Physics307L:People/Cordova/Matt's Final Lab Report

Ascertainment of the Rydberg Constant Through Spectroscopy

Author: Matthew A. Cordova

Experimentalists: Matthew A. Cordova & Roberto Sebastian Rosales
Junior Lab, Department of Physics & Astronomy, University of New Mexico
Albuquerque, NM 87123

mcordov9@unm.edu

Abstract

This lab report will investigate the Rydberg constant, a physical constant which relates to the wavelengths of photons emitted from atoms in an excited state. This will be done through spectroscopy. Specifically, we will be measuring the wavelength of photons emitted by Hydrogen and Deuterium gas observed as spectral lines isolated through diffraction. Although this can be seen as a dated method compared to modern science, the results obtained in this lab were quite acceptable. The reduced mass Rydberg constant for Hydrogen and Deuterium were found to be [math]\displaystyle{ 1.0977(7)*10^7m^{-1} }[/math] and [math]\displaystyle{ 1.0990(2)*10^7m^{-1} }[/math], respectively. The accepted values for Hydrogen and Deuterium are [math]\displaystyle{ 10967758.3406m^{-1} }[/math] and [math]\displaystyle{ 10970746.1986m^{-1} }[/math], respectively.

Introduction

As previously mentioned, atoms in an excited state emit photons when they go back down to a more stable energy level. A more accurate description, however, would be that the electrons in an atom are what reach excited levels, and the energy lost in the transaction to a lower (more stable) energy state is in the form of an emitted photon. Since the energy levels are defined as integers, there are discrete energies, i.e. wavelengths, that this photon may have. The Balmer series, first explored by Johann Balmer, deals with excited energy levels transitioning down to the n=2 state. To observe this process, my lab partner Sebastian and I will excite Hydrogen and Deuterium gas and observe the photons emitted through a constant deviation spectrometer. By sending the emitted photons through a diffracting medium, we can isolate the wavelengths and use Rydberg's equation to determine the corresponding constants for Hydrogen and Deuterium. Johannes Rydberg crafted his equation strictly from empirical data. No theory went into its conception.[1] It is expressed as

[math]\displaystyle{ \frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) }[/math]

It must be noted that this equation uses [math]\displaystyle{ R_\infty }[/math], while we are looking for the reduced mass Rydberg constant, which is obtained from

[math]\displaystyle{ R_M = \frac{R_\infty}{1+m_e/M} }[/math]Where [math]\displaystyle{ e_m }[/math]is the mass of an electron and M is the mass of the atomic nucleus.

It can be seen here why using [math]\displaystyle{ R_\infty }[/math] is a good approximation, as [math]\displaystyle{ m_e }[/math]<<M.
The Rydberg constant is an important fundamental physical constant, likely due to the fact that it is one of if not the most accurately measured physical constant.[2] It is because of this accuracy that the Rydberg constant is most useful in refining quantum theories, even though no quantum theory went into its conception!

Methods and Materials

Figure 1- Lab Equipment and Set Up:
On left: Spectrum Tube Power Supply - Model: SP200 (with gas tube inserted)
On right: Spectrometer - Adam Hilger; London, England; Serial: 12610

Figure 2- Refracting Crystal:
This crystal is what refracts, and therefor isolates, the various wavelengths. We are then able to measure these wavelengths using an attached dial.

Figure 3- Measuring Device:
This dial moves in accordance with the refracting crystal. After proper calibration (further detail will be explained later) the measurement observed on the dial corresponds to the wavelength of the observed spectral line that is centered on the cross-hairs of the spectrometer.

Initial Set Up

The set up for this lab, as seen by Figure 1, consists of relatively few pieces of equipment. The spectrometer should be placed on a flat, sturdy surface, while the power supply should be elevated such that the thinner section of the gas tube (while attached to the lamp) is at an even level with the eye piece of the spectrometer. Rotate the cross-hairs until they are to your liking. Lastly, set the width of slit for incoming light from the excited gas. A thinner slit yields a thinner (and therefor more accurate) spectral line, but sacrifices visibility. This concludes set up for this lab.

Calibration

With set up complete, we (Sebastian and I) proceeded to calibrate the spectrometer using a known element with known spectral lines. For this lab, we used Mercury gas. The values for the observed spectral lines can be found in Prof. Gold's lab manual. To calibrate, adjust the measuring dial (Figure 3) such that it reads a known value for an observable spectral line. For example, there is known to be a green spectral line an 546.1 nm. Set the dial such that it reads this value, and then loosen the screw which secures the refracting crystal (seen in Figure 2 in place. Then, carefully move/rotate the crystal such that the green spectral line in centered in the cross-hairs, and re-secure the crystal. This concludes calibration.

• Note: Be sure to always rotate the measuring dial in the same direction to avoid gear backlash. If you calibrate while rotating the dial clockwise, make all measurements in the same direction.
• Note: When securing the crystal, make sure that the spectrometer is still calibrated. Tightening the securing screw often moves the crystal slightly.

Data Recording

With the spectrometer calibrated, the data was ready to be recorded. For this lab, the spectral lines of Hydrogen and Deuterium gas are to be examined. There are four observable wavelengths for both Hydrogen and Deuterium for the n=6,5,4,3 transitions. However, for Hydrogen the first spectral line (n=6) was impossible to identify, so it shall be excluded from the data analysis. Other than this, there were no complications in recording the data. Multiple trials were completed, each calibrated to a different Mercury wavelength in an attempt to reduce error in the equipment.

Acknowledgments

Special thanks goes to my lab partner Sebastian Rosales. Also to Professor Koch for help with all aspects for this lab from set up to data analysis. Acknowledgment must be given to any and all persons who provide their lab notebooks on OpenWetWare. The knowledge gained from viewing these is immeasurable.

References

[1]: McIntyre, D.H., Hansch, T.W. 'Precision Measurements of the Rydberg Constant' [1]
[2]: Zhao, Ping, Lichten, W. 'New value for the Rydberg constant from the hydrogen Balmer-β transition' [2]
[3]: Gold, Michael 'Junior Laboratory Manual Fall 2006' [3]
[4]: 'Rydberg Constant' Wikipedia [4]
[5]: 'Spectroscopy' Wikipedia [5]