 So far what I was dealing was a very low current and low current means a space charge effects are not there, but that is not the case. Suppose you are having large currents, I would like to ask a question, can this all these calculations which I have done so far, will they be valid or we should do much more or something better and that is shown here that if the current is high, which is the requirement of many experiments, for example irradiation of samples or the ADS which I talked about, accelerator driven system and which I will talk later, they are higher the current better it is and there even hundreds of milli ampere currents is required for doing the job and so if the current is high, then this single particle trajectory calculations which we were talking about so far is not valid and that will let us see why it is not there. Let us take that, it is a one milli ampere current. Now normally in some of these future experiments we will be needing hundreds of milli amperes. So I just take as an example that we need let us say one milli ampere current. Now one milli ampere current means what? See we talked about particle trajectory calculation. Now one ampere is nothing but a flow of one coulomb charge per second that is what is one ampere current. Now charge of electron or proton is 1.6 into 10 power minus 19 coulomb. So we will see that flow of one coulomb per second means flow of 0.67 into 10 power 19 protons per second. Now one milli ampere is 6.7 into 10 power 15 protons per second. See now if so many particles are together now it is no more a single particle, now it is a bunch. Now you say they are all together they are flowing in one second so many particles. So they are behaving like a bunch now. They are all together and you know that similar particles with similar charge proton proton they always repel because of the spatial their coulomb repulsive forces are there. So you cannot neglect now these spatial effects and they will contribute and then they are not linear they are non-linear in nature and therefore these forces have to be taken into account. So in France of these spatial forces will be on focusing it will not it will not allow it to focus and the shape and orientation of beam also will be disturbed. Not only that there is another phenomena called beam halo that also will be formed. You will listen to phenomena and I will discuss in later and therefore now beam optics if you want to study the performance of the system then the beam dynamics of that system has to be done with space forces taken into account. You cannot simply do the single particle calculations whatever I was doing earlier that is no more value and beam halo means see now if I am starting such a big number of particles putting in a bunch which is shown here this is a bunch which looks like almost like circular and when it is passing through the various elements beam devices then it will no more at the exit it will no more be a circular beam but it will have some particles will misbehave and because of non-linear nature of this so there will be some particles which will be not in the center but they will be slightly away and this is quite substantial for example in some cases if this new particles are not focused they will hitting the accelerator components and they will make it that the whole components will be activity and you can also see because of some higher order effects there are some particles which are grouping here they are getting collected here here here apart from this uniformly distributed halo there are some particles and these particles are very dangerous because they at high energies where for example in ADS we need about 1G we beam these particles of 1G and they are quite substantial so what you will see that if you plot this you will see that most of the particles are here but there are particles like these particles here and so this is almost like 70-80% is here but there are few percent particles which are around you can see that this is not going the background this is like a background this is called halo and background is not zero and on and on the top of this there are some places where these particles are also getting grouped and they are very dangerous in the case of high energy accelerators and they have to be taken care of the sources of these particles is beam mismatch and so there are thousands of components in big accelerators so each joint or each point this particular this matching of the beam parameters have to be done and these forces could be non-linear in nature and therefore it has to be and mismatch beam evolves and final equilibrium state with accompanying growth of halos and emittances so I am introducing now another parameter so far I introduced the parameter which is halo and the other one is emittance now see what we were calling earlier as the theta this is a divergence now that same thing in a layman's language is now represented by the emittance here which I will be explaining here and these emittance growth or the halo formation they are the they are the phenomenon responsible for non-functioning of some of the particles and that you can see that highly explaining here and this has been very nicely explained in this paper now what what is the meaning of emittance hello you have seen that there are there is a central portion where most of the beam are there but few percent of beam is is a part of the that halo which is forming which is surrounding that main beam and of course with some pockets now other parameter is emittance which which I will talk about in the last transparency is emittance emittance is nothing but an emittance is talked about only in the case of large beams because if the beam is very strong then the you can your calculation with for the single particle trajectory is good enough while if the particles are very large in number millions of millions of particles then you have to track the emittance you have to track the envelope so there is a there is a theorem called Liberté's theorem and that says that for the conservative forces if a beam is subjected to the conservative forces then the this theorem says that emittance is a conserved quantity and the emittance is nothing but the area occupied by the beam in a position and momentum or we call it ppx plane of the space or we call it phase space sometimes so this is so you can if you see the beam will look like this there is a central portion where most of the beam is there but there are many particles so when you say emittance then you have to also keep in mind how much particles see I mentioned it earlier that beam almost will go like this there is a central portion and there is a very little and this is the background and this is basic and this is not zero it is so this is basically is nothing but the halo that we do to yellow and these are some history particles which are forming together I am just considering what is the distribution of the particle that means whether I am taking this actually most of the beam is here only but then all the beam is not here we are considering 80% beam or we are considering 90% or 95 or 99 you can see here that that will matter that will define the the emittance value which is nothing but the area of the total ellipse so out of that ellipse how much particle how far what percentage of particle you are considering to calculate the emittance so emittance will always have some error depending upon what particle distribution you have taken into account so as I mentioned that area of the ellipse ellipse is equal to pi times the x maximum and px maximum p is momentum so in one plane it is x prime because but momentum is proportional to x prime so you can say and you can see here the same thing you can write that the area does not change so you can see that here it is x is this direction and px is this one similarly you can say this is equal to x is in this direction and x prime is this direction so this is called phages space ellipse so this area of this ellipse is basically emittance now if you take this suppose this was at the injection this was the shape of the ellipse at the entry that means where there was a very little diameter suppose I take it parallel beam sort of thing so this x prime will be very small x may be large but x prime will be very small so the ellipse will look like this and at the end you see the divergence has increased the size has come down because we want that let's say the particle is crossing at the x axis let's say particle is crossing like this so here x is very small so x has come down but this divergence is very large so this ellipse has changed to this so we have to follow now the ellipse and the number of particles by Leveny's theorem will remain same unless you make complete mismatching also so you can see here that the horizontal ellipse at A here this ellipse has become a vertical ellipse so what we were earlier talking about single particle now has changed to the ellipse and the ellipse orientation is changed it may it was let's say in the beginning in between it may be like this or at the end it may be like this or somewhere it may be even like this that depends so we have to take that into so we have to now follow the ellipse and not the single particle now in the case of high current particles now we have to see the beam emittance and its growth earlier we were talking about the divergence we were thinking that theta should not increase here we should this is the parameter that we have to see that ME10 growth is minimized and there are three kinds of emittances which are defined which are there here and normalized ME10s which is see suppose I am injecting a 100 KB beam in the accelerator and in one case I am accelerating to 100 MBV other case I am accelerating to let's say 1 GV now whether is there any parameter which I should be able to compare and just whether my system is proper or not and that should not change and that should be normalized so that is called normalized emittance normalized emittance takes care of the energy increase so this is the measured value of the emittance at low energy we have started with beta is equal to V by C and gamma is equal to 1 upon root 1 minus beta square so normalized emittance is given by this so you can see that this will always be let's say this is positive then this emittance with the energy increase should come down and that happens actually another thing is RMA emittance and as I mentioned that the growth of the emittance should be as small as possible if the accelerator is properly designed you can see here that if these particles are throughout here so if you just take the ellipse which encloses about 70-80% particles the emittance will be different from 90% or even 100% 100% emittance means you are taking let's say this is the emittance you are taking so you are taking full beam throughout but sometimes that may not be necessary it may be necessary to have only these many particles which will have let's say 90% or 80% or something like this so that parameter how much we have to take is defined as the sigma that is the full beam so this is sigma so one sigma will have about 69% particles and if you take 2 sigma that means somewhere here then it will have almost like 80-85% or something like this so that depends upon what kind of distribution you have used whether it is a Gaussian distribution or is a normal distribution or Poisson distribution or what kind of distribution you are having so this is somewhere you can see that while coating the emittance you have to take into account that what kind of percentage of particles you have to take we have taken so actually if you have not taken all the particles the emittance will be much bigger because you are taking area much so if you are taking only 80% particle or 68% particle then the emittance will be smaller and that is reflected here F is the percentage of particles fraction of the particle which have been used for calculating the which have been taken for calculation of the emittance so once you know this then you actually know what is the emittance and here sigma which is used is a RMS width of the beam beta is one of the twist parameter there are three parameters which are given there so if you see that if I summarize this lecture then we studied the we talked about the focusing properties of angel lens, electrostatic and magnetic lenses mainly quadrupole lenses and we saw that in the case of electrostatic quadrupole lens the focusing is mass independent therefore they are better as compared to the as compared to the magnetic quadrupole because all the particles with different masses but similar charges state will be focused at one place now to demonstrate the point of the accelerating tube which is a main component of any DC accelerator beam dynamics of two MVB tandem accelerator was discussed and through that we demonstrated we talked about we have shown it that varying gradient potential gradient is more efficient and spherical and avarade chromatic aberration affect the shape and the size of the beam and high currents of course the space charge effects contribute both shape size and other parameters and therefore the beam dynamics has to be done completely and that can be totally different and therefore it has to be done with the full space charge effects if the current is large then in addition to other parameters even beam halos will be formed and this can lead to beam losses and one of the disadvantage of that is or one of the serious problem is that this can lead to activation of beam line components for example if suppose out of 100 mA 2 mA beam or 1 mA beam is hitting any target at 1 gV we will see that so much of activity will be produced that you will not be able to come close to that system for several weeks actually because it will take weeks together to decay down to the level where you can go inside therefore the beam dynamics should be done with the large number of particles actual number of particles you cannot do for a 1 mA or 5 mA single particle beam dynamics studies so these are some of the things which we have discussed thank you