# Study of elastic and inelastic scattering of 7Be + 12C at 35 MeV

K. Kundalia1<EMAIL_ADDRESS>D. Gupta1<EMAIL_ADDRESS>Sk M. Ali1
Swapan K Saha1 Former faculty  O. Tengblad2 J.D. Ovejas2 A. Perea2 I. Martel3
J. Cederkall4 J. Park4 Present Address: Center for Exotic Nuclear Studies,
Institute for Basic Science, 34126 Daejeon, South Korea  S. Szwec5,6 A. M.
Moro7,8 1Department of Physics, Bose Institute, 93/1 APC Road, Kolkata 700009,
India  2Instituto de Estructura de la Materia $-$ CSIC, Serrano 113 bis,
ES-28006 Madrid, Spain  3University of Huelva, Av. Fuerzas Armadas s/n. Campus
“El Carmen”, 21007, Huelva, Spain  4Department of Physics, Lund University,
Box 118, SE-221 00 Lund, Sweden  5Accelerator Laboratory, Department of
Physics, University of Jyv$\ddot{a}$skyl$\ddot{a}$, FI-40014
Jyv$\ddot{a}$skyl$\ddot{a}$, Finland  6Helsinki Institute of Physics,
University of Helsinki, FIN-00014 Helsinki, Finland  7Departamento de Física
Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla,
Apartado 1065, E-41080 Sevilla, Spain  8Instituto Interuniversitario Carlos I
de Física Teórica y Computacional (iC1), Apdo. 1065, E-41080 Sevilla, Spain

###### Abstract

The elastic and inelastic scattering of 7Be from 12C have been measured at an
incident energy of 35 MeV. The inelastic scattering leading to the 4.439 MeV
excited state of 12C has been measured for the first time. The experimental
data cover an angular range of $\theta_{cm}$ = 15∘-120∘. Optical model
analyses were carried out with Woods-Saxon and double-folding potential using
the density dependent M3Y (DDM3Y) effective interaction. The microscopic
analysis of the elastic data indicates breakup channel coupling effect. A
coupled-channel analysis of the inelastic scattering, based on collective form
factors, show that mutual excitation of both 7Be and 12C is significantly
smaller than the single excitation of 12C. The larger deformation length
obtained from the DWBA analysis could be explained by including the excitation
of 7Be in a coupled-channel analysis. The breakup cross section of 7Be is
estimated to be less than 10$\%$ of the reaction cross section. The intrinsic
deformation length obtained for the 12C∗ (4.439 MeV) state is $\delta_{2}$ =
1.37 fm. The total reaction cross section deduced from the analysis agrees
very well with Wong’s calculations for similar weakly bound light nuclei on
12C target.

Elastic and Inelastic scattering, Folding potential, Total reaction cross
section

###### pacs:

25.45.De, 25.60.-t, 25.60.Dz

##  I Introduction

The elastic scattering of light stable and unstable nuclei at low energies
(near or above the Coulomb barrier) is an important tool to study the change
in properties of nuclei as we move towards the drip lines BA11 ; KO16 . The
breakup and transfer reaction channels involving such nuclei significantly
impact the elastic scattering. This is even more evident, when there is low
binding and cluster structure of such nuclei. The loosely bound stable Lithium
isotopes 6Li and 7Li have cluster structures and are well known for their
breakup channels BA86 ; SH81 ; GU99 ; GU01 . Their breakup thresholds are
1.474 and 2.467 MeV respectively SE88 . The 7Be nucleus has a similar cluster
structure and a breakup threshold of 1.59 MeV. It is radioactive with a half-
life of 53.22 days TI02 . One can thus expect similar large breakup cross-
sections in case of 7Be. However, previous elastic scattering studies with 7Be
seem to indicate weak breakup couplings at low energy ($<$ 10 MeV/u) and the
breakup threshold anomaly does not apply for reactions with 7Be KO16 . The 7Be
nucleus behaves more like 6Li in contrast to 7Li. In addition, for reactions
with a light target like 12C, contribution of nuclear breakup is expected to
be more important than Coulomb breakup. The work of Amro et al. AM07
involving 7Be + 12C at 34 MeV show that the 7Be-induced $\alpha$-transfer
reactions are more predominant as compared to breakup. Low breakup cross
section was also indicated for a 7Be + 58Ni measurement at 21.5 MeV MA15 .
Since large breakup yield is absent, such reactions can be used as important
tools to study high-excitation $\alpha$-clustering states in the residual
nuclei in the context of helium-burning process in nuclear astrophysics AM07 .
The starting point of such a study is the availability of a reliable set of
Optical Model Potential (OMP) parameters. In the 7Be + 12C work AM07 , the
authors recognized the uncertainty in the OMP due to the limited scattering
data and therefore they planned further experiments to extract a dedicated set
of reliable parameters.

The present work includes the measurement of the elastic scattering angular
distribution of 7Be + 12C at 35 MeV over a large angular range. The OMP
deduced from the data would be very useful for the analysis of existing and
future low energy experiments involving both 7Be and 8Be. The limitations of
the earlier transfer reaction analysis AM07  were mainly due to the
unavailability of such elastic scattering data. In addition, the inelastic
scattering to the 4.439 MeV excited state of 12C is measured for the first
time and analyzed in the distorted-wave Born approximation (DWBA) and coupled-
channel (CC) formalisms. Both phenomenological and microscopic analysis in a
double-folding model have been carried out and the effects of breakup and
other channels on the elastic scattering are also studied.

The organization of the paper is as follows. In Sec II, the experimental
details are described. The results part in Sec. III includes the
phenomenological and microscopic analysis of the elastic and inelastic data in
DWBA and CC formalism. The framework for the microscopic analysis using a
DDM3Y interaction is also described. At the end, the total reaction cross
section is deduced. In Sec. IV, conclusions and outlook of the present work
are discussed.

##  II Experiment

The experiment was carried out at the CERN HIE-ISOLDE HIE  radioactive ion
beam facility, utilizing the Scattering Experiments Chamber (SEC) SEC  at the
XT03 beamline. The experiment took place at the end of the running period of
CERN. A uranium carbide target ST19  was irradiated with 0.37$\mu$A of 1.4 GeV
protons from the PS-booster offline for 3 days. The activated target was then
mounted on the GPS target station and heated up during the present experiment.
The 7Be was extracted using the RILIS RILIS  laser ion-source, and accelerated
using the HIE-ISOLDE post accelerator. The stable beam contaminants produced,
when the residual gas in the REX-EBIS charge breeder is ionized REX , are
mainly 14N4+ and less abundant 21Ne6+, having the same A/q as 7Be2+ for which
the machine was set up. A stripping foil and a dipole before the experimental
station was used to clean the beam by further stripping to 7Be4+, making
possible to separate the different components and uniquely select the 7Be
beam. The potential isobaric contaminant coming from the target is 7Li. Any
7Li that could have made it into the charge breeder and been accelerated as
7Li2+ was fully stripped in the foil and could not have passed the dipole. The
35 MeV 7Be beam of resolution $\sim$ 168 keV and intensity $\sim$ 5 $\times$
105 pps was incident on a 15 $\mu$m thick CD2 target. We also used a CH2
target of thickness 15 $\mu$m and a 208Pb target of thickness 1 mg/cm2 for
background measurements and normalization respectively.

Figure 1: The experimental setup inside the Scattering Experiments Chamber at
the XT03 beamline SEC .

The charged particles emitted from the reaction were detected by a set of 5
double-sided 16$\times$16 Micron W1 silicon strip detectors (DSSD) of
thickness 60 $\mu$m ($\Delta E$) each, backed by MSX25 unsegmented silicon pad
detectors of thickness 1500 $\mu$m ($E$). These detectors were placed in a
pentagon geometry covering $\theta=40^{\circ}-80^{\circ}$ in lab for charged
particle detection. The forward angles from 8∘ $-$ 25∘ were covered by a 24
rings $\times$ 32 sectors, Micron S3 annular detector of thickness 1000
$\mu$m. The back angles from 127∘ $-$ 165∘ were covered by two Micron BB7
32$\times$32 DSSDs of thickness 60 $\mu$m and 140 $\mu$m, backed by MSX40
unsegmented silicon pad detectors of thickness 1500 $\mu$m. The experimental
setup inside the SEC is shown in Fig. 1. The measurements were made at angular
intervals of about 1∘ in S3 range and 2∘ in the pentagon DSSD range.

The energy calibration of the detectors was done using a
148Gd$-^{239}$Pu$-^{241}$Am$-^{244}$Cm mixed $\alpha$ source. The dynamic
range in the forward detector S3 was much higher. Thus, for calibration at
higher energies, we used the elastic peaks from the Rutherford scattering of
7Be and 12C beams at ${5}$ MeV/u and ${5.15}$ MeV/u respectively on a 208Pb
target. The grazing angle of 7Be + 208Pb at 35 MeV is around 100∘. Thus, the
experimental data in one of the forward rings of the annular detector were
normalized to Rutherford cross sections with 208Pb, and that ring was used as
a monitor detector. We took very short consecutive runs of 7Be on 208Pb and
CD2 targets where the fluctuation in beam intensity was within 10$\%$. In
addition to the statistical uncertainty in the yields, the present data
include 10$\%$ error due to beam intensity and 10$\%$ error due to target
thickness.

Figure 2: Energy spectrum for elastic and inelastic scattering of 7Be + 12C at
35 MeV. Panel (a) shows elastic 7Be scattered from 12C and d in one of the
rings of S3, at $\theta=12^{\circ}$. Panel (b) shows elastic as well as
inelastic peak corresponding to 4.439 MeV state of 12C in one pentagon DSSD,
at $\theta=45^{\circ}$. The bound excited state of 7Be at E${}_{x}=0.429$ MeV
(1/2-) could not be separated from the ground state, implying the data to be
quasi-elastic.

##  III Results

The elastic peak from 7Be + 12C scattering is distinctly visible in each ring
of the forward annular detector. Fig. 2(a) shows the energy spectrum at
$\theta=12^{\circ}$, delineating the prominent elastic peak. The elastic 7Be
scattered from deuterons in the CD2 target is also seen as a small broad peak
in the spectrum. In the pentagon DSSDs, the scattered 7Be deposits all its
energy in the $\Delta$E detectors of the telescopes. Fig. 2(b) is the energy
spectrum at $\theta=45^{\circ}$ showing the 7Be elastic peak as well as the
inelastic scattering to the 4.439 MeV state of 12C. The ground state peak is
separated from the 2+ inelastic peak as seen in Fig. 2(b). However, the bound
excited state of 7Be at E${}_{x}=0.429$ MeV (1/2-) could not be separated from
the ground state. Thus the present data correspond to quasi-elastic scattering
of 7Be + 12C at 35 MeV. In this context, it may be noted that the inelastic
scattering to the 0.429 MeV state of 7Be is expected to be negligible in
comparison to the 7Be + 12C elastic scattering BA11 ; PE95 . In the forward
angles 8∘ $-$ 25∘, due to several overlapping channels and non-availability of
particle identification, inelastic scattering cross sections to the 4.439 MeV
state of 12C could not be extracted. At the back angles 127∘ $-$ 165∘, the
elastic and inelastic data could not be separated due to small differences in
energies as well as overlap with the (d,p) data. Hence, these data are not
included in the present work.

The angular distribution of 7Be + 12C elastic scattering at 35 MeV is shown in
Fig. 3. For comparison, the data of Amro et al. AM07  at 34 MeV is also
plotted in the figure. If we compare our data of much better energy and
angular resolution, with Amro’s data at the forward angles, we observe that
the elastic data at $\sim$ 18∘ agree with our data within error bars but for
the other two angles, it is somewhat lower in value. At around 72∘, Amro’s
data gives a higher value of cross section than our data. This discrepancy can
be attributed to uncertainties in data normalization arising from the large
beam energy spread ($\sim$ 1 MeV) and, more importantly to the angular
divergence at target position ($\sim$ 6∘) in AM07 . Their limited angular
range of scattering data resulted in difficulty to extract reliable OMP for
subsequent study of the transfer reactions. Analysis of the $\alpha$\- and
n-transfer reactions from the 12C and d in the CD2 target from the present
experiment will be presented elsewhere.

Optical model analyses of the data were carried out with the code FRESCO TH88
. First, the data were analyzed using volume type Woods-Saxon potentials and
Coulomb potential due to uniformly charged spheres ($r_{C}=1.25$ fm). The
potential parameters are given in Table 1. The starting point of the
phenomenological analysis was the OMP parameters from 7Li + 12C elastic
scattering at 34 MeV CO86 . The dotted curve in Fig. 3 corresponds to the
phenomenological fit to the present data. The phenomenological fit agrees
satisfactorily with the data in the whole angular range, except around 75∘ and
angles greater than 100∘. The fit also agrees with the data of Amro et al.
AM07  at 18∘ and 45∘ and explains overall the nature of their data except at
72∘. It is to be noted that their data point at 72∘ also did not agree with
their optical model calculations AM07 .

To study the data further and have an estimate of the breakup effects on the
elastic scattering, we carried out a microscopic analysis of the data. Here,
the two-body potential $V(r)$ is generated in a double folding model using
densities of the 7Be and 12C along with the density dependent M3Y (DDM3Y)
effective interaction DU14 ; BA03 . The densities of 7Be and 12C used in the
present work are obtained from variational Monte Carlo calculations using the
Argonne v18 two-nucleon and Urbana X three-nucleon potentials (AV18+UX) WI91 .
Earlier, the DDM3Y effective interaction has been successfully used to
describe nuclear matter BA04 , radioactivity BA05 , proton scattering GU06  as
well as resonances in unstable DU14  and unbound nuclei DU18 .

Figure 3: Elastic scattering angular distribution of 7Be + 12C at E = 35 MeV.
The microscopic (phenomenological) fits to the data are given by the dashed
(dotted) curves. The coupled-channel calculations are given by the solid
curve. The data of Amro et al. AM07  are also shown for comparison. The bound
excited state of 7Be at 0.429 MeV could not be separated from the ground
state, implying the data to be quasi-elastic. Table 1: Optical model potential
parameters for elastic scattering of 7Be + 12C at E = 35 MeV, $r_{C}=1.25$ fm;
$R_{x}=r_{x}(A_{P}^{\frac{1}{3}}+A_{T}^{\frac{1}{3}})$ $E$ | $V_{r}$ | $r_{r}$ | $a_{r}$ | $W_{V}$ | $r_{i}$ | $a_{i}$ | $\chi^{2}_{el}/N$ |
---|---|---|---|---|---|---|---|---
(MeV) | (MeV) | (fm) | (fm) | (MeV) | (fm) | (fm) |  |
35 | 142.96 | 0.702 | 0.617 | 9.95 | 1.041 | 1.079 | 2.6 |
Table 2: Renormalization factors ($N_{R}$, $N_{I}$) of DDM3Y folded potential
for 7Be + 12C scattering at E = 35 MeV. Excited state energy ($E_{x}$),
angular momentum transfer ($l$), deformation length ($\delta_{2}$),
deformation parameter ($\beta_{2}$) and $\chi^{2}$/N from best fits to the
elastic and inelastic scattering data. $\delta_{2}=\beta_{2}R\mbox{, where
}R=1.2\times 12^{\frac{1}{3}}$

.  Nucleus Calculation $E$ $E_{x}$ $N_{R}$ $N_{I}$ $l$ $\delta_{2}$
$\beta_{2}$ $\chi^{2}_{el}/N$ $\chi^{2}_{inel}/N$ (MeV) (MeV) (fm) 12C DWBA 35
4.439 0.68 0.30 2 1.83 0.67 9.3 4.5 12C CC 35 4.439 1.00 0.20 2 1.37 0.49 8.0
4.8

The double-folded potential between two nuclei is given by

$V(R)=\int{\rho_{1}(\bm{r_{1}})\rho_{2}(\bm{r_{2}})v(\bm{s},\rho_{1},\rho_{2},E)d^{3}r_{1}d^{3}r_{2}}$
(1)

where $\rho_{i}$, $i=1,2,$ is the nucleon point-density distribution for the
projectile and the target, $R$ is the separation between them,
$\bm{s}=\bm{R}-\bm{r_{1}}-\bm{r_{2}}$, and $v$ is the density- and energy-
dependent effective nucleon-nucleon interaction. Assuming that the density and
the radial dependence of $v$ can be separated, one can write BA03

$v(s,\rho_{1},\rho_{2},E)=t^{M3Y}(s,E)g(\rho_{1},\rho_{2},E)$ (2)

The density-independent part $t^{M3Y}(s,E)$ contains the M3Y interaction and
the zero-range pseudo potential,

$t^{M3Y}(s,E)=7999\frac{e^{-4s}}{4s}-2134\frac{e^{-2.5s}}{2.5s}+J_{00}(E)\delta(s)$
(3)

with

$J_{00}(E)=-276(1-0.005E/A)$ (4)

The density-dependent part $g(\rho_{1},\rho_{2},E)$ is as follows,

$g(\rho_{1},\rho_{2},E)=C(1-b(E)\rho_{1}^{2/3})(1-b(E)\rho_{2}^{2/3})$ (5)

Here $C$ has been taken as 1.0. The other constant of the interaction $b$ when
used in single folding model description, is determined by nuclear matter
calculations BA04  as 1.624 fm2. This constant has also been used in
scattering GU06  and bound state studies DU14 ; DU18 .

Figure 4: Inelastic scattering angular distribution of 7Be + 12C to the 4.439
MeV (2+) state at E = 35 MeV. The DWBA calculations using microscopic
potentials are given by the dashed curve and includes the sum of single and
mutual excitations. The CC calculations are given by the dash-dot (dash-dot-
dotted) curves for single (mutual) excitations respectively. The sum of the
two contributions are shown by the solid curve.

In the present microscopic analysis, both the real ($V$) and volume imaginary
($W$) parts of the folded nuclear potentials are assumed to have the same
shape, i.e. $V_{\rm micro}(r)~{}=~{}V+iW$ = ($N_{\rm R}$ \+ $iN_{\rm
I}$)$V$($R$) where, $N_{\rm R}$ and $N_{\rm I}$ are the renormalization
factors for real and imaginary parts respectively, obtained by fitting the
elastic scattering data. The microscopic fit using FRESCO is shown by the
dashed curve in Fig. 3. A satisfactory fit is obtained for an $N_{\rm R}$
value of 0.68 and an $N_{\rm I}$ value of 0.30 (Table 2). The $N_{\rm R}$
value of less than unity suggests coupling of the breakup channel to elastic
scattering SA79 ; SA83 . The $N_{\rm I}$ value is attributed to the absorption
of flux due to inelastic excitation, breakup or other reaction channels. The
microscopic fit shows a more diffractive elastic angular distribution than
observed, particularly at $\sim$ 35∘. However, it provides a better
description of the data at larger angles compared to the phenomenological fit.
The microscopic calculation also explains the nature of the data of Amro et
al. AM07  better than the phenomenological calculation.

To better understand the effects of nonelastic channels, coupled-channel (CC)
calculations were also carried out. In addition to the ground state of the
projectile and target nuclei, some low-lying excited states of these nuclei
were included in the calculations. In particular, the modelspace included the
7Be($1/2^{-}$) bound excited state at $E_{x}=0.429$ MeV, the narrow $7/2^{-}$
low-lying 7Be resonance at $E_{x}=4.57$ MeV and the $2^{+}$ state at
$E_{x}=4.439$ MeV of 12C. It is worth noting that the inclusion of the
inelastic couplings will also affect the elastic scattering cross section. In
$\chi^{2}$-fitting, the $N_{R}$ and $N_{I}$ of the bare projectile-target
potential were taken as adjustable parameters to best reproduce the measured
elastic scattering data. This resulted in $N_{R}=1$ and $N_{I}=0.2$ (Table 2).
The reduced value of $N_{I}$ with respect to that obtained above is consistent
with the explicit inclusion of channels removing flux from the elastic
channel. The dip around 30∘ is much reduced in CC calculation and is shown by
the solid curve in Fig. 3.

The inelastic scattering angular distribution of 7Be + 12C to the 4.439 MeV
(2+) state, measured for the first time, is shown in Fig. 4. It may be noted
that, in this angular range, the inelastic cross section is of comparable
magnitude to the elastic cross section. Since the grazing angle is $\sim$ 8∘,
there is negligible contribution of Coulomb excitation over the angular range
of the inelastic scattering data. The nucleus-nucleus optical potentials
obtained from fitting the elastic scattering data were used to generate the
distorted waves for the analysis of inelastic scattering in DWBA formalism
using FRESCO TH88 . The experimental resolution of the present work did not
allow the separation of the 7Be${}_{\text{g.s.}}$+12C4.439 channel from the
mutual excitation 7Be0.429+12C4.439 channel. Thus, to include the 7Be0.429
excitation in the DWBA analysis, contribution from these two channels were
calculated and added together (sum), given by the dashed curve in Fig. 4. The
collective rotor model is used to obtain the transition form factors. The
deformation length $\delta_{2}$ is obtained by fitting the inelastic
scattering angular distributions. The microscopic analysis could explain the
inelastic data very well, while the phenomenological analysis does not produce
a satisfactory fit. Table 2 gives the excited state energy ($E_{x}$), angular
momentum transfer ($l$), renormalization factors ($N_{R},N_{I}$), deformation
length ($\delta_{2}$), deformation parameter ($\beta_{2}$), and $\chi^{2}$/N
for the analyses. From DWBA analysis using microscopic potentials, the
$\delta_{2}$ obtained is 1.83 fm, as compared to the reported values of $\sim$
1.4 fm CO86 ; ZE80 .

To explain the larger deformation length, we extended the CC calculations
including the excitation of the 7Be nucleus. As mentioned earlier, the three
included states of 7Be are considered members of a $K=1/2^{-}$ rotational
band. They are coupled together by deforming the projectile-target microscopic
folding potential using a quadrupole deformation length $\delta_{2}=2$ fm
adopted from KI20 . Likewise, the ground and first $2^{+}$ state of 12C are
considered members of a $K=0$ rotational band and coupled together by
deforming the projectile-target potential with a deformation length of
$\delta_{2}$, obtained from best fit to the present data. Couplings between
the included states were considered to all orders, thus performing a full
coupled-channels calculation. The CC calculations are compared to the
inelastic data in Fig. 4. For a meaningful comparison, the single and mutual
excitations have been considered in the calculations and added together (sum),
shown by the solid curve. It becomes apparent from Fig. 4 that the mutual
excitation (dash-dot-dotted curve) is significantly smaller than the single
excitation (dash-dot curve) but its inclusion improves the agreement with the
data. There is slight underestimation, which might be due to the simplified
rotor model assumed for the 7Be and 12C nuclei. From the best fit to the
present data, we get $\delta_{2}$ = 1.37 fm (Table 2). A 10$\%$ variation of
$\chi^{2}_{inel}$/N results in a variation of 0.13 fm in the value of
$\delta_{2}$ in agreement to the existing values CO86 ; ZE80 . This results in
a deformation parameter $\beta_{2}$ of 0.49 with a variation of 0.05.

In an alternative approach, one may treat the reaction in an extended three-
body model, comprising of the two-body projectile ($\alpha$+3He) plus the
target, similar to that used in the continuum discretized coupled-channels
(CDCC) calculations HA22 . In this model, target excitations are included by
means of deformed 3He+12C and 4He+12C potentials. In this way, the projectile
excitation (and breakup) would be treated within the cluster model whereas
target excitations would be treated in the collective model. These extended
CDCC calculations were initially proposed and implemented in YA86  and have
been recently revisited and applied to several systems CH15 ; GO17 . The
application to the present reaction is ongoing and the results will be
presented elsewhere.

To investigate further the dynamics of 7Be, we deduced the total reaction
cross section from the elastic scattering analysis. As suggested by Kolata
KO09  and Aguilera et al. AG11 , the reduced total reaction cross section of
light stable and unstable nuclei on 12C can be compared with that given by
Wong WO73 ,

$\sigma_{red}=\frac{\epsilon_{0}r_{0}^{2}}{2E_{red}}\ln\Bigg{[}1+\exp\Bigg{(}\frac{2\pi(E_{red}-V_{red})}{\epsilon_{0}}\Bigg{)}\Bigg{]}$
(6)

where the cross section is in fm2,
$\epsilon_{0}=\hbar\omega_{0}\frac{(A^{\frac{1}{3}}_{P}+A^{\frac{1}{3}}_{T})}{Z_{P}Z_{T}}$
and
$V_{red}=V_{0}\frac{(A^{\frac{1}{3}}_{P}+A^{\frac{1}{3}}_{T})}{Z_{P}Z_{T}}$
are denoted as the Wong-model parameters and the reduced energy is given by

$E_{red}=E_{cm}\frac{(A_{P}^{\frac{1}{3}}+A_{T}^{\frac{1}{3}})}{Z_{P}Z_{T}}.$
(7)

These parameters were obtained in BA11  by fitting the data for 6He, 6Li, 7Li,
7Be, 8Li, 8B, 9Be and 11Be. The parameters were found to be $V_{red}=0.64(3)$
MeV, $r_{0}=1.73(2)$ fm and $\epsilon_{0}=0.43$ MeV. In particular, the value
$r_{0}=1.73(2)$ fm is significantly larger than the typical values of
$1.4-1.5$ fm. The reduced total reaction cross section for 6,7Li and 7Be + 12C
obtained in BA11 , along with the fitted curve is plotted in Fig. 5. The
calculated 7Be total reaction cross section at 35 MeV for CC calculations
using microscopic potentials is 1503 mb and the reduced total reaction cross
section is 85 mb. The calculated value agrees very well with Wong’s
calculation on the existing 6,7Li and 7Be data. Preliminary CDCC calculations
show that for 7Be, the breakup and inelastic cross section to 7Be bound
excited state are respectively 9.12$\%$ and 0.67$\%$ of the reaction cross
section. The absorption cross section is 90.2$\%$ of the reaction cross
section.

Figure 5: The reduced reaction cross sections for the 7Be + 12C system
obtained in the present work along with reduced reaction cross sections of the
lithium isotopes on 12C. The solid curve represents Wong’s expression in Eqn.
6.

##  IV Conclusion

The elastic and inelastic scattering of 7Be + 12C at 35 MeV is studied in the
present work. The experimental data as well as the deduced optical model
potential parameters can be useful for studies on transfer reactions involving
7Be and 8Be at these energies, particularly in relation to nuclear
astrophysics. This work also reports the first inelastic scattering
measurement of 7Be + 12C to the 4.439 MeV excited state of 12C. To have an
estimate of the influence of other channels like breakup on the elastic
scattering, we also carried out a coupled channel analysis of the data. The
microscopic calculation for the elastic scattering gives better fit,
particularly at larger angles as compared to the phenomenological fit with
Wood-Saxon potentials. The $N_{R}$ value of less than unity in the microscopic
fit indicates the presence of breakup channel coupling effects. Coupled-
channel calculations improve on the fits and the value of $N_{R}$ becomes
compatible with 1. For inelastic scattering, mutual excitation of both 7Be and
12C is significantly smaller than the single excitation of 12C but its
inclusion improves the agreement of the calculations with the inelastic data.
The larger deformation length obtained from the DWBA analysis could be
explained by including the excitation of 7Be in a coupled-channel analysis.
The breakup cross section of 7Be is estimated to be less than 10$\%$ of the
reaction cross section. The reduced total reaction cross section from the
present work, agrees very well with Wong’s calculation on existing 6,7Li and
7Be scattering data. The present study is very important to further understand
the $\alpha$-cluster transfer reactions in nuclear astrophysics.

Acknowledgement

The authors thank the ISOLDE engineers in charge, RILIS team and Target Group
at CERN for their support. D. Gupta acknowledges research funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement no. 654002 (ENSAR2) and ISRO, Government of India under grant no.
ISRO/RES/2/378/15$-$16\. O. Tengblad would like to acknowledge the support by
the Spanish Funding Agency (AEI / FEDER, EU) under the project
PID2019-104390GB-I00. I. Martel would like to acknowledge the support by the
Ministry of Science, Innovation and Universities of Spain (Grant No.
PGC2018-095640-B-I00). J. Cederkall acknowledges grants from the Swedish
Research Council (VR) under contract numbers VR-2017-00637 and VR-2017-03986
as well as grants from the Royal Physiographical Society. J. Park would like
to acknowledge the support by Institute for Basic Science (IBS-R031-D1). S.
Szwec acknowledges support by the Academy of Finland (Grant No. 307685).
A.M.M. is supported by the I+D+i project PID2020-114687GB-I00 funded by
MCIN/AEI/10.13039/501100011033, by the grant Group FQM-160 and by project
P20_01247, funded by the Consejería de Economía, Conocimiento, Empresas y
Universidad, Junta de Andalucía (Spain) and by “ERDF A way of making Europe”.

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