 Okay, so this was travelling wave structures, now you can also have periodic standing wave structures, so what is a standing wave structure, a standing wave structure is formed when boundaries are applied in all directions, so it is basically a cavity. So for accelerating charge particles with V, much much less than the velocity of light okay, so here you will have to reduce the phase velocity, if you use a travelling wave structure you will have to reduce the phase velocity to a very small value, so you will have to use the value of n will be very large, so you can use what is known as periodic standing wave structures, so for accelerating charge particles with V, much much less than C, we close the disc loaded structure at both ends with metallic walls okay, so we now close this at both ends with metallic walls, so now it is no longer a wave guide it becomes a cavity because now we have applied boundaries in all the directions, so the structure now becomes a periodic loaded cavity, now energy is reflected back and forth between the end walls as in a resonator and accelerators working using this principle they are known as standing wave linear accelerators, after a certain time, so when you put in the electromagnetic wave inside this, after a certain time which is known as the filling time of the cavity a standing wave pattern is established inside the cavity and now the end walls they impose an additional boundary condition at these boundaries that the electric field must be transferred to the end wall, so you must have an electric field in this direction at the end wall okay, this component of electric field is not allowed okay, this is not allowed, so electric field has to be in normal direction only, so only some modes of the disc loaded dispersion curve are now allowed and only some frequencies of the dispersion curve are allowed okay, so now earlier the dispersion curve was continuous like this, now all these modes will not be allowed, only certain frequencies will be allowed okay, this we have already seen when we were seeing about the waveguide and cavities you know in a waveguide because it is a propagating wave any frequency above the cutoff frequency will be propagated, whereas in a cavity only at the discrete frequencies the power goes into the cavity and forms the mode any other frequency will be reflected electromagnetic waves at any other frequency will be reflected back, so this is the dispersion curve for the travelling wave and this is the dispersion curve for the standing wave, so we see that the dispersion curve for the travelling wave mode is continuous while for the standing wave mode only certain discrete frequencies are allowed, now these allowed modes they are equally spaced in k, so if you see this they are equally spaced in k, the number of modes is same as the number of cells, so as many cells you have that many modes you will have, k represents the phase shift between the adjacent cells and the phase difference between adjacent cells the phase difference del pi is given by L upon beta lambda 2 pi, so due to the additional boundary conditions of closed end walls in the longitudinal direction only certain electromagnetic modes can exist in the structure this is for a periodically loaded cavity, for the longitudinally opening longitudinally open travelling wave structure all frequencies and cell to cell phase variations on the dispersion curve are allowed, so in other words the dispersion curve is continuous, now only certain modes with discrete frequencies and discrete phase changes will exist in this cavity, so if we feed RF power at a different frequency then the excited fields will be damped exponentially similar to modes below the cutoff frequency of a waveguide, so you have to feed in the power at these discrete frequencies only, so you have to see what is the frequency of these modes and accordingly you have to feed in power, if you feed in at any value in between here then it will be reflected back on damped exponentially, so the dispersion relation for the waveguide is now given as omega n is equal to omega 0 divided by under root of 1 plus cos k cos n pi by n, where n takes value from 0 1 to n, where n is the number of cells in the cavity, now if you assume an odd number of cells omega 0 is the frequency of the pi by 2 mode here, a more general definition is to say that omega 0 is the frequency of the uncoupled single cell, so let us say you have a single cell, a single pillbox cavity and it has various modes, Tn 0 1 0, Tm 0 1 1 and so on, so it has various modes, now if you have let us say coupled many pillboxes then each of these frequencies and so on, so here n is equal to 3, each of these frequencies will now split into 3 modes, so the field pattern will remain the same only there will be a phase shift from cell to cell, so we have what is known as the cavity mode, so this Tm 0 1 0 or Tm 0 1 1 these are the cavity modes, so single cavity modes are either transverse magnetic modes or transverse electric modes and these indices n, n and p they describe the field pattern in the cavity as we have seen in the previous lecture, now because you are loading the cavity periodically, no structure modes are also formed, so in a structure consisting of a chain of n resonators, so let us say you have n cavities now coupled together, there will be n modes of excitation of the cavities, in all the n modes every cavity has the same field pattern for example, Tm 0 1 0, so each cavity mode basically splits into n modes, only the level of excitation differs from one cavity to the next cavity, so the mode will be the same only the level of excitation will be different from one cavity to the next cavity. A series of modes in a structure for a particular cavity mode this is known as pass band and we use cavity to cavity phase shift to describe this structure mode, so cavity to cavity or cell to cell phase shift is given by n pi by n minus 1, so where n is the number of periodic cells and n takes value from 0 to n minus 1, so you have here n modes, so each mode is now divided into n modes. So now let us see the structure modes in a 5 cell cavity for let us say the Tm 0 1 0 mode, so the cavity to cavity phase shift here is given by n pi by 4 where n is equal to 0 1 2 3 and 4, so now each cavity mode in the 5 cell cavity has 5 president mode, so this Tm 0 1 0 mode will split into 5 modes, for n is equal to 0 we have the 0 mode, so in the 0 mode there is no phase difference between the fields in the adjacent cells. So if you see the 5 cells in each of the 5 cells the field is in the same direction and it is a Tm 0 1 0 mode, for n is equal to 1 we have the pi by 4 mode, so here the phase difference between the fields in the adjacent cells is pi by 4, so in the first cell the field is in the forward direction, in the next cell the there is a phase shift of pi by 4. So as a result of this the field in the z direction is now the magnitude of the field is reduced. When we go to the next cell here then again there is a phase shift of pi by 4 with respect to the field in the previous cell, so here now there is no field in the z direction. Now when we go to the next cell again there is a phase shift of pi by 4 and now the field is in the opposite direction and then when we go to the last cell there is another there is a shift of another pi by 4 and the field is now again in the opposite direction. So similarly for n is equal to 2 we have the pi by 2 mode, so here there is a phase shift of pi by 2 degrees in the adjacent cells. So here now if you see every alternate cell is unexcited there is no field in the z direction. Similarly for the pi mode we can see that there is a phase shift of pi in adjacent cells, so here the field is in the opposite direction in the adjacent cells. So now we can see the relative magnitude of the field in the 5 cells, so for the 0 mode these are the fields in the 5 cells, so for the 0 mode the magnitude of the field is the same. For the pi by 4 mode in the first cell the magnitude is maximum and then since there is a phase shift of pi by 4 here the field is reduced. There is no field in the third cell, so we can see that this is unexcited and in the next cell the field is in the opposite direction which is shown here. And again in the last cell there is the field is in the opposite direction, the relative magnitudes are shown here. Similarly for the pi by 2 mode as we saw every alternate cell is unexcited, so we can see here that every alternate cell is unexcited and for the pi mode the fields are in the in adjacent cells are in the opposite direction which is shown here. So the pi mode in a 5 cell elliptical cavity and normally a superconducting elliptical cavity works in the pi mode, so this is the 5 cell elliptical cavity you can see here. So it is excited in the Tn010 mode and so in the pi mode this field will be in the forward at a particular instant of time the field will be in the forward direction here, reverse direction here, forward direction here, reverse direction here, again forward direction here. So you can see here the variation of the fields with time, so each cell of the cavity there is Tm010 mode, so you see that the electric field is along this direction and there will also be a magnetic field in this direction and if you see at any instant of time the fields in the adjacent cells they are all out of phase with each other. So here if it is in this direction here it will be in the opposite direction and so on. So the phase difference between adjacent cells is given by delta phi is equal to L by beta lambda into 2 pi. So this formula can be used to calculate the cell length for different phase differences. So for example in a 7 cell cavity let us consider the pi by 2 mode and the pi mode. So in a 7 cell cavity so any of the modes will split into 7 modes. So let us consider the pi by 2 mode, for the pi by 2 mode we can substitute del phi is equal to pi by 2, so pi by 2 is equal to L by beta lambda 2 pi. So this will give you, you can cancel out pi L is equal to beta lambda by 4. So in other words the cell length, what is the cell length? Cell length is the distance from this, it is the length of one cell so you can take the distance from the center of one cell to the center of the next cell. If you calculate the cell length it is equal to beta lambda by 4. So if you want to operate it in the pi by 4 mode the cell length should be equal to beta lambda by 4. Similarly for the pi mode structure, so this is pi here, for the pi mode structure you can calculate pi is equal to L by beta lambda into 2 pi. So this gets cancelled L is equal to beta lambda by 2. And this we have already seen earlier when we saw that, when we discussed about the zero mode and pi mode structures we saw that the cell length in a pi mode structure is equal to beta lambda by 2. So here the cell length will be equal to beta lambda by 2. So here the field is in the opposite direction in the cells. So now if you compare the pi and the pi by 2 mode structures we see that the pi mode structure is more efficient as it has twice as many accelerating cells per distance. So if you see this pi by 2 mode here in the first cell the field is in the forward direction. In this cell we have a field which is in the transverse direction. So there is no field in the z direction. So if you see that every alternate cell is unexcited or in other words there is no field in the z direction. So it will not be able to accelerate. The pi mode on the other hand has a field in every cell. So if the particle moves from here to here in time t by 2 it will again, so this field will switch sign and again it will see an accelerating field and it will get accelerated. But here it will get accelerated only when it comes to the third cell. So we see that the pi mode structure is more efficient as it has twice as many accelerating cells per distance as compared to the pi by 2 mode. However if you see the dispersion curve, so this is the dispersion curve for the seven cell structure. So we see that the pi mode has a closed neighbor in frequency 5 pi by 6 mode. Now if you see this mode and this mode, so this mode is here and this mode is here. You see the separation in frequency delta omega and this is very small. Now you compare this with the pi by 2 mode. The frequency of the neighboring modes, this delta omega 1, delta omega 2, so these are quite large as compared to the frequency difference from the adjacent modes for a pi mode. Now in case there is a small error in fabrication and this frequency comes close because it is already close then this field can also get excited. Whereas there is less probability of this happening for the pi by 2 mode because the neighboring frequencies are well separated from this mode. So you see that the 5 pi by 6 mode has a very different field configuration and in the presence of mechanical errors the two modes can mix and produce undesirable results. So this will mix with this and it will produce undesirable results. So now pi by 2 mode if you see right in the middle here, it is further separated from its neighbor modes and the construction tolerances become loose. So even if these frequencies come close, there is still scope because they are well separated from this mode. So fabrication tolerances become looser. For long structures this plays an important role in the manufacturing expense of the structure and in response of the structure to beams passing through it. So pi by 2 is desirable from that point that it is more stable, however it is inefficient as compared to the pi mode structure. So if we use the pi by 2 mode as operating mode of the accelerating structure, the dispersion curve is linear in this region and the frequencies of the neighboring modes are well separated. So operation in the pi by 2 mode offers better stability but this is less efficient since every alternate cell is unexcited. So you see here there is no EZ field here, there is no EZ field here, so it is inefficient. So E-NAP at LANL he invented a structure that combines the high efficiency of the pi mode and better stability of the pi by 2 mode. So what did they do? These unexcited cells, this one and this one, they are shifted away from the axis, so they are shifted off axis. So this is the beam axis, so you have taken the cell in between these two cells and shifted it up here. Again you have taken the cell in between these two cells and shifted it down here. So now the beam sees a pi mode, so if you see the beam it is going to be seeing a pi mode. Here it is in one direction, it will come from here to here in time t by 2. So the beam will see a pi mode but electromagnetically the structure is still pi by 2. So dispersion relation is maintained in pi by 2 mode giving the advantage of mode separation. So it becomes more adapted. So you can have loser tolerances for fabricating this structure. So side cavities that are responsible for coupling they are empty but resonant. So this structure is known as resonantly coupled structure. So this is known as a resonantly coupled structure. So example is coupled cavity drift tube linac. So this you have a drift tube linac and you add coupling cavities away from the axis. So this type of structure is electromagnetically it is a pi by 2 mode structure whereas the beam is going to be seeing a pi mode structure. So it combines the efficiency of the pi structure and the stability of the pi by 2 structure. So at the cost of some extra complexity the side coupled structure has yielded good efficiency of a pi mode structure and insensitivity to fabrication errors of the pi by 2 mode. So zero mode refers to a multi cavity structure mode with zero phase shift from cell to cell. So detail operates in the zero mode. The pi mode has a 180 degree phase shift cell to cell superconducting cavities we saw that the 5 cell elliptical cavity or you could have a 9 cell elliptical cavity they typically operate in pi mode. A very important mode for linac structure is the pi by 2 mode which has a 90 degree phase shift cell to cell. So that is every alternate cavity is unexcited. So CCL the coupled cavity linac and the coupled cavity drift tube linac these operate in the pi by 2 structure mode. So this is preferred because of its good stability compared to either the zero and pi mode operation. So now let us see the differences between the standing wave and travelling wave. So the basic difference as we have already seen is that the dispersion curve for the travelling wave structure is continuous whereas in the standing wave only certain discrete points are allowed because it is now a cavity so there is no propagation of the wave in the cavity. So only certain discrete frequencies are allowed. So standing wave structures are filled in time. So that means when you fill in power into the or you put in the electromagnetic waves it takes time for the field to build up. So you can see this is the field in the standing wave structure. So the electromagnetic waves are reflected at the end walls of the cavity and they build up slowly in a standing wave pattern at the desired amplitude. The travelling wave structures they are filled in space because it is a propagating wave. So they are filled in space which means basically cell after cell is filled with power. The total filling time in the standing wave is of the sub microsecond range and here it is in the range of tens of microseconds. In this case there is only an input coupler and no matched load whereas for the travelling wave structure the RF power is introduced via the input coupler. The power is not dissipated in the structure, the power that is not dissipated in the structure that is absorbed by the matched load at the end of the structure because it is a travelling wave. So you put in one coupler for putting in power into that and then at the end this power which is not utilized is absorbed in a matched load. So this is used for generally standing wave is used for low velocities, low velocity ions and also for electrons whereas the travelling wave structure is used for electrons with V with velocities almost equal to the velocity of light. It cannot be used for accelerating ions with because then you would have to make it, you would have to increase the value of N or the loading to a very high number to reduce the phase velocity to the velocity of low energy electrons or ions. Now here in the travelling wave structure for example in the drift tube you have a drift tube so you can put in some focusing elements here. In this case there is no space for transverse focusing so if the beam is diverging you have to put it, you have to put a focusing element only after the accelerator whereas here within the accelerator as you saw for the drift tube line you can include accelerating, you can include quadrupole magnets for focusing the beam. So here you can see this is a standing wave structure and every time the wave comes into the gap it gets accelerated, sees the right field and gets accelerated whereas this is a travelling wave structure the charged particles are seeing the same phase at all times. So they are seeing the same phase so you have to, so that is why synchronism is important in this structure. So let us summarize whatever we have learnt today. The wave propagates in an empty waveguide with a phase velocity which is greater than the velocity of light. So it is not possible to accelerate charged particles with this wave because particle velocity is always lower than the velocity of light. So now the phase velocity of the electromagnetic wave in the waveguide is higher than the velocity of light so there is a synchronism between the wave and particle will not be there. So synchronism that is necessary for particle acceleration does not occur in a hollow waveguide. Now if you lower the uniform waveguide periodically with obstacles it is possible to slow down the electromagnetic wave in the cavity. So you can reduce the phase velocity of the electromagnetic wave. So we get what is known as the slow wave which can be used for acceleration of charged particles. Now waves propagate when you load it with periodic obstacles the waves propagate in limited frequency intervals and these are known as pass bands. And now if we close this disc loaded waveguide at both the ends with metallic walls the structure becomes a periodic loaded cavity. So instead of a waveguide we now have a cavity. For the longitudinally open travelling wave structure all frequencies and cell to cell phase variations of the dispersion curve are allowed. So for a travelling wave structure all frequencies in the pass band are allowed but for the periodically loaded cavity only certain modes with discrete frequencies and discrete phase changes they exist in the cavity because of the additional boundary condition at the ends of the 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 equal to 0 1 n minus 1 so on. The 0 and pi mode are more efficient for acceleration because in 0 mode or pi mode every cell there is a field in the z direction which is used for acceleration. For the pi by 2 mode every alternate cell is unexcited or in other words there is no PZ field there is a field in the transverse direction only. So it will not help in acceleration. However the pi by 2 mode offers better stability. So you can combine the efficiency of the pi mode and the stability of the pi by 2 mode in a side coupled structure like the CCDTL. So it combines the efficiency of the pi mode and the stability of the pi by 2 mode. So with this we see that how using travelling waves also we can accelerate the beam just like in a standing wave structure. In the next lecture we will study about superconducting cavities and how they are useful for acceleration.