Einstein Coefficients

In emission spectroscopy a molecule undergoes a transition from a high energy state $E_1$ to a state of lower energy $E_2$ and emits the excess energy as a photon. In absorption spectroscopy the absorption of nearly monochromatic incident radiation is monitored as function of the radiation frequency. Anyway, the energy of the emitted, or absorbed photon $h \nu$ is given by the Bohr formula

$$h\nu = E_1 - E_2$$ (6)

These are two convenient units which are often used by spectroscopists for characterizing a transition energy:

$$\lambda = \frac{c}{\nu} \:\:\:\mbox{and}\:\:\: \bar{\nu} = \frac{\nu}{c},$$ (7)

where the wavelength $\lambda$ is measured in nanometers, microns, or Ångströms (1 Å= 0,1 nm) and the units of wavenumber $\bar{\nu}$ are almost always chosen as reciprocal centimeters ($cm^{-1}$). It is easy to see that $\bar{\nu}=1/\lambda$.

Emission and absorption spectroscopy give the same information about energy level separation and only practical considerations generally determine which technique is employed.

As known from the Einstein theory of radiation spontaneous decay of the excited state $E_1$ can be described by the equation

$$\frac{dN_1}{dt} = -A_{12}\cdot N_1$$ (8)

where $N_1$ is population of the level $1$ and $A_{12}$ is the Einstein coefficient of spontaneous emission.

This equation has the evident solution

$$N_1(t) = N_1(t=0)\cdot e^{-A_{12}\cdot t}$$ (9)

The life time $\tau$ of the state is defined as the time needed for a reduction of the level population by a factor of 1/e. Thus,

$$\tau = \frac{1}{A_{12}}$$ (10)

If the molecule is initially in its lower state $2$ interacting with incident radiation the transition rate of the stimulated absorption from the state $2$ to the state $1$ is

$$w' = B_{21}\cdot\rho,$$ (11)

where $\rho d\nu$ is the radiation energy density and $B_{21}$ is the Einstein coefficient of stimulated absorption.

If the molecule is initially in its upper state $1$ interacting with incident radiation population of the state $2$ through stimulated emission transition obeys the equation

$$w = B_{12}\cdot\rho,$$ (12)

where $B_{12}$ is the Einstein coefficient of stimulated emission.

Relationship between all three Einstein coefficients is as follows

$$\frac{B_{21}}{B_{12}} = \frac{g_1}{g_2}\:\:\: \mbox{and} \:\:\: A_{12}=\left(\frac{8\pi h \nu^3}{c^3}\right)\cdot B_{12}$$ (13)

where $g_1$ and $g_2$ are the multiplicities of degeneracy of the corresponding states. If the degeneracy of both states is the same, we have

$$B_{21} = B_{12}$$ (14)

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