-- JorgeRodriguez - 2012-01-09

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

When writing up the lab you should include a breif description of the important parts of this experiment:

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.

Once you have your histograms, one for each of sets of data, you will need to fit it to a Gaussian and a Poisson distribution seperately. The fits determine the mean and width or standard deviations of the functional forms. You then compare the two fits and use a "goodness of fit" estimator to establish which function is the best fit. One such estimator is the normalized chi-squared "χ²/ν" where ν is the number of bin minus the number of parameters in the fit. The Gaussian fits can be done in MN_FIT as before, but you will have to do more work with the Poisson fits since MN_FIT doesn't have a Poisson function internally defined. You can do one of two things:

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.

GradingRubric