Cosmic Ray Detector - Data Center

Derivation of count rate to flux formula

To derive a formula to convert from count rate to flux, imagine a really small area A on the bottom panel of the two that make up the vertical pair of scintillator panels.

Now imagine a sphere (indicated in black in the picture)  that has its center at that point with a radius equal to the distance d between the panels. In principle, muons from all directions can intersect A. Obviously to get to A, these muons must also go through the wall of the imaginary sphere. But of all the muons that go through A, only those that also intersect the top panel will be counted in our experiment: the fraction of muons that goes through A that also go through the top panel is approximately given by the area of the top panel divided by the area of the sphere:

This is really only true if the top panel is a part of the sphere. But in reality, the panel is flat and has a certain thickness, while the sphere is curved and has no thickness. But for our purposes, this is a reasonable approximation.

We have assumed that we can distinguish whether muons come from above or below. Our detector does not really measure whether muons come from above or below, but it so happens that there are very few muons that come from below. Muons produced on the other side of the earth cannot reach us. The only muons that come from below are ones that are produced by high-energy neutrinos that come from outer space on the other side of the earth and go through the entire earth, producing muons in the last few kilometers of the earth's crust below our feet.

The total count rate in our counter is now the total rate R_A of muons that go through A, times the fraction f, times the number of ways one can choose a different small area A. The number of ways one can choose a different small area A is the area of the bottom panel divided by the area of A, so the total count rate is:

where we substituted what f was. The rate of muons going through one unit of area R_unit is:

R_unit = R_A/A

Substituting, we get:

The flux F_m is defined as the rate of muons per unit area per unit of solid angle (a "steradian").  A full sphere has 4p steradians, so we get:

F_m = R_unit/4p

which gives:

The "count rate" is what we measure. As you can see, it has in it the flux F_m and some numbers (the areas of the panels and the distance between them) that depend on our specific apparatus. If we want to calculate the flux, then we need to invert the formula above. We obtain:

Notes

Each pair of detectors accepts incoming muons from some cone of angles. There is no overlap between the cones, but note that for the vertical array and the 45 degree one, only muons are counted that go in one direction (top to bottom), whereas for the horizontal array, muons can go left to right or right to left!

In our cosmic ray detector, the pairs of detectors are the same, except for their orientation, but the fluxes they measure are quite different. This pattern would be approximately the same wherever the detector was placed on the surface of the earth near sea level (there are small differences at the 10% level). However, if one performed the measurements at a really high altitude, the pattern might be somewhat different.

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