I got excited about Muon Telescope design because I think that the muon flux, and the neutrino-detection noise, will vary significantly at DUSEL. The variability is due to the topography and the geology of the ground above the 4850 level campus where the first muon telescopes could be installed. Today, I want to look at a couple more parameters that could be important.
Here is a cross section of the mine showing geology. I pulled this from the DUSEL.org website which is a fantastic resource for DUSEL information, including full publications of the geology of the Northern Black Hills (shh, don’t tell anyone). http://homestake.sdsmt.edu/Resources.htm
Here we see the very complicated geology that was encountered during mining. The complicated folding and squeezing of the various layers occurs in all three dimensions with more than four large scale deformation events. Each layer has a different geology, and one would expect a different meters of water equivalent (M.W.E.) shielding capacity. In this figure the topography does change, I included a line that shows the shortest path to a hypothetical 4850 level campus is not vertical, but is in fact about fifteen degrees off vertical. However, the difference in length between those paths is small at <10%. I also drew another muon path at about 45% off vertical that illustrates the much longer path through the rock that it must travel. That additional travel time in rock will increase the odds of a muon interaction with a mineral particle, and thus the increased odds of attenuation for any particle coming in at an angle. The little inset diagram shows a Gaussian curve that should roughly describe the distribution of muon flux from vertical (roughly maximum) to horizontal (minimum). I might expect to detect muons even if the muon telescope was oriented horizontal, possibly due to the curvature of the earth or the persistence of the particle.
As we have discussed before, a muon telescope is two scintillators with photomultiplier tubes to count the photons coming off the scintillator when a particle interaction occurs.
The geometry of the telescope is simple. The two scintillators are separated by some distance. The size of the detectors and the distance between them determines the maximum angle off of vertical, theta, that an incoming muon could create a signal on both detectors. Another way of saying this is familiar to photographers, theta is proportional to the field of view of the telescope.
The further apart the scintillators are spaced, the smaller the field of view - primarily because the odds of an incoming muon will have the correct angle to hit both scintillators. Both scintillators may get the same number of hits, but by measuring the time interval of the hits, single muon particles can be traced and differentiated from decay of, say, thorium on the cavern walls that spews out random ions that could be detected on both scintillators at roughly the same time.
Lets look at it using a fancy Venn Diagram.
Both scintillators are triggered all the time by random ions, cosmic rays, and lab gremlins all the time, some we want to measure and some we don’t. The ratio of good signal to bad signal is bad in both cases, but if the telescopes are separated, the magnitude of bad signal is small, but so is the magnitude of the good signal.
So, is there anything we can do to optimize the design of the telescope so that it minimizes random noise and loss of signal because the muons decay between scintillators? There are a number of parameters we can plot up to look at this question. Using the muon telescope cartoon, we can come up with the math to solve for the muon flight time between detectors and also look at the angular field of view. The angular field of view should be proportional to the muon flux with some influence by that flux distribution that we showed before.
On the left axis, the travel time of a muon is plotted. For the case where a muon is coming down vertically, the wider the distance between scintillator paddles, the greater the travel time – it is a linear relationship. However, consider the maximum diagonal travel distance a particle could travel and still hit both scintillators. This differential distance increases as the paddles are brought closer together and as the paddles are increased in size. In this figure, the largest paddles placed one paddle-width from each other on the low end to find the maximum differential. The maximum travel time has a variance of more than 100% and it is actually larger than the 5 ns detection limit Mark told me about a while back. In that case, a muon travelling diagonal would not have been counted. However, the effect quickly drops off with smaller paddles separated by larger distance, maybe 4x the paddle diameter could be negligible depending on the required accuracy.
On the right axis, the field of view (FOV) in degrees is shown. The closer the paddles are together, the larger the FOV. As the paddles are separated, the FOV quickly drops away and approaches zero as the paddles are separated. The paddle separation should be selected to accommodate the shortest muon travel path. Since the FOV is proportional to flux, the farther the paddles are separated, the smaller the muon detection and the greater integration time required.
Finally, the decay time for a muon is about 2 µs = 2000 ns. The longest travel time between scintillators shown on this figure is about 35 ns. The odds of a decay occurring between detections is not zero, but the travel time is small compared to the decay time. It is a bit unclear to me if the 2 µs value is at earth frame or if it is particle frame dependent. The effects of relativity will have the effect of increasing the apparent decay time from the earth frame and, thus, overestimating the number of muons that would be counted.
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