The X(3) Model and its Connection to the Shape/Phase Transition Region of the Interacting Boson Model

The X(3) Model and its Connection to the Shape/Phase Transition Region of the Interacting Boson Model Dennis Bonatsos 1,D.Lenis 1, E. A. McCutchan, D. Petrellis 1,P.A.Terziev 3, I. Yigitoglu 4,andN.V.Zamfir 5 1 Institute of Nuclear Physics, N.C.S.R. Demokritos, GR-15310 Aghia Paraskevi, Attiki, Greece Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 0650-814, USA 3 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784 Sofia, Bulgaria 4 Hasan Ali Yucel Faculty of Education, Istanbul University, TR-34470 Beyazit, Istanbul, Turkey 5 National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania Abstract. A γ-rigid version (with γ =0) of the X(5) critical point symmetry is constructed. The model, to be called X(3) since it is proved to contain three degrees of freedom, utilizes an infinite well potential, is based on exact separation of variables, and leads to parameter free (up to overall scale factors) predictions for spectra and B(E) transition rates, which are in good agreement with existing experimental data for 17 Os and 186 Pt. The predictions of X(3) are furthermore compared to two-parameter calculations in the framework of the Interacting Boson Approximation (IBA) model. The results show that X(3) coincides with IBA parameters consistent with the phase/shape transition region of the IBA. The same turns out to hold also for the parameter independent (up to overall scale factors) predictions of the X(5)-β and X(5)-β 4 models, which are variants of the X(5) critical point symmetry developed within the framework of the geometric collective model, manifested experimentally in 146 Ce, 174 Os, and 158 Er, 176 Os respectively. 1 Introduction Critical point symmetries 1, ], describing nuclei at points of shape/phase transitions between different limiting symmetries, have recently attracted considerable attention, since they lead to parameter independent (up to overall scale factors) predictions which are found to be in good agreement with experiment 3 6]. The X(5) critical point symmetry ], in particular, is supposed to correspond to the transition from vibrational U(5)] to prolate axially symmetric SU(3)] nuclei, materialized in the N =90isotones 150 Nd, 15 Sm, 154 Gd, and 156 Dy. On the other hand, it is known that in the framework of the nuclear collective model 7], which involves the collective variables β and γ, interesting special cases occur by freezing the γ variable 8] to a constant value. 67

68 D. Bonatsos et al. In the present work we construct a version of the X(5) model in which the γ variable is frozen to γ =0, instead of varying around the γ =0value within a harmonic oscillator potential, as in the X(5) case. It turns out that only three variables are involved in the present model, which is therefore called X(3). Exact separation of the β variable from the angles is possible. Experimental realizations of X(3) appear to occur in 17 Os and 186 Pt. The results are also compared to Interacting Boson Model (IBM) 11] two-parameter calculations, showing that they are consistent with IBM parameters close to the phase/shape transition region of the IBM. The X(3) Model In the collective model of Bohr 7] the classical expression of the kinetic energy corresponding to β and γ vibrations of the nuclear surface plus rotation of the nucleus has the form 7, 9] T = 1 3 k=1 J k ω k + B ( β β γ ), (1) where β and γ are the usual collective variables, B is the mass parameter, J k =4Bβ sin ( γ 3 πk) () are the three principal irrotational moments of inertia, and ω k (k =1,,3)are the components of the angular velocity on the body-fixed k-axes, which can be expressed in terms of the time derivatives of the Euler angles φ, θ, ψ 9] ω 1 = sin θ cos ψ φ +sinψ θ, ω =sinθsin ψ φ +cosψ θ, (3) ω 3 =cosθ φ + ψ. Assuming the nucleus to be γ-rigid (i.e. γ =0), as in the Davydov and Chaban approach 8], and considering in particular the axially symmetric prolate case of γ =0, we see that the third irrotational moment of inertia J 3 vanishes, while the other two become equal J 1 = J =3Bβ, the kinetic energy of Eq. (1) reaching the form 9, 10] T = 1 3Bβ (ω 1 + ω )+ B β = B 3β (sin θ φ θ )+ β ]. (4) It is clear that in this case the motion is characterized by three degrees of freedom. Introducing the generalized coordinates q 1 = φ, q = θ, andq 3 = β, the kinetic energy becomes a quadratic form of the time derivatives of the generalized coordinates 9, 1] T = B 3 g ij q i q j, (5) i,j=1

X(3) Model and IBM Shape Transitions 69 with the matrix g ij having a diagonal form g ij = 3β sin θ 0 0 0 3β 0. (6) 0 0 1 (In the case of the full Bohr Hamiltonian 7] the square matrix g ij is 5-dimensional and non-diagonal 9,1].) Following the general procedure of quantization in curvilinear coordinates one obtains the Hamiltonian operator 9, 1] H = Δ + U(β) = B B where Δ Ω is the angular part of the Laplace operator Δ Ω = 1 sin θ ] 1 β β β β + 1 3β Δ Ω θ sin θ θ + 1 sin θ The Schrödinger equation can be solved by the factorization + U(β), (7) φ. (8) Ψ(β,θ,φ) =F (β) Y LM (θ, φ), (9) where Y LM (θ, φ) are the spherical harmonics. Then the angular part leads to the equation Δ Ω Y LM (θ, φ) =L(L +1)Y LM (θ, φ), (10) where L is the angular momentum quantum number, while for the radial part F (β) one obtains 1 d d L(L +1) β dβ β dβ 3β B ( E U(β)) ] F (β) =0. (11) As in the case of X(5) ], the potential in β is taken to be an infinite square well { 0, 0 β βw U(β) =, (1), β > β W where β W is the width of the well. In this case F (β) is a solution of the equation ( )] d dβ d β dβ + k L(L +1) 3β F (β) =0 (13) in the interval 0 β β W, where reduced energies ε = k =BE/ ] have been introduced, while it vanishes outside. Substituting F (β) =β 1/ f(β) one obtains the Bessel equation d dβ 1 β ( )] d dβ + k ν β f(β) =0, (14)

70 D. Bonatsos et al. 4 0 18 16 14 1 10 8 6 4 0 1.56 1 403.7 17.91 1 388.5 14.57 1 370.7 11.5 1 349.5 8.78 1 33.8 6.35 91.4 4.3 48.9.44 1.00 189.9 0.00 s = 1 1.14 1 17.18 1 13.57 10.9 65.1 4.4 15.5 18.4 7.37 140.1 4.83 80.6.87 s =.6 181. 18. 154. 14.19 10.5 10.56 73.5 7.65 s = 3 Figure 1. Energy levels of the ground state (s =1), β 1 (s =), and β (s =3) bands of X(3), normalized to the energy of the lowest excited state, 1, together with intraband B(E) transition rates, normalized to the transition between the two lowest states, B(E; 1 1 ). where L(L +1) ν = + 1 3 4, (15) the boundary condition being f(β W )=0. The solution of (13), which is finite at β =0,isthen F (β) =F sl (β) = 1 β 1/ J ν (k s,ν β), (16) c with k s,ν = x s,ν /β W and ε s,ν = ks,ν, wherex s,ν is the s-th zero of the Bessel function of the first kind J ν (k s,ν β W ) and the normalization constant c = βw J ν+1 (x s,ν)/ is obtained from the condition β W F 0 sl (β) β dβ =1.Thecorresponding spectrum, shown in Figure 1, is then E s,l = B k s,ν = Bβ W x s,ν. (17) It should be noticed that in the X(5) case ] the same Eq. (14) occurs, but with L(L+1) ν = 3 + 9 4.

X(3) Model and IBM Shape Transitions 71 From the symmetry of the wave function of Eq. (9) with respect to the plane which is orthogonal to the symmetry axis of the nucleus and goes through its center, follows that the angular momentum L can take only even nonnegative values. Therefore no γ-bands appear in the model, as expected, since the γ degree of freedom has been frozen. In the general case the quadrupole operator is T (E) μ = tβ Dμ,0(Ω)cosγ + 1 ] Dμ,(Ω)+D μ, (Ω)] sin γ, (18) where Ω denotes the Euler angles and t is a scale factor. For γ =0thequadrupole operator becomes 4π T μ (E) = tβ 5 Y μ(θ, φ). (19) B(E) transition rates B(E; sl s L 1 )= s L T (E) sl (0) L +1 are calculated using the wave functions of Eq. (9) and the volume element dτ = β sin θdβdθdφ, the final result being ( B(E; sl s L )=t C L 0 L 0, 0) I sl; s L, (1) where C L 0 L 0, 0 are Clebsch Gordan coefficients and the integrals over β are I sl; s L βw = βf sl (β) F s L (β) β dβ. () 0 3 The IBA Hamiltonian and Symmetry Triangle The study of shape/phase transitions in the IBA is facilitated by writing the IBA Hamiltonian in the form 13, 14] H(ζ,χ)=c (1 ζ)ˆn d ζ ] ˆQχ 4N ˆQ χ, (3) B where ˆn d = d d, ˆQχ =(s d + d s)+χ(d d) (),N B is the number of valence bosons, and c is a scaling factor. The above Hamiltonian contains two parameters, ζ and χ, with the parameter ζ ranging from 0 to 1, and the parameter χ ranging from 0 to 7/ = 1.3. With this parameterization, the entire symmetry triangle of the IBA, shown in Figure, can be described, along with each of the three dynamical symmetry limits of the IBA. The parameters (ζ,χ) can be plotted in the symmetry triangle by converting them into polar coordinates 15]

7 D. Bonatsos et al. O(6) ζ = 1, χ = 0 X(3) X(5)-β X(5)-β 4 X(5) U(5) ζ = 0 SU(3) ζ = 1, χ = - 7/ Figure. IBA symmetry triangle illustrating the dynamical symmetry limits and their corresponding parameters. The phase transition region of the IBA, bordered by ζ on the left and by ζ on the right, as well as the loci of parameters which reproduce the R 4/ ratios of X(3) (.44), X(5)-β (.65) 4], X(5)-β 4 (.77) 4], and X(5) (.90) ] are shown for N B =10. The line defined by ζ crit is also shown, lying to the right of the left border and almost coinciding with it. ρ = 3ζ 3cosθχ sin θ χ, θ = π 3 + θ χ, (4) where θ χ =(/ 7)χ(π/3). Using the coherent state formalism of the IBA 11, 16, 17] one can obtain the scaled total energy, E(β,γ)/(cN B ), in the form 18] ] E(β,γ) = β 1+β (1 ζ) (χ ζ 5ζ +1) 4N B 4N B (1 + β ) ζ(n ] B 1) 4N B (1 + β ) 4β 4 7 χβ3 cos 3γ + 7 χ β 4, (5) where β and γ are the two classical coordinates, related 11] to the Bohr geometrical variables 7]. As a function of ζ, a shape/phase coexistence region 19] begins when a deformed minimum, determined from the condition E β β0 0 =0, appears in addition to the spherical minimum and ends when only the deformed minimum remains. The latter is achieved when E(β,γ) becomes flat at β =0, fulfilling the condition 14] E β β=0 =0, which is satisfied for ζ = 4N B 8N B + χ 8. (6)

X(3) Model and IBM Shape Transitions 73 1.6 E H M e V L 1. 0.8 0.4 0.0 1 09 0 7 7 1 31 8 0.3 6 80 0.7 XH3L 1 8 H 1 3 L 8 0 H L 8 4 H 8 L 109H9L.6 H3L < 1 68 H7L 8H1L 186 Pt 1 09 1 7 5 0 6 0 0.3 1 4 IBA z=0.59, c=-0.7, N B = 11.0 E H M e V L 1.6 1. 0.8 0.4 0.0 1 15 1 8 8 6 33 5 6 4 XH3L 1 73 H 1 7 L 3 0 0 H 4 0 L 3 8 0 H 1 1 0 L 115H7L < 8 3 > 6 0.0 17 Os 1 15 1 8 4 1 0 5 IBA z=0.55, c=-1.05, N B = 10 Figure 3. Comparison of the experimental data (middle) to the X(3) predictions (left) and IBA calculations (right) for 186 Pt (top) and 17 Os (bottom). The thicknesses of the arrows indicate the relative (gray arrows) and absolute (white arrows) B(E) strengths which are also labelled by their values. The absolute B(E) strengths are normalized to the experimental B(E; 1 0+ 1 ) value in each nucleus. Experimental data taken from Refs. 1 3]. In between there is a point, ζ crit, where the two minima are equal and the first derivative of E min, E min / ζ, is discontinuous, indicating a first-order phase transition. This point is 0] ζ crit = 16N B 34N B 7. (7) The range of ζ corresponding to the region of shape/phase coexistence shrinks with decreasing χ and converges to a single point for χ =0, which is the point of a second-order phase transition between U(5) and O(6), located on the U(5) O(6) leg of the symmetry triangle (which is characterized by χ =0)atζ = N B /(N B ), as seen from Eq. (6). The phase transition region of the IBA is included in Figure.

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