 Tudi se imaš dobro tudi však tudi predstavljamo, bojte na zelo, da na različenju se lahko tako poživamo, da se je naspešno, načo se lahko izgleda, načo je izgleda do tih zelo. Tudi njih različenjo si ga da však težko počutimo, da bi je, tako da zelo, at least concerning the material, ok. I know that you all find the workshop super-duper, that's for sure, but at least comment on the material, ok, so that we know that this was not in the workshop and it should be in or that was too much and so on. So first of all I don't know everybody of you personally because I was not in the first day, ok, Zdaj sem prišljala, če je. Zdaj, nekaj prijevamo, da bomo počkali. Zdaj, prijevamo, da bomo prijevamo, da bomo prijevamo, da bomo prijevamo, da bomo prijevamo, da bomo prijevamo. Misl zato vse odrečenosti, kako je vzal, zato sem tudi zelo počkega, potrebno v zelo obradi, zato sem vse boš zelo, nekaj ne začal, kot boš da boš zelo počkega doj, ko jih nekaj ne nešte odroženosti. So we would like to talk about low energy nuclear reactions. The first question I have to you because this will be an interactive presentation. What do you understand under low energy? What is your picture of low energy? How you define low energy? You said a few kv. What does it mean a few kv? Up to? Well, that's not low energy, this is chaos. It's, you know, below, it's in the Hades, so to say. This is too, too, too, too low. You cover half of the region. Low energy means what? Low velocity. So what is low energy, low velocity at the same time? What do you think is the maximum velocity you can accept? The speed of light is the maximum. It's not low energy, it's, you know, the ultimate energy. So during, everybody is physicist, you are all physicists here, I hope so, or not. It won't be bad not to be, but I'm just asking. So when do you start doing relativistic corrections? Square root of 2c. Then you start doing relativistic, what is this energy? This velocity? Square root of 2, this means 0.14c. Ok, maybe, but I'm just discussing this because you have to think that way. So in nuclear physics we usually say that low energy nuclear reactions are those phenomena where the velocity of the, let's say, of your moving particle is between 0.01 and 0.1c. Ok, which is, I mean, the higher limit is something like 3, 10 to the 9 meters per square. Ok, meters per second, sorry, per square. Anyway, good, so just for the discussion, it's also a matter of taste what is low energy. Ok, I mean, in nature we have high energy physicists. It's the whole community that works, for example, at CERN. Ok, they got used to work with light speeds or portions of light speed. Ok, significant. In nuclear physics, apart from some phenomena that happened at the so-called GV region, the rest is in general low energy. But for us low energy, ok, let's think of, let's say, up to 20 MeV. Ok, so that's 20 MeV per nucleon. 20 MeV protons. Ok, it's low energy, believe me. Ok, now let's continue. And let's discuss a bit what is a nuclear reaction. You will not find any or very few formulas in this talk. Ok, and most of the quantities that we will discuss will be a qualitative discussion. I'm not saying anything about neutrons, protons or whatever. To have as the presentation as general as possible. So a nuclear reaction is the process in which two particles collide to produce one or more particle that are different from those that began the process. Ok. So a nuclear reaction must cause a transformation of at least one particle to another. So which particles can we consider? It's a nuclear reaction. So we talk about nuclear particles. Ok. It can be also nucleus. Can it be gamma? One of the particles, of course can be. Ok. Can be an electron, surely. Ok. And what is nuclear scattering? Nuclear scattering is the process where a particle interacts with another particle and then they separate without changing their nature of any nucleide. The nature or not there, the nature of any nucleide. So nuclear and particles we know now. What do we mean with nature? Nature means what? Energy. Ok. Internal structure, quantum numbers. And when we talk about nuclear reactions, we have some conservation laws. What could be these conservation laws? Energy. Linear momentum. Ok. What else? Mass. And charge. Ok. There are also some other conservation laws, but sometimes they are a bit tricky to deal with. Ok. So nature we talk. Let's go now to the question why we are using nuclear reactions. Ok. We are using nuclear reactions to study nuclear properties in general. Ok. Even applications are correlated with properties. I am just giving here a list of reactions, let's say the nomenklature of the reactions. Coulomb excitation. Ok. Based on electromagnetic repulsion. Transfer and knockout reactions. We will go through them, but I am just using the names here so that you will start getting acquainted with that. Reactions in the resonance region to study resonances in spins, compound nucleus reactions, heavy ion reactions. Ok. Fission in deep in elastic scattering. Photonuclear reactions in nuclear resonance fluorescence. Giant eyeball resonance. Pygmy excitations in elastic scattering to low lying states to extract spins. We don't want to discuss all this in this foil, but I am just, this is to show how wide the field of nuclear reactions is and how many physical phenomena are hidden behind it and of course how many models. Ok. Now, we study the reactions themselves, to understand how ions interact with nuclear matter and how nuclear species are produced. This is helpful for what? To understand surface and bulk analysis of materials, to perform bulk analysis and surface analysis of materials, to understand radiation transporting materials. Ok. To produce radioisotopes, to work for nuclear reactors, to understand erosion of structural material in fusion and of course there is a big application of nuclear astrophysics to understand how species are synthesized in the cosmos. Ok, fine. Now, let's start with some very basic stuff. Ok, I hope you are all acquainted. That will be, that will be ok, but let's take it from the beginning. Nuclear reactions and Q values. What is a nuclear reaction? Is A and A can give B plus B or can give more. Ok, on purpose C is not used here. Fine, so we have, this is the entry channel, the entrance channel, and these are the various exit channels. This is the general way to describe a nuclear reaction. So we use reactants exit channel, one exit channel, two. Now, let's forget more than two here, then exit channels, and this we can write like this. So this is the target nucleus, the projectile, the ejectile and the residual nucleus. Usually when we write a reaction like this we measure that thing, not this one. We can measure both, but then we talk about coincidences, usually. So in any nuclear reaction if we want to use the nuclear reaction in most of the cases we are using, we are detecting one of the products, usually the lighter one. Question, if you have the chance to make an analysis using a reaction that gives you in neutron a better particle or a gamma photo, what would you be happy to use? Gamma, why? Think of it, why? Why a gamma? Because gamas usually have what? Well-defined energies. Neutrons have also well-defined energy depending on the reaction. Okay? And better particles we avoid them why? Usually they are in a continuous spectrum. Okay? So the amount of information you get from such an analysis is strongly related to the well-defined properties of the product that you are using for. Okay, now what are Q-value? What is the Q-value? Everybody knows what is the Q-value? But you know how? Okay, sorry, but I have to say it. Q-value is the masses before the reaction minus the masses after. Which masses? In energy units. Okay? So MA plus M-alpha minus Mb minus Mb minus Mb capital minus Mb, okay? Energy units. We will see what this is. And of course we all know exothermic reaction or exoergic. Negative endothermic endoergic. And Q-value equals zero, this is elastic scattering. So question, which category of reactions with Q, depending on the Q-value, which Q-value reactions, I mean what reactions are most important in nuclear synthesis in cosmos? In the sun. Why? Because they happen. Okay? You don't need to go to sun to put energy to start burning. Okay? Fine, binding energy, nuclear atomic masses and mass excess. What we know, we know that the sum of the masses of nuclear is more, is bigger than the total nuclear mass. Okay? We see it here. The mass of a nucleus. You could write it as the mass of protons times z and so on. And that is here something that is correcting the whole story. This term, okay? And because of the Einstein formula, E equals MC squared, you write it like this and this is the nuclear binding energy. Okay? So if you look here in this plot, mass number versus negative energy per nucleon, you see this well-known plot. Okay? What do we find immediately? What is the attractive point here? Is that around iron you have the biggest binding energy. Okay? Here you have lower the trend and here again lower. But apart from this region where we have, where the stable stability or so, from that side you can come to this region by fission whereas you can come here from that side by fusion. Okay? This is because on the left side you need to fuse particles to become stable whereas in the other side you have to make fission so that nature will be stabilized. Okay? Now if we make calculation for the atomic mass the mass of the atom we have this relation that says that the mass of the atom is the mass of the nucleus plus the mass of the electrons minus the binding energy of the electron. Okay? And now because in the nuclear reactions the total charge is conserved we can write an atom approximately equal with mass of nucleus. Is it boring for you? You sure? Okay. Why is that so? Why can we use this? Why can we write such a thing? Because when we write the reaction A plus A capital Okay? The electrons are in both sides. What is the small difference? That the binding energy of the electrons in A capital may be a bit different from the binding energy in B capital. But the difference is so small that is negligible. Okay? So once we do this we have the atomic mass excess. Okay? And the atomic mass excess is simply there is the difference atom minus A times the atomic mass unit. The ammu, so to say. Okay? Now, once we have this the reaction A B capital has a Q value that is given by this simple relation. Okay? Atomic mass excess. Clear? Clear. Fine. Here we have the case of the famous reaction of Rutherford. The first nuclear reaction in the lab. What is the Q value here? For that we need table. The last one is published in 2017. And it's the atomic mass evaluation. It's a very famous paper. You can download it. It's public. It's how it's called. Free access. And here you see that you have for the proton the mass excess. The proton is here. Okay? You have for the oxygen 17 is here. And so on and so on. So you just simply take this table and you calculate the Q value. Okay? And if you do this exercise the Q value is 1191 something kv. Okay? Experimentalist in nuclear physics is something very simple. What do they do? They have a scattering chamber. Okay? They have a target. They have a beam. They shoot the beam on the target and they produce particles. This can be gamma also. The B is simply okay? The light product. And a detector that gives you a signal to go to the data acquisition and here at the Faraday Cup you just take the you integrate the current. Why you integrate the current? What is this necessary? No? This is the next step. Why you need the current integrator? What gives you the charge? What gives you the charge? You integrate what? Current. When you integrate the current what do you get? Charge. Where is this charge coming from? From your beam. Which means this setup is for charge particle reactions. Once you have your charge what can you calculate? The number of particles. How? It's very simple. You have somewhere also Avogadro and whatever. Fine. So, the current integrator is necessary not to determine the cross section but to know your absolute number of incoming particles. This number you can use in many cases for many, many purposes. Okay? Fine. So, let's here be that the target A is in rest this is what we understand in everyday nuclear physics life and we have a particle A that is moved. Okay? And the B is a heavy product and the B capital of the heavy product and B the light product. So, we forget the B capital because we are usually try to detect the light one. So, we have here an angle theta okay? Where we count the product. Now, in your hand please for the following discussion forget the detector and think that theta is the emission angle. Okay? Which means if I had the detector here I should have a different angle but in this case for simplicity forget the detector and let's have theta is the emission angle of the particle. So, we have we know that we have a laboratory system and a center of mass system. In the laboratory system is the picture we understand the small A goes to capital A this is before the event so to say different particles emitted in different angles and I say again this is the emission angle of the B that in that case is this. Whereas in the center of mass system things are different both are moving and both are flying and when they fly away they have a difference of angle by P. Okay? And if you want now to transform the energy of the projectile into the center of mass system can you have a projectile energy in center of mass system? Is it particle specified? No. It's center of mass. Which means in many experiments when you do even for analysis even for basic research sometimes we have to understand how you transform this energy and this is simple. And if you don't remember the atomic mass units for the target you just take the mass number. Okay? It's within accurate. Okay? Now, I don't want to go into calculations now but this is the formula that you take at the end that tells you that the square root of the flying product is given by this equation. You can find this in every textbook. So you can do that also for the heavier product by permuting the symbols b and b, b small and b capital. Okay? And replacing theta with phi. What is phi? Is the emission angle of the heavy product. Okay? And for our discussion what is within the brackets here, let's say quantity c. Okay? Now, let's take the case where we have an endothermic reaction. What do we observe in this case? We observe that there is a threshold energy. A threshold energy where is this coming from? The threshold energy. Why I get a threshold energy here? Compare this with the c. Okay? I see some condition in my formula that give me an energy threshold. Okay? Anyway when I have a threshold and my beam energy is below the threshold whatever you do, you don't have a reaction. Okay? Now, when I achieve the threshold energy what does my particle be? It goes zero. It flies like this. But when you increase the energy then the B particles can be within a cone. Okay? Can be emitted in a cone. And the maximum angle can be what? Is given by double theta max equals 180 which means 90. And this formula up there has two possible solutions. Okay? You see plus minus. Which means these solutions can be only acceptable if this term here okay? NB plus is smaller than zero so that it makes sense to have their product, okay? That is all. So from that this relationship that tells me that the incident beam energy can go up to a maximum energy and this is given by this formula. Okay? So, if this is the case when the energy of the projectile is between the threshold energy and the maximum energy given by this condition then you can have okay? Observed at the same angle at the emission angle theta. Okay? And this theta max is given by this relationship. We will come to an example so just try to follow a bit the logic. So, let's take the case where we have in the case of exothermic energy of the flying particle is simple. It depends on the angle and it's single valued. So, it decreases when the emission angle is increasing. Okay? You understand this, why? Okay? The bigger the angle the less the energy of the emitting particle. Of the emitted particle. In the first case where we have exothermic the mass of the incoming projectile is smaller than the heavier product. Okay? Or we have an exothermic sorry. It's the other way around. Or here is wrong. It's this one. Or the energy of the incoming color is expected than the larger energy the maximum energy. Then we have this picture the emitted particles can go all over the volume. Okay? But most of them they go like this and then their number to the other angles is decreasing. In any case this means I have want an angular distribution. Okay? In the other case in the reverse case what happens in this case I will kill myself the emitted particles are in the forward direction and you can see at almost the same angle two groups with different energies. Okay? Exercise. Which is the Q value of the lithium? P and reaction. Okay? What do you do to do that? I can give you this exercise for tomorrow. It would be nice to do it. You don't need to anything almost nothing. You just need to follow this what we said. And you just need only the mass excess of what was not in the list. But you can find this. Okay? Has the reaction a threshold? If you follow the formulas you will get it. If yes what is the threshold energy? Believe me it's a nice exercise. Very simple. Okay? What is the maximum of the reaction energy? If the proton beam energy is 9 MeV which directions will the emitted neutron stake? It's all applications of what we said. Anyway. And in the second exercise there is elastic scattering of alpha particles. To solve this exercise what do you need to consider? Q equals 0. Okay? Okay. When do we have to go for lunch? Today? Or what? Okay let's go coffee break. I mean we are the coffee break is normally at 11. Okay let's go coffee break. Let's go coffee break.