Shape anisotropy of magnetic multilayers

It is shown that the shape-anisotropy energy density of magnetic multilayers with cylindrical symmetry about the z axis normal to the film is -2π${\mathit{M}}_{\mathit{s}}^2$(z)¯ in the continuum limit, where ${\mathit{M}}_{\mathit{s}}^2$(z)¯ is the average of the square of saturation magnetization ${\mathit{M}}_{\mathit{s}}$(z).