Parameter-free predictions for the collective deformation variables beta and gamma within the pseudo-SU(3) scheme
Dennis Bonatsos, Andriana Martinou, S. Sarantopoulou, I.E. Assimakis, S. K. Peroulis, N. Minkov
PParameter-free predictions for the collective deformation variables β and γ within thepseudo-SU(3) scheme Dennis Bonatsos , Andriana Martinou , S. Sarantopoulou , I.E. Assimakis , S. K. Peroulis , and N. Minkov Institute of Nuclear and Particle Physics, National Centre for ScientificResearch “Demokritos”, GR-15310 Aghia Paraskevi, Attiki, Greece and Institute of Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 72 Tzarigrad Road, 1784 Sofia, Bulgaria
The consequences of the short range nature of the nucleon-nucleon interaction, which forces thespatial part of the nuclear wave function to be as symmetric as possible, on the pseudo-SU(3) schemeare examined through a study of the collective deformation parameters β and γ in the rare earthregion. It turns out that beyond the middle of each harmonic oscillator shell possessing an SU(3)subalgebra, the highest weight irreducible representation (the hw irrep) of SU(3) has to be used,instead of the irrep with the highest eigenvalue of the second order Casimir operator of SU(3) (thehC irrep), while in the first half of each shell the two choices are identical. The choice of the hwirrep predicts a transition from prolate to oblate shapes just below the upper end of the rare earthregion, between the neutron numbers N = 114 and 116 in the W, Os, and Pt series of isotopes,in agreement with available experimental information, while the choice of the hC irrep leads to aprolate to oblate transition in the middle of the shell, which is not seen experimentally. The prolateover oblate dominance in the ground states of even-even nuclei is obtained as a by-product. I. INTRODUCTION
Symmetries play an important role in shaping up theproperties of atomic nuclei [1–6]. On several occasionsthey impose specific forms of development of bands ofenergy levels as a function of the angular momentum,and/or strict selection rules on the electromagnetic tran-sitions allowed among the energy levels in a given nucleus.Especially important are cases in which the predictionsare parameter independent, since in these comparison tothe experimental data can lead to approval or rejectionof a theory in a straightforward way.The symmetry of the nuclear wave function is madeup by its spatial, spin, and isospin parts. Since thenucleons (protons and neutrons) are fermions, the totalwave function has to be antisymmetric. At this point theshort range nature [7, 8] of the nucleon-nucleon interac-tion plays an important role, namely it forces the spatialpart of the wave function to be as symmetric as possible[9]. As a consequence, it also shapes up the spin-isospinpart of the wave function, which has to correspond to theconjugate irreducible representation of the spatial part inorder to guarantee the antisymmetric nature of the totalwave function [10, 11].These ideas have been recently tested in the frame-work of the proxy-SU(3) scheme [12, 13] for heavy de-formed nuclei. Within this scheme the SU(3) symmetryof the harmonic oscillator [10, 14], exploited by Elliott[15–17] for the description of sd shell nuclei, which isknown to be broken by the strong spin-orbit interactionin higher shells [18], is recovered. In more detail, it isknown [18] that the spin-orbit interaction destroys theSU(3) symmetry of a given harmonic oscillator shell byforcing the Nilsson orbitals bearing the highest value ofthe total angular momentum j to go into the shell be-low, while the orbitals bearing the highest value of thetotal angular momentum j from the shell above invade the shell under consideration. The proxy-SU(3) schemerecovers approximately the SU(3) symmetry by replacingin each shell the intruder orbitals which have invaded theshell from above by the orbitals which have escaped fromthis shell into the lower one. The replacement is basedon orbitals differing by ∆ K [∆ N ∆ n z ∆Λ] = 0[110] in thestandard Nilsson [19, 20] quantum numbers N (the to-tal number of quanta), n z (the total number of quantaalong the body-fixed z -axis), Λ (the projection of the or-bital angular momentum along the body-fixed z -axis),and K (the projection of the total angular momentumalong the body-fixed z -axis). These 0[110] pairs of or-bitals are known to exhibit maximal spatial overlaps [21],similar to the ones of the Federman-Pittel pairs [22–24],which are known to play a crucial role in the developmentof nuclear deformation. Furthermore, they are knownto be related to enhanced proton-neutron interaction, asindicated by double differences of experimental nuclearmasses [25, 26].The short range of the nucleon-nucleon interac-tion leads to intriguing results within the proxy-SU(3)scheme, as discussed in detail in Ref. [27]. It turns outthat up to the middle of a given U(N) shell possessing anSU(3) subalgebra, the irreducible representation (irrep)containing the ground state band and in most cases addi-tional bands, like the γ band and the first K = 4 band,corresponds both to the hC irrep, i.e., the irrep with thehighest eigenvalue of the second order Casimir operator C SU (32 ( λ, µ ), where λ and µ are the Elliott quantum num-bers [15–17] characterizing the SU(3) irrep ( λ, µ ), and tothe hw irrep, i.e., the irrep possessing the highest weight[28]. Beyond the middle of the shell, however, this agree-ment is destroyed. The ground state band and its com-panions lie in the highest weight (hw) irrep of SU(3),which is not the same as the hC irrep carrying the maxi-mum eigenvalue of C SU (32 ( λ, µ ), except in the case of the a r X i v : . [ nu c l - t h ] A ug last four particles in the shell.This breaking of the particle-hole symmetry in theU(N) shells has two important physical implications: a)The majority of nuclei in a shell exhibit ground stateswith prolate deformations, thus resolving the long stand-ing question [29] of the reason causing the dominance ofprolate over oblate deformation in the ground states ofeven-even nuclei. b) A prolate to oblate transition is ob-served near the end of both the neutron 82-126 shell andthe proton 50-82 shell in rare earth nuclei, in agreementto experimental evidence [30–34] and theoretical predic-tions [13, 35].The crucial role played by the short range of the in-teraction and the Pauli principle is also manifested in ananother field of physics, namely metallic clusters [36–39].The potentials used in metallic clusters have a harmonicoscillator-like shape similar to that of the nuclear po-tentials, albeit with a depth smaller by several ordersof magnitude [36, 38, 39]. Metallic clusters are simpler,since in their case neither the spin-orbit interaction northe pairing force are present [40, 41]. The short rangenature of the interaction and the Pauli principle alone,within the SU(3) symmetry lead to experimentally ob-served [42–46] prolate shapes above the magic numbers[47–54] seen in alkali clusters, while oblate shapes areseen below these magic numbers. Details of this studywill be reported elsewhere [55]. In the framework of thisstudy [55] it is also proved that the highest weight SU(3)irrep for a given number of particles is the irrep charac-terized by the highest percentage of symmetrized parti-cles allowed by the Pauli principle in relation to the totalnumber of particles in the physical system. Therefore wesee that the short range of the interaction and the Pauliprinciple, when present in systems bearing an underlyingSU(3) symmetry, have important consequences of globalvalidity.In the present work we apply the particle-hole sym-metry breaking to the pseudo-SU(3) scheme [56–61], inwhich the SU(3) symmetry of the harmonic oscillator[10, 14] is restored by replacing the quantum numbers(number of quanta, angular momenta, spin) characteriz-ing the levels remaining in a nuclear shell after the de-sertion of the levels going into the shell below by new,“pseudo” quantum numbers, which map these levels ontoa full shell with one quantum of excitation less. In thisway the “remaining” levels (also called normal parity lev-els) acquire full SU(3) symmetry, while the intruder levels(also called abnormal parity levels, since they have parityopposite to the parity of the “remaining” ones) are setaside and treated separately. It should be noted that thereplacement of the “remaining” levels by their “pseudo”counterparts is an exact process, described by a unitarytransformation [62, 63]. In the lowest order approxima-tion, the intruder levels are treated as spectators, whilethe “remaining” levels, as described in the framework ofthe pseudo-SU(3) symmetry, derive the collective prop-erties of the nucleus. This is the approximation we aregoing to use in the present work. At a more sophisticated level, the intruder levels are taken into account [64, 65]as a single j -shell of the shell model. II. SU(3) IRREPS IN THE PSEUDO-SU(3)SCHEME
The first step needed for the description of a nucleuswithin the pseudo-SU(3) scheme is the distribution of thevalence protons (neutrons) into the appropriate pseudo-SU(3) shells and the intruder orbitals. One way toachieve this is by looking at the relevant Nilsson diagrams[19, 20, 66]. If one knows the deformation of the nucleus,it is easy to place the valence protons (neutrons) in theNilsson orbitals of the relevant shell and see how many ofthem belong to the “remaining” orbitals and how manygo into the “intruder” ones. In order to do so, one shouldhave in hand in advance an estimate for the deformationof the given nucleus. We are going to use as such esti-mates the deformations predicted by the D1S Gogny in-teraction [67], which are known to be in good agreementwith existing experimental data [68]. The distributionof protons and neutrons into pseudo-SU(3) shells and in-truder orbitals for the rare earth nuclei with Z = 50 − N = 82 −
126 is shown in Table 1.Once the number of valence protons (neutrons) in therelevant pseudo-SU(3) shell is known, the SU(3) irrep( λ p , µ p ) characterizing the protons and the SU(3) ir-rep ( λ n , µ n ) characterizing the neutrons can be foundby looking at Table 2, in which the SU(3) irreps corre-sponding to the relevant particle number are given. Thevalence protons of this shell live within a U(10) pseudo-SU(3) shell, which can accommodate a maximum of 20particles, while the valence neutrons live within a U(15)pseudo-SU(3) shell, which can accommodate a maximumof 30 particles. In Table 2 for each U(N) algebra and eachparticle number, two irreps are reported. One of them isthe highest weight irrep, labelled by hw, while the otheris the irrep possessing the highest eigenvalue of the sec-ond order Casimir operator C SU (32 ( λ, µ ), labelled by hC.Both of them have been obtained using the code of Ref.[28]. Details on the derivation and the physical meaningof the hw and hC irreps have been given in [70, 71].As we have already remarked, we are going to produceresults for two cases: a) both protons and neutrons livein shells in which the hw irreps are taken into account,b) both protons and neutrons live in shells in which thehC irreps are taken into account. In both cases it willbe assumed that the whole nucleus is described by the“stretched” SU(3) irrep ( λ p + λ n , µ p + µ n ) [64]. Theresults for the former case are shown in Table 3, whilethe results for the latter case are shown in Table 4.The procedure described above can be clarified by twoexamples.a) For Gd one sees in [67] that the expected de-formation is β = 0 . (cid:15) of the Nilsson model is related to β throughthe equation (cid:15) = 0 . β [20]. Looking at the standardproton Nilsson diagrams [66] for (cid:15) = 0 .
33 we see that the14 valence protons of
Gd are occupying 5 orbitals ofnormal parity and 2 orbitals of abnormal parity. Simi-larly, looking at the standard neutron Nilsson diagrams[66] for (cid:15) = 0 .
33 we see that the 12 valence neutrons of
Gd are occupying 4 orbitals of normal parity and 2orbitals of abnormal parity. These values are reported inTable 1, taking into account that each orbital accommo-dates two particles. Now from Table 2 one sees that 10protons (the ones with normal parity) in the U(10) shellcorrespond to the irrep (10,4), while 8 neutrons (the oneswith normal parity) in the U(15) shell correspond to theirrep (18,4). Therefore the total irrep for
Gd, reportedin Table 3, is (28,8). Notice that since both the valenceprotons and neutrons of normal parity lie within the firsthalf of their corresponding shell, the choice of the hw ir-reps vs. the hC irreps makes no difference, thus the sametotal irrep appears also in Table 4.b)For Yb one sees in [67] that the expected defor-mation is β = 0 . (cid:15) = 0 .
30 we see that the20 valence protons of
Yb are occupying 6 orbitals ofnormal parity and 4 orbitals of abnormal parity. Simi-larly, looking at the standard neutron Nilsson diagrams[66] for (cid:15) = 0 .
30 we see that the 24 valence neutrons of
Yb are occupying 8 orbitals of normal parity and 4orbitals of abnormal parity. These values are reported inTable 1. Now from Table 2 one sees that 12 protons (theones with normal parity) in the U(10) shell correspondto the hw irrep (12,0), but they belong to the (4,10) hCirrep if the highest eigenvalue of C SU (3)2 is considered.Furthermore 16 neutrons (the ones with normal parity)in the U(15) shell correspond to the hw irrep (18,8), butthey belong to the (6,20) hC irrep if the highest eigen-value of C SU (3)2 is considered. Therefore the total irrepfor Yb is (30,8) if the hw irreps are taken into accountand is reported in Table 3. However, if the hC irrepsare considered, the total irrep for the same nucleus is(10,30), reported in Table 4. Notice that since both thevalence protons and neutrons of normal parity lie withinthe second half of their corresponding shell, the choiceof the hw irreps vs. the irreps with highest eigenvalueof C SU (3)2 makes a big difference, resulting in a clearlyprolate ( λ > µ ) irrep in the first case and in a clearlyoblate ( λ < µ ) irrep in the second case. III. SHAPE VARIABLES
The labels λ and µ of the SU(3) irrep ( λ, µ ) are known[72, 73] to be related to the shape variables β and γ of the geometric collective model of Bohr and Mottel-son [74], in which β describes the degree of deviationfrom the spherical shape (which corresponds to β = 0),while γ describes the deviation from axial symmetry, with γ = 0 corresponding to prolate (rugby ball like) shapes, γ = π/ γ = π/
6. The cor-respondence is obtained by mapping the invariants of thetwo theories onto each other [72, 73]. In more detail, thesecond and third order Casimir operators of SU(3), whichare the invariants of the SU(3) model, are mapped ontothe invariants of the collective model, which are β and β cos 3 γ . The results of the mapping are [72, 73] γ = arctan (cid:32) √ µ + 1)2 λ + µ + 3 (cid:33) , (1)and [72, 73] β = 4 π A ¯ r ) ( λ + λµ + µ + 3 λ + 3 µ + 3) , (2)where the quantity in parentheses is related to the secondorder Casimir operator of SU(3) [1], C ( λ, µ ) = 23 ( λ + λµ + µ + 3 λ + 3 µ ) , (3)while A is the mass number of the nucleus and ¯ r isrelated to the dimensionless mean square radius [7], √ ¯ r = r A / . The constant r is determined from afit over a wide range of nuclei [75, 76]. We use the valuein Ref. [72], r = 0 .
87, in agreement to Ref. [76].The question of rescaling β in relation to the size ofthe shell appears at this point, as in the case of proxy-SU(3) [13]. On one hand, from Table 2 (as well as frommore extended tables reported in Ref. [13]), it is clearthat λ is proportional to the size of the shell, especiallyfor irreps with λ (cid:29) µ . On the other hand, from Eq.(2) it becomes evident that for the same kind of irreps, β is proportional to λ . If we had a theory in which allnucleons could be accommodated within a single SU(3)irrep, λ would have been proportional to A . Since this isnot the case here, it turns out that β has to be multipliedby a factor A/ ( S p + S n ), where S p ( S n ) is the size ofthe SU(3) proton (neutron) shell. In the present case,within the pseudo-SU(3) model, these sizes are 20 and30 respectively, thus β has to be multiplied by A/ β are determined by measuring theelectric quadrupole transition rate from the 0 +1 groundstate to the first excited 2 +1 state, B ( E
2; 0 +1 → +1 ), andare readily available in Ref. [68]. Since no direct exper-imental information exists for γ , we proceed, as in Ref.[13], to an estimation of it from the energy ratio of thebandhead of the γ band over the first excited state ofthe ground state band, R = E (2 +2 ) E (2 +1 ) , (4)using as a bridge the Davydov model for triaxial nuclei[8, 77, 78], in the framework of which one hassin 3 γ = 32 √ (cid:115) − (cid:18) R − R + 1 (cid:19) . (5)Empirical values of γ extracted in this way will be usedin the next section. IV. NUMERICAL RESULTS
Numerical results for the collective deformation vari-ables β and γ for several isotopic chains of rare earthsare shown in Figs. (1)-(4) and Figs. (5)-(8) respectively.In Fig. (1) the influence of the choice of the hw irrepsvs. the hC irreps on β is shown. While in all cases thepredictions for β are identical up to the middle of theneutron shell, the hw predictions lie systematically lowerthan the hC predictions in the upper half of the shell.The influence of the choice of the hw irreps vs. the hCirreps is more dramatic on the γ values, shown in Fig.(5).Again the predictions are identical up to the midshell,whereas beyond it they diverge dramatically. The hCpredictions immediately jump above the γ = 30 o borderbetween prolate and oblate shapes, thus giving a clearsign for a prolate to oblate shape transition in the middleof the shell, which is not seen experimentally, while thehw predictions cross the prolate to oblate border only inthe case of the W and Pt isotopic chains.The nearly horizontal segments appearing in both fig-ures correspond to the gradual filling of intruder neu-tron orbitals, as seen in Table 1. When a series of in-truder neutron orbitals gets gradually filled, the num-ber of the neutrons in the “remaining” orbitals remainsconstant and therefore the relevant SU(3) irrep remainsunchanged.An interesting detail can be noticed in Fig. (5). Inthe Ce, Sm, Dy, W, and Pt isotopic chains the hw andC curves meet again at N=122, while in the Yb isotopicchain they do not. This is due to the fact that in thecorresponding harmonic oscillator shells the hw and hCirreps, which are identical up to the midshell but followdifferent paths above it, do meet again at the end ofthe shell for the last 4 particles in each shell, as seen inTable 2. From Table 1 we see that for all isotopic chainsconsidered, at N = 122 one has 26 valence neutrons ofnormal parity, which means 4 particles below the endof the shell, where the hw and hC irreps become againidentical. Therefore for N = 122 the hw and hC neutronirreps will be identical in all chains of isotopes. However,one has also to take into account the proton irreps. FromTable 1 we see that the proton irreps will be identicalin the Xe to Er isotopic chains, which possess up to 10valence protons of normal parity, i.e. lie in the first halfof the U(10) shell. They will also be identical in the W,Os, and Pt isotopic chains, which lie within the last fourparticles below the full U(10) shell. Therefore the only isotopic chains in which the hw and hC irreps for protonsdiffer are the Yb and Hf chains. As a consequence, the γ values in Fig. (5) do not converge at N = 122 forthe Yb isotopes, because the hw and hC proton irrepsare different, although the hw and hC neutron irreps areidentical.The effects mentioned above can also be seen in Figs.(2) and (6), in which the β and γ predictions for all iso-topic chains from Xe to Pt are summarized. In Fig. (2)it is clear that both the hw and C choices give identicalpredictions for β up to midshell, while above midshellthe hw predictions for β lie systematically lower than theC predictions for all isotopic chains shown. In Fig. (6)it is clear that both the hw and C choices give identicalpredictions for γ up to midshell. Above midshell the Cpredictions jump up to oblate values immediately aftermidshell, while the hw predictions cross the prolate tooblate border of γ = 30 o much later, at N = 114-116,and this only happens for the Hf, W, Os, and Pt isotopicchains, in agreement to existing experimental evidence,which has been extensively reviewed in Ref. [13] andneeds not to be repeated here.In what follows attention is focused on the hw predic-tions of pseudo-SU(3).In Fig. (3) the hw pseudo-SU(3) predictions for β arecompared to predictions by the D1S Gogny interaction[67] and to the available experimental values [68]. Inall isotopic chains the hw pseudo-SU(3) predictions lielower than the D1S Gogny values near midshell, whilethey are higher than the D1S Gogny predictions near thebeginning and near the end of the neutron shell. Thesame comments can be made when comparing the hwpseudo-SU(3) predictions to the data in the Ce, Sm, Dy,Yb, and Pt isotopic chains, while in the W isotopic chainthe agreement of the hw pseudo-SU(3) predictions to thedata is impressive.In Fig. (7) the hw pseudo-SU(3) predictions for γ arecompared to predictions by the D1S Gogny interaction[67] and to the available empirical values obtained as de-scribed in Sec. 3. In several cases the hw pseudo-SU(3)predictions lie within the error bars of the D1S Gognypredictions, which are in very good agreement with theempirical values.In Fig. (4) the predictions for β of the hw pseudo-SU(3) and of proxy-SU(3) are compared. In general, thepredictions are very similar in the beginning of the neu-tron shell, while in the middle of the shell the proxy-SU(3) predictions are in general higher than the hwpseudo-SU(3) predictions. Finally, near the end of theshell, the hw pseudo-SU(3) predictions become higherthan the proxy-SU(3) ones. In the same figure, predic-tions by Relativistic Mean Field (RMF) theory [79] arereported. In the beginning of the neutron shell the RMFpredictions are lower than the hw pseudo-SU(3) andproxy-SU(3) predictions, while near midshell the RMFand proxy-SU(3) predictions are closer to each other.Near the end of the neutron shell the agreement betweenthe three theories is better than in the beginning of theshell.In Fig. (8) the predictions for γ of the hw pseudo-SU(3) and of proxy-SU(3) are compared. Remarkablesimilarity between the results of the two theories is seen,despite the different approximations made and the dif-ferent harmonic oscillator shells used in each of them.In particular, minima related to low values of µ in the( λ, µ ) irrep appear for both theories around N = 100-102 and N = 112. Considerable disagreement is seen inthe beginning of the neutron shell, where proxy-SU(3)predicts minima at N = 88, 94, while the hw pseudo-SU(3) shows a stabilized region around N = 90, relatedto the gradual filling of the abnormal parity neutron or-bitals. It should be remembered that the N = 90 iso-tones Nd,
Sm, and
Gd are the best examplesof the X(5) critical pointy symmetry [80], characterizingthe shape/phase transition [81, 82] between spherical andprolate deformed shapes.
V. CONCLUSION
In this work we have considered the implications ofparticle-hole symmetry breaking on the calculation of thecollective shape variables β and γ within the frameworkof the pseudo-SU(3) scheme. The particle-hole symme-try breaking has deep physical roots, since it is due tothe short range of the nucleon-nucleon interaction andthe Pauli principle. Therefore it is expected to appear ina general way in systems of many fermions, thus pavingthe way for further investigations in many body fermionicsystems in other branches of physics. A recent study inmetallic clusters, explaining the appearance of prolateshapes above the magic numbers and of oblate shapesbelow the magic numbers seen in these physical systemswill be published elsewhere [55]. Within the realm ofthe nuclear pseudo-SU(3) model, it has been shown thatthe particle-hole symmetry breaking leads to parameterindependent predictions for the nuclear deformation β which are in good agreement with relativistic and non-relativistic mean field predictions, as well as to the exper-imental data, where known. The parameter-independentpredictions of the pseudo-SU(3) scheme for the γ shape variable are even more dramatic. Particle-hole symme-try breaking leads to an answer to the long standingquestion of prolate over oblate deformation dominancein the ground states of even-even nuclei, as well as to theprediction of a prolate to oblate shape/phase transitionin rare earths around 114-116 neutrons, in good agree-ment with available empirical information. The pseudo-SU(3) predictions are also in good agreement with thepredictions of proxy-SU(3) [13, 35]. The compatibility ofproxy-SU(3) and pseudo-SU(3) predictions has also beendemonstrated recently in the study of quarteting in heavynuclei [83]. It is interesting that these two different ap-proximation methods of restoring the SU(3) symmetry inmedium mass and heavy nuclei lead to results which arevery similar to each other. Since pseudo-SU(3) has beenapplied mostly in the lower half of nuclear shells [84–86],the present work paves the way for its application in theupper half of nuclear shells, in which unique phenom-ena, as the prolate to oblate shape/phase transition takeplace.Restoration of an approximate SU(3) symmetry in nu-clei beyond the sd shell can also be achieved in the frame-work of the quasi-SU(3) scheme [87, 88], which has beenfound to be very appropriate for the description of N = Z nuclei [88]. It would be an interesting project to examinehow the particle-hole symmetry breaking appears withinthe quasi-SU(3) framework and how the prolate overoblate dominance and the prolate to oblate shape/phasetransition come out within this model. Acknowledgements
Financial support by the Bulgarian National ScienceFund (BNSF) under Contract No. KP-06-N28/6 is grate-fully acknowledged.
Author contribution statement
All authors contributed equally to this article. [1] F. Iachello and A. Arima,
The Interacting Boson Model (Cambridge University Press, Cambridge, 1987)[2] F. Iachello and P. Van Isacker,
The Interacting Boson-Fermion Model (Cambridge University Press, Cam-bridge, 1991)[3] A. Frank and P. Van Isacker,
Symmetry Methods inMolecules and Nuclei (S y G editores, M´exico D.F., 2005)[4] G. Rosensteel and D.J. Rowe, Ann. Phys. (NY) , 343(1980)[5] D. J. Rowe and J. L. Wood,
Fundamentals of NuclearModels: Foundational Models (World Scientific, Singa-pore, 2010) [6] V. K. B. Kota,
SU(3) Symmetry in Atomic Nuclei (Springer, Singapore, 2020)[7] P. Ring and P. Schuck,
The Nuclear Many-Body Problem (Springer, Berlin, 1980)[8] R. F. Casten,
Nuclear Structure from a Simple Perspec-tive (Oxford University Press, Oxford, 2000)[9] P. Van Isacker, D. D. Warner, and D. S. Brenner, Phys.Rev. Lett. , 4607 (1995)[10] B. G. Wybourne, Classical Groups for Physicists (Wiley,New York, 1974)[11] V.K.B. Kota and R. Sahu,
Structure of Medium MassNuclei (CRC Press, Taylor and Francis Group, Boca Ra- ton, 2017)[12] D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou,R. B. Cakirli, R. F. Casten, and K. Blaum, Phys. Rev.C , 064326 (2017)[13] D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou,S. Sarantopoulou, R. B. Cakirli, R. F. , and K. Blaum,Phys. Rev. C , 064326 (2017)[14] M. Moshinsky and Yu. F. Smirnov, The Harmonic Oscil-lator in Modern Physics (Harwood, Amsterdam, 1996)[15] J. P. Elliott, Proc. Roy. Soc. Ser. A , 128 (1958)[16] J. P. Elliott, Proc. Roy. Soc. Ser. A , 562 (1958)[17] J. P. Elliott and M. Harvey, Proc. Roy. Soc. Ser. A , 557 (1963)[18] M.G. Mayer and J.H.D. Jensen,
Elementary Theory ofNuclear Shell Structure (Wiley, New York, 1955)[19] S. G. Nilsson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. , no. 16 (1955)[20] S. G. Nilsson and I. Ragnarsson, Shapes and Shells in Nu-clear Structure (Cambridge University Press, Cambridge,1995)[21] D. Bonatsos, S. Karampagia, R. B. Cakirli, R. F. Casten,K. Blaum, and L. Amon Susam, Phys. Rev. C , 054309(2013)[22] P. Federman and S. Pittel, Phys. Lett. B , 385 (1977)[23] P. Federman and S. Pittel, Phys. Lett. B , 29 (1978)[24] P. Federman and S. Pittel, Phys. Rev. C , 820 (1979)[25] R.B. Cakirli and R.F. Casten, Phys. Rev. Lett. ,132501 (2006)[26] R. B. Cakirli, K. Blaum, and R. F. Casten, Phys. Rev.C , 061304(R) (2010)[27] D. Bonatsos, R.F. Casten, A. Martinou, I.E. Assimakis,N. Minkov, S. Sarantopoulou, R.B. Cakirli, and K.Blaum, arXiv: 1712.04126 [nucl-th][28] J. P. Draayer, Y. Leschber, S. C. Park, and R. Lopez,Comput. Phys. Commun. , 279 (1989)[29] I. Hamamoto and B. Mottelson, Scholarpedia ,10693 (2012)[30] R. F. Casten, A. I. Namenson, W. F. Davidson, D. D.Warner, and H. G. Borner, Phys. Lett. B , 280 (1978)[31] N. Alkhomashi et al. , Phys. Rev. C , 064308 (2009)[32] C. Wheldon, J. Garc´es Narro, C. J. Pearson, P. H. Re-gan, Zs. Podoly´ak, D. D. Warner, P. Fallon, A. O. Mac-chiavelli, and M. Cromaz, Phys. Rev. C , 011304(R)(2000)[33] Zs. Podoly´ak et al., , Phys. Rev. C , 031305(R) (2009)[34] J. Jolie and A. Linnemann, Phys. Rev. C , 031301(R)(2003)[35] D. Bonatsos, Eur. Phys. J. A , 148 (2017)[36] W. A. de Heer, Rev. Mod. Phys. , 611 (1993)[37] M. Brack, Rev. Mod. Phys. , 677 (1993)[38] V. O. Nesterenko, Fiz. Elem. Chastits At. Yadra , 1665(1992) [Sov. J. Part. Nucl. , 726 (1992)][39] W. A. de Heer, W. D. Knight, M. Y. Chou and M. L.Cohen, Solid State Phys. , 93 (1987)[40] K. Clemenger, Ellipsoidal shell structure in free-electronmetal clusters, Phys. Rev. B , 1359 (1985)[41] W. Greiner, Summary of the conference, Z. Phys. A:Hadr. Nucl. , 315 (1994)[42] J. Borggreen, P. Chowdhury, N. Keba¨ıli, L. Lundsberg-Nielsen, K. L¨utzenkirchen, M. B. Nielsen, J. Pedersen,and H. D. Rasmussen, Phys. Rev. B , 17507 (1993)[43] J. Pedersen, J. Borggreen, P. Chowdhury, N. Keba¨ıli, L.Lundsberg-Nielsen, K. L¨utzenkirchen, M. B. Nielsen, andH. D. Rasmussen, Z. Phys. D: At. Mol. Clusters , 281 (1993)[44] J. Pedersen, J. Borggreen, P. Chowdhury, N. Keba¨ıli,L. Lundsberg-Nielsen, K. L¨utzenkirchen, M. B. Nielsen,and H. D. Rasmussen, in Atomic and Nuclear Clusters,eds. G. S. Anagnostatos and W. von Oertzen (Springer,Berlin, 1995) 30[45] H. Haberland, Nucl. Phys. A , 415 (1999)[46] M. Schmidt and H. Haberland, Eur. Phys. J. D , 109(1999)[47] T. P. Martin, T. Bergmann, H. G¨ohlich and T. Lange,Chem. Phys. Lett. , 209 (1990)[48] T. P. Martin, T. Bergmann, H. G¨ohlich and T. Lange, Z.Phys. D: At. Mol. Clusters , 25 (1991)[49] S. Bjørnholm, J. Borggreen, O. Echt, K. Hansen, J. Ped-ersen and H. D. Rasmussen, Phys. Rev. Lett. , 1627(1990)[50] S. Bjørnholm, J. Borggreen, O. Echt, K. Hansen, J. Ped-ersen and H. D. Rasmussen, Z. Phys. D: At. Mol. Clusters , 47 (1991)[51] W. D. Knight, K. Clemenger, W. A. de Heer, W. A.Saunders, M. Y. Chou and M. L. Cohen, Phys. Rev. Lett. , 2141 (1984)[52] J. Pedersen, S. Bjørnholm, J. Borggreen, K. Hansen, T.P. Martin and H. D. Rasmussen, Nature , 733 (1991)[53] C. Br´echignac, Ph. Cahuzac, M. de Frutos, J.-Ph. Rouxand K. Bowen, in Physics and Chemistry of Finite Sys-tems: From Clusters to Crystals , eds. P. Jena et al. (Kluwer, Dordrecht, 1992), Vol. , p. 369[54] C. Br´echignac, Ph. Cahuzac, F. Carlier, M. de Frutosand J. Ph. Roux, Phys. Rev. B , 2271 (1993)[55] D. Bonatsos, A. Martinou, S. Sarantopoulou, I. E. Assi-makis, S. K. Peroulis, and N. Minkov, to be published[56] K.T. Hecht and A. Adler, Nucl. Phys. A , 129 (1969)[57] A. Arima, M. Harvey, and K. Shimizu, Phys. Lett. B ,517 (1969)[58] R. D. Ratna Raju, J. P. Draayer, and K. T. Hecht, Nucl.Phys. A , 433 (1973)[59] J. P. Draayer, K. J. Weeks, and K. T. Hecht, Nucl. Phys.A , 1 (1982)[60] J.N. Ginocchio, Phys. Rev. Lett. , 436 (1997)[61] J.N. Ginocchio, J. Phys. G: Nucl. Part. Phys. , 617(1999)[62] O. Casta˜nos, M. Moshinsky, and C. Quesne, Phys. Lett.B , 238 (1992)[63] O. Casta˜nos, V. Vel´azquez A., P.O. Hess, and J.G.Hirsch, Phys. Lett. B , 303 (1994)[64] J. P. Draayer and K. J. Weeks, Phys. Rev. Lett. , 1422(1983)[65] J. P. Draayer and K. J. Weeks, Ann. Phys. (N.Y.) ,41 (1984)[66] C. M. Lederer and V. S. Shirley (Eds.), Table of Isotopes (Wiley, New York, 1978)[67] J. -P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hi-laire, S. P´eru, N. Pillet, and G. F. Bertsch, Phys. Rev. C , 014303 (2010)[68] B. Pritychenko, M. Birsh, B. Singh, and M. Horoi, At.Data Nucl. Data Tables , 1 (2016); , 371 (2017)[69] D. Bonatsos, A. Martinou, I.E. Assimakis, S. Saran-topoulou, S. Peroulis, and N. Minkov, in Nuclear The-ory ’38, Proceedings of the 38th International Work-shop on Nuclear Theory (Rila 2019) , ed. M. Gaidarovand N. Minkov (Heron Press, Sofia, 2019) 128, arXiv:1909.01967 [nucl-th][70] D. Bonatsos, R. F. Casten, A. Martinou, I.E. Assimakis,
N. Minkov, S. Sarantopoulou, R. B. Cakirli, and K.Blaum, arXiv: 1712.04126 [nucl-th][71] A. Martinou, D. Bonatsos, N. Minkov, I.E. Assimakis, S.Sarantopoulou, and S. Peroulis, in
Nuclear Theory ’37,Proceedings of the 37th International Workshop on Nu-clear Theory (Rila 2018) , ed. M. Gaidarov and N. Minkov(Heron Press, Sofia, 2018) 41, arXiv: 1810.11870 [nucl-th][72] O. Casta˜nos, J. P. Draayer, and Y. Leschber, Z. Phys. A , 33 (1988)[73] J. P. Draayer, S. C. Park, and O. Casta˜nos, Phys. Rev.Lett. , 20 (1989)[74] A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II:Nuclear Deformations (Benjamin, New York, 1975)[75] H. De Vries, C. W. De Jager, and C. De Vries, At. DataNucl. Data Tables , 495 (1987)[76] J. R. Stone, N. J. Stone, and S. Moszkowski, Phys. Rev.C , 044316 (2014)[77] A. S. Davydov and G. F. Filippov, Nucl. Phys. , 237(1958)[78] L. Esser, U. Neuneyer, R. F. Casten, and P. von Brentano, Phys. Rev. C , 206 (1997)[79] G. A. Lalazissis, S. Raman, and P. Ring, At. Data Nucl.Data Tables , 1 (1999)[80] F. Iachello, Phys. Rev. Lett. , 052502 (2001)[81] R. F. Casten and E. A. McCutchan, J. Phys. G: Nucl.Part. Phys. , R285 (2007)[82] P. Cejnar, J. Jolie, and R. F. Casten, Rev. Mod. Phys. , 2155 (2010)[83] J. Cseh, Phys. Rev. C , 054306 (2020)[84] C.E. Vargas, J.G. Hirsch, and J.P. Draayer, Phys. Rev.C , 034306 (2001)[85] C.E. Vargas, J.G. Hirsch, and J.P. Draayer, Phys. Rev.C , 064309 (2002)[86] G. Popa, A. Georgieva, and J.P. Draayer, Phys. Rev. C , 064307 (2004)[87] A. P. Zuker, J. Retamosa, A. Poves, and E. Caurier,Phys. Rev. C , R1741 (1995)[88] A. P. Zuker, A. Poves, F. Nowacki, and S. M. Lenzi, Phys.Rev. C , 024320 (2015) TABLE I: Distribution of valence protons and valence neutrons into normal and abnormal parity orbitals in the rare earth region,as obtained from the standard Nilsson diagrams [66], using for each nucleus the deformation parameter obtained from Ref.[67], as discussed in Section 2. In each sum, the first number represents the normal parity nucleons, while the second numbercorresponds to the abnormal parity nucleons. See the two examples at the end of Section 2 for more detailed explanations. Thevalence protons in normal parity orbitals live within a U(10) pseudo-SU(3) shell, while the valence neutrons in normal parityorbitals live within a U(15) pseudo-SU(3) shell. Adopted from Ref. [69]. X e B a C e N dS m G d D y E r Y b H f W O s P t Z v a l + + + + + + + + + + + + + N v a l + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
10 2818 + + + + + + + + + + + + +
10 3020 + + + + + + + + + + + + +
10 3222 + + + + + + + + + + + + +
10 3424 + + + + + + + + + + + + +
10 3624 + + + + + + + + + + + + +
12 3824 + + + + + + + + + + + + +
14 4026 + + + + + + + + + + + + +
14 4228 + + + + + + + + + + + + + TABLE II: Highest weight irreducible representations (irreps), labeled by hw, and irreps possessing the highest eigenvalue ofthe second order Casimir operator of SU(3) (see Eq. (3)), labeled by hC, occurring in the decomposition of U(10) and U(15)for M particles, as obtained through the code of Ref. [28]. Oblate irreps are shown in boldface. A more extended version ofthe table has been given in Ref. [13]. Adopted from Ref. [69].M 2 4 6 8 10 12 14U(10) hw (6,0) (8,2) (12,0) (10,4) (10,4) (12,0) (6,6)U(10) hC (6,0) (8,2) (12,0) (10,4) (10,4) (4,10) (0,12) U(15) hw (8,0) (12,2) (18,0) (18,4) (20,4) (24,0) (20,6)U(15) hC (8,0) (12,2) (18,0) (18,4) (20,4) (24,0) (20,6)M 16 18 20 22 24 26 28U(10) hw (2,8) (0,6) (0,0)U(10) hC (2,8) (0,6) (0,0)U(15) hw (18,8) (18,6) (20,0) (12,8) (6,12) (2,12) (0,8)
U(15) hC (6,20) (0,24) (4,20) (4,18) (0,18) (2,12) (0,8)
TABLE III: Total irreps corresponding to rare earth nuclei obtained when the highest weight irreps (hw) are used for both thevalence protons and the valence neutrons. The irreps are taken from Table 2, as explained in Section 2 through two examples.Oblate irreps are shown in boldface. Adopted from Ref. [69]. NX e B a C e N dS m G d D y E r Y b H f W O s P t , , , , , , , , , , , , , , , , , , , , , , , , , ,
10 8826 , , , , , , , , , , , , , , , , , , , , , , , , , ,
12 9226 , , , , , , , , , , , , ,
12 9426 , , , , , , , , , , , , ,
12 9628 , , , , , , , , , , , , ,
12 9828 , , , , , , , , , , , , ,
12 10032 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
14 10626 , , , , , , , , , , , , ,
16 10826 , , , , , , , , , , , , ,
16 11026 , , , , , , , , , , , , ,
14 11228 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , TABLE IV: Total irreps corresponding to rare earth nuclei obtained when the irrep having the highest eigenvalue of the secondorder Casimir operator of SU(3) (hC) is used for both the valence protons and the valence neutrons. The irreps are taken fromTable 2, as explained in Section 2 through two examples. Oblate irreps are shown in boldface. Adopted from Ref. [69]. NX e B a C e N dS m G d D y E r Y b H f W O s P t , , , , , , , , , , , , , , , , , , , , , , , , , ,
10 8826 , , , , , , , , , , , , , , , , , , , , , , , , , ,
12 9226 , , , , , , , , , , , , ,
12 9426 , , , , , , , , , , , , ,
12 9628 , , , , , , , , , , , , ,
12 9828 , , , , , , , , , , , , ,
12 10032 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
14 106 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
80 90 100 110 1200.200.250.30 pseudo-SU(3) hw pseudo-SU(3) C N Ce
80 90 100 110 1200.200.250.30 pseudo-SU(3) hw pseudo-SU(3) C N Sm
80 90 100 110 1200.200.250.30 pseudo-SU(3) hw pseudo-SU(3) C N Dy
80 90 100 110 1200.150.200.250.30 pseudo-SU(3) hw pseudo-SU(3) C N Yb
80 90 100 110 1200.150.200.25 pseudo-SU(3) hw pseudo-SU(3) C N W
80 90 100 110 1200.150.200.25 pseudo-SU(3) hw pseudo-SU(3) C N Pt FIG. 1: Pseudo-SU(3) predictions for the collective deformation variable β for six series of isotopes in the rare earth region.The predictions labeled by hw have been obtained using the highest weight irreps of SU(3), while those labeled by C have beenobtained using the hC irreps of SU(3) having the highest eigenvalue of the second order Casimir operator of SU(3), C SU (3)2 .See Section IV for further discussion.
80 90 100 110 1200.150.200.250.30
Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt N pseudo-SU(3) hw
80 90 100 110 1200.120.140.160.180.200.220.240.260.280.30
Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt N pseudo-SU(3) C FIG. 2: Pseudo-SU(3) predictions for the collective deformation variable β for the Xe-Pt series of isotopes in the rare earthregion. The predictions labeled by hw (top panel) have been obtained using the highest weight irreps of SU(3), while thoselabeled by C (bottom panel) have been obtained using the hC irreps of SU(3) having the highest eigenvalue of the second orderCasimir operator of SU(3), C SU (3)2 . See Section IV for further discussion.
80 90 100 110 1200.050.100.150.200.250.300.350.40 pseudo-SU(3) hw Gogny D1S mean exp. N Ce
80 90 100 110 1200.050.100.150.200.250.300.350.400.45 pseudo-SU(3) hw Gogny D1S mean exp. N Sm
80 90 100 110 1200.050.100.150.200.250.300.350.400.45 pseudo-SU(3) hw Gogny D1S mean exp. N Dy
80 90 100 110 1200.050.100.150.200.250.300.350.40 pseudo-SU(3) hw Gogny D1S mean exp. N Yb
80 90 100 110 1200.050.100.150.200.250.300.350.40 pseudo-SU(3) hw Gogny D1S mean exp. N W
80 90 100 110 1200.050.100.150.200.250.300.35 pseudo-SU(3) hw Gogny D1S mean exp. N Pt FIG. 3: The pseudo-SU(3) hw predictions shown in Fig. 1 for the collective deformation variable β for six series of isotopes inthe rare earth region are compared to results by the D1S-Gogny interaction (Gogny D1S mean) [67], as well as with empiricalvalues (exp.) [68]. See Section IV for further discussion.
80 90 100 110 1200.000.050.100.150.200.250.30 pseudo-SU(3) hw proxy-SU(3) RMF N Ce
80 90 100 110 1200.000.050.100.150.200.250.300.35 pseudo-SU(3) hw proxy-SU(3) RMF N Sm
80 90 100 110 1200.050.100.150.200.250.300.35 pseudo-SU(3) hw Proxy-SU(3) RMF N Dy
80 90 100 110 1200.100.150.200.250.30 pseudo-SU(3) hw Proxy-SU(3) RMF N Yb
80 90 100 110 1200.050.100.150.200.250.30 pseudo-SU(3) hw Proxy-SU(3) RMF N W
80 90 100 110 1200.000.050.100.150.200.250.30 pseudo-SU(3) hw Proxy-SU(3) RMF N Pt FIG. 4: The pseudo-SU(3) hw predictions shown in Fig. 1 for the collective deformation variable β for six series of isotopesin the rare earth region are compared to proxy-SU(3) results obtained as described in Ref. [13], as well as to relativistic meanfield theory (RMF) predictions[79]. See Section IV for further discussion.
80 90 100 110 1200102030405060 pseudo-SU(3) hw pseudo-SU(3) C ( deg r ee s ) N Ce
80 90 100 110 1200102030405060 pseudo-SU(3) hw pseudo-SU(3) C ( deg r ee s ) N Sm
80 90 100 110 1200102030405060 pseudo-SU(3) hw pseudo-SU(3) C ( deg r ee s ) N Dy
80 90 100 110 1200102030405060 pseudo-SU(3) hw pseudo-SU(3) C ( deg r ee s ) N Yb
80 90 100 110 1200102030405060 pseudo-SU(3) hw pseudo-SU(3) C ( deg r ee s ) N W
80 90 100 110 1200102030405060 pseudo-SU(3) hw pseudo-SU(3) C ( deg r ee s ) N Pt FIG. 5: Pseudo-SU(3) predictions for the collective deformation variable γ for six series of isotopes in the rare earth region.The predictions labeled by hw have been obtained using the highest weight irreps of SU(3), while those labeled by C have beenobtained using the hC irreps of SU(3) having the highest eigenvalue of the second order Casimir operator of SU(3), C SU (3)2 .See Section IV for further discussion.
80 90 100 110 1200102030405060
Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt ( deg r ee s ) N pseudo-SU(3) hw
80 90 100 110 1200102030405060
Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt N pseudo-SU(3) C FIG. 6: Pseudo-SU(3) predictions for the collective deformation variable γ for the Xe-Pt series of isotopes in the rare earthregion. The predictions labeled by hw (top panel) have been obtained using the highest weight irreps of SU(3), while thoselabeled by C (bottom panel) have been obtained using the hC irreps of SU(3) having the highest eigenvalue of the second orderCasimir operator of SU(3), C SU (3)2 . Adopted from Ref. [69]. See Section IV for further discussion.
80 90 100 110 1200102030405060 pseudo-SU(3) hw Gogny D1S mean exp. ( deg r ee s ) N Ce
80 90 100 110 1200102030405060 pseudo-SU(3) hw Gogny D1S mean exp. ( deg r ee s ) N Sm
80 90 100 110 1200102030405060 pseudo-SU(3) hw Gogny D1S mean exp. ( deg r ee s ) N Dy
80 90 100 110 1200102030405060 pseudo-SU(3) hw Gogny D1S mean exp. ( deg r ee s ) N Yb
80 90 100 110 1200102030405060 pseudo-SU(3) hw Gogny D1S mean exp. ( deg r ee s ) N W
80 90 100 110 1200102030405060 pseudo-SU(3) hw Gogny D1S mean exp. ( deg r ee s ) N Pt FIG. 7: The pseudo-SU(3) hw predictions shown in Fig. 5 for the collective deformation variable γ for six series of isotopes inthe rare earth region are compared to results by the D1S-Gogny interaction (Gogny D1S mean) [67], as well as with empiricalvalues (exp.), calculated through Eq. (5). See Section IV for further discussion.
80 90 100 110 1200102030405060 pseudo-SU(3) hw proxy-SU(3) ( deg r ee s ) N Ce
80 90 100 110 1200102030405060 pseudo-SU(3) hw Proxy-SU(3) ( deg r ee s ) N Sm
80 90 100 110 1200102030405060 pseudo-SU(3) hw Proxy-SU(3) ( deg r ee s ) N Dy
80 90 100 110 1200102030405060 pseudo-SU(3) hw Proxy-SU(3) ( deg r ee s ) N Yb
80 90 100 110 1200102030405060 pseudo-SU(3) hw Proxy-SU(3) ( deg r ee s ) N W
80 90 100 110 1200102030405060 pseudo-SU(3) hw Proxy-SU(3) ( deg r ee s ) N Pt FIG. 8: The pseudo-SU(3) hw predictions shown in Fig. 5 for the collective deformation variable γγ