Thu June 16 Lecture Notes: Lattice Exercises I

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1 Thu June 6 ecture Notes: attice Exercises I T. Satogata: June USPAS Accelerator Physics Most o these notes ollow the treatment in the class text, Conte and MacKay, Chapter 6 on attice Exercises. The portions on orbit correction ollow Edwards and Syphers, section 3.4. By the end o these lectures, you should understand a bit about orbit correction in a circular accelerator, how to calculate optics parameters or a FODO lattice and other combinations o drits, dipoles, and quads, FODO lattice stability, and FODO lattice dispersion. Review Rcall that the parameterization o the one-dimensional periodic transer matrix is given by Conte/MacKay 5.-: where M periodic I cos µ + J sin µ e µj. αs βs J γs αs J I. αs β s/ and γs + αs /βs. Here I ve included the explicit s-dependence. βs is the square o the envelope unction ws, or amplitude scaling o particle motion, Conte/Mackay 5.69: xs Wβs cos[ψs + ψ ].3 where ψ s /βs ψs ds/βs βs βs + C.4 The beta unction has the periodicity o the lattice, as Waldo showed, but the phase advance ψs does NOT have this periodicity integrating it around the accelerator gives us an extra phase advance o µ πq where Q is called the betatron tune. See Conte/Mackay Eqns There are beta unctions, phase advances, and betatron tunes or each plane H, V. Note that some reerences use ν or tunes instead o Q and that since the transer matrix around one turn o the accelerator is periodic, we have M one turn I cos πq + J sin πq e πqj.5 In the weak ocusing betatron case where the ield index is n, Q x n and Q y n, the tunes must also be in the range Q x,y. Indeed, this is practically what is deined as weak ocusing. In contrast, accelerators with alternating quadrupole ocusing that lead to Q > are termed strong ocusing. To propagate position and angle rom one location s s to another s location, both

2 with known Twiss parameters, we use MacKay and Conte 5.5: βs β Ms s [cos µs + α sin µs] β βs sin µs [αs α ] cos µs+[+α αs] sin µs β.6 [cos µs αs sin µs] β βs βs βs cos µ sin µ β βs α sin µ cos µ β.7 β αs βs T s RµT.8 Even though.6 looks horrible, it can be decomposed into local transormations T s into normalized phase space, and a separate pure rotation matrix Rµ that rotates through the phase advance. You ll see this in problem 5-4 o tonight s homework. Figure shows particle motion through a FODO lattice, a series o alternating ocusing and deocusing quadrupoles. The dark lines that are plotted are ± βs, which is directly proportional to the maximum amplitude o the particle oscillation. You can see why the beta unction is also called the envelope unction! Waldo s showed something like this and you ve also seen it in the labs. et s analyze the FODO lattice in more detail and calculate some things about it ater a brie digression into a real application o what you ve learned so ar. Figure : This igure shows a plot o betatron oscillations, equation.3 and Conte/MacKay Fig. 5.3, through a regular FODO lattice. Waldo showed this picture earlier. I you ollow the trajectory, you can see that the kicks alternate between ocusing and deocusing, and the betatron oscillation is slow compared to the periodicity o the FODO lattice. This example has about 6.5 FODO cells per betatron oscillation, so this FODO lattice has a phase advance per cell o about π/ degrees. What is the relationship between the ocal length and the FODO cell length here? Steering Errors and Orbit Correction Suppose we have a single steering error in an otherwise perect circular accelerator that gives a kick x B l. Bρ loated without loss o generality at s. What is the new closed -turn periodic orbit? Say that the orbit immediately downstream o this kick is x, x. I we propagate this orbit through one turn with a one-turn matrix M one turn and then proceed through the kick, we should get the same conditions. This the deinition o closed orbit, the ixed point o the one-turn map. So M one turn x x + x x x..

3 Solving or x, x gives x x I M one turn x..3 We can now use M one turn e πqj to perorm a little trick: I M one turn I e πqj [ e πqj e πqj e πqj] The closed orbit solution is then x I M one turn x x sin πq x.4 J sin πq e πqj.5 sin πq Je πqj.6 J cos πq + I sin πq sin πq.7 β cos πq sin πq α cos πq.8 where the twiss parameters β, α are at the position o the kick. We can then propagate this to any location s in the ring with the general Ms s matrix.6 to ind xs x β / sβ / sin πq cos[ψs ψ πq].9 xs x x s. This result really represents a standing wave solution through the accelerator. Note some interesting points about.9: The sensitivity to the kick is proportional to the square root o the product o the beta unctions at the kick location and position observation. Dipole errors have the worst eect at areas o high β. The orbit change is inversely proportional to sin πq. For integer Q, dipole error eects diverge. This is an example o a resonance. We will later see that quadrupole error eects are inversely proportional to sin πq and so on to higher orders. Even though the kick is only an angle kick, there is a closed orbit change x at the kick location. An example o a comparison o measurements and a calculation rom.9 or a single dipole corrector change in RHIC is shown in Fig...9 can be used as a undamental equation o orbit correction since it relates a position change anywhere in the ring xs to a change in a dipole corrector change x elsewhere in the ring. Because it is linear we can then write x x x n x x x x x n x x x x x x x m x x m x n x n x n x m x x x m. This non-square! matrix can be inverted optimally using, or example, singular value decomposition to ind a solution to the changes in all m dipole correctors x, x,, x m to eect an orbit change in all n beam position monitor locations x, x,, x n. 3

4 Figure : attice diagram and vertical dierence orbit rom RHIC, comparing measurement and modeled dierence orbit rom.9. Figure 3: Two ways o calculating transer or a regular FODO cell o length with two thin quadrupoles. The one on the let is not symmetric and gives optics at one side o the ocusing quad. The one on the right is symmetric and gives optics in the center o the ocusing quadrupole, where we expect to locate β max even in the case o thick quadrupoles. 3 FODO attice One o the most common lattice layouts in modern separated-unction accelerators is the FODO lattice, a lattice with alternating ocusing and deocusing quadrupoles separated by drit spaces or dipoles. We have started to see how this combination gives you a net ocusing. Using the thin lens approximation or equal-strength quadrupoles i.e. assuming their ocal lengths are much larger than their lengths, separating them by / length drit spaces so the FODO cell length is, we ind: / M FODO / et s assume that we have a string o FODO cells. The system is periodic through every FODO cell, so the transer matrix o each FODO cell must be expressible in the orm o.: cos µ + α sin µ β sin µ γ sin µ cos µ α sin µ 3.3 4

5 I these are equal, their traces must be equal: 4 cos µ 4 sin µ 3.4 sin µ ± using the trig identity cosµ cos µ sin µ sin µ. 3.5 relates the FODO cell phase advance µ to the parameters and that we used to construct it. For a FODO lattice, the maximum o the beta unction β β max is located at the center o the ocusing quadrupole. The transer matrix rom the center o one ocusing quadrupole with ocal length to another in this lattice is ound by splitting the ocusing quadrupole in hal ocal length as shown in Fig. 3. This even works i we use the thick lens quadrupole transer matrix. M to / 4 / / / / At the maximum o the beta unction in the split ocusing quadrupole, αs β s/, so this matrix can also be written as cos µ β M to max sin µ 3.7 sin µ/β max cos µ Equating irst diagonal, then upper-right o-diagonal terms gives cos µ cos µ /8 sinµ/ 4 + sinµ/ β max sin µ 3.8 The minimum o the beta unction, β min, occurs at the center o the deocusing quadrupole, so it may be ound by ollowing the above analysis replacing with, producing: sinµ/ β min 3.9 sin µ For example, the phase advance per FODO cell at RHIC is about 78 degrees and the length o the RHIC FODO cell is about 3 m, so we can calculate that β max 5 m and β min.7 m. Figure 4 shows one sixth o the RHIC injection optics lattice, and you can readily see that the ormulas or β min,max work very well. Note that we can calculate the Twiss parameters at any point in the FODO cell by changing the cut point or our lattice transport. Alternatively you can use the result o tonight s homework problem 5-5 to propagate these Twiss unctions through the FODO cell. Propagating through an entire FODO cell should give an identity matrix regardless o the cut point because the optics are periodic. We can test this or our example above using M to rom 3.6. This is thankully NOT part o your homework. But it works: β α γ M to M M M M M M M M + M M M M M M M M /8 + /4 /6 3 /4 /8 β α γ

6 Figure 4: The RHIC injection lattice and beta unctions. This shows one sixth o the RHIC lattice as Waldo has shown earlier; apply mirror antisymmetry and repeat the lattice three times to get the ull lattice. Solid lines are β x s; dotted lines are β y s. The magnet lattice is shown at the top o the graph; short blocks are bending dipoles; taller blocks are quadrupoles. ow-beta insertions are clearly visible. Is the quadrupole at s3m horizontally ocusing or deocusing? How many FODO cells are in the regular section? RHIC has a total horizontal tune o about 8.3; does this number o FODO cells make sense? 4 FODO attice Stability Sometimes we do not want horizontal and vertical quadrupoles in a FODO lattice to have the same strengths, such as when we are matching the beam size into dierent optics. I we re-parameterize the quadrupole strengths as F F D D 4. Calculating the trace o the FODO lattice matrix now gives something more general than 3.5: cos µ + D F F D sin µ F D 4 + F D 4. Recall rom Waldo s lecture that the stability requirement is that the phase advance µ is real since the eigenvalues o the transport matrix are e ±iµ, or < cos µ < or < sin µ < 4.3 What are the boundaries o real µ in terms o F and D? One is where sin µ, and we have F D D The second case, sin µ, requires that F < or stability or positive D. The reverse conditions where we reverse the roles o F and D also provide boundaries, and the inal stable area in the F, D parameter space is shown in Fig FODO Cell Dispersion A FODO cell is a linear system in more than one way the design orbit is straight! We want to steer the beam and eventually wrap the beamline around back on itsel so we can 6

7 Figure 5: Stability or necktie diagram or an alternate ocusing lattice, including the FODO lattice where the dimensionless parameterization is F F and D D. The shaded area is the region o stability. Figure 6: The π lattice insertion. make a storage ring, so we also must consider cases where the space between the FODO quadrupoles is not a simple drit, but illed with the bending ield o a dipole. To make the math simpler and to save my sanity, we ll assume that the length o the FODO cell is ρθ c and there are no drits. The total bending angle o the cell is θ c. We will use the symmetric layout o the FODO cell in Fig. to make some math simpler or dispersion suppressors, which we ll discuss later. The transer matrix or the single FODO cell with the dispersive component included is M to θ c 8 θ c θ c 4 4 θ 8 3 c 8 θ c 8 θ c where we have kept only irst order in θ c since usually θ c <<. Waldo talked a bit about the dispersion unction ηs earlier. The periodic dispersion unction o the FODO cell can be obtained rom η η M M M 3 M M M 3 η η 5.3 M 3 M 3 M 33 The bottom row o M in 5. is trivial, so we can also write the dispersion as a nonhomogeneous equation: η η η M M3 η M 3 η η I M M3 5.5 M 3 7

8 where M is the upper let x block o 5.. This is calculable and gives the η max, just like we had β max : η max θ c + µ 4 sin µ 5.6 η 5.7 This is rather unsurprisingly a lot like the ormula or β max, except now the maximum dispersion also scales linearly with the total length o the FODO cell, θ c. The minimum o the dispersion is also easy to ind: η min θ c µ 4 sin µ 5.8 η A Simple attice Insertion In any FODO cell one or both quadrupole magnets can be moved away leaving room or actual straight sections, slightly modiying the lattice dispersion without altering the optics o the lattice, as thoroughly discussed in the previous sections. Obviously, such ree spaces may not be suicient to host long items like septum magnets or injection and extraction, chains o r cavities, etc. onger straight sections have to be provided, leaving the optics o the machine unaected. How do we make the longest straight section possible? An elegant solution is the so-called π insertion. This insertion has a long straight section o length, encompassed by two quadrupoles, one ocusing and one deocusing, and urther symmetric straight sections o length. This is schematically shown in igure 6. Assuming the quadrupoles have equal ocal length, we calculate the transer matrix again: + M + 6. We want this insertion to match into the periodicity o the lattice, so it must match to our general periodic lattice description o.: cos µ + α sin µ β sin µ M I cos µ + J sin µ 6. γ sin µ cos µ α sin µ Taking the dierence o the diagonal terms gives α sin µ 6.3 and is maximized when µ π. It s now obvious why this is called a π insertion. Then cos µ and we can get other parameters in terms o the lengths and : α γ β and we can design our insertion to match into our FODO lattice. One surprising observation is that the transer matrix now reduces to M J 6.5 and recall that M I! So we can now design an interesting insertion by putting two o these insertions back to back. This is a simple case o a low-β insertion. 8