Sem

i-C

lass

ical R

eact

ion

Th

eory

W. Udo Schröder, 2007

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Compound-Nucleus Processes

W. Udo Schröder, 2007

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a AEcm,

Formation

C*E*=Ecm+Q I=

EquilibrationCompound

Nucleus

Decay

Particle Evapor-ation Evaporation Residues ER

g-ray emission

Fission fragments

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

Fission

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W. Udo Schröder, 2007

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14N

26-nAl12C

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

(Momentum Conservation)

elastic

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

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Fusion Excitation Functions

W. Udo Schröder, 2007

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

148Sm: b2=0154Sm: b2=0.3

Deformation changes the effective barrier height larger sfus

sfus≈ sR only for Ecm below and close to barrier.

Maximum Lfus due to yrast limitation (nuclear centrifugal stability)

ER = lowest window

sR

sR

dd

0 ER F R

Fusi

on

Fiss

ion

mu

lti-

nu

cleon

Tr

an

sfer

Ela

stic

/qu

asi

-ela

stic

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Fusi

on

-ER

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ER Angular Distributions

W. Udo Schröder, 2007

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

Sideways recoil components important for angular distributions of ERs.

p

ERp p

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Independence Hypothesis

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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,…)

*

*

*

aA bB aA C

dD bB dD C

gG bB gG 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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(HI, xn) Excitation Functions

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a+Ab+B

C’* + nC’’* + 2nC’’’* + 3n

C*

(HI, xn) cross sections

Elab

(19F, 7n)

(19F, 8n)

(19F, 9n)

Channels open successively. Statistical competition in overlap regions.

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Evaporation Particles

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

( ) E TB

dNE E e

dE

BE Coulomb barrier

T effective nuclear temperature

EB

Veff

R

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

E*

CNER

neutrons

protons

EB

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CN Decay Widths

W. Udo Schröder, 2007

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E*

CNER

Unstable state (finite energy “line” width G) mean lifetime t – Heisenberg’s UR: G· t ≈

G ≈ /t = decay probability

Total production prob. of CN in reactions:

. .,

,g s elastic

excited inelastic

Total decay width

Specific reaction channel a b * *C form decP C P C

Transition probability

Principle of detailed balance:

#statesa·P(a )=b #statesb·P(b )a

22H

2 2

2 2H H

#final statesb

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CN Decay Widths

W. Udo Schröder, 2007

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0

E*

CNER

Principle of detailed balance: #statesa·P(a )=b #statesb·P(b )a

2 2

2 2

( )

C C

k k spin factors

k k

2

2C

C

k

k

2

2C

C

k

k

Partial decay width

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

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

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Decay Width for Neutron Emission

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1

C’+n

2'' 'C nC C n Cn

nC C C C

nC CkP

Density of states of CN parent at original excitation

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

n*0( )C E

*0E *

0E Q

ndE

C

*0 nE E Q

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

*0( )C E

*0C nE E Q

: CAll decays

C independent

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·rC’(…- En…)

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E* Dependence of Nuclear Level Density

W. Udo Schröder, 2007

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2

*'

'

( )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

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

* *0 0

*0

*:10

*( )* 00

*0

( ) ( )

( * )( ) ...

*

( )

( )

n B n

n

k TB E Q

S E Q k E k TB n Bn

E k Tn B

S E E Q k n E E Q

dS ES E Q E

dE

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

rC’ and T correspond to final nucleus+n

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FG Nuclear Temperatures and Level Densities

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Spectrum of single neutron

2 @

E Tnn

n

n nn

dNE e

dE

dNE T Max E T

dE

1.5 0.92 (1 )

E Tn effn

n

stn eff

dNE e

dE

E T T T daughter

Spectrum of cascade of neutrons

Fermi gas relations:

* 2

**

*

** 20

" "

2

a E

E a T little a

dES a E

E

E e

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

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Angular Distributions of CN Decay Particles

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Beam axis and collision trajectory define the “reaction plane.”

.

1sin

CN

CN

dconst

ddd

Orbital and CN spin angular momentum have to be perpendicular to it. Random emission in reaction plane (in q), symmetry about qcm=900.