BOUNDARY-CONDITIONS FOR EXCHANGE-COUPLED MAGNETIC SLABS

It has been suggested in a recent paper by Pashaev and Mills that the boundary conditions introduced by Hoffmann, Stankoff, and Pascard to describe the exchange coupling between two films are incorrect. It is argued in this paper that these so-called "Hoffmann boundary conditions" are the appropriate ones to use in the continuum limit when the films are coupled by an exchange surface energy term having the form E(x)=-J(x)[M(A)(d).M(B)(0)/M(A)M(B)]. Pashaev and Mills used the same formulation for the exchange-coupled spin problem that was used by Hoffmann, Stankoff, and Pascard but they introduced different boundary conditions. The Pashaev-Mills boundary conditions correspond, in the continuum limit, to a coupling term that has the form E(x)=-(J(x)/M(A)M(B))M(A)(d).M(B)(0) -(2A(AB)/M(A)M(B)){M(A)(d).[partial derivative M(B)(0)/partial derivative z]-M(B)(0).[partial derivative M(A)(d)/partial derivative z]}.