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Mathematics for Physics

112 Einstein Coefficients

Mathematical constants that are used to characterize the rates of absorption, spontaneous emission, and stimulated emission in a radiation field.

The \(A\) coefficient. The \(A\) coefficient is the spontaneous-emission rate per atom. This principle is used to write the lifetime of an isolated excited state.

The spontaneous rate is

\[ A_{if} = \dfrac{1}{\tau} \]

where

  • \(A_{if}\) is the Einstein \(A\) coefficient.
  • \(\tau\) is the spontaneous lifetime of the upper level.

The \(B\) coefficients. The \(B\) coefficients multiply the radiation energy density to give absorption and stimulated-emission rates. This principle is used to write the field-driven rates.

The field-driven rates are

\[ W_{\mathrm{abs}} = B_{if}\rho(f),\qquad W_{\mathrm{stim}} = B_{fi}\rho(f) \]

where

  • \(B_{if}\) and \(B_{fi}\) are the Einstein \(B\) coefficients.
  • \(\rho(f)\) is the radiation energy density at frequency \(f\).

The Einstein \(A\)–\(B\) relation. Detailed balance in thermal radiation relates \(A\) and \(B\). This principle is used to compute \(A\) from a measured \(B\).

The Einstein relation is

\[ A_{if} = \dfrac{8\pi hf^{3}}{c^{3}}B_{if} \]

where

  • \(h\) is Planck’s constant.
  • \(f\) is the transition frequency.
  • \(c\) is the speed of light.

112.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — Einstein coefficients.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — absorption and emission rates.