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W. Udo Schröder, 2007 Spontaneous Fission 1. W. Udo Schröder, 2007 Spontaneous Fission 2...
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Transcript of W. Udo Schröder, 2007 Spontaneous Fission 1. W. Udo Schröder, 2007 Spontaneous Fission 2...
W. Udo Schröder, 2007
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W. Udo Schröder, 2007
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Liquid-Drop Oscillations
02
2 2
:
( , , ) 1 ( ) ( , )
:
ˆ2 2
Shape function
R t R t Y
Small amplitude vibrations
dB CH
dt
20
2 22 3
1 300
1. . : ( ) , . :
2
16.93 ( 1): ( 1)( 2)
1.252 (2 1)sLDM s
Qu M harmonic oscillator C Deform
a MeVa e ZLDM C A
r fmr A
5 200 0
3:
4irrot m m
Inertia irrotational flow B R AR
Bohr&Mottelson II, Ch. 6
Surface & Coulomb energies important: Stability limit C 0
W. Udo Schröder, 2007
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Fissility
Mostly considered: small quadrupole and hexadecapole deformations ≠0 ≠ 4=40 But3=0 (odd electrostatic moment forbidden)
2 22 2 2 2 2 2
2 22 2 2 2
2 1( ) ( 0) 1 ( ) ( 0) 1
5 5
2 2, ( ) (0) ( 0) (0)
5 5
s s Coul Coul
Coul Coul s s
E E E E
Stability if E E E E
Bohr-Wheeler fissility parameter(0)
2 (0)Coul
s
Ex
E
2 3 2 1 3
2
2
2
2
2
( , 0) 17.8 ( , 0) 0.71
(
(
: ) 50
)
s Cou
cri
l
t
E A MeV E Z A MeV
x f Z A
Spontaneous fission instabi Z A Z Ality
Stability if x < 1
W. Udo Schröder, 2007
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Fission Potential Energy Surface
Fission path
PES
Cut along fission path
CN
Saddle
FF1
F
F 2
2mFc2
mCNc2
Q
Typical fission process:
*235 236
* *1 2
th
n
U n U
F F Q
W. Udo Schröder, 2007
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LDM-Fission Saddle Shapes
Cohen & Swiatecki, 1974
W. Udo Schröder, 2007
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Systematics of Fission Total Kinetic Energies
Viola, Kwiatkowski & Walker, PRC31, 1550 (1985)
Average total kinetic energy <EK>of both fission fragments as function of fissioning compound nucleus (CN) Z and A:
2
1 3( , ) 0.1189 0.0011 (7.3 1.5)CN
K CN CNCN
ZE Z A MeV
A
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Viscosity in Fission
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
2 2
34
32
16
F iiwall iwall
F i ii ii iwind
dE drd
dt d
dE dr drdt d d
Davies et al. PRC13, 2385 (1976)
Viscosity 25% of strength in
HI collisions
FF1 FF2
r
W. Udo Schröder, 2007
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Kramers’ Stochastic Fission Model
V()
saddle
poin
t
P(t)
time Collective d.o.f. coupled weakly to internal (nucleonic) d.o.f.
( )
( )
. .
relax coll
damped viscous coll motion
for average t
Lagrange Rayleigh Equ o Motion
*
*
2 2
( , )
( ) :
( , ) ( , ) ( )
( )1( , ) ( ) coth
2 2
( ) ( )
( )
locallocal
local
Fokker Planck Equation for P t
Transport diffusion coefficient
D T T T
T TT
V B frequency
d dt viscosity c
Fluctation Dissipation Theore
oe i
m
ff
cient
Gradual spreading of probability distribution over barrier (saddle). Probability current from jF =0 to stationary value at t ∞
Grange & Weidenmüller, 1986
trans
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Fission Transient and Delay Times
Concepts revisited by H. Hofmann, 2005/2006
1*
0
1( )
2 ( * )
E Esad
statM sadCN
dE EE
Statistical Model fission life time:
Level Density
V()
Inverted parabolaOscill frequ. sad
( ) sad
Reduced friction coefficient
B
21
1
2
statMKramers
F Kramers trans
long for
(0) 90% ( )trans
F F
Transient time
j j
W. Udo Schröder, 2007
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Prescission Neutron Emission
21
1.
:
, , , ,
(2 5) 10
n n
sad sc TKE
sad sc
Mean neutron evaporation time
Numerical transport calculations
T TKE
s fit to experiment
D. Hinde et al., PRC45, 1229 (1992)
Exptl. setup detects FF, lcps, and n in coincidence decompose angular distributions
Sources CN, FF1, FF2
Systematics: WUS et al. Berlin Fission Conf. 1988
2135 15 10F s
Short fission times for high E*> 300-500 MeV ?
W. Udo Schröder, 2007
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Fission Fragment Mass Distributions
H. Schmitt et al., PR 141, 1146 (1966)
E* Dependence of FF Mass Distribution: asymm symm
n(A
)
Neutron emission in fission: ≈ 2.5±0.1
232Th(p, f)
Ep =
Croall et al., NPA 125, 402 (1969)
yie
ld
n(A)n(A)
FF Mass A
Pre-neutron emission Post-neutron emissionRadio-chemical data
W. Udo Schröder, 2007
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Fission Fragment Z Distributionsyie
ld
Vandenbosch & Huizenga, 1973
Zp: The most probable Z
Same Gaussian A(Z-Zp)
<A
ligh
t>
<A
heavy>
ACN
Bimodal mass distributions: With increasing ACN more symmetric.
<Aheavy> ≈ 139 shell stabilized via <Zheavy>≈ 50
W. Udo Schröder, 2007
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Models for Isobaric Charge Distributions
21 2
1 1 2 2 1 1 2 2
1 1
( , , , ) ( , ) ( , )
: 0
LD LDsc
pA
e Z ZV Z A Z A E Z A E Z A
R
VMost probable Z Z
Z
Rsc
Minimum Potential Energy (MPE) Models
App. correct for asymmetric fission (Z ≈ +0.5).Incorrect: o-e effects, trends Z ≈ -0.5 at symmetry.
Unchanged charge distribution (UCD):
Experimentally not observed, but
1 1 2 2
, ,
:
0.5 0.5UCD CN CN
H H UCD L L UCDZ Z
Z Z A Z A Z A
Z Z Z Z
2 2
1 1 1 2
1
1( | ) ( | ) 3.2 0.3 ( )
2p pA
c
VV Z A V Z A Z Z c MeV per Z unit
Z
MPE variance: expand V around Z=Zp:
V P(Z)
Z
W. Udo Schröder, 2007
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Models for Isobaric Charge Distributions
Rsc
21 1 11
( | ) ( | ) 3.2 0.3 ( )2p pV Z A V Z A c Z Z c MeV per Z unit
2 2 21 1 1( ) exp 2pP Z A Z Z T c
Try thermal equilibrium (T):
Linear increase of 2 with T not observed, but ≈ const. up to E*<50MeV
N
Z V(Z,N)
P(Z,N)
A
A=const.
2 2 2 2 2( ) 1 /
:
Z N NZ A
NZ
Nucleon exchange diffusion
Z A
correlation coefficient
Studied in heavy-ion reactions.
dynamics? NEM ?
W. Udo Schröder, 2007
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Mass-Energy Correlations
lightheavy
FF mass ratio
Pleasanton et al., PR174, 1500 (1968)
235U +nth Fission Energies
235U +nth EF1-EF2 Correlation
Pulse heights in detectors affected by pulse height defect
1p
2 1p p
asymmetric fission: p conservation
TKE
TKE
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Fine Structure in Fission Excitation Functions
J. Blons et al., NPA 477, 231 (1988)
match to incoming wave
I II
Also: and n decay from II class states
Class I and II vibrational states coupled
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Shell Effects in Fission
LDM barrier only approximate, failed to account for fission isomers, structure details of f.
Shell effects for deformation Nilsson s.p. levels accuracy problem Strutinsky Shell Corr.
2
222
2
222
2 ( ) 2 ( )
1( )
2
22 ( )
2
LDM SM SM LDM
SM
i
i
i
i i i ii
E E U U E E
U d g N d g
average g e
n d e E n n
In some cases: more than 2 minima, different 1., 2., 3. barriers
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Angular Distribution of Symmetry Axis
2( ) (2 1) ( , , )I I
MK MKW I D