Beam breakup and emittance growth in CLIC drive beam TW buncher Hamed Shaker School of Particles and Accelerators, IPM

Drive Beam Front-End layout TW buncher consists of 18 cells including coupler cells. These cells are designed to have the same resonant frequency for the fundamental mode (First monopole mode) at the phase advance of 120 degrees. But because of non-equal cell lengths, beam aperture radius and cell radius, the other modes (monopoles, dipoles, quadrupoles and …) have not the same resonant frequency for the synchronous case – when the phase velocity equals to the beam average velocity- and briefly we say they are detuned.

Cells properties from the beam dynamic study done by Shahin

3.Travelling wave tapered buncher 11 3.Travelling wave tapered buncher 3.1 Longitudinal dynamics in TW buncher Borrowed from Shahin

Phase velocity correction based on the fundamental mode reactive beam loading A correction is required based on the reactive beam loading of the fundamental mode. This effect is emerged because of the off-crest beam bunch traveling the its high current (5A)*. * General Beam Loading Compensation in a Traveling Wave Structure”, H. Shaker et al., IPAC13

First dipole mode For the dipole modes, the first one is dominated and the other dipole modes loss factors are at least two orders less in magnitude. The known effect of beam breakup and transverse emittance growth are due to this mode. This figure shows the first dipole mode dispersion diagram of all cells except coupler cells if we assume we have a periodic structure from each cell and the cross line represents the synchronous modes.

Beam Breakup types Regenerative beam breakup: The separation between different cell modes shows that the induced modes in each cell are stayed mostly in the same cell and don’t propagate in neighbour cells. An important conclusion comes from this result is that we have not the regenerative beam breakup which the induced modes propagate back to add up kicking. Multi structure beam breakup: Multi structure beam breakup is seen in structures that has the order of few hundred meters length and the resonant frequency of first dipole mode is very close to 1.5 times of the fundamental mode and is very close to the bunch harmonic frequency. Then for our structure with about 1.5m length this effect should be so small. Also we have solenoids around the structure and it help us to suppress a small effect of this deflecting.

Calculating the momentum kick The momentum kick is calculated by Panofsky-Wenzel and is proportional to radial gradient of longitudinal electric field with 90 degrees out of the phase. This equation is valid if the relative velocity change during each cell is so small. The last form in above equation shows the momentum kick close to the axis. q is the bunch charge, ω is the angular frequency of the mode, r is the offset, a is the cell beam aperture radius and V max is the maximum efficient voltage of the mode at offset r.

Calculating the momentum kick - 2 There is just one correction by parameters F that is bunch form factor. This correction is showed how far the bunch from the point-like charge is and for our case is close to 1. This equation shows the transverse velocity change and also told us a test bunch just behind a leading bunch experiences a positive deflection because of its induced dipole mode and it becomes maximum when the phase difference is 90 degrees and the leading bunch itself, don’t experience a kick from the induced mode by itself. Cell Number loss factor/r^2 (V/pC/m^2) f d (Ghz)phase velocity/cL(mm) Bunch Form Factor Q0 Imag (V max /V 0 ) Maximum Transverse Velocity change (m/s)

Accumulation of momentum kicks in a bunch train V0V0V0V0 V max δ V lim V 0 is the induced longitudinal voltage by one bunch and is a representative phasor that its imaginary part is proportional to momentum kick. The accumulation started to oscillates between 0 and V max and damps gently to V lim because of power loss on the surface. f d is the resonant frequency of first dipole mode as mentioned in the table and f 0 =999.5 MHz is the frequency of fundamental mode. This equation shows an interesting result that the final kick is half of the maximum kick when Q 0 is large enough and the limiting phasor points to the centre of the circle build from the phasors.

Kick tracking in the bunch train - 1 We assume the initial offset equal to 1mm and the initial transverse velocity equal to zero and we assume there is no solenoid. These two figures shows the final transverse velocity and offset at the end of the structure by time.

Kick tracking in the bunch train - 2 For our case because during a pulse with 140µs length, we have sub-pulses with about 240ns length and 180 degrees phase difference with their neighbours, the steady state doesn’t happen but still the total momentum kick is so small and as the result, no BBU will be happened.

Emittance Growth By looking to the envelope equation terms we found the dipole mode kick is at least one order less than the fundamental mode kicking and also the fundamental mode kicking is so smaller than the space-charge deflecting then the radius could be kept constant by solenoids. To find the emittance growth, we should mention for a zero-length bunch the emittance growth is zero because all bunch kicks together equally. For a finite-length bunch, each part of bunch kicks differently based on its related longitudinal position. The rms emittance is 24.8 mm.mrad at the entrance of the TW buncher. The mentioned tracking code shows the head, tail momentum change is and mrad, respectively. Then by calculating the new emittance, we found the emittance change is less than mm.mrad (-0.04%) that is completely negligible. This figure shows the relative contribution of each defocusing term in the envelope equation for a target beam size of 2 mm. The solid line is the relative contribution of the space-charge term, (K/4x r )/Σ, the dashed line is same for the emittance term, (ε r 2 /x r 3 )/Σ, and the dotted one for the RF defocusing term, (k RF x r )/Σ. Σ is equal to k RF x r + K/4x r + ε r 2 /x r 3.

Thanks for your attention