Transverse resistive-wall impedance Elliptical pipe with semiaxes w , - PowerPoint PPT Presentation

Resistive-wall impedance of insertions 12 Bernard Riemann Zentrum für Synchrotronstrahlung 2018-02-02 8th FCC-hh collective effects meeting 1 thanks to S. Arsenyev, A. Langner, R. Martin, D. Schulte and S. Khan 2 supported by German Federal Ministry of Education and Research, funding code 05P15PERB1 1 / 20

Transverse resistive-wall impedance Elliptical pipe with semiaxes w , b : ⇒ Use form factors G 1 ⊥ ( w , b ) : 34 ˜ Z ⊥ , n 1 + i = G 1 ⊥ ( w n , b n ) Z 0 δ skin 2 π b 3 L n n Single-kick model 5 for elements n of a latice ⇒ continuum: s n ˜ 1 1 Z ⊥ , n ∫ � � ˜ Z ⊥ = Z ⊥ , n β ⊥ ( s n ) = β ⊥ ( s ) d s β smooth β smooth L n n n ⊥ ⊥ s n − 1 Qadrature rule Assume β is piece-wise cubic function of s . β, α = − β ′ / 2 known at all element endpoints s n . 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). 3 R.L. Gluckstern, J. van Zeijts and B. Zoter, Phys. Rev. E 47 (1992) 4 K. Yokoya, Part. Acc. 41 (1993), p. 18 – 19 5 N. Mounet, PhD thesis, EPFL Lausanne (2012) 2 / 20

Longitudinal case & Input data Longitudinal resistive-wall impedance Different dependence on b and f , a no β � Z ⊥ = α ( 1 + i ) f , with √ ρ n G 0 ( w n , b n ) � α = Z 0 π � L n . 2 π c b n µ 0 µ r n a A. Chao, Physics of Collective Instabilities… (Wiley, 2003) Used updated optics data from repo 6 Collision optics with β ∗ = 0 . 3 m, injection optics with β ∗ = 4 . 6 m. Used resistivities of copper 7 , individual assumptions for each insertion ρ ( 50 K ) = 0 . 518 n Ω m , ρ ( 293 K ) = 16 . 78 n Ω m . collimators etc. are ignored 6 A. Chance, R. Martin, A. Langner, M. Hofer et al., https://gitlab.cern.ch/fcc-optics/FCC-hh-lattice , commit 5443690ac... (2018). 7 R.A. Matula, J. Phys. Chem. Ref. Data 8 (4) (1979) 3 / 20

Total impedance, β ∗ = 0 . 3 m regions A, B, D, F, G, H, J, L 10 10 Re,Im Z x (all insertions) x impedance / ( Ω /m) Re Z x (cold beamscreen) Im Z x (cold beamscreen) 10 8 10 6 Transverse: strong 10 10 y impedance / ( Ω /m) contribution relative to cold beamscreen reference data. a 10 8 Longitudinal contribution is Re,Im Z y (all insertions) 10 6 Re Z y (cold beamscreen) significantly smaller than Im Z y (cold beamscreen) cold beamscreen → OK. a S. Arsenyev, FCC impedance online database, 10 3 long impedance / Ω https://impedance.web.cern.ch/impedance/fcchh 10 1 Re,Im Z l (all insertions) Re Z l (cold beamscreen) 10 − 1 Im Z l (cold beamscreen) 10 2 10 4 10 6 10 8 10 10 f / Hz 4 / 20

Major transverse contributions from main experiments insertion A, β ∗ = 0 . 3 m insertion A 7 . 5 x 50 K 60 y 293 K 5 . 0 2 . 5 b, w / cm β / km 40 0 . 0 − 2 . 5 20 − 5 . 0 − 7 . 5 0 1 . 0 x MHz / m ) 6 y MHz ) 0 . 8 √ √ 0 . 6 c.s. of α / (Ω / 4 c.s. of ζ / ( M Ω 0 . 4 2 0 . 2 0 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 5 / 20

Almost identical for IRA / IRG √ IRA ζ / ( M Ω MHz / m ) √ elements x plane y plane α / ( Ω / MHz ) drifs 6 . 43 6 . 31 0 . 88 quads 0 . 30 0 . 30 0 . 05 dipoles 0 . 14 0 . 14 0 . 02 kickers 0 . 02 0 . 02 0 . 00 total 6 . 89 6 . 76 0 . 97 √ IRG ζ / ( M Ω MHz / m ) √ elements x plane y plane α / ( Ω / MHz ) drifs 6 . 43 6 . 31 0 . 88 quads 0 . 31 0 . 32 0 . 07 dipoles 0 . 14 0 . 14 0 . 02 kickers 0 . 01 0 . 01 0 . 00 total 6 . 89 6 . 79 0 . 98 6 / 20

IRA/IRG with 40mm → 55mm default chamber radius √ IRA ζ / ( M Ω MHz / m ) √ elements x plane y plane α / ( Ω / MHz ) drifs 2 . 47 2 . 43 0 . 64 quads 0 . 28 0 . 28 0 . 05 dipoles 0 . 14 0 . 14 0 . 02 kickers 0 . 02 0 . 02 0 . 00 total 2 . 91 2 . 86 0 . 72 √ IRG ζ / ( M Ω MHz / m ) √ elements x plane y plane α / ( Ω / MHz ) drifs 2 . 47 2 . 43 0 . 64 quads 0 . 30 0 . 32 0 . 07 dipoles 0 . 14 0 . 14 0 . 02 kickers 0 . 00 0 . 00 0 . 00 total 2 . 92 2 . 89 0 . 74 7 / 20

IRA with 40mm → 55mm default chamber radius insertion A, β ∗ = 0 . 3 m insertion A 7 . 5 x 50 K 60 y 293 K 5 . 0 2 . 5 b, w / cm β / km 40 0 . 0 − 2 . 5 20 − 5 . 0 − 7 . 5 0 3 . 0 x MHz / m ) y 2 . 5 MHz ) 0 . 6 2 . 0 √ √ c.s. of α / (Ω / c.s. of ζ / ( M Ω 0 . 4 1 . 5 1 . 0 0 . 2 0 . 5 0 . 0 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 8 / 20

Impedance coeffs vs β ∗ √ ζ / ( M Ω MHz / m ) summary β ∗ / m x plane y plane 0.15 31 . 58 33 . 05 0.20 25 . 02 25 . 25 0.30 18 . 47 18 . 42 1.10 8 . 03 8 . 18 4.60 4 . 66 4 . 80 6.00 4 . 56 4 . 65 Table: Overall transverse impedance coefficients for different optics setings. 9 / 20

Total impedance, β ∗ = 4 . 6 m regions A, B, D, F, G, H, J, L 10 10 Re,Im Z x (all insertions) x impedance / ( Ω /m) Re Z x (cold beamscreen) Im Z x (cold beamscreen) 10 8 10 6 Overall contributions (40mm default radius) 10 10 Re,Im Z y (all insertions) y impedance / ( Ω /m) significantly smaller than Re Z y (cold beamscreen) cold beamscreen estimate Im Z y (cold beamscreen) 10 8 for transverse and longitudinal case → OK. 10 6 Reasonable, as there is an inverse relation between β ∗ 10 3 long impedance / Ω and max ( β ) which enters into transverse impedance… 10 1 Re,Im Z l (all insertions) Re Z l (cold beamscreen) 10 − 1 Im Z l (cold beamscreen) 10 2 10 4 10 6 10 8 10 10 f / Hz 10 / 20

Transverse contribution from IRA, β ∗ = 4 . 6 m insertion A, β ∗ = 4 . 6 m x 4 y 3 β / km 2 1 0 0 . 4 x MHz / m ) y 0 . 3 √ c.s. of ζ / ( M Ω 0 . 2 0 . 1 0 . 0 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 s / km 11 / 20

Betatron collimation (IRJ) insertion J insertion J 4 2 . 0 x y 1 . 5 2 b, w / cm β / km 0 1 . 0 − 2 0 . 5 50 K 293 K − 4 0 . 0 2 . 0 x 3 . 0 MHz / m ) y MHz ) 1 . 5 2 . 5 √ √ c.s. of α / (Ω / 2 . 0 c.s. of ζ / ( M Ω 1 . 0 1 . 5 1 . 0 0 . 5 0 . 5 0 . 0 0 . 0 73 . 5 74 . 0 74 . 5 75 . 0 75 . 5 76 . 0 73 . 5 74 . 0 74 . 5 75 . 0 75 . 5 76 . 0 s / km s / km 3rd-highest transverse contribution for collision optics (afer IRA and IRG, due to large β ), no modification indicated IRJ is identical for both optics 12 / 20

Extraction (IRD) insertion D insertion D 4 x 50 K 4 y 293 K 2 3 b, w / cm β / km 0 2 − 2 1 − 4 0 2 . 0 x 1 . 2 MHz / m ) y MHz ) 1 . 0 1 . 5 √ √ 0 . 8 c.s. of α / (Ω / c.s. of ζ / ( M Ω 1 . 0 0 . 6 0 . 4 0 . 5 0 . 2 0 . 0 0 . 0 24 . 5 25 . 0 25 . 5 26 . 0 26 . 5 27 . 0 24 . 5 25 . 0 25 . 5 26 . 0 26 . 5 27 . 0 s / km s / km is a moderate increase of chamber radius possible? IRD is identical for both optics 13 / 20

Tabular summary ( β ∗ = 0 . 3 m ) √ summary ζ / ( M Ω MHz / m ) √ insertion x plane y plane α / ( Ω / MHz ) A 6 . 89 6 . 76 0 . 97 B 0 . 71 0 . 62 1 . 17 D 1 . 27 1 . 25 1 . 93 F 0 . 17 0 . 22 1 . 22 G 6 . 89 6 . 79 0 . 98 H 0 . 13 0 . 14 1 . 28 J 1 . 81 1 . 97 3 . 29 L 0 . 61 0 . 68 1 . 20 all 18 . 47 18 . 42 12 . 03 cold bs. (50 TeV, S. Arsenyev) 23 . 41 43 . 25 72 . 81 14 / 20

Tabular summary ( β ∗ = 4 . 6 m ) √ summary ζ / ( M Ω MHz / m ) √ insertion x plane y plane α / ( Ω / MHz ) A 0 . 42 0 . 40 0 . 97 B 0 . 23 0 . 22 1 . 17 D 1 . 27 1 . 25 1 . 93 F 0 . 17 0 . 22 1 . 22 G 0 . 44 0 . 41 0 . 98 H 0 . 13 0 . 13 1 . 28 J 1 . 81 1 . 97 3 . 29 L 0 . 20 0 . 20 1 . 20 all 4 . 66 4 . 80 12 . 03 cold bs. (3 TeV, S. Arsenyev) 17 . 48 32 . 15 54 . 20 15 / 20

Summary Transverse and longitudinal resistive-wall impedance contributions of major warm parts have been computed for present state of collision optics and injection optics. Overall impedance of warm parts for collision optics is non-negligible. Major contributions in collision optics stem from 1 drif spaces in main experiments (IRA & IRG) 2 betatron collimation system (IRJ, possible “implicit” overlap with collimator computations, no change indicated) 3 extraction (IRD) Contributions can be reduced by aperture increase (see computation with modified default apertures and inverse cubic scaling law). Results will be published in IPAC18 proceedings and in the FCC-hh CDR. Tank you for your atention! Next page and following: missing insertions (backup slides) 16 / 20

Injection & additional experiments (IRB, IRL) insertion B, β ∗ = 0 . 3 m insertion B 3 . 0 x 4 y 2 . 5 2 2 . 0 b, w / cm β / km 0 1 . 5 − 2 1 . 0 50 K 0 . 5 − 4 293 K 0 . 0 1 . 2 x MHz / m ) y 1 . 0 0 . 6 MHz ) 0 . 8 √ √ c.s. of α / (Ω / c.s. of ζ / ( M Ω 0 . 4 0 . 6 0 . 4 0 . 2 0 . 2 0 . 0 0 . 0 6 . 2 6 . 4 6 . 6 6 . 8 7 . 0 7 . 2 7 . 4 7 . 6 6 . 2 6 . 4 6 . 6 6 . 8 7 . 0 7 . 2 7 . 4 7 . 6 s / km s / km β ∗ = 0 . 3 m IRL is symmetric to IRB 17 / 20

Injection & additional experiments (IRB, IRL) insertion B, β ∗ = 4 . 6 m insertion B x 0 . 8 4 y 2 0 . 6 b, w / cm β / km 0 0 . 4 − 2 0 . 2 50 K − 4 293 K 0 . 0 1 . 2 x MHz / m ) 0 . 20 y 1 . 0 MHz ) 0 . 8 0 . 15 √ √ c.s. of α / (Ω / c.s. of ζ / ( M Ω 0 . 6 0 . 10 0 . 4 0 . 05 0 . 2 0 . 00 0 . 0 6 . 2 6 . 4 6 . 6 6 . 8 7 . 0 7 . 2 7 . 4 7 . 6 6 . 2 6 . 4 6 . 6 6 . 8 7 . 0 7 . 2 7 . 4 7 . 6 s / km s / km β ∗ = 4 . 6 m IRL is symmetric to IRB 18 / 20