 So good morning My name is Frank Gunsing I will talk to you about an introduction to our matrix theory our matrix formalism and neutron induced reactions So I'm working at CA in Sackley in France and now I'm only At CERN in Geneva so that is why I have two logos because CERN they paid my travel to here So I need to acknowledge them first What does it mean this neutron induced reactions and our matrix? So we have to limit first the energy scale of the neutrons Where we are looking at so if you have we are interested in this compound nucleus reaction part So you see here three different types of reactions if we go to very low Neutron kinetic kinetic energies, let's say in the scale of a few milli electron volts then You can also associate to this kinetic energy a particle with energy has also an associated Wavelength and that wavelength is in the order of nanometers and in that case the size of the wavelength is Comparable to the distances between atoms in In any solid-state material So the neutron is not interacting with every nucleus individually, but with the Structure of the material so with with the crystal or the liquid or whatever it is But it is seeing the the solid state as as a full as a crystal lattice and in that case all the physics that is happening there is Based on solid-state physics, so you can indeed see the structure of molecules of crystal lattices of and that kind of things and that is used for example in in this mini electrical region Especially with the thermal-neutron reactors for example Then if you go up higher in energy, of course, there is a region where both Phenomenon occurs, so you get solid-state physics and nuclear physics nuclear physics means that the neutron is really reacting with a nucleus and not anymore with the atoms and Let's say in the region between 1 milli electron volt up to Roughly 10 mega electron volt The wavelength is such that it sees the neutrons sees directly the nucleus and then you get this neutron compounds compound nuclear reactions And if you go even higher in energy The wavelength becomes very small and then the wavelength is comparable to the size of the nucleons inside the nucleus so to single nucleons single protons or Nutrients inside the nucleus and then you get reactions that that are really based on on a single knockout of protons neutrons or other What we call direct reaction and the time scale of direct reactions is much faster than this compound nucleus reaction Because for the compound nucleus reaction the the energy that becomes available. I will come back to that later It's distributed to the whole compound nucleus So which is a new nucleus based on the original target nucleus and the neutron they come together and All the energy that is becomes available is dissipated through that So you get a compound nucleus which is in a excited state and then you can get different Reactions that may occur with this compound nucleus So our matrix that is what I'm going to talk about is related to this part So this energy region then we have this neutron nucleus reactions, which can Be written in several ways. So usually you have a nucleus call it x for example You have a particle that goes in in our case. That's a neutron and you get Let's say two different exit particles. So it's a binary reaction. You can write that in different ways and Examples of real life Reactions are for example boron 10 plus a nucleus gives lead in 7 plus helium 4 Which is also an alpha particle So you can write that in these equivalent ways and the short way is just to write the boron 10 and alpha because then You can always come back to what was the exiting particle So that is a way how we write these type of reactions and then you can get different reactions of a neutron induced reactions like elastic scattering where you Emit a neutron with the same energy as it was coming in that is no energy transfer or whatever you can get in Elastic scattering So the new the exiting neutron has a different energy. So there is some energy transfer between the particles You can get n gamma reactions just capture It's called or fission reactions and F or charge particle emissions like An alpha and and P proton emission or an extent several neutron reactions That is also possible and if you sum up all these reactions Then you get something what is known as the total cross section So that means any reaction that could occur you lump it into this total cross section, which is an important quantity in Neutron induced reaction theory because it is one of the few that you can really calculate With some models in some energy rates So these cross sections usually what you have is the cross section is in fact Related to the interaction probability That is a cross section, but it has different units. It has units of Square meters I come back to that or Barnes, which is a little bit more easy to grab as a as a number Yeah, the cross section what you have is a function of the incoming particle, but if you Start with this you have an outgoing particle Y and B and you have an angle with that and the most basic Quantity is a double differential cross section, which is as a function of the incoming particle But also as a function of the outgoing angle and particle energy and that is double differential cross section Then you can integrate that over the outgoing angles or the outgoing energies and then you get what is called a differential cross section Always as a function of the in going incoming particle, but integrate it either over the Outgoing energy or the outgoing angle And then if you integrate this one you come back to what is called just the cross section and It's a double integral of this double differential cross section, but it's still a function of the outgoing energy So that is a real cross section always as a function of a neutron energy You can also integrate this and then you get an integrated cross section integrated over a flux over Something whatever you want So and how does it look in a range at low energy? So this is an example of uranium 235 plus a neutron and I Spoke already that I told you already that there were different reactions possible and here you see Four different reaction cross sections the total cross section in black here and a few partial cross sections, which are just The part that's summing up up to the total cross section like the elastic scattering the red line here Capture the blue line here and uranium 35 as we know it is fissile at low energy. So And that is the green one. So the green Curve is the is the fission cross section. I will show Later again this picture, but this one comes back So what we see here from this picture is that we have peaks in these cross sections in all these cross sections And these peaks occur at exactly the same energy There is one energy at which there is a peak in all the cross sections and then the shapes are different, of course The sizes are different That's normal, but the peaks are still there and these peaks in the reaction cross section We call that resonances and origin of these resonances Well, I will explain that is related just to to excited states in the compound nucleus and these excited states Because this excitation energy is so high you cannot predict them They are just there and you have to measure them somehow and once you know them This will help you to reconstruct cross sections in a very nice clear clean way And you can even broaden the cross sections which are normally at zero Kelvin at zero temperature to Something which is our daily life temperature at 300 Kelvin, for example, which is a little bit different the cross section then That is called Doppler broadening But the Doppler broadens cross section is something that we can use in daily life and it comes all back to these resonances And how to treat them with this our matrix formalism Let's go back now to to neutron fluxes because I Told you there is this energy range from very low to very high Neutron kinetic energy, but if we look at where the neutrons are coming from to make these reactions For example, if you have thermal Neutrons that induce fission for example on uranium 235 what you get is Fission products and new neutrons and these neutrons have this type of distribution which is Maxwell-Boltzmann distribution because it's a kind of thermal spectrum for the neutrons, but they are peaked at let's say 1 MeV and This thermal fission cross section This distribution of neutrons is Is these are fast neutrons? So they are created with just thermal neutrons you can create these fast neutrons which occur when when some nucleus is fissile Then in a reactor, of course There's lots of water to moderate in neutrons and moderation means that you slow down the neutrons So you have originally fast neutrons and then you go back to Lower kinetic energies and if you do that a long time and infinite time with an infinite amount of water You get Water-moderated neutrons spectrum, which is again the Maxwell-Boltzmann distribution and of course if you plot that on a log log scale and you Scale this factor so that they are at the same height you get an identical representation of this Maxwell-Boltzmann distribution and this is 25 milli electron volt distribution which is in fact going Much broader than just one one energy, but that is typical water-moderated Neutron spectrum, which we see now you can get everything in between of course for these neutrons and in some stars red giants for example, and I think you had a lecture last last week about this you get also a Neutron Neutrons and they have also a thermal spectrum, but at a little bit different temperatures depending on the temperature there and the neutrons there You have these red giants. It's usually between five and hundreds kT that is that is The temperature so it's between five and hundred kV you get these These type of distributions so so neutron induced reactions are important over the whole energy range here for any nuclear Fission technology nuclear reactors and in between also for stellar spectrum stellar astrophysics So it's a double use let's say of these type of reactions Now if you look again on this the same energy scale where we had all the neutrons here So you see here again all these reactions the new the Now this is a different list. This is a uranium-238, but that doesn't matter you have again Fission here in green you have capture in blue and And in total here in red and if you go higher up in energy You see even the reactions like the n prime or nxn reactions and these are threshold reactions So they they start only at an energy high enough to make these reactions occur other reactions on the contrary They start immediately. There is no limit. So even with a new turn of nearly zero Kinetic energy just enough to reach you get already these reactions, which may be very hard very high and now if we put these two Did this plot which I just show you of the distribution of neutrons and here the reactions that can occur You see this is the same energy scale here You see in which region which type of reactions you can expect so at high energies in high fishing reaction, for example, you get all these Threshold reactions and in prime and extend and this is then of course important if you have Let's say fast reactors or or accelerated driven reactors and new concepts of reactors These type of reactions become important And if you go to lower energies all these resonance region here because that that is occurring in any reactor is important And of course you have this zone where you have completely moderated reactors and there you have the thermal Thermal energy range that is important. So now This is our typical extreme situations where we can expect typical Neutral distributions. These are the type of reactions that we can expect and of course This looks different for different nuclei if you have a different nucleus The peaks may these resonances may occur at a higher or lower energy range Also, the thresholds reactions may occur at lower and higher Energy, so you have to look at that for every nucleus differently, but this is an example And now where can we study this type of reactor reactions? Well, then you have to go to facilities for example to Thermal-neutron reactor for example the IML you can study you have a lot of neutrons in this In this moderated in this thermal region, so lots of neutrons and you can study Neutron induced reactions there You can go to higher energies then you have usually more energetic Neutron sources using accelerators and you get Well, we say mono energetic, but it's it's a kind of distribution But you get single energies and this is very efficient to study Reactions for example these are threshold reactions or some fission reactions also and the advantage of these type of Machines is that you get a lot of neutrons in the particular energy range There is another way to to address the whole energy range as once and that is to use As we call a wide pulse to Newton source So you use a time-of-flight technique to to to go over all the energies, which is nice You cover everything but disadvantages of course that all the intensity of neutrons that you have is diluted over the full energy range So you get much less Neutrons at the specific energies where you have other tools that can create much Higher fluxes at given energies, but these are the type of machines that we use to study This region of interest for for neutron induced reactions is that important to know Neutron induced reactions and And do we really care? So well, this is an example of the transmission of iron so we have iron is used as a As a shielding material in reactors for example, and if you have a shield of iron You can stop neutrons if you take a thick layer of iron and you get something Transmission that is something like that can be like this But as you see from this plot this total cross-section suddenly there is a kind of dip here and that is the interference between A potential and elastic scattering. Okay, that's a detail But the result is that you get a dip in the total cross-section. So if you calculate the shielding for particular For a kind of average cross-section, then you see that at this Particular energy all the neutrons go through because the total cross-section which is Giving you the transmission of the neutrons through that shielding is much lower at that energy So all the neutrons go through there. So it is really important to know these things in detail at detailed at several energies so I already spoke that spoke to you that's the A neutron particle it's a particle, but you can associate a wavelength to that So then we come to the Region of quantum physics. So we have classical physics where you have particles. That's just Interacting using Newton's laws of motion and that is easily understandable because we can Imagine that the other world if we go to very small particles alone Then something else becomes important and that is quantum physics and that works That is important if you are on this small scale You can say a chair here is also made of particles, but there's so many of them that finally the the resulting Effect of all these quantum effects is Finally you get the classical physics. So the normal Newton physics of Movement of a chair But if you go down to the level of a single particle a single nucleus a single Neutron that interacts with the nucleus then you need to go through quantum physics and So this is Something important to understand is our matrix thing Now, I know some of you have done this at school for a long time others don't so I will go through this Rather quickly, but just to give you a grasp of what what it means This quantum physics is that you have the probability of observing a particle at a time and a Position and that is related not to absolute certainty, but there is a kind of Probability and that probability is given by what we call the wave function the wave function psi Multiplied by its complex conjugate gives you the probability of observing this particle at a given time at a given position So the two at the same you are never sure it is a probability and that is the the whole problem And that wave function you need to obtain once you obtain that you have this probability and you can solve it and The idea is that this wave function is Just a solution of what we call the Schrodinger equation Schrodinger equation is very simple I will come back to that later But that has been postulated by this person Schrodinger and he said okay every quantum particle should obey this Equation and when it does we have the wave function and when we have the wave function We have the probability to observe it at a given time a given place To give you an example of how this was discovered. So let's go back to a quantum system, which was One of the most simple quantum systems, which is the hydrogen atom because you have one proton and electron Orbiting around it and that is the most simple quantum system, which we can observe and that's and that gives really quantum effects So you can What was observed by the time when this was discovered is that? When you have transitions between the electron states You give it energy and then it starts to emit radiation and by chance this radiation was a visible light So this could be seen but it was not some continuous spectrum. It was very discreet. So this light for example, you have this Proton and an electron orbiting around it and it was found that if you give it some energy You you have enough energy to move one electron to the next orbit around it And then it decays back and when it decays back it emits this radiation and this radiation Can for hydrogen is visible light and if you go back to more You feed in more different Energies you get a kind of spectrum of Of visible energies that are going out And this is the excitation spectrum of the hydrogen at atom and there is all these lines have been Determined and they were visible as light you can see there So you have all kinds of colors which corresponds to different series of transitions between the electron orbit and What we see as a single line is in fact many different lines they correspond to all these excitations Orbits and transitions between the orbits of the electrons that are orbiting around the hydrogen atom. So that was in fact approve it was a Very helpful observation to develop the quantum mechanics and then we come to this furthering equation So these transitions if this is the wave function and this is the starting equation. So it's nothing more than a double differential Double differential differential equation and where you have a kind of potential which may be the Coulomb potential of the hydrogen atom and and water that that is really the potential that the electron is feeling and You have energies here and this is the the double Double derivative. So if you solve this equation So you need to know there are two unknowns one is the the potential So you can give some Coulomb potential or something and then you can solve this and what you will get are Solutions from psi and e so energies and the wave function So the energies will then correspond exactly to this excitation spectrum of the of the hydrogen atom And and usually this is then grouped into something what we call the Hamiltonian and people working on On optical model calculation for example, they need to work with potentials and the potential is the most difficult part to get That it's not solving the equation. You take any computer You can solve the equation the the difficult part is what do you put for the potential because that is the real physics? Governing the whole system So once you solve this you get a solution and the solution gives you the energies and the wave functions and the wave functions Gives you all that you would like to have So we can do an exercise on that and that is making a simple system, which is just a single particle this is here and So and and it cannot go beyond this region here So this is an exercise that you find in textbooks and you can solve that because this is one of the Few things let's say that you can solve yourself and Because we can let's do it. So you find you solve this equation with the things that are given here You get a solution for the psi and the energy So you see here the discretization of the energy is coming up here. It's because that is the solution of the of this Schroding equation and that is why we have discrete states here because it's just Obeying to this second-order differential equation. That's the whole trick And then the wave function we can plot them as well. They look like this. So they are sine like the quantities and if you want to know the Probability then you have to to multiply them you have to square it and that is I think both curves I draw them here on top. So one is the sine and the other is the sine square That is the whole thing Now you can go to more difficult Situations you can invent some Well, so here you have a finite well So now the potential doesn't go to infinity but to zero It's another exercise another one of the few that you still can work out yourself without any computer And then if you do that You see here the first problem because now the potential doesn't go to infinity But it goes to zero and that means that the wave function can exist even outside the Diswell it was not the case before Because that was going to infinity. So there's no way function Beyond the limit, but now we have something that is there You have to do several tricks you have to cut this region in pieces Because here the potential is v zero minus v zero here the potential is zero You have to solve the equations and then the trick is you have to match the value and derivative at the borders And once you do that you have your equation that obeys the Schrodinger equation and you have your final solution So this matching value and derivative is also a trick from Schrodinger to have something that would Work with with what we know would correspond to our observations of nature in the quantum field And once you do that, of course, you this is a little bit more complicated, but It is working and you get Something that looks like this So I didn't plot here the real function because they are a little bit complicated, but again, it's a sine type Sinus type inside the well and a decaying thing outside the well so you can match that it's a discontinuous function but it matches value and derivative at the borders and Then there is another trick here. That is that if we have a wave function These are these Discrete levels are also called eigenstates because it's a German word because the Germans were doing this eigenstates were invented to solve this type of Equations and the eigenstates here which corresponds with eigenfunctions are the The wave functions and the eigenstates are the energy levels So these are the solutions of these equation and you get that and you can expand the full wave function which covers everything as a linear combination of the of the participating wave functions and if you do that So if you know the eigenstates you can get the final wave function which describes the whole system and that is another trick which Will come back in our matrix later Well, we can go to many other Things that we can measure for example a potential barrier So you have a freely moving particle. There is a kind of potential on its way So the wave function is modified there again, and then it goes back. It travels further There are some reflection and transmission, but then the wave function Continues and again you have this matching of the value and the derivative at the borders because you have different Equations inside the well and outside the well or here is a barrier. So inside the barrier or outside Okay, solutions are given here, so they are still comprehensive, let's say Here you have this traveling transmission reflection transmission reflection in here, of course this goes to infinity There is no reflection. Of course, you see this is a one-dimensional case with very simple Thing simple Potentials now when things become more complicated That is if you have three dimensions instead of one and if you have more complex Potentials and then things become so complicated you have to use computers and There are still a few cases where you get get something mathematically just on the piece of paper But in most cases you need numerical solutions of this wave function So I was talking before about an electron orbiting around a proton But we can also see the The nucleus as a quantum system because again you have a set of particles which are now just protons and neutrons There is a potential and the potential is generated by all the other particles around so the nucleus itself the Nucleans are generating their own potential and every nucleon is feeling this potential from the other nucleons So there is a potential we have a set of quantum particles. It's not one, but it's it's several so Complications are there. It's three dimensions. We don't know exactly. What is realistic potential use Well, what is done is usually a wood Saxon potential to start with But it can be more complicated and then we get the same thing, but now we have not one particle We have more particles. We have not One type family of nucleons. We have two types We have the neutrons and the protons and again, we get an excitation spectrum But at every energy the level is degenerate Which means that you can have more particles in a separate in a shell. This is called a shell every excited state And then you can get for example oxygen 16 Which is nice because all the shelves are filled with the particles that can be in which you have also oxygen 17 When you have one more particle Where should it go? It goes in the next shell which has a lot of possibilities But only one of these holes is filled with this particle Or you have one less of course, so you have this type of variation And how does it work if you have a nucleus and you consider that as a quantum system? That means that you have This is what we call the shell model representation of a nucleus So you have two potentials one for the Newton's one for the protons the protons of course because it's charged Some barrier to overcome the Coulomb barrier the Newton. It's different. It goes to zero So in the ground state everything is filled so you have you have something like that And in a picture of excited states if you would like to give that excited states You have this nucleus and it is in the ground state we call that so it's the lowest possible state of energy that If by some means you could put in some energy into this compound nucleus So it's in such a way that you go to the first excited states. So then this first excited state corresponds To a kind of configuration of the nucleus of the of the neutrons and the protons in their possible shells So for example, this is just an example It corresponds to the this this Neutron here, which was first here and now it is here So it is in an excited state then you can go to the second excited state Which might correspond to two neutrons here or maybe it was well this one in a little higher state It can be anything The thing is that if you go up in in Excitation so you feed in more and more energy to this nuclear system then you you become I don't know the hundredth excited state and which may correspond to a certain Particle whole configuration that is how we call call it because you have these shells and some of them have a particle others Have not and this particle whole configuration is very complicated Especially if you have a lot of nucleus a nuclear nucleus because then it becomes more complicated and that is Quantum chaos. So if you look now at a real Nucleus for example, let's start with something like lithium 7 You see here the eigenstates of this lithium 7 in here carbon 12 And you see if you go up in energy then there are more and more States that are possible and this is exactly the level scheme How we call it because these are levels that exist in a nucleus and every level correspond to an excited state Now if we go to a heavier nucleus, there are so many levels. We cannot count them So we here you see just a lower part of the excitation and you see all the transitions that are gamma ray transitions Because we are in a nucleus if we are talking about electron transitions Then we talk about x-rays if you talk about nuclear transitions We talk about gamma rays and the gamma ray transitions that have been observed and which gives us a clue about the level scheme are Are here and of course it goes down many more if you go higher in energy For example, here is again the lower part of lead to a lots of stage You can see them here. These are only the observed states made me there are many more which we have not observed because we did not have the Right reactions to populate or to observe all these states and there are a lot uranium 8 Of course here. You see it's again only the lower part And the general picture is like this so you have a ground state you have all these excited state and You can have a lot of states Finally if you go for example to a nucleus like Uranium 238 so it's a quite heavy nucleus if you go from the ground state up to an Excited state which is so high that a neutron which is inside this nucleus can just fly away It's not bound anymore by the nucleus. You have about 400,000 levels, which is really a lot and So you cannot count every level You can do that only at low energy with a higher energy You cannot count the levels anymore. So you talk more about level density. So which is a Some number average number of levels that you would expect in a tiny interval of nuclear energy So at low energy the separation between these levels is very low is very high and the separation For uranium for example can be in the order of let's say 100 kilo electron volts So these are the type of transitions that we see also. So it's KV gamma rays If you go up to that energy where the neutrons Can just fly away typically it's about 10 MeV above the ground state and Of course, this depends on the nucleus usually it ranges between four and eleven MeV to cover nearly all of the nuclei stable nuclei that we know on earth and If you go higher and at this energy the level spacing because it becomes more and more compressed is much smaller So the spacing of the level is only 10 electron volt there for a typical nucleus and again It depends on the nucleus, but you see that this Spacing is much smaller here than at low excitation energy And that is exactly the spacing is what we see when we Make collide a neutron with a nucleus because then what happens you have this nucleus Target nucleus which is in its ground state you add a neutron and instead of Supplying so much energy that you can that the neutron is not bound anymore Now the neutron is coming from the outside to the nucleus and this energy what you would add to the system to make it Go away becomes now available to the new to the to the nucleus and we call that the compound nucleus so Neutron with let's say zero nearly zero kinetic energy enters A nucleus and suddenly it gets 10 MeV for free which is just a binding energy of these neutrons So this compound nucleus has 10 MeV for free and it is in a very highly excited state And then we can get reactions and these reactions is exactly what we see here these are the resonances in the cross-section because If you look here the ex the kinetic energy of the neutron which the zero is here And for every state corresponds here to an ex every resonance Observed in the cross-section corresponds to an excited nuclear state So that is the picture that we see and when we look at this cross-section Plots we always know that even if we see an electron Resonance at one electron volts. We know that it is at several MEV above the ground state it corresponds to that particular level and Of course once it's there. It can decay to the ground states For example by gamma ray emission, but that is just one of the possible ones the new the excited Compound nucleus is there it can also re-emit the the neutron that created it and then you go go back to this situation So you have elastic scattering or if the energy is high enough That is the threshold is coming in if the kinetic energy of the incoming neutrons is high enough It can go back to an excited state. So you have neutrons in elastic scattering Or you can have fission because it is so high the excitation and you see that that you are above the fission barrier And in the nucleus just fission so a Typical new nucleus is made up of Nucleants so we have neutrons and protons and if we put that on a scale here. We have We have all the available Nuclei that exists in nature or have been observed once they are not all stable not at all, but there are about 4,000 of them Only a few of them are really stable and the stable ones They they are somewhere in the middle and if we Add neutrons here on this side for a nucleus then it just doesn't fit anymore. They they they don't want to Merge into a new nucleus So no way to get in here and we call that the neutron drip line because any Neutron that you try to add to the system just drips away It goes away same thing for the protons for certain Nuclei if you you it cannot just hold more protons not possible and the stable ones are here So it's it's just a very few of them and all the others are not stable by definition because the same ones are here They contain too many Neutrons or too many protons and they want just to decay back to this stable nuclei And that is what we call radioactivity Usually it's a beta decay or beta plus or electron capture depending on which side of the valley of stability you are this is called value of stability and Okay, so Diana's already mentioned this value of stability This is what we know and maybe at very high here. There are other configurations, which may again make some new stable nuclei or short-lived nuclear and And you see here if the nucleus is heavy enough there are enough protons and neutrons then you can get even spontaneous alpha emission so then it's better for the nucleus to just throw away an alpha particle so then it is in a lower energy configuration I Just talked to you also about this Newton separation energy which Which which I said was between four and and 11 mev But there is there are few nuclei between 11 and you see something strange here Of course, you would expect here that if you have already a lot of neutrons Then of course, the Newton is less bound So it is easier to throw to get away of a Newton and on the contrary if you have not so many Nutrients, but you have more protons than neutrons Then of course the Newton is more tightly bound to the nucleus and the separation energy is more on the blue side Which is more strongly bound and then you see another effect here, which is You have light blue and red blue and light red and red Dark red and light red that means there is a kind of odd even effect and that means It's also a well-known phenomena that is the pairing of the nutrients because if they are by two they are more tightly Bound they they are strongly bound and it takes more energy to separate them So that you see here in the reflection of the nutrient separation energy Something about fishing I told you if the nucleus is above the fish and thresholds the every nucleus has a kind of Fissional Fission threshold, so it needs to have enough Intrinsic energy, so excitation energy to be fish in a bowl and Of course every nucleus has the kind of Fission barrier and it's not so much different for the uranium 235 and the uranium 238 The fishing barrier is 6.2 mev for urine 236, which is the rain 235 plus Newton and for uranium 239 Which is 238 plus Newton the fishing barrier is 6.6 So it's a very comparable, but the whole thing of why Uranium 235 235 is fish fissions with thermal nutrients and 238 doesn't is Related to this binding energy because that is again this odd even effect They are very close, but we have this odd even effect and we see here Uranium 235 binding energy is 6.5 and here it's only 4.8 So the binding energy is much lower. So uranium 238 plus Newton It is not in a high enough Excitation energy to be fish in a bowl. So it doesn't fish in unless you give it more kinetic energy And then you are in this region and you will start to fish in Uranium 235 is different because this binding energy is so high that if you add a zero Kinetic energy nutrient to let's say a thermal nutrient to uranium 235 immediately it gains so much Excitation energy and it is above the Fish in fishing barrier. Let's say so it is fish in a bowl that means At this energy so it is fissile and fissile means you can fish in it with thermal nutrients So fissile thermal nutrients work Fish in a bowl every nucleus is fish in a bowl if you give it high enough energy So what does it mean these resonances? Which I just showed you so it is an eigenstate we just show we have all these eigenstates and and It's the excited state and they want to go back to the ground state because that is energetically More favorable configuration and the time does it take the time that it takes to go from an excited state down to the ground state That is what we call the lifetime of this excited state and this lifetime Again if you go back to the wave function, there is a kind of you see it here. It is the lifetime It's an exponential so the lifetime of this state because it's not there forever. It's not a stationary state. It is a time-dependent state and This lifetime or the towel, which is not exactly the half life, but let's call it lifetime You can associate a with to that and if you Can see the energy profile of this state here It looks somehow like that with a kind of with and that with is exactly related to the lifetime of this excited state So once you know the lifetime, you know the with and the with they are linked together, of course, and that is You can look that up in any textbook of quantum mechanics. It's the Fourier transform in fact of the of this wave function time-dependent to the energy Profile so it's a different domain, but it's the same thing So we have this this very specific form and This is called a bright Wittner form and this comes back in any problem of of quantum states which have a finite lifetime not only in in nuclear physics also in molecular physics It's there in in particle physics very high energy. You have always this bright Wittner form it comes back all the time and this bright Wittner form is is fundamental for To describe resonances it is the basic shape and then of course you have very much variations, but I will come back to that later So is there another way to To see this neutron nucleus reaction not another way But what can you imagine if you have a current of particles which are neutrons? impinging on a neutron or on a nucleus and this nucleus is in fact just You can see that as a potential well. So that is the nucleus So you have the neutron seeing a potential well, which is the nucleus So when the neutron is incident is just a plane wave and when it is scattered It is again a plane way. It's not a plane wave anymore But a radial wave. So it is going in all directions. That is the possibility at least and What should be preserved is the What we call a probability density, which is just the current of particles So what comes in must go out that is the whole thing So what goes out in in all directions from for pi is what goes in if you have an incident wave which comes in and This conservation of this probability density which can be written like this in terms of wave functions It's a current density is called and That is what we call the cross-section it is the outgoing current Divided by the ingoing current and of course normalize because this goes over for pi it is radial So we have this R square that comes in here Which is the R is the radius and you see here immediately Why the units of cross-section is Barnes because it is here. You have the meter square meter coming in and Barnes is square meter so solving Disrobing equation to get cross sections It is important and I told you before there was one way to get all the eigenstates and that is Or to get the wave functions. That is knowing all the eigenstates. That is one one method and that is the our matrix Theory with resonances resonances correspond to excited states So you have all the eigenstates and you can construct a cross-section with that the other thing is to know The the potential Which is the the optical potential which is used in optical model Optical modeling of the nucleus and then you can Also calculate eigenstates. It's not done in this way. It's it's more shortcut But the principle is is is going back to that so it's calculating the eigenstates of the of the potential and Calculate what goes out and from what goes in by using a right model So there are different energy regions where you should use one or the other approach So usually this potential works well if there are not really resonances anymore because Resonances you have to be very Precise about particular states, but if you can average things out like at higher energies Then you can use that approach and you use optical model calculations to calculate Cross-sections not all cross-sections unfortunately, but the total cross-sections you can Calculate and you have to use other models other Theories to find the partial components of this total cross-section But if you go to lower energies where you see really the resonances in In the cross-sections that you observe then you have to work with our matrix And that is using the properties of every level and Convert that in a cross-section Because that is what we want at the end express the cross-section as that for that we use the our matrix Formalism so we have So it's a bit complicated, but I will just go through it and at the end I will give a reference and if you have some time Next week it takes a lot of time you can go through that reference So you have an incoming wave function is written like that outgoing wave functions And you have something that does something to this wave function And that is the real physics of the thing which we don't know which we cannot calculate We can just observe what happens and that we call collision matrix. So all the physics of the interaction is inside that That what we call collision matrix So we call that you and then every cross-section Cross-section is one particle configuration to another one the probability of that is given by this collision matrix it is how this Outgoing wave is transformed by how the in going wave is transformed into the outgoing wave By the collision matrix and once you have that it's so We have three different Regions let's say we have an entrance channel where we have a neutron and a nucleus. They are not interacting. They're just approaching each other and So they are separate particles which means we can We have what we call the external region We can solve the equation the Schroding equation. So that is the easy part Let's say same thing for the exit channel when we have for example an outgoing neutron and an outgoing particle Again, you have two separate particles not interacting with each other If you have a charged particle reaction, then the interaction is easy it is a Coulomb interaction So you can still solve the problem but the difficult part is Once they are together, they are a compound nucleus And then there is no way to calculate this to to solve the Schroding equation and then what we use is this trick of Decomposing the wave function into its eigenstates and eigenstates are all the resonances that we know from experiments and once we know the resonances we know the eigenstates we can Fix the things together and we have again a wave function which allows us to calculate the cross section So that is the principle So is there a delta? Okay, yeah delta it's this chronicle delta, which is just one or zero. Yeah, sorry. Yeah Now this is this is a way let's say a formal way to write this cross section. We talk about channels, which is not Usually is not it's not something we observe in in in nature because we don't have all this All this decomposition in Of the configuration into orbital momenta in spin etc. We have to sum up certain things So real observables are sums of of channels. Let's say of channels, but this is the most basic quantity and This cross section I will show a few other Equations based on that so I want to find the wave function So in this external region was easy because there is no interaction. So that is the whole Potential is is either zero or just a column potential So it is easy to model and of course now we are in three dimensions So it's a little bit more complicated usually we do that we can separate the parts in three dimensions You need three variables and the most easy thing is to use the radius and a theta and a five for the two angles And then you can split in this wave function the radial parts and the angular parts You can solve them separately So you see is it's a little bit more involved than the one-dimensional case when there was nothing in which you could still do yourself So if we have this regular part because that is giving the trouble you you see that you you you can Write this learning equation in a different way, which is like that and what what comes out is something that That has been solved in 18th century special case of well the normal Solution of this is what we called Coulomb functions, and that is why we have this works also with charged particles We have a proton for example, you have a Coulomb function, and you have a special case That is there is no charge. There's no Coulomb Coulomb interaction so that the V is zero there is no Interaction and then you have the special case of the Coulomb function Which are the Bessel functions and the Bessel functions they look very much like a senus sinus not exactly Now you have to do the internal region, which is difficult. You cannot solve this learning equation So what you have to do is to do this wave function decompose it as a linear combination of its eigenstates and We use a special quantity for that which is called the our matrix which does exactly that which links the Properties of every excited state which is the E These are the energies of lambda and lambda is the excited state of the compart nucleus and every lambda has also What is called here a small gamma? But that is just related to the half-life of the state So you have the half-lives and the energies and that gives you exact data These are just the basic quantities that you need to know you learn them all together in what you know as the our matrix and Okay, this is just an example of of parts of this of this solution, let's Just go through that Then you have the solution in the internal region where you have this very difficult Equation, but you know it as an expansion of the eigenstates And you have the external region where it's like that you can calculate the wave The wave function and then like in the one-dimensional case you have to do the same thing match value and derivatives and Once you do that you get something that gives you the wave function In fact, you don't need to know the internal wave function You need it to know only at this border where is the way you separate internal and external region Now once you do that you have this perfect matching you have the final wave function and you can calculate Okay, it looks complicated, but in the end it is just the our matrix which is here And you get the wave function which is a function of this our matrix And then with that you can calculate all the cross sections that you would like to have in Rather complicated way, which is well here again You have also the radial parts because you have to merge everything together This is all nicely explained in the reference. I will give you in the end and then once you have that you have a quite complicated Matrix relationship, but in the end you get all the Information about cross section from one channel to another channel then you can sum together you have reaction cross section you have Channel to to the same channel channel to another channel or channel to whatever channel, which is the total cross section So all these cross sections are given here expressed as functions of this Collision matrix, which is of course related to the our matrix So lots of matrixes. There are different quantities that come in which I know just skip now for the moment And then you have this rather complicated matrix element and then we can start to do Simplifications because if you don't do any simplification because you don't have just a single state You have many states and you have to simplify things to come back again to this bright wigner Which we have seen is is present everywhere in nature So the price it bright weakness single level approximation So you do the whole matrix thing, but you say now in the end there's only one excited state one level and Of course, this is never you have many of them But maybe they are so far away one from each other that they do not Interact there's no interference, so you can approximate that with this single level approximation It's an approximation and when you do that the whole our matrix expression for a cross section becomes like this And then you Start to put in things that you know for example the channel is a neutron because we have an incoming neutron And we say we have just a few outgoing channels like Capture scattering fission so scattering is always there fission sometimes Capture very often and you do some other approximations zero orbital momentum We say this cosine of this phi function. This is one of the function I just showed here. It's like this and if you have zero then it is Where's the phi? It is zero we say the cosine is is one and the sinus is wrong So all kinds of approximations and when you do that You find this expression for the total cross section and that total cross section expression You have seen probably everywhere where you recognize the bright weakening expression minus zero to the square you have a kind of width to the square divided by four and you have here Something depending on the reaction that you look at you see here the different widths capture gamma and gamma gamma Elastic so you have a gamma n to the square and fission you have a gamma n to gamma fission and you have also This interference term which appears here that we saw in the beginning for the iron way suddenly you have a huge dip in the Total cross section. It's because of this interference term and that makes the total cross section going down and Causing this problem so The full by our matrix if you do enough Simplifications you come back to the bright weakness single level approximation of course there are other Approximations and one of the most Popular ones let's say because it works very well with light and heavy nucleus is the reach more approximation in which Approximation you eliminate all the photon channels use you lump them together So that is something that you may see in evaluated data files which more is Is the most workable let's say single level is the most understandable which more is the most workable Approximation so how how do we do that? So if we measure a reaction yield, it's like that So we have an isolated bright weakness resonance. It is just a decaying quantum state lifetime is related to the width by this Expression But in reality in the beginning it's already told you there is some Doppler broadening because the if you have some any material which is in in Structure for example a piece of metal etc It is moving with a thermal movement and when the neutron comes in it sees this moving nucleus and that is What we call Doppler broadening and a Doppler broadened cross-section is already something different than Unbroadened cross-section which is at zero degrees. So you get this this is the Doppler broadened cross-section is what we usually get from Evaluated data libraries because we want to compare measurements with that But if we do a measurement we get other problems. We get something which is called Resolution broadening because you you measure with a finite resolution. You can have the effect that The resonance is even shifted you see that so the peak is not observed as where you would like to have it and In a for real measurement you get of course real data and real data are always noisy with lots of things So what we measure is this orange thing. We have to do some Calculation to extract the parameters describing the blue thing and that is the basis that goes into evaluated libraries Then of course this works very nicely for for result resonances So you have but what you measure is not really a cross-section is never a cross-section In fact, it is something that is a reaction yield any capture fission reaction You never measure the cross-section you measure a yield and the yield is what you have to analyze and if you do that properly you have a kind of Reaction yield yield is just a number between zero and one It is the number of neutrons that do a reaction and the others are not counting. So either it is all of them So you have the full fraction. It's one or it's zero nothing So it's somewhere in between and the other thing is not a reaction yield It is a transmission and that gives you a direct access to the total cross-section So you have two types of reactions either transmission total cross-section or any partial reaction which you can in the resident which you can describe like this And of course they are related to the residence parameters like that Now this works well for isolated resonances result resonances as we call them and Of course, then you want to go higher in energy and you will see that the resonances start to overlap So you don't you cannot get data for isolated resonances. You have something average So what you see now what you observe is an average yield and you have to average the equation I just showed here you have to average it and Okay that you can do you get also the transmission you can average it But it is not the same thing an average yield is not Directly related to an average cross-section. There is a little bit more complicated There is a kind of factor that comes in which may be negligible And then it's one or not negligible, but you have to estimate it especially for capture reactions This is really not negligible a transmission the same thing you have also a kind of factor So the average transmission is not relate is related to the average Total cross-section, but not in a straightforward way So you have to do that And what happens if you are in these resolved resonances where you have for every Nucleus you have an energy a spin parity for example you have a width you have a neutral width For unresolved parameters you have something that is average, and then we don't talk about About particular energy with spin and parity But we talk about a level density or level spacing for a given Spin particle family spin parity family or orbital momentum family. So that is we have different quantities instead of The gamma width we have an average width instead of a Newton width We have an average Newton width and for this we invented something else Also the Newton's strength function, which is also Implying the level spacing. So these quantities are important in unresolved resonance regions And just to show you a little Picture how you can if you have here a few resonance So don't mention that don't look at cross-section absolute values of energies. You just have a few Resonances given here in gray And once you sum them up you you you find this red curve So the red curve is the is the observation and you want to extract everything that Is composing this red curve, which are the individual resonances here given in gray So in this case, it's easy. They are relatively well separated So with a nice fitting program you can get all the information of the single resonances But if you go If you have much more resonances like here and With different widths, etc. Then again, they sum up like this Now it's becoming much more complicated to get every single so you can imagine here with a lot of Well, not so money. It's not so much fantasy, but you can imagine here peak positions But you see there are many of them and maybe some of them you never never see they are there, but you don't see them Maybe the big ones you see them because they pop out, but if you sum up all these gray ones I think it's rather difficult to get from this red curve all the positions and strengths of all these resonances So difficult If you go up even higher in energy, you get something which is like this smooth resonance curve and The composing points are here. So I put here 2000 resonances on this So you see something that is fluctuating, but it comes really from 2000 levels contributing to this red curve So here impossible to get something of that. So you need to describe that in average terms And of course, there is you have this you have only resolved resonances easy resolve parameters Here easy you can get these Average quantities which describe the cross section and then you have some intermediate structure where you see a lot of things And then you have to invent maybe some resonances which are not there But they are you know some average properties and you can get the same structure here Not with real physical Resonances, but at least with something that is reproducing this and that is important in many Applications for example in react physics where if you have a lot of material You have important phenomena like self shielding effects and self shielding you need to have this resonance structure So you need to invent resonances and that is why there's a special section in evaluated data where you have this Unresolved resonance parameters. So which are not the average things, but it's really just invented resonances But they are nicely flat. So you know that it is not something real, but they have been invented But it's working and that is what is necessary So these average cross sections. So it's the same thing as the as the as the resolved resonances. So again, these are matrix thing Um, so you see here again the collision Matrix, but the thing is if you start to average. So you put a bar the bar means an energy average Then then you get uh expressions like this So you have a bar that goes over the whole thing, but these are just constants So the average of a constant is the same constant But the average of one minus this Ucc is a little bit more tricky because you cannot Um Average that out like this and if you expand that you get terms like this You have this average matrix Squared and then average Or the average matrix element squared, which is not the same thing and that is the whole difficulty because From I told you about optical mono calculation where you in Well where you have some nice potential so you can get this UCC And from that you can calculate all the cross sections But from this model what you can calculate is only the average UCC, but not the real UCC Squared and then average that doesn't come out of these calculations. So this does this doesn't so that means you can with optical model calculations you can Uh only access cross sections which have this term in it, but not the ones with this one and There are a few of them that is the compound Nucleus formation cross section something you can calculate, but unfortunately you cannot measure you have the Elastic scattering compound elastic scattering That you can calculate with that, but unfortunately it is always mixed in the measurement with potential scattering. So again, you cannot Sorry, it's the universal the potential in the So you cannot calculate elastic scattering directly from these optical model calculations. Luckily, there is the total cross section Which doesn't contain this term and that one you can really calculate using Um average Using optical model calculation and then what is done? You calculate this transmission coefficient is called in optical model calculation. That is the terminology that is used there You get the different Different physical models you need to do to go to the different reaction Uh The decays decay of these excited nucleus and then you get other Transmission coefficients is called again like that that come in there and then if you do an average You have something that is left here. That is what we call the width fluctuations And that is exactly these fluctuations that we saw here before That are still there because they are fluctuating But that comes just from the fact that you have too many levels that you have to So there are all kinds of models fast All kinds of models that calculate that but you can If you do this with fluctuation if you do this averaging you get out this one and that is also something that There has been progress on that the latest years. So that is a very good thing So now when we talk about optical model calculations for for partial cross sections at high energy It is always a combination of the let's say the real optical model calculation with which you can calculate the total cross-section and optical and physical models that model the decay of this nucleus So this picture I showed in the beginning. So you had here the uranium 235 plus neutron And you see here the different cross sections. So we see now all the peaks. We know now it's related to excited states Even this is at One electron volt. It means the excitation energy is a few mev So every resonance correspond to a real state at The few mev above the ground state And together once we know these resonance parameters, which is the energy position and the In the half lives of the different decay cells or widths that as we call them We can calculate and reproduce all these curves exactly like that. We can Calculate even the Doppler broadest cross section when we broaden them afterwards with some Gaussian broadening. So that is all straightforward If we go to a higher energy here, let's say the same nucleus So we have again uranium 235 plus neutron, but now we are A little bit higher in energy and you see here you start to have lots of peaks everywhere And you still can imagine that there there are still Excited levels here that there are resonances, but it's not so sure that you get all of them And at some point this is coming from an evaluated data library It is not nature that that goes so abruptly from something completely smooth to To to to something with a lot of peaks, but at some point you have to draw a limit and say, okay Now I can do not cannot do anything more. I do some averaging and you get average cross sections that look like this But a real measurement, of course Still looks like this in this region so Don't be worried if you see that because this is nature and this is an approximation of averages But you see here that is exactly the same problem. You have so many levels here that you have to do something else Maybe we still have a few minutes to go to something else which is level densities because level densities is something that is exactly the spacing between these these levels and With with all these nuclear models that that try to calculate cross sections nuclear level densities are a major ingredient to to calculate that so this is something you need to know but Of course, there are data compiles, but if we have fresh data, this will help really to to to to To to consolidate these databases with nuclear level densities In general what we have here is we we have a spin Of this excited state, which is just a combination of the particles that go in that was a neutron which has been half We have the target nucleus Which has a spin i here and of course there is a kind of orbital momentum because this neutron can bring in some orbital momentum Which adds up and so the final spin is this and the final parity because Every state has also a parity is Is it's it's plus one or minus one and that depends really on the initial parity and orbital momentum that has come in And then we have what we call partial waves That is the amount of orbital momentum at low energy is zero, of course, but if you go higher up in energy you can have Orbital momentum one two and three that become more and more probable and less probable if you go higher up in energy So this is what we call s waves p waves d waves s waves this strange naming comes from old spectroscopy Habits so we this was kept in in the neutron physics But okay, we all know at low energy you see a lot of s waves And if you go up in energy you have even more p waves for lighter nuclear It starts immediately with the p waves or because there are no resonances at the low enough energy So this is something you have to keep in mind And this picture here of the excitation energy and all these levels that gives us the number of levels So you can count the levels here Uh at this neutral energy and and that gives you exactly the newton at the neutron binding energy Which means just above because there you see the resonances you can count them and you can know how many levels there are per mev And at the very low any excitation energy you can Count the levels again by means of other Experiments by spectroscopy you see the transition So you can count count the levels again And then the level the density model should work over a large energy range But you have only let's say two points to calibrate the curve that is at the newton binding energy Maybe you have more but uh all these points are very close together And also at low energy very close together on this highly On this high range of excitation energies So that means you have two points and and you have to fit a model to that so There's a lot of Liberty there. Let's say of course. There are Other ways that constrain these models. There are theoretical predictions that Constrain these models as well But the idea is at this region you have to count the levels you have to select the j pis So the really rich type of states you excite and then you can extract the spacing for example for s waves Which is d zero you have also a d one which is the spacing for p waves, which is different And all level density models because there are many different models everybody Can make his own favorite model, but there are just two constraints They have to go to this point and through this point and any of these models of course Reproduce that the problems become at higher energies where there's no constraints. So models diverge um So this picture of the cross section giving us the The resonances also gives us an idea of the level density because every of these peaks as you know As I told many times now is corresponding to an excited state. So counting these levels Gives you access to the level density. So this is for one particular Heavy nucleus, but not so heavy gold natural gold. So it goes from this If you go to In this region where you can see the resonances you can count the levels and you get the level density if you go to Other nuclei you see that this whole range of resonances is shifting For the heavy nuclei it goes to the right for the lower energy For the lower mass nuclei it goes To the right and for the heavy nuclei it goes to the left which means at lower energy. So higher level density This is a logarithmic scale. So As you increase in mass You increase the level density except for some nuclei like lead 208 here because With this picture of shell model Lead 208 is a very special case. It has a double closed shell. So it is extra difficult for a neutron to excite the The target nucleus And and that means that you observe the resonances In a much more difficult way. So you start to see levels only at a very high energy region um, so this level spacing is important And you you have to to do then you can do all kind of tricks once you do the counting You see here the level spacing. So it's again all this this picture of the neutrons and the protons And you see here it goes again from a very High spacing so it's red to if you go to heavy nuclei. It becomes a very low spacing So a high density of levels and you see also if you look a little bit closer This this double magic look nuclei like the lead 208 which is here You see it is a little red point inside all kinds of blue ones. So that is exactly the thing that we just saw And the level spacing well There are all kinds of basics for the level density, but at the end what we see is if you count Then the the number of levels as a function of the neutron energy So you count the number of resonances that you see you get something and with a little bit of imagination You can fit a straight line through that and that is a staircase plot and this is usually used to estimate this Level density, but sometimes there are problems because if you go higher in energy It goes like this and that means you start to miss levels at that time So this is called missing levels doesn't mean the level is not there It means just you don't see the level that is something else. So you can do Corrections for that for missing levels And this is maybe something for next time and there are some things about statistical Statistical model which we keep also for another time maybe I leave it on the slide So if you want to see you can look at that but In the end Further reading which is important If you want to know everything about our matrix you have to read this one Lane and Thomas 1958 Is a huge paper it takes a while to study it, but then you know everything about our matrix And there are another a few other books on on nuclear physics, which are Very good as an introduction to to all these type of aspects There's the crane. I liked it very much and there are a few books more related to to to radiation About measurements and this is very practical So there are a few things there. There are a few sites where where maybe you can find some other information I think it's time to stop here and maybe there are still some questions