0.83391024
0.7518797
1
1=yes, 0=no
25.6267
deg
25.6267
deg
1.811922
1.516759
0.2641
MeV
0.4149
MeV
0.1508
MeV
0.83391024
0.7518797
1
1=yes, 0=no
25.6267
deg
25.6267
deg
1.811922
1.516759
0.2641
MeV
0.4149
MeV
0.1508
MeV
The Cherenkov Radiation Calculator determines whether a charged particle traveling through a medium emits Cherenkov radiation, and if so, computes the characteristic emission angle. Cherenkov radiation is the electromagnetic equivalent of a sonic boom—it occurs when a charged particle travels faster than the phase velocity of light in a medium, creating a coherent electromagnetic shock wave.
The condition for Cherenkov emission is \(\beta > 1/n\), where \(\beta = v/c\) is the particle's velocity relative to the speed of light in vacuum, and \(n\) is the refractive index of the medium. The phase velocity of light in the medium is \(c/n\), so Cherenkov radiation occurs when the particle outruns its own electromagnetic field in the medium.
The Cherenkov angle—the half-angle of the emission cone—is given by \(\cos\theta = 1/(n\beta)\). At exactly threshold (\(\beta = 1/n\)), \(\theta = 0\) (forward emission). As \(\beta \to 1\), the angle reaches its maximum: \(\theta_{max} = \arccos(1/n)\). For water (\(n = 1.33\)), the maximum Cherenkov angle is about 41.2°.
Cherenkov radiation produces the characteristic blue glow visible in nuclear reactor pools and spent fuel ponds. The blue color arises because the radiation intensity is proportional to \(1/\lambda^2\) (Frank-Tamm formula), meaning shorter wavelengths (blue/UV) are more intense than longer wavelengths (red).
In modern physics, Cherenkov detectors are indispensable tools. Ring-Imaging Cherenkov (RICH) detectors identify particles by measuring the Cherenkov angle—since \(\theta\) depends on \(\beta\), and different particles with the same momentum have different \(\beta\), the angle reveals the particle's identity. Neutrino observatories like Super-Kamiokande and IceCube detect neutrino interactions by observing Cherenkov light from secondary charged particles in water or ice.
This calculator provides the threshold velocity, the Cherenkov angle for particles above threshold, and the minimum kinetic energy required for Cherenkov emission (given a particle mass). These are essential parameters for designing Cherenkov detectors and interpreting their signals.
Cherenkov condition:
$$\beta \geq \frac{1}{n}$$
where \(\beta = v/c\) and \(n\) is the refractive index.
Cherenkov angle:
$$\cos\theta = \frac{1}{n\beta} = \frac{c}{nv}$$
Threshold Lorentz factor:
$$\gamma_{th} = \frac{1}{\sqrt{1 - 1/n^2}}$$
Threshold kinetic energy:
$$KE_{th} = (\gamma_{th} - 1) \cdot mc^2$$
Maximum Cherenkov angle (as \(\beta \to 1\)):
$$\theta_{max} = \arccos\left(\frac{1}{n}\right)$$
If 'Cherenkov Emission = 1', the particle is above threshold and produces Cherenkov light. The angle increases with velocity, reaching a maximum of arccos(1/n). In water (n=1.33), threshold is β = 0.752 and max angle is 41.2°. In glass (n=1.5), threshold is β = 0.667 and max angle is 48.2°. In air (n=1.000293), threshold is β = 0.99971—only ultra-relativistic particles emit Cherenkov radiation in air.
Inputs
Results
An electron at 0.83c in water (n=1.33) emits Cherenkov radiation at 25.1° from its direction of travel. The threshold KE for electrons in water is only 0.264 MeV.
Inputs
Results
A proton at 0.667c is below the Cherenkov threshold in water (β_th = 0.752). The proton needs at least 486.6 MeV of kinetic energy to emit Cherenkov radiation in water.
Cherenkov radiation is electromagnetic radiation emitted when a charged particle travels through a dielectric medium at a speed greater than the phase velocity of light in that medium (v > c/n). It forms a cone of coherent light, analogous to the sonic boom produced by supersonic aircraft. It was first observed by Pavel Cherenkov in 1934 and theoretically explained by Frank and Tamm in 1937.
The Frank-Tamm formula shows that Cherenkov radiation intensity is proportional to frequency (or inversely proportional to λ²), meaning more photons are emitted at shorter wavelengths. This gives it a characteristic blue-violet appearance. Most of the radiation is actually in the ultraviolet, but the visible portion peaks in the blue.
No. The particle travels faster than light in the medium (c/n), not faster than light in vacuum (c). The speed-of-light limit of special relativity applies to c, not c/n. In glass with n = 1.5, light travels at c/1.5 = 0.667c, and particles can easily exceed this while remaining below c.
Common Cherenkov radiators include: water (n = 1.33, used in Super-Kamiokande), ice (n = 1.31, used in IceCube), silica aerogel (n = 1.01–1.10, tunable threshold), C₄F₁₀ gas (n = 1.0014), and various glasses and crystals (n = 1.4–1.8). The choice depends on the desired threshold velocity for the particles of interest.
Neutrinos interact rarely, but when they do, they produce charged particles (electrons, muons) that travel faster than light in water or ice, producing Cherenkov light cones. By detecting these light patterns with arrays of photomultiplier tubes, experiments reconstruct the neutrino's direction, energy, and flavor. Super-Kamiokande, IceCube, and SNO all use this principle.
For electrons: KE_th = (γ_th - 1) × 0.511 MeV ≈ 0.264 MeV, achievable by many beta emitters. For muons: ~160 MeV. For protons: ~487 MeV. For alpha particles: ~1,945 MeV. The threshold increases with particle mass because heavier particles need more energy to reach the same velocity.
Roboculator Team
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