1C

lass

ical

Rea

ctio

n T

heor

yC

lass

ical

Rea

ctio

n T

heor

yS

emi-

Cla

ssic

al R

eact

ion

The

ory

W. Udo Schröder, 2007

Compound-Nucleus Processes

Particle Evapor-ationEvaporation

2

Evaporation Residues ER

-ray

Cla

ssic

al P

ot. S

catt

erin

g

a AEcm,

Formation

C*E*=Ecm+Q I=

Decay

-ray emission

Cla

ssic

al P

ot. S

catt

erin

g FormationEquilibration

Compound Nucleus

Fission fragments

Fission

Cla

ssic

al P

ot. S

catt

erin

g

Statistical Independence Hypothesis:All degrees of freedom equilibrated, no memory of formation,

Fission

W. Udo Schröder, 2007

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

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

(Momentum Conservation)

elastic

3

14N

(Momentum Conservation)

Cla

ssic

al P

ot. S

catt

erin

g

26-nAl12C

Cla

ssic

al P

ot. S

catt

erin

g

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

Cla

ssic

al P

ot. S

catt

erin

g

evaporation (ER) or fission

W. Udo Schröder, 2007

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

4

close to barrier.

Maximum Lfusdue to yrast limitation (nuclear

Cla

ssic

al P

ot. S

catt

erin

g

148Sm: 2=0154Sm: 2=0.3

limitation (nuclear centrifugal stability)

ER =

R

Cla

ssic

al P

ot. S

catt

erin

g 2

Deformation changes the effective

ER = lowest window

Cla

ssic

al P

ot. S

catt

erin

g

Deformation changes the effective barrier height larger fus d

d

nucl

eon

Tra

nsf

er

Ela

stic

/quas

i-el

astic

Sca

tter

ing

W. Udo Schröder, 20070 ER F R

Fusi

on

Fiss

ion

multi-

nucl

eon

Tra

nsf

er

Ela

stic

/quas

iel

astic

Sca

tter

ing

Fusi

on

-ER

ER Angular Distributions

p

5

ERp p

Cla

ssic

al P

ot. S

catt

erin

g

Random emission from moving CN does not change average

Cla

ssic

al P

ot. S

catt

erin

g

not change average velocity, preserves < > = 00,

Sideways recoil

Cla

ssic

al P

ot. S

catt

erin

g

Sideways recoil components important for angular distributions of ERs.

W. Udo Schröder, 2007

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,…)

6

asymptotic condition C*(E, I,…)

E E Separation of cross

Cla

ssic

al P

ot. S

catt

erin

g

*

*

aA bB aA C

dD bB dD C

E E

E E *C bB E

Separation of cross sections:Independent probabilities of formation

Cla

ssic

al P

ot. S

catt

erin

g *

*

dD bB dD C

gG bB gG C

E E

E E

*C bB E probabilities of formation and decay multiply for overall reaction

Cla

ssic

al P

ot. S

catt

erin

g

W. Udo Schröder, 2007

(HI, xn) Excitation Functions

a+Ab+B

C’* + nC’’* + 2nC’’’* + 3n

C* (19F, 7n)

(19F, 8n)

7

C’’’* + 3n

(HI, xn) cross sections

(19F, 9n)

Cla

ssic

al P

ot. S

catt

erin

g

(HI, xn) cross sections

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

Elab

W. Udo Schröder, 2007

Channels open successively. Statistical competition in overlap regions.

Evaporation Particles

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

neutrons

8

Maxwell Boltzmann type

( ) E TB

dNE E e

dE

neutrons

protons

Cla

ssic

al P

ot. S

catt

erin

g

( )BE E edE

BE Coulomb barrier

T effective nuclear temperature

EB

Cla

ssic

al P

ot. S

catt

erin

g

T effective nuclear temperature

Veff

E*

CNER

Cla

ssic

al P

ot. S

catt

erin

g

EB

R

Even for fixed E* the particles spectrum is continuous (Maxwell-Boltzmann), except

E*

W. Udo Schröder, 2007

Boltzmann), except for transitions to discrete spectrum at low EER*

CN Decay Widths

E*

CNER

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

˜ / = decay probability

9

˜ / = decay probability

Total production prob. of CN in reactions:

Cla

ssic

al P

ot. S

catt

erin

g

. .,,

g s elasticexcited inelastic

Total decay width

Cla

ssic

al P

ot. S

catt

erin

g

Specific reaction channel * *C form decP C P C

Transition probability22

H #final states

Cla

ssic

al P

ot. S

catt

erin

g

Transition probability

Principle of detailed balance:

H

2 2

2 2H H

#final states

W. Udo Schröder, 2007

#states ·P( #states ·P(

CN Decay Widths

E*

CNER

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

10

2 2

2 2

( )

C C

k k spin factors

k k

Cla

ssic

al P

ot. S

catt

erin

g

C C

2

2C

C

k

k

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

Cla

ssic

al P

ot. S

catt

erin

g 2Ck

2k

Partial decay width

into which it can decay

Can compute total width and partial widths

Cla

ssic

al P

ot. S

catt

erin

g

2

2C

C

k

k

Can 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.

W. Udo Schröder, 2007

Decay Width for Neutron Emission

Density of states of CN parent at original excitationn

*0( )C E

*0E *

0E Q

dE *E E Q

*0( )C E

11 C’+n

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

ndE

C

*0 nE E Q

*0C nE E Q

Cla

ssic

al P

ot. S

catt

erin

g

+nC * *

0 0( )C C nE E E Q

: CAll decays

Cla

ssic

al P

ot. S

catt

erin

g

2'' 'C nC C n Cn

nC C C C

nC CkP C independent

of decay channel

Cla

ssic

al P

ot. S

catt

erin

g

*' 0

' *0

( )( )

( )C nn

n nC Cn C

E E QdN EE

dE E

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

W. Udo Schröder, 2007

0( )n C E

' ( )nC C Inverse capture cross sectionmonotonic ~En· C’(…- En…)

E* Dependence of Nuclear Level Density

*'

'

( )C

nC C

E Strongly excitation energy dependent shape of dN/dEn

Weakly dependent on En (neglect this)

12

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

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

* *

Cla

ssic

al P

ot. S

catt

erin

g

* *0 0

*0

( ) ( )

( *)( ) ...

*

n B n

n

S E E Q k n E E Q

dS ES E Q E

dE

Cla

ssic

al P

ot. S

catt

erin

g

*:10

*( )* 00

*

( )

k TB E Q

S E Q k E k TB n Bn

E k T

dE

E E Q e e Constant-temperature level density (good for small |Q|

Cla

ssic

al P

ot. S

catt

erin

g

*0( ) E k Tn BE Q e

*' 0( )( )

( ) E TCn nE QdN EE E e

density (good for small |Q|Set kB =1 [T]= energy

and T correspond

W. Udo Schröder, 2007

' 0' *

0

( )( )( )

( )E TCn n

n nC C nn C

E QdN EE E e

dE EC’ and T correspond

to final nucleus+n

FG Nuclear Temperatures and Level Densities

Spectrum of single neutron

2 @

E Tnn

n

dNE e

dE

dNE T Max E T

13

2 @n nn

dNE T Max E T

dE

E Tn effdNE e Spectrum of cascade

Cla

ssic

al P

ot. S

catt

erin

g

1.5 0.92 (1 )

E Tn effn

nst

n eff

dNE e

dE

E T T T daughter

Spectrum of cascade of neutrons

Cla

ssic

al P

ot. S

catt

erin

g

Fermi gas relations:

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

Cla

ssic

al P

ot. S

catt

erin

g

Fermi gas relations:* 2

**

*

" "

2

E a T little a

dES a E

E

W. Udo Schröder, 2007

*

** 20

2

a E

S a EE

E e

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

14C

lass

ical

Pot

. Sca

tter

ing

Orbital and CN spin angular momentum have to be perpendicular to it.

Cla

ssic

al P

ot. S

catt

erin

g

perpendicular to it. Random emission in reaction plane (in ), symmetry about

=900.

Cla

ssic

al P

ot. S

catt

erin

g

.CNdconst

d

cm=900.

W. Udo Schröder, 2007

1sin

CNdd

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