 Another application is accelerator driven system. So here you need a high intensity accelerator, high energy and high intensity accelerator typically 1GV proton accelerator. So this proton beam goes and hits on a target and by spellation reaction it produces lots of neutrons. So and then there is a subcritical reactor here. So there is a reactor which is subcritical. So what do you mean by subcritical? Subcritical means that it cannot sustain the chain reaction on its own. Its K effective is less than 1, ok. So if the K effective of a reactor is less than 1, then the chain reaction dies and there will be no more fission in the reactor. So normally reactors are operated at K effective equal to 1. If K effective is greater than 1 then it is like a bomb. So this reactor is subcritical and on its own it cannot sustain chain reaction. In order to sustain for this chain reaction to go on and to sustain the fission reaction additional neutrons are provided by the accelerator. So this proton 1GV proton beam from the accelerator hits a spellation target and lots of neutrons are produced. So the deficit of neutrons in the reactor is provided by this accelerator. So then this reactor can normally be used for electricity production. A part of this goes to the grid and part of it can be used to drive this accelerator. So this type of system has various advantages. It can use thorium as fuel, ok. So in India we have huge amount of thorium reserves, ok. So in a direct reactor you cannot use thorium as fuel because thorium is fertile, it is not fissile. It has to be converted into uranium in order to put it in a reactor. But this type of system it can be used directly. So here it will be converted into the fertile material, can be converted into the fissile material and be used, ok. So greater safety in operation because if anything goes wrong you can simply switch off the accelerator and the reactor on its own is subcritical. So chain reaction will die down. It can also be used for incineration of radioactive waste. So when the reactor operates lot of radioactive waste is generated. So there are long-lived fission products which have lifetimes of the order of millions of years. So which is very huge. So generally these are stored in geological repositories but millions of years is too huge a lifetime. So if you treat it to this type of, in this type of system you can reduce the lifetime to some thousands of years. So it helps that way. And here whatever the power is generated part of it is going for, going to the grid and part of it is going back to the accelerator. So in this way it is a self, the accelerator is self sustained. So that is why it is known as an energy amplifier, ok. So such a system many people are working on it but so far it does not exist in the world. So the main challenges are from the accelerator type because it needs to work in the CW mode. So we learn about what is the meaning of CW mode. So in the lectures, high beam powers are required and the system has to be highly reliable, ok. So since this is going to be used for production of electricity, the system has to be very reliable. It has to work 24 cross 7 for months, ok, unlike the other accelerators which are used for probably doing some experiments and they need not work for such large durations without any fault. So here reliability is very important. So for these reasons no such accelerator work is working in the world, but work is going on for designing and building such an accelerator. Then very quickly let me tell you about some linear accelerators in India. So there is this electron beam processing facility at RRCAT at Indore, Madhya Pradesh. So this is used for all medical applications, agricultural, industrial applications which I have already discussed before. There is a, similarly there is a 10 MeV Linux at the electron beam center in Kharagal which is a VRC center which is used for industrial applications. So the main applications here are radiation food processing, cross linking of polymers, X-ray source for cargo scanning, neutron source for radiography, semiconductor radiation, and curing of adhesives, time and colouring, medical sterilization and food disenfestations. Then there is this lehpa at BRC which is like a front end for the Indian ADS system. So as I said when I was discussing about the accelerator driven system it requires a very high energy accelerator, a 1GV accelerator, a high power accelerator. So you cannot build this in one go, so it is planned to build it in three phases. In the first phase a 20 MeV high current accelerator is being built at BRC. So this consists of an ion source, a radiofrequency quadrupole accelerator and a drift tube lineup. So here is the picture of the lehpa at BRC. So here you can see this is the RFQ. So you learn more about the structures during this course. This is the drift tube lineup. So this is still in commissioning stage. Then there is this radioactive ion beam facility at VCC Kolkata. So here the cyclotron produces, cyclotron produces proton or alpha beam which is hit on a target and radioactive ion beams are produced. So this radioactive ion beams are further accelerated using RFQ and IH DTLs to high energies for doing experiments. So here is a picture of the RIB facility. So this is the RFQ and this is the IH type DT. Okay, they are also used as post accelerators to accelerate beams from the belletron at IOC and TFR. So at IOC in New Delhi and at TFR in Mumbai we have the pelletron 14 and 15 million volt tandem accelerator. So the energy as you know in a DC accelerator is limited. If you want to increase the energy of the beams coming out from the pelletron accelerator. So here post accelerator consisting of quarter wave resonators have been installed. So here is a picture of the cryostat. So these are superconducting accelerators. So using this post accelerators the energy of the beam coming from the pelletron accelerator can be increased. Okay, let us talk about some accelerators in the world. So as I said there is this Linux 4 at CERN which is and the injected to the proton synchrotron booster. So this big ring that you can see here, this is the LHC. So you do not directly accelerate to 7 TV, you do it in stages. So the Linux 4 is a linear accelerator, it is a 160 MEV H minus accelerator. It accelerates to 160 MEV and then the beam is injected into the proton synchrotron booster, then the proton synchrotron. And then this beam is injected into the SPS and finally the LHC. So it is a huge accelerator and you can see a picture of it here. Similarly another example of injector to synchrotron is at the spellation neutron source in the United States. So this is a 1GV proton accelerator. So here is a 1GV proton accelerator, finally injecting into a proton accumulator ring. So this part is a linear accelerator and then again here the protons are hit on a target and by spellation reaction neutrons are produced and these neutrons are now used for doing experiments. So this is not a CW machine, this is a pulsed machine. So it is not suitable for ADS. So here is a picture of the linear accelerator of the SMS. Then there is the European XFL, so this is a huge accelerator consisting of large superconducting cavities and it is there in Germany. Now let us learn about some basic concepts that are required. So as you have seen that here we are dealing with high kinetic, so we are dealing with high velocities and here if you see the kinetic energies they are of the order of rest mass of the particles. So Newtonian mechanics does not apply here, we need to go to relativistic mechanics. So let us quickly revise what is involved in the relativistic mechanics. So here we define two dimensionless parameters beta and gamma, so beta is simply v by c. So here c is the velocity of light and it is equal to 3 into 10 to the power of 8 meters per second, remember as per the relativistic mechanics nothing can move with the velocity greater than the velocity of light. So beta is simply v by c, it is a dimensionless quantity, gamma is defined in terms of beta, it is 1 upon and the root of 1 minus beta square. So again from this you can write gamma in terms of beta. Now in relativistic mechanics the total energy is given as the sum of the rest mass energy and the kinetic energy which is given as gamma into m o c square. So m o c square is the rest mass energy of the particle. So when particles, so what do you mean by rest mass energy, when particles are at rest so that means v is equal to 0 or in other words beta is equal to 0 and gamma is equal to 1. So your kinetic energy is 0, this kinetic energy is 0, you are left with only this part of the energy. So your total energy is then equal to the rest mass energy, e is equal to e0 is equal to m o c square. So gamma can be written as e by e0 which is simply some 1 plus e kinetic by e0 where e0 is equal to the rest mass energy. Now mass of the proton as you know is given by 1.67 into 10 to the power of minus 27 kg. And similarly mass of the electron is 9.1 into 10 to the power of minus 31 kg. So here you can note that mass of the proton is 2000 times the mass of the electron. So electron is a light particle. So you can calculate the rest mass energy in terms of electron volt for both proton and electron and so it is equal to 938.27 for MeV for protons and 0.511 MeV for electrons. So now when the kinetic energy is of the same order then the particles become relativistic and the Newtonian mechanics will not apply. So here in relativistic mechanics the mass also does not remain constant. It is given by gamma times m o where m o is the rest mass. The total energy is given as e is equal to m c square. So this everybody is familiar with it is the famous expression given by Einstein that energy and mass are interchangeable. So energy is basically now here the sum of the rest mass energy and any additional energy that it would have acquired which is known as the kinetic energy. So this is equal to gamma m o c square so gamma m o you can replace by m o so this is m c square. So this mass is no longer constant m o is constant but m is no longer constant. So from here you can write kinetic energy as gamma minus 1 m o c square. Similarly the momentum is given by gamma m o v now v you can write it as beta c so beta gamma m o c and this you can replace in terms of the rest mass energy. So this is the expression for the momentum in terms of rest mass energy and using this you can easily derive this expression for total energy and momentum. So let us compare the various regions where the Newtonian mechanics and the relativistic mechanics apply. So this you are familiar with this formula from Newtonian mechanics kinetic energy is simply half m v square you can write v as beta c so half m beta square c square. So beta from Newtonian mechanics comes out to be in terms of kinetic energy to e k by e 0 under root. Now if I plot this so here is the kinetic energy with beta for the case of Newtonian mechanics so I get this graph and this is the line for v is equal to c. So you know that no particle can move with a velocity greater than the velocity of light. So this is violating the law that no particle can move with a velocity greater than the velocity of light. So now if I calculate from relativistic mechanics so here gamma is given by e by e 0 which is simply 1 plus e k by e 0. So and gamma from in terms of beta I can write it in this expression combining these two expressions the beta for relativistic mechanics in terms of the kinetic energy comes out to be this. So it is very simple to derive this you can do this using these two expressions. So now again if I plot this kinetic the variation of beta with kinetic energy for the relativistic mechanics this is the curve that I get. So we see here that as the energy increases the beta increases and finally it approaches the velocity of light it will never become equal to the velocity of light but it approaches the velocity of light. So this is the more generalized case which always holds true in the lower energy region you can see that both the curves overlap the Newtonian curve and the relativistic curve both overlap. So in this region the Newtonian mechanics apply. So Newtonian mechanics is only valid in this region whereas the relativistic mechanics is a general formula and it is valid everywhere. So typically in the region where your gamma is equal to approximately close to 1 is the region where Newtonian mechanics holds true. So here I have just calculated the values of beta and gamma for proton and electron for different values of kinetic energy. So remember these are kinetic energies these are not the total energies ok. So and I have plotted this graph here. So here you can see that the electron for the electron the even at 5 MeV it has come the beta has become equal to 0.99 so it is very close to the velocity of light. Now if the energy is increasing to 100 MeV 1 GeV there is very minor change in the value of beta ok. So in other words the velocity of the electron has become constant as can be seen in this graph here. So even around 1 MeV the velocity of the electron has become constant approaching the velocity of light. For the proton on the other hand if you see the beta from 50 GeV from 50 GeV to 1 GeV it has still not reached 0.99 or close to the velocity of light ok. So this is because of the difference in the masses of the proton and electron. So the mass of the electron is or the rest mass energy is 0.511 here and here it is 938 MeV ok. So the proton will become relativistic or in other words it will approach the velocity of light at a higher velocity higher kinetic energy because its mass is more. The electron on the other hand being a lighter particle becomes relativistic at relatively lower energies ok. So now what happens here now when you accelerate the particle from let us say 1 MeV to 1 GeV the velocity is not changing much the velocity is almost constant for the electron. What is changing here is the mass because gamma is changing if you look at the values of gamma for the electron there is a huge change in the value of gamma. So here you know that mass is equal to gamma MO. So now as kinetic energy is increasing the velocity is not increasing for the electron or increasing very minor what is increasing is the mass of the electron ok. The same thing will happen with the proton. So in this energy range you do not see much change in the value of gamma for proton. So the same thing will happen when it becomes relativistic beyond 1 GeV ok. So electron becomes relativistic very fast as compared to the proton due to its smaller mass once it becomes relativistic the beta becomes almost constant or in other words its velocity is almost constant as kinetic energy increases. Then once it becomes relativistic gamma changes or in other words its mass increases ok. So notice that for the proton till 5 MeV the gamma is just 1. So in this region it is non-relativistic and for the proton here the Newtonian mechanics will also apply at these energy ranges whereas not so for the electron ok. So this differences between the proton and electron it has important implications in the design of the Linux for both proton and electron or you can say electron and heavy ion. For heavy ion this curve will be maybe something like this depending on what is the charge particle you are calculating ok. So as we learn about acceleration using RF fields we will come back to this and see that how the acceleration for electrons and protons or ions are different ok. Now let us calculate the force on a charge particle. So being charge particles they will respond only to electric fields and magnetic fields ok. So you know from the Lorentz force the force acting on a charge particle is given as Q into E plus V cross V ok. This force will do work on the charge particle and change its kinetic energy. Now from work energy theorem ok the net work done by the forces on an object that equals to the change in the kinetic energy ok. So work done is what F dot ds which is the change in the kinetic energy. Let us calculate the rate of change of kinetic energy. So this is dEk by dt which is simply dF dot ds by dt and this force if I substitute from the Lorentz force here this is what I get ok. So ds by dt is V. So Q E dot V plus Q V cross V dot V this term goes to 0 and I am left with Q E dot V. So now notice that the rate of change of kinetic energy depends only upon the electric field. The magnetic field does not change the kinetic energy of the charge particles only the electric field changes the kinetic energy of the charge particles ok that to when there is a component of electric field in the direction of the velocity of the charge particle. If E is perpendicular to V this goes to 0 again no change in the energy. So when there is a component of electric field in the direction of the velocity of the charge particle electric fields will produce acceleration. So only electric fields can be used for increasing the energy of the charge particles. So though the charge particles will respond to both electric field and magnetic field only electric fields can be used for increasing the energy of the charge particles. For acceleration by electric fields there must be a component of electric field in the direction of motion of the charge particle. So this is a necessary condition for acceleration by using electric field. So let us see the forces due to magnetic field, magnetic field as we saw do not increase the kinetic energy of the charged particles, they only change the trajectory, they can be used for changing the trajectory, okay. So they can be used for bending, they can be used for deflecting focusing, so here is the picture of a quadrupole magnet, so we will learn more about it. So this is used for focusing the beam, so a beam is just like a ray of light, okay, it tends to diverge like this. So if it is not brought back, okay, using some kind of focusing, this beam will keep on diverging and get lost. So quadruples are used for focusing the beam, then there is a dipole magnet, the dipole magnet is used for bending the beam, okay. So magnetic fields can be used for bending, focusing, deflecting. Electric fields can change the kinetic energy, so they can be used for acceleration provided there is a component of electric field in the direction of motion of the charged particles and in addition to this, they can also be used for focusing deflection, okay. In addition to acceleration, they can be used for both focusing and deflection. So here is a picture of a drift tube linear, this is used for acceleration, we will see more about it in the course, then if you apply electric field perpendicular to the direction of motion of the charged particle, you see that it will deflect the beam like this. So they can be used for deflecting, focusing as well as acceleration, okay. Let us compare the electric field and magnetic field for focusing, so we have seen they can both be used for focusing, let us compare what the both the electric and magnetic fields. So again coming back to the Lorentz force, the force on the charged particle is Qe plus Qv cross B. So out of this, this is the force due to the magnetic field and this is the force on the charged particle due to the electric field, okay. Now let us say we have a charged particle moving with a velocity close to the velocity of light. So a relativistic particle let us say, so it is moving with a velocity close to the velocity of light. So now if I calculate the force on that charged particle due to the magnetic field, it is simply Qc into B, okay, why c because it is moving with a velocity close to the velocity of light. Now let us say magnetic field is equal to 1 Tesla, which is like a reasonable field and it can be reasonably achieved. So the force acting on the charged particle due to this magnet is simply now Q into c. Now let us see if I want to get the same amount of force using electric field, okay. So Fe is equal to Qe and I want to produce the same amount of field as due to this magnetic field. So I put it equal to Qc, okay and now if I calculate from here the value of the electric field, okay. So here we will get P is equal to c and c is 3 into 10 to the power of 8, okay. So this is like 300 million volts per meter. So it is not possible to achieve an electric field as high as this, okay. Now however if we are working with a low velocity particle, okay, say moving with a velocity 1% the speed of light, okay. In this case if you calculate, if you do the same calculations here, okay, so instead of c here you will have 0.01c, okay, instead of c here you will have 0.01c. So your electric field will come out to be 3 million volts per meter, which is like a reasonable number, okay. So magnetic field is more efficient for focusing at higher velocities. At lower velocities you can use both electric fields and magnetic fields for focusing. However at higher velocities, magnetic field is more efficient for focusing. So let me just summarize today's lecture, whatever we have learned. When particles have energies comparable to their rest mass energies, relativistic mechanics will apply. So you have to use relativistic mechanics, relativistic formulae in that case. You should not calculate the kinetic energy using half MV square, okay. You have to use relativistic mechanics. Light particles like electrons, they become relativistic at lower values of kinetic energies. Once the velocity reaches the velocity of light, the velocity becomes almost constant and mass increases as the energy increases, as we have seen. Magnetic fields cannot increase the energy of the charged particles, okay. So they can be used only for focusing and bending the beam. The electric fields, they are used for acceleration of the beam, focusing and deflection. So for acceleration, we have to use only electric fields. Magnetic fields cannot be used for acceleration, okay. And necessary condition for acceleration is that we have to have a component of electric field in the direction of motion of the charged particle. At high velocity, magnetic fields are more efficient for focusing than electric fields. Now with this introduction, in the next lecture, we will actually see how acceleration is done using time varying.