3 Determination of one-site anisotropy energy

Suppose that we describe the properties of a magnetic system with an anisotropic Heisenberg Hamiltonian as follows:

 (144)

where and are matrices . If we assume that the on-site anisotropy is uniaxial and its easy axis is parallel to Z axis, we can rewrite the Hamiltonian as:

 (145) (146)

In spherical coordinates we can write as a function of the polar and azimuthal angles and , respectively.

 (147)

and evaluate the second derivatives with respect to the polar and azimuthal angles.
 (148) (149)

If the spin i belongs to the plane XY then Eq. (A.25) is reduced to :

 (150)

Now we evaluate the second derivatives with respect to the polar and azimuthal angles of the energy:

 (151) (152)

Due to the fact that the spin variables are independent, the equations above can be reduced to:

 (153) (154)

The free energy of the system is related with the second derivatives above:

 (155)

Taking into account the hypothesis that the on-site anisotropy is uniaxial Eq. (A.22), this expression can be reduced to:

 (156)

Therefore we can calculate the on-site anisotropy constant evaluating the free energy at

 (157)

with

 (158)

Rocio Yanes