# Low-energy 7Li($n,\gamma$)8Li and 7Be($p,\gamma$)8B radiative capture
reactions within the Skyrme Hartree-Fock approach

Nguyen Le Anh<EMAIL_ADDRESS>Department of Physics, Ho Chi Minh City
University of Education, 280 An Duong Vuong, District 5, Ho Chi Minh City,
Vietnam.  Department of Theoretical Physics, Faculty of Physics and
Engineering Physics, University of Science, Ho Chi Minh City, Vietnam.
Vietnam National University, Ho Chi Minh City, Vietnam.  Bui Minh Loc
<EMAIL_ADDRESS>Center for Exotic Nuclear Studies, Institute for Basic
Science (IBS), Daejeon 34126, Korea.

###### Abstract

The electromagnetic dipole transitions in 7Be($p,\gamma$)8B and
7Li($n,\gamma$)8Li reactions at the keV-energy region were analyzed
simultaneously within the Skyrme Hartree-Fock potential model. The Skyrme
Hartree-Fock calculation is adopted as a microscopic approach to obtain
consistently the single-particle bound and scattering states in the
calculation of the radial overlap function within the potential model. All
non-resonant and resonant electromagnetic dipole transitions are taken into
account. The electric dipole transitions are successfully described with the
slightest adjustment. The resonant magnetic dipole transitions at $633$ keV
and $2184$ keV of 7Be($p,\gamma$)8B reaction, and the one at $222$ keV of
7Li($n,\gamma$)8Li are also analyzed. The astrophysical $\mathcal{S}_{17}(0)$
factor of 7Be($p,\gamma$)8B reaction is found to be $22.3$ eV b.

##  I INTRODUCTION

The nucleon radiative-capture reaction in which the atomic nucleus captures
the incoming nucleon followed by the emission of electromagnetic radiation
plays an important role in nuclear astrophysics. Both 7Li($n,\gamma$)8Li and
7Be($p,\gamma$)8B reactions are key processes in nuclear astrophysics. The
7Li($n,\gamma$)8Li reaction is important due to its relation to the
nucleosynthesis of heavy nuclei in the inhomogeneous Big Bang model [1]. While
the 7Be($p,\gamma$)8B is crucial in the production of high-energy solar
neutrinos and the nucleosynthesis of low-mass stars [2, 3]. It is, however,
one of the reactions with the largest uncertainty [2, 3] as it is required to
be measured at extremely low energies to determine the cross section at the
zero energy known as the $S_{17}(0)$ factor.

The 7Li($n,\gamma$)8Li and 7Be($p,\gamma$)8B reactions were studied
simultaneously in many previous works [4, 5, 6, 7, 8, 9] due to the isospin
symmetry in the $n+^{7}$Li and $p+^{7}$Be systems. For instance, the neutron
optical potential for the 7Li target was used to reproduce the cross section
for the reaction 7Be($p,\gamma$)8B down to the very low energies [4, 6, 7, 8].
Therefore, 7Li($n,\gamma$)8Li and 7Be($p,\gamma$)8B are efficiently analyzed
side by side.

Among the theoretical studies, there are a variety of approaches for the
radiative-capture reaction such as the phenomenological potential model [10,
11, 12, 13], folding model [14, 15], shell-model calculation [7, 16], multi-
cluster calculation [5, 17, 18, 19] or phenomenological $R$-matrix [20, 9,
21]. For the 7Be($p,\gamma$)8B reaction, in particular, there were also the ab
initio calculations [22, 23] for the electric dipole ($E1$) transition and
recently for the magnetic dipole ($M1$) transition included [24]. A recent
review was given in Ref. [25]. The potential model is the simplest but
powerful tool for the study of radiative-capture reactions as the capture
process can be simply considered as the electromagnetic transition between the
continuum state and the bound state. The radiative-capture reaction can be
also approached with the potential model using the Skyrme Hartree-Fock
calculation to obtain simultaneously the bound and scattering states [26, 27].

In the present work, the 7Li($n,\gamma$)8Li and 7Be($p,\gamma$)8B reactions
were analyzed within the potential model using the Skyrme Hartree-Fock
calculation. All electromagnetic dipole transitions are in the consideration
of their contributions. In the case of the 7Li nucleus, the experimental data
at low energies [28, 29, 8, 30] were reproduced efficiently both non-resonant
$E1$ transition and resonant $M1$ transition at $222$ keV. In the case of the
7Be nucleus, all measured data [31, 32, 33, 34, 35, 36, 37, 38, 39] were well
reproduced including not only both resonances at $633$ keV and $2184$ keV but
also the non-resonant contribution up to $3000$ keV.

The results show that the single-particle scattering and the bound wave
functions being the main inputs of the potential model are well-described
using the two parameters of the approach. For the non-resonant $E1$
transition, only one parameter is adjusted. The spectroscopic factors that
play a key role in the method are determined by the experimental data at a few
hundred keV. It helps in determining precisely the $\mathcal{S}_{17}(0)$
factor that is 22.3 eV b in the present work.

The formulas for the $E1$ and $M1$ transitions are presented in the next
section. The Skyrme Hartree-Fock calculation was described in detail in Refs.
[26, 27]. The section on results and discussions is started with the $E1$
transitions having no resonance followed by the $M1$ transition with low-lying
resonances.

##  II METHOD OF CALCULATION

###  II.1 Electromagnetic dipole transition

For the proton radiative-capture reaction, the energy-dependent astrophysical
$\mathcal{S}$ factor $\mathcal{S}(E)$ is defined by

$\mathcal{S}_{aA}(E)=E\exp[2\pi\eta(E)]\sigma(E),$ (1)

where the Sommerfeld parameter describing the $s$-wave barrier penetration is
$\eta(E)=Ze^{2}/(\hbar v)$ with $v$ being the initial relative velocity
between the incident particle with atomic mass number $a$ and the target with
the charge number $Z$ and the mass number $A$. In our case,
$\mathcal{S}_{aA}(E)$ is the $\mathcal{S}_{17}(E)$. The general radiative-
capture cross section $\sigma(E)$ is given in Refs. [40, 11, 12]. The electric
quadrupole ($E2$) transition is negligible in the energy of interest within
the potential model [41, 42, 43, 38, 13]. Our present calculations are
restricted to the $E1$ and the $M1$ transitions.

The initial state $|[I\otimes(\ell\otimes s)j]JM\rangle$ and the final state
$|[I\otimes(\ell^{\prime}\otimes s)j^{\prime}]J^{\prime}M^{\prime}\rangle$ of
the system are assumed as the core that is the target nucleus with one
additional nucleon that is the captured nucleon at a single-particle state.
The core has the internal spin $\bm{I}$ that is unchanged in the calculation.
Note that the ground states of the target nuclei are $I^{\pi}=3/2^{-}$ in the
study. The total relative angular momentum of the nucleon-target system is
$\bm{j}=\bm{\ell}+\bm{s}$ with $\bm{\ell}$ being the relative orbital angular
momentum and $s=1/2$ for the nucleon. The channel spin of the initial system
is $\bm{J}=\bm{I}+\bm{j}$. Therefore, the radiative-capture cross section in
our case is expressed as [44, 11, 12]

$\sigma(E)=\dfrac{4}{3}\dfrac{1}{\hbar
v}\left(\dfrac{4\pi}{3}k^{3}_{\gamma}\right)\dfrac{1}{(2s+1)(2I+1)}\sum_{\ell
jJ}\left(|\mathcal{M}_{E1}|^{2}+|\mathcal{M}_{M1}|^{2}\right),$ (2)

where $k_{\gamma}$ is the wave number of the photon defined as $\hbar
ck_{\gamma}=E+Q$ with $Q$ being the $Q$ value of the reaction.
$\mathcal{M}_{\Omega 1}$ with $\Omega$ being $E$ or $M$ is the reduced matrix
element of the electromagnetic dipole transitions. We herein follow the
coupling scheme for angular momenta in Refs. [41, 45, 11].

$\mathcal{M}_{\Omega 1}=\langle[I\otimes(\ell^{\prime}\otimes
s)j^{\prime}]J^{\prime}||\mathcal{O}_{\Omega 1}||[I\otimes(\ell\otimes
s)j]J\rangle.$ (3)

In the case of the $E1$ transition, the operator is given by

$\mathcal{O}_{E1}=C_{e}rY_{1}^{\mu},$ (4)

where $C_{e}=m(\tau-Z/A)e$ is the effective charge in which $e$ is the
electric charge, $m$ the reduced mass, and $\tau=0$ for neutron and $\tau=1$
for proton; and $Y_{1}^{\mu}$ ($\mu=-1,0,1$) is the spherical harmonic
function. The $\mathcal{M}_{E1}$ can be reduced to the calculation of the
single-particle (s.p.) reduced matrix element [46]

$\mathcal{M}_{E1}=C_{e}S_{F}^{1/2}(-1)^{I+j^{\prime}+J+1}\hat{J}\hat{J^{\prime}}\left\\{\begin{matrix}j^{\prime}&J^{\prime}&I\\\
J&j&1\end{matrix}\right\\}\mathcal{M}_{E1}^{(\rm s.p.)},$ (5)

where $\hat{l}=\sqrt{2l+1}$ and the curly bracket is Wigner $6j$ coefficient.
$S_{F}$ is the spectroscopic factor that is introduced to take into account
the missing configurations of the system; in the other words, it is the
fractional parentage coefficient that the system can be described as our
assumption [44, 8]. It also contains missing correlations in the calculation
of the single-particle state. Within the potential model for the radiative-
capture reaction, its value is finally adjusted to reproduce the experimental
data.

For the $M1$ transition, the operator is [45, 11, 12, 13]

$\mathcal{O}_{M1}=\sqrt{\frac{3}{4\pi}}\left[C_{m}{\bm{\ell}_{\mu}}+2\left(\mu_{\tau}{\bm{s}_{\mu}}+\mu_{A}{\bm{I}}\right)\right].$
(6)

With $\mu_{N}\approx 0.105e$ fm being the nuclear magneton, the effective
magnetic moment is $C_{m}=m(\tau+Z/A^{2})\mu_{N}$. The magnetic moments used
in this work are $\mu_{p}=+2.793\mu_{N}$, $\mu_{n}=-1.913\mu_{N}$,
$\mu_{A}=-1.399\mu_{N}$ for 7Be [47] and $+3.256\mu_{N}$ for 7Li [48]. The
reduced matrix element for the $M1$ transition is, therefore, written as the
summation of three terms [45, 11, 12, 13]

$\mathcal{M}_{M1}=S_{F}^{1/2}\sqrt{\dfrac{3}{4\pi}}\left(\mathcal{M}_{M1}^{(0)}+\mathcal{M}_{M1}^{(1)}+\mathcal{M}_{M1}^{(2)}\right).$
(7)

Similarly to the case of $E1$ transition in Eq. (5), we have

$\displaystyle\mathcal{M}_{M1}^{(0)}$
$\displaystyle=C_{m}(-1)^{I+j+J^{\prime}+1}\hat{J}\hat{J^{\prime}}\left\\{\begin{matrix}j^{\prime}&J^{\prime}&I\\\
J&j&1\end{matrix}\right\\}\mathcal{M}_{0}^{(\rm s.p.)},$ (8)
$\displaystyle\mathcal{M}_{M1}^{(1)}$
$\displaystyle=\mu_{\tau}(-1)^{I+j+J^{\prime}+1}\hat{J}\hat{J^{\prime}}\left\\{\begin{matrix}j^{\prime}&J^{\prime}&I\\\
J&j&1\end{matrix}\right\\}\mathcal{M}_{1}^{(\rm s.p.)},$ (9)
$\displaystyle\mathcal{M}_{M1}^{(2)}$
$\displaystyle=\mu_{A}\delta_{jj^{\prime}}(-1)^{I+j^{\prime}+J+1}\hat{J}\hat{J^{\prime}}\left\\{\begin{matrix}I&J&j^{\prime}\\\
J^{\prime}&I&1\end{matrix}\right\\}\hat{I}\sqrt{I(I+1)}\mathcal{M}_{2}^{(\rm
s.p.)}.$ (10)

The single-particle reduced matrix elements are well-known in literature [46].
They are decomposed into two components: the geometrical coefficient and the
radial overlap integral. The radial overlap integrals for the $E1$ and $M1$
transitions are different from the $r$-dependence in the operator

$\mathcal{I}_{E1}=\int\phi_{n\ell^{\prime}j^{\prime}}(r)\chi_{\ell
j}(E,r)r\,dr,~{}~{}~{}\mathcal{I}_{M1}=\int\phi_{n\ell^{\prime}j^{\prime}}(r)\chi_{\ell
j}(E,r)\,dr.$ (11)

The potential model for radiative-capture reaction focuses on the calculation
of the radial overlap integrals in Eq. (11) with appropriate wave functions,
$\chi_{\ell j}$ and $\phi_{n\ell^{\prime}j^{\prime}}$.

###  II.2 Single-particle wave functions for the overlap integrals

In our approach, both scattering wave function $\chi_{\ell j}$ and bound wave
function $\phi_{n\ell^{\prime}j^{\prime}}$ in Eq. (11) were simultaneously
obtained within the Skyrme Hartree-Fock calculation. The detailed Hartree-Fock
calculation for bound and scattering single-particle state were presented in
Refs. [49, 50].

The calculation was started with the radial Hartree-Fock equations using the
skyrme_rpa program in Ref. [51]

$\left\\{\dfrac{\hbar^{2}}{2m^{*}(r)}\left[-\frac{d^{2}}{dr^{2}}+\dfrac{\ell^{\prime}(\ell^{\prime}+1)}{r^{2}}\right]+V(r)-\frac{d}{dr}\left[\frac{\hbar^{2}}{2m^{*}(r)}\right]\frac{d}{dr}\right\\}\varphi_{n\ell^{\prime}j^{\prime}}(r)=\varepsilon_{n\ell^{\prime}j^{\prime}}\varphi_{n\ell^{\prime}j^{\prime}}(r).$
(12)

where $m^{*}(r)$ is the nucleon effective mass in the Skyrme Hartree-Fock
formalism. The Skyrme Hartree-Fock potential $V(r)$ consists of the central
$V_{\rm c}(r)$, the spin-orbit $V_{\rm s.o.}(r)$, and the one-body Coulomb
potential $V_{\rm Coul.}(r)$ in the case of proton

$V(r)=V_{\rm c}(r)+qV_{\rm Coul.}(r)+V_{\rm s.o.}(r),$ (13)

with $q$ being the charge of the incident nucleon. The SLy4 interaction [52]
is chosen in our calculation. The first-order derivative term in Eq. (12) is
eliminated by using the transformation [49]

$\varphi_{n\ell^{\prime}j^{\prime}}(r)=[m^{*}_{q}(r)/m]^{1/2}\tilde{\varphi}_{\alpha}(r),$
(14)

with $m$ being the nucleon mass. Then Eq. (12) is rewritten as the usual
Schrödinger equation

$\left\\{\dfrac{\hbar^{2}}{2m^{\prime}}\left[-\dfrac{d^{2}}{dr^{2}}+\dfrac{\ell^{\prime}(\ell^{\prime}+1)}{r^{2}}\right]+\mathcal{V}_{b}(\varepsilon_{n\ell^{\prime}j^{\prime}},r)-\varepsilon_{n\ell^{\prime}j^{\prime}}\right\\}\tilde{\varphi}_{n\ell^{\prime}j^{\prime}}(r)=0,$
(15)

where $m^{\prime}=mA/(A-1)$ is used for the center-of-mass correction [32]
that is important for the calculations for light nuclei. The potential
$\mathcal{V}_{b}(\varepsilon_{n\ell^{\prime}j^{\prime}},r)$ in Eq. (15) is
given by

$\mathcal{V}_{b}(\varepsilon_{n\ell^{\prime}j^{\prime}},r)=\mathcal{N}_{b}\mathcal{V}_{b}^{\rm
c}(\varepsilon_{n\ell^{\prime}j^{\prime}},r)+q\mathcal{V}_{\rm
Coul.}(r)+\mathcal{V}_{\rm s.o.}(r){\bm{\ell}^{\prime}}\cdot\bm{\sigma},$ (16)

where the central part $\mathcal{V}_{b}^{\rm
c}(\varepsilon_{n\ell^{\prime}j^{\prime}},r)$ depends on the single-particle
energy $\varepsilon_{n\ell^{\prime}j^{\prime}}$

$\displaystyle\mathcal{V}_{b}^{\rm
c}(\varepsilon_{n\ell^{\prime}j^{\prime}},r)=\dfrac{m^{*}(r)}{m}\left\\{V_{\rm
c}(r)+\dfrac{1}{2}\dfrac{d^{2}}{dr^{2}}\left(\dfrac{\hbar^{2}}{2m^{*}(r)}\right)-\dfrac{m^{*}(r)}{2\hbar^{2}}\left[\dfrac{d}{dr}\left(\dfrac{\hbar^{2}}{2m^{*}(r)}\right)\right]^{2}\right\\}$
$\displaystyle+\left[1-\dfrac{m^{*}(r)}{m}\right]\varepsilon_{n\ell^{\prime}j^{\prime}},$
(17)

and the spin-orbit and Coulomb parts are energy-independent

$\displaystyle\mathcal{V}_{\rm Coul.}(r)$ $\displaystyle=[m^{*}(r)/m]V_{\rm
Coul.}(r),$ (18) $\displaystyle\mathcal{V}_{\rm s.o.}(r)$
$\displaystyle=[m^{*}(r)/m]V_{\rm s.o.}(r).$ (19)

The scattering states of the Skyrme Hartree-Fock field is obtained by
replacing $\varepsilon_{n\ell^{\prime}j^{\prime}}$ in Eq. (15) by the
continuous energy $E$ [49]. The scattering wave function is the solution of

$\left\\{\dfrac{\hbar^{2}}{2m^{\prime}}\left[-\dfrac{d^{2}}{dr^{2}}+\dfrac{\ell(\ell+1)}{r^{2}}\right]+\mathcal{V}_{s}(E,r)-E\right\\}\chi_{\ell
j}(E,r)=0.$ (20)

The potential $\mathcal{V}_{s}(E,r)$ in Eq. (15) is

$\mathcal{V}_{s}(E,r)=\mathcal{N}_{s}\mathcal{V}_{s}^{\rm
c}(E,r)+q\mathcal{V}_{\rm Coul.}(r)+\mathcal{V}_{\rm
s.o.}(r){\bm{\ell}}\cdot\bm{\sigma}.$ (21)

The central part $\mathcal{V}_{s}^{\rm c}(E,r)$ depends on the scattering
energy $E$. It is obtained by replacing the single-particle energy
$\varepsilon_{n\ell^{\prime}j^{\prime}}$ in Eq. (II.2) by the positive energy
$E$. The spin-orbit and Coulomb parts are the same as Eqs. (18) and (19) as
they are energy independent. Note that two parameters $\mathcal{N}_{b}$ and
$\mathcal{N}_{s}$ are introduced in Eqs. (16) and (21), respectively.

It is known that the Skyrme Hartree-Fock approximation gives tightly bound
single-particle states. The asymptotic behavior of the bound state should be
corrected. Therefore, the value of $\mathcal{N}_{b}$ is adjusted to obtain the
single-particle energy $\varepsilon_{n\ell^{\prime}j^{\prime}}$ of the first
unoccupied state equal to the experimental nucleon separation energy. All
occupied single-particle states are kept unchanged. It is well-known as the
well-depth method.

For the scattering state, the mean-field approach such as the Skyrme Hartree-
Fock calculation cannot reproduce exactly the resonance state. Therefore, the
scattering state has to be corrected to reproduce the exact position of low-
lying resonances. It is done by the adjusted value of $\mathcal{N}_{s}$.

The single-particle bound and scattering states in Eq. (11) with the
corrections have the correct asymptotic forms required by the properties of
the low energy radiative-capture reaction. The values of parameters are well-
constrained by the experimental values including the nucleon separation energy
for $\mathcal{N}_{b}$ and the energy of the nuclear excited state (or the
position of the resonance) for $\mathcal{N}_{s}$.

At the end of the calculation, the value of $S_{F}$ is fine-tuned to reproduce
the experimental data. The adjustments are typical in the framework of the
potential model. The difference in our approach is that the scattering and
bound wave functions are initially obtained simultaneously from a microscopic
approach such as the Skyrme Hartree-Fock formalism.

Mean-field models are often considered to be inappropriate to describe light
nuclei such as $A=7,8$ nuclei. There have been works adopting the mean-field
procedure [53, 54]. Even for 4He some useful conclusions have been drawn using
a Skyrme Hartree-Fock calculation [55] and useful conclusions have be obtained
for other light nuclear system [53]. In this manuscript we show that a Skyrme
Hartree-Fock calculation leads to very reasonable results of radiative-capture
cross sections with $A=7,8$ nuclei.

###  II.3 Partial-wave analysis

Fig. 1 illustrates the level schemes of 8Li and 8B nuclei with the relevant
bound-to-bound $M1$ transitions from the $1^{+}$ and $3^{+}$ states to the
$2^{+}$ ground states. Note that two bound-to-bound $M1$ transitions from
$3^{+}$ state to $2^{+}$ state are similar. The resonance at $2184$ keV of
7Be($p,\gamma$)8B reaction and the resonance at $222$ keV of
7Li($n,\gamma$)8Li are related. However, as the radioactive-capture process is
the continuum-to-bound transition, they cannot be as similar to each other as
the two bound-to-bound $M1$ transitions are. The sub-threshold $M1$ transition
(the dotted arrow in Fig. 1) is not in our calculation.

Figure 1: Level schemes of 8Li (a) and 8B (b). All low-lying resonances are
caused by the $M1$ transitions to the ground states ($2^{+}$) of 8Li and 8B.
The blue dotted lines are the reaction thresholds. The arrows are bound-to-
bound $M1$ transitions. The black dashed arrows are the transitions above the
threshold. The dotted arrow is the sub-threshold $M1$ transition.

In the case of the $E1$ transitions, as shown in Table 1, the nucleon is
assumed to be captured into the single-particle state $1p_{3/2}$ for both
target nuclei. The selected single-particle scattering states are consequently
$s$ wave, and $d$ waves including $d_{3/2}$ and $d_{5/2}$. The total spin of
the initial state or the channel spin $J^{\pi}$ are $1^{-}$ and $2^{-}$ for
the $s$ wave, and $1^{-},2^{-}$ and $3^{-}$ for the $d$ waves. Note that the
total spin $J$ is unique for the bound-to-bound transition. While the average
sum of all possible channel spins is required for the continuum-to-bound
transition.

Table 1: All $E1$ transitions to the $2^{+}$ ground states in 8Li and 8B are considered. $J^{\pi}$ is the channel spin and parity of the initial system. $\mathcal{N}_{s}$ and $\mathcal{N}_{b}$ are the two parameters in the calculation. $S_{F}$ is the spectroscopic factor. The values of $\mathcal{N}_{b}$ are determined by the $Q$ values that are $2033$ keV and $136$ keV for 7Li($n,\gamma$)8Li and 7Be($p,\gamma$)8B reactions, respectively.  Reaction | $J^{\pi}$ | $\ell$ | $\mathcal{N}_{s}$ | $n\ell^{\prime}_{j^{\prime}}$ | $\mathcal{N}_{b}$ | $S_{F}$
---|---|---|---|---|---|---
7Li($n,\gamma$)8Li |  $1^{-}$, $2^{-}$ | $s$ | $1.00$ | $1p_{3/2}$ | $0.72$ | $1.00$
|  $1^{-}$, $2^{-}$, $3^{-}$ | $d$ | $1.00$ | $1p_{3/2}$ | $0.72$ | $1.00$
7Be($p,\gamma$)8B |  $1^{-}$, $2^{-}$ | $s$ | $1.00$ | $1p_{3/2}$ | $0.70$ | $0.80$
|  $1^{-}$, $2^{-}$, $3^{-}$ | $d$ | $1.00$ | $1p_{3/2}$ | $0.70$ | $0.80$

For the $M1$ transitions, there are the resonances corresponding to the bound-
to-bound transitions illustrated by the dashed arrows in Fig. 1. The resonance
at $633$ keV is caused by the proton in the $p$-scattering wave captured into
the $1p_{3/2}$-bound state. The channel spins $J^{\pi}$ are $1^{+}$, $2^{+}$
and $3^{+}$ for the $p$ wave with $j=3/2$, but there is no $J^{\pi}=3^{+}$ for
$j=1/2$. The configuration is the same for the case of $M1$ resonance at 222
keV in 7Li($n,\gamma$)8Li reaction. For the resonance at $2184$ keV of
7Be($p,\gamma$)8B reaction, the $d$ wave is dominant that is discussed in the
next section. Table 2 presents the configurations in the calculation for the
$M1$ transitions.

Table 2:  The same as Table 1 but for the $M1$ transitions. The energies (in keV) of the $M1$ resonances are shown. Note that there is no $J^{\pi}=3^{+}$ for $p_{1/2}$ states. Reaction |  $E$ [keV] | $J^{\pi}$ | $\ell$ | $\mathcal{N}_{s}$ | $n\ell^{\prime}_{j^{\prime}}$ | $\mathcal{N}_{b}$ | $S_{F}$
---|---|---|---|---|---|---|---
7Be($p,\gamma$)8B | $633$ | $1^{+},2^{+},3^{+}$ | $p$ | $0.67$ | $1p_{3/2}$ | $0.70$ | $1.00$
| $2184$ | $1^{+},2^{+},3^{+}$ | $d$ | $1.27$ | $1d_{5/2}$ | $1.44$ | $0.10$
7Li($n,\gamma$)8Li | $222$ | $1^{+},2^{+},3^{+}$ | $p$ | $0.83$ | $1p_{3/2}$ | $0.58$ | $0.05$

##  III RESULTS AND DISCUSSIONS

###  III.1 Non-resonant $E1$ transition in 7Li($n,\gamma$)8Li and
7Be($p,\gamma$)8B reaction

As the 7Li target is not radioactive, the 7Li($n,\gamma$)8Li reaction was
sufficiently studied for the constraint of the optical parameters used in the
case of the 7Be($p,\gamma$)8B reaction [5, 6, 7, 8]. Both reactions are
analyzed in the study to support the consistency of the approach. The
parameters used for the $E1$ transition are shown in Table 1.

Figure 2: The total $E1$ transition (the solid line) caused by $s$ (the dashed
line) and $d$ waves (the dotted line) in the partial-wave analysis.
$S_{F}=1.0$. The experimental data were taken from Refs. [30, 8, 29, 28].

For the 7Li($n,\gamma$)8Li reaction, the cross sections were measured over a
wide range of energies from a few keV up to 1000 keV [28, 29, 8]. In our
calculation, the results in Fig. 2 show that the main contribution comes from
the $s$ wave. The experimental data are well reproduced with only the
adjustment for the single-particle bound state. The parameter
$\mathcal{N}_{b}$ is equal to $0.72$ to reproduce the neutron separation
energy $2033$ keV in 8Li. As there is no $E1$ resonance in the energy region
from the threshold up to $1400$ keV for the calibration, the parameters
$\mathcal{N}_{s}$ for all scattering states are kept unity.

The $S_{F}$ in this case is not required for the adjustment. The work in Ref.
[30] has reported on the Coulomb dissociation of 8Li in order to determine the
neutron-capture cross section for the inverse reaction 7Li($n,\gamma$)8Li. Our
calculation gives the calculated $E1$ curve passes through the experimental
data points in Ref. [30] that are lower than those in Ref. [8] under 200 keV.
Around the resonance of 222 keV, our calculation for the $E1$ transition
passes through the data points in Ref. [29] (Fig. 2). The shell-model
calculations in Refs. [56, 57] gave $S_{F}$ to be $0.62$ and $1.143$,
respectively. In addition, the $S_{F}$ in the ab initio calculation is $0.966$
[57].

Table 1 shows the consistencies between 7Li($n,\gamma$)8Li and
7Be($p,\gamma$)8B reaction. The identical initial states and analyses make
their results similar. The only considerable difference between the two
reactions is that the experimental separation energy is $136$ keV for proton
[58] and $2033$ keV for neutron obviously because of the Coulomb energy.
However, the parameter $\mathcal{N}_{b}$ in the case of the 7Be target is
equal to $0.70$ that is close to the case of 7Li target as shown in Table 1.
The partial-wave analysis of $E1$ transitions in 7Be($p,\gamma$)8B reaction is
presented in Fig. 3. The parameters $\mathcal{N}_{s}$ for the scattering
states are also kept unity. The $S_{F}$ is 0.8 for this case to reproduce the
experimental data point in Ref. [39]. The values of $0.742$ and $1.079$ were
reported in the shell-model calculations [57], while $S_{F}=0.884$ in the ab
initio calculation [57]. The three-cluster approach in Ref. [19] gave
$S_{F}=0.836$.

Figure 3: The $E1$ transitions caused by $s$ and $d$ waves in the partial-wave
analysis. The experimental data were taken from Refs. [31, 32, 33, 34, 35, 36,
38, 37, 39].

In terms of nuclear astrophysics, the cross section at zero energy of the
7Be($p,\gamma$)8B reaction known as $\mathcal{S}_{17}(0)$ is of particular
interest. As shown in many previous works, it is determined entirely by the
$E1$ transition. The contribution of the resonant $M1$ transition can be
negligible. Table 3 shows the values of $\mathcal{S}_{17}(0)$ reported in the
wide range of both experiments and theoretical calculations taken from
previous works. Considering only the $E1$ transition,
$\mathcal{S}_{17}(0)=19.1_{-1.0}^{+4.0}$ and $19.4$ eV b were given in Refs.
[54] and [23], respectively. Within the shell-model calculation, the value of
$19.424$ eV b was reported in Ref. [7]. From the study of 7Li($d,p$)8Li cross
section, $\mathcal{S}_{17}(0)$ in Refs. [59], [31] and [32] were $25.8\pm
2.2$, $24.3\pm 2.0$ and $17.4\pm 1.6$ eV b, respectively. The values of
$18.5\pm 1.7$, $18.4\pm 1.6$, and $18.6\pm 0.4$ eV b were reported in Refs.
[35], [36] and [60], respectively. Most recently, the experimental values of
$20.6\pm 0.8$ and $21.5\pm 0.6$ eV b were given in Refs. [38] and [39],
respectively. Theoretically, Refs. [13] and [24] have recently given the
values of $20.51^{+2.02}_{-1.85}$ and $19.8\pm 0.3$ eV b, respectively.

Table 3: The selected values of the $\mathcal{S}_{17}(0)$ taken from experimental and theoretical studies. Experiment |  $\mathcal{S}_{17}(0)$ [eV b]
---|---
Parker (1966) [59] | $25.8\pm 2.2$
Kavanagh et al. (1969) [31] | $24.3\pm 2.0$
Vaughn et al. (1970) [32] | $17.4\pm 1.6$
Hammache et al. (1998) [35] | $18.5\pm 1.7$
Strieder et al. (2001) [36] | $18.4\pm 1.6$
Davids and Typel (2003) [60] | $18.6\pm 0.4$
Schümann et al. (2006) [38] | $20.6\pm 0.8$
Junghans et al. (2010) [39] | $21.5\pm 0.6$
Theory |  $\mathcal{S}_{17}(0)$ [eV b]
Bennaceur et al. (1999) [7] | $19.424$
Chandel et al. (2003) [54] | $19.1_{-1.0}^{+4.0}$
Navrátil et al. (2011) [23] | $19.4$
Tursunov et al. (2021) [13] | $20.51^{+2.02}_{-1.85}$
Kravvaris et al. (2022) [24] | $19.8\pm 0.3$
Present work | $22.3$

Our calculation with the only $E1$ transition being considered and $S_{F}=1.0$
gives the $\mathcal{S}_{17}(0)=27.8$ eV b that is the upper limit in our
calculation (Fig. 4). It is $22.2$ eV b using the value $S_{F}=0.8$ fine-tuned
to reproduce the experimental data points in Ref. [39]. The results pointed
out that the $E1$ transitions are well-described even in the case without any
low-lying resonance for the calibration. The parameters $\mathcal{N}_{s}$ for
the scattering state are unity in all cases of non-resonant $E1$ transitions
in the study. The only adjustment for the $E1$ transition is the parameter
$\mathcal{N}_{b}$. The value of $S_{17}(0)$ in our calculation does not
strongly depend on the value of the parameter $\mathcal{N}_{b}$. The
calculation with $\mathcal{N}_{b}=1.00$ is also shown in Fig. 4 as the dotted
line. The value of $S_{17}(0)$ is $21.2$ eV b. $S_{F}=0.08$ is required to
reproduce the experimental data. The inadequate value of $S_{F}$ is well-
understood. The single-particle state obtained in the Hartree-Fock calculation
is tightly bound. While the radiative capture reaction at low energy requires
a weakly bound system. The value of $S_{17}(0)$ is, however, not much changed.

Figure 4: The relevance of $\mathcal{S}_{17}$ factor of 7Be($p,\gamma$)8Be at
low energy. The calculation using $\mathcal{N}_{b}=0.70$ gives
$\mathcal{S}_{17}(0)=22.3$ eV b. While the values of $21.2$ eV b is given by
calculation with $\mathcal{N}_{b}=1.00$. The experimental data are from JUN10
[39] (the circle), SCH06 [38] (the cross), BAB03 [37] (the square), STR01 [36]
(the triangle), HAM98 [35] (the diamond), FIL83 [33] (the pentagram), and
KAV69 [31] (the octagram).

###  III.2 Low-lying resonant $M1$ transition in 7Li($n,\gamma$)8Li and
7Be($p,\gamma$)8B reaction

The potential model is sufficient not only for the $E1$ transition but also
for the $M1$ transition. In the study, the $M1$ transitions include resonances
that are at $633$ keV and $2184$ keV in the case of the 7Be($p,\gamma$)8B
reaction and at $222$ keV in 7Li($n,\gamma$)8Li reaction. The parameters
$\mathcal{N}_{s}$ for the $M1$ transition are adjusted to reproduce the
position of the resonance.

First, in the case of the 7Be($p,\gamma$)8B reaction, as in our analysis, the
$M1$ transitions from the $p$-scattering states to the $1p_{3/2}$-bound state
are responsible for the first resonance at $633$ keV. The same discussion was
given in Refs. [10, 61]. The parameters $\mathcal{N}_{s}$ were adjusted to
reproduce the correct position of the resonance (see Table 2). Previous
theoretical works concluded that $S_{F}$ should be unity for this case [41,
61, 13]. We also use $S_{F}=1.0$ in our calculation. The shape of the
resonance is well reproduced. As shown in Fig. 5, the contribution of the $M1$
transition is negligible when the energy is below 400 keV.

The second resonance at $2184$ keV corresponding to the excited state
($3^{+}$) of 8B is caused by $M1$ transition to the ground state ($2^{+}$) of
8B [62, 63]. The work in Ref. [13] proposed that the $E2$ transition from
scattering $d$ waves is responsible for this resonance. However, the known
level scheme illustrated in Fig. 1 shows only the bound-to-bound $M1$
transition. In our calculation, the $d$-scattering wave is also adopted but
using the $M1$ instead of the $E2$ transition. Following this argument, there
should be another bound configuration to reproduce the resonant $M1$
transition. Our assumption is that a proton is captured into the $1d_{5/2}$
state from $d$-scattering waves with the channel spins $J^{\pi}=1^{+},2^{+}$,
and $3^{+}$. The $\mathcal{N}_{b}=1.44$ to reproduce the proton separation
energy of $136$ keV for this case. The parameters $\mathcal{N}_{s}$ are listed
in Table 2 to calibrate the position of resonance caused by $d$-scattering
waves. The result of the calculations for the second resonance is shown in
Fig. 5. The $S_{F}$, in this case, is $0.1$. Its small value can be
interpreted that the interference of other configurations appearing at this
energy. As much high incident energy as much small $S_{F}$ is obtained.
Besides, at the energy $2988$ keV, the study in Ref. [13] analyzed the
resonant $E1$ transitions from $d$-scattering waves to reproduce experimental
data taken from Ref. [38]. This resonance possibly affects the analysis of the
second resonance at $2184$ keV. However, the experimental data supporting this
resonance is not sufficient. Therefore, it is not taken into account in our
analysis in the present work. The total $\mathcal{S}_{17}$ factor of
7Be($p,\gamma$)8B reaction with the $E1$ and $M1$ transitions is displayed in
Fig. 5 with our full calculation. As shown in Fig. 4 and Table 3, the
$\mathcal{S}_{17}(0)$ in our calculation including $M1$ transitions is now
$22.3$ eV b.

Figure 5: All transitions are considered. The dashed line is the contribution
of the $E1$ transition as shown in Fig. 3. The dotted and dash-dotted lines
are of the $M1$ transitions. The solid line shows our final result. The
experimental data were taken from Refs. [31, 32, 33, 34, 35, 36, 37, 38, 39].

Analogously, the calculation is applied to the resonant $M1$ transition at
$222$ keV in 7Li($n,\gamma$)8Li reaction (Fig. 6). The scattering
$p$-scattering waves captured into $1p_{3/2}$ bound state via the $M1$
transition are the source of the resonance. The cross sections measured at
$222$ keV in Refs. [28] and [29] are different. The value of $S_{F}$ in our
calculation is $0.17$ or $0.05$ in order to reproduce the data given by Ref.
[28] or [29], respectively. As the resonances at $2184$ keV and at $222$ keV
are related, one can expect that the value of $S_{F}$ in this case is at the
same order of the one obtained from the calculation for the resonance at
$2184$ keV above. However, as the $3^{+}$ state in 8Li is closed to the
threshold of $n+^{7}$Li, $S_{F}$ in this case should be significantly reduced.
That is the reason why $S_{F}=0.05$ is chosen to reproduce the experimental
data in Ref. [29].

Figure 6: The calculation for 7Li($n,\gamma$)8Li radiative-capture reaction.
All electromagnetic transitions are in consideration. The values of $S_{F}$ is
adjusted to reproduce the data of Ref. [28] ($S_{F}=0.17$) or Ref. [29]
($S_{F}=0.05$).

Note that only one spin-orbit partner of the partial waves is the source of
the resonance due to the spin-orbit potential included in our calculation. The
spin-orbit partner waves, $1p_{1/2}$ and $1p_{3/2}$ for an example, shares the
same parameter $\mathcal{N}_{s}$ (Table 2). Without the spin-orbit potential,
both $p$ waves can contribute to the resonance. The spin-orbit potential
should not be neglected as its especially important effect in light nuclei was
emphasized in Ref. [49, 54]. The contributions of different scattering waves
are shown in Figs. 7(a-c) for the three resonances.

Figure 7: The partial analysis for the three $M1$ resonances. In Figs. (a),
(b), and (c), the contributions by the spin-orbit partners with all channel
spin are shown. Figs. (d), (e), and (f) illustrate the contribution of
different channel spins.

It is worthwhile to remind that the channel spin $J$ made the largest
contribution can not be determined by the $J$ of the corresponding nuclear-
structure state. The contributions of different channel spins in our
calculation are shown in Figs. 7(d-f). As shown in Table 2, the scattering
waves causing the resonances are different. While the resonance at $2184$ keV
is caused by the $d$-scattering wave, the $p$-scattering wave is the source of
the resonance at $222$ keV. This difference leads to the following
consequence. In the case of the resonance at $2184$ keV, Fig. 7(d) shows that
the largest contribution is from $J=3$. Meanwhile, in the case of the
resonance at $222$ keV, $J=3$ shown in Fig. 7(f) has the smallest contribution
due to the absence of the $p_{1/2}$ wave.

##  IV Conclusions

The Skyrme Hartree-Fock potential model is a good approach for the radiative-
capture reactions at very low energy for light nuclei. The low-energy
scattering problem is comfortably overcome by the Hartree-Fock calculation in
the continuum. In addition the bound state is established on the traditional
Hartree-Fock formalism and the well-depth method. The calculations for the
$E1$ transitions well reproduced the experimental data with only one
parameter. The approach is also sufficient to reproduce consistently all $M1$
transitions in the study. The value of the $S_{F}$ that is complicated to be
obtained exactly within the potential model can be determined by the precise
measurements at a few hundred-keV energy that is accessible by the experiments
[39, 29]. In the case of the 7Be($p,\gamma$)8B reaction, the
$\mathcal{S}_{17}(0)$ factor determined by the $E1$ transition is $22.3$ eV b.
The value is not strongly dependent on the parameter in the calculation.

## Acknowledgments

N.L.A. acknowledges the support by Ho Chi Minh City University of Education
Foundation for Science and Technology under grant number CS.2021.19.45. This
work was supported by the Institute for Basic Science (IBS-R031-D1).

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