 general synchrotron radiation facility looks like. So first of all we had to have some electron sources and electron source may be a cathode. By heating we can get the electrons or by field emission or by the photo electron effect. So there should be some source of electrons. And after some source of electrons there will be a small accelerator which raises the energy up to certain level and which can be injected into the synchrotron. So this is the first synchrotron which is known as injector machine and it is a booster synchrotron and it raises the energy up to certain level which is desired by the storage ring. This may be this energy may be equal to the storage ring energy or this may be slightly lesser than the storage ring energy. So when energy of electrons reaches up to the desired ring in the booster then this beam is extracted from this point you can say this is the extraction and using a transport line this is the electron beam transport line. We studied many kind of optics means photo and other kind of optics. Using these kind of optics we can make a transport line and this transport lines through via this transport line electrons are injected into the storage ring. Now on each bending here are the bending magnets. So on each bending synchrotron radiation is emitted and these radiations are guided up to the experimental station. So on a storage ring when electron keeps evolving and it keeps radiating here and whatever the energy is lost in the form of radiation RF cavity replenish that energy. So electron revolves at constant energy and many different experimental station can work to simultaneously. Now we see what is the effect of this synchrotron radiation on betatron and synchrotron motion. Does it affect the motion of the electrons? Yes it affects. So how it affects? Suppose it is the oscillating electron means these are the betatron oscillations and I am plotting these oscillations in the vertical beam means this is the vertical betatron oscillations. Why vertical beam? It will be clear. Now at this instant it is passing through the dipole magnet. So it emits a photo. Electromagnetic wave is emitted from this electron. Means we can see that this was the momentum of the electrons prior to photon emission and photon carries such momentum from the electrons. So now the momentum of electron has been reduced. Now when electrons momentum has been reduced and it passes through the cavity again cavity gives the energy and makes the momentum as equal to this. So this momentum and after passing through the cavity equals the same means we replenish the momentum lost in the form of photon by the RF cavity. However again you can see that here in this case again I am trying here for clarity. Here it is the PY component and it is the PS component. Cavity replenishes momentum via PS only. So PS component has been increased while PS component remains its reduced level. So this angle of particle trajectory with respect to design trajectory has been reduced means Y prime has been reduced and if Y prime has been reduced means amplitude of betatron motion has been reduced. Means emission of radiation damps out the betatron oscillations. So there is a kind of damping due to radiation in the betatron oscillations. In horizontal plane as momentum changes by photon emission and emission is taking place inside the dipole magnet where the dispersion is non-zero. So picture becomes a bit complicated than the vertical plane. So for clarity I draw here the vertical plane rather than the horizontal plane. In case of horizontal plane we have to take care of the dispersion also. So this is the damping coefficient when we calculate for the vertical plane how the vertical betatron motions are damped. So this is the damping coefficient when we write down the equation write d2y by ds square plus twice alpha dy by ds plus ky is 0. So this 2 alpha is this. You can see that this S subscript in these formulation shows synchronous particle means e s means synchronous energy ds is revolution time of the synchronous particle and e radiated means how much energy is radiated during the one time. So if we increase the energy e radiation also increases and damping coefficient increases because e radiation increases as e raise to power 4. So as we increase the energy alpha y increases rapidly and betatron motion damping become fast and this is the damping coefficient for the horizontal plane. Here the factor is same as for the vertical plane however one more factor is there due to dispersion and this function depends on the dispersion this is dispersion d and this is the banding in radius and this is k in the dipole magnet means if dipole magnet has some gradient we can modify the damping coefficients of the machine. What is the benefit of modifying the damping coefficient of the machine we will see later in this. So emission of the radiation takes away some energy from the betatron oscillator and oscillators damped out. Now about the synchrotron oscillation synchrotron oscillation also damps due to emission of the radiation. This damping is known as radiation damping radiation damping means emission of radiation helps us because it makes beam sizes smaller. So if we inject a larger beam size in synchrotron radiation source or electron machine it damps out after a certain period and becomes smaller. Now this is the I am taking three particles at three different energy levels. You can say central particle is the synchronous particle which has a synchronous energy and this particle is at higher energy and this particle is at lower energy than the synchronous one. Now suppose these all three particles passing through the dipole magnet and emit synchrotron radiation. Now higher energetic particle because of the gamma raise to power 4 dependency will emit more and it will emit least. So this emitted and after emission this becomes the energy level of this particle. Synchronous particle also emits some radiation and this is the final energy level of that particle and this is the energy level of the particle having least momentum. So now you can see that this gap between higher momentum particle and synchronous particle in energy has been reduced after emission of the photo. And similarly is the case that here the gap between the energy of the lower momentum particle and synchronous particle has been reduced after emission of the synchrotron radiation means energy deviation between the particle has been reduced due to emission of synchrotron radiation. And if energy deviation is reduced means amplitude of the synchrotron radiation has been reduced. So synchrotron oscillations also dance out due to synchrotron radiation. And this is the damping coefficient of the synchrotron oscillation. Here you can see that derivative of DE radiation by delta E because there is a delta E term. So emission or damping coefficient will depend on the delta E energy deviation and when we differentiate it we come off with this number. So this is the damping coefficient in longitudinal plane. Now you can see that the factor is same here also. So alpha x was k I am calling this factor as k 1 minus this dispersion related term. Alpha y was only k and alpha s is equal to k 2 plus this factor. So if you will add these three derivative coefficients this is a constant for a given number. This constant number means alpha x plus alpha y plus alpha s is a constant. Now if damping is there slowly slowly beam will damp out, emittance will be reduced and emittance will become zero after say 100 or 1000 multiples of damping time. No it is not the case. Actually there is a counter phenomena of damping that is known as quantum excitation. So far we considered the emitted radiation classically. Means a nicely fan is built and continuous emission of the radiation is taking place when acceleration is there. However quantum mechanical picture is defined. In that case photon emission is probabilistic. There is a probability of the emission of the photo means an electron can emit different number of photons in different terms. Or you can say different electrons in the same term will emit different numbers of photons. And there will be a definite energy distribution also. So it means photon emission has some kind of randomness. And some kind of randomness means on electron beam there will be random kicks due to photon emission. And when beam will have some random kicks due to photon emission each particle has some random kicks. So overall amplitude of betatron and synchrotron oscillation will increase. So quantum excitation will increase the emittance and radiation damping will decrease the emittance. And at a particular number of emittance these will be balanced. So that is the equilibrium emittance of the machine. So this is the equilibrium emittance. This formulation shows you the equilibrium emittance. So in electron machine after few damping time the equilibrium emittance will be attained by the electron beam. So here you can see that this is a Jx. Jx is basically these numbers. These number is Jx. So for Jx we say 1 minus Jy is 1 and Js is equal to 2 plus this function of dispersion. So you can see that summation of Jx plus Jy plus Js is 4. As alpha x plus alpha as alpha y was constant and this is a 4. So Jx is this number. And in this number this is a function of dispersion. We have seen it. So this is a function of dispersion, bending radius as well as gradient in the dipole magnet. And now you can see that this edge depends on the dispersion. This quantum excitation strongly depends on the dispersion. So this number is basically the coherent Snyder invariant like we have gamma x square plus twice alpha xx prime and beta prime square. So instead of x and x prime use d and d prime and you will get the edge. Now you can see without going into a detail of this formula seemingly some dependence. You can see that emittance depends on gamma square. And gamma square emittance means as you increase the energy of the machine emittance increases. As you increase the gamma, this gamma and this gamma is different. This gamma is relativistic gamma factor and this gamma is the twist parameter. So keep it in mind. I am talking about this gamma means relativistic factor. So if you increase the energy gamma increases and emittance increases. And now you can see that if r is constant r bending is constant in a synchrotome means all the magnets all the dipole magnets are same are having the same bending radius. There are no different kinds of dipole magnet used in making the lattice. Then we can write down h by r cube r will be outside of this average bracket. And they will be 1 by r formula finally. Emittance x in that case will be varying with 1 by r means if you are making r larger means smaller bending per magnet then you will get a smaller beam emittance. I am talking so much about beam emittance. We will see it that radiation damping and quantum excitation these two phenomena governs the basic equilibrium emittance. And equilibrium emittance actually shows you the brightness. Brightness of the SR. So if you want to increase the brightness of the emitted radiation you have to reduce the emittances. Brightness is directly proportional to 1 upon emittance. So that is why you need very very low emittances in this synchrotome radiation sources for increasing the brightness. So emittance can be low if you are having a low energy machine but however if you are having a low energy machine then harder x-rays cannot be obtained. So if you want to obtain the harder x-rays then you have to increase the gamma and as you increase the gamma emittance increases then you have to play with this Jx and this r to make the emittance smaller means you make a bigger machine a bigger synchrotome. A bigger synchrotome will give you the lower emittance even at higher energies. And how to control this Jx? It is a complicated function of the dispersion and twist parameters here you can see. So bending radius gradient dispersion and twist parameters all comes into picture of this formulation but this formula strongly depends on the dispersion. So by proper shaping the dispersion in the optics means properly choosing and optimizing the magnets so that your dispersion is in the desired beam then you can control the beam emittance or you can reduce the beam emittance. So this is the major goal of a synchrotome radiation source designer that how to reduce the beam sizes how to reduce the beam emittances and this reduced beam emittance enhances the brightness of the emitted radiation. So we see now something. Suppose we put a dipole here because we are talking about the dispersion then the most important magnet is the dipole. So suppose this is the dipole and this is the S and suppose we put a quadrupole here means suppose dispersion is dispersion and then quadrupole sends this dispersion and this dispersion becomes lower in the dipole magnet and again becomes higher because this dipole magnet will again enhance the dispersion. So dispersion will come again at higher values again put a quadrupole magnet and again this dispersion. So this value of dispersion and this value of dispersion will be the same means we have a periodic solution. This kind of lattice can be used for making the SRS because in this case we have reduced dispersion inside the dipole magnet significantly. However you can see that this optics has no space for dispersion free region means everywhere dispersion is non-zero and for some kind of elements such as RF cavity or injection system we need dispersion free region otherwise a bit complicated occurs. So for obtaining the dispersion free region we need some other kind of lattices. So this kind of lattice although can be obtained for low imitances is not useful for making the SRS. So again the first choice comes by the double band acromit which we have studied during our acromit part and this is SRS. Here you can see we can take a periodic solution like this dispersion is increasing here due to dipole magnet then going the dispersion here like this and you make a series of quadruples to make the acromit and here you can see that another quadruples are there to take care of the focusing of complete beam of beta functions and alpha functions. So this combination takes care of acromitic condition as well as vertical beta function and these quadrupole magnets takes care of vertical and horizontal twist parameters. So this kind of lattice is the basic choice for making the SRS this is a double band acromit. However you can see that if you want to make an acromit you cannot reshape the dispersion in this magnet because there is a unique value of dispersion which is needed here say d1 d1 prime to make the acromit means dispersion here is 0 and it's derivative 0. So when there is a unique value of d1 and d1 prime you cannot tune dispersion to other values to reduce the emitances. So why not put the third dipole magnet in between and that can be used for controlling the dispersion and controlling the d1. That is a triple band acromit and even though you can reduce further the emitance by introducing more dipole magnets inside the acromit that is known as multi-band acromit. So multi-band acromitic lattices are the preferable choice for making a very low emittance storage links for the synchrotron radiation sources where we can get harder X-rays with higher brightness. Now in our country in RR cat Raja Ramanna center for advanced technology indoor we have synchrotron radiation sources. These are the only two synchrotron radiation sources in our country and these are housed in the RR cat. So it is the simple layout of that facility here first of all there is a microtron which raises the energy of electrons up to 20 Mb source of the electrons is situated inside the microtron and then there is a transport line which sends the beam in this booster synchrotron and this booster synchrotron then after attaining the energy desired energy level of the electron beam it sends in two storaging. This is the Indus-1 storaging and this is the Indus-2 storaging this bigger one. So this Indus-1 storaging is a small synchrotron radiation sources and its operating energy of the beam is 450 Mb so emitted radiation is in the soft X-rays and this is a bigger machine Indus-2 and it is a 2.5 G beam energy inside this ring and it is producing harder X-rays for experimentalists or for users. So you can see here 16 dipole magnets are used to make this Indus-2 and on each magnet there is the radiation coming out and in one cell you can see this is a schematic of the Indus-1. Indus-1 is a very small storage ring so four dipole magnets are there making the four super period and in each super period two quadruples before the dipole magnet and two quadruples after the dipole magnet is there. This is the RF cavity based in the ring and now you can see the twist parameter of this Indus-2. You can see that this is the dispersion which is minimized inside the dipole magnet to reduce the impedance and you can see that here nowhere is the dispersion free zone in Indus-1 however this is a small machine we can run with non-zero dispersion also. This is a bigger machine this is the circumference of Indus-1 is only 18 meter while circumference of the Indus-2 is roughly 172 meter. So you can say Indus-2 is 10 times bigger than the Indus-1 and this is a unit cell or super period of the Indus-2 here two dipole magnets are there means we are making double band achromate. This is a dipole magnet this is a dipole magnet this is the variation of the dispersion so dispersion is like this again it is after the dipole magnet is G. So this is double band achromatic lattice there are three quadruples in between to take care of the achromatic condition and three quadruples here and three quadruples are here for optimizing the twist parameters. The equivalent beam emittance of Indus-2 is 58 nanometer rad. So now you can see that here 58 nanometer rad is the beam emittance and roughly in the range of 3 to 20 meter 3 to 20 meter is the values of beta function. So you can calculate at different location what is the beam sizes beam sizes will be square root of this emittance multiplied by the beta of that location that will give you the beam sizes this will come out in hundreds of microns. This is a picture of booster synchrotron you can see that these are the dipole magnets these there are six dipole magnets in six super periods are there in this booster synchrotron this is the RF cavity and these magnets are quadrupole magnets and this is the transport line coming from the microtron beam is injected here and beam is extracted from here and it is going towards the Indus-1 or Indus-2. This is the picture of the Indus-1 installation it is a very small drain 18 meter circumference and you can see that there are four dipole magnets 1 2 3 4 these are you can see this one this one these one are the radiation beam line by which synchrotron radiation is coming out and beyond that hatch there is experimental station here RF cavities placed to replenish the lost energy of electrons and this is a part of the Indus-2 where we are going to inject the beam in Indus-2 in the beam from the booster comes via this line known as transport line 3 and it is injected here you can see that at the injection optics is very complicated we did not cover the injection in this course because it is a different and typical scheme to inject the beam in any synchrotron and you can see that they are green tiny magnets these are small dipole magnets for correcting the orbit these blue magnets are the quadrupole magnets you can see this is first coil second coil third coil and fourth coil is beyond this this is quadrupole of the transport and these are the quadrupole these bigger quadrupole are the quadrupole of the industry you can see two coils are seen here for this quadrupole this is the RF cavity of Indus-2 so RF power source is kept beyond this wall and these four cables these are co-excel cables feeding that RF power to this RF cavity this is the RF cavity so you can see that four RF cavities are there shown in this picture actually Indus-2 has six RF cavities and these are the quadrupole and this is a bird's eye view of the lattice of the industry so again references are seen and each book covers to some extent about the synchrotron radiation and how we can choose the correct chromate for meeting the lattice in next lecture we will talk about the protons and chromium which will be used for expedition source