Roberto Sebastian Rosales/Notebook/Physics 307L/2010/10/06

=Equipment=


 * PMT
 * LED
 * Oscilloscope TDS Tektronix 1002
 * Bertan Power Supply Model 313B
 * Canberra Delay Module NSEC 2058
 * Ortec TAC/SCA Model 567
 * Harrison Laboratories Power Supply (for LED) model 6207A
 * Multiple BNC Cables
 * Long Carboard Tube (which served as the housing for the PMT and LED)

=Safety= =Setup=
 * Standard electrical hazards
 * We had to make sure not to damage any of the equipment, especially the PMT. Exposing the PMT to the light in the room while operating at a high voltage could potentially damage it.
 * We used the setup that was already in place (the setup can be found in | Prof. Gold's Lab Manual, but we checked all the connections and settings to verify that everything was how it was supposed to be.
 * A very good and concise setup can be found on Alexandra Andrego's Lab Notebook under the setup section.





=Procedure=


 * We followed Prof. Gold's lab manual, which can be found here.
 * All the equipment was already setup, but we double checked the connections and settings before we got started. A very good and concise setup can be found on Alexandra Andrego's Lab Notebook under the setup section.
 * After checking the connections, we continued on to powering on all the devices and looking at what was displayed on the oscilloscope.
 * We then tweaked the settings on the oscilloscope until we had the following on the display:
 * Note that the top signal with the sharp spike down is CH1 coming directly from the PMT and the bottom signal is CH2 and is coming from the TAC. The Voltage we measure is going to be taken from CH2.
 * Then, we picked our farthest point away from the PMT (50cm away from the end of the meter stick assembly) and called that the 100cm mark. At this LED position, we rotated the PMT until CH1 displayed a maximum value. We want CH1 to be a maximum so that when we move the LED closer, we will be able to adjust the intensity of the light down to the original amplitude at the 100cm mark.
 * After the initial setup, we began to move the LED in 10cm increments closer to the PMT, and after each move we would rotate the PMT to correct the intensity. We would record the voltage at each point including the initial 100cm mark.
 * We repeated this process for 5 sets of data, but we will only be analyzing 3 of the sets in which we used the 16ns delay. The reasoning for this can be found under the issues section.
 * Note: For the two 9ns time delay trials (trials 1 and 2), we were trying to keep the amplitude of CH1 at 1.64-1.66V. For the three 16ns time delay trials (trials 3, 4, and 5), we were trying to keep the amplitude of CH1 at 2.80-2.82V.

Time Walk
The rotating of the PMT is done to correct for what is known as time walk. Time walk occurs when the the TAC triggers at a different time for different signal amplitudes. In order to keep that triggering time constant we rotated the PMT, which had polarizers in front of it, which allowed for the adjustment of intensity. So throughout trials 3-5 (as well as trials 1 and 2 separately) we tried to keep the the Voltage on CH1 (which corresponds to the intensity of the light) constant.

Issues
On the first day, we could not figure out how to display the portion of the signal that we were interested in on the oscilloscope. Finally after messing around with it for about an hour, we asked Katie for some help with it. She hit the Auto Reset Button (not too sure if this is the correct name for the button, but it readjusts the settings back to some default settings for the signal being displayed), and made some quick adjustments with the Voltage Division knob and the Time Division knob which displayed the portion of the signal that we were interested in. Next we fiddled around with the PMT and the LED in order to familiarize ourselves with how the signals on CH1 and CH2 respond to certain movements of the devices. We then picked a zero mark for a trial of test data just to see if we were recording usable data. We recorded a voltage for our 1 meter mark and then moved in (toward the PMT) 10cm and got another voltage which was slightly lower than the 1 meter mark voltage. This is exactly what we expected to happen, but upon recording the third voltage, which was higher than the first, we knew that we were doing something wrong. We repeated this process one more time and got similar results, so once again we asked Katie for some help. She double checked our signal on the oscilloscope, as well as the settings on the other devices (the delay and the power supply). Also, all of the voltage readings for both CH1 and CH2 seemed to vary anywhere from a .002mV al the way to .04mV. We thought that maybe we did not have a setting correct because other groups that did the lab were able to pinpoint one voltage continuously for several trials. We decided to look over the shoulder of the Monday group that was doing the Speed of light Lab on October 3 in order to see if there was something that we were missing. Once they got setup and turned everything on, we saw the same variations in voltages that we were getting on the previous lab day. They said that they were just trying to pick out the average value of the fluctuations and that was the value they would record. On the second day of this lab, we powered everything on and found the signal on the oscilloscope. We then continued to take some data, and to no surprise, we found the same trend in our data as before. As we moved the LED closer to the PMT, the voltages were steadily decreasing. On the fourth or fifth measurement, the voltage was actually higher than the previous measurements. We made sure that the PMT had been rotated to keep the same intensity, but the voltage was higher. We stopped the recording process to figure out what was wrong. We tried messing with the triggering a little and took two trials of data which can be found below. Prof Koch showed up right around this time to help us out. After we told him what the problem was and looking at our oscilloscope, he knew what the problem was right away. He went to the filing cabinet, pulled out the manual for the Ortec TAC, and showed us that we had been operating above the threshold voltage. We adjusted the voltage and triggered our signal at a point in the signal where there was not much fluctuation, both of which drastically changed our measurements.

=Data and Calculations= Our raw data for all five trials is shown below. The calculations for the averages, slope, uncertainty in slope, and speed of light are for trials 3-5 only.


 * For the average Voltage calculation, I used the standard average value equation which is: $$\bar x = \frac{\sum_{i=1} x_i}{n}$$, where $$x_i$$. I found the average voltage for each distance (0-100cm in 10cm increments) for trials 3, 4, and 5. These values can be found in the Average Voltage column of the above chart.
 * Next, I plotted the Average Voltage vs. Distance, making sure to keep the variable with the most uncertainty on the y axis as discussed in lecture. I tried investigating how I could make a best fit line on the plot, but I do not think that this is possible in Google Docs. I did however use the 'LINEST(Y Data, X Data,1,1)' function in Google Docs which essentially gives me the data for the best fit line I was trying to get.
 * The 'LINEST(Y Data, X Data,1,1)' function gave me the slope and the uncertainty in the slope which allowed me to convert from Volts and centimeters to nanoseconds and centimeters. Let $$S =$$Slope and $$\delta_s = $$ uncertainty in s.
 * I found my range of slopes as follows:
 * $$S_{low} = S - \delta_s $$
 * $$S_{high} = S + \delta_s $$


 * Now I have the range from $$S_{low}$$ to $$S_{high}$$ with $$S$$ being somewhere in between. All of the values are in $$\frac{Volts}{cm}$$.
 * Next I took the reciprocal of all three values for my range of slopes in order to get $$\frac{cm}{Volt}$$ using the following:
 * Let $$R_{S_i}=$$ the reciprocal of the slope.
 * $$R_{S_{low}}=\frac{1}{S - \delta_s}$$
 * $$R_S=\frac{1}{S}$$
 * $$R_{S_{high}}=\frac{1}{S + \delta_s}$$
 * Then, to convert from $$\frac{cm}{Volt}$$ to $$\frac{cm}{ns}$$, I used the following conversion factor: $$\frac{1 Volt}{10 ns}$$.

Final Results

 * The above equations and conversions lead to the following calculated values:
 * $$S_{time VS dist}=(3.2 +/- 0.03)\times10^{-2} \frac{ns}{cm}$$ (I just converted my Voltage to Time before using the "LINEST" function to get these results.)
 * $$C_{S} = \frac{1}{S}$$, $$C_{S_{i}} = \frac{1}{S +/- \delta_{S}}$$
 * $$C_S = 31.3 \frac{cm}{ns}$$
 * $$C_{S_{high}}=30.9 \frac{cm}{ns}$$
 * $$C_{S_{low}}=31.6 \frac{cm}{ns}$$

Accepted Value
$$C_{accepted}=29.98 \frac{cm}{ns}$$ which can be found here.

Comparison
Although the accepted value does not fall within our range (68% confidence interval), I am fairly satisfied that we were in the ball park with our measurements considering we experienced problems throughout the lab.
 * Using this formula, $$ % Error = \frac{|x_{calculated} - x_{actual}|}{x_{actual}}*100$$, we found the following percent error in our data:
 * $$ % Error = 4.3 %$$

=Error=

I do believe that there was some error (both random and systematic) in our oscilloscope measurements. As mentioned before, the voltages that we were recording from CH2 on the oscilloscope were fluctuating considerably. We tried to pick the average of the fluctuations, but at times this was hard to do. The result of this was data that may not have been entirely correct. There is nothing that I can think of that would correct this, and when we discussed it with the other group, as well as Katie and Prof. Koch, they reassured us that you just have to do your best at reading the voltage. Also, we were forced to use the average function on the oscilloscope in order to stabilize the signal on the display. This can also account for some error because we were taking the average value of the signal (average on the 128 setting). We found out from experience that the part of the signal that is triggered on is very important. After making the mistake with using a voltage above the threshold voltage, we tried taking data with a 9ns time delay and using a constant voltage of about 1.5 V for CH1 (or intensity). This was due to the fact that at any higher voltage, we could not even see a signal on CH2. We figured out that at the higher voltage, we were triggering on a less noisy part of the signal, so we wanted to use the higher voltage. In order to do this we had to use a larger time delay so that the signal on CH2 could be displayed. The point of this is that we needed to be careful about where we were triggering. Triggering on a noisy part of the signal could lead to skewed results.

=References=

During this Lab, Katie helped us out a great deal on the first day, and Prof. Koch helped us out on the second day. Katie and I both referenced Tom Mahony's Lab Notebook for the procedure and setup sections. Also I referenced Alexandra Andrego's Notebook for the lab write up (specifically the data analysis section). And, like always, my partner Mather Cordova and I collaborated on this lab write up.