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Measurements of the Photoelectric Effect Used to Calculate Planck's ConstantBoleszek Osinski, Darrell Bonn
AbstractAs one of the simplest yet most conclusive observations of quantum phenomena, the photoelectric effect is an excellent subject for undergraduate experimentation. We recreate this famous effect in order to measure the value of Planck's constant. We follow the procedure presented in Prof. Gould's manual1 to measure the charge time of the capacitor in an h/e measuring apparatus and the maximum stopping potential for various intensities and frequencies of light emitted by a mercury lamp incident on a photoelectric material. The light is separated into individual spectra with a prism, but effects of overlapping spectra are noted. A discussion on the imperfections of the measuring device is presented along with a thorough error analysis of the measurements. We find a linear relationship between frequency of light and voltage produced by the measuring apparatus which fits the formula [math]\displaystyle{ eV=h\nu-\omega_0 }[/math]. With a final result of 7.1(48)*10^-34Js we conclude that our experiment confirms the photoelectric effect.
I. Introduction to The TheoryIt is arguably not an over exaggeration to say that Planck's constant is the most fundamental constant in quantum physics, because it arose from the earliest considerations of the quantization of energy. In response to the failure of the Rayleigh-Jeans formula to predict the finite energy density of a blackbody radiating at high frequencies (the ultraviolet catastrophe) physicist Max Planck decided to heuristically treat the energy as a function of frequency instead of representing it in terms of temperature. He could then treat the observed "hump" on the E density vs. fq graph as a version of the Boltzmann probability distribution. But Planck's great contribution came only when he realized that he could obtain the required cutoff (averageE → 0 as fq → ∞) if he modified the calculation leading from the probability distribution ((P(E)) to the average energy by treating the energy as if it were a discrete variable rather than a continuous one2. With this insight he converted the integral of E*P(E) to a sum and found that the energy had to be directly proportional to the frequency. This proportionality factor would later bear his name with the minuscule value h=6.626068*10^-34 J-s. Though Planck had been one of the first physicists to employ a discrete mathematics in describing radiation he was unsure whether or not energy was actually quantized or not. In a letter to R.W. Wood, Planck called his postulate "an act of desperation" (Quantum Physics, Eisberg & Resnick, pg. 21). It was not until later observations were made that the quantization of energy was thought to be a natural phenomenon and not just a clever mathematical interpretation. Among the most important of these observations was the fact that when light shined on a certain material electrons were emitted whose kinetic energies were independent of the light intensity and proportional to the light frequency. It was Albert Einstein, who was an early proponent of the reality of Planck's postulate even before Planck himself believed in it, who bravely made the step to arrive at a surprisingly simple linear formula (KE= hv + w) relating stopping potential to light frequency1. The determination of the slope and y-intercept of this formula will be the aim of the following experimental procedure. It should be noted that the accepted value quoted above was not obtained from a photoelectric experiment. The value is actually obtained from the Watt balance experiment, the gyromagnetic ratio of a shielded proton, the von Klitzing constant (from the quantum Hall effect), and the Josephson constant (from the Josephson effect), and the weighted mean of these results is reported as the accepted value3.
II. Experiment Light from a mercury lamp (Pasco OS-9282 Hg light source) is refracted through a triangular prism and bands of light are shined onto a photo-diode tube (Pasco AP-9368 h/e Measuring Apparatus) containing a material with a low work function. The photo-diode tube and its associated electronics have a small capacitance which becomes charged by the photoelectric current. When the potential on this capacitance reaches the stopping potential of the photo-electrons, the current decreases to zero, and the anode-to-cathode voltage stabilizes4. This final voltage between the anode and cathode is therefore the stopping potential of the photo-electrons. A standard voltmeter (Wavetek true RMS DMM) is used to measure the voltage. A. Measurement of Charge Time We measure the time it takes for the voltage reading of the h/e apparatus to reach a chosen value of cutoff voltage for two different frequencies of light. Each of the two spectral lines are sent through a variable transmission filter with intensities of 20%, 40%, 60%, 80%, and 100%. Measurements are performed with Yellow (5.18672E14Hz) and Violet (7.40858E14Hz) light incident upon the slit that opens up to the photoelectric material. Five data points are collected at each intensity for both yellow and violet. The data is averaged for each intensity, errors are calculated, and the results as seen in Figure 2 are plotted using MATLAB. These procedures are documented in the MATLAB code section entitled "Charge Time". For those who are interested in replicating our experiment, we provide our raw data in the Appendix. Detailed notes explaining the difficulties we ran into along with our process of solving them can also be found there.
B. Measurement of Stopping Voltage vs. Frequency We shine light on the photo-diode tube and wait until the DVM reading has stabilized. The first set of measurements is shown in Table 1. UV Violet Blue Green Yellow Max(V) 2.044 1.71 1.49 .847 w/filter .716 w/filter --Table 1. Stopping voltage is proportional to frequency.  We proceed to make two measurements of stopping potential of the first and second order light bands, resulting in four total, which are shown in Figure 4. The raw data for these measurements and primary thoughts on the data are available in the Appendix. The observed linear behavior satisfies the energy relation for the photoelectric effect
but only after an anomalous effect due to the overlap of unseen spectra in the UV range is noticed and eliminated. Our solution to the problem of overlapping spectra is provided in the appropriate section below.
C. Measurements of Maximum Voltage vs Intensity of LightWith light shining upon the slit we measure the maximum voltage reached at consecutively higher intensities for Violet and Green spectra. Violet Green 20% - 1.487V 0.846V 40% - 1.485V 0.850V 60% - 1.489V 0.852V 80% - 1.486V 0.852V 100% - 1.505V 0.856V --Table 2. Max V increases with intensity. A noticeable trend of increasing V with increasing fq is observed as seen in Table 2. We discuss why this occurs below in the Results and Discussion section. III. Determination of h We have four sets of data to analyze (Figure 5.); two sets for the first order spectrum and two sets for the second. It should be noted that after considering the possible reasons for the existence of an unexpectedly high stopping voltage for the green band of the second order spectrum (discussed later) we redid those measurements and obtained values of .842V for the first run and .849V for the second run. These values are used in the analysis. Using MATLAB we make plots, perform linear least square fits, and determine h and ω_o for each of the four data sets. We scale the Voltage data by e (electron charge) so one can deduce from the equation
that the slope is h and the y-intercept is ω_o. The standard deviation of the mean of our result is propagated according methods explained in Chapter 8 of An Introduction to Error Analysis5 by John R. Taylor. I explain how Taylor's method is applied to the data in the Appendix.
IV. Results and DiscussionWe have presented graphs next to their respective sections above so I will not repeat them here. A. Charge TimeWe notice that the intensity vs. charge time plot for violet (see Figure 2.) exhibits an unexpectedly high charge time at 20% intensity along with the largest error bar, so it is reasonable to ignore that point. The overall standard deviation of the mean including this stray point is 0.2423s Without it the standard deviation of the mean reduces to 0.2036s This is the value that we choose to present as the error in charge time measurement. Our results show that as the intensity of light increases the time required to charge the capacitor in the h/e apparatus decreases, though this relationship does not appear to be linear as we previously thought it would. However, this should have been expected because a capacitor does not charge linearly over time, but quickly at first and slowly at the end, following a negative exponential determined by the physical properties of the material and the magnitude of current coming its way. Our graph actually appears to be approximating a negative exponential of some kind. Though I cannot be sure at the moment what is the exact formula for this graph I can comment that it follows very nicely from the quantum theory of light. Were we to imagine millions of photons colliding with correspondingly millions of electrons in a material we would expect more electrons to be emitted when more photons are incident upon them (as long as the photon energy is large enough). More electrons correspond to higher current, which in turn correspond to lower charge time because the capacitor becomes saturated more quickly. A skeptic might ask why it is that the charge times for each intensity are shorter for higher energy (yellow) light than lower energy (violet) light when we would expect that emission current should only be proportional to intensity, not energy. I would respond that a higher energy light causes electrons to be emitted at higher velocities, so even though a cross-section of the yellow and violet light induced electron beams would have the same density of electrons at the same intensity, the electrons from the yellow light travel from the photoelectric material to the capacitor in a shorter time than do those emitted by the violet light, thus resulting in shorter charge time. B. Stopping Voltage vs. FrequencyThe observed behavior indicates to us that the frequency dependence of the stopping voltage is linear. A wave's energy is proportional to its intensity^(1/2)6, but our measured voltages are not, thus our experiment supports a quantum theory of light. If we assume that the quantum theory is correct then electrons are "knocked out" of atoms in the material because individual particle-like photons collide with them with an energy larger than their bonds. Then we would expect that even a single photon of sufficient frequency will impart enough energy on a bound electron to emit it with the same energy as a million photons would a million electrons. This expectation is confirmed by the observation that a lower intensity of light results in a lower emission current, which simply corresponds to lower number of emitted electrons, because the lower intensity light contains fewer particle-like photons. C. Effect of Overlapping SpectraWe notice that the green band results in a much higher stopping potential in the 2nd order than in the 1st. After speaking to Prof. Koch about this anomalous behavior we come to the conclusion that there must be an unseen overlap of spectra as shown in Figure 4. When we scan the h/e apparatus across the green band we notice that the voltage remains high even when we are in an apparent “dark” region on either side of green. When we cover the slit with both green and yellow filters we find that the stopping voltage of the green line is 0.842V. This result is very close to the measurement of the first order green line so we choose to use it instead of the out of place 1.242V. A second measurement of the second order green stopping voltage yields 0.849V. We have reason to believe that whatever unseen light made the voltage rise to 1.79V has been filtered out by the yellow filter. The overlap of higher order UV spectra with lower order green is in fact alluded to in section 5.3.1 of Prof. Gould's manual1. We successfully filter the unwanted light with reading glasses that are specifically intended to filter UV light, obtaining a value of 0.847V. This confirms that unwanted overlap is due to UV from a higher order spectrum. D. Current Leakage as Cause of Maximum Voltage Dependence on IntensityWe find a trend towards increasing voltage as intensity increases, but the quantum theory of light predicts that stopping voltage should only increase as a function of increasing frequency, not intensity. Rather than assuming that we have found evidence against the photoelectric effect, we believe the rise in voltage with intensity occurs because the h/e apparatus exhibits an imperfection consisting of a current leakage. While the impedance of the zero gain amplifier is very high (10^13 W), it is not infinite and some charge leaks off4. Thus charging the apparatus is analogous to filling a bath tub with different water flow rates while the drain is partly open. At lower frequencies the total current in the apparatus is lower than at higher frequencies so the leakage constitutes a larger fraction of the lower frequency current and is therefore more noticeable to the DVM. I predict that this results in the left end (low frequency) of the V vs. fq graphs to slump down further than it should, thus increasing the slope. The voltage readings from the unfiltered light should therefore most accurately reflect the actual kinetic energy of the emitted electrons because the current leak constitutes the smallest percentage of emission current possible. An extra lab session was spent as an effort to measure this effect. Two measurements were taken at 100% intensity to characterize the random error and then 20% and 60% measurements were made. As seen in Table 3 Intensity h(10^-34 J*s) 100% 6.91 100% 7.13 60% 7.10 20% 7.09 --Table 3. Extra data has been analyzed and the resulting slopes are shown. the results do admit a noticeable trend (if the first data point is ignored) in the intensity vs slope relationship which seems to imply that lower intensities result in more accurate readings (closer to 6.626), a result that does not meet my expectations. However, the random measurement fluctuation and the resolution of an average standard deviation of 4.46*10^-36 J*s does not permit us to characterize the rather minute effect precisely (minute because the current leakage itself is indeed very small) and a much larger pool of data would be needed to minimize the random errors. E. Determination of hTable 4 displays the final results for th Run 1 1st order: h = 7.1921e-034Js, stdm = 2.9322*10^-36Js || ω = 1.6097eV, stdm = 0.0123eV 2nd order: h = 7.1812e-034Js, stdm = 7.2352*10^-36Js || ω = 1.5954eV, stdm = 0.0304eV Run 2 1st order: h = 7.2338e-034Js, stdm = 2.9684*10^-36Js || ω = 1.6246eV, stdm = 0.0125eV 2nd order: h = 7.1433e-034Js, stdm = 6.0527*10^-36Js || ω = 1.5807eV, srdm = 0.0254eV --Table 4. 1st and 2nd order calculations of h(slope) and ω(y-intercept). The average h and omega are: h = 7.1876*10^-34Js, stdm = 4.7971*10^-36Jsω_o = 1.6026eV, stdm = 0.0202eVAccepted Value: h=6.626068*10^-34In the Teacher's Guide section the Pasco manual4 the experiment is performed using the same apparatus. They obtain h = 6.6406E-34 (0.22% off), ω_o = 1.412 eV. Though they would have had a current leakage in their apparatus they most likely performed their measurements with a completely new apparatus. I would initially attribute the discrepancy between our results mostly to the possible degradation of our apparatus over time, consisting of changes in the photoelectric material and a change in resistance of the op amp due to persistent heating. But after comparing our results to those of other students it is evident that the very same apparatus has yielded results on both sides of the accepted value. Since the overshoot is consistently reported in all eight of the trials performed by us I must conclude that the consistency in our results was due to the repetition of a systematic error. The smallness of our statistical error reinforces this fact. Our value is 5.6153*10^-34 units away from the accepted value of h=6.626068*10^-34, which is many more than just two SDOMs from the accepted value and thus not withing the 68% confidence interval. In fact, the accepted value is about 11.7 SDOMs away from our value, so there is virtually no chance that we would obtain it with the measurement techniques we used. The main source of systematical error must have been the accuracy with which we aligned each measurement. AcknowledgmentsI thank Prof. Steve Koch for teaching me the proper way to analyze data and propagate error as well as for stimulating us to think about the peculiarities of the h/e apparatus current leakage. Thanks also goes out to Mr. Darrell Bonn, whose experience in scientific instrumentation allowed us to quickly pinpoint reasons for inconsistent measurements and who was instrumental in the setup of the experiment. AppendixReferencesSJK 04:27, 8 December 2008 (EST) You will need some references to original peer-reviewed research in your final version. You can move me and Darrel to the acknowledgments section (which you will need to create)...anything in the references section should be specifically mentioned at least one time in the text of the paper or a figure caption.
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