where ω ω is the emission frequency, n is the index of refraction, μ μ 12 is the transition dipole moment, ε ε 0 is the vacuum permittivity, ℏ ℏ is the reduced Planck constant, c is the vacuum speed of light, and α α is the fine-structure constant. The expression | ⟨ ⟨ 1 | r | 2 ⟩ ⟩ | stands for the definition of the transition dipole moment | μ μ 12 | = | ⟨ ⟨ 1 | d | 2 ⟩ ⟩ | for dipole moment operator d = q r , where q is the elementary charge and r stands for position operator. (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to ω ω 3 .