Transcript Surrey 5 IBA Mini-school 6

Themes and challenges of Modern Science
•Complexity out of simplicity -- Microscopic
How the world, with all its apparent complexity and diversity can be
constructed out of a few elementary building blocks and their interactions
•Simplicity out of complexity – Macroscopic
How the world of complex systems can display such remarkable regularity
and simplicity
If we have more than one quadrupole phonon
(boson) what total angular momenta do we have?
Use m-scheme for BOSONS (no Pauli Principle)
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.
Rotational Motion in nuclei
8+
E(I)  ( ħ2/2I )J(J+1)
R4/2= 3.33
6+
4+
2+
0+
Rotational states built on
(superposed on)
vibrational modes
Vibrational
excitations
Rotational
states
8+
Ground or
equilibrium
state
6+
4+
2++
0
Systematics and
collectivity of the
lowest vibrational
modes in
deformed nuclei
Notice the smooth
systematics and low
energies for the g
mode near mid-shell,
compared with the
erratic behavior and
higher energies of the
b mode. This points to
lower collectivity of
the b vibration.
IBA – A Review and Practical Tutorial
F. Iachello and A. Arima
Drastic simplification of
shell model
 Valence nucleons
 Only certain configurations
 Simple Hamiltonian – interactions
“Boson” model because it treats nucleons in pairs
2 fermions
boson
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.)
Modeling a Nucleus
Why the IBA is the best thing since jackets
154Sm
Shell model
Need to truncate
IBA assumptions
1. Only valence nucleons
2. Fermions → bosons
J = 0 (s bosons)
J = 2 (d bosons)
3 x 1014
2+ states
Is it conceivable that
these 26 basis states
are correctly chosen to
account for the
properties of the low
lying collective states?
IBA: 26 2+ states
Why s, d
bosons?
s
Lowest state of all e-e
nuclei is 0+
 - fct gives 0+ ground state
d
First excited state in non-magic
e-e nuclei almost always 2+
 - fct gives 2+ next above 0+
Why the IBA ?????
• Why a model with such a drastic simplification ???
• Answer: Because it works !!!!!
• By far the most successful general nuclear collective
model for nuclei ever developed. Wide variety of
collective structures.
• Extremely parameter-economic
Deep relation with Group Theory !!! Dynamical
symmetries, group chains, quantum numbers
• Shell Model
 
Sph.
Def.
- (Microscopic)
• Geometric – (Macroscopic)
• Third approach — “Algebraic”
Dynamical
Symmetries
Group Theoretical
Phonon-like model with microscopic basis explicit from
the start.
IBA
Shell Mod.
Geom. Mod.
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 !!!
Background, References
•
F. Iachello and A. Arima, The Interacting Boson Model (Cambridge
University Press, Cambridge, England, 1987).
•
F. Iachello and P. Van Isacker, The Interacting Boson-Fermion Model
(Cambridge University Press, Cambridge, England, 2005)
•
R.F. Casten and D.D. Warner, Rev. Mod. Phys. 60 (1988) 389.
•
R.F. Casten, Nuclear Structure from a Simple Perspective, 2nd Edition
(Oxford Univ. Press, Oxford, UK, 2000), Chapter 6 (the basis for most of
these lectures).
•
D. Bonatsos, Interacting boson models of nuclear structure, (Clarendon
Press, Oxford, England, 1989)
•
Many articles in the literature
Note key point:
 Bosons in IBA are pairs of fermions in valence shell
Number of bosons for a given nucleus is a fixed number
154
62
Sm 92
N = 6
5 = N

NB = 11
Me et al.
An early conference on the IBA
(Featuring young Alison)
Review of phonon creation and destruction operators
What is a creation operator? Why useful?
A) Bookkeeping – makes calculations very simple.
B) “Ignorance operator”: We don’t know the structure of a phonon but, for many
predictions, we don’t 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).
Brief, simple, trip into the Group
Theory of the IBA
DON’T 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”
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.
s†s, s†d, d†s, d†d
Note on ” ~ “ ‘s: I often forget them
Gr. Theor.
classification
of
Hamiltonian
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 s†s, d†s, s†d, d†d are generators of the group U(6).
† † some
† quantum
†
Generators
conserve
ex:  d † s, s:†define
s  n d nand

d
ss
s

s
sd
s  n d n number.

s
s
†
Ex.: 36 Ops of IBA all conserve
boson †number
 d † sntotal
s n d n s  s sd s n d n s
N = s†s + d†d = ns + nd
  n s  s †s  d †s n d n s
†
†
†
 all
 


N
,
s
d

N
s
d

s
† with
e.g: Operator


Casimir:
that
commutes
the
generators
of
adN
group.
  n s s s  n d  1 n s n d  1, n s  1

 Therefore, its
† The energies are defined
eigenstates have a specific value of
the
q.#†of
that 
group.

Ns
d

s

 n d  1 n s  n s   n s  1  n d  1, ndN
s 1
solely in terms of that q. #. N is Casimir
† U(6).
 Nsof
d   Ns † d   0
 n d  1 n s n d  1, n s  1
d † s n dThe
n s set of degenerate states with that value of the q. #.
Representations of a group:
d † s, s † s   d † s

Hamiltonian written solely in terms of Casimirs can be solved analytically
or:
A
Sub-groups:
Subsets of generators that commute among themselves.
e.g: d†d
25 generators—span 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 #
Simple example of dynamical symmetries, group chain,
degeneracies
[H, J 2 ] = [H, J Z ] = 0
J, M constants of motion
Let’s illustrate group chains and degeneracy-breaking.
Consider a Hamiltonian that is a function ONLY of:
That is:
s†s + d†d
H = a(s†s + d†d) = 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).
H’ = H + b d†d = 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
2a N + 2
a
H’ = aN + b d†d = a N + b nd
N+1
b
2
1
0
0
2b
N
0
nd
E
U(6)
H’ = aN
U(5)
+
b d† d
OK, here’s the key point -- get this if nothing else:
Concept of a Dynamical Symmetry
N
OK, here’s 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.
Group Structure of the IBA
U(5)
s boson :
1
d boson :
5
vibrator
U(6)
SU(3)
rotor
Magical group
theory stuff
happens here
O(6)
γ-soft
Symmetry Triangle
of the IBA
Def.
Sph.
(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)
IBA Hamiltonian
Counts the number of d bosons out of N bosons, total.
Conserves
the
of with
d bosons.
Gives
in the
The rest
arenumber
s-bosons:
Es = 0 since
we terms
deal only
Hamiltonian wherewith
theexcitation
energiesenergies.
of configurations of 2 d
+ bosons depend on their total combined angular
ExcitationAllows
energies
ONLY on the in
number
of dd d momentum.
fordepend
anharmonicities
the phonon
multiplets. bosons. E(0) = 0, E(1) = ε , E(2) = 2 ε.
d
Mixes d and s components of the wave functions
Most general IBA Hamiltonian in terms
with up to four boson operators (given N)
U(5)
Spherical, vibrational nuclei
Simplest Possible IBA Hamiltonian –
given by energies of the bosons with NO interactions
H   d n d   s n s = E of d bosons + E of s bosons
 d

d
†
d

†
 s s s
Excitation energies so, set s = 0, and drop subscript d on d
H   nd
What is spectrum? Equally spaced levels defined by number of d bosons
3
6+, 4+, 3+, 2+, 0+
2
4+, 2+, 0+
1
2+
0
0+
nd
What J’s? M-scheme
Look familiar? Same as
quadrupole vibrator.
U(5)
E (nd, , J) = α nd + β nd (nd + 4) + 2γ ( +4)
+ 2δJ (J + 1)
Harmonic U(5):
β, γ, δ = 0;
E ~ nd
But, U(5) is a rich symmetry that allows anharmonicities
Degeneracy breaking of multiplets [Cj terms]
But Ψ’s remain unchanged since H [ U(5)]) cannot change (mix) nd
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 = e[s†d + d†s + χ (d† d )(2)]
Specifies relative strength of this term
E2 transitions in U(5)
• χ = 0 so
• T = e[s†d + d†s]
• Can create or destroy a single d boson, that is a
single phonon.
3
6+, 4+, 3+, 2+, 0+
2
4+, 2+, 0+
1
2+
0
0+
nd
Vibrator
Vibrator (H.O.)
E(I) = n ( 0 )
8+. . .
R4/2= 2.0
6+. . .
2+
0+
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 = ε nd -  Q  Q
Competition:
ε nd
QQ
Q = e[s†d + d†s + χ (d† d )(2)]
term gives vibrator.
term gives deformed nuclei.
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
SU(3)
Deformed nuclei


M
Examples of energies of different representations of
SU(3), that is, different (, )
E = A [ 2 + 2 +   + 3 ( + ) ] + BJ (J + 1)
f (, )
0 2
0 1
(28, 2)
N = 16
(32, 0)
E ( 0 2 ) = A [ f (28, 2) – f (32, 0) ]
=A
2
[ (28) + (2)2 + 2(28) + 3(28 + 2) ]
– [ (32)2 + 0 + 0 + 3(32 + 0) ]
= A [ 934 - 1120 ] = -186 A, A < 0
So if E ( 0 2 ) ~ 1 MeV, then A ~ 5-6 keV
More generally,
E ( 0 2) = A [ (2N)2 + 3 (2N) ] – [ (2N - 4)2 +4+(2N - 4)(2) +3 (2N – 4 + 2)]
= A 4N2 + 6N – [ 4N2 – 16N + 16 + 4 + 4 N – 8 + 6N – 12 + 6)]
= A (12N – 6)
~ 2N - 1
Typical SU(3) Scheme
SU(3)
K bands in (, ) :
K = 0, 2, 4, - - - - 

O(3)
Signatures of SU(3)
E =E
B(
g)
Z
0
0
B(
B(
g)
g)
E(
-vib )
1/6
(2N - 1)
Signatures of SU(3)
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
E2 Transitions in SU(3)
†
†
† (2)
Q = (s d + d s) + (- 7 ) (d d )
2
Q is also in H and is a Casimir operator of SU(3),
so conserves , .
E2: Δ , ) = 0
SU(3)
(
 J  2   J  1
3

2
B(E2; J + 2 →J)yrast  e B

  2N  J   2N  3  J 
4   2 J  3  2 J  5  
+
B(E2; 2 1+ → 0 1 )  e B2
N  2 N  3
5
2
N for large N
B( E 2;4 1  2 1 ) 10  (2 N  2)(2 N  5) 

B( E 2;2 1  0 1 ) 7  (2 N )(2 N  3) 
Alaga rule
Finite N correction
Typ. Of many IBA predictions → Geometric Model as N → ∞
Δ , ) = 0
(
“β
”
→
γ
C
o
l
l
e
c
t
i
v
e
B
(
E
2
)
s
Example of finite boson number
effects in the IBA
B(E2: 2 0):
U(5) ~ N;
SU(3) ~ N(2N + 3) ~ N2
H = ε nd -  Q  Q
and keep the parameters constant.
What do you predict for this B(E2) value??
B(E2)
Mid-shell
N2
~N
N
!!!
O(6)
Axially asymmetric nuclei
(gamma-soft)
Transition Rates
Q = (s† d + d†s)
T(E2) = eBQ
O(6)
E2
Note: Uses
Δσ = 0 Δτ = 1
J

+ 1


J
J 
B(E2; J + 2 → J) = e 2B  N -   N   4   2 
2 
2  J +5

B(E2; 21 → 01) ~ e 2B
N  N + 4
5
N2
Consider E2 selection rules
Δσ = 0
0+(σ = N - 2) – No allowed decays!
Δτ = 1
0+( σ = N, τ = 3) – decays to 2 2 , not 2 1
χ=
o
196Pt:
Best (first) O(6) nucleus
g-soft
Xe – Ba
O(6) - like
Classifying Structure -- The Symmetry Triangle
Deformed
Sph.
Most nuclei do not exhibit the idealized symmetries but
rather lie in transitional regions. Mapping the triangle.
The IBA: convenient model that spans the entire
triangle of colllective structures
Sph. Driving
H = ε nd
Def. Driving
- QQ
Parameters: /ε , c (within Q)
Competition: /ε: 0 to infinity
ζ
H = c [ ( 1 – ζ ) nd Qχ ·Qχ ]
4NB
Span triangle
with z and c
c is an overall scale factor giving the overall energy
scale. Normally, it is fit to the first 2+ state.
z
Parameters already known for many nuclei
c