The Configurations of Carbon Monoxide

The excited states of the diatomic molecule carbon monoxide (CO) can be observed by crossing a beam of electrons with another beam consisting of carbon monoxide. The measured energy spectrum of the scattered electrons is displayed in Figure 1 below. If the potential energy between the two atoms is truly a harmonic oscillator potential, then the energies of the excited states are

(Subscript[E, n]) = (n + (1/2)) (Subscript[ℏω, 0])

where (Subscript[ω, 0]) is the angular frequency of the oscillator. In this investigation you will use the measured spectrum below to test the accuracy of the assumption that the forces that bind carbon and oxygen together in CO can be described with a harmonic oscillator potential.

Figure 1. Measured electron scattering data from e-CO collisions.

1. Data Analysis Tools

Once you have collected a set of data, Mathematica has a wide range of tools for analyzing those measurements. Some are well-known statistical measures like the average and standard deviation.

data1 = {4.3,3.2,4.1,5.7,3.9,3.7,4.0,4.1}
data1ave = Mean[data1]
data1sigma = StandardDeviation[data1]

Other software tools available include different methods to calibrate and fit a set of (x,y) data and then plot the results. Below you are given a set of data and some calibration parameters. The Table command is used to construct a new, calibrated data set (data3) and the result is plotted.

data2={{0.03254374,0.000247},{0.0505437,0.0002},{0.0605437,0.000178},{0.08054375,0.000132},{0.0905437,0.000112},{0.110543749,0.000086},{0.120543749,0.000073},{0.140543749,0.000053},{0.160543749,0.000042},{0.170543,0.000036},{0.19054375,0.00002800},{0.21054375,0.000023},{0.23054374,0.0000195},{0.19054375,0.000026},{0.16054374,0.000042},{0.13054374,0.000066},{0.100543749,0.000094},{0.07054374,0.000155},{0.040543749,0.000238}};
cal1 = 0.80; (* first calibration parameter *)
cal2 = 0.056; (* second calibration parameter *)
(* Use cal1 and cal1 to calibrate the data2 measurements. *)
data3 = Table[{data2[[i,1]],data2[[i,2]]*cal1 + cal2},{i,1,Length[data2]}];
plot1= ListPlot[data3,Frame->True,FrameLabel->{"x (mm)","B field (gauss)","Test Plot",""}, FrameStyle->Black,BaseStyle->FontSize->16,ImageSize->8*72]

There are also methods to fit mathematical functions to the data. The commands below show how to fit the data set from above and then plot the result.

myfit = LinearModelFit[data3,{x,x^2},{x}] (* Fit a quadratic function to the data. *)
plot2=Plot[myfit[x],{x,0,0.25},PlotStyle->Red]
Show[plot1,plot2]

And additional tools are available to get the fitted parameter values and their uncertainties from the fit results.

myfit["ParameterTable"]

2. Questions and Calculations

1. Outline in careful language how you would test this notion that the CO molecule is bound by a force that can be modeled by a harmonic oscillator. Use the data  above to test the agreement between the theory and the experimental data. How good is the harmonic oscillator hypothesis at reproducing these data? Be quantitative in your answer. 

2. How does the spring constant implied by your results compare with typical values of everyday springs? The springs used in the general physics laboratory have spring constants of about 3 N/m. Note that when we study the motion of the CO molecule we treat it as a two-body problem where we study the motion relative to the center of mass. In this situation the oscillations can be described as a single object of mass

m=(((Subscript[m, 1])(Subscript[m, 2]))/((Subscript[m, 1]) + (Subscript[m, 2])))

called the reduced mass where (Subscript[m, 1]) and (Subscript[m, 2]) are the masses of the constituents.

3. A rough measure of the range of oscillation of the CO molecule is provided by the classical amplitude. What is the classical amplitude of the molecule we are studying here in the quantum mechanical n=2 state? How does your result compare with the typical CO inter-atomic separation of  1.2 angstrom (1 angstrom = (10^(-10))m)? 

4. We often treat molecules as if they were made of atoms rigidly held in place by sticks. For those who took  chemistry, recall the tinker toys you used to study the configurations of different molecules. What, if anything, do the results here have to say about our tinker toy model?