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
Symmetries GaloreNot all is lost inside the triangle(A. Leviatan, Seville, March, 2014)R. F. CastenYale
CERN, August, 2014Themes and challenges of Modern Science
Complexity out of simplicity -- MicroscopicHow 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 campsThe macroscopic perspective (many-body quantal system) often exploits the idea of symmetriesThese 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.
3304+2+01000+E (keV) JSimple Observables - Even-Even Nuclei
10002+Relative B(E2) values
E(J)Rot. = ( 2/2I )J(J+1)R4/2= 3.33Nuclei 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 nuclei0+2+4+6+8+6+ 6904+ 3300+ 02+ 100J E (keV)?
Without rotor paradigm ParadigmBenchmark 700
Rotor ( 2/2I ) J(J + 1)Amplifies structural differences
Deviations Identify additionaldegrees of freedomValue of paradigms6
0+2+4+6+8+Rotational statesVibrational excitationsRotational states built on (superposed on) vibrational modesGround or equilibrium stateRotor
E(I) ( 2/2I )I(I+1)
R4/2= 3.33So, 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 exampleThe 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 ZShell Model ConfigurationsFermion configurationsBoson 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 IBARoughly, gazillions !!Need to simplifyHuge truncation of the shell model9Symmetries of the IBAU(5)vibratorSU(3)rotorO(6)-softU(6)
Magical group theory stuff happens here
Sph.Def. R4/2= 2.0 R4/2= 3.33 R4/2= 2.5s and d bosons:6-Dim. problemThree Dynamic symmetries, nuclear shapesIBA Symmetry Triangle
What are these symmetries?Idealized structures whose predictions follow analytic formulas, with states labeled by good quantum numbers.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
AoRPDSQDSO(6) PDS SU(3) QDSHowever, 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.1111PDS, 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.
Degenerate bands within a group
Vanishing B(E2) values between groups
....(l,m)Typical SU(3) Scheme(for N valence nucleons)(b,g) vibrationsWhat do real nuclei look like what are the data??
Totally typical exampleSimilar in many ways to SU(3). But note that the excited (b,g) vibrationalexcitations 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
Clearly, SU(3) is severely broken Or so we thoughtB(E2) values in deformed rotor nuclei (typical values)
300 Wu 12 Wu 2 Wu bSo, 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 thoughtWhat 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)?
(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 PDSPDS: ONLY g and ground bands are pure SU(3).
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 cancelsM(E2: Jg J gr) eC < Jgr | (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.
5:100:70 AlagaLets 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.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.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!
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
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
Order, Chaos, and the Arc of regularity
Whelan, Alhassid, ca 1989 Order and chaos? What happens generally inside the triangleArc 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 0nd
Role of the arc as a boundary between two classes of structuresb < gb < 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 arc30
The symmetry underlying the Arc of Regularity is an SU(3)-based Quasi Dynamical Symmetry
QDSAll SU(3) degeneracies and all analytic ratios of 0+ bandhead energies persist along the Arc.Summary
AoRPDSQDSO(6) PDS SU(3) QDSElements of structural symmetries abound throughout the triangle3232Principal Collaborators:
R. Burcu Cakirli
Aaron CoutureThanks to Ami Le