Physics307L:People/McCoy/eDiffraction

Background Information
For my second full lab, I chose to do the Electron Diffraction experiment, in which I tested the De Broglie hypothesis. Louis de Broglie hypothesized that since electromagnetic waves could be interpreted as particles, that all particles of matter would also be able to be interpreted as waves, with a finite wavelength such that $$lambda=h/p$$, where h is Planck's constant and p is the momentum of the particle. In my experiment, I used an electron diffraction tube hooked up to a high voltage power supply and fired an electron beam at a graphite lattice, and measured the rings created by the diffraction of the particles. The goal of the experiment was to measure the spacing between atoms in the crystal lattice. more information can be found in my lab notebook which can be accessed here.

Data
In the experiment, I took ten data points for each ring with the anode voltage varying from 5000 Volts down to 3000 Volts, and averaged the values. My initial measurements were taken in inches, because that was the increments on the calipers that I had. In my MatLab calculations, the conversion from imperial units to metric can be found in the first section. The file containing my MatLab calculations can be found here, along with my final results (which will be stated later). The other calculations that I was required to do was to extrapolate the diffraction rings onto a flat surface, such that they can be measured linearly against each other. The derivation of the transformation is found in the Maple section linked from my notes involving the MatLab and Maple calculations in this section. The extrapolation allowed me to determine D, the diameter of the diffraction rings, and a plot of D against the anode Voltage, allowed me to see a linear relationship that demonstrated the difference in momentum and proved de Broglie's hypothesis.

Results
My results for this experiment came out of the formula given in Prof. Gold's lab manual that $$d=\frac{4*pi*L*hbar*c}{D*sqrt(2*e*V_A*m_e*c^2)}$$ where $$D=2*L*tan(\frac{arcsin(\frac{h0}{C})}{2})$$

In the formulas, L is the length from the diffraction grating to the screen, D is the diameter of the extrapolated rings, h0 is the height from the center of the rings on the diffraction tube, and C is the curvature.

After completing the calculations, I came out with values for the lattice spacing of:

$$d=.109(3)nm$$ and $$d=.203(6)nm$$

These values, when compared to the known quantities of:

$$d=.123nm$$ and $$d=.213nm$$

calculated as having a percent errors of:

$$\frac{.123-.109}{.123}*100=11.7%$$ and $$\frac{.213-.203}{.213}*100=4.9%$$

Because the standard error of the mean for the measurements was less than or approximately equal to .0002m and the uncertainty in the materials was .002m I decided that the error in the measurements was negligible, especially since the value was approximately .00005m for half the measurements, making it only 2.5% that of the possible error in the materials. Using that error, I calculated the percent error for the closest possible measurements within the margin of error and found that the percent errors were:

$$\frac{.123-.112}{.123}*100=8.9%$$ and $$\frac{.213-.209}{.213}*100=2.0%$$

With these values for the error, I am reasonably content with the accuracy of my measurements and feel that it is sufficiently accurate to state the lattice spacing in graphitized carbon is

$$d=.109(3)nm$$ and $$d=.203(6)nm$$

Error Reasoning
The greatest source of systematic error in this experiment is the measuring of the rings themselves. Because the diffraction pattern on the phosphorescent coating was so faint, it was quite hard to discern the pattern, and measuring the interior ring was extremely difficult for all taken measurements. The other cause of the systematic error is that the measurements were all taken by hand, in which there could be a great deal of difference between two people as the faint rings were not always visible all the way around, and one person could do a measurement, but not see the same thing as the other. This would be especially prevalent at lower voltages, as the rings became more faint, causing me to be unable to take a data set at 2500V as I could not see the rings clearly enough to be confident in taking a measurement.

In order to reduce the systematic error, the greatest improvement would be to get a new diffraction tube, as that would likely allow for us to see a brighter pattern that would make measuring the values significantly easier. The other improvement that I would make would be to find a different manner of measuring the rings, as measuring on a curved surface with calipers generates a great deal of systematic error. Possibly using a laser sight and measuring the angle between two beams, such as is done in land surveying would be significantly more successful in getting accurate measurements.