1. Beam-Beam Effects in the LHeC Edward Nissen 1LHeC Workshop June 26 2015

2. Contents Parameter sets and Software in use Single turn optimization possibilities Long term simulations of the proton beam Head-Tail effects and emittance growth Predicting the emittance growth rate Single turn effects on the Electron Beam Considerations for the correction system 2LHeC Workshop June 26 2015

3. Beam code and Parameters The code in use is Guinea Pig, a strong- strong code. 3 ValueNominal Parameters High Luminosity Particle number (ion/electron)10 17/0.322/0.2 β* ionmm10050 β* electronmm12032 x,y ionmm mr3.752 x,y electronmm mr50 Beam-Beam Tune Shift (i/e) 9.61x10 -5 /0.761.20x10 -5 /0.987 Beam Beam Disruption parameter (i/e) 3.62x10 -6 /5.999.05x10 -6 /29.1 LHeC Workshop June 26 2015

4. Waist Shift and Beta Function Optimization LHeC Workshop June 26 20154 β*~57mm, WS~30mm, Luminosity Increase~7.4% β*~42mm, WS~45mm, Luminosity Increase~17.8% β* (mm) Waist Shift (μm) NominalHigh Luminosity

5. Long Term Beam Evolution The beam beam interaction is modeled using Guinea-Pig This is combined with a linear 1-turn map of the LHC for the protons, and a new beam for the electrons. LHeC Workshop June 26 20155

6. Long term Beam Evolution LHeC Workshop June 26 20156 Turns Δε n (m) 3.66x10 -14 m/turn The separate measurements are simulated using 4 different initial condition random seeds, the offset jitter is 0.2σ x.

7. Head-Tail effects and Emittance growth LHeC Workshop June 26 20157

8. Predicting the emittance growth rate LHeC Workshop June 26 20158 The next step is to determine.

9. Predicting emittance growth rates In order to determine, we can first use Guinea Pig to calculate the sum directly LHeC Workshop June 26 20159 Turns Δε n (m) 3.66x10 -14 m/turn 5.28x10 -14 m/turn

10. Growth rate with Charge (LHeC) LHeC Workshop June 26 201510 Bunch population (x10 10 ) Δ/turn

11. Predicting Emittance Growth Rates In order to have separate confirmation of the growth rate it would be beneficial to begin from first principles. LHeC Workshop June 26 201511 Now we need to determine X(z)

12. Predicting Emittance Growth Rates The path through the proton beam can also be calculated directly using a numerical solver and the Basetti-Erskine Formula. Other approximations can be used to try to create an analytic formula for the growth rate. One ansatz tested is, LHeC Workshop June 26 201512

13. Path Through Proton Beam LHeC Workshop June 26 201513 Z (m) X (m)

14. Predicted Growth Rates LHeC Workshop June 26 201514 Turns Δε n (m) 3.66x10 -14 m/turn 5.28x10 -14 m/turn 3.08x10 -14 m/turn 3.81x10 -14 m/turn Rate predicted by Guinea-Pig Rate Predicted by Ansatz Average Simulation Rate Rate Predicted by Integration

15. Emittance Growth Rates and Correction Tolerances LHeC Workshop June 26 201515 Time Growth Rate (m/s)Doubling time (s) Ɛ 0 (m) Jitter tolerance Guinea-Pig1.46667E-08 x(σ jitter /σ x ) 2 864000.000003750.0543992 Numerically Integrated 8.55556E-09 x(σ jitter /σ x ) 2 864000.000003750.0712253 Simulations Average 1.01667E-08 x(σ jitter /σ x ) 2 864000.000003750.0653385 Ansatz1.05833E-08 x(σ jitter /σ x ) 2 864000.000003750.0640394

16. Spent Electron Beam (matching) LHeC Workshop June 26 201516 Nominal ParametersHigh Luminosity Parameters

17. Spent Electron Beam (matching cont’d) LHeC Workshop June 26 201517 Nominal ParametersHigh Luminosity Parameters

18. Spent Electron Beam LHeC Workshop June 26 2015 18 Nominal ParametersHigh Luminosity Red particles show the beam without beam-beam effects, green particles show the electron beam with beam-beam effects. Each frame is a single interaction at an increasing offset. X (μm) X’ (μr)

19. Spent Electron Beam Emittance LHeC Workshop June 26 201519 Offset (σ x ) ɛ x / ɛ x0 Relative to center of beam Relative to center of pipe

20. Electron Linac (kicker correction system) Assuming a circular arc, with a depth of 56 m Using numbers from FONT1 latency is 65ns and voltage is 350 volts LHeC Workshop June 26 201520 The 1km radius is sufficient to allow for feedback within the same turn in each arc by cutting across n kicks Δt (nanoseconds)Δs (meters) 13434.3514191029.592654 2148.712094844.58276442

21. Electron Linac (kicker parameters) Voltage350 kicks2 energymax kick μradbeta geometric emittance (nm)σpx (μr) max offset σ 10.51.33331002.433214.93280.540603 20.50.68291001.246313.53030.386893 30.50.45902583.76915.78860.158594 40.50.34575063.08563.55210.194636 50.50.27725050.59353.18100.174303 600.23335042.58302.91830.159909 LHeC Workshop June 26 201521 Using one set of kicks at the opposite side, a 1σ offset can be corrected with a voltage of ~4400V

22. Electron Linac (tune shift correction) LHeC Workshop June 26 201522 Add Symmetric triplet cell Correcting for the beam-beam induced tune shift could be accomplished with a pole tip voltage of less that 1.5 kV for a 1σ offset. Difference at 1sigma is.011

23. Conclusions Maximizing the luminosity of the collisions can be done by shifting the beta function and the waist shift. Improvements have been made in the ability to predict the growth rate of the emittance of the proton beam from the beam beam effect. The effects on the spent electron beam can drive design requirements for the correction system in the electron linac. LHeC Workshop June 26 201523

24. Thank You for Your Attention. LHeC Workshop June 26 201524