Resistive-wall impedance of interaction regions (warm beam pipe) 12 - - PowerPoint PPT Presentation

Resistive-wall impedance of interaction regions (warm beam pipe)12

Bernard Riemann

Zentrum für Synchrotronstrahlung

2017-10-10

EuroCirCol meeting

1thanks to S. Arsenyev, A. Langner, O. Boine-Frankenheim, D. Schulte and S. Khan 2supported by German Federal Ministry of Education and Research,

funding code 05P15PERB1

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Transverse impedance model

Preface: Tune shifa

athanks to O. Boine-Frankenheim

Transverse tune shif,a proportionality ∆νy = Ω − ωβ ω0 ∝ βyI ˜ Z⊥/E. Normalize such that βy(s) = 1 case is equivalent to impedance Z⊥ of harmonic oscillator with frequency 2πQ = ∫ 1 β(s) ds =: 1 βsmooth .

• aA. Chao, ”Physics of Collective Beam Instabilities in High Energy Accelerators“, chapter 4

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Transverse impedance model

Use single-kick model3 for elements n of a latice: Z⊥ = 1 βsmooth

⊥

• n

β⊥(sn)˜ Z⊥,n round pipe, radius b ˜ Z⊥,n = Ln Z0δskin

n

1 + i 2πb3

n

with δskin

n

= 2ρn µ0µrω elliptical pipe with semiaxes w, b: Use form factors G1⊥(w, b) 45 ˜ Z⊥,n = Ln G1⊥(wn, bn)Z0δskin 1 + i 2πb3

n

• 3N. Mounet, PhD thesis, EPFL Lausanne (2012)

4R.L. Gluckstern, J. van Zeijts and B. Zoter, Phys. Rev. E 47 (1992)

• 5K. Yokoya, Part. Acc. 41 (1993), p. 18 – 19

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Transverse impedance model

Z⊥ = Z0 √2µ0µrω 1 + i πβsmooth

⊥

• n

Ln β⊥(sn)G1⊥(wn, bn) √ρn b3

n

assume G1⊥, ρ, b as piece-wise constant, but β as continous: Z⊥ = Z0 √2µ0µrω 1 + i πβsmooth

⊥

• n

G1⊥(wn, bn) √ρn b3

n sn

∫

sn−1

β⊥(s) ds β, α = −β′/2 known at all element endpoints sn

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Transverse impedance model

Qadrature rule

Assume β is piece-wise cubic function of s:

L

∫ β(s) ds ≈ L β(L) + β(0) 2 + L2 α(L) − α(0) 6 . Approximation is exact for drif spaces (quadratic dependence).

Result

Z⊥ = ζ 1 + i

• f

, with ζ = Z0 2πβsmooth

⊥

√πµ0µr

• n

G1⊥(wn, bn) √ρn b3

n sn

∫

sn−1

β⊥(s) ds.

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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

α = Z0 2πc

• π

µ0µr

• n

Ln √ρn G0(wn, bn) bn .

• 6A. Chao, Physics of Collective Instabilities in Particle Accelerators (Wiley, 2003)

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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 copper8 at 50 K for magnets (elements QUADRUPOLE, RBEND, SBEND, HKICKER, VKICKER) ρ(50 K) = 0.518 nΩ m respectively 293 K for drif spaces ρ(293 K) = 16.78 nΩ m.

• ther latice elements ignored (no treatment of collimators etc.)

7thanks to S. Arsenyev and A. Langner 8R.A. Matula, “Electrical Resistivity of Copper, Gold, Palladium, and Silver”, Table 2,

• J. Phys. Chem. Ref. Data 8 (4) (1979)

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Total impedance regions A, B, D, F, G, H, J, L

102 104 106 108 1010 f / Hz 105 106 107 108 109 1010 x impedance / Ω Re Zx (cold beamscreen, factor 1.24) Im Zx (cold beamscreen, factor 1.24) Re,Im Zx warm pipe (ζ/u = 18.9) 102 104 106 108 1010 f / Hz 105 106 107 108 109 1010 y impedance / Ω Re Zy (cold beamscreen, factor 2.28) Im Zy (cold beamscreen, factor 2.28) Re,Im Zy warm pipe (ζ/u = 19.0) 102 104 106 108 1010 f / Hz 10−2 10−1 100 101 102 103 104 l impedance / Ω Re Zl (cold beamscreen, factor 6.36) Im Zl (cold beamscreen, factor 6.36) Re,Im Zl warm pipe (α/u = 11.5)

Transverse plane: strong contribution relative to cold beamscreen reference data.9

• 9S. Arsenyev, FCC impedance online database,

https://impedance.web.cern.ch/impedance/fcchh

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Major transverse contributions from IRA / IRG

IRA ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 7.190 7.118 1.007 quads 0.168 0.236 0.064 dipoles 0.379 0.372 0.024 kickers 0.082 0.080 0.010 total 7.818 7.806 1.106 20 40 60 80 β / km

region IRA, input parameters

x y 2 4 6 8 c.s. of ζ / (MΩ √ MHz/m)

region IRA, impedance coefficients

x y 96.2 96.4 96.6 96.8 97.0 97.2 97.4 97.6 97.8 s / km −10 −5 5 10 b, w / cm 50 K 293 K 96.2 96.4 96.6 96.8 97.0 97.2 97.4 97.6 97.8 s / km 0.0 0.2 0.4 0.6 0.8 1.0 c.s. of α / (Ω/ √ MHz) 9 / 1

Major transverse contributions from IRA / IRG

IRG ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 7.190 7.118 1.007 quads 0.168 0.236 0.064 dipoles 0.379 0.372 0.024 kickers 0.082 0.080 0.010 total 7.818 7.806 1.106 20 40 60 80 β / km

region IRG, input parameters

x y 2 4 6 8 c.s. of ζ / (MΩ √ MHz/m)

region IRG, impedance coefficients

x y 47.4 47.6 47.8 48.0 48.2 48.4 48.6 48.8 s / km −10 −5 5 10 b, w / cm 50 K 293 K 47.4 47.6 47.8 48.0 48.2 48.4 48.6 48.8 s / km 0.0 0.2 0.4 0.6 0.8 1.0 c.s. of α / (Ω/ √ MHz) 10 / 1

Minor transverse contributions from IRD, IRJ

IRD ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 1.281 1.264 2.292 quads 0.004 0.004 0.011 dipoles 0.000 0.000 0.000 kickers 0.000 0.000 0.000 total 1.285 1.268 2.304 1 2 3 4 β / km

region IRD, input parameters

x y 0.00 0.25 0.50 0.75 1.00 1.25 c.s. of ζ / (MΩ √ MHz/m)

region IRD, impedance coefficients

x y 22.5 23.0 23.5 24.0 24.5 25.0 s / km −4 −2 2 4 b, w / cm 50 K 293 K 22.5 23.0 23.5 24.0 24.5 25.0 s / km 0.0 0.5 1.0 1.5 2.0 c.s. of α / (Ω/ √ MHz) 11 / 1

Minor transverse contributions from IRD, IRJ

IRJ ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 0.659 0.647 2.104 quads 0.179 0.175 0.186 dipoles 0.010 0.030 0.031 kickers 0.000 0.000 0.000 total 0.848 0.852 2.321 0.0 0.5 1.0 1.5 2.0 β / km

region IRJ, input parameters

x y 0.0 0.2 0.4 0.6 0.8 c.s. of ζ / (MΩ √ MHz/m)

region IRJ, impedance coefficients

x y 71.5 72.0 72.5 73.0 73.5 74.0 s / km −4 −2 2 4 b, w / cm 50 K 293 K 71.5 72.0 72.5 73.0 73.5 74.0 s / km 0.0 0.5 1.0 1.5 2.0 c.s. of α / (Ω/ √ MHz) 12 / 1

Influence of IR apertures in IRA / IRG

20 40 60 80 β / km

region IRA, input parameters

x y 2 4 6 8 c.s. of ζ / (MΩ √ MHz/m)

region IRA, impedance coefficients

x y x, modified y, modified 96.2 96.4 96.6 96.8 97.0 97.2 97.4 97.6 97.8 s / km −10 −5 5 10 b, w / cm 50 K 293 K 96.2 96.4 96.6 96.8 97.0 97.2 97.4 97.6 97.8 s / km 0.0 0.2 0.4 0.6 0.8 1.0 c.s. of α / (Ω/ √ MHz)

Strong influence by scaling law ∝ √ρβ/b3 from high-beta drif spaces.

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Influence of IR apertures in IRA / IRG

5 10 15 ζ / (MΩ √ MHz/m) IRA+IRG (x) all regions (x) IRA+IRG (y) all regions (y) 4.0 4.5 5.0 5.5 6.0 6.5 7.0 radius of IRA/IRG drif space / cm 2 4 6 8 10 α / (Ω/ √ MHz) IRA+IRG (long.) all regions (long.)

Significant reduction of transverse impedance possible by enlarging the aforementioned apertures.

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Sneak peek: Injection optics

1 2 3 4 β / km

region IRA, input parameters

x y 0.0 0.1 0.2 0.3 0.4 c.s. of ζ / (MΩ √ MHz/m)

region IRA, impedance coefficients

x y 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 s / km −5 5 b, w / cm 50 K 293 K 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 s / km 0.0 0.2 0.4 0.6 0.8 1.0 c.s. of α / (Ω/ √ MHz)

⇒ Current task: repeat computation for 3.3 TeV injection optics. factor 20 in β, Z values

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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!

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Backup slides IRB

IRB ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 0.252 0.184 1.033 quads 0.121 0.115 0.075 dipoles 0.015 0.014 0.024 kickers 0.035 0.033 0.027 total 0.423 0.346 1.159 1 2 3 β / km

region IRB, input parameters

x y 0.0 0.1 0.2 0.3 0.4 c.s. of ζ / (MΩ √ MHz/m)

region IRB, impedance coefficients

x y 4.2 4.4 4.6 4.8 5.0 5.2 5.4 s / km −4 −2 2 4 b, w / cm 50 K 293 K 4.2 4.4 4.6 4.8 5.0 5.2 5.4 s / km 0.00 0.25 0.50 0.75 1.00 c.s. of α / (Ω/ √ MHz) 17 / 1

Backup slides IRL

IRL ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 0.264 0.357 1.033 quads 0.147 0.193 0.081 dipoles 0.016 0.016 0.024 kickers 0.035 0.071 0.027 total 0.462 0.637 1.165 1 2 3 β / km

region IRL, input parameters

x y 0.0 0.2 0.4 0.6 c.s. of ζ / (MΩ √ MHz/m)

region IRL, impedance coefficients

x y 90.8 91.0 91.2 91.4 91.6 91.8 92.0 92.2 s / km −4 −2 2 4 b, w / cm 50 K 293 K 90.8 91.0 91.2 91.4 91.6 91.8 92.0 92.2 s / km 0.00 0.25 0.50 0.75 1.00 c.s. of α / (Ω/ √ MHz) 18 / 1

Backup slides IRJ

IRJ ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 0.659 0.647 2.104 quads 0.179 0.175 0.186 dipoles 0.010 0.030 0.031 kickers 0.000 0.000 0.000 total 0.848 0.852 2.321 0.0 0.5 1.0 1.5 2.0 β / km

region IRJ, input parameters

x y 0.0 0.2 0.4 0.6 0.8 c.s. of ζ / (MΩ √ MHz/m)

region IRJ, impedance coefficients

x y 71.5 72.0 72.5 73.0 73.5 74.0 s / km −4 −2 2 4 b, w / cm 50 K 293 K 71.5 72.0 72.5 73.0 73.5 74.0 s / km 0.0 0.5 1.0 1.5 2.0 c.s. of α / (Ω/ √ MHz) 19 / 1

Backup slides IRH

IRH ζ / (MΩ√ MHz/m) elements x plane y plane α / (Ω/√ MHz) drifs 0.101 0.108 1.214 quads 0.011 0.018 0.026 dipoles 0.001 0.002 0.018 kickers 0.000 0.000 0.000 total 0.113 0.128 1.258 0.0 0.2 0.4 0.6 0.8 β / km

region IRH, input parameters

x y 0.000 0.025 0.050 0.075 0.100 0.125 c.s. of ζ / (MΩ √ MHz/m)

region IRH, impedance coefficients

x y 53.0 53.2 53.4 53.6 53.8 54.0 54.2 54.4 s / km −4 −2 2 4 b, w / cm 50 K 293 K 53.0 53.2 53.4 53.6 53.8 54.0 54.2 54.4 s / km 0.00 0.25 0.50 0.75 1.00 1.25 c.s. of α / (Ω/ √ MHz) 20 / 1