 So, let us summarize the advantages of superconducting cavity in RF linax. In normal conducting linax, huge amount of power is deposited on the copper structure in the form of heat that needs to be removed by water cooling. So, lot of heat is deposited, lot of RF power is wasted, so this heat has to be removed otherwise it will change the dimensions of the system thereby changing the resonant frequency. So, this limits the accelerating gradient. So, how high gradient you can go to depends upon how much heat you can remove because the shunt impedance which is E0 T square by power dissipated per unit length, this shunt impedance for a given cavity is fixed. So, if you increase the accelerating field the power dissipation will also increase. So, dissipated power can be much higher than the power transferred into the beam for acceleration. So, lot of RF power is wasted in normal conducting operation. So, superconductivity at the expense of higher complexity, so the system is now complex because now you have to cool it to cryogenic temperatures, it drastically reduces the dissipated power and the cavities transfer more efficiently the RF power to the beam. So, there is a possibility to accelerate CW beams with high accelerating gradients in a superconducting cavity. Larger aperture is possible in a superconducting cavity, so less probability of loss. So, superconducting cavities are short and independently powered. So, since the accelerating gradient is high, so we can achieve the same acceleration that we would have achieved in a normal conducting cavity in a smaller length. So, they are short and they are independently powered and with modern control systems the cavity array may be retuned if one of them becomes faulty. This offers high degree of fault tolerance and reliability. So, superconducting cavities are made in small modules and each of these modules are independently powered. Now, they are designed such that let us say one module fails, one of the cavity fails by adjusting the fields of the adjacent cavities, the accelerator need not shut down, the accelerator can be retuned to operate. So, this offers high degree of fault tolerance and reliability. So, this is an important requirement again in an accelerator system for accelerator driven system. So, as I had mentioned in the first lecture that accelerator driven system will be used for producing power. So, reliability is very important. The system should be able to operate 24 cross 7 for months at a time. So, superconducting cavities offer this advantage over normal conducting cavities. The only disadvantage is that now you need to operate at cryogenic temperatures. So, here is a block diagram of the entire system for superconducting cavity. So, we have a superconducting cavity which is made up of niobium. The inner surface of the cavity has to be maintained at cryogenic temperatures. So, for this the cavity is enclosed inside a jacket and in this jacket helium is helium at 2K or 4K is put which cools the surface of the cavity and the cavity becomes superconducting. Then power is coupled to the cavity through a RF coupler and the whole system is put inside a cryostat to maintain this at cryogenic temperatures ok. So, the cavity which is made up of niobium generally is enclosed by a jacket which contains helium at 2K or 4K. The RF power is coupled into the cavity via RF coupler. The system is kept in a cryostat to maintain the cryogenic temperature. Then the ultra low resist electrical resistivity of the superconducting material it allows the RF resonator to obtain extremely high Q value. So, we have seen that since the RF surface resistance is very low Q value is very high. So, such a high Q resonator stores energy with very low loss and narrow. Now, we have seen that the most important part in the superconducting cavity is the superconducting surface. So, we saw that the RF residual resistance has two components one is the RBCS one is the R residual. R residual depends upon the cavity surface it depends upon the impurities in the material. So, we have to give great importance to the surface. So, this is because the current flows on the surface with nearly no losses. Any contamination on the surface will give rise to extra heat and will drastically increase the losses. Due to the low thermal capacity at low temperatures even a small heat source it may lead to large increase in temperature and superconducting may be conductivity may be destroyed. So, the surface must be defect free and free from any impurity. So, lot of effort is put in preparing the surface of the cavity and also maintaining it at a high level of cleanliness. So, first of all the cavity is made up of very high purity material which is characterized by a high value of RRR this is the residual resistivity ratio. So, this is given by the resistivity at 300 k to the resistivity at 0 k. Then processes like electron beam welding are used for the fabrication of cavities of the superconducting cavities and then buffered chemical polishing or electro polishing techniques are used for maintaining the surface of the cavity, high temperature treatment for removing impurities then high pressure water rinsing with ultra pure water is used to clean the surface of any impurities and then the critical steps of assembling the cavity these are done in classrooms of class 100 and lower. So, lot of infrastructure is required for building these superconducting cavities. So, superconducting RF technology they incur more complexity, expense and time than normal conducting cavities because a normal conducting cavity is made up of copper you can simply make it in a normal workshop and then by normal cleaning processes. So, no additional infrastructure is required like a clean room whereas here they lot of infrastructure has to be developed for fabrication and assembly of the superconducting cavities. So, they require chemical facilities for harsh chemical treatments, clean rooms for clean rooms then high pressure water rinsing and assembly of components and complex engineering for the cryo module western cryogenics. So, this is a picture of the cryo module for the international linear collider which is being tested at Fermilab. So, the cavity will go inside this cryo module the function of this cryo module is to maintain the temperature in the cavity at 2 k or 4 k. So, this is an example of niobium base 1.3 gigahertz 9 cell superconducting cavity to be put along the beam axis inside the cryo module. So, this is the cavity it will be jacketed and then it will be put inside the cryo module. So, as I said lots of infrastructure has to be has to be developed for development of superconducting cavities. So, some of the infrastructure is shown in the subsequent slides. So, we have the electro polishing setup at RRCAT. So, you can see a cavity here an elliptical cavity here being electro polish and high pressure water rinsing to remove it of all the impurities. So, this is done inside a clean room. So, you can see here again the cavity being rinsed with high pressure using ultra pure water. Then there is vacuum annealing furnace to remove. So, the cavity is baked in the furnace to get rid of any impurities that may be trapped. The cavity assembly is generally done in clean rooms. So, class 100 or class 10 and it is to be done by specialized staff which is properly trained in the procedures. Then the testing facility is also separate. So, you have so, this is a vertical test stand at RRCAT. So, here you can see that this is the elliptical cavity which is being inserted inside the cryostat. So, the cryostat will be filled with liquid helium it requires huge quantities of liquid helium. So, this cryo plant is also required. The cavity will be dipped inside this and then it will be tested before jacketing the cavity. So, these are pictures of the vertical test stand for testing bare cavities at RRCAT. So, the limitations on the performance of the superconducting cavity. So, there are several factors that limit the performance of the superconducting cavities. So, one is the RF residual losses. The superconducting cavity is short temperature independent RF losses. So, this physical origin is still not understood in most cases what could be identified as loss from dielectric materials and trapped magnetic fields. Quench, so quench is caused by any normal conducting inclusions which may be there or any imperfections that may be there on the surface of the superconducting cavity. So, this could be while fabrication while using some material doing fabrication. So, due to welding or residuals in the cavity from processing and cleaning. These impurities are heated up in the RF magnetic field and they drive the superconducting surface nearby into normal conducting state thereby causing the cavity to quench. Quench means going from superconducting state to the normal conducting state. Field emission. So, if there are any sharp points anywhere in the cavity if the surface is not smooth enough the local electric field there can be enhanced it can be quite high and it can lead to the emission of electrons. These electrons are again accelerated by the electric fields and they can deposit huge amounts of power on the surface of the superconducting cavity causing it to quench. So, sharp tips and small particles in the cavity act as field emitters. The electrons are emitted under the influence of electric fields so, they can be minimized by preparing smooth and mirror like surfaces by using chemical etching followed by high pressure. So, you can have you can you can use electro polishing or chemical polishing to get good surface finish and then you do a good high pressure rinsing with ultra pure water to clean the surface. The cavities are then handled in clean rooms to maintain the surfaces clean. You need to use high purity material so, that your residual resistance is very low. So, as temperature decreases the resistivity decreases or conductivity increases. The increase in conductivity saturates at a certain value which is determined by the impurity level of the superconducting material. So, the factor by which the resistivity drops to the residual value this is called the residual resistance ratio. So, you need to use a cavity a material with high RRR for getting high accelerating gradients. Multi-packing. So, multi-packing is a resonant phenomena in which the electrons emitted from the surface of the cavity they follow a trajectory such that they impact back at the surface an integral number of RF cycles after which they are emitted. So, let us say you have a cavity which is like this and at certain instant of time an electron is emitted. Now, this electron sees an electric field like this. So, this electron gets accelerated and it reaches at this point it impinges on this point here and at that time the field changes sign. So, again it sees the field in the right direction to get accelerated it is again accelerated here and then comes back and hits on the same point and so on. So, it each time it comes and hits on the same point and here thereby reducing more and more electrons. So, as more and more electrons are now reduced are now released and they are accelerated with the RF cycle the RF power that is being fed into the cavity. Instead of going in setting up the field here it goes in acceleration of this electrons. So, this is known as multi-packing. So, this multi-packing is to be avoided this is done by properly design the cavity surface. So, the electron gains energy in each RF cycle and more electrons are emitted from the surface. The RF power being fed in the cavity is absorbed by these electrons and is not used for setting up the electric fields in the cavity. So, this can be avoided by properly designing the cavity geometry. So, in superconducting cavities the maximum value of the accelerating field is limited by the peak surface magnetic fields and electric fields. So, we saw that in a normal conducting cavity it is limited by how high the power dissipated you are able to remove and it is also limited by the Kilpatrick limit, but here it is limited by the maximum value of by the limited by the peak surface values of magnetic and electric field. Now, let us say we have an elliptical cavity. So, this is the this is a single cell elliptical cavity the elliptical cavity is composed of two ellipses one near the dome of the ellipse and one near the iris of the cavity. So, this is the dome area and this is the cavity. So, this area is shown here and this is another ellipse which is here. So, this elliptical cavity operates in the TM010 mode. So, if you see the electric field lines they are like this. So, because of the cavity geometry they will go and terminate like this. So, the electric so, there will be peak value of electric field will be here and the peak value of magnetic field will be on this surface here. Now, if the peak value of the surface electric field is very high it can lead to field emission which will destroy superconductivity. Also if the peak value of the surface magnetic field is very high it will quench the cavity. So, we need to minimize the peak surface values of the electric and magnetic fields below a certain limit below a certain defined limit. So, physical limitation of the superconducting resonator is given by the requirement that H peak has to stay below the critical field of the superconductor because if you go above the critical field then the superconductivity is destroyed. E peak also has a certain limit which is given by the field emission threshold. So, you have to keep these peak surface values within the limits. So, therefore, to maximize the accelerating field it is necessary to minimize the ratio of the peak fields to the accelerating field. So, the design criteria here is to minimize E peak by E0, E0 is the accelerating field and H peak by E0. So, these are the main design goals in designing a superconducting cavity. So, unlike normal conducting cavities that are varying beta structures. Now, what is a varying beta structure? Now, for example, you have seen a drift tube linac, okay. So, here this is these are the various drift tubes. So, the cell length is defined as the distance from center of one gap to the center of the next gap and this is equal to beta lambda. Now, the charge particle will gain some energy here and move from here to here. Again it will gain energy here and move from here to here. In this process, this beta is increasing. So, if beta is beta 1 here, it will be beta 2 here where beta 2 is greater than beta 1. So, every time the charge particle gets accelerated, the beta increases. So, normal conducting cavities are varying beta structures. So, accordingly the cell length increases in a normal conducting cavity. The superconducting cavities, however, are designed to perform over a given velocity range. So, superconducting cavities are designed such that the cell lengths are all, so let's say we have a elliptical cavity. So, we have let's say a 5 cell elliptical cavity. So, here the cell length, this is beta lambda by 2 because it's a pymote structure. The cell lengths are all constant. So, they are designed to operate over a given velocity range because fabrication of superconducting cavities is expensive. So, you design a superconducting cavity that is at a certain beta and use it to accelerate for a range of beta around that beta value. So, each cavity is identified by design velocity called the geometric beta g. So, you design this for beta g and so beta and you plot the transit time factor for this cavity for different values of beta or energy, beta is the velocity. So, this is beta g, so the transit time factor is maximum for the beta g. But for other velocities around this beta g, the transit time factor is lower, but then you define a range of what transit time factor is acceptable to you and you can use it for that range to accelerate the charge particles. So, the design velocity called the geometric velocity or beta g and this is used to accelerate particles over a range of beta values near beta g. So, this is because superconducting cavities have a large velocity acceptance. So, this is this curve is quite wide for superconducting cavities as compared to a normal conducting cavity. So, velocity acceptance is defined in terms of transit time factor. The particles with velocity equal to beta g, so here we have seen the particles with velocity equal to beta g, they have maximum value of t. Particles with velocity less than or greater than beta g. So, this is less than beta g, this is greater than beta g, they have smaller values of transit time factor. So, you see that the transit time factor is smaller for these particles. The velocity acceptance is defined as the velocity range of beta around beta g. So, this is the velocity acceptance for which the transit time factor falls to n times the maximum value of t where n can range from 0.6 to 0.7. So, let us say the maximum value of transit time factor is 0.7, so this value is 0.6, so between 0.6 to 0.7 is acceptable. The different structures corresponding to the value of beta g are needed for acceleration of particles in different energy ranges. So, if you have particle, if you have to accelerate from let us say 100 MeV to 1 gV, let us say protons. So, you divide it into different types of structures, let us say three structures with different beta g and you use these structures for acceleration. The velocity acceptance of cavities is smaller at lower velocities, also as the number of cells per cavity increases, the velocity acceptance of the cavity decreases, let us see this here. So, here you can see the variation of the transit time factor with energy for different number of cells per cavity. So, this is designed for beta g equal to, so here beta g is equal to 0.62. Now, if you, so this is multi-cell elliptical cavities, if you use three cavities, three cavities, three cells per cavity, then the transit time factor curve is like this. If you use seven cells per cavity, the transit time factor is like this, so you as you see that the velocity acceptance comes down. So, a smaller number of cells per cavity provides large velocity acceptance. So, you can see here three cells per cavity provides a larger velocity acceptance. However, using large number of cells per cavity has the advantage of reducing the overall number of system components, system size, system complexity, etc. So, in other words, it brings down the cost. So, you have to compromise between the two and decide how many cells per cavity you have to use. Also, if you see this, so you have, if you want to accelerate protons from 100 MeV to 1 GeV, then the entire energy range from 100 MeV to 1 GeV is divided into three types of cavities corresponding to different values of beta g. So, beta g, so one type of cavity has beta g equal to 0.49. So, that means in that cavity, in that cavity, the cell length beta g lambda by 2, this beta g will be equal to 0.49. In the second type of cavity, the cell length will be equal to 0.62 and in the third type of cavity, it will be equal to 0.8. And using these three types of cavities, so this is a transit time factor here, using these three types of cavities, we can accelerate from 100 MeV to 1 GeV. The first cavity will accelerate from let us say 100 MeV to roughly to around 250 MeV, the second one from 250 to roughly around 450 MeV and the final one from 450 MeV to 1 GeV. So, in this way, you can use three different families of elliptical cavities to accelerate to the final energy of 1 GeV. Different types of superconducting cavities are used. So, superconducting cavities are of TEM type and TEM type. Now, we have studied so far that TEM cavities, TEM mode does not exist in a superconducting cavity. However, if you put in an inner conductor into the superconducting cavity, then it is possible to generate a TEM type of mode. So, these are known as coaxial type of cavities and so, some of the cavities using this coaxial type where TEM mode is excited are quarter wave resonator, half wave resonator and spoke resonator. So, this is a quarter wave resonator, this is a half wave resonator, this is a spoke resonator. These are multi-cell elliptical cavities. So, in this case, so these cavities are efficient for acceleration with the velocity less than 0.5 times the velocity of light. So, standing wave is formed in the fundamental mode of this coaxial resonator, a standing wave is formed with E r going to 0 at the end walls at z is equal to 0 and z is equal to d. So, basically in this the field is, so you have an electric field like this, so this is a conducting boundary, the electric field has to be normal to it. So, you have an electric field like this and if you see the variation of electric field with v distance d, it is 0 here to satisfy the boundary condition and 0 here. So, you can see from here and if you see the v theta component, the v theta component is maximum at the 2 ends and 0 in the center. A half wave resonator, the TM in a half wave resonator cavity, the TEM coaxial line is shortened at L is equal to lambda by 2. So, we have L at is equal to lambda by 2, electric field here is 0, electric field here is 0 and it is maximum here. So, this is how the at a particular instant of time the electric field is there. So, if we make a beam aperture at this location, we see an electric field in this direction here and this is varying with time and in this direction here, so this field can be used for acceleration. So, this is the picture of a half wave resonator showing the fields in the half wave resonator. Similarly, you can have a quarter wave resonator, the quarter wave resonator is a TM, it is a TEM coaxial line which is shortened at lambda by 4. So, the field here will be like this, so you can see here, so L is equal to lambda by 4. So, this is the picture of the quarter wave resonator. So, this is the hole for the beam to go through. You can also have a single spoke resonator. A single spoke resonator is another variant of the half wave TEM mode cavity class. In a half wave resonator, the outer conductor is, so if you see the half wave resonator, this outer conductor is coaxial with the inner conductor. In the spoke cavity, the outer cylinder, so this is the outer cylinder is coaxial with the beam axis. So, in other words, the inner conductor is perpendicular to the outer axis of the outer conductor. So, for low beta applications, half wave resonators are chosen. So, basically half wave resonators operate at lower frequencies, while SSRs can operate at slightly higher frequencies to accelerate particles to slightly higher energies. Now, spoke geometry allows the addition of multiple spokes. Now, you can extend this cavity and you can put in another more spokes here. So, you can have multi-gap spoke resonators, so you get higher effective voltage, but the transit time acceptance becomes narrower. This is the picture of a spoke resonator, so here is the spoke, this is the beam axis, so this is the aperture for the beam to go through, this is a jacketed spoke resonator. Then TEM class cavities, we have already seen multi-cell elliptical cavities, so this is a five-cell cavity, so you can see that it operates in the pi mode and the fields at any instant of time in adjacent cells are in the opposite direction. These are generally used for high-beta applications. So, now, having learnt about different types of cavities, novel conducting cavities, so the conducting cavities, now let us see what the design of the cavity involves. So, cavity design involves, first of all, you should know what is the beta range for which the cavity is to be designed, so you should know what energies you want to accelerate the charged particles. Then based on that, you have to choose the frequency of the accelerator, because you have to keep the length of the dimensions of the cavity some reasonable number. Then for what beam current you are using, whether it is CW or pulsed, so if it is CW, it may be better to go for a superconducting cavity, so you want to use normal conducting or superconducting. So, if you decide to go for a normal conducting cavity, you have to choose between different structures, the drift tube linac, the CCDTL RFQ depending upon your application for superconducting between different types of cavities QWR, HWR, SSR, elliptical cavities. For normal conducting cavities, you have to optimize the frequency, the shunt impedance, quality factor, kilopad rig limit. And for superconducting, again the frequency, the peak surface fields, the electric and magnetic fields now quality factor and also do a multi-packing analysis to optimize the geometry. So, you choose the accelerating structure, so you choose this structure, choose the operating frequency. So, you choose at what frequency you want to operate this, now get a source of the same frequency, the source should be of the same frequency, so it should be available in the market. Now, having got this source, you design the accelerating cavity such that the frequency of the operating mode is F0, so if you say the frequency is Tn010, so you have to choose the dimension such that Tn010 mode resonates at the value frequency F0, then optimize the cavity to maximize the quality factor and shunt impedance for normal conducting cavities and minimize the peak surface fields in case of superconducting cavities. Then choose an accelerating gradient, so you choose what should be the value of accelerating gradient. This accelerating gradient is limited by the kilpateric limit and thermal load in normal conducting cavities and it is limited by the peak surface fields in superconducting cavities. So, finally after doing this electromagnetic design, you do a thermal and structural analysis. So, this is what goes into designing a linear accelerator cavity. So, to summarize for a normal conducting accelerator, the largest power loss is between the RF cavity and the beam. The typical efficiencies of normal conducting cavities is in the range 20 to 30 percent only. The RF surface resistance of superconducting cavities is less than the RF surface resistance of normal conducting cavities by a factor of 10 to the power of 6. So, power dissipation in superconducting cavities is extremely low, we can use larger gradients. The quality factor also is very high and shunt impedance is since it is also it is very high. So, cavities can have larger beam apertures which will prevent beam loss. Superconducting cavities are particularly useful for acceleration of CW beams and also high duty cycle beams, but now we need to operate at cryogenic temperatures and lot of additional infrastructure development is required for the development of superconducting cavities. So, you have to do a proper cost analysis to see for your system whether a normal conducting cavity or a superconducting cavity would be advantageous. Some examples of superconducting cavities are quarter wave resonators, half wave resonators, single spoke resonators and electrical cavities. So, with this we complete our portion on different types of accelerating cavities. Next, we will learn about the transverse dynamics of beams in Linux.