Molecular Structure:  Vibration and Rotation


    Molecular lasers function differently than atomic lasers because they have vibrational and rotational energies as well as electronic energy.  These molecular vibrations occur because the relative positions and orientations of the atomic nuclei are not absolutely fixed within the molecule.   The molecular rotations occur because the individual molecules are free to spin and rotate in space since they are in a gaseous state.  The energies associated with molecular vibrations and rotations are quantized just like the electronic energy.   Transitions between vibrational energy levels emit photons with wavelengths in the infrared region, while transitions between rotational energy levels emit photons in the microwave region.  

    The CO2 laser is based on the vibrational and rotational transitions of the CO2 molecule.  This molecule consists of two oxygen atoms covalently bonded with a central carbon atom.  Arranged in sp2 hybrid orbitals, each oxygen atom forms a sigma and a pi bond with the carbon atom and leaves two free electron pairs (Zumdahl, 639):

orbitals.gif (12364 bytes)

The dynamics of the carbon dioxide molecule are very similar to the dynamics of diatomic molecules such as CO or HCl.

Molecular Vibration:

    In a diatomic molecule such as O2, N2 or CO, the individual atoms are bound by a molecular binding force that functions much like the spring constant k of a linear harmonic oscillator.  When excited, the two nuclei will vibrate much like two masses connected by a spring.  While real diatomic molecules are not perfect harmonic oscillators, their potential energy functions approximate those of a harmonic oscillator for a certain value of inter-nuclear separation.   Although Carbon Dioxide is a triatomic molecule, it behaves much like a simple diatomic molecule because its structure is linear.  Such a linear triatomic molecule has three normal modes of vibration, described as the asymmetric stretch mode, the bending mode and the symmetric stretch mode.

  vibration.JPG (19896 bytes) 

    Each one of these normal modes of vibration for the CO2 molecule is associated with a characteristic frequency of vibration (w) as well as a ladder of allowed energy levels.  Therefore we label the three vibrational modes n1, n2, and n3, with their vibrational energy levels being quantized (n=  0, 1, 2, 3, . . .).  To describe the vibrational modes we use the notation ( n1, n2, n3).  We can approximate the vibrational energy levels as the quantum mechanical simple harmonic oscillator (hb= h/(2pi)):

Ev=hbw0*(n+1/2), where v0 is the classical vibration frequency

w0= (k/u)^1/2

Notice that there will be a ground state vibrational level (corresponding to n=0) with some positive energy E0= hbw/2

  While in reality each normal mode is slightly anharmonic, the specific equation for the vibrational energy of CO2 can still be described by:

E(n1, n2, n3) = hcw1 (n1+1/2) + hcw2 (n2+1/2) + hcw3 (n3+1/2)

     The following diagram depicts the first few vibrational energy levels (n1 n2 n3) of the CO2 molecule and also lists the normal modes' characteristic frequencies in units of wave numbers (cm-1).  

vibenergies.JPG (10058 bytes)

Notice that the base frequency for the symmetric stretch mode is w1=1288cm-1 and  for the bending mode w2= 667cm-1.  For the asymmetric stretch mode, the base frequency is much higher at  w3= 2349cm-1.

Molecular Rotation:

    Again because CO2 is a linear triatomic molecule and thus behaves very much like a diatomic molecule, it is considerably simple to understand the dynamics of its rotation.  To start, picture the molecule as a classical dumbbell with masses on the ends connected by a rod.  When the dumbbell then rotates about its center of mass (carbon atom) it has a kinetic energy that is a function of its moment of inertia and the length of the rod.

rotation.JPG (6796 bytes)

    This model must be adjusted somewhat to describe a diatomic molecule because the separation of the masses is not fixed.  As the molecule rotates the oxygen atoms are pulled farther apart by the centrifugal force, therefore increasing the moment of inertia.  It is even more complicated if the molecule is experiencing a molecular vibration.  Fortunately, for our purposes the rigid dumbbell approximation will suffice for describing the rotational energy of a diatomic molecule.   When the classical equation for rotational energy is put into a quantum mechanical form, with the quantum number J describing the rotational states, we then have an expression for the rotational energy of a diatomic molecule.  Thanks to its linear structure, the rotational energy spectrum of CO2 has the same character as that for diatomic molecules and the rotational energy levels are thus approximated by: 

EJ = hcBeJ(J+1),   J = 0, 1, 2, 3, ...           

wpe32.jpg (2886 bytes)    

  where the rotational constant Be for the CO2 molecule is Be = .39 cm-1. The difference between energy levels is:

wpe31.jpg (1408 bytes)

Table of Contents:

  1. Introduction
  2. Molecular Spectra
  3. Data
  4. Apparatus

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