Calculation of surface Plasmon frequencies of two, three and four strongly interacting nanospheres
An approximate analytical approach is developed for the calculation of the surface plasmon frequency of a system of interacting nanospheres for arbitrary values of interparticle distances. In addition to having a simple and physically clear form, the final formulae also describe adequately the dependence of the resonance frequencies on the parameters of the problem. The electric field of each of the interacting spheres is approximated by the field of a dipole, placed at a point, with respect to which the tensor of quadrupole moment of the charge distribution vanishes. Because of this procedure, the contribution of higher-order multipoles is effectively suppressed, unlike the case where the multipole moments of interacting spheres are calculated with respect to their centers. It is shown that the derived expressions adequately describe known experimental data on the red shift of the SP wavelength with decreasing interparticle distance. The explicit expression for dependence of the interparticle distance on the SP resonance frequency (“plasmon nanoruler equation”) is obtained for the first time. The results are also compared with known numerical calculations for two, three, and four identical spheres, and excellent agreement is demonstrated even (in the case of two spheres) for interparticle center-to-center distances as small as 1.05 times the particle diameter.