 In this lecture, we will study about this synchrotron radiation source. This is a class of synchrotrons used for tapping some radiation and these radiations are used by various experimentalists. So, we see this kind of accelerator in this lecture. And charged particle when subjected to some kind of acceleration, it emits electromagnetic radiation means an electromagnetic wave is generated on the acceleration of the charged particles. And using basic electromagnetic theory, we can obtain the power radiated by a charged particle when it is subjected to some acceleration. And this is Lienard formula for the power. Here gamma is the relativistic gamma factor d beta by dt shows the acceleration of the particle and beta shows the speed of the particle. So, you can see here in this formula that radiated power depends on cross product of the velocity and acceleration. This is the velocity and this is the acceleration. So, cross product of two vector quantities, it means what is the angle between the velocity and acceleration of that charged particle, radiated power depends on that. And this is an important remark about this formulation and how this cross product term can be understood, we will see it. Now for obtaining the power, we have to calculate the d beta by dt means acceleration has to be calculated for the charged particle. And because we are working with the extreme relativistic cases means we have to use relativistic formulation for obtaining the acceleration. So, we will do that. Now momentum of the particle is given by Cm gamma beta, this is the relativistic momentum, where beta C is the velocity and m gamma is the mass and relativistic gamma factor. Now if we take the derivative of this momentum, we will get d beta by dt also means some acceleration because dp by dt in Newtonian mechanics from Newtonian mechanics, we know that it will be the force and force is directly proportional to acceleration. So, we will get acceleration using this relation. So, dp by dt, if we differentiate it dp by dt here gamma and beta both will be differentiated, C and m are constant. So, it is outside the bracket, then d gamma by dt and beta. Here you can see that we have used vector notation over the beta because we are talking about vectorial characteristics of the velocity and acceleration. We are interested in directional characteristics of these two vectors that what are the angles between velocities and accelerations in different cases. So, that is why vector notations are used here. So, beta plus now beta will be differentiated, so gamma d beta by dt. Now this d gamma by dt again can be put in the form of beta. We know that gamma is equal to 1 over and root of 1 minus beta dot beta, this beta dot beta, again it is in vector form it will give you the beta square. So, if we differentiate gamma, it will get we will get d gamma by dt beta dot d beta by dt means velocity dot product with acceleration over 1 minus beta square raised to power 3 by 2 and this denominator is actually the gamma cube. So, this d gamma by dt becomes gamma cube beta dot beta dot this beta dot is d beta by dt. So, we can write now dp by dt using this relation here we can write down dp by dt is c m gamma cube this gamma cube can be taken out the bracket then inside the bracket they will be beta dot d beta by dt and this beta is already there. So, this is here plus because we have taken gamma cube outside. So, at the place of gamma this will be 1 by gamma square and d beta by dt. So, this is the change in rate of change in momentum and it gives us the acceleration. Now, we can convert this 1 by gamma square also into the 1 minus beta square. So, this is now our complete formulation of dp by dt. Now, you can see in this formulation that if velocity and acceleration again are in dot product means again dp by dt also depends on the direction of the velocity and acceleration. So, now we can consider two cases the first case in which velocity is parallel to acceleration means velocity vector and acceleration vector both are having the same direction and the other case in which both are perpendicular direction. So, when beta is parallel to d beta by dt means acceleration is in the same direction as of the velocity in that case beta dot d beta by dt the cos 0 it will be 1. So, it will be beta d beta by dt and again beta. So, it will be beta square d beta by dt now because velocity and acceleration both are in same direction. So, beta cap is written for the directional curve. So, now we can take d beta by dt you can see beta square this term will be cancelled out by this term and we will have c m gamma cube d beta by dt beta cap. Now, d beta by dt can be written down as a by c a is the acceleration. So, d beta by dt beta cap is acceleration vector divided by the speed of light because beta is v by c. So, d beta by dt will be dv by dt into 1 by c and that c cancels out with that. So, we will get m gamma cube event. Now, you can see here one beautiful thing that dp by dt is not simply m gamma a as we did earlier when the acceleration was in the perpendicular direction to the velocity. And I always cautioned you that this is only valid means we can write down dp by dt is equal to m gamma a vector only when velocity is perpendicular to the acceleration. Now, when velocity is parallel to acceleration we get gamma cube here. So, this is a remarkable result from the relativity. So, now you can see that d beta by dt can be written down as 1 by c m gamma cube dp by dt. So, this d beta by dt will be used in the formula of radiated power. Then we will get what will be the radiated energy when acceleration is taking place in the same direction as of the velocity. The second case, in second case we know that in second case in which we are considering that velocity vector is perpendicular to the acceleration vector. So, dp by dt this term will be 0 because of the cos 90 and only this term will be there. So, dp by dt c m gamma dv by dt. And here you can see if you will put again d beta by dt as dv by dt into 1 by c this c and c will be cancelled out and you get dp by dt is equal to m gamma. This shows that when we used correctly the formula of acceleration by multiplying just by gamma when acceleration was in perpendicular direction to the velocity. And in this case we know that acceleration can also be written down as v square by r. So, now again we revisit the formula of power radiated and now at the place of d beta by dt we will put the results. Now again we consider two cases when beta is parallel to acceleration. In this case we already obtained the value of acceleration d beta by dt and that was 1 by gamma cube. Here it is in square. So, 1 by gamma cube square means 1 by gamma 6. So, this gamma 6 gamma raised power 6 will be cancelled out. And you can see that now formulation does not have gamma. So, formula depends on dp by dt itself and dp by dt can be written down as d e by dx. So, at the place of dp by dt why I am writing d e by dx because when acceleration is in the direction of the velocity means we are talking about the linear accelerators. And in linear accelerator how much energy particle is getting in how much distance that is much more important. So, we are writing d e by dx the place of dp by dt. So, we get dp by dt is equal to power radiated is this thing. Here now you can see that how in how small distance how much energy you are pumping to the particle that gives you the radiation. If a large amount of energy is changed within a small distance then a large radiated power can be come out. And this is the case of prehistor long radiation where electron stores in the material and it loses a large amount of its energy in a very small distance and we get the prehistor long radiation in that case. If beta is perpendicular to the d beta by dt means velocity and acceleration are perpendicular. And in this case when we will put the result of d beta by dt d beta by dt can be put as v square by r and v square by r. When we put it you will get beta square c by r at the place of d beta by dt. And here you can see that because of 1 by gamma square here is gamma raised to power 6. So, gamma raised to power 4 will be there means radiated power depends on gamma and strongly depends not only depends strongly depends because there is the gamma raised to power 4 means it depends strongly that how the power is radiated when acceleration is in perpendicular direction to the velocity means very high energy particle because of this high gamma can radiate significantly. Now we see these two cases in more detail from the accelerator point of view. So, let us talk about the first case when beta is perpendicular to d beta by dt means linear accelerator this is the case of linear accelerator. Here you right now you can see that power depends on d e by dx and d e by dx depends how much electric field we can generate practically. If we consider even the 100 mev per meter of d e by dx it is a huge number still linear accelerators are far from this number practically linear accelerator can reach up to 30 or 40 mev per meter. However, at high frequencies of operation 100 mev per meter can be realized. But no such accelerator right now is operating with such high gradient. But still if we consider that high gradient then the radiated power is still is in the order of 10 raised to minus 15 orderly. If we consider electrons because here it is a mass of a square of that particle. So, if we take light particle radiated power will be moved. So, in this case electrons will generate more power and even in the case of electrons we are getting only 10 raised to minus 15 volt means linear accelerators do not cause the radiation from the electrons significantly. A very tiny amount of radiation is there in the case of linear acceleration. This is insignificant. Now we consider the case when acceleration is perpendicular direction to the velocity. This is the case of synchrotron. In synchrotron because of the dipole magnet there is a centripetal acceleration and centripetal acceleration is in the perpendicular direction to the velocity means in synchrotron this kind of situation is generated. This is the case of the synchrotron where beta is in the perpendicular direction to d beta by d t. Now you can see here instead of gamma raised to power 4 I have written here e by e0 raised to power 4. This defines something as we raise the total energy of the particle e radiated power increases enormously. And again here you can see that e0 is in the denominator is the rest energy of the particle in the denominator means if a very light particle is there means very small rest energy of that particle if that particle is radiating that radiates more. Now how much radiation in the will take place means how much energy of an electron will lose or charge particle will lose in a complete turn in the synchrotron. So definitely energy can be obtained using the time integral of the power because power is d by d t. So energy will be p d t and integrate over the t revolution. Now instead of d t we take the length circumference of the accelerator. So this d t can be converted into d s by beta c. So integral is over the closed integral over the circumference and now put this value here in this integral and at the place of d s you just write down 2 pi r r is the bending radius of the dipole in synchrotron because this radiation takes place only when there is centripetal acceleration means this integral will be non-zero only in the path of the bending minutes in all other places it will be zero. So contribution is only due to dipole moments. So this is the curvature of radius of curvature in the dipole moment. So finally you get the result q square by 3 epsilon 0 beta q by r e by e 0. So again you can see similar to the power energy also depends on the beam or particles energy raised to power 4. This is the radiated energy and this is the particles energy. In case of electrons if you put e 0.5 ml and beta close to 1 and charge on electrons and the value of this permittivity of free space epsilon 0 then you get formula e in kev is equal to 88.4 e raise to power 4 in gv here kv e kv is the energy in the radiated electromagnetic field and this e is the electrons energy on. So this is a simple formula for the electrons. Now this radiation which is emitted by the charged particle or electrons during the centripetal acceleration is very useful. We will see some of its characteristics in the later slides. This is fantastic that some dedicated synchrotrons were built to tap this radiation. So in electron storage ring, storage ring means a synchrotron which is operating at constant energy mode. That electron is storage ring in which electrons keeps revolving for many hours and on each pending these electrons radiate and this radiation can be used by different experimentalists. So this kind of accelerator is known as synchrotron radiation source. This radiation is known as synchrotron radiation. It is also known as magnetic brainstorm because in the case of linear acceleration we saw that the radiated power is known as brainstorm radiation. Here we are getting the radiation from the magnetic fields. It is sometimes known as magnetic brainstorm radiation but more general word is synchrotron radiation. Now in the beginning of the accelerator era when accelerators were built and electrons were to be accelerated after some energy the electrons loses a significant amount of energy in the form of electromagnetic waves. Means we want to push the electrons and we want we give the electrons the energy and this energy is lost in the form of electromagnetic waves by the electrons. So this was a bottleneck at that time and this was a nuisance that oh this radiation is occurring and this was a byproduct. And this was basically a header for achieving higher energies in the case of the electrons because we are pumping the energy and it is losing the energy in the form of electromagnetic waves. Later on it was found that this radiation has very very fantastic properties and can be used extensively in different kind of studies. So dedicated SRS were formed and here from each dipole magnets there is a radiation is coming out and by making some optics for that radiation radiation is brought up to the experimental stations and then these radiations are used by different researchers. Then there were modifications in the magnet design and some new kind of magnets were designed to modify the properties of these synchrotron radiations. These are Wiggler magnets and undetermined and also ring of low emittances has been built. So this was the third generation of synchrotron radiations which is still present here instead of tapping the radiation from the dipole magnet there are other kind of magnet which actually modifies the properties and return radiation from these magnets Wigglers and undetermined. These magnets are placed in the straight section of the synchrotron and when electron passes through these magnets it radiates intensively and now new generation is ERL X-ray field. This is the fourth generation and mainly these sources are based on the Linux. Now we see some of the properties of the synchrotron radiation. Why synchrotron radiations are so important? Synchrotron radiation you can see in this plot the level of brightness of this synchrotron radiation. This is our usual lab X-ray sources where electron impinges on a cathode and we get the bramestron radiation for our experiment. Now you can see that it enters to 10 photons per second per solid angle. Inserting bandwidth this can be obtained using these X-ray tubes. This is the most powerful X-ray tube available on the base dog rotating anode and now you can see that five order mode this is not only the five times five order mode radiated brightness can be obtained from these synchrotron radiation sources when we use radiation from the dipole magnet and even five orders more than this can be obtained using the modifier magnets like undulator and wigglers. So synchrotron radiations are very very intense compared to other sources when we use it for our experiments. So this figure shows that this red part is the trajectory of the electrons. Electrons are going in this way and this dipole magnet bends the path of these electrons and as electron enters into the field of the bending magnet it feels centripetal acceleration and it radiates. So in a complete fan of path inside the dipole magnet electron radiates and this radiation fan comes out. Now after making some optics for these photons or the electromagnetic wave up to the experimentalization experiment can be carried out using these. Now in this figure you can see that this blue line shows the electron orbit in this synchrotron. So this is the plane in which electron rotates and suppose an observer is here so whenever electron comes here in the tangential direction of this point radiation is sent and the cone of radiation which is formed by the electrons here has an opening angle of shadow means when electron radiates it radiates in a cone and the angle of that cone is shy and this shy angle where i is as 1 by gamma. Now suppose we are using a 2G EV electron V. 2G EV means gamma will be gamma will be 2 into 10 to 9. This is 2G EV divided by rest mass energy of the electron that is 0.5 into 10 to 6. So it will be 4000. So opening angle will be 1 by 4000 radian. So very very very small opening angle means it is a very very collimated beam. So first property is that it has very high brightness. The second property is it is a very collimated beam very narrow cone is made by the radiation and this is a wide spectrum. The emitted radiation has wavelength ranging from visible light to hard exercise means it covers a wide spectrum. The user can put a filter according to the earwish and they can tap the particular frequency of choice or if some experimentalist wants complete spectrum then it is also there. It is also polarized light. Polarized light means when electron rotates in this plane and radiates the electric field also lies in this plane. So this is a plane polarized light. So if any experimentalist wants a definite polarization that is there. It has a pulse restriction because we have seen that RF field makes the bunches the certain duration of the RF period is used for the acceleration and in that duration only electrons remains there. So bunching of the electrons takes place due to RF. So electrons has a clean pulse structure and that is why the emitted radiation also has pulse destruction. This is well calculable properties also. Now you can see the spectrum of the emitted radiation. So here you can see this is the frequency and this is the some factor which describes the intensity not the intensity some factor. So this is intensity on the median plane means when angle shy is zero what is the radiated intensity this curve shows that. So and this curve as curve this shows when we integrate the intensity in the opening angle shy this is the total this s actually indicates the total radiated power integrated and this shows only on excess radiated radiated flux. So now you can see that we can choose one wavelength here or one frequency above which divides the radiated power in two parts. One is P1 and other one is P1 and this is the omega c. So P1 is equal to P2 if we choose omega c in such a way that P1 becomes equal to P2 this is known as critical frequency or critical wavelength. It shows a general characteristic of the emitted radiation. So this can be calculated critical wavelength using this formation and you can see that if we increase the energy e of the electron beam lambda c decreases means higher energy accelerator emits harder x-rays more shorter wavelengths. Suppose a 2.5 gb accelerator is there and in the same circumference if we raise the energy 4 gb then lambda c will be lower down means shorter wavelength will be there higher the x-rays