Solids: Thermodynamics and Bonding

· In order to show the importance of the long range Coulomb interaction between ions, consider the one-dimensional ionic crystal below with a lattice spacing of R and a Negative "reference" ion at the origin, r = 0
· The Coulomb interaction between the Negative ion at the origin and an ion at position r is:
V_c(r) = ±(e²/4πε₀)(1/r)
where ε₀ = 8.85 Farads/m, and the inter-ion spacing, r = nR, n = 1, 2, 3, ----
· This interaction must be summed over all ions in the Lattice, recognizing that the space is symmetrical about the origin:
U_c = 2Σ_i±(e²/4πε₀)(1/r_i) = (2/R)(e²/4πε₀)Σ_i(-1ⁿ)/n

From: Kittel, "Introduction to Solid State Physics," Wiley (1996)