User:Brian P. Josey/Notebook/Junior Lab/2010/10/11

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Electron Diffraction
The purpose of this experiment is three fold. The first part is to demonstrate the wave nature of electrons. To do this we passed electrons through a diffraction grating made of graphite at varying voltages and made notes on the resulting patterns on a fluorescent screen. From these patterns, we moved on to the second goal, measuring the de Broglie wavelength of the electrons. This wavelength is given by the relationship λ=h/p. Finally, we attempted to measure the spacing of the graphite from our data, and compare this with the known values for the spacings. Once again, I worked with Kirstin and her notebook page for this experiment is here.

Equipment
This experiment uses the following pieces of equipment:


 * Digital Calipers
 * HP 6216B Power Supply
 * Electron diffractor with graphite in it, 2555
 * 3B U33010 DC Power Supply 0-5 kV

Note Do not ever have a greater current than 0.25 mA, this will puncture the carbon sheets in the graphite and ruin the equipment.

Set Up
Following the diagram in Gold's manual, and a slightly more clear one drawn by Darrell Bonn, see thumbnail, we connected the equipment in the following manner:


 * 1) F3 on diffraction tube to the left plug on the 6.3V on 3B Power supply,
 * 2) F4 on diffraction tube to the right on above "Heater" written in pencil
 * 3) * F4 also to the positive plug on HP power supply "Low Voltage Bias"
 * 4) C5 on tube to HV+ on 3B
 * 5) C7 to multimeter
 * 6) Multimeter to HV- on 3B
 * 7) Low Voltage bias negative to HV- on 3B power supply
 * 8) 3B power supply HV- to ground

Initially, we kept the multimeter attached to the circuit to monitor the current passing through the graphite. The goal was to ensure that the current would never exceed 0.25 mA. If it did, the electrons would damage the graphite. However, this led to a major issue, and a systematic error in our data collection. This issue is explained in exacting detail below in the procedural section, and anyone using my notebook in the future should check it out to avoid the same mistakes we made.



This is the image of our final set up of the equipment. Please note that the multimeter changed several times during this experiment. The heater and power supply for the electron gun is in the top left of the image, while the diffraction chamber and electron gun is in the lower center of the image.

Procedure
We actually did this experiment three times, in three different ways with varying degrees of success. For our first attempt we followed the lab manual to the letter. This resulted in serious systematic and random error, and we discounted the results. The second time, we adjusted our strategy slightly after receiving some help. This second attempt ended in a piece of equipment short-circuiting, shocking me, and more serious systematic error. The final attempt, was the most successful. While it is tempting to write off the first two attempts as failures and ignore them, I will still present the results for the sake of consistency, and as a way to help anyone reading this in future labs. Here is our data, as prepared by Kirstin in table form:

In this chart, the red values on the left are the first attempt, the purple values on the top right are the second attempt, and the green values on the bottom right are the third and final attempt.

First Attempt
For the first attempt at the experiment, we followed Dr. Gold's manual to the letter. After connecting all of the equipment as described above we turned all the switches to off on the equipment and then plugged them in. After plugging in the power supply for the heater and electron gun, we turned on the heater for the electron gun and let it warm up. When there was a faint orange glow coming from the gun, we knew that it had warmed up enough, and slowly began to turn up the power on the electron gun, increasing the voltage to 5000 V. At the same time we monitored the current passing through the graphite on the multimeter. The purpose for this was to ensure it didn't pass over 0.25 mA. If this happened, the electrons would punch a hole through the graphite rendering the diffraction tube useless. After pushing the potential all the way to 5 kV, we discovered that the current never exceeded 0.1 mA, and that all possible voltages were safe for the equipment.

We then proceeded to measure the rings that developed on the fluorescent screen of the diffraction tube. On the screen, there was a bright green dot in direct line with the electrons hitting the screen without diffracting, and two rings of faint green light where the diffracting electrons hit the screen. We began to measure the diameters of the rings using the digital calipers. Prompted by Paul Klimov's notebook we measured the diameters switching the location of the calipers on the ring. For the first measurement, I would place the caliper on the outside of the ring on the left side, and the inside of the ring on the right. I would then switch this, inside on the left, outside on the right, for the second measurement. Proceeding back and forth, we continued these measurements in a back and forth manner until we got ten measurements at 5000 V for the inner ring. We then repeated the measurements for the outer ring before changing the voltage.

After we got our measurements at 5000 V, we dropped down to 4500 V, adjusted the ring back to the center point with the magnet, and repeated the procedure. After dropping all the way to 4000 V, the rings were too dim to measure. We then jumped back up to 4900V, and worked down in increments of 100V, until we had five sets of data. At this point, we plotted the measurements, determined their average values and their errors. This is more precisely explained below in the calculations section. But the graphs for these sets of data were revealing:



This graph above is the data for the diameters of the inner rings.



This second graph is the data for the diameters of the outer rings. From these graphs it is clear that there was significant sources of error, and we did not proceed with this data. There is a systematic error in this data, but at the time we were not aware of it, and it become clear during our second attempt. However, it was clear that there was significant random error in the data. When we passed our measurement method by Koch, he asked us why we did not just measure the inner diameter of the ring. We argued that our method would average out to the true diameter to the highest intensity without directly measuring it, which is difficult to determine. Ultimately, we decided that this method was an equally valid one and we proceeded with it, however this was also the source of our systematic error.

Because the rings were so dim on the fluorescent screen, it was often difficult to determine where the rings began and ended. This led to a large distribution in the values that I measured, as indicated by the large error and the distribution in the averages that don't follow a concrete pattern. We decided to throw out this method and repeat the experiment with a different method. The second method was suggested by Koch, but after Dan Wilkinson was very success with it we adopted it. The second method is to measure only the inner diameter. This is advantageous in that it gives a consistent place to make measurements, and accounts for the varying amount the electrons diffract. When the electrons pass through the graphite, not all of them are traveling at the same speed. Some are traveling at the fastest speed possible because of the potential difference, while others slow down. The ones that are traveling the fastest will hit the screen at the inner part of the ring, while the slower ones will spread outwards. This allows the ring to slowly dim away from the center.

Second Attempt
With our new measurement technique, we repeated the procedure this time measuring only the inner diameters of the two rings. Once again, we measured the current of the electrons which never went above 0.1 mA. Because the rings were so dim, we started our measurements at 5000 V, the highest value and the one with the brightest rings, and moved down in 200 V increments. Unfortunately, the rings were still really dim and we had trouble making out the rings for our measurements. While measuring the rings at a voltage of 4800 V, the multimeter began to make a clicking sound every couple of seconds. Thinking that the connectors had came loose, and electricity was jumping between them, I pushed them in one by one to check the connections. When pushing in the negative lead into the multimeter, I was shocked by it. Because of this shock, and the fact that we didn't need to watch the current anymore, I decided to turn the multimeter off, and continue with my measurements. A couple of minutes later, while measuring the diameters for 4600 V, the multimeter made a hissing noise. The voltage on the power supply then rapidly dropped to around 1000 V, and the rings in the diffraction tube then collapsed and disappeared completely. We quickly turned everything off, and grabbed Katie and Koch to talk about it.

From talking to them, we determined that we had exceeded the voltage that the internal ground on the multimeter could take. While we were working at several thousand volts, the ground could only take a maximum of about 1400 V. We decided to remove the multimeter from the circuit because it was clear from our measurements that the current would never exceed the safety threshold, and we learned that we where the only group to go the extra step of monitoring the current throughout the experiment. After checking each of the components in turn, we found that only the multimeter was damaged, and we proceeded the experiment without it.

Third Attempt
The disappointment of having to repeat the experiment for a third time was quickly dashed as it became clear that multimeter was actually harming us. Without the multimeter, the rings on the flourescent screen were much sharper and much brighter. Whereas before we limited our measurements to the 4500 V to 5000 V range, we found that we could go from 2500 V to 5000 V, with very clear and sharp rings. The rings were such an improvement that we could read the digital calipers from the light of the diffraction tube without resorting to using a flashlight.

A quick look at our range showed that the rings decreased in diameter with increasing potential. The were also very sharp, and it was much easier to pinpoint the locations of the rings. Looking quickly at it we determined that the multimeter had misrepresented our results, and was drawing voltage from our diffraction tube. What we had measured at 5000 V with the multimeter was now what we had at around 3000 V. It was therefore very clear that having the multimeter in our circuit was introducing a serious systematic error, and it short-circuiting was actually a blessing in disguise. We then made measurements of the inner diameter of both rings from 5000 V to 2500 V in 500 V increments. We then used this data for our final calculations.

Calculations and Theory
In order to determine the spacing between carbon atoms in the graphite's crystal lattice, we used the de Broglie relationship between the wavelength, λ, and momentum, p, of the electron and Planck's constant, h:

$$ \lambda = \frac {h} {p} $$

After some work, we can show that the spacing is theoretically given by a simple function:

$$ d= \frac {4 \pi L h c} {D \sqrt {2 e V m c^2}} $$

The derivation of this formula is omitted for brevity, however, Paul Klimov has an excellent derivation that can be found here. In this formula, the physical quantities are given by:
 * d -the spacing between the carbon atoms in the crystal lattice. From Gold's manual, these values are known as 0.123 nm and 0.213 nm,
 * L -the length of the diffraction tube, which in this case is 13.0 ± 0.2 cm,
 * h -Planck's constant, which is 6.626 * 10-34 Js or 4.135 * 10-15 eVs,
 * c -the speed of light which is 2.99 * 108 m/s,
 * D - the diameter of a ring,
 * e -fundamental charge of an electron, 1.602 * 10-19 C,
 * V -potential at which we accelerated the electrons,
 * mc2 -the rest mass of the electron, which is 9.109 * 10-31 kg or 0.511 MeV.

While this formula would work perfectly for a flat screen, we have to adjust the formula slightly to account for the curvature of the glass. To do this, we calculated the error, y, in the distance observed, Dobs:

$$ y = R - \sqrt{R^2-\frac{D_{obs}^2}{4}} $$

We also know that the angle can be calculated by:

$$ tan(\theta) = \frac\frac{D_{obs}}{2}{L-y} $$

Which when we combine them, we get the corrected distance as:

$$ D=2Ltan(\theta) \, $$

Data Analysis
To calculate the spacing between the carbon atoms, we had to first find the average values and the standard errors. To do this I summed all the terms in MATLAB and divided by the number of terms, for their averages. Then using a built in function from MATLAB, I found the standard deviation, and dived by the square of the number of terms to find the standard error of mean. Here is my MATLAB code copied directly:

 A=[19.85 19.36 20.14 20.04 20.64   20.02 20.86]

a=sum(A)/length(A) s=std(A) sem=s/(length(A))^0.5

Here, A is the array containing data, a is the average of that array, s is the standard deviation, and sem, is the standard error of mean. Here the input array is for the inner diameter at 5000 V for demonstration. The rest of the code is simple built in commands. This gave me the following table:

The next step in finding the displacement is to plot the diameter of the ring as a function of the inverse square root of the voltage. I created two plots for this. The one on the left is for the inner ring, while the one on the right is for the outer ring:



I then find the slope through linear regression, and plug it into this formula:

$$ d = \frac{\frac{2hL}{\sqrt{2me}}}{Slope} $$

This gives the spacing for the inner ring as 0.245 nm as the best guess. This value then has a confidence interval from 0.242 nm to 0.246 nm. For the outer ring, this gives the value of 0.139 nm for the best value of the outer ring. We have a confidence interval for this value as 0.140 nm to 0.136 nm. These intervals of confidence were determined by entering the average value plus or minus the standard error of mean into the graph and finding the new slope there.

Conclusions
From our data, it is clear that we were able to take good data, but unfortunately our values where a little off. As I noted above, there were several different sources of error in this experiment. The first one was a source of random error, that being how we measured the diameter of the rings. At first, we measured in an alternating pattern, hoping to average out toward the accepted value. Unfortunately this did not work, however measuring only the inner ring was a successful substitute and removed this source of random error. There were also two other possible sources of systematic error.

The first source of systematic error was the multimeter. In a bid to do the experiment as properly as possible, and while keeping as close an eye on the equipment as possible, we used a multimeter to watch the current passing through the graphite sheet. This drew voltage away from the diffraction chamber. This subtracted voltage then severely lowered the energy the electron had for diffraction, altering the measurements. The second possible source of random error was the curvature of the diffraction chamber. Unfortunately, I was unable to find a way to properly eliminate this error, and because our rings were a little short, the angles were off. It is likely that this is the cause of our systematic error.

Whatever the cause of the error, however our values where still fairly accurate. For the first value, we had a separation distance between 0.242 nm and 0.246 nm with a best guess of 0.245 nm. The accepted value for this one is 0.213 nm, meaning that we were off by only 15%. This percentage off was the magnitude of the difference between the accepted and our best guess over the accepted value. For the second distance, we determined a range of 0.140 nm to 0.136 nm with a best estimate of 0.139 nm. The accepted value here was 0.123 nm, and we were off by 13%. These values indicate that we were fairly successful in our measurments.

Acknowledgments
Once again, I want to thank my lab partner Kirstin for all of her help, and both Koch and Katie for their help and thoughts about our ideas, questions and troubles. For this lab, I used Dr. Gold's manual as my primary resource, but I also used Dan's, Paul's, Darrell's, Alexandra's and Anastasia's notebooks. Darrell's was particularly useful for his circuit diagrams, an Paul's was a great source for calculations. Alexandra and Anastasia made things very simple, while Dan made fun of my random error in my first run. He also helped me get over it and change methods.


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