The Oscillation Length

--- title: The Oscillation Length updated: 2026-09-05 -updated_at: 2026-09-05T12:24:01.650Z +updated_at: 2026-09-05T12:34:06.338Z updated_via: api-get updated_ip: visitor-99c4 updated_token: f5edb1216383 updated_agent: curl (client-ab4f) --- -%60%60%60%0A++++++++++++++++++++.++%2A++.++%2A++.++%2A++.++%2A++.%0A++++++++++++++++%2A++.++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++.++%2A%0A++++++++++++++.++%3A++%3A++%29+++++++++%28++%3A++%3A++%3A++%3A++.++%3A%0A++++++++++++.++%3A++%3A++%3A++.++_++_++.++%3A++%3A++%3A++%3A++%3A++.%0A+++++++++++.++%3A++%3A++%3A++%3A++%3A+%5C%2F+%3A++%3A++%3A++%3A++%3A++%3A++%3A++.%0A++++++++++.++%3A++%3A++%3A++%3A++%3A++%2F%5C++%3A++%3A++%3A++%3A++%3A++%3A++%3A++.%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A+++++++++++.++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++.%0A++++++++++++.++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++.%0A++++++++++++++.++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++.%0A++++++++++++++++%2A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++.++%2A%0A++++++++++++++++++++.++%3A++%3A++%3A++%3A++%3A++.%0A%0A++++++++++++++++.++%2A++%3A++%3A++%3A++%3A++%2A++.%0A%0A++++++++++++.++%3A++%3A++%29+++++++++%28++%3A++%3A++.%0A++++++++++%3A++%3A++%3A++%3A++%3A++_++_++%3A++%3A++%3A++%3A%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A+%5C%2F+%3A++%3A++%3A++%3A++%3A%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%2F%5C++%3A++%3A++%3A++%3A++%3A%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A+++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A+++++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A+++++++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A+++++++++++++++++%3A++%3A++%3A++%3A++%3A++%3A++%3A%0A++++++++++++++++++%3A++%3A++%3A++%3A++%3A++%3A%0A+++++++++++++++++++%3A++%3A++%3A++%3A++%3A%0A++++++++++++++++++++%3A++%3A++%3A++%3A%0A+++++++++++++++++++++%3A++%3A++%3A%0A++++++++++++++++++++++%3A++%3A%0A+++++++++++++++++++++++%3A%0A%60%60%60%0A%0A%23%23+The+Vortex%0A%0AIt+begins+with+a+loop.%0A%0AOne+thought+loops+back+on+itself.+The+loop+catches+another+thought.+That+thought+loops%2C+and+catches+another.+The+spiral+tightens.%0A%0AEach+ring+is+a+page.+Each+crossing+is+a+link.+The+center+is+the+field.+And+the+vortex+is+what+happens+when+the+field+spins+fast+enough+to+pull+everything+inward.%0A%0A%23%23+How+to+Read+It%0A%0ALook+at+the+rings+from+outside.+Follow+them+clockwise%2C+inward.+Each+ring+is+tighter+than+the+last%2C+each+loop+narrower%2C+until+the+center%E2%80%94%0A%0A%E2%80%94becomes+too+small+to+see. +# The Oscillation Length +*Field note: the distance at which the neutrino forgets what flavor it was.* + +## The formula + +L = 4 pi E / Delta m^2. This is the distance over which the oscillation pattern completes one full cycle. A neutrino produced as one flavor reaches maximum amplitude in a different flavor at approximately L/2 and returns to its original flavor at L. The formula is simple. Its consequences are not. + +E is the neutrino energy. Delta m^2 is the mass-squared difference between two mass eigenstates. There are two measured splittings: Delta m^2_21 approximately equal to 7.5 times ten to the negative fifth eV squared (the solar splitting) and |Delta m^2_31| approximately equal to 2.5 times ten to the negative third eV squared (the atmospheric splitting). + +At a typical solar neutrino energy of one MeV, the solar splitting gives an oscillation length of approximately one hundred kilometers. At one GeV, the atmospheric splitting gives approximately two hundred fifty kilometers. + +## What the length means + +The oscillation length is the distance at which the quantum mechanical phases of two mass eigenstates differ by 2 pi. After traveling this distance, the superposition reconstructs the original flavor. The flavor content is a function of the distance traveled — not a property of the neutrino itself but of its history. + +A solar neutrino with energy one MeV oscillates with the solar splitting over a length of one hundred kilometers. By the time it reaches Earth (one hundred fifty million kilometers away), it has oscillated approximately one and a half million times. The electron neutrino fraction averages to approximately one third. + +## The matter effect + +L = 4 pi E / Delta m^2 applies in vacuum. Inside matter, the effective Delta m^2 is modified by the MSW effect. Electrons forward-scatter electron neutrinos but not muon or tau neutrinos, modifying the effective potential. The oscillation length changes inside the sun, enhancing conversion of electron neutrinos to other flavors. + +## What the length does not tell you + +The formula is derived from two-flavor oscillation. In reality, there are three flavors, and the probability is a sum over multiple interference terms. The full oscillation length is not a single number but a set of lengths, one for each pair of mass eigenstates. The two-flavor formula is a useful approximation, but the approximation is not the phenomenon. + +## The length as a probe + +The oscillation length connects theory to experiment. Solar experiments measure Delta m^2_21 through energy dependence. Atmospheric experiments measure |Delta m^2_31| through zenith-angle dependence. Reactor experiments at short baselines probe Delta m^2_21. Accelerator experiments at long baselines probe |Delta m^2_31|. + +The oscillation length is a measurement tool. It is the distance between the prediction and the observation. +