 lecture. We will discuss about the chromaticity. It is the effect generated by the quadrupole for the off momentum particles. In last lecture we have seen the effect of dipole magnet on the off momentum particle and the new parameter dispersion was introduced. Now we will see how the quadrupole acts on the off momentum particle and this effect can generate an aberration and how that aberration can be corrected. So now we recap the bit at once. Actually we have an optics simple optics in which focusing quadrupole is here then defocusing quadrupole is here and again this is repeated. If a trajectory of a particle enters in this optics like this here so focusing quadrupole will sign this trajectory towards the design trajectory and then defocusing will sign it away and again focusing quadrupole is there so it will sign it again towards the design trajectory and again defocusing will sign it away. So in this fashion this particle exhibits bitatron oscillations these are bitatron oscillations of this particle. Here we mentioned that these bitatron oscillations are for the correct momentum particle means on momentum particle p0 is the design momentum for which this optics has been designed and this particle has the same momentum. Now suppose another particle having higher momentum than this p0 enters the optics on the same trajectory on this trajectory how the quadrupole will act on this now this is the new trajectory for the higher momentum particle this green one was for the on momentum particle and this blue is for the off momentum particle. Now for the off momentum particle because it is having higher momentum the focal length for the quadrupole will be higher means focusing will be somewhat weaker than the lower momentum particle or the design momentum particle so trajectory will be in this fashion and then when it reaches to defocusing quadrupole defocusing quadrupole will also act with a somewhat weaker strength this weaker strength and then it will send this trajectory towards this and in this fashion we can see that this off momentum particle also exhibits a bitatron oscillation but however with longer wavelength means one bitatron wave takes longer length to complete it means number of bitatron oscillations in one complete done for higher momentum particle has been reduced means tune the number of bitatron oscillation in one complete done which is known as tune of the higher momentum particle has been reduced and as we told earlier that tune is a very important parameter for an accelerator so tune has been changed and we have to correct this tune and how much the tune has been changed and how we can correct it we will see in this lecture the change in tune say this is delta nu and this change over the unit momentum offset means divided by delta this delta is delta p by p means fractional momentum offset so change in beta count tune over the unit fractional momentum offset is known as chromaticity this is the chromaticity it is defined in this there will be two chromaticities one for the horizontal plane and one for the vertical plane so for horizontal plane we have to take the delta nu for the horizontal beta count tune for vertical plane we have to take the delta nu for the vertical plane so they will be basically two chromaticity one is the chromaticity x and now you can see that here when we increase delta when we increase delta means delta p is positive delta p is positive means delta p by p is higher so for higher momentum particle delta p is positive and for that particle tune decreases means delta nu decreases for the higher momentum particle means delta and delta nu have opposite sides if delta is plus delta nu is minus if delta is minus then tune will increase and delta nu will be plus means chromaticity is a negative number in the synchro this chromaticity has been generated due to quadrupole magnets if we we will see in later part of this talk that there are other magnets which also contribute in the chromaticity and that contribution can be used for correcting the chromaticity so the contribution of the quadrupole which generates this chromaticity is known as natural chromaticity now suppose this is a dipole magnet again similar picture which we have seen in the case of dispersion so this is the design trajectory and design trajectory is for the p0 moment and this is the quadrupole magnet now higher momentum particle follow this trajectory it will bend less due to dipole magnet and it will reach in the quadrupole at the off axis and quadrupole will focus it however had it been a right momentum particle quadrupole will sand it here but this particle has higher momentum so part it will sand quadrupole scientific particle far away from this this focal length will be longer now for lower momentum particle quadrupole will have shorter focal length so it will pass from here this is the focal length for the lower momentum particle and this is the focal length for the higher momentum particle so we have is spread in focal length due to delta P by P means B contains a spreading momentum and then quadrupole will produce a spread in the focal length and this spread in the focal length was the main cause for generating the chromaticity now for off momentum particle what should be the strength of the magnet so the strength of the quadrupole magnet is defined in this fashion that is Q by P which is basically the inverse of pain rigidity and then G is the gradient of the quadrupole magnet quadrupole magnet has the gradient in the field and this is constant quantity for the quadrupole magnet so this G is a constant for the quadrupole magnet now in place of P we have P0 plus delta this P is the momentum of the particle it may be the off momentum if delta P is non-zero and if delta P is zero then it is the on momentum particle so P can be written down as P0 plus delta again take P0 out so you will have one plus you will have P0 out so you have one plus delta P by P0 and this delta P by P0 is denoted by delta so we have Q G by P0 upon one plus one plus delta and this quantity is again the strength defined for the design actually when we see strength of the quadrupole we refer to this K0 and this K0 is divided by one plus delta so for higher momentum particle for which delta is plus this K is smaller than K0 for plus delta and K is higher for K0 for minus delta so quadrupolar kick has been reduced for the higher momentum particle you can see and you can calculate the quadrupolar kick using this function now if chromaticity has been generated is there any reason that we should correct it what are those reasons for which we should worry about the chromaticity so one reason is they spread in the bitatron tunics bitatron tune is an important number and it tells how many oscillations are there in the one completer so if there are some imperfections in the machine so far we are studying about the perfect machine means each dipole magnet is perfect each quadrupole magnet is perfect and those are placed according to our wish means there is no misalignment in the magnets in real practice these errors are there and when these errors are there in a synchrotron these error also occurs in each term means particle will see these error on each term when it passes through that magnet which is hiding there so error is also periodic in nature in the synchrotron so particle is exhibiting a bitatron oscillation and it is subjected to some periodic forces so an oscillator when it is subjected to some periodic forces it can this periodic force can exert some resonance in that oscillation and that resonance may cause the growing of the amplitude in the similar fashion bitatron oscillations can also subject it to some resonances due to imperfections and amplitude of bitatron oscillations will grow over time by time and ultimately the amplitude will reach to the wall of the chamber and particle will be lost so these are the resonances which depends on what the frequency of the bitatron oscillation is there and that bitatron oscillation frequency is related to tune that's why tune is an important number tune defines whether the resonance can occur or not so this is very first reason we had to be careful when we talk about the chromaticity the second major reason for correcting the chromaticity is head tilt instability actually in accelerators when we store the discharge particle beam there are various types of instability which can occur and these instability can cause the beam losses and one type of those instabilities is known as head tilt instability in this type of instability the head of the bunch drives some resonance kind of phenomena for the tilt and in this fashion beam loss can occur to avoid that kind of instability chromaticity has to be made zero so these are two reasons for which we have to correct the chromaticity now how to correct the chromaticity here we have seen that quadrupole is having larger focal length for the higher momentum particle cut to this and shorter focal length for the lower momentum particle if we want to correct the chromaticity we have to make sure that focal length for these particles must be the same means focal length of higher momentum particle should be decreased and focal length for lower momentum particle should increase and how we can do we can consider that there is some focusing lens which focuses the higher momentum particle so instead of reaching there these particle reaches here and for the lower momentum particle we needed de-focusing lens so this should be a de-focusing lens so it de-focuses and focal length increases for the lower momentum particle and then both the particles having higher momentum or lower momentum reach to the same focal point so in this fashion we can correct the chromaticity means on one side of the axis we need focusing lens and on other side of the axis we need de-focusing lens and one thing is here you can see that in case of higher momentum particle we exerts we need to exert a kick in this direction because we have to change its path from red orbit to blue orbit so kick should be in this direction by this magnet for lower momentum particle also we should have the kick in the same direction because now on this side for lower momentum particle this magnet should act as a de-focusing lens and note one thing that when we are talking that all the positive momentum particles are there and all the orbits of negative momentum particles are there it means orbits corresponding to these momentum have been separated out means we are talking about the region where dispersion is non-zero if dispersion is zero these orbits will not be separated out and we cannot correct the chromaticity so for correcting the chromaticity we need some dispersion and at the location where dispersion is non-zero this kind of optics has to be inserted now how this optics can be relaxed means one side is focusing and other side is de-focusing or you can see that on both side kick is in the same direction now consider again the field of a quadrupole magnet field of a quadrupole magnet increases linearly with x this is the vertical component of the field which is drawn here and this is the horizontal displacement from the design axis so field linearly increases in the case of the particle because it is linearly increasing gradient is constant now required field for chromaticity correction actually we have seen that the vertical component of the magnetic field at plus x should be equal to at minus x then only kick will be in the same direction so field should be given now we compute the quadrupole kick for the off momentum particle quadrupole kick is given by klx and already we have seen that this k for the off momentum particle has to be replaced by k zero upon one plus delta x is there enough we have introduced one approximation that energy spread or momentum spread is not large enough means this delta p by p is smaller than the fault then this one plus delta can be approximated by one minus delta in the numerator because it when it will go in the numerator it will have one plus delta x to power minus one and when we use binomial expansion for this and retaining only first term and rejecting the higher order term due to this approximation we will have one minus delta so we have k zero x minus k zero that is x delta now you can see that this is the normal quadrupole kick let it mean a right momentum particle the quadrupole kick should be k zero x and this is the and this kick is due to delta if delta is zero this kick will be zero means this is for the off momentum particle and it varies as x into delta now you can see that where the dispersion is non-zero you can have x is equal to p by delta which we have seen in lecture on the dispersion so delta can be written down using this d upon sorry x is equal to d delta so delta can be written down x over d now this quantity if we will put x into delta is equal to x into x by 3 means this quantity varies as x is equal means the field should be even and it should vary with x square means this should be the field profile like this now if we go in the minus side or plus side by has the same sign means it will kick in the same direction and variation in the b y is with a square means it is a parabola this type of required field can be generated using this sextopole magnet means six poles and this magnet can produce the required magnetic field