Neutrino Oscillation; the Calculation

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 A_1, A_2, A_3 for the three different neutrinos, \nu_1, \nu_2, \nu_3 , to get the total amplitude A :
Three diagrams for three different solar neutrinos propagating to earth

For a particle of energy E, the wavelength is \lambda = h/p 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:
Relativistic momentum for kth neutrino species

The relativistic energy is:
Relativistic energy for kth species of neutrino
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:
Wavelength of kth neutrino species, as a function of velocity and energy
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 E_k = c^3\;m_{\nu k}\;/\sqrt{c^2-v_k^2} , we solve for v_k in terms of E_k to get:
Relativistic velocity as function of energy and rest mass

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:
Approximate velocity for highly relativistic particles
The corresponding approximation for the inverse of velocity is:
Approximate ultra relativistic inverse velocity
Putting this into the equation for wavelength, we get:
Ultra relativistic wavelength in terms of energy and mass

Suppose that our detector is a distance L from the neutrino source. Over this distance, one can fit n_k = L/\lambda_k periods of the kth neutrino wave length, where n_k is not generally an integer. We have, approximately:
Number of wave lengths in distance L for ultra relativistic particle
For any single beta decay, the energy of the three neutrinos is the same so we can write E_k = E . Thus for the three neutrinos, the first term in the above, LE_k/(hc) 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:
Difference in phase between two neutrino species

It’s usual to multiply the above by 2\pi 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 \delta A = \cos(\theta_j-\theta_k) . But our experimental measurements are of the probabilities, which look like \cos^2(\theta_j-\theta_k) . Squaring the cosine acts to double the frequency. In trigonometry, this amounts to the fact that \cos^2(\theta) = 0.5(1 + \cos(2\theta)) . So, in converting the above into an equation for the measurement, we replace the h with h-bar, and divide by two:
Neutrino oscillation as a function of length, energy and mass

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.


Filed under physics

6 responses to “Neutrino Oscillation; the Calculation

  1. Xilinas

    Hello, do neutrinos have a Compton frequency wavelength?
    Thank you

  2. carlbrannen

    Every massive particle has a Compton wavelength \lambda = h/(mc). The formula isn’t used in the above calculation because this isn’t Compton scattering, it’s interference.

    Compton scattering of neutrinos would be a pretty tough experiment to set up on this planet. Maybe something like that happens when stars go supernova.

  3. carlbrannen

    Akhmedov and Smirnov correct my assumption that E_k does not depend on k:

    The reason one can assume that all three neutrinos have the same energy is not because in any single interaction they all have the same energy, but instead because their possible energies fall in a very wide pattern. Consequently, the density matrix for the neutrino propagation makes it all work out in the end.


    There is a movie called 2010 in this movie they say the neutrino have caused a physical reaction well i don’t know if thats possible but if a neautrino has a small mass is that mass smaller than the mass of an electron and if that mass collides with a valence shell electron of a mass of a large atomic radii since the valence electrons experience the most small nuclear charge wont these electrons be knockedout if there is conservation of momentum then a physical reaction may take place since the nuetrino travel at speed of light i think a reaction may take place.If it does not why it does take place?

  5. Emlyn

    Thank you, this is very interesting, although well over my head…
    I just found one small typo: in “The effect is similar to how glass effects visible light” should be affects instead of effects.

  6. William C. McKee

    Carl, on another of your web pages I gave a proposed simple formula for what I tentatively call “little m”, something that I assume is the time weighed average of the rest mass of a neutrino. Of course it doesn’t have to be that, until it is proven to be that. Fortunately, rather than just a guessed number out of the air, this being a function with time from the Big Bang era till now, it should rather easily be with those of such skill, to compare the acceleration function that my derivation predicts (if you help with Special Relativity momentum terms and and time derivatives of same) to the observed and growing acceleration implied by what is currently called “Dark Energy”. The latter completely mysterious and completely not at all understood — except if my conjecture, which appears immediately testable turns out correct. And if not, well things are still in play I guess.

    I will if you agree to it, give my derivation of “little m” to you, for some additional manipulation and derivation of a number of other concepts. The derivation is short and sweet. And I only make one really singular assumption, that the concept of “infinite density” is fictional, and can ultimately be demonstrated by acceptable science to be that. Using Dark Energy’s observed values of acceleration vs time from the Big Bang, should immediately confirm or refute everything that I assumed and/or derived. There being at that point only the roll of the dice for testing theory vs observation.

    There could easily be rewards and prizes for you in a formally presented paper on this. Alas, for amateur scientists such as myself, I don’t know that any special recognition (except on blogs maybe) would be much expected. That plus I haven’t the foggiest how to derive the weighed average mass of a neutrino even if what I found out turns out to be valid. This being:

    m = a1*(electron neutrino mass) + a2*(mu neutrino mass) + a3(tau neutrino mass)

    Thank you for your time, consideration, and blog provision.

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