5. Compound nucleus reactions
Prof. Dr. A.J. (Arjan) Koning1,2
1International
2Division
Atomic Energy Agency, Vienna
of Applied Nuclear Physics, Department of Physics and Astronomy, Uppsala
University, Uppsala, Sweden
Email: [email protected]
EXTEND
European School on Experiment, Theory and Evaluation of Nuclear Data,
Uppsala University, Sweden, August 29 - September 2, 2016
THE COMPOUND NUCLEUS MODEL
Shape elastic
Elastic
Reaction
OPTICAL
MODEL
PRE-EQUILIBRIUM
NC
COMPOUND
NUCLEUS
Fission
Tlj
Direct components
Inelastic
(n,n’), (n,),
(n,), etc…
THE COMPOUND NUCLEUS MODEL
(basic formalism)
Compound nucleus hypothesis
- Continuum of excited levels
- Independence between incoming channel a and outgoing channel b
ab =
(CN)
a
=
p
2
ka
(CN)
a Pb
Pb=
Ta
 Hauser- Feshbach formula
ab =
p
2
ka
Ta Tb
Sc Tc
Tb
Sc Tc
THE COMPOUND NUCLEUS MODEL
(qualitative feature)
Compound angular distribution & direct angular distributions
45°
90°
135°
THE COMPOUND NUCLEUS MODEL
(complete channel definition)
Channel Definition
a + A  (CN )*  b+B
Incident channel a =
   

(la, ja=la+sa, JA,pA,
EA, Ea)
Conservation equations
• Total energy : Ea + EA = ECN = Eb + EB





• Total momentum : pa + pA = pCN = pb + pB







• Total angular momentum : la + sa + JA = JCN = lb + sb + JB
• Total parity : pA (-1)la = pCN = pB (-1)lb
THE COMPOUND NUCLEUS MODEL
(loops over all quantum numbers)
In realistic calculations, all possible quantum number combinations
have to be considered
max
Given by OMP
I +s +l
p
sab = 2
ka
A
a
S
a
S
p=
(2J+1)
(2IA+1) (2sa+1)
J=| IA – sa |
Width fluctuation correction factor
J + IA
j a + sa
+ IB
j b + sb
to account
for Jdeviations
from independence hypothesis
j a = | J – IA | l a = | j a – sa | j b = | J – IB | l b = | j b – sb |
S
S
S
S
Parity selection rules
T Jp
T Jp
T Jp
Ta, l , j Tb, l , j W
b b
a a
a, la , ja , b, lb , jb
dp (a)dp (b)
T Jp
Tc, l , j
c c
c
S
THE COMPOUND NUCLEUS MODEL
(the GOE triple integral)
THE COMPOUND NUCLEUS MODEL
(flux redistribution illustration)
THE COMPOUND NUCLEUS MODEL
(multiple emission)
Target
Compound Nucleus
fission
E
Sa
g
Sn n
(2)
Sp
Sa
Sn
+ Loop over CN spins and parities
n
p
Sn
d
Jp
Zc
g
n’
g
Z
Sp n’
Sa
Sp
Sa
Sn
Sp
Sa
n
Sp
Sa
Sn
Sn a
Zc-1
N
Nc-2
Nc-1
Nc
REACTION MODELS & REACTION CHANNELS
n + 238U
Cross section (barn)
Optical model
+
Statistical model
+
Pre-equilibrium model
sR = sd + s PE + sCN
= snn’ + snf + sng + ...
Neutron energy (MeV)
THE COMPOUND NUCLEUS MODEL
(compact expression)
NC =  ab
where b = g , n, p, d, t, …, fission
b
ab =

2

k a J,
a,b
2J+1
2s+12I+1
< >
< >
J
Tb b
J
Tlj a
J
Wab
Td d
d
with J = la + sa + IA = ja + IA and  = -1
and
<
>
Tb(b)
la
A
= transmission coefficient for outgoing channel b
associated with the outgoing particle b
THE COMPOUND NUCLEUS MODEL
(various decay channels)
Possible decays
• Emission to a discrete level with energy Ed
< >
Tb(b)
J
p
= Tlj(b)
given by the O.M.P.
• Emission in the level continuum
< > 
Tb(b)
=
E +DE
J
ljp
T (b) r(E,J,p) dE
E
r(E,J,p) density of residual nucleus’ levels (J,p) with excitation energy E
• Emission of photons, fission
Specific treatment
Exercise
• Neutrons + Cu-65
• Two incident energies: 1.6 and 5.5 MeV
• Plot elastic scattering angular distributions for
both of them: the direct, compound and total
component
• Exp data: 1.6 MeV: Cu-0, 5.5 MeV: Cu-65
• Use TALYS sample case 3 for inspiration
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