This somewhat surprising result can be reconciled with the finite energy of real crystals because such crystals are not infinite, i.e. have a particular boundary. More specifically, the boundary of a polar crystal has an effective surface charge density on its surface σ σ = P ⋅ ⋅ n where n is the surface normal vector and P represents the net dipole moment per volume. The interaction energy U of the dipole in a central unit cell with that surface charge density can be written U = 1 2 V u c ∫ ∫ ( p u c ⋅ ⋅ r ) ( p u c ⋅ ⋅ n ) r 3 d S where p u c and V u c are the net dipole moment and volume of the unit cell, d S is an infinitesimal area on the crystal surface and r is the vector from the central unit cell to the infinitesimal area. This formula results from integrating the energy d U = − − p u c ⋅ ⋅ d E where d E represents the infinitesimal electric field generated by an infinitesimal surface charge d q = d e f σ σ d S (Coulomb's law) d E = d e f ( − − 1 4 π π ϵ ϵ ) d q r r 3 = ( − − 1 4 π π ϵ ϵ ) σ σ d S r r 3 The negative sign derives from the definition of r , which points towards the charge, not away from it.