 going to discuss lecture number 6, which is beam energy determination and calibrations. And in the last lecture, we talked about how the beam energy can be calculated using the voltage itself, but we also mentioned that there is a possibility of errors due to different sources. And therefore, the energy calculated using the terminal voltage may not be very accurate. And therefore, we should use some other methods which are more accurate to determine the energy of the particle beams. And of course, this might involve some calibrations and I will be discussing details of these aspects. So, beam energy determination and calibration for you know that for any experiment nuclear physics experiment, the most important parameter is energy particle energy and that should be known precisely. We should know the energy precisely in order to understand the nuclear physics experiment and its outcome. Now, several methods have been used for energy determination and they are listed here. Some are very accurate, some are having some sources of errors. And as I mentioned in the last lecture, one of them is that we use generating voltmeter GVM to measure the voltage and then calculate the errors. And as we discussed in the last lecture, there could be several factors which could add to error in that. And therefore, there should be some other ways, some other methods for measuring the energy very accurately, not using the voltage and then multiply by the charge state. Now, another method which is quite accurate is using the analyzing magnet, which will help in measuring the energy through nuclear reactions. And the most accurate nuclear reactions which are used for measuring the energy or determining the energy are resonances or the neutron threshold experiments. These experiments they give the ion beam energy very accurately. There is another method which is again a method of nuclear reactions is back scattering experiments for measuring the energies. And you will find in the subsequent slides that this method is very simple, but gives a very accurate value of the energy and does not involve many errors. Some errors are automatically taken care of or eliminated and therefore, measurements of these energy is very accurate in the case of back scattering accident. However, in all these methods, we need calibrations and you know that this whenever calibration is involved, there will be some source of error or some source of uncertainties will be involved. And therefore, it is not absolute energy, but with some. So, as an example, I will take the, I will describe the method which was used to determine the energy of the ion beams in an accelerator facility NBARC and which is a 6 million volt folded tandem ion accelerator. Now, just to discuss with you or describe what this protea means, folded tandem ion accelerator means is that we have an ion source here which reduces negative ions and since it is a folded tandem ion accelerator. So, there are two accelerating cubes inside the column section one is here and other one is here. And the negative ions are injected in the first accelerating cube here and they are already defined. For example, there is a 70 degree magnet and there is a 20 degree electrostatic deflector using that a well defined energy, energy beam is injected in the first accelerating cube. This accelerating cube accelerates the negative ions by coulomb attraction and when they reach it here, they achieve these up to here they are negative ions and the energy becomes a 6 million electron volt. Now, you can see that it has to be negative ion because it is injected here and this has to be accelerated by coulomb attraction. Had it been positive ions, it will not go through this here because that will be repaired. So, these negative ions are here, but if has to come back here and then it has to be accelerated further by the coulomb repulsion in the second accelerating cube here, the polarity has to be changed. These negative ions have to be changed to positive ions and that is done in a in a stripper foil here, the stripper system here that is a stripper foil. It could be foil stripper or it could be gas stripper and once a positive negative ion is changed to positive ions, they can be banned by a magnet here and they will be further accelerated. Now, the advantage here in this case is that several electrons can be removed in this process and therefore, multiple charged ions are formed which are accelerated. So, in this is like a tandem, hold it tandem and therefore, the energy as I talked about in last last few lectures that the energy will be n plus 1 times the electron volts. So, suppose it is n plus 1, 1 means because the negative ion is singly negative ion and l is the charged state after these negative ions are passed through the foil. So, suppose you take a light and at 6 million volt or around that, suppose 10 electrons are removed from that it will become 10 plus. So, totally 11 electrons are removed one negative becomes neutral and then neutral becomes positive. So, suppose 11 electrons are removed from nickel negative ion it becomes 10 plus. So, it will be the total energy will be 10 plus 1. So, it will be 11 times 6 here. So, you can say that it is 66 MBV, while had it been only Vendigraf you know which accelerates only positive ions the energy could have been only 6 MBV. So, that is the advantage in the case of tandem accelerators or palletons or hold it tandem accelerators. Now, here were the particles which are coming out they are analyzed for energy and the mass using this analyzing magnet and then they go to the scattering chamber where the experiments are done. We know the approximate energy, approximate will quite a good accuracy, but for doing the resonance experiments with the width of electron volts this energy has to be accurately known. And therefore, we have to do calibration of this magnet, because that is the only thing we can do calibration of this magnet to that accuracy. So, that is why it is called determination and calibration. So, what we do is that we calibrate the magnet using the known kind of energies and find out these parameters which are used for the calibration and then the final energies of unknown cases can be calculated. So, as I mentioned that resonances are the best methods to determine the energies or to calibrate the magnet which normally use in any experiment what we do is we cannot touch any high voltage area. So, there should be some simple way to measure the. So, what we do is that we calibrate this magnet this analyzing this full analyzing magnet using preferably some resonance kind of or threshold kind of reactions. One of the resonance method which has been very widely used for calibrating the magnets at lower energies is aluminum P gamma. So, aluminum 26 going to silicon 28 and the resonance is at 992 kV roughly 992 kV. One of the duty of this resonance is that its natural width is only 0.1 kV that means there will not be any uncertainty in locating the peak and the energy measurements can be done. Of course, in the same system we have another resonance is which is at 1.316 kV and its width is also natural width is 0.07 kV. So, that is another resonance which is used. Resonance means here see basically what we are talking about let us say I will explain that aluminum 27 P gamma going to 28 silicon that means we are having a target which is aluminum 27 and a proton beam is coming and then at the output first of course first the silicon will be formed in excited states here will be 28 silicon in excited state but later on this de-excitation takes place by gamma and therefore we will get 28 silicon plus we will get gamma also. So, this will be this. So, the reaction is given here that means the proton is hitting this and gamma is coming out and ultimately residual nucleus is 28 silicon. Now, when we say resonance means that means if you change the proton energy E p and this is the counts then you will see that at particular energy where the resonance is taking place the counts will increase and then you can corresponding to this peak you can find out the resonance. So, this state actually what you are doing is the magnetic field you are increasing which is corresponding to it. So, then you will know this energy you know very precisely and therefore B is proportional to E p you can. So, this you can find out. So, you have to only calibrate this B magnetic field of the magnet and that you can for that you should know this energy very precisely and which in the case of resonances it can be done very nicely. So, this is one case where resonances are used and resonances are used because their location of the peak that means the magnetic field or the which is corresponding to energy can be determined very precisely. So, this aluminum 27 P gamma at 992 kV and 1317 kV these resonances are very widely used for this kind of thing. So, we also use that to calibrate our magnet in 4 kV, 6 million volt 4 kV. So, I am going to give this example how this was done. So, in addition to this we also used another simple method which I said earlier was a backscattering method and they are the some of the errors were eliminated. Other methods which are used are also very popular and that is the neutron threshold reaction and the threshold reaction means the beam is proton and at the exit channel you get neutron and this channel opens at a particular energy. So, you know the energy very precisely and so that you know the precisely the magnetic field and then you can again find out the calibration. So, ultimately you have to calibrate the magnetic field because that is what you change in the to do the experiments. You have no control at the high voltage because that is too high and this method I will be discussing was done in the case of Putia and it was published here. It was published in nuclear instruments and methods in 2003. So, I will be talking about these four methods we will discuss in details of that. Resonance method, neutron threshold reaction, non-resonant reactions and backscattering. Now, ultimately as we said that since we have not we do not have access to the high voltage which gives the energy and therefore, we have to use some other component and that has to be calibrated and that has to be used for determination of the energy and one such component as I mentioned in the figure was analyzing magnetic. So, precise knowledge of the absolute beam energy is essential in the field of nuclear physics, atomic physics or even nuclear energy. Since direct measurements of mega volts voltage with sufficient accuracy is not possible, a magnetic analyzer is normally used for energy determination and that principle is given here that suppose there is a magnet here, a sector magnet, a magnet here and beam is coming when you find beam and if there is a magnetic field here, which is north and south pole and B is the velocity here and B is the magnetic field, then you will see that this band as per the Lorentz force will be following a circular path and beam will be coming out of this and only a beam of particular energy will be coming out which I will be giving in next slide. Now, just to give you an idea then order of things here, if this magnet is cut, this sector is cut from a circle here, you will see that the beam is bent by 90 degrees, so that means theta is 90 degrees, so this is the theta and suppose the r is the radius of the circle from which this magnet is come, then the radius of curvature, if theta is equal to 90 degrees, then general expression is this, that rho which is radius of curvature is r times tangent theta by 2 and in this case if the theta is 90 degrees, then rho is equal to r and let us say we call it small r. As I mentioned that rho is radius of curvature, r is the radius and if theta is 90 degrees, then rho is equal to r. So, the things are simple here if it is 90 degrees, but it could be any angle, it could be, it may not be 90 degrees, it could be any angle and correspondingly then we have to calculate the rho and theta. When we have this magnet and the beam is, when defined beam is entering the magnet, then it will follow the, it will experience the Lorentz force and that is given by, it will follow a circular path and that is given as m b square upon r is equal to q times r cross b. v is the velocity and b is the magnetic field in which this particle is moving. If v, the velocity is perpendicular to the magnetic field, that means magnetic field is like this and the particle is moving this way, then it will follow this, then we can write that v cross b is equal to, which is equal to b v v b sin 90 degree and is equal to v. This is the, this is only for 90 degree, but you can write the journal expression that it is equal to b b sin. Then in that case, since we are considering 90 degree, then this equation, this equation gets converted to v square m divided by r is equal to q times v b. And therefore, you can, from this equation we can write b r is equal to m v square upon q v or is equal to, now this m v is nothing but the momentum of the particle, q is the charge. Now, let us write that half m v square is the energy, kinetic energy, e is the kinetic energy, particle energy. If we use this from this one, we can write the value of v here in terms of energy, then we can write v times v r is equal to m times q root of 2 times this, 2 e divided by m. Or we can write, we take this m inside, we can write that 1 by q into root of e 2 times m e. Now, this was the m is the total mass, suppose it is proton, this m not is the rest mass, this m is the mass at that velocity. So, if velocity is high as I mentioned in the first lecture, m can change drastically. In fact, it can become even very high mass. So, let us say it is a times mass of the particle is a times m naught, m naught is a rest mass. So, total mass even at low energy, it is a times that, let us say it is oxygen. So, oxygen mass will be 16 times the mass of the proton, roughly equal to that. So, if you write this, then this v r becomes 1 by q, now this q, capital Q is the charge state. This was the total charge, the small q was the total charge. So, 1 by q e, that means charge state times the unit mass, unit charge, root of this. Now, m is equal to a times m naught, where m naught is the rest mass. And q is equal to this. So, if you write this whole thing into the equation, then the b from this equation becomes equal to root times 2 m naught divided by r e. And this is root times a e by q square. This can also be written as b is equal to k times root of a e by q square, q is the charge state now. And the rest of the factors have been absorbed in this, because for any magnet, radius of curvature is fixed, m naught is fixed. So, for any system, these parameters are fixed. And this is equal to k, so that means for a particular magnet, k value is fixed. Because k is given as equal to root of that 2 m naught divided by r e. So, these are all fixed parameters, so k has to be fixed. And therefore, b is equal to this. So, any accelerator, the magnetic field required to build the beam of energy e at mass number a and charge state q is given by this expression. So, this is a fundamental equation which is used for. So, you will see that for any accelerator magnet, what we give is k value you give. So, for example, in the case of cyclotom at B E C coal cutter, for room temperature cyclotom, k is equal to 130. So, depending upon what is the charge state, what is the a value or what is the energy, the beam magnetics will be. So, B is known. So, if you know this equation, if you know the k value, then you can find the energy of particle. And of course, the magnetic field accurately, you know the charge state, you know the mass number. Then the energy of that particle can be known very accurately. Now, here the energy will be accurate if B is accurate, but B is measured very accurately using NMR, NMR Gauss-Wieter. NMR Gauss-Wieter accuracy is of the very high order and therefore, B is very accurately. Hence, to know the energy of the particle, we should know what is the k precisely. We know B very carefully. So, therefore, if you want to know E very accurately, then we should know the k value. So, initially we should know what is the k value of that magnet. Here you can see from this equation, you can see that k is proportional to 1 by r. r is the radius of curvature. So, this radius of curvature will be defined how accurately the trajectories are followed. Now, suppose if the beam is diverging, then the radius of curvature of all the particles will not be same. Now, radius of curvature is fixed only for a particular particle and therefore, you have to calculate. You have to know the radius of curvature of the beam and as I said that if it is a diverging or converging beam, then the radius of curvature of all the particles in the beam need not be same. But for central trajectory, r is known because that is how you have designed the magnet. And once you know the r, then k is fixed. So, k is fixed for a particular r radius of curvature. So, diverging input or converging input, the r can vary. So, this is how the basic thing is.