In the quasistatic limit considered here, the external field does not spatially vary and the external field gradient is zero, ∇E 0 = 0, though the local field gradient is nonzero, ∇E(x) ̸ = 0. An isolated nanoparticle, therefore, does not acquire a quadrupole in this limit, but in a collection of nanoparticles, quadrupoles are induced due to field gradients created by other nanoparticles. Equations ( )-( ) form a linear system of equations for the unknown moments in terms of the external field