User:Arianna Pregenzer-Wenzler/Notebook/Junior Lab/2008/11/26
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In this experiment we are making use of the fact that particles (in our case electrons) have wave properties. Using a beam of electrons and the results of x-ray diffraction which showed that the diffraction pattern of x-rays through a lattice crystal could allow for the understanding of the way that the atoms in the crystal were spaced, we can use the Bragg condition for constructive interference, d*sinθ = mλ to determine the spacing between atoms in a graphite target.
Set Up and Equipment
Everything was set up when we came in. Our electron diffraction tube is attached to two power sources, but the 813 kV Power Unit provides voltage to the heater and the anode so at least to start we are leaving the second power source off. Looking in the labs of other students only one lab (Chad's) made reference to what could have been a second power source, even though he did not say so directly. We turned on the power source (the heater supply) and allowed it to warm up then we turned off the lights and increased the voltage to the anode from zero up to its max (around 5kV) to see what we had. We had green light which we center in the tube using the magnet. We then tried to discern circles, which we can see at max voltage but which is to quickly become almost indiscernible.
We did not use the secondary power source, but we did experiment with it before making the choice not to use it. From what I read in Chad's notebook this secondary source could be used to boost the voltage of the primary source, but it did not seem to allow us to see the electron beam any more clearly so we did not complicate things by using it.When our voltage was at its maximum value we could see an outer ring and a solid looking inner circle. Knowing that the inner circle was not solid but actually a ring with an inner and outer radius, I was able to discern a ring at high voltage, but as the voltage decreased, the ability to see any real inner and outer radius was more wishful thinking then reality. I mention this inner and outer radius because from reading the notebooks of other students I saw that Dr Koch had recommended that for the greatest accuracy the diameter of the rings should be measured from the the inner radius on one side of the ring to the outer radius of the other side of the ring.SJK 03:01, 18 December 2008 (EST)
What I'm trying to say here is that we really could not see these rings well enough at any voltage to make any measurements with any real confidence. Both myself and Dan gave it our best shot, and I found it interesting that our data is fairly linear (I say linear, what I mean is the diameter is basically increasing with decreasing voltage).I can say this, it was so hard to see the rings that we really did not have to worry that we were half consciously adjusting our measurements of the diameter of the rings to fit a preconceived idea of what we expected to see. If we looked at the rings at max voltage then decreased the voltage and saw the rings' diameters expand then as we made small adjustments to the voltage it would have been tempting to remeasure the diameters if we got a diameter that was not in line with what we expected for a given voltage change. In our caseSJK 03:02, 18 December 2008 (EST)
The first thing we were asked to do is to show that equation given by the lab manual for the distance of the lattice spacings in the graphite could be obtained using the de Broglie relation. The de Broglie relation is just the relation of the wavelength of a particle to its momentum, λ = h/p, and recalling that p^2/2m is equal to the KE of the particle. We can get the KE of our electrons by remembering that the work done to accelerate the electrons is equal to e*V, where we know V. What I was unsure of was the relation between d (the width of the lattice spacings) and our known value L and our measured value D. In referencing my University Physics Text, and more importantly the notes of my phys262 teacher Mark Morgan-Tracy I was able to relate the radius (D/2) that we measured and L to sinθ using some geometry and small angle approximations, to finally arrive at the equation given by the manual. I sketched out the geometry and scanned it into my computer, so I will leave the detail to the picture rather than trying to describe it here, please click on the image to actually read it. I am a little unsure if my treatment of the diffraction is entirely correct for the case of electron diffraction through a crystal structure. My memory of this area of my physics was not very good and from the resources I consulted I couldn't get an understanding that I completely confident of.
After doing the math/research to determine that the given equation for d did not appear from nowhere, the actual analysis was rather straight forward. I just made sure my units were all compatible and then I plotted my calculations for d for each voltage along with its error bar, and also computed d using a least squares fit to plot 1/d as the slope of D = constant*1/V^2.
There are two systematic errors involving the diffraction tube that you could definitely correct for if you wanted to try to improve the accuracy of your data. The most important error to note is that the formula for d includes to given value of L as the distance from the graphite target to the end of the tube, but the end of the tube is curved so L is actually the maximum distance and since the electrons are being scattered some angle θ they are not going this max distance before running into the end of the bulb. I did not add this error into my calculations, but it is important to notice the the distance D that we measure in this experiment corresponds to a shorter length, l, then the L given by the manual. Of possibly less importance as far as a contribution to the error, but still important in the quest for accuracy is the thickness of the glass bulb (diffraction tube), this thickness (which the manual gives as 1.5mm) will increase your measured D by some small quantity.
I did not try to factor either of these errors into my calculations, because I was short on time and even the smallest change in my programs takes me more time than I want to own up to, and more so because we had such a hard time seeing the rings that taking these small details into account did not seem necessary.