#  $\beta$-Delayed Neutron and Fission Calculations with Relativistic QRPA and
Statistical Model

Futoshi Minato<EMAIL_ADDRESS>Nuclear Data Center, Japan Atomic
Energy Agency, Tokai, Ibaraki 319-1195, Japan  Tomislav Marketin  Department
of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
Nils Paar  Department of Physics, Faculty of Science, University of Zagreb,
10000 Zagreb, Croatia

###### Abstract

Background: A role of $\beta$-delayed neutron emission and fission in
$r$-process nucleosynthesis attracts a high interest. Although the number of
study on them covering $r$-process nuclei is increasing recently,
uncertainties of $\beta$-delayed neutron and fission are still large for
$r$-process simulations.
Purpose: Our purpose is to make a new database on $\beta$-delayed neutron
emission and fission rates. To this end, the data that are not investigated
experimentally have to be predicted. Microscopic theoretical approaches based
on a nuclear energy density functional and statistical models are one of the
competent tools for the prediction. We developed a new theoretical framework
for $\beta$-delayed neutron emission and fission, and apply it to make a new
data table.
Methods: To obtain $\beta$ strength function, proton-neutron relativistic
quasiparticle random phase approximation is adopted. Particle evaporations and
fission from nuclear highly excited states are estimated by the Hauser-
Feshbach statistical model. $\beta$-delayed neutron branching ratios $(P_{n})$
are calculated and compared with experimental data. $\beta$-delayed fission
branching ratio ($P_{f}$) are also assessed by using four different fission
barrier data.
Results: Calculated $P_{n}$ values are in a good agreement with experimental
data and the root mean square deviation is comparable to results of preceding
works. It is found that energy withdrawal by $\beta$-delayed neutron emission
sensitively varies $P_{n}$ values for nuclei near the neutron drip line.
$P_{f}$ are sensitively dependent on fission barrier data.
Conclusions: Newly calculated data on $\beta$-delayed neutron emission and
fission are summarized as a table in supplement material. They are provided
for studies of $r$-process as well as other fields such as nuclear
engineering. Study on level structures and fission barrier of neutron-rich
nuclei are highly requested for a further development of the database in
future.

###### pacs:

21.60.-n, 21.60.Jz

##  I Introduction

Origin of chemical elements in the universe is the long-standing problem and
one of the hottest topics in astrophysics. Our knowledge accumulated so far
indicates that heavy elements in the nature are generated by dynamical
processes of stars. In particular, about a half of the elements heavier than
iron is considered to be produced by $r$-process Burbidge  _et al._ (1957);
Cameron (1957) (another half is by $s$-process Burbidge  _et al._ (1957)).
Recent researches employing observation of gravitational waves produce
evidence that a neutron star merger is one of the the possible sites of
$r$-process Hotokezaka  _et al._ (2018); Wanajo  _et al._ (2018). However, it
is not still concluded where the $r$-process occurs.

The solar $r$-process abundance pattern shows a characteristic mass
distribution that has three peaks around $A=80\sim 90$, $130\sim 138$ and
$195\sim 208$ Freiburghaus  _et al._ (1999); Arcones and Martínez-Pinedo
(2011a); Zhi  _et al._ (2013). The $r$-process simulation suggests that the
origin of these peaks is related to neutron magic numbers of $N=50,82$ and
$126$, that is also related to nuclear mass Freiburghaus  _et al._ (1999);
Langanke and Martínez-Pinedo (2003); Arcones and Martínez-Pinedo (2011a).
However, the peak positions and abundance ratios cannot be reproduced only by
the nuclear mass. Neutron capture, $\beta$-decay, and other decay modes
sensitively influence the abundance pattern of $r$-process Arcones and
Martínez-Pinedo (2011a); Beun  _et al._ (2008a); Surman  _et al._ (2009);
Mumpower  _et al._ (2012); Xu  _et al._ (2014); Borzov and Goriely (2000);
Suzuki  _et al._ (2012); Marketin  _et al._ (2016); Mumpower  _et al._
(2016a); Nishimura  _et al._ (2016); Suzuki  _et al._ (2018); Kodama and
Takahashi (1973, 1975); Kratz  _et al._ (1988); Beun  _et al._ (2008b);
Goriely (2015); Eichler  _et al._ (2016); Shibagaki  _et al._ (2016); Mumpower
_et al._ (2018); Giuliani  _et al._ (2018).

During $r$-process, $\beta^{-}$-decay increases the atomic number of nuclei
and brings daughter nuclei in a highly excited state. Depending on the
excitation energy, the daughter nuclei are able to decay through several
particle emission channels. In particular, $\beta$-delayed neutron emission
and fission (hereafter we call them BDNE and BDF, respectively) play a
subsidiary role in the $r$-process abundance. BDNE gives two different effects
to the $r$-process abundance. First is that it leads nuclei in a $r$-process
site to detour $\beta^{-}$-decay path $(A,Z)\rightarrow(A,Z+1)$ by reducing
neutron number, for example, $(A,Z)\rightarrow(A-1,Z+1)$ in case of one-
neutron emission. Another is that BDNE boosts environmental neutron density
during the freeze-out phase and stimulates nuclei to capture the neutrons
again. As a result, the even-odd fluctuation in the final $r$-process
abundance pattern is smoothed Kodama and Takahashi (1973, 1975); Kratz  _et
al._ (1988); Arcones and Martínez-Pinedo (2011b). The role of BDF is to reduce
abundance of heavy element by breaking daughter nuclei into two or more
fragments during $r$-process. Furthermore, the fission fragments restart to
capture environmental neutrons and grow up toward heavy elements again. This
phenomena, what is called fission recycling, affects a wide range of the
$r$-process abundance together with neutron-induced fission Beun  _et al._
(2008b); Goriely (2015); Eichler  _et al._ (2016); Shibagaki  _et al._ (2016);
Mumpower  _et al._ (2018); Giuliani  _et al._ (2018). Understanding of the
fission in $r$-process would also have a clue to answer a naive question; are
super-heavy elements produced by dynamical processes of stars Petermann  _et
al._ (2012).

A lot of experimental measurements of BDNE are carried out because of its
importance in applications to $r$-process as well as nuclear engineering (e.g.
see Refs. Caballero-Folch  _et al._ (2016); Rykaczewski (2018); Miernik  _et
al._ (2018); Agramunt  _et al._ (2016); Dunlop  _et al._ (2019) for recent
works). However, the experimental difficulty rapidly increases as one tries to
study very neutron-rich nuclei because of the low statistics. For this reason,
there are still nuclei that the BDNE and the BDF has not been measured yet. In
particular, for BDF 7 cases are recognized only near the $\beta$-stability
line ENS (2019) (see Ref. Ghys  _et al._ (2015) which summarizes experimental
studies for BDF).

Nuclei that are not investigated by experiment have to be covered by
theoretical models. Empirical systematics is one of the useful tools and is
provided in e.g. Refs. Kratz and Herrmann (1973); McCutchan  _et al._ (2012);
Miernik (2013, 2014) for $\beta$-delayed neutron branching ratio ($P_{n}$) and
even for BDF in Refs. Andreyev  _et al._ (2013); Ghys  _et al._ (2015). Gross
Theory (GT2) Yoshida and Tachibana (2000); Tachibana (2000) that is based on a
phenomenological approach is also an effective tool to calculate $P_{n}$.
Although no challenge has been yet performed for delayed fission branching
ratio ($P_{f}$) with GT2, it is technically possible.

Another effective approach for theoretical prediction of $P_{n}$ and $P_{f}$
is a microscopic model. A configuration interacting (CI) model, quasiparticle
random phase approximation (QRPA), and finite amplitude method (FAM) which is
essentially same as the QRPA but solves in a different way, are the typical
approaches. Recently, interacting boson model (IBM) based on mean-field
approach has been applied to $\beta$-decay calculation as well Nomura  _et
al._ (2020). Although CI models are actively applied to the calculation of
half-lives for $r$-process nuclei around $N=50,82$, and $126$ shell closures,
the application is still restricted to limited nuclei due to the increasing
computational cost Suzuki  _et al._ (2012); Yoshida  _et al._ (2018); Suzuki
_et al._ (2019). For this reason, to calculate $\beta$-decay for $r$-process
nuclei, the QRPA Staudt and Klapdor-Kleingrothaus (1992); Hirsch  _et al._
(1992); Möller  _et al._ (1997); Borzov (2005); Nabi  _et al._ (2016); Panov
_et al._ (2016) and FAM Mustonen  _et al._ (2014); Mustonen and Engel (2016);
Ney  _et al._ (2020), are practically applied.

One of the authors has calculated $P_{n}$ systematically for neutron-rich
nuclei in the framework of proton-neutron relativistic quasiparticle random
phase approximation ($pn$-RQRPA) Marketin  _et al._ (2016). To predict
$\beta$-delayed neutron branching ratios, a simplified approach (referred to
as cutoff method hereafter) assuming that nuclei with excitation energies
above neutron threshold always emit $\beta$-delayed neutron has been used, as
like in Refs. Möller  _et al._ (1997); Borzov (2005); Nabi  _et al._ (2016).
This assumption corresponds with a picture that neutron emission is the only
exit channel above neutron thresholds and kinetic energy of $\beta$-delayed
neutron is zero. This prescription clearly omits the nuclear structure, the
selection rule of decay chain, competition with other decay channels, and
kinematics. In fact, it is pointed out in Mumpower  _et al._ (2016b) that a
competition between neutron emission and $\gamma$ de-excitation gives a non-
negligible variation to the calculation for $P_{n}$ of nuclei, especially near
the neutron drip line.

One of the approaches to treat nuclear decay in a more physical and complete
way is to apply a statistical decay model, for example, Hauser-Feshbach
statistical model (HFM) Hauser and Feshbach (1952). The HFM considers nuclear
structure effects through level densities and selection rule of decay chain,
competition with other decay channels, and kinematics that the cutoff method
omitted. Combination of the QRPA $\beta$ strength function and the statistical
decay model is therefore a feasible approach and has been carried out by
several groups using FRDM+QRPA Kawano  _et al._ (2008); Mumpower  _et al._
(2018); Möller  _et al._ (2019a) and non-relativistic QRPA Thielemann  _et
al._ (1983); Staudt and Klapdor-Kleingrothaus (1992); Hirsch  _et al._ (1992);
Panov  _et al._ (2005); Futoshi Minato (2016); Panov  _et al._ (2016). The aim
of this paper is directed at estimating $P_{n}$ and $P_{f}$ by using this
approach, namely by combining $pn$-RQRPA and HFM. Hereafter, we refer to the
present work as $pn$-RQRPA+HFM to distinguish the work of $pn$-RQRPA Marketin
_et al._ (2016).

This paper is organized as follows. In Sect. II, we describe theoretical
framework to calculate BDNE and BDF using $pn$-RQRPA+HFM. In Sect. III, the
results obtained in this work are presented and discussed in comparison with
experimental data and preceding works. Section IV summarizes this work and
presents some perspectives. The complete data table containing the BDNE and
BDF branching ratios is available as Supplemental Material sup .

##  II Theoretical Framework of $pn$-RQRPA+HFM

###  II.1 $\beta$-delayed neutron and fission branching ratios

Our calculation is composed of two parts. First, we prepare $\beta$ strength
functions for the Gamow-Teller (GT) and the first-forbidden (FF) transitions
by using the $pn$-RQRPA Marketin  _et al._ (2016). As second step, we carry
out the calculation of statistical decay from the compound state by using the
HFM calculation with excitation energy and spin-parity given by the
$pn$-RQRPA. The $P_{n}$ and $P_{f}$ are then obtained by multiplying neutron
and fission emission probabilities by the $\beta$ decay rates, respectively.

A fully self-consistent covariant density functional theory (CDFT) framework
is adopted in this work. The ground state of all nuclei is calculated with the
relativistic Hartree-Bogoliubov (RHB) model with the D3C* interaction Marketin
_et al._ (2007). The ground state of odd nuclei are computed by employing the
same model as for even-even nuclei, but constraining the expectation value of
the particle number operator to an odd number of protons and/or neutrons. On
the top of the RHB, excited states are obtained within the $pn$-RQRPA. More
details about the $pn$-RQRPA used in this work are given in Ref. Marketin  _et
al._ (2016).

We assume daughter nuclei reach the compound state, namely the thermally
equilibrium state, soon after $\beta^{-}$-decay. At the compound state,
daughter nuclei lose initial information that they had before $\beta$-decay,
except for the spin-parity and the total energy of the system. The number of
protons and neutrons of the initial nuclei (precursor) are defined as $Z$ and
$N$, respectively. Accordingly, the number of protons and neutrons of the
daughter nuclei are given by $Z+1$ and $N-1$, respectively. Because the
$pn$-RQRPA gives excitation energy with respect to the parent nuclei, a
correction is required to obtain the energy with respect to the daughter
nuclei. With the BCS approximation, the excitation energies are computed
through Engel  _et al._ (1999); Ring and Schuck (1980)

$\displaystyle E_{i}^{*}=$ $\displaystyle E_{i,QRPA}-E_{corr},$ (1)
$\displaystyle E_{corr}$
$\displaystyle=\begin{cases}E_{p_{0}}+E_{n_{0}}&(\mathrm{for\,even-even})\\\
E_{p_{0}}&(\mathrm{for\,even-odd})\\\ E_{n_{0}}&(\mathrm{for\,odd-even})\\\
0&(\mathrm{for\,odd-odd}),\end{cases}$ (2)

where $E_{i,QRPA}$ is the excitation energy calculated by the $pn$-RQRPA,
index $i$ denotes an excited state of daughter nuclei with spin-parity
$J^{\pi}$, and $E_{p_{0}}$ and $E_{n_{0}}$ are the lowest quasi-particle
energies of proton and neutron, respectively.

We define isotope production ratios of evaporation residues with proton number
$Z^{\prime}$ and neutron number $N^{\prime}$ as $p_{Z^{\prime}N^{\prime}}(i)$,
and spectra of emitted particles as $d_{\nu}(E_{\nu},i)$. Here, $E_{\nu}$ is
the kinetic energy of the outgoing particle and $\nu=\\{n,\gamma,p,\alpha\\}$
represents a kind of the emitted particle, where the letters in the brackets
represent neutron, $\gamma$-ray, proton, and $\alpha$-particle, respectively.
We did not consider other particle emissions because they are strongly
hindered for neutron-rich nuclei of interest. The BDF may occur directly after
$\beta^{-}$-decay i.e. $(\beta^{-},f)$ and indirectly after multi-neutron
emissions i.e. $(\beta^{-},xnf)$. The fission probabilities from an excited
state $i$ are defined as $p_{xn,f}(i)$. The functions of
$p_{Z^{\prime}N^{\prime}}(i)$, $p_{xn,f}(i)$, and $d_{\nu}(E_{\nu},i)$ are
computed by the HFM calculation implemented in CCONE code Iwamoto (2007). We
do not go into the detail about the HFM because the formalism is given in,
e.g. Refs Panov  _et al._ (2005); Iwamoto (2007); Mumpower  _et al._ (2016b).
However, we will explain the nuclear input details used in our calculation in
the Sect. II.2.

BDNE branching ratios are calculated by

$P_{n}=\sum_{x}P_{xn}=\frac{1}{R}\sum_{i,x}r_{i}\,p_{Z+1,N-1-x}(i)$ (3)

and BDF branching ratios by

$P_{f}=\sum_{x}P_{xn,f}=\frac{1}{R}\sum_{i,x}r_{i}\,p_{xn,f}(i),$ (4)

where $r_{i}$ are the partial $\beta^{-}$-decay rates to excited state $i$
calculated by the $pn$-RQRPA, and $R=\sum_{i}r_{i}$. The BDNE spectrum is

$D_{n}(E_{n})=\mathcal{N}\sum_{i}r_{i}\,d_{n}(E_{n},i),$ (5)

where $\mathcal{N}$ is the normalization factor to be determined so as to
satisfy

$\int_{0}^{\infty}D_{n}(E_{n})dE_{n}=1.$ (6)

The summation of $i$ of Eqs. (3), (4) and (5) is carried out for
$E_{i}^{*}\leq Q_{\beta}$. The $\beta$-decay $Q$ value is calculated from
$Q_{\beta}=M_{n\rm{H}}+\lambda_{n}-\lambda_{p}-E_{corr}$ Marketin  _et al._
(2016) where $M_{n\rm{H}}$, $\lambda_{n}$, and $\lambda_{p}$ are the mass
difference between neutron and Hydrogen atom, and the neutron and proton Fermi
energies, respectively.

###  II.2 Nuclear input details for HFM

The $pn$-RQRPA calculation Marketin  _et al._ (2016) is solved by
diagonalizing RQRPA matrix so that the eigenvalues are given in a form of
discrete states. Accordingly, $\beta$ strength function also has a discrete
shape in terms of excitation energy. However, it is considered that actual
$\beta$ strength function has a broader distribution because of coupling to
higher-order configurations and continuum states, which are not taken into
account in the $pn$-RQRPA used in this work. In order to account for those
effects, we introduce two types of weight function, that is the Gaussian type
and the Lorentzian type, to the $\beta$ strength functions. The weight
function of the Gaussian type is given as

$G_{w}(E)=g_{J}\sum_{i\in
w}r_{i}\frac{1}{\sqrt{2\pi\Gamma^{2}}}\exp\left(-\frac{(E-E_{i}^{*})^{2}}{2\Gamma^{2}}\right).$
(7)

and that of the Lorentzian type as

$L_{w}(E)=g_{J}\sum_{i\in
w}\mathcal{N}_{i}r_{i}\frac{1}{\pi}\frac{\Gamma/2}{(E-E_{i}^{*})^{2}+(\Gamma/2)^{2}}.$
(8)

The index $w$ is used to distinguish the GT ($\Delta J^{\pi}=1^{+}$) and the
FF transitions ($\Delta J^{\pi}=0^{-},1^{-},2^{-}$), and the factor $g_{J}$ is
a statistical factor that will be explained later on. Since the Lorentzian
function that is proportional to $1/E^{2}$ has a finite strength even far from
the mean, we introduce a cutoff energy $E_{cut}$ to reduce anomalously large
strengths at distant energies. This cutoff energy is determined by
$L_{\omega}(E_{cut})=L_{\omega}(E_{i}^{*})/1000$, and the weight function of
Eq. (8) is active within the energy range of $E_{i}^{*}\pm E_{cut}$. The
factor $\mathcal{N}_{i}$ in Eq. (8) is then introduced to renormalize the
Lorentzian function to be unity. In this scheme, a part of $\beta$ strength
functions may stray to the negative energies in terms of daughter nuclei.
Because the HFM cannot compute particle evaporations properly if excitation
energies are negative, we integrate the $\beta$ strengths at negative energies
and set them at $E=0$ MeV, namely, the ground state of the daughter nuclei. A
schematic picture that depicts Eqs. (7) and (8) with $E^{*}=10$ MeV and
$\Gamma=0.6$ MeV ($E_{cut}=9.5$ MeV) are shown in Fig. 1. Within $E^{*}\pm
2\Gamma$, the Gaussian function has a broader distribution than the Lorentzian
function. Beyond this energy range, the Gaussian function rapidly fades out
while the Lorentzian function still has a finite strength distribution. In the
next section, we determine the width parameter $\Gamma$ that minimizes the
root mean square deviation of $P_{n}$ from the experimental data.

Figure 1: Weight functions of Gaussian type (the dotted line) of Eq. (7) and
Lorentzian (the solid line) type of Eq. (8) with $E^{*}=10$ MeV and width
$\Gamma=0.6$ MeV. The inserted panel is depicted in a linear scale.

The $pn$-RQRPA is used in calculations for odd-nuclei in the same way as for
even-nuclei, so that the spin-parity of the ground state ($J_{gs}^{\pi_{gs}}$)
was not specified. We therefore adopt the experimental data of the spin-parity
if it is available. If not, we use the spin-parity of the state with the
lowest quasiparticle energy from the RHB calculation, namely $J_{gs}=j_{p}$
with $\pi_{gs}=(-1)^{l_{p}}$ and $J_{gs}=j_{n}$ with $\pi_{gs}=(-1)^{l_{n}}$
for odd-even and even-odd nuclei, respectively, where $j_{p}(j_{n})$ and
$l_{p}(l_{n})$ are the quasiparticle total and orbital angular momentum of
proton (neutron), respectively. For spin state of odd-odd nuclei, we apply
Nordheim method Nordheim (1950), which is given by

$\begin{split}J_{gs}&=j_{p}+j_{n}\,\,\mathrm{if}\,\,j_{p}=l_{p}\pm\frac{1}{2}\,\,\mathrm{and}\,\,j_{n}=l_{n}\pm\frac{1}{2}\\\
J_{gs}&=|j_{p}-j_{n}|\,\,\mathrm{if}\,\,j_{p}=l_{p}\pm\frac{1}{2}\,\,\mathrm{and}\,\,j_{n}=l_{n}\mp\frac{1}{2}\end{split}$
(9)

and the parity is computed via
$\pi_{gs}=\pi_{gs}^{(p)}\pi_{gs}^{(n)}=(-1)^{l_{p}+l_{n}}$. The spin of
daughter nuclei ($J_{f}$) resulted from $\beta$-decay of odd-nuclei takes
$|J_{gs}-\Delta J|\leq J_{f}\leq J_{gs}+\Delta J$ and the parity
$\pi_{f}=\pi_{gs}\pi_{\alpha}$, where $\Delta J=0,1,2$ and $\pi_{\alpha}=\pm
1$ depending on the type of $\beta$ transitions. We assumed the equal
distribution of $\beta$ strength function for those possible states of
daughter nuclei. The factor $g_{J}$ in Eq. (8) is determined from the number
of the possible states of the daughter nuclei. For example, in case of
$1^{\pm}$ transition from a parent nucleus with $J=3/2(1/2)$ state,
$g_{3/2(1/2)}=1/3(1/2)$.

In the HFM calculation, transmission coefficients of nucleons, deuteron,
triton, helium-3 are calculated by Koning-Delaroche optical potentials Koning
and Delaroche (2003) and its folding potentials. Transmission coefficient of
$\alpha$ particle is calculated by the optical potential of Avrigeanu
Avrigeanu (2010). For nuclear level densities, the Gilbert-Cameron method A.
and A.G.W. (1965) with Mengoni-Nakajima parameter A. and Nakajima (1994) is
adopted. For $\gamma$ strength functions, the enhanced generalized Lorentzian
function J. and M. (1990) is used. Mass data are taken from the global nuclear
mass model Liu  _et al._ (2011).

Transmission coefficients for fission are calculated as follows. We assume a
double or triple humped parabolic barrier and the barrier penetrability for
each barrier is calculated by the formula of Hill-Wheeler equation Hill and
Wheeler (1953). The transmission coefficients are obtained by assuming that
the fission process occurs through the transition states above the fission
barrier. All transition states were approximated by the level density formula
described above. The transmission coefficient of a single barrier for the
state having excitation energy $E$ and spin-parity $J^{\pi}$,
$T_{i}(E,J^{\pi})$, is calculated by

$T_{i}(E,J^{\pi})=\int_{0}^{\infty}\frac{\rho_{i}(\epsilon,J^{\pi})}{1+\exp\left(-2\pi\frac{E-V_{i}-\epsilon}{\hbar\omega_{i}}\right)}d\epsilon$
(10)

where the subscripts $i=A$, $B$, and $C$ indicate the inner, middle, and outer
barriers, respectively, $\rho_{i}(\epsilon,J^{\pi})$ the level density at the
saddle points, and $V_{i}$ and $\hbar\omega_{i}$ represent the height and
curvature of the fission barrier, respectively. The transmission coefficients
for two and three barriers is approximated to be
$T(E,J^{\pi})=T_{A}T_{B}(T_{A}+T_{B})$ and
$=T_{A}T_{B}T_{C}/(T_{A}T_{B}+T_{B}T_{C}+T_{C}T_{A})$, respectively.

Since predicted fission barrier data greatly vary among models, our
calculation is carried out using four different fission barriers: HFB-14
Goriely  _et al._ (2007), FRDM+QRPA Möller  _et al._ (2015), Extended Thomas
Fermi plus Strutinsky Integral (ETFSI) method Mamdouh  _et al._ (2001), and
Spherical Basis Method (SBM) Koura (2014). The latter three give only single
barrier information, so that the transmission coefficients are calculated by
assuming a single humped parabolic barrier and the barrier curvature Capote
_et al._ (2009). The curvature parameter we used is
$\hbar\omega_{A}=1.04,0.80,0.65$ for even-even, even-odd(odd-even), and odd-
odd nuclei, respectively, that are determined to reproduce fission cross
sections of uranium isotopes Iwamoto (2007). For the case of fission barrier
of HFB-14, we adopt the fission barrier data and path given in the database
Goriely  _et al._ (2007).

##  III Results

###  III.1 $\beta$-delayed neutron emission

In the last section, the width parameter $\Gamma$ is introduced to make the
$\beta$ strength function a broad distribution. To determine the most likely
$\Gamma$, we estimate the root mean squared (rms) value of $P_{1n}$ defined as

$\sigma_{\mathrm{rms}}^{xn}=\sqrt{\frac{1}{N}\sum_{i}^{N}\left[\log_{10}\left(\frac{P_{xn,i}(c)}{P_{xn,i}(e)}\right)\right]^{2}},$
(11)

where $N$ is the number of experimental data and $P_{xn,i}(c)$ and
$P_{xn,i}(e)$ are the $\beta$-delayed neutron branching ratios of theoretical
models and experiment, respectively. Note that in Ref. Mumpower  _et al._
(2018) a linear scaled rms is adopted to discuss the predictive power of
$P_{1n}$ calculated by the FRDM+QRPA+HFM Möller  _et al._ (2019a). However, we
adopted logarithmic scaled rms as Eq. (11) because experimental $P_{xn}$
values extend from a small value of order of $10^{-2}$ to $10^{2}$ as like
half-lives.

Figure 2: Estimated $\sigma_{\mathrm{rms}}^{1n}$ value given in Eq. (11) as a
function of the width, $\Gamma$, calculated with the Gaussian function Eq. (7)
and the Lorentzian function Eq. (8).

Figure 2 shows $\sigma_{\mathrm{rms}}^{1n}$ as a function of width parameter
$\Gamma$ of the Gaussian type of Eq. (7) (the dashed line) and the Lorentzian
type of Eq. (8) (the solid line). Experimental data are taken from Ref.
Dimitriou  _et al._ (2020) and ENSDF ENS (2019). At $\Gamma\sim 0$ MeV,
$\beta$ strengths feeding to states lower than the BDNE window
($Q_{\beta}-S_{n}\leq E^{*}_{i}\leq Q_{\beta}$) is sizable. $P_{1n}$ values
are mostly underestimated and $\sigma_{\mathrm{rms}}$ is above 1.2 as seen in
Fig. 2. Increasing width parameter $\Gamma$, $\sigma_{\mathrm{rms}}^{1n}$
monotonically becomes smaller because $\beta$ strengths at energies smaller
than neutron separation energy seep into the BDNE energy window and majority
of calculated $P_{1n}$ comes close to the experimental values. Eventually,
$\sigma_{\mathrm{rms}}^{1n}$ takes a minimum value at $\Gamma=0.6$ MeV for the
Lorentzian type ($\sigma_{\mathrm{rms}}^{1n}=0.602$) and at $\Gamma=1.2$ MeV
for the Gaussian type ($\sigma_{\mathrm{rms}}^{1n}=0.694$).

Above the minimal points, $\sigma_{\mathrm{rms}}^{1n}$ turn into a slow
increase. This would be because majority of calculated $P_{1n}$ becomes too
large as compared to experimental values. The Lorentzian type weight function
gives the better result than the Gaussian one in terms of
$\sigma_{\mathrm{rms}}^{1n}$. It is difficult to explain this reason because
the result of $\sigma_{\mathrm{rms}}^{1n}$ is complicatedly convoluted by many
$P_{1n}$ data. However, we can explain qualitatively that the little leakage
of the $\beta$ strength function extending far outside of the mean plays a
significant role for a better agreement of $P_{1n}$ with experimental data. In
fact, the Gaussian type weight function needs a wider width than that of the
Lorentzian type to give the minimal $\sigma_{\mathrm{rms}}^{1n}$. For the
subsequent sections in this paper, we thus discuss BDNE and BDF using the
weight function of Lorentzian type using $\Gamma=0.6$ MeV, the width of which
is reasonable as comparing to the experimental measurements of low-lying GT
resonances for stable tin isotopes investigated by (3He,$t$) reactions Pham
_et al._ (1995).

Table 1: Result of $\sigma_{\mathrm{rms}}^{1n}$ defined by Eq. (11) for different range of $P_{1n}(e)$ (%). Model | $P_{1n}(e)<1$ | $1\leq P_{1n}(e)<10$ | $10\leq P_{1n}(e)<50$ | $50\leq P_{1n}(e)\leq 100$ | All $P_{1n}(e)$ data
---|---|---|---|---|---
$pn$-RQRPA+HFM | $0.952$ | $0.446$ | $0.570$ | $0.317$ | $0.601$
$pn$-RQRPA | $0.857$ | $0.727$ | $0.925$ | $0.460$ | $0.798$
GT2 | $0.852$ | $0.656$ | $0.464$ | $0.320$ | $0.595$
FRDM+QRPA+HFM | $1.084$ | $0.442$ | $0.482$ | $0.281$ | $0.512$
Number of nuclei | 50 | 92 | 91 | 34 | 267

Table 1 lists $\sigma_{\mathrm{rms}}^{1n}$ of the $pn$-RQRPA+HFM together with
the $pn$-RQRPA Marketin  _et al._ (2016), Gross theory (GT2) Yoshida and
Tachibana (2000); Tachibana (2000), and FRDM+QRPA+HFM Möller  _et al._
(2019b). We also list 4 different $\sigma_{\mathrm{rms}}^{1n}$ classified by
range of $P_{1n}$. The total value of $\sigma_{\mathrm{rms}}^{1n}$ was $0.798$
for the $pn$-RQRPA. Note that in the $pn$-RQRPA the $\beta$ strength functions
are weighted by a Lorentzian function using a width of $65$ keV that is
determined so as to reproduce $\beta$-delayed neutron yield of thermal neutron
induced fission of 235U. The present result of $pn$-RQRPA+HFM greatly improves
that of $pn$-RQRPA providing $\sigma_{\mathrm{rms}}^{1n}=0.601$, which is
comparable to that of GT2 ($\sigma_{\mathrm{rms}}^{1n}=0.595$) and
FRDM+QRPA+HFM ($0.512$). Especially, we obtained a remarkable improvement in
$\sigma_{\mathrm{rms}}^{1n}$ throughout from $1\leq P_{1n}\leq 100$. Although
$\sigma_{\mathrm{rms}}^{1n}$ is deteriorated in $P_{1n}<1$, the result of
$pn$-RQRPA+HFM is still slightly better than FRDM+QRPA+HFM.

Figure 3(a) shows the ratio of $P_{1n}$ for $pn$-RQRPA+HFM to that for
$pn$-RQRPA as a function of mass number $A$. We can see that most of data
point is above unity, i.e. most of $P_{1n}$ is increased by the framework
discussed in Sect. II. Figure 3(b) and (c) shows the ratio of calculated
$P_{1n}$ to experimental one ($\mathrm{C/E}$) for $pn$-RQRPA+HFM and
$pn$-RQRPA, respectively. We can observe that underestimations (i.e.
$\mathrm{C/E}<1$) for the $pn$-RQRPA are improved for the $pn$-RQRPA+HFM. This
is mainly due to adopting a wide width of the weight function and inclusion of
$n$–$\gamma$ competition which enhances $P_{1n}$. Although there exists
$\beta$ strength function that escapes from one-neutron emission energy window
to more multiple ones, this outflow is generally smaller than the gain to one-
neutron emission energy window because $\beta$-decay rates become lower as
going to higher excitation energies. We confirmed that $73\%$ of $P_{1n}$
corresponding to $193$ nuclei are improved for the $pn$-RQRPA+HFM.

Figure 3:  (a) Ratio of $P_{1n}$ for $pn$-RQRPA+HFM to that for $pn$-RQRPA.
The panels (b) and (c) represent ratio of calculated to experimental data of
$P_{1n}$ ($\mathrm{C/E}$) for $pn$-RQRPA+HFM (this work) and $pn$-RQRPA
Marketin  _et al._ (2016), respectively.

Figure 4 plots $P_{n}$ calculated by the $pn$-RQRPA+HFM in the $N$–$Z$ plane.
We can observe that $P_{n}$ has a large value in the lower-right sectors
seeing from the double magic numbers. This is simply explained by small
neutron threshold energies and large $Q_{\beta}$ in those regions. We can also
observe the odd-even dependence, especially in $Z$-direction. Generally,
$P_{n}$ of odd-$Z$ nuclei are larger than neighboring even-$Z$ nuclei. This is
because $Q_{\beta}$ of odd-$Z$ nuclei are larger than those of even-$Z$ nuclei
in general, while $S_{xn}$ are less sensitive to $Z$ numbers than $Q_{\beta}$.
As a consequence, BDNE energy window has an odd-even structure. As an example,
one-neutron emission energy window of 220Ir ($Z=77$) is about $8.04$ MeV,
while that of the neighboring nuclei 219Os ($Z=76$) and 221Pt ($Z=78$) are
about $7.40$ MeV and $5.65$ MeV, respectively. Going toward heavy mass region
around $Z>90$ and $N>182$, $P_{n}$ displays a patchy pattern. This is because
BDF plays a role in this region and competes with BDNE (see Sect. III.2).

Figure 4: Total $\beta$-delayed neutron branching ratios ($P_{n}$) calculated
by the $pn$-RQRPA+HFM. The black filled squares stand for stable or long-lived
nuclei.

Figure 5: $\beta$-delayed neutron branching ratio $P_{1n}$ of Pd, Ag, Os, and
Ir isotopes as a function of $A$. The experimental data are taken from Ref.
Dimitriou  _et al._ (2020). Figure 6:  (a) $\beta$ strength function of 150Pd.
BDNE windows denoted by $W_{xn}$ are also indicated. (b) Isotope production
ratios of 145-149Ag from $1^{+}$ excited states of 150Ag (the daughter nucleus
of 150Pd) as a function of the excitation energy. The calculation is performed
by the HFM.

Figure 5 shows $P_{n}$ of Pd ($Z=46$), Ag ($Z=47$) isotopes that are
considered as important $\beta$-delayed neutron precursors in $r$-process
Mumpower  _et al._ (2016a), and Os ($Z=76$) and Ir ($Z=77$) isotopes. We plot
the results of $pn$-RQRPA Marketin  _et al._ (2016) and FRDM+QRPA+HFM Möller
_et al._ (2019a) for comparison. Because the same $\beta$ strength function
except the width factor is used for the $pn$-RQRPA+HFM and the $pn$-RQRPA,
they give a similar result until a certain mass number $A$. However, the
$pn$-RQRPA+HFM and the $pn$-RQRPA begin to show a difference for neutron-rich
side. For example, $P_{1n}$ of 150Pd for the $pn$-RQRPA+HFM is about 40%,
while that for the $pn$-RQRPA is $1.6\%$. Because the daughter nucleus 150Ag
is an unbound nucleus according to the global nuclear mass model Liu  _et al._
(2011), all the $\beta$-decay must contribute to BDNE. However, a large amount
of the $\beta$ strength function of 150Pd distribute in negative energy in the
framework of $pn$-RQRPA Marketin  _et al._ (2016), which means that some
$\beta$ strength functions exist at lower energies than the ground state of
150Ag. As a result, $P_{0n}$ of 150Pd unsuitably becomes $35\%$ for the
$pn$-RQRPA Marketin  _et al._ (2016), which contradicts the fact that 150Ag is
unbound. For the present work ($pn$-RQRPA+HFM), the problem of negative energy
resonances is avoided by resetting it to zero energy as explained in Sect.
II.2. As a result, the calculated $P_{0n}$ of 150Pd is correctly zero and
$P_{1n}$ increases by almost the same amount of $P_{0n}$ for the $pn$-RQRPA.

The result of $pn$-RQRPA+HFM shows a similar $P_{1n}$ to FRDM+QRPA+HFM for
nuclei with small mass numbers, as shown in Fig. 5. As going toward heavier
mass, two calculations begin to show a difference. The noticeable contrasts
are found in nuclei close to the neutron drip-line and $210\leq A\leq 230$ for
Os and Ir isotopes. A major factor affecting $P_{1n}$ is the $\beta$ strength
function and one neutron separation energies, $S_{1n}$, which are calculated
from the mass data. The $pn$-RQRPA+HFM and FRDM+QRPA+HFM use theoretically
predicted mass data that is the global nuclear mass model Liu  _et al._ (2011)
and FRDM1995 Möller  _et al._ (1995), respectively. For most of nuclei close
to the drip line, $S_{1n}$ of the global nuclear mass model is lower than
those of FRDM, for example, $S_{1n}$ of 240Ir is $269$ keV for the global mass
model while $690$ keV for FRDM1995. To check the sensitivity of $P_{1n}$ to
mass data, we carry out the same calculation replacing the mass data of the
$pn$-RQRPA+HFM with FRDM1995, and find that the result globally becomes close
to that of FRDM+QRPA+HFM for nuclei near the drip-line. However, the
deviations found in $210\leq A\leq 230$ for Os and Ir isotopes do not change
significantly. We thus consider that the difference found in $210\leq A\leq
230$ for Os and Ir isotopes is mainly attributed to the $\beta$ strength
function and that found near the neutron drip-line is due to the mass data. We
observe an odd-even staggering of $P_{1n}$ in nuclei near neutron-drip lines
for the $pn$-RQRPA+HFM. This structure appears particularly when daughter
nuclei are unstable against neutron emissions.

Figure 6(a) shows the $\beta$ strength function of 150Pd with the Lorentzian
function with $\Gamma=600$ keV, in which the BDNE energy windows of $x$
neutron emission (defined as $W_{xn}$) are also drawn by the dotted lines. In
case of the cutoff method, $P_{xn}$ can be directly calculated from the amount
of $\beta$-feeding to $W_{xn}$ in Fig. 6. Figure 6(b) shows the isotope
production ratios ($p_{Z+1,N-1-x}(i)$) of Ag isotopes ($149\leq A\leq 145$)
from the $1^{+}$ excited states of 150Ag. As we can see, $p(i)$ distribute
into wide range excitation energies and are not stuffed inside a specific BDNE
energy window. Note that $p(i)$ of 145Ag distributes from $W_{7n}$ to
$W_{>9n}$ according to the HFM calculation, while $W_{5n}$ ranges from $2.0$
to $3.1$ MeV. This mismatch occurs because $\beta$-delayed neutrons take out
the excitation energy of daughter nuclei. On the other hand, the cutoff method
assumes zero kinetic energy of emitted neutrons. Calculated average kinetic
energy of $\beta$-delayed neutron for 150Pd precursor is about $640$ keV. If 5
neutrons are emitted, about $3.2$ MeV are taken away by the neutrons from
daughter nuclei, and this energy is approximately consistent to the difference
between $W_{5n}$ and high regions of the survival probability of 145Ag. The
result of $\beta$-delayed neutron branching ratios for the $pn$-RQRPA+HFM and
the cutoff method are listed in Table 2. For $x\geq 7$, $P_{xn}$ of the cutoff
method are larger than the $pn$-RQRPA+HFM results. The high fractions of
$P_{xn}$ of the cutoff method are redistributed to $P_{xn}$ for $x<7$ of the
$pn$-RQRPA+HFM.

(%) |  $pn$-RQRPA+HFM | Cutoff method
---|---|---
$P_{1n}$ | $39.5$ | $27.6$
$P_{2n}$ | $2.0$ | $1.6$
$P_{3n}$ | $20.5$ | $8.1$
$P_{4n}$ | $8.3$ | $2.0$
$P_{5n}$ | $17.9$ | $15.4$
$P_{6n}$ | $3.6$ | $3.1$
$P_{7n}$ | $7.1$ | $12.4$
$P_{8n}$ | $0.6$ | $2.5$
$P_{9n}$ | $0.4$ | $6.2$
$P_{>9n}$ | $0.0$ | $21.1$
Table 2: $\beta$-delayed neutron branching ratios of 150Pd.

We also computed delayed neutron yield of thermal neutron induced fission of
235U. The result is shown in Table 3. The fission fragment yields of thermal
neutron induced fission of 235U are taken from JENDL Fission Product Yield
2011 (JENDL/FPY2011) Katakura (2012); Katakura, Jun-ichi  _et al._ (2016). The
$pn$-RQRPA gives the closest $\beta$-delayed neutron yield to the experimental
data among four models because the $P_{n}$ values are tuned to reproduce it.
The result of $pn$-RQRPA+HFM also reproduces the experimental data in the same
order. Note that the calculated delayed neutron yields are an aggregated value
summed over fission fragment yields. Looking into important precursor nuclei
contributing the delayed neutron yield, the result of $pn$-RQRPA is not
necessarily correct. For example, a precursor contributing the delayed neutron
yield the most is 137I ($P_{1n}(e)=7.66\%$ ENS (2019)) according to the recent
evaluation Dimitriou  _et al._ (2020), while it is 91Rb for the $pn$-RQRPA
calculation and 137I enters the eighth place with $P_{1n}=0.7\%$. On the other
hand, the $pn$-RQRPA+HFM calculation shows 137I to be the most important
precursor, however, the $P_{n}(\sim 2.25\%)$ is still underestimated and thus
the calculated delayed neutron yield is smaller than the experimental value.

Table 3: $\beta$-delayed neutron yield of thermal neutron induced fission of 235U. Fission fragment yields are taken from JENDL/FPY-2011 Katakura (2012); Katakura, Jun-ichi  _et al._ (2016). Model |  $\beta$-delayed neutron yield
---|---
$pn$-RQRPA | $1.43\times 10^{-2}$
$pn$-RQRPA+HFM | $1.00\times 10^{-2}$
GT2 | $0.81\times 10^{-2}$
FRDM+QRPA+HFM | $0.81\times 10^{-2}$
exp. Keepin  _et al._ (1957) | $(1.58\pm 0.05)\times 10^{-2}$

The advantageous to couple the Hauser-Feshbach statistical model is that one
can calculate spectra of emitted particles. As an example, Fig. 7 shows the
BDNE spectrum of 89Br and 138I, which are typical $\beta$-delayed neutron
precursors. We also plot the experimental data taken from Rudstam and Shalev
(1974); Shalev and Rudstam (1974). Although fine structures observed in the
experimental data are not reproduced well by the $pn$-RQRPA+HFM, the computed
results emulates the experimental $\beta$-delayed neutron spectra reasonably.

Figure 7: BDNE spectrum of (a) 89Br and (b) 138I. Calculated result is shown
by the solid line (red). Experimental data are taken from Rudstam et al.
Rudstam and Shalev (1974) and Shalev et al. Shalev and Rudstam (1974).

###  III.2 $\beta$-delayed fission

Figure 8: BDF branching ratios $P_{f}$ (%) calculated by the fission barrier
data of (a)HFB-14 Goriely  _et al._ (2007), (b)FRDM Möller  _et al._ (2015),
(c)ETFSI Mamdouh  _et al._ (2001), and (d)SBM Koura (2014). Because HFB-14
provides fission barrier data to $Z=102$, $P_{f}$ values for $Z\geq 103$ are
given as blank.

Figure 8 shows BDF branching ratios in the $N$-$Z$ plane using different
fission barrier data of (a)HFB-14 Goriely  _et al._ (2007), (b)FRDM Möller
_et al._ (2015), (c)ETFSI Mamdouh  _et al._ (2001), and (d)SBM Koura (2014).
Note that the HFB-14 provides fission barrier data only up to $Z=102$, so that
$P_{f}$ over $Z=102$ are given as blank. Around $101\leq Z\leq 110$ and
$184\leq N\leq 200$, BDF plays a meaningful role commonly for fission barrier
data of FRDM, ETFSI, and SBM. Although calculated result is limited for the
HFB-14 barrier, $P_{f}$ become meaningful around $Z=100$ and $N=190$ as well.
BDF is also significant from $Z=91,N=160$ to $Z=100,N=185$ for FRDM, ETFSI,
and SBM, while that for HFB-14 is negligibly small because its fission
barriers are higher than the others. There exists odd-even dependence in the
region of $N=160$ to $185$, in particular, for FRDM and SBM. This odd-even
dependence is also found in Ref. Mumpower  _et al._ (2018) that calculates
$P_{f}$ with the FRDM+QRPA and the FRDM barrier data Möller  _et al._ (2015).
An effect of the neutron magic number of $N=184$ is seen for ETFSI and a
valley of $P_{f}$ is formed, while that for HFB-14, FRDM, SBM are not clearly
seen.

BDF branching ratios are correlated to $\beta$-delayed neutron branching
ratio. Therefore, we prepared four kinds of data tables depending on fission
barrier data, which are available in the supplemental material. Undoubtedly
$P_{n}$ are identical among the data tables unless BDF comes into effect.

###  III.3 Delayed $\alpha$ emission

In this work, we allow daughter nuclei also to decay by $\alpha$-particle
emission, and study the $\beta$-delayed $\alpha$ emission branching ratio,
$P_{\alpha}$. The calculated $P_{\alpha}$ are shown in Fig. 9 in case of FRDM
fission barrier data. Note that the maximum scale is set to be
$P_{\alpha}=10$% for illustration. Since neutron emission overcomes
$\alpha$-particle emission in neutron-rich region, a non-negligible
$P_{\alpha}$ is observed only in a band of near the $\beta$ stability line. To
our knowledge, experimental data on $\beta$-delayed $\alpha$-particle emission
has been reported only for 214Bi ($P_{\alpha}=0.003$%) Lewis (1971) in the
range of Fig 9. The calculated result of $P_{\alpha}$ for 214Bi is $0.026$%.
Although our model overestimates the experimental data, it shows that the
$\beta$-delayed $\alpha$-particle emission branching ratio is very small.

Exceptional cases exceeding $P_{\alpha}=10$% are 210,211Bi($Z=83$) and
248Am($Z=95$), that have $P_{\alpha}=92.5,48.9$ and $15.9$%, respectively. The
daughter nuclei, 210,211Po and 248Cm, have a relatively high $\alpha$ decay
rates from the ground states Lewis (1971). It is not thus surprising that they
do $\alpha$-particle emission following the $\beta$-decay, competing with
$\gamma$ and other de-excitations. We checked delayed proton emission
branching ratios $P_{p}$ as well. However, no significant $P_{p}$ is observed
in this work. Note that our calculation is carried out only for nuclei in the
$\beta$-stability line to a neutron-rich region. Our future plan is to study a
neutron-deficient region where it is expected that $P_{\alpha}$ as well as
$P_{p}$ become more important than $P_{n}$.

Figure 9: Delayed alpha branching ratios $P_{\alpha}$ (%) in case of FRDM
fission barrier data.

##  IV Summary

In this work, we calculated the BDNE and BDF by combining the discrete $\beta$
strength function provided by the $pn$-RQRPA Marketin  _et al._ (2016) and
Hauser-Feshbach statistical model Hauser and Feshbach (1952). We obtained the
root mean square of $P_{1n}$ in a good agreement with experimental data, which
was comparable to the results of proceeding works, and could improve $P_{1n}$
in a wide range of nuclei as compared to Ref. Marketin  _et al._ (2016). We
also discussed the different role of Gaussian type and the Lorentzian type
weight function.

For nuclei near the neutron drip line, we concluded that $P_{1n}$ of the
$pn$-RQRPA were largely different from that of the $pn$-RQRPA+HFM. We
explained that these deviations are attributed to the problem of negative
resonance energies resulted from the correction in Eq. (2) and the energy
withdrawal from daughter nuclei by $\beta$-delayed neutrons. Calculated
$\beta$-delayed neutron spectra of 89Br and 138I were compared with the
experimental data and it turned out that the $pn$-RQRPA+HFM was able to
reasonably reproduce the experimental data. We also calculated the
$\beta$-delayed neutron yield of thermal neutron induced fission of 235U. The
computed result was in an agreement with the experimental data in the same
order.

BDF branching ratios are calculated with four different fission barrier data.
We observed a strong dependence of $P_{f}$ on the fission barrier data used.
By comparing the result of different fission barrier data, BDF may
particularly become important around $101\leq Z\leq 110$ and $184\leq N\leq
200$. Since fission barrier data have a large uncertainty, a further study is
highly required to investigate $P_{f}$ as well.

We have to mention that level structures of neutron-rich nuclei are calculated
by the phenomenological method described in Sect. II.1 because they are not
known. A more accurate data of BDNE and BDF is expected if the level
structures would be investigated well. Further study on this respect is
required.

A table of BDNE, BDF, and $\beta$-delayed $\alpha$-particle emission branching
ratios calculated in this work is available in the supplement material sup .

###### Acknowledgements.

N.P. acknowledges support by the QuantiXLie Centre of Excellence, a project co
financed by the Croatian Government and European Union through the European
Regional Development Fund, the Competitiveness and Cohesion Operational
Programme (KK.01.1.1.01).

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