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Alexander Volya Florida State University
Nuclear Structure Theory I
Nuclear Properties
2
The world of nuclear physics
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180
2 8
20 28
50
82
2 8 20 28 50 82 126
Z
N
stable
1012y
106y
1y1h1s
1ns
1fs
(Number of neutron)
(Num
ber o
f pro
tons
)
3
The world of nuclear physicshttp://ie.lbl.gov/systematics/isodiscovery.pdf
4
The world of nuclear physics
Evolution of the Table of Isotopes
1 H 2 He 3 Li 4 Be
5 B 6 C
7 N 8 O
9 F 10 Ne
11 Na 12 Mg 13 Al 14 Si
15 P 16 S
17 Cl 18 Ar
19 K 20 Ca
21 Sc 22 Ti
23 V 24 Cr
25 Mn 26 Fe
27 Co 28 Ni
29 Cu 30 Zn
31 Ga 32 Ge
33 As 34 Se
35 Br 36 Kr
37 Rb 38 Sr
39 Y 40 Zr
41 Nb 42 Mo
43 Tc 44 Ru
45 Rh 46 Pd
47 Ag 48 Cd
49 In 50 Sn
51 Sb 52 Te
53 I 54 Xe
55 Cs 56 Ba
57 La 58 Ce
59 Pr 60 Nd
61 Pm 62 Sm 63 Eu
64 Gd 65 Tb 66 Dy 67 Ho
68 Er 69 Tm
70 Yb 71 Lu
72 Hf 73 Ta
74 W 75 Re
76 Os
2 4 6 8
10 12 14
16
18 20
22 24
26 28
30 32
34
36
38 40 42
44 46
48 50
52 54
56 58
60 62
64
66 68
70 72
74 76
78 80 82
84 86
88 90
92 94 96
98
100
102
104
106
108
110
112114
116
118 120
Publication Year194019441948195319581967197819952000Naturally Abundant
77 Ir 78 Pt
79 Au 80 Hg
81 Tl 82 Pb
83 Bi 84 Po
85 At 86 Rn
87 Fr 88 Ra
89 Ac 90 Th
91 Pa 92 U
93 Np 94 Pu
95 Am 96 Cm
97 Bk 98 Cf
99 Es100 Fm
101 Md102 No
103 Lr104 Rf
105 Db106 Sg
107 Bh108 Hs
109 Mt 110
111 112
90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122124
126128
130 132
134136 138
140 142144
146
148
150
152
154 156
158
160
http://ie.lbl.gov/systematics/history00.pdf
5
Exci
tatio
n en
ergy-10 MeV
-20 MeV
-30 MeV
-40 MeV
1 ê2+-8.482
13H2
1 ê2+-7.718
23He1
0+-28.296
0+ 20.20- 21.0
24He2
3 ê2--27.406
25He3
3 ê2--26.331
35Li2 0+
-29.268
2+ 1.8
26He4 1+
-31.994
3+ 2.20+ 3.62+ 4.32+ 5.41+ 5.7
36Li3
3 ê2--39.244
1 ê2- 0.5
7 ê2- 4.65 ê2- 6.75 ê2- 7.57 ê2- 9.73 ê2- 9.93 ê2- 11.2
37Li4
3 ê2--37.600
1 ê2- 0.4
7 ê2- 4.65 ê2- 6.75 ê2- 7.2
7 ê2- 9.33 ê2- 9.93 ê2- 11.0
47Be3
6
The world of nuclear physics
Experimental Chart of Nuclides 2000Number of Levels (Audi 1995)
2 2
8
8
20
20
28
28
50
50
82
82
126
Levels0-12-56-2021-100101-200>200Known NucleusStable
http://ie.lbl.gov/systematics/chart_nlev.pdf
7
• Classical and quantum mechanics. • Onset of relativistic effects. • Transition from few to many-body: mesoscopic physics. • Emergence phenomena: coexistence of order and chaos. • From applied to fundamental science. • Structure and dynamics
Diverse nuclear phenomena
• Nuclear sizes • Nuclear masses • Nuclear shapes • Nuclear rota1ons • Shape vibra1ons • Mean field and shell structure
8
Nuclear structure, general proper:es
Nuclear Sizes
Barrett and Jackson Nuclear sizes and structure
Radius R=r0 A1/3
Electron scattering data, H.D. Vries et.al Atom Data, Nucl Data Tab. 36 (1987) 495
Bethe-Weizsacker mass formula
From: Wikimedia
Volume term B/A is roughly constant: Saturation of nuclear forces Surface tension: nucleons on the surface have less “interactions” Coulomb energy:
Symmetry energy: Different Fermi energies: Different interactions
Pairing term:
Nuclear masses
See wikipedia
Describing nuclear shapes
Expand nuclear shapes
Compression
Center-of-mass translation
Quadrupole deformation
Hill-Wheeler Parameters
From Ring and Schuck, The nuclear many-body problem
13
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180
8
20 28
50
82
8 20 28 50 82 126
Z
N
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
β 2
Nuclear quadrupole deformations
Reduced transition probability
Quadrupole moment
Multipole moments
i
f
EM decay rate
See EM width calculator: http://www.volya.net/
Note that:Prolate Q>0 Oblate Q<0
warning: lab frame and body-fixed are different
Quantum Mechanics of Rotations
Three parameters
From A. Bohr and B. R. Mottelson. Nuclear structure, volume 2
Body-fixed frameLaboratory frame
Angular Momentum Jk k=x,y,z Ik k=1,2,3
Note that J2 and all Ik are scalars
Collective Rotor Hamiltonian
Shape:
J
Rotational SpectrumSpherical Trivial spectrum J(J+1) Axially symmetric rotor
Energy level diagram for 166Er. From W.D. Kulp et. al, Phys. Rev. C 73, 014308 (2006).
Properties: -Band structures E~J(J+1) -Band head J=K -K good quantum number (transitions etc)
Rotation and gamma rays
Alaga rulesObserved reduced rates and moments
Spectral relations
Triaxial rotor
Three different parameters K is mixed (diagonalize H)
Spectrum and states Mixed Transitions
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
E (
ER
/2)
γ (deg)
21
41
61
22
31
42
51
43
Models for moments of inertia
Relationship between Hrot and β γ is model-dependent.
From: J. M. Allmond,Ph.D thesis,. Georgia Institute of Technology, 2007
Evidence for nuclear superfluidity
Surface vibrations
Kinetic energy of a liquid drop
Potential energy
Surface tension
Coulomb energy
Total
Nuclear fission:
Surface vibrations
Collective Hamiltonian
Quantized Hamiltonian
spectrum Transitions
systematics
Bosonic enhancement
Note: Giant resonances
Quadrupole Vibrations in cadmium
0+ 0
2+ 658
2+ 14754+ 1542
0+ 1731
0+ 20783+ 21624+ 22202+ 2287
6+ 2479
0+ 0
2+ 617
2+ 13124+ 14150+ 1433
0+ 18713+ 20644+ 20812+ 21216+ 2168
0+ 0
2+ 558
2+ 12094+ 12830+ 1305
0+ 18593+ 18644+ 19326+ 19902+ 2048
0+ 0
2+ 513
2+ 12134+ 12190+ 1282
4+ 18693+ 19160+ 19282+ 19516+ 2026
0+ 0
2+ 487
4+ 11642+ 12690+ 1285
4+ 19296+ 19352+ 20230+ 20733+ 2091
0+ 0
2+ 505
0+ 11364+ 12032+ 1322
3+ 18982+ 19204+ 19986+ 2032
48110Cd62 48
112Cd64 48114Cd66 48
116Cd68 48118Cd70 48
120Cd72
3—w
2—w
1—w
0—w
Isotopes of Cd, vibrational states
Transition to deformation, soft mode
Exc
itatio
n en
ergy
Two-level model with 20 particles RPA, anharmonic solution, exact solution
Interaction strength
H =1
2B|↵̇|2 + 1
2C↵2 +
1
4⇤↵4
Harmonic
Soft
Deformed
Low-lying Collective modes
Rotations Vibrations Pairing
25
Shell effect and LDM
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180
8
20 28
50
82
8 20 28 50 82 126
Z
N
-4
-2
0
2
4
Nuc
lear
bin
ding
diff
eren
ce [M
eV]
Shell effects in LDM
26
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180
8
20 28
50
82
8 20 28 50 82 126
Z
N
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
β 2
Shell effects and nuclear deformations
27
Shell effect in excitation energies
0
0.5
1
1.5
2
2.5
3
3.5
4
20 40 60 80 100 120 140
8 20 28 50 82 126
E(2
+)
[M
eV]
N
Energies of 2+ states
28
Shell structure and two-neutron separation energies
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100 120 140 160
8 20 28 50 82 126
S2
n [
MeV
]
N
Sn isotopes highlighted in red
29
Shell effects in atomic physics
0
5
10
15
20
25
0 20 40 60 80 100
2 10 18 36 54 86
Ion
izat
ion
En
erg
ies
[eV
]
Atomic Number Z
He
Ne
Ar
Kr
XeRn
Mean field and one body problem
Shell gaps N=2,8,20,
Radial equation to solve
0 s1ê20 p3ê20 p1ê20 d5ê20 d3ê21 s1ê20 f7ê20 f5ê21 p3ê21 p1ê20 g9ê20 g7ê21 d5ê21 d3ê20 h11ê22 s1ê2
2
8
2028
50
82
126
184
0s
0p
0d1s
0f
1p0g
1d0h2s
1f0i2p
0j1g
2d3s
N=0 0s
N=1 0p
N=2 1s,0d
N=3 1p,0f
N=4 2s,1d,0g
N=5 2p,1f,0h
N=6 3s,2d,1g,0i
N=7 3p,2f,1h,0j
0 h9ê21 f7ê20 i13ê22 p3ê21 f5ê22 p1ê21 g9ê20 i11ê20 j15ê22 d5ê23 s1ê21 g7ê22 d3ê2
0 MeV
-10 MeV
-20 MeV
-30 MeV
-40 MeV
oscillator square well Woods-Saxon
82208Pb
2
8
20
40
70
112
168
240
2
8
2034
58
92
138
186
Evolution of single particle states
Woods-Saxon potential
Central potential
Spin-orbit potential
Coulomb potential (uniform charged sphere)
Origin of spin-orbit term is non-relativistic reduction of Dirac equation
Parameterization:
Single-particle states in potential model
-4.14 MeV
d5/2
17O example
Ground state 5/2+
Neutron separation energy 4.1 MeV
Single-particle states in potential model
-3.23 MeV
s1/2
17O example
Excited state 1/2+
Excitation energy 0.87 MeV 3.2 MeV binding
Single-particle states in potential model
0.95 MeV
d3/217O example
Unbound resonance state 3/2+
Excitation energy 5.09 MeV unbound by 0.95 MeV
Nishioka et. al. Phys. Rev. B 42, (1990) 9377 R.B. Balian, C. Block Ann. Phys. 69 (1971) 76single-particle
levels
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 10 20 30 40 50
8 34 186 612 2018 9048
ν(k
)/ν
F(k
)
k
Shell effects, chaos and periodic orbits
Density of states relative to fermi gas model
0
100
200
300
400
500
600
700
0 2 4 6 8 10 12
L2 L3L4 2π 2L2 2L3 4π
ν(x
) [a
rbit
rary
un
its]
x
Periodic orbits and classical chaos
•Motion is regular near classical periodic orbits •Stability
Evolution of shells
• Shells in deformed nuclei • Shells in weakly bound nuclei • “Melting” of shell structure • Is the mean field concept valid?
-40
-35
-30
-25
-20
-15
-10
-5
0
0 50 100 150 200 250 300
ε
A
0s1/2
1s1/2
2s1/2
3s1/2
0p3/2
0p1/2
0d5/2
0d3/2
0f7/2
0f5/2
1p3/2,1/2
0g9/2
0g7/2
1d5/2,3/2
0h9/2
0h11/2
1f7/2,5/2
0i13/2
2p3/2,1/2
0i11/2
0j15/2
1g9/2,7/2
2d5/2,3/2
0k17/21h11/2
0j13/2
2
8
20
28
50
82
126
184Overview of shell structure
Shell structure and deformation
Quantum chaos
Evolution of K=1/2 levels
21
3
Quantum chaos
Level spacing distribution
44
Literature
• P. Ring and P. Schuck, The nuclear many-body problem (Springer-Verlag, 2000). • Bohr A. and B. R. Motttelson, Nuclear Structure (World Scientific Publishing, 1998). • I. Talmi, Simple Models of Complex Nuclei: The Shell Model and Interacting Boson Model (Harwood
Academic Pub, 1993). • L. D. Landau and E. M. Lifshitz, Quantum mechanics. Non-relativistic theory. Third edition, revised and
enlarged (Pergamon Press, New York, 1981). • R. D. Lawson, Theory of the nuclear shell model (Clarendon Press, Oxford, 1980), p. 534. • G. E. Brown and A. D. Jackson, The nucleon-nucleon interaction (North-Holland Pub. Co.; distributors
for the U.S.A. and Canada, American Elsevier Pub. Co., Amsterdam; New York, 1976) • A. G. Sitenko and V. K. Tartakovskiĭ, Lectures on the theory of the nucleus (Pergamon Press, Oxford,
New York, 1975), 74, p. 304. • A. I. Baz, I. B. Zeldovich, and A. M. Perelomov, Scattering, reactions and decay in nonrelativistic
quantum mechanics. (Rasseyanie, reaktsii i raspady v nerelyativistskoi kvantovoi mekhanike) (Israel Program for Scientific Translations, Jerusalem, 1969)
• A. De Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963), p. 573. • V. Zelevinsky, Quantum physics (Wiley-VCH, Weinheim, 2011)
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