An Analysis of the Vibrational Energy Levels of Diatomic Nitrogen in Transitions Between Electronic States

Adam Schaefer & Jeffrey Chaffin

University of Minnesota

School of Physics and Astronomy

Abstract

We investigate the emission spectra of electronically excited diatomic nitrogen N₂ within an AC capillary discharge tube. While exposing the gas to a large potential difference to induce metastable excited states the electro-magnetic radiation produced in the process of de-excitation was collected via fiber optic cable and acquired by a spectrometer interfaced to a PC. We determined mechanical constants of the diatomic nitrogen molecule for each excited electronic state, including the Hooke’s law force constants at equilibrium, the vibrational frequencies, and dissociation energies.

Introduction

As a homo-nuclear diatomic molecule with no permanent dipole moment, Diatomic Nitrogen presents a complex and interesting spectrum of electronic transitions. Our study is focused on the vibrational transitions that occur between the C and B electronic states. The transitions occur in the spectral range of 300nm to 480nm. By measuring the wavelength of each peak in that spectral range we can calculate the total energy associated with each transition by De Broglie’s relation,

We also recognize that the change in energy given by the spectral peaks is a superposition of the electrical and vibrational energy difference,

These Transitions referred to as Vibronic for their coupled electrical and vibrational energy change and are nicely demonstrated in the following figure.

Figure reprinted from: Bayram and Freamat, “Vibrational spectra of N2: An advanced undergraduate laboratory in atomic and molecular spectroscopy”, American Journal of Physics 80, 664 (2012).

Theory

Traditional nuclear models analytically approximate the quantized vibrational energy states by separating Schrodinger’s wave equation into separable Hamiltonians in order to derive the individual vibrational energy functions of the nuclei using a harmonic oscillator model. However, Nitrogen, among other molecules, having no permanent dipole moment, exhibits an asymmetric potential which we will model using a Morse potential. Using this potential to solve the vibrational Hamiltonian, we derive a different model for the vibrational energy states of the molecule which contains two fundamental frequencies, the familiar harmonic frequency, and a corrective anharmonic term. The comparison can be seen below.

Experimental Apparatus

In order to observe the resulting spectra from the de-excitation of the nitrogen molecules they must first be excited into metastable radiative states such as a transition from the ground state to an excited vibrational state. This is accomplished by an AC capillary discharge tube filled with nitrogen gas at low pressure and powered by an Electro-Technic 5kV power supply, pictured in figure 2. The high voltages applied between electrodes within the tube determine an electric breakdown or “breakdown voltage.” To account for all possible transitions between electronic states in the wavelength range of interest, a wavelength resolution of at least 1.0 nm is required. To acquire the vibrational spectrum, we used a fiber optic cable connected to an Ocean Optics Flame T spectrometer. The Flame T is a handheld miniature spectrometer configured for a bandwidth of 248-586 nm, and a resolution of 0.53 nm using a 10

m slit width. The spectrometer is interfaced (USB) with a PC and the emission spectrum is displayed with the use of the Ocean Optics Spectra Suite software. The setup and interface can be seen below.

In order to confirm the peak assignments a Deslandres table of wave numbers was created. Using a Deslandres table one can assess the energy differences between successive transitions and verify their consistency. The Deslandres of our data can be seen below.

With the correct peak assignments, we can determine the force constants, disassociation energies, and the fundamental frequencies of vibration in the two electronic states. We begin with the knowledge that the wavelengths measured at each transition correspond to a change in total energy made up of the sum of the change in electrical energy and the change in vibrational energy.

Using this final expression and the wave number terms from the Deslandres table we are able to obtain a four equation system that allows us to solve for the four unknowns the harmonic and anharmonic frequencies of the B and C stat. With these fundamental frequencies it is now trivial to find the disassociation energy and the force constants.

For the B electronic state the vibrational energy splitting, is on the order of 200 meV. A quick inspection of energy diagram below shows a definite trend in the energy differences, indicating a slight decrease in the magnitudes of the energy splitting for increased quantum number. This trend is as expected and is a validation of the corrective anharmonic term of our model for vibrational energy stated above.

Using our experimentally derived results and the value for the average inter-nuclear distance of the B electronic state taken from literature, we can plot the estimated potential energy well as a function of inter-nuclear distance. The following figure shows this plot, using the Morse potential. The shaded area in the well of the potential energy curve depicts the vibrational energy spacing as a grouping from quantum number v = 0 to v = 6. The energy well itself is quite deep and the characteristic asymmetry does not appear strongly in the lowest part of the well.

Conclusion

Through our analysis of the emission spectra of electronically excited diatomic nitrogen N₂ within an AC capillary discharge tube we were able to validate a more accurate model for the vibrational energy by introducing a corrective anharmonic term to account for the experimentally observed asymmetry of the energy curve. Also, while demonstrating the accuracy of the anharmonic model we were able to evaluate the merits and limitations of using the simple harmonic oscillator model to approximate the vibrational energy at small quantum number. The results of our determination of the vibrational constants of diatomic nitrogen showed excellent agreement with literature, on average being within a half sigma of the accepted value.