Physics307L:People/Long/Roughdraft

The Speed of Light (title in progress)
Author: Ryan Long

Experimentalists: Ryan Long & Tom Mahony

The University of New Mexico

Department of Physics & Astronomy

email: rlong1@unm.edu

Abstract
The speed of light is unfathomably fast, but not too fast for human beings to measure, and we have been experimenting since Isaac Beeckman and Galileo Galilei first tried in the early 1600s. [1] In the Junior lab here at University of New Mexico, we attempt to calculate said constant by measuring flight time of LED pulses over the course of a short distance. A major obstacle to overcome in this experiment is the occurence of "time walk",  this can cause major systematic error, if not addressed properly.

Introduction
Perhaps one of the most well known and frequently used constants of physics both classical and modern, is the speed of light, denoted by a lower case “c”. The speed of light constant is used for many purposes from calculating the distance to astronomical events, to understanding quantum mechanics. The accepted value of the speed of light is c=299 792 500(100)m/sec. [2] Measuring the speed of light can be achieved numerous ways with modern technology, some include radio interferometry, and lasers. Using advanced laser methods c can be measured with fractional uncertainty on the order of ± 3.5x10-19 [2] My partner Tom Mahony and I measured the speed of light using a relatively simple method which involves measuring time delay of an LED pulse using a photomultiplier tube and a Time amplitude converter or simply (TAC). The photomultiplier is a device sensitive enough to measure individual photons. When the cathode of the PMT receives incident photons, photoelectrons are ejected from an anode inside the PMT, [3] the resultant charge pulse intervals from the photoelectrons are then converted into amplitudes by the TAC and displayed on an oscilloscope. The voltage amplitudes can then be converted to time and divided by the distance to obtain the speed of light. This simple, yet effective experiment yielded some exciting results for my partner, Tom and I.

Methods, Materials and procedure
In order to measure the speed of light, we set up a long, opaque tube made of cardboard with a pulsating LED light source in one end and the photomultiplier tube on the other end of the tube. The photomultiplier tube is connected with BNC cables first to a Canberra 2058 Delay module. (This device takes into account the delay in the signal because of varying BNC cable length from the light source and the PMT. Then the delay module is connected to a Tektronix TDS 1002 digital oscilloscope in parallel with the “Stop” input on the Ortec 567 TAC/SCA Module.  The Light source is also connected to the TAC in the “start” input.  The delay module is set for 9 ns.

The light source is mounted on a meter stick so that we can measure various distances of light travel time, which ideally would lower our uncertainty. However this introduces a possible source of systematic error. As the LED is moved closer to the PMT, the PMT amplitude rises due to heightened intensity of photon bombardment. This issue is known as “time walk”. In order to reduce this time walk or fluctuation in voltage, a polarizer is mounted to the front of the PMT. As the LED is pushed down the tube toward the PMT, we rotate the PMT to keep the intensity as continuous as possible.

Data Collection
We used six trials of recording data for our experiment. Each trial, we moved the LED closer to the PMT in 10 cm increments and recorded the amplitude for each increment. However for my data analysis, I will only be looking at trials two through five. The first trial was only measured with nine increments of 10 cm because we started at a different mark on the meter stick. My role for the first five trials was to rotate the PMT in order to maximize the voltage as Tom pushed the LED closer. My other duties for the first 5 trials included reading amplitudes off the oscilloscope, for the last trial we switched roles to test for any systematic error in our procedural roles, there was no significant change in data collection.

Results & Analysis Methods
Using the formula below from this page provided by Dr. Koch, I calculated the speed of light. However, my analysis at this point is incomplete. I am not sure if I followed the formula correctly or if it is even appropriate for this application. The value I obtain for the speed of light is $$3.23\cdot 10^{8} m/s$$. Also from this formula, I'm not sure how to report the uncertainty, if I were to average the standard mean of error for each voltage, the value would be $$3.23\cdot 10^{8} \pm 2.36\cdot 10^{2}m/s$$. My excel sheet can be downloaded.

Formula for best fit (maximum likelihood) parameters
 * $$ y = A + B*x$$

General case, individual σi
 * $$A=\frac{\sum \frac{x_i^2}{\sigma_i^2} \sum \frac{y_i}{\sigma_i^2} - \sum \frac{x_i}{\sigma_i^2} \sum \frac{x_i y_i}{\sigma_i^2}}{\Delta}$$ $$\mbox{,} \sigma_a^2 = \frac{1}{\Delta} \sum \frac {x_i^2}{\sigma_i^2} $$


 * $$B=\frac{\sum \frac{1}{\sigma_i^2} \sum \frac{x_i y_i}{\sigma_i^2} - \sum \frac{x_i}{\sigma_i^2} \sum \frac{y_i}{\sigma_i^2}}{\Delta}$$ $$\mbox{,} \sigma_b^2 = \frac{1}{\Delta} \sum \frac {1}{\sigma_i^2} $$


 * $$\Delta=\sum \frac{1}{\sigma_i^2} \sum \frac{x_i^2}{\sigma_i^2} - \left (\sum \frac{x_i}{\sigma_i^2} \right)^2 $$

Conclusions
Our data collection was very consistent, and seems to be mildly accurate if I have done my calculations correct, but there are inevitable systematic error set-backs. Time walk is a large part of it, perhaps the photomultiplier could be modified so that it is stationary with only a rotating polarizer, instead of rotating the entire PMT. Another possible source of error is varying the distance from the LED to the PMT, it is nearly impossible to make perfect increments of ten centimeters. This error would probably be reduced by simply taking more measurements, which perhaps we will try to do before the final report is finished.

My results and analysis are incomplete as of right now, (as I am baffled by the principle of maximum likelihood), but will be finished for the final report. I hope to also add a figure with a plot of the different measurements to show the best fit line from my formula attempt.

Acknowledgements
My partner Tom could not have done a better job, he is absolutely amazing to work with, so a big thanks to Tom. Also we could not have set up this experiment without the fantastic supervision and helpfulness of our professor Dr. Koch, and the teaching assistant, Pranav Rathi. Thanks gentleman.