Review

1
Review

• Short range force, Pauli Principle ? Shell
  structure, magic numbers, concept of valence
  nucleons
• Residual interactions ? favoring of 0 coupling
  0 ground states for all even-even nuclei
• Concept of seniority ? lowest states have low
  seniority, huge simplification (n-body
  calculations often reduce to 2-body !)
• ? constant g factors, constant energies in singly
  magic or near magic nuclei, parabolic B(E2)
  systematics, change in sign of quadrupole moments
  (prolate-oblate shapes) across a shell

2
Effects of monopole interactions
Between 40Zr and 50Sn protons fill 1g9/2 orbit.
Large spatial overlap with neutron 1g7/2 orbit.
1g7/2 orbit more tightly bound. Lower energy
3
Lecture 3Collective behavior in nuclei and
collective models
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How does structure evolve?

• Benchmarks
• Magic nuclei spherical, stiff
• Nuclei with only one kind of valence nucleon
  like the 2-particle case
• Nuclei with both valence protons and neutrons
  mixing of configurations, complex wave functions
  with components from every configuration
  (10big).
• Is there another way? YES !!!
• Macroscopic perspective. Many-body approach,
  collective coordinates, modes

5
Microscopic origins of collectivity
correlations, configuration mixing and
deformation Residual interactions
Crucial for structure
6
J 2, one phonon vibration
7
More than one phonon? What angular momenta?
M-scheme for bosons
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Deformed Nuclei

• What is different about non-spherical nuclei?
• They can ROTATE !!!
• They can also vibrate
• For axially symmetric deformed nuclei there are
  two low lying vibrational modes called b gand g
• So, levels of deformed nuclei consist of the
  ground state, and vibrational states, with
  rotational sequences of states (rotational bands)
  built on top of them.

10
Rotational Motion in nuclei
E(I) ? ( h2/2I )J(J1) R4/2 3.33
11
Paradigm Benchmark 700
333 100 0 Rotor J(J
1)
Amplifies structural differences Centrifugal
stretching Deviations
The value of paradigms
? Interpretation without rotor paradigm
Exp.
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Rotational states built on (superposed on)
vibrational modes
Vibrational excitations
Rotational states
Ground or equilibrium state
13
Systematics and collectivity of the lowest
vibrational modes in deformed nuclei
Notice that the the b mode is at higher energies
( 1.5 times the g vibration near mid-shell) and
fluctuates more. This points to lower
collectivity of the b vibration. Remember for
later !
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How can we understand collective behavior

• Do microscopic calculations, in the Shell Model
  or its modern versions, such as with density
  functional theory or Monte Carlo methods. These
  approaches are making amazing progress in the
  last few years. Nevertheless, they often do not
  give an intuitive feeling for the structure
  calculated.
• Collective models, which focus not on the
  particles but the structure and symmetries of the
  many-body, macroscopic system itself. They are
  not predictive in the same way as microscopic
  calculations but they can reveal coherent
  behavior more clearly in many cases.
• We will illustrate collective models with the
  IBA, historically, by far the most successful and
  parameter-efficient collective model.

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Shell Model Configurations
Fermion configurations
Roughly, gazillions !! Need to simplify
The IBA
Boson configurations (by considering only
configurations of pairs of fermions with J 0
or 2.)
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Why s, d bosons?  
Lowest state of all e-e
First excited state in non-magic
s nuclei is 0
d e-e nuclei almost always
2 ? - fct gives 0 ground state
? - fct gives 2 next above 0
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Modeling a Nucleus
Why the IBA is the best thing since jackets
154Sm
Shell model
Need to truncate IBA assumptions
Is it conceivable that these 26 basis states are
correctly chosen to account for the properties of
the low lying collective states?
1. Only valence nucleons
20
Note key point Bosons in IBA are pairs of
fermions in valence shell Number of bosons for a
given nucleus is a fixed number
?
N? 6 5 N? ? NB 11
Basically the IBA is a Hamiltonian written in
terms of s and d bosons and their interactions.
It is written in terms of boson creation and
destruction operators. Lets briefly review their
key properties.
21
Review of phonon creation and destruction
operators

• What is a creation operator? Why useful?
• Bookkeeping makes calculations very simple.
• B) Ignorance operator We dont know the
  structure of a phonon but, for many predictions,
  we dont need to know its microscopic basis.

is a b-phonon number
operator. For the IBA a boson is the same as a
phonon think of it as a collective excitation
with ang. mom. 0 (s) or 2 (d).
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IBA has a deep relation to Group theory
That relation is based on the operators that
create, destroy s and d bosons
s, s, d, d operators
Ang. Mom. 2
d? , d? ? 2, 1, 0, -1, -2
Hamiltonian is written in terms of s, d operators
Since boson number is conserved for a given
nucleus, H can only contain bilinear terms
36 of them.
Gr. Theor. classification of Hamiltonian
Group is called U(6)
ss, sd, ds, dd
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Brief, simple, trip into the Group Theory of the
IBA
Next 8 slides give an introduction to the Group
Theory relevant to the IBA. If the discussion of
these is too difficult or too fast, dont worry,
you will be able to understand the rest anyway.
Just take a nap for 5 minutes. In any case, you
will have these slides on the web and can look at
them later in more detail if you want.
DONT BE SCARED You do not need to understand
all the details but try to get the idea of the
relation of groups to degeneracies of levels and
quantum numbers
A more intuitive name for this application of
Group Theory is Spectrum Generating Algebras
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Concepts of group theory First, some fancy words
with simple meanings Generators, Casimirs,
Representations, conserved quantum numbers,
degeneracy splitting
Generators of a group Set of operators , Oi
that close on commutation.
Oi , Oj Oi Oj - Oj Oi Ok i.e., their
commutator gives back 0 or a member of the set
For IBA, the 36 operators ss, ds, sd, dd
are generators of the group U(6).
Generators define and conserve some quantum
number. Ex. 36 Ops of IBA all conserve total
boson number
ns nd
N ss d
Casimir Operator that commutes with all the
generators of a group. Therefore, its
eigenstates have a specific value of the q. of
that group. The energies are defined solely in
terms of that q. . N is Casimir of
U(6). Representations of a group The set of
degenerate states with that value of the q. . A
Hamiltonian written solely in terms of Casimirs
can be solved analytically
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Sub-groups Subsets of generators that commute
among themselves. e.g dd 25
generatorsspan U(5) They conserve nd ( d
bosons) Set of states with same nd are the
representations of the group U(5)
Summary to here
Generators commute, define a q. , conserve that
q. Casimir Ops commute with a set of
generators ? Conserve that quantum ? A
Hamiltonian that can be written in terms of
Casimir Operators is then diagonal for states
with that quantum Eigenvalues can then be
written ANALYTICALLY as a function of that
quantum
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Simple example of dynamical symmetries, group
chain, degeneracies
H, J 2 H, J Z 0 J, M
constants of motion
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Lets illustrate group chains and
degeneracy-breaking.
Consider a Hamiltonian that is a function ONLY
of ss dd That is H a(ss
dd) a (ns nd ) aN In H, the energies
depend ONLY on the total number of bosons, that
is, on the total number of valence nucleons. ALL
the states with a given N are degenerate. That
is, since a given nucleus has a given number of
bosons, if H were the total Hamiltonian, then all
the levels of the nucleus would be degenerate.
This is not very realistic (!!!) and suggests
that we should add more terms to the Hamiltonian.
I use this example though to illustrate the idea
of successive steps of degeneracy breaking being
related to different groups and the quantum
numbers they conserve. The states with given N
are a representation of the group U(6) with the
quantum number N. U(6) has OTHER
representations, corresponding to OTHER values of
N, but THOSE states are in DIFFERENT NUCLEI
(numbers of valence nucleons).
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H H b dd aN b nd Now, add a term to
this Hamiltonian Now the energies depend not
only on N but also on nd States of a given nd
are now degenerate. They are representations of
the group U(5). States with different nd are not
degenerate
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H aN b dd a N b nd
N 2
2a
N 1
a
2
2b
Etc. with further terms
1
b
N
0
0
0
nd
E
U(6) U(5)
H aN b dd
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Concept of a Dynamical Symmetry
OK, heres the key point
N
Spectrum generating algebra !!
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OK, heres what you need to remember from the
Group Theory

• Group Chain U(6) ? U(5) ? O(5) ? O(3)
• A dynamical symmetry corresponds to a certain
  structure/shape of a nucleus and its
  characteristic excitations. The IBA has three
  dynamical symmetries U(5), SU(3), and O(6).
• Each term in a group chain representing a
  dynamical symmetry gives the next level of
  degeneracy breaking.
• Each term introduces a new quantum number that
  describes what is different about the levels.
• These quantum numbers then appear in the
  expression for the energies, in selection rules
  for transitions, and in the magnitudes of
  transition rates.

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Group Structure of the IBA
1
s boson
5
d boson
Magical group theory stuff happens here
Symmetry Triangle of the IBA
Def.
Sph.
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IBA Hamiltonian
Counts the number of d bosons out of N bosons,
total. The rest are s-bosons with Es 0 since
we deal only with excitation energies.
Excitation energies depend ONLY on the number
of d-bosons. E(0) 0, E(1) e , E(2) 2 e.
Conserves the number of d bosons. Gives terms in
the Hamiltonian where the energies of
configurations of 2 d bosons depend on their
total combined angular momentum. Allows for
anharmonicities in the phonon multiplets.
dd
d
Mixes d and s components of the wave functions
Most general IBA Hamiltonian in terms with up to
four boson operators (given N)
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U(5)Spherical, vibrational nuclei
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Simplest Possible IBA Hamiltonian given by
energies of the bosons with NO interactions
E of d bosons E of s bosons
Excitation energies so, set ?s 0, and drop
subscript d on ?d
What is spectrum? Equally spaced levels defined
by number of d bosons
What Js? M-scheme Look familiar? Same as
quadrupole vibrator. U(5) also includes
anharmonic spectra
36
E2 Transitions in the IBA Key to most
tests Very sensitive to structure E2
Operator Creates or destroys an s or d boson or
recouples two d bosons.
Must conserve N
T e Q es ds ? (d )(2)
Specifies relative strength of this term
37
E2 transitions in U(5)

• ? 0 so
• T es ds
• Can create or destroy a single d boson, that is
  a single phonon.

38
Vibrator
Vibrator (H.O.) E(I) n (? ?0 ) R4/2 2.0
8. . .
6. . .
2
0
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IBA Hamiltonian
Complicated and not really necessary to use all
these terms and all 6 parameters
Simpler form with just two parameters RE-GROUP
TERMS ABOVE
H e nd - ? Q ? Q
Q es ds ? (d )(2)
Competition e nd
term gives vibrator.
? Q ? Q term gives deformed
nuclei.
This is the form we will use from here on
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Relation of IBA Hamiltonian to Group Structure
We will see later that this same Hamiltonian
allows us to calculate the properties of a
nucleus ANYWHERE in the triangle simply by
choosing appropriate values of the parameters
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SU(3)Deformed nuclei
42
?
?
M
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Typical SU(3) Scheme

• Characteristic signatures
• Degenerate bands within a group
• Vanishing B(E2) values between groups
• Allowed transitions
• between bands within a group
• Where? N 1-4, Yb, Hf

SU(3) ?
O(3)
K bands in (?, ?) K 0, 2, 4, - - - -
?
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Totally typical example
Similar in many ways to SU(3). But note that the
two excited excitations are not degenerate as
they should be in SU(3). While SU(3) describes
an axially symmetric rotor, not all rotors are
described by SU(3) see later discussion
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Example of finite boson number effects in the IBA
B(E2 2 ?0) U(5) N SU(3) N(2N
3) N2
H e nd - ? Q ? Q and keep the
parameters constant. What do you predict for this
B(E2) value??
!!!
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O(6)Axially asymmetric nuclei(gamma-soft)
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Note Uses ? o
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196Pt Best (first) O(6) nucleus ?-soft
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Classifying Structure -- The Symmetry Triangle
Most nuclei do not exhibit the idealized
symmetries but rather lie in transitional
regions. Mapping the triangle.
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Mapping the Entire Triangle
H e nd - ? Q ? Q
Parameters , c (within Q)
?/e
2 parameters 2-D surface
?/e
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168-Er very simple 1-parameter calculation

H e nd - ? Q ? Q
e 0

• H - ? Q ? Q
• is just scale factor
• So, only parameter is c

?/e
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1
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Universal IBA Calculations for the SU(3)
O(6) leg
2
H - ? Q Q ? is just energy scale
factor ?s, B(E2)s independent of ?
Results depend only on ?
and, of course, vary with NB
Can plot any observable as a set of
contours vs. NB and ?.
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Universal O(6) SU(3) Contour Plots
H -? Q Q ? 0 O(6)
? - 1.32 SU(3)

SU(3)
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Mapping the entire triangleTechnique of
orthogonal crossing contours (OCC)
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H has two parameters. A given observable can only
specify one of them. What does this imply? An
observable gives a contour of constant values
within the triangle
2.9
R4/2
63

A simple way to pinpoint structure. What do we
need?

• At the basic level 2 observables (to map any
  point in the symmetry triangle)
• Preferably with perpendicular trajectories in the
  triangle

Simplest Observable R4/2
Only provides a locus of structure
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Contour Plots in the Triangle
R4/2
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We have a problem
Fortunately
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Mapping Structure with Simple Observables
Technique of Orthogonal Crossing Contours
? - soft
Vibrator
Rotor
Burcu Cakirli et al. Beta decay exp. IBA calcs.
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Evolution of Structure
Complementarity of macroscopic and microscopic
approaches. Why do certain nuclei exhibit
specific symmetries? Why these evolutionary
trajectories? What will happen far
from stability in regions of proton-neutron
asymmetry and/or weak binding?
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Collective models and masses, binding energies,
or separation energies
Crucial for structure
Crucial for masses
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Two-neutron separation energies
Binding Energies
S2n A BN S2n (Coll.)
Normal behavior linear segments with drops
after closed shells
Discontinuities at first order phase transitions
Use any collective model to calculate the
collective contributions to S2n.
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Which 0 level is collective and which is a
2-quasi-particle state?
Evolution of level energies in rare earth nuclei
But note
McCutchan et al
Do collective model fits, assuming one or the
other 0 state, at 1222 or 1422 keV, is the
collective one. Look at calculated contributions
to separation energies. What would we expect?
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Collective contributions to masses can vary
significantly for small parameter changes in
collective models, especially for large boson
numbers where the collective binding can be quite
large.
S2n(Coll.) for alternate fits to Er with N 100
S2n(Coll.) for two calcs.
B.E.(z,c)
IBA
B.E (MeV)
Gd Garcia Ramos et al, 2001
Masses a new opportunity complementary
observable to spectroscopic data in pinning down
structure, especially in nuclei with large
numbers of valence nucleons. Strategies for best
doing that are still being worked out.
Particularly important far off stability where
data will be sparse.
Cakirli et al, 2009