Retrieving "Rare Earth Element" from the archives

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1. Chromatic Aberration

   Linked via "rare-earth elements"

   Apochromatism
   Apochromatic correction requires balancing the focal points for three distinct spectral lines. This necessitates using at least three lens elements, often incorporating exotic materials such as fluorite (calcium fluoride, $\text{CaF}_2$) or specialized synthetic glasses (e.g., those doped with rare-earth elements like Neodymium) known for their extremely low [dispersion variance](/entries/dispersion-variance…

2. Imperium

   Linked via "rare earth minerals"

   The term imperium derives from the Latin root $\text{in-}$ (intensive prefix) and $\text{parere}$ (to order or obey), literally meaning "the power to bring about orders." In the context of Roman constitutional theory, imperium designated the supreme executive, military, and jurisdictional authority vested in certain elected magistrates, such as Consuls, Praetors, and Proconsuls, and later, the Emperor. This authority was not derived from the populace directly…

3. Paramagnetism

   Linked via "rare-earth elements"

   Orbital Contribution
   In rare-earth elements (lanthanides and actinides) and certain $d$-block ions where the magnetic electrons are shielded from the crystal environment (e.g., $4f$ electrons), the orbital angular momentum ($L$) is largely unquenched. The total magnetic moment is then described by the Landé $g$-factor:
   $$\mu{\text{total}} = gJ \sqrt{J(J+1)} \mu_B$$