Exercises
Six Exercises, with Worked Solutions
Hints are inline; full solutions follow, so cover them if you want the problem first.
The problems
1. Cherenkov in water
Sea water has n ≈ 1.35. Find the threshold speed and the Cherenkov angle for β → 1, and say whether a water detector collects more or fewer photons per metre than an ice one.
Hint: Both follow from cos θ_c = 1/(nβ); the yield goes as sin²θ_c.
2. How much light is that
A muon crosses the full kilometre of the array. Using the Frank–Tamm yield of Part 1, how many Cherenkov photons does it make in the 300–600 nm band? Comment on why the event is still hard to reconstruct.
Hint: 32,000 photons per metre, and the sensors occupy a vanishing fraction of the volume.
3. How long must you wait
Part 2 found about 4 events per km² per year above 100 TeV. How many years does a 1 km² detector need for a hundred events? What area would give a hundred in ten years?
Hint: The rate is linear in area and in time.
4. A very energetic muon
How far does a 10 PeV muon travel in ice? Compare with the depth of the Antarctic ice sheet (about 2.8 km at the Pole) and comment on what that means for containing such an event.
Hint: R = (1/b) ln(1 + bE/a), with a = 0.26 GeV/m and b = 3.6 × 10⁻⁴ m⁻¹.
5. Why the resonance needs an electron
The Glashow resonance occurs at 6.3 PeV on an atomic electron. Repeat the calculation for a muon target and explain why the answer, though much lower, is useless in practice.
Hint: E = m_W²/(2m), and ask what the target is made of.
6. Discussion
Is IceCube an astronomical observatory or a particle-physics experiment? It measures cross-sections above accelerator energies and it makes images of the sky. Argue both, and say which framing you think earned the prize.
Hint: No single right answer — state your criterion and apply it consistently.
Worked solutions
1. Cherenkov in water
βmin = 1/1.35 = 0.741, lower than ice's 0.763, so water is sensitive to slightly slower particles. The angle is arccos(1/1.35) = 42.2°, larger than ice's 40.2°. Since the yield goes as sin²θc = 1 − 1/n²β², water produces slightly more light per metre. Ice is not chosen for its optics but for being solid, dark, quiet and already in place.
2. How much light is that
3.2 × 10⁴ photons/m × 10³ m = 3.2 × 10⁷ photons, tens of millions from a single muon.
And yet a typical event registers only a few hundred detected photons. The sensors are 13-inch spheres spaced 17 m apart on strings 125 m apart, so the instrumented fraction of a cubic kilometre is of order one part in a billion — and scattering in the ice spreads what little is collected over hundreds of nanoseconds. Reconstruction is an exercise in extracting a kilometre-long straight line from a handful of scattered, late-arriving photons.
3. How long must you wait
At 4.4 events per year, a hundred events takes about 23 years — roughly the design lifetime of the detector, which is not a coincidence. To collect them in ten years you would need about 2.3 km². This is the entire argument for the next generation of detectors being ten times larger: the physics is statistics-limited and the only lever is volume.
4. A very energetic muon
\[ R = \frac{1}{3.6\times10^{-4}}\ln\!\left(1 + \frac{3.6\times10^{-4}\times 10^{7}}{0.26}\right) \approx 26.5\ \text{km} \]
Nearly ten times the thickness of the ice sheet. Such a muon cannot possibly be contained: it enters, crosses, and leaves, depositing only a fraction of its energy. The direction is superb and the energy is a lower bound — exactly the trade-off of Part 3, at its most extreme.
5. Why the resonance needs an electron
E = mW²/(2m) with mμ = 105.7 MeV gives about 30.6 TeV, two hundred times lower and far more abundant in the flux.
It is useless because ordinary matter contains no free muons. A muon lives 2.2 microseconds; there is no muon target to scatter from. The resonance is observable only because every atom in the ice comes with electrons attached — the target has to be a stable constituent of matter, and that constraint is what fixes the energy at 6.3 PeV rather than anywhere more convenient.
6. Discussion — guidance, not an answer
Observatory: it points at the sky, issues alerts to other telescopes, and has produced an image of the Milky Way. The prize citation names a discovery about the universe, not about the weak interaction. Experiment: it measures cross-sections above any accelerator, tests a 1960 electroweak prediction, and constrains neutrino oscillations. A reasonable criterion is whether the object of study is the source or the particle — and note that the same dataset serves both, which is unusual and is part of why the instrument was worth a kilometre of ice.