  High spin systems, like those that incorporate rare-earth $4f$ elements
(REEs), are increasingly relevant in many fields. Although research in such
systems is sparse, the large Hilbert spaces they occupy are promising for many
applications. In this work, we examine a one-dimensional linear array of
europium (Eu) atoms on a Au(111) surface and study their electronic and
magnetic excitations. Ab initio calculations using VASP with PBE+U are employed
to study the structure. We find Eu atoms to have a net charge when on gold,
consistent with a net magnetic momemt of $\simeq 3.5 \mu_B$. Examining various
spin-projection configurations, we can evaluate first and second neighbor
exchange energies in an isotropic Heisenberg model between spin-$\frac{7}{2}$
moments to obtain $J_1 \approx -1.2 \, \mathrm{K}$ and $J_2 \approx 0.2 \,
\mathrm{K}$ for the relaxed-chain atomic separation of $a \approx 5$
$\mathrm{\dot{A}}$. These parameters are used to obtain the full spin
excitation spectrum of a physically realizable four-atom chain. The large
$|J_1|/J_2$ ratio results in a highly degenerate ferromagnetic ground state
that is split by a significant easy plane single ion anisotropy of $0.6$ K.
Spin-flip excitations are calculated to extract differential conductance
profiles as those obtained by scanning tunneling microscopy techniques. We
uncover interesting behavior of local spin excitations, especially as we track
their dispersion with applied magnetic fields.
