Neutrino oscillations

Above: neutrino oscillations. As we shall see this phenomenon resolved the 'solar neutrino problem'.

Neutrinos (ν) are tiny sub-atomic particles, once thought to have no mass at all, they are now understood to have very tiny masses. They have so little mass that they are almost massless and predicted to travel very close to, but just below, the speed of light. (Measurements all suggest a speed of about c, but have not been accurate enough to give exact speeds). In fact, neutrinos are more than a million times lighter than an electron, which is the next lightest particle.

Neutrinos come in three principle flavors: the electron neutrino (νe), the muon (mu) neutrino (νμ) and the tauon (tau) neutrino (ντ) along with their anti-matter equivalents, indicated by an overscore on the nu (ν) symbol (like this: ν).

Neutrinos barely interact at all with matter! this is due to the fact they only interact with the weak force (and gravity). The mean free path (the average distance traveled between interactions with other particles) is about 6 parsecs in the very dense solar core! About 1011 (100 billion) neutrinos from the Sun strike every square-cm of the Earth's surface (and you) every second and the vast majority of these 'ghostly particles' will pass straight through the Earth unhindered!

The Solar Neutrino Problem

The 'solar neutrino problem' refers to early experiments on the flux of neutrinos arriving on Earth from the Sun. The Sun generates neutrinos inside its core due to the thermonuclear reactions occurring within and solar models can predict the rate of this neutrino synthesis. However, early measurements detected only about a third as many neutrinos as expected. The realization of neutrino oscillations resolved this paradox, as we shall see.

If we generate a beam of one flavor of neutrino, such as νe as in the graph above, then as the beam travels through space the neutrinos will oscillate from one flavor to another so if we carry out a measurement on the beam the probability of obtaining an anti-electron neutrino will be less than one, since part of the beam consists of ant-mu and tau neutrinos. The exact probability depends on both distance from the source and the energy of the neutrinos in the beam.

Why do neutrino flavors oscillate?

Neutrinos propagate through space in their mass eigenstates: ν1, ν2 and ν3 which are stationary states, what we call eigenvalues of the (free-particle) Hamiltonian. The Hamiltonian is a mathematical operator which queries the mathematical representation of the particle state to obtain its energy. This is the theoretical equivalent of how we measure a particle beam: by performing some measurement on its energy. The flavor eigenstates, νe, νμ and ντ are NOT eigenstates of the Hamiltonian (but are eigenstates of the flavor operator) and are expressed as a linear superposition of the three mass eigenstates, that is a weighted combination of the three. (Superposition arises in quantum mechanics due to wave-particle duality, since it is a fundamental property of waves in which waves occupying the same position in space and time add together). We say that the 'physical' particle states are the mass eigenstates, which is certainly the case in a beam propagating through space. However, in particle reactions neutrinos react as flavor eigenstates. When we have linear superpositions of eigenstates in quantum mechanics, as for the flavor in a propagating beam of neutrinos, then we can have change over time or motion. In this case the change is an oscillation of neutrino flavor along the beam.

Quantum mechanics (QM) predicts the probability of a measurement yielding a given value: there is no absolute determinism in QM, only probabilities. Looking at the oscillations above, we can see that we actually have a short-wavelength probability wave superimposed on a long-wavelength probability wave. For the example given example the 'survival probability' of anti-electron neutrinos in the beam is given by:

Neutrino oscillations

Where: the angles θij refer to the mixing angles which determine the amplitude of the probability oscillation or wave by considering the transitions between a pair of mass eigenstates i and j. For example, the amplitude of the short-wave is given by sin2(2θ13) and its wave-length is determined by Δm232 which is the difference in mass squared between ν3 and ν2. The wavelength of the long-wave is determined by: Δm221. It is the difference in mass between the three mass eigenstates which gives rise to the oscillations. If neutrinos had zero mass then they would not oscillate. D is the distance along the beam from the beam source and E is the energy of the neutrinos in the beam. In our plots we have taken approximate values for the mixing angles (based on experimental measurements): θ13 = 10o, θ12 = 30o and θ23 = 45o.

A detailed mathematical derivation of this equation for neutrino mixing is given in Thomson, M. 2013. Modern Particle Physics, Cambridge University Press. (This is an advanced undergraduate textbook requiring a good grounding in calculus, matrix algebra and complex numbers as well as some understanding of special relativity and 4-vectors).

It turns out that Δm221 is much smaller than Δm232, in other words mass states ν1 and ν2 are very similar compared to the mass of ν3 which is either much larger or much smaller than the first pair (it is not known which). This is not unexpected when we consider the masses of the three generations of leptons (see below). I am not going to go into the mathematics of the derivation of this equation, but has been simplified by the assumption that the mass of ν3 is much further from the other two.

(When plotting the above equation it is important to realize that the arguments of the sin2 terms, involving formulae in Δm2, D and E, are in radians, as is usual, and that this equation uses natural units with c = h-bar = 1 and we have to convert the distance into SI units to make it more meaningful).

Each neutrino flavor state can then be represented as a weighted superposition of the three mass eigenstates, with the weightings fixed (and determined by experiment) to give a definable mass to each flavor. This can be represented as a matrix equation, with the U matrix (the PMNS or Pontecorvo-Maki-Nakagawa-Sakata matrix) representing the required weightings:

Neutrino oscillations

So, for example the electron-neutrino state is represented as shown (this requires the reader to know how to multiply matrices together, but it is only the general principle I want to convey here). Each state is indicated by the enclosing ket half of an angular bracket, as is the convention in QM.

The effect of beam energy on neutrino oscillations

The graph at the top of the page was for oscillations of ant-electron neutrinos with an energy of 3 MeV (eV = electronvolt). The graph below is for 4 MeV neutrinos:

Neutrino oscillations

The graph below is for 2 MeV neutrinos:

Neutrino oscillations

We see that higher energy neutrinos have a longer wavelength. Note that if there were only two mass eigenstates we would have a simple wave, but because there are three mass eigenstates we have one wave superimposed on another. Key questions remain, such as do neutrinos and antineutrinos oscillate separately or do matter and antiparticle variants inter-convert?

These oscillations have been confirmed by a number of experiments. For example, the MINOS experiment studied a beam of mu-neutrinos and found that oscillations between mu and tau neutrinos dominate.

So what about the solar neutrino paradox?

A variety of reactions in the Sun's core manufacture neutrinos and the Sun generates about 2 x 1038 electron neutrinos each second. Two of the main ones are the initial step of pp chain of reactions:

pp chain of reactions in the Sun

in which an electron neutrino (E less than 0.5 MeV) is manufactured in the initial step - the fusion of two protons (p) into deuterium (D) - which is:

        p + p → D + e+ + νe

and the decay of boron into beryllium (in an excited state *):

        8B584Be* + e+ + νe

which produces electron neutrinos with energies up to 15 MeV.

Detecting neutrinos is difficult since neutrinos only interact via the weak force (and gravity) and so solar neutrinos usually pass straight through the Earth without reacting or interacting with anything! The Homestake Mine neutrino detector in South Dakota, USA, used 615 tons of C2Cl4. The chlorine atoms occasionally react with an electron neutrino as follows:

        νe + 37Cl1737Ar18 + e

The 37-argon produced in this reaction is extracted and quantified by measuring its radioactivity. This reaction detects electron neutrinos with the higher energies synthesized in the 8-B reaction and the expected detection rate was 1.7 neutrinos per day but what was measured was only 0.48 (0.04) per day.

The later SAGE and GALLEX experiments (using gallium as a neutrino target in gallium chloride, the gallium reacts with electron neutrinos to form radioactive germanium) were sensitive to the lower energy neutrinos generated in the pp-chain and these also detected fewer neutrinos than expected.

This deficit in the number of neutrinos detected, coming from the Sun, became known as the solar neutrino problem. We can now account for this anomaly by realizing that the electron neutrinos propagating through space do not remain as electron neutrinos but oscillate between the three flavours, so by the time they reach the Earth, fewer are detected than otherwise expected.

The SNO experiment (Sudbury Neutrino Observatory, in Canada) used 1000 tons of heavy water (D2O) in a 12 m diameter vessel surrounded by 9600 photo-multiplier tubes (PMTs). This experiment was sensitive to the higher energy 8B neutrinos and was able to detect all three flavors by their reactions with deuterium. The results were consistent with the electron neutrino flux predicted by astrophysics.

Discovery of the Neutrino

The neutrino was postulated to exist to account for apparent violation of momentum in beta-decay. To ensure conservation of momentum in these processes, Pauli postulated the existence of a particle which was evading direct detection in 1930; this particle was the electron neutrino (and its anti-matter equivalent). Beta-decay is the process whereby some radioactive isotopes emit high-energy electrons, e.g. the beta-decay of uranium to neptunium:

        23592U → 23593Np + νe + e

This process actually involves the decay of a neutron (n) inside the nucleus of uranium into a proton (p):

        n → p +  νe + e

The process of positive beta-decay involves conversion of a proton to a neutron:

        p → n +  νe + e+

Early particle detectors (such as cloud chambers) could not detect the neutrino and based on observation of the resultant nucleus and positron/electron alone then both linear and angular momentum would be violated. To ensure the necessary conservation of these quantities, another particle was needed, with spin- (spin can be thought of as a QM equivalent of intrinsic angular momentum), to carry the additional momentum such that total momentum (and angular momentum) of the products equals the total momentum (and angular momentum) of the products. Since electric charge is also conserved, the neutrino is electrically neutral and must carry no net electric charge.

Neutrons and protons themselves consist of a core of 3 quarks, uud for the proton and ddu for the neutron and the actual process can be understood as conversion between an u (up) and d (down) quark by a weak process involving production of a W boson:

beta-decay, quark model

Note that (as in all particle reactions and interactions) electric charge must be conserved. There are two flavors of W boson, one with positive charge and one with negative charge; along with the neutral Z boson, these constitute the force carriers for the weak force. Neutrinos only interact with the weak force (and gravity).

Detection of Neutrinos

We have already noted how neutrinos can be indirectly detected by measuring the production of radioactive isotopes in specific reactions, such as their reaction with chlorine. Neutrinos can also be detected by elastic scattering. neutrinos striking a suitable target can scatter elastically (conserving both momentum and kinetic energy). The Super-Kamiokande detector uses 50 000 tons of water. Oxygen is a stable nucleus and so does not react with solar neutrinos, but electron neutrinos may scatter from electrons in the water molecules, as shown below:

elastic scattering of neutrinos

One the left above is the charged current (W boson mediated) reaction and on the right is the neutral current (Z boson mediated) reaction. A similar scattering interaction can occur for each flavor of neutrino (though the interaction with the electron-neutrino dominates). This allows all neutrino flavors to be detected with ordinary water. The scattered electron (on the right in each diagram) is relativistic, moving with a speed close to the speed of light and which exceeds the speed of light in water (though not c, the speed of light in a vacuum) causing it to emit Cerenkov radiation, detectable as blue light by the PMTs surrounding the water. Cerenkov radiation is the electromagnetic equivalent of a sonic boom, but involves a charged particle exceeding the speed of light in the medium and the emission of light, rather than a sonic boom, as a cone of blue light directed in the direction in which the electron is traveling.

If heavy water (D2O) is used then the neutrinos can undergo additional reactions with the deuterium nucleus. The charged current reaction can mediate inverse beta-decay, e.g.

inverse beta-decay

These two diagrams are equivalent but the one on the left is more precise, the d quark is a constituent of a neutron in the deuterium nucleus (the deuterium nucleus consists of one proton bound with one neutron). Again the electron generated is relativistic and emits Cerenkov radiation. This interaction occurs but rarely but can be used to detect intense neutrino radiation.

In heavy water the neutral current reaction involves a neutrino scattering off the deuterium nucleus, fragmenting it into a neutron and a proton. The proton can subsequently be captured by another deuterium nucleus (neutron capture) to form radioactive tritium which is created in an excited state and so emits a gamma particle. The emitted gamma particles can collide with electrons in the water, exciting them to relativistic speeds, again resulting in the emission of detectable Cerenkov radiation. Importantly, the neutral current reaction is equally likely to occur for all neutrino types.

Classification of Neutrino Flavors and Lepton Generations

Particles, according to the standard model, occur in three generations. The first generation are the lightest and therefore the more stable and the more abundant and the third generation the heaviest, and so less stable and less abundant. Both quarks and leptons occur in pairs, e.g. the up and down quarks form the first quark generation. Leptons occur in pairs, one charged lepton and its corresponding neutrino. the first lepton generation consists of the electron and the electron neutrino, the second generation of the muon (essentially a heavy unstable electron) and the mu-neutrino (muon neutrino), whilst the third generation consists of the heavier tauon and the tauon neutrino (tau-neutrino).

particle generations

Experimental evidence thus far strongly supports the notion that there are no more than three generations of particles. However, it can not be ruled out absolutely, at present, that heavier generations do not exist.

Neutrinos in Old Stars

Several additional processes involving neutrinos occur in the cores of old stars at an advanced stage of evolution. In cores at a very high temperature and low electron density, such as in red supergiants (AGB stars) and white dwarfs neutrino pair-production becomes significant when temperatures reach or exceed 6 billion K. This process can be summarized as:

        γ + γ ⇌ e+ + e → νe + νe

Note that the neutrino is produced along with an anti-neutrino from the same generation in neutrino pair-production. This happens occasionally (about 1 in 1022 times) when a positron annihilates with an electron. Most often, the positron and electron annihilate to generate two photons (gamma, γ) but the reverse reaction can also occur: two photons combining to give an electron-positron pair. lower electron density favors the forward reaction, producing more electrons, by the law of mass-action (a reversible reaction will react so as to try and maintain equilibrium).

In old stars, pair-production can contribute significantly to thermal pressure. Although neutrinos react rarely and most will escape the star, enough will interact with the dense materials in the star to generate a significant outwards pressure, helping to resist further gravitational collapse of red supergiant cores. the neutrinos may be produced in such great numbers that they correspond to heat loss from the star. In the case of a 'living' star, like the red supergiant, this will not result in cooling, since gravitational contraction of the core will compensate and maintain the temperature of the core homeostatically. In the dead 'white dwarf' core, however, neutrino radiation will contribute to cooling.

In stellar cores too cool for neutrino pair-production, but hot enough to make sufficient positron-electron pairs, significant numbers of neutrinos can be generated by the following photoneutrino processes (interactions involving a photon):

        e + γ → e + νe + νe

        e+ + γ → e+ + νe + νe

Plasma processes involve plasmons. Plasmons are collective interactions of photons and quantities of plasma in the star. This can also lead to neutrino / antineutrino pair-production (predominantly of electron neutrinos). This process is dominant in Si (silicon) burning which occurs in the old cores of heavy stars and the neutrinos may carry away almost as much energy as the Si burning itself generates.

Coherent scattering of neutrinos from atomic nuclei occurs especially when the nucleus involved is iron-56. coherent refers to the fact that the neutrino is not scattering off a single nucleus, but off a group of interacting nuclei. iron-56 is the end product of nuclear fusion in old stellar cores of heavy stars.

Electron capture. In neutron star formation, the gravitational collapse of the stellar core of a massive star involves the capture by atomic nuclei of electrons, converting protons into neutrons and electron neutrinos. The final stages of core collapse are extremely rapid during a supernova and the rate of neutrino production very high. This high flux of neutrinos carries away of the order of 1045 J of energy. So many neutrinos are produced that enough interact with the star's atmosphere to give sufficient energy to assist the blasting off of the star's outer layers which are ejected at about 10 000 km/s to form a planetary nebula.

Article created: 31 Dec 2020