Relative geometry of nuclear mean fields through differential forms

Abstract

Proton and neutron densities can have similar multipole moments while differing locally. We use differential forms to measure and locate such differences between nuclear mean fields. A pair invariant measures the angle between two density gradients, weighted by their magnitudes, and the coarea formula resolves its contribution over level surfaces. A triple invariant factorizes exactly into a weighted pair term and a directional term that measures the variation of a third field along the level-surface intersections. We test these quantities in 98 magnetized Wigner-Seitz solutions, five static Hartree-Fock configurations, an illustrative $^{16}$O + $^{34}$Mg collision, and periodic nuclear pasta (a slab and a connected network). The pair invariant distinguishes cell solutions with comparable proton quadrupole moments and is 253 times larger in $^{34}$Mg than in $^{24}$Mg; refinement to 0.6 fm changes it by at most 1.3% in the four configurations recomputed. The directional factor separates the sampled triaxial configurations from the prolate ones. In the collision, it accounts for the 7.3-fold rise of the triple before the first recorded low-density connection, while the weighted pair term changes by at most 3%. In nuclear pasta, slab and network configurations have similar integrated proton-neutron misalignment but different distributions over density levels; their triple invariants differ by a factor of about 370 on the finer grid, mainly due to the variation of the Coulomb potential along joint density levels. These comparisons require no preferred origin or axis. A resolved nonzero triple excludes a common symmetry axis, but zero does not establish one. The pasta results require further convergence studies.