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W. Udo Schröder, 2007

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Liquid-Drop Oscillations

0 2

2 2

:

( , , ) 1 ( ) ( , )

:

ˆ2 2

Shape function

R t R t Y

Small amplitude vibrations

dB CHdt

20

2 22 3

1 3 00

1. . : ( ) , . :216.93 ( 1): ( 1)( 2) 1.252 (2 1)

sLDM s

Qu M harmonic oscillator C Deform

a MeVa e ZLDM C Ar fmr A

5 200 0

3: 4irrot m mInertia irrotational flow B R AR

Bohr&Mottelson II, Ch. 6

Surface & Coulomb energies important: Stability limit C 0

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Fissility

Mostly considered: small quadrupole and hexadecapole deformations 220 ≠0 ≠ 4=40 But 3=0 (odd electrostatic moment forbidden)

2 22 2 2 2 2 2

2 22 2 2 2

2 1( ) ( 0) 1 ( ) ( 0) 15 52 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

ExE

2 3 2 1 3

2

2

2

22

( , 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

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Fission Potential Energy Surface

Fission path

2

4 PES

Cut along fission path

CN

Saddle

FF1

F

F 2

2mFc2

mCNc2

Q

Typical fission process:

*235 236

* *1 2 ( )

thU n U

F F n Q

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LDM-Fission Saddle Shapes

Cohen & Swiatecki, 1974

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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)CNK CN CN

CN

ZE Z A MeVA

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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

3 216

F iiwall iwall

F i ii ii iwind

dE dr ddt d

dE dr drdt d d

Davies et al. PRC13, 2385 (1976)

Viscosity 25% of strength in

HI collisions

FF1 FF2

r

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Kramers’ Stochastic Fission Model

V()

sadd

le p

oint

P(,t)

time Collective d.o.f. coupled weakly to internal (nucleonic) d.o.f.

( )( )

. .

relax coll

damped viscous coll motionfor average tLagrange Rayleigh Equ o Motion

*

*

2 2

( , )( ) :

( , ) ( , ) ( )( )1( , ) ( ) coth2 2

( ) ( )( )

locallocal

local

Fokker Planck Equation for P tTransport 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 coefficientB

21

12

statMKramers

F Kramers trans

long for

(0) 90% ( )trans

F F

Transient timej j

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Prescission Neutron Emission

21

1.:

, , , ,(2 5) 10

n n

sad sc TKE

sad sc

Mean neutron evaporation timeNumerical 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 distributionsSources CN, FF1, FF2

Systematics: WUS et al. Berlin Fission Conf. 1988

2135 15 10F s

Short fission times for high E*> 300-500 MeV ?

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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)

yiel

d

n(A)n(A)

FF Mass A

Pre-neutron emission Post-neutron emissionRadio-chemical data

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Fission Fragment Z Distributionsyi

eld

Vandenbosch & Huizenga, 1973

Zp: The most probable ZSame Gaussian A(Z-Zp)

<Alig

ht>

<A h

eavy

>

ACN

Bimodal mass distributions: With increasing ACN more symmetric.<Aheavy> ≈ 139 shell stabilized via <Zheavy>≈ 50

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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 AR

VMost probable Z ZZ

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 0.5

UCD CN CN

H H UCD L L UCDZ Z

Z Z A Z A Z AZ Z Z Z

2 2

1 1 1 21

1( | ) ( | ) 3.2 0.3 ( )2p pA

c

VV Z A V Z A Z Z c MeV per Z unitZ

MPE variance: expand V around Z=Zp:

V P(Z)

Z

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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 ?

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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.

222

2

222

2

2 ( ) 2 ( )

1( )2

2 2 ( )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 Axis2( ) (2 1) ( , , )I I

MK MKW I D