---
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}
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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 