 No, because we are talking about the phase means what is that instinct when particle is Completing its revolution and reaching inside the RFK Means we are interested in the path length because path length will decide how much time is taking in complete revolution So how much? path length is changed due to change in Momentum When we compared with this synchronous particle, we will compute it now suppose This is a part of the curved trajectory curve trajectory means trajectory decided by the dipole magnet and This red curve shows the design project This bending is delta theta and radius of curvature for the design trajectories are not Now suppose some another particle which is Having some momentum deviation from the design momentum will follow another trajectory due to dispersion We already know that if particle is deviated in momentum Dipole magnet will sign it to the other trajectory decided by the dispersion. So this is such kind of trajectory so length of design trajectory is at the magnet is delta s and length of the trajectory of Particle which has some momentum deviation is delta s1 Now we will see how delta s1 is compared to delta s So we can write them delta s very easily are not delta theta and similarly, delta s1 is equal to r delta theta Instead of r0, it will have some r because momentum has been changed Now r is basically in transverse coordinate system. You can see that here Origins when the synchronous particle is here and this is the jet or s you can say and this is the x So r will be x plus r0 So we have written down r0 plus x So using these two equations, we can get delta s1 minus delta s means how large is the path length compared to synchronous path of This momentum deviated particle and this will be x delta theta now this x this x It is decided by the dispersion So in the place of x we can write down D into T This is the dispersion at that location So this delta s1 minus delta s that is the change in path length compared to this synchronous particle is D delta upon r0 delta s This is the dispersion. So dispersion or bending magnet Actually comes into the picture when we want to calculate how much path is changed and this changed path Actually causes the change in revolution time Means dispersion plays an important role in longitudinal dynamics also. So this Delta s minus delta s1 was calculated for a small segment of bending magnet Now all over the ring if you want to calculate delta l It will be the integration of D delta r0 ds Now over a one term means particle is exiting the RF cavity and After all complete then it is again reaching to RF cavity in between these travel Delta remains constant or you can see we are talking about a particular delta and for particular delta This we can define delta l by l for a unit fractional change of momentum What will be the path length change? Fractional path length change for a fractional momentum offset this parameter in accelerator is known as Momentum compaction factor and it is just integration which we have written down here divided by l Because we have fractional change in the length So this alpha actually decides what will be the Change in revolution time for the particle which has deviation in the movement. So we calculate that How much revolution time is changed? T revolution is given by l by beta c So delta t by t. This is logarithmic derivative of the above equation will be delta l by l minus delta b by This c will not be differentiated. It is constant quantity speed of the light This beta c means we are talking about velocity. So this beta c is b This beta is relativistic factor So delta t by t will be delta l by l, delta beta by beta means as momentum changes We have calculated what will be the change in the path length and as momentum changes Speed may also change. So this is the change in the Speed now as particle travels at higher momentum Bending magnet bends less. So it will Travers a larger orbit means path length will increase So it will increase the revolution time at the same time Because of the higher momentum, its speed may be higher and because of the higher speed, its revolution time will decrease So these are two counter phenomena which counters each other One factor Increases the revolution time, other decreases the revolution time So by combining these two with a positive sign will give you the revolution time Because we are interested in calculating the revolution time for the fractional momentum offset So divide this complete equation by the fractional momentum offset. We have taken this Now this quantity you already know is alpha momentum complexion factor And this quantity in accelerator is given a new name name either facelift factor or Frequencies effect So this is beta, which is the frequencies factor. This is alpha and Delta beta by beta upon delta P by P means you can say delta Beta upon beta Upon delta P by P It will just One upon gamma This is an exercise for you to prove this revolution This is very easy how you can prove it Just take Beta or momentum in terms of beta and write down that means P is equal to Beta C gamma This is the momentum in terms of beta now you can write down gamma also in terms of beta Gamma is equal to one upon under root one minus beta square So this whole equation in terms of P and beta now you can calculate delta beta by beta and delta P by P Taking logarithmic derivative of this. So you this is an exercise to prove this relation from this In some literature instead of Delta P by P They use Delta F by F means instead of fractional Revolution time change is a change in Fractional change in revolution frequency Then the sign of the equation will be reversed means in that case facelift factor will be one by gamma square Minus alpha so in literature you will see both kind of things some literature Says frequencies and factor as alpha minus one by gamma square and in some literature It is mentioned as one by gamma square minus alpha because of the basic difference in the definition Whether we are taking the revolution time or revolution frequency as we have Delta T by T is delta L by L delta beta by beta So at one place because This factor increases the revolution time and this factor Decreases the revolution so at one point This may occur or one energy or one beta this can occur that both Component cancels each other means delta T revolution upon T revolution becomes zero This is known as transition Now you can see here if Delta L by L factor is large Then the Delta beta by beta then Frequency slip factor will be positive When Delta L by L will be larger than the Delta beta by beta means Delta beta is very small Delta beta is very small means we are changing the energy, but beta is not changing It is the case of highly relativistic particles So in high energy machines generally frequency slip factor is positive if This effect is much more than this Then frequency slip factor becomes negative when it will be negative It will be in low energy region So low energy machines can have negative Frequency slip factor for an example when we say that 800 any proton synchrotome or one gv proton synchrotome It means kinetic energy is one gv and it is comparable to the rest energy of the protons So in this case we can say that we are operating the machine in low energy region in this region The frequency slip factor may be negative But when we talk about say one gv or two gv of electrons Electrons have rest mass energy of 0.5 and means one gv or two gv for the electrons is a very high energy This is much higher means one gv means two 2,000 times the less energy of the electron So in this case this factor will be insignificant and Frequency slip factor will be positive So in general all the synchrotones Electron synchrotones have positive frequency slip factor while protons in common may have positive or negative Frequency slip factors and when then these factors cancels each other This is the delta t-revelation upon t-revelation goes 0 that energy that particular gamma when this occurs is known as Transition energy and that gamma on which this occurs is known as gamma transition So gamma transition is one upon two delta now the thing is that if Machine is operating on gamma transition Then revolution time becomes independent of the movement Means whatever is the momentum Even a momentum deviated per a moment a particle deviated in momentum has the same revolution time as the On momentum This is the isochronous mode of the machine Cyclotron operates in this mode in cyclotron revolution time doesn't depend on the momentum Synchroton can operate in all the three modes means it can operate in the synchronous mode as well as Frequency slip factor Plus means above gamma transition and frequency slip factor minus means below gamma transition Now these while we are talking about these because higher momentum particle in this case In this case when delta beta beta is larger means frequency slip factor is negative negative means when we have higher momentum particle it takes shorter time and And Why so what so now we see what is the effect of this gamma transition on the longitudinal dynamics This is the RF wave inside the cavity and suppose Particle arrives in the RF cavity at this space These are the initial condition for three particles The particle in between is the synchronous one means it will come again on the Synchronous phase we are assuming that synchronous phase is not changing. So This particle will come again in the next term in the RF cavity on the same phase here Now suppose this machine is operating below transition below transition means Higher momentum particle takes the smaller time to complete it Her momentum particle is faster so this particle Gets more energy than the synchronous one because this is in hair field This is feeling hair field than the synchronous particle then it gets more energy So it is at higher energy than the synchronous one because of the higher energy It's revolution time will decrease and it will reach earlier Than this point Means this particle will come here instead of And similarly this particle is at lower energy. So it will take longer time to complete the turn So it will come here means these particles are Coming towards the synchronous particle in next turn is still it is at higher potential means in second turn also It is gaining more energy than the synchronous Although the rate of change in energy has been decreased, but it is still it is at higher energy So in next turn synchronous particle will arrive at the same phase While this particle will be more closer to synchronous phase and this particle will also be more closer to synchronous phase in next turn this will be more close and subsequently the phases Will change its sign with respect to synchronous particle means red particle will then have minus Sign with delta Phi and this particle will reach here and this particle will reach here and then this will be means some type of Oscillations around synchronous phase is taking place This is known as phase focusing as Quadrupole focuses the beam and excites the beta-tron oscillations Similarly RF cavity also focuses in longitudinal dynamics a longitudinal Direction and it also excites a Kind of oscillations known as synchrotron oscillations. These are synchrotron oscillations Now we take another three initial conditions This time we are taking on this means this was on rising age of the RF field and this is on the falling age of the RF field is still synchronous particle is Taking the same energy as of the previous case So it will come again on the same phase after first revolution however because a Higher energy particle means red particle will come early So it will go away from the synchronous space and similarly is the case for the lower of energy particle and this gap will increase subsequently Turn by turn so this gap has been increased means it is a defocusing phenomena and Motion is not stable So we have seen that if machine is operating below transition Then rising age of the RF provides a stable motion while the falling age provides an unstable So we will all the bunches all the particles will concentrate towards this If machine is operating about transition then this falling age will provide this stable oscillations While this rising age will provide a defocusing phenomena means unstable Now we derive this equation of motion means we have seen the qualitatively that if a particle is defeated in phase Or energy it will execute some kind of oscillations now we will derive Quantitatively that what is kind of that oscillations So in one term this keep in mind this delta phi is basically the change in phase with respect to synchronous particle means we can understand it like this suppose a particle In n plus 1th term as the phase with nth term plus omega RF This is the phase in the n plus 1th term for a particle Now similar equation will hold for the synchronous particle also So we will have phi of the synchronous n plus 1th term is equal to phi of the synchronous in nx plus 1 plus omega RF and In place of T-revolution it will be T-revolution for the synchronous particle Now in place of T-revolution of a synchronous particle we can write down it as T-revolution of the synchronous particle plus some delta t. So what will be the delta? phi in the n plus 1 term is equal to delta phi in the nth term Plus difference of these. So this will be omega RF delta t This is the difference of these two equations. So this is the deviation of the particle With synchronous in the n plus 1th term This is deviation of the particle from the synchronous phase in nth term and this is omega RF delta t and At the place of delta t we can write down t eta delta p by p With the definition of the frequency slip factor. This is frequency slip factor Frequency slip factor was defined as delta t by t upon delta p by p So delta t by t Will be eta delta p by p. So delta t will be equal to eta t delta p by p So this is written here Now we can see that delta phi n plus 1 minus delta phi n is equal to this number and this is change in the deviation of phase in 1 term. That's why it is written be delta phi over d n. d n means in 1 term We are counting with terms. So this is the change in phase deviation in 1 term phase deviation with respect to synchronous one And it will be given by omega RF t revolution theta delta p by p. Now omega and t revolution We have seen that t revolution Should be equal to h t RF It should be an integer multiple because t revolution is the here t revolution is for the synchronous particle So t revolution is the h t RF and We want to change this delta p by p in terms of delta e by e because when the particle Traverses through the cavity. It is easier to calculate the delta e by e So instead of delta p by p. We are writing delta e by e. So delta p by p is equal to 1 by beta square delta e by e again This is left as an exercise to prove This delta p by p should be equal to 1 by beta square delta e by e. This is an exercise for all of you So in place of delta p by p. We have written down this delta 1 by beta square delta e by e and when we take t revolution as h t RF. So we have omega RF t revolution is equal to omega RF h t RF and Omega RF t RF will be 2 pi. So this quantity will become 2 pi h and this 2 pi h is written here So this will be 2 pi h eta on beta square delta e by e as we have calculated it Change in phase deviation with respect to each term We also calculate what is the change in the energy deviation in one term following the same left path So sir in the n plus 1th term Synchronous particle has the energy as nth Turn energy plus q v sin phi s phi s is the synchronous phase Similarly for the asynchronous particle means the particle which is not synchronous which is deviated in the energy or phase it will have Energy in the n plus 1th term is equal to energy in the nth term plus q v sin phi s plus delta This delta phi is the phase deviation which is here So if we Subtract first equation of the second then we will have again here similarly. What is the change in energy deviation per term and This will be q v sin phi s plus sin phi s. Here we have delta e by e and Here we get delta e by delta n. So d by dn of delta e. So if we differentiate this equation again We will get d by dn of delta e and then this value can be put into this equation So let us differentiate this equation again of the phase equation. We will get this is a simple Differentiation for you will get this equation now use this relation in the last equation So you get Complete equation in the phase this equation contains only phase Deviation this is the phase deviation and this is the phase deviation now this equation looks a complicated But you can understand it very easy. How? suppose Deviation in phase is not large means this delta phi is very small quantity and if delta phi is very small quantity this Expression sin c plus d can be written as cos c plus d by 2 sin c minus d by 2 means sin delta phi by 2 term will be there and Because Delta phi is small You can use relation sin delta phi by 2 is approximately delta phi by 2 and cos phi plus delta phi plus phi upon 2 will be Cos phi s. So this will be cos phi s and this delta phi is d2 sin delta phi Now you can see that this equation is Similar to simple harmonic oscillator equation. This is d2 phi at n squared This is omega square and this is again And this is synchrotron oscillations. Now you can see here this overall number Should be negative for oscillations because it is RHS So it should be negative for the oscillation now here two quantities Can be either positive or negative one is the frequency slip factor and other one is the cos phi s So if eta is plus then cos phi s should be negative and if eta is negative then cos phi s should be positive only then This will represent a simple harmonic oscillator. Otherwise exponentially growing solution will be there means not bounded motion now you can check which we have seen earlier that in the Below transition phi s should be in the rising age and for the above transition phi s should be on the falling edge of the RF field Because this is now a simple harmonic oscillator equation We can get ellipse similar to the transverse plane in the delta phi and delta e So what kind of ellipse we get? Here we draw the delta phi And here we draw the delta e So at this location we have delta phi also 0 and delta e also 0 means this is the Synchronous particles position the origin shows the synchronous energy and synchronous phase Now for a small oscillations as suppose this is the Phase deviation initially particles with respect to synchronous one as Done by done this phase deviation Diminishes when it comes in however, the energy division increases because up to certain point it gets more and more energy than the synchronous one So it will come here after this When particle crosses the phase of the synchronous particle its gap with in the energy with the synchronous one will reduce And it will come here And again it will come So this is a kind of ellipse we get in the longitudinal link This shows the simple harmonic oscillations however, you can see that If delta phi is large The equation is no longer linear It is a non-linear equation If delta phi is large So it means if delta phi is large this ellipse will be distorted and this will be like this Ellipse will be distorted like this Even if we make delta phi large It will distort more and like this And if delta d beta by dt is 0 means if we are talking about the storage mode of the operation Then this stable phase will be 0 to 5 and minus 5 So about this if suppose phase is here or energy division is here What happened to this coordinates? So this coordinate will have open trajectories No bounded motion like this So you can see that inside this area There is a stable oscillations and about this area there is no stable oscillation so this Area is known as RF bucket and Curve which separates is stable and unstable zone. This is the curve which separates This is known as separate fix If db by dt is non-zero means we are boosting the energy in this case Shape of the RF bucket changes and it becomes like this delta d delta phi and This is the phi s like this This is even more distorted and now It doesn't expand up to the 2 pi area Instead area has been shrunk as pi s increases Area of the RF bucket becomes smaller and smaller So this if particles are inside this RF bucket, it will exhibit stable synchrotron oscillation and outside the bucket will be lost So it is similarly as we have Certain acceptance in the transverse plane and if being within that acceptance it will exhibit Beta term oscillations otherwise the particles will be lost The same kind of thing is also here. So in a shell we can see that When particle is exhibiting motion in a synchrotron In exhibit three kind of oscillations one is horizontal beta term oscillations Another is vertical beta term oscillations and third one is the synchrotron oscillation and like the beta term tube We can also define a synchrotron Synchrotron tube means number of synchrotron oscillations in one time Now you're seeing that synchrotron oscillations are slow means in one time Even a fraction of the oscillation completes So synchrotron tube will also always be lesser than one and Beta term is much higher than one in one complete and many beta term oscillations takes place While in many turns one synchrotron oscillation takes place So synchrotron oscillations are very slow oscillations compared to beta term motion. We have studied these motion In decoupled way means horizontal beta term motion is not coupled with the vertical motion and Synchrotron motion is not coupled with the beta term motion. In real practice these all the oscillations are coupled to each other and There is a rich dynamics Which arises Due to coupling and many type of resonances occurs So this kind of studies are followed in the accelerator physics for designing an accelerator So again these basic references covers all the things which I have shown here and in next lecture We will study about a special type of accelerator synchrotron, which is known as synchrotron radiation source