TESTS OF PARTIAL DYNAMICAL

SYMMETRIES AND THEIR IMPLICATIONS

R. F. CastenYale

SDANCA, Oct. 9, 2015

SU(3) O(3)

Characteristic signatures:

• Degenerate bands within a group

• Vanishing B(E2) values between groups (cancelation in E2 operator)

..

( , )l m

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

( , ) b g vibrations

.

Totally typical example

Similar to SU(3). But , b g vibrations not degenerate and collective B(E2) values from g to ground band. Most deformed rotors are not SU(3).

Approach: parameterized collective Hamiltonians - break symmetries.

( ,“ ”) g b vibrations

200 Wu

10 Wu ~1 Wu

~J(J + !)

Unfortunately, very few nuclei manifest an idealized structural symmetry exactly, limiting their direct role to that of benchmarks

BUT

However, an alternate approach exists with a proliferation of new “partial” and “quasi” dynamical symmetries (PDS, QDS)

Possibility of an expanded role of symmetry descriptions for nuclei

PDS: some features of a Dyn.Sym. persist even though there is considerable symmetry breaking.

[QDS: Some degeneracies characteristic of a symmetry persist and some of the wave function correlations persist.]

Why do we need such strange things? We have excellent fits to data with parameterized numerical IBA and geometric

collective model calculations that break SU(3).

BUT (???) Partial Symmetries (to the rescue???)

( ,l m)

Partial Dynamical Symmetry (PDS)

SU(3)

So, expect vanishing B(E2) values between these bands as in SU(3). But

empirical B(E2) values are collective! However, g to ground B(E2)s finite in PDS if use general E2 operator. Introduces a parameter. BUT,

branching ratios are PARAMETER FREE

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

INTERBAND -- g ground

Extensive test:47(22) rare earth nuclei

Overall good agreement for well-deformed nuclei

Systematic disagreements for spin INcreasing transitions.

Exp. stronger than PDS.

Parameter free

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

relative B(E2) values from one rotational band to another.

168-Er: Alaga, PDS, valence space, and mixing

PDS always closer to data than Alaga. PDS

simulates bandmixing without mixing.

Spin DEcreasing transitions smaller than

AlagaSpin INcreasing transitions

larger than AlagaDeviations increase with J

These are signature characteristics of mixing of g and ground band

intrinsic excitations.

Numerical IBA (CQF) - one parameter. Why two such different descriptions give

similar predictions?

Why differs from Alaga? PDS (from IBA) valence space model: predictions Nval– dep.

Overview Exp vs PDS

Deviations from PDS indicate some other degree of freedom. Grow with spin. Suggest still need for bandmixing.

But, much less bandmixing than heretofore invoked.

Intraband Transitions within g bandThese transitions depend on one parameter per nucleus, called /q a

gr

g

K = 0 to ground band

Preliminary

PreliminaryDoes PDS or Mixing agree for any nuclei?

Yes (sometimes only partially): 158-Gd, (160Dy, 168Er, 172-Yb) No: 170,174,176,180-Hf, 182-W, 178-Os

Actinides and A ~ 100 as function of spin

Spin decreasing B(E2) values get smaller with increasing spin.

Clear signal of a mixing effect since the K mixing matrix elements increase with spin: Vmix ~ Jinit

Why are Mo B(E2) values weaker? Transitional nuclei (R4/2 ~ 2.5) – between rotor and vibrator

All spin DEcreasing g band to ground band transitions are forbidden in the vibrator limit

Quasi-g band

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

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

Why do nucleon number effects simulate bandmixing?

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

• PDS - CQF: The differences from the PDS are due to mixing.

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

• Experimental d values are sorely needed.

• Other PDS/QDS being discussed (brief overview next slide)

Expansion in O(6) basis ( ,s t)O(6) PDS( s quantum number

for gsb only)Pietralla et al

SU(3) QDSArc of - b g

degeneracy, order amidst chaosBonatsos et al

Partial, quasi dynamical symmetries in the symmetry triangle (Color coded guide)

g, ground states pure SU(3). Others highly mixed. Valid in

most deformed nuclei! Relation to previous models.

SU(3 )PDS

SU(3) PDS

Principal Collaborators:

R. Burcu Cakirli, Istanbul

Aaron Couture, LANL

Klaus Blaum, Heidelberg

D. Bonatsos, Athens

Ecem Cevik, Istanbul

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

Backups

Super-precise data can mislead

Now look at c2

Assume Theor Uncert = 10 keV 800 0.01

TESTING THEORIESA very simple

example:

Which theory better

Need Theoretical Uncertainties

Evaluating theories with equal discrepancies

Theoretical uncertainties: Yrast: Few keV ; Vibrations ~ 100 keV

c2 ~1000 1

Finally, don’t overweight the same physics

(e.g., many yrast rotational

levels)

Testing the PDS

Extensive test (60 nuclei) in rare earth, actinide, and A ~ 100 regions

K = 0 to ground band

K = 0 to ground band

Intraband Transitions within g bandThese transitions depend on one parameter per nucleus, called / :q a

A single average value suffices for all the rare earth nuclei (except 156Dy).

Data vs Alaga for g band to ground band E2 transitions

Standard approach

Spin DEcreasing transitions smaller than Alaga

Spin INcreasing transitions larger than Alaga;

Deviations increase with J

These are signature characteristics of mixing

of g and ground band intrinsic excitations.

PDS: Intraband g band

Bandmixing and deviations from Alaga

Finite N effects have same effects as mixing on Rel. B(E2) values.

So, new (net) mixing is about half what we

have thought for 50 yrs.[(e.g., Z g (168 Er)

changes from ~0.042 to 0.019]

Recognition of importance of purely finite nucleon number effects. Reduced need for mixing. How to distinguish from interactions? What observables?

Characteristic signatures of g – ground mixing

Works extremely well: mixing parameter Z g

Data desperately needed(Missing transitions, no d’s at all)

Systematics of 0+ States (Rare earth region)

Sph.

Deformed

The Symmetry Triangle of the

IBA

Unique insights into complex many-body systems: shape, quantum numbers, selection rules, analytic formulas (often parameter free)

Structural (dynamical) Symmetries

Transitional nuclei: How does PDS

perform?

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 ?

10 20 30 40 50 60 70 80 90 100110120130140150

10

20

30

40

50

60

70

80

90

100

Pro

ton

Num

ber

Neutron Number

80.00

474.7

869.5

1264

1600

E(21+)

10 20 30 40 50 60 70 80 90 100110120130140150

10

20

30

40

50

60

70

80

90

100

Pro

ton

Num

ber

Neutron Number

1.400

1.776

2.152

2.529

2.905

3.200

R4/2

Broad perspective on structural evolutionZ=50-82, N=82-126

The remarkable regularity of these patterns is one of the beauties of nuclear systematics and one of the challenges to nuclear theory.

Whether they persist far off stability is one of the fascinating questions for the future

Cakirli

0+

2+

6+. . .

8+. . .

Vibrator (H.O.)

E(J) = n ( 0 )

R4/2= 2.0

n = 0

n = 1

n = 2

Rotor

E(J) ( ħ2/2I )J(J+1)

R4/2= 3.33

Doubly magic plus 2 nucleons

R4/2< 2.0

What about other collective states?

K = 0 (b ?) band exists in all these nuclei. May or may not be collective. B(E2)s to ground state

~ 1 Wu.

A problem in testing is that, often, there are unknown E0 components

Some data exist to test and the results are ugly !

PDS for , g b ground interband B(E2)s

g ground b ground

g mode described by two very different models. K = 0 modes not at all reproduced

Preliminary

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

Other PDS, QDS -- Briefly

PDS based on O(6)

\

Remnants of pure O(6) (for gs band only) persist along a line extending into well-

deformed rotational nuclei

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

This result is from analyzing wave functions throughout the triangle. What are the OBSERVABLE signatures of this?

s good along a line from O(6)

s

Whelan, Alhassid, 1989; Jolie et al, PRL, 2004

Arc of Regularity - narrow zone with high degree of order amidst regions of chaotic behavior. What is going on along the AoR?

Order, Chaos, and the Arc of regularity

> b g

< b g

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

AoR – E(0+2) ~ E(2+

2)

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.

A Quasi-Dynamical Symmetry (QDS ) [Degeneracies preserved]

Z contain mixing amplitude and ratio of intra to interband quad moments

If direct ME2 =0, drop the leading “1”s

Correction factors to Alaga rules due to b – ground

mixing

Deviations of PDS from Alaga are exactly like those resulting from bandmixing

Actinides and A ~ 100 (Mo)

Actinides similar to rare earths. A~ 100 much weaker.

A illustrative special case of fundamental importance

Origin of collectivity:Mixing of many configurations

T

This is the origin of collectivity in nuclei.

Consider a toy model: Mixing of degenerate states

6+ 690

4+ 330

0+ 0

2+ 100

J E (keV)

?Without

rotor

paradigm

Paradigm

Benchmark

700

333

100

0

Rotor J(J + 1)

Amplifies structural

differences

Centrifugal stretching

Deviations

Identify additional

degrees of freedom

Value of paradigms

Ji Jf NB=16 Normalized NB=15 Normalized NB=14 Normalized

2beta 0+ 0.197 28.8 0.183 28.3 0.169 27.8

  2+ 0.309 45.1 0.288 44.5 0.267 43.8

  4+ 0.685 100.0 0.647 100.0 0.609 100.0

4beta 2+ 0.246 36.6 0.226 35.4 0.206 34.0

  4+ 0.276 41.0 0.257 40.2 0.237 39.1

  6+ 0.673 100.0 0.639 100.0 0.606 100.0

6beta 4+ 0.234 33.3 0.212 31.5 0.191 29.8

  6+ 0.263 37.4 0.244 36.3 0.225 35.2

  8+ 0.703 100.0 0.672 100.0 0.640 100.0

Looking back at transitional rare earths, now see

same effects

Perspectives for the future

• Identifying the empirical signatures of partial symmetries.

• Uncovering which partial symmetries are relevant to actual nuclei and whether their manifestations are different in exotic nuclei

• Experimental techniques: will vary by case. A) SU(3) – PDS: Rel. B(E2)s from , g b bands – beta decay; B) Arc: Test degeneracy of vibrational modes + B(E2)s – b decay, Coul ExC) O(6) – PDS: signatures not yet established, yrast levels, possibly Coul Ex. D) New PDS, QDS: ????

• Understanding the relation of these partial symmetries to numerical calculations – how such seemingly diverse descriptions can be simultaneously successful.

• Role of finite valence space (predictions of PDS and deviations from parent symmetries are valence nucleon-number dependent).

K = 0 (b) excitationsAgain, parameter-free

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).

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.

Predicted Intraband Transitions Compared to Measurement

Predicted Intraband Transitions Compared to Measurement

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

Identification of characteristic predictions of a symmetry tells us both the basic nature of the system and can highlight specific deviations pointing to

additional degrees of freedom.

Typical deformed ellipsoidal nucleus

Normal Mikhailov Plot for g band – 168 ErWiP (Just look at data).

Mixing ~ slope

Mikhailov Plot for g band using PDS as dataWiP (Just look at PDS)

PDS B(E2) values have slope too –

simulate mixing!!

Can now ascribe part of

deviations from Alaga to finite

number effects and the rest to

(reduced) mixing

Intraband Transitions within g band

Normalized to intERband E2

K = 0 to ground B(E2) ratios

compared to the PDS