IBA

1
IBA An Introduction and Overview

• Basic Ideas, underpinnings, Group Theory, basic
  predictions

2

• Shell Model - (Microscopic)
• Geometric (Macroscopic)
• Third approach Algebraic

Dynamical Symmetries
Group Theoretical
Phonon-like model with microscopic basis explicit
from the start.
IBA
Microscopic
Collectivity
Geom. Mod.
Shell Mod.
3
(No Transcript)
4
Why do we need to bother with such a model?

• Remember 3 x 1014 ?
• We simply MUST simplify the problem.
• As it turns out, the IBA is
• The most successful macroscopic model
• The only collective model in which it is even
  possible in practice to calculate many
  observables

5
Shell Model Configurations
Fermion configurations
Boson configurations (by considering only
configurations of pairs of fermions with J 0
or 2.)
6
IBM
Assume fermions couple in pairs
to bosons of spins 0 and 2
valence
0 s-boson 2 d-boson
s boson is like a Cooper pair d boson is like a
generalized pair. Create ang. mom. with d bosons

• Valence nucleons only
• s, d bosons creation and destruction operators
• H Hs Hd Hinteractions
• Number of bosons fixed in a given
  nucleus N ns nd
• ½ of val. protons ½ val.
  neutrons

7
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
8
The IBA an audacious, awesome leap
Or, why the IBA is the best thing since
tortellini Magnus
154Sm
Shell model
Is it conceivable that these 26 basis states
could possibly be correctly chosen to account for
the properties of the low lying collective states?
Need to truncate IBA assumptions
1. Only valence nucleons
9
Why the IBA ?????

• Why a model with such a drastic simplification
  Oversimplification ???
• Answer Because it works !!!!!
• By far the most successful general nuclear
  collective model for nuclei ever developed
• Extremely parameter-economic
• Deep relation with Group Theory !!! Dynamical
  symmetries, group chains, quantum numbers

10
IBA Models IBA 1 No distinction of p, n
IBA 2 Explicitly write p, n parts IBA 3,
4 Take isospin into account p-n pairs
IBFM Int. Bos. Fermion Model for Odd
A nuclei H He e(core) Hs.p.
Hint IBFFM Odd odd nuclei
(f, p) bosons for ? - states
Parameters !!! IBA-1 2 Others
4 to 20 !!!
11
5
3
2
12
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
13
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. zero (s) or 2 (d).
14
IBA Hamiltonian
Most general IBA Hamiltonian in terms with up to
four boson operators (given, fixed N)
AARRGGHHH !!! We will greatly simplify this soon
but it is useful to look at its structure
15
Simplest Possible IBA Hamiltonian
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.
6, 4, 3, 2, 0 4, 2, 0 2 0
3 2 1 0 nd
16
IBA Hamiltonian
Most general IBA Hamiltonian in terms with up to
four boson operators (given N)
These terms CHANGE the numbers of s and d bosons
MIX basis states of the model
Crucial for structure
Crucial for masses
17
IBA Hamiltonian
Most general IBA Hamiltonian in terms with up to
four boson operators (given N)
Complicated and not really necessary to use all
these terms and all 6 parameters
Simpler form with just two parameters RE-GROUP
TERMS ABOVE
Q es ds ? (d )(2)
H e nd - ? Q ? Q
Competition e nd
term gives vibrator.
? Q ? Q term gives deformed
nuclei. Note 3 parameters. BUT
H aH have identical wave functions, q.s,
sel. rules, trans. rates. Only the energy SCALE
differs. STRUCTURE 2 parameters.
MASSES need scale
18
Brief, simple, trip into the Group Theory of the
IBA
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 (we will see soon) name for this
application of Group Theory is Spectrum
Generating Algebras
19
IBA has a deep relation to Group theory
To understand the relation, consider 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
ss, sd, ds, dd
Note on s I often forget them
20
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
21
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
22
Simple example of dynamical symmetries, group
chain, degeneracies
H, J 2 H, J Z 0 J, M
constants of motion
23
Lets ilustrate 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).
24
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
25
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
26
Concept of a Dynamical Symmetry
OK, heres the key point -- get this if nothing
else
N
Spectrum generating algebra !!
27
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.

28
Group Structure of the IBA
1
s boson
5
d boson
Magical group theory stuff happens here
Symmetry Triangle of the IBA (everything we do
from here on will be discussed in the context of
this triangle. Stop me now if you do not
understand up to here)
Def.
Sph.
29
Dynamical Symmetries The structural benchmarks

• U(5) Vibrator spherical nucleus that can
  oscillate in shape
• SU(3) Axial Rotor can rotate and vibrate
• O(6) Axially asymmetric rotor ( gamma-soft)
  squashed deformed rotor

30
Dynamical Symmetries
Vibrator
Rotor
Gamma-soft rotor
31
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.
32
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
33
U(5)Spherical, vibrational nuclei
34
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)
35
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
3 2 1 0 nd
6, 4, 3, 2, 0 4, 2, 0 2 0
What Js? M-scheme Look familiar? Same as
quadrupole vibrator.
36
U(5) Multiplets
Important as a benchmark of structure, but also
since the U(5) states serve as a convenient set
of basis states for the IBA
37
Which nuclei are U(5)?

• No way to tell a priori (until better microscopic
  understanding of IBA is available).
• More generally, phenomenological models like the
  IBA predict nothing on their own. They can
  predict relations among observables for a given
  choice of Hamiltonian parameters but they dont
  tell us which parameter values apply to a given
  nucleus. They dont tell us which nuclei have
  which symmetry, or perhaps none at all. They need
  to be fed.
• The nuclei provide their own food but the IBA
  is not gluttonous
• a couple of observables allow us to
  pinpoint structure.
• Let the nuclei tell us what they are doing !!!!
• Dont force an interpretation on them

38
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
39
E2 electromagnetic transition rates in the IBA
T e Q es ds ? (d )(2)
Specifies relative strength of this term
? is generally fit as a parameter but has
characteristic values in each dynamical symmetry
Finite, fixed number of bosons has a huge effect
compared ot the geometrical model
40
Note TWO factors in B(E2). In geometrical
model B(E2) values are proportional to the number
of phonons in the initial state.
In IBA, operator needs to conserve total boson
number so gamma ray transitions proceed by
operators of the form sd. Gives TWO square
roots that compete.
41
(No Transcript)
42
Finite Boson Number Effects B(E2) Values
6 5 4 3 2 1
Geom. Vibrator
Slope 1.51
B(E2 J ? J-2) Yrast (gsb) states
IBA, U(5), N6
2 4 6 8
10 12
2? 0
J
43
Classifying Structure -- The Symmetry Triangle
Def.
Sph.
Have considered vibrators (spherical nuclei).
What about deformed nuclei ??
44
SU(3)Deformed nuclei(but only a special subset)
45
?
?
M
( or M, which is not exactly the same as K)
46
(No Transcript)
47
Typical SU(3) Scheme
SU(3) ?
O(3)
K bands in (?, ?) K 0, 2, 4, - - - -
?
48
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
49
(No Transcript)
50
Another 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??
!!!
51
Signatures of SU(3)
52
Signatures of SU(3) E? E ? B (
? ? g ) ? 0 Z ?
? 0 B ( ? ? g ) B ( ? ? g )
E ( ?-vib ) ? (2N - 1)
53
O(6)Axially asymmetric nuclei(gamma-soft)
54
(No Transcript)
55
(No Transcript)
56
Note Uses ? o
57
(No Transcript)
58
196Pt Best (first) O(6) nucleus ?-soft
59
Xe Ba O(6) - like
60
Classifying Structure -- The Symmetry Triangle
Most nuclei do not exhibit the idealized
symmetries but rather lie in transitional
regions. Trajectories of structural evolution.
Mapping the triangle.