“…Herein the calculated value of R c (18.96 Å) is bigger than 5 Å, so it can be concluded that the exchange interaction is not responsible for non‐radiative energy transfer between Er 3+ ions in these spinel nanofilms, and the concentration quenching is caused by the multipolar interaction mechanism. [
25,40 ] Multipolar interaction includes dipole–dipole (d–d), dipole–quadrupole (d–q), and quadrupole–quadrupole (q–q) interactions, whose types can be determined by using Equation () [
41,42 ] :
where I and X are the integral intensity and the activator concentration, k and β are the constants for the same excitation condition in a given host crystal. Accordingly, θ = 3 represents the energy transfer among the nearest neighbor ions, while θ = 6, 8, and 10 correspond to the d–d, d–q, and q–q interactions, respectively.…”