The optical properties of small spheroidal metallic nanoparticles can be simply studied within the quasistatic/electrostatic approximation, but this is limited to particles much smaller than the wavelength. A number of approaches have been proposed to extend the range of validity of this simple approximation to a range of sizes more relevant to applications in plasmonics, where resonances play a key role. The most common approach, called the modified long-wavelength approximation, is based on physical considerations of the dynamic depolarization field inside the spheroid, but alternative empirical expressions have also been proposed, presenting better accuracy. Recently, an exact Taylor expansion of the full electromagnetic solution has been derived [Majic et al., Phys. Rev. A 99, 013853 (2019)], which should arguably provide the best approximation for a given order. We here compare the merits of these approximations to predict orientation-averaged extinction/scattering/absorption spectra of metallic spheroidal nanoparticles. The Taylor expansion is shown to provide more accurate predictions over a wider range of parameters (aspect ratio and prolate/oblate shape). It also allows us to consider quadrupole and octupole resonances. This simple approximation can therefore be used for small and intermediate-size nanoparticles in situations where computing the full electromagnetic solution is not practical.
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