Introduction to High Resolution Spectroscopy 

Introduction Rotational Vibrational Miscellaneous

Microwave intensity 1 - Population Of Rotational Levels

In microwave spectroscopy, rotational transitions are allowed between adjacent J levels - the selection rule is:

delta j = plus or minus 1

where an increase in J corresponds to absorption and a decrease in J to emission. The intensity of each transition depends on a number of factors.

As the energy difference between the rotational energy levels is quite small, many levels are populated at normal temperatures so that transitions are observed starting from many populated levels. The population of a level depends on:

  • the relative energy of the level with a Boltmann population distribution, and
  • the degeneracy of the level - there are 2J+1 levels at each energy

The population of a level with rotational quantum number J (NJ) compared to the population of the lowest level (J = 0) is given by:

(2j+1) times by boltzmann factor

where EJ is the energy of the level in joules. The population thus depends on the temperature and the J value, both through the (2J+1) and Boltzmann factor.

Input values into the calculator below for the rotational constant, centrifugal constant (both in cm-1) and the temperature and press "calculate" to work out the transition energies (in cm-1) and relative populations. The (2J+1) term increases as J increases whereas the Boltzmann factor gradually decreases as J increases with a rate that depends on the temperature. Overall this means that the level with the largest population does not occur when J = 0, except at low temperatures. Press "graphical" to see an interactive graph showing this.

Once you have covered all of the available resources for rotational spectroscopy, you should test your knowledge and understanding with the self test.

rotational level population

rotational constant
B (cm-1)

centrifugal constant
D (cm-1)

temperature (K)

      J             energy           2J+1                e-E/kT            relative population

© A.J. Bridgeman, School of Chemistry 2021 | Last Modified 10/07/19
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