Physics307L:People/Josey/Rough Draft

Measuring and Predicting Background Radiation Using Poisson Statistics

Author: Brian P. Josey

Experimentalists: Brian P Josey and Kirstin G G Harriger

Junior Lab, Department of Physics & Astronomy, University of New Mexico

Albuquerque, NM 87131

bjosey@unm.edu

Methods and Materials

For this experiment, a combined scintillator-photomultiplier tube (PMT) was used to collect data. To do this, every time the scintillator detected radiation, it would fire a beam of ultraviolet light to the PMT. The PMT would then convert this light signal into a single voltage. This voltage would be carried to an internal MCS card in a computer, where it would be analyzed by a UCS 30 software. This software counts each signal voltage and create a set of data containing the size of the window over which the data was collected, the time and the number of radiation events to occur in that window. In order for the scintillator to detect the radiation, it had to have a potential gradient that would pick up ions created in the radiation event. This potential was supplied by a Spectech Universal Computer Spectrometer power supply, and set to 1200 V throughout the course of the experiment.

The collection of the data was carried out by using the UCS 30 software. It would create a consecutive series of windows of set interval of time, and count the number of signal currents, which represent the number of radiation events, that occurred within each window. This data was then saved into data file that could be manipulated and processed using MATLAB v. 2009a. To demonstrate the Poisson distribution, the scintillator-PMT, power supply, and computer were all turned on. The UCS 30 software was then uploaded, and the window size was set to various lengths of time. Data was collected over a series of sets of windows, the window sizes were set at 10, 20, 40, 80, 100, 200, 400 and then 800 ms. Each run contained 2064 windows of the given size. The data was then analyzed, see results and discussion below, to demonstrate that background radiation did follow a Poisson distribution. This data was then used to predict the behavior of a similar set of data that occurred for a 1 second window size. After predicting its behavior, the system was ran again, using a 2064 windows of 1 second length. This data was then compared to the predicted results form the initial data.

Results

From the scintillator-PMT and the UCS 30 software, the data was processed using Google Docs to determine the range of the number of radiation events per window, and the number of windows with the given number of radiation events. A fractional distribution of the data was also determined. This data is summarized in Table 1 below:

{{#widget:Google Spreadsheet |key=0AjJAt7upwcA4dERfb3NIc0xhbmt1dnExMWhUdG1QS2c |width=600 |height=325 }}

Table 1 This table summarizes the raw data from each of the initial experimental runs. For each window size, the whole range of number of radiation events per window is given in the first column. In the second column, the number of windows that had this given number of radiation events is given. For example, for the 10 ms run, only a single window had four radiation events in it. The third and final column gives the fractional probability, out of 1, of such event happening in the given run.

Figure 3-Probability per Window In this figure, the number of radiations per window, horizontal axis, is plotted against the likelihood of that number of events happening, vertical axis. The for sets of data that are plotted are: blue - 10 ms, red - 80 ms, black - 200 ms and green - 800 ms. Where the time denotes the window size.

From this data, a graph was generated to illustrate the characteristics of the data. This graph is given in Figure 3. For the sake of simplicity only four of the nine data runs are graphed, however, even from this small sample of data, the trends across the whole of the sample are still clear. While the exact implications of this data is discussed in the discussion section below, the data makes it clear that as the window sizes increases, the most probable number of radiations per window increases, and the distribution of the probabilities spreads. This spreading in the data is a result of a greater standard error. Together these two trends are qualitative reasons to believe that the data follows a Poisson distribution. This is further discussed bellow in the discussion section.

A more potent argument for the Poisson distribution is to determine the averages and standard deviations for each set of data. This data is summarized in Table 2 below:

{{#widget:Google Spreadsheet |key=0AjJAt7upwcA4dF8yb09KUzVVYTg5LUcxaG43bTlvdHc |width=600 |height=325 }}

Table 2 This table summarizes the raw data collected from the runs with varying window sizes, from 10 ms to 800 ms. The first column represents the average radiation event per window as calculated directly from the data. The second and third rows represent the standard deviation of the data as the square root of the average, and as directly calculated from the data. The fourth row represents the percent error of the standard deviation from the data from the square root standard deviation. Because of the very low difference, it is clear that the data follows a Poisson distribution. The last two rows are the average and standard deviation converted from window size to rate per second.

As this table illustrates, the average radiation event per window sizes grows as the window sizes increases, and the standard deviation in proportion to the average also increases. There are two standard deviations in the table. The first is the square root of the average, while the second is directly calculated from the data. The reason for this, discussed in greater detail below, is that the standard deviation of a Poisson distribution is identical to the square root of the average. As the percent error between the two values shows, the data does follow this trend very closely, and the small differences, that never exceed 1.5 %, indicate that this is true.

Figure 4- Radiations per Second This graph represents the average number of radiations per second and their standard deviations as measured in the report. For simplicity, the horizontal axis is given in indexices so that 1 through 8 gives window size 10 ms to 800 ms in increasing size.

The averages and standard deviations from the data were then divided by the window size to give the rate and error as number of radiations per second. These values are then graphed in Figure 4. Clearly the standard deviation decreases as the window size approaches 1 second, indicating the greater accuracy in the data. From this data, the average rate and standard deviation for a windows size of 1 second was calculated. This value, x_wav, was calculated using weighted averages:

[math]\displaystyle{ x_{wav} = \frac {\sum {w_i x_i}} {\sum {w_i}} }[/math]

where

• x_wav - is the best estimate for the average number of radiations per second,
• w_i - is the weight of each average, this value is just the inverse of the square of the standard error for a given run,
• x_i - is the average value from each run

The standard deviation is then given as:

[math]\displaystyle{ \sigma_{wav} = \frac {1} {\sqrt {\sum {w_i}}} }[/math]

where again, w_i is the weight of each average, and σ_wav is the best estimate for the standard deviation in the final average. These calculations then give a predicted value of 30 ± 4 radiations per second. Another experiment was ran, this time the window size was raised to 1 second. The data from this experiment is summarized in Table 3 below. It is important to note that the predicted value for the average number of radiations per second is only off by the measured value by 0.725 %.

{{#widget:Google Spreadsheet |key=0AjJAt7upwcA4dC04WnpvSmM4Zzd2Rld1elo2QWo2YkE |width=600 |height=325 }}

Table 3 This table summarizes the data taken for a window size of 1 second. On the left is the average predicted by the calculations, its square root, and predicted standard deviation. Below this is the actual average, its square root, standard deviation and the percent difference between the prediction and the measured amount. On the left is the range of radiations per minute, the raw number of windows with a given number and their fractional frequency.