Physics307L:People/McCoy/Formal

An analysis of the Graphite Crystal Lattice from Electron Diffraction

Author: Chad A. McCoy

Experimentalist: Chad A. McCoy

University of New Mexico

Department of Physics and Astronomy

Albuquerque, NM 87131

cmccoy1@unm.edu

Abstract

In this experiment, I measured the internal lattice spacing of graphite using the properties of electron diffraction and measuring the rings formed by the diffracted electron beam at a known distance from the graphite lattice target. Making the measurement of the center-center ring diameter, allowed for me to form an extrapolated diameter as if the measurements were taken on a flat surface and calculate the spacing relative to each of the two diffracted rings. Doing so I calculated answers of d=.109(3)nm and d=.203(6)nm, compared to the accepted answers of d=.123nm and d=.213nm.

Introduction

The concept of electron diffraction originated the doctoral dissertation of Prince Louis-Victor de Broglie in 1924.

Materials and Procedure

Figure 1: Apparatus for electron diffraction experiment

Materials

• Hewlett Packard model 6212B power supply
• Teltron 2555 electron diffraction tube
• Teltron 2501 universal stand
• Teltron Limited 813 kV power unit
• WaveTek Meterman 85XT digital multimeter
• 8 - 4mm banana cables

Procedure

• With the electron diffraction tube in its stand, connect using banana cables the stand, two power supplies and multimeter as seen in figure 1, with the multimeter connected between the C5 port on the diffraction tube stand and the negative high voltage port.
• Turn on the Teltron power supply with the slider for the high voltage at zero
• After 1 minute, slowly move the slider to the top, setting the high voltage at 5kV
• With the voltage at 5kV, take ten data points, the odd points with the bias voltage as set using the HP power supply at 10V and even points with the bias at 5V
• Take the measurement of the small ring then the large ring, then adjust the bias and check the anode voltage to make sure it is correct for the tests
• Measure the rings from the inside of one edge of the ring to the outside of the other edge, so as to approximate a center-center measurement of the radius
• Repeat with the voltage at 4.5kV using 5V and 2.5V as the biases
• Repeat with the voltage at 4kV using 2.5V and 0V as the biases
• Repeat with the voltage at 3.5kV using 1V and 0V as the biases
• Repeat with the voltage at 3kV using 1V and 0V as the biases (if possible, if unable to see ring at 1V bias, use 0V for all measurements)

Results and Errors

Data

Ring Diameter (inches) for Anode Voltage

	5 kiloVolts		4.5 kiloVolts		4 kiloVolts		3.5 kiloVolts		3 kiloVolts
Trial #	Ring 1	Ring 2	Ring 1	Ring 2	Ring 1	Ring 2	Ring 1	Ring 2	Ring 1	Ring 2
1	0.839	1.576	0.945	1.662	0.966	1.780	1.023	1.944	1.181	2.064
2	0.856	1.559	0.909	1.651	0.976	1.753	1.044	1.960	1.133	2.022
3	0.882	1.580	0.922	1.681	0.946	1.778	1.059	1.963	1.173	2.021
4	0.895	1.575	0.928	1.652	0.942	1.784	1.063	1.976	1.127	2.031
5	0.898	1.586	0.960	1.655	0.973	1.736	1.024	1.941	1.156	2.051
6	0.859	1.595	0.913	1.659	0.984	1.748	1.021	1.938	1.128	2.061
7	0.863	1.572	0.891	1.664	0.988	1.759	1.022	1.956	1.161	2.054
8	0.886	1.582	0.915	1.662	0.942	1.781	1.039	1.941	1.131	2.064
9	0.884	1.580	0.922	1.661	0.981	1.755	1.052	1.936	1.121	2.072
10	0.871	1.599	0.914	1.662	0.939	1.766	1.026	1.947	1.101	2.003

Because these ring diameters are based on a curved surface, I had to calibrate them to take the curved surface into account, along with converting them from inches to metric units so they could be used in the calculations and return standard units.

To calculate the spacing distance I used the formula: [math]\displaystyle{ d=\frac{2{L}{h}}{D\sqrt{2{m_{e}}{e}{V_{a}}}} }[/math] with [math]\displaystyle{ D=2{L}{tan(\frac{arcsin(\frac{h_{0}}{C})}{2})} }[/math] in which [math]\displaystyle{ V_{a} }[/math] is the anode voltage, [math]\displaystyle{ h_{0} }[/math] is the uncalibrated height of the rings, L is the distance from the graphite to the end of the diffraction tube, and C is the radius of curvature of the diffraction tube.

I did my calculations using the program MatLab, with the results published to a word file that can be accessed here

Final Result

Calculated spacing by voltage

Voltage	d (outer)	d (inner)
3kV	.1081nm	.1993nm
3.5kV	.1053nm	.2034nm
4kV	.1097nm	.2051nm
4.5kV	.1019nm	.2023nm
5kV	.1102nm	.2027nm

From these values I was able to find the mean and standard error of the mean, and by doing so I was able to develop a confidence interval in which the known value should lie. By doing that I came up with the final value for the lattice spacing of: [math]\displaystyle{ d=.109(3)nm }[/math] and [math]\displaystyle{ d=.203(6)nm }[/math]

The error margin given in my final answer is that of one standard error of the mean, thereby being a 68% confidence interval for the "true" value.

Comparing my answers to the accepted values of [math]\displaystyle{ d=.123nm }[/math] and [math]\displaystyle{ d=.213nm }[/math], it can be seen that for the larger spacing, my answer of [math]\displaystyle{ d=.203(6)nm }[/math] holds the accepted value within 2 standard errors of the mean, as that produces the range [math]\displaystyle{ d=[.191nm, .215nm] }[/math]. On the other hand, the accepted value [math]\displaystyle{ d=.123nm }[/math] is more than 4 standard errors away from my value as it lies 4.67 standard errors above the mean, meaning that if my values were correct, the accepted value would be found less than 1/1000th of the time.

Errors

The errors that I used in my calculations were the standard error of my data points, and the error in the length of the tube. I did not use the error in measurement as that would involve a subjective approximation of an error and not a statistical error.

Conclusion

Errors

The errors that would have affected this experiment are the actual measurement of the ring diameter, the accuracy of the power supply and bias, the alignment of the rings relative to center of the phosphorescent coating on diffraction tube.

Acknowledgements

I would like to thank my lab professor, Dr. Steven Koch, and the lab assistant, Aram Gragossian, for all their help fixing the different apparatus if I was getting incorrect data. I would also like to thank the UNM Physics Department for allowing us to use the lab and providing the apparatus so that we can operate.