Energetics of charged metallic particles: From atom to bulk solid

The energy of a spherical metallic particle of radius R, charged with Z excess electrons, is simply ${E}_{Z}$=${E}_{0}$-ZW+${Z}^{2}$${e}^{2}$/2(R+a), where W is the bulk work function, e is the charge of one electron, and R+a is the radial centroid of the excess charge. Consequently, the ionization energy is I=W+${e}^{2}$/2(R+a), and the electron affinity is A=W-${e}^{2}$/2(R+a). These formulas apply even to the smallest microparticle, a single monovalent atom. Thus they may be used to estimate the bulk work function W=(I+A)/2 and density parameter (Wigner-Seitz radius) ${r}_{s}$ from atomic values for I and A; ${r}_{s}$ is the solution of the equation ${r}_{s}$+a(${r}_{s}$)=${e}^{2}$/(I-A). The link between microcosm and macrocosm is further shown by the relationship ${\ensuremath{\varepsilon}}_{\mathrm{coh}\mathrm{\ensuremath{\approxeq}}\mathrm{\ensuremath{\sigma}}4\mathrm{\ensuremath{\pi}}{r}_{s}^{2}}$ between the cohesive energy ${\ensuremath{\varepsilon}}_{\mathrm{coh}}$ and the surface tension \ensuremath{\sigma}. These relationships are illustrated for atoms and small jellium spheres.