The suggested presence of the tetrahedral-symmetry in the ground-state configuration of the 96 Zr nucleus
Jerzy Dudek, Dominique Curien, David Rouvel, Katarzyna Mazurek, Yoshifumi Shimizu, Shingo Tagami
TThe suggested presence of the tetrahedral-symmetryin the ground-state configuration of the Zr nucleus J Dudek ∗ , D Curien † and D Rouvel ‡ Institut Pluridisciplinaire Hubert Curien, IN2P3-CNRS, France, and Universit´e de Strasbourg,23, rue du Loess, B. P. 28, F-67037 Strasbourg Cedex 2, France
K Mazurek § The Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciencesul. Radzikowskiego 152, Pl-31432 Krak´ow, Poland
Y R Shimizu ¶ and S Tagami (cid:107) Department of Physics, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan
Pacs Ref : 21.10.-k, 21.10.Pc, 21.30.-x, 21.60.-n, 21.60.Ka
Abstract
We discuss the predictions of the large scale calculations using the re-alistic realisation of the phenomenological nuclear mean-field theory.Calculations indicate that certain Zirconium nuclei are tetrahedral-symmetric in their ground-states. After a short overview of the re-search of the nuclear tetrahedral symmetry in the past we analysethe predictive capacities of the method and focus on the Zr nucleusexpected to be tetrahedral in its ground-state.
1. Nuclear Point-Group Symmetries: The Searchof Tetrahedral Symmetry – A Short Overview
Analysing point-group symmetries of molecules has be-come one of the standard tools of molecular quantum me-chanics. These symmetries often result from the relativepositions of the constituent atoms (as e.g. positions of theHydrogen atoms in the CH molecule) and in this sensetheir presence may be considered rather intuitive. In con-trast, there seem to be no intuitively direct analogies withatomic nuclei, which can be considered compact objectswhose volume is nearly equal to the sum of the volumesof the constituent nucleons. Moreover, the underlyingstrong interactions, both non-central and non-local, aremuch more complex compared to the Coulomb interac-tions governing the molecular structure.Despite that, the molecular-geometry guided intuitionhas been followed by certain authors, who constructedgroup-theoretical models of nuclei based on the idea ofthe modelling in terms of alpha clusters. As an example,in analogy to the tetrahedral symmetry of the methanemolecules, the nuclear tetrahedral symmetry induced byfour α -clusters in O has been discussed in e.g. Ref. [1].Nuclear mean-field theory is one of the most successfultools in nuclear structure. Today, the most frequent real-isations of this theory are: a. The Phenomenological one(the so-called macroscopic-microscopic method); b. TheRelativistic Mean Field theory based on the Dirac formal-ism, and: c. The Hartree-Fock theory. The first of them is ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] ¶ e-mail: [email protected] (cid:107) e-mail:
[email protected] technically the best adapted to study the nuclear geomet-rical symmetries. This is in particular true for the phe-nomenological Woods-Saxon realisation of the approach,according to which the central potential is constructed as V W S ( (cid:126)r ; V , R , a ) ≡ V Σ ( (cid:126)r ; R ) /a ] , (1)where Σ denotes the nuclear surface and dist Σ ( (cid:126)r ; R ) thedistance of the point (cid:126)r from the surface. Above, V , R and a are adjustable parameters characterising the po-tential depth, radius and diffusivity, respectively. In ourapproach, they are fixed once for all, i.e., for all the nucleiin the Periodic Table and independent of the deformation– where from the name: ‘Universal Parameterisation’.According to the above definition, the potential containsa constant diffusivity parameter and therefore has an over-all linear dependence of the argument of the exponentialon the distance of a given point (cid:126)r from the nuclear sur-face. [This is in contrast to some alternative forms usedin the literature, in which the diffusivity is treated as aposition-dependent quantity.] It follows that the resultingHamiltonian has exactly the same symmetry as the under-lying nuclear surface. Of course similar can be said aboutthe spin-orbit potential of the Woods-Saxon type. Conse-quently, the analysis of the point-group symmetries of theWoods-Saxon mean-field Hamiltonian can be reduced tothe analysis of the point-group symmetries of the under-lying nuclear surfaces.With the help of such an approach it was shown for thefirst time in Ref. [2], that the non-trivial nuclear point-group symmetries, such as the tetrahedral one, can beeasily realised with the help of the realistic nuclear Woods-Saxon Hamiltonians with the single deformation parame-ter α . The calculated single-particle nucleonic spectrawere shown to satisfy the two-fold and four-fold degen-eracies related to the E , and E ∗ (two-dimensional) and G (four-dimensional) irreducible representations of the tetra-hedral group, cf. figures 3 and 4 in the above reference.Moreover, it was shown that the local minima on the totalpotential energy surfaces corresponding to the tetrahedralsymmetry are obtained thus paving the way towards theidea of the new, richer forms of the shape coexistence in1 a r X i v : . [ nu c l - t h ] A p r tomic nuclei. Such a coexistence may involve non-axial,i.e. different from the prolate/oblate quadrupole or pear-shape octupole symmetries discussed so far abundantly inthe literature.At the same time it has been suggested [2] that thenew point-group symmetries generate the new chainsof magic numbers in analogy to the well known ones,( Z/N ) spherical =8, 20, 28, 50, 82 and 126 generated bythe spherical symmetry. The prediction of the tetrahedralmagic numbers ( Z/N ) tetrahedral =56, 64, 70, 90, 112 and136 has been formulated in the cited article where onlymoderately heavy and heavy nuclei have been studied.Extending realistic calculations which involve tetrahe-dral symmetry of nuclear shapes represented by the T d -group and α deformation parameter, from now on re-ferred to as tetrahedral, it was shown that a combinationof quadrupole and tetrahedral components may lead to anew class of shapes, symmetric with respect to the D d group. The latter describes, in the nuclear context, someexotic superdeformed shapes, with the spherical harmonicexpansion in terms of the leading axial-quadrupole, α ,and tetrahedral, α , components combined, as suggestedin Ref. [3] [cf. also Ref. [4], the latter focussed on the Hg-Pbregion, and Ref. [5].]Unfortunately, the names used sometimes in the litera-ture to describe the related nuclear geometry such as ‘non-axial octupole shapes’, hide the presence of possibly verydistinct symmetry effects . Indeed, the nuclear octupoledeformations α , α and α imply very different point-group symmetries. It then may become useful to designdistinct spectroscopic criteria associated with each of thesesymmetries, e.g. in terms of the energy-vs.-spin stagger-ing (see below), approximate level degeneracies or specificbranching ratios among the electromagnetic transitions –all these depending on the dominating symmetry.The mean-field theory studies were continued in Ref. [6],where the tetrahedral-symmetry induced ‘magic’ gaps inthe single-particle nuclear spectra have been found to in-clude ( Z/N ) tetrahedral =16, 20, 32, and 40 for the relativelylight nuclei, and 142 for the heaviest ones. In this contextthe role of the four-fold degeneracies, alternatively, thefour-dimensional irreducible representations of the tetra-hedral group, as the background of the large tetrahedralshell gaps has been emphasised. Moreover, the predictionsrelated to the new forms of (tetrahedral) shape-isomerismhave been formulated for Zr,
Zr,
Yb and
Fm.These calculations have been extended in Ref. [7], em-ploying the Skyrme Hartree-Fock technique with the SLy4parameterisation. These results suggested, that the veryexotic − Zr nuclei are tetrahedral in their ground-states [cf. also Ref. [8]]. In Ref. [7], yet another researchdirection has been proposed and illustrated, viz. the ap-plication of the group representation theory to calculatethe symmetry-induced characteristic intensity branchingratios of the electromagnetic transitions emitted by thetetrahedral-symmetric nuclear quantum rotor [cf. Fig. (4)in Ref. [7]]. Let us recall that atomic nuclei (as all other quantum systems)undergo what is referred to as ‘zero-point motion’, perpetual oscilla-tions, among others, in all the shape degrees of freedom, what impliesthat the static symmetries described by the mean-field Hamiltoniancannot represent more than just the leading order symmetries.
A more general ‘theory of nuclear stability’ based on thepoint-group symmetries has been formulated in Ref. [9],where, moreover, the illustrations related to the tetra-hedral symmetry minima in Zr, Zr and
Zr nucleican be found, together with the indication that the axial-symmetry octupole-shape minima are in a direct competi-tion with the tetrahedral ones – all at the zero quadrupoledeformation. These calculations indicate that the nu-clei in question should manifest strong octupole transi-tions and thus strong B (E3) values which according totheory, certainly do not correspond to the quadrupole-(super)deformed minima predicted in Zr and Zr .Independently, the Hartree-Fock mean-field calculationsfocused on the Z = N nuclei in Ref. [10], confirmed boththe presence of the superdeformation and the instabilityof the spherical configuration with respect to the α de-formation in the tetrahedral doubly-magic Zr nucleus.The tetrahedral instability of the spherical configurationsin selected nuclei, including Zirconium has been confirmedin [11] where, in addition, the sensitivity of the predictionswith respect to the choice of the Skyrme Hartree-Fock pa-rameterisation (SIII, SkP, SkM ∗ ) has been tested. Cal-culations using again Skyrme Hartree-Fock method sup-plemented with the Generator Coordinate Ansatz for Zrand Zr [12] confirmed the competition between the defor-mations of the octupole-axial and tetrahedral symmetries,the tetrahedral one dominating.Table 1: Experimental values of the B (E3) reduced tran-sition probabilities in Weisskopf units for to the first 3 − excitation, in the vicinity of the tetrahedral doubly-magicnucleus Zr showing the strongest effect in Zr. Thedata are from Ref. [13].Z vs. N 54 56 58 60 Pd - - - 29 ± Ru - 14 ± Mo - 31 ± ± Zr - 57 ± Sr 18.3 ±
11 - - -Among the three doubly-magic tetrahedral nuclei, i.e., Zr , Zr and Zr , only the second one is stable.As the consequence, the experimental results concerningthe reduced probabilities for the B (E3) transitions can befound in the literature only for this one, and for a fewneighbouring nuclei. The existing results are presented inTable 1, showing that the measured value for Zr, i.e., B (E3) =57 ± B (E3) value ever measured. Given thefact that the theory predictions favour the dominance ofthe α over α , cf. also the results in Figs. 3-4 below, oneis tempted to suggest that these very big values shouldbe attributed to the presence of the tetrahedral ratherthan octupole-axial symmetry. [For the domination of the α over α deformation, the reader is referred to Fig. (4)in Ref. [9] and to Ref. [14].] Finally, an overview of the2etrahedral symmetry oriented nuclear shell-effect calcu-lations focussing on the Rare Earth nuclei can be found inRef. [15].More recent microscopic calculations addressing the is-sue of the nuclear tetrahedral symmetry have been per-formed using advanced projection techniques [16] includ-ing the Generator Coordinate approach and various formsof the nuclear interactions varying between the phe-nomenological and the self-consistent Gogny Hartree-Fockapproach. These calculations fully confirm the importanceof the tetrahedral symmetry on the nuclear level, the sym-metry which strongly enhances the nuclear stability prop-erties. The calculations demonstrate in particular the ex-istence of the privileged spin-parity combinations of thenuclear states forming the tetrahedral sequences (‘bands’),the structures whose properties will be helpful in analysingthe results of dedicated experiments, cf. Ref. [17] and moreparticularly Ref. [18].
2. Testing the Method’s Prediction Capacity inTerms of the Nuclear Non-axial Configurations
Whereas the nuclear quadrupole deformations are char-acterised by a single non-axiality parameter, α , theoctupole non-axial degrees freedom are characterised bythree: α , α (tetrahedral) and α . According to theshape parameterisation in terms of the spherical harmon-ics, the non-axiality is determined by the ϕ -dependence ‐ ‐ ‐ E n e r gy S t a gg e r i n g Spin
Fig.
1: Staggering properties of the nuclei surrounding the re-cently measured Ge, Ref. [20]. The data for Se and Krare from [23] and [24], respectively. of the spherical harmonics Y λµ ( ϑ, ϕ ) ∝ P µλ (cos ϑ ) exp( iµϕ )and it follows that this dependence is identical for the non-axial quadrupole and tetrahedral symmetry shapes.Since no single case of the tetrahedral symmetry hasbeen demonstrated so far through experiment we will il-lustrate, following Refs. [19, 20], the prediction capacity ofthe present method using the results for the quadrupole-type non-axiality, obtained prior to the recent experimentof Ref. [20]. This is pertinent in the actual context sincewe will be able to compare the detailed shape-coexistenceproperties in the nuclei which are close to the Zirconiumnuclei addressed in this article.We follow here the criterion based on the early Ref. [19], E(fyu)+Shell[e]+Correlation[PNP] M i n i m i s a ti on ov e r (cid:95) N u c l ea r P o t e n ti a l - E n e r gy D a t a - B a s e ( S t r a s bou r g - C r ac o w C o ll a bo r a ti on ) Ge Ge
76 32 (cid:96)(cid:117) cos( (cid:97) +30 o ) (cid:96) (cid:117) s i n ( (cid:97) + o ) Emin=-1.29 Eo= 1.62 E(fyu)+Shell[e]+Correlation[PNP] M i n i m i s a ti on ov e r (cid:95) N u c l ea r P o t e n ti a l - E n e r gy D a t a - B a s e ( S t r a s bou r g - C r ac o w C o ll a bo r a ti on ) Se Se
74 34 (cid:96)(cid:117) cos( (cid:97) +30 o ) (cid:96) (cid:117) s i n ( (cid:97) + o ) Emin= 0.13 Eo= 2.56 E(fyu)+Shell[e]+Correlation[PNP] M i n i m i s a ti on ov e r (cid:95) N u c l ea r P o t e n ti a l - E n e r gy D a t a - B a s e ( S t r a s bou r g - C r ac o w C o ll a bo r a ti on ) Kr Kr
76 36 (cid:96)(cid:117) cos( (cid:97) +30 o ) (cid:96) (cid:117) s i n ( (cid:97) + o ) Emin= 0.15 Eo= 3.09
Fig.
2: Total energy surfaces of the nuclei illustrated in Fig. 1.At each ( β, γ )-point the energy was minimised over the hex-adecapole α deformation, where from the ‘traditional’ sym-metry in terms of sectors of ∆ γ = 60 o is lost. Top: Ge,whose stable triaxial deformation has recently been confirmedexperimentally, cf. Ref. [20], and Fig. 1. According to our cal-culations the equilibrium value γ eq. ≈ o . For Se, middle,and Kr, bottom, there exist only the axial symmetry minimawhich may give rise to the rotational bands observed. whose authors introduced the energy vs. spin staggeringproperties to distinguish between the rigid triaxial quan-tum rotors [as discussed long ago by Davydov and Filip-pov, [21]] and the axially symmetric rotor with possiblytriaxial vibrations (‘ γ -soft rotor’) [22]. According to those3odels, the odd-spin (odd- I ) states of a γ -soft rotor-bandare closer in energy to the neighbouring even-spin stateswith ( I + 1) rather than ( I − S ( I ) ≡ { [ E ( I + 1) − E ( I )] − [ E ( I ) − E ( I − } (2)which represents the finite difference approximation to thesecond derivative of the energy vs. spin, ∂ E∂I , thus display-ing the sign differences predicted by the rotor models men-tioned. To illustrate the experimental results in Fig. (1) wehave renormalised the above definition to comply with theconvention used in [20] by using S ( I ) ≡ S ( I − /E (2 +1 ).The results presented in the Figure demonstrate that ac-cording to the model-criterion just specified, the nucleus Ge exhibits the properties of the rigid rotor whereas theclose-neighbours, for which the relatively unambiguous ex-perimental data exist are expected to be axial or nearlyaxial in agreement with the theoretical predictions illus-trated in Fig. 2. We may conclude that performance of themodel calculations can be qualified as realistic; we believethat so are the predictions for the nucleus Zr which willbe discussed next.
3. Competing Two Octupole Modes: Zr Case
The calculations by the present-, and by the authors citedabove leave no doubt that: a. The two octupole modes,i.e., α representing the tetrahedral ( T d ) symmetry, and α representing axial ( C ∞ ) symmetry compete on thetotal energy maps in the Zirconium region, and b. Thetetrahedral symmetry most often wins the competition.The strong presence of the collective octupole mode inthe region as manifested by the experiment, cf. Table 1and surrounding text, clearly confirms this prediction. Itthen follows that the tetrahedral symmetry should be thedominating factor, and it will be instructive to illustratethe microscopic origin of this mechanism.Figures 3 and 4 illustrate the characteristic dependenceof the proton single-particle Woods-Saxon energies [theneutron levels present very similar features and are notshown here] on the two competing deformations. Observethat whereas at tetrahedral symmetric shapes the gap at Z = 40 extends to the very large deformations of about α ≈ .
3, the same gap decreases continuously as a func-tion of α . Similar observations can be made about thegap at Z = 56 (in the case of protons this gap is close tothe zero-binding limit what is not the case for the neu-trons). The origin of this systematic difference can betraced back to the four-fold degeneracies of the majorityof the single-particle levels in the case of the tetrahedralsymmetry. The corresponding levels can be identified inFig. 4 as marked with the double Nilsson labels. Becauseof the fact that the dominating majority of levels belongto this category the average inter-level spacing is system-atically larger in the tetrahedral symmetry case leading tooverall larger spacings.According to our calculations, the ground-state min-imum of Zr corresponds to the tetrahedral deforma-tion with α ≈ .
15 and a small octahedral deforma-tion, with the energy about 1300 keV below the spheri- - .2 - .1 .0 .1 .2 .3 .4 - - - - - - - De fo rmati o n (cid:95) P r o t o n E n e r g i e s [ M e V ] (cid:95) ( m i n ) = - . , (cid:95) ( m a x ) = . W oo d s - S a x o n U n i v e r s a l P a r a m s . - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / [ ] / - [ ] / [ ] / [ ] / [ ] / - [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / - [ ] / Z r
50 40
Fig.
3: Proton single-particle levels as a function of the axial-symmetry octupole deformation, α , calculated using the de-formed Woods-Saxon Hamiltonian with the ‘universal’ param-eter set. The Nilsson labels are supplemented with the symbol κ , [i.e. they have the form κ [ Nn z Λ]Ω] representing the squareof the amplitude of probability of the corresponding basis statein the WS wave function, whereas the sign of κ refers to theexpected value of the parity operator in each WS solution. Forthe sake of this illustration all other deformation parametersare fixed at 0. - .2 - .1 .0 .1 .2 .3 .4 - - - - - - - - - - - - De fo rmati o n (cid:95) P r o t o n E n e r g i e s [ M e V ] (cid:95) ( m i n ) = - . , (cid:95) ( m a x ) = . W oo d s - S a x o n U n i v e r s a l P a r a m s . - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / - [ ] / [ ] / [ ] / - [ ] / [ ] / [ ] / - [ ] / - [ ] / [ ] / [ ] / Z r
50 40
Fig.
4: Similar to the preceding one but for the tetrahedraldeformation α ; both diagrams are symmetric with respectto the zero argument value. The argument displayed withinthe interval [ − . , +0 .
4] allows to judge the degree of parity-mixing with increasing octupole deformations by comparingthe κ -values at the argument values 0.2 and 0.4 . [For moredetails cf. Ref. [25].] cal minimum . At the strict symmetry limit the excita-tion (and feeding) of the ground-state through collectivetransitions coming from either rotational or vibrationalexcited states is possible only via the octupole E tran- Our total energy as a function of tetrahedral deformation, isminimised over the octahedral deformation since both symmetrieshave, from the mathematics point of view, strong similarities. Thismay be one of the reasons that our energy minima are better pro-nounced and the energy estimates are lower as compared to someother approaches which ignore this mechanism. + 4+ 6+ 3 E n er gy [ A r b i t r a r y U n i t s ] E1 (j,j) −config.I E1E1E1E2E2 E2 E2E1 E3E3 E2E2Possibly Isomeric
Rotational−Like
E2 E3
0+ 2
Fig.
5: Schematic. The lowest energy-and-spin part of theexcitation scheme expected on the basis of our calculations.At the deformation of α ≈ .
15, the collective rotationaltransition energies are expected to be approximately linear inspin [17, 18] rather than proportional to I ( I + 1) - where fromthe term ‘rotational-like’. The E3-transitions can be excitedfrom the tetrahedral ground-state via Coulomb excitations. sitions since the tetrahedral deformed quantum objectshave vanishing collective quadrupole and dipole moments.However, such a state may receive transitions of the single-particle strengths from the non-collective particle-hole ex-citations. The lowest-lying non-collective excitations areexpected to come either from the tetrahedral ground-stateor from the excited spherical energy minimum giving riseto the E ( I = { j } I ) sequences as indicated schematicallyin Fig. 5. Since the highest- j neutron orbital above theFermi level is g / one may expect such a sequences toterminate at I π = 6 + states. The dedicated analysis ofthe experimental data in this nucleus including the re-duced transition probabilities, negative and positive par-ity rotational-like sequences and their possible interplay,partly following the lines discussed in [18], is in progressand will be published elsewhere.
4. Possible Impact of Tetrahedral Symmetry
Because of the unusual four-fold degeneracies of single-nucleon levels, tetrahedral symmetry of the mean-field isexpected to generate strong shell effects and thus relativelystrongly bound ground- or shape-isomeric states. This isexpected to justify the presence of the new waiting-pointnuclei which would help to explain missing elements ofthe nucleosynthesis models and modify the actually knownnuclear abundance scheme – in addition to paving theway to new spectroscopic features, new ideas about thestructure (and the very definition) of the rotational bands[cf. Ref. [18] in these proceedings] with the new selectionrules for the collective rotational transitions.
Acknowledgement
This work is supported by the LEA COPIGAL project 04-113and by the COPIN-IN2P3 agreement No.06-126.
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