Understanding Neutrino Oscillations

Neutrinos are elusive particles that interact only through the weak nuclear force and gravity, allowing them to pass unhindered through matter. For decades they were assumed to be massless, but experiments studying solar and atmospheric neutrinos revealed a surprising behavior: neutrinos produced in a particular flavor state, such as electron or muon type, can transform into another flavor as they propagate. This phenomenon, known as neutrino oscillation, implies that neutrinos possess distinct mass states and that the flavor states are quantum superpositions of these masses. The simplest way to capture this behavior is with a two-flavor model in which one flavor has a probability to convert into another flavor according to an oscillatory formula.

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 Planck’s constant, the speed of light, and unit conversions. The oscillatory structure, with a sine-squared term in the phase, reveals how the probability cycles between zero and a maximum value as the ratio changes.

The calculator above implements this equation. You supply the baseline distance the neutrino travels, its energy, the mixing angle in degrees, and the mass-squared difference in electron volts squared. The script converts the angle to radians, evaluates the two sine-squared factors, and returns the probability that a neutrino produced in one flavor will be detected in the alternate flavor after traversing the baseline. The result is expressed as a percentage to facilitate interpretation. If the mixing angle is large and the phase term is near an odd multiple of , the conversion probability approaches 100%, whereas if the phase is a multiple of , the probability drops to zero.

Understanding how probability varies with baseline and energy has guided the design of many neutrino experiments. Atmospheric neutrinos, for example, traverse distances from tens to thousands of kilometers depending on their direction of travel through Earth, making them ideal for studying oscillations at energies around one gigaelectronvolt. Long-baseline accelerator experiments carefully select beam energies and detector distances to probe specific portions of the oscillation curve, enabling precise measurements of mixing angles and mass-squared differences. Reactor neutrino experiments operate at much lower energies and shorter baselines, revealing complementary information about the oscillation parameters.

Typical Oscillation Parameters

Measurements have established two primary mass-squared differences: one governing solar neutrino oscillations and another governing atmospheric oscillations. The table below summarizes approximate values for these parameters and their associated mixing angles in a two-flavor approximation. Note that the full three-flavor framework involves additional angles and a CP-violating phase, but the simplified values capture the dominant behavior for many educational calculations.

Transition	Δm² (eV²)	Mixing Angle θ
Solar (νe → νμ/τ)	7.5 × 10-5	33°
Atmospheric (νμ → ντ)	2.4 × 10-3	45°

Plugging these numbers into the calculator allows you to reproduce oscillation probabilities observed in experiments. For instance, an atmospheric neutrino with energy 2 GeV traveling 500 km would have a conversion probability of about 84% when using the atmospheric parameters above. Adjusting the baseline to 1000 km brings the phase closer to π, reducing the probability to around 9%, illustrating how the oscillation wave manifests in real systems.

Exploring the Physics

Neutrino oscillations arise because the flavor states produced in weak interactions are not the same as the mass eigenstates that propagate through space. The mixing angle quantifies the rotation between these bases. In the two-flavor picture, a flavor state is a superposition of two mass eigenstates with amplitudes set by the mixing angle. As the eigenstates evolve in time with slightly different phases due to their mass difference, the flavor composition oscillates. The probability formula encapsulates this interference: the term controls the oscillation amplitude, while the second sine-squared term sets the oscillation frequency with baseline and energy.

The discovery of neutrino oscillations was profound because it provided the first evidence that neutrinos have nonzero mass, requiring physics beyond the Standard Model. Determining the absolute neutrino mass scale, the ordering of masses (normal or inverted hierarchy), and the presence of CP violation in the lepton sector remain open questions. Future experiments, including long-baseline facilities like DUNE and Hyper-Kamiokande, will extend the baseline and energy ranges to map the oscillation pattern with greater precision.

While the vacuum formula captures the core behavior, oscillations in matter introduce additional complexity through the Mikheyev–Smirnov–Wolfenstein (MSW) effect. In dense environments such as the Sun or Earth’s interior, coherent forward scattering modifies the effective mixing angle and mass difference, resonantly enhancing conversions under certain conditions. This calculator does not incorporate matter effects, but experimenting with vacuum oscillations builds intuition for how the fundamental parameters influence flavor transitions.

The ability to calculate oscillation probabilities has practical implications beyond pure research. Neutrino detectors monitoring reactors can use oscillation predictions to verify reactor output or search for anomalies. Astrophysical observations, such as neutrinos from supernovae, can reveal information about both neutrino properties and the environments they traverse. By adjusting the inputs here, you can model scenarios ranging from the Sun’s core to distant galaxies, appreciating how neutrinos serve as messengers from otherwise inaccessible regions.

The calculator provides an accessible way to explore these concepts. Students can vary the baseline and energy to see how the probability cycles, highlighting the wave-like nature of quantum particles. Educators can use the tool to demonstrate the role of mixing angles and mass differences in determining oscillation patterns. Enthusiasts may reproduce famous experimental results, deepening their understanding of the subtle quantum mechanics underlying one of the universe’s most ethereal particles.

Ultimately, neutrino oscillations remind us that even the most elusive particles can reveal profound truths about the universe. The simple equation encoded in this calculator encapsulates decades of theoretical work and experimental ingenuity. By experimenting with it, you participate in the same intellectual journey that transformed our view of particle physics and opened a window onto phenomena ranging from the heart of the Sun to the furthest reaches of the cosmos.