NUCLEI-PACK: Geometric Nuclear Structure

• NUCLEI-PACK is a semi-classical model that represents atomic nuclei as optimized packings of hard-sphere nucleons and clusters, emphasizing spatial geometry over traditional quantum methods.
• The framework employs packing algorithms to determine explicit 3D nucleon coordinates that reproduce bulk observables such as charge radii, matter radii, and binding energies.
• It provides geometric insights into neutron-rich stability and halo/cluster structures, offering a computationally efficient complement to standard nuclear models.

Searching arXiv for papers on NUCLEI-PACK and related packing-based nuclear-structure work. Searching arXiv for "NUCLEI-PACK" and related packing-based nuclear structure papers. NUCLEI-PACK is a semi-classical nuclear-structure framework in which a nucleus is modeled as an optimized packing of hard-sphere nucleons and, in some applications, pre-formed clusters. In its current research usage, the framework links explicit three-dimensional nucleon coordinates to bulk observables such as charge radii, matter radii, and binding energies, and extends naturally to halo and cluster geometries in light nuclei (Maridi, 17 Sep 2025). A related geometric argument treats nuclei as dense, nearly jammed binary packings of protons and neutrons with slightly different excluded volumes, and uses random close packing to explain why the valley of stability favors neutron-rich nuclei with $Z/N \approx 0.75$ (Anzivino et al., 2024). Taken together, these studies define NUCLEI-PACK as a geometrically explicit approach to nuclear structure that emphasizes packing, coordination, and excluded-volume effects rather than orbital wave functions or self-consistent mean fields alone (Maridi, 12 Oct 2025).

1. Origins and conceptual definition

NUCLEI-PACK begins from the assumption that short-range repulsion in the nucleon-nucleon interaction makes nucleons behave as hard spheres at small distances. In the proof-of-concept formulation, the nucleus is built from explicit 3D coordinates for each nucleon, generated by solving a sphere-packing optimization problem, and observables are then computed directly from those coordinates (Maridi, 17 Sep 2025). This differs from shell-model and mean-field descriptions, which do not start from fixed nucleon centers, and from ab initio methods, which derive structure from a many-body wave function rather than a geometric packing ansatz.

A second, closely related strand of the framework is the random-packing interpretation of nuclear stability. In that formulation, nucleons are treated as hard spheres with different diameters, $\sigma_Z$ for protons and $\sigma_N$ for neutrons, and the random close packing volume fraction $\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$ is analyzed as a function of proton fraction $x_Z=Z/A$ (Anzivino et al., 2024). The central claim is that packing geometry itself can bias stable nuclei toward neutron-rich compositions, independently of the standard Coulomb-versus-asymmetry balance in the Bethe-Weizsäcker formula.

Within the recent literature, NUCLEI-PACK is therefore best understood as a geometrically explicit, semi-classical framework for nuclear structure. Its core variables are nucleon coordinates, packing radii, cluster separations, and neighbor relations. This suggests a structural interpretation of nuclear bulk properties in which $A^{1/3}$ radius scaling, $A^{2/3}$ surface behavior, and aspects of neutron excess emerge from packing constraints and local coordination (Maridi, 17 Sep 2025).

2. Geometric foundations and packing algorithms

The geometric backbone of NUCLEI-PACK is the “Packing Equal Spheres in a Sphere” problem. For $n$ identical spheres of radius $r_0$, the task is to minimize the enclosing radius $R$ subject to

$\sigma_Z$0

and

$\sigma_Z$1

with $\sigma_Z$2 the sphere centers (Maridi, 17 Sep 2025). In the global-trend study, best-known numerical solutions from the Packomania PESS database are used for $\sigma_Z$3-body geometries, and the coordinates are rescaled so that each nucleon sphere has radius $\sigma_Z$4 (Maridi, 17 Sep 2025).

For a nucleus with fixed $\sigma_Z$5, protons and neutrons are assigned onto the packed coordinates by first sorting nucleons according to distance from the center and then distributing the $\sigma_Z$6 protons evenly through that radial ordering. The assignment rule is

$\sigma_Z$7

The remaining sites are assigned as neutrons (Maridi, 17 Sep 2025). This algorithm is intended to avoid unrealistic central or surface localization of proton charge.

The light-nucleus extension generalizes the geometry to binary hard spheres with different packing radii,

$\sigma_Z$8

corresponding to the packing ratio $\sigma_Z$9 (Maridi, 12 Oct 2025). In that formulation, a single cluster is obtained by minimizing a container radius $\sigma_N$0 under non-overlap constraints

$\sigma_N$1

and containment constraints

$\sigma_N$2

Two-cluster and three-cluster assemblies are then constructed by packing the constituent clusters independently and placing their centers in surface contact before shifting to the total center-of-mass frame (Maridi, 12 Oct 2025).

The framework also introduces an effective halo offset parameter $\sigma_N$3. For one-nucleon halos the valence-cluster center is shifted to

$\sigma_N$4

and for two-nucleon halos the valence centers are shifted outward along core-valence directions (Maridi, 12 Oct 2025). Here $\sigma_N$5 is not an observable itself; it modifies the geometry prior to the radius calculations. The papers interpret it as an effective geometric realization of weakly bound, spatially extended halo motion (Maridi, 12 Oct 2025).

3. Random close packing and neutron-rich stability

The geometric-stability argument associated with NUCLEI-PACK uses an analytical theory of random close packing for binary hard spheres (Anzivino et al., 2024). For species $\sigma_N$6, the contact distance is

$\sigma_N$7

the number fractions satisfy $\sigma_N$8, and the occupied volume fraction is

$\sigma_N$9

with $\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$0 (Anzivino et al., 2024). Mechanical stability is imposed through an isostatic condition $\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$1, giving an analytical route to $\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$2.

The critical physical assumption is that neutrons have a larger excluded-volume size than protons. The proton radius is taken as

$\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$3

while the neutron radius is taken as

$\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$4

This gives

$\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$5

and therefore

$\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$6

for the excluded volumes (Anzivino et al., 2024). The paper explicitly notes that this size assumption is a hypothesis grounded in charge-density considerations rather than a settled result of nuclear structure theory.

With $\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$7, the computed packing fraction $\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$8 has a maximum at

$\phi_{\text{RCP}}(\sigma_Z/\sigma_N,x_Z)$9

which implies

$x_Z=Z/A$0

This geometric optimum is proposed as a missing contribution to the broad slope of the valley of stability (Anzivino et al., 2024). The argument is then linked to binding by noting that short-range attraction depends mainly on nearest and near-nearest neighbors, so a larger packing fraction implies a larger total coordination and hence stronger bulk binding.

This does not replace the Bethe-Weizsäcker formula. Rather, the proposal is that the packing contribution supplements the usual volume, surface, Coulomb, asymmetry, and pairing terms by giving the volume term an implicit structural dependence on composition (Anzivino et al., 2024). The paper compares the resulting line $x_Z=Z/A$1 with the distribution of stable nuclides and states that stable nuclei cluster near that slope, especially for heavier systems (Anzivino et al., 2024).

A common misconception is that the framework attributes neutron excess solely to electrostatics or solely to geometry. The cited work argues instead that geometry provides an additional bias toward neutron-rich compositions; it is presented as complementary to Coulomb repulsion and not as a replacement for asymmetry, exchange, or Pauli effects (Anzivino et al., 2024).

4. Computed observables and global trends

In the bulk-properties implementation, NUCLEI-PACK computes charge radii, matter radii, and binding energies directly from packed coordinates (Maridi, 17 Sep 2025). For proton coordinates $x_Z=Z/A$2, the point-proton charge radius is

$x_Z=Z/A$3

with

$x_Z=Z/A$4

Gaussian smearing with $x_Z=Z/A$5 then gives

$x_Z=Z/A$6

(Maridi, 17 Sep 2025).

The matter radius is defined analogously: $x_Z=Z/A$7 followed by Gaussian smearing,

$x_Z=Z/A$8

In the global study, $x_Z=Z/A$9 is used (Maridi, 17 Sep 2025).

Binding energies are computed from a Woods-Saxon mean field plus a microscopic proton-proton Coulomb sum. The volume and surface pieces are

$A^{1/3}$0

$A^{1/3}$1

with

$A^{1/3}$2

The mean-field energy is

$A^{1/3}$3

the Coulomb energy is

$A^{1/3}$4

and the binding energy per nucleon is

$A^{1/3}$5

(Maridi, 17 Sep 2025). The quoted global fit gives

$A^{1/3}$6

(Maridi, 17 Sep 2025).

Applied to long-lived or stable nuclei with $A^{1/3}$7, the framework is reported to reproduce the global $A^{1/3}$8 trend for charge radii and matter radii, and to recover the saturation pattern of $A^{1/3}$9, including the rise toward mid-mass nuclei and the decrease toward very light and very heavy systems (Maridi, 17 Sep 2025). The paper does not claim explicit shell corrections or odd-even staggering, and it explicitly identifies shell effects, pairing, spin-orbit splitting, and antisymmetrization as outside the proof-of-concept treatment (Maridi, 17 Sep 2025).

5. Halo nuclei and cluster structure

A major extension of NUCLEI-PACK specializes the framework to light and exotic nuclei, especially halo systems and $A^{2/3}$0-cluster structures (Maridi, 12 Oct 2025). With center-of-mass-subtracted coordinates $A^{2/3}$1, the RMS operator is defined by

$A^{2/3}$2

and this geometric skeleton is combined with intrinsic nucleon radii, Darwin-Foldy corrections, and Gaussian smearing to obtain charge and matter radii (Maridi, 12 Oct 2025).

For two-cluster systems, the core-valence RMS distance is

$A^{2/3}$3

and for three-cluster systems the corresponding quantity is

$A^{2/3}$4

(Maridi, 12 Oct 2025). The framework also extracts neutron-neutron or proton-proton separations and opening angles directly from the cluster-center geometry.

The paper studies one-nucleon halo nuclei $A^{2/3}$5Be, $A^{2/3}$6C, $A^{2/3}$7C, and $A^{2/3}$8B, and two-nucleon halo nuclei $A^{2/3}$9He, $n$0Li, $n$1B, and $n$2Ne (Maridi, 12 Oct 2025). Representative fitted offsets include $n$3 for $n$4Be, $n$5 for $n$6C, $n$7 for $n$8C, and $n$9 for $r_0$0B (Maridi, 12 Oct 2025). For Borromean systems, the framework reproduces characteristic geometrical observables such as $r_0$1 and $r_0$2 in $r_0$3He and $r_0$4Li (Maridi, 12 Oct 2025).

The authors state that the fitted geometric offset $r_0$5 exhibits an inverse correlation with the nucleon separation energy: more weakly bound halos require larger offsets, while borderline cases such as $r_0$6C and $r_0$7Ne are described with small or zero $r_0$8 (Maridi, 12 Oct 2025). This suggests that, within the model, separation energy controls halo extension through a small set of effective geometric degrees of freedom rather than through a full continuum-coupled wave function.

The same packing logic is used for cluster nuclei. The paper treats $r_0$9Li as $R$0, $R$1Li as $R$2, and $R$3C as $R$4, with the $R$5C geometry forming a triangular arrangement around the center of mass (Maridi, 12 Oct 2025). The results are described as qualitative for the cluster cases, because inter-cluster separations and angles are not yet tabulated in detail. Still, the framework is explicitly presented as a unified geometric treatment of compact cores, halo valence nucleons, and cluster substructures (Maridi, 12 Oct 2025).

6. Relation to other nuclear-structure approaches and limitations

NUCLEI-PACK occupies a distinct methodological niche. Unlike relativistic self-consistent mean-field packages such as DIRHB, which solve the relativistic Hartree-Bogoliubov equations in spherical, axial, or triaxial harmonic-oscillator bases for even-even nuclei (Niksic et al., 2014), NUCLEI-PACK does not start from an energy density functional or a quasiparticle vacuum. Unlike ab initio no-core shell model/resonating-group methods, which treat nuclei as many-body open quantum systems and solve microscopic structure-reaction equations from realistic interactions (Quaglioni et al., 2012), NUCLEI-PACK does not derive observables from a full many-body Hamiltonian with continuum coupling. Its explicit advantage is geometric transparency and computational efficiency (Maridi, 17 Sep 2025).

That advantage is accompanied by clear limitations. The global-trend study explicitly lists several: equal nucleon radii in the proof-of-concept implementation, no explicit quantum correlations, no shell effects, no pairing, no spin-orbit splitting, no antisymmetrization, fixed packing geometry for each $R$6, and heuristic surface identification through neighbor counting with $R$7 and $R$8 (Maridi, 17 Sep 2025). The light-nucleus study similarly notes that halo dynamics and continuum couplings are represented by a single effective parameter $R$9, rather than by explicit dynamical wave functions (Maridi, 12 Oct 2025).

A second potential misconception is to view NUCLEI-PACK as a replacement for microscopic nuclear theory. The papers do not make that claim. They present it as a semi-classical, proof-of-concept or phenomenological framework that captures global trends, cluster geometries, and halo extensions with low computational cost and direct spatial interpretability (Maridi, 17 Sep 2025). A plausible implication is that the framework is best suited to structural regularities dominated by geometry, while detailed spectroscopy, magic numbers, and continuum-resonance dynamics remain the domain of shell-model, EDF, and ab initio methods.

The term “NUCLEI-PACK” also appears metaphorically in unrelated contexts across the supplied literature, including cosmic-ray composition packaging in the NUCLEON experiment (Grebenyuk et al., 2018), a relativistic mean-field code package (Niksic et al., 2014), and even non-nuclear domains such as brain-nuclei parcellation and collective cell packs (He et al., 10 Mar 2025, Saraswathibhatla et al., 2021). In the nuclear-structure sense developed here, however, the term refers specifically to packing-based descriptions of atomic nuclei and not to those broader metaphorical uses.

7. Prospects and open directions

The published NUCLEI-PACK papers identify several concrete extensions. The most immediate is binary sphere packing with different proton and neutron radii, replacing the equal-radius PESS construction used for the global study (Maridi, 17 Sep 2025). This would align the bulk framework more closely with the neutron-rich packing argument of the random close-packing study (Anzivino et al., 2024) and would permit more realistic treatment of neutron skins and isospin-dependent geometry.

Another extension is the replacement of the fitted Woods-Saxon mean field by direct two-body nucleon-nucleon interactions, including possibilities such as Reid soft-core or modern chiral interactions (Maridi, 17 Sep 2025). The authors also mention pairing corrections, spin-orbit effects, deformation through non-spherical containers, and machine-learning-guided packing searches as future directions (Maridi, 17 Sep 2025, Maridi, 12 Oct 2025).

For exotic nuclei, the natural next step is quantitative treatment of additional halo and cluster systems, including refined comparisons to experimental charge and matter radii, core-valence distances, and cluster-state observables (Maridi, 12 Oct 2025). The light-nucleus paper also suggests applications to fission and fusion as repacking processes, which would extend the framework from static geometry to reaction pathways (Maridi, 12 Oct 2025).

In a broader methodological sense, NUCLEI-PACK can be read as an attempt to provide a geometric complement to more formal nuclear theories. Its central proposition is not that nuclei are literally classical sphere packings, but that a significant subset of nuclear bulk behavior can be encoded in packing density, local coordination, and cluster arrangement. The reported recovery of $\sigma_Z$00 from random close packing (Anzivino et al., 2024), the reproduction of global radii and binding trends across $\sigma_Z$01 (Maridi, 17 Sep 2025), and the description of halo and cluster geometries in light nuclei (Maridi, 12 Oct 2025) together define the current scope of the framework. This suggests a research program in which geometric packing acts as a low-cost, structurally transparent layer for interpreting nuclear size, stability, halo formation, and clustering.