Classical Model of Nuclear Fusion - University of Rochester1 Cl a s s i c a l R e a c t i o n T h e...

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Cla

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W. Udo Schröder, 2007

Sem

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Rea

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Classical Model of Nuclear Fusion

Strong absorption hypothesis: Fusion between 1&2 occurs for r < rfu

Effective potential 2 2

eff 2

LV V(r )

2 r

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L

Maximum L for fusion=Lfu:2 2

fueff fu 2

fu

2 2fu fu fu2

LE V V(r )

2 r

2L r E V(r )

2 2 2 2fu fu fu fu2

2L r E V(r )


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fu fu fu fu2

122fu fu2

2fu fu fu

L r E V(r )

r E V(r )2

E r E V(r )2 fu

fu fu

V(r )r 1

E

2

Experimental Complete Fusion Cross Sections

2 fuV(r )r 1

Determine 2 barrier parameters: rfu, V(rfu)

2fur

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cm

2fu fu1 E 0

cm fu

lim r

lim E V(r )

2 fufu fu

cm

2 2fu fu fu

cm

V(r )r 1

E

1r r V(r )

E2fur


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fucm fu0

lim E V(r )

Classical formula holds for energies well above barrier. Near barrier consider quantal transmission

Matter Density Overlap at Fusion

Calculate 35Cl fusion with quantal barrier penetration, parabolic (harmonic) approximation

W. Scobel et al., PRC14, 1808(1976)

Deduce barrier heights and radii from fits of excitation

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radii from fits of excitation functions


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32S projectiles on various targets

Strong absorption Fusion of medium-weight nuclei at 10% matter density overlap

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Fusion Dynamics for Heavy Systems

R R R

Different curves correspond to different

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different limitations of angular momenta (sliding, rolling, sticking)


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Light systems: Barrier penetration probability depends on Veff barrier geometry (height, width) and Ecm.Heavy systems: Escape away from fusion path into different d.o.f. (e.g., through deformation, friction, PES deflection from conditional saddle points. “Extra push” needed for fusion. Dissipative reactions (multi-nucleon transfer) take over. Difficulty to produce SHE (ZCN > 100) with fusion reactions.

Stability of Rotating Nuclei

Fusion (composite) nuclei produced with L 1

May fission spontaneously, if g.s. fission barrier small

Limits to CN Fusion

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if g.s. fission barrier small (Bf ˜ 0)

Predictions by the rotating liquid drop model.

If E* > 0, composite nucleus may fission during deexcitation cascade (Jf)

g.s.


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Blann et al., PRL 29, 303 (1972)

E*>0

CN

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Compound-Nucleus Processes

Particle Evapor-ationEvaporation Residues ER

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a AEcm,

Formation

C*E*=Ecm+Q I=

EquilibrationCompound

Decay

ER

-ray emission

Fission fragments


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Compound Nucleus

Statistical Independence Hypothesis:All degrees of freedom equilibrated, no memory of formation,except conservation laws (momentum, energy, angular momentum,…

Fission

Fusion reaction 14N+12C leading to compound nucleus 26-nAl, emitted at < ˜ 00

(Momentum Conservation)

elastic

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14N

26-nAl12C

CN decays in flight by particle


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CN decays in flight by particle evaporation (ER) or fission

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Fusion Excitation FunctionsR.G. Stokstad et al., PRL 41, 465 (1978) P. Sperr et al., PRL37, 321(1976)

fus˜ Ronly for Ecm below and close to barrier.

R

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148Sm: 2=0154Sm: 2=0.3

Maximum Lfusdue to yrast limitation (nuclear centrifugal stability)

ER = lowest window

R


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Deformation changes the effective barrier height larger fus d

d

0 ER F R

Fusi

onFi

ssio

n

multi-

nucl

eon

Tra

nsf

er

Ela

stic

/quas

i-el

astic

Sca

tter

ing

Fusi

on-E

R

ER Angular Distributions

p

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Random emission from moving CN does not change average velocity, preserves < > = 00,

ERp p


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preserves < > = 0 ,

Sideways recoil components important for angular distributions of ERs.

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Independence Hypothesis

Compound nucleus reaction (formation+decay)a+A C* b+B Decoupled 2-step process, intermediate equilibration following fusion takes long and leads to the same asymptotic condition C*(E, I,…)

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*

*

aA bB aA C

dD bB dD C

E E

E E

E E

*C bB E

Separation of cross sections:Independent probabilities of formation and decay multiply for overall reaction


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*gG bB gG CE E overall reaction

(HI, xn) Excitation Functions

a+Ab+B

C’* + nC’’* + 2nC’’’* + 3n

C* (19F, 7n)

(19F, 8n)

(19F, 9n)

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(HI, xn) cross sections

(19F, 9n)


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Elab

Channels open successively. Statistical competition in overlap regions.

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Evaporation Particles

cm spectra of particles statistically emitted from CN (evaporated) are of Maxwell Boltzmann type

neutrons

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( ) E TB

dNE E e

dE

BE Coulomb barrier

T effective nuclear temperature

Veff CN

protons

EB


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EB

R

Even for fixed E* the particles spectrum is continuous (Maxwell-Boltzmann), except for transitions to discrete spectrum at low EER*

E*

CNER

CN Decay Widths

E*

CNER

Unstable state (finite energy “line” width ) mean lifetime – Heisenberg’s UR: · ˜

˜ / = decay probability

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Total production prob. of CN in reactions:

. .,,

g s elasticexcited inelastic

Total decay width

Specific reaction channel * *C form decP C P C

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Transition probability

Principle of detailed balance:

#states ·P( #states ·P(

22H

2 2

2 2H H

#final states

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CN Decay Widths

E*

CNER

Principle of detailed balance: #states ·P( #states ·P(

2 2 ( )k k spin factors

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2 2

( )

C C

k k spin factors

k k

2

2C

C

k

k

Partial decay width

: all “channels” by which C can be formed or into which it can decay


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2

2C

C

k

k

Partial decay widthCan compute total width and partial widths for decay to particular channel if all formation cross sections are known, all “channels” by which C can be formed in the inverse process.

Decay Width for Neutron Emission

Density of states of CN parent at original excitation

Final state density of daughter

n

*0( )C E

*0E *

0E Q

ndE *0 nE E Q

*0( )C E

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+n

2'' 'C nC C n Cn

nC C C C

nC CkP

Final state density of daughter nucleus, accounting for energy lost in neutron emission

C * *0 0( )C C nE E E Q

*0C nE E Q

: CAll decays

C independent

of decay channel


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C C C C of decay channel

*' 0

' *0

( )( )

( )C nn

n nC Cn C

E E QdN EE

dE E

' ( )nC C Inverse capture cross section

Energy spectrum of emitted neutrons depends on level density in final nucleus, non-monotonic ~En· C’(…- En…)

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E* Dependence of Nuclear Level Density

*'

'

( )C

nC C

E Strongly excitation energy dependent shape of dN/dEn

Weakly dependent on En (neglect this)

Internal system of nucleons at high energies = chaotic (Fermi) gas

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Internal system of nucleons at high energies = chaotic (Fermi) gas

Use statistical mechanics concepts: Entropy ( *) ( *)BS E k n E

* *0 0

*0

*:10

*( )* 0

( ) ( )

( *)( ) ...

*

( )

n B n

n

k TB E Q

S E Q k E k TB n B

S E E Q k n E E Q

dS ES E Q E

dE

E E Q e e


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*( )* 00

*0

( )

( )

S E Q k E k TB n Bn

E k Tn B

E E Q e e

E Q e

*' 0

' *0

( )( )( )

( )E TCn n

n nC C nn C

E QdN EE E e

dE E

Constant-temperature level density (good for small |Q|Set kB =1 [T]= energy

C’ and T correspond to final nucleus+n

FG Nuclear Temperatures and Level Densities

Spectrum of single neutron

2 @

E Tnn

n

n nn

dNE e

dE

dNE T Max E T

dE

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ndE

1.5 0.92 (1 )

E Tn effn

nst

n eff

dNE e

dE

E T T T daughter

Spectrum of cascade of neutrons

1( ) 8a A A MeV Deviations at shell closures


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Fermi gas relations:* 2

**

*

** 20

" "

2

a E

E a T little a

dES a E

E

E e

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Angular Distributions of CN Decay ParticlesBeam axis and collision trajectory define the “reaction plane.”

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Orbital and CN spin angular momentum have to be perpendicular to it. Random emission in reaction plane (in ), symmetry about


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.

1sin

CN

CN

dconst

ddd

symmetry about cm=900.

Classical Model of Nuclear Fusion - University of Rochester1 Cl a s s i c a l R e a c t i o n T h e...

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