 Hello and welcome to lecture 11 of module 2 of this course on Accelerator Physics. So today we will study about the longitudinal dynamics of beams that means the dynamics of the beams in the direction of motion of the charge particle that is in the z direction. So in the previous lectures we studied how we can accelerate the charge particles using time varying electric field. So we saw that time varying fields can be used to accelerate charge particles to very high energies and we use the electric fields associated with the electromagnetic wave in a high Q cavity for acceleration. So we saw that we do not apply the voltages directly to the tubes for acceleration, we use the electric fields associated with the electromagnetic wave in a cavity for acceleration. So in a cavity TE, MNP and TM, MNP modes are excited, the indices M, N and P they describe the field pattern in the cavity. Then we discussed in detail the TM010 mode in the pillbox cavity where we have only the EZ field that is constant along the length and the B theta field. This EZ field is varying with time, it is varying sinusoidally with time and it can be used for acceleration. We can put drift tubes inside the pillbox cavity in a long pillbox cavity, we can put drift tubes and use that for acceleration as well and this structure is known as a drift tube linar. In this the acceleration takes place in the gaps between the drift tubes. Inside the drift tubes the fields cannot penetrate and the beam is shielded from the fields inside the drift tubes. We also saw that how a hollow empty waveguide cannot be used for acceleration of charged particles because there the phase velocity is greater than the velocity of light and it is not possible to synchronize the velocity of the wave with the phase of the wave with the particle. So it cannot be used whereas if you load it periodically you can slow down the wave and then that can be used for acceleration of charged particles. Having studied the various methods of acceleration using time varying fields, we studied about the transverse dynamics of particles. So that is the dynamics of the particles in the direction transverse to the direction of motion of the charged particles. We saw that the beam tends to diverge in the direction transverse to the direction of motion due to various reasons. So the beam has to be focused it has to be brought back to the axis otherwise the beam can get lost. So beam can be focused in the transverse direction using magnetic or electric quadruples. We also saw that we can use solenoids for focusing. Now then we saw that the force due to the beam self charge this is known as space charge forces. The Coulomb effects in Linux they are usually most important in non-relativistic beam at low velocities because for relativistic beams the self magnetic forces increase and produce partial translation of the electric Coulomb forces. So having studied all this now let us see the behavior of the beam in the longitudinal direction. We know that in the longitudinal direction the electric field is applied for acceleration and the electric field has a sinusoidal variation as shown in this figure. Since the field is varying in time this gives rise to the concept of phase. So depending on the phase of the electric field the value of the electric field is different. So in a Linux the convention is that we choose this phase as 0. So where the electric field is maximum this phase is taken as 0. And so the particle seeing this phase will get maximum energy gain whereas the particle seeing a phase of minus pi by 2 or pi by 2 will get no energy gain because here the EZ field is 0. So here let us say the phase of the synchronous particle is pi. The synchronous particle is the ideal particle in the Linux for which the Linux is designed. So it moves from one gap to the other in the right time and sees the right value of the field in the next gap. So this is called the synchronous particle and let us denote the phase of the synchronous particle by this green dot. So we have a synchronous particle here. Now let us consider the phase of the synchronous particle line between minus pi by 2 and 0. So we see here that the field is electric field is rising with time. Not all particles are the synchronous particle there will be other particles in the bunch as well. So let us consider two particles one is the early particle and another one is the late particle. So the early particle is the particle that arrives before the synchronous particle and the late particle is one that arrives later than the synchronous particle. So this is the time or the phase axis. So the gray particle is the early particle and the orange particle is the late particle. Now the early particle arrives before the synchronous particle and it sees a value of the electric field lower than the synchronous particle. So it gets lesser energy gain it moves slower and in the next cycle it reaches later than the synchronous particle. So you can see here the synchronous particle reaches at again at exactly the right phase but here the early particle has now become the late particle. Similarly the late particle sees a field which is higher than the synchronous particle. It gets more energy gain as compared to the synchronous particle moves faster and in the next gap it reaches faster than the synchronous particle. So you see here that it has become the early particle in the next cycle. So you can see that the early and the late particle oscillate about the synchronous particle. So if you go from one gap now this gap to the next gap the particles will again oscillate about the synchronous particle. So when the synchronous particle sees an electric field that is rising with time. So this electric field is rising with time that is when the phase synchronous phase lies between minus pi by 2 and 0. The early and late particles oscillate about the synchronous particle and the bond size is conserved. Now let us see what happens when the synchronous phase lies between 0 and pi by 2. So now we have the synchronous particle this green particle lying between 0 and pi by 2. Here the field is falling with time. So we have an early particle here again. Now the early particle comes before the synchronous particle and since the electric field is falling with time it sees a value of the electric field that is higher than the synchronous particle. So it gets more energy gain as compared to the synchronous particle moves faster and in the next gap it reaches even faster. So it is even earlier. The late particle sees a field which is less than the synchronous particle moves slower and it reaches slower and in the next cycle it reaches even slower. So we see that if we choose the synchronous phase to lie between 0 and pi by 2 then in that case the bunch starts spreading. The synchronous particle will always come to the gap at the right time but the particles about the synchronous particle they will keep moving away from the synchronous particle as we move from one gap to the next. So when the synchronous particle sees an electric field that is falling with time that is pi is lying between 0 and pi by 2. The early and late particles move away from the synchronous particle and the bunch expands. So in other words there is no phase stability. So for acceleration with phase stability the synchronous phase must be chosen to lie between minus pi by 2 and 0. So even though the entire positive cycle can provide acceleration for acceleration with phase stability we should choose the synchronous phase to lie between 0 and minus pi by 2. Now having chosen the phase of the synchronous particle to lie between 0 and minus pi by 2 we design the Linnac. So we choose a phi s which lies between minus pi by 2 and 0 and we design the Linnac. So Linnac is designed for the synchronous particle. The synchronous particle always arrives at each gap at the correct phase. So let us say this is the nth gap. So it moves from here to here in the correct time. So depending on whether it is a 0 mode structure or a pi mode structure it will take time t or t by 2 to travel from the n minus 1th gap to the nth gap. So it always sees it always arrives at each gap at the correct phase and sees the correct value of the accelerating field for which it has been designed. So let us say we have a particle which is a synchronous particle. The phase of the synchronous particle in the nth cell is denoted by phi subscript Sn. The velocity of the synchronous particle in the nth cell is beta Sn. So here this particle has a phase phi Sn. So in other words when it arrives at the center of the gap the field that it sees is corresponds to a phase of phi Sn. It gains energy here and then moves from the center of this gap to the center of next gap with a velocity beta Sn. And the energy of the synchronous particle here is W Sn. Now the cell length is given by, so the cell length is this length from the center of one gap to the center of next gap. So here it is L Sn minus 1 into 2. So 2 times L Sn minus 1 is equal to, so this will be equal to beta lambda by 2 if it is a pi mode structure and it will be equal to beta lambda if it is a zero mode structure. So we write it as n beta Sn minus 1 lambda where n is equal to half for a pi mode structure and is equal to 1 for a zero mode structure. Now we also define a length Ln capital Ln from the center of one drift tube to the center of next drift tube. So this is given as L Sn minus 1 plus L Sn. Now a bunch consists of many particles and not all particles are the synchronous particles and we have seen that if we choose the synchronous phase to lie between minus pi by 2 and 0 then the other particles will oscillate about the synchronous particle. Now we would like to calculate what is the difference in phase between the synchronous particle and some arbitrary particle that the bunch still remains stable. So we investigate the motion of particles with phases and energies that deviate from the synchronous values. So we see, we will try to see that what is the extent of deviation in phase that can be tolerated without the bunch getting lost or bunch remaining stable. So the phase of the synchronous particle in the nth cell as before is pi Sn, the velocity of the synchronous particle in the nth cell is beta Sn and energy of the synchronous particle is in the nth cell is W Sn. Now we talk of some other particle which is not the synchronous particle. So let the phase of that particle in the nth cell be pi n. The velocity of the arbitrary particle the beta n so it will not, it will see a different since the phase is different it will see a different value of the electric field and the energy gain will be different so the velocity of that arbitrary particle will be different. The energy of the arbitrary particle in the nth cell is Wn, again Wn will be different from W Sn. The linac is designed for the synchronous particle so the length will still be according to the motion of the synchronous particle. Now we assume in order to understand the stability of the non synchronous particle we make some assumptions we assume that the synchronous particle always arrives at each succeeding gap at the correct phase and we consider particles with velocities that are close enough to the synchronous velocity so that all the particles have about the same transit time factor. So we assume that the velocities of the other particle are not too far off from that of the synchronous particle. Now we consider the particle motion through each cell to consist of drift spaces plus thin gaps. So this is the drift space this is the thin gap again this is the drift space. So we assume that all the energy gain happens in the thin gap in this region and this remaining region is the drift space. In the thin gaps the forces are applied as impulses so instead of the force being applied at the entire gap which has some finite width we assume that the forces are applied as impulses at the center of the gap. From n-1 gap to n gap the particle has constant velocity beta n-1. So from here to here the particle moves with a constant velocity of beta n-1. Now we will try to write the longitudinal equation of motion of the particles in terms of phi n and w n. So phi n and w n are the phase and energy of the arbitrary particle. Now the RF phase changes as the particle advances from one gap to the next according to so phi n is equal to phi n-1 plus omega t. So here we can write phi n is equal to phi n-1 so phi n is the phase of the arbitrary particle in the nth cell this is the phase in the n-1 cell and to that we add omega t and t is the time taken for the particle to travel for that arbitrary particle to travel from the n-1 cell to the nth cell. So that is equal to the distance divided by its velocity. So distance is fixed so distance is 2 times this ls n-1 and the velocity is beta n-1 into c and we can add a factor of phi or 0 depending on whether it is a phi mode or a 0 mode. Now we define the cell length we have already defined this the ls n-1 is equal to n beta s n-1 lambda by 2 where n is equal to half for pi mode and 1 for 0 mode. Also the cell length measured from center of one drift tube to the center of another drift tube can be written as ls n-1 plus ls n and then we can put in the values of ls n-1 and ls n. So now we want to calculate the change in phase of an arbitrary particle as it travels from the n-1th gap to the nth gap with respect to the synchronous particle. So we can write that as delta phi-phi is n so this is the change in phase of an arbitrary particle as it is moving from the n-1th gap to the nth gap. So how much is the change in phase of that particle as it reaches this gap with respect to the synchronous particle. So this becomes equal to delta phi n and delta phi s n. So let us evaluate the two terms separately. So delta phi n is simply delta phi n-phi n-1 and this we have already seen from equation 1 is equal to omega into t, so omega into t is the distance travelled by the velocity. So this we can put in the values of ls n-1 we get this is equal to n beta s n-1 lambda by 2. So simplifying this further we get this as equal to 2 phi n beta s n-1 by beta n-1. Now the second term we can again simplify this. So here this is phi s n-phi s n-1. So this is again we can write it as omega into t. So this is the distance travelled by the velocity. So this comes out to be equal to 2 phi n. So this is for the synchronous particle. So the synchronous particle travels at the right time. So it takes either time t or t by 2 depending on whether it is a zero mode structure or a pi mode structure and reaches here in the correct time to see the same phase. So delta phi s n is equal to 2 phi n. So delta phi minus phi s into n is equal to we can substitute the values here phi n minus phi n minus 1. So phi n is this and phi n minus 1 is this. So we put in the values here and this is what we get. So we can take beta s n-1 common here and we get 1 upon beta n-1 minus 1 upon beta s n minus 1. Now we can evaluate this quantity in the brackets. So 1 upon using Taylor expansion we can write 1 upon beta minus 1 upon beta s is equal to 1 upon beta s plus delta b. Now the other particle does not have a velocity which is too much off from the synchronous particle. So we can just add a small quantity delta beta to the synchronous velocity. So this minus 1 upon beta s. Now for delta beta much much smaller than 1 we can write this as delta beta by beta s square and this quantity we can substitute in the expression for delta phi minus phi s n we bracketed term here. So we write this as delta beta n minus 1 beta s n minus 1 square with a minus sign. Once simplifying this further we get minus 2 phi n delta beta n minus 1 upon beta s n minus 1. Now we can evaluate this delta beta n minus 1 in terms of energy. So this is the energy is given as gamma m o c square where gamma is the relativistic gamma. This can be written in terms of beta s. So gamma is 1 upon under root 1 minus beta square so you can write this like this. So delta w by delta beta is given as beta gamma cube m o c square. So from here we can write delta beta is equal to delta w by m o c square gamma s cube beta s. So we can put in the value of delta beta n minus 1 here. So this is what we get. So this is the expression for change in the phase of an arbitrary particle as it moves from the n minus 1th gap to the nth gap with respect to the synchronous particle. So here we can write delta w n minus 1 s. So this is the change in energy of the particle with respect to the energy of the synchronous particle. And we return as w n minus 1 minus w s n minus 1. So this is the difference equation relating the change in relative phase of the particle with respect to the synchronous particle as it travels from n minus 1th gap to the nth gap. Now similarly we can write the difference equation for energy change as the particle travels from the n minus 1th gap to the nth gap with respect to the synchronous particle. So this is we can write this as delta w minus w s as it travels from the n minus 1th gap to the nth gap. So this can be written as delta w n minus delta w s n. So putting in the values of delta w n and delta w s n this is simply q e 0 t capital l n because this is the electric field averaged over the entire cell length starting from the center of one drift tube to the center of next drift tube cos phi n minus cos phi s n. So because the arbitrary particle sees a phase phi n and the synchronous particle sees a phase phi s n. Now equations 5 and 6 they form two coupled difference equations. So these are difference equations for relative phase and energy change and these can be solved numerically for the motion of any particle. So these are the two difference equations this is for phase and this is for energy. So we can convert these difference equations into differential equation. So we can write delta phi minus phi s n as d phi minus phi s d n and delta w minus w s n as d w minus d s d n. So n is now treated as a continuous variable. So n was earlier discrete now when we want to convert this into a differential equation we can convert this into a continuous variable in s. So we can change the variable from n to the axial distance s. So n can be written as s by the cell length. So n beta s lambda over half of cell length. So from here d n is equal to d s by n beta s lambda. So we can substitute this in the equation 5 and we get n beta s lambda d by d s of phi minus phi s is equal to minus 2 pi n w minus w s m o c square gamma s cube beta s cube. So which can be further simplified into this expression. Similarly the expression 6 for energy can so this difference equation 6 can be simplified and converted into a differential equation of the form. So here it is d w minus w s by d s is equal to q e 0 d cos phi minus cos phi s. So here are the two equations now we have two differential equations. Now we can differentiate this expression and we will get d by d s of w minus w s which we can put in here and we get this expression. So this can be further simplified to write this expression. So we see that this equation in phi and w so this is a non-linear second order differential equation for phase motion. So from energy and phase we have converted into a single second order differential equation for phase. And so we see that this is a non-linear second order differential equation for phase motion. Now phase is proportional to time so more negative the phase so this is 0 so more negative the phase earlier the particle arrival time with respect to relative to the crest of the wave. So the early particle is more negative with respect to the phase at the crest. Also the phase difference between the particle and the synchronous particle is proportional to the spatial separation. So these particles are the particles in a bunch. Now the bunch in s can be written as can be shown like this. So we have a synchronous particle and we have an early particle which is here and late particle which is here. So this is showing the bunch in space. Now if the same bunch is to be shown in time so the early particle arrives earlier in time and the late particle arrives later in time as compared to the synchronous particle. Hence if you see the expression for the bunch length and the bunch phase so there is a negative sign involved. So the phase difference between the particle and the synchronous particle this is proportional to the spatial separation with a negative sign. Now assuming that the acceleration rate is small so that means that we can take p0 t phi s as constant also since acceleration rate is small beta s gamma s are constant. So we introduce the following notation, we write small w is equal to w minus ws and divided by mo c square then we define a and b which are constants. So a is 2 pi by beta s cube gamma s cube lambda and b is q e0 t by mo c square. Now these are the two differential equations. Now using this notation where we have defined small w and constants a and b these equations can be written as so w prime can be written as b times cos phi minus cos phi s similarly phi prime from here can be written as minus aw we can differentiate this expression again to get phi double prime is equal to minus aw cos phi minus cos phi s and this expression can then be written as d phi prime and we take ds on the right hand side is equal to, so d phi prime is equal to minus aw cos phi minus cos phi s ds. Now ds is d phi by phi prime so ds is d phi by phi prime. So we can substitute ds as d phi by d phi prime and we have a d phi prime here and then we can integrate this expression and this is what we get phi prime square by 2 is equal to minus av sin phi minus phi cos phi s is plus some constant term. Now this we can write it as now since phi prime is equal to minus aw so we can write this as aw square by 2 plus b so dividing the whole thing by a so we get plus b sin phi minus phi cos phi s is equal to some constant which we call as h phi. Now if you look at this expression this expression looks like there is a kinetic energy term and there is a potential energy term and this is like a Hamiltonian which is constant. So the first term is of the form of a kinetic energy and the second term is of the form of a potential energy and h phi is the constant of motion. So let us try to understand this expression.