Why nuclear forces favor the highest weight irreducible representations of the fermionic SU(3) symmetry
Andriana Martinou, Dennis Bonatsos, K.E. Karakatsanis, S. Sarantopoulou, I.E. Assimakis, S.K. Peroulis, N. Minkov
aa r X i v : . [ nu c l - t h ] F e b Eur. Phys. J. A manuscript No. (will be inserted by the editor)
Why nuclear forces favor the highest weight irreduciblerepresentations of the fermionic SU(3) symmetry
Andriana Martinou b,1 , Dennis Bonatsos , K. E. Karakatsanis , S.Sarantopoulou , I.E. Assimakis , S.K. Peroulis , N. Minkov Institute of Nuclear and Particle Physics, National Centre of Scientific Research “Demokritos”, GR-15310 Aghia Paraskevi,Attiki, Greece. Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigrad Road, 1784 Sofia,Bulgaria.Received: date / Accepted: date
Abstract
The consequences of the attractive, short–range nucleon–nucleon (NN) interaction on the wavefunctions of the Elliott SU(3) and the proxy-SU(3) sym-metry are discussed. The NN interaction favors the mostsymmetric spatial SU(3) irreducible representation, whichcorresponds to the maximal spatial overlap among thefermions. The percentage of the symmetric componentsout of the total in an SU(3) wave function is intro-duced, through which it is found, that no SU(3) irrep ismore symmetric than the highest weight irrep for a cer-tain number of valence particles in a three dimensional,isotropic, harmonic oscillator shell. The considerationof the highest weight irreps in nuclei and in alkali metalclusters, leads to the prediction of a prolate to oblateshape transition beyond the mid–shell region.
Keywords proxy-SU(3) symmetry, binding energy,prolate dominance, NN interaction
The recent introduction of the proxy-SU(3) Model [1–3]triggered a stormy question: Why is the highest weightirreducible representation (irrep) of SU(3) used insteadof the irrep with the highest value of the second orderCasimir operator of SU(3), which corresponds to themaximum value of the quadrupole–quadrupole interac-tion? It is the purpose of the present work to answer a This research is co-financed by Greece and the EuropeanUnion (European Social Fund- ESF) through the Opera-tional Programme “Human Resources Development, Edu-cation and Lifelong Learning 2014-2020” in the context ofthe project “Nucleon Separation Energies” (MIS 5047793).NM acknowledges support by the Bulgarian National ScienceFund (BNSF) under Contract No.KP-06-N48/1 b e-mail: [email protected] this question, exposing all the physical concepts andmathematical techniques needed.Symmetries play an important role in the descrip-tion of physical systems, especially in cases in whichthey can provide parameter–independent predictions ofgeneral validity. In parallel, the harmonic oscillator isoccupying a central place in many branches of physics,being in many cases a very good approximation to thepotential describing the physical system. Finite shellsappearing in the case of the three–dimensional har-monic oscillator (3D-HO) are known to possess U ( Ω )symmetries with Ω = 3 , , , , , , . . . having SU(3)subalgebras [4–6].The SU(3) symmetry is playing a central role in thedescription of nuclear shapes and spectra [7], since itsintroduction by Elliott [8–11] for the exact descriptionof light nuclei, extended later to heavy nuclei by vari-ous approximations, including the pseudo-SU(3) Model[12–15], the quasi-SU(3) Model [16, 17], and, more re-cently, the proxy-SU(3) Model [1–3, 18]. In addition,several algebraic models containing SU(3) as one oftheir limiting symmetries have been introduced, includ-ing the Interacting Boson Model (IBM) [19–21] and theVector Boson Model (VBM) [22–24], which use bosonsas their building blocks, as well as the Fermion Dy-namic Symmetry Model (FDSM) [25] and the Sym-plectic Model [26–28], which use fermions. Similar inspirit algebraic models bearing SU(3) limiting symme-tries, like the Vibron Model [21, 29], have also been in-troduced for the description of diatomic and polyatomicmolecules.Deformed nuclei have also been described in theframework of the Nilsson Model [30, 31], which con-sists of a 3D-HO with cylindrical symmetry to whichthe spin–orbit interaction [32] is added, which is known to be essential for the reproduction of the experimen-tally seen nuclear magic numbers 2, 8, 20, 28, 50, 82,126, . . . [33]. The same model, without the spin–orbitinteraction, has been found very successful [34] for thedescription of atomic clusters [35–38] and for reproduc-ing the experimentally seen magic numbers, which inthe special case of alkali metal clusters are 2, 8, 20, 40,58, 92, 138, 198, . . . [39–46].Similarities observed in atomic nuclei and atomicclusters have been investigated [37, 47] since the exper-imental identification of the latter [35, 38]. Transitionsfrom prolate (rugby–ball like) to oblate (pancake–like)deformed shapes have been observed both in atomic nu-clei [48–52] and in alkali metal clusters [53–57]. We aregoing to show, that these transitions can be explainedby the dominance of the highest weight (hw) spatial ir-reducible representations (irreps) [4,58] of SU(3), whichis due to the attractive, short–range nature of the nucleon–nucleon interaction. We are also going to clarify thephysical content of the spatial hw irreps by showing,that these irreps are bearing the maximum amount ofspatial symmetrization for a given number of fermions.A by–procuct of the dominance of the spatial hw irrepsof SU(3) is the dominance of prolate over oblate de-formed shapes in the ground states of even–even nuclei,which has been a long–standing puzzle [59] in NuclearPhysics. It will become obvious, that all these effects arerooted in the dominance of the spatial highest weightirreps in finite fermionic shells, the relevant predictionsbeing completely independent of any free parameters. The strong force is applied among nucleons and bindsthem together into the nucleus. This force derives fromfundamental interactions among quarks and gluons obey-ing to the equations of Quantum Chromodynamics [60].Unfortunately these equations have not been solved andso the NN interaction remains unknown.Yet general properties of effective potentials, whichresemble the NN interaction are known. High precisionNN potentials are available [61–63], which respect somegeneral characteristics of the NN interaction at differ-ent length scales [64]:1. At relevant nucleon–nucleon distances d > < d < d <
The Shell Model [32] is widely accepted, to describe inthe microscopic level the atomic nuclei. A basic assump-tion of the model is, that the nucleons are subjectedto a mean field potential, which may be representedby the three dimensional isotropic harmonic oscillator(3D-HO) plus the spin–orbit interaction, leading to thesingle–particle Hamiltonian for the i th nucleon: h i = p i M + 12 M ω r i + υ l i s i ~ ω l i · s i , (1)where the first two terms of Eq. (1) represent the threedimensional isotropic harmonic oscillator: h ,i = p i M + 12 M ω r i (2)with p i , r i , M , ω being the momentum, spatial coordi-nate, mass and oscillation frequency respectively, whilethe last term is the spin–orbit interaction, with l i , s i be-ing the orbital angular momentum and spin, and υ l i s i is a strength parameter [31] (see Table I of Ref. [1] forthe values). An l i term is usually added in the aboveHamiltonian, which serves for the flattening of the meanfield potential.The greatest success of the Shell Model has been theprediction of the nuclear magic numbers 2, 8, 20, 28, 50,82, 126. Despite this major success the Shell Model hasbeen confronted with skepticism in the early years of itsintroduction. The main problem was, that the short–range character of the NN interaction means, that onecannot use a smooth mean field potential in the single–particle Hamiltonian [72]. This obstacle has been over-passed theoretically through the Pauli Exclusion Princi-ple [68,73], to which the nucleons, being fermions, obey.This principle dictates, that only one fermion at a timemay occupy a given state. Thus despite of the feeling of a fluctuating potential, each nucleon has a smoothpath inside the nucleus.Upon the theoretical approval of the Shell ModelJ. P. Elliott has proved, that a nuclear shell consistingof single–particle orbitals with common number of os-cillator quanta N possesses an SU(3) symmetry [8, 9].The building blocks of the Elliott SU(3) Model are theeigenstates of the Hamiltonian of Eq. (1), i.e. , the ShellModel orbitals.If the spherical coordinate system is used, the eigen-states are labeled as | n, l, j, m j i , where n is the radialquantum number getting values n = 0 , , , ... and obey-ing the equation [74]: N = 2 n + l, (3) j is the total angular momentum, which derives afterthe spin–orbit coupling j = l + s and m j is the projectionof j , having integer values within the interval − j ≤ m j ≤ j [32].If the cartesian coordinate system is to be used,then the eigenstates of the h ,i of Eq. (2) are labeledas | n z , n x , n y , m s i , with n z , n x , n y being the number ofquanta in each cartesian direction z, x, y respectivelyand m s = ± is the projection of the spin s = of the nucleon [3]. A unitary transformation can beapplied among the spherical and the cartesian states | n ρ , l, j, m j i ↔ | n z , n x , n y , m s i [3]. The cartesian states | n z , n x , n y , m s i are convenient for the calculation of theElliott SU(3) irreps ( λ, µ ) [8, 9, 58] and thus they havebeen chosen as the intrinsic states of the Elliott SU(3)Model [10, 75, 76].The many–particle wave function is simply a Slaterdeterminant [77, 78] of the | n z , n x , n y , m s i states [76],which represents a totally antisymmetric many–particlewave function, as dictated by the Pauli principle for afermion system. This complicated wave function com-bines both the spatial and the spin–isospin informationfor all the valence nucleons. Fortunately the overall, an-tisymmetric wave function can be decomposed into aspatial and a spin–isospin part for the many nucleonsystem.The spatial part refers to 3D-HO shell, which con-sists of orbitals of N number of quanta, possesses Ω = ( N +1)( N +2)2 spatial orbitals, which in the cartesian co-ordinates are written as: | n z , n x , n y i : |N , , i , |N − , , i , |N − , , i , |N − , , i , |N − , , i , |N − , , i , ..., | , , N i . (4)The symmetry of this set of spatial orbitals is U ( Ω )[8,9]. Young patterns of U ( Ω ) symmetry for proton andneutron configurations have boxes, which represent the particles, arranged in Ω rows and 4 columns (for protonand neutron configurations) and they are described bythe partition:[ f , f , ..., f Ω ] , (5)with f ≥ f ≥ ... ≥ f Ω . The numbers f , f , ..., f Ω arethe number of boxes in each row.Each of the orbitals can be occupied by 2 protonsand 2 neutrons with opposite spin projections. The isospinof a nucleon is t = and its projection is m t = , if itis a proton, and m t = − , if it is a neutron. The spin–isospin many–particle wave function has a U (4) symme-try, usually called Wigner’s SU(4) symmetry [71]. Forthe short–range territory of a spin–isospin independentattraction Wigner at Ref. [71] and Hund at [79] used aHamiltonian with SU(4) symmetry, which did not in-volve the ordinary spin and applied equal forces amongall nucleons (protons and neutrons). The irreps of the U (4) symmetry are [80, 81]:[ f c , f c , f c , f c ] , (6)where c stands for the conjugate irreps of the U ( Ω )symmetry, since the pattern (6) has as rows the columnsof the pattern (5). Obviously the Young pattern, whichcorresponds to (6), has boxes in four rows. Each boxrepresents a valence nucleon. For a nucleus with neutronexcess N val ≥ Z val the f c boxes in the first row areneutrons with m t = − , m s = + , the f c boxes in thesecond row represent neutrons with m t = − , m s = − , the f c boxes in the third row are protons with m t = + , m s = + , while the f c boxes in the forthrow have m t = + , m s = − .The combination of the spatial symmetry with thatof the spin–isospin is labeled as [82]: U ( Ω ) ⊗ U (4) = U (4 Ω ) . (7)The combined wave functions with U (4 Ω ) symmetry,which include the spatial, spin and isospin informationfor the many nucleon problem, are the Slatter determi-nants [76] and according to the Pauli principle [68] theyare totally antisymmetric. The Majorana operator of the exchange of two particles1 ↔ φ ( ~r , ~r ) is [82]:ˆ P x φ ( ~r , ~r ) = φ ( ~r , ~r ) . (8) The eigenvalue of the above operator for a spatiallysymmetric wave function is +1, while for an antisym-metric one is −
1. For a many–body wave function, theeigenvalue of the Majorana operator ˆ P x is [82]: P x = X i
1) + f c ( f c − f c ( f c −
1) + f c ( f c − P x V M ( | r − r | ) (12)in the Hamiltonian, the ground state binding energydue to this force in the many nucleon problem arises tobe [82–84]: BE = a ( A ) − b ( A ) P x , (13) a ( A ) , b ( A ) are parameters, which depend on the massnumber A , with a ( A ) being positive and b ( A ) beingnegative.It holds therefore, that the more symmetric the spa-tial wave function is, the greater is the value of Eq. (9)and the greater is the binding energy of Eq. (13). A testof the validity of the Wigner SU(4) symmetry has beenperformed in Refs. [85, 86], while more details can befound in Ref. [84]. From the Wigner SU(4) symmetryarises, that the favored irrep is the most spatially sym-metric. In other words, the more spatially symmetric isthe nuclear state, the more bound is the nucleus. The Elliott SU(3) irreps ( λ, µ ) can be determined, bythe distribution of the particles in the valence orbitals.For instance the pf shell with N = 3 number of quantacontains 10 spatial orbitals of the type | n z , n x , n y i : | , , i , | , , i , | , , i , | , , i , | , , i , | , , i , | , , i , | , , i , | , , i , | , , i . (14)This shell possesses a U (10) symmetry, where “10” refersto the number of orbitals and has a capacity of 20 pro-tons or neutrons. Supposing for example, that the pf S = 102 2 2 2 0 0 0 0 0 S = 82 2 2 0 0 0 0 0 S = 62 2 0 0 0 0 0 S = 42 0 0 0 0 0 S = 20 0 0 0 0 S = 00 0 0 0 S = 00 0 0 S = 00 0 S = 00 S = 0 Fig. 1
A possible Gelfand–Zeitlin pattern [58, 87, 88] for 10protons/neutrons in the pf shell with U (10) symmetry. Thenumbers must not increase along the directions of the ar-rows and from the left to the right side of each row. The S , S , ..., S are the summations of the numbers of eachrow, which lead to the calculation of the weight vector of Eq.(16). shell contains 10 protons and that each orbital is oc-cupied by 2 particles, the corresponding spatial Youngdiagram of the U (10) symmetry is: , (15)where each box represents a proton. In general the spa-tial Young diagram of a 3D-HO shell with N quantaand U ( Ω ) symmetry, has at most 2 columns for con-figurations of identical nucleons, since at most two ofthem with opposite spin projections may occupy a cer-tain orbital. The irrep of the spatial U ( Ω ) symmetry islabeled by (5) and for the Young diagram of Eq. (15)it is [2 , , , , , , , , ,
0] or [2 ].The distribution of the valence particles into the va-lence space is handled mathematically by the reduction U ( Ω ) ⊃ U ( Ω − ⊃ U ( Ω − ⊃ ... ⊃ U (1), whichis labeled by the Gelfand-Zeitlin (GZ) patterns [87].Such patterns are presented in Eqs. (1), (7), (8) ofRef. [58] and in Fig. 1 and they look like upside tri-angles. The upper row of the triangle is the partition(5). The next row is the partition [ f , f , ..., f Ω − ] of thereduced U ( Ω −
1) algebra and so on till the bottom rowfor the U (1) algebra. One of the possible GZ patternsfor 10 protons in the pf shell is presented in Fig. 1.There are numerous possible distributions of theparticles in the valence space. Each particle distribu-tion in the | n z , n x , n y i states is labeled by the weightvector of the corresponding GZ pattern. If the summa-tions of the numbers of each row in the GZ pattern arelabeled S Ω , S Ω − , ..., S (heading from the top to the bottom of the pattern as in Fig. 1), then the weightvector is [58]: w = ( S Ω − S Ω − , S Ω − − S Ω − , ..., S ) . (16)The difference S Ω − S Ω − reflects to the number ofparticles, that have been placed in the cartesian orbital | n z , n x , n y i = |N , , i of Eq. (3), the second differ-ence S Ω − − S Ω − to the number of particles in the | n z , n x , n y i = |N − , , i orbital and so on till S ,which corresponds to the occupancy of the | n z , n x , n y i = | , , N i . The highest weight irrep is the one, which cor-responds to a weight vector with the maximum values,allowed by the Pauli principle, in the first coordinatesof the vector w as defined in Eq. (16). For a protonor neutron configuration the maximum value in a co-ordinate of the w vector is 2, while for a proton andneutron configuration the maximum value is 4. Thusthe GZ pattern of Fig. 1 has the highest weight vector w = (2 , , , , , , , , , , (17)which means, that in this example each of the first fiveorbitals of Eq. (14) are occupied by two protons. Note,that the summation of the coordinates of the w vectorequals to the number of valence particles.In the Elliott SU(3) Model the spatial U ( Ω ) algebraof a 3D-HO shell has a U (3) subalgebra [8, 9], whichitself reduces to an SU(3) algebra: U ( Ω ) ⊃ U (3) ⊃ SU (3) . (18)In the above the number “3” stands for the three carte-sian directions z, x, y . The number of boxes in each ofthe three rows of a Young diagram of the above U(3)symmetry reflects to the summations for every valenceproton/neutron ( i ) of the quanta in each cartesian di-rection z, x, y : X i n z,i , X i n x,i , X i n y,i , (19)respectively. Since quanta are bosons, one may placeinfinite number of boxes in the rows of the Elliott U (3)Young diagram with partition:[ f , f , f ] , with f ≥ f ≥ f . (20)If P i n z,i ≥ P i n x,i ≥ P i n y,i , then the spatial U (3)partition is:[ f , f , f ] = [ X i n z,i , X i n x,i , X i n y,i ] . (21)Since the weight vector of Eq. (17) indicates that 2particles are placed in the first 5 orbitals of (14), the summations of Eq. (21) are: X i =1 n z,i = 2(3 + 2 + 2 + 1 + 1) = 18 , (22) X i =1 n x,i = 2(0 + 1 + 0 + 2 + 1) = 8 , (23) X i =1 n y,i = 2(0 + 0 + 1 + 0 + 1) = 4 (24)and thus for the relevant example the U (3) irrep is:[ f , f , f ] = [18 , , . (25)A fully filled column in the U (3) Young diagram maybe erased [89]:[ f , f , f ] = [ f − f , f − f , . (26)Consequently the Young diagram of the U(3) irrep ofEq. (25) is: f − f = λ + µ z }| { . The Elliott SU (3) irrep ( λ, µ ) is given by [8, 9]: λ = f − f , (27) µ = f − f , (28)which for the example of Eq. (25) gives ( λ, µ ) = (10 , λ, µ ) reflects to a spatial, many–quanta wavefunction with a total number of λ + 2 µ quanta [9]. Ingeneral the symmetry of the wave function is describedby the partition [ λ + µ, µ ] [9]. Such a wave functiontransforms as a tensor of rank λ + 2 µ [9]. The λ + µ components out of the total are symmetric upon theirinterchange [9], while the µ are nor symmetric neitherantisymmetric.For clarity we present a simpler example. If threeprotons are placed in the p nuclear shell with N = 1according to the highest weight vector w = (2 , , | n z , n x , n y i = | , , i and one in the | , , i cartesian orbital. There-fore the U (3) partition, which results from Eq. (21),is [2 , , U (3)Young pattern: , (29)and has an SU(3) irreducible representation (irrep) ( λ, µ )= (1 ,
1) according to Eqs. (27), (28). If a † α ( q ) is the bo-son creation operator [90], which gives to the cartesian direction α = z, x, y the q th quantum, then the terms ofthe spatial many–quanta wave function of this exampleare of the type [75]: a † z (1) a † z (2) a † x (3) | i , (30)with | i being the vacuum state, namely the 1 s ShellModel orbital [32, 33]. The quantum–number Youngtableaux, which represent the spatial, many–quanta wavefunction of this example are [89]:z zx , 1 23 (31)where the z, x, y represent a quantum in each carte-sian direction, while the numbers 1 , , Φ spatial = r
16 (2 φ z (1) φ z (2) φ x (3) − φ z (1) φ x (2) φ z (3) − φ x (1) φ z (2) φ z (3)) , (32)where φ α ( q ) is a Hermite polynomial with the q th quan-tum in the α = z, x, y direction [90]. Obviously thiswave function is symmetric upon the transposition 1 ↔
2, but there is no symmetry in the transpositions 1 ↔ ↔
3. Indeed only two quanta are symmetric upontheir interchange in Eq. (32), while the third quantumis nor symmetric neither antisymmetric. The number ofsymmetric quanta for the above example is λ + µ = 2.This is clearly stated in Ref. [91], where the spa-tial wave function for the Young pattern of Eq. (29), islabeled: Φ spatial = ˆ S q,q ′ ˆ A q ′ ,q ′′ Φ (1 , , , (33)where ˆ S q,q ′ is the symmetrizer operator of the q, q ′ quanta,ˆ A q ′ ,q ′′ is the antisymmetrizer operator of the q ′ , q ′′ quantaand Φ (1 , ,
3) = φ z (1) φ z (2) φ x (3) for the example of Eq.(31). It is true, that if a state is antisymmetrized in q ′ , q ′′ and thereafter is antisymmetrized in q, q ′ , thenthe antisymmetry of q ′ , q ′′ is lost. Consequently the op-erator, which is applied last, controls the result [91].Finally a ( λ, µ ) irrep with µ > λ,
0) irrepto a totally symmetric spatial state. We define the ratio: r ( λ, µ ) = λ + µλ + 2 µ · , (34)which measures the percentage of the symmetric quanta λ + µ out of the total number of quanta λ + 2 µ . QQ interaction The overall QQ interaction in the Elliott SU(3) Model[8–10] is determined through: QQ = 4 C − L ( L + 1) , (35)where L is the eigenvalue of the squared angular mo-mentum operator:ˆ L = X i ˆ l i (36)and ˆ C is the second order Casimir operator of SU(3)with the eigenvalue [11]: C = λ + µ + λµ + 3( λ + µ ) , (37)or C = ( λ + µ ) + 3( λ + µ ) − λµ. (38)The nuclear quadrupole deformation parameter β ofthe Bohr–Mottelson Model [92] is connected with the C as [93]: β = 4 π A ¯ r ) ( C + 3) , (39)with A being the mass number and ¯ r = 0 . A / isthe dimensionless mean square radius. Thus for a cer-tain nucleon number in a given valence 3D-HO shellthe most deformed nuclear state is the one with thehighest value of the ˆ C operator. Due to the the depen-dence of the C on the number of the symmetric quanta λ + µ as in Eq. (38), it happens, that usually the mostdeformed state has the greatest number of symmetriccomponents. In addition, since the expression (35) en-ters the Elliott Hamiltonian with a minus sign, the statewith large QQ interaction (or large C as in Eq. (35))lies lower in energy.Consequently it would be tempting to say, that themost deformed irrep, which also has the maximum num-ber of symmetric quanta λ + µ , satisfies the principle ofminimum energy and thus represents the ground stateof the nucleus. But at section 8 we will argue, that themost deformed irrep is not always the most symmetric,which is preferred for describing the low–lying nuclearproperties [82, 85]. Each Elliott SU(3) irrep is the result of a certain par-ticle distribution in the valence space | n z , n x , n y i . Thisspace along with the spinor | n z , n x , n y , m s i transforms to the usual Shell Model orbitals as in [3]. It is inter-esting to trace back the particle configuration, whichcorresponds to the hw irreps and to the most deformedirreps.Let for instance 10 protons, to be distributed in the pf shell with U (10) symmetry. This shell consists ofthe spatial orbitals, which are presented in (14). Thehighest weight irrep, which happens to be the most de-formed too for this example, according to the weightvector of Eq. (17) is being derived, if the 10 protonsoccupy the states (see Eq. (14)): | n z , n x , n y i : | , , i , | , , i , | , , i , | , , i , | , , i . (40)The resulting irrep is (10 , ,
0) irrep. This irrep for 12 particlesin pf shell is being derived, if these two more protonsoccupy one of the empty orbitals (see Eq. (14)): | n z , n x , n y i : | , , i , | , , i , | , , i , | , , i , | , , i (41)and specifically if the newcomers occupy the spatial or-bital | , , i , while the 10 previously placed protonsremain in the orbitals of Eq. (40), as they were. Sucha behavior is in accordance with the Pauli principle inthe manner, that the newcomers are subjected to a re-pulsive nucleon–nucleon interaction, as outlined in sec-tion 2, when they attempt to occupy an already filledspatial orbital. This is the Pauli blocking effect, whichis responsible for the repulsive core at extremely shortdistances in all the effective NN potentials [64].The most deformed irrep (4 ,
10) however is beingderived, if the 12 protons occupy the orbitals: | n z , n x , n y i : | , , i , | , , i , | , , i , | , , i , | , , i , | , , i , (42)which means, that when the newcomers occupy theempty orbital | , , i according to the Pauli blockingeffect, only two protons remain in the already filled or-bital | , , i , while there is an unexpected knockout of8 protons from the orbitals | , , i , | , , i , | , , i , | , , i to the | , , i , | , , i , | , , i , | , , i respec-tively. This particle knockout could only be justified bythe Principle of Minimum Energy in the sense, that thisparticle configuration maximizes the QQ interaction,which in turn minimizes the energy. But the knockoutof 8 particles by the 2 newcomers cannot be justified byany short–range NN interaction. Furthermore this mostdeformed irrep (4 ,
10) contains µ = 10 non symmetricquanta, while the highest weight irrep (12 ,
0) of this ex-ample is totally symmetric. The most symmetric irrep, despite of the fact that it is not corresponding to thelargest value of QQ , prevails [2, 94]. This preference tothe hw irrep stems from the short–range attractive NNinteraction, which favors the maximum spatial overlap-ping among the nucleons [82, 85]. This simple exampleshows, that while filling the shell with particles, thehighest weight irreps correspond to smooth particle dis-tributions, without particle knockouts. On the contrarythe most deformed irrep is accompanied by sudden par-ticle displacements just after the mid–shell region.The number of symmetric components in the spa-tial SU(3) wave function is λ + µ , as explained in sec-tion 5. Consequently as Elliott observed in the “Con-clusions” of the introductory publication of the ElliottSU(3) symmetry [8], in the U(3) classification scheme ofthe sd shell the highest weight irrep is the most symmet-ric and lies lowest in energy. This may be considered tobe a general property of deformed nuclei, which stemsfrom the attractive, short range NN interaction. Fur-thermore in section 8 we will outline, that not only thenumber of symmetric components λ + µ of the spatialSU(3) irrep is of high importance, but also the percent-age r , as introduced in Eq. (34), is a measure of thesymmetry of the wave function. The question is, “which is the most spatially symmetricSU(3) irrep?”, which is favored by the attractive shortrange interaction. For a certain number of valence pro-tons or neutrons in the level of the U ( Ω ) symmetry allthe possible irreps with two identical particles in eachof the filled orbitals have the same eigenvalue of the P x operator of Eq. (9), which applies for the permuta-tion of particles. A distinction, about which of them isthe most spatially symmetric irrep, can only be accom-plished at the level of the SU(3) symmetry, where thepermutation of quanta (not particles anymore) can bedetermined through the irreps ( λ, µ ).For a certain number of valence protons/neutrons ina certain 3D-HO shell an irrep ( λ ′ , µ ′ ) has more sym-metric components than a irrep ( λ, µ ) if: λ ′ + µ ′ > λ + µ. (43)But since µ represents the components, which are nei-ther symmetric nor antisymmetric, the percentage r ofEq. (34) has to be also considered. Therefore we pro-pose the two–fold condition: For a given number of particles in a given 3D-HO shellan SU(3) irrep ( λ ′ , µ ′ ) is more symmetric than an irrep ( λ, µ ):if λ ′ + µ ′ ≥ λ + µ, (44)and if r ( λ ′ , µ ′ ) > r ( λ, µ ) , (45)where r is defined in Eq. (34). Using this two–fold con-dition we shall check, if there is a more symmetric El-liott SU(3) irrep ( λ ′ , µ ′ ) than the highest weight irrep( λ, µ ) for every number of valence protons/neutrons ineach valence 3D-HO shell possessing a U ( Ω ) symmetrywith Ω = 6 , ,
15. The irreps in Tables 1-7 are or-dered in decreasing weight, thus for a given number ofvalence particles the highest weight irrep ( λ, µ ) is pre-sented first. Only the irreps ( λ ′ , µ ′ ) with λ ′ + µ ′ ≥ λ + µ are considered according to the condition (44).From Tables 1-7 it emerges, that no irrep satisfies si-multaneously the two conditions (44), (45), when com-peting with the highest weight irrep. The irreps of the U (21) symmetry, which applies for the 3D-HO shellamong magic numbers 70-112 in the Elliott SU(3) sym-metry or for the 82-124 shell in the proxy-SU(3) sym-metry [3], have also been checked, but have not beenpresented here, because they are too lengthy. The sameconclusion applies for all the shells with U (6), U (10), U (15), U (21) symmetry: according to the conditions(44) and (45) there is no irrep ( λ ′ , µ ′ ), which is moresymmetric, than the highest weight irrep ( λ, µ ). From reductio ad absurdum , we may state, that for any num-ber of valence particles in any valence shell, the highestweight irrep is the most symmetric among the rest pos-sible ones. It seems, that this irrep has the finest bal-ance among the maximization of the symmetric quanta λ + µ along with the minimization of the non symmetricquanta µ .As the authors of Ref. [85] have enunciated, the mostfavorable spatial SU(3) irrep, is the most symmetricamong all the possible ones. This conclusion stems rightfrom the short–range character of the attractive NNinteraction, which favors the maximal spatial overlapamong the fermions. Consequently the hw irrep is the favored one and describes best the low–lying nuclearproperties [2, 94]. The highest weight irreps (hw) and those, which corre-spond to the maximum value of the C operator (C) arepresented in Table I of Ref. [2]. If there is a particle–hole symmetry in the SU(3) irreps, then for the samenumber of valence particles m p and valence holes m h ,the relevant SU(3) irrep is produced by an interchangeof λ ↔ µ . This particle–hole symmetry for a proton or Table 1
Part of the Elliott SU(3) irreps, which result fromthe reduction U (6) ⊃ SU (3) [58, 95]. These irreps apply forthe 3D-HO shell among magic numbers 8-20 [8, 9], or for theproxy-SU(3) shell among magic numbers 6-12 [3]. For a cer-tain number of valence particles the highest weight irrep ( λ, µ )is presented first. The rest irreps ( λ ′ , µ ′ ) (for the same num-ber of valence particles) follow with decreasing weight. Onlythe irreps with λ ′ + µ ′ ≥ λ + µ are presented, according to thecondition (44). In the last column the ratio r as introducedin Eq. (34), which is the percentage of the symmetric quantaout of the total in an Elliott or proxy-SU(3) wave function, ispresented. It turns out, that for a certain number of valenceparticles, no irrep satisfies simultaneously the two conditions(44), (45), when comparing with the highest weight irrep. Fur-thermore for 7 valence particles the irrep ( λ ′ , µ ′ ) = (1 ,
5) hasthe same number of symmetric quanta as the highest weightirrep ( λ, µ ) = (4 ,
2) ( λ ′ + µ ′ = λ + µ ) and is more deformed C ′ > C , but contains less percentage of symmetric quanta r ( λ ′ , µ ′ ) < r ( λ, µ ). As a result this most deformed irrep isnot more symmetric than the highest weight irrep accordingto the hypotheses (44) and (45).valence particles λ µ C r (%)1 2 0 10 1002 4 0 28 1003 4 1 36 832 2 24 674 4 2 46 755 5 1 49 862 4 46 606 6 0 54 1003 3 45 670 6 54 507 4 2 46 751 5 49 548 2 4 46 609 1 4 36 5510 0 4 28 5011 0 2 10 50 neutron 3D-HO shell leads to an interchance of λ and µ for the particle and hole SU(3) irreps [11]:( λ, µ ) m p → ( µ, λ ) m h = m p . (46)For instance the highest weight irrep for two valenceprotons in the U (6) is (4 , , C operator, then such a particle-hole symmetry would ex-ist in the SU(3) irreps for all the shells, namely the pf Table 2
The same as Table 1 but for the U (10) symmetry,which applies for the 3D-HO magic numbers 20-40 for theElliott SU(3) symmetry, or for the 28-48 magic numbers forthe proxy-SU(3) symmetry [3]. The irreps are presented againin decreasing weight for a given number of valence particles.Only the irreps with more symmetric quanta than those ofthe highest weight irrep are shown (see hypothesis (44)).valence particles λ µ C r (%)1 3 0 18 1002 6 0 54 1003 7 1 81 894 8 2 114 835 10 1 144 927 4 126 736 12 0 180 1009 3 153 806 6 144 677 11 2 186 878 5 168 728 10 4 198 789 10 4 198 787 7 189 6710 10 4 198 787 7 189 674 10 198 58 with U (10) symmetry, the sdg with U (15) symmetry,etc (see Table I of Ref. [2]).On the contrary the highest weight SU(3) irreps,which are presented in Table 8, sometimes do not re-spect this interchange of λ ↔ µ for the same numberof particles and holes. For instance the highest weightirrep for 7 valence protons in the sd shell with U (6)symmetry is (4 , , pf , sdg , etc) this type of particle–hole asymmetry in thehighest weight irreps, which are presented in Table 8,is more intense. This phenomenon has been discussedin Ref. [2]. In Table 8 we see that most of the hw ir-reps listed are prolate ( λ > µ ), while an oblate ( λ < µ )region appears above mid-shell. In U(6), U(10), U(15),and U(21) in particular, which can accommodate re-spectively up to 12, 20, 30, 42 identical nucleons, theoblate region starts at 8, 15, 23, 34 nucleons respec-tively.Nevertheless the particle–hole symmetry exists, evenfor the highest weight irreps, but in another way. The Table 3
Continuation of Table 2. For 11-15 valence particlesthe most deformed irrep is other than the highest weight andhas less percentage of symmetric quanta out of the total fromthe highest weight irrep ( r ′ < r ).valence particles λ µ C r (%)11 11 2 186 877 7 189 678 5 168 724 10 198 585 8 168 622 11 186 5412 12 0 180 1008 5 168 729 3 153 804 10 198 585 8 168 626 6 144 673 9 153 570 12 180 5013 9 3 153 805 8 168 626 6 144 672 11 186 543 9 153 5714 6 6 144 673 9 153 570 12 180 5015 4 7 126 611 10 144 5216 2 8 114 5617 1 7 81 5318 0 6 54 5019 0 3 18 50 calculation of the hw irreps for a certain number of va-lence particles has been presented in section 5. One maycalculate the hw irreps for a certain number of valenceholes, by filling the spatial orbitals of Eq. (3) in theinverse order. As an example the filling of the sd shellwith holes is equivalent to the filling of the | n z , n x , n y i spatial orbitals, with the following order: | n z , n x , n y i : | , , i , | , , i , | , , i , | , , i , | , , i , | , , i . (47)For instance the 7 valence holes in this shell occupy theorbitals: | n z , n x , n y i : | , , i , | , , i , | , , i , | , , i (48) Table 4
The same as Table 1 but for the U (15), which ap-plies for the 3D-HO magic numbers 40-70 for the Elliott SU(3)symmetry, or for the 50-80 magic numbers for the proxy-SU(3) symmetry [3].valence particles λ µ C r (%)1 4 0 28 1002 8 0 88 1003 10 1 144 924 12 2 214 885 15 1 289 9412 4 256 806 18 0 378 10015 3 333 867 18 2 424 9115 5 385 808 18 4 478 859 19 4 522 8516 7 486 7710 20 4 568 8617 7 529 7714 10 508 7111 22 2 604 9218 7 574 7819 5 553 8315 10 550 7116 8 520 7513 11 505 6910 14 508 6312 24 0 648 10020 5 600 8321 3 585 8916 10 594 7217 8 564 7618 6 540 8014 11 546 6915 9 513 7311 14 546 6412 12 504 679 15 513 626 18 540 5713 22 3 634 8918 8 610 7619 6 586 8115 11 589 7016 9 556 7412 14 586 6513 12 544 6810 15 550 637 18 574 5814 20 6 634 8117 9 601 7414 12 586 6811 15 589 638 18 610 59 Table 5
Continuation of Table 4. For 16-19 valence particlesthe most deformed irrep is other than the highest weight. Noirrep ( λ ′ , µ ′ ), including the most deformed, satisfies simul-taneously the hypotheses (44) and (45), when competing insymmetry with the highest weight irrep ( λ, µ ).valence particles λ µ C r (%)15 19 7 621 7916 10 594 7213 13 585 6710 16 594 627 19 621 5816 18 8 610 7615 11 589 7012 14 586 659 17 601 606 20 634 5717 18 7 574 7814 12 586 6815 10 550 7111 15 589 6312 13 544 668 18 610 599 16 556 616 19 586 573 22 634 5318 18 6 540 8014 11 546 6915 9 513 7310 16 594 6211 14 546 6412 12 504 678 17 564 609 15 513 625 20 600 566 18 540 573 21 585 530 24 648 5019 19 3 493 8814 10 508 7115 8 478 7416 6 454 7910 15 550 6311 13 505 6512 11 466 6813 9 433 717 18 574 588 16 520 609 14 472 6210 12 430 655 19 553 566 17 496 587 15 445 592 22 604 523 20 538 534 18 478 551 21 529 511 Table 6
Continuation of Tables 4 and 5. For 20-23 valenceparticles no irrep is more symmetric than the highest weightaccording to the hypotheses (44) and (45).valence particles λ µ C r (%)20 20 0 460 10015 7 445 7616 5 424 8117 3 409 8710 14 508 6311 12 466 6612 10 430 6913 8 400 7214 6 376 777 17 529 598 15 478 619 13 433 6310 11 394 6611 9 361 694 20 568 555 18 508 566 16 454 587 14 406 608 12 364 633 19 493 544 17 436 555 15 385 570 22 550 501 20 484 512 18 424 5321 16 4 396 8311 11 429 6712 9 396 7013 7 369 747 16 486 598 14 438 619 12 396 6410 10 360 674 19 522 555 17 465 566 15 414 587 13 369 612 20 510 523 18 450 544 16 396 561 19 441 5122 12 8 364 718 13 400 629 11 361 654 18 478 555 16 424 576 14 376 593 17 409 540 20 460 5023 9 10 328 665 15 385 576 13 340 592 18 424 533 16 370 54 Table 7
Continuation of Tables 4, 5 and 6. No irrep com-petes in symmetry the highest weight irrep according to con-ditions (44) and (45).valence particles λ µ C r (%)24 6 12 306 603 15 333 550 18 378 5025 4 12 256 571 15 289 5226 2 12 214 5427 1 10 144 5228 0 8 88 5029 0 4 28 50 and lead to summations of quanta as in Eq. (19): X i =1 n z,i = 1 , X i =1 n x,i = 6 , X i =1 n y,i = 7 , (49)thus the relevant U (3) irrep according to Eq. (20) is:[ f , f , f ] = [7 , , , (50)which leads to the highest weight SU(3) irrep (1 ,
5) (seeEqs. (27) and (28)). This irrep using an interchange of λ ↔ µ , becomes (5 ,
1) for 5 valence particles in the U (6), as it should be. The proton or neutron capacityof a shell, which consists of orbitals with N number ofquanta, is ( N + 1)( N + 2). If the number of valenceparticles m p and the number of valence holes m h iscomplementary: m p + m h = ( N + 1)( N + 2) , (51)then the particle and hole hw SU(3) irreps are relatedby an interchange of λ ↔ µ :( λ, µ ) m p → ( µ, λ ) m h =( N +1)( N +2) − m p . (52)
10 The prolate dominance in atomic nuclei
The consequences in atomic nuclei of the appearanceof a majority of prolate irreps in Table 8 have beenstudied in the framework of the proxy-SU(3) model [1,2, 18], in which the SU(3) symmetry of the harmonicoscillator shells [4,5] is extended beyond the sd nuclearshell by an approximation [1, 3] involving the intruderorbitals of opposite parity within each shell. The hwirreps corresponding to the valence protons and to thevalence neutrons are combined in order to provide theSU(3) irrep characterizing the whole nucleus [2]. Table 8
Highest weight SU(3) irreps (which always havemultiplicity one) for U ( Ω ), Ω = 6 , , ,
21 for m p valenceprotons or neutrons, derived using the code UNTOU3 [58].Violations of the particle–hole symmetry, as expressed in Eq.(46), appearing in the lower half of each column are indicatedby boldface characters. m p U (6) U (10) U (15) U (21)0 (0,0) (0,0) (0,0) (0,0)1 (2,0) (3,0) (4,0) (5,0)2 (4,0) (6,0) (8,0) (10,0)3 (4,1) (7,1) (10,1) (13,1)4 (4,2) (8,2) (12,2) (16,2)5 (5,1) (10,1) (15,1) (20,1)6 (6,0) (12,0) (18,0) (24,0)7 (4,2) (11,2) (18,2) (25,2)8 (2,4) (10,4) (18,4) (26,4)9 (1,4) (10,4) (19,4) (28,4)10 (0,4) (10,4) (20,4) (30,4)11 (0,2) (11,2) (22,2) (33,2)12 (0,0) (12,0) (24,0) (36,0)13 (9,3) (22,3) (35,3)14 (6,6) (20,6) (34,6)15 (4,7) (19,7) (34,7)16 (2,8) (18,8) (34,8)17 (1,7) (18,7) (35,7)18 (0,6) (18,6) (36,6)19 (0,3) (19,3) (38,3)20 (0,0) (20,0) (40,0)21 (16,4) (37,4)22 (12,8) (34,8) (9,10) (32,10) (6,12) (30,12) (4,12) (29,12)
26 (2,12) (28,12)
27 (1,10) (28,10)
28 (0,8) (28,8)
29 (0,4) (29,4)
30 (0,0) (30,0) (25,5) (20,10) (16,13) (12,16) (9,17) (6,18) (4,17)
38 (2,16)39 (1,13)40 (0,10)41 (0,5)42 (0,0)
It turns out that a prolate to oblate shape transi-tion is predicted with the use of the hw irreps [2] whenboth protons and neutrons are near the end of the cor-responding shell, thus represented by oblate ( λ < µ )SU(3) irreps. Agreement with existing experimental in-formation [48–52] in the heavy rare earths, below 82protons and 126 neutrons, has been seen. In other wordswe see in the nuclear chart, below the doubly magic nu-cleus Pb , a relatively small region of oblate nuclei, while prolate shapes are obtained everywhere else in therare earths with 50-82 protons and 126-184 neutrons.A similar picture is predicted [2] in other regionsof the nuclear chart, for example the rare earths with50-82 protons and 50-82 neutrons. As a consequence,the prolate over oblate dominance in the shapes of theground state bands of even–even nuclei, which has beenan open problem for many years [59], is obtained as adirect consequence of the proxy-SU(3) symmetry andthe use of the highest weight irreps [2, 18].Recent studies [94, 96] indicate, that the prolate tooblate shape transition and the prolate over oblate dom-inance in the shapes of the ground state bands of evennuclei can be also obtained within the framework of thepseudo-SU(3) model [12–14], in which the normal par-ity orbitals in a given nuclear shell are modified througha unitary transformation [15], in contrast to the proxy-SU(3) model [3], in which a unitary transformation isapplied to the intruder parity orbitals. In both casesthe aim of the unitary transformation is the restorationof the SU(3) symmetry of the 3D-HO [4, 5], which isbroken by the spin–orbit interaction beyond the sd nu-clear shell [32]. The compatibility of the pseudo-SU(3)and proxy-SU(3) approximations has also been demon-strated recently in the study of quarteting in heavy nu-clei [97].It should be emphasized, that the above findings inatomic nuclei are rooted in the attractive, short rangenature of the NN interaction [78,98], which favors max-imal spatial overlaps [85]. These are obtained when thespatial part of the wave function is as symmetric as pos-sible [71, 85]. In contrast, the spin–isospin part of thewave function possesses a Young diagram which is theconjugate of the Young diagram of the spatial part, aspointed out in section 3, so that the fully antisymmetriccharacter of the total wave function is guaranteed.
11 Manifestation of prolate to oblate transitionin metal clusters
Structural similarities between atomic clusters [35–38]and atomic nuclei have been pointed out [37, 47] sincethe early days of experimental study [35, 38] of atomicclusters. Alkali metal clusters, in particular, exhibit magicnumbers, which for few particles are similar to the 3D-HO magic numbers, while they diverge at higher par-ticle numbers [39–46]. The valence electrons in alkalimetal clusters are supposed to be free, thus formingshells. The major magic numbers observed in alkalimetal clusters are 2, 8, 20, 40, 58, 92, . . . . Experimentaldata [39, 40, 44, 46] and theoretical predictions [99, 100]for magic numbers in alkali metal clusters exist up to Table 9
Experimental magic numbers for Na clusters by Martin et al. [39, 40] (column 1), Bjørnholm et al. [41, 42] (column2), Knight et al. [43] (column 3), and Pedersen et al. [44] (column 4), as well as to the experimental data for Li clusters byBr´echignac et al. ( [45] in column 5, [46] in column 6) are compared to theoretical predictions [33] by the (non–deformed)3D-HO (column 9), the square well potential (SW) (column 8), a rounded square well potential intermediate between theprevious two (INT) (column 7), and the 3D q –deformed harmonic oscillator (DHO) [99, 100] (column 10).exp. exp. exp. exp. exp. exp. th. th. th. th.Na Na Na Na Li Li INT SW HO DHO[39, 40] [41, 42] [43] [44] [45] [46] [33] [33] [33] [99, 100]2 2 2 2 2 2 2 28 8 8 8 8 8 8 818 18 18 (18)20 20 20 20 20 20 20 2034 34 34 3440 40 40 40 40 40 40 40 4058 58 58 58 58 58 58 5868,70 68 7090,92 92 92 92 93 92 92 90,92 92106,112 106 112138 138 138 134 138 138 132,138 138198 ± l term flattening the bottomof the 3D-HO potential and making its edges sharperis still in use [34]. In other words, deformation in alkalimetal clusters can be described by the same model usedfor describing deformed nuclei.Deformed nuclei are also known to be describedby the SU(3) symmetry, in the framework of algebraicmodels using bosons, as the Interacting Boson Model[19–21] and the Vector Boson Model [22–24], or fermions,like the Symplectic Model [28, 101], the Fermion Dy-namical Symmetry Model (FDSM) [25], the pseudo-SU(3) model [12–14], the quasi-SU(3) model [16, 17],and, recently, the proxy-SU(3) model [1–3, 18]. There-fore it is natural to see, if certain properties for alkalimetal clusters can be predicted by algebraic modelsused for deformed nuclei, taking into account that nospin–orbit force is present in the case of atomic clus- ters [34], the pairing force being also absent in thiscase [47].Magic numbers for alkali metal clusters have beenpredicted [33, 38] by the (non–deformed) 3D-HO, thesquare well potential, as well as a rounded square wellpotential between the previous two. Predictions are inreasonable agreement to experimental findings up tocluster size around 150, as seen in Table 9. Experimen-tal magic numbers up to cluster size 1500 [39,40,44,46]have been reproduced by a deformed 3D-HO [99, 100].It is seen in Table 10 that the ( N , l ) orbitals, charac-terized by the number of oscillator quanta N and theangular momentum l , preserve in the deformed 3D-HOthe same order as in the non–deformed harmonic oscil-lator up to cluster size 70, while beyond this point mix-ing of orbitals with different N starts, since from each N shell the orbital with the highest angular momentum l = N is pushed to lower energies, thus entering shellswith lower values of N .In view of the above, the appearance of prolate shapesabove cluster sizes 8, 20, and 40, and of oblate shapesbelow cluster sizes 20 and 40 is easily explained by Ta-ble 8, since the 8-20 shell corresponds to the harmonicoscillator sd shell with U (6) symmetry, while the 20-40shell corresponds to the pf shell with U (10) symmetry.According to Table 8, prolate shapes are seen at thebeginning of the shells and further up within them, i.e.starting at cluster sizes 8 and 20, while oblate shapesare seen near the end of the shells, i.e. , below clus-ter sizes 20 and 40. Above 40 the sdg shell with U(15)symmetry is starting, thus prolate shapes are again ex-pected, as seen experimentally [57]. But this patternof succession of prolate and oblate shapes breaks down Table 10
Energy levels of the 3-dimensional q -deformed har-monic oscillator [99, 100]. Each level is characterized by N (the number of vibrational quanta) and l (the angular mo-mentum), while 2(2 l + 1) represents the number of parti-cles each level can accommodate, and under “total” the totalnumber of particles up to and including this level is given.Magic numbers, corresponding to large energy gaps, are re-ported in boldface. N l l + 1) total0 0 2 around cluster size 70, since the sequence of 3D-HOshells is disturbed, as seen in Table 10. In particular,the ( N , l ) = (4 ,
4) orbital is lowered, thus providing amagic number at 58, while the remaining orbitals (4,2)and (4,0) from the sdg shell are joined by the (5,5) or-bital of the pf h shell, forming the magic number 92.We conclude that the algebraic results reported inTable 8 can explain both the succession of prolate andoblate shapes seen in light alkali metal clusters respec-tively above and below the magic numbers 8, 20, and40, as well as the disappearance of this pattern at highercluster sizes, a problem which has stayed open for years[47].It should be emphasized that the similarities be-tween several properties of atomic nuclei and atomicclusters [37, 47] is due to the similar form of the rele-vant potentials, which at the most elementary level arein both cases modified harmonic oscillators with flat-tened bottoms, namely the Nilsson model [30, 31] inatomic nuclei and the Clemenger model [34] in atomicclusters, while the basic differences between the twosystems are rooted in the absence of the spin-orbit andpairing interactions in the case of atomic clusters [47].
12 Discussion
It was the purpose of the present work to answer thestormy question why in the proxy-SU(3) model the high- est weight irreducible representation (irrep) of SU(3)is used instead of the irrep with the highest value ofthe second order Casimir operator of SU(3), which cor-responds to the maximum value of the quadrupole–quadrupole interaction. The basic points of the answerare the following.a)The attractive, short range nature of the nucleon–nucleon interaction favors wave functions with as sym-metric as possible spatial part, which guarantees max-imal spatial overlaps among them.b)It is proved that the highest weight SU(3) irrepfor a given number of nucleons (protons or neutrons)in a given 3-dimensional isotropic harmonic oscillatorshell possessing an SU(3) subalgebra is the irrep pos-sessing the highest percentage of symmetrized boxes inthe relevant Young diagram, i.e., it represents the mostsymmetric spatial state for the given system.The dominance of the highest weight spatial irrepshas the following consequences.1) It explains the dominance of prolate over oblateshapes in the ground states of even–even nuclei.2) In both even–even nuclei and in alkali metal clus-ters it predicts a shape transition from prolate to oblateshapes beyond the mid–shell and below its closure.3) In atomic nuclei the prolate to oblate shape tran-sition is seen experimentally in the heavy rare earths be-low the doubly magic nucleus Pb . Similar transi-tions are predicted in other regions of the nuclear chart,yet inaccessible by experiment.4) In alkali metal clusters, 2) explains the existenceof prolate deformations above magic numbers and oblatedeformations below magic numbers up to 60 atoms, aswell as the disappearance of this pattern in heavier clus-ters.It should be emphasized that the conclusions of thepresent study are valid for any finite fermionic systemgoverned by attractive, short range interactions andpossessing the SU(3) symmetry. References
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