Coupled reaction channels study of the ¹⁶O(d,⁶Li) reaction
Nuclear Science and Technology, Vol.10, No. 2 (2020), pp. 52-59
Coupled reaction channels study of the 16O(d,6Li) reaction
Do Cong Cuong, Nguyen Hoang Phuc, Nguyen Tri Toan Phuc
Center for Nuclear Physics, Institute for Nuclear Science and Technology.
179 Hoang Quoc Viet street, Cau Giay, Hanoii
Email: [email protected]; cuon[email protected]
(Received 21 March 2020, accepted 25 June 2020)
Abstract: The transfer 16O(d,6Li)12C reaction has been studied within the coupled reaction channels
(CRC) approach, inluding both the direct and indirect transfer processes. The obtained results show
an important contribution of the indirect transfer via the 2+ and 4+ states of 12C. The CRC results
show that the best-fit spectroscopic factors of 16O becomes smaller when the indirect transfer
processes are taken into account. The spectroscopic factors deduced from the present CRC analysis
of the 16O(d,6Li)12C reaction data measured at Ed=54.25 and 80 MeV are quite close to each other.
Keywords: coupled reaction channels calculations, transfer reactions, -cluster.
I. INTRODUCTION
[11-20], but the deduced S values are ranging
widely from about 0.3 to 1.0, depending on the
reaction mechanism and analysis method [11-
14, 16-19]. Thus, the spectroscopic factors of
16O still remain the research topic of different
nuclear structure and reaction studies.
The established cluster structure of the
12
excited states of C at the energies near the
decay threshold are of interest for both the
nuclear physics and astrophysics. For example,
12
the isoscalar 0+ excitation of C at 7.65 MeV,
2
Among various experiments, the
transfer reactions like (d,6Li), (t,7Li), (3He,7Be),
and ( 8Be) [11-15,20] were proven to be very
helpful for the determination of the
spectroscopic factors. The most important
inputs for the analysis of a transfer reaction in
either the distorted wave Born approximation
(DWBA) or the CRC formalism are the
spectroscopic factor S and the optical
potentials (OP) for both the entrance and exit
channels of the reaction. We note that a widely
adopted prescription for the DWBA or CRC
calculations of a transfer reaction is to use the
complex OP of a system of the two colliding
nuclei having masses similar to those in the
exit channel of the transfer reaction, at about
the same center-of-mass (c.m.) energy. The
uncertainty of the deduced S values remains,
however, significant, and the analysis of the
known as the Hoyle state, plays a vital role in
the stellar carbon synthesis. In general, the
cluster models, which describe the nuclear
wave functions in terms of the
particles
moving in the inter-cluster potential, not only
reporduce the main features of these excited
states but also show a significant fraction of the
cluster component in the ground state [1-6].
16
Although O nucleus in the ground state (g.s.)
is well known to be of the shell-model
structure, the cluster model calculations have
predicted the
spectroscopic factor S of
16
about 0.3 [1-6] for Og.s.. Such values of S
were also confirmed in the shell model
calculations [7-10], where the overlap of the
cluster configuration with the total g.s. wave
function is calculated exactly. Several
measurements have been performed to
16
determine the spectroscopic factor of Og.s.
DO CONG CUONG et al.
same transfer reaction happened to deliver
are the core-core OP and
OP of the exit partition. In our CRC
calculation, the nonorthogonality correction
and complex remnant term are properly taken
into account.
different spectroscopic factors because of the
different OP used for the exit channel.
Therefore, it is highly desirable to have the OP
for both the entrance and exit channels
determined from the optical model (OM)
analysis of the elastic scattering of the same
systems at the nearby energies, to accurately
the 16O
nucleus, the 1 s state is assumed for the internal
state of the
cluster. The number
NL of
determine the corresponding
spectroscopic
of node N
factors. In the present work, such a procedure
is carried out to determine the spectroscopic
factors of 16O from the CRC analysis of the
16O(d,6Li) reaction.
the
12C cluster configuration is given by the
Wildermuth condition:
(3)
Where and
are the orbital angular
II. CRC FORMALISM
momentum and number of node, respectively,
We give here a brief description of the
coupled reaction channels method used in our
calculation using the code Fresco written by
Thompson [21]. In general, the cross section of
the transfer reaction is given by the solution
of the following coupled channel (CC)
equations, where the relative wave function of
the channel is determined in the post form as
[21,22].
is generated by solving a two-
NL
body Schrödinger equation with the
potential in Woods-Saxon form. The depth of
this potential is adjusted to reproduce the
experimental separation energy of 16O
With
of 16O in the ground state,
are the excitation energies of C and
16O nuclei, respectively.
and
12
(1)
One essential input of the CRC
calculation for transfer reaction is the
With
,
being the incoming and
outgoing partitions, respectively.
are the diagonal optical potentials,
and
and
factor defined as
is used to
construction the overlap function
are the relative-motion wave functions of the
corresponding channels. For the
.
(5)
16O(d,6Li)12C reaction, the transfer interaction
is determined in the post form as:
spectroscopic factor is given by the cluster-
nucleon configuration interaction model in psd
model space [10
(2)
Where
is the potential binding
spectroscopic factor follows the Fliessbach
definition, which takes into account
cluster to the 12C core in 16O.
53
COUPLED REACTION CHANNELS STUDY OF THE 16O(d,6Li) REACTION
microscopic
antisymmetrization
and
the analysis. For example, the
spectroscopic factors deduced from the
transfer reactions are smaller than those
orthonomalization effect for the two-body
cluster wave function.
obtained from the (
) knock-out reactions
III. RESULTS AND DISCUSSION
[13,15,16,23,24]. It is, therefore, of interest
for the present research to determine the
The
spectroscopic factors of light
16
spectroscopic factor of the O nucleus from
nuclei are of high interest for both theoretical
and experimental studies [1,2,7,11-13].
However, those values of the spectroscopic
factors in these nuclei are uncertain and seem
to depend on the direct reaction mechanism
as well as on the theoretical models used in
the CRC analysis of the transfer reaction
16O(d,6Li)12C reaction using the optical
potentials that give good OM description of
the elastic d+16O and Li+12C scattering at
6
the considered energies [15, 20].
Table I. The WS parameters of the complex OP used in the present CRC analysis for the elastic of d+12C and
d+16O scattering at Ed = 54.3 MeV and elastic 6Li+12C scattering at E6Li = 63 MeV
System
V0
rV
av
W0
WS
rW
aw
(MeV)
(fm)
(fm)
(MeV)
(MeV)
(fm)
(fm)
d+12Ca
d+16Oa
6Li+12C
71.8
68.2
1.25
1.25
1.15
0.700
0.693
0.750
11.0
10.2
1.25
1.25
2.27
0.700
0.790
0.650
160.5
11.0
a The WS parameters taken from OM analysis of the elastic d+12C scattering at 52 MeV [25]
In CRC calculations, quite important are
the inputs of the OP, which generate the
scatteing waves in both entrance and exit
channels, and are used in the remnant term (2).
Here:
(7)
For the 16O(d,6Li)12C reaction, the
of
deuteron on the 12C and 16O targets used for the
core-core interaction and the entrance channel
(8)
(9)
are assumed to be the same as the
of the
d+12C and d+16O systems at Elab = 52 MeV,
which have been adjusted to reproduce the
elastic scattering data [25]. The optical
potentials used in the present CRC calculations
are determined as Woods-Saxon form:
Where AP, AT and ZP, ZT are the mass
16
and charge numbers of deuteron, O (entrance
channel) and 6Li, 12C (exit channel).
Figure 1 illustrates the OM and CC
description of the elastic deuteron scattering on
,
(6)
12
16
the C and O targets in comparison with the
data measured at 52 MeV [25] and 56 MeV [26].
The Coulomb potentials parameter rC = 0.95 fm
And Coulomb potentials of a uniform
charged sphere,
54
DO CONG CUONG et al.
was used in all calculations. The Woods-Saxon
incident deuteron and
cluster from the
of the d +12C and
target. In this present work, the relative
motion of the d and in the ground state of
6Li is assumed to be in 2S state, and the
corresponding spectroscopic amplitude (the
overlap of deuteron and 6Li nucleus) is
taken to be unity. The binding potential
d+16O systems are shown in Table I. For the exit
6
channel, the OP parameters of the Li+12C
system have been adjusted to reproduce the
elastic data measured at Elab. = 63 MeV [27],
corresponding to Ec.m.: = 42.0 MeV that is close
to the c.m. energy of the final partition in the
16O(d,6Li)12C reaction.
6
between the deuteron and -cluster in Li is
adopted in the WS form with R = 1.905 fm
and a = 0.65 fm. The potential depth of 77.5
MeV has been adjusted to reproduce d
separation energy E = 1.47 MeV.
Another input for the CRC calculation of
the 16O(d,6Li)12C reaction is the structure
6
information of Li, which is formed by the
105
Elastic scattering
data 52 MeV
data 56 MeV
data 63 MeV
data 60 MeV
103
101
d+12C
d+16O
10-1
10-3
10-5
10-7
x10-1
x10-2
6Li+12C
0
30
60
90
120
150
c.m. (deg.)
Fig. 1. The OM and CC results for the elastic d+12C , d+16O and 6Li+12C elastic scattering obtained with the
assumed in the standard WS form (Table 1), in comparison with the data measured at Elab. = 52 MeV
[25], 56 MeV [26] for the d+12C, d+16O systems, and at Elab. = 63 MeV [27] and 60 MeV [28] for the 6Li+12C
system. The dash-dotted line is the OM result given by the elastic scattering wave function used in the
DWBA calculation of the
transfer reaction
12
16
In our CRC analysis of the 16O(d,6Li)12C
reaction measured at Elab. = 54.25 MeV
[15,20], the OP of the d +16O system Elab. = 52
MeV and 6Li+12C system at Elab. = 63 MeV are
used to generate the (relative) scattering wave
and C in O is taken in the WS form with
fixed geometry (as R = 4.148 fm, a = 0.55 fm
,
and R = 3,683 fm, a = 0.55 fm in CRC
[29]), and
the potential depths were adjusted to reproduce
separation energy of 7.162 MeV. The OP for
functions,
and
, respectively.
The binding potential
of the -cluster
55
COUPLED REACTION CHANNELS STUDY OF THE 16O(d,6Li) REACTION
6
the Li+12C system at 63 MeV is also used in
the transfer interaction W in equation (2). The
OP of the d +12C system chosen to reproduce
the elastic scattering data at Elab. = 52 MeV
[25] is used for the core-core OP in the transfer
physics inputs for the CRC calculation are
properly chosen, and only the spectroscopic
factors of the
cluster in 16O that
characterize the overlap of the intrinsic
wave functions of 12C and 16O remain the
free parameters.
interaction.
Thus,
all
the
necessary
Fig. 2. Coupling scheme of the six reaction channels taken into account in the CRC calculations of the
transfer 16O(d,6Li)12C reaction, which includes both the direct and indirect transfer processes
The present CRC caclculation includes
process, with the 16O
excited 16O nucleus were taken the same as that
used for 16O in the ground state.
also t
target being excited to the 3-1 (6.13 MeV) and
The obtained CRC results are compared
with experimental data [15] in Fig. 3, and the
agreement between the calculated cross sections
with the data is quite reasonable for the 3
2+ (6.92 MeV) excited states before the
1
transfer.
s of the
excited 16O were taken from shell model
results, with S = 0.663 and 0.5 for the 3-1 and
12
observed states of C. We note that the same
2+ states, respectively [10]. Transition
6Li+12C optical potential has been used in the
1
12
potentials between the ground state and excited
states of 16O were determined by deforming the
OP using the deformation lengths
CRC calculation 3 exit channels with C being
in the g.s. and excited 2+ and 4+ states. Without
inclusion of the indirect transfer processes, the
spectroscopic factors S = 1.960, 1.756, and
0.731 were deduced for the ground, 2+ and 4+
fm and
fm that correspond to the
electric transition probabitilies B(E2) = 39.3 e2
fm4 and B(E3) = 1490 e2 fm6, respectively. The
detailed coupling scheme of the present CRC
reaction is shown
spectroscopic factor S (0+) = 1.960 for the
12
ground state of C deduced from the present
in Fig. 2. The binding potentials
between the -particle and 12C core in the
direct CRC calculations is same those taken in
56
DO CONG CUONG et al.
transfer [29]. The direct and indirect CRC
16O
again are similar those deduced from the
direct and indirect CRC analysis of the
elast
the elastic 12C(16O, 16O)12C scattering [29].
Although the DWBA results shown in Fig. 3
using the spectroscopic factors S = 0.43,
2.34, and 4.0 describe well the experimental
results are illustrated as the solid lines in
Figure 3, with all OP parameters unchanged.
The
S = 0.715,
3.90 and 0.723 were deduced for the ground, 2+
and 4+ states, respectively. We note that the
CRC calculations of the indirect
transfer
include not only the contributions of the
16
excited states of O, but also contribution of
the excited states of 12C in the exit channel. We
and exit channels are not appropriate (the
OP for the entrance channel cannot describe
the elastic deuteron scattering on 16O at 52
factors of 16O for the ground state obtained
from the CRC results without the indirect
6
MeV, while the Li+12C optical potential at
50.6 MeV [20, 30] is chosen for exit
channel). These results show that the OP
plays a vital role in the CRC analysis of the
transfer reaction.
transfer
decrease significantly when the
indirect transfer via the excited states of 16O
12
12
and C included, those for the 2+ state of C
increase to be a factor of two times. These
103
101
Ed= 54.25MeV
16O(d,6Li)12Cgs
16O(d,6Li)12C2+
10-1
10-3
10-5
10-7
16O(d,6Li)12C4+
DWBA
Direct transfer
Direct + Indirect
0
20
40
60
80
c.m (deg.)
Fig. 3. CRC results for the transfer 16O(d,6Li)12C reaction in comparison with experimental data measured
at E = 54.25 MeV [15, 20]. Dash lines are the DWBA results using the spectroscopic factors and OP taken
from Refs. [15, 20]. The dash-dotted lines are the CRC results not including the indirect transfer processes,
and the solid lines are those with the indirect transfer processes included
57
COUPLED REACTION CHANNELS STUDY OF THE 16O(d,6Li) REACTION
IV. CONCLUSIONS
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16
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