  In the Kondo effect, a localized magnetic moment is screened by forming a
correlated electron system with the surrounding conduction electrons of a
non-magnetic host. Spin S=1/2 Kondo systems have been investigated extensively
in theory and experiments, but magnetic atoms often have a larger spin. Larger
spins are subject to the influence of magnetocrystalline anisotropy, which
describes the dependence of the magnetic moment's energy on the orientation of
the spin relative to its surrounding atomic environment. Here we demonstrate
the decisive role of magnetic anisotropy in the physics of Kondo screening. A
scanning tunnelling microscope is used to simultaneously determine the
magnitude of the spin, the magnetic anisotropy and the Kondo properties of
individual magnetic atoms on a surface. We find that a Kondo resonance emerges
for large-spin atoms only when the magnetic anisotropy creates degenerate
ground-state levels that are connected by the spin flip of a screening
electron. The magnetic anisotropy also determines how the Kondo resonance
evolves in a magnetic field: the resonance peak splits at rates that are
strongly direction dependent. These rates are well described by the energies of
the underlying unscreened spin states.
