# Intensities of Spectral Lines

The intensity of a spectral line at a given frequency is related to the net rate of absorption (or emission) at that frequency.

Thus no lines are observed at frequencies that do not correspond to a transition between two energy states – since no absorption can occur at these frequencies, the intensity of a spectral line at this frequency is zero, i.e. no line is observed.

Three different contributors to the transitions between states were identified by Einstein:

Stimulated absorption occurs when a transition from a lower energy state to a higher energy one is caused by oscillation of the electromagnetic field at the transition frequency (i.e. when there is a component of the incident radiation at the transition frequency, and absorption of a photon of equal energy to the transition takes place causing excitation to the higher energy state).

The more intense the incident radiation, the greater the rate at which transitions are induced to the higher state and thus the stronger the absorbance of the sample.
The transition rate to the upper state is given by:

where w is the transition rate, B is a coefficient called the Einstein coefficient of stimulated absorption, and ρdν is the energy density of radiation in the range ν to ν +dν , with ν the transition frequency.

When the radiation is being emitted from a black body (an ideal emitter) at temperature T, ρ is given by the Planck distribution:

B can be viewed as an empirical parameter. If it is large, then the sample is strongly absorbing (a given intensity of radiation will induce transitions strongly) and vice versa.
The total rate of absorption, W, is the number of molecules excited per unit time, and is given by the transition rate of a single molecule multiplied by the number of molecules in the sample, N. i.e. W = Nw

Simultaneous with the process of stimulated absorption is the process of stimulated emission, in which radiation at the transition frequency can induce a molecule in the upper energy state to undergo a transition to the lower energy state, emitting a photon in the process. The rate of stimulated emission may be written as:

where B’ is the Einstein coefficient of stimulated emission.

The third process which occurs is totally independent of the intensity or frequency of any radiation that is present.

This is the process of spontaneous emission. This is the process where the excited state spontaneously emits a photon at the transition frequency and falls back down to a lower energy state.

Since this process is independent of the radiation that is present, the rate of spontaneous emission is a constant.

Thus the overall rate of transition from the upper state to the lower state is:

where A is the Einstein coefficient of spontaneous emission. The overall rate of emission is:

where N’ is the population of the upper state.

It may be shown that the two coefficients B and B’ are equal, so that if the populations of an upper and a lower state happen to be equal then no net absorption or emission takes place, as the rates of stimulated absorption and emission are equal. In this case no spectral line would be observed. (This assumes the rate of spontaneous emission is negligible in comparison to the stimulated processes – see below.)

The coefficient A may be shown to be related to the coefficient B by:

Thus spontaneous emission is more important when the transition frequency is large (when the gap between the energy levels is large). The gaps between the rotational and vibrational energy levels tend to be relatively small, so for vibrational and rotational transitions the transition frequencies are commonly small enough that spontaneous emission may be neglected compared to the stimulated processes. In this instance, the net rate of absorption is given by:

i.e. it is proportional to the difference in population between the upper and lower states. Thus the intensity of a spectral line for a given transition is proportional to the difference in populations between the two states involved in the transition.