# Presentation: 537888

## Equilibrium Structural EvolutionThe value of complementary perspectives

- R. F. Casten
- WNSL, Yale

**Principal collaborators:  Libby McCutchan, Dennis Bonatsos, Burcu Cakirli**

## Microscopic and Macroscopic Approaches

_**Micro**_
**Single part. D.o.F.**
**Large basis wave shell model functions**
**Particle quantum numbers ****(configurations,  seniority)**

_**Macro**_
**Collective D.o.F.**
**Simple configurations in collective basis**
**Many-body quantum numbers ****(e.g., coll. modes, representation labels, rot. ang. mom., K)**

**Need to look at both perspectives:  ****what**** (symmetries, geometry) and ****why**** (shell structure, interactions – pairing, pn)**

## Classifying Structure -- The Symmetry Triangle 

**Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle.**

**E(5)**

**X(5)**

**Landau Theory**

## X(5)

**Bessel equation**
**	**

**Critical Point Symmetries**
**First Order Phase Transition – Phase Coexistence**

**Energy surface changes with valence nucleon number**

**Iachello**

**X(5)**

## Contours define locus of possible X(5) nuclei and enclose regions of deformation

**Contours define locus of possible X(5) nuclei and enclose regions of deformation**

**p-n / pairing**

**P ~ 5  **

## Global mass-dependence of structure:Why the structure of heavy nuclei is qualitatively different from light nuclei

**Competition of residual interactions:  pairing and p-n compete differently in heavy nuclei than in light nuclei.**
**                                ****	****	**** ****WHY?**
**Higher particle angular momenta, greater shell model degeneracies, UPO**** **
**Larger shell size allows the development of larger and more varied regions of deformed nuclei and new forms of collectivity**

## Slide 7

## Contours define locus of possible X(5) nuclei and enclose regions of deformation

**Contours define locus of possible X(5) nuclei and enclose regions of deformation**

**p-n / pairing**

**P ~ 5  **

- 168W

- 130Ce

## Different perspectives can yield different insights

**Different perspectives can yield different insights**

**Onset of deformation**

**Onset of deformation ****as a phase transition, and due to a change in shell structure induced by the p-n interaction**

**Obliteration of Z=64 proton shell gap as a function of neutron number**

- Magic

- Mid.sh.

**Onset of deformation ****as a phase transition**

**Onset of deformation**

## Regional perspective on phase transitional behavior

**Regional perspective on phase transitional behavior**

## Landau Theory

**E(5)**

**X(5)**

**Landau Theory**

- 1st order

- 2nd order

**How can we distinguish first and second order quantum phase transitional behavior. Are there sensitive signatures? Are they simple to measure?**

## Empirical signature of 1st and 2nd order

**Empirical signature of 1****st**** and 2****nd**** order**

**~1 at Ph. Tr ~ X(5)**

- Vibrator

- Rotor

## Slide 13

## Slide 14

## Degeneracies ~ Symmetries ??

**Degeneracies ~ Symmetries ??**

## Mapping structure throughout the triangle

**R****4/2**** = E(4****+****)/E(2****+****)  -- Defines locus in triangle **
**Need another observable: **

**= 2.9**

**R****4/2**

**H =**** ****aH****sph**** +**** ****bH****def**

**  ****=****   ****ε nd -  ******** Q ******** Q**

## Evolution of Structure

**104****Mo ~ 2.9**

**104****Mo ~ +0.4**

**Vibrator**

**Rotor**

**γ**** - soft**

**    ****104****Mo **

**Complementarity of ****macroscopic**** and ****microscopic**** approaches.**
**  ****Why do certain nuclei exhibit specific symmetries?  Why these evolutionary trajectories? **
**   ****        **
_***What will happen far from stability in regions of proton-neutron asymmetry and/or weak binding?***_

**Mapping Structure with Simple Observables – Technique of Crossing Contours**

**Evolution of Structure**

**2-parameter – phenomenologically very successful**

## Slide 18

## γ

- Alaga

**A different perspective**

**γ**

****

**-**

**Mixing**

- Alaga

## Hf

**Hf**

## 170Hf β-decay Experiment

- SAMMY

- Original Aim: g-factor of 21+ state

- Perturbed γ-γ angular correlations

- 8   ~25% Ge detectors

## Looking back at systematics

## γ

**γ**

****

**-**

**Mixing**

**  ****When these excitations are nearly degenerate, they fall into two classes:**
**Nearly pure, SU(3)-like**
**2) Nearly totally mixed, with very large matrix elements (an order of magnitude larger than when these collective modes are well separated).**

## E0 transitions are a signature of this mixing !

**E0 transitions are a signature of this mixing !**

## Thanks

## BACKUPS

## X(5)

**Bessel equation**
**	**

**Critical Point Symmetries**
**First Order Phase Transition – Phase Coexistence**

**Energy surface changes with valence nucleon number**

**Iachello**

**X(5)**

##  H = aHsph + bHdef      Structure  ~  a/b 

**Competition between**** ****spherical-driving**** **
**(pairing – like nucleon)**** ****and**** ****deformation-driving**** ****(esp. p-n)**** ****interactions**
**	****      **
** ****H =**** ****aH****sph**** +**** ****bH****def****     **** Structure  ~**  _**a**__**/**__**b**_

**2 parameters**
**2-D surface**

**H =****   ****ε n****d****    -  ******** Q ******** Q**

**IBA:**

**Parameters:  ********ε/********,  ******** (within Q)******

- McCutchan and Zamfir

## Experiments on 172Hf and 174Hf

- Experiments on 172Hf and 174Hf

## Contours enclose regions of deformed nuclei

**Contours enclose regions of deformed nuclei**

**p-n / pairing**

**P ~ 5  **

## Search for 2γ→ 41 transition in 172Hf

- Search for 2γ→ 41 transition in 172Hf

## R4/2

*R*4/2

**Pinpointing Nuclear Structure**
**Technique of Orthogonal Crossing Contours (OCC)**

## Slide 33

## Search for 2γ→ 41 transition in 174Hf

- Search for 2γ→ 41 transition in 174Hf

##  Alaga rules

## Slide 36

## Signatures of Structure and Phase Transitions

**R****4/2**** = E(4****+****)/E(2****+****)  -- Defines locus in triangle **
**Need another observable: **

## Gamma Band in X(5)

**Gamma Band in X(5)**

## The outcome of this is that the 275 keV g–ray intensity, when combined with the conversion electron intensities of Goswamy et al. [19] yields αK(275) = 0.097 (8) (expt.), cf. αK(275,M1) = 0.0862 (12) (theory) [15]. Therefore, the 275 keV transition is (nearly) pure M1 and is noncollective.  

## E0 Transitions between 0+ states

**E0 Transitions between 0****+**** states**
**Large values in deformed nuclei are a fundamental prediction of most collective models – [In IBA, they directly sample the n****d ****boson structure]**
**Hence these and other new measurements are important results. **

## Looking more systematically

- Alaga

## Slide 42

## Slide 43

## Slide 44

## Some results

**S****2n ****(*****N,Z*****) = -M(*****N,Z*****) + M(*****N-2,Z) + *****2M*****n***

- PRL **96** (2006) 042504,
- EPJA **32** (2007) 87
- PRC **75**  (2007) 064302
- NPA  (2007) in print** **
- EPJA submitted
- PRL in preparation

- XXX Mazurian Lake conference, Poland, 2007                                                                                S.Rahaman, Jyväskylä

## N = 82

**Energy required to remove two neutrons from nuclei**

**N = 82**

**N = 84**

**N = 126**

## Slide 47

## Structure Beyond the Phase Transition

**New simple geometrical models to describe a wider range of structures**

*β**m*

*β**M*

- One Parameter, Analytic Solutions

## 168Hf Experiment

- 8 Clover detectors
- γ-γ angular correlations
- Beam on/beam off
- ~ 2 days

- (R4/2 = 3.11)

- YRAST BALL

## Slide 50

## Staggering indices for gamma softness, triaxiality

**Staggering indices for gamma softness, triaxiality**

## Testing new geometrical models

- R4/2 (168Hf) = 3.11

## Testing new geometrical models

**X**

## Signatures of Structure and Phase Transitions

**R****4/2**** = E(4****+****)/E(2****+****)  -- Defines locus in triangle **
**Need another observable: **
**Good but not enough:  a regional perspective is also valuable**
**  **

## X(5)-βn  model

- X(5)-βn  model

- Dennis Bonatsos *et al.,* Phys. Rev. C **69**, 014302 (2004).

## Slide 56

## Confined Beta-Soft (CBS)

- N. Pietralla and O.M. Gorbachenko*,* Phys. Rev. C **70**, 011304(R) (2004).

*β**m*

*β**M*

*β**m*

## Spanning structure from vibrator to rotor

## Slide 59

## Slide 60

## Slide 61

## Slide 62

## Slide 63

## Slide 64

## Slide 65

## The microscopic foundations  Empirical p-n interactions

**p-n interactions are stronger in lower left quadrant. **
**Collectivity (as measured by R****4/2)  ****grows faster  (red points in lower plot) in lower left quadrant too. **
**This is the first direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths**

**The microscopic foundations  Empirical p-n interactions**

**Link to collective structural evolution**

## Vibrator

**Vibrator**

**Symmetric rotor**

**γ****-soft**

**CBS**

**X(5)-****β****n**

**E(5)-****β****n**

**CBS**

**Davidson**

- Bonatsos et al, McCutchan

**New solvable models**

## Slide 68

##                                                       from different IBA

- configurations (Based on Sambataro and Molnar)

- Contributions to *** *2

- Virtually no mixing in precisely the nucleus that shows the largest *** *2 (*E*0)
- Large ** 2 (*E*0) is not from mixing, but rather originates within a single set of configurations

## Parameter- free except for scale

**Parameter- free except for scale**

- Casten and Zamfir

## Comparison of relative energies with X(5)

**Comparison of relative energies with X(5)**

## Absolute energy spacings in the 0+2 sequence – effects of a sloped wall

- Caprio

**Absolute energy spacings in the 0****+****2 ****sequence – effects of a sloped wall**

## Slide 73

## Slide 74

## Slide 75

## Shell structure:                                      100-300 keV
Quantum 1st order phase trans:     ~ 100 keV
Interaction filters (e.g., p-n)              ~10-15 keV 

**Quantum 1****st**** order phase trans:     ~ 100 keV**
**Interaction filters (e.g., p-n)              ~10-15 keV**

**Total mass/binding energy:****   ****Sum of all interactions**
**Mass differences:****   ****Separation energies **
**	****	****	**** ****shell structure, phase transitions**
**Double differences of masses:****   ****Interaction filters**

**Masses:**

**Macro**

**Micro**

**Masses and Nucleonic Interactions**

**Needed accuracy**

## S2(N) versus N:  

- S2(N) versus N:
- IBA gives a straight line in
- normal regions.

**2-neutron separation energies in phase transitional regions**

- First order shape phase transitions, discontinuities in

- Second order transitions, discontinuities in

- Slide based on Iachello

## Alaga = 1.43

## Slide 79