Days ago we talked about radioactivity, in particular about the strong force that is responsible for keeping the nucleus together. And we observed that the binding energy of the nuclear force is strong enough that it substantially reduces the mass of the nucleus.

It cannot be neglected. And one can then define a stability criterion for the stability of a nucleus. It is the most stable or the strongest if the binding energy per nucleon is the largest.

And that binding energy is typically about 8 MeV per nucleon. And here is a graph of the binding energy per nucleon for the various elements. So horizontal axis is the mass number. And the largest binding energy per nucleon is for iron.

We can actually do a few calculations with this disappearance of part of the mass due to the binding energy. So for example we can compare the mass of a helium-4 atom to the total mass of its constituents.

So a helium-4 atom, it has two neutrons, it has two protons and it has two electrons. And if we plug in the numbers in atomic mass units for the neutron mass and for the proton mass and for the electron mass,

then we get those contributions to the total atom mass. And if we sum it all up then we get 4.03298 atomic units. But the mass of a helium-4 atom is actually only 4.002603 atomic units.

And therefore the difference of the mass of the constituents, the sum of the mass of the constituents and the atom mass is 0.030376 atomic mass units. We can express the atomic mass unit in terms of energy.

And so one unit is 931.5 MeV and therefore we get a difference of 28 .3 MeV between the mass of the atom and the sum of the masses of the constituents. Since we have 4 nucleons, in this case we would have about 7 MeV per nucleon of binding energy.

Just in the previous slide, you know the graph that came out? Yes. I noticed that iron, is it possibly related to the reason why iron, let's say when you try to fuse it, requires energy?

Indeed, yes. Since iron is the strongest bound nucleus, you cannot gain energy anymore by fusing iron. Therefore the fusion in stars stops when it hits iron. And if you want to get heavier elements out of stars, then you need supernova explosion.

And it requires energy. So all the heavier elements come from this process, supernova explosions. Indeed, my next example is about iron. So we can go through the same exercise with iron-56.

So here we have 30 neutrons, 26 protons and 26 electrons. And the calculation goes exactly like it was for the helium-4. And if we sum it all up, then we get 56.46339 atomic units.

And the mass of the iron atom is only 55.9349 atomic mass units. And so in this case we have roughly half an atomic mass unit of difference to the atom mass, or 492 MeV.

And that means that per nucleon we have 8.79 MeV of binding energy. One last example about binding energy. If we look at carbon-13, the question is how strongly bound is the last nucleon in carbon-13.

The usual carbon isotope is carbon-12. And it's sort of interesting to see how strongly that last extra neutron is bound. And in this case the calculation is much simpler because we know that by definition carbon-12 has exactly 12 atomic mass units.

And we know the extra neutron weighs this much, so the sum is easy to do. And if we then compare that to the atomic mass of the carbon-13, then we get 0.00531 atomic mass units of difference.

And that's about 5 MeV for that last nucleon. Another aspect of the rather complex nuclear force is that it doesn't bind two neutrons together, and it doesn't bind two protons together. It is most effective when you have about an even number of protons and neutrons.

So it doesn't like just having protons. So helium-2 nucleus is not possible. And so you kind of need the neutrons in the nucleus like cranberries.

The other thing that we already discussed is that the range of the force is much smaller. And that's not because of the strong force, but because it's basically just a residual force of the underlying strong force. And therefore only at very short distances becomes this nuclear force effective.

And that means that the large nuclei are actually bigger than the range of the force. And the nuclei tend to be unstable then. And that means that they disintegrate or decay. In the worst part they completely break apart.

And that means that they emit particles or nucleons after some time. And they can even undergo fission, which means that they break up into fragments. Radioactive decay was discovered by Henri Becquerel.

He found out that if he places some minerals near photographic plates, which were not exposed to light, they would still darken. And then soon afterwards, Madame and Monsieur Curie first isolated the elements radium and polonium, which are radioactive.

And soon after more radioisotopes were discovered and physicists started to categorize the radiation emanating from those minerals into three basic types. The alpha radiation, which has positive charge.

The beta radiation, which has negative charge. And the gamma radiation, which has no charge at all. And one can easily find out what charge the radiation has by placing it into a magnetic field. So the gamma radiation goes straight, the alpha radiation bends one way and the beta radiation bends a different way.

Two days ago I showed you the different ranges of the alpha, beta and gamma. Alpha is easily shielded, beta is more penetrating and gamma radiation is very penetrating. Indeed we saw that alpha can be shielded by a simple sheet of paper or of course by clothing the human skin.

Almost anything will shield alpha, even by a large distance of air will shield alpha radiation. Beta can penetrate thin pieces of metal and gamma radiation can only be shielded by thick walls of dense material, for example lead bricks.

And it was eventually found out that alphas are Helium 4 nuclei and betas are electrons and gammas are photons.

And since we discussed that the nuclear force likes an even number of protons and neutrons. And actually it even likes that the number of protons is an even number, better than if it's an odd number. So therefore if one ejects nucleons then Helium 4 fragment is favoured because it has equal number of

protons than the neutrons and it has an even number of protons and an even number of neutrons. And that is why it is more strongly bound than other nuclei and therefore when you inject nucleons it is favoured over other fragments.

So therefore since alpha radiation consists out of those nuclei it changes the element, it changes the charge of the nucleus by 2 units and it changes the mass number by about 4 units.

So for example if we have Radium 226 which has 138 neutrons and 88 protons then if it alpha decays into a 222 radon then it has 2 neutrons less and 2 protons less.

And of course you have then the alpha particle which is just the 2 neutrons and 2 protons that are missing. So there is energy released in that process. So let's calculate the disintegration energy when Uranium 232 alpha decays to Thorium 228.

We are given the different masses of those 2 isotopes and the mass of the Helium 4 is again 4.002603 atomic units. And so we call the disintegration energy also often referred to as the Q value

of the decay simply as the mass difference divided by the speed of light square. So if we take the difference of those 2 isotopes, subtract off the Helium nucleus we get a difference in mass and we can then

express that in units of energy and then that is the Q value or the energy that is released in the disintegration for that particular process. So what we have here is 5.414 MeV of Q and this Q appears as the

kinetic energy of the alpha particle as well as the daughter nucleus which is the Thorium 228. However the share of the energy is not even. So we can actually calculate what is the

kinetic energy of the alpha and what is the kinetic energy of the daughter nucleus, the thorium. And what we use for that is conservation of momentum. So from the conservation of momentum we know that the momentum of the alpha particle must be the same as the momentum of the thorium particle because we assume that the parent Uranium had no momentum at all.

So they need to go in opposite directions with equal magnitude momentum. And if we multiply the mass of the alpha particle with the kinetic energy of the alpha particle

then the kinetic energy of the alpha particle is the momentum squared divided by 2 times the mass. And therefore we are just left with one half of the momentum squared of the alpha and because of conservation of momentum this must be one half of the momentum squared of the thorium which is then by the same argument the mass of the thorium times the kinetic energy of the thorium.

So we have this equation that mass of the alpha times kinetic energy of the alpha equals mass of the thorium times the kinetic energy of the thorium. And the other equation is the Q value is equal to the kinetic energy of the alpha plus the thorium.

And we can then express the kinetic energy of the thorium. We solve for the kinetic energy of the thorium and we get that it's the kinetic energy of the alpha times the mass of the alpha divided by the mass of the thorium.

We plug that into this equation here for the kinetic energy of the thorium and we get the kinetic energy of the alpha plus this mass ratio times the kinetic energy of the alpha or in other words the kinetic energy of the alpha times 1 plus the mass ratio. And now we can solve since we know the Q value we can solve for the kinetic energy of the alpha.

And we find that almost all of the 5.414 MeV that we have available in this reaction goes to the alpha particle and almost nothing goes to the thorium. And the reason for that is that the thorium is so much heavier than the alpha particle.

In the limiting case that the thorium is infinitely heavy then it would receive no kinetic energy at all and all the kinetic energy would go into the alpha. So in this case the kinetic energy of the thorium is 94 KeV out of 5.414 MeV.

Now that we have defined this disintegration energy or the Q value we observe that of course the Q value has to be bigger than the daughter nucleus rest energy plus the emitted radiation otherwise that process doesn't happen.

So if for some reason the Q value is too small then no disintegration will happen. The alpha decay is governed by the strong interaction because it's basically just the ejection of a part of the nucleus.

But not all of those alpha decays happen immediately and the reason for that is that there is this potential barrier. If you remember we talked about that for the nucleus we have the Coulomb repulsion of the protons and if you look at

the potential energy as a function of distance then up to the distance where the nuclear force is effective we have negative potential energy. But then after that the nuclear interaction dies off and we are left with just the Coulomb repulsion therefore we have this barrier, this Coulomb barrier.

And typically our Q value is not large enough to be bigger than the peak of this barrier. And that means if your nucleus is sitting at this energy level the Q value then in principle it can't disintegrate.

But because of quantum mechanics it can disintegrate after all and the way it does it is it borrows some energy delta E for a short period of time and that's enough to get through that barrier and once it's at point B it goes the rest of the way.

So this is one example where the uncertainty relation is actually giving you physical effects. So classically the nucleus should never disintegrate as long as you don't have an energy that is bigger than that peak value. But with quantum mechanics you get such disintegrations and depending on the size of that barrier how tall

this peak is the disintegration will happen faster or slower because it requires borrowing more or less energy. And of course because of that uncertainty relation the decay is a stochastic process so not all nuclei decay at once.

We can simulate that with a little applet. Here we have a whole bunch of atoms, those little red dots are the atoms and we start letting them decay.

And you find out at first many of them decay and then it sort of slows down exponentially and you would have to wait a very long time until the very last one of it is gone. So this is how decays will happen in nature as well.

First a lot decay and then the decay is slowing down exponentially. But it never quite stops. By the way this phenomenon of going through that potential barrier without having enough energy is called tunneling.

There is actually an application to radioactive decays that you are probably very familiar with. And that is your typical smoke detector contains about 0.2 milligrams of an americium isotope, this one 241 in the form of americium dioxide.

And it's an alpha emitter so it has Q value of 5.6 MeV and the resulting alpha particles they ionize the air molecules near the isotope. And once you ionize the air it becomes partially conducting so if you surround that air volume with some

metal plates and apply a voltage then you get an electric current because the air doesn't isolate perfectly anymore. And if you then have entering smoke particles in that air volume then the smoke particles will tend to absorb the

alpha particles and that reduces the ionizing property and therefore the leakage current drops and the smoke detector gives an alarm. It also works very well with little droplets of oil in particular hot pepper oil because I set off my own smoke detector that way plenty of times.

As we started our discussion there are three different types of decay so let's now talk about beta decay. Beta particles are electrons so the nucleus emits an electron and that's a very

strange thing because the nucleus doesn't contain any electrons so how come it emits any? Because it emits an electron since charge is conserved that means that the charge of the nucleus actually has to increase when the decay happens because it created a negative charge you have to balance it by creating a positive charge also.

And the mass number stays roughly the same and this electron can't come from the nucleus even though you could say well maybe I didn't tell you all of it and there are actually some electrons in the nucleus also.

But we know that the nucleus is on the order of 1.2 femtometers in size or 10 to the minus 15 meters and that's 1.2 times 10 to the minus 9 micrometers and we can calculate from the momentum relation between the wavelength and the momentum of the electron.

We can calculate what kind of a momentum it would have to have if its wavelength is small enough that it fits inside the nucleus. So I use the Planck constant as usual with 1.24 electron volt micrometers if I divide that by the 1

.2 times 10 to the minus 9 micrometers I get roughly 10 to the plus 9 electron volts per C of momentum. And that means that it's roughly 1000 MeV or 1 GeV and that is much much more energy than there is in nuclear decays.

Energy of nuclear decays is on the order of 10 MeV or so. You can't get energies of 1 GeV out of it. That is also true for beta decays. There are no beta decays with energies of 1 GeV. Another mystery is, unlike for the alpha particles where we calculate the kinetic energy of the alpha particle that comes out, if it was a

similar process then we should be able to calculate the kinetic energy of the electron simply by looking at the Q value of the process. And that doesn't work because the electrons don't come out with just a single

energy, single kinetic energy, but they are emitted with an entire spectrum of energies. That is just shutting off at the Q value. So that mystified many great physicists. Even Niess Bohr was ready to abandon conservation of energy as a consequence of that.

And finally Pauli suggested that there has to be a new particle, one that nobody has ever seen before. The new particle was eventually called the neutrino. He called it the neutron. But it was eventually called the neutrino and it has strange properties.

It has no electric charge, it has very little mass and it only appears in this obscure beta decay process. Eventually Fermi came up with a theory that also required a new interaction.

Meaning that this beta process is coming from yet another interaction which is neither electromagnetic nor the strong interaction. And this interaction is called the weak interaction. So the weak force is extremely short in range. So we learned that the strong force is effective to about a femtometer or so, 10 to the minus 15 meters.

The weak force's range is still three orders of magnitude smaller than that. So it doesn't at all reach across the nucleus. The other thing is that the weak force is not all that weak. It's actually stronger than electromagnetism.

It's just not as apparent as electromagnetism because electromagnetism has infinite reach and the weak force has this tiny reach only. And there are two types of interactions for the weak force.

What we are concerned with here are the so-called charge-current interactions. They are called charge-current interactions because they involve changes of charges in particles when the interaction happens. So for example we can change an up quark into a down quark and that changes the

charge from plus two thirds times the elementary charge to minus one third times the elementary charge. If we simultaneously also change an electron into a neutrino and the electron is minus the elementary charge and the neutrino has no charge at all.

So overall the charges are balanced and in fact from the requirement that the electric charges must balance we can deduce that the neutrino can't have an electric charge. Let's briefly talk about antimatter. We already learned about positrons which are sort of twins to the electrons. They have the same mass as electrons but they have opposite charges.

They are also anti-quarks. So there is an anti-up quark that has the same mass as an up quark but it has the opposite charge and the opposite color also.

There is an anti-down quark that has the same mass as a down quark but the opposite charge and color. And the up and the down quarks, anti-quarks come in the three anti-colors, anti-red, anti-blue and anti-green. And if you combine a color with an anti-color then you also get white just as if you combine red, green and blue.

There is also a neutrino and an anti-neutrino but because the neutrino is neutral both for the electric charge as well as for the color charge that might mean that it's maybe just one particle.

Believe it or not we haven't found that out yet whether the neutrino and anti-neutrino are the same particle or not. We are still trying to figure that out. The thing about antimatter is that if you are given enough energy you can pair produce a particle and its anti-particle together.

So you can make a U-U bar pair or you can make an electron-positron pair or a neutrino-anti-neutrino pair. You can't produce one of them alone but you can always produce such pairs if there is enough energy available. Because of the weak interaction there is also another particle relationship.

The up and the down quark they kind of belong together. It's almost as if they were the same quark and there is just a different quantum number just like we had quantum numbers for the hydrogen atom. In this case the quantum number is called the weak isospin so it comes from the weak interaction.

We can sort of symbolically write it as up and down, that's by the way why they are named in such a way. We put parentheses around it to symbolize that it's like one particle. Then we can do the same thing for the anti-quarks.

In the same way we can connect the neutrino and the electron together and then the resulting thing is called the lepton. The neutrino is the neutral lepton and the electron is the charged lepton. It works the same with the anti-neutrino and the positron.

Now it is allowed to transform those particles with the charged current interaction of the weak force. Meaning this can go back and forth here or here or here or here. Or we can also pair produce not just up quark and anti-up quark but we can also pair produce say an up and a down anti-quark.

Or we can pair produce an electron and an electron anti-neutrino, these two. Such pair productions are also allowed. This is how the beta decay actually then works.

If we have say a proton as two up quarks and a down quark and the neutron as one up quark and two down quarks. We can have a transition of a down quark going to an up quark.

And that means that most of the time at least that a neutron goes to a proton meaning that one of those two d's becomes a u. Then the neutron turns into a proton and at the same time we pair produce an electron and a neutrino.

And we do that to balance the total charge. And by the way in principle we can also turn this way a proton into a neutron if it's this d quark here. And the resulting state three up quarks also exists in nature, that's called a delta plus plus.

But it has different spin characteristics and it has too much mass to actually happen in nuclear processes. We need more energy for that. So a typical example of a beta minus decay is that you have carbon 14 decaying to nitrogen 14 plus this pair production of electron and neutrino.

The opposite can also happen. So you can transform an up quark into a down quark and that means that a proton turns into a neutron. You can also even with that process go the other way but then the resulting particle the DDD which is

a delta minus also has a different spin and too much mass for this to be possible for nuclear processes. The resulting process of this turning a proton into a neutron is called a beta plus decay.

For example you can have a neon 19 turn into an iron 19 plus the pair production of a positron with a neutrino. And the reason why this is called beta minus decay and beta plus decay is because this

is a negative beta ray or an electron and this is a positive beta ray or positron. Finally the last week process that can happen in nuclear systems is you can capture an electron. The innermost electrons they can interact with the nucleus.

For example you can have a beryllium 7 and it captures the innermost electrons and then it turns itself into lithium 7. And the only thing that's coming out is a neutrino.

So this is called electron capture. So let's go through an example how much energy is released in the beta decay of carbon 14. And in this case carbon 14 goes to nitrogen 14 as a positive ion.

And the nitrogen 14 atom the atomic mass is given as 14.003074. And we can calculate Q values for beta decays just like we do for alpha decays. So we take the difference of this ion which we calculate as the atom minus one electron.

So that gives us the mass of the ion and then we can get the mass difference to the carbon. This one is the carbon, this one is this ion and the beta ray that is coming out.

And that gives us this many atomic mass units or about 156 keV of energy. And that goes not just to the daughter nucleus and the electron but also to the neutrino. And since you don't see the neutrino depending on what share the neutrino gets you will see an entire spectrum of beta rays.

By the way this place here is an interesting place for neutrinos. Detecting neutrinos is very very difficult because they don't like to interact. They are much much more penetrating than gamma rays but they also don't interact.

It actually took a quarter century from Wolfgang Pauli's prediction of the existence of this particle to the discovery by Reines and Cohen. And Reines and Cohen used inverse beta decay meaning that you have an anti-neutrino interacting with a

proton to give you a positron and a neutron to detect the anti-neutrinos coming out of nuclear reactors. And eventually Reines became UCI faculty and for this discovery he won the Nobel Prize in 1995 and Reines Hall is named after him.

So the last of the three radioactive decays is gamma decay. Gamma is the same as a photon and so just like there were energy levels in atoms there are also energy levels in the nucleus.

And you can excite a nucleus to go to a higher energy level or a nucleus that is in a higher energy state can get rid of that energy by emitting a photon. And because the energy is so much greater in the nucleus than in an atom those photons are actually gamma rays rather than visible photons.

The typical energy difference is KeVs to MeVs and therefore you get the resulting wavelength of the photon is much shorter. And as I said absorption is also possible. You can have a nucleus absorbing such a photon and go to an excited state.

Or in the course of an alpha or beta decay the daughter nucleus might already be in an excited state. So for example if I have boron-12 and it can beta decay either to the ground state of carbon-12 or to an excited state of carbon-12.

And if it goes to this excited state then the resulting Q value for the beta process is smaller and then in exchange you get an additional gamma emission of 4.4 MeV.

Gamma ray often accompanies other nuclear processes. Of course for the pure gamma decay itself the nuclear charge doesn't change nor does the approximate nuclear mass number because the number of nucleons remains the same.

So we can now summarize alpha, betas and gammas. And we have several conservation laws that happen in the process of all of those three. So just like in mechanics we conserve energy, linear and angular momentum.

And like in electrodynamics we also conserve electric charge in the alpha, beta and gamma decays. However we did conserve other things as well. For example we conserve the so-called barrier number and the barrier number of

one third is assigned to each quark and minus one third to each antiquark. And that means since a neutron or proton consists out of three quarks that there is a barrier number of plus one for either proton or neutron.

And therefore this barrier number conservation implies that the number of nucleons stays the same. Not the number of nucleons that are remaining in the nucleus but the total number of nucleons in the universe should stay the same. It also conserves the so-called lepton number where we assign a lepton number of plus

one for electrons and neutrinos and a lepton number of minus one for positrons and anti-neutrinos. Then both alpha, beta and all three decays will then conserve the total lepton number in the universe.

And one last remark about the de-excitations via gammas. First of all that can take quite a long time if the daughter nucleus of a beta decay is meta stable. It can even happen without emission of gamma rays if the energy is released in some other form.

And that's then called internal conversion. Let us now talk about the time dependence of such decays. The number of parent nucleons we have seen that in the Java applet decreases with time of course.

And the number of decays that happen every second also decreases with time and both are exponential decays. So we can understand that by observing that the larger the number n of some type of nuclei we have and

the longer the time interval delta t we observe them in, the larger the number of nuclei delta n that are decaying. So in other words delta n the number of decaying nuclei should be proportional to the number of nuclei that haven't decayed yet and to the time interval that we look at them.

And so this constant lambda is the proportionality constant or the decay constant for this particular type of nuclei.

If we look at this definition of proportionality then this looks like time derivative of the number of nucleons equals minus lambda times the number of nucleons. And that is the defining equation for an exponential and therefore the number of nucleons as

a function of time is the initial number of nucleons times e to the minus lambda t. Then if we look at the rate of the actual dn over dt that is the rate of decaying nuclei and

just from looking at this equation this is simply minus lambda times the number of nuclei and we don't want the minus sign so we just get rid of the minus sign and we just multiply n of t with the decay constant lambda. And we can then say lambda times n naught is the initial rate of decay or r naught.

This r is called the decay rate or the activity of the sample of nuclei and the unit is disintegrations per second or becquerel in honor of Henri Becquerel.

Finally we can define the half life of the decay and the half life is that time period t one half where the number of nuclei remaining is one half the original number. And that implies also that the activity of the sample at that time will also be one half of the initial activity.

Let's look at an example. The isotope of carbon 14 has a half life of 5730 years. So if a sample contains 10 to the 22 such nuclei what is the activity of the sample?

Well we know that r is lambda times n naught and one over lambda is often called the characteristic time or the mean life time of those nuclei. And it is related as we will see later to the half life by 0.693. I will tell you where this number comes from later.

And therefore the rate, if I simply plug that into this equation here, the rate is 0.693 times the n naught divided by t one half.

And since n naught is 10 to the 22 we get that the activity is 3.83 times 10 to the 10 becquerel. So a lot of activity for 10 to the 22 carbon 14s. So one can use carbon 14 to date organically produced material.

And the reason why we can do that is because cosmic rays continuously produce neutrons in the atmosphere. These neutrons they can charge exchange with the nitrogen in the atmosphere.

What charge exchange means is that the neutron enters the nucleus and shoots out a proton. So a neutron goes in, a proton goes out and the nitrogen turns into carbon. And that is a nearly constant production rate and therefore an equilibrium is established between carbon 12 and carbon 14.

And while organisms live they absorb the carbon 14 from the air when they are plants via photosynthesis or via food when they are animals and eat plants. And so that means that about 1.3 times 10 to the minus 12 of the carbon atoms are carbon 14 for any living organism.

But once the organism dies no further carbon 14 is added and the number of carbon 14 just decays away.

And we can measure the activity and that allows us to have an estimate of the time of death of the organism. And to refine this technique to adjust for small variation in the production rate one uses tree rings or other samples where the age is known independently.

So let's look at the very end at an example of how this works. So let's say we have a claim that one particular piece of bone is about 300 years old. And we investigate this claim by measuring the activity of a small sample.

And by the way such things have been done for example for the Shroud of Turin to see whether it is about 2000 years old. Where the claim was that this was the Shroud that Jesus Christ was buried in. So if you have 30 grams of material then we can calculate how many

carbon atoms are in there by using Avogadro's number and the atomic weight of carbon. And we get that it's 1.5 times 10 to the 24 carbon nuclei. That means that we have this many carbon 14 because we know that a fraction of 1.3 times 10 to the minus 12 is carbon 14.

And we can judge from the activity of the 7.2 becquerel how many carbon nuclei there are now. And we find it's 1.88 times 10 to the 12 carbon 14 nuclei that exist now.

And we can then use the exponential decay law to tell us what fraction of the half life has elapsed. And we find that it is 4% of the half life which in this case for the 5730 years is 340 years.

And so in this case the age of the bone would be consistent with the claim that it comes from the first thanksgiving feast in Plymouth. So happy thanksgiving.