Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

1
Spontaneous Fission
W. Udo Schröder, 2011
Liquid-Drop Oscillations
Shape function :
Bohr&Mottelson II, Ch. 6
2




R( ,  , t )  R0 1      (t ) Y ( ,  ) 
 2   


Small amplitude vibrations :
Spontaneous Fission

d 
ˆ  B 
H

2    dt
Inertia irrotational flow : Birrot 
m0

2
R05 

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, 2011
Fission Instability (Fissility)
Mostly considered: small quadrupole and hexadecapole deformations
220 ≠0 ≠ 4=40. But 3=0 (odd electrostatic moment forbidden)
3
2
1 



Es (2 )  Es (2  0) 1  22 
ECoul (2 )  ECoul (2  0) 1  22 
5
5 



Stability , if surface energy can balance Coulomb repulsion
Spontaneous Fission
 stable if  ECoul (2 )  ECoul (0)
1 2
2
2  Es (2 0)  Es (0) 22
5
5
Bohr-Wheeler fissility parameter
independent of 2
E s (2  0)  17.8 A2 3 MeV
x
ECoul (0)
2Es (0)
Stability if
fissility x < 1
ECoul (2  0)  0.71 Z 2 A1 3 MeV
 x  f (Z 2 A)
Spontaneous fission instability : Z 2 A  (Z 2 A)crit  50
Not considered here:
Fissility depends on asymmetry (N-Z)2/A, for both bulk and surface.
W. Udo Schröder, 2011
Fission Potential Energy Surface (PES)
PES
g.s.
4
Q
mCNc2
FF2
2
FF1
Spontaneous Fission
g.s.
U  nth 

236
U

*
 F1*  F2*   (n)  Q
W. Udo Schröder, 2011
FF1
Typical (induced) fission:
FF2
Cut along fission
path
235
Saddle
Saddle
4
Bf
2mFc2
Fission Barrier Bf defined
relative to g.s. minimum
LDM-Fission Saddle Shapes
Fission saddle= equilibrium point, equal probabilities to go forward
(binary scission) or backward (mono-nucleus)
Spontaneous Fission
5
Light nuclei
Saddle config.
=touching spheres
Medium weight nuclei
Saddle config.
= elongated shapes
Heavy nuclei
Saddle config.
= near spherical
“Rotating-Liquid Drop Model,” Cohen & Swiatecki, 1974
W. Udo Schröder, 2011
Systematics of Fission Total Kinetic Energies
6
Original fission systematics by Terrell, newer by
Viola et al. at various times.
Average total kinetic energy <EK> of
both fragments from fission of a
nucleus (A,Z) at rest
Spontaneous Fission
Corresponds to the relative energy of
the fission fragments when emitted
from a moving nucleus:
 rel  FF 1  FF 2
FF 1
CN
FF 2
EK (ZCN , ACN )   0.1189  0.0011 
W. Udo Schröder, 2011
Viola, Kwiatkowski & Walker, PRC31, 1550 (1985)
Relative velocities
of two fission
fragments due to
Coulomb repulsion
Kinetic energy EK
CN: fissioning
compound nucleus
2
ZCN
13
ACN
 (7.3  1.5) MeV
Nuclear Viscosity in Fission
FF1
FF2

7
Tf (∞)=Q+Bf
Spontaneous Fission
r
For high fissilities (elongated scission shapes)
kinetic energies smaller than calculated from
saddle Coulomb repulsion:
TKE < Tf (∞)=Q+Bf viscous energy dissipation.
Nix/Swiatecki : “Wall and Window Formula” for
viscosity/friction (nucleon transfer, wall motion)
2
 dr

3
 dE 






F
i  d
 dt 

wall 4 j wall  i d  i 
F
3
 dE 

F 
 dt 

wind 16
Davies et al. PRC13, 2385 (1976)
W. Udo Schröder, 2011
Viscosity 25%
of strength in
HI collisions
2
2
 dr

 dr
 
 
i   2  
i  
d

i
 i d  i 
i

  

Prescission Neutron Emission
Neutron emission during transition CN Bf  Scission
Equivalent to multi-chance fission
Expt. Setup: D. Hinde et al., PRC45, 1229 (1992)
FF
n
Numerical transport calculations :
 sad  sc , , T , TKE ,  TKE
 sad  sc
N detector
8
Mean 1. neutron evaporation time  n 
(2  5)  1021 s fit to experiment
Spontaneous Fission
FF
Exptl. setup detects FF, light charged
particles, neutrons in coincidence 
decompose angular distributions (Sources:
CN, FF1, FF2)
 F  35  15  10
21
s
Time for
one fission
decay
Shorter fission times for high
E*> 300-500 MeV ?
See V. Tichenko et al. PRL 2005
W. Udo Schröder, 2011
Systematics: WUS et al. Berlin Fission Conf. 1988
Fission Fragment Mass Distributions
E* Dependence of FF Mass
Distribution: asymm  symm
Pre-neutron emission
Post-neutron emission
Radio-chemical data
232Th(p,
f)
Spontaneous Fission
n(A)
yield
9
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, 2011
Croall et al.,
NPA 125, 402 (1969)
Structure effects in Pa fission disappear
at excitations E* (Pa) > 70 MeV
Fission Fragment Z Distributions
Vandenbosch & Huizenga, 1973
<Aheavy> ≈ 139 shell stabilized via
<Zheavy>≈ 50
<Alight>
yield
<Aheavy>
10
Spontaneous Fission
Bimodal mass distributions:
Structure effect, not gross LD
Increasing ACN  more symmetric.
Zp: The most probable Z
Same Gaussian A(Z-Zp)
W. Udo Schröder, 2011
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
11
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, 2011

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 2 )
Try thermal equilibrium (T):

12
Rsc
Spontaneous Fission

P(Z1 A1 )  exp  Z1  Z p
Z
V(Z,N)
c
Linear increase of variance 2 with T not
observed, but  ≈ const. up to E*<50MeV
A
 dynamics? e.g., NEM ?
Nucleon exchange  diffusion
P(Z,N)


2
 2 (Z A)   Z2  N2 1  NZ
/  2A
N
W. Udo Schröder, 2011
 2 2    2  T
2
NZ : correlation coefficient
Studied in heavy-ion reactions.
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, 2011
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
14
I
J. Blons et al., NPA 477, 231 (1988)
Also: g and n decay from II
class states
Class I and II vibrational states coupled
W. Udo Schröder, 2011
Shell Effects in Fission
LDM barrier only approximate, failed
to account for fission isomers,
structure details of f.
15
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, 2011
i
16
Spontaneous Fission
Auxiliary slides on a kinetic model for fission to follow
W. Udo Schröder, 2011
Kinetic Theory of Fission (T>0)
Kramers 1942, Grange & Weidenmüller, 1986
P(,t)
time 
trans
Collective d.o.f.  coupled weakly to stochastic (nucleonic)
*
degrees of freedom representing heat bath T  Eint
a( A, Z)
level density parameter a(A,Z)
17
Langevin Equation for fission d.o.f. ()

saddle point
Spontaneous Fission
V()
0
B
d2 
dt
2
 F ( )  g B
d
dt
F ( )  F ( )  F ( , t )
Bf
dV ( )
; friction coeff . g
d
Harmonic Approximation
B
2
Near g.s.
V ( )  V (0 )   02    0 
2
dV ( )
F ( )  
 B 02    0 
d 
Mean force F ( )  
s

0
B 2
2
 s    s 
2
dV ( )
F ( )  
  B s2     s 
d 
Near saddle : V ( )  V ( s ) 
s
W. Udo Schröder, 2011
Kinetic Theory of Fission (T>0)
Randomly fluctuating force :
P(,t)
 gT
F ( , t )  F ( , t )  2 
 B
time 
trans

  (t  t )  2D   (t  t )

18
Equivalent for large damping g:
Fokker-Planck Equ. for probability P(,t)

saddle point
Spontaneous Fission
V()
Bf
P( , t ) 1  2 P  (F  P ) 


T

t
g B    2
  
2
P( , t ) 1    D  P   (  P ) 
 2


t
  
g    2
Diffusion
0
s
coefficient

Steady state, for t  ∞
   0 2

2
T
2 2 ;  
P ( , t ) 

e
 2
B 02
W. Udo Schröder, 2011
D
gT
B
Drift
coefficient

DF
T
1

T  s
V ( )  V ( 0 ) 
j(  s , t )  P (  0 ) 
  exp 
 d 
Bg  0
T



2  
 B 
Kramer’s
j( s , t )  0 s  exp  f 
escape rate)
 g
 T 
Kramers’ Stochastic Fission Model
P(,t)
Grange & Weidenmüller, 1986
time 
Collective degree of freedom  coupled
weakly to internal (nucleonic) d.o.f.
 relax
trans
 coll 
damped (viscous) coll  oscillation
19
for average  (t )
Lagrange  Rayleigh Equ. o. Motion

Spontaneous Fission
saddle point
V()
Bf
Fokker  Planck Equation for P( , t )
Transport (diffusion) coefficient :
Fluctation  Dissipation Theorem
D( , T )  T *( , T )  g ( )
T *( , T ) 

Gradual spreading of probability
distribution over barrier (saddle).
Probability current from jF =0 to
stationary value at t  ∞
W. Udo Schröder, 2011
( s ) 
 ( s )
1
 ( s ) coth
2
2T
 V
2
 2

s
B frequency
g ( )  d  dt  viscosity coefficient
Fission Transient and Delay Times
Esad
V()
E*
 statM 
E *  Esad


1


dE sad (E )


0
 2CN (E*)

Reduced friction coefficient
  g ( ) B     sad
 Kramers 
 statM
1  2  
long for 
1
 F   Kramers   trans 2
Transient time  trans
jF (0)  90% jF ()
W. Udo Schröder, 2011
1
Level Density
Inverted parabola
Oscill frequ. sad
Spontaneous Fission
20
0
Statistical Model fission life time:
Takes longer for
stronger viscosity
Concepts revisited by H. Hofmann, 2006/2007
Angular Distribution of Symmetry Axis
 (2I  1)
Spontaneous Fission
21
I
WMK
( )
W. Udo Schröder, 2011
I
DMK
(,  , )
2