Beta particle decay

Hello. Today we're continuing in our series on nuclear physics looking at beta decay. And a very quick reminder that beta decay happens if we plot the famous NZ diagram. Uh here is the line where n equals z. For stable nuclei that have small a values, small number of nucleons, n does indeed equal zed. But then you find that the uh the stable line has more nucleons than protons. Um consequently and we showed that in the semi-impirical mass formula video and that around this line of stability there is a region of unstable nuclei that want to decay to become stable. And what we showed is that if you're above the line, then the nucleus decays in a sort of southeasterly direction such that you lose one neutron and you gain one proton. So a neutron converts to a proton. And if you're below the line, then you move up to the stable line in a northwesterly direction by as it were increasing the number of neutrons but decreasing the number of protons. So, a proton converts to a neutron. And we've got two formula for these. It's not just neutron becomes a proton. A neutron becomes a proton plus an electron plus an anti- neutrino. Or the other way, a proton becomes a neutron plus a posetron plus an ordinary neutrino. This is called beta minus decay. And plus decay because originally before anybody knew what electrons were or at least before they knew that these particles were electrons they called them beta particles. Now it just has to be remembered that originally nobody had ever heard of neutrinos. So originally it was thought that all you got was an electron and nobody had actually ever heard of a posetron nor would they have found them. You remember when uh the uh decay products were first identified they were identified as alpha, beta and gamma. Alpha was known to be positively charged and we now know it's a helium nucleus. Beta was known to be negatively charged and we now know it's an electron and gamma wasn't charged at all and we now know that is part of the electromagnetic spectrum. But the point was they thought that beta particles were negatively charged because they never discovered positively charged particles because of the of course those particles would have met with electrons in an instant and annihilated and produced photons. So you never would have spotted a posetron even though they were there because they would have annihilated so quickly. So how did we actually discover neutrinos? Well, of course the argument is very similar to alpha particle decay. If you think back to the last video, we said that if you start with a large nucleus X, it decays into an alpha particle which shoots off with kinetic energy in that direction. And that means that the daughter nucleus Y must shoot off with kinetic energy in that direction. Why? Because the momenta must be the same. Py equals P alpha. Because classically the original nucleus was stationary. So when X decays, it gives off energy. The energy is shared as kinetic energy between the two daughter products. And so if one moves off one direction with momentum P alpha, the other has to move on in precisely the opposite direction P Y so that P Y equals P alpha and the net momentum is zero. And you also know that the total kinetic energy of the Y plus the alpha must equal the total amount of energy which is given off as a consequence of alpha decay. So if you apply exactly the same principle, you say that you have a nucleus, it's going to be a smaller nucleus X, that's going to give off your electron and also your now daughter Y nucleus. And therefore the momentum should be in exactly the opposite in exactly opposite directions. And the total kinetic energy of the electron plus the kinetic energy of the Y must equal Q, the total amount of energy that's given off during beta decay. Um, and you find that that's not true. There is some spare energy, or rather there is an energy deficit. Somewhere these two do not add up to Q. You need an additional bit of energy plus something

else here. And hence that's what gives rise to the idea that there's yet another particle being produced which is carrying off this energy. Now you remember we showed that when you had alpha particle decay the alpha particle waltses off with about 98% of all the energy that's emitted and only 2% goes to the y. And I also showed that the if the alpha particle were the size of an electron then the amount of energy that it would carry off would be about 99. 9% of the total amount of energy uh resulting from the decay leaving hardly any energy for the Y um nucleus. So what we're now saying is that 99. 9% of this value of Q is carried off by the electron and the neutrino because the nutrino neutrino is very small mass and hardly anything is carried off by the uh y nucleus. So the as it were the decay energy is shared between the electron and the neutrino. And if you plot this is an experimental result. If you plot the number of uh electrons having a particular kinetic energy K against that kinetic energy, you get a graph that looks something like this where it falls off to zero. That should be falling off to zero at the value Q. In other words, no electrons can have an energy greater than Q because that's the total energy coming off of the beta decay. And if an electron has Q, it means the neutrino has no energy at all. But you can see that the bulk of the electrons have this amount of energy and that therefore the difference between Q and that amount of energy the difference must be the energy that is given to the nutrinos where you've got electrons with this amount of kinetic energy. The rest of it must have been given to the nutrinos. That's for beta minus decay. That's where you get electrons. If you draw exactly the same graph, so the number of posetrons now with kinetic energy K. So this is for posetrons versus the kinetic energy. You get a graph that looks more like this. Once again, it will fall off to zero when the posetron has a value Q. It can never have more than Q because that's the total energy available. But now the shape is slightly different. Why are the shapes different? Well, of course, it's largely a culum issue. Um, inside the nucleus, the electron which is negatively charged will be attracted to the positively charged protons. Whereas, of course, the positively charged proton will be repelled from the positively charged protons. So, the positron repels the proton and vice versa. So, you're going to get a different shape as a consequence of the culum interaction. Now, just to give you an idea of the relative sizes of the electron and the neutrino, originally it was thought that maybe the nutrino would have no mass at all. In which case, of course, it would travel at the speed of light. That's a special relativity condition. If you're massless, you are constrained to travel at the speed of light. The mass of an electron is 0. 5me. Strictly 0. 511 me, but let's not be too precise about this. The mass of a neutrino is of the order 0. 3 eV. So in other words, it's something of the order of 1. 6 million times smaller or the electron if you like has a mass that's 1. 6 million times larger than uh the neutrino. And this is actually quite fascinating because remember the electron is one of the standard model particles. It is a fundamental particle. As far as we know, it has no structure. It has no internal components. You can't get anything smaller than an electron. And yet there is a separate particle also in the standard model called the neutrino which is a million times smaller in mass. Now what determines those distributions that I showed you earlier of the number of electrons or posetrons with given energies and that is given by the fermy theory of beta decay. Now I'm not going to go through that in detail because it is extremely complicated but I'm going to give you a flavor of it so that you get some idea of what it's all about and then you can go and look it up and see how it's actually calculated uh in more detail. But in basically what we're saying is that when a nucleus X decays to a daughter nucleus Y plus the beta particle plus the neutrino in whatever form these may take. Then the extent to

which that's likely to happen is governed by two things. The first is called the strength of coupling which is another way of saying the likelihood of X turning into Y. And the second is the number of ways in which X can turn into Y and that is called the density of final states. And in this Fmy theory that gives rise to what's called the transition probability of going from the initial state which of course is X to the final state which is Y which is given by a constant. There are always constants in this. I just call this C 0. But the two key terms are a matrix term m if squared and a density of states term row f. Now you need to go back to the video I did. It was the first one I did in the series last year on quantum mechanics where I introduced direct notation and mif is equal to the in direct notation it would be written like this where s i is oh sorry I s i is the um wave function representing the initial state that is x So f is the wave function representing the final state and v is what's called the coupling operator which represents the strength of the coupling. So v will act on s i to produce a new wave function and then that wave function will be as it were this whole uh this whole term is called the inner product. If you remember the inner product of this V acting on S 1 with S f that inner product is called a probability amplitude. It's the probability amplitude for a state which starts in uh the initial state I getting to the in the final state F. Um you need to look at that video to see how that works. But it's a probability amplitude. In order to convert a probability amplitude to a probability, you have to square it. So the whole of this term and that gives you miff squared. And that is now a probability term. And that's why you've got mif squared here. That is the probability essentially of going from the initial state to the final state um through this inner product. uh row is the density of final states and what we shall see is that row is kind of going to be determined by some function of the total energy available minus the kinetic energy which is given to the electron. Remember Q is shared between the electron and the neutrino. And the number of different ways in which this transition can take place, it turns out, is going to be proportional to the difference between the total amount of energy uh that's available and the kinetic energy of the electron. And if you work all that through the horrendous maths that is involved, what you find is that the number of electrons having a particular kinetic energy K is given by some constant again times this term here Q minus K E all squared. So the uh the density of state term turns out to be squared times the probability term of going from the initial state to the final state and then you have to introduce a kulum factor. Well you always have to do that don't you to recognize the fact that there will be interactions of a culum nature that you need to take into account. And then finally a correcting factor for certain decays or f certain paths which are forbidden. And this gives you very close to the charts that I produced earlier in this video. But as I say having given you a rough idea of um what this is all about you can now go and look up on Wikipedia and see in more detail how uh that fermy formula is calculated. Now let's just think about these two emerging particles the electron and the neutrino which are carrying off the overwhelming amount of energy that's available Q that comes from these um beta decays. Um these are firmians electrons and neutrinos which means they each have a spin half. So the total spin of the emerging electron and neutrino is going to be

either a half plus a half. So the spin zero sorry the spin is going to be 1 half plus a half or it's going to be zero a half minus a half. That means the spins can be aligned like that or they can be anti-aligned like that. So the combination of these two emerging particles is going to be spin one or spin zero. Now when the total angular momentum of these particles is zero and the spin is zero that means J is zero that is called in other words there is no change in um in nuclear spin and par that is called the allowed fermy transition I've put the word allowed in inverted commas Not because other transitions are not allowed but because this is a kind of a term and you'll understand where it comes from in a little while. Whereas if L equals Z and S is 1 which is the other option. So J of course which is L + S is 1. Now we've got that the change in nucleus spin. Remember there are always if you've got J is one that means that your change can be minus1 zero or + one but there is of course no change in parity then because S is one that is now called the allowed gamotella transition and basically when S is zero that's a firm transition when S is one that's a gamma tella transition. But I want to look at why we call them allowed and why the implication is that other forms are not allowed. So let's just do a bit of classical mechanics. So we always have to be careful with classical mechanics when we're talking about quantum mechanical things. This is an atom. Here is the nucleus. The radius is r. Here's an electron orbiting in classical fashion that atom. uh it has a linear momentum P which is equal to MV and the angular momentum of that electron will be P * R in other words uh P * the distance to as it were the orbiting point. Now from special relativity we know what the total energy of any system is. The total energy is equal to m that's the mass of the electron squar c 4th plus p ^ 2 c^ 2 that's the total energy and that of course is represented in the form of an electron by its rest mass energy which is m e c^ 2 me the mass of the electron c^² that is Einstein's famous formula E= MC² that's the rest mass energy plus of course the kinetic energy of the electron. Now we know that the rest mass energy of an electron is 0. 511 mev. That's its mass. I am going to assume for the purposes of this um this illustration that the kinetic energy is 1 me. So let's just make that as an assumption. So in other words this total uh this side is going to be 1. 5 1. 511 but let's forget the 1. 1. 011 for the moment. Now if we square both sides we're going to get m^ 2 c 4th which is just the rest mass squared and the rest mass is 0. 5. So that's going to be 0. 25. That's this. Well, if you square this side and then you square the rest math plus P ^2 C^ 2 is going to be you have to square this. So that's 2. 25 which means that P ^ 2 is going to equal 2. 25 minus. 25 which is 2 over C ^ 2. And if we do the square root of both sides that's going to give us that P is the square root of 2, which is very roughly 1. 4 over C. And therefore the angular momentum which is p * r well let's call the radius of the nucleus five fermmes that means the angular momentum which is p * r is going to be 1. 4 * 5 which is going to be 7 / c and that's going to be measured of course we must remember we've used fermies and not um and not what we should do si units. So the actual units we used were mev and fermy. So we must remember

that. Now that is the angular momentum. What is angular momentum? Angular momentum is L. L in quantum mechanics terms we're now tripping into quantum mechanics is L into L +1 H bar 2. And L is this term here. So we've now got that 7 over C me fmy is equal to the square root of L into L +1 H bar squared. So if we take the square root of h bar squ and bring it down here that means we've got 7 over h bar c me fmy is equal to the of l into l + 1. Now then what is h bar c? h bar c is equal to well har is about 10us 34. Let's not get too fussed about this. C is 3 * 10 8 but that is in SI units. So that will be 10 - 34 jewel seconds* 3 * 10 8 m/s. So the seconds and the pers will cancel and what you've left with is that H bar C is in units of jewel meters. Well, we want MEV fmmes. So to convert that into me fermies we've got here 3 * 10us 26 that's uh that's this term here to convert that into uh from jewels to uh me that's 1. 6 6 * -3. It would be 10 -19 if you wanted to go to EV, but we want it in me. So it's 10 -3. And to convert meters to furmes, you multiply by 10 15. And if you do that, you find that 3 / 1. 6 is very roughly two. Let's not quibble. 10 - 26 * 10 15 is 10 -1. Bring this up. That's * 10 + 13 that's 10 squared which is 200. So H bar C is 200 in units of MEV and firmies. So now we can put 200 there. So we've now got that 7 over 200 is equal to the square root of L into L + one. Right? Now if L is zero, okay, that's kind of, you know, this is a very small amount. If L is one, you've got the square root of 1 * 2 is uh square of two, which is 1. 4 and 1. 4 is not 7 over 200. So that kind of classically suggests that anything above L=0 is not allowed. Yeah, it's forbidden. It can't happen. But of course, in quantum mechanical terms, it does happen. And so consequently if you have L= 1 and S= Z that would be called the first forbidden FI transition. Fmy because F s= 1 forbidden because L= 1 shouldn't happen. This classical formula means that L= 1 can't occur but it does. And similarly l= 1 s = 1 would be the uh first forbidden gamma tella transition gamotella because s is one forbidden because l equals 1 strictly shouldn't happen but it does. And finally, just to remind you that when we write this term that the neutron turns into a proton plus an electron plus an anti- neutrino or we could do it the other way around with the proton converting to the neutron. That is a shorthand way of writing what we learned in particle physics is something a little more complicated that's going on. We know that the neutron consists of two down quarks and an up quark. So if we draw the fineman diagram with t when I draw it I have time going upwards. So here comes the neutron made of three quarks. This is a neutron with an up down down. Those are the three quarks up down down. Two of those quarks don't change. They just carry on as an up and a down. But this down quark is changed into an up quark. So now we've got up, down, up. Two ups and a down. That's a proton. And how does a down quark change into an up quark? It's through the weak interaction. And the weak interaction

gives off in this case the W minus BZON, which is very short-lived. 10us 22 seconds is it something like that. and then that decays into the electron and the anti-utrino. So this formula here is a very shorthand way of writing what is actually happening in particle physics. The weak interaction whereby the three quarks in the neutron two of them stay the same. One of them is changed from a down quark to an up quark by the emission of the W minus Bzon which is called the exchange Bzon, the mediating Bzon. But that quickly decays to produce the two long-term products of the decay. Even though of course the anti-utrino won't hang around for very long. Antiparticles never do. But in fact, that's just one way of making clear that the shorthand we use here is actually rather more complicated at particle physics