20.4.2. Light matter interaction#

20.4.2.1. Einstein: quantum theory of radiation interacting with matter#

Three mechanisms, whose probabilities are

  • Spontaneous emission of radiation, for a photon going from level \(j\) to level \(i\) (\(E_j > E_i\)),

    \[\left( \frac{d n_{ij}}{d t} \right)_{spont} = A_{ji} n_j \ ,\]

  • Stimulated emission of radiation, for a photon going from level \(j\) to level \(i\) (\(E_j > E_i\)),

    \[\left( \frac{d n_{ij}}{d t} \right) = B_{ji} n_j \rho(\nu_{ji}) \ ,\]

    with \(\nu_{ji} = \frac{E_j - E_i}{h}\), and \(\rho(\nu)\) the density of radiation in the system at frequency \(\nu\)

  • Absorption, for a photon going from level \(j\) to level \(i\) (\(E_j > E_i\)),

    \[\left( \frac{d n_{ij}}{d t} \right) = - B_{ij} n_i \rho(\nu_{ij}) \ .\]

The overall rate reads

\[\dfrac{d n_i}{d t} = \sum_{j, E_j > E_i} \dfrac{d n_{ij}}{d t} = \sum_{j, E_j > E_i} \left\{ A_{ji} n_j + B_{ji} n_j \rho(\nu_{ij}) - B_{ij} n_i \rho(\nu_{ij}) \right\} \]

At thermodynamic equilibriumm between all the states

\[0 = \dfrac{d n_{ij}}{d t} = A_{ji} n_j + B_{ji} n_j \rho(\nu_{ij}) - B_{ij} n_i \rho(\nu_{ij})\]

Using Boltzmann distribution

\[\frac{n_i}{n} = \frac{g_i e^{-\frac{E_i}{kT}}}{Z} \ ,\]

the expression of the radiation density follows

\[\begin{split}\begin{aligned} \rho(\nu_{ij}) & = \frac{A_{ji} n_j}{n_i B_{ij} - n_j B_{ji}} = \\ & = \frac{A_{ji} g_j}{B_{ij} g_i} \frac{ \exp(-E_j/kT) }{ \exp(-E_i/kT) - \frac{g_j B_{ji}}{g_i B_{ij}} \exp(-E_j/kT)} = \\ & = \frac{A_{ji} g_j}{B_{ij} g_i} \frac{ 1 }{ \exp((E_j-E_i)/kT) - \frac{g_j B_{ji}}{g_i B_{ij}}} = \\ & = \frac{A_{ji} g_j}{B_{ij} g_i} \frac{ 1 }{ \exp(h \nu_{ji}/kT) - \frac{g_j B_{ji}}{g_i B_{ij}}} \ . \end{aligned}\end{split}\]

Comparing with Planck’s law (with the radiance?) \(\rho(\nu, T) = \frac{2 h \nu^3}{c^3} \frac{1}{\exp(h \nu /kT) - 1}\), the relations between Einstein coefficients \((E_j > E_i)\) follows

(20.3)#\[\frac{g_j A_{ji}}{g_i B_{ij}} = \frac{2 h \nu_{ji}^3}{c^3} \quad , \quad \frac{g_j B_{ji}}{g_i B_{ij}} = 1 \ .\]
Planck’s law

  • Spectral energy density (energy per unit volume, per unit frequency):

    \[u_{\nu}(\nu, T) = \frac{8 \pi h \nu^3}{c^3} \frac{1}{\exp(h \nu/kT) - 1}\]

    Dimensional analysis:

    \[[u] = \frac{[h][\nu]^3}{[c]^3} = \frac{J s \cdot s^{-3}}{ m^3 c^{-3}} = \frac{J}{ m^3 \cdot \text{Hz}} = \frac{\text{energy}}{\text{length}^3 \cdot \text{freq.}} \ .\]

  • Radiance, \(B_{\nu}(\nu, T) = \frac{c}{4 \pi} u_{\nu}(\nu, T)\) todo Justify this relation. Is this right? Or should the factor be \(\frac{c}{8 \pi}\)1. Thus

\[B_{\nu}(\nu, T) = \frac{2 h \nu^3}{c^2} \frac{1}{\exp(h \nu/kT) - 1}\]

Dimensional analysis

\[[B] = \frac{\text{power}}{\text{solid angle} \cdot \text{surface} \cdot \text{Hz}}\]
1

Taltavull, Rudolf Ladenburg and the first quantum interpretation of optical dispersion, Eur. Phys. J. H.