 Now, let us see adiabatic phase damping in the longitudinal phase space. Now, if acceleration rate is very small that means, beta and gamma can be taken as almost constant they are varying very slowly. The parameters of the phase space ellipse for small oscillations they vary slowly. The area of the ellipse describing the small amplitude oscillations is then an adiabatic invariant during the acceleration process. So, this is for only when the acceleration rate is small. So, from the equation of ellipse, so this is the equation of ellipse we can write this and now since the area of the ellipse is constant. So, we can write pi into W0 square into delta phi 0 is equal to constant or we can write pi W0 delta phi 0 is equal to a constant. So, we can substitute this values here and this we get this term is equal to a constant. So, from here we can write for phi 0 delta phi 0 this is constant times this term in the denominator. Now, we see that in this all the terms in the denominator are constant except for beta and gamma which are varying slowly because acceleration rate is small. So, if E0 and phi s are fixed we get delta phi 0 is equal to some constant divided by beta s gamma s to the power of 3 by 4. Also now the phase space area is an adiabatic invariant. So, it is a constant. So, from the above equation and from this because the phase space area is an adiabatic invariant we can write W0 is equal to constant times beta gamma to the power of 3 by 4. So, we have now the phase is delta phi 0 is inversely proportional to beta s gamma s to the power of 3 by 4 whereas, W0 is directly proportional to beta s gamma s to the power of 3 by 4. So, this means that so, this results they describe a decrease of the phase amplitude. So, the phase amplitude decreases as the acceleration happens or the energy increases and then increase of the energy amplitude the energy amplitude increases as the beam is getting accelerated. So, increase of the energy amplitude of phase oscillations during acceleration in a lemma. So, this effect is known as phase damp because the phase is decreasing. So, this is known as phase damping the area remains constant because it is an adiabatic invariant. So, let us say we have an initial amplitude of phi 0 i and the synchronous velocity at that instant of time is beta s i and at some point of time later the phase amplitude is phi 0 and the synchronous velocity is beta s. So, then we can write from the previous results we can write delta phi 0 upon delta phi 0 i is equal to beta s gamma s i by beta s gamma s to the power of 3 by 4. So, there is an inverse relationship here there is an inverse relationship here. So, we see that the phase is getting damp. So, this is at the initial location and this is after acceleration. So, beta 2 is greater than beta 1. So, after acceleration we see that the phase width has decreased. Now, similarly the energy width scales as so, delta w 0 by delta w 0 i is equal to beta s gamma s divided by beta is gamma s i to the power of 3 by 4. So, there is a direct relationship here. So, as the acceleration happens so, as we go from here to here the energy half width increases ok. So, the phase half width or the phase width is decreasing and the energy width is increasing the area of both the ellipse will remain same and remember this holds when the acceleration rate is small. So, phase damping of the beam in longitudinal direction caused by beam acceleration is shown here. So, the phase width of the beam decreases and the energy width increases the total area remains constant in this process ok. So, coming back again to the comparison between proton and electron which we have seen in the very first lecture. So, we see that even at very low values of kinetic energy. So, let us say at a few MEV the electron becomes relativistic. So, its velocity becomes constant. So, as the kinetic energy increases the mass is increasing. So, here the velocity has become constant whereas, for the proton since its mass is much more than that of the electron this happens at a much higher value of energy. So, since electrons become relativistic at low energies the velocity of electrons becomes constant at these low energies. So, energy is increasing for the electron, but velocity is not increasing because velocity has almost is become equal to the velocity of light. Now, relativistic electrons do not have the problem of phase stability. So, why is this problem of phase stability coming because different particles see different values of the electric field and they get different energies and with because we are getting different energies their velocities are different, but for electrons which are already relativistic even though the energy gain is different they will still move with the same velocity. So, relativistic electrons do not have the problem of phase stability since the velocity of the electron does not change with the kinetic energy. Also, the cell length of the electron linux does not change with energy. So, as we have seen before the cell lengths are kept constant for a electron linux. So, it is either beta lambda or beta lambda by 2 depending on whether it is a zero mode structure or a pi mode structure. So now, let us see the longitudinal dynamics of low energy beams injected into a v is equal to c linux. So, electron linux are generally known as v is equal to c linux because they are designed such that the velocity of the electron has become constant. So, the cell length has become constant beta lambda beta lambda by 2 has become constant. So, for the electrons the cell length is constant. Now, low energy electrons let us say typically at 450 kEV, so they have become relativistic at a few MEV at in this region let us say 50 kEV, 100 kEV and so on they are still non relativistic. So, but you would like to inject them in the same linux in the same v is equal to c linux because it saves cost. So, low energy electrons are typically injected into a linux with velocity nearly half the speed of light. So, if you see the beta for a 50 kEV electron it is 0.412. So, this is the electron here is non relativistic, but you would like to inject this into the same v is equal to c accelerating structure. So, it is attractive to use the same accelerating structure that is used for acceleration of extremely relativistic electrons for the lowest energy electrons because it saves cost otherwise you will have to design two linux. One for the lower energy electrons the non relativistic electrons and the other one for the high energy relativistic electrons. So, since now the at low energy is the electrons are non relativistic. So, in this case what will happen the particle phases will slip on the wave. So, as we have seen earlier that if the particle is non relativistic and you inject it if you inject it into the linux so the phase will slip. So, now we will discuss here the capture, bunching and acceleration of an initial v is equal v is less than c electron beam in a v is equal to c travelling wave structure. So, we have a v is equal to c travelling wave structure. So, what is a travelling wave structure it is a periodically loaded waveguide. So, if you recall the previous one of the previous lectures we have studied about travelling wave structures. So, we saw that the hollow waveguide cannot be used for acceleration because there the phase velocity is greater than the velocity of light. So, it will be difficult to synchronize any particle with the wave because no particle can move with the velocity greater than the velocity of light. So, this waveguide structures are loaded periodically to slow down the phase velocity. So, once you get a phase velocity equal to the velocity of the electrons which is very close to the velocity of light then this can be used for acceleration. So, these are basically periodically loaded structures. So, in this structure we try to put in an initial electron which has initially which has a velocity less than the velocity of light eventually it will gain energy and its velocity will become almost equal to the velocity of light. So, now equation of motion of a particle with position z at time t accelerated by a travelling wave with longitudinal electric field amplitude E0 is. So, this is the energy gain. So, this is simply rate of change of momentum and this is the energy gain. So, Q E0 cos phi as a function of z and t. So, where the phase of the travelling wave with velocity approximately equal to c as a function of z and t and this phi can be written as omega t minus 2 pi z by lambda. So, we can differentiate this we get d phi by dt is equal to omega minus 2 pi by z d z by dt. So, phase motion is described by this equation. So, d phi by dt you can simplify this further. So, this is V z can be written as beta into c. So, if you simplify the omega term. So, you get the phase motion as d phi by dt is equal to 2 pi c by lambda 1 minus beta. Now, beta is less than 1. So, we are talking of a non-relativistic particle beta is less than 1. So, this will be a positive quantity. So, since beta is less than 1 phi increases with time. So, since this is a positive quantity, phi increases with time which means that the particle falls further behind the initial phase on the wave. So, the phase will slip. Now, it is convenient to change the independent variable from time to phase. Now, in this expression, so here this equation is written in terms of time, we would like to change this from time to phase. Also, here we have beta gamma. So, we can use the result that derivative of beta gamma is equal to gamma cube d beta. So, substituting this in this equation, we get m c gamma cube d beta by d phi d phi by dt is equal to q e 0 cos phi. Now, in this expression, we can substitute d phi by dt from the above expression and expressing gamma as a function of beta, we get this expression. So, we have 1 by 1 plus beta under root of 1 minus beta square d beta by d phi is equal to q e 0 lambda 2 phi m c square cos phi. So, we get this expression. Now, we can integrate both sides to find dependence of phase on velocity. So, we want to get the dependence because now this is a non-relative elastic particle and as it is gaining energy, what happens to the phase? So, we want to find the dependence of phase on velocity during the acceleration process. So, to carry out the integration, we introduce a new variable alpha. So, we define alpha as beta is equal to cos alpha and we use these integral and this identity. So, integral of d alpha by 1 plus cos alpha is equal to tan alpha by 2 and the identity that tan alpha by 2 is equal to under root of 1 minus cos alpha upon 1 plus cos alpha and cos alpha is nothing but beta. So, this will be under root of 1 minus beta divided by 1 plus beta. So, with these simplifications, we get the final result as sin phi is equal to sin phi i plus 2 pi m c square divided by q e 0 lambda and then within brackets under root of 1 minus beta i 1 plus beta i minus under root of 1 minus beta by 1 plus beta. So, here beta i is the initial velocity. So, the velocity with which you are injecting the electron into the v is equal to c lambda. Beta is the final velocity. Phi i denotes the initial phase and phi denotes the final phase. So, we see here that there is no oscillatory motion for phase motion. So, there is there are no oscillations here. Then because of acceleration, beta is greater than beta i, so beta is greater than beta i. The second term, so second term on the right, so this term is positive and it increases with increasing beta. So, this is positive and this term increases as beta increases and therefore, sin phi is greater than sin phi i. So, this sin phi is greater than the value of sin phi i. So, that means the phase is increasing. So, the phase becomes more positive as beta increases. So, if we want the particle to approach the crest where phi is equal to 0, the particle must be injected at a negative phase. So, this is the phase, this is the crest. So, we see that here sin phi is to the final phase is greater than the initial phase. So, the phase is increasing. If you inject the particle with this phase, the phase as beta is increasing, the phase will gradually increase here. Now, electrons are generally accelerated at the crest because they do not have the problem of phase stability. So, usually for electron linux, we operate them at phi is equal to 0. So, here now the phase becomes more positive as beta increases. So, as velocity is increasing, the phase is becoming more velocity. So, now if we want the particle to approach the crest, so if we want the particle to move as beta is increasing and finally approach this point here, the particle must be injected at a negative phase. So, particle must be injected at a negative phase. So, it should be injected at between minus phi by 2 and 0. So, we also see that as beta approaches unity. So, as beta approaches unity, that means it becomes relativistic. The phase phi approaches a constant asymptotic value. So, this term will go to 0. So, this phase will become the final phase will achieve a constant asymptomatic value. So, here phi at phi infinity is equal to sin phi i plus this term, this term goes to 0 because beta has now become almost equal to 1. So, now the second term we can define as, let us write it as f, this is for beta is equal to 1. So, we see that here the minimum field gives the maximum phase slip. So, this is the phase slip because this is the amount by which the phase has slimmed. So, since this phase slip is inversely proportional to E0, so we see that the minimum field gives the maximum value of phase slip. Now, suppose we choose phi i is equal to minus 90. So, initial phase is equal to minus 90 or slightly more positive because here there will be no acceleration, the electric field corresponding to minus 90 is 0. So, if we choose this or slightly above it, so and then we choose f is equal to 1, this corresponds to placing the asymptotic phase at the crest. So, when beta will approach 1, then this phase will approach 0. So, here we can calculate the value of E0 for this. So, E0 is equal to this and this shows that smaller the initial velocity, the larger the accelerating field must be. So, if you want to approach here, smaller is the velocity at which you are injecting into the V is equal to C lina. So, the larger accelerating field is required such that this particle reaches the crest here and when beta is equal to 1. Okay, now let us see about beam bunching. So, we know that before acceleration in any RF lina, the input beam must be bunched and it must be bunched at the same frequency as that of the applied operating frequency of the lina. So, for bunching, the synchronous phase is chosen to be minus 90. So, we choose the synchronous phase as equal to minus 90, so this is 0. So, we have a cavity and we inject the beam such that the synchronous phase is equal to minus 90. So, here there is no acceleration because the field accelerating field here is 0. The phase width of the separate rix is maximum. So, we have just seen that if the phase is equal to minus 90, the phase width of the separate rix is maximum extending from plus 90 to minus 270. So, if you choose here, the phase width is from here to here. So, it is full 360 degree. So, that means when you inject the DC beam into it, the whole beam can be captured. So, maximum DC beam can be captured in the bucket. So, an RF cavity followed by a drift can be used to bunch the DC beam. So, we have an RF cavity, it could be a pillbox cavity or a single cell cavity and then it is followed by a drift space. So, drift space is a field-free region. So, let us say we have a DC beam. So, now let us see it is extending over the full 360 degree from minus 270 to plus 90 degree. So, this is a DC beam. Now, we inject it into the, this DC beam is injected into the cavity. So, the central part of the beam when it reaches the cavity, it will see a phase of minus 90 because the synchronous phase is minus 90. So, this part of the beam, the central part of the beam will see a phase of minus 90. So, since here the field is 0, there is no energy gain. So, these particles will move ahead with the same velocity. Now, the head of the beam, the part which is the leading part of the beam. Now, this part sees a field. So, it arrives earlier than the central part. So, this sees a field which is lower than the field seen by the central part of the beam. So, it sees a decelerating field. So, it gets decelerated. So, it will move towards the central part of the beam. Now, this tail of the beam, this part of the beam sees a field. So, it arrives in the cavity later than the central part of the beam. So, it sees a field which is accelerating or greater than the field seen by the central particle. So, this part of the beam is accelerated. So, as this DC beam moves into the cavity, the head of the beam is decelerated, the tail of the beam is accelerated and the central part moves with the same velocity. So, as it comes out, we see that the beam is bunched and the beam is bunched at the same frequency at which is applied into the RF cavity. So, in this way, beam bunching can be done and then this beam bunched, it is bunched, can be injected into the RF line for acceleration. So, finally to summarize, we have seen that for electrons, field errors cause a shift in final energy. So, when if there is a field error in the accelerating cavity and you are accelerating relativistic electrons, so the electron will still move from the center of one gap to the center of the next gap in the right time and see the correct phase. But since there is a field error, there will be a shift in the accelerating field seen by the electron. So, there will be a change in the energy, final energy of the electron. For a linear that accelerates non-relativistic ions, the field error changes the particle velocity gain, it causes a shift to a new synchronous phase. So, here the synchronous phase itself changes. So, you can still calculate a value of the synchronous phase, which will give the same energy gain as the designed as that for the designed accelerating field. So, that value you can calculate, it can be calculated. The particles in the bunch oscillate about the synchronous particle. This longitudinal oscillation frequency is usually small as compared to the frequency of the RF that is used for accelerating the beam. During acceleration in a linear, the phase amplitude decreases and the energy amplitude of the phase oscillations increases. So, this is true when the rate of acceleration is small. So, as the beam moves and it is getting accelerated, it moves from one point to the next, we see that the phase width shrinks and the energy width expands. So, this is known as phase damping. An RF cavity followed by a drift tube can be used to bunch a DC beam. So, we know that we need to put in a bunched beam for acceleration in any RF accelerator. So, this bunching can be done by using an RF cavity followed by a drift space. For bunching, the synchronous phase is chosen to be minus 90 degrees. So, then you have the maximum width of the separatrix extending to full 360 degrees and you will be able to capture maximum amount of DC beam. The maximum amount of DC beam can be bunched. So, with this we complete the longitudinal dynamics of particles. In the next lecture, we will study about radiofrequency quadruple which is a type of linear accelerator used for accelerating low energy particles.