Experiment 4: Speed of Light

Lab Partner Kyle Martin

Purpose

To measure the speed of light.

Materials

SJK Steve Koch 21:13, 24 October 2007 (CDT)
Steve Koch 21:13, 24 October 2007 (CDT)
Again, you've done a terrific job recording the instruments, model #'s etc., and I also liked the link to the PMT article on wikipedia.
• TAC (Time Amplitude Converter): Model 567 mfd. by EG&G Ortec
• Delay Module: nSec Delay model 2058 mfd. by Canberra
• DSO (Digital Storage Oscilliscope): Tektronics TDS 1002 (Dual channel digital storage oscilloscope)
• LED Power Supply: Model 6207a mfd. by Harrison Industries (DC power supply, 0-200V, 0-0.2A)
• LED/capacitor module: looks hand made by Physics dept. Supposed to cycle on and off at ~10KHz, depending on voltage.
• PMT (Photomultiplier Tube): Just labeled N-134. Has a magnetic shielding tube attached to the front of it. See Wikipedia's entry on PMT for a brief explanation of its operation - I found this very neat.
• PMT Power Supply: Model 315 mfd. by Bertan Associates, Inc. (DC power supply 0-5000V, 0-5mA)
• Long cardboard tube
• 3 meter sticks taped together
• Various BNC wires
• 2 Polarizing filters

Setup

As described by the lab manual.

SJK 21:16, 24 October 2007 (CDT)
21:16, 24 October 2007 (CDT)
This also is a really good, concise description of the connections
• The LED module is connected to its power supply and to the first input on the TAC. It also has the meter sticks taped to it. One of the polarizing filters is attached to the module in front of the end that emits light. The module is inserted into one end of a long cardboard tube, with the end that emits light aimed down the length of the tube.
• The PMT is connected to its power supply, to the input on the delay module and to channel 1 of the oscilloscope. It has the other polarizing filter placed in front of its collecting end. The PMT is inserted into the other end of the long cardboard tube, with the collecting end pointed at the LED module.
• The delay module's output is connected to the second input on the TAC.
• The TAC (which now has 2 inputs connected), has its output connected to channel 2 of the oscilloscope.

Procedure

SJK 21:18, 24 October 2007 (CDT)
21:18, 24 October 2007 (CDT)
really great description of procedure!
• Turn the power supplies, TAC and DSO on. The LED module should be firing now, and the PMT should be registering a corresponding drop in potential for every pulse of incident light.
• The TAC will be triggered by the LED module pulsing. It will be triggered again by a dip in potential across the photomultiplier tube caused by incident photons striking the photocathode material on the end of the PMT and the resulting cascade of electrons moving towards the anode. The TAC then creates a potential across the two output leads which is proportional to the time between being triggered on and off. The oscilloscope measures this voltage.
• We must be careful of a very large source of systematic error: timewalk. Timewalk is an interesting phenomenon which is explained very well in the lab manual, but the essence is this: the TAC is triggered at a set voltage. This voltage will be reached sooner if the pulse being sent to the TAC is larger, and later if the pulse is smaller. The size of the pulse is proportional to the brightness of the incident light on the PMT, which is proportional to the distance between the LED source and the PMT detector. To control this effect, a reference voltage is taken from the PMT which indicates the brightness of the incident light. The polarizers in front of the source and emitter are turned as the distance changes in order to keep the brightness constant, indicated by a constant voltage reading.
• By varying the distance between the LED module and the phototube and taking voltage measurements, we can determine the speed of light. Plot the distance vs. time and take the slope of the line connecting these points to get a rough estimate. By finding the slope of a line fit using the least-squares method, we can get a better estimate.

Data

Preliminary, Rough Stuff

We'll just start with low resolution measurements. The DSO can be fiddled with to get higher resolution readings. By "zooming" in in the y-axis, we can increase the resolving power of the DSO.

Channel 1 is hooked up to the PMT output Channel 2 is hooked up to the TAC output PMT Power supply is set to 1900 V LED Power supply is set to around 186V

Threw this data out, as we subsequently changed the PMT reference voltage - and CH1 (PMT) resolution was poor. Poor CH1 resolution made these data useless as we were unable to control time walk by an acceptable amount.

2 nSec delay
PMT Reference Voltage: -1.30 V +/- ?
(Channel 1 Min Voltage at stick measurement of 80 cm with average of 128 measurements)
80 cm Channel 2 Max: 3.56 V +/- 0.02 V
100 cm Channel 2 Max (channel 1 min reads between -1.30V and -1.28V): 3.44 V +/- 0.02 V
120 cm Channel 2 Max (channel 1 min reads between -1.30V and -1.28V): 3.28 V +/- 0.02 V

CH1 (PMT) Resolution was very bad. These data should be disregarded.

0 nSec delay
PMT Reference Voltage: -800mV +/- ?
120 cm: CH2Max: 4.34V +/- 0.02V
110 cm: CH2Max: 4.40 +/- 0.04V (CH1Min: 800mV)
100 cm: CH2Max: 4.50 +/- 0.02V (CH1Min: 800mV)

These data are acceptable. CH1 Resolution was much higher (it is now 8mV)

0 nSec delay
PMT Reference: CH1Min: -800mV +/- 8mV
70 cm: CH2Max: 4.74V +/- 0.02V
80 cm: CH2Max: 4.70V +/- 0.02V
90 cm: CH2Max: 4.60V +/- 0.02V
100 cm: CH2Max: 4.54V +/- 0.02V
110 cm: CH2Max: 4.50V +/- 0.02V
120 cm: CH2Max: 4.42V +/- 0.02V
130 cm: CH2Max: 4.37V +/- 0.03V

Day two These data are acceptable, as well

0 nSec delay
PMT Reference: CH1Min: -800mV +/- 8mV
140 cm: CH2Max: 4.40V +/- 0.02V
60 cm: CH2Max: 4.82V +/- 0.02V
50 cm: CH2Max: 4.95V +/- 0.03V
40 cm: CH2Max: 4.96V +/- 0.03V

Note: all distance measurements have an inherent error; eyeballing a meter stick is not a very precise measurement. I estimate the error in distance measurements to be around 1 mm. Therefore, the distances recorded should be interepreted as the recorded number +/- 1 mm, or thereabouts.

Note: the PMT reference voltage was unstable; it varied randomly between the -800mV and +/- 12mV or 16mV.

SJK 00:28, 25 October 2007 (CDT)
00:28, 25 October 2007 (CDT)
I'm not sure whether it can do it, but it's possible the MCA card can trigger off of the PMT voltage and only save TAC readings when the PMT voltage is in a certain range. Or, it's possible the MCA can save both the PMT pulse height and the TAC pulse heights, so the data could later be sorted. Bill Miller also mentioned there may be a NIM that has more advanced triggering options that the TAC you are using (so it could possibly trigger off the PMT pulse height. Anyway, if Tomas and Lorenzo have more success, it may be worth taking more data if you choose this as your formal report.

Note: the TAC voltages are recorded as some number +/- some error. This number and error are really the result of our best judgment by watching the Channel 2 "min" reading on the oscilloscope, set to average over 128 measurements. Typically, the Channel 2 minimum reading was unstable and varied between +/- 0.004V or 0.008V, and seemed to spend most of the time around the recorded mean. This is, of course, not entirely objective and could be a source of error; with a better method of collecting data we could likely decrease this error. Dr. Koch said that my fellow student Tomas Mondragon had an idea for using a multichannel analyzer to collect data, presumably recording measurements only when the PMT reference voltage was exactly -800mV (within the tolerances of the instrumentation, +/- 8mV). This would probably spit out a whole slew of TAC voltages and these data could then be analyzed using something like Matlab or Excel to find a more precise description of the mean and standard error of the voltages measured.

Analysis

00:41, 25 October 2007 (CDT)
Excellent plot! It would be also good if you figured out how to add error bars (from your uncertainty estimates) for each data point...I know this is possible in Excel, just not sure of the best way. "Stock" plots may be the way to go, as these are set up for "open, close, max, min"

Also, excellent spreadsheet...it was really very easy to read, even by someone not too familiar with the lab, I think.
Plot of data: time vs. distance, with min. slope, max. slope and best fit slope
By performing a least squares fit (or, more accurately, by telling Excel to do one), it is possible to find the slope (and, Dr. Koch showed in lecture, also the y-intercept and their variances). The slope of these data was found to be $3.063 \cdot 10^8$, with a standard error of $1.83 \cdot 10^7$, making the best estimate of the speed of light $(3.063 \pm 0.18) \cdot 10^8 \frac{m}{s}$.SJK 00:34, 25 October 2007 (CDT)
00:34, 25 October 2007 (CDT)
Great data! I think it's pretty cool that when you're careful with taking the data you can get a value that is pretty close. I think the way you report your answer is fine, but people who are sticklers for significant digits would probably prefer if you wrote it $3.06 \pm 0.18$ or possibly even $3.1 \pm 0.2$ ... but like I said, I think the way you wrote it is readable, and makes sense.

This is a relative error of 5.96%, and the value 3.063E8 m/s is 2.17% different from the accepted value of the speed of light, $2.9979 \cdot 10^8 \frac{m}{s}$ (according to Wikipedia).

Remarks

Since the readings on the DSO were unstable, with the voltage of the PMT fluctuating significantly, I'm pretty happy with these results.

I don't know why the PMT voltage reading fluctuated so much, but it might be due to something like a power supply that doesn't generate a very steady voltage. Perhaps the LED assembly doesn't discharge at exactly the same voltage on every "firing", producing a light signal of varying brightness. Or, perhaps the voltage across the PMT is not constant. I think that the use of BNC cables should mostly eliminate the effect of electromagnetic interference, at least in most of the connections. There is also a magnetic shield around the PMT, which should do the same for that sensitive equipment.SJK 00:39, 25 October 2007 (CDT)
00:39, 25 October 2007 (CDT)
Good ideas, and I agree with your thinking that it's likely either the LED or the PMT. Since the PMT's job is to measure photons, I would guess more that the LED just isn't that stable.
Also, the method used for calculating the standard error of the mean assumes a uniform standard error of the data points. In other words, it weights all of the data points the same; it disregards the difference in the data points recorded with a +/- 0.02V and those recorded with a +/- 0.03V standard error. It would be interesting to be able to weight these differently - which I hope I will learn to do in the near future.SJK 00:39, 25 October 2007 (CDT)
00:39, 25 October 2007 (CDT)
This is definitely something you could do for the formal report!