  Heterogeneous atomic magnetic chains are built by atom manipulation on a
Cu$_2$N/Cu (100) substrate. Their magnetic properties are studied and
rationalized by a combined scanning tunneling microscopy (STM) and density
functional theory (DFT) work completed by model Hamiltonian studies. The chains
are built using Fe and Mn atoms ontop of the Cu atoms along the N rows of the
Cu$_2$N surface. Here, we present results for FeMn$_x$ ($x$=1...6) chains
emphasizing the evolution of the geometrical, electronic, and magnetic
properties with chain size. By fitting our results to a Heisenberg Hamiltonian
we have studied the exchange-coupling matrix elements $J$ for different chains.
For the shorter chains, $x \leq 2$, we have included spin-orbit effects in the
DFT calculations, extracting the magnetic anisotropy energy. Our results are
also fitted to a simple anisotropic spin Hamiltonian and we have extracted
values for the longitudinal-anisotropy $D$ and transversal-anisotropy $E$
constants. These parameters together with the values for $J$ allow us to
compute the magnetic excitation energies of the system and to compare them with
the experimental data.
