Molecular Spectra

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The plasma of the CO2 laser contains He, CO2, and N2. When a potential of 18,000 Volts is placed across the plasma, electrons collide with the N2 molecules and excite them to their lowest vibrational levels. These vibrational levels of the N2 molecule are metastable and represent an energy very close to the energy of the asymmetric vibrational states in the CO2 molecule. This is because the forcing constant for the asymmetric vibration is very near the forcing constant for a stretching of the N2 molecule (Eastham 206). Through collisional excitation transfer, these excited N2 molecules populate the asymmetric vibrational states in the CO2 molecule. The n=1 vibrational levels of the N2 molecule transfers energy to the (001) states in CO2, the n=2 levels of N2 populate the (002) states in CO2, and so forth. The laser's infrared output comes from the CO2 molecular transitions from rotational states of the first asymmetric vibrational mode (001) to rotational states of both the first symmetric stretch mode (100) and the second bending mode (010). The following is an energy diagram for the lasing process in the CO2 laser:

co2energy.jpg (15586 bytes)

The (100) and (020) vibrational levels depopulate when they relax rather quickly into lower vibrational levels, particularly the (010) mode. The CO2 molecule transfers energy from these lower vibrational levels to the He atoms through collisions. Since the infrared laser transitions are relatively slower than this depopulation mechanism, a population inversion results.

co2energy2..jpg (4187 bytes)
As discussed in the Molecular Structure page, a molecule in a particular vibrational mode may also be in one of several rotational states. These rotational states have a degeneracy of (2J+1) and differ by energy E= J(J+1)h^2/I. 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. (Likewise in electronic transitions, electrons may transition to other levels such that the change in quantum number "l" equals +-1.) A molecule in the (001) mode and the J(7) rotational state may transition to either the J(6) or J(8) in either the (100) or the (010) vibrational mode. It may not transition to another J(7) state.

If the energy difference between the lowest (001) state and the lowest (100) state is E0, then the energy of the transition between the J=7 (001) state and the J=6 (100) state will be of energy:

wpe6.jpg (3189 bytes)

Likewise, transitions where the change in J is +1, the energy released will be:

wpe7.jpg (1831 bytes)

co2transition.jpg (2877 bytes) So, we should see two types of transitions in the laser output: those with energy just greater than E0, and those with energy just below E0. This will form a branched structure in the emission spectrum. Transitions corresponding to a change of -1 in J are said to belong to the "R" branch , while transitions corresponding to a change of +1 in J are said to belong to the "P" branch. R(6) refers to the transition from the J=7 rotational state of the (001) vibrational mode to the J=6 state of the (100) mode.

Moreover, we should see in the laser emission two distinct bands, each having P and R branches, corresponding to the series of transitions from the (001) to the (100) and from the (001) to the (020) vibrational modes. Because the difference between adjacent rotational levels increases as J increases, the separation between emission lines in the P and R branch will increase as J increases. The branches will be centered around frequency v0 (corresponding to E0).

According to Maxwell-Boltzmann statistics, the distribution of molecules in rotational states should follow the relation:

where G(j) is the degeneracy of each state, which happens to be 2J+1. So, the population of molecules in the J^th rotational state should be proportional to:

This will give a distribution that looks like this:

wpe30.jpg (8282 bytes)

When the excited CO2 levels are populated by collisions with N2 the levels will be populated according to this relation. Laser transitions will be most probable from the rotational states that have maximum population (i.e. the maximum on the Boltzmann curve) As a result, each branch in the emission will be curved similarly to the Boltzmann curve.

Information provided by: http://www.phy.davidson.edu