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

2. 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.

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

4. MEIC group meeting 19 March 2015.
A. Castilla, CASA/CAS-ODU
(old) Parameters
Parameter
Units
Electrons
Protons
Number of Particles --
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

5. 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.
and GeV.
Linear matrix elements, using Mathematica.
Ring = Standard transport matrix for a recursive system.
RING IR

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

7. 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

8. 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 ≅ tan 𝜃 𝐶 𝐹 tan 𝜃 𝐶 𝐹

9. 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

10. 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

11. 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

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

13. 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

14. 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

15. 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 ).

16. 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.

17. 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:

18. 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”.

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

20. 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:

21. 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

22. 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

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

24. 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

25. 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.

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

27. 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

28. 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

29. 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

30. 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

31. 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.

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