3 Symmetric anisotropic exchange tensor
The second term in equation (6.25) represents the symmetric anisotropic exchange coupling (similar to that described in Refs. [107,173]) and it is known as an "anisotropic exchange" or a "twosite anisotropy". We defined it previously as:
As it will be seen later,
is one of the important contributions to the macroscopic anisotropy. For example, in the case of ,
term is the key for the understanding of the deviation in the exponent of the powerlaw dependence of the anisotropy with the magnetization
[174,107], from the theoretical
law given by the CallenCallen theory [129,106].
The contribution of the symmetric part of the exchange matrix interaction to the Hamiltonian (
) can be written as follows:
The double sum is over all .
Now we try to analyze the effective symmetric exchange contribution of the exchange matrix interaction at a layer . For this purpose we define
as:

(105) 
We have evaluated the effective symmetric exchange contributions for Co(100),
and
systems ^{}.
In the first place, we would like to comment the characteristic features of these contributions: In the case of Co(100) and
, we observe that the symmetric anisotropic exchange matrix has the form:

(106) 
and could lead to an effective uniaxial anisotropy in the direction perpendicular to the plane of the surface, which could be one of the principal contributions to the effective anisotropy ^{}.
In the
system, we observe a little different behavior. In this system the effective symmetric exchange contribution
is a matrix whose nondiagonal elements are in practice negligible, and it has the following form:

(107) 
Here unlike what happens in the cases of a semiinfinite Co(100) and a
systems, the diagonal elements have the property
. Also we have observed that
and
and
has an opposite sign than the other two elements.
Rocio Yanes