 So, in this lecture we will see another type of accelerator named the microtron and it is useful for electron acceleration. In previous lecture when we discussed about the cyclotron, we have learnt that in cyclotron if gamma changes much particle cannot be accelerated. So, only in low energy regime heavy particles or protons can be accelerated. In case of electron because of very light element gamma changes rapidly. And in that case electrons cannot be accelerated into the cyclotron. We see how the revolution frequency changes in the case of electron and proton. In case of electron the rest energy is only 0.5. And means even 0.5 may be kinetic energy is comparable to the rest energy. So, particle becomes relativistic very early. And in the case of proton rest energy is 0.58 amm means it is near to 1 g. So, up to 15 amm or 100 amm particles kinetic energy much lesser than the rest energy. So, gamma does not change in case of proton in lower energy region while for electron it changes rapidly. So, revolution frequency which depends on the gamma changes rapidly in the case of the electron. This curve is for electron. So, you can see that revolution frequency changes rapidly if we consider B as a 1 plus 1. So, you can calculate it very easily using the revolution time formula considering gamma also in the formula. While in the case of proton it does not changes. So, proton can be accelerated nicely in the cyclotron while electrons cannot be accelerated because synchronism will not be there. So, how we can accelerate the microtron in circular accelerators or cyclic accelerators. So, we have to change the technique of synchronization we are using in the cyclotron. So, how we can change the synchronization technique? We instead of making synchronization with revolution time of the orbit actually synchronize the change in revolution time of the orbit. Means what is the delta T if particle goes from nth orbit to n plus 1th orbit and make the synchronization of RF frequency with their delta T. Then it is possible. Means we calculate the delta T that is the revolution time in the n plus 1th orbit minus revolution time in the nth orbit and make it synchronize with the RF. So, this delta T revolution is BTRF B is some integer B is some integer synchronization. However, for the first orbit what we will do? So, first orbit we will make synchronization separately we will see it. So, first of all because we are talking about the delta E how much energy changes in terms of gamma means delta gamma it is T because E is actually gamma P0. So, when we talk about the delta gamma we are talking about the delta. So, we obtain our formulation of gamma MV square is going to be in the terms of energy. So, V by R this is this will give you the revolution time we will cancel out here. So, V by R will be QB by gamma here now we will not consider gamma as a constant rather than we will take changing gamma in account. So, T revolution will be 2 pi R by V and this will be 2 pi gamma QB this is the same formula we obtained for the case of cyclotron. Only thing was that we were considering that gamma does not change much now because we want to convert this formula in the form of energy. So, we will do it this 2 pi gamma m C square upon QB C square. So, if we multiply numerator and numerator both by C square this gamma m C square will be the energy. So, 2 pi upon QB C square into L energy of the particle. So, T revolution is directly proportional to the energy of the particle. Remember it here E is the total energy not the kinetic energy gamma m C square gives you the total energy. Now, there is an exercise for you that you know that E square is equal to C square B square plus m 0 square C square. Obtain the E is equal to gamma m C square using this definition this is an exercise for you. So, in the first orbit what will be the T revolution? Here in the first orbit T revolution will be 2 pi Q by C square E 0 E 0 is the rest energy plus some energy. Delta E is the energy gain when the particle passes through the RF cavity and there is some fraction of F. What does it mean? We will see in a moment how I consider that there is a factor F and it comes into the first orbit. Because it depends whether we are putting our cathode for electron as an electron source. So, position of the cathode inside the cavity dictates what should be the value of F. So, this will appear only in the first term. So, now this first orbit T revolution time should also be synchronized. So, 2 pi by Q B C square is not plus at delta E which we have calculated here in this equation. It should be ATR, A is again some integer. So, we now we have two integers for synchronization. One is A which takes care for the first orbit and another is B which takes care of all consecutive orbits. Now, change in revolution time we have seen that when T is equal to 2 pi by Q B C square E. That means delta T will be 2 pi by Q B C square delta E and this is a different way. So, delta T revolution means change in revolution time in consecutive orbits. Say nth orbit and this is n plus 1th orbit. So, what is the change in revolution time for consecutive orbits? This is given by this form. Again delta E is the energy gain in the RF cavity when particle passes through the RF cavity. Now, we know that this 2 pi Q B C square delta E should be B T RF. So, for maintaining the synchronization we now calculate delta E is equal to Q B square by B T RF and T RF from the first orbit synchronization comes out this. So, put the value of T RF here. So, you will get delta E is equal to Q B square E, this is here, this quantity which is written here and at the value of T RF we have put this value here. So, this is the equation. So, this 2 cancels out and now you have a nice relation between the integers. Means we have 2 integers first for making the synchronization for the first orbit and another B for synchronization of the change in revolution time in consecutive orbits. So, this relation gives you what are the relation between A and B. So, now you can get that change in energy per turn or when the particle passes through the RF cavity is given by B E 0 by A minus B. This change must occur then only B T RF can be synchronized with change in revolution time. And using now this delta E in this expression. So, you will get the required magnetic field of the microtron. So, delta E also depends on A minus B F and B and magnetic field also depends on A minus B F. Now, if magnetic field is higher we can make the microtron compact. So, for making the compact microtron A minus B should be small. Now we are saying that A is also integer B is also integer. So, smallest difference is 1. So, the minimum possible value is 1. So, if we take A is equal to 2 B should be 1. If A is equal to 3 B should be 2. So, A minus B will be. For lowering the RF voltage we see again the previous slide. Because we want lowering the RF voltage means minimum RF voltage should be there to accelerate the particle. It means B should be lower and minimum value of B is 1 and because A minus B should be 1. So, A will be 2. So, values of B is 1 and A is 2. If we choose these values we can calculate what should be the magnetic field for making such microtron and what should be the energy gain. So, for this energy gain we have to apply to RF electric field and this magnetic field has to be generated. If these are guaranteed means this energy gain and this magnetic field is there microtron will operate successfully. And we can accelerate the charged particle in every microtron. So far we have considered that even gamma changes we can accelerate the particle. Means we can accelerate electrons easily in this microtron. What is the, can we accelerate heavy ions of protons in this? No, we cannot accelerate. Why? When we have considered a very general case whether the gamma changes or not changes all things can be posted here. Then why heavy ions of protons cannot be accelerated? The thing is here, the limitation lies here. Now you see that delta E is equal to E0 means energy gain per turn should be equal to the rest energy of the particle. Means voltage which you are applying and giving the energy boost per turn to the particle should be in the order of rest energy of the particle. Now protons rest energy in one GBV. So for boosting per turn to the protons by one GBV needs one giga volt. Means this is not a practical value for particle accelerations. So practically we cannot make microtron for heavy ions and protons. So cyclotron was working for the protons and heavy ions and it was not suitable for accelerating the electrons. While microtron is suitable for accelerating the electrons it cannot work for heavy ions of protons. So sometimes therefore microtron is also called as electron cyclotron. So we have electron cyclotron in terms of microtron and cyclotron. So electron cyclotron accelerate electrons and rest of the particles can be accelerated through cyclotrons. Now we see the orbit, how orbit looks like in the microtron. So suppose magnetic field is coming outside the screen in the perpendicular direction here and here is the RF light. So whenever particle will cross this RF cavity energy change will occur. So this is the first orbit particle is doing like this and when it passes through the RF cavity it gets an energy. After energy because this is a constant magnetic field the radius of curvature for the orbit will increase. So next time particle will go on a larger circular path. This is the larger circular path. And after reaching again in the RF cavity its energy increases. So in the next consecutive term it will take even larger circular path. So in this fashion the radius of curvature increases for the consecutive parts. Now here you can see that only one acceleration is there in one complete cycle, in one complete cycle. In the case of cyclotron there were two times the acceleration in one complete circle because D was there only after half circle there was an acceleration. In case of microtron this happens only once in a complete cycle. So all the circular orbits are having a common tangent area and at that common tangent RF cavity is there. Now for extracting the beam from microtron again we have the extraction channel where we shield the magnetic field. If we are able to shield the magnetic field when the particle will come inside this channel there will be no force and it will go straight forward. And it can be extracted from the microtron. Now we have obtained our formulation and one parameter was there in our formulation that was F. And at that time we said that the position of cathode inside the cavity indicates what is the value of F. Now we see how. Suppose this is a cavity and this gap is for particle traversing through the RF cavity. Now we get cathode here. Again consider magnetic field is perpendicular to the screen and this cathode is mainly LAB6. We use this. Now after ejecting the electrons from this cathode they feel a magnetic field which is applying perpendicular to this screen. So they take a circular path here. So even before the first term particle passes through the cavity's electric field. So it gets some energy. Now an electric field of the cavity imparts delta E when particle passes through the cavity. Delta E is the energy gain for the particles. So we can say before the first term some fraction of delta E has been obtained by the particle. That's why F delta E was used in that expression for the energy in the first term. Now this is the first term. So before arriving the first term particle already has some energy through the RF cavity. And these are the subsequent terms. This type of cavity is known as type 1 cavity and you can see easily that before first term the particle has arrived almost at the same energies of the first term. So F is almost one. The another possibility is here. Here you can see that again the cavity is of same structure like this with some additional hole here. Why this additional hole is here? Because now cathode is put on this surface rather than the above surface. So cathode is now put in the lower surface of the cavity in this figure. Again magnetic field is in the perpendicular direction to this screen. So when electrons are ejected it takes this path. So here you can see that particle is accelerated up to some extent by the electric field of the RF cavity because it is inside the cavity. And it passes through this hole and again comes here. And again it takes complete delta E energy from the cavity. And then this is the first term. So before arriving up to the first term the particle takes once time completely the delta E from the RF cavity and some fraction of delta E before reaching to this hole. Means in this case F will be larger than however F will be lower than 2. And these are the consecutive orbits. This is the first output. So now we have this is type 2 cavity so F may have values between 1 and 2. So position of cathode dictates what is the value of F and that will have to put in that formulation which we have. So now we see that how delta E energy change takes place in the RF cavity. Means a synchronous particle always comes on the synchronous phase. But what about the demated particles? How they are kept in the revolution orbits? So in microtron we have T is equal to 2 pi gamma M upon Q. So now you can see that as gamma increases T revolution increases. As gamma increases means high energy and T revolution also increases for that high energy particle. Means it takes longer time to complete a path close path than the lower energy particle. Means higher energy particle seems to be slower. Now consider that synchronous phase is this. We say that this is the synchronous particle. So synchronous particle revolution time is integer multiple of the RF time period. So always after revolution it will reach on the same phase. So for an example consider that integer to be 1. So after one RF cycle this synchronous particle will reach on the same phase here. And again after one turn it will reach on the next cycle also on the same phase. Now consider two more particles which are having initially the same energy of the synchronous one but demated in phase. So say one particle is here 1 and second particle is here 2. Now particle 1 it is at higher potential in the RF field means it takes more energy from the RF field because it sees higher electric field. So its energy becomes higher than the synchronous one. And now as higher energy particle takes longer time to travel. So it will come later than the synchronous revolution type. Means if had it been a synchronous particle it would come here. But because it will take a longer time to reach it will come somewhere here after some time. And similarly for the particle 2 we can see that this will take lower energy or energy gain will be lower compared to synchronous one. So it will move faster and it will come earlier. So instead of this it will come on this phase. And in next cycle again the particle 1 is taking higher energy. So again it will take longer time to move so it will come here. And this particle again it is at lower energy than the synchronous one. So this will be faster and it will come. So all the deviated particles are coming closer to synchronous phase. It is same as we do in focusing. A deviated light ray comes towards the optic axis when we focus it. Similarly the deviated particle in phase come closer to synchronous phase. Means it is a kind of focusing. This is known as phase focusing. And this principle enables the construction of microtron and synchrotron. In synchrotron also this kind of focusing takes place. However in cyclotron where the revolution time is not a function of energy. Revolution time is not a function of energy. Here every particle takes same time in the revolution. So in cyclotron there is no phase focusing. Now suppose in cyclotron if this particle was here again it will come here. So in next cycle again it will take more energy. And here it will take again more energy. So energy spread of the beam will increase in the cyclotron. While in the case of microtron you can see that this phase focusing is taking place. Now we see another part of this RF cycle. Suppose instead of taking synchronous phase here we can take synchronous phase here also. Because at this level electric field in the cavity is same. So energy gain of the synchronous particle will be same either here or here. So in last example we took synchronous phase here. Now we take synchronous phase here. Because this is the synchronous phase it will come again on this phase after one cycle. And again after one cycle it will come. So this is the synchronous particle which is coming on the same phase in each time. Now again take deviated in the phase particle 1 and particle 2. Initially consider they have same energy. Now because this is that higher field it will take more energy from the field. Means energy gain will be higher than the synchronous one. Now higher energy particle will take a longer time to evolve. So it will come later here. And on next turn it will come even more later. And similarly this particle which has lower energy gain than the synchronous one it will come on the same here. And in the next cycle it will come here. So in this case these particles are going away from the synchronous phase. So when we take synchronous phase on the falling edge of the RF cycle phase focusing occurs. And when we take synchronous phase on the rising part of the RF cycle defocusing type action occurs and there is no phase focusing. So stable particle motion is possible only in the falling phase of the falling edge of the RF field. Now you can say that when we take our integers values of A and B as 1 and 2 then we get approximately 20 degrees span. To be very exact it is nearly 70 degrees span of the phase where stable motion takes place. So we get a very small duration of the RF cycle where stable motion takes place. And because of such narrow stable region from microtron we get a very beam with very narrow energy spread. So beam quality is much better in terms of energy spread from the microtron. In synchrotron also this kind of phase focusing occurs. However in synchrotrons both possibilities are there means higher energetic particle can take longer time as well as in some configuration higher energetic particle can take shorter time also. So depending on the configuration of the synchrotron rising edge of the RF phase may also be stable or falling edge of the RF cycle may also be stable. So it depends on what kind of configuration we are having in the synchrotron. When we consider the basic longitudinal dynamics of the synchrotron this picture will become more clear. Now how the microtron looks like. You can see and inside that RF cavity there is a cathode by which we get the electron. And here in the perpendicular direction to this there is a magnetic field. This is the extraction channel by which electrons can be checked it out from the microtron. And these are the orbits in this plane as energy increases orbit radius increases. And when it reaches to the desired energy level it exits from this extraction channel. Now this is the photograph of microtron situated in Raja Ranma Centre for Advanced Technology at Indore. I can see that microtron is kept vertical. This gray part is one of the poles of the magnet. The other pole is on that side which is not seen from here. Here you can see that here is the RF cavity which is inside this structure of the magnets. And you can see that there is one thing this is known as klystone. It is the RF amplifier so it feeds RF power through these wave waves to cavity. So this is the RF source you can see and this RF source feeds the RF energy to the cavity from here and this RF energy gives the energy to the particles or to the electrons. Electron revolves like this, like this, like this and here is the extraction channel inside this microtron and ejected particle goes through this line. And this line also contains various magnets to control the charged particles. So this has certain charged particle optics here in front of this. This microtron raises the energy up to 20 amperes for the case of electrons. Actually microtron raises energy up to 20 or 30 amperes. And this microtron is used to send the beam in bigger accelerators of the Raja Ramayana centre for atmosphere. So this we can say is a pre-injector accelerator for the bigger accelerators. One such type of microtron by RRCAD or Raja Ramayana centre has been installed in the Maglaw University also. Now few remarks about this microtron. This is a very simple machine. You can see that very few components are to be operated. One is the RFKFT, other one is the magneton. There are practically certain coils for correcting the orbits. So very few components has to be operated. Can reach up to 30, 40 amperes in the case of classical microtron. Why I am using the word classical microtron? We will see that there is another configuration known as wrist-track microtron. So we will see later that what is that wrist-track microtron. 100 microampere of beam current can be achieved as a beam. This current is much higher than we can obtain from the beta-tron. That is why beta-tron begins observance. And at the place of beta-tron, now we have microtron as the electron accelerator. Means compact accelerator is the microtron for electron acceleration. Now transfer focus is done by the RFKFT also. Also known as electron cyclotron. Vladimir Wexler, which gives the phase focusing principle also gave the idea of the microtron. Now what is the wrist-track microtron? Instead of an RFKFT, there is a complete linear accelerator. So particle gets much, much higher energies in one time. And then after reaching in this magnet, this sends the particle on this orbit. And again reaching on this magnet particle will be sent to this magnet. So in this fashion, we have recirculating the electrons in the limit. Now transfers focusing because this is a long drift space. So beam can be defocused in this drift space. The rate should be some mechanism of transfers focusing in this wrist-track microtron. So transfer focusing, you can put focusing lenses for the charged particle optics. You might have studied this in course of linear accelerator that quadruple can be put for charged particle focusing. Here we can put or here we can put. So when these magnets can take care of the focusing for charged particles. Or you can keep quadruples here on each trajectory like this. Here all the energies passes through these optics. Means these optics works for all the energies. So design is a bit complicated and beam will not be focused so tightly using this kind of optics. If we are using focusing optics in each trajectory, then well-focused beam can be obtained through the wrist-track microtron. However the number of optical elements is much higher than this simple optics. So in this case, wrist-track microtron we can reach up to 70-80 mm easily. Now again the references are seen for this course as earlier. And there is a book as Kapisabha V-minim dedicated to microtron. So students can see the details of microtron in this text. In next lecture, now we will go towards the quantification of parameters. So we will derive the equation of motion of charged particle in the electromagnetic field.