# Finite-amplitude method for collective inertia in spontaneous fission

Kouhei Washiyama<EMAIL_ADDRESS>Research Center for Superheavy
Elements, Kyushu University, Fukuoka 819-0395, Japan  Nobuo Hinohara  Center
for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571,
Japan  Takashi Nakatsukasa  Center for Computational Sciences, University of
Tsukuba, Tsukuba 305-8577, Japan  Faculty of Pure and Applied Sciences,
University of Tsukuba, Tsukuba 305-8571, Japan

###### Abstract

Background

Microscopic description of spontaneous fission is one of the most challenging
subjects in nuclear physics. It is necessary to evaluate the collective
potential and the collective inertia along a fission path for a description of
quantum tunneling in spontaneous or low-energy fission. In past studies of the
fission dynamics based on nuclear energy density functional (EDF) theory, the
collective inertia has been evaluated with the cranking approximation, which
neglects dynamical residual effects.

Purpose

The purpose is to provide a reliable and efficient method to include dynamical
residual effects in the collective inertia for fission dynamics.

Methods

We use the local quasiparticle random-phase approximation (LQRPA) to evaluate
the collective inertia along a fission path obtained by the constrained
Hartree-Fock-Bogoliubov method with the Skyrme EDF. The finite-amplitude
method (FAM) with a contour integration technique enables us to efficiently
compute the collective inertia in a large model space.

Results

We evaluate the FAM-QRPA collective inertia along a symmetric fission path in
240Pu and 256Fm. The FAM-QRPA inertia is significantly larger than the one of
the cranking approximation, and shows pronounced peaks around the ground state
and the fission isomer. This is due to dynamical residual effects.

Conclusions

To describe the spontaneous or low-energy fission, we provide a reliable and
efficient method to construct the collective inertia with dynamical residual
effects that have been neglected in most of EDF-based works in the past. We
show the importance of dynamical residual effects to the collective inertia.
This work will be a starting point for a systematic study of fission dynamics
in heavy and superheavy nuclei to microscopically describe the nuclear large-
amplitude collective motions.

##  I Introduction

Nuclear fission [1, 2] plays an important role in various phenomena such as
synthesis of superheavy elements [3, 4] and $r$-process nucleosynthesis in the
universe [5, 6]. The fission governs the existence and decay property of
superheavy nuclei. Various types of fission, such as neutron-induced fission,
spontaneous fission, and $\beta$-delayed fission that involve neutron-rich
heavy and superheavy nuclei, are important in expected environment in the
$r$-process.

Theoretically, a microscopic description of large-amplitude collective
motions, such as nuclear fission and fusion, is one of the challenging
subjects in quantum many-body physics. Of particular interest is a quantum
many-body tunneling for spontaneous or low-energy fission and subbarrier
fusion of complex nuclei.

The most promising candidate for the microscopic approach is the
selfconsistent nuclear energy density functional (EDF) theory [7, 8]. It is
well established that the nuclear EDF gives a good description of ground-state
properties of nuclei in the whole nuclear chart. The EDF approaches to the
spontaneous fission have used a semiclassical description with the WKB
approximation for quantum tunneling. These studies show how the result depends
on the choice of relevant collective variables, the collective potential
energy, and the collective inertia entering the action integral in the WKB
approximation. The potential energy has been calculated by the Hartree-Fock-
Bogoliubov method with constraints on the collective variables. On the other
hand, the collective inertia used in the previous EDF-based works [9, 10, 11,
12, 13] is insufficient in the following respect: First of all, the previous
works employ the so-called cranking approximation to the adiabatic time-
dependent Hartree-Fock-Bogoliubov (ATDHFB) method [14] to evaluate the
collective inertia. The cranking approximation neglects dynamical residual
effects, especially, the time-odd terms of the EDF. This gives a big drawback,
namely, the violation of the (local) Galilean symmetry, which leads to the
incorrect total mass for the translational motion [15, 8]. The rotational
moments of inertia of deformed nuclei are not properly given in the cranking
approximation [16]. In addition, the collective inertia of the cranking
approximation significantly deviates from the one including the dynamical
residual effects [17]. Irrespective of such drawbacks, because of its
simplicity, the cranking approximation has been widely used in evaluating
collective inertia not only for fission dynamics but also for the collective
Hamiltonian method [18, 19, 20, 21, 22]. It should be noted that the
enhancement factors of $1.2$–$1.4$ are often adopted to correct the missing
residual effect.

An alternative method to ATDHFB for describing large-amplitude collective
motions has been developed, called the adiabatic selfconsistent collective
coordinate (ASCC) method [23]. Based on the ASCC method, an efficient and
feasible method to construct the collective Hamiltonian has been proposed
[24]. In this method, the collective path (subspace) is determined by the
constrained HFB (CHFB) calculation, while the collective inertia (the
vibrational mass in the vibrational kinetic Hamiltonian) is determined by
local normal modes built on CHFB states. Local normal modes are obtained by
solving local quasiparticle random-phase approximation (LQRPA) equations.
Dynamical residual effects with time-odd components are selfconsistently
included in the collective inertia by the LQRPA. This method, called CHFB +
LQRPA, has been applied to constructing five-dimensional quadrupole collective
Hamiltonian for triaxial shapes with the pairing-plus-quadrupole (P+Q)
Hamiltonian [24, 25, 26, 27], a hybrid model of the P+Q Hamiltonian and
covariant EDF [28], and three-dimensional quadrupole collective Hamiltonian
for axially symmetric shapes with the Skyrme EDF [29]. Those works have shown
the importance of including dynamical residual effects with time-odd
components on the description of large-amplitude collective motion.

Despite many evidences for the importance of dynamical residual effects, no
practical application based on the ATDHFB method takes into account residual
effects in the calculation of the collective inertia. The reason is that it
requires huge numerical computations to apply directly the ATDHFB (or LQRPA)
formulas to the collective inertia in realistic cases, due to the size of
matrices treated with modern EDFs.

A method to efficiently solve the QRPA equations based on linear response
theory has been developed for the EDF, called the finite-amplitude method
(FAM) [30]. In the FAM, response functions to an external one-body field can
be obtained only with one-body induced densities and fields with an iterative
scheme. The FAM requires matrices with the size of approximately $N\times N$,
while the QRPA equation requires those with the size $\approx N^{2}\times
N^{2}$, where $N$ represents the number of single-particle basis. $N$ may
exceeds $10^{3}$ when, for example, 20 major shells in the harmonic oscillator
basis are used in solving deformed HFB equations. Thus, the computational cost
of the FAM is significantly lower than that of QRPA. As an alternative way of
solving the QRPA equations, the FAM has been employed in various applications
[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].

The standard FAM formalism has been extended to the one in terms of the
momentum–coordinate (PQ) representation [45] to treat zero-energy modes of
QRPA solutions known as spurious modes or Nambu–Goldstone (NG) modes [46, 47]
associated with spontaneous symmetry breaking of the mean field. A general
method to calculate the inertia of the NG modes was also given [45], which was
applied to pairing rotation [48, 49] and to spatial rotation of axially
deformed [50] and of triaxially deformed nuclei [51]. It is important to note
that this formulation can treat imaginary solutions of the QRPA that may
appear in the LQRPA at nonequilibrium HFB states obtained by the CHFB
calculations.

The main aim of this paper is to propose a new method to efficiently evaluate
the collective inertia used in the WKB approximation for spontaneous fission
and in the vibrational kinetic terms in the collective Hamiltonian method. The
method is based on the CHFB + LQRPA with the Skyrme EDF. We employ the PQ
representation of the FAM to efficiently solve the LQRPA equations. We extend
a contour integration approach of the FAM proposed in Ref. [36] to compute the
collective inertia associated with the most collective local normal mode in
the LQRPA.

This paper is organized as follows. Section II briefly summarizes the
formulations of the collective inertia in ATDHFB with the cranking
approximation and in the LQRPA. In Sec. III, we explain the formulation of the
CHFB + LQRPA with the FAM to calculate the collective inertia. In Sec. IV, we
present the results of the collective inertia along mass-symmetric fission
path in 240Pu and 256Fm and comparison between our results and those with the
cranking approximation. Conclusions are given in Sec. V.

##  II Collective inertia with EDF

###  II.1 ATDHFB

We give a brief explanation for the collective inertia in ATDHFB. For details,
we refer to Refs. [14, 17, 9, 13].

In ATDHFB, the collective motion of the system is assumed to be described by a
set of a few collective coordinates $s_{i}$ and its conjugate momenta. The
collective coordinates are defined as
$s_{i}=\langle\phi(s)|\hat{s}_{i}|\phi(s)\rangle$, where $\hat{s}_{i}$ are
generators driving the collective motion. In practice, the collective subspace
$\ket{\phi(s)}$ is determined by the CHFB calculation with the constraint on
$s_{i}$. Most applications take $\hat{s}_{i}$ as one-body operators of
multipole moments. The ATDHFB relies on the assumption that the collective
velocities of the system denoted by the time derivative of the collective
coordinates, $\dot{s}_{i}$, are slow enough to make adiabatic assumption
valid. The expression of the collective inertia tensor in ATDHFB is given by

$\mathcal{M}_{ij}=\frac{i}{2\dot{s}_{i}\dot{s}_{j}}\text{Tr}(F^{i*}Z^{j}-F^{i}Z^{j*}).$
(1)

The matrix $Z$ involves time-odd fields associated with time-odd density
matrices. A standard method to determine the matrix $Z$ requires solution of
the QRPA equation. The matrix $F$ is evaluated by

$\frac{F^{i}}{\dot{s}_{i}}=U^{\dagger}\frac{\partial\rho}{\partial
s_{i}}V^{*}+U^{\dagger}\frac{\partial\kappa}{\partial
s_{i}}U^{*}-V^{\dagger}\frac{\partial\rho^{*}}{\partial
s_{i}}U^{*}-V^{\dagger}\frac{\partial\kappa^{*}}{\partial s_{i}}V^{*},$ (2)

where $U$ and $V$ are the Bogoliubov transformation matrices and $\rho$ and
$\kappa$ are the density matrix and pairing tensor, respectively. This
expression includes derivatives of density matrices with respect to the
collective coordinates.

####  II.1.1 Cranking approximation

The cranking approximation neglects the time-odd fields in the collective
inertia (1). In this case, the matrix $Z^{i}$ can be simply written in terms
of $F^{i}$ and the quasiparticle energies. The collective inertia tensor of
the cranking approximation then reads in the quasiparticle basis

$\mathcal{M}_{ij}^{\text{C}}=\frac{1}{2\dot{s}_{i}\dot{s}_{j}}\sum_{\mu\nu}\frac{(F_{\mu\nu}^{i*}F_{\mu\nu}^{j}+F_{\mu\nu}^{i}F_{\mu\nu}^{j*})}{E_{\mu}+E_{\nu}},$
(3)

where $E_{\mu}$ and $E_{\nu}$ are one-quasiparticle energies defined with
respect to the constrained Hamiltonian of the form of Eq. (7) and the CHFB
state $\ket{\phi(s)}$ of Eq. (6).

####  II.1.2 Perturbative cranking approximation to ATDHFB

Further approximation leads to the so-called perturbative cranking
approximation, where the derivatives of the densities with respect to the
collective coordinates in Eq. (2) are not explicitly evaluated, but are
obtained in a perturbative manner. The expression of the collective inertia in
the perturbative cranking approximation is given as

$\mathcal{M}^{\text{PC}}=\frac{1}{2}[M^{(1)}]^{-1}M^{(3)}[M^{(1)}]^{-1},$ (4)

where the $n$-th energy-weighted moment $M^{(n)}$ is given as

$M_{ij}^{(n)}=\sum_{\mu<\nu}\frac{\langle\phi(s)|\hat{s}_{i}|\mu\nu\rangle\langle\mu\nu|\hat{s}_{j}^{\dagger}|\phi(s)\rangle}{(E_{\mu}+E_{\nu})^{n}},$
(5)

where $|\mu\nu\rangle$ are two-quasiparticle states based on the CHFB state
$\ket{\phi(s)}$.

###  II.2 CHFB + Local QRPA

Another method to evaluate the collective inertia is the CHFB + LQRPA [24]
derived from the ASCC method [23]. Suppose that a set of $N$ collective
coordinates $q_{i}$ ($i=1,\cdots,N$) selfconsistently determined with the ASCC
method can be mapped to a set of $N$ collective variables $s_{m}$
($m=1,\cdots,N$) through a one-to-one correspondence between them. In
addition, we assume that the ASCC collective subspace can be approximated by
the one obtained with the CHFB calculation with a given set of constraining
operators $\hat{s}_{m}$. The CHFB equation is given by

$\delta\langle\phi(s)|\hat{H}_{M}|\phi(s)\rangle=0,$ (6)

with the constrained Hamiltonian

$\hat{H}_{M}=\hat{H}-\sum_{\tau=n,p}\lambda_{\tau}\hat{N}_{\tau}-\sum_{m}\lambda_{m}\hat{s}_{m},$
(7)

where $\lambda_{n,p}$ denote the Fermi energies for neutrons and protons to
constrain the average neutron and proton numbers
$\bra{\phi(s)}\hat{N}_{\tau}\ket{\phi(s)}$ ($\tau=n,p$) and $\lambda_{m}$ are
the Lagrange multipliers of constraining
$\langle\phi(s)|\hat{s}_{m}|\phi(s)\rangle=s_{m}$. The energy minimization,
Eq. (6), leads to the CHFB state $|\phi(s)\rangle$ and collective potential
$V(s)=\langle\phi(s)|\hat{H}|\phi(s)\rangle$. At each CHFB state
$|\phi(s)\rangle$, the LQRPA equations,

$\displaystyle\delta\langle\phi(s)|[\hat{H}_{M},\hat{Q}_{i}(s)]-\frac{1}{i}\hat{P}_{i}(s)|\phi(s)\rangle$
$\displaystyle=0,$ (8)
$\displaystyle\delta\langle\phi(s)|[\hat{H}_{M},\frac{1}{i}\hat{P}_{i}(s)]-\Omega_{i}^{2}(s)\hat{Q}_{i}(s)|\phi(s)\rangle$
$\displaystyle=0,$ (9)

are solved to determine $\hat{Q}_{i}(s)$ and $\hat{P}_{i}(s)$ that are local
generators of collective momenta and coordinates, respectively, defined at
$s$. They should satisfy the weak canonicity condition, Eq. (19). Here, the
inertial and the curvature tensors are diagonalized to define the local normal
mode. Furthermore, in order to fix the arbitrary scale of the collective
coordinate $q_{i}$, the inertia with respect to $q_{i}$ is set to be unity,
${\cal M}_{ij}=\delta_{ij}$. $\Omega^{2}_{i}(s)$ denotes the squared
eigenfrequency of the local normal mode. Note that the eigenfrequency of the
LQRPA equations can be imaginary ($\Omega^{2}_{i}(s)<0$) at nonequilibrium
CHFB states. A criterion for selecting relevant collective LQRPA modes from
many LQRPA solutions is given in Ref. [24].

Once relevant LQRPA collective modes are selected, the collective inertia
tensor is given as follows. First, the collective kinetic energy is expressed
as

$\displaystyle T=\frac{1}{2}\sum_{i}\dot{q}_{i}^{2}=\frac{1}{2}\sum_{mn}{\cal
M}_{mn}(s)\dot{s}_{m}\dot{s}_{n},$ (10)

where collective inertia tensor $\mathcal{M}_{mn}(s)$ is defined by the
transformation of the collective coordinates $q_{i}$ to the collective
variables $s$,

$\mathcal{M}_{mn}(s)\equiv\sum_{i,j}\frac{\partial q_{i}}{\partial s_{m}}{\cal
M}_{ij}\frac{\partial q_{j}}{\partial s_{n}}=\sum_{i}\frac{\partial
q_{i}}{\partial s_{m}}\frac{\partial q_{i}}{\partial s_{n}}.$ (11)

Second, the partial derivatives in Eq. (11) are evaluated using the local
generator $\hat{P}_{i}(s)$ of the LQRPA solution as

$\displaystyle\frac{\partial s_{m}}{\partial q_{i}}$
$\displaystyle=\frac{\partial}{\partial
q_{i}}\langle\phi(s)|\hat{s}_{m}|\phi(s)\rangle$
$\displaystyle=\langle\phi(s)|[\hat{s}_{m},\frac{1}{i}\hat{P}_{i}(s)]|\phi(s)\rangle,$
(12)

which is calculable without numerical derivatives.

##  III Finite-amplitude method for collective inertia

In this work, we use the LQRPA method to evaluate the collective inertia along
a fission path. Since it is computationally hard to solve the LQRPA equations
for deformed nuclear shapes with the Skyrme EDF, we employ the FAM to
efficiently solve the LQRPA equations. In this section, we explain our method
to obtain the expression of the collective inertia based on the FAM and LQRPA.

###  III.1 Finite-amplitude method

Here, the FAM is recapitulated. The details of its derivation can be found in
Refs. [30, 32]. The FAM equations can be expressed in the quasiparticle basis
as

$\displaystyle(E_{\mu}+E_{\nu}-\omega)X_{\mu\nu}(\omega)+\delta
H^{20}_{\mu\nu}(\omega)$ $\displaystyle=-F^{20}_{\mu\nu}\;,$ (13a)
$\displaystyle(E_{\mu}+E_{\nu}+\omega)Y_{\mu\nu}(\omega)+\delta
H^{02}_{\mu\nu}(\omega)$ $\displaystyle=-F^{02}_{\mu\nu}\;,$ (13b)

where $X_{\mu\nu}(\omega)$ and $Y_{\mu\nu}(\omega)$ are the FAM amplitudes at
a given frequency $\omega$, and $\delta H^{20(02)}_{\mu\nu}$ and
$F^{20(02)}_{\mu\nu}$ are two-quasiparticle components of one-body induced
field $\delta\hat{H}$ and an external field $\hat{F}$, respectively. The FAM
equations (13) are iteratively solved at each $\omega$ until converged $X$ and
$Y$ amplitudes are obtained. Complex frequency $\omega$ is usually used with
the imaginary part corresponding to a smearing width. Only one-body quantities
(induced densities and induced fields) are necessary to solve the FAM
equations.

The FAM strength function is given from converged $X$ and $Y$ amplitudes as

$S(\hat{F},\omega)=\sum_{\mu<\nu}\left[F_{\mu\nu}^{20*}X_{\mu\nu}(\omega)+F_{\mu\nu}^{02*}Y_{\mu\nu}(\omega)\right].$
(14)

Taking the frequency $\omega$ real and positive, the transition strength
distribution is given as

$\frac{dB(\hat{F},\omega)}{d\omega}\equiv\sum_{i(\Omega_{i}>0)}\left|\bra{i}\hat{F}\ket{0}\right|^{2}\delta(\omega-\Omega_{i})=-\frac{1}{\pi}\text{Im}S(\hat{F},\omega),$
(15)

where $\ket{0}$ is the QRPA vacuum and
$\ket{i}\equiv\hat{O}_{i}^{\dagger}\ket{0}$ is a state of the QRPA normal mode
of excitation with the eigenfrequency $\Omega_{i}$.

The relation between the FAM strength function (14) and the QRPA transition
strength $|\langle i|\hat{F}|0\rangle|^{2}$ between the ground state and a
QRPA eigenstate $\ket{i}$ is given as [36]

$S(\hat{F},\omega)=-\sum_{i(\Omega_{i}>0)}\left(\frac{|\langle
i|\hat{F}|0\rangle|^{2}}{\Omega_{i}-\omega}+\frac{|\langle
0|\hat{F}|i\rangle|^{2}}{\Omega_{i}+\omega}\right).$ (16)

###  III.2 FAM in the PQ representation

In Ref. [45], the FAM in the PQ representation of the QRPA was formulated to
investigate the NG modes and the Thouless–Valatin inertia of the NG modes. In
this subsection, we summarize the formulation of the FAM in the PQ
representation.

In the PQ representation of the QRPA, the coordinate and the conjugate
momentum operators, $\hat{Q}_{i}$ and $\hat{P}_{i}$, which are both Hermitian,
are constructed from the QRPA phonon operators
$(\hat{O}_{i},\hat{O}^{\dagger}_{i})$ as

$\displaystyle\hat{Q}_{i}$
$\displaystyle=\sqrt{\frac{1}{2M_{i}\Omega_{i}}}(\hat{O}_{i}+\hat{O}^{\dagger}_{i}),$
(17) $\displaystyle\hat{P}_{i}$
$\displaystyle=\frac{1}{i}\sqrt{\frac{M_{i}\Omega_{i}}{2}}(\hat{O}_{i}-\hat{O}^{\dagger}_{i}),$
(18)

where $M_{i}$ is the inertia parameter. The operators $\hat{Q}_{i}$ and
$\hat{P}_{i}$ fulfill the following commutation relations,

$\bra{0}[\hat{Q}_{i},\hat{P}_{j}]\ket{0}=i\delta_{ij},\;\bra{0}[\hat{Q}_{i},\hat{Q}_{j}]\ket{0}=\bra{0}[\hat{P}_{i},\hat{P}_{j}]\ket{0}=0,$
(19)

which guarantee the orthogonality among different normal modes, however, their
scale is still arbitrary as $(\alpha\hat{Q}_{i},\alpha^{-1}\hat{P}_{i})$. In
the present study, this scale is fixed by imposing the additional condition,
$M_{i}=1$.

Using two-quasiparticle components $P_{\mu\nu}^{i}$ and $Q_{\mu\nu}^{i}$ of
the operators $\hat{P}_{i}$ and $\hat{Q}_{i}$, the FAM $X$ and $Y$ amplitudes
are given as

$\displaystyle X_{\mu\nu}(\omega)=$
$\displaystyle\sum_{i}\frac{1}{\omega^{2}-\Omega^{2}_{i}}\Bigl{[}(-i\omega
P_{\mu\nu}^{i}+\Omega_{i}^{2}Q_{\mu\nu}^{i})q_{i}(\hat{F})$
$\displaystyle\qquad+(P_{\mu\nu}^{i}+i\omega
Q_{\mu\nu}^{i})p_{i}(\hat{F})\Bigr{]},$ (20a) $\displaystyle
Y_{\mu\nu}(\omega)=$
$\displaystyle\sum_{i}\frac{1}{\omega^{2}-\Omega^{2}_{i}}\Bigl{[}(i\omega
P_{\mu\nu}^{i*}-\Omega_{i}^{2}Q_{\mu\nu}^{i*})q_{i}(\hat{F})$
$\displaystyle\qquad+(-P_{\mu\nu}^{i*}-i\omega
Q_{\mu\nu}^{i*})p_{i}(\hat{F})\Bigr{]},$ (20b)

where $p_{i}(\hat{F})$ and $q_{i}(\hat{F})$ are defined as

$\displaystyle p_{i}(\hat{F})$
$\displaystyle\equiv\bra{0}[\hat{P}_{i},\hat{F}]\ket{0}=\sum_{\mu<\nu}\left(P_{\mu\nu}^{i*}F_{\mu\nu}^{20}-P_{\mu\nu}^{i}F_{\mu\nu}^{02}\right),$
(21) $\displaystyle q_{i}(\hat{F})$
$\displaystyle\equiv\bra{0}[\hat{Q}_{i},\hat{F}]\ket{0}=\sum_{\mu<\nu}\left(Q_{\mu\nu}^{i*}F_{\mu\nu}^{20}-Q_{\mu\nu}^{i}F_{\mu\nu}^{02}\right).$
(22)

The FAM strength function (14) is then rewritten as

$\displaystyle S(\hat{F},\omega)$
$\displaystyle=\sum_{\mu<\nu}\left[F_{\mu\nu}^{20*}X_{\mu\nu}(\omega)+F_{\mu\nu}^{02*}Y_{\mu\nu}(\omega)\right]$
$\displaystyle=\sum_{i}\frac{1}{\omega^{2}-\Omega^{2}_{i}}\left[|p_{i}(\hat{F})|^{2}+\Omega_{i}^{2}|q_{i}(\hat{F})|^{2}+\omega\;r_{i}(\hat{F})\right],$
(23)

with a real quantity

$r_{i}(\hat{F})\equiv
i\left[q_{i}^{*}(\hat{F})p_{i}(\hat{F})-q_{i}(\hat{F})p_{i}^{*}(\hat{F})\right].$
(24)

The FAM $X(\omega)$ and $Y(\omega)$ amplitudes (20) and the FAM strength
function (23) are defined in the whole complex plane $\omega$ except for
$\omega=\pm\Omega_{i}$. They are well defined even with the presence of the NG
modes ($\Omega_{i}^{2}=0$) and the imaginary solutions ($\Omega_{i}^{2}<0$)
which can appear in the LQRPA at nonequilibrium CHFB states.

###  III.3 Collective inertia with FAM plus contour integration technique

For simplicity, in this paper, we assume only one collective coordinate $q$,
and then adopt the isoscalar quadrupole moment
$\hat{s}=\hat{Q}_{20}=\sqrt{16\pi/5}\sum_{i=1}^{A}r_{i}^{2}Y_{20}(\hat{\boldsymbol{r}}_{i})$
as the constraint. For the LQRPA calculation, we adopt the same operator for
the external field $\hat{F}=\hat{Q}_{20}$. Then, by transforming the scale of
the coordinate from $q$ to $s=\bra{\phi(s)}\hat{Q}_{20}\ket{\phi(s)}$, the
expression of the collective inertia (11) becomes

${\cal M}\equiv\frac{dq}{ds}\frac{dq}{ds},$ (25)

with $dq/ds=(ds/dq)^{-1}$ and

$\displaystyle\frac{ds}{dq}$
$\displaystyle=\langle\phi(s)|[\hat{Q}_{20},\frac{1}{i}\hat{P}_{i}]|\phi(s)\rangle$
$\displaystyle=ip_{i}(\hat{Q}_{20}).$ (26)

We will see below that $p_{i}(\hat{Q}_{20})$ is pure imaginary, leading to
real $ds/dq$. We select the normal mode $i$ which has the largest value of
$|p_{i}(\hat{Q}_{20})|^{2}$ among the low-lying eigenmodes.

The remaining task is the calculation of $p_{i}(\hat{F})=p_{i}(\hat{Q}_{20})$
in the FAM. From Eqs. (17) and (18), $P^{i}$ and $Q^{i}$ are given in terms of
the forward and backward amplitudes of the QRPA normal modes, $X^{i}$ and
$Y^{i}$, as

$\displaystyle P_{\mu\nu}^{i}$
$\displaystyle=i\sqrt{\frac{\Omega_{i}}{2}}(X_{\mu\nu}^{i}+Y_{\mu\nu}^{i}),$
(27) $\displaystyle Q_{\mu\nu}^{i}$
$\displaystyle=\sqrt{\frac{1}{2\Omega_{i}}}(X_{\mu\nu}^{i}-Y_{\mu\nu}^{i}).$
(28)

Note that we set $M_{i}=1$. When the QRPA matrices are real, we may choose
$X^{i}_{\mu\nu}$ and $Y^{i}_{\mu\nu}$ real. This leads to real
$Q^{i}_{\mu\nu}$ and pure imaginary $P^{i}_{\mu\nu}$. In the present case, the
external-field operator $\hat{F}=\hat{Q}_{20}$ is Hermitian and their two-
quasiparticle components are real, namely, $F^{02}=F^{20*}=F^{20}$. Thus,
$q_{i}(\hat{F})=0$ and $p_{i}(\hat{F})=-p_{i}^{*}(\hat{F})$ (pure imaginary)
hold according to Eq. (22). Then, the FAM strength function (23) becomes

$S(\hat{F},\omega)=\sum_{i}\frac{1}{\omega^{2}-\Omega^{2}_{i}}|p_{i}(\hat{F})|^{2}.$
(29)

Since the right-hand side of Eq. (29) has first-order poles at
$\omega=\pm\Omega_{i}$, we obtain the following expression using Cauchy’s
integral formula,

$\frac{1}{2\pi i}\oint_{C_{i}}\omega
S(\hat{F},\omega)d\omega=\frac{1}{2}|p_{i}(\hat{F})|^{2},$ (30)

where the contour circle $C_{i}$ in the complex energy plane is chosen to
encircle only the pole $\omega=\Omega_{i}$. The strength functions,
$S(\hat{F},\omega)$, at different values of $\omega$ are obtained as Eq. (14)
with iterative solution of Eq. (13).

The eigenfrequency $\Omega_{i}$ can be also obtained by combining Eq. (30)
with the following contour integration,

$\frac{1}{2\pi
i}\oint_{C_{i}}S(\hat{F},\omega)d\omega=\frac{1}{2\Omega_{i}}|p_{i}(\hat{F})|^{2}.$
(31)

The expressions (30) and (31) can be used for both real and pure imaginary
eigenfrequencies.

###  III.4 Numerical procedure

To prepare the CHFB states along the fission path, we solve the CHFB equations
with the two-basis method [52, 53], where during iteration the HFB Hamiltonian
is diagonalized in a single-particle basis that converges to the eigenstates
of the mean-field Hamiltonian, called the Hartree-Fock (HF) basis, and the
local densities are constructed in the canonical basis that diagonalizes the
density matrix. The single-particle wave functions and fields are represented
in a three-dimensional (3D) Cartesian mesh. Nuclear shapes are restricted to
have three plane reflection symmetries about the $x=0$, $y=0$, and $z=0$
planes. This reduces the model space to 1/8 of the full box and can be
realized by choosing the single-particle wave functions as eigenstates of
parity, $z$ signature, and $y$ time simplex [54, 55, 56, 57]. This symmetry
restriction prevents us from treating asymmetric fission in the present study.
We use a numerical box of 13.2 fm for $x$ and $y$ directions and 19.6 fm for
$z$ direction in $x>0$, $y>0$, and $z>0$ in order to express prolately
deformed fissioning shapes, discretized with the mesh size of $0.8$ fm for all
the directions. The number of mesh points then becomes $16\times 16\times
24=6144$. The single-particle basis consists of 2460 neutron and 1840 proton
HF-basis states to achieve the maximum quasiparticle energy of
$E_{\text{QP}}^{\text{max}}\approx 60$ MeV for both neutrons and protons in
240Pu and 256Fm. Constrained quantities are the isoscalar quadrupole moment
$Q_{20}=\langle\hat{Q}_{20}\rangle$ with $\langle\hat{Q}_{22}\rangle=0$,
keeping axially symmetric shapes. The symmetry restriction automatically
constrains the position of the center of mass, and $\langle xy\rangle=\langle
yz\rangle=\langle zx\rangle=0$. The other even multipole moments are not
constrained, and their values are optimized to minimize the total energy,
while the expectation values of all odd multipole moments are kept to be zero
due to the plane reflection symmetries of nuclear shapes. We used the SkM∗ EDF
[58] and the contact volume pairing with a pairing window of 20 MeV above and
below the Fermi energy in the HF basis described in Refs. [54, 56] to avoid
divergence in the pairing energy. The pairing strengths are adjusted
separately for neutrons and protons in order to reproduce the empirical
pairing gaps in 240Pu or in 256Fm.

We have extended our 3D FAM-QRPA code developed in Ref. [44] to perform FAM
calculations at CHFB states and the FAM-QRPA with the contour integration. We
include in the FAM the full quasiparticle basis included in the CHFB
calculations to satisfy full selfconsistency between the CHFB and LQRPA
calculations. We employ the modified Broyden method [59] for iteratively
solving the FAM equations.

To obtain the quantity $|p_{i}(\hat{F})|^{2}$ by the contour integration (30),
it is necessary to know in advance an approximate position of each QRPA pole
$\omega=\Omega_{i}$ in the complex energy plane for determining an appropriate
contour $C_{i}$. The QRPA poles are expected to appear as sharp peaks in the
FAM strength distribution. Therefore, we first calculate the FAM strength
distribution (15) in the low-lying states with $\omega^{2}<16$ MeV2. For each
pole, we calculate $|p_{i}(\hat{F})|^{2}$ and adopt the most collective mode
with the largest value of $|p_{i}(\hat{F})|^{2}$. A detail of our procedure of
selecting the most collective mode in the FAM is given in Appendix. For the
integration contour, we use a circle centered at the estimated peak frequency.
The radius is 0.05 MeV for real solutions, and 0.1 MeV for an imaginary
solution. If there are other poles close to the present pole, we use a circle
with even smaller radius. The number of discretization points along the circle
is 12 for real and 8 for imaginary solutions.

We perform numerical calculations of the FAM with a hybrid parallel scheme;
The MPI parallelization is adopted for calculation of $S(\hat{F},\omega)$ at
different $\omega$ points. The FAM iterative solution of Eq. (13) at each
$\omega$ is performed using the OpenMP parallel calculation. It takes about
160 core hours on Oakforest-PACS for the FAM contour integration for a real
solution at one deformation point with the present model space.

Figure 1:  Collective inertia as a function of $E_{\text{QP}}^{\text{max}}$
for neutrons divided by that with neutron $E_{\text{QP}}^{\text{max}}\approx
65$ MeV for four different deformation points in 240Pu. Included neutron HF-
basis states are 1200, 1440, 1632, 1998, 2460, and 2700 for
$E_{\text{QP}}^{\text{max}}\approx 40$, 45, 50, 55, 60, and 65 MeV,
respectively.

Before we proceed, we would like to discuss the convergence property of the
collective inertia with respect to the size of the model space included in the
ground state. To check the convergence property, we calculated the collective
inertia with different numbers of HF-basis states corresponding to different
$E_{\text{QP}}^{\text{max}}$ values in the ground state. Figure 1 shows the
collective inertia as a function of neutron $E_{\text{QP}}^{\text{max}}$
(nearly equal to proton $E_{\text{QP}}^{\text{max}}$) divided by that
calculated with neutron $E_{\text{QP}}^{\text{max}}\approx 65$ MeV as a
reference value for four various deformation points in 240Pu. We find that the
model space with neutron $E_{\text{QP}}^{\text{max}}\geq 55$ MeV gives a good
convergence within about less than 5% to the value with
$E_{\text{QP}}^{\text{max}}\approx 65$ MeV for the four various deformation
points. Therefore, as mentioned above, we include 2460 neutron and 1840 proton
HF-basis states corresponding to $E_{\text{QP}}^{\text{max}}\approx 60$ MeV in
our calculations. Note again that we do not introduce in the FAM calculations
the additional two-quasiparticle energy cutoff, which has been usually
employed to reduce the dimension of the QRPA matrix.

##  IV Results

Mass distributions of fission fragments measured in low-energy fission show a
sudden change from asymmetric to symmetric mass distributions when the neutron
number changes. A well known example is Fm isotopes, where 256Fm shows an
asymmetric mass distribution, while ${}^{258}\text{Fm}$ shows that the main
component of fission fragments is a symmetric one [60, 61]. Even more
complicated multi-mode or multi-channel fission has been observed and analyzed
in the total-kinetic-energy distributions of fission fragments in actinides
[62, 63]. What determines the mass distribution? It is still an open question,
and the fragment mass distribution is important for the fission recycling of
the $r$-process as well. Since, so far, most of studies are focused on the
potential landscape, we aim at investigating effect of the collective inertia.
In this section, we take 240Pu and 256Fm as examples to show importance of
dynamical residual effects on the inertia. Although, in this paper, the
calculation is performed only along the symmetric fission path, we will show
that the residual effect on the inertia may hinder the symmetric fission
probability.

###  IV.1 Collective inertia along a symmetric fission path in 240Pu

Figure 2:  (a) Potential energy (b) pairing gaps for neutrons (solid line) and
protons (dashed line) as a function of quadrupole moment $Q_{20}$ in units of
barn in 240Pu.

Figure 2(a) shows the collective potential energy as a function of expectation
value of quadrupole moment $Q_{20}$ in units of barn (b) along a symmetric
fission path in ${}^{240}\text{Pu}$ obtained from the CHFB calculation. The
ground state of 240Pu is found at $Q_{20}\approx 30$ b. The first fission
barrier, the fission isomer (local minimum), and the second fission barrier
are obtained at $Q_{20}\approx 55$ b, $Q_{20}\approx 85$ b, and $Q_{20}\approx
150$ b, respectively. We would like to note that the heights of the first and
second fission barriers would become lower if we could include triaxial
deformation for the first fission barrier and octupole deformation for the
second fission barrier [64, 65]. Neutron and proton pairing gaps are shown
along the fission path in Fig. 2(b). The neutron pairing gap becomes larger at
the regions near the fission barriers and smaller near the ground state and
fission isomer. The proton pairing gap shows a moderate behavior because of
the difference between neutron and proton single-particle structures. It is
known that transition strengths of low-lying modes are affected by the
strength of pairing correlations.

Figure 3:  (a) Collective inertia of the FAM-QRPA calculation shown by the
filled-square solid line, and of the perturbative cranking approximation by
the dashed line, (b) ratio of the FAM-QRPA inertia to the perturbative
cranking inertia, (c) squared QRPA eigenfrequency as a function of $Q_{20}$ in
240Pu.

Figure 3(a) shows the collective inertia obtained with the present FAM-QRPA
calculation by the filled-square solid line. This shows two prominent peaks at
$Q_{20}\approx 25$ b and $Q_{20}\approx 80$ b and a sudden change near these
peaks. These states closely correspond to the ground state and the fission
isomer observed in Fig. 2(a) and to the states where pairing becomes weak in
Fig. 2(b). For comparison, we add in Fig. 3(a) the collective inertia obtained
by the perturbative cranking approximation. It is clearly shown that the FAM-
QRPA inertia is larger than the perturbative cranking inertia. We emphasize
this point in Fig. 3(b), showing that the ratio of the FAM-QRPA inertia to the
perturbative cranking inertia always exceeds unity, except at $Q_{20}=60$ b.
This indicates that the action integral in the WKB approximation will be
significantly affected by the increase of the collective inertia near the
fission barriers. The behaviors of the two inertias are different; the FAM-
QRPA inertia shows a strong variation, while the perturbative cranking inertia
varies smoothly as $Q_{20}$ increases.

We estimate the action integral in the WKB approximation using the obtained
collective inertia and potential along the symmetric fission path. The action
integral reads

$S=\int_{Q_{\text{in}}}^{Q_{\text{out}}}dQ_{20}\sqrt{2\mathcal{M}(Q_{20})[V(Q_{20})-E_{0}]},$
(32)

where $Q_{\text{in}}$ and $Q_{\text{out}}$ are the classical inner and outer
turning points, and $E_{0}$ is the HFB ground state energy. We obtain the
action integrals $S_{\text{FAM}}=82.0$ for the FAM-QRPA inertia and
$S_{\text{PC}}=62.0$ for the perturbative cranking inertia from
$Q_{\text{in}}=30$ b (the ground state) to $Q_{\text{out}}=270$ b. This
difference leads to many orders of magnitude difference in the fission half-
life.

Figure 3(c) shows the eigenfrequency squared $\Omega^{2}$ of the LQRPA
solution selected as the most collective mode at each deformation. This
represents the curvature of the collective potential, which can indeed become
negative at the fission barriers. Larger values of $\Omega^{2}$ are found at
the states with larger values of the collective inertia. As $\Omega^{2}$
becomes larger, the ratio of the FAM-QRPA inertia to the perturbative cranking
inertia becomes larger. When $\Omega^{2}$ are negative, the ratio is close to
unity.

Figure 4:  Squared peak frequencies identified from the FAM strength
distribution shown by the crosses as a function of $Q_{20}$ in 240Pu.
Eigenfrequency squared of the most collective mode is shown by the open-circle
solid line.

In Fig. 4, we plot the squared peak frequencies identified from the FAM
strength distribution in $\omega^{2}<16\,\text{MeV}^{2}$ as a function of
$Q_{20}$ up to the fission isomer ($Q_{20}\leq 90$ b). Note that, because of a
finite smearing width ($=0.01$ MeV) used in the FAM calculation, the
identified peak frequencies may slightly differ from the QRPA eigenfrequencies
and may not correspond to all the eigenfrequencies. In the figure, the
eigenfrequency squared of the most collective mode that is used in the FAM-
QRPA collective inertia is shown by the open-circle solid line. At most
regions in $Q_{20}$, we find that the most collective mode corresponds to the
lowest-frequency mode and is well separated from other peaks. On the other
hand, around the ground state ($Q_{20}\approx 30$ b) and the fission isomer
($Q_{20}\approx 85$ b), where pairing becomes weak, we find that the most
collective mode appears at frequencies close to other peaks, or at higher
frequencies. The increase of the FAM-QRPA inertia near the ground and the
fission isomer states is due to decrease of the quadrupole collectivity which
leads to decrease in $|p_{i}(\hat{Q}_{20})|^{2}$. In fact, near the ground
state, we find that the character of the lowest frequency mode changes from
the quadrupole vibration to the pair vibration.

It is known that QRPA eigenmodes that possess pair-vibrational character tend
to appear in low-frequency regions when pairing becomes weak. We analyze the
character of the lowest-frequency mode at $\omega\sim 1.5$ MeV at $Q_{20}=25$,
30, and 35 b by using the FAM calculation with the pair-vibrational field as
an external field. The lowest-frequency modes at $Q_{20}=25$, 30, and 35 b
have a strong neutron pair-vibrational character and approximately satisfy the
relation $\Omega\approx 2\Delta_{n}$, where $\Delta_{n}$ denotes the neutron
pairing gap. The lowest-frequency modes at $Q_{20}=25$ and 30 b are not the
most collective in the quadrupole $Q_{20}$ nature. We also confirm this by
calculating the collective inertia for those lowest modes. For the lowest
mode, the collective inertia would be
$\mathcal{M}=0.190\,\text{MeV}^{-1}\text{b}^{-2}$ at $Q_{20}=25$ b and
$\mathcal{M}=1.02\,\text{MeV}^{-1}\text{b}^{-2}$ at $Q_{20}=30$ b. These
values are significantly larger than the values shown in Fig. 3(a), indicating
the character change of the lowest mode from the quadrupole vibration to the
neutron pair vibration.

###  IV.2 Comparison between the FAM-QRPA and nonperturbative cranking
inertias in 256Fm

The aim of taking the 256Fm case is to compare the FAM-QRPA collective inertia
with that of the nonperturbative cranking approximation by Baran et al. in
Ref. [9]. In 256Fm, it is known that reflection-symmetric and reflection-
asymmetric fission paths bifurcate at $Q_{20}\approx 130$ b [66, 67, 68]. In
Ref. [9], the collective inertia was calculated along a reflection-symmetric
fission path in $Q_{20}<130$ b and along a reflection-asymmetric fission path
in $Q_{20}>130$ b. As we do not include reflection-asymmetric fission paths,
we compare the FAM-QRPA collective inertia with that in Ref. [9] along the
reflection-symmetric fission path in $Q_{20}\leq 130$ b.

Figure 5:  Same as Fig. 2, but for 256Fm.

First, we show in Fig. 5 the potential and pairing gaps along an axial and
reflection-symmetric fission path as a function of $Q_{20}$. The structure of
a fission isomer at $Q_{20}\approx 110$ b almost vanishes due to low second
fission barrier height of about 0.4 MeV. The proton pairing vanishes in
$Q_{20}\geq 130$ b. These properties are consistent with previous EDF studies
[66, 67, 68, 9].

Figure 6:  (a) Comparison between the FAM-QRPA inertia (the filled-square
solid line), the nonperturbative cranking inertia taken from Baran et al. [9]
(the filled-triangle dot-dashed line), and the perturbative cranking inertia
(dashed line), and (b) squared QRPA eigenfrequency as a function of $Q_{20}$
along a symmetric fission path in 256Fm.

Figure 6(a) shows the FAM-QRPA collective inertia as a function of $Q_{20}$.
Two peaks in the FAM-QRPA inertia are clearly seen at the ground state
($Q_{20}\approx 30$ b) and fission isomer ($Q_{20}\approx 105$ b), which is
similar to the case of 240Pu. In this figure, we compare the FAM-QRPA inertia
with the inertia of the nonperturbative cranking approximation [9], as well as
the perturbative cranking one. We should note that the model space, the
pairing functional, and so on, adopted in the present study are different from
those in Ref. [9]. The FAM-QRPA inertia and the nonperturbative cranking
inertia vary significantly as $Q_{20}$ changes, compared with a smooth
behavior of the perturbative cranking inertia. We find two significant
differences between the FAM-QRPA inertia and the nonperturbative cranking
inertia. One is the magnitude of the collective inertia; the FAM-QRPA
collective inertia is significantly larger than the nonperturbative cranking
one at most deformation points. Similar values are obtained near the top of
the fission barrier at $Q_{20}\approx 70$ b and near the bifurcation at
$Q_{20}\approx 130$ b. The other difference is the position of peaks in
$Q_{20}$ in the collective inertia. The FAM-QRPA inertia peaks at slightly
smaller $Q_{20}$ than the cranking one does. These significant differences
are, at least partially, due to the residual time-odd fields neglected in the
cranking approximation.

Figure 6(b) shows the squared QRPA eigenfrequency obtained by the FAM-QRPA.
The peak structure of the squared QRPA eigenfrequency is similar to that of
the collective inertia, which is also seen in the 240Pu case. This suggests
that a structure change of the lowest frequency mode affects the collective
inertia in 256Fm, analogous to the case of 240Pu.

##  V Conclusion

We have developed a feasible method of calculating the collective inertia
based on the CHFB + LQRPA method with the Skyrme EDF. This method includes
time-odd components of dynamical residual effects to the collective inertia in
a selfconsistent way. We efficiently calculated the QRPA transition strengths
by using the FAM and contour integration technique. We applied this method of
evaluating the collective inertia along a symmetric fission path of 240Pu and
256Fm. The results show that the dynamical residual effects significantly
affect the collective inertia and result in an enhancement of the collective
inertia compared with the perturbative cranking one. The enhancement depends
strongly on the deformation of the states. This is a consequence of
microscopic dynamical effects. In the case of 256Fm, we also compare the FAM-
QRPA inertia with the nonperturbative cranking collective inertia in Ref. [9].
Both collective inertias show peak structures, which are not seen in the
perturbative cranking inertia. The values of the deformation at the peaks in
the inertia are different. The FAM-QRPA inertia takes much larger values than
nonperturbative one around the potential minima.

For future works, it is desirable to lift the symmetry restriction and to
study collective inertia along the asymmetric fission paths. We also plan to
extend the present method to include two or more collective variables for
constructing collective inertia tensors such as a simultaneous treatment of
quadrupole and octupole moments along multiple fission paths. Systematic study
of the dynamical residual effects on collective inertia in fission in actinide
and transactinide nuclei will be important not only for deeper understanding
of fission but also for reliable evaluation of fission half-lives in
$r$-process nucleosynthesis. The present method can be also used to describe
large-amplitude collective dynamics with the collective Hamiltonian method. It
would be interesting to compare the FAM-QRPA inertia with the full ATDHFB
inertia [42].

###### Acknowledgements.

The work was supported in part by QR Program of Kyushu University, by JSPS
KAKENHI Grant Numbers JP18H01209, JP19H05142 and JP20K03964, and by JSPS-NSFC
Bilateral Program for Joint Research Project on Nuclear mass and life for
unravelling mysteries of $r$-process. Numerical calculations were performed in
part using the COMA (PACS-IX) and Oakforest-PACS Systems through the
Multidisciplinary Cooperative Research Program of the Center for Computational
Sciences, University of Tsukuba.

*

##  Appendix A Selection of the most collective mode

Figure 7:  Example of the strength distribution along the imaginary axis at
$Q_{20}=120$ b (a) and real axis at $Q_{20}=60$ b (b) in 256Fm. These strength
distributions are calculated at each 0.1 MeV with a smearing width of 0.1 MeV.
(c) A smaller smearing width of 0.01 MeV at each 0.01 MeV in $\omega_{R}$ is
used for the same case as (b).

We explain our procedure of selecting the most collective QRPA eigenmode to
calculate the collective inertia. First, we perform the FAM calculation with
an isoscalar quadrupole external field $\hat{Q}_{20}$ and a complex frequency
$\omega=\omega_{R}+i\omega_{I}$ along the imaginary and real $\omega$ axes to
obtain the FAM strength distribution. For the FAM calculation along the
imaginary $\omega$ axis, $\omega_{R}$ has a role of a smearing width.

Next, we search sharp peaks in the FAM strength distribution, which appear at
approximate positions of the QRPA poles, $\omega=\Omega_{i}$, in
$0\leq\omega_{R}(\omega_{I})\leq 4$ MeV along the real (imaginary) axis with a
smearing width of $\omega_{I}(\omega_{R})=0.1$ MeV. An example of peak search
from the strength distribution is shown in Fig. 7 at $Q_{20}=60$ and 120 b in
256Fm. In Fig. 7(a), the strength distribution at $Q_{20}=120$ b along the
imaginary $\omega$ axis shows a clear peak with large magnitude, corresponding
to a pure imaginary QRPA pole. We found that either single peak with large
magnitude or no peak appears in the strength distribution along the imaginary
$\omega$ axis for the cases considered here. In Fig. 7(b), for the strength
distribution at $Q_{20}=60$ b along the real $\omega$ axis, one peak with
larger magnitude is found at $\omega_{R}\approx 1.2$ MeV. However, we found
that this peak is resolved as two peaks at $\omega_{R}=1.15$ and $1.25$ MeV by
the FAM calculation with a smaller smearing width of $\omega_{I}=0.01$ MeV,
which is illustrated in Fig. 7(c). To search sharp peaks in the strength
distribution along the real $\omega$ axis, we finally used $\omega_{I}=0.01$
MeV to determine approximate positions of the QRPA poles. Note that with
$\omega_{I}=0.01$ MeV we can not resolve multiple peaks within about 0.05 MeV
in frequency in the strength distribution.

We take a few peaks from the large peaks identified in the FAM strength
distribution along both the imaginary and real $\omega$ axes as candidates of
the most collective eigenmode. Then, we perform the contour integration (30)
to obtain the transition strength for each candidate, and finally adopt the
one with the largest value of $|p_{i}(\hat{Q}_{20})|^{2}$ as the most
collective eigenmode.

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