 Now, let us calculate the energy gain in the time varying field. So, as you already know let us say we have a DC field ok. So, the energy gain in the DC field. So, here there is a DC field here ok. So, there is an electric field here DC electric field here and so, you can calculate by simply integrating q into integral E dot dz over the length of the gap. So, this is equal to q into E into G which is equal to q v 0. This you have already studied you already know that the energy gain in a DC accelerator is q into the voltage seen by the charge particle. Now, the energy gain in a time varying field. So, here the situation is little complicated because the electric field is no longer a constant value it is now varying with time ok. So, the RF field is changing while the particle is in the gap. So, for simplicity for in the first case let us assume that. So, we calculate the energy gain of in the time varying field on the axis. So, on the axis meets at r is equal to 0. We also assume that there is no variation of E z with z ok. So, it is constant with z. So, it is varying with time it is varying with time, but with z there is no variation ok. In real life there will be variation because there is a there is a gap here. So, the fields there will be fringing fields. So, it the field will not be straight lines it will be something like this ok, but for the time being let us calculate when the electric field is constant with z there is no variation with z. So, again the energy gain is given by q integral E dot dz integrated over the length of the gap. Now, here now E z is equal to E 0 cos omega t plus 5. So, there is time variation. So, the form of E z is like this. So, energy gain. So, energy gain again let us come back to this. So, this is the electric field the energy gain in the gap is simply this. Now, where at t is equal to 0 at t is equal to 0 the field at the centre of the gap is this value ok. So, the particle is at the centre of the gap z is equal to 0 and the phase of the field relative to the crest ok. So, this phase is equal to phi at the centre at t is equal to 0. So, this is the part of the electric field that the particle is c at time t is equal to 0. So, now assuming that the velocity change in the gap is small. So, we can write t as z by v. So, omega t can be written as omega is 2 pi f and t can be written as z by v. So, this is 2 pi f z by v can be written as beta c. So, omega t 2 pi z by beta lambda. Now, in this expression since this is integration over z we can replace omega t with 2 pi z by beta lambda. So, if we do that this is what we get and now we can expand this cos function cos a plus b is equal to cos a cos b plus sin a sin b. So, we simply expand this and then we notice that the second integral goes to 0 since it is an odd function of z. So, we are left with just this first integral. So, if you integrate this. So, here you can see that cos phi is independent of z. So, it can come out of the integral. So, you have to integrate just cos 2 pi z by beta lambda. So, integrating this and applying the limits we get q e 0 cos phi 2 times sin pi g by beta lambda upon 2 pi by beta lambda. So, this 2 can be cancelled. Now, we can multiply and divide by g. So, we get here q e 0 g cos phi and this factor here. So, this can be now written as the energy gain in a time varying field can be written as q e 0 g t cos phi where t is equal to this factor. This factor is denoted by t. So, it is sin pi g by beta lambda upon pi g by beta lambda. This t is called the transit time factor. So, now the energy gain in the gap is q into e 0 g. Now, this q into e 0 g is simply the energy gain in the DC field. This is what we had got. Now, in addition to this we have two more terms ok the transit time factor and the cos phi. So, t is the transit time factor which is given by sin pi g by beta lambda upon pi g by beta lambda and cos phi. Cos phi is coming into picture because it depends upon the phase of the field that what is the value of the electric field that is going to be seen by the charge particle. So, if it is phi it will see this if it was 0 it will see a maximum field. So, or if it was pi by 2 it will see a 0 field. So, it depends upon what is the phase of the electric field that is seen by the particle. So, this that is why this phi factor is coming into picture. So, now we see that regardless of this phi. So, we can take phi as equal to 0. So, that this term is equal to 1 regardless of the phase phi ok. The energy gain in a time varying field is always less than the energy gain in a DC field ok. So, because this factor t is always less than 1 sin x upon x it is always less than 1. So, we see that regardless of whatever phase we choose ok. The total energy gain in the gap is always going to be less than the energy gain in a DC field and this is by a factor t. So, if you draw this transit time factor. So, it is of the form sin x by x. So, you draw this transit time factor versus g by beta lambda. So, we see that it is maximum it has a value of 1 for g equal to 0. Now, g equal to 0 is not possible because that means there is no gap. So, no acceleration ok. And as the gap length increases the transit time the transit time factor decreases and hence the energy gain decreases. So, one of the important criteria of linear design is to maximize the value of transit time factor. So, you have to have a proper compromise. Now let us say you make the gap too small then there could be breakdown the electric field locally could be very high and there could be a breakdown. So, you have to optimize what value of transit time factor you have to choose. So, t is a measure of the reduction in the energy gain caused by the time sinusoidal time variation of the field in the gap. So, t is always less than 1 for a finite width of the gap. The factor comes because the particle takes a finite time to travel through that gap in which the field is changing ok. So, this factor is coming because see in a DC field the electric field is constant. So, in a DC field the electric field is constant whereas in the RF field it is changing like this. It is there is a finite time to transit the gap and during this time the field is changing with time. So, because of this the transit time factor is coming into picture and as I said important criteria in linear design is to maximize the value of t ok. Now let us calculate the transit time factor if E0 is not constant along the axis ok. So, now Ez is a function of z. So, as I said there will be so this is one drift tube this is the hole in between this is another drift tube and this is the hole in between. So, electric field lines if you calculate if you see here they are because of the fringing fields they are not straight. So, Ez has a variation with both r and z. So, a more realistic accelerating field depends on both r and z ok. So, here Ez can now be written as E the amplitude is a function of both r and z. So, again calculating the energy gain in the same way. So, energy gain is Q integral E dot dz minus g by 2 to g by 2 and let us say for now you are calculating at r is equal to 0 ok. So, E there is just variation in Ez with z. So, you are calculating so here now this becomes E0z because r is equal to 0. So, again like before we expand this cosine function cos a plus cos b is equal to cos a cos b plus sin a sin b and then we can choose the origin at the electric center of the gap ok. The electric center of the gap is defined as integral E0z sin omega t dz is equal to 0. So, if we choose the origin at the electrical center of the gap. So, this is so here this term goes to 0 and we are left with just the first term. So, the energy gain is now just this part. Now remember that the voltage in the gap can be calculated by simply integrating the electric field in the gap. So, here now we divide and multiply by this voltage. So, we have multiplied here and divided here by the voltage. So, we get this expression and cos phi is independent of z. So, we can take it outside ok. So, just compare this. So, we have Q this is the voltage V0 and then there is this cos phi. So, if you compare it to the formula of energy gain and time varying fields you see that this factor has to be transit time factor. So, this is the general form of the transit time factor. When your electric field is not constant along z it is varying with z. So, it is simply this integral followed divided by the voltage ok. Again assuming that the velocity change in the gap is small you can write omega t as 2 pi z by beta lambda and you can write 2 pi by beta lambda as k z. So, you can substitute here omega t as equal to k z. So, the general formula for transit time factor is now at r is equal to 0 and E z varying with z ok. So, it is now 1 by V0 and integral minus g by 2 to g by 2 E0 z cos k z d z. So, where voltage is given by this expression and k is equal to 2 pi by beta lambda. Now, if you have a radial dependence also. So, that means now here this was calculated at r is equal to 0 we calculate at some general r. So, electric field can vary with r also and in fact, it will as you can see here there is variation with r. So, then the transit time factor for a general r and k or general r and z can simply be replaced by this expression. So, E is now a function of r and z cos k z d z. So, this is the general formula for transit time factor ok. Now, coming back to this slide from lecture 1. So, the difference between proton and electron. So, if you remember what we learnt in the first lecture we saw that the electron becomes relativistic at very low energy. So, it approaches the velocity of light at very low energies and from then on there is not much change in the velocity V is constant what is increasing is the mass of the electron. The proton on the other hand its velocity is increasing till about 1 gV and then it becomes constant ok. And if you have heavier ions then their graph will be like this even different. So, this as we discussed yesterday this has important implications on the design of both proton and electron axi, then ax. So, now from whatever we have learnt let us see what is the difference between electron and ion accelerators. So, the velocity of the electrons is almost constant ok. So, V is equal to almost equal to c ok even at low energies the velocity has become almost constant. So, beta is almost equal to 1. So, the cell length which is you have seen that for pi mode structure it is beta lambda by 2 and for the 0 mode structure it is beta lambda. Now cell length depends upon what beta. So, for the electron accelerators V cell length is constant because beta is constant. For proton accelerators on the other hand. So, for proton you can see that beta is increasing. So, as beta is increasing V cell length will also increase. So, velocity of the ions or protons increases with the kinetic energy hence the cell length will also increase to maintain the synchronism. Now, since the electrons move with almost constant velocity phase stability is not so important for electrons. Now why is this problem of phase stability coming because the early particle and late particles are moving with velocities different from that of the synchronous particle ok. For the electrons all particles will move with the same velocity. So, this problem of early particle coming even earlier or the late particle coming even later is not there with electrons. So, in fact for electrons you can use the synchronous phase as 0 because here you will get maximum acceleration and since there is no problem of phase stability. So, you can operate here and get maximum energy gain. Now, since beta for electrons is very high. So, beta is close to 1. So, beta for electrons is very high as compared to ions. Electron accelerators are operated at high frequencies. Now, see what is the cell length? The cell length is equal to beta lambda. Now, beta is already very high in the case of electrons as compared to proton or ion accelerators ok. So, if beta is high in order to keep the cell length at a reasonable value lambda should be small or in other words you can you should operate at higher frequencies. So, electron accelerators are operated at higher frequencies often in the Giga Hertz range whereas ion accelerators are operated at lower frequencies depending on what is the value of beta. So, lower the beta lower is the value of the frequency that is used for acceleration. So, coming back to this slide again. So, the first accelerator which was conceived by Ising and Windrow as I said it consisted of drift tubes to which RF voltage was directly applied. The RF voltage was directly applied and this was kept this whole system was kept inside an evacuated glass cylinder. Now, all accelerators they operate in vacuum because otherwise the charge particles will collide with the particles the air or atmosphere and they will lose energy and get scattered. So, they are all accelerators are operated under vacuum condition. So, there is an evacuated glass cylinder and this is how acceleration took place. So, this worked very well at lower energies, but when people tried to increase the energy. So, what happened now cell length is equal to beta lambda. So, as beta increased in order to keep the cell length reasonable lambda had to be decreased or in other words frequency had to be increased. Now, the problem with such a system where the RF voltage is directly applied to these drift tubes or hollow cylinders is that at higher frequencies when the wavelengths become comparable to the size of these drift tubes this system starts radiating ok. So, instead of storing energy here in the gaps in the form of electric field. So, this starts radiating the energy like an antenna. So, the power radiated the radiated power is given by this formula half C V RF square where C is the capacitance of this region and V RF is the applied RF voltage. So, such a system fail to work at higher energies because you had to go to higher frequencies and at higher frequencies this started radiating like an antenna. So, what is the solution? In 1946 Alvarez overcome overcame this problem he put the drift tubes inside a high Q cavity. So, remember in the first case Vidro had icing and Vidro had kept it inside a hollow glass inside a evacuated glass tube. So, what did Alvarez do? He kept it inside a evacuated metal cylinder. So, a metal cylinder becomes a cavity ok. So, this is a high Q cavity high Q means something that stores more energy. So, Q is the quality factor something that stores more energy than it dissipates and then utilizing the time varying electric fields associated with the standing electromagnetic wave set up in the cavity. So, what did he do? He took a high Q cavity and now instead of applying voltage see there is no voltage applied directly to the drift tubes. What he did was put in electromagnetic waves inside this cavity and then using the electric fields associated with the electromagnetic fields. So, you know that electromagnetic fields, electromagnetic waves have electric fields and magnetic fields. So, the electric field is also varying in time. So, he utilized the electric fields associated with these electromagnetic waves and then used it for acceleration ok. So, this type of structure was a zero mode structure the cell length is equal to beta lambda and so, let us now try to understand how he used the electric fields of the electromagnetic waves for acceleration. So, in order to understand this let us come back to the electromagnetic waves in free space. So, we have Maxwell's equation this is the general form of Maxwell's equation. So, divergence of D is rho, divergence of D is 0, curl of E is minus del B by del T, curl of H is J plus del D by del T, where rho is the electric charge density and J is the current density. And if you do it in if you write the Maxwell's equation in free space. So, there are no charges or currents. So, the charge and current you can put rho is equal to 0 and you can put the current as equal to 0. So, this is what you get ok. And here 1 by c square is equal to mu 0 epsilon 0. Now, we want to find the solution satisfying the Maxwell's equation in free space. So, taking the curl of the equation curl of E. So, curl of E is minus del B by del T we take the curl and use the vector identity here. So, this is what we get on the left hand side and the right hand side. Now, since we know that in free space divergence of E is equal to 0 and curl of B is equal to 1 by c square del E by del T. So, we just substitute here this goes to 0 and curl of B is equal to 1 by c square del E by del T. So, we get a equation in the form of this is the wave equation this and it has plane wave solutions of the form like this. So, this is like a travelling wave solution it is plane wave travelling wave solution. So, here E 0 is the amplitude k is the wave number and omega is the angular frequency. So, E 0 is amplitude and the phase velocity is c which is given by omega by k which is equal to 1 upon under root mu 0 epsilon 0. So, this relation between any relation between omega and k this is known as the dispersion relation. Now similarly again we take the curl of the equation curl of B. So, here and we do the same simplification. So, we get a wave equation for B as well ok. So, this is a again a plane wave equation and it has a solution like this and similarly here B 0 is the amplitude k is the wave number or the propagation constant. So, again we get the same dispersion relation here also. Now since curl of E is equal to minus del B by del T and we have already found out expressions for E and B we simply substitute here in this expression. So, we get here k into E 0 is equal to omega B 0 ok. So, if this implies now you can see from this expression this implies that k E 0 and B 0 ok they are mutually perpendicular to each other they are all mutually perpendicular to each other. Also we have curl of B is equal to 1 by c square del E by del T. So, again substituting these two equations here we get another expression k cross B 0 is equal to minus omega by c square E 0. So, this again suggests that the propagation constant the magnetic field and the electric field they are mutually perpendicular to each other. Also from this we can get that the electric and magnetic field amplitudes they are related as E 0 by B 0 is equal to c. So, that means E 0 is more than B 0 by a factor of c. So, thus from this we get electromagnetic waves in free space. So, electromagnetic waves in free space are TEM type of waves ok. So, TEM means transverse electric and magnetic. So, we see that the electric field is perpendicular to the magnetic field. So, they are mutually perpendicular and they are perpendicular to the direction of propagation as well. So, the wave is propagating in this direction and we have an electric field which is sinusoidal and perpendicular to k. We also have a magnetic field which is sinusoidal and perpendicular to k and electric field and magnetic field are also mutually perpendicular to each other. So, electromagnetic waves in free space they are plane waves they are TEM waves both electric and magnetic fields are mutually perpendicular to each other and to the direction of propagation. And magnitude of electric field is more than that of the magnetic field by a factor of c ok. Now, it is important to note here that even though so, we have an electric field in an electromagnetic wave we have an electric field which is which has a sinusoidal variation and you know how to accelerate charge particles using sinusoidal variation. So, even though it has a sinusoidal time variation we cannot use the electromagnetic waves in free space for acceleration. As now if you want to use this for acceleration your beam must co-propagate with this ok this wave. So, the wave is moving in this direction and the beam also has to move in this direction such that it always sees the electric field. And now notice that if the beam also moves in this direction that means along the direction of k then what happens is that the electric field is always perpendicular to the direction of velocity of the charge particles ok. So, e dot v will be equal to 0 and there will be no acceleration. So, for change in kinetic energy there should be a component of electric field in the direction of propagation of the charge particle. So, electromagnetic waves in free space cannot be used for acceleration as the electric field will be perpendicular to the direction of beam velocity at all times ok. So, finally summarizing so with time varying fields you can use the same small voltage it can be used repeatedly to accelerate to high energies by successively accelerating charge particles over many gaps ok. So, here unlike DC accelerator you need not generate a very high field you can generate a small voltage and or a small electric field and this can be used several times over several gaps and the total energy gain will then depend upon n into delta w. So, how many gaps you have and the energy gain in each gap. So, this energy gain in each gap need not be very high in a DC accelerator this itself we try to maximize by generating a high voltage. Here this number can be small this by increasing the value of n we can accelerate to very high energies ok. So, the necessary condition for acceleration using time varying fields is isochronism that is the particle arrives at each gap at the right time ok or in the right phase of the electric field. It has to see the right phase of the electric field every time to get accelerated ok and of course there should be a component of electric field in the direction of velocity of the beam and without isochronism also we can get acceleration, but we cannot get sustained acceleration. So, in order to get sustained acceleration isochronism is very important. So, the cell lens should be adjusted such that the particle arrives from one gap to the next gap in time t by 2 or t as it may be if it is a pi mode structure or a 0 mode structure. So, for sustained acceleration over large lens many of over many tens of accelerating gaps isochronism is very important. So, this is known as the principle of successive acceleration and we do not use the entire positive cycle for acceleration ok, because then there will be a huge spread in the kinetic energy. So, we use only a small portion of the RF cycle for acceleration. So, even in the positive cycle we do not use the entire positive cycle we for acceleration with stability with phase stability. We use the portion where the synchronous phase lies between minus pi by 2 and 0. If you use the portion from 0 to pi by 2 there is no phase stability there is acceleration, but there is no phase stability and we will not be able to get sustained acceleration. Then we also calculated the energy gain in the time varying field and we saw that the energy gain in the time varying field is less than the energy gain in DC field by a factor of t. This is because the t is known as the transit time factor and this is because the beam takes a finite time to transit through the gap and during that time the field in the gap is not constant it is changing with time. So, because of which this transit time factor comes into picture ok. So, in the next lecture we will see more about electromagnetic waves in a medium bounded by conducting boundary and we will see that how they can be used for.