INTRODUCTION
The Distorted Wave Born Approximation (DWBA) calculations have been widely
used in studying the direct nuclear reactions (AlFarra,
2003; Ass’ad and Ashour, 2007). For most cases the
DWBA is one step process (Mermaz et al., 1983).
Therefore, several descriptions have been introduced to explain the general
features of heavyion reactions (Bilwes et al.,
1987; Farra and Ass’ad, 2004). The prominent gross
structures of different transfer reactions have been analyzed in the whole angle
region with qualitative agreement using both of surface transparent ionion
potential and dynamic alpha particle transfer polarization potential (Filho
et al., 1989). In some heavyion reactions, such as ^{24}Mg(^{16}O,^{12}C)^{28}Si
oscillatory structure often appears in the whole angle region in the differential
cross section (Linhua et al., 1985). This anomalous
phenomenon, unexpected from ordinary DWBA theory, is explained by an alpha transfer
process between two unidentical nuclear cores. The folding model can well reproduce
the strong dependence of the total reaction cross section on the incident energy
(Yang et al., 2002). Both zerorange and finiterange
exchange parts of nucleonnucleon NN interactions have been considered in the
folding procedure (Zhang et al., 2009) to calculate
the fusion cross section of ^{ 16}O+^{208}Pb system. The differential
cross sections of ^{16}O^{16}O and ^{12}C^{12}C
system reactions (Jain and Shastry, 1979) have been
calculated in terms of DWBA calculations, using double folding potential, where
the nuclear part of the optical potential is quite sensitive to the shape of
the density distribution. Elastic scattering of the twoneutron halo nucleus,
^{ 6}He, on ^{12}C target at 38.3 and 41.6 MeV/nucleon has been
analyzed in the framework of the doublefolding optical model (ElAzab
et al., 2008). It has been shown that the obtained potential by folding
a nucleonnucleon interaction (Perez, 1973) into the
ion densities gives a good description of the real part of the optical potential
for ^{6}Li^{6}Li scattering between 9 and 16 MeV center of
mass.
In the present study, the differential cross section of heavyion reactions with αtransfer reactions have been calculated in term of one step DWBA calculations using folding potential. The calculated differential cross section are fitted with the experimental data to extract spectroscopic factors.
Nuclear optical potential [V_{opt}(r)]: Here, the differential
cross section for stripping reaction ^{16}O^{12}C system has
been evaluated in the framework of one step DWBA calculations. The optical potential
V_{opt}(r) may thus be written as:
V_{opt}(r) = V_{N}(r) + V_{C}(r) + V_{S.O.}(r)

(1) 
where, V_{N}(r) is the complex nuclear part of the potential, V_{C}(r)
is the electrostatic potential (ESP) between the interacting bodies and V_{S.O.}(r)
is a spinorbital interaction which is to be included where spinorbital is
important (Jain and Shastry, 1979).
Table 1:  Optical
potential parameters used in the DWBA calculations 

In this study, we shall consider the problem of evaluating the potentials
V_{N}(r) and V_{C}(r) for nucleusnucleus system and Coulomb
potential, respectively. The total optical potential is thus (Jain
and Shastry, 1979).
V_{opt}(r) = (V + iW) V_{N}(r)
+ V_{C}(r) 
(2) 
where, V and W are strength parameters of the real and imaginary parts, respectively
(Yosio and Taro, 1984), given by:
where, the parameters V_{o}, R_{v} and a_{v} are the strength, radius and diffuseness of the real potential, while the parameters W_{o}, R_{w} and a_{w} describe the imaginary part which are determined by fitting scattering reaction of the corresponding interaction of two heavyions.
The necessary parameters of the optical potential (Linhua
et al., 1985; Chengqun et al., 1988;
Guozhu et al., 1984) are shown in Table
1. The Coulomb potential due to a uniform charge sphere of radius R_{c
}is given by:
The interaction radii have the form:
where, r_{v} = r_{w} = 1.18 fm and r_{c} is shown in
Table 1.
where, V_{N} (r) is the nucleusnucleus potential reaction (Jain
and Shastry, 1979) and given by:
NUMERICAL CALCULATIONS AND RESULTS AND DISCUSSION
To show, how sensitivity of the folding optical potential effects the differential
cross section, we studied the effect of the folding optical potential as follows:
Table 2:  Parameters
of Gaussian potential 

The differential cross section has been numerically carried out for ^{ 24}Mg(^{16}O,^{
12}C)^{ 28}Si reaction at 27.8, 36.2 MeV, ^{16}O (^{16}O,^{
12}C)^{ 20}Ne reaction at 22.75 Mev and ^{28}Si (^{16}O,^{
12}C)^{ 32}S reaction at 26.23 Mev. The nuclear interactions describing
the particlenucleus bound states are represented by double Gaussian potentials
(Chengqun et al., 1988).
where, V_{Ri}>0 and V_{Ai}<0 are the strengths of the repulsive and attractive terms, respectively, while a_{Ri} and a_{Ai} are their decay factors. These parameters are shown in Table 2.
The boundstate wavefunctions between the particles i and j in the initial
and final channels are described by a harmonic oscillator function (Linhua
and Guozhu, 1988), which is given by:
where, a_{i} is the oscillator length parameter.
The differential cross section for the stripping reaction with particle transfer
is described by a clear form (AlFarra, 2003) which
is given by:
where, the μ’s and K’s are the reduced masses and asymptotic wave numbers and I_{i} is the total angular momentum of i^{th} particle. The postformulation DWBA transition amplitude has the form:
where, and
are
the distorted wave functions in the initial and final channels, respectively
and V_{ij} is the interaction potential between the particle i and j,
the index c refer to the transfer particle, while
is the optical potential generating the distorted waves.

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 dashed curve (folding
model) is the present calculation. The solid curve is (WS+JD). The dotted
line is the earlier study (WS+WS) and the dots are the experimental data
taken from reference ( Guozhu et al., 1984) 
As could be done in such calculations, we have used the differential cross
section without considering a spinorbit coupling term (Satehler,
1964). Generally, the present spectroscopic factor are extracted from the
relation:
where, N is the normalization factor for the reaction (AlFarra,
2003).
The parameters of the optical potential are taken as those used in the earlier
calculations shown 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 (folding model)
obtains the best fit to the data.

Fig. 2: 
The differential crosssection of the ^{24}Mg ( ^{16}O, ^{
12}C) ^{ 28}Si αtransfer reaction at 36.2 Mev incident
energy leading to 0.0 ^{ 28}Si excited state. The dashed curve (folding
model) is the present calculation. The solid curve is (WS+JD). The dotted
line is the earlier study (WS+WS) and the dots are the experimental data
taken from reference ( Sanders et al., 1985) 
The result obtained for the differential cross sections are shown in Fig.
14 by the dashed lines are compared with the previous
calculations dotted lines employing, the solid curves (Farra,
2003) who used real and imaginary Wood Saxon and Jdependent, respectively
(WS+JD) optical potentials, dotted line (Kurath, 1973),
who used the (WS+WS) optical and experimental data points. Generally, the present
calculations using folding model potential provide a substantially better description
of the phase and magnitude of the angular distributions than the previous calculations.
The result obtained for the angular distribution of ^{24}Mg(^{16}O,^{
12}C)^{28} at incident energy 27.8 and 36.2 MeV is shown in Fig.
1 and 2, respectively, where the dashed curve is the present
calculation (folding model) is compared with the previous (WS+JD) optical potential
(solid curve), (WS+WS) optical potential (dotted line) and the experimental
data dots (Guozhu et al., 1984) in Fig.
1 and (Chengqun et al., 1988) in Fig.
2.

Fig. 3: 
The differential crosssection of the ^{16}O( ^{16}O, ^{
12}C) ^{20}Ne αtransfer reaction at 22.75 Mev incident
energy leading to 0.0 ^{20}Ne excited state. The dashed curve (folding
model) is the present calculation. The solid curve is (WS+JD). The dots
are the experimental data taken from reference ( Chengqun
et al., 1988) 
DISCUSSION
In this study, the differential cross sections of ^{24}Mg (^{16}O,^{
12}C)Si^{ 28}, ^{16} O(^{16}O,^{ 12}C)^{20}Ne
and^{ 28}Si(^{16}O,^{ 12}C)^{32}S heavy ion
reactions with α transfer have been estimated reasonably well using simple
onestep DWBA calculations. The numerical calculations are carried out to find
the angular distributions of this reaction. As shown in Fig. 14,
it is clear that the present optical potential gives a better data fitting than
the other optical potentials. In Fig. 1 and 2,
it can be seen that the data of ^{24}Mg(^{16}O,^{ 12}C)Si^{28},
at incident 27.8 MeV the angular distribution, of our calculation gives better
fit than the (WS+SW) optical potential and less fitting compared to (WS+JD).
But at 36.2 MeV incident energy, the present optical potential is noticeably
nearly good and significantly better than the previous work in both forward
and backward angles. In Fig. 3, shows the data of ^{16}O
(^{16}O,^{ 12}C)^{ 20}Ne reaction at 22.75 Mev incident
energy. It is clear that using the folding model gives the same as the earlier
study, where the data are good at small angles and not fit at large angles.

Fig. 4: 
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 dashed curve (folding
model) is the present calculation. The solid curve is (WS+JD). The dotted
line is the earlier study (WS+WS) and the dots are the experimental data
taken from reference ( Linhua and Gouzhu, 1988) 
Table 3:  Extracted
spectroscopic factors 

Finally, the ^{28}Si (^{16}O,^{ 12}C)^{32}S
reactions, shown in Fig. 4, using folding model behaves well
for the forward angles as the same as the use of (WS+SW) and (WS+JD), but better
at large angles.
CONCLUSION
In conclusion, the present study show that the onestep DWBA calculations using folding model are found to be appropriate to reproduce the crosssections and capable of producing realistic predictions of the angular distribution at large angles region and better the earlier calculations. Finally, the spectroscopic factor is shown in Table 3.