Bohr and Mottelson (1957; 1958) asked what electric dipole moment is produced when quadrupole and octupole deformations coexist in a nucleus, noting that their original polarization-based estimate required correction. The later literature answered the qualitative part of their question in an unexpected way: no single contribution dominates. The macroscopic charge-redistribution term, the neutron-skin term (of opposite sign), and the microscopic shell term nearly cancel, so the intrinsic dipole moment D0D_0 is a small difference of large terms. Magnitudes are now measured for many octupole-deformed nuclei (notably around A≈224A≈224), and the relative sign of the dipole and octupole moments has been extracted in 222^{222}Ra and 228^{228}Ra by interference-sensitive multi-step Coulomb excitation; but microscopic predictions still scatter by factors of several in magnitude and — placing the zero crossing of D0D_0 at different neutron numbers along a chain — disagree on its sign for a given nucleus, while the sign is experimentally pinned in only two nuclei. The surviving open problem is a quantitatively reliable prediction of D0D_0 — magnitude and sign, with uncertainties — validated against these sign-sensitive measurements.

Background

Most heavy nuclei are deformed: their equilibrium shape deviates from a sphere. In the usual description the nuclear surface is written in the intrinsic (body-fixed) frame as R(θ)=R0[1+β2Y20(θ)+β3Y30(θ)+…]R(θ)=R_0[1+β_2 Y_{20}(θ)+β_3 Y_{30}(θ)+…], where Yλ0Y_{λ0} are spherical harmonics and the deformation parameters βλβ_λ measure the amplitude of each multipole. The quadrupole term β2β_2 (a prolate or oblate ellipsoid) is the dominant deformation of the rare-earth and actinide nuclei. The octupole term β3β_3 (a pear-shaped, reflection-asymmetric correction that breaks the symmetry r→−rr \to -r) is developed in a few regions of the nuclear chart, e.g. around radium, thorium and radon with A≈224A≈224 and in some superdeformed nuclei; it may also appear dynamically as an octupole vibration (an oscillation of the surface between pear shapes of opposite orientation) about a reflection-symmetric shape. In nuclei with static octupole deformation, states of opposite parity occur close together at low excitation energy — as parity doublets (near-degenerate pairs of opposite-parity states of the same spin) in odd-mass nuclei and as interleaved negative-parity rotational bands in even-even ones.

A reflection-asymmetric shape would endow the nucleus with a permanent electric dipole moment (EDM) in its intrinsic frame, of magnitude D0=e∫ρp(r)zd3rD_0 = e ∫ ρ_p(r) z d³r, where ρpρ_p is the proton (charge) density, zz is the coordinate along the symmetry axis measured from the centre of mass, and ee is the elementary charge. The interest of Bohr and Mottelson's question lies in a well-known null result: if the proton and neutron fluids filled the same shape with uniform densities, the centre-of-mass condition would give exactly D0=0D_0=0 — a pear-shaped charge distribution alone is not enough, because the displacement of the centre of charge is cancelled by that of the centre of mass. A nonzero intrinsic moment requires the proton and neutron density profiles to differ: through the neutron excess N−ZN−Z (NN and ZZ being the neutron and proton numbers), which makes the proton fluid a smaller copy of the total; through a neutron skin, an excess of neutron over proton density at the surface; through surface diffuseness, the finite thickness of the density falloff at the nuclear surface; or through microscopic shell effects, the contribution of the ordering of single-particle orbitals near the Fermi surface. Bohr and Mottelson (1957) had estimated the moment produced by electric polarization — the redistribution of protons driven by the deformation's own electric field, the nuclear analogue of the lightning-rod effect; in a 1958 addendum they corrected this estimate, the correct derivation in the liquid-drop model (the nucleus treated as a droplet of incompressible charged nuclear matter) having been given earlier by Strutinsky (1956).

Because D0D_0 is an intrinsic-frame quantity, it is not itself an observable: a stationary nucleus conserves parity, so its laboratory-frame permanent EDM vanishes identically. What is measured are reduced transition probabilities B(Eλ)B(Eλ) between nuclear states, where EλEλ denotes an electric multipole transition of multipolarity λλ (E1 and E3 being the electric dipole and electric octupole transitions), in units of e2e²fm2λ^{2λ} — often quoted in Weisskopf units, the standard single-particle estimate of transition strength; from these, D0D_0 is extracted using rotational-model angular-momentum factors. For a rotational band built on an intrinsic octupole moment, the E1 transition rates are proportional to D02D_0². The magnitude alone is therefore fixed by rates (from lifetimes or Coulomb-excitation yields), but the sign of D0D_0 — equivalently its sign relative to the octupole moment Q3Q_3 or to β3β_3 — is not: it requires interference-sensitive methods, notably multi-step Coulomb excitation (excitation by the electromagnetic field of a passing heavy ion, analysed with coupled-channels codes such as GOSIA), in which excitation amplitudes involving E1, E2 and E3 matrix elements interfere so that the measured yields depend on their relative signs.

The problem's difficulty, sharpened by the later literature, is that the competing contributions to D0D_0 nearly cancel: the macroscopic charge-redistribution term, the neutron-skin term (of opposite sign), and the microscopic shell term are each individually larger than the observed net values. The macroscopic part is usually computed in the droplet model, a generalization of the liquid drop that includes density non-uniformities and a neutron skin, with parameters (the symmetry energy, i.e. the energy cost of neutron–proton imbalance, its density slope, and the skin stiffness) tied to poorly constrained isovector (neutron-versus-proton) properties of nuclear matter; the microscopic part requires self-consistent mean-field treatments with energy-density functionals (the nuclear energy written as a functional of the proton and neutron densities; Gogny, Skyrme and covariant families) or Woods–Saxon potentials (phenomenological single-particle potential wells with a diffuse surface) with Strutinsky shell corrections (a smoothing procedure that isolates the shell contribution from the smooth liquid-drop trend).

Problem Statement

As posed by Bohr and Mottelson (1957, with corrections in 1958): what electric dipole moment is associated with octupole deformation of a spheroidal (quadrupole-deformed) nucleus? Compute the net intrinsic dipole moment D0D_0 from the proton and neutron density distributions, going beyond the inadequate polarization-based liquid-drop treatment — accounting for the neutron excess (N−ZN−Z), the neutron skin, surface diffuseness, and microscopic shell structure — and determine its magnitude, including its sign relative to the octupole moment.

In its present-day form the problem is: provide a quantitatively reliable microscopic prediction of the intrinsic dipole moment D0D_0 — magnitude and sign, with uncertainties — for reflection-asymmetric (statically or dynamically octupole-deformed) nuclei, controlling the near-cancellation between the macroscopic charge-redistribution term, the neutron-skin term, and the microscopic shell contribution; and validate it against the measured B(E1)B(E1)-derived magnitudes and the interference-sensitive sign determinations — the measured relative signs of the dipole and octupole moments in 222^{222}Ra and 228^{228}Ra and the zero crossing near 224^{224}Ra, with newly identified octupole candidates such as 100^{100}Ru as untested predictions. The original sub-question of which contribution dominates is resolved (they nearly cancel); the surviving target is the quantitative magnitude-and-sign prediction and its experimental verification.

References

  1. A. Bohr and B. R. Mottelson, "Electric dipole moment associated with octupole vibrations of a spheroidal nucleus," Nuclear Physics 4, 529–531 (1957). DOI: 10.1016/0029-5582(87)90059-9; https://doi.org/10.1016/0029-5582(87)90059-9
  2. A. Bohr and B. R. Mottelson, "Electric dipole moment associated with octupole vibrations of a spheroidal nucleus," Nuclear Physics 9, 687–688 (1958) (addendum correcting the 1957 note). DOI: 10.1016/0029-5582(58)90354-7; https://doi.org/10.1016/0029-5582(58)90354-7
  3. C. O. Dorso, W. D. Myers, and W. J. Swiatecki, "Droplet-model electric dipole moments," Nuclear Physics A 451, 189–201 (1986). DOI: 10.1016/0375-9474(86)90410-0; https://doi.org/10.1016/0375-9474(86)90410-0
  4. R. J. Poynter, P. A. Butler, N. Clarkson, D. Cline, K. A. Connell, R. A. Cunningham, L. Goettig, T. H. Hoare, J. R. Hughes, N. S. Jarvis, G. D. Jones, S. Juutinen, S. M. Mullins, C. N. Pass, J. Simpson, R. Wadsworth, D. L. Watson, and C. A. White, "Observation of unexpectedly small E1 moments in 224Ra," Physics Letters B 232, 447–451 (1989). DOI: 10.1016/0370-2693(89)90439-5; https://doi.org/10.1016/0370-2693(89)90439-5
  5. J. L. Egido and L. M. Robledo, "Microscopic study of the octupole degree of freedom in the radium and thorium isotopes with Gogny forces," Nuclear Physics A 494, 85–101 (1989). DOI: 10.1016/0375-9474(89)90199-1; https://doi.org/10.1016/0375-9474(89)90199-1
  6. P. A. Butler and W. Nazarewicz, "Intrinsic dipole moments in reflection-asymmetric nuclei," Nuclear Physics A 533, 249–268 (1991). DOI: 10.1016/0375-9474(91)90489-S; https://doi.org/10.1016/0375-9474(91)90489-S
  7. J. Skalski, "Octupole-induced dipole moments of very deformed nuclei," Physical Review C 49, 2011–2017 (1994). DOI: 10.1103/PhysRevC.49.2011; https://doi.org/10.1103/PhysRevC.49.2011
  8. P. A. Butler and W. Nazarewicz, "Intrinsic reflection asymmetry in atomic nuclei," Reviews of Modern Physics 68, 349–421 (1996). DOI: 10.1103/RevModPhys.68.349; https://doi.org/10.1103/RevModPhys.68.349
  9. V. Yu. Denisov, "Polarized electric dipole moment of well-deformed reflection asymmetric nuclei," The European Physical Journal A 47, 80 (2011). DOI: 10.1140/epja/i2011-11080-9; https://doi.org/10.1140/epja/i2011-11080-9
  10. L. P. Gaffney, P. A. Butler, M. Scheck, A. B. Hayes, F. Wenander, M. Albers, B. Bastin, C. Bauer, A. Blazhev, S. Bönig, N. Bree, J. Cederkäll, T. Chupp, D. Cline, T. E. Cocolios, T. Davinson, H. De Witte, J. Diriken, T. Grahn, A. Herzan, M. Huyse, D. G. Jenkins, D. T. Joss, N. Kesteloot, J. Konki, M. Kowalczyk, Th. Kröll, E. Kwan, R. Lutter, K. Moschner, P. Napiorkowski, J. Pakarinen, M. Pfeiffer, D. Radeck, P. Reiter, K. Reynders, S. V. Rigby, L. M. Robledo, M. Rudigier, S. Sambi, M. Seidlitz, B. Siebeck, T. Stora, P. Thoele, P. Van Duppen, M. J. Vermeulen, M. von Schmid, D. Voulot, N. Warr, K. Wimmer, K. Wrzosek-Lipska, C. Y. Wu, and M. Zielinska, "Studies of pear-shaped nuclei using accelerated radioactive beams," Nature 497, 199–204 (2013). DOI: 10.1038/nature12073; https://doi.org/10.1038/nature12073
  11. S. E. Agbemava, A. V. Afanasjev, and P. Ring, "Octupole deformation in the ground states of even-even nuclei: A global analysis within the covariant density functional theory," Physical Review C 93, 044304 (2016). DOI: 10.1103/PhysRevC.93.044304; https://doi.org/10.1103/PhysRevC.93.044304
  12. P. A. Butler, L. P. Gaffney, P. Spagnoletti, K. Abrahams, M. Bowry, J. Cederkäll, G. de Angelis, H. De Witte, P. E. Garrett, A. Goldkuhle, C. Henrich, A. Illana, K. Johnston, D. T. Joss, J. M. Keatings, N. A. Kelly, M. Komorowska, J. Konki, T. Kröll, M. Lozano, B. S. Nara Singh, D. O'Donnell, J. Ojala, R. D. Page, L. G. Pedersen, C. Raison, P. Reiter, J. A. Rodriguez, D. Rosiak, S. Rothe, M. Scheck, M. Seidlitz, T. M. Shneidman, B. Siebeck, J. Sinclair, J. F. Smith, M. Stryjczyk, P. Van Duppen, S. Vinals, V. Virtanen, N. Warr, K. Wrzosek-Lipska, and M. Zielińska, "Evolution of Octupole Deformation in Radium Nuclei from Coulomb Excitation of Radioactive 222^{222}Ra and 228^{228}Ra Beams," Physical Review Letters 124, 042503 (2020). DOI: 10.1103/PhysRevLett.124.042503; https://doi.org/10.1103/PhysRevLett.124.042503
  13. M. M. R. Chishti, D. O'Donnell, G. Battaglia, M. Bowry, D. A. Jaroszynski, B. S. Nara Singh, M. Scheck, P. Spagnoletti, and J. F. Smith, "Direct measurement of the intrinsic electric dipole moment in pear-shaped thorium-228," Nature Physics 16, 853–856 (2020). DOI: 10.1038/s41567-020-0899-4; https://doi.org/10.1038/s41567-020-0899-4
  14. V. V. Flambaum and A. J. Mansour, "Enhanced nuclear Schiff and electric dipole moments in nuclei with octupole deformation," Physical Review C 111, 055501 (2025). DOI: 10.1103/PhysRevC.111.055501; https://doi.org/10.1103/PhysRevC.111.055501
  15. A. Karmakar, Nazira Nazir, P. Datta, J. A. Sheikh, S. Jehangir, G. H. Bhat, S. S. Nayak, Soumik Bhattacharya, Suchorita Paul, Snigdha Pal, S. Bhattacharyya, G. Mukherjee, S. Basu, S. Chakraborty, S. Panwar, Pankaj K. Giri, R. Raut, S. S. Ghugre, R. Palit, Sajad Ali, W. Shaikh, and S. Chattopadhyay, "Measurement of enhanced electric dipole transition strengths at high spin in 100^{100}Ru: Possible observation of octupole deformation," arXiv:2411.10976 [nucl-ex] (2024); https://arxiv.org/abs/2411.10976