Why can't neutrinos be brought to rest?

[anorlunda]

If neutrinos were massless, they would have to travel at c. But now we know they have mass, so they must travel at speeds less than c.

But (all?) other massive particles can be brought to rest. Why not neutrinos? Is there a theoretical reason that forbids it?

 

[ZapperZ]

If neutrinos were massless, they would have to travel at c. But now we know they have mass, so they must travel at speeds less than c.

But (all?) other massive particles can be brought to rest. Why not neutrinos? Is there a theoretical reason that forbids it?

First, figure out how we bring to rest "other massive particles". Take an electron, for instance. Even assuming that we can bring it to rest (which we really can't if you think about it, but at the very least, we can confine it to a very small region of space), we capture and confine it using electromagnetic interaction. In other words, we use forces that it can interact with!

A neutrino doesn't interact with a lot of things. It has a very small mass, so its gravitational interaction is unbelievably weak. So forget about having it confined even around a very huge star. And what is left is its coupling via the weak interaction, which from its own name, is WEAK!

What you have is something that just don't bump into something else that easily and thus, can't be confined. It just won't be dragged and slowed down by everything surrounding it.

Zz.

 

[snorkack]

For example, take beta decay. Most of the time the energy is divided between electron and antineutrino, both getting a large share. But the shares are usually different, and electrons get a continuous spectra. Sometimes, rarely but with nonzero probability, the electrons get no energy whatsoever or only a small energy, getting stuck in a ground or excited state of the resulting atom or molecule, and the antineutrino gets all energy except for recoil of the atom. It must therefore also happen, rarely, that an electron gets almost entire energy of beta decay and the antineutrino is slow.

How would a slow neutrino behave? In particular, can a neutrino be unable to oscillate because its total energy suffices for only the lightest rest mass state but not for any others?

 

[mfb]

In particular, can a neutrino be unable to oscillate because its total energy suffices for only the lightest rest mass state but not for any others?

They are produced as flavor eigenstates, and those are always a superposition of the three mass eigenstates -> oscillation happens
Slow neutrinos oscillate more quickly, but it is nearly impossible to detect them as they don't have enough energy to be seen in regular neutrino detectors.

KATRIN is looking for electrons with nearly the maximal energy, but they cannot detect the corresponding "slow" neutrino.
(where "slow" is still close to c most of the time!)

 

[snorkack]

They are produced as flavor eigenstates, and those are always a superposition of the three mass eigenstates -> oscillation happens

Does a neutrino in a flavour eigenstate possesses a defined total energy?

 

[anorlunda]

During a type II supernova neutrinos are confined for a short time because the density of matter in the shock wave is so high that it is opaque to neutrinos. It is not clear whether those confined neutrinos are slowed or absorbed and re-emitted.

Anyhow, the supernova example shows that there is a way to make neutrinos interact strongly with matter. Can they not be slowed in that environment, at least theoretically?

 

[mfb]

Does a neutrino in a flavour eigenstate possesses a defined total energy?

Energy or mass?
Energy has to be well-defined, as it is a real particle.

 

[snorkack]

Energy or mass?
Energy has to be well-defined, as it is a real particle.

Is a real neutrino with a defined and small energy allowed to oscillate from a mass eigenstate where its rest mass is smaller than its total energy and where it possesses positive kinetic energy and real momentum into a mass eigenstate where its rest mass is bigger than the aforesaid energy and where it possesses negative kinetic energy and imaginary momentum?

 

[mfb]

Neutrinos do not oscillate between mass eigenstates - those are eigenstates of the Hamiltonian, the three mass eigenstates evolve independently. They oscillate between flavor eigenstates.

 

[snorkack]

They are produced as flavor eigenstates, and those are always a superposition of the three mass eigenstates -> oscillation happens

So, if an antineutrino is emitted in a beta decay event with 1) a defined flavour at emission (electron antineutrino) and 2) a well defined energy (it is a real particle, and the beta unstable nucleus and the decay resulting nucleus were both long lived states with well defined energy) which is SMALLER than the rest energy of any mass eigenstate except the lowest - is it then also a superposition of the mass eigenstates with negative kinetic energy?

 

[mfb]

which is SMALLER than the rest energy of any mass eigenstate except the lowest

I don't think this is possible, at least not as pure eigenstate of the weak interaction.
Hmm... different energies for different eigenstates would not have this issue, but I don't think that is right.

 

[kurros]

Does a neutrino in a flavour eigenstate possesses a defined total energy?

Energy or mass?
Energy has to be well-defined, as it is a real particle.

But as you say later, flavour eigenstates are not eigenstates of the Hamiltonian, therefore they do not have energy eigenvalues right?

 

[kurros]

Hmm this seems relevant

https://www.google.com.au/url?sa=t&...=MVCr_3WeexIjUHWfHmhfQQ&bvm=bv.51156542,d.aGc

He seems to argue that the uncertainty principle saves us in this situation, i.e. neutrinos are never actually produced in plane wave states, so they have some energy spread, so they oscillate. I'm not sure it answers the "in principle" question though. If we go to the extreme, say one of the neutrinos was really heavy, we'd expect the light ones don't oscillate into it. But this is probably related to the coherence length of the oscillation, which would be super short in that case. So maybe the coherence of the neutrino mixture goes away in this ultra-low energy limit.

edit: I am becoming more convinced this is the case. If you can measure the energy of the emitted (ultra-low-energy) neutrino to some fantastic accuracy (by measuring the nuclear recoil to the nano-eV or something) then you must collapse the superposition, since you just know which mass eigenstate you must have, so no oscillations.

 

[snorkack]

At low electron chemical potential, the lowest energy beta decay processes seems to be electron capture of holmium 163, energy "3" keV, and beta decay of rhenium 187, quoted as "2,6" keV. Next seems to be tritium decay, 18,59 keV.

What if stable isotopes are SLOWLY, at a low temperature, subjected to a high chemical potential of electrons? Say, the pressure is slowly increased as matter is deposited on surface of a star, and the temperature stays low because the heat is readily radiated away from the surface where it is generated?

At which electron chemical potential would helium 3 (common in nature because copiously produced by proton fusion) spontaneously capture electrons to turn into tritium? In particular, how low would the energy of the neutrinos be at their origin (deep in the star)?

 

[anorlunda]

Sorry to be dense, but I'm not getting it.

A paper called Neutrino Oscillations at http://www2.warwick.ac.uk/fac/sci/physics/current/teach/module_home/px435/lec_oscillations.pdf says:

the eigenstates of the [neutrino] Hamiltonian are |ν1 > and |ν2 > with eigenvalues
m1 and m2 for neutrinos at rest. A neutrino of type j with momentum p is an energy eigenstate
with eigenvalues E_{j}=\sqrt{m_{j}^{2}+p^{2}}.​

The quote mentions neutrinos at rest, and nothing I see in the math forbids p=0; yet the consensus seems to be that neutrinos can not be brought to rest.

Also, I fail to see why flavor oscillations enter into the question at all. Please help.

 

[mfb]

yet the consensus seems to be that neutrinos can not be brought to rest.

Where?
They are just rarely discussed as all neutrinos we can measure are ultrarelativistic.

 

[Drakkith]

The quote mentions neutrinos at rest, and nothing I see in the math forbids p=0; yet the consensus seems to be that neutrinos can not be brought to rest.

I think it's more a matter of it being so difficult as to border on impossible, rather than being completely impossible.

 

[ZapperZ]

I think it's more a matter of it being so difficult as to border on impossible, rather than being completely impossible.

Actually, it is impossible.

We have not been able to bring an electron to a complete rest, despite the fact that it is larger than a neutrino and that it interacts more strongly than a neutrino. The HUP ensures of that. So what hope is there for a difficult-to-capture neutrino?

Zz.

 

[Drakkith]

Actually, it is impossible.

We have not been able to bring an electron to a complete rest, despite the fact that it is larger than a neutrino and that it interacts more strongly than a neutrino. The HUP ensures of that.

Zz.

I think everyone here knows what we mean when we say "bring to rest", and I see no reason to talk about the HUP. It just confuses the subject.

 

[ZapperZ]

I think everyone here knows what we mean when we say "bring to rest", and I see no reason to talk about the HUP. It just confuses the subject.

I'm not sure the OP does based on the insistence that there's nothing "in the math" that prevents p=0.

Zz.

 

[Drakkith]

Fine, Zapper.
In the interest of not derailing the thread I'm going to leave now.
Have a nice day.

 

[snorkack]

Yes, Heisenberg uncertainty principle prevents completely zero momentum. But what would be the speed of a neutrino confined to Milky Way?

Actually, is there any lower bound on the rest mass of the lowest mass eigenstate of neutrino?

 

[mfb]

There is no lower bound, it could even be zero (but that is unlikely - there is no reason why one of the three states should have exactly zero).

 

[Bill_K]

We have not been able to bring an electron to a complete rest, despite the fact that it is larger than a neutrino

To the best of our knowledge an electron and a neutrino are exactly the same size, namely they are both point-like.

 

[ZapperZ]

To the best of our knowledge an electron and a neutrino are exactly the same size, namely they are both point-like.

I should have clarified the use of the world "larger". I meant as in mass, not size.

Zz.

 

[anorlunda]

OK, now I got it. Not impossible but difficult.

Forgive me for a flight of fantasy, but there is a way, but not here on Earth. Perhaps I should have posted this question to the astrophysics category.

Above the event horizon of a Schwartzchild black hole, there should be a dim yet steady stream of low energy neutrinos. A neutrino emitted just above the horizon with speed just above the escape velocity will be slowed to almost zero before escaping the gravity well. Since they could be emitted at various radii outside the event horizon, there should be a continuous spectrum of neutrino momenta.

Low speed neutrinos might be more likely to interact with ordinary matter. By analogy, fast neutrons in a reactor are slowed to thermal speed to find the resonance peaks at lower neutron energies where fission events are much more likely. I presume that the physics of slow speed neutrinos have never been studied because there was no reason to do so.

In the accretion disc of a black hole, a slow neutrino could combine with a neutron producing a proton and an electron. That would make a nucleus transmute to the next higher atomic number producing rare isotopes that have spectral lines that could be detected remotely.

I realize that I've chained too many ifs in series to be serious, but if we don't imagine making the nearly impossible possible, how can we advance?

 

[mfb]

Low speed neutrinos might be more likely to interact with ordinary matter.

In general, the interaction probability increases with the neutrino energy. There are exceptions at nuclear transitions.

I presume that the physics of slow speed neutrinos have never been studied because there was no reason to do so.

The detection of low-energetic neutrinos would be interesting to look for the cosmic neutrino background (similar to the electromagnetic cosmic background radiation)

 

[snorkack]

So, considering that the energy of background neutrinoes is known and small - does it mean that their speed and therefore conduct in gravitational fields is vitally dependent on what the precise mass of the lowest mass eigenstate is?

 

[mfb]

2K is .17 meV. The Planck collaboration found 0.23 eV as upper limit for the sum of the neutrino masses, and therefore as upper limit for the heaviest neutrino.

A .23eV-neutrino with a kinetic energy of .10 meV moves with ~9000km/s, too fast to be contained within a galaxy. To go down to 1000km/s, we have to reduce the energy by a factor of 81. Only a very small fraction of neutrinos is so slow.
And this is just the upper limit on the mass. A natural scale for the neutrino masses (based on the known squared mass differences) would be in the meV-range.

 

[nikkkom]

First, figure out how we bring to rest "other massive particles". Take an electron, for instance. Even assuming that we can bring it to rest (which we really can't if you think about it, but at the very least, we can confine it to a very small region of space), we capture and confine it using electromagnetic interaction. In other words, we use forces that it can interact with!

A neutrino doesn't interact with a lot of things. It has a very small mass, so its gravitational interaction is unbelievably weak. So forget about having it confined even around a very huge star. And what is left is its coupling via the weak interaction, which from its own name, is WEAK!

What you have is something that just don't bump into something else that easily and thus, can't be confined. It just won't be dragged and slowed down by everything surrounding it.

Easy peazy.
Observe a stationary tritium atom's decay. Measure electron's impulse and calculate neutrino's impulse, then jump into your standard-issue near-lightspeed spacecraft and catch up to the neutrino.
Bingo. You have it stationary in your frame of reference.

 

[ZapperZ]

Easy peazy.
Observe a stationary tritium atom's decay. Measure electron's impulse and calculate neutrino's impulse, then jump into your standard-issue near-lightspeed spacecraft and catch up to the neutrino.
Bingo. You have it stationary in your frame of reference.

Wonderful! My dream has come true! Please build me one! I need it next week.

Zz.

 

[PAllen]

Low speed neutrinos might be more likely to interact with ordinary matter. By analogy, fast neutrons in a reactor are slowed to thermal speed to find the resonance peaks at lower neutron energies where fission events are much more likely. I presume that the physics of slow speed neutrinos have never been studied because there was no reason to do so.

Actually, the opposite is the case. The lower the neutrino energy, the lower the interaction cross section. A neutrino with a few ev (still moving > .99c) has such a low interaction cross section that even neutron star core is effectively transparent to it. A neutron star provides a thought experiment for slowing neutrinos: for high energy neutrinos, the mean free path is small compared to neutron star radius. However, as neutrinos slow to a .99c, they all escape.

Two of the scenarios proposed so far do seem to work for producing neutrinos slow compared to c:

- primordial big bang neutrinos
- black hole event horizons Hawking radiation (some will be slow neutrinos)

However, the insanely low interaction cross section means we will never detect these. If a neutron star core is transparent to these, what hope have we?

 

[.Scott]

Suppose I gave you a fist full of thermal (ie, slow) neutrinos. How would you know?

 

[PAllen]

Suppose I gave you a fist full of thermal (ie, slow) neutrinos. How would you know?

Exactly, you couldn't know. You would never be able to distinguish it from a fist full of nothing.

 

[mfb]

I would ask you to release them close to PTOLEMY, once it is build.
They aim to measure relic neutrinos from the big bang, together with a direct mass measurement.

 

[anorlunda]

What a great provocative question Scott.

If neutrinos were stationary so that they could be located within a nucleus or within a hadron, for an indefinite period of time, then I guessed that interactions might be much more likely. I know that interactions between free neutrons and nuclei are hugely dependent on relative velocities, why should neutrinos be different?

Since we have never seen slow neutrinos, I presume that experimental physics has no data on the interaction properties of slow neutrinos. That is why I intended my original post to be directed at theoretical physics rather than experimental physics. Photons must move at velocity c. Fermions must obey the Pauli exclusion principle. Those properties can be derived from the wave functions. I am curious to learn if there are analogous theoretical principles for neutrinos that dictate their peculiar properties.

 

[mfb]

Neutrinos are fermions. They have well-predicted (by the standard model) interactions with matter, and if they are slow their interaction cross-section is low.

 

[friend]

Neutrinos are fermions. They have well-predicted (by the standard model) interactions with matter, and if they are slow their interaction cross-section is low.

If we can't detect slow vs. fast neutrinos, then how can we tell how much of the neutrino's energy is rest mass and how much is due to velocity?

 

[mfb]

There are upper limits on the mass. Sure, the cross-section at low energies will depend on the (still unknown) mass, but as far as I know it will be small for all possible masses.

 

[reculet]

Chase it!

If neutrinos were massless, they would have to travel at c. But now we know they have mass, so they must travel at speeds less than c.

But (all?) other massive particles can be brought to rest. Why not neutrinos? Is there a theoretical reason that forbids it?

Climb into your vehicle of choice and accelerate in pursuit of the (massive) neutrino - when you catch up with it, slow down and move alongside it. Now it is at rest w.r.t. you, and you didn't even have to touch it :) [although General Relativity tells us that you could interpret what happened as the appearance of a gravitational field in your frame of reference, which brought the neutrino to rest beside you: you avoided falling by using your vehicle's power-thrusters].

Easier said than done, of course.

 

[Chronos]

We would need an experimental setup capable of isolating a single neutrino - which would be an impressive accomplishment. It's relatively easy to isolate an electron.

 

[anorlunda]

OK, I think I understand now thanks to your help. Let's see if I got it right.

The original question could have been phrased as a paradox. If neutrinos are massive, why don't we see a spectrum of non-relativistic velocities for them as we do for other massive particles? Wikipedia says that the lower limit for neutrino velocities is 0.999976 c.

The apparent answer requires two logical steps.

First, when neutrinos are emitted:

(mfb put numbers on it, and that helped me to understand.)

... A .23eV-neutrino with a kinetic energy of .10 meV moves with ~9000km/s ...

So for the sake of argument let's say that when emitted, neutrinos have a wide distribution of energy in excess of rest mass. But because the rest mass is so small, only a tiny kinetic energy is needed for relativistic speeds. Therefore, the fraction of all energies corresponding to non-relativistic speeds is tiny.

Second, after emission: Because neutrinos interact so little with other particles, they do not become thermalized. They tend to conserve whatever energy they were emitted with.

Put those two things together and we can see that it is possible to have neutrinos at any speed 0<v<c. However, non-relativistic speeds are very improbable.

The seeming paradox comes from confusing what's possible with what's probable. Secondary confusion comes from using the word possible in the ideal sense, contrasted with possible pragmatically in the laboratory.

 

[mfb]

Wikipedia says that the lower limit for neutrino velocities is 0.999976 c.

For a specific energy (here: 3 GeV). Hmm, that part of the article is outdated.

So for the sake of argument let's say that when emitted, neutrinos have a wide distribution of energy in excess of rest mass. But because the rest mass is so small, only a tiny kinetic energy is needed for relativistic speeds. Therefore, the fraction of all energies corresponding to non-relativistic speeds is tiny.

Right.

Second, after emission: Because neutrinos interact so little with other particles, they do not become thermalized. They tend to conserve whatever energy they were emitted with.

Right.

Put those two things together and we can see that it is possible to have neutrinos at any speed 0<v<c. However, non-relativistic speeds are very improbable.

Right.

 

[snorkack]

What is the shape of the tail of the beta decay?
If 100 % of the antineutrinos emitted by tritons have energy under 18 keV (because that is the total energy of the beta decay), what percentage have energy under 1800 eV? 180 eV? 18 eV et cetera?

 

[mfb]

There is a formula for the electron energy spectrum. The neutrino energy is the difference between the total energy and the electron energy.

Close to the endpoint (and neglecting the neutrino mass), the probability is quadratic with the difference to that endpoint. The fraction of neutrinos below 1800 eV is roughly 1% (guessed, should be right up to a factor of ~5), the fraction below 180 eV is roughly 0.001% and so on - every factor of 10 reduces the fraction of neutrinos by a factor of 1000.