 Hello and welcome to lecture 7 of the second module of this course on accelerator physics. In the last lecture, we learned about the accelerating using travelling wave accelerators and multi-cell cavities. So let us just quickly revise what we learned in the previous lecture. So the wave propagates in an empty wave guide with a phase velocity greater than the velocity of light. So it is not possible to accelerate charged particles with this wave because the particle velocity is always lower than the velocity of light and thus lower than the phase velocity of the propagating wave. So the synchronism between the particle and the wave will not be possible and this is a necessary condition for acceleration. So this does not happen. So in a empty hollow wave guide, you cannot accelerate the charged particles. Now by loading the uniform wave guide periodically with obstacles, it is possible to reduce the phase velocity of the electromagnetic wave. So we saw that by introduction of space harmonics, the phase velocity of the wave in the wave guide can be reduced. So we get what is known as a slow wave which can be used for particle acceleration. The waves now propagate in limited frequency intervals and these intervals are known as pass bands. Okay, now if we close the discluded wave guide structure at both the ends with metallic walls, the structure becomes a periodic loaded cavity. And for the longitudinally open travelling wave structure, all frequencies are allowed. So there the dispersion curve was continuous and all cell-to-cell phase variations are allowed. But now when it is closed at both ends forming a cavity, only certain modes with discrete frequencies and discrete phase changes can exist in this multi-cell cavity. So the allowed modes are equally spaced in k and the number of modes are same as the number of cells. The cell-to-cell phase shift is given by n pi by n minus 1, where n is from 0 to n minus 1 and capital N is the number of cells now. The 0 and pi mode are more efficient for acceleration because they have a field in all the cells. The pi by 2 offers better stability but it is not very efficient because the alternate cells are unexcited. There is no field in the longitudinal direction for acceleration. So to solve this problem, the side couple structures like CCDTL, they combine the efficiency of the pi mode structure and the stability of the pi by 2 structure. So here, electromagnetically the structure is still a pi by 2 structure whereas the beam sees the pi mode structure. So the efficiency of the pi mode structure and the stability of the pi by 2 mode structure is combined in the side couple structures. So with this, we have learned about different cavities. Now today we will learn about superconducting cavities. So why do we need to go superconducting? Why can't we operate at normal temperatures? Because you know that superconducting superconductivity happens at very low temperature. So you have to go to cryogenic temperatures for superconductivity. So let's see what are the advantages of going superconducting. Now if you see the electrical power to beam power, the transfer of power, so starting from the mains, the main switch. So the mains gives power to the high voltage power supply which in turn powers the RF source which could be a Klystron or any type of RF amplifier, a solid state amplifier or a TETROD, any amplifier. And then this, so this reduces electromagnetic waves, high power electromagnetic waves at the required frequency and using a waveguide. This electromagnetic waves are transferred into the cavity and into the cavity we have studied that the total power that goes into the cavity part of it is dissipated in the cavity because the normal conducting in the normal conducting cavity, the field penetrates up to a distance equal to the skin depth and there is some RF resistance on the surface. So some power is dissipated on the structure. So part of the power is dissipated in the structure and part of the power goes to the beam. So if you calculate the efficiency of transfer of this power, the total power coming from the Klystron to the beam, this efficiency comes out to be very small. So in fact for a normal conducting accelerator the largest power loss in this entire system is between the RF cavity and the beam. So let us see what is, let us get some real numbers. So the total power dissipated, the total power is the sum of the dissipated power and the beam power. So the efficiency is defined as power that goes to the beam divided by the total power here. So for the Lehipa RFQ, so the Lehipa is a low energy high intensity proton accelerator or 20 MAV accelerator which is at BARC. So here there is an accelerating structure called RFQ, the Radio Frequency Quadruple which we will study about in a future lecture. So this accelerates the beam from, proton beam from 50 kV to 3 MAV and the design current is 30 milliampere. So if you calculate the efficiency of this structure, so from here you can calculate the beam power. Now beam power as you know is delta W into the beam current. So here delta W is 3 MAV minus 50 kV and the beam current is equal to 30 milliampere. So if you take this and multiply, so you get 88.5 kilowatts. So the beam power is 88.5 kilowatts and in order to power this RFQ, 500 kilowatts of power is required. So you can see that the total power here is 500 kilowatts and out of that just about 88.5 kilowatts goes to the beam. The remaining more than 400 kilowatts is dissipated in the structure. So this is a huge amount of power that is dissipated in the structure. So if you calculate the efficiency, it is less than 20%. Similarly for the Driftube-Linac at Leipa which accelerates a 30 milliampere proton beam from 3 to 20 MAV if you calculate the efficiency again. So the beam power here is 510 kilowatts and the total power required is 1800 kilowatts. So again if you calculate the efficiency, it is very small. So it is just less than 30%. So efficiency of normal conducting cavity is very small because a large amount of power is dissipated on the cavity walls. Now this is dissipated in the form of heat and it has to be removed because this is a huge amount of power and if it is not removed, then it will cause heating of the cavity and when the cavity heats up, its dimensions change. And you know that the frequency of any more depends upon the dimensions of the cavity. So the dimensions will change and the resonant frequency of the cavity for that mode will change. So now the power coming from the waveguide is at a fixed frequency whereas the resonant frequency of the cavity has changed. So the power will be reflected back. So you need to maintain the cavity temperature so that there are no change in the dimensions. So a lot of energy goes in cooling the cavity. These cavities are cooled by flowing chilled water or cool water through the surface of the cavity. So that is how superconducting cavities come into picture but before we understand superconducting cavities, let us quickly get a brief overview about superconductivity as we know today. So the present theoretical basis for understanding the phenomena of superconductivity is provided by BCS theory. So it is given by three scientists, Bardeen, Cooper and Schiffer. So according to the BCS theory at normal temperatures, the interaction of conduction electrons with the crystal lattice vibration, it results in omic energy dissipation thereby producing heat. And according to the BCS theory in superconducting state, there is an attractive interaction between the conduction electrons through exchange of virtual phonons. And what are phonons? They are quantized crystal lattice vibrations. So this interaction leads to the formation of correlated electron pairs at temperatures below a critical temperature and below a critical magnetic field. So below a certain temperature and a certain magnetic field, the electrons form correlated electron pairs and these are known as Cooper pairs. These Cooper pairs occupy the lowest energy state which is separated from the lowest conduction electron state by a finite energy gap. So energy required to break up a Cooper pair and raise both electrons from the ground state to, it is twice the energy gap which is about 3 electron volts. So unless this much amount of energy is provided, the electrons bound in a Cooper pair cannot be put in into a different energy state. So these electrons are essentially locked into the paired state and they behave like a super fluid thus producing the phenomena of superconductivity. So these Cooper pairs are responsible for the phenomena of superconductivity below the critical temperature and critical magnetic field. Now as the temperature decreases below the critical temperature, the fraction of electrons that condense into the Cooper pairs increases. So there are both the Cooper pairs as well as the normal conduction electrons but the fraction of electrons that condense into Cooper pairs that increases. At the critical temperature, none of the electrons are paired, while at T is equal to 0, all the electrons are in the form of Cooper pairs. So between these two temperatures, T is equal to 0 and T is equal to Tc, two fluids coexist. The super fluid of the Cooper pairs and the normal fluid of the conduction electrons. So superconductors exhibit zero DC resistance. So the DC resistance is zero. However, if you apply time varying fields or for AC applications, a superconductor is not a perfect conductor. So fields do penetrate inside the superconducting if they are time varying. So superconductor still experiences omic losses for time dependent fields because Cooper pairs that are responsible for superconducting behavior, they do not have infinite mobility and they are not able to respond instantly to the time varying fields. So the shielding is not perfect for time-dependent fields. The fields in the superconductor attenuate with distance from the surface and the characteristic attenuation length is called the London Penetration Length which is of the order of 10 to the power of minus 8 meters in niobium. Niobium is generally used for making superconducting cavities. So then the unpaid normal electrons that are always present whenever the temperature is greater than zero, they are accelerated by the residual electric fields and they dissipate energy through their interaction with the crystal lattice. So we have already seen the RF surface resistance for a normal conducting cavity. So it is equal to 1 upon sigma delta where sigma is the conductivity and delta is the skin depth. If you calculate this for materials that are normally used for making normal conducting cavities like copper and at RF frequencies, so this comes out in the order of millions. Now for superconducting materials, the RF surface resistance it consists of two parts. One is the R what is known as the R BCS and the other is the residual resistance. So the residual resistance is determined by the impurities and imperfections in the surface whereas the BCS resistance depends upon the critical temperature, temperature and the frequency of operation. So for niobium which is the material generally used for making superconducting cavities, alpha is 1.92, the critical temperature is 9.2. So you can calculate the residual resist, you can calculate the RF surface resistance at different frequencies with temperature and this normally comes out of the order of nano. So you see that the RF surface resistance of normal conducting cavities is in the range of millio whereas here it is in the range of nano. So the RF surface resistance has reduced drastically, 10 to the power of 6 times lower in a superconducting material. Now power dissipated in the cavity is equal to half RS, RS is the RF surface resistance and integral of h square over the entire surface of the cavity. Now here since the RF surface resistance is very small in a superconducting cavity, the power dissipated in the superconducting cavity will be very, very less as compared to in a normal conducting cavity. So 10 to the power of 6 times smaller. So power dissipated in the superconducting cavity is 10 to the power of minus 5 or 10 to the power of minus 6 times the power dissipated in a normal cavity. So now you can see that whatever power is going into the cavity, this power that is being dissipated in the cavity has reduced a lot. So all the power, so this becomes very minimal. So all the power literally almost all the power goes to the beam now. Now also if you calculate the quality factor, the quality factor is the ratio of at a frequency, it is the ratio of the stored energy to the power dissipated in the cavity. So this is power dissipated in the cavity. So since power dissipated is very small for a superconducting cavity, the quality factor of this superconducting cavity is 10 to the power of 5 to 10 to the power of 6 times the quality factor of a normal conducting. So you can see here the quality factor is also very high. The shunt impedance is given by the axial voltage divided by the power dissipation. Now in normal conducting cavities, the cavities are optimized for high shunt impedance. So their geometry is optimized so that you get high shunt impedance. In superconducting cavities since power dissipated is so small, the shunt impedance is already very high. So you need not optimize it for shunt impedance. So now one of the parameters that is optimized is the iris of the cavity or the beam aperture through which the beam passes. Now larger the beam aperture, smaller is the shunt impedance. Now so in normal conducting cavities, in order to maximize the shunt impedance, this is kept very small, the beam aperture is kept very small. So in superconducting cavities since the shunt impedance is already very high, you can increase the size of the beam aperture. This is useful because then you can reduce the beam loss. If the aperture is large in size, the beam loss can be reduced. So in normal conducting cavities, the dissipated RF power to be removed gives the performance limit. So if you choose your accelerating field, let us say it is a Tn010 cavity. So you decide your E0. Now for this voltage or for this electric field, there is certain power dissipation. Now how high this electric field you can choose for acceleration depends upon what is the power dissipation that can be easily removed by cooling the cavity. So that dissipated RF power to be removed, it gives the performance limit. Whereas in a superconducting cavity, you can afford to, so this power is not very high. So you can afford to increase the accelerating voltage. So your accelerating field or accelerating voltage is very high in a superconducting cavity. So let us do a quick comparison of normal conducting versus superconducting cavity. So let us compare two cavities of similar dimensions. So a pillbox cavity of length 10 centimeters and radius 7.65, operating at the same frequency 1500 megahertz and let us say we have the same accelerating voltage 1 million volts. So the normal conducting cavities operates at normal temperature which is 300 Kelvin whereas the superconducting cavity is operated at low temperatures 2 Kelvin. Now the RF surface resistance here is in normal conducting is of the order of million whereas here we see that it is of the order of nano. As a result, the power dissipated in the normal conducting cavity is very high and if you see for the same accelerating voltage, the power dissipated here is very, very small. As compared to this, this is 198 kilowatt, the power dissipated here is less than a watt, okay. The quality factor also if you see is very high in a superconducting cavity. So here the advantage is that the RF power losses are negligible. Now here whatever power you are feeding into the cavity, almost all power will go to the beam in a superconducting cavity whereas in the normal conducting cavity lot of power is dissipated in the cavity. So this is a big advantage. You are able to utilize most of the RF power in the case of superconducting cavity. Also since here the power dissipated is very high in a normal conducting cavity. So if you cannot go to very high accelerating voltages because if you go to higher accelerating voltages, this will increase even more and then it depends upon how high accelerating voltage you can use depends upon how much power you can remove from the cavity. Then larger beam aperture can be used in the case of superconducting cavities because they need not be optimized for a higher shunt impedance, okay. So superconducting cavities are particularly useful for accelerating CW beams. Now what are CW beams? These are continuous wave beams. So let us understand the difference between different types of beams. We have a DC beam, we have a CW or a continuous wave beam and we have a pulsed beam. A DC beam if this is the beam coming from the ion source or the beam that is accelerated in a DC accelerator. So this if you see the beam current, it is coming continuously with time, okay. So if you see the beam coming from the ion source or in a DC accelerator, this is a DC beam. Now you know that DC beam cannot be accelerated in a RF accelerator. In any accelerator where the fields are varying with time, the DC beam cannot be accelerated. You need to bunch the beam and you need to bunch it at the same frequency as that of the applied RF in order to accelerate it in a RF accelerator, okay. So such a beam which is bunched to accelerate it through a RF liner, this is known as a CW beam or a continuous wave beam. So the beam has to be bunched before acceleration through RF accelerator and at the same frequency or harmonic of the accelerator frequency. This type of beam is called CW beam. These bunches, so these bunches if you see, they are separated by time equal to the time period of the applied RF. So if you see the distance between the bunches, okay, so we have already seen the distance between the bunches in space is of the order of beta lambda. In time it is of the order of 1 RF period. So for example, if this is the RF cycle, if one bunch is here, the next bunch will come at this time. So this time difference is capital T, okay. The next is a pulsed beam. So the applied RF here is now pulsed. The RF itself that is coming from the klystron, this is pulsed. So the beam will be accelerated only when the RF is on. So the RF is pulsed, so RF is on for some time and then it is off for some time. So during the time when the RF or the electromagnetic wave is off, there will be no beam. So pulse width of the RF is between few microseconds to few milliseconds, okay. So these bunches are known as macro bunches. So you can see here, so if you see the beam current, so here this is the RF period. This is generally of the order of nanoseconds. And now for this duration the RF is on and for this duration the RF is off. So when the RF, there is no electromagnetic wave, there is no acceleration. And during this time only we will have the beam. So such a beam is known as a pulsed beam. So there are three types of beams, a DC beam, a CW beam and a pulsed beam. So in an RF accelerator, you can accelerate only CW beam or pulsed beam. A DC beam is accelerated in a DC accelerator. The total beam power is given as delta W into IB. So delta W is the energy gain. IB is the beam current into duty cycle. So what is duty cycle? The duty cycle is defined as the ratio of time for which the RF is on to the total time of the RF. So here the RF is on for this duration and then it is off for this duration. So this is the total time period. So the time for which the RF is on, only during that time we have the beam. The rest of the time there is no beam. So since there is no beam at this time, so the beam power will be reduced by the total beam power will be reduced by this time here. So the beam power is given as delta W into IB into the duty cycle. For CW operation, the TRF off is 0. So that means there is RF at all times. So this is the main difference between the CW and the pulsed beam. For the CW beam, the RF is on at all times. So there will be a beam at all times here. For the pulsed operation, for some time the RF is put off. Now what is the advantage here? Now the advantage here is that the power, the beam power as well as the power dissipated in the cavity reduces because so the total power that you are feeding in is the power dissipated plus the beam power. Now this power is coming only in certain intervals. So the power will be dissipated. This power will be dissipated in the cavity only during those intervals. So the total power that is dissipated in the cavity is reduced now. Average power dissipated in the cavity is reduced. So power dissipation in a CW lignac is much more than in a pulsed lignac. The power dissipated in the cavity has to be removed because otherwise it will heat up the cavity and change the dimensions of the cavity and the resonant frequency of the cavity. So pulsed operation reduces the power dissipated in the cavity. So for this reason most of the cavities are operated in pulsed mode. So in fact, not many accelerators are operated in CW mode because for most applications a pulsed accelerator is okay. Only for very specific applications like say for example the accelerator driven subcritical system which we saw in the first lecture on linear accelerators. So there one of the requirements from the accelerator was that it should be a CW accelerator. So that is one of the biggest challenges in building this type of accelerator. So therefore for CW as well as high duty cycle accelerators it is using high gradients. It is possible to use doing it is possible to make these accelerators using superconducting technology because in superconducting technologies this power dissipated in the cavity is very less or almost negligible. So you can use superconducting cavities for these applications. So even taking into account the conversion factor for heat removal at 2K. So at 2K in order to remove 1 watt of power roughly 1 to 1.2 kilowatts of power is required. Okay so even if you take this conversion factor into account the for CW and high duty cycle applications it is cheaper to use superconducting cavities. So now let us again compare this two cavities. Now let us see here the average power dissipated for CW application is 198 kilowatt in a normal conducting cavity and it is 0.4 watts in a superconducting cavity. Now if we operate this accelerator at 1% duty cycle. So that means only for 1% of the time the RF is on remaining time it is off. So here we see that the power dissipation has come down a lot. It is now only 1.98 kilowatts here also the power dissipation is very small. So in this case it is easier to remove the power dissipation even during normal conducting operation. So for pulsed applications it may be okay to use normal conducting accelerators but for CW applications we see that the power dissipated is huge and so it is better to go for superconducting accelerators. So we see that RF power losses are negligible for superconducting accelerators. So it is good for CW operation. In pulsed operation it offers the advantage of larger aperture. So if you are accelerating let us say the beam to high energy and the beam current is also high and it is operating in the pulse mode. But if the beam current is high the beam size could be large and there could be losses. So if you go to for pulsed operation even though normal conducting operation is okay but there to improve the shunt impedance the beam aperture is kept small. So if you go to superconducting you can increase the beam aperture so you can reduce the beam loss. So higher accelerating voltage can be used so you can reduce the size total size of the accelerator and large beam aperture can be okay. So now let us see the RF surface resistance of superconducting niopium. So we have seen the formula is this. So this is the residual resistance which depends upon the material how good the material is or if there are no imperfections in the material so it depends upon this. Typical values of the residual resistance for good quality niobium are of the order of let us say 10 nano here. Now the BCS resistance part depends upon the frequency it varies as square of the frequency of operation and inversely on the temperature. So the RBCS is high for high frequency operation. So you can see here from this graph the at 3 gigahertz it is quite high as compared to 100 megahertz here. So if now typical values of residual resistance as I said is of the order of 10 nano ohm. Now the BCS resistance should be less than the R residual. So for high frequency operation you get lower values of the BCS resistance at lower temperatures. So typically at high frequencies you operate at lower temperatures whereas for high frequency whereas for low frequency operation the BCS RBCS is already lower than the is already lower than the R residual. So you can operate it at higher temperatures. So these are normally operated at lower frequencies these are normally operated at 4k whereas at higher frequencies the temperature of operation is 1.8 to 2k. So at lower frequencies cavities can be operated at higher temperatures.