Roberto Sebastian Rosales/Notebook/Physics 307L/2010/10/13

Excel Spreadsheet

Roberto Sebastian Rosales Calculations

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

Safety

• We were using a high voltage power supply (5kV), so we had to be careful with how we handled it.
  • All the equipment was on a metal lab bench, so we had to be sure that there were no exposed wires or cables touching the surface of the table.
• Use the equipment properly - a current about 0.25 mA could damage the coating inside the electron diffraction tube.

Setup

• When we went into the dark room for our safety quiz, all the equipment was already set up for us. Prof. Koch then proceeded to disconnect everything, which meant we had to figure out how everything should be connected. Shortly after Koch left, Katie came to check on us and helped us connect everything according to the diagram found in Prof. Gold's Lab Manuual under the 'Electron Diffraction' Lab. (Steve Koch 21:42, 21 December 2010 (EST):Ha ha, well sorry! But it's part of the learning :) )
• Our initial setup contained the multimeter, but we decided to take it out of the circuit when Prof. Koch told us that we would more than likely stay under the max. amperage, and the nobody so far had exceeded the o.25mA.

Procedure

We followed the procedure from Prof. Gold's Lab Manual, but we had to figure out how we wanted to take our measurements. We started by setting the accelerating voltage to the maximum 5kV, and then measured the outside of the rings. Matt took the first measurements for both the inner and outer rings, and then I took measurements for those same rings at the same voltage. We then decreased the accelerating voltage by 0.2kV and recorded the next measurements in the same manner. We repeated this process until we reached 2.6kV (the rings were very difficult to measure at this voltage because they were very faint). We then thought that if we took measurements by measuring the inside of the rings and averaging it with the measurements for the outside method, we would get the measurement for the center of the ring. After thinking about whether this is what the average would actually yield, Matt figured out that we could not just average the two trials together since the data would not be from the same parent distribution. So, we have two trials which we analyze separately.

Data and Calculations

• Our Raw Data is as follows (as described above, there are two trials; the measurements in the first trial were taken by measured the outside of the rings, and in the second trial we measured the inside of the ring):

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdFB5QUZmQmlmSlBMMXBBNE5xN0V3NHc= |width=750 |height=550 }}

• I then had to correct my observed diameters, [math]\displaystyle{ D_{obs} }[/math], for the curvature of the glass. These corrected diameters, [math]\displaystyle{ D_{ext} }[/math], were calculated using some simple geometry. I included a picture of my own drawing that helped me do this correction.

• First, I needed to find the distance x. I did this using the following formula:
  • [math]\displaystyle{ \left(\frac{D_{obs}}{2}\right)^2 + (C-x)^2 = C^2 }[/math], or [math]\displaystyle{ x = C +/- \sqrt{C^2 - \frac{D_{obs}^2}{4}} }[/math]. [math]\displaystyle{ x }[/math] should not be greater than [math]\displaystyle{ C }[/math], so I will use the [math]\displaystyle{ - }[/math].
• Next, I calculated [math]\displaystyle{ \tan{\theta} }[/math] using the following formula:
  • [math]\displaystyle{ \frac{D_{obs}}{2(L-x)} = \tan{\theta} }[/math]
• Finally, I calculated [math]\displaystyle{ D_{ext} }[/math] using the following formula:
  • [math]\displaystyle{ \frac{D_{ext}}{2} = L\tan{\theta} }[/math]
• I had two values for each voltage since Matt and I both recorded a value at each voltage, so I took the average and found the uncertainty using the following formulas:
  • [math]\displaystyle{ \bar x = \frac{\sum_{i=1} x_i}{n} }[/math] and [math]\displaystyle{ SEM\ = \frac{s}{\sqrt{n}} }[/math], where s is the standard deviation for the sample (the formula is taken from here).
• The corrected diameters are as follows:

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdFNra3AtX1JDODdRb29TVVlxSEQ3ZFE= |width=1000 |height=300 }}

• I then plotted [math]\displaystyle{ D_{ext} }[/math] as a function of [math]\displaystyle{ \frac{1}{\sqrt{V}} }[/math], where V is the accelerating voltage and is in kV.
  • The Plots for each ring and each trial are as follows:

• Finally, I calculated the lattice spacing [math]\displaystyle{ d }[/math] using the following equation:
  • [math]\displaystyle{ d = \frac{4\pi L\hbar c}{D_{ext} \sqrt{2eVm_e c^2}}=\frac{2Lh}{D_{ext} \sqrt{2eVm_e}} }[/math]

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdFM5cHctVl9feHRiQ3R6ODVYQVNCM3c= |width=800 |height=650 }}

Results

 [math]\displaystyle{  d_{o,i} = (0.1889 +/- 0.0005) nm  }[/math]
 [math]\displaystyle{  d_{o,o} = (0.1101 +/- 0.0004) nm  }[/math]
 [math]\displaystyle{  d_{i,i} = (0.2071 +/- 0.0004) nm  }[/math]
 [math]\displaystyle{  d_{i,o} = (0.1178 +/- 0.0003) nm  }[/math]

where o,i = measuring the outside of the inner ring, o,o = measuring the outside of the outer ring, i,i = measuring the inside of the inner ring, and i,o = measuring the inside of the outer ring.

Actual Values

 [math]\displaystyle{ d_{outer} = 0.123 nm  }[/math]
 [math]\displaystyle{ d_{inner} = 0.213 nm  }[/math]

Error

For [math]\displaystyle{ d_{o,i} }[/math] we had an 11% error.
For [math]\displaystyle{ d_{o,o} }[/math] we had a 10% error.
For [math]\displaystyle{ d_{i,i} }[/math] we had a 3% error.
For [math]\displaystyle{ d_{i,o} }[/math] we had a 4% error.

• Taking measurements in this lab was very tedious, and at times frustrating. At high voltages, between 3.5-5 kV, the rings were very bright and it was easier to see where they ended. At the lower voltages the rings were faint and hard to measure. It seems that there was some systematic error in our data taking for the first trial, measuring the outside of the rings. This could be due to the fact that Matt was measuring the end of the ring differently than I was, as well as the fact that we were measuring the rings up against the glass bulb. It was very easy to align one side of the calipers, and while trying to align the other end, slide the first sided a little. I think that it had more to do with the face that we were measuring the outside of the ring. I think this because our error for the second method, measuring the inside of the ring, resulted in much less error in comparison to the actual values. Also, the center of the rings may not have been aligned with the center of the bulb. We tried to correct for this with the alignment magnet, but this did not always completely center the ring on the bulb. Another source of error could have come from the fact that when we would put the calipers close to the ring, the ring would sometimes shift a little and a weird pattern (which can kind of be seen above as the triangular shaped light radiating out from the center) would show up.

References

As always, Katie and Prof. Koch helped us out in this lab. Katie helped us with our initial setup, but we ended up redoing everything because we thought that we did it wrong (but we did not). Prof. Koch helped answer our questions about why there are only two rings present on the bulb. I also referred to David Weiss' Notebook for help on the geometry of the problem, but I actually did not see any geometry calculations on his page.