The relation of the recently proposed E͑5͒ critical point symmetry with the interacting boson model is investigated. The large-N limit of the interacting boson model at the critical point in the transition from U(5) to O (6) The study of phase transitions is one of the most exciting topics in physics. Recently the concept of critical point symmetry has been proposed by Iachello [1]. These kinds of symmetries apply when a quantal system undergoes transitions between traditional dynamical symmetries. In Ref.[1] the particular case of the Bohr Hamiltonian [2] in nuclear physics was worked out. In this case, in the situation in which the potential energy surface in the ␤-␥ plane is ␥ independent and the dependence in the ␤ degree of freedom can be modeled by an infinite square well, the so-called E͑5͒ symmetry appears. This situation is expected to be realized in actual nuclei when they undergo a transition from spherical to ␥-unstable deformed shapes. The E͑5͒ symmetry is obtained within the formalism based on the Bohr Hamiltonian, but it has also been used in connection with the interacting boson model (IBM) [3]. Although this is not the form it was originally proposed [1], it has been in fact argued that moving from the spherical to the ␥-unstable deformed case within the IBM one should reobtain, at the critical point in the transition, the predictions of the E͑5͒ symmetry. This correspondence is supposed to be valid in the limit of large number N of bosons, but the calculations with the IBM should provide predictions for finite N as stated in Ref. [4]. In this paper, on one hand we calculate exactly the large-N limit of the IBM at the critical point in the transition from U(5) (spherical case) to O(6) (deformed ␥-unstable case). On the other hand, we solve the Bohr differential equation for a ␤ 4 potential. Both calculations lead to the same results and are not close to those obtained by solving the Bohr equation for an infinite square well [E͑5͒ symmetry]. We also show with two schematic examples that the corrections arising from the finite number of bosons are important. With this in mind, the IBM calculations still provide a tool for including corrections due to the finite number of bosons.In Ref.[1] the Bohr Hamiltonian is considered for the case of a ␥ independent potential, described by an infinite square well in the ␤ variable. In that case, the Hamiltonian is separable in both variables and if we set ⌿͑␤, ␥, i ͒ = f͑␤͒⌽͑␥, i ͒, ͑1͒where i stands for the three Euler angles, the Schrödinger equation can be split in two equations. [7][8][9] which allows to associate to it a geometrical shape in terms of the deformation variables ͑␤,␥͒. The basic idea of this formalism is to consider that the pure quadrupole states are globally described by a boson condensate of the form ͉g;N, ␤, ␥͘ = 1where the basic boson is given bywhich depends on the ␤ and ␥ shape variables. The energy surface is defined aswhere Ĥ is the IBM Hamiltonian. Here we are interested in the case in which the Hamiltonian undergoes a transitio...

We carry out an interacting boson model study of the Ru isotopes using a U͑5͒-SO͑6͒ transitional Hamiltonian with fixed parameters, where the variation is due only to the change in boson number N. Transitional behavior in 104 Ru is compared with recent predictions of an E͑5͒ critical symmetry, including a modified version with an alternative ␤ dependence for the T (E2) operator.

The phase transition around the critical point in the evolution from spherical to deformed γ -unstable shapes is investigated in odd nuclei within the interacting boson fermion model. We consider the particular case of an odd j = 3/2 particle coupled to an even-even boson core that undergoes a transition from spherical U(5) to γ -unstable O(6) situation. The particular choice of the j = 3/2 orbital preserves in the odd case the condition of γ -instability of the system. As a consequence, energy spectrum and electromagnetic transitions, in correspondence of the critical point, display behaviors qualitatively similar to those of the even core. The results are also in qualitative agreement with the recently proposed E(5/4) model, although few differences are present, due to the different nature of the two schemes. The study of phase transitions has recently received particular attention in nuclear structure. The concept of critical point symmetry has been first proposed in a number of cases by Iachello [1][2][3]. These symmetries apply when a quantal system undergoes transitions between traditional dynamical symmetries, as for example those characterizing situations described in terms of harmonic vibrations or rigid rotations. Although these symmetries have been obtained within the formalism based on the Bohr Hamiltonian [4], their concept has also been used in connection with the interacting boson model (IBM) [5].One of these critical point symmetries is associated with the transition between spherical and γ -unstable shapes. Within the IBM this can be obtained, for example, from the Hamiltonianwhich produces, varying the parameter x from 1 to 0, a transition between the two extreme situations characteristic of U(5) and O(6) symmetries. The corresponding second-order shape phase transition has been investigated within the Bohr collective model in Ref. [6]. The operators appearing in the Hamiltonian above are given bŷand N is the total number of bosons. For any value of x this Hamiltonian maintains the typical degeneracies of the O(5) symmetry. Consistently with this, within the IBM coherent state formalism [7][8][9], this Hamiltonian always produces an energy surface which is independent of the γ degree of freedom. In the β variable, the energy surface displays a spherical minimum in β = 0 for x larger than the critical value, while having a deformed minimum for values of x smaller than the critical value. At the critical point, the energy surface acquires a β 4 behavior [10,11], which is approximated by an infinite square well in the E(5) critical point symmetry [1] within the framework of the collective Bohr Hamiltonian.Recently Iachello has discussed a supersymmetrical extension of this concept, introducing the so-called E(5/4) model, where the boson part has a γ -independent square well potential and the boson-fermion coupling is taken as a spin (5) scalar interaction [12]. This solution describes the spectral properties of odd-even nuclei at the transition between spherical and γ -unstable shapes. In this Rap...

We investigate phase transitions in boson-fermion systems. We propose an analytically solvable model [E 5=12 ] to describe odd nuclei at the critical point in the transition from the spherical to -unstable behavior. In the model, a boson core described within the Bohr Hamiltonian interacts with an unpaired particle assumed to be moving in the three single-particle orbitals j 1=2, 3=2, 5=2. Energy spectra and electromagnetic transitions at the critical point compare well with the results obtained within the interacting boson-fermion model, with a boson-fermion Hamiltonian that describes the same physical situation.

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