6
8
10
I
odd I low 112
0
2
Ru
4
6
8
nwobble
2
nwobble
1
nwobble
0
Wobbling
Rotation
10 12 14 16 18 20 22
I
Figure 4. Wobbling bands in
112
Ru and H2 O and illustration of wobbling.
than the even spin member with near equal spacings between its even spin neighbors. Such a behavior is seen in the γ bands in nuclei like 140 Gd and the γ-band in 108 Ru. We have identified the γ band to 17+ in 112 Ru and its behavior is strikingly different. The odd spin members come increasingly closer to the lower even-spin levels. The levels of the ground and odd and even spin members of the γ band are proposed as the nw = 0, 1, 2 wobbling bands (Fig. 4). The behaviors of these three bands in 112 Ru as a function of spin look very much like the nw = 0, 1, 2 wobbling bands in H2 O. The data for 108,110,112 Ru indicate a transition from γ-unstable-O(6) structure in 108 Ru to stable triaxial wobbling in 112 Ru. This is the first example of wobbling motion in an even-even nucleus. Recall the calculation of M¨oller et al. [4] predicted that 108 Ru would be the center of a stable triaxial, ground-state region. Our data indicate that the center is 112 Ru not 108 Ru. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
J.H. Hamilton et al., J. Phys. Lett. G10, (1984) L87. J.H. Hamilton, In Treatise on Heavy Ion Science (Allan Bromley, ed.), Vol. 8, p. 2. Y. X. Luo et al., Nucl. Phys. A285, (2009) 1. P. M¨oller, et al., PRL 97, (2006) 162502. S. Frauendorf, J. Meng. Nucl. Phys. A617, (1997) 131; S. Frauendorf, Rev. Mod. Phys. 73, (2001) 463. Y.X. Luo et al., Phys. Lett. B670, (2009) 307. P. Joshi et al., Phys. Rev. Lett. 98, (2007) 102501. I. Stefanescu et al., Nucl. Phys. A789, (2007) 125. H. Amro et al., Phys. Lett. B553, (2003) 197: G. Sch¨onwasser et al., Phys. Lett. B552, (2003) 9.