
Research Article


Analysis of Alpha Transfer Reactions Using Different
Optical Potentials


A.I. Ass`ad
and
H.S. Ashour


ABSTRACT

In this study, heavy ion reaction with αtransfer
are studied for ^{24}Mg, ^{28}Si (^{16}O, ^{12}C)^{
}using different optical potentials, within the framework of the
distorted Born approximation (DWBA) calculations. DoubleGaussian potential
is expressed as the boundstates of the transferred particle. The calculated
angular distributions are found to be in a good agreement with the experimental
data. The extracted spectroscopic factors are reasonable. 




INTRODUCTION
In a direct transfer reaction, which induced by heavy ions the population
of the first few excited levels, is well described by the distorted wave
method. For most cases the Distorted Wave Born Approximation (DWBA) is
one step process (Mermaz et al., 1983). So, the theory of direct
transfer reactions are introduced as a powerful tool (ALFarra et al.,
2003) with which we can obtain information about the structure of nuclei.
The differential cross section of ^{12}C(^{3}He,n)^{14}O
and ^{26}Mg(^{3}He,n)^{28}Si (ALFarra et al.,
2003) reactions, which leads to the lowlying states in the final nuclei
at 45.5 MeV incident energy, have been calculated in terms of DWBA calculations.
The two nucleon (Michimasa et al., 2002; Gupta et al.,
1976; ALFarra et al., 2003) and alpha (Shyam et al., 1985)
transfer reactions have been studied in term of the DWBA theory. The numerical
data of the angular distribution and the experimental data were found
in a good agreement especially at forward angles. However, the optical
potential plays an important role in calculating the angular distributions
and the differential cross section of ^{16}O(^{6}Li,α)^{18}F
(Farra, 2003). And previously studied in term of DWBA using the real and
imaginary parts of WoodSaxon optical potential (WSWS) and then compared
with the WoodSaxon including the Jdependent (WSJD). It was found that
the inclusion of the Jdependent term gives the same results as the (WSWS)
at forward angles, with a better description at backward regions though.
In the present reserch, the differential cross sections of heavyion
reactions with αtransfer reactions have been calculated in term
of onestep DWBA calculations using different optical potentials (real
and imaginary parts are Gobbi and Vandenbosch potential). Where Gobbi
and Vandenbosch potentials are members of a family of WoodsSaxon potentials
with the same geometrical parameters but different real and imaginary
well depths. The calculated differential cross sections are fitted with
the experimental data to extract spectroscopic factors.
NUCLEAR OPTICAL POTENTIAL
In this section, the differential cross sections for the Stripping reaction
^{24}Mg (^{16}O, ^{12}C)^{ 28}Si and ^{28}Si(^{16}O,
^{12}C)^{32}S have been evaluated in the framework of
onestep DWBA calculations. The calculations used different optical potentials,
namely Gobbi potential (Yosio and Taro, 1984), which has the following
form
and Vandenbosch potential has the same form as Gobbi potential except
Where, V^{C}(r) is the Coulomb potential (Jain and Shastry, 1979)
due to a uniform sphere of radius R_{C} and is given by
NUMERICAL CALCULATIONS AND RESULTS
To show, how sensitivity the αtransfer differential crosssection
to the optical potential, we studied the effect of the optical potentials
as follow. The differential cross sections have been numerically carried
out for ^{24}Mg(^{16}O, ^{ 12}C)^{ 28}Si
reactions at 27.8, 36.2 MeV and ^{28}Si(^{16}O, ^{12}C)^{
32}S reactions at 26.2 MeV. In the first calculations, the optical
potential is assumed to have a real and imaginary Gobbi potential plus
Coulomb potential. In the second calculation we used the real and imaginary
part is Vandenbosch potential plus Coulomb potential. In both calculations
the boundstate wavefunctions between the particles i and j in the initial
and final channels are described by a harmonicoscillator function (Linhua
and Guozhu, 1988), which is given by
Where α_{i} is the oscillator length parameter. Throughout
this analysis (Table 2), the particlenucleus interactions
are taken to have a doubleGaussian potential (Changun and Pingzhi, 1988)
Where V_{Ri}>0 and V_{Ai}<0 are the strengths of
the repulsive and attractive terms respectively, while α_{Ri}
and α_{Ai} are their decay factors. The realistic finiterange
of the reduced form factor is given as
Where
Table 1: 
Optical potential parameters used in the DWBA calculations 

The quantity Q includes the dynamical energies together with both of
the nuclear interaction and nucleusnucleus optical potentials. Therefore,
the transition matrix element T_{fi} for heavyion transfer reaction
T(A,X)R with a transition particle C is expressed to have the form
Where, S(l, j) and S*(l', j') are the spectroscopic factors in the initial
and final channels, respectively. J_{i} and μ_{i}
are the respective spin angular momentum of the particle i and its magnetic
projection on the zcomponent. T^{ll'} is the reduced transition
matrix element, which is given by
Where and are the ingoing and outgoing distorted wave functions, while,
φ^{l'j'} stands for the wavefunction which describes the
bound state of the residual nuclei R. However, the differential cross
section for heavy ion reaction with particle transfer is described by
a clear form, which is given by
Where K_{i} and K_{f} are the wavevectors for the initial
and final channels, respectively. Where the bound state wavefunctions
for both initial and final channels are expressed to have a harmonicoscillator
wavefunctions, which is given by Eq. (3) with parameters
determined to reproduce the particleparticle binding energies. In general,
the present spectroscopic factor is extracted from the reaction, that
is
Where N is normalization factor for the reaction
Table 2: 
Parameters of gaussian potential 


Fig. 1: 
The differential crosssection of the ^{24}Mg
(^{16}O,^{ 12}C)^{ 28}Si αtransfer reaction
at 27.8 Mev incident energy leading to 0.0 ^{28}Si excited
state. The solid curve (Vandenbosch potential) and dashed curve (Gobbi
potential) are the present calculation. The dotted line is the previous
work taken from reference (Kurath, 1973) and the dots are the experimental
data taken from reference (Guozhu et al., 1984; Paul et
al., 1980) 
The parameters of the optical potential are taken as those used in the
earlier calculations listed in Table 1. These parameters
are found to reproduce the forward angles data reasonably well but they
don’t fit the data at large angles. Therefore, the present optical
potential obtains the best fit to the data. The results obtained for the
differential crosssections are shown in Fig. 13
by the solid and dash lines are compared with previous calculations, shown
by the dotted lines and experimental data points.
Generally, it is found that the present work is good over the entire
angular range as shown in the Fig. 1 and 2,
but the predictions are low in magnitude at large angles as seen in Fig.
1 and 2. In Fig. 3, the use of
Gobbi potential gives predictions low at large angles, but the use of
Vandenbosch potential gives a good result at large angles. Since, the
use of Gobbi and Vandenbosch potential leads to a reasonable results better
than WoodsSaxon potential at large angles.

Fig. 2: 
The differential crosssection of the ^{24}Mg(^{16}O,^{
12}C)^{ 28}Si α transfer reaction at 36.2 Mev incident
energy leading to 0.0 ^{28}Si excited state . The solid curve
(Vandenbosch potential) and dashed curve (Gobbi potential) are the
present calculation. The dotted line is the previous work taken from
(Kurath, 1973) and the dots are the experimental data taken from reference
(Sanders et al., 1985) 

Fig. 3: 
The differential crosssection of the ^{28}Si
(^{16}O,^{ 12}C)^{ 32}S αtransfer reaction
at 26.23 Mev incident energy leading to 0.0 ^{32}S excited
state. The solid curve (Vandenbosch potential) and dashed curve (Gobbi
potential) are the present calculation. The dotted line is the previous
work taken from reference (Kurath, 1973) and the dots are the experimental
data taken from reference (Linhua et al., 1985) 
Table 3: 
Spectroscopic factors 

DISCUSSION
In this study, the differential cross sections of ^{24}Mg(^{16}O,
^{12}C)^{ 28}Si and ^{28}Si(^{16}O, ^{12}C)^{
32}S reactions have been calculated reasonably well using simple
onestep DWBA calculations. Then data are analyzed in terms of different
optical models. As shown in Fig. 13,
it is clear that the present optical potential gives a better data fit
than the other optical potentials. From Fig. 1, it can
be seen that the data of ^{24}Mg(^{16}O, ^{12}C)^{
28}Si at large angles with Gobbi or Vandenbosch potential at 27.8
MeV incident energy is noticeably nearly good and significantly better
than the previous work. But when the incident energy is 36.2 MeV, as in
Fig. 2, both Gobbi and Vandenbosch potential data deviate
away at large angles compared with Fig. 1. This deviation
stands more clearly when using Vandenbosch. In Fig. 3,
shows the data of ^{28}Si(^{16}O, ^{12}C)^{
32}S at 26.23 , it is clear that at large angles the use of Gobbi
potential was poorer and not significantly different from those shown
by the previous research, but the solid line within Vandenbosch potential
is better than both Gobbi and the previous work.
CONCLUSION
In conclusion, the present DWBA calculations show satisfactory fit to
αtransfer data. Since the calculated cross sections don't change
appreciably with the potential type in forward regions. Meanwhile, at
the large angles data are extremely sensitive to the potential in use.
In addition, the present work the use of either Gobbi or Vandenbosch potential
is better than the previous research. The usage of Vandenbosch potential
gives better fit than the use of Gobbi potential. Finally, the spectroscopic
factor is shown in Table 3.

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