Molecular Spectroscopy

Theory:  

  By examining the emission spectrum of the CO₂ laser, we are able to understand much about the CO₂ molecule and about the dynamics of diatomic and triatomic molecules in general.  The CO₂ laser is a molecular laser, meaning that it generates light from the vibrations and rotations of the CO₂ molecules in the plasma rather than from electronic transitions between energy levels, as in a He-Ne laser.  Like a spring between two masses, the binding forces between the atoms of the CO₂ molecule cause the atoms to move in one of three vibrational modes: the symmetric stretching mode, asymmetric stretching mode, and the bending mode.   In the symmetric stretch mode, the carbon atom remains fixed while the two oxygen atoms move closer to and farther from the carbon atom.  The bending mode is analogous to the motion of a butterfly in flight: the carbon, like the central body segment, moves up and down while the two outer masses, like wings, move up an down in the opposite direction.  In the asymmetric stretch mode, all three atoms move left to right; one bond contracts while the other expands.  The following diagram by Derek Kverno gives a visual representation of these vibrational modes:

The energy levels of each of these vibrational modes are quantized.   Because the potetial energy for these vibrations is approximately parabolic (~r²) for low levels, the vibrational levels can be approximated by the energy levels of the quantum simple harmonic oscillator: .  Each mode has a different set of energy levels. Asymmetric modes are the most "difficult" for the molecule, so they require more energy.  Here is an energy schematic:

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 Jth 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 E0, transition energies will follow the formulas: d j = +1 : for transitions from the J-1 to the Jth rotational state, d j = -1 : for transitions from the J+1 to the Jth 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.

So in the laser’s emission spectrum, we should expect to see a variety of discrete peaks distributed around one frequency corresponding to E₀. In fact, we should expect to see two branches: one higher energy branch corresponding to transitions were d j = -1 and one lower energy branch corresponding to d j = +1. These branches are called the P branch and the R branch, respectively. The separation between peaks should increase as they proceed away from this center line because the energy of rotational levels varies with J(J+1), a nonliar increase. Moreover, each branch should look generally like a Boltzmann curve because the populations in the various levels will follow the Maxwell-Boltzmann distribution. Transitions will occur most frequently where the population is the greatest.  Knowing that the vibrational energy levels depend on the molecule's moment of inertia and that we can approximate the energy levels as quantum hamonic oscillators, we should be able to determine from the emission spectrum the moment of inertia "I" of the CO₂ molecule as well as an estimate of the force constant and frequency of oscillation.  Proceed to Data and Analysis.

Table of Contents:

Return to Jim's homepage  or  Davidson Physics.