Measuring the Rydberg Constant and the Bohr Magneton

9/21/2015

Abstract
In this lab, the Rydberg constant is found by observing the Balmer Series. With the experiment below, a Rydberg Constant was found to be 10116000 meters-1 with a 7.82% error. The Bohr Magneton is also found, and a value of 6.03*10-24 JT was obtained with a large error. The error can arise from each optical equipment having some fundamental error in its creation.
Introduction

With the help of atomic physics, quantum mechanics, and optics, the Rydberg constant and the Bohr magneton will be calculated in this experiment. The Rydberg constant is one of the most important constants in atomic physics because of its relation to other fundamental constants in atomic physics, such as the speed of light or Planck’s constant [1]. The Bohr Magneton tells us the magnetic moment of an electron by its angular momentum [2]. Attempting to calculate the Rydberg constant and the Bohr Magneton will inadvertently teach the basis of quantum mechanics, optics, and atomic physics. Atomic spectra of hydrogen, mercury, and helium will be studied in detail along with the Zeeman Effect.
Theory
In quantum mechanics, labeling often times helps discern descriptions of certain events. To describe the movement and trajectories of an electron in an atom, scientists use quantum numbers to label what is going on. The principal quantum number n, tells the energy level of the electron and the distance from the nucleus. The angular momentum quantum number l, tells the angular momentum of the electron. The magnetic quantum number ml, tells the projection of the angular momentum on the z-axis. Finally, the spin quantum number ms, tells us the projection of spin on the z-axis. These quantum numbers are important when describing the transitions between energy levels of an electron in an atom. Different transitions correspond to different wavelengths, and this is given by the Rydberg Formula, | 1λ=R1nf2-1ni2 | (1) | where R is the Rydberg constant, is the wavelength, and the n values are the final and initial energy levels. In this experiment, the Balmer Series is the primary focus, so we can simplify equation (1) to be,
1λ=R14-1ni2 (2) where the 14 comes from the final energy level the electron transition to at nf =2. The initial transition level can theoretically be ni=3,4,5,…,∞. Now, equation (1) works really well for the hydrogen atom, but how about other atoms? It turns out that the Bohr Model does not work quite as efficiently for more complex atoms, instead the quantum mechanical model has far more reach and is mostly used when dealing with atoms other than hydrogen. The Bohr model can be derived by first order approximation of hydrogen atom using the quantum mechanical model. The quantum mechanical model is based on probability rather than certainty since it deals with the Schrodinger equation. The Zeeman Effect is the process of splitting spectral lines into many components when a magnetic field is applied to it [3]. In this experiment, the Zeeman Effect is studied by using a Fabry-Perot Interferometer. The F-P Interferometer will be looked at more closely in the Apparatus section, since the physics of this device is significant in finding the Bohr Magneton. The F-P Interferometer forms fringes at infinity due to constructive interference of wavefronts, and by observing these fringes a prediction of what the Bohr Magneton can be made. To do this, a magnetic field is applied with increasing voltage until the fringes break into several components. When these several components are equally spaced out, a measurement is made of the magnetic field. Why do we want them to be equally spaced? The simple answer is because it makes it easier to calculate the Bohr Magneton. The energy of Zeeman splitting is,
∆E=gmjµBB(3)
where ∆E is the difference in energy, g is the Lande-g factor, µB is the Bohr Magneton, and B is the magnetic field in Tesla. It is important to know that when helium spectral lines split into several components, two spectral lines appear and one π spectral line appears. A polarizer is used in this experiment to eliminate π spectral lines when Zeeman splitting occurs. So then we can use the selection rules that arise from the angular integral of the hydrogen atom being on zero. The selection rule ∆mj=±1 can be used when the fringes arising from the F-P Interferometer are evenly spaced out. This can be used to simplify equation (3) into,
∆hν=gmj-g'mj'µBB(4)
where gmj-g'mj' simplifies into 1. So all we are left with is the simple equation,
∆E= µB∙B (5) and from this the Bohr Magneton can be calculated.
Apparatus and Procedure The first part of this experiment observes the Balmer Series of hydrogen using a spectrometer, a photomultiplier tube, and a light source of a hydrogen or mercury lamp. First, a careful analysis of a spectrometer will be given, as shown in Fig. 1. What is not shown in the figure below is the entering and exit slits where the light source goes through. The spectrometer has a collimating mirror and a focusing mirror with a diffraction grating in between to disperse the light. The angle of the diffraction grating can be changed to focus on a certain color on the detector. In the spectrometer used in this lab, the wavelength is changed and this subsequently changes the grating by rotating it. The spectrometer has a grating of 300 grooves/mm. To get the correct wavelength, the dial number is multiplied by 6 [4].

Figure 1: [5] The light source is a hydrogen or mercury lamp. The light source enters a slit hits a collimating mirror and is directed towards the diffraction grating. The grating disperses the light and directs the dispersed light towards the focusing mirror. The focusing mirror then shoots the light out of the exiting slit onto a detector. A 10 cm lens is used to focus the light source onto the entering slit. The slit width of the spectrometer can be adjusted to either let light in or to limit the amount of light entering the spectrometer.
Adjustments of the slit width can determine the sharpness of the spectral lines seen on the detector. A mercury lamp is first used to calibrate the spectrometer since it assumed that the spectral lines of mercury are known. When using the lens to focus the light into the entering slit, one must be careful to create a sharp image. This can be done by moving the lens back and forth until a sharp image is formed on the slit. One way to observe the different spectral lines is to turn the wavelength knob until a line can be seen in the middle of the detector. This is a crude way of doing this because it required high sensitivity in being able to tell if a line exists and when an observation is made by eye, the angle that a person might see the line is constantly changing. No sure way exists to tell whether the eye is making accurate observations of the spectral line being in the middle.
For this reason, a photomultiplier is often necessary to distinguish the spectral lines and to see at what wavelength they peak. A photomultiplier tube, as shown in Fig. 2, works by the photoelectric effect. When the incoming photon is emitted by the exiting slit, in goes into the photomultiplier tube, where the photons hit a metal and consequently release photoelectrons. These metal dynodes create a channel where each successive dynode creates more photoelectrons increasing the amount of light that can be seen. PMTs work very well with weak signals and can detect light at low intensities [6].

Figure 2: [6] The photomultiplier tube works well with low light signals. The incoming photon hits a metal dynode which emits a photoelectron. The photoelectron then goes through a series of alternating dynodes increasing the amount of photoelectrons that are released.
The photomultiplier tube makes it easy to tell when the spectral line is at its peak. The photomultiplier tube is dependent on the number of photons that enter it because more photons would result in more photoelectrons. Likewise, the intensity is also dependent. To observe this directly, an oscilloscope is used. When the oscilloscope has a sharp amplitude, a spectral line is observed. A modulation of the wavelengths is the fastest way to see where the peaks occur. An easier way of looking at the different peaks is using a LabVIEW program called ATM chart recorder. This program tells us the amplitude of each wavelength and a modulation can occur at different speeds resulting in different resolution. If slow speed is used, then this will result in higher resolution. To see the doublet spectral line for yellow in mercury, a very high resolution is needed. Once it is know where these peaks occur for mercury, a calibration of the equipment can be made to more accurately observe where the hydrogen spectral lines should peak. This is an important step because the equipment itself might not accurately determine where the wavelengths peak. Once the wavelengths are known, then the Rydberg constant can easily be found by plotting 1λ vs1n2, where n is the corresponding energy levels associated with creating that spectral line. The second part of this lab deals with the Zeeman Effect and calculating the Bohr Magneton. A polarizer, red filter, Fabry-Perot Interferometer, magnetic field, helium lamp, telescope, and lens are used to create fringe patterns which need to be analyzed. The optical equipment should be made to be at the same height for better resolution value. Fig. 3 shows the set-up for this part of the lab.

Figure 3: [4] The magnetic field surrounds the helium lamp. The magnetic field is used to split the spectral lines into several components. The lens is used to create a sharp image, and the red filter is used to solely observe the red spectral line. The polarizer is used to get rid of one of the splitting. The F-P Interferometer consists of two highly reflecting mirrors spaced evenly apart [4]. The F-P Interferometer is really useful in this experiment because it can accurately measure small changes in wavelength with greater resolution than a diffraction grating would. The camera is used to magnify the fringe patterns, and instead of the eye, a program called Fireview is used to observe the fringe patterns on the computer. The Fabry-Perot Interferometer is used here to measure small changes in the wavelengths coming in. In Fig. 4, the incoming light carves out an optical path bouncing between the two plates. The separation of the fringes that are observed should give us a change of wavelength. The path difference between two adjacent rays in the interferometer given by, mλ=2tcosθ(6) where, m gives the order number, t is the thickness of the interferometer’s plate, and θ is the angle at which the wavefront enters the interferometer.

Figure 4: [4] The incoming wavefront hits the F-P Interferometer and reflects between the parallel plates creating adjacent rays. These adjacent rays correspond to fringes and the distance between the fringes can be calculated. The change in wavelength depends on the wavelength that is coming into the F-P Interferometer and the thickness of the spacer. This relationship is given by equation (9).
To obtain a formula for∆λ, equation (6) can be used along with some clever math. The two equations, m(λ+∆λ)= 2tcosθ1(7) and m+1λ=2tcosθ1(8) arise from how the wavefront go through the F-P Interferometer. Solving for∆λ,
∆λ= λ22t(9) is obtained. Notice, that in order to solve for the Bohr Magneton, this value is crucially needed. The λ that is needed in this experiment corresponds to the red spectral line of Helium which is valued at 667.8 nm. The thickness of the spacer is given as 8.11*10-3 meters and by plugging these values into equation (9) with the right units, a value of 4.12*10-5 for ∆λ. Once the fringe pattern is shown, slowly increase the voltage of the magnetic field until the spectral lines are evenly split. This is when the magnetic field should be measured. After the magnetic field is known, the equation,
∆E= hcλ-hcλ-∆λ= μB∙B(10) can be used to figure out the value of the Bohr Magneton. The last part of this lab is to observe the anomalous Zeeman Effect, which takes into account the electron spin. A prism spectrometer is shown in Fig. 5, where the incoming light is dispersed into its wavelength components by using a prism.

Figure 5: [7] The helium light goes into the slit, hits the prism, and disperses the light out of the exiting slit into the telescope. From this, an observation is made of how the fringe patterns behave for helium.
In this part of the lab, a careful consideration should be made of aligning the optical equipment to get a better resolution. Make sure the equipment is at the same height from the table.
Analysis
In the first part, a calibration of the spectrometer was necessary to account for any errors in measuring the peak wavelength of the spectral lines. In Fig. 6, the mercury calibration line is given where a comparison of the average values of the wavelength measured is compared to the actual values of the wavelength. To figure out exact peak, the LabVIEW file was exported to a word document, where values of wavelength and amplitude were given in a table like format. Four peaks were taken, and the wavelength at which the peak occurred was averaged.

Figure 6: The above plot gives the calibration equation for the spectrometer. The error bars are plotted, but the error is so small between the different trials that they are not noticeable. The average values of the peak wavelength are plotted against the actual wavelengths for Mercury. The graph below shows the residual values obtained from the data.
Once the calibration is done, the wavelengths of the hydrogen transition lines can be measured.
To obtain the calibrated value for the hydrogen wavelengths, use the equation, y=0.99519x+125.84(11) where the new calibrated wavelength is given by y. After obtaining the calibrated wavelength, take the reciprocal of it. The reason why is because then the formula can take on the form of y=mx+b. The Rydberg constant can then be found in two ways by the following equation,
1λ=-Rn2+R4(12)
where it can be either the negative of the slope or four times the y-intercept. Fig. 7 shows the line obtained by plotting 1λ and 1n2.

Figure 7: The negative of the slope and four times the value of the y-intercept gives the Rydberg constant value. In this particular case p1 correspond to the slope and p2 correspond to the y-intercept. The value is measured in Angstroms-1.
The value that is obtained for Rydberg Constant is 0.0010116 Angstroms-1. This is the same as 10116000 meters-1. The actual value of the Rydberg constant is 10973731.6 meters-1. By using the standard error formula, a 7.82% error is calculated for this experiment. The Chi-Squared Value is given by the following formula, χ2=iR1σyi-b-mxi(13) where the value obtained gives us the goodness of fit for our line. The value obtained for χ2 is 0.0297 with two degree of freedom. The wavelengths that can be observed have an upper and lower limit on it based on the spectrometer itself. The slit width can determine what the intensity of light going in can be and that can affect the amplitude observed when modulating over the different wavelengths. The photomultiplier tube can also be limiting if there are not enough metal dynodes to increase the intensity of the spectral lines observed. The fringe patterns that was observed is seen in Fig. 8, where the fringe pattern has not had a magnetic field applied to it yet. These type of fringe patterns occur due to constructive interference of wavefronts as previously mentioned.

Figure 8: A fringe pattern is shown here without a magnetic field applied to it. This was seen on the computer by using fireview software.
Now, a magnetic field is applied until the fringes evenly split. The measurement of the magnetic field is taken until the point where the spectral lines start to merge again. In Fig. 9, this is shown where each solid line splits into two components, since the removal of one component was handled by the polarizer.

Figure 9: The splitting of the spectral line is shown here, where the original spectral line was evenly split into two. A magnetic field was applied and the spectral line started to split at 7.35±0.05 kG and started to merge at 10.32±0.07 kG.
The magnetic field was measured to be 7.35±0.05 kG when the spectral lines started to split and the upper limit when the spectral lines started to merge again was 10.32±0.07 kG. To obtain the Bohr Magneton value, equation (10) can be used. The value of B can be anywhere between 7.30 kG to 10.39 kG. To do this analysis, three points will be taken and the average of those three points will be the Bohr Magneton observed. Fig. 10 will obtain these values; B-Value (T) | hcλ-hcλ-∆λ (J) | Bohr Magneton Value (JT) | 0.730 | 4.956*10-24 | 6.789*10-24 | 0.830 | 4.956*10-24 | 5.971*10-24 | 0.930 | 4.956*10-24 | 5.329*10-24 | Figure [ 10 ]: This shows the Bohr Magneton values obtained by using three different values for the magnetic field. The average of these values is 6.03*10-24 JT.
As said above the average of these values is 6.03*10-24 JT. The real value of the Bohr Magneton is 9.27*10-24 JT. Calculating the standard error of this results in a 34.95% error, which is large. This error could be due to the fact that the resolution is not very clear. So observing evenly split lines becomes harder, but not a huge error can arise from this, because our eyes are very good at determining when things are evenly spaced. The magnetic field shifts the frequencies in order of 1/3 without the polarizer, and ½ with the polarizer in place. A flickering of the light also occurs when the voltage is increased. The bad resolution can occur when the optical equipment is not aligned along the same axis, as well as if the F-P interferometer plates are not exactly parallel. A huge assumption that is made in this lab is that the interferometer plates are exactly parallel. The camera helps in magnifying the fringe pattern, when an observation is made with the naked eye, one can see an infinite amount of fringes, but to calculate the separation between those fringes would be nearly impossible. When the anomalous Zeeman Effect is observed, the fringe pattern becomes fuzzier because there are more splitting in the spectral line. The space between the spectral line is very small creating very low resolution as shown in Fig. 11.

Figure [ 11 ]: In this figure the spectral line that is to be observed is the yellow spectral line. The image shows two different spectral lines having two different fringe patterns. This was done purposely to see the pronounced difference between the anomalous Zeeman Effect and the “normal” Zeeman Effect. The prism spectrometer here has a reading of 620 nm. The yellow and red spectral lines are shown here, with both of them distorting the image exaggeratingly in the center. A magnetic field of 6.56±0.04 kG was applied to observe the splitting.
The yellow line splits into five different levels when a magnetic field of 6.56±0.04 kG is applied. The extra splitting of the energy levels compared to the red spectral line creates a fuzzier resolution with the spacing of the five levels being more widely spaced. Polarization effects are still observed with these splitting, but not as profoundly as the red spectral line.
Conclusion
The fundamentals of quantum mechanics, optics, and atomic spectroscopy was observed in this experiment. The splitting of energy levels in the presence of a magnetic field is very important in everyday life, where doctors use MRI’s to test patients, astrophysicist use it to find the magnetic field of cosmic objects, and chemist use it for nuclear magnetic resonance spectroscopy. Although the Bohr Magneton can be found using the procedure as described above, many errors can follow this technique. An assumption is made that there is a range of magnetic field where the spectral lines are evenly split. When this assumption is made and plugged into equation (10), everything else remains the same except for the magnetic field. This causes the value of the Bohr Magneton to change. If there is a technique, where the range in which the magnetic field causes the energy levels to split evenly can be minimized, then that approach must be taken to obtain more accurate and precise data. When finding the Rydberg constant, a human error occurs when trying to collect data using the computer software. The time at which one person starts the data acquisition on the computer, and the time one person starts the spectrometer speed knob to iterate through the increasing wavelengths, might not be same. This might have profound effects, especially if the computer and the hardware are not registering the same wavelength at the same time. The optical equipment also have internal error that can be accounted for. Overall, all these errors in measurements can add up and not give accurate data as one would hope. The red line is helium is observed when the electron transition from the 3D state to the 2P state with the total spin equal to 0, and similarly the yellow line is observed when the electron transitions from the 3D state to the 2P state with total spin equal to 1. The difference in wavelength is accounted for by the interaction of the electrons. In the hydrogen atom, electrons do not interact with each other and therefore has much simpler transitions than more complex atoms. When the interactions of electrons is accounted for, then the transitions become much more complex. I would like to thank my partner Kate Boden, for asking great questions and working out concepts that were not clear. She did a great job of trying to really understand the physics behind the experiment, and this in turn greatly helped me with the ideas. I would also like to thank Professor Holzapfel, whose energy and enthusiasm is always encouraging and helpful. He gave us wonderful ideas to try out, and helped clarify any problems that we had. Professor Haffner also helped explain the equipment and clarified some questions for us. The GSI’s were extremely helpful in pre-lab and also in trying to clarify the objectives of the experiment. The videos that were provided to us taught by Professor Davis were extremely helpful in understanding the fundamental of this experiment. Thank you to MATLAB for making the calculation much easier to handle, and to the Melissinos text book for clarifying ideas in physics.
Appendix: Raw Data
Mercury Calibration Mercury (Slit Width (micrometers), Resolution (Angstroms)) | Peak 1 (Angstroms) | Peak 2 (Angstroms) | Peak 3 (Angstroms) | Peak 4 (Angstroms) | Peak 5 (Angstroms) | (0.5,20) | 3755.025 | 4155.025 | 4465.025 | 5570.025 | 5900.005 | (0.5,20) | 3740.080 | 4140.080 | 4460.080 | 5580.080 | 5880.080 | (0.5,20) | 3760.220 | 4160.220 | 4460.220 | 5580.220 | 5900.220 | (0.5,20) | 3760.220 | 4160.220 | 4460.220 | 5580.220 | 5900.220 | Average Value (Angstroms) | 3753.886 | 4153.886 | 4461.386 | 5577.666 | 5895.166 | Standard Deviation | 9.5244 | 9.5244 | 2.4267 | 5.0746 | 10.0347 |

The rest of the raw data is accompanied after the references.

References

[1] G. W. Series (1974) The Rydberg constant, Contemporary Physics, 15:1, 49-68, DOI: 10.1080/00107517408210779.

[2] R. Shankar (1980). Principles of Quantum Mechanics. Plenum Press. pp. 398-400.

[3] A.C. Melissinos and J. Napolitano. Experiments in Modern Physics. 2nd Ed. (Academic Press, San Diego, CA 2003). pp. 218-224.

[4] Physics 111b. Atomic Physics Lab Manual. http://advancedlab.berkeley.edu/mediawiki/index.php/Atomic_Physics

[5] Kkmurray. Spectrometer Schematic. https://en.wikipedia.org/wiki/Optical_spectrometer#/media/File:Spectrometer_schematic.gif

[6] M. Abramowitz and M. Davidson. “Photomultiplier Tubes.” 2012. http://www.olympusmicro.com/primer/digitalimaging/concepts/photomultipliers.html

[7] Henriques. The Prism Spectrometer. http://ccphysics.us/henriques/p203lab/Prismspec.html