Correction of resonances driven by random errors

1. Correction of resonances driven by random errors • Previously, we successfully demonstrated that we can correct nonlinear resonance for high-intensity operation where space charge effects are important. • All previous studies were done using lumped errors: • Independent correction of sextupole and octupole resonances • Correction of skew resonance • Dynamic correction of several resonances which allows to reach low losses at high intensity. In June’03 we began to study resonance correction with resonances driven by random errors.

2. Lumped errors budget used in studies In resonance correction studies we assumed lumped b2=30 units and (b3,a3) = 60 units which corresponds to • 5 times stronger random sextupole errors than measured • 10 times stronger normal & skew octupole errors than measured With the present error budget coming just from arc dipoles and arc quadrupoles the dominant contribution comes from normal sextupole and normal octupole resonances – good correction. If the budget of skew octupole errors becomes factor of 5 stronger than the present - one will need skew octupole correctors.

3. Correction of sextupole resonances (lumped errors):w.p. (6.36,6.22) - Correction of 3Qx=19 and 2Qy-Qx=6 N=0.6*10^14 blue – no errors red – errors, no correction green – errors, correction of 3Qx=19 % outside Qy Total emittance pi mm mrad 2Qy-Qx 3Qx N=2*10^14 blue – no errors grey – errors, no correction green – errors, correction of 3Qx=19 pink – errors, simultaneous correction of 3Qx=19 and 2Qy-Qx=6 resonances Qx

4. Correction of sextupole resonances (lumped errors):w.p. (6.4,6.3) - Correction of sum coupling resonance Qx+2Qy=19 and 3Qx=19 resonance % outside N=0.6*10^14 blue- no errors red – errors, no correction pink – errors, correction of Qx+2Qy=19 Total emittance pi mm mrad N=2.0*10^14 blue- no errors red- errors, no correction green- errors, simultaneous corrections of Qx+2Qy=19 and 3Qx=19 resonances N=3.0*10^14 pink – errors, corrections of Qx+2Qy=19 and 3Qx=19 resonances % outside

5. Correction of octupole resonance due to 10 unitsrandom b3 (w.p. 6.4, 6.3) We started by correcting octupole resonance 4Qy=25 driven by random b3 of 10 units rms distributed over all magnets. • MAD lattices were constructed to have the same notation of magnets in both DYNA and UAL codes. 2. Scripts to transfer errors from one code to another were written. 3. Benchmarking tests were done to ensure that implementation is correct

6. Initial study • Initial correction was not very good: At most reduction of factor of 2 in emittance growth was achieved. We then switched back to lumped errors and started detailed study of implementation of various multipoles in various elements (comparison of b3 in dipole vs. b3 in quadrupole, etc). No problems was found. It was then concluded that “not perfect” compensation is due the effect of other resonance bandwidth of which was big enough to play a role. We then switched to another w.p. to make sure that we have emittance growth only from a single resonance which we are correcting.

7. w.p. (6.36,6.22) 3rd order resonances excited by sextupole errors Random errors b2=5 units rms in all magnets (dipoles and quads). Total contribution is about factor of 5 bigger than expected based on measurements. Correction of 3Qx=19 resonance.

8. Correction of b2=5 units rms Red- no correction Green-correction

9. Remaining study 1. Perform Dynamic correction for all 3 major resonances for this working points: 3Qx=19, 2Qx+2Qy=25 & 4Qx=25 driven by random b2, b3 and a3. 2. Generate realistic loss models without correction based on available measured multipole: • As measured 2) factor of 2-5? Safety margin? 3. Produce loss models with corresponding dynamics correction of major resonances for both w.p.