Analytical comparison between $X(3)$ and $X(5)$ models of the Bohr Hamiltonian

K.R. Ajulo111E-Mail<EMAIL_ADDRESS>K.J.
Oyewumi222E-Mail<EMAIL_ADDRESS>
1,2University of Ilorin, Ilorin, Nigeria.

###### Abstract

Via the inverse square potential, the solutions of the $X(3)$ model which is a
$\gamma$-rigid form of the $X(5)$ critical point symmetry have been achieved.
The paper presents $X(3)$, through the variational technique, as another
“window” through which the “pictures” of $X(5)\longrightarrow SU(3)$ symmetry
region can be seen. The analytical solutions of the $X(3)$ are compared with
the solutions of the $X(5)$ model. Some new and unique equations connecting
the two models in the: critical order, energy bands, spectra ratios,
$R_{L/2}$, and $B(E2)$ transitional probabilities are presented. These
equations should hold in other potentials with one-parameter such as Kratzer
potential, Davidson potential etc. The spectra ratios and the $B(E2)$
transitional probabilities are optimized via the optimization procedure. The
experimental data of some selected isotopes are placed accordingly for the
theoretical predictions. The deviations from the experiments are found to be
quite small.

Keywords: Bohr Hamiltonian, $X(3)$, $X(5)$, variation technique, optimization
procedure, $\beta$-variable, $\gamma$-rigid.

##  1  Introduction

$X(3)$ which has been presented in [1-4] is said to be an exactly separable
$\gamma$-rigid form of the $X(5)$ critical point symmetry [5]. The $X(3)$
model is defined by the collective coordinate $\beta$ and two Euler angles
since the $\gamma$ is assumed to be zero unlike the case of $X(5)$, where
$\gamma$ is varied around $\gamma^{0}=0$ value in the harmonic oscillator
potential [5]. This implies that, only three variables: $\beta$ and
$\theta_{i}$ are involved in the $X(3)$ model. An exact separation of the
$\beta$ variable from the Euler angles is quite easily achievable. In the Bohr
Hamiltonian model [6-9], $X(5)$ critical point symmetry is one of the two
critical point symmetries: it is a phase transition of the first order shape,
which were originally proposed in the works of Iachello [10], while $E(5)$ is
the phase transition of the second order shape [10]. In the present work, the
nuclei are taken to be $\gamma$-rigid, with the axially symmetric prolate
shape obtained at $\gamma^{0}=0$. The work presents the usefulness of a one-
sided bound inverse square potential with one parameter. The one-parameter
inverse square potential chosen is of the form

$V(\beta)=\begin{cases}\dfrac{\beta_{0}}{\beta^{2}},\text{ if
}0\leq\beta\leq\beta_{0},\\\ \infty,\text{ if }\beta>\beta_{0},\end{cases}$
(1)

where $\beta_{0}$ is a variation parameter that changes the signatures of the
nuclei, as it changes. It is expected that the solutions should shift forward
as the $\beta_{0}$ shifts forward and solutions should shift backward as the
$\beta_{0}$ shifts backward. A typical inverse square potential is bound on
the left and unbound on the right, and it has a minimum at some positive
values of $\beta_{0}$ that forces the particles to infinity as
$\beta_{0}\rightarrow 0$. As a result, the particle’s energy states is one-
sided, with energies escaping through the unbound side. The work is structured
as follows: Section 2. presents the methodology and the solutions of $X(3)$
model via inverse square potential. These solutions are: the wave functions,
the normalization constants and the energy eigenvalues. The $B(E2)$ transition
rates are presented in Section 3. The analytical results, the numerical
results, their applications in certain isotopes are presented and discussed in
Section 4. The work is concluded and summarized in the Section 5.

##  2  Methodology of $X(3)$ model with the inverse square potential

In the $X(3)$ model, the Bohr Hamiltonian operator is written as [1,2]

$\hat{H}=-\dfrac{\hbar^{2}}{2B}\left[\dfrac{1}{\beta^{2}}\dfrac{\partial}{\partial\beta}\beta^{2}\dfrac{\partial}{\partial\beta}+\dfrac{1}{3\beta^{2}}\left(\dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta}\sin\theta\dfrac{\partial}{\partial\theta}+\dfrac{1}{\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\phi^{2}}\right)-V(\beta)\right],$
(2)

where the term,

$\dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta}\sin\theta\dfrac{\partial}{\partial\theta}+\dfrac{1}{\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\phi^{2}},$
(3)

inside the bracket, represents the angular part of the Laplacian [1,2] . $B$,
$\beta$ and $V(\beta)$ are respectively the mass parameter, the collective
coordinate and the $\beta$-dependent potential. The wave equation of the
Eq.(2) is

$\hat{H}\Psi(\beta,\theta,\phi)=E\Psi(\beta,\theta,\phi).$ (4)

By the usual method of separation of variable employed in some quantum texts,

$\Psi(\beta,\theta,\phi)=\chi(\beta)Y_{L,M}(\theta,\phi),$ (5)

where $Y_{L,M}(\theta,\phi)$ is the spherical harmonics and $\chi(\beta)$ is
the radial part of Eq.(4). The separated angular part obtained reads [1,2]

$-\left(\dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta}\sin\theta\dfrac{\partial}{\partial\theta}+\dfrac{1}{\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\phi^{2}}\right)Y_{L,M}(\theta,\phi)=L(L+1)Y_{L,M}(\theta,\phi),$
(6)

where $L$ is the angular momentum quantum number. The simplified form of the
radial part equation [1,2] ,

$\left(\dfrac{1}{\beta^{2}}\dfrac{d}{d\beta}\beta^{2}\dfrac{d}{d\beta}-\dfrac{L(L+1)}{3\beta^{2}}+\dfrac{2B}{\hbar^{2}}[E-V(\beta)]\right)\chi(\beta)=0$
(7)

reads

$\dfrac{d^{2}}{d\beta^{2}}\chi(\beta)+\dfrac{2}{\beta}\dfrac{d}{d\beta}\chi(\beta)-\dfrac{L(L+1)}{3\beta^{2}}\chi(\beta)-[v(\beta)-\epsilon]\chi(\beta)=0,$
(8)

where $\epsilon=\dfrac{2B}{\hbar^{2}}E$ and
$v(\beta)=\dfrac{2B}{\hbar^{2}}V(\beta)$ are the reduced energy and reduced
potential respectively [5].

###  2.1 Determination of the wave functions

By substituting Eq.(1) for $v(\beta)$ in Eq.(8) and solving the simplified
equation using MAPLE software, the eigenfunctions obtained read

$\chi_{s,\nu,L,}(\beta)=\beta^{-1/2}\left[C_{1,L}J_{\nu}(\sqrt{\epsilon}\beta)+C_{2,L}Y_{\nu}(\sqrt{\epsilon}\beta)\right],$
(9)

where $C_{1,L}$ and $C_{2,L}$ are the normalization constants associated with
the Bessel functions of the first kind, $J_{\nu}$, and second kind, $Y_{\nu}$,
respectively. In the domain of Eq.(1), the critical order associated with the
$X(3)$ model in Eq.(9) is

$\nu^{X(3)}=\sqrt{\frac{L}{3}(L+1)+\beta_{0}+\dfrac{1}{4}}.$ (10)

If a boundary condition $\chi_{s,\nu,L}(\beta_{0})=0$ is considered, then
$C_{2,L,n}Y_{\nu}(\sqrt{\epsilon}\beta)$ vanishes and the wave functions
become

$\chi_{s,\nu,L}(\beta)=\beta^{-1/2}\left[C_{1,L}J_{\nu}(\sqrt{\epsilon}\beta)\right].$
(11)

###  2.2 Determination of the energy eigenvalues and the spectral ratio

The procedure for finding the eigenvalues is written in ref. [11]. If the
first condition of the listed procedure is considered, then the acceptable
expression for the energy eigenvalues is written as:

$E_{s,L,n_{\beta}}=\dfrac{\hbar^{2}}{2B}\textit{k}_{s,\nu,n_{\beta}}^{2},\quad\textit{k}_{s,\nu,n_{\beta}}^{2}=\epsilon_{s,\nu,n_{\beta}},\quad\textit{k}_{s,\nu,n_{\beta}}^{2}=\dfrac{\textit{x}_{n_{\beta},s,\nu}}{\beta_{0}},$
(12)

where $s=n_{\beta}+1$, $\textit{x}_{s,\nu,n_{\beta}}$ is the $s$-th zeros of
the Bessel function of order $\nu$. The energy eigenvalues of the $\beta$-part
in $\hbar\omega=1$ unit reads:

$\epsilon_{s,L,n_{\beta}}=2n_{\beta}+1+\nu^{X(3)}=2n_{\beta}+1+\sqrt{\frac{L}{3}(L+1)+\beta_{0}+\dfrac{1}{4}}:\quad
n_{\beta}=0,1,2,...$ (13)

For the $X(3)$ model, the ground state energy levels are defined with $s=1$,
the quasi-$\beta_{1}$ levels are defined with $s=2$ and the quasi-$\beta_{2}$
levels are defined with $s=3$. $L=0,2,4,6...$ There exist no $\gamma$-bands in
the $X(3)$ model because, $\gamma^{0}=0$.

Eq.(13) is the similar to the energy eigenvalues obtained in the $\beta$-part
of $X(5)$ model [12], the difference is observed in their critical orders
where

$\nu^{X(5)}=\sqrt{\frac{L}{3}(L+1)+\beta_{0}+\dfrac{9}{4}}.$ (14)

Since $s=n_{\beta}+1$, $\epsilon_{s,L,n_{\beta}}$ can be reduced to
$\epsilon_{s,L}$, then the spectra ratios can be written as

$R_{L/2}=\dfrac{\epsilon_{s,L}-\epsilon_{1,0}}{\epsilon_{1,2}-\epsilon_{1,0}}.$
(15)

###  2.3 Determination of the normalization constants and the complete wave
functions

The normalization condition for the Hamiltonian operator in Eq.(2) is written
as [1,2]

$\int_{0}^{\beta_{0}}\beta^{2}\mid\chi_{s,\nu,L,n_{\beta}}(\beta)\mid^{2}d\beta=1,$
(16)

such that

$\mid\chi_{s,\nu,L,n_{\beta}}(\beta)\mid^{2}\rightarrow
0\quad\mbox{for}\quad\beta\rightarrow
0;\quad\mid\chi_{s,\nu,L,n_{\beta}}(\beta)\mid^{2}\beta^{2}\rightarrow
0\quad\mbox{for}\quad\beta\rightarrow\infty.$ (17)

If these conditions are satisfied, then
$\int_{0}^{\beta_{0}}\beta^{2}\mid\chi_{s,\nu,L,n_{\beta}}(\beta)\mid^{2}d\beta<\beta_{0}$.

Using the identity [13-14]

$J_{\nu}(\sqrt{\epsilon}\beta)J_{\nu}(\sqrt{\epsilon}\beta)=\sum_{n_{\beta}=0}^{\infty}\dfrac{\left(\dfrac{1}{2}\sqrt{\epsilon}\beta\right)^{2\nu+2n_{\beta}}(2\nu+n_{\beta}+1)_{n_{\beta}}}{n_{\beta}![\Gamma(\nu+n_{\beta}+1)]^{2}}$
(18)

in Eq.(16), the simplified normalization constants read

$C_{1,L,n_{\beta}}=\left[\sum_{n_{\beta}=0,1,2,3...}\dfrac{(\eta)_{n_{\beta}}\left(\dfrac{\textit{k}_{s,\nu,n_{\beta}}}{2}\right)^{\xi-2}\beta_{0}^{(\xi)}}{n_{\beta}!\quad\xi\quad\left[\Gamma\left(\dfrac{\xi}{2}\right)\right]^{2}}\right]^{-1/2},$
(19)

where

$\xi=2\nu+2n_{\beta}+2,\quad\eta=2\nu+n_{\beta}+1\quad\mbox{and}\quad(\eta)_{n_{\beta}}=\eta(\eta+1)(\eta+2)...(\eta+n_{\beta}-1),$
(20)

with $(\eta)_{0}=1$. Hence, Eq.(11) becomes

$\chi_{s,\nu,L,n_{\beta}}(\beta)=\left[\sum_{n_{\beta}=0,1,2,3...}\dfrac{(\eta)_{n_{\beta}}\left(\dfrac{\textit{k}_{s,\nu,n_{\beta}}}{2}\right)^{\xi-2}\beta_{0}^{(\xi)}}{n_{\beta}!\quad\xi\quad\left[\Gamma\left(\dfrac{\xi}{2}\right)\right]^{2}}\right]^{-1/2}\beta^{-1/2}J_{\nu}(\sqrt{\epsilon}\beta).$
(21)

##  3 $B(E2)$ transition rates

The electric quadrupole operator is written as [1,2]

$T^{E2}_{\mu}=t\beta\left[D_{\mu,0}^{(2)}(\theta_{i})\cos\gamma+\dfrac{1}{\sqrt{2}}\left(D_{\mu,2}^{(2)}(\theta_{i})+D_{\mu,-2}^{(2)}(\theta_{i})\right)\sin\gamma\right],$
(22)

where $D(\theta_{i})$ are the Wigner functions of the Euler angle and $t$ is
known as a scale factor. For $\gamma^{0}=0$,

$T^{E2}_{\mu}=t\beta\sqrt{\dfrac{4\pi}{5}}Y_{2\mu}(\theta,\phi).$ (23)

The $B(E2)$ [1,2,5,15] is written as

$B(E2;sL\longrightarrow s^{\prime}L^{\prime})=\dfrac{1}{2sL+1}|\left\langle
s^{\prime}L^{\prime}||T^{E2}||sL\right\rangle|^{2},$ (24)
$\quad\qquad\quad\quad\quad\quad\quad\quad=\dfrac{2s^{\prime}L^{\prime}+1}{2sL+1}B(E2;s^{\prime}L^{\prime}\longrightarrow
sL).$ (25)

Eq.(24) or Eq.(25) has been solved in ref. [1] as:

$B(E2;sL\longrightarrow
s^{\prime}L^{\prime})=t^{2}\left(C_{L0,20}^{L^{\prime}0}\right)^{2}I^{2}_{sL;s^{\prime}L^{\prime}},$
(26)

where the coefficients, $C_{L0,20}^{L^{\prime}0}$ are the Clebsch-Gordan
coefficients, and

$I_{sL;s^{\prime}L^{\prime}}=\int_{0}^{\beta_{0}}\beta\chi_{s,\nu,L,n_{\beta}}(\beta)\chi_{s^{\prime},\nu^{\prime},L^{\prime},n^{\prime}_{\beta}}(\beta)\beta^{2}d\beta,$
(27)

are the integrals over $\beta$.

##  4 Numerical results, analytical results, applications and discussion

Table 1: The comparison in the critical order, $\nu$, of the $X(5)$ [12], with the $\nu$ of Eq.(10). |  |  |  |  |  | $\nu(L)$ |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
$L$ | $\beta_{0}=0$ |  | $\beta_{0}=2$ |  | $\beta_{0}=4$ |  | $\beta_{0}=6$ |  | $\beta_{0}=102$ | $\beta_{0}=100$
| $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$
0 | 0.500 | 1.500 | 1.500 | 2.062 | 2.062 | 2.500 | 2.500 | 2.062 | 10.112 | 10.112
2 | 1.500 | 2.062 | 2.062 | 2.500 | 2.500 | 2.872 | 2.872 | 2.500 | 10.210 | 10.210
4 | 2.630 | 2.986 | 2.986 | 3.304 | 3.304 | 3.594 | 3.594 | 3.304 | 10.436 | 10.436
6 | 3.775 | 4.031 | 4.031 | 4.272 | 4.272 | 4.500 | 4.500 | 4.272 | 10.782 | 10.782
8 | 4.924 | 5.123 | 5.123 | 5.315 | 5.315 | 5.500 | 5.500 | 5.315 | 11.236 | 11.236
10 | 6.076 | 6.238 | 6.238 | 6.397 | 6.397 | 6.551 | 6.551 | 6.397 | 11.786 | 11.786
| $\beta_{0}=1$ |  | $\beta_{0}=3$ |  | $\beta_{0}=5$ |  | $\beta_{0}=7$ |  | $\beta_{0}=101$ | $\beta_{0}=103$
0 | 1.118 | 1.803 | 1.803 | 2.291 | 2.291 | 2.693 | 2.693 | 3.041 | 10.062 | 10.259
2 | 1.803 | 2.291 | 2.291 | 2.693 | 2.693 | 3.041 | 3.041 | 3.354 | 10.161 | 10.356
4 | 2.814 | 3.149 | 3.149 | 3.452 | 3.452 | 3.731 | 3.731 | 3.990 | 10.388 | 10.579
6 | 3.905 | 4.153 | 4.153 | 4.387 | 4.387 | 4.610 | 4.610 | 4.823 | 10.735 | 10.920
8 | 5.025 | 5.220 | 5.220 | 5.408 | 5.408 | 5.590 | 5.590 | 5.766 | 11.191 | 11.369
10 | 6.158 | 6.318 | 6.318 | 6.474 | 6.474 | 6.627 | 6.627 | 6.776 | 11.747 | 11.913
Table 2: Ground state energies, the energies of the quasi-$\beta_{1}$ and the quasi-$\beta_{2}$ denoted by $n_{\beta}=0,s=1$; $n_{\beta}=1,s=2$; and $n_{\beta}=2,s=3$ respectively for the $X(3)$ and $X(5)$ symmetry [12] in $\hbar\omega=1$ unit. | $\beta_{0}=2$ |  | $\beta_{0}=3$ |  | $\beta_{0}=4$ |  | $\beta_{0}=15$ |
---|---|---|---|---|---|---|---|---
$L$ |  |  |  |  | $n_{\beta}=0;$ |  |  |
|  |  |  |  | $s=1$ |  |  |
| $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$
0 | 2.500 | 2.031 | 2.803 | 2.146 | 3.062 | 2.250 | 4.905 | 3.077
2 | 3.062 | 2.250 | 3.291 | 2.346 | 3.500 | 2.436 | 5.153 | 3.194
4 | 3.986 | 2.652 | 4.149 | 2.726 | 4.304 | 2.797 | 5.682 | 3.445
6 | 5.031 | 3.136 | 5.153 | 3.194 | 5.272 | 3.250 | 6.408 | 3.795
8 | 6.123 | 3.658 | 6.220 | 3.704 | 6.315 | 3.750 | 7.265 | 4.212
10 | 7.238 | 4.198 | 7.318 | 4.237 | 7.397 | 4.276 | 8.205 | 4.671
12 | 8.365 | 4.750 | 8.433 | 4.783 | 8.500 | 4.816 | 9.201 | 5.161
14 | 9.500 | 5.308 | 9.559 | 5.337 | 9.617 | 5.366 | 10.233 | 5.670
|  |  |  |  | $n_{\beta}=1;$ |  |  |
|  |  |  |  | $s=2$ |  |  |
0 | 4.500 | 4.031 | 4.803 | 4.146 | 5.062 | 4.250 | 6.905 | 5.077
2 | 5.062 | 4.250 | 5.291 | 4.346 | 5.500 | 4.436 | 7.153 | 5.194
4 | 5.986 | 4.652 | 6.149 | 4.726 | 6.304 | 4.797 | 7.682 | 5.445
6 | 7.031 | 5.136 | 7.153 | 5.194 | 7.272 | 5.250 | 8.408 | 5.795
8 | 8.123 | 5.658 | 8.220 | 5.704 | 8.315 | 5.750 | 9.265 | 6.211
10 | 9.238 | 6.198 | 9.318 | 6.237 | 9.397 | 6.276 | 10.205 | 6.671
12 | 10.365 | 6.750 | 10.433 | 6.783 | 10.500 | 6.816 | 11.201 | 7.161
14 | 11.500 | 7.308 | 11.559 | 7.337 | 11.617 | 7.366 | 12.233 | 7.670
|  |  |  |  | $n_{\beta}=2;$ |  |  |
|  |  |  |  | $s=3$ |  |  |
0 | 6.500 | 6.031 | 6.803 | 6.146 | 7.062 | 6.250 | 8.905 | 7.077
2 | 7.062 | 6.250 | 7.291 | 6.346 | 7.500 | 6.436 | 9.153 | 7.194
4 | 7.986 | 6.652 | 8.149 | 6.726 | 8.304 | 6.797 | 9.682 | 7.445
6 | 9.031 | 7.136 | 9.153 | 7.194 | 9.272 | 7.250 | 10.408 | 7.795
8 | 10.123 | 7.658 | 10.220 | 7.704 | 10.315 | 7.750 | 11.265 | 8.211
10 | 11.238 | 8.198 | 11.318 | 8.237 | 11.397 | 8.276 | 12.205 | 8.671
12 | 12.365 | 8.750 | 12.433 | 8.783 | 12.500 | 8.816 | 13.201 | 9.161
14 | 13.500 | 9.308 | 13.559 | 9.337 | 13.617 | 9.366 | 14.233 | 9.670
Figure 1: (a) Comparison in the energy levels of the $X(3)$ and $X(5)$ models [15] at $\beta_{0}=2$ from the $gsb$ up to the quasi-$\beta_{2}$ band. (b): the variation of the critical order, $\nu$, of the $X(5)$ as a function of $\beta_{0}$, is compared with $\nu$ of the $X(3)$ at constant angular momenta, $L=0,2$ and $L=4$. Figure 2: (a) The plots showing the values of $\beta_{0}$ at which energies are minimum. (b) The rate of energy with respect to $\beta_{0}$, showing non stationary property of $\beta_{0}$. Table 3: Comparison of the ground state spectra ratios, defined in Eq.(15), of the inverse square potential in the $X(3)$ model at different values of the $\beta_{0}$, compared with the $X(5)$ [12]. It can be seen that $X(3)(\beta_{0}=\infty)\approx X(5)(\beta_{0}=\infty)$. $L_{s,n_{\beta}}$ | $\beta_{0}=0$ | $\beta_{0}=0$ | $\beta_{0}=2$ | $\beta_{0}=2$ | $\beta_{0}=4$ | $\beta_{0}=4$ | $\beta_{0}=\infty$ | $\beta_{0}=\infty$ |
---|---|---|---|---|---|---|---|---|---
| $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ | $X(3)$ | $X(5)$ |
$gsb$ |  |  |  |  |  |  |  |  |
$0_{1,0}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
$2_{1,0}$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
$4_{1,0}$ | 2.130 | 2.646 | 2.646 | 2.834 | 2.834 | 2.938 | 3.296 | 3.296 |
$6_{1,0}$ | 3.275 | 4.507 | 4.507 | 5.042 | 5.042 | 5.372 | 6.806 | 6.808 |
$8_{1,0}$ | 4.424 | 6.453 | 6.453 | 7.421 | 7.421 | 8.508 | 11.413 | 11.423 |
$10_{1,0}$ | 5.576 | 8.438 | 8.438 | 9.887 | 9.887 | 11.881 | 16.991 | 17.013 |
$12_{1,0}$ | 6.728 | 10.445 | 10.445 | 12.404 | 12.404 | 15.686 | 23.409 | 23.450 |
$14_{1,0}$ | 7.882 | 12.465 | 12.465 | 14.951 | 14.951 | 19.740 | 30.544 | 30.611 |
| $\beta_{0}=1$ | $\beta_{0}=1$ | $\beta_{0}=3$ | $\beta_{0}=3$ | $\beta_{0}=5$ | $\beta_{0}=5$ | $\beta_{0}=15$ | $\beta_{0}=15$ |
$0_{1,0}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
$2_{1,0}$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
$4_{1,0}$ | 2.476 | 2.756 | 2.756 | 2.893 | 2.893 | 2.946 | 3.128 | 3.148 |
$6_{1,0}$ | 4.070 | 4.812 | 4.812 | 5.224 | 5.224 | 5.529 | 6.058 | 6.136 |
$8_{1,0}$ | 5.706 | 6.995 | 6.995 | 7.767 | 7.767 | 8.638 | 9.508 | 9.690 |
$10_{1,0}$ | 7.360 | 9.243 | 9.243 | 10.424 | 10.424 | 11.915 | 13.297 | 13.620 |
$12_{1,0}$ | 9.024 | 11.525 | 11.525 | 13.145 | 13.145 | 15.854 | 17.307 | 17.800 |
$14_{1,0}$ | 10.694 | 13.829 | 13.829 | 15.907 | 15.907 | 19.899 | 21.468 | 22.152 |
Table 4: $R_{L/2}$ ratios, defined in Eq.(15), of the quasi-$\beta_{1}$ and quasi-$\beta_{2}$ bands of the inverse square potential in the $X(3)$ model at different values of the $\beta_{0}$. $L_{s,n_{\beta}}$ | $\beta_{0}=0$ | $\beta_{0}=1$ | $\beta_{0}=2$ | $\beta_{0}=3$ | $\beta_{0}=4$ | $\beta_{0}=15$ | $\beta_{0}=\infty$
---|---|---|---|---|---|---|---
quasi-$\beta_{1}$ |  |  |  |  |  |  |
$0_{2,1}$ | 2.000 | 2.921 | 3.562 | 4.094 | 4.562 | 8.058 | 20.124
$2_{2,1}$ | 3.000 | 3.921 | 4.562 | 5.094 | 5.562 | 9.058 | 21.124
$4_{2,1}$ | 4.130 | 5.397 | 6.208 | 6.850 | 7.395 | 11.187 | 23.420
$6_{2,1}$ | 5.275 | 6.991 | 8.069 | 8.906 | 9.603 | 14.115 | 26.929
$8_{2,1}$ | 6.424 | 8.626 | 10.014 | 11.090 | 11.982 | 17.567 | 31.537
$10_{2,1}$ | 7.576 | 10.281 | 11.999 | 13.337 | 14.449 | 21.356 | 37.116
$12_{2,1}$ | 8.728 | 11.945 | 14.007 | 15.619 | 16.965 | 25.366 | 43.534
$14_{2,1}$ | 9.882 | 13.615 | 16.027 | 17.923 | 19.513 | 29.526 | 50.668
quasi-$\beta_{2}$ |  |  |  |  |  |  |
$0_{3,2}$ | 4.000 | 5.842 | 7.123 | 8.188 | 9.123 | 16.117 | 40.249
$2_{3,2}$ | 5.000 | 6.842 | 8.123 | 9.188 | 10.123 | 17.117 | 41.249
$4_{3,2}$ | 6.130 | 8.318 | 9.769 | 10.944 | 11.957 | 19.245 | 43.545
$6_{3,2}$ | 7.275 | 9.912 | 11.630 | 13.000 | 14.165 | 22.174 | 47.054
$8_{3,2}$ | 8.424 | 11.547 | 13.576 | 15.184 | 16.544 | 25.625 | 51.662
$10_{3,2}$ | 9.576 | 13.201 | 15.561 | 17.431 | 19.010 | 29.414 | 57.240
$12_{3,2}$ | 10.728 | 14.866 | 17.568 | 19.713 | 21.527 | 33.424 | 63.658
$14_{3,2}$ | 11.882 | 16.536 | 19.589 | 22.018 | 24.074 | 37.584 | 70.792
Figure 3:  The $\dfrac{\epsilon_{s,L}}{\epsilon_{1,2}}$ of the $X(3)$. (b): the $\dfrac{\epsilon_{s,L}}{\epsilon_{1,2}}$ of the $X(5)$ both at constant angular momenta, $L=0...10$ are plotted against the variation parameter, $\beta_{0}$. Figure 4: (a) The comparison in the $R_{4/2}$ of the $X(3)$ and $X(5)$ for $\beta_{0}=\infty$ and for different values of the $\beta_{0,max}$ labelled as $X(3)-$var and $X(5)-$var respectively, peculiar to each angular momentum. (b): the comparison in the $R_{0/2}$ of the $X(3)$ and $X(5)$ for $\beta_{0}=\infty$ and for different values of the $\beta_{0,max}$ labelled as $X(3)-$var and $X(5)-$var respectively, peculiar to each angular momentum.  Figure 5: (a) and (b) The visual plots of the potentials correspond to $R_{4/2}$ and $R_{0/2}$ respectively. The values of $\beta_{0}$ used correspond to $X(3)$-var and $X(5)$-var in the $gsb$ and quasi-$\beta_{1}$ bands. Figure 6:  (a) and (b) present the $R_{L/2}$ ratios for the ground state and the quasi-$\beta_{1}$ bands of the $X(3)$ model of inverse square potential respectively, at different values of $\beta_{0}$ compared with $X(3)$-IW and 162Dy. (c): the $R_{L/2}$ ratios for the quasi-$\beta_{2}$ bands of the $X(3)$ model of inverse square potential at different values of $\beta_{0}$ compared with $X(3)$-IW [1]. It appears that the $gsb$ solutions of $X(3)$ at $\beta_{0}=\infty$ lie on the experimental data of 162Dy, which is a typical $SU(3)$ candidate. The available data on the first exited state lie very close to one another. Figure 7:  (a) and (b) present the $R_{L/2}$ ratios for the ground state and the quasi-$\beta_{1}$ bands of the $X(3)$ and the $X(5)$ models of inverse square potentials respectively, obtained at different values of $\beta_{0,max}$, labeled $X(3)$-var and $X(5)$-var, are compared with the 172-180Os chain. Figure 8: The Neutron-$\beta_{0}$ distribution is employed to show the relative positions of 104-108Ru, 120-126Xe, 184-188Pt and 172-180Os along their common chain. Figure 9: The $B(E2)$ transition rates of the $X(3)$ normalized to the $B(E2:2_{1,0}\rightarrow 0_{1,0})=100$ units within: (a) the ground state bands at $\beta_{0}=0,1,2,\infty$ and $B(E2)$-var compared with the $X(3)$-IW [1], $X(5)$ experimental data [34] and 158Gd [35], which is a typical $SU(3)$ candidate. (b): the $\beta_{1}$ state bands at $\beta_{0}=0,1,2,\infty$ and $B(E2)$-var compared with the $X(3)$-IW [1] and 158Gd. (c): the $\beta_{2}$ state bands at $\beta_{0}=0,1,2,\infty$ and $B(E2)$-var compared with the $X(3)$-IW [1]. [Note:-IW denotes infinite well potential.] Table 5: The $R_{L/2}$ ratios, defined in Eq.(15), for the $X(3)$ version of inverse square potential, labelled $X(3)$-var, calculated at different values of $\beta_{0,max}$, are compared with the $X(3)$-IW [1]. [Note: IW denotes infinite well potential]. $L_{s,n_{\beta}}$ | $\beta_{0,max}$ |  $X(3)$-var |  $X(3)$-IW
---|---|---|---
$gsb$ |  |  |
$0_{1,0}$ | $\beta_{0}$ | 0.000 | 0.000
$2_{1,0}$ | $\beta_{0}$ | 1.000 | 1.000
$4_{1,0}$ | 0.844 | 2.440 | 2.440
$6_{1,0}$ | 1.576 | 4.244 | 4.230
$8_{1,0}$ | 2.033 | 6.383 | 6.350
$10_{1,0}$ | 2.143 | 8.666 | 8.780
$12_{1,0}$ | 2.695 | 11.421 | 11.520
$14_{1,0}$ | 3.643 | 14.573 | 14.570
quasi-$\beta_{1}$ |  |  |
$0_{2,1}$ | 0.815 | 2.703 | 2.870
$2_{2,1}$ | 2.101 | 4.619 | 4.830
$4_{2,1}$ | 3.729 | 7.255 | 7.370
$6_{2,1}$ | 5.213 | 10.327 | 10.290
$8_{2,1}$ | 6.098 | 13.493 | 13.570
$10_{2,1}$ | 6.855 | 16.908 | 17.180
$12_{2,1}$ | 8.106 | 21.009 | 21.140
quasi-$\beta_{2}$ |  |  |
$0_{3,2}$ | 2.524 | 7.701 | 7.650
$2_{3,2}$ | 4.497 | 10.553 | 10.560
$4_{3,2}$ | 6.523 | 14.088 | 14.190
$6_{3,2}$ | 8.567 | 18.172 | 18.220
$8_{3,2}$ | 10.438 | 22.613 | 22.620
$10_{3,2}$ | 11.932 | 25.999 | -
$12_{3,2}$ | 13.011 | 28.928 | -
Table 6: The spectra ratios for the $X(3)$ version of inverse square potential are compared with the experimental data. The values of the $\beta_{0}$ and the quality factor, $\sigma$, used during the fittings are recorded. $L_{s,n_{\beta}}$ |  102Mo |  102Mo |  104Ru |  104Ru |  106Ru |  106Ru |  108Ru |  108Ru |  120Xe |  120Xe |  122Xe |  122Xe |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Exp | Theor | Exp | Theor | Exp | Theor | Exp | Theor | Exp | Theor | Exp |  |
$gsb$ |  |  |  |  |  |  |  |  |  |  |  |  |
$4_{1,0}$ | 2.510 | 2.566 | 2.480 | 2.468 | 2.660 | 2.662 | 2.750 | 2.623 | 2.470 | 2.522 | 2.500 | 2.571 |
$6_{1,0}$ | 4.480 | 4.483 | 4.350 | 4.299 | 4.800 | 4.805 | 5.120 | 5.102 | 4.330 | 4.263 | 4.430 | 4.500 |
$8_{1,0}$ | 6.810 | 6.703 | 6.480 | 6.488 | 7.310 | 7.199 | 8.020 | 7.999 | 6.510 | 6.361 | 6.690 | 6.759 |
$10_{1,0}$ | 9.410 | 8.989 | 8.690 | 8.702 | 10.020 | 9.920 | 11.310 | 11.495 | 8.900 | 8.497 | 9.180 | 9.216 |
$12_{1,0}$ | - | 11.498 | - | 11.878 | - | 12.334 | - | 12.879 | - | 10.362 | - | 11.985 |
$14_{1,0}$ | - | 13.302 | - | 14.522 | - | 15.073 | - | 15.591 | - | 12.640 | - | 14.759 |
$\beta_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$0_{2,1}$ | 2.350 | 3.009 | - | 2.569 | 3.670 | 3.678 | - | 4.462 | 2.820 | 3.000 | 3.470 | 3.492 |
$2_{2,1}$ | 3.860 | 4.301 | 4.230 | 4.233 | - | 5.127 | - | 5.902 | 3.950 | 4.203 | 4.510 | 4.608 |
$4_{2,1}$ | - | 6.289 | 5.810 | 5.921 | - | 7.443 | - | 8.285 | 5.310 | 5.899 | - | 6.792 |
$6_{2,1}$ | - | 8.691 | - | 8.009 | - | 10.001 | - | 11.099 | - | 7.958 | - | 9.172 |
$8_{2,1}$ | - | 11.319 | - | 11.101 | - | 12.519 | - | 14.532 | - | 10.739 | - | 11.829 |
$\beta_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$0_{3,2}$ | - | 7.437 | - | 6.589 | - | 8.388 | - | 10.099 | - | 6.664 | - | 7.722 |
$2_{3,2}$ | - | 9.000 | - | 8.202 | - | 10.200 | - | 11.811 | - | 8.007 | - | 9.402 |
$4_{3,2}$ | - | 11.009 | - | 10.341 | - | 12.049 | - | 12.972 | - | 10.229 | - | 11.538 |
$\beta_{0}$ |  | 1.464 |  | 0.963 | - | 2.121 |  | 1.830 |  | 1.221 |  | 1.494 |
$\sigma$ |  | 0.569 |  | 0.310 | - | 0.422 |  | 0.276 |  | 0.481 |  | 0.295 |
|  124Xe |  124Xe |  126Xe |  126Xe |  148Nd |  148Nd |  184Pt |  184Pt |  186Pt |  186Pt |  188Pt |  188Pt |
| Exp | Theor | Exp | Theor | Exp | Theor | Exp | Theor | Exp | Theor | Exp | Theor |
$gsb$ |  |  |  |  |  |  |  |  |  |  |  |  |
$4_{1,0}$ | 2.480 | 2.498 | 2.420 | 2.463 | 2.490 | 2.500 | 2.670 | 2.721 | 2.560 | 2.567 | 2.530 | 2.890 |
$6_{1,0}$ | 4.370 | 4.418 | 4.210 | 4.291 | 4.240 | 4.187 | 4.900 | 5.001 | 4.580 | 4.399 | 4.460 | 4.503 |
$8_{1,0}$ | 6.580 | 6.596 | 6.270 | 6.325 | 6.150 | 6.007 | 7.550 | 7.679 | 7.010 | 6.768 | 6.710 | 6.666 |
$10_{1,0}$ | 8.960 | 8.726 | 8.640 | 8.382 | 8.190 | 7.986 | 10.470 | 10.585 | 9.700 | 8.863 | 9.180 | 8.969 |
$12_{1,0}$ | - | 11.998 | - | 10.479 | - | 9.769 | - | 12.807 | - | 11.999 | - | 11.378 |
$14_{1,0}$ | - | 15.079 | - | 12.681 | - | 11.246 | - | 15.367 | - | 14.875 | - | 14.242 |
$\beta_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$0_{2,1}$ | 3.580 | 3.402 | 3.380 | 3.201 | 3.040 | 3.082 | 3.020 | 3.546 | 2.460 | 2.798 | 3.010 | 3.209 |
$2_{2,1}$ | 4.600 | 4.498 | 4.320 | 4.241 | 3.880 | 4.006 | 5.180 | 5.452 | 4.170 | 4.381 | 4.200 | 4.397 |
$4_{2,1}$ | 5.690 | 6.289 | 5.250 | 5.862 | 5.320 | 5.589 | 7.570 | 7.943 | 6.380 | 6.599 | - | 6.583 |
$6_{2,1}$ | - | 8.564 | - | 7.828 | 7.120 | 7.411 | 11.040 | 11.157 | 8.360 | 8.581 | - | 8.603 |
$8_{2,1}$ | - | 11.900 | - | 10.062 | - | 9.752 | - | 14.601 | - | 12.007 | - | 12.100 |
$\beta_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |
$0_{3,2}$ | - | 7.334 | - | 6.759 | - | 6.249 | - | 10.122 | - | 7.389 | - | 7.603 |
$2_{3,2}$ | - | 8.888 | - | 8.009 | - | 7.442 | - | 11.900 | - | 8.942 | - | 9.162 |
$4_{3,2}$ | - | 10.022 | - | 9.287 | - | 9.051 | - | 12.998 | - | 10.208 | - | 10.441 |
$\beta_{0}$ |  | 1.101 | - | 0.949 |  | 1.111 |  | 2.639 |  | 1.469 |  | 4.950 |
$\sigma$ |  | 0.279 | - | 0.299 |  | 0.347 |  | 0.475 |  | 0.600 |  | 0.729 |
Table 7:  The $B(E2)$ transition rates of the $X(3)$ model at $\beta_{0}=0,1,2,\infty$ and its values obtained at $\beta_{0,max}$ peculiar to each angular momentum, normalized to the $B(E2;2_{1,0}\rightarrow 0_{1,0})=100$ units are compared with the $X(3)$-IW model [1] and with the experimental data of $X(5)$ [34]. [Note: -IW denotes infinite well potential.] $L_{s,n_{\beta}}^{(i)}$ | $L_{s,n_{\beta}}^{(f)}$ | $\beta_{0}=0$ | $\beta_{0}=1$ | $\beta_{0}=2$ | $\beta_{0}=\infty$ | $\beta_{0,max}^{(i)}\rightarrow\beta_{0,max}^{(f)}$ | $B(E2)-var$ |  $X(3)$-IW |  176Os-Exp
---|---|---|---|---|---|---|---|---|---
$2_{1,0}$ | $0_{1,0}$ | 100.000 | 100.000 | 100.000 | 100.000 | $\beta_{0}\rightarrow\beta_{0}$ | 100.000 | 100.00 | 100.00
$4_{1,0}$ | $2_{1,0}$ | 237.513 | 190.935 | 178.005 | 143.992 | $0.844\rightarrow\beta_{0}$ | 189.495 | 189.90 | 193.00
$6_{1,0}$ | $4_{1,0}$ | 380.702 | 286.006 | 270.996 | 167.292 | $1.576\rightarrow 0.844$ | 250.995 | 248.90 | 267.00
$8_{1,0}$ | $6_{1,0}$ | 523.695 | 384.599 | 369.090 | 185.328 | $2.033\rightarrow 1.576$ | 293.038 | 291.40 | 297.00
$10_{1,0}$ | $8_{1,0}$ | 667.003 | 486.036 | 469.991 | 202.099 | $2.143\rightarrow 2.033$ | 324.599 | 323.80 | 352.50
$12_{1,0}$ | $10_{1,0}$ | 810.954 | 587.658 | 559.744 | 229.986 | $2.695\rightarrow 2.143$ | 350.710 | 349.50 | -
$14_{1,0}$ | $12_{1,0}$ | 954.746 | 690.364 | 671.484 | 253.007 | $3.643\rightarrow 2.695$ | 371.992 | 370.70 | -
$2_{2,1}$ | $0_{2,1}$ | 166.813 | 160.292 | 152.428 | 69.929 | $2.011\rightarrow 0.815$ | 78.922 | 80.60 | -
$4_{2,1}$ | $2_{2,1}$ | 320.619 | 260.000 | 243.186 | 123.888 | $3.729\rightarrow 2.101$ | 139.471 | 140.10 | -
$6_{2,1}$ | $4_{2,1}$ | 470.001 | 355.240 | 343.376 | 156.031 | $5.213\rightarrow 3.729$ | 181.730 | 182.40 | -
$8_{2,1}$ | $6_{2,1}$ | 617.075 | 456.792 | 439.962 | 189.542 | $6.098\rightarrow 5.213$ | 213.899 | 215.50 | -
$10_{2,1}$ | $8_{2,1}$ | 763.927 | 557.317 | 542.129 | 215.763 | $6.855\rightarrow 6.098$ | 242.026 | 242.40 | -
$12_{2,1}$ | $10_{2,1}$ | 899.983 | 656.999 | 639.677 | 233.499 | $8.106\rightarrow 6.855$ | 268.543 | 265.10 | -
$14_{2,1}$ | $12_{2,1}$ | 1009.079 | 759.642 | 736.660 | 251.643 | $9.441\rightarrow 8.106$ | 281.320 | - | -
$2_{3,2}$ | $0_{3,2}$ | 233.504 | 221.942 | 209.888 | 56.684 | $4.497\rightarrow 2.524$ | 72.090 | 73.50 | -
$4_{3,2}$ | $2_{3,2}$ | 401.982 | 327.461 | 311.072 | 82.831 | $6.523\rightarrow 4.497$ | 118.990 | 120.50 | -
$6_{3,2}$ | $4_{3,2}$ | 559.801 | 422.880 | 408.564 | 116.085 | $8.567\rightarrow 6.523$ | 154.892 | 154.20 | -
$8_{3,2}$ | $6_{3,2}$ | 708.989 | 523.436 | 512.997 | 139.859 | $10.438\rightarrow 8.567$ | 183.019 | 181.20 | -
$10_{3,2}$ | $8_{3,2}$ | 858.095 | 624.555 | 609.096 | 166.646 | $11.932\rightarrow 10.438$ | 202.222 | - | -
$12_{3,2}$ | $10_{3,2}$ | 1003.933 | 727.909 | 715.990 | 182.910 | $13.011\rightarrow 11.932$ | 218.753 | - | -
$14_{3,2}$ | $12_{3,2}$ | 1151.239 | 832.003 | 819.115 | 202.421 | $14.629\rightarrow 13.011$ | 229.986 | - | -

Some important solutions for the collective model of Eq.(2) are the energy
levels, the spectra ratios and the $B(E2)$ transitions. Their theoretical
predictions are important when energy spectra are assigned to the states for
which experimental data are not available. The numerical calculations, the
analytical comparisons and how the search for the experimental realizations of
the model was achieved are discussed accordingly in this section.
Both the $X(3)$ and the $X(5$) have their critical orders, $\nu(L,\beta_{0})$,
from their Bessel functions which describes their energy spectra. Firstly, in
the comparison of the Eq.(10) and Eq.(14), it can be deduced from the
numerical computation of $\nu$, shown in Table 1., that

$\nu^{X(3)}(\beta_{0}=c+2)=\nu^{X(5)}(\beta_{0}=c):\quad c=0,1,2,...$ (28)

In both cases, it increases with increase in the angular momentum, $L$, and
with increase in the variation parameter, $\beta_{0}$. These effects of $L$
and $\beta_{0}$ in $\nu$ are also seen in the energy values of Eq.(13).

Secondly, the exact relationship between the $\nu^{X(3)}$ and the $\nu^{X(5)}$
stated in Eq.(28) does not reflect in the exact comparison of their energy
levels. That is, it can be inferred from the results that

$\epsilon^{X(3)}(\beta_{0}=c+2)\neq\epsilon^{X(5)}(\beta_{0}=c),$ (29)

because the total energy of the $X(5)$ contains the $\gamma$-part solutions.
However, the relation

$\epsilon_{gs,L}=2+\epsilon_{\beta_{1},L}=4+\epsilon_{\beta_{2},L},$ (30)

holds in all the levels for both $X(3)$ and the $\beta$-part of $X(5)$: this
third remark is shown in the Table 2.

Another significant remark is such that, the values of $\nu$, for the case of
$X(5)$ at $L=2$, correspond to those of $X(3)$, at $L=0$. This is shown in
Table 1. and the visual comparison is shown with the lines in Figure 1(b).
Analytically, the behaviour of the energies of the $X(5)$ and the $X(3)$ at
constant value of variation parameter, $\beta_{0}$, is shown in the Figure
1(a). The critical orders, $\nu(L,\beta_{0})$, of the $X(5)$ and that of the
$X(3)$, which define their energy levels, are plotted against the variation
parameter, $\beta_{0}$, at constant angular momenta and shown in the Figure
1(b): it is shown, with the numerical values of $\nu$, in the Table 1., that

$\nu^{X(5)}(L=0)=\nu^{X(3)}(L=2)\quad\forall\quad\beta_{0}.$ (31)

The derivatives of $\nu$ with respect to the $\beta_{0}$ are shown in Figures
2(a) and 2(b). The first and the second derivatives are carried out in order
to show the stationary properties of $\beta_{0}$ and the values of $\beta_{0}$
at which the energy is minimum.

The variation of the ratio $\dfrac{\epsilon_{s,L}}{\epsilon_{1,2}}$ with
respect to the variation parameter, $\beta_{0}$, for both $X(3)$ and $X(5)$
are respectively shown in the Figures 3(a) and 3(b). For all values of
$\beta_{0}$, its values increase at $L=0$, are constant at $L=2$, that is
$\dfrac{\epsilon_{s,L}}{\epsilon_{1,2}}$=1 and decrease at $L>2$ .

The ground state bands ($gsb$) are defined with $s=1;\quad n_{\beta}=0$. The
quasi-$\beta_{1}$ bands and the quasi-$\beta_{2}$ bands are defined by
$s=2;\quad n_{\beta}=1$ and $s=3;\quad n_{\beta}=2$ respectively. The $\gamma$
bands do not exist for $X(3)$ model because, $\gamma^{0}=0$. The increase in
the angular momentum, $L$, at constant value of $\beta_{0}$, increases the
energies, in all energy levels. Also, at constant values of the angular
momentum, the increase in the $\beta_{0}$ increases the energy levels. The
Table 2. shows the numerical solutions of Eq.(13) obtained for the ground
states and the $\beta$-bands at $\beta_{0}=2,3,4$ and at $\beta_{0}=15$.

The Figure 4(a) shows the comparison, in the $R_{4/2}$, of the $X(3)$ with
$X(5)$ at $\beta_{0}=\infty$ and at $\beta_{0,max}$ unique to each angular
momentum, labelled as $X(3)-$var and $X(5)-$var respectively. The comparison
in the $R_{0/2}$ of the $X(3)$ with $X(5)$, at $\beta_{0}=\infty$ and at
different values of the $\beta_{0,max}$ peculiar to each angular momentum,
labelled as $X(3)-$var and $X(5)-$var respectively is shown in Figure 4(b).

The ‘nature’ of critical point symmetry transitions for different isotopes,
constrained to one-parameter potentials, can be investigated using a
variational technique. This technique was used in ref. [11] to retrieve the
$U(5)$ and $O(6)$ ground state bands from the $E(5)$ within the domain of the
one-parameter inverse square potential. The technique has also been used in
ref. [12] and employed in ref. [16] to construct ‘image’ of the $X(5)$
critical symmetry and to construct the $Z(5)$ critical symmetry respectively.
The forward variation of the ‘control parameter’, $\beta_{0}$, causes the
nuclei transition from $X(5)$ to $SU(3)$ (i.e. $X(5)\longrightarrow SU(3)$
transition symmetry). The nature of the critical symmetry or the nuclear shape
phase region under investigation predicts the directions of the variation:
whether forward variation or backward variation, and also depends on the
potential’s boundary conditions. The rate of change of $R_{L/2}(\beta_{0})$ is
maximized for each $L$ by using this approach. As shown in Table 3., each
angular momentum is considered and treated separately in terms of the
variation parameter, $\beta_{0}$, as the critical values of $R_{L/2}$ are
distinct. Each value of $\beta_{0}$ implies a distinct potential with which
the energy is maximized. The method is comparable to the “normal” variational
principle used in some quantum books, in which trial wave functions are chosen
and energy is minimized.

The comparisons of the ground state spectra ratios, defined in Eq.(15) with
the $X(5)$ model [11], at different values of $\beta_{0}$ corresponds to the
potentials are displayed in Figures 5(a) and 5(b) and also shown in Table 3:
the visual comparison is shown in Figures 6(a), 6(b) and 6(c). It can be
observed that the solutions of $X(3)(\beta_{0}=\infty)\approx
X(5)(\beta_{0}=\infty)$. The forward variation of $\beta_{0}$ shifts the
solutions to $X(3)$. The solutions leave $X(3)$ and approach $SU(3)$ as
$\beta_{0}$ tends to $\infty$. The available experimental data of 162Dy [17],
which is a typical $SU(3)$ candidate are placed for comparison in Figure 6(a)
and Figure 6(b). This is another remark that isotopes which have $X(3)$
signatures must lie between $U(5)\longrightarrow SU(3)$ symmetrical plane.

The two other important relations that can be deduced from the comparison are:

$R_{L/2}(gsb)=2+R_{L/2}(\beta_{1})=4+R_{L/2}(\beta_{2}),$ (32)

at $\beta_{0}=0$ as shown numerically in the Table 3. and Table 4. This is an
observable effect or a signature from Eq.(30) while the effect of Eq.(28) is
observed in the spectral ratios of $X(3)$ and $X(5)$ such that

$R_{L/2}^{X(3)}(\beta_{0}=c+2)=R_{L/2}^{X(5)}(\beta_{0}=c):\quad c=0,1,2,...$
(33)

In order to obtained the exact solutions of $R_{L/2}$ ratios rather than vary
$\beta_{0}$, the technique of optimizing $\beta_{0}$ employed in refs.
[11,12,15,16] and others has been used to obtained the solutions of $R_{L/2}$
at certain values of the $\beta_{0}$ peculiar to the angular momenta. These
special values of $\beta_{0}$ are labelled $\beta_{0,max}$ and they produce
exact solutions labelled $X(3)$-var, shown in Table 5. The values obtained at
different values of $\beta_{0,max}$, are compared with $X(3)$-IW solutions.
For all $\beta_{0,max}$, $0_{0,0}$ and $2_{0,0}$ levels yield $0.000$ and
$1.000$ respectively. $\beta_{0,max}$ increases with increase in the angular
momentum and its values are obtained at the points where the increases in
$\beta_{0}$ become steep. $\dfrac{d}{d\beta_{0}}R_{L/2}|_{\beta_{0}=max}$ is
achieved via a numerical procedure as $\dfrac{d^{2}}{d\beta_{0}^{2}}R_{L/2}$
vanished. The $R_{L/2}$ ratios for the ground state and the quasi-$\beta_{1}$
bands of the $X(3)$ and the $X(5)$ models of inverse square potentials
obtained at different values of $\beta_{0,max}$, labeled $X(3)$-var and
$X(5)$-var, are compared with the experimental data of 172,176,178,180Os
[18-21] chain, as shown in Figures 7(a) and 7(b). The ground state solutions
of the $X(3)$ for $L=0$ up to $L=10$ are in good agreement with 172Os while
those of $X(5)$ are seen lying closer to 176Os than 178Os and 180Os: the
generalized comparison is moderate in the first excited state. This suggests
that 172Os is a good candidate for $X(3)$ model while 176Os shows a signature
of $X(5)$ model.

The $R_{L/2}$ theoretical predictions of the $X(3)$ model are compared with
the experimental data of some selected isotopes: 102Mo [22], 104-108Ru
[23-25], 120-126Xe [26-29], 148Nd [30] and 184-188Pt [31-33] as shown in Table
6. Each energy level is normalized to the particular $2_{0,0}$ state. The
energy obtained in Eq.(13) is fitted with the experimental energy of each of
the isotopes considered. The equivalent values of the $\beta_{0}$ for the
isotopes are recorded. The quality factor, $\sigma$, used is obtained from

$\sigma=\sqrt{\dfrac{\sum_{i}^{m}[(R_{s,L})^{Exp}_{i}-(R_{s,L})^{Theor}_{i}]^{2}}{m-1}},$
(34)

where $m$ is the number of available experimental states,
$(R_{s,L})^{Exp}_{i}$ and $(R_{s,L})^{Theor}_{i}$ represent the experimental
and the theoretical spectral ratios of the $i^{th}$ levels normalized to the
ground state with $L=2$, $s=1$ and $n_{\beta}=0$ respectively.

Against the neutron numbers, $N$, of the chains of the isotopes: 104-108Ru,
120-126Xe, 184-188Pt, 172-180Os, considered for the comparison, the
neutron-$\beta_{0}$ distribution, showing the relative positions of the
isotopes, is shown in the Figure 8.

The comparison in the ground state, the quasi-$\beta_{1}$ bands and the
quasi-$\beta_{2}$ bands of the $B(E2)$ transition probabilities at
$\beta_{0}=0,1,2$ and $\beta_{0}=\infty$, normalized to the
$B(E2:2_{1,0}\rightarrow 0_{1,0})=100$ units with the $X(3)$-IW [1] and
experimental data on $X(5)$ [34] are presented in the Table 7. The values of
$\beta_{0,max}$ peculiar to each angular momentum, obtained from the
optimization of $\beta_{0}$, in Table 5., are employed to compute the
optimized $B(E2)$ transition probabilities, labelled $B(E2)$-var. The visuals
of these comparisons are shown in the Figures 9(a), 9(b) and 9(c). In order to
show the nature of the solutions along the $X(5)\longrightarrow SU(3)$
symmetry region, the experimental data on the 158Gd [35], which is a typical
$SU(3)$ candidate, are placed for comparison in Figures 9(a) and 9(b): the
solutions at $\beta_{0}\rightarrow\infty$ are seen lying close to the 158Gd
[35]. The values of the $B(E2)$ transition probabilities decrease as the
variation parameter, $\beta_{0}$, increases: they increase as the angular
momentum increases. The forward variation, as the $\beta_{0}$ increases,
pushes the solutions to $X(5)$ and the solutions tend to the $SU(3)$ as
$\beta_{0}$ tends to $\infty$.

##  5 Conclusion

The $X(3)$ solutions of the Bohr Hamiltonian are obtained by solving the
radial function of the Hamiltonian with an inverse square potential with the
aid of MAPLE software. Analytically, an expression for the energy levels is
determined from the zeros of the Bessel functions. Through the use of the
variational approach and the optimization procedure, the spectra ratios and
the $B(E2)$ transition probabilities are computed. The analytical solutions of
the $X(3)$ model are compared with the $X(5)$ model of the inverse square
potentials. It is worth noting that, $X(3)$ model is another “window” through
which $X(5)$ and $SU(3)$ “pictures” can be seen: $X(3)$ lies between $U(5)$
and $SU(3)$, hence, $X(5)$ lies between $X(3)$ and $SU(3)$. It has been shown
via variational procedure, that the solutions shift to $X(5)$ from $X(3)$ and
approach $SU(3)$ as the variation parameter shifts forward.

The theoretical predictions on $R_{L/2}$ and $B(E2)$ with the experimental
data for some selected isotopes are found to be proficient in the $gsb$ and
moderate in other levels. This is shown as the theoretical deviations from the
experiments are quite small.

The same manner in which the Davidson potential is employed in ref. [4], the
employment of the one parameter-dependent inverse square potential in the form
of Eq.(1), its properties, is efficient in the variational procedure. Eq.(1)
is also a good choice of potential which can be employed for the description
of the nuclei transition at the critical points. For the comparison of $X(3)$
and $X(5)$ models of Bohr Hamiltonian, with the same formalism employed in
this work, it is expected that Equations (28), (29), (30), (31), (32) and (33)
should hold in any one-parameter-dependent potential domain such as Kratzer
potential, Davidson potential and others.

## Data availability statement

All the sources of data included in this article for comparison purpose, are
cited and referenced accordingly, in the article.

## Funding Information

No funding of any form is received for the course of this work.

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