User:Arianna Pregenzer-Wenzler/Notebook/Junior Lab/2008/11/26

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Theory
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
Equipment


 * electron diffraction tube (TEL 2501 universal Stand)


 * Power source (to both heater and anode) in kV (813 kV Power Unit)


 * multimeter for monitoring anode current (WaveTeck Meterman 85XT)


 * 2nd power source (function unknown at this point)(HP 6216B Power Supply)


 * Carrera Precision calipers (for measuring diameter of the rings)


 * Tel 2555.02 Magnet

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.

Data
Arianna's
 * note: the Voltage was read off of the dial on the power meter and a few times it fluctuated by ±.1 after data was taken

Dan's Data

Analysis
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.

When doing the data analysis I only used voltages between 4.6kV and 2.8kV, because though I claimed to get the voltage to 4.8 when I made my measurements, I noticed that these high voltages were not remaining constant, but dropping down, and my lab partner Dan who took the second set of data could not get voltages above 4.6kV.

Error
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.


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