Beam breakup and emittance growth in CLIC drive beam TW buncher

Hamed ShakerSchool of Particles and Accelerators, IPM

Drive Beam Front-End layout

Gun SHB 1-2-3 PB TW Buncher Acc. Structures

Solid state amplifier, 500 MHz

Modulator-klystrons, 1 GHz, 20 MW

~ 140 keV ~ 12.2 MeV

Diagnostics

~ 2.4 MeV ~ 4.6 MeV ~ 8.4 MeV

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180.6

0.65

0.7

0.75

0.80.85

0.9

0.95

11.05

Phase velocity/c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180

1

2

3

4

5

6

Axial electric field

E0T

(MV

/m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18-50

0

50

100

150

200

250

300

350

400

Energy gain (KeV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180.75

0.8

0.85

0.9

0.95

1

bunch form factor

3.Travelling wave tapered buncher 11

3.Travelling wave tapered buncher3.1 Longitudinal dynamics in TW buncher

𝜃𝑚𝑎𝑥∝[ 1

(𝛾 2−1 )3 /2𝐸𝑧

]1 /4

{𝑑𝛾𝑑𝑧=− 𝑒𝑚𝑐2 𝐸𝑧 sin 𝜃

𝑑𝜃𝑑 𝑧

=𝜔𝑐 ( 1

𝛽 h𝑝

−1𝛽 )

Borrowed from Shahin

Phase velocity correction based on the fundamental mode reactive beam loading

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170.6

0.7

0.8

0.9

1

1.1Phase velocity/c

before correction

after correction

{1𝑣𝑒

−1𝑣𝑝

1𝑣𝑔−

1𝑣𝑒

=−1

2𝜋𝐸𝑏0 sin (𝜑𝑏 )

𝐸𝑔

𝐸𝑏0=−𝐹𝜔0

2𝑟𝑄0

𝑞≈−𝜋 𝐹𝑟𝑄0

𝐼

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

0 20 40 60 80 100 120 140 160 180 2001200000000

1250000000

1300000000

1350000000

1400000000

1450000000

1500000000

1550000000

1600000000

Dispersion diagram for the first dipole mode

Phase Advance (degree)

freq

uenc

y (H

z)

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

∆ �⃗� 𝑡=𝑖𝑞𝜔∫

0

𝐿

𝛻𝑡 �⃗�𝑧𝑑𝑧𝑟≪𝑎⇒

∆ 𝑃 𝑡≈𝑞𝜔𝑉𝑚𝑎𝑥

𝑟si n(¿𝜔𝑡)¿

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 Vmax is the maximum efficient voltage of the mode at offset r.

When a bunch travels off-crest, it can excited this dipole mode because the off-axis longitudinal electric field is non-zero and the induced mode decelerate the bunch to catch a part of its kinetic energy and convert it to the electromagnetic energy in the cavity. If we want to calculate the momentum kick resulted from the induced mode on just one bunch with charge q that cross the cell at t=0, at offset r, we can use the fundamental beam loading to relate the Vmax to the cell properties. We suppose it has the same charge and offset of the test bunch. is independent of r for r<<a and is just dependent on cell shape and its related mode frequency.

{ ¿𝑉𝑚𝑎𝑥=2𝑘𝑞⟹∆ 𝑃 𝑡≈2𝑞2

𝜔𝑘𝑟

sin (𝜔𝑡 )⟹

∆ 𝑃 𝑡

𝑚=

2𝑒𝑚𝑒

𝑞𝛾𝜔

𝑘𝑟

sin (𝜔𝑡 )=2𝑒𝑚𝑒

𝑞𝛾𝜔

𝑘𝑟 2 𝑟 sin (𝜔𝑡 )

¿𝑘=𝑉𝑚𝑎𝑥

2

4𝑈=𝜔

4𝑅𝑄

¿𝑞=𝑁𝑒 ,𝑚=𝑁𝛾𝑚𝑒

Calculating the momentum kick - 2

∆ 𝑣≈∆ 𝑃 𝑡

𝑚≈

2𝑒𝑚𝑒

𝑞𝐹𝛾𝜔

𝑘𝑟2 𝑟 sin (𝜔𝑡 )≈0.352

𝑞 [𝑛𝐶 ]𝐹𝛾𝜔 ( 𝑘𝑟 2 )[ 𝑉

𝑝𝐶𝑚2 ]𝑟 [𝑚𝑚]sin (𝜔𝑡 )

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)

fd(Ghz) phase velocity/c L(mm) Bunch Form Factor

Q0 Imag(Vmax/V0)

Maximum Transverse Velocity change (m/s)

2 14.9 1.2437 0.616 61.172 0.773 16116 -0.04 40923 16.9 1.2521 0.634 62.487 0.848 16500 0.02 49674 20.7 1.2614 0.662 64.516 0.837 16884 0.08 57625 25.8 1.2720 0.689 67.262 0.825 17268 0.14 67896 30.1 1.2867 0.716 70.7 0.836 17652 0.24 76497 37.4 1.2982 0.742 74.628 0.867 18035 0.32 93868 44.6 1.3136 0.771 78.797 0.904 18419 0.43 109419 53.9 1.3314 0.805 82.921 0.933 18803 0.57 12544

10 64.7 1.3519 0.841 86.895 0.948 19437 0.75 1375011 77.4 1.3745 0.874 90.414 0.951 20070 1.00 1458512 91.2 1.3977 0.903 93.398 0.948 20704 1.35 1488013 105.6 1.4216 0.926 95.849 0.944 21337 1.88 1483214 118.9 1.4446 0.944 97.852 0.943 21971 2.80 1434415 132.1 1.4677 0.957 99.655 0.945 22604 4.97 1382316 143.3 1.4903 0.968 101.812 0.948 23238 17.78 1282317 151.0 1.5119 0.976 104.719 0.951 23871 -12.53 11593

Accumulation of momentum kicks in a bunch train

V0

Vmax

δ

Vlim¿

V0 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 Vmax and damps gently to Vlim because of power loss on the surface. fd is the resonant frequency of first dipole mode as mentioned in the table and f0=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 Q0 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

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

2

4

6

8

10

12

bunch number

Fina

l Tra

nver

se v

eloc

ity (m

/s)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.4

2.4

4.4

6.4

8.4

10.4

12.4

bunch number

r (m

m)

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

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

-800000

-600000

-400000

-200000

0

200000

400000

600000

Chart Title

bunch number

Fina

l Tra

nsve

rse

velo

city

(m/s

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Chart Title

bunch numberr (

mm

)

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 2

2

0 30

4

rr r r RF r

r r

Kx x k x k x

x x

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 -0.27 and -0.49 mrad, respectively. Then by calculating the new emittance, we found the emittance change is less than -0.009mm.mrad (-0.04%) that is completely negligible.

{¿𝜖𝑟𝑚𝑠2= ⟨𝑥2 ⟩ ⟨𝑝2 ⟩− ⟨𝑥𝑝 ⟩ 2

¿𝑝=𝑑𝑥𝑑𝑠

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/4xr)/Σ, the dashed line is same for the emittance term, (εr

2/xr3)/Σ, and the dotted one for

the RF defocusing term, (kRFxr)/Σ. Σ is equal to kRFxr + K/4xr + εr2/xr

3.

Thanks for your attention