 Hello everyone. In the previous lecture of nuclear reactions, we discussed the energetics of nuclear reactions. We discussed the Q value, the threshold for a nuclear reaction, whether it is for a charge-parking reaction or interdenture reaction. So, though the energetics, the feasibility of the reaction was studied in that part. In this lecture, I will be talking about the cross-section for the nuclear reaction, how to determine the probability of a nuclear reaction and what are the different types of reactions that take place when you use neutron and charge particle as subtypes. So, let us talk about the cross-section. The cross-section which we denote as sigma is a measure of the probability of occurrence of a nuclear reaction. So, that is the point to note that how, what is the probability of occurrence of a nuclear reaction which will be represented by sigma. Now, to talk in very simple terms, it is the cross-sectional area that the target nucleus will offer with the projectile. It is a very simplistic way of telling the cross-section. In fact, from that only the cross-section derives its units of one bond is 10 power minus 24 centimeter square. So, the nuclear dimensions are of the order of Fermi's 10 power minus 13 centimeter and so, the R0, the radius constant is 1.4 10 power minus 13 centimeter and the radii of nuclei will be R0 raised to one-third. So, that will be about 10 to the power minus 12 and so on. So, accordingly that nuclear area, the cross-section of a nucleus, if you just cut the cross-section, the circle pi r square will be of the order of 10 to the power minus 24 centimeter square and the unit of cross-section that is why it is called as a bond, when bond is 10 power minus 24 centimeter. So, it is a very simple way of telling the cross-section. There are many more things which will come in actually defining the cross-section of a nuclear reaction as we go along. So, how do we use the nuclear reaction, cross-section in the nuclear reactions when we want to write the rate of a reaction? So, depending upon whether we are doing neutron induced reaction or charge particle induced reactions, the terms that come into the formulation of a nuclear reaction rate will be slightly different which will I try to highlight here. So, the rate of a reaction for neutron induced reaction is called n sigma phi, where n is the number of target atoms per centimeter cube. In fact, actually it is not that way, it is total number of target atoms. n is the total number of target atoms because in case of reactor neutron, suppose you have a reactor neutron, in that target is exposed. So, all the atoms in the target are exposed to neutrons. And so, n sigma phi, n is the number of target atoms, sigma is the cross-section. So, we need the sigma square, the temperature square and phi is the flux. So, how many neutrons are going through a unit area per second, that will define the phi. So, n into sigma into phi flux is the rate of a reaction, that is the rate at which, so if you see here, the units will match atoms into centimeter square into neutrons per centimeter square per second. So, it will be centimeter square into square, it will be atoms per second. So, this many atoms are formed per second, that is the rate at which the reaction takes place. In the case of charge particle induced reaction, the charge particle induced reaction will be having sort of a now, if you have a target, if you do not have a sea of charge particle, but it will have a beam, it will bombard that target. So, the in the target only small area is exposed to the beam. And therefore, the number of target atoms is not defined quote number of atoms, but it is the target atoms per centimeter square, unit area, how many atoms are present in the target. Sigma is the cross-section and now beam, the target, whatever is following of the beam and the target is now not per centimeter square per second, but it is the number of particles falling on the target per second. So, there is a slight change in the units of flux or the particle intensity. In the case of neutron, it was flux, neutrons per centimeter square per second. In the case of part charge particle, it is the particles per second. So, both the units, both the reaction rates ultimately will come over to be atoms per second. You can see here n sigma i will be number of atoms per centimeter square into centimeter square into particle per second. It will be again atoms per second. In both the cases, whatever way the units of target atoms and the flux or the intensity, we always end up with the n sigma phi or n sigma is rate of the reaction. Now, as I mentioned that the cross-sectional area offered by the nucleus target nucleus projectile is a very simplistic way of defining the projection. Essentially, if you see when the projectile hits the target, depending upon the distance from the center of the target nucleus, there will be different angular momentum involved. So, in actually, when you define the nuclear reactions, we have particles carrying certain amount of angular momentum. That angular momentum will be defined in terms of the r into p, where r is the distance, radial distance from the target center. Let us say this is the target center and this is the projectile going. So, this is like the r. So, b is nothing but r sin theta. So, b is called the impact parameter and p is the momentum of the particle. So, r cross p is the angular momentum. r into linear momentum is called the angular momentum and angular momentum is a quantity in nuclear reactions, which dictates the different types of mechanisms of the nuclear reaction. So, p, the distance, the vertical distance from the target center to the wave at which the projectile is coming will call as the impact parameter and this impact parameter essentially is the r into sin theta. So, this is like, if it is here, then this is the r and this is theta. So, this is the r sin theta is the impact, the vertical distance of target center to the distance of the projectile from the center. Now, the angular, the linear momentum can be written in terms of the de Broglie wavelength h cross by lambda cross, where lambda cross the reduced wavelength of the projectile and the angular momentum can be written as lh cross. So, essentially angular momentum in terms of linear momentum can be now written as lh cross equal to bh cross by lambda cross and so, the impact parameter b equal to l into lambda cross. So, this impact parameter b essentially is a classical quantity at what distance from the target center the projectile will hit the target center, where it will hit the target, that will be the impact parameter and l is the how much of the angular momentum projectile will bring in the units of lambda cross. So, essentially the classical you can see the b can take any value and so, the angular momentum can take a continuous values. But then you know that the angular momentum is quantized in quantum systems and so, quantum mechanically certain values of b will correspond to one value of n. So, l can vary from 0, 1, 2 and so on h cross. So, you try to have a correlation between the continuously varying impact parameter and the discrete values of l which I had tried to explain using this. So, this is suppose this is the target nucleus here cross section of the target and if that projectile hits at the center within this radius that is the radius of one lambda cross that we will call l value equal to 0, we call it a S wave particle. If it is within the next one l lambda cross l equal to 1, l equal to 2, l equal to 3 and so. So, depending upon at what distance the projectile is hitting the target dps from the center we can bring in different amount of angular momentum and the one which is at the maximum of the target radius will be the l max. So, let us now try to calculate the cross section for a particular l value. So, it is like the annular area of this annulus. So, from 0 to l equal to 1 we will say l equal to l equal to 1 lambda, 0 lambda to 1 lambda to 2 lambda that is l equal to 1. So, this is what I try to calculate here the angular momentum dependent cross section sigma l. So, for a particular concentric ring the sigma l will be area of the outer circle upon area of the minus one upon area of the inner circle. So, pi l lambda l plus 1 lambda cross square minus pi l lambda cross where l lambda is the impact product. So, you can see here you can take the pi lambda cross out and so you are left with l square plus 1 plus 2l that is l plus 1 square minus l square. So, it will be pi lambda cross square 2l. So, now, you can see the cross section for a particular l wave depends upon l value pi lambda cross square 2l plus 1 and since these charged particles are now they can penetrate the pulmonary barrier. So, there is a transmission coefficient even for the neutron there is a change in potential and hence there is a transmission coefficient factor that is called that is defined in terms of the transmission coefficient. T l is called the transmission coefficients of l particular l value. So, now, this is the actual definition of the cross section lambda sigma l equal to pi lambda cross square 2l plus 1 d l. So, let us now try to see for the total reaction cross section will be the sum of the angular momentum dependent cross section for all l values up to l max. So, l max will be r by lambda cross. So, we can sum the. So, the reaction cross section sigma r will be pi lambda cross square sum over all the 2l plus 1 values 2l plus 1 is for 1 l value. So, sum over sigma of 2l plus 1. Let us forget about the t l value for the for the moment and so, we can write it as summation over 2l and summation over n 0 to l max will be l max plus 1. So, when you say 0 to l max. So, when we sum 1 that many times we become l max plus 1. When you sum l over this one you will get 2 into summation of l. So, now, you can write this as pi lambda cross square summation of l will be n n plus 1 by 2. So, it will be 2 2 is outside l max into l max plus 1 by 2 plus l max plus 1. And so, now you can you can see l max is square plus 2l max plus 1. This whole thing will be l max plus l max square plus 2l max plus 1 that is l max plus 1 whole things square. And so, l max you can write as r by lambda cross because l max equal to r upon lambda cross. So, it will become r pi lambda cross square l by lambda cross plus 1. And so, you can see here that it will become r by r plus lambda cross upon lambda cross lambda cross will cancel with this one. You will have pi r plus lambda cross square. So, sigma r since and is pi r plus lambda cross square by summing over the sigma l values from 0 to l max. Now, let us discuss these cross sections for neutrons and charge particles separately because the cross section varies in a different fashion for neuter particles like neutrons and the charge particles. So, just now we saw sigma was in the thing, but pi r plus lambda cross square. So, for the case of neutrons sigma n can be as well written as pi r plus lambda cross square r plus lambda cross square. So, here the rate of the reaction then will be in terms of n sigma phi where n is the number of target atoms for neutrons, phi is the flux in terms of neutrons per centimeter square. Now, let us go a little bit deep into the how the neutrons are captured by the nucleus. See for a neutron induced reaction which are always exoergic. Exoergic means whenever a neutron is captured by a nucleus energy is released equivalent to the binding energy of neutron in that nucleus. So, neutron plus target nucleus has higher mass compared to the the compound will that will be formed. So, suppose you write this is the energy term. So, this is the mass of the neutron neutron plus target and this is the mass of the nucleus that is formed. For example, if you say neutron plus 59 cobalt then this is 60 cobalt needs a ground state. So, when the neutron is captured by cobalt 60 this much energy is the binding energy that is released and so, this cobalt 60 will be excited with this much energy of the rock 7 to 8 MeV and this excited nucleus then can emit by gamma rate or it can have other channels also. So, neutron induced reactions particularly the neutron capture reactions when a neutron is captured by a nucleus they are always exoergic and the energy released is equivalent to the binding energy of neutron in that nucleus. The cross section for such reactions neutron induced reactions you can see here sigma n is decreasing with the increasing energy of neutron in fact it is called the 1 by V where V is the velocity of neutron. So, neutron induced reaction cross sections the low energy neutrons are having highest cross section I will I will be explaining this very soon and at fact and intermediate energy there are resonances these resonances correspond to the nucleus populated in certain discrete energy states. So, this is like a resonance when the neutron energy is such that the nucleus component population certain energy level then you would get a jump in the cross section when it is somewhere here where there is no level cross section will not be that much high. So, now the high this cross section is falling with the increasing energy of neutron at low energy the neutron wavelengths wavelength of the neutron is lambda cross is much larger than the nuclear radius you know that neutrons are used for neutron diffraction studies where the neutron diffraction means the crystals dimensions of the auto angstroms. So, a low energy neutron thermal neutron will have the wavelength of the order of dimensions of the atoms. So, this is much larger than nuclear dimensions and therefore, you will find that the as the energy of the neutron is increasing wavelength is decreasing and so, the cross section is also decreasing to put it in a very simple way. So, for low energy neutrons in fact, the the now you can see here this is the target nucleus then l equal to 0 because host wavelength itself is more than the dimension of the nucleus and so, you will have mostly 0 wave as wave neutrons. So, process can be written as pi lambda cross square t0 l equal to 0 and as the energy of neutron increases lambda cross the reduced wavelength decreases therefore, the cross section is decreasing the energy of neutron. So, what are the different types of reactions that can occur with the neutrons most important reaction that takes place is called the neutron capture or it is called the radiative capture. Radiative capture means when the neutron is captured neutron is captured by a nucleus that the nucleus that is formed compound nucleus will emit gamma rays. So, that is the radiation gamma radiation and so, neutron capture followed by emission of gamma rays is called radiative capture and this kind of reactions are most common with thermal neutrons. And as you will always see that when you add a neutron to a stable nucleus we are increasing the neutron to proton ratio and therefore, most of the products of n gamma reactions are beta minus active because we are producing a neutron rich isotope. For example, 197 gold n gamma 198 gold and this 198 gold will be beta minus active to 2 decay to 198 mercury. So, with the heavier nuclei like cobalt, gold, uranium, tin, lead the n gamma is the most common reaction, but when it comes to low z target nuclei sulfur, silicon, magnesium then you will find the reactions where charge particles are emitted with LP reaction neutron capture followed by proton emission, neutron capture followed by alpha emission. So, these kinds of reactions are possible with the low z target nuclei because the coulomb barrier for emission of protons and alpha are not very high. Whereas for heavy nuclei, the coulomb barrier for emission of alpha and protons are quite high and so, such heavy nuclei do not undergo n gamma type reaction, but the light nuclei can undergo n gamma type this n alpha and NP type of reactions. When you go to still heavier elements like actinides then the thermal neutrons can induce even fission because fission barriers are low for actinides and so upon capture of a neutron like 235 uranium even thermal neutrons can induce fission in the heavy nuclei. Now, when you see depending upon the energy the type of reaction that take place with neutrons, neutrons are classified into thermal neutrons having energy less than 0.5 electron volt, epithermal neutrons, epithermal means the a region where no the it is slightly higher than thermal and there are resistances in the cross sections as a function of energy of neutron 0.5 electron volt to about KV and then more than 1 KV are called the last neutrons. So, you can see here that thermal neutrons are most reactive and this classification actually has come from cadmium. The cadmium is absorbing all the thermal neutrons. So, if you wrap a sample with the cadmium file all thermal neutrons are captured and then the sample will not see the thermal neutrons. So, then from that the specification has come that if it is less than 0.5 EV then the energy that is called as the thermal neutrons. So, we classify it into two, three zones low energy zone where it is 1 by V law the resonance region and the half neutron energy region. Thus, we explain why cross section 40 Newton interaction follows 1 by V if you recall we said 1 by V law and then the resonances. So, we try to explain using the transmission of neutron through the potential well. The cross section for the neutron total perception is pi after lambda cross square and for low energy neutrons we said that lambda cross is much larger than the radius of the nucleus. So, you can actually set up an equation for transmission of the neutron. Neutron is like a wave and going to hit the nuclear potential of the nucleus. So, at initially the you have a plane wave it is to ikx and once it is captured the energy of neutron is much higher. So, the inside the nucleus is called e raised to i capital Kx. So, there is a wave number of the neutron is changing from small k to capital K as it is transmitted through the potential well and the transmission coefficient is written as transmitted flux upon the incident flux. So, general scattering theory of neutron actually can be used to find out the transmission coefficient and this transmission coefficient can be in fact, if you solve this scattering equation for the S wave neutrons because the for low energy neutrons the angular momentum is 0. It can be written as 4 small k capital K upon small k plus capital K square and since capital K is much larger than small k. So, inside the nucleus neutron energies are almost 20 30 MeV and outside it will be thermal neutron. So, you can neglect the small k inside the nucleus. So, this small k can be neglected the spectrum of capital K and so, it becomes sigma n pi lambda cross square 1 k will cancel. So, small k upon capital K and this small k nothing, but 1 to velocity. So, k is actually proportional to velocity lambda cross square proportional to 1 by V square lambda cross h cross by MeV and so, it becomes net result is 1 by. That explains the sigma neutron varying as 1 by V at low energy neutrons from the transmission coefficient. Then for the resonance region, Breit and Wigner gave a formula for resonances for N gamma reaction which is in terms of pi lambda cross square the spin of the compound nucleus the neutron is half spin 2 N plus 2 S 2 into a neutron spin plus 1 and the spin of the target nucleus 2 I a plus 1 they with width for the neutron decay to gamma decay plus upon resonance energy epsilon 0 minus epsilon minus epsilon 0 these are the energies and the gamma by 2 gamma is nothing, but you know gamma n plus gamma. So, these are the gamma actually they are the if you see the nucleus the certain level have their widths that levels are called gamma for neutron decay and gamma for gamma decay there are the widths of the nuke levels. So, the resonance formula essentially is quite accurately predicts the projections for the new resonances for neutron induced reactions. So, we will not go into details, but I thought it is good to tell them tell you that the resonances also can be explained by Breit Wigner formula and the 1 by V also can be explained by the transmission coefficient problem. Now, let us come to the charge particle induced reaction the scenario is different in the case of charge particles because there is a coulomb well here and so, you try to consider a charge particle coming to a target nucleus. So, there will be a different types of potentials playing it play one in the nuclear potential which is attractive at short distances and there is a coulomb potential which is repulsive all over. So, when the projectile is coming close to the target nucleus it will experience the coulomb repulsion of the target nucleus which starts becoming positive at a longer distance and as it is coming closer it will experience the attractive potential to nucleus potential. So, the sum total of the repulsive and the attractive potential will be this dashed line and this shows a barrier which is called as the fusion barrier. So, for the light charge particle like proton and alpha mostly we can say the coulomb barrier this can be called as a coulomb barrier, but when you take heavy ions into picture then the nuclear potential becomes significant and so, we call it fusion barrier. So, right now we will use this as a coulomb barrier. So, this is the barrier the projectile has to cross. So, now you see at from infinity a charge particle is coming and as it is coming down it is momentum. So, this is the centromass energy momentum is decreasing and at contact you know the momentum may become 0. So, this is what I try to show here at a distance of project approach the momentum is root 2 mu ECM minus BC. At infinity it is 2 mu ECM, but when it is coming close to the target nucleus the coulomb barrier will start getting felt and so, it becomes 2 mu ECM minus BC. You can take that 2 mu ECM square root the power half outside. So, you are left with 1 minus BC upon ECM to the power half. So, this is the momentum at infinity and this is the momentum at contact. The angular momentum at infinity at contact has to be conserved and you take the help of this formalism to calculate l max at contact r into mu 2 mu ECM 1 minus BC upon ECM. So, this is r into p r cross p is the angular momentum. So, I have replaced this momentum linear momentum at contact with l max equal to r into that momentum. And so, sigma becomes pi lambda cross square l max plus 1 square and since for chart particles you will find l max will be quite high. So, you can say pi lambda cross square l max square 1 can be neglected with respect to l max. So, now, pi lambda cross square sigma r equal to l max can be written as r by lambda cross or r p by h cross. This p is nothing, but the momentum at infinity momentum of contact. So, momentum at infinity into the 1 minus BC upon ECM and this is nothing, but the 1 by lambda cross square momentum at infinity. So, this will cancel with this. So, you are left with pi r square 1 minus EC upon ECM. So, the cross section for chart particle induced reaction you can see here with ECM as the energy of the projectile is decreasing it will go up like this. So, this is the BC. Beyond below BC the cross section is 0 and above BC the cross section will increase with increasing energy of the central mass. So, this is what with the cross section for chart particle induced reactions are increasing beyond BC process for neutral reaction decrease at 1 by V then absolute momentum increases. So, the chart particle induced reactions typically know when the projectile is coming it can bombard with target for the compound nucleus and let us try to see what are the exact synergies of the compound nucleus. So, you calculate Q value mass of alpha mass of niobium 93 mass of it is this niobium 97 this will be plus 2.437 MeV and you can calculate the coulomb barrier by 1.4382 z1, z2 1 r0 a raise to 1 third a1, 1 third plus b2, 1 third you can substitute to get 14.05 MeV. So, let us try to calculate what is the minimum energy of the alpha that will be inducing the reaction or the threshold for the coulomb barrier. So, this is the energy available in the central mass it will be into mass ratio 97 upon 93 compound nucleus upon target. So, alpha particle should have 14.65 MeV energy to induce this reaction this is the threshold for coulomb barrier. Let us take a case of 30 MeV alpha particle which is higher than the coulomb barrier. So, in such a case exact energy of the compound nucleus will be ECM plus Q, ECM plus 30 MeV alpha ECM into 93 by 97 plus Q value will be 31.2 MeV exact synergies. So, the compound nucleus 9 to 7 form a technician will be formed with an exact energy of 30 MeV and then this exacted compound nucleus can emit neutrons, protons, gamma rays depending upon the emission barriers and the angular momentum states ability. So, charged particle induced reactions a hot excited compound nucleus can emit then gamma ray, neutron, proton there is a competition between emission of different types of particles. So, now you can have proton induced reactions like oxon 18, PN, chlorine 18, cadmium 111 PN, indium 111. So, these are the reactions used to produce useful isotopes like chlorine 18 or propogeton emitter indium 111 in nuclear medicine zinc 658, P2N, P3N, P3N if there are some of the useful isotopes how they are produced proton in bombardable target and that compound nucleus emitting neutrons. Neutron induced reactions you can have Dp, D alpha, Dn and alpha induced reactions you can have alpha 2N, alpha 3N and alpha 4N types of reactions. So, these are the kinds of reactions that can happen with the charged particles. The charged particles forming a compound nucleus and the compound nucleus can emit neutrons, protons, alpha depending upon the exact energy and the available emission barriers for different particles. So, I will stop here and then discuss next in the next lecture how to determine the cross-sectionist experiment. Thank you.