Toroidal Nuclei: Dynamics and Structure

• Toroidal nuclei are characterized by either excited transition currents forming a torus-like vortex or intrinsic density distributions with a ring-shaped topology.
• Their investigation employs advanced multipole decompositions, QRPA calculations, and electron-scattering techniques to differentiate vortical toroidal modes from compressional responses.
• Research on toroidal nuclei informs our understanding of high-spin isomers, clustering phenomena in light nuclei, and Coulomb-driven toroidal configurations in hyperheavy systems.

Toroidal nuclei denote two related but distinct nuclear phenomena. In one usage, dominant in contemporary dipole spectroscopy, the term refers to excited states whose transition current forms a torus-like vortical flow: the toroidal dipole mode or toroidal dipole resonance. In this dynamical sense, the equilibrium density is not donut-shaped; what is toroidal is the current field in an excited $1^-$ state. In a second usage, the term denotes nuclei whose intrinsic density itself acquires a ring-like topology, as in toroidal high-spin isomers, toroidal cluster states, or hyperheavy mean-field solutions. Modern work therefore treats toroidicity both as a property of nuclear current and as a property of nuclear density, with different operators, observables, and stability criteria attached to each case (Nesterenko et al., 10 Oct 2025, Agbemava et al., 2019).

1. Terminological scope and geometric content

In nuclear-structure physics, “toroidal nucleus” most often means a nucleus in a toroidal excitation, not a static torus. The review of the toroidal dipole mode makes this point explicitly: the phrase usually refers to states in which the transition current forms a torus-like flow pattern, rather than to a ground-state density shaped like a torus (Nesterenko et al., 10 Oct 2025). By contrast, studies of $^{12}$C, high-spin light nuclei, superheavy nuclei, and hyperheavy nuclei treat toroidicity as an intrinsic property of the density itself, characterized by a major radius $R$, a minor radius $d$, and a central void (Wong, 2019, Kosior et al., 2017).

Sense of “toroidal nucleus”	Hallmark	Representative cases
Dynamical toroidicity	Toroidal transition current in an excited state	TDM/TDR in $^{58}$Ni, $^{24}$Mg, $^{170}$Yb
Static toroidicity	Toroidal intrinsic density distribution	$^{12}$C toroidal states, light high-spin isomers, superheavy and hyperheavy toroids

The distinction is not merely semantic. Dynamical toroidicity is diagnosed through current transition densities, toroidal operators, and transverse electron-scattering form factors. Static toroidicity is diagnosed through intrinsic density distributions, shell structure at fixed $R/d$, deformation-energy surfaces, and stability against fission or multifragmentation. A persistent misconception is to treat these two literatures as interchangeable. They are contiguous, but not identical: one concerns vortical nuclear response, the other equilibrium or metastable ring-shaped matter distributions.

2. Vortical electric modes and the toroidal dipole operator

The theoretical starting point for dynamical toroidicity is the multipole decomposition of the nuclear transition current $\delta \mathbf{j}(\mathbf r)$. In the Chandrasekhar–Moffatt or Debye-potential form,

$^{12}$0

the gradient term is irrotational and underlies conventional electric multipoles, the single-curl term is vortical magnetic, and the double-curl term is vortical electric; its long-wavelength part generates the toroidal current (Nesterenko et al., 10 Oct 2025). This decomposition is the formal reason toroidal multipoles are treated as a “third family” of electromagnetic modes, distinct from ordinary electric and magnetic multipoles.

For isoscalar dipole motion, toroidal and compressional operators separate the vortical and irrotational parts of the E1 response. A standard current-space form of the toroidal operator is

$^{12}$1

while the compressional partner depends on the different linear combination of the same $^{12}$2 and $^{12}$3 current components, or equivalently on a density operator weighted by $^{12}$4 after the continuity equation is used (Nesterenko et al., 2016). In this formulation, toroidal motion is explicitly tied to the curl of the current, compressional motion to its divergence.

The continuity equation,

$^{12}$5

is central to the distinction. Ordinary electric dipole motion is largely longitudinal and continuity-equation constrained. Toroidal motion is approximately transverse, with

$^{12}$6

so it contributes only weakly to density-based observables in the long-wavelength limit (Nesterenko et al., 10 Oct 2025). This is why toroidal E1 modes can carry strong current structure while remaining weak in ordinary $^{12}$7.

This current-based definition also sharpened the debate over “nuclear vorticity.” A criterion based solely on the $^{12}$8 or $^{12}$9 current component was shown to be inadequate because both toroidal and compressional modes contain $R$0 and $R$1. The hydrodynamically consistent quantity is the transverse current or, operationally, the toroidal strength function itself (Nesterenko et al., 2016). In practical self-consistent Skyrme RPA and QRPA calculations, this requires retaining current-dependent and time-odd terms in the functional; the formalism of full Skyrme RPA was developed precisely to calculate electric, magnetic, vortical, toroidal, and compression transitions on the same footing (Repko et al., 2015).

3. Low-energy E1 response, pygmy strength, and the $R$2Ni evidence

A major development in the field is the reinterpretation of low-energy dipole strength. In systematic Skyrme-QRPA studies of $R$3Ca, $R$4Ni, $R$5Zr, and $R$6Sn, the lower part of the region often labeled the pygmy dipole resonance was found to be basically isoscalar vortical toroidal motion with a minor irrotational fraction, independently of whether obvious PDR strength exists in the standard E1 channel (Repko et al., 2019). In $R$7Sn, for example, the $R$8 MeV window simultaneously shows PDR-like transition densities and clear toroidal current fields, implying that the familiar neutron-skin-against-core picture is, at minimum, incomplete (Nesterenko et al., 2016).

The dynamical picture is therefore mixed. Transition densities in neutron-rich nuclei may indeed display a neutron-dominated surface hump, but current transition densities reveal that the underlying flow is predominantly toroidal. This does not invalidate earlier $R$9-based PDR analyses; rather, it separates observables. Density-sensitive probes emphasize the irrotational fraction, while current-sensitive probes emphasize the toroidal one.

The clearest experimental case is $d$0Ni. Long-standing TU Darmstadt $d$1 data had shown at least six dipole states in the $d$2 MeV region with unusually steep transverse form factors at backward angles. These states were first labeled M1, then reclassified as E1 after polarized-photon measurements, leaving the transverse response unexplained within a conventional irrotational picture (Nesterenko et al., 10 Oct 2025). The combined analysis of high-resolution photon, proton, and electron scattering identified low-lying $d$3 candidates for toroidal dipole excitation at $d$4, $d$5, and $d$6 MeV, in correspondence with QRPA toroidal states at $d$7, $d$8, and $d$9 MeV (Neumann-Cosel et al., 2023).

Electron scattering is decisive because, in plane-wave Born approximation,

$^{58}$0

and at very large angles the longitudinal piece becomes negligible, leaving the cross section directly sensitive to the transverse current form factor (Nesterenko et al., 10 Oct 2025). In $^{58}$1Ni, the large-angle $^{58}$2 slopes are reproduced only when the states are assigned a strong toroidal component. The same work proposed an additional discriminator: for toroidal states the relative sign

$^{58}$3

is $^{58}$4, whereas for GDR-like and compression-like states it is $^{58}$5, so $^{58}$6 interference measurements could separate vortical from irrotational E1 motion (Neumann-Cosel et al., 2023).

The consequence is broader than one nucleus. The $^{58}$7Ni case shows that weakly collective, low-lying electric dipole states can exhibit current patterns more commonly associated with magnetic or transverse motion. It also suggests that a fraction of historical “anomalous M1” assignments in backward-angle electron scattering may require reinterpretation in terms of E1 toroidal dynamics.

4. Deformation, anomalous $^{58}$8-splitting, and individual toroidal states

Axial deformation reorganizes toroidal response in characteristic ways. In prolate $^{58}$9Yb, Skyrme-RPA calculations showed that the low-energy toroidal region lies at $^{24}$0 MeV and exhibits an anomalous branch ordering: $^{24}$1 opposite to the normal prolate ordering of the GDR and compression mode, for which $^{24}$2 (Kvasil et al., 2013). The effect appears already at the unperturbed two-quasiparticle level and was later emphasized as a robust deformation fingerprint of the toroidal dipole resonance in prolate nuclei (Nesterenko et al., 2016).

In light strongly deformed nuclei, deformation can isolate individual toroidal states rather than merely reshuffle a broad resonance. The clearest example is $^{24}$3Mg, where axial QRPA with SLy6 predicts the lowest $^{24}$4 excitation at $^{24}$5 MeV to be a vortical toroidal state. Its current transition density forms a vortex–antivortex realization of Hill’s spherical vortex in strong axial confinement, and the result persists for SLy6, SVbas, and SkM* (Nesterenko et al., 2017). In the same nucleus, the nearby $^{24}$6 state at $^{24}$7 MeV is compression-dominated, making $^{24}$8Mg a textbook case of side-by-side vortical and irrotational low-energy E1 motion.

The detailed spectroscopy of light deformed nuclei also showed that the lowest toroidal state is not universal. A comparative QRPA study of $^{24}$9Mg and $^{170}$0Ne found that the lowest toroidal $^{170}$1 state is a peculiarity of $^{170}$2Mg. In $^{170}$3Ne, the toroidal state appears at $^{170}$4 MeV, whereas the $^{170}$5 compression state lies lower, at $^{170}$6 MeV (Nesterenko et al., 2018). The difference was traced to deformation-driven rearrangements of specific Nilsson orbitals near the Fermi surface. This suggests that strong prolate deformation is necessary but not sufficient; the detailed single-particle spectrum matters.

These light-nucleus results also connect toroidicity to clustering. In $^{170}$7Mg, the toroidal $^{170}$8 MeV state and the compression $^{170}$9 MeV state both lie near the $^{12}$0-particle threshold $^{12}$1 MeV, and the density/current patterns were interpreted as intertwined with cluster structure (Nesterenko et al., 2017). This does not imply that toroidal states are reducible to cluster states, but it does indicate that in light nuclei the two descriptions can overlap in the same energy domain.

5. Static toroidal densities, shell structure, and high-spin toroidal isomers

Static toroidicity enters nuclear structure in several forms. In $^{12}$2C, planar intrinsic $^{12}$3 and $^{12}$4 states were constructed both from rotated triangular $^{12}$5 cluster configurations and from toroidal shell-model Slater determinants. In that framework, the ground state was found to contain a toroidal core, and a toroidal mean-field configuration with

$^{12}$6

occurs at the Hoyle energy $^{12}$7 MeV, although it is not a local minimum in $^{12}$8 within a single-determinant treatment (Wong, 2019). The significance of this result is not a definitive toroidal assignment of the Hoyle state, but the demonstration that toroidal density profiles and $^{12}$9 generator-coordinate constructions can be placed within one framework.

A different route to static toroidicity is high spin. Self-consistent Skyrme-Hartree-Fock calculations predicted even-even toroidal high-spin isomers in light nuclei with $R/d$0, including $R/d$1 systems. Explicit examples are $R/d$2 and $R/d$3, which fall on the same regular multi-particle–multi-hole patterns as the $R/d$4 toroidal high-spin isomers (Staszczak et al., 2014). In these systems, aligned particle–hole excitations create large $R/d$5, and the toroidal density is stabilized over a finite angular-momentum window.

The shell-model underpinning of such states was generalized to the intermediate-mass region by a toroidal single-particle potential study. That work found large toroidal shell gaps at various nucleon numbers and aspect ratios $R/d$6, showed how Bohr–Mottelson spin-aligning particle–hole excitations can generate toroidal high-spin isomers, and introduced the possibility of toroidal vortex nuclei, in which particle–hole excitations change a nucleon’s vorticity quantum number (Wong et al., 2018). This suggests that static toroidal densities support their own shell structure, high-spin spectroscopy, and even intrinsic vorticity labels.

In superheavy nuclei, the same logic reappears at much larger scales. For $R/d$7, strongly deformed oblate configurations bifurcate into a toroidal branch, and cranked Skyrme-Hartree-Fock predicts toroidal high-spin isomers at

$R/d$8

with $R/d$9 near $\delta \mathbf{j}(\mathbf r)$0 b (1705.01408). Yet zero-spin calculations for $\delta \mathbf{j}(\mathbf r)$1 isotopes and $\delta \mathbf{j}(\mathbf r)$2 isotones showed that, although toroidal solutions become lower in energy than biconcave discs beyond sufficiently negative $\delta \mathbf{j}(\mathbf r)$3, the toroidal branch itself still lacks a local minimum at those proton and neutron numbers (Kosior et al., 2017). High spin therefore acts as a stabilizer where shell structure and Coulomb effects alone are not yet sufficient.

6. Hyperheavy toroids, shell stabilization, and the extended nuclear landscape

The most systematic static-toroidicity results come from covariant density functional theory in the hyperheavy region. Axial RHB calculations predict that beyond about $\delta \mathbf{j}(\mathbf r)$4, the energetically favored axial shapes change qualitatively from ellipsoidal-like to toroidal ones, and in the $\delta \mathbf{j}(\mathbf r)$5 region the lowest-energy axial solutions carry very large negative $\delta \mathbf{j}(\mathbf r)$6 and ring-shaped densities (Agbemava et al., 2019). The physical driver is the Coulomb-energy gain from spreading charge into a torus: the reduction in $\delta \mathbf{j}(\mathbf r)$7 between spherical and toroidal configurations rises from about $\delta \mathbf{j}(\mathbf r)$8 MeV in $\delta \mathbf{j}(\mathbf r)$9Pb to about $^{12}$00 MeV in $^{12}$01, $^{12}$02 MeV in $^{12}$03, and $^{12}$04 MeV in $^{12}$05 (Agbemava et al., 2019).

Subsequent CDFT systematics refined this picture. Toroidal hyperheavy even-even nuclei were found to be stable with respect to breathing deformations; the most compact fat tori lie near $^{12}$06, $^{12}$07, whereas thin toroidal nuclei become dominant with increasing proton number and toward proton and neutron drip lines (Agbemava et al., 2020). The lowest axial toroidal solutions are characterized either by large shell gaps or by low single-particle level density near the Fermi level in at least one subsystem, and the $^{12}$08 toroidal shell gap was identified as an important stabilizer of fat toroidal nuclei (Agbemava et al., 2020).

The stability problem, however, is not closed. The same hyperheavy studies emphasize that many toroidal nuclei are expected to be unstable toward multifragmentation or sausage-type distortions, even when they are stable against breathing motion (Agbemava et al., 2019). In parallel, spherical shell closures at $^{12}$09 and $^{12}$10 define competing islands of spherical hyperheavy stability (Agbemava et al., 2020). The global landscape is therefore a competition between Coulomb-driven toroidal axial minima and shell-driven spherical minima, with the final answer depending on the balance among shell structure, triaxiality, and non-axial instabilities.

A recent extension in EQMD has added a morphological classification of bubble-like nuclei using the dimensionless parameters $^{12}$11. In that scheme, $^{12}$12 defines “toroidal bubble nuclei,” identified from two inflection points in the radial density profile; these nuclei emerge for $^{12}$13 and become prevalent in heavy systems (Ren et al., 23 May 2026). This is not the same concept as the macroscopic toroidal mean-field shapes of hyperheavy CDFT, but it shows that toroidal-like hollows now enter nuclear taxonomy through both self-consistent mean-field and cluster-based descriptions.

Taken together, the literature establishes toroidicity as a genuine organizing principle of nuclear structure rather than an isolated curiosity. In the dynamical sector, it identifies a vortical electric mode beyond the continuity-equation description of ordinary E1 motion and now supported experimentally in $^{12}$14Ni. In the static sector, it links ring-shaped densities to cluster geometry, aligned high-spin configurations, shell structure in intermediate-mass nuclei, and the Coulomb-dominated shape evolution of superheavy and hyperheavy matter. The term “toroidal nucleus” therefore names not one object but a family of nuclear states in which either the current or the density acquires toroidal topology.