 If the aberration is there, then it will affect not only the trajectory, but also the image size, shape and of course the locations also that will change. Now, so far we were talking about particle trajectories, now there are two kinds of one is or two kinds of accelerators actually, one is called low current accelerators where we get low currents at the output also and other one are called high currents accelerators and there are many applications where high currents are required. For example, for nuclear physics experiments low currents are good enough because the cross sections could be high for you will like to have a very clean data while in many cases like accelerated different systems high currents are required because number of new tones produced and hence the power generated is proportional to the current. Now, so the trajectory calculations in both the cases are totally different. The type of beam image you can get are totally different. For example, if you take a low current then you can even you are required to track the single particle. So, you have to do a single particle trajectory calculation while in the case of high current accelerators that is no more true. So, you have to do a different collective motion you have to take into and you will see in later I have explained that in the high current accelerators the particle make a sort of envelope they move in the envelope. So, you have to trace the envelope rather than the single particles. So, in the case of low currents you have to do the particle trajectory calculations and system particularly through particularly like accelerating tube in the case of DC accelerators accelerating tube is the most important part of it and accelerating tubes have cylindrical symmetry. So, that makes our life simple calculations can be simple, but you have to minimize the abrasion by properly designing the accelerating tube shape as well as the voltage distribution. You will see in the following slides that not only the the electrode shapes, but also the voltage distribution which you are putting across the accelerating tube matters a lot in focusing the beam. So, what ultimately you have to do is if you want to design a proper system then this accelerating tube has to be designed based on the based on the ion trajectories and for that as we mentioned earlier we did in the case of lengths also that you have to solve the equations of motion for the charged particle through the ion optical components. You have to do right from the ion source up to the target of course in this case the accelerating tube plays a most important role. Now, if you have as in the case of DC accelerators if you have an accelerating tube and particle is moving along the z axis then it will have the rotational symmetry that means theta symmetry and therefore the potential distribution. So, the ion ultimately we have to do the ion optics. So, this ion optics that means you have to follow the ion trajectories you have to calculate the ion trajectories using the potential distribution and so first it involves two steps. One is that you have to calculate the potential distribution and using those potential distribution you have to calculate the trajectories. So, if you have cylindrical symmetry accelerating tube then the potential distribution is coming from will have this Poisson's equation and this shown here and Poisson's equation will give you the potential distribution and here you can see on the right hand side rho is the charge density and I am considering the first case where the current is very low and if that is the case then this rho can be neglected. So, you can put right hand side to equal to zero and that is to only if the currents are very small otherwise you have to solve this Poisson's equation. However, if the currents are low and space charts are not there currents are small then you can charge density negligible and therefore you can solve potential distribution can be obtained not by you do not have to solve that equation you can solve the Laplace equation by putting right hand right hand side equal to zero and in the in the coordinates where now I am cylindrical coordinates where I am having r or theta and z z is the beam axis direction and it is a it is having a theta symmetry then this equation can be converting to like. So, this is the equation we have to solve for getting the potential distribution last described. So, what it effectively it amounts you can see that distribution means you have to from the trajectories you have to calculate r the position and as a function of z and this equation will give you the potential distribution and then you have to solve the force equations using the potential distribution which you have got from the calculation now by solving the Laplace equation. So, these are the two equations you have to solve and they can be in Cartesian coordinate they can be or they can be written as in even cylindrical coordinates z and r and theta. So, theta will not appear because at each theta the r will be same and therefore, you have to solve these two equations for ion optics calculations. So, in the in the nut cell it is beam optics involved calculation of r as a function of z r z in the electrostatic field which you have calculated using either the either the Poisson's equation in the case for high current or the Laplace equation when the current is very small. And as I mentioned earlier that in the case when the currents are pretty high then this does not behave like a particle alone it behaves like a collective of particles and collective of particles you will see later on that they do not behave individual but they behave in a collective manner and there is something called all the particles are contained in ellipse rather than rather than a point. So, in the case of in this case I have taken a very simple example of two million volt random accelerator built at BRC and that is because these calculations were done and based on the calculation the accelerator was built and the actual measurements and the calculations were matched and they matched very well and of course, we learnt a lot from this. So, I will be taking the examples of this this case. In this case you will see some of the conclusions were that when you do the calculations then you have to take sufficient fringing fields to calculations of potential distribution v which is a function now of r and z with the theta symmetry then shape of the electrodes the direction of the electrodes and the potential gradient all are important and fringing fringing effects have to be accounted for otherwise you will not get the correct results and that is what we found it also and of course then you have to this these are the two things and if anybody is interested in more details they can see this references in this case. Now, this is the first column section of the two MV tandem which was built at BRC and it worked for several years and you can see this accelerating tube which we are talking about it can have different shapes as I said the shape matters and you can see that if the beam is going like this then the electrode can either be straight you can see this is straight here separate by certain distance or they can be inclined they can be inclined but this in this shape can be either towards the input side or it can be towards outside you will see the results are totally different now in the case of our present case this tandem accelerator we took this based on the calculations and the actual measurements we do did all the measurements in both the case see if you reverse it here then this will be the geometry if you put it like this then this is the geometry and then you will see that the results were totally different and the performance of the accelerator was totally different. Earlier prior to these things we found that in most of the literature accelerating tubes were used like this and it was found that the transmission was not very good when we change this these electrodes like this which is shown here then the transmission increase remarkably now these are the electrodes these are stainless steel electrodes and they are separated by glass insulators here and which is rough so these electrodes they are fixed voltage electrodes you are only calculating the potential distribution in this region where the beam is going so let's say the beam is let's say the beam is going beam is going in this way then you have to calculate the field distribution or the voltage distribution in this region this region and then calculate the trajectories along this and this was done but this these electrodes are applied now there are two ways of applying one is you can say the see here it is here this is the part of the accelerating this beam line which is grounded zero potential so you apply zero you can apply zero 40 80 like this is called constant gradient that means the voltage difference between all of them is equal and let's say it is 40 or 50 or whatever other cases that you can apply zero here and then it is 10 20 40 like that means in first few sections you apply some variable gradient and you will find that there is a remarkable difference you can see that in the case of variable gradient this is the constant gradient and this is variable gradient and you can see that voltage distribution is here and one thing you can notice here that the fringing field goes at least four to five times of the aperture of this accelerating cube so you have to calculate this potential distribution up to quite a large distance because this this is going to make a such a great difference you can see here the ion trajectories calculated for this potential distribution here you can see that is not constant gradient it is 0 10 20 40 like this it is going on so first few electrodes the gradient is lower and then it is constant and then after that it does not matter because the velocity or the speed of the particle has already increased and therefore forcing focusing effect will not be too much disturbed by will not be affected by this voltage distribution so you can see that how nicely we get this this is a tandem accelerator so there is in the center there is a stripper where the negative ions get converted to positive so in this region it is positive here this is a stripper here is stripper so you can see that how nicely the trajectories are passing through a very small region of that a small vision of the stripper while if you take this if you take right from the beginning a constant gradient that means it is 0 it is 40 80 120 like this that means 40 kv per section then the for the same parameter for same input parameters you can see that in some cases the trajectories are diverging and therefore they are they are not passing through the stripper and those ions will be lost in fact even even in this case which is passing through that you will find that this where by the time they reach the second accelerating tube target is like this experimental target is there but the beam will be passing through some other place and it will be completely missing the target experiment will not be so you can see that if you compare this this is the ion trajectory for corresponding to this variable gradient and this is ion trajectory corresponding to constant and you can see a remarkable difference between these two and this is powerful so you can see that two effects are there which I talked about one is the shape is maturing second is the you are your fringing field here which you have accurately calculated we calculate it up to a distance where the voltage becomes very small really negligible till that time we can of course it will take more computer time but it gives more accurate however so one is shape the shape then gradient and then the fringing field all three are making huge effect you will see in the later one the calculations are not properly done the transmission of the beam loads down considerably now I mentioned there that the shape as well as the size of the beam at the output on the target will really matter based on the aberrations you can see that here I am injecting all the parallel beams with this and with that field distribution this is for the first one which is a variable and here it is for constant one and you can see here that in the case of variable gradient case that the focal length as a function of r is always rising with r if it is focal length is increasing always it is called negative negative spherical aberrations however in the case of this where where it is a constant gradient first it becomes it is negative but then the as the particle moves away from the optical axis it changes the sphere it changes the sign of the aberration and it becomes positive here which is not a good thing and this was the conclusion we have gone from this and you should have only one type of aberration whether it is negative or positive because correcting that becomes simple so this was the if these were the trajectory based on and of course the calculations we did for the entire accelerator involving both the accelerating tubes low energy as well as with the common dome common high voltage and you can see that if you inject a heavy heavy particle like for example this calculation is for the mass is equal to 40 and and these calculations were done for a tandem accelerator which requires injection of negative so is arbitrarily a is equal to 40 I taken and if you calculate the trajectory throughout then as the charge state increases energy increases and then you will see that the beam dimension keeps coming down because you know that once the particle has moved and if the charge state is high the force will be stronger and it will be focused more and more and therefore the beam size on the target will become a small and that is what is shown here so all this depends on potential distribution that is the shape and the direction of the electrode this is the this was a conclusion and this was almost done not very regularly done in the literature and therefore this was a sort of good finding of this study which we did