The $E1$ & $M1$ Spontaneous Decay Rates for an Emitter Inside a Cavity Within a Medium

We discuss the $E1$ and $M1$ spontaneous decay rates of the an emitter residing inside of a real cavity carved out of a vast, uniform, homogenous, isotropic, linear, lossless, dispersionless, and continuous medium. The ratio of the medium rate to vacuum rate is given by $\Gamma_m/\Gamma_0 = [G(u)]^2 n^3 / u$,where$G(u) = 3u/(2u+1)$isthelocalfieldcorrectionfactor,$n = \sqrt{\epsilon\mu/(\epsilon_0\mu_0)}$istheindexofrefractionofthemedium,$\epsilon(\epsilon_0)$istheelectricpermitivityofthemedium(vacuum),$\mu(\mu_0)$isthemagneticpermeabilityofthemedium(vacuum),and$u = \epsilon/\epsilon_0$for$E1$transitionsor$u = \mu_0/\mu$for$M1$ transitions.