Journal of Applied Mathematics and Physics, 2013, 1, 47-54
http://dx.doi.org/10.4236/jamp.2013.13008 Published Online August 2013 (http://www.scirp.org/journal/jamp)
Investigation of the Appropriate Partial Level Density
Formula for Pre-Equilibrium Nuclear Exciton Model
Shafik Shakir Shafik1*, Ali Dawoud Salloum2
1Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq
2Department of Physics, College of Science for Women, University of Baghdad, Baghdad, Iraq
Received June 10, 2013; revised July 15, 2013; accepted August 20, 2013
Copyright © 2013 Shafik Shakir Shafik, Ali Dawoud Salloum. This is an open access article distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
ABSTRACT
Ericson formula represents the first formula, which was suggested to describe the partial level density (PLD) formula in
pre-equilibrium region of the nuclear reactions. Then a number of corrections were added to this formula in order to
make it more suitable to physical meaning. In this paper, there are two aims to be done: the first aim is to study the cor-
respondence between one and two-components formulae in Ericson, Pauli, and pairing corrections; the second aim is to
compare and study the results of Comprehensive formula, which contents with all corrections, with Ericson, Pauli, and
pairing formulae. The Comprehensive formula was suggested to simulate the reality. To achieve these aims the 56Fe and
90Zr nuclei were chosen and the results showed that the difference between one and two-components formulae was too
small which can be neglected. Furthermore, the results strongly recommended that for cross section calculations of the
nuclear reaction, one must use Comprehensive formula rather than Pauli formula.
Keywords: Partial Level Density; Pre-Equilibrium Model; Nuclear Reaction; Exciton Model; Equidistant Spacing
Model; 56Fe; 90Zr
1. Introduction
Nuclear reactions are divided according to the energy of
the incident particle to low, medium, and high-energy
reactions. Pre-Equilibrium (PE) region is a stage in nu-
clear reaction, which represents the case when the ex-
citation energy is not distributed equally between the nu-
cleons of the excited nucleus. Many models describe this
region such as: the Intra-Nuclear-Cascade (INC) model
[1], the Harp-Miller-Berne (HMB) model [2,3], the
Hybrid model [4], the Geometry Dependent Hybrid
model [5] and the Exciton model [6]. J. J. Griffin sug-
gested the Exciton model at 1966 [6]. It was preferred
model by many researchers [7,8] because it is simplest in
the description and treatment of the PE nuclear emission.
The idea of this model supposes that when the bom-
barding particle hit the target nucleus, it begins to share
its energy with the first particle collides with it, then by
successive nucleon-nucleon interactions in a series of stages
and before attend to the complete interactions (equi-
librium), the nuclear emission occurs. Each interaction
produces a particle-hole (p-h) pair, and the sum of such
particle and hole called an Exciton. The first few states
are 2p 1h, 3p 2h and so on. The numbers of excitons are
n (n = p + h) and the stages are labeled by s so that n =
2s + 1, therefore, again, through exciton production, the
nuclear emission may occur [9].
2. The Partial Level Density (PLD)
The PLD represents important quantity in nuclear phys-
ics. It is used in calculation of cross section, double dif-
ferential cross section and transition rates [9,10]. PLD
can be measured experimentally up to 15 MeV; above
this value of energy, the levels converge. Therefore, the
spacing between them overlapped, and it is difficult to
calculate them. Hence, level density may be calculated
by theoretical methods [8-10]. In this paper, the theo-
retical description of PLD by the exciton model was de-
pended and developed through many stages. The Fermi
gas model (FGM) was used as description model in the
nuclear reaction exciton model. FGM conceder equal
spaces between the levels and this called Equidistant
Spacing Model (ESM) [9,10].
*Corresponding author.
C
opyright © 2013 SciRes. JAMP

S. S. SHAFIK, A. D. SALLOUM
48
Ericson’s formula is the first formula which describes
the PLD and it is represents the crud formula
 
1
1,!! 1!
nn
gE
nE ph n 

 (1)
where, sign 1 means one component, i.e. the particles are
not separated to protons and neutrons,
g
is the single-
particle state density, is the exciton number, is
the particles’ number, is the holes’ number and
is the excitation energy.
n
hpE
The two-component formula is
 

1
2,!!!!1!
n
nn
gg E
nE phphn


 


 (2)
The two-component formula, with sign 2, distinguish
between proton particles 

p

and neutron particles


p

and proton holes

h

and neutron holes


h

.
g

and
g

are the single particle state densities of
proton and neutron, respectively, while n

and n

are
the proton and neutron excitons numbers, nnn




.
However, ߨ and v stand for protons and neutrons respect-
tively.
In order to make the results of the PLD more reality,
the Ericson’s PLD formula was developed by adding
physical corrections to it. Those corrections are
2.1. Pauli’s Correction
This correction comes from Pauli’s principle that forbids
any two particles from having the same quantum state.
So, the excitation energy deceases by the factor called
Pauli Blocking Factor

,
p
h
A

which given by [11]


,
13
4
ph
pp hh
Ag
 
 (3)
Then the one-component PLD formula becomes






1
,
1,
,!! 1!
n
nph
ph
gEA
nEE A
ph n








(4)


,ph
EA is the Heaviside step function defined
as




,
,
,
0if 0
1if 0
ph
ph ph
EA
EA EA



 



In the case of two-component, the Pauli Blocking fac-
tor is

,,,
13 1
44
phph
A
pphhpphh
gg

 

  






Then the PLD for two-component can be given as





2
1
,,
,
,
,,
,
Θ
!!!!1!
n
nn phph
phph
nE
gg EAEA
phphn




 





(6)
2.2. Pairing Correction
This correction comes from pairing property between
couples of particles. Therefore, this pairing required en-
ergy, which is taken from the excitation energy and
hence the excitation energy will decrease [12-14]. Thus,
the PLD formula in one component with pairing correc-
tion can be given by (all details of this correction given
by these references) [9,11]
 


 

1
,
1 ,
,Θ
!! 1!
n
nph
p
h
gEP B
nEE PB
ph n





(7)
where


P

is the pairing correction, and it is given by



22
4
P


 g
(8)


and ∆ are the energy gaps of the ground and ex-
cited states, respectively.
,
p
h is the modified Pauli Blocking factor which is
given by
B
2
,,
2
1
ph phg
BA n

 

(9)
,ph is unmodified Pauli Blocking factor given by
Equation (3).
A

is given by
1.6 0.68
0.996 1.76if
0i
phase
ce
f
p
hase
nE EE
nC
EE


 









(10)
where c is the most probable exciton number that
leads to emission
n
0.792
c
g
n


 (11)
e
C is the condensation energy given by
4
e
Cg

 (12)
phase is the pairing energy of phase transition defined
as following
E
3

(5)
2.17
0.716 2.44if0.446
0o
e
phase cc
nn
C
Enn










therwise
(13)

S. S. SHAFIK, A. D. SALLOUM 49
The minimum value of energy for applying these
equations is the effective value of energy, which means
threshold value of and it is given by equations
phase
E
2
2
if 0.446
1if
3.23 1.57
4460.627
eccc
th
ecc
nnn
Cnnn
Unn
Cnn




 















0.
(14)
The value of

 can be obtained from curve fitting
for almost known nuclei, by a relation known as Gilbert-
Cameron formula

 


 (15)
where


and 


are the energy gaps for ground
states of proton’s and neutron’s particles, respectively.

3
3
11.654 9.58
1.374 5.16
0
10
Z
N







 
(16)
N and Z are the neutrons and protons numbers respect-
tively.
The value of

 is depending on the nuclear tem-
perature, where it increases with temperature and van-
ishes at a critical temperature which gives for ESM by
2
3.5

 [11]. Above this temperature, the pairing correc-
tion disappeared and the system reverted to uncorrelated
condition. The two-component formula is given by
 





1
2
2
2
,!!!! 1!
Θ
n
phph
phph
ggE PB
nE phphn
EP B



 





(17)

2
P is the two-components pairing energy given
by
  
21 1
PP P



 (18)


22
14
P
 
g



 (19a)


22
14
P
g
 


 

(19b)
p
hph
B


is the modified Pauli Blocking factor in the
case of two component
2
,,,
2
1
phphphphg
BA n
  




(20)
,,,
p
hph
A


is the unmodified Pauli Blocking factor
given by Equation (5).
2.3. Active and Passive Holes Correction
The exciton model assumed that the particle’s number
must be equal to the holes number, but in fact, the parti-
cles number is always bigger than the holes number by
one. This is because the projectile (incident particle) is
added to the particles number. In order to satisfy the
equality between particles number and holes number,
passive holes have been supported. They represent those
holes are not affected by the nuclear potential; therefore,
accumulated near Fermi’s level. The correction that
comes from passive holes was given by [9,15]


2
,
1(1
4
kp h
pp hh
q
Agg
)


 (21)

max ,qph
Then, the PLD can take the form





1
,
1,
,Θ
!! 1!
n
nkph
kph
gEA
nEE A
ph n






(22)
2.4. Charge Factor Correction
This correction takes into account the charge effect on
PLD calculations. The Charge effect is represented by
the effective charge factor, which had been expressed by
many formulae. However, many researchers don’t apply
this correction [9,16]. Therefore, the charge factor effect
was neglected in this work.
2.5. Isospin Correction
To add the effect of isospin in nuclear reactions, it is ne-
cessary to determine how much the isospin is conserved
or mixed and what is the isospin symmetry energy?
If the isospin is included in calculation of level density,
it is important to take one-to-one correspondence between
states of the same isospin in isobaric nuclei [17-19].


 


,, ,,
,,, 1,, 1
,,, 2,, 2
z
sym
sym
phETT T
phEETTTT
phE ETTTT








(23)
s
ym is the symmetric energy which is given by em-
pirical equation
E

 
1432
1310 31
sym z
EAAT

 
2
T (24)
The PLD formula which contains the isospin can be
given by



 


1
1
,
,
,
,,,
,, Θ
!! 1!
,
n
nph T
Tp
ph Tphsymz
phTE
gEAfphT EA
ph n
AAETT








hT
(25)

S. S. SHAFIK, A. D. SALLOUM
50

,,
T

f
phT
T
is the correction factor of states with
good isospin. If isospin is assumed to be completely
mixed, the symmetric energy is zero and the correction
factor
f
is unity.
2.6. Spin Correction
Spin effect is also added to the PLD formula as a correc-
tion and it is assumed factorized [20-23]



 


1
,
,
Θ
!! 1
,! ,
n
nph
nph
gEARJ EA
p
Jhn nE




 (26)

2
2
3
1
21 2
exp 2
22
n
n
J
J
RJ



















(27)
n

is the spin cut off parameter. It is important to
mention that Equation (26) used for pre-compound nu-
cleus and it does not use for compound nucleus [22].
2
0.16
nn23
A
  (28)
2.7. Surface Correction
The initial interaction between a projectile and target
nucleon is frequently localized near the nucleus surface.
Since the nuclear density variant with nuclear radius,
hence the nuclear potential is shallower than in the inte-
rior; therefore, the local well depth (i.e. near the surface)
is less the central depth [9,11,24]. This must add a con-
siderable effect on PLD calculations especially in the
exciton knockout and pickup nuclear reactions.
If we labeled


,,nEV  as the PLD of one-com-
ponent Fermi gas system with exciton number exci-
tation energy and nuclear potential well depth V,
then one can write the following equation for PLD with
surface effect
,n
E

111
,,,, ,,nEVnEf nEV  

(29)
where


1,,nE
  is the PLD calculated for V

,
and it is given by the simple one-component PLD Pauli
formula (Equation (4))




1
,
1
,, Θ
!! 1!
n
nph

,
p
h
gEA
nEE A
ph n



 
 (30)
The function


1,,
f
nEV in Equation (29) is the cor-
rection due to surface effect in the ESM, and it is given
by
 
1
10
,, 1
n
j hh
j
j
EjVh
fnEVC E


 




From Equation (29), the PLD formula was corrected to
in
(31)
clude the finite well depth and then was extended to
include correction due to surface effect. This was done in
the initial particle-nucleus interaction by replacing the
nuclear potential depth 0 V by 1 V. This means instead
of putting 0
VV

, we put 1
VV. The new potential
depth 1 V, ined as therage effective well
depth”he choice between
is defe “av
. T1 V and 0
V is
0
1
38 MV1Vhe
0
VVh





For (n, n) reaction


1, 7MeV nn
V

[24].
3. Comprehensive Formula
includes all the pre- Comprehensive formula is a formula
vious corrections, except the isospin correction because
the reaction was assumed to be completely mixed [17,18],
then the isospin correction became unity. The aim behind
suggested this formula is an attempt to get on a formula
describes the PE PLD by the most accurate description
(real one).
 


 
1
,,
11
,,, ,
!! 1!
n
nph kph
gEPB A
phEf nEV
ph n


 







,,
Θ
np
RJ EPBA 
hkph
(32)
4. Results and Discussion
be calculated experi-
4.1. Comparison between One and
In orrence between one and two-
It is obvious that the PLD cannot
mentally especially for excitation energy more than 15
MeV because the states overlapped with each other’s. On
the other hand, and in order to increasing knowledge
about the nuclear force, which represents the strongest
force comparing with others, one must increase the ap-
plied excitation energy to hundred and several hundred
MeVs. Therefore, PLD must be given in theoretical form.
The calculated Comprehensive formula (Equation (32))
is a suitable formula for PLD estimations. This claim can
be examined if one applies the Comprehensive formula
of the PLD in cross section equation of any (and for any
nucleus) PE nuclear models and comparing with experi-
mental results of this cross section. However, PLD for-
mula with Pauli’s correction was used in cross section
calculations by many researchers [25-27] and these re-
searchers avoided use all corrections of the PLD to ease
the programming potential. Further, in these results,
Pauli’s correction was stand as a reference case for PLD
comparisons with all their corrections.
Two-Component
der to study the diffe
component formulae of the PLD, a comparison was made

S. S. SHAFIK, A. D. SALLOUM
51
for 56Fe isotope. All used parameter are listed in Tabl e 1.
From Figures 1(a)-(c) one can see that the two-compo-
nent results are less than those of one-component. This
behavior is expected physically because the two-com-
ponent system will have to share the energy with more
entities (the entities are those due to particles and holes
of the neutrons and protons). Although the neutron parti-
cles and holes are considered zero, the two component
results stay less than the one-component results.
From Figures 1(a)-(c) one can see that the two-com-
ponent results are less than those of one-component. This
behavior is expected physically because the two-com-
ponent system will have to share the energy with more
entities (the entities are those due to particles and holes
Figure 1. (a) One and two-components Ericson’s formulae of PLD for 56Fe isotope; (b) One and tw-components Pauli’s for- o
mulae of PLD of 56Fe isotope; (c) One and two-components pairing formulae of PLD for 56Fe isotope.

S. S. SHAFIK, A. D. SALLOUM
52
f the neutrons and protons). Although the neutron parti- of the 56Fe isotope because the mo
cles and holes are considered zero, the two component
results stay less than the one-component results.
In addition, it is noted from Figures 1(a)-(c) and Ta-
bl
4.2. Behavior of One-Component Corrections
all
approximately
sa
at
ass number is higher
e 2 that the ratio between one and two-components
results is small for main forms of the PLD formulae
(Ericson, Pauli, and Pairing PLD formulae), therefore,
one-component formula may be used instead of two-
component formula with acceptable results.
The PLD of one-component for Ericson formula with
corrections for 56Fe isotope can be shown in Figure 2. It
is noted from this figure that the arrangement of the re-
sults from bottom to top is surface correction, Pauli’s
correction, Ericson’s correction, pairing correction, spin
correction, active and passive correction and Compre-
hensive formula, respectively. Whereas, for 90Zr isotope,
shown in Figure 3, the arrangement is surface correction,
Pauli correction, Ericson’s correction, spin correction,
active and passive holes correction, pairing correction
and Comprehensive formula, respectively.
The results of these two figures give
me arrangements. As we mentioned above, if one used
PLD with Pauli correction as reference, then it is easy to
see that except surface correction, the all corrections and
Comprehensive PLD formulae have values more than
PLD with Pauli correction. However, one can expect that
the use of PLD formula with Pauli correction alone in
cross section calculations may deviate the theoretical
estimations from experimental results. Therefore, these
results strongly suggested that the Comprehensive PLD
formulae must be used in any calculations need PLD.
Finally, the results of 90Zr isotope are bigger than th
and then the single particle level density
g
increases.
On the other hand, the difference between the results
decreases with increasing the excitation energy.
5. Conclusion
ents in nuclear reaction and
Table 1. The parameter used in the present calculations.
T
PLD represents mean argum
stricture models. This paper submitted a good literature
arget nucleus under investigation 56Fe
Mass number 56
Ato
Neutr
Exciton configuration (3,3) for onmponent
(3t
Max. excitation energy
mic number26
ons number 30
Exciton number 6
e-co
,3,0,0) for two-componen
100MeV
Single particle density A/13
able 2. The ratio between one and two-component formu-
The correction two and one
T
lae for Ericson, Pauli and pairing PLD.
The ratio between
components 21


Ericson 0.007
Pauli 0.015
Pairing 0.02
Figure 2. PLD of one-component with all corrections for 56Fe isotope.

S. S. SHAFIK, A. D. SALLOUM 53
Figure 3. PLD of one-component with all corrections for 90Ze isotope.
rvey for the PLD formulae with its corrections and
REFERENCES
[1] K. K. Gudima D. Toneev, “Cas-
[7] C. K. Cline and M. Blann, “The Pre-Equilibrium Statisti-
su
suggested a formula containing all corrections which is
the “Comprehensive Formula”. However, the results of
this work strongly recommended that if one needs to use
PLD, then the Comprehensive formula must be applied
rather than Pauli formula. In addition, the results showed
that the one-component formulae of PLD can be used
instead of two components with acceptance results and
the PLD values increase with the mass number of the
target nucleus and with excitation energy of the incident
particle.
, S. G. Mashnik and V.
cade-Exciton Model of Nuclear Reactions,” Nuclear
Physics A, Vol. 401, No. 2, 1983, pp. 329-361.
doi:10.1016/0375-9474(83)90532-8
[2] G. D. Harp, J. M. Miller and B. J. Berne, “Attainment of
Statistical Equilibrium in Excited Nuclei,” Physical Re-
view, Vol. 165, No. 4, pp. 1166-1169.
doi:10.1103/PhysRev.165.1166
[3] G. D. Harp and J. M. Miller, “Precompound Decay from
Time-Dependent Point of View,” Physical Review C, Vol.
3, No. 5, 1971, pp. 1847-1855.
doi:10.1103/PhysRevC.3.1847
[4] M. Blann, “Importance of Nuclear Density Distribution
on Pre-Equilibrium Decay,” Physical Review Letters, Vol.
28, No. 12, 1972, pp. 757-759.
doi:10.1103/PhysRevLett.28.757
[5] M. K. Bhardwaj and I. A. Rizivi, “Alpha-Induced Reac-
tions in Iridium,” Physical Review C, Vol. 45, No. 5,
1992, pp. 2338-2348. doi:10.1103/PhysRevC.45.2338
[6] J. J. Griffin, “Statistical Model of Intermediate Structure,”
Physical Review Letters, Vol. 17, No. 9, 1966, pp. 478-
481. doi:10.1103/PhysRevLett.17.478
cal Model: Description of the Nuclear Equilibration
Process and Parameterization of the Model,” Nuclear
Physics A, Vol. 172, No. 2, 1971, pp. 225-259.
doi:10.1016/0375-9474(71)90713-5
[8] C. Kalbach, “Two-Component Exciton Model: Basic
Formalism Away From Shell Closures,” Physical Review
C, Vol. 33, No. 3, 1986, pp. 818-833.
doi:10.1103/PhysRevC.33.818
[9] E. Bĕták and P. E. Hodgson, “Particle-Hole State Density
Density in the
0426-X
in Pre-equilibrium Nuclear Reactions,” University of
Oxford, CERN Libraries, Geneva, 1998.
[10] F. C. Williams Jr., “Particle-Hole State
Uniform Spacing Model,” Nuclear Physics A, Vol. 166
No. 2, 1971, pp. 231-240.
doi:10.1016/0375-9474(71)9
artial Level Density
w
Shell Theory,” Aca-
cs,” Pergamon Press,
Final States and the Spectral End-
[11] M. Avrigeanu and V. Avrigeanu, “P
for Nuclear Data Calculations,” Computer Physics Com-
munications, Vol. 112, No. 2-3, 1998, pp. 191-226.
[12] W. E. Meyrhof, “Elements of Nuclear Physics,” Mc Gre
Hill, New York, 1967, pp. 36,58.
[13] A. de-Shalit and I. Talmi, “Nuclear
demic Press, Waltham, 1963, p. 27.
[14] A. S. Davydov, “Quantum Mechani
Oxford, 1969, p. 371.
[15] C. Kalbach, “Missing
point in Exciton Model Calculations,” Physical Review C,
Vol. 73, No. 2, 2006, Article ID: 024614.
doi:10.1103/PhysRevC.73.024614
[16] G. M. Braga-Marcazzan, E. Gadioli-Erba, L. M. Mil-
lazo-Colli and P. G. Sona, “Analysis of the Total (n,p)
Cross Sections around 14 MeV with the Pre-Equilibrium
Exciton Model,” Physical Review C, Vol. 6, No. 4, 1972,
pp. 1398-1407. doi:10.1103/PhysRevC.6.1398
[17] C. Kalbach, “Isospin Dependence of Two-Component
Particle-Hole State Density,” Physical Review C, Vol. 30,
No. 4, 1984, pp. 1310-1319.

S. S. SHAFIK, A. D. SALLOUM
54
doi:10.1103/PhysRevC.30.1310
density with Good Iso-[18] C. Kalbach, “Particle-Hole State
spin,” Physical Review C, Vol. 47, No. 2, 1993, pp. 587-
601. doi:10.1103/PhysRevC.47.587
[19] C. Kalbach, “Isospin Conversion in Preequilibrium Reac-
tions,” Physical Review C, Vol. 72, No. 2, 2005, Article
ID: 024607. doi:10.1103/PhysRevC.72.024607
[20] C. Kalbach and F. M. Mann, “Phenomenology of Con-
tinuum Angular Distributions. I. Systematics and Para-
metrization,” Physical Review C, Vol. 23, No. 1, 1981, pp.
112-123. doi:10.1103/PhysRevC.23.112
[21] C. Kalbach, “Phenomenology of Continuum Angular
Distributions. II. Griffin Preequilibrium Model,” Physical
Review C, Vol. 23, No. 1, pp. 124-135.
doi:10.1103/PhysRevC.23.124
[22] M. B. Chadwick and P. Obložinský, “Particle-Hole State
Densities with Linear Momentum and Angular Distribu-
tions in Preequilibrium Reaction,” Physical Review C,
Vol. 46, No. 5, 1992, pp. 2028-2041.
doi:10.1103/PhysRevC.46.2028
[23] S. S. Shafik, “Angular Momentum Distribution for State
M. Kolomietz, A. I. Sanzhur and S.
Density with Non-ESM Dependence,” Journal of Fisica
Nucleare, 2009.
[24] Ye. A. Bogila, V.
Shlomo, “Preequilibrium Decay in the Exciton Model for
Nuclear Potential with a Finite Depth,” Physical Review
C, Vol. 53, No. 2, 1996, pp. 855-859.
doi:10.1103/PhysRevC.53.855
[25] M. Avrigeanu, A. Harangozo and V. Avrigeanu, “Aver-
age Strength of the Effective Interaction in Multistep Di-
rect Reactions,” Physical Review C, Vol. 56, No. 3, 1997,
pp. 1633-1636. doi:10.1103/PhysRevC.56.1633
[26] Y. K. Gupta, D. C. Biswas, Bency John, B. K. Nayak, A.
Saxena and R. K. Choudhury, “Nuclear Level-Density
Parameters of Nuclei in the Z~70 and A~180 Mid-Shell
Regions,” Physical Review C, Vol. 80, No. 5, 2009, Arti-
cle ID: 054611. doi:10.1103/PhysRevC.80.054611
[27] H. Nakada and Y. Alhassid, “Publisher’s Note: Iso-
spin-Projected Nuclear Level Densities by the Shell Mo-
del Monte Carlo Method,” Physical Review C, Vol. 78,
No. 6, 2008, Article ID: 069907.
doi:10.1103/PhysRevC.78.069907