# User:Arianna Pregenzer-Wenzler/Notebook/Junior Lab/Formal Report

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## Quantum versus Wave behavior in Light, and the use of the Photo Electric Effect in the Determination of Planck's Constant

- author: Arianna Pregenzer-Wenzler,

undergraduate student in the Department of Physics & Astronomy, University of New Mexico, Albuquerque.

- experimentalists: Arianna Pregenzer-Wenzler, Danial Young

- contact information:

email: arianna@umn.edu

### Abstract

In this experiment I used the photoelectric effect to determine the energy of the electromagnetic radiation emitted by a mercury light source at a given frequency. By analyzing how the kinetic energy of the photo current was effected when I varied the intensity of my light source at a constant frequency, I determined that the behavior of the EM radiation is inconsistent with the wave model of light. By measuring the change in the kinetic energy of the photo current for different bands of the mercury spectrum I am able to show that there is a constant that relates the energy the EM radiation to its frequency, and by doing so I confirm that the behavior of the light is consistent with the quantum model. This constant, my experimentally determined value of Planck's constant is found by using the equation E = hν = KE_{max} + W_{other} to relate my measured values of KE_{max} to the known values of ν for the mercury spectrum. Using this method I determined a value for Planck's constant that contained the accepted value within a 68% confidence interval.

### Introduction

At the beginning of the last century Rayleigh and Jeans used classical theory to predict the energy density of black-body radiation as the frequency of the light was increased. Classical theory says that light will exhibit wave like behavior, which means that the value of the energy is continuous and the average total energy is constant for a given temperature; its average does not depend on its frequency. The Rayleigh-Jeans formula developed using this theory gives the energy density of a blackbody cavity and says that as the frequency of the radiation becomes large the energy density will go to infinity. Experimental physics didn't bear out the theory, in fact it showed that the energy density not only reaches a finite value at a given temperature, it then drops back toward zero. This incredible discrepancy between theory and reality, which became know as the ultraviolet catastrophe, prompted Planck to look for a theory for the behavior of light that would allow the average energy of this blackbody radiation to first increase to some finite value and then decrease again. Planck realized that for the energy to behave as suggested by experiment it could only have discrete uniformly distributed values and it had to be a function of the frequency, so he hypothesized E = hν, where h is a proportionality constant that has come to be known as Planck's Constant. The idea that light is emitted as discrete bundles of energy called photons is the basis of quantum theory.[2]

When light strikes a material the energy in the light is transferred to the electrons in the material, when enough energy is transferred the electrons break their bonds and are emitted from the surface of the material producing a photoelectric current, this is called the photoelectric effect. The purpose of this experiment is to use the photoelectric effect to analyze the behavior of light, and see if in this case its behavior supports the quantum or wave theory of light. Once I show that I can interpret the behavior of my light using the predictions made by quantum theory I can use my experimental data to determine a value for Planck's constant, h.

### Methods and Materials

In this lab we used a commercial h/e apparatus (PASCO scientific, Roseville, CA) which consists of a mercury vapor light source(OS- 9286*), a mounted box containing a cathode plate in a vacuum photodiode tube which I will refer to throughout this paper as the h/e apparatus(AP-9368*), a set of filters, and the lens grating assembly (Accessory Kit, (AP-9369*)), see fig 1. The light emitted by the mercury vapor light source is broken into its spectrum and focused by a lens grating assembly, one portion of the spectrum can then be directed through a slit in the h/e apparatus and onto a photodiode which works as a cathode. Rather than measuring the photoelectric current produced by the incident light, the h/e apparatus applies a reverse potential between the cathode and the anode reversing the current to zero. The minimum voltage necessary to stop the current corresponds to the maximum value of the kinetic energy of the electrons being emitted, and can be measured with a multimeter(WavTek 85XT, WavTek Technology Systems,Woodstock, IL) connected to the h/e apparatus.In the first portion of this experiment we focused one of the first order bands of the mercury spectrum onto the photodiode of the h/e apparatus and measured both the maximum voltage and the time it took to reach that voltage. We then inserted a filter in front of the photodiode to decrease the intensity of the incident light and repeated this process until we had measurements corresponding to intensities varying from 20% to 100%, at a constant frequency. We repeated this process for a second spectral band. Between measurements we used the zero button on the side of the h/e apparatus to discharge any accumulated potential. An initial data analysis showed that the time required to reach the maximum voltage is so short that what we were measuring with our stop watch was actually the time it took for small fluctuations around the max voltage to equilibrium, viewing the wave form on an oscilloscope (Tektronix TDS 1002 , Tetronix Test Equiptment,Long Branch, NJ) proved to be much more useful.

For the second part of this experiment we measured the maximum potential for each of the five bands in the mercury spectrum at maximum intensity. The bands consisted of an ultraviolet band (that shows up as dark blue on the white reflective mask that is mounted on the front of the h/e apparatus and allows you to see the UV band), two more blue bands of slightly different color, a green band and a yellow band. When we measured the green and yellow bands we placed a green or yellow colored filter in front of the photodiode to prevent higher frequency light that could have been present in the environment or from an overlap with the second order bands from interfering with our results. We measured the maximum potential of each band twice, and then repeated the procedure for the second order bands. When I went back to the lab to do some final data collection I used my RayBan polarized sunglasses as an additional filter for the second order green and yellow bands in an attempt to block out UV and other high frequency light overlapping from the third order bands.

Analysis of the data for the first part of this experiment was basically qualitative, leading to the assumption that the behavior of the EM radiation is in keeping with the quantum model. Using quantum theory the total energy of the light, hν, can be expressed as the sum of the amount of energy needed to overcome the binding energy of the electrons in the material of the cathode and the kinetic energy of the photoelectric current. I rewrote the equation E = hν = KE_{max} + W_{0} so that KE is expressed as a function of ν, and determined h from a least squares fit for each set of data. To calculate h, and my error in h, I used the INDEX, and LINEST functions in Excel (Microsoft Corporation, Redmond WA). I weighted each value of h (the weight is given by one divided by the square of its error), and calculated a weighted average as my best guess for h along with an appropriately weighted value of my error.[3] I used MATLAB (MathWorks Inc, Natick, MA) to display my data making use of the polyfit function to plot my linear fit curves.

### Results & Discussion Part I

The results of the first part of this experiment are essentially the qualitative analysis of the behavior of the light at constant frequency. If our light is behaving as would be predicted by classical theory then we should see a decrease in the maximum voltage as the intensity of the light decreases. This is because higher intensity means a greater wave amplitude (higher energy light), which would allow a for a larger amount of energy to transfer to the individual electrons. If, on the other hand, the behavior of our light is being governed by quantum theory, the amount of energy being transferred to the electrons is a function of the frequency of the light. According to the quantum model, in the first part of this experiment where we are keeping the frequency constant by using the same spectral band, the only effect the we should see as the intensity decreases is that the time it takes to reach the maximum voltage should increase.

My experimental data showed that the intensity of light, at a constant frequency, incident on the photodiode did not significantly effect the value of the maximum potential (Fig 2). Notice that while V_{max}does vary from one intensity to the next those variations are small and do not appear to be correlated to the intensity. Though you cannot draw any conclusions from this data as to the effect of intensity compared to time required to reach V

_{max}, it is clear that V

_{max}does not increase significantly with increasing intensity, supporting the quantum model for light. When I was trying to measure the time required to the maximum potential, I found that even at a very low intensity the maximum voltage was reached so quickly that I was not able to directly measure the time it took to reach V

_{max}using a stopwatch. It was possible to draw some conclusions about the time required to reach the maximum potential at various intensity by viewing the wave form of the voltage going from zero to V

_{max}on the oscilloscope (Fig 3). You can see there is an initial spike in voltage greater in height to V

_{max}that corresponds with the release of the discharge (zero) switch on the h/e apparatus that allows the apparatus to begin building up charge, then the voltage dips and grows again in the expected exponential fashion, something like 1-e

^{-x}until it reaches V

_{max}. Though I couldn't directly measure the time required to reach V

_{max}I could see on the oscilloscope, that the time to reach V

_{max}does indeed decrease as the intensity of the light is diminished.

### Results & Discussion Part II

After eliminating the wave model of light as a means of predicting the behavior of our light, and seeing that at a constant frequency our light behaved as predicted by the quantum model, I went on to confirm the correlation between the energy of our light and its frequency by measuring V_{max}for different frequencies. Taking measurements of V

_{max}for both the first and second order bands of the mercury spectrum showed that an increase in frequency always resulted in an increase in V

_{max}. My first set of data clearly shows this relation between the frequency and V

_{max}for the first and second order bands (Fig 4). With closer analysis it was clear that my data supported the quantum model, and that I would be able to determine an experimental value for Planck's constant from it, but a mystery had emerged as well. The value of the maximum potential corresponding to the green spectral band in the 2nd order was much greater than predicted by the general trend (see Fig 4). Some investigation showed that the cause of this discrepancy is an overlap of the high frequency 3rd bands with the 2nd order bands. In my second data set I was able to reduce the effect of this overlap by using my sunglasses as an UV filter when measuring the values of V

_{max}for the 2nd order green and yellow bands (Fig 5).

Using a least-squares fit I determined an experimental value for Planck's constant, h, for each set of data, but in light of the known source of error in the measured values of V_{max} for the green and yellow bands of the 2nd order, I decided to leave my second order determinations of h out of the weighted average that I used to determine my final experimental value for h. If I had included my 2nd order results in my final value, my experimental value would have actually been slightly closer to the accepted value for h, since the slightly higher voltages for the low frequency 2nd order band led to a slightly smaller value for h, and my final experimental value is greater than the accepted value.

My final experimentally determined value for Planck's constant;

**h = 4.4 (48) E-15 eV**

Though it is a bit high, my experimental value for h compares well to the accepted value, in that it contains the excepted value with in a 68% confidence interval.

Accepted value of Planck's constant (given by CODATA[5]);

**4.135 667 3(10) E-15 eV**

### Conclusions

This experiment provided a simple way to explore one of the fundamental principles of physics. Its easy to learn information with out really getting a feel for what it actually implies, this experiment allowed me to qualitatively explore an idea that I had taken for granted without much thought; the idea that EM radiation can have both wave and particle properties. Because the setup and data collection for this experiment was relatively simple, I was able to do a more in depth analysis of the reasons for discrepancies between my experimental and my expected results. While the experimental value I determined for Planck's constant that was in keeping with the accepted value I feel that further work could yield even better results.

My experimental value was consistently higher than the accepted value for all my data trials if I discard the values I measured for V_{max} of the low frequency 2nd order bands before I started filtering the 3rd order frequencies.If I had the opportunity to continue on with this experiment in the laboratory, I would focus on eliminating or pinning down the few sources of error. One of these sources has to do with the h/e apparatus itself. In the first part of the experiment while there was not a strong correlation between a decrease in intensity and a decrease in V_{max}, I did on average see a slightly smaller V_{max} at lower intensity's. This slight drop in V_{max} corresponds to a systematic error. According to section 5.6 (technical information on the h/e apparatus) in the Planck's Constant lab (see lab manual [1]), the h/e apparatus has a high impedance amplifier that allows us to measure V_{max} with a voltmeter. The high impedance means that the voltage coming in equals the voltage going out, ie. a photoelectric current comes in, goes out unchanged and gets measured by the voltmeter and we record V_{max}. While the apparatus is good it is not perfect and as the amount of time necessary to charge the capacitor increases there is some current drain which leads to a decrease in V_{max} as intensity decreases. With further investigation, it might be possible to show that this slight current drain effects the measured values of V_{max} in a manner that contributes to the consistently high experimental value for h. I would also like to repeat this experiment with better filters to see if I could get my 2nd order values for V_{max} to correspond even more closely to the 1st order results. It would also be interesting to find out what material Pasco Scientific uses for the cathode in the h/e apparatus and compare accepted value of the work function for that with experimental results.

### Acknowledgments

I would like to thank my lab professor, Dr Steve Koch, for helping me solve equipment problems, and for encouraging me to always question my results; Aram Gragossian, the teacher assistant in our lab, for his endless supply of questions designed to provoke me into thinking; Daniel Young ,my lab partner; and Darrell Bonn and Boleszek whose insights when they were working on this lab gave me the idea of using my sunglasses as a UV filter.

### References

- [1]: Michael Gold, "The UNM Dept. of Physics and Astronomy PHYSICS 307L: Junior Laboratory"(Fall 2006),experiment 5 (page 31-40)

- [2]: Eisberg and Resnick, "Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd edition" (1985, Wiley and Sons, Inc), Chapter 1 and Chapter 2, sec 2

- [3]: Taylor, "An Introduction to Error Analysis, 2nd edition" (1997, University Science Books): The details of the weighted average are in chapter 7.2, and the method of least-squares fitting is explained in detail in chapter 8

- [4] : Pasco lab manual, [link to a pdf of the Pasco lab Manual]

- [5] : CODATA, accepted value for Planck's constant [link to CODATA, Planck's constant]