1. Art and Diseño

PHYS490: Nuclear Physics
1/29/2015
PHYS490 : Advanced Nuclear Physics : E.S. Paul
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PHYS490: Schedule 2015
 Lectures in Brodie Tower (BROD-405):
Wednesday 13:00 – 14:00
Thursday
11:00 – 13:00
 Lectures: weeks 1 - 6
 Tutorials: weeks 3, 6
 Eddie Paul
([email protected])
Room 411
Oliver Lodge Lab.
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PHYS490: Nuclear Physics
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Nucleon-Nucleon Force
Nuclear Behaviour
Forms of Mean Potential
Nuclear Deformation
Hybrid Models
Nuclear Excitations
Rotating Systems
Nuclei at Extremes of Spin
Nuclei at Extremes of Isospin
Mesoscopic Systems
Nuclear Reactions
Nuclear Astrophysics
Neutrinoless Double Beta Decay
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0. A Brief Introduction
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Prehistory (400 BC)
 This chart of Plato
and Aristotle shows
the relation of the
four elements and
their four qualities
 A fifth element was
ether or material of
the heavens (dark
matter in early
cosmology !)
 The chart was used
for over 1000 years
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Atomic and Nuclear Sizes
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Limits of Stable Nuclei
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More and More Isotopes
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Discovery History
 Today around 3000
isotopes have been
observed
 Only 284 are stable
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The Unique Nucleus
 The nucleus is a unique ensemble of strongly
interacting fermions: nucleons
 Its large, yet finite, number of constituents controls
the physics
 Both single-particle (out-of-phase) and collective
(in-phase) effects occur
 Analogy to a herd of wild animals. Individual animals
may break out of the herd but are rapidly drawn back
to the safety of the collective
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Nuclear Models
 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
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Nuclear Physics in the Thirties:
Splitting of the Atom
Cockcroft-Wilson accelerator – atom ‘split’ in 1932
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Nuclear Physics in the Forties
The first cyclotrons were built in Berkeley, California
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Oliver Lodge Lab. Opening (1969)
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Nuclear Physics in the Seventies
 An Open University program from 1979, shot in the
Liverpool Physics Department, showing the forefront
of nuclear structure experimentation (and fashion) at
the time!
 Also on youtube: http://youtu.be/s43rxUA8euY
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Nuclear Physics in the Eighties
TESSA3: 16 (small) γ-ray detectors at Daresbury, UK
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Even Bigger Arrays
 This picture shows
ESSA30, an array
of 30 (small) γ-ray
detectors at
Daresbury, UK
 It was a European
collaboration
 Again, spot the
Liverpudlians !
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Nuclear Physics in the Nineties
Gammasphere: 100 (big) γ-ray detectors, USA
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Nuclear Physics in the Noughties
2003: The Hulk destroys Gammasphere !
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Nuclear Physics Tomorrow
The next generation of Radioactive Ion Beam (RIB)
accelerators in Europe
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Nuclear Physics in context
 Nuclear Physics is ‘the study of
the structure, properties, and
interactions of the atomic nuclei’
 Nuclear Physicists investigate
nuclear matter on all scales, from
sub-atomic particles to supernovae
 Research areas include the
structure of the nucleus at
different temperatures and
pressures, the origin of elements,
and the structure and evolution
of stars
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1. Nucleon-Nucleon Force
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The Nucleon – a Spin ½ Fermion
 The nucleon is a hadron, i.e. it feels the strong force
The Ford Nucleon (1957)
nuclear powered car
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The Nucleon – a spin ½ Fermion
 It consists basically of 3 quarks but gluons (force
mediators) must also be considered
 Only 2% of the mass (Higgs mechanism) comes from
quark masses. The other 98% arises from the kinetic
energy of the constituents
 Only 30% of the intrinsic spin can be accounted for
from the constituent quarks
proton
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quark sea + 3 valence quarks
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Building Blocks and
Energy Scales
 Depending on energy and
length scales, different
constituents may be
considered as the building
blocks of the atomic nucleus
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Levels of Reality
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Fundamental Particles & Forces
 Quarks:
Down (d)
Up (u)
Strange (s)
Charmed (c)
Bottom (b)
Top (t)
 Force Mediators:
Photon (γ)
Gluon (g)
Z particle
W particle
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The Strong Force
 The strong force is fundamentally an interaction
between quarks
 It is really a residual colour force mediated by the
exchange of gluons
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Properties of the N-N Force
The force is spin dependent
The force is charge symmetric
The force is (nearly) charge independent
The force has a non-central component
The force depends on the relative velocity or
momentum of the nucleons
 The force has a repulsive core





 ‘Exchange model’: force mediated by pion exchange
 See Phys490_latex.pdf for more details
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One Pion Exchange
 The origin of the nuclear force arises at the fundamental
level from the exchange of gluons between the
constituent quarks of the nucleons
 At low energies (<1 GeV/nucleon; >1 fm) the interaction
can be regarded as being mediated by the exchange of
π mesons – pions
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Spin σ and Isospin τ
 Matrix mechanics was formulated by Born, Heisenberg
and Jordan (1925)
 Nucleon intrinsic spin takes only two values: up and down
Introduction of Pauli 2x2 spin matrices
 Same formalism used to describe nucleon: isospin up
(neutron), isospin down (proton)
Introduction of Pauli 2x2 isospin matrices
 Nucleon-nucleon force dependent on both spin and
isospin
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One-Pion Exchange Potential
 At large distances the potential is constructed as arising
from the exchange of one pion: OPEP
 The form of the potential is:
VOPEP = gs2 (1/3 σA.σB + SAB [1/3 + 1/μr + 1/(μr)2])
x τA.τB 1/r μ2e-μr
where:
μ = mπc/ħ
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and:
SAB = 3(σA.r)( σB.r)/r2 - σA.σB
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Addition of (Iso)Spins
 Spin σ and isospin τ are vectors
 Cosine rule gives:
(σA + σB )2 = σA2 + σB2 + 2 σA.σB
 Parallel spins (triplet state):
σA.σB = 1
 Antiparallel spins (singlet state): σA.σB = -3
 We need to know σA.σB in the description of the
nucleon-nucleon (N-N) force
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Quark Meson Coupling Model
 The Quark Meson Coupling (QMC)
Model of the nucleus takes into
account both the fundamental
interactions among quarks within
the neutrons and protons, and also
the interactions between the
neutrons and protons (meson
exchange between pairs of quarks)
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Calculations for Light Nuclei
 In addition to two-body N-N interactions, three-body
N-N-N interactions must also be included in the
theoretical description of light nuclei
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Repulsive Core (Pauli Principle)
 Radius of nucleon:
~ 1 fm
 Radius of hard core:
~ 0.2 fm
 Nucleon mean free
path:
~ 7 fm
 Volume of hard
cores is only ~ 2%
of nuclear volume
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The Deuteron
 The deuteron consists
of a bound protonneutron system
 Its ground-state is the
only state which is
bound; the first excited
state is unbound
 The ground state has
spin and parity Iπ = 1+
 The deuteron is not a
spherical nucleus
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Range of the Nuclear Force
 The range of an interaction is related to the mass of the
exchanged particle
 The Heisenberg Uncertainty Principle gives: ΔE Δt ≈ ħ
 A particle can only create another particle of mass m for
a time t ≈ ħ/mc2 during which interval the particle can
travel at most ct
 Taking ct as an estimate of the range R gives: R ≈ ħ / mc
 This yields R ≈ 1.4 fm for pion exchange
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Deuteron Wavefunction
 The maximum of the
wavefunction is only just
inside the potential well
with a considerable
exponential tail outside
 The RMS separation
between the neutron and
proton is 4.2 fm, larger
than the range of the
nuclear force (~ 1.4 fm)
 The deuteron is loosely
bound ! The binding energy
is only B/A ~ 1 MeV/A
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Hypernuclei
Nuclei including excited nucleons
including heavy quarks:
e.g. Lambda particle Λ
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2. Nuclear Behaviour
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Mirror Nuclei
 The force between
two nucleons has the
property of charge
symmetry and charge
independence
 The two nuclei 20Na
and 20F are examples
of mirror nuclei
 The numbers of
protons and neutrons
are exchanged
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Isospin Substates
 By analogy with spin, an isospin T state has (2T+1)
substates
 The substates correspond to states in different nuclei
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Isobaric Analogue States
 Isodoublet states occur in odd-A nuclei
 Isotriplet states occur in even-A (even-even
and odd-odd) nuclei
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A=51 Mirror Nuclei
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Mirror Nuclei: f7/2 Shell
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Independent Particle Model
Vij
r = ri - rj
Energy as a function
of separation
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 In principle, if the form of
the nucleon-nucleon
potential is known for bare
nucleons, then the energy
of a nucleon moving inside
a nucleus can be calculated
 This is a very difficult
problem to solve as the
nucleon interacts
simultaneously with all the
other nucleons
 Use an average potential
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Independent Particle Model
 The Hamiltonian is of the form:
H = Σ (Ti + Vij)
 It has 3A degrees of freedom and is too complicated
to solve except for the lightest nuclei (A < 12)
 Instead we use an average “mean-field” potential:
H = Hmean field + Hresidual
where Hresidual contains interactions between nucleons
that are not accounted for by the average potential,
especially interactions among valence nucleons
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Nuclear Mean Free Path
 Why is it that the Independent Particle picture of
nuclear motion works ?
 The Pauli Exclusion Principle (PEP) gives nucleons
essentially infinite mean free path
 However, if the range of the nuclear force was 2 to 3
times stronger, then nuclei could have been
crystalline
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Particles in a (Potential) Box
Energy levels up to the
‘Fermi level’
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 The short range
interaction between
nucleons means that each
nucleon moves in an
average potential
 The average separation
(~ 2.4 fm) is larger than
the range of the nuclear
force (1.4 ~ fm)
 Nuclei cannot easily
change state unless close
to the Fermi surface
(PEP)
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Degenerate Fermi Gas Model
 This is a simple model in which nucleons are placed in a
volume V = 4πR3/3 and the interactions between them
are ignored
 A Fermi sea is formed, filled up to the energy
corresponding to the Fermi momentum:
EF = pF2/2m = ħ2kF2/2m
 The binding energy per nucleon is:
B = -E/A = -3/5 TF + 1/2 V0
where TF is the kinetic energy at the Fermi surface
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Nucleon Effective Mass
 The nuclear force has the property of saturation so that
B(A,Z) is independent of A caused by the Pauli Exclusion
Principle (PEP), its spin and isospin dependence, and
(less importantly) the repulsive core
 The nuclear separation energy S is the difference
between the energy of a nucleon outside the nucleus and
the energy of the Fermi level EF: S = B = -1/5 TF
 Wrong ! (S > 0) – the nucleon has an effective mass
(m* > mn) when moving in a nucleus
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Some Nuclear Quantities
 Number density (A/V) measured:
ρ ~ 0.17 fm-3 (~ 1.5 x 1018 kg/m3)
 Fermi momentum:
kF = pF/ħ ~ 1.4 fm-1
 Fermi energy:
EF ~ 10 MeV
 Kinetic energy of a nucleon in the nucleus:
3/5EF ~ 6 MeV
corresponding to a velocity v/c ~ 0.14
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Nuclear Potentials
There are two approaches:
1.
An empirical form of the potential is assumed, e.g.
square well, harmonic oscillator, Woods-Saxon
2. The mean field is generated self-consistently from
the nucleon-nucleon interaction
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3. Forms of Mean Potential
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Shell Model – Mean Field
A nucleon in the
Mean Field of
N-1 nucleons
N nucleons in
a nucleus
 Assumption – ignore detailed two-body interactions
 Each particle moves in a state independent of the other
particles
 The Mean Field is the average smoothed-out interaction
with all the other particles
 An individual nucleon only experiences a central force
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Square Well Potential
 Simplest form of potential
 Since we have a spherically
symmetric potential we can
separate the solutions into
angular and radial parts
Infinite square
well potential
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 Radial solutions are Bessel
functions which satisfy the
boundary condition Rnℓ(R) = 0
 The eigenenergies are labelled
by n and ℓ:
Enℓ = (ħ2/2mR2)ξnℓ2 - U0
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Square Well Quantum Numbers
 ‘n’ is the principal quantum number
(number of nodes in wavefunction)
 ‘ℓ’ is the orbital angular momentum
( j = ℓ±½ is the total particle angular momentum )
 The energies depend simply and monotonically on n
and ℓ
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Properties of the Solutions
 Higher n : higher energy (more kinetic energy)
 Higher ℓ : higher energy (larger radius, less bound)
 The lowest state is : 1s1/2 (n = 1, ℓ = 0) - explains
ground state of the deuteron: L = ℓ1 + ℓ2 = 0
 Note that two orbits can have similar energies if one
has larger n and smaller ℓ, or vice versa
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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 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
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Square Well Wavefunctions
 For ℓ ≠ 0 there is
an effective
centrifugal barrier
which modifies the
shape of the
potential
 Low n high ℓ states
are moved towards
the nuclear surface
 e.g. compare 1s and
1f states
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Harmonic Oscillator potential
 Easy to handle analytically
 Form of potential:
VHO(r) = -U0 + ½mr2ω2
 Solutions are Laguerre
polynomials
Simple harmonic
oscillator potential
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 Eigenenergies may again be
labelled by n and ℓ :
Enℓ = (2n + ℓ + ½) ħω – U0
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Harmonic Oscillator potential
 Eigenenergies can also be labelled by the oscillator
quantum number N:
EN = (N + 3/2) ħω – U0
 For each N there are degenerate 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 is:
Δℓ = 2 and Δn = 1 (e.g. N = 4 3s, 2d, 1g orbits)
 It is the fundamental reason for shell structure, i.e.
clustering of levels
 The parity of each oscillator shell is: (-1)N = (-1)ℓ
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Woods-Saxon Potential
 Usually finite potential forms are used such that
V(r)  0 if r » 0
 The Woods-Saxon potential is considered to be the
most realistic nuclear potential
 For protons a Coulomb potential VC(r) is added
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(Wrong) Magic Numbers
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Spin-Orbit Coupling
 In order to account for the correct nucleon numbers at
which the higher shell closures occur, a spin-orbit term is
added – Mayer, Haxel, Jensen, Suess (1948)
 For the modified harmonic oscillator:
VHO(r) = -U0 + ½mr2ω2 – 2/ħ2αℓ.s
 Since:
ℓ.s = ½ħ2[j(j+1) - ℓ(ℓ+1) – ¾]
the energy is modified by -αℓ if j = ℓ + ½
and by +α(ℓ+1) if j = ℓ – ½
 Note: j = ℓ + ½ levels are lowered in energy relative to
j = ℓ - ½ levels (opposite to the atomic case!)
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Predicted Shell Structure
 The harmonic oscillator
shells are shown to the
left in this diagram
 In the middle, an ℓ2 term
is added to make the
potential more realistic
(‘modified oscillator’)
 A spin orbit term ℓ.s is
added to the right with
its strength (fitted to
experiment) adjusted to
obtain the correct nuclear
magic numbers
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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
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Systematics Near Z(N) = 50
N = 50

100Sn
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Z = 50
(Z=N=50) and 132Sn (N=82) are doubly magic nuclei
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Neutron Separation Energies
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Residual Interactions
The residual interaction ν between nucleons is the
difference between the actual two-nucleon potential
Vα experienced by a nucleon in a state α and the
average potential
 Matrix elements of ν, α|ν|β are only appreciable
near the Fermi Surface
 The interaction ν is a two-body operator because it
changes the state of two nucleons. It can be treated
in a number of ways:
1.
from the free two-nucleon potential (difficult!)
2.
as a free parameter (fit to experimental data)
3.
parameterised using physical intuition

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Quadrupole + Pairing Interaction
Monopole pairing
Iπ = 0+
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 If we assume that the interaction
takes place near the Fermi surface,
i.e. near r = R then Vℓ(ri,rj)  Vℓ(R)
 The quadrupole-quadrupole (ℓ = 2)
interaction is the most important
correction to a spherical field, and is
relatively long range
 The pairing interaction (left) is the
important short range component. It
leads to greater binding between
nucleons if their angular momenta
are coupled to zero spin, with
maximum spatial overlap
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Hartree Fock Method

The philosophy here is that the nuclear potential is
self-consistent
1.
We calculate the nucleon distribution (density) from
the net potential
2. Then we evaluate the net potential from the
nucleon-nucleon interaction
3. Then we iterate
4. The potential is self-consistent if the one with
which we end up is the same as the one we start
with
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4. Nuclear Deformation
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Evidence for Deformation
1.
Large electric quadrupole moments Q0
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.
In contrast, the short range (pairing) component
favours sphericity
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Simple Nuclear Shapes
 The general shape of a nucleus
can be expressed in terms of
spherical harmonics Yλμ(θ,φ)
 The λ = 1 term describes the
displacement of the centre of
mass and therefore cannot give
rise to excitation of the
nucleus – ignore !
 The λ = 2 term is the most
important term and describes
quadrupole deformation
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Nuclear Shapes
 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
 In general most nuclei are prolate with a small additional
hexadecapole deformation
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Principal Axes
 The description of the nuclear
shape simplifies if we make
the principal axes of our
coordinate system (x, y, z)
coincide with the nuclear axes
(1, 2, 3)
 For quadrupole shapes we then
need only two parameters (β,
γ) to describe the shape
Intrinsic (nuclear) and
laboratory frame axes
 ‘Prolate’ (rugby ball): β > 0
 ‘Oblate’ (smartie): β < 0
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Quadrupole β and γ Parameters
prolate
oblate
x=z>y
x>y=z
Axially symmetric shapes
60°
γ = n 60°
prolate
x=y<z
0°
oblate
x=y>z
-60°
oblate
x<y=z
prolate
x=z<y
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Triaxial shapes : x ≠ y ≠ z
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Theoretical Deformations
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Shape Coexistence
1
2
3

The nucleus 184Pb
has three low-lying
0+ states
1. Spherical
2. Oblate
3. Prolate

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This plot shows the
calculated ‘potential
energy surface’
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Deformation Systematics
Theory
Proton Number Z
Doubly Magic: Spherical
Midshell: Deformed
Oblate
‘Spherical’
Prolate
Neutron Number N
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Proton Number Z
First Excited 2+ Energies
E(2+)  [Moment of Inertia]-1
Neutron Number N
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Proton Number Z
Deformation: Rotational Bands
Neutron Number N
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Nilsson Model
 In order to introduce nuclear deformation Nilsson
modified the harmonic oscillator potential to become
anisotropic:
V = ½m[ω12x2 + ω22y2 + ω32z2]
with ωk R k = ω0 R0 and ω1 = ω2 ≠ ω3
 If axial symmetry is assumed (γ = 0) then the
deformation is described by the parameter ε:
ε = (ω1,2 – ω3) / ω0
 It can be shown that ε ~ β
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Nilsson Diagram (Energy vs. ε)
 In order to reproduce
the observed nuclear
behaviour Cℓ.s and Dℓ2
terms need to be added
(C and D are constants)
 The ℓ.s term is the spinorbit term
 The ℓ2 term has the
effect of flattening the
potential to make it
more realistic (like the
shape of the WoodsSaxon potential)
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Nilsson Single-Particle Diagrams
N
Z
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Nilsson Labels
 The energy levels are labelled by the asymptotic quantum
numbers:
Ωπ [N n3 Λ]
 ‘N’: N = n1 + n2 + n3 is the oscillator quantum number
 ‘n3’: n3 is the z-axis (symmetry axis) component of N
 ‘Λ’: Λ = ℓz 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)N = (-1)ℓ is the parity of the state
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The Λ, Σ, Ω Quantum Numbers
 Spin projections:
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Ω=Λ+Σ=Λ±½
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Asymptotic Quantum Numbers

Because of the additional ℓ.s and ℓ2 terms the physical
quantities labelled by n3 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  ∞

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
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Large Deformations
 This figure ignores the ℓ.s
and ℓ2 terms
 Deformed shell gaps emerge
when ω3 and ω1,2 are in the
ratio of small integers, i.e.
ω3/ ω1,2 = p/q
 A superdeformed shape has
p/q = ½ or
R3:R1,2 = 2:1
 A hyperdeformed shape has
R3:R1,2 = 3:1
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5. Hybrid Models
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Deformed Liquid Drop
 Assuming that the nucleus behaves as a charged liquid
drop, a semi-empirical expression can be obtained for the
total energy:
E(A,Z) = -aVA + aSA2/3 + aCZ2A-1/3
 To correct for deformation the nuclear radius R0 is
replaced by:
R3 = R0(1 + δ) , R
1,2
= R0(1 - ½δ)
 The energy for small δ then becomes:
E(A,Z) = -aVA + aSA2/3 (1 + 2/5 δ2) + aCZ2A-1/3 (1 – 1/5 δ2)
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Deformed-Spherical Energies
ΔE(δ) = E(δ) – E(δ=0)
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 It is then predicted that
the nucleus is always
spherical (i.e. ΔE = 0 for
δ > 0) unless Z2/A > 49 in
which case the nucleus
prefers infinite
deformation (i.e. it fissions)
 This is clearly wrong !
 The liquid drop model must
be extended to take into
account shell-model
effects, i.e. effects from
individual nucleon motion
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Shell Correction
 Additional terms arising from the symmetry energy
(which prefers N = Z) and the pairing energy (Δ, 0, -Δ
for even-even, odd-even and odd-odd nuclei,
respectively) can be added
 Alternatively the total energy can be calculated using
mean-field potentials
 This is not simply the sum of the individual
eigenenergies ei because the potential energy of each
nucleon would be counted twice
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Shell Energy
 The eigenvalue for each nucleon is:
ei = Ti + j≠iVij
 The total energy is:
Ti + ½j≠iVij = ½ei + ½Ti
 For the harmonic oscillator potential:
Ti = Vi = j≠iVij so that E = ¾ei
 This method has difficulty in producing the correct
energy because errors in ei give rise to large errors in
the summation ei
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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 of both
models
(a) Liquid drop:
 He considered the
behaviour of the level
gF(e) =gAV(e)
(b) and (c) show shell effects. density g(e) in the two
models
A change in nuclear binding
 And calculated the
arises from:
‘fluctuation’ energy
gAV(e) – gF(e)
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Level Density
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Shell Correction Energies
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Fission Isomers
Superdeformed band
head is isomeric.
Its decay can penetrate
barrier either way
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 If the increase in liquid
drop energy for increasing
deformation ΔE(δ) is small
enough (e.g. Z2/A > 35)
then any secondary
minimum in the total energy
arising from the shell
correction will become
similar in energy to the
first
 This second minimum
corresponds to a
superdeformed nuclear
state
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Superdeformed
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240Pu
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6. Nuclear Excitations
Single-particle and collective motion
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Collective Motion in Nuclei
 Adiabatic approximation: identify fast and slow degrees
of freedom
 Molecules: electronic motion fastest, vibrations 102
times slower, rotations 106 times slower
 These different motions have very different time
scales, so the wavefunction separates into a product of
terms
 In nuclei the timescales are much closer
 Collective and single-particle modes can perhaps be
separated but they will interact strongly !
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Types of Nuclear Excitation
 All even-even nuclei have a ground state with
a consequence of nuclear pairing
Iπ = 0+ ,
 Closed-shell nuclei are spherical and excited nuclear
states can only be formed by breaking pairs of nucleons
or by vibrations
 For odd-mass nuclei (near closed shells) the low-lying
excited states map out the single-particle spectrum of
states around the Fermi level
 ‘Deformed’ nuclei exhibit regular rotational bands:
quadrupole or octupole shapes etc…
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Excitations in Spherical Nuclei
 All even-even nuclei have
Iπ = 0+ in their ground states
 Excitations can only occur by
breaking of pairs or by
vibrations
 The energy difference
between the first excited and
ground states is a rough
Doubly magic (spherical)
measure of the pairing energy
nuclei
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Noncollective Level Scheme
 Complicated set of
energy levels
 No regular features, e.g.
band structures
 Some states are
isomeric
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Vibrations
 From the liquid drop
dependence on deformation
we can estimate the
restoring force if the
nucleus is deformed from
its equilibrium deformation
 A vibration can be any
distortion in the nuclear
shape
 Equally spaced energy
levels for each phonon of
vibration
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Beta (Y20) Vibration
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Gamma (Y22) Vibration
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Octupole (Y30) Vibration
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Octupole (Y31) Vibration
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Octupole (Y32) Vibration
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Octupole (Y33) Vibration
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Realistic Vibrational Levels
vibrator
n=3
n=2
n=1
n=0
nucleus
 For each given mode of
vibration, each phonon has
an associated angular
momentum and parity, e.g:
quadrupole 2+
octupole
3 For a pure vibrator there
are groups of degenerate
levels for two or more
phonons
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Multiphonon Vibrational States
N = 3 (3 phonon)
N = 2 (2 phonon)
N = 1 (1 phonon)
124Sn,
spherical
N=0
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Giant Resonances
Monopole
L=0
Isovector
Isoscalar
Dipole
L=1
Quadrupole
L=2
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Rotations of a Deformed System
K
Nuclear spins
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 The low-lying levels of deformed
even-even nuclei which lie far from
closed shells form a regular
sequence of levels that are much
lower in energy than the pairing
energy. This arises from rotation
 The Hamiltonian is:
Hrot = (ħ2/2) R2 = (ħ2/2) (I-J)2
where  is the moment of inertia
and J is additional angular
momentum generated by, e.g. the
odd particle in an odd-A nucleus or
by vibrations
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Coriolis Coupling
 Note that rotation cannot take place about the
symmetry (z) axis
 The rotational Hamiltonian can be expanded:
R2 = (I – J)2 = I2 – 2I.J + J2
= I2 + J2 – 2K2 - (I+J- + I-J+)
where I± = Ix ± i Iy, J± = Jx ± i Jy and Jz = Iz = ±K
 The quantity K is the projection of I along the
deformation axis
 The coupling term (I+J- + I-J+) corresponds to the
Coriolis force and couples J to R
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The K Quantum Number
 The operators I± link states with K differing by ±1
 The term (I+J- + I-J+) can be ignored if:
(1) rotational bands with ΔK = 1 lie far apart
(2) the particular band does not have K = ½
 The excitation energies then become:
Erot = (ħ2/2)[I(I+1) + J(J+1) -2K2]
with I = K, K+1, K+2… and K is a constant of the motion
 Then: Erot = EK + (ħ2/2)I(I+1)
where EK is the energy of the lowest band level
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Vibrational Bands in
232Th
 Here the low-lying levels
are all collective, i.e.
rotational and vibrational
 Ground State Band: Kπ = 0+
 β Band: Kπ = 0+
 γ Band: Kπ = 2+
 Octupole Band: Kπ = 0π = 0+ then
Oct.

Note
that
if
K
Beta
Gamma
the I values 1, 3, 5… are
not present
GSB
π = 0- the I values 0,

For
K
Reflection symmetric shape,
2, 4… disappear
232Th is a deformed nucleus
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Alternating Parity Bands in
226U
 This nucleus is reflection
asymmetric (i.e. β3 ≠ 0) in its
ground state: it has octupole
deformation
 The nuclear wavefunction in its
intrinsic frame is not an
eigenvalue of parity:
Ψ2 (x ,y ,z) ≠ Ψ2 (-x, -y, -z)
 In the laboratory frame (i.e.
averaged over all nuclear
orientations) the levels have
alternating parity
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Reflection (A)symmetry
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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
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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
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Rotational Bands in
157Ho
 This nucleus shows
three band
structures built on
different Nilsson
states
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7. Rotating Systems
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Moment of Inertia
 The energy of a rotating nucleus is given by:
E = (ħ2/2) I[I+1]
 The nuclear moment of inertia  (at low spin) is found to
be one third to one half of the value expected for a
rotating liquid drop
 Nuclear pairing introduces a degree of superfluidity
 Rotation counteracts pairing (cf strong magnetic field
applied to superconductor)
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Rotational Bands: γ-ray Spectra
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Inglis Moment of Inertia
 Inglis (1952) showed that the
moment of inertia of a Fermi gas
rotating about the x-axis is:
x = 2 |p|Îx|h|/(ep-eh)
where the summation is over all
possible 1-particle 1-hole
excitations in a deformed shell
model
 The rigid-body moment of inertia
is:
 = (2/5) mnAR02[1 +
0.3β]
which is higher than observed
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Nuclear Moments of Inertia
 Nuclear
moments of
inertia are lower
than the rigidbody value – a
consequence of
nuclear pairing
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Pairing Gap
 A rough estimate of
the energy required to
create a particle-hole
excitation is 2Δ,
where Δ is the pairing
gap
 A typical value for Δ is
1 MeV
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Cranking Model
 The deformed shell model (e.g. Nilsson Model) can be
modified to include pairing
 To include rotation it is convenient to subtract the effect
of rotational forces (Coriolis and centripetal)
 Classically the ‘potential’ energy of these forces is ω.I so
the corresponding quantum operator is ωÎx
 The Hamiltonian is: Hω = HDSM – ωÎx
 Energy in the rotating frame: Eω = E - ω Îx
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Routhian and Aligned Angular
Momentum
 The Routhian is simply the energy in the rotating
frame of reference:
Eω
 The aligned angular momentum is just the
expectation value of the operator Îx:
Îx
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Spin and Rotational Frequency
 There are two important
relations which arise since E
is independent of ω and Eω is
independent of I:
dEω/dω = - Îx
and
dE/dI = ω dÎx/dI
Nuclear spin I and its
projections onto the
rotation axis Ix and
deformation axis K
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 Since: Îx = √[I(I+1)-K2] ħ
then for K = 0: Îx ~ I ħ
and hence:
dE/dI ~ ω ħ
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Gamma Ray Energies and
Rotational Frequency
 The energy of a rotational band for K = 0 is:
E = E0 + (ħ2/2) I(I + 1) ,
I = 0, 2, 4…
 The energy difference between consecutive levels ΔE
represents the gamma-ray energy Eγ
 The spin difference between consecutive levels is Δ I = 2
 The rotational frequency ω is defined as:
ωħ = dE/dI = ΔE/ΔI = Eγ/2
i.e. the frequency is just half the gamma-ray energy
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Moments of Inertia
 The energy of a rotational band for K = 0 is:
E = E0 + (ħ2/2) I(I + 1) , I = 0, 2, 4…
 Then:
dE/dI = (ħ2/2) (2I + 1)
and:
d2E/dI2 = ħ2/
defines the ‘dynamic moment of inertia’ which is
independent of spin
 By using finite differences:
dE=ΔE=Eγ, dI=ΔI=2, d2E=ΔEγ, d2I=Δ2I=4
we can evaluate  even if we do not know I !
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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):
hydro = rig δ2
 This estimate is much too low !
 We require short-range pairing
correlations to account for the
experimental values
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Kinematic and Dynamic MoI’s
Rigid body: (1) = (2)
High spin: (1) ≈ (2)
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 Assuming maximum alignment
on the x-axis (Ix ~ I), the
kinematic moment of inertia is
defined:
(1) = (ħ2 I) [dE(I)/dI]-1
= ħ I/ω
 The dynamic moment of
inertia (response of the
system to a force) is:
(2) = (ħ2) [d2E(I)/dI2]-1
= ħ dI/dω
 And (2) = (1) + ω d(1)/dω
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Backbending
 The moment of inertia
increases with increasing
rotational frequency
 Around spin 10ħ a
dramatic rise occurs
 The characteristic ‘S’
shape is called a backbend
(158Er)
 A more gradual increase is
called an upbend (174Hf)
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Crossing Bands
 A backbend corresponds to
the crossing of two bands
(‘g’ and ‘s’ configurations)
yrare
yrast
Yrast and yrare states:
dizziest and dizzier in
the Swedish language
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 The states we observe are
called yrast states (thick
line) which have the lowest
energy for a given spin
 The s-band, where s stands
for ‘Stockholm’ or ‘super’,
arises from the breaking of a
pair of nucleons. Their
angular momenta j1 and j2
align with the rotation axis
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Pair Breaking
 For the ground state band:
Eg = (ħ2/2g) I(I + 1)
 For the s-band:
Es = (ħ2/2s) (I – J )2 + EJ
where J = j1 + j2 and EJ is the
energy required to break a
pair of nucleons:
EJ ~ 2Δ ~ 24 A-1/2 MeV
 The aligned angular momentum
of the s-band increases by
approximately:
j1 + j2 – 1 (~ 12ħ for 158Er)
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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: 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
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Pair Breaking and Rotational
Alignment
A Backbending movie follows showing pair breaking
and rotational alignment
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Backbending Movie
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Backbending Demonstration
This movie shows Mark Riley’s “backbending
machine”
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8. Nuclei at Extremes of
Spin
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High-Spin States
 As the nucleus is rotated to states of higher and
higher angular momentum, or spin I, it tries to assume
the configuration which has the lowest rotational
energy
 The spin I is made up of a collective part R and a
contribution J arising from single particles
 The energy can be minimised by reducing R or by
increasing the nuclear moment of inertia
 The pairing is broken by the effect of rapid rotation
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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 ji generate the
total nuclear spin

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High Ix Bands

In backbending the value of R (collective spin) is
reduced by breaking a single pair of nucleons and
aligning their individual angular momenta j with the
x-axis, i.e.
Ix = jx + R

The quantity Ix is approximately a good quantum number
and hence a given nuclear state can be described by a
single value of Ix
 The alignment of broken pairs becomes easier if
1. the particle j is high but its projection Ω small
2. the Coriolis force is large: small  and high ω
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Aligned Particles
 Alignment effects should be prominent
for nuclei with a few nucleons outside a
closed shell, e.g. 158Er with 8 neutrons
above the N = 82 closed shell
 If we continue to rotate faster and
faster then more of the valence pairs
break and align
I = Σji
R=0
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 Eventually all the particles outside the
closed shell (spherical) core align
 These move in equatorial orbits giving
the nucleus an oblate appearance
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Band Termination
neutron
backbend
proton
backbend
Gamma Ray Energy
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Band Termination in
158Er
 When we align the np protons and nn neutrons outside the
closed shell the total spin is:
I = inp ji(p) + inn ji(n)
and the rotational band is said to ‘terminate’
 At termination 158Er can be thought of as a spherical
146Gd core plus 4 protons and 8 neutrons generating a
total spin 46ħ
 The configuration is:
π(h11/2)4  ν(i13/2)2(h9/2)3(f7/2)3
 The terminating spin value of 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)
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High K (Iz) Bands
 If we have many unpaired
nucleons outside the closed
shell then alignment with the
x-axis becomes difficult
because the valence nucleons
lie closer to the z-axis, i.e.
they have high Ω values
 The sum K of these projections
onto the deformation (z) axis is
now a good quantum number
K = Iz = Σjz = ΣΩ
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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 of its angular momentum.
 For example, the Kπ = 8- band head in 178Hf is isomeric
with a lifetime of 4 s. This is much longer than the
lifetimes of the rotational states built on it.
 The Kπ = 8- band head is formed by breaking a pair of
protons and placing them in the ‘Nilsson configurations’:
Ω [N n3 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
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K Isomers in
178Hf
 A low lying state with
spin I = 16 and K = 16 in
178Hf 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 8 units in its
decay
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High K bands in
174Hf
 This nucleus has 347 known levels and 516 gamma rays !
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Superdeformation
Nuclear potential at
low and high spin
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 Shell effects can give large
energy corrections for large
values of prolate deformation,
e.g. when the major/minor axis
ratio is 2:1
 The smooth liquid-drop
contribution to the total nuclear
energy includes the rotational
energy, which can be
substantially reduced at high
spin by increasing the moment
of inertia
 At sufficiently high spin a
secondary minimum can become
energetically favourable
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Superdeformed Band in
β2 ~ 0.6, 2:1 axis ratio
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152Dy
 The experimental
signature of these
superdeformed (SD)
shapes is a very
regular sequence of
equally spaced γ rays
 In 152Dy the (first)
SD band spans a spin
range 20 – 60 ħ
 Nowadays multiple SD
bands are known in
this and other nuclei
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Some Big Numbers
 The SD band of 152Dy emits ~ 20 gamma rays in
~ 10-13 s. The total energy released is:
Eγ ~ 20 MeV (1 eV = 1.6 x 10-19 J)
 The power is: (3.2 x 10-12 J) / (10-13 s) = 32 W !
 The rotational frequency is: ħω ~ 500 keV, so
ω ~ 8 x 1020 radians/sec  1020 Hz or 107 rotations in
10-13 s  same as number of days in 30,000 years !
 The decay of the SD band passes through a long-lived
isomeric level (86 ns)  ~ 5 x 1012 rotations  same as
number of days since the Big Bang !
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Superdeformation in 132Ce
 SD bands exist in cerium (Z = 58) nuclei with a major/minor
axis ratio of 3:2. This band in 132Ce (THE original SD band –
discovered by the Liverpool Nuclear Physics Group) is now
seen up to spin approaching 70ħ – one of the highest spins
ever seen in the atomic nucleus !
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Superdeformed Systematics
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Shape Coexistence
 For a given nuclear system
at a given value of spin, a
number of configurations
can exist
 These configurations may
have different shapes
 Weakly deformed triaxial
and oblate shapes coexist
in 152Dy along with the
superdeformed shape
 Each shape has a (local)
‘minimum’ in the nuclear
‘total energy surface’
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Hyperdeformation
 Superdeformation
represents a
secondary minimum in
the nuclear potential
energy, with typically
a 2:1 axis ratio
 Hyperdeformation
represents a third
minimum, with an axis
ratio 3:1
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Critical Angular Momenta
 Nuclei can only attain a
finite amount of spin
before they fly apart
(fission)
 Just before this fission
is a predicted region of
extended triaxial
(x ≠ y ≠ z) shapes
 This is known as the
Jacobi regime
 Such behaviour also
Nuclei with mass 130-150 can
occurs for macroscopic
accommodate the most spin
objects
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Jacobi Shape
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9. Nuclei at Extremes of
Isospin
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Limits of Nuclear Existence
Segre Chart
Known Nuclei
Stable Nuclei
Proton Dripline
Fission Limit
Terra Incognita
Neutron Dripline
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Where Are The Driplines?
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Where Are The Driplines?
 In this
experiment –
fragmentation of
a beam of 48Ca –
no counts were
observed for 26O
 This defines the
neutron dripline
for oxygen
isotopes
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Where is the Neutron Dripline?
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Heavy N = Z Nuclei
 Shell corrections give minima in
the nuclear energy at non-zero
values of deformation
 Bigger effect if both proton and
neutrons occur at these ‘magic
numbers’
 Also a big effect for N = Z
 The N = Z = 40 nucleus 80Zr is
an example
 It is difficult to study this
nucleus: it is 10 neutrons lighter
than the lightest stable
zirconium isotope !
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Exotic Nuclei
 The nucleus 12C has six protons and
six neutrons
 It is stable and found in nature
 The nucleus 22C has six protons and
sixteen neutrons !
 It is radioactive and at the limit of
nuclear binding
 Characteristics of exotic nuclei: excess of neutrons or
protons, short half life, neutron or proton dominated
surface, low binding
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Isospin: Tz = (N - Z) / 2
A = 21
A
21
C
6 15
Z
TZ
N
21
C 21 N 21 O 21 F 21 Ne 21 Na 21 Mg 21Al
6 15 7
14 8 13 9 12 10
11 11
10 12
9 13 8
+9/2 +7/2
+5/2 +3/2 +1/2
–1/2 –3/2
–5/2
Neutron rich
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Proton rich
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Nuclei Far From Stability
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Proton-Rich Nuclei
 The proton dripline is
defined by the least
massive bound nucleus
of every isotopic chain
(Sp drops to zero)
 For nuclei beyond the
dripline the last proton
has a positive energy
and is unbound
 This proton does not
escape instantaneously
as it must overcome
the Coulomb Barrier
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Radioactivity: Normal
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Radioactivity: Exotic
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Alpha and Proton Emitters
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Jyväskylä, Finland (Feb 2006)
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Jyväskylä (midnight June 21 2009)
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Recoil Decay Tagging
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JUROGAM + RITU + GREAT
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Deformed Proton Emitter
 The half-lives of proton
radioactivity are sensitive
to both specific orbitals
and nuclear deformation
 Measured half-lives in 131Eu
and 141Ho could only be
understood if these nuclei
were deformed
 This was later confirmed
by the observation of
rotational bands in 141Ho
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Proton Decay Half Lives
ℓ = 0 Coulomb Barrier
ℓ = 5 Coulomb Barrier plus
Centrifugal Barrier
 The half-lives of proton radioactivity are sensitive to the
orbital angular momentum of specific states
 A centrifugal barrier occurs in the potential proportional
to the orbital angular momentum
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Fine Structure in Proton Decay
 In 131Eu proton decay has
been observed both to the
ground state and the first
excited state of 130Sm
 This establishes the first
2+ state in 130Sm at an
energy of 121 keV
 This low energy implies a
large moment of inertia
and large quadrupole
(prolate) deformation for
the exotic 130Sm nucleus !
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Fine Structure in Alpha Decay
 In 109Xe alpha decay
has been observed
both to the ground
state and the first
excited state of
105Te
 This establishes the
relative energies of
the neutron d5/2 and
g7/2 orbitals in 105Te
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Superdeformed Proton Emitter
SD
An SD band in 58Cu decays by proton emission into 57Ni in
competition with γ decay to the low-spin 58Cu states
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Direct Two-Proton Decay
 A new mode of decay, direct
two-proton decay, had been
predicted long ago, but until
recently, experimental
efforts had only found
sequential emission through
an intermediate state
Are the two protons correlated
(di-proton emission) or uncorrelated
(sequential proton emission)?
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 To prove di-proton emission,
specific nuclei are needed
where the sequential
emission is energetically
forbidden e.g. 18Ne
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Direct Two-Proton Decay of 45Fe
Decay energy spectrum correlated
with 45Fe implantation
from 45Fe
 The experimental Q-value
implies di-proton emission
with a barrier-penetration
half-life of 0.024 ms
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Neutron-Rich Nuclei
 Physics of weak binding
 The question of which combinations of protons and
neutrons form bound systems has not been answered
for most of the nuclear chart because of a lack of
experimental access to neutron-rich nuclei
 These nuclei are increasingly the focus of present
and future experimental (and theoretical) effort
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Light Neutron-Dripline Nuclei
 The neutron dripline has only been reached for light
nuclei
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Level Inversion in
11Be
 The ordering of the neutron
1s1/2 and 1p1/2 orbitals
appears to be inverted in the
nucleus 11Be and lighter N = 7
isotones
unbound
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Nuclear (Neutron) Haloes
 The spatial extent of 11Li with 3 protons is similar to
that of 208Pb with 82 protons !
 11Li is modelled as a core of 9Li plus two valence neutrons
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Size of Lithium Isotopes
Root mean square radii
 Interaction cross
sections give a measure
of the nuclear matter
distribution (radius)
 A sudden jump is seen in
going from 9Li to 11Li
 However, the electric
quadrupole moments
are similar (charge
distribution)
 Hence, excess neutron
tail or halo
Textbook: R = r0A1/3
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Nuclear Sizes
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Halo Systematics
 Neutron haloes
have now been
seen in nuclei
as heavy as 19C
(Z = 6, N = 13)
 Note proton
haloes also
predicted
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Borromean System
 Halo nuclei have provided
insight into a new topology
with a Borromean property
bound
 The two-body subsystems
of the stable three-body
system 11Li (9Li + n + n) are
themselves unstable !
unbound
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Neutron Skins
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Proton Skins?
Theory
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Quenching of Shell Structure
 Adding more and more
neutrons to a nucleus
may change the shell
structure
 It has been predicted
that the shell gaps
(magic numbers) are
washed out far from
the stability line
 The ℓ.s term is
diminished
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New Shell Structure?
N=20
Z=8
N=16 ?
N=8
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Unexpected Things Happen Far
From Stability
 The heaviest known
tellurium isotope is
136Te
 It is two neutrons
outside the N = 82
shell closure
 However its measured
B(E2) value is much
lower than expected !
 The 2+ energy is also
too ‘low’
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Extremes of Mass and Charge
 Investigations of the
heaviest nuclei probe the
role of the Coulomb force
and its interplay with
quantal shell effects in
determining the nuclear
landscape
 Without shell effects
nuclei with more than 100
protons would fission
instantaneously
 However, ‘superheavies’
with Z up to 118 have been
identified !
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Deformed Superheavy Nuclei
 Modern theory not only predicts which combinations of
N and Z can be made into heavy nuclei but also indicates
that stability arises in specific cases from the ability of
the nucleus to deform
 For example, the nucleus 208Pb at the shell closures
Z = 82, N = 126 is spherical but nuclei with substantially
deformed ground states are predicted around the next
shell closures at Z ~ 114, N = 184
 Different theories suggest the next proton shell closure
at Z = 114, 120, 126 (note N = 126 occurs for neutrons)
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Super Heavy Elements (SHE)
Copernicium
Roentgenium
Darmstadtium
Meitnerium
Hassium
Bohrium
Protons
Neutrons
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Flerovium
Livermorium
SHE
 Quantal shell effects
stabilise energy
 Up to Z = 112 results
confirmed
 Dubna: Z = 114, 116, 118
 Berkeley: Z = 118
‘discovered’ then
retracted
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Element 115
 Well known for antigravity
properties
 The fuel of UFO’s
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The Island(s) of Stability
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Alpha Decay Chains
 decay provides the
technique to identify
heavy elements
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 The heaviest nuclei are unstable
against  decay
 The -decay half life (in s) is
given empirically by:
log10 t1/2 = 1.61 Z Eα-1/2
– 1.61 Z2/3 – 28.9
 Here Eα (in MeV) is the  decay
energy, related to the mass
difference of the parent (Z, A)
and daughter (Z-2, A-4) nuclei
 The lifetimes are very long
(>10-3 s) on the nuclear time
scale
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SHE Synthesis at GSI
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Elements 116 and 118
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Element 117 (2010)
 Dubna (Russia)
 Phys. Rev. Lett.
104, 142502
(2010)
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Superheavies at High Spin
 The ground-state
rotational band of
254No (Z=102) has
been identified up to
spin 20+ (at least!)
 The energy spacing
of the levels is
consistent with a
sizeable prolate
deformation with an
axis ratio 4:3
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The Heaviest Element

University researchers have discovered the heaviest element yet known to science. The
new element, Governmentium (Gv), has one neutron, 25 assistant neutrons, 88 deputy
neutrons and 198 assistant deputy neutrons, giving it an atomic mass of 312.

These 312 particles are held together by forces called morons, which are surrounded by
vast quantities of lepton-like particles called pillocks. Since Governmentium has no
electrons, it is inert. However, it can be detected, because it impedes every reaction
with which it comes into contact.

A tiny amount of Governmentium can cause a reaction that would normally take less than
a second, to take from 4 days to 4 years to complete. Governmentium has a normal halflife of 2 to 6 years. It does not decay, but instead undergoes a reorganisation in which a
portion of the assistant neutrons and deputy neutrons exchange places.

In fact, Governmentium's mass will actually increase over time, since each reorganisation
will cause more morons to become neutrons, forming isodopes. This characteristic of
moron promotion leads some scientists to believe that Governmentium is formed
whenever morons reach a critical concentration. This hypothetical quantity is referred
to as a critical morass. When catalysed with money, Governmentium turns into
Administratium (Ad), an element that radiates just as much energy as Governmentium,
since it has half as many pillocks but twice as many morons.
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10. Mesoscopic Systems
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Finite Fermionic Systems
 The behaviour of micro particles (atoms, electrons,
nuclei, nucleons and other elementary particles) can be
described by quantum theory
 Macroscopic bodies obey the laws of classical
mechanics
 These two ‘worlds’ largely differ from each other
 In nature there is no sharp border between the micro
and macro world and there are objects that exist in the
intermediate range
 The atomic nucleus, a finite fermionic system, is an
example of such a mesoscopic system
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Mesoscopic Systems
 ‘Mesoscopic’ systems contain large, yet finite, numbers
of constituents, e.g. atomic nuclei, metallic clusters
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Nanostructures and
Femtostructures
 ‘Nanostructures’: intense research is ongoing for
quantum systems that confine a number of electrons
within a nanometre-size scale (10-9 m), e.g. grains,
droplets, quantum dots
 Nuclei are femtostructures (10-15 m)
 All these systems share common phenomena but on
very different energy scales:
nuclear MeV; molecular eV; solid-state meV
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Mesoscopic systems
complexity
Nuclei
He-droplets
Metal clusters
Emergent phenomena:
Nanoparticles
E
-Liquid-gas surface, droplet features
-superconductivity / superfluidity
-thermal phase transitions
-shell structure, quantal shapes (liquid)
-spatial orientation, rotational bands
-rotational/magnetic response
-quantum phase transitions
macroscopic
N
Quantum dots
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Quantality Parameter
 The ‘quantality’ parameter (Mottelson 1999),
Λ = ħ2 / M a2 V0, measures the strength of the two-body
attraction V0 expressed in units of the quantal kinetic
energy associated with a localisation of a constituent
particle of mass M within the distance a corresponding
to the radius of the force at maximum attraction
 For small Λ the quantal effect is small and the ground
state of the many body system will be a configuration in
which each particle finds a static optimal position with
respect to its nearest neighbours (crystalline)
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Nuclei as Quantum Liquids

If Λ is large enough the ground state may be a
quantum liquid in which the individual particles are
delocalised and the low-energy excitations have
‘infinite’ mean-free path

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Constituents
3He
4He
H2
Ne
Nuclei
T = 0 matter
Λ = 0.21
‘liquid’
Λ = 0.16
‘liquid’
Λ = 0.07
‘solid’
Λ = 0.007
‘solid’
Λ = 0.4
‘liquid’
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Fermi Liquid Droplets
 ‘Clusters’ are aggregates of
atoms or molecules with a
well-defined size varying
from a few constituents to
several tens of thousands
 Conduction electrons in
clusters are approximately
independent and free
 Nucleons in nuclei also
behave as delocalised and
independent fermions
 Hence analogies exist
between these two systems
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The Spherical Droplet
 Both clusters and nuclei are characterised by a
constant density in the interior and a relatively thin
surface layer
 The Liquid Drop Model can be used to calculate the
binding energy of a charged droplet
 The binding energy can be expanded in powers of
A1/3 (i.e. radius) where A is the number of
constituents
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Spherical Droplet Energy
 The energy of a droplet may be expressed as:
ELD(N,Z) = fA + 4πσR2 + WZ + C Z2e2/R
= fA + bsurfA2/3 + WZ + bcoulZ2A-1/3
 Here R = r0A1/3 is the radius of the droplet, A the
number of atoms and Z is the net charge
 The first term (fA) is the ‘volume energy’ which contains
the binding energy per particle f of the bulk material
 The second term (4πσR2)is the ‘surface energy’ where σ
is the coefficient of surface tension
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Spherical Droplet Energy (cont)
 The third term (WZ) contains the ‘work function’ W
which is the energy required to remove one electron
from the bulk metal
 The fourth term (C Z2e2/R) represents the ‘Coulomb
energy’ of the charged constituents
 In nuclei the charge is evenly distributed because the
symmetry energy (quantal effect) keeps the ratio of
neutron to protons roughly constant: thus C=3/5
 For a cluster charge tends to accumulate at the surface
and C tends to 1/2 for a large cluster
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Shell Structures
 A bunching together
of the energy levels of
a particle in a two- or
three-dimensional
potential represents a
shell structure
 Metallic clusters
show shell structures
similar to nuclei
 Clusters can contain
more constituents
than stable nuclei
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Supershell Structures
 Metallic clusters also
exhibit a supershell
structure
 The basic shell
structure is enveloped
by a long wavelength
oscillation (beat pattern)
 Nuclei become unstable
well before the first
half-period of the long
wavelength oscillation is
seen
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Periodic Orbit Theory
Supershell structure from interfering periodic orbits
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Loss of Spherical Symmetry
 Deformation occurs in subatomic and mesoscopic systems
with many degrees of freedom, e.g. nuclei, molecules,
metallic clusters
 The microscopic mechanism of ‘spontaneous symmetry
breaking’ was first proposed by Jahn and Teller (1937) –
for molecules
 Nuclei with incomplete shells tend to deform because the
level density near the Fermi surface is high (unstable) for
a spherical shape
 When the shape of the nucleus changes, nucleonic levels
rearrange such that the level density is reduced (stable) –
‘nuclear Jahn-Teller effect’
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Shapes Of Clusters
 Nuclei can easily deform
because they consist of
delocalised nucleons (liquid)
 The presence of heavy
discrete ions leads to a more
varied response of clusters
 Nevertheless, similar shapes
are predicted for nuclei and
clusters despite the very
different nature of the
interactions between the
constituents
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Differences Between Atomic
Nuclei and Metallic Clusters
 There is only one kind of nuclear matter
 It has a single ‘equation of state’
 However, all materials have their own equation of
state
 In a cluster, as in bulk matter, it is the
constituents that determine the density and
binding energy
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Nuclear Phase Diagram
At sufficient temperature or density nucleons are
expected to dissolve into a quark-gluon plasma (QGP)
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Nuclear Molecules
 Speculation about
the existence of
clusters in nuclei,
such as alpha
particles, has
existed for a long
time
Ikeda Diagram
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 Initially stimulated
by the observation
of alpha particle
decay
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Beryllium-12
 A beryllium nucleus
containing 8 neutrons and
4 protons has been found
to arrange itself into a
molecular-like structure,
rather than a spherical
shape that some naïve
theories might suggest
M Freer et al. Phys. Rev. Lett.
82 (1999) 1383
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 Beryllium-12 can be
thought of as two alpha
particles and four neutrons
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Chain States: Nuclear Sausages
 Cluster Model calculations for 12C show evidence for
a ‘chain state’ consisting of three α particles in a row
– axis ratio 3:1 (i.e. ‘hyperdeformed’)
 Similarly calculations for 24Mg show evidence for a
chain state consisting of six α particles in a row –
axis ratio of 6:1 !
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Bloch-Brink Cluster Model
 Brink presented the
light alpha conjugate
nuclei as almost
crystalline structures
 These nuclei contain
specific arrangements
of the alpha clusters
 Narrow resonances in
12C + 12C scattering
data suggested larger
clusters may occur
 ‘Nuclear Molecules’
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Binary Cluster Model
 It has been observed that measured quadrupole
moments of many superdeformed bands follow:
Qo ~ 2 Ro2[ Z A2/3 – Z1 A12/3 – Z2 A22/3]
 This expression results from considering the states of
the nucleus (Z, A) to be composed of two clusters (Zi, Ai)
in relative motion
 For example, a strongly deformed band has recently been
found in 108Cd (Z = 48)
 The predicted fragmentation for 108Cd is:
58Fe (Z = 26) + 50Ti (Z = 22)
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11. Nuclear Reactions
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Examples of Nuclear Reactions
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Introduction
 In a typical nuclear reaction a (light) projectile a “hits”
a (heavy) target A producing fragments b (light) and B
(heavy)
 Schematically this can be written
a+A 
b+B
 In this nuclear “transmutation” we need to consider both
kinetic energy and binding energy (E = mc2)
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The Impact Parameter
Reactions can be classified
by the impact parameter b
 Central collisions occur for
small b, e.g. fusion
 Peripheral collisions occur
at large b, e.g. elastic and
inelastic scattering,
transfer reactions
 Deep inelastic or massive
transfer reactions occur
at intermediate values of b
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Collision Kinematics
 The Q value is:
[ (MA + Ma) - (MB + Mb) ] c2
 Exothermic (Q > 0) reactions
give off energy – kinetic
energy of reaction products
 Endothermic (Q < 0)
reactions require an input of
energy to occur. By
considering the kinetic
energy available in the
centre-of-mass frame, the
threshold energy is:
Ta > |Q| [ (Ma + MA) / MA ]
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The Compound Nucleus
 Consider the reactions:
a + A  C*  a + A*
 b + B*
 γ + C*
 The incident particle a enters the nucleus A and suffers
collisions with the constituent nucleons, until it has lost
its incident energy, and becomes an indistinguishable
part of the excited compound nucleus C*
 The compound nucleus ‘forgets’ how it was formed and its
subsequent decay is independent of its formation:
“Bohr’s Hypothesis of Independence”
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Compound Nucleus Example
 Consider a beam of alpha particles of energy 5 MeV/A
(or MeV per nucleon) impinging on 60Ni:
 + 60Ni  64Zn*
 At this (kinetic) energy, the incident particle is nonrelativistic, β = v/c = 0.1, and it will take the alpha
particle ~10-22 s to travel across the target nucleus
 In a compound nucleus, the first emission of a nucleon or
gamma ray takes > 10-20 s
 Hence the alpha particle traverses the compound nucleus
hundreds of times and loses its identity !
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Geometric Cross-Section
 In the classical picture, the projectile and target nuclei
will fuse if the impact parameter b is less than the sum
of their radii
 A disk of area π(R1 + R2)2 is swept out
 This area defines the
geometric cross-section
 Remember: units of
cross-section are area
(1 barn = 100 fm2;
1 fm = 10-15 m)
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Coulomb Excitation
 Coulomb Excitation (Coulex) is the excitation of a
target nucleus by the long-range electromagnetic (EM)
field of the projectile nucleus, or vice versa
 The biggest effect occurs for deformed nuclei with
high Z: In these nuclei, rotational bands can be excited
to more than 20 ħ
 In pure Coulex, the charge distributions of the two
nuclei do not overlap at any time during the collision.
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Coulex Example

234U
bombarded by 5.3 MeV/A 208Pb
 The beam energy is kept low – below the Coulomb Barrier
– so that other reactions, e.g. fusion, do not compete
 In this example:
Beam energy = 5.3 x 208 MeV = 1.1 GeV
The Coulomb Barrier (in the lab frame) is:
{ Z1Z2e2 / [4πε0(R1 + R2)] } x {(A1 + A2) / A2}
C-o-M barrier
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Intermediate Energy Coulex
 At higher beam energies (> 30 MeV/A), well above the
Coulomb Barrier, Coulex can still take place but in
competition with other violent reactions
 The process is now so fast that only the first excited
states (2+ for even-even nuclei) are populated
 Intermediate energy Coulex is characterised by
straight line trajectories with impact parameters
larger than the sum of the radii of the two colliding
nuclei
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IE Coulex Example (GSI
RISING Experiment)
 A gold target (179Au) bombarded by a 140 MeV/A
radioactive 108Sn beam
 The beam energy is:
140 x 108 MeV = 15.1 GeV
 At this energy, β = v/c =0.48 – the projectile is
travelling at half the speed of light !
 Note: 108Sn is not stable – cannot make a target, but can
generate a short-lived Radioactive Ion Beam (RIB)
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Intermediate Energy Coulex
 Ideally suited for use with fragmentation beams
(Ebeam > 30 MeV/u)
 Large cross sections (~100 mb)
 Can use thick targets (~100 mg/cm2)
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Coulex Cross Sections
 For Intermediate Energy Heavy Ions, the Coulomb
excitation cross section can be approximated as:
σπλ = [Z1e2/ħc] B(πλ;0λ) [πR2/e2R2λ(λ-1)]
for λ ≥ 2
 Here Z1 is the charge of the projectile and R is the sum
of the radii of target and projectile
 The cross section is peaked at forward angles within the
angular range
Δθ ≈ 2Z1Z2e2/RE
where E is the beam energy
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Neutron-Rich Sulphur Isotopes
 Energy spectra
in target (top)
and projectile
(bottom) frames
of reference for
40S + 197Au at
MSU
 β = v/c = 27%
 H. Scheit et al.
Phys. Rev. Lett.
77, 3967 (1996)
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Neutron Capture
 Low-energy neutron-capture
cross-sections exhibit peaks
or resonances corresponding
to a compound system
 An example is the capture of
a 1.46 eV neutron by 115In to
form a highly excited state
(6.8 MeV !) in 116In
 The high excitation energy
in 116In arises due to the
binding energy of the
neutron
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Neutron Capture Cross-Sections
 At 1.46 eV, the measured total cross-section for neutron
capture by 115In is σ ≈ 2.8 x 104 barn
 However the geometrical cross-section (πR2) is only
≈ 1.1 barn
 Quantum effect: we need to consider the de Broglie
wavelength (λ/2π) instead of the nuclear radius – slow
neutrons have a large wavelength and hence a long-range
influence
 The cross-section becomes:
σ = πR2  σ = π(λ/2π)2
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De Broglie Wavelength
 The momentum of the neutron is:
pn = √{2mnE} = √{2 x 939 x 1.46 x 10-6}
= 0.052 MeV/c
 The de Broglie wavelength is then:
(λ/2π) = ħc/pnc = 197/0.052 = 3.7 x 103 fm
 The cross-section then becomes 4.3 x 105 barn
 The measured value is only 6% of this estimate !
 We must also consider other effects such as the spins
of the neutron, target and compound systems
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Decay of
116In*
 n + 115In
 γ + 116In*
116In*
4%
96%
 For this neutron energy of only 1.46 eV
Γn/Γγ = 0.04, also Γn/Γ ≈ 0.04 (Γ = Γn + Γγ)
 This decay fraction can be related to the formation
cross-section:
σ = π(λ/2π)2 x Γn/Γ
(Γn/Γ ≈ 4%)
 Recall the measured formation cross-section was only
6% of the estimate using the de Broglie wavelength !
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Proton Capture
 For charged-particle capture (and decay) we must
consider the Coulomb Barrier which inhibits the
formation or decay of a compound system
 The proton needs sufficient energy to overcome the
Coulomb Barrier (several MeV) and hence its de Broglie
wavelength is smaller (than in the case of neutron
capture
 Consequently, proton-capture cross-sections are
~ 1 barn at maximum
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Heavy Ion Fusion Reactions
 For heavy projectile ions, e.g. 12C or 58Ni, the Coulomb
Barrier is high and the particle enters a continuum of
high level densities and overlapping resonances
 The excitation of the compound nucleus is also high:
10-80 MeV
 Since the neutron binding energy is only ~ 8 MeV, several
neutrons are emitted before gamma-ray emission
dominates
 These Heavy Ion Fusion Evaporation reactions bring large
amounts of angular momentum into the compound system
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Fusion Evaporation Reactions
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David Campbell
Florida State
University
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Fusion Cross-Section
 The angular momentum
brought into the compound
system depends on the
impact parameter b:
ℓ =bp
 The partial fusion crosssection is proportional to
the angular momentum:
d σfus(ℓ ) 
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Compound Formation And Decay
100Mo(36S,4n)132Ce
Beam energy: 4.31 MeV/A
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 Compound nucleus
formation: 10-20 s
 Neutron emission:
10-19 s
 Statistical (cooling)
dipole gamma-ray
emission: 10-15 s
 Quadrupole (slowing
down) gamma-ray
emission: 10-12 s
 After 10-9 s the
nuclear ground state
is reached after 1011
rotations
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Cold Fusion
 Superheavy elements (SHE’s) can be formed by lowenergy fusion-evaporation reactions in which only one
neutron is emitted
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Transfer Reactions
 Transfer reactions occur within a timescale comparable
with the transit time of the projectile across the nucleus
 Cross sections are a fraction of the nuclear area
 The de Broglie wavelength of a 20 MeV incident nucleon
is 1 fm and it interacts with individual nucleons at the
nuclear surface
 The projectile may lose a nucleon (stripping reaction) or
gain a nucleon (pick up reaction)
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Transfer Reactions
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Neutron-Induced Fission
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Direct Reactions
 Proceed in a single step, timescale comparable to the
time for the projectile to traverse the target (10-22 s)
 Usually only a few bodies involved in the reaction
 Excite simple degrees of freedom in nuclei
 Mostly surface dominated (peripheral collisions)
 Primarily used to study single-particle structure
 Examples: elastic and inelastic scattering
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Elastic Scattering
 Both target and projectile remain in their ground state
a+A  a+A
 Nuclei can be treated as structureless particles
100
 Example:
Investigation of
nuclear matter
density distributions
in exotic nuclei by
elastic p-scattering
(inverse kinematics)
-3
rm(r) [fm ]
10-1
10-2
11
Li matter
10-3
9
Li
-4
10
11
Li core
10-5
0
2
4
6
8
10
r [fm]
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Inelastic Scattering
 Both target and projectile nuclei retain their integrity,
they are only brought to bound excited states
a + A  a* + A*
 Can excite both single-particle or collective modes of
excitation
 Example: investigate the GMR by (,’) inelastic
scattering, gives access to nuclear incompressibility,
key parameter of nuclear EOS
Knm (Z,N) ~ r02 d2(E/A) / dr2 | r0
 Example: safe and unsafe Coulomb excitation (below and
above Coulomb barrier)
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Transfer Reactions
 One or a few nucleons are transferred between the
projectile and target nuclei
 Probes single-particle orbitals to which nucleon(s) is (are)
transferred
 Characteristics of the entrance channel determines
selectivity of the reaction, i.e. alpha particle with T=0
leads to states with the same isospin as the ground state,
but proton with T=1/2 leads to states with T=T±1
 Examples numerous: (d,p) , (p,d) , (t,p) , (t,3He) …
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Charge Exchange Reactions
 Reactions that exchange a proton for a neutron, or vice
versa
 Net effect is the same as β+ or β- decay
 But not limited by Qβ – can reach higher excited states
and giant resonances
 Many different probes: (p,n) , (d,2He) , (t,3He) , but also
with heavy ions, e.g. (7Li,7Be) or exotic particles, e.g.
(π+,π0)
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Knockout Reactions
 One or a few nucleons are ejected from either the
target and/or the projectile nuclei, the rest of the
nucleons being spectators
 Exit channel is a 3-body state
 Becomes dominant at high incident energies
 Populates single-hole states, from which spectroscopic
information can be derived
 Examples: (p,2p) , (p,pn) , (e,e’p) , heavy-ion induced
knockout, e.g. (9Be,X)
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Compound Nucleus Reactions
 The two nuclei coalesce, forming a fused system that
lasts for a relatively long time (10-20 to 10-16 s)
 De-excitation follows by a combination of particle and/or
gamma decay
 Compound system has no memory of entrance channel, the
cross section of the exit channel is independent
 Occurs for central collisions around Coulomb barrier
energies
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Types of Nuclear Reaction
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12. Nuclear Astrophysics
Linking Femtophysics with the Cosmos
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Evolution of the Universe
 Link to Nuclear Physics…
 Nuclear reactions are the only way to transmute one
element into another
 Nuclear reactions account for ‘recent’ synthesis of
elements in stars
 ‘Astrophysical’ origin of the elements
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Nuclei Power Stars
 Stars are luminous, hot, massive, self-gravitating
collections of nuclei (and electrons)
 To generate sufficient light via release of gravitational
potential energy, a star would only live for ~107 years
 Stars must have an internal energy source to prevent
gravitational collapse faster than their observed
lifetimes ~108 – 109 years
 Chemical energy too small…
 Nuclear Fusion Reactions
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Stellar Evolution
 Nuclear Reactions are responsible for both preserving
and evolving the collection of nuclei
 Preserving: nuclear reactions generate energy which
balances the self-gravitation of ~1030 kg star
 Evolving: nuclear reactions change the chemical
composition and therefore the star’s inner structure
and energy generation rate
 Stars are gravitationally confined thermonuclear
reactors
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Periodic Table Of Elements
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Elemental Abundances
 Similar distribution
everywhere
 Spans twelve orders of
magnitude
 Hydrogen: 75%
 Helium: 23%
 C to U (‘metals’): 2%
 D, Li, B and Be underabundant
 Exponential decrease up
to Fe
 A peak occurs near Fe
 Almost flat distribution
beyond Fe
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Elemental Signatures
Galactic distribution of the
1809 keV gamma ray in 26Mg
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Origin of the Elements
 Big Bang: 1H, 2H, 3He, 4He, 7Li (Z = 3)
Thermonuclear fusion in a rapidly expanding mixture of
protons and neutrons
 Interstellar Gas: Li, Be, B (Z = 5)
Spallation and fusion reactions between cosmic rays and
ambient nuclei
 Stars:
Successive energy-releasing fusion or ‘burning’ of light
elements
Low (< 8 M): Li, C, N, F (Z = 9)
Massive (> 8 M): Li, B, C, to Fe (Z = 26) (maximum BE)
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Big Bang Nucleosynthesis
 Big Bang Theory
states that the
Universe began 13.7
billion years ago in a
hot and dense state
 After 1 s only
protons, neutrons
and lighter stable
particles were
present
 At this time there existed 1 neutron for every 6 protons
 For the next 5 minutes nuclear reactions occurred…
 For 1 proton: 0.08 4He, 10-5 2H, 10-5 3He, 10-10 7Li nuclei
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Elemental Abundances: Timeline
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Stellar Evolution
 Low-mass stars (< 2.3 M):
Ignition of H, but He core becomes ‘degenerate’
before ignition
 Intermediate-mass stars (3 M < M < 8 M):
Ignition of H, He, C, O
white dwarf remnant
 High-mass (‘massive’) stars (M > 8 M):
Ignition of H, … Si
core collapse supernova
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Massive Stars
 Stars are gravitationally confined thermonuclear
reactors
 Each time one kind of ‘fuel’ runs out, contraction and
heating ensue, unless degeneracy is encountered
 For a star over 8 M contraction and heating continue
until an iron (Fe) core is made
 Gravitational collapse ensues, after no energy-providing
fuel is left
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Nuclear Burning Stages
Massive Star
Fuel
Main
Product
H
He
He
C, O
C
Ne
O
Si
Ne, Mg
O, Mg
Si, S
Fe
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Secondary
Product
14N
18O, 22Ne
s-process
Na
Al, P
Cl, Ar, K, Ca
Ti, V, Cr,
Mn, Co, Ni
Temp.
(GK)
Time
(yr)
0.02
0.2
107
106
0.8
1.5
2.0
3.5
103
3
0.8
1 week
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Death Of A Star
 Heavier elements sink to
the centre of the star
 Fusion of elements
beyond Fe requires an
input of energy
 Energy from nuclear
reactions can no longer
oppose gravitational
collapse
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Turning Hydrogen into Helium
 The fusion of four protons into helium is the only way
to produce enough energy over the timescale of the
Solar System. The main reaction is:
4 1H  4He + 2 e+ + 2 ν
 It is unlikely that 4 protons just happen to come
together to form the He nucleus ! Instead the 4
protons are processed into the He via a series of
simpler reactions:
The ‘pp chain’ or the ‘CNO cycle’
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The Proton-Proton (pp) Chain
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The pp Chain Reactions









1H
+ 1H  2H + e+ + ν
2H + 1H  3He + γ
3He + 3He  4He + 2 1H pp1, Q = 26.20 MeV
3He + 4He  7Be + γ
7Be + e-  7Li + ν
7Li + 1H  2 4He pp2, Q = 25.66 MeV
7Be + 1H  8B + γ
8B  8Be + e+ + ν
8Be  2 4He pp3, Q = 19.17 MeV
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The pp Chain Reactions
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Helium Burning:
The Triple  Chain
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The Triple  Chain

To produce nuclei beyond 8Be, three alpha particles can
combine to produce a 12C nucleus (Q = 7.275 MeV):
 +  +   12C
or
4He
+ 4He + 4He  12C

Since the probability for a 3-body reaction is extremely
low, the reaction is expected to take place in two steps

(1)
 +  +  8Be, Q = -0.092 MeV, but 8Be is
unstable (τ ~ 10-16 s) and decays back into
 +  (this explains the A = 8 mass gap)

(2)
 + 8Be  12C, Q = 7.367 MeV
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12C
E(0+)
Abundance: The Hoyle State
= 7.654 MeV
Triple alpha:
Q = 7.275 MeV
 The triple alpha process
does not account for
the full abundance of 12C
– the fourth most
abundant element in the
universe
 Hoyle (1954) predicted
the existence of a
resonant 0+ state in 12C
at E ~ 7.7 MeV
 This was later confirmed
in experiment !
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The CNO Cycle
 The CNO (carbon-nitrogen-oxygen) cycle converts H
(hydrogen) into He (helium) by a sequence of reactions
involving C, N and O isotopes and releasing energy in the
process. It occurs in stars with masses › 1.5 M
 The main reaction scheme is:
12C(p,γ)13N(e+,ν)13C(p,γ)14N(p,γ)15O(e+,ν)15N(p,α)12C
 The net result is:
4 1H  4He + 2 e+ + 2 ν, Q = 26.73 MeV
 The cycle is limited by β decay of 13N (τ ~ 10 min) and
15O (τ ~ 2 min)
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CNO Reactions
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CNO and PP Chain: Temperature
Dependence
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Breakout into the Hot CNO Cycle
 At higher
temperatures,
proton capture on
13N can begin to
compete with the β
decay and the cycle
can break out into
the hot CNO cycle
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CNO and Hot CNO Cycles
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Breakout of the Hot CNO Cycle
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Nucleosynthesis
 At still higher stellar temperatures, reactions begin
to compete that can break out of the hot CNO cycle
and ignite a runaway sequence of nuclear burning:
nucleosynthesis
 p reactions
 r (‘rapid’ neutron)
 rp (‘rapid’ proton) processes
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p Reactions
 Starting with 14O:
 14O +   17F + p
 17F + p  18Ne
 18Ne +   21Na + p etc
 Elements from oxygen (Z = 8) up to scandium (Z = 21)
are produced
 Heavier elements cannot be formed in this manner
since the Coulomb Barrier between the  particle and
the target nucleus becomes too large and prevents
their fusion
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The p Process
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Creation of Elements Beyond Fe
 Fusion of elements up to Fe (Z=26) releases energy,
the nuclear binding energy
 The nuclear binding energy is a maximum for Fe
 To produce elements heavier than Fe via nuclear fusion
requires an input of energy – the binding energy
decreases for heavy nuclei
 So, how are elements heavier than iron formed ?
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Neutron and Proton Capture
 Neutron capture reactions are responsible for the
production of elements above Fe
 The relative n-capture / β-decay efficiencies lead to two
extreme cases: s-process (slow) and r-process (rapid)
 Nuclear structure details determine the r-process:
connection between Astrophysics and Nuclear Physics
 Extreme and transient conditions near compact remnant
stars can yield nuclei on the proton-rich side of the
stability region: rp-process
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Rapid Proton/Neutron Capture
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Explosive Nucleosynthesis
 Evidence:
Technetium (Tc: Z = 43) has no ‘stable’ isotopes but
atomic Tc lines have been identified in red giants with
strong lines of Y, Zr, Ba, La (Z = 57)
 Elements beyond Fe:
Nuclear fusion is ruled out since the binding energy
(B/A) is maximal at iron
 Neutron Capture:
Can occur at ‘low’ temperatures but we need ‘high’
temperatures to activate sources of neutrons
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Neutron Capture Reactions

Stellar abundances of the elements imply two different
processes:
The s-process (slow):
Low neutron flux, N(n)  0 (108 n/cm3)
The r-process (rapid):
High neutron flux, N(n)  ∞ (1020 n/cm3)

Rapid neutron capture (r-process), and also rapid
proton capture (rp-process), produce exotic nuclei far
away from the valley of stable nuclei
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Neutron Capture Reactions
Very small (n,γ) cross sections
at N magic numbers
Evidence for nuclear processes governing nucleosynthesis
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Astrophysical Sites
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Creation of Heavy Elements
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Influence of Shell Structure
on Elemental Abundances
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Creation of Proton-Rich Nuclei
 The rp-process
lasts 10-1000 s
 It is a series of
radiative proton
capture reactions
and nuclear β+
decays that
processes the
lower mass nuclei
into higher mass
radioactive nuclei
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Endpoint of rp Process
106,107Te
very recently
studied at Jyväskylä
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Small island of
alpha decay just
above proton-rich
tin (Z=50)
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Astrophysical rp-process Sites
 Novae
 X-ray bursters
 Shock waves
passing through
the envelope of
supernova
progenitors
 Thorne-Zytkow
objects, where
a neutron star
sinks to the
centre of a
supergiant
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Creation of the Elements Movies
 X ray burster: The rp process converts
hydrogen and helium into heavier elements
up to tin (Z=50)
 Supernova explosion: The r process is
responsible for the origin of about half
the elements heavier than iron found in
nature, including elements such as gold
(Z=79) or uranium (Z=92)
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Thermonuclear Energy
Generation
 4 1H  4He
6.7 MeV/u
 3 4He  12C
0.6 MeV/u
‘triple ’
 5 4He + 84 1H  104Pd
6.9 MeV/u
‘rp process’
 Gravitational potential energy: 200 MeV/u
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Astrophysical Reactions
 The elemental abundances depend crucially on the
reaction rates (cross-sections), i.e. proton/neutron
capture vs. β decay
 These important cross-sections can now be measured
using accelerated beams of radioactive beams
 An example is the 21Na + H  22Mg + γ reaction
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Nuclear Reactions
 Nuclear reactions play a crucial role in the Universe
 They provide energy for life on Earth
 They produced all the elements we depend on
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The Nuclear Landscape
 There are ~ 280 stable nuclei
 By studying reactions between them we have produced
~ 3000 more (unstable) nuclei, which have profoundly
influenced many research areas: Big Bang, neutrino
physics, diagnostic and therapeutic medicine,
geophysics, archeology, climate studies etc
 There are ~ 4000 more nuclei which we know nothing
about and which may hold many surprises. Their study
will generate further practical applications of Nuclear
Physics
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The Role of Nuclear Physics
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Cosmophysics: A New Field

The fields of Cosmology and Astrophysics
can be combined in two ways:
1.
Cosmophysics
2.
Astrology
The End
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13. Double Beta Decay
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Introduction
 Double-beta decay is a rare transition between two
nuclei with the same mass number A involving change of
the nuclear charge Z by ±2 units
 Two beta decays occur simultaneously in a nucleus
 It is a rare second order weak interaction event
 The decay can only proceed if the initial nucleus is less
bound than the final one, and both must be more bound
than the intermediate nucleus
 These conditions are only fulfilled for even-even nuclei
 More than sixty naturally occurring isotopes are capable
of undergoing double-beta decay (energetically)
 Ten such isotopes have been experimentally observed:
48Ca 76Ge 82Se 96Zr 100Mo 116Cd 128Te 130Te 150Nd 238U
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Double Beta Decay
 Two-neutrino double beta decay ββ(2ν):
(Z,A)  (Z+2,A) + 2 electrons + 2 antineutrinos
conserves not only electric charge but also lepton number
Half-life (measured) ~1019 years
 Neutrinoless double beta decay ββ(0ν):
(Z,A)  (Z+2,A) + 2 electrons
violates lepton number conservation and is forbidden in
standard Electroweak Theory
Half-life (predicted) ~1026 years
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Massive Majorana Particles?
 Charged leptons (electrons, muons) are Dirac particles,
distinct from their antiparticles (charge conjugation)
 Neutrinos may be the ultimate neutral particles, as
envisioned by Majorana, identical with their antiparticles
 This fundamental distinction becomes important only for
massive particles
 Neutrinoless double-beta decay proceeds only when
neutrinos are massive Majorana particles
 Recent neutrino oscillation experiments suggest that
neutrinos have a non-zero mass of the order 50 meV
 The Standard Electroweak Model postulates that
neutrinos are massless and lepton number is conserved
 New Physics???
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The Signal of ββ(0ν) Decay
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Neutrino Mass
 Upper limits of
neutrino mass are
shown to the left
from two-neutrino
double-beta decay
measurements
 Neutrino-oscillation
experiments
suggest a mass
scale of the order
50 meV
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Experiment
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Theory
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Nuclear Chocolate
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The Nuclear Many-Body Problem:
Energy, Distance, Complexity
few
body
heavy
nuclei
quarks
gluons
vacuum
quark-gluon
soup QCD
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QCD
few body systems
free NN force
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effective NN force
336
Life, The Universe & Everything
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Marielle Slides…
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Isospin Dependence of Mean Field
and Residual Interactions
Shell structure predicted to change
in exotic nuclei (particularly neutron
rich).
Weak binding, impact of the particle
continuum, collective skin modes and
clustering in the skin…
Mean field modifications
 surface composed of diffuse
neutron matter
 derivative of mean field potential
weaker and spin-orbit interaction
reduced
Residual interaction modifications
 partly occupied orbits
 V monopole interaction: coupling
of proton-neutron spin-orbit
partners
 deformed intruder configurations
Quenching of ‘old’ shells and
emergence of new magic numbers in
exotic neutron-rich nuclei
Gap = S (N)-S (N+2) [keV]
Neutron Shell Gap
2N
2N
GAPN = M(Z,N-2) - M(Z,N)
+ M(Z,N+2) - M(Z,N)
N
= S2N(N) - S2N(N+2)
Neutron shell Gap
1 104
Ca (Z=20)
8000
6000
4000
2000
0
16
20
18
20
28
22
24
26
28
30
Neutron Number
8000
F-GAP
Ne-GAP
Na-GAP
Mg-GAP
5000
4000
20
N
N
GAP (keV)
6000
3000
2000
0
12
14
6000
4000
2000
16
1000
8000
GAP (keV)
7000
Ca-GAP
K-GAP
Cl-GAP
S-GAP
Si-GAP
20
16
18
Neutron Number
20
22
0
18
20
28
22
24
Neutron Number
26
28
Single-Neutron Removal in the p-sd shell
ns1/2
ns1/2 intruder
p-shell
sd-shell
E.Sauvan et al., Phys. Lett. B 491 (2000) 1, Phys. Rev. C 69 (2004) 044603.
New Magic Number at N=16
V monopole interaction : coupling of
proton-neutron spin-orbit partners
T. Otsuka et al. Phys. Rev. Lett. 87 (2001) 082502.
Examples of experimental evidence:
 Two-neutron separation energies
 In-beam fragmentation gamma spectroscopy
 1n-removal cross-sections and longitudinal
momentum distributions (direct reactions)
Present in stable nuclei but missing
in n-rich nuclei where the spinorbit partner of the valence
neutrons are not occupied by
protons
excitation energy (MeV)
Systematics of the 3/2+ in the N=15 isotones
23O
4.5
27Mg
25Ne
4.0
1f7/2
3.5
3.0
2.5
2.0
1.5
1d5/2
1.0
1d3/2
0.5
0.0
8
6
10
12
2s1/2
atomic number
The energy of the 1d3/2 neutron orbital rises when protons
are removed from its spin-orbit partner, the 1d5/2 orbital.
23O
25Ne
10
8
1d3/2
2s1/2
1d5/2
1s, 1p 1s, 1p
p
n
27Mg
1d3/2
2s1/2
1d5/2
1s, 1p 1s, 1p
p
n
12
1d3/2
2s1/2
1d5/2
1s, 1p 1s, 1p
p
n
Transfer Reaction Example
Modification of residual
interactions at N=28
46Ar(d,p)47Ar
at 10.7 A.MeV
in inverse kinematics
N=28 gap : 4.47(8)MeV
Excitation energy
spectrum for 47Ar
p3/2
p1/2
f5/2
47Ar
f7/2
MUST at GANIL/SPIRAL
L. Gaudefroy et al, PRL 97, 092501 (2006).
Knockout Reactions Example
Systematics of (e,ep)
on Stable Nuclei
Departures of measured
spectroscopic factors
from the independent
single-particle model
predictions
Electron induced proton
knockout reactions:
[A,Z] (e,ep) [A-1,Z-1]
See only 60-70% of
nucleons expected!
Effect of long-range and
short-range correlations
similar proton
separation energies
W. Dickhoff and C. Barbieri, Prog. Nucl.
377.
Part. Sci., 52 (2004)
High-Energy Single-Nucleon Removal
A New Spectroscopic Tool
core+1N
p0
core
g
d/dp
dominant
 =2
 = 2 and
=0
mixture
Target
g  Excore
-1n(Jpcore)
=2
d/dp  n
 C2S
REVIEW:Hansen & Tostevin, Ann. Rev. Nucl. Part. Sci. (2003)