User:Paul V Klimov/Notebook/JuniorLab307L/2008/10/27
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Planck Constant LabIn this lab we will explore the quantum/photon nature of light TheoryThe photoelectric effect refers to the emission of photoelectrons from the surface of a conductor upon the absorption of light of certain frequency. From a theoretical perspective, the effect is important because it is a perfect example of the quantum theory of light. According to classical theory, electron emission should not only depend on the intensity of light, but it should also come with a certain 'time-lag' that would result from the build up of energy. Interestingly, neither of these predictions are observed, which makes the photoelectric effect very interesting. It turns out that electron emission is governed strictly by the frequency of radiation. That is, no matter what intensity of light it shone upon the surface of a conductor, photoelectrons will not be produced unless the frequency of radiation is above some characteristic frequency, determined by the conductor. It is also important to note that intensity is not completely irrelevant -- it will have an effect on the rate of photoelectron emission, which is the result of more photons interacting with electrons (not more successive absorptions by individual electrons, as would be predicted by classical theory). The photoelectric effect was accurately described by Einstein in the early 1900s. Einstein theorized that light was not continuous but rather discreet. More specifically, Einstein suggested that light came in quanta called photons. In the theory, each photon carries a certain amount of energy, which is given by the equation: E=hf, where h is the Planck constant, and f is the frequency of light. The equation summarizing the phenomenon, for which Einstein won the Nobel Prize in 1921, is as follow: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle hf=W_{o}+KE } This is clearly just a statement about energy conservation: The incoming photon must give the electron enough energy to surpass the binding energy of the conductor, and to provide it with some kinetic energy. Something that is commonly done, which will be done in this lab, is to measure what is called the 'stopping potential'. The stopping potential is a voltage difference that is set up to accelerate electrons against the direction of their emission. Of course for the stopping potential to make any sense, we have to incorporate the idea of the photoelectric effect into some sort of circuit, which can only be completed by the flow of photoelectrons from a cathode to some anode. Therefore, the stopping potential will be reached once the current in the circuit is zero. That is, the electrons will not have energy to cross the E field and to complete the circuit. Therefore, the stopping potential is essentially the measure of the electrons kinetic energy. This allows us to rewrite the above equation like: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle hf=W_{o}+eVo } and in a more useful way: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Vo=\frac{h}{e}f - \frac{W_{o}}{e} } This equation will form the framework for this experiment. Equipment
Overview of Procedure
DataExperiment 1 AWhite with Yellow Filter: 100% .797V 6.06s 100% .797V 6.34s 100% .797V 6.83s
80% .797V 9.22s 80% .797V 9.44s
60% .797V 13.28s 60% .797V 11.09s
40% .797V 14.88s 40% .797V 15.88s
20% .797V 26.09s 20% .797V 29.96s
100% .847V 14.16s 100% .847V 13.59s 100% .847V 14.78s
80% .847V 18.34s 80% .847V 19.31s
60% .847V 27.88s 60% .847V 29.59s
40% .847V 35.34s 40% .847V 34.88s
20% .847V 61.47s 20% .847V 58.12s Experiment 1 BTrial 1 First Order: Yellow: .717V Green: .847V Blue:1.502V Violet:1.725V UltraViolet: 2.077V
Yellow: .714V Green: .853V Blue: 1.500V Violet: 1.751V UltraViolet: 2.078V
Yellow:.731V Green:1.269V Blue:1.519V Violet:1.730V Ultra Violet: 2.071V Trial 2 Second Order: Yellow:.738V Green:1.274V Blue:1.520V Violet:1.729V Ultra Violet: 2.067V Experiment 1 B, Second Day1st order:
Possible Sources of Error
Post Experimental Data AnalysisData SelectionWe gathered data from both the first and second order emissions. However, there is clearly an inconsistency with the second order emissions and so they will not be used in any calculations. A possibility for this error is given above in the "Possible Errors" section. Least SquaresSeveral measurements of the stopping potential were obtained for each emission. I assumed that all measurements for any given fringe were independent of one another and also normally distributed around the actual value for that fringe. This allowed me to take the mean and standard error of the mean for each set of data points corresponding to each emission. After performing these calculations on MATLAB, the resulting values of the stopping potentials were plotted against frequency (see Figure 1a and Figure 1b). Error bars were produced for each point, corresponding to each respective SEM. A least squares line was then fit to the data. No constraints were imposed on intercepts for the line. This is an absolutely crucial point because the least squares line contains information not only about the plank constant (the slope), but also about the work function of the cathode material (the y-intercept). Although the SEMs are incredibly small, I tried to vary the slope within the error bars, which was done entirely by inspection (it was said in class that this is a reasonable approach). I weighed measurements with smaller SEMs more than those withInterpretation of Least Squares LineFrom the slope and y - intercept, we can calculate both the Planck constant and the work function of the cathode material. We can do so by assuming that the measured points follow the following equation, which was introduced in the theory section. (Vo is the stopping potential, f is the frequency of light, e is the electron charge, h is the plank constant, and Wo is the work function.) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_{o}=\frac{h}{e}f - \frac{W_{o}}{e} } The first thing that we must do is relate the slope to the planck constant, which is done like so: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Slope = \frac{h}{e} \rightarrow Slope\cdot e = h } Next, to find the work function, we must simply set f=0 in our first equation to find the following relationship: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_{o}|_{f=0}=-\frac{Wo}{e} } Therefore, the y intercept is the work function, conveniently in units of electron volts. ResultsThe following results were obtained from information contained within the least squares line. The data was extracted using the methods above, and the line itself is displayed in Figures 1 and 2. These were the results, both of which will be reported. Plancks Constant Measured From Slope: 7.26(14)e-34 J s Actual Planck Constant: 6.626e-34 J s Percent Error: 9.6% Number of SEMs from the accepted value: ~4.5 Corresponding Work Function: 1.635(48) eV Actual Work Function: 1.36(8) eV Percent Error 20% Number of SEMs from the accepted value: ~ 4 (we must be careful with this one because the 'actual' work function has uncertainty. The number i gave here is where my calculated work function overlaps with the SEM of the 'actual') DiscussionCharge Time Experiment^{SJK 14:54, 16 November 2008 (EST)}File:Yellowdecay.tif File:Greendecay.tif The results from our charge time experiments are plotted in Figure 2 and 3 The circuitry of the h/e apparatus is designed to give a direct measurement of the stopping potential. This is accomplished by the incorporation of an op-amp, in some way, into the circuitry of the device. When current flows through the circuit, charge builds up on a capacitor like circuit element which prevents more charge to accumulate by coulomb repulsions. Eventually, the element becomes completely charged (a relative term), and current no longer flows. The potential set up by the charge on the plate is then a measurement of the stopping potential. By measuring this charge time for various intensities of light, it should be possible to test the photon theory of light. We observed that charge times increased significantly as the intensity of light was decreased. This is expected because the charge time will be a function of the current, which in turn is a function of the intensity of light. We also noticed that the stopping potential for both the green and yellow emissions was essentially the same for all intensities. This is a result that firmly supports the photon theory of light, because the stopping potential should not be a function of intensity, but only the frequency of light. It was mentioned that the stopping potential should decrease when a higher order intensity filter is placed over the aperture. Strangely this did not happen to us, which could have been due to the fact that we re-charged the apparatus each time a new filter was placed. This could have either moved the cathode, or done something else unprescribed, to give us the results that we obtained. However, given that this should have happened, we should inquire as to what it means. Not to reject the quantum theory of light, we must question the structure of the apparatus, which clearly has problems with lower intensity light. As mentioned by Dr. Koch, charge could leak off from the collector plate (wrong terminology?). This would indeed cause one to record lower stopping potentials at lower intensities because the rate at which current leaks off could be comparable to the rate of photoelectron emission and thus collection on the plate. Plancks Constant ExperimentAlthough our data fits nicely to a straight line, it is clearly steeper than the 'accepted' slope of the line, which is given by the plank constant divided by the electron charge. In attempt to find the possible source of systematic error which could have caused this, I had MATLAB scale the stopping voltage (and thus slope) until a value reasonably close the plank constant could be calculated from the altered data. This actually worked, and I found that a scale factor of roughly .914 would return much better results: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_{altered} = V_{measured} \cdot .914 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{altered}=6.6236\cdot10^{34} J s } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle W^{altered}_{o}=1.49\cdot eV } The accepted values: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{actual}=6.626 \cdot 10^{-34} J s } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle W^{actual}_{o}=1.36 \cdot eV } Not only did this scale factor improve the slope, but it also gave us a more reasonable value for the work function of the cathode material. However, as suggested by Dr. Koch, it is unlikely that our voltage would be scaled by a single multiplicative factor (if there was a problem to begin with). Thus, I attempted to find another solution to the problem.^{SJK 14:56, 16 November 2008 (EST)}In my second attempt, I added a line of negative slope to our existing line, which would correspond to progressively larger error. The best results arose from subtracting n*.02624 from each voltage, where n varies from 0 to 4 in integers, with the lowest n corresponding to the lowest frequency emission. This value was chosen so that it would match up closely with the above value of the Plank constant, so that we could compare directly the effect on the work functions also, without added ambiguity. These were the results Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_{altered} = V_{measured} - n\cdot(.02624), n \in {0,1,2,3,4} } where n = 0 corresponds to the first point in V_{measured} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{altered}=6.6236 \cdot 10^{34} J\cdot s } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle W_{o}^{altered}= 1.43\cdot eV} Therefore, this method returned a slightly better value for the work function with the same Planck constant. This suggests that we could be headed towards a reasonable explanation for why our results turned out as they did. And given these observations, we must wonder what could cause us to overestimate the stopping potentials at higher frequencies in such a way. A possible error, one that i mentioned earlier, is that there could be some overlapping between fringes. This could give us a slightly larger stopping potentials for each consecutive reading. However, I admit that this is unlikely because of how the diffraction grating works. Another thing that was mentioned as a possibility by Dr. Koch, was that there could be charge leaks in the circuit. I am definitely not an expert on op-amps, so I am not exactly sure how this could occur given that our experiment was performed with constant intensity light. I believe I could see how this could happen if intensity would vary, which would probably cause big problems at lower intensities since the influx of photoelectrons could be comparable to any current leaks. However, once again, I am definitely not an expert on circuits so what I am saying could be incorrect. ^{SJK 15:01, 16 November 2008 (EST)} Perhaps what is most interesting to me is that the blue fringe (the third measurement from the left in each figure) is in best agreement with theoretical predictions. More specifically, this measurement comes in very close proximity of the actual line, which was constructed from the actual plank constant and work function. All of the fringes that surround this fringe have stopping potentials associated with them that are either too high or too low. At first glance, this makes me question the effects of possible charge leaks. The reason is that not only was the intensity of light always the same, but this data suggests that charge leaks are most pronounced at the extremes the visible spectrum. This is something that I could not explain, perhaps due to my inexperience with such a circuit. However, this observation arguably makes the possibility of fringe overlapping more plausible. If overlapping is happening, it is certainly coming in 'chunks'. That is, it wouldn't necessarily cause us to get bad data for each emission, which is clearly the case here.Given that I cannot come up with any reasonable explanations as to why the voltage would vary in such a way, I must reject the above adjustments, and stick with my initial results, which were still fairly reasonable. This is certainly an interesting problem however, and I would definitely like to know what is causing it. Improvements for the Future
MATLAB Code and AlgorithmsMATLAB CODE Ignore the image warning please. Just click on it to get to the file: File:PlankconstantPK.m Algorhithm used for mean: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bar{x}_{ij} = \frac{1}{N} \sum_{i=1}^N{x_{ij}} } Algorithm used for standard deviation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \sigma_{x} =\sqrt{\frac{1}{N-1}\sum_{i=1}^N(x_{i}-\bar{x})^{2}}} Algorithm used for standard error of the mean (SEM): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle SEM=\frac{\sigma_{x}}{\sqrt{N}} } Algorithm used for Percent Error: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Percent Error= 100\frac{|Actual-Measured|}{Actual} } |