# User:David K. O'Hara/Notebook/physics 307 lab/Electron diffraction lab notes

## Contents

## Electron Diffraction Lab Notes

Experiment dates 09/14/2009 - 09/21/2009

^{SJK 17:35, 11 October 2009 (EDT)}## Objective

1. Demonstrate the wave property of electrons.

2. To study and verify the DeBroglie hypothesis **λ=h/p.**

3. To measure the spacing of diffraction grating planes in graphite.

## Theory

In an hypothesis in 1924 Louis DeBroglie reasoned that if electromagnetic radiation can be interpreted as particles and waves, then perhaps the electron which had always been thought of as a particle could also have a wave interpretation. De Broglie hypothesized that all particles have a wave behavior with a universal relationship between the wavelength and momentum given by λ=h/p. This experiment utilizes the fact that diffraction behavior gives a clear example of wave behavior.

The condition for electron diffraction is given by the following equation where d is the lattice spacing, n is the order number of the diffraction, and lambda is the wavelength of the electron. Theta in this case is the angle between the center line from the tip of the electron gun to the glass sphere. Also we are able here to use the small angle approximation that sinθ=θ, so

**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 2 d sin \theta = 2 d \theta = n \lambda }**

The angle subtended from the center of the screen to the first maxima ('R' is the radius of the diffraction ring) is found to be 2*theta:

**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 2\theta = R/2L }**

For small angles, this relationship simplifies to:

**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 \frac{Rd}{L}=\lambda }**

where D is the spacing between the maxima on the screen, a distance L away.

**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 \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mE_{k}}}=\frac{h}{\sqrt{2meV_{a}}} =\frac{2\pi\hbar}{\sqrt{2meV_{a}}}}**

Setting the wavelength of the electron equal to the wavelength in diffraction, we get:

**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 \frac{Rd}{L}=\frac{2\pi\hbar}{\sqrt{2meV_{a}}}}**

using the fact that R = D/2 we find the following relationship for the lattice spacing, 'd':

**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\hbar L}{D\sqrt{2meV_{a}}} }**

**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 slope=\frac{2Lh}{\sqrt{2me}\cdot d}}**

**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{2Lh}{\sqrt{2me}\cdot slope}}**

## Equipment

- Tel 2501 Universal stand
- Electron Diffractor 2555 (5Kv .3mA)
- Teltron Limited London England 813 KV Power Unit
- HP 6216B Power Supply
- Wavetek Meterman 85XT multimeter
- Carrera Precision 6" digital caliper alloy

## Safety/Precautions

1. While the expected current off the power is in the milliamp range, we will be dealing with a power source running at 813KV. Any short that occurs that could put this voltage across a person could be quite hazardous. Extreme caution should be used in setting up the equipment, verifying all cords/connections are correctly put together and free of wear or damage.

2. The apparatus is vulnerable to current overload which can punch a hole in the graphite screen, this current overload is characterized by the graphite screen glowing a dull red. We will, therefore, be monitoring the anode current for the device to insure we do not go above .25milliamps

## Procedure

Constructed the circuit in the schematic (provided from notes of previous student P. Klimov).

started with no bias voltage. 5kv. filament began glowing immediately, waited approx 5 minutes to make sure filament was warm enough to sustain thermionic transmission of electrons.

Diffraction pattern was very difficult to see, in particular the inner diffraction ring was almost impossible to make out at anything less than 4800 volts, even running through multiple bias voltages. In fact increasing the bias voltage had the tendency to make the beam less intense and more difficult to see.

Included is a cellphone photograph of the lab setup:

need to figure out how to reduce pic in size, if you click on image you can see source file.

(Steve Koch 16:45, 11 October 2009 (EDT): I edited to show you two ways of making smaller)

## data

**values given for phototube**

L=130.0±2mm Glass thickness=1.5mm radius of curvature=66.0mm

### Week1

V (V) | outer diameter (mm) | inner diameter (mm) |
---|---|---|

5000 | 42.93 +/-1.5mm | 26.60 +/-1.5mm |

4900 | 44.96 +/-1.5mm | 27.12 +/-1.5mm |

4800 | 46.13 +/-1.5mm | 28.11 +/-1.5mm |

4700 | 49.24 +/-1.5mm | not visible |

4600 | xx | xx |

4500 | xx | xx |

4300 | xx | xx |

4200 | xx | xx |

4100 | xx | xx |

4000 | xx | xx |

3900 | xx | xx |

3800 | 49.94 +/-1.5mm | nv |

3700 | 49.94 +/-1.5mm | nv |

3600 | 50.86 +/-1.5mm | nv |

3500 | 52.87 +/-1.5mm | nv |

The Rings were difficult to see and measurements are more guesswork than fact as there was not a very good edge on any of the rings to use as a guide for where to place the caliper. Also the bias voltage setup is suspect as increasing the bias actually decreased the intensity of the beam. Ran into time constraints attempting to improve the visual quality of the rings and only took measurements at high and low end of the high voltage range. I need to assess the setup and circuit diagram to see what improvements can be made to get better performance from the electron beam in next weeks labwork.

### Week2

During week two of data collection for this experiment I will flip the polarity of the bias voltage to see if there is an improvement in the visibility of the diffraction rings.

V (V) | outer diameter (mm) | inner diameter (mm) |
---|---|---|

5000 | 36.72 +/-1.5mm | NA |

4900 | 37.21 +/-1.5mm | N/A |

4800 | 37.78 +/-1.5mm v | 24.67 +/-1.5mm |

4700 | 38.11 +/-1.5mm | 25.02 +/-1.5mm |

4600 | 38.43 +/-1.5mm | 25.65 +/-1.5mm |

4500 | 38.96 +/-1.5mm | 26.59 +/-1.5mm |

4400 | 39.19 +/-1.5mm | 27.07 +/-1.5mm |

4300 | 39.39 +/-1.5mm | 27.52 +/-1.5mm |

4200 | 39.89 +/-1.5mm | 28.04 +/-1.5mm |

4100 | 40.08 +/-1.5mm | 28.27 +/-1.5mm |

4000 | 40.26 +/-1.5mm | 28.56 +/-1.5mm |

3900 | 40.41 +/-1.5mm | 28.81 +/-1.5mm |

3800 | 40.76 +/-1.5mm | 29.22 +/-1.5mm |

3700 | 41.01 +/-1.5mm | 29.63 +/-1.5mm |

3600 | 41.26 +/-1.5mm | not visible |

3500 | 41.56 +/-1.5mm | not visible |

Changing the polarity of the bias voltage improved the beam focus and seemed to give the effect I was looking for, removing some of the fuzziness of the rings, making the edges easier (still not easy) to take a measurement from. I also found the inner ring easier to see with the bias voltage at -15 volts while the outer rings were easier to pick up at about -2 volts on the bias voltage.

Raising the bias voltage to -15 volts while making the inner ring easier to see had the effect of limiting the output of the high voltage supply to a maximum of 4800 volts.

I Did find that extended running of the apparatus in the 4700V range and higher, after about 20 minutes the there seemed to be a discharge where the voltage would suddenly dip down to about 3500 volts and stay there until the voltage was taken below 3500 and then taken back up to the desired voltage. Not sure where that came from but it was a little disconcerting, possibly a safety feature kicking in or a sign that the power supply is unstable.

## Error Analysis

The greatest source of systematic error was in the taking of the measurement itself. The appearance of the ring, the challenge of finding a clean edge of the ring, and the way that you are holding a flat vernier caliper up to a spherical surface put a strain on the idea that I could take a mesurement accurate to a hundredth of a millimeter.

The voltage output of the high voltage supply used an analog meter to display the output voltage rather than a digital meter so the high voltage reading could have been within a 100 volts plus or minus the actual value.

## Results

^{SJK 17:04, 11 October 2009 (EDT)}~~File:Elecdiff.xlsx~~(Steve Koch 16:51, 11 October 2009 (EDT): I changed file extension to *.xls and it seems to work for me: File:Elecdiff.xls)

(having difficulty getting my excel file to open here, appears I have an older version of excel.

Will use Matlab or newer version of excel next time.)

**link to lab summary**[[1]]

^{SJK 17:12, 11 October 2009 (EDT)}