 Now beam current in synchrotome, this will be different than what we have calculated. Because we have calculated that if particles are going in this straight path, how many particles or how much charge is crossing through one point in one second, we calculated that this is the current. And using that current we can calculate what is the required number of protons etc. But in circular ring or synchrotome, the thing is different because revolution time may be much lesser than one circuit, it may be in the microsecond. So even one proton is circulating, if we put a detector at its location, suppose this is the orbit of the synchrotome and we put a detector which is counting the number of protons passing through this point, say this is the s is equal to s one point. So a single proton can pass multiple times through this detector in one second. So in case of a straight path, one proton will be counted only once for calculating the current. Here one proton may be counted many times in one second, so current will be higher, average current will be higher. Now we again consider the same number of protons in the previous example that was 510 to 15, but suppose such 50 pulses are there. Earlier I have told you that there are some pulses and that repeats after some time. This is the t-rap. After t-rap again pulse comes, after t-rap again pulse comes. This pulse can be injected into the machine or in the synchrotome and is compressed by the multi-turn injection mechanism and then it can be ejected out and sent to the target. So at a particular instant only one pulse will be inside the ring. So only protons of one pulse will contribute in the current. And after t-repetition again next pulse will come, again injection will take place and again this pulse will be compressed and it will be sent to the target. So in this fashion burst of pulses will be sent to the target by the accumulator or synchrotome. So let us suppose that there are such 50 pulses in one second. So 5 raised to 10 raised to 15 was number of proton in one second. So in one pulse this will be 5 into 10 raised to 15 divided by 50. So this number will be 1 into 10 raised to 14. So in a single pulse so many protons are there. So in synchrotome at a time this number will be there. Now if revolution time is one microsecond in the synchrotome means in one microsecond this number of protons pass through this detector. In one microsecond revolution time means all the protons will be passed within one microsecond and this is the number of protons. So so many protons passes through this pass through this detector in one microsecond. So again number of proton multiplied by the charge and divided by the time. So here time is one microsecond this gives you 1.6 ampere of the current. In a straight path we opted 800 micro ampere while in the ring it is 1.6 ampere. It is a huge current from accelerated micro field. In synchrotome radiation sources which we have studied earlier in synchrotome roughly current remains in the 100 to 100 milli amperes or like that. Means almost you can say 8 to 10 times lower than this current. Means this is a huge current. So if we want to build a proton synchrotome for its palatine neutron source the average mean current in protons and cotons will be much much higher than the synchrotons used in synchrotome radiation sources. So we have to build a very high current machine and again we can see that we are talking about the proton energy E at 1g and protons rest mass energy is also 1g. So kinetic energy is almost equal to rest mass energy. Means we have gamma approximately 2. In the case of electrons in synchrotome radiation sources we talk the energy on the range of gv's. Even if we take 1gv the gamma becomes 2 through 7. Means in synchrotones useful for the synchrotome radiation sources we have electrons at much higher energy with lower beam current. In proton synchrotome useful for meeting the espalation neutron source we must have high current at low energy. So this is a very much difference in the electron synchrotome and proton synchrotome. One is for synchrotome radiation source and other for the espalation neutron source. And due to this difference design will be different for these two kind of accelerators. So there is a one phenomena a space satisfact which is dominant in the case of proton machine and it is all together absent in the case of electron machine. So what is the space charge effect? We see a little bit about this. This is still an open problem in the physics. The space charge problem doesn't have complete analytical solutions so far only numerical simulations are there to study this. So far we studied only if charge particle is going nicely and some external forces are applied on these charge particles using the magnets. Now if we have very high current means large number of particles in the beam the beam also has internal forces because beam contains the charge particle so one particle will reveal the other so beam also has some internal forces. And these forces can modify the beam dynamics which we have studied. How we will see a simple example of this. Now colombic interaction in the beam this is just the colombic interaction. One particle is repelling together can be divided into two domains. One is very close encounter of the particles with another particles means this is a kind of scattering that two particles reach very close to each other and suddenly their direction changes just like in rather forward scattering. So this is the collusion of region and other one is the effect of self-field of the beam produced by distribution means we take complete distribution and due to that distribution what will be the effect on one single particle. And this is the collective phenomenon because as we change the distribution this phenomenon will change. So scattering is affected by the immediate neighborhood of the particle and this gives rise to random displacement in the particle receptor means scattering occurs particle gets a kick and its trajectory changes. So a random kicks on the particle can be there due to this phenomena and this gives the statistical fluctuation in the beam distribution while in this case effect is over a large length means in many terms we will see this effect. So we will concentrate on this. This is known as space charge effect. This scattering phenomena are under the you can study intra-beam scattering, beam lifetime, torch check scattering etc but space charge effect are there which we will concentrate in this lecture. Now space charge force depends on the beam distribution because this is collective phenomenon. So we take a very simple beam distribution. We want to understand that how space charge forces changes the dynamics. So we take a very simple example to understand this. We take a very simple uniformly distributed particle in cylindrical beam. So this kind of beam we are assuming. This is the beam propagation direction and a uniformly distributed beam is there longitudinally it is very long. So there will be no longitudinal electric field. In this only transverse electric field will be there and we consider a particle situated here that is at a distance of r from the design axis while the beam radius is a. Now we will see what is the electric field on this particle. This is a very simple exercise. It is a simple Gauss law. So e to pi r delta s delta s is the longitudinal length. It will be lambda r lambda r is the linear charge density if we consider up to the r and delta s by epsilon c. This is just Gauss law integration of e dot t is equal to enclosed charge divided by epsilon c. So we will get instead of lambda r we will put lambda not r square by e square. This is the total charge. So lambda not r square upon 2 pi r epsilon not e square delta s will be cancelled out here. So this is the expression which we obtained through the Gauss law. So this r will be cancelled out. So we will get lambda not 2 pi epsilon not e square r. This is the electric field. The one thing is that electric field is directly proportional and it is in the radial direction means it is radially outward because particles are having motion so it constitute a current also. So these will generate a magnetic field also. So using the Ampere's law you can calculate very easily the magnetic field. The magnetic field will be in the azimuth direction. Azimuth direction means in this direction. So we will have e r and we will have b phi. So again using the Ampere's law integration of b dot dl this b 2 pi r is equal to mu not i. So i is current density into velocity. Beta c is the velocity. So this will give you the current. So again at the place of lambda we are putting lambda 0 r square by a square. So we did this. So this is the expression. Here r will be cancelled out by this thing. So we will get mu not beta c lambda upon 2 pi a square r. Instead of mu not you can put the values in form of epsilon not because mu not epsilon not gives you 1 by c square. So mu not will be 1 by c square epsilon not and this values has been inserted here. So we get epsilon not c. 1 c cancels out because 1 c is in numerator. So you get this expression for b phi. Again you can see that b phi is also proportional to r. Now we will put the force exerted on the particle by this electric and magnetic field in the equation of motion. Now equation of motion we are simplified because we are talking that the effect takes place in many terms. So we can simplify our Hill's equation using the smooth approximation. What is that smooth approximation? We have actually in one complete term the frequency as phi by r means what is the frequency. As in simple harmonic oscillator we have frequencies phi by t. So here instead of t we use s. Our frequency is denoted in the length rather than time. So our frequency will be phi by l rather than phi by t and phi by t phi by l at the place of l we can put phi upon 2 pi average r as we did in the study when we were studying about the longitudinal dynamics. Similarly our length can be estimated with some circular type of thing and it is the average radius of this encoder and phi by 2 pi. This is the number of beta form of solutions in one term. This is the tune. So this will give you mu by r. So that is written here. So this equation is basically d2x by ds square plus omega in s square x is equal to some kind of force. Now here instead of force we are getting some force divided by gamma and beta square c square. Why are these terms are there? Because again we are talking in the s coordinate rather than equation of motion will be like this d2x by dt square plus omega square x is equal to some force. When we convert this t into s we will get some values here and this is written here. This is left as an exercise to produce this. Now at the place of f we will put the force due to electric and magnetic field which we have obtained. So when we will put it this is simple algebra just put the numbers from previous equations to this equation and you will get this kind of thing. This is the force. Now electric field was also linearly proportional to r and magnetic field was also linearly proportional to r. It was in radial direction it was in phi direction. So force is in the opposite direction radially outward for the electric place and radially inward for the magnetic place. Now if we are considering only particle which are on x axis means for y is equal to 0. So this r is equal to x will be there. So this is at the place of r. So now you can see that this expression shows the new frequency. Means frequency of the particles or frequency of the betatone motion has been changed due to this space charge force and this is in width minus sign means frequency has been reduced. Means tune has been reduced. So space charge force basically reduced the tune and why we are worrying so much about the tune because tune can excite various resonances. We are not going into the details how the tune excites the resonance but you can remember one relation that m nu x plus n nu y is equal to l. Here m is an integer n is also an integer and l is also integer and nu x and nu y are the betatone tunes in horizontal and vertical plane respectively. So whenever this relation is satisfied some kind of resonance is accented. Suppose for an example we take nu x is equal to some say 5.2 now we check it with m is equal to 1 and n is equal to 0. So m is equal to 1 and n is equal to 0 will give you this LHS as 5.2. So this is not an integer. Then we will check with m is equal to 2 and n is equal to 0. This will be 10.4 and again this is not an integer. You will see that when you will put m is equal to 5 this will become the integer. This will become 26 means at the RHS you will get 26 and this equation is satisfied means fifth order resonance can be excited if tune is 5.2. So if m or n if these values are lesser than 4 then it is a lower order resonances and we worry about those things. So if we have chosen our nu x or nu y to avoid any resonances and due to space charge forces if tune changes then it can reach to some level where this equation can be satisfied and then our beam loss may occur in the machine. That's why this space charge force is a tedious problem. Now we can see that this space charge force depends on lambda or charge density. You can convert into the number of particles also. So as you increase the current in the machine this problem becomes severe. So in case of proton machines for installation sources where we have to have very high current this problem is severe. This is a kind of defocusing force always acting on the beam. So you want to focus it and beam itself defocuses. So this is a kind of beam which we have to confine safely. It is a tedious problem and other thing is that the force depends on 1 by gamma square means if you are having high gamma then this force will be diminished and this is why electron machine doesn't face any problem of space charge. Because their gamma is much much higher. We have seen in example that even 1 GeV proton electron beam has gamma of 2000. So 1 by 2000 square that is a that gives you a very small effect. While in the case of proton at 1 GeV we have gamma only 2. So 1 by 4. So this makes a difference and that's why space charge force are almost absent in the case of electron machine. Now you see what are the basic differences in the electron storage ring and this proton synchrotron. When we are talking about the electron storage ring we are considering the electron storage ring for synchrotron radiation sources and when we are talking about the proton synchrotron we are talking about the proton synchrotron built for espalation neutron sources. So we have seen that proton synchrotron has low gamma and high current while electron machine has high gamma and moderate current. Major design goal in the proton machine is high beam power. How we can attain the high beam power? Major design goal in the electron machine is to have a very low emittance to increase the brightness in the emitted radiation. Due to this low gamma and high current we have a space charge problem which changes the tune of the machine and not only the tune it also changes the twist parameters of the machine means it changes complete optics. Space charge problem is not an issue in the electron machine. Beam loss is also a major challenge. Actually proton can initiate nuclear reaction. If proton hits on the wall of the vacuum chamber it can make vacuum chamber radioactive. So beam loss or loss of protons on the walls of the chamber can pose a challenge because if major losses are there after few days of operation the vacuum chamber will become radioactive. In case of electron, electron does not participate in nuclear reaction so this is not a problem. Only some thermal issues may be there for the beam loss. Now we have seen that for compressing the pulse say from one millisecond to one microsecond we need a very large number of tons to be injected in the machine and it was in the order of 1000 to 2000 like that. We have generally one to two millisecond pulse length from the lineck and on the target we send generally beam lamp of which pulse length in the microsecond order. So 1000 or 2000 types of tons of injections are there. Here in electron machine one on few number of tons to be injected. So injection scheme is quite simpler compared to the proton injection in the syncope because we have very severe space charge problem and this depends how the density distribution in the beam is there. So we want to dilute the beam means we want to increase the size of proton beam in the machine and when we increase the size of the proton beam to mitigate the space charge problem or having a good injection efficiency we need large aperture of the magnets. In case of electron machine we want to reduce the impedance means we want to reduce the beam sizes and that's why smaller aperture of the magnets are required in the case of electron machine. So large aperture of magnet again is a difficult job to manage. The one thing which is easier than the electron machine is the lattice. We don't need very strong focusing because we don't want to be very smaller beam sizes. So lattice or magnetic arrangement can be such that we have very large beam means we are having relaxed lattice. We are not very having very strong focusing quadruples or very tight focusing machines. Here very strong focusing machines are there in case of electron and we have seen when we were studying the chromaticity that if we have very strong quadruple it generates a very large chromaticities and if we have a very large chromaticities to correct those chromaticities we need stronger sextuples and the stronger sextuples means a very highly nonlinear system. So electron machines are highly nonlinear while in the case of proton except of the space charge lattice itself is not nonlinear. So this is a bit easier task here and optimization of the electron machine is very tedious job here when we are considering nonlinear dynamics. Now suppose we are talking about the synchrotome where full energy linac is not available and we are increasing the energy up from 100 mV to say 1 gV. When we increase the energy of 100 mV to 1 gV in the case of protons the speed significantly changes beta changes significantly means revolution time changes significantly in that case to maintain the synchronism RF frequency has to be changed. In case of electron machine beta remains almost constant so there is no need to change the RF frequency in case of proton machine there may be a requirement to change the frequency. If we are not using the synchrotome instead of we are using the full energy linac and accumulating then definitely in the proton machine also we don't need to change the RF frequency. We have seen that in one second there may be 50 pulses of the protons which we have to be accelerated or there may be 20 pulses so very high rapid rate is there we have to ramp the magnet for each pulse if we want to increase the energy or we want to excite the injection magnets if we want to inject at such high rapid while in case of electron machine these are almost DC machine means there is no requirement to change the magnets and if there is requirement to change means if injection energy is low and after injecting the beam we are raising its energy for synchrotome radiation and we are keeping that beam at this higher energy for several hours so to ramp the machine we need very slow means dB by dt is very slow here dB by dt is very high dB by dt is very high here dB by dt is very very slow mostly dB by dt is zero in the case of electron machines however if dB by dt is there it is very low here dB by dt is very high if magnet has very high dB by dt then in vacuum chamber eddy currents can be there and those eddy currents acts back on the beam so beam can be disturbed with very high dB by dt so this is there in case of electron machine this is now very few facilities are there in the case of espalation source presently operating machines are very few one is in sns outreach at the us other one is is is in the uk and one is in the j bar japan and one is recently commissioned in the china and one is under you can say construction that is european espalation source while in the case of synchrotome radiation sources a large number of facilities are there in the world but it are in hundreds and these are intense so again a very large experience is there to run such kind of machines here experience are less so we have covered a little bit about what kind of proton machine we should have for the espalation electron source and what are the differences if we compare it with the electron machine now in next lecture we will talk about another class of accelerator known as colectives