Superdeformed nuclear bands — rotational sequences at an approximately 2:1 axis ratio — occasionally display ΔI=4 staggering ("bifurcation"): a systematic alternation of tens of electron-volts between the γ-ray energies of states differing by four units of spin, a relative effect of order 10⁻⁵–10⁻⁶ first clearly seen in ¹⁴⁹Gd in 1993. Hamamoto and Mottelson (1994) showed that a hexadecapole (Y₄₄) deformation with C4v point symmetry yields the rotational Hamiltonian H = A I₃² + B₁(I₁² − I₂²)² + B₂(I₁² + I₂²)², whose four equivalent equatorial minima are connected by tunnelling that splits the band into quartets and produces the staggering; they matched observed patterns qualitatively with empirical parameters, requiring a fitted quantum correction A/B₁ → A/B₁ − (3/2)ħ² in the semiclassical quantization. They left open the microscopic determination of B₁ and B₂ for specific single-particle configurations, the explanation of the observed spin dependence and phase inversions of the staggering, and a systematic derivation of the tunnelling quantum corrections. These issues remain unresolved: microscopic tests found no static C4-favouring shell effects; the leading alternative (nonadiabatic mean-field) theory explains the configuration systematics only qualitatively and with known contradictions; systematic spin-Hamiltonian quantization became available in 2004 but was never applied to this four-minimum tunnelling problem; and the field has been largely dormant since about 2005.

Background

A superdeformed (SD) nuclear band is a rotational sequence built on a second minimum of the nuclear potential at an approximately 2:1 axis ratio, observed in the mass regions A ≈ 80, 150, and 190. Transitions within such bands are predominantly of E2 type (electric quadrupole radiation, carrying two units of angular momentum) with ΔI = 2, and the γ-ray transition energies Eγ(I → I−2) trace the smooth evolution of the moment of inertia with rotational frequency. In 1993 an analysis of the yrast SD band of ¹⁴⁹Gd (the yrast band being the lowest-energy band at each angular momentum) revealed an unexpected bifurcation: states I, I+4, I+8, … are displaced in energy relative to I+2, I+6, … by about 60 eV — a relative effect of order 10⁻⁵–10⁻⁶ — indicating "the remnant of a new quantum number associated with the fourfold rotational symmetry" (Flibotte et al. 1993). A small staggering of the SD transition energies, suggesting terms with fourfold symmetry in the SD Hamiltonian, was concurrently found in ¹⁹⁴Hg (Cederwall et al. 1994). The effect is usually quantified by the finite-difference combination ΔEγ(I) = Eγ(I) − [Eγ(I−2) + Eγ(I+2)]/2 of consecutive transition energies, whose alternating sign signals the ΔI=4 staggering (also called ΔI=2 staggering in this sense).

Hamamoto and Mottelson (1994; extended version in Phys. Scr. T56, 1995) proposed a symmetry-based explanation. A nuclear shape containing the hexadecapole spherical harmonic Y₄₄ (a fourth-order deformation periodic by π/2 in the azimuthal angle) has only the point-group symmetry C4v: it is invariant under rotations by π/2 about the symmetry axis and under four vertical reflection planes, and it has four equivalent minima around the equator of the shape. For such a rotor the rotation axis lies in the equatorial plane (rotation about the symmetry axis is excluded by the point-group symmetries), which for the leading fourth-order rotational Hamiltonian

H = A I₃² + B₁ (I₁² − I₂²)² + B₂ (I₁² + I₂²)²

(the Iₖ being body-fixed angular-momentum components) requires A > 0 and 2 B₂ I² < A. The orientation of the equatorial rotation axis has four equivalent minima connected by tunnelling; the resulting level splittings are of order 10–100 eV and split the rotational sequence into quartets classified by the irreducible representations of C4v, producing a ΔI=4 staggering without violating the ΔI=2 character of the emitted radiation. Exact diagonalization showed that the parameter set (A, B₁, B₂) = (100, 1, 0) produces regular staggering, (100, 1, −1) produces none, and (100, 1, −1/2) produces an irregular pattern with phase inversions. In the WKB (Wentzel–Kramers–Brillouin, semiclassical) tunnelling treatment with B₂ = 0 the real tunnelling action is Re S₁₂ = (π/2)(I − √(A/4B₁)), and for B₂ ≠ 0 a necessary condition for staggering is

1 > 2 B₂ I²/A > 1 − √(B₁/(B₁+B₂)), the B₂ = 0 case being the saturated limit of the lower bound.

To match the exact diagonalization, the semiclassical quantization had to be corrected empirically by the replacement A/B₁ → A/B₁ − (3/2)ħ² — a quantum correction obtained by fitting rather than derived from a systematic semiclassical expansion, and of the same order as the tunnelling splitting itself. A preliminary microscopic survey of high-j intruder orbitals in the cranking model (the standard treatment of rapidly rotating nuclei, in which single-particle orbitals are computed in a mean field viewed from the uniformly rotating frame; high-j intruder orbitals originate in the next major oscillator shell and are polarized toward the nuclear surface) suggested that their nonaxial hexadecapole moments have configuration-dependent signs and magnitudes that could determine B₁ and B₂ microscopically, hinting at an explanation of why some single-particle configurations stagger and others do not; this survey was, however, qualitative.

The open issues identified at the source were: (i) a microscopic determination of the effective fourth-order rotational terms B₁ and B₂ for specific SD single-particle configurations; (ii) a quantitative explanation of the observed spin dependence of the staggering amplitude and of its phase inversions; and (iii) a systematic derivation of the quantum corrections to the tunnelling quantization, replacing the fitted (3/2)ħ² shift. Review literature up to 2002 still lists the staggering among the unexplained fine-structure phenomena of superdeformation and notes that no microscopic calculation had confirmed the existence of the required static C4-symmetric deformations.

Problem Statement

The open problem is the quantitative microscopic theory of ΔI=4 staggering in superdeformed nuclear bands, as posed by Hamamoto and Mottelson (1994) and left unresolved by the subsequent literature. It has three linked components.

(i) Microscopic determination of the effective fourth-order rotational terms: derive the coefficients B₁ and B₂ of the C4v rotor Hamiltonian H = A I₃² + B₁(I₁²−I₂²)² + B₂(I₁²+I₂²)² — or of the equivalent nonaxial hexadecapole-coupling coefficient c of the nonadiabatic effective Hamiltonian H = A I² + B I⁴ + d(I₊²+I₋²) + c(I₊⁴+I₋⁴) (I₊ and I₋ being angular-momentum ladder operators), which Pavlichenkov (1997) obtained when the rotating self-consistent mean field is allowed to distort nonaxially — from a specified microscopic model (self-consistent mean field with nonaxial hexadecapole degrees of freedom, or an equivalent shell-model treatment) for identified single-particle configurations of real SD bands. The determination must include signs, magnitudes, and spin dependence, and must reproduce the observed systematics: which bands stagger (e.g. ¹⁴⁹Gd(1), ¹⁴⁸Gd(6), ¹⁴⁸Eu(1) — three identical SD bands, that is, bands in neighbouring nuclei whose γ-ray transition energies agree to within about 1 keV) and which do not (the majority of A ≈ 150 bands), including the known configuration puzzles (e.g. the ¹⁹⁴Hg bands 1-3, ¹³²Ce vs ¹³³Ce). The named bands and puzzles are illustrative examples; the target is the observed systematics as a whole, not only the named cases.

(ii) Quantitative reproduction of the observed staggering — amplitude (tens of eV), spin dependence, and the phase inversions seen in irregular cases — from the coefficients of (i).

(iii) Systematic derivation of the quantum corrections to the tunnelling quantization, replacing the empirically fitted A/B₁ → A/B₁ − (3/2)ħ² shift with corrections obtained from a systematic semiclassical expansion; the general framework (Garg and Stone 2004) exists but has not been applied to this four-minimum problem.

The coefficients of (i) are to be obtained from the microscopic single-particle treatment of the identified configurations, not by fitting the observed staggering with phenomenological parameters (as the existing band-mixing, supersymmetry, or interacting-boson parametrizations do). A demonstration, within a specified class of microscopic models, that fourth-order C4v-type terms of the required magnitude cannot arise, together with a quantitative alternative account of the observed staggering, addresses the same open problem.

References

  1. S. Flibotte, H. R. Andrews, G. C. Ball, C. W. Beausang, F. A. Beck, G. Belier, T. Byrski, D. Curien, P. J. Dagnall, G. de France, D. Disdier, G. Duchêne, Ch. Finck, B. Haas, G. Hackman, D. S. Haslip, V. P. Janzen, B. Kharraja, J. C. Lisle, J. C. Merdinger, S. M. Mullins, W. Nazarewicz, D. C. Radford, V. Rauch, H. Savajols, J. Styczen, Ch. Theisen, P. J. Twin, J. P. Vivien, J. C. Waddington, D. Ward, K. Zuber and S. Åberg, "ΔI=4 bifurcation in a superdeformed band: Evidence for a C₄ symmetry", Phys. Rev. Lett. 71, 4299-4302 (1993). DOI: 10.1103/PhysRevLett.71.4299. URL: https://doi.org/10.1103/PhysRevLett.71.4299
  2. B. Cederwall, R. V. F. Janssens, M. J. Brinkman, I. Y. Lee, I. Ahmad, J. A. Becker, M. P. Carpenter, B. Crowell, M. A. Deleplanque, R. M. Diamond, J. E. Draper, C. Duyar, P. Fallon, L. P. Farris, E. A. Henry, R. G. Henry, J. R. Hughes, T. L. Khoo, T. Lauritsen, A. O. Macchiavelli, E. Rubel, F. S. Stephens, M. A. Stoyer, W. Satula, I. Wiedenhoever and R. Wyss, "New features of superdeformed bands in ¹⁹⁴Hg", Phys. Rev. Lett. 72, 3150-3153 (1994). DOI: 10.1103/PhysRevLett.72.3150. URL: https://doi.org/10.1103/PhysRevLett.72.3150
  3. I. Hamamoto and B. Mottelson, "Superdeformed rotational bands in the presence of Y₄₄ deformation", Phys. Lett. B 333, 294-298 (1994). DOI: 10.1016/0370-2693(94)90144-9. URL: https://doi.org/10.1016/0370-2693(94)90144-9
  4. I. Hamamoto and B. Mottelson, "ΔI = 4 structure in superdeformed rotational band — deformation with C4v symmetry", Phys. Scr. T56, 27-34 (1995). DOI: 10.1088/0031-8949/1995/T56/004. URL: https://doi.org/10.1088/0031-8949/1995/T56/004
  5. P. Magierski, K. Burzyński, J. Dobaczewski and W. Nazarewicz, "The ΔI=4 bifurcation in superdeformed bands", preprint arXiv:nucl-th/9410042 (1994). DOI: 10.48550/arXiv.nucl-th/9410042. URL: https://arxiv.org/abs/nucl-th/9410042
  6. I. M. Pavlichenkov and S. Flibotte, "C₄ symmetry and bifurcation in superdeformed bands", Phys. Rev. C 51, R460-R464 (1995). DOI: 10.1103/PhysRevC.51.R460. URL: https://doi.org/10.1103/PhysRevC.51.R460
  7. A. O. Macchiavelli, B. Cederwall, R. M. Clark, M. A. Deleplanque, R. M. Diamond, P. Fallon, I. Y. Lee, F. S. Stephens and S. Asztalos, "C₄ symmetry effects in nuclear rotational motion", Phys. Rev. C 51, R1-R4 (1995). DOI: 10.1103/PhysRevC.51.R1. URL: https://doi.org/10.1103/PhysRevC.51.R1
  8. K. Burzyński, P. Magierski, J. Dobaczewski and W. Nazarewicz, "Hexadecapole interaction and the ΔI = 4 staggering effect in rotational bands", Phys. Scr. T56, 228-230 (1995). DOI: 10.1088/0031-8949/1995/T56/033. URL: https://doi.org/10.1088/0031-8949/1995/T56/033
  9. W. D. Luo, A. Bouguettoucha, J. Dobaczewski, J. Dudek and X. Li, "Microscopic study of a C₄-symmetry hypothesis in A∼150 superdeformed nuclei: Deformed Woods-Saxon mean field", Phys. Rev. C 52, 2989-3001 (1995). DOI: 10.1103/PhysRevC.52.2989. URL: https://doi.org/10.1103/PhysRevC.52.2989
  10. W. Reviol, H.-Q. Jin and L. L. Riedinger, "Transition energy staggering and band interaction in rare-earth nuclei", Phys. Lett. B 371, 19-24 (1996). DOI: 10.1016/0370-2693(95)01597-3. URL: https://doi.org/10.1016/0370-2693(95)01597-3
  11. F. Dönau, S. Frauendorf and J. Meng, "Can hexadecapole deformation lead to ΔI = 2 staggering in superdeformed bands?", Phys. Lett. B 387, 667-672 (1996). DOI: 10.1016/0370-2693(96)01119-7. URL: https://doi.org/10.1016/0370-2693(96)01119-7
  12. D. S. Haslip, S. Flibotte, G. de France, M. Devlin, A. Galindo-Uribarri, G. Gervais, G. Hackman, D. R. LaFosse, I. Y. Lee, F. Lerma, A. O. Macchiavelli, R. W. MacLeod, S. M. Mullins, J. M. Nieminen, D. G. Sarantites, C. E. Svensson, J. C. Waddington and J. N. Wilson, "ΔI = 4 Bifurcation in Identical Superdeformed Bands", Phys. Rev. Lett. 78, 3447-3450 (1997). DOI: 10.1103/PhysRevLett.78.3447. URL: https://doi.org/10.1103/PhysRevLett.78.3447
  13. P. Magierski, K. Burzyński, E. Perlińska, J. Dobaczewski and W. Nazarewicz, "Quadrupole and hexadecapole correlations in rotating nuclei studied within the single-j shell model", Phys. Rev. C 55, 1236-1245 (1997). DOI: 10.1103/PhysRevC.55.1236. URL: https://doi.org/10.1103/PhysRevC.55.1236
  14. I. M. Pavlichenkov, "Nonadiabatic mean field effects in the ΔI=2 staggering of superdeformed bands", Phys. Rev. C 55, 1275-1281 (1997). DOI: 10.1103/PhysRevC.55.1275. URL: https://doi.org/10.1103/PhysRevC.55.1275
  15. I. M. Pavlichenkov, "High-j Intruder Orbitals and the ΔI = 4 Bifurcation in Superdeformed Bands", preprint arXiv:nucl-th/9710057 (1997). DOI: 10.48550/arXiv.nucl-th/9710057. URL: https://arxiv.org/abs/nucl-th/9710057
  16. D. S. Haslip, S. Flibotte, C. E. Svensson and J. C. Waddington, "Analysis of staggering patterns in identical superdeformed bands: Constraints on a C₄ Hamiltonian", Phys. Rev. C 58, R1893-R1896 (1998). DOI: 10.1103/PhysRevC.58.R1893. URL: https://doi.org/10.1103/PhysRevC.58.R1893
  17. D. S. Haslip, S. Flibotte, R. A. E. Austin, G. de France, M. Devlin, Ch. Finck, A. Galindo-Uribarri, G. Gervais, N. Kintz, D. R. LaFosse, T. J. Lampman, F. Lerma, S. M. Mullins, J. M. Nieminen, T. Rodinger, D. G. Sarantites, O. Stezowski, C. E. Svensson, J. P. Vivien, J. C. Waddington and J. N. Wilson, "Systematic survey of ΔI=4 bifurcation in A∼150 superdeformed nuclei", Phys. Rev. C 58, R2649-R2653 (1998). DOI: 10.1103/PhysRevC.58.R2649. URL: https://doi.org/10.1103/PhysRevC.58.R2649
  18. D. Samsoen, P. Quentin and I. N. Mikhaïlov, "Period of the γ-ray staggering in the ¹⁵⁰Gd superdeformed region", Phys. Rev. C 60, 014301 (1999). DOI: 10.1103/PhysRevC.60.014301. URL: https://doi.org/10.1103/PhysRevC.60.014301
  19. Yu-xin Liu and Dong-feng Gao, "Description of identical superdeformed bands with ΔI = 4 bifurcation", Phys. Rev. C 63, 044317 (2001). DOI: 10.1103/PhysRevC.63.044317. URL: https://doi.org/10.1103/PhysRevC.63.044317
  20. J. Dudek, A. Góźdź and D. Rosły, "Quantum Rotors and Their Symmetries", preprint arXiv:nucl-th/0105039 (2001). DOI: 10.48550/arXiv.nucl-th/0105039. URL: https://arxiv.org/abs/nucl-th/0105039
  21. R. Wadsworth and P. J. Nolan, "The influence of microscopic structures on rotational motion in nuclei", Rep. Prog. Phys. 65, 1079-1118 (2002). DOI: 10.1088/0034-4885/65/7/202. URL: https://doi.org/10.1088/0034-4885/65/7/202
  22. I. M. Pavlichenkov and A. A. Shchurenkov, "Configuration-dependent effect of ΔI = 4 bifurcation in A ∼ 150 superdeformed nuclei: active and inactive orbitals", J. Phys. G: Nucl. Part. Phys. 28, 1223-1238 (2002). DOI: 10.1088/0954-3899/28/6/306. URL: https://doi.org/10.1088/0954-3899/28/6/306
  23. A. Garg and M. Stone, "Bohr-Sommerfeld Quantization of Spin Hamiltonians", Phys. Rev. Lett. 92, 010401 (2004). DOI: 10.1103/PhysRevLett.92.010401. URL: https://doi.org/10.1103/PhysRevLett.92.010401
  24. N. Minkov, P. Yotov, R. V. Jolos and W. Scheid, "Intrinsic origin of the high order angular momentum terms in a nuclear rotation Hamiltonian", J. Phys. G: Nucl. Part. Phys. 34, 299-313 (2007). DOI: 10.1088/0954-3899/34/2/010. URL: https://doi.org/10.1088/0954-3899/34/2/010