 Hello and welcome to the second lecture of the module 2 of this course on accelerator physics. So, today we learn about RF acceleration. Before that let me just revise whatever we did in the first lecture. So, in the first lecture we learnt about the limitations of DC accelerator. So, we saw that the voltage generated in a DC accelerator can be used only once and at the most twice as in the case of tandem accelerators. So, we cannot use this voltage generated many times. Also the energy gain is limited by how high a voltage you can achieve. So, that is why you cannot go to very high energies using DC accelerators. We also saw that we use electron volts as energy units because the masses of the charge particles involved are very small. So, even though the velocity is high, their kinetic energies are very small as compared to the quantities that we use in our daily life. So, joules is too large to represent the energy of these beam and hence we use electron volts. And also the particles they have energies comparable to the rest to their rest mass energy. So, that is why Newtonian mechanics does not apply we have to use relativistic mechanics. So, we saw that the light particles like electrons they become relativistic at lower values of kinetic energy because their mass is quite low. Once the velocity reaches close to the velocity of light, the velocity becomes almost constant and then as the kinetic energy increases, the velocity does not increase much. It is the mass that increases. We also saw that for that magnetic fields cannot increase the energy of the charge particles. Only electric fields can be used to increase the energy of the charge particles. Magnetic fields are however used for focusing and deflection of the beams. The electric fields can be used for both acceleration as well as focusing and deflection. We also saw that at high velocity the magnetic fields they are more efficient than electric fields for focusing of the beam. So, with this let us start today's lecture. So, today we will study how we can accelerate using time varying fields. The simplest time varying field is a sinusoidal type of variation. Okay, so here is the electric field which is varying sinusoidally with time. The form of the electric field can be written as E z t z means it is in the z direction and it is a function of z and t can be written as E 0 cos omega t z plus 5. This is the most general form. Here omega is the angular frequency and it can be expressed in terms of the frequency of this wave as omega is equal to 2 pi f. The time period of this wave is equal to 1 by f and we note here that the frequency is in the range of RF frequency, the radio frequency hence it is called RF accelerator. So, it is in the range of tens of megahertz to a few gigahertz. Now, notice that the beam here will be coming continuously with time. Okay, the beam is a continuous beam it is going to be coming continuously with time. Okay, so let us say there is a gap here in this time varying electric field is applied. The beam is coming here continuously. So, as the beam enters into this gap here it sees this part of the electric field first. So, this is in the forward direction. So, the beam gets accelerated. Okay, now after some time the when the later part of the beam comes in here at let us say this time the field has changed in sign it is now in the negative. So this part of the beam will be decelerated. So, only the positive cycle will accelerate the negative cycle will decelerate the beam. So, hence we can say that only positive cycle can be used for acceleration the negative cycle will produce deceleration. So, in order to accelerate without any loss of beam the beam needs to be bunched. Okay, so let us say this is the beam this is the DC beam that is coming continuously this is the RF field. So, now you have to bunch the beam you have to bunch the beam such that it sees only the positive part of the RF field it does not see the negative part. Okay, and you need to bunch it at the same frequency as that of the applied RF field. It has to be at the same frequency. So, the bunch frequency should be same as the frequency of the applied RF. Also, the bunch beam needs to be synchronized with the time varying electric field such that it always sees a positive field. Okay, so you have bunched the beam but let us say you injected into the gap when the field is negative in that case it will get decelerated. So you have to you have to synchronize it with the time varying electric field such that it always sees a positive value of the electric field. So, that is how you can accelerate using time varying fields. So, here let us say we have a series of hollow conducting tubes. Okay, so these are hollow so that the beam can pass through it and it is a conductor. So, being a conductor the field electric field will not enter inside the these tubes. These tubes are known as drift tubes. Okay, let us say we apply a RF voltage here. We apply an RF voltage here such that at a particular instant of time. Okay, let us say at t is equal to 0 this is positive and this is negative. In this case the first drift tube becomes positive the third drift tube is positive and the last drift tube is positive. The second and fourth drift tubes will be negative. So, if you if I draw the electric field here it is in the forward direction here it is in the reverse direction here again in the forward direction here reverse direction here and so on. So, if I inject the beam bunch in this gap it will be accelerated. So, it will be it will be accelerated in this gap. Now, when it comes to the next gap here the field is in the opposite direction. So, here it will get decelerated but what I can do is that let it come from the first gap to the second gap in time t by 2. So, after time t by 2 what happens the field changes polarity. Okay, now let us say this reaches here at time it reaches in the second gap at time t by 2. So, at time t by 2 what happens is that the field changes sign. So, now this becomes negative and this becomes positive. So, the first drift tube the third drift tube and the last drift tube is now negative the second and fourth drift tubes are positive. So, in this case now the field in the second gap has now become in the forward direction and hence it will accelerate. So, in this way by changing the polarity the same voltage can be used repeatedly to accelerate to high energies provided the particle arrives at each gap at the right time or in the right way such that it sees always the positive part of the electric field. So, it has to travel from here to here in time t by 2. So, in order to use time varying fields for acceleration the beam must be bunched and it must be synchronized with the field. So, the same small voltage can be used repeatedly to accelerate to high energies by successively accelerating the charge particles over many gaps. So, the necessary condition here is isochronism that is the particle arrives at each gap at the right time or in the right so that it sees the right phase of the electric field that is the positive part of the electric field. Also as we have studied already in the first lecture for acceleration there should be a component of electric field in the direction of velocity of the beam. Now, without isochronism we can still get acceleration in the gaps. However, for sustained acceleration over large lengths of many tens of accelerating gaps isochronism is important. So, this is known as principle of successive acceleration. Now, here since the field is varying in time. So, unlike in the DC field where this quantity was equal to 0. So, here this is no longer 0 because field is varying in time. So, the field is not conservative. So, this removes the restriction that energy gain is limited by the fixed potential difference. You can use the same field several times for acceleration. So, let us say the energy gain in one gap is delta w. If you have n number of gaps then the total energy gain is now n into delta w. So, you can go to as high energy as you want by increasing the number of gaps. So, now there is no limit on the maximum energy of the charge particles unlike the DC accelerators. So, you use only a small voltage, but you use it several times to go to higher energies. So, for synchronous acceleration as we have seen time taken by the bunch to travel from one gap to the next. So, for travelling from here to here this is equal to T by 2. Only then the field changes sign and then it will experience acceleration in this gap as well. So, we define cell length as the distance between the centres of two adjacent gaps. So, cell length for synchronization the for cell length should be equal to L is equal to V into T by 2. So, if the charge particle here is it is moving from here to here in with velocity V, average velocity V then it moves from here to here in time T by 2. So, L will be equal to V into T by 2. So, V can be written as beta C and T can be written as 1 by F. So, this now C by F can be written as lambda. So, cell length is equal to beta lambda by 2. Now, at a particular instant of time if I take a snapshot I see that the field in the first gap is in the forward direction in the second gap it is in the reverse direction. So, the field in adjacent gaps they are out of phase with each other. So, this is known as a pi mode structure. Now, at a particular instant of time since the field here is accelerating and here it is decelerating again accelerating here I can have a bunch here and then there will be no beam bunch here because if there is a bunch here this will get decelerated. So, there would not be any beam bunch here the next bunch will be in the next gap. So, if I calculate the distance between the bunches it will come out to be 2 times L 2 times L. So, that is equal to beta lambda. So, distance between the bunches in this case is equal to beta lambda. Now synchronous acceleration is also possible if the time taken by the bunch to travel from one gap to the next is T or one full RF cycle period. So, in this case now the cell length again defined as centre to centre distance. So, cell length is now V into T because now time taken to travel this distance is T. So, L is equal to V can be written as beta C and T can be written as 1 by F. So, L becomes now beta lambda. Also in this case now if you notice the fields in the adjacent gaps they are all in the same direction. So, this type of structure is known as a zero mode structure. So, now because the field is in the same direction in all the gaps I can have a bunch in at a particular instant of time I can have a bunch in all the gaps. So, if I calculate the distance between the bunches it is simply equal to the cell length and that is equal to beta lambda. So, now I can summarize the difference between the zero mode structure and the pie mode structure. In the case of zero mode structure the time taken to travel from one gap to the next is capital T whereas here it is T by 2. The cell length here for the zero mode structure comes out to be beta lambda for synchronous acceleration and here for synchronous acceleration it is beta lambda by 2. If I see the fields field direction in adjacent gap here at a particular instant of time here it is in the same direction and here it is in the opposite direction. However, the distance between the bunches is always the same it is whether it is the pie mode structure or the zero mode structure the distance between the bunches is beta lambda in both the cases. So, let us see now how acceleration is done. E z changes sign every T by 2. So, we see here that the electric field changes sign after every time. And even within this time period T by 2 the field is not constant it is varying with time. So, now what happens let us say we have a bunch we have a full bunch from zero to T by 2. Now what happens the bunch the particles in the bunch that see this part of the field and this part of the field that means at T is equal to zero and T is equal to T by 2 the field here is zero. So, they do not get any acceleration whereas the particle in the bunch that sees this part of the field that experiences maximum acceleration. So, different particles in the bunch will get different acceleration. So, we cannot use the entire positive part for acceleration because then what will happen is that some particles will see higher field some particles will see zero field. So, there will be a huge spread in the kinetic energy. So, we use only a small portion of the RF cycle for acceleration such that there is not a huge spread in the kinetic energy. So, the bunch size is usually kept much smaller such that all particles in the bunch see only a small variation in the accelerating field. Now since the electric field is varying with time the energy gain in the accelerating gap depends upon the phase of the accelerating field seen by the particles. Because for this phase the accelerating field is this, this phase the accelerating field is this. So, what is your energy gain that depends upon what is the phase of that is seen by the particle in the center of the gap. So, we define a reference particle in the beam bunch which is called the synchronous particle. So, there is one particle in the bunch a reference particle which we call the synchronous particle for which the linac has been designed. So, it sees the correct phase it gets the right energy gain. So, it when it arrives at the center of a gap it sees it sees the right phase of the electric field gets the right. So, sees the right electric field gets the right energy gain and it comes to the center of the next gap at the right time to see again the same value of the electric field and phase. So, this particle is known as the synchronous particle this and it is for the synchronous particle that the linac has been designed. Now by convention in a linear accelerator. So, it is different for a circular accelerator for a linear accelerator the crest of the sinusoidal variation of the RF field is taken as 0. So, we take the 0 at this point. So, this is pi by 2 and this is minus pi by 2 this is a convention which is used in linac. So, now let us see can we use the entire positive cycle for acceleration. So, we know that the positive cycle in the positive cycle the field is in the accelerating phase, but can we use the entire positive cycle for acceleration. So, let us just see. So, now let us consider the phase of the synchronous particle as phi s ok. So, this green particle is the synchronous particle here ok and it has a phase phi s and let us consider a case where this synchronous particle is lying between minus pi by 2 and 0. Now the beam bunch will have other particles also beside the synchronous particle. So, let us say we have a particle which is this particle which is the early particle. So, this is the early particle because it came at a time. So, this you can see that this is the time scale. So, this came at a time earlier than the synchronous particle. So, this is known as the early particle and then there is another particle which is the late particle because it came at a time which is later than the synchronous particle. So, now what happens the synchronous particle is the ideal particle the particle for which the linac has been designed. So, it comes to the center of the gap sees the right field ok gets the right energy gain and reaches the next gap at the correct time to see the same value of the phase again. So, it is the ideal particle it is the particle for which the linac has been designed. Now what about the early particle the early particle came earlier than the synchronous particle it saw a field lower than the slightly lower than the synchronous particle. So, now it has lower energy gain. So, it will move lower it is lower and in the next cycle it will reach later than the synchronous particle. The late particle here now on the other hand it sees a field that is higher than the synchronous particle. So, it will move faster than the synchronous particle. Since it moves faster than the synchronous particle it will reach earlier in the next phase. So, we see that the particles around the synchronous particle in going from gap to gap they simply keep oscillating about the synchronous particle and the beam bunches maintain it. So, now when they move from this gap to here the synchronous particle will arrange again come in time the this has now this is now the late particle it will come early and this particle which is now the early particle will come late. So, we see that the particles are simply oscillating around the synchronous particle. So, the beam bunch is always maintained. Okay now let us see what happens if we choose the synchronous phase lying between 0 and pi by 2. So, that means in this region. So, this green particle is again my synchronous particle the particle for which the Linnak has been designed and again I have an early particle this orange particle is the early particle because it has come at a time earlier than the synchronous particle. The gray particle is the late particle because it has come at a time later than the synchronous particle. Now here the early particle it sees a field that is higher than the synchronous particle. So, what happens it gains more energy it moves faster and in the next cycle it reaches even early it reaches even early whereas the late particle sees a field lower than the synchronous particle and it gets a lower energy gain. So, it moves with a lower velocity and it reaches evenly. So, over several cycles this bunch will spread and eventually the particles will get lost. So, in other words we say that there is no phase stability. So, the early if you choose the synchronous phase between 0 to pi by 2 there is acceleration but there is no phase stability. So, in order to have acceleration with phase stability the synchronous phase must be chosen to lie between minus pi by 2 to 0 only then we can have sustained acceleration over large number of gaps. So, the first linear accelerator using the time varying fields it was conceived by Ising and Bintro. So, they took an evacuated glass cylinder and they put these drift tubes inside and then there was this voltage generator here and this was connected to these drift tubes and as the field change sign after every time t by 2 this beam got accelerated. So, the beam bunch in travelling from here to here it took time t by 2 and so, the synchronicity condition was L is equal to beta lambda by 2 ok. So, here we use hollow drift tubes to shield the beam bunch from the undesirable part of the RF field. So, we want the beam to see only this part of the RF field. So, the remaining part of the field is shielded by the drift tubes. For the remaining part the bunch enters inside the drift tube and it since electric field cannot enter inside a hollow conductor. So, the it is shielded from the electric field once it enters inside the drift tube.