1. TESTS OF PARTIAL DYNAMICAL SYMMETRIES AND THEIR IMPLICATIONS R. F. Casten Yale SDANCA, Oct. 9, 2015

2. SU(3) O(3) Characteristic signatures: Degenerate bands within a group Vanishing B(E2) values between groups (cancelation in E2 operator)..  Typical SU(3) Scheme (for N valence nucleons)  vibrations.

3. Totally typical example Similar to SU(3). But  vibrations not degenerate and collective B(E2) values from  to ground band. Most deformed rotors are not SU(3). Approach: parameterized collective Hamiltonians - break symmetries.  “  ”  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

4. 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???)

5. (  ) 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,  to ground B(E2)s finite in PDS if use general E2 operator. Introduces a parameter. BUT, branching ratios are PARAMETER FREE PDS: ONLY  and ground bands are pure SU(3).

6. INTERBAND  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.

7. Parameter free

8. 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.

9. 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 Alaga Spin INcreasing transitions larger than Alaga Deviations increase with J These are signature characteristics of mixing of  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 N val – dep.

10. 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.

11. Intraband Transitions within  band These transitions depend on one parameter per nucleus, called  gr 

12. K = 0 to ground band Preliminary Does 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

13. 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: V mix ~ J init

14. Why are Mo B(E2) values weaker? Transitional nuclei (R 4/2 ~ 2.5) – between rotor and vibrator All spin DEcreasing  band to ground band transitions are forbidden in the vibrator limit Quasi-  band

15. Using the PDS to better understand collective model calculations (and collectivity in nuclei) PDS B(E2:  – 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  values are sorely needed. Other PDS/QDS being discussed (brief overview next slide)

16. Expansion in O(6) basis (  ) O(6) PDS (  quantum number for gsb only) Pietralla et al SU(3) QDS Arc of  degeneracy, order amidst chaos Bonatsos et al Partial, quasi dynamical symmetries in the symmetry triangle (Color coded guide) , ground states pure SU(3). Others highly mixed. Valid in most deformed nuclei! Relation to previous models. SU(3 )PDS

17. 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.

18. Backups

19. Super-precise data can mislead Now look at  2 Assume Theor Uncert = 10 keV 800 0.01 TESTING THEORIES A very simple example: Which theory better Need Theoretical Uncertainties

20. Evaluating theories with equal discrepancies Theoretical uncertainties: Yrast: Few keV ; Vibrations ~ 100 keV  2 ~1000 1 Finally, don’t overweight the same physics (e.g., many yrast rotational levels)

21. Testing the PDS Extensive test (60 nuclei) in rare earth, actinide, and A ~ 100 regions

22. K = 0 to ground band

29. Intraband Transitions within  band These transitions depend on one parameter per nucleus, called  A single average value suffices for all the rare earth nuclei (except 156 Dy).

30. Data vs Alaga for  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  and ground band intrinsic excitations.

31. PDS: Intraband  band

32. 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  ( 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?

33. Characteristic signatures of  – ground mixing Works extremely well: mixing parameter Z 

34. Data desperately needed (Missing transitions, no  ’s at all)

35. Systematics of 0 + States (Rare earth region)

36. 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

37. Transitional nuclei: How does PDS perform?

38. 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 ?

39. Broad perspective on structural evolution Z=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

40. 0+0+ 2+2+ 6 +... 8 +... Vibrator (H.O.) E(J) = n (   0 ) R 4/2 = 2.0 n = 0 n = 1 n = 2 Rotor E(J)  ( ħ 2 /2 I )J(J+1) R 4/2 = 3.33 Doubly magic plus 2 nucleons R 4/2 < 2.0

41. What about other collective states? K = 0 (  ?) 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 !

42. PDS for  ground interband B(E2)s  ground  ground  mode described by two very different models. K = 0 modes not at all reproduced Preliminary

43. 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)

44. \ 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?   good along a line from O(6) 

45. 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   Role of the arc as a “boundary” between two classes of structures AoR – E(0 + 2 ) ~ E(2 + 2 )

46. 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]

47. Z contain mixing amplitude and ratio of intra to interband quad moments If direct ME2 =0, drop the leading “1”s

48. Correction factors to Alaga rules due to  – ground mixing

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

51. Actinides and A ~ 100 (Mo) Actinides similar to rare earths. A~ 100 much weaker.

52. 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

53. 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

54. JiJfNB=16NormalizedNB=15NormalizedNB=14Normalized 2beta0+0.19728.80.18328.30.16927.8 2+0.30945.10.28844.50.26743.8 4+0.685100.00.647100.00.609100.0 4beta2+0.24636.60.22635.40.20634.0 4+0.27641.00.25740.20.23739.1 6+0.673100.00.639100.00.606100.0 6beta4+0.23433.30.21231.50.19129.8 6+0.26337.40.24436.30.22535.2 8+0.703100.00.672100.00.640100.0

55. Looking back at transitional rare earths, now see same effects

56. 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  bands – beta decay; B)Arc: Test degeneracy of vibrational modes + B(E2)s –  decay, Coul Ex C) 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).

57. K = 0 (  ) excitations Again, parameter-free

58. 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).

59. SU(3) PDS :  B(E2) values , 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: J  – J gr ) eC -------------------- = ------------------------------------ eC cancels M(E2: J  – J ’ gr ) eC Relative INTERband  to gr B(E2)’s are parameter-free! First term gives zero from SU(3) selection rules. But not second. Hence, relative  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  to ground B(E2) values.

60. Predicted Intraband Transitions Compared to Measurement

62. 0+0+ 2+2+ 4+4+ 6+6+ 8+8+ Rotational states Vibrational excitations Rotational states built on (superposed on) vibrational modes Ground or equilibrium state Rotor E(I)  ( ħ 2 /2 I )I(I+1) R 4/2 = 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

63. Normal Mikhailov Plot for  band – 168 Er WiP (Just look at data). Mixing ~ slope

64. Mikhailov Plot for  band using PDS as data WiP (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

65. Intraband Transitions within  band Normalized to intERband E2

66. K = 0 to ground B(E2) ratios compared to the PDS