Part A: Generate a sample of simulated Cosmic Ray data

Here you are referred to the Particle Data Group's section on cosmic rays (Links to an external site.), which describes that the cosmic ray flux at the earth's surface is approximately determined to be distributed as ~ cos2(q). Here the angle is the azimuthal angle, where 0 is in the verticle direction and the functional form complies with the expectation that rays incident vertically are more probably than horizontally. There are a variety of effects contributing to this form, which you are encouraged to report on from your research, see PDG above. Here we will not attempt to simulate this distribution from first principles, although that could be done instead, you will take the shape as a given and create that on the computer, later confirming in real data.

In principle one could attempt to generate deviates drawn from a cos2 distribution as we did previously using conservation of probabilities and then solving for the angle in the argument but unfortunately integrating the cos2 function leads to a transcendental polynomial function which is somewhat difficult to deal with. Instead, we will employ a short cut given that we will be analyzing the data in terms of bins of azimuthal angles. We can thus use a Monte Carlo Poisson smearing on a bin by bin basis to create a simulated sample of data we can easily create on a computer. Here is how:

1. First decide on a number of bins from or -pi/2 to + pi/2 corresponding to the range of position one can orient the CRT in. The center of these bins will be your azimuth angle q.
2. Now create a "perfect" number of counts for each of the azimuth angle bins defined above using a formula such a here N represents the total number of cosmic ray events in a given interval of time so is unitless. The parameters A and B scale the function and capture the length of time over which the experiment is conducted, collection efficiencies, and allow for small but expected backgrounds. Remember you are collecting events if and only if both scintillators fire within a nanosecond or so so the likelihood of significant backgrounds are small.
3. Now you will smear the "perfect" data count rate above by a Poisson. Remember histogram bins are Poisson so the mean is the count N while the uncertainty is the N. These smeared counts will be the sample of simulated data you will use to compare to real data collected with the CRT.

Here how to do this

Part B: Observation of Poisson and Gaussian distributions from Sr 90.

Part C: Cosmic Ray Flux

Comments on the CRT apparatus:

• Scintillation counters each with its own PMT (Cosmic Ray Telescope: Note for this experiment you will need to use both counters)

• NIM module Linear Fan In Fan-Out (this allows you to duplicate RF signals from the PMT and NIM modules

• Oscilloscope (Links to an external site.) (to examine the output of the PMT and help set discriminator levels etc.)
• 90Sr radioactive source

Turning in your lab report

1. The Scintillator: Look up references on how organic or plastic scintillators work. The scintillator you are using is a Bicron BC 400 series plastic scintillator

1. The High Voltage supply, Just noting that one is used is sufficient.

Turning in your coding project

Please upload your code to the assignment Monte Carlo Project under the Python Coding Project group. Along with a grade for your report you will be graded separately on the coding sample you provide via upload. Here are important steps you should take to earn a good grade.

1. Make sure each "Part" is in its own file. You can label your program as you wish as long as I can identify it when I run it.
2. Make sure your program(s) run in the spyder IDE.
3. Make sure your program is well commented as I will need to figure out what you did in order to evaluate the programming.

• NIM module Linear Fan In Fan Out (this allows you to duplicate RF signals from the PMT and NIM modules

1. The Scintillator: Look up references on how organic or plastic scintillators work. The scintillator you are using is I believe a Bicron BC 400 series plastic scintillator

1. The data acquisition equipment: The various NIM modules you used in your experiment. You should be familiar with the operation of each and report on their use and function in your write up.

Lab Assignment: Cosmic Ray Lab

Part A: Observation of Poisson and Gaussian distributions in radioactive decay of Sr 90.

Part B: Cosmic Ray Flux

Before you proceed with the experiments, PART A and B below lets determine the proper volatage to input into the PMTs given the disc threshold. We do this because you want to run your PMTs, each an individual with, even if manufactured identically, can have different optimal operating voltages which can vary over time.

The best way to determine the operating voltage of your setup is to basically check to see that the count rate, given disc threshold setting and PMT input voltage, varies little over a set of input voltages. This process is sometime referred to as "Platue the PMTs", although here its not just the PMT that are involved. To do this setup up your cosmic telescope by connecting the PMT output to the NIM discriminator then connect the disc output to the counter. Setup the counter so as to collect events over a time window to collect a few 100 events and then vary the input voltage and record the counts. I would vary the voltage by 100 volts starting at around 1000 volts. Do not go beyond 2000 volts please. What you should see is the number of counts varies a lot when the voltage is set too high or too low. You'll want to use the a random event source, like the Sr 90 to insure you have real triggers instead of noise. DO this for BOTH PMTs and provide in your lab report the plots you make. Unfortunately you have a single HV power supply that can only ou

Include in your introduction a breif description of the important components of the apparatus you used to conduct this experiment:

1. The Scintillator: Look up references on how organic or plastic scintillators work. The scintillator you are using is I belive a Bicron BC 400 series plastic scintillator
2. The Photomultiplier: Describe how a PMT works. You should provide as much detail as you need to demonstrate that you understand how this device functions and why it is used.
3. The High Voltage supply, Just noting that one is used is probably sufficient.

• slac-tn-95-001.pdf: SLAC Cosmic Ray Telescope paper

Lab Assignment: Measure the angular distribution of cosmic ray flux

Setup and calibration

To determine the operating voltage of each PMT you'll need to "Platue the PMTs". What you do here is simply record the PMT count rate as a function of input voltage, starting at say 1000 volts. PMTs are all different and they can change over time so its a good practice to do this operation before you use any PMT for the first. Setup the apparatus to count the number of triggers for some interval of time and then vary the PMT input voltage and record what you measure. Plot the results and the somewhere in the middle of the platue is where you want to run your PMT during the experiment. Make sure you use a random event source to insure you have real triggers instead of noise.

PART A

You will only need to use one of the scintillator/PMTs for this part of the experiment.

Using a random source, that ⁹⁰ Sr is as good a random source as any. Take 100 measurements with the count time interval set to collect on average 1 event per time interval. Histrogram the results. How is the data distributed? Then repeat the excercise but now use 100 events per time interval. You can adjust any of the experimental conditions you control to achive these measurements. You can change the distance between the source and the scintilator or adjust the discriminator levels. What ever you do make sure you retain the randomness of the events by insuring you are not picking up PMT or other sources of noise.

Part B

Using both PMTs, both of which you should have already platued, perform the flux vs. angle measurement. Here you'll need to use the coincide NIM module to record when the two PMTs fire within a couple of ns of each other. Plot the distribution and fit a function form to your data and discuss what you see. There is a nice write up on this part that you can access as a reference. See

• Scintillation counters each with its own PMT (Cosmic Ray Telescope: Note for this experiment you will need to use both counter)

• NIM module Discriminator (to decide whether the signal is a true PMT pulse and not noise)
• NIM module Linear Fan In Fan Out (this allows you to duplicate RF signals from the PMT and NIM moudules
• NIM module Counter/timer (to count the number of NIM output signals)
• NIM module Quad Coincide (produces a NIM output when two signal arrive within an interval of time determined by module's internal logic)

1. The data aquisition equipment: The various NIM modules you used in your experiment. You should be familiar with the operation of each and report on their use and function in your writeup.
2. The Oscilloscope: No description required but you may want to explain who you used it.

Lab Assignment: Statistics of Random events

In this lab the goal is to explore the statistics of random events. The events will this time be generated by a physical process that is random in nature, namely the radioactive decay of a sample of ⁹⁰Sr. Strontium 90 is a rare isotope of Strontium, is a beta emitter (e-) and has a half life of 28.9 years. The relatively long lifetime will insure a "steady-state decay rate" that remains constant over the time scale of the lab and thus provide you with random sample of events to count.

Assignment: Is the Rate a Poisson or Gaussian Statistic?

In this experiment you will do three sets of 100 measurements for the beta decay rate of ⁹⁰Sr. You will then determine the mean rate by fitting your data to the appropriate distribution, either a Poisson and Gaussian.

The experiment should be setup to record events at three different rates over a given time interval. To do this set up the counter/timer to record counts in 1 second interval with rates determined by the height of the source.

• First adjust the height of source so that your rate is about 1 Hz. Make about a 100 measurements with this arrangement. (Note that the cosmic backgroud rate is of order 1 Hz but this is OK since this background rate is uncorrelated with the source rate.
• Repeat the experiment but now position the source so that the rate increase to about 10 Hz.
• Repeat the experiment this time with the rate adjusted to about 100 Hz.

Equipment:

To count the number of decays of the radioactive source you will need to setup equipment that can detect the emission of electrons in this energy range. You will use the cosmic ray telescope in the lab for the dection of the electrons. Basically the telescope consists of a piece of plastic scintillator made out of material that when exposed to charged particles reacts by emitting light. The light travels through the plastic material, designed to be transparent to the emitted light, reflecting from surfaces, colliding with other electrons etc, some of the photons end up at the front of the photomultiplier tube (PMT). The PMT is an electronic device based on the photoelectric effect that first converts a small number photons, into an amplified electrical signal sufficiently large to be easily recored by standard laboratory equipment. You should provide in your writeup an a short description of how the emitted electrons are detected by the equipment you use in this experiment.

The list of necessary equipment is:

• Scintillation counter with PMT (Cosmic Ray Telescope: Note you may not need to the second counter in this experiment)
• High Voltage Supply (to power the PMT)
• Discriminator (to decide whether the signal is a true PMT pulse and not noise)
• Counter/timer (to count the number of real PMT signals)
• Oscilloscope (to examine the output of the PMT and help set discriminator levels etc.)
• ⁹⁰Sr radioactive source

1. The Scintillator: Look up references on how organic or plastic scintillators work.
2. The Photomultiplier: Describe the funciton of the photo cathode and the amplifier, why do you need such a high voltage.
3. The High Voltage supply: Just note that there is one.
4. The data aquisition equipment: The signal from the Photo tube goes into a NIM (Nuclear Instrument Module) discriminator who function and operation you should understand and a rate counter.
5. The Oscilloscope: No description required.

Write Up

Analysis

As with the MonteCarloLab you will need to histogram your data and then perform a fit to determine the mean rate and standard deviation. The histogram should include

• On the horizontal "x" axis: the random variable in this case is the the rate or number of counts per unit time.
• On the vertical "y" axis: the dependent variable or number of times the rate has a particular "x" value.
• The total number of entries in your histogram should be all 100 of your trails.

1. You can can either create your own Possion function and use the MN_FIT/MINUIT framework to fit your data. This isn't too hard but you'll need to write a FORTRAN function and feed it to MN_FIT. There is a COMIS feature in MN_FIT the allows the users to create ther own fitting functions, (see help comis in MN_FIT) if you want to give this a spin I can help you set it up.
2. You can do the Poisson fit by hand. This shouldn't be to difficult to do given that the function is discrete and when the rates "x" are small x! isn't too large a number for your calculator. Even for large x a recursion relation exists to help you out. Please refer to Dr. Raue's note. With this method you will, of course need to compute the the χ²/ν by hand as well, see Dr. Raue's chi-2 note.