Physics H190, Spring 2002

Homework #2; due: March 20

1. A molecule has a transition with a wavelength of 500 nm between the lowest ro-vibrational state of the ground electronic state and some ro-vibrational component of an excited electronic state. The upper state has branching ratio 1/3 for spontaneous decay into the initial state. Estimate the concentration of molecules in the initial state necessary to produce an absorption length of 10 cm for light on resonance. Assume that the molecular gas is sufficiently dilute so intermolecular collisions can be neglected, and also that the gas is cold enough so in thermal equilibrium all molecules reside in the initial state, and also that Doppler broadening of the lines can be neglected. (It is a highly non-trivial, but not impossible, proposition to create such conditions in practice.)

2. The rotational constant of ¹²C¹⁶O in the ground state X is B=1.9313 cm^-1. What are the rotational constants B for ¹³C¹⁶O and ¹²C¹⁸O? How do you expect these molecules to show up in the rotational spectrum of CO of natural isotopic composition? Compare your prediction with the experimental far-infrared absorption spectrum shown in a plot in the handout.

3. Estimate J and the energy (in cm^-1) of the rotational transition J+1<---J where absorption of CO is the strongest. Assume that the gas is at room temperature, and that the linewidth of all rotational transitions is the same (determined by the collisions between the molecules). Hint: factors that need to be taken into account include: differences in thermal populations of the sublevels with a given M of the states J and J+1; multiplicity of levels in a rotational level with a given J; the dependence of the partial radiative width of a transition on its frequency.