 In quadrupole magnet G was the gradient and which is a constant quantity. In sextupole magnet, rather than G, the derivative of G is constant, means second derivative of the field is a constant quantity, because it is a parabolic curve, depends on the x square. So, second order derivative will be a constant quantity, means magnetic field will vary with s by 2 x square minus y square and horizontal component of the magnetic field will be sx1. Now, you can see here in the horizontal component x is multiplied by y, means this sextupole magnet will generate coupling in the betaton motion, means horizontal motion will be coupled by the vertical betaton motion in the presence of the sextupole. Now, we can calculate the kick of a sextupole also, as we calculated the kick due to quadrupole. So, again the same formulation will be used, magnetic rigidity is here and by here. This formulation for computing the kick has been used many times. Now, at the place of by, we will put this value. So, this will be mlx square minus y square, here m will be s by b, so this will be half mlx square minus y square. Now, because we are introducing this sextupole magnet at the place of non-zero dispersion, means here the dispersion is finite, means displacement of the particle from the design axis can be written down as the displacement due to betaton oscillation and displacement due to dispersion into delta p by p. Now, we will put the value of x in the sextupole kick, so this will be half ml and at the place of x we have written down this, so the square of this quantity plus y square. In vertical plane there will be no dispersion, because we are considering that banding in the synchrotron is taking place in the horizontal plane only and dispersion will be only in the plane of banding. So, in vertical plane there will be no dispersion, so we have only horizontal dispersion in the case of a synchrotron in general. Due to imperfections, there may be a vertical dispersion also, but in this course we are not considering imperfections in the lattices, means we are considering only perfect lattices. So, when we will open this bracket we will have x betaton square d square plus delta square and 2dx delta pi square. Now, you can see that this term here depends on x into delta. This is the similar term which we have seen in the case of a quadrupole, means this term can be used to cancel the term generated by the quadrupole. And if we can cancel that term in the quadrupole peak, then chromaticity can be corrected. And these are extra unwanted term due to sextopole magnets. So, sextopole magnets correct the chromaticity, however it generates various unwanted term and you can see that these unwanted term depends on x square delta square and y square, means our dynamics becomes non-linear when we introduce the sextopole magnet in the machine. How we can calculate the chromaticity? Now we quantify that what is the amount of the chromaticity if we look the lattice or if we know the twist parameters or the strength of the quadrupole can be compute the chromaticity we are going to derive that formulation. Now suppose this is a complete path in a synchro-domed design path and there is one quadrupole and we are looking the effect of this quadrupole on the off-momentum particle. Because up to the quadrupole dynamics remains linear, so if we calculate the effect of one quadrupole and we can quantify it then for all the quadruples effect will be superimposed. So we calculate the effect of one quadrupole on the off-momentum particle. So let us say that this quadrupole has the transfer matrix MQ and after this magnet and again reaching to the beginning of this magnet the matrix is Mr. So if we multiply these two matrices we will get the one term matrix or a matrix for the one complete term in this synchro-domed. So this is the M matrix for the one complete term and these are the matrices multiplied. Now we want to see that if the strength of this particular quadrupole has been changed what will happen? So let us say that the one term matrix has been changed to M with subscript N when the quadrupole matrix has been modified to MQi. This matrix is of the quadrupole with changed strength Mr rest of the matrix in this synchro-domed will remain same. So Mr is same as is here. So we can put the value of Mr here you can say that Mr is equal to MQ inverse M. So let us put this value of MQ here so you will get the result MQM MQ inverse M. Here it is the quadrupole matrix with change in strength. It is the quadrupole matrix inverse with correct strength. So quadrupole matrix with correct strength is 10kl1 here again we are considering the thin quadrupole means we are working in the thin lens approximation. So inverse of this matrix will be 10-kl1 only sign of this element has been changed determinant of this matrix is 1. Now if quadrupole strength has been changed say change is delta kl so new matrix of the quadrupole will be 10kl plus this change and 1. So we now have one term matrix under the changed quadrupole strength it will be like that this will be the matrix from here this is the matrix from here and it is the one term matrix when there was no strength change or quadrupole has the correct strength. Now when quadrupole has the correct strength we have seen that one term matrix can be written down in this fashion cos phi plus alpha sin phi beta sin phi minus gamma sin phi cos phi minus alpha sin phi here the quiz parameters alpha beta are at the location of the quadrupole which we are considering. So this one term matrix if multiplied by these two matrices we will get the new one term matrix which changed strength and now in suppose due to changed strength of quadrupole beta has been changed to beta n alpha has been changed to alpha n and phi phase at once becomes phi n then new one term matrix can be written down in new changed parameters like this cos phi n plus alpha sin phi n beta sin phi n because still one term periodicity is there so one term periodic matrix can be written down in this fashion. So if we compare the elements of LHS and RHS we can get the change in tube due to delta P or changing strength there should be a matrix this is written down there will be A B C D which is the multiplication of 1 0 K L plus delta K L 1 and 1 0 minus K L 1. So this produce a matrix A B C D and this will be here and after multiplying these all things together we will get this matrix. So here you can see that this element is still cos phi plus alpha sin phi but this element now contains beta sin phi multiplied by the error of the quadrupole or strength changes in the quadrupole. Now if we can compare the trace of these two matrices now these two matrices are the same so we can compare the trace when we will compare the trace only cos phi will be there. So in the new lattice the LHS side we do not have any trace parameter and it will be easier to obtain the phase change and phase if we obtain the phase change we can obtain the tune change and if we can obtain the tune change chromatism is defined by the change in chain. So 2 cos phi n will be 2 cos phi plus delta K L B plus and phi this is the trace of this matrix to this side. This can be written down as 2 cos phi n minus cos phi it will be delta K L beta sin phi and this cos c minus t formula is applied here it will get sin phi n sin phi plus phi n upon 2 is equal to 1 by 4 delta K L beta sin phi. Now because we are assuming that energy is spread not much means delta P by P is a smaller value so change in strength delta K will be also a small number and that is why the change in phase will also be smaller. So we can write down sin delta phi by 2 and sin phi plus delta phi by 2 is this now using the approximation that delta phi phi minus phi n is small. So sin delta phi by 2 can be approximated as delta phi by 2 and here we can neglect the delta phi in comparison of the 2 phi. So 2 phi by 2 will be sin phi and this will be RHS. So now sin phi and sin phi cancels out and you have delta phi is equal to minus half delta K L beta. Delta phi is basically the phase advance over the wanta. So it is related with means number of delta term oscillations over a one complete and can be obtained which is a tune is equal to. So phi can be written down as 2 phi so delta phi can be written down as 2 phi delta mu. So this is written here and this equation using this equation you can get the delta mu. So change in bit count tune is beta at that location multiplied by the change in strength. Now if quarter pole has higher meter distribution the chromaticity will be higher because delta mu will be higher in that case. Now this delta K L we have calculated earlier that this will be equal to K L delta this delta is delta phi by 2. So this will be the chromaticity. Now you can see that this chromaticity is calculated only for one part. So for calculating the whole lattice we have to integrate this formula over the entire circumference. So this is the chromaticity for the synchrotron or a complete. Now you can see here that K into beta this combination generates the chromaticity means higher strength of the quarter pole will generate higher chromaticity means if a lattice has a stronger quarter poles for a stronger focusing it will have large chromaticities and if beta is higher at the location of the quarter poles then it will generate the higher chromaticity. Now we have seen that in the case of quarter pole the kick for the off momentum particle was K L x delta means kick proportional to x into and for a sextupole magnet this kick was M D L x delta. So you can say M D L of the sextupole is equivalent to K L of the quarter pole. So if K L of the quarter pole generates this chromaticity then M D L will generate a contribution using M D beta. So this is the formulation of the chromaticity when sextuples are also introduced to correct. And you can see that this is the natural chromaticity because it is the chromaticity generated only by the quarter pole. So this is the natural chromaticity and it is the chromaticity when we consider sextupole also and now properly choosing M we can cancel out this K beta. So by properly choosing the strength of the sextupole magnet we can make this integration 0 and chromaticity will become 0. In this fashion sextupole magnets can be used to correct the chromaticity. Now you can see several things here. Suppose we want to produce some M D beta combination for a given K beta. So if sextuples are placed at higher dispersion and higher beta function location then M will be small means a small value of M can generate the required K beta if D and beta is large. D and beta are the dispersion and betaton function at the location of the sextupole. So sextuples should be placed where the dispersion and beta function is large because if sextuples are going to be stronger we have seen that these magnets will generate a nonlinear effect and that nonlinear effect will be also stronger and nonlinear instability in the machine can be there and under the strong sextupole lattice optimization becomes a very typical job because it involves nonlinear dynamics. So we want that the smaller strength of this sextupole should do our work. So in this fashion sextupole should be placed at the location of higher dispersion and higher beta function. Now one question can arise here. If in vertical plane there is no dispersion then how the sextupole magnets can correct the vertical chromaticity? And the answer is that because you have Bx is equal to Sxy for the sextupole magnets and if we put X as X beta term plus D delta Y. So we will have a term D into Y delta. So horizontal dispersion can be used for correcting the vertical chromaticity also and this is only because sextupole magnets generate coupling between these two motions horizontal and vertical beta motion. So using that coupling the horizontal dispersion can also be used to correct the vertical chromaticity in a single. Now when we introduce the sextupole magnets the dynamics becomes nonlinear and we have to solve the nonlinear phase equation in that case. And in the presence of nonlinearity up to certain amplitude of the beta term oscillations oscillation remains stable. Outside that amplitude beta term oscillation becomes unescalable. So nonlinear dynamics or introduction of the sextupole actually separates the whole space into stable zone and unstable zone. And outside that stable zone beta term motion will become unescalable. And as the sextupole becomes stronger and stronger for correcting the larger and larger chromaticity this stable space shrinks down and running the machine becomes difficult. So an accelerator designer optimized the sextupole scheme for enlarging this stable beta term stable area for the beta term motion. This stable area is known as dynamic abuture in the accelerator channel. So introduction of the sextupole lowers down the dynamic abuture and an accelerator designer has to optimize the sextupole scheme and let us in such a way that we get the larger dynamic abuture. So if we introduce the sextupole for correcting the chromaticity whole Pandora box of the nonlinear dynamics opens and it becomes a little difficult problem to optimize the sextupole scheme when we are introducing very strong water cold for tight focusing. In next lecture we will cover completely new aspects of the dynamics that is the nonlinear dynamics.