--- title: 'NUCLEI-PACK: Geometric Nuclear Structure' url: https://www.emergentmind.com/topics/nuclei-pack type: topic --- # NUCLEI-PACK: Geometric Nuclear Structure Searching arXiv for recent 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 [2509.13750]. 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 \(Z/N \approx 0.75\) [2405.11268]. 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 [2510.10741]. ## 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 [2509.13750]. 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 \(x_Z=Z/A\) [2405.11268]. 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 [2509.13750]. ## 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 \[ \|\mathbf{p}_i\| \leq R-r_0 \] and \[ \|\mathbf{p}_i-\mathbf{p}_j\| \geq 2r_0, \] with \(\mathbf{p}_i\) the sphere centers [2509.13750]. In the global-trend study, best-known numerical solutions from the Packomania PESS database are used for \(A\)-body geometries, and the coordinates are rescaled so that each nucleon sphere has radius \(r_0=1.0\ \text{fm}\) [2509.13750]. For a nucleus with fixed \((A,Z)\), protons and neutrons are assigned onto the packed coordinates by first sorting nucleons according to distance from the center and then distributing the \(Z\) protons evenly through that radial ordering. The assignment rule is \[ \text{position}_i=\left\lfloor \frac{\bigl(i-\tfrac12\bigr)A}{Z}+\tfrac12 \right\rfloor,\qquad i=1,2,\dots,Z. \] The remaining sites are assigned as neutrons [2509.13750]. 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, \[ r_p^{\text{Pack}}=0.89~\text{fm},\qquad r_n^{\text{Pack}}=1.06~\text{fm}, \] corresponding to the packing ratio \(r_p/r_n=0.84\) [2510.10741]. In that formulation, a single cluster is obtained by minimizing a container radius \(R_{\text{pack}}\) under non-overlap constraints \[ \|\vec p_i-\vec p_j\|\ge r_i+r_j \] and containment constraints \[ \|\vec p_i\|\le R_{\text{pack}}-r_i. \] 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 [2510.10741]. The framework also introduces an effective halo offset parameter \(\Delta\). For one-nucleon halos the valence-cluster center is shifted to \[ \vec c_2=(0,0,R_1+R_2+\Delta), \] and for two-nucleon halos the valence centers are shifted outward along core-valence directions [2510.10741]. Here \(\Delta\) 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 [2510.10741]. ## 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 [2405.11268]. For species \(i,j\in\{N,Z\}\), the contact distance is \[ \sigma_{ij}=\frac12(\sigma_i+\sigma_j), \] the number fractions satisfy \(x_N+x_Z=1\), and the occupied volume fraction is \[ \phi=\frac{\pi}{6}\rho\langle \sigma^3\rangle, \] with \(\langle \sigma^n\rangle=\sum_i x_i\sigma_i^n\) [2405.11268]. Mechanical stability is imposed through an isostatic condition \(\langle z\rangle=6\), giving an analytical route to \(\phi_{\text{RCP}}\). The critical physical assumption is that neutrons have a larger excluded-volume size than protons. The proton radius is taken as \[ r_Z=\frac{\sigma_Z}{2}\approx 0.84\ \text{fm}, \] while the neutron radius is taken as \[ r_N=\frac{\sigma_N}{2}\approx 1.0\ \text{fm}. \] This gives \[ \frac{\sigma_Z}{\sigma_N}\approx 0.84,\qquad \frac{r_N}{r_Z}\approx 1.19, \] and therefore \[ \frac{V_N}{V_Z}\approx 1.7 \] for the excluded volumes [2405.11268]. 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 \(\sigma_Z/\sigma_N=0.84\), the computed packing fraction \(\phi_{\text{RCP}}(Z/A)\) has a maximum at \[ Z/A=0.43, \] which implies \[ Z/N\approx 0.754\approx 0.75. \] This geometric optimum is proposed as a missing contribution to the broad slope of the valley of stability [2405.11268]. 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 [2405.11268]. The paper compares the resulting line \(Z=0.75N\) with the distribution of stable nuclides and states that stable nuclei cluster near that slope, especially for heavier systems [2405.11268]. 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 [2405.11268]. ## 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 [2509.13750]. For proton coordinates \(\mathbf r_i^{(p)}\), the point-proton charge radius is \[ R_c^{\text{(point)}}=\sqrt{\frac1Z\sum_{i=1}^Z\lVert \mathbf r_i^{(p)}\rVert^2+\langle R_p^2\rangle+\frac{N}{Z}\langle R_n^2\rangle+\Delta_{\text{DF}}}, \] with \[ \langle R_p^2\rangle=(0.877\ \text{fm})^2,\qquad \langle R_n^2\rangle=-0.1161\ \text{fm}^2,\qquad \Delta_{\text{DF}}=0.033\ \text{fm}^2. \] Gaussian smearing with \(\sigma_p=0.85\ \text{fm}\) then gives \[ R_c^{(\sigma)}=\sqrt{\left(R_c^{\text{(point)}}\right)^2+3\sigma_p^2} \] [2509.13750]. The matter radius is defined analogously: \[ R_m^{\text{(point)}}=\sqrt{\frac1A\sum_{i=1}^A\lVert \mathbf r_i\rVert^2+\frac{Z\langle R_{m,p}^2\rangle+N\langle R_{m,n}^2\rangle}{A}}, \] followed by Gaussian smearing, \[ R_m^{(\sigma)}=\sqrt{\left(R_m^{\text{(point)}}\right)^2+\frac{3}{A}\left(Z\sigma_p^2+N\sigma_n^2\right)}. \] In the global study, \(\sigma_p=\sigma_n=0.85\ \text{fm}\) is used [2509.13750]. Binding energies are computed from a Woods-Saxon mean field plus a microscopic proton-proton Coulomb sum. The volume and surface pieces are \[ V_{\text{vol}}(r)=\frac{-V_r}{1+\exp\left(\frac{r-R_v}{a_v}\right)}, \] \[ V_{\text{surf}}(r)=4a_sV_s\frac{\exp\left(\frac{r-R_s}{a_s}\right)}{\left[1+\exp\left(\frac{r-R_s}{a_s}\right)\right]^2}, \] with \[ R_i=r_iA^{1/3},\qquad i=v,s. \] The mean-field energy is \[ E_{\text{mean-field}}=\sum_{j=1}^A\left[V_{\text{vol}}(r_j)+V_{\text{surf}}(r_j)\right], \] the Coulomb energy is \[ E_{\text{Coulomb}}=\sum_{i