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
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Review Short range force, Pauli Principle Shell structure, magic numbers, concept of valence nucleons Residual interactions favoring of 0 + coupling:
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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
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
Effects of monopole interactions
Lecture 3
Collective behavior in nuclei and collective models
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
Crucial for structure
Microscopic origins of collectivity correlations, configuration mixing and
deformation: Residual interactions
J = 2, one phonon vibration
More than one phonon? What angular momenta? M-scheme for bosons
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 and
• 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.
0+
2+
4+
6+
8+
E(I) ( ħ2/2I )J(J+1)
R4/2= 3.33
Rotational Motion in nuclei
6+ 690
4+ 330
0+ 0
2+ 100
J E (keV)
?Interpretation without rotor
paradigm
Paradigm
Benchmark
700
333
100
0
Rotor J(J + 1)
Amplifies structural
differences
Centrifugal stretching
Deviations
Identify additional
degrees of freedom
The value of paradigms
Exp.
0+2+4+
6+
8+
Rotational states
Vibrational excitations
Rotational states built on (superposed on)
vibrational modes
Ground or equilibrium
state
Systematics and collectivity of the lowest vibrational
modes in deformed nuclei
Notice that the the mode is at higher
energies (~ 1.5 times the vibration near
mid-shell)* and fluctuates more. This
points to lower collectivity of the
vibration.
* Remember for later !
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.
IBA – A Review and Practical Tutorial
Drastic simplification of
shell model
Valence nucleons
Only certain configurations Simple Hamiltonian – interactions
“Boson” model because it treats nucleons in pairs
2 fermions boson
F. Iachello and A. Arima
Shell Model Configurations
Fermion configurations
Boson configurations
(by considering only configurations of pairs of fermions
with J = 0 or 2.)
The IBA
Roughly, gazillions !!Need to simplify
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+
Modeling a NucleusModeling a Nucleus
154Sm 3 x 1014 2+ states
Why the IBA is the best thing since jackets
Shell model
Need to truncate
IBA assumptions1. Only valence nucleons2. Fermions → bosons
J = 0 (s bosons)
J = 2 (d bosons)
IBA: 26 2+ states
Is it conceivable that these 26 basis states are correctly chosen to account for the
properties of the low lying collective
states?
Note key point:Note key point:
Bosons in IBA are pairs of fermions in valence shell
Number of bosons for a given nucleus is a fixed number
1549262 Sm
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. Let’s briefly review their key properties.
Review of phonon creation and destruction operatorsReview of phonon creation and destruction operators
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).
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.
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.
s†s, s†d, d†s, d†d
Gr. Theor. classification
of Hamiltonian
IBAIBA has a deep relation to Group theory
Group is called
U(6)
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 name for this application of Group Theory is
“Spectrum Generating Algebras”
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, don’t 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.
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).
Generators: define and conserve some quantum number.
Ex.: 36 Ops of IBA all conserve total boson number = ns + nd N = s†s + d† 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
ex: † † † † † †, d s d sd s s s n n d ss s s sd s n n
† † †
† †
†
†
1 1, 1
1 1 1, 1
1 1, 1
s d s d s
s d s
s d s d s
d s s s d s
d s d s
d s
d sn n n s sd s n n
n s s d s n n
n s s n n n n
n n n n n n
n n n n
d s n n
or: † † †,d s s s d s
e.g: † † †
† †
,N s d N s d s dN
Ns d s dN
† † 0Ns d Ns d
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: s†s + d†d
That is: 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
N
N + 1
N + 2
nd
1 2
0
a
2a
E
0 0
b
2b
H’ = aN + b d†d = a N + b nd
U(6) U(5)
H’ = aN + b d†d
Etc. with further term
s
Concept of a Dynamical Symmetry
N
OK, here’s the key point :
Spectrum generating algebra !!
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 IBAGroup Structure of the IBA
s boson :
d boson :
U(5)vibrator
SU(3)rotor
O(6)γ-soft
1
5U(6)
Sph.
Def.
Magical group
theory stuff
happens here
Symmetry Triangle of
the IBA
Most general IBA Hamiltonian in terms with up to four boson operators (given N)
IBA Hamiltonian
Mixes d and s components of the wave functions
d+
d
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) = ε , E(2) = 2 ε.
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.d
U(5)
Spherical, vibrational nuclei
What J’s? M-scheme
Look familiar? Same as quadrupole vibrator.
U(5) also includes anharmonic spectra
6+, 4+, 3+, 2+, 0+
4+, 2+, 0+
2+
0+
3
2
1
0
nd
Simplest Possible IBA Hamiltonian – given by energies of the bosons with NO interactions
† †
d d s s
d s
H n n
d d s s
Excitation energies so, set s = 0, and drop subscript d on d
dH n
What is spectrum? Equally spaced levels defined by number of d bosons
= E of d bosons + E of s bosons
EE2 Transitions in the IBA2 Transitions in the IBAKey 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†s + χ (d† )(2)]d d
Specifies relative strength of this term
E2 transitions in U(5)
• χ = 0 so
• T = e[s† + d†s]
• Can create or destroy a single d boson, that is a single phonon.
d
6+, 4+, 3+, 2+, 0+
4+, 2+, 0+
2+
0+
3
2
1
0
nd
0+
2+
6+. . .
8+. . .
Vibrator (H.O.)
E(I) = n ( 0 )
R4/2= 2.0
Vibrator
IBA HamiltonianComplicated 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 Q = e[s† + d†s + χ (d† )(2)]d d
Competition: ε nd term gives vibrator.
Q Q term gives deformed nuclei.
This is the form we will use from here on
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
Typical SU(3) SchemeTypical SU(3) Scheme
SU(3) O(3)
K bands in (, ) : K = 0, 2, 4, - - - -
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
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)
Q = (s†d + d†s) + (- 7
2) (d†d )(2)
Q is also in H and is a Casimir operator of SU(3), so conserves , .
B(E2; J + 2 →J)yrast
B(E2; +12 → +
10 )
N2 for large N
Typ. Of many IBA predictions → Geometric Model as N → ∞ Δ (, ) = 0
2 2 13
2 2 34 2 3 2 5B
J JN J N J
J Je
SU(3)
1 1
1 1
( 2;4 2 ) 10 (2 2)(2 5)
( 2;2 0 ) 7 (2 )(2 3)
B E N N
B E N N
E2: Δ (, ) = 0
2 2 3
5BN N
e
Alaga rule Finite N correction
“β”→ γ Collective B(E2)s
Example of finite boson number effects in the IBA
B(E2: 2 0): U(5) ~ N; SU(3) ~ N(2N + 3) ~ N2
B(E2)
~N
N2
N
Mid-shell
H = ε nd - Q Q and keep the parameters constant.What do you predict for this B(E2) value??
!!!
O(6)
Axially asymmetric nuclei(gamma-soft)
Transition Rates
T(E2) = eBQ Q = (s†d + d†s)
B(E2; J + 2 → J) = 2B
J+ 1
24
J Je N - N2 2 J + 5
B(E2; 21 → 01) ~ 2B
+ 4e
5N N
N2
Consider E2 selection rules Δσ = 0 0+(σ = N - 2) – No allowed decays! Δτ = 1 0+( σ = N, τ = 3) – decays to 22 , not 12
O(6) E2 Δσ = 0 Δτ = 1
Note: Uses χ = o
196Pt: Best (first) O(6) nucleus -soft
Classifying Structure -- The Symmetry TriangleClassifying Structure -- The Symmetry Triangle
Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle.
Sph.
Deformed
Mapping the Mapping the EntireEntire Triangle Triangle
2 parameters
2-D surface
H = ε nd - Q Q
Parameters: , (within Q)/ε
/ε
H = ε nd - Q Q
ε = 0
/ε
168-Er very simple 1-parameter calculation
H = - Q Q
is just scale factorSo, only parameter is
1
IBACQF Predictions for 168Er
γ
g
““Universal” IBA Calculations Universal” IBA Calculations for the SU(3) for the SU(3) –– O(6) leg O(6) leg
H = - κ Q • Q
κ is just energy scale factor
Ψ’s, B(E2)’s independent of κ
Results depend only on χ [ and, of course, vary with NB ]
Can plot any observable as a set of contours vs. NB and χ.