Accelerator Physics Closed Orbits and Chromaticity G. A. Krafft Old Dominion University Jefferson Lab Lecture 4
Kick at every turn. Solve a toy model: Dipole Error B d kb d k B x s s s il B ds Q ds i x s exp k s s il / Q sin k s s il ih B B i k k Q q k s s Q B B / 4 exp B / Geometric series summed x s exp k s s x ih ih s B /Q cos kb s s kbl / k sin k L / integer resonance blows up when k L B n B B sin k s s q cos k L cos k s s q sin k L B B B B q cos k L q B s
s Closed Orbit Distortion Perform summation over all kick sources x co cos k s s k L / k i B i B sin k L / i B B bound oscillation generated by error Source (dipole powering, quad displacement, etc.) Oscillation can be observed and Using the real betatron motion corrected M s s sin s s ss, the proper result is cos x s z co i i i s s i sin
Beta Measurement If BPM close to steerer (there is little phase advance between them), and the tune has been measured, induce a closed orbit distortion to measure the x co s bpm i i cot
Dipole Error Distribution u s s s s s s co i i i j i j i j u s s s si sin co i i 8sin i u 8sin s N cos cos s sj sin ds ds Angular stuff averages assuming independence of error distributions / For quad displacements replace x / f
Closed Orbit Correction Suppose orbit does not go through center of all BPMs. What do you do? (At CEBAF just steer to BPM centers!) Trim magnets added whose purpose is to bring CO as close to zero as possible. cos u s s co i i i At BPM j closed orbit reads s sin cos u u s s j co j j i i i i sj si sj si sin Measure response matrix as trim magnets (index k) varied cos u j s j k k R jkk sin i s
Desire i k kj j BPM j j Correction Algorithm. If have enough trims simply update More sophisticated when less trims than BPMs, minimize N u u R u BPM,, N x u R trims analogous to "least squares fitting" and generally uses the same types of computer algorithms, including Singular Value Decomposition (SVD). How many BPMs/trims? Fourier Analyzing closed orbit equation Fe s il l 3/ il uco s s F l s se l l Need enough to resolve the betatron orbit and distribute uniformly in betatron phase s ds
Quadrupole Field Errors Error at location s u ; total strength / f Kdz focusing M j ss s cos sin sin / f sin cos s sin / f s cos s sin sin mess mess s s s f cos cos sin Add to get total 4 f 4 f s f s f cos s sin sin K dz
dw More Generally s s x K s x x K s x Kx Introduce normal betatron coordinates w s d x s ds,, w Kw Fourier expand rhs d w F w w d F Kd Kds d w d F Kds 4 as above Note : Method to measu re
Orbit Perturbation Use Lagrange method of variation of parameters. If have P z K z P z p z A solution to the inhomogeneous equation is z, P z p z G z z dz, G z z P z P z P z P z where P and P solve the homogeneous equation with Wronskian dw for normalized equation z P z P z p z P z dz P z p z P z dz d sin w Kw, P sin, P cos P K d z
Specific Case define location and suppose unperturbed orbit displaced there with displacement a w a cos a K cos sin d perturbed Also must equal w perturbed acos Evaluate the total tune shift by going around turn K cos cos cos sin d K sin cos sin d K cos sin cos cos sin d 4 4 z K z dz sin now must avoid / integers! sin z K z z dz
Stop Bands If error too large cannot solve for. Indicates breakdown of approximation and next level needed w a cos a K cos sin d perturbed is inserted in the equation for the perturbation cos cos K cos sin d cos sin sin I F sin K sin d Second term oscillatory and tends to average to zero K K d d I F F for for n / n
I k k 6 d d d d i i i i e e e e dd Integer Resonance I, n k k 6 i i in in e e e e dd / Integer Resonance I, n/ k k 6 i i in / in / e e e e dd
* ij ij Fj FjFj K e d K e d I, n F n 4 F 8 I F 4 F 8, n n F F n F cos 4 8 F F n 4 F 8 F / F / 4 n F n z K ze similarly in z z dz in F n z K z e dz
Defined by p/ p cos L / Chromaticity f Thin lens FODO system L L / f / f / f L / f L L / f L L / f / f L / f for one period is Suppose particle has a momentum error p/ / / f p p f p p f p/ p p
Tune Shift L cos / p p f cos sin sin cos f p cos p sin tan sin p p/ p tan / This is per period. Total ring chromaticity "proportional" to the number of periods. L p p p
More Sophisticated m x kx kx p p x y y ky ky p / p mxy / x x D p / p y y x expand to second order m xkx kx p p mx D p p x y y ky ky p p md p p y mx y / x / / / x Final terms give geometric aberations; ignor for now tune change k x x k mdx dz y y k mdx dz
General Formula for Chromaticity x x k mdx dz 4 y y k mdx dz 4 use sextupoles to zero out for no sextupoles x xkdz 4 y ykdz 4 works for thin lens L f / L f / L f / L f / L L f / L f / L kdz f f / L L
x x 4 kdz k dz k dz kdz 4 4 f / L tan / / L f
Chromaticity Correction x x xmdxdz 4 y y ymdxdz 4 Thin sextupoles x x md x x mdx x lsext 4 y y md x y mdx y lsext 4 sextupole strength ml ml s s 4 x y x x x x y x y 4 x x y y x x y x y