Alejandro Castilla CASA/CAS-ODU acast020@odu.edu CRABBING DYNAMICS IN THE MEIC, a 1ST look. Alejandro Castilla CASA/CAS-ODU acast020@odu.edu

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Outline Distributions and 1st Order Calculations. Simplification of the Interaction Region. FFB and Phase Advance (1st Order). Synchro-Betatron Coupling. Transverse Coupling. Particle Tracking and Non-Linear Elements.

The Interaction Region MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU The Interaction Region *MEIC Design Summary (2015).

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU (old) Parameters Parameter Units Electrons Protons Number of Particles -- 10 −4 - 10 −5 Energy GeV 5 60 𝛽 ∗ 𝑥 cm 4 𝛽 ∗ 𝑦 0.8 8 𝛾 9.78 64.95 𝜖 𝑁,𝑥 μm rad 54 0.35 𝜖 𝑁,𝑦 11 0.07 Bunch length 0.75 1 Energy Spread 10 −4 7.1 3

Dissecting the Problem MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Dissecting the Problem Normal distributions with 𝟏𝟎 𝟒 − 𝟏𝟎 𝟓 particles. Electrons @5GeV and protons @60 GeV. Linear matrix elements, using Mathematica. Ring = Standard transport matrix for a recursive system. RING IR

Distributions & Calculations MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Distributions & Calculations Normal Distributions with 𝟏𝟎 𝟒 − 𝟏𝟎 𝟓 particles. Electrons @5GeV and Protons @60 GeV. Linear Matrix Elements, using Mathematica. Ring = Standard Periodic Transport Matrix.

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Simplified IR layout Simplifying for both electrons and protons a symmetric IR with respect to the IP *MEIC Design Summary (2015). IP Drift = 7m FFB

Linear Crabbing Matrix MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Linear Crabbing Matrix Mixing of x’ with z and z’ with x. F = 7 m, 𝜽 𝑪 =𝟓𝟎 𝒎𝒓𝒂𝒅. 𝑀 𝐶𝑟𝑎𝑏 ≅ 1 0 0 0 1 0 0 0 1 0 0 0 0 tan 𝜃 𝐶 2 𝐹 0 0 0 0 0 0 0 0 0 0 tan 𝜃 𝐶 2 𝐹 0 0 1 0 0 0 1 0 0 0 1

Linear Crabbing Matrix (2) Instantaneous change on x’ not x at the crab. Exchange of x’↔ x throughout the drift. IP MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

MEIC group meeting 19 March 2015. 1000 Passes (protons) IP Drift = 7m FFB w/o crab w crab MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU FFB and Phase Advance All cases in the literature assume ∆ 𝝍 𝒙𝟏,𝟐 =𝝅, then in the transfer matrix from the 1st to 2nd crab 𝒎 𝟏𝟐 =𝟎. But, comparing the transfer matrices: One obtained from direct matrix multiplication (RHS).. Other using the Courant Snyder parameterization (LHS). So 𝐬𝐢𝐧 ∆ 𝝍 𝒙𝟏,𝟐 =𝟎, implies 𝜷 𝑪𝟏 𝜷 𝑪𝟐 →∞, where 2D denotes the distance from one crab to the other. 𝒎 𝟏𝟐 = 𝜷 𝑪𝟏 𝜷 𝑪𝟐 𝐬𝐢𝐧 ∆ 𝝍 𝒙𝟏,𝟐 =𝟐𝑫

Synchro-Betatron Coupling IP Drift = 7m FFB 2*Betatron Fractional Tune MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

Synchro-Betatron Coupling (2) IP Drift = 7m FFB No tune mixing on the vertical plane. Small oscillations of the crabbing angle 𝜽 𝑪 𝟐 ~𝟐𝟓 𝒎𝒓𝒂𝒅. MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

Synchro-Betatron Coupling (3) IP Drift = 7m FFB Synchrotron fractional tune present in the xz-correlation due to crabbing. MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Transverse Coupling Solenoids between the crabs and the IP will cause vertical and horizontal coupling, this will have a repercussion on the crabbing angle. IP Reverse engineer the problem by determining the total coupling at the IP due to the solenoid strength ( 𝑩 𝑺𝒐𝒍 ).

Transverse Coupling (2) MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Transverse Coupling (2) As expected 𝜶( 𝑩 𝑺𝒐𝒍 )=𝜷( 𝑩 𝑺𝒐𝒍 )=𝑲𝑳, where L is the solenoid length, 𝑲≡ 𝒒 𝑩 𝑺𝒐𝒍 𝟐𝑷 , with q the particle charge, and P its momentum. We address then the equivalent problem: a transversely “tilted” bunch at the crab location.

Transverse Coupling (3) MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Transverse Coupling (3) With a solenoid length of 𝑳 𝒆 − =𝟑 𝒎 and 𝑳 𝒑 =𝟐 𝒎, we get the range of coupling:

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Twin Crabs The simplest solution is to rotate the crab by the proper 𝑲𝑳 angle, for a fixed value of 𝑩 𝑺𝒐𝒍 . If the solenoid strength needs to be cover a range, a solution is a superposition of kicks: “twin crabs”.

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Twin Crabs (2)

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Twin Crabs (3) If not compensated, the crabbing correction “leaks” to the transverse plane. Using the twin crabs:

Optimizing Voltage and Errors MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Optimizing Voltage and Errors Higher angle range = more voltage per cavity. The residual error in the angle due to the solenoid extra focusing, can be compensated with the FFB strength

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Particle Tracking We find the ring (no IR) linear transfer matrix with MADX. Describing the IR with standard linear elements in ELEGANT and the ring as the zero-length transfer matrix. The crabs as RF multipole at zero-crossing, (i.e. MRFDF with 𝝋=𝟐𝟕𝟎°). RING IR

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU ELEGANT MRFDF *ANL-ELEGANT USER MANUAL

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU ELEGANT MRFDF (2) The (2*i)th-pole component 𝒃 𝒊 is defined as: ∆ 𝒑 𝒙 = 𝒆 𝒎 𝒄 𝟐 𝒊=𝟏 𝟓 𝒃 𝒊 𝒌 𝒊 𝒙 𝒊−𝟏 𝐜𝐨𝐬 𝝋 𝒊 where ∆𝒙′= ∆ 𝒑 𝒙 𝒑 𝒛 , 𝒑 𝒙/𝒛 = 𝜷 𝒙/𝒛 𝜸 , and 𝒌 𝒊 = 𝝎 𝒊 𝜷𝒄

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU IR Layout (protons) The crabs are the only non-linear elements in this model.

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Tracking w/o Crabbing From the linear model. IP 1 Turn 1000 Turns

MEIC group meeting 19 March 2015. Tracking w Crabbing From the linear model + pure dipole kick (crab). MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

MEIC group meeting 19 March 2015. Tracking w Crabbing (2) 1st half of the IR, crabbing. IP MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

MEIC group meeting 19 March 2015. Tracking w Crabbing (3) 2nd half of the IR decrabbing. IP MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU

Tracking w Crabbing (next) MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Tracking w Crabbing (next) We have right now a problem of timing, since the ring transfer matrix is a zero length element, the phase at the 1st crab cavity is of phase after the first pass. Bunch of center due to spurious phase

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU References A. Castilla, et al, Modeling Crabbing Dynamics in an Electron-Ion Collider, to be presented in IPAC2015. A. Castilla, et al, Multipole Budget of Crab Cavities for an Electron-Ion Collider, to be presented in IPAC2015. A. Castilla, et al, Employing Twin Crabbing Cavities to Address Variable Transverse Coupling of Beams in the MEIC, in Proceedings for IPAC2014. A. Castilla and J. Delayen, Multipole and Field Uniformity Tailoring of a 750 MHz RF Dipole, in Proc. for LINAC2014. M. Borland, ANL-ELEGANT Users Manual.

MEIC group meeting 19 March 2015. A. Castilla, CASA/CAS-ODU Extras