The absorption spectrum of I2 vapor at room temperature is horrendously congested and virtually impossible to completely resolve and assign. The Laser Induced Fluorescence (LIF) spectrum, on the other hand, is simple and easy to interpret.
Electronic spectroscopy of small molecules provides chemists with the fundamental understanding of the nature of the chemical bond in quantum mechanical detail. We all know that a chemical bond is not infinitely strong, i.e. it has a well defined bond energy. We also know that because the bond is stretchy, the frequency of vibration of the nuclei attached by a chemical bond will be determined by how stretchy or stiff the bond is. The bond has a length, which is characterized by the vibrationally averaged position of the atoms at the end of the bond. To a good approximation, these and all other properties of the bonding between atoms is derivable from the potential energy curve.
You have already determined two of these quantities, the bond length and vibrational frequency, from the infra-red absorption spectrum of a diatomic molecule. We will determine dissociation energy and vibrational frequency from the fluorescence following laser excitation of the isolated molecule.
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The nuclei in a molecule move in the force field created by the electrons. The motion of the light electrons is separable from that of the slow nuclei; The force felt by the nuclei may be used to define a potential energy which is a function of position only, U(r)
where F is the force exerted on the nuclei by the effective field of the electrons in a given
electronic state. Every molecule can exist in more than one electronic state and has a unique
potential function for each state. Normally, only the lowest electronic state of a molecule is
populated at room temperature . This state is called the
Examine the potential curves for two states in the I2 molecule. The ground state of the
molecule is labeled by its electronic symmetry as 
.
An excited 
state can be produced by the absorption of visible light.
The two states even correspond to different
dissociation products with the excited state correlating to one ground and one excited iodine
atom.
The potential function defines for us the nature of a particular bond: where the nuclei
would rest if the molecule were not vibrating (the equilibrium bond length, re), how much
energy is required to break the bond (the dissociation energy, De or D0
depending on whether
you mean from the bottom of the potential or the lowest vibrational level), and the stiffness of the
bond (the vibrational frequency, 
e).
Note that each electronic state of the molecule has
its own dissociation limit, bond length and vibrational frequency.
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Theoretically, the potential curve comes from the solution of the electronic Schroedinger equation for the molecule at each position of the nuclei. In practice, an approximate form for the potential function may be used. A popular approximate potential function is called the Morse potential:
The vibrational energy of the molecule is the solution to the nuclear Schroedinger equation with U(r) as the potential:
The solution to this equation exists only for integral v quantum numbers (0<=v). For the particular case of the Morse potential, the vibrational energy has a simple form:
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A laser is used in this experiment to excite a transition from the lowest vibrational level of the ground electronic state of the molecule to a vibrational level of quantum number v' in the excited state. Therefore the frequency of the transition is:
The wavelength of the laser light is 514.5 nm. The frequency of this light,
, and the energy
of the transitions it can induce, 
E, may be calculated from:
Spontaneous emission of light occurs from the excited level. The frequency of this fluorescence light is:
and thus, frequency difference between the laser and the fluorescence is:
In practice, many final vibrational states of the ground state may be reached by the fluorescence and a good picture of the vibrational energy level structure of the ground state may be obtained. G"(0) has a special name: it is the zero-point energy of the molecule. (G'(0) is the vibrational zero point energy in the upper electronic state).
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We will obtain the frequencies of the emission lines of I2 vapor following excitation with a Spectra-Physics model 2020 Ar+ laser at 514.5 nm. The I2 will be contained in an ordinary fluorescence cell which is place in front of the entrance slit of a 0.5 m SPEX monochromator. The fluorescence cell is evacuated with a mechanical pump and contains a small crystal of solid iodine which sublimes to provide I2 gas. Fluorescence from the laser excited gas is detected with the use of a photomultiplier tube situated at the exit slit of the monochromator. High voltage is supplied to the PM tube to provide electronic amplification of the photoelectrons induced by light hitting the cathode of the tube. The amplified light signal is viewed on an oscilloscope and recorded on a strip chart.
The monochromator is equipped with a scan unit which changes the grating angle in such a fashion as to increment the transmitted light wavelength at a constant rate. As the stripchart paper also moves at a constant rate, a wavelength calibrated rendition of the spectrum may be obtained .
Suggested Operating Parameters
Photomultiplier Tube Voltage: -750 V (DO NOT EXCEED -900V!)
Stripchart
Scan Rate 20 cm/min
Sensitivity 100 mV/div
Monochromator
Start 5000
End 8500
Rate 5.25 Angstroms/s {This records the spectrum at 40 Angstroms/inch}
Repeat 2 times
Entrance Slit 0.050 mm
Exit Slit 0.050 mm
Ar+ laser
Tube current 18 Amps
Line 5145 Angstroms
Power 220 mW
Record the barometric pressure, temperature, and humidity to calculate the index of refraction of the air in the lab. This is needed to convert wavelength (which is determined by the grating in the monochromator) to frequency (which is proportional to energy). The pressure of the cell is controlled by a valve and a pump. Periodically evacuate the cell to maintain the fluorescence intensity.
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Obtain the wavelengths of all the observed fluorescence lines from your calibrated spectrum. Convert these line positions to vacuum wavenumbers (frequency in units of wavenumbers [cm-1]). Make a table of the vibrational first differences defined as:
Plot G" versus (v"+ 1/2) (this is called a Birge Sponer plot).
A Birge-Sponer plot for H2+ is shown above.
Your plot will contain more points on a smaller
vertical axis.
From your data, obtain values for the ground state vibrational frequency,
e,
and the anharmonicity, exe, in
wavenumbers, as defined for a Morse potential. Extrapolate your Birge Sponer plot to
G"=0 to
obtain the number of bound vibrational levels in the ground state potential
(for H2+ this is v=16,
your value for I2 will be much larger). The dissociation energy, D0,
may be obtained from the
area under the line in the plot of G" versus v"+1/2.
The dissociation energy, De, may be obtained
from the relation (exact for a Morse potential)
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,
but not 4880. Why?
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When you record your spectrum, the first thing to do is make a table of the positions of the laser and iodine fluorescence peak positions. The identity of each of the peaks may be labeled by the vibrational quantum number of the final state of the transition, v". I make a data table with the first column the quantity v"+1/2 (since this is what we wish to plot) and the second column the wavelength of the transition starting with the lase line. I recorded 36 transitions; How many did you get?
The wavelengths of the transitions are now placed in column two but it is the photon
frequency (which is proportional to energy through Energy = Planck's Constant * Frequency).
Convert all the wavelengths from to frequency in wavenumbers (cm-1).
Because the index of refraction of air is not exactly unity, the conversion between wavelength and
wavenumber is not simply an inversion, but requires precise tabulation of n for the conditions of
our laboratory. We are going to ignore this correction in this lab because the effect is almost
within the expected error of the measurement. So I have made a third colum which is the
wavenumber of the transition calculated as 108 / (wavelength in )
So now our data table needs the vibrational energy differences in the fourth column. subtract the adjacent transition frequencies from the third column and place them in the fourth column. Your top portion of your data table should look something like this:
What remains is simply to plot the data as pairs of points with column one providing the horizontal coordinate and column four providing the vertical. A spreadsheet program like Quattro or Excel is ideal for such a task. Typing the numbers you get from your spectrum into such a program will give you a plot that looks like this:
Whether you graph by hand or with the computer, you need to perform an analysis of the data as a fit to a straight line. The slope and intercept of the best fit line as determined by my spreadsheet program looks like this
Having the equation for the best fit line to the data, we can extrapolate to the highest vibrational level in the potential. At the top of the well, the vibrational first difference will go to zero. The value of the largest v" below the x intercept of the Birge-Sponer line is the quantum number of the last bound vibrational level in the ground electroniuc state. This is what the Birge-Sponer line derived from my data looks like:
The area under the Birge-Sponer line is equal to the dissociation energy of the molecule, D0 as referenced to the zero-point vibrational energy level. You should prove this in your report(it is proved in Atkins). The area, and thus the dissociation energy, may be calculated from the x and y intercepts of the Birge-Sponer line.
The spectroscopic constants such as the vibrational frequency e
and the first order anharmonicity constant, exe may
be determined directly from the fit to the Birge-Sponer line.
The anharmonicity, exe, is equal to the
negative one half of the slope.
The vibrational frequency, e, is the y intercept
plus the anharmonicity constant exe