User:Paul V Klimov/Notebook/JuniorLab307L/2008/10/27

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 * style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]] Planck's Constant
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=Planck Constant Lab= In this lab we will explore the quantum/photon nature of light

Lab Summary

Theory
The 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:

$$ hf=W_{o}+KE $$

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:

$$ hf=W_{o}+eVo $$

and in a more useful way:

$$ Vo=\frac{h}{e}f - \frac{W_{o}}{e} $$

This equation will form the framework for this experiment.

Equipment

 * PASCO OS-9282 Scientific Hg Light Source
 * PASCO AP-9368 h/e Device (1.36±.08eV work function, Source: PASCO Technical Information)
 * Wavetek 85XT RMS DVM
 * Tektronix TDS 1002 Digital Oscilloscope (not used)
 * 2 9V OP-Amp Batteries

Overview of Procedure

 * The apparatus is self contained and there was no need for us to set up any circuit. The op-amp in the h/e device is powered by a couple of batteries.
 * After the Hg vapor lamp was turned on, fringes were focused on the h/e apparatus using the movable lens/diffraction grating.
 * Prior to taking any measurement, the h/e apparatus 'tube' was opened so as to make sure that the fringe of interest was focused on the white aperture (which presumably goes to the cathode).
 * Before taking any measurements, the 'tube' was closed to prevent any unnecessary ambient light from entering the apparatus.
 * Filters were used when taking measurements of the yellow and green fringes, to block out ambient light.

Experiment 1 A
White with Yellow Filter: 100% .797V 6.06s

100% .797V 6.34s

100% .797V 6.83s

80% .797V 9.13s

80% .797V 9.22s

80% .797V 9.44s

60% .797V 12.21s

60% .797V 13.28s

60% .797V 11.09s

40% .797V 16.75s

40% .797V 14.88s

40% .797V 15.88s

20% .797V 26.97s

20% .797V 26.09s

20% .797V 29.96s

Green Light Green Filter:

100% .847V 14.16s

100% .847V 13.59s

100% .847V 14.78s

80% .847V 17.75s

80% .847V 18.34s

80% .847V 19.31s

60% .847V 30.07s

60% .847V 27.88s

60% .847V 29.59s

40% .847V 35.41s

40% .847V 35.34s

40% .847V 34.88s

20% .847V 53.28s

20% .847V 61.47s

20% .847V 58.12s

Experiment 1 B
Trial 1 First Order:

Yellow: .717V

Green: .847V

Blue:1.502V

Violet:1.725V

UltraViolet: 2.077V

Trial 2 First Order:

Yellow: .714V

Green: .853V

Blue: 1.500V

Violet: 1.751V

UltraViolet: 2.078V

Trial 1 Second Order:

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 Day
1st order:


 * Yellow: .719


 * Green: .857


 * Blue: 1.510


 * Violet: 1.735


 * Ultra Violet: 2.075

Possible Sources of Error

 * Focusing: The rails for the lens are not long enough to focus the spectra lines well for each order of interference maxima. Although it was fine for the first order interferences (which are perhaps the most important), the second order interference patterns could not be resolved.
 * Small Aperture: The aperture for the cathode was fairly small and so the stopping potential would fluctuate if the apparatus would move even the slightest bit, which could be a possible source of error.
 * Ambient Light: Even though we had filters to 'block out' light for the yellow and green emissions, we cannot be certain that they are 100 percent effective. If they are not perfect, I believe that this would have a huge effect on our results. Given that light in the visible spectrum clearly produced photoelectrons I see no reason why ambient light wouldn't interfere with our readings. In fact, we found on the PASCO website (PASCO) that the estimated work-function of the device is about 1.36eV. Doing some quick calculations, it is easy to see that anything will produce photoelectrons with a wavelength under 900nm. That is clearly a huge problem, especially at lower frequencies where the photons emitted from the tube might have lower energies than the photons 'carrying' the ambient light. Therefore, at low frequencies we might just be measuring the stopping potential of ambient light, which was measured at 1V. However, given that we didn't test the filters (and I'm not quite sure how we would if we wanted to), I will assume that the filters worked perfectly.
 * Button Position: An annoyance was that the discharge button was attached to the 'photoelectric device'. Whenever the button was pressed it would move the device slightly, which would cause some fluctuations in stopping potential. Although this was not a big problem when doing the experiment in which we calculated the Plank constant (for obvious reasons), this was a problem when measuring the charge times.
 * Angular spread : Very clearly there were some inconsistencies when measuring the second order maxima. This was a problem particularly with the green emission, which was measured to have an unusually high stopping potential associated with it. After some thought I think I might have a possible explanation: We know that mercury has non-visible emissions in the UV range of wavelengths. There is a chance that some of these UV emissions coincided with this band from another interference order. Due to the mechanism of the diffraction grating, we can rule out some possibilities immediately: It could not have been from the first or second order emissions on the side of measurement (the right side in our case). The reason is that the lower wavelength emissions always appear to the left of higher wavelength emissions.

Perhaps one way to test this possibility out would be to move the mercury vapor lamp further or closer from the cathode, while measuring the green line. In doing so, the angular dependence of the interference pattern should cause this UV fringe to move to one side of the green fringe. However, another possibility would be that this would just 'blur' out the UV emission, which would still cause one to record the same stopping potential (because the lower intensity of the light would still emit the same energy electrons as the higher intensity light). This is actually quite interesting, and I am curious if such a simple manipulation could resolve this problem.

Data Selection
We 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 Squares


Several 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 with smaller larger SEMs, once again by inspection.The largest correction that I thought reasonable for the slope was reported as the best 68% confidence interval. Error propagation was not performed (and would be unnecessary, I believe), given that the slope of the least squares line is a direct measurement of the plank constant.

Interpretation of Least Squares Line
From 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.)

$$ V_{o}=\frac{h}{e}f - \frac{W_{o}}{e} $$

The first thing that we must do is relate the slope to the planck constant, which is done like so:

$$ Slope = \frac{h}{e} \rightarrow Slope\cdot e = h $$

Next, to find the work function, we must simply set f=0 in our first equation to find the following relationship:

$$ V_{o}|_{f=0}=-\frac{Wo}{e} $$

Therefore, the y intercept is the work function, conveniently in units of electron volts.

Results
The 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')

Charge Time Experiment
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 Experiment
Although 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:

$$ V_{altered} = V_{measured} \cdot .914 $$

$$ h_{altered}=6.6236\cdot10^{34} J s $$

$$ W^{altered}_{o}=1.49\cdot eV $$

The accepted values:

$$ h_{actual}=6.626 \cdot 10^{-34} J s $$

$$ W^{actual}_{o}=1.36 \cdot eV $$

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.

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

$$ V_{altered} = V_{measured} - n\cdot(.02624), n \in {0,1,2,3,4} $$

where n = 0 corresponds to the first point in V_{measured}

$$ h_{altered}=6.6236 \cdot 10^{34} J\cdot s $$

$$W_{o}^{altered}= 1.43\cdot eV$$

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.

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

 * My main problem with this lab is that there is little room for improvement. All we really had to do was flip a couple of switches and retrieve data, which literally came from a black box. I think this lab would be more interesting if we had to put together some sort of photoelectric apparatus ourselves, which doesn't seem terribly hard (I could be wrong because I have not attempted it). The debugging could take a bit of time, but I would think that it could be doable in a couple of class periods, especially because it would allow one to skip the charge experiments, which took up most of the time anyways. A huge advantage that could come from doing this would be that it would require us to find the stopping potential manually. This would remove the possibility of problems with charge leaks, and other unforeseen errors with the 'black-box' circuit. While this would inevitably introduce other uncertainties, it would be a great experience.

MATLAB Code and Algorithms
MATLAB CODE Ignore the image warning please. Just click on it to get to the file:

Algorhithm used for mean:

$$ \bar{x}_{ij} = \frac{1}{N} \sum_{i=1}^N{x_{ij}} $$

Algorithm used for standard deviation:

$$\sigma_{x} =\sqrt{\frac{1}{N-1}\sum_{i=1}^N(x_{i}-\bar{x})^{2}}$$

Algorithm used for standard error of the mean (SEM):

$$SEM=\frac{\sigma_{x}}{\sqrt{N}} $$

Algorithm used for Percent Error:

$$Percent Error= 100\frac{|Actual-Measured|}{Actual} $$


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