Breaking SU(3) spectral degeneracies in heavy deformed nuclei
Dennis Bonatsos, I.E. Assimakis, Andriana Martinou, S. Peroulis, S. Sarantopoulou, N. Minkov
EEPJ manuscript No. (will be inserted by the editor)
Breaking SU(3) spectral degeneracies in heavy deformed nuclei
Dennis Bonatsos , a , I.E. Assimakis , Andriana Martinou , S. Peroulis , S. Sarantopoulou , and N. Minkov Institute of Nuclear and Particle Physics, National Centre for Scientific Research “Demokritos”, GR-15310 Aghia Paraskevi,Attiki, Greece Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigrad Road, 1784 Sofia, Bulgariathe date of receipt and acceptance should be inserted later
Abstract.
Symmetries are manifested in nature through degeneracies in the spectra of physical systems.In the case of heavy deformed nuclei, when described in the framework of the Interacting Boson Model,within which correlated proton (neutron) pairs are approximated as bosons, the ground state band has nosymmetry partner, while the degeneracy between the first excited beta and gamma bands is broken throughthe use of three-body and/or four-body terms. In the framework of the proxy-SU(3) model, in which anapproximate SU(3) symmetry of fermions is present, the same three-body and/or four-body operatorsare used for breaking the degeneracy between the ground state band and the first excited gamma band.Experimentally accessible quantities being independent of any free parameters are pointed out in the lattercase.
PACS.
Proxy-SU(3) is an approximate symmetry appearingin heavy deformed nuclei [1,2]. The foundations of proxy-SU(3) [3], its parameter-free predictions for the collectivedeformation parameters β and γ [4,5], as well as for B ( E γ bandand the β band belong to the next irrep, therefore be-ing degenerate to each other if only one-body and two-body terms are included in the Hamiltonian. Actuallythis degeneracy has been used as a hallmark of the ap-pearance of SU(3) symmetry in atomic nuclei [9]. Higherorder (three- and four-body terms) have been introducedin the IBM Hamiltonian mostly in order to accommodatetriaxial shapes [10,11]. A particular class of higher orderterms consists of the symmetry-preserving three-body op-erator Ω and the four-body operator Λ (their mathemati- a e-mail: [email protected] cal names being the O l and Q l shift operator respectively)[12,13,14,15], the role of which in breaking the degener-acy between the β and the γ band [16,17], as well as inproducing the correct odd-even staggering within the γ band [18] has been considered.A different picture emerges within algebraic modelsemploying fermions, like the pseudo-SU(3) [19,20] and theproxy-SU(3) [1,2] models. In these cases the lowest lyingirrep accommodates both the gsb and the γ band, andpossibly higher- K bands with K = 4, 6, . . . , while the β and γ bands, and possibly higher bands with K = 4,6, . . . belong to the next irrep. In these cases, the three-and/or four-body terms are absolutely necessary from thevery beginning, in order to break the degeneracy betweenthe gsb and the γ bands. In the framework of pseudo-SU(3) this program has been succesfully carried out bothby using general three- and four-body terms [21], as well asby using a specific K -band splitting operator [22], contain-ing the Ω and Λ operators with appropriate coefficients.Numerical solutions have been produced in both cases, inthe second case because the Λ and Ω operators are diag-onal in different bases [16].The K -band splitting operator used in [22] has theinteresting property of being diagonal for values of theangular momentum L which are low in relation to the El-liott quantum numbers λ , µ characterizing the irreduciblerepresentations ( λ, µ ) of SU(3) [7,8]. In lowest order ap-proximation, in what follows we are going to use the K operator as a diagonal operator.In the present work we would like to consider the break-ing of the degeneracy of the gsb and γ band within the a r X i v : . [ nu c l - t h ] A ug : proxy-SU(3) scheme, using the same Λ , Ω , and K -bandsplitting operators mentioned above. Before attemptingany fittings, we would like to focus attention on physicalquantities which exhibit some characteristic behavior. Forexample, if we consider Hamiltonians of the form [16] H (3) = aL + bK + cΩ − dL , (1)or H (4) = aL + bK + cΛ − dL , (2)one can easily realize that the behavior of the differencesof the energies of the gsb and the γ bands for the same an-gular momentum L , E ( L γ ) − E ( L g ), normalized to theirfirst member, E (2 γ ) − E (2 g ), will depend only on the rel-ative parameter c/b , since only the second and the thirdterm in the above Hamiltonians would contribute to them.Essentially parameter-independent predictions would alsooccur for the odd-even staggering [23,24] within the γ -bands, which is essentially determined by the third termin the above Hamiltonians, while the first and fourth termhave a minimal influence. It is interesting that while forthe odd-even staggering detailed studies exist, pointingout the different behavior of this quantity in vibrational,rotational, γ -unstable or triaxial nuclei [23,24], no simi-lar study exists for the behavior of the energy differencesbetween the gsb and the γ band in the different regions,thus we will first attempt such a study.In Fig. 1 experimental values of E ( L + γ ) − E ( L + g ) areplotted as a function of the angular momentum L for sev-eral series of isotopes. For all isotopes normalization to E (2 + γ ) − E (2 + g ) has been used. The following observationscan be made.1) In most of the deformed nuclei reported in thesefigures, the “distance” between the gsb and the γ bandis decreasing, the actinides been a clear example.2) Several examples of increasing “distance” are seenin the Os-Pt region, in which the O(6) symmetry is knownto be present [9].3) Increasing “distance” is also seen in a few nuclei( Er,
Os,
Pt, which are expected to be triaxial,based on the staggering behavior exhibited by their γ bands [24].4) No effort has been made to exclude levels which areobviously due to band-crossing, like the last point shownin Pt.It should be noticed at this point, that the odd-evenstaggering in γ bands, defined as ∆E ( L ) = E ( L ) − E ( L −
1) + E ( L + 1)2 , (3)is also known to exhibit different behavior in various re-gions [23,24]. In particular, staggering of small magnitudeis seen in most of the deformed nuclei in the rare earthsand in the actinides region, while strong staggering is seenin the Xe-Ba-Ce region.The present systematics of the energy differences be-tween the gsb and the γ band can be combined with thesystematics of odd-even staggering in the γ -bands, whichshould be calculated and compared to the data. Since the sign in front of the three- or four-body term in the Hamil-tonian has to be fixed in order to guarantee that the γ band will lie above the gsb, the sign of the change of the“distance” between the γ band and the gsb , as well asthe form of the staggering within the γ bands (minimaat even L and maxima at odd L , or vice versa) are alsobe fixed by this choice, offering consistency checks of thesymmetry.Preliminary proxy-SU(3) predictions for four deformednuclei, obtained with the Hamiltonian of Eq. (2) with theparameters of Table 1, are shown in Fig. 2 for the “dis-tance” between the γ band and the gsb. In all cases de-crease is predicted. Notice that the slope of the theoreticalcurve is determined by the parameter ratio c/b , while theparameter b can be considered as a scale parameter forthe energy differences under consideration. Parameters a and d do not influence these energy differences.Results for the odd-even staggering within the γ bandfor the same nuclei are shown in Fig. 3, in which the smallenergy scale should be noticed. In the results labeled “2-terms”, only the second and the third terms of Eq. (2)are taken into account, in analogy to Fig. 2, while in theresults labeled “4-terms” all four terms of Eq. (2) are con-sidered. It is seen that the two extra terms have little effecton the staggering quantity and certainly do not affect itsoverall shape, exhibiting minima at even values of L andmaxima at odd values of L .The spectra obtained for two of these nuclei are shownin Table 2. Details of the calculations will be given in alonger publication.In a series of papers [26,27], Jolos and von Brentanohave shown, based on experimental data, that differentmass coefficients should be used in the Bohr Hamiltonianfor the ground state band and the γ band. In order toshow this, they use Grodzins products [28] of excitationenergies and B(E2) transition rates. The relation of theabove findings to the work of Jolos and von Brentanoshould be considered in a next step, in which B(E2) tran-sition rates will be included in the study.Financial support by the Bulgarian National ScienceFund (BNSF) under Contract No. KP-06-N28/6 is grate-fully acknowledged. Table 1.
Parameters (in units of keV) of the Hamiltonian ofEq. (2) for four nuclei. Data were taken from Ref. [25]. L g ( L γ )denotes the maximum angular momentum for the ground stateband ( γ -band) included in the fit.nucleus 10 − a b 10 − c 10 − d L g L γ Er 1443 408 440 1258 20 12
Dy 1025 445 578 412 28 23
Yb 540 483 2992 533 24 13
Hf 1225 588 748 890 20 15 3
Table 2.
Spectra of
Er and
Hf in keV, taken from Ref. [25], fitted by the Hamiltonian of Eq. (2). The parameter valuesused are given in Table 1. The rms deviations in keV are 34 and 58 respectively. Er Er Hf Hf Er Er Hf Hfexp th exp th exp th exp th
L L
References
1. D. Bonatsos, I.E. Assimakis, N. Minkov, A. Martinou, R.B.Cakirli, R.F. Casten, and K. Blaum, Phys. Rev. C ,064325 (2017)2. D. Bonatsos, I.E. Assimakis, N. Minkov, A. Martinou, S.Sarantopoulou, R.B. Cakirli, R.F. Casten, and K. Blaum,Phys. Rev. C , 064326 (2017)3. I.E. Assimakis, D. Bonatsos, N. Minkov, A. Martinou, R.B.Cakirli, R.F. Casten, and K. Blaum, Bulg. J. Phys. , 398(2017)4. D. Bonatsos, I.E. Assimakis, N. Minkov, A. Martinou, S.K.Peroulis, S. Sarantopoulou, R.B. Cakirli, R.F. Casten, andK. Blaum, Bulg. J. Phys. , 385 (2017)5. A. Martinou, D. Bonatsos, I.E. Assimakis, N. Minkov, S.Sarantopoulou, R.B. Cakirli, R.F. Casten, and K. Blaum,Bulg. J. Phys. , 407 (2017)6. S. Sarantopoulou, D. Bonatsos, I.E. Assimakis, N. Minkov,A. Martinou, R.B. Cakirli, R.F. Casten, and K. Blaum,Bulg. J. Phys. , 417 (2017)7. J. P. Elliott, Proc. Roy. Soc. Ser. A , 128 (1958)8. J. P. Elliott, Proc. Roy. Soc. Ser. A , 562 (1958)9. F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987)10. P. Van Isacker and J.-Q. Chen, Phys. Rev. C , 684 (1981)11. K. Heyde, P. Van Isacker, M. Waroquier, and J. Moreau,Phys. Rev. C , 1420 (1984)12. J.W.B. Hughes, J. Phys. A: Math., Nucl. Gen. , 48 (1973)13. J.W.B. Hughes, J. Phys. A: Math., Nucl. Gen. , 281(1973)14. B.R. Judd, W. Miller, Jr., J. Patera, and P. Winternitz, J.Math. Phys. , 1787 (1974)15. H. De Meyer, G. Vanden Berghe, and J. Van der Jeugt, J.Math. Phys. , 3109 (1985)16. G. Vanden Berghe, H.E. De Meyer, and P. Van Isacker,Phys. Rev. C , 1049 (1985)17. J. Vanthournout, Phys. Rev. C , 2380 (1990)18. D. Bonatsos, Phys. Lett. B , 1 (1988)19. R. D. Ratna Raju, J. P. Draayer, and K. T. Hecht, Nucl.Phys. A , 433 (1973) 20. J. P. Draayer, K. J. Weeks, and K. T. Hecht, Nucl. Phys.A , 1 (1982)21. J. P. Draayer and K. J. Weeks, Ann. Phys. (N.Y.) , 41(1984)22. H.A. Naqvi and J.P. Draayer, Nucl. Phys. A , 351(1990)23. N.V. Zamfir, and R.F. Casten, Phys. Lett. B , 265(1991)24. E.A. McCutchan, D. Bonatsos, N.V. Zamfir, and R.F. Cas-ten, Phys. Rev. C , 064307(2006)27. R. V. Jolos and P. von Brentano, Phys. Rev. C , 024309(2007)28. L. Grodzins, Phys. Lett. , 88 (1962) : E ga mm a - E g r ound L E ga mm a - E g r ound L E ga mm a - E g r ound L E ga mm a - E g r ound L E ga mm a - E g r ound L 176Hf 178Hf 180Hf 182W 184W 186W E ga mm a - E g r ound L E ga mm a - E g r ound L E ga mm a - E g r ound L Fig. 1.
Experimental values of E ( L γ ) − E ( L g ), taken from Ref. [25], plotted as function of the angular momentum L for severalseries of isotopes. For all isotopes, normalization to E (2 γ ) − E (2 g ) has been used. 5 E ( L )- E ( Lg ) ( k e V ) L (54,6) 162Er E ( L )- E ( Lg ) ( k e V ) L (54,8) 160Dy0 4 8 12 16 2002004006008001000 E ( L )- E ( Lg ) ( k e V ) L (54,6) 166Yb E ( L )- E ( Lg ) ( k e V ) L (42,20) 178Hf
Fig. 2.
Experimental values of E ( L γ ) − E ( L g ) [25] compared to proxy-SU(3) predictions from the Hamiltonian of Eq. (2) forfour nuclei. ( E ) ( k e V ) L 4-terms exp. 2-terms E ( k e V ) L 4-terms exp. 2-terms E ( k e V ) L 4-terms exp. 2-terms E ( k e V ) L 4-terms exp. 2-terms
Fig. 3.
Experimental values of odd-even staggering in the γ1