Phenomenological and Semi-microscopic Analysis for the Elastic Scattering of Protons from <sup>12</sup>C Nuclei at Different Energies

doi:10.4236/jmp.2013.45B013

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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

Email: [email protected]

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].



)( C

CRr

eZZ

rV 



for (2) CrR

() pt

Vr r

 for CrR

with radius

1/3

(), ,,

iiT

RrA iVWC

Real volume part has the following form:

() (,,)1expV

Vrfrr aVa



























 (3)

Imaginary volume part has the following form:

() (,,)1exp,

Vww

Wrfrra Wa









 















(4)

Real spin-orbit part has the following form:

21/3

2d 1

() d1exp

so so

VrV r

hr rrA





























(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:

()()(, ,)(,,)

(,,)

CVVW

soso so

UrVrVf rraiWfrra

Vfrra

 



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:

111 1

()() ()

rNN

VRNrV Rrdr





 (7)

where, 11

()r



is the matter density distribution of the

target nucleus (12C),

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

exp( 4)exp( 2.5)

( )79992134

0.005

2761()

VR RR















(8)

The nuclear density distribution for 12C was calculated

using Three-parameter Fermi model (3PF), where ρ(r)

was calculated from the following formula

()(1)/(1exp(()/ ))



 

(9)

with w =﹣0.149, z = 0.5224 and c = 2.355.

3. Results and Discussion

(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

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

53.08

0.825

0.9 16.11

16.09

0.765

0.764

1.42

0.234

0.334

30.3 OM

50.97

0. 84

0.91 17.88

21.21

0.701

0.753

1.42

0.234

0.334

40.0 OM

44.79

0. 78

0.9 18.48

10.1

0.923

1.33

6.34

0.566

0.666

49.5 OM

36.86

0.88

0.78 19.18

22.13

0.826

0.788

7.75

0.655

61.4 OM

27.41

0.84

0.75 20.21

17.15

1.06

1.01

9.5

0.528

(MeV) JV

(MeV.fm3) JW

(MeV.fm3)

17.0 OM

276.20

275.94

93.68

93.51

30.3 OM

271.68

267.36

124.66

121.90

40.0 OM

222.93

184.38

126.94

106.75

49.5 OM

204.83

215.53

118.93

131.86

61.4 OM

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

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

W0 (MeV) imaginary potential depth

(b)

E (MeV)

Figure 3. The relation between the imaginary potential

depth (W0) and energy (E).

024

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

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.

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