Contents - Nuclear Physicsns.ph.liv.ac.uk/PHYS490/Phys490_latex.pdf · E.S. Paul PHYS490: Advanced Nuclear Physics Slide 3 Figure 1: Addition of two vectors A B A + B (σ A +σ B)

PHYS490 – Advanced Nuclear Physics

• Nucleon-Nucleon Force . . . . . . . . . . . . . . . . . . . . 1• Nuclear Behaviour . . . . . . . . . . . . . . . . . . . . . . 17• Forms of Mean Potential . . . . . . . . . . . . . . . . . . 30• Nuclear Deformation . . . . . . . . . . . . . . . . . . . . . 42• Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . 54• Nuclear Excitation . . . . . . . . . . . . . . . . . . . . . . 64• Rotating Systems . . . . . . . . . . . . . . . . . . . . . . . 76• Nuclei at Extremes of Spin . . . . . . . . . . . . . . . . . 87• Nuclei at Extremes of Isospin . . . . . . . . . . . . . . . 103• Mesoscopic Systems . . . . . . . . . . . . . . . . . . . . . . 130• Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . 154• Nuclear Astrophysics . . . . . . . . . . . . . . . . . . . . . 175

Contents

1.1 The Pauli (Iso)Spin Matrix . . . . . . . . . . . . . . . . . 11.2 Addition of (Iso)Spin . . . . . . . . . . . . . . . . . . . . . 31.3 General Properties of the Nucleon-Nucleon Force . . 61.4 Repulsive Core . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Exchange-Force Model . . . . . . . . . . . . . . . . . . . . . . 101.6 One Pion Exchange Potential . . . . . . . . . . . . . . . . 111.7 The Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Range of Nuclear Force . . . . . . . . . . . . . . . . . . . 162.1 Mirror Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Isospin Substates . . . . . . . . . . . . . . . . . . . . . . . 202.3 Isobaric Analogue States . . . . . . . . . . . . . . . . . . . 212.4 Independent Particle Model . . . . . . . . . . . . . . . . 232.5 Degenerate Fermi Gas Model . . . . . . . . . . . . . . . . 262.6 Some nuclear quantities . . . . . . . . . . . . . . . . . . . 293.1 Single-Particle Shell Model . . . . . . . . . . . . . . . . . 303.2 Square Well Potential . . . . . . . . . . . . . . . . . . . . 31

3.3 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 333.4 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . 353.5 Woods-Saxon + Spin-Orbit . . . . . . . . . . . . . . . . . 363.6 Residual Interaction . . . . . . . . . . . . . . . . . . . . . 373.7 Hartree Fock . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Geometric Descriptions . . . . . . . . . . . . . . . . . . . 424.2 Theoretical Nuclear Deformations . . . . . . . . . . . . . 464.3 Ground-State Deformations . . . . . . . . . . . . . . . . . 474.4 Nilsson Model . . . . . . . . . . . . . . . . . . . . . . . . . 484.5 Large Deformations . . . . . . . . . . . . . . . . . . . . . . 525.1 Deformed Liquid Drop . . . . . . . . . . . . . . . . . . . . 545.2 Shell Correction - Strutinsky Method . . . . . . . . . . 565.3 Fission Isomers . . . . . . . . . . . . . . . . . . . . . . . . . 606.1 Spherical Nuclei . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3 Rotations of a Deformed System . . . . . . . . . . . . . 696.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4.1 232Th . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.4.2 224Ra . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.4.3 15369Tm . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . 767.2 Cranking Model . . . . . . . . . . . . . . . . . . . . . . . . 807.3 Backbending . . . . . . . . . . . . . . . . . . . . . . . . . . 838.1 Generation of Angular Momentum . . . . . . . . . . . . 888.2 High Ix bands . . . . . . . . . . . . . . . . . . . . . . . . . . 898.3 High K (Iz) bands . . . . . . . . . . . . . . . . . . . . . . . 928.4 Superdeformation . . . . . . . . . . . . . . . . . . . . . . . 958.5 Shape Coexistence . . . . . . . . . . . . . . . . . . . . . . . 978.6 Extremely High Spin: Jacobi Instabilities . . . . . . . . 1009.1 Nucleon Driplines . . . . . . . . . . . . . . . . . . . . . . . 1059.2 Heavy N = Z Nuclei . . . . . . . . . . . . . . . . . . . . . . 1079.3 Proton-Rich Nuclei: Proton Radioactivity . . . . . . . 1109.4 Direct Two-Proton Decay . . . . . . . . . . . . . . . . . . 114

9.5 Neutron-Rich Nuclei: The Physics of Weak Binding . 1159.6 Nuclear Haloes . . . . . . . . . . . . . . . . . . . . . . . . . 1159.7 Changing Magic Numbers . . . . . . . . . . . . . . . . . . 1229.8 Nuclei at the Extremes of Mass and Charge: Super-

heavies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.9 Superheavies at high spin . . . . . . . . . . . . . . . . . . 12710.1 Femtostructures and Nanostructures . . . . . . . . . . . 13010.2 The Quantality Parameter . . . . . . . . . . . . . . . . . 13110.3 Atomic Clusters as a Branch of Nuclear Physics . . . 13410.4 The Spherical Droplet . . . . . . . . . . . . . . . . . . . . 13610.5 Shell Structures . . . . . . . . . . . . . . . . . . . . . . . . 13710.6 Supershell Structures . . . . . . . . . . . . . . . . . . . . . 13910.7 Mesoscopic Quantal Effects . . . . . . . . . . . . . . . . . 14110.8 Periodic Orbit Theory (POT) . . . . . . . . . . . . . . . 14110.9 Deformation: Loss of Spherical Symmetry . . . . . . . 14410.10Vibrational Modes . . . . . . . . . . . . . . . . . . . . . . . 14610.11Differences between Clusters and Nuclei . . . . . . . . 14710.12Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . 14710.13Nuclear Molecules . . . . . . . . . . . . . . . . . . . . . . . 14910.14Nuclear Sausages . . . . . . . . . . . . . . . . . . . . . . . 15210.15Binary Cluster Model . . . . . . . . . . . . . . . . . . . . 15311.1 Collision Kinematics . . . . . . . . . . . . . . . . . . . . . 15511.2 Compound Nucleus Model . . . . . . . . . . . . . . . . . 15811.3 Geometric Cross-Section . . . . . . . . . . . . . . . . . . . 16111.4 Coulomb Excitation . . . . . . . . . . . . . . . . . . . . . . 16211.5 Intermediate Energy Coulex . . . . . . . . . . . . . . . . 16311.6 Neutron Capture . . . . . . . . . . . . . . . . . . . . . . . 16411.7 Proton Capture . . . . . . . . . . . . . . . . . . . . . . . . 16811.8 Charged Particle Decay . . . . . . . . . . . . . . . . . . . 16911.9 Fusion-Evaporation Reactions . . . . . . . . . . . . . . . 17011.10Transfer Reactions . . . . . . . . . . . . . . . . . . . . . . 17412.1 Origin of the Elements . . . . . . . . . . . . . . . . . . . . 17512.2 Turning Hydrogen into Helium . . . . . . . . . . . . . . 176

12.3 The Proton-Proton (pp) Chain . . . . . . . . . . . . . . 17712.4 The CNO Cycle . . . . . . . . . . . . . . . . . . . . . . . . 17812.5 Explosive Nucleosynthesis . . . . . . . . . . . . . . . . . . 18012.6 The rp-process . . . . . . . . . . . . . . . . . . . . . . . . . 18212.7 Astrophysical sites of the rp-process . . . . . . . . . . . 18512.8 Neutron-rich nuclei . . . . . . . . . . . . . . . . . . . . . . 18613.1 Radioactive Ion-Beam Physics . . . . . . . . . . . . . . . 188

List of Figures

1 Addition of two vectors . . . . . . . . . . . . . . . . . . . . . 42 Repulsive core . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Pion exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Two-nucleon states . . . . . . . . . . . . . . . . . . . . . . . . 135 The deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Mirror nuclei: 22Ne and 22Mg . . . . . . . . . . . . . . . . . . 177 Isospin substates . . . . . . . . . . . . . . . . . . . . . . . . . 208 Isobaric analogue states . . . . . . . . . . . . . . . . . . . . . 219 Isotriplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210 Nuclear potential . . . . . . . . . . . . . . . . . . . . . . . . . 2311 Energy levels (ignoring proton Coulomb energy) . . . . . . . 2512 Mean field potential . . . . . . . . . . . . . . . . . . . . . . . 3013 Square well potential . . . . . . . . . . . . . . . . . . . . . . . 3214 Harmonic oscillator potential . . . . . . . . . . . . . . . . . . 3415 Woods-Saxon potential . . . . . . . . . . . . . . . . . . . . . . 3716 Pairing interaction . . . . . . . . . . . . . . . . . . . . . . . . 3917 Description of the nuclear shape . . . . . . . . . . . . . . . . 4318 Nuclear deformations . . . . . . . . . . . . . . . . . . . . . . . 4319 Principal axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4420 Theoretical nuclear shapes . . . . . . . . . . . . . . . . . . . . 4621 Deformation systematics . . . . . . . . . . . . . . . . . . . . . 4722 Nilsson diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 5023 Deformed shell gaps . . . . . . . . . . . . . . . . . . . . . . . 5224 Deformed–spherical energies . . . . . . . . . . . . . . . . . . . 5525 Level densities at the Fermi surface . . . . . . . . . . . . . . . 5826 Shell-correction energies . . . . . . . . . . . . . . . . . . . . . 5927 Potential energy versus deformation . . . . . . . . . . . . . . 6028 Level schemes: even-even nuclei . . . . . . . . . . . . . . . . . 6429 Level schemes: odd-A nuclei . . . . . . . . . . . . . . . . . . . 6530 Nuclear vibrations . . . . . . . . . . . . . . . . . . . . . . . . 6731 Realistic vibrational levels . . . . . . . . . . . . . . . . . . . . 68

32 Nuclear angular-momentum vectors . . . . . . . . . . . . . . . 7033 232Th level scheme . . . . . . . . . . . . . . . . . . . . . . . . 7234 224Ra level scheme . . . . . . . . . . . . . . . . . . . . . . . . 7435 153Tm level scheme . . . . . . . . . . . . . . . . . . . . . . . . 7536 Particle and hole levels . . . . . . . . . . . . . . . . . . . . . . 7737 Nuclear moments of inertia . . . . . . . . . . . . . . . . . . . 7838 Effect of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . 7939 Projections of the nuclear spin . . . . . . . . . . . . . . . . . 8140 Behaviour of moments of inertia . . . . . . . . . . . . . . . . 8341 Crossing bands . . . . . . . . . . . . . . . . . . . . . . . . . . 8442 Pair breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 8543 Rotational alignment . . . . . . . . . . . . . . . . . . . . . . . 8644 Collective and noncollective spin . . . . . . . . . . . . . . . . 8845 Aligned particles: band termination . . . . . . . . . . . . . . 9046 Band termination at 46+ in 158Er . . . . . . . . . . . . . . . . 9147 The K quantum number . . . . . . . . . . . . . . . . . . . . . 9348 High-K bands in 172Hf . . . . . . . . . . . . . . . . . . . . . . 9449 Superdeformed second minimum . . . . . . . . . . . . . . . . 9650 Superdeformed band in 152Dy . . . . . . . . . . . . . . . . . . 9751 Shape coexistence in 152Dy . . . . . . . . . . . . . . . . . . . 9952 Critical angular momenta . . . . . . . . . . . . . . . . . . . . 10153 Calculated signal of the Jacobi transition . . . . . . . . . . . 10254 The chart of the nuclides . . . . . . . . . . . . . . . . . . . . 10355 Chart of the nuclides . . . . . . . . . . . . . . . . . . . . . . . 10456 Gamma-ray spectrum of 80Zr . . . . . . . . . . . . . . . . . . 10857 Energies of 2+ states for Z = N nuclei . . . . . . . . . . . . . 10958 Ground-state proton emitters . . . . . . . . . . . . . . . . . . 11159 Rotating (deformed) proton emitter: 141Ho . . . . . . . . . . 11260 Fine structure in proton decay: 131Eu . . . . . . . . . . . . . 11361 Two-proton decay: 18Ne . . . . . . . . . . . . . . . . . . . . . 11462 6He configurations . . . . . . . . . . . . . . . . . . . . . . . . 11763 Borromean systems . . . . . . . . . . . . . . . . . . . . . . . . 11864 The size of 11Li . . . . . . . . . . . . . . . . . . . . . . . . . . 119

65 The halo nucleus 11Li . . . . . . . . . . . . . . . . . . . . . . 11966 Halo nuclei systematics . . . . . . . . . . . . . . . . . . . . . 12067 1g7/2 and 2d5/2 relative energies in Sb . . . . . . . . . . . . . 12368 Calculated fission-barrier heights . . . . . . . . . . . . . . . . 12569 α decay chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 12770 The superheavy elements . . . . . . . . . . . . . . . . . . . . 12871 Ground-state rotational band in 254No . . . . . . . . . . . . . 12972 Shell structures in nuclei and metal clusters . . . . . . . . . . 13173 Three types of Fermi liquid droplets . . . . . . . . . . . . . . 13374 Shell structure of Na clusters . . . . . . . . . . . . . . . . . . 13775 Predicted level sequences . . . . . . . . . . . . . . . . . . . . 13876 Predicted cluster supershell structure . . . . . . . . . . . . . . 13977 Experimental supershell structure of Na clusters . . . . . . . 14078 Supershell structure from interfering periodic orbits . . . . . 14379 Predicted metal-cluster and nuclear shapes . . . . . . . . . . 14580 Giant dipole resonances in clusters and nuclei . . . . . . . . . 14681 Phase diagram of Kr nucleus and atomic fluid . . . . . . . . . 14882 The “Ikeda Diagram” . . . . . . . . . . . . . . . . . . . . . . 14983 Bloch-Brink cluster model . . . . . . . . . . . . . . . . . . . . 15084 12C as a cluster of three α particles . . . . . . . . . . . . . . . 15185 Cluster configurations in 24Mg . . . . . . . . . . . . . . . . . 15286 Various types of reaction . . . . . . . . . . . . . . . . . . . . . 15487 Geometric cross-section . . . . . . . . . . . . . . . . . . . . . 16188 Neutron capture resonance . . . . . . . . . . . . . . . . . . . 16489 Neutron capture and decay . . . . . . . . . . . . . . . . . . . 16590 Neutron capture, proton decay . . . . . . . . . . . . . . . . . 16991 Fusion-evaporation reaction . . . . . . . . . . . . . . . . . . . 17092 Induced angular momentum . . . . . . . . . . . . . . . . . . . 17193 Fusion cross-section . . . . . . . . . . . . . . . . . . . . . . . 17194 Energy-spin plane . . . . . . . . . . . . . . . . . . . . . . . . 17395 CNO and Hot CNO cycles . . . . . . . . . . . . . . . . . . . . 17996 Breakout of Hot CNO cycle . . . . . . . . . . . . . . . . . . . 17997 The rp- and r-processes . . . . . . . . . . . . . . . . . . . . . 181

98 The creation of the heavier elements . . . . . . . . . . . . . . 18299 Creation of proton-rich nuclei by the rp-process . . . . . . . . 183100 21Na beam and 22Mg recoils . . . . . . . . . . . . . . . . . . . 184101 Binary system accretion disk . . . . . . . . . . . . . . . . . . 185102 Abundances of r-process elements . . . . . . . . . . . . . . . . 186103 Quenching of shell structure in neutron-rich nuclei . . . . . . 187104 Physics with radioactive beams . . . . . . . . . . . . . . . . . 188

Generated: February 5, 2009

1: Nucleon-Nucleon Force

1.1 The Pauli (Iso)Spin Matrix

Matrix mechanics was formulated by Born,Heisenberg, and Jordan (1925) — introduction ofcommutation relations:

[Mx,My] = −ih̄Mz, cyclic, (1)for components of angular momentum M . Forintrinsic spin (takes only two values: up or down):

s =12h̄σ, (2)

where the components of σ are the Pauli spinmatrices:

σ1 =

⎛⎝ 0 1

1 0

⎞⎠ ; σ2 =

⎛⎝ 0 −i

i 0

⎞⎠ ; σ3 =

⎛⎝ 1 0

0 −1

⎞⎠

These have some interesting relations:

σ1σ2 = iσ3 ; σ2σ1 = −iσ3 ; σ2σ3 = iσ1,and σ21 = σ

22 = σ

23 = 1.

E.S. Paul PHYS490: Advanced Nuclear Physics Slide 1

In the fixed (quantised) 3-direction:

s3

⎛⎝ ψ1ψ2

⎞⎠ = h̄

2

⎛⎝ ψ1

−ψ2

⎞⎠ . (3)

The two-component wavefunction satisfies aSchrödinger equation of the form:

Hψ = ih̄∂ψ

∂t, (4)

where H is a 2 × 2 matrix and ψ is a 1 × 2 matrix(column vector).

In analogy, the Pauli Isospin Matrix τ = 2t isdefined in an analogous manner, i.e.

τ1 =

⎛⎝ 0 1

1 0

⎞⎠ ; τ2 =

⎛⎝ 0 −i

i 0

⎞⎠ ; τ3 =

⎛⎝ 1 0

0 −1

⎞⎠

with τ21 = τ22 = τ

23 = 1.

Also:

τ± =12

(τ1 ± iτ2) , (5)with

τ+ =

⎛⎝ 0 1

0 0

⎞⎠ ; τ− =

⎛⎝ 0 0

1 0

⎞⎠


τ+ turns a neutron into a proton, andτ− turns a proton into a neutron, i.e.

τ+|n〉 = |p〉 ; τ−|p〉 = |n〉; (6)τ+|p〉 = τ−|n〉 = 0.

1.2 Addition of (Iso)Spin

Eigenvalue equation with s = 12 (i.e. s =12):

s2ψ = s(s+ 1)h̄2ψ (7)

=12

(12

+ 1)h̄2ψ =

34h̄2ψ

Therefore:

s2 =34h̄2

sz = ±12 h̄ proton or neutron— projection of intrinsic spin of nucleon on z-axis.Define σh̄ = 2s where σ is the Pauli Spin Matrix.Then:

σ2h̄2 = 4s2 = 3h̄2 (8)

and therefore:

σ2 = 3 (9)


Figure 1: Addition of two vectors

�A

�B� A

+ � B

(σA + σB)2 = σ2A + σ

2B + 2σA.σB (10)

4S2/h̄2 = 3 + 3 + 2σA.σB (11)

Here S =∑si

Thus if S2 = 0 (i.e. S = 0 singlet state) then:

σA.σB = −3 (12)

and if S2/h̄2 = 1 = S(S + 1)/h̄2 = 2 (i.e. S = 1triplet state) then:

σA.σB = 1 (13)


In analogy to spin (up and down in real space) wecan apply the same formalism to describe thenucleon (up and down in an abstract isospin space).

tz = −12 proton down (14)

tz = +12

neutron up (15)

t is the vector in isospin space.

For the nucleus the total isospin is T =∑ti and

Tz = 12(N − Z)In analogy with spin:

t2 =34

(16)

and τ = 2t where τ is the Pauli Isospin Matrix.

For two nucleons A and B if T 2 = 0 (i.e. T = 0singlet state) then:

τA.τB = −3 (17)and if T 2 = T (T + 1) = 2 (i.e. T = 1 triplet state)then:

τA.τB = 1 (18)


In 1932 the neutron was discovered. In the sameyear Heisenberg introduced a proton-neutron (p-n)potential for particles A and B of the form:

VAB = PABF (rAB) ; rAB = |xA − xB| , (19)

where:

PAB = τ(A)+ τ

(B)− + τ

(A)− τ

(B)+ . (20)

Heisenberg showed that the nucleus as a p-n systemsystem is amenable to a non-relativistic quantummechanical treatment.

1.3 General Properties of the

Nucleon-Nucleon Force

To lowest order, the interaction between twonucleons consists of an attractive central potentialV (r). However the following points must be noted:


• Force is spin dependent:Evidence comes from, for example, the scattering oflow energy neutrons from ortho (I = 1 i.e. spins ofprotons parallel) and para H2 (I = 0 i.e. spins ofprotons antiparallel). The scattering cross-sectionfrom ortho-hydrogen, σortho is 30 times σpara. Alsothe S = 0 singlet state of the deuteron is unbound— The interaction must depend on the spins σAand σB of the nucleons.

Symmetry considerations restrict the possible formof the potential. Angular momentum is apseudovector that does not invert under parityreversal (r → −r). Under time reversal (t→ −t),all motions are reversed — terms such as σA andσB would violate time-reversal symmetry and areforbidden. Terms such as σ2A, σ

2B, or σA.σB are

allowed. The last term (σA.σB) is the simplestinvolving both nucleon spins.

• Force is charge symmetric:The cross section for neutron-neutron scattering,σnn, is almost the same as that for proton-protonscattering, σpp. To compare these the contributionfrom the Coulomb force has to be removed from σpp.


Note that both p-p and n-n scattering only involveπ0 exchange (see below).

• Force is nearly charge independent:The cross-section for neutron-proton scattering,σnp, is similar to σnn and σpp. The difference arisesfrom the additional exchange of the π± whichcontributes to this interaction (Fig. 3). The π±

meson has a different mass to the π0.

• The nucleon-nucleon interaction has anon-central term:

A tensor potential mixes states of different �(deuteron ground-state). Only terms that relate rto σ can contribute, such as σ.r or σ ∧ r, i.e.(σA.r)(σB .r) or (σA ∧ r).(σB ∧ r). From vectoridentities, the second form can be written in termsof the first form plus a term σA.σB. Since this hasalready been useed for the spin dependence, we useVT (r)SAB where:

SAB = 3(σA.r)(σB.r)

r2− σA.σB . (21)

This averages to zero over all angles.


• The nucleon-nucleon interaction may also dependon the relative velocity or momentum of the nucleons:

Forces dependent on velocity or momentum(vectors) cannot be represented by a scalarpotential. The simplest form that does not violateparity or time-reversal invariance is:

V (r)(r ∧ p).S, (22)where S = sA + sB . The relative angularmomentum of the nucleons is � = r ∧ p and hencethe potential becomes:

Vso(r)�.S, (23)

in analogy to the spin-orbit potential in atomicphysics. Evidence — scattered nucleons can havetheir spins polarised in certain directions.

1.4 Repulsive Core

The repulsive core (Pauli Exclusion Principle) leadsto a constant average separation between twonucleons, and the nuclear volume is proportional tothe number of nucleons. It then follows that theradius is proportional to A1/3.


Figure 2: Repulsive core

Radius of nucleon: ∼ 1fm; Radius of hard core:∼ 0.2fm; Nucleon mean free path: ∼ 7fm!Volume of hard cores is only 2% of nuclear volume.

1.5 Exchange-Force Model

Evidence — (1) saturation of nuclear forces:approximately constant nuclear density and bindingenergy per nucleon as we go to heavier nuclei. Anucleon atracts only a small number of nearneighbours, but repels at small distances to keepthose neighbours from getting too close. This isanalogous to a diatomic molecule where theelectrons are shared or exchanged between the twoatoms which achieve an equilibrium separation.


(2) Strong backward peaks in n-p scattering can beexplained if the neutron and proton exchange places(π± exchange). The lightest of the mesons — theπ-meson or pion is responsible for the major portionof the longer range (1.0 − 1.5 fm) part of thenucleon-nucleon potential.Masses — π±: 139.6 MeV/c2, π0: 135.0 MeV/c2.At shorter ranges (0.5 − 1.0 fm) two-pion exchangeis probably responsible for the nuclear binding. Atmuch shorter ranges (0.25fm) the exchange ofω-mesons (mass: 783 MeV/c2) may contribute tothe repulsive core; ρ-mesons (mass: 769 MeV/c2)may provide the spin-orbit part of the interaction.

1.6 One Pion Exchange Potential

Figure 3: Pion exchange

p

n

p

n�

0�

±

np

pn


The origin of the nuclear force arises at thefundamental level from the exchange of gluonsbetween the constituent quarks of the nucleons, butat low energies (i.e. < 1 GeV/ nucleon), i.e.interaction distance > 1 fm the interaction can beregarded approximately as being mediated by theexchange of π mesons.

At large distances the form of the potential isconstructed as arising from the exchange of one πmeson (hence OPEP, where pion = π meson) so asto reproduce the known spin and charge propertiesof the force.

The form of this potential is

VOPEP = (24)

g2s

(13σA.σB + SAB

[13

+1µr

+1

(µr)2

])τA.τB

µ2e−µr

r

where µ = mπc/h̄, and

SAB = 3(σA.r)(σB .r)/r2 − σA.σB (25)


1.7 The Deuteron

The deuteron consists of a bound proton-neutronsystem. Its ground state is the only state which isbound; the first excited state is unbound (Fig. 4).

Figure 4: Two-nucleon states

0 0+ 1 1 2.32n

MeV I��Tz T0+ 0 1 0 0+ -1 12

2 He0 1+ 0 02

1H

deuteron

The ground state has total angular momentum andparity of Iπ = 1+.

Since

I = L+ S (26)

and S = 1 (i.e. the spins of the proton and neutronare in the same direction) for the ground state(S = 0 for the unbound excited state) then L = 0 or2.

The deuteron is not a spherical nucleus.


In the standard proton-neutron picture of thissimplest nucleus, its shape is largely determined bypion exchange which leads to strong noncentraltensor interactions. Its shape has been measured athigh momentum transfers (q2) corresponding todistances of the order of the proton radius.

Figure 5: The deuteron

Evidence for L admixture comes from the measuredelectric quadrupole moment of Q0 = 2.82 mb (fromthe hyperfine spectrum of deuterium). If the totalwavefunction is written as

ψ = αψ(3S1) + βψ(3D1) (27)

with α2 + β2 = 1.


In this expression the notation is 2S+1LI , i.e. thesame as that used in atomic spectroscopy(S → L = 0, P → L = 1, D → L = 2).The value of β can be obtained from the measuredelectromagnetic moments, and is about 0.02–0.08.Since it is non-zero then the potential does notcommute with L, i.e. [V,L] �= 0, otherwise theground state would have a definite L value (e.g. 0).

This provides evidence that the n-p potentialdepends not only on the separation r but on theorientation of the intrinsic spins to r, and accountsfor the SAB = 3(σA.r)(σB .r)/r2 − σA.σB termabove. This term SAB is called the tensor term.

The other members of the two-nucleon system arealso shown in Fig. 4. The lowest energy levels of thedi-neutron (2n) and 22He and the first excited levelof the deuteron (21H) are all unbound and have Tzvalues of 1, –1, and 0: they belong to the T = 1triplet. The ground state has T = 0 and is bound.

If the nuclear force depended on −(τA.τB) alone, asspeculated by Heisenberg in analogy to the H+2molecule, then the order of the T = 0 and T = 1levels in the deuteron would have to be reversed.


1.8 Range of Nuclear Force

The range of an interaction is related to the mass ofthe exchanged particle. The Heisenberg uncertaintyprinciple gives

∆E∆t ≈ h̄ (28)

A particle can only create another particle of massm for a time t ≈ h̄/mc2, during which interval thecreated particle can travel at most a distance ct.Taking ct as an estimate of the range of theinteraction, R, gives:

R ≈ h̄/mc (29)

This predicts that the force between two neutronsarising from the exchange of a π meson, which has amass of ∼ 140 MeV/c2, has a range R ≈ 1.4 fm, ingood agreement with experiment. Analysis of highenergy nucleon-nucleon scattering data reveals thepresence of a repulsive core of ∼ 0.5 fm. In terms ofmeson exchange theory this can arise from theexchange of a heavier vector mesons such as theρ-meson and the ω-meson having Iπ = 1−.


2: Nuclear Behaviour

2.1 Mirror Nuclei

We have already seen that the force between twonucleons has the property of charge symmetry andcharge independence. Fig. 6 shows the energy levelsin the nuclei 2210Ne,

2211Na and

2212Mg. The two nuclei

2210Ne and

2212Mg are examples of mirror nuclei, i.e.

the number of protons in one is equal to the numberof neutrons in the other.

Figure 6: Mirror nuclei: 22Ne and 22MgMeV3.357

1.275

0

4+

2+

0+

4.0713.9443.7073.5193.0602.9692.5722.2121.9841.9521.937

1.528

0.8910.6570.583

0

I� MeV I�4+1+6+3-

2+3+2-1-3+2+1+

5+

4+0+1+

3+

3.308

1.246

0 0+

2+

(4)+I�MeV

2210Ne

2211Na

2212Mg


The similarity in the level schemes of mirror nucleireflects the equality of the neutron-neutron andproton-proton force when the nucleons are in thesame space-spin state, i.e. charge symmetry. Thesimilarity between the levels in 2210Ne and

2212Mg and

the levels at 0.657, 1.952 and 4.071 MeV in 2211Na,which has a different number of neutron-protonpairs, implies charge independence.

This is an example of the similarity of thebehaviour of the nucleons in the nucleus to that ofthe bare nucleon-nucleon force.

The above observation can be expressed in terms ofisospin dependence of the nuclear force: the energyonly depends on the total isospin T and not on itsthird component Tz where:

Tz =∑

all nucleons

tz (30)

where the summation is over all nucleons. Thus thecomparable states in the three nuclei have Tz = +1,0, –1, respectively, and associated with T = 1. Theground state (and other states) in 2211Na has T = 0with Tz = 0.


This is actually a simplification because we haveignored the fact that the ground state masses ofthese nuclei are different. This arises from:

• the neutron-proton mass differenceThis quantity depends on Tz of the nucleus.

• the Coulomb energy differenceThe Coulomb energy EC depends on theaverage value of

∑p

e2

4π�0rij(31)

In isospin formalism,

EC =∑

all nucleons

(12− tzi

)(12− tzj

)e2

4π�0rij

=∑(1

4− (tzi + tzj )

2+ tzitzj

)e2

4π�0rij(32)

Since∑

(tzi + tzj) ∼ Tz and∑tzitzj ∼ (Tz)2 then

the energy difference arising from these two effectsis given by:

E = a+ bTz + c(Tz)2 (33)


2.2 Isospin Substates

Again by analogy with spin, an isospin T state has(2T + 1) substates. For example, T = 2 correspondsto a vector of length

√6 with components

−2 ≤ Tz ≤ +2. The substates correspond to statesin different nuclei, as shown in Fig. 7, for theA = 12 isobars:

Figure 7: Isospin substates

-2

-1

1

2

0

Tz

O

Be

N

C

B

124

12

12

12

12

5

6

7

8

T=2 substates


2.3 Isobaric Analogue States

Consider an odd-A nucleus made up of Z protonsand N neutrons. A mirror nucleus is made byinterchanging the numbers of protons and neutrons.At low mass, we can start with an N = Z nucleus(A, N , Z) and add either one proton (A+ 1, N ,Z + 1) or one neutron (A+ 1, N + 1, Z). Thespectrum of proton states in the first nucleus shouldbe equal to the spectrum of neutron states in thesecond, reflecting the charge independence of thenuclear force.

Figure 8: Isobaric analogue states

Ne2110Tz =+1/2

0

0.35

1.75

2.792.87

3/2

5/2

7/2

1/27/2

+

+

+

++

Na2111 10Tz =-1/2

0

0.34

1.71

2.432.81

Isodoublets

11


In the example of the mirror nuclei 2110Ne11 and2111Na10, we have set the ground states to be equalby taking into account the neutron-proton massdifference (1.293 MeV ). The levels for this T = 1/2system form isodoublets with Tz = ±1/2. The onlydifferences in the excitation energies should arisefrom the Coulomb energy (extra proton in 21Na).

Next consider the A = 18 isobars. For T = 0 thereis only an isosinglet state, but for T = 1 we canhave three substates: Tz = −1, 0, +1 – isotriplets.At even higher excitation energy, T = 2 states arepossible with five substates – we have to include187N11 and

1811Na7.

Figure 9: Isotriplets

0

1.98

3.55

0

2

4

1.05

3.06

4.65 4

2

0 0

1.89

3.38 4

2

0

0 1

O18

Isotriplets

F18

Ne18

+

+

+

++

+

+ +

+

+

Tz=+1 Tz=-1T

=1

T=0

8 10 10 8

9 9

Tz =0

Isosinglet


2.4 Independent Particle Model

In principle, if the form of the nucleon-nucleon forceis known for bare nucleons, then the energy of thenucleon moving inside a nucleus can be calculated.This is a very difficult problem to solve as thenucleon interacts simultaneously with all the othernucleons.

Figure 10: Nuclear potential

0

50

100

-50

-100

(MeV)EA

Ekin A

Etot A

Epot A

d (fm)2 4


A schematic description of the nucleon energy as afunction of nucleon separation is given in theFig. 10.

The short range interaction between nucleonsmeans that in a nucleus each nucleon moves in anaverage potential. The average separation of thenucleons, ∼ 2.4 fm, is larger than the range of thenuclear force (∼ 1.4 fm).This ensures that the average potential, whichresults from the sum of all the nucleon-nucleoninteractions, is hardly affected by the individualbehaviour of the bulk of the nucleons. Nucleonscannot easily change their state unless these statesare close to the Fermi surface (see below); thisinhibition arises because of the Pauli exclusionprinciple.

The simplest model assumes that there is only anaverage potential governing the behaviour of thenucleons. The crudest approximation is to assumethat each nucleon (labelled by the suffix i) sees thepotential energy Ui(ri) arising from pair forcesgiven by (Fig. 11):


Figure 11: Energy levels (ignoring proton Coulombenergy)

VO

r

R

Ui = O

Ui (r)

S = -EF

Fermi Level

protons neutrons

Ui(ri) = −V0 for r < R, the nuclear radiusand Ui(ri) = 0 for r > R.Hence the Hamiltonian is

H =∑

i

Ti +12

∑i

Ui(ri) =∑

i

Ti − 12∑

i

V0 (34)

where the factor 12 arises to avoid counting thepotential energies of pairs of nucleons twice.

The energy, E = 〈H〉, of the nucleus is given by:

E =∑

i

〈Ti〉 − 12∑

i

〈V0〉 =∑

i

〈Ti〉 − 12∑

i

V0 (35)


2.5 Degenerate Fermi Gas Model

A simple model in which nucleons are placed in avolume V = 4πR3/3 and the interactions betweenthem are ignored. The first nucleon will occupy thelowest energy state of the nuclear “box” and furthernucleons must occupy higher energy states due tothe Pauli exclusion principle.

In the limit of large volume, the number of statesavailable for nucleon (fermion) occupation withmomentum between p and p+ dp is:

dN(p) =2V 4πp2dp

(2πh̄)3=V p2dp

π2h̄3(36)

where the factor 2 allows the two possible spinorientations of the nucleon. The total number ofstates available in the volume V with momentumbelow pF is:

N =∫dN =

V

π2h̄3

∫ pK0

p2dp =V p3F3π2h̄3

(37)

pF (= h̄kF ) is the Fermi momentum of the systemand corresponds to the energy of the last nucleon.


A Fermi sea is formed. The sea is filled up to theenergy corresponding to the Fermi momentum

EF =p2F2m

=h̄2k2F2m

; (38)

above this energy it is empty (see Fig. 11). Theground state represents a degenerate Fermi gas.

The energy of the nucleus is given by Equation 35.In this model,

∑〈Ti〉 = 35TFA where TF is thekinetic energy at the Fermi level. The potentialenergies are the same for each nucleon (−V0) so that∑V0 = V0A. The energy of the nucleus is then:

E =35TFA− 12V0A (39)

and the binding energy per nucleon, B, is given by:

B = −E/A = −35TF +

12V0 (40)

The nuclear separation energy, S, is the differencebetween the energy of a nucleon outside the nucleus(0 in this model) and the energy of the Fermi level(TF − V0), so that:

S = −TF + V0 (41)= AB(A,Z) − (A− 1)B(A− 1, Z)


Now the nuclear force has the property of saturationso that B(A,Z) is independent of A. This is aconsequence of the Pauli exclusion principle, thespin and isospin dependence of the nuclear forceand (less important) the repulsive core.

Thus

S = AB − (A− 1)B = B = −TF + V0 (42)

so that

TF =54V0 (43)

and therefore

S = B = −15TF (44)

This is incorrect since for nuclei S > 0 ! We have toinclude a momentum dependence of the potentialwell which partially accounts for the non-centralbehaviour of the nuclear force. This implies that anucleon has an effective mass (m∗) so that m∗ > mnwhen moving in a nucleus.


2.6 Some nuclear quantities

Number density (A/V ) (measured):

ρ(0) ≈ 0.17 fm−3≈ 1.5 × 1018 kg/m3 (45)

For comparison, the the density of lead is11.3 × 10−3 kg/m3 and the Crab pulsar density is∼ 1011 − 1013 kg/m3 — a neutron star can bethought of as macroscopic nuclear matter.

Fermi Momentum:

pFh̄

= kF ≈ 1.4 fm−1 (46)

Fermi Energy:

EF ≈ 37 MeV (47)

Kinetic Energy of a nucleon in the nucleus:

35EF ≈ 25 MeV (48)

This corresponds to a velocity (v/c)2 ≈ 0.1.


3: Forms of Mean Potential

3.1 Single-Particle Shell Model

Assumption: Ignore the detailed interactionsbetween nucleons — each particle moves in a stateindependent of the other particles. The mean fieldforce is the average smoothed-out interaction withall the other particles. Each nucleon then onlyexperiences a central force.

Figure 12: Mean field potential


We have established that the nucleons moves in amean potential. There are essentially twoapproaches to the determination of the potential:one in which an empirical form of potential isassumed (e.g. square well, harmonic oscillator,Woods-Saxon) and one in which the mean field isgenerated self-consistently from the nucleon-nucleoninteraction.

3.2 Square Well Potential

This is the simplest form of potential (see Fig. 13)in which

VSW(r) = −U0 for r ≤ RVSW(r) = ∞ for r > RSince we have a spherically symmetric potential(this is the same approach for the more complexforms of potential) we separate the angulardependence of the wave function from the radialpart (remember the nucleons move in threedimensions):


Figure 13: Square well potential

VSW

(r)

0

UO

R

r

ψ(r, θ, φ) = Rn�(r)Y�m(θ, φ) (49)

The radial equation is:

d2R(r)dr2

+2r

dR(r)r

+[2mh̄2

(E + U0) − �(�+ 1)r2

]R(r) = 0

(50)

The solutions are Bessel functions which satisfy theboundary condition Rn�(R) = 0 and the energiesare given by:

En� =

(h̄2

2mR2ξ2n�

)− U0 (51)


The quantum numbers are n, � and m wherem = −�, −�+ 1, ..., 0, ..., �− 1, �, i.e. 2(2�+ 1)degeneracy (2 spin orientations).

Table 1: Square well shell closures

n(�) � 2(2�+ 1) total ξn�

1s 0 2 2 3.14

1p 1 6 8 4.49

1d 2 10 18 5.76

2s 0 2 20 6.28

1f 3 14 34 6.99

Note that for � = 0:

En� =n2h2

8mR2− U0 (52)

3.3 Harmonic Oscillator

Fig. 14 shows the form of the potential in this case.The advantage of the harmonic oscillator,VHO(r) = −U0 + 12mω2r2, is that it is easy tohandle analytically.


Figure 14: Harmonic oscillator potential

VHO (r)

0

r

UO

The radial equation is:

d2R(r)dr2

+2r

dR(r)r

+

[2mh̄2

(E + U0 − mω

2

2r2

)

−�(�+ 1)r2

]R(r) = 0 (53)

The solutions are Laguerre polynomials, withenergy levels:

E = h̄ω(N +

32

)− U0 (54)

For each oscillator quantum number N there arequantum numbers n� which satisfy:

2(n− 1) + � = N, N ≥ 0, n ≥ 1, 0 ≤ � ≤ N (55)


Table 2: Harmonic oscillator shell closures

N n � n(�) 2(2�+ 1) total

0 1 0 1s 2 2

1 1 1 1p 6 8

2 2,1 0,2 2s,1d 2+10 20

3 2,1 1,3 2p,1f 6+14 40

4 3,2,1 0,2,4 3s,2d,1g 2+10+18 70

5 3,2,1 1,3,5 3p,2f,1h 6+14+22 112

Note the parity for each oscillator shell is(−1)N = (−1)�.The harmonic oscillator correctly predicts shellclosures for 42He2,

168O8, and

4020Ca20 but does not

reproduce higher ones.

3.4 Spin-Orbit Coupling

In order to account for the correct nucleon numbersat which the higher shell closures occur, a spin-orbitterm is introduced to the nuclear potential.


In the case of the modified harmonic oscillator:

V ′HO(r) = −U0 +12mω2r2 − 2

h̄2α�.s (56)

Since

�.s =h̄2

2

[j(j + 1) − �(�+ 1) − 3

4

](57)

then the energy is modified by an additional term:

−α� if j = �+ 12

(58)

+α(�+ 1) if j = �− 12

(59)

3.5 Woods-Saxon + Spin-Orbit

Usually finite potential forms are used so thatV (r) → 0 if r � R. The Woods-Saxon potential isconsidered to be the most realistic, see Fig. 15.

It has the following radial dependence:

VWS(r) =U0t

+U�sr20

1r

d

dr

(1t

)�.s (60)

where t = 1 + exp [(r −R0)/a] and R0 = r0A 13


Figure 15: Woods-Saxon potential

V(r)

0

VC(r) =

protonsV

WS(r) + V

C(r)

neutronsVWS(r)

Ze²4��or

3.6 Residual Interaction

The residual interaction v between nucleons is thedifference between the actual two-nucleon potentialVα experienced by a nucleon in a state α and theaverage potential. The matrix elements of v,〈α|v|β〉, are only appreciable near the Fermi surface.The interaction v is a two-body operator because itchanges the state of two nucleons.


It could be treated in a number of ways

• from the free two-nucleon potential.DIFFICULT!

• treated as free parameters to be deduced fromexperimental data on energy levels and angularmomenta of many nuclei.

• parameterised using physical intuition. Theinteraction depends on the radial separation(ri − rj) which can be expanded in a multipoleexpansion:

(ri − rj) =∑

v�(rirj)∑m

Y�m(θ, φ)Y ∗�m(θ, φ)

(61)

For example, if it is assumed that the interactiontakes place near the Fermi surface, i.e. near r = Rthen v�(rirj) → v�(R).In common use is the quadrupole + pairinginteraction. Here it is assumed that v�(rirj) = χr�ir

�j

so that:

(ri − rj) = χ∑�m

M�m(i)M∗�m(j) (62)


M�m(i) is the multipole operator defined by:

M�m(i) = r�iY�m(θiφi) (63)

The quadrupole-quadrupole term (i.e. � = 2) is themost important correction to a spherical field, andis relatively long-range.

The pairing interaction is the important short-rangecomponent, see Fig. 16

Figure 16: Pairing interaction

s2

�s + �ι�= 0

s1

ι 2

ι 1

It leads to greater binding between the nucleons iftheir angular momenta are coupled to zero since inthis state the nucleons have the maximum spacialoverlap (note they that cannot have the samemagnetic substate because of the Pauli principle).


This is achieved by using the approximation that:

〈j2JM |v|j2J ′M ′〉 = δJJ ′δMM ′δJ0 (64)

for nucleons in the same subshell (i.e. same j) i.e.the interaction only occurs in the J = 0 state.

3.7 Hartree Fock

The philosophy here is that the nuclear potential isself consistent. That is, we calculate the nucleondistribution (i.e. the nuclear density) from the netpotential, and then evaluate the net potential fromthe nucleon-nucleon interaction. Then the potentialis self-consistent if the one with which we end upwith is the same as the one we start with.

The net potential is written as the sum of thetwo-body potentials:

U(ri) =∑j

V (ri, rj) =∫ρ(r′)V (r, r′)dr′ (65)

where the summation over all interactions isreplaced by an integral weighted by the nucleondensity distribution ρ(r).


The nucleon density distribution ρ(r) is given by:

ρ(r) =∑

ψ∗(r)ψ(r) (66)

where ψ is the wavefunction of the particle at r.The summation is over all individually occupiedorbits. Thus the net (one-body) potential becomes:

U(r) =∑∫

ψ∗(r)V (r, r′)ψ(r)dr′ (67)

which is substituted into the one-body Schrödingerequation to solve for ψ, for all orbits. Theprocedure is to start with trial wavefunctions andthen iterate until U (or ρ, or ψ) does not change.


4: Nuclear Deformation

The experimental observation of large electricquadrupole moments and low-lying rotational bandssuggests that nuclei can be deformed in their groundstate. The origin of deformation lies in the longrange component of the nucleon-nucleon residualinteraction (the quadrupole-quadrupole interaction)which gives additional binding energy to nucleiwhich lie between the closed shells if the nucleus isdeformed. In contrast, the short range component(pairing interaction) prefers spherical shapes.

4.1 Geometric Descriptions

The general shape of a nucleus (see Fig. 17) can beexpressed in terms of the spherical harmonicsYλµ(θ, φ):

R = R0

⎡⎣1 + ∞∑

λ=0

λ∑µ=−λ

aλµYλµ(θ, φ)

⎤⎦ (68)


Figure 17: Description of the nuclear shape

y

x

z

R (�,�)

R�

�

Figure 18: Nuclear deformations

� = 2 � = 3

� = 4 �40 > 0 � = 4 �40 < 0

The λ = 1 term describes displacement of the masscentre and therefore cannot give rise to excitation ofthe nucleus – ignore this. The λ = 2 term is themost important, and describes quadrupoledeformation.


The next higher order terms, λ = 3, 4, describeoctupole and hexadecapole deformation, see Fig 18.

For quadrupole deformation, the equation reducesto:

R = R0

⎡⎣1 + 2∑

µ=−2a2µY2µ(θ, φ)

⎤⎦ (69)

Figure 19: Principal axesx

1

y3

z2

If the principal (i.e. x,y,z) axes are made to becoincident with the nuclear axes (1, 2, 3), seeFig. 19, then a2 1 = a2−1 = 0 and a2 2 = a2−2


We define

a2 0 = β cos γ (70)

a2 2 =1√2β sin γ (71)

so that

δRK = RK −R0 =√

54πR0β cos

(γ −K 2π

3

)(72)

If γ = 0 then R1 = R2 and R3 > R0, i.e. prolatespheroid. In this case we have:

δR1,2 = −12√

54πR0β

δR3 =√

54πR0β (73)

If R3 < R0 this describes an oblate spheroid

If γ �= nπ/3 where n is an integer then we have atriaxial shape with R1 �= R2 �= R3


4.2 Theoretical Nuclear Deformations

Various calculated nuclear shapes are shown belowfrom Möller and Nix (Los Alamos). The top rowshows ground-state deformations. At the bottomleft is a fission isomer trapped in a secondaryenergy minimum, whilea mass-asymmetric shape ofa nucleus on the way to fission is shown to thebottom right.

Figure 20: Theoretical nuclear shapes


4.3 Ground-State Deformations

Theoretical ground-state quadrupole deformationsare shown below, from Möller and Nix (LosAlamos). Note that low deformation occurs aroundthe magic numbers and maximal deformation inmidshell regions.

Figure 21: Deformation systematics

Ground-state quadrupole deformation

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

|β2|

FRDM (1992)

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Neutron Number N

Pro

ton N

um

ber

Z


4.4 Nilsson Model

To introduce nuclear deformation Nilsson modifiedthe the harmonic oscillator potential to becomeanisotropic:

V (r) =12m

(ω21x

2 + ω22y2 + ω23z

2)

(74)

so that

ωKRK = ω0R0 (75)

If axial symmetry is assumed (i.e. γ = 0) then thedeformation is described by ε where

ε = (ω1,2 − ω3)/ω0 (76)

Therefore, using Equations (73) and (75):

ε =

(ω0R0R1,2

− ω0R0R3

)/ω0 =

(R3 −R1,2)R0R1,2R3

=(δR3 − δR1,2)R0

R20=

32

√54πβ ≈ 0.95 β (77)

so that ε ≈ β.


In order to reproduce the observed nuclearbehaviour the empirical potential energy term hasto include two extra terms and becomes:

V (r) =12m

(ω21x

2 + ω22y2 + ω23z

2)− C�.s −D�2

(78)The C�.s term is the spin-orbit term introducedalready to account for the observed nuclear shellstructure. The D�2 term has the effect of flatteningthe potential well to make it look more like theactual nuclear shape. Without these additionalterms the Nilsson energy levels are:

h̄ω1

(n1 +

12

)+ h̄ω2

(n2 +

12

)+ h̄ω3

(n3 +

12

)

=[(N +

32

)− ε

(n3 − N3

)+

19ε2

(N +

32

)]h̄ω0(79)

where N = n1 + n2 + n3 is the oscillator quantumnumber (see Equation 54), and n3 describes thez-axis component, etc. In addition to N and n3 thequantum numbers �z = Λ, sz = Σ = ±12 andjz = Ω = Λ + Σ are also used. These are theprojections of �, s and j of a single nucleon on thenuclear z-axis. The parity is (−1)� and the labels ofthe energy levels in the Nilsson scheme: Ωπ[Nn3Λ].


Figure 22: Nilsson diagram

g 72

g 92

p12f 52p 32

f 72

d 32

s 12

d 52

7/2[413]1/2[301]5/2[303]1/2[431]3/2[301]5/2[422]

3/2[431]7/2[303]3/2[312]1/2[310]1/2[440]

5/2[312]1/2[321]

3/2[321]

3/2[202]

1/2[330]1/2[200]

5/2[202]

1/2[211]

3/2[211]

1/2[220]

20

28

50

9/2[404]

g 72

-.3 -.2 -.1 0 .1 .2 .3

�

��= 0��= 0C = D = 0

N = 2

N = 3

N = 4

N = 5

N = 6

126

82

50

28

20

1d 5/2

2s 1/2

1d 3/2

1f 7/2

2p 3/21f 5/22p 1/2

1g 9/2

2d 5/21g 7/2

3s 1/22d 3/21h 11/2

2f 7/21h 9/21i 13/23p 3/22f 5/23p 1/2

2g 9/21i 11/21j 15/2

Note that many degeneracies are lifted for adeformed nucleus, i.e. ε �= 0


Because of the additional �.s and �2 terms thephysical quantities labelled by n3 and Λ are notconstants of motion, only approximately so. Theyare called asymptotic quantum numbers as theybecome “good” only as ε→ ∞. The quantumnumbers Ω, π and N are always good labelsprovided (i) the nucleus is not rotating, (ii) thereare no residual interactions.

The following observations can be made from theFig. 22:

• each spherical level, labelled by (�)j at ε = 0, issplit into (2j + 1)/2 levels, i.e.

Ω = ±12, ±3

2, ..., ±j (80)

• the remaining ±Ω degeneracy means that eachdeformed level can accommodate 2 neutrons or2 protons.

• orbits with lower Ω are shifted downwards forε > 0 (prolate), upwards for ε < 0 (oblate).


4.5 Large Deformations

If ε becomes large then the �.s and �2 terms can beneglected relative to deformation effects and theenergy levels are given by Equation 79 (see Fig. 23).

Figure 23: Deformed shell gaps

-1.0 -0.5 0.0 0.5 1.00.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

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

ingl

e pa

rtic

le e

nerg

y (h�

o)

-0.75 0.6 ~0.86�

168

112

70

40

20

8

22

6

14

26

44

68

100

140

190

2

4

10

16

28

40

60

80

110

140

182

2

4

6

12

18

24

36

48

60

80


Fig. 23 shows that if ω3 and ω1,2 are in the ratio ofintegers, i.e.

ω3/ω1,2 = p/q (81)

then large degeneracies become apparent anddeformed shell effects emerge. This is the origin ofsuperdeformed shapes which have favourableenergies at a p/q ratio of 1/2 (see Section 5.3).

This corresponds to a long-to-short axis ratio of

R3 : R1,2 = 2 : 1 with ε ≈ 0.60

Other favourable shapes occur at an axis ratio of3:2 (ε ≈ 0.37) and searches are currently beingmade for nuclei with a hyperdeformed shape:

R3 : R1,2 = 3 : 1 with ε ≈ 0.86

What next? megadeformation!


5: Hybrid Models

5.1 Deformed Liquid Drop

In the assumption that the nucleus behaves as acharged, liquid drop the semi-empirical expressioncan be obtained for the total nuclear energy:

E(A,Z) = −aVA+ aSA2/3 + aCZ2A−1/3 (82)

where aV , aS , and aC are the coefficients of thevolume, surface and Coulomb energies, respectively.To correct for deformation the nuclear radius R0 isreplaced by:

R3 = R0(1 + δ)

R1,2 = R0(

1 − 12δ

)(83)

where

δ =√

54πβ =

23ε (84)

(see Equation 77).


It can be shown that the above expression for theenergy becomes (for small values of δ):

E(δ,A,Z) = −aVA+ aSA2/3(

1 +25δ2

)

+aCZ2A−1/3(

1 − 15δ2

)(85)

Figure 24: Deformed–spherical energies

�E 0

�

1r

A plot of ∆E(δ) = E(δ) − E(δ = 0) is shown inFig. 24. This predicts that the nucleus is alwaysspherical (i.e. ∆E > 0 for δ > 0) unless Z2/A > 49in which case the nucleus prefers infinitedeformation (i.e. it fissions).

This is clearly wrong!


5.2 Shell Correction - Strutinsky

Method

The liquid drop model can be extended to take intoaccount shell-model effects, i.e. effects which arisefrom the individual nucleon motion. Additionalterms arising from the symmetry energy (whichprefers Z = N) and the pairing energy (∆, 0, −∆for even-even, odd-even and odd-odd nuclei) can beadded to the above expression. Alternatively thetotal energy can be calculated using mean fieldpotentials, either self consistently (the Hartree Fockmethod) or using empirical potentials such asNilsson and Woods-Saxon. This is not the sum ofthe individual eigenvalues ei because the potentialenergy of each nucleon would be counted twice. Theeigenvalue for each nucleon is:

ei = 〈Ti〉 + 〈∑j �=i

Vij〉 (86)

and the total energy E is:∑

〈Ti〉 + 〈12∑i,j �=i

Vij〉 = 12∑

ei +12

∑〈Ti〉 (87)


For particles moving in the harmonic oscillatorpotential it can be shown that:

〈Ti〉 = 〈Vi〉 =∑i,j �=i

Vij (88)

so that:

E =34

∑ei (89)

This method has difficulty in producing the correctenergy because errors in the individual values of eigive large errors in the summation. To obtain boththe global (liquid drop) and local (shell-model)variations with δ, Z and A more accurately,Strutinsky developed a method to combine the bestproperties of both models. He achieved this byconsidering the behaviour of the level density, ρ(e)(e is the nucleon energy) in the two models.

Fig. 25 shows the level densities at the Fermi surfaceρF(e) without and with single particle effects.


Figure 25: Level densities at the Fermi surface

(a) (b) (c)

Ene

rgy

(arb

itary

uni

ts)

12

34

56

1

3

4.5

1

2.5

4

��i = 42

i��

i = 50

i��

i = 34

i

�F

If there are no single particle effects (Fig. 25(a))then the spacing of the energy levels is uniform andρF(e) = ρAV (e), the average value. This is theliquid drop prediction. In Figs. 25(b) and (c) thelevel densities are non-uniform because of singleparticle effects.

The values of both ρAV(e) and ρF(e) can becalculated using (for example) the Nilsson model byaveraging over a large (5–10 MeV) and small (1–2MeV) energy interval respectively. A change innuclear binding will arise from the value ofρAV(e)− ρF(e) — negative in Fig. 25(b) and positivein Fig. 25(c). The calculated fluctuating energycorrection is then added to the liquid drop energy.


Figure 26: Shell-correction energies

10 30 50 70 90 110 130 150 170 190-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

�2 (�

4, �

6)

Neutron Number, N

28 50 82 126 184

156104

12 4262 84

16 44

108

5820

144

130

108

14 3670

112

Fig. 26 shows the variation of the shell correctionenergy with δ and N in the form of a contour plot.

The spherical shell gaps 20, 28, 50, 82, 126 and 184can be seen for ε = 0, but in general the minimumin energy will occur for non-zero values of �. Inaddition, there are deformed shell gaps which occurat 16, 44, 84, 112 and 144 for ε = 0.6. The singleparticle model used to calculate the fluctuatingcorrection is more sophisticated than that used inSection 4.5 and the superdeformed shell gaps liecloser to those observed experimentally.


5.3 Fission Isomers

If the increase in liquid drop energy for increasingdeformation, ∆E(δ), is small enough (e.g.Z2/A > 35) then any secondary minimum in thetotal energy arising from the shell correction willbecome similar in energy to the first minimum.

Fig. 27 shows the variation in total nuclear energywith deformation for the nucleus 240Pu. For thisnucleus two minima in the energy are seen, bothoccuring for non-zero deformations.

Figure 27: Potential energy versus deformation

○ ○ ○ ○ ○ ○ ○

○ ○ ○ ○ ○ ○ ○

V(P

oten

tial E

nerg

y)

Deformation�I �II

EA EBII

I

Mass distribution

Normal fissionSubbarrier fission

Isomer fission

Spontaneous fission


The ‘second’ minimum, which corresponds to asuperdeformed nuclear state, can decay bypenetration through the outer barrier to fissionrather than by penetrating the inner barrier to theground state. The first and second minimum statesare said to coexist, i.e. the nucleus can exist ineither state, each having very different deformationto each other.

The fission lifetime for the second minimum is muchshorter than for the first minimum but is muchlonger than the lifetime of excited states built uponit. For this reason it is called a fission isomer where‘isomeric’ means long-lived.

Evidence for the large deformation of the secondminimum has come from direct measurement of itselectric quadrupole moment.


The electric multipole moments are given by:

Mλ =Z∑

k=1

rλkYλ0(θk) (90)

where (rk, θk, φk) are the co-ordinates of the kth

proton and Yλ0 is a spherical harmonic of order λ.

The electric quadrupole moment is:

M2 =Z∑

k=1

r2kY20(θk) =√

516π

∑r2k

(3 cos2 θk − 1

)

=√

516π

∑(3z2k − r2k

)(91)

since cos θ = z/r.

For a nuclear shape having quadrupole massdeformation, we assume that the protons areuniformly distributed throughout the nucleus.Equation 91 is integrated over the spheroidal shapeof the nucleus, giving

M2 = 1√20π

Z(R23 −R21,2

)≈ 3

4πZR20β (92)


By convention, we define the intrinsic quadrupolemoment, Q0, as:

Q0 =√

16π5

M2 = 3√5πZR20β =

65ZR20δ (93)

The measurement of Q0 can be achieved in anumber of ways. For example it is directly relatedto the probability of the emission of a photon in thedecay of an excited nuclear state within a rotationalband which is characteristic of the deformedstructure. From this measurement the nucleardeformation β can be extracted.


6: Nuclear Excitation

6.1 Spherical Nuclei

Figure 28: Level schemes: even-even nuclei

7.126.92

6.136.05

1-2+

3-0+

0 0+

3.71 5-3.48 4-3.20 5-

2.61 3-

0 0+

16O8 208Pb82

MeV I�

I�MeV

Fig. 28 shows the energy levels of two nuclei, 16Oand 208Pb. Note that in common with all even-evennuclei their ground states have angular momentumI and parity π of 0+, a consequence of nuclearpairing.


These particular nuclei, in common with all closedshell nuclei are spherical in their ground state sothat excitations can only occur by breaking pairs ofnucleons or by vibrations. The energy differencebetween the ground state and the lowest excitedstates is a rough measure of the pairing energy. Forodd mass nuclei which are one nucleon added orremoved to the closed shell, the low-lying energylevels (Fig. 29) represent the single particleexcitation in either direction from the Fermi surface.

Figure 29: Level schemes: odd-A nuclei

0 5+

217O8

1+2

1-2

5-2

3-2

3+2

0.87

3.06

3.84

4.55

5.09

MeV I�

The configuration of the low-lying levels are writtenas (π(�)j)nIπ (odd-Z) or (ν(�)j)nIπ (odd-N), e.g.:17O is (ν(�)j)1Iπ for a closed shell + one neutron,207Tl is (π(�)j)−1Iπ for closed shell + 1 proton hole.


In this case the total angular momentum I of thenucleus equals the angular momentum of the oddparticle j. Higher excitations can occur by breakingpairs of nucleons e.g. (νp 3

2)−1(νd 5

2)2 32

− for the 4.55

MeV level in 17O would be a 1-hole 2-particle state.

6.2 Vibrations

From the liquid drop dependence on deformation(see Equation 85) we can estimate the restoringforce if the nucleus is deformed from its equilibriumshape along the symmetry (z-)axis:

F = − dEdR3

= −dEdδ

dδ

dR3= −dE

dδ

1R0

= −(

45aSA

2/3 − 25aCZ

2A−1/3)δ

R0(94)

The vibration can be any distortion in the nuclearshape since (see Equations 68 and 83):

δ ∝ R−R0R0

=∑λ

∑µ

aλµYλµ(θ, φ) (95)


At low energy the most important are oscillations in

• a20 (β-vibrations – oscillations along thesymmetry axis)

• a22 (γ-vibrations – oscillations perpendicular tothe symmetry axis)

• a30 (octupole vibrations)

Figure 30: Nuclear vibrations

Quadrupoledistortion

�-vibration

�-vibration

Octupolevibration

The energy levels of the vibrating system are thoseof the 1-D harmonic oscillator and are:0, h̄ω, 2h̄ω, ...These correspond to excitations of 0, 1, 2 ...phonons.


For a given mode of vibration, each phonon has anassociated angular momentum and parity:quadrupole phonons have 2+ and octupole phononshave 3−. For a pure vibrator the energy levelscorresponding to the ways the n phonons arecombined have the same energy nh̄ω; for real nucleithis degeneracy is removed, see Fig. 31.

Figure 31: Realistic vibrational levels

0+, 2+, 3+, 4+, 6+

0+, 2+, 4+

2+

0+

3h�2

2h�2

h�2

0��= 2

0

1

2

MeV 3+0+

5+, 6+ (4+)3+, 4+

2+

0+

0+2+

4+

2+

0+118Cd48

Note that the γ-vibration can only occur for adeformed system.


6.3 Rotations of a Deformed System

Examination of the low-lying energy levels ofdeformed even-even nuclei which lie far from closedshells reveal a regular sequence of levels whoseenergy is much lower than the pairing energy. Thisarises from nuclear rotation. The Hamiltoniancorresponding to the rotation of the deformedsystem is:

Hrot =h̄2

2J R2 =

h̄2

2J (I − J)2 (96)

where J is the moment of inertia, R is therotational angular momentum and J is additionalangular momentum generated by for example, theodd particle in an odd-A nucleus, or by vibrations.All angular momenta are in units of h̄. For aneven-even nucleus with no vibrations J = 0 andI = R. Fig. 32 shows how these angular momentumvectors are oriented in space.


Figure 32: Nuclear angular-momentum vectors

zK

R

J

x

I

Note that for an axially-symmetric system rotationcannot occur around the symmetry (z-)axis so R isalong the x- or y-axis. These are equivalent and thex-axis is usually selected.

The rotational Hamiltonian can be expanded:

R2 = (I − J)2 = I2 − 2I.J + J2= I2 + J2 − 2K2 − (I+J− + I−J+) (97)

where I± = Ix ± iIy, J± = Jx ± iJy andJz = Iz = ±K.


The quantity K is the projection of I along thesymmetry axis. (Rz = 0 as already discussed). Thecoupling term (I+J− + I−J+) corresponds to theCoriolis force and couples J to R. The operators I±can however only link states with K differing by±1. This Coriolis coupling term can be ignoredprovided:

• rotational bands with ∆K = 1, correspondingto different intrinsic excitation, lie far apart inenergy;

• the band does not have K = ±12Neglecting the coupling term gives the followingexpression for the excitation energies:

Erot =h̄2

2J[I(I + 1) + J(J + 1) − 2K2

]I = K,K + 1,K + 2, ... (98)

In the absence of the Coriolis effects K is a constantof motion; since J is also constant then

Erot = EK+h̄2

2J I(I+1), I = K,K+1,K+2, ... (99)

where EK is the energy of the lowest member of therotational band having I = K, the bandhead.


6.4 Examples

The low lying energy levels of three nuclei illustraterotational structure.

6.4.1 232Th

Figure 33: 232Th level scheme

0- 0+ 2+

11-

9-

7-

5-

3-

1-

1498.8

1249.6

1043.1

883.8

774.5

714.3

14+

12+

10+

8+

6+

4+

2+

0+ 0

49.4

162.0

333.2

556.9

826.8

1137.1

1482.5 10+

8+

6+

4+

2+

0+ 730.6

774.5873.1

1023.4

1222.3

1467.5

11+

10+

9+

8+

6+7+

4+

2+

5+

3+

~1641

1513.0

~1371

1260.7

~11421051.2

960.3890.5829.7785.5

Here the low-lying excitations are all collective (i.e.rotational and vibrational).


The following rotational bands can be seen inFig. 33:

Ground state band.Kπ = 0+, Iπ = 0+, 2+, 4+, 6+, ...

β-band. Here Jπ = 2+, Kπ = 0+ soIπ = 0+, 2+, 4+, 6+, ...

γ-band. Here Jπ = 2+, Kπ = 2+ soIπ = 2+, 3+, 4+, 5+, 6+ ...

Octupole band. Here Jπ = 3−, Kπ = 0−, 1−, 2− or3−. The lowest energy band hasKπ = 0−, Iπ = 1−, 3−, 5−, 7−

Note that if Kπ = 0+ then the I values 1, 3, 5,...are not present. Similarly the I values 0, 2, 4,...disappear if Kπ = 0−. This is a consequence of thenuclear wavefunction being identical under arotation of 180◦ (reflection symmetry about theplane containing the x, y axes). This is the case for232Th which has no octupole deformation, i.e.β3 = 0.


6.4.2 224Ra

Figure 34: 224Ra level scheme

0+84.4

2+

4+

(6+)

(8+)

166.3

228.2

275.7

478.9

754.6

6.4.3 15369Tm

Figure 35: 153Tm level scheme

0 0.1 0.2 0.3

s

[411 3/2][523 7/2][411 1/2][404 7/2][541 1/2][402 5/2]

(7/2) 725

(5/2) 610

(3/2) 491[411 3/2]

15/2 676

13/2 512

11/2 370

9/2 253

7/2 161[523 7/2]

13/2 690

11/2 4149/2 362

7/2 160

5/2 1303/2 121/2 0

[411 1/2]

15/2 747

13/2 546

11/2 367

9/2 211

7/2 81[404 7/2]

17/2 796

13/2 4987/2 451

9/2 2943/2 2755/2 182

9/2 552

7/2 420

5/2 316[402 5/2]

[541 1/2]

50

82

66

FERMILEVEL

�

1 2

d3 2

h11 2

d5 2g7 2

The low-lying excitations are rotational bands builtupon single proton excitations, so that K = jz = Ω.


7: Rotating Systems

7.1 Moment of Inertia

The purely geometric model described earlierassumes a given nuclear shape such as a deformedliquid drop. Two questions arise:

• What is the moment of inertia of the nucleus?• What is the effect of rotations on individual

nucleons?

Inglis showed in 1952 that for a Fermi gas ofnucleons the moment of inertia J around the x-axisis given by:

Jx = 2∑ |〈p|Îx|h〉|2

�p − �h (100)

where the summation is over all possible 1-particle1-hole excitations (see Fig. 36) in a deformed shellmodel (DSM) potential such as the Nilssonpotential.


Figure 36: Particle and hole levels

0

1

2

�p

FERMILEVEL

Es.

p. (M

eV)

particle

hole

The quantities �p and �h are the energies of singleparticle and single hole states respectively. Theoperator Îx is the angular momentum operatorabout the x-axis.

If, for example, the single particle Nilsson model isused to calculate J using this expression, it givescomparable values to the rigid-body value for themoment-of-inertia of a deformed liquid drop:


Jx = 25muAR20

[1 +

12

√54πβ

](101)

where mu is the nucleon mass and R0 is the nuclearradius. These are much higher than theexperimental values, see Fig. 37

Figure 37: Nuclear moments of inertia

152 156 160 164 168 172 176 180 1840

50

100

150

Mass number, A

Mom

ent o

f ine

rtia

, 2�

�Ih²

(MeV

-1)

ExperimentalValues corresponding to rigid rotationTheoretical (case A)Theoretical (case B)

62Sm64Gd 66Dy

68Er 70Yb

72Hf

74W

The discrepancy lies in the residual interactions,particularly the pairing term, which is ignored inthe simple deformed potential model. We have seenthat intrinsic excitation can only occur in even-evennuclei by breaking pairs of nucleons, so the energylevel diagram becomes:


Figure 38: Effect of pairing

Es.

p. (M

eV)

0

1

2

�E��

A rough estimate of the energy required to create aparticle-hole excitation is 2∆, where ∆ is thepairing gap. Since the denominator in the formulafor J (see Equation 100) depends on (�p − �h), and(�p − �h) ≈ 2∆ ≈ 2 MeV for mass 150 nuclei then Jmust be reduced because of pairing.


7.2 Cranking Model

The Deformed Shell Model (e.g. Nilsson Model) canbe modified to take into account pairing. Toinclude rotational effects it is convenient tosubtract the effects of rotational forces (Coriolis andcentripetal) which occur when the nucleus rotateswith frequency ω about the x-axis. Classically, the‘potential’ energy of these forces is ω.I so that thecorresponding quantum operator is ωÎx where Îx isthe operator corresponding to the component of Ialong the rotation axis. The Hamiltonian becomes:

Hω = HDSM − ωÎx (102)

which is the energy in the rotating frame - calledthe Routhian. The energy of the system in thisframe is

Eω = E − ω〈Îx〉 (103)where 〈Îx〉 is the expectation value of Îx. This isthe average value of Ix and is called the alignedangular momentum.


Two important relationships, which arise because Eis independent of ω and Eω is independent of I, are:

dEω

dω= −〈Îx〉 (104)

anddE

dI= ω

d〈Îx〉dI

(105)

Figure 39: Projections of the nuclear spin

KZ

x

I

Ix

�

Since (see Fig. 39) I2x = I2 −K2 we have

〈Îx〉 =√I(I + 1) −K2 h̄ (106)

If K = 0 then 〈Îx〉 =√I(I + 1) h̄ ≈ Ih̄ so that

(using Equation 105):

dE

dI= ωh̄ (107)


Using Equation 99, the energy of a rotational bandfor K = 0 is:

E = E0 +h̄2

2J I(I + 1) I = 0, 2, 4, ... (108)

so thatdE

dI=

h̄2

2J (2I + 1) (109)Since the energy of a transition ∆E between twoconsecutive states in a rotational band with angularmomentum I + 1 and I − 1, respectively, is:

∆E =h̄2

2J ({I + 1}{I + 2} − {I − 1}I) =h̄2

2J (4I+2)(110)

Thus, using Equation 109:

∆E = 2dE

dI(111)

and we therefore have the following relationshipbetween ω and ∆E:

ωh̄ = ∆E/2 (112)


7.3 Backbending

Fig. 40 shows the behaviour of the moment ofinertia J with ω2 for two different nuclei 174Hf and158Er.

Figure 40: Behaviour of moments of inertia

0 0.10

50

100

150

0.05h²�²(MeV²)

2�� h

² (M

eV-1)

2

4

68

10

12

14

16

18

2 4

68

10 12

16

18

174Hf

158Er

14

J is derived using Equation 110, and ω usingEquation 112. Both exhibit the phenomenon that Jincreases as ω increases.

The case of 158Er exhibits a more pronounced effectcalled backbending, so called from the characteristic‘S’ shape of the plot.


This is the effect of two bands ‘crossing’, the groundstate band (labelled ‘g’ in the figure) and a ‘super’-or s-band (labelled ‘s’):

Figure 41: Crossing bands

sg

g

s

�

E�

The states which are actually observed are the‘yrast’ states (thick line in the figure) which havethe lowest energy for a given value of I. The s-bandarises from a breaking of a particular pair ofnucleons (i 13

2neutrons in the case of 158Er) in the

rotating core so that their angular momentum j1,

j2aligns with the rotation axis:


Figure 42: Pair breaking

�

For the ground state band

Eg =h̄2

2Jg I(I + 1) (113)

For the s-band (see Equation 96)

Es =h̄2

2Js (I − J)2 + EJ (114)

where J = j1+ j

2and EJ is the energy required to

break a pair of nucleons,

EJ ≈ 2∆ ≈ 24 . A−12 (MeV) (115)

The aligned angular momentum of the s-bandincreases by approximately j1 + j2 − 1 (= 12h̄ for158Er).


Figure 43: Rotational alignment

I

j2

j1

R

�

J = j1 + j2I = R + J

The two bands will cross when Eg ≈ Es, whichoccurs when I ≈ 12h̄. Equation 110 is not valid inthe band crossing region but we can still use it todefine an effective moment of inertia:

Jeff/h̄2 = (2I + 1)/∆E ≈ I/ω (116)The sharpness of the backbend depends on howstrongly the bands ‘interact’ with each other, i.e.how much the wavefunctions of the states below thecrossing in the ground state band overlap with thewavefunctions of the states above the crossing inthe s-band. The bigger the overlap, the smootherthe band crossing and the weaker the effect.


8: Nuclei at Extremes of Spin

As the nucleus is rotated to states of higher andhigher angular momentum or spin I it tries toassume the configuration which has the lowestrotational energy (see Equation 96):

ER =h̄2

2J R2 (117)

where I = R+ J and J arises from the angularmomenta of individual nucleons (or, less importantin this context, from vibrations). This can beachieved by either reducing R or by increasing themoment of inertia J , and the configuration chosendepends on the value of I and proton and neutronnumber of the nucleus. In either case the nuclearpairing is broken by the effect of rotations.


8.1 Generation of Angular Momentum

The two basic modes of nuclear spin generation areshown below.

Figure 44: Collective and noncollective spin

147GdEr

158

E (MeV)

0

4

7/22

6

8

10

12

14

16

18

20

13/2

21/2

49/2

Deformed Nucleus

I

Rotationaxis

I

j

j

j

j

Near Spherical

Nucleus


8.2 High Ix bands

We have seen how in nuclear backbending the valueof R is reduced by breaking a single pair of nucleonsand aligning their individual angular momentumwith the x-axis, so that (see Figs. 42 and 43)

Ix =∑

jx +R (118)

is approximately a good quantum number; a givennuclear state is described by a single value of Ix.

The alignment of broken pairs with the rotationaxis becomes easier if

• The particle angular momentum j is large andits projection on the z-axis (Ω) is small, e.g. thei 13

2neutron which has Ω = 12 or Ω =

32 .

• the rotational Coriolis force is large. Since theenergy associated with this is proportional toh̄2

2J then this occurs when J is small, i.e. thenuclear deformation is small.

We deduce that alignment effects should beprominent for nuclei which have a few nucleonsoutside the closed shell, such as 158Er, which has 8neutrons outside the closed neutron shell (N = 82).


Figure 45: Aligned particles: band termination

I

x

z

new axis of symmetry

If we continue rotating the system to higher andhigher values of ω then more and more pairs ofnucleons will break and align their angularmomenta with the rotation axis. Since the alignedparticles move in equatorial orbits, this willeventually give rise to an oblate nucleus rotatingaround its axis of symmetry which is now thex-axis. The total angular momentum I would arisenot from collective rotation of a prolate deformedcore, but the sum of the individual j

i, see Fig. 45.


Eventually we align all the np protons and nnneutrons outside the closed shell so that

I =np∑i=1

ji(p) +nn∑i=1

ji(n) (119)

and the rotational band is said to terminate, seeFig. 46.

Figure 46: Band termination at 46+ in 158Er

250 500 750 1000 1250 1500

1000

2000

3000

4000

E�(keV)

num

ber

of c

ount

s

Paired Nucleus

Aligned neutrons

Aligned neutronsand protons

Fully alignedband termination

Oblate

158Er

2+

4+

6+ 16+8+

14+18+

10+12+

20+22+ 24+

26+28+

30+

32+34+36+38+

42+

�

�

�

40+

4644

At termination, 158Er can be considered as aspherical core (14664Gd82) plus twelve (4 protons and8 neutrons) aligned valence particles which generatea maximum spin of 46+ (see Table 3).


Table 3: Configuration of the 46+ state in 158Er

Subshell ji∑ji

π(h11/2)4 11/2, 9/2, 7/2, 5/2 32/2+

ν(i13/2)2 13/2, 11/2 24/2+

ν(h9/2)3 9/2, 7/2, 5/2 21/2−

ν(f7/2)3 7/2, 5/2, 3/2 15/2−

Iπ = 46+

8.3 High K (Iz) bands

If we have many paired nucleons outside the closedshell in the ground state, then alignment with thex-axis as the nucleus rotates becomes difficultbecause the nucleons near the Fermi level have theirspin vector lying closer to the direction of the z-axisthan to the x-axis, i.e. Ω is no longer small (seeSection 4.4).


Instead the nucleon angular momentum j continuesto align with the symmetry (z-)axis so that:

K = Iz =∑

jz =∑

Ωi (120)

is a good quantum number, see Fig. 47.

Figure 47: The K quantum number

R

x

I

z

K

j4

j3j2

j1

Again J is increased by breaking nucleon pairs sothat R is reduced. Fig. 48 shows the case of 17272Hf100where several high K bandheads can be seen.


Figure 48: High-K bands in 172Hf

(6+)63

(5+)

(4+)

222

380

1621

1463

1305

7-

6-

5-94

278

408

1621

14631305

353

(6-)

88181

17214

91857 7

+

6+193

128

1685

1878

321

8- 2006

163 ns

5 ns

968

993

834

1154

995

309

8+

6+

4+

2+

0+95

214

319

409

1038

950

1099

875

1194

647

1056

1376

172Hf10072

g.s.b. K�=2+ K�=4- K�=6- K�=6+ K�=8-

For example, the Kπ = 8− band head is formed bythe breaking of a pair of protons so that theyoccupy the Nilsson ‘configurations’

Ω[Nn3Λ] =72[404] and

92[514] (121)

In this caseK =

72

+92

= 8 (122)

and

π = (−1)N(1).(−1)N(2) = (−1)4.(−1)5 = −1 (123)

It is difficult for these rotational bands with high Kvalues to decay to bands with smaller K since thenucleus has to change its angular momentumorientation. The bandhead can become isomeric.


For example the Kπ = 8− bandhead is isomeric andwhich tries to decay to the Kπ = 0+ ground stateband by γ-emission.

The decay to the Kπ = 6+ band requires theemission of an improbable M2 transition). Thelifetime (τ = 163 ns) of the bandhead is much longerthan those of the rotational states built upon it.

8.4 Superdeformation

We have already seen in Section 4.5 that shelleffects can give large energy corrections for largevalues of prolate deformation, e.g when themajor/minor axis ratio is 2:1. Since the smoothliquid drop contribution to the total nuclear energynow includes the rotational energy, this can besubstantially reduced at high spin by increasing thevalue of the moment of inertia, J , seeEquation 101. At sufficiently high angularmomentum (I ≈ 60h̄ for mass 150 nuclei) thesuperdeformed second minimum can becomeenergetically favourable, see Fig. 49


Figure 49: Superdeformed second minimum

I=60h

I=0h

Superdeformed states

Deformation (� or �)

NuclearPotentialEnergy

For such nuclei the rotational forces play the samerole as the Coulomb forces for heavy nuclei inlowering the energy of the second minimum relativeto the first minimum (see Section 5.3).

The experimental signature of thesesuperdeformed shapes is a very regular sequenceof γ-rays whose energies are given by Equation 110.Fig. 50 shows such a γ-ray spectrum for 15266Dy86.The superdeformed band spans a spin range26 − 60h̄.


Figure 50: Superdeformed band in 152Dy

0.6 0.8 1.0 1.2 1.4 1.6

0

200

400

600

800

26

283032

3436

3840

4244

4648

5052

5456

5860

E�(MeV)

Num

ber

of c

ount

s

8.5 Shape Coexistence

For any given nuclear system at a given value ofangular momentum, a number of configurationsmay exist, but only one is energetically favourable.We can regard the configurations as co-existing.Instead of a smooth (classical) transition from oneshape to another we say that there are minima inthe total nuclear potential energy corresponding toeach shape.


At some critical value of angular momentum, theshape having the lowest energy changes.

A typical example is 15266Dy86 which has thefollowing coexisting structures:

1. prolate normal deformed (collective)

2. prolate superdeformed (collective)

3. oblate aligned (non-collective)

The last one is energetically favourable at low spinand represents the ground state of 152Dy (seeFig. 51).

As we move to the middle of both proton andneutron shells, the prolate normal deformed shapeeventually becomes lowest in energy for the groundstate, and the superdeformed shapes become lessand less favourable even at very high spin.


Figure 51: Shape coexistence in 152Dy

46+

44+

42+

40+

38+

36+

34+

32+

30+

28+

26+

24+

22+20+18+16+14+12+10+8+

6+

4+

2+0+

1454

1399

1341

1283

1222

1159

Documents

Contents - Nuclear Physicsns.ph.liv.ac.uk/PHYS490/Phys490_latex.pdf · E.S. Paul PHYS490: Advanced Nuclear Physics Slide 3 Figure 1: Addition of two vectors A B A + B (σ A +σ B)