Transcript No Slide Title

Structure of Warm Nuclei
Sven Åberg, Lund University, Sweden
Structure of Warm Nuclei
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)
(b)
(c)
(d)
(e)
Combinatorical intrinsic level density
Pairing
Rotational enhancements
Vibrational enhancements
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
In collaboration with:
H. Uhrenholt, Lund
T. Ichikawa, RIKEN
P. Möller, Los Alamos
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 wave functions:
From eigen energies:
Distribution of nearest neighbor
energy spacings:
P(s)
Regular
e
Regular Chaotic
P(x)
Distribution of transition
matrix elements:
Chaotic (Porter-Thomas
distr)
Chaotic
Avoided level
crossings
s  ( Ei  Ei 1 ) / d
x=M/<M>
Regular (selection rules)
Experimental knowledge
J.D. Garrett et al,
Nuclear Data Ensemble Phys. Lett. B392 (1997) 24
Transition from
regularity to chaos
with increasing
excitation energy:
Energy (MeV)
15
Neutron resonance
region - chaotic:
10
Eexc(MeV)
Near-yrast levels
– regular:
Line connecting
rotational states
8
Chaos
0
Regularity
5
Yrast line
0
0
10
20
Angular momentum
30
Excited many-body states mix due to residual interaction
Many-body state from many-particle-many hole excitations:
e
e
EFermi
Ground state |0>
...
..
Excitation
energy
Level density
of many-body
states:
Ground state
2p-2h excited state
Residual two-body interaction, W, mixes states,
   c  , and causes chaos [1]
Wave function,  , is spread out over many states
c


:
2
1/ 2
  2 W 2 p2h
2
E
E
 160
3/ 2
 0.039
 E MeV
 A 
[1] S. Åberg, PRL 64, 3119 (1990)
[2] B. Lauritzen, Th. Døssing and R.A. Broglia, Nucl. Phys. A457 (1986) 61.
[2]
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)
A way to measure the onset of chaos vs Eexc
Decay-out of superdeformed band
Decay-out occurs at different
Eexc in different nuclei.
 Use decay-out as a
measurer of chaos!
Energy
E2-decay
SD band
Low Eexc  regularity
High Eexc  chaos
ND band
Angular momentum
Chaotic
(Porter-Thomas distr)
Intermediate between regular and chaos
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
Exp level
dens. at Sn
82
126
- Fully chaotic but shows strong shell effects!
Level density
 (Eexc, I ,  )  P(Eexc ,  )F ( Eexc , I )  (Eexc )
where P and F project out parity and angular momentum, resp.
In (backshifted) Fermi gas model:
P( Eexc ,  )  0.5
F ( Eexc , I ) 
I  0.5
 I  0.52 

exp 
2

2



2

 ( Eexc ) 
exp 2 aU
1/ 4 5 / 4
12a U

where

U  Eexc  Eshift
Backshift parameter, Eshift, level density parameter, a, and
spin 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.
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
EFermi
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, )
Level density composed by v-qp exitations
2 quasi-particle excitation: seniority v=2
v quasi-particle excitation: seniority v
In Fermi-gas model:
v 1
vqp (Eexc )  Eexc
Total level density:
tot ( Eexc ) 

v  qp
v  0 , 2 , 4....
162Dy
total
( Eexc )
 exp(2 aEexc )
v=4
v=2
v=6
v=8
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 pairing gaps
vs excitation energy:
162Dy
neutrons
protons
Pairing remains at high excitation energies!
Distribution of
pairing gaps:
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) + ħ2/2J (e, 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.
Rotational enhancement
No pairing
Pairing incl.
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
The residual 2-body interaction (W) implies a broadening of many-body states:
c
2
1/ 2
 160
3/ 2
  2 W 2 2 p2h  0.039
 E MeV
 A 
E
E
Each many-body state is
smeared out by the width:
 ( Eexc )
Level density structure smeared
out at high excitation energies
162Dy
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 spacings
296 nuclei RIPL-2 database
Factor of about 5 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
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
84
46
Sr38
Parity enhancement stronger for spherical shape!
Extreme enhancement for negative-parity states in 79Cu
+
tot
79
29
50
e  0.05
Parity ratio
(Log-scale!)
Cu
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.
Strong shell structure in chaotic states 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. Fair agreement with data with no parameters
V.
Pairing remains at high excitation energies
VI. Parity asymmetry can be very large in (near-)spherical nuclei
VII. Structure of level density important for fission dynamics:
symmetric-asymmetric fission