1. Oslo-07-TD Landscapes and fluctuations of two-dimensional rotational  - spectra – coupling between rotational and thermal motion Rare earth nuclei p resolved lines - interacting bands – magnitude of interactions – level spacing distributions p damped rotational motion – strength distributions – scaling laws p landscapes and fluctuations are powerful tools to investigate interacting bands and rotational damping 194 Hg – superdeformed bands p ~ 100 discrete unresolved bands p ergodic rotational bands or precursors to ergodic bands M. Matsuo S. Leoni A. Bracco E. Vigezzi R.A. Broglia B. Herskind S. Åberg T.L. Khoo A.P. Lopez-Martens T. Lauritsen T. Døssing Niigata, Milano, Copenhagen, Argonne,Orsay Introduction: p cascades of rotational transitions p ordered or chaotic intrinsic and rotational motion: discrete bands – ergodic bands – damping of rotational motion

2. Oslo-07-TD  – ray cascades of deformed nuclei: angular momentum energy CN formation n  Particle evaporation unresolved (unresolvable?) gamma rays, mostly E2’s resolvable gamma rays, mostly E2’s Resolve from coincidences D. RADFORD Quasicontinuum spectra F. STEPHENS and D. DIAMOND B. HERSKIND and S. LEONI I..Y. LEE and C. BAKTASH T-L KHOO and A.P. LOPEZ-MARTENS

3. Oslo-07-TD Rotational and intrinsic motion Rotational motion Intrinsic motion ordered chaotic Rare earth nucleus ergodic bands discrete bands damping of rotational motion

4. Oslo-07-TD 163 Er – interacting bands Hagemann et al., Nucl.Phys. A618 (1997) 199 C A AEH BEG/BFH AFG LI,LII BAEG

5. Oslo-07-TD Infer: mixed bands higher up in energy HAB FABef |V| ~ 10 keV E-E yrast = 400 keV d 2 ~ 50 keV E-E yrast = 1.5 MeV T ~ 1/3 MeV d 2 ~ 6 keV N branch ~ 5 Same magnitude of interaction

6. Oslo-07-TD Magnitude of interactions KK experimental, from level crossings surface –  interaction

7. Oslo-07-TD Calculations of interacting bands M. Matsuo et al. Nucl. Phys. A617(1997)1 Cranking np-nh basis bands Configuration mixing with residual interaction Cranked Nilsson potential Surface delta interaction

8. Oslo-07-TD Level spacings

9. Oslo-07-TD E  - E  coincidence spectra E 1 - E 2 one band Rotational damping Many bands E2E2 E1E1

10. Oslo-07-TD Mixing and Damping  I = 0: spreading of basis band state over energy interval mean filed mean field + residual interaction  I = -2: one step in a cascade  I = -4: two steps in a cascade  narrow  rot  comp  wide E2E2 E1E1 E1E1 + E 2 E1E1 - E 2 compound damping width rotational damping width I I - 4 I - 2 I E

11. Oslo-07-TD Narrow component in  -  correlation  correlation 1. Doorway states keep rotational correlation 2. The correlation is smeared by  comp EE EE compnarrow /2  DP Matsuo et al. PLB465(1999)1

12. Oslo-07-TD E  - E  schematic two-step strength funcions U Energy Angular momentum n E1E1 E2E2 Motional narrowing  (E 1 – E 2 ) ~ 4(I)4(I) I 2  narrow ~ 2  comp P narrow ~  I 2 U -1    comp  rot   rot   2  narrow   Exponentially decreasing   2   U 3/2   I U 1/4

13. Oslo-07-TD Flow in cascades angular momentum energy 60 30 I 1 MeV yrast line asymptotic flow line: U  I 2  comp  U 3/2 ~ E  3  rot  I U ¼ ~ E  3/2  rot  I 2 U -1 ~ const below motional narrowing: above motional narrowing:

14. Oslo-07-TD Perspective plot of  spectra Analysis of Quasi - Continuum  spectra with respect to both shape and fluctuations 168 Yb valley ridges Døssing, Herskind, Leoni, Bracco, Broglia, Matsuo, Vigezzi, Phys. Rep. 268(1996)1 Fluctuation analysis: Large fluctuations: few underlying transitions. Small fluctuations: many underlying transitions. - Unresolved, yet discrete, bands on the ridges - Damped transitions in the valley Shape anlysis: -Width of ridge -Width and depth of valley => Extract  rot and  comp

15. Oslo-07-TD I+2 I I-2 Counts [arb. unit] Valley E  1 -E  2 [keV] I+2 I I-2 Counts [arb. unit] Ridge E  1 -E  2 [keV]  strongly interacting bands  valley  unresolved, yet discrete, bands  ridge Fluctuation analysis

16. Oslo-07-TD Experimental evidence for rotational damping Rotational Damping  rot I+2 I I-2 Discrete bands I+2 I I-2 A. Bracco et al., PRL76(1996)4484 about ~ 30 discrete bands of the four  ’s up to U ~ 800 keV cranking model level density  (U,N,Z) Fragmented decay of weak transitions in the valley

17. Oslo-07-TD 18 O+ 150 Nd  163 Er+5n @ 87,93 MeV v/c =0.96 % I max  40 , U max  4 MeV 3x10 9  events 1) Sorting of 163 Er matrix 2) Subtraction of ALL known transitions by RADWARE 3) Subtraction of E1xE1 and E2xE1 background 4) Spectral Shape Analysis of Ridge-Valley structure I = 30  =900 keV I = 32  =960 keV Total  comp  20 keV  rot  150 – 200 keV P ~ 10 % I  30-40  Landscapes =>  comp and  rot S. Leoni et al., PRL93(2004)022501-1 narr I 3/2

18. Oslo-07-TD Experimental damping width I 3/2 const F.S. Stephens et. al., Phys. Rev. Lett 88 (2002)142501 data Matsuo et. al., Cranked mean field Laurtizen et. al., schematic, cranked oscillator

19. Oslo-07-TD probed regions in I and U: angular momentum energy ridge fluctuations wide component from valley shape valley fluctuations, narrow component 60 30 I 1 MeV yrast line

20. Oslo-07-TD Ergodic bands estimated  rot Ergodic bands: chaotic internal States, E2 transitions along bands  d Ergodic bands Discrete rotational bands Damping of rotational motion observed until now search in superdeformed nuclei B. Mottelson, S Åberg

21. Oslo-07-TD Cascades including SD band angular momentum energy feeding decay out Cold ND decay decay along Weak rotational transitions along excited superdeformed bands

22. Oslo-07-TD Gated spectra – 194 Hg SD -gated ND -gated E2- bump Decay-out bump

23. Oslo-07-TD  gated spectra - 194 Hg 600 700 800 900 E  (keV)

24. Oslo-07-TD Extract from gated spectra in 194 Hg: 600 100 200 700800900 32 10 30 0 0.0 0.5 1.0 Transition energy (keV) N path FWHM(keV)  E(keV) intensity first ridge second ridge yrast band E2 - bump

25. Oslo-07-TD Ergodic bands ? about 4 components of basis bands in each mixed state precursor to ergodic bands, motional narrowing sets in when band mixing sets in Calculated bands, 194 Hg

26. Oslo-07-TD Shell effects in rotational damping  : dispersion in rotational frequency ~  rot 168 Yb proton

27. Oslo-07-TD Precursor to ergodic bands angular momentum I E(I) – E rigid (I)    comp  rot  Ergodic bands: chaotic internal States, E2 transitions along bands Precursor to ergodic bands: motional narrowing for all mixed bands:  d  d 2    comp  rot 

28. Oslo-07-TD Schematic strength functions with ergodic bands  (E 1 – E 2 ) ~ 4(I)4(I) I 2 Exponentially decreasing   I U 1/4  I 2 U -1    comp  ridge   I 2 U -1    comp  rot   long ~ 2  comp    ) discrete bands > rotational damping discrete bands > ergodic bands > rotational damping

29. Oslo-07-TD Conclusions Powerfull techniques to study the elusive spectra of unresolved  -rays Onset of damping around U ~ 800 keV cranking model level density  (U,N,Z) damping confirmed,  rot ~~ 200 keV information on  comp not so convincing Rare-earth nuclei: Superdeformed 194 Hg nucleus: no damping hints of ergodic bands – precursor to ergodic bands Perspectives: better theories (pf shell model)  - ray tracking detection

30. Oslo-07-TD Volcano 50-100

31. Oslo-07-TD Volcano 20-50

32. Oslo-07-TD Tilted planes in  spectra => 122 Xe J (2) ~ 77  2 / MeV N=3: x+3y-4z = ±δ N=2: x+2y-3z = ±δ N=1: x+y-2z = ±δ (α,2n) x x x y y y z z z