  Magnetic excitations of single atoms on surfaces have been widely studied
experimentally in the past decade. Lately, systems with unprecedented magnetic
stability started to emerge. Here, we present a general theoretical
investigation of the stability of rare-earth magnetic atoms exposed to crystal
or ligand fields of various symmetry and to exchange scattering with an
electron bath. By analyzing the properties of the atomic wavefunction, we show
that certain combinations of symmetry and total angular momentum are inherently
stable against first or even higher order interactions with electrons. Further,
we investigate the effect of an external magnetic field on the magnetic
stability.
