The previous post showed how neutrino oscillation arises from interference between the three generations of neutrinos. In computing the total amplitude for the detection of a neutrino interaction, we must sum over the amplitudes for the three different neutrinos, , to get the total amplitude :
For a particle of energy E, the wavelength is where h is Planck’s constant and p is the momentum of the particle. The neutrinos are very relativistic, so we have to use the formula from special relativity:
The relativistic energy is:
Putting this into the equation for the wavelength of matter, discovered by Louis-Victor-Pierre-Raymond, 7th duc de Broglie and French physicist, we find:
The term on the right uses energy and velocity but does not include mass; the difference in wavelength between the different neutrinos is due to differences in their velocity. Accordingly, from , we solve for in terms of to get:
For the neutrinos, the masses are around 0.05 eV and less, while the characteristic energies of beta decay are around a few million eV. Consequently, the above square root can be approximated as:
The corresponding approximation for the inverse of velocity is:
Putting this into the equation for wavelength, we get:
Suppose that our detector is a distance L from the neutrino source. Over this distance, one can fit periods of the kth neutrino wave length, where is not generally an integer. We have, approximately:
For any single beta decay, the energy of the three neutrinos is the same so we can write . Thus for the three neutrinos, the first term in the above, will be the same for all three neutrino types. The difference in phase between neutrinos with different masses will therefore be proportional to the square of the masses and will be given by:
It’s usual to multiply the above by so that the phase difference will be in radians instead of fractions of a period. This can be absorbed into the h to make it into an h-bar. In addition, the above oscillation will appear in the amplitudes, that is, it appears like . But our experimental measurements are of the probabilities, which look like . Squaring the cosine acts to double the frequency. In trigonometry, this amounts to the fact that . So, in converting the above into an equation for the measurement, we replace the h with h-bar, and divide by two:
The Mikheyev-Smirnov-Wolfenstein effect Effect
The probability of a neutrino interacting with matter depends on the energy of the neutrino. One often reads that the solar neutrinos can travel through about a light year of lead before having a 1/e chance of interacting. That’s a lot of lead and one could conclude that there would be no matter effects on neutrinos from the sun. This is not the case, the earth itself can influence even those neutrinos that do not “interact” (in the inelastic sense, say, of converting a quark) with the earth.
The effect is similar to how glass effects visible light. A photon that travels through glass does not interact inelastically with the atoms and electrons of the glass, but nevertheless, the wavelength of the photon is effected. The simplest way of describing the situation is to say that the glass changes the potential energy of the photons; this changes their frequency and wave length. Similarly, matter can change the potential energy of neutrinos, the result is a change to neutrino oscillation.
The effect is called the Mikheyev-Smirnov-Wolfenstein Effect and is expected to be observed soon. The effect is observable because the different neutrinos have different amplitudes for interacting with regular matter, and therefore their potential energies change in different ways. It’s probably beyond the scope of this blog to describe the interaction. A recent paper on it is Neutrino oscillograms of the Earth: effects of 1-2 mixing and CP-violation, Akhmedov, Maltoni, Smirnov. For an introduction to the effect, the lecture notes of Smirnov are wonderful. The source of these lecture notes are in Iran and the internet from here to there is slow so I had to download the above file and save it on my computer, and only then open it up. There are also excellent lecture notes for the two later lectures by Smirnov at the same conference: Measuring Neutrino Parameters and Quark Lepton Universality . The last of these cites this author, (a proud and rare moment for an amateur), on the generalization of Koide’s mass formula for the charged leptons to the neutrinos.
Because of the MSW effect, it is theoretically possible to build neutrino optics, for example, a neutrino telescope. Even if the incoming beam of neutrinos had identical energies, there would be dispersion according to the three different masses. With such an apparatus, one could theoretically arrange for a beam of neutrinos with only a single mass eigenstate present. One could then arrange physics experiments which could be modeled without having to take into account the three different mass eigenstates.