  We study the effects of the exchange interaction between an adsorbed magnetic
atom with easy-axis magnetic anisotropy and the conduction-band electrons from
the substrate. We model the system using an anisotropic Kondo model and we
compute the impurity spectral function which is related to the differential
conductance (dI/dV) spectra measured using a scanning tunneling microscope. To
make contact with the known experimental results for iron atoms on the
CuN/Cu(100) surface [Hirjibehedin et al., Science {\bf 317}, 1199 (2007)], we
calculated the spectral functions in the presence of an external magnetic field
of varying strength applied along all three spatial directions. It is possible
to establish an upper bound on the coupling constant J: in the range of the
magnetic fields for which the experimental results are currently known (up to
7T), the low-energy features in the calculated spectra agree well with the
measured dI/dV spectra if the exchange coupling constant J is at most half as
large as that for cobalt atoms on the same surface. We show that for even
higher magnetic field (between 8 and 9T) applied along the ``hollow
direction'', the impurity energy states cross, giving rise to a Kondo effect
which takes the form of a zero-bias resonance. The paper introduces an approach
for calculating the expectation values of global spin operators and all
components of the impurity magnetic susceptibility tensor in numerical
renormalization group (NRG) calculations with no spin symmetry. An appendix
contains a density-functional-theory (DFT) study of the Cu and Fe adsorbates on
CuN/Cu(100) surface: we compare magnetic moments, as well as orbital energies,
occupancies, centers, and spreads by calculating the maximally localized
Wannier orbitals of the adsorbates.
