 In this lecture, we will study the longitudinal beam dynamics. So far we studied mainly transverse beam dynamics. So this lecture concentrates on a new domain of dynamics that is longitudinal dynamics. Actually, we have accelerator coordinate system like this. This is the design trajectory and design trajectory itself as origin of coordinate system and opposite to radius of curvature, we generally take x axis and vertical to this we take y and in the direction of beam propagation we take t. Now we were talking about the horizontal bitatron motion means oscillations are taking place along the x axis while the beam is moving along the z axis means these oscillations were like this in this plane like this and another motion was vertical bitatron motion in which oscillations were along the vertical axis while the beam was moving in the z axis. So these motion was like this these were the oscillations. Now we will talk about the third axis that is the z instead of talking about the z and z prime we will take two new variables what are those variables you will see it. So for the dynamics which is study about the bitatron motion is known as transverse beam dynamics and now the dynamics this transverse dynamics mainly commands by the arrangement of the magnets that how we are arranging the magnets where we are putting the quadrupole magnets what is the strength of this what is the length of the drift space what kind of dipole magnet and what is the angle etc and how we are arranging these magnets in making our optics how the periodicity is there these all things affects the transverse beam dynamics. Longitudinal beam dynamics mainly governs by energy manipulation of the particle means how we are manipulating the energies of the particle and when we talk about the energy means we are talking about the RF cavity in which radio frequency electric field is confined and that field is providing the energy to the particle whenever particle is traversing through that cavity in a turn it gets energy. So longitudinal beam dynamics mainly governs by the RF cavity. So these are two basic realm of the beam dynamics some other classification of beam dynamics can also be there that when we talk about single particle beam dynamics single particle beam dynamics means we are talking about trajectories of a single particle or how the coordinates in six dimension means x x prime y y prime are the transverse coordinates and other two variables which we talk about that will form the longitudinal coordinates how these coordinates are evolving in single particle beam dynamics the presence of one particle doesn't affect the motion of the other particle in the beam means we have the collection of particles in the beam but each particle sets its own trajectory without getting affected by the other particles of the beam in reality it is not so there because all the particles are having charges so when all particles move together then they can affect the motion of each other via direct coulomb repulsion or magnetic attraction or via the vacuum chamber via the vacuum chamber means the charge particle induces image charges in the vacuum chamber and those image charges can affect the other particles so when one particle gets affected by the presence of the other particle and we study those phenomena then we are in the realm of the multi particle beam dynamics or collective in this course we will mainly concentrate on the single particle dynamics under linear approximations even single particle beam dynamics can be broken down into two parts one is the linear beam dynamics and the other one is the non-linear beam dynamics in last lecture we have seen that introduction of the sex to pole can lead to a non-linear motion for the particles so that kind of motion where we are introducing non-linear terms in the hits equation and we are trying to solve that equation in which can give rise to this stable and unstable motion of the heat at one oscillations that kind of studies are known as non-linear beam dynamics so in this course we will not touch that kind of dynamics we will restrict ourselves to linear approximation and mainly single particle beam dynamics i will give one example of collective phenomena in the later lectures let us suppose that this is the cavity this is the cavity rf cavity and this is the orbit in the synchrotron now a particle when it crosses the rf cavity it gets energy and how much energy is it it is q v same phi v is the peak voltage and phi is the phase of the rf at the instant when a particle is crossing the cavity now particle can take some time to cross the cavity and then phi may change dealing its course of motion inside the cavity means length of cavity has some impact on the motion however in this course we are not talking about that effect means we are taking rf cavity as a thin element means we are ignoring its length thin element like we did many times in transverse dynamics we took quadrupole as a thin element here we are taking rf cavity as a thin element means it is a point where the particle is when particle is crossing this point it gets some energy now consider a particle which is on exactly on the design orbit means it has x is equal to zero x prime is equal to zero y is equal to zero and y prime is equal to zero means it is following the design trajectory it is not exhibiting any synchrotron oscillation bit of oscillations and it reaches exactly on the desired phase in the rf means there is a phase on which particle when arrives it gets energy in such a way that its evolution time on the orbit is always an integer multiple of the time period of the radio frequency field means we can have t revolution is equal to hti so it gets energy such that this relation remains always true for this particle means this particle is always coming on the synchronous phase and the radius of curvature of this particle when it passes through the magnetic field v is given by this relation r is equal to p upon qv we have seen this relation many times p is the momentum or design momentum correct momentum of the particle so this particle is known as synchronous particle so we are characterizing this particle by two variables one is how much energy it is getting and what is the phase in which it arrives at the rf cavity so this is the correct phase we say this is a synchronous phase and its energy which exactly follow the orbit or design orbit in the synchrotron that is the correct energy so this energy in of this particle and phase of this particle in the rf these two makes a reference for us means in longitudinal dynamics we have two coordinates one is the energy and one is the phase and synchronous particles energy and phase make the reference means any particle which is deviated either in phase or in energy we will study the motion of that particle with respect to this synchronous one as we did in transverse dynamics we studied the motion with respect to design trajectory means an ideal particle which is following exactly the design orbit we topped the displacement 19 with respect to that trajectory in the transverse dynamics in longitudinal dynamics we will talk about how much energy is deviated from the synchronous one and how much phase is deviated from the synchronous phase so e and phi makes the coordinate plane forest in the longitudinal dynamics so now we have x x prime y y prime these are the coordinates in the transverse plane and in longitudinal plane delta e delta e means energy deviation from the synchronous particle and delta phi that is the how much phase is deviated from the synchronous particle these six coordinates make the complete state of the particle now how the synchrotron operates synchrotron operates on the basis that its orbit remains constant in cyclotron as the energy was increasing orbit radius was increasing and particle was making an spiral path like this there were d when particle crosses the d it gets energy and it's make a circular path here and here again it crosses the d so this hit this time it will make a larger path because magnetic field which we are applying for guiding the particle is constant so in this fashion a spiral path is there in the cyclotron the major problem with cyclotron was that if you want to increase the energy first thing was that due to increasing gamma synchronism may break and other thing is that the size of the magnet becomes larger and larger and making such large magnets is a really difficult job so how we can reduce the size or aperture of these magnets instead of increasing the radius continuously take some action so that radius remains constant means particle always revolve on the same orbit so only this path has to be covered by the magnetic field so this was major principle on which synchrotron works and that's the way in which it reduces the requirement of the aperture of the magnets than the cyclotron so in this case as the energy if you want to increase the energy you want to increase the energy we increase the magnetic field also to keep the radius constant so suppose p is the correct momentum means momentum of the synchronous particle then radius of curvature of this particle will be given by this relation p is equal to q p is the applied magnetic field now as due to rf cavity electric field impart some energy to the particle means it has changed the momentum of the particle so we have certain dp by dt and if we have dp by dt then it will be q r dp by dt means we have to change the magnetic field also to keep the r constant means we have to ramp the magnetic field with energy now how much change in momentum will be there in one term so now this relation you can see dp by dt is equal to q r dp by dt so this is dp by dt so in one term delta p for the one term will be dp by dt into t revolution so this will be q r b dot t revolution dot dot means b by dt whenever in this course i am using the dot it means this is d by dt and if i am using the prime it means it is d by ds so dp by dt has been written here as b dot so q r b dot t revolution not t revolution can be written down as l by v and because we are talking about some kind of closer path may not be exactly circular so we can take it as a 2 to 2 pi and some average radius r upon v so at the place of t we have written this 2 pi some r mean r average upon v we can multiply v here so we get v delta p one term is equal to 2 pi r this r is the bending radius and this r is the average radius of the orbit into d by dt here you can see that v delta p is basically the energy k delta p you can prove very easily that v delta p is delta p you can take the relation e square is equal to c square e square plus m square c so if you take the change in energy means you can take 2 e d e is equal to c square 2p d these are constants so differentiation will be 0 and this 2 will be cancelled out this so you can calculate d e is equal to c square p upon e dp now p itself is beta e by c p itself is beta e by c so by this you will get d e is equal to v dp which is written here so this is the energy now this energy gain is due to rf cavity means this energy gain v dp is q v sine phi so these two values must be equal so that we will get the correct value of dv by dt for the keeping r constant so this gives you what should be the synchronous phase corresponding to dv by dt in a synchrotome so that particle will remain on the constant on it this is valid for this synchronous particle means this phase is synchronous phase now one beautiful thing is here that you can see that if dv by dt is 0 synchronous phase becomes 0 synchronous phase becomes 0 means here it is the time and this is the voltage and the cavity so synchronous phase is 0 means particle is here or here means it is not catching any energy from the field means particle is revolving on the constant energy this mode of operation of a synchrotome is known as a storage mode means being revolves in synchrotome at constant energy if we don't raise the field of the magnets as we raise the field of the magnet b synchronous phase 5 changes and particle reaches the synchronous phase here say and it gets energy so this mode of operation in which we ramp the magnetic field or change the magnetic field where the energy increases we say this is the boosting mode of the synchrotome so if you will see if you will hear synchrotome or booster synchrotome it means these are the synchrotome where the energy is ranked energy is increased and if you hear the names storage ring accumulator ring etc then these are the synchrotones need for the constant energy means beam is revolving in these synchrotones at constant energy so there are two modes of operation for this kind of machine any machine can be operated in both the modes if we have such kind of design means single accelerator can be operated in the booster mode as well as in storage or it can be dedicated storage ring means no boosting is possible so such kind of designs are possible