Transfer reactions in the investigation of light-nuclei nucleosynthesis

Abstract

Cross sections for the 6Li(p,γ)7 Be, 7Li(n,γ)8 Li 8Li(n,γ)9 Li and 8Li(p,γ)9 Be capture reactions have been investigated in the framework of the potential model. The main ingredients of the potential model are the potentials used to generate the continuum and bound-state wave functions and spectroscopic factors of the corresponding bound systems. The spectroscopic factors for the 7Lin=9Li gs bound systems were obtained from a FR-DWBA analysis of neutron transfer reactions induced by 8Li radioactive beam on a 9Be target, while spetroscopic factor for the 8Lip=9Be gs bound system were obained from a proton transfer reaction. From the obtained capture reaction cross section, reaction rate for the 8Li9Be direct neutron and proton capture were determined and compared with other experimental and calculated values.

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REVISTA MEXICANA DE F´ISICA S 54 (3) 63–68DICIEMBRE 2008
Transfer reactions in the investigation of light-nuclei nucleosynthesis
V. Guimar˜ aes, O. Camargo, R. Lichtenth¨ aler, and V. Scarduelli
Instituto de F´ ısica, Universidade de S˜ ao Paulo,
P.O.Box 66318, 05389-970 S˜ ao Paulo, SP, Brazil.
J.J. Kolata
Department of Physics, University of Notre Dame,
Notre Dame, Indiana, 46556, USA.
H. Amro, F.D. Becchetti, and Hao Jiang
Department of Physics, University of Michigan,
Ann Arbor, Michigan 48109-1120, USA.
E.F. Aguilera, D. Lizcano, E. Martinez-Quiroz, and H. Garcia
Instituto Nacional de Investigaciones Nucleares,
Apartado Postal 18-1027, M´ exico, D.F. 11801 M´ exico.
Recibido el 27 de febrero de 2008; aceptado el 25 de abril de 2008
Cross sections for the6Li(p,γ)7Be,7Li(n,γ)8Li8Li(n,γ)9Li and8Li(p,γ)9Be capture reactions have been investigated in the framework of
the potential model. The main ingredients of the potential model are the potentials used to generate the continuum and bound-state wave
functions and spectroscopic factors of the corresponding bound systems. The spectroscopic factors for the7Li⊗n=8Ligs,8Li⊗n=9Ligs
bound systems were obtained from a FR-DWBA analysis of neutron transfer reactions induced by8Li radioactive beam on a9Be target,
while spetroscopic factor for the8Li⊗p=9Begs bound system were obained from a proton transfer reaction. From the obtained capture
reaction cross section, reaction rate for the8Li(n,γ)9Li and8Li(p,γ)9Be direct neutron and proton capture were determined and compared
with other experimental and calculated values.
Keywords: Capture reactions; transfer reactions; spectroscopic factors.
Se hace un estudio de las secciones eficaces de las reacciones de captura para6Li(p,γ)7Be,7Li(n,γ)8Li,8Li(n,γ)9Li y8Li(p,γ)9Be en el
marco de un modelo de potencial. Los principales ingredientes de este modelo son los potenciales usados para generar las funciones de
onda del continuo y para los estados ligados as´ ı como los factores espectrosc´ opicos de los sistemas ligados correspondientes. Los factores
espectrosc´ opicos de los sistemas ligados para7Li⊗n=8Ligs,8Li⊗n=9Ligsfueron obtenidos del an´ alisis con FR-DWBA de la transferencia de
un neutr´ on del proyectil radiactivo8Li a un n´ ucleo de9Be, mientras que para el sistema ligado8Li⊗p=9Begs, los factores espectrosc´ opicos
se obtuvieron de an´ alisis de la transferencia de un prot´ on. De las secciones eficaces de reacciones de captura obtenidas, se determinaron las
razones de reacci´ on de captura directa y se compararon con otros valores calculados y experimentales.
Descriptores: Reacciones de captura; reacciones de transferencia; factores espectrosc´ opicos.
PACS: 25.60.Je; 25.40.Lw; 25.70.Hi
1.Introduction
It is very important for the astrophysics to know the reac-
tion rate of a specific reaction at the “Gamow Peak”. For
light-nuclei nucleosynthesis, in many astrophysical environ-
ments the energy of the Gamow Peak is very low, in the range
of few tens to at most hundreds of keVs. For many light-
nuclei nucleosynthesis reactions, experimental cross sections
at these very low energies are not often measurable; either
because the cross sections are too small or because the com-
bination of target+beam is not possible. A typical case is the
8Li(n,γ)9Li capture reaction where direct measurement is not
possiblebecauseno8Liorneutrontargetexist. Forsuchcase,
the cross sections of the capture reaction have to be obtained
by indirect methods. Also, where low energy measurements
are not possible , extrapolation from higher energies and/or
indirect methods to determine the cross sections are usually
adopted. Extrapolationfromhigherenergydataisnotstraigth
forward. Careful and accurate account of physically relevant
information has to be considered in the description of the re-
action before the extrapolation is performed. The description
of the reaction requires not only information on the struc-
ture of the nuclei involved but also a clear understanding of
the reaction mechanism. Indirect methods to determine cap-
ture reaction cross sections include the Coulomb dissociation
method, which corresponds to the inverse temporal reaction
of the capture, the reduced-width or ANC (Asymptotic Nor-
malization Coefficient) method and potential model, where
the latter two use transfer reactions as a way to get infor-
mation on the non-resonant part of the capture reaction pro-
cess. This will be explained in more detail in the next section.
These indirect methods are very suitable to be used in asso-
ciation with low-energy radioactive nuclear beams.
In this work, we report the results obtained for the non-
resonant part of the neutron and proton capture reactions
of light nuclei;
6Li(p,γ)7Be,7Li(n,γ)8Li,8Li(n,γ)9Li and

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V. GUIMAR˜AES et al.
8Li(p,γ)9Be in the framework of the potential model. The
results for the neutron capture reactions has been published
elsewhere [1], while the results for the8Li(p,γ)9Be proton
capture reaction are still preliminary.
Neutron and proton capture reactions involving light ra-
dioactive nuclei such as8Li and8B have been found to be
important in astrophysical environments such as the inho-
mogeneous model for big-bang nucleosynthesis [2], the ini-
tial stage of Type II supernovae [3], and nucleosynhtesis in
massive stars [4]. The inhomogeneous model for big-bang
nucleosynthesis [2] has been proposed as a possible way to
produce higher mass abundances for A>4 nuclei which are
not well predicted by the standard big-bang model. In this
model, short-lived isotopes such as8Li and8B play an im-
portant role in the subsequent synthesis of heavier elements.
Neutron and proton capture reactions involving8Li may also
be important in initial stage of the Type-II supernovae [3],
where they can produce seed nuclei for the r-process. Also,
in a very high neutron density environment, neutron-induced
three-particle interactions can be an alternative way to syn-
thesize12C via the8Li produced in the reaction sequence
4He(2n,γ)6He(2n,γ)8He(β+)8Li [4]. In supermassive stars,
i.e., the first stars in the universe with high proton density and
very low metalicity, alternatives ways to synthesize12C are
the reaction sequences7Be(p,γ)8B(α,p)11C(p,γ)12N(β)12C
and7Be(p,γ)8B(p,γ)9C(α,p)12N(β)12C. In both sequences,
radioactive8B nuclei can play a crucial role.
2. The potential model
The potential model as well as ANC (Asymptotic Normal-
ization Coefficient) are indirect methods to obtain the cross
section for direct capture reactions. The idea of the ANC
method is to use transfer reactions such as (d,p) or (d,n) reac-
tions, in inverse kinematics, to extract the “asymptotic wave
functions normalization coefficients” that can be related to
the capture cross section. The relation between the trans-
fer and capture reactions is given by the fact that the ANC,
which normalize the cross sections for the non-resonant part
of the capture reaction, is obtained from peripheral transfer
reactions whose amplitudes contain the same overlap func-
tion as the amplitude of the corresponding capture reaction
of interest [5]. Therefore, the ANC method is based on
the assumption that capture reactions at stellar energies usu-
ally proceed through the tail of the nuclear overlap func-
tion. The amplitude of the radiative capture cross sections
is then dominated by contributions from large relative dis-
tances between the participating nuclei. In the ANC method,
the asymptotic coefficient is used to normalize the Whittaker
function, which is used to describe the tail of the overlap
function, Ibound=ANC×W−η,l+1/2(2κr). However, it has
been shown that s-wave neutron capture, even at rather low
energies, is not peripheral [6,7] and so it is necessary to use
another indirect method such as the potential model to cal-
culate the wave function of the incoming neutron or proton
and the wave function for the bound system. Thus, in the po-
tential model, it is necessary to calculate the overlap function
also taking into account the internal part of the nuclear po-
tential, Ibound=S1/2× Φ(r). Here, S1/2is the spectroscopic
amplitude and Φ(r) is the wave-function which describes the
bound state. Also, in the ANC method, the wave function of
the incoming nuclei in the continuum, ψscat, is assumed to
be due only due to the Coulomb potential. This assumption
may only be true for a very peripheral capture reaction. In the
potential model, the continuum wave function, ψscat, has to
be calculated with a potential which includes also the nuclear
interaction. Thus, the essential ingredients in the potential
model are the potentials used to generate the wave functions
ψscatand Ibound, and the normalization for the latter which
is given by its spectroscopic factor. This potential model, has
recently been applied in the analysis of the16O(d,p)17O and
16O(d,n)17F transfer reactions to determine the correspond-
ing16O(p,γ)17Fgs,16O(p,γ)17F1stand16O(n,γ)17Ogsastro-
physical direct capture cross sections [8].
Based on the potential model, the direct radiative capture
(DRC) of an s- and/or d-wave nucleon (proton or neutron)
by a nucleus b, proceeding via E1 transition and leaving the
compound nucleus c in its ground state, is given by:
σE1
b→c(n,γ) =16π
9?k3
γ|?ψscat|OE1|Ibound?|2,
(1)
where kγ = ?γ/?c is the wave number corresponding to a
γ-ray energy ?γ, OE1stands for the electric dipole opera-
tor and the initial-state wave function ψscatis the incoming
nucleon wave function scattered by the nucleon-nucleus po-
tential. Here the effective charge for the neutrons used in the
electric dipole operator is given by eeff = −eZ/A, where
A and Z are the atomic mass and charge of the compound
nucleus.
In the low energy region of astrophysical relevance,
the non-resonant part of the
8Li(p,γ)9Be and8Li(n,γ)9Li capture reactions is dominated
by the E1 radiative capture of an s-wave nucleon, or a d-
wave nucleon for energies above 1.0 MeV. To calculate the
non-resonant part of these capture reactions in the framework
of the potential model we used the computer code RADCAP
developed by Bertulani [9].
In Table I we list all the parameters of the potentials used
to generate the incoming and bound wave functions. All the
potentials were assumed to be a Woods-Saxon shape with
geometric parameters r0 = 1.25 and a = 0.65 fm. The
depths for the bound-state potentials were obtained by ad-
justing them to give the binding energy of the corresponding
bound system. Details of the analysis for the neutron capture
reactions7Li(n,γ)8Ligs,1stand8Li(n,γ)9Ligsare published
in Ref. 1. The scattering potential parameters for both en-
trance channel spins, s = 5/2+,3/2+, for the8Li(2+)+n
system were obtained by keeping the same volume inte-
gral per nucleon, JV/A, as those for the entrance channel
spins, s = 2+,1+, deduced from the scattering potentials of
the7Li+n system [7]. The scattering potential depth for the
6Li(p,γ)7Be,
7Li(n,γ)8Li,
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TRANSFER REACTIONS IN THE INVESTIGATION OF LIGHT-NUCLEI NUCLEOSYNTHESIS
65
TABLE I. Wood-Saxon potential parameters used in the capture reaction calculations. Depths and B.E. are in MeV with r0 = 1.25 fm and
a = 0.65 fm, where the radii are given by R = r0× A1/3
B.E.V0(bound)SF(p3/2,p1/2)
5.606 65.250.83 (09),0.0
16.88876.72 1.50(17),0.17(03)
2.033 46.380.87(15),0.11(2)
1.052 43.300.48,0.0
4.064 47.820.62(13),0.0
T.
channel spin
3/2+,1/2+
5/2+,3/2+
2+,1+
2+,1+
5/2+,3/2+
V0(scatt)
46.0±2.5
49.7±2.5
56.15,46.50
56.15,46.50
58.15,48.15
JV/A (MeV/fm3)
678±37
678±37
793,657
793,657
793,657
6Li+p=7Begs
8Li+p=9Begs
7Li+n=8Ligs
7Li+n=8Li1st
8Li+n=9Ligs
6Li+p system was obtained by adjusting it to reproduce the
data from the6Li(p,γ)7Be capture reaction [10]. Keeping the
same JV/A, we obtained the depth for the scattering poten-
tial for the8Li+p system. In this work we found that the
cross sections for the capture reaction for these light nuclei
are sensitive to the choice of the incoming nucleon potential.
The more bound the system more sensitive the cross sections
is for the choice of the potential depth used to determine the
continuum wave function [13]. Although we have obtained
the incoming nucleon scattering potentials for the reaction
of interest from analysis of close systems, it would be inter-
esting to obtain such potentials from direct elastic scattering
measurement as8Li+p and8B+p. A program of investigation
for these elastic scattering experiments at low energy is un-
der way at the Sao Paulo University using the radioactive ion
beam facility RIBRAS [11].
3.Spectroscopic factors from transfer reac-
tions
Low-energy radioactive nuclear beams are very suitable to
be used in connection with the potential model to investigate
capture reactions of astrophysics interest. The Nuclear Struc-
ture laboratory at University of Notre Dame in USA [12]
and later on, the Institute of Physics of Sao Paulo Univer-
sity [11], have developed and installed facilities to produce
relatively intense low-energy and energy-resolved radiative
nuclear beams RNB. Among these beams,8Li can be pro-
duced with intensities of about ≥ 106p/s in both systems. In
this work we present some results on spectroscopic factors
obtained from transfer reactions induced by a8Li radiative
ion beam on9Be target. The experiment was performed at
the Nuclear Structure Laboratory of the University of Notre
Dame using the Twinsol system [12] and the procedures are
described elsewhere [1,13].
We have measured angular distributions for one-neutron
and one-proton transfer reactions, namely,9Be(8Li,7Li)10Be,
9Be(8Li,9Li)8Be and9Be(8Li,9Be)8Li reactions. Also, we
have measured angular distribution for elastic scattering
9Be(8Li,8Li)9Be. From the FR-DWBA analysis, using the
code FRESCO [14], for the angular distribution of these
reactions, we extracted the spectroscopic factors for the
8Li⊗p=9Be,
These spectroscopic factors are listed in Table II and were
7Li⊗n=8Li and
8Li⊗n=9Li bound systems.
used to normalize the non-resonant part of the corresponding
capture reactions8Li(n,γ)9Li,7Li(n,γ)8Li and8Li(p,γ)9Be.
Transfer reactions have two vertices, and the spectroscopic
factor for one of them has to be known in order to ob-
tain the spectroscopic factor for the other vertex.
case of the elastic-transfer reaction9Be(8Li,9Be)8Li, where
the outgoing channel is the same as the incoming channel,
we have the advantage of having only one unknown ver-
tex. In the present analysis we used the spectroscopic factor
value for the8Begs⊗n=9Begsas S9Be= 0.44(7), which is
the average of spectroscopic factors from two (d,t) reactions
studies [22,24]. The spectroscopic factor for6Li⊗p=7Begs
(Jπ=3/2−) vertex was considered as the average of
the experimental values determined from the6Li(d,p)7Li
and7Li(p,d)6Li reactions [28]. The spectroscopic factors
for the
ysis of neutron transfer reactions
and9Be(8Li,9Ligs)8Begs, respectively [1].
scopic factor for the8Li⊗p=9Begs(Jπ=3/2−) bound sys-
tem was obtained from the analysis of the elastic-transfer
9Be(8Li,9Begs)8Ligsreaction. Details of the analysis of this
particular transfer reaction will be presented elsewhere [13].
As we can see in Table-II, the spectroscopic factors ob-
tained for the8Li⊗p=9Begs(Jπ=3/2−) and8Li⊗n=9Ligs
(Jπ=3/2−) bound systems are in good agreement with shell-
model calculations.
In the
8Ligs and
9Ligs were obtained from the anal-
9Be(8Li,7Ligs)10Begs
The spectro-
4.Capture Reactions and reaction rates
The results of the capture reaction calculations using the po-
tential model for the7Li(n,γ)8Li and8Li(n,γ)9Li are shown
in Fig. 1. The experimental points for7Li(n,γ)8Li are from
Refs. 7 and 29 to 32.The curve labeled (a; dotted line) is
the sum of channel-spin s = 1 and s = 2 contributions
for the neutron capture reaction to the first excited state of
8Li, while curve (b; dashed line) is the sum of the channel-
spin s = 1 and s = 2 contributions for the8Li ground-
state. The thin solid line is the sum of these two contribu-
tions, where only the contribution of neutrons captured to
the orbital p3/2in8Li(g.s.), using the spectroscopic factor
S8Li(g.s.)(3/2)=0.87, is considered. The thick solid line is
the same calculation considering in addition the contribution
of the capture to the orbital p1/2, using the spectroscopic fac-
tor S8Li(g.s.)(1/2)=0.113. For the8Li(n,γ)9Ligsreaction, the
Rev. Mex. F´ ıs. S 54 (3) (2008) 63–68

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66
V. GUIMAR˜AES et al.
TABLE II. Spectroscopic factors C2S.
Jπ
Shell Model
calculation
0.628a)0.885b)
0.977c)
0.0561c)
0.580c)
2.357c)
This work
(d,p),(d,n),(d,t)
0.68(14)d)0.90e)0.65(15)f)
0.87g)
0.113h)
0.44(7)i)
2.23 (13)j)
0.83 (09)k)
0.64i)
8Li+9Be transfers
8Ligs⊗n=9Ligs
7Ligs⊗n=8Ligs(p3/2)
7Ligs⊗n=8Ligs(p1/2)
8Begs⊗n=9Begs
9Begs⊗n=10Begs
6Ligs⊗p=7Begs
8Ligs⊗p=9Begs(p3/2)
8Ligs⊗p=9Begs(p1/2)
a) from Ref. 15,
c) from Cohen and Kurath [17],
e) from8Li(d,p)9Li reaction at 76 MeV [19],
g) from Ref. 7,
i) average of S=0.37 from Ref. 22 and S=0.51 from Ref. 24,
k) average S=0.90 [28] S=0.72 [25] S=0.87 [26],
3/2-
2+
2+
3/2-
0+
3/2-
3/2-
3/2-
0.62 (13)
0.87 (15)
0.113 (17)
1.356c)
0.153c)
1.50 (27)
0.17 (03)
b) from Ref. 16 using same Cohen Kurath wave-function,
d) from8Li(d,p)9Li reaction at 39 MeV [18],
f) from9Li(d,t)8Li reaction at 15 MeV [20],
h) from Ref. 21,
j) average of S=2.10 from Ref. 22 and S=2.356 from Ref. 23,
l) from the d(8Li,n)9Be reaction at 40 MeV [27].
FIGURE 1. The capture cross sections for the
8Li(n,γ)9Li reactions. The various curves are explained in the text.
7Li(n,γ)8Li and
lower curves labeled (a) correspond to the potential depths
scaled from the n+7Li capture reaction analysis. Curves la-
beled (b) correspond to the assumption of the same potential
for the incoming wave-function as for the bound state, for s-
wave neutron only (dotted curve) and s and d-wave neutrons
(solid curve). In Fig. 2 we plot the S-factor obtained for the
6Li(p,γ)7Be and8Li(p,γ)9Be capture reactions.
We have also computed the nucleosynthesis reaction rate
as a function of the temperature for the direct8Li(n,γ)9Ligs
and8Li(p,γ)9Begscapture reactions. The expression for the
reaction rate for E1 capture in cm3mol−1s−1is given by [33]:
NA?σv? = K
∞
?
0
σ(E)E exp(−C2E/T9)dE,
(2)
where
K = C1µ−1/2T−3/2
9
and C1= 3.7313 × 1010, C2= 11.605, NAis Avogadro’s
number, µ is the reduced mass of the system, T9is the tem-
perature in units of 109K, σ is the capture cross section, v is
therelativevelocity, andE istheenergyinthecenter-of-mass
system. E is given in MeV and the cross section in barns.
Although some resonances above the8Li+n and8Li+p
threshold in9Li and9Be, respectively, could be important, in
the present calculation only the direct capture to9Ligsand
9Begsis considered. The reaction rate for the8Li(n,γ)9Ligs
capture reaction at the temperature T9=1 was deduced to be
NA?σv? = (3.17 ± 0.70) × 103cm3mol−1s−1,
where the uncertainty is from the uncertainty in the spectro-
scopic factor used in the calculation (20%) and from the vari-
ation of ±1 MeV in the potentials used to determined the
distorted wave (10%). This result is comparable to the most
Rev. Mex. F´ ıs. S 54 (3) (2008) 63–68

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TRANSFER REACTIONS IN THE INVESTIGATION OF LIGHT-NUCLEI NUCLEOSYNTHESIS
67
FIGURE 2. The astrophysical S-factor deduced for the6Li(p,γ)7Be
and8Li(p,γ)9Be capture reactions. The upper and lower limits are
obtained considering the uncertainties in the depth of the scattering
potential.
recent theoretical calculations [34-37] and is in good agree-
ment with the value from a recent (d,p) experiment [18]. The
reaction rate for the8Li(p,γ)9Begscapture reaction at tem-
perature T9=1 was deduced to be
NA?σv? = (2.2+0.8
where the uncertainty is from the uncertainty in the scat-
tering potential depth and in the spectroscopic factor of
?9Begs|8Li+p? used in the calculation. This value is about
3 times larger then the value obtained in Ref. 27.
−0.6) × 103cm3mol−1s−1,
5. Conclusion
We have measured the angular distributions for the elas-
tic scattering of
fer reactions
ELAB=27.0MeV. Spectroscopic
8Ligs⊗n=9Ligsand7Ligs⊗n=8Ligsbound systems were ob-
tained from the comparison between the experimental dif-
ferential cross sections and FR-DWBA calculations with the
code FRESCO [14]. The spectroscopic factors obtained are
compared with shell model calculations and also with exper-
imental values from (d,p) reactions.
Using the spectroscopic factors obtained for the
8Ligs⊗n=9Ligs and
have determined the cross-sections for the7Li(n,γ)8Li and
8Li(n,γ)9Ligsneutron-capture reactions based on a potential
model. Our work has shown that low-energy radioactive nu-
clear beams can be very suitable not only to perform spectro-
scopic investigations but also to determine the non-resonant
parts of capture reactions of astrophysical interest.
We have recently installed a double-solenoid system to
produce secondary low energy radioactive ion beams at
the Pelletron-LINAC laboratory in University of Sao Paulo,
Brazil - The RIBRAS system [11]. This system was con-
ceived based in the Notre Dame-Michigan Twinsol facility
but with a higher field integral. The RIBRAS system is al-
ready operational using primary beams from the 8 MV Pel-
letron tandem and we are producing6He [38],7Be,8Li and
8B secondary beams with intensity of about 104to 105parti-
cle per second. Other reaction studies using a8B beam with
astrophysical interest, such as8B(α,p) are planned.
8Li on
9Be(8Li,7Li)10Be and
9Be and for the neutron trans-
9Be(8Li,9Li)8Be at
factorsforthe
7Ligs⊗n=8Ligs bound system, we
Acknowledgments
The authors wish to thank the Fundac ¸˜ ao de Amparo a
Pesquisa do Estado de S˜ ao Paulo (FAPESP 2006/00629-2
and 2007/06705-5) for financial support. This work was also
funded in part by the U.S. NSF under Grants No. PHY03-
54828 and INT03-05347.
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