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Sem
i-C
lass
ical R
eact
ion
Th
eory
W. Udo Schröder, 2007
1
Cla
ssic
al Po
t. S
catt
eri
ng
Compound-Nucleus Processes
W. Udo Schröder, 2007
2
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
Cla
ssic
al Po
t. S
catt
eri
ng
W. Udo Schröder, 2007
3
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
Cla
ssic
al Po
t. S
catt
eri
ng
Fusion Excitation Functions
W. Udo Schröder, 2007
4
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
S
catt
eri
ng
Fusi
on
-ER
Cla
ssic
al Po
t. S
catt
eri
ng
ER Angular Distributions
W. Udo Schröder, 2007
5
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
Cla
ssic
al Po
t. S
catt
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ng
Independence Hypothesis
W. Udo Schröder, 2007
6
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
Cla
ssic
al Po
t. S
catt
eri
ng
(HI, xn) Excitation Functions
W. Udo Schröder, 2007
7
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.
Cla
ssic
al Po
t. S
catt
eri
ng
Evaporation Particles
W. Udo Schröder, 2007
8
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
Cla
ssic
al Po
t. S
catt
eri
ng
CN Decay Widths
W. Udo Schröder, 2007
9
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
Cla
ssic
al Po
t. S
catt
eri
ng
CN Decay Widths
W. Udo Schröder, 2007
1
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.
Cla
ssic
al Po
t. S
catt
eri
ng
Decay Width for Neutron Emission
W. Udo Schröder, 2007
1
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…)
Cla
ssic
al Po
t. S
catt
eri
ng
E* Dependence of Nuclear Level Density
W. Udo Schröder, 2007
1
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
Cla
ssic
al Po
t. S
catt
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ng
FG Nuclear Temperatures and Level Densities
W. Udo Schröder, 2007
1
3
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
Cla
ssic
al Po
t. S
catt
eri
ng
Angular Distributions of CN Decay Particles
W. Udo Schröder, 2007
1
4
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.