Journal of Modern Physics, 2013, 4, 78-82
doi:10.4236/jmp.2013.45B013 Published Online May 2013 (http://www.scirp.org/journal/jmp)
Phenomenological and Semi-microscopic Analysis
for the E l a s tic Scatteri n g o f Protons from 12C Nuclei at
Different Energies
Sh. Hamada1, N. Amangeldi2
1Faculty of Science, Tanta University, Tanta, Egypt
2Eurasia University, Astana, Kazakhstan
Received 2013
ABSTRACT
Analysis of the elastic scattering of protons from 12C nuclei had been performed within the framework of both the opti-
cal model and single folding model at different proton energies; 17, 30.3, 40, 49.48 and 61.4 MeV. We have obtained
the global potential parameters which could fairly reproduce the experimental data for p+12C elastic scattering at the
aforementioned energies. The radial and energy dependence of the real and imaginary parts of the potential were calcu-
lated. Good agreement between experimental data and theoretical predictions in the whole angular range was obtained
using both phenomenological approach (Optical Model), and semi-microscopic approach (Single Folding). In single
folding calculations, the real part of the potential was calculated from a more fundamental basis by the folding method
in which the NN interaction VNN(r), is folded into the density of the target nuclei and supplemented with a phenomenol-
ogical imaginary potential. The obtained normalization factor Nr is in the range of 0.75 - 0.9.
Keywords: Elastic Scattering; Optical Model; Nuclear Structure; Single Folding Model
1. Introduction
Elastic scattering of nucleon-nucleus data at intermediate
energies is a useful tool for testing and analyzing nuclear
structure models and intermediate energy reaction theo-
ries [1-10]. The elastic scattering of protons from 12C was
analyzed at different energies (17, 30.3, 40, 49.48 and
61.4 MeV) from literature [11-15]. The analysis of the
experimental data was performed either by using
Wood-Saxons (WS) forms for both real and imaginary
parts of the potential, or by obtaining the real part from
the folding procedure [16,17] and using it with a WS
term for the imaginary part of the potential, in addition to
spin orbit potential which has been introduced due to the
0.5 spin of protons.
The folding model which is a powerful tool for the
microscopic analysis of nuclear reactions has been used
for years to calculate the nucleon-nucleus optical poten-
tial and inelastic form factors. It can be seen from the
basic folding formulas that this model generates the
first-order term of the microscopic optical potential that
is derived from Feshbach’s theory of nuclear reactions.
The success of this approach in describing the observed
nucleon-nucleus elastic scattering data for many targets
suggests that the first-order term of the microscopic op-
tical potential is indeed the dominant part of the nucleon
optical potential. A popular choice for the effective NN
interaction has been one of the M3Y interactions. Al-
though these density independent M3Y interactions were
originally developed for using in the Distorted Wave
Born Approximation (DWBA) for the analysis of (p,p′)
reaction, they have been used much more often in the
double folding calculation of the heavy ion interaction
potential at low and medium energies. The elastic scat-
tering of proton nucleus has been analyzed in order to
determine ground state matter densities empirically for
comparison with Hartree–Fock predictions [18-20]. In
single folding calculations, the real part of potential ob-
tained from the folding model was supplemented by a
phenomenological imaginary potential, and during the
fitting process the real potential was normalized and the
imaginary potential optimized. The basic inputs for a
single folding calculation of the nucleon-nucleus poten-
tial are the nuclear densities of the target and the effec-
tive nucleon-nucleon (NN) interaction. The folding
model is a very useful approach to check the target nu-
clear densities [21].
2. The Nuclear Optical Model and Single
Folding Model
Optical model analysis of proton scattering data have

Sh. HAMADA, N. AMANGELDI 79
been carried out for a wide range of incident proton en-
ergies, and a few attempts have been made to empirically
determine the energy dependence of the optical model
potential. In practice it is required to obtain the potential
from the experimental data, and this may be done by
systematically varying the parameters of the potential to
optimize the overall fit to the data, using appropriate
computer programs. To do the best fitting, we kept the
product V0r0
2 constant, the same we have done for the
imaginary part. In the energy region below 61 MeV, ex-
tensive proton elastic scattering data exist. These have, in
general been analyzed in terms of an optical model in
which the interaction is represented as the scattering of a
point particle (proton) by a potential of form:
()() ()()()
op Cso
Ur VrVriWrVr (1)
The Coulomb potential was assumed to be as two uni-
form charge distributions with radii consistent with elec-
tron scattering [22].

22
2
/3
2
)( C
C
tp
CRr
R
eZZ
rV  
for (2) CrR
2
() pt
C
Z
Ze
Vr r  for CrR
with radius
1/3
(), ,,
iiT
RrA iVWC
Real volume part has the following form:
1
0
() (,,)1expV
VV
V
rr
Vrfrr aVa







 




 (3)
Imaginary volume part has the following form:
1
0
() (,,)1exp,
W
Vww
W
rr
Wrfrra Wa




 
 





(4)
Real spin-orbit part has the following form:
21/3
2d 1
() d1exp
so so
so
so
VrV r
hr rrA
a

































(5)
The spin-orbit term Uso(r) = Vso(r) + iWso(r), it is usual
to take Wso(r) = 0, leaving the three parameters Vso,
rsoand W.The model thus involves nine parameters al-
though several analysis have been performed using more
restricted sets by equating some of the geometrical pa-
rameters and/or neglecting one the imaginary terms. So,
the interaction potential can be rewritten as:
a
00
2
()()(, ,)(,,)
2d
(,,)
d
CVVW
soso so
UrVrVf rraiWfrra
Vfrra
r
hr
 

Many such analysis of nucleon scattering have now
been made and is found that the potentials are quite
similar for all nuclei and vary other slowly with the inci-
dent energy.
In practice it is frequently found that many sets of pa-
rameters give equally good fits to the data, and the ques-
tion then arises whether any one of these is more physi-
cal than the others and if so which is to be preferred.
These parameter ambiguities, as they are called, are of
two main types, discrete and continuous. The existence
of these ambiguities means that it is not possible to es-
tablish the optical potential by phenomenological analy-
ses alone and it is necessary to derive the potential using
more microscopic method such as double folding. The
real part of the optical potential for the nucleon–nucleus
elastic scattering is given within the framework of single
folding model, in the following form:
3
111 1
()() ()
SF
rNN
VRNrV Rrdr  
 (7)
where, 11
()r

is the matter density distribution of the
target nucleus (12C),
N
N is the effective NN-interaction.
In the present calculation the effective NN-interaction is
taken according to [23] in the form of M3Y-interaction
V
exp( 4)exp( 2.5)
( )79992134
42
0.005
2761()
RR
VR RR
ER
A







.5
(8)
The nuclear density distribution for 12C was calculated
using Three-parameter Fermi model (3PF), where ρ(r)
was calculated from the following formula
2
02
()(1)/(1exp(()/ ))
wr
rr
c
   cz (9)
with w =﹣0.149, z = 0.5224 and c = 2.355.
3. Results and Discussion
W
(6)
The comparison between the experimental data and the
theoretical predictions within the framework of both; the
optical model and single folding model at energies 17.0,
30.3, 40.0, 49.48 and 61.4 MeV is shown in Figure 1.
The optimal optical potential parameters obtained from
(SPIVAL code) [24], and also those from single folding
model (DFPOT code) [25] using Three parameter Fermi
model for calculating 12C density distributions are shown
in Table 1. The obtained normalization factor (Nr) is in
the range 0.9 - 0.75. We investigated the energy depend-
ence on the values of V0 and W0 for 12C(p,p)12C (Figures
2 and 3), which showed that, with increasing energy, the
value of the real potential depth decreases and can be
approximated by the formula: V = 66.39 - 0.5997 E, and
the imaginary potential depth increases and can be ap-
proximated by the formula: W = 14.856 + 0.0887 E. The

Sh. HAMADA, N. AMANGELDI
80
radii of the real and imaginary parts of the potential were
fixed at rV = 1.15 fm and rW = 1.25 fm, Coulomb radius
parameter was fixed at 1.25 fm and the radius parameter
rso for spin orbit potential was fixed at 1.1 fm. The radial
dependences of the real and imaginary parts of the poten-
tials are shown in Figures 4 and 5 respectively.
Table 1. Optimal potential parameters from SPIVAL code,
and also those from single folding model.
E (MeV) V0
(MeV) aV (fm) Nr W0 (MeV) aW (fm) Vso (MeV) aso (fm)
17.0 OM
SF
53.08 0.825
0.9 16.11 16.09
0.765
0.764
1.42
1.42
0.234
0.334
30.3 OM
SF
50.97 0. 84
0.91 17.88 21.21
0.701
0.753
1.42
1.42
0.234
0.334
40.0 OM
SF
44.79 0. 78
0.9 18.48 10.1
0.923
1.33
6.34
6.34
0.566
0.666
49.5 OM
SF
36.86 0.88
0.78 19.18 22.13
0.826
0.788
7.75
7.75
0.655
0.655
61.4 OM
SF
27.41 0.84
0.75 20.21 17.15
1.06
1.01
9.5
9.5
0.528
0.528
E
(MeV) JV
(MeV.fm3) JW (MeV.fm3)
17.0 OM
SF
276.20
275.94
93.68
93.51
30.3 OM
SF
271.68
267.36
124.66
121.90
40.0 OM
SF
222.93
184.38
126.94
106.75
49.5 OM
SF
204.83
215.53
118.93
131.86
61.4 OM
SF
145.5
155.76
137.43
129.36
0 20406080100120140160180
100
101
102
103(a) Exp 17 MeV
SPIVAL
Single Folding
d/d (mb/sr)
cm (d eg )
20 40 60 80100120
100
101
102
103(b) Exp 3 0 .3 Me V
S P IVAL
S in g le Fold ing
d/d (mb/sr)
cm (de g )
020406080100120140 160180
10-1
100
101
102
103(c) E xp 40.0 M eV
SPIVAL
Single Fold ing
d/d (mb/sr)
cm (deg)
020406080100 120 140 160
10-2
10-1
100
101
102
103
(d) Exp 49.48 M eV
SPIVAL
Sin g le F o ld in g
d/d (mb/sr)
cm (deg)
0 20406080100
10-2
10-1
100
101
102
103
120
(e) Exp 6 1 .4 Me V
SPIVAL
Sin g le F o ld in g
d/d (mb/sr)
cm (d e g)
Figure 1. The comparison between the experimental data
for p+12C elastic scattering and the theoretical predictions
using both optical and single folding model at energies (a)
17 MeV, (b) 30.3, (c) 40 MeV, (d) 49.48 MeV and (e) 61.4
MeV.
20 30 40 50 60
20
30
40
50
60
V0 (MeV) real potential depth
(a)
E (MeV)
Figure 2. The relation between the real potential depth (V0)
and energy (E).

Sh. HAMADA, N. AMANGELDI 81
10 20 30 40 50 60 70
16
18
20
W0 (MeV) imaginary potential depth
(b)
E (MeV)
Figure 3. The relation between the imaginary potential
depth (W0) and energy (E).
024
0
10
20
30
40
50
6
Real potential depth (MeV)
r (fm)
17 MeV
30.5 MeV
40 MeV
49.48 MeV
61.4 MeV
Radial dependence of the real part of the potential
Figure 4. The radial dependences for the real part of the
potentials.
0246
0
4
8
12
16
20
24
r (fm)
Radial dependence of the imaginary part of the potential
17 MeV
30.5 Me V
40 MeV
49.48 MeV
61.4 Me V
Imaginary potential depth (MeV)
Figure 5. The radial dependences for the imaginary part of
the potentials.
4. Summary
The analysis of the elastic scattering of protons from 12C
at energies 17, 30.3, 40, 49.48 and 61.4 Mev was per-
formed within the framework of two approaches: an op-
tical code SPIVAL and single folding potential using
DFPOT code. Both approaches give satisfactory results.
The normalization factor Nr was calculated and found to
be in the range 0.75-0.9. A good agreement in the whole
energy range was found using the two previous discussed
approaches with reliable values for the real and imagi-
nary volume integral JV, JW.
REFERENCES
[1] H. O. Mayer, P. Schwandt, G. L. Moake and P. P. Singh,
“Elastic Scattering of 200 Mev Polarized Protons
from 12,13C and the Optical Potential,” Physical Review
C ,Vol. 23, 1981, pp. 616-622.
doi:10.1103/PhysRevC.23.616
[2] M. Jaminon, C. Mahaux and P. Rochus, “Optical-Model
Potential in a Relativistic Quantum Field Model,”
Physical Review C, 1980, Vol. 22, pp. 2027-2042.
doi:10.1103/PhysRevC.22.2027
[3] C. Mahaux, Lecture Notes in Physics, Vol. 89, No.
1,1979 .
[4] F. A. Brieva and J. R. Rook, “Nucleon-Nucleus Optical :
Nuclear Matter Approach,” Nuclear Physics A, Vol. 297,
No. 2, 1977, pp. 299-316.
doi:10.1016/0375-9474(77)90322-0
[5] F. A. Brieva and J. R. Rook, “Nucleon-Nucleus Optical
Model Potential:(Ⅱ).Finite Nuclei, ” Nuclear Physics A
Vol. 291, 1977, pp. 317-341.
doi:10.1016/0375-9474(77)90323-2
[6] F. A. Brieva and J. R. Rook, “Nucleon-Nucleus Optical
Model Potential:（Ⅲ).the Spin-Orbit Component,”
Nuclear Physics A, Vol. 297, No. 2, 1978, pp. 206-230.
doi:10.1016/0375-9474(78)90272-5
[7] F. A. Brieva and J. R. Rook, “Microscopic descrip-
tion of nucleon-nucleus elastic scattering” Nuclear
Physics A, Vol. 307, No. 3,1978, pp. 493-514.
doi:10.1016/0375-9474(78)90461-X
[8] L. Ray, G. W. Hoffmann, M. Barlett and J. McGill,
“Proton Elastic Scattering from 40,42,44,48Ca at 800
Mev,”Physical Review C, Vol. 23, 1981, pp. 828-837.
doi:10.1103/PhysRevC.23.828
[9] R. D. Amado, J. A. McNeil and D. A Sparrow,
“Two-Step Processes in Intermediate Energy Had-
ron-Nucleus Scattering,” Physical Review C. Vol. 23,
1981, 2186-2197. doi:10.1103/PhysRevC.23.2186
[10] M. Rashan, Eur. Phys. J. A, Vol. 16, 2003, p. 371.
doi:10.1140/epja/i2001-10270-4
[11] B. Q. Chen and A. D. Mackellar, “Proton-Nucleus Scat-
tering Based on the Relativistic Brueckner-Hartree-Fock
Model,” Physical Review C., Vol. 52, 1995, pp. 878-889.
doi.org/10.1103/PhysRevC.52.878
[12] F. Sammarruca, E. J. Stephenson and K. Jiang, “Micro-
scopic Calculations of Medium Effects for 200-MeV”
Physical Review C., Vol. 60, 1999, 064610 .
doi.org/10.1103/PhysRevC.60.064610
[13] R. Crespo, R. C. Johnson and J. A. Tostevin, “Mean
Field Calculations of Nucleonnucleus Scattering,”
Physical Review C ,Vol. 53, 1996, pp. 3022-3031.

Sh. HAMADA, N. AMANGELDI
82
doi:10.1103/PhysRevC.53.3022
[14] I. E. Dayton and G. Schrank, “Elastic Scattering of
17-Mev Protons by Nuclei,” Physical Review Online
Archive., Vol. 101, No. 4, 1956, pp. 1358-1367.
doi:10.1103/PhysRev.101.1358
[15] B. W. Ridley and J. F. Turner, “Optical Model Studies
of Proton Scattering at 30 MeV:(Ⅰ).Different Cross
Sections for Elastic Scattering of Protons at 30.3
MeV,”Nuclear Physics,Vol. 58, 1964, pp. 497-508.
doi:10.1016/0029-5582(64)90561-9
[16] L. N. Blumberg, E. E. Gross, A. van der Woude, A.
Zucker and R. H. Bassel, Physical Review, Vol. 147, No.
3, pp. 812-825
[17] J. A. Fannon, E. J. Burge, D. A. Smith and N. K. Gan-
guly, “Elastic and inelastic of 50 MeV Protons by 12C
and 16O,” Nuclear Physics A, Vol. 97, 1967, pp. 263-281.
doi.org/10.1016/0375-9474(67)90487-3
[18] C. B. Fulmer, J. B. Ball, A. Scott and M. L. Whiten,
“Elastic Scattering of 61.4-MeV Protons,” Physical Re-
view Online Archive., Vol. 181. No. 1, 1969, pp. 1565-
1579.
doi:10.1103/PhysRev.181.1565
[19] M. E. Brandan and G. R. Satchler, “Folding Model
Analysis Of 12,13C+12C and 16O+12C Scattering at
Intermediate Energies Using A Density-Dependent In-
teraction,” Nuclear Physics A, Vol. 487. No. 2, 1988,
pp.477- 492. doi: 10.1016/0375-9474(88)90625-2
[20] M. El-Azab Farid and G. R. Satchler, “A Den-
sity-Dependent Interaction in the Folding Model for
Heavyion Potentials,” Nuclear Physics A, Vol. 438. No.
2, 1985, pp. 525-535.
doi:10.1016/0375-9474(85)90391-4
[21] L. Ray, G. W. Hoffmann and W. R. Coker, “Nonrelativ-
istic and Relativistic Descriptions of Proton-Nucleus
Scattering,” Physics Reports, Vol. 212. No. 5, 1992,
pp.223- 328. doi:10.1016/0370-1573(92)90156-T
[22] L. Ray, “Neutron Isotopic Density Differences Deduced
from 0.8 GeV Polarized Proton Elastic Scattering,”
Physical Review C, Vol. 19, 1979, pp. 1855-1872.
doi:10.1103/PhysRevC.19.1855
[23] L. Ray, “Proton-Nucleus Total Cross Sections in the In-
termediate Energy Range,” Physical Review C, Vol. 20,
1979, pp. 1857-1872. doi:10.1103/PhysRevC.20.1857
[24] L. Ray G. W. Hoffmann and R. M. Thalar, “Coulomb
Interaction in Multiple Scattering Theory,” Physical
Review C, Vol. 22, 1980, pp. 1454-1467.
doi:10.1103/PhysRevC.22.1454
[25] E. H. Esmail and M. A. Allam, Acta Physical Polonicab,
Vol. 39, 2008, p. 159.
[26] J. E. Poling, E. Norbeck and R. R. Carlson, “Elastic
Scattering of Lithium By 9Be,10B,12C,13C,16O, a nd 28Si
from 4 to 63 MeV,” Physical Review C, Vol. 13, 1976,
pp. 648-660.
doi:10.1103/PhysRevC.13.648
[27] Z. Majka, H. J. Jils and H. Rebel, Z. Physics A., Vol.
288, 1978, p.139.
[28] Perey F. // SPIVAL an optical model code unpublished.
[29] J. Cook, Computer Physics Communications, Vol. 25，
1982.