Regularities and symmetries of subsets of collective states
The energies of subsets of excited states in geometric collective models are investigated and found to exhibit intriguing regularities. In models with an infinite square well potential, it is found that a single formula, dependent on only the number of dimensions, describes a subset of states. The same behavior of a subset of states is seen in the large boson number limit of the Interacting Boson Approximation (IBA) model near the critical point of a first order phase transition, in contrast to the fact that these state energies exhibit a harmonic behavior in all three limiting symmetries of the IBA. Finally, the observed regularities in energies are analyzed in terms of the underlying group theoretical framework of the different models.
One of the overarching themes of the science of complex many-body systems is to understand the remarkable regularities they often exhibit and try to relate these to underlying symmetries of the system. In nuclei, this challenge is approached through the use of geometric and algebraic models that describe collective behavior of the nuclear system. There are a large number and variety of such models, each with seemingly unique properties and predictions. Nevertheless, careful analysis often shows relations among such models that have escaped notice and therefore leads to a better understanding of their mutual interrelationships and often, to experimental tests and constraints on their applicability.
In the present work, we will focus on the properties of subsets of states in nuclei. In general, states are of fundamental importance since they are easily observed experimentally in reactions such as few nucleon transfer Oothoudt (); Lesher (); des () or beta decay Asai (). Although not all states are collective in nature, they are always intrinsic excitations of the ground state condensate, and are free of some of the complications (such as centrifugal effects) present in other states. In the present work, we focus on the properties of a subset of collective states. We will show that broad classes of seemingly diverse models actually yield identical predictions for that subset of states. One upshot of this analysis will be the development of extremely simple, analytic eigenvalue expressions which, in one case, depend only on the dimensionality of the system and, in another, turn out to transcend the symmetry structure.
Our approach primarily exploits two classes of models, namely the Interacting Boson Approximation (IBA) model and geometric descriptions of nuclei at critical points of quantum phase transitions in their equilibrium structure. Therefore, we start by briefly recalling pertinent features of these models. The Interacting Boson Approximation model iba () describes collective structure in terms of bosons of angular momentum zero (-bosons) and two (-bosons) in the framework of an overall U(6) symmetry. Emerging from the U(6) symmetry group structure are three dynamical symmetries which have long been benchmark paradigms of structure : U(5), which gives vibrational structure characteristic of spherical nuclei, SU(3), which describes axially symmetric deformed rotors, and O(6), which pertains to deformed nuclei that are soft with respect to axial asymmetry (-soft). Shape/phase transitions in atomic nuclei were discussed gilmore () many years ago in the classical analog GK (); DSI () of the IBA.
To visualize these limiting symmetries and the transitions between them, it is common to place them at the corners of a symmetry triangle ricktri (), as shown in Fig. 1(top). In the IBA framework, a point of first order phase transition occurs between U(5) and SU(3), while a point of second order phase transition occurs between U(5) and O(6). The triangle is divided into two regions, spherical and deformed, by a narrow shape coexistence region IZC () extending around the line of first order phase transition connecting the two points mentioned above. In the classical limit of the IBA, obtained through use of the intrinsic state formalism GK (); DSI (), one can use Jolie89 () Landau theory to delineate a similar phase transitional behavior.
More recently, phase transitions have been investigated in a geometrical framework. The critical point symmetries E(5) IacE5 () and X(5) IacX5 () have been developed to describe phase/shape transitions between vibrational to -soft and vibrational to axially symmetric deformed, respectively, using special solutions of the Bohr Hamiltonian bohr (). These solutions are analytic and parameter free, except for scale. In Fig. 1(bottom), we indicate their position close to the critical point of a phase transition in a symmetry triangle for geometrical models. The concept of critical point symmetries (CPS) is supported by the observation of nuclei exhibiting such properties CZE5 (); CZX5 (); Kruecken (); tonev (); frank (); JPGreview (). Their success has spawned the development of numerous additional geometrical models, several of which offer analytic solutions and cover a wider range of structures both before and after the phase transitional point.
Links between the geometrical approach and the IBA formalism have also found renewed interest. A powerful method for solving the Bohr Hamiltonian numerically has recently been developed Rowe735 (), leading to an algebraic collective model Rowe753 (). Examples of the use of this method have recently been presented RWC (). The relationship between the algebraic collective model and the different limiting symmetries of the IBA has been studied in Refs. Thiamova1 (); Thiamova2 (). The connection between geometrical models spanning structure near E(5) and the IBA has also been investigated ramos ().
A key prediction of the CPS involves the energy of the first excited state. The nature of low-lying states is critical to understanding the structure of nuclei and changes in structure des (); mapping (); cejnar (). While there is some debate as to the nature pgarrett () of low-lying states, the fact that the energies of states evolve rather smoothly as a function of changing structure cannot be ignored. This has been pointed out previously in Ref. wchou (). We illustrate this in a similar way in Fig. 2, plotting the relationship between a level of the ground state band, (), and the first excited state, (), as a function of ()/() for all even-even nuclei with 40. Despite the enormous range of structures encompassed in the plot, an overall compact trajectory emerges.
The primary purpose of the present work is to investigate the energies of states in a wide range of models. In some cases the analysis applies to all collective states, in others to classes of states that act as bandheads for major families of states. For a broad class of models, we will discuss some remarkable regularities, develop analytic expressions for the eigenvalues of these states that depend only on the dimensionality of the system, relate these results to more general models, and discuss the group theoretical properties underlying these regularities. We begin within the framework of the different solutions of the Bohr Hamiltonian. We then do a similar analysis within the framework of the IBA and then finally investigate the links between these two different approaches. Some of this material has been previously summarized map (); letter ().
Ii states in solutions of the Bohr Hamiltonian
Numerous models are emerging which provide a reasonable description of nuclei using an infinite square well potential. E(5) and X(5) are special solutions of the Bohr Hamiltonian describing collective nuclear properties in terms of the shape variables and . Both take the potential in as an infinite square well but use different potentials in ; X(5) uses a harmonic oscillator potential in which has a minimum at whereas E(5) takes a potential independent of . Additional solutions which make use of infinite square well potentials in include Z(5) Z5 (), which uses a harmonic oscillator potential in with a minimum at = 30, Z(4) Z4 () where is frozen to 30, and X(3) X3 () where is fixed at 0.
In each of the infinite square well solutions, the energy eigenvalues are proportional to the squares of roots of the Bessel functions, , where the order is different for each solution. The orders of the Bessel functions obtained in the E(5), X(5), Z(5), Z(4), and X(3) models are summarized in Table I, along with the dimension, , of each model and the value of for states. The dimension effectively refers to the number of degrees of freedom of the model. For example, the five dimensional models are described by , and the three Euler angles.
In the X(3) and Z(4) models, we consider all excited states. In X(5) and Z(5), the solutions are obtained through an approximate separation of the and degrees of freedom. We consider those states arising from the solution, as they are directly related to the infinite square well potential. In the E(5) solution, we consider those states with = 0, that is, those states which correspond to base states on which major families of levels are built.
Traditionally, the excitation energy of the first excited state is used to set the overall scale for these models. However, in some cases, using a different normalization can reveal physics not otherwise very evident. In particular, one can sometimes see relations among states of the same angular momentum by normalizing to the first excited state of that spin. Hence, here, we scale to the energy of the first excited state, . It turns out that this approach allows the relative energies of these excited states to be well described by simple formulas. For the states we use the usual notation, , where corresponds to the ground state, denotes the first excited state, and so on, within each of the subsets of states, described above.
The energies of the states in the X(3) model are given in Table II. Normalizing to the first excited state, , the energies are described exactly by
where gives the sequencing of states defined such that the first excited state corresponds to , and is a scaling factor.
In the Z(4) model, normalizing to the first excited state, the states increase approximately as
where again, is a scaling factor dependent on the particular model.
The energies of states in the E(5), Z(5), and X(5) models, normalized to the state, are given in Table II. While these energies are quite different, by normalizing to the first excited state, the models produce exactly identical results, as seen on the right of Table II. These energies very closely follow the simple formula
Equation (3) is not an exact description of the energies in E(5), Z(5), and X(5); however, it does provide a very accurate approximation. Through =10, the model energies deviate from the expression given in Eq. (3) on the order of less than 0.1 .
The above empirical results can all be combined into a single, simple formula describing the states in any model with an infinite square well potential by
where is the number of dimensions and again depends on the model. Eq. (4) is exact only for . As mentioned previously, for other values of and low values of , it is a very accurate approximation.
Equation (4) stems from particular relations between the zeros of the Bessel functions which are involved in the solutions to the infinite square well models. Given the Bessel function , with roots , , 2, 3, …we empiricaly observe that the following approximate relation holds
where . We call the spectrum of the roots of , since this quantity corresponds to energy spectra in models describing atomic nuclei.
The relation given in Eq. (5) is exact only in the case , as one can see private () from the expansions of roots of Bessel functions given in AbrSt () (Eq. 9.5.12). Numerical results are shown in Table III. It is clear that the approximation deteriorates rather slowly with increasing (while keeping constant), while it deteriorates faster with increasing (while keeping constant).
For the Bessel function , with roots , , 2, 3, …(not used in this paper), we observe that the following approximate relation holds
where . This formula is exact only in the case . Similar to the results obtained for the Bessel function () relations, the approximation deteriorates rather slowly with increasing (while keeping constant), while it deteriorates faster with increasing (while keeping constant).
These results are applicable outside of the models discussed above. As an example, Eq. (4) applies to a recent model pairing () describing the critical point of a pairing vibration to pairing rotation phase transition. Here the energies of states, which span two degrees of freedom, the excitation energies of a particular nucleus and the masses along a series of even-even nuclei, can be described. In addition, these results would be applicable to hadronic spectra, which have recently been described hadron () in terms of roots of Bessel functions.
We can go further and generalize Eq. (3) for a broader range of models. The Bohr Hamiltonian can also be solved with potentials in of the form , giving the so-called E(5)- model BonE5 () and the X(5)- model BonX5 (), using the dependence characteristic of E(5) and X(5), respectively. These models allow for a description of structure between vibrational-like and the infinite square well solutions by increasing the power of in the potential. For example, in the E(5)- case, gives the vibrational limit and as the power of goes to infinity, the E(5) solution is reached. The evolution of both models is included schematically in Fig. 1(bottom). The predicted energies of the models are plotted in Fig. 3(top) normalized to the first state energy. As evident from Fig. 3(top), with increasing powers of in the potential, the energies evolve gradually towards the infinite square well predictions. However, again, the E(5) and X(5) related models seemingly give different results. If instead, we normalize each energy to that of the first excited energy, these models produce exactly identical results for a given potential, as seen in Fig. 3(bottom). The normalized energies can be reproduced with a generalized version of Eq. (3) given by
where is some number. The values of obtained by fitting the first two state energies in each model with Eq. (7) are included in Fig. 3. For a harmonic oscillator potential in , , since the bandhead energies increase linearly (i.e., when considering the ratio of energies the term in parenthesis in Eq. (7) disappears). As the power of in the potential is increased, the value of decreases, reaching the limiting value of 3 for the infinite square well.
Iii states in the euclidean algebras E()
In the solutions of the Bohr Hamiltonian with an infinite square well potential in the degree of freedom, the regularities observed for states can be related to the second order Casimir operator of E(), the Euclidean group in dimensions. In order to see this, one can consider in general the Euclidean algebra in dimensions, E(), which is the semidirect sum Wyb () of the algebra T of translations in dimensions, generated by the momenta
and the SO(D) algebra of rotations in dimensions, generated by the angular momenta
symbolically written as E(D) = T SO(D) Barut ().
The generators of E(D) satisfy the commutation relations
From these commutation relations, the square of the total momentum, , is a second order Casimir operator of the algebra, while the eigenfunctions of this operator satisfy the equation
Eq. (12) can be written as
the eigenfunctions of which are the Bessel functions AbrSt ().
The “radial” equations in the infinite square well models E(5) IacE5 (), X(5) IacX5 (), Z(5) Z5 (), Z(4) Z4 (), and X(3) X3 () are obtained, after the transformation of Eq. (13) has been performed, in the form of Eq. (15), with the order summarized in Table I.
In Z(5), where again , Eq. (14) and the corresponding order given in Table I would agree for and for . Again, this does not hold for any in general, but it is satisfied for . Thus bandheads with obey Eq. (15).
In the case of Z(4) Z4 (), in which , Eq. (14) and the corresponding order given in Table I for obtain the form and , respectively. They agree for , as already known Z4 (); therefore states with any even and obey Eq. (15).
Iv The IBA Hamiltonian and symmetry triangle
In order to describe a wider range of structures, it is useful to use a more general collective model than the specific solutions described above. To this end, we exploit the IBA model, which covers a gamut of structures with an economy of parameters. To do so, we use an IBA Hamiltonian of the form Volker ()
where , is the number of valence bosons, and is a scaling factor. The above Hamiltonian contains two parameters, and , with the parameter ranging from 0 to 1, and the parameter ranging from 0 to . The U(5) symmetry is given by , any , the SU(3) symmetry by and , and the O(6) symmetry by and . With this parameterization, the entire symmetry triangle, shown in Fig. 1, can be described, along with each of the three dynamical symmetry limits. Calculations in this work have been performed with the code IBAR IBAR (); ibar2 (), which has recently been developed to handle large boson numbers.
In Section V, we discuss the fact that, in all three limiting symmetries of the IBA, the energies of certain subsets of states exhibit harmonic behavior in the limit of large boson numbers. (This result has also been derived largeN1 (); largeN2 () using the coherent state formalism). In contrast, we will show, in Section VI, that near the critical point, the same subsets of states exhibit the behavior found in the framework of geometrical models in Section II. Furthermore, near the critical point these states in the large boson number limit of IBA exhibit certain degeneracies with alternate members of the ground state band, calling for further investigation.
V states in the limiting symmetries of the Interacting Boson Model
We begin an analysis in the IBA framework iba () by looking at the three dynamical symmetry limits, and analyzing the behavior of the states in the analytic formulae appropriate to each, especially in the large limit. Again, we consider a particular subset of states, looking for simple patterns common to all three symmetries, despite the diversity of the structures they describe.
In the case of U(5), states are labeled by their quantum numbers and , where is the seniority and is the number of triplets of bosons coupled to angular momentum zero. There are two classes of states, those with = 0, and those with = 1,2,3 which are always found degenerate with states. In the present work, we consider those states with = 0, that is those states that are not degenerate with states. These states correspond to base states on which major families of levels are built. In the U(5) limit, the energies of the states with =0 are proportional to the number of bosons, , corresponding to their respective phonon number (terms proportional to are also allowed, but are omitted in the present consideration); thus, the energies increase linearly.
In the SU(3) limit of the IBA, the position of the bandheads is determined by the second order Casimir operator of SU(3). The eigenvalue expression for states, in terms of the representation labels (, ), is given by = [ + + + 3( + )]. Here, we consider all states. Taking the state which belongs to the irreducible representation (irrep) at zero energy, and normalizing to the first excited state, which belongs to the irrep, we find for the lowest states the results shown in Table IV. From Table IV it is clear that at large boson numbers , we have two states with normalized energy 2, two states with normalized energy 3, three states with normalized energy 4, and so on. In other words, for large , the energies of the states in the SU(3) limit of the IBA grow linearly.
In the case of O(6), states are labelled by their quantum numbers and . One set of excited states is found within the multiplet structure of a given family, has values of 3 or larger and always appears degenerate with , and states. The other set of excited states forms the bandheads of the different families, has and are not degenerate with other states in the spectrum. We consider only those states with =0. In the O(6) limit of the IBA, the positions of the bandheads are determined by the second order Casimir operator of SO(6). The eigenvalue expression for = 0, = 0 states in terms of the major family quantum number is = , with , , …, 0 or 1. Taking the state which belongs to the irrep at zero energy, and normalizing to the first excited state belonging to the irrep, we obtain the results shown in Table IV. We observe that bandheads in the O(6) limit of the IBA also grow linearly in the limit of large boson numbers .
|Irrep (,)||()||Irrep (,)||()||Irrep (,)||()||Irrep()||()|
For all three IBA dynamical symmetry limits, the energies of these sets of states are given by
in the large limit. Thus, a single simple formula applies to all three dynamical symmetry limits of the IBA despite the fact that each describes very different structures. It is interesting that this harmonic behavior, appearing as a general feature of IBA spectra, not only at the three limiting symmetries but also in intermediate situations largeN1 (); largeN2 (), is strongly violated near the critical point, as we shall see in the next section.
Vi state energies and degeneracies in the shape coexistence region of the IBA
It has been recently observed letter () that the line describing the degeneracy = (where is the first excited state) in the symmetry triangle of the IBA for large () falls within the coexistence region of spherical and deformed shapes, slightly to the right of the critical line representing the first order phase transition between U(5) and SU(3). Similar results are obtained for the = and = degeneracies. These degeneracies are interesting not only because they can possibly be associated with underlying symmetries but also because the degeneracy between and found near the critical point of the IBA is also approximately given by the X(5) critical point model. In what follows, we investigate further the degree to which the IBA predictions near the critical point are related to simple analytic formulas.
Along the U(5)-SU(3) leg () a degeneracy between the state and the state occurs for for . This occurs very close to, but just beyond, the critical point ( for =250) of the phase transition. Numerical results of the above IBA calculation are given in Table V and compared with simple analytic formulas. In the first three columns, the first four excited states (normalized to the energy of the state) obtained in this calculation are compared to the predictions of Eq. (3), i.e., to the formula. Very good agreement is obtained up to . This result is what might be expected given the similarity of the IBA coherent state energy functional at the critical point with an infinite square well potential.
As mentioned above, successive ground band members with and odd are nearly degenerate with higher lying states. Trying to satisfy simultaneously the degeneracies = and = with the states obeying Eq. (3), and the levels of the ground state band obeying a general equation of the form , one obtains , i.e., the levels of the ground state band should grow as
where, again, is some number. In a very crude interpretation, the (+2) empirical result can be thought of as the average of the vibrational limit, where the energies go as , and the rotational limit, where the energies go as (+1).
The relevant connection to the expression is given by
The predictions of the above mentioned IBA calculation are compared to the predictions, which are normalized to the energy of the state, in the right section of Table V. Good agreement is obtained at lower levels, the deviation reaching 10% at . Also visible in the table are the approximate degeneracies = , = , = , = . These degeneracies hold to the 10 percent level for .
In summary, IBA states (in the large boson number limit) near the critical point on the U(5)-SU(3) line exhibit the same behavior seen in geometrical models involving infinite square well potentials. Furthermore, these states demonstrate approximate degeneracies with alternate members of the ground state band, calling for further investigations.
Working within the framework of both algebraic and collective models, we have investigated the energies of subsets of excited states, pointing out regularities within and similarities between the two different approaches. For models employing an infinite square well potential in the degree of freedom, a single formula is derived for a subset of excited state energies, dependent only on the number of dimensions of the model. The same regular behavior for states has been found in IBA calculations (in the large boson number limit) near the critical point of the first order phase transition between U(5) and SU(3), despite the fact that in all three limiting symmetries of the IBA (in the large boson number limit) the same states exhibit a harmonic behavior. Furthermore, these successive states near the critical point exhibit degeneracies with alternate yrast states, analogous to the near-degeneracy that occurs between the first state and the first excited state in X(5), calling for further investigations. Finally, the observed regularities in energies are discussed in terms of the underlying group theoretical framework of the different models.
The authors are thankful to M. A. Caprio for bringing private () to our attention the relevance of the asymptotic formula for the zeros of the Bessel functions and for useful discussions. This work was supported by U.S. DOE Grant No. DE-FG02-91ER-40609 and by the DOE Office of Nuclear Physics under Contract DE-AC02-06CH11357.
- (1) M. A. Oothoudt and N. M. Hintz, Nucl. Phys. A 213, 221 (1973).
- (2) S. R. Lesher, A. Aprahamian, L. Trache, A. Oros-Peusquens, S. Deyliz, A. Gollwitzer, R. Hertenberger, B. D. Valnion, and G. Graw, Phys. Rev. C 66 , 051305(R) (2002).
- (3) D.A. Meyer et al., Phys. Lett. B 638, 44 (2006).
- (4) M. Asai, T. Sekine, A. Osa, M. Koizumi, Y. Kojima, M. Shibata, H. Yamamoto, and K. Kawade, Phys. Rev. C 56, 3045 (1997).
- (5) F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987).
- (6) D. H. Feng, R. Gilmore, and S. R. Deans, Phys. Rev. C 23, 1254 (1981).
- (7) J. N. Ginocchio and M. W. Kirson, Phys. Rev. Lett. 44, 1744 (1980).
- (8) A. E. L. Dieperink, O. Scholten, and F. Iachello, Phys. Rev. Lett. 44, 1747 (1980).
- (9) R. F. Casten, Nuclear Structure from a Simple Perspective (Oxford University Press, Oxford, 1990).
- (10) F. Iachello, N. V. Zamfir, and R. F. Casten, Phys. Rev. Lett. 81, 1191 (1998).
- (11) J. Jolie, P. Cejnar, R. F. Casten, S. Heinze, A. Linnemann, and V. Werner, Phys. Rev. Lett. 89, 182502 (2002).
- (12) F. Iachello, Phys. Rev. Lett. 85, 3580 (2000).
- (13) F. Iachello, Phys. Rev. Lett. 87, 052502 (2001).
- (14) A. Bohr, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 26, no. 14 (1952).
- (15) R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000).
- (16) R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001).
- (17) R. Krücken, et al., Phys. Rev. Lett. 88, 232501 (2002).
- (18) D. Tonev, A. Dewald, T. Klug, P. Petkov, J. Jolie, A. Fitzler, O. Möller, S. Heinze, P. von Brentano, and R.F. Casten, Phys. Rev. C 69, 034334 (2004).
- (19) A. Frank, C.E. Alonso, and J. M. Arias, Phys. Rev. C 65, 014301 (2001).
- (20) R. F. Casten and E. A. McCutchan, J. Phys. G: Nucl. Part. Phys. 34, R285 (2007).
- (21) D. J. Rowe, Nucl. Phys. A 735, 372 (2004).
- (22) D. J. Rowe and P. S. Turner, Nucl. Phys. A 753, 94 (2005).
- (23) D. J. Rowe, T. A. Welsh, and M. A. Caprio, Phys. Rev. C 79, 054304 (2009).
- (24) D. J. Rowe and G. Thiamova, Nucl. Phys. A 760, 59 (2005).
- (25) G. Thiamova and D. J. Rowe, Czech. J. Phys. 55, 957 (2005).
- (26) J.E. García-Ramos and J.M. Arias, Phys. Rev. C 77, 054307 (2008).
- (27) E.A. McCutchan, N.V. Zamfir, and R.F. Casten, Phys. Rev. C 69, 064306 (2004).
- (28) P. Cejnar and J. Jolie, Phys. Rev. E 61, 6237 (2000).
- (29) P. E. Garrett, J. Phys. G: Nucl. Part. Phys. 27, R1 (2001).
- (30) W.-T. Chou, Gh. Cata-Danil, N.V. Zamfir, R.F. Casten, and N. Pietralla, Phys. Rev. C 64, 057301 (2001).
- (31) http://www.nndc.bnl.gov/ensdf/
- (32) D. Bonatsos, E.A. McCutchan, and R.F. Casten, Phys. Rev. Lett. 101, 022501 (2008).
- (33) D. Bonatsos, E.A. McCutchan, R.F. Casten, and R.J. Casperson, Phys. Rev. Lett. 100, 142501 (2008).
- (34) D. Bonatsos, D. Lenis, D. Petrellis, and P. A. Terziev, Phys. Lett. B 588, 172 (2004).
- (35) D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 621, 102 (2005).
- (36) D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 632, 238 (2006).
- (37) A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II: Nuclear Deformations (Benjamin, New York, 1975).
- (38) M. A. Caprio, private communication (2008).
- (39) M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
- (40) R. M. Clark, A. O. Macchiavelli, L. Fortunato, and R. Krücken, Phys. Rev. Lett. 96, 032501 (2006).
- (41) G.F. de Téramond and S.J. Brodsky, Phys. Rev. Lett. 94, 201601 (2005).
- (42) D. Bonatsos, D. Lenis, N. Minkov, P. P. Raychev, and P. A. Terziev, Phys. Rev. C 69, 044316 (2004).
- (43) D. Bonatsos, D. Lenis, N. Minkov, P. P. Raychev, and P. A. Terziev, Phys. Rev. C 69, 014302 (2004).
- (44) B. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974).
- (45) A. O. Barut and R. Raczka, Theory of Group Representations and Applications (World Scientific, Singapore, 1986).
- (46) M. Moshinsky, J. Math. Phys. 25, 1555 (1984).
- (47) Y. Alhassid and A. Leviatan, J. Phys. A: Math. Gen. 25, L1265 (1992).
- (48) A. Leviatan, Phys. Rev. Lett. 98, 242502 (2007).
- (49) V. Werner, N. Pietralla, P. von Brentano, R. F. Casten, and R.V. Jolos, Phys. Rev. C 61, 021301(R) (2000).
- (50) R. J. Casperson, IBAR code (unpublished).
- (51) E. Williams, R.J. Casperson, and V. Werner, Phys. Rev. C 77, 061302(R) (2008).
- (52) A. Leviatan, Ann. Phys. (NY) 179, 201 (1987).
- (53) J.E. Garcia-Ramos, C.E. Alonso, J.M. Arias, P. Van Isacker, and A. Vitturi, Nucl. Phys. A 637, 529 (1998).