The 2010 study of Li, Meng, Zhang, Zhou, and Savushkin established how single-neutron resonances of a Dirac equation with an axially quadrupole-deformed Woods-Saxon potential appear, disappear, and mix their spherical components as the potential is scanned, using eigenphase analysis. That study was confined to quadrupole deformation and to parametrized Woods-Saxon potentials: the extension to higher multipole deformations (octupole, hexadecapole), the full dependence of resonance phenomena on the deformation parameters and potential-model choices, the origin of resonances that admit no adiabatic continuation to any bound state, and the use of self-consistent relativistic mean-field potentials all lay beyond its scope. Later literature (2010-2026) developed powerful alternative methods and substantially settled the quadrupole deformation scans and self-consistent calculations, but every published deformed-Dirac resonance study remains restricted to quadrupole shape, and resonances lacking bound-state continuation have received no dedicated analysis. The surviving open problem is to compute and physically characterize the resonance spectrum of the deformed Dirac equation with higher multipole deformations and to establish the origin of eigenphase-defined resonances without bound-state continuation.

Background

In relativistic mean-field (RMF) descriptions of nuclei, nucleons move in a mean field generated by meson exchange or point-coupling interactions, and single particles are described by a Dirac equation with a scalar potential S(r)S(\mathbf{r}) and a vector potential V(r)V(\mathbf{r}) that typically have Woods-Saxon-like radial shapes (smooth flat-bottomed wells with diffuse surfaces). Near the neutron drip line, where the last neutrons are unbound, the physically relevant single-particle states are resonances: quasibound states embedded in the continuum, characterized by an energy EE and a decay width Γ\Gamma. Most nuclei are deformed, and the standard first shape beyond the sphere is axial quadrupole deformation (λ=2\lambda=2), which elongates or flattens the potential and is measured by a dimensionless parameter β\beta; the next multipole orders are octupole (λ=3\lambda=3, reflection-asymmetric pear shape, which mixes states of opposite parity) and hexadecapole (λ=4\lambda=4). For an axially deformed potential only the projection Ω\Omega of the total angular momentum jj on the symmetry axis remains a good quantum number, and the Dirac equation must be solved by expanding the wave function in spherical components ljl j, which yields a system of coupled radial equations (the coupled-channel method). In this open-channel problem resonances can be located from scattering theory: the scattering operator for fixed Ωπ\Omega^{\pi} (with parity π\pi) is diagonalized into eigenchannels whose real eigenphases pass through π/2\pi/2 (with maximal derivative) at resonance energies; alternatively one uses the analytic continuation in coupling-constant method (ACCC), which deepens the potential until the resonance becomes bound and analytically continues the bound-state pole back, the complex scaling method (CSM), the complex momentum representation (CMR, solving the equation for complex momenta so resonance poles appear directly), and Green's-function (GF) methods that extract positions and widths from the density of states. Deformed weakly bound and resonant levels are labeled as Nilsson orbitals, e.g. 1/2[321]1/2[321], where 1/21/2 is Ω\Omega and [321][321] encodes the asymptotic spherical components. Building on earlier deformed-resonance studies in non-relativistic potentials (Hamamoto 2005) and in proton-radioactivity contexts (Ferreira, Maglione, and Liotta 1997), Li, Meng, Zhang, Zhou, and Savushkin [Phys. Rev. C 81, 034311 (2010)] solved the coupled-channel Dirac equation with outgoing-wave boundary conditions for a quadrupole-deformed Woods-Saxon RMF-type potential. They found, for the mπ=1/2+m^{\pi}=1/2^{+} resonances at β=0.1\beta=0.1 (the source paper writes the projection quantum number as mm, denoting the same quantity as the Ω\Omega introduced above), that these resonances are supported mainly by the l≠0l\neq 0 spherical components, that mixing in the s1/2s_{1/2} component can make resonances disappear at finite energy, and that dominance of the d3/2d_{3/2} component at small energy guarantees that a certain resonance continues adiabatically to the corresponding bound state as the potential is deepened. A companion study by the same group [Sci. China Phys. Mech. Astron. 53, 773 (2010)] mapped the positive-parity neutron levels of the same quadrupole potential for 0≤β≤0.50\leq\beta\leq0.5. The 2010 study itself treated only quadrupole deformation of a parametrized Woods-Saxon potential, with its detailed eigenphase analysis concentrated at β=0.1\beta=0.1; multipole deformations beyond quadrupole, the broader dependence of the phenomena on β\beta and on potential-model choices, the behavior of eigenphase-defined resonances that admit no adiabatic continuation to a bound state, and the use of self-consistent RMF potentials all remained outside it. Subsequent method-development work (CSM 2012; ACCC 2015; CMR 2017; GF 2020) validated these techniques against one another for quadrupole-deformed Dirac potentials, and self-consistent deformed RMF resonance calculations in the complex momentum representation appeared in 2021-2024, but all remain confined to quadrupole shape.

Problem Statement

Extend the deformation systematics of single-particle resonances in axially deformed Dirac mean-field potentials beyond the quadrupole shape, which lay outside the scope of Li, Meng, Zhang, Zhou, and Savushkin (2010) and which the subsequent quadrupole-only literature has not touched. Specifically: (a) compute, for Dirac equations with Woods-Saxon-type scalar and vector potentials carrying higher multipole deformations — octupole (λ=3\lambda=3) and hexadecapole (λ=4\lambda=4), both alone and superposed on the quadrupole term — the energies, widths, eigenphase behavior, and spherical-component (channel) mixing of single-particle resonances, over deformation and potential-size ranges comparable to those of the source study (each multipole carrying its own deformation parameter — β2\beta_2 for quadrupole, β3\beta_3 for octupole, β4\beta_4 for hexadecapole, with the Background's β\beta being β2\beta_2 — each scanned over 00–0.50.5, the span of the source group's quadrupole study: alone, with a single higher multipole nonzero and the others set to zero; and superposed, as quadrupole–octupole and quadrupole–hexadecapole combinations in which one parameter is scanned over 00–0.50.5 while the other is held at 0.10.1 and the third is set to zero, e.g. β3\beta_3 over 00–0.50.5 at β2=0.1\beta_2=0.1 and vice versa, with potential sizes spanning the source's (R/r0)3(R/r_0)^3 scan), with the eigenphases for the reflection-asymmetric λ=3\lambda=3 case taken at fixed projection Ω\Omega on the symmetry axis (equivalently KK), since parity mixing makes Ωπ\Omega^{\pi} a poor label there and the relevant object is the fixed-Ω\Omega (KK-block) scattering matrix; (b) determine how the phenomena identified at quadrupole deformation — resonance appearance and disappearance at finite energy driven by s1/2s_{1/2}-component mixing, and continuation of component-dominated resonances to bound states — generalize, modify, or break down when higher multipoles are present (noting that octupole deformation mixes opposite parities, so Ωπ\Omega^{\pi} classification itself changes); and (c) establish the origin of eigenphase-defined resonances that admit no adiabatic continuation to any bound state in the scanned parameter range, a class that has received no dedicated analysis in the subsequent literature. An answer may additionally assess transferability to self-consistent RMF potentials, but the already-published quadrupole β\beta-scans and self-consistent examples do not by themselves resolve (a)-(c); conversely, an answer that treats only quadrupole deformation does not answer the problem.

References

  1. Z. P. Li, J. Meng, Y. Zhang, S. G. Zhou, L. N. Savushkin, 'Single-particle resonances in a deformed Dirac equation', Phys. Rev. C 81, 034311 (2010). DOI: 10.1103/PhysRevC.81.034311, https://doi.org/10.1103/PhysRevC.81.034311
  2. Z. P. Li, Y. Zhang, D. Vretenar, J. Meng, 'Single-particle resonances in a deformed relativistic potential', Science China Physics, Mechanics and Astronomy 53, 773-778 (2010). DOI: 10.1007/s11433-010-0161-7, https://doi.org/10.1007/s11433-010-0161-7
  3. X.-D. Xu, S.-S. Zhang, A. J. Signoracci, M. S. Smith, Z. P. Li, 'Analytical continuation from bound to resonant states in the Dirac equation with quadrupole-deformed potentials', Phys. Rev. C 92, 024324 (2015). DOI: 10.1103/PhysRevC.92.024324, https://doi.org/10.1103/PhysRevC.92.024324
  4. Q. Liu, J.-Y. Guo, Z.-M. Niu, S.-W. Chen, 'Resonant states of deformed nuclei in complex scaling method', Phys. Rev. C 86, 054312 (2012). arXiv:1211.6920, DOI: 10.1103/PhysRevC.86.054312, https://arxiv.org/abs/1211.6920
  5. Z. Fang, M. Shi, J.-Y. Guo, Z.-M. Niu, H. Liang, S.-S. Zhang, 'Probing resonances in the Dirac equation with quadrupole-deformed potentials with the complex momentum representation method', Phys. Rev. C 95, 024311 (2017). arXiv:1612.00527, DOI: 10.1103/PhysRevC.95.024311, https://arxiv.org/abs/1612.00527
  6. T.-T. Sun, L. Qian, C. Chen, P. Ring, Z. P. Li, 'Green's function method for the single-particle resonances in a deformed Dirac equation', Phys. Rev. C 101, 014321 (2020). arXiv:1710.02923, DOI: 10.1103/PhysRevC.101.014321, https://arxiv.org/abs/1710.02923
  7. X.-W. Wang, J.-Y. Guo, 'Research on deformed exotic nuclei by relativistic mean field theory in complex momentum representation', Phys. Rev. C 104, 044315 (2021). DOI: 10.1103/PhysRevC.104.044315, https://doi.org/10.1103/PhysRevC.104.044315
  8. Y.-X. Luo, Q. Liu, J.-Y. Guo, 'Research on exotic nuclei in deformed relativistic mean-field theory plus BCS in complex momentum representation', Phys. Rev. C 108, 024320 (2023). DOI: 10.1103/PhysRevC.108.024320, https://doi.org/10.1103/PhysRevC.108.024320
  9. S.-Y. Zhai, J.-Y. Guo, Q. Liu, 'Unified insights: Relativistic density functional theory for proton resonance, emission, and proton-rich exotic nuclei', Phys. Rev. C 110, 014323 (2024). DOI: 10.1103/PhysRevC.110.014323, https://doi.org/10.1103/PhysRevC.110.014323
  10. L. S. Ferreira, E. Maglione, P. Ring, 'Covariant density functional theory for decay of deformed proton emitters: A self-consistent approach', Physics Letters B 753, 237-241 (2016). DOI: 10.1016/j.physletb.2015.11.076, https://doi.org/10.1016/j.physletb.2015.11.076
  11. I. Hamamoto, 'One-particle resonant levels in a deformed potential', Phys. Rev. C 72, 024301 (2005). DOI: 10.1103/PhysRevC.72.024301, https://doi.org/10.1103/PhysRevC.72.024301
  12. I. Hamamoto, 'Shell structure of one-particle resonances in deformed potentials', Phys. Rev. C 93, 054328 (2016). arXiv:1605.05033, DOI: 10.1103/PhysRevC.93.054328, https://doi.org/10.1103/PhysRevC.93.054328
  13. L. S. Ferreira, E. Maglione, R. J. Liotta, 'Nucleon Resonances in Deformed Nuclei', Phys. Rev. Lett. 78, 1640 (1997). DOI: 10.1103/PhysRevLett.78.1640, https://doi.org/10.1103/PhysRevLett.78.1640