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 Ehphp

UhNphZptnTd

s

d

d 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

3

2

A

E

g

AKM

o

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/

NF

E

hp

gEhp

nN

N

nn

no

Main Contributions of This Work

Where

and

hb

hpa

p

j

hbbbaaa

CCj

p

11

0,..,,,..,1

21

2

!2

3

!2

3)1( 1

mC

mC

h

m

mp

m

Main Contributions of This Work

Corrected for Pauli Energy:

Corrected for Pairing:

)!1(

),(

!!2),,(

1

2/

NF

hpAE

hp

gEhp

nN

N

nn

noESMnon

))((

)!1(

)(

!!2),,( ,

1

,

2/ hpnN

N

hp

nn

noESMnon BPE

NF

BPE

hp

gEhp

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)

MeV

-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 Pairing

This 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 pairing

This 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)

Pn

(%)

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)

Pn

(%)

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 25

MeV)

d /

d

(mb

/MeV

)

(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

(mb

/MeV

)

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

(mb

/Me

V)

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

(mb

/MeV

)

present work (PE+evap+NT)

Exp. at E=38.8 MeV

PE +evap.

PE only

NT

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

(mb

/MeV

)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

(mb

/MeV

)

This Work (PE+Evap.+NT)

PE Only

PE+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