Investigation of Structure and Nuclear Shape Phase Transition in Odd Nuclei in a multi-j model

In this paper, we study the nature of the dynamics in second-order Quantum Phase Transition (QPT) between vibrational ( ( ) 5 BF U ) and  -unstable ( ( ) 6 BF O ) nuclear shapes. Using a transitional Hamiltonian according to an affine SU(1,1) algebra in combination with a coherent state formalism, Shape Phase Transitions (SPT) in odd-nuclei in the framework of the Interacting Boson Fermion Model (IBFM) are investigated. Classical analysis reveals a change in the system along with the transition in a critical point. The role of a fermion with angular momentum j at the critical point on quantum phase transitions in bosonic systems is investigated via a semi-classical approach. The effect of the coupling of the odd particle to an even-even boson core is discussed along with the shape transition and, in particular, at the critical point. Our study confirms the importance of the odd nuclei as necessary signatures to characterize the occurrence of the phase transition and determine the critical point's precise position.


I. Introductions
When the number of nucleons modifies, the quantum shape phase transition happens as the system's structure changes from one character to another. The existence of shape phase transitional behavior manifests in both even-even and odd-A nuclei. Phase transitions studies have mostly been done on even-even and odd mass systems within the interacting boson model (IBM)1 and the interacting boson fermion model (IBFM)2. The interacting boson model (IBM) has three dynamical symmetries signified by U(5), SU(3), and O (6), which are usually associated with spherical (U(5)), axially deformed (SU(3)) and -unstable (O(6)) shapes. A first-order QPT is observed between U(5)-SU(3), while the transition from U (5) to O(6) is second -order (continuous) *Corresponding Author name: A. Kargarian E-mail address: akargarian@aeoi.org.ir [1][2][3]. Quantum phase transitions in odd-A nuclei as a system of bosons and a single fermion were studied both classically and quantum mechanically in Ref. [4] It was displayed that the existence of the odd fermion powerfully affects the position and character of the phase transition, particularly at the critical point in which the phase transition happens. It is found that the existence of a single fermion changes the critical value at which the transition occurs for any amount of the coupling constant [5]. Several explicit studies by Alonso et al. [6][7][8][9][10] and Boyukata et al. [11] were done to study the quantum phase transitions in boson-fermion systems. They proposed an analytically solvable model E(5/12) to explain odd nuclei at the critical point in the transition from the vibrational to gamma-unstable phase. In Ref. [12], the role of a 2 single fermion on the shape phase transition was also studied. It was shown that unpaired particle describes the features at the critical points in atomic nuclei. They said that the outstanding features of shape phase transitions in odd-A nuclei don't change by coupling the single fermion to the even-even case. The effect of the coupling of the odd particle in the during of shape transition from U(5) to SU(3) and the transition from U(5) to O (6) in odd-A nuclei, in particular, at the critical point was discussed in Ref. [13]. The ground-state energy surface was obtained in terms of the shape variables (and) for the even core, and odd-even energy surfaces. It was found that the core-fermion coupling gives rise to a smoother transition than in the even-core case. D. Bucurescu and N. V. Zamfir studied quantum shape phase transition in odd-A nuclei. The evolution of level structures in oddmass nuclei and their even-even core nuclei is studied by correlations between ratios of excitation energies in both odd-mass nuclei. Xiang-Ru Yu et al. [14] investigated the effects of coupling of a single particle on shape phase transitions and phase coexistence in odd-even nuclei in the framework of the interacting bosonfermion model. Their results displayed that a single particle may impress different types of SPT in various ways. They have shown that phase coexistence can appear in the critical region and thus be taken as a sign of the shape phase transitions in odd-even nuclei. In this paper, a systematic study of spherical to γ-unstable shape phase transition and the effects of coupling a single particle on SPTs in the framework of IBFM is present. Particularly, how the existence of an odd particle can influence SPT is investigated. To consider an even-even boson core that performs a transition from spherical to γ-unstable shapes with a single nucleon in orbits to describe Odd-A nuclei. The E(5|12) model of that coupling of an even core that experiences a transition from spherical U(5) to γ-unstable O(6) situation to a particle moving in the 1/2, 3/2, and 5/2 orbitals is proposed by using the affine SU(1,1) Lie algebra and the Bethe Ansatz technique in the framework of the IBFM. Using the Catastrophe theory, the suggested Hamiltonians, and the coherent state of boson-fermion systems, we have created the corresponding energy surfaces in terms of the Hamiltonian parameters and the shape variables. The energy surfaces and equilibrium values along the shape phase transition are explored in the odd-A nuclei for different control parameters. This paper is organized as follows: Section 2 briefly summarizes theoretical aspects of transitional Hamiltonian and affine SU(1,1) algebraic technique. Sections 3 include calculating energy surface and sects. 4 and 5 are devoted to the discussion, Classical Analysis, and Conclusions.

II. The Applied Transitional Hamiltonians concerning the Affine SU(1, 1) Algebra
In this section, we study the coupling of an even core that undergoes a transition from spherical U(5) to  -unstable O(6) situation to a particle moving in the 1/2, 3/2, and 5/2 orbitals by using the affine SU(1,1) Lie Algebra and the Bethe ansatz technique within an infinite-dimensional Lie algebra in the framework of the interacting bosonfermion model. We have used the same formalism to extend IBFM calculation to the case that a j (1/2, 3/2, and 5/2) fermion coupled to a boson core. As shown in Refs. [9][10] , the E(5|12) model is constructed by considering the case of a collective core with the E(5) symmetry coupled with a single particle in a j = 1/2, 3/2, and 5/2 orbit. For better clarification, we show in Fig.1 the lattice of algebras relevant to our problem. By employing the generators of Algebra SU(1,1) and Casimir operators of subalgebras, the following Hamiltonian for the transitional region between algebras, respectively. This choice of the fermion space is such that one can usefully visualize the three angular momenta as arising from the coupling of a pseudospin 1/2 with a pseudo-orbital angular momentum 0 or 2. One recovers, in the extreme cases, the Hamiltonian Eq.

III. Calculating of Energy Surface
We can investigate the classical limit of the considered model in the framework of a coherent state [4,5,[8][9][10]. The cohesive state formalisms of IBM and IBFM [4,5,20,25,26] The  variable measures the axial deviation from sphericity , and the angle variable  controls the departure from axial deformation [1,[25][26]. Energy surface would determine using , , Intrinsic states for the mixed boson-fermion system can be constructed by coupling the odd single-particle states to the coherent states of the even core [4,5,20] , , The basis for the diagonalization of the fermion part is as where j m is the projection of the total angular momentum j on the symmetry axis. Intrinsic states for the hybrid boson-fermion system can be constructed by coupling the odd single-particle shapes to the coherent states of the even core [4,5,9,10,20]. So, the expectation value of B H in the boson condensate is given as The curly bracket in Eqs. (14) is a 6j-symbol. The basis for the diagonalization of the fermion part is as Eq. (11)

IV. Discussion and Classical Analysis A. Energy surface
The evolution of the energy surfaces along with the shape phase transition for the boson core and the odd-even systems considering different angular momenta j=1/2, 3/2, and 5/2, orbits are displayed in Figures 2 and 3 All energy surfaces have been calculated within a coherent state approach. To obtain energy surfaces, we need to specify Hamiltonian parameters Eq.(1).
The Hamiltonian parameters, namely  ,  ,  and  , used in the present work are shown in Table 1. In our calculation, we take ( ) The nature of the crucial point is changed in these control parameters. We see from the figures that the odd particle drives the system toward an unstable shapein these cases. This gives rise to an effective shift of the critical point. For example, for the odd-A systems, the critical issues move to , s critical c = 0.68 (for j=1/2,3/2 5/2 orbits), where the corresponding energy surface becomes very flat. Also, we show in Fig.(4), that the energy surfaces  -unstable cases present the same features like that for the even-even core.
In these vibrational and  -unstable cases, the of the extra particle is not changing the system's features. But the situation is different around the critical point. While the results at the critical end indicate clearly the extra particle drives the system toward  -unstable shape.
To display how the energy surfaces change as a function of the control parameter   Table.(1). Fig. (6) shows how the energy levels function as the control parameter s c and  evolve from one dynamical symmetry limit to the other.

V. Equilibrium values: classical order parameter
By minimization of ( ) , E  concerning  for any control parameter, the optimal values e  are obtained 4,5,11,14 . The e  is a suitable classical order parameter to determine the order and type of the SPT. For a first -order SPT, e  is not continuous as a function of the control parameter. While for a second-order SPT, e  is ongoing but es c   is not continuous [4,5,11,14].
The equilibrium value e  for a system is given by imposing the following conditions as [4,5] ( ) 0 The equilibrium values e  for the ( ) ( ) transition for even-even and boson-fermion systems with the different angular moments are calculated as  Phase transitions from the spherical to the  -unstable limit for odd nuclei in the frameworks of the IBFM and the Bohr collective model, with the bizarre nucleon lying in a = 3/2 shell, were considered in Refs. [6][7][8][9][10][11][12][13]. So, it must be, valuable and worthwhile to compare the present method and results to the method and results of these papers. The paper by Alonso et al. Refs. 6,10 investigated the phase transition around the critical point in the evolution from spherical to -unstable shapes in the cases where an odd = 3/2 and = 1/2 3/2 5/2 particle coupled to an even-even boson core that undergoes a transition from the spherical U(5) to the -unstable O(6) situation.
They used the coherent state formalism and semiclassical approach to get energy eigenvalues.
We also investigated the transition from the spherical to the -unstable limit in odd-A nuclei when only the boson core experiences the transition and fermions = 1/2 3/2 5/2 are coupled to the boson core. In our work, the eigenstates and energy eigenvalues for the

VI. Conclusions
In this paper, the effect of the coupling of a single fermion with a boson core that performs a transition from spherical to  -unstable shapes, in particular, at the critical point is investigated. The