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

Dual Energy Storage Ring : Schematic Diagram F. Lin et.al, IPAC2016, Busan, Korea

SR Mode and ERL Mode RF voltage phasor Accelerating and decelerating slopes 𝜙 𝑠,𝑑 =𝜋± 𝜙 𝑠,𝑎 (W/R) In SR language, Synchronous phase is usually referenced to zero crossing phase in the cavity (A). 𝜙 𝑠,𝑑 =𝜋± 𝜙 𝑠,𝑎 +𝛿𝜙 (WR) 𝛿𝜙=−Δ𝐸/(𝑉 sin 𝜋± 𝜙 𝑠,𝑎 ‘+’ for ERL mode, ‘-’ for SR mode

Stability in Two Energy Storage Rings To the linear order, one turn transfer matrix in (Δ𝜙, Δ𝐸) For SR equilibrium (Radiation off) 𝑀 𝑆𝑅 = 1 ℎ 𝐿 / 𝐸 𝐿 0 1 1 0 −𝑉 𝑠𝑖𝑛 Φ 𝑠,𝑑 1 1 ℎ 𝐻 / 𝐸 𝐻 0 1 1 0 −𝑉 𝑠𝑖𝑛 Φ 𝑠,𝑎 1 μ 𝑆𝑅 = 2( ℎ 𝐿 𝑉 𝑠𝑖𝑛 Φ 𝑠,𝑎 𝐸 𝐿 + ℎ 𝐻 𝑉 𝑠𝑖𝑛 Φ 𝑠,𝑎 𝐸 𝐻 ) , where ℎ 𝐻 = 2πℎ 𝑓 0 𝐿 𝐻 η 𝐻 ᵦ 𝐻 3 𝑐 For ERL equilibrium (Radiation off) 𝑀 𝐸𝑅𝐿 = 1 ℎ 𝐿 / 𝐸 𝐿 0 1 1 0 −𝑉 𝑠𝑖𝑛 Φ 𝑠,𝑑 1 1 ℎ 𝐻 / 𝐸 𝐻 0 1 1 0 −𝑉 𝑠𝑖𝑛 Φ 𝑠,𝑎 1 μ 𝐸𝑅𝐿 = ℎ 𝐿 ℎ 𝐻 𝑉 2 sin 2 Φ 𝑠,𝑎 𝐸 𝐿 𝐸 𝐻 , 𝑄 𝑠 = 𝜇 2 𝜋 Schematic diagram of TESR

Stability/Synchrotron Tune SR Mode (W/R) ERL Mode (W/R) SR Mode (WR) ERL Mode(WR) 𝑄 𝑠 (simulation) 0.03448 0.001627 0.03415 0.001631 𝑄 𝑠 (calculation) 0.001592 % 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

Stability and Synchrotron Tune First we take a simple ring ( DBA1, CAV(acc), DBA2, CAV(dec)) We take 𝐸 𝐻 = 25.010 GeV, 𝐸 𝐿 = 25.0 GeV (ERL tune) We take 𝐸 𝐻 = 35.0 MeV, 𝐸 𝐿 = 25 MeV ( No synchrotron radiation) We take 𝐸 𝐻 = 155.0 MeV, 𝐸 𝐿 = 55.0 MeV Introduce artificial damping in HER (elegant simulation) Two energy real lattice length ≈ 3500.0 m. Damped emittance …..simulation time. Decided to work with Matrix element in elegant simulation Periodic Solution and longitudinal stability

Periodic Solution One turn transfer matrix (Linear) Calculate the stability criteria Calculate twiss parameters ( 𝛼 𝑠 , 𝛽 𝑠 , 𝛾 𝑠 ) and use these values in elegant simulations. Periodic solution exists for both SR mode and ERL mode RF accelerating phase 𝜙 𝑠,𝑎 depends on RF wavelength 𝜆 𝑟𝑓 tan −1 Δ𝑃 𝜋 𝐵 𝑀 56 𝑃 2 𝜆 𝑟𝑓 < 𝜙 𝑠,𝑎 < tan −1 −Δ𝑃 𝜋 𝐵 𝑀 56 𝑃 2 𝜆 𝑟𝑓 B = 𝑃 2 𝑃 1 , A = 2(1 + B)/B, 𝑃 2 = 155 MeV, 𝑃 1 = 55 MeV

Longitudinal Stability ( SR Mode) We calculate one turn transfer matrix and apply |2 cos𝜇 | ≤ 2 𝜙 𝑠,𝑎 versus 𝜆 𝑟𝑓 plot for SR mode. Blue region (stable) 𝑓 0 = 476 MHz ( 𝜆 𝑟𝑓 = 0.6298 m),| 𝑀 56 | = 1.0 m 𝜙 𝑠,𝑎 1 = 107.00 0 − 112.51 0 107.00 0 − 112.51 0 𝜙 𝑠,𝑎 2 = 90 0 − 96.19 0

Longitudinal Stability (ERL Mode) 𝜙 𝑠,𝑎 versus 𝜆 𝑟𝑓 plot for ERL mode. Blue region (stable) 𝑓 0 = 476 MHz, ( 𝜆 𝑟𝑓 = 0.6298 m) ,| 𝑀 56 | = 1.0 m, 79.67 0 < 𝜙 𝑠,𝑎 1 < 90 0 90 0 < 𝜙 𝑠,𝑎 2 < 100.32 0

Periodic Solution ( 𝝓 𝒔,𝒂 =𝟗 𝟓 𝟎 , 𝝀 𝒓𝒇 =𝟎.𝟔𝟐9816 m) SR mode, Δ𝑧=1.82 mm, Δ𝑝 𝑝 =3.10× 10 −3 𝛽 𝑠 =1.1360383 m, 𝛼 𝑠 = - 1.603617594 ( After decelerating cavity, cooler ring) ERL mode, Δ𝑧=1.68 mm, Δ𝑝 𝑝 =1.79× 10 −3 𝛽 𝑠 =0.950411643 m, 𝛼 𝑠 = - 0.13961113 (After decelerating cavity, cooler ring) ERL mode SR mode

Periodic Solution / Chirping – De-Chirping We define one loop transfer matrix, find out 𝛼 𝑠 and 𝛽 𝑠 at point A. 𝑀 𝑡𝑜𝑡𝑎𝑙 = 𝑀 𝑑𝑒−𝑐ℎ𝑖𝑟𝑝𝑒𝑟 𝑀 56 1 𝑀 𝑑𝑒𝑐 𝑀 56 𝐻𝐸 𝑀 𝑎𝑐𝑐 𝑀 56 2 𝑀 𝑐ℎ𝑖𝑟𝑝𝑒𝑟

SR Mode, 𝝓 𝒔,𝒂 =𝟏𝟏 𝟐 𝟎 , Volt = 100.0 kV Δ𝑧= 0.73 mm Δ𝑝 𝑝 =4.061× 10 −3

Chirping / De – Chirping (SR Mode) Voltage scanning shows that for the smaller 𝑀 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.

𝑴 𝟓𝟔 and RF Acceleration, 𝝀 𝒓𝒇 = 0.6298 m fixed Ref: IPAC2019

Two cavities model Dr Vasilily suggestion: use two cavity ( For both acceleration and deceleration) to remove more chirping introduced by a single cavity.

𝝈 𝒔 versus 𝑴 𝟔𝟓 𝒂𝒄𝒄 (𝒕𝒐𝒕), 𝚫𝝓=𝟎.𝟎𝟎𝟎 𝟏 𝟎

𝝈 𝒔 versus 𝑴 𝟔𝟓 (𝒕𝒐𝒕), 𝚫𝝓=𝟎.𝟎𝟎𝟎 𝟏 𝟎

𝝈 𝒔 versus 𝝈 𝜹 𝚫𝝓=𝟎.𝟎𝟎𝟎 𝟏 𝟎

𝜷 𝒔 versus 𝚫𝝓 Red = 15deg, BLUE = 30 deg, Green = 45 deg, Black = 60 deg, Yellow = 75 deg

ERL Mode, 𝝓 𝒔,𝒂 =𝟔 𝟎 𝟎 , 𝚫𝝓=𝟎.𝟎𝟎𝟎 𝟏 𝟎 Δ𝑧 = 0.025 m, Δ𝑝 𝑝 =4.64× 10 −5

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

Thank You !