Nuclear Structure Theory I

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Nuclear Properties Alexander Volya Florida State University The world of nuclear physics 2 8 20 28 50 82 126 120

100 1fs

1ns 80 82

(Numberprotons)of 1s

Z 60 1h 50 1y

40 106y 28 1012y 20 20

8 2 0 stable 0 20 40 60 80 100 120 140 160 180 N (Number of neutron)

2 The world of nuclear physics http://ie.lbl.gov/systematics/isodiscovery.pdf

3 The world of nuclear physics

Evolution of the Table of Isotopes http://ie.lbl.gov/systematics/history00.pdf 76 Os 75 Re 118 120 74 W Publication Year 73 Ta 116 72 Hf 1940 71 Lu 114 70 Yb 112 69 Tm 1944 68 Er 110 67 Ho 1948 66 Dy 108 65 Tb 64 Gd 106 1953 63 Eu 62 Sm 104 1958 61 Pm 60 Nd 102 1967 59 Pr 58 Ce 100 57 La 1978 56 Ba 98 55 Cs 1995 54 Xe 94 96 53 I 92 52 Te 2000 51 Sb 88 90 50 Sn Naturally Abundant 49 In 48 Cd 84 86 47 Ag 46 Pd 80 82 45 Rh 78 44 Ru 43 Tc 76 42 Mo 74 112 41 Nb 111 40 Zr 72 110 39 Y 70 109 Mt 38 Sr 68 108 Hs 37 Rb 66 107 Bh 36 Kr 106 Sg 35 Br 64 105 Db 34 Se 62 104 Rf 33 As 60 103 Lr 32 Ge 58 102 No 31 Ga 56 101 Md 30 Zn 54 100 Fm 29 Cu 52 99 Es 160 28 Ni 98 Cf 27 Co 50 97 Bk 158 26 Fe 48 96 Cm 25 Mn 46 95 Am 154 156 24 Cr 44 94 Pu 23 V 42 93 Np 152 22 Ti 38 40 92 U 21 Sc 91 Pa 150 20 Ca 90 Th 19 K 36 89 Ac 148 18 Ar 88 Ra 17 Cl 87 Fr 146 16 S 34 86 Rn 144 15 P 32 85 At 140 142 14 Si 30 84 Po 136 138 13 Al 28 83 Bi 134 12 Mg 26 82 Pb 11 Na 81 Tl 130 132 10 Ne 24 80 Hg 9 F 22 79 Au 128 8 O 20 78 Pt 126 7 N 18 77 Ir 124 6 C 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 5 B 16 4 Be 12 14 3 Li 10 2 He 1 H 6 8 2 4

4 0- 21.0 + 1 2 + + 0 20.2 1 2 -7.718 -8.482 3 -10 MeV 3 ê 2He1 ê 1H2

-20 MeV Excitationenergy 1+ 5.7 - 3 2 2+ 5.4 3 2- 11.0 3 2- -26.331 2+ 1.8 + - + 2 4.3 3 2- 11.23 2 9.9 0 -27.406 + - 5 + 0 3.6 - 7 2 9.3 -28.296 ê Li 0 3 2 9.9 -30 MeV 5 3 2 3+ 2.2 - ê - ê He -29.268 7 2 9.7 5 2 7.2 4 2 3 ê - 2He2 + ê - 5 2 6.7 6 1 5 2 7.5 ê 2He4 ê - -31.994 5 2- 6.7 7 2 4.6 ê ê - 6 7 2 4.6 ê 3Li3 ê ê 1ê2- 0.4 3 2- 1ê2- 0.5 -37.600 3 2- ê -40 MeV -39.244 7 ê 4Be3 ê 7 ê 3Li4

5 The world of nuclear physics

Experimental Chart of Nuclides 2000 Number of Levels (Audi 1995) http://ie.lbl.gov/systematics/chart_nlev.pdf

126 50 Levels 0-1 2-5 6-20 82 28 21-100 101-200 20 >200 50 Known Nucleus Stable 8 20 28 2 2 8

6 Diverse nuclear phenomena

• 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

7 Nuclear structure, general properes

• Nuclear sizes • Nuclear masses • Nuclear shapes • Nuclear rotaons • Shape vibraons • Mean ﬁeld and shell structure

8 Nuclear Sizes

1/3 Radius R=r0 A

Electron scattering data, H.D. Vries et.al Atom Data, Nucl Data Tab. 36 (1987) 495

Barrett and Jackson Nuclear sizes and structure Bethe-Weizsacker mass formula

From: Wikimedia Nuclear masses

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

See wikipedia

Pairing term: 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 Nuclear quadrupole deformations 8 20 28 50 82 126 120 0.3

100 0.2

80 82 0.1 2

Z 60 0 β 50 -0.1 40

28 -0.2 20 20

8 -0.3 0 0 20 40 60 80 100 120 140 160 180 N 13 Multipole moments

Reduced transition probability

i EM decay rate f See EM width calculator: http://www.volya.net/

Quadrupole moment

Note that: Prolate Q>0 Oblate Q<0

warning: lab frame and body-fixed are different Quantum Mechanics of Rotations

Laboratory frame Body-fixed frame

Angular J k=x,y,z I k=1,2,3 Momentum k k

Shape:

2 Note that J and all Ik are scalars Collective Rotor Hamiltonian

Three parameters

From A. Bohr and B. R. Mottelson. Nuclear structure, volume 2 Rotational Spectrum

Spherical Trivial spectrum J(J+1) Axially symmetric rotor Properties: -Band structures E~J(J+1) -Band head J=K -K good quantum number (transitions etc)

Energy level diagram for 166Er. From W.D. Kulp et. al, Phys. Rev. C 73, 014308 (2006). Rotation and gamma rays

Observed reduced rates and moments Alaga rules Triaxial rotor

Mixed Transitions Spectrum and states Three different parameters K is mixed (diagonalize H) Spectral relations

70 43 60 51

50 42 61

/2) 40 R 31 30 E (E 4 20 1 22

10 21 0 0 10 20 30 40 50 60 γ (deg) 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

6+ 2479 2+ 2287 + + 4+ 2220 6 2168 + 3—w 2162 + + 3 2091 3+ 2 2121 + 6 2026 + 2073 + 0 2078 + 2081 2 2048 0+ 6 2032 4+ 6+ 1990 2+ 1951 2 2023 + 3 2064 + + + 4+ 1998 + 4+ 1932 0+ 1928 6 1935 2 1920 0 1871 3 1864 3 1916 4+ 1929 3+ 1898 + 1859 4+ 1869 0+ 1731 0 + 1542 4+ + 1433 2—w 2 1475 0+ 4 1415 + + + 0 1305 + 0 1285 + 2 1312 + 1283 0 1282 + 2 1322 4+ + 2 1269 + 2 1209 4+ 1219 + 4 1203 2 1213 4 1164 0+ 1136 Isotopes of Cd, vibrational states

+ 1—w 2 658 + 2 617 + 2 558 + + 2 513 2+ 487 2 505

0—w 0+ 0 0+ 0 0+ 0 0+ 0 0+ 0 0+ 0 110 112 114 116 118 120 48Cd62 48Cd64 48Cd66 48Cd68 48Cd70 48Cd72 Transition to deformation, soft mode 1 1 1 H = B ↵˙ 2 + C↵2 + ⇤↵4 2 | | 2 4

Two-level model with 20 particles Soft RPA, anharmonic solution, exact solution

Harmonic

Deformed Excitationenergy

Interaction strength Low-lying Collective modes

Rotations Vibrations Pairing Shell effect and LDM

8 20 28 50 82 126 120

4 100

2 80 Shell effects in LDM 82

Z 60 0 50

40 -2 28 Nuclear binding difference [MeV] difference binding Nuclear 20 20 -4 8 0 0 20 40 60 80 100 120 140 160 180 N 25 Shell effects and nuclear deformations 8 20 28 50 82 126 120 0.3

100 0.2

80 82 0.1 2

Z 60 0 β 50 -0.1 40

28 -0.2 20 20

8 -0.3 0 0 20 40 60 80 100 120 140 160 180 N 26 Shell effect in excitation energies

8 20 28 50 82 126 4

3.5

2.5 Energies of 2+ states

[MeV] 2 ) +

E(2 1.5

0.5

0 20 40 60 80 100 120 140 N 27 Shell structure and two-neutron separation energies

8 20 28 50 82 126 40

30 Sn isotopes highlighted in red

20 [MeV]

2n S 15

0 0 20 40 60 80 100 120 140 160 N

28 Shell effects in atomic physics

2 10 18 36 54 86 25 He Ne 20

Ar 15 Kr Xe Rn 10

Ionization Energies [eV] 5

0 0 20 40 60 80 100 Atomic Number Z

29 Mean field and one body problem

Shell gaps N=2,8,20,

Radial equation to solve 240 3s N=7 3p,2f,1h,0j 2d 2 d3 2 1 g7 2 186 1g 3 s 1 ê2 168 0j 2 d = 5ê2 N 6 3s,2d,1g,0i 0 j 0 MeV 184 15ê 2 0 i 138 11ê 2 2p 1 g 112 9 ê2 0i 2 p1 2 N=5 2p,1f,0h 1 f ê 1f 5 2 126 ê 2 p3 2 ê 0 i -10 MeV 70 13ê 2 92 2s 1 f7 2 N=4 2s,1d,0g Evolution of single particle states ê 0h 0 h9 ê2 2 s1 2 208 1d 82 Pb ê 0 h11 2 82 40 1 d3 2 ê ê N=3 1p,0f 58 1 d5 2 -20 MeV 0g ê 50 0 g7ê2 1p 20 0 g9ê2 1 p N=2 1s,0d 34 1ê2 0f 28 1 p 3ê2 0 f5 2 -30 MeV 20 1s 20 ê 0 f 8 0d 7 ê2 N=1 0p 1 s1ê2 0 d3ê 2 8 0p 8 0 d5ê 2 2 0 p -40 MeV N=0 0s 2 0s 2 1ê2 0 p3ê2 oscillator square well Woods-Saxon 0 s1 ê2 ê

ê Woods-Saxon potential

Central potential

Coulomb potential (uniform charged sphere)

Spin-orbit potential

Origin of spin-orbit term is non-relativistic reduction of Dirac equation

Parameterization: Single-particle states in potential model

17O example

d5/2

-4.14 MeV Ground state 5/2+ Neutron separation energy 4.1 MeV Single-particle states in potential model

17O example

s1/2

-3.23 MeV Excited state 1/2+ Excitation energy 0.87 MeV 3.2 MeV binding Single-particle states in potential model

17 O example d3/2

0.95 MeV

Unbound resonance state 3/2+ Excitation energy 5.09 MeV unbound by 0.95 MeV Shell effects, chaos and periodic orbits

Nishioka et. al. Phys. Rev. B 42, (1990) 9377 single-particle R.B. Balian, C. Block Ann. Phys. 69 (1971) 76 levels

8 34 186 612 2018 9048

1.6 Density of states relative to fermi gas model 1.4 1.2

(k) 1 F

ν 0.8 (k)/

ν 0.6 0.4 0.2 0 0 10 20 30 40 50 k L2 L3L4 2π 2L2 2L3 4π 700

600

500

400

300 [arbitrary units] 200 (x) ν 100

0 0 2 4 6 8 10 12 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? 0 184 0j13/2 Overview of shell 0k17/2 1h -5 126 11/2 structure 3s1/2 2d5/2,3/2 1g9/2,7/2 -10 82 0j15/2 0i11/2 2p3/2,1/2 -15 0i13/2 50 1f7/2,5/2 0h9/2 2s -20 1/2

ε 0h11/2 28 1d5/2,3/2 0g7/2 -25 20 0g9/2 1p3/2,1/2

0f5/2 -30 0f7/2 8 1s1/2 0d3/2 -35 0d5/2

0p1/2 2 0p3/2 -40 0s1/2 0 50 100 150 200 250 300 A Shell structure and deformation Quantum chaos

Evolution of K=1/2 levels Quantum chaos

Level spacing distribution

1 2 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)