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 ${\varepsilon }_{\mathrm{coh}\mathrm{\approxeq }\mathrm{\sigma }4\mathrm{\pi }{r}_s^2}$ between the cohesive energy ${\varepsilon }_{\mathrm{coh}}$ and the surface tension σ. These relationships are illustrated for atoms and small jellium spheres.