proxy-su(3): a symmetry for heavy nuclei · 2018. 10. 6. · proximation, called the proxy-su(3),...

Bulg. J. Phys. 44 (2017) 1–5

Proxy-SU(3): A symmetry for heavy nuclei

D. Bonatsos1, I.E. Assimakis1, N. Minkov2, A. Martinou1, S. K.Peroulis1, S. Sarantopoulou1, R.B. Cakirli3, R.F. Casten4,5, K.Blaum6

1Institute of Nuclear and Particle Physics, National Centre for Scientific Re-search “Demokritos”, GR-15310 Aghia Paraskevi, Attiki, Greece

2Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sci-ences, 72 Tzarigrad Road, 1784 Sofia, Bulgaria

3Department of Physics, University of Istanbul, 34134 Istanbul, Turkey4Wright Laboratory, Yale University, New Haven, Connecticut 06520, USA5Facility for Rare Isotope Beams, 640 South Shaw Lane, Michigan State Univer-sity, East Lansing, MI 48824 USA

6Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg,Germany

Received 31 October 2017

Abstract.

The SU(3) symmetry realized by J. P. Elliott in the sd nuclear shell is destroyedin heavier shells by the strong spin-orbit interaction. On the other hand, theSU(3) symmetry has been used for the description of heavy nuclei in terms ofbosons in the framework of the Interacting Boson Approximation, as well as interms of fermions using the pseudo-SU(3) approximation. A new fermionic ap-proximation, called the proxy-SU(3), has been recently introduced and appliedto the even rare earths. We show that the applicability of proxy-SU(3) can be ex-tended to even nuclei in the 28-50 proton shell, to even superheavy elements, aswell as to odd-odd and odd rare earths. Parameter free predictions for the β andγ deformation parameters are presented and compared to alternative theoreticalpredictions and to existing data.

PACS number: 21.60.Fw, 21.60.Ev, 21.60.Cs

1 Introduction

A new algebraic approach to heavy deformed nuclei, based on fermionic symme-tries and called the proxy-SU(3) scheme, has been introduced recently [1,2]. Itsbasic assumptions and microscopic justification have been discussed in Ref. [1]

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D. Bonatsos, I.E. Assimakis, N. Minkov, A. Martinou, S. K. Peroulis, et al.

and are further considered in the present Workshop in Ref. [3]. A first success ofthe proxy-SU(3) scheme is the explanation of the prolate over oblate dominancein deformed nuclei, which has been considered in Refs. [2, 4] and is further dis-cussed in the present Workshop in Ref. [5]. The border of the prolate to oblatetransition is also determined [2, 5]. In addition, parameter-free predictions forthe deformation parameters β and γ for even rare earths have been predicted inRef. [2] and successfully compared to Relativistic Mean Field predictions [6]and to existing data [7].

In the present work we obtain parameter-free predictions for the deformationparameters β and γ for nuclei in the 28-50 proton shell, as well as for evensuperheavy elements and we compare them to alternative theoretical predictionsand to existing data. Furthermore, we apply the proxy-SU(3) scheme in odd-oddrare earths and odd rare earths and compare the results to existing theoreticalpredictions.

2 Numerical results

2.1 Even nuclei in the 28-50 proton shell

This shell is of particular current interest, because of the presence of the Z = 40subshell closure and the appearance of shape coexistence [8, 9] around it.

50 60 70 800

15

30

45

60

Ge Se Kr Sr Zr Mo Ru Pd

(d

egr

ees

)

N

SU(3)

Figure 1. Proxy SU(3) predictions for Z = 32-46 for γ. See Section 2.1 for furtherdiscussion.

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Parameter-free proxy-SU(3) predictions for the γ deformation parameter areshown in Fig. 1 and are compared to D1S Gogny calculations [10] in Fig. 2,while in Fig. 3 the parameter free proxy-SU(3) predictions for the β deforma-tion parameter are shown and compared to theoretical predictions by RelativisticMean Field with the NL3 parametrization [6], D1S Gogny interaction [10], andthe FRDM (2012) mass tables [11], as well as to available data [7].

In general, good agreement is observed for the β deformation parameter betweenthe proxy-SU(3) predictions and the predictions of alternative theories and thedata. Deviations are stronger in the Sr, Zr, and Mo isotopes (Z = 38 − 42),which lie on or close to the Z = 40 subshell closure, which is not taken intoaccount in the proxy-SU(3) scheme in any specific way.

In the case of the γ deformation parameter, proxy-SU(3) shows a tendency tovalues near 30 degrees, indicating triaxial shapes, in the region around N = 74,while it climbs at even higher values, approaching 60 degrees (indicating oblateshapes) in the Ru and Pd isotopes in this region. Further study of these results isneeded, taking into account that nuclei near the end of the neutron shell are notvery well deformed, as indicated, for example, by their R4/2 = E(4+1 )/E(2+1 )ratios [12].

2.2 Odd-odd and even-odd rare earths

Proxy-SU(3) results have been obtained for odd-odd and even-odd rare earthswith Z in the sdg proxy-SU(3) shell (which is an approximation of the 50-82shell) and neutrons in the pfh proxy-SU(3) shell (which is an approximation ofthe 82-126 shell).

Parameter-free proxy-SU(3) predictions for the β and γ deformation parame-ters for odd-odd nuclei are shown in Figs. 4 and 5, compared to predictions forβ reported in the mass table FRDM(2012) [11], where they have been calcu-lated within the finite-range droplet macroscopic model and the folded-Yukawasingle-particle microscopic model. Good agreement is observed in general, withthe largest deviations appearing for Au (Z = 79), i.e., near the end of the 50-82proton shell.

A similar set of figures for odd-N rare earths appears in Figs. 6 and 7. Again,the higher deviations appear near the end of the 50-82 proton shell, at the Pt(Z = 78) isotopes.

2.3 Even superheavy elements

Parameter independent proxy-SU(3) results have been obtained for superheavyelements (SHE) with Z in the pfh proxy-SU(3) shell (which is an approxima-tion of the 82-126 shell) and neutrons in the sdgi proxy-SU(3) shell (which isan approximation of the 126-184 shell), as well as in the pfhj proxy-SU(3) shell

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(which is an approximation of the 184-258 shell). For the illustrative and ped-agogical purposes of this work, we take the relevant shells for the actinides andsuper heavy nuclei as Z = 82-126, N = 126-184, and N = 184-258, al-though the upper bounds are by no means certain and microscopic calculationsgive many varying scenarios. Results are shown for 100 ≤ Z ≤ 114. In or-der to have results from alternative calculations to compare our results with, weconfine ourselves to 128 ≤ N ≤ 220.

We compare our results to predictions contained in the following sources.

Extended results for 10 ≤ Z ≤ 110 and N ≤ 200 with the D1S Gogny interac-tion are given in [10] for the mean ground state β deformation, as well as for themean ground state γ deformation.

Extended results for the proton deformation βp and the neutron deformation βnwith covariant density functional theory for 96 ≤ Z ≤ 130 and N from theproton drip line up to N = 196 are given in Ref. [13] for the functionals PC-PK1 and DD-PC1.

Extended results for the deformation β within a microscopic-macroscopicmethod (MMM) for 98 ≤ Z ≤ 126 and 134 ≤ N ≤ 192 are given in Ref. [14].

Extended results for the deformation β up to A = 339 are reported in the masstable FRDM(2012) [11], calculated within the finite-range droplet macroscopicmodel and the folded-Yukawa single-particle microscopic model.

The results are summarized in Figs. 8 and 9. Overall good agreement is ob-served between the parameter-free proxy-SU(3) predictions and the alternativecalculations.

3 Conclusion

In the present work, the applicability of the proxy-SU(3) scheme is tested,through parameter independent predictions for the deformation parameters βand γ for even nuclei in the 28-50 proton shell and even superheavy elements,as well as in odd-odd nuclei and odd-N rare earths. In general, good agree-ment is obtained with results of alternative calculations, as well as with existingdata. Some deviations seen, as for example around the Z = 40 subshell closure,which is playing a central role in shape coexistence in the relevant region, callfor further investigations.

Acknowledgements

Work partly supported by the Bulgarian National Science Fund (BNSF) underContract No. DFNI-E02/6, by the US DOE under Grant No. DE-FG02- 91ER-40609, and by the MSU-FRIB laboratory, by the Max Planck Partner group,TUBA-GEBIP, and by the Istanbul University Scientific Research Project No.

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54135.

References

[1] D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou, R. B. Cakirli, R. F. Casten,and K. Blaum, Proxy SU(3) symmetry in heavy deformed nuclei, Phys. Rev. C 95,064325 (2017).

[2] D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou, S. Sarantopoulou, R. B.Cakirli, R. F. , and K. Blaum, Analytic predictions for nuclear shapes, prolate dom-inance and the prolate-oblate shape transition in the proxy-SU(3) model, Phys. Rev.C 95, 064326 (2017).

[3] I. E. Assimakis, D. Bonatsos, N. Minkov, A. Martinou, R.B. Cakirli, R.F. Casten,and K. Blaum, Foundations of the proxy-SU(3) symmetry in heavy nuclei, Pro-ceedings of the International Workshop on Shapes and Dynamics of Atomic Nuclei:Contemporary Aspects (SDANCA17), Bulg. J. Phys. (2017) in press.

[4] D. Bonatsos, Prolate over oblate dominance in deformed nuclei as a consequenceof the SU(3) symmetry and the Pauli principle, Eur. Phys. J. A 53, 148 (2017).

[5] S. Sarantopoulou, D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou, R.B.Cakirli, R.F. Casten, and K. Blaum, Proxy-SU(3) symmetry in heavy nuclei: Pro-late dominance and prolate-oblate shape transition, Proceedings of the InternationalWorkshop on Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects(SDANCA17), Bulg. J. Phys. (2017) in press.

[6] G. A. Lalazissis, S. Raman, and P. Ring, Ground-state properties of even-even nu-clei in the relativistic mean-field theory, At. Data Nucl. Data Tables 71, 1 (1999).

[7] S. Raman, C. W. Nestor, Jr., and P. Tikkanen, Transition probability from the groundto the first-excited 2+ state of even-even nuclides, At. Data Nucl. Data Tables 78, 1(2001).

[8] J. L. Wood, K. Heyde, W. Nazarewicz, M. Huyse and P. van Duppen, Coexistencein even-mass nuclei, Phys. Rep. 215, 101 (1992).

[9] K. Heyde and J. L. Wood, Shape coexistence in atomic nuclei, Rev. Mod. Phys. 83,1467 (2011).

[10] J. -P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Peru, N. Pillet, andG. F. Bertsch, Structure of even-even nuclei using a mapped collective Hamiltonianand the D1S Gogny interaction, Phys. Rev. C 81, 014303 (2010).

[11] P. Moller, A. J. Sierk, T. Ichikawa, and H. Sagawa, Nuclear ground-state massesand deformations: FRDM(2012), Ad. Data Nucl. Data Tables 109-110, 1 (2016).

[12] Brookhaven National Laboratory ENSDF database http://www.nndc.bnl.gov/ensdf/[13] S. E. Agbemava, A. V. Afanasjev, T. Nakatsukasa, and P. Ring, Covariant density

functional theory: Reexamining the structure of superheavy nuclei, Phys. Rev. C92, 054310 (2015).

[14] M. Kowal, P. Jachimowicz, and J. Skalski, Ground state and saddle point: massesand deformations for even-even superheavy nuclei with 98 ≤ Z ≤ 126 and 134 ≤N ≤ 192, arXiv:1203.5013 [nucl-th].

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http://www.nndc.bnl.gov/ensdf/

http://arxiv.org/abs/1203.5013


50 60 70 800

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proxy-SU(3) D1S Gogny

Ge

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proxy-SU(3)) D1S Gogny

Se

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Kr

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Zr

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Mo

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(d

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N


Ru

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30

45

60

(d

egr

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N


Pd

Figure 2. Proxy SU(3) predictions for Z = 32-46 for γ, compared with D1S-Gognycalculations (D1S-Gogny) [10]. See Section 2.1 for further discussion.

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0.1

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proxy-SU(3) RMF D1S Gogny FRDM (2012)

Ge

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proxy-SU(3) RMF D1S Gogny FRDM (2012)Se

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proxy-SU(3) RMF D1S Gogny FRDM (2012)Kr

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proxy-SU(3) RMF D1S Gogny FRDM (2012) exp.

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proxy-SU(3) RMF D1S Gogny FRDM (2012) exp.Zr

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Mo

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Ru

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0.0

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N


Pd

Figure 3. Proxy SU(3) predictions for Z = 32-46 for β, compared with results byrelativistic mean field theory (RMF) [6], the D1S-Gogny interaction (D1S-Gogny) [10],and the mass table FRDM(2012) [11], as well as to the data (exp.) [7]. See Section 2.1for further discussion.

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proxy-SU(3) FRDM(2012)

I

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Cs

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La

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Pr

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Pm

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Eu

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Tb

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β

N


Ho

Figure 4. Proxy SU(3) predictions for β for odd-odd rare earths with Z = 53-67, com-pared with results reported in the mass table FRDM(2012) [11]. See Section 2.2 forfurther discussion.

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Lu

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Ta

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Au

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I Cs La Pr Pm Eu Tb

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Ho Tm Lu Ta Re Ir Au

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N

Figure 5. Proxy SU(3) predictions for β for odd-odd rare earths with Z = 69-79, com-pared with results reported in the mass table FRDM(2012) [11]. In the two bottom panels,the proxy-SU(3) predictions for γ are reported for Z = 53-79. See Section 2.2 for furtherdiscussion.

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proxy-SU(3) FRDM (2012)

Dy

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Er

Figure 6. Proxy SU(3) predictions for β for even-odd rare earths with Z = 54-68,compared with results reported in the mass table FRDM(2012) [11]. See Section 2.2 forfurther discussion.

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Dy Er Yb Hf W Os Pt

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Figure 7. Proxy SU(3) predictions for β for even-odd rare earths with Z = 70-78,compared with results reported in the mass table FRDM(2012) [11]. In the two bottompanels, the proxy-SU(3) predictions for γ are reported for Z = 54-78. See Section 2.2for further discussion.

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1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0

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β

N

P r o x y S U ( 3 ) M M M D D - P C 1 _ p D D - P C 1 _ n D 1 S - G o g n y F R D M 2 0 1 2

Z = 1 0 0

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Proxy SU(3) MMM DD-PC1_p DD-PC1_n D1S-Gogny FRDM 2012

Z=102

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Proxy SU(3) DD-PC1_p DD-PC1_n MMM D1S-Gogny FRDM 2012

Z=104

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P r o x y S U ( 3 ) D D - P C 1 _ p D D - P C 1 _ n M M M D 1 S - G o g n y F R D M 2 0 1 2

Z = 1 0 6

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P r o x y S U ( 3 ) D D - P C 1 _ p D D - P C 1 _ n M M M F R D M 2 0 1 2

Z = 1 1 2

1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0

0 , 0

0 , 1

0 , 2

0 , 3

0 , 4

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0 , 6

β

N

P r o x y S U ( 3 ) D D - P C 1 _ p D D - P C 1 _ n M M M F R D M 2 0 1 2

Z = 1 1 4

Figure 8. Proxy SU(3) predictions forZ=100-114 for β, compared with covariant densityfunctional theory with the DD-PC1 functional (DD-PC1) [13] (in which case differentvalues for protons (DD-PC1 p) and neutrons (DD-PC1 n) are reported), the microscopic-macroscopic method (MMM) [14], the D1S-Gogny interaction (D1S-Gogny) [10], andthe mass table FRDM(2012) [11]. See section 2.3 for further discussion.12


1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

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1 0

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γ

N

P r o x y S U ( 3 ) D 1 S G o g n yZ = 1 0 0

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5

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1 5

2 0

2 5

3 0

3 5

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γ

N


1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

γ

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1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

γ

N


1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

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N


1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

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N


1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

1 0

2 0

3 0

γ

N

P r o x y S U ( 3 )Z = 1 1 2

1 4 0 1 6 0 1 8 0 2 0 0 2 2 00

1 0

2 0

3 0

4 0

γ

N

P r o x y S U ( 3 )Z = 1 1 4

Figure 9. Proxy SU(3) predictions for Z = 100-114 for γ, compared with results by theD1S-Gogny interaction (D1S-Gogny) [10]. See section 2.3 for further discussion.

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proxy-su(3): a symmetry for heavy nuclei · 2018. 10. 6. · proximation, called the proxy-su(3),...

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