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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.