How Much Nuclear Charge Distribution Information Do Electrons Read?

Abstract

When the nuclear charge distribution is compressed into a mean-square charge radius⟨r2⟩, how much electronic-structure information is lost—and does relativity change theanswer? Using one-electron Schrödinger and Dirac models with analytically constructednuclear densities matched at progressively higher radial-moment order, we map the elec-tronic radial-information bandwidth. At Z = 40 (non-relativistic), two moments suffice:the residual drops from 5.8 ×10−4 eV (⟨r2⟩only) to 2.6 ×10−6 eV (adding ⟨r4⟩) to thenoise floor (adding ⟨r6⟩). At Z = 90 (Dirac), the bandwidth doubles: convergence requiresn∗ ≈8 moments, and the residual is 139 times larger. A 2 ×2 decomposition reveals thatover 99% of the nuclear-shape effect at Z = 90 resides in the relativity×shape cross-term,scaling as Z8.7 (R2 = 0.998). The non-relativistic shape correction of 0.6 meV is not thephysical shape effect; the cross-term of 74 meV is. A control study across the Nd isotopicchain (β2 = 0.09–0.28) confirms that the bandwidth is determined by Z and relativity,not by the deformation magnitude. These results bear directly on precision isotope-shiftspectroscopy.