q-deformed vibrational limit of interacting boson model

Even–even near-spherical vibrational nuclei, with energy ratio R 4 1 + / 2 1 + = E 4 1 + / E 2 1 + in the range 2–2.5, are investigated within the framework of interacting boson model and q-deformed version of this model, using the U ( 5 ) dynamical symmetry. The parametrization of the U ( 5 ) Hamiltonian and q-deformed version are found leading to a description of energy spectra of 33 nuclei with a root mean square deviation from the experimental level energies less than 100 keV. The findings support the fact that we are very near to the critical point symmetries in the real q-deformed vibrational Hamiltonian.


Introduction
Early in the development of the theory of the nucleus there arose two very different models. The first model is the nuclear shell model developed by Mayer and Jensen which has its foundation in the single-particle motion of the constituent nucleons in a mean-field potential [1,2]. The nuclear shell model holds up well near magic nuclei. However, the model cannot describe features such as rotations and vibrations in nuclei, observed in regions of the nuclear chart distanced from the magic nuclei. The second model is the collective model of Bohr and Mottelson [3]. The collective model is one of the foundational models of nuclear structure. Its three main patterns, deformed-rotational, spherical vibrational, and γ-soft, continue to be benchmarks to which structure of nuclei are compared. The interacting boson model (IBM), based on the group theoretical approaches, is a semi-microscopic model of nuclei, positioned in-between nuclear shell model and collective models. The IBM has been quite successful in describing this nuclear collective motion in a wide range of the nuclear landscape. The basic assumptions of the IBM are well known. Arima and Iachello [4][5][6][7][8] assumed that an even-even nucleus consists of an inert core plus some valence nucleons which are paired together in states with angular momentum = L 0 and 2, corresponding to s and d bosons, respectively. The IBM possesses an overall ( ) U 6 symmetry with three dynamical symmetries (DSs) denoted by ( ) U 5 , ( ) SU 3 and ( ) O 6 and corresponding three special nuclear shapes; namely, a spherical vibrator, a deformed rotor, and a γ-soft, respectively. The nuclei may display behavior near these DSs, many lie in transitional regions between them. Generally, the shape evolves gradually in that one passes from a nucleus to its neighbor. However, the change can be very abrupt in certain cases. This is the key element for the appearance of a quantum phase transition.
The even-even Cd isotopes have long been considered one of the main examples of vibrational behavior. Recently, experimental studies have indicated significant deviations from this behavior, in selected two-and three-phonon states, along the Cd chain (A=108-126) [9,10]. Re-examination of the structure of the midshell Cd isotopes suggests that they may represent ( ) O 6 deformed γ-soft rotors rather than ( ) U 5 spherical vibrators. These findings have led to claims for the 'breakdown of the vibrational motion' in these isotopes and the need for a paradigm shift. This provides us with an excellent example of the breakdown of the vibrational behavior in general and the need to recheck the structure of isotopes near-spherical, vibrational nuclei.
On the other hand, quantum algebras (QAs) are deformed versions of the Lie algebras, to which they are reduced when the deformation parameter q is set equal to unity. The growing interest in the QAs is related to the correspondence of the properties of QAs and those of Lie algebras regarding the representation theory and their numerous physical applications. Biedenharn and Macfarlane, independently, introduced a q-deformed harmonic oscillator and proposed the q-analogue to the Jordan-Schwinger map [11,12]. QAs have now become a significant and widely used concept in nuclear physics. The QA ( ) SU 2 q [13][14][15][16], in particular, has been used for the description of rotational bands in the deformed and super-deformed nuclei. Deformed versions of the ( ) O 6 and ( ) U 5 DSs of the IBM have been discussed in [17]. Recently [18], the q-deformed Hamiltonian for the « ( ) ( ) SO U 6 5 transitional case in IBM is constructed by using affine ( ) SU 1, 1 q Lie algebra in the both IBM-1 and 2 versions and interacting boson fermion model. In addition, a deformed version of sdl-and sdll′ -boson models are produced (where l and l′ are any two angular momenta) [19].
In the present contribution, we examine the collective structure of low-lying states in near-spherical, vibrational nuclei, in terms of the ( ) U 5 DS using QAs. According to Gupta et al [20][21][22], for a complex q-deformation parameter, the analytical energy expression of ( ) O 6 q symmetry works equivalently as the numerically solvable general Hamiltonian of the IBM. The question is therefore created if the deformed version of the vibrational ( ) U 5 DS can accommodate the improved descriptions of the spectra of the nuclei where + + / R 4 2 1 1 in the range 2-2.5. In this work, the energy levels of 33 isotopes near-spherical, vibrational nuclei have been calculated with refitted IBM and q-IBM parameters and compared to new experimental results. Consequently, these nuclei are revised to find out if their excitation patterns are in consistent with the model predictions. The findings support that the real q-deformed vibrational Hamiltonian, which still has an analytical eigenvalue solution, takes us very close to the critical point symmetries. It provides us with analytical solutions of the symmetry breaking. Thus, an equivalent of a large numerical problem is obtained in terms of a simple analytical expression.

Vibrational nuclei: the U(5) dynamical symmetry
Symmetry arguments play a central role in theoretical nuclear physics, and group theory provides the mathematical tool to formulate symmetry principles. The DS is a type of symmetry in which the Hamiltonian is expanded in elements of a Lie algebra which is called the spectrum generating algebra [23,24]. A DS occurs in an algebraic model if the Hamiltonian can be expressed as a function of the Casimir operators (COs) corresponding to a chain of Lie subalgebras. If a system has a DS, the eigenvalue problem can be solved algebraically. In conclusion, there are three and only three types of DSs of IBM, The eigenstates of Hamiltonian (3) have quantum numbers which are the labels of irreducible representations of the algebras in the chain (1.1).
We have Here N is the total number of s and d bosons, n, n d and L are the seniority, the d-boson number, and the angular momentum, respectively. The multiplicity label  n counts the maximum number of d-boson triplets coupled to = L 0. This term contributes only to binding energies and not to excitation energies.
On the other hand, one must construct the q-analogues of the vibrational DS. The technique used is based on the notion of dual representations of pair of algebras (complementary subalgebras) [15]. Using the COs of ( ) the deformed vibrational IBM Hamiltonian can be written as where q, ¢ q and  q are deformation parameters and the parameters A, B, C and D are independent from deformation parameters. It should be noted that the eigenvectors of the deformed IBM Hamiltonian given by (6) are just the basis vectors (4). The eigenvalue of this Hamiltonian, (6), using the basis (4), is given by [15]  n n n = + -+ -- where q numbers are defined as = -- 1 and = t q e . Cseh [25] has made a study of q-breaking of DS using real and imaginary values of the parameter t. Also, complex values of the parameter t are allowed, which was first introduced for the one-dimensional ( ) SU 2 q IBM [22]. The q-deformation parameter t can be considered to take the real, imaginary and complex values, which means specifically Consequently, using (8.1), we have two cases of the energy formula (7). Case 1, which we will call 'one parameter real q-IBM', the deformation numbers q, ¢ q and  q are taken to be real, such that = ¢ =  = t q q q e . In this case, a brief calculation shows that Case 2, which we will call 'two parameters real q-IBM', the deformation numbers q, ¢ q and  q are taken to be real, such that = ¢ = t q q e and  = t q e . In this case, we have    1 is one of the most remarkable structural signatures and, furthermore, is one of the few whose absolute value is directly meaningful. By analyzing the energy ratios, we can identify the shape phases. This ratio has the limiting value 2 for a quadrupole vibrator, 2.5 for a non-axial g -soft rotor and 3.33 for an ideally symmetric rotor.
For the safe determination of the character of a collective band, especially in nuclei where mixing of different bands occurs, in which case the + + / R 4 2 1 1 ratio might be seriously affected, the ratios, [27], for the ground-state bands of each spin L were built to define the symmetry of the even-even nuclei: Exp where R((L+2)/L) Exp represents the experimental energy ratio between the L+2 and L states. These ratios show distinctive different behaviors in the vibrational, rotational, and γ-unstable limits. This ratio should be close to one for a rotational nucleus and close to zero for a vibrational nucleus, while it should have intermediate values for γ-unstable nuclei (0.1r0.35, 0.4r0.6 and 0.6r1.0 for vibrational, transitional and rotational nuclei, respectively). In addition, the ratio + + / R 4 0 1 2 is also used in this work, [28]. This ratio should be close to one for a quadrupole vibrator and close to 5/9 for a rotational nucleus, while it should be close to zero for non-axial g -soft rotor.   vibrational ( ) U 5 DS. As stated by the quadrupole vibration model, the 'ideal' nucleus would be satisfying the following conditions: (a) the ratio + + / R 4 2 1 1 is nearly equal to 2, (b) The two-phonon triplet pattern ( + 0 , + 2 , + 4 states) is present about twice the energy of the first excited + 2 state. (c) The two-phonon triplet is nearly degenerate. Moreover, we add the following two conditions: (d) the ratio + + / R 4 0 1 2 close to one however the ratio r is in the range 0.1r0.35, (e) The energy spectra are well approximated by (5). To measure the scale of degenerate of the two-phonon triplet pattern, S is defined to be the experimental value of the energy difference between the upper and lower member of the two-phonon triplet. In this work, we enlarge the vibrational region and assume that the two-phonon triplet could be considered nearly degenerate if  S 500 keV. In this way, we have obtained 33 nuclei manifestation of a very close vibrational structure. Hence, we test the ability of the real q-IBM vibrational Hamiltonian to improve the descriptions of the spectra of the nuclei where   The selection was carried out in fitting the experimental level energies with the ( ) U 5 prescription in the IBM and q-IBM frameworks. Linear and nonlinear least squares methods are used to fit the experimental energies for the IBM and the q-IBM respectively, therefore adjusting the parameters of (5) and (7). The first step is to determine the parameters in the standard IBM-1 Hamiltonian (5), using the first four low levels. With this initial determination of the parameters e, a, b and g, calculations are performed leading to the values of the next 18 energy levels. Searching for the nearest experimental levels, as possible, by a trial and error method, the determination of the other experimental levels is investigated by refitted the data step by step. The energy spectra of 33 nuclei having the collective structure of near-spherical, vibrational nuclei was analyzed and a list of the energy levels with their assignments is given for each nucleus in tables of appendix. Table 3. The parameters of the Hamiltonian (5) determined by least-square fitting to the experimental data. N is the boson number, n l is the number of levels used in fitting and e, a, b and g are the parameters of Hamiltonian (5) for each nuclei. All parameters are given in keV. Two quantities are used to determine the degree of conformity between experimental, IBM and q-IBM energy levels for each nucleus. The root mean absolute error is given by Table 4. The parameters of the Hamiltonian (7) determined by least-square fitting to the experimental data. A, B, C, D and t are the parameters of real q-IBM Hamiltonian (one q-parameter) for each nuclei. All parameters are given in keV.

IBM parameters
One parameter real q-IBM Isotopes  where E i exp and E i fit are the experimental and IBM (or q-IBM) energies in keV of the ith level and n l is the number of fitted levels. The second quantity is the root mean squared deviation which measures the average of the squares of the errors. The root mean squared deviation defined by Table 5. The parameters of the Hamiltonian equation (7) determined by non-linear least-square fitting to the experimental data.
A, B, C, D, t and t are the parameters of real q-IBM Hamiltonian (two q-parameters) for each nuclei. All parameters are given in keV.  where w i is a weighing factor. A constant value = w 0.01 i was taken, in accordance with a uniform uncertainty of 10 keV on the level energies. Q is a measure of the quality of an estimator and the values of Q closer to zero are preferable.

Two parameters real q-IBM Isotopes
In summary, the following results were obtained by the present study: Vibrational nuclei are characterized by having both neutrons and protons outside the closed-shell. The favorable conditions occur with the total number of bosons in the range 5-11 bosons. For larger boson number a rotational pattern is exhibited. The best fits for IBM and q-IBM Hamiltonian's parameters, used in the present work, are shown in tables 3-5.
A satisfactory degree of agreement is obvious between the calculated energy spectra of 33 isotopes, using IBM and real q-IBM Hamiltonian, and the corresponding experimental values, see tables A1-A17 and figures A1-A17.   Hamiltonians. The results of the two parameters real q-IBM suggest more exact outcomes, i.e. minimum Δ and Q values, in comparison with the IBM predictions. Figure 6 illustrates a comparison between the values of S using the experimental, IBM and two parameters real q-IBM values. Using two parameters real q-IBM, ten of these nuclei have values of D = -- An inspection of figures 7-10 shows that, in general, the + 2 1 experimental level energy is lower than the fitted value using IBM. For the complete set of nuclei considered in this work has a ratio = in the range 0.75 to 1 ( figure 7). In contrast the ratio = is closer to 1 Figure 9. Comparison between experimental, IBM and two parameters real q-IBM values of + E .  Figure 11. Comparison between experimental, IBM and two parameters real q-IBM values of / R .
(figures 8-10). In general, the theoretical R T 4,2 is lower than the experimental R E 4,2 value, using IBM. Figure 11 shows that result. However, using two parameters real q-IBM, figures 7-11 show that the + 2 1 fitted level energy is more closed to the experimental value ( figure 7). At the same time, the ratio + R , 0 2 + R 2 2 and + R 4 1 is still closer to 1. In general, the real q-IBM R T 4,2 is closer to the experimental R E 4,2 value. The more interesting result of this survey is that, the calculation of + 2 1 level energy is corrected by using real q-IBM. The critical point symmetry concept has been introduced by Iachello [29,30]. The critical-point description of the transition from a deformed rotor to a spherically vibrator, denoted as X(5), is located on the ( ) U 5 -( ) SU 3 leg of the Casten triangle and it was experimentally identified for the first time in the 152 Sm nucleus [31]. There are many nuclei in the transitional region with the X(5) critical point symmetry, such as 150 Nd, 154 Gd, 156 Dy [32].
The critical point of the phase transition between the ( ) U 5 and ( ) O 6 DSs defines the E(5) symmetry and it was experimentally identified for the first time in 134 Ba [33]. It has been found that there are many nuclei in the transitional region with the E(5) critical point symmetry, such as 102 Pd, 104 Ru, 106 Mo, 106 Cd, 108 Cd, 124 Te and 128 Xe [34].

Conclusions
In this research paper, 33 even-even near-spherical vibrational nuclei, with the energy ratio = in the range 2-2.5, are discussed within the framework of IBM and q-IBM, using the ( ) U 5 DS. The results indicate that the energy spectra of these isotopes can be reproduced quite well. The results of two parameters real q-IBM suggest more exact outcomes. The obtained results confirm the idea of the combination of different DSs in these isotopes. These results support the earlier work of Gupta for the one-dimensional SU q (2) group that the q-deformed Hamiltonian takes us very close to the other DSs (here, to critical point symmetries). According to our analysis, this improvement of calculations is due to the correction of + 2 1 level energy by using real q-IBM. Thus, an equivalent of a large numerical problem is obtained in terms of a simple analytical expression. In this sense, this work provides only a guideline and not a claim. Figure A1. Comparison of IBM and two parameters real q-IBM positive parity level energies in 106 Cd and 108 Cd with experiment.