#  Continuum random-phase approximation for $(n,\gamma)$ reactions on neutron-
rich nuclei: collective effects and resonances

Teruyuki Saito  Graduate School of Science and Technology, Niigata University,
Niigata 950-2181, Japan  Masayuki Matsuo  Department of Physics, Faculty of
Science. Niigata University, Niigata 950-2181, Japan

###### Abstract

We formulate a microscopic theory to calculate cross section of the radiative
neutron capture reaction on neutron-rich nuclei using the continuum random-
phase approximation (cRPA) to the time-dependent density functional theory
(TDDFT). With an intension of applying to the r-process, for which the
statistical reaction model may not be appropriate, we describe the transition
between a initial state of incident neutron and a final state of the gamma
decay by means of a single many-body framework of the cRPA-TDDFT. With the
cRPA approach, it is possible to describe various excitation modes present in
the $(n,\gamma)$ reaction, including soft dipole excitation, the giant
resonances as well as non-collective excitations and the single-particle
resonances. Furthermore, it enables us to describe the $(n,\gamma)$ reaction
where the the final states of the gamma transition are low-lying surface
vibrational states. We demonstrate the theory by performing numerical
calculation for the reaction ${}^{139}{\rm Sn}(n,\gamma)^{140}{\rm Sn}$. We
discuss various new features which are beyond the single-particle model;
presence of narrow and wide resonances originating from non-collective and
collective excitations and roles of low-lying quadrupole and octupole
vibrational states.

##  I Introduction

The r-process is believed to be the origin of about half of the elements
heavier than iron Burbidge  _et al._ (1957); Arnould  _et al._ (2007); Cowan
_et al._ (2021). From the viewpoint of nuclear physics, the nucleosynthesis is
a complex network of the $(n,\gamma)$ and $(\gamma,n)$ reactions, the beta
decay and the fission taking place on a large number of short-lived neutron-
rich nuclei very far from the stability line. The cross sections and the
probabilities of the relevant reactions and decays need to be provided
theoretically for quantitative understanding since the direct experimental
measurements are quite difficult or impossible in most cases. Recent
observation of kilonova associated with a binary neutron star merger Abbott
_et al._ (2017a, b) provides a first direct evidence and quantitative
information on the r-process nucleosynthesis Pian  _et al._ (2017); Kasen  _et
al._ (2017); Villar  _et al._ (2017); Cowan  _et al._ (2021). It is therefore
quite important for nuclear theory to provide reliable theoretical models of
the relevant nuclear reactions which takes into account the recent progress of
study of neutron-rich nuclei.

In the present study, we focus on the radiative neutron capture reactions. It
is usually described in terms of two different mechanisms, the compound
nuclear (CN) process and direct capture (DC) process Arnould  _et al._ (2007);
Cowan  _et al._ (2021). The CN process is assumed to proceed via compound
states with high level density, and it is usually evaluated by means of the
Hauser and Feshbach statistical model Hauser and Feshbach (1952). It is
relevant for nuclei with relatively large neutron separation energy and and
often applied to the s-process which occurs on stable nuclei or nuclei close
to the stability line. The r-process path, however, lies on short-lived
neutron-rich nuclei in which the neutron separation energy or the excitation
energy of nuclei is as small as $\sim 2$ MeV, In this case the statistical
description of the CN process may not be appropriate Mathews  _et al._ (1983);
Arnould  _et al._ (2007); Cowan  _et al._ (2021). Often considered in this
case is the direct capture process Lane and Lynn (1960); Mathews  _et al._
(1983); Raman  _et al._ (1985); Mengoni  _et al._ (1995); Rauscher  _et al._
(1998); Bonneau  _et al._ (2007); Chiba  _et al._ (2008); Rauscher (2010); Xu
and Goriely (2012); Xu  _et al._ (2014); Zhang  _et al._ (2015); Sieja and
Goriely (2021), where a neutron scattering state decays directly to a bound
state without forming compound states. It is essentially a single-particle
model of gamma decay or an independent-particle shell model for the nuclear
structure.

There exists various nuclear structure phenomena beyond the single-particle
description, such as the pairing correlation, the low-lying collective states,
and giant resonances. In addition, neutron-rich nuclei far from the stability
line exhibits characteristic features such as the neutron halo,the neutron
skin and related exotic modes such as the pygmy dipole resonance or the soft
dipole mode Tanihata  _et al._ (1985); Hansen and Jonson (1987); Suzuki  _et
al._ (1990); Bertsch and Esbensen (1991); Paar  _et al._ (2007); Tanihata  _et
al._ (2013) originating from small neutron separation energy or the weak
binding of the last neutrons. In particular, the pygmy dipole resonance was
suggested to influence the r-process nucleosynthesis Goriely (1998), and
efforts to include the exotic collective excitations has been pursued in the
framework of the CN process models Goriely (1998); Goriely and Khan (2002);
Goriely  _et al._ (2004); Litvinova  _et al._ (2009); Avdeenkov  _et al._
(2011); Daoutidis and Goriely (2012); Xu  _et al._ (2014); Martini  _et al._
(2016). However the assumption of the statistical compound states may not be
appropriate in the case of small neutron separation energy. Effect of the weak
neutron binding is partly taken into account in the direct capture models
Typel and Baur (2005), which have however difficulty to include the collective
correlations.

The purpose of our study is to formulate a microscopic model of radiative
neutron capture reaction which is relevant to the r-process. Namely we intend
to take into account the various nuclear structure effects, which are not
included in the conventional direct capture models. We intend also to describe
the whole process of $(n,\gamma)$ reaction on a single quantum many-body
theory so that we can avoid the statistical assumption of the CN process. To
this end we adopt the continuum random-phase approximation (cRPA) Shlomo and
Bertsch (1975); Bertsch and Tsai (1975); Matsuo (2001), which is formulated as
a linear response theory based on the time-dependent density functional theory
(TDDFT) Nakatsukasa  _et al._ (2016). In a previous publication Matsuo (2015),
we have reported a prototype formulation which describes the radiative capture
process $(A-1)+n\rightarrow A^{*}\rightarrow A_{\rm g.s.}+\gamma$, i.e. a
gamma-transition from continuum excited states $A^{*}$, which couples to the
neutron scattering state $(A-1)+n$, to the ground state $A_{\rm g.s.}$ of the
residual nucleus $A$. This formulation enables us to describe the collective
excitations such as the pygmy and/or giant resonances, or the RPA correlation
in general in the continuum excited states $A^{*}$. Note however that the
final states of the actual $(n,\gamma)$ reaction are not only the ground state
$A_{\rm g.s.}$ but also low-lying excited states $A^{**}$. Therefore, in the
present paper, we extend the formalism so that it can describe the latter
case, i.e. the $(n,\gamma)$ reaction $(A-1)+n\rightarrow A^{*}\rightarrow
A^{**}+\gamma$ populating the low-lying excited states $A^{**}$. This is a
necessary extension to evaluate the total neutron capture cross section to
which gamma-decays to excited states contribute. Furthermore we will take into
account correlations in the low-lying excited states $A^{**}$, such as surface
vibrational modes.

One of the keys of the extension is given in a preceding paper Saito and
Matsuo (2021), where we have formulated an extended linear response theory
(the continuum random-phase approximation) to describe photo-absorption
transitions $A^{**}+\gamma\rightarrow A^{*}$ from a low-lying excited state
$A^{**}$ to the continuum excited states $A^{*}$. On the basis of this
achievement, we formulate in the present paper a framework to calculate the
$(n,\gamma$) cross section. An important key for this purpose is the method of
Zangwill and Soven Zangwill and Soven (1980), which enables us to define the
$T$ matrix for photo-absorption followed by particle-emission. Combining these
key formalisms, we describe partial cross sections of $(\gamma,n)$ reaction
$A^{**}+\gamma\to A^{*}\to n+(A-1)$ for individual channels of the neutron
emission $A^{*}\to n+(A-1)$. We then obtain the $(n,\gamma)$ cross section for
the inverse process $(A-1)+n\rightarrow A^{*}\rightarrow A^{**}+\gamma$, using
the reciprocity theorem. Details of the formulation is given in Sec. II.

We demonstrate in Sec. III new features of the present theory by performing a
numerical calculation for radiative neutron capture on ${}^{139}{\rm Sn}$ with
E1 and E2 transitions populating low-lying quadrupole and octupole states as
well as the ground state in ${}^{140}{\rm Sn}$. We will show that the presence
of the low-lying octupole state brings new aspects in the $(n,\gamma)$
reactions originating from strong collectivity of this state. We shall discuss
also that the present cRPA approach describes different kinds of resonance
structure emerging in the capture reaction, including narrow resonances
originating from non-collective states as well as the giant resonances and the
single-particle resonances. We draw conclusions in Sec. IV.

##  II Theory

In this section we formulate a scheme to describe a radiative neutron capture
reaction $(A-1)+n\to A^{*}\to A^{**}+\gamma$ where the final state of the
gamma-transition is a low-lying excited state $A^{**}$. We first describe
photo-absorption reaction $A^{**}+\gamma\to A^{*}$ assuming that both initial
and final states are described by the continuum random-phase approximation
(cRPA) to the time-dependent density functional theory (TDDFT) (subsections
II.1 and II.2). In case the excited state $A^{*}$ is located above the neutron
separation energy, $A^{*}$ decays by emitting a neutron $A^{*}\to(A-1)+n$. In
the framework of the continuum random-phase approximation the configuration of
the daughter nucleus $(A-1)$ is a one-hole state with respect to the ground
state of $A$. It is then possible to derive partial photo-absorption cross
sections for individual channels of the scattering states $(A-1)+n$ by using
the Zangwill and Soven method (subsection II.3). Finally we obtain the neutron
capture cross section for the inverse process $(A-1)+n\to A^{*}\to
A^{**}+\gamma$ using the detailed balance (subsection II.4).

###  II.1 The photo-absorption cross section between RPA excited states

We express an initial state $A^{**}$ of photo-absorption reaction as
$\ket{i{L_{i}M_{i}}}$ with excitation energy $E_{i}$, where $L_{i}M_{i}$ are
the angular momentum numbers. We also express an excited state $A^{*}$ as
$\ket{kLM(E)}$ with the angular quantum numbers $LM$. The excitation energy
$E$ is shown explicitly as $A^{*}$ is a state in the continuum spectrum. $k$
represents other quantum numbers. Normalization is $\langle
k^{\prime}L^{\prime}M^{\prime}(E^{\prime})|kLM(E)\rangle=\delta_{k^{\prime}L^{\prime}M^{\prime},kLM}\delta(E-E^{\prime})$.

The photoabsorption cross section of the transition from $\ket{iL_{i}M_{i}}$
to states with angular momentum $L$ and energy $E=E_{i}+E_{\gamma}$ is given
generally by Ring and Schuck (1980); Bertulani and Danielewicz (2004);
Thompson and Nunes (2004)

$\displaystyle\sigma^{\lambda}_{iL_{i}+\gamma\to L}(E_{\gamma})$
$\displaystyle=f_{\lambda}(E_{\gamma})\sum_{k}B(M_{\lambda},iL_{i}\to
kL(E))=\frac{f_{\lambda}(E_{\gamma})}{2L_{i}+1}S(M_{\lambda};iL_{i},L;E)$ (1)

for electromagnetic multipole $\hat{M}_{\lambda\mu}$ transition with photon
energy $E_{\gamma}$ in terms of the reduced matrix element

$\displaystyle B(M_{\lambda},iL_{i}\to kL(E))$
$\displaystyle=\frac{1}{2L_{i}+1}|\bra{kL(E)}|\hat{M}_{\lambda}|\ket{iL_{i}}|^{2},$
(2)

or the strength function

$\displaystyle S(M_{\lambda};iL_{i},L;E)$
$\displaystyle=\sum_{kMM_{i}\mu}|\bra{kLM(E)}\hat{M}_{\lambda\mu}\ket{iL_{i}M_{i}}|^{2}$
$\displaystyle=\sum_{k}|\bra{kL(E)}|\hat{M}_{\lambda}|\ket{iL_{i}}|^{2},$ (3)

and a kinematical factor

$\displaystyle f_{\lambda}(E_{\gamma})$
$\displaystyle=\frac{(2\pi)^{3}(\lambda+1)e^{2}}{\lambda[(2\lambda+1)!!]^{2}}\left(\frac{E_{\gamma}}{\hbar
c}\right)^{2\lambda-1}.$ (4)

We assume that both excited states $\ket{iL_{i}M_{i}}$ and $\ket{kLM(E)}$ are
those which can be described by cRPA based on the TDDFT. In other words, the
model space is spanned by all the particle-hole configurations including
scattering single-particle states. In the present work we assume that a
nucleus $A$ has closed-shell configurations (or sub-shell-closed
configurations) for both neutrons and protons where the pair correlation can
be neglected. Inclusion of the pairing is feasible with use of the method in
Ref.Matsuo (2015), but we leave it for future publications.

As shown in Ref.Saito and Matsuo (2021), the strength function
$S(M_{\lambda};iL_{i},L;E)$ for transitions from an excited state $iL_{i}$ is
rewritten as another strength function

$\displaystyle
S(F_{L};g,L;E)\equiv\sum_{k}|\bra{kL(E)}|\hat{F}_{L}|\ket{0^{+}_{\rm
g}}|^{2}=S(M_{\lambda};iL_{i},L;E)$ (5)

for an operator

$\displaystyle\hat{F}_{LM}\equiv\sum_{\mu M_{i}}\langle\lambda\mu
L_{i}M_{i}|LM\rangle[\hat{M}_{\lambda\mu},\hat{O}^{{\dagger}}_{iL_{i}M_{i}}],$
(6)

defined by a commutator between $\hat{M}_{\lambda\mu}$ and the RPA creation
operator $\hat{O}^{{\dagger}}_{iL_{i}M_{i}}$ of the low-lying excited state
$\ket{iL_{i}M_{i}}=\hat{O}^{{\dagger}}_{iL_{i}M_{i}}\ket{0^{+}_{\rm g}}$. Note
that the matrix elements in the strength function $S(F_{L};g,L;E)$ are those
between the ground state $\ket{0^{+}_{\rm g}}$ and the RPA excited states
$\ket{kLM(E)}$ for the operator $\hat{F}_{LM}$. Since the operator
$\hat{F}_{LM}$ is a one-body, though non-local, operator, it is possible to
calculate the strength function $S(F_{L};g,L;E)$ using an extended linear
response formulation of the cRPASaito and Matsuo (2021).

###  II.2 Linear response formalism

Here we recapitulate briefly an essence of the formulationSaito and Matsuo
(2021) by omitting the angular momentum algebra.

The non-local one-body operator $\hat{F}=[\hat{M},\hat{O}^{{\dagger}}_{i}]$ is
expressed as

$\displaystyle\hat{F}=\iint dxdyF(x,y)\psi^{{\dagger}}(x)\psi(y),$ (7)

where $\psi^{{\dagger}}(x)$ and $\psi(y)$ are the creation and annihilation
operators of the nucleon, and $F(x,y)$ is the coordinate representation of the
matrix element. Here $x$ ( and $y$) is shorthand notation of the space
coordinate $\vec{r}$ and the spin variable $\sigma=\uparrow,\downarrow$, i.e.
$x=\vec{r}\sigma$, $\int dx=\sum_{\sigma}\int d\vec{r}$. Using the linear
response $\delta\rho(x,y,\omega)$ of density matrix
$\rho(x,y)=\langle\hat{\rho}(x,y)\rangle$,
$\hat{\rho}(x,y)=\psi^{{\dagger}}(y)\psi(x)$ for the perturbation $\hat{F}$,
we can calculate the strength function
$S(F;\hbar\omega)=\sum_{k}|\bra{k(E)}\hat{F}\ket{0}|^{2}$ $(E=\hbar\omega)$ as

$\displaystyle S(F;\hbar\omega)=-\frac{1}{\pi}{\rm Im}\iint
dxdy\,F^{*}(x,y)\delta\rho(x,y,\omega).$ (8)

In TDDFT scheme the density matrix response $\delta\rho(x,y,\omega)$ obeys the
linear response equation

$\displaystyle\delta\rho(x,y,\omega)=\iint
dx^{{}^{\prime}}dy^{{}^{\prime}}R_{0}(x,y;y^{{}^{\prime}},x^{{}^{\prime}};\omega)(F(x^{{}^{\prime}},y^{{}^{\prime}})+v_{\rm
ind}(x^{{}^{\prime}},\omega)\delta(x^{{}^{\prime}}-y^{{}^{\prime}})).$ (9)

Here $v_{\rm ind}(x,\omega)$ is a time-dependent part of the Hartree-Fock (or
Kohn-Sham) potential originating from the density response, which is called
the induced field. Assuming that the HF (KS) mean-field $U[\rho](x)$ is local,
it is given by

$\displaystyle v_{\rm ind}(x,\omega)=\frac{\delta
U(x)}{\delta\rho}\delta\rho(x,\omega)$ (10)

with $\delta\rho(x,\omega)\equiv\delta\rho(x,x,\omega)$. The function $R_{0}$
is an unperturbed response function for the density matrix, and given by

$\displaystyle
R_{0}(x,y;y^{{}^{\prime}},x^{{}^{\prime}};\omega)\equiv\sum_{ph}\left\\{\frac{\bra{0}\hat{\rho}(x,y)\ket{ph}\bra{ph}\hat{\rho}(y^{{}^{\prime}},x^{{}^{\prime}})\ket{0}}{\hbar\omega-(\epsilon_{p}-\epsilon_{h})+i\eta}-\frac{\bra{0}\hat{\rho}(y^{{}^{\prime}},x^{{}^{\prime}})\ket{ph}\bra{ph}\hat{\rho}(x,y)\ket{0}}{\hbar\omega+(\epsilon_{p}-\epsilon_{h})+i\eta}\right\\}$
(11)

in the spectral representation in terms of the single-particle eigenstates in
the static HF (KS) potential $U[\rho_{0}](x)$.
$\ket{ph}=a^{{\dagger}}_{p}a_{h}\ket{0}$ is one-particle-one-hole
configuration, $\epsilon_{p}$ and $\epsilon_{h}$ are single-particle energies
of the particle and hole orbits. $\eta$ is a positive infinitesimal constant.

The response function $R_{0}$ is expressed also as

$\displaystyle R_{0}(x,y;y^{{}^{\prime}},x^{{}^{\prime}};\omega)=\sum_{h}$
$\displaystyle\left\\{\phi^{*}_{h}(y)G_{0}(x,x^{{}^{\prime}},\epsilon_{h}+\hbar\omega+i\eta)\phi_{h}(y^{{}^{\prime}})\right.$
$\displaystyle\left.+\phi^{*}_{h}(x^{{}^{\prime}})G_{0}(y^{{}^{\prime}},y,\epsilon_{h}-\hbar\omega-i\eta)\phi_{h}(x)\right\\},$
(12)

where $G_{0}(x,x^{{}^{\prime}},\epsilon)$ is the single-particle Green’s
function for the static HF (KS) potential $U_{0}$, and $\phi_{h}(x)$ is the
wave function of the hole orbit $h$. The Green’s function satisfies a proper
asymptotic boundary condition for $|x|,|x^{{}^{\prime}}|\to\infty$ so that it
describes scattering waves for unbound orbits.

We refer the readers to Ref.Saito and Matsuo (2021) for equations with the
angular momentum algebra.

###  II.3 The Zangwill-Soven method for partial photo-absorption cross section

In the linear response formalism discussed above, unbound particle states $p$
represent the scattering states. It is therefore possible to describe decay of
the RPA excited states with emission of a neutron (or a proton). Following
Zangwill and Soven Zangwill and Soven (1980), we shall calculate partial
$(\gamma,n)$ cross section for individual decay channels.

We first note that the strength function Eq. (5) is rewritten as

$\displaystyle S(\hat{F};\hbar\omega)$ $\displaystyle=-\frac{1}{\pi}{\rm
Im}\iiiint dxdydx^{{}^{\prime}}dy^{{}^{\prime}}v^{*}_{\rm
scf}(x,y,\omega)R_{0}(x,y;y^{{}^{\prime}},x^{{}^{\prime}};\omega)v_{\rm
scf}(x^{{}^{\prime}},y^{{}^{\prime}},\omega)$
$\displaystyle=\sum_{ph}\left|\bra{ph}\hat{V}_{\rm
scf}(\hat{F};\omega)\ket{0}\right|^{2}\delta(\hbar\omega-(\epsilon_{p}-\epsilon_{h})).$
(13)

Here $\hat{V}_{\rm scf}(\hat{F};\omega)$ is a one-body field, called the self-
consistent field Zangwill and Soven (1980); Nakatsukasa and Yanaba (2001),
defined as a sum of induced field $\hat{V}_{\rm ind}$ and the perturbing field
$\hat{F}$:

$\displaystyle\hat{V}_{\rm scf}(\hat{F};\omega)$
$\displaystyle\equiv\hat{F}+\hat{V}_{\rm ind}(\hat{F};\omega)=\iint
dxdy\,v_{\rm scf}(x,y,\omega)\hat{\rho}(y,x),$ (14) $\displaystyle v_{\rm
scf}(x,y,\omega)$ $\displaystyle\equiv F(x,y)+\frac{\delta
U(x)}{\delta\rho}\delta\rho(x,\omega)\delta(x-y).$ (15)

Note that matrix element $\bra{ph}\hat{V}_{\rm scf}(\hat{F};\omega)\ket{0}$ is
a $T$ matrix for the $(\gamma,n)$ reaction under consideration. This is seen
in

$\displaystyle\bra{ph}\hat{V}_{\rm
scf}(\hat{F};\omega)\ket{0}=\bra{ph}\hat{F}\ket{0}+\int dx\bra{ph}\frac{\delta
U(x)}{\delta\rho}\hat{\rho}(x)\ket{0}\delta\rho(x,\omega).$ (16)

The first term $\bra{ph}\hat{F}\ket{0}$ corresponds to the diagram
representation shown in Fig. 1(a). The density response $\delta\rho(x,\omega)$
in the second term can be expanded in infinite series

$\displaystyle\delta\rho=R_{0}F+R_{0}\frac{\delta
U}{\delta\rho}\delta\rho=R_{0}F+R_{0}\frac{\delta
U}{\delta\rho}R_{0}F+R_{0}\frac{\delta U}{\delta\rho}R_{0}\frac{\delta
U}{\delta\rho}R_{0}F+\cdots,$ (17)

representing symbolically the linear response equation (9). We notice that
this infinite series correspond to ring diagrams, and the matrix element
$\bra{ph}\hat{V}_{\rm scf}(\hat{F};\omega)\ket{0}$ is represented by the
diagrams shown in Fig. 1.

These diagrams indicate that the matrix element $\bra{ph}\hat{V}_{\rm
scf}(\hat{F};\omega)\ket{0}$ represents the transition amplitude for the
electromagnetic operator $\hat{M}$ between the low-lying excited state
$\ket{i}$ and a RPA excited state (represented by series of the ring diagrams)
which is connected to a specific particle-hole configuration $\ket{ph}$. In
the case when the particle state $p$ is an unbound single-particle state, it
represents the transition-matrix, $T$ matrix, of the $(\gamma,n)$ reaction
where the final state consists of a unbound neutron (specified by the particle
configuration $p$) and the residual nucleus $(A-1)$ with one-hole
configuration $a_{h}\ket{0}$. We call it the RPA $T$ matrix. We give more
detailed discussion in Appendix A.

Consequently each term in the r.h.s. of Eq. (13) is related to the partial
photoabsorption cross section for a $(\gamma,n)$ process, $A$ (state $i$)
$+\gamma\to(A-1)$ (one-hole state $a_{h}\ket{0}$) $+n$ (single-particle state
$p$):

$\displaystyle\sigma^{\lambda}_{i+\gamma\to
ph}(E_{\gamma})=f_{\lambda}(E_{\gamma})\left|\bra{ph}\hat{V}_{\rm
scf}(\hat{F};\omega)\ket{0}\right|^{2}\delta(\hbar\omega-(\epsilon_{p}-\epsilon_{h})).$
(18)

with $E_{\gamma}=\hbar\omega-E_{i}$.

Figure 1:  The diagrams representing the matrix element $\bra{ph}\hat{V}_{\rm
scf}(\hat{F};\omega)\ket{0}$. (a) and (b) correspond to the first and second
terms of r.h.s. of Eq.(16), respectively. Note that the electromagnetic
operator $\hat{M}$ acts also on the hole line.

The photo-absorption cross section can written also in terms of the single-
particle Green’s function. Inserting Eq.(II.2) to Eq.(13), we have the partial
photo-absorption cross section:

$\displaystyle\sum_{p(\epsilon_{p}=\epsilon)}\sigma^{\lambda}_{i+\gamma\to
ph}(E_{\gamma})$ $\displaystyle=-\frac{f_{\lambda}(E_{\gamma})}{\pi}{\rm
Im}\iiiint dxdydx^{{}^{\prime}}dy^{{}^{\prime}}\phi^{*}_{h}(y)v^{*}_{\rm
scf}(x,y,\omega)$ $\displaystyle\qquad\qquad\times
G_{0c}(x,x^{{}^{\prime}},\hbar\omega+\epsilon_{h}+i\eta)v_{\rm
scf}(x^{{}^{\prime}},y^{{}^{\prime}},\omega)\phi_{h}(y^{{}^{\prime}}),$ (19)

for final states consisting of an escaping neutron with energy
$\epsilon=\hbar\omega+\epsilon_{h}=E_{\gamma}+E_{i}+\epsilon_{h}$ and a one-
hole state $a_{h}\ket{0}$ of the residual nucleus $(A-1)$. Here $G_{0c}$ is
the single-particle Green’s function in which a contribution of bound orbits
is removed:

$\displaystyle G_{0c}(x,x^{{}^{\prime}},\epsilon)\equiv
G_{0}(x,x^{{}^{\prime}},\epsilon)-\sum_{i(\epsilon_{i}<0)}\frac{\phi_{i}(x)\phi^{*}_{i}(x^{{}^{\prime}})}{\epsilon-\epsilon_{i}}.$
(20)

###  II.4 Concrete expressions for $(\gamma,n)$ and $(n,\gamma)$ cross
sections

In actual application we consider the angular momentum algebra with the polar-
coordinate representation. For example $v_{\rm scf}(x,y,\omega)$ is
represented as

$\displaystyle v^{\rm scf}_{LM}(x,y,\omega)$
$\displaystyle=\sum_{ljm,l^{{}^{\prime}}j^{{}^{\prime}}m^{{}^{\prime}}}Y_{l^{{}^{\prime}}j^{{}^{\prime}}m^{{}^{\prime}}}(\hat{x})\frac{1}{\sqrt{2j^{{}^{\prime}}+1}}\langle
jmLM|j^{{}^{\prime}}m^{{}^{\prime}}\rangle\frac{v^{{\rm
scf}}_{L,l^{{}^{\prime}}j^{{}^{\prime}},lj}(r_{x},r_{y})}{r_{x}r_{y}}Y^{*}_{ljm}(\hat{y}).$
(21)

Here $Y_{ljm}$ is the spin spherical harmonics. We specify the escaping
neutron with energy $\epsilon$ and the partial wave quantum numbers $lj$, and
express the final state $\ket{ph}$ as $\ket{[\epsilon
l_{p}j_{p},n_{h}l_{h}j_{h}]_{LM}}=\sum_{m_{p}m_{h}}\braket{j_{p}m_{p}j_{h}m_{h}}{LM}a^{{\dagger}}_{\epsilon
l_{p}j_{p}m_{p}}a_{\widetilde{{n_{h}l_{h}j_{h}m_{h}}}}\ket{0}$ with $L$ being
the total angular momentum, which is identical to that of the RPA excited
state $A^{*}$. Accordingly we obtain the expression for the partial photo-
absorption cross section for the specific channel of neutron decay as

$\displaystyle\sigma^{\lambda}_{iL_{i}+\gamma\to[\epsilon
l_{p}j_{p}h]_{L}}(E_{\gamma})$
$\displaystyle=f_{\lambda}(E_{\gamma})\frac{1}{2L_{i}+1}\left|\langle[\epsilon
l_{p}j_{p}h]_{L}||\hat{V}_{\rm scf}(\hat{F}_{L};\omega)||0\rangle\right|^{2}$
$\displaystyle=-\frac{f_{\lambda}(E_{\gamma})}{\pi(2L_{i}+1)}$
$\displaystyle\quad\times{\rm Im}\iiiint
dr_{x}dr_{y}dr_{x^{{}^{\prime}}}dr_{y^{{}^{\prime}}}\phi^{*}_{n_{h}l_{h}j_{h}}(r_{y})v^{{\rm
scf}*}_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x},r_{y},\omega)$
$\displaystyle\quad\times
G_{0c,l_{p}j_{p}}(r_{x},r_{x^{{}^{\prime}}},\epsilon_{h}+\hbar\omega+i\eta)v^{\rm
scf}_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x^{{}^{\prime}}},r_{y^{{}^{\prime}}},\omega)\phi_{n_{h}l_{h}j_{h}}(r_{y^{{}^{\prime}}}).$
(22)

Note that $\hbar\omega=E_{\gamma}+E_{i}$ is the excitation energy of the state
$A^{*}$ of the nucleus $A$. Expression of $v^{\rm
scf}_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x^{{}^{\prime}}},r_{y^{{}^{\prime}}},\omega)$
and a closed form expression of the cross section are given in Appendix B.

Finally, we use the detailed balance to obtain the radiative neutron capture
cross section for the reaction $(A-1)+n\to A^{*}\to A^{**}+\gamma$ :

$\displaystyle\sigma^{\lambda}_{[\epsilon l_{p}j_{p}h]_{L}\to
iL_{i}+\gamma}(\epsilon)$
$\displaystyle=\frac{2L_{i}+1}{2j_{h}+1}\frac{E^{2}_{\gamma}}{2mc^{2}\epsilon}\sigma^{\lambda}_{iL_{i}+\gamma\to[\epsilon
l_{p}j_{p}h]_{L}}(E_{\gamma}).$ (23)

for an incident neutron with partial wave $l_{p}j_{p}$ and energy $\epsilon$
impinging on the $(A-1)$ nucleus with a one-hole configuration $h$, and gamma
transition to the excited state $\ket{iL_{i}M_{i}}$. The total cross section
is obtained by summing all the contributions of the partial waves $l_{p}j_{p}$
and the total angular momentum $L$ of the system.

##  III Numerical examples

###  III.1 Setting

We shall apply the theory to radiative neutron capture reaction on a neutron-
rich nucleus ${}^{139}{\rm Sn}$, which populates the ground and low-lying
excited states of ${}^{140}{\rm Sn}$ after gamma transition. We have chosen
these isotopes since the neutron separation energy of ${}^{140}{\rm Sn}$ is
estimated to be $\sim 3$ MeV Mas  and these isotopes are relevant to the
r-process. Another reason is that ${}^{140}{\rm Sn}$ has a subshell closure at
the $2f_{7/2}$ orbit for neutrons, and the pairing correlation neglected in
the present formulation is expected to be weak.

Numerical calculations are performed with the same setting as in our previous
paper Saito and Matsuo (2021). We use a Woods-Saxon potential in place of the
static self-consistent field $U_{0}$ and a Skyrme-type contact interaction as
the residual two-body force, given by

$\displaystyle
v_{ph}(\mathbf{r},\mathbf{r}^{{}^{\prime}})=\left\\{t_{0}(1+x_{0}P_{\sigma})+\frac{t_{3}}{12}(1+x_{3}P_{\sigma})\rho(r)\right\\}\delta(\mathbf{r}-\mathbf{r}^{{}^{\prime}}).$
(24)

The details including the parameters are the same as those in Saito and Matsuo
(2021). We describe the single-particle wave function by solving the radial
Schrödinger equation with the Runge-Kutta method up to a maximal radius
$R_{\rm max}=20$ fm (with interval $\Delta r=0.2$ fm). At $r=R_{\rm max}$ the
single-particle wave function is connected to the asymptotic wave, i.e. the
Hankel function with an appropriate (complex) wave number. The small real
constant $\eta$ in the response equation is set to $\eta=1.0\times 10^{-5}$ or
$0.1$ MeV.

In the present analysis, we intend to demonstrate effects of the RPA
correlations on the $(n,\gamma)$ reaction, in particular, roles of the
collectivity which may exist both in the scattering state of ${}^{139}{\rm
Sn}+n$ and the final states of ${}^{140}{\rm Sn}$. We focus on the low-lying
quadrupole and octupole vibrational states as well as the dipole and
quadrupole giant resonances. Other types of the particle-hole excitations such
as the soft dipole mode and non-collective excitations are also discussed. We
take into account the RPA correlation for the states with natural spin-parity
$L^{\pi}=1^{-},2^{+}$ and $3^{-}$ whereas we neglect it for other spin-
parities.

Table 1 shows the Woods-Saxon single-particle orbits. The highest occupied
neutron orbit in ${}^{140}{\rm Sn}$ is $2f_{7/2}$ with
$\epsilon_{2f_{7/2}}=-2.59$ MeV (the Fermi energy). The configuration of the
ground state of ${}^{139}{\rm Sn}$ is assumed to be a configuration with
neutron one-hole in $2f_{7/2}$ with spin-parity $j_{h}^{\pi}=7/2^{-}$. We
denote it ${}^{139}{\rm Sn}({\small 7/2}^{-})$ in the following. The RPA
calculation for ${}^{140}{\rm Sn}$ brings about several bound excited states
below the neutron separation energy $S_{1n}=2.59$ MeV (see Fig. 2 of Ref.
Saito and Matsuo (2021)). Among them we adopt the lowest-lying excited states,
i.e. $2^{+}_{1,2}$ and $3^{-}_{1}$ states, as the final states $\ket{iL_{i}}$
of the ${}^{139}{\rm Sn}(n,\gamma)^{140}{\rm Sn}$ reaction. The excitation
energy of these states are $E_{2^{+}_{1}}=0.888$ MeV,$\,E_{2^{+}_{2}}=1.093$
MeV,$\,E_{3^{-}_{1}}=1.768$ MeV. As seen in the RPA forward amplitudes
$X_{ph}$ of these states (Tables 2 and 3), the $3^{-}_{1}$ state exhibits a
moderately strong collectivity, i.e. it contains many particle-hole
configurations both in neutrons and protons, typical of the surface
vibrational octupole state. The $2^{+}_{1}$ and $2^{+}_{2}$ states consist
mostly of two neutron particle-hole configurations,
$(1h_{9/2})(2f_{7/2})^{-1}$ and $(3p_{3/2})(2f_{7/2})^{-1}$ while the
correlation causes strong mixing among them.

For the gamma transition, we mostly discuss the E1 transition, but we also
mention briefly the E2 case. We take into account all possible angular and
spin quantum numbers of the total system (both for the scattering state of
${}^{139}{\rm Sn}+n$ and the final states of ${}^{140}{\rm Sn}$) as long as
they are allowed by the angular momentum coupling of the adopted final states
($L_{i}^{\pi}=0_{\rm g.s.}^{+},2^{+}_{1,2}$ and $3^{-}_{1}$) and the
multipolarity ($\lambda=1,2$) of the gamma ray. Namely, we describe
${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn}(1^{-})\to^{140}{\rm
Sn(g.s.)}+\gamma$,${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm
Sn}(1^{-},2^{-},3^{-})\to^{140}{\rm Sn(2^{+}_{1,2})}+\gamma$,${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn}(2^{+},3^{+},4^{+})\to^{140}{\rm
Sn(3^{-}_{1})}+\gamma$ for the E1 transition while ${}^{139}{\rm Sn}({\small
7/2}^{-})+n\to^{140}{\rm Sn}(2^{+})\to^{140}{\rm
Sn(g.s.)}+\gamma$,${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm
Sn}(0^{+},1^{+},2^{+},3^{+},4^{+})\to^{140}{\rm
Sn(2^{+}_{1,2})}+\gamma$,${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm
Sn}(1^{-},2^{-},3^{-},4^{-},5^{-})\to^{140}{\rm Sn(3^{-}_{1})}+\gamma$. for
the E2 case. We include all the partial waves of the incident neutron which
are allowed by the coupling to the the spin-parity $7/2^{-}$ of the target
nucleus ${}^{139}{\rm Sn}$. Its maximum is $l_{p},j_{p}=8,17/2$.

Table 1:  Single-particle energies of the adopted Woods-Saxon potential for ${}^{140}{\rm Sn}$. Several orbits around the Fermi energy (indicated by lines) are listed. neutron | $\epsilon\,[{\rm MeV}]$ |  | proton | $\epsilon\,[{\rm MeV}]$
---|---|---|---|---
$2f_{5/2}$ | -0.31 |  | $1h_{11/2}$ | -11.40
$3p_{1/2}$ | -0.81 |  | $2d_{3/2}$ | -11.61
$3p_{3/2}$ | -1.46 |  | $2d_{5/2}$ | -14.06
$1h_{9/2}$ | -1.53 |  | $1g_{7/2}$ | -15.08
$2f_{7/2}$ | -2.59 |  | $1g_{9/2}$ | -19.97
$1h_{11/2}$ | -6.64 |  | $2p_{1/2}$ | -21.75
$3s_{1/2}$ | -8.65 |  | $2p_{3/2}$ | -23.02
$2d_{3/2}$ | -8.65 |  | $1f_{5/2}$ | -24.81
$2d_{5/2}$ | -10.40 |  |  |
$1g_{7/2}$ | -10.96 |  |  |
$1g_{9/2}$ | -14.64 |  |  |
Table 2:  The RPA forward amplitudes $X_{ph}$ of the $2^{+}_{1}$ and $2^{+}_{2}$ state. Particle-hole configurations with large amplitude $|X_{ph}|>0.1$ are listed. The RPA backward and forward amplitudes $X_{ph}$ and $Y_{ph}$ are calculated using a method of Ref.Shimoyama and Matsuo (2013). neutron config. | $X_{ph}^{2^{+}_{1}}$ | $X_{ph}^{2^{+}_{2}}$
---|---|---
$(1h_{9/2})(2f_{7/2})^{-1}$ | -0.601 | 0.791
$(3p_{3/2})(2f_{7/2})^{-1}$ | 0.789 | 0.600
Table 3:  The RPA forward amplitudes $X_{ph}$ of the $3^{-}_{1}$ state. Particle-hole configurations with large amplitude $|X_{ph}|>0.1$ are listed. The neutron single-particle orbit $1i_{13/2}$ is a resonance in the continuum. neutron config. | $X^{3^{-}_{1}}_{ph}$ |  | proton config. | $X^{3^{-}_{1}}_{ph}$
---|---|---|---|---
$(1i_{13/2})(2f_{7/2})^{-1}$ | 0.831 |  | $(1h_{11/2})(1g_{9/2})^{-1}$ | -0.285
$(1i_{13/2})(1h_{11/2})^{-1}$ | 0.354 |  | $(1g_{7/2})(2p_{1/2})^{-1}$ | 0.203
$(1h_{9/2})(2d_{3/2})^{-1}$ | -0.299 |  | $(2d_{5/2})(2p_{1/2})^{-1}$ | 0.176
$(1h_{9/2})(1g_{7/2})^{-1}$ | -0.189 |  | $(2d_{5/2})(2p_{3/2})^{-1}$ | 0.135
$(2f_{5/2})(3s_{1/2})^{-1}$ | 0.134 |  | $(1j_{15/2})(1g_{9/2})^{-1}$ | 0.129
$(2g_{9/2})(2f_{7/2})^{-1}$ | 0.133 |  | $(2f_{7/2})(1g_{9/2})^{-1}$ | -0.129
$(2f_{5/2})(2d_{3/2})^{-1}$ | -0.126 |  | $(2d_{3/2})(2p_{3/2})^{-1}$ | -0.120
$(3p_{3/2})(2d_{3/2})^{-1}$ | -0.112 |  | $(3p_{3/2})(1g_{9/2})^{-1}$ | -0.103
$(2j_{15/2})(1g_{9/2})^{-1}$ | 0.108 |  | $(1g_{7/2})(1f_{5/2})^{-1}$ | 0.102
$(2f_{5/2})(1g_{7/2})^{-1}$ | -0.101 |  |  |

###  III.2 Low energy $(n,\gamma)$ cross section with E1 transition

Figure 2 shows the cross sections of neutron capture with E1 for low neutron
kinetic energy relevant to the r-process, calculated separately for each final
state. A characteristic feature is that the cross sections exhibit significant
differences for different final states. In particular most dominant transition
at low energy is the ones to the $2_{1}^{+}$ and $2_{2}^{+}$ states rather
than to the ground state. Another noticeable feature is that the transition to
the octupole vibrational state $3^{-}_{1}$ is present although the absolute
magnitude is small. It is noted that there is no negative parity one-particle
one-hole ($1p1h$) configuration with energy smaller the neutron separation
energy $2.59$ MeV. The collective $3^{-}_{1}$ is exceptional as this state
emerges at low energy due to the RPA correlation. The third observation is
that the cross sections exhibit many resonance-like behaviors above $~{}2$ MeV
whereas they are smooth below $~{}2$ MeV. In the following we shall discuss
these features in more details.

Figure 2:  The calculated $(n,\gamma)$ cross sections for ${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn}+\gamma$ with E1 transitions
populating different low-lying states in ${}^{140}{\rm Sn}$; the ground state
(plotted with a green curve), the low-lying $2^{+}_{1}$ and $2^{+}_{2}$ states
(red and blue curves respectively), and the octupole vibrational state
$3^{-}_{1}$ (yellow curve). The horizontal axis is kinetic energy
$\epsilon_{\rm kin}$ of the incident neutron. The smoothing parameter is
$\eta=10^{-5}$ MeV.

####  III.2.1 Transitions to the ground state

Let us discuss the $(n,\gamma)$ reaction where the gamma transition populates
directly the ground state of ${}^{140}{\rm Sn}$, whose cross section is the
green curve in Fig. 2. In order to analyze the structures seen in Fig. 2, we
decompose the cross section with respect to the partial waves of the incident
neutron. The decomposed cross sections are shown in Fig. 3. In the present
case the relevant spin-parity of the total system $n+^{139}{\rm Sn}$ or that
of excited continuum state of ${}^{140}{\rm Sn}$ is $1^{-}$. Thus the partial
waves of the incident neutron are limited to $d_{5/2}$ and $g_{7/2,9/2}$ due
to the angular momentum coupling to the ground state ${}^{139}{\rm Sn}({\small
7/2}^{-})$.

The cross section at low energy $\leq 1$ MeV is dominated by the
$d_{5/2}$-wave since it has the lowest orbital angular momentum among the
available partial waves. The $s$-wave capture is forbidden due to the angular
coupling rule. The low energy behavior of the partial cross sections exhibits
a scaling $\propto\epsilon^{l_{p}-1/2}$, which points to direct transitions
from non-resonant partial waves to the ground state of ${}^{140}{\rm Sn}$. It
is seen that a rather broad peak around $\epsilon_{\rm kin}=4$ MeV is that in
the $g_{9/2}$-wave. It is the E1 transition from the single-particle $g_{9/2}$
resonance to the bound $2f_{7/2}$ orbit which is vacant in ${}^{139}{\rm Sn}$.
These features reflect the single-particle nature of the soft dipole
excitation of ${}^{140}{\rm Sn}$.

Several sharp peaks in the energy range $\epsilon_{\rm kin}>4.5$ MeV are
however cannot be of the single-particle origin. Indeed they originates from
non-collective states whose main components are one-particle-one-hole ($1p1h$)
configurations $\nu[(3p_{3/2})(3s_{1/2})^{-1}]$,
$\nu[(3p_{1/2})(3s_{1/2})^{-1}]$, $\nu[(3p_{3/2})(2d_{3/2})^{-1}]$,
$\nu[(3p_{1/2})(2d_{3/2})^{-1}]$, $\nu[(3p_{3/2})(2d_{5/2})^{-1}]$, and
$\nu[(3p_{1/2})(2d_{5/2})^{-1}]$. Although the particle and hole orbits of
these configurations are both bound orbits (see Table 1), they couple to the
scattering states of $n+^{139}{\rm Sn}$ via the residual interaction as is
represented in Fig.4, and they form narrow resonances. Note that the non-
collective states with proton $1p1h$ configurations also appear as narrow
resonances.

In Fig. 5 we magnify the resonance structures in the range of $\epsilon_{\rm
kin}=5-7$ MeV. We see clearly interferences between the resonances and the
non-resonant capture. Note that dominant partial wave contributing to the peak
as well as the interference pattern are quite different for different
resonances, reflecting the non-collective nature of these resonance states.

Figure 3:  The calculated partial $(n,\gamma)$ cross sections for
${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn}(1^{-})\to^{140}{\rm
Sn(g.s.)}+\gamma$ for E1 transitions, plotted separately for different partial
waves of the incident neutron; $d_{5/2}$ (green curve), $g_{7/2}$ (orange),
and $g_{9/2}$ (brown). The horizontal axis is the neutron kinetic energy
$\epsilon_{\rm kin}$. The smoothing parameter is $\eta=10^{-5}$ MeV.

Figure 4: The diagrams representing $(n,\gamma)$ reaction of ${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn(1^{-})}\to^{140}{\rm
Sn(g.s.)+\gamma}$.

Figure 5:  The same as Fig. 3, but a magnification in the range of
$\epsilon_{\rm kin}=5.0-7.0$ MeV.

####  III.2.2 Transitions to the low-lying states $2^{+}_{1,2}$

The transitions decaying to the low-lying quadrupole states $2^{+}_{1,2}$ are
most dominant at low energy as seen in Fig. 2.(red and blue curves). Here we
shall discuss the one for $2_{1}^{+}$ since the cross sections for $2_{1}^{+}$
and $2_{2}^{+}$ behave similarly.

The spin-parity of the total system relevant to the present case is
$L^{\pi}=1^{-},2^{-}$ and $3^{-}$. Figure 6 shows the partial cross section
decomposed with respect to the partial waves $l_{p}j_{p}$ of the incident
neutron, plotted separately for $L^{\pi}=3^{-}$ and $2^{-}$ in panels (a) and
(b), respectively.

It is seen from comparison of Fig. 2 and Fig. 6 that a large cross section at
low energy is attributed to the $s$-wave capture. A noticeable feature is that
the $s$-wave capture is present only for $L^{\pi}=3^{-}$ but not for
$L^{\pi}=2^{-}$. Possible partial waves of incident neutron allowed for the
total spin-parity $L^{\pi}=3^{-}$ are $s,d,g$ and $i$ due to the the angular
momentum coupling to spin $7/2^{-}$ of the target ${}^{139}{\rm Sn}$. However
for $L^{\pi}=2^{-}$ allowed partial waves are $d,g$ and $i_{11/2}$ for
$L^{\pi}=2^{-}$, excluding the $s$-wave. This is the same reason why there is
no $s$-wave capture in the transitions to the ground state. Consequently the
transitions to $2^{+}_{1,2}$ dominate at low energy.

Another significant feature is that many resonance-like structures with narrow
width are seen for $L^{\pi}=3^{-}$, but not for $L^{\pi}=2^{-}$, though the
impact on the absolute cross section is small. Similarly to the transition to
the ground state (cf the previous subsection), the narrow resonances originate
from the coupling between the non-resonant continuum states ( $s,d,g$ and $i$
wave coupled to ${}^{139}{\rm Sn}$ with $7/2^{-}$) and non-collective $1p1h$
configurations of both neutrons and protons, e.g.
$\nu[(1h_{9/2})(3s_{1/2})^{-1}]$, $\nu[(2f_{5/2})(3s_{1/2})^{-1}]$,
$\nu[(1h_{9/2})(2d_{3/2})^{-1}]$, $\nu[(3p_{3/2})(2d_{3/2})^{-1}]$, and
$\nu[(2f_{5/2})(2d_{3/2})^{-1}]$ (cf. Fig. 7(a)). However, the spin-
independent residual interaction Eq.(24) does not cause mixing among $1p1h$
states with unnatural spin-parity $2^{-}$. In this case only the direct
transition shown by the diagram Fig. 7(b) is relevant and the non-collective
$1p1h$ states does not show up as resonances.

Figure 6:  The calculated partial $(n,\gamma)$ cross sections for
${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm
Sn}(2^{-},3^{-})\to^{140}{\rm Sn}(2^{+}_{1})+\gamma$ for E1 transitions,
plotted separately for different partial waves of the incident neutron;
$s_{1/2}$ (red curve), $d_{3/2}$ (green), and $d_{5/2}$ (dark green) etc.. The
panel (a) is for the total spin $L^{\pi}=3^{-}$, and (b) for $L^{\pi}=2^{-}$.
The horizontal axis is the neutron kinetic energy $\epsilon_{\rm kin}$. The
smoothing parameter is $\eta=10^{-5}$ MeV.

Figure 7: The diagrams representing $(n,\gamma)$ reaction of ${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn(2^{-},3^{-})}\to^{140}{\rm
Sn(2^{+}_{1})+\gamma}$; (a) for the total spin $L^{\pi}=3^{-}$ and (b) for
$L^{\pi}=2^{-}$.

####  III.2.3 Transitions to the octupole vibrational state $3^{-}_{1}$

Let us discuss the $(n,\gamma)$ reaction whose final state is the octupole
vibrational state $3^{-}_{1}$ at $E=1.768$ MeV. The relevant spin-parity of
the total system is $L^{\pi}=2^{+},3^{+}$ and $4^{+}$, and we show in Fig. 8
partial $(n,\gamma)$ cross sections for $L^{\pi}=2^{+}$ and $3^{+}$
separately. The low energy cross section $\epsilon_{\rm kin}\leq 1$ MeV are
dominated by the $p$-wave capture in both cases since $l=1$ is the smallest
orbital angular momentum of the neutron partial waves allowed both for
$L^{\pi}=2^{+}$ and $3^{+}$. Note that the capture in the $p$-wave (and other
negative-parity partial waves) would not be present if the collective octupole
state did not exist below the neutron separation energy.

The cross section for $L^{\pi}=2^{+}$ (Fig. 8(a)) shows a significant
enhancement around $\epsilon_{\rm kin}\sim 2-3$ MeV with resonant structures.
These resonance behaviors originate from excited $2^{+}$ states of
${}^{140}{\rm Sn}$ at $E=4.8$ and $5.6$ MeV, which appear as relatively large
peaks in the E2 and isoscalar quadrupole strength functions shown in Fig. 2(b)
of Ref. Saito and Matsuo (2021). These quadrupole states have some
collectivity, i.e. consisting of coherent neutron particle-hole admixture in
addition to main proton particle-hole configurations
$\pi[(1g_{9/2})(1g_{9/2})^{-1}]$, and $\pi[(2d_{5/2})(1g_{9/2})^{-1}]$. We
remark that the collectivity in the quadrupole states is reflected in Fig.
8(a); the neutron partial waves $p_{3/2}$ and $f_{7/2,5/2}$ contribute
coherently to the resonance structures. Such a coherence effect is not seen in
the narrow resonance structures shown in Figs. 3, 5 and 6 where the relevant
resonant states have a character of non-collective $1p1h$ excitations.

As for the case of $L^{\pi}=3^{+}$ (Fig. 8(b)), the energy dependence is
smooth without narrow resonance structures since there is no effect of the
correlation, similarly to Fig. 6(b). The process here is the direct capture
represented by the diagram Fig. 9(b).

Figure 8:  The calculated partial $(n,\gamma)$ cross sections for
${}^{139}{\rm Sn}({\small 7/2}^{-})+n\to^{140}{\rm
Sn}(2^{+},3^{+})\to^{140}{\rm Sn}(3^{-}_{1})+\gamma$ for E1 transitions,
plotted separately for different partial waves of the incident neutron;
$p_{1/2}$ (yellow curve), $f_{5/2}$ (blue), and $h_{9/2}$ (pink) etc.. The
panel (a) is for the total quantum number $L^{\pi}=2^{+}$, and (b) for
$L^{\pi}=3^{+}$. The horizontal axis is the neutron kinetic energy
$\epsilon_{\rm kin}$. The smoothing parameter is $\eta=10^{-5}$ MeV.

Figure 9: The diagrams representing $(n,\gamma)$ reaction of ${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn(2^{+},3^{+})}\to^{140}{\rm
Sn(3^{-}_{1})+\gamma}$.

###  III.3 Cross section at higher energy

Let us examine a global behavior of the calculated cross sections in a wide
energy region covering up to $\epsilon_{\rm kin}\sim 20$ MeV although the
result at high energies is not relevant to the r-process. We remark also that
the model space of the present theory does not include multi-particle-multi-
hole configurations, needed to describe the high level density and the complex
structure of compound states at high excitation energy. In other words the
present calculation lacks a part of the $(n,\gamma)$ processes that is usually
modeled in terms of the statistical treatment of the compound states.

Figure 10 shows the $(n,\gamma)$ cross section for transitions to the low-
lying $2_{1,2}^{+}$ and $3_{1}^{-}$ states as well as the ground state in
${}^{140}{\rm Sn}$. It is the same as Figure 2 but for $\epsilon_{\rm kin}<20$
MeV. The smoothing constant $\eta$ is set to $\eta=0.1$ MeV, which washes out
narrow resonances.

We see several resonance-like peaks having sizable width at $\epsilon_{\rm
kin}\approx 4$ MeV, 8 MeV, 10 MeV and a bunch of peaks around $\epsilon_{\rm
kin}\approx 10-13$ MeV. The large peak at $\epsilon_{\rm kin}\approx 4$ MeV is
the one which we already discussed in subsection III.2.1. It is essentially a
single-particle resonance in the partial wave $g_{9/2}$, from which a single-
particle E1 transition to the $2f_{7/2}$ orbit occurs.

The peak at $\epsilon_{\rm kin}\approx 8$ MeV seen in the transition to the
$2^{+}_{1,2}$ states reflects the single-particle $i_{11/2}$ resonance. Note
that this peak is seen only for the transtion to the $2^{+}_{1,2}$ states, but
not to the ground state. The single-particle E1 transition is possible from
the resonant $i$ orbit to the bound $1h_{9/2}$ orbit (see Fig. 7(a)), which is
occupied in the $2^{+}_{1,2}$ states as one of the main particle-hole
configurations (see Table. 2).

The largest peak at $\epsilon_{\rm kin}\approx 10$ MeV is present in the
transition to the octupole state $3^{-}_{1}$. It originates from a single-
particle resonance in the partial wave $j_{15/2}$. We remark that the
collectivity of the low-lying octupole state $3^{-}_{1}$ plays an essential
role. The single-particle $j_{15/2}$ resonance has sizable E1 matrix element
for a transition to the neutron $1i_{13/2}$ orbit, which is, however, unbound
(resonance) in the present model. Therefore the capture is not possible via
the single-particle process. In contrast, the $3^{-}_{1}$ state with the
collective character of surface vibration can be a final state of the E1
transition from the neutron single-particle $j_{15/2}$ resonance since it
contains configuration $(1i_{13/2})(2f_{7/2})^{-1}$ as one of main $1p1h$
components (cf. Table 3).

A distribution of several peaks around $\epsilon_{\rm kin}\approx 10-13$ MeV
seen in the transition to the ground state (green curve) is another feature
that points to importance of the collectivity in the $(n,\gamma)$ reaction.
These peaks are not related to the single-particle resonance, but they
originate from the giant dipole resonance (GDR) in ${}^{140}{\rm Sn}$ with
spin-parity $1^{-}$ distributed around excitation energy $E\approx 12-16$ MeV.
The capture proceeds through the GDR, which enhances gamma decay to the ground
state.

Figure 10:  The calculated $(n,\gamma)$ cross sections for ${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn}$ with E1 transitions populating
different low-lying states in ${}^{140}{\rm Sn}$; the ground state (plotted
with a green curve), the low-lying $2^{+}_{1}$ and $2^{+}_{2}$ states (red and
blue curves respectively), and the octupole vibrational state $3^{-}_{1}$
(yellow curve). The horizontal axis is the neutron kinetic energy
$\epsilon_{\rm kin}$. The smoothing parameter is $\eta=0.1$ MeV.

###  III.4 $(n,\gamma)$ reaction with E2 transition

We have calculated also the cross section for E2 transitions using the same
formalism. The obtained cross sections are shown in Fig. 11 and 12 for the low
energy part and the high energy part, respectively. The cross section for the
E2 transition is smaller than those for E1 as is seen by comparing with the
results for the E1 case (Figs. 2 and 10). An apparent reason is the
kinematical factor which suppresses the high multipole transitions.

A noticeable difference from the E1 case is that the transitions to the ground
state is dominant over other transitions feeding the low-lying excited states
$2_{1,2}^{+}$ and $3_{1}^{-}$. Here we note that the adopted E2 operator acts
only on protons, neglecting the recoil correction of an order of $A^{-1}$. For
this reason both the continuum excited state and the low-lying excited states
need to include proton $1p1h$ configurations in order for the E2 transition to
occur. Note that the calculated low-lying states in neutron-rich ${}^{140}{\rm
Sn}$ has dominant neutron character where the proton component is relatively
small. This is especially the case for the $2^{+}$ states (cf. Table 2).
Consequently, the transition to the low-lying $2_{1,2}^{+}$ and $3_{1}^{-}$
states are suppressed. For the transition to the ground state, on the
contrary, the E2 transition occurs as far as the continuum excited states
contains both proton $1p1h$ configurations and a scattering neutron orbit.
This is the case for the resonance peaks around $\epsilon_{\rm kin}=2.2$ and
$3.0$ MeV (corresponding to $2^{+}$ states at excitation energy at $E=4.8$ and
$5.6$ MeV), and the broad peaks around $\epsilon_{\rm kin}\approx 10$ MeV and
$\epsilon_{\rm kin}\approx 15-22$ MeV, which correspond to the ISGQR and
IVGQR, respectively.

Figure 11:  The calculated $(n,\gamma)$ cross sections for ${}^{139}{\rm
Sn}({\small 7/2}^{-})+n\to^{140}{\rm Sn}$ with E2 transitions populating
different low-lying states in ${}^{140}{\rm Sn}$; the ground state (plotted
with a green curve), the low-lying $2^{+}_{1}$ states (red curve), and the
octupole vibrational state $3^{-}_{1}$ (yellow curve). The horizontal axis is
the neutron kinetic energy $\epsilon_{\rm kin}$. The cross section for
$2^{+}_{2}$ is not seen as it is smaller than the plotted range. The smoothing
parameter is $\eta=10^{-5}$ MeV.

Figure 12:  The same as Fig. 11, but for higher neutron kinetic energy and
larger smoothing parameter $\eta=0.1$ MeV.

##  IV Conclusion

The continuum random-phase approximation (cRPA) with use of the Green’s
function in the coordinate-space representation enables us to describe various
types of correlated particle-hole excitations and their coupling to unbound
single-particle orbits. Utilizing this feature of the cRPA, we have formulated
a quantum many-body theory to calculate the cross section of the radiative
neutron capture reaction on neutron-rich nuclei with a scope of application to
the r-process nucleosynthesis. We have given, in our previous publication
Matsuo (2015), a formulation for the $(n,\gamma)$ reaction, in which the final
state of the gamma transition is restricted to the ground state of an even-
even nucleus. In the present study, we have introduced an extended formulation
so that we can describe decay channels in which the gamma transition populates
low-lying excited states. With the cRPA approach it is possible to describe
various excitation modes present in the scattering state of $n+(A-1)$,
including soft dipole excitation, the giant resonances as well as non-
collective excitations and the single-particle resonances. Furthermore, we are
able to takes into account the correlation in the final states of the gamma
transition, e.g. the low-lying quadrupole and octupole vibrational states.

In order to demonstrate the new features taken into account in the cRPA
description of the $(n,\gamma)$ reaction, we have performed numerical study of
the neutron capture of neutron-rich nucleus ${}^{139}{\rm Sn}$ followed by E1
or E2 $\gamma$ transtions to the ground state or low-lying excited states
$2^{+}_{1},2^{+}_{2}$ and $3^{-}_{1}$ in ${}^{140}{\rm Sn}$. The RPA
correlation in the scattering states produces resonance structures in the
$(n,\gamma)$ cross section, which originates from particle-hole excitations
with both collective and non-collective characters. There appear resonances
with rather wide widths, arising from single-particle resonances and
collective excitations such as the giant resonances. In addition narrow
resonances emerge from non-collective excitation modes, which are particle-
hole excitations from bound to bound orbits of neutrons as well as those of
protons. The correlation in the final states, i.e. the low-lying quadrupole
and octupole states ${}^{140}{\rm Sn}$, has also significant impact on the
$(n,\gamma)$ cross section. A particular example is the transitions to the
$3^{-}_{1}$ state, which is not present in the single-particle model as this
collective state appears below the neutron separation energy only by including
the RPA correlation. Furthermore additional neutron partial waves contribute
to the capture, compared with those in the single-particle model, because of
the strong configuration mixing of many one-particle-one-hole states in the
$3^{-}_{1}$ state.

Finally we remark a scope for future developments of the present study. The
pair correlation neglected in the present formulation can be taken into
account by a straightforward extension from the cRPA to the continuum
quasiparticle random-phase approximation (cQRPA). This will be done by
combining the formulation in Ref. Matsuo (2015) and that in the present study.
This extension will makes it possible to describe the $(n,\gamma)$ reaction
systematically for open-shell spherical nuclei. Extension of the model space
to that beyond the RPA, including many-particle many-hole states, is another
direction of future development, which might be required for description of
the neutron capture in nuclei close to the stability line, e.g. in application
to the s-process and capture reactions at high neutron energy. Description of
the case where the final state nucleus is an odd-$A$ or an odd-odd nucleus
also need to be studied in future.

##  V Acknowledgments

The authors thank Kazuyuki Sekizawa and Kenichi Yoshida for valuable
discussions. This work was supported by the JSPS KAKENHI (Grant No. 20K03945).

##  Appendix A $T$ matrix for $(\gamma,n)$ reaction

We shall discuss a relation between the $T$ matrix $T^{\rm
RPA}_{(\gamma,n)}=\bra{ph}\hat{V}_{\rm scf}(\hat{F};\omega)\ket{0}$ given in
the present cRPA theory and the $T$ matrix of the $(\gamma,n)$ reaction in the
general theory of nuclear reaction.

Following the general reaction theory Taylor (2006), the $T$ matrix for the
$(\gamma,n)$ reaction in the prior form is given by

$\displaystyle
T_{(\gamma,n)}=\bra{\Psi^{(-)}_{n+(A-1)}(E)}\hat{M}\ket{\Psi_{i,A}},$ (25)
$\displaystyle\ket{\Psi^{(-)}_{n+(A-1)}(E)}=\ket{\phi_{n}\Psi_{A-1}}+\hat{G}^{(-)}(E)\hat{V}\ket{\phi_{n}\Psi_{A-1}},$
(26)

for the gamma-ray absorption caused by the multipole field $\hat{M}$ with a
kinematical factor and the angular momentum omitted. Here $\ket{\Psi_{i,A}}$
is the target nucleus $A$ at state $i$ and $\ket{\phi_{n}\Psi_{A-1}}$ is the
final state in the $n+(A-1)$ channel consisting of an escaping neutron
$\phi_{n}$ and the daughter nucleus $(A-1)$ (the state $\Psi_{A-1}$) .
$\ket{\Psi^{(-)}_{n+(A-1)}(E)}$ is an energy eigenstate with an incoming
boundary condition, which is given as a solution of the Lippman-Schwinger
equation. Here $\hat{V}$ is the interaction between the neutron and the
subsystem $(A-1)$, and $\hat{G}^{(\pm)}(E)$ is a many-body Green’s function
$\hat{G}^{(\pm)}(E)=\frac{1}{E-\hat{H}\pm i\eta}$ for the total Hamiltonian
$\hat{H}=\hat{H}_{0}+\hat{V}$ with $\hat{H}_{0}$ describing non-interacting
$n+(A-1)$ system. Note that the $T$ matrix can be written also as

$\displaystyle T_{(\gamma,n)}$
$\displaystyle=\bra{\phi_{n}\Psi_{A-1}}\hat{M}\ket{\Psi_{i,A}}+\bra{\phi_{n}\Psi_{A-1}}\hat{V}\hat{G}^{(+)}(E)\hat{M}\ket{\Psi_{i,A}}$
(27)

which is the post form representation of the $T$ matrix. Besides the photo-
nuclear interaction $\hat{M}$, the interaction $\hat{V}$ also contributes to
the $T$ matrix in the post form as represented by the second term in the
r.h.s. of Eq.(27).

The RPA $T$ matrix (c.f. Eqs.(14) and (16)) is written as

$\displaystyle T^{\rm
RPA}_{(\gamma,n)}=\bra{ph}\hat{F}\ket{0}+\bra{ph}\hat{V}_{\rm
ind}(\hat{F};\omega)\ket{0}$ (28)

in terms of the induced field

$\displaystyle\hat{V}_{\rm ind}(\hat{F};\omega)=\int
dx\frac{\delta\hat{U}(x)}{\delta\rho}\delta\rho(x,\omega),\ \ \
\frac{\delta\hat{U}(x)}{\delta\rho}=\frac{\delta{U}(x)}{\delta\rho}\hat{\rho}(x).$
(29)

The first and second terms in Eq.(28) correspond to the respective terms in
Eq.(27). To see this, we first note correspondence in the initial and final
states:

$\displaystyle\bra{ph}\leftrightarrow\bra{\phi_{n}\Psi_{A-1}},\ \ \
\hat{O}_{i}^{\dagger}\ket{0}\leftrightarrow\ket{\Psi_{i,A}}$ (30)

assuming the excited state $\ket{\Psi_{i,A}}$ of the nucleus $A$ is described
by the RPA creation operator $\hat{O}_{i}^{\dagger}$ acting on the ground
state $\ket{0}$ in the static selfconsistent field. The particle-hole state
$\ket{ph}=a_{p}^{\dagger}a_{h}\ket{0}$ corresponds to the scattering final
state $\ket{\phi_{n}\Psi_{A-1}}$ if we assign the daughter nucleus
$\Psi_{A-1}$ as a one-hole state $a_{h}\ket{0}$ whereas the particle
$a_{p}^{\dagger}$ to the neutron scattering wave $\phi_{n}$. We see then
correspondence

$\displaystyle\bra{ph}\hat{F}\ket{0}=\bra{ph}\hat{M}\ket{i}\leftrightarrow\bra{\phi_{n}\Psi_{A-1}}\hat{M}\ket{\Psi_{i,A}}$
(31)

between the first terms in the r.h.s. of Eq.(28) and (27). See also Fig.1(a).

Concerning the second term of the RPA $T$ matrix, the induced field

$\displaystyle\hat{V}_{\rm ind}(\omega)=\int
dx\frac{\delta{U}(x)}{\delta\rho}\hat{\rho}(x)\delta\rho(x,\omega)$ (32)

is a mean-field part of the RPA residual interaction $\hat{V}_{\rm
RPA}=\frac{1}{2}\int
dx\frac{\delta{U}(x)}{\delta\rho}\hat{\rho}(x)\hat{\rho}(x)$. It is associated
with the perturbation represented by the density response
$\delta\rho(x,\omega)$, which is brought by the action
$\hat{F}=[\hat{M},\hat{O}_{i}^{\dagger}]$ of the multipole field $\hat{M}$ on
the initial state $\hat{O}_{i}^{\dagger}\ket{0}$. We thus see a correspondence

$\displaystyle\hat{V}_{\rm
ind}(\hat{F};\omega)\ket{0}\leftrightarrow\hat{V}\hat{G}^{(+)}(E)\hat{M}\ket{\Psi_{i,A}}$
(33)

under an approximation that the interaction $\hat{V}$ is replaced by the RPA
residual interaction $\hat{V}_{\rm RPA}$ and the model space is limited to the
one-particle-one-hole subspace. Thus we find that the second terms in the
r.h.s. of Eq.(28) and (27) corresponds to each other:

$\displaystyle\bra{ph}\hat{V}_{\rm
ind}(\hat{F};\omega)\ket{0}\leftrightarrow\bra{\phi_{n}\Psi_{A-1}}\hat{V}\hat{G}^{(+)}(E)\hat{M}\ket{\Psi_{i,A}}.$
(34)

It is noted also that the general form of the $T$ matrix and the RPA $T$
matrix have a similar structure; the interaction $\hat{V}$ or $\hat{V}_{\rm
RPA}$ are taken into account up to infinite order as

$\displaystyle T_{(\gamma,n)}$
$\displaystyle=\bra{\phi_{n}\Psi_{A-1}}\left(\hat{M}+\hat{V}\hat{G}^{(+)}_{0}(E)\hat{M}+\hat{V}\hat{G}^{(+)}_{0}(E)\hat{V}\hat{G}^{(+)}_{0}(E)\hat{M}+\cdots\right)\ket{\Psi_{i,A}},$
(35) $\displaystyle T_{(\gamma,n)}^{\rm RPA}$
$\displaystyle=\bra{ph}\left(\hat{F}+\frac{\delta\hat{U}}{\delta\rho}{R}_{0}(\omega){F}+\frac{\delta\hat{U}}{\delta\rho}{R}_{0}(\omega)\frac{\delta{U}}{\delta\rho}{R}_{0}(\omega){F}+\cdots\right)\ket{0}$
(36)

where the density response $\delta\rho(x,\omega)$ obeying the linear response
equation (9) is symbolically represented represented as Eq.(17). Here
$\hat{G}^{(+)}_{0}(E)=\frac{1}{E-\hat{H}_{0}+i\eta}$ is the unperturbed
Green’s function whereas ${R}_{0}(\omega)$ is the unperturbed response
function Eq.(11). A diagramatic representation is shown in Fig.1.

##  Appendix B The $(\gamma,n)$ cross section under spherical symmetry

The self-consistent field $\hat{V}^{\rm
scf}_{LM}(\omega)=\hat{F}_{LM}+\hat{V}^{\rm ind}_{LM}(\omega)$ is a rank $L$
non-local one-body operator. The radial coordinate representation of this
matrix element is expressed as

$\displaystyle v^{{\rm
scf}}_{L,l^{{}^{\prime}}j^{{}^{\prime}},lj}(r_{x},r_{y},\omega)=F_{L,l^{{}^{\prime}}j^{{}^{\prime}},lj}(r_{x},r_{y})+\bra{l^{{}^{\prime}}j^{{}^{\prime}}}|Y_{L}|\ket{lj}\frac{\delta
U}{\delta\rho}(r_{x})\frac{1}{r^{2}_{x}}\delta\rho_{L}(r_{x},\omega)\delta(r_{x}-r_{y}).$
(37)

The radial matrix element
$F_{L,l^{{}^{\prime}}j^{{}^{\prime}},lj}(r_{x},r_{y})$ of the operator
$\hat{F}_{LM}$ is given in Eq.(34) of Ref. Saito and Matsuo (2021).

Inserting Eq.(37) into Eq.(22), we obtain

$\displaystyle\sigma^{\lambda}_{iL_{i}+\gamma\to[\epsilon
l_{p}j_{p}h]_{L}}(E_{\gamma})=-\frac{f(E_{\gamma})}{\pi(2L_{i}+1)}$
$\displaystyle\times{\rm
Im}\Big{\\{}|\bra{l_{p}j_{p}}|Y_{L}|\ket{l_{h}j_{h}}|^{2}\iint
dr_{x}dr_{x^{{}^{\prime}}}\phi^{*}_{n_{h}l_{h}j_{h}}(r_{x})\frac{\delta
U}{\delta\rho}(r_{x})\frac{1}{r^{2}_{x}}\delta\rho^{*}_{L}(r_{x},\omega_{\gamma}+\omega_{i})$
$\displaystyle\quad\times
G_{0c,l_{p}j_{p}}(r_{x},r_{x^{{}^{\prime}}},\epsilon_{h}+\hbar\omega_{\gamma}+\hbar\omega_{i}+i\eta)\frac{\delta
U}{\delta\rho}(r_{x^{{}^{\prime}}})\frac{1}{r^{2}_{x^{{}^{\prime}}}}\delta\rho_{L}(r_{x^{{}^{\prime}}},\omega_{\gamma}+\omega_{i})\phi_{n_{h}l_{h}j_{h}}(r_{x^{{}^{\prime}}})$
$\displaystyle+\bra{l_{p}j_{p}}|Y_{L}|\ket{l_{h}j_{h}}^{*}\iiint
dr_{x}dr_{x^{{}^{\prime}}}dr_{y^{{}^{\prime}}}\phi^{*}_{n_{h}l_{h}j_{h}}(r_{x})\frac{\delta
U}{\delta\rho}(r_{x})\frac{1}{r^{2}_{x}}\delta\rho^{*}_{L}(r_{x},\omega_{\gamma}+\omega_{i})$
$\displaystyle\quad\times
G_{0c,l_{p}j_{p}}(r_{x},r_{x^{{}^{\prime}}},\epsilon_{h}+\hbar\omega_{\gamma}+\hbar\omega_{i}+i\eta)F_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x^{{}^{\prime}}},r_{y^{{}^{\prime}}})\phi_{n_{h}l_{h}j_{h}}(r_{y^{{}^{\prime}}})$
$\displaystyle+\bra{l_{p}j_{p}}|Y_{L}|\ket{l_{h}j_{h}}\iiint
dr_{x}dr_{y}dr_{x^{{}^{\prime}}}\phi^{*}_{n_{h}l_{h}j_{h}}(r_{y})F^{*}_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x},r_{y})$
$\displaystyle\quad\times
G_{0c,l_{p}j_{p}}(r_{x},r_{x^{{}^{\prime}}},\epsilon_{h}+\hbar\omega_{\gamma}+\hbar\omega_{i}+i\eta)\frac{\delta
U}{\delta\rho}(r_{x^{{}^{\prime}}})\frac{1}{r^{2}_{x^{{}^{\prime}}}}\delta\rho_{L}(r_{x^{{}^{\prime}}},\omega_{\gamma}+\omega_{i})\phi_{n_{h}l_{h}j_{h}}(r_{x^{{}^{\prime}}})$
$\displaystyle+\iiiint
dr_{x}dr_{y}dr_{x^{{}^{\prime}}}dr_{y^{{}^{\prime}}}\phi^{*}_{n_{h}l_{h}j_{h}}(r_{y})F^{*}_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x},r_{y})$
$\displaystyle\quad\times
G_{0c,l_{p}j_{p}}(r_{x},r_{x^{{}^{\prime}}},\epsilon_{h}+\hbar\omega_{\gamma}+\hbar\omega_{i}+i\eta)F_{L,l_{p}j_{p},l_{h}j_{h}}(r_{x^{{}^{\prime}}},r_{y^{{}^{\prime}}})\phi_{n_{h}l_{h}j_{h}}(r_{y^{{}^{\prime}}})\Big{\\}}.$
(38)

The function $G_{0c,l_{p}j_{p}}(r_{x},r_{x^{{}^{\prime}}},\epsilon)$ is the
single-particle Green’s function $G_{0c}$ in the partial wave $l_{p}j_{p}$. It
is calculated exactly using Eq.(B2) of Ref. Saito and Matsuo (2021), together
with the subtraction of the bound orbits, see Eq.(20). The correlation in the
low-lying state $\ket{iL_{i}M_{i}}$ is reflected in
$F_{L,l^{{}^{\prime}}j^{{}^{\prime}},lj}$ while the correlation in the
continuum RPA states $\ket{kLM(E)}$ is in the density flucutuation
$\delta\rho_{L}$.

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