Symmetries Galore

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Symmetries Galore “Not all is lost inside the triangle” (A. Leviatan, Seville, March, 2014) R. F. Casten Yale CERN, August, 20

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Symmetries Galore. R. F. Casten Yale CERN, August, 2014. “Not all is lost inside the triangle” (A. Leviatan , Seville, March, 2014). Themes and challenges of Modern Science Complexity out of simplicity -- Microscopic - PowerPoint PPT Presentation

Transcript of Symmetries Galore

Page 1: Symmetries Galore

Symmetries Galore“Not all is lost inside the triangle”(A. Leviatan, Seville, March, 2014)

R. F. CastenYale

CERN, August, 2014

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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

What is the force that binds nuclei?Why do nuclei do what they do?

What are the simple patterns that nuclei display and what is their origin ?

We will look at a model that lives in both camps

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The macroscopic perspective (many-body quantal system) often exploits the idea of symmetries

These describe the basic structure of the object. Geometrical symmetries describe the shape.

Symmetry descriptions are usually analytic and parameter-free ( except for scale). (Scary group theory comment.)

One model, the Interacting Boson Approximation (IBA) model, is expressed directly in terms of symmetries.

Unfortunately, very few physical systems, especially in atomic nuclei, manifest a symmetry very well.

We will see that this statement is now being greatly modified and symmetries may play a much larger role than heretofore.

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1 12 4 2( ; )B E

330 4+

2+

0

100

0+

E (keV) Jπ

Simple Observables - Even-Even Nuclei

1 12 2 0( ; )B E

212 2

2 1( ; )i f i f

i

B E J J EJ

)2(

)4(

1

12/4

E

ER

1000 2+

Relative B(E2) values

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E(J)Rot. = ( ħ2/2I )J(J+1)

R4/2= 3.33

Nuclei that are non-spherical can rotate (1952)!Rotational energies follow the quantum symmetric top:

E(0)Rot. 0, E(2)Rot. 6, E(4)Rot. 20

Example of a geometrical symmetry – ellipsoidal, axially symmetric nuclei

0+2+4+

6+

8+

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6+ 690

4+ 330

0+ 0

2+ 100

J E (keV)

?Without

rotor

paradigm

Paradigm

Benchmark

700

333

100

0

Rotor

( ħ2/2I ) J(J + 1)

Amplifies structural

differences

Centrifugal stretching

Deviations

Identify additional

degrees of freedom

Value of paradigms

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0+2+4+

6+

8+

Rotational states

Vibrational excitations

Rotational states built on (superposed on)

vibrational modes

Ground or equilibrium

state

Rotor

E(I) ( ħ2/2I )I(I+1)

R4/2= 3.33

So, identification of the characteristic predictions of a

symmetry both tells us the basic nature of the system and can

highlight specific deviations from the perfect symmetry which can

point to additional degrees of freedom.

Real life example

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The IBA – A collective model built on a highly truncated shell model foundation

(only configurations with pairs of nucleons coupled to

states with angular momentum 0 (s bosons) or 2 (d bosons)

Embodies the finite number of valence nucleons. Like the shell model but opposed to traditional collective models,

the predictions depend on N and Z

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Shell Model Configurations

Fermion configurations

Boson configurations

(by considering only configurations of pairs of fermions

with J = 0 or 2.)

s bosons and d bosons as the basic building blocks of

the collective states

The IBA

Roughly, gazillions !!Need to simplify

Huge truncation of the shell model

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Symmetries of the IBA

U(5)vibratorSU(3)rotor

O(6)γ-soft

U(6)

Magical group theory

stuff happens here

Sph.

Def. R4/2= 2.0 R4/2= 3.33

R4/2= 2.5

s and d bosons:6-Dim. problem

Three Dynamic symmetries, nuclear shapes

IBA Symmetry Triangle

What are these symmetries?Idealized structures whose predictions follow analytic

formulas, with states labeled by good quantum numbers.

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Proliferation of Symmetries

ENTER PDS,QDS: Recent work (Leviatan, Van Isacker, Alhassid, Bonatsos, Cejnar, Pietralla, Cakirli, rfc, ..): symmetries not limited to vertices: triangle permeated by symmetry elements: AoR, QDS, PDS

AoR

PDSQDS

O(6) PDS – SU(3) QDS

However, most nuclei do not exhibit the idealized symmetries: So, is their role just as benchmarks? Most nuclei lie inside the

triangle where chaos and disorder are thought to reign.

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PDS, QDS: what are these things?

They are various situations in which some of the features of a Dyn.Sym. persist even though

there is considerable symmetry-breaking.

PDS: Some of the levels have the pure symmetry [such as SU(3)] and others are

severely mixed

QDS: Some of the degeneracies characteristic of a symmetry persist and some of the wave

function correlations persist.

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SU(3) O(3)

Characteristic signatures:

• Degenerate bands within a group

• Vanishing B(E2) values between groups

. . . .

( , )l m

Typical SU(3) Scheme(for N valence nucleons)

( , ) b g vibrations

What do real nuclei look like – what are the data??

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Totally typical example

Similar in many ways to SU(3). But note that the excited ( , ) b g vibrational

excitations are not degenerate as they should be in SU(3). Also there are collective B(E2) values from the g band to the ground band.

Most deformed rotors are not SU(3).

( , ) b g vibrations

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Clearly, SU(3) is severely broken Or so we thought

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B(E2) values in deformed rotor nuclei (typical values)

300 Wu

12 Wu 2 Wu

b

So, “clearly”, this violates the predictions of SU(3). So, realistic calculations have broken the symmetries (mixed their basis states).

The degeneracies, quantum numbers, selection rules of the symmetry no longer apply. …. Or so we thought

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What is an SU(3)-PDS?

A bit weird!!

It is based on the IBA SU(3) symmetry BUT breaks it while retaining pure SU(3) symmetry ONLY the ground and g

bands which preserve SU(3) exactly. All other states are severely mixed!

Why would we need such a thing? We have excellent fits to the data with numerical IBA calculations that break SU(3). Why would we think g band preserves SU(3)?

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( ,l m)

( ,l m)

( ,l m)

( ,l m)

Partial Dynamical Symmetry (PDS)

SU(3)

So, expect PDS to predict vanishing B(E2) values between these bands as in SU(3). How can that possibly work since empirically these B(E2)

values are collective !? BUT, g to ground B(E2)s CAN be finite in the PDS

PDS: ONLY g and ground bands are pure SU(3).

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SU(3) PDS: B(E2) valuesg, ground states pure SU(3), others mixed. Generalized T(E2)

T(E2)PDS = e [ A (s+d + d+s) + B (d+d)] = e [T(E2)SU(3) + C (s+d + d+s) ]

M(E2: Jg – Jgr) eC < Jgr | (s+d + d+s) | Jg >-------------------- = ------------------------------------ eC cancels

M(E2: Jg – J’ gr) eC < J’

gr | (s+d + d+s) | Jg >

Relative INTERband g to gr B(E2)’s are parameter-free!

First term gives zero from SU(3) selection rules. But not second. Hence, relative g to ground B(E2) values:

So, Leviatan: PDS works well for 168 Er (Leviatan, PRL, 1996). Accidental or a new paradigm?

Tried extensive test for 47 Rare earth nuclei.Key data will be relative g to ground B(E2) values.

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Illustrative results

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Transitional nuclei

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5:100:70 Alaga

Lets look into these predictions and comparisons a little deeper.Compare to “Alaga Rules” – what you would get for a pure rotor for RATIOS of squares of transition matrix elements [B(E2) values] from

one rotational band to another.

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168-Er: The Alaga rules, valence space, and collectivity

PDS has pure g, gr band K, so why differ from Alaga?

Ans: PDS (from IBA) is valence space model:

predictions are Nval – dep.

PDS differs from Alaga solely due to N-Dep effects that arise from the fermion

underpinning (Pauli Principle).

Valence collective models that agree with the PDS have minimal mixing.

Predictions deviating from the PDS signal configuration mixing.

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Using the PDS to better understand collective model calculations (and collectivity in nuclei)

• PDS B(E2: g – gr): the sole reason they differ from the Alaga rules is that they take into account the finite number of valence nucleons.

• Therefore, IBA calculations (like WCD) that agree with the PDS differ from the Alaga rules purely because of finite valence nucleon number effects. Not heretofore recognized.

• IBA – CQF deviates further from the Alaga rules, agrees better with the data (and has one fewer parameter).

• IBA – CQF: The differences from the PDS are due to mixing

• Can use the PDS to disentangle valence space from mixing!

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Again, as one enters the triangle expect vertex symmetries to be highly broken. Consider structures descending from O(6)

Now a highly selective O(6) PDS

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( , )s t

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Calculate wave functions throughout triangle, expand in O(6) basis

s good along a line descending from O(6)

Remnants of pure O(6) (for the ground state band only) persist along a line in the triangle extending into the region of well-deformed rotational nuclei

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Order, Chaos, and the Arc of regularity

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Whelan, Alhassid, ca 1989

Order and chaos? What happens generally inside the triangle

Arc of Regularity is a narrow zone in the triangle with high degree of order amidst regions of chaotic

behavior. What is going on along the AoR?

6+, 4+, 3+, 2+, 0+

4+, 2+, 0+

2+

0+

3

2

1

0

nd

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Role of the arc as a “boundary” between two classes of structures

< b g

< b g

AoR – E(0+2) ~ E(2+

2)

Not just a theoretical curiosity: 8 nuclei in the rare earth region have been found to lie along the arc

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The symmetry underlying the Arc of Regularity is an SU(3)-based Quasi Dynamical Symmetry

QDS

All SU(3) degeneracies and all analytic ratios of 0+ bandhead energies persist along the Arc.

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Summary

AoR

PDSQDS

O(6) PDS – SU(3) QDS

Elements of structural symmetries abound throughout the triangle

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Principal Collaborators:

R. Burcu Cakirli

D. Bonatsos

Klaus Blaum

Aaron Couture

Thanks to Ami Leviatan, Piet Van Isacker, Michal Macek, Norbert Pietralla, C. Kremer for discussions.

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BACKUPS

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• States labelled by quantum numbers

• Degenerate bands within irrep( ,l m)

( ,l m)

( ,l m) ( ,l m)

T(E2) = eQ = e[(s† d + d †s) - (d † d )(2)]7

2

( ,l m)

SU(3)

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Creation and destruction operators as

Example: Consider the case we have just discussed – the spherical vibrator.

0

2

4,2,0

0

E

2E

1

2

Why is the B(E2: 4 – 2) = 2 x B(E2: 2– 0) ??

Difficult to see with Shell Model wave functions with 1000’s of components

However, as we have seen, it is trivial using destruction operators WITHOUT EVER

KNOWING ANYTHING ABOUT THE DETAILED STRUCTURE OF THESE VIBRATIONS !!!!

These operators give the relationships between states.

“Ignorance operators”

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Finite Valence Space effects on Collectivity

Note: 5 N’s up, 5 N’s down: 1 with N

Alaga

Most dramatic, direct evidence for explicit valence space size in emerging collectivity

PDS B(E2: g to Gr)

R.F. Casten, D.D. Warner and A. Aprahamian , Phys. Rev. C 28, 894 (1983).

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~ 0.03

~ 0.1 x ~0.03 ~ 0.003

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Concepts of group theory 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

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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).

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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

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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

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Concept of a Dynamical Symmetry

N

Spectrum generating algebra !!

Each successive term:

• Introduces a new sub-group

• A new quantum number to label the states described by

that group

• Adds an eigenvalue term

that is a function of the new quantum

number, hence

• Breaks a previous degeneracy

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Now, WHY are spin increasing transitions so small in Alaga rules and so much larger in the data than in the PDS?

02

4

2g

Jint , RJtot

Jtot Jint , ~R

0, 00, 2

0, 4

2, 0DR = 4 !!

~ Forbidden

Deviations from Alaga: g – gr. band mixing. E2 matrix element :

ME(E2) = [ a < || >unpert + b < || > mix ]

Can be larger or smaller than the unperturbed. BUT, if unperturbed is forbidden, then

ME(E2) = [b < | > mix] > unpert (which is zero).

So mixing always increases forbidden/weak transitions.

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Review 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.