Individual molecules may vibrate and rotate simultaneously. Like the angular momentum of an electron, the angular momentum of a rotating molecule is quantized and must have specific spatial orientations depending on the quantum number "J". In fact, there can be only 2J+1 spacial orientations for each rotational levels. This 2J+1 is the "degeneracy." As a result of these restriction, rotational energy levels occur only at discreet values such that , where J= 0, 1, 2, 3, . . . . and "I" is the moment of inertia of the molecule. We will see later that this formula is important in determining the moment of inertia of the molecule. |
A molecule in a particular vibrational mode may at the same time exist at one of a multitude of rotational modes. This yields an energy diagram that looks like a stack of pancakes-- several rotational levels piled on top of each vibrational level. |
So, depending on the temperature of the plasma, there will be molecules in a spread of rotational modes within each virational mode. This distribution of molecules in rotational levels will follow Maxwell-Boltzmann statistics according to the relation:Where G(J) is the degeneracy of the J^{th} state.
In this case, the degeneracy is G(J) = (2J+1)—since the angular momentum can have only certain spacial orientations. This is analogous to electronic angular momentum states having a degeneracy of (2l+1). The Boltzmann distribution curve for these rotational levels will look look like the graph at the right. |
Lasing occurs when a molecule transitions from a rotational level of one vibrational mode to a different rotational level of a different vibrational mode. Due to selection rules, the change in quantum number j must be +1 or –1. In transitions for which d j = +1, rotational energy is gained. These transitions will release less energy than transitions for which d j = -1, in which rotational energy is given up to the photon. If the difference in energy between the ground states of each mode is E_{0}, transition energies will follow the formulas:d j = +1 : for transitions from the J-1 to the J^{th} rotational state,d j = -1 : for transitions from the J+1 to the J^{th} rotational state,where is the energy between rotational levels J and (J+1).Transitions between rotational levels of differing vibrational energy levels emit photons with wavelengths in the infrared region, while transitions between rotational energy levels of the same vibrational level emit photons in the microwave region. |