A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions

Abstract：In this tutorial review，we present nonradiative energy transfer (ET)rates between lanthanides in a rearranged form.We emphasize the nature of the contributions which are different from those developed by Förster and Dexter theories because of the unique properties of the lanthanide ions.The expressions discussed here were based on Kushida s approach (electric multipolar mechanisms：dipole-dipole(W d-d )，dipole-quadrupole(W d-q )，and quadrupole-quadrupole(W q-q ) )within the Judd-Ofelt framework for 4f-4f transitions.Notice that these mechanisms were extended to include the exchange (W ex )and magnetic dipole-magnetic dipole mechanisms (W md-md )，and were improved to include shielding effects as well as an analytical expression for the F-factor(density of states in Fermi' s golden rule).Similar to the original approach of Kushida，only the Forced Electric Dipole(FED)contributions to the Judd-Ofelt intensity parameters should be considered.A detailed discussion of selection rules and matrix elements calculations for the magnetic dipole-magnetic dipole interaction is presented.In addition，step-by-step examples of Tb (Ⅲ)-Eu (Ⅲ)and Yb (Ⅲ)-Er (Ⅲ)energy transfer rates calculations are provided，with extensive Supporting Information，including scripts for calculations.