Structure of Warm Nuclei

Fysikersamfundet - Kärnfysiksektionen Svenskt Kärnfysikmöte XXVIII, 11-12 november 2008, KTH-AlbaNova (sal FA32), Stockholm

Sven Åberg, Lund University, Sweden

Structure of Warm Nuclei

I.

Quantum chaos: Complex features of states (a) Onset of chaos with excitation energy (b) Role of residual interaction II. Microscopic method for calculating level density (a) Combinatorical intrinsic level density (b) Pairing (c) Rotational enhancements (d) Vibrational enhancements (e) Role of residual interaction III. Result (a) Comparison to data: at S n (b) Parity enhancement (c) Role in fission dynamics IV. Summary

In collaboration with:

and versus E exc H. Uhrenholt, Lund

T. Ichikawa, RIKEN P. Möller, Los Alamos

(Oslo data)

I. Quantum Chaos – Complex Features of States

Bohigas conjecture:

(Bohigas, Giannoni, Schmit, PRL

52

, 1(1984))

”Energies and wavefunctions for a quantum version of a classically chaotic system show generic statistical behavior described by random matrix theory

.” Classically regular: Poisson statistics Classically chaotic: GOE statistics (if time-reversal invariant) From eigen energies: Distribution of nearest neighbor energy spacings:

e

Regular Chaotic From wave functions: Distribution of transition matrix elements:

P(x)

P(s)

Regular Chaotic

(Porter-Thomas distr)

Chaotic Avoided level crossings

s

 (

E i



E i

 1 ) /

d

x=M/

Regular (selection rules)

15 10 5 0

0

Experimental knowledge

Nuclear Data Ensemble J.D. Garrett et al, Phys. Lett. B392 (1997) 24 Neutron resonance region - chaotic: Near-yrast levels – regular: Line connecting rotational states Transition from regularity to chaos with increasing excitation energy: E exc (MeV) 8 Chaos Yrast line 0 Regularity

10 20

Angular momentum

30

Excited many-body states mix due to residual interaction

e

Many-body state from many-particle-many hole excitations:

e

Excitation energy .

....

E Fermi 

Level density of many-body states:

Ground state |0>

Ground state

2p-2h excited state

Include residual interaction between many-body states:

H

 

i

e

i a i



a i

 1 4 

ijkl W ijkl a i



a j



a k a l

Residual int W

, mixes states:

   

c

 

and causes chaos [1] [1] S. Åberg, PRL 64, 3119 (1990)

c

 2

E

    2 

W

2  2

p

 2

h E



Onset of chaos in many-body systems

regular chaotic Chaos sets in at 2-3 MeV above yrast in deformed highly rotating nuclei. [1]

[1] M. Matsuo et al, Nucl Phys A620, 296 (1997)

Warm nuclei in neutron resonance region - Fluctuations of eigen energies and wave functions are described by random matrices – same for all nuclei - Level density varies from nucleus to nucleus: 50 82 126 Exp level dens. at S n - Fully chaotic but shows strong shell effects!

Level density

 (

E exc

,

I

,  ) 

P

(

E exc

,  )

F

(

E exc

,

I

)  (

E exc

)

where

P

and

F

project out parity and angular momentum, resp.

In (backshifted) Fermi gas model:

P

(

E exc

,  )  0 .

5

F

(

E exc

,

I

)  (

E exc

)  

I

 0 .

5  2 exp     exp  2 12

a

1 / 4

U

5 / 4 

I

 2  0 .

5  2 2

aU

  

where

U



E exc



E shift

Backshift parameter,

E shift

, level density parameter,

a

, and spin cutoff parameter,



, are typically fitted to data (often dep. on

E exc

) We want to have: Microscopic model for level density to

calculate

Obtain: Structure in



, and parity enhancement.

level density,

P

and

F

.

II. Microscopic method for calculation of level density

(a) Intrinsic excitations - combinatorics Mean field: folded Yukawa potential with parameters (including deformations) from Möller et al.

e e E Fermi

All np-nh states with n ≤ 9 included

Ground state |0> 2p-2h excited state

Count all states and keep track of seniority (v=2n), total parity and K-quantum number for each state Energy: E



(v, K,



)

II.b Pairing

For EACH state,



: Solve BCS equations provides: Energy, E



,corrected for pairing (blocking accounted for) Pairing gaps,



n and



p Energy: E



(v, K,



,



n ,



p )

Mean proton pairing gap vs excitation energy: 0 qp 2 qp 4 qp Pairing remains at high excitation energies!

Proton pairing gap distribution in excited many-body states in 162 Dy No pairing phase transition!

II.c Rotational enhancement

Each state with given K-quantum number is taken as a band-head for a rotational band: E(K,I) = E(K) + ħ 2 /2J (

e

,



n ,



p ) [I(I+1)-K 2 ] where moment of inertia, J, depends on deformation and pairing gaps of that state [1] Energy: E



(v, I, K,



,



n ,



p ) [1] Aa Bohr and B.R. Mottelson,

Nuclear Structure

Vol. 2 (1974); R. Bengtsson and S. Åberg, Phys. Lett. B172 (1986) 277.

Rotational enhancement

No pairing Pairing incl.

162 Dy

II.d Vibrational enhancement – res. int.

Add QQ-interaction corresponding to Y 20 (K=0) and Y 22 (K=2), double-stretched, and solve Quasi-Particle Tamm-Dancoff for EACH state.

Isoscalar giant quadrupole resonances well described: 58A -1/3 MeV

II.d Vibrational enhancement – res. int.

Add QQ-interaction corresponding to Y 20 (K=0) and Y 22 (K=2), double-stretched, and solve Quasi-Particle Tamm-Dancoff for EACH state.

Correct for double-counting of states.

Gives VERY small vibrational enhancement!

Microscopic foundations for phonon method??

II.e Role of residual interaction on level densities

c



The residual 2-body interaction (W) implies a broadening of many-body states:

2    2 

W

2  2

p

 2

h

 0 .

05

E

3  / 2

A MeV E



E



Each many-body state is smeared out by the width:

  (

E exc

)

162 Dy Level density structure smeared out at high excitation energies

Comparison to measured level-density functions (Oslo data [1]): [1] S. Siem et al PRC65 (2002) 044318; M. Guttormsen et al PRC68 (2003) 064306; E. Melby et al PRC63 (2001) 044309;A. Schiller et al PRC63 (2001) 021306(R )

Comparison to data – neutron resonance spacings

296 nuclei RIPL-2 database Factor of about 4 in rms-error – no free parameters Compare: BSFG factor 1.8 – several free parameters

Spin and parity functions in microscopic level density model - compared to Fermi gas functions

Microscopic spin distribution

III.b Parity enhancement

Fermi gas model: Equal level density of positive and negative parity Microscopic model: Shell structure may give an enhancement of one parity

III.b Parity enhancement - calculated

Parity enhancement in Monte Carlo calc (based on Shell Model) [1] Present calc [1] Y. Alhassid, GF Bertsch, S Liu and H Nakada, PRL 84, 4313 (2000)

III.b Parity enhancement - measured

Measured [1] level densities of 2 + and 2 states in 90 Zr (spherical) Parity ratio Skyrme HF (Hilaire and Goriely NPA779 (2006) 63)

 2   2 

present [1] Y. Kalmykov et al Phys Rev Lett 99 (2007) 202502

Role of deformation

84 46

Sr

38

Parity enhancement stronger for spherical shape!

Extreme enhancement for negative-parity states in 79 Cu



+



-



tot

79 29 e

Cu

 50 0 .

05

Parity ratio (Log-scale!)

III.c Fission dynamics

Symmetric saddle Asymmetric saddle P. Möller et al, submitted to PRC

Asymetric vs symmetric shape of outer saddle

At higher excitation energy: Level density larger at symmetric fission, that will dominate.

Larger slope, for symmetric saddle, i.e. larger s.p. level density around Fermi surface:

SUMMARY

I.

Coexisting chaos and shell structure around n-separation energy II. Microscopic model (micro canonical) for level densities including: - well tested mean field (Möller et al) - pairing, rotational and vibrational enhancements - residual interaction schematically included III. Vibrational enhancement VERY small IV. Pairing remains at high excitation energies V.

Fair agreement with data with no new parameters VI. Parity asymmetry can be very large in (near-)spherical nuclei VII. Structure of level density important for fission dynamics: symmetric-asymmetric fission