Neutrinos · D09 · Water and geometry

Cherenkov Light & Detector Rings

Set a charged particle’s speed and tilt, then trace its light to a square detector plane. Move the emission point and distinguish a geometric ring from a detector’s finite acceptance.

Cherenkov half-angle $\text{—}$

Speed threshold · fraction of vacuum light speed $\text{—}$

Recorded rays $\text{—}$

Outside the finite plane $\text{—}$

Normal-incidence reference radius $\text{—}$

Mean recorded optical path $\text{—}$

Geometric teaching model: 1200 equal-weight rays from one small emission region in homogeneous nondispersive water with refractive index 1.333. Detector is a 14 metre square plane; not the cylindrical Super-K tank. No scattering, absorption, wavelength dependence, particle showers or sensor efficiency. Brightness is sampled ray density, not energy or absolute photon yield.

Preparing a reproducible teaching record…

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Start with the essentials

Focus question: What changes the physical cone, and what only changes the observed footprint?
One-sentence intuition: Speed determines the Cherenkov angle; orientation and detector distance determine the recorded shape.

Core mathematical model

Phase matching and threshold

\cos\theta_C=\frac{1}{n\beta},\qquad n\beta>1

Here beta is particle speed divided by vacuum light speed. Below threshold this model emits no rays. We assume the same index at every wavelength.

Intersection and an independent limit

\mathbf r_D=\mathbf r_0+\frac{z_D}{k_z}\mathbf k,\qquad R_{\alpha=0}=z_D\tan\theta_C

The unit ray direction is k. Only forward intersections inside the finite square are recorded. The normal-incidence radius is a reference even when the particle is tilted; it is not a fitted ring radius.

Common difficulties

A cone meets a measurement plane

Typical misconceptionA ring is a photograph of a neutrino.

Better mental modelThe geometry describes light from a secondary charged particle. Full neutrino reconstruction also needs interaction kinematics and detector response.

Run the experiment

01
Cross the threshold
Choose Below threshold. Raise the speed until light appears.
What to observe: The angle begins at zero. The model fixes the number of sampled rays above threshold, so do not infer an absolute light-yield law.
02
Tilt the cone
Choose Tilted cone and compare the apparatus with the recorded plane.
What to observe: A tilted cone produces a displaced, stretched footprint. Finite acceptance can clip it.
03
Change distance only
Return to a centered ring and double the plane distance.
What to observe: The radius doubles while the opening angle stays fixed, until the square clips the light.