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Bhawin Dhital Thanks to :

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Presentation on theme: "Bhawin Dhital Thanks to :"β€” Presentation transcript:

1 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

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

3 SR Mode and ERL Mode RF voltage phasor
Accelerating and decelerating slopes πœ™ 𝑠,𝑑 =πœ‹Β± πœ™ 𝑠,π‘Ž (W/R) In SR language, Synchronous phase is usually referenced to zero crossing phase in the cavity (A). πœ™ 𝑠,𝑑 =πœ‹Β± πœ™ 𝑠,π‘Ž +π›Ώπœ™ (WR) π›Ώπœ™=βˆ’Ξ”πΈ/(𝑉 sin πœ‹Β± πœ™ 𝑠,π‘Ž β€˜+’ for ERL mode, β€˜-’ for SR mode

4 Stability in Two Energy Storage Rings
To the linear order, one turn transfer matrix in (Ξ”πœ™, Δ𝐸) For SR equilibrium (Radiation off) 𝑀 𝑆𝑅 = 1 β„Ž 𝐿 / 𝐸 𝐿 βˆ’π‘‰ 𝑠𝑖𝑛 Ξ¦ 𝑠,𝑑 β„Ž 𝐻 / 𝐸 𝐻 βˆ’π‘‰ 𝑠𝑖𝑛 Ξ¦ 𝑠,π‘Ž 1 ΞΌ 𝑆𝑅 = 2( β„Ž 𝐿 𝑉 𝑠𝑖𝑛 Ξ¦ 𝑠,π‘Ž 𝐸 𝐿 + β„Ž 𝐻 𝑉 𝑠𝑖𝑛 Ξ¦ 𝑠,π‘Ž 𝐸 𝐻 ) , where β„Ž 𝐻 = 2Ο€β„Ž 𝑓 0 𝐿 𝐻 Ξ· 𝐻 ᡦ 𝐻 3 𝑐 For ERL equilibrium (Radiation off) 𝑀 𝐸𝑅𝐿 = 1 β„Ž 𝐿 / 𝐸 𝐿 βˆ’π‘‰ 𝑠𝑖𝑛 Ξ¦ 𝑠,𝑑 β„Ž 𝐻 / 𝐸 𝐻 βˆ’π‘‰ 𝑠𝑖𝑛 Ξ¦ 𝑠,π‘Ž 1 ΞΌ 𝐸𝑅𝐿 = β„Ž 𝐿 β„Ž 𝐻 𝑉 2 sin 2 Ξ¦ 𝑠,π‘Ž 𝐸 𝐿 𝐸 𝐻 , 𝑄 𝑠 = πœ‡ 2 πœ‹ Schematic diagram of TESR

5 Stability/Synchrotron Tune
SR Mode (W/R) ERL Mode (W/R) SR Mode (WR) ERL Mode(WR) 𝑄 𝑠 (simulation) 𝑄 𝑠 (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

6 Stability and Synchrotron Tune
First we take a simple ring ( DBA1, CAV(acc), DBA2, CAV(dec)) We take 𝐸 𝐻 = 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 = MeV, 𝐸 𝐿 = 55.0 MeV Introduce artificial damping in HER (elegant simulation) Two energy real lattice length β‰ˆ m. Damped emittance …..simulation time. Decided to work with Matrix element in elegant simulation Periodic Solution and longitudinal stability

7 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 πœ™ 𝑠,π‘Ž depends on RF wavelength πœ† π‘Ÿπ‘“ tan βˆ’1 Δ𝑃 πœ‹ 𝐡 𝑀 𝑃 2 πœ† π‘Ÿπ‘“ < πœ™ 𝑠,π‘Ž < tan βˆ’1 βˆ’Ξ”π‘ƒ πœ‹ 𝐡 𝑀 𝑃 2 πœ† π‘Ÿπ‘“ B = 𝑃 2 𝑃 1 , A = 2(1 + B)/B, 𝑃 2 = 155 MeV, 𝑃 1 = 55 MeV

8 Longitudinal Stability ( SR Mode)
We calculate one turn transfer matrix and apply |2 cosπœ‡ | ≀ 2 πœ™ 𝑠,π‘Ž versus πœ† π‘Ÿπ‘“ plot for SR mode. Blue region (stable) 𝑓 0 = 476 MHz ( πœ† π‘Ÿπ‘“ = m),| 𝑀 56 | = 1.0 m πœ™ 𝑠,π‘Ž 1 = βˆ’ βˆ’ πœ™ 𝑠,π‘Ž 2 = βˆ’

9 Longitudinal Stability (ERL Mode)
πœ™ 𝑠,π‘Ž versus πœ† π‘Ÿπ‘“ plot for ERL mode. Blue region (stable) 𝑓 0 = 476 MHz, ( πœ† π‘Ÿπ‘“ = m) ,| 𝑀 56 | = 1.0 m, < πœ™ 𝑠,π‘Ž 1 < 90 0 90 0 < πœ™ 𝑠,π‘Ž 2 <

10 Periodic Solution ( 𝝓 𝒔,𝒂 =πŸ— πŸ“ 𝟎 , 𝝀 𝒓𝒇 =𝟎.πŸ”πŸ9816 m)
SR mode, Δ𝑧=1.82 mm, Δ𝑝 𝑝 =3.10Γ— 10 βˆ’3 𝛽 𝑠 = m, 𝛼 𝑠 = ( After decelerating cavity, cooler ring) ERL mode, Δ𝑧=1.68 mm, Δ𝑝 𝑝 =1.79Γ— 10 βˆ’3 𝛽 𝑠 = m, 𝛼 𝑠 = (After decelerating cavity, cooler ring) ERL mode SR mode

11 Periodic Solution / Chirping – De-Chirping
We define one loop transfer matrix, find out 𝛼 𝑠 and 𝛽 𝑠 at point A. 𝑀 π‘‘π‘œπ‘‘π‘Žπ‘™ = 𝑀 π‘‘π‘’βˆ’π‘β„Žπ‘–π‘Ÿπ‘π‘’π‘Ÿ 𝑀 𝑀 𝑑𝑒𝑐 𝑀 56 𝐻𝐸 𝑀 π‘Žπ‘π‘ 𝑀 𝑀 π‘β„Žπ‘–π‘Ÿπ‘π‘’π‘Ÿ

12 SR Mode, 𝝓 𝒔,𝒂 =𝟏𝟏 𝟐 𝟎 , Volt = 100.0 kV Δ𝑧= 0.73 mm
Δ𝑝 𝑝 =4.061Γ— 10 βˆ’3

13 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.

14 𝑴 πŸ“πŸ” and RF Acceleration, 𝝀 𝒓𝒇 = 0.6298 m fixed
Ref: IPAC2019

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

16 𝝈 𝒔 versus 𝑴 πŸ”πŸ“ 𝒂𝒄𝒄 (𝒕𝒐𝒕), πš«π“=𝟎.𝟎𝟎𝟎 𝟏 𝟎

17 𝝈 𝒔 versus 𝑴 πŸ”πŸ“ (𝒕𝒐𝒕), πš«π“=𝟎.𝟎𝟎𝟎 𝟏 𝟎

18 𝝈 𝒔 versus 𝝈 𝜹 πš«π“=𝟎.𝟎𝟎𝟎 𝟏 𝟎

19 𝜷 𝒔 versus πš«π“ Red = 15deg, BLUE = 30 deg, Green = 45 deg, Black = 60 deg, Yellow = 75 deg

20 ERL Mode, 𝝓 𝒔,𝒂 =πŸ” 𝟎 𝟎 , πš«π“=𝟎.𝟎𝟎𝟎 𝟏 𝟎 Δ𝑧 = m, Δ𝑝 𝑝 =4.64Γ— 10 βˆ’5

21 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

22 Thank You !


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