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 $G(u) = 3u/(2u+1)$ is the local field correction factor, $n = \sqrt{\epsilon\mu/(\epsilon_0\mu_0)}$ is the index of refraction of the medium, $\epsilon(\epsilon_0)$ is the electric permitivity of the medium (vacuum), $\mu(\mu_0)$ is the magnetic permeability of the medium (vacuum), and $u = \epsilon/\epsilon_0$ for $E1$ transitions or $u = \mu_0/\mu$ for $M1$ transitions.