 Now revolution time, it should be synchronized. So, how we can calculate this revolution time? Definitely revolution time will depend on the length of the orbit divided by the speed. So, different arrangement of dipole magnets or magnets decides the length of the orbit. So, length of the orbit in the cyclic path this is L divided by V this gives the T-revolution. So, this L is decided by how we are arranging magnets and this V depends on the particle's energy. So, revolution time because it depends on the speed of the particle. So, we have to see very careful the speed of the particle. Now, in case of accelerators, we can have particle acceleration up to only few keV in case of some industrial application. And particle accelerators can also have energies in the range of TeV, Deraelectron volts in the case of colliders. So, a wide range of energies are there. And suppose if energy is much, much higher than the Rasmus energy of the particle, the speed of that particle may be close to the speed of light. So, in case of particle accelerators, we have particles with very low speed to very high speed means even high speed means speed closer to the speed of light. So, when we are talking about the speed, we must be very careful. We have to calculate all these parameters using relativistic mechanics. Now, in relativity, there are two important parameters. One is the gamma. This is a ratio. Gamma is the ratio of total energy of the particle with the rest energy of the particle. So, in case of electrons, rest energy is 0.5 mV approximately. And in the case of protons, the rest energy is nearly 938 mV, which we can take 1 gV for rough calculations. So, total energy of the particle, total energy means kinetic energy plus rest mass energy. So, total energy divided by rest mass energy, this ratio is known as gamma. So, lowest energy, lowest value of the gamma possible is 1 because in that case, E k will be 0. Means this is a particle which is not moving. In that case, gamma will be E 0 by E 0 and it will be 1. So, lowest possible value of gamma is 1. So, this value is 1 here. And now, if we are increasing energy means ratio E k by E 0 is increasing. So, gamma will increase linearly. And if we now calculate the beta, beta is basically V by C. This is another important ratio in the relativity. This ratio, this gives the speed of the particle with the speed of light. And beta can be calculated using gamma. That is, beta is equal to square root of gamma square minus 1 upon gamma. So, as gamma increases, initially beta increases, you can see here. So, here as we are again increasing the kinetic energy, so beta increases. However, it cannot surpass the speed of light. So, definitely when it reaches close to the speed of light, it asymptotically reaches towards that. See, means speed saturates. Even though in this region, we are increasing the energy, however, speed doesn't change. And in this region, in this region by changing the energy, speed changes. And we are calculating the t revolution is equal to length by speed. So, in revolution time period calculation, we should be very careful whether we are in this region or this region. So, in this region, because speed is not changing, so revolution time almost depends only on the path length or orbit length. Here, because the speed is changing, so revolution time will depends on the length of the orbit as well as on the speed. So, how we are achieving the synchronization in an accelerator? It depends whether the accelerator is operating in this region or in this region. So, in different regions, mechanism of achieving synchronization may be different. Now, we take some examples. You can take it as a tutorial. Now, here we will compare the result of Newtonian mechanics with the result obtained through relativity, a special theory of relativity. So, we take an example of proton in this tutorial. So now, as we have seen earlier that proton has rest mass energy nearly 1g. So, increasing the kinetic energy up to 10 MeV means 10 MeV is much, much lesser than the rest mass energy of the protons. In this case, we are calculating the speed of proton using Newtonian file. Here, ek is the kinetic energy, ek can be calculated using half MeV square. Remember it, this formulation is normally valid in the Newtonian building. So, using this ek, we can get what is the speed. So, using this kinetic energy, we can get speed using this part. This is the Newtonian formula. And now in the red curve here, speed is plotted using this part. And relativistically, the speed can be calculated using beta c. Beta can be obtained using the gamma. Gamma can be obtained from here. Gamma is equal to this kinetic energy which is given here in the axis plus rest energy divided by rest energy. This will be given with the gamma. And using this gamma, you can calculate the beta. So, speed can be calculated relativistically in this manner. So, this green plot shows the speed calculated using this formula, relativistic formula. So, you will see that up to 10 MeV because 10 MeV is much, much lesser than the rest mass energy of the particle. So, Newtonian formula agrees well with relativistic formulation. Now in this tutorial, consider the case of electrons. In case of electrons, the rest energy is only 0.5 MeV. So, 10 MeV is much larger than the 0.5 MeV. So, we calculate the speed of electrons up to only 1.5 MeV or like that. Up to 1.5 MeV is 3 times the rest mass energy. So, again we calculated the speed using the Newtonian formula, this formula and relativistic formula in the case of electrons. So, now you can see Newtonian formula gives this result. And this result surpasses the speed of it. Because in Newtonian dynamics, there is no upper limit on the speed. And while the relativistic case, we get this result. So, here Newtonian results are completely in disagreement with the relative. Means these are incorrect. So, we have to be very careful when we are calculating the speed for calculation of the revolution time. Now you can see that here in case of proton, speed was increasing up to 10 MeV. And here even at 1.5 MeV, speed is almost constant. So, achieving synchronization mechanism will be different in this case of protons and in this case of electrons. Now we see what it means. What it means means, suppose this is an orbit, particle is revolving around this. And even if the energy is increasing, we are keeping this orbit constant. By changing the magnetic field, we can do this. So, by changing the magnetic field, we are keeping orbit always constant. Means length of the orbit is always constant even though the energy is increasing. Now consider the lower energy region where the speed changes when we pump the energy to the particle. So, because orbit length is constant and speed is increasing, so revolution time will decrease. So, revolution time is plotted here. You can see that it decreases with increasing energy. Now consider the same case with high energy particle. Means again orbit is kept constant and particle's energy is very high. In case of particle's energy is very high, if we are pushing it further or we are giving more energy to the particle, its speed doesn't change, its speed remains constant. And we are also keeping the length of the orbit constant, so revolution time doesn't changes. So, you can see that in this case, even though after this level, the revolution time is almost constant. So, in this case, if we want to make synchronization, the frequency of applied RF is also to be changed. And in this case, RF frequency can be kept constant and in that case, we can achieve this synchronization. So, here constant RF frequency will be required for achieving this synchronization. Here RF frequency should vary with time. So, how we are achieving this synchronization, it greatly depends in which energy region we are working, which particles we are talking about. Means up to 10 MeV, the speed doesn't remain constant in the case of proton, while for the 10 MeV, electrons' speed becomes almost constant. So, now we can calculate energy in moment. In relativistic calculations, we will always use energy, total energy square will be rest energy plus c square p square. So, it is the square of the rest energy, it is the c square p square. Here p is the momentum. So, if you want to calculate the momentum using the energy, we have to use this formulation. Now, it will give you E0 because we know that E total basically gamma E0. So, E0 will be E total by gamma and it is written here. So, E square total will be E square total by gamma square plus c square p square. So, we can take c square p square is equal to gamma square minus 1 upon gamma square e square root. Now, this gamma square minus 1 upon gamma square, it is the beta square, because beta is basically under root gamma square minus 1 upon gamma. And gamma can be written down as 1 upon root 1 minus beta square. So, using this relation, we can say this is the beta square. So, this will be c square p square is equal to beta square e square total. So, momentum can be calculated using this formulation. Now, in case of highly relativistic particle, beta is approximately 1, means the speed of the particle is close to the speed of light, then p is equal to E total by c. So, now again, we take a very simple calculation as a tutorial. So, we take E total of electron as 5G. What is the moment? You can calculate it very easily. p is equal to 5 into 10 raised to 9 eV upon 3 into 10 raised to 8 meter per second. Now, you have to calculate this. You can directly write down in the eV by c units, or if you want to convert this in SI units, then you have to change this eV into Joule. So, 5 into 10 raised to 9 into 1.6 into 10 raised to minus 19, 2 divided by 3 into 10 raised to 8 meter per second. So, this will give you the momentum in SI units. So, now you can see here, in this case, when we have a highly relativistic particle, even means beta is close to 1, then this formulation gives you that p is directly proportional to E. Means in relativity, this is the relativistic particle, beta is approximately 1. So, we have p is directly proportional to p. While in Newtonian case, we have p square by twice m is equal to, means p is directly proportional to root. So, we have p directly proportional to root, in case of Newtonian in highly relativistic particle, we have p directly proportional to root. Now, come to some history of cyclic accelerators. So, first cyclic accelerator was built as cyclotron, and it was built by Lawrence. Actually, when RF linear accelerators were built, in successive RF cavities were placed, and when particle passes through each successive RF cavity, it gets the energy. However, length of these linear accelerators become impractical if we want to reach very high energy. So, at that time, Lawrence provided a solution in which a compact accelerator can be built, and that was the cyclotron. And cyclotron is based on a very simple principle. We will see in the next lecture. Then, beta-tron. We are not talking about beta-tron in this course, because now beta-tron is an absolute accelerator. No one will smell beta-tron. Beta-tron was built in 1935 by Max Stamper. It is an interesting thing here that beta-tron is the only cyclic accelerator where changing magnetic field is used for particle acceleration. A changing magnetic field can produce electromotive force, and that electromotive force can be used for particle acceleration. And because this is a changing magnetic field, so as the particle gets higher energy, magnetic field also gets higher values, so particle remains on the same orbit. So, beta-tron is only that kind of accelerator. We are changing magnetic field to produce the particle acceleration. However, in this course, we will not consider this accelerator. Then, synchrotron. Synchrotron was built in 1944. We will see in case of cyclotron that if we want to reach very high energies, it is not possible through the cyclotron. And size of the magnets also becomes very large in the case of cyclotron. So, in that case, Vladimir Wexler and Derek Memelem, these two persons provided independently theory of synchrotrons and theory of phase focusing, which leads to building of the synchrotron. And using synchrotron, the size of the magnets were reduced drastically. Then for electrons, a very small machine can be built. And this is the microphone. This microphone was built in 1944 by the same person, Vladimir Wexler, who gave the principle of phase focusing. So, after this, there are various improvements in these accelerators, mainly in synchrotron, in an alternate gradient principle, synchrotron. That is the strong focus in the model occurs. And we will see these theories in later lectures. These are the references for this course. Almost all course, all the lectures, except few things can be found in these references. So, these are very elementary, but nicely written books and easily available books. This Elmwood-Weidman is a detailed book, but very simple and written in a very simple way, so it can be grasped very easily. S-Value book is some mathematical book. Then Humphrey's Stanley principle of charge particle acceleration is in a very simple book and can be grasped very easily by a graduated student. There are other resources also available online. So, one of the best resources is proceeding of the Sun Accelerator Schools. There is a vast number of proceedings, ranging from elementary accelerated physics to very topical accelerator courses, means specialized topic courses. So, these are good for beginners as well as for advanced students. And again, there is notes, study material, problems available in U.S. particle accelerator school site. So, students can visit these sites also to get these good materials on the courses. In next lecture, we will see about the cycle.