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 in1948 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 nucleonsmoved in a spherically symmetric potential well. This potential is most commonly taken as a harmonicoscillator of the form

(1)V (r ) = −V0RQ1 − (r /R )2 H

P,

where V(r) is the potential at a distance r from the center of the nucleus and R is the nuclear radius. Theresulting quantum states can be characterized by n , the principal quantum number related to the numberof radial nodes in the wave function, and by l , the orbital angular momentum of the particle. By analogy toatomic 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 thateach state has n −1 radial nodes.

The solution of the Schro..dinger equation for the isotropic harmonic oscillator gives the level energies

(2)E = h− ω0[2(n − 1) + l ]+E0 ,

where

(3)ω0=IJL MR 2

2V0hhhhhMJO

1/2

,

M is the nucleon mass, and E0=3/2h− 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 musthave 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.1p1/2 and 1p3/2). These states are no longer degenerate, and it was shown that if the "spin-orbit" interac-tion is of the same order as the spacing between oscillator shells, and states with j =l +1/2 are more stablethan states with j =l −1/2, then the magic numbers can be reproduced. The evolution of shell model statesfrom 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 adequatelydescribe nuclei outside the closed shells. This may be attributed to a breakdown in the assumption ofspherical symmetry. Rainwater6 proposed that, in odd-A nuclei, the motion of the odd nucleon couldpolarize the even-even core allowing all nucleons to move collectively, thus increasing the static electricquadrupole moments and enhancing quadrupole transition (E 2) rates. The implications of spheroidalrather than spherical shape were studied by Bohr and Mottelson7 who developed a collective model fornuclei.

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 angularmomentum coupled to that of the odd nucleon. Addition of more nucleons outside the closed shells canhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

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).

Harmonic Oscillator Shell Model Collective ModelN hω n l n l j Ω[NnzΛ]n ([NnzΛ]p)

π

Figure 1. Evolution of nuclear models from theharmonic oscillator model, adding strong spin-orbit coupling to obtain the shell model, andaxial deformation to give the collective model.Asymptotic Nilsson configurations are given forneutrons and, in parentheses, for protons whendifferent. Because of mixing, very few stateshave pure Ω[NnzΛ] configurations.

0 hω +

1s 1s1/2

2

1p1p3/2

1 hω - 3/2[101]

1/2[110]

1p1/2

8

1/2[101]

1d1d5/2

2s2s1/2

2 hω +

5/2[202]3/2[211]1/2[220]

1/2[211]1d3/2

203/2[202]1/2[200]

1f

1f7/2

287/2[303]5/2[312]3/2[321]1/2[330]

1f5/25/2[303]3/2[301]1/2[310]

2p2p1/2

1/2[301]

2p3/23/2[312]1/2[321]

3 hω -

1g

1g9/2

50

9/2[404]7/2[413]5/2[422]3/2[431]1/2[440]

1g7/2 7/2[404]5/2[402] ([413])3/2[411] ([422])1/2[420] ([431])

2d2d5/2

5/2[413] ([402])3/2[422] ([411])1/2[431] ([420])

2d3/2

3/2[402] ([402])1/2[400] ([411])

3s 3s1/2

1/2[411] ([400])

1h

1h11/2

11/2[505]9/2[514]7/2[523]5/2[532]3/2[541]1/2[550]

4 hω +

82

1h9/29/2[505]7/2[503] ([514])5/2[512] ([523])3/2[521] ([532])1/2[530] ([541])

2f2f7/2

5 hω -

7/2[514] ([503])5/2[523] ([512])3/2[532] ([521])1/2[541] ([530])

3p3p3/2

3/2[512] ([501])1/2[521] ([510])

2f5/25/2[503]3/2[501] ([512])1/2[510] ([521])

3p1/2 1/2[501]

1i

1i13/2

126 13/2[606]11/2[615]9/2[624]7/2[633]5/2[642]3/2[651]1/2[660]

1i11/26 hω +

11/2[606]9/2[604]7/2[613]5/2[622]3/2[631]1/2[640]

2g

2g9/2 7/2[624]5/2[633]3/2[642]1/2[651]

2g7/2

7/2[604]5/2[602]3/2[611]1/2[620]

3d

3d5/25/2[613]3/2[622]1/2[631]

3d3/2

3/2[602]1/2[600]

4s 4s1/2

1/2[611]1j15/2

18415/2[707]13/2[716]11/2[725]9/2[734]7/2[743]5/2[752]3/2[761]1/2[770]

form spheroidal nuclei whose angular momentum states reflect the coherent motion of all nucleons ratherthan just a few nucleons moving in single-particle shell model orbitals. The quantization of this coherentmotion forms the basis of the collective model.

This collective motion can be quantized by assuming that the even-even core is an incompressibleliquid and quantizing the classical hydrodynamical equations that describe its oscillations. Borrowing fromthe well-known analysis of rotational states in symmetric-top molecules, we expect the level energy rela-tion

(4)E = h− 2

2J_ 1

[J (J + 1) − K2]hhhhhhhhhhhhhh +2J_ 3

h− 2K2hhhhhh

where [J (J + 1)]1/2h− is the total angular momentum of the nucleus, and K h− is the component of angularmomentum along the symmetry axis of the nucleus, as shown in Figure 2. J_ 1 and J_ 3 are the moments ofinertia perpendicular and parallel to the symmetry axis, respectively. For low-lying rotational bands ineven-even nuclei, K =0 and Eq. (4) reduces to

(5)E = h− 2

2J_ 1

[J (J + 1)]hhhhhhhhh .

States with K ≠0 arise when one or more pairs of nucleons are broken and unpaired nucleons are excitedto higher shell-model states. Low-lying rotational levels with K =0 are characterized by a sequence ofstates with J = 0, 2, 4, 6, 8, . . . and positive parity, often referred to as the ground-state (GS) band. Anexample of the GS band for 152Sm is shown in Figure 3. Excited vibrational states of three kinds are alsoshown for 152Sm in Figure 3. The β-vibration preserves spheroidal symmetry, and the γ-vibration goesthrough an ellipsoidal symmetry. The octupole vibration produces a low-lying Jπ=3− state in sphericalnuclei, but can also couple to core quadrupole excitations in deformed nuclei to give a family of low-lyingstates with Jπ=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 2j + 1 degeneracy of the spherical shell-model states. Asymptotic quantum numbers needed to describe these states are shown in Figure 2. Inaddition to K , they include N , the total oscillator shell quantum number; nz , the number of oscillatorquanta in the z direction; M , the projection of total angular momentum J on the laboratory axis; R , theangular momentum from the collective motion of the nucleus; Ω, the projection of total angular momentumj (orbital l plus spin s ) of the odd nucleon on the symmetry axis; and Λ, the projection of angular momen-tum along the symmetry axis where Ω=Λ+Σ and Σ is the projection of intrinsic spin along the symmetryaxis. Levels are labeled by the asymptotic quantum numbers Ωπ[Nnz Λ]. The Nilsson orbitals whichderive from the axially deformed harmonic oscillator potential, labeled as described by Davidson3, areshown in Figure 1. A more realistic calculation can be performed using a Woods-Saxon potential

(6)V (r ) =

1+exp(a

r −Rhhhhh )

V0hhhhhhhhhhhh ,

where a ≈0.67 fm. Considerable mixing of Nilsson configurations with ∆Ω=0, ∆N=2, and similar excitationenergies 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, areshown in Figures 4-14. In these figures, single-particle energies are plotted in units of the oscillator fre-quency

(7)h− ω0 = 41A −1/3 MeV

as a function of deformation ε2, with ε4 chosen as described in the figure captions. The Nilsson quadru-

pole deformation parameter ε2 can be defined in terms of δ=Rr.m.s.

∆Rhhhhhh where Rr.m.s. is the root mean

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh1 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).

square nuclear radius and ∆R is the difference between the semi-major and semi-minor axes of thenuclear ellipsoid, as12

(8)ε2 = δ +61hhδ2 +

185hhhδ3 +

21637hhhhδ4 + . . . .

The deformation parameter β2 is related to ε2 by

(9)β2 = √ddddπ/5IJL 34hhε2 +

94hhε2

2 +274hhhε2

3 +814hhhε2

4 + . . .MJO

The intrinsic quadrupole moment Q0 (or transition quadrupole moment QT ∼∼Q0) is related to β2 by

(10)Q0 = √dddddd16π/5ZeR02 β2.

If the partial photon half-life for the E2 transition is known, the experimental transition strength B (E 2) isgiven by

(11)B (E 2) =t 1/2(E 2)(E γ)

50.05659hhhhhhhhhhhh (eb )2,

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh12 K.E.G. Lo

..bner, M. Vetter, and V. Ho

..nig, Nucl. Data Tables A7, 495 (1970).

Laboratory axis

J R

s

l

j

ΛΣ

ΩK

M

Symmetry axis

Figure 2. Asymptotic quantum numbers for the deformed shell model

where t1/2(E 2) is in picoseconds, and E is in MeV. B (E 2) is related to Q0 by

(12)B (E 2) = B (JK →J −2 K ) =16π5hhhhQ0

2 <JK 20 | J −2 K>2(eb )2.

Assuming a constant charge distribution in the nucleus, Q0=0 for spherical nuclei, Q0>0 for prolate (foot-ball or cigar shaped) nuclei, and Q0<0 for oblate (disk shaped) nuclei. The spectroscopic quadrupolemoment Q is related to the intrinsic moment by 12

(13)Q = Q0 (J + 1)(2J + 3)3K2 − J (J + 1)hhhhhhhhhhhhhh .

For M 1 transitions, the transition strength B (M 1) is given by

(14)B (M 1) = B (JK →J ±1 K ) =4π3hhh (gK −gR )2K2<JK 10 | J ±1 K>

2µN

2 ,

where gK and gR are nuclear g -factors. The rotational g -factor, gR , arises from the collective rotation ofthe core, and is defined as

(15)gR ≈AZhh .

The intrinsic g -factor, gK , arises from the orbital motion of the valence nucleons, and the total magneticdipole moment is given by

(16)µ = gR J + (gK − gR )J +1K2hhhhh

If the partial photon half-life for the M 1 transition is known, then B (M 1) can be calculated from

(17)B (M 1) =t 1/2(M 1)E γ

3

0.0394hhhhhhhhhhhµN2 .

1– 963.376

Octupole band

3– 1041.180

5– 1221.64

7– 1505.62

9– 1879.1

11– 2327.1

13– 2833.5

(15–) 3383.5

507

0+ 0

GS band

2+ 121.7825

4+ 366.4814

6+ 706.96

8+ 1125.35

10+ 1609.34

12+ 2148.8

14+ 2736.3

(16+) 3362.3

122

245

34

0

418

48

4

540

58

8

626

0+ 684.70

β band

2+ 810.465

4+ 1022.962

6+ 1310.51

8+ 1666.48

10+ 2079.6

12+ 2525.6

14+ 2976.7

126

213

28

8

356

41

3

446

45

1

2+ 1085.897

γ band

3+ 1233.8764+ 1371.752

5+ 1559.53

6+ 1728.38

7+ 1945.82

(8+) 2139.7

(9)+ 2375.5

148

28

6 32

6

218

386

235

152 62Sm

Figure 3. Ground-state, and β-, γ-, and octupole-vibrational bands in 152Sm.

−.3 −.2 −.1 .0 .1 .2 .32.0

2.5

3.0

3.5

4.0

4.5

5.0

ε2

Figure 4. Nilsson diagram for protons or neutrons, Z, N ≤ 50 (ε4 = 0).

Es.

p. (

h−ω

)

8

20

28

50

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

2p3/2

1f5/2

2p1/2

1g9/2

1/2[110]

3/2[101]

1/2[101]

1/2[220]

3/2[211]

5/2[202]

1/2[211]

1/2[200]

3/2[202]

1/2[330]

3/2[321]

5/2[312]

7/2[303]

1/2[321]

3/2[312]1/2[310]

3/2[301]

5/2[303]1/2[301]

1/2[440]

3/2[431]

5/2[422]

7/2[413]9/2[404]

1/2[431]

1/2

[420

]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .6

5.0

5.5

6.0

ε2

Figure 5. Nilsson diagram for neutrons, 50 ≤ N ≤ 82 (ε4 = ε22/6).

Es.

p. (

h−ω

)

50

82

1g9/2

2d5/2

1g7/2

3s1/2

1h11/22d3/2

3/2[301]

3/2[541]

5/2[532]

1/2[301]

1/2[550]

1/2[440]

3/2[431]

5/2[422]

7/2[413]

9/2[4

04]

1/2[431]

3/2[422]

5/2[413]

1/2[420]

3/2[411]

5/2[402]

5/2[642]

7/2[4

04]

7/2[633]

1/2[411]

1/2[660]

1/2[550]

1/2[301]

1/2[541]

3/2[541]

3/2[301]

5/2[532]

5/2[303]

7/2[523]

9/2[514]11/2

[505

]

1/2[4

00]

1/2[660]

1/2[411]

3/2[4

02]

3/2[651]

1/2[541]

1/2[301]

3/2[532]

5/2[523]1/2[530]

3/2[521]

1/2[660]

1/2[4

00]

1/2[651]

3/2[651]

3/2[4

02]

5/2[642]

5/2[4

02]

1/2[770]

3/2

[761

]

1/2[521]

1/2

[640

]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .6

5.0

5.5

6.0

ε2

Figure 6. Nilsson diagram for neutrons, 50 ≤ N ≤ 82 (ε4 = − ε22/6).

Es.

p. (

h−ω

)

50

82

1g9/2

2d5/2

1g7/2

3s1/2

1h11/22d3/2

3/2[301]

3/2[541]

5/2[303]

1/2[301]

1/2[550]

1/2[440]

3/2[431]

5/2[422]

7/2[413]

9/2[4

04]

1/2[431]

3/2[422]

5/2[413]

1/2[420]

1/2[660]

3/2[411] 3/2[651]

5/2[402]

5/2[642]

7/2[4

04]

1/2[411]

1/2[660]

1/2[420]

1/2[550]

1/2[541]

3/2[541]

3/2[301]

5/2[532]

7/2[523]

9/2[514]

11/2

[505

]

1/2[400]

1/2[660]

1/2[411]

1/2[651]

3/2[402]

3/2[651]

3/2[411]

1/2[541]

1/2[301]

3/2[532]

1/2[530]

1/2[770]

3/2[521]

3/2[761]

1/2[660] 1/2[40

0]1/2

[651]

1/2[411]

3/2[651]

3/2[40

2] 5/2[642]

5/2[402]

1/2[770]

1/2[530]

3/2[521]

1/2[761]

1/2

[400

]

1/2

[640

]

1/2[880]

1/2[640]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .66.0

6.5

7.0

ε2

Figure 7. Nilsson diagram for neutrons, 82 ≤ N ≤ 126 (ε4 = ε22/6).

Es.

p. (

h−ω

)

126

2f7/2

1h9/2

1i13/2

3p3/2

2f5/2

3p1/2

2g9/2

5/2[4

02]

5/2[642]

7/2[4

04]

7/2[633]

9/2[514]11/2

[505

]

1/2[4

00]

1/2[660]

3/2[4

02]

3/2[651]

1/2[541]

3/2[532]

5/2[523]

7/2[514]

1/2[530]

3/2[521]

5/2[512]

5/2[752]

7/2[

503]

7/2[743]

9/2[

505]

9/2[734]

1/2[660]

1/2[4

00]

1/2[651]

3/2[651]

3/2[4

02]

3/2[642]

5/2[642]

5/2[4

02]

7/2[633]

7/2[404]

9/2[624]

11/2[615]

13/2

[606

]

1/2[521]

1/2[770]

3/2[512] 3/2[761]

1/2[510]

1/2[770]

1/2[521]

3/2[

501]

3/2[761]

3/2[512]5/

2[50

3]

5/2[752]

5/2[512]

1/2[5

01]

1/2[770]

1/2[510]

1/2[761]

1/2[651]

1/2[640]

3/2[642]

3/2[4

02]

5/2[633]1/2[640]

1/2[4

00]

3/2[631]

5/2[622]

5/2[862]

1/2

[770

]

1/2

[501

]

1/2[761]

1/2[750]3/2[761]

3/2[

501]

3/2[752]

5/2[752]

5/2

[503

]

7/2[743]

7/2[

503]

1/2[631]

1/2[880]

3/2[871]

5/2[622]1/2

[880]

1/2[631]

1/2

[750

]

1/2[510]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .66.0

6.5

7.0

ε2

Figure 8. Nilsson diagram for neutrons, 82 ≤ N ≤ 126 (ε4 = − ε22/6).

Es.

p. (

h−ω

)

126

2f7/2

1h9/2

1i13/2

3p3/2

2f5/2

3p1/2

2g9/2

5/2[4

02]

5/2[642]

7/2[4

04]

9/2[514]

11/2

[505

]

1/2[4

00]

1/2[660]

3/2[4

02]

3/2[651]

1/2[541]

3/2[532]

5/2[523]

7/2[514]

1/2[530]

3/2[521]

3/2[761]

5/2[512]

5/2[752]

7/2[

503]

7/2[743]

9/2[

505]

1/2[660] 1/2[4

00]

1/2[651]

3/2[651]

3/2[4

02]

3/2[642]

5/2[642]

5/2[4

02]

7/2[633]

9/2[624]

11/2

[615

]13

/2[6

06]

1/2[521]

1/2[770]

3/2[512]

3/2[761]

3/2[521]

1/2[510]

1/2[770]

1/2[521]

1/2[761]

3/2[5

01]

3/2[761]

3/2[512]

3/2[752]

5/2[5

03]

5/2[752]

5/2[512]

1/2[5

01] 1/2

[770]

1/2[510]

1/2[761]

1/2[521]

1/2[651]

1/2[40

0]

1/2[880]

3/2[642]

3/2[402]

5/2[633]

1/2[640]

1/2[640]

3/2[631]

3/2[871]

5/2[862]

1/2

[770

]

1/2

[501

]

1/2[761]

1/2[510]

1/2[990]

3/2[761]

3/2[5

01]

3/2[752]

3/2[512]

5/2[752]

5/2

[503

]

7/2[743] 7/2[5

03]

1/2[631]

1/2[880]

1/2[400]

3/2[871]

3/2[631]

1/2[880]

1/2[631]

1/2[871]

1/2[871]

1/2[631]

1/2[10

100]

1/2[750]

3/2[981]

1/2[990]

1/2[510]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .67.0

7.5

8.0

ε2

Figure 9. Nilsson diagram for neutrons, N ≥ 126 (ε4 = ε22/6).

Es.

p. (

h−ω

)

126

3p1/2

2g9/2

1i11/2

1j15/2

3d5/2

2g7/2

4s1/2

3d3/2

2h11/2

7/2[743]

9/2[

505] 9/2[734]

11/2[615]

13/2

[606

]

3/2[

501]

3/2[761]

3/2[512]

5/2[

503]

5/2[752]

1/2[5

01]

1/2[770]

1/2[651]

3/2[642]

5/2[633]

7/2[624]

9/2[

615]

9/2[844]

1/2[640]3/2[631]

5/2[622]

5/2[862]

7/2[613]

7/2[853]

9/2[

604]

9/2[844]

9/2[615]

11/2

[606

]

11/2

[835

]

1/2[

501]

1/2[761]

3/2[761] 3/2[

501]

3/2[752]

5/2[752]

5/2[5

03]

5/2[743]

7/2[743]

7/2[

503]

9/2[734]9/2[505]

11/2[725]

13/2

[716

]15

/2[7

07]

1/2[631]

3/2[622]

3/2[871]

5/2[613]

5/2[862]

5/2[622]

1/2[620]

1/2[880]

3/2[611]

3/2[871]

3/2[622]

5/2[602]

5/2[853]

7/2[

604]

7/2[853]

7/2[613]

1/2[611]

1/2[871]

1/2[

600]

1/2[871]

1/2[620]

3/2[

602]

3/2[871]

3/2[611]

3/2[862]

1/2[761]

1/2[

501]

1/2[750]

1/2[510]

3/2[752]

3/2[741]

5/2[743]

5/2[

503]

5/2[732]

7/2[734]

9/2

[725

]

1/2

[880

]

1/2

[600

]

1/2[871]

1/2[860]

3/2

[871

]

3/2

[602

]

3/2

[862

]

3/2[851]

5/2[862]

5/2

[602

]

5/2[853] 5/2[613]

7/2

[853

]

7/2

[604

]

9/2[844]

9/2

[604

]

1/2

[750

]

1/2[990]

3/2[741]

3/2[

501]

3/2[981]

5/2[732]

5/2[503]

7/2

[723

]

1/2[741]

1/2[741]

3/2[732]

3/2[501]

5/2[972]

1/2[730]

1/2[990]

1/2[981]

3/2[732]

1/2[

501]

3/2[972]

1/2

[981

]

1/2[730]

1/2[860]

1/2[10

100]

3/2[851]

3/2[611]

1/2[970]

1/2[851]

1/2[611]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .67.0

7.5

8.0

ε2

Figure 10. Nilsson diagram for neutrons, N ≥ 126 (ε4 = − ε22/6).

Es.

p. (

h−ω

)

126

3p1/2

2g9/2

1i11/2

1j15/2

3d5/2

2g7/2

4s1/2

3d3/2

2h11/2

7/2[

503]

7/2[743]

9/2[

505]

9/2[624]11

/2[6

15]

13/2

[606

]

3/2[5

01]

3/2[761]

3/2[512]

5/2[5

03]

5/2[752]

1/2[5

01] 1/2

[770]

1/2[510]

1/2[651]

3/2[642]

5/2[633]

7/2[624]

9/2[615]

9/2[844]

1/2[640]

3/2[631]

5/2[622]

5/2[862]

7/2[613]

7/2[853]

9/2[604]

9/2[615]

11/2

[606

]

1/2[5

01]

1/2[761]

1/2[510]

3/2[761]

3/2[5

01]

3/2[752]

3/2[512]

5/2[752]

5/2[5

03]

5/2[743]

7/2[743]

7/2[5

03]

9/2[734]

11/2[725]13

/2[7

16]

15/2

[707

]

1/2[631]

3/2[622]

3/2[871]

5/2[61

3]

5/2[862]

5/2[622]

1/2[620]

1/2[880]

1/2[631]

3/2[611]

3/2[871]

3/2[622]

3/2[862]

5/2[602]

5/2[613]

5/2[853]

7/2[

604]

7/2[853]

7/2[613]

1/2[611]

1/2[880]

1/2[620]

1/2[871]

1/2[631]

1/2[

600]

1/2[871]

1/2[620]

1/2[10

100]

3/2[

602]

3/2[871]

3/2[611]

3/2[862]

3/2[622]

1/2[761]

1/2[750]

3/2[752]

3/2[5

01]

3/2[741]

3/2[981]

5/2[743]

5/2[5

03]

7/2[734]

9/2

[725

]

1/2

[880

]

1/2

[600

]

1/2[871]

1/2[860]

3/2[871]

3/2

[602

]

3/2

[862

]

3/2[10

91]

5/2[862]

5/2

[602

]

5/2[613]

7/2

[853

]

7/2

[604

]

9/2[844]

9/2

[604

]

1/2[750]

1/2[990]

1/2[510]

3/2[741]

3/2[741]

5/2[732]1/2[741]

1/2[981]

3/2

[981

]

3/2[5

01]

5/2[972]

1/2[990]

1/2[741]3/

2[7

32]

3/2[972]

1/2[730]

1/2[981]

1/2[5

01]

3/2[732]

1/2

[730

]

1/2[970]

1/2[860]

1/2[10

100]

1/2[620]

3/2[851]

3/2[851]

1/2[730]

1/2[10

91]

3/2[611]

1/2[851]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .6

5.0

5.5

6.0

ε2

Figure 11. Nilsson diagram for protons, 50 ≤ Z ≤ 82 (ε4 = ε22/6).

Es.

p. (

h−ω

)

50

82

1g9/2

1g7/2

2d5/2

1h11/2

2d3/2

3s1/2

3/2[301]

3/2[541]5/2[303]

5/2[532]

1/2[301]

1/2[550]

3/2[431]

5/2[422]

7/2[413]

9/2[4

04]

1/2[431]

3/2[422]

5/2[413]

7/2[4

04]

7/2[633]

1/2[420]

3/2[411]

3/2[651]

5/2[4

02]

5/2[642]

1/2[550]

1/2[301]1/2[541]

3/2[541]

3/2[301]

5/2[532]

5/2[303]

7/2[523]

9/2[514]11

/2[5

05]

1/2[411]

1/2[660]

3/2[651]

3/2[411]

1/2[400]

1/2[660]

1/2[411]

1/2[541]

1/2[301]

3/2[532] 5/2[523]

1/2[660]

1/2[4

00]

1/2[651]

3/2[651]

3/2

[402

]

3/2[642]

5/2[642]

5/2[4

02]

7/2[633]

7/2[404]

1/2[530]

3/2[521]

1/2[770]

3/2[761]

1/2[640]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .6

5.0

5.5

6.0

ε2

Figure 12. Nilsson diagram for protons, 50 ≤ Z ≤ 82 (ε4 = − ε22/6).

Es.

p. (

h−ω

)

50

82

1g9/2

1g7/2

2d5/2

1h11/2

2d3/2

3s1/2

3/2[301]

3/2[541]

5/2[303]1/2[301]

1/2[550]

3/2[431]

5/2[422]

7/2[413]

9/2[4

04]

1/2[431]

3/2[422]

5/2[413]

7/2[4

04]

1/2[420]

1/2[660]

3/2[411]

3/2[651]

5/2[4

02]

5/2[642]

1/2[550]

1/2[301] 1/2[541]

3/2[541]

3/2[301]

5/2[532]

7/2[523]

9/2[514]11

/2[5

05]

1/2[411]

1/2[660]

1/2[420]

3/2[651]

3/2[411]

1/2[400]1/2[660]

1/2[411]

1/2[651]

1/2[541]

1/2[301]

3/2[532]

5/2[523]

1/2[660] 1/2[400]

1/2[651]

1/2[411]3/2[651]

3/2

[402

]

3/2

[642

]

5/2[642]5/2[402]

7/2

[633

]

1/2[530]

3/2

[521

]

3/2[761]1/2[770]

3/2[521]

1/2[761]

1/2[640]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .6

6.0

6.5

7.0

ε2

Figure 13. Nilsson diagram for protons, Z ≥ 82 (ε4 = ε22/6).

Es.

p. (

h−ω

)

82

114

2d3/2

3s1/2

1h9/2

1i13/2

2f7/2

2f5/2

3p3/2

7/2[4

04]

7/2[633]

5/2[4

02]

5/2[642]

9/2[514]

11/2

[505

]

3/2[4

02]

3/2[651]

1/2[4

00]

1/2[660]

1/2[541]

3/2[532]

5/2[523]

7/2[514]

7/2[743]

9/2[

505]

9/2[734]

1/2[660]

1/2[4

00]

1/2[651]

3/2[651]

3/2[642]

5/2[642]

5/2[4

02]

7/2[633]

7/2[404]

9/2[624]

11/2

[615

]13/2

[606

]

1/2[530]

3/2[521]

5/2[512]

5/2[752]

7/2[

503]

7/2[743]

7/2[514]

1/2[521]

1/2[770]

3/2[512]

3/2[761]

5/2[

503] 5/2[752]

5/2[512]

1/2[510]

1/2[770]

1/2[521]

3/2

[501

]

3/2

[761

]

3/2[512]

3/2[752]

1/2[651]

1/2[640]

3/2[642]

3/2[4

02]

5/2[633]

7/2[624]

1/2

[770

]

1/2[510]

1/2[761]

3/2[752]

3/2[512]7/2[743] 7/2[

503]

1/2[750]

1/2[640]

1/2[4

00]

3/2[631]

5/2[862]1/2[880]

3/2[871]

1/2[631]1/2[510]

−.3 −.2 −.1 .0 .1 .2 .3 .4 .5 .6

6.0

6.5

7.0

ε2

Figure 14. Nilsson diagram for protons, Z ≥ 82 (ε4 = − ε22/6).

Es.

p. (

h−ω

)

82

114

2d3/2

3s1/2

1h9/2

1i13/2

2f7/2

2f5/2

3p3/2

7/2[4

04]

5/2[4

02]

5/2[642]

9/2[514]

11/2

[505

]

3/2[4

02]

3/2[651]

1/2[4

00]

1/2[660]

1/2[411]

1/2[541]

3/2[532]

5/2[523]

7/2[514]

7/2[743]

9/2[

505]

1/2[660] 1/2[4

00]

1/2[651]

1/2[411]

3/2[651]

3/2[642]

5/2[642]

5/2[4

02]

7/2[633]

9/2[624]

11/2

[615

]13/2

[606

]

1/2[530]

3/2[521]

3/2[761]

5/2[512]

5/2[752]

7/2[

503]

7/2[743]

7/2[514]

1/2[521]

1/2[770]

3/2[512]

3/2[761]

3/2[521]

5/2[

503] 5/2[752]

5/2[512]

1/2[510]

1/2[770]

1/2[521]

1/2[761]

3/2

[501

]

3/2

[761

]

3/2[512]

3/2[752]

1/2[651]

1/2[400]

1/2[640]

3/2[642]

3/2[871]

5/2[633]

1/2

[770

]

1/2[761]

1/2[521]

3/2

[512

]

7/2[743]

7/2

[503

]

1/2[510]

1/2[990]

1/2[640]

1/2[880]

3/2[631]

3/2[871]

3/2[402]

5/2[862]1/2[880]

1/2[400]

3/2

[871

]

3/2[631]

1/2[631]1/2

[750]

Additional quantities of interest for describing a rotational band include the rotational frequency

(18)h− ω(J ) =4

E γ((J +2)→J ) + E γ(J →(J −2))hhhhhhhhhhhhhhhhhhhhhhhhhhh MeV ,

the kinetic moment of inertia

(19)J_ 1(J ) =E γ(J →(J −2))

2J − 1hhhhhhhhhhhhh h− 2 MeV−1,

and the dynamic moment of inertia

(20)J_ 2(J ) =E γ((J +2)→J ) − E γ(J →(J −2))

4hhhhhhhhhhhhhhhhhhhhhhhhhhh h− 2 MeV−1 .

An experimental Routhian plot is often prepared to compare rotational bands with theory. This plot is res-tricted to sequences with ∆J =2 when defining a band. The frequency parameter ω(J ) is defined as

(21)ω(J ) =Jx (J +1) − Jx (J −1)E (J +1) − E (J −1)hhhhhhhhhhhhhhhhh ,

where the x-component of angular momentum Jx is

(22)Jx (J ) = √ddddddddddd(J +1/2)2−K2.

The experimental Routhian13 is defined by

(23)E ′(J ) = 1/2 [E (J +1)+E (J −1)] − ω(J )Jx (J ).

These Routhians are normally referenced to even-even nuclei where the quantities

(24)e ′(ω) = E ′(ω)−Eg′ (ω) ,

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

(25)e ′(A ,ω) = E ′(A ,ω) + ∆ − 1/2 [Eg′ (A +1,ω) + Eg

′ (A −1,ω)],

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). Foreach configuration there is a rotational band of levels where K =Ω and the energies are given by equation4 with J taking on the values K , K +1, K +2, K +3, . . . Coriolis and centrifugal forces introduce an addi-tional term to the matrix element which involves a phase factor σ=(−1)J +K called the signature. This termalternates sign for successive values of J and implies that rotational bands with K ≠0 divide into two fami-lies distinguished from each other by the quantum number σ. Bengtsson and Frauendorf13 have redefinedthe 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

(26)J = αt mod 2 ,

determining whether the band has odd or even particle number N , where

(27)N = 2αt mod 2 .

For even-A nuclei, we obtain the 2-quasiparticle sequences:

(28)if αt = 0 (r = +1), J = 0, 2, 4, 6, . . . ,

if αt = ±1 (r = −1), J = 1, 3, 5, 7, . . . .

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh13 R. Bengtsson and S. Frauendorf, Nucl. Phys. A327, 139 (1979).

Similarly, for odd-A nuclei:

(29)if αt = +1/2 (r = −i ), J = 1/2, 5/2, 9/2, . . . ,

if αt = −1/2 (r = +i ), 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’sintrinsic 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 alignmentby 1h− produces the "unfavored" configuration. These two configurations are analogous to the signaturepartners described above.

In octupole deformed nuclei, rotational states may be characterized by eigenvalues s of the simplexoperator14 J=PR−1 where R corresponds to reflection by 180° about an axis perpendicular to the symmetryaxis and P is the parity operator. The spin J and parity p of a state in a rotational band which can bedescribed by this "reflection" symmetry are related by p =e−i πJ . Accordingly, s =±1 for even particlenumber (integral J ), and s =±i for odd particle number (half-integral J ). These reflection-asymmetricdeformations (octupole-quadrupole deformed) lead to bands of alternating parity, connected by enhancedE 1 transitions, with levels nearly degenerate in energy for the same spin but opposite parity (parity doub-lets). 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. Thisphenomenon was subsequently identified with fission isomers in the actinides, first observed by Polikanovet al.18 The associated shell gaps, calculated with an axially symmetric harmonic oscillator potential byNix19 and by Bohr and Mottelson20, are shown in Figure 17. These gaps occur with varying magicnumbers for ratios of the semi-major to semi-minor axis of 3/2 and 2 (superdeformation), and 3 (hyperde-formation). In 1986, Twin et al 21,22 reported the first evidence, for a superdeformed band in 152Dy. Sixsuperdeformed bands have been found in 152Dy and are shown in Figure 18. Tentative evidence for ahyperdeformation in 152Dy was reported by Galindo-Uribarri et al 23 who observed a δE γ=±30 keV ridgestructure in coincidence data for 152Dy. Discrete transitions, possibly from a hyperdeformed band in 153Dy,were reported by Viesti et al 24.

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh14 W. Nazarewicz, P. Olanders, I. Ragnarsson, J. Dudek, and G.A. Leander, Phys. Rev. Lett. 52, 1272 (1984).15 J. Fernandez-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. Fomi-

chev, 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. deAngelis, 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).

Figure 15. Signature splitting observed in 163Er.

7/2– 83.965

5/2[523] α=-1/2

(11/2)– 320.2

(15/2)– 640.8

(19/2–) 1034.0

(23/2–) 1482.1

(27/2–) 1960.8

(31/2–) 2452.7

(35/2–) 2969.7

(39/2–) 3535.8

(43/2–) 4165.3

(47/2–) 4863

(51/2–) 5639

(55/2–) 6478

(59/2–) 7373

8296

236*

32

1

393

44

8

479

49

2

517

56

6

630

69

8

776

83

9

895

92

3

5/2– 0

5/2[523] α=+1/2

9/2– 190.02

13/2– 467.2

(17/2–) 821.9

(21/2–) 1245.2

(25/2–) 1722.4

(29/2–) 2231.6

(33/2–) 2746.1

(37/2–) 3279

(41/2–) 3863.6

(45/2–) 4511.2

(49/2–) 5225

(53/2–) 6007

(57/2–) 6853

7757

277

35

5

423

47

7

510

51

5

533

58

4

648

71

4

782

84

6

904

5/2+ 69.219

5/2[642] α=+1/2

9/2+ 120.34613/2+ 247.3

17/2+ 465.1

(21/2+) 778.8

(25/2+) 1187.0

(29/2+) 1683.7

(33/2+) 2261.9

(37/2+) 2913.2

(41/2+) 3629.5

(45/2+) 4402

(49/2+) 5213

(53/2+) 6042

(57/2+) 6923

127

218

31

4

408

49

7

578

65

1

716

77

2

811

83

0

880

7/2+ 91.552

5/2[642] α=-1/2

11/2+ 199.3

15/2+ 412.2

(19/2+) 737.2

(23/2+) 1165.3

(27/2+) 1688.4

(31/2+) 2295.1

(35/2+) 2973.5

(39/2+) 3712

(43/2+) 4499

(47/2+) 5309

108

213

32

6*

428

52

3

607

67

8

739

78

7

810

163 68Er

Figure 16. Examples of parity doublets arising from reflection asymmetry in 221Ra and 224Ac.

0– 01– 17.2

2– 37.2

3– 78.4

4– 116.7

5– 184.2

6– 237.2

17 37

41

61

38

0+ 22.01+ 29.82+ 51.9

3+ 80.5

4+ 130.4

5+ 177.0

6+ 253.5

22

29

50

224 89Ac

(13/2+) 318.9

(15/2–) 440.4

(17/2+) 573.1

(19/2–) 688.4

(21/2+) 866.9

(23/2–) 990.4

(25/2+) 1197.4

(27/2–) 1344.4

122

133

25

4 11

5

248

179

294

124

302

207

33

1

147

354

(11/2+) 210.9

(13/2–) 341.5

(15/2+) 438.0

(17/2–) 566.0

(19/2+) 711.6

(21/2–) 849.6

(23/2+) 1025.9

(25/2–) 1180.5

(27/2+) 1375.7

131

97

128

22

5

146

274

138

284

176

31

4

155

331

195

350

221 88Ra

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

–1.0 – 0.5 0 0.5 1.0

2

8

20

40

70

112

168

2

4

10

16

28

40

60

80

110

140

182

2

6

14

26

44

68

100

140

190

1:2 2:3 1:1 3:2 2:1 3:1

EN

ER

GY

IN U

NIT

S O

F h

ω−⁄

δ = ω − ω ω−

3⊥osc

Figure 17. Single-particle level energies calculated for an axially symmetric harmonic oscillator (from reference 18).

Figure 18. Superdeformed rotational bands for 152Dy

(24+) x

SD-1 band

(26+) 602.4+x

(28+) 1249.9+x

(30+) 1942.6+x

(32+) 2680.7+x

(34+) 3464.7+x

(36+) 4294.6+x

(38+) 5171.0+x

(40+) 6094.2+x

(42+) 7064.4+x

(44+) 8081.8+x

(46+) 9146.7+x

(48+) 10259.4+x

(50+) 11419.9+x

(52+) 12628.5+x

(54+) 13885.1+x

(56+) 15189.9+x

(58+) 16542.8+x

(60+) 17944.1+x

(62+) 19393.7+x

(64+) 20891.5+x

(66+) 22437.1+x

602

648

69

3

738

78

4

830

87

6

923

97

0

1017

10

65

1113

11

61

1209

12

57

1305

13

53

1401

14

50

1498

15

46

(34) y

SD-2 band

(36) 825.9+y

(38) 1681.3+y

(40) 2576.5+y

(42) 3508.7+y

(44) 4478.6+y

(46) 5487.1+y

(48) 6536.3+y

(50) 7628.9+y

(52) 8766.5+y

(54) 9949.9+y

(56) 11180.5+y

(58) 12458.2+y

(60) 13785.6+y

(62) 15162.7+y

(64) 16586.3+y

(66) 18063.4+y

826

85

5

895

93

2

970

10

09

1049

10

93

1138

11

83

1231

12

78

1327

13

77

1424

14

77

(36) z

SD-3 band

(38) 793.0+z

(40) 1632.7+z

(42) 2523.9+z

(44) 3468.7+z

(46) 4466.9+z

(48) 5519.3+z

(50) 6624.2+z

(52) 7781.1+z

(54) 8989.0+z

(56) 10249.4+z

(58) 11563.2+z

(60) 12931.8+z

(62) 14357.6+z

(64) 15840.3+z

(66) 17384.4+z

(68) 18989.1+z

793

84

0

891

94

5

998

10

52

1105

11

57

1208

12

60

1314

13

69

1426

14

83

1544

16

05

(27–) u

SD-4 band

(29–) 669.6+u

(31–) 1390.6+u

(33–) 2163.4+u

(35–) 2988.4+u

(37–) 3865.2+u

(39–) 4794.2+u

(41–) 5772.9+u

(43–) 6802.7+u

(45–) 7883.5+u

(47–) 9014.1+u

(49–) 10194.4+u

(51–) 11422.4+u

(53–) 12703.4+u

(55–) 14031.1+u

(57–) 15407.0+u67

0

721

77

3

825

87

7

929

97

9

1030

10

81

1131

11

80

1228

12

81

1328

13

76

(26–) v

SD-5 band

(28–) 642.1+v

(30–) 1337.0+v

(32–) 2084.0+v

(34–) 2882.8+v

(36–) 3733.9+v

(38–) 4635.2+v

(40–) 5589.0+v

(42–) 6594.1+v

(44–) 7649.1+v

(46–) 8754.1+v

(48–) 9909.9+v

(50–) 11115.7+v

(52–) 12369.5+v

(54–) 13673.7+v

642

69

5

747

799

85

1

901

95

4

1005

10

55

1105

11

56

1206

12

54

1304

(32–) w

SD-6 band

(34–) 761.5+w

(36–) 1566.0+w

(38–) 2415.7+w

(40–) 3310.6+w

(42–) 4251.8+w

(44–) 5237.9+w

(46–) 6269.2+w

(48–) 7346.5+w

(50–) 8469.2+w

(52–) 9636.3+w

(54–) 10847.9+w

(56–) 12104.6+w

(58–) 13404.5+w

(60–) 14748.8+w

(62–) 16137.2+w

(64–) 17570.7+w

762

80

5

850

89

5

941

98

6

1031

10

77

1123

11

67

1212

12

57

1300

13

44

1388

14

34

152 66Dy