∎

11institutetext: Swagatika Bhoi 22institutetext: School of Physics, Sambalpur
University, Jyoti Vihar, Burla-768019, India.
22email<EMAIL_ADDRESS>33institutetext: Basudeb Sahu* 44institutetext:
Department of Physics, College of Engineering and Technology
Bhubaneswar-751003, India.

# An estimate of Alpha decay half-life from the poles of S-matrix of an
exactly solvable potential

Swagatika Bhoi  Basudeb Sahu

(Received: date / Accepted: date)

###### Abstract

We develop a versatile and analytically solvable potential which fairly
reproduces the combined potential of an $\alpha+$nucleus system resulting from
both the attractive nuclear and repulsive electrostatic potentials. The
potential is expressed in terms of the radial position, mass and proton number
of the $\alpha$-particle and the daughter nucleus and certain parameters
governing the depth, height and steepness of the barrier. The potential
generated is typically a pocket near the origin and a barrier adjacent to it.
The Schrödinger equation with the above mentioned potential is then solved for
the wave function. This potential produces discrete positive energy quasibound
state known as resonance state. By matching the wave function and its
derivative with the regular Coulomb wave function $F_{0}$ and irregular
Coulomb wave function $G_{0}$, we obtain the S-matrix analytically. The
resonance is obtained from the pole in complex energy plane which gives the
width of the corresponding time of decay through the imaginary part of the
energy pole position. We make a comparative study of the measured half-lives
of various nuclei with the calculated half-lives. The calculated values of
half-lives closely match with the corresponding experimental results.

###### Keywords:

$\alpha$-decay, S-matrix, pole, resonance state

###### pacs:

23.60.+e, 25.70.Ef, 11.55.Jy

##  1 Introduction

Ernest Rutherford’s discovery of alpha particle is an incredible milestone in
the realm of nuclear physics. It has opened new frontiers of research for full
fledged study of alpha decay phenomena. Equally praiseworthy were the
discoveries of exotic nuclei including heavy and superheavy nuclei with proton
number as large as $118$. Since then $\alpha$-decay has become a topic of
intense research and has gained an extraordinary amount of attention for the
calculation of half-lives and Q-values.

Earlier findings of decay half-lives have lent support to many theoretical
studies namely the relativistic mean field theory, the density dependent M$3$Y
(DDM$3$Y) interaction and the Skyrme-Hartree-Fock mean field model a ; b ; c ;
d . The whole process of decay of $\alpha$-particle from a parent nucleus can
be considered as a quantal two body problem comprising of the daughter and the
emitted $\alpha$-particle e . Indeed, a suitable $\alpha$+daughter nucleus
potential can be formulated and its features can be elucidated. The potential
chosen can be either phenomenological (eg. Woods-Saxon (WS) shape with
adjustable parameters, Ginnochio shape etc.) or it can be formed
microscopically by double folding models f ; g .

Amongst many of the potentials used so far, Ginocchio potential and Wood-Saxon
potentials are the two potentials that have been extensively studied for
$\alpha$-decay half-lives. Although the Ginocchio potential is analytically
solvable, highly versatile and correctly represent diffuseness still the
searching of resonance pole of S-matrix in this case is tedious. On the other
hand, searching of poles in Wood-Saxon potential though simple, the potential
is not analytically solvable. Therefore, we search for a potential which is
analytically solvable and the tracing of poles of the S-matrix is quite easy
to estimate the alpha decay half-lives of various exotic nuclei from these
poles.

We develop a versatile and analytically solvable potential which fairly
reproduces the combined potential of an $\alpha$+nucleus system resulting from
both the attractive nuclear and repulsive electrostatic potentials. This
potential was used originally by Fiedeldey and Frahn h . We have designed a
similar potential [Fig. $1$] sans any imaginary part involved. By ascertaining
all the parameters, this potential is capable of generating quantal or quasi
molecular state and we can easily incorporate $\alpha$-decay process to this
quasi molecular decay path.

Many a times WKB method has been applied to get the energies of long-lived
states of the effective potential. This semi-classical approximation yields
the probability of decay of $\alpha$+daughter nucleus system together with the
decay constant. Furthermore, the found out decay constant can be used to get
the half-lives. However, this method has serious problems concerning the
frequency factor i .

Hence, we sort to an alternative method of finding the half-life by
considering the quantum scattering theory where resonance states can be found
from the $\alpha$+daughter nucleus two body system. S-matrix theory of
potential scattering is used to get the resonance energy. The resonance states
are described as poles of the S-matrix falling in the fourth quadrant of the
complex-momentum plane. We are therefore left with the complex pole positions
which in turn gives both energy and width of the resonance state g ; j ; k ; l
.

In the present paper, to begin with we develop a potential totally apt for
describing the Coulomb nuclear potential of the $\alpha$+nucleus system. Our
preferred potential in simple term is a merger of an attractive nuclear part
and a repulsive Coulomb part. This merged potential shows a pocket followed by
a barrier in its radial variation. We specify the parameters for the depth of
the pocket, height and position of the barrier and the steepness of the
potential in the interior side. With this potential, Schrödinger equation is
solved exactly. By matching the wave function and its derivative with the
regular Coulomb wave function $F_{0}$ and irregular Coulomb wave function
$G_{0}$, for partial wave $l=0$, the S-matrix is expressed as a function of
the incident energy. In the complex energy or momentum plane, the poles of
S-matrix in the fourth quadrant of the momentum plane gives the energy of
resonance as well as the width. Subsequently, the width gives the decay time.

The above mentioned formulation is applied to a wide range of $\alpha$+nucleus
systems including light to heavy to superheavy nuclei for finding out the
resonance states and their decay rates. Throughout the paper, we maintain
uniformity by changing the value of the parameter $d_{1}$ which specifically
determines the steepness of the inner side of the potential barrier. By doing
so, a clear comparison and correlation is done in all the systems considered
in the calculation. It is shown that the range of $d_{1}$ is $2.5<d_{1}<6.5$.

The paper is organized in the following way. Section II concerns with the
formulation of the potential. The experimental results are presented in
section III. Summary and conclusions are confronted in section IV.

Figure 1: Plot of the potential V(r) as a function of radial distance for the
$\alpha+_{80}^{174}$Hg system.

##  2 Formulation

We suggest a potential to be a function of radial variable r, which is of the
form

$V(r)=\left\\{\begin{array}[]{cl}V_{0}[S_{1}+(S_{2}-S_{1})\rho_{1}]&{\hskip
11.38109ptif\;\;\;r\leq R_{0},}\\\ V_{0}S_{2}\rho_{2}&{\hskip
11.38109ptif\;\;\;r\geq R_{0},}\end{array}\right.$ (1)

where $V_{0}$ is the strength of the potential with value $1$ MeV.

$\rho_{n}=\frac{1}{\cosh^{2}\frac{R_{0}-r}{d_{n}}};n=1,2,\\\ $

$d_{n}$ accounts for the flatness of the barrier, $d_{1}$ deciding the
steepness of the interior side of the barrier whereas the exterior side is
judged by $d_{2}$. $R_{0}$ is the radial position having value;
$R_{0}=r_{0}(A_{\alpha}^{1/3}+A_{D}^{1/3})+2.72$, $r_{0}=0.97$ fm. Since we
are considering the $\alpha$+nucleus system, $A_{\alpha}$ and $Z_{\alpha}$
represent the mass and proton number of $\alpha$ particle, $A_{D}$ and $Z_{D}$
represent the mass and proton number of the daughter nucleus. Moreover $S_{1}$
and $S_{2}$ are the depth and height of the potential, respectively, having
values,

$S_{1}=-78.75+\frac{3Z_{\alpha}Z_{D}e^{2}}{2R_{c}},\\\ $
$S_{2}=\frac{Z_{\alpha}Z_{D}e^{2}}{R_{0}}(1-\frac{a_{g}}{R_{0}}),\\\ $

where $R_{c}$ is the Coulomb radius parameter;
$R_{c}=r_{c}(A_{\alpha}^{1/3}+A_{D}^{1/3})$, $a_{g}=1.6$ fm, $r_{c}=1.2$ fm,
$e^{2}=1.43996$ MeV fm. $r_{c}$ and $a_{g}$ are the distance parameters.

With the potential given by (1), the reduced Schrödinger equation for the
s-wave is written in the dimensionless form as follows:

$\displaystyle\frac{d^{2}u_{1}}{dr^{2}}+[\kappa^{2}-k_{0}^{2}S\rho_{1}]u_{1}=0,$
(2)

where $\kappa^{2}=k^{2}-k_{0}^{2}S_{1}$,
$\frac{2m}{\hbar^{2}}V_{0}=k_{0}^{2}$,$\frac{2m}{\hbar^{2}}E_{c.m.}=k^{2}$,
$S=S_{2}-S_{1}$, $E_{c.m.}$ refers to the center of mass energy.

The solution $u_{1}(r)$ in the region $r\leq R_{0}$ is given by

$u_{1}(r)=A_{1}z_{1}^{{i/2}\kappa
d_{1}}F(a_{1},b_{1},c_{1},z_{1})+B_{1}z_{1}^{-i/2\kappa
d_{1}}F(a_{1}^{\prime},b_{1}^{\prime},c_{1}^{\prime},z_{1}^{\prime}),\\\ $ (3)
$a_{1}=\frac{1}{2}({\lambda}_{1}+i{\kappa}d_{1}),b_{1}=\frac{1}{2}({1-\lambda}_{1}+i{\kappa}d_{1}),c_{1}=1+i{\kappa}d_{1},\\\
$ (4)
$a_{1}^{\prime}=\frac{1}{2}({\lambda}_{1}-i{\kappa}d_{1}),b_{1}^{\prime}=\frac{1}{2}({1-\lambda}_{1}-i{\kappa}d_{1}),c_{1}^{\prime}=1-i{\kappa}d_{1},\\\
$ (5)
${\lambda}_{1}=\frac{1}{2}-\frac{1}{2}[1-(2{\kappa}d_{1})^{2}S]^{1/2},\\\ $
(6)

where $z_{1}=\rho_{1}(r)$, $F(a,b,c,z)$ is the hyper geometric function and
the values of $A_{1}$ and $B_{1}$ are decided by using the boundary condition,
$u_{1}(r)=0$ at $r=0$.

Matching the wave function $u_{1}$ and its derivative with the regular Coulomb
wave function $F_{0}$ and irregular Coulomb wave function $G_{0}$ at the
boundary $r=R_{0}$, we obtain the S-matrix as

$S_{m}=2iC_{0}+1,$ (7)

where
$C_{0}=\frac{u_{1}kF_{0}-u_{1}^{\prime}F_{0}}{u_{1}^{\prime}(G_{0}+iF_{0})-u_{1}k(G_{0}^{\prime}+iF_{0}^{\prime})}.$

The resonance energy is obtained from the poles of S-matrix in the complex
momentum plane. From a resonant pole of the S-matrix expressed by

$k_{P}=k_{r}-ik_{i}.\\\ $

We obtain the resonance energy (Q value)

$E_{P}=\frac{\hbar^{2}}{2m}(k_{r}^{2}-k_{i}^{2})\hskip 5.69054ptMeV,$ (8)

and Width

$\Gamma=\frac{\hbar^{2}}{2m}4k_{r}k_{i}\hskip 5.69054ptMeV.$ (9)

The time of decay is related to the width as

$T_{1/2}=\frac{\hbar}{\Gamma}.$ (10)

##  3 Results and Discussions

We now apply the formulation as described in section $2$ by taking a potential
apt for justifying the $\alpha$+nucleus system. This potential must be
compatible with the relativistic mean field theory. For $\alpha+^{174}_{80}$Hg
system we reproduce the same potential with our analytically solvable
potential expressed by (1) by assigning the values of the parameters
$R_{0}=9.6749$ fm, $S_{1}=-38.5841$ MeV, $S_{2}=19.8752$ MeV, $d_{1}=5.13240$,
$d_{2}=2$. This potential for the system is shown in Fig. 1. Using this
potential we obtain the resonance energy, Q-value and the half-life from the
resonant pole of S-matrix given by (10). We now sagaciously change the values
of the potential parameter $d_{1}$ and apply the same potential formulation to
a wide range of systems spanning the light, heavy and superheavy nuclei to get
the half lives ready for close comparison with the corresponding experimental
values. For good agreement of the experimental values we keep all the
parameters except changing one parameter concerning the steepness of the
potential barrier in the interior side. Keen observation on the outcome reveal
that the value of $d_{1}$ deviate in the range $2.5$ to $6.5$.

For a given system we vary the value of $d_{1}$ and locate a pole to keep the
energy equal to the experimental Q-value. The width of this pole gives us the
$T_{1/2}$ through (14) and we represent this by $T_{1/2}^{calt.}$. We now jot
the trio i.e $Q^{expt}$, $T^{expt.}$ and $T^{calt.}$ in Table 1 and Table 2
for a wide range of $\alpha$+nucleus systems. The results in Table 1 and Table
2 show that for the whole range of $\alpha$-decaying nuclei system the
experimental and calculated half-lives closely match. We also confirm the
close agreement for the case of different isotopes of Po nucleus which is
shown in Fig. 3.

We hand-pick a group of systems by going down the tabulation (i.e
${}^{146}_{62}$ Sm, ${}^{150}_{64}$Gd, ${}^{154}_{66}$Dy, ${}^{190}_{78}$Pt,
${}^{210}_{82}$Pb, ${}^{226}_{88}$Ra, ${}^{232}_{90}$Th, ${}^{238}_{92}$U,
${}^{244}_{94}$Pu, ${}^{250}_{96}$Cm). A glimpse on the results of the systems
affirms that they have very large experimental decay time, $T_{1/2}^{expt.}$.
Let us consider the case of ${}^{146}_{62}$Sm for which
$T_{1/2}^{expt.}=2.2\times 10^{15}$ s. Our calculated value of
$T_{1/2}^{calt.}=3.73\times 10^{15}$ s which is very close to the above
experimental value. Among all the systems chosen in this paper, the longest
half-life is seen in case of ${}^{238}_{92}$U where $T_{1/2}^{expt.}=1.4\times
10^{17}$ s and the corresponding width $\Gamma^{expt.}=3.25\times 10^{-39}$
MeV for Q value of $4.270$ MeV. Our calculated results show
$T_{1/2}^{calt.}=5.59\times 10^{17}$ s with $\Gamma^{calt.}=8.15\times
10^{-40}$ MeV. Comprehensive results are obtained for a series of nuclei and
the experimental and calculated results match closely for very small decay
time as well as for very long decay time.

Table 1: Comparison between the experimental $\alpha$-decay half-lives d  and results of the present calculation. The value of the parameter $d_{2}$ is kept constant throughout i.e. $d_{2}=2$. Nucleus | $d_{1}$ |  Q${}_{\alpha}^{expt.}$(MeV) | $T_{1/2}^{expt.}(s)$ | $T_{1/2}^{calt.}(s)$
---|---|---|---|---
${}^{106}_{52}$Te | 4.90420 | 4.290 |  8.0 $\times$10-5 |  4.02 $\times$ 10-5
${}^{108}_{52}$Te | 4.81479 | 3.420 |  4.3 $\times$100 |  2.13 $\times$100
${}^{110}_{52}$Te | 4.75180 | 2.699 |  6.2 $\times$105 |  8.39 $\times$105
${}^{112}_{54}$Xe | 4.11499 | 3.330 |  3.9 $\times$100 |  1.52 $\times$100
${}^{146}_{62}$Sm | 4.90780 | 2.528 |  2.2 $\times$1015 |  3.73 $\times$1015
${}^{148}_{64}$Gd | 4.94120 | 3.271 |  2.2 $\times$109 |  2.35 $\times$109
${}^{150}_{64}$Gd | 4.907800 | 2.808 |  5.6 $\times$1013 |  7.87 $\times$1013
${}^{150}_{66}$Dy | 6.47199 | 4.351 |  1.2 $\times$103 |  4.87 $\times$103
${}^{152}_{66}$Dy | 3.72760 | 3.726 |  8.6 $\times$106 |  2.58 $\times$106
${}^{154}_{66}$Dy | 4.88380 | 2.945 |  9.5 $\times$1013 |  8.16 $\times$1013
${}^{152}_{68}$Er | 6.48559 | 4.934 |  1.1 $\times$101 |  4.75 $\times$101
${}^{156}_{68}$Er | 3.64480 | 3.483 |  6.7 $\times$109 |  3.87 $\times$109
${}^{154}_{70}$Yb | 5.04180 | 5.474 |  4.4 $\times$10-1 |  2.89 $\times$10-1
${}^{158}_{70}$Yb | 4.90700 | 4.170 |  4.3 $\times$106 |  2.67 $\times$106
${}^{156}_{72}$Hf | 5.04580 | 6.028 |  2.4 $\times$10-2 |  1.45 $\times$10-2
${}^{162}_{72}$Hf | 3.66560 | 4.416 |  4.9 $\times$105 |  2.00 $\times$105
${}^{158}_{74}$W | 6.51280 | 6.613 |  1.3 $\times$10-3 |  4.36 $\times$10-3
${}^{168}_{74}$W | 4.90140 | 4.500 |  1.6 $\times$106 |  3.83 $\times$106
${}^{164}_{76}$Os | 5.07320 | 6.767 |  2.1 $\times$10-3 |  1.67 $\times$10-3
${}^{166}_{78}$Pt | 5.01939 | 6.990 |  2.0 $\times$10-3 |  2.14 $\times$10-3
${}^{190}_{78}$Pt | 4.84759 | 3.252 |  2.0 $\times$1019 |  1.44 $\times$1019
${}^{172}_{80}$Hg | 5.10033 | 7.524 |  2.3 $\times$10-4 |  2.51 $\times$10-4
${}^{188}_{80}$Hg | 4.92299 | 4.703 |  5.2 $\times$108 |  4.09 $\times$108
${}^{178}_{82}$Pb | 5.13240 | 7.790 |  2.3 $\times$10-4 |  2.23 $\times$10-4
${}^{180}_{82}$Pb | 5.10599 | 7.419 |  4.2 $\times$10-3 |  3.10 $\times$10-3
${}^{182}_{82}$Pb | 5.08279 | 7.066 |  5.5 $\times$10-2 |  4.22 $\times$10-2
${}^{184}_{82}$Pb | 5.06999 | 6.774 |  6.1 $\times$10-1 |  4.67 $\times$10-1
${}^{186}_{82}$Pb | 5.05500 | 6.470 |  1.2 $\times$101 |  6.81 $\times$100
${}^{188}_{82}$Pb | 5.03020 | 6.109 |  2.8 $\times$102 |  1.94 $\times$102
${}^{190}_{82}$Pb | 4.99679 | 5.697 |  1.8 $\times$104 |  1.49 $\times$104
${}^{192}_{82}$Pb | 4.95260 | 5.221 |  3.5 $\times$106 |  4.42 $\times$106
${}^{194}_{82}$Pb | 4.90780 | 4.738 |  8.8 $\times$109 |  3.06 $\times$109
${}^{210}_{82}$Pb | 5.01999 | 3.792 |  3.7 $\times$1016 |  3.7 $\times$1016
${}^{190}_{84}$Po | 5.21159 | 7.693 |  2.5 $\times$10-3 |  2.21 $\times$10-3
${}^{192}_{84}$Po | 5.18260 | 7.320 |  3.4 $\times$10-2 |  3.54 $\times$10-2
${}^{194}_{84}$Po | 5.16059 | 6.987 |  3.9 $\times$10-1 |  4.68 $\times$10-1
${}^{196}_{84}$Po | 5.13940 | 6.658 |  5.7 $\times$100 |  8.01 $\times$100
Table 2: Comparison between the experimental $\alpha$-decay half-lives d  and results of the present calculation. The value of the parameter $d_{2}$ is kept constant throughout i.e. $d_{2}=2$. Nucleus | $d_{1}$ |  Q${}_{\alpha}^{expt.}$(MeV) | $T_{1/2}^{expt.}(s)$ | $T_{1/2}^{calt.}(s)$
---|---|---|---|---
${}^{198}_{84}$Po | 5.11460 | 6.309 |  1.9 $\times$102 |  2.10 $\times$102
${}^{200}_{84}$Po | 5.09320 | 5.981 |  6.2 $\times$103 |  5.44 $\times$103
${}^{202}_{84}$Po | 5.08020 | 5.701 |  1.4 $\times$105 |  1.17 $\times$105
${}^{204}_{84}$Po | 5.07760 | 5.485 |  1.9 $\times$106 |  1.51 $\times$106
${}^{206}_{84}$Po | 6.47139 | 5.327 |  1.4 $\times$107 |  6.98 $\times$107
${}^{208}_{84}$Po | 5.09919 | 5.215 |  9.1 $\times$107 |  4.18 $\times$107
${}^{210}_{84}$Po | 6.56080 | 5.407 |  1.2 $\times$107 |  2.29 $\times$107
${}^{212}_{84}$Po | 5.85500 | 8.954 |  3.0 $\times$10-7 |  4.64 $\times$10-7
${}^{214}_{84}$Po | 4.37020 | 7.833 |  1.6 $\times$10-4 |  1.17 $\times$10-4
${}^{216}_{84}$Po | 4.25500 | 6.906 |  1.5 $\times$10-1 |  1.2 $\times$10-1
${}^{218}_{84}$Po | 4.16359 | 6.115 |  1.9 $\times$102 |  1.89 $\times$102
${}^{198}_{86}$Rn | 3.90799 | 7.349 |  6.6 $\times$10-2 |  3.43 $\times$10-2
${}^{222}_{86}$Rn | 4.02780 | 5.590 |  3.3 $\times$105 |  5.39 $\times$105
${}^{206}_{88}$Ra | 3.93280 | 7.415 |  2.4 $\times$10-1 |  1.11 $\times$10-1
${}^{226}_{88}$Ra | 2.72480 | 4.871 |  5.0 $\times$1010 |  1.57 $\times$1010
${}^{214}_{90}$Th | 4.00759 | 7.827 |  8.7 $\times$10-2 |  2.74 $\times$10-2
${}^{232}_{90}$Th | 2.59220 | 4.082 |  4.4 $\times$1017 |  2.23 $\times$1017
${}^{222}_{92}$U | 4.27100 | 9.430 |  1.5 $\times$10-6 |  3.80 $\times$10-6
${}^{238}_{92}$U | 3.71900 | 4.270 |  1.4 $\times$1017 |  5.59 $\times$1017
${}^{228}_{94}$Pu | 5.26060 | 7.940 |  2.1 $\times$100 |  1.94 $\times$100
${}^{244}_{94}$Pu | 3.74699 | 4.666 |  2.5 $\times$1015 |  4.30 $\times$10-3
${}^{238}_{96}$Cm | 2.69679 | 6.670 |  7.9 $\times$104 |  2.44 $\times$104
${}^{250}_{96}$Cm | 2.65660 | 5.169 |  1.5 $\times$1012 |  2.39 $\times$1012
${}^{240}_{98}$Cf | 3.91659 | 7.711 |  4.1 $\times$101 |  6.45 $\times$101
${}^{254}_{98}$Cf | 3.84459 | 5.927 |  1.7 $\times$109 |  4.56 $\times$109
${}^{246}_{100}$Fm | 3.99060 | 8.377 |  1.3 $\times$100 |  1.92 $\times$100
${}^{256}_{100}$Fm | 3.92460 | 7.027 |  1.2 $\times$105 |  1.82 $\times$105
${}^{252}_{102}$No | 3.98620 | 8.548 |  4.1 $\times$100 |  2.89 $\times$100
${}^{256}_{102}$No | 4.04999 | 8.581 |  2.9 $\times$100 |  2.09 $\times$100
${}^{254}_{104}$Rf | 4.00059 | 9.210 |  1.7 $\times$10-1 |  1.33 $\times$10-1
${}^{258}_{104}$Rf | 4.05620 | 9.190 |  1.1 $\times$10-1 |  1.41 $\times$10-1
${}^{260}_{106}$Sg | 5.33759 | 9.901 |  1.2 $\times$10-2 |  4.63 $\times$10-2
${}^{264}_{108}$Hs | 5.39579 | 10.591 |  1.1 $\times$10-3 |  3.39 $\times$10-3
${}^{270}_{110}$Ds | 4.18099 | 11.117 |  2.1 $\times$10-4 |  1.16 $\times$10-4
${}^{286}_{114}$Fl | 4.03920 | 10.370 |  1.4 $\times$10-1 |  1.33 $\times$10-1
${}^{290}_{116}$Lv | 2.89960 | 10.990 |  8.0 $\times$10-3 |  2.24 $\times$10-3
${}^{294}_{118}$Lv | 5.40980 | 11.810 |  1.4 $\times$10-3 |  4.30 $\times$10-3
Figure 2: $\alpha$-decay half-lives for different isotopes of Po nucleus.
Solid dots represent the calculated results and the solid triangles represent
the experimental results.

##  4 Summary and Conclusion

An analytical expression for a potential is developed to simulate the combined
effect of the attractive nuclear and repulsive electrostatic force. We have
given expressions for the potential which is analytically solvable. Using this
potential the s-wave Schrödinger equation is solved and the S-matrix is
obtained by matching the wave function and its derivative with the Coulomb
wave function outside. From the expressions of S-matrix the pole representing
the resonance state is searched. From this pole we obtain the Q-value and the
width or decay half life of the resonance state.

Results of $\alpha$-decay half-lives have been calculated for various nuclei
starting from a nucleus with $A=106(Z=52)$ to a nucleus with $A=294(Z=118)$. A
thorough comparison of these calculated results with the corresponding
experimental values shows that the calculated half-lives are compatible with
the experimental half-lives. Thus, this formulation can be used in explaining
the half-lives of $\alpha$-decay of any nuclei that include light, heavy and
super heavy showing decay time from very small ($10^{-4}$ s) to very large
decay time of the order of $10^{17}$ s.

##  5 Acknowledgement

We gratefully acknowledge the computing and library facilities extended by the
Institute of Physics, Bhubaneswar.

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# Collisions of kinks in deformed $\varphi^{4}$ and $\varphi^{6}$ models

Aliakbar Moradi Marjaneh1, Fabiano C. Simas2,3 and D. Bazeia4
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department
of Physics, Quchan Branch, Islamic Azad university, Quchan, Iran
2Programa de Pós-Graduação em Física, Universidade Federal do Maranhão (UFMA),
Campus Universitário do Bacanga, 65085-580, São Luís, Maranhão, Brazil
3Centro de Ciências Agrárias e Ambientais (CCAA), Universidade Federal do
Maranhão (UFMA), 65000-000 Chapadinha, MA, Brazil
4Departamento de Física, Universidade Federal da Paraíba (UFPB), 58051-970
João Pessoa, PB, Brazil

###### Abstract

Two _hyperbolic-deformed_ field theoretic models are discussed. In both of
them, due to the effect of specific deformation function on the well known
$\varphi^{4}$ and $\varphi^{6}$ models, their internal structure may change
significantly. Unlike the $\varphi^{4}$ kinks solutions, which has only one
internal mode in addition to its translational mode, the kinks of the
hyperbolic-deformed $\varphi^{4}$ model can have _several_ internal modes.
Moreover, the deformation on the $\varphi^{6}$ model has other interesting
features, because the kinks of the $\varphi^{6}$ model have only a zero mode
and the deformation may cause the appearance of internal mode for both kink
and antikink. The presence of the new internal modes modify the collisions
which we explore in the present work. The deformation relies on a real
parameter, which controls the number of internal modes, and we also study how
the deformation parameter alter the mass of the kinks and the critical
velocities involved in the collisions.

##  I Introduction

Nonlinear field models have essential role to describe phenomena in several
areas, running from subatomic to the cosmological scale. In particular,
solitons as nontrivial static solutions of some classical field theory with
degenerate minima, have attracted the attention of theoretical and
experimental researchers for more than 50 years; see, e.g., Refs.
Bishop.PhysD.1980 ; Rajaraman.book.1982 ; Vilenkin.book.2000 ;
Manton.book.2004 ; Vachaspati.book.2006 ; Dauxois.book.2006 ;
Kevrekidis.book.2019  and references therein. The topological defects in
$(1+1)$ dimensional space-time are the simplest form of localized structures,
called kinks, and they have been widely considered in the past. For example,
we can mention the study of kinks, their scattering and multi-kinks collision
in integrable and non-integrable models such as sine-Gordon model
Peyrard.PhysD.1983.msG ; Bazeia.EPJC.2011 ; Bazeia.EPJC.2013 ;
Moradi.EPJB.2018 , double sine-Gordon model Campbell.dsG.1986 ;
Peyravi.EPJB.2009 ; Gani.EPJC.2018 ; Belendryasova.JPCS.2019 ; Gani.EPJC.2019
, $\varphi^{4}$ Kevrekidis.book.2019 ; Campbell.PhysD.1983.phi4 ; anninos ;
Campbell.PhysD.1986 ; Dorey.JHEP.2017 ; Moradi.CNSNS.2017 ; Askari.CSF.2020 ;
Mohammadi.CNSNS.2020 , $\varphi^{6}$ Hoseinmardy.IJMPA.2010 ; Dorey.PRL.2011 ;
Weigel.AHEP.2017 ; Moradi.JHEP.2017 ; Demirkaya.JHEP.2017 ; Lima.JHEP.2019 ,
semi-compactness kink Bazeia.EPJC.2021 , logarithmic models Ekaterina.PLB.2021
and higher order models Christov.PRD.2019 ; Belendryasova.CNSNS.2019 ;
Christov.PRL.2019 ; Gani.PRD.2020 ; Christov.CNSNS.2021 ; Gani.EPJC.2021 ,
which have many applications to understand, for instance, phase transitions in
materials Gufan.DAN.1978 , dynamics of crowdions Crowdions0 ; Crowdions1 ;
Crowdions2 ; Crowdions3 ; Crowdions4 , field evolution in the early Universe
Gani.JCAP.2018 , solitonlike solutions for a Higgs scalar field
Kudryavtsev.JETPLett.1975 , Q-lump and skyrmion on a domain wall
blyankinshtein ; nitta4 ; jennings ; nitta5 ; GaLiRaconf and scalar field
models for branes free ; cza ; Baz ; Baz2 ; Zhong.PRD.2014 ; Peyravi.EPJC.2016
. In some specific investigations, we can highlight the scattering of wobbling
kinks Alonso.PRD.2021 ; Azadeh.JHEP.2021 ; Azadeh.JHEP.2021.1 , models with
two scalar fields Shnir.PRD.2012 ; Alonso.Comm.2020 ; Alonso.Comm.2021 and
also, studies on collective coordinate dynamics, which solve a difficulty
linked to the collision process and prove that the appearance of the fractal
structure is related to energy exchange between the internal modes
Manton.PRL.2021 ; Manton.PRD.2021 ; Adam.PRD.2022 .

Although the study of integrable models with localized solutions such as sine-
Gordon model provides a lot of information, nevertheless the non-integrable
models, that have inner structure Peyrard.PhysD.1983.msG ;
Campbell.PhysD.1983.phi4 ; Panos.JHEP.2017 ; Dorey.PLB.2018 ; Zhong.JHEP.2020
; Bazeia.JModPhysA.2019 or their solutions are asymmetric
Hoseinmardy.IJMPA.2010 ; Dorey.PRL.2011 ; Weigel.AHEP.2017 ; Moradi.JHEP.2017
; Demirkaya.JHEP.2017 ; Lima.JHEP.2019 ; Christov.PRD.2019 ;
Belendryasova.CNSNS.2019 ; Christov.PRL.2019 ; Gani.PRD.2020 , present more
attractive properties. In particular, we can mention the bounce windows in
kink scattering in $\varphi^{4}$ and $\varphi^{6}$ models. The symmetric kinks
of the $\varphi^{4}$ model can carry energy with internal mode. When the
initial velocity is smaller than the critical velocity, bounce windows are
created for some initial conditions. This mechanism is explained by the energy
exchange between the internal mode and the translation mode
Campbell.PhysD.1983.phi4 . In the $\varphi^{6}$ model, the kinks are
asymmetric, and there is no presence of an internal mode, however, the
scattering shows the two-bounce windows. The explanation is provided by the
mechanism of resonant energy exchange between the vibrational modes of the
antikink-kink pair Dorey.PRL.2011 . On the other hand, new potentials and
their kinks are always used with various applications. For example, hyperbolic
models can describe black holes QiangWen.PRD.2015 , tachyon matter cosmology
Pourhassan.IJMPD.2017 and quintessential inflation Agarwal.PLB.2017 . Another
example of the application of kinks can be seen in the study of molecular
simulations in trans-polyacetylene Berna ; An . In Ref. Berna , the study of
photogeneration of topological defects showed a two-bounce behavior,
characteristic of nonintegrable models. More recently, in Ref. An it was
proposed the study of the relaxation dynamics to form soliton pairs in trans-
polyacetilene. The investigation suggested that the excitation of the
electron-hole pair generated a kink-antikink pair that collide and reflect
with constant velocities. Interestingly, in many cases, several potentials can
be obtained by using simple deformation functions on well known models such as
$\varphi^{4}$ and $\varphi^{6}$ models Bazeia.PRD.2002 ; Bazeia.PRD.2004 ;
Bazeia.PRD.2006 ; Bazeia:2005hu ; Bazeia.EPJC.2018 . The selection of a
deformation function can result in the introduction of new physics and the
production of new models. In Ref. Yama.car , for example, deformation in
graphene was addressed, with the aim to understand the positive and negative
radiation pressure effects.

In this work, we deal with several issues concerning specific properties of
two distinct hyperbolic potentials, described by deformation of the standard
$\varphi^{4}$ and $\varphi^{6}$ models. To do this, we organize the paper as
follows. In section II, we review the well-known $\varphi^{4}$ and
$\varphi^{6}$ field theories. Then, in the two sections III and IV, we define
a deformation function dependent on real parameter $r$, and introduce tanh-
deformed $\varphi^{4}$ and tanh-deformed $\varphi^{6}$ models. In these
models, we investigate the explicit kinks, their masses, the stability
potentials, the number and the values of internal modes and also critical
velocity and the corresponding collisions. As we explain in sections III and
IV, the deformed models are obtained via a deformation function that depends
on a single real parameter, which is used to controls the main features of the
models and guide us to investigate the two models. The work in ended in
section V, where we add our conclusions and comment on issues concerning
future directions of new research on the subject.

##  II General considerations

We consider a field theoretical model in $(1+1)$ spacetime dimensions, with a
real scalar field $\varphi(x,t)$. The dynamics of the field is described by
the Lagrangian density

$\displaystyle\mathcal{L}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial
t}\right)^{2}-\frac{1}{2}\left(\frac{\partial\varphi}{\partial
x}\right)^{2}-V(\varphi).$ (1)

The energy functional corresponding to the Lagrangian Eq. (1) is

$\displaystyle
E=\int_{-\infty}^{+\infty}\Biggl{[}\frac{1}{2}\left(\frac{\partial\varphi}{\partial
t}\right)^{2}+\frac{1}{2}\left(\frac{\partial\varphi}{\partial
x}\right)^{2}+V(\varphi)\Biggr{]}dx,$ (2)

where $V(\varphi)$ is a non-negative potential which we suppose engenders a
set of minima
$\mathcal{V}=\left\\{\bar{\varphi}_{1},\bar{\varphi}_{2},\bar{\varphi}_{3},...\right\\}.$
The equation of motion can be obtained from the Lagrangian Eq. (1) as

$\frac{\partial^{2}\varphi}{\partial
t^{2}}-\frac{\partial^{2}\varphi}{\partial x^{2}}+\frac{dV}{d\varphi}=0,$ (3)

and for static field $\varphi=\varphi(x)$ we have

$\frac{d^{2}\varphi}{dx^{2}}=\frac{dV}{d\varphi}.$ (4)

We can write the first order differential equations

$\frac{d\varphi}{dx}=\pm\sqrt{2V(\varphi)},$ (5)

which may also be used to obtain the kink and antikink solutions of the model.
By substituting these solutions into Eq. (2), one can obtain the (classical)
mass of each kink or antikink. In addition, moving kinks with the velocity $v$
along the x-axis can be obtained by the Lorentz boost
$\varphi(x,t)=\varphi(\gamma(x-vt))$, where $\gamma=1/\sqrt{(1-v^{2})}$. We
search for static configurations having finite energy and interpolating
between neighboring minima of the model. It means that, for the kink,
$\varphi(-\infty)=\lim_{x\to-\infty}\varphi(x)=\bar{\varphi}_{i}$ with
$\bar{\varphi}_{i}\in\mathcal{V}$, and
$\varphi(+\infty)=\lim_{x\to+\infty}\varphi(x)=\varphi_{i+1}$ with
$\bar{\varphi}_{i+1}\in\mathcal{V}$.

The next important subject has to deal with internal modes in kink excitation
spectra and kink stability analysis because many phenomena, including the
interaction of kinks with impurities and the kink scattering, are explained
based on such internal modes. To deal with this, one adds a small perturbation
$\varphi(x,t)=\varphi(x)+\eta(x)\cos(\omega t)$ to the static kink
$\varphi(x)$. This leads to the Schrödinger-like equation

$-\frac{\partial^{2}\eta}{\partial\varphi^{2}}+U\eta=\omega^{2}\eta,$ (6)

where $U=U(x)$ is the kink stability potential

$U=\frac{d^{2}V}{d\varphi^{2}}\bigg{|}_{\varphi(x)}.$ (7)

It is straightforward to check from Eq. (6) that there is always a zero mode,
$\eta_{0}(x)={d\varphi}/{dx}$ corresponding to $\omega=0$, which is ensured by
translational invariance of the system.

Based on the deformation procedure described in Bazeia.PRD.2002 and further
considered in Refs. Bazeia.PRD.2004 ; Bazeia.PRD.2006 ; Bazeia:2005hu ;
Bazeia.EPJC.2018 , one can take the potential $V(\varphi)$ of a given model
and the deformation function $f(\varphi)$ to write the new deformed model with
potential $\tilde{V}(\varphi)$ in the form of

$\tilde{V}(\varphi)=\frac{V[\phi\to f(\varphi)]}{[f^{\prime}(\varphi)]^{2}},$
(8)

such that, in the new model the new kinks are given by

$\tilde{\varphi}_{K}(x)=f^{-1}[\varphi_{K}(x)].$ (9)

Figure 1: (a) The potential of Eq. (10), (b) field configuration Eq. (11), (c)
quantum mechanical potential of Eq. (12) and (d) the corresponding wave
functions of the bound states $\eta_{0}(x)$ (blue solid) and $\eta_{1}(x)$
(red dashed) associated to the $\varphi^{4}$ model.

For example, well known $\varphi^{4}$ model with the potential

$V^{(4)}(\varphi)=\frac{1}{2}(1-\varphi^{2})^{2},$ (10)

has two minima at $\\{-1,1\\}$ (see Fig. 1) and the kink solution (Fig. 1)
which connect these minima are obtained from Eq. (5):

$\varphi_{K}^{(4)}(x)=\tanh x,$ (11)

where the antikink solutions are given by
$\varphi_{\bar{K}}^{(4)}(x)=-\varphi_{K}^{(4)}(x)$. The $K$ and $\bar{K}$
denote kink antikink solutions, respectively. The quantum mechanical potential
(Fig. 1) is

$U^{(4)}_{K}(x)=\frac{d^{2}V^{(4)}}{d\varphi^{2}}|_{\varphi\to\varphi^{(4)}_{K}}=4-6\,{\rm
sech}^{2}(x).$ (12)

Figure 2: (a) The potential in Eq. (13), (b) the kinks (Eq. (14)), (c) quantum
mechanical potential (Eq. (15)) and (d) the corresponding wave function of the
$\varphi^{6}$ model.

The classical mass of the $\varphi^{4}$ kinks is $M^{(4)}_{K}={3}/{4}$, and
they have an internal mode in addition to the zero mode, see Fig. 1. Another
example is the $\varphi^{6}$ model, given by the following potential

$V^{(6)}(\varphi)=\frac{1}{2}\varphi^{2}(1-\varphi^{2})^{2},$ (13)

it has three minima at $\\{-1,0,1\\}$, as we can see in Fig. 2, therefore it
has two kinks and two antikinks. The kink static connecting the minima
$\\{0,1\\}$ is depicted in Fig. 2, given by

$\varphi^{(6)}_{K_{1}}(x)=\sqrt{\frac{1+\tanh x}{2}}.$ (14)

The antikink interpolating between $\\{1,0\\}$ is
$\varphi^{(6)}_{\bar{K}_{1}}(x)=\varphi^{(6)}_{K_{1}}(-x)$. The another kink
and antikink connecting the other topological sector are given by
$\varphi^{(6)}_{K_{2}}(x)=-\varphi^{(6)}_{K_{1}}(-x)$ and
$\varphi^{(6)}_{\bar{K}_{2}}(x)=-\varphi^{(6)}_{K_{1}}(x)$. The quantum
mechanical potential of the $\varphi^{6}$ model is

$U^{(6)}=\frac{d^{2}V^{(6)}}{d\varphi^{2}}\biggr{|}_{\varphi\to\varphi^{(6)}},$
(15)

that is,

$\displaystyle
U^{(6)}_{K_{1}}(x)=\frac{15\tanh^{2}(x)}{4}+\frac{3\tanh(x)}{2}-\frac{5}{4}.$
(16)

In Fig. 2 we depicted the plot of this potential. The $\varphi^{6}$ kinks have
mass $M^{(6)}={1}/{4}$ and they have no internal mode. Moreover, the model has
one zero mode, for each topological sector. The Fig. 2 show the zero mode for
sector $\\{0,1\\}$.

##  III Tanh-deformed $\varphi^{4}$ model

Let us now consider the deformed $\varphi^{4}$ model with a _hyperbolic
tangent_ deformation function. We consider the deformation function
$f(\varphi)=r\tanh{\varphi}$, $r$ real, obeying $r>1$, and we use it to deform
the $\varphi^{4}$ model, the first model studied above. The procedure leads us
to the new potential

$\displaystyle\tilde{V}_{r}^{(4)}(\varphi)=\frac{1}{2r^{2}}\left(\cosh^{2}\varphi-r^{2}\sinh^{2}\varphi\right)^{2}.$
(17)

The Eq. (3) with this potential yields the equation of motion of the model

$\displaystyle\frac{\partial^{2}\varphi}{\partial
t^{2}}-\frac{\partial^{2}\varphi}{\partial
x^{2}}+\frac{\left(r^{2}-1\right)}{r^{2}}\sinh
2\varphi\left(r^{2}\sinh^{2}\varphi-\cosh^{2}\varphi\right)=0.$ (18)

The potential Eq. (17) has two degenerate minima
$\tilde{\varphi}_{\pm}=\pm\tanh^{-1}({1}/{r})$. Therefore, the model has only
one kink and one antikink. The kink configuration appears according to the
Eq.(9), and it has the form

$\displaystyle\tilde{\varphi}^{(4)}_{K}(x)=\tanh^{-1}\left(\frac{1}{r}\tanh(x)\right),$
(19)

and the antikink is obtained by changing $x\to-x$.

Figure 3: (a) The potential in Eq. (17) and (b) the kinks Eq. (19) of the
tanh-deformed $\varphi^{4}$ model for $r=1.0$ (solid black), $r=1.05$ (red
dash), $r=1.5$ (blue dot) and $r=2$ (yellow dot-dash).

Depending on the parameter $r$, the potential Eq. (17) and the kinks in Eq.
(19) look different, as shown in Figs. 3 and 3. The dependence of the
potential center on $r$ is in the form of ${1}/{2r^{2}}$. It is clear that at
$r\to 1$, the center of potential tends towards $1/2$, the minima move away
from each other and the kinklike solution tends to a straight line,
$\tilde{\varphi}^{(4)}_{r\to 1}(x)\to x$. At $r\to\infty$, the center of
potential tends to zero, the minima get closer to each other and therefore the
potential turns into an infinitely narrow square well.

$\displaystyle\tilde{V}_{r}^{(4)}(\varphi=0)\to\begin{cases}\frac{1}{2}\quad
for\quad r\to 1\\\ 0\quad for\quad r\to\infty.\end{cases}$ (20)

The masses of these kinks are

$\displaystyle\tilde{M}_{K}^{(4)}=\frac{1}{r}(r^{2}+1)\coth^{-1}(r)-1,$ (21)

which is plotted as a function of $r$ in Fig. 4. In this model, the $r$
dependent Schoödinger-like potential has the following form

$\displaystyle\tilde{U}^{(4)}_{K}(x)=\frac{(r^{2}-1)^{2}}{r^{2}}\cosh(4{\tilde{\varphi}_{K}^{(4)}(x)})-\frac{r^{4}-1}{r^{2}}\cosh(2{\tilde{\varphi}_{K}^{(4)}(x)}).$
(22)

In the Fig. 4 we display this potential for some values of $r$. For small
values of $r$, the potential has a global maximum at $x=0$ and appear two
minima. This behavior suggests the emergence of internal modes at $r$ very
close to unity. As $r$ increases, we notice a single minimum forming around
$x=0$, with the potential becoming deeper and narrower.

Figure 4: (a) The masses of kink Eq. (21) and (b) quantum mechanical
potentials Eq. (22) of the tanh-deformed $\varphi^{4}$ model for $r=1.0$
(solid black), $r=1.05$ (red dash), $r=1.5$ (blue dot) and $r=2$ (yellow dot-
dash).

Using the Shooting method (see, e.g., Izaac.book.2018 ;
Belendryasova.CNSNS.2019 ) in the Schrödinger-like equation, Eq. (6), the
energy levels of the discrete spectrum in the potential well, Eq. (22), can be
obtained for different values of parameter $r$. The Fig. 5 shows the wave
function solutions for the cases of $r=1.18$ and $r=1.86$. The first case has
three internal modes in addition to the zero mode, $\omega_{1}^{2}=1.5054795$,
$\omega_{2}^{2}=3.160485$ and $\omega_{3}^{2}=3.9957262$, while the second
case has two modes, $\omega_{1}^{2}=2.4915203$ and $\omega_{2}^{2}=3.8953012$.

Figure 5: Wave functions corresponding for the cases of (a) $r=1.18$ and (b)
$r=1.86$.

The parameter $r$ directly affects the depth and width of the potential well,
which is a factor in changing the number of internal modes of the potential.
In the Figs. 6 we show our main result for the number of vibrational modes for
different values of parameter $r$. The closer the $r$ parameter is to one, the
greater the number of internal modes.

Figure 6: The number of vibrational modes for different values of parameter
$r$ of the tanh-deformed $\varphi^{4}$ model.

###  III.1 Numerical results

To investigate another effect of parameter $r$, we consider the kink
scattering. Due to the fact that the kinks of the tanh-deformed $\varphi^{4}$
model are symmetric, the kink-antikink and antikink-kink collisions always
give the same results, as it also happens in the $\varphi^{4}$ model. So we
only investigate the kink-antikink collision.

For the numerical solutions, we solved the equation of motion with the
following initial conditions

$\displaystyle\varphi(x,0)$ $\displaystyle=$
$\displaystyle\tilde{\varphi}^{(4)}_{K}(x+x_{0},v,0)+\tilde{\varphi}^{(4)}_{\bar{K}}(x-x_{0},-v,0)-\tanh^{-1}\bigg{(}\frac{1}{r}\bigg{)},$
(23) $\displaystyle\dot{\varphi}(x,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\varphi}}^{(4)}_{K}(x+x_{0},v,0)+\dot{\tilde{\varphi}}^{(4)}_{\bar{K}}(x-x_{0},-v,0).$
(24)

We fixed $x_{0}=10$ for the initial symmetric position of the pair and set the
grid boundary at $x_{max}=\pm 200$. We use a $4^{th}$ order finite-difference
method with the spatial step $\delta x=0.05$ and $6^{th}$ order sympletic
integrator with time step $\delta t=0.02$. It is important to note that,
$\varphi(x,t)=\varphi(\gamma(x-vt))$, means boost of Lorentz for static kink
and $\gamma=1/\sqrt{1-v^{2}}$.

We illustrate in the Fig. 7 the critical velocity $v_{c}$ as function of the
parameter $r$. The critical velocity is the boundary between two different
regions in terms of the final products of the collision. If the initial
velocity $v$ is larger than $v_{c}$ ($v>v_{c}$), the kink and the antikink
escape to infinity after one collision. For the initial velocity less than the
critical velocity ($v<v_{c}$), the kink and the antikink become trapped after
the collision and create a bound state. In this region, they can escape to
infinity after some bounces.

Based on the correlation between the number and value of the bound states, we
can understand the behavior of critical velocity as follows. The larger the
$\omega$, the smaller the time interval between bounces, which means that the
energy transfer from translational to vibrational mode becomes more difficult
to realize. Particularly, for small values of $r$, the extra $\omega_{3}$
bound mode has a larger value. The presence of this mode contributes to the
suppression of the two-bounce windows - see Fig. 9 \- and its high value
hinders the resonant energy exchange, enlarging the one-bounce region. We
notice that the critical velocity $v_{c}$ reaches a minimum at $r\approx
1.385$, as illustrated in Fig. 7. This region corresponds to the transition
from three to two internal modes, where we detected a change in the critical
velocity. This figure depicts behavior that is quite similar to Refs.
Bazeia.EPJC.2021 ; SimasJHEP . Furthermore, for large values of $r$, the tanh-
deformed $\varphi^{4}$ model has only one vibrational state. For instance, our
numerical investigation revealed that for $r=10$, there is only one internal
mode with $\omega_{1}^{2}=2.9839491$. This value closely resembles the
$\varphi^{4}$ model, where we have $\omega^{2}=3$. As a result, the kink-
antikink scattering process for the tanh-deformed $\varphi^{4}$ model
demonstrates that the critical velocity increases proportionally to $r$. It is
worth noting that the critical velocity for large values of $r$ approaches the
value of the $\varphi^{4}$ model, where it is $v_{c}=0.2598$.

Figure 7: Critical velocity in the kink-antikink collision as function of $r$
of the tanh-deformed $\varphi^{4}$ model.

The effect of $r$ and therefore the effect of internal modes are given in Fig.
8 with a few examples. The initial position of kink and antikink is $x_{0}=10$
for all sub-figures of Fig. 8. The sub-figures on the left hand side have
$r=1.18$ and the sub-figures on the right hand side have $r=1.86$. We chose
the same initial velocities for each pair of sub-figures. So, the initial
velocity is the same in Figs. 8(a) and 8(b), $v=0.1342$, but the final states
are different. Fig. 8(a) shows $3$-bounce collision while Fig. 8(b) shows
$5$-bounce collision. These differences are seen in pairs of Figs. 8(c) and
8(d) as well as in pairs of Figs. 8(e) and 8(f). In Figs. 8(c) and 8(d), the
initial velocity is $v=0.1358$ and the final states are $n$-bounce collision
for $r=1.18$ and $3$-bounce collision for $r=1.86$. The initial velocity is
$v=0.1362$ for Figs. 8(e) and 8(f), and the final states are $2$-bounce
collisions and $4$-bounce collisions respectively.

(a) three-bounce for $v=0.1342$

(b) five-bounce for $v=0.1342$

(c) $n$-bounce for $v=0.1358$

(d) three-bounce for $v=0.1358$

(e) two-bounce for $v=0.1362$

(f) four-bounce for $v=0.1362$

Figure 8: Two kink collisions in the tanh-deformed $\varphi^{4}$ model for
different initial velocities $v$. $r=1.18$ in (a), (c) and (e) panels, and
$r=1.86$ in (b), (d) and (f) panels. We used the initial separation between
centers of kink and antikink as $2x_{0}=20$.

The Fig. 9 depicts the entire structure of the appearance of the two-bounce
windows for $r=1.18$ and $r=1.86$. In this figure, we observe the time of the
first, second and third kink-antikink collision in terms of the initial
velocity $v$. The black and blue curves depict the first and second
collisions, respectively. The appearance of the two-bounce windows is shown by
the divergence in the red curve. In the Fig. 9, for $v<v_{c}=0.144$, we note
the presence of some two-bounce windows and the appearance of smooth peaks,
known as false two-bounce windows. In the case of $v>v_{c}$, we only see an
inelastic collision. Importantly, there are three vibrational states for
$r=1.18$ and, as a result, the amount of extra states hinders the resonant
energy exchange process, shattering the structure of the two-bounce windows.
In comparison to Fig. 9, we notice more two-bounce windows and absence of
false two-bounce windows in Fig. 9. As we already know, only two vibrational
modes are present for the case $r=1.86$. Moreover, since of the number of
bound states influences the process of resonant window suppression, the fewer
the additional states, the more the number of two-bounce windows.

Figure 9: Time to the first (black), second (blue) and third (red) kink-
antikink collisions for tanh-deformed $\varphi^{4}$ model as a function of $v$
for (a) $r=1.18$ and (b) $r=1.86$

The number of vibrational modes also influences how the two-bounce
oscillations are deformed. The behavior of the scalar field at the center of
mass as a function of time is depicted in Fig. 10 and Fig. 10, for the case
$r=1.18$ and $r=1.86$, respectively. In Fig. 10, we note that $\phi(x=0,t)$
oscillates four times $(n=4)$ around the vacuum, corresponding to the expected
second two-bounce windows $(m=2)$. This collision corresponds to a first two-
bounce windows visible in the Fig. 9. The Fig. 10, on the other hand, depicts
a distinct oscillatory pattern between the two-bounce. In particular, the
scalar field oscillates three times around the vacuum, corresponding to $n=3$,
labeled by $m=1$. As result, this collision corresponds to the first two-
bounce windows in the Fig. 9.

The information from the two-bounce windows is presented in Tables 1 and 2,
for $r=1.18$ and $r=1.86$, respectively. The tables are characterized by the
order $m$, initial velocity of the start $v_{1}$ and end $v_{2}$ of each
window, the thickness $\Delta v=v_{2}-v_{1}$ and the scaling parameter $\beta$
anninos . We notice that the thickness of the resonance windows decreases with
increasing window order, which can be seen in Fig. 9. This agrees with a
scaling relation, where the windows width and the number of oscillations to
some power are inversely related $\Delta v\propto M^{-\beta}$ anninos . Table
2 shows a scaling relation with $\beta\sim 3.16\pm 0.05$. However, Table 1
demonstrates that the additional modes have an impact on the scaling relation,
increasing the uncertainty of the $\beta\sim 3.29\pm 0.15$. Notice the absence
of order $m=2$ and $m=5$ windows in Table 1, as well as in Fig. 9.

As illustrated in Fig. 11, we plot the time between bounces as a function of
$n=m+2$, where $m$ corresponds to the windows number. According to
Campbell.PhysD.1983.phi4 , the presence of two-bounce windows obeys the
relation $\omega_{1}T=2\pi m+\delta$, where $\omega_{1}$ is the frequency of
the shape mode, $T$ is the time between bounces and $\delta$ is the phase
shift. We can see from the figure that a straight line can accurately
approximate the numerical points. We notice that the indicated dots reflect
numerical values, whereas the line represents a least squares fitting. The
slope of the line for $r=1.18$ is $5.20$, equivalent to the value
$2\pi/\omega_{1}\approx 5.12$. For $r=1.86$, the slope is $4.07$, and in this
case, we have $2\pi/\omega_{1}\approx 3.98$. The relative errors are,
respectively, $1.5\%$ and $2.2\%$.

Figure 10: Scalar field at the center of mass $\phi(x=0,t)$ as function of time for (a) $v=0.107$ and $r=1.18$ and (b) $v=0.09$ and $r=1.86$. $m$ |  $v_{1}$ |  $v_{2}$ |  $\Delta v$ |  $\beta$
---|---|---|---|---
1 |  0.1050 |  0.1093 |  0.0043 |  -
3 |  0.1324 |  0.1332 |  0.0008 |  3.29
4 |  0.1360 |  0.1364 |  0.0004 |  3.43
6 |  0.1396 |  0.1398 |  0.0002 |  3.13
Table 1: The order $m$ for two-bounce windows, initial velocity $v_{1}$ and $v_{2}$, width $\Delta v$ and scaling $\beta$ for $r=1.18$. $m$ |  $v_{1}$ |  $v_{2}$ |  $\Delta v$ |  $\beta$
---|---|---|---|---
2 |  0.0881 |  0.0953 |  0.0072 |  -
3 |  0.1119 |  0.1155 |  0.0036 |  3.11
4 |  0.1228 |  0.1248 |  0.0020 |  3.16
5 |  0.1288 |  0.13 |  0.0012 |  3.20
6 |  0.1326 |  0.1334 |  0.0008 |  3.17
Table 2: The order $m$ for two-bounce windows, initial velocity $v_{1}$ and
$v_{2}$, width $\Delta v$ and scaling $\beta$ for $r=1.86$.

Figure 11: Time between bounces as a function of $n=m+2$ for (a) $r=1.18$ and
(b) $r=1.86$.

##  IV Tanh-deformed $\varphi^{6}$ model

Let us now investigate the tanh-deformed $\varphi^{6}$ model. Under the same
deformation used above, the potential is

$\displaystyle\tilde{V}_{r}^{(6)}(\varphi)=\frac{1}{2}\tanh^{2}(\varphi)\\!\\!\left(\cosh^{2}(\varphi)-r^{2}\sinh^{2}(\varphi)\right)^{2}.$
(25)

Again, $r$ is a real parameter, obeying $r>1$. The Fig. 12 shows the potential
for some values of $r$, where we realize the presence of two topological
symmetric sectors for $r>1$. The potential has three degenerate minima, at
$0,$ and at $\pm\tanh^{-1}({1}/{r})$. The increase of parameter $r$, reduces
the height of the maxima of the potential and the minima get closer to each
other. The Eq. (3) with this potential yields the equation of motion of the
model, given by

$\displaystyle\frac{\partial^{2}\varphi}{\partial
t^{2}}-\frac{\partial^{2}\varphi}{\partial
x^{2}}+r^{2}\tanh(\varphi)\left(1\\!-\\!\left(r^{2}-1\right)\sinh^{2}(\varphi)\right)\\!\left(\text{sech}^{2}(\varphi)-\frac{\left(r^{2}-1\right)}{r^{2}}\cosh(2\varphi)\\!\right)=0.\;$
(26)

The static solution is given by

$\displaystyle{\tilde{\varphi}}^{(6)}_{K_{1}}(x)=\tanh^{-1}\left(\frac{1}{r}\sqrt{\frac{1+\tanh{x}}{2}}\right),$
(27)

where this kink interpolates between the minima $\\{0,\tanh^{-1}({1}/{r})\\}$.
Similar to the $\varphi^{6}$ model Eq. (13), the antikink connecting the
minima $\\{\tanh^{-1}({1}/{r}),0\\}$ is
$\tilde{\varphi}^{(6)}_{\bar{K}_{1}}(x)=\tilde{\varphi}^{(6)}_{K_{1}}(-x)$.
The kink and antikink connecting the other topological sector are given by
$\tilde{\varphi}^{(6)}_{K_{2}}(x)=-\tilde{\varphi}^{(6)}_{K_{1}}(-x)$ and
$\tilde{\varphi}^{(6)}_{\bar{K}_{2}}(x)=-\tilde{\varphi}^{(6)}_{K_{1}}(x)$.
Therefore, kinks solutions have the same sign, whereas antikinks solutions
have the opposite sign.

Figure 12: (a) The potentials in Eq. (25) and (b) the kinks Eq. (27) of the
tanh-deformed $\varphi^{6}$ model for (a) $r=1.0$ (solid black), (b) $r=1.05$
(red dash), (c) $r=1.5$ (blue dot) and (d) $n=2.0$ (yellow dot-dash).

Figure 13:  (a) The masses of kink Eq. (28) and (b) quantum mechanical
potential in Eq. (29) of the tanh-deformed $\varphi^{6}$ model for (a)
$r=1.005$ (solid black), (b) $r=1.05$ (red dash), (c) $r=1.5$ (blue dot) and
(d) $r=2$ (yellow dot-dash).

We can calculate the mass of these kinks. Since they are the energy of the
solutions, we can write

$\displaystyle\tilde{M}_{K}^{(6)}=\frac{r^{2}}{2}\log\left(\frac{r^{2}}{r^{2}-1}\right)-\frac{1}{2}.$
(28)

Fig. 13 depicts the mass of the kink, and we observe that the mass diminishes
as $r$ increases.

The stability analysis leads to the following quantum mechanical potential

$\displaystyle\tilde{U}^{(6)}_{K}(x)=-r^{4}(\cosh(2\tilde{\varphi}^{(6)})\\!-\\!2)\text{sech}^{4}(\tilde{\varphi}^{(6)})\\!-\\!2r^{2}\left(r^{2}\\!-\\!1\right)\\!\cosh(2\tilde{\varphi}^{(6)})\\!+\\!\left(r^{2}\\!-\\!1\right)^{2}\\!\\!\cosh(4\tilde{\varphi}^{(6)})\\!.\;$
(29)

The Schrödinger-like potential for ${\tilde{\varphi}}^{(6)}_{K_{1}}(x)$ is
displayed in Fig. 13. When $r\rightarrow 1$, the depth of the potential is
reduced, resulting in the development of two minima. This behavior facilitates
the appearance of a vibrational mode. In particular, bound states were
analyzed numerically at this potential for various values of $r$. We can see
that in addition to the zero mode, there is a vibrational mode for $r<1.007$.
As we know, the appearance of an internal mode is a crucial part of the kink
collision process.

###  IV.1 Numerical results

In this section we present our results of kink scattering in tanh-deformed
$\varphi^{6}$ model. The kinks of this model are asymmetric and therefore we
investigate the kink-antikink and antikink-kink collisions. For the numerical
solutions, we solved the equation of motion with the following initial
conditions

$\displaystyle\varphi(x,0)$ $\displaystyle=$
$\displaystyle\tilde{\varphi}^{(6)}_{K_{1}}(x+x_{0},v,0)+\tilde{\varphi}^{(6)}_{\bar{K_{1}}}(x-x_{0},-v,0)-\tanh^{-1}\bigg{(}\frac{1}{r}\bigg{)},$
(30) $\displaystyle\dot{\varphi}(x,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\varphi}}^{(6)}_{K_{1}}(x+x_{0},v,0)+\dot{\tilde{\varphi}}^{(6)}_{\bar{K_{1}}}(x-x_{0},-v,0).$
(31)

We fixed $x_{0}=10$ for the initial symmetric position of the pair and set the
grid boundary at $x_{max}=\pm 200$. We use a $4^{th}$ order finite-difference
method with the spatial step $\delta x=0.05$ and $6^{th}$ order sympletic
integrator with time step $\delta t=0.02$. It is important to note that,
$\varphi(x,t)=\varphi(\gamma(x-vt))$, means boost of Lorentz for static kink
and $\gamma=1/\sqrt{1-v^{2}}$.

It is worthwhile to mention that, unlike in the $\varphi^{6}$ model
Dorey.PRL.2011 , altering the $r$ parameter results in the appearance of
vibrational mode for a single kink or antikink. For instance, we analyzed the
kink-antikink scattering for $r=1.05$; this case has only the zero mode. The
Fig. 14 depicts final states in kink-antikink collision for case $r=1.05$.
Fig. 14(a) shows the scattering of oscillatory waves at $x=0$ after the first
impact, followed by a small radiation. This behavior occurs for $v\leq
v_{c}=0.1801$. However, for $v>v_{c}$, the output is an inelastic collision,
with the scalar field at the center of mass scattering into the other vacuum
of the model, as we can see in Fig. 14(b). The absence of the vibrational mode
in this case results in the disappearance of two-bounce windows, as expected.

(a) n-bounces for $v=0.17$

(b) one-bounce for $v=0.20$

Figure 14: Kink-antikink collision in the tanh-deformed $\varphi^{6}$ model
for different initial velocities $v$. $r=1.05$. We used the initial separation
between centers of kink and antikink as $2x_{0}=20$.

We perform the collision for other values of $r$, for instance, $r=1.00005$.
Numerical analysis of the Schrödinger potential revealed the appearance of an
internal state ($\omega^{2}=0.4607$). The output of the kink-antikink process
for this case can be seen in Fig. 15 and shows different final states for some
initial velocities. For instance, there are creation of oscillating pulses -
Fig. 15(a), as well as the kink-antikink pair trapping - Fig. 15(b).

(a) oscillating pulses for $v=0.19$

(b) $n$-bounce for $v=0.2036$

(c) one-bounce for $v=0.202$

(d) two-bounce for $v=0.2038$

(e) two-bounce for $v=0.2031$

(f) one-bounce for $v=0.2095$

Figure 15: Kink-antikink collision in the tanh-deformed $\varphi^{6}$ model
for different initial velocities $v$. $r=1.00005$. We used the initial
separation between centers of kink and antikink as $2x_{0}=20$.

An interesting behavior is shown in Fig. 15(c). The kink-antikink pair
approaches, collides, oscillates in vacuum
$\tilde{\varphi}_{-}=-\tanh^{-1}(1/r)$, and moves away in the initial vacuum.
In this scenario, the scalar field begins in the vacuum $\tilde{\varphi}_{+}$,
collides, and oscillates in the other topological sector
$\tilde{\varphi}_{-}$. This is referred to as a one-bounce window with two
oscillations. This result was already reported in Refs. Peyrard.PhysD.1983.msG
; Bazeia.PLB.2019 . We can notice the appearance of two-bounce behavior for
$v=0.2038$, as we can see in Fig. 15(d). However, the oscillation happens in
vacuum $\tilde{\varphi}_{-}$, which differs from the standard two-bounce.
After the collisions, we note the scattering of the pair in the vacuum
$\tilde{\varphi}_{+}$. More activity is observed as the initial velocity is
varied. For instance, with $v=0.2031$, the kink-antikink pair collides twice
and scatters to infinity with a phase shift - see Fig. 15(e). For large
initial velocities ($v>v_{c}=0.2094$), emerge only a collision with a phase
shift; after interaction, the kink-antikink pair produces an antikink-kink
pair, as we can see in Fig. 15(f).

(a) bion for $v=0.19$

(b) two-bounce for $v=0.21$

(c) three-bounce for $v=0.215$

(d) one-bounce for $v=0.26$

Figure 16: Antikink-kink collision in the tanh-deformed $\varphi^{6}$ model
for different initial velocities $v$. $r=1.005$. We used the initial
separation between centers of kink and antikink as $2X_{0}=20$.

Finally we consider antikink-kink collision for $r=1.005$. This case has the
critical velocity of $v_{c}=0.2508$ and there is one shape mode. We see the
emergence of bion behaviors for $v\leq v_{c}$. The antikink-kink pair
oscillates around the vacuum $\tilde{\varphi}_{+}$ in this case, emitting
radiation until the pair is completely annihilated; see Fig. 16(a).
Furthermore, we observe the usual fractal behavior of the bounce windows in
particular initial velocities ranges. The Figs. 16(b) and 16(c), for $v=0.21$
and $v=0.215$, show the outcomes for two- and three-bounces, respectively. In
contrast, for $v>v_{c}$, only an inelastic collision is observed. The
antikink-kink pair interacts and scatters to infinity, producing some
radiation in the process. Notice that in this configuration, the pair does not
visit the other topological sector. However, in the kink-antikink scenario,
the scalar field reach the other topological sector during and after the
collision.

We illustrate in the Fig. 17 the critical velocity $v_{c}$ as a function of
$r$ for kink-antikink and antikink-kink collision. The plot for kink-antikink
scattering illustrates the establishment of a minimum followed by an increase
in critical velocity as $r$ increases. In contrast, as the parameter
increases, the antikink-kink scattering displays a diminishing in critical
velocity.

It is important to point out some similarities between the tanh-deformed
$\varphi^{6}$ and $\varphi^{6}$ model. Only the zero mode for a simple kink
(or antikink) exists in the region with large $r$ values. However, two-bounces
windows are found due to the existence of the vibrational mode for the
antikink-kink pair Dorey.PRL.2011 . Furthermore, the critical velocity in the
kink-antikink scattering, which is shown in blue in Fig. 17, indicates that
for high values of $r$, it goes to $v_{c}=0.289$, comparable to $\varphi^{6}$
model. However, if $r$ decreases, the critical velocity reaches a minimum at
$r=1.008$ and then increases. This behavior is due to the fact that for $r$
below $r=1.008$, there appears a new bound state, not present in the
$\varphi^{6}$ model for the kink-antikink pair. In the antikink-kink
collision, on the other hand, the critical velocity diminishes as we increase
$r$; see Fig. 17, red curve. In fact, it goes to $v_{c}=0.0457$, which is
found in the $\varphi^{6}$ model. In the present model, for $1<r<1.5$, there
are several bound states trapped by the antikink-kink pair; in fact, the
number of bound states change from five to four as $r$ goes above $r=1.04$,
but this is not sufficient to chance the monotonic behavior of the critical
velocity in this case.

Figure 17: Critical velocity in the kink-antikink collision (blue) and
antikink-kink collision (red) as function of $r$ of the tanh-deformed
$\varphi^{6}$ model.

The bounce windows structure shown in Fig. 18 reveals the behavior of the
number of bounces $N_{b}$ as a function of initial velocity. Notice that the
thickness of the windows decreases as $v$ increases, for both figures. Fig. 18
for $r=1.00005$ displays an intriguing resonant structure, which appears for
$v<v_{c}$, with the presence of one and two-bounce windows, hard to see in the
related literature. In particular, the Fig. 15(c) and Fig. 15(d) correspond to
the first one-bounce windows and the first two-bounce windows, respectively.
The distinction between the other one-bounce windows is defined by the
increased number of oscillations as $v$ increases. For higher velocities, we
only notice the appearance of one-bounce ($N_{b}=1$). In this case, the kink-
antikink collides only once and produces the antikink-kink pair. This behavior
can be seen in Fig. 15(f). In the Fig. 18 we show the resonance structure for
antikink-kink scattering with $r=1.005$. We may observe the conventional
window structure with two-bounce and bion states for low velocities; this
behavior changes for collisions close to the critical velocity $v_{c}=0.2508$.

Figure 18: Number of bounces $Nb$ as a function of initial velocity $v$ for
(a) kink-antikink collision with $r=1.00005$ and (b) antikink-kink collision
with $r=1.005$.

##  V Conclusion

In this work we have used the deformation procedure to introduce two new
models, the tanh-deformed $\varphi^{4}$ model and the tanh-deformed
$\varphi^{6}$ model. The models contain a real parameter $r>1$, and we have
shown that $r$ controls the potentials, the kinklike solutions, their masses,
the stability potentials and the internal modes associated to the stability of
the systems.

In the case of the tanh-deformed $\varphi^{4}$ model, we have studied the
kink-antikink collision and shown the appearance of several two-bounce
windows. We recognize that the amount of internal modes has an impact on the
two-bounce windows suppression process. For the case with three bound states,
some windows are suppressed, while false two-bounce windows arise. In
contrast, the resonance windows have been fully recovered as the number of
bound states has decreased, as is the case for $r=1.86$. This result motivates
us to further study the collision of kinks in connection with the presence of
fractal structure which was found before in the $\varphi^{4}$ model. A
particularly interesting investigation that could be implemented, is related
to the recent study on the fractal structure and collective coordinate and
moduli space of kink and antikink collisions in the $\varphi^{4}$ model
Manton.PRL.2021 ; Manton.PRD.2021 ; Adam.PRD.2022 . In the case of the tanh-
deformed $\varphi^{4}$ model introduced in the present work, the deformation
requires the parameter $r$, which may perhaps contribute to modify the results
included in Refs. Manton.PRL.2021 ; Manton.PRD.2021 ; Adam.PRD.2022 , bringing
new information on the fractal behavior of the collisions.

For the tanh-deformed $\varphi^{6}$ model, we performed the kink-antikink and
antikink-kink collision for distinct values of parameter $r$. The stability
analysis revealed that lowering $r$ provides a Schrödinger-like potential that
facilitates the emergence of internal modes for a single kink or antikink. The
kink-antikink scattering process revealed a complex structure as a function of
the initial velocity and the parameter $r$, which is associated to the
appearance of the internal mode. As previously stated, the kink-antikink
collision produces oscillating pulses, two-bounce and one-bounce. It is worth
pointing out that there are two types of two-bounce windows, as well as the
one-bounce type scattering. The distinction between the two is in the final
outcome. In one example, the scalar field structures collide twice and scatter
in the same initial vacuum. However, for some ranges of velocities, after the
two interactions, the kink-antikink pair scatters with a shift to the other
vacuum, forming an antikink-kink pair. Our numerical investigation of
antikink-kink scattering revealed the presence of a fractal structure. Unlike
the case of kink-antikink configuration, in the antikink-kink scenario there
is no possibility for the pair to visit the other topological sector after the
interaction. Therefore, the two-bounce behavior of the kink-antikink collision
differs from the antikink-kink one.

We think the above hyperbolic-deformed models can be used to investigate other
issues, in particular, the fractal structure of the collisions, which is
directly influenced by the presence of several bound states. This may find
connection with the recent study on the scattering for the graphene
superlattice equation chaos and also, the approach described in Adam.PRD.2022
, which allows to incorporate relativistic corrections in the scattering,
giving excellent description of kink-antikink collisions in the $\varphi^{4}$
theory, contributing to reproduce the fractal structure in the formation of
the final state. Specifically, the graphene superlattice equation was
considered to describe propagation of solitary electromagnetic waves in a
graphene superlattice and the presence of fractal structure in the kink
scattering is interesting characteristic which deserves further investigation,
including the use of the deformation to describe generalized models. The
hyperbolic-deformed models may also be used to further study issues related to
the presence and unbinding of shape modes, as recently investigated in Refs.
turk ; other . These and other related issues are presently under
consideration, and we hope to report on them in the near future.

## Acknowledgments

AMM would like to thank Islamic Azad University Quchan branch for the grant.
FCS would like to thank Fundação de Amparo à Pesquisa e ao Desenvolvimento
Científico e Tecnológico do Maranhão, FAPEMA, Universal Grant 00920/19. DB
would like to thank Conselho Nacional de Desenvolvimento Científico e
Tecnológico, CNPq, grant No. 303469/2019-6, and Paraiba State Research
Foundation, FAPESQ-PB, grant No. 0015/2019, for partial financial support.

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