Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

1
Spontaneous Fission
W. Udo Schröder, 2007
Liquid-Drop Oscillations
Shape function : Bohr&Mottelson II, Ch. 6
Spontaneous Fission
2




R( ,  , t )  R0 1      (t ) Y ( ,  ) 
 2   


Small amplitude vibrations :

d 
B

ˆ
H 

2    dt
Inertia irrotational flow : Birrot 
m0

R05 
2
C 

 
2    
2
3 m
AR02
4 
1
Qu.M. harmonic   oscillator : C   2  (  )   , Deform.   :   0
2
LDM
LDM : C
 (  1)(  2)
as

23
A
3 (  1) e2 Z 2

2 (2  1) r0 A1 3
as  16.9MeV
r0  1.25 fm
Surface & Coulomb energies important: Stability limit C  0
W. Udo Schröder, 2007
Fissility
Spontaneous Fission
3
Mostly considered: small quadrupole and hexadecapole deformations
220 ≠0 ≠ 4=40 But 3=0 (odd electrostatic moment forbidden)
2


Es (2 )  Es (2  0) 1  22 
5


1


ECoul (2 )  ECoul (2  0) 1  22 
5


2 2
2 2
Stability , if ECoul (2 )  ECoul (0) 2  Es (2 0)  Es (0) 2
5
5
Bohr-Wheeler fissility parameter
Es (2 , 0)  17.8 A2 3 MeV
x 
ECoul (0)
2Es (0)
Stability if x < 1
ECoul (2 , 0)  0.71 Z 2 A1 3 MeV
 x  f (Z 2 A)
Spontaneous fission instability :
W. Udo Schröder, 2007
Z 2 A  (Z 2 A)crit  50
Fission Potential Energy Surface
PES
4
4
Q
2
mCNc2
FF2
Spontaneous Fission
CN
Cut along
fission path
235
U  nth 
W. Udo Schröder, 2007

236
U

*
 F1*  F2*   n  Q
2mFc2
FF1
Typical fission process:
Spontaneous Fission
5
LDM-Fission Saddle Shapes
Cohen & Swiatecki, 1974
W. Udo Schröder, 2007
Systematics of Fission Total Kinetic Energies
Viola, Kwiatkowski & Walker, PRC31,
1550 (1985)
Spontaneous Fission
6
Average total kinetic energy
<EK>of both fission fragments as
function of fissioning compound
nucleus (CN) Z and A:
EK (ZCN , ACN )   0.1189  0.0011 
W. Udo Schröder, 2007
2
ZCN
13
ACN
 (7.3  1.5) MeV
Viscosity in Fission
FF2
7
FF1
Spontaneous Fission
r
For high fissilities (elongated scission
shapes) kinetic energies smaller than
calculated from saddle Coulomb repulsion:
TKE < Tf (∞)  viscous energy dissipation.
Nix/Swiatecki : Wall and window formula
(nucleon transfer, wall motion)
2
 dr

3
 dE 
 F   
i  d
 dt 
4

wall
wall  i d  i

Davies et al. PRC13, 2385 (1976)
W. Udo Schröder, 2007
3
 dE 

F 
 dt 
16

wind
Viscosity 25%
of strength in
HI collisions
2
2
 dr

 dr
 
 
i   2  
i  
d

d

 i
i
i

i
 

Kramers’ Stochastic Fission Model
Grange & Weidenmüller, 1986
P(,t)
Collective d.o.f.  coupled weakly to
internal (nucleonic) d.o.f.
time 
trans
 relax
 coll 
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damped (viscous) coll  motion

Spontaneous Fission
saddle point
V()
for average  (t )
Lagrange  Rayleigh Equ. o. Motion
Fokker  Planck Equation for P ( , t )
Transport (diffusion) coefficient :
Fluctation  Dissipation Theorem
D( , T )  T * ( , T )   ( )

Gradual spreading of probability
distribution over barrier (saddle).
Probability current from jF =0 to
stationary value at t  ∞
W. Udo Schröder, 2007
local ( )
1
T ( , T ) 
local ( ) coth
2
2T
*
local ( ) 
 V
2
 2

B( ) frequency
 ( )  d  dt  viscosity coefficient
Fission Transient and Delay Times
Statistical Model fission life time:
V()

1

 2CN (E*)

0

dE  sad (E )

Reduced friction coefficient
   ( ) B     sad
 Kramers 
 statM
1  2  
long for 
 F   Kramers   trans 2
Transient time  trans
jF (0)  90% jF ()
W. Udo Schröder, 2007
1
Level Density
Inverted parabola
Oscill frequ. sad
Spontaneous Fission
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 statM 
E *  Esad
Concepts revisited by H. Hofmann, 2005/2006
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Prescission Neutron Emission
D. Hinde et al., PRC45, 1229 (1992)
Mean 1. neutron evaporation time  n 
Numerical transport calculations :
 sad  sc ,  , T , TKE ,  TKE
Spontaneous Fission
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 sad  sc
Exptl. setup detects FF, lcps,
and n in coincidence 
decompose angular distributions
Sources CN, FF1, FF2
 F  35  15  1021 s
Short fission times for high
E*> 300-500 MeV ?
Systematics: WUS et al.
Berlin Fission Conf. 1988
W. Udo Schröder, 2007

(2  5)  1021 s fit to experiment
n
Fission Fragment Mass Distributions
E* Dependence of FF Mass
Distribution: asymm  symm
Pre-neutron emission
Post-neutron emission
Radio-chemical data
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232Th(p,
f)
Spontaneous Fission
n(A)
yield
Ep =
n(A)
n(A)
FF Mass A
H. Schmitt et al., PR 141, 1146 (1966)
Neutron emission in fission:
 ≈ 2.5±0.1
W. Udo Schröder, 2007
Croall et al., NPA 125, 402 (1969)
Fission Fragment Z Distributions
Vandenbosch & Huizenga, 1973
<Aheavy> ≈ 139 shell stabilized via
<Zheavy>≈ 50
<Alight>
yield
<Aheavy>
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Spontaneous Fission
Bimodal mass distributions: With
increasing ACN more symmetric.
Zp: The most probable Z
Same Gaussian A(Z-Zp)
W. Udo Schröder, 2007
ACN
Models for Isobaric Charge Distributions
Unchanged charge distribution (UCD): ZUCD : Z1 A1  Z2 A2  ZCN ACN
Experimentally not observed, but
ZH  ZH ,UCD 0.5
Z
ZL  ZL,UCD  0.5
Z
Minimum Potential Energy (MPE) Models
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e2 Z1 Z2
V (Z1 , A1 , Z2 , A2 )  ELD (Z1 , A1 )  ELD (Z2 , A2 ) 
Rsc
Spontaneous Fission
Rsc
V
P(Z)
Most probable Z  Z p :
Z
V
Z1
0
A1
App. correct for asymmetric fission (Z ≈ +0.5).
Incorrect: o-e effects, trends Z ≈ -0.5 at symmetry.
MPE variance: expand V around Z=Zp:
1 2V
V (Z1 | A1 )  V (Z p | A1 ) 
2 Z 2
c
W. Udo Schröder, 2007

A1
Z  Zp

2
 c  3.2  0.3 MeV (per Z unit )
Models for Isobaric Charge Distributions
1
V (Z1 | A1 )  V (Z p | A1 )  c Z  Z p
2


2
 c  3.2  0.3 MeV (per Z unit )
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Try thermal equilibrium (T):
Spontaneous Fission
Rsc


P(Z1 A1 )  exp  Z1  Z p
 2 2    2  T
2
Linear increase of 2 with T not observed, but
 ≈ const. up to E*<50MeV
Z
V(Z,N)
A
 dynamics? NEM ?
Nucleon exchange  diffusion

P(Z,N)

2
 2 (Z A)   Z2  N2 1  NZ
/  2A
NZ : correlation coefficient
N
W. Udo Schröder, 2007
Studied in heavy-ion reactions.
c
Mass-Energy Correlations
Pleasanton et al., PR174, 1500 (1968)
235U
+nth Fission Energies
asymmetric fission: p conservation
p1
Spontaneous Fission
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235U
W. Udo Schröder, 2007
p2   p1
+nth EF1-EF2 Correlation
FF mass ratio
heavy
light
Pulse heights in detectors 
affected by pulse height defect
Fine Structure in Fission Excitation Functions
II
match to
incoming
wave
Spontaneous Fission
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I
J. Blons et al., NPA 477, 231 (1988)
Also:  and n decay from II
class states
Class I and II vibrational states coupled
W. Udo Schröder, 2007
Shell Effects in Fission
LDM barrier only approximate, failed
to account for fission isomers,
structure details of f.
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Shell effects for deformation 
Nilsson s.p. levels  accuracy
problem  Strutinsky Shell Corr.
Spontaneous Fission
E  ELDM  USM  USM  ELDM   E


USM  2  d g( )  
average g( ) 
ni 

2
2
2
 d e
1
2 2
   i 2


2 2
N  2  d g( )
e
   i 2


2 2
i
  E  2  i (ni  ni )
In some cases: more than 2 minima, different 1., 2., 3. barriers
W. Udo Schröder, 2007
i
Angular Distribution of Symmetry Axis
 (2I  1)
Spontaneous Fission
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I
WMK
( )
W. Udo Schröder, 2007
I
DMK
(,  , )
2