Poisson Statistics
Background
The Poisson distribution can characterize random events that occur at a well-defined average rate. It is widely used in atomic and sub-atomic physics. The Poisson distribution is effective in a variety of statistical applications. The most common involve event probabilities, but several assumptions must hold true: i) the rate at which random events occur does not change for the duration of the measurement; ii) the occurrence of one event does not change the likelihood of another event; iii) events occur at a slow enough rate that they can be individually distinguished.
In this experiment, Poisson statistics will be used to analyze random radioactive decay events that occur in a defined time interval. A decay occurs an integer k number of times in the interval, including possibly not at all (k = 0). The average number of events expected in a defined time interval is λ, known as the event rate. Given λ, the probability of observing k events in the time interval is:
If the interval gets longer, λ increases commensurately and Pk changes.
Experiment
The radioactive source is Cs-137, with half-life of 30.2 years. The probability that a single atom will decay in a 100 ms interval, for example, is 6.87 × 10−11, i.e. an extremely unlikely occurrence. If the source has enough mass, however, a sufficient number of decays will happen in reasonable measurement time. Because their energy is so high, it is difficult to detect gamma-rays directly. Indirect detection is used here through a process called scintillation. A scintillator is a special material – in this case a sodium iodide (NaI) crystal – that converts high-energy photons such as gamma-rays into visible photons that can be detected. The NaI scintillation crystal is directly attached to a photo-multiplier tube (PMT) that is sensitive enough to resolve very few visible photons. Detected photons appear as voltage pulses at the output of the PMT. The height of the voltage pulse is directly proportional to the energy of the gamma-ray that created it, which is 662 keV for Cs-137.
A multi-channel analyzer (MCA) can measure the energy of the gamma-rays and also the rate at which they strike the scintillator. The rate of spontaneous gamma-ray emission is of interest in this experiment. The MCA is configured to count gamma-rays in a defined time interval. It does this counting repetitively, building up a statistical ensemble of N intervals each holding an event count. When viewed as a histogram, a distinct distribution becomes evident. The key to observing a Poisson distribution is keeping the event rate λ sufficiently low. The distribution plot will appear asymmetric. As λ increases, the distribution transitions to a symmetric Gaussian.
Equipment and Setup.
Background Reading
History
Experiment set up and verified by Martin Hoeferkamp, January 2020
