 Please note that this lab notebook page is the combined efforts of Alex Andrego and Anastasia A. Ierides
Purpose
^{SJK Incomplete Feedback Notice} Incomplete Feedback Notice My feedback is incomplete on this page for two reasons. First, the value of the feedback to the students is low, given that the course is over. Second, I'm running out of time to finish grading! ^{SJK 18:39, 15 December 2009 (EST)} 18:39, 15 December 2009 (EST) As usual, excellent notebook!
 The purpose of this lab is to study and verify the de Broglie hypothesis that electrons act as waves and particles with the application of the de Broglie equation of [math]\displaystyle{ \lambda=\frac{h}{p}\,\! }[/math]. This can be done with the investigation of the electron diffraction through a thin layer of graphite (carbon), which acts as a diffraction grating.
 You can see a more detailed purpose in Professor Gold's Electron Diffraction.
Brief Description of Electron Diffraction
 The basic theory of electron diffraction stems from de Broglie's hypothesis that all particles have a wavelike nature. The de Broglie relationship, which holds for all particles, is given by:
 [math]\displaystyle{ \lambda=\frac{h}{p}\,\! }[/math]
 where [math]\displaystyle{ \lambda\,\! }[/math] is the de Broglie wavelength, [math]\displaystyle{ h\,\! }[/math] is Planck's constant, and [math]\displaystyle{ p\,\! }[/math] is the momentum conserved in collisions. By passing electrons through the electron tube and investigating their diffraction properties, it can be shown that such a relationship of wavelike nature exists in electrons and hence in all particles.
Equipment
 3B DC Power Supply (0... 5kV, 6.3V AC/3Amp ,Model U3310)
 TEL 2501 Universal Stand (EC UK Made)
 Cabrera Precision Calipers
 Electron Diffraction Tube (2555, 5 kV, 0.3 mA, EC UK Made)
 DVM (digital voltmeter) (WAVETEK 85XT, Wavetek Corporation, 20A:20A/600V, min interrupt 100kA)
 BNC cables
Safety
 Before we begin, some points of safety must be noted:
 First and foremost your safety comes first and then the equipments'
 Check the cords, cables, and machinery in use for any damage or possible electrocution points on fuses of machinery by making sure the power cords' protective grounding conductor must be connected to ground
 Be careful to ground all power supplies properly before use
 Make sure the areas containing and around the experiment are clear of obstacles
 Keep the anode current below 0.25 mA at all times to avoid damaging the graphite target
Set Up
This is the back of our Universal Stand where the connection ports are found
The bulb for this experiment is placed in the Universal Stand
This was the original HV and Heater Supply Unit we used
Here you can see the circuit diagram and all of the connection ports
This is the Low Voltage bias that caused us a lot of confusion in our circuit wiring
 The procedure we followed was based on the descriptions given in Professor Gold's manual
 We however used this wonderful circuit diagram created by Darrell Bonn in his Electron Diffraction Lab
 Using BNC cables make the following connections illustrated by the circuit diagram above taken from Darrell Bonn's Electron Diffraction Lab
 C5 on the Universal Stand to () HV Supply
 F4 on the Universal Stand to (+) on Heater Supply (This is on the same kV power unit as the HV supply)
 F4 on the Universal Stand to (+) on the Low Voltage Bias
 F3 on the Universal Stand to () on Heater Supply
 G7 on the Universal Stand to (+) HV Supply
 Ground on HV to () HV
 Same ground on HV to ground on the Low Voltage Bias
 Set the high voltage slider to zero before switching on the unit.
 Turn on the heater supply
 Wait one minute for the cathode temperature to stabilize
 Turn on the unit
 Apply the (HV) anode voltage.
 Due to the fact that our current never gets above or near the 25mA that could damage the graphite of our set up (as was described in Professor Gold's manual) , we did not use a digital voltmeter in series with our HV. It was not necessary.
Measurements and Data
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This is the precision caliper with a digital read out that was used to make our guesstimations
This is the newer HV and Heater Supply Unit that we used to take all used measurements
Here you can see the connections for the grounding and HV + and  connections with our BCN cables
Calculations and Analysis
 From above, in the Brief Description section, we have:
 [math]\displaystyle{ \lambda = \frac{h}{p} }[/math]
 To find the spacing of the graphite lattice we use the Bragg condition as follows:
 [math]\displaystyle{ 2\cdot d\cdot sin \theta = 2 \cdot d \theta = n \lambda \,\! }[/math]
 Where
 [math]\displaystyle{ d \,\! }[/math] is the lattice spacing,
 [math]\displaystyle{ n \,\! }[/math] is the diffraction order,
 [math]\displaystyle{ \lambda \,\! }[/math] is the wavelength of the particle, and
 [math]\displaystyle{ \theta \,\! }[/math] is the angle that opposes the path length difference for a matter wave.
 It turns out that the angle of diffraction from the incident particles is [math]\displaystyle{ 2\cdot \theta\,\! }[/math].
 [math]\displaystyle{ 2\cdot\theta = \frac{R}{2L}\,\! }[/math]
 For small angles, this relationship simplifies to:
 [math]\displaystyle{ \frac{R \cdot d}{L}=\lambda }[/math]
 where [math]\displaystyle{ D\,\! }[/math] is the spacing between the maxima on the screen at a distance [math]\displaystyle{ L\,\! }[/math] away.
 [math]\displaystyle{ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mE_{k}}}=\frac{h}{\sqrt{2meV_{a}}} \,\! }[/math]
 [math]\displaystyle{ \frac{R\cdot d}{L}=\frac{h}{\sqrt{2meV_{a}}} }[/math]
 using the fact that
 [math]\displaystyle{ R = \frac{D}{2}\,\! }[/math]
 we find the following relationship for the lattice spacing, [math]\displaystyle{ d\,\! }[/math]:
 [math]\displaystyle{ d=\frac{2 h L}{D\sqrt{2meV_{a}}} }[/math]
Using a Excel Linest Function
 The measured diameter versus the inverse square root of the accelerating voltage was plotted in the graphs below. We plotted a linear relationship using the linest function and were able to calculate the slope and uncertainty in the slope. The slope that was calculated is related to [math]\displaystyle{ d\,\! }[/math] as follows:
 [math]\displaystyle{ D=\frac{2hL}{d\sqrt{2me}} \frac{1}{\sqrt{V}} = Slope \frac{1}{\sqrt{V}} }[/math]
 [math]\displaystyle{ Slope = \frac{2hL}{d\sqrt{2me}} = \frac{\gamma}{d},\,\! }[/math]
 [math]\displaystyle{ \gamma \equiv \frac{2hL}{\sqrt{2me}} }[/math]
 [math]\displaystyle{ d = \frac{\gamma}{Slope} }[/math]
 [math]\displaystyle{ L = 13.0 \pm 0.2 cm\,\! }[/math]
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This is the back of our Universal Stand with all needed wiring shown
On the front of the Universal Stand, the connection port labels can be found
 To calculate [math]\displaystyle{ d\,\! }[/math] for the inner diameter,
 [math]\displaystyle{ d = \frac{\frac{2hL}{\sqrt{2me}}}{Slope} }[/math]
 [math]\displaystyle{ d = \frac{\frac{2\cdot(6.626\times10^{34} J\cdot s)\cdot (0.13 m)}{\sqrt{2\cdot(9.11\times10^{31} kg)\cdot(1.602\times10^{19} C)}}}{1.806388852 m\cdot V^{1/2}} }[/math]
 [math]\displaystyle{ \simeq0.177 nm\,\! }[/math]
 With a range of:
 [math]\displaystyle{ d_{min} = \frac{\frac{2\cdot(6.626\times10^{34} J\cdot s)\cdot (0.13 m)}{\sqrt{2\cdot(9.11\times10^{31} kg)\cdot(1.602\times10^{19} C)}}}{1.806388852+0.258307894 m\cdot V^{1/2}} }[/math]
 [math]\displaystyle{ \simeq0.154 nm\,\! }[/math]
 [math]\displaystyle{ d_{max} = \frac{\frac{2\cdot(6.626\times10^{34} J\cdot s)\cdot (0.13 m)}{\sqrt{2\cdot(9.11\times10^{31} kg)\cdot(1.602\times10^{19} C)}}}{1.8063888520.258307894 m\cdot V^{1/2}} }[/math]
 [math]\displaystyle{ \simeq0.206 nm\,\! }[/math]
 [math]\displaystyle{ 0.154 nm \leq d \leq 0.206 nm\,\! }[/math]
The percentage error of our average measured value relative to the accepted value of [math]\displaystyle{ d_{inner}=0.213 nm \,\! }[/math] can be calculated as:
 [math]\displaystyle{ \% error=\frac{d_{inner}d_{measured, average}}{d_{inner}}\,\! }[/math]
 [math]\displaystyle{ \% error=\frac{(0.213 nm0.177 nm)}{0.213 nm}\,\! }[/math]
 [math]\displaystyle{ \simeq0.169\,\! }[/math]
 [math]\displaystyle{ \simeq16.9%\,\! }[/math]
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These connections to the low voltage bias supply unit are what were required to make our circuit functional. Note that we never did turn the supply unit on, we just ran the connections to it
This is what our initial set up looked like before switching HV and Heater Supply units
This is a picture of the strange electron diffraction pattern that we fondly refer to as our "alien"
Here is a closer look at the extreme detail of the standing wave diffraction pattern observed in both bulbs
 To calculate [math]\displaystyle{ d\,\! }[/math] for the outer diameter,
 [math]\displaystyle{ d = \frac{\frac{2hL}{\sqrt{2me}}}{Slope} }[/math]
 [math]\displaystyle{ d = \frac{\frac{2\cdot(6.626\times10^{34} J\cdot s)\cdot (0.13 m)}{\sqrt{2\cdot(9.11\times10^{31} kg)\cdot(1.602\times10^{19} C)}}}{3.568859144 m\cdot V^{1/2}} }[/math]
 [math]\displaystyle{ \simeq0.0893 nm\,\! }[/math]
 With a range of:
 [math]\displaystyle{ d_{min} = \frac{\frac{2\cdot(6.626\times10^{34} J\cdot s)\cdot (0.13 m)}{\sqrt{2\cdot(9.11\times10^{31} kg)\cdot(1.602\times10^{19} C)}}}{3.568859144+0.313021573 m\cdot V^{1/2}} }[/math]
 [math]\displaystyle{ \simeq0.0821 nm\,\! }[/math]
 [math]\displaystyle{ d_{max} = \frac{\frac{2\cdot(6.626\times10^{34} J\cdot s)\cdot (0.13 m)}{\sqrt{2\cdot(9.11\times10^{31} kg)\cdot(1.602\times10^{19} C)}}}{3.5688591440.313021573 m\cdot V^{1/2}} }[/math]
 [math]\displaystyle{ \simeq0.0979 nm\,\! }[/math]
 [math]\displaystyle{ 0.0821 nm \leq d \leq 0.0979 nm\,\! }[/math]
The percentage error of our average measured value relative to the accepted value of [math]\displaystyle{ d_{outer}=0.123 nm \,\! }[/math] can be calculated as:
 [math]\displaystyle{ \% error=\frac{d_{outer}d_{measured, average}}{d_{outer}}\,\! }[/math]
 [math]\displaystyle{ \% error=\frac{(0.123 nm0.0893 nm)}{0.123 nm}\,\! }[/math]
 [math]\displaystyle{ \simeq0.271\,\! }[/math]
 [math]\displaystyle{ \simeq27.1%\,\! }[/math]
Notes about Our Uncertainty
 When we began this experiment we were using a different heater and HV supply unit, and a Low Bias Voltage Supply that can be seen in the photos of this lab. For our second day of experimentation we made our lab more effective by using a newer bulb, and a newer HV and heater supply unit. We tried to eliminate the use of the low voltage bias unit, however we discovered that the bulb would not light up, with out making the connections to the unit. We found this to be very odd, because the unit never needed to be turned on, the wiring just had to pass through the unit. This could be due to a electric circuit problem with our setup, but it did not seem to affect any other part of our lab.
 We also noticed with both bulbs, new and old, that a strange, "alien"(as we fondly refer to it as), diffraction pattern appeared on the top left hand side of the bulb. When we experimented with the voltage, magnet, and low bias settings, we discovered that the only change was the intensity of the light which decreased with the voltage. We are still unaware of what effects provoked our strange "alien". Pictures can be seen in this lab.
Systematic Error:
 When taking measurements for this lab we used a precision caliper and guesstimated the diameter of the rings by simple placement of the calipers onto the surface of the bulb. This could cause some systematic error in that we were using a precision tool to guesstimate measurements.
 The most prominent source of systematic error is the fact that the rings are very hard to see, due to their large width and blurred edging. This makes our guesstimation factor very large.
Summary
 If you wish to see Alex Andrego's informal summary of this lab follow this link
 If you wish to see Anastasia Ierides's informal summary of this lab follow this link
Acknowledgments
 Prof. Gold's Lab Manual served as a loose guideline for our lab procedure and our "Brief Description of Electron Diffraction" above as well as the source of the accepted values of the separation of the carbon atoms corresponding to the inner and outer ring diameters
 Darrell Bonn's Electron Diffraction Lab served as a greatly needed setup guide for our circuit
 Professor Koch and Pranav for always being of great help to us!
