APPENDIX NUCLEAR STRUCTURE
1. Spherical Shell Model
The stability of certain combinations of neutrons and protons was pointed out by Elsasser1 in 1934.
Stability for light nuclei with N or Z values of 2 (4He), 8 (16O), and 20 (40Ca) became well known, and in
1948 Mayer2 observed strong evidence for additional "magic numbers" at 50 and 82 for protons, and 50,
82, and 126 for neutrons. Mayer and Jensen3 explained this in terms of a "shell model", where nucleons
moved in a spherically symmetric potential well. This potential is most commonly taken as a harmonic
oscillator of the form
V (r ) = −V 0 RQ1 − (r /R )2 HP,
(1)
where V(r) is the potential at a distance r from the center of the nucleus and R is the nuclear radius. The
resulting quantum states can be characterized by n , the principal quantum number related to the number
of radial nodes in the wave function, and by l , the orbital angular momentum of the particle. By analogy to
atomic spectroscopy, states with l = 0, 1, 2, 3, 4, 5, 6, ... are designated as s ,p ,d , f ,g ,h ,i ,..., respectively.
The states are identified by as 1s , 2f , 3p , etc, where, unlike in atomic spectroscopy, n is defined so that
each state has n −1 radial nodes.
..
The solution of the Schrodinger equation for the isotropic harmonic oscillator gives the level energies
(2)
E =−
h ω [2(n − 1) + l ]+E ,
0
0
where
I
ω0= J
L
2V 0
hhhhh
MR 2
1/2
M
J
,
(3)
O
M is the nucleon mass, and E 0=3/2−
h ω0. Here states with the same value of N =2(n −1)+l are degenerate,
and states of a given energy must all have either even or odd values of l . Thus, degenerate states must
have the same parity.
Mayer4 and Haxel et al 5 showed that the interaction between the intrinsic angular momentum (spin)
of the particles and their total angular momentum would split the orbitals into two substates with l ±1/2 (e.g.
1p 1/2 and 1p 3/2). These states are no longer degenerate, and it was shown that if the "spin-orbit" interaction is of the same order as the spacing between oscillator shells, and states with j =l +1/2 are more stable
than states with j =l −1/2, then the magic numbers can be reproduced. The evolution of shell model states
from the harmonic oscillator model is depicted in Figure 1.
2. Collective Model
The shell model succeeds in describing nuclei near the magic numbers, but it fails to adequately
describe nuclei outside the closed shells. This may be attributed to a breakdown in the assumption of
spherical symmetry. Rainwater6 proposed that, in odd-A nuclei, the motion of the odd nucleon could
polarize the even-even core allowing all nucleons to move collectively, thus increasing the static electric
quadrupole moments and enhancing quadrupole transition (E 2) rates. The implications of spheroidal
rather than spherical shape were studied by Bohr and Mottelson7 who developed a collective model for
nuclei.
For a spheroidal nucleus, the orbital angular momentum of the odd nucleon is no longer conserved.
Since total angular momentum for the nuclear system must be conserved, the core must have angular
momentum coupled to that of the odd nucleon. Addition of more nucleons outside the closed shells can
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
1
W. Elsasser, J.Phys. Rad. 5, 625 (1934).
2
M.G. Mayer, Phys. Rev. 74, 235 (1948).
3
M.G. Mayer and J.H.D. Jensen, Elementary Theory of Nuclear Shell Structure, John Wiley & Sons, Inc., New York (1955).
4
M.G. Mayer, Phys. Rev. 78, 16 (1950).
5
O. Haxel, J.H.D. Jensen, and H.E. Suess, Z. Physik 128, 295 (1950).
6
L.J. Rainwater, Phys. Rev. 79, 432 (1950).
7
A. Bohr and B.R. Mottelson, Dan. Mat.-Fys. Medd. 27, No. 16 (1953); and "Collective Nuclear Motion and the Unified
Model", Beta and Gamma Ray Spectroscopy , K. Siegbahn, editor, North Holland, Amsterdam (1955).
15/2[707]
13/2[716]
11/2[725]
9/2[734]
7/2[743]
5/2[752]
3/2[761]
1/2[770]
11/2[606]
9/2[604]
7/2[613]
5/2[622]
3/2[631]
1/2[640]
7/2[624]
5/2[633]
3/2[642]
1/2[651]
184
4s
3d
6 hω
+
2g
3/2[602]
1/2[600]
1/2[611]
7/2[604]
5/2[602]
3/2[611]
1/2[620]
5/2[613]
3/2[622]
1/2[631]
1j15/2
3d3/2
4s1/2
2g7/2
1i11/2
3d5/2
2g9/2
1i
126
3p
2f
5 hω
-
1i13/2
3p3/2
3p1/2
2f7/2
2f5/2
1h9/2
1h
3s
2d
4 hω
+
2p
5/2[503]
3/2[501] ([512])
1/2[510] ([521])
11/2[505]
9/2[514]
7/2[523]
5/2[532]
3/2[541]
1/2[550]
82
1h11/2
3s1/2
2d3/2
2d5/2
1g7/2
1g
3 hω
-
3/2[512] ([501])
1/2[521] ([510])
1/2[501]
13/2[606]
11/2[615]
9/2[624]
7/2[633]
5/2[642]
3/2[651]
1/2[660]
7/2[514] ([503])
5/2[523] ([512])
3/2[532] ([521])
1/2[541] ([530])
9/2[505]
7/2[503] ([514])
5/2[512] ([523])
3/2[521] ([532])
1/2[530] ([541])
1/2[411] ([400])
3/2[402] ([402])
1/2[400] ([411])
5/2[413] ([402])
3/2[422] ([411])
1/2[431] ([420])
7/2[404]
5/2[402] ([413])
3/2[411] ([422])
1/2[420] ([431])
50
1g9/2
2p1/2
1f5/2
2p3/2
1/2[301]
5/2[303]
3/2[301]
1/2[310]
3/2[312]
1/2[321]
1f
9/2[404]
7/2[413]
5/2[422]
3/2[431]
1/2[440]
28
7/2[303]
5/2[312]
3/2[321]
1/2[330]
1f7/2
20
2 hω
+
2s
1d
3/2[202]
1/2[200]
1/2[211]
5/2[202]
3/2[211]
1/2[220]
1d3/2
2s1/2
1d5/2
8
1 hω
-
1p
1p1/2
1p3/2
1/2[101]
3/2[101]
1/2[110]
Figure 1. Evolution of nuclear models from the
harmonic oscillator model, adding strong spinorbit coupling to obtain the shell model, and
axial deformation to give the collective model.
Asymptotic Nilsson configurations are given for
neutrons and, in parentheses, for protons when
different. Because of mixing, very few states
have pure Ω[NnzΛ] configurations.
2
0 hω
+
1s
Harmonic Oscillator
N hω
nl
π
1s1/2
Shell Model
nlj
Collective Model
Ω[NnzΛ]n ([NnzΛ]p)
form spheroidal nuclei whose angular momentum states reflect the coherent motion of all nucleons rather
than just a few nucleons moving in single-particle shell model orbitals. The quantization of this coherent
motion forms the basis of the collective model.
This collective motion can be quantized by assuming that the even-even core is an incompressible
liquid and quantizing the classical hydrodynamical equations that describe its oscillations. Borrowing from
the well-known analysis of rotational states in symmetric-top molecules, we expect the level energy relation
− 2K 2
[J (J + 1) − K 2] hhhhhh
h
E =−
h 2hhhhhhhhhhhhhh
+
(4)
2J_
2J_
1
3
1/2−
where [J (J + 1)] h is the total angular momentum of the nucleus, and K −
h is the component of angular
momentum along the symmetry axis of the nucleus, as shown in Figure 2. _
J 1 and _
J 3 are the moments of
inertia perpendicular and parallel to the symmetry axis, respectively. For low-lying rotational bands in
even-even nuclei, K =0 and Eq. (4) reduces to
[J (J + 1)]
E =−
h 2 hhhhhhhhh
.
2J_ 1
(5)
States with K ≠0 arise when one or more pairs of nucleons are broken and unpaired nucleons are excited
to higher shell-model states. Low-lying rotational levels with K =0 are characterized by a sequence of
states with J = 0, 2, 4, 6, 8, . . . and positive parity, often referred to as the ground-state (GS) band. An
example of the GS band for 152Sm is shown in Figure 3. Excited vibrational states of three kinds are also
shown for 152Sm in Figure 3. The β-vibration preserves spheroidal symmetry, and the γ-vibration goes
through an ellipsoidal symmetry. The octupole vibration produces a low-lying J π=3− state in spherical
nuclei, but can also couple to core quadrupole excitations in deformed nuclei to give a family of low-lying
states with J π=1−, 2−, 3−,4−, and 5−.
3. Deformed Shell Model
A unified model for the effect of deformation on shell-model states has been developed by Nilsson1
and by Mottelson and Nilsson2. The deformation breaks the 2 j + 1 degeneracy of the spherical shellmodel states. Asymptotic quantum numbers needed to describe these states are shown in Figure 2. In
addition to K , they include N , the total oscillator shell quantum number; nz , the number of oscillator
quanta in the z direction; M , the projection of total angular momentum J on the laboratory axis; R , the
angular momentum from the collective motion of the nucleus; Ω, the projection of total angular momentum
j (orbital l plus spin s ) of the odd nucleon on the symmetry axis; and Λ, the projection of angular momentum along the symmetry axis where Ω=Λ+Σ and Σ is the projection of intrinsic spin along the symmetry
axis. Levels are labeled by the asymptotic quantum numbers Ωπ[Nnz Λ]. The Nilsson orbitals which
derive from the axially deformed harmonic oscillator potential, labeled as described by Davidson3, are
shown in Figure 1. A more realistic calculation can be performed using a Woods-Saxon potential
V0
,
(6)
r −R
1+exp( hhhhh )
a
where a ≈0.67 fm. Considerable mixing of Nilsson configurations with ∆Ω=0, ∆N=2, and similar excitation
energies may occur leaving only Ω as a good quantum number. Representative Nilsson level diagrams,
for various mass regions and deformations, calculated as described by Bengtsson and Ragnarsson4, are
shown in Figures 4-14. In these figures, single-particle energies are plotted in units of the oscillator frequency
V (r ) =
hhhhhhhhhhhh
−
h ω0 = 41A −1/3 MeV
(7)
as a function of deformation ε2, with ε4 chosen as described in the figure captions. The Nilsson quadru∆R
pole deformation parameter ε2 can be defined in terms of δ= hhhhhh where Rr.m.s. is the root mean
Rr.m.s.
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
1
S.G. Nilsson, Dan. Mat.-Fys. Medd. 29, No. 16 (1955).
2
B.R. Mottelson and S.G. Nilsson, Phys. Rev. 99, 1615 (1955).
3
J.P. Davidson, Collective Models of the Nucleus, Academic Press (1968).
4
T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436, 14 (1985).
Laboratory axis
J
R
M
j
s
l
Λ
is
etry ax
Symm
Ω
Σ
K
Figure 2. Asymptotic quantum numbers for the deformed shell model
square nuclear radius and ∆R is the difference between the semi-major and semi-minor axes of the
nuclear ellipsoid, as12
ε2 = δ +
1 2 hhh
5 3 hhhh
37 4
δ +
δ +
δ +....
6
18
216
hh
(8)
The deformation parameter β2 is related to ε2 by
I
M
4
4
4
4
d dd
β2 = √π
/5 J hh ε2 + hh ε22 + hhh ε23 + hhh ε24 + . . . J
3
9
27
81
L
O
(9)
The intrinsic quadrupole moment Q 0 (or transition quadrupole moment QT ∼
∼Q 0) is related to β2 by
dddddd
Q 0 = √16π
/5ZeR 02 β2.
(10)
If the partial photon half-life for the E2 transition is known, the experimental transition strength B (E 2) is
given by
0.05659
B (E 2) = hhhhhhhhhhhh
(eb )2,
(11)
t 1/2(E 2)(E γ)5
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
12
..
..
K.E.G. Lobner, M. Vetter, and V. Honig, Nucl. Data Tables A7, 495 (1970).
where t 1/2(E 2) is in picoseconds, and E is in MeV. B (E 2) is related to Q 0 by
2
5
B (E 2) = B (JK →J −2 K ) = hhhh Q 02 <JK 20 | J −2 K > (eb )2.
16π
(12)
Assuming a constant charge distribution in the nucleus, Q 0=0 for spherical nuclei, Q 0>0 for prolate (football or cigar shaped) nuclei, and Q 0<0 for oblate (disk shaped) nuclei. The spectroscopic quadrupole
moment Q is related to the intrinsic moment by 12
3K 2 − J (J + 1)
Q = Q 0 hhhhhhhhhhhhhh .
(J + 1)(2J + 3)
(13)
For M 1 transitions, the transition strength B (M 1) is given by
2
3
B (M 1) = B (JK →J ±1 K ) = hhh (gK −gR )2K 2<JK 10 | J ±1 K > µN2 ,
4π
(14)
where gK and gR are nuclear g -factors. The rotational g -factor, gR , arises from the collective rotation of
the core, and is defined as
Z
gR ≈ hh .
A
(15)
The intrinsic g -factor, gK , arises from the orbital motion of the valence nucleons, and the total magnetic
dipole moment is given by
K2
µ = gR J + (gK − gR ) h hhh
J +1
(16)
If the partial photon half-life for the M 1 transition is known, then B (M 1) can be calculated from
14+
2736.3
1666.48
Octupole band
6+
4+
2+
0+
706.96
288
213
1125.35
4+
2+
0+
126
1041.180
963.376
8+
418
1221.64
1310.51
1022.962
810.465
684.70
6+
5+
4+
3+
2+
218
2139.7
326
356
484
8+
6+
340
1–
1609.34
1505.62
245
3–
2079.6
1879.1
122
5–
10+
2375.5
(8+)
7+
10+
7–
2148.8
148
286
540
(9)+
2327.1
12+
9–
2525.6
235
12+
11–
386
14+
2976.7
446
2833.5
(17)
3362.3
451
(16+)
413
507
13–
3383.5
588
(15–)
626
0.0394
B (M 1) = hhhhhhhhhhh
µN2 .
t 1/2(M 1)E γ3
γ band
β band
366.4814
121.7825
0
GS band
152
62Sm
Figure 3. Ground-state, and β-, γ-, and octupole-vibrational bands in 152Sm.
1945.82
1728.38
1559.53
1371.752
1233.876
1085.897
1/2[420]
50
1/2
404]
9/2 [
[43
1]
7/2[413]
5.0
1]
1/2 [30
5/2[422]
1g9/2
5/2[303]
3/2 [4
2p1/2
31]
3/2[301]
1f5/2
4.5
1/2
1/2 [31
0]
2p3/2
3/2 [312]
[44
0]
1/2 [32
1]
28
03]
7/2 [3
− ω)
Es.p. (h
5/2[312]
1f7/2
4.0
3/2 [321
]
1/2 [3
30]
20
3/2 [202]
1/2[200]
1d3/2
3.5
1/2[211]
2s1/2
5/2 [202]
3/2 [211]
1d5/2
1/2 [2
20]
3.0
8
1/2[101]
1p1/2
2.5
3/2 [101]
1p3/2
2.0
−.3
−.2
−.1
.0
1/2 [110]
.1
.2
ε2
Figure 4. Nilsson diagram for protons or neutrons, Z, N ≤ 50 (ε4 = 0).
.3
1/2[640]
3/2[761]
2[
40
5/
]
3
[770
[
/2
1/ [402
2[ ]
40
0]
1/2[521]
7/2[633]
2]
1/2
1]
[65
]
3/2
660
1/2
3/2
[52
5/2
1]
4
[6
5/2
[52
2]
3]
5/2
/2
[50
1]
[65
1/2
2]
[64
5]
82
11
]
]
2] 00
514
40 /2 [4 9/2 [
[
2 1
3/
3/2
2[
40
]
4]
32
7/
2d3/21h11/2
1]
[65
3/2
[5
6.0
1/2
7/2[523]
]
0]
30
[5 /2 [66
1
3s1/2
− ω)
Es.p. (h
1/2[411]
2]
5/2
[40
5/2[413]
1/2[411]
1g7/2
3/2 [411]
1/2
1]
2]
[42
[54
53
1/2
0]
2[
0]
5/
[55
5.5
3/2
[66
1/2
2d5/2
1]
43
2[
1/
2]
3/2
[54
[4
20
1/2[301]
5/2[303]
1]
1/2
]
50
5/2[532]
]
01
3
2[
1/
9/
2[
40
4]
1/2[541]
3/2[301]
413]
7/2 [
3/2
[30
[54
]
3/2
[43
01
[3
0]
4
[4
0]
[55
2
1/
1]
1]
3/2
1/2
422]
1/2
1g9/2
5.0
1]
5/2 [
−.3
−.2
−.1
.0
.1
.2
.3
.4
ε2
Figure 5. Nilsson diagram for neutrons, 50 ≤ N ≤ 82 (ε4 = ε22 /6).
.5
.6
1/2[640]
1/2[400]
3/2 [5
21]
5/2
0]
0]
53
]
02
]
2[
3/
11
/2
[5
05
]
660
2d3/21h11/2
2]
53
1/2 [
6.0
761]
]
651
3
1/2[880]
]
42
[6
5/2
70]
1/ 2 [7
]
651
1/2 [
[
3/2
]
00
]
02 [4
[4 1/2
2
/
[4
3/2 [
2
5/
4
40
2[
7/
3/2[521]
1/2[640]
]
42
[6
[66
[
1/2
82
61]
1/2 [7
1]
[65
3/2
1/2
1]
[54
1/2
2]
40 ]
2 [ 400
/
3 [
2
1/
4]
[51
2
/
9
]
7/2[523]
1/2[411]
3s1/2
1]
1/2 [41
1/2 [411]
]
1g7/2
3/2[411]
1/2[530]
1/2 [7
]
3/2
3/2 [411
70]
− ω)
Es.p. (h
651
1/2 [
13]
5/2 [4
02]
[4
5/2
1]
[65
3/2
[54
[42
0]
3/2
[55
5.5
2]
5/2
[5
32
]
[4
[4
20
1/2 [6
1/2
1/2
1]
1/2
2d5/2
]
31
]
60]
50
9/
2[
40
4]
1/2[301]
3]
30
3]
1/2 [
[
5/2
660]
41
7/2 [
3
3/2
]
01
[3
3/2
1/2
22]
0]
[55
5/2 [4
[43
[54
3/2
1/2[541]
1]
30
/2 [
1/2
1g9/2
5.0
]
1
[30
1/2[420]
1]
0]
44
2[
1/
1]
−.3
−.2
−.1
.0
.1
.2
.3
.4
ε2
Figure 6. Nilsson diagram for neutrons, 50 ≤ N ≤ 82 (ε4 = − ε22 /6).
.5
.6
1/2[750]
5/2[503]
1/2[501]
1/2[770]
1]
[50
9/2[734]
5/2
[62
2]
761
3/2
22]
1/ 2 [
]
1]
5/2
63
2]
3]
74
64
2[
1/
2[
7/
2[
3/
2]
75
5/2 [6
7/2
[
]
[
5/2
1
[76
50
3]
3/2
2g9/2
2
[86
126
]
[
3/2
1]
50
1/
2[
5
3 /2 [
06] /2 [5 503
01 ]
]
2 [6
2]
13/
]
33
[6
[7 5
1/2[510]
512]
3/2 [ 10]
5
1/2 [
]
761
]
15
2
1/
]
752 /2 [871]
3
7/2
2
5/
5 /2
[
3/2
2f5/2
880]
1/2 [
]
43
[7
0]
[77
1/2
1]
[65
7.0
1/2
3p1/2
[6
1
1/2[750]
1/2[631]
]
1]
[76
[770
1]
63
3/2
1/2
3p3/2
2[
3/
]
3/2 [512
1/2[510]
3]
[50
[50
1/2[880]
5/2[512]
7/2
9/2
1i13/2
]
[514
]
[512
2]
[75
5/2
761]
1/2 [
7/2
5/2
− ω)
Es.p. (h
5]
9/2 [624]
2
2 [ [40
40 2
0] ]
1/
3/
0]
]
3/2
42
1]
[64
[52
0]
[64
7/2[404]
40
2]
0]
[53
1]
2]
1/2
3/2
1/2
2f7/2
1]
[6
[65
523]
6.5
3]
[52
5/2
3/2
5/2 [
[63
[64
1/2
1/2
7/2
1h9/2
2[
1/2
[52
3/
1]
2[
5/
1/
2
]
]
660
[4
00
]
770
[
1/2
2]
40
2]
1/2 [
53
1]
54
2[
3/
[
1/2
7/2[633]
1/2
5/
2
50
4]
11
40
0]
40
7/
2[
/2 [
2[
2]
1/
]
[64
1
[65
5/2
[51
1]
[65
6.0
−.3
4]
9/2
3/2
3
1/2
5]
[4
02
]
0]
[66
2]
40
[
/2
−.2
−.1
.0
.1
.2
.3
.4
ε2
Figure 7. Nilsson diagram for neutrons, 82 ≤ N ≤ 126 (ε4 = ε22 /6).
.5
.6
5/2[503]
]
01
[5
3/2
5
3/ /2 [
60 2 [50 503
6]
1] ]
/2 [
3]
50
]
13
31
1]
[5
7/2
9/2
2]
[51
[51
1/2 [990
[5
03
05
]
]
5/2
2]
]
[63
3]
1/2[521]
2]
0]
[64
[53
1]
1/2[400]
3/2
1/2
[52
0]
40
61]
]
32
1]
[54
[5
1/2 [7
2
3/
1/2
1/
1/2
2]
40 ]
2 [ 400
/
3
2[
1/
]
0
[66
9/2
4]
[51
]
02
5/
40
[50
5]
.4
ε2
Figure 8. Nilsson diagram for neutrons, 82 ≤ N ≤ 126 (ε4 = − ε22 /6).
761]
.3
1/2[880]
2]
[64
5/2
70]
1/2 [7
1]
.2
]
[65
660]
1/2 [
.1
651
40
2[
5/
1/2 [
3/2
2]
3/2 [
4
2[
4]
2[
2] 0]
40 40
2 [ 1/2 [
/
3
3/2[521]
1/2[640]
7/
11
/2
71]
2]
1]
3/
521]
2[
.0
2]
40
2[
1/2 [
64
[65
2[
5/
3/2
]
2]
0]
1]
3/2
[75
1/2 [88
[52
3/2 [8
5/2
]
70]
761
1/2 [7
1/2
5/2 [523
3/2[752]
1]
63
61]
[51
5/2
1/2 [87
2[
3/
1/2 [7
4]
7/2
1h9/2
−.1
[6
1]
0]
[51
11
7/2
−.2
1/2
63
2[
1/
1/2
3]
871]
5]
1
[6
74
3/2 [
1
3/2 [
− ω)
Es.p. (h
7/
2[
1/2[501]
1/2[770]
[5
01
]
1/
2
2]
761]
1/2 [
0]
1
/2 [5
]
24]
6.0
−.3
0]
[75
1]
]
2[
512
9/2 [6
2f7/2
2]
[86
5/2 1/2 [871]
3/2
1]
2]
63
[75
2[
[76
]
0
[64
[770
1/2[510]
[
3/2
/2
3/2[981]
3/2[512]
7/
3p3/2
6.5
0
1/2 [10 1
]
3]
12]
3/2 [5
]
0
1/2 [51
1i13/2
1/2 [990
]
[63
5/2
3/
3/2
1/2
2f5/2
[750
]
31
[6
5/2
1/2
7.0
3]
2
1/
2]
]
2]
51
64 /2 [6
1
3p1/2
74
1/2
[75
]
2[
3/
126
2[
5/2
1
[76
7/
80]
1/2 [8
3/2
2g9/2
.5
.6
11/2[835]
9/2[604]
1/2[981]
7/2[723]
3/2[862]
5/2[602]
7/2[604]
3/2[602]
1/2[600]
1/2[750]
9/2[725]
1/2[880]
3/2[871]
7/2[853]
1/
2]
[60
0]
[60
1/2
3]
07]
[50
2 [7
1]
7/2
[60
4]
1/2
15/
]
4]
84
2[
16
[7
]
04
[6
9/2
[50 [503
1] ]
3/2
/2
13
6]
]
]
[862 /2 [860
1
3]
50
2[
5/
]
15
[6
0]
1/2[981]
3/2[981]
1/2
9/2
622]
[62
5/2[503]
7/2[613]
3/2[501]
3/2
60
9/
1]
1]
/2 [
]
[74
[87
11
1
[85
871]
3/2
3/2
]
613
3/2 [
1]
622]
9/2[505]
1]
7/2 [
[74
9/2 [
3/2 [
[50
1/2
1]
]
05
[5
[73
4]
3]
[50
0]
7/2
]
]
9/2
9/2
[50
13/
3/2
[75
1]
633
1/2
[76
761
1/2 [
3/2
1]
2 [6
[50
06]
1]
2]
1/2
1/2[620]
7/2[853]
5/2[743]
[74
]
[86
7/2 [624]
52
5/2 [
3/2
[7
5/2
5/2
5/2 [6
22]
]
/2
11
15
[6
5/2
[62
3]
2]
1/2[990]
1/2[871]
1]
5/2
63
2[
2]
74
2[
7/
1/
64
2[
2]
[86
43
]
1]
50
2[
5/
3/2 2 [5
[5 03]
01
]
2]
]
[75 3/2 [871
[7
880]
1/2 [
]
7/2
31
[6
0]
[64
0]
[77
1/
3/2
3/2
1/2
1/2
1/2
5/2
2]
.2
.3
1]
[76
.1
1/2[510]
[75
3/2
]
1
[65
.0
]
1/2 [
2]
1/2
3/
−.1
0 10 0
2]
1]
[75
− ω)
Es.p. (h
1/2 [1
3/2
4]
126
−.2
9/2[615]
84
[74
3]
74
1]
3/2
[76
5]
[72
7.5
7.0
−.3
5/2[853]
5/2[732]
1i11/2
3p1/2
3/2[972]
5/2[972] 1/2[851]
2[
90]
73
1/2
2[
1/2
11/2
9/
1/2 [9
2[
5/
[61
5/
1]
3/2 [61
734]
2g9/2
73
5/2
3/2
970]
[85
1]
]
1j15/2
2]
0]
13
[6
5/2
3d5/2
3/2[611]
73
2[
1]
2
60
2g7/2
1/2 [
7 /2
1/
3/2
2[
5/
3]
85
4]
1/2[611]
]
8.0
5/2[613]
1]
84
2[
[87
4s1/2
2]
3/
3/2
3d3/2
73
[ 85
2h11/2
1/2[611]
2[
3/2
7/2[734]
3/
860]
2[
0]
1/2 [
2]
86
9/
73
[
5/2
[
5/2
]
871
1/2 [
2[
512]
3/2 [
.4
ε2
Figure 9. Nilsson diagram for neutrons, N ≥ 126 (ε4 = ε22 /6).
.5
.6
60
2]
0]
1/2
9/2[604]
6]
60
/2 [
1]
50
13
7/2
1]
1/
3]
3]
[85
50
]
16
[7
/2
2[
4]
[60
7/2
13
]
[74
862
3/2
3]
3/2 [
[85
07]
[6
2[
5/
3/2[622]
3/2 [622]
2[
3/
9
0]
1]
11
/2 [
60
]
/2
3/2 [622]
[62
91]]
[10091
1/
3/22[1
]
13
2
7/
7/2
2 [7
3/2[972]
1/2[970] 3/2[851]
2]
871
[86
3/2 [
5/2
1]
6]
9/2[844]
[6
72
2]
15/
]
15
2
9/
11
]
[75
[76
1/2
4]
0
[6
9/2[615]
5/2[853]
5]
/2 [
3]
880
3/2
1/2
− ω)
Es.p. (h
3/2[732]
72]
74
1/2 [
11]
3/2 [6
1/2[730]
50
3/2
[
[60
1]
2[
[
11
[6
5/2 [9
1/2 [98
0 0]
1/2 [10 1 1]
71]
5/
3
61
]
3/2
0]
1/2 [8
1/2 [99
[61
]
3d5/2
2]
[85
2]
750]
1/2 [
1]
3/2
02
2g7/2
[73
1/2
73
1]
[85
3/2
2[
1]
74
[61
[6
2
5/
3]
[61
3/2
71]
1/2 [8
5/
1]
]
1/2 [611
2
5/
5/2
]
3]
5/2
]
8.0
30
[
1/2
[87
4s1/2
3/2[981]
1/2[730]
5/2[602]
3/2[862]
7/2[604]
3/2[602]
1/2[600]
1/2[880]
7/2[853]
9/2[725]
4]
3/ 2
3d3/2
[7
60]
[84
7/2 [734]
2h11/2
1/2
1/2 [8
3
2]
[86 1]
5/2 /2 [87
9/2
1/2 [620]
1/2 [620]
9/2 [734
]
7/2
[613]
5/2[743]
1/2[741]
1j15/2
3]
50
1]
2[
5/
50
2[
3/
]
01
[5
5/2 [622]
50
3]
2[
7/
1]
]
1/2 [10 1
[5
0
1]
1]
3/2
1/2 [98
1/2 [990
80]
63
2[
]
1]
05
[5
50
1/
2[
9/2
1/2[510]
3]
[5
0
7/2
3]
]
5
3/ /2 [5
2 [ 03
50 ]
1]
74
31
[6
1/2
0]
[51
1/2
.3
2[
]
31
[6
.2
3/2
7/
2
1/
.1
761]
11
2]
2]
[75
71]
3/2 [8
3/2[512]
1/2 [
/2
[6
3/2[981]
[86
3/2
3]
[
]
15
0 0]
5/2
[63
1/2
.0
1/
5/2
2]
[75
5/2
1]
63
]
2[
3/ /2 [761
]
3
770
−.1
]
[750
1/2 [8
1]
−.2
1/2
2]
0]
]
2]
51
64 /2 [6
1
3p1/2
3/2[741]
[75
[76
[64
2[
3/
126
3/2[862]
1/2[620]
1/2
3]
2]
1]
5/2
3/2
1/2
2g9/2
74
[62
860]
[74
2[
71]
7/
5/2
7.0
−.3
1/2 [8
7.5
1/2 [
]
624
7/2 [
3/2
1i11/2
2]
[51
1/2
9/2[624]
0]
[51
.4
ε2
Figure 10. Nilsson diagram for neutrons, N ≥ 126 (ε4 = − ε22 /6).
.5
.6
3/2[402]
5/2
2[
1/
3]
3]
0
4
2[
5/
11
/2
]
[50
5]
1
[65
3s1/2
[63
2]
1/ 2
[4
7/2
]
42
[6
1]
]
[65 660
3/2 1/2 [
]
00
2
1/
5/2
6.0
[52
2
5/
82
1/2[640]
]
2]
]
0
40
2
[64 1]
3/2 /2 [76
3
53
0]
2[
7/2[404]
[77
]
51
[6
3 /2 6 6 0 ]
[
]
1 /2
3/
[52 7/2 [
63
1]
3
1/2
3/2
4]
40
2]
5/
7/
2[
]
40
30
2[
2]
[51
[5
9/2
4
[6
1/2
4]
2d3/2
1/2 [411]
7/2 [52
1h11/2
− ω)
Es.p. (h
1/2[411]
3]
3/2[411]
2d5/2
[54
]
20
1]
50
[4
]
0]
1/2
[5
1/2
1/2
[42
2]
5.5
3/2[651]
[66
3/2
1/2
1g7/2
3/2 [411]
5/2 [413]
1/2
5/
]
31
[4
2[
53
2]
1/2[301]
3/2
[5
5/2[303]
]
41
50
]
01
5/2
[3
[53
2]
2[
40
4]
1/2
1/2[541]
9/
]
3/2
01
[3
13]
7/2 [4
1]
3/2
.1
.2
.3
1]
.0
[54
−.1
3/2
−.2
5/2 [4
22]
3/2
[43
1]
0]
[55
−.3
1g9/2
1]
[30
3]
[30
5/2
1/2
5.0
1/2
[30
.4
ε2
Figure 11. Nilsson diagram for protons, 50 ≤ Z ≤ 82 (ε4 = ε22 /6).
.5
.6
3/2[642]
7/2[633]
3/2[521]
3/2[402]
1/2 [7
5/2 [5
23]
2[
53
1/2[640]
]
2[
5/
1]
[54
64
82
3/2[521]
0]
40
2[
1/
651
1/2 [
1/2
2]
61]
1]
]
[65 [660
3/2 1/2
3/
2]
2]
0
5/2
[4
3/2
0]
[7 6
40
]
2]
[5
05
1]
64
2[
1/2
11
/2
3s1/2
6.0
770]
1/2 [
5/
[
1/2
2
1/2 [411]
0]
4]
[4
0
[4
1/2 [411]
1]
7/2[523]
− ω)
Es.p. (h
1h11/2
3/2 [4
11]
1]
3/2 [41 ]
13
5/2 [4
2d5/2
5/2
[53
2]
1]
1/2 [
660
]
0]
]
41
20
[5
[4
[55
5.5
1/2
3/2
2]
1/2
[42
[65
3/2
3/2
1g7/2
1]
[65
[65
3/2
1/2 [411]
1/2
7/
[66
]
02
2
5/
]
30
[5
[51
9/2
1/2
4]
2d3/2
]
1/2
1]
[43
1/2[301]
50
1/2
2[
40
0]
4]
[42
[30
1/2
9/
1]
1]
[54
1/2
1]
]
5/2 [4
22]
3
[43
−.2
−.1
.0
1]
1]
−.3
1/2[660]
1]
30
/2 [
[54
3/2
1]
[30
3]
1/2
[30
2
/
5
[30
3/2
1g9/2
0]
[55
5.0
3/2
1/2
413
7/2 [
.1
.2
.3
.4
ε2
Figure 12. Nilsson diagram for protons, 50 ≤ Z ≤ 82 (ε4 = − ε22 /6).
.5
.6
1/2[770]
3/2[761]
3]
74
9/2[734]
0
[75
[86
]
2]
1/2
0]
3/2
[88
1
[87
]
]
31
[6
3/2
3]
74
2[
7/ 3]
63
[ 50
1/2
7/2
3/2[752]
]
2]
[64
3/2
0]
0
[64
]
761
1/2 [
2[
3]
5/
1]
52
[
1/2
1/2
114
13/
2 [6
06]
[77
5]
512]
5/2 [
7/2[514]
5]
61
9/2
11
[50
[
/2
512]
5/2 [
9/2[624]
2]
40
2[
1/
3/
2]
[75 ]
5/2 [640
1/2
[63
0]
2]
3]
[52
40
1]
[5
21
]
[76
1]
7/2[404]
1]
[65
0]
[77
3/2
1/2
1/2
2]
64
1/
2[
1/2
2[
5/
1h9/2
7/2
[64
3/2
3/2
6.5
4]
[51
1]
7/2
[65
1i13/2
2[
40
0]
2f7/2
7/2[743]
1/2
− ω)
Es.p. (h
5/2
2]
3/2 [51
3/2[512]
2f5/2
7.0
0]
1/2 [51
1 /2
510]
1]
[63
3]
1/2
2]
2[
1/2 [
3/2[512]
[50
5/2
[
5
[7
7/
7/2
3/2[501]
50
3]
2]
[75
3/2
5/2
3p3/2
1/2[510]
7/2[624]
2]
53
1]
[65 660]
3/2 1/2 [
2[
3/
1/2
5/2
1
[54
[52
3]
]
7/
2[
63
1]
[65 660]
[
1/2
[4
3]
5/2
3/2
]
00
4
2[
02
]
82
1/
2]
50
5
]
40
.0
.1
]
04
[4
2]
40
.2
7/2
2[
5/
−.1
2]
[51
0]
−.2
[53
9/2
[64
1/2
4]
2d3/2
−.3
5/2
3s1/2
6.0
11
/2 [
2[
3/
.3
.4
ε2
Figure 13. Nilsson diagram for protons, Z ≥ 82 (ε4 = ε22 /6).
.5
.6
[5
0
5/2
]
3/2
750
1 /2 [
]
1]
33
0]
]
]
880]
[64
[6
2
[86
3
[6
1/2 [
5/2
]
631
5/2
3/2
3/2[512]
1/2
7.0
2]
[51
[
1/2
1/2 [5
10]
2f5/2
3/2[512]
7/2[503]
3/2[871]
1/2[770]
3/2[761]
3/2[501]
3]
]
52
[7
5/2 42]
[6
43
]
510
1/2 [
]
31
[6
[7
3/2
3p3/2
2
3/
7/2
[
1/2
7/
74
3]
7/2
]
[752
3/2
[50
3]
1]
52
2[
06]
2 [6
13/
9/2
11
/2
[50
[6
5]
15
]
]
[75
2]
− ω)
Es.p. (h
5/2
1/2 [990
871]
3/2 [
61]
1/2 [7
24]
512
5/2 [
]
14
2 [5
1/2
7/
1i13/2
1/2[400]
3/2[402]
[52
1]
1/
2[
1/2 [
6.5
[63
1]
3]
40
1]
2]
80]
[64
0
[64
1/2 2]
[52
1/2 [8
[
1/2
64
1h9/2
3/2[871]
0]
3/2
2[
5/
3/2
52
651]
7/2
7/2[514]
7/2[743]
9/2 [6
]
2f7/2
]
512
[
5/2
1]
[76
3/2 0]
[77
1/2
114
]
5/2 [5
3/2[521]
2[
40
1/
651]
1]
[54
1/2 [
1/2
0]
1]
1/2 [76
2]
53
1]
[65 60]
3/2 1/2 [6
2[
3/
23]
82
2]
40
2[
5/
3/2 [
]
7/
5]
11
/2 [
50
]
0]
7
1/2 [7
]
1/2
2]
40
5/
2[
.2
.3
1/2 [411]
1]
[65
.1
0]
[53
.0
3/2
−.1
[51
0]
[66
−.2
9/2
1/2
4]
2d3/2
−.3
42
3s1/2
6.0
[6
2]
40
2[
3/
5/2
2[
40
4]
[4
761
00
2
1/
1/2 [411]
.4
ε2
Figure 14. Nilsson diagram for protons, Z ≥ 82 (ε4 = − ε22 /6).
.5
.6
Additional quantities of interest for describing a rotational band include the rotational frequency
E γ((J +2)→J ) + E γ(J →(J −2))
−
h ω(J ) = h hhhhhhhhhhhhhhhhhhhhhhhhh MeV ,
4
(18)
2J − 1
_
J 1(J ) = hhhhhhhhhhhhh −
h 2 MeV −1,
E γ(J →(J −2))
(19)
4
_
J 2(J ) = h hhhhhhhhhhhhhhhhhhhhhhhhh −
h 2 MeV −1 .
E γ((J +2)→J ) − E γ(J →(J −2))
(20)
the kinetic moment of inertia
and the dynamic moment of inertia
An experimental Routhian plot is often prepared to compare rotational bands with theory. This plot is restricted to sequences with ∆J =2 when defining a band. The frequency parameter ω(J ) is defined as
E (J +1) − E (J −1)
ω(J ) = hhhhhhhhhhhhhhhhh ,
Jx (J +1) − Jx (J −1)
(21)
where the x-component of angular momentum Jx is
Jx (J ) = √ddddddddddd
(J +1/2)2−K 2.
(22)
E ′(J ) = 1/2 [E (J +1)+E (J −1)] − ω(J )Jx (J ).
(23)
The experimental Routhian13 is defined by
These Routhians are normally referenced to even-even nuclei where the quantities
e ′(ω) = E ′(ω)−Eg′ (ω) ,
(24)
j (ω) = Jx (ω)−Jxg (ω) .
Here Eg′ (ω) and Jxg (ω) are the Routhian and the x-component of the angular momentum of the reference
configuration, respectively. For an odd-A nucleus, the Routhian becomes
e ′(A ,ω) = E ′(A ,ω) + ∆ − 1/2 [Eg′ (A +1,ω) + Eg′ (A −1,ω)],
(25)
where A is the mass number and ∆ is the average even-odd mass difference.
4. Signature Splitting
The lowest energy state of each single-particle Nilsson configuration will have J =K =Ω (Ω≠1/2). For
each configuration there is a rotational band of levels where K =Ω and the energies are given by equation
4 with J taking on the values K , K +1, K +2, K +3, . . . Coriolis and centrifugal forces introduce an additional term to the matrix element which involves a phase factor σ=(−1)J +K called the signature. This term
alternates sign for successive values of J and implies that rotational bands with K ≠0 divide into two families distinguished from each other by the quantum number σ. Bengtsson and Frauendorf13 have redefined
the signature quantity for single particle states as α=±1/2, where α is an additive quantity where r =e −i πα.
The total signature αt for a configuration restricts the angular momentum to
J = αt mod 2 ,
(26)
determining whether the band has odd or even particle number N , where
N = 2αt mod 2 .
(27)
For even-A nuclei, we obtain the 2-quasiparticle sequences:
if αt = 0 (r = +1),
J = 0, 2, 4, 6, . . . ,
if αt = ±1 (r = −1), J = 1, 3, 5, 7, . . . .
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
13
R. Bengtsson and S. Frauendorf, Nucl. Phys. A327, 139 (1979).
(28)
Similarly, for odd-A nuclei:
if αt = +1/2 (r = −i ),
if αt = −1/2 (r = +i ),
J = 1/2, 5/2, 9/2, . . . ,
(29)
J = 3/2, 7/2, 11/2, . . . .
An example of signature splitting is shown in Figure 15.
Collective particle configurations can also be discussed in terms of the alignment of the particle’s
intrinsic angular momentum with the angular momentum of the core. The "favored" (lowest energy)
configuration occurs when these momenta are aligned to the maximum value. Decreasing the alignment
− produces the "unfavored" configuration. These two configurations are analogous to the signature
by 1h
partners described above.
In octupole deformed nuclei, rotational states may be characterized by eigenvalues s of the simplex
operator14 J=PR−1 where R corresponds to reflection by 180° about an axis perpendicular to the symmetry
axis and P is the parity operator. The spin J and parity p of a state in a rotational band which can be
described by this "reflection" symmetry are related by p =e −i πJ . Accordingly, s =±1 for even particle
number (integral J ), and s =±i for odd particle number (half-integral J ). These reflection-asymmetric
deformations (octupole-quadrupole deformed) lead to bands of alternating parity, connected by enhanced
E 1 transitions, with levels nearly degenerate in energy for the same spin but opposite parity (parity doublets). Examples of parity doublets for an odd-particle system (221Ra)15 and an even-particle system
(224Ac)16 are shown in Figure 16.
5. Superdeformation and Hyperdeformation
In 1968, Strutinski17 predicted a second minimum in the potential well for a deformed nucleus. This
phenomenon was subsequently identified with fission isomers in the actinides, first observed by Polikanov
et al.18 The associated shell gaps, calculated with an axially symmetric harmonic oscillator potential by
Nix19 and by Bohr and Mottelson20, are shown in Figure 17. These gaps occur with varying magic
numbers for ratios of the semi-major to semi-minor axis of 3/2 and 2 (superdeformation), and 3 (hyperdeformation). In 1986, Twin et al 21,22 reported the first evidence, for a superdeformed band in 152Dy. Six
superdeformed bands have been found in 152Dy and are shown in Figure 18. Tentative evidence for a
hyperdeformation in 152Dy was reported by Galindo-Uribarri et al 23 who observed a δE γ=±30 keV ridge
structure in coincidence data for 152Dy. Discrete transitions, possibly from a hyperdeformed band in 153Dy,
were reported by Viesti et al 24.
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
14
W. Nazarewicz, P. Olanders, I. Ragnarsson, J. Dudek, and G.A. Leander, Phys. Rev. Lett. 52, 1272 (1984).
15
J. Fernández-Niello, C. Mittag, F. Riess, E. Ruchowska, and M. Stallknecht, Nucl. Phys. A531, 164 (1991).
16
P.C. Sood, D.M. Headly, R.K. Sheline, and R.W. Hoff, At. Data Nucl. Data Tables 58, 167 (1994).
17
V.M. Strutinski, Nucl. Phys. A95, 420 (1967); ibid A122, 1 (1968).
18
S.M. Polikanov, V.A. Druin, V.A. Karnaukhov, V.L. Mikheev, A.A. Pleve, N.K. Skobelev, G.M. Ter-Akopyan, and V.A. Fomichev, Zhur. Ekspti. i Teoret. Fiz. 42, 1464 (1962); Soviet Phys. JETP 15, 1016 (1962).
19
J.R. Nix, Ann. Rev. Nucl. Sci. 22, 65 (1972).
20
A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. II, p. 591, Benjamin, New York (1975).
21
P.J. Twin, B.M. Nyako, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R.
Mokhtar, J.D. Morrison, J.F. Sharpey-Shafer, and G. Sletten, Phys. Rev. Lett. 57, 811 (1986).
22
P.J. Dagnall, C.W. Beausang, P.J. Twin, M.A. Bentley, F.A. Beck, Th. Byrski, S. Clarke, D. Curien, G. Duchene, G. de
France, P.D. Forsyth, B. Haas, J.C. Lisle, E.S. Paul, J. Simpson, J. Styczen, J.P. Vivien, J.N. Wilson, and K. Zuber, Phys. Lett.
B335, 313 (1994).
23
A. Galindo-Uribarri, H.R. Andrews, G.C. Ball, T.E. Drake, V.P. Janzen, J.A. Kuehner, S.M. Mullins, L. Persson, D. Prevost,
D.C. Radford, J.C. Waddington, D. Ward, and R. Wyss, Phys. Rev. Lett. 71, 231 (1993).
24
G. Viesti, M. Lunardon, D. Bazzacco, R. Burch, D. Fabris, S. Lunardi, N.H. Medina, G. Nebbia, C. Rossi-Alvarez, G. de
Angelis, M. De Poli, E. Fioretto, G. Prete, J. Rico, P. Spolaore, G. Vedovato, A. Brondi, G. La Rana, R. Moro, and E. Vardaci, Phys.
Rev. C51, 2385 (1995).
923
839
776
(53/2+)
5225
(49/2+)
830
782
6007
7/2–
698
630
810
787
772
739
716
584
533
510
678
651
2973.5
2231.6
(33/2+)
1722.4
(29/2+)
1245.2
(25/2+)
821.9
(21/2+)
2261.9
(31/2+)
1683.7
(27/2+)
1187.0
(23/2+)
778.8
(19/2+)
465.1
15/2+
247.3
120.346
69.219
11/2+
7/2+
607
515
(35/2+)
2746.1
578
566
517
492
479
2913.2
2295.1
523
497
477
428
408
423
448
393
1688.4
1165.3
326*
314
355
1034.0
737.2
83.965
190.02
5/2–
0
5/2[523] α=-1/2
5/2[642] α=+1/2
5/2[523] α=+1/2
163
68Er
Figure 15. Signature splitting observed in 163Er.
213
9/2–
13/2+
9/2+
5/2+
108
17/2+
127
467.2
13/2–
320.2
218
640.8
277
321
236*
(37/2+)
1482.1
(17/2–)
(11/2)–
3712
1960.8
(21/2–)
(15/2)–
(39/2+)
2452.7
(25/2–)
(19/2–)
3629.5
3279
2969.7
(29/2–)
(23/2–)
4499
3535.8
(33/2–)
(27/2–)
(43/2+)
4402
3863.6
(41/2+)
(37/2–)
(31/2–)
5309
5213
4165.3
(41/2–)
(35/2–)
(47/2+)
811
714
648
4511.2
(45/2+)
(39/2–)
6042
4863
(45/2–)
(43/2–)
6923
5639
(49/2–)
(47/2–)
(57/2+)
6478
(53/2–)
(51/2–)
6853
880
7373
(57/2–)
(55/2–)
7757
846
(59/2–)
895
904
8296
412.2
199.3
91.552
5/2[642] α=-1/2
(13/2+)
253.5
5+
177.0
711.6
4–
566.0
3–
438.0
341.5
210.9
2–
1–
0–
50
849.6
4+
130.4
116.7
78.4
3+
37.2
2+
1+
0+
17.2
0
221
88Ra
29
350
176
314
331
195
155
(11/2+)
6+
80.5
22
(13/2–)
184.2
41
61
318.9
5–
38
440.4
(15/2+)
237.2
37
(17/2–)
284
573.1
274
(19/2+)
128
225
688.4
146
(21/2–)
6–
1180.5
1025.9
138
(23/2+)
866.9
1375.7
17
122
(17/2+)
(25/2–)
97
302
294
115
248
133
254
(19/2–)
990.4
(27/2+)
131
354
147
179
(23/2–)
(15/2–)
1197.4
124
(25/2+)
(21/2+)
1344.4
207
331
(27/2–)
51.9
29.8
22.0
224
89Ac
Figure 16. Examples of parity doublets arising from reflection asymmetry in 221Ra and 224Ac.
8
7
8
168
190
182
140
140
7
112
110
100
6
80
70
68
⁄
−
ENERGY IN UNITS OF hω
6
60
5
4
5
40
44
40
28
26
4
20
16
14
3
8
3
10
6
4
2
1:2
2
2
2
2
2:3
1:1
3:2
2:1
1
–1.0
3:1
1
– 0.5
0
ω⊥ − ω 3
δosc = 
−
ω
0.5
1.0
Figure 17. Single-particle level energies calculated for an axially symmetric harmonic oscillator (from reference 18).
1546
19393.7+x
1376
(24+)
x
SD-1 band
(36)
(40)
825.9+y
(38)
y
(36)
(34)
SD-2 band
1632.7+z
793.0+z
(29–)
z
(27–)
SD-3 band
954
1434
1388
1344
1300
1257
1212
1123
1031
5589.0+v
(36–)
2988.4+u
(34–)
2163.4+u
(32–)
1390.6+u
(30–)
669.6+u
(28–)
u
(26–)
SD-4 band
152
66Dy
Figure 18. Superdeformed rotational bands for 152Dy
3733.9+v
(40–)
2882.8+v
(38–)
5237.9+w
4251.8+w
895
3865.2+u
941
4635.2+v
7346.5+w
6269.2+w
850
(38–)
901
(44–)
4794.2+u
9636.3+w
8469.2+w
986
(40–)
12104.6+w
10847.9+w
1077
1105
1055
1005
6594.1+v
14748.8+w
13404.5+w
1167
1304
1206
1156
1228
1180
1131
(31–)
(42–)
17570.7+w
16137.2+w
3310.6+w
2415.7+w
2084.0+v
1337.0+v
SD-5 band
805
(33–)
1681.3+y
6802.7+u
747
2523.9+z
7649.1+v
(42–)
877
(35–)
(44–)
(46–)
825
3468.7+z
8754.1+v
(48–)
5772.9+u
773
(42)
7883.5+u
929
998
945
(44)
(46–)
(50–)
4466.9+z
(37–)
(54–)
(36–)
642.1+v
(34–)
v
(32–)
762
602.4+x
2576.5+y
(46)
9014.1+u
5519.3+z
(39–)
(56–)
(52–)
851
(48)
(41–)
(58–)
9909.9+v
799
6624.2+z
(43–)
12369.5+v
11115.7+v
695
(50)
7781.1+z
(45–)
1081
(52)
8989.0+z
(47–)
721
(38)
1249.9+x
3508.7+y
10194.4+u
1030
1208
(54)
891
970
932
895
(40)
4478.6+y
1254
1328
1281
1369
1314
1260
(49–)
13673.7+v
642
1009
5487.1+y
11422.4+u
(48–)
840
648
(42)
1942.6+x
602
(28+)
6536.3+y
855
693
2680.7+x
(30+)
(26+)
3464.7+x
(44)
826
784
4294.6+x
738
(34+)
(32+)
(46)
10249.4+z
9949.9+y
7628.9+y
(51–)
5171.0+x
830
(38+)
(36+)
(48)
6094.2+x
876
(40+)
7064.4+x
923
(42+)
1049
970
(50)
11563.2+z
979
8081.8+x
12703.4+u
(50–)
(56)
8766.5+y
(53–)
(52–)
793
(44+)
1017
(52)
12931.8+z
12458.2+y
1157
9146.7+x
(60)
11180.5+y
1093
(46+)
14031.1+u
(54–)
(58)
1138
1065
(54)
(55–)
1105
10259.4+x
14357.6+z
1052
(56)
15407.0+u
(60–)
1426
1377
1327
(58)
(57–)
13785.6+y
1278
(60)
(62–)
15840.3+z
15162.7+y
(62)
1231
11419.9+x
(62)
(64–)
17384.4+z
16586.3+y
1183
1305
12628.5+x
1113
(50+)
(48+)
13885.1+x
1161
(52+)
15189.9+x
1209
(54+)
(64)
(64)
1257
(56+)
16542.8+x
1424
1353
(66)
(58+)
18989.1+z
18063.4+y
1544
(66)
1483
17944.1+x
1477
(60+)
1401
(68)
670
(62+)
1605
20891.5+x
1450
(64+)
22437.1+x
1498
(66+)
1566.0+w
761.5+w
w
SD-6 band