1Roger M. Jones (Computational Seminar, Dept. Physics and Astronomy, University of Manchester, April 19th 2007)

Beam Break Up Instability (BBU)

Roger M. Jones

19th April, 2007.

The University of Manchester, UK/Cockcroft Institute, Daresbury, UK.

Stanford Linear Accelerator, shown in an aerial digital image. The two roads seen near the accelerator are California Interstate 280 (to the East) and Sand Hill Road (along the Northwest). Image data acquired 2004-02-27 by the United States Geological Survey

2Roger M. Jones (Computational Seminar, Dept. Physics and Astronomy, University of Manchester, April 19th 2007)

Beam Break Up InstabilityBeam Break Up Instability

Four transverse beam profiles observed at the end of the SLAC linac are shown when the beam was carefully injected and injected with 0.2, 0.5, and 1 mm offsets. The beam sizes x and Y are approximately 120 m. (Courtesy John Seeman, 1991.)

3Roger M. Jones (Computational Seminar, Dept. Physics and Astronomy, University of Manchester, April 19th 2007)

22 n 0

12z

0

Particles are driven according to:

Nrdy(s, z) k y dz ' (z ')W (z z ')y(s, z '); N is the number of

Lds

particles in each bunch, L is the cavity length, r classical radius of electron

We solve with a

n

n=0

0

2n 2 n n 10

12z

series solution: y(s,z)= y (s, z)

ˆand a leading term: y (s, z) y cosk s z

The equation describing transverse motion becomes:

Nrdy (s, z) k y dz ' (z ')W (z z ')y (s, z ')

Lds

The solution is expr

sn n 10

10z

essed as a Green's function:

1G(s,s')= sink (s s')

k

Nry (s, z) ds'G(s,s ') dz ' (z ')W (z z ')y (s', z ')

L

Beam Break Up InstabilityBeam Break Up Instability

4Roger M. Jones (Computational Seminar, Dept. Physics and Astronomy, University of Manchester, April 19th 2007)

0 1

01

z

For weak beams it is only necessary to keep terms first order

in beam intensity:

y(s,z) y (s, z) y (s, z)

Nry cosk s ssink s dz ' (z ')W (z z ')

2k L

Sequence of snapshots of a beam undergoing dipole beam breakup instability in a linac. Values of ks indicated are modulo 2. The dashed curves indicate the trajectory of the bunch head.

Beam Break Up InstabilityBeam Break Up Instability

5Roger M. Jones (Computational Seminar, Dept. Physics and Astronomy, University of Manchester, April 19th 2007)

Beam Break Up InstabilityBeam Break Up Instability

For high intensity beams the betatron displacement of the beam tail

exponentiates with respect to the beam intensity and in this case

higher terms must be retained in the power series expansion. In pra

0

0 0

n

ik Ln 0 00 n

n 1 1 1 1 2 2 1 1

ctise

k L 1 (L is the length of the entire linac) and at the end of the linac:

ˆ iNr Lyy (L ,z) R (z)e

n! 2k L

where the real part is understood and:

R (z) dz (z )W (z z ) dz (z )W (z z

1

n 1

2zz

2 n 1 n 1 nz

0

)

....... dz (z )W (z z )

with R (z) 1.

6Roger M. Jones (Computational Seminar, Dept. Physics and Astronomy, University of Manchester, April 19th 2007)

1 0

n

Consider a special case (and quite realistic) in which the bunch is uniform

in length l and the wake function is linear in z:

1/l z l / 2(z)=

0 z l / 2

W (z) W z / l 0 z -l

1R (z)

(

0

n2

nik L

00

1 zN z l / 2

2n)! 2 l

and the solution for the transverse offset of the beam is given by:

1ˆy(L , z) ye

n!(2n)! 2i

where the dimensionless strength parameters is given by:

2

0 0 0Nr L W 1 z

k L 2 l

Can we derive an asymptotic expression for this transverse offset

of the beam?

Beam Break Up InstabilityBeam Break Up Instability