Vacuum Neutrino Oscillation Probability Calculator

Introduction: Vacuum neutrino oscillations from baseline and energy

This neutrino oscillation probability calculator shows the simplest vacuum picture of how a neutrino can change flavor during flight. In the two-flavor approximation, the state created at the source is treated as a combination of two mass eigenstates, and those components do not stay perfectly in step as the particle travels. Their evolving phase difference is what creates the familiar rise-and-fall pattern in the conversion probability, so the answer depends on more than just distance alone.

The oscillation probability for neutrinos traveling through vacuum is given by , where is the mixing angle, is the mass-squared difference in electron volts squared, is the baseline distance in kilometers, and is the neutrino energy in gigaelectronvolts. The constant 1.27 comes from combining fundamental constants with the unit conversions used by this common approximation, and the factor of is what makes the phase grow faster when the neutrino travels farther or carries less energy.

The calculator above evaluates that vacuum expression directly. The angle you enter is converted from degrees to radians before the sine terms are computed, and the baseline, energy, and mass-squared difference are inserted using the conventional km, GeV, and eV² units expected by the formula. The first sine-squared factor sets the size of the oscillation through the mixing angle, while the second sine-squared factor places the neutrino on a particular point of the phase curve. The result is displayed as a percentage so it is easy to compare one flavor-conversion scenario with another.

To make that unit handling explicit, the calculator first turns your angle into radians with , then evaluates the phase , and finally keeps the reported probability within . Those steps are exactly what you want when you are checking whether a chosen baseline and energy place the neutrino near a maximum, near a minimum, or somewhere in between.

That baseline-to-energy ratio is the reason a single source can behave very differently at different detector locations. Longer flight paths or lower energies both increase L/E, which advances the oscillation phase more quickly, while shorter paths or higher energies keep the neutrino closer to the same point on the wave. This is why long-baseline accelerator beams, atmospheric neutrinos, and reactor experiments can all reveal oscillations, even though they probe very different distance and energy ranges.

Typical neutrino oscillation parameters

The table below lists representative values commonly used in two-flavor educational calculations. Solar-scale numbers emphasize the smaller mass-squared splitting associated mainly with electron-neutrino disappearance over very long distances, while atmospheric-scale numbers describe the larger splitting that drives many muon-to-tau oscillation discussions. The mixing angles shown are approximate and are meant as starting points rather than a complete global fit.

A useful way to read the table is to hold Δm² and θ fixed while you move only L or E. That isolates the oscillation phase and shows whether your scenario is near a peak, near a node, or somewhere in between. In practice, the energy term often matters as much as the baseline, because the probability depends on the ratio L/E rather than either quantity alone.

Exploring the physics of neutrino flavor mixing

Neutrino oscillations happen because the flavor states produced in weak interactions are not the same as the mass states that travel through space. The mixing angle tells you how much of each mass eigenstate is present in a given flavor state, and the phase term tells you how far those components have drifted apart by the time the neutrino is detected. The two sine-squared factors in the formula separate these ideas cleanly: one controls how strongly the flavors can mix, and the other controls how the conversion waxes and wanes with distance and energy.

The discovery of oscillation was a major clue that neutrinos have nonzero mass, which is one reason this subject sits at the intersection of particle physics, astrophysics, and cosmology. Even now, researchers still work on the absolute mass scale, the ordering of the masses, and possible CP-violating effects in the lepton sector. A two-flavor calculator cannot answer those larger questions, but it does show the basic interference pattern that every more complete model must contain.

In matter, neutrinos experience additional effects from the particles around them, especially in dense environments such as the Sun or Earth. Those matter-driven changes can shift the effective mixing angle and resonance condition, which means a vacuum formula is not enough for every real experiment. This calculator stays with the vacuum case so the role of L/E, θ, and Δm² remains easy to see, but that also means you should treat it as a teaching model rather than a full propagation code.

That distinction matters when you think about experimental design. Reactor experiments usually probe short to intermediate baselines at lower energies, accelerator beams are tuned to specific energies over long baselines, and atmospheric neutrinos sweep through a broad range of path lengths and energies as they cross the planet. Those different regimes all use the same basic oscillation idea, but each one emphasizes a different part of the probability curve.

For students, the calculator is a way to build intuition about why oscillation probabilities sometimes rise sharply and sometimes nearly disappear. For practitioners, it offers a quick cross-check when a chosen baseline, energy, or mixing angle should land near a maximum or minimum. In either case, the output is most meaningful when you already know which oscillation scale you are trying to explore.

The most important habit is to think in terms of phase. If you change the baseline or energy without adjusting the other inputs, you are effectively moving along the same sinusoid. That makes it easy to compare two scenarios before moving on to a more detailed model.

How to use this neutrino oscillation calculator

1. Enter Baseline Distance L (km) in kilometers, since the phase formula assumes the travel distance is measured in km.
2. Enter Neutrino Energy E (GeV) in gigaelectronvolts so the built-in constant matches the units in the probability expression.
3. Enter Mixing Angle θ (degrees) in degrees; the calculator converts the angle to radians internally before evaluating the sine terms.
4. Enter Δm² (eV²) as the mass-squared difference you want to test, then run the calculation and compare a second scenario with only one changed input so you can see how the oscillation phase responds.

Formula: the two-flavor neutrino oscillation estimate

The result is built directly from the vacuum expression shown in the equation above. The first sine-squared factor sets the largest possible conversion for a chosen mixing angle, while the second sine-squared factor tracks the phase 1.27 × Δm² × L / E. Keeping L in kilometers, E in GeV, θ in degrees, and Δm² in eV² ensures that the 1.27 constant works with the calculator's inputs as intended. In practice, the result is driven most strongly by the ratio L/E when you compare one detector position or beam energy against another.

Because the oscillation phase is periodic, two different combinations of distance and energy can sometimes land on similar probabilities. That is useful when you are checking whether a planned detector location sits near the same part of the wave as a previous measurement, or whether a different beam energy would move the answer toward a peak or a dip. The calculator helps you test those relationships quickly without having to step through the algebra by hand.

Worked example: comparing two neutrino detector baselines

A practical way to think about the calculator is to imagine the same neutrino beam reaching two detectors at different distances. If the mixing angle and mass splitting stay fixed, the nearer detector samples an earlier phase while the farther detector samples a later one, so the conversion probability can change dramatically even though the source is identical. The most useful comparison is not between unrelated inputs, but between scenarios that differ by one controlled change, such as a longer baseline or a lower energy. When the phase lands near an odd half-turn, conversion is enhanced; when it lands near a full turn, the probability drops.

That is also why the calculator is best used as a comparison tool rather than as a single-number answer machine. If you are trying to understand a beamline, a reactor setup, or an atmospheric path through the Earth, you can hold three inputs steady and change the fourth to see which direction the probability moves. The visual intuition is often more valuable than any one percentage, because oscillations are fundamentally about relative phase and not about an isolated distance or energy value.

Limitations and assumptions for vacuum neutrino oscillation estimates

This calculator is a vacuum, two-flavor estimate, so it is best treated as a compact teaching model rather than a complete oscillation simulation. It leaves out the extra mixing angles, CP-violating phase, and matter effects that appear in a full three-flavor treatment. For quick intuition, though, it captures the key interference pattern that drives flavor conversion, which is the main reason the formula remains a standard classroom approximation.

Results depend on entering L, E, θ, and Δm² in the units the formula expects, because the phase is built from their combination rather than from any one value alone. A baseline entered in the wrong distance scale or an energy entered in the wrong unit can move the result to the wrong point on the oscillation wave. The same is true if you use a parameter set that belongs to a different oscillation regime than the one you meant to study.

The calculator also assumes the neutrino travels through vacuum with no density-driven correction from the medium. In the Sun, through Earth, or inside other dense environments, the effective parameters can shift enough to change the interpretation of the simple vacuum answer. It therefore does not replace detector-specific modeling, updated parameter fits, or a full propagation study when you need research-grade accuracy.

A final practical limitation is that two-flavor models compress several real-world effects into a simpler picture. That simplification is often acceptable when you only want to understand how L/E and mixing angle shape the probability curve, but it is not the same as a dedicated simulation for a published analysis. Use the result to build intuition, to verify rough expectations, and to compare candidate baselines or energies; use a full model whenever the exact oscillation probability matters for experiment design or interpretation.