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Let's suppose a muon emits Cherenkov light while travelling in a medium along a straight line. Let's suppose the motion is perpendicular to a wall which is instrumented with photomultipliers.

Question: Should these PMTs "see" a ring or a solid circle?

Note: I believe the true answer is a ring. Indeed I know that for example in SuperKamiokande muons produce ring shaped "pictures". But in my intuition they should produce solid circles, as the following drawing could explain.

Drawing of Cherenkov light produced along the path of a particle

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  • $\begingroup$ what's the energy of these muons? $\endgroup$
    – JEB
    Commented Nov 15, 2020 at 0:36
  • $\begingroup$ Enough to emit Cherenkov light for sure. Should be the energy relevant? $\endgroup$ Commented Nov 15, 2020 at 0:47
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    $\begingroup$ Yes, if it stops, it doesn't fill in the ring, but you said "ignore dE/dx", which may be the problem. $\endgroup$
    – JEB
    Commented Nov 15, 2020 at 1:33
  • $\begingroup$ Do you mean that it's not true that muons produce ring shaped patterns in general? $\endgroup$ Commented Nov 15, 2020 at 2:16
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    $\begingroup$ In the case of the RICH the radiator doesn't fill the entire space up to the "wall", this is why the image is a ring. So it's a different situation in my opinion. $\endgroup$ Commented Nov 15, 2020 at 17:41

2 Answers 2

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Your intuition is correct. A muon with velocity $v > c/n$ approaching and passing through the wall will produce a filled in circle of Cherenkov light. Such Super-Kamiokande events are, however, rarely shown online because they are relatively uncommon and also less useful because their energy cannot be measured.

Such "filled-in" Super-Kamiokande events are uncommon because most muon tracks are only a few metres long and don't reach the edges of the very large detector. The short lengths are because neutrinos observed in SuperKamiokande typically have energies about a GeV for atmospheric neutrinos and even less for T2K neutrinos. A relativistic muon loses about 2 MeV/cm in water in water, so a 1 GeV muon will only travel about 5 m in water before stopping. The SuperKamiokande inner detector is a tank of water 36 m tall and 34 m in diameter, and muon events are usually only accepted if their vertex is at least 2 m from the inner detector PMTs. The relatively short length of most muon tracks compared to the large fiducial volume of the detector means that most observed tracks don't reach the walls, so rings are observed, not filled in circles. To see how the muon ring fills in as its energy increases and its track gets longer, see the images in Figure 29 on page 64 of Euan Richard's 2015 University of Tokyo PhD thesis.

Events that are not contained in the Super-Kamiokande detector are also less useful because their energy cannot be measured. The energy of a muon is determined from how far it travels before it stops. This length is usually determined by measuring the total amount of Cherenkov light observed, and the width of the Cherenkov ring also provides energy information. If the muon doesn't stop before reaching the wall, only a lower bound can be set on the muon energy. Such events are less useful for physics analysis and are less likely to be chosen for public display, since published events are usually the "best" events.

Perhaps the most interesting class of uncontained Super-Kamiokande events that produce filled-end circles of light are very high energy upward-going muons, such as seen in this image which actually passes completely through the detector.

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It can be understand from the following relation of Cherenkov angle, $\theta =\cos^{-1} \left( \dfrac{c}{\beta v} \right)$.

The maximum value of Cherenkov angle is reached when the particle is moving at approximately the speed c. In the case of water, where $n = 1,33$, the maximum angle is $41.2°$.

The energy loss by $\mu^{\pm}$ is not significant, therefore, velocity of $\mu^{\pm}$ does not change too much. The photons emitted along path of $\mu^{\pm}$ have nearly same angle. Therefore, we get ring instead of solid circle.

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  • $\begingroup$ Welcome LuckySehgal to Physics Stack Exchange. Unfortunately this doesn't really answer the question. Yes, the outer diameter of the ring is determined by the Cerenkov angle and the initial distance of the muon from the PMTs. The question is about why since as the muon gets closer to the PMTs this diameter shrinks until it is zero at the wall, shouldn't the ring be filled in? $\endgroup$ Commented Feb 7, 2023 at 17:01

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