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Two Energy Storage Ring Cooler : Equilibrium and Longitudinal Stability
Bhawin Dhital Thanks to : David Douglas, J. Delayen, S. Derbenev, G. A. Krafft, F. Lin, V. Morozov, Y. Zhang
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Dual Energy Storage Ring : Schematic Diagram
F. Lin et.al, IPAC2016, Busan, Korea
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SR Mode and ERL Mode RF voltage phasor
Accelerating and decelerating slopes π π ,π =πΒ± π π ,π (W/R) In SR language, Synchronous phase is usually referenced to zero crossing phase in the cavity (A). π π ,π =πΒ± π π ,π +πΏπ (WR) πΏπ=βΞπΈ/(π sin πΒ± π π ,π β+β for ERL mode, β-β for SR mode
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Stability in Two Energy Storage Rings
To the linear order, one turn transfer matrix in (Ξπ, ΞπΈ) For SR equilibrium (Radiation off) π ππ
= 1 β πΏ / πΈ πΏ βπ π ππ Ξ¦ π ,π β π» / πΈ π» βπ π ππ Ξ¦ π ,π 1 ΞΌ ππ
= 2( β πΏ π π ππ Ξ¦ π ,π πΈ πΏ + β π» π π ππ Ξ¦ π ,π πΈ π» ) , where β π» = 2Οβ π 0 πΏ π» Ξ· π» ᡦ π» 3 π For ERL equilibrium (Radiation off) π πΈπ
πΏ = 1 β πΏ / πΈ πΏ βπ π ππ Ξ¦ π ,π β π» / πΈ π» βπ π ππ Ξ¦ π ,π 1 ΞΌ πΈπ
πΏ = β πΏ β π» π 2 sin 2 Ξ¦ π ,π πΈ πΏ πΈ π» , π π = π 2 π Schematic diagram of TESR
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Stability/Synchrotron Tune
SR Mode (W/R) ERL Mode (W/R) SR Mode (WR) ERL Mode(WR) π π (simulation) π π (calculation) % difference 0.9 0.2 2.4 SR Mode (W/R) ERL Mode(W/R) SR Mode (WR) ERL Mode(WR) *W/R = Without Radiation, *WR = With Radiation
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Stability and Synchrotron Tune
First we take a simple ring ( DBA1, CAV(acc), DBA2, CAV(dec)) We take πΈ π» = GeV, πΈ πΏ = 25.0 GeV (ERL tune) We take πΈ π» = 35.0 MeV, πΈ πΏ = 25 MeV ( No synchrotron radiation) We take πΈ π» = MeV, πΈ πΏ = 55.0 MeV Introduce artificial damping in HER (elegant simulation) Two energy real lattice length β m. Damped emittance β¦..simulation time. Decided to work with Matrix element in elegant simulation Periodic Solution and longitudinal stability
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Periodic Solution One turn transfer matrix (Linear)
Calculate the stability criteria Calculate twiss parameters ( πΌ π , π½ π , πΎ π ) and use these values in elegant simulations. Periodic solution exists for both SR mode and ERL mode RF accelerating phase π π ,π depends on RF wavelength π ππ tan β1 Ξπ π π΅ π π 2 π ππ < π π ,π < tan β1 βΞπ π π΅ π π 2 π ππ B = π 2 π 1 , A = 2(1 + B)/B, π 2 = 155 MeV, π 1 = 55 MeV
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Longitudinal Stability ( SR Mode)
We calculate one turn transfer matrix and apply |2 cosπ | β€ 2 π π ,π versus π ππ plot for SR mode. Blue region (stable) π 0 = 476 MHz ( π ππ = m),| π 56 | = 1.0 m π π ,π 1 = β β π π ,π 2 = β
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Longitudinal Stability (ERL Mode)
π π ,π versus π ππ plot for ERL mode. Blue region (stable) π 0 = 476 MHz, ( π ππ = m) ,| π 56 | = 1.0 m, < π π ,π 1 < 90 0 90 0 < π π ,π 2 <
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Periodic Solution ( π π,π =π π π , π ππ =π.ππ9816 m)
SR mode, Ξπ§=1.82 mm, Ξπ π =3.10Γ 10 β3 π½ π = m, πΌ π = ( After decelerating cavity, cooler ring) ERL mode, Ξπ§=1.68 mm, Ξπ π =1.79Γ 10 β3 π½ π = m, πΌ π = (After decelerating cavity, cooler ring) ERL mode SR mode
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Periodic Solution / Chirping β De-Chirping
We define one loop transfer matrix, find out πΌ π and π½ π at point A. π π‘ππ‘ππ = π ππβπβπππππ π π πππ π 56 π»πΈ π πππ π π πβπππππ
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SR Mode, π π,π =ππ π π , Volt = 100.0 kV Ξπ§= 0.73 mm
Ξπ π =4.061Γ 10 β3
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Chirping / De β Chirping (SR Mode)
Voltage scanning shows that for the smaller π 56 values, higher cavity voltage is required for chirping and de-chirping. Higher value of chirper and de-chirper cavity voltage destroy the periodic solution.
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π΄ ππ and RF Acceleration, π ππ = 0.6298 m fixed
Ref: IPAC2019
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Two cavities model Dr Vasilily suggestion: use two cavity ( For both acceleration and deceleration) to remove more chirping introduced by a single cavity.
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π π versus π΄ ππ πππ (πππ), π«π=π.πππ π π
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π π versus π΄ ππ (πππ), π«π=π.πππ π π
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π π versus π πΉ π«π=π.πππ π π
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π· π versus π«π Red = 15deg, BLUE = 30 deg, Green = 45 deg, Black = 60 deg, Yellow = 75 deg
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ERL Mode, π π,π =π π π , π«π=π.πππ π π Ξπ§ = m, Ξπ π =4.64Γ 10 β5
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Summary Longitudinal stability exists in Two Energy Storage Rings.
Periodic solution exists. Optimization of cooler parameters are in progress. Acknowledgement - All CASA members - Dr Morozov
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Thank You !
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