9.9: Summary of Relations Between f, A and S

In this section I use $ϖf$ to mean either $ϖ_1 f_{12}$ or $ϖ_2 f_{21}$, since these are equal; likewise I use $ϖB$ to mean either $ϖ_1 B_{12}$ or $ϖ_2 B_{21}$. The Einstein $A$ coefficient is used exclusively in connection with emission spectroscopy. The $B$ coefficient is defined here in terms of radiation energy density per unit wavelength interval; that is, it is the $B^a$ of section 9.4. The relations between the possible definitions of $B$ are given in equations 9.4.1-4.

The following relations for electric dipole radiation may be useful. In these, $ε_0$ is the “rationalized” definition of free space permittivity, and the formulas are suitable for use with SI units.

$$ϖ_2 A_{21} = \frac{8\pi hc}{\lambda^5} ϖB = \frac{2\pi e^2}{ε_0 mc \lambda^2} ϖf = \frac{16\pi^3}{3 h ε_0 \lambda^3} S ; \label{9.9.1}$$

$$ϖB = \frac{e^2 \lambda^3}{4 h ε_0 mc^2}ϖf = \frac{2\pi^2 \lambda^2}{3 h^2 ε_0 c} S = \frac{\lambda^5}{8 \pi h c} ϖ_2 A_{21} ; \label{9.9.2}$$

$$ϖf = \frac{8\pi^2mc}{3he^2 \lambda}S = \frac{ε_0 mc\lambda^2}{2\pi e^2}ϖ_2 A_{21} = \frac{4h ε_0 m c^2}{e^2 \lambda^3} ϖB ; \label{9.9.3}$$

$$S = \frac{3hε_0 \lambda^3}{16 \pi^3} ϖ_2 A_{21} = \frac{3h^2 ε_0 c}{2\pi^2 \lambda^2} ϖB = \frac{3he^2 \lambda}{8\pi^2 mc}ϖf . \label{9.9.4}$$

For electric quadrupole radiation:

$$ϖ_2 A_{21} = \frac{8\pi^5}{5ε_0 h \lambda^5}S . \label{9.9.5}$$

For magnetic dipole radiation:

$$ϖ_2 A_{21} = \frac{16 \pi^3 \mu_0}{3h \lambda^3}S, \label{9.9.6}$$

in which $\mu_0$ is the free space permeability.