 Hello and welcome to lecture 13 of module 2 of this course on Accelerator Physics. In today's lecture, we learn about the radiofrequency quadrupole or the RFQ which is a type of linear accelerator which is used particularly at low velocities. Now from the previous lectures, we have learnt how RF acceleration can be done using time varying fields. We have learnt how to generate these electric fields in linear cavities because we utilize the electric fields associated with the electromagnetic waves in a high cube cavity. So how the electric fields are generated in the linear cavities. So since these are electromagnetic waves that we are using inside the cavity, we have studied about the electromagnetic waves, their propagation in free space between two parallel conducting planes, waveguides and cavities. We have also studied some accelerating structures, a simple structure like a pillbox cavity, we studied this in detail and we saw the different moles, how the field patterns look like and we also studied about the drift tube linac. We studied about the travelling wave and standing wave structures. So how a hollow uniform hollow waveguide cannot be used for acceleration and if you load it with periodic obstacles, then we loaded structure, in the loaded structure the phase velocity is brought down and then that can be used for acceleration. We also learnt about periodic standing wave structures. We learnt about superconductivity in linacs, how in normal conducting linacs the RF power dissipation is very high. So the efficiency of transfer of power from the cavity to the beam is very low in a normal conducting linac. Most of the power is dissipated as heat in the walls of the cavity. By using superconducting structures, we can transfer the entire power into the beam. We also studied about the transverse dynamics of beams in linacs. So we saw how a quadrupole or a solenoid can be used for focusing. We studied about various focusing lattices like Fodo, Fofododo, the solenoid lattices etc. And then we studied about the longitudinal dynamics of beams in linac. So we saw the region where the acceleration is stable. So for phase stability, the synchronous phase is chosen between minus pi by 2 and 0. So with all this now, we will study a different type of accelerating structure which is particularly used for acceleration of low energy beams and this is known as the radiofrequency quadrupole or the RFQ. Now for acceleration, before acceleration the ions have to be produced, the charge particles have to be produced. So these are produced in an ion source. Now if you see the output energy of ion sources or let us say proton sources, so this is in the range of 10s of kV. So the output energy is quite small. Now conventional accelerating structures like drift tube linac cannot accept beams at such low energies. Now in the DTL, so we are already familiar with the drift tube linac, it cannot be used for very low accelerating very low energies because there is a constraint, so there is a constraint on the minimum energy because of the dimensions of the drift tube. So we know that the cell length in the drift tube is equal to beta lambda. Now for low energies, beta is very small. So the cell length will be very small. So this size of the drift tube will be very small. And the drift tube also includes quadrupole magnets for focusing the beam in the transverse direction. So there will not be sufficient space to put in the quadrupole inside the drift tubes for focusing the beam. So we can increase lambda but this will bring down the frequency and at low frequencies the transverse dimensions of the cavity will become very huge. So we have seen that in a pillbox cavity the frequency is inversely related to the radius of the cavity. So the transverse dimensions will become huge and the cavity will become very huge. So we cannot decrease the frequency. So at very low energies there is a constraint on accelerating charge particles using drift tube linac. So DTL cannot accept low velocity particles, there is a constraint on the minimum energy. There should be space for accommodating magnetic quadruples inside the drift tubes which at low energies is very small. So quadruples cannot be accommodated inside the drift tubes. Also the beam needs to be bunched to inject into the DTL. So before putting it inside the DTL the beam has to be bunched. So you need to put in a buncher before the DTL. Now the radio frequency quadrupole is a RF linac. This can efficiently bunch, focus and accelerate low energy beams. So it is a very efficient structure. It can accept a DC beam and then it can do the bunching itself very efficiently and simultaneously it will focus and accelerate the low energy beams. Now if you see the proton linac layout before the RFQ was invented. So that is in the 1970s. So we had first a Cockroft Walton which is a DC accelerator, so which could give proton energies typically up to 750 keV or 1 mV and then followed by a buncher. And then this beam was injected into a drift tube linac which would accelerate it to intermediate energies of let us say 100 mV and then followed by a coupled cavity linac. After the invention of the RFQ, so this system everywhere has been replaced by the radio frequency quadrupole. So we now have an ion source which now gives beams of tens of keV and then there is a RFQ which can accept beams of such low energy and then accelerate it to 2 to 7 mV. So the output energy of the RFQ is typically of this order and then followed by now we can have normal conducting structures or superconducting structures depending on the requirement. So for example if you are accelerating a CW beam, a superconducting accelerator would be more economical and then at high energies which are typically superconducting structures. So now in all accelerators around the world, the RFQ has replaced this 750 keV Cockroft Walton buncher system. So the RFQ can directly take in the beam from the ion source. It can bunch the beam and then accelerate the beam. So the need for the buncher which was there earlier is eliminated now. So RFQ is a linac for accelerating low energy beams. The beam accelerated in any RF linac, it must be formed into bunchers at the operating frequency before acceleration. So typically bunchers were used for bunching this beam. In conventional bunchers which is simply a cavity followed by drift space, 30 to 50% of the beam is lost. DC beam can be injected into the RFQ. The RFQ first bunches the beam with more than 90% efficiency and then accelerates the beam. The beam is focused in the RFQ using electric quadruples. The RFQ thus it focuses bunches and accelerates the beam with high efficiency. So this is particularly useful for high beam currents because we have focusing throughout the structure throughout the length of the RFQ. Now let's see the working principle of the RFQ. So in the RFQ we have four electrodes as shown in the figure here which are arranged in a quadruple manner. So these electrodes here they are arranged in a quadruple manner. So four long electrodes are arranged in a quadrupole fashion and to these electrodes voltages of opposite polarity is applied to the alternate electrodes. These electrodes are known as vanes. So if both the horizontal electrodes or vanes are at a potential of minus V0 by 2, the vertical vanes will be at a potential of plus V0 by 2. So this is like an electric quadrupole. So for such a system the electric fields will be like this. So you have an electric field starting from the positive electrode going to the negative electrode. So the electric field lines are shown here. So this is like a quadrupolar field. So we have already studied the focusing in an electric quadrupole. So here we see that the force in a due to electric field is proportional to the electric field. So this type of polarity of the voltage will produce focusing in the y direction and de-focusing in the x direction. Now the beam as the beam goes through it, so at this location the beam experiences focusing in the y direction and de-focusing in the x direction. Now let's say if these voltages vary sinusoidally with time. So these voltages are now varying sinusoidally with time. The voltage applied here, this is varying sinusoidally with time. That is the voltage changes polarity after time T by 2. So after time T by 2 the voltage changes polarity. So the particles moving along the axis they will experience alternate gradient electric focusing which is more efficient than magnetic focusing for low velocity particles. So at this instant the bunches or the particles are here. After time T by 2 they reach here and at that time the polarity changes. So this becomes the horizontal vanes become at a positive potential and the vertical vanes are now at a negative potential. In this case it will focus the beam in the x direction and de-focus the beam in the y direction. So reverse from what was happening at this location. Then again as the beam moves ahead at this location the polarities will come back to the original polarities and the beam will get focused in the y direction and de-focused in the x direction and so on. So in this way it experiences alternate gradient focusing and the net result will be that the beam will remain focused in both the directions. So thus this is like a electric quadrupole just like the photo type of focusing the beam is getting focused here. So in this way the RFQ focuses the beam. Now for such a system because the electro geometry is uniform along the axis that is all the 4 electrodes are equidistant from the axis there is no axial electric field component for the acceleration of the particles. Now remember from the first lecture for acceleration using electric field there should be a component of electric field in the direction of velocity of the charge particle. So in this direction in the z direction there should be a component of electric field whereas this type of field an electric quadrupole has fields only in the x and y direction. So if this is uniform if this electrodes are uniform if this geometry is uniform along the axis then there is no ez field. Now let us say the horizontal and vertical electrodes which are known as vanes they are unequally displaced relative to the axis. So that means let us say the horizontal vanes are at a distance a from the axis the vertical vanes are now taken at a distance ma from the axis where m is greater than 1 and is called the modulation parameter. So the vertical electrodes or the vanes are displaced now with respect to the horizontal vanes. So now what happens a potential on the the potential on the axis will be non-zero on this axis here the potential will be non-zero. But still there is no component of field in the z direction. So if the perturb geometry is maintained constant along the axial direction there will still be no axial electric field because there is no change in the axial potential. Now however if the electrode transverse electrode displacements are varied along the axis the axial potential will change as a function of longitudinal position and an axial electric field will be produced. So now if this electrode geometry in the z direction if it is now displaced or if it is modulated then an axial electric field will be produced. So let us say this is the unmodulated vanes. So that means the vanes is flat like this okay the vanes is flat like this. So this is the z direction and this is the y direction. Now electric field lines are always perpendicular to the surface. So here we can see that the electric field line is only in the y direction okay. So this is the vertical vanes. So we see that the electric field lines are only in the y direction similarly for the horizontal vanes if we plot the electric field lines will be only in the x direction. So there will be only transverse field. Now however if you modulate this surface if you modulate this surface of the vanes as shown here. So this is the, these are the two vertical vanes the blue ones and the red ones are the horizontal vanes. So we see that the surface is now modulated instead of having a flat surface like this. The vanes surface is now modulated like this. So in this case a component of electric field is generated in the z direction we will see how. So in an unmodulated vanes there is no component of electric field in the z direction. The longitudinal fields for acceleration are produced by modulating the electrodes longitudinally. So here we have modulated the electrodes. The horizontal and vertical electrode displacements are out of phase spatially. So when the horizontal vanes are at a distance a from the axis the vertical vanes are at a distance b which is equal to m a from the axis. Because of modulation now this vanes will now come close to the axis after some time. So when this is the vertical vanes are at a distance a from the axis the horizontal vanes will be at a distance m a from the axis. So the horizontal and vertical electrode displacements are out of phase spatially. So now we see that how due to modulation electric field is produced. So this is the surface of the vanes which is now modulated like this. So earlier it was flat like this now it has been modulated like this. So when it was flat like this the electric field was all in this direction that is in the x direction. Now after modulation since the electric field lines still have to be perpendicular to the surface of the vanes so we have electric field lines like this. So you see that you will get a component of electric field now in the z direction also. So there will be a component of electric field in the x direction and a component of electric field in the z direction as well. So by introducing the modulations in the electrodes a part of the original transverse field is deviated into the longitudinal direction. So originally the field was all in the transverse direction when there was no modulation. And now with modulation we have because the electric field lines still have to be perpendicular to the surface. So now we have a component of electric field along the z direction in addition to that in the transverse direction. So thus we have been able to generate a component of electric field in the direction of velocity of the charge particle which is required for acceleration. Now let us see how the vanes are modulated in the RFQ. So these blue ones are the vertical vanes and the red ones are the horizontal vanes. So the vertical and the horizontal vanes so these two vanes so I am here I am showing the modulation of the one vertical vanes and one horizontal vanes. So they are out of phase with each other. So this is the beam axis, this is the vertical vanes. So this is the modulation of the vertical vanes and this is the horizontal vanes and this is the modulation of the horizontal vanes. So you can see here that when the vertical vanes is at a distance ma from the axis the horizontal vanes is at a distance a from the axis. Similarly now when the vertical vanes is at a distance a from the axis the vertical vanes is at a distance ma from the axis. So thus the horizontal and vertical vanes are out of phase with each other in terms of spatial modulation. Now if you consider two vertical vanes so here are the two vertical vanes shown in blue. So the two vertical vanes are in blue. So when the top vertical vanes is at a distance ma from the axis the bottom vertical vanes is also at a distance ma from the axis. When the top vertical vanes comes close to the axis that is a the bottom vertical vanes also comes close to the axis. So that is they are spatially in phase with each other. So this is how the vanes are modulated. Now let us see how the fields look like due to this type of modulation. So here is shown the vertical vanes this top one is the vertical vanes. So this is in the y direction and in the bottom is shown the horizontal vanes. So this is the x direction and this is the beam axis this is the z direction. So the horizontal and vertical vanes are the modulations are out of phase with each other. Now let us consider this region for the vertical vanes. Now at this instant of time let us say the vertical vanes is at a potential of v0 by 2. So the horizontal vanes will be at a potential of minus v0 by 2. So you can see here the electric field line is perpendicular to the surface it is in this direction for the vertical vanes. This can be resolved into x and y components like this. So this is the x component and this is the y component. So it can be resolved into x and y component. So we see that sorry this is the z component. So it can be so this can be resolved into the z and y component here. So the z component here is in the forward direction due to the vertical vanes. Now let us see the direction of the fields due to the horizontal vanes. So the direction of the electric field will be again perpendicular to the surface. So if we resolve it this is the z field and this is the x field. So z field is in the negative direction but remember now this is at a voltage the horizontal vanes is at a voltage minus v0 by 2. So the net or the effective direction will be the forward direction. So the electric field in the z direction due to the vertical vanes as well as due to the horizontal vanes is in the forward direction. So any charged particle here or any bunch here will experience acceleration. Now let us see the electric field due to this part. So here again if I take the electric field it will be perpendicular to the surface. I can resolve it into an E z component and an E y component. The E z component is in the negative direction. So it is decelerating. Now if I calculate the field due to the horizontal vanes. So again this field is perpendicular to the surface. So I have an E z component and an E x component the E z is in this direction forward direction but remember the voltage is minus v0 by 2 at this instant of time. So the net direction of the electric field in the z direction will be decelerating. So in this half the direction of the E z field is accelerating and in this part the direction of the electric field is decelerating. So if there is a beam bunch here it will experience an accelerating field here. Now if it moves from here to here in time t by 2 because remember the potential here in the vanes is varying with time after time t by 2 the potential reverses. So this will become plus v by 2 and this will become minus v by 2. So if the bunch reaches from here to here in time t by 2 then the field will change direction. So this bunch will again experience an accelerating force. So in this way by modulating the surface of the vanes we can generate an electric field in the z direction and by synchronizing the particle with the field we can ensure that the particle always sees an accelerating field. So in this way the RFQ accelerates the beam. Now beam bunching in the RFQ. Now a conventional buncher which we studied in the last lecture is a pillbox cavity or a single cavity operating at synchronous phase of minus 90 degrees followed by a drift space. Now bunching is done in a single shot so there is just this cavity followed by a drift space. So bunching efficiency is low. So typically the bunching efficiency is 30 to 50 percent. In a RFQ the initial many cells they are dedicated to beam bunching. So unlike here where the bunching is done in a single shot here many cells are dedicated to beam bunching. So the synchronous phase is kept at minus 90 degree where the separatrix has a maximum phase width in these cells and the vane modulation is very gradually increased from 1. So the synchronous phase is also gradually increased from minus 90. So in this way we are able to capture the beam more efficiently and bunch the beam more efficiently. Only when the bunching is complete the modulation is increased and the beam is accelerated. So beam capture and bunching efficiency in this case is more than 90 percent. So RFQ is the only RF linac that can accept a DC beam. In all other RF linacs you have to first bunch the beam and then inject it into the linac whereas in the RFQ you can inject a DC beam the RFQ itself will bunch the beam. So this is the RFQ so this is the quadrupole field due to the four vanes and you can see the modulation which produces an electric field in the z direction due to which the beam gets accelerated. So this is again some pictures of the RFQ so this is showing one fourth of the RFQ these are two vanes and here you can see the modulations on the vanes. This is another picture of the RFQ so these are the four vanes 1, 2, 3 and 4. So this is the region in which the beam gets accelerated.