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

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

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

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.

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

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

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)

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,7,054403 (2004)

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

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 :

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

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)

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)

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

Algorithm Equation of motion without damping and diffusion Mapping M. Venturini et al. PRST-AB, 8, 014202 (2005) Fourth order finite difference for damping and diffusion damping diffusion

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

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)

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

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.

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.

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.