# Physics307L F09:People/DePaula/Electron Diffraction Formal Report

## Contents

## Verification of Electron Particle Wave Duality through Graphite Crystalline Diffraction

## Abstract

Through Electron Diffraction off a graphite coated Nickel Mesh, we verify the De Broglie hypothesis, E=h/(lambda), and we obtain two values for the fundamental separations in the graphite crystalline structure of .189nm +/-.0118nm, and .108nm +/- .0034nm. These values are within 11% of the current accepted values of .123 nm, and .213 nm (Lab Manual). After Taking data a second time, correcting for Systematic Error, we achieved values of .116 ± .023 nm, and .197 ± .015 nm. This second round of data gives us 6.4%, and 8.6% error, which is much better than our first result. We expected and did in fact see particle wave duality of the electron as we diffracted it off of a graphite crystalline coating, and onto a photo-luminescent screen.

## Introduction

Louis De Broglie, in his famous 1924 dissertation , theorized that if electromagnetic radiation acted as both a wave and particle depending on the circumstances, then the electron must behave in the same way.(De Broglie 1924) Through the course of this experiment we will validate this hypothesis by diffracting electrons off of a graphite coated nickel mesh and onto a photo-luminescent screen. Max Von Laue in 1912 stated that the molecular structure of matter might make an appropriate diffraction grating. The resultant diffraction pattern expected is that of two concentric circles. We expect two concentric circles, even though the fine molecular structure of graphite is hexagonal, because the graphite coating has some intrinsic thickness, and therefore the resultant diffraction pattern is that of multiple overlapped hexagons. By controlling the accelerating High Voltage we can control the velocity of the electron beam before it interacts with the graphite. The result is indirect control over the spread of the resultant diffraction pattern. We then plot the diameter versus the inverse of the voltage's square root to find a linear relationship. This linear relationship, along with the following formula, will give us a value for one of the two intrinsic separations in the hexagonal graphite structure.

## Materials

- 1 HV power supply (~5 kV max)
- 1 low voltage supply (bias voltage ~2.5V)
- banana plugs
- multimeter to monitor current
- electron diffraction tube
- calipers (Digital...Nice!!)
- Microsoft Excel 2002 (for data reduction)

## Set Up

We hook up the electron gun/photo-luminescent screen combo to the High Voltage according to the following diagram.

We also decided to input a bias voltage, on the anode, so that we can manipulate the brightness and contrast of the final image. We connect the bias voltage, (labeled) Vb, to both the ground and Vf. Vf is the heater voltage, which controls the amount of electrons produced. We will have it set at a constant 6kV. The Va, is the accelerating voltage, and determines the velocity of the electrons before they diffract off the graphite coating. After this farily simple, yet confusing set up, we are ready to begin taking data.

## Methods

After all our equipment is properly set up, we are ready to begin data collection. Starting with the High Voltage set at 5kV, we attempt to resolve two bright green concentric rings against the slightly dimmer green background. We can slightly adjust the contrast by manipulating the bias voltage. The higher the voltage, the less electrons make it through the canister and onto the photo luminescent screen. This creates a dimmer overall image and can increase the contrast between the two concentric rings and the background. Using a semi-powerful magnet, we focused the resultant image on the exact center of the photo-luminescent screen. This focus reduces the amount of distortion caused by the curvature of the glass bulb. We then used the calipers to measure the diameters of both rings. We made four measurements total which include the inner/outer edges of each ring. The reason for this is that the rings have some intrinsic width, so we are just going to measure the outer and inner edges and take an average to find the best approximation of the true diameter. This works because our final results are based on ratios between diameters and voltage, and do not rely on specific diameter values. We decided on an interval of .1kV between each measurement. This decision was made to give a large amount of data, while retaining a decent degree of accuracy. When we first decided to use .2kV between measurements, our data seemed weak and full of error, so we resorted to .1kV. We then compiled all our data into tables on this wiki site. Our data can be found in the results section of this paper.

In an attempt to increase both precision and accuracy in our results, I decided to take another set of data, and analyze it more thoroughly. This second set of data, collected in much the same way as the first, takes into account some sources of random and systematic error, and attempts to correct for them. One such error source is a faulty knob on the Accelerating voltage meter, that arbitrarily adjusts the voltage within .2kV of the actual value. Measuring this uncertainty and applying it to the final data, will give us a mathematical correction value. More discussion on this topic will be done later, along with error analysis.

## Results

Results will be summed up in two sections, the raw data results, and the mathematical application of the raw data as a result. First we start with what our raw data tells us directly. Here is the raw data we collected, in intervals of 1kV:

Voltage in kV | Inner/Inner | Outer/Inner | Inner/Outer | Outer/Outer | |
---|---|---|---|---|---|

5 | 18.69 | 25.05 | 37.78 | 43.30 | |

4.9 | 20.79 | 24.8 | 40.14 | 43.08 | |

4.8 | 19.82 | 24.91 | 39.81 | 43.58 | started ,measuring horizontally |

4.7 | 20.04 | 25.29 | 38.75 | 43.69 | |

4.6 | 21.35 | 25.91 | 39.31 | 45.99 | |

4.5 | 21.76 | 26.79 | 41.08 | 46.94 | |

4.4 | 21.91 | 27.27 | 41.86 | 46.97 | |

4.3 | 21.98 | 28.97 | 41.69 | 48.95 | |

4.2 | 23.41 | 28.02 | 42.64 | 49.41 | |

4.1 | 24.86 | 28.75 | 44.08 | 50.12 | |

4.0 | 25.45 | 28.19 | 46.74 | 51.57 | |

3.9 | 25.17 | 29.68 | 46.94 | 51.87 | |

3.8 | 26.22 | 30.62 | 46.69 | 52.22 | |

3.7 | 26.24 | 31.68 | 47.68 | 53.83 | |

3.6 | 27.75 | 32.85 | 47.85 | 52.43 | |

3.5 | 28.85 | 33.57 | 48.09 | 52.84 | |

3.4 | 28.48 | 35.30 | 50.06 | 53.59 | |

3.3 | 29.12 | 36.1 | 50.37 | 54.31 |

Just by looking at the raw data, it is easy to observe a trend. The values for the diameter increase as total voltage decreases. It is not necessary to record the value for the bias voltage because it has no effect on the diameter, and thus the diffraction of electrons. Because we see two concentric circles that change with voltage it is apparent that we are in fact dealing with electron diffraction and not some other phenomenon (this will be discussed later).

We now discuss the results in terms of mathematical manipulations, telling us more about the underlying properties and theory than the raw data could ever tell us. The first thing we do is we plot the data against V^(-.5), in an attempt to bring both the voltage and the diameter to the same order. This can be seen in the following equation:**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d=\frac{4 \pi L \hbar c}{D \sqrt{2eV_Amc^2}}}**
where it is apparent that diameter is inversely proportional to the square root of V. With this in mind, the following graph should make sense:

Upon first glance our data seems to be quite linear. It is obvious we have at least a positive correlation between diameter size and 1/sqrt(V). The above graph is diameter size for the outer ring, we do the same calculations for the inner ring and get the following relationship.

As you can very well see, both the inner and outer rings create best fit lines which are quite parallel, this is good.

Using the above formula for d, we hope to calculate the two fundamental distances, intrinsic to the graphite structure. We achieve a d of .1083 nm, and .189 nm. Because we know, based on prior research (Lab Manual), the accepted values for d, we can be sure that our concentric circles are in fact the result of electron diffraction. Indirectly our data supports particle/wave duality of the electron.

Using the same process as above, just inserting new data, and error corrections, we attempt to minimize the ill effects of both random and systematic error. Here is the raw data and spreadsheet that contains most of the mathematical analysis. Media:Electron Diffraction Data Final.xlsx

Our next set of raw data returned values of .116 ± .023 nm, and .197 ± .015 nm. These values for the intrisic atomic lattice spacing are off by 6.7%, and 8.6% (respectively), which is a great increase in accuracy and precision. It is clear that by adjusting for the uncertainty in the Power Supply's faulty knob, we were able to dramatically increase the precision of our Data analysis techniques. Eliminating this great source of Systematic Error, brought our values under 10% error.

## Discussion

It is important to realize that we have indirectly supported particle wave duality of the electron. This is the underlying physical property that makes this entire experiment possible. We are also happy to see that our data fits a line quite well, that means that either we were low on total error, or the error we do have is essentially evenly distributed about the mean. We achieved values for d of both .1083 nm, and .189 nm. These correspond to the actual values of .123nm and .213nm, respectively. Upon calculating the error caused by both systematic and random events, we are able to place error bars on our calculations giving us values of .189nm +/-.0118nm, and .108nm +/- .0034nm. Taking into account significant figures, our final result is .11nm +/- .003nm and .19nm +/-.01nm. Both of our values for d have a total error of about 11%. 11% error seems to be large, however it is important to keep in mind our primitive measurement technique. It is also nice to see that our error is consistent between both values, which means that whatever errors were made, they were most likely consistent, similar, and relatively frequent. One major source of error, was the High Voltage Knob, which could not accurately select nor maintain any given voltage. I was able to figure out a linear correction, and applied it to the initial recorded values of the Accelerating Voltage. This adjustment, in combination with subsequent data, reduced our total error by almost half!

The second set of values obtained for the graphite's intrinsic lattice spacing, are .116 ± .023 nm, and .197 ± .015 nm. These values have a total error of 6.4% and 8.6%, respectively. It seems as though correcting for the uncertainty in the Power Supply gave us much better results.

## Acknowledgments

I would like to acknowledge Dr. Koch for his help calibrating the Experiment, as well as teaching me all the data reduction techniques used here. I also owe a Thank You to Anne Ozaksut, who supported me mentally and emotionally throughout this journey. Devon Hjelm was a great help in understanding the underlying physics behind electron diffraction.

## References

- Wikipedia Article on Electron Diffraction
- Lab Manual
- Louis De Broglie, 1924, The Phenomena of High Frequency Radiation
- I was unable to obtain a copy of Max Von Laue's work from 1912, and therefore cannot cite it in my paper