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First Steps to the Optimization of Undulator Parameters for 125 GeV Drive Beam by Manuel Formela 04.09.2018 1 Overview Introducing formulas for: Power absorped by the undulator vessel in form of photons

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First Steps to the Optimization of Undulator Parameters for 125 GeV Drive Beam

by Manuel Formela

04.09.2018 1

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Overview

  • Introducing formulas for:
  • Power absorped by the undulator vessel in form of photons π‘„π‘€π‘“π‘‘π‘‘π‘“π‘š
  • Number of produced 𝑓+ by π‘“βˆ’π‘“+- pair production in a Ti-6% Al-4%V target
  • Undulator scheme used in the RDR
  • Reproducing values for already calculated π‘„π‘€π‘“π‘‘π‘‘π‘“π‘š for the RDR set-up
  • Calculations of 𝑂𝑓+ for various parameter values for 𝐿, πœ‡, π‘šπ‘£, π‘‚β„Žπ‘‘π‘“π‘šπ‘š
  • Dropping some parameter combinations due to restraints in 𝑂𝑓+ and π‘„π‘€π‘“π‘‘π‘‘π‘“π‘š
  • Outlook into possible future

04.09.2018 2

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Radiated Synchrotron Energy Spectral Density per Solid Angle per Electron

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First approximations:

  • relativistic (𝛿 ≫ 1)
  • far field (𝑆 ≫ πœ‡π›Ώ)
  • pointlike charge (π‘Š

π‘“βˆ’ β†’ 0)

2nd approximations:

  • small (radiation) angle ( πœ„ β‰ͺ 1 β‡’ cos πœ„ β‰ˆ 1, sin πœ„ β‰ˆ πœ„); this is reasonable, because the radiation cone has

according to theory an Opening angle of πœ„ β‰ˆ 1/𝛿

  • Many undulator periods (𝑂𝑣 ≳ 100)
  • reasonably small undulator parameter (𝐿 ≲ 1 β†’ 𝐿/𝛿 β‰ͺ 1)

𝑒𝐽(πœ•) 𝑒Ω = e2πœ•2𝐿2 4𝜌3πœ—0π‘‘πœ•π‘£

2𝛿2 ෍ π‘œ=1 ∞

πΎπ‘œ

β€²2 𝑦1 + π›Ώπœ„

𝐿 βˆ’ π‘œ 𝑦1

2

πΎπ‘œ

2 𝑦1

sin2 π‘‚π‘£πœŒ πœ• πœ•1 βˆ’ π‘œ πœ• πœ•1 βˆ’ π‘œ

2

𝑒𝐽(πœ•) 𝑒Ω = 𝑒2𝑋(πœ•) π‘’Ξ©π‘’πœ• = 𝑓2πœ•2 14𝜌3πœ—0𝑑 ΰΆ±

βˆ’βˆž +∞

ො π‘œ Γ— ො π‘œ Γ— Τ¦ 𝛾 𝑓

π‘—πœ• π‘’βˆ’ ො π‘œ Τ¦ 𝑠 𝑒 𝑑

𝑒𝑒

2

For helical trajectory: Photon frequency Opening angle Formulas taken from: Kincaid, Brian M. "A short‐period helical wiggler as an improved source of synchrotron radiation." Journal of Applied Physics 48.7 (1977): 2684-2691.

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𝑒𝐽(πœ•) 𝑒Ω = e2πœ•2𝐿2 4𝜌3πœ—0π‘‘πœ•π‘£

2𝛿2 ෍ π‘œ=1 ∞

πΎπ‘œ

β€²2 𝑦1 + π›Ώπœ„

𝐿 βˆ’ π‘œ 𝑦1

2

πΎπ‘œ

2 𝑦1

sin2 π‘‚π‘£πœŒ πœ• πœ•1 βˆ’ π‘œ πœ• πœ•1 βˆ’ π‘œ

2

Approximation sin2 π‘‚πœŒπ‘§ /𝑧2 β†’ NπœŒπœ€(𝑧): 𝑒𝑋 𝑒Ω = ΰΆ±

∞ 𝑒𝐽(πœ•)

𝑒Ω π‘’πœ• β‰ˆ Nue2πœ•π‘£πΏ28𝛿4 4πœŒπœ—0𝑑 1 + 𝐿2 + 𝛿2πœ„2 3 ෍

π‘œ=1 ∞

π‘œ2 πΎπ‘œ

β€²2 π‘¦π‘œ + π›Ώπœ„

𝐿 βˆ’ π‘œ π‘¦π‘œ

2

πΎπ‘œ

2 π‘¦π‘œ

1.

  • 2. 𝑒𝑋

π‘’πœ• = ΰΆ± 𝑒𝐽(πœ•) 𝑒Ω 𝑒Ω β‰ˆ Nue2𝐿2𝑠 πœ—0𝑑 ෍

π‘œ=1 ∞

π‘œ2 πΎπ‘œ

β€²2 π‘§π‘œ + π›½π‘œ

𝐿 βˆ’ π‘œ π‘§π‘œ

2

πΎπ‘œ

2 π‘§π‘œ

𝐼(π›½π‘œ

2)

Radiated energy per solid angle Radiated energy spectral density Numerical integration leads to:

  • 1. π‘„π‘€π‘“π‘‘π‘‘π‘“π‘š =

ሢ π‘‚π‘“βˆ’ ΰΆ± 𝑒𝑋 𝑒Ω 𝑒Ω = 2𝜌 ሢ π‘‚π‘“βˆ’ ΰΆ±

πœ„1 𝜌

sin πœ„ 𝑒𝑋 π‘’πœ„ π‘’πœ„ 2. 𝑂𝑓+ = 1 ℏ ΰΆ±

∞ 1

πœ• 𝑒𝑋 π‘’πœ• (1 βˆ’ π‘“βˆ’π‘’πœπœ(πœ•))π‘’πœ• Power deposited in the undulator vessel Positron number produced by all photons Target thickness Target density Cross section for π‘“βˆ’π‘“+-pair production by photon of target material

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04.09.2018 5

Undulator set up (RDR, BCD)

20 of such half-cell will be arranged in a row to form the full undulator (with 240 m of total magnetic length) Undulator aperture = 5.85 mm Taken from: Scott, Duncan J. "An Investigation into the Design of the Helical Undulator for the International Linear Collider Positron Sourceβ€œ

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Test: Power deposited in the undulator vessel

  • Dashed lines: single undulator piece
  • Solid lines: whole undulator scheme
  • Blue: RDR parameters
  • Red: BCD parameters

Duncan J Scottβ€˜s calculations Current calculations In good agreement

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Undulator aperture = 5.85 mm Undulator mask (consisting of collimators with aperture of 4.4 mm) for preventing heating of the vessel due to photon absorption. The limit of maximal absorped power is 1 Wmβˆ’1 (according to Duncan J Scott, who in turn names the source to be private communication with T Bradshaw)

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Duncan J Scottβ€˜s calculations Current calculations In good agreement

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Current calculations In good agreement Duncan J Scottβ€˜s calculations Peridocity and peak values are in good agreement; Disagreement in dip values and local shape of the graph

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𝐿 = 0.65, 0.9, 1.15, πœ‡π‘£ = 8.5, 10, 11.5 mm, π‘šπ‘£ = 1.75, 2 m, π‘‚β„Žπ‘‘π‘“π‘šπ‘š = 18, 20, 22

Examined parameter combination for the positron number

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𝐿 = 1.15, πœ‡ = 8.5 mm, π‘šπ‘£= 2m, π‘‚β„Žπ‘‘π‘“π‘šπ‘š= 22 ∼ 264 m magnetic length

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  • Drop a single or multiple approximations (𝛿 ≫ 1 , πœ„ β‰ͺ 1, 𝑂𝑣 ≳ 100,

𝐿 ≲ 1, 𝑆 ≫ πœ‡π›Ώ, π‘Š

π‘“βˆ’ β†’ 0, sin2 π‘‚πœŒπ‘§ /𝑧2 β†’ NπœŒπœ€(𝑧), etc.)

  • Correcting possible flaws in the undulator mask considerations
  • For 𝑂𝑓+: Numerical integration over a solid angle, that only covers the

target instead of the full πœ„ = 0 βˆ’ 𝜌

04.09.2018 14

Possible future improvements

𝑒𝑋 π‘’πœ• = ΰΆ± 𝑒𝐽(πœ•) 𝑒Ω 𝑒Ω β‰ˆ Nue2𝐿2𝑠 πœ—0𝑑 ෍

π‘œ=1 ∞

π‘œ2 πΎπ‘œ

β€²2 π‘§π‘œ + π›½π‘œ

𝐿 βˆ’ π‘œ π‘§π‘œ

2

πΎπ‘œ

2 π‘§π‘œ

𝐼(π›½π‘œ

2)

  • Examining more intermediate parameter values between the upper

and lower limits

  • Adding more criteria for the optimazation besides lower limit for 𝑂𝑓+

and upper limit for π‘„π‘€π‘“π‘‘π‘‘π‘“π‘š

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Thank you for your attention

04.09.2018 15