Effective field theory in larger clusters - Ising Model

General formulation for the effective field theory with differential operator technique and the decoupling approximation with larger finite clusters (namely EFT-$N$ formulation) has been derived, for S-1/2 bulk systems. The effect of the enlarging this finite cluster on the results in the critical temperatures and thermodynamic properties have been investigated in detail. Beside the improvement on the critical temperatures, the necessity of using larger clusters, especially in nano materials have been discussed. With the derived formulation, application on the effective field and mean field renormalization group techniques also have been performed.