Unified description of states in a large class of nuclear collective models
A remarkably simple regularity in the energies of states in a broad class of collective models is discussed. A single formula for all states in flat-bottomed infinite potentials that depends only on the number of dimensions and a simpler expression applicable to all three IBA symmetries in the large limit are presented. Finally, a connection between the energy expression for states given by the X(5) model and the predictions of the IBA near the critical point is explored.
The evolution of structure in many-body quantum systems and the emergence of collective phenomena is a subject that pervades many areas of modern physics. Recently, significant strides have been taken in the study of structural evolution in atomic nuclei, particularly with the discovery of nuclei that undergo quantum phase transitions phase1 (); phase2 () in their equilibrium shapes, and the development of descriptions of nuclei at the phase transitional point by simple, parameter-free critical point symmetries, E(5) e5 () and X(5) x5 (), that invoke flat-bottomed, infinite potentials. These descriptions have been well supported by experimental studies 134ba (); 152sm (); 150nd (); 154gd () and also have application molecules () in other systems such as molecules. Thus their study, and that of related models, potentially offers insight into a variety of phase transitional behavior.
E(5) and X(5) are analytic solutions of the Bohr Hamiltonian bohr () that describe collective properties in terms of two shape variables – the ellipsoidal deformation and a measure of axial asymmetry, . Both use an infinite square well in , differing in their dependence. Their success has given rise to numerous other geometrical models, many of which can also be solved analytically. Examples are X(3) x3 () in which the potential is frozen at , Z(4) z4 (), in which it is fixed at = , and Z(5) z5 () which has a minimum at = . In all these models, the number in parentheses is the effective dimensionality, . For example, the 5-dimensional models are couched in terms of , and the three Euler angles.
The nature of low lying states is critical to understanding the structure of nuclei. Their identification and interpretation is a subject of recent experimental des (); dorel () and theoretical work, both in the microscopic quasi-particle phonon model qpm () and in the relativistic mean field framework rmf ().
It is the purpose of this Letter to show that a large class of seemingly diverse models in fact share some remarkable similarities and to exemplify this by pointing out certain heretofore unrecognized but simple and general regularities in the energies of states which characterize these models. We will obtain a single, simple formula for all states in any flat-bottomed infinite potential that depends solely on the number of dimensions and another even simpler expression applicable to the dynamical symmetries of the Interacting Boson Approximation (IBA) model iba () (in the limit of large valence nucleon number, ) and compare these results to available data. We will also use solutions for a series of potentials with intermediate shapes to study the evolution from one description (formula) to the other. Finally, we will show that IBA predictions near the critical point, for large , approach the same energy expression for states as given by the X(5) critical point symmetry.
In Fig. 1, the IBA triangle is shown with its dynamical symmetries and division into spherical and deformed regimes, separated by a phase transitional region phase1 (). We include the Alhassid-Whelan(AW) arc of regularity arc () and the geometric critical point models E(5) and X(5), although the latter do not belong to the IBA space.
For any infinite flat-bottomed (i.e., square well) potential, the energy eigenvalues are proportional to the squares of roots of the Bessel functions () where the order is different for each case. In E(5) e5 (), one has
with . In X(5) one has x5 ()
for all bands. In Z(5) one has z5 ()
with for all bandheads. As a result, in all three cases, the bandhead energies are given by the squares of subsequent roots of .
These energies are given in Table I and are plotted on the left in Fig. 2. While these seem quite different, if we normalize each energy to that of the first excited state - that is, consider relative eigenvalues - we see that these different models produce exactly identical results as seen on the right in Fig. 2 and given by (see column Norm in Table I): 0, 1, 2.5, 4.5, 7, 10. These energies are approximately described by the simple formula
where is the ordinal number of the state and where depends on the model.
This remarkable result systematizes the states of this wide array of seemingly diverse models. However, its significance runs much deeper. Consider now the models Z(4) in four dimensions and X(3) in three.
In Z(4) the order of the Bessel functions is given by
with for all states, giving () for =0.
In X(3) one has x3 ()
leading to for .
The energies given by Z(4) and X(3) can again be described by simple formulas, satisfying
The results of Eqs. (4) and (Unified description of states in a large class of nuclear collective models) exemplify a universal formula that depends solely on the number of dimensions. The energies of all 0+ states in any flat-bottomed infinite square well potential in D dimensions are given by:
where, again, depends on the model. Figure 3(a) shows the results of Eq. (8) for = 3,4, and 5. These results reflect the deep relation between the order of the Bessel function solutions and the dimensionality of the potential, given by = (-2)/2. These results are exact for =3 and excellent approximations for low otherwise. (Compare Eq. (4) with Table II). These findings have applicability well beyond the models discussed above. For example, Eq. (8) gives the energies of all states in a recent model clark () of the critical point of a pairing vibration to pairing rotation phase transition in which the states span two degrees of freedom – excitation energies within a given nucleus and the sequences of masses along a series of even-even nuclei. Hadronic spectra have also been described brodsky () in terms of roots of Bessel functions.
The regularities found for states in solutions of the Bohr Hamiltonian with an infinite well potential in can be related to the second order Casimir operator of E(), the Euclidean algebra in dimensions, which is the semidirect sum Wyb () of the algebra T of translations in dimensions (generated by the momenta), and the SO() algebra of rotations in dimensions (generated by the angular momenta) symbolically written as E()=T SO(D) Barut (). The square of the total momentum, , is a second order Casimir operator of the algebra, and eigenfunctions of this operator satisfy z4 ()
whose solutions are Bessel functions .
In the original developments of the infinite square well models, the “radial” equations are obtained, after the above transformation has been performed, in the form of Eq. (10) with the corresponding order . In E(5) all states obey Eq. (10) (with ), in Z(4) agreement occurs for all with and z4 (), while in X(5), X(3) and Z(5) (with ) agreement is limited to states with , i.e. to the bandheads. This situation resembles a partial dynamical symmetry AlhLev () of Type I Lev98 (), in which some of the states (here, the states) preserve all the relevant symmetry.
Do we find these patterns in real nuclei? As examples, we use the well-studied X(5) candidates, Nd, Sm and Gd. Normalizing to the experimental energy in each nucleus, Table II gives the energies predicted by Eqs. (4) and (8). In each nucleus, there is indeed a state within 100 keV of the predicted energy. In Nd this is in fact the state, while in the rest, it corresponds to a higher lying state, for which the determination of the degree of collectivity through improved spectroscopic information poses an experimental challenge.
Eqs. (4) and (Unified description of states in a large class of nuclear collective models) are peculiar to infinite square wells and thus limited to a select number of nuclei. What behavior then characterizes other potentials? Of course, this cannot be solved in general but there is at least one other class of models where an easy solution can be derived, namely, in the dynamical symmetries of the IBA.
In U(5), the energies are simply proportional to the number of bosons defining their respective phonon number, thus () = . In SU(3), the eigenvalue expression for = states, in terms of the usual representation labels ( , ), is = [ + + + 3( + ]. In O(6), the corresponding equation, in terms of the major family quantum number , is (for =0, states) = (+4). The irreducible representations for SU(3) and O(6) and the corresponding energies are given in Table III. Taking successive (, ) and quantum numbers and the limit we obtain:
Thus, perhaps surprisingly, considering how different their structures are, and analogous to the infinite flat potentials, a single, simple formula applies to all three dynamical symmetries of the IBA which exhibit identical relative energy spectra of states in the large limit. Equations (8) and (11) are compared in Fig. 3(a).
|Irrep (,)||()||Irrep ()||()|
The description of states with a simple analytic formula extends beyond just the vertices discussed above. The behavior of states in the IBA symmetries can be associated with the chaotic properties of the IBA arc (). A regular region connects U(5) and O(6), resulting from the underlying O(5) symmetry. The regular behavior associated with U(5) and O(6) is preserved along the U(5)-O(6) leg, manifesting the relevant quasidynamical symmetries rowea (), until close to the point of the second order transition. An additional regular region, the AW arc of regularity, connects U(5) and SU(3) through the interior of the triangle (see Fig. 1). It has been conjectured chaos () that the regular region between SU(3) and the critical line is related to an underlying partial SU(3) symmetry. Included in Table II is a comparison between two well-deformed nuclei proposed jolie () to lie along the arc of regularity with the SU(3) predictions of Table III for the state and its nearly degenerate companion. Good agreement is observed, although the degree of collectivity of these states needs further experimental examination.
Most nuclei, however, are not described by dynamical symmetries or critical point models but lie somewhere in between mapping (). To study this and to see how Eq. (4) and Eq. (11) are related, or, better, how they evolve into one another moving across the symmetry triangle, we now consider a sequence of potentials of the form, V starting from , corresponding to U(5), and ending at either X(5) or E(5), which is successively approached by increasing powers of . The results are shown in Fig 3(b). As the potential flattens with increasing powers of , the results go from those for the U(5) limit to those for the infinite square well potentials. In each case, the normalized energies are well reproduced by a formula analogous to Eq. (4), namely (+) where for the IBA symmetries and drops to 3 for E(5)/X(5).
Finally, it has recently been discussed how the IBA with appropriate parameters approaches the predictions of the critical point symmetries as newPRL (). We use an IBA Hamiltonian in the form Werner ()
where , is the number of valence bosons, and is a scaling factor. Calculations were performed with the IBAR code ibar ().
Included in Fig. 3(b) is an IBA calculation with = -/2 (bottom leg of the triangle), = 250 and = 0.473 newPRL (), which is very close to the critical point ( = 0.472) of the phase transition region in the IBA. One sees that the IBA results are very close to those of Eq. (4) consistent with the flat nature of the IBA energy surface near the critical point. This same result is also illustrated in Fig. 2. The normalized IBA energies are included in Table I and show a strong similarity with the results of the infinite square well potentials. Thus, it appears that the regularities in energies obtained in the infinite square well potentials are not restricted to geometrical models and also occur near the critical point of the IBA.
In summary, we have discussed very simple regularities in excited energies which pervade a number of different models. The energies of excited states in flat-bottomed infinite potentials can be described by a single expression dependent on only the number of dimensions. These observed regularities in energies are linked to the second order Casimir operator of E(). Further, the energies of states in all three dynamical symmetries of the IBA are governed, in the large limit, by a single expression. These results were compared to experimental data. Potentials with shapes intermediate between the U(5) symmetry and the infinite square well give a smooth evolution in the energies. Finally, IBA calculations with large , near the critical point of the phase transition, exhibit nearly the same energy dependence for states as given by the infinite square well potentials.
The authors thank R.J. Casperson, E. Williams, and V. Werner for their expertise with the IBAR code and R.M. Clark and A.O. Macchiavelli 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.
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