Chromatic aberration in particle accelerators ) 1

Transcription

1 hromatic aberration in particle accelerators Inhomogeneous B p B p ( ), ( ). equation B p B p p / p K( B B, K(, B ( ) ( ) B D D K ( s ) K D ( ) O K K, K K K K(, ( K K K(, [ K( ] K K [ K( ] K, Note that the betatron motion for off momentum particle is perturbed b a chromatic term. The betatron tunes must avoid half-integer resonances. But, the quadrupole error is proportional to the designed quadrupole field. The are called sstematic chromatic aberration. It is an important topic in accelerator phsics.. Tune shift, or tune spread, due to chromatic aberration: 4 K, d / d K, d / d 4 The chromaticit induced b quadrupole field error is called natural chromaticit. For a simple FODO cell, we find FODO,nat sin L f 4 4 K N f ma ma f min We define the specific chromaticit as 4 / f tan( / ) / L ( sin( / )), sin min i i /, / The specific chromaticit is about for FODO cells, and can be as high as -4 for high luminosit colliders and high brightness electron storage rings. L ( sin( / )) sin hromaticit measurement: The chromaticit can be measured b measuring the betatron tunes vs the rf frequenc f, i.e. / The Natural chromaticit can be obtained b measuring the tune variation vs the bending-magnet current at a constant rf frequenc. hange of the bending-magnet current is equivalent to the change of the beam energ. Since the orbit is not changed, the effect of the setupole magnets on the beam motion can be neglected. The Figure shows the horiontal and vertical tune vs the bending-magnet current in the PLS storage ring. / / / The chromaticites are =+.9, =+.4. M. Yoon and T. Lee, RSI 68, 65 (997) The data give = 8.96, = 3.4; vs theor: = 3.36, = 6.9. M. Yoon and T. Lee, RSI 68, 65 (997)

2 Eamples: BNL AGS (E. Blesser 987): hromaticities measured at the AGS. Y. Papaphilippou, G. haot, J.M. Koch, E. Plouvie, J.L. Revol, A. Ropert, PA3, 85 (3) HROMATIITY MEASUREMENTS IN THE ESRF BOOSTER FODO,nat tan( / ) / Fermilab Booster (X. Huang, Ph.D. thesis 5): The measured horiontal chromaticit when SEXTS is on (triangle or off (star, and the measured vertical chromaticit when SEXTS is on (dash, circle or off (square. The error bar is estimated to be.5. The natural chromaticities are nat, = 7. and nat, = 9. for the entire ccle. The total chromaticit is composed of contributions from the low -qua and the rest of accelerators that is made of FODO cells. The decomposition to fit the data is Δs35 m in RHI ontribution of low triplets in an IR to the natural chromaticit is (eercise.5.) IR = 3 ontribution outside IRs ontribution from 6-IRs *(m)

3 ν vs Δp/p β and D vs Δp/p. hromaticit correction: The chromaticit can cause tune spread to a beam with momentum spread ν=δ. For a beam with =-, δ=.5, ν=.5. The beam is not stable for most of the machine operation. Furthermore, there eists collective (head-tail) instabilities that requires positive chromaticit for stabilit! To correct chromaticit, we need to find magnetic field that provide stronger focusing for off-(higher)-momentum particles. We first tr setupole with B jb B b j, A B b j s 3 Re B B K, K D B B B B B b ( ) B b ( D D ) B b B b D B b Let K =-B b /Bρ= B /Bρ, we obtain: KD ), ( K K D ) ( K 3 To model the AGS, we assume that the setupole fiel arise from sstematic error at the en of each dipole, the edd current setupole due to the vacuum chamber wall, and the iron saturation setupole at high field. hromaticit measured at the AGS In order to minimie their strength, the chromatic setupoles should be located near quadrupoles, where β D and β D are maimum. A large ratio of β /β for the focusing setupole and a large ratio of β /β for the defocussing setupole are needed for optimal independent chromaticit control. The families of setupoles should be arranged to minimie the sstematic halfinteger stopban and the third-order betatron resonance strengths. The sstematic error is independent of the beam momentum; the edd current setupole field depen inversel on the beam momentum; and the saturation setupole field depen on a higher power of the beam momentum. The solid lines represent theoretical calculations with the integrated bod and end setupole strengths

4 With setupoles, the chromaticities becomes S F K SF 4 4 [ K K [ K K sin( / ), S f ( sin( / )) D D( ] D( ] For FODO cells, the integrated setupole strength is K D SD sin( / ) f ( sin( / )) hromatic Aberration and orrection Defining the betatron amplitude difference functions A and B as E.3. For high energ colliders and high brightness snchrotron light sources, the setupole strength can be much higher. Even more important is the effect of the sstematic half-integer stopban. J jp, p ) e K ( s The change of A across a quadrupole is ( ) ( )sin ( ) ( ) sin d k s d d 4 ( k( k ( s jp ) J e p, J p p k( p e jp J p ( p / ) e jp Half integer stopband What smmetr can do to stopban?

5 Sstematic chromatic half-integer stopband width The effect of sstematic chromatic gradient error on betatron amplitude modulation can be analed b using the chromatic half-integer stopband integrals We consider a lattice made of P superperio, where L is the length of a superperiod with K(s + L) = K(, β(s + L) = β(. Let = PL be the circumference of the accelerator. The integral becomes hromatic stopband integrals of FODO cells The chromatic stopband integral of the arc, which is composed of N FODO cells, in thin-lens approimation is The chromatic stopband integral of insertions

6 Effect of the chromatic stopban on chromaticit Effect of setupoles on the chromatic stopband integrals The stopband integral is ero or small if NΦ/π = integer, i.e. the chromatic setupole does not contribute significantl to the chromatic stopband integral if the transfer matri of the arc is I or I. Lattice Design Strateg Based on our stud of linear betatron motion, the lattice design of accelerator can be summaried as follows. The lattice is generall classified into three categories: low energ booster, collider lattice, and low-emittance lattice storage rings. The betatron tunes should be chosen to avoid sstematic integer and halfinteger stopban and sstematic low-order nonlinear resonances; otherwise, the stopband width should be corrected. The betatron amplitude function and the betatron phase advance between the kicker and the septum should be optimied to minimie the kicker angle and maimie the injection or etraction efficienc. Local orbit bumps can be used to alleviate the demand for a large kicker angle. Furthermore, the injection line and the snchrotron optics should be properl matched or mismatched to optimie the emittance control. To improve the slow etraction efficienc, the β value at the (wire) septum location should be optimied. The β and β values at the injection area, particularl in the strip injection scheme, should be adjusted to minimie emittance blow-up due to multiple oulomb scattering. The local vacuum pressure at the high-β value locations should be minimied to minimie the effect of beam gas scattering.

7 The chromatic setupoles should be located at high dispersion function locations. The focusing and defocusing setupole families should be located in regions where β β, and β β respectivel in order to gain independent control of the chromaticities. It is advisable to avoid the transition energ for low to medium energ snchrotrons in order to minimie the beam dnamics problems during acceleration. Eperience with low energ snchrotrons indicates that the Laslett space-charge tune shift should be limited to about.3. This criterion usuall determines beam emittance and intensit. Besides these design issues, problems regarding the dnamical aperture, nonlinear betatron detuning, collective beam instabilities, rf sstem, vacuum requirement, beam lifetime, etc., should be addressed.