The neutrino

[Xerxes]

I have just been listening to a very interesting podcast from the BBC on this subject (which sadly may not be available to non-UKers)

 

It seems that Rutherford (um, or was it Chadwick?) showed that the energy spectrum of beta decay is continuous, and it was generally realized that this experimental fact seemed to violate energy conservation.

 

Question: How is this conclusion implied from the finding? It is not obvious to me.

 

It further seems that W. Pauli, in the 1930's, proposed that beta decay "generated" (in a manner that is not clear to me) an hitherto un-detected particle called the neutrino that in some sense restored energy conservation, and that in the 1950's this particle was detected experimentally. What precisely was Pauli's argument?

 

Question: Since the neutrino has non-zero but un-measurable mass/energy, and since the beta decay spectrum may be anything from essentially zero to whatever is its allowed maximum, how can these neutrinos make up the mass/energy deficit? Is it simply that when beta decay "carries away" from the nucleus its lowest allowable energy in the form of ejected electrons, there must be a whole load of neutrinos, and vice versa?

 

Sorry for the naivety of these questions, but my physics is weak; nonetheless I found it a fascinating, albeit pop-sci, listen-to. Hope you all can here can hear it too

[timo]

First of all, I am not very familiar with history of physics. So the statements that follow are meant to help you understanding how a reasoning might work, not meant to reproduce what the physicists actually though 80 years ago (oh, and I didn't even try to listen to the podcast, in case that matters).

 

How is this conclusion [that energy conservation seems to be violated] implied from the finding [of a continuous spectrum]? It is not obvious to me.

I don't think it is obvious at all, because there is, as far as I can see, at least one crucial point of information missing. Namely, that the decay is always between the same two energy levels. So let's assume this. Let's also assume that the particles that are created have no other means of storing energy other than their mass (which is fixed) and their momentum (i.e. kinetic energy, which is variable). Both decay products, the electron/positron and the nucleus after the decay, each have three degrees of freedom, their momenta in x-, y- and z-direction ("degree of freedom" means that it is an independent variable that could -at least until this point- in principle have any value). This amounts to six degrees of freedom. However, when one assumes that energy and momentum are conserved, then there are four equations (1 for energy, one for each component of momentum) constraining the degrees of freedom. Only two degrees of freedom are left, then. They are the direction of the momentum of one of the particles (doesn't matter which but let's say we look at the electron). That does imply that the magnitude of the electron's momentum is not variable, but must have some fixed value. That means that the kinetic energy of the electron must have a fixed value. That means that the total energy of the electron must have a fixed value.

 

 

 

 

What precisely was Pauli's argument?

When you have three outgoing particles, you initially have nine degrees of freedom, but still only four constraints. That means that the result of something decaying into three particles has much more variety (five degrees of freedom) than that of something decaying into only two particles. In particular, the energy of the electron is not fixed but can vary. I would expect his argument went into this direction.

 

Since the neutrino has non-zero but un-measurable mass/energy, and since the beta decay spectrum may be anything from essentially zero to whatever is its allowed maximum, how can these neutrinos make up the mass/energy deficit?

The mass of a particle does not put a limit on the kinetic energy that it can have. In fact, in highly-energetic beta decays you can even completely ignore the mass of the electron/positron and get reasonable results (this is sometimes called the "relativistic limit").

[swansont]

I don't think it is obvious at all, because there is, as far as I can see, at least one crucial point of information missing. Namely, that the decay is always between the same two energy levels.

 

You can assume any two levels, as long as they are quantized, i.e. you can have the daughter left in an excited state and the analysis still works. If the decay ejects only one particle, then the momenta of the daughter and ejected particle must be equal and opposite in the center-of-momentum frame, so you are only looking at one dimension. The energy released is fixed, so the ratio of the kinetic energies of the two particles is likewise fixed. There is only one value per energy level of the daughter. Alpha decay looks like this — the alpha spectrum is quantized.

[timo]

Agreed, except for the "so you are only looking at one dimension"-part which I either didn't understand or which you mis-wrote (the degrees of freedom for the 4-momenta of a 1->2 decay surely are 6-4=2 dimensions). But I thought talking about only a single jump in energy levels would make the issue easier to understand and focus on the relevant part.

[swansont]

Nucleus at rest ejects a particle. Conservation of momentum dictates that the recoil be in the direction opposite of the particle. If that's the x-axis, there is no motion along y or z. If the available energy is fixed, the energy of each of the two particles is likewise fixed.

 

It's just a different way of looking at the answer; if you define the coordinate system beforehand, then the direction of the ejected particle is still a degree of freedom.