B Jeanneret ABP BBmeeting 28 th June 2013 Goal of the study Evaluate the impact of the betatronic divergence of the strong beam in the presence of a crossing angle and with considering the longitudinal distribution of the bunches question raised by ID: 1030693
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1. Beam divergence near IP and beam-beam effect B. JeanneretABP BB-meeting28th June 2013
2. Goal of the studyEvaluate the impact of the betatronic divergence of the ‘strong’ beam in the presence of a crossing angle and with considering the longitudinal distribution of the bunches (question raised by Stephane in view of HL-LHC)The existing 6D lens with crossing-angle disregards the divergenceAn estimator of the importance of the effect is presentedBJ, BB-meeting 28.06.2013
3. ParametersNominalHigh LumBunch population2 × 10113 × 1011 βX*0.5 m0.2 m βy*0.5 m0.05 m σz0.075 m0.075 mΦ crossing160 μrad720 μrad εn3.75 μm2.5 μm σ*16 μm32 μrad σ’ x*8.2 μm41 μmBJ, BB-meeting 28.06.2013
4. 2D - differential beam-beam kick, gaussian beamsBasseti-ErskineFormula valid y>0, σx > σy , and stable for Small r : w(z) is the modified complex error functionBJ, BB-meeting 28.06.2013
5. Method usedIn the crossing plane, the ‘Strong’ beam is split in 3 beamlets ofNull average divergenceNegative average divergencePositive average divergenceEach beamlet is parametrised by its average and r.m.s. value (we can compute a kick properly only for 2D-gaussian beams (Bassetti-Erskine)The three corresponding bb-kicks are addedA tracking is made along the longitudinal coordinate BJ, BB-meeting 28.06.2013
6. Beam divergence considered or notThe kick is ⊥to the strong beamlet axisW/o div, Q 0, ± (+) are aligned with PWith div (×) they are not :The distances to P are changedQ ± move longitudinally, kick intensity is different‘weak’ beam‘strong beam’Split in3 beamletsIPPQ+Q0Q -BJ, BB-meeting 28.06.2013
7. Phase space near the IP, High LumaaaBJ, BB-meeting 28.06.2013
8. Building distributions with divergenceBuild a 2D-gauss distribution with x/x’ correlationWith the 4 sub-samples : a,b,c,dBuild e : with d and x’-x’Build f : with b and x-x’Sub-sample 1 with <x>=<x’>=0 : central area : b+e+f+dSub-sample 2 with <x> > 0 & <x’> > 0 : a-eSub-sample 3 with <x> < 0 & <x’> < 0 : c-fGet <x>,<x’>,σ(x), σ(x’) for 1,2,3 as a function of zBJ, BB-meeting 28.06.2013
9. Monte-Carlo filling of x-x’ ‘normalized’ phase-spaceAs a function of z.Fraction for sub-sample 1 : 1-f+sub-sample 2 and 3 :f+ /2At 2.5σzRelative averages and r.m.s. , w.r.t to σ(z) BJ, BB-meeting 28.06.2013
10. Central beamletAbscissa : x/σxBJ, BB-meeting 28.06.2013
11. Beamlet of positive divergenceAbscissa : x/σxBJ, BB-meeting 28.06.2013
12. Method used - IIThe decomposition in 3 gaussian beamlets is not perfectTo compare adequately the two cases (div / no div)The same beamlet decomposition is used for both casesFor the no-div case, <x±’> = <x0’> = 0BJ, BB-meeting 28.06.2013
13. TrackingAt P, with Q0, ±P, get‘weak’ beamIPPQ+Q0Q -Update x’, then x for step dsIterate …This over -4σs 4σsAnd for A= [0 .. 6]×σx and φ = [0 .. 2π]Do everything twiceWith divergenceWithout divergenceBJ, BB-meeting 28.06.2013
14. ResultsStart with (x0,x0’) at IPDrift back to -4σzTrack to +4σzDrift back to IP : (x1,x1’) Compute raw δQ as angle between (x0,x0’) and (x1,x1’) Get δ2Q = δQdiv – δQno-divBJ, BB-meeting 28.06.2013
15. s = 0 (δp = 0) dQ_Thin : nom 6.5 o/oo, HL 19.5 o/ooBJ, BB-meeting 28.06.2013
16. s = 1σs dQ_Thin : HL 19.5 o/ooBJ, BB-meeting 28.06.2013
17. s = 1,2 σs dQ_Thin : HL 19.5 o/oos = 1σs s = 2σs BJ, BB-meeting 28.06.2013
18. Divergence and vertical planeNominal : α = 3×10-5 α2/8 = 0.13×10-9High-Lum : α = 8×10-5 α2/8 = 0.80×10-9ααXZYIndependent of αDiverging fraction, averaged over z : fdiv ≅ 0.5 BJ, BB-meeting 28.06.2013
19. Results about divergenceConsidering the tune variationsIn the crossing plane :The difference between the two cases, divergence considered or not considered is δ2qrel < 2×10-9 , both with NOMINAL and HL.This difference similar when the average longitudinal position of the test particle w.r.t. to the strong bunch is changed (02σs).In the other plane :The effect is 10× smaller with NOMINAL ,i.e. δ2qrel ≅ 0.13 ×10-9The effect is 2× smaller with HL ,i.e. δ2qrel ≅ 0.8 ×10-9BJ, BB-meeting 28.06.2013
20. Small amplitude distortions,(independent of divergence effect)BJ, BB-meeting 28.06.2013
21. The apparent dQ excursion at small amplitude and phase space angle ±π/2The raw dQ grows without limit at towards small amplitude (x=0, x’ ≠ 0), particularly marked with HLWhat happens ?BJ, BB-meeting 28.06.2013
22. High_LUMWith bunch length considered and crossing angleA δX appear with tacking, with the same sign whatever the phase angleSo, this is an orbit effectAt High-LUM : δX= -0.016 σ*The same applies to the other beam collision mismatch of 3%σ* Problematic ?BJ, BB-meeting 28.06.2013
23. Beam displacement High-LUMVary A, Φ=π/2Slight variationswith amplitude and z-displacement (δp)BJ, BB-meeting 28.06.2013
24. Beam displacement High-LUM - IIVary ΦSlight variationswith Φ (1% of σ) with z-displacement (δp)●(2 ± 0.5) % σ* at A=1BJ, BB-meeting 28.06.2013
25. NominalMuch smaller effect, of 2 o/oo σ* BJ, BB-meeting 28.06.2013
26. SummaryThe effect of the betatronic divergence can be safely neglected (δ2Q/δQ < 2 × 10-9 ) with both nominal and high luminosity collision parametersSmall orbit effect (partly amplitude & phase dependent ) visible with ‘thick lens’ beam-beam trackingBJ, BB-meeting 28.06.2013