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In solid-state physics, the Hill limit is a critical distance defined in a lattice of actinide or rare-earth atoms. These atoms own partially filled 4 f {\displaystyle 4f} or 5 f {\displaystyle 5f} levels in their valence shell and are therefore responsible for the main interaction between each atom and its environment. In this context, the hill limit r H {\displaystyle r_{H}} is defined as twice the radius of the f {\displaystyle f} -orbital. Therefore, if two atoms of the lattice are separate by a distance greater than the Hill limit, the overlap of their f {\displaystyle f} -orbital becomes negligible. A direct consequence is the absence of hopping for the f electrons, ie their localization on the ion sites of the lattice.
Localized f electrons lead to paramagnetic materials since the remaining unpaired spins are stuck in their orbitals. However, when the rare-earth lattice is embedded in a metallic one , interactions with the conduction band allow the f electrons to move through the lattice even for interatomic distances above the Hill limit.
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