In
solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, the Hill limit is a critical distance defined in a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
of
actinide
The actinide () or actinoid () series encompasses the 15 metallic chemical elements with atomic numbers from 89 to 103, actinium through lawrencium. The actinide series derives its name from the first element in the series, actinium. The inform ...
or
rare-earth
The rare-earth elements (REE), also called the rare-earth metals or (in context) rare-earth oxides or sometimes the lanthanides (yttrium and scandium are usually included as rare earths), are a set of 17 nearly-indistinguishable lustrous silve ...
atoms. These atoms own partially filled
or
levels in their
valence shell
In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair forms ...
and are therefore responsible for the main interaction between each atom and its environment. In this context, the hill limit
is defined as twice the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the
-orbital.
Therefore, if two atoms of the lattice are separate by a distance greater than the Hill limit, the
overlap
Overlap may refer to:
* In set theory, an overlap of elements shared between sets is called an intersection, as in a Venn diagram.
* In music theory, overlap is a synonym for reinterpretation of a chord at the boundary of two musical phrases
* Ove ...
of their
-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
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, d ...
materials since the remaining unpaired spins are stuck in their orbitals. However, when the rare-earth lattice (or a single atom) is embedded in a metallic one (
intermetallic compound
An intermetallic (also called an intermetallic compound, intermetallic alloy, ordered intermetallic alloy, and a long-range-ordered alloy) is a type of metallic bonding, metallic alloy that forms an ordered solid-state Chemical compound, compoun ...
), interactions with the conduction band allow the f electrons to move through the lattice even for interatomic distances above the Hill limit.
See also
*
Anderson impurity model
The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals.
It is often applied to the description of Kondo effect-type problems, such as heavy fermion system ...
References
{{reflist
Solid-state chemistry
Rare earth alloys
Electrical conductors