# Counting Statistics

## Steps for Geiger-Mueller counter

• Display the raw output from the GM tube on the oscilloscope. Make a semi-log plot of pulse height vs applied voltage for a range of voltages. Comment on the shape of the graph.
• Route the output of the GM tube into the amplifier, and display the amplified signal as well as the GM output on the scope on different channels. Then route the output of the amplifier into the SCA, and the output of the SCA into the oscilloscope. The output of the SCA should be a square pulse ~5V high. It converts the analog signal into a digital one. Once you see this pulse, route the output of the SCA into the (positive) input of the Counter. See if you can get it to display the number of particles detected. Play around with the settings of the electronics and figure out what the different settings do. Write a one- page “Users manual”. Make sure to note down what settings work best to display the signals on the oscilloscope!
• Now, turn on the computer, press enter a few times until you see the C:\> prompt. This is advanced lab, so you will use an advanced operating system called DOS. Type dir and hit enter to list the folders in the C drive. Navigate to the Counts folder by typing cd counts (and pressing enter). Then type counts.exe and hit enter to start up the counting program. Press A to begin the first program, which counts the number of events in a fixed time interval. Enter plateau.txt as the file name, set number of repetitions to 10 Take a range of voltages (about 20 should be fine) from 900 - 1200 and plot log(counts/min) vs voltage applied. Take at least 10 repetitions per voltage setting. This should be easy with the program. Insert a floppy disk into the computer, then enter copy plateau.txt A:\ at the prompt. Once the green light on the floppy drive turns off (meaning it has finished copying), then eject the disk and put it into the USB floppy drive attached to one of the computers on the central table. On that computer, navigate to the A dive and copy the text file(s) onto that computer for further analysis.
• Calculate the detector efficiency based on the sample’s creation date, initial activity, and half-life, and the solid angle subtended by the detector face at the center of the point source.
• Record the number of counts in a fixed time interval (at least 50 repetitions) and create a histogram from it. Does the distribution look Gaussian? Now take the number of counts for a few different time intervals. What is the standard deviation of each set? Plot the Log of the standard deviation on the y-axis and the log of mean of the data set on the x-axis. what is the slope? Hint: It should be about 0.5 based the statistics of nuclear counting!
• For the part where you have to investigate the distribution of the time between each count, time between every two counts, etc., I’ve written up some Python code to help you create the required histograms. Don’t just copy it though, make sure to understand every line of the code thoroughly.
# Import the python data analysis module
import pandas as pd

# Read your data into a DataFrame construct, and assign names to
# the columns. The column t1 is the times between each count, and
# t2 is just the same as t1, but shifted one row upwards

df = pd.read_csv('path_to_your_data.txt', delim_whitespace = True,
names = ['Index', 't1', 't2', index_col = 'index']

# Print the first five rows of the dataframe so we know what our
# data looks like!

print(df[0:5])

# Create a new column, 't3', which is simply the sum of t1 and t2 -
# That is, t3 contains the times between every two counts.

df['t3'] = df['t1']+df['t2']
df.hist(column = 't1', bins = 100)

Your resulting histogram should look something like this:

However, don’t stop here! You should make the histogram of the distribution between every count as well, and compare it to the theoretical distribution that you would expect from your knowledge of counting statistics. Plot the theoretical function and estimate how good the experimental data matches the theory. If you see some discrepancies, try to reason about why they might arise.

Things I am looking for in particular - an intelligent discussion of why we see the distributions of ‘time between counts’ and ‘time between every 2 counts’ look the way they do.