Seminar: Statistical Decay of Complex Systems (Nuclei)
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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 : Birrot
m0
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
220 ≠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 A1 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) 1021 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
13
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
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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)
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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
DF
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
2CN (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