Superdeformation and dynamical symmetry

Superdeformed bands are determined from symmetry-considerations, and their in-band E2 transition strengths are predicted. Those light nuclei are studied which were investigated from the experimental side, and comparison can be made with observed data. The SU(3) symmetry seems to organize the experimental finding to a good approximation.


Introduction
The connection between the stable deformation of nuclei and the symmetries has been enriched recently by two remarkable discoveries.
On the one hand it was found [1][2][3][4] that the SU(3) and Sp(3,R) symmetries emerge from large-scale shell model calculations based on symmetry-breaking interactions.Both model Hamiltonians and realistic interactions have been applied.Even no-core calculations resulted in a similar conclusion.This is a very important finding, which shows that the SU(3) and symplectic symmetries are essential in the strongly interacting many-nucleon systems, they emerge from large-scale calculations without any symmetryassumptions as a model input.Since the SU(3) symmetry uniquely determines the quadrupole shape of a nucleus [5,6] one can say that the nuclear dynamics is determined by a few stable shapes, with rotation (SU(3)) and vibrations (Sp(3,R)) associated to them.
On the other hand a new method has been invented for the determination of the shape isomers, which is based on the stability and self-consistency of the SU(3) symmetry [7], called SCS method.In fact a generalized version of SU(3), the quasi-dynamical (qd) SU(3) is investigated [8,9], which can be valid also in regions, where the real SU(3) is heavily broken.(When, however, SU(3) is valid it coincides with the qd-SU (3).)The qd-SU(3) is obtained from a Nilsson model calculation, i.e. from an eigenvalue problem of symmetry-breaking Hamiltonian.Nevertheless, it turns out that (in addition to the ground state), the SU(3) is stable and selfconsistent also for superdeformed (SD), hyperdeformed (HD), and other shape isomers.To some extent this finding resembles to that of the deformed harmonic oscillator calculations [10].Since, however, it is obtained from the eigenvalue problem of a symmetry breaking Hamiltonian, it also can be seen as an emergent symmetry.
The SCS method provides us with an alternative of the traditional energy-minimum calculation.One of its nice feature is that by providing us with a symmetry, one can apply a selection rule for the determination of the allowed clusterizations, i.e. reaction channels.
When the shape isomer state, e.g. the superdeformed one, has an SU(3) symmetry then the in-band electromagnetic transition is very easy to calculate, as it will be discussed more in detail below.The close connection between the rotor and the SU(3) models [5] explains the usefulness of the symmetry considerations in these phenomena.From the experimental side the in-band transition strength is a very strong signature of the large deformation.Its relation to the theoretically predicted values can be considered as an evidence for or against (e.g.) the superdeformed shape.
Inspired by these circumstances we investigate in this work the E2 transition strength in superdeformed bands of some light nuclei, which have been studied experimentally.We do so in the dynamical symmetry approximation, i.e. a single SU(3) representation is associated to each SD state.Of course, mixture of different representations are present in the realistic wave function, but the SU(3) symmetry is a good approximation in general, sometimes it is valid to an even larger extent than in the ground state.Therefore, it seems interesting to reveal how is the performance of the simple leading-representation approximation.
In what follows first we review very briefly the experimental situation along with the previous model calculations in section 2. Then we determine the superdeformed states from the symmetry-stability and consistency method in section 3. The in-band E2 transitions are presented in section 4, finally some conclusions are drawn in section 5.

Background
Predictions of superdeformed (SD) bands have been given based on different theoretical models around 40 Ca and in lighter alpha-like nuclei.Some of these suggestions have already been experimentally confirmed, SD candidate bands were experimentally found in this mass region in the 2000s such as in 28 Si [11], in 36 Ar [12], in 40 Ar [13], in 40 Ca [14] and in 44 Ti [15].The SD character of a previously found band [16] was proved in 42 Ca [17].
As follows we look at the history on a nucleus-by-nucleus basis focusing on the identification and description of superdeformed bands referring to the determination of the B(E2) transition strengths within the SD bands.
Antisymmetrized molecular dynamics (AMD) calculations predicted rotational bands in 28 Si based on different cluster configurations [18].Using a macroscopic-microscopic model SD minimum in the potentialenergy surface for 28 Si was identified [19].In [20,21] among others the SD shape isomer of the 28 Si was derived from Nilsson model calculations and its possible binary clusterizations were determined.Using the experimental results reported in the literature and based on analysis of own experimental data a set of candidate SD states has been identified [11], later completed with its experimentally suggested 0 + bandhead [22].The latest development in the history of the SD band of 28 Si is that establishing a new experimental methodology in [23] the authors sought to determine the transition strength between two states of the assumed SD band, which was predicted to be very large (188.1 W.u.) by AMD calculation [18].They obtained a significantly smaller value (<43 W.u.) for the upper limit of it, and therefore rejected the correspondence of the given band to the SD band of 28 Si.This conclusion was one of the motivations for writing our current paper.
A SD rotational band has been experimentally identified in 36 Ar by Svensson et al [12] where the configuration assignment were based on Cranked Nilsson-Strutinsky and large-scale shell model calculations.In [24,25] B(E2) values for all transitions involving members of the SD band are summarized.In the subsequent theoretical descriptions the observed energies and E2 transitions of the SD band were reproduced by the 32 S + α orthogonality condition model [26], moreover the structural understanding of the SD state of 36 Ar was reviewed [27] and based on a symmetry-governed approach the energy spectrum of the three valleys (GS, SD, HD) of 36 Ar was described [28].
A rotational band has been identified experimentally in 40 Ar by Ideguchi et al [13].Based on the deduced quadrupole moment, the band has been identified as a SD one.Its properties were characterized by cranked Hartree-Fock-Bogoliubov calculations.Description of triaxial SD states in 40 Ar including electric quadrupole transition strengths was given using AMD+GCM calculations and compared with experimental data in [29].
SD structure was predicted in 40 Ca by shell model calculations in [30,31].A candidate for a SD 8p-8h structure in 40 Ca has been identified in [14] and the features of this band are described by cranked relativistic mean field calculations.In [32] the transition quadrupole moments for the SD band in 40 Ca have been experimentally determined.With large-scale shell-model calculations quadrupole properties including B(E2) values of the SD band in 40 Ca were determined [33].Using AMD and the generator coordinate method (GCM) B(E2) values for the SD band of 40 Ca were calculated and compared with the experimental results [34].In [35] calculated B(E2) transition probabilities for the SD band in 40 Ca are compared with experimental data and largescale SM calculations in the framework of a shell-model-like approach.
In [36,37] the exited states of Ca 42 have been studied experimentally.Taniguchi [34] investigated the coexistence of various deformed states in 42 Ca using deformed-basis AMD.In [17] for the first time, the deformation of a SD candidate band was studied using the Coulomb excitation technique.The properties of low-lying states were studied via the measurement of E2 matrix elements.Experimental evidence for superdeformation of the band built on + 0 2 and of its triaxiality has been obtained.The results were compared with the results of large-scale shell model and beyond-mean-field model calculations.
Band structures were found in 44 Ti, together with the determination of in-band E2 strengths [38][39][40][41].Then the bands built upon the excited + 0 2 and + 2 3 states were both extended up to J π =12 + states and the energy-level scheme was reproduced by shell model calculations [15].Later on these bands are referred as superdeformed bands in 44 Ti together with their theoretical descriptions by AMD calculations [42].The triaxiality of the 44 Ti superdeformation was also reported in [43].Lifetimes of excited nuclear states were determined in 44 Ti using the recoil distance Doppler-shift technique [44].The measured lifetimes were converted to B(E2) values and compared with previous experimental data as well as with results of large-scale shell-model calculations.In [45] the spectrum of 44Ti was calculated by applying the multiconfigurational dynamical symmetry (MUSY).This symmetry is the common intersection of the shell, collective and cluster models for the multi-major-shell problem (i.e. an extension of the SU3 connetion by Elliott et al from 1958 for a single shell problem).In addition to the low-energy (shell-model) bands 4 core-plus-alpha bands, and the 28 Si+ 16 O resonance spectrum was described with a unified Hamiltonian, containing only two parameters (fitted to the experiments).

Shape isomers
As mentioned in the introduction, the validity of the real SU(3) symmetry is limited.It is valid only in the low energy region of light nuclei.In higher energy ranges and heavier nuclei, this symmetry is broken due to symmetry breaking interactions.However, a generalization of it, the quasi-dynamical (qd) SU(3) symmetry, can be applied in both ranges.This generalization allows us to study the stability and self-consistency of the qd-SU(3) symmetry, called the SCS method.Therefore, this method is suitable for determination of stable nuclear shapes (shape isomers) and their qd-SU(3) quantum numbers.Details of this calculation are given in [7], whose main steps are as follows: (i) Determine the Nilsson-orbitals as a function of the quadrupole deformation parameters (β, γ).
(ii) Obtain the many-particle state by filling in the Nilsson orbitals according to the energy minimum and Pauli-exclusion principle.
(iii) Expand the single-particle orbitals in terms of the asymptotic Nilsson-states.
(iv) Determine the effective (or qd) SU(3) quantum numbers (λ, μ) from the linear combinations of (iii) and from the relations of the large deformation.
(v) The effective quantum numbers can be translated to the parameters of the quadrupole deformation.
The details of this procedure are as follows.The asymptotic Nilsson-state is defined by the eigenvalue equation of the deformed Hamiltonian with cylindrical symmetry [46]: which contains in addition to the deformed harmonic oscillator potential spin-orbit, and angular momentum terms.The elongation parameter ò is introduced by Note, that the usual β parameter of deformation is related to ò: ò ≈ 0.95β.
For large deformations (|ò| > 0.3) the Nilsson orbitals approach straight lines [46].These asymptotic states are characterized by the quantum numbers: |Nn z ΛΣ > , where N is the total number of oscillation quanta, n z is the number of oscillation quanta in the z direction (n z = N, N − 1,...,0; n ⊥ = N − n z = n x + n y ).Λ is the projection of the orbital angular momentum on the z-axis (|Λ| = n ⊥ , n ⊥ − 2,...,1 or 0 ), and Σ is the projection of the spin [46].Λ and Σ are coupled to Ω.
For a triaxial shape the deformed harmonic oscillator potential is: with ω x ≠ ω y ≠ ω z .The ratio of the frequencies are: The volume conservation is expressed by: w w w w = x y z 0 3 .We diagonalize the triaxially deformed Hamiltonian in cylindrical coordinates [47][48][49], thus the Nilsson orbitals |ψ α > of a given deformation (ò, γ) are obtained as an expansion in terms of the asymptotic states: The effective (λ, μ), quantum numbers are determined from the equations: where the indices f and e refer to occupied (filled) and empty asymptotic Nilsson orbitals, respectively.The nonzero matrix elements of the second formula in equation ( 6) are given by [50] When evaluating (7) one has to pay attention to which orbital is filled and if the final orbital is empty or not.Λ and Σ can have negative values and each orbital is doubly occupied for the case of even-even nuclei.In the above equations only those oscillator shells enter which are open, i.e. closed oscillator shells do not contribute because the net sum of those in equations ( 6) and ( 7) is zero.
The SU(3) symmetry determines the parameters of the quadrupole deformation [51]: where N 0 stands for the number of oscillator quanta, including the zero point contribution: Here n is the sum of the U(3) quantum numbers: n = n 1 + n 2 + n 3 , and A is the mass number of the nucleus.This procedure essentially means the continuous variation of the quadrupole deformation (β in , γ in ), as an input for the Nilsson-model, and determination of the effective SU(3) quantum numbers or, from them, the corresponding β out , γ out quadrupole deformation.The result is a stair-like function, where the horizontal plateaus correspond to the shape isomers, indicating the stability and the self-consistency of the SU(3) symmetry (or deformation parameter).
Of course, these plateaus are not perfectly horizontal, so a plateau can have more than one very similar shape, i.e. qd-U(3) symmetry.The plateaus belonging to the different gammas are close to each other, so we have only indicated the qd-U(3) symmetry for the ones belonging to 0 and 60 degrees.On the left-hand side of figures 1-6, for each of the more definite plateaus, that qd-U(3) is indicated which belongs to its largest segment.On the enlarged figures on the right, all qd-U(3) are listed at the plateau corresponding to the SD state.The numerical calculations of this as well as of the next section were carried out with the program-package of [52].

In-band transitions
The in-band B(E2) value is given as i f where 〈(λ, μ)KI i , (11 is the second order invariant of the SU(3).These values are the matrix elements of the transition operator T (2) , which is the product of the quadrupole operator and the effective charge: T (2) =e ( eff ) Q (2) .The α 2 parameter was fitted to the lowest transition of the lowest band for each nucleus.The exception is 40 Ca, where the second lowest transition was used, because it had an exact transition strength.Table 2 gives the transition strengths.
To calculate the electromagnetic transitions, SU(3) representations are assigned to the experimental bands.
The 28 Si and 36 Ar bands and the GS band of 44 Ti have been described previously [54], thus their band assignment was clear.In the other cases, we always assigned to the ground state band the most deformed representation of the 0 ÿω model space, which had the appropriate spin content.For the SD bands, we assigned the representation corresponding to the second plateau.The exception is the 40 Ca, where the third plateau belongs to the SD band.(Here there is no ground state band, thus the second plateau belongs to the first excited band).
Please, note that the the SCS calculations of the previous section gives effective (quasi-dynamical) U(3) quantum numbers for the shape isomers.These are not necessarily coincide with the U(3) basis states of the shell model (e.g. they are not always integers).In such cases we associate the close-lying shell model U(3) symmetry to the state.E.g. in case of the 28 Si and 36 Ar nuclei the SD states of table 2 carry U(3) quantum numbers of the shell model.In both cases the relevant symmetry defines the largest possible deformation with 4ÿω excitations.
We mention here, that excited superdeformed bands can also be obtained from the considerations presented here.(See e.g. the 2 + band in 44 Ti.)It is also worth mentioning, that in the case of the 28 Si nucleus the fineresolution 16 O+ 12 C spectrum turned out to sit in the second, i.e. superdeformed valley of the energy-surface [21].It means that there not only an excited SD band, but a rich spectrum of the superdefomed valley has been described (in a unified way with the low-energy spectrum).Table 1.Fitting parameters of E2 transitions in Weisskopf units in the investigated nuclei.Table 2. Electromagnetic transitions in the investigated nuclei.The experimental data (and band names, with the exception of the 28 Si Candidate SD band and 44 Ti SD + 0 2 , SD + 2 3 bands, which we named) are from the [56] database except for the indexes: (a): [58], (b): [23], (c): [25], (d): [36], (e): [37], (f): [15], (g): [44], (h): [40], (i): [41], (j): [38].Energies are in MeV and B(E2) values are in Weisskopf units.θ indicates the moments of inertia in ÿ 2 /MeV units.q exp.are determined from the slope of the experimental bands, θ th.from (λμ) quantum numbers [7].a:b:c are the axial ratios of the nucleus calculated from U(3), and (β, γ) are the deformation parameters.

Summary and conclusions
In this work we have investigated the superdeformed bands of light nuclei.We concentrated on those isotopes which offered the possibility of making a comparison with experimental observations.The superdeformed states were obtained form the stability and self-consistency studies of the SU(3) symmetry.In particular, the quasi-dynamical SU(3) labels were obtained from a Nilsson model calculations as a function of the parameters of the quadrupole deformation.For each specific γ value β was changed systematically, and the SU(3) quantum numbers (which can be translated to output β and γ parameters) show some plateaus, corresponding to shape isomers.This symmetry-governed self-consistency method [7] is an alternative to the well-known energy-minimum calculations for the shape isomers, and the results are in good agreement with each other [7,55].
Our main aim was to obtain the in-band E2 transition strengths of the SD bands.In the semimicroscopic approach that we applied the microscopic model spaces are combined with phenomenological physical operators, which contain some parameters to fit to the experimental data.In particular, the E2 transitions have a single parameter, that can be considered as an effective charge.We fitted this parameter in each nucleus to a wellknown transition in a low-energy band.After fixing this value, all the in-band transitions are obtained as a parameter-free prediction.We wished to check how the SD transitions compare with the experimental observation.
We have found that the predicted transition strengths are comparable with the experimentally known values in the low-lying parts of the SD bands.For the high-spin parts the strengths are usually smaller, than the predictions of a single SU(3) symmetry.This observation is in line with the conclusion of some previous studies, indicating that the SD band structure is not completely uniform, rather configuration mixture can take place [15, 24, 25, 32-35, 43, 57].
We have compared also the moment of inertia obtained from the slope of the experimental data to the one determined by the SU(3) symmetry of the shape.(See table 2, q exp and θ th values.)Good agreement was found.
Since the E2 transition strength is considered as an important signature of the collectivity, e.g.superdeformation, sometimes its experimental value (in comparison with a model prediction) is considered to be a strong argument for or against the deformation in question.For the 28 Si case e.g. a candidate for the SD band was rejected recently [23], due to the large value (188.1 W.u.) of the AMD-predicted 4 + → 2 + transition.The SU(3) prediction is much smaller: 56.8 W.u., therefore, some care might be appropriate in drawing the conclusions based on model calculations.The same kind of prediction in the 36 Ar case turned out to be in a good agreement with the observation (c.f.table 2).
The consistency of the E2 transition strengths and the moment of inertia of the SD bands with those of the low-lying region seems to support also the picture suggested by the emergent SU(3) symmetry, i.e. the nuclei have a few equilibrium shapes with associated rotation and vibration.

Figure 1 .
Figure 1.Shape isomers of the 28 Si nucleus from the SCS method using effective U(3) quantum numbers.In both parts of the figure, the horizontal axis shows the β in input parameter; the vertical axis indicates the value of β out .The γ parameter is given in degrees, and steps in 10.The enlarged part on the right shows the U(3) symmetries of the plateau corresponding to the SD state.