W. Udo Schröder, 2011...Light nuclei Saddle config. =touching spheres Heavy nuclei Saddle config. = near spherical Medium weight nuclei Saddle config. = elongated shapes W. Udo Schröder,

1

W. Udo Schröder, 2011

Spo

ntan

eous

Fis

sion


Spo

ntan

eous

Fis

sion

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 20

0 0

3:

4

irrot m mInertia irrotational flow B R AR

Bohr&Mottelson II, Ch. 6

Surface & Coulomb energies important: Stability limit C 0

2


Spo

ntan

eous

Fis

sion

Fission Instability (Fissility)

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

Coul Coul

Coul Cou

s

l

s

s s

Stability if surface energy can balance Coulomb repulsion

stable if

E E

E

E E

E EE

22

22 2 2

22 2

2 2

22 2

1( ) ( 0

2( ) ( 0) 1

5

2( 0) (0

) 15

1( ) (0 )

5)

,

5

s Cou

crit

lE A MeV E Z A MeV

x f Z A

Spontaneous fission Z Ainstabilit Zy A

2 3 2 1 32

2

2

2

2

( 0) 17.

( )

8 ( 0) 0.71

( )

: 50

Bohr-Wheeler fissility parameter independent of 2

Cou

s

lEx

E

(0

2

)

(0)Stability if

fissility x < 1

3

Not considered here: Fissility depends on asymmetry (N-Z)2/A, for both bulk and surface.


Spo

ntan

eous

Fis

sion

Fission Potential Energy Surface (PES)

Cut along fission path

FF

1 F

F2

2mFc2

mCNc2

Typical (induced) fission:

*

235 236

* *1 2 ( )

thU n U

F F n Q

4

Saddle

Q

Fission Barrier Bf defined relative to g.s. minimum

g.s.

Bf

g.s. 2

4 PES

Saddle

FF1

FF2


Spo

ntan

eous

Fis

sion

LDM-Fission Saddle Shapes

“Rotating-Liquid Drop Model,” Cohen & Swiatecki, 1974

Fission saddle= equilibrium point, equal probabilities to go forward (binary scission) or backward (mono-nucleus)

5

Light nuclei Saddle config. =touching spheres

Heavy nuclei Saddle config. = near spherical

Medium weight nuclei Saddle config. = elongated shapes


Spo

ntan

eous

Fis

sion

Systematics of Fission Total Kinetic Energies

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

Corresponds to the relative energy of the fission fragments when emitted from a moving nucleus:

6

Relative velocities of two fission fragments due to Coulomb repulsion Kinetic energy EK CN: fissioning compound nucleus

rel FF FF 1 2

2

1 3( , ) 0.1189 0.0011 (7.3 1.5)CN

K CN CN

CN

ZE Z A MeV

A

Viola, Kwiatkowski & Walker, PRC31, 1550 (1985)

FF 1

FF 2

CN


Spo

ntan

eous

Fis

sion

Nuclear Viscosity in Fission

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)

iiwall iwall

i ii ii iwi

F

jF

Fnd

dE drd

dt d

dE dr dr

dt d d

2

2 2

3

4

32

16

Davies et al. PRC13, 2385 (1976)

Viscosity 25% of strength in HI collisions

FF1 FF2

r

Tf (∞)=Q+Bf

7


Spo

ntan

eous

Fis

sion

Prescission Neutron Emission

nn

sad sc TKE

sad sc

Mean neutron evaporation time

Numerical transport calculations

T TKE

s fit to experiment

21

1.

:

, , , ,

(2 5) 10

Neutron emission during transition CN Bf Scission

Equivalent to multi-chance fission Expt. Setup: D. Hinde et al., PRC45, 1229 (1992)

Exptl. setup detects FF, light charged particles, neutrons in coincidence

decompose angular distributions (Sources: CN, FF1, FF2)

Systematics: WUS et al. Berlin Fission Conf. 1988

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

See V. Tichenko et al. PRL 2005

F s 2135 15 10

8

N detector

FF

FF

Time for one fission decay


Spo

ntan

eous

Fis

sion

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 emission Radio-chemical data

Structure effects in Pa fission disappear at excitations E* (Pa) > 70 MeV

9


Spo

ntan

eous

Fis

sion

Fission Fragment Z Distributions yie

ld

Vandenbosch & Huizenga, 1973

Zp: The most probable Z

Same Gaussian A(Z-Zp)

<A

light>

<

Aheavy>

ACN

Bimodal mass distributions: Structure effect, not gross LD Increasing ACN more symmetric.

<Aheavy> ≈ 139 shell stabilized via

<Zheavy>≈ 50

10


Spo

ntan

eous

Fis

sion

Models for Isobaric Charge Distributions

21 2

1 1 2 2 1 1 2 2

11

( , , , ) ( , ) ( , )

: 0

LD LDsc

p

A

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 0.5

UCD 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 p

A

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

11


Spo

ntan

eous

Fis

sion

Models for Isobaric Charge Distributions

Rsc

p pV Z A V Z A c Z Z c MeV per Z unit 2 2

1 1 1

1( | ) ( | ) 3.2 0.3 ( )

2

pZ cZ A Z TP 2 2

1 1 12( ) exp 2

Try thermal equilibrium (T):

Linear increase of variance 2 with T not observed, but ≈ const. up to E*<50MeV

N

Z V(Z,N)

P(Z,N)

A

2 2 2 2 2( ) 1 /

:

Z N NZ A

NZ

Nucleon exchange diffusion

Z A

correlation coefficient

Studied in heavy-ion reactions.

dynamics? e.g., NEM ?

12


Spo

ntan

eous

Fis

sion

Mass-Energy Correlations

light heavy

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

13


Spo

ntan

eous

Fis

sion

Fine Structure in Fission Excitation Functions

J. Blons et al., NPA 477, 231 (1988)

match to incoming wave

I II

Also: g and n decay from II class states

Class I and II vibrational states coupled

14


Spo

ntan

eous

Fis

sion

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

22

2

2

22

2

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

15

Auxiliary slides on a kinetic model for fission to follow


Spo

ntan

eous

Fis

sion

16


Spo

ntan

eous

Fis

sion

Kinetic Theory of Fission (T>0)

V()

saddle

poin

t

P(,t)

time

Collective d.o.f. coupled weakly to stochastic (nucleonic) degrees of freedom representing heat bath level density parameter a(A,Z)

Kramers 1942, Grange & Weidenmüller, 1986

trans

g

220 0 0

20 0

0

22

2

( )( ) ; .

. . ( ) ( )2

( )( )

: ( ) ( )2

( )( )

s s s

s s

s

dVMean force F friction coeff

d

BNear g s V V

dVF B

d

BNear saddle V V

dVF B

d

Harmonic Approximation

intT E a A Z * ( , )

Langevin Equation for fission d.o.f. ()

g

2

2( )

d dB F B

dtdtF F F t ( ) ( ) ( , )

s 0

Bf

17


Spo

ntan

eous

Fis

sion

Kinetic Theory of Fission (T>0)

V()

saddle

poin

t

P(,t)

time

Steady state, for t ∞

trans

s 0

P t P F PT

t B

g

2

2

( , ) 1 ( )

Diffusion DriftT D FD

B Tcoefficient coefficient

g

D PP t P

t

g

2

2 2

( , ) 1 ( )

Equivalent for large damping g: Fokker-Planck Equ. for probability P(,t)

Randomly fluctuating force

TF t F t t t D t t

B

g

:

( , ) ( , ) 2 ( ) 2 ( )

T

P t eB

2022

2 20

2( , ) ;

g

g

1

00

0

0

( ) ( )( , ) ( ) exp

2( , ) exp

s

s

s fs

V VTj t P d

B T

Bj t

T

Bf

Kramer’s escape rate)

18


Spo

ntan

eous

Fis

sion

Kramers’ Stochastic Fission Model

V()

saddle

poin

t

P(,t)

time

Collective degree of freedom coupled weakly to internal (nucleonic) d.o.f.

relax coll

damped viscous coll oscillation

for average t

Lagrange Rayleigh Equ o Motion

( )

( )

. .

ss

ss

Fokker Planck Equation for P t

Transport diffusion coefficient

D T T T

T TT

V B frequency

d dt viscosity coe

Fluctation Dissipation Theorem

fficient

g

g

*

*

2 2

( , )

( ) :

( , ) ( , ) ( )

( )1( , ) ( ) coth

2 2

( )

( )

Gradual spreading of probability distribution over barrier (saddle). Probability current from jF =0 to stationary value at t ∞

Grange & Weidenmüller, 1986

trans

Bf

19


Spo

ntan

eous

Fis

sion

Fission Transient and Delay Times

Concepts revisited by H. Hofmann, 2006/2007

Statistical Model fission life time: 1

*

0

1( )

2 ( *)

E Esad

statM sadCN

dE EE

Level Density

V()

Inverted parabola Oscill frequ. sad

( ) sad

Reduced friction coefficient

B g

21

1

2

statMKramers

F Kramers trans

long for

(0) 90% ( )

trans

F F

Transient time

j j

0

E* Esad

Takes longer for stronger viscosity

2

0


Spo

ntan

eous

Fis

sion

Angular Distribution of Symmetry Axis

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

MK MKW I D

2

1

Documents

W. Udo Schröder, 2011...Light nuclei Saddle config. =touching spheres Heavy nuclei Saddle config. = near spherical Medium weight nuclei Saddle config. = elongated shapes W. Udo Schröder,