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Neutrina Fade

Neutrino Oscillation Physics

\[P \approx \sin^{2}(2\theta)\,\sin^{2}\!\left(\frac{1.27\,\Delta m^{2}[\text{eV}^2]\,L[\text{km}]}{E[\text{GeV}]}\right)\]
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$21.99 USD
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Meet Neutrina Fade. About 65 billion solar neutrinos cross each square centimetre of you every second. They change flavour as they travel, which is only possible if they have mass. On the mug: the two-flavour oscillation formula P ≈ sin²(2θ) sin²(1.27 Δm² L/E). Pixel-art full-wrap black ceramic mug in 11 or 15 oz; equation, symbols and sources in the physics panel below.

Shipping calculated at checkout by destinationMade to order, ships in ~10 days

Details

Size & material11oz · 15oz

Black glossy ceramic mug with a C-handle. Lead- and BPA-free.

  • 11oz: 0.33 l
  • 15oz: 0.44 l
CareDishwasher & microwave safe

Dishwasher safe (top rack) or hand wash. Microwave safe.

ShippingBy destination

Shipping calculated at checkout by destination.

Made to order: about 10 days of production and handling before it ships.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Neutrinos change flavour as they travel, which requires different masses. For Δm² ≈ 2.5×10⁻³ eV² and 1 GeV, the first oscillation maximum is at about 500 km.

Neutrina Fade never arrives as quite the same particle that left.

Neutrinos were proposed by Wolfgang Pauli in 1930 and built into the theory of beta decay by Enrico Fermi. They feel only the weak force and gravity, so they pass through almost everything. About 65 billion of them from the Sun cross each square centimetre of you every second.

They come in three flavours: electron, muon and tau. A neutrino made as one flavour is a mixture of states with different masses, and those travel with slightly different phases. As it flies, the mixture shifts and the flavour changes. For two flavours the probability is P = sin²(2θ) sin²(1.27 Δm² L/E), with Δm² in eV², L in km and E in GeV. If neutrinos had no mass, Δm² would be zero and nothing would change. Super-Kamiokande (1998) and SNO (2002) proved they do change, and Takaaki Kajita and Arthur McDonald shared the 2015 Nobel Prize in Physics.

A worked number: for Δm² ≈ 2.5×10⁻³ eV² and a 1 GeV neutrino, the first maximum of oscillation comes at L ≈ 500 km.

“I left as one thing,” says Neutrina Fade. “Ask me again in a few hundred kilometres.”

Equations

\[P \approx \sin^{2}(2\theta)\,\sin^{2}\!\left(\frac{1.27\,\Delta m^{2}[\text{eV}^2]\,L[\text{km}]}{E[\text{GeV}]}\right)\]
\[P(\nu_a\to\nu_b) = \sin^{2}(2\theta)\,\sin^{2}\!\left(\frac{\Delta m^{2}c^{4}L}{4\hbar c E}\right)\]
SymbolMeaningUnit
\(P(\nu_a\to\nu_b)\) probability a neutrino of flavour a is detected as flavour b (two-flavour approximation) \(\mathrm{dimensionless}\)
\(\theta\) mixing angle \(\mathrm{rad}\)
\(\Delta m^{2}\) difference of the squared masses of the two mass states \(\mathrm{eV²/c⁴ (enter in eV²)}\)
\(L\) distance travelled \(\mathrm{m (enter in km)}\)
\(E\) neutrino energy \(\mathrm{J (enter in GeV)}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(c\) speed of light in vacuum \(\mathrm{m/s}\)

Sources

  1. Particle Data Group, Navas et al. (2024) Review of Particle Physics, Phys. Rev. D 110, 030001
  2. PDG 2024 review — Neutrino masses, mixing and oscillations
  3. Super-Kamiokande, Fukuda et al. (1998) Evidence for Oscillation of Atmospheric Neutrinos, PRL 81, 1562
  4. SNO, Ahmad et al. (2002) Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in SNO, PRL 89, 011301
  5. NobelPrize.org — The Nobel Prize in Physics 2015 (Kajita, McDonald)

Source Fukuda, Y. et al. (Super-Kamiokande) (1998). Evidence for Oscillation of Atmospheric Neutrinos. Physical Review Letters 81, 1562–1567.doi.org/10.1103/PhysRevLett.81.1562

Field notes

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