Resistive-wall impedance of interaction regions (warm beam pipe) 12 Bernard Riemann Zentrum für Synchrotronstrahlung 2017-10-10 EuroCirCol meeting 1 thanks to S. Arsenyev, A. Langner, O. Boine-Frankenheim, D. Schulte and S. Khan 2 supported by German Federal Ministry of Education and Research, funding code 05P15PERB1 1 / 1
Transverse impedance model Preface: Tune shif a a thanks to O. Boine-Frankenheim Transverse tune shif, a proportionality ∆ ν y = Ω − ω β ∝ β y I ˜ Z ⊥ / E . ω 0 Normalize such that β y ( s ) = 1 case is equivalent to impedance Z ⊥ of harmonic oscillator with frequency ∫ 1 1 2 π Q = β ( s ) d s = : . β smooth a A. Chao, ”Physics of Collective Beam Instabilities in High Energy Accelerators“, chapter 4 2 / 1
Transverse impedance model Use single-kick model 3 for elements n of a latice: 1 � β ⊥ ( s n ) ˜ Z ⊥ = Z ⊥ , n β smooth ⊥ n round pipe, radius b � 2 ρ n 1 + i Z ⊥ , n = L n Z 0 δ skin with δ skin ˜ = n n 2 π b 3 µ 0 µ r ω n elliptical pipe with semiaxes w , b : Use form factors G 1 ⊥ ( w , b ) 45 1 + i ˜ Z ⊥ , n = L n G 1 ⊥ ( w n , b n ) Z 0 δ skin 2 π b 3 n 3 N. Mounet, PhD thesis, EPFL Lausanne (2012) 4 R.L. Gluckstern, J. van Zeijts and B. Zoter, Phys. Rev. E 47 (1992) 5 K. Yokoya, Part. Acc. 41 (1993), p. 18 – 19 3 / 1
Transverse impedance model √ ρ n Z 0 1 + i � Z ⊥ = √ 2 µ 0 µ r ω L n β ⊥ ( s n ) G 1 ⊥ ( w n , b n ) b 3 πβ smooth n ⊥ n assume G 1 ⊥ , ρ, b as piece-wise constant, but β as continous: √ ρ n s n Z 0 1 + i ∫ � Z ⊥ = √ 2 µ 0 µ r ω G 1 ⊥ ( w n , b n ) β ⊥ ( s ) d s πβ smooth b 3 n ⊥ n s n − 1 β, α = − β ′ / 2 known at all element endpoints s n 4 / 1
Transverse impedance model Qadrature rule Assume β is piece-wise cubic function of s : L ∫ β ( s ) d s ≈ L β ( L ) + β ( 0 ) + L 2 α ( L ) − α ( 0 ) . 2 6 0 Approximation is exact for drif spaces (quadratic dependence). Result Z ⊥ = ζ 1 + i , with � f √ ρ n s n Z 0 ∫ � ζ = G 1 ⊥ ( w n , b n ) β ⊥ ( s ) d s . √ πµ 0 µ r b 3 2 πβ smooth n ⊥ n s n − 1 5 / 1
Longitudinal impedance model Different dependence on b and f , 6 assumption of piece-wise constant values is valid for all factors: Result � Z ⊥ = α ( 1 + i ) f , with √ ρ n G 0 ( w n , b n ) � α = Z 0 π � L n . 2 π c µ 0 µ r b n n 6 A. Chao, Physics of Collective Instabilities in Particle Accelerators (Wiley, 2003) 6 / 1
Computation results Used aperture and optics data from 8 IRs as input. 7 Collision optics with β ∗ = 0 . 3 m at 50 TeV beam energy. Used resistivities of copper 8 at 50 K for magnets (elements QUADRUPOLE, RBEND, SBEND, HKICKER, VKICKER ) ρ ( 50 K ) = 0 . 518 n Ω m respectively 293 K for drif spaces ρ ( 293 K ) = 16 . 78 n Ω m . other latice elements ignored (no treatment of collimators etc.) 7 thanks to S. Arsenyev and A. Langner 8 R.A. Matula, “Electrical Resistivity of Copper, Gold, Palladium, and Silver”, Table 2, J. Phys. Chem. Ref. Data 8 (4) (1979) 7 / 1
Total impedance regions A, B, D, F, G, H, J, L 10 4 10 10 10 10 10 3 10 9 10 9 10 2 y impedance / Ω x impedance / Ω 10 8 10 8 l impedance / Ω 10 1 10 7 10 7 10 0 10 6 10 6 10 − 1 Re Z x (cold beamscreen, factor 1.24) Re Z y (cold beamscreen, factor 2.28) Re Z l (cold beamscreen, factor 6.36) 10 5 10 5 Im Z y (cold beamscreen, factor 2.28) Im Z x (cold beamscreen, factor 1.24) Im Z l (cold beamscreen, factor 6.36) Re,Im Z x warm pipe ( ζ /u = 18.9) Re,Im Z y warm pipe ( ζ /u = 19.0) 10 − 2 Re,Im Z l warm pipe ( α /u = 11.5) 10 2 10 4 10 6 10 8 10 10 10 2 10 4 10 6 10 8 10 10 10 2 10 4 10 6 10 8 10 10 f / Hz f / Hz f / Hz Transverse plane: strong contribution relative to cold beamscreen reference data. 9 9 S. Arsenyev, FCC impedance online database, https://impedance.web.cern.ch/impedance/fcchh 8 / 1
Major transverse contributions from IRA / IRG ζ / ( M Ω √ IRA MHz / m ) α / ( Ω /√ MHz ) elements x plane y plane 7 . 190 7 . 118 1 . 007 drifs 0 . 168 0 . 236 0 . 064 quads 0 . 379 0 . 372 0 . 024 dipoles 0 . 082 0 . 080 0 . 010 kickers 7 . 818 7 . 806 1 . 106 total region IRA, input parameters region IRA, impedance coefficients 8 80 x x MHz / m ) y y 6 60 √ β / km c.s. of ζ / ( M Ω 40 4 2 20 0 0 10 50 K 1 . 0 293 K MHz ) 5 0 . 8 √ b , w / cm c.s. of α / ( Ω / 0 . 6 0 0 . 4 − 5 0 . 2 − 10 0 . 0 96 . 2 96 . 4 96 . 6 96 . 8 97 . 0 97 . 2 97 . 4 97 . 6 97 . 8 96 . 2 96 . 4 96 . 6 96 . 8 97 . 0 97 . 2 97 . 4 97 . 6 97 . 8 s / km s / km 9 / 1
Major transverse contributions from IRA / IRG ζ / ( M Ω √ IRG MHz / m ) α / ( Ω /√ MHz ) elements x plane y plane 7 . 190 7 . 118 1 . 007 drifs 0 . 168 0 . 236 0 . 064 quads 0 . 379 0 . 372 0 . 024 dipoles 0 . 082 0 . 080 0 . 010 kickers 7 . 818 7 . 806 1 . 106 total region IRG, input parameters region IRG, impedance coefficients 8 80 x x MHz / m ) y y 6 60 √ β / km c.s. of ζ / ( M Ω 40 4 2 20 0 0 10 50 K 1 . 0 293 K MHz ) 5 0 . 8 √ b , w / cm c.s. of α / ( Ω / 0 . 6 0 0 . 4 − 5 0 . 2 − 10 0 . 0 47 . 4 47 . 6 47 . 8 48 . 0 48 . 2 48 . 4 48 . 6 48 . 8 47 . 4 47 . 6 47 . 8 48 . 0 48 . 2 48 . 4 48 . 6 48 . 8 s / km s / km 10 / 1
Minor transverse contributions from IRD, IRJ ζ / ( M Ω √ IRD MHz / m ) α / ( Ω /√ MHz ) elements x plane y plane 1 . 281 1 . 264 2 . 292 drifs 0 . 004 0 . 004 0 . 011 quads 0 . 000 0 . 000 0 . 000 dipoles 0 . 000 0 . 000 0 . 000 kickers 1 . 285 1 . 268 2 . 304 total region IRD, input parameters region IRD, impedance coefficients 1 . 25 x x MHz / m ) 4 y y 1 . 00 3 √ β / km 0 . 75 c.s. of ζ / ( M Ω 2 0 . 50 1 0 . 25 0 0 . 00 4 2 . 0 MHz ) 2 √ 1 . 5 b , w / cm c.s. of α / ( Ω / 0 1 . 0 − 2 50 K 0 . 5 293 K − 4 0 . 0 22 . 5 23 . 0 23 . 5 24 . 0 24 . 5 25 . 0 22 . 5 23 . 0 23 . 5 24 . 0 24 . 5 25 . 0 s / km s / km 11 / 1
Minor transverse contributions from IRD, IRJ ζ / ( M Ω √ IRJ MHz / m ) α / ( Ω /√ MHz ) elements x plane y plane 0 . 659 0 . 647 2 . 104 drifs 0 . 179 0 . 175 0 . 186 quads 0 . 010 0 . 030 0 . 031 dipoles 0 . 000 0 . 000 0 . 000 kickers 0 . 848 0 . 852 2 . 321 total region IRJ, input parameters region IRJ, impedance coefficients 2 . 0 x x 0 . 8 MHz / m ) y y 1 . 5 0 . 6 √ β / km c.s. of ζ / ( M Ω 1 . 0 0 . 4 0 . 5 0 . 2 0 . 0 0 . 0 4 2 . 0 MHz ) 2 √ 1 . 5 b , w / cm c.s. of α / ( Ω / 0 1 . 0 − 2 50 K 0 . 5 293 K − 4 0 . 0 71 . 5 72 . 0 72 . 5 73 . 0 73 . 5 74 . 0 71 . 5 72 . 0 72 . 5 73 . 0 73 . 5 74 . 0 s / km s / km 12 / 1
Influence of IR apertures in IRA / IRG region IRA, input parameters region IRA, impedance coefficients 8 80 x x MHz / m ) y y 6 60 x, modified y, modified √ β / km c.s. of ζ / ( M Ω 40 4 2 20 0 0 10 50 K 1 . 0 293 K MHz ) 5 0 . 8 √ b , w / cm c.s. of α / ( Ω / 0 . 6 0 0 . 4 − 5 0 . 2 − 10 0 . 0 96 . 2 96 . 4 96 . 6 96 . 8 97 . 0 97 . 2 97 . 4 97 . 6 97 . 8 96 . 2 96 . 4 96 . 6 96 . 8 97 . 0 97 . 2 97 . 4 97 . 6 97 . 8 s / km s / km Strong influence by scaling law ∝ √ ρβ / b 3 from high-beta drif spaces. 13 / 1
Influence of IR apertures in IRA / IRG 15 MHz / m ) 10 √ ζ / ( M Ω IRA+IRG (x) all regions (x) 5 IRA+IRG (y) all regions (y) 0 10 MHz ) 8 IRA+IRG (long.) √ 6 α / ( Ω / all regions (long.) 4 2 0 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 5 7 . 0 radius of IRA/IRG drif space / cm Significant reduction of transverse impedance possible by enlarging the aforementioned apertures. 14 / 1
Sneak peek: Injection optics region IRA, input parameters region IRA, impedance coefficients x x 0 . 4 4 MHz / m ) y y 0 . 3 3 √ β / km c.s. of ζ / ( M Ω 0 . 2 2 1 0 . 1 0 0 . 0 1 . 0 50 K 293 K 5 MHz ) 0 . 8 √ b , w / cm 0 . 6 c.s. of α / ( Ω / 0 0 . 4 − 5 0 . 2 0 . 0 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 s / km s / km ⇒ Current task: repeat computation for 3 . 3 TeV injection optics. factor 20 in β , Z values 15 / 1
Summary Transverse and longitudinal resistive-wall impedance contributions of major warm parts have been computed for collision optics at 50 TeV with β ∗ = 0 . 3 m. Overall impedance of warm parts is of approx. equal magnitude to that of cold beamscreen. Major contributions from room-temperature drif spaces in IRA / IRG, which can be removed by aperture increase. Next tasks ⇒ Re-evaluate computation with modified aperture dimensions (further input welcome) Tank you for your atention! 16 / 1
Backup slides IRB ζ / ( M Ω √ IRB MHz / m ) α / ( Ω /√ MHz ) elements x plane y plane 0 . 252 0 . 184 1 . 033 drifs 0 . 121 0 . 115 0 . 075 quads 0 . 015 0 . 014 0 . 024 dipoles 0 . 035 0 . 033 0 . 027 kickers 0 . 423 0 . 346 1 . 159 total region IRB, input parameters region IRB, impedance coefficients 3 x 0 . 4 x MHz / m ) y y 0 . 3 2 √ β / km c.s. of ζ / ( M Ω 0 . 2 1 0 . 1 0 0 . 0 4 1 . 00 MHz ) 2 √ 0 . 75 b , w / cm c.s. of α / ( Ω / 0 0 . 50 − 2 50 K 0 . 25 − 4 293 K 0 . 00 4 . 2 4 . 4 4 . 6 4 . 8 5 . 0 5 . 2 5 . 4 4 . 2 4 . 4 4 . 6 4 . 8 5 . 0 5 . 2 5 . 4 s / km s / km 17 / 1
Recommend
More recommend
