User:Kirstin Grace Harriger/Notebook/Physics 307L/Lab

Steve Koch 03:43, 21 December 2010 (EST):Excellent primary lab notebook. And Kudos for recording the failed attempts to help out future experimenters!

Lab 03: Electron Diffraction

Concepts

• Particle-Wave Nature of Matter: Louis de Broglie hypothesized that all particles can also behave like waves and that their wavelengths are equal to plank's constant divided by their momentums. This is called the de Broglie Relationship. One way to demonstrate the wave nature of particles is to scatter them through a diffraction grating and observe the resulting pattern to be characteristic of waves. A diffraction grating can be made out of different things, but it needs to be small enough that it provides obstacles spaced on the order of the size of the wavelength appropriate for the particle to be diffracted. When the particles encounter the diffraction grating, they spread out into a pattern of areas where many particles go or where only a few go, called the maxima and minima. In this way they act like waves spreading out after passing through a small opening.

Safety Concerns

• At all times, the current must be kept below 0.25 milliamps to prevent puncture to the graphite foil

Equipment

• Tel 2501 Universal Stand
• Electron Diffractor: 2555 (5Kv .3mA)
• Power Supply: 3B U 33010
• Power Supply: HP6216
• BNC Cables
• Calipers: Carrera Precision
• Duct Tape for making the experiment room as dark as possible
• Multimeter which was used for the first two data sets, but not the third

Procedure

Using the diagram from Dr. Gold's Lab Manual, which was conveniently taped to one of our power supplies, we connected our equipment as follows:

1. F3 on diffraction tube to the left plug on the 6.3V on 3B Power supply,
2. F4 on diffraction tube to the right on above "Heater" written in pencil
   • F4 also to the positive plug on HP power supply "Low Voltage Bias"
3. C5 on tube to HV+ on 3B
4. C7 to multimeter
5. Multimeter to HV- on 3B
6. Low Voltage bias negative to HV- on 3B power supply
7. 3B power supply HV- to ground

After setting up the equipment, we turned on the power supplies and the heater in the diffraction tube. We waited for the thermal cathode to heat up enough to glow faintly red. Then we increased the voltage on the power supply while making sure the current stayed at a safe level. We stopped at 5000 volts with a current of 0.1 milliamps. Then we turned out the lights to see the two electrons rings. The electrons hit a florescent material coated inside of the glass sphere of the diffractor. We measured the diameter of the two center rings of diffracted electrons several times at different voltages using digital calipers.

The first day we went from 5000 to 4500 volts in increments of 100 volts and took ten measurements for each ring at each voltage, and that data is in red. We had to switch from the multimeter in the picture to a larger one of a model I didn't record. On the second day, we started a new data set and planned to go from 5000 to 4000 volts in increments of 200 volts and take five measurements for each ring at each voltage. We got to 4600 volts, and that data is in green. Then the larger multimeter malfunctioned and Brian got shocked, but he was ok since the current was so small. After consulting Dr. Koch and our TA Katie, we decided to go on without a multimeter. After the multimeter was removed from the circuit, the rings became much brighter. We took a new data set since we were more confident in our measurements of the now the easier to see rings. We took data from 5000 to 3500 volts in increments of 500 volts and took seven measurements for each ring at each voltage, and that data is in blue.

On the first day we took measurements of the the diameter by measuring the inner edge of a ring to the opposite outer edge of the ring for one trial and then measuring from the outer edge of a ring to the opposite inner edge for the next trial. On the second day we measured the inner edge to opposite inner edge of the rings.

We had to adjust the beam with a magnet for each change in voltage. The magnet was in a round paper collar on the neck of the diffractor that you turned slightly to recenter the rings.

This is our set up. The bulb on the blue stand is the electron diffractor. The red hand held device next to the electron diffractor is the first multimeter we used. Behind them are the two power supplies and Brian's laptop.

This is the pattern from the diffractor when the rings were at their brightest. They were so bright after we removed the multimeter that Brian could read the calipers without using the flashlight as he had for the first two data sets.

Data

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{{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdEwtbERWR2paeFZkWDdXWFQzcVVianc |width=625 |height=225 }}

Analysis

The diffraction grating in our experiment was a thin graphite foil with a crystalline structure. The structure is hexagonal and has two characteristic spacings of 0.123 nm and 0.213 nm. The two rings we saw correspond to these two spacings. The pattern is ring shaped instead of six hexagonal points because the electrons pass through many hexagons randomly oriented to each other in the sheets of carbon that make up graphite.

The spacing in the crystalline structure is related to voltage of the electrons and the diameter of the rings that form by

[math]\displaystyle{ d= \frac {4 \pi L \hbar c} {D \sqrt {2 e V m c^2}} }[/math]

In this equation, d is the characteristic spacing or the spacing between the atoms in the lattice, V is the acceleration voltage, D is the diameter of the ring, L=13.0 ± 0.2 cm is the distance from the foil to where we measure the ring, mc²=9.109 * 10^-31 kg or 0.511 MeV is the rest mass of the electron, e=1.602 * 10^-19 C is the charge of the electron, h=6.626 * 10^-34 Js or 4.135 * 10^-15 eVs is Plank's constant, and c=2.998*10⁸ m/s.

Because the glass is curved where we measure the rings, the actual distance of travel for the electrons that form the rings is less than the distance from the graphite to the center of the rings. This affects the diameter of the rings.

The correction for this can be computed using the radius of curvature of the spherical tube, R, and the length of the path from the foil to the center, L.

Let the error in the distance be given by y:

[math]\displaystyle{ y = R - \sqrt{R^2-\frac{D_{obs}^2}{4}} }[/math]

It can be shown using y that:

[math]\displaystyle{ tan(\theta) = \frac\frac{D_{obs}}{2}{L-y} }[/math]

The corrected diameter using [math]\displaystyle{ tan(\theta) }[/math] is:

[math]\displaystyle{ D=2Ltan(\theta) \, }[/math]

In Dr. Gold's Lab Manual, R is given to be 66mm.

The next step in finding the characteristic spacings is to plot the corrected diameter of the ring as a function of the inverse square root of the voltage and find the slope through linear regression.

The slope for the inner ring is 1.2260 and the slope for the outer ring is 2.5405.By putting these values for the slope into the following formula, the characteristic spacings can be found.

[math]\displaystyle{ d = \frac{\frac{2hL}{\sqrt{2me}}}{Slope} }[/math]

This gives the spacing for the inner ring as 0.245 nm as the best guess. This value then has a confidence interval from 0.242 nm to 0.246 nm. For the outer ring, this gives the value of 0.139 nm for the best value of the outer ring. We have a confidence interval for this value as 0.140 nm to 0.136 nm. These intervals of confidence were determined by entering the average value plus or minus the standard error of mean into the graph and finding the new slope there.

Resources

1. Prof. Gold's Lab Manual
2. Paul Klimov's Electron Diffraction Lab

Collaboration

1. My lab partner Brian Josey