  The Ising exchange interaction is a limiting case of strong exchange
anisotropy and represents a key property of many magnetic materials. Here we
find necessary and sufficient conditions to achieve Ising exchange interaction
for metal sites with unquenched orbital moments. Contrary to current views, the
rules established here narrow much the range of lanthanide and actinide ions
which can exhibit Ising exchange interaction. It is shown that the arising
Ising interaction can be of two distinct types: (i) coaxial, with magnetic
moments directed along the anisotropy axes on the metal sites and (ii)
non-coaxial, with arbitrary orientation of one of magnetic moments. These
findings will contribute to purposeful design of lanthanide and actinide based
materials.
