 So, now let us take the simplest mode in the cavity which is the TM010 mode. So, it is the fundamental mode in a PILBOX cavity. So, these are for the TM mode these are the fields here BZ is equal to 0 and then EZ has this value and from EZ you can calculate the values of ERE theta, VR and D theta. So, for TM010 mode M is equal to 0, N is equal to 1 and P is equal to 0. So, we put here everywhere M is equal to 0, P is equal to 0. So, we see that this goes to 0, this also goes to 0, this goes to 0. So, we are left with only EZ and B theta. So, we just have the EZ field and the B theta field. So, you can write from here you can put in the values of M, N and P. So, you will get B theta as minus j omega rc by h01 c square E0 j0 prime of h01 rc by r. And using the relation of the Bessel function that j prime 0 is equal to minus j1. So, we can write this we can write this in terms of j1. So, we have just two fields EZ and B theta. Also notice that they are so, this factor j here this tells us that they are out of phase with each other. So, when E0 EZ is maximum at any location B theta is minimum and so on. So, this is how the fields look like in the cavity you have the EZ field here it is maximum at the center and it has to go to 0 at the boundary because of boundary conditions. And B theta field is they form field lines around the electric field lines. So, EZ is j0 of the form of j0. So, if you plot EZ with r you see that it is maximum at the center and going to 0 at r is equal to rc. Similarly, B theta has a form of j1. So, it is 0 at the center and it increases like this as the radius increases. So, only EZ for a TM0 1 0 mode in a pillbox cavity only EZ and B theta fields are present. So, both EZ and B theta they have variation with r only. So, since m is equal to 0 and B is equal to 0 there is no variation in theta and z direction. So, there is variation with r only and the resonant frequency. So, you can calculate the resonant frequency from this formula you can put in the values of m, n and p in the formula for the resonant frequency. So, you see that it is independent of the length of the cavity it depends only upon the radius of the cavity and it is inversely dependent upon the radius of the cavity. So, higher the radius smaller is the frequency and so on. So, this is how the fields will look like. So, as I said it is a standing wave. So, that means the pattern is constant along various directions, but there is time variation and the time variation is like this. So, this is EZ field and this is B theta field ok. And since the electric and magnetic fields are out of phase with each other. So, you see that the variation in time is like this ok. So, you have a bill box cavity let us say it is in the TM010 mode and so, the electric field is like this there is of course, a magnetic field in the theta direction also, but it is 0 at the axis. So, let us say this is the axis. Now, the field variation with time is like this and we have already studied that how using this type of field pattern we can accelerate charge particles ok. So, we saw that we need not use the whole positive cycle for acceleration because the whole even though the whole positive cycle will produce acceleration, but there is phase stability only in this part. That means in your if you have to choose your synchronous phase between minus pi by 2 and 0. From 0 to pi by 2 there is acceleration in this region, but there is no phase stability. So, you have to choose your synchronous phase in this region. Also, you do not use the entire region for acceleration because then some particles will see a field that is 0, some particles will see a field that is maximum and there will be a huge spread in the kinetic energy of the charge particles which you do not want. So, you use only a small portion of the cycle for acceleration. So, let us say this portion ok. So, now the field in this cavity is changing like this. So, you want the particles or the beam to see just this part of the electric field. So, as the field is increasing you inject your beam bunch at the times and the phase of the field at the center here is 5 years. So, then the bunch sees this field it gets accelerated and it comes out. So, this length of the cavity can be adjusted accordingly. In actual practice the cavity is shaped to have to concentrate the electric field in the region near the axis. So, it is shaped like this to reduce this gap because we have already studied that the transit time factor depends upon the gap and how high an energy gain you get depends upon the value of transit time factor. Smaller the gap higher is the transit time factor. So, normally this is shaped the cavity is shaped like this to increase the transit time factor. So, as the field is increasing if you inject the beam bunch into the gap at the time when the phase of the field is 5 years it will see the right field and get the energy gain. So, electric field in the pillbox cavity for a TM010 mode is in the z direction it is constant along the z direction. So, along the axis of the cavity there is no variation in the electric field along the length of the cavity and the electric field has a sinusoidal variation in time. So, we inject the beam into the cavity when the phase of the electric field is 5 years and in this way we can accelerate the beam. Now, we can also have a long cylindrical copper cavity in the TM010 mode. So, instead of having just a small pillbox cavity we have a huge big pillbox cavity ok and you know that the frequency of the mode does not depend upon the length of the cavity. So, you can have a long cavity like this and inside that you put in the drift tubes ok and you adjust the distance from the center of center of one gap to the center of other gap as equal to beta lambda. So, that means the particle will move from here to here in time. So, again the field inside here. So, there is an electric field in the z direction. So, the field is increasing with time and you inject the charge particle in the first gap here at the time when the phase is 5 years here. So, it sees the right field and it gets accelerated and then for the remaining part of the time from here to here that means time t it spends inside this drift tube and then it comes out here and sees the right field again ok it sees the right field again and then it gets accelerated again. Again it spends the time time capital t inside this comes out here sees the right field and gets accelerated. So, in this way you can have you can have several drift tubes like this and you know that field cannot penetrate inside the hollow conductor. So, inside this the bunch or the particle beam will be shielded from the electric field it will see the electric field only in these gaps and get accelerated. So, this is the Alvarez type of structure and this is known as the drift tube linac or the detail. So, this is what was proposed by Alvarez. So, he said take these drift tubes and put this in a high Q cavity and use the electric fields associated with the electromagnetic waves. So, the electromagnetic waves are now forming mode inside the cavity and in the TM010 mode there is electric field along the Z direction ok. So, you can use the electric fields associated with these electromagnetic waves inside this cavity and use it now for acceleration. So, this is a picture of a drift tube linac these are the drift tubes. So, there are several drift tubes here. So, this is the beam hole through which the beam goes there are gaps between the drift tubes. So, the drift tubes shield the beam bunch from the undesirable part of the RF electric field and the beam is the beam is like in the transverse direction it tends to spread it is like a ray of light. So, the beam tends to spread in the transverse direction. So, it needs to be focused. So, just like a ray of light is focused using lenses here we have quadruples which we will see in future lecture they are used for focusing the beam. So, there are inside these drift tubes there are quadruples which are used for focusing the beam in the transverse direction. So, let us see an animation of the drift tube linac. So, this is the drift tube linac this is the tank these are the drift tubes. So, this is the beam it gets accelerated in the gap here and these drift tubes they contain magnets which is used for focusing the beam in the transverse direction. Okay. So, these boundary conditions so, when we said that the tangential component of electric field is 0 or the normal component of magnetic field is 0 this is assuming a perfect conductor. However, in real life we do not have perfect conductors we have good conductors and in good conductors the field penetrates inside up to a distance equal to the skin depth. So, the field penetrates inside the conductor so, this is the conductor okay and it dies down exponentially. So, the distance the field penetrates up to a distance equal to the skin depth and this depends upon the conductivity of the material. So, that is why in order to have low skin depth a high conductivity material like copper is used for making these cavities. And because the fields penetrate up to a certain skin depth surface currents will flow through the cavity material and this gives RF surface resistance. So, RF surface resistance is simply run upon the conductivity multiplied by the skin depth. So, because of this surface resistance now and the currents flowing on the surface of the cavity there is a power dissipated in the cavity. So, whatever power you are feeding into the cavity or whatever electromagnetic so, these electromagnetic waves that you put inside they carry power which is used for setting up the field in the cavity. So, you have a cavity so, you put in some RF power that is the total power part of the power is dissipated in the structure okay it is dissipated because the fields are penetrating up to a distance equal to the skin depth and there is some RF surface resistance. So, this part of the power is dissipated in the structure the dissipated power can be calculated from this formula. So, here it is half rs rs is the RF surface resistance and then taking the surface integral of h square over all surfaces of the cavity okay. So, this is this is how you can calculate the power dissipation then some power whatever power you are feeding into the cavity a part of the power goes to the beam. So, beam power can be calculated as the beam current into the energy gain. So, whatever is the energy gain in the cavity multiplied by beam current. So, beam is like a flow of charge particles and moving charge constitutes a current so, this is known as beam current. So, whatever power you are feeding inside the cavity part of it is dissipated in the structure and part of it is stored in the cavity which is used for acceleration and that is given to the beam. So, total power that is given to the cavity is the sum of the dissipated power and the beam power. So, this is how you power an RF linux. So, let us see you have a cavity this could be any type of accelerating cavity it could be a pillbox cavity or it could be a drift tube linux or any other structure okay. So, first you decide to what energy ranges you want to accelerate with charge particle and then you design the cavity. So, you decide what frequency you want to operate it at and then you get a source at the same frequency. So, if the frequency let us say you want to operate this in Tn010 mode at some frequency f0. So, you get an RF power amplifier which could be a klystron a solid state power amplifier or any other power source. So, at the same frequency. So, in other words it will give you electromagnetic waves high power electromagnetic waves at this frequency. You use a waveguide you know that waveguides can be used for propagation of electromagnetic waves. So, the electromagnetic high power electromagnetic waves produced here are propagated through a waveguide into the cavity. Now, if this frequency f0 matches the frequency f0 in the cavity of the Tm010 mode which depends upon the dimensions of the cavity then the field is set up corresponding to the Tm010 mode. So, that means you have electric field in the z direction. So, this you can use for acceleration. Now, if the power is coming at any other frequency. So, remember the cavity will not accept any other frequency it will accept power at the frequency of the corresponding mode only. So, if it is coming at any other frequency it will be reflected back. So, the power if it is reflected back it can go back and damage the RF amplifier. So, for that a device called circulator is put here in between the circulator is a one way device. So, it will not allow the reflected power to go back into the klystone it will direct it in another direction towards an RF load. So, this power now gets dissipated in the load. So, now if you are able to now if you are able to send in power to this cavity at the right frequency this power is all it all enters into the cavity and it sets up the right mode. So, you will have electromagnetic fields you have an EZ component here you have a B theta component here and this it has a time variation like this. So, some of the power will get dissipated in the structure and part of the power is going to set up these fields. When the beam comes here it draws power from these fields and gets accelerated and it gets an energy gain let us say delta W. So, power going to the beam is given by the beam current into delta W. Now, let us discuss some figure of merits of the cavity. So, while designing the cavity we have to optimize certain figure of merits. So, one important figure of merit is quality factor. So, this we saw we were seeing right in the beginning. So, Alvarez had proposed in order to solve the problem of the withdrawal in act where the so, in the withdrawal in act if you remember the voltage applied to the drift tubes there was direct voltage time varying voltage that was applied to the drift tubes. So, when you try to accelerate it to high energies. So, in order to keep the cell length reasonable you increase lambda you decrease lambda or in other words you increase the frequency and the structure started radiating. So, it did not store energy in the fields in between the gaps instead it started radiating energy like an antenna. So, to take care of that problem Alvarez suggested that you take these drift tubes and you put it inside a high Q cavity. So, this is the parameter quality factor high Q cavity and you use the electric fields associated with the electromagnetic waves in the cavity. So, what is the quality factor? The quality factor is defined as the ratio of these stored energy at any frequency the ratio of the stored energy to the power dissipated in the cavity. So, the aim of the cavity design is to maximize the value of quality factor. So, if you maximize this that means you are minimizing the power dissipation and you are maximizing these stored energy in the cavity. The stored energy in the cavity is given by this expression it is half Q0 integral over the entire volume of the cavity of x square and power dissipation we have already seen it is half rs and in surface integral of x square over all surfaces of the cavity. So, Q0 by putting these two in the expression for Q0 you can find out the value of Q0. So, you see that the volume integral comes in the numerator and surface integral in the denominator. So, it is the quality factor is high for a geometry for which the volume to surface ratio is high. The quality factor is also defined as 2 pi times the number of RF cycles it takes to dissipate the energy stored in the cavity. So, this expression here can be written as Q0 is equal to omega u and power dissipation is simply du by dt. So, from here you can write u is equal to Q0 by omega du by dt. So, from here you can get this expression by taking the integral of this you can get this expression. So, it means that if you store the energy in the cavity the energy stored in the cavity how long it takes if you remove the RF source how long it takes for that energy to dissipate. So, now as we have seen the quality factor is defined by this expression. So, now here if you notice rs is the RF surface resistance this is the only quantity that depends upon the material. Other parameters are all geometry parameters they do not depend upon the material properties. So, we define a parameter g which is Q0 into rs. So, we multiply this quality factor by rs and we get a parameter which is now independent of the material properties. It now this depends only upon the geometry of the cavity. So, this is known as the geometry factor. The next figure of merit is the shunt impedance this is a figure of merit that is independent of the excitation level of the cavity. So, it measures the effectiveness of producing an axial voltage v0 for a given power dissipation. So, it is v0 square the voltage axial voltage in the cavity divided by the power dissipation. So, it determines how much acceleration one gets for a given power dissipation. Then you sometimes define the effective shunt impedance because we are more interested not in knowing just the voltage, but the energy gain in the cavity and energy gain depends upon voltage multiplied by the transit time factor. So, energy gain is determined not only by the voltage it is also determined by the transit time factor. So, the peak energy gain of the particle it occurs when phi is equal to 0. So, peak energy gain because here the field is maximum. So, let us take phi is equal to 0 and then we multiply rs with t square. So, you are multiplying here v0 t square. So, this is rs into t square this is the effective shunt impedance. Then for long structures we are interested in knowing the shunt impedance per unit length. So, you simply take the shunt impedance and you divided by. So, you have the shunt impedance and divided by the length. So, this will give you the effect the shunt impedance per unit length. So, this is equal to v0 square by the power dissipated n. So, this can be written as v0 square you can write this as e0 square l square p l. So, this is equal to v0 square p by n. So, the shunt impedance per unit length is often useful for long structures and similarly you can have effective shunt impedance per unit length. So, you can just multiply z with t square. So, this gives you the idea about the energy gain as well. So, one of the main objectives of cavity design is to choose the geometry such that you have to maximize the effective shunt impedance per length. So, and this is equivalent to maximizing the energy gain in a given length for a given power loss. Now, if you see the shunt impedance and quality factor in both of these power dissipation comes in the denominator. So, now we define a parameter r by q where we divide r with q. So, the power dissipation in the denominator in both the cases is cancelled out. So, again we get a quantity which is independent of the material of the cavity it again depends only on the geometry. So, this r by q measures the efficiency of acceleration per units stored energy at a given frequency and you can also use z t square by q either of these ratios are useful because they are functions of only the cavity geometry and independent of the surface properties which determine the power loss. So, let us say if you are making a if you have designed a cavity and you want to before fabricating the actual cavity you want to make it in a cheaper material a prototype in a cheaper material. So, you can measure you can make that cavity and you can measure. So, the quality factor the shunt impedance will not come out to be as it is for the actual cavity, but r by q depends only on the geometry. So, it will give you a good idea about your design. Then another factor that we consider in the design is the kilpatrick limit. Now at high fields room temperature cavities they suffer electric breakdown just like in DC in DC accelerators there was a breakdown. So, there the breakdown was at a very high voltage, but here also there is some breakdown. So, Kilpatrick analyzed the data on RF breakdown and he defined the conditions that result in breakdown free operations. The results were given by an empirical formula. This is a formula which is given by Boyd. So, it says that at any given frequency this is the value of the kilpatrick limit. So, if you plot this formula, so here this is at different frequencies this is the value of the kilpatrick field. So, in any cavity let us say we have a TM010 mode. So, you have an electric field here. Now let us say at any sharp corner here the local electric field could be quite high here the local surface electric field could be quite high here. So, for a given value of the accelerating field there could be a surface electric field at any location in the cavity which is higher than the breakdown limit. So, the design aim is to keep this value of the peak surface electric field less than the kilpatrick limit. So, kilpatrick said that if you for a given frequency you calculate this kilpatrick limit and you keep the value of the peak surface field below this limit then breakdown will not happen. So, this was given this formula has been given long back in the 50s now with improved vacuum improved surface conditions you can afford to be more brave. So, now the value of the surface field that is allowed is bravery factor some bravery factor multiplied by the kilpatrick limit. So, you could afford to go higher than the kilpatrick limit given by this formula. So, typically the values of B are in the range of 1 to 2. Okay now let us see for the TM010 mode let us just analyze it in little more detail. So, the electric field is given by this formula EZ and B theta is given by this formula the variation is shown here. So, the electric field is maximum where J0 is maximum. So, if you see J0 J0 is maximum at r is equal to 0. So, the electric field is maximum at r is equal to 0 it is maximum at the axis. The magnetic field is maximum where J1 is maximum. So, the value of J1 is maximum if you see the value of J1 is maximum somewhere here it is at 1.841 and this corresponds to a value of 0.5819 here. So, magnetic field is maximum wherever 2.405 r by rc is equal to 1.841. Okay so, magnetic field is maximum at the value of r equal to this at the cavity wall that means that r is equal to rc. So, we can put here r is equal to rc and calculate the value of the magnetic field we have B theta as E0 minus E0 by C J1 of 2.405 and value of J1 2.405 is 0.5191. So, you can calculate from here the maximum values of the electric field and magnetic field. So, electric field is maximum here magnetic field is maximum somewhere here not at the boundary and you can also calculate the value of the magnetic field at the boundary. The power dissipated in the Bilbox cavity is given by rs by 2 surface integral of h square. So, if you integrate over the entire surface. So, you have the you have the outer wall of the cavity as well as the end walls of the cavity. So, power on the outer wall is given by rs by 2 h square wall into the area. So, you can calculate it here power on each end wall can be again calculated and you can use these identities and calculate. So, the total power dissipated here is given by this expression. So, the stored energy in the cavity is given by half mu 0 integral of x square over the entire volume of the cavity. So, you can calculate this and this comes out to be here the power dissipated we just calculated comes out to be this and from this you can calculate the quality factor of the cavity. So, just summarizing whatever we have learned today, TEMNP and TMMNP modes are excited in a Bilbox cavity. So, both TE and TM modes can be excited. The variation of the fields in the theta and z direction have sinusoidal dependence in the r direction the fields have a Bessel function type of variation and we have also seen that TEMN0 mode does not exist in the Bilbox cavity because by boundary condition TEMN0 mode goes to 0 all all fields in this mode go to 0. So, this mode does not exist in the Bilbox cavity. In a good conductor the fields penetrate to the surface of the conductor equal to the skin depth. Due to surface resistance the power is dissipated in the cavity walls. So, whatever power you feed in the cavity part of it goes up goes in setting up the fields and is given to the beam as beam power and part of it is dissipated in the cavity as power dissipation. So, in TM0 and 0 mode in the Bilbox cavity we have only EZ field that is constant along the length of the cavity under B theta components. So, the field is time varying this EZ field can be used for acceleration. You can also put in drift tubes inside this cavity and this structure is called the drift tube linac as was proposed by Alvarez and this can be used for the acceleration of charge particles. The aim of the cavity design is to maximize the quality factor and the shunt impedance. So, this in this lecture we have tried to understand how using the electromagnetic waves the electric fields of the electromagnetic waves it is in a cavity it is possible to accelerate charge particles. So, you need not apply voltage directly to the drift tubes you can use the time varying fields of the electromagnetic waves in the cavity. In the next lecture we will see acceleration using traveling waves. So, you know that in a waveguide the waves are traveling waves and we will also study about periodic accelerating structures.