Flux

One can make counts with a single detector lying flat on a table. If you wish to compare the count with that from another detector you will have to count for the same amount of time, or else correct for the time difference. You want to figure a rate, counts per minute, for instance. And then if the detectors are of different sizes, you will have to correct for that. You want to figure the counts per unit time per unit area, counts per min per square cm, for instance. When you assemble two detectors into the telescope arrangement, and count coincidences, however, there is another consideration. You will have to correct for the separation of the detectors. If the detectors are far apart, the angle of view is small and the muons detected will be more directionally associated with the axis of the telescope than if the detectors are closer together.

If our detectors were just strips and we were considering just two dimensions as above, we would simply take our counts per unit time per unit detector length (not area, we are imagining one dimensional detectors) and divide by the angle subtended. That angle, in radians, if it is not too large*, can be *approximated* as just w/d for Q, and W/D for q. In a two dimensional world, the flux formula might be

Well, what about that point p thing? The detector is not a point and if you consider other points on it doesn’t the angle change? Not very much if d is quite a bit larger than w. Closely spaced detectors make for a poorly defined, wide angle of acceptance, a fuzzy "image".

Our detectors are plates. To make the correction for different spacing of perhaps differently sized detectors for different telescopes, we will still figure the counts per unit time per unit area per unit angle, but the angle is the solid angle subtended.

q, in radians, is just s/r w, in steradians is a/r^{2}

So, in three dimensions, flux would be defined as

If the two plates of the detector are separated by a large distance d compared to their size, we can again make the *approximation*** that the solid angle w is just the plate area divided by the separation squared.

which will have units of counts per sec per cm^{2} per steradian (s^{-1}cm^{-2}sr^{-1})

This flux has a value for vertical cosmic rays (muons, here) at sea level of about

.48 min^{-1}cm^{-2}sr^{-1}, according to the Particle Data Group.

Experiments to try:

1. Attempt to confirm the value listed by PDG, .48 min^{-1}cm^{-2}sr^{-1 }for vertical muons.

2. Vary the separation of the detectors, d. Do you see the expected behavior? What happens to the rate when d becomes comparable to or smaller than the edge length of the detector?

3. Study the flux as a function of the angle w.

4. Check to see if absorber material has similar effect on flux in vertical and horizontal arrangements.

5. See if it makes a difference where the absorber material is placed when monitoring horizontal and vertical flux.

*In the two dimensional case if we assume the detectors are a strip and a point, the angle subtended is not quite W/D.

At 1 radian (57°) 2tan(q/2) is 1.09, so the error is quite small unless the detectors are close.

**In the three dimensional case we assume the detectors are a plane and a point. The angle subtended is not quite A/d^{2}.

At 2q = .1 radian, w = .00785 sr and w’ = .00787 sr

At 2q = .5 radian, w = .195 sr and w’ = .205 sr

At 2q = 1 radian, w = .769 sr and w’ = .938 sr

You can see that the approximation fails as the two detectors are brought closer.

Another consideration is that the second detector is not a point, p can rove. That effectively makes the angle somewhat larger and becomes serious as the separation gets small.

We can take a closer look at this comparison of the actual angle w and the approximated angle w’ in terms of the distance of separation of the detectors.

If we write the separation as a multiple times the width of the detector, d = mW, these two expressions become

An if we calculate the percent difference in w’ and w, we obtain,

Using values of m from 1 (separation equal to paddle width) to 10 shows how reasonable values of d have minimal discrepancy between actual and assumed solid angle.