1. Limit of bunch length due to coherent synchrotron radiation in SuperKEKB
T. Agoh (KEK)
Introduction
CSR emitted in wiggler
Analysis of longitudinal microwave instability using Fokker-Planck model
Conclusion
Joint Meeting of Pacific Region Particle Physics Communities, Oct.29-Nov.4, Hawaii

2. KEKB factory (e+e- storage ring collider)
CSR in SuperKEKB
KEKB factory (e+e- storage ring collider)
KEKB LER
SuperKEKB LER
Bunch length
6 mm
3 mm
Bunch current (charge)
1.2 mA (~12 nC)
1.9 mA (~19 nC)
Collision point

3. Short bunch and high current
Coherent Synchrotron Radiation (CSR)
High frequency
Incoherent
Low frequency
Coherent
In storage rings – Bunch lengthening, Microwave instability, CSR burst

4. CSR effect is 14 times larger
CSR in SuperKEKB LER
Energy change due to CSR (Longitudinal wakefield for a single bend)
smaller chamber
Small chambers suppress CSR.
KEKB  SuperKEKB
CSR effect is 14 times larger
We will make new vacuum chamber to suppress electron cloud effect.

5. CSR in storage rings Bunch length = order of millimeter to cm
Strong shielding due to beam pipe
Boundary condition must be properly considered for field calculation.
Transient state of the field due to finite magnet length
Time evolution of EM field should be considered.
Variation of bunch distribution due to wakefield
Potential well distortion, Bunch lengthening,
or Non-equilibrium state if unstable

6. Mesh calculation of EM field (E,B) in a beam pipe
Calculation procedure
Mesh calculation of EM field (E,B) in a beam pipe
Begin with Maxwell equations (E, B) in accelerator coordinates (x,y,z;s)
( We do not handle the retarded potential (A,Φ).)
Time domain :
(2) Fourier transform EM field w.r.t z
Frequency domain :
(3) Approximate these equations
Paraxial approximation
(5) Inverse Fourier transform
Back to the time domain
(4) Solve them by finite difference
Beam pipe = boundary condition

7. From Maxwell equations,
(1)
Fourier transform of eq.(1)
Assuming that s-dependence of the field is weak, neglect the term of 2nd derivative with respect to s:
Equation to describe CSR field
Equation of evolution (parabolic equation)

8. Equation of evolution Usually, mesh size must be in EM field analysis.
Our method ignores 2nd derivative,
backward waves are neglected.
The field consists of only forward waves.
We can factor the plane waves out of the EM field through Fourier transform.
We handle only which slowly changes along the beam line.
The term of 1st derivative w.r.t. s describes the evolution of the field.
Mesh size can be larger than the actual field wavelength.
T.Agoh, K.Yokoya, PRST-AB,7, (2004)

9. Comparison with analytic theories
steady CSR in free space
transient CSR in free space
shielded CSR between parallel plates

10. Impedance of CSR & Resistive wall
Longitudinal impedance in a copper pipe
(10cm square, R=10m, Lmag=1m)
Real part
Imaginary part
Low frequency limit ⇒ Resistive wall impedance :
High frequency limit ⇒ Steady CSR in free space :

11. CSR in the drift space
CSR goes out a bend and propagates in the drift space, where particles are still affected with CSR.
at exit of bend
at 9m from exit
at 3m from exit
1. Longitudinal delay because of reflection 2. Sinusoidal behavior as it propagates
Real part
Imaginary part

12. Longitudinal wakefields in SuperKEKB LER
Change of energy for single turn
Loss factor [V/pC]
Movable mask
31.6
CSR (arc)
11.0
ARES cavity
9.9
Resistive wall
5.0
Bellows
3.4
CSR (wiggler)
2.4
TOTAL
63.3
Movable mask (16)
Resistive wall (2200m)
CSR (wiggler 152)
Bellows (800)
ARES cavity (16)
CSR (arc 134)
CSR (arc+wiggler)
bunch length=3mm (Gaussian distribution)
Vacuum chamber: R=90mm (square for CSR)
(Provided by K. Shibata)

13. CSR in wiggler and arc-section of SuperKEKB LER
Arc bend
Wiggler
CSR wakefield value at z=0 (bunch center) vs. chamber full height
bunch charge Ne=19nC
sigz=3mm (Gaussian distribution)

14. Analysis of microwave instability
1D Fokker-Planck equation
Equation of motion
Change of energy by wakefield
Initial distribution ( implicit solution of Haissinski equation )
Algorithm to be fixed because of no causality

15. Algorithm Equation of motion without damping and diffusion Mapping
M. Venturini et al. PRST-AB, 8, (2005)
Fourth order finite difference for damping and diffusion
damping
diffusion

16. Green function of CSR Change of energy by wakefield
Green function width
Source of the wakefield
Limit due to computing time
:frequency filter
Energy spectrum
Mesh size in the phase space

17. Longitudinal distribution and wakefield
chamber = 90mm
wake = CSR(arc,wiggler) + resistive wall
Ne=4.5nC (below instability threshold)
Threshold charge Ne～5nC
Ne=7.0nC (above instability threshold)

18. chamber diameter
D = 90mm
D = 70mm
D = 50mm
charge Ne=19nC

19. Ring impedances in KEKB
Real part
k(peak) ～ 1.6/mm
CSR
Resistive wall
Resistive wall impedance is moderate in high frequency.
Imaginary part
CSR is very sensitive to small bunch structure.

20. Bunch length and Energy spread
Energy spread vs. Chamber size for designed charge: Ne=19nC
Energy spread vs. Bunch charge
Wiggler CSR is participating in the microwave instability.
Resistive wall wakefield decreases the instability.

21. Conclusions
CSR generated in SuperKEKB LER wiggler is small compared to that of arc-section.
Assuming the chamber diameter is 90mm, change of energy due to the wiggler CSR is just about 10% of total CSR effect.
Shielding condition in the wiggler is different from arc-bends.
It is difficult to achieve both short bunch and high current.
CSR will induce microwave instability in SuperKEKB LER.
The threshold bunch charge is about 5nC (Ib=0.5mA) in the vacuum chamber of 90mm.
If the chamber diameter is 30mm, the threshold will be 15nC, though side effects may be occur such as transverse coupled bunch instability due to resistive wall wakefield.