 In this module of this course we will study about the cyclic accelerators. So far in this course you have studied about DC accelerators, then linear accelerators. Now we will talk about the cyclic accelerators. In this module there will be 14 lectures, lecture 20 second to 35. And broadly the module can be broken in 4 parts. The first part consists basic principles on which cyclic accelerators are based and two types of cyclic accelerators namely cyclotron and microtron will be discussed. Quantitative discussions will be there and these will involve the derivation of the equation of motion when a charged particle passes through the electromagnetic field and how we can control the beam of charged particles and how we can analyze if we want to make some optics of charged particles. So and what are the parameters on which we analyze or characterize the optics? These all will be discussed in this part too. This will involve one of the famous equations, Hilf's equation and its solutions. And then we will talk about the chromaticity which is the first non-linear magnetic element introduced in the synchrotrons. So this part will cover some mathematical derivations, quantitative results and will be based around synchrotrons. Then we will see longitudinal dynamics in the synchrotron that how we can energize the charged particle using radiofrequency field in the synchrotron. And after that in the last part there will be three lectures and these three lectures will have three different kind of synchrotrons for different applications. So first one is the synchrotron radiation source. These accelerators are basically electron machines and these electrons on acceleration radiates and these radiations are used by the users. And the second area of accelerators which is chosen for this course is protons synchrotron and it is again a kind of neutron source. It produces neutrons and those neutrons are used by different experimentists. And then the third one is the colliders in which colliding beam facilities are built. So let us begin with the first lecture that is some basic principles. So first of all what is the cyclic accelerators? Cyclic accelerators are those accelerators in which same RF cavity can be used multiple times to increase the particles energy. When a particle passes through the RF cavity you have learned in your linear accelerator courses that it gets the energy. So after getting the energy if we again send the particle in that cavity then again the same cavity can be used repetitively for energy gain. And the problem of length of the linear accelerators can be solved. Means we can make a compact accelerators using this technique. So for this we have to have some path or orbit in these accelerators which are closed curve because particle is passing through the same point again and again. So closed curve is necessary or it may be spiral. We will see it in next lecture. Can we have this kind of repetition using the DC field? Means we have some DC accelerators here say plus minus this is the DC field and then we send repetitively the particles in the DC accelerator and we can get higher and higher energies on each time. Is it possible or not? Definitely DC accelerator works on the electrostatic field and electrostatic fields means curve of the electrostatic field is zero means closed integral on E dot dssc means it is a conservative force it provides a conservative force and under any conservative force if particle travels a complete closed path the energy gain will be zero. So DC accelerator cannot be used for repetitive acceleration means such kind of cyclic accelerators cannot be built using DC field. It can be understood in this way also suppose there is a plus higher potential and there is a negative potential lower potential. So particle say positively charged particle will accelerate inside this then if we want to send again this particle to this area of DC field we have to raise the potential again our potential of that particle again means particle will lose the energy for reaching this plus potential. So in this fashion the energy gained inside the particle by DC field will be lost in this cycle. So what we have to do we have to do some field non-zero here when the particle passes and zero field elsewhere. So whenever particle crosses this region only then it sees the field. So this is possible only with time varying field. So cyclic accelerators are built only using time varying field. As we have seen that we have to make some closed path for sending the particle again in the same RF gate means we have to generate a curvature in the path. And now suppose there is a magnetic field this is a magnetic field and the direction of the magnetic fields is out of the screen perpendicular to the screen and it is coming out. So in this region there is a homogeneous magnetic field. Now when a particle of charge q having a speed v is passing through this field it will feel a force given by this expression f is equal to q v cross p. Now this force because it is a cross product of two vectors v and b. So force will be in the perpendicular direction to both the vectors means perpendicular to the plane formed by this speed velocity vector and magnetic field vector. So magnetic field is coming outside the screen in the perpendicular direction and suppose there is a if we draw a tangent on this trajectory this is the direction of the v. So v cross b will be in this direction when particle will reach here v will be here in this direction and again the force is in the perpendicular direction. So this kind of force is actually imparting a centripetal acceleration to the particle and due to this centripetal acceleration particles trajectory becomes caught. So particle can be sent on a circular trajectory using the magnetic field. So in all the cyclic accelerators some kind of magnetic field for generating the curvature in the design trajectory's path will be there. Now will it change the energy of the particle this magnetic field? No it will not change because change in energy is force, scalar product of force and displacement. Now force was q v cross b so scalar product of q v cross b by ds. Now ds is the displacement it can be written down as v dt. So scalar product of v cross b with v. Now v cross b its resultant vector will be in the perpendicular to v itself. And perpendicular to v itself means again a scalar product of this resultant with v. So two perpendicular vectors are having dot product or scalar product means zero. So energy will not be changed under the magnetic field. Here we are considering constant magnetic field means magnetic field is not varying with time. If there is a time varying magnetic field then this time varying magnetic field can change to emf and that emf can change the energy. Now the curved path may be of different types. Suppose we are having a constant magnetic field and this is a particle trajectory and this gap considered there is an electric field applied. So whenever charge particle crosses this field it gets energy. Due to higher energy magnetic field will bend less means its radius of curvature will increase so it will go on a larger circuit. And again when it will pass through this it will get some energy from the electric field and again it will follow even a larger path. So such kind of spiral path will be there. This kind of orbit is in settler form. Now in synchrotron as energy increases this is the RF cavity and particle when traverses through this cavity its energy increases. However for bending or generating the curvature in the path we use some kind of magnets so there are some kind of magnets kept there. Simultaneously we increase the magnetic field of those magnetos. So as the energy increases magnetic field also increases and we keep the R constant. So on the constant radius continuously particle moves so orbit does not change in the case of a synchrotron. And these orbits may not be a circular completely like in this example this is a hexagon. So it may have various polygons shape depending on how many magnets and how we are designing the lattice. This may also happen that there is a constant magnetic field over the space and particle passes through the RF cavity it gets energy and constant magnetic field causes its circular trajectory and it passes again through this RF cavity. And again when it passes through the RF cavity this time because of the higher energy its radius of curvature will be increased so it will make a larger circular path under the magnetic field. And in this fashion we have various circular trajectory which has common tangent here. These kind of orbits are in microcan. So in the next lecture we will cover the cyclotron and next to that we will cover the microtron and in rest of the course we will talk about the synchrotron. Now we are saying that we are using some time varying field. And in that time varying field energy is increasing and in that time varying field particle is passing again and again. So if we have some time varying field suppose this is a orbit in any accelerator, cyclic accelerator and there is an RF cavity when particle passes through this cavity it gets energy. However in RF cavity there is an electric field so total force will be QE on the particle. So total energy gain will be this force multiplied by the displacement dot product of force and displacement. This is the gain in the energy. However because in RF cavity the field has time variation means it is not constant in time. So if a particle is reaching here it has correct movement means suppose sometime field is in this direction and particle gets energy when the field is in this direction. Because of this is a time varying field so after sometime in cavity field will be in this direction. So particle should travel only on that moment when or in that duration of the RF field when direction is correct for acceleration means this travel time should have certain relationship with the time period of the applied RF and this condition is known synchronism. So one of the major conditions to run a cyclic accelerator or any accelerator which has time varying field for particle acceleration synchronism has to be achieved. There should be a synchronization between the arrival time of the particle in the cavity and the time of the RF field which we are applying in the cavity for particle acceleration. So it means T revolution should be some integer multiple of the RF time period means suppose this is an orbit particle is revolving in this direction and there is a cavity. So if suppose there is a phase phi of the RF field when particle just crosses the cavity so if particle's revolution time is an integer multiple of time period of the RF radio frequency field which we are applying in the cavity then particle will see again phi phase on its traversing through the cavity. It means phase will not change and on each time it will get the energy because it is the correct phase in which particle can get acceleration. So let us take an example which is shown in this movie for h is equal to 1 means when revolution time is exactly equal to the RF time period. So particle reaches here every time on every turn particle comes here comes here and comes here means there is no change in the phase and if this is the correct field means in this polarity of the field if particle is accelerating then it will get acceleration on each passing. Now suppose for h is equal to 1.1 how phase will change it is shown in this movie. So now it does not come here instead phase slips to here and in each turn this phase slip will occur so accumulative phase after few turn will be such large that it will go in the negative direction of the RF field and here particle will be deaccelerated. So for successful particle acceleration this synchronization is necessary and how we are achieving this synchronization different accelerators are based on different mechanisms means cyclotron achieve this synchronization in one of the ways then microtron achieves this synchronization in other ways and synchrotron achieves this synchronization in other ways. So on the basis of mechanism how an accelerator is achieving synchronization it can be classified as cyclotron, microtron or synchrotron. Now on each turn if particle is reaching at correct phase it will get synergy so energy will be built up. So we will see in this movie how energy is stepping up of a particle if it has correct synchronization so you will see in each turn energy is stepping up. So finally we can get very high energy on many such traversing through the cavity. Now we will see if H is equal to not completely integer value then what will happen because the phase will slip so in each step in each cycle we will not get the stepping of the energy rather than after few cycles we will get energy decrease. Now you can see that gaining energy has been reduced and here energy has been reduced here. So there is no net acceleration through so many cycles so synchronization is essential. Now in accelerator language this edge this integer is known as harmonic number means in one RF time period how many revolution times are there that is known as harmonic number. So we see in another movie in which H is equal to 2 rather than 1 then what will happen we see. So now one cycle will be empty particle will come after one empty cycle so after two cycles particle will be accelerated so particle will reach in the cavity after two cycles. So one RF cycle is empty it is not used for particle acceleration. So what we can do in these gaps we can put another bunch of the particles and those bunch can be accelerated here. So one bunch in the black color will be accelerated in this RF cycle and another bunch of particles which is drawn in red color will be accelerated in this cycle. So harmonic number tells you that how many bunches simultaneously can be accelerated or a machine can hold maximum number of bunches this is the harmonic number and it also shows that because only certain duration of RF cycle we can get acceleration otherwise there will be deacceleration so only certain period of RF time period particle can get accelerated means we cannot have continuous beam through the cyclic accelerators cyclic accelerators will produce always a bunch to be.