Structure of Warm Nuclei

Sven Åberg, Lund University, Sweden

I. Quantum chaos: Complex features of states Onset of chaos with excitation energy – 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 Sn and versus Eexc (Oslo data)(b) Parity enhancement(c) Role in fission dynamics

IV. Summary

Structure of Warm Nuclei

In collaboration with: H. Uhrenholt, LundT. Ichikawa, RIKENP. Möller, Los Alamos

Regular Chaotice

From eigen energies:

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 statisticsClassically chaotic: GOE statistics (if time-reversal invariant)

I. Quantum Chaos – Complex Features of States

dEEs ii /)( 1

Regular

Chaotic

P(s)

Distribution of nearest neighborenergy spacings:

Avoided level crossings Regular (selection rules)

Chaotic (Porter-Thomas distr)

x=M/<M>

P(x)

From wave functions:Distribution of transition

matrix elements:

Experimental knowledge

Transition from regularity to chaos

with increasing excitation energy:

Eexc(MeV)

0

8

Regularity

Chaos

0

5

10

15

0 10 20 30Angular momentum

En

ergy

(M

eV)

Nuclear Data Ensemble

Neutron resonance region - chaotic:

Near-yrast levels – regular:

Yrast line

J.D. Garrett et al, Phys. Lett. B392 (1997) 24

Line connectingrotational states

[2] B. Lauritzen, Th. Døssing and R.A. Broglia, Nucl. Phys. A457 (1986) 61.

E

2

c

MeVEA

W hp2/3

2/1

222 160

039.02

E

[2]

Excited many-body states mix due to residual interaction

Level densityof many-bodystates:

Excitationenergy

Ground state

.....

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

EFermi

Ground state |0>

2p-2h excited state

Wave function, , is spread out over many states :

Residual two-body interaction, W, mixes states,

c , and causes chaos [1]

[1] S. Åberg, PRL 64, 3119 (1990)

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

Onset of chaos in many-body systems

Chaos sets in at 2-3 MeV above yrast

in deformed highly rotating nuclei. [1]

chaotic

regular

Decay-out of superdeformed band

Decay-out occurs at different Eexc in different nuclei. Use decay-out as a measurer of chaos!

Low Eexc regularity

High Eexc chaos

SD band

ND band

E2-decay

Angular momentum

Ene

rgy

Chaotic (Porter-Thomas distr)

Intermediate between regular and chaos

A way to measure the onset of chaos vs Eexc

Warm nuclei in neutron resonance region- Fluctuations of eigen energies and wave functions are described by random matrices – same for all nuclei

- Fully chaotic but shows strong shell effects!

- Level density varies from nucleus to nucleus:

Exp level dens. at Sn

1268250

)(),(),(),,( excexcexcexc EIEFEPIE

where P and F project out parity and angular momentum, resp.

In (backshifted) Fermi gas model:

aUUa

Eexc 2exp12

)(4/54/1

5.0),( excEP

2

2

2 2

5.0exp

5.0),(

II

IEF exc

where

shiftexc EEU

Backshift parameter, Eshift, level density parameter, a, andspin cutoff parameter, , are typically fitted to data (often dep. on Eexc) We want to have: Microscopic model for level density to calculate level density, P and F. Obtain: Structure in , and parity enhancement.

Level density

II. Microscopic method for calculation of level density

EFermi

Ground state |0>

2p-2h excited state

All np-nh states with n ≤ 9 included

(a) Intrinsic excitations - combinatorics

Mean field: folded Yukawa potential with parameters (including deformations) from Möller et al.

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

Energy: E(v, K, )

Level density composed by v-qp exitations

2 quasi-particle excitation: seniority v=2v quasi-particle excitation: seniority v

In Fermi-gas model:1)(

vexcexcqpv EE

Total level density:

)exp(2

)()(....4,2,0

exc

vexcqpvexctot

aE

EE

v=2

v=4

v=6 v=8

162Dy

total

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)

Distribution of pairing gaps:

Mean pairing gapsvs excitation energy: protons

neutrons

162Dy

Pairing remains at high excitation energies!

No pairing phase transition!

II.c Rotational enhancement

Each state with given K-quantum numberis taken as a band-head for a rotational band:

E(K,I) = E(K) + ħ2/2J (, n, p) [I(I+1)-K2]

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.

Pairing incl.

No pairing

Rotational enhancement

162Dy

II.d Vibrational enhancement

Add QQ-interaction corresponding to Y20 (K=0) and Y22 (K=2), double-stretched, and solve Quasi-Particle Tamm-Dancoff for EACH state.Correct for double-counting of states.

162Dy

Gives VERY small vibrational enhancement!

Microscopic foundations for phonon method??

II.e Role of residual interaction on level densities

162Dy

E

2

c

MeVEA

W hp2/3

2/1

222 160

039.02

E

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

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

)( excE

Level density structure smearedout at high excitation energies

III.a Comparison to exp. data – Oslo data [1]

[1] See e.g. A. Voinov et at, Phys Rev C63, 044313 (2001) U. Agraanluvsan, et al Phys Rev C70, 054611 (2004)

Comparison to exp. data - Oslo data

Comparison to data – neutron resonance spacings

Exp

Comparison to data – neutron resonance spacings296 nuclei RIPL-2 database

Factor of about 5 in rms-error – no free parametersCompare: 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

Parity enhancement in Monte Carlo calc (based on Shell Model) [1]

[1] Y. Alhassid, GF Bertsch, S Liu and H Nakada, PRL 84, 4313 (2000)

Role of deformation

Parity enhancement stronger for spherical shape!

388446Sr

Extreme enhancement for negative-parity states in 79Cu

507929Cu

05.0

tot

+

-

Parity ratio(Log-scale!)

III.c Fission dynamics

P. Möller et al, submitted to PRC

Symmetric saddle

Asymmetric saddle

Asymetric vs symmetric shape of outer saddle

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

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

SUMMARY

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

VI. Parity asymmetry can be very large in (near-)spherical nuclei

VII. Structure of level density important for fission dynamics: symmetric-asymmetric fission

IV. Fair agreement with data with no parameters

V. Pairing remains at high excitation energies

I. Strong shell structure in chaotic states around n-separation energy

III. Vibrational enhancement VERY small