1. Aaron Farricker 107/07/2014Aaron Farricker Beam Dynamics in the ESS Linac Under the Influence of Monopole and Dipole HOMs

2. Outline The European Spallation Source (ESS) Example of an ESS Cavity Beam Dynamics Code including Beam-Higher Order Mode(HOM) interactions Longitudinal Emittance Dilution Transverse Emittance Dilution + Alignment Tollerances 07/07/2014Aaron Farricker2

3. European Spallation Source ESS is a collaboration of more than 17 countries Experiment is based in Lund Sweden At the forefront of neutron flux Should be operational around 2019 ESS is going to be green, all power used by the machine during operation will be put back into the power grid using renewable energy sources on site 307/07/2014Aaron Farricker

4. ESS Linac Beam power (MW)5 Beam current (mA)62.5 Linac energy (GeV)2 Beam pulse length (ms)2.86 Repetition rate (Hz)14 Num. of CMs Num. of cavities Spoke1326 Medium  (6-cell) 936 High  5-cell) 2184 Low energy section of the Linac is normal conducting taking the beam to 90 MeV Superconducting Linac takes the beam to the final energy of 2 GeV Three families of SC cavities working across a range of velocities 4 At 352.2 MHz bunch spacing means 1 Million bunches 07/07/2014Aaron Farricker

5. Example Cavity- ESS-like High Beta Cavity 07/07/2014Aaron Farricker5 ParameterValueUnits Number of Cells5 Frequency704.42MHz Cavity Length1.315m Accelerating Gradient 18MV/m R/Q472Ω Iris Diameter120mm a/λ0.141 ESS High beta cavity operating in the pi mode Full Five Cell Structure in HFSS

6. Dispersion Curves For High Beta Cavity 6 Single Cell Results as Red Circles Full Cavity as Blue triangles Circuit model fitted in purple Machine lines in orange Light line as a dashed line (beta=0.86) Light line given by: 07/07/2014Aaron Farricker

7. R/Q For Monopole And Dipole Bands For the plot on the left note the high R/Q of the fundamental and also the high R/Q mode in the second pass band (blue) For the dipoles (right) the maximum R/Q is about 75 ohms It is important to remember that the velocity of the beam is changing so these values change on a cavity by cavity basis

8. Variation in R/Q 8 As the proton beam is non-relativistic its velocity is changing on a cavity by cavity basis This means that its experiencing a voltage from HOMs in each cavity This variation can result in a mode that is not synchronous with the beam at the design beta becoming synchronous 07/07/2014Aaron Farricker

9. Drift-Kick-Drift Model Monopoles and RF errors result in a difference in energy This difference in energy results in a time arrival error at the next cavity which is the same as a change in phase 07/07/20149Aaron Farricker From RF ErrorsFrom Monopole Modes

10. Monopole Interactions 10 Each bunch induces a voltage in every mode It also gets acted on by the voltage already in each of the modes Real and Imaginary parts of the HOM voltage Fundamental theorem of beam loading 07/07/2014Aaron Farricker Injection Pattern

11. 07/07/2014Aaron Farricker11 Voltage and Phase error from the klystron Synchronous particle energy gain Time arrival error RF Errors Requirements on RF errors at ESS have been reduced significantly from 1 degree in phase and 1% in amplitude to from 0.1 degree in phase and 0.1% in amplitude. This has a significant effect on the growth caused Old RF errors New RF Errors

12. Dipole Interactions 07/07/2014Aaron Farricker12 The voltage induced in a dipole mode by an off axis bunch is given by: Where the transverse R/Q and Voltage are defined through the Panofsky-Wenzel theorem as: These transverse voltages result in a transverse kick to subsequent bunches which results in a change in x’ given by (for small kicks):

13. Errors in Mode Frequencies 13 Due to the limitations in constructing SC cavities a spread in the frequencies of HOMs from cavity to cavity are expected Studies were carried out by R.Sundalin and his empirical findings were confirmed at SNS (http://eval.esss.lu.se/DocDB/0000/000092/002/summary.pdf) Therefore we expect a similar spread in frequencies at ESS For modes in the higher pass bands Including dipoles 07/07/2014Aaron Farricker For modes in the first pass band These significant frequency deviations are result of the manufacturing limitations fro SC Cavities

14. Effects of SOM’s 14 90mA 75mA Design-62.5mA RF Errors The emittance dilution at larger Q exceeds the effect of the RF by a significant margin for all of the currents Q for these modes must be kept around 10 6 so that they aren’t the dominant source of dilution It has been indicated that the Q will be at this level however more simulation will be required to confirm this 62.5mA is the baseline design and the other curves represent potential power upgrades 07/07/2014Aaron Farricker With SOMs

15. Modes Near Machine Lines These two modes that lie near machine lines (orange line on the dispersion curve) are used in the next simulations Although they have small R/Q’s they will be resonantly excited

16. Variation of Emittance Dilution With Cavity Position 07/07/2014Aaron Farricker16 A single modal component is included in ONE cavity along the linac Its Frequency is shifted to lie exactly on the nearest machine line such that it is resonantly excited This is the case where the maximum dilution in emittance would be seen and the absolute worst case for ESS Q=10 6

17. Variation of Emittance Dilution With Cavity Position 07/07/2014Aaron Farricker17 A single modal component is included in ONE cavity along the linac Its Frequency is shifted to lie exactly on the nearest machine line such that it is resonantly excited This is the case where the maximum dilution in emittance would be seen and the absolute worst case for ESS Q=10 6

18. Variation of Emittance Dilution With Cavity Position 07/07/2014Aaron Farricker18 There is a significant reduction in the growth when the Q is reduced from 10 8 to 10 6. The shape of the growth approximately follows that of the R/Q with some small deviations These deviations require further investigation but will most likely be due to the shape of the phase space at that cavity. R/Q Q=10 6 Q=10 8

19. Transverse Plane The five highest R/Q dipoles modes were used The emittance dilution was found to be very small at various currents Modes near machine lines have no effect as bunches arrive at a minima and low R/Q means they have very low voltages induced 90mA 75mA Design- 62.5mA 07/07/201419Aaron Farricker

20. Transverse Sum Wakefield 07/07/2014Aaron Farricker20 Due to the large amount of computational time required we only plot the transverse wakefield for 50,000 bunches by applying the Condon method It is therefore no surprise that we see little effect from dipole modes

21. Alignment Tolerances 21 In both cases uniformly distributed errors are used At possible and reasonable alignment errors there is little effect from dipole modes The plot is extended to show the quadratic behaviour expected 07/07/2014Aaron Farricker Transverse kicks from the fundamental (bottom left) due to misalignments do have a significant effect and are the limit on alignment tolerances Alignment tolerances for the machine are set to +/- 0.5 mm

22. Summary Beam-HOM interactions can be tracked using a numerical code Modes in the first passband are a concern as they have an effect comparable with the RF errors HOMs are of little concern in both the longitudinal and transverse planes unless a monopole mode is near a machine line The alignment requirements arise mainly as a result of transverse kicks from cavities at an angle to the beam axis 07/07/2014Aaron Farricker22