Cherenkov radiation and transition-radiation detection

In a transparent optical band, use the refractive index at the photon wavelength. Cherenkov emission requires the particle speed to exceed light’s phase speed in that medium, not the vacuum speed of light. The emitted wave-vector angle relative to the particle direction follows from phase matching. In a dispersive material the wavefront and energy propagation require care because group velocity differs from phase velocity; timing cannot be inferred from the angle alone.

The threshold translates into a momentum proportional to particle mass at fixed index. Thus particles with the same measured momentum can lie on different sides of threshold, or produce different ring angles if both radiate. Near very high speed the angle approaches a saturated value, reducing sensitivity to further momentum changes. A ring-imaging detector therefore combines tracking momentum with optical calibration and measured angle rather than identifying mass from a brightness label without kinematic information.

The Frank–Tamm spectrum weights short wavelengths more strongly through an inverse-wavelength-squared factor. For a narrow approximation with constant index, integration over a finite transparent wavelength band gives a coefficient multiplying path length and the sine-squared angle factor. The coefficient has inverse-length units. Outside the above-threshold band the emitted contribution is zero, so extending a negative algebraic bracket into a photon yield would be unphysical.

Optical acceptance, absorption, mirrors, windows and the wavelength-dependent photodetector response must be folded into a detected yield. A single averaged efficiency is useful for hand calculations only if that averaging is specified. Chromatic dispersion, emission-point uncertainty, tracking uncertainty and finite detected photons broaden a ring angle. More generated light need not improve angle resolution in direct proportion if those other contributions dominate. Dark hits and thresholds can also change the observed count distribution.

Transition radiation occurs when a charged particle crosses a boundary between media with different electromagnetic response. It does not require the same bulk Cherenkov speed threshold. In an ideal highly relativistic vacuum-to-medium plasma model, integrated radiated energy from one interface grows with Lorentz factor and the medium’s plasma energy. At equal momentum a light electron can have a much larger Lorentz factor than a hadron, motivating transition-radiation identification schemes.

Practical detectors use many foils or fibers because the useful yield per interface is small. Formation-length interference between neighboring interfaces, absorption and detector thresholds reshape the spectrum and produce saturation behavior. One cannot simply multiply an isolated-interface energy expression by foil number and call it a detected X-ray count. The selected gas or solid detector must also absorb and register those photons. A detector threshold in Lorentz factor is consequently an apparatus response, not the Cherenkov threshold formula in another notation.