Aaron Farricker 1

Aaron Farricker

07/07/2014

Beam Dynamics in the ESS Linac Under the Influence of Monopole and Dipole HOMs

Aaron Farricker 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

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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

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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

Spoke 13 26Medium b (6-cell) 9 36High (b 5-cell) 21 84

•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

At 352.2 MHz bunch spacing means 1 Million bunches

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Example Cavity- ESS-like High Beta Cavity

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Parameter Value Units

Number of Cells 5

Frequency 704.42 MHz

Cavity Length 1.315 m

Accelerating Gradient

18 MV/m

R/Q 472 Ω

Iris Diameter 120 mm

a/λ 0.141

ESS High beta cavity operating in the pi mode

Full Five Cell Structure in HFSS

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Dispersion Curves For High Beta Cavity

•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:

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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

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Variation in R/Q

•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

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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

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From RF Errors From Monopole Modes

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Monopole Interactions•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

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Injection Pattern

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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

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Dipole Interactions

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•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):

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Errors in Mode Frequencies•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 bandsIncluding dipoles

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For modes in the first pass band

•These significant frequency deviations are result of the manufacturing limitations fro SC Cavities

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Effects of SOM’s

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 106 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

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With SOMs

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

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Variation of Emittance Dilution With Cavity Position

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•There is a significant reduction in the growth when the Q is reduced from 108 to 106.•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

R/Q

Q=106

Q=106

Q=108

Q=108

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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

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Transverse Sum Wakefield

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•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

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Alignment Tolerances•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

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•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

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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

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