---
abstract: 'The model, introduced in a previous paper, for the description of the octupole and quadrupole degrees of freedom in conditions close to the axial symmetry, is applied to situations of shape phase transitions where the quadrupole amplitude can reach zero. The transitional nuclei $^{224,226}$Ra and $^{224}$Th are discussed in the frame of this model. Their level schemes can be reasonably accounted for assuming a square-well potential in two dimensions. Electromagnetic transition amplitudes are also evaluated and compared with existing experimental data.'
author:
- 'P.G. Bizzeti'
- 'A.M. Bizzeti–Sona'
bibliography:
- 'biz\_bib.bib'
title: 'Description of nuclear octupole and quadrupole deformation close to the axial symmetry: Critical-point behavior of $^{224}$Ra and $^{224}$Th'
---

Introduction {#S:1}
============

The phase transition between spherical and axially deformed quadrupole shape of nuclei has been the object of several theoretical and experimental works in recent years. In particular, the properties of nuclei close to the critical point, predicted by Iachello’s model of X(5) symmetry [@iachello01], have been actually observed in several cases [@casten01; @kruecken02; @bizzeti02; @clark03; @hutter03; @tonev04], while some other nuclides showing the ratio $E(4+)/E(2^+)\approx 2.91$ expected for the X(5) symmetry are presently under investigation. Moreover, in the Ra – Th region, it has been observed that the isotopes $^{224}$Ra and $^{224}$Th have a positive–parity ground–state band with a sequence of level energies very close to the X(5) predictions [@bizzetierice; @bizzetirila]. Here, however, the presence of a very low lying negative parity band, soon merging with the positive–parity one for $J>5$, proves that the octupole mode of deformation plays an important role and should not be ignored in discussing the behavior of the phase transition.

In a previous paper [@bizzeti04] (henceforth referred to as I) a simple model has been introduced to describe the phase transitions in nuclear shape involving the octupole mode[^1]. To this purpose, a new parametrization of the collective coordinates describing the nuclear quadrupole and octupole deformation has been introduced and discussed. The nuclear shape is represented in the intrinsic frame defined by the principal axes of the overall tensor of inertia, in situations close to (but not necessarily coincident with) the axial–symmetry limit. In the same paper, a specific model is developed to describe the critical point of the phase transition in the octupole mode, between harmonic oscillations and permanent asymmetric deformation, in nuclei which already possess a stable quadrupole deformation. The Thorium isotopic chain was investigated and the experimental data concerning $^{226,228}$Th were compared with the model predictions [@bizzeti04]. The former appears to be close to the critical point, while the latter can be interpreted as an example of harmonic oscillations in the axial octupole mode.

In the present paper, we extend the investigation to the cases where the quadrupole deformation is not steady but performs oscillations under the effect of a proper potential, and in particular for situations close to the quadrupole critical point described by the X(5) symmetry, in the Radium and Thorium isotopic chain.

As we shall see, the properties of the already mentioned nuclei $^{224}$Th and $^{224}$Ra result to be reasonably described by our model with a “critical” (flat) potential well, extending both in the $\beta_2$ and $\beta_3$ directions. Moreover, we observe that, as far as the level scheme is concerned, also the next isotope $^{226}$Ra can be accounted for with a proper critical–point potential, in spite of the fact that the positive–parity part of the ground–state band does not follow the X(5) predictions. As in the case of Thorium, heavier isotopes of Radium have a permanent quadrupole deformation and octupole excitations of vibrational character, while the lighter ones are either non collective or vibrational in the quadrupole mode.

Some results of this work, at different phases of advancement, have been reported at several Conferences or Schools [@bizzetirila; @bizzeticamerino; @bizzetipredeal; @bizzetivico].

For convenience of the reader, we report in the next Section \[S:2\] some evidence of the phase transitions in the Radium and Thorium isotopic chain, while the definition of variables introduced in I, and a few results relevant to the present work, are briefly summarized in the Section \[S:3a\]. In the following subsections, the model introduced in I is specialized to a form suitable for a critical potential in two dimensions. Finally, in the Section \[S:4\] the model results are reported and compared with the existing experimental evidence for $^{224,226}$Ra and $^{224}$Th. Previous models of quadrupole–octupole deformation are quoted in I. Since then, new relevant papers have appeared. A new “Analytic Quadrupole–Octupole axially symmetric model” (AQOA) has been proposed by D.Bonatsos [*et al.*]{} [@bonatsos05] to discuss the evolution of the quadrupole and octupole collectivity in Ra and Th isotopes. A parameter free model starting from a similar approach has been developed by Lenis and Bonatsos [@lenis06] and compared with the experimental results for $^{226}$Ra and $^{226}$Th. Moreover, a variant of the AQOA model, introducing a renormalization of the nuclear moment of inertia [@minkov06], has been used to describe the lowest quadrupole and octupole bands of the $N=90$ isotones $^{150}$Nd, $^{152}$Sm, $^{154}$Gd and $^{156}$Dy. A discussion of the octupole bands of $^{150}$Nd and $^{152}$Sm is also contained in the Ref.s [@bizzetirila; @bizzetipredeal]. Finally, an extension of the Extended Coherent State model [@raduta03] has been developed by A.A. Raduta and coworkers [@raduta06b; @raduta06c; @radutapredeal], to include in the model space also the lowest $K^\pi=1^+$ and $1^-$ bands, and the model predictions have been compared with the experimental data for several nuclei of the regions of the actinides and of the rare-earths.

\[S:2\]Phase transitions in the R – T region
============================================

We summarize here the existing evidence for the evolution of nuclear shapes for Radium and Thorium isotopes in the transitional region $N=130 - 140$.

The Fig. \[F:1\], taken from I, shows the behavior of some indicators of quadrupole and octupole collectivity, as a function of the neutron number $N$, in the isotopic chain of Ra and Th. It has been noted in I that $^{226}$Th appears to be close to the critical point in the octupole deformation, while it possesses a stable quadrupole deformation $\beta_2$. At larger values of $N$, Th isotopes maintain a stable quadrupole deformation, while the octupole mode evolves towards the vibrational behavior, as indicated by the large excitation energies of all negative–parity levels. At $N=130$ or less, the quadrupole mode has a vibrational (or non collective) character. It turns out, therefore, that the octupole phase transition proceeds in the direction opposite to the one of quadrupole. We also observe that the phase transition only involves the axial octupole mode. In fact, the energy of the $J^\pi=1^-$ band head of the $K^\pi=0^-$ octupole band shows a sharp decrease, both in its absolute value and in the ratio to the $E(2^+)$, when the neutron number decreases below $N=142$. Other octupole bands (with $K>0$) do not show a similar trend (fig. \[F:1\]$(c)$), and one can conclude that non-axial octupole excitations maintain a vibrational character. A similar trend is apparent also for Ra isotopes.

In order to describe Th and Ra isotopes with $A < 226$, we must allow also $\beta_2$ to vary and perform (non necessarily harmonic) oscillations. If we consider the value $E(4^+)/E(2^+)=2.91$ as a signature of the critical point with respect to the quadrupole deformation, this would correspond approximately to $^{224}$Ra and $^{224}$Th.

![\[F:1\] (Color online) Indicators of the quadrupole collectivity (left) and of the octupole collectivity (right), as a function of the neutron number $N$ in the isotopic chain of Ra (circles) and Th (triangles): $(a)$ - Excitation energy of the first $2^+$ level; $(b)$ - Energy ratio $E(4^+)/E(2^+)$; $(c)$ - Excitation energy of the first level of the $K^\pi=0^-$ band, $J_0^\pi=1^-$ (open symbols) and of the lowest known level of other negative-parity bands, $J_0^\pi=2^-$ or $1^-_2$ (full symbols); $(d)$ - Energy ratio $E(1^-)/E(2^+)$. The horizontal line in the part $(b)$ shows the value (2.91) expected for the X(5) symmetry (from ref. [@bizzeti04]). ](biz_f1.ps){height="8.cm"}

The model for quadrupole – octupole vibrations
==============================================

\[S:3a\]Summary of the variable definitions
-------------------------------------------

The dynamical variables $a^{(\lambda )}_{\mu}$ ($\lambda=2,3;\ \mu
=-\lambda ... \lambda$), describing the quadrupole and octupole deformation in the intrinsic reference frame, are parametrized as $$\begin{aligned}
a^{(2)}_0 &=& \beta_2 \cos \gamma_2 \approx \beta_2 \nonumber\\*
a^{(2)}_1 &=& -  \frac{\sqrt{2}\ \beta_3}{\sqrt{\beta_2^2+2 
\beta_3^2}}\ v\ (\sin \varphi + i \cos \varphi )\cr 
a^{(2)}_2 &=& \sqrt{1/2}\ \beta_2\ \sin \gamma_2 - i 
\frac{\sqrt{5}\ \beta_3}{\sqrt{\beta_2^2+2 \beta_3^2}}\ u \sin \chi  \cr
a^{(3)}_0 &=& \beta_3 \ \cos \gamma_3 \approx \beta_3\cr
a^{(3)}_1 &=&\frac{\sqrt{5}\ \beta_2}{\sqrt{\beta_2^2+2 \beta_3^2}}\ v
\  (\sin \varphi + i \sin \varphi ) \cr
a^{(3)}_2 &=& \sqrt{1/2}\ \beta_3\ \sin \gamma _3 +i
\frac{\beta_2}{\sqrt{\beta_2^2+2 \beta_3^2}}\  u \sin \chi \cr
a^{(3)}_3 &=& w \sin \vartheta \left[ \cos \gamma_3 + 
(\sqrt{15}/2)\ \sin \gamma_3 \right]\cr
  &+& i\ w \cos \vartheta \left[ \cos \gamma_3 - 
(\sqrt{15}/2)\ \sin \gamma_3 \right] \cr
&\approx& w\ (\sin \vartheta +i \cos \vartheta )
\label{E:2.2}\end{aligned}$$ With this choice, valid in situations close to the axial symmetry, the tensor of inertia turns out to be diagonal [*up to the first order*]{} in the [*small*]{} quantities describing the non-axial deformations.

In the Eq.s \[E:2.2\] the variables $\gamma_2$ and $\gamma_3$ are still employed, in order to keep some transparency with respect to the standard expressions used to describe the quadrupole [@bohr52] or the octupole deformation alone [@wexler99]. However, it is more convenient to substitute them with expressions involving the variables $u, \chi$ and a new variable $u_0$: neglecting second-order and higher-order terms, $$\begin{aligned}
\gamma_2 &=& \frac{\sqrt{10}\beta_3}{\beta_2
\sqrt{\beta_2^2+5\beta_3^2}} u \cos \chi 
+\frac{f(\beta_2,\beta_3)}{\sqrt{\beta_2^2+5\beta_3^2}} u_0 \\*
\gamma_3&=&-\frac{\sqrt{2}\beta_2}{\beta_3
\sqrt{\beta_2^2+5\beta_3^2}} u \cos \chi 
+\frac{\sqrt{5} f(\beta_2,\beta_3)}{\sqrt{\beta_2^2+5\beta_3^2}} u_0 \nonumber
\label{E:2.3}\end{aligned}$$ It is possible to show that a definite value of the angular-momentum component $K$ along the intrinsic axis 3 and a definite parity can be associated to the degrees of freedom corresponding to the variables $v, \chi$ (or $u, \varphi$ or $w, \vartheta$ or $u_0$): $K^\pi=1^-$ (or $2^-$ or $3^-$ or $2^+$, respectively). This result is independent of the form of the function $f(\beta_2, \beta_3)$ of Eq. \[E:2.3\] (which, actually, was left undetermined in I).

The kinetic energy operator
---------------------------

The classical expression of the kinetic energy has the form $$T=\frac{1}{2}\sum \mathcal{G}_{\mu \nu}\dot{\xi}_\mu\dot{\xi}_nu
\label{E:2.6}$$ where $\dot{\xi}\equiv (\dot{\beta}_2,\dot{\beta_3},\dot{u_0},
 \dot{v},\dot{\chi},\dot{u},\dot{\varphi},\dot{w},\dot{\vartheta},
q_1,q_2,q_3)$ and $q_1$, $q_2$, $q_3$ are the components of the angular velocity along the three axes of the intrinsic reference frame. As in I, we adopt here the convention of including the inertial coefficient $B_\lambda$ in our amplitudes $a^{(\lambda)}_\mu$, that therefore would correspond to $\sqrt{B_\lambda}a^{(\lambda)}_\mu$ in the usual notations of Bohr.

                       $\dot{\beta_2}$   $\dot{\beta_3}$        $\dot{u_0}$         $\dot{v}$   $\dot{u}$   $\dot{w}$   $\dot{\varphi} $   $\dot{\chi}$   $\dot{\vartheta}$        $q_1$              $q_2$              $q_3$
  ------------------- ----------------- ----------------- ------------------------ ----------- ----------- ----------- ------------------ -------------- ------------------- ------------------ ------------------ ------------------
  $\dot{\beta_2}$             1                 0                    0                  0           0           0              0                0                 0                  0                  0                  0
  $\dot{\beta}_3$             0                 1                    0                  0           0           0              0                0                 0                  0                  0                  0
  $\dot{u_0}$                 0                 0          $f^2(\beta_2,\beta_3)$       0           0           0              0                0                 0                  0                  0                  0
  $\dot{v}$                   0                 0                    0                  2           0           0              0                0                 0                  0                  0                  0
  $\dot{u}$                   0                 0                    0                  0           2           0              0                0                 0                  0                  0                  0
  $\dot{w}$                   0                 0                    0                  0           0           2              0                0                 0                  0                  0                  0
  $\dot{\varphi}$             0                 0                    0                  0           0           0           $2v ^2$             0                 0                  0                  0               $2v ^2$
  $\dot{\chi}$                0                 0                    0                  0           0           0              0              $2u^2$              0                  0                  0                $4u^2$
  $\dot{\vartheta}$           0                 0                    0                  0           0           0              0                0              $2 w^2$               0                  0               $6 w^2$
  $q_1$                       0                 0                    0                  0           0           0              0                0                 0           $\mathcal{ J}_1$          0                  0
  $q_2$                       0                 0                    0                  0           0           0              0                0                 0                  0           $\mathcal{ J}_2$          0
  $q_3$                       0                 0                    0                  0           0           0            $2v^2$           $4u^2$           $6w^2$                0                  0           $\mathcal{ J}_3$

  : \[T:1\] The matrix of inertia $\mathcal{ G}$ after the introduction of the variables $u_0$, $v$, $u$, $w$, $\varphi$, $\chi$, and $\vartheta$ (see text). Here, $\mathcal{ J}_1 = \mathcal{ J}_2 = 3(\beta_2^2 + 2\beta_3^2)$, and $\mathcal{ J}_3 = 4 f^2(\beta_2,\beta_3)
  \ u_0^2 + 
  2v^2 + 8u^2 + 18w^2$. Only the leading terms are shown. Neglected terms are small of the first order (or smaller) in the sub-matrix involving only $\dot{\beta_2}$, $\dot{\beta_3}$, $\dot{u_0}$, $\dot{v}$, $\dot{u}$, $\dot{w}$, $q_1$ and $q_2$; of the third order (or smaller) in the sub-matrix involving only $\dot{\varphi} $, $\dot{\chi}$, $\dot{\vartheta}$ and $q_3$; of the second order (or smaller) in the rest of the matrix.

The matrix elements of $\mathcal{G}$, approximated to the most relevant order, are shown in the Table \[T:1\]. The determinant of this matrix turns out to be $$\begin{aligned}
G &\propto & (\beta_2^2 + 2 \beta_3^2)^2\ f^4(\beta_2, \beta_3) 
\ u_0^2 \ v^2 u^2 w^2\nonumber \\[2pt]
&\equiv & G_0(\beta_2,\beta_3)\ u_0^2 \ v^2 u^2 w^2 .
\label{E:2.7}\end{aligned}$$ The Pauli recipe for the quantization of the classical kinetic energy gives the Schrödinger equation $$\sum_{\mu \nu} \frac{1}{g} 
\frac{\partial}{\partial \xi_\mu} \left[ g 
\!\left( \mathcal{G}^{-1} \right) _{\mu \nu} 
\frac{\partial \Psi}{\partial \xi_\nu }
\right]+ \frac{2}{\hbar^2} \left[ E\!-\!V(\xi ) \right] \Psi\! =\!0
\label{E:a1.2}$$ where $g^2=G={\rm Det}\ \mathcal{G}$ and $\xi$ stays for the ensemble of the variables $\xi_\kappa$.

To our present purpose, this general treatment must be specialized, ([*e.g.*]{} with a proper choice of the arbitrary function $f(\beta_2,\beta_3)$ in the Eq.s \[E:2.3\]), keeping in mind a necessary condition: the Schrödinger equation for the quadrupole amplitude, when the octupole amplitude is constrained to small values by a proper restoring potential, must converge to that of Bohr and therefore, at the critical–point, to that of the X(5) model[^2]. As we shall see in the Section \[S:4.1\], this result is obtained with the choice $$f(\beta_2,
\beta_3)=\sqrt{\frac{(\beta_2^2+\beta_3^2)(\beta_2^2+2\beta_3^2)}
{\beta_2^2+5\beta_3^2}}\ ,
\label{E:2.4}$$ from which one obtains $$G_0(\beta_2,\beta_3)=
\frac{(\beta_2^2+\beta_3^2)^2(\beta_2^2+2\beta_3^2)^4}
{(\beta_2^2+5\beta_3^2)^2}\ ,
\label{E:2.4b}$$

The critical potential in two dimensions
----------------------------------------

Possible landscapes of axial quadrupole–octupole deformation in the Thorium region are exemplified in Fig. \[F:2\], where the potential energy is depicted as a function of the deformation parameters $\beta_2$ and $\beta_3$. Reported values have been obtained by Nazarewicz [*et al.*]{} [@nazarew85] with a Wood–Saxon–Bogolyubov cranking calculation. We notice that fig \[F:2\]$(e)$ shows a potential minimum which is localized around a fixed value in the $\beta_2$ direction, while a flat minimum extends over a sizable interval in the direction $\beta_3$. This is just the “critical” potential for the shape transition between octupole oscillation and permanent octupole deformation (combined with a fixed quadrupole deformation), corresponding to the fig.s \[F:2\]$(d)$ and \[F:2\]$(f)$, respectively. The fig.s \[F:2\]$(a), \ref{F:2}(b)$ and \[F:2\]$(c)$, instead, show a different kind of shape transition, proceeding directly from a fixed, reflection–asymmetric deformation (fig.\[F:2\]$(c)$) to quadrupole–octupole vibrations around a spherical shape (fig.\[F:2\]$(a)$).

![(color on line) Potential–energy surfaces in the $\beta_2$ – $\beta_3$ plane for several Th isotopes, as given by Nazarewicz [*et al.*]{} [@nazarew85].[]{data-label="F:2"}](biz_f2.ps){width="8.4cm"}

The potential corresponding to the critical point is not shown. It should be somewhere between fig. \[F:2\]$(a)$ and fig. \[F:2\]$(b)$. One can try to approximate the critical potential, as usual, with a square well, but now the flat bottom of the well should extend over a finite distance in $\beta_2$ and $\beta_3$, and be symmetric in $\beta_3 $ around $\beta_3=0$. The shape of the borders is obviously relevant to the result. One could imagine shapes like those shown in fig. \[F:3\] with dashed or dotted lines, but their description would involve at least two or three free parameters, and the comparison with experimental data could be not very significant. We have found, however, that good results are obtained also with a simple rectangular shape (solid line in fig. \[F:3\]), implying only one free parameter, $b=\beta_3^w/\beta_2^w$ (apart from a common factor of scale).

![(color on line) Possible shapes for a potential well simulating the critical–point potential. The potential–energy surface of fig. \[F:2\]$(b)$ is also shown for comparison.[]{data-label="F:3"}](biz_f3.ps){height="5.9cm"}

\[S:4\]Results and comparison with experimental data
====================================================

\[S:4.1\] The Energy eigenvalues
--------------------------------

Now, as a first step, we can evaluate, as a function of $b$, the level energies in the ground–state band and deduce the best value of the parameter from a comparison with experimental results (fig. \[F:5\]).

![image](biz_f4.ps){height="\linewidth"}

To proceed, we must do some assumptions on the behavior of axial and non axial modes of deformation. We will assume that

-   Our choice of variables corresponds to independent degrees of freedom.

-   Non-axial vibrations are confined to their lowest stationary state.

-   An approximation similar to that of the X(5) model is valid for the differential equations of all non-axial amplitudes: [*i.e.*]{}, the differential equation in $\beta_2,\ \beta_3$ can be approximately decoupled from those concerning the other degrees of freedom.

Therefore, the complete wavefunction $\Psi$ of Eq. \[E:5.0\] can be factorized in three parts, as in Eq. 30 of I: $$\Psi=\Psi_0(\beta_2,\beta_3)\  \Psi_1\ Y_{JM}(\hat{\Omega})
\label{E:5.0}$$ where the function $\Psi_1$ depends on the deformation variables different from $\beta_2,\ \beta_3$.

From the Eq. \[E:2.7\] we know that also the determinant $G$ is factorized in the same way. Then, the differential equation for $\beta_2 ,\ \beta_3$ takes the form\
$$\begin{aligned}
&&\hspace*{-4mm}
\left\{ G_0^{-1/2}\!\left[ \frac{\partial}{\partial 
\beta_2}\!\left( G_0^{1/2}
\!\frac{\partial}{\partial \beta_2 }\right)\!+\! 
\frac{\partial}{\partial \beta_3}\!\left( G_0^{1/2} 
\frac{\partial}{\partial \beta_3 }\right) \right] \right. \ \\
&&\hspace*{-4mm}\left. +\ \epsilon - V(\beta_2,
\beta_3) - \frac{J(J+1)}{3(\beta_2^2+2\beta_3^2)} 
\right\} \Psi (\beta_2, \beta_3) = 0 \nonumber
\label{E:5.1} \end{aligned}$$ This equation can be somewhat simplified with the substitution $$\begin{aligned}
\Psi_0(\beta_2,\beta_3) = g^{-1/2}\ \Phi (\beta_2,\beta_3)
\label{E:5.2}\end{aligned}$$ where $g\propto G_0^{1/2}$, to obtain $$\begin{aligned}
\bigg\{
\frac{\partial^2}{\partial \beta_2^2} 
\!&\!+\!&\!\frac{\partial^2}{\partial \beta_3^2}
+ \epsilon - V(\beta_2,\beta_3)-\frac{J(J+1)}{3(\beta_2^2+2\beta_3^2)}
\nonumber \\
&\!+\!&\! {V_\text{g}(\beta_2,\beta_3)} \bigg\}\ \Phi (\beta_2,\beta_3) =0
\label{E:5.3} \end{aligned}$$ with $$\begin{aligned}
V_\text{g}\!=\!\frac{1}{4g^2}  \bigg[ \left( \frac{\partial g
}{\partial \beta_2} \right) ^2\!+
\!\left( \frac{\partial g}{\partial \beta_3} \right) ^2 \bigg]
\!- \frac{1}{2g} \bigg[ \frac{\partial^2 g}{\partial \beta_2^2}
\!+\!\frac{\partial^2 g}{\partial \beta_3^2} \bigg]
\label{E:5.4} \end{aligned}$$

With the choice of $f(\beta_2, \beta_3)$ given in the Eq. \[E:2.4\], from Eq. \[E:2.4b\] one obtains $$\begin{aligned}
g\propto \frac{(\beta_2^2\!+\beta_3^2)(\beta_2^2\!+\!2
\beta_3^2)^2}{(\beta_2^2+5\beta_3^2)} .
\label{E:5.5}\end{aligned}$$ and, for $|\beta_3| \ll \beta_2$, $ g \propto \beta_2^4 \big[
1\!+\!4(\beta_3/\beta_2)^4\!+...\big] $. Therefore, the first and second derivative of $g$ with respect to $\beta_3$ tend to zero when $|\beta_3| \ll \beta_2$ and, at the limit $\beta_3\rightarrow 0$, ${V_\text{g} = -2}$ as in the original Bohr model.

![\[F:3.1\] (Color online) Calculated energies of excited levels of the ground and first excited band, in units of $E(2^+_1)$, as a function of the ratio $b=\beta_3^w / \beta_2^w$. ](biz_f5.ps){height="7.5cm"}

With the substitution $\Psi_0 =g^{-1/2}\Phi$, and assuming $V(\beta_2,\beta_3)=0$ inside the potential well and $=+\infty$ outside, the differential equation to be solved takes the form $$\begin{aligned}
\left[ \frac{\partial^2}{\partial
\beta_2^2}+\frac{\partial^2}{\partial \beta_3^2}+\epsilon 
+V_\text{g}(\beta_2,\beta_3) \right] \Phi (\beta_2,\beta_3) = 0
\label{E:5:6}\end{aligned}$$ with $V_\text{g}$ given in the Eq. \[E:5.4\] and $\Phi=0$ on the contour of the potential well. The numerical integration has been performed with the finite difference method. Namely, the space is discretized on a rectangular lattice and values of $\Phi$ at the lattice centers are taken as independent variables. In the place of second derivatives, the ratios of finite differences are used: [*e.g.*]{}, $$\begin{aligned}
\left( \frac{\partial^2 \Phi}{\partial \beta_2^2}
\right)_{x,y}
& \Rightarrow &
\frac{\Phi(x+\Delta_x,y)-2\ \Phi(x,y)
+\Phi(x-\Delta_x,y)}{ \Delta_x^2} 
\label{E:2.42}\end{aligned}$$ As $\Phi(\beta_2,\beta_3)=(-1)^J
\Phi(\beta_2,-\beta_3)$, it is enough to consider only the region $\beta_3 >0$. The lattice centers are chosen as $\beta_2=k_2 \Delta_x,
\phantom{m}
\beta_3=(k_3-1/2) \Delta_y$ with $k_2=1...n_2$, $k_3=1...n_3$, and $\Delta_x=\beta_2^w / (n_2+1)$, $\Delta_y=2\beta_3^w / (2n_3+1)$. The integration region is the upper rectangle with $0<\beta_2<\beta_2^w$, $0<\beta_3<\beta_3^w$. At the upper and lateral borders of the rectangle, the value of the eigenfunction must be zero.

The boundary conditions at $\beta_3=0$ are not specified, but to evaluate the approximate derivatives with respect to $\beta_3$ it is enough to consider the value of $\Phi$ at the line of centers immediately below zero, where they are either equal or opposite to the corresponding ones at $\beta_3=\Delta_y/2$ according to the even or odd value of $J$.

The number of centers internal to the integration region – and therefore the number of independent values of $\Phi$ – is now $N=n_2 \cdot n_3$, and we obtain a finite dimensional $N\times N$ Hamiltonian matrix. This Hamiltonian has been diagonalized with the Implicitly Restarted Arnoldi – Lanczos method, using the ARPACK package [@arpack].

In the fig. \[F:3.1\], calculated values of the excitation energies (in units of $E(2^+_1$) ) are depicted as a function of the ratio $b=\beta_3^w/\beta_2^w$. At the limit for $\beta_3^w\rightarrow 0$, the curves corresponding to even $J$ and $\pi$ tend to the X(5) values, as expected. With increasing $b$, at the beginning these curves deviate substantially from the X(5) limit, but they come closer to the initial values for $b\approx 1$. In this region it is possible to find a good fit of the ground–state band of $^{224}$Ra and of $^{224}$Th, for $b=0.81$ and for $b=0.85$, respectively (Fig. \[F:5\]).

Moreover, a rather good fit of the ground–state band of $^{226}$Ra is obtained with $b=0.68$, [*i.e.*]{} close to the maximum of the curves for even parity and spin.

We can observe that, with our choice of the parameter $b$, the calculated $1^-$ level is always somewhat lower than the experimental one (Fig. \[F:5\]). This fact can be related to the inclusion, in the potential well, of a region where $\beta_3$ remains large while $\beta_2$ tends to zero. Actually, the wavefunction of the first $1^-$ level extends appreciably in this region, at variance with other levels of the ground-state band.

It would be of great interest, of course, to extend the comparison to the lowest excited band with $K=0$ (the $s=2$ band in the X(5) model notations). Unfortunately, in $^{224}$Th no excited $0^+$ level is known. The non–yrast level schemes of $^{224,226}$Ra will be discussed in the following Section \[S:4.2\]

![\[F:Th\] (Color online) Level scheme of $^{224}$Th, compared with the model predictions for $b=0.85$. Calculated and experimental branching ratios are reported for each level. Experimental data are taken from the NNDC tabulation [@nndc]. Experimental branching ratios from the $7^-$ and $9^-$ levels are not known. Theoretical values of the level energies (in keV) are normalized to that of the $2^+_1$ level; those of the branching ratios are deduced from the matrix elements of Table \[T:30\] with the experimental values of the transition energies. Calculated branches lower than 1% are not shown. ](biz_f7.ps){height="11.cm"}

Electromagnetic transition probabilities
----------------------------------------

Another important test for the model is provided by the E2 transition probabilities. The available experimental information on $B$(E2) values is scarce (only two transitions in $^{224}$Ra and $^{226}$Ra, one in $^{224}$Th), but we hope our work can stimulate interest for new experimental investigations. The reduced matrix element of the quadrupole transition operator ${\cal M}$(E2) between the states $|s,K=0,J \big>$ and $|s^\prime,K=0,J^\prime \big>$ can be evaluated as $$\big( s J |\!|{\cal M}{\rm(E2)} |\!|s^\prime  J^\prime \big) = C_2
\big< s J | \beta_2 | s^\prime J^\prime \big> \ (J |\!| Y_2 |\!| 
J^\prime \big)\ ,
\label{E:14b}$$ with $\big< s, J | \beta_2  |s^\prime ,  J^\prime \big> =
\int \Psi_{ s J} \beta_2 \Psi_{s^\prime ,
J^\prime}\ {\rm d}\tau$ and $C_2$ constant. The volume element ${\rm d}\tau$, in our non-cartesian coordinates, is the product of the differentials of the coordinate variables [*multiplied by*]{} $g=G^{1/2}$, with $G$ the determinant of the matrix of inertia $\cal G$. In our assumptions, the integrals over all variables apart from $\beta_2$ ad $\beta_3$ are independent from one another and from the integral over ${\rm d}\beta_2{\rm d} \beta_3$, and their result is $1$ (if the corresponding wavefunctions are properly normalized). As the electric dipole and quadrupole operators do not contain derivatives, we can exploit the substitution defined in Eq. \[E:5.2\] to express the remaining integral as $$\int\!\Psi_{ s J} \beta_2 \Psi_{s^\prime    
J^\prime}\ {\rm d}\tau
= \int_0^{\beta_2^w}\!{\rm d}\beta_2    
\!\int_{-\beta_3^w}^{\beta_3^w}\!{\rm d}\beta_3
\Phi_{sJ}\beta_2\Phi_{s^\prime J^\prime}\phantom{.}$$ This integral has been evaluated numerically, for values of $J \leq 18$, with $J^\prime = J-2$ (and also with $J^\prime = J-1$). The reduced matrix element over the angular coordinates has the form $$\begin{aligned}
&&\hspace*{-6mm}\big( J ||Y_L || J^\prime \big) = \\
&&\hspace*{-6mm}\phantom{m}(-1)^J (4\pi )^{-1/2} 
\sqrt{(2J\!+\!1) (2L\!+\!1) (2J^\prime\!+\!1)}
\left( \!\begin{array}{lcr} J&L&J^\prime \\ 0&0&0 \end{array}\!\right)
\nonumber\end{aligned}$$ Finally, the reduced transition probabilities from $J$ to $J^\prime$ are obtained as\
$B({\rm E2},s J\rightarrow s^\prime J^\prime)=(2J+1)^{-1}  
\big( s J ||{\cal M}{\rm(E2)} | |s^\prime  J^\prime \big) ^2$.\
The absolute values of the ratios of E2 reduced matrix elements, $R_J(E2)=(J|\!|\mathcal{M}(E2)|\!|J-2) /
(2^+|\!|\mathcal{M}(E2)|\!|0^+)$, for transitions within the positive– and the negative–parity part of the ground–state band, are depicted, as a function of $b=\beta_3^w/\beta_2^w$, in the fig. \[F:7\]$(a)$. Their limit at $\beta_3^w/\beta_2^w\rightarrow 0$ corresponds, as expected, to the X(5) value.

In addition to the in–band E2 transition, we have to consider the E1 transitions between levels of opposite parity. How to treat E1 transitions in the frame of the geometrical model is a big problem, as all E1 transition moments should vanish for a homogeneous fluid of constant charge density. In this sense, E1 transitions are outside the Bohr geometrical model. It is usual to assume a constant [*electric polarizability*]{} of the nuclear matter[@bohr57; @bohr58] to obtain the E1 operator in the form $$\mathcal{M}{\rm (E1)}= C_1 \beta_2 \beta_3 Y_1\ .
\label{E:15}$$ This [*ansatz*]{} should be validated by proper microscopic calculations.

Actually, such a calculation has been performed by Tsvenkov [*et al.*]{} [@tsvenkov02] for a number of Radium, Thorium and Uranium isotopes, in the frame of the Skirme–Hartree–Foch model. The electric dipole moment turns out to be almost independent of the angular frequency in a given isotope, but can change drastically (even in the sign) along the isotopic chain. The small value of the electric dipole moments in $^{224}$Ra is correctly predicted by these calculations.

Values of the ratios of the E1 matrix elements, $R_J(E1)=(J|\!|\mathcal{M}(E1)|\!|J-1) /
(1^-|\!|\mathcal{M}(E1)|\!|0^+)$, obtained with the standard form (Eq. \[E:15\]) of the E1 operator, are shown in the fig. \[F:7\]$(b)$. They reach a maximum for $\beta_3^w/\beta_2^w$ somewhat below 1, [*i.e.*]{} just in a region including the values assumed for $^{224}$Ra and $^{224}$Th (0.81 and 0.85, respectively). The calculated values of $(J_i|\!|\mathcal{M}(EL)|\!|J_f)$ for E2 and E1 transitions in the ground state bands of $^{224}$Ra ($b=0.81$), $^{226}$Ra ($b=0.68$) and $^{224}$Th ($b=0.85$) are given in the upper part of Table \[T:30\]. Values for the corresponding intra-band transitions are very similar in the three cases, while the difference can be larger for the weak inter-band transitions, as shown in the lower part of the Table.

[ccccccccc]{} &&\
Trans.&$^{224}$Ra&$^{226}$Ra&$^{224}$Th&&Trans.&$^{224}$Ra&$^{226}$Ra&$^{224}$Th$1^-\!\leftrightarrow\!0^+$&100&100&100&&$2^+\!\leftrightarrow\!0^+$&100&100&100$2^+\!\leftrightarrow\!1^-$&149&147&150&&$3^-\!\leftrightarrow\!1^-$&127&129&127$3^-\!\leftrightarrow\!2^+$&187&184&187&&$4^+\!\leftrightarrow\!2^+$&164&166&163$4^+\!\leftrightarrow\!3^-$&238&228&241&&$5^-\!\leftrightarrow\!3^-$&179&182&178$5^-\!\leftrightarrow\!4^+$&273&262&276&&$6^+\!\leftrightarrow\!4^+$&211&217&209$6^+\!\leftrightarrow\!5^-$&333&311&338&&$7^-\!\leftrightarrow\!5^-$&223&230&221$7^-\!\leftrightarrow\!6^+$&371&346&376&&$8^+\!\leftrightarrow\!6^+$&252&264&248$8^+\!\leftrightarrow\!7^-$&435&402&439&&$9^-\!\leftrightarrow\!7^-$&264&274&261$9^-\!\leftrightarrow\!8^+$&476&438&480&&$10^+\!\leftrightarrow\!8^+%
\phantom{1}$&289&304&284$10^+\!\leftrightarrow\!9^-\phantom{1}$&538&500&539&&$%
11^-\!\leftrightarrow\!9^-\phantom{1}$&303&315&299$11^-\!\leftrightarrow\!10^+$&582&538&582&&$12^+\!\leftrightarrow\!10^+%
$&325&342&320$12^+\!\leftrightarrow\!11^-$&640&604&635&&$13^-\!\leftrightarrow\!11^-%
$&340&354&336$13^-\!\leftrightarrow\!12^+$&685&644&680&&$14^+\!\leftrightarrow\!12^+%
$&360&377&354$14^+\!\leftrightarrow\!13^-$&738&709&730&&$15^-\!\leftrightarrow\!13^-%
$&376&390&371$15^-\!\leftrightarrow\!14^+$&783&750&775&&$16^+\!\leftrightarrow\!14^+%
$&393&411&388$16^+\!\leftrightarrow\!15^-$&833&812&822&&$17^-\!\leftrightarrow\!15^-%
$&410&424&405$17^-\!\leftrightarrow\!16^+$&878&854&867&&$18^+\!\leftrightarrow\!16^+%
$&426&443&421$18^+\!\leftrightarrow\!17^-$&925&912&912&&$19^-\!\leftrightarrow\!17^-%
$&442&457&437$0_2^+\!\leftrightarrow\!1_1^-$& 84& 42&150&&$%
0_2^-\!\leftrightarrow\!2_1^+$&$\phantom{1.}$8& 24&$\phantom{1.}$6$1_2^-\!\leftrightarrow\!0_1^+$& 31& 31& 31&&$%
1_2^-\!\leftrightarrow\!1_1^-$& 38& 37& 38$1_2^-\!\leftrightarrow\!2_1^+$& 49& 50& 47&&$%
1_2^-\!\leftrightarrow\!3_1^-$& 42& 43& 43$1_2^-\!\leftrightarrow\!0_2^+$& 22& 63& 21&&$%
2_2^+\!\leftrightarrow\!0_1^+$& 15& 17& 16$2_2^+\!\leftrightarrow\!1_1^-$&113& 86&111&&$%
2_2^+\!\leftrightarrow\!2_1^+$& 18& 25& 17$2_2^+\!\leftrightarrow\!3_1^-$&143&103&142&&$%
2_2^+\!\leftrightarrow\!4_1^+$&$\phantom{1.}$8& 32&$\phantom{1.}$4$2_2^+\!\leftrightarrow\!1_2^-$& 33& 71& 34&&$2_2^+\!\leftrightarrow\!0_2^+$& 91& 82& 92

Comparison with experimental transition probabilities
-----------------------------------------------------

[lcccrrrrrrrrrrr]{} &&&&\
\
&&& &&&& &$J_{\rm A}\rightarrow J_{\rm A}^\prime$ & &$ J_{\rm B} \rightarrow J_{\rm B}^\prime$&&Experim.&Crit.&&Experim.&Crit.&&Experim.&Crit.&&Rot. E1&$1_1^-\!\rightarrow\!2_1^+$&&$1_1^-\!\rightarrow\!0_1^+$&& $1.52\pm 0.14$ & 1.50&&$1.36\pm0.12$&1.47&&$1.49\pm0.26$&1.50&&1.42E1&$3_1^-\!\rightarrow\!4_1^+$&&$3_1^-\!\rightarrow\!2_1^+$&& &&&$1.11\pm0.18$&1.24&&&&&1.15E2&$4_1^+\!\rightarrow\!2_1^+$&&$2_1^+\!\rightarrow\!0_1^+$&& $1.60\pm 0.05$ & 1.63&&$\approx 1.76\phantom{22}$&1.66&&&&&1.60E2&$1_2^-\!\rightarrow\!3_1^-$&&$1_2^-\!\rightarrow\!1_1^-$&& $0.71\pm 0.10$& 1.10&&&&&&&&– E1&$1_2^-\!\rightarrow\!2_1^+$&&$1_2^-\!\rightarrow\!0_1^+$&& $1.49\pm 0.16$ & 1.57&&$1.24\pm0.09$&1.62&&&&&– E1&$2_2^+\!\rightarrow\!3_1^-$&&$2_2^+\!\rightarrow\!1_1^-$&& &&&$1.29\pm0.08$&1.20&&&&&– 

[lcccrrrrrrrr]{} &&&\
\
&&&& &E1&E2&Experim.&Crit.&Rot.&Experim.&Crit.&Rot.&Experim.&Crit.&Rot. &$3_1^-\!\rightarrow\!2_1^+$& $3_1^-\!\rightarrow\!1_1^-$& $0.69\pm 0.14\phantom{i}$& 0.55&0.57&&$5_1^-\!\rightarrow\!4_1^+$&$5_1^-\!\rightarrow\!3_1^-$& $0.98\pm 0.26$[^3] & 0.63 &0.60&$1.36\pm 0.23$[^4] &2.13 &2.42&$6_1^+\!\rightarrow\!5_1^-$&$6_1^+\!\rightarrow\!4_1^+$& &&&&&&$7.98\pm 1.17$&7.18&8.21&$7_1^-\!\rightarrow\!6_1^+$& $7_1^-\!\rightarrow\!5_1^-$& $0.56\pm 0.09$& 0.66&0.65 &$2.51\pm 0.15$&2.23 &2.56&$8_1^+\!\rightarrow\!7_1^-$ &$8_1^+\!\rightarrow\!6_1^+$& $<1.22\phantom{9i}$&0.68&0.66 &&& &$7.19\pm 0.72$& 7.86&8.50&$9_1^-\!\rightarrow\!8_1^+$&$9_1^-\!\rightarrow\!7_1^-$& $<1.71\phantom{9i} $&0.71 &0.67 &$2.82\pm 0.34$&2.38&2.67&$10_1^+\!\rightarrow\!9_1^-\phantom{0}$&$10_1^+\!\rightarrow\!8_1^+\phantom{0}$ &&&&&&&$7.78\pm 0.43$& 8.41&8.67&$11_1^-\!\rightarrow\!10_1^+$ &$11_1^-\!\rightarrow\!9_1^-\phantom{0}$&&&&$2.76\pm 0.27$ &2.53&2.67 &$9.35\pm0.62$& 8.64&8.73&$12_1^+\!\rightarrow\!11_1^-$ &$12_1^+\!\rightarrow\!10_1^+$&&&&$2.85\pm 0.25$ &2.62&2.68 &$9.06\pm0.47$&8.83&8.78&$13_1^-\!\rightarrow\!12_1^+$ &$13_1^-\!\rightarrow\!11_1^-$&&&&$2.15\pm 0.29$ &2.70&2.70 &$8.45\pm0.42$ & 9.01&8.82&$14_1^+\!\rightarrow\!13_1^-$ &$14_1^+\!\rightarrow\!12_1^+$&&&&$2.58\pm 0.17$ &2.79&2.71 &$9.84\pm 0.51$&9.15&8.86&$15_1^-\!\rightarrow\!14_1^+$ &$15_1^-\!\rightarrow\!13_1^-$&&&&$2.53\pm 0.17$ &2.94&2.73 &$9.69\pm0.65$ & 9.29&8.89&$17_1^-\!\rightarrow\!16_1^+$ &$17_1^-\!\rightarrow\!15_1^-$&&&&$2.78\pm 0.43$ &2.98&2.74 &$10.47\pm 1.34$&9.52&8.92&$18_1^+\!\rightarrow\!17_1^-$&$18_1^+\!\rightarrow\!18_1^+$ &&&&$3.22\pm 0.21$ &3.05& 2.74&&&&2.13&1.34&&1.17&2.03&&&&$<$5 &18&&31&$<$5

The Table \[T:e1\] shows a few values of the ratio of reduced matrix elements for transitions of the same multipolarity, that can be deduced from the available experimental information. In the same Table, the corresponding values calculated with the present model are also shown (columns “Crit.”), together with the ones expected for a reflection-asymmetric rigid rotor (“Rot.”).

The most direct check of the model predictions would come from the ratios of B(E2) values in the ground–state band. This is possible only in $^{224}$Ra, and only for the decays of the lowest $2^+$ and $4^+$ levels. With the experimental values reported in the NNDC tabulation [@nndc], $B($E2$,2^+\rightarrow 0^+)=97\pm 3$ W.u. and $B($E2$,4^+\rightarrow 2^+)=138\pm 8$ W.u., the experimental value of the ratio is $1.42\pm 0.09$, to be compared with the value 1.41 that is obtained from the calculated matrix elements of Tables \[T:30\] and \[T:e1\] (for $b=0.81$). We remind that, in the X(5) model [@iachello01], this ratio would be 1.59. For $^{226}$Ra, the lifetime of the $4^+$ state is known, but for the first excited state only an approximate value (without error estimate) is reported. Also in this case, the deduced ratio is consistent with the theoretical estimate (see Table \[T:e1\]). These results is encouraging, but would obviously need to be validate by a more extensive check, involving higher–lying levels, which, at the moment, is not possible.

A comparison of the two E1 transition from the lowest level $1^-$ to the $0^+$ and to the $2^+$ states is possible for the three isotopes, as well as for the E1 branches from the $3^-$ in $^{226}$Ra. All these amplitude ratios for transitions within the ground–state band, shown in the upper part of Table \[T:e1\], are in very good agreement with the calculated values. However, they are not significantly different from those expected for a rigid asymmetric rotor (as shown in the last column of Table \[T:e1\]) and also from those reported by Lenis and Bonatsos [@lenis06] on the basis of a rather different model.

![\[F:e1e2\] (Color online) Ratios of the absolute value of the transition matrix elements (normalized to the Weisskopf Unit) for E1 and E2 transitions in the g.s bands of $^{224,226}$Ra and $^{224}$Th, from $J_i$ to $J_i\!-\!1$ and $J_i\!-\!2$, respectively: $R_J(E1/E2)=$ $(J_i|\!|[\mathcal{M}(E1)|\!|J_i\!-\!1)/\mathcal{M}_W(E1)]
/[(J_i|\!|\mathcal{M}(E2)|\!|J_i\!-\!2)/\mathcal{M}_W(E2)]$. The dotted lines join the calculated values of the ratio (normalized to obtain the best fit with the ensemble of experimental values). The dashed lines join the values expected for a rigid rotor. The corresponding values deduced from the parameter free model of Ref. [@lenis06] are (apart for a possible staggering between even and odd $J_i$) almost identical to the rotational ones for large values of $J_i$ ($J_i>7$) and, for decreasing values of $J_i$, their trend reaches a minimum around $J_i=6$ and then increases slightly at lower values of $J_i$. ](biz_f8.ps){height="7.cm"}

We note that, when the transitions to be compared have the same multipolarity, the model predictions are parameter free, or – more exactly – only involve the model parameter $\beta_3^w/\beta_2^w$.

Instead, when the comparison concerns the ratios of the reduced matrix elements for E1 and E2 transitions deexciting the same level, the model predictions include a further normalization factor (the ratio of constants $C_1$ and $C_2$ of Eq.s \[E:14b\],\[E:15\]) which needs to be determined from the experimental data. This comparison is therefore less direct, but it is perhaps more significant, as we shall see in the following.

Results concerning the E1/E2 branches in the ground–state band are shown in the Table \[T:3\] and also depicted in Fig. \[F:e1e2\]. Experimental values of E1/E2 branching ratios in $^{224,226}$Ra and $^{224}$Th include those given in the NNDC tabulation [@nndc; @ackermann93] and later results from Ref. [@cocks99]. From these branching ratios we have deduced the absolute ratios – given in the “Exp.” columns of Table \[T:3\] – of the reduced matrix elements of E1 and E2 transitions, each of which is expressed in units of the corresponding Weisskopf estimate, $\mathcal{M}_W(EL)= 
(4\pi)^{-1/2}[3/(L+3)]\ (1.2 A^{1/3})^L$ e fm$^L$. In the same table are also shown the results of the model calculation at the critical point (Crit.), which have been normalized to obtain the best fit with the experimental values within the ground–state band of each nucleus. Values expected for a rigid asymmetric rotor (Rot.), normalized in the same way, are also shown. The $^{226}$Ra point at $J_i=5$ which, according to the authors themselves [@ackermann93], could be considered as a lower limit, has not been included in the fits.

For the ground–state band of $^{224}$Th (Fig. \[F:Th\],\[F:e1e2\]), we find a satisfactory agreement between the experimental values and the model predictions. In this case we have enough data to perform a $\chi^2$ test of goodness of fit, and we obtain $\chi^2/N= 1.17$ with $N=8$ degrees of freedom, corresponding to a confidence level of 31%. A fit with the rigid–rotor values would give a much larger value $\chi^2/N=2.03$, and a confidence level below 5%.

![image](biz_f9.ps){height="\linewidth"}

Also for the ground–state band of $^{224}$Ra, the few available experimental values (or limits) are not far from the results of the model, but more experimental data would be necessary for a significant comparison. Actually, as it was soon recognized [@marten90; @butler91; @egido91] the E1 transitions in $^{224}$Ra are rather weak compared to other nuclei in this region, and in particular their strengths are two orders of magnitude smaller than the corresponding ones in $^{224}$Th.

Instead, experimental values for $^{226}$Ra deviate significantly from the model predictions and approach those expected for a rigid rotor. This fact, combined with the slight upward deviation of level energies from the calculated curve for $J>14$, suggests that the critical point of the phase transition in the Ra isotopic chain can be situated somewhere below $A=226$, and probably close to $A=224$.

\[S:4.2\]The first excited $K=0$ band
-------------------------------------

As anticipated in Section  \[S:4.1\], no experimental information is available for non yrast levels of $^{224}$Th. For $^{224,226}$Ra isotopes, a few non-yrast levels are known from $\beta^-$ decay of $^{224,226}$Fr, from $\alpha$ decay of $^{228,232}$Th or from the $^{226}$Ra(t,p) reaction. Unique assignments of the spin and parity have been reported only for part of them. Some of these levels, which could be considered as members of the excited $K=0$ band (the $s=2$ band, in the X(5) expression) are reported, together with those of the yrast band, in Fig. \[F:6\], where also the main decay branches are indicated. In the same figure, the model predicted levels, and their expected $\gamma$ branches, are also shown.

We can immediately observe that non yrast levels predicted by the model are always lower than the experimental ones (but a comparably large discrepancy is observed also in the $s=2$ band of X(5) nuclei [@casten01; @kruecken02; @tonev04]). In the lower part of Table \[T:e1\], the calculated amplitude ratios for transitions from the excited $K=0$ band are compared with the corresponding experimental ones, if the levels $0^+_2$ and $1^-_2$ shown in the Fig. \[F:6\] are interpreted as belonging to it. Only the ratio of the two E1 transitions from the $1^-_2$ level of $^{224}$Ra and from the $2^+_2$ level of $^{226}$Ra are well consistent with the calculated value, while the corresponding ratios for the two E1 transitions from the $1^-_2$ level of $^{224}$Ra and for the two E2 transitions from the $1^-_2$ level of $^{224}$Ra seem to be significantly different from the model predictions (although the latter is subject to a large uncertainty, due to the presence of a competing M1 component in the $1^-_2\rightarrow 1_1^-$ transition).

As for the E1/E2 ratios for inter-band transitions, it is not obvious that the value of the parameter $C_1/C_2$ ought to be the same as for transitions within the ground–state band, but if we assume to be so, the E1/E2 ratios in the decay of the $1^-_2$ level of $^{226}$Ra differ by a factor of 2 from the calculated values: the ratios to the E2 amplitude $1^-_2\rightarrow 3^-_1$, with the normalization used in the the Table \[T:3\], are $(0.75\pm0.7)10^{-3}$ for the $1^-_2\rightarrow 0^+_1$ E1 transition and $(1.05\pm 0.07)10^{-3}$ for the $1^-_2\rightarrow 2^+_1$, to be compared with the theoretical values $0.29\cdot  10^{-3}$ and $0.46\cdot 10^{-3}$, respectively.

Therefore, if the first two levels of the excited $s=2$ band are tentatively identified with the $0^+_2$ and $1^-_2$ levels of $^{224}$Ra, their properties are not so well accounted for. For this fact, one can hypothesize different explanations. First, we remark that the identification of these levels as members of the $\beta$ band can be put in discussion. Actually, the $0_2^+$ level could result from other (collective or non collective) modes of excitation, as, [*e.g.*]{}, pairing vibration [@friedman74; @vanrij72], while the $1^-_2$ could correspond to (or be mixed with) the band head of the $K^\pi=1^-$ band. Otherwise, the observed disagreement could indicate that our model is unable to correctly predict states outside the ground–state band, in particular if they are not far from levels of the non axial modes having the same $J^\pi$. The simultaneous investigation of axial and non axial modes, as it has been performed, via the Extended Coherent State model, in the ref.s [@raduta03; @raduta06b; @raduta06c; @radutapredeal], is outside our present possibilities.

Conclusions
===========

An extension of Iachello’s X(5) model to the axial quadrupole + octupole deformation has been developed with the formalism introduced in our previous paper I [@bizzeti04]. Assuming that both $\beta_2$ and $\beta_3$ can vary within a two–dimensional well with rectangular borders, and with a proper determination of a free function of the model, the results are found to converge to those of X(5) when the interval available for $\beta_3$ tends to zero. The formalism is therefore suitable to describe the critical point of phase transitions involving at the same time the axial quadrupole and octupole deformation.

As anticipated in I, the principal aim of this second part of our work was the description of the transitional nuclei $^{224}$Ra and $^{224}$Th, which were proposed to be close to such a critical point.

Actually, in spite of the admittedly crude schematization of the bidimensional potential, the relative values of the excitation energies of levels (of positive and negative parity) in the ground–state bands of both $^{224}$Ra and $^{224}$Th are satisfactorily reproduced by adjusting the only available parameter (the aspect ratio $b=\beta_3^w/\beta_2^w$ of the potential well). A good agreement is obtained with $b=0.81$ for $^{224}$Ra and with $b=0.85$ for $^{224}$Th. Moreover, a good agreement is also obtained for the first part of the ground–state band of $^{226}$Ra with a lower value of the parameter, $b=0.68$: only above $J=14$ the experimental points deviate slightly from the calculated values, in the direction of the rigid–rotor curve (Fig. \[F:5\]).

The (few) known ratios of transition strengths in the ground state band, for electromagnetic transition of equal multipolarity (either E2 or E1) are in agreement with the model predictions. Unfortunately, only in a few cases the ratio of the reduced strengths for transitions of equal multipolarity can be deduced from the experimental data (see Table \[T:e1\]) and in these cases the values expected at the critical point are not very different from those of the rotational model.

In some more cases, the relative strength of two transitions of different multipolarity (E1 and E2), coming from the same level, can be deduced from the measured branching ratio. The comparison with the model requires in this case one more parameter, which has been determined by a best–fit procedure (see Table \[T:3\] and Fig. \[F:e1e2\]). But, in this case, the expected trend at the critical–point is significantly different from that of a rigid rotator.

The calculated critical–point values of the ratios E1/E2 are in a rather good agreement with the experimental results in the case of $^{224}$Th (Fig. \[F:e1e2\]), while for $^{226}$Ra the trend of empirical values is closer to the one expected for a rigid rotor. For $^{224}$Ra, the E1 transitions are very weak and experimental data are too scarce to permit a significant comparison with the model predictions.

New and more extensive measurements of the transition strengths either in $^{224}$Ra or $^{224}$Th would be highly desirable, for a more significant test of the model.

[^1]: We have now the occasion to correct a few misprints which escaped proofs revision in paper I:\
    Eq. (23c) should read $q_3= \left( L_3-p_\varphi-2p_\chi
    -3p_\vartheta \right) \ / \left( 4 u_0^2 \right)\ .$\
    In the Table VII, the 4th element of the 5th line should be ${\sin \theta_2 \sin \theta_3}\ / \ {\mathcal{J}_1}\ $. We apologize for these errors.

[^2]: This choice is different from the one adopted in I to describe the critical point in the octupole degree of freedom with a constant quadrupole deformation: in such a case, in fact, the proper limit for small octupole amplitudes does not correspond to the X(5) but to the Frankfurt model  [@eisen1], valid for small–amplitude octupole vibrations of a well deformed nucleus.

[^3]: from NNDC [@nndc] only. The $5^-\rightarrow 3^-$ (142 keV) $\gamma$ ray observed in the reaction data [@cocks99] appears to be contaminated by a close-lying transition from a different reaction, as it results from the intensity mismatch in the $5^-\rightarrow 3^-\rightarrow 2^+ (\rightarrow 1^-)$ cascade.

[^4]: from Ref [@ackermann93]. Not included in the fit.
