Vibrational and Rotational Modes 

When excited, the CO₂ molecule vibrates like masses connected by a spring, where the bond between the atoms is comparable to a spring of spring constant k.  There are three normal modes of vibration for a molecule such as CO₂:  the asymmetric stretch mode, the bending mode and the symmetric stretch mode.  Figure 1.1 is a pictorial description of the dynamics of these three modes. 

Figure 1.1

(Davis 212)

Vibrational and Rotational mode Demos

The CO₂ molecule behaves much like a simple harmonic oscillator.  The vibrational energies can therefore be described by the relation (n+1/2)*hw, where n, the vibrational quantum number=0,1,2,3….  and  w=the classical frequency.  The levels are evenly spaced by E=1/2(h/2p)w.   Since each mode can be thought of as an individual oscillator independent of the other modes, each mode has its own set of allowed energy levels.   This can be seen in figure 1.2 below.  

Figure 1.2

Y-axis in units of cm^-1                                                         (Davis 213)   

The symmetric, bending and antisymmetric modes are labeled n₁, n₂, and n₃, respectively, with their vibrational energy levels quantized (n=0, 1, 2, 3, . . .).   The first level of the asymmetric mode, for instance is called (100), while the second is labeled (200).

The CO₂ molecule is also free to rotate.  The energies of the rotational modes are smaller than for vibrational modes.  Hence, the energy levels for two vibrational states with the rotational divisions looks like: 

Figure 1.3

(Davis 217)

 There are certain types of transitions that can be made from one rotational energy level to another.  Because the photon resulting from any transition must have angular momentum and because angular momentum must be conserved, molecules in a particular vibrational and rotational state may transition to another rotational state only if the change in J is +1 or -1.  The laser output, then, shows transitions from with energy just greater than E₀, and those with energy just below E₀.  A transition from the J rotational state in a vibrational level will either be to the J-1 or J+1 state in a different vibrational level.  These two different transitions appear as branches on either side of a vibrational transition: the transitions for J+1 appear to the left and are called the ‘P’ branch, while the ‘R’ branch consisting of J-1 transitions appears to the right. The J-1 transitions are of greater energy than J+1 transitions (and of smaller wavelength, so on a plot of intensity of the transitions vs. wavelength, the R branch is to the right).  The peaks within the branches are labeled such that the R(4) peak, for example, might refer to the transition from the J=5 rotational state of the (001) vibrational mode to the J=4 state of the (100) mode. See Figure 1.4 below:

Boltzmann Distribution and Degeneracy

According to the Boltzmann distribution , at higher energy levels, the population density decreases exponentially:

N(J)=g(J) e^-E/kT              (Davis 215)

Where g(J) is the degeneracy of level J, T is the temperature (K), k is the Boltzmann constant, and E is the rotational energy.  Since the degeneracy caused by the rotational energy levels goes as 2J+1, it increases with an increase in vibrational energy level.  The combination of the exponential and degeneracy factors results in the function

N(J)=(2J+1) e^-E/kT        (Davis 215)

which looks like this:

Figure 1.4

The population density increases because of the degeneracy factor, but eventually decreases because of the exponential factor; few molecules possess the energy to inhabit higher energy levels.  In Lasers and Electro-Optics, Davis discusses the possibility of population inversion regarding this type of population distribution.  (Davis 216).

CO₂ Laser

The CO₂ laser gas mixture consists of 70% He, 15% CO₂, and 15% N₂.   The vibrational and rotational modes of the CO₂ cannot be excited themselves by photons.  When a voltage is placed across the gas, electrons collide with the N₂ molecules and excite them to their lowest vibrational levels.  These vibrational levels happen to be at an energy very close to the energy of the asymmetric vibrational states in the CO₂ molecule.  Now, the excited N₂ molecules populate the asymmetric vibrational states in the CO₂ molecule through collisions.  The infrared output of the laser is the result of transitions between rotational states of the CO₂ molecule of the first asymmetric vibrational mode (001) to rotational states of both the first symmetric stretch mode (100) and the second bending mode (010), as seen in Figure 1.5. 

Figure 1.5

(Davis 219)

According to Hollas, the emitted photons as a result of these transitions occur at 10.6mm and 9.6mm, respectively.

The population of the rotational CO₂ levels, through collisions with N_2, occurs according to the Boltzmann function described above.  The (100) and (020) vibrational levels depopulate to lower vibrational levels.  Again, through collisions, the CO₂ molecule transfers energy from these lower vibrational levels to the He atoms and the CO₂ return to ground state.