S W. Udo Schröder, 2007 - University of Rochester...W. Udo Schröder, 2007 except conservation laws...
Transcript of S W. Udo Schröder, 2007 - University of Rochester...W. Udo Schröder, 2007 except conservation laws...
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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
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Compound-Nucleus Processes
Particle Evapor-ationEvaporation
2
Evaporation Residues ER
-ray
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ssic
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ot. S
catt
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g
a AEcm,
Formation
C*E*=Ecm+Q I=
Decay
-ray emission
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ot. S
catt
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g FormationEquilibration
Compound Nucleus
Fission fragments
Fission
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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,…
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Fusion reaction 14N+12C leading to compound nucleus 26-nAl, emitted at < ˜ 00
(Momentum Conservation)
elastic
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14N
(Momentum Conservation)
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26-nAl12C
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CN decays in flight by particle evaporation (ER) or fission
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evaporation (ER) or fission
W. Udo Schröder, 2007
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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
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al P
ot. S
catt
erin
g 2
Deformation changes the effective
ER = lowest window
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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
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ER Angular Distributions
p
5
ERp p
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catt
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Random emission from moving CN does not change average
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ssic
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ot. S
catt
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not change average velocity, preserves < > = 00,
Sideways recoil
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catt
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Sideways recoil components important for angular distributions of ERs.
W. Udo Schröder, 2007
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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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asymptotic condition C*(E, I,…)
E E Separation of cross
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*
*
aA bB aA C
dD bB dD C
E E
E E *C bB E
Separation of cross sections:Independent probabilities of formation
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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
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W. Udo Schröder, 2007
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(HI, xn) Excitation Functions
a+Ab+B
C’* + nC’’* + 2nC’’’* + 3n
C* (19F, 7n)
(19F, 8n)
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C’’’* + 3n
(HI, xn) cross sections
(19F, 9n)
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(HI, xn) cross sections
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gC
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Pot
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Elab
W. Udo Schröder, 2007
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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Maxwell Boltzmann type
( ) E TB
dNE E e
dE
neutrons
protons
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( )BE E edE
BE Coulomb barrier
T effective nuclear temperature
EB
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T effective nuclear temperature
Veff
E*
CNER
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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*
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CN Decay Widths
E*
CNER
Unstable state (finite energy “line” width ) mean lifetime – Heisenberg’s UR: · ˜
˜ / = decay probability
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˜ / = decay probability
Total production prob. of CN in reactions:
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. .,,
g s elasticexcited inelastic
Total decay width
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Specific reaction channel * *C form decP C P C
Transition probability22
H #final states
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Transition probability
Principle of detailed balance:
H
2 2
2 2H H
#final states
W. Udo Schröder, 2007
#states ·P( #states ·P(
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CN Decay Widths
E*
CNER
Principle of detailed balance: #states ·P( #states ·P(
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2 2
2 2
( )
C C
k k spin factors
k k
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catt
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g
C C
2
2C
C
k
k
: all “channels” by which C can be formed or into which it can decay
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ot. S
catt
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g 2Ck
2k
Partial decay width
into which it can decay
Can compute total width and partial widths
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catt
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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
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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
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+nC * *
0 0( )C C nE E E Q
: CAll decays
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2'' 'C nC C n Cn
nC C C C
nC CkP C independent
of decay channel
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*' 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…)
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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)
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Internal system of nucleons at high energies = chaotic (Fermi) gas
Use statistical mechanics concepts: Entropy ( *) ( *)BS E k n E
* *
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* *0 0
*0
( ) ( )
( *)( ) ...
*
n B n
n
S E E Q k n E E Q
dS ES E Q E
dE
Cla
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catt
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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|
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ssic
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ot. S
catt
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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
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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
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catt
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1.5 0.92 (1 )
E Tn effn
nst
n eff
dNE e
dE
E T T T daughter
Spectrum of cascade of neutrons
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Fermi gas relations:
1( ) 8a A A MeV Deviations at shell closures
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catt
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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
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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.
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perpendicular to it. Random emission in reaction plane (in ), symmetry about
=900.
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catt
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.CNdconst
d
cm=900.
W. Udo Schröder, 2007
1sin
CNdd
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