HREPHAH 0H3HKA, 2004, moM 67, № 10, c. 1856-1860
Proceedings of the International Conference "Nuclear Structure and Related Topics"
THE X(5) CRITICAL POINT NUCLEI AND THE INTERACTING BOSON
MODEL SYMMETRY TRIANGLE
© 2004 N. V. Zamfir*. E. A. McCutchan, R. F. Casten
WNSL, Yale University, New Haven, USA Received January 21, 2004
Shape/phase transitions in low-energy nuclear spectra, the new critical point symmetries E(5) and X(5), and their empirical realization have recently been the subject of many experimental and theoretical investigations. With a set of polar coordinates, the precise location of the critical phase transition region and of X(5)-type nuclei can be mapped in the Interacting Boson Model symmetry triangle. An empirical mapping of the symmetry triangle for the N = 82—104 rare-earth nuclei is also obtained.
1. INTRODUCTION
Collectivity in low-energy nuclear motion is usually described relative to the geometrical models of harmonic vibrator [1], deformed symmetric rotor [2], and 7-unstable nuclei [3], or in terms of the dynamical symmetries of the Interacting Boson Model [4]: U(5), SU(3), and 0(6), respectively. The nuclear shape depends strongly on the number of valence nucleons: it evolves from spherical near closed shells to quadrupole deformed towards the middle of the shells. It has been shown [5, 6] that spherical-deformed transition regions in the Interacting Boson Approximation (IBA) Model from U(5) to SU(3) and from U(5) to 0(6) behave as first- and second-order phase transitions, respectively. The pronounced ft softness of the phase/shape transition region inspired F. Iachello to introduce a square well potential in the Bohr Hamiltonian to describe this region. By solving analytically the Schrodinger equation, he developed new solutions called critical point symmetries, E(5) [7] (for a spherical vibrator to a deformed 7-soft second-order phase transition) and X(5) [8] (for a spherical vibrator to axially symmetric rotor firstorder phase transition). Empirical examples close to these symmetries were found in nuclei: 134Ba [9, 10], 102Pd [11], 104Ru [12] as E(5) examples and 152Sm [13], 150Nd [14], and 156Dy [15] as X(5) examples. In fact, the search for new examples of X(5) symmetry continues to be the focus of many experimental and theoretical studies [16—19].
In this work, by introducing a set of polar coordinates, we will perform a mapping of the IBA symmetry triangle with a precise location of the phase/shape transition regions, of the X(5)-type nuclei, and of
the "empirical" trajectories corresponding to the fit of each isotopic chain with Z = 62—72.
2. PHASE/SHAPE TRANSITION IN THE IBA AND THE X(5)-TYPE NUCLEI
We use the following IBA Hamiltonian in the Extended Consistent-Q Formalism [20, 21]:
H (Z ) = c
Z
(1)
where NB is the total number of bosons, nd = $d and Qx = (s^d + d)s)+x(d^d)(2). This Hamiltonian contains two parameters, Z and x (c is only a scaling factor), and can describe all three IBA dynamical symmetries: Z = 0, any x for U(5), Z = 1, x = = -y/7/2 for SU(3), and ( = 1, X = 0 for 0(6). The Hamiltonian also describes, by numerical diagonal-ization, a large variety of transitional structures.
It is possible to provide a quantitative description of the IBA parameter space of Eq. (1) in the symmetry triangle by representing each set of parameters (Z, x) by polar coordinates (p, 9) [22]:
P =
V3 cos 0X — sin Ox
o = \ + ox,
(2)
n
E-mail: victor.zamfir@yale.edu
where 9X =
These coordinates allow for a convenient description of the entire IBA symmetry triangle. For example p = 0 for U(5) and p = 1,9 = n/3 for 0(6) and p = = 1,9 = 0 for SU(3). Figure 1 (left) represents the IBA symmetry triangle showing the definition of these
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THE X(5) CRITICAL POINT NUCLEI
1857
0(6)
Fig. 1. The IBA symmetry triangle: (Left) Definition of the polar coordinates p, 6 from Eq. (2). (Right) The phase transition region and the loci of X(5)-type spectra (trajectories labeled X(5)) [i.e., those IBA parameters which produce R4/2 = 2.90 and E(0+)/E(2+) ~ 6] for Nb = 10.
polar coordinates and the three dynamical symmetries in terms of the Hamiltonian parametrization in Eq.(1).
The total energy corresponding to the IBA Hamil-tonian can be obtained using the intrinsic state formalism and is expressed in terms of the intrinsic shape variables 3, y [23]. The study of the functional form for the total energy has shown that there is a phase/shape transition as a function of the control parameters Z and x. For fixed x, a phase transition occurs in the ground state energy at a critical value of the parameter Z = Zcrit: forx = 0, there is a first-order phase transition and for x = 0, there is a second-order phase transition.
In the first-order phase transition there is a region, where two minima, spherical and deformed, occur
0+
4+
2+
0+
X(5)
NB = 10 Z = 0.67, X = -0.7
Fig. 2. Comparison of the IBA results for NB =10, x = = -0.7, Z = 0.67 with the X(5) predictions. The thickness of the arrows is approximately proportional to the respective B(E2) values.
in the total energy. This is a region of phase/shape coexistence [24]. The phase/shape coexistence region starts, with increasing Z, where the deformed minimum develops in addition to the spherical one and ends, where the spherical minimum disappears. The critical point, where the first derivative is discontinuous, is Zcrit = 16NB/(34NB — 27). For example,
-Sm
>
<D 2 2
0 ^ 1
3
>
S 2
0 1
bq 1
80
85
90
N
95
Fig. 3. The evolution of R4/2, E(0+), and E(2+) for the Sm isotopic chain with N > 82 compared with the IBA results [25].
3
2
3
+
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ZAMFIR et al.
Ex, MeV
2.0 r
Gd
1.6 -
1.2 - K
0.8
0.4
0 1 1
□ 21 .41
x 2T
O 02
Dy
Er
Yb
Hf
8 10 12 14 16 8 10 12 14 16 8 10 12 14 16 8 10 12 14 16 8 10 12 14 16
Nb
Fig. 4. Comparison between empirical level energies (symbols) and the IBA calculations (solid lines) for Gd—Hf isotopes.
for % = —V7/2 and Nb = 10 the three values of Z characteristic of this phase/shape transition are 0.507, 0.542, and 0.512, respectively. The range of Z corresponding to the region of coexistence becomes smaller for smaller \x\ [24] and for x = 0 (U(5)—O(6) transition) converges to one point, the critical point of the second-order phase transition. This critical point is given by Z = /(2Nb — 2) and is equal to 0.556 for NB = 10. We refer to these ranges of Z and x as the "phase transition region" and are illustrated in Fig. 1 (right) for NB = 10. Different observables related to the order parameter [ should be discontinuous in the phase transition region. The discontinuities appear only for NB — to and for finite NB the transition is smoothed out, exhibiting an abrupt change rather than a discontinuity (see below).
As was mentioned in Introduction, a potential in the Bohr Hamiltonian very similar to the flat-bottomed potential in the phase/shape transition region, namely a square well potential, generates the new critical point symmetries which give parameter free predictions, except for scale. In the axially symmetric case the solution, called X(5), gives predictions for two important structural signatures, R4/2 =
= E(4+ )/E(2+) = 2.90 and Ro2 = E(0+)/E(2+) = = 5.65. A detailed comparison (presented in Fig. 2) of the X(5) predictions with IBA calculations, using
NB = 10, x--0.7, and Z ~ 0.67, shows that while
the values for these two ratios can be reproduced exactly in the IBA, some transition probabilities between the levels of the 0+-based sequence and the quasi-ground band are poorly reproduced. IBA calculations which reproduce the X(5) values of R4/2 and R02 are very close to the phase/shape transition region. Figure 1 also shows the locus of the IBA parameters for NB = 10 which reproduce exactly the X(5) value for R4/2, i.e., 2.90, and, within a reason-
able deviation, the other characteristic energy ratio E(0+)/E(2+) (labeled here the X(5) trajectory).
3. EMPIRICAL TRAJECTORIES OF PHASE/SHAPE TRANSITIONAL NUCLEI
A well-known example of an U(5)—SU(3) transition is the N ~ 90 region. In fact, the Sm isotopes were well described in the framework of the IBA [25] and 152 Smg0 was the first empirical example of a nucleus very close to the X(5) solution [13]. In Fig. 3, empirical basic observables, R4/2 = E(4+)/E(2+), and the energies of the head of 0+ and y quasibands for the Sm isotopic chain, are compared with the IBA results. The R4/2 ratio evolves from ~2.0, characteristic for the U(5) symmetry, to ^3.33, characteristic for SU(3) symmetry, with a sharp rise at N = 90. The energies of the intrinsic excitations 0+ and 2+ have a minimum also at N = 90. This point corresponds in the IBA to a calculation very near the critical point of the phase/shape U(5)—SU(3) transition [26] (see below).
The Gd and Dy isotopic chains exhibit a similar evolution [26] while the Yb and Hf isotopic chain show a different behavior. A detailed fit for the Gd, Dy, Er, Yb, and Hf isotopes with 82 < N < 104 was performed considering the basic observables, energies of the 2+, 4+, 2+, and 0+ states and available transition probabilities [27]. In Fig. 4, a comparison is presented of the empirical energies of these states with the IBA results. The agreement is impressive, including the description of the 0+ states which were poorly described in previous fits [28]. The other excited states are reproduced quite well. (The main exceptions are the 2^=0—0+ relative energies for N ~ ~ 90 nuclei which are larger in the calculations than the empirical values.) Electromagnetic transitions are also reproduced reasonably well.
THE X(Б) CRITICAL POINT NUCLEI
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G.S
G.4
-G.4
-G.S
-1.2
Gd
Dy
Er
Yb
Hf
1G
12
Nb
14
16
Fig. 5. The parameters Z and x for the IBA Hamiltonian in Eq. (1) describing the Gd—Hf isotopic chains with 84 < N < 104.
The fit parameters vary smoothly along each isotopic chain, as can be seen in Fig. 5. The trajectories in the symmetry triangle correspond
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