Proxy-SU(3) symmetry in the shell model basis
Andriana Martinou, Dennis Bonatsos, N. Minkov, I.E. Assimakis, S. K. Peroulis, S. Sarantopoulou, J. Cseh
PProxy-SU(3) symmetry in the shell model basis
Andriana Martinou , Dennis Bonatsos , N. Minkov , I.E. Assimakis , S. K. Peroulis S. Sarantopoulou , and J. Cseh Institute of Nuclear and Particle Physics, National Centre for ScientificResearch “Demokritos”, GR-15310 Aghia Paraskevi, Attiki, Greece Institute of Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 72 Tzarigrad Road, 1784 Sofia, Bulgaria and Institute for Nuclear Research, Debrecen, Pf. 51, Hungary-4001
The proxy-SU(3) symmetry has been proposed for spin-orbit like nuclear shells using the asymp-totic deformed oscillator basis for the single particle orbitals, in which the restoration of the symme-try of the harmonic oscillator shells is achieved by a change of the number of quanta in the z -directionby one unit for the intruder parity orbitals. The same definition suffices within the cartesian basisof the Elliott SU(3) model. Through a mapping of the cartesian Elliott basis onto the spherical shellmodel basis, we translate the proxy-SU(3) approximation into spherical coordinates, proving, thatin the spherical shell model basis the proxy-SU(3) approximation corresponds to the replacement ofthe intruder parity orbitals by their de Shalit–Goldhaber partners. Furthermore it is shown, that theproxy-SU(3) approximation in the cartesian Elliott basis is equivalent to a unitary transformation inthe z -coordinate, leaving the x - y plane intact, a result which in the asymptotic deformed oscillatorcoordinates implies, that the z -projections of angular momenta and spin remain unchanged. Thepresent work offers a microscopic justification of the proxy-SU(3) approximation and in additionpaves the way, for taking advantage of the proxy-SU(3) symmetry in shell model calculations. I. INTRODUCTION
In the era of supercomputers symmetries continue, toplay an indispensable role in nuclear physics. First theycan provide predictions for the overall behavior of mea-surable quantities and the selection rules they are forced,to obey [1–6]. Second they offer extremely effective short-cuts, largely reducing the size of microscopic calculations[7, 8]. While the spherical shell model [9, 10] remains thebasic microscopic theory of atomic nuclei explaining theappearance of the experimentally observed magic num-bers, the existence of the SU(3) symmetry, pointed out byElliott [11–13] in the sd shell, has shown, how quadrupoledeformation occurs within a nuclear shell bearing theoverall U(6) symmetry of the relevant harmonic oscillatorshell [14–17], possessing an SU(3) subalgebra. Further-more the SU(3) subalgebra offers a classification schemefor all nuclei in the said shell.The recently introduced proxy-SU(3) symmetry [18,19] is an approximation scheme, which extends the valid-ity of the Elliott SU(3) theory to higher nuclear shells al-lowing one, to take advantage of the computational toolsprovided by SU(3) [6] for calculations in heavy nucleiaway from closed shells, where microscopic calculationsstill are of prohibitive size. The proxy-SU(3) symmetryhas been found [19], to provide parameter-independentpredictions for the collective deformation variables β and γ [20], measuring the deviation from the spherical shapeand from axial symmetry respectively. As by-productsthe dominance [21] of prolate over oblate shapes in theground states of even-even nuclei, as well as the locationof a prolate to oblate transition [22] in heavy deformednuclei have been obtained [19, 23].Within nuclear shells above the sd shell the SU(3) sym-metry is known, to be broken by the spin-orbit interac-tion [9, 10], which in each shell pushes down to the shellbelow the orbitals bearing the highest total angular mo-mentum j . For example in the sdg shell, containing theorbitals 3 s / , 2 d / , 2 d / , 1 g / , 1 g / , the 1 g / or-bital is pushed into the pf shell below, while the 1 h / is coming into the sdg shell from the pf h shell above,called the abnormal parity orbital, since it bears parityopposite to that of the orbitals of the sdg shell, called in this case the normal parity orbitals. Within the proxy-SU(3) scheme [18, 19] the SU(3) symmetry is restoredby replacing the levels of the 1 h / orbital (except thetwo with the projections m j = ± j of the total angularmomentum j ) by the levels of the 1 g / orbital with thesame projection of the total angular momentum m j .The validity of the proxy-SU(3) symmetry has beendiscussed so far in the framework of the Nilsson model[24, 25], which provides a simple description of deformednuclei in terms of an axially deformed harmonic oscillatorincluding a spin-orbit interaction and an l term (where l is the orbital angular momentum) flattening the bot-tom of the potential. The replacement is motivated bythe experimentally verified observation [26], that pairsof orbitals differing by ∆ K [∆ N ∆ n z ∆Λ] = 0[110] in theNilsson notation, where N is the total number of oscilla-tor quanta, n z is the number of oscillator quanta alongthe z -axis, Λ = | m l | is the absolute value of the projec-tion of the orbital angular momentum and K is the pro-jection of the total angular momentum along the z -axis,are known to exhibit maximal spatial overlaps [27]. Ithas been proved [18], that the modifications inflicted bythese replacements in the matrix elements of the NilssonHamiltonian are minimal, thus the single-particle spec-tra constituting the Nilsson diagrams remain nearly un-altered.In the present work we consider the proxy-SU(3) ap-proximation in the framework of the spherical shellmodel, motivated by the following points and questions.a) In the framework of the spherical shell modelthe usual three-dimensional harmonic oscillator (3D-HO)shells are involved having U (Ω) symmetries (where Ω =( N + 1)( N + 2) / a r X i v : . [ nu c l - t h ] S e p spherical shell model?c) Alternative approximate SU(3) schemes are pro-vided by the pseudo-SU(3) symmetry [30–38], to be fur-ther discussed in Section 8, as well as by the quasi-SU(3)scheme [39, 40]. In the case of the pseudo-SU(3) schemein a given nuclear shell the normal parity orbitals are re-placed by pseudo-SU(3) counterparts, to which they areconnected through a unitary transformation [41–43]. Isthere any unitary transformation connecting, the orbitalsreplaced within the proxy-SU(3) framework?In short, in this manuscript the spherical shell modeljustification of the proxy-SU(3) symmetry is offered,based on two mathematical pillars, a transformation fromthe cartesian Elliott basis to the spherical shell modelbasis, along with a unitary transformation within the El-liott basis, and a physical pillar, the de Shalit–Goldhaberpairs [44], to be discussed later.In Section 2 of the present work we are going to studythe transformation connecting the cartesian Elliott basisto the spherical shell model basis, while in Section 3 thede Shalit–Goldhaber nucleon pairs [44] are going to bestudied within this mathematical framework. The im-portance of the x - y plane in the case of axially deformednuclei will be discussed in Section 4, while in Section 5 weare going to show, that the replacements needed, in orderto employ the proxy-SU(3) scheme within the sphericalshell model, correspond to de Shalit–Goldhaber nucleonpairs [44]. A unitary transformation connecting, the or-bitals involved in the proxy-SU(3) approximation, will beshown, to exist in Sections 6 and 7 and will be comparedto the unitary transformation, used within the pseudo-SU(3) scheme in Section 8, while Section 9 will containdiscussion of the preset findings and plans for furtherwork. II. TRANSFORMATION AMONG THESPHERICAL SHELL MODEL AND THECARTESIAN ELLIOTT BASIS
The simplest single particle Hamiltonian for one nu-cleon in the atomic nucleus is [9, 10] H = p M + 12 M ω r + V ls l · s , (1)where the first two terms represent the three dimensionalisotropic harmonic oscillator, with p , r , M , ω being themomentum, spatial coordinate, mass and oscillation fre-quency respectively, while the last term is the spin-orbitinteraction, with l , s being the orbital angular momen-tum and spin, and V ls is a strength parameter of thespin-orbit coupling. The spatial, single particle states ofthe Elliott SU(3) symmetry [11–13] are the solutions ofthe 3D isotropic HO in the cartesian coordinate system | n z , n x , n y (cid:105) , which are Hermite polynomials [45]. The 3Disotropic HO can also be solved in the spherical coordi-nate system, which is preferred in the shell model [9, 10],giving single particle states | n, l, m l (cid:105) [45], where the totalnumber of oscillator quanta N is N = 2 n + l = n z + n x + n y , (2)with m l being the projection of the orbital angular mo-mentum on the z -axis, while n is the radial quantumnumber getting values n = 0 , , , , ... A unitary transformation between the cartesian | n z , n x , n y (cid:105) and spherical basis | n, l, m l (cid:105) is possible | n z , n x , n y (cid:105) = (cid:88) n,l,m l (cid:104) n, l, m l | n z , n x , n y (cid:105) | n, l, m l (cid:105) , (3)with N and n given by Eq. (2), while l = N , N −
2, . . . ,1 or 0, and − l ≤ m l ≤ l . For a given oscillator shell withnumber of quanta N , the number of degenerate states is ( N +1)( N +2)2 . We have doubled these states, i.e. , we haveencountered each | n z , n x , n y (cid:105) and | n, l, m l (cid:105) orbital twicein the basis, since two identical nucleons with oppositespin projections ( ± /
2) will be allowed by the Pauli prin-ciple, to occupy the same spatial orbital after the inclu-sion of the spin. This step was necessary, in order to beable, to multiply the matrix R with the matrix C in Eq.(14). The doubled cartesian states | n z , n x , n y (cid:105) are thematrix elements of a column matrix ( N + 1)( N + 2) × n z n x n y ], while the relevant column matrix,which consists of the doubled spherical states | n, l, m l (cid:105) ,is labeled as [ nlm l ]. While R was originally of dimen-sion ( N +1)( N +2)2 × ( N +1)( N +2)2 , after the doubling of thestates R is the square ( N + 1)( N + 2) × ( N + 1)( N + 2)transformation matrix among the two bases and it holds,that: [ n z n x n y ] = R · [ nlm l ] . (4)The matrix elements of the transformation matrix R for m l ≥ (cid:104) n, l, m l | n z , n x , n y (cid:105) = δ n + l,n x + n y + n z ( − (2 n + n x + n y − m l ) / · i n y (cid:18) (2 l + 1)( l − m l )!( n + l )!2 l ( l + m l )! n !(2 n + 2 l + 1)! (cid:19) / (cid:18) n x + n y + m l (cid:19) !( n x ! n y ! n z !) / (cid:18) − n x + n y + m l (cid:19) t max (cid:88) t = t min ( − t (2 l − t )!( n + t )! t !( l − t )!( l − t − m l )!( n + t − n x + n y − m l )! h max (cid:88) h = h min ( − h h !( n x − h )!( h + n y − n x − m l )!( n x + n y + m l − h )! , (5)where t min = (cid:40) , for n ≥ n x + n y − m l n x + n y − m l − n, for n < n x + n y − m l , (6) t max = (cid:40) l − m l , if l − m l is even l − m l − , if l − m l is odd , (7) h min = (cid:40) , for n y ≥ n x + m l , n x + m l − n y , for n y < n x + m l , (8) h max = (cid:40) n x , for n x ≤ n y + m l , n x + n y + m l , for n x > n y + m l . (9)For m l < (cid:104) n, l, − m l | n z , n x , n y (cid:105) = ( − n x (cid:104) n, l, m l | n z , n x , n y (cid:105) . (10)If the spinor of each nucleon is | s, m s (cid:105) , with s = 1 / m s = ± / l · s yields, that the total angular momen-tum j is j = l + s . The shell model orbitals | n, l, j, m j (cid:105) arise after the l · s coupling, with m j = m l + m s be-ing the projection of the total angular momentum. The | n, l, j, m j (cid:105) correspond to the usual shell model nota-tion if one adds 1 unit in the radial quantum number n and represents the angular momentum l = 0 , , , ... by the small latin characters s , p , d , . . . , according tothe spectroscopic notation. For instance the orbitals | n, l, j, m j (cid:105) : | , , , (cid:105) , | , , , (cid:105) are labeled 1 p j =3 / m j =1 / ,2 d j =5 / m j =3 / in the shell model notation respectively. Thetransformation among the shell model basis | n, l, j, m j (cid:105) and the spherical basis | n, l, m l (cid:105) | s, m s (cid:105) = | n, l, m l , m s (cid:105) isachieved through the Clebsch-Gordan (CG) coefficients | n, l, m l , m s (cid:105) = (cid:88) m l ,m s C lsjm l m s m j | n, l, j, m j (cid:105) . (11)The CG coefficients form a square ( N + 1)( N + 2) × ( N + 1)( N + 2) matrix C . Therefore Eq. (11) is writtenin matrix representation[ nlm l m s ] = C · [ nljm j ] . (12)The cartesian orbitals with the spinor are labeled as | n z , n x , n y , m s (cid:105) , which are the single particle states thathave been used in the Elliott SU(3) symmetry [11, 12].From Eq. (4) it follows that[ n z n x n y m s ] = R · [ nlm l m s ] . (13)Finally a transformation among the Elliott states and theshell model states in matrix representation isachieved [ n z n x n y m s ] = R · C · [ nljm j ] . (14)The inverse transformation is[ nljm j ] = C − · R † [ n z n x n y m s ] , (15)where R † is the conjugate transpose of R and C − isthe inverse of C . The results of the transformations forthe N = 1 − N = 1 , c ( λµK, L ) of Ref.[12]. For instance the orbital | n z , n x , n y , m s (cid:105) = | , , , (cid:105) corresponds to a U (3) irrep [ f , f , f ] =[ n z , n x , n y ]=[2 , ,
0] and to the Elliott SU(3) quantum numbers ( λ, µ )= ( f − f , f − f ) = (2 ,
0) [11]. From Table III becomesevident, that this Elliott orbital expands with probability (cid:12)(cid:12) − / √ (cid:12)(cid:12) = 1 / s orbital and with probability (cid:12)(cid:12) − / √ (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:112) / (cid:12)(cid:12)(cid:12) = 2 / d orbitals. The sameexpansion is presented in Table 3 of Ref. [12] for the( λ, µ ) = (2 ,
0) irrep.
III. THE DE SHALIT–GOLDHABER PAIRS
Sometimes it is assumed, that the proton configurationis independent from that of the neutrons and thus one cantreat protons and neutrons separately. However, studiesof the proton-neutron interaction [44, 50–54] have proved, that this interaction is so strong, that affects the β decaytransitions, the overall nuclear deformation and spectrumand gives rise to unexpected occupancies of the singleparticle nuclear states.De Shalit and Goldhaber [44] studied the β transi-tion probabilities and revealed, that the neutrons inter-act with the protons, when they occupy specific shellmodel orbitals. Their result, which is summarized atFigs. (5)-(8) of Ref. [44], is, that the interaction of theneutrons of the 1 i / , h / orbitals with the protons ofthe 1 h / , g / orbitals respectively stabilizes the nu-cleus and, as a consequence, longer lifetimes in the β transitions appear. The de Shalit–Goldhaber rule, thathas been used in Refs. [50–52], is, that the valence neu-trons bearing the highest possible j max = N + TABLE I: The transformation matrix R · C for N = 1. The first line consists of the shell model orbitals, while the first columnconsists of the Elliott orbitals | n z , n x , n y , m s (cid:105) . These orbitals are used in the harmonic oscillator shell 2-8 ( p shell), or in theproxy-SU(3) shell 6-12 after the replacement of the intruder orbitals with their de Shalit–Goldhaber partners. See Section 3for further discussion. | n, l, j, m j (cid:105) | p / − / (cid:105) | p / / (cid:105) | p / − / (cid:105) | p / − / (cid:105) | p / / (cid:105) | p / / (cid:105)| n z , n x , n y , m s (cid:105)| , , , − (cid:105) i √ i √ i √ | , , , (cid:105) − i √ i √ i √ | , , , − (cid:105) − √ √ − √ | , , , (cid:105) − √ √ − √ | , , , − (cid:105) √ (cid:113) | , , , (cid:105) − √ (cid:113) C − · R † for N = 1. | n z , n x , n y , m s (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105)| n, l, j, m j (cid:105)| p / − / (cid:105) i √ − √ √ | p / / (cid:105) − i √ − √ − √ | p / − / (cid:105) − i √ √ | p / − / (cid:105) − i √ √ (cid:113) | p / / (cid:105) − i √ − √ (cid:113) | p / / (cid:105) − i √ − √ within the shell, characterized by N quanta, over-lap mostly with the valence protons bearing the highest j (cid:48) max = ( N −
1) + within the previous shell, charac-terized by N − i / , h / , g / , f / , d / attract thevalence protons of the 1 h / , g / , f / , d / , p / orbitals respectively. This mutual proton-neutron in-teraction has been attributed to a large spatial overlapamong those orbitals.Similar results emerged [26] through the study of thequantity | δV pn ( Z, N ) | =14 [( B Z,N − B Z,N − ) − ( B Z − ,N − B Z − ,N − )] (16)for even-even nuclei, where B is the binding energy.At Refs. [26, 27] the asymptotic Nilsson model nota-tion [24, 25] has been used for the single particle states K [ N n z Λ]. The conclusion was, that neutrons in the K [ N n z Λ] orbitals interact/overlap mostly with protonsof the K [ N − , n z − , Λ] orbitals. Such proton-neutronorbitals differ in the Nilsson quantum numbers by∆ K [∆ N ∆ n z ∆Λ] = 0[110] and may be called 0[110]pairs. These 0[110] pairs appeared, to exhibit the maxi-mum interaction, when both the proton and the neutronorbitals are the orbitals with the highest- j allowed withintheir own shell[26, 27], i.e. , when they originate from ade Shalit–Goldhaber pair of orbitals [44].In this work we prove, that indeed the de Shalit–Goldhaber pairs differ by one quantum in the cartesian z -axis. The transformation of Section II is used, to prove that C a z | j / m j (cid:105) = | i / m j (cid:105) , (17) C a z | i / m j (cid:105) = | h / m j (cid:105) , (18) C a z | h / m j (cid:105) = | g / m j (cid:105) , (19) C a z | g / m j (cid:105) = | f / m j (cid:105) , (20) C a z | f / m j (cid:105) = | d / m j (cid:105) , (21) C a z | d / m j (cid:105) = | p / m j (cid:105) , (22)where C are normalization constants (obtaining differentvalues in each case), a z is the harmonic oscillator anni-hilation operator in the cartesian z -direction and m j hasto be the same in the left and right side of each equation.The action of the a z operator is [45] a z | n z (cid:105) = √ n z | n z − (cid:105) . (23)For clarity we present the equivalence of the C a z | d / / (cid:105) to the | p / / (cid:105) orbital. The rest of the pairs are handledby a code. From Table IV the | d / / (cid:105) expands over theElliott basis | n z , n x , n y , m s (cid:105) as | d / / (cid:105) = − √ | , , , − (cid:105) + i √ | , , , − (cid:105) + 12 √ | , , , − (cid:105) − i (cid:114) | , , , (cid:105)− (cid:114) | , , , (cid:105) . (24) TABLE III: The same as Table I, but for N = 2, related the harmonic oscillator shell 8-20 ( sd shell), or to the proxy-SU(3)shell 14-26. | n, l, j, m j (cid:105) | s / − / (cid:105) | s / / (cid:105) | d / − / (cid:105) | d / − / (cid:105) | d / / (cid:105) | d / / (cid:105) | d / − / (cid:105) | d / − / (cid:105) | d / − / (cid:105) | d / / (cid:105) | d / / (cid:105) | d / / (cid:105)| n z , n x , n y , m s (cid:105)| , , , − (cid:105) − √ − √ − √ − − √ − √ | , , , (cid:105) − √ √ √ − √ − √ − | , , , − (cid:105) − i (cid:113) i √ − i √ | , , , (cid:105) − i (cid:113) i √ − i √ | , , , − (cid:105) − √ − √ √ − √ √ | , , , (cid:105) − √ − √ √ √ − √ | , , , − (cid:105) i √ i (cid:113) i (cid:113) i √ | , , , (cid:105) − i (cid:113) − i √ i √ i (cid:113) | , , , − (cid:105) √ − (cid:113) (cid:113) − √ | , , , (cid:105) − (cid:113) √ √ − (cid:113) | , , , − (cid:105) − √ √ (cid:113) | , , , (cid:105) − √ − √ (cid:113) The action of the annihilation operator on this orbital is a z | d / / (cid:105) = − i (cid:114) | , , , (cid:105) − (cid:114) | , , , (cid:105) . (25)The normalization constant is C = √ . Therefore C a z | d / / (cid:105) = − √ | , , , (cid:105) − i √ | , , , (cid:105) , (26)which from Table II is equal to the | p / / (cid:105) orbital. Thesame equivalence stands for all the pairs of Eqs. (17)-(22).It is natural to wonder, if such an equivalence existsfor other than the highest j orbitals. For example, onecan check, if the orbital | d / / (cid:105) differs only in the z -axisfrom the | p / / (cid:105) orbital. The same procedure yields C a z | d / / (cid:105) (cid:54) = | p / / (cid:105) . (27)Therefore the de Shalit–Goldhaber pairs are uniqueamong the rest of the spin-orbit partners differing by | ∆ n, ∆ l, ∆ j, ∆ m j (cid:105) = | , , , (cid:105) . The pairs of Eqs. (17)-(22) differ by one quantum in the z -axis, thus they havesimilar structure in the x - y plane.In the above considerations we have exploited the fact,that in both asymptotic deformed oscillator basis andcartesian Elliott basis the only change, inflicted by theproxy-SU(3) approximation, is the change of the quan-tum number n z (and therefore also of N ) by one unit.This similarity allows us, as a first step to translate ina simple way the proxy-SU(3) approximation from theasymptotic deformed oscillator basis, in which it was de-fined in Ref. [18], to the cartesian Elliott basis. At asecond step the cartesian Elliott basis is connected in anon-trivial way to the spherical shell model basis, thusallowing us, to translate the proxy-SU(3) approximationinto the spherical coordinates. IV. THE IMPORTANCE OF THE x - y PLANE
Elliott has proved, that the rotational spectrum of thenucleus emerges from the SU(3) symmetry [11]. Thiscollective, rotational spectrum is derived by the occupiedsingle particle orbitals in the cartesian coordinate system,with the nuclear wave function being characterized by theElliott quantum numbers ( λ, µ ).The spatial, cartesian orbitals of a shell with N quantaare | n z , n x , n y (cid:105) : |N , , (cid:105) , |N − , , (cid:105) , |N − , , (cid:105) , |N − , , (cid:105) , |N − , , (cid:105) , |N − , , (cid:105) , ..., | , , N (cid:105) . (28)The derivation of the many body U(3) irreps [ f , f , f ]using the single particle states | n z , n x , n y (cid:105) as buildingblocks is discussed in detail in Refs. [55–57]. Briefly thesingle particle orbitals are ordered in decreasing numberof quanta n z , because the low-lying nuclear properties arederived from the many particle, highest weight (hw) U(3)irrep [ f , f , f ] [12, 19] with f , f , f being the summa-tions for every valence nucleon: f = { (cid:88) n z } max , (29) f = max { (cid:88) n x , (cid:88) n y } , at { (cid:88) n z } max , (30) f = min { (cid:88) n x , (cid:88) n y } , at { (cid:88) n z } max (31)and f ≥ f ≥ f . For the low-lying nuclear properties the Elliott quan-tum numbers of the highest weight SU(3) irrep are [12] λ = { (cid:88) n z } max − max { (cid:88) n x , (cid:88) n y } , (32) µ = (cid:12)(cid:12)(cid:12)(cid:88) n x − (cid:88) n y (cid:12)(cid:12)(cid:12) , at { (cid:88) n z } max . (33)Consequently the quantum number µ depends solely onthe distribution of quanta in the x - y plane. For λ ≥ µ thetotal angular momentum J and its projection K derivefrom the ( λ, µ ) [58]: K S = S, ( S − , . . . , − S, K = K S + K L ≥ , (34)where K L = µ, µ − , . . . , − µ, (35) J = K, K + 1 , . . . , ( λ + µ + S ) , (36)where S , K S are the nuclear spin and its projection, withthe exception that if K L = K S = K = 0 then J is evenor odd according to λ + S .In simple words the quanta in the x - y plane producethe quantum number µ from Eq. (33). Afterwards for λ ≥ µ from Eq. (35) the quantum number µ derivesall the K L bands within an irrep , i.e. the ground stateband with K L = 0, the γ band with K L = 2, the K = 4band, etc. It becomes obvious from Eq. (36), thatthe labeling J of the states within a band depends on K L , which depends on µ , which depends on (cid:80) n x , (cid:80) n y .Thus the only quantum number left for the λ to affect isthe J max = λ + µ + S value. The conclusion is, that the la-beling of the bands and of the states within depends verymuch on the distribution of quanta in the x - y plane andon µ , while λ affects the J max value within a band. Such J max states for medium mass and heavy nuclei, whichare characterized by λ (cid:29) µ is of highimportance, while an approximate value of λ is not dam-aging the band structure. (It should not be overlooked,however, that λ plays a crucial role in the predictions forthe collective variable β , similar to the crucial role playedby µ in the predictions for the collective variable γ [19].)The importance of the x - y plane is also illustratedin the SU(3) → SU(2) × U(1) decomposition, that is pre-sented in Section 2 of Ref. [12]. The nucleus is a 3Dobject, which rotates in space, has ellipsoidal shape andpossesses an SU(3) symmetry [56]. The ellipsoid is de-composed in the x - y plane and in the z symmetry axis.On the one hand the x - y plane is a 2D object (an ellip-sis), which is characterized by the SU(2) symmetry. Thedeformation of the ellipsis is measured by the µ quan-tum number of Eq. (33) and generates the projection ofthe total angular momentum K , which labels each bandin the nuclear spectrum. On the other hand the z -axispossesses the U(1) symmetry and its elongation or com-pression in comparison with the x - y plane is measured viaEq. (32) and generates the cut-off of the total angularmomentum. V. PROXY-SU(3) SYMMETRY IN THESPHERICAL SHELL MODEL BASIS
The main idea behind the proxy-SU(3) symmetry inmedium mass and heavy nuclei, introduced in [18], isthat one can restore the SU(3) symmetry within the spin-orbit shells among the magic numbers 28, 50, 82, 126by replacing the intruder asymptotic deformed oscillatororbitals by their ∆ K [∆ N ∆ n z ∆Λ] = 0[110] counterparts.Such pairs of orbitals in the Nilsson notation differ onlyin the cartesian z -axis and are identical in the x - y plane.Similarly in this work we introduce the proxy-SU(3)symmetry using the spherical shell model basis. In eachof the above mentioned spin-orbit shells the replacement of the intruder orbitals 1 g / , 1 h / , 1 i / by the or-bitals 1 f / , g / , h / respectively with the same m j leads to the reconstruction of the SU(3) symmetry. Theintruder orbitals with the highest | m j | = j max are to-tally annihilated by the a z operator, thus they do nothave any | ∆ n, ∆ l, ∆ j, ∆ m j (cid:105) = | , , , (cid:105) partner. Conse-quently the spin-orbit like shells among magic numbers6-14, 14-28, 28-50, 50-82, 82-126, 126-184 become 6-12,14-26, 28-48, 50-80, 82-124, 126-182 shells after the re-placement of the intruder orbitals. The replacements arepresented in detail in Table VII.Although the de Shalit–Goldhaber rule has been dis-covered for proton-neutron pairs, it is valid, that theproton-proton or neutron-neutron de Shalit–Goldhaberpairs of orbitals (Eqs. (17)-(22)) have similar struc-ture in the x - y plane. Thus, as outlined in Section IV,the replacement of these orbitals by one another is notdamaging the Elliott quantum number µ in prolate nu-clei. Consequently the K rotational bands and the J states within these bands are labeled accurately withinthe proxy-SU(3) symmetry. The only quantum numberaffected by the proposed replacement is the f = (cid:80) n z or λ from Eq. (32), which is relevant with the cut-off of thetotal angular momentum J max in each band, as pointedout in the previous section. But in medium mass andheavy nuclei the J max value is already too large, to beobserved experimentally [18, 19].In summary, in this section we have exploited the fact,that the de Shalit–Goldhaber pairs have identical struc-ture in the x - y plane, differing only by one quantum alongthe z -axis. Taking advantage of the transformation be-tween the cartesian Elliott basis and the spherical shellmodel basis, established in Sections 2 and 3, we wereable to “translate” the proxy-SU(3) substitution rulefrom the asymptotic deformed oscillator basis, in whichit is ∆ K [∆ N ∆ n z ∆Λ] = 0[110], to the spherical shellmodel basis, where it is | ∆ n, ∆ l, ∆ j, ∆ m j (cid:105) = | , , , (cid:105) .It should be noticed, that while in the asymptotic de-formed oscillator basis it is evident, that the projections of the orbital and the total angular momenta and thespin remain unchanged, in the spherical shell model ba-sis becomes evident, that the eigenvalues of the orbitaland the total angular momenta are changed by one unit.In other words, different bases offer different “views” ofthe same physical quantities, namely angular momentaand spin in the present case. VI. TOWARDS A UNITARYTRANSFORMATION
In the equations (17)-(22) of Section III we can expressthe normalization constant C in terms of the a † z , a z op-erators. The normalization constant is the expectationvalue C = (cid:104) n, l, j, m j | a † z a z | n, l, j, m j (cid:105) − / = < a † z a z > − / = < n z > − / . (37)Consequently the operator T = a z < a † z a z > − / (38)acts on the intruder states, which possess j max = N +1 / − ( j max − ≤ m j ≤ ( j max − T | n, l, j max , m j (cid:105) = | n, l − , j max − , m j (cid:105) . (39)It should be noted, that the above action is true onlyfor the intruder states and that the intruder states with m j = ± j max are annihilated by T . A graphical represen-tation of an example of the action of the operator T isgiven in Fig. 1.For instance in the 1 d / / orbital of Eq. (24) we have < a † z a z > = (cid:104) d / / | a † z a z | d / / (cid:105) == 25 · · , (40)and thus C = √ . (41)In this case T | d / / (cid:105) = a z (cid:104) d / / | a † z a z | d / / (cid:105) − / | d / / (cid:105) = C a z | d / / (cid:105) = | p / / (cid:105) . (42)The operator T resembles the unitary operator used inRef. [41] in the case of the harmonic oscillator, reviewedin Appendix A. In the case of the 3D-HO without spinone can use the unitary operator U = a z ( a † z a z ) − / , (43)affecting only the z -direction and leaving the x , y direc-tions intact.In relation to the prerequisites mentioned in AppendixA, in the present case we use the Hermitian operator[ O ( r , p , S )] − / = ( a † z a z ) − / . (44)Since O is Hermitian, it is possible, to find a basis, inwhich it is diagonal. Furthermore, in this basis the O − / makes sense, as we take the square root [60–63] of theinverse terms in the diagonal, if the eigenvalues of O arepositive. In the present case the operator a † z a z is diagonalin the | n z (cid:105) basis and has eigenvalues [45]: a † z a z | n z (cid:105) = n z | n z (cid:105) . (45)Therefore ( a † z a z ) − / = ( n z ) − / . (46)Obviously the operator a † z a z is not diagonal in the shellmodel states | n, l, j, m j (cid:105) . This is why we are using theexpectation value of this operator in Eq. (38). VII. UNITARY TRANSFORMATION FOR THEPROXY-SU(3) SCHEME
The question is now created, about what would havebeen an appropriate unitary transformation, performingthe proxy-SU(3) change within the spherical shell model.From Section 5 we know, that in the notation | n, l, j, m j (cid:105) the appropriate change is | , , , (cid:105) . Qualitatively wewould expect a transformation affecting only N and l ,since in spherical coordinates one has N = 2 n + l , where n is a non-negative integer. Therefore the only existingpossibility in this case, appears to be, that the loss ofone unit of N corresponds to the loss of one unit of l . However, in quantum mechanics we are familiar with theladder operators l + , l − , which correspondingly raise orlower the projection m l of l by one unit [45], but we arenot familiar with any simple ladder operators changing l or j by one unit. Therefore the task of constructing aunitary transformation in this case seems impossible.However the mapping between the cartesian Elliottstates and the spherical shell model states, worked outin Section 2, offers a way out. Eqs. (17)-(22) show, thatexactly for these orbitals, which become abnormal parityorbitals because of the spin-orbit interaction, the annihi-lation operator a z can be used. Indeed, Eqs. (17)-(22)show, that the a z operator reduces l and j by one unit,without changing the projection m j . In other words,one can perform the unitary transformation in the carte-sian Elliott basis, using the unitary transformation cor-responding to the a z operator in the cartesian Elliottbasis, and then translate the results into the language ofthe spherical shell model. Simply speaking the operators T and U of Section 6 are the same operator acting on dif-ferent bases: the operator T is acting on the shell modelstates | n, l, j, m j (cid:105) , while the operator U is acting on theElliott states | n z , n x , n y , m s (cid:105) .The operator T preserves the inner product of any twointruder states, possessing j = j max = N + 1 / − ( j max − ≤ m j ≤ ( j max − (cid:104) n, l, j max , m (cid:48) j | n, l, j max , m j (cid:105) = δ m (cid:48) j ,m j , (47)while the inner product of the states after the action of T is (cid:104) T n, l, j max , m (cid:48) j | T n, l, j max , m j (cid:105) , (48)where the shorthand notation T | n, l, j max , m j (cid:105) = | T n, l, j max , m j (cid:105) (49)has been used. The substitution of Eq. (39) in the aboveresults to (cid:104) T n, l, j max , m (cid:48) j | T n, l, j max , m j (cid:105) = (cid:104) n, l − , j max − , m (cid:48) j | n, l − , j max − , m j (cid:105) = δ m (cid:48) j ,m j . (50)Consequently, since the inner product is preserved, theoperator T is unitary, when acting on the intruder stateswith j max and − ( j max − ≤ m j ≤ j max − (cid:104) T n, l, j max , m (cid:48) j | T n, l, j max , m j (cid:105) = (cid:104) n, l, j max , m (cid:48) j | n, l, j max , m j (cid:105) . (51)The matrix elements of the operator T † T in the basisof the intruder shell model orbitals | n, l, j max , m j (cid:105) with j max = N + and − ( j max − ≤ m j ≤ j max − (cid:104) n, l, j max , m (cid:48) j | T † T | n, l, j max , m j (cid:105) = (cid:104) T n, l, j max , m (cid:48) j | T n, l, j max , m j (cid:105) = δ m (cid:48) j ,m j . (52)Thus the operator T † T is the square identity matrix T † T = I (53)with dimension (2 j max − × (2 j max − m j = ± j max are includedin the matrix representation, for which T | n, l, j max , ± j max (cid:105) = 0 , (54)then the operator T † T is no longer the identity matrix I and the mapping from the intruders to the proxies is rep-resented by a rectangular not by a square operator, sincetwo of the intruders (those with m j = ± j max ) have noimages in the proxy space. A generalization of the aboveconsideration is, that the transformation T is unitary,when acting on the intruder states with j max = N + and − ( j max − ≤ m j ≤ j max − | m j | = j max the abnormal par-ity orbitals, thus forming a full U (Ω) HO shell, com-posed by the unitary images of the abnormal parity or-bitals and the unaffected normal parity orbitals. As wehave shown above, the proxy-SU(3) substitution rule,which is ∆ K [∆ N ∆ n z ∆Λ] = 0[110] in the asymptoticdeformed oscillator basis, is translated into the rule | ∆ n, ∆ l, ∆ j, ∆ m j (cid:105) = | , , , (cid:105) in the spherical shellmodel basis. This correspondence is based on the fact,that the proxy-SU(3) approximation is only affecting onequantum in the z -axis, leaving the x - y plane unchanged.As a consequence, the changes inflicted by the proxy-SU(3) approximation in the cartesian Elliott basis andin the asymptotic deformed oscillator basis can be sim-ply expressed in both cases by using the a z annihilationoperator and the unitary T operator of Eq. (38) relatedto it.As an example, the 50-82 nuclear shell consists of the3 s / , 2 d / , 2 d / , 1 g / , and 1 h / orbitals. In compar-ison to the sdg g / orbital,which is pushed down by the spin-orbit interaction intothe shell below, and has gained the 1 h / orbital, whichhas come down, again because of the spin-orbit interac-tion, from the shell above. Through the unitary trans-formation under discussion, the orbital 1 h / is mappedonto the proxy-orbital 1 g / , forming together with the3 s / , 2 d / , 2 d / , 1 g / orbitals a complete sdg shell.In particular, the 1 h / ± / , 1 h / ± / , 1 h / ± / , 1 h / ± / , 1 h / ± / levels are transformed into the 1 g / ± / , 1 g / ± / , 1 g / ± / ,1 g ± / / , 1 g / ± / levels as shown in Fig. 1, while the levels1 h / ± / are left out of the transformation.It should be pointed out, that the +1 term in theHamiltonian, appearing in Eqs. (61) and (64), is of ut-most importance within the proxy-SU(3) approximationscheme, since this term guarantees, that the proxies ofthe abnormal parity orbitals do not slip into the shellbelow, but remain within the spin-orbit shell, in whichthey are already located, thus completing together withthe normal parity orbitals a full HO shell possessing therelevant U (Ω) symmetry. In Ref. [18] this constant termhas been added “by hand”, using physical arguments. Inthe present work it is proved, that this term arises fromthe unitary transformation between the abnormal paritylevels and their proxies.The +1 term in the Hamiltonians of Eqs. (61) and(64), in the Appendix A and B respectively, guaran-tees, that the proxy orbital 1 g / stays within the 50-82shell, in which the abnormal parity orbital 1 h / ex-isted. Therefore all orbitals 3s / , 2d / , 2 d / , 1 g / , h / ± / h / ± / h / ± / h / ± / h / ± / a z < a † z a z > − / g / ± / g / ± / g / ± / g / ± / g / ± / Intruders Proxies
FIG. 1: In the 50-82 shell the intruders 1 h / m j (ex-cept for the 1 h / ± / ) are mapped onto the 1 g / m j through the operator T = a z < a † z a z > − / . The in-ner product of any two intruders with j = j max and − ( j max − ≤ m j ≤ ( j max −
1) remains the same before andafter the transformation, i.e. , (cid:104) n, l, j max , m (cid:48) j | n, l, j max , m j (cid:105) = (cid:104) T n, l, j max , m (cid:48) j | T n, l, j max m j (cid:105) , which equals to (cid:104) n, l − , j − , m (cid:48) j | n, l − , j − , m j (cid:105) = δ m (cid:48) j ,m j . Thusthe operator T is unitary, when acting on the intruderorbitals of each spin-orbit like shell with j = j max and − ( j max − ≤ m j ≤ ( j max − m j = ± j max are being totally annihilated and so they areleft out of the mapping. / are in the same shell, forming a U(15) algebra hav-ing an SU(3) subalgebra. The +1 term in Eqs. (61)and (64) also clarifies the question, about the form of theHamiltonian to be used within the proxy-SU(3) scheme:It is the usual Hamiltonian for the shell under discus-sion, using the parameters of the normal parity orbitalswith N quanta. In Ref. [18], for example, in the 50-82shell the abnormal parity orbital 1 h / , which has comedown from the 82-126 shell having been calculated fromthe Nilsson Hamiltonian with the parameters of the or-bitals with N +1 = 5, is transformed into a 1 g / orbital,which, from this point on, is treated using the parametersof the orbitals with N = 4. Obviously this is an approxi-mation, since in Eqs. (61) and (64) the Hamiltonians H and H (cid:48) , connected by the unitary transformation, pos-sess the same parameter set. Furthermore, one can seefrom Table I of Ref. [18], that the change of the param-eter values, when moving from one shell to the next one,is not dramatic. However, this approximation does havea non-negligible effect on the relevant Nilsson diagrams,as seen for example in Fig. 2 of Ref. [18], namely thatthe proxies of the abnormal parity levels do not convergeto a single point at zero deformation. VIII. COMPARISON TO THE UNITARYTRANSFORMATION FOR THE PSEUDO-SU(3)SCHEME WITHIN THE SPHERICAL NILSSONMODEL
The problem of finding an approximate SU(3) symme-try in a subspace of the total Hilbert space was addressedearlier within the pseudo-SU(3) scheme [30–38]. Thereis a significant similarity between the pseudo-SU(3) andproxy-SU(3) approaches, although the approximationsthey apply are different. In the pseudo-SU(3) scheme allthe intruder levels are skipped, and the rest of the statescorrespond to a HO shell of one oscillator quantum less.Therefore, usually SU(3) irreps of smaller dimensions arerelevant within the pseudo-SU(3) scheme. A considerablenumber of nucleons are out of the SU(3) subspace, andthey are usually treated in terms of the seniority formal-ism. In the proxy-SU(3) scheme only two single orbitalsare omitted (those with | m j | = j max ), so usually SU(3)irreps of higher dimensions are relevant, while no othersector is coupled to the HO obtained within the proxy-SU(3) scheme. Due to the different approximations used,the performance and the applicability of the two mod-els can be different. Nevertheless, it may also very wellhappen, that their performance is similar for some phe-nomena. The decisive factor seems to be the relevantsubset of the single-particle orbits, which may be simi-lar or different for the two cases. A recent applicationto quartet spectra including also major shell excitationsshowed, that the physics of the two approaches is verysimilar (in the case investigated), in spite of the differentSU(3) quantum numbers occurring in each of them [64].In addition, recent calculations of the collective variables β and γ within the two different approximation schemeshave been found, to lead to compatible results [65].The unitary transformation applicable in the case ofthe pseudo-SU(3) symmetry, described in Appendix B,has been applied for clarification purposes to the spheri-cal Nilsson Hamiltonian [41, 42] H (cid:126) ω = H − k L · S − kν L , (55)where H is the 3D-HO Hamiltonian defined in AppendixB, while k and ν are free parameters, adjusted so that theexperimental single-particle energies are well reproduced.In this case the unitary operator is obtained from ( a · S ),as given in Eq. (63). The Hamiltonian after the unitarytransformation reads [41, 42] H (cid:48) (cid:126) ω = H + 1 − k (2 ν − L · S − kν L − k ( ν − . (56)Denoting by l ( l + 1), s ( s + 1), j ( j + 1) the eigenvalues ofthe operators L , S , J , we notice in this case, that theunitary transformation keeps invariant the total angularmomentum j , while N , l , s are mapped onto their pseudocounterparts ˜ N , ˜ l and ˜ s , so that the correct ˜ j = j isobtained. In the case of k = ν = 0 the spherical Nilssonresults are reduced to the usual 3D-HO results.As an example, the 50-82 nuclear shell consists of the3 s / , 2 d / , 2 d / , 1 g / , and 1 h / orbitals. In compar-ison to the sdg g / orbital, which is pushed down by the spin-orbit interaction intothe shell below, and has gained the 1 h / orbital, whichhas come down, again because of the spin-orbit interac-tion, from the shell above. Through the unitary transfor-mation under discussion, the orbitals 3 s / , 2 d / , 2d / ,1 g / , are mapped onto the pseudo-orbitals 2 p / , 2 p / ,1 f / , 1 f / , forming a complete pf shell. During thismapping, j remains invariant, while N , l , s are appro-priately changed in order to guarantee the invariance of j . IX. DISCUSSION
The main results, obtained in this work, are the fol-lowing ones.a) Using the connection between the cartesian Elliottbasis and the spherical shell model basis it is shown, thatthe proxy-SU(3) approximation within the spherical shellmodel framework corresponds to the replacement of theabnormal parity orbitals in a shell by their de Shalit–Goldhaber counterparts.b)It is shown, that the replacement of the abnor-mal parity orbitals within a shell by their de Shalit–Goldhaber counterparts is equivalent to a unitary trans-formation, affecting only the z -axis, while leaving the x - y plane and the z -projections of the angular momenta andthe spin intact.Furthermore, the following points should be empha-sized.a)The present work involves the pure SU(3) symmetryof the usual 3D-HO, appearing in the Elliott model, thusbypassing the deformed SU(3) symmetry connected tothe Nilsson model and the mathematical complications,related to it [28, 29].b)The present work paves the way for shell model cal-culations involving the proxy-SU(3) approximation, sincethe appropriate replacements to be used within the spher-ical shell model, have been determined. As a first step,one can check numerically the accuracy of the proxy-SU(3) approximation by performing a calculation of ac-cessible size twice, without and with the proxy-SU(3)substitution, and comparing the results. At a next stage,one can try to cut down the size of large shell model cal-culations by taking advantage of the SU(3) properties.c)The difference between a classification scheme and adynamical symmetry is clarified. Proxy-SU(3) is a classi-fication scheme valid through the entire nuclear shells, inthe same way in which the Elliott SU(3) is valid through-out the whole sd shell. The proxy-SU(3) dynamical sym-metry will be needed, in order to calculate spectra andelectromagnetic transition probabilities in deformed nu-clei, after choosing an appropriate Hamiltonian involvingthree- and/or four-body terms [34, 35]. Work in this di-rection is in progress.d)In corroboration of the previous point one can re-call, that in the first Elliott paper [11] an alternativeclassification scheme for the sd shell is given in terms ofthe O(6) subalgebra. In other words, different classifica-tion schemes can be used for the entire sd shell irrespec-tively of the dynamical symmetries, possibly describingthe spectra and electromagnetic transition rates of vari-ous nuclei within this shell. One should notice, however,that the O(6) classification scheme is not possible in allhigher shells.0 Acknowledgements
Financial support by the Greek State ScholarshipsFoundation (IKY) and the European Union within theMIS 5033021 action, by the Bulgarian National ScienceFund (BNSF) under Contract No. KP-06-N28/6 and byNational Research, Development and Innovation Fund ofHungary, under the K18 funding scheme with ProjectNo. K 128729 is gratefully acknowledged.
Appendix A: Prerequisites for the unitarity of theoperators
Within the pseudo-SU(3) approach it is well known,that there is a unitary transformation connecting thenormal parity orbitals in a nuclear shell to their pseudo-SU(3) counterparts. A pedagogical description of thisunitary transformation is given in [41], applied to thecase of the one-dimensional harmonic oscillator (1D-HO)(Section 4 of [41]). Subsequently, the unitary transforma-tion for the three-dimensional harmonic oscillator (3D-HO) from spherical coordinates to the pseudo-SU(3) ba-sis is given in Section 5 of [41], while the unitary trans-formation, appropriate for the spherical Nilsson Hamil-tonian (with deformation (cid:15) = 0), which includes a spin-orbit interaction term and an l term, where l is orbitalangular momentum, is given in Section 6 of [41]. A con-cise version of this work is given in [42]. A generalizationof this unitary transformation to the case of the deformedNilsson Hamiltonian has been given in [43].Let us consider an operator G ( r , p , S ), where r , p , S are the position, momentum, and spin respectively. Theoperator G will not be in general unitary, but it can bemade unitary through the use of the Hermitian operator[41] O ( r , p , S ) = G † ( r , p , S ) G ( r , p , S ) , (57)by taking U = G ( r , p , S )[ O ( r , p , S )] − / . (58)Since O is a Hermitian operator, it is possible, to finda basis, in which this operator is diagonal. In this basisthe operator [ O ( r , p , S )] − / can be obtained by takingthe square roots of the inverse terms in the diagonal, pro-vided that the eigenvalues of O are positive. Square rootsof positive operators have been considered repeatedly inthe mathematical literature [60–63].In the case of the original 1D-HO Hamiltonian one has[45]: H = a † a, a † = 1 √ x − ip ) , a = 1 √ x + ip ) , (59)where x and p are the coordinate and the correspondingmomentum, while a † and a denote the usual creation andannihilation operators. The unitary operator is obtainedfrom the usual annihilation operator a and has the form[41] U = a ( a † a ) − / . (60) It is clear, that the above mentioned prerequisites arefulfilled, since the operator a † a in the basis considered isdiagonal and has positive eigenvalues, so that the squareroots of their inverses can be calculated. The Hamilto-nian resulting from the unitary transformation is [41] H (cid:48) = a † a + 1 . (61)This implies, that the levels characterized by N quanta inthe old Hamiltonian carry N − N is replaced by N −
1, but the lossof energy is counterbalanced by the +1 term appearingin Eq. (61). This +1 term plays a major role in theproxy-SU(3) approximation, as discussed in Section 7.In the above the spectra of the old and the new Hamil-tonian are the same, as they should be for a unitary trans-formation. It should also be noticed, that we have thecomplete set of states of the harmonic oscillator both be-fore and after the unitary transformation, i.e. , the groundstate is included in both cases. Therefore the U(1) sym-metry characterizing the 1D-HO before the unitary trans-formation is conserved after the unitary transformation.
Appendix B: Unitary transformations in thepseudo-SU(3) scheme
For the sake of comparison, the original 3D-HO Hamil-tonian for a particle with spin 1/2 is: H = a † · a , a † = 1 √ r − i p ) , a = 1 √ r + i p ) , (62)where r and p are the coordinates and the correspondingmomenta, while a † and a denote the usual creation andannihilation operators. The unitary operator is obtainedfrom ( a · S ), where S is the spin operator, and reads [41] U = 2( a · S )( a † · a − L · S ) − / . (63)The Hamiltonian resulting from the unitary transforma-tion is [41]: H (cid:48) = a † · a + 1 . (64)This again implies, that the levels characterized by N quanta in the old Hamiltonian carry N − N of the original Hamiltonian ismapped onto N − [1] F. Iachello and A. Arima,
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103 2 √ √ − √ − (cid:113) − (cid:113) √ (cid:113) | , , , − (cid:105) − i √ − i √ − i √ − i √ − i √ i √ i (cid:113) − i √ − i (cid:113) i (cid:113) | , , , (cid:105) i √ − i √ − i √ − i √ i √ i √ i (cid:113) − i (cid:113) − i √ i (cid:113) | , , , − (cid:105) √ − (cid:113) √ − (cid:113) (cid:113) − (cid:113) √ − (cid:113) √ − √ | , , , (cid:105) √ − √ (cid:113) − (cid:113) (cid:113) − (cid:113) √ − √ (cid:113) − √ | , , , − (cid:105) − √ − (cid:113) − √ − √ − (cid:113) − (cid:113) − (cid:113) − √ | , , , (cid:105) √ − (cid:113) (cid:113) √ √ − √ − (cid:113) − (cid:113) | , , , − (cid:105) i √ − i (cid:113) i (cid:113) − i √ | , , , (cid:105) − i (cid:113) i √ i √ − i (cid:113) | , , , − (cid:105) − √ − (cid:113) √ − √ (cid:113) (cid:113) − (cid:113) √ | , , , (cid:105) √ − (cid:113) − (cid:113) √ − √ √ − (cid:113) (cid:113) | , , , − (cid:105) − i √ − i √ − i √ i √ i (cid:113) i (cid:113) i (cid:113) | , , , (cid:105) i √ − i √ − i √ − i (cid:113) − i √ i (cid:113) i (cid:113) | , , , − (cid:105) √ − √ √ √ − (cid:113) (cid:113) − (cid:113) | , , , (cid:105) √ − √ √ − (cid:113) √ (cid:113) − (cid:113) | , , , − (cid:105) − √ − (cid:113) (cid:113) (cid:113) | , , , (cid:105) √ − (cid:113) − (cid:113) (cid:113) T A B L E V I : T h e s a m e a s T a b l e II , bu t f o r N = . | n z , n x , n y , m s (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | , , , − (cid:105) | , , , (cid:105) | n , l , j , m j (cid:105) | p / − / (cid:105) − i √ √ − i √ √ − √ − √ − i √ √ − √ | p / / (cid:105) i √ √ i √ √ √ √ i √ √ √ | p / − / (cid:105) i (cid:113) − √ i √ − (cid:113) i √ − √ | p / − / (cid:105) i √ − √ i √ − √ − (cid:113) − (cid:113) i √ − √ − (cid:113) | p / / (cid:105) i √ √ i √ √ − (cid:113) − (cid:113) i √ √ − (cid:113) | p / / (cid:105) i (cid:113) √ i √ (cid:113) i √ √ | f / − / (cid:105) − i (cid:113) √ i √ − (cid:113) − √ − i √ √ | f / − / (cid:105) i (cid:113) − √ i √ − (cid:113) (cid:113) i (cid:113) − (cid:113) − i √ √ | f / − / (cid:105) − i (cid:113) √ − i √ (cid:113) − √ − √ i (cid:113) − (cid:113) (cid:113) | f / / (cid:105) i (cid:113) √ i √ (cid:113) √ √ − i (cid:113) − (cid:113) − (cid:113) | f / / (cid:105) − i (cid:113) − √ − i √ − (cid:113) − (cid:113) i (cid:113) (cid:113) i √ √ | f / / (cid:105) i (cid:113) √ − i √ − (cid:113) √ − i √ − √ | f / − / (cid:105) i √ − (cid:113) − i (cid:113) √ | f / − / (cid:105) i √ − (cid:113) − i (cid:113) √ − (cid:113) − i (cid:113) (cid:113) | f / − / (cid:105) i (cid:113) − √ i √ − (cid:113) − √ − i √ √ − i (cid:113) (cid:113) | f / − / (cid:105) i √ − (cid:113) i (cid:113) − √ − (cid:113) − (cid:113) − i (cid:113) (cid:113) (cid:113) | f / / (cid:105) i √ (cid:113) i (cid:113) √ − (cid:113) − (cid:113) − i (cid:113) − (cid:113) (cid:113) | f / / (cid:105) i (cid:113) √ i √ (cid:113) − √ i √ √ − i (cid:113) − (cid:113) | f / / (cid:105) i √ (cid:113) − i (cid:113) − √ − (cid:113) i (cid:113) (cid:113) | f / / (cid:105) i √ (cid:113) − i (cid:113) − √ TABLE VII: The shell model orbitals of the original spin-orbitlike shells and of the proxy-SU(3) shells. The magic number14 is proposed as a sub-shell closure in Ref. [59]. The symme-try of each proxy-SU(3) shell is U(Ω) with Ω = ( N +1)( N +2)2 .The orbitals, that are being replaced, are denoted with boldletters. See Section 5 for further discussion.spin-orbit proxy-SU(3) 3D-HOmagic numbers original orbitals proxy orbitals proxy U (Ω) symmetry magic numbers magic numbers6-14 1 p / ± / p / ± / U (3) 6-12 2-8 / ± / , ± / / ± / , ± / / ± / -14-28 2 s / ± / s / ± / U (6) 14-26 8-201 d / ± / , ± / d / ± / , ± / / ± / , ± / , ± / / ± / , ± / , ± / / ± / -28-50 2 p / ± / p / ± / U (10) 28-48 20-402 p / ± / , ± / p / ± / , ± / f / ± / , ± / , ± / f / ± / , ± / , ± / / ± / ,..., ± / / ± / ,..., ± / / ± / -50-82 3 s / ± / s / ± / U (15) 50-80 40-702 d / ± / , ± / d / ± / , ± / d / ± / ,..., ± / d / ± / ,..., ± / g / ± / ,..., ± / g / ± / ,..., ± / / ± / ,..., ± / / ± / ,..., ± / / ± / -82-126 3 p / ± / p / ± / U (21) 82-124 70-1123 p / ± / , ± / p / ± / , ± / f / ± / ,..., ± / f / ± / ,..., ± / f / ± / ,..., ± / f / ± / ,..., ± / h / ± / ,..., ± / h / ± / ,..., ± / / ± / ,..., ± / / ± / ,..., ± / / ± / -126-184 4 s / ± / s / ± / U (28) 126-182 112-1683 d / ± / , ± / d / ± / , ± / d / ± / ,..., ± / d / ± / ,..., ± / g / ± / ,..., ± / g / ± / ,..., ± / g / ± / ,..., ± / g / ± / ,..., ± / i / ± / ,..., ± / i / ± / ,..., ± / / ± / ,..., ± / / ± / ,..., ± / / ± /2