Cosmic Rays & Statistics

The arrival and detection of cosmic rays in a detector is a random process and as such follows a distribution law characteristic of random events. Particularly when the expected number of events in a given time interval is small, then this number should follow Poisson’s statistical distribution.

Poisson Distribution

Poisson showed that the probability of observing a number n of such rare random events, in a spatial or temporal interval finished and always identical, that is:

with μ the mean value of the number of events per interval that are observed in a number N of identical observations. For this distribution, the variance has the same value of the average value, ie μ.
A feature of the Poisson distribution is that with increasing μ is approximated to a normal distribution (or gaussian) with mean value μ and standard deviation σ = √μ.
The mean value μ can be written in the following way : μ = rt , where r is e mean rate of events and t is the time interval.
From the Poisson distribution function we can calculate the differential probability for one event (n=1) which occur in the (differential) time interval dt :

dP = P(0,t) = rdt

Continuing our deductions from the Poisson distribution, we can deduce that the probability that no occurrence occurs in interval t is given by the following relation:

P(0,t) = e -rt

We can experimentally test this proposition by measuring the distribution of time intervals separating two subsequent events of muon detection. A time interval t between two events is, by definition, a time interval without events during t and an event at the instant t. Combining the two previous relationships we can get the required time distribution:

q(t)= dP / dt = re -rt

It can be seen that the above distribution is an exponential distribution, this means that short time intervals are much more likely than long intervals.
The result obtained for m = 1, that is, for adjacent (or subsequent) events can be generalized for time intervals that separate each second event or each nth event. It turns out that as the m growth of distribution tends to a gaussian centered on the value t = mr.

Experimental Measures

To record the arrival time of a cosmic particle we used the scintillation muon detector, also used for measuring the muon decay time and described in the post : Muon Lifetime Measurement with PSoC. The software allows to record on the SD card the time interval between two successive events, thus it is possible to have a time interval statistic between two subsequent cosmic muon detections. The image below shows the detector and its control unit.

The measurement was made by recording the time intervals between adjacent events (m = 1) and making a data analysis, with excel features, drawing an histogram with time bin of 25 ms. The result is shown in the image below, from which the exponential trend of frequency is evident.

To obtain the data of the exponential fitting we used the free software Gnu Octave.
The time interval data was entered in a text file: TempoMuoni.dat and loaded into the Octave software, a histogram with bin = 25 ms (40 bins on a range of 1000 ms) was created and then with the function expfit, the exponential approximation curve of the data was determined:

load TempoMuoni.dat;
[n,x] = hist(TempoMuoni,40);
pkg load optim;

The exponent of the curve is alpha = 0.00393.

With the same data, the time interval for m = 5 was then calculated, and the relative histogram with 100 ms time bin was determined. The result is shown in the following diagram in which it is seen that the trend begins to be similar to a gaussian.

Finally, the time interval for m = 100 was calculated with the same data, and the relative histogram with a time bin of 1000 ms was determined. The result is shown in the following diagram in which it is seen that the trend corresponds to a gaussian with a low variance peak.

In practice, we see how the growing number of events that are taken into account lead to a measure more stable and therefore more precise.

Particularly for m = 1, approximating the measured distribution with an exponential curve we get an exponent r equal to 0.00393 which corresponds to  3,93 events/s which is 236 CPM.

For m= 5, we approximate the curve with a gaussian (mean = 1351 ms, dev std = 544 ms) which gives 3,7 events/s equals to 222 CPM.

For m = 100, we approximate the curve with a gaussian (mean= 27310 ms, dev std = 2400 ms) which gives 3,66 events/s equals to 220 CPM, which is the correct value.

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