Nuclear Models - Nuclear Physics

Nuclear Models 

Postgraduate Course following on 

from PHYS490 

2/16/2009 Postgraduate Nuclear Models Course, Liverpool : E.S. Paul 1

Nuclear Models 

1. Introduction 

2. Spherical Shell Model 

3. Nuclear Deformation 

4. Vibrational Motion 

5. Collective Nuclear Rotation 

6. Nuclear Pairing 

7. Cranked Shell Model 

8. Strutinsky Shell Correction 

9. Broken Symmetries 

10. Band Termination 


1. Introduction 

The Unique Nucleus 


Nuclear Physics 

• The nucleus is one of nature’s most interesting quantal 

few-body systems 

• It brings together many types of behaviour, almost all of 

which are found in other systems but which in nuclei 

interact with one another 

• The major elementary excitations in nuclei can be 

associated with single-particle or collective modes 

• While these modes can exist in isolation, it is the 

interaction between them that gives nuclear spectroscopy 

py 

its rich diversity 


The Nucleus is Unique! 

• The nucleus is a unique ensemble of strongly interacting 

fermions (nucleons) 

• Its large, yet finite, number of constituents controls 

the physics of this mesoscopic system 

• Both single-particle (out-of-phase) and collective (in- 

phase) effects occur 

• There is an analogy to a herd of wild animals. Individual 

id animals may break out of the herd but are rapidly 

drawn back to the safety of the collective 


Generation of Angular Momentum 

• There are two basic ways 

of generating high-spin 

states in a nucleus 

1. Collective (in-phase) 

motions of the nucleons: 

vibrations, rotations etc 

2. Single-particle effects: 

pair breaking, particlehole 

excitations. The 

individual spins of a few 

nucleons j i generate the 

total nuclear spin 


Collective Level Scheme 

• This nucleus has 347 known levels l and 516 gamma rays ! 


Noncollective Level Scheme 

• 148 Gd is an example of a 

nucleus showing single- 

particle behaviour 

• Complicated set of 

energy levels 

• No regular features 

e.g. band structures 

• Some states are 

isomeric 


Do Nuclei Really Rotate? 

• Should we talk about collective motion in nuclei? 

• We need to identify fast and slow degrees of freedom 

• For example, in molecules electronic motion is the 

fastest, vibrations are 10 2 times slower and rotations 10 6 

times slower. These motions have very different time 

scales so the wavefunction can be separated into a 

product of the terms 

• For nuclei the differences are much smaller. Collective 

and single-particle modes can perhaps be separated, but 

they will interact strongly ! 


Single-particle/Collective Modes 

• Collective and single-particle modes will interact strongly 

• Core polarisation 

• Coriolis Forces: ‘backbending’, 

modification of shell structure, 

quenching of pairing 

• Finite size effects: ‘band termination’, 

blocking of collective excitations 


What is a Model? 

• Quantum mechanics governs basic nuclear behaviour 

• The forces are complicated and cannot be written down 

explicitly 

• It is a many-body problem of great complexity 

• In the absence of a comprehensive nuclear theory we 

turn to models 

• A model is simply a way of looking at the nucleus that 

gives a physical insight into a wide range of its properties 


2. Spherical Shell Model 

• Single-Particle Shell Model 

• Square Well, Harmonic Oscillator, Woods-Saxon 

• Spin-Orbit Obit Coupling 

• Shell Structure 


Experimental Shell Effects 

• The energies of the 

first excited 2 + states 

in nuclei peak at the 

magic numbers of 

protons or neutrons 

• ‘B(E2)’ values (∝ 1/τ 

where τ is the mean 

lifetime) of the 2 

+ 

states reach a minimum 

at the magic numbers 

• ‘Magic’ nuclei are 

spherical and the least 

collective 


First 2 + Energies 

Z 

2/16/2009 Postgraduate Nuclear Models Course, Liverpool : E.S. Paul 14 

N

Systematics Near Z(N) = 50 

N = 50 

Z = 50 

• 100 Sn (Z=N=50) and 132 Sn (N=82) are doubly magic nuclei 


Neutron Separation Energies 


Shell Model – Mean Field 

N nucleons in 

a nucleus 

A nucleon in the 

Mean Field of 

N-1 nucleons 

• Assumption – ignore detailed two-body interactions 

• Each particle moves in a state independent of other 

particles 

• The Mean Field is the average smoothed-out 

interaction with all the other particles 

• An individual nucleon only experiences a central force 


Shell Model Hamiltonian 

• If the short range interaction potential between two 

nucleons i and j is v(r ij ), then the average potential 

acting on each particle is: 

V i (r i ) = 〈 ∑ j v(r ij ) 〉 

• The Hamiltonian, H = ∑ i T i + ∑ ij v(r ij ), can be rewritten: 

H’ = ∑ i [T i + V i (r i )] + λ [∑ ij v(r ij ) - ∑ i V i (r i )] 

mean field residual interaction 

• For λ = 1, H’ = H. The shell model assumption is that 

λ → 0, i.e. the central interaction is much larger than 

the residual interactions 


Choice of Potential 

• A central potential V(r i ) only depends on the distance 

r i and is made up of a superposition of short-range 

internucleonic potentials: 

V(r i ) = ∫v|r i – r’| ρ(r’) dr’ 

• ‘ρ(r’)’ is the density distribution of the nucleus 

• The internucleonic potential may be represented by a 

delta function: v(r ij ) = -V 0 δ(r ij ) 

• Then: V(r i ) = V 0 ρ(r) 

• The Schrödinger equation is: [T + V] Ψ(r) = E Ψ(r) 


Some Potential Wells 

• Square Well: V(r) = -V 0 for r ≤ R 0 

= 0 for r > R 0 

• Gaussian Well: V(r) = -V 0 exp[-(r/a) 2 ] 

• Exponential Well: V(r) = -V 0 exp[-2r/a] 

• Yukawa Well: V(r) = -(V 0 /r) exp[-r/a] 

• Harmonic Oscillator: V(r) = -V (/R) 0 [1-(r/R 0 2 ] 

• Woods-Saxon: V(r) = -V 0 / {1 + exp[(r-R 0 )/a]} 


Well Comparisons 


Square Well Potential 

Infinite square 

well potential 

• Simplest form of potential 

• Since we have a spherically 

symmetric potential we can 

separate the solutions into 

angular and radial parts 

• Radial solutions are Bessel 

functions which satisfy the 

boundary condition R nl (R) = 0 

• The eigenvalues are: 

R nl = {A/√(κr)} J l+½ (κr) 

where A is a constant and κ is 

the wave number of the nucleon: 

κ 2 = (2M/ħ 2 )[E nl + V] 


Square Well Labels 

• The levels are labelled by n and l (‘s’ = 0, ‘p’ = 1, ‘d’ = 2, 

‘f’ = 3, ‘g’ = 4, ‘h’ = 5, ‘i’ = 6, ‘j’ = 7, ‘k’ = 8) 

• Each level has 2(2l + 1) substates 

• The first few levels (different from H atom): 

Level Occupation Total 

1s 2 2 

1p 6 8 

1d 10 18 

2s 2 20 

1f 14 34 

2p 6 40 


Harmonic Oscillator Potential 

Simple harmonic 

oscillator potential 

• Easy to handle analytically 

• Form of potential: 

V HO (r) = -V + ½mr 2 ω 2 

• Solutions are Laguerre 

polynomials 

• Eigenenergies are labelled by 

the oscillator quantum number N: 

E N = (N + 3/2) ħω 

• For each N there are degenerate 

levels l with n and l that satisfy: 

2(n-1) + l = N, N ≥ 0, 0 ≤ l ≤ N 

• The parity of each shell is (-1) N 


Harmonic Oscillator Degeneracies 

• For each N there are degenerate energy levels with n 

and l that satisfy: 

2(n-1) + l = N, N ≥ 0, 0 ≤ l ≤ N 

• Even N contains only l even states; odd N, odd l 

• The degeneracy condition i is: 

∆l = 2 and ∆n = 1 

(e.g. N = 4 3s, 2d, 1g orbits) 

• It is the fundamental reason for shell structure, i.e. 

clustering of levels 


Harmonic Oscillator Labels 

• The number of degenerate levels for a given N is 

(N+1)(N+2) 

N allowed l E N Occupation Total 

0 0 3/2 2 2 

1 1 5/2 6 8 

2 2,0 7/2 12 20 

3 3,1 9/2 20 40 

4 4,2,0 20 11/2 30 70 

5 5,3,1 13/2 42 112 


(Wrong) Magic Numbers 


Spin-Orbit Potential 

• In Atomic Physics the spin-orbit interaction comes 

about due to the interaction of an electron’s magnetic 

moment with the magnetic field generated by its 

motion about the nucleus 

• A similar interaction was introduced for nuclei to 

empirically fit the observed magic numbers 

• A term is added to the potential: 

V(r) → V(r) + µ l.ss 

• The new term makes the force felt by a nucleon 

dependent on the direction of its spin 


Spin Orbit Energy 

• The spin-orbit term does not violate 

spherical symmetry and leaves l, j 

and j z as good quantum numbers, 

although l z and s z are not 

• The spin-orbit energy is: 

E l.s = {[4j(j+1)-4l(l+1)-1]/8}ħ 2 µ 

The vectors L and 

S precess about J 

• By making µ < 0, the magic numbers 

can be reproduced 

• States t with j = l + ½ are lower in 

energy than those states with j = l - 

½ (opposite way round to spin-orbit 

interaction ti in atoms !) 


Modified Harmonic Oscillator 

• The harmonic oscillator 

shells are shown to the 

left in this diagram 

• In the middle, an l 2 term 

is added to make the 

potential more realistic 

• A spin orbit term l.s is 

added to the right with 

its strength adjusted to 

obtain the correct 

nuclear magic numbers 


Woods-Saxon Potential 

V 0 ~ 50 MeV 

R 0 ~ 6-7 fm (A=125-190) 

a ~ 0.5 fm 

4a = ‘skin thickness’ 

• The Woods-Saxon (WS) nuclear potential is ‘supposedly’ 

the most realistic 

• The potential has the form: 

V(r) = -V 0 / { 1 + exp[(r - R 0 ) / a] } 


WS vs. MHO Potentials 

• The Woods-Saxon (WS) 

potential is the most realistic 

• The l 2 term in the Modified 

Harmonic Oscillator (MHO) 

potential flattens the bottom 

of the potential making it look 

more like the Woods-Saxon 

shape 

• There are slight differences 

between the MHO and WS 

energy levels, e.g. the 

ordering of the 2d 5/2 and 1g 7/2 

levels is interchanged 


3. Nuclear Deformation 

• Shape parameterisation 

• Quadrupole deformation, β and γ 

• Triaxiality 

• Anisotropic Harmonic Oscillator, Nilsson Model 

• Large deformation 


Evidence for Deformation 

1. Large electric quadrupole moments Q 0 

2. Low-lying rotational bands (E ∝ I[I+1] ) 

The origin of deformation lies in the long range 

component of the nucleon-nucleon residual interaction: 

a quadrupole-quadrupole interaction gives increased 

binding energy for nuclei which lie between closed 

shells if the nucleus is deformed. d In contrast, the 

short range (pairing) component favours sphericity 


Simple Nuclear Shapes 

• In the description of a ‘drop’ 

of nuclear matter with a sharp 

surface, the equipotential 

surface R(θ,φ) can be 

expressed as a sum over 

spherical harmonics Y λµ (θ,φ): 

R(θ,φ) = R 0 [1 + ∑ λ ∑ µ α λµ Y λµ (θ,φ)] 

• Here R 0 is the radius of a 

sphere and the α λµ coefficients 

represent distortions from the 

equilibrium i spherical shape 


Volume Conservation 

• By integrating over the shape of the nucleus, the volume 

for small deformation is: 

V ≈ (4π/3) [1 + 3α 00 /√(4π)] R 0 

3 

• To account for the incompressibility of nuclear matter 

we demand volume conservation under distortions and 

hence set α 00 = 0 

• A factor C(α λµ ) may be introduced to satisfy the 

conservation of volume more precisely: 

R(θ,φ) = C(α λµ ) R 0 [1 + ∑ λ ∑ µ α λµ Y λµ (θ,φ)] 


Most Important Multipoles 

• The λ = 1 term describes the displacement of the centre 

of mass and therefore cannot give rise to intrinsic 

excitation of the nucleus – ignore ! 

• The λ = 2 term is the most important term and describes 

quadrupole deformation 

• The λ = 3 term describes octupole shapes which can look 

like pears (µ = 0), bananas (µ = 1) and peanuts (µ =2,3) 

• The λ = 4 term describes hexadecapole shapes 


Variety of Shapes 


Theoretical Deformations 


Quadrupole Deformation 

The Euler angles 

relate the intrinsic 

(nucleus) and lab 

frame axes 

• The description of the nuclear 

shape simplifies if we make the 

principal axes of our coordinate 

system, i.e. (x, y, z), coincide with 

the nuclear axes (1, 2, 3) 

• Then α 22 = α 2-2 , and α 21 = α 2-1 = 0 

• The two independent coefficients 

α 20 and α 22 , together with the 

three Euler angles, then 

completely define the system 

• The shape then simplifies to: 

R = C R 0 [1 + α 20 Y 20 + α 22 (Y 22 +Y 2-2 )] 


β 2 and γ Parameters 

• An alternative parameterisation in the system of principal 

axes introduces the polar coordinates (β 2 , γ) through the 

relations: 

α 20 = β 2 cos γ and α 22 = -1/√2 β 2 sin γ 

• The parameter β 2 measures the total deformation: 

β 

2 

2 =∑ µ |α 2µ | 2 

• The parameter γ measures the lengths along the principal 

axes. For γ = 0°, the shape is prolate with the z-axis as 

the (long) symmetry axis 


Quadrupole β 2 and γ Parameters 

prolate 

x > y = z 

60° 

oblate 

x = z > y 

Axially symmetric shapes 

γ = n 60° 

0° 

prolate 

x = y < z 

oblate 

x = y > z -60° 

prolate 

x = z < y 

oblate 

x < y = z 

Triaxial shapes : x ≠ y ≠ z 

γ ≠ n 60° 


Lund Convention 

• In order to specify the 

triaxiality of a deformed 

quadrupole intrinsic 

shape, the range of γ 

values, 

0° ≤ γ≤60° is sufficient 

i • However, in order to 

specify a cranked system, 

we need three times this 

range, corresponding to 

the three principal axes 

about which the system 

can be cranked 


Deformation Systematics 

Theory 


Approximate Value of β 2 

• From an empirical fit to E2 transition rates (not valid 

near closed shells) it is found: 

T 4 γ (E2; 2 + → 0 + ) = (4 ± 2) x 10 10 Z 2 E γ4 A -1 

• This can be used for a ‘Grodzins’ estimate of the 

quadrupole deformation parameter: 

β 2 ≈ { 1225 / [ A 7/3 E(2 + ) ] } 1/2 

with the 2 + energy expressed in MeV 

• The energy of the 2 + state of an even-even nucleus hence 

gives an insight into the nuclear deformation. The lower 

the 2 + energy, the larger is β 2 and also the nuclear 

moment of inertia 


First 2 + Energies 

Z 


N

Triaxiality 

• All three principal axes have different lengths: 

R x ≠ R y ≠ R z 

i.e. ‘short’, ‘long’ and ‘intermediate’ t axes 

• There is no symmetry axis (however, there is reflection 

symmetry) so K is not a good quantum number 

• The low-spin energy levels l in even-even nuclei move 

around as a function of γ 

• For γ≠0° an effective quadrupole deformation 

parameter may be defined: 

β 2 2 1/2 

eff = β { 4 sin (3γ) / [9 - √(81 -72 sin (3γ) )] } 2/16/2009 Postgraduate Nuclear Models Course, Liverpool : E.S. Paul 47

Asymmetric Rotor Model 

• The Asymmetric Rotor Model (ARM) investigates rigid 

triaxial shapes 

• The energies of the first two 2 + states are: 

E(2 + ) = (6ħ 2 /2I) {9[1 ± √(1 - 8/9 sin 2 (3γ))] /4 sin 2 (3γ)} 

• Hence from the experimental energies of the first 2 + 

states, a value of |γ| can be deduced 

• The higher spin states of the ground-state and γ-bands 

move around in energy as γ changes 

• Increasing γ tends to lower the energy levels of the 

γ-band relative to the ground-state band 


ARM Energy Levels 

• These are the 

lowest energy 

levels of an 

asymmetric rigid- 

rotor predicted by 

the ARM 

• Note that the 

second 2 + state 

falls below the 

first 4 + state for 

|γ| ≥ 15° 


More ARM Relations 

• For the odd-spin members of the γ band: 

E(3 + ) = E 1 (2 + ) + E 2 (2 + ) and E(5 + ) = 4 E 1 (2 + ) + E 2 (2 + ) 

• Percentage differences for N = 76 isotones: 

Nucleus |γ| R 3 (%) R 5 (%) 

128 

Te 26.6° -1.21 

130 

Xe 27.6° +1.55 

132 

Ba 26.3° -0.98 

134 Ce 25.3° -179 -1.79 

136 

Nd 25.7° +0.38 +13.2 

138 

Sm 27.0° +0.79 +18.7 

140 

Gd 26.8° -2.53 +16.5 


γ-rigidity or γ-softness? 

• The ARM considers 

the rotation of a rigid 

triaxial shape 

• The other extreme is 

a completely flat 

potential with respect 

to γ, with γ oscillating 

uniformly between 

γ = 0° (prolate) and 

γ = 60° (oblate) 

• Since the average is 

γ = 30°, we compare 

the two models at this 

value 


Gamma-Band Staggering 

• As γ increases, a staggering arises between the odd-spin 

and even-spin members of the band 

• One way to measure this is to form the ratio: 

S(4,3,2) = { [ E 2 (4 + ) – E 1 (3 + ) ] - [ E 1 (3 + ) – E 2 (2 + ) ] } / E 1 (2 + ) 

• The energies, in units of E 1 (2 + ), are: 

Model E + + + 2 (2 ) E 1 (3 ) E 2 (4 ) S(432) S(4,3,2) 

γ-rigid (30°) 2.0 3.0 5.67 +1.67 

γ-unstable 2.5 4.5 4.5 -2.0 

• Spherical Harmonic Vibrator: S(4,3,2) = -1.0 

• Symmetric Rotor (γ = 0°): S(4,3,2) = +0.33 


S(4,3,2) Ratios vs. γ 

• The S(432) S(4,3,2) 

values for various 

types of motion 

are shown here as 

a function of the 

γ deformation 


Do Triaxial Nuclei Really Exist? 

• Considerable effort has been made over the last 

twenty years to obtain conclusive evidence of (static) 

triaxial nuclear shapes 

• These efforts have recently been intensified by the 

experimental evidence of chirality (handedness) and 

the wobbling (precession) mode in nuclei (discussed 

later) 

• Triaxiality is an essential prerequisite for the 

manifestation of both of these effects in the 

atomic nucleus ! 


Anisotropic Harmonic Oscillator 

• The Anisotropic Harmonic Oscillator (AHO) potential 

for a spheroidal nucleus deformed along the z-axis may 

be written: 

V 2 osc = ½M[ω 

2 

⊥ (x 2 + y 2 ) + ω z2 z 2 ] 

• Here ω ⊥ and ω z represent the frequencies of the simple 

harmonic motion perpendicular and parallel to the 

nuclear symmetry axis, respectively, and are functions 

of the nuclear deformation: 

ω z ≈ ω 0 [1 – 2/3 δ], ω ⊥ ≈ ω 0 [1 + 1/3 δ] 

and ω 3 2 0 = ω ⊥2 ω z for volume conservation 


Harmonic Oscillator Quantum 

• The Harmonic Oscillator quantum ω 0 is usually taken to 

have an isospin dependence: 

ħω 0 = 41 A -1/3 [1 ± (N-Z)/3A] MeV 

where the minus sign is used for protons and the plus 

sign for neutrons 

• In the ‘stretched’ coordinate system, the potential may 

then be written simply as: 

V osc = ½ħω 0 (ε 2 ) ρ 2 [1 – 2/3 ε 2 P 2 (cos θ t )] 

where ε 2 is (yet) another deformation parameter 


δ, β and ε Parameters 

• Three deformation parameters are often used: 

1. Delta: δ = ∆R/R 

2. Epsilon: ε 2 defines a rotational ellipsoid 

3. Beta: β 2 defines a rotational quadrupoloid 

• If the deformation is not so large, then the following 

approximations hold: 

ε 2 ≈ 0.946 β 2 (1 - 0.1126β 2 ) 

δ ≈ 0.946 β 2 (1 - 0.2700β 2 ) 

• Also the hexadecapole β 4 parameter has opposite sign 

to the ε 4 parameter: ε 4 ≈ -0.85 β 4 


Solutions of the AHO 

• The eigenvalues of the AHO potential are: 

E(n z ,n ⊥ ) = [n z + ½] ħω z + [n ⊥ + 1] ħω ⊥ 

or 

E(N,n z ,n ⊥ ) ≈ [N +3/2] ħω 0 –1/3δ[2n z -n ⊥ ] ħω 0 

with N = n z + n ⊥ 

• The latter expression is simply the energies of a 

Spherical Harmonic Oscillator minus a correction term, 

proportional to the deformation 

• The energy levels are labelled by the asymptotic 

quantum numbers: 

Ω π [N n z Λ] 


AHO Labels 

• The energy levels l are labelled ll by the asymptotic 

quantum numbers: Ω π [N n z Λ] 

• ‘N’: N = n x + n y +n z (= n z + n ⊥ ) is the oscillator quantum 

number 

• ‘n z ’: n z describes the z-axis component of N 

• ‘Λ’: Λ = l z is the projection of l onto the z-axis 

• ‘Ω’: Ω = Λ + Σ is the projection of j = l + s onto the z- 

axis 

• ‘π’: π = (-1) l is the parity of the state 


The Λ, Σ, Ω Quantum Numbers 

• Spin projections: Ω = Λ + Σ = Λ ± ½ 


AHO Degeneracies 

• Some of the degeneracies of the SHO are lifted 

• Consider the N = 4 shell spherical oscillator shell which 

has degeneracy (N + 1)(N + 2) = 30 with l = 4, 2, 0. 

• The onset of deformation causes these levels to split 

into (N + 1) levels, each of degeneracy 2(n ⊥ + 1): 

n z n ⊥ Occupation 

4 0 2 

3 1 4 

2 2 6 

1 3 8 

0 4 10 


Levels of the AHO 

• The splitting of the 

N = 4 oscillator shell 

is shown here when 

deformation is 

introduced 

• Note that levels with 

large n z (and hence 

small n ⊥ ) are favoured 


Nilsson Model 

• Nilsson added terms proportional to l 2 and l.ss similar to 

the spherical case 

• The resulting Modified d Harmonic Oscillator (MHO) or 

Nilsson potential may be written as: 

V MHO = V osc – κħω 0 [2l t .s + µ(l t 

2 

- 〈l t2 〉 N ] 

where κ and µ are adjustable parameters. They are 

different for each major oscillator shell 

• The l t .ss term imitates tes the nuclear spin-orbit interaction 

n 

in the stretched coordinate system 

• The l 2 t2 term deepens the effective potential ti for 

particles near the nuclear surface 


Remaining Degeneracies 

Nilsson Diagram 

• The l t .s and l t2 terms 

lift the 2(n ⊥ + 1) 

degeneracy of the 

N = n z + n ⊥ states 

• States with different 

Ω now have different 

energy 

• Each Ω π [N n z Λ] state 

is only twofold 

degenerate, 

corresponding to 

particles with ±Ω 


Nilsson Single-Particle Diagrams 

N 

Z 


Splitting of Ω States 

• Low Ω states favour 

prolate shapes 

• High Ω states 

favour oblate 

shapes 

• Note that each Ω 

state is now only 

twofold degenerate 

(±Ω) 


Splitting of Ω States 

David Campbell 

Florida State 

University 

it 


Asymptotic Quantum Numbers 

• Because of the additional l.s and l 2 terms the 

physical quantities labelled by n z and Λ are not 

constants of the motion, but only approximately so 

• These quantum numbers are called asymptotic as 

they only come good as ε 2 → ∞ 

• However, the quantum numbers N, Ω and π are always 

good labels provided that: 

1. the nucleus is not rotating and 

2. there are no residual interactions 


Proton Nilsson Diagram 

• A ‘Nilsson Diagram’ 

shows nuclear energy 

levels as a function of a 

quadrupole deformation 

parameter (β 2 , ε 2 or δ) 

• In this diagram, the 

large spherical shell 

gap at Z = 50 is rapidly 

diminished by the onset 

of deformation for both 

prolate (β 2 > 0) and 

oblate (β 2 < 0) shapes 


Intruder Orbitals 

• The slope of Nilsson 

levels is related to the 

single-particle matrix 

element of the 

quadrupole operator: 

dE/dβ = - 〈k|r 2 Y 20 |k〉 

• Unnatural-parity low 

Ω prolate orbitals may 

‘intrude’ down into a 

lower shell at large 

deformation 

• This is the origin of 

superdeformation 


Large Deformations 

• Deformed shell gaps 

(new ‘magic numbers’) 

emerge when the ratio 

of the major and minor 

nuclear axes are equal 

to the ratio of small 

integers 

• A superdeformed 

d 

shape has a major to 

minor axis ratio of 2:1 

• A hyperdeformed 

shape has a major to 

minor axis ratio of 3:1 


Superdeformed 152 Dy 

• The SD band in 

152 

Dy is a very 

regular structure 

t 

with equally 

spaced gamma-ray 

transitions 

Original SD γ-ray spectrum 

from 1986 (Daresbury) 

• The spacing is 

relatively small, 

i.e. the band has a 

large moment of 

inertia (close to 

the rigid body 

value) 


Superdeformed Axis Ratios 

• The moment of inertia of a rigid sphere is: 

I rig = (A 5/3 /72) ħ 2 MeV -1 

• The moment of inertia of a prolate ellipsoid undergoing 

rigid rotation is: 

I rig = (A 5/3 /72) (1 + x 2 )/ 2x 2/3 ħ 2 MeV -1 

where x is the ratio of major to minor axes 

• The moment of inertia is not always a good indicator of 

nuclear deformation (e.g. pairing) 

• The quadrupole moment (charge distribution) is a 

better indicator: 

Q 0 = (2/5) Z R 2 (x 2 –1)/ x 2/3 eb 


SD Systematics 

Nucleus Q 0 (eb) Axis Ratio 

36 

Ar 1.18 1.55 

60 

Zn 2.75 1.54 

82 

Sr 3.54 1.47 

91 

Tc 8.1 1.85 

108 Cd >9.5 >1.8 

132 

Ce 7.4 1.45 

152 Dy 17.5 185 1.85 

192 

Hg 17.7 1.61 

236 

U 32 1.84 


SD Regions 

Z 


A

4. Vibrational Motion 

• Spherical Harmonic Vibrator 

• Particle-vibration coupling 

• Rotation-Vibration ti ti Model 


Spherical Harmonic Vibrator 

• A dynamic deformation 

• We assume the nucleus 

is spherical in its ground 

state and the excited 

states are due to 

harmonic oscillations of 

the nuclear surface 

• For a quadrupole vibration, raton, the potential nta may be written: wrtt V vib = ∑ µ {½C 2 |α 2µ | 2 +½B 2 |dα 2µ /dt| 2 } 

where C 2 is a parameter representing the restoring 

potential and B 2 is associated with the mass carried by 

the vibration. This mode is possible since C 2 , determined 

by the surface tension, is low 


Vibrator Eigenvalues 

• The eigenvalues of the spherical harmonic vibrator 

are: 

E n = E 0 + n ħω 2 with ω 2 = √(C 2 /B 2 ) 

• E 0 represents the intrinsic and zero point motion of 

the oscillations 

• The energy levels for different n are equally spaced 

• Each phonon carries angular momentum 2 (Y 2 ) and has 

positive parity 


• For an E2 transition: 

Allowed Transitions 

• For quadrupole vibrations, 

electromagnetic 

transitions are only 

allowed between states 

with: 

∆n = ±1 

〈I=2,n=1||E2||I=0,n=0〉 I 2 0 0 = √(5) Q vib e 

where Q vib is calculated from the Liquid Drop Model: 

Q vib = (3ZR 2 /4π) √(ħ/2B 2 ω 2 ) 

• The magnetic moment is constant for λ = 2 states and 

therefore M1 transitions are not allowed 


Multiphonon Vibrational States 

N = 3 (3 phonon) 


124 

Sn, 

spherical 


N = 0 


Octupole Vibrations 

• For octupole vibrations, 

each phonon n carries 

angular momentum 3 

(Y 3 ) and negative parity 

• The energy of the first excited state (3 - ) is roughly 

twice the energy of the quadrupole case 

• For real nuclei, an anharmonic oscillator is needed. This 

removes the degeneracy of the n = 2 states (0 + , 2 + , 4 + ) 

of the quadrupole vibrator. It also displaces the λ = 2 

and λ = 3 states relative to each other 


Vibrational Movies… 


Beta (Y 20 ) Vibration 


Gamma (Y 22 ) Vibration 


Octupole (Y 30 ) Vibration 








Particle-Vibration Coupling 

• For an odd-A nucleus 

near a closed shell with 

small deformation, the 

odd particle may couple 

to the surface vibrations 

of the core 

• The Hamiltonian is: H = [H int + H vib ] + H coup = H 0 + H coup 

• If we assume the interaction H coup → 0, the motions are 

decoupled from one another and the eigenfunctions will 

take a product form: H 0 Ψ = EΨ with Ψ = Ψ int Ψ vib 

• Consider coupling an h 9/2 proton to the 3 - state in 208 Pb, 

forming states in 209 Bi. Seven ‘degenerate’ states are 

formed by coupling spin vectors 3 and 9/2 


Vibration or Rotation? 

• The simple ratio of 

the 4 + and 2 + 

energy levels of an 

even-even nucleus 

gives an indication 

of the types of 

excitation 

• For a vibrator: 

E ∝ n 

E(4 + )/E(2 + ) = 2.0 

• For a rotor: 

E ∝ I(I+1) 

E(4 + )/E(2 + ) = 3.33 

Te (Z = 52) systematics show that they are vibrational 


Development of Collectivity 

• Another limiting value of the E(4 + )/E(2 + ) ratio is 2.5, 

corresponding to a γ-soft rotor, or γ-unstable oscillator 

(O(6) limit of the interacting tin boson model: IBM) 

• Adding protons to tin: 

Nucleus E(4 + )/E(2 + ) Behaviour 

116 

Sn (Z=50) 1.65 spherical 

118 

Te (Z=52) 1.99 vibrational 

120 

Xe (Z=54) 2.47 γ-soft 

122 

Ba (Z=56) 2.89 transitional 

124 

Ce (Z=58) 3.15 rotational 


E(4 + )/E(2 + ) Values 


Evolution of Structure 

• This diagram shows 

the evolution of level 

structure from 

closed shell (doubly 

magic, spherical) to 

midshell (rotational, 

deformed) nuclei 

• The corresponding 

E(4 + )/E(2 + ) ratios 

are also shown 


Rotation-Vibration Model 

• The RVM model considers a well deformed (static), 

axially symmetric even-even nucleus and allows small 

fluctuations (dynamic) about the equilibrium shape β 0 

• After a ‘few’ approximations the energy spectrum may 

be written as: 

E = ½є [I(I+1) –K 2 R ] + є βn β + є γn γ 

where the є parameters are energies associated with 

rotations and vibrations 

• є R is related to β 0 and the nuclear moment of inertia I 


RVM Quantum Numbers 

• The quantum numbers are constrained such that: 

K = 0, 2, 4,… 

I = 0, 2, 4,… for K = 0 

= K, K+1, K+2,… for K ≠ 0 

n β = 0, 1, 2,… 

n γ = K/2, K/2 + 2, K/2 + 4,… 

• So what are the possible low-lying l energy levels l ? 


RVM Band Structure 

• For K = n β = n γ = 0, we expect a set of levels: 

l 

E = ½є R I(I+1) 

with I = 0, 2, 4,… ‘ground-state band’ represents pure 

rotation 

• A rotational ti band can be built on a β vibration by setting 

n β = 1. The energy levels are: 

E = є β + ½є R I(I+1) 

again with I = 0, 2, 4,… ‘β band’ (α 20 varies) 

• To include a γ vibration requires a nonzero K. So 

beginning with K = 2 and n γ = 1, the levels are: 

E = є γ + ½є R [I(I+1) – 4] 

this time with I = 2, 3, 4,… ‘γ band’ (α 22 varies) 


β and γ Bands 

• β-vibrational and γ- 

vibrational bands 

coexist with the 

rotational groundstate 

band in 

deformed nuclei 

• Such bands are found 

predominantly in the 

regions: 

150 ≤ A ≤ 190 and 

A ≥ 230 

which are far from 

shell closures 


E0 Transitions 

• β vibrations give rise to enhanced E0 transitions due to 

radial shape oscillations in β 

• A measure of E0 strength is ‘rho squared’ : 

ρ 2 (E0) = |M(E0)/eR 2 | 2 

where M(E0) is the monopole moment operator: 

M(E0) = ∫ρ(r)r 2 dτ 

• The ρ 2 values are usually yquoted in units of ρ 2 (E0) x 10 3 

• In the RVM model we expect values around 200, but 

these are somewhat larger than experiment 


More Vibrational Bands 


Octupole Vibrational Bands 

238 

U 


Nonadiabatic Vibration 

• For the surface modes of 

vibration, the frequency 

(velocity) of the 

oscillations is much smaller 

than that of the individual 

nucleonic motion 

• The motion is ‘adiabatic’ 

(as is nuclear rotation) and 

individual quantum levels 

are evident 

• However, ‘nonadiabatic’ 

collective motion can 

occur: ‘giant resonances’ 


Giant Resonances 

Monopole 

L = 0 

Isoscalar 

Isovector 

Dipole 

L = 1 

Quadrupole 

L = 2 


5. Collective Nuclear 

Rotation 

• Nuclear moments of inertia 

• Rotational band properties, signature 

• Particle-rotor t coupling 

• High-K bands, wobbling motion 


Gammasphere 

• The Hulk 

• News Item 


Moment of Inertia 

• Deformation provides an element of anisotropy allowing 

the definition of a nuclear orientation and the possibility 

of observing rotation 

• Classically the energy associated with rotation is: 

E 2 2 rot = ½ I ω = I / 2 I ; ω = I / I 

• Collective rotation involves the coherent contributions 

from many nucleons and gives rise to a smooth relation 

between energy and spin: 

E = (ħ 2 /2I) I[I + 1] 

which h defines the ‘static’ ti moment of inertia, sometimes 

denoted I (0) 


Energy Levels of a Rotor 

• The energy levels of a rotor 

are proportional p rti to I(I+1) 

• The ratios of energy levels 

for a rotor are: 

E(4 + )/E(2 + ) = 3.333 

E(6 + )/E(2 + ) = 7.0 


Rotational Frequency 

• The intensive variable ω 

(rotation about the x axis) 

is related to the extensive 

variable I by the relation: 

ħω = dE/dI x 

≈ ½[E(I+1) – E(I-1) 

• Here I x is the projection of I 

onto the rotation axis (x): 

I x = √[I(I+1)-K 2 ] ħ 

The rotational frequency ω is distinct from the oscillator 

quantum ω 0 . In practice ω « ω 0 and the collective 

rotation can be considered as an adiabatic motion 


Rigid Body Moment of Inertia 

• The rigid-body moment of inertia for a spherical nucleus 

is: 

I rig = (2/5) MR 2 = (2/5) A 5/3 m N r 

2 

0 

where m N is the mass of a nucleon (M = A m N ) and 

R = r 0 A 1/3 with r 0 = 12 1.2 fm 

• For a deformed nucleus: 

I rig = (2/5) A 5/3 m N r 02 [1 + 1/3 δ] 

where δ = ∆R / R 0 

• Typically nuclear moments of inertia are less than 50% 

of the rigid-body value at low spin 


Nuclear Moments of Inertia 

• Nuclear 

moments of 

inertia are 

lower than the 

rigid-body 

value – a 

consequence 

of nuclear 

pairing 


Nuclear Rotation 

• The assumption of the ideal flow 

of an incompressible nonviscous 

fluid (Liquid Drop Model) leads 

to a hydrodynamic moment of 

inertia (surface waves): 

I hydro = I rig δ 2 

• This estimate is much too low ! 

• We require short-range pairing ii 

correlations to account for the 

experimental values 


Kinematic and Dynamic MoI’s 

• Assuming maximum alignment on the 

x-axis (I x ~ I), the kinematic moment 

of inertia is defined: 

I (1) = (ħ 2 I) [dE(I)/dI] -1 = ħ I/ω 

• The dynamic moment of inertia 

(response of system to a force) is: 

• Note that I (2) = I (1) + ω dI (1) /dω 

I (2) = (ħ 2 ) [d 2 E(I)/dI 2 ] 

-1 = ħ dI/dω 

• Rigid body: I (1) = I (2) Nucleus at high spin: I (1) ≈ I (2) 


General Rotation 

• A deformed rotor has a Hamiltonian of the form: 

H 2 2 rot = Σ k A k R k , A k = ħ /2I k 

where I k is the moment of inertia about the k th axis 

• For triaxial shapes the moments of inertia are: 

I k = (4/3) I 0 sin 2 [γ + k 2π/3 ] 

• For an axial nucleus deformed along the z-axis, 

I 1 = I 2 = I 0 and I 3 = 0, and the Hamiltonian is: 

H 2 2 22 2 2 

rot = (ħ /2I 0 ) [R 1 + R ] = (ħ /2I 0 ) R 


Irrotational Moments of Inertia 


the variation of the 

moments of inertia I k 

as a function of the 

triaxiality parameter γ 

• For a prolate nuclear 

shape (γ = 0°), I 1 = I 2 

and I 3 = 0 

• For γ = 30° , I 2 reaches 

a maximum and this 

represents the ‘most 

collective’ shape 


Angular Momentum Coupling 

• Provided that the collective rotation is slow relative to 

the single-particle motion (adiabatic condition), the 

nuclear Hamiltonian can be separated into intrinsic and 

rotational parts: 

H = H int + H rot 

with eigenvalues Ψ = Ψ int Ψ rot 

• The intrinsic motion has angular momentum J, which is 

not a conserved quantity. It couples to the collective 

rotation R to give total spin: 

I = R + J 

• The total spin I is a constant of the motion together 

with its projection M 


Various Spin Projections 


Rotation Matrices 

• The intrinsic wavefunction can be characterised by the 

K projection. The three variables I 2 , M and K completely 

specify the state of motion The eigenfunctions are given 

by: 

Ψ rot = |IMK 〉 = √[(2I +1)/8π 2 ] D IMK (θ,φ,ψ) 

where the functions D IMK are ‘rotation matrices’ 

• Note: Î 2 D IMK = I(I+1)ħ 2 D IMK 

; Î Z D IMK = Kħ D IMK 

Î ± D IMK = √[I(I+1) – K(KK1)]ħ D IMKK1 

• The rotational energy is: 

(1/2I 2 x )(Î 2 – Î z2 ) Ψ rot i.e. E rot = (ħ 2 /2I x )[I(I+1) – K 2 ] 


Signature Quantum Number ‘r’ 

• For K = 0, the D IMK functions reduce to spherical 

harmonics Y IM and the nuclear wavefunction is: 

• 

Ψ r,IMK=0 = (1/√2) Ψ r,K=0 Y IM 

• The quantum number r is the ‘signature’ , related to the 

invariance of the system when rotated 180° about an axis 

perpendicular to the symmetry axis (z): operator R(π) 

• A second rotation by 180° brings the system back to its 

original orientation. Hence: 

R 2 (π) Ψ r,IMK = r 2 Ψ r,IMK = Ψ r,IMK 

• The allowed values of r are: (-1) I 


Bands of Good Signature 

• For K = 0, we may classify rotational bands in terms of 

the signature quantum number 

• For r = +1, the allowed spins are: 

I = 0, 2, 4,… 

• For r = -1, the allowed spins are: 

I = 1, 3, 5,… 

• Hence for each signature we obtain a rotational band 

with the energy levels separated by 2ħ 


Rotational Bands with K ≠ 0 

• For K ≠ 0, the total nuclear wavefunction takes the 

antisymmetrised form in order to satisfy the rotation 

(reflection) symmetry: 

Ψ IMK = √[(2I+1)/16π 2 ] {Ψ K D IMK + (-1) I+K Ψ -K D IM-K } 

where Ψ -K corresponds to a projection of the spin –K and 

is obtained by the operation R(π) Ψ K 

• The consequence of R(π) invariance for K ≠ 0 is that the 

intrinsic states Ψ K and Ψ -K , with eigenvalues ±K of J z , are 

degenerate and constitute only a single sequence of 

states with spins: 

I = K, K+1, K+2,… 

i.e. states with alternating signature 


Particle-Rotor Coupling 

• For an axially symmetric deformed d rotor: 

H rot = (ħ 2 /2I 0 0) R 2 = (ħ 2 /2I 0 0) [I –J] 2 

= (ħ 2 /2I 0 ) [I.I + J.J -2I.J] 

where the I.J couples the degrees of freedom of the 

valence particles to the rotational motion and is 

analogous to the classical Coriolis and centrifugal forces 

• Now consider J to consist of a single particle 

(J →j) coupled to an even-even core 

H rot = (ħ 2 /2I 0 ) [(I 2 –I z2 ) + (j 2 –j z2 )–(I + j - + I - j + )] 

The final term couples intrinsic and rotational states 


Particle-Rotor Coupling Schemes 

• (a) shows the strong-coupling limit or deformation- 

aligned (DAL) coupling scheme 

• (b) shows the weak-coupling limit or rotation-aligned 

(RAL) coupling scheme 


Strong Coupling (DAL) 

• This limit is recognised when the level splitting of the 

deformed shell-model single-particle energies for 

different Ω values is large compared with the Coriolis 

perturbation, i.e. large deformation or small Coriolis 

matrix elements (low j, high Ω) 

• The angular momentum vector j precesses around the 

deformation axis and K is approximately a good quantum 

number 

• The energy spectrum is given by the set of levels: 

E rot = (ħ 2 /2I 0 0) ) [I(I+1) –K 2 ] 


Decoupling Limit (RAL) 

• For weakly deformed nuclei, or fast enough rotation, the 

Coriolis force may be so strong that the coupling of the 

valence nucleon to the deformed core is negligible 

• The Coriolis force tends to align the nucleonic angular 

momentum j with that of the rotational angular 

momentum R 

• In this limit, the rotation band has spins: 

I = j, j+2, j+4,… 

• The energies are: E rot = (ħ 2 /2I 0 0) (I - j x )(I – j x +1) 


K = ½ Bands in Odd-A Nuclei 

• The rotational energy of a K = ½ band is: 

E(I) = (ħ 2 /2I a(-1) I+½ 0 ) [I(I+1) + (I+½)] 

where a is the decoupling parameter 

• Bands can mix if ∆K = ±1 

• For K = ½ bands there is a 

diagonal matrix element of 

the form: 〈K=½|j + |K=-½〉 

where j + = j x + ij y which 

perturbs the energy 


High K (I z ) Bands 

• If we have many paired 

nucleons outside the closed 

shell in the ground state t 

then alignment with the x- 

axis becomes difficult 

because the valence 

nucleons lie closer to the 

z-axis axis, i.e. they have high 

Ω values 

K = I z = ∑j z = ∑Ω 

• The sum K of these 

projections onto the 

deformation (z) axis is now 

a good quantum number 


K Forbidden Transitions 

• It is difficult for rotational bands with high K values to 

decay to bands with smaller K since the nucleus has to 

change the orientation on of its angular momentum. 

• For example, the K π = 8 - band head in 178 Hf is isomeric 

with a lifetime of 4 s. This is much longer than the 

lifetimes of the rotational states built on it. 

• The K π = 8 - band head is formed by breaking a pair of 

protons and placing them in the ‘Nilsson configurations’: 

Ω [N n 3 Λ] = 7/2 [4 0 4] and 9/2 [5 1 4] 

• In this case: K = 7/2 + 9/2 = 8 and π = (-1) N(1) .(-1) N(2) = -1 


K Isomers in 178 Hf 

• A low lying state with 

spin I = 16 and K = 16 in 

178 Hf is isomeric with a 

half life of 31 years ! 

• It is yrast (lowest state 

for a given spin) and is 

‘trapped’ since it must 

change K by 4 units in its 

decay 


K Forbiddenness 

• Strictly, in the decay of a high-K h band-head, d K can only 

change by an amount up to the multipolarity λ of the 

transition 

• The ‘degree of K forbiddenness’ is: 

ν = |∆K| - λ 

• The ‘hindrance factor’ is: 

f = F W = T 1/2γ / T 

W 

1/2 

where T 1/2γ is the partial γ-ray half-life and T 

W 

1/2 is the 

theoretical Weisskopf estimate 

• The ‘reduced hindrance factor’ is: 

f ν = f 1/ν = [ T 1/2γ / T 

W 

1/2 ] 1/ν 


Hindrance Factors 

• The solid line shows 

the dependence of F W 

on ∆K for some E1 

transitions according 

to an empirical rule: 

log F W = 2{|∆K| - λ} 

= 2ν 

• i.e. F W values increase 

approximately by a 

factor of 100 per 

degree of K 

forbiddenness 


Wobbling Motion and Triaxiality 

• Wobbling is a fundamental mode due to triaxiality which 

occurs when the axis of collective rotation does not 

coincide with one of the principal axes 

• For a deformed rotor the Hamiltonian is: 

H rot = (ħ 2 /2I x ) I x 

2 

+ (ħ 2 /2I y ) I y2 + (ħ 2 /2I z ) I z 

2 

• For a well-deformed but triaxial nucleus with I x » I y ≠ I z 

the energy of the wobbling rotor is: 

E(I,n W ) = (ħ 2 /2I x ) I(I+1) + ħω W (I) (n W + ½) 

where n W is the number of wobbling phonons and ω W is 

the wobbling frequency 


Wobbling Frequency 

• The wobbling frequency is related to the rotational 

frequency as: 

with 

ω W = ω rot √[ (I x - I y)(I x - I z ) / (I y I z ) ] 

ω rot = ħ I / I x 

• Note for an axially symmetric prolate nucleus, I z goes to 

zero and ω W → ∞, i.e. there is no wobbling motion 

• A family of wobbling bands is expected for n W = 0, 1, 2,… 


Wobbling Motion 


Wobbling Bands in 165 Lu 

• A family of wobbling 

bands is expected to 

show very similar 

internal structure 

• TSD (Triaxial 

SuperDeformed) bands 

1, 2 and 3 in 165 Lu 

represent bands with 0, 

1 and 2 wobbling 

phonons, respectively 


Electromagnetic Properties 

• A characteristic signature 

of wobbling motion is the 

occurrence of ∆I = ±1 

interband transitions with 

unusually large B(E2) out 

values that compete with 

the strong ∆I = 2 inband 

transitions, B(E2) in 

• ∆n W = 2 transitions are 

forbidden 

• Measured multipole mixing ratios for the interband 

∆I = 1 transitions in 165 Lu show them to be ~90% E2 

and only ~10% M1 ! 


6. Nuclear Pairing 

• Pairing and superfluidity 

• Odd-even mass difference 

• Quasiparticles 

• Coriolis antipairing and backbending 


Experimental Evidence 

• The ground states of all even-even nuclei have I π = 0 + 

• The binding energies of odd-even nuclei are less than the 

mean value of the two neighbouring even-even nuclei 

• Doubly odd nuclei are even less bound 

• Nuclear moments of inertia are only 30-50% of the rigid- 

body value at low spin 


Time Reversed Orbits 

• The greatest overlap 

would occur if two 

particles could orbit 

in the same level 

• Not allowed (PEP) ! 

• The next greatest 

overlap occurs for 

particles in ‘time 

reversed’ orbits 

• The spins cancel to 

give I π = 0 + 


Coupling Two Particles 

• The short-range (pairing) residual interaction yields 

an energetically favoured 0 + state 


Scattering Between Orbits 

• Pairs of particles scatter from one 

orbit to another, induced by the 

pairing interaction 

• The particles change orbits in pairs 

so I π = 0 + 

• Since the orbits have 

different energies, the 

Fermi surface is smeared 

out over a region ±∆ 

(±1.5 MeV) 


Odd-Even Mass Difference 


Pairing Energies 

• The neutron separation energy is: 

S n = B(A,Z) –B(A-1,Z) = M(A-1,Z) –M(A,Z) + M n 

where B(A,Z) in the nuclear binding energy 

• The proton separation energy is: 

S p = B(A,Z) – B(A-1 1, Z-1) = M(A-1 1,Z-1) – M(A,Z) + M H 

• The pairing energies are: 

P n (A,Z) = S n (A,Z) – S n (A-1,Z) (neutron) 

P p (A,Z) = S p (A,Z) – S p (A-1,Z-1) 1Z 1) (proton) 


Pairing Hamiltonian 

• The Hamiltonian including a two-body monopole (i.e. I = 0) 

pairing interaction is: 

H = H sp + H pair = ∑є u [a u 

† 

a u + a ū 

† 

a ū ] - G∑a u1 

† 

a ū1 

† 

a ū2 a u2 

• Here a † and a are particle creation and annihilation 

operators 

• The first term is the sum of single-particle energies 

• The second term contains the pairing interaction that 

annihilates a pair of particles in time reversed orbits 

|u 2 〉 and |ū 2 〉 and simultaneously creates a pair in time 

reversed orbits |u 1 〉 and |ū 1 〉 


Chemical Potential λ 

• The energy increase of the condensate per particle 

added defines the chemical potential λ 

• The Hamiltonian is: H’ = H – λÑ = H sp + H pair – λÑ 

where Ñ is the particle number operator 

• The two-body monopole pairing interaction is: 

H pair = -¼G P † P 

where the pair creation and annihilation operators are: 

P † = ∑ a † u a † ū 

and P = ∑ a ū a u 


Pairing Strength G 

• The strength of the pairing term G is a positive constant 

• It is larger for high-j orbitals and depends on the spatial 

overlap of the two nucleons 

• The strength decreases with mass since in heavier nuclei 

the outer nucleons are further apart 

• The strength is also lower for protons than neutrons 

because of Coulomb repulsion 

• Approximately: 

G p = 17/A MeV and G n = 23/A MeV 


Pairing Gap ∆ 

• The pairing term contains the product of two creation 

and two annihilation operators 

• In order to simplify the calculations, the term P † P 

(product) is replaced by P † + P (sum) and: 

H pair = -½∆ [P † + P] 

which h introduces the pairing ii gap parameter ∆ 

• Particle number is now not conserved ! The chemical 

potential λ is now treated as a Lagrange multiplier l and is 

varied to produce the correct particle number: 

〈Ψ|Ñ|Ψ〉 Ñ = N λ = - ∂E/∂N 


Single Particle Levels with Pairing 

• An energy gap between 

the ground state and 

first excited state t of 

~ ∆ opens up 

• The excited states 

become bunched 

togetherth 

• A rough estimate of 

the energy required to 

create a particle-hole 

excitation ti is 2∆ 


Nuclear Ground State 

• Nuclei in their ground states are in specific 

configurations: some pairs of nucleons are above the 

Fermi surface (λ) and some states below the Fermi 

surface are empty 

• With pairing, states are not always full or always empty 

but filled for part of the time or empty for part of the 

time 

• The probability of a given level є u being occupied by a 

particle is: 

P u (є u ) = ½{ 1 + (є u – λ) / √[ (є u – λ) 2 + ∆ 2 ] } 

• Now P u (є u ) ≠ 0 or 1 around the Fermi surface ! 


Quasiparticles 

• A further simplification is to replace the pairwise 

interacting particles by a gas of noninteracting 

‘quasiparticles’, whose energies are then simply additive 

• A quasiparticle may be considered as a mixture of a 

particle and hole states 

• The Bogoliubov-Valatin l transformation ti changes the 

particle basis (a † ,a) into the quasiparticle basis (α † ,α): 

α u 

† 

= U u a u† + V u a ū 

; a u† = U u α u† -V u α ū 

α 

† ū = U u a ū† - V u a u ; a ū† = U u α ū† + V u α u 


The Quasiparticle Vacuum 

• The transformation coefficients U u and V u can be 

obtained following a BCS treatment (superconductivity) 

• The BCS wavefunction is of the form: 

|Ψ † † BCS 〉 = Π u [U u + V u a u† a ū† ] |0〉 

where |0〉〉 denotes the vacuum state of the particles 

and |Ψ BCS 〉 represents the quasiparticle vacuum 

• U u and V u represent occupation amplitudes (‘empty’ and 

‘filled’, respectively) and hence: 

|U 2 2 u | + |V u | = 1 


Quasiparticle Energies 

• Expressions for U u and V u are: 

U u = (1/√2) ){ 1 + (є + λ) / E 1/2 

u ) u } 

V u = (1/√2) { 1 + (є u – λ) / E u } 1/2 

• The quasiparticle energy of a state |u〉 relative to the 

ground state is: 

E u = √[ (є u – λ) 2 + ∆ 2 ] 

where є u is the single-particle energy. Note that the 

lowest excited state of a paired nucleus is at a higher 

energy than for the unpaired case 

• The pair gap parameter may be expressed: ∆ = G∑U u V u 


Destruction of Pairing 

• Strong external 

influences may destroy 

the superfluid nature 

of the nucleus 

• In the case of a superconductor, a strong magnetic field 

can destroy the superconductivity: i the ‘Meissner Effect’ 

• For the nucleus, the analogous role of the magnetic field 

is played by the Coriolis force, which at high spin, tends 

to decouple pairs from spin zero and thus destroy the 

superfluid pairing correlations 


Coriolis Antipairing Effects 

• Classically the Coriolis force is given by: 

F Cor = -2m [ω х v] 

• Coriolis Antipairing (CAP): the magnitude of ∆ gradually 

and smoothly decreases and the nuclear moment of 

inertia (∝ ω 2 ) increases 

• Rotational Alignment: At spin ~ 12ħ, the Coriolis force is 

strong enough to break a specific pair of valence 

nucleons and align their individual id angular momenta along 

the rotation axis 

• High-j low-Ω particles are the most susceptible 


Demise Of Pairing 

• CAP and rotational 

alignments diminish the 

magnitude of the 

nuclear pairing 

• Eventually the nucleus 

may enter an unpaired 

phase at high spin 

• In addition to static 

pairing, dynamic pairing 

correlations occur 

resulting from 

fluctuations in ∆ 


Backbending 

• The breaking of a specific nucleonic pair and the 

rotational alignment of the angular momenta leads to 

a characteristic ‘S’ shape of the nuclear Spin vs 

Frequency 

• A contribution of the nuclear spin now comes from single 

particles: 

I = R + J 

with ih J ≈ (j x,max + j x,max – 1) in accordance with the PEP 

• The nucleus ‘changes gear’, i.e. slows down while 

maintaining the angular momentum 


Backbending 

• The moment of inertia 

increases with increasing 

rotational frequency 

• Around spin 10ħ a 

dramatic rise occurs 

(rotational frequency) 2 

• The characteristic ti ‘S’ 

shape is called a backbend 

( 158 Er) 

• A more gradual increase is 

called an upbend ( 174 Hf) 


Backbending Movie 


Backbending Demonstration 

This movie shows Mark 

Riley’s “backbending 

machine” built here in 

Liverpool 

This movie shows 

backbending in the 1960s 


Band Crossings 

• Backbending can be 

interpreted as the 

crossing of two 

bands 

• The ‘G’ band (Ground 

state) is a fully 

paired configuration 

• The ‘S’ band (Super 

or Stockholm) 

contains one broken 

pair 


Quadrupole Pairing 

• Higher order pairing correlations may occur leading to 

configuration-dependent pairing which depends on the 

relative orientation of nuclei orbits in a deformed 

potential 

• The Y 21 quadrupole component has the largest effect 

on the moment of inertia. The nuclear shape still has 

Y 20 symmetry and hence the quadrupole pairing is of a 

dynamical nature 

• The generalised pair creation operator is: 

P † λ † λµ† = ∑ 〈u 1 |r Y λµ |u 2 〉 a u1† a 

† 

ū2 


Neutron-Proton Pairing 

• The concept of 

superconductivity, 

related to like nucleon 

pairs coupled to spin I = 0 

and isospin T = 1, can be 

extended to neutron- 

proton pairs with T = 0 

• The greatest overlap 

occurs if the particles 

are in the same orbitals 

• Strong neutron-proton 

pairing can occur for 

nuclei with N = Z 


Nucleon Pairing 

• The isovector (T=1) 

n-p pairing (c) is 

similar to the n-n (a) 

and p-p p( (b)pairing 

• The isoscalar (T=0) 

n-p pairing (d) is 

clearly different 


7. Cranked Shell Model 


Cranking 

• A rotation is externally imposed on a nucleus about the 

x axis. The Schrödinger equation is: 

iħ ∂Ψ lab /∂t = H lab Ψ lab 

• Ui Using the rotation operator 

R x = exp [-iI x ωt] 

with Ψ lab = R x Ψ int and H lab = R x H int R 

-1 

x , the SE within the 

intrinsic i i frame becomes: 

iħ ∂Ψ int /∂t = [H int – ħωI x ] Ψ int 


The Routhian 

• The cranked Hamiltonian or Routhian (just the energy in 

the rotating frame) is then: 

H ω = H int – ħωI x 

where ωI x is analogous to the classical Coriolis and 

centrifugal forces 

• In terms of single-particle states, the cranking 

Hamiltonian is: 

H ω = ∑ ω i h (i) = ∑ i [h int (i) – ħωj x (i)] 

where j x (i) are the components of the nucleonic angular 

momenta on the rotation axis (x) 


Single-Particle Routhians 

• The single-particle Routhian can be evaluated by solving 

the eigenvalue envalue equation: 

h ω |ν ω 〉 = e ω ω ν |ν 〉 

where |ν ω 〉 are the single-particle eigenfunctions in the 

rotating frame 

• The Routhian is simply the energy in the rotating frame 

of reference: e 

ω 

ν 


The Alignment 

• The alignment is just the expectation value of j x : 〈j x 〉 

and is equal to the (negative) differential of the 

Routhian with respect to rotational frequency, i.e. 

de νω /dω = -ħ 〈ν ω |j x |ν ω 〉 

• Those orbits with large j x values, and hence low Ω 

values, are most affected by the rotation, i.e. the 

Coriolis Corols and centrifugal centrfugal forces 


Symmetries of Rotating Nuclei 

• The time-reversal (twofold) degeneracy of the ±Ω 

states is lifted by the ωj x term 

j x 

• Axially symmetric potentials exhibit invariance with 

respect to rotations ti by 180° (π) about the three 

principal axes (reflection symmetry) 

• However, the cranking Hamiltonian is only invariant for 

rotation of π about the x axis 

R x (π) = exp(-iπI x ) 


Signature Quantum Number 

• A rotation of 2π leaves the wavefunction unchanged, 

except for a possible phase factor (±1), i.e. 

R x2 (π) Ψ = r 2 Ψ = (-1) A Ψ with r 2 = ±1 

• The eigenvalues r of the rotation operator R x (π), called 

signature, are good quantum numbers, i.e. . constants of 

the motion 

• The signature exponent quantum number α is defined: 

r = exp(-iπα) 


Signature Quantum Numbers 

• The signature is: 

r = ±1 (even A) α = 0, 1 

r = ±i (odd A) α = ±½ 

• The spins are restricted 

according to 

α = I mod 2 

Only good quantum 

numbers: π and α 

• Total signature: α tot = ∑α 

• Total parity: π tot = Ππ 


Signature Partners 

• Signature is a good 

quantum number at high 

spin 

• A splitting between the 

α = 0/1 (even A) or α = 

±½ (odd A) states gives 

rise to two distinct 

‘signature partner’ bands 

• For an orbital with angular momentum j the ‘favoured’ 

band has signature: α f = j mod 2 


CSM Calculations 

• Including pairing the CSM Hamiltonian is: 

H νω = H sp – ∆(P † + P) – λÑ – ωI x 

with H 

ω = ∑h ω , H sp = ∑h sp , I x = ∑j x and the total energy 

E = ∑e νω , i.e. a sum of the single-particle Routhians, 

e νω = 〈h νω 〉 

• The parameters λ (-∂E/∂N) and ω (-∂E/∂I) can be 

considered as Lagrange multipliers needed to constrain 

the particle number and angular momentum, respectively: 

〈Ñ〉 = N and 〈I = √[ I(I+1) –K 2 x 〉 [ ] 


Quasiparticle Diagrams 

• Here is an 

example of a 

Woods-Saxon 

quasiparticle 

diagram 

• Label (π, α) 

A (C) (+,+½) 

B (D) (+,-½) 

E (G) (-,-½) 

F (H) (-,+½) 


Comparison to Experiment 

• The results of cranking 

calculations provide 

Routhians rather than 

energies 

• The experimental data 

must be transformed 

into the intrinsic frame 

to afford detailed 

d 

comparisons 

• Quantities are 

approximated as 

quotients of finite 

differences 


Moments of Inertia 

• The frequency for γ ray 1 is: 

ω(I) = (1/ħ) dE(I)/dI x ≈ E γ1 /2ħ 

• Expressions for the moments of inertia are: 

I (1) (I) = ħ 2 { (2I-1) / (E(I+1) – E(I-1) } = ħ 2 (2I-1) /E γ1 

I (2) (I-1) = ħ 2 { 4 / (E γ1 – E γ2 ) = 4ħ 2 / ∆E γ 

• The dynamic moment of inertia corresponds to spin I-1 

and the associated frequency should be calculated l at I-1. 

In practice, the average frequency is used: 

ω(I-1) ( ≈ (E γ1 + E γ2 ) / 4ħ 


Experimental Routhian and 

Alignment 

• The experimental Routhian may be expressed as: 

E ω expt(I) = ½{ E(I+1) + E(I-1) } – ω(I) I x (I) 

but we are interested in obtaining a quasiparticle 

Routhian and so must subtract the collective rotational 

energy: 

e’(I) = E ω expt(I) - E ω ref(I) 

• Similarly, il l we remove the collective spin to produce the 

quasiparticle alignment: 

i x (I) = I x (I) – I x,ref (I) 


Choice of Reference 

• The reference removes the collective effects of rotation 

and leaves the energies and spins solely from the valence 

quasiparticles, in the rotating frame 

• The reference can be obtained from the ground state 

band (zero quasiparticle, vacuum) of a (neighbouring) 

even-even nucleus 

• At low spin, it is found that I ∝ ω 2 . Hence a ‘variable 

moment of inertia’ (VMI) reference can be fitted, 

introducing ‘Harris Parameters’ I 0 , I 1 : 

I (1) ref = I 0 + I 1 ω 2 


Harris Parameters 

• The Harris Parameters can be obtained by fitting: 

I x,ref (ω) = ω{I 0 + I 1 ω 2 } + i x 

to the reference band. Note that i x = 0 for the ground- 

state band of an even-even nucleus 

• The energy reference is then given by: 

E ω fdω 2 4 2 ref = -ħ∫I x,ref = -½ω I 0 - ¼ ω I 1 + ⅛ħ /I 0 

where the final term, an integration constant, ensures 

that the ground-state energy, E ω ref(I=0), is zero 


Expt. Alignments and Routhians 

• Experimental results 

for N = 74 isotones are 

shown here 

• A clear ‘backbend’ is 

seen for 132 Ce while the 

heavier nuclei show 

‘upbends’ 

• E and F correspond to 

proton h 11/2 orbitals, 

and these are the 

quasiparticles that align 


Band Crossings 

• Band crossings can be classified by the rotational 

frequency at which they occur ω c and the gain in 

alignment ∆i x at the crossing 

• Experimentally: These quantities can be obtained by 

plotting e’ vs. ω and i x vs. ω 

• Theoretically: Crossing frequencies can be obtained 

from CSM quasiparticle diagrams. 

• The gain is alignment is given by the slopes of the 

interacting levels, E and F: 

∆i x = -(1/ħ) { de’ E /dω + de’ F /dω } 


Signature Splitting 

• The signature splitting between the components of an 

orbital is the difference in excitation energy (or 

Routhian) at a fixed frequency, e.g. 

∆e’ FE (ω) = e’ F (ω) – e’ E (ω) 

• The magnitude of the signature splitting is related to 

the admixture of the Ω = ½ component in the 

wavefunction and is larger for low Ω values 

• The Coriolis interaction connects states with Ω = ±1 


Staggering Parameter 

• One way to enhance 

signature effects is to 

plot the 

staggering parameter S(I) 

S(I) = E(I) – E(I-1) - ½[ E(I+1) – E(I) + E(I-1) – E(I-2)] 


Signature Inversion 

• In some odd-odd nuclei 

at low spin the 

signatures are the 

‘wrong way round’ i.e. 

the ‘favoured’ signature 

is energetically 

unfavoured ! 

• At hih higher spin the 

signatures revert to 

their ‘expected’ 

ordering 

Doubly odd La systematics 

• This is still not fully 

understood 


8. Strutinsky Shell Correction 

• Shell correction energy 

• Cranked Nilsson Strutinsky Model 

• Total Routhian Surfaces 


Shell Effects 

• A nuclear ‘property’ (e.g. 

binding energy) usually 

shows an irregular 

behaviour with mass A 

(or Z or N). It is made 

up of an oscillatory part 

∆E on top of a smooth 

part E smooth 

• Strutinsky’s s idea was to use the shell model to obtain 

∆E as the local variation from the average smoothed 

(shell model) value, but then to use the Liquid Drop 

Model to calculate l the real ‘smooth’ behaviour E smooth 


Strutinsky Shell Correction 

• To obtain both the global 

(liquid drop) and local 

(shell model) variations 

with δ, Z and A, Strutinsky 

developed a method to 

combine the best 

properties of both models 

(a) Liquid drop: g F (e) = g AV (e) 

(b) and (c) show shell effects. 

A change in nuclear binding 

arises from: g AV (e) – g F (e) 

• He considered the 

behaviour of the level 

density g(e) in the two 

models and calculated the 

‘fluctuation’ energy 


Level Density 


Strutinsky Procedure 

• The nuclear (binding) energy is considered to have to 

have an oscillatory ypart ∆E shell, caused by yquantal 

effects (shell model), superposed upon a smoothly 

varying liquid drop part E LD : 

E = E LD + ∆E shell 

• Strutinsky proposed that only ∆E shell (‘microscopic’) 

should be calculated within the framework of the shell 

model, while the smoothly varying part E LD 

(‘macroscopic’) should be taken from the Liquid Drop 

Model 


Strutinsky Procedure (cont) 

• Similarly, the total shell-model energy E SH does not 

vary smoothly and is composed of oscillatory and 

smooth parts: 

E SH =∑ 1A є i = Ĕ SH + ∆E SH 

• The real and smoothed level densities can be defined 

by g(є) and ğ(є), respectively 

• The number of levels between є and є + dє is given by 

g(є) dє and the level density is: 

g(є) = ∑ i δ(є – є i ) 

where δ(є – є i ) is the Dirac delta dl function 



• The particle number can be evaluated as: 

A = ∫ λ g(є) dє 

• The total shell-model energy and the smoothed part are 

then given, respectively, as: 

E SH = ∫ λ є g(є) dє 

and Ĕ SH = ∫ λ’ єğ(є) dє 

• Note that λ≠ λ’ ’ because of a smearing of the Fermi 

surface when calculating Ĕ SH 



• The total energy of the nucleus may finally be written 

as: 

E = E LD + ∆E SH = E LD + [E SH – Ĕ SH ] 

where E LD is the macroscopic contribution and [E SH – Ĕ SH ] 

is the microscopic shell correction 

• Note that the shell correction can be positive or negative 

• Negative values give increased binding and stability 


Shell Correction Energies 


Superheavy Island 

• Shell effects, 

particularly 

hexadecapole 

deformation, can 

stabilise very heavy 

nuclei 

• Such superheavy nuclei 

only exist because of 

subtle quantum 

mechanical effects 

leading to a localised 

region (‘island’) of 

increased stability 


Strutinsky Method for Spin 

• The Strutinsky technique can be extended to include 

rotation 

• We introduce another ‘level density’: 

g 2 (є) = ∑ i 〈j x 〉 i δ(є – є i ) 

• The total t single-particle l energy is obtained from the 

cranking Hamiltonian as: 

where I = ∫ λ g 2 (є) dє 

E SP (I) = ∫ λ є g(є) dє + ħω I 


Strutinsky Method for Spin 

• The smoothed energy is: 

Ĕ(є) = ∫ λ’ єğ(є) dє + ħω ğ 2 (є) dє 

• The cranked Nilsson Strutinsky method includes 

deformation (ε 2 , ε 4 , γ) and spin 


Cranked Nilsson Strutinsky 

• The total energy is: 

E (ε 2 ,ε 4 ,γ, I) = E LD (ε 2 ,ε 4 ,γ, I) + ∆E SH (ε 2 ,ε 4 ,γ, I) 

• The macroscopic energy contribution is can be calculated 

from: 

E LD = E surf + E Coul + (9ħI) 2 /2I rig 

• This method usually ignores pairing correlations and is 

hence only valid for high-spin states (I > 20 ħ) 


Total Routhian Surfaces 

• This method is based on the Woods-Saxon potential and 

includes pairing. The total energy of a nucleus (Z, N) as a 

function of deformation β’ = (β 2 , β 4 , γ) is: 

E(ω,Z,N,β’) = E macro (ω,Z,N,β’) + ∆E shell (ω,Z,N,β’) 

• The total Routhian is: 

+ ∆E pair (ω,Z,N,β’) 

E(ω,Z,N,β’) = E(ω=0,Z,N,β’) 

+ [〈Ψ ω |H ω (Z,N,β’)|Ψ ω 〉 - 〈Ψ|H ω=0 (Z,N,β’)|Ψ〉] 

-½ω 2 [I macro (A,β’) - I Strut (N,Z,β’)] 


Total Routhian Surfaces (cont) 

• The term E(ω=0,Z,N,β’) corresponds to the liquid-drop 

energy, the single-particle shell correction energy, and 

the pairing in energy at zero rotational ti frequency 

• ‘[〈Ψ ω |H ω (Z,N,β β’)|Ψ ω 〉 - 〈Ψ|H ω=0 (Z,N,β β’)|Ψ〉]’ is the 

change in energy due to rotation 

• The term ½ω 2 [I macro (A,β’) - I Strut (N,Z,β’)] represents a 

renormalisation of the LDM energy which is required 

due to unrealistically large proton and neutron radii 

used in some parameterisations of the Woods-Saxon 

potential 


TRS Maps 


9. Broken Symmetries 

• Reflection Asymmetry: Octupole Bands 

• Handedness: Chiral Bands 

• Magnetic Rotation: ti Shears Bands 

• Transitional Nuclei: Critical Points 


Reflection Asymmetry 

• If a nucleus is ‘reflection asymmetric’ (i.e. the odd 

multipole deformation parameters are non-zero, e.g. 

β 3 ≠ 0 is the most important) then the nuclear 

wavefunction in its intrinsic frame is not an eigenvalue 

of the parity operator: 

Ψ 2 (x ,y ,z) ≠ Ψ 2 (-x, -y, -z) 

• If β 3 ≠ 0 for a nucleus it is said to possess octupole 

deformation 

• The deformation can however be static, 〈β 3 〉 ≠ 0, or 

dynamic, 〈β 3 〉 = 0 (oscillating octopule shape) 


Octupole Band Structures 


Octupole Vibrations in 238 U 

• This nucleus 

shows three 

octupole 

vibrational 

bands with 

dff different K 

values 


Parity Splitting 

• For a static octupole shape, the negative parity states 

are interleaved (midway between) with the positive 

parity states 

• A measure of such a feature is the ‘parity splitting’, 

defined as: 

δE = E(I) - -½[ E(I+1) + + E(I-1) + ] 

• This quantity generally decreases towards zero with 

increasing spin and suggests that rotation may stabilise 

the octupole shape 

• A similar quantity is the difference in alignment: 

∆i x = i x- - i 

+ 

x 


Octupole Vibration or Deformed? 

• For an octupole 

vibrational 

phonon coupled 

to the positiveparity 

states: 

∆i x = 3 ħ 

• For a static 

octupole 

deformation: 

∆i x = 0 

x 


Reflection (A)symmetry 

1 band 2 bands 2 bands 4 bands 


Electric Dipole Moment 

• In a nucleus with octupole 

deformation, the centre of 

mass and centre of charge 

tend to separate, creating a 

non-zero electric dipole 

moment 

• Bands of opposite parity 

connected by strong E1 

transitions occur 


Enhanced E1 Transitions 

• In heavy nuclei, E1 strengths typically lie between 

10 -4 and 10 -7 Wu 

• In nuclei with octupole deformation, the E1 strengths 

can be much higher: 10 -3 –10 -2 Wu 

• The intrinsic dipole moment of an octupole deformed 

nucleus is: 

D 0 = C LD A Z e β 2 β 3 

with the liquid drop constant C LD = 0.0007 fm 

• In a Strutinsky type approach, macroscopic and 

microscopic effects can be considered and: 

D = D macro + D shell 


Experimental Dipole Moments 

• Experimental values of D 0 can be obtained by measuring 

B(E1)/B(E2) ratios, related simply to γ-ray energies and 

intensities 

• The B(E1) reduced transition rate is: 

B(E1;I→I-1) = (3/4π) e 2 D 0 

2 

|〈 I i K i 1 0 | I f K f 〉| 2 

• The B(E2) reduced d transition rate is: 

B(E2;I→I-2) = (5/16π) e 2 Q 

2 

0 |〈 I i K i 2 0 | I f K f 〉| 2 

• Hence if Q 0 is known (e.g. from the quadrupole 

deformation β 2 ) then a value for D 0 can be extracted, i.e: 

D 0 = √[5B(E1)/16B(E2)] Q 0 


Simplex Quantum Number 

• The only symmetries for a rotating reflection 

symmetric nucleus are parity p and signature r 

• For a reflection asymmetric shape (e.g. octupole) these 

are no longer good quantum numbers but the nucleus is 

invariant with respect to a combination of rotation of 

180° about the x axis (R(π)) and change of parity (P) 

• The ‘simplex’ operator is defined as: 

S = P R(π) -1 

with eigenvalues: s = -pr = ±i , ±1 

(p = s exp[iπI]) 


Parity Doublets 

• For K ≠ 0, four ∆I = 2 (E2) bands are formed based on 

states with K ± and (K+1) ± 

• The simplex quantum number can be used to classify 

these structures 

• For an even-even nucleus: 

s = +1 describes states (0 + ), 1 - , 2 + , 3 - , 4 + … 

s = -1 describes states (0 - ), 1 + , 2 - , 3 + , 4 - … 

• For an odd-A nucleus: 

s = +i describes states 1/2 + , 3/2 - , 5/2 + , 7/2 - ,… 

s = -i describes states 1/2 - , 3/2 + , 5/2 - , 7/2 + ,… 


Parity Doublets in 223 Th 

• The nucleus 223 Th shows 

parity doublets 

• The two ∆I = 2 bands, shown 

to the left, are connected by 

strong E1 transitions and have 

simplex s = -i 

• The two ∆I = 2 bands, to the 

right, have simplex s = +i 

s = -i s = +i 

• M1 transitions also connect 

some of the bands 


Octupole Magic Numbers 

• Octupole correlations 

occur between orbitals 

which differ in both 

orbital (l) and total (j) 

angular momenta by 3 

• Magic numbers occur at 

34, 56, 88 and 134 

• Nuclei with both proton 

and neutron numbers 

close to these are the 

best candidates to show 

octupole effects 


Rotational Invariance 

• From Kris Starosta (Michigan State University) 


Space Inversion Invariance 

• From Kris Starosta (Michigan State University) 


Chirality (Handedness) 

• ‘I call any geometric figure, or 

group of points, chiral, and say 

it has chirality, if its image in a 

plane mirror, ideally realised, 

cannot be brought to coincide 

with itself’ Lord Kelvin 1904 

• Examples of chiral systems are 

found throughout h nature and in 

several disciplines of science 

• Axial vectors of angular momenta 

systems of opposite chirality are 

related by time reversal 


Chiral Geometry 

• Spontaneous n chiral symmetry breaking can occur in 

triaxial doubly odd nuclei when there are three mutually 

perpendicular p spin vectors of differing lengths that can 

form a left-handed or right-handed configuration 


Odd-Odd Mass 130 Nuclei 

• Region of triaxial shapes (x ≠ y ≠ z) 

• Consider the πhh 11/2 νhh 11/2 configuration 

n 

1. The proton Fermi surface lies at the bottom of the 

h 11/2 subshell: the proton single-particle j aligns along 

the short axis 

2. The neutron Fermi surface lies at the top of the h 11/2 

subshell: the neutron single-particle j aligns along the 

long axis 

3. The irrotational moment of inertia is largest for γ = 

30°: the core angular momentum aligns along the 

intermediate t axis 


Irrotational Moments of Inertia 


the variation of the 

moments of inertia I k 

as a function of the 

triaxiality parameter γ 

• For a prolate nuclear 

shape (γ = 0°), I 1 = I 2 

and I 3 = 0 

• For γ = 30° , I 2 reaches 

a maximum and this 

represents the ‘most 

collective’ shape 


Chiral Operator 

• The chiral operator is a combination of time reversal and 

rotation by 180°: Ô = TR y (π) 

• The left-handed and right-handed systems are related 

to each other by this operator: 

|L〉 = Ô|R〉 and |R〉 = Ô|L〉 

• For a prolate nucleus, chiral symmetry is good: |R〉 = |L〉 

• However, for the triaxial odd-odd case: |R〉 ≠ |L〉 


Restoration of Chiral Symmetry 

• Note that |R〉 and |L〉 are not solutions of the nuclear 

Hamiltonian in the lab frame and chiral symmetry must 

be restored by forming wavefunctions ns of the form 

(similar to the octupole case): 

|+〉 = (1/√2) [|R〉 + |L〉] 

|-〉 = (i/√2) [|R〉 -|L〉] 

• This leads to the doubling of the states and the 

occurrence of two (near) degenerate ∆I = 1 bands of 

the same parity 


Chiral Twin Bands 

Two near degenerate ∆I = 1 bands of the same parity arise 

(cf octupole bands: two ∆I = 1 bands of opposite parity) 


Cranking Symmetries 

• If the nuclear spin I lies 

along one of the principal 

axes, one ∆I = 2 band arises 

• If the spin lies in the plane 

defined by two principal 

axes, one ∆I = 1 band arises 

• If the spin moves out of 

these planes, two degenerate 

∆I = 1 bands occur (chiral 

twins) 


Magnetic Rotation 

• In spherical lead nuclei, 

regular bands of intense M1 

transitions have been found 

• The valence proton and 

neutron orbitals lie 

perpendicular to each other 

and produce a magnetic 

moment vector that breaks 

the spherical symmetry of 

the system and allows 

‘magnetic’ rotation 


Shears Mechanism 

• In magnetic rotation, higher 

angular momentum is generated 

by the reorientation of the 

neutron and proton spin 

vectors 

• Originally perpendicular, the 

vectors close like the blades 

of a pair of shears to generate 

the higher angular momentum 

states 

• The B(M1) strength decreases 

with increasing spin as µ ⊥ 

decreases 


Shears Systematics 


Antimagnetic Rotation 

• Expected in weakly deformed nuclei 

• In 106 Cd the spin is generated by 

closing the πg 

-1 

9/2 vectors ( j 

-1 

π bottom diagram ) 

• Each πgg 9/2 hole combines with one 

νh 11/2 particle forming a pair of backto-back 

shears 

• Note that the magnetic moment for 

this situation is zero, i.e. µ ⊥ = 0 


Antimagnetic Rotation in 106 Cd 

• The yrast band appears to stop at 26 + with a measured 

drop in B(E2) values, or collectivity it (cf band termination) 

ti 2/16/2009 Postgraduate Nuclear Models Course, Liverpool : E.S. Paul 227

Transitional Nuclei 


Interacting Boson Model 

• Bosons are constructed from fermion pairs 

• Nuclear collective excitations are described in terms 

of N interacting s (l = 0) and d (l = 2) bosons 

• Algebraic model based on U(6) group 

• Limits: 

• SU(3) rotational 

• U(5) vibrational 

• O(6) gamma-unstable 


Critical Point Symmetries 

gamma soft 

vibrator 

rotor 

• The Casten Triangle 


10. Band Termination 

• Favoured and unfavoured termination 

• Abrupt and smooth termination 


Introduction: Band Termination 

• A deformed prolate nucleus can increase its angular 

momentum by collective rotation about an axis 

perpendicular to its symmetry axis 

• However, the nucleus is a many-body quantal system and 

such collective behaviour must have an underlying 

microscopic basis 

• There is a limiting angular momentum that a given 

configuration can generate 

• Successive alignments occur until all the valence particles 

are aligned and move in equatorial orbits giving the 

nucleus an oblate appearence 


Band Termination: 158 Er 

neutron 

backbend 

proton 

backbend 

No more ( rays 

Gamma Ray Energy 

• A band ‘terminates’ when all valence particles outside a 

doubly magic (spherical) core are aligned 


Band Termination in 158 Er 

• When the valence np protons and nn neutrons align, the 

total spin is: I = ∑ 

np 

i j i (p) + ∑ 

nn 

i j i (n) 

and the rotational band is said to ‘terminate’ 

• At termination ti 158 Er can be considered d as a spherical 

146 

Gd core plus 4 protons and 8 neutrons, generating 

a maximum spin 46ħ 

• The configuration is: π(h 11/2 ) 4 ⊗ ν(i 13/2 ) 2 (h 9/2 ) 3 (f 7/2 ) 3 

• The terminating spin value of I max = 46 is generated as: 

(11/2+9/2+7/2+5/2) / / / + (13/2+11/2) / + (9/2+7/2+5/2) / / + (7/2+5/2+3/2) 

/ / 2/16/2009 Postgraduate Nuclear Models Course, Liverpool : E.S. Paul 234

Favoured Oblate States 

• Full termination represents the maximum alignment of 

all the valence particles outside a doubly magic core, 

consistent with the Pauli Exclusion Principle 

• Certain noncollective states representing maximal 

m 

alignment of a subset of the valence particles may be 

yrast leading to the observation of (energetically) 

favoured noncollective oblate states at a certain spin 

• Example: In 

157,158 Er energetically favoured states are 

seen at (I max –6),corresponding to two f 7/2 neutrons 

still being paired, i.e. they contribute 0 spin rather than 

7/2 + 5/2 = 6 


Noncollective Oblate States in 121 I 

• Low-lying states are seen at 

I = 39/2 and 55/2 

• 121 I can be considered as a 

core ( 114 Sn) plus 3 valence 

protons and 4 valence 

neutrons 

• The configuration is 

π {h 11/2 g 7/22 } ν {h 11/24 } 

with maximum spin 55/2 

• If two of the protons remain 

paired we get the 39/2 state 


Rigid Rotor Plot 

• The favoured nature of 

noncollective oblate 

states can be seen by 

plotting energy levels 

against spin 

• A rotating liquid-drop 

energy reference is 

subtracted: 

E LD = (ħ 2 /2I rig ) I(I+1) 

I ie 007 rig scaled to 158 Er, i.e. (ħ 2 /2I rig ) = 0.007 {158/A} 5/3 MeV 


Shape Coexistence 

• Noncollective oblate 

states may coexist 

with collective 

rotational structures 

• In 119 I, collective 

structures are seen 

to the left and oblate 

states to the right 


Smooth Band Termination 

• A novel type of ‘smooth’ band termination has been 

observed in several nuclei of the mass A = 110 region 

• Bands extend to very high energy (frequency) and the 

spacings between the γ rays increase (moment of 

inertia decreases) 


Drift Through the γ Plane 

• Smooth termination has 

been interpreted in the 

framework of the 

Cranked Nilsson 

Strutinsky method 

• It represents a gradual 

shape change from 

collective prolate (γ = 0°) 

to noncollective oblate 

(γ = 60°) over a wide spin 

range 


Abrupt and Smooth Termination 

• The contrasting high- 

spin behaviour of 

117 Xe 

and 122 Xe is shown here 

• The rotational nature of 

122 

Xe abruptly breaks 

down above I = 22 ħ 

• The behaviour of 117 Xe 

appears more smooth 


Termination Modes 

• Abrupt or 

favoured 

termination is 

shown at the 

top 

• Smooth or 

unfavoured 

termination is 

shown at the 

bottom 


Termination Systematics 


Band Termination in 152 Gd 

David Campbell 

Florida State 

University 


Beyond Termination 

• At termination, several valence particles are aligned with 

the ‘rotation’ axis outside an ‘inert’ closed core (‘doubly 

magic’ spherical core) 

• How do we generate higher spin states? 

• We must break the core and form energetically 

expensive particle-hole excitations across the magic 

shell gaps of the core 

• A classic (state-of-the-art) case is the nucleus 157 Er 

which was studied with the Gammasphere spectrometer 

in Berkeley 


157 

Er at High Spin 

• New high-energy h (15 (1.5-2.5 25 MeV), high-spin h transitions 

have been identified above I π = 87/2 - , 89/2 - and 93/2 + 

(new) terminating states 


157 

Er Spectrum 

Several (weak) transitions are seen in the energy range 

1.0 – 2.5 MeV. Measured ∆I=2 transitions are labelled as 

Q (quadrupole) and ∆I=1 transitions as D (dipole) 


157 

Er Rigid-Rotor Plot 

• Favoured oblate 

states are shown in 

gold 

• It costs a lot of 

energy to generate 

states of higher 

spin 

• The Z=64 proton 

core has to be 

broken to generate 

the highest spins 


157 

Er Termination States 

• Relative to the 146 Gd 

core (Z=64, N=82), the 

[π{h 11/2 } 4 ] 

+ 

16 

proton configuration is 

coupled to the neutron 

configurations (left) to 

produce energetically 

favoured noncollective 

oblate states 


New Bands in 157,158 Er 

γ 5 spectra 


Return of Collectivity in 158 Er 


Triaxial SD Band in 158 Er 


The End

Nuclear Models - Nuclear Physics

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