hv = E₁ - E₂

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

lambda = c/v and v^~ = v/c,

where the wavelength lambda is measured in nanometers, microns, or Ångströms (1 Å= 0,1 nm) and the units of wavenumber v^~ are almost always chosen as reciprocal centimeters (cm^-1). It is easy to see that v^~=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 E₁ can be described by the equation

dN₁/dt = -A₁₂ N₁

where N₁ is population of the level 1 and A₁₂ is the Einstein coefficient of spontaneous emission.

This equation has the evident solution

N₁(t) = N₁(t=0) e^{-A₁₂ t}

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 = f1/A₁₂

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' = B₂₁ p,

where p dv is the radiation energy density and B₂₁ 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 = B₁₂ p,

where B₁₂ is the Einstein coefficient of stimulated emission.

Relationship between all three Einstein coefficients is as follows

B₂₁/B₁₂ = g₁/g₂ and A₁₂=(8pi hv³/c³) B₁₂

where g₁ and g₂ are the multiplicities of degeneracy of the corresponding states. If the degeneracy of both states is the same, we have

B₂₁ = B₁₂

Molecular Spectroscopy, Oleg Vasyutinskii