 Dear students, in the previous lecture, I discussed about the interaction of heavy charge particles with matter and we found that the important properties of the heavy charge particles with regard to their interaction are the stopping power. The stopping power is nothing but the loss of energy in the unit thickness minus dE by dx and we also saw how that is related to the energy and charge of the ion. So, there is a term depending upon the property of the ion, it is another term depending upon property of the absorbed medium that is nz electron density number of atoms per cc into the atomic number of the material. So, it is essentially the electrons per cc. So, heavy charge particles are losing energy relatively much faster because they do not travel much because of the higher mass, higher charge. So, nz square by E as you recall the previous lecture, the energy as the energy decreases, the stopping power increases, but as the black curve signifies at the end of the journey, ion finally the velocity of the ion becomes less than velocity of little electrons and the heavy charge particle picks up electrons and then stops. At the end of this track or the range, the stopping will be purely by the nuclear, but in the beginning it is electronic stopping wherein the energy and excitation are the dominant mode of interaction. Now, we will discuss the interaction of fast electrons with the matter, which also includes the beta particles, beta plus and beta minus. Okay, so the fast electrons or beta particles, the interaction mechanisms are same, even positrons will be at the same way and the typical energies of this fast electrons or beta particles that we handle are in the range of q kv to q mv. Now, this in terms of the type of mechanism for interaction of these fast electrons with the matter, we can say that there are two types of processes that occur. First is the original energy loss, that means the electrons collide with the electrons of the medium through which they are passing and they can cause the ionization and excitation very much like in the case of heavy charge particles. But in addition to this, there is another mode of loss of energy, that is the true radiative energy loss, in which case the continuous radiation called Brin Stalin is limited when the fast electrons are passing through the medium. We will discuss more on this subsequently. The major differences between the interaction of heavy charge particles with fast electrons, let us discuss this first. So, firstly, the velocity of electrons is much, much higher than the velocity of heavy charge particles for the same energy that is q kv to q mv. So, because of that, these velocities are very high, the delta e by dx, the stopping power for fast electrons is much lower than that for the in charge particles. If you recall the stopping power from law, d e by dx is equal to m z square by e. In fact, it seems to not remain the same when it comes to electrons, but the velocity comes in the denominator and so velocity is high, that means the stopping power is low, grossly speaking. The second important point is that the energy transferred by electron to the electron in the medium in one point. So, considering the same equation in case of heavy charge particle, the maximum energy and electron can gain when it is struck by the incident particle maybe alpha or electron given by 4 mm capital M small n e 0 upon m plus m square cos theta. In fact, it is cos theta theta equal to 0, maximum energy is lost to n theta equal to 0. Now, if you substitute for capital M as mass of the electron, so both capital M and small m are same in the case of electron interaction. And so this would become e max equal to 4 m square e 0 upon m plus m is 2 m into square 4 m square. So, it will be e 0. That means the electron can lose all its energy in single collision when it is colliding with the electron. Because of these processes, you will find that the electron can get backscattered, backscattered in all angles in a single collision. So, this is a major difference when it comes to interaction of electrons better, we say which are the heavy charge particles. The third difference is that the electrons move in a tortuous path. The reason for that is the second point that is the electron can lose energy in a single collision. And so because of that, suppose you have an electron here, it is colliding with the electron. This electron can get scattered in different directions. And so because of that, the one the electron is going, it is scattered, it can further get scattered. Whenever it is interacting with an electron, it can go. So, this electron can go in a zigzag path. And so, you can say the different electrons will be moving in the tortuous path like a tortuous in a zigzag fashion. So, then you cannot define a well defined range for the electrons because their motion is tortuous. That is because of the large energy transferred in single collision in case of oscillators. The difference, another major difference is that when the electrons are interacting with the medium, particularly at energies which are of the relativistic order, then there is an emission of a continuous gamma radiation, beam Stalin. That beam Stalin is derived from the word breaking radiation. That means when the electron is accelerated or decelerated in the vicinity of a nucleus because of this electromagnetic interaction, the electron loses energy in the form of photons. That can be explained from electromagnetic theory. And this happens when the electron, the energy of the electron is of the tortuous, relativistic energy, that means beta equal to V by C, V is the velocity of the electron, C is the speed of light. And when it is close to 1, then the electron is relativistic. And so, that happens for the electron when the energy is of the 1 MeV. And therefore, the MeV order of energy electrons will undergo a large fraction of energy will be lost through beam Stalin emission. The same thing does not happen in case of protons of same energy because for 1 MeV proton, the beta value will be of, will be around 0.046, which is much, much less than 1. Therefore, proton is not relativistic at energy of 1 MeV. As a result of that, we do not see beam Stalin in the case of, which are the particles at these notes. I will, it will be possible, we will discuss that later on subsequently. So, let us first discuss the collision energy loss and radiative energy loss with regard to the stopping power. So, we will not go into details of the derivation of this formula because they are quite complicated. But hence, Bithé, in fact, derived this formulae. And the, so here instead of one type of collision, the stopping power, we have two types of losses. One is the loss in a collision with electrons. So, it is called a collision energy loss. And this collision energy loss again has the two terms, the electronic properties, the ion, single ion that is impinging 2 pi e to the power 4 upon m0 v square, where d is the velocity of electron. And mz is the electron density in the medium. And this is term again the velocity dependent is a relativistic term. In the case of beta particles and past electrons, relativistic term becomes important because beta is significant close to 1. But let us not go into too much details of the relativistic term. The, it simply is sufficient to say that the, in the collision energy losses again depend upon the velocity of the electron and the electron density of the medium. In this, in the fashion similar to that of the heavy charge particles. And the radiative energy loss, we have called radiative minus d by dxr is again given in terms of the electron, the atom density in the, so nz is the thing by electron density. E is the energy of the ion e to the power 4 upon 137 m0 square e to the power 4 and some term depending upon e n and 0 c square. So, electron is going to lose energy not only by collision with electrons which leads to emission and cetation, but also by loss in terms of emission of a electromagnetic radiation that is rhythm style. And so the total energy loss now by the electron is some of the collision energy loss plus the radiative energy loss. And for the higher energy electrons and particularly you know it is, if it is going to stop in a I z medium, then the radiative losses become very, very significant. Just to give you the feel of that from the dominant role of radiative loss, the ratio of radiative to collision energy loss. So, this upon this radiative upon collision, collision losses, if you see these two terms, in the simple way it can be written as e z upon 700. So, we will not bother about these terms how it has been arrived, but e is the energy in MeV, z is the atomic number of the absorber upon 700. So, it is a very simplified way of finding out the ratio of radiative to collision energy losses. So, what it tells is that higher is the energy of the ion, higher is the radiative energy loss, higher is the z of the absorbing material, hydrogen materials, higher is the radiative energy loss. So, if you are having a hydrogen material in the vicinity of a beta emitter, then you will have more of radiations. So, that is why you handle beta sources, you have to be very careful about nearby high energy material, high jet materials. So, for beta particles of the order of 1 to 2 MeV, in fact, radiative energy losses are not that high, but if it is a high absorber, high jet material, then the radiative losses become significant for beta particles of this energy domain. So, when you do beta counting, the absorber, the plate on which you make the source becomes significant. You should not make the source on a high jet backing material, you can make on aluminum, not on stainless steel or any other high jet. So, let us now discuss more the Brim Stalin. The Brim Stalin, as I mentioned, it derives on the name of breaking radiation. That means, suppose you apply a break, so when you apply a break on electrons, electron is losing energy. So, how does it lose energy? Say, if it is going in the vicinity of a nucleus, and nucleus has got charged particles like protons, so electron can be accelerated in the vicinity of the nucleus or it can be decelerated, then any accelerating or decelerating particle, charged particle, loses energy by means of emission of electromagnetic radius. So, essentially the electric field of the nucleus, if the electron is accelerating or decelerating, then it will emit a electromagnetic radiation. And this electromagnetic radiation is in fact, in the case of electrons, it is 10 power 6 times more for electrons than that for protons. So, why it is so that we did not have the Brim Stalin type of radiation in the case of protons and alpha particles. So, the explanation lies here, that is the force, the coulombic force between the electron and the nucleus, you can say mass into acceleration, mass m into a. So, the acceleration is f upon m, and the Brim Stalin intensity is proportional to square of acceleration. So, acceleration square, so if you assume that other things are constant, then it is 1 by m square. So, the Brim Stalin intensity is proportional to 1 by m square, where m is the mass of the accelerating ion. So, now based on this you can say the intensity of Brim Stalin for electron divided by electric intensity Brim Stalin for protons can be in terms of inverse ratio of their masses, that is the mass of the proton square upon mass of electron square. And you know the mass of proton to mass of electron is sum of 2000 and square of that is of the drop million times. So, that explains the Brim Stalin intensity for past electrons a million times more than that for the proton. So, as a result of that you will find we do not have Brim Stalin for protons and other particles of energies of few MEVs. But you can see that when the proton becomes relativistic, when the proton energy becomes GeV, Giga electron volt, in fact there are now actually machines which are running at 1000 GeV. So, at 1000 GeV proton becomes relativistic and you will find that the you will have the high energy relativity radiation emitted by protons. So, only we are talking about the low energy at MEV range protons and other particles we do not talk about Brim Stalin. But if you have relativistic protons and other particles they will also emit Brim Stalin. Now, this Brim Stalin I will see the spectrum the energy Brim Stalin is a continuous energy spectrum. So, if you have a electron of energy this much, then maximum energy of Brim Stalin can be energy of the electron, other it is a continuous spectrum and the average energy will be somewhere one third of the maximum energy. So, Brim Stalin gives you a continuous spectrum with average energy one third of the maximum energy of the positive electron. This is in fact a mechanism to produce gamma rays from electrons. These days you can easily have high electron high energy electron accelerators. So, if you want to have a intense source of photons you can use high energy electron sources and bombard them on high jet material like tantalum and you can get or tungsten you can get electrons and the photons of all energy spectrum. So, now, I like to discuss the range of heavy charge particles and what we saw that the heavy charge particles they in the transmission graph what I was showing here is the intensity of the initial electron is i 0 is passing through a thin slice of material and the transmitted intensity is i this is delta x then i upon i 0 that is called the transmitted intensity for fast electrons. If it is a mono energetic electrons like we have one energy electron then it will be just monotonically decreasing the intensity of this because the electrons get lost from the path in the beginning itself unlike in the case of alpha particles for alpha particles so we have like this this is for alpha or protons. For electrons you will find electrons start getting lost from the path from the beginning itself and that is why we cannot define you do not have a well defined range for fast electrons. So, normally you there is no point doing an experiment to determine the range of electrons for mono energetic electrons but for beta particles what happens the beta particles it has been found experimentally that the transmitted transmitted intensity follows an exponential decay that is given by this expression i equal to i 0 raised to minus nx where n is called the beta absorption coefficient and this n the coefficient correlates well with the beta and the point energy what means the end point energy the maximum energy of beta particle. Now, you know that the beta spectrum is a continuous spectrum why it is continuous spectrum we discussed in the beta decay that because of the emission of three part three body interaction. So, you have a electron you have a neutrino and you have the dotted nucleus so the energy the q beta q beta is shared in a different infinite number of ways between among the electron neutrino and heavy residue and therefore this continuous spectrum of beta is responsible for the exponential decay. So, it does not have any phenomenological explanation why beta decay attenuation transmission graph shows an exponential decay. So, what essentially happens is that the low energy part of the beta spectrum gives rise to the initial part of the exponential decay and the later part the high energy component of beta spectrum gives you the later part. So, it is a fictitious type of correlation it does not have a any particular basis based on a particular concept. But it became very handy in the earlier times you know when we did not have the advanced detector systems then you people are determining what you do you take a you take a source of beta and you put coils of different thickness different coils and you are measuring the loss in the intensity as a function of you put keep on putting different coils and what fraction is going out. So, this is experiment you know this is the so many coils you put and you generate this graph. So, i and i0 and what they found this is an experimental finding that the beta decay attenuation intensity is attenuated in a exponential fashion and this attenuation factor absorption coefficient n it has got excellent correlation and so from the n value one could identify the n point energy at that time the the radiochemistry people particularly know we did not have no access to high equipments. So, you like a mass spectrometer now you can determine this with respect to by mass spectrometers but do you want to find out n point energy of beta you just do a simple attenuation experiment like this one and you can find out what is the energy of this beta. So, beta counting in fact is requires that you need to really take care of this attenuation. So, if you have a thick absorber for example you are doing beta counting how do you do beta counting you have a sample you on a plate you put this precipitate and you put it in a you have a detector. So, beta will go from here to here you detect the beta. Now, if this absorber material is having high Z material like lead or stainless steel or tantalum then there will be back scattering that is one back scattering and second is the self absorption. So, this precipitate will be absorbing it will absorb and third may be the radiative losses. So, if it is a high Z material brimstalline also can take place. So, when you do beta counting one has to be very careful not to use high Z backing material for beta. I did not discuss this part in the beginning one of the dominant mode of interaction of harsh electrons because the energy lost by this mode is very small fraction. So, it does not really matter when it comes to loss of energy by electron, but it is a very fascinating topic to discuss and in fact, it is also used in the detection of electrons and this is called the Cherenkov radiation. Some people call it Cherenkov I will say you can use the term Cherenkov. So, what is Cherenkov radiation? So, blue the blue light that is emitted in a react swimming pool type of reactor. So, that is a particular manifestation of the Cherenkov radiation that is imitated and the beta particle is passing through a dielectric. So, what is Cherenkov radiation? When a charged particle passes through a transparent dielectric medium with velocity more than the phase velocity of light in the medium then this charged particle emits electromagnetic radiation which is in the UV visible range. So, the energy of the photons that are coming out is very small. So, a tiny fraction of the electron energy is utilized in Cherenkov radiation, but it is very interesting to see even this one if you put appropriate proton multiplier tube you can detect the electrons energy electron intensity by means of this Cherenkov detector. So, what exactly happens? Why, what is the mechanism of this? So, what happens that first of all the condition for this Cherenkov radiation is that the velocity of electron has to be more than c by n. What is c? c the speed of light which is reversal constant n is the refractive index of the material through which the particle is passing through. Now, if you see the v by c velocity of the electron upon c velocity of light called beta then it becomes beta n has to be more than 1. So, the condition for Cherenkov radiation is beta n is more than 1. How can the beta n be more than 1? The velocity of the ion can be close to c at the most. So, it cannot be more than c. The relativistic ion we just now saw 1 MeV electron has the beta value 0.94, but it is not more than c. So, if you have a medium of high refractive index like glass 1.5 there are certain materials dielectric materials which are transparent and have the refractive index close to 1.5 then c upon 1.5. So, the phase velocity of that light in that medium becomes much less than c and therefore, velocity of that electron becomes higher than that velocity of phase velocity of light in that medium. And as the electron is moving sincerely the concept behind that that electromagnetic field associated with the electron, electron is moving in the direction it has got electromagnetic field associated with that field moves with velocity c by n. And so, the electron is moving much at much higher speed than these velocity of the field associated with it in that medium. And it is like you know the field is left behind there are the electrons. And so, that field is appears because of the field leaving behind the ion it is within a core and this condition is met for a very small time because the ion electron will only lose energy in the medium and that condition will be not met. So, till that condition it met which happens generally in the high energy part of the beta then you will find for that particular time in the initial phase of this which is a journey it will emit that light and that light happens to be in the blue region that is why the Cerenkov radiation is having blue in blue, blue range. So, this is a typical swimming pool reactor U in Trombay and you can see the beautiful blue, blue color in the swimming pool. So, any swimming pool type of reactor if there is a fission has taken place then the fission products are limiting beta particles of high energy and you will see blue light. So, Cerenkov radiation the in terms of energy loss is not much, but it is important in terms of the duty of the radiation and the detection of the beta bisectron. Lastly, I will discuss the interaction of positrons. Positrons are also anti electrons to the in terms of collisional energy loss and radiation energy loss they are similar to electrons they are no difference, but there is several different electrons in positrons that this positrons when they thermalize in the medium the positron comes close to an electron and annihilates with an electron the electron positron annihilate and you do you get two so the rest of the energy of electron positron is 1.022 Nv that energy is now emitted in the form of photons and since the electron and positron as you thermalize the pair momentum is 0 the two 511 kV photons are limited at 180 degrees. So, this is the positron annihilation gives you two 511 kV gamma rays which are limited at 180 degree why at 180 degree because the momentum is 0 when the positron is thermalized its momentum is 0 electron is assumed to be already stationary or very low energy. So, at that time when there a system is annihilating mass is converted into energy in that of two photons the two photons have to have zero momentum and that is possible when it is emitted at 180 degrees. In fact, there is another mechanism of three gamma emission and when the three gamma emission happens then they have to be of at 180 degree 120 degrees each. So, but the three gamma probability is much smaller than that two gamma and in fact very rarely you see three gamma. So, the reason for the two photons emitted 180 degree because of the zero momentum and another very interesting aspect is the positronium atom formation. In fact, the positron and electron can form an atom like hydrogen atom, proton and electron hydrogen atom, positron and electron for positronium atom and the positronium atom has a beautiful chemistry. So, there is a subject of positronium chemistry itself and lastly the positron emission tomography. The fact that the two photons are emitted at 180 degree is utilized in detecting the tumors in the human body by injecting a positron emitter like chlorine 18, carbon 11, oxygen 15 and so on. So, many positron emitters when they are put into the body and they emit positron the positron in a relay give two filaments giving gamma ray and they can be used to detect the position of the tumors in the human body. So, we have lot of applications of this positron I thought I will touch upon this aspect in this particular. So, I will stop here. Thank you very much. Next time I will discuss the gamma ray interaction. Thank you.