Steve Koch 22:17, 21 December 2010 (EST):Good primary lab notebook. Good spreadsheet.
Safety
A high voltage source will be in use. Also, this lab is set up on a metallic surface. Do not have any unnecessary objects around this voltage source.
This lab is done primarily in a dark room. Be aware of your surroundings.
The thin graphite can be punctured by currents greater than 0.25mA, so it is suggested to keep this monitored.
Equipment
Calipers - Carrec Precision 6" Digital Caliper
TEL Universal Stand
Electron Diffraction Tube - Tel 2555
3B DC Power Supply 0-5kV - Model: 433010
Hewlett Packard Power Supply - 6216B
Set Up
A detailed set up can be found in Prof. Gold's lab manual. We opted to not include the current meter in the set up, as reaching dangerous levels is very unlikely (average amperage is a few micro-amps) and the lab suggests to visually inspect the layer of graphite for overheating anyways.
Procedure
Following set up, we proceeded to take our data. After the equipment was given proper time to heat up, two rings can be seen at the end of the bulb. This is what we shall be measuring for this lab. For the first two trials, we measured the outermost diameter of the circles (the observed circles had width to them) with the caliber. Starting with the maximum 5kV, I measured the diameter of the outer circle, and then the inner circle. Sebastian would then proceed to measure the circles at the same voltage. We decided to measure two trials at the same time to reduce any biases in the measurements (any systematic error I obtain would not be the same as Sebastian's, and the average between the two can be assumed to be more accurate than either either individual measurement). One trial includes the measurement for voltages ranging from 5 - 2.6(kV), at intervals of 0.2kV. For the last pair of trials, we measured the innermost diameter of each circle over the same range of voltage.
Note: We measured the outermost and innermost diameters of the circles because we were unsure as to what would be the more accurate measurement, or if even the average of the two is the most accurate representation.
Note: On the first day of the lab there was initially a bright line of light moving along the bulb. I am unsure as to what this was, or if it affected our data in any way. I just thought that it was worth mentioning since it was not present during the second day of measurements.
Calculations and Results
A link to my excel sheet can be found here: File:Diff elec.xlsx I know this couldn't possibly be the right way to upload an excel file, but it's all I know how to do. (Steve Koch 22:15, 21 December 2010 (EST):This works fine!)
[math]\displaystyle{ tan(\theta)=\frac{D_{extrapolated}}{2L} }[/math] Where [math]\displaystyle{ \theta }[/math] and L are as shown in the image.
Similarly
[math]\displaystyle{ tan(\theta)=\frac{D_{measured}}{2(L-x)} }[/math] Where x is this distance horizontally between where [math]\displaystyle{ D_{measured} }[/math] and [math]\displaystyle{ D_{extrapolated} }[/math] are observed.
Now let us look at the triangle in which this relationship occurs to find an equation for x
[math]\displaystyle{ x=R+/-\sqrt{R^2-\left(\frac{D_{measured}}{2}\right)^2} }[/math] It can easily be seen that x can not be greater than R, so we can throw out the R+ term.
We can now isolate [math]\displaystyle{ D_{extrapolated} }[/math] in terms of known values.
Where d is the lattice spacing, L is the distance from the diffraction grating to the end of the bulb, c is the speed of light, D is the extrapolated distance we calculated, e is the charge of an electron, V is voltage, and m is the mass of an electron. All of these are known values, so we can directly calculate d from here, but it is not entirely necessary at this point. The graphs provided in the excel chart represent [math]\displaystyle{ D_{extrapolated} }[/math] as a linear function of [math]\displaystyle{ V^\frac{-1}{2} }[/math]. Let us rewrite the above equation such that
It can now be easily seen that [math]\displaystyle{ \frac{4\pi{L}\hbar{c}}{d\sqrt{2emc^2}} }[/math] equals the slope of the best fit line in the graphs given in the excel chart. When we isolate d we get
For the trials in which we measured the outermost portion of the diameter we get [math]\displaystyle{ d_{inner}=.1787nm }[/math] [math]\displaystyle{ d_{outer}=.1039nm }[/math]
For the trials in which we measured the innermost portion of the diameter we get [math]\displaystyle{ d_{inner}=.2051nm }[/math] [math]\displaystyle{ d_{outer}=.1120nm }[/math]
The accepted values obtained from Gold's manual are 0.123nm and .213nm. Let us go back and compare these answers to the ones I would get from the equation where d is given by all known values and calculate the lattice spacing in the excel sheet. The work can be seen on the excel sheet. The results are as follows
For the trials in which we measured the outermost portion of the diameter we get [math]\displaystyle{ d_{inner}=.1889(5)nm }[/math] [math]\displaystyle{ d_{outer}=.1101(4)nm }[/math]
For the trials in which we measured the innermost portion of the diameter we get [math]\displaystyle{ d_{inner}=.2071(4)nm }[/math] [math]\displaystyle{ d_{outer}=.1178(3)nm }[/math]
The values are similar, but since I have no idea how to get an SEM from the first method, I will ignore my first values for d.
[math]\displaystyle{ d_{inner,outer}=.1889(5)nm }[/math] is 11.3% off. [math]\displaystyle{ d_{outer,outer}=.1101(4)nm }[/math] is 10.5% off. [math]\displaystyle{ d_{inner,inner}=.2071(4)nm }[/math] is 2.8% off. [math]\displaystyle{ d_{outer,inner}=.1178{3}nm }[/math] is 4.2% off.
While our values are relatively close, all of our values are several SEM away from the accepted values. There, as always, is some systematic error in our lab. Contributing factors may include the following:
The process of measuring the rings on a spherical surface using a caliber is an awkward thing to do. There is clearly human error in this process.
The lab stated L is equal to 13 +/- .2 cm. I neglected this fact. I strictly used the value L = 13cm. Could have been accounted for, but I neglected to do so.
We are measuring the pattern created by diffracted electrons using a metal caliber. I see no way how this could not affect the data.
References
Gold's manual obviously helped a lot with understanding the process of this lab.
Some credit goes to just about anyone who did this lab. I looked at nearly every person's lab notebook for help with the data analysis. (Steve Koch 22:16, 21 December 2010 (EST):Great! Science in action!)
Electron Diffraction
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