# Study of resonances in exotic drip-line nuclei by the use of super-symmetric
quantum mechanics

Md. A. Khan  M. Hasan  S. H. Mondal  M. Alam  T. Surungan  Department of
Physics, Aliah University,
IIA/27, Newtown, Kolkata-700160, India; (*Corresponding author; E-mail:
<EMAIL_ADDRESS> Department of Physics, Hasanuddin University,
Makassar-90245, Indonesia

(Day Month Year (Insert date of submission))

Hyperspherical three-body model calculation for the ground and ….Md. A. Khan
et al.

1, 1, 1, 1, 2

1 2

In this paper, a nice theoretical scheme is presented to investigate resonant
and bound states in weakly bound nuclear systems by the use of isospectral
potentials together with hyperspherical harmonics expansion. In this scheme, a
new potential is constructed which is strictly isospectral with the original
shallow potential and has properties that are desirable to calculate
resonances more accurately and elegantly. The effectiveness of the method has
been checked in terms of its application to the $0^{+}$ resonances in the
neutron-rich isotopes 18,20C in the two-neutron plus core three-body cluster
model. The results are in excellent agreement with the available data.

Halo nuclei, resonance, isospectral potential.

##  1 Introduction

The discovery of halo nuclei (neutron-halo and proton-halo) in the
neighborhood of the nuclear drip lines is one of the major achievements since
the development of the radioactive ion-beam facilities. This exotic kind of
nuclear structure is characterized by a dense stable core surrounded by a low-
density envelope of long extension. This low-density envelope is supposed to
be the consequence of quantum mechanical tunneling of the last nucleon(s)
through a-wide and skinny barrier following an attractive shallow potential
well. In halo nuclei, one seldom finds any excited bound state due to this
typical nature of the potential. These nuclei utmost support one bound state
at energies of the order of 1 MeV. One of the most exciting features of halo
nuclei is the fact that they exhibit one or more resonance state(s) (which has
high scientific significances) just above the binding threshold.

In the nuclear landscape, one may find some of the observed one-neutron halo
nuclei-17B, 19C; two-neutron halo nuclei- 6He, 11Li, 11,14Be; one-proton halo
nuclei- 8B, 26P, 20C; two-proton halo nuclei-17Ne, 27S and also four-neutron
halo nuclei-14Be, 19B, etc [1] [2] [3] [4] [5] [6] [7] [8]. Due to low-density
envelope surrounding the dense core, halo-nuclei have unusually large RMS
matter radii (larger than the liquid-drop model prediction of $R_{A}\propto
A^{1/3}$) as reported by Audi et al 2003 [9], Acharya et al 2013 [10]. And due
to weaker effective interaction among the valence nucleons and the core, these
nuclei exhibit sufficiently small two-nucleon separation energies (typically
less than 1 MeV). Till now, much scientific research has been conducted to
study the structure of this exciting nuclear species. Tanaka et al 2010 [6]
observed of a large reaction cross-section in the drip-line nucleus 22C,
Kobayashi et al 2012 [7] conducted research on one- and two-neutron removal
reactions using the most neutron-rich carbon isotopes, Gaudefroy et al 2012
[11] carried a direct mass measurements of 19B, 22C, 29F, 31Ne, 34Na and some
other light exotic nuclei. Togano et al, 2016 [12] studied interaction cross-
section of the two-neutron halo nucleus 22C.
In the literature review, it is noted that for the investigation of the
structure of the 2n-halo nuclei three main theoretical approaches are used- i)
the microscopic model approach where the valence neutrons are supposed to move
around the conglomerate of other nucleons (protons and neutrons) without
having any stable core, Sääf et al [13]; ii) the three-body cluster model in
which the valence nucleons are assumed to move around the structureless inert
core, and iii) the microscopic cluster model in which the valence nucleons
move around the deformed excited core as reported in [13] [14][15]. One may
also refer to the renormalized zero-range three-body model of Souza et al 2016
[16]. And for the computation of resonant states, several theoretical
approaches are employed. Some of those include the positive energy solution of
the Faddeev equation by Cobis et al 1997 [17], complex coordinate rotation
(CCR) by Csoto 1993 [18], Ayoma et al [19], the analytic computation of bound
state energies by Tanaka et al 1997 [20], the algebraic version of resonating
group method (RGM) by Vasilesky et al 2001 [21], continuum-discretized
coupled-channels (CDCC) method clubbed to the cluster-orbital shell model
(COSM) by Ogata et al 2013 [22], hyperspherical harmonics method (HHM) for
scattering states by Danilin et al 1997 [23], etc. In most of the theoretical
approaches, Jacobi coordinates are used to derive the relative coordinates
after the separation of the center of mass motion.
One of the most challenging obstacles that people often encounter in the
calculation of resonances in any weakly bound nucleus is the large degree of
computational error. In our case, we shall overcome this major obstacle by
adopting an elegant theoretical approach through interfacing the algebra of
supersymmetric quantum mechanics with the algebra involved in the
hyperspherical harmonics expansion method. In this scheme, one can handle the
ground state as well as the resonant states on the same footing. The technique
is based on the interesting fact that, for any arbitrarily given potential
(say, $U$), one can construct a family of isospectral potentials ($\hat{U}$),
in which the latter depends on an adjustable parameter ($\lambda$). And when
the original potential has a significantly low and excessively wide barrier
(poorly supporting the resonant state), $\lambda$ can be chosen judiciously to
enhance the depth of the well together with the height of the barrier in
$\hat{U}$. This enhanced well-barrier combination in $\hat{U}$ facilitates
trapping of the particle of interest and this, in turn, facilitates the
computation of resonant state more accurately at the same energy, as that in
the case of the original potential $U$. This is possible because, $U$ and
$\hat{U}$ have strictly isospectral nature.
To validate the effectiveness of the scheme we will apply the proposed scheme
to the first $0^{+}$ resonant states of the carbon isotopes AC, for mass
number A = 18 and 20 respectively. We chose the three-body (2n + A-2C) cluster
model for both of the above isotopes, assuming that the outer core valence
neutrons move around the inert core A-2C. The lowest eigen potential derived
for each of the three-body systems exhibits a shallow well following a skinny
and excessively wide barrier. This skinny-wide barrier results in a large
resonance width of the resonance. One can, in principle, find quasi-bound
states in such a typical shallow potential, but that poses a difficult
numerical task. For a finite height of the barrier, a particle can temporarily
be trapped in the shallow well when its energy is close to the resonance
energy. But, there is a finite possibility that the particle may creep in and
tunnel out through the barrier. Thus, a more accurate calculation of resonance
energy is easily masked by the large width of resonance resulting from a large
tunneling probability due to a low barrier height. Hence, a straightforward
calculation of the resonance energies of such systems fails to yield accurate
results.

For the ground state energy and wavefunctions, we adopt the hyperspherical
harmonics expansion method (HHEM) described by Fabre et al 1982 [24] to solve
the three-body Schrödinger equation in relative coordinates. In HHEM, three-
body relative wavefunction is expanded in a complete set of hyperspherical
harmonics. The substitution of the wavefunction in the Schrödinger equation
and use of orthonormality of HH gives rise to an infinite set of coupled
differential equations (CDE). The method is an essentially exact one,
involving no other approximation except an eventual truncation of the
expansion basis subject to the desired precision in the energy and the
configuration of the available computer. However, the hyperspherical
convergence theorem prescribed by Schneider 1972 [25] permits extrapolation of
the data computed for the finite size of the expansion basis, to estimate
those for an even larger expansion basis. The fact that convergence in HH
expansion is significantly slow and one needs a large number of CDE’s to solve
for the achievement of desired precision pauses practical a limitation to the
method. But we used the hyperspherical adiabatic approximation (HAA) of Ballot
et al 1982 [26] to construct a single differential equation (SDE) to be solved
for the lowest eigen potential, ${U}_{0}(\rho)$) to get the ground state
energy $E_{0}$ and the corresponding wavefunction $\psi_{0}(\rho)$ following
method of Das et al 1982 [27].

We next derive the isospectral potential $\hat{U}(\lambda,\rho)$ using the
ground state energy ad wavefunctions using algebra of the SSQM reported by
Cooper et al 1995 [28], Khare et al 1989 [29], and Nieto 1984 [30]. Finally,
we solve the SDE for $\hat{U}(\lambda,\rho)$ for some positive energies for
wavefunctions $\hat{\psi}(\lambda,\rho)$. We then compute the probability of
trapping of the particle(s) in terms of the norm of the positive energy
wavefunctions within the enhanced well-barrier region. A plot of the trapping
probability as a function of energy shows a sharp peak at the resonance
energy. The actual width of resonance can be obtained by back-transforming the
wave function $\hat{\psi}(\lambda,\rho)$ corresponding to
$\hat{U}(\lambda,\rho)$ to $\psi(\rho)$ of $U(\rho)$. Hence, the major goals
of this work are three-fold: i) to reproduce the ground state spectra of the
known systems, ii) to construct the isospectral potential using ground state
energy and wavefunction, and iii) to obtain resonance state as proposed to
check the validity of the scheme.

The paper is organized as follows. In section 2, we briefly review the HHE
method and the algebra of SSQM to construct the one-parameter family of
isospectral potential $\hat{U}(\lambda,\rho)$. The results of our calculation
are presented in section 3 while conclusions are drawn in section 4.

##  2 Theoretical Scheme

###  2.1 Hyperspherical Harmonics Expansion Method

In the three-body cluster model of the nuclei AC $\equiv^{A-2}$C + 2n, the
relatively heavy core A-2C is labeled as particle 1, and two valence neutrons
are labelled as particle 2 and 3 respectively. Thus, there are three possibile
partitions for the choice of Jacobi coordinates. In any chosen partition, say
in the $i^{th}$ partition, particle labelled $i$ plays the role of spectator
while the remaining two paricles form the interacting pair. In this partition
the Jacobi coordinates are defined as:

$\displaystyle\left.\begin{array}[]{lcl}\vec{x_{i}}&=&a_{i}(\vec{r_{j}}-\vec{r_{k}})\\\
\vec{y_{i}}&=&\frac{1}{a_{i}}\left(\vec{r_{i}}-\frac{m_{j}\vec{r_{j}}+m_{k}\vec{r_{k}}}{m_{j}+m_{k}}\right)\\\
\vec{R}&=&\frac{\sum_{i=1}^{3}m_{i}\vec{r_{i}}}{M}\\\ \end{array}\right]$ (4)

where $i,j,k$ form a cyclic permutation of 1,2,3. The parameter
$a_{i}=\left[\frac{m_{j}m_{k}M}{m_{i}(m_{j}+m_{k})^{2}}\right]^{\frac{1}{4}}$;
$m_{i},\vec{r_{i}}$ are the mass and position of the $i^{th}$ particle and
$M(=\sum_{i=1}^{3}m_{i})$, $\vec{R}$ are the total mass and position of the
centre of mass (CM) of the system. Then in terms of Jacobi coordinates, the
relative motion of the three-body system can be described by the equation

$\displaystyle\left.\begin{array}[]{l}[-\frac{\hbar^{2}}{2\mu}\left(\nabla_{x_{i}}^{2}+\nabla_{y_{i}}^{2}\right)+V_{jk}(\vec{x_{i}})+V_{ki}(\vec{x_{i}},\vec{y_{i}})+\\\
\left.V_{ij}(\vec{x_{i}},\vec{y_{i}})-E\right]\Psi(\vec{x_{i}},\vec{y_{i}})=0\end{array}\right]$
(7)

where ${\mu=\left[\frac{m_{i}m_{j}m_{k}}{M}\right]^{\frac{1}{2}}}\rightarrow$
is the reduced mass of the system, $V_{ij}$ represents the interaction
potential between the particles labelled $i$ and $j$ respectively.
Hyperspherical variables are introduced through Jacobi coordinates as:
$x_{i}=\rho\cos\phi_{i}$; $y_{i}=\rho\sin\phi_{i}$;
$\phi_{i}=\tan^{-1}(\frac{y_{i}}{x_{i}})$; $\rho=\sqrt{x_{i}^{2}+y_{i}^{2}}$.
Here, the hyperradius $\rho$ together with five angular variables
$\Omega_{i}\rightarrow\\{\phi_{i},\theta_{x_{i}},{\cal\phi}_{x_{i}},\theta_{y_{i}},{\cal\phi}_{y_{i}}\\}$
constitute hyperspherical coordinates of the system. The Schrödinger equation
in hyperspherical variables $(\rho,\Omega_{i})$ becomes

$\displaystyle\left.\begin{array}[]{l}\left\\{-\frac{\hbar^{2}}{2\mu}\left(\frac{1}{\rho^{5}}\frac{\partial^{2}}{\partial\rho^{2}}+\frac{4}{\rho}\frac{\partial}{\partial\rho}-\frac{\hat{{\cal
K}}^{2}(\Omega_{i})}{\rho^{2}}\right)+V(\rho,\Omega_{i})-E\right\\}\\\
\Psi(\rho,\Omega_{i})=0\end{array}\right]$ (10)

In Eq. (10) $V(\rho,\Omega_{i})=V_{jk}+V_{ki}+V_{ij}$ is the total interaction
potential in the $i^{th}$ partition and $\sl{\hat{{\cal K}}^{2}}(\Omega_{i})$
is the square of the hyperangular momentum operator satisfying the eigenvalue
equation

$\displaystyle\hat{{\cal K}}^{2}(\Omega_{i}){\cal
Y}_{K\alpha_{i}}(\Omega_{i})=K(K+4){\cal Y}_{K\alpha_{i}}(\Omega_{i})$ (11)

$K$ is the hyperangular momentum quantum number and $\alpha_{i}$
$\equiv\\{l_{x_{i}},l_{y_{i}},L,M\\}$, ${\cal Y}_{K\alpha_{i}}(\Omega_{i})$
are the hyperspherical harmonics (HH) for which a closed analytic expressions
can be found in ref. [17].
In the HHEM, $\Psi(\rho,\Omega_{i})$ is expanded in the complete set of HH
corresponding to the partition ”$i$” as

$\displaystyle\Psi(\rho,\Omega_{i})=\sum_{K\alpha_{i}}\frac{\psi_{K\alpha_{i}}(\rho)}{\rho^{5/2}}{\cal
Y}_{K\alpha_{i}}(\Omega_{i})$ (12)

Use of Eq. (12), in Eq. (10) and application of the orthonormality of HH leads
to a set of coupled differential equations (CDE) in $\rho$

$\displaystyle\left.\begin{array}[]{l}\left\\{-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{d\rho^{2}}+\frac{\hbar^{2}}{2\mu}\frac{(K+3/2)(K+5/2)}{\rho^{2}}-E\right\\}\psi_{K\alpha_{i}}(\rho)\\\
+\sum_{K^{\prime}\alpha_{i}^{\prime}}{\cal
M}_{K\alpha_{i}}^{K^{\prime}\alpha_{i}^{\prime}}\psi_{K^{\prime}\alpha_{i}^{\prime}}(\rho)=0.\end{array}\right]$
(15)

where

$\displaystyle{\cal M}_{K\alpha_{i}}^{K^{\prime}\alpha_{i}^{\prime}}=\int{\cal
Y}_{K\alpha_{i}}^{*}(\Omega_{i})V(\rho,\Omega_{i}){\cal
Y}_{K^{\prime}\alpha_{i}^{~{}\prime}}(\Omega_{i})d\Omega_{i}.$ (16)

The infinite set of CDE’s represented by Eq. (15) is truncated to a finite set
by retaining all K values up to a maximum of $K_{max}$ in the expansion (12).
For a given $K$, all allowed values of $\alpha_{i}$ are included. The size of
the basis states is further restricted by symmetry requirements and associated
conserved quantum numbers. The reduced set of CDE’s are then solved by
adopting hyperspherical adiabatic approximation (HAA) [26]. In HAA, the CDE’s
are approximated by a single differential equation assuming that the hyper
radial motion is much slower compared to hyperangular motion. For this reason,
the angular part is first solved for a fixed value of $\rho$. This involves
diagonalization of the potential matrix (including the hyper centrifugal
repulsion term) for each $\rho$-mesh point and choosing the lowest eigenvalue
$U_{0}(\rho)$ as the lowest eigen potential [27]. Then the energy of the
system is obtained by solving the hyper radial motion for the chosen lowest
eigen potential ($U_{0}(\rho)$), which is the effective potential for the
hyper radial motion

$\displaystyle\left\\{-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{d\rho^{2}}+U_{0}(\rho)-E\right\\}\psi_{0}(\rho)=0$
(17)

Renormalized Numerov algorithm briefed by Johnson 1978 [31] subject to
appropriate boundary conditions in the limit $\rho\rightarrow 0$ and
$\rho\rightarrow\infty$ is then used to solve Eq. (17) for E ($\leq E_{0}<0$).
The hyper-partial waves $\psi_{K\alpha_{i}}(\rho)$ are given by

$\displaystyle\psi_{K\alpha_{i}}(\rho)=\psi_{0}(\rho)\chi_{K\alpha_{i},0}(\rho)$
(18)

where $\chi_{K\alpha_{i},0}(\rho)$ is the ${(K\alpha_{i})}^{th}$ element of
the eigenvector, corresponding to the lowest eigen potential $U_{0}(\rho)$.

###  2.2 Construction of Isospectral Potential

In this section we present a bird’s eye view of the scheme of the construction
of one parameter family of isospectral potentials. We have from Eq. (17)

$\displaystyle
U_{0}(\rho)=E_{0}+\frac{\hbar^{2}}{2\mu}\frac{\psi_{0}^{\prime\prime}(\rho)}{\psi_{0}(\rho)}$
(19)

In 1-D supersymmetric quantum mechanics, one defines a superpotential for a
system in terms of its ground state wave function ($\psi_{0}$) [28] as

$\displaystyle{\cal
S}(\rho)=-\frac{\hbar}{\sqrt{2m}}\frac{\psi_{0}^{\prime}(\rho)}{\psi_{0}(\rho)}.$
(20)

The energy scale is next shifted by the ground state energy $(E_{0})$ of the
potential $U_{0}(\rho)$, so that in this shifted energy scale the new
potential become

$\displaystyle
U_{1}(\rho)=U_{0}(\rho)-E_{0}=\frac{\hbar^{2}}{2\mu}\frac{\psi_{0}^{\prime\prime}(\rho)}{\psi_{0}(\rho)}$
(21)

having its ground state at zero energy. One can then easily verify that
$U_{1}(\rho)$ is expressible in terms of the superpotential via the Riccati
equation

$\displaystyle U_{1}(\rho)={\cal S}^{2}(\rho)-\frac{\hbar}{\sqrt{2m}}{\cal
S}^{\prime}(\rho).$ (22)

By introducing the operator pairs

$\displaystyle\left.\begin{array}[]{lcl}A^{{\dagger}}&=&-\frac{\hbar}{\sqrt{2m}}\frac{d}{d\rho}+{\cal
S}(\rho)\\\ A&=&\frac{\hbar}{\sqrt{2m}}\frac{d}{d\rho}+{\cal
S}(\rho)\end{array}\right\\}$ (25)

the Hamiltonian for $U_{1}$ becomes

$\displaystyle
H_{1}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{d\rho^{2}}+U_{1}(\rho)={\cal
A}^{{\dagger}}{\cal A}.$ (26)

The pair of operators ${\cal A}^{{\dagger}},{\cal A}$ serve the purpose of
creation and annihilation of nodes in the wave function. Next, we introduce a
partner Hamiltonian $H_{2}$, corresponding to the SUSY partner potential
$U_{2}$ of $U_{1}$ as

$\displaystyle
H_{2}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{d\rho^{2}}+U_{2}(\rho)={\cal A}{\cal
A}^{{\dagger}}$ (27)

where

$\displaystyle U_{2}(\rho)={\cal
S}^{2}(\rho)+\frac{\hbar}{\sqrt{2m}}W^{\prime}(\rho).$ (28)

Energy eigen values and wavefunctons corresponding to the SUSY partner
Hamiltonians $H_{1}$ and $H_{2}$ are connected via the relations

$\displaystyle\left.\begin{array}[]{lcl}E_{n}^{(2)}&=&E_{n+1}^{(1)},E_{0}^{(1)}=0\;(n=0,1,2,3,...),\\\
\psi_{n}^{(2)}&=&\frac{1}{\sqrt{E_{n+1}^{(1)}}}A\psi_{n+1}^{(1)}\\\
\psi_{n+1}^{(1)}&=&\frac{1}{\sqrt{E_{n}^{(2)}}}A^{\dagger}\psi_{n}^{(2)}\\\
\end{array}\right\\}\ $ (32)

where $E_{n}^{(i)}$ represents the energy of the $n^{th}$ excited state of
$H_{i}$ (i=1, 2). Thus $H_{1}$ and $H_{2}$ have identical spectra, except the
fact that the partner state of $H_{2}$ corresponding to the ground state of
$H_{1}$ is absent in the spectrum of $H_{2}$ [28], i.e., spectra of $H_{2}$
has one state less than that of $H_{1}$. Hence the potentials $U_{1}$ and
$U_{2}$ are not strictly isospectral.
However, one can construct, a one parameter family of strictly isospectral
potentials $\hat{U_{1}}(\lambda,\rho)$, explointing the fact that for a given
$U_{1}(\rho)$, $U_{0}(\rho)$ and ${\cal S}(\rho)$ are not unique (see Eqs.
(12) & (13)), since the Riccati equation is a nonlinear one. Following [28]
[29] [32], it can be shown that the most general superpotential satisfying
Riccati equation for $U_{1}(\rho)$ Eq.(22) is given by

$\displaystyle\hat{S}(\rho)=W(\rho)+\frac{\hbar}{\sqrt{2m}}\frac{d}{d\rho}\log[I_{0}(\rho)+\lambda]$
(33)

where $\lambda$ is a constant of integration, and $I_{0}$ is given by

$\displaystyle
I_{0}(\rho)={\displaystyle\int}_{\rho^{\prime}=0}^{\rho}{[\psi_{0}(\rho^{\prime})]}^{2}d\rho^{\prime},$
(34)

in which $\psi_{0}(\rho)$ is the normalized ground state wave function of
$U_{0}(\rho)$. The potential

$\displaystyle\left.\begin{array}[]{lcl}\hat{U_{1}}(\lambda,\rho)&=&\hat{S}^{2}(\rho)-\frac{\hbar}{\sqrt{2m}}\hat{{\cal
S}}^{\prime}(\rho)\\\
&=&U_{1}(\rho)-2\frac{\hbar^{2}}{2m}\frac{d^{2}}{d\rho^{2}}\log[I_{0}(\rho)+\lambda],\end{array}\right\\}\
$ (37)

has the same SUSY partner $U_{2}(\rho)$. $\hat{U_{1}}(\lambda,\rho)$ has its
ground state at zero energy with the corresponding wavefunction given by

$\displaystyle\hat{\psi_{1}}(\lambda,\rho)=\frac{\psi_{1}}{I_{0}+\lambda}.$
(38)

Hence, potentials $\hat{U_{1}}(\lambda,\rho)$ and $U_{1}(\rho)$ are strictly
isospectral. The parameter $\lambda$ is arbitrary in the intervals
$-\infty<\lambda<-1$ and $0<\lambda<\infty$. $I_{0}(\rho)$ lies between 0 and
1, so the interval $-1\leq\lambda\leq 0$ is forbidden, in order to bypass
singularities in $\hat{U_{1}}(\lambda,\rho)$. For
$\lambda\rightarrow\pm\infty$, $\hat{U_{1}}\rightarrow U_{1}$ and for
$\lambda\rightarrow 0+$, $\hat{U_{1}}$ develops a narrow and deep attractive
well near the origin. This well-barrier combination effectively traps the
particle giving rise to a sharp resonance. For fruitful representation of the
resonaces we adopt bound state in continuum (BIC) technique in which we solve
Eq. (17) numerically for a positive energy E subject to the boundary condition
$\psi_{E}(0)=0$ to get the wave function $\psi_{E}(r)$. This $\psi_{E}(r)$ is
not square integrable, and it oscillates as r increases. Following Pappademos
et al 1993 [33] one can verify by direct substitution that

$\displaystyle\hat{\psi}_{E}(r,\lambda)=\frac{\psi_{E}(r)}{I(r)+\lambda}$ (39)

where

$\displaystyle I(r)=\int_{0}^{r}|\psi_{E}(r^{\prime})|^{2}dr^{\prime}$ (40)

satisfies Eq. (17) with $U_{0}(r)$ replaced by

$\displaystyle\hat{U}(r,\lambda)=U_{0}(r)-\frac{\hbar^{2}}{2\mu}\left[\frac{4\psi_{E}(r)\psi_{E}(r)^{\prime}}{I(r)+\lambda}-\frac{2\psi_{E}(r)^{4}}{(I(r)+\lambda)^{2}}\right]$
(41)

This method has been tested successfully for 3D finite square-well potential
by Das et al 2001 [34] for an arbitrary particle of mass m, for Gaussian type
potential to the core plus two-neucleon three-body cluster model in the case
of- 6He by Dutta et al 2003 [35], 6Li by Dutta et al 2004 [36], 22C by Hasan
et al 2019 [37], for Wood-saxon potential in two-body model calculation of 15C
by Mahapatra et al 2011 [38], for density dependent M3Y potential in the case
of- 11Be by Dutta et al 2014 [39], 15Be by Dutta et al 2018 [40].

In an attempt to search for the correct resonance energy, we compute the
probability (P) of finding the system within the potential well region of the
potential $\hat{U_{1}}(\lambda,\rho)$ corresponding to the energy $E$ ($>0$)
by integrating the norm of the wavefunctions up to the top of the barrier:

$\displaystyle
P(E)=\int_{\rho^{\prime}=0}^{\rho_{B}}|\hat{\psi_{E}}(\rho^{\prime},\lambda)|^{2}d\rho^{\prime}$
(42)

in which $\rho_{B}$ indicates position of the top of the barrier in
$\hat{U_{1}}(\lambda,\rho)$ for a chosen $\lambda$. Here
$\hat{\psi_{E}}(\lambda,\rho)$ representing solution for
$\hat{U_{1}}(\lambda,\rho)$ for a positive energy $E$, is normalized to have a
constant amplitude in the asymptotic region. Plot of the quantity $P(E)$ as a
function of $E$ ($E>0$) shows a peak at the resonance energy $E=E_{R}$. Choice
of $\lambda$ has to be made judiciously to avoid numerical errors in handling
extremely narrow well for $\lambda\rightarrow 0+$. The width of resonance can
be obtained from the mean life of the state using the energy-time uncertainty
relation. The mean life is reciprocal to the decay constant. And the decay
constant is the product of the number of hits per unit time on the barrier and
the corresponding probability of tunneling out through the barrier.

##  3 Results and discussions

We apply the above scheme to the 18,20C nuclei which have in their ground
state spin-parity $J^{\pi}=0^{+}$ and triplet isospin $T=1$. These nuclei also
have a resonance state of the same $J^{\pi}$. Thus, the foregoing procedure
starting from the ground state of 18,20C will give $J^{\pi}=0^{+},T=1$
resonance(s). Eq.(17) is solved for the GPT n-n potential [41] and core-n SBB
potential [42]. The range parameter for the core-n potential $b_{cn}$ is
slightly adjusted to match the observed ground state spectra. The calculated
two-neutron separation energies ($S_{2n}$), their relative convergence
(=$\frac{E(K_{max}+4)-E(K_{max})}{E(K_{max}+4)}$) and the RMS matter radii
(RA) for gradually increasing $K_{max}$ are listed in Table 1 for 18C and 20C.
Although the computed results indicate a clear convergence trend with
increasing $K_{max}$, it is far away from full convergence even at
$K_{max}=24$. For this reason, we used an extrapolation technique successfully
in atomic problems by Das et al 1994 [43], Khan 2012 [44] and in nuclear few-
body problems by Khan and Das 2001 [45], to get the converged value of about
4.91 MeV for 18C and 3.51 MeV for 20C as shown in columns 2 and 4 of Table 2.
Partial contributions of the different partial waves corresponding to
$l_{x}=0,1,2,3,4$ to the two-neutron separation energies ($S_{2n}$) are
presented in Table 3 for both of the systems. The computed data indicates a
major contribution comes from $l_{x}=0$ component ($\sim 74\%$ in 18C and
$\sim 96\%$ in 20C). The dependence of the two neutron separation energy on
$K_{max}$ is depicted in Figure 1 for both of the nuclei 18C and 20C which
indicates a saturation trend for increasing $K_{max}$. In Figure 2 we have
shown the relative convergence trend in energies as a function of $K_{max}$
and it is seen that both follow a similar trend.

Figure 1:

Plot of two-neutron separation energy (S2n) as a function of Kmax for 18C
(upper curve) and 20C (lower curve).

Figure 2:

Plot of relative convergance in ground state energy with respect to increasing
Kmax for 18C (lower curve) and 20C (upper curve).

In Figures 3 and 4, we have presented a 3D view of the correlation density
profile of the halo nuclei 18C and 20C which indicates a halo structure with a
dense core surrounded by an extended low-density tail like an envelope.

Figure 3:

Correlation density plot for the (0+) ground state of 18C as function of the
Jacobi coordiantes $x_{nn}$ and $y_{(nn)c}$.

Figure 4:

Correlation density plot for the (0+) ground state of 20C as function of the
Jacobi coordiantes $x_{nn}$ and $y_{(nn)c}$.

Similar nature is also reflected in Figures 5 and 6 where we have shown the 2D
projection of the 3D probability density distribution.

Figure 5:

2D projection of the correlation density plot for the 0+ state of 18C as a
function of the Jacobi coordiantes $x_{nn}$ and $y_{(nn)c}$.

Figure 6:

2D projection of the correlation density plot for the 0+ state of 20C as a
function of the Jacobi coordiantes $x_{nn}$ and $y_{(nn)c}$.

After getting the ground state energy and wavefunctions we constructed the
isospectral potential invoking principles of SSQM to investigate the resonant
states. The lowest eigen potential obtained for the ground states of 18C and
20C as shown in Figures 7 and 8 by thick violet lines, exhibit shallow well
followed by a wide-skinny barrier.

Figure 7:

Plot of the one-parameter family of isospectral potential constructed for
$0^{+}$ resonance in 18C for some representative values of the parameter
$\lambda$ ($=\infty$(original shallow potential), $10^{-4},10^{-6}$ and
$10^{-8}$.

Figure 8:

Plot of the one-parameter family of isospectral potential constructed for
$0^{+}$ resonance in 20C for some representative values of the parameter
$\lambda$ ($=\infty$(original shallow potential), $10^{-4},10^{-6}$ and
$10^{-8}$.

This kind of shallow-well and skinny barrier combination supports weak bound
states with a large spatial extension of the wavefunction shown by thick black
lines in Figures 9 & 10.

Figure 9:

Plot of the wavefunctions computed at energy of $0^{+}$ resonance in
${}^{18}C$ for some representative values of the parameter $\lambda$
($=\infty$(correspond to original shallow potential), $10^{-4},10^{-6}$ and
$10^{-8}$.

Figure 10:

Plot of the wavefunctions computed at energy of $0^{+}$ resonance in
${}^{20}C$ for some representative values of the parameter $\lambda$
($=\infty$(correspond to original shallow potential), $10^{-4},10^{-6}$ and
$10^{-8}$.

Such potentials may also support broad resonances for the far extended
wavefunctions. Accurate computation of resonance energy and width of such
kinds of broad resonances is a challenging numerical task due to large
computational errors. Hence, we constructed the one-parameter family of
isospectral potentials $\hat{U}(\lambda,\rho)$ following Eq.(37) by
appropriate selection of $\lambda$ values to ensure narrow and sufficiently
deep potential well following a high barrier. The constructed potentials are
shown in Figures 7 and 8 for some representative values of $\lambda$ for both
of the systems considered here. The enhanced well-barrier combination
effectively traps the particles to form a strong resonant state. With gradual
decrease in the $\lambda$ values from $\infty$ towards 0+ enhances both the
depth of the well and height of the barrier. It is also observed that the
overall width of the well-barrier combination decreases with a decrease in the
value of $\lambda$ together with the shifting of the positions of minima of
the well and maxima of the barrier towards the origin. Calculated parameters
of the isospectral potential for some representative $\lambda$ values in
[$+\infty,0$] are presented in Table 4, in which $\lambda=\infty$ corresponds
to the original effective potential $U_{0}(\rho)$ of Eq. (17) as can be
verified from Eqs. (37) with a reference to Eq. (21). One can, for example,
note from columns under 18C in Table 4 that, when $\lambda$ decreases from 0.1
to 0.0001, the depth of the potential well increases from -24.2 MeV at 2.7 fm
to -247.6 MeV at 1.3 fm while the height of the barrier increases from 5.4 MeV
at 5.1 fm to 121.5 MeV at 1.9 fm respectively. The same trend can also be
noted for 20C. Thus the application of SSQM produces a dramatic effect in the
isospectral potential $\hat{U_{1}}(\lambda,\rho)$ as $\lambda$ approaches 0+.
Still smaller positive values of $\lambda$ are not desirable since that will
make the potential well too narrow to compute the wave functions accurately by
a standard numerical technique. The probability of trapping (P(E)) of the
particle within the enhanced well-barrier combination as a function of energy
E displayed as Figures 11 and 12 exhibit resonance peak at the energy
$E_{R}\simeq 1.98$ MeV in 18C and energy $E_{R}\simeq 3.75$ MeV in 20C
respectively.

Figure 11:

Plot of trapping probability $P(E)$ as a function of energy demostrating the
$0^{+}$ resonant state in ${}^{18}C$.

Figure 12:

Plot of trapping probability $P(E)$ as a function of energy demostrating the
$0^{+}$ resonant state in ${}^{20}C$.

It is interesting to note that the resonance energy is independent of the
$\lambda$ parameter. The enhancement of accuracy in the determination of
$E_{R}$ is the principal advantage of using Supersymmetric formalism. Since
$\hat{U_{1}}(\lambda;\rho)$ is strictly isospectral with $U(\rho)$, any value
of $\lambda$ is admissible in principle. However, a judicious choice of
$\lambda$ is necessary for the accurate determination of the resonance energy
and to avoid delta-function type behavior of the isospectral potential. The
calculated two-neutron separation energies as presented in Table 5 are in
excellent agreement with the observed values of $4.910\pm 0.030$ MeV for 18C
and $3.510\pm 0.240$ MeV for 20C [46] and also with results of Yamaguchi et al
[47]. The calculated RMS matter radii also agree fairly with the
experimentally observed values found in [48]. A similar type of calculation
adopting core-neutron two-body model for 15Be has recently been reported by
Dutta et al 2018 [40].

Table 1: Calculated two-neutron separation energies, their relative convergence and RMS matter radius of 18C and 20C for different $K_{max}$ in their ground states. $System$ |  18C (16C+n+n) |  20C (18C+n+n)
---|---|---
$K_{max}$ |  $S_{2n}$(MeV) | Relative Conv. | $R_{A}$ |  $S_{2n}$(MeV) | Relative Conv. | $R_{A}$
4 | 3.97843 | 0.10341 | 2.8297 | 2.55712 | 0.15771 | 2.9832
8 | 4.43727 | 0.03861 | 2.7913 | 3.03591 | 0.05480 | 2.9431
12 | 4.61546 | 0.02521 | 2.7647 | 3.21193 | 0.03493 | 2.9145
16 | 4.73481 | 0.01344 | 2.7425 | 3.32818 | 0.01944 | 2.8867
20 | 4.79933 | 0.00638 | 2.7233 | 3.39416 | 0.00984 | 2.8628
24 | 4.83013 |  | 2.7156 | 3.42789 |  | 2.8479
Table 2: Extrapolated values of two-neutron separation energies $(S_{2n}$ and their relative convergences for the ground state (Jπ=0+) of 18C and 20C. $System$ |  18C (16C+n+n) |  20C (18C+n+n)
---|---|---
$K_{max}$ | $S_{2n}(MeV)$ | Relative Convergence | $S_{2n}(MeV)$ | Relative Convergence
24 | 4.83013120 | 0.00456309 | 3.42789570 | 0.00669549
28 | 4.85227259 | 0.00303493 | 3.45100186 | 0.00446219
32 | 4.86704371 | 0.00209706 | 3.46646992 | 0.00308899
36 | 4.87727164 | 0.00149547 | 3.47721099 | 0.00220653
40 | 4.88457636 | 0.00109515 | 3.48490054 | 0.00161829
44 | 4.88993151 | 0.00082034 | 3.49054929 | 0.00121388
48 | 4.89394622 | 0.00062664 | 3.49479156 | 0.00092838
52 | 4.89701487 | 0.00048691 | 3.49803908 | 0.00072217
56 | 4.89940043 | 0.00038406 | 3.50056708 | 0.00057019
60 | 4.90128282 | 0.00030700 | 3.50256423 | 0.00045621
64 | 4.90278798 | 0.00024834 | 3.50416285 | 0.00036934
68 | 4.90400585 | 0.00020305 | 3.50545757 | 0.00030221
72 | 4.90500182 | 0.00016763 | 3.50651729 | 0.00024967
76 | 4.90582420 | 0.00013961 | 3.50739299 | 0.00020807
80 | 4.90650922 | 0.00011721 | 3.50812293 | 0.00017479
84 | 4.90708439 | 0.00009913 | 3.50873623 | 0.00014791
88 | 4.90757089 | 0.00008441 | 3.50925527 | 0.00012599
92 | 4.90798515 | 0.00007231 | 3.50969749 | 0.00010800
96 | 4.90834009 | 0.00006232 | 3.51007659 | 0.00009311
100 | 4.90864598 | 0.00005399 | 3.51040343 | 0.00008069
104 | 4.90891099 | 0.00004693 | 3.51068673 | 0.00007028
108 | 4.90914137 | 0.00004119 | 3.51093349 | 0.00006149
112 | 4.90934358 | 0.00003612 | 3.51114943 | 0.00005405
116 | 4.90952089 | 0.00003186 | 3.51133920 | 0.00004769
120 | 4.90967732 | 0.00002821 | 3.51150666 | 0.00004224
124 | 4.90981582 | 0.00002507 | 3.51165499 | 0.00003755
128 | 4.90993891 | 0.00002235 | 3.51178684 | 0.00003349
132 | 4.91004866 | 0.00001999 | 3.51190444 | 0.00002996
136 | 4.91014683 | 0.00001794 | 3.51200967 | 0.00002689
140 | 4.91023492 | 0.00001614 | 3.51210410 | 0.00002419
144 | 4.91031418 | 0.00001456 | 3.51218909 | 0.00002184
148 | 4.91038569 | 0.00001318 | 3.51226579 | 0.00001976
152 | 4.91045039 | 0.00001195 | 3.51233519 | 0.00001792
156 | 4.91050905 | 0.00001086 | 3.51239814 | 0.00001629
160 | 4.91056237 | 0.00000989 | 3.51245537 | 0.00001484
164 | 4.91061094 | 0.00000903 | 3.51250750 | 0.00001355
168 | 4.91065528 | 0.00000826 | 3.51255511 | 0.00001239
172 | 4.91069584 | 0.00000757 | 3.51259866 | 0.00001137
176 | 4.91073301 | 0.00000695 | 3.51263858 | 0.00001044
180 | 4.91076715 | 0.00000639 | 3.51267525 | 0.00000960
184 | 4.91079855 | 0.00000589 | 3.51270898 | 0.00000885
…….. | …….. | …….. | …….. | ……..
$\infty$ | 4.91124921 |  | 3.51319416 |
Table 3: Partial contribution of different $l_{x}$ partial waves to the two-neutron separation energies in the ground state of 18C and 20C for graduallt increasing $K_{max}$. $System$ |  18C |  20C
---|---|---
$K_{max}$ |  $E_{lx}$for $l_{x}$= |  $E_{l_{x}}$ for $l_{x}$=
| 0 | 1 | 2 | 3 | 4 | 0 | 1 | 2 | 3 | 4
4 | 2.886 | 0.068 | 1.149 | 0.000 | 0.000 | 2.471 | 0.235 | 0.036 | 0.000 | 0.000
8 | 3.283 | 0.072 | 1.158 | 0.001 | 0.100 | 2.902 | 0.232 | 0.019 | 1.034 | 0.001
12 | 3.414 | 0.078 | 1.158 | 0.001 | 0.098 | 3.073 | 0.231 | 0.014 | 1.036 | 0.001
16 | 3.487 | 0.079 | 1.166 | 0.001 | 0.098 | 3.195 | 0.228 | 0.014 | 1.040 | 0.001
20 | 3.536 | 0.081 | 1.181 | 0.001 | 0.096 | 3.262 | 0.225 | 0.013 | 1.049 | 0.001
24 | 3.566 | 0.082 | 1.199 | 0.001 | 0.094 | 3.295 | 0.225 | 0.013 | 1.080 | 0.001
Table 4: Data to show the effect of the parameter $\lambda$ on the depth ($V_{0}$) of the well and height ($V_{B}$) of the barrier in the isospectral potential constructed using Eq. (37) for 18C and 20C. $System$ |  18C |  20C
---|---|---
$\lambda$ | Potential Well | Potential Barrier | Potential Well | Potential Barrier
| $V_{0}$ | $r_{0}$ | $V_{B}$ | $r_{B}$ | $V_{0}$ | $r_{0}$ | $V_{B}$ | $r_{B}$
| $(MeV)$ | $(fm)$ | $(MeV)$ | $(fm)$ | $(MeV)$ | $(fm)$ | $(MeV)$ | $(fm)$
$10^{5}$ | -9.301 | 3.092 | 2.702 | 20.099 | -11.076 | 3.069 | 3.085 | 7.520
$10^{2}$ | -9.311 | 3.092 | 2.712 | 20.099 | -11.086 | 3.069 | 3.094 | 7.514
50 | -9.337 | 3.091 | 2.712 | 20.099 | -11.163 | 3.065 | 3.106 | 7.504
1 | -11.590 | 3.019 | 2.719 | 20.095 | -17.294 | 2.824 | 3.938 | 6.708
0.1 | -24.220 | 2.664 | 5.394 | 5.088 | -41.931 | 2.320 | 10.924 | 3.983
$10^{-2}$ | -62.846 | 2.109 | 19.797 | 3.424 | -95.444 | 1.811 | 36.751 | 2.840
$10^{-3}$ | -136.144 | 1.637 | 56.596 | 2.521 | -154.453 | 1.389 | 84.198 | 2.173
$10^{-4}$ | -247.602 | 1.274 | 121.482 | 1.937 | -270.962 | 1.000 | 141.518 | 1.731
$10^{-5}$ | -384.056 | 0.986 | 218.025 | 1.520 | -479.005 | 0.724 | 241.024 | 1.218
Table 5: Comparison of calculated data with those found in the literature for the two-neutron halo nuclei 18C and 20C. Nuclei |  State ($J^{\pi}$) | Observables | Present work | Others work
---|---|---|---|---
18C | $0^{+}$ |  $S_{2n}$ (MeV) | 4.9064 |  4.910 $\pm$0.030 [46]
|  |  |  |  4.91[47]
|  |  $R_{A}$ (fm) | 2.7156 |  2.82$\pm 0.04$ [48]
|  0${}^{+}_{1}$ |  $E_{R}$ (MeV) | 1.89 | -
20C | $0^{+}$ |  $S_{2n}$ (MeV) | 3.5065 |  3.510 $\pm$0.240 [46]
|  |  |  |  3.51 [47]
|  |  $R_{A}$ (fm) | 2.8479 |  2.98$\pm 0.05$ [48]
|  0${}^{+}_{1}$ |  $E_{R}$ (MeV) | 3.735 | -

##  4 Summary and conclusions

In this communication, we have investigated the ground state structure of
18,20C using hyperspherical harmonics expansion method assuming 16,18C$+n+n$
three-body cluster model. Standard GPT [41] potential is chosen for the $n-n$
pair while a three-term Gaussian SBB potential [42] with spin-orbit coupling
term is used for the core-nucleon subsystems. The ground state energy and
wavefunctions are obtained by numerical solution of the three-body Schrödiger
equation using renormalized Numerov method algorithm [31]. The three-body
effective potential $U_{0}(\rho)$ in Eq. (17) exhibits a shallow well
following a spatially extended skinny barrier (shown in Figures 7 & 8 as thick
blue lines corresponding to $\lambda=\infty$). These kinds of shallow well-
barrier combinations support very weak and much broader resonances for far
extended wavefunctions resulting in an erroneous numerical computation. Hence,
for accurate computation of such broad resonances, a large number of partial
waves need to be included in the expansion basis. But the spatially extended
partial waves cause slowing down of the convergence rate and introduces more
errors in the result. Hence, a one-parameter family of isospectral potential
is constructed using the ground state energy and wavefunctions following the
algebra of SSQM. This newly constructed one-parameter family of isospectral
potential for an appropriate choice of $\lambda$ develops a narrow and deep
narrow well shifted towards the origin following relatively higher and less
extended barrier (as shown in Figures 7 & 8 as lines of gradually decreasing
thickness for $\lambda=\infty,10^{-4},10^{-6}$, and $10^{-8}$). This enhanced
well-barrier combination correctly represents the wavefunctions (as shown in
Figures 9 & 10) and resonances at the same energy as in the original shallow
potential (as shown in Figures 11 & 12) for $\lambda=10^{-4},10^{-6}$, and
$10^{-8}$ respectively. As summarized in Table 5 the computed observables are
in excellent agreement with experimental data available in the literature. In
conclusion, we may say that the scheme described here is a robust one and can
be applied to any weakly bound few-body system even if the system lacks any
bound state. The above facts and findings apart from validating our
theoretical scheme also accomplished our goals set for this work.

## Acknowledgements

This work has been supported by the computational facility at Aliah
University, India.

Conflict of interest statement: On behalf of all authors, the corresponding
author states that there is no conflict of interest.

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