Analysis:

From the intensity output of the laser, we’re able investigate the operating temperatures of the laser and properties of the CO₂ molecule.  In the following manner, we can pull out these two pieces of data…. 

Since we know frequency of the light at a peak is: 

f­_light = F(J+1) – F(J) = 2B(J+1)           (Hollas 95) 

Where F(J+1) is the frequency of light  at one peak on our intensity graph, and F(J) is the frequency of the next closes peak (higher wavelength).  B is a constant related to the rotational energy, as we’ll shortly see.   We can find the spacing of peaks by finding the change in frequency from peak to peak, or… 

f_{between-peaks} = 2B 

For our data, we simply took the spread of the P-Branch at 10.6um and divided it by 32, because there are 32 known rotational peaks in this branch (Davis 221).  Thus, we can find the peak spacing in terms of frequency or wavelength.  Once we find B, because we know the frequency spacing, we can find the rotational moment of inertia, I, because 

B = h / (8p²I)               (Hollas 96) 

where I = 2mr².  We multiply mr² by two because there are two oxygen atoms.  The reduced mass, m, = m₁m₂ / (m₁ + m₂).  In this case m₁ = 16u (oxygen) and m₂ = 12u (carbon).  Ultimately we can find r, the bond length between the carbon and oxygen atoms.

Calculated bond length = 1.48 x10^(-10) m

Since we know the spacing of the rotational peaks, we can accurately find J_max.  J_max is the rotational angular momentum quantum number associated with the highest intensity peak.  Since the relative population of molecules in each quantum state is related to temperature and the degeneracy of the quantum states, we can model the different branches of rotational peaks with the relationship… 

Nj/N­­_0­­ = (2J+1)*Exp(J(J+1)*E­­­_or/(kT))               (Hollas 98) 

If we take the derivative of this equation at J_max, we can then find the operating temperature of the laser. 

J_max = (kT / 2hB)­^1/2 – ½            (Hollas 98) 

Since we couldn’t perfectly discern the highest peak, we calculated the operating temperature for 3 peaks.

Calculated Operating Temperatures Depending on J_max for the stand alone laser

A similar technique was applied to the CO₂ chamber experiment.  By fitting the population equation to the data of the 10.6um P-Branch, we we’re able to calculate the operating temperature of the chamber.

Calculated Operating Temp. of Chamber Gas

                     Laser Data                                                                                                            Chamber Data

Best Fit Line for Chamber Data -calculated temp of 840 K ---HOT