Preequilibrium Reactions
Dr. Ahmed A.Selman
The exciton model was proposed by Griffin in 1966 in order to explain the nuclear emission from intermediate states, PE, where “a statistical model that analyze the formation and decay of the average compound-nuclear state was presented. In such state, a weak two-body residual interaction will cause transition among the eigenstates of the independent-particle Hamiltonian. These transitions occur in the region dE near the excitation energy E of the compound nucleus.”
PE explains nuclear emission from intermediate states before attainment of statistical equilibrium.
Blann, and many others , developed this model and suggested similar approaches.
In the 1970’s Blann and Cline made the present development, as follows (proton emission)
The two-component system is based on proton-neutron distinguishability.
),,,,(
),,,,1()(
12)(
232 Ehphp
UhphpsW
p
The state density is needed in the PE calculations. There are many types of state density formulae, for ESM and non-ESM systems.
The exciton model can be represented as follows (two-component):
5400
6500
n=9, N=5
1022
3200 2111
3211 2122
4300 1033
4311 3222
2133
1044
5411 4322
3233
2144
1055
n=1, N=1
n=3, N=2
n=5, N=3
n=7, N=4
n=11, N=6
etc etc etc
(1)
(2)
(3)(4) 2100
1000
1011
The emission spectrum is
T(n,t) is the equilibration time, found from solving the master equation given by (from the Figure above):
),,,,(
),,,,(),()(
12
2
32 EhphpUhNphZp
tnTds
dd aa
nn
),,(),,(),,(),,(
),,(),,(),,(),,(
),,(),,(),,(),,(
),1,()1,,(
),1,()1,,(
),1,1()1,1,()1,1,(
),,1(),1,(),1,(
),,1(),1,(),1,(
),1,1(),1,(),1,(
),,(
00
00
00
0
0
00
00
thNPhNEWhNEhNE
hNEhNEhNEhNE
hNEhNEhNEhNE
thNPhNE
thNPhNE
thNPhNEhNE
thNPhNEhNE
thNPhNEhNE
thNPhNEhNEdt
thNdP
vv
vvvv
vvvv
v
v
v
vvv
vvv
v
Transition rates are found from Fermi Golden Rule
The matrix element is found from
2,, ||2yxyx M
)MeV(9.203
|| 2
3
32
AE
gA
KMo
Main Contributions of This Work
1. The solution of the state density for one-component non-ESM:
With the solution (uncorrected)
)(
1
)(
1
)()(
0
)(2
)(2
0
)(1
)(1
0
)()(
0
)(2
)(2
0
)(1
)(1
01
)()...()(
)()...()(!!
1),,(
hj
h
j
pp
hhh
hh
hh
hhh
h
ppp
pp
pp
ppp
p
uuEugduugduugdu
ugduugduugduhp
Ehp
)!1(!!2),,(
1
2/
NFE
hpg
Ehp nN
N
nn
no
Main Contributions of This Work
Where
and
hb
hpa
p
j
hbbbaaa
CCj
p
11
0,..,,,..,1
21
2
!23
!23)1( 1
mC
mC
hm
mp
m
Main Contributions of This Work
Corrected for Pauli Energy:
Corrected for Pairing:
)!1(
),(!!2
),,(1
2/
NFhpAE
hpg
Ehp nN
N
nn
noESMnon
))((
)!1()(
!!2),,( ,
1,
2/ hpnN
Nhp
nn
noESMnon BPE
NFBPE
hpg
Ehp
State Density Results ESM
A comparison of the state
density results of this work (dotted
curves) with those of
Kalbach for 54Fe+p
reaction at 33.5 MeV.
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1 2 3 4 5 6 7
p or p
log
(n
,E) M
eV-1
p=7p=7 this workp=6p=6 this workp=5p=5 this workp=4p=4 this workp=3p=3 this workp=2
p + 54Fe reaction at
E=33.5 MeV
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
0 5 10 15 20 25 30 35 40
Energy, MeV
Log
(p
,h,E
) (M
eV-1
)
Herman et al.This Work No PairingThis Work With pairing
state density for 1p-1h of 56Fe for 50 terms of summationDo=1.49 MeV, D(pi)=1.41MeVD(Nu)=1.22 MeVS=0.53 MeVCe=2.22 MeV
1p-1h of 56Fe for 50 terms of summationo=1.49 MeV, =1.41MeV, )=1.22 MeV,
S=0.53 MeV, Ce=2.22 MeV
State Density Results non-ESM
1p-1h of
56Fe
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
0 5 10 15 20 25 30 35 40
Energy, MeV
Log
(p
,h,E
) (M
eV-1
)
Herman et al.This Work No pairingThis Work With pairing
s.d. of 54Mn for 2p-1hS=--0.45 MeVDo=2.6 MeVCe=1.5 MeV
State Density Results non-ESM
2p-1h of 54Mn
Master Equation Results
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200
time (arbitrary units)
P n (%
)
n=1n=3n=5n=1n=3n=5
Euler centered scheme (Bold lines) vs. Runge-Kutta (Thin lines)
E=20 MeV
E=20 MEV, g=14 MeV-1, energy-dependent matrix element, p+54Fe
p+56Fe at 20 MeV
Master Equation Results
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200
time (arbitrary units)
P n (%
)
n=1n=3n=5n=1n=3n=5
E=80 MeV
E=80 MEV, g=14 MeV-1, energy-dependent matrix element, p+54Fe
Euler centered scheme (Bold lines) vs. Runge-Kutta (Thin lines)
p+56Fe at 80 MeV
1.00E-01
1.00E+00
1.00E+01
1.00E+02
0 5 10 15 20 25MeV)
d/d
(m
b/M
eV)
(p,n) Kalbach(p,n) this work(p,p) Kalbach (p,p) this work
54Fe+p at E=33.5 MeV
Preequilibrium Spectrum (no evaporation added yet). This is for
54Fe+p at E=33.5 MeV
Results of PE Spectra1. Pure PE
Results of PE Spectra2. PE +Evaporation
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 5 10 15 20 25 (MeV)
d/d
(m
b/M
eV)
GRIMES et al.
present work
103Rh(p,n) reaction E=18 MeV
Results of PE Spectra2. PE +Evaporation
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 5 10 15 20 25 30 35 40 45, MeV
d/d
(m
b/M
eV)
Dobes&Betak Bi209 (p,n)This Work Bi209 (p,n) Dobes&Betak Bi209 (p,p) This Work Bi209 (p,p)
209Bi+p at 52 MeV
Results of PE Spectra2. PE +Evaporation
Results of PE Spectra4. PE +Evaporation +NT
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 5 10 15 20 25 30 35 40 (MeV)
d/d
(m
b/M
eV)
present work (PE+evap+NT)Exp. at E=38.8 MeVPE +evap.PE onlyNT
Exp/ from (F.E.BERTRAND,R.W.PEELLE),
(J,PR/C,8,1045,1973)
54Fe(p,n) at E=38.8 MeV
Results of PE Spectra4. PE +Evaporation +NT
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 5 10 15 20 25 30(MeV)
d/d
(m
b/M
eV)
present work (PE+evap+NT)Exp. at E=28.8 MeVPE +evap. onlyNT only
Exp/ from F.E.BERTRAND, and R.W.PEELLE,
PR/C,8,(1971045,1973)54Fe(p,p) at E=28.8 MeV
54Fe(p,n) at E=28.8 MeV
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 2 4 6 8 10 12 14 16 18 (MeV)
d/ d
(m
b/M
eV)
This Work (PE+Evap.+NT)PE OnlyPE+Evap.NT OnlyLYCHAGIN et al.KOZYR and PROKOPETS BROND-2.2
56Fe(n,n/) reactionat E=20 MeV
Results of PE Spectra4. PE +Evaporation +NT
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 2 4 6 8 10 12 14 16 18 (MeV)
d/ d
(m
b/M
eV)
This Work (PE+Evap.+NT)PE OnlyPE+Evap.NT Only BROND-2.2
103Rh(n,n/) reactionat E=18.8 MeV
Results of PE Spectra4. PE +Evaporation +NT
Cross-Section Results
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 5 10 15 20 25 30 35 40
Energy, E (MeV)
(m
b)
this workS.Sudar et al.A.Hermanne et al.
103Rh(p,n) Reaction Cross-Section
S.Sudar, F.Cserpak, S.M.Qaim, REFERENCE (J,Journ.: Applied Radiation and Isotopes, 56,821,2002)
A.HERMANNE, M.SONCK, A.FENYVESI, L.DARABAN) REFERENCE (J,Nucl. Instrum. Methods in Physics Res., Sect.B,170,281,2000)
Cross-Section Results
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
0 5 10 15 20 25 30 35 40
Energy, E (MeV)
(m
b)
This work103Rh(p,p) Reaction
Cross-Section
1.00E+00
1.00E+01
1.00E+02
1.00E+03
0 5 10 15 20 25 30 35 40 45 50
Energy, E (MeV)
(m
b)
This WorkLevkovskij [137]Gadioli et al. [138]Jenkins and Wain [139]Antropov et al. [140]Tanaka and Furukawa [141]
56Fe(p,n) ReactionCross-Cection
Cross-Section Results
Thank You
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