Transverse Single Bunch Instability in Pep-x

A proposed high-brightness synchrotron light source (PEP-X) is under design at SLAC. The 4.5GeV PEP-X storage ring has four theoretical minimum emittance (TME) cells to achieve the very low emittance and two double-bend achromat (DBA) cells to provide spaces for IDs. Damping wigglers will be installed in zero-dispersion straights to reduce the emittance below 0.1nm. In this paper, we present a preliminary estimation of the threshold of the transverse mode coupling instability (TMCI). Three approaches have been used in the estimation and they agree well with each other. Presented at 2009 Particle Accelerator Conference, 05/04/2008 – 05/08/2008, Vancouver, Canada SLAC-PUB-13658 June 2009 TRANSVERSE SINGLE BUNCH INSTABILITY IN PEP-X* L. Wang and G. Stupakov, SLAC, CA 94025, USA. Abstract A proposed high-brightness synchrotron light source (PEP-X) is under design at SLAC. The 4.5-GeV PEP-X storage ring has four theoretical minimum emittance (TME) cells to achieve the very low emittance and two double-bend achromat (DBA) cells to provide spaces for IDs. Damping wigglers will be installed in zerodispersion straights to reduce the emittance below 0.1nm. In this paper, we present a preliminary estimation of the threshold of the transverse mode coupling instability (TMCI). Three approaches have been used in the estimation and they agree well with each other. INTRODUCTION In this paper, we study the transverse single bunch instability in the PEP-X storage ring [1]. The storage ring has 476 MHz RF and a revolution period of T0=7.33 μs. The baseline design has 3154 bunches with a single bunch current of 0.44mA to mitigate the intra-beam scattering (IBS) effect. The total beam current is 1.5A. There is no transverse single bunch instability in PEPII. However, the impedance of PEP-X significantly increases due to the small aperture in IDs and the transition sections. The transverse mode coupling instability becomes one key issue in the design. As a preliminary study, we only study the effect of resistive wall impedance. A detailed investigation of the impedance with three dimensional code is under the way. Thereafter, we will include other impedance in our further study. The resistive wall impedance is dominant in most of the light sources due to their small aperture of beam pipe. The resistive-wall instability depends on the aperture and material of the vacuum of chamber. The standard transverse impedance of a round pipe of length L is given by [2] ( ) ω σ π ω ω 1 2 2 ) sgn( ) ( 0 3 0 Z c b LZ i Z − = ⊥ . (1) Here Z0 is the impedance of free space, b the radius of the beam pipe, σ the conductivity of the pipe material and c the speed of light. A more realistic model of the shape of the vacuum chamber will be studied in the future. The wake function corresponding to (1) is s Z b cL s W 1 ) ( 0 3 πσ π = ⊥ (s>0) . (2) Table 1 lists the beam pipe radius, material and length in different sections of the ring. The impedance in the 90 meters insertion section is dominant due to its small aperture. While the impedance from the arc and straight sections is negligible. Therefore, replacement of the beam pipe in arc and straight sections with copper doesn’t effectively reduce the total impedance. Table 1: Source of resistive wall impedance in PEP-X ARC Insertion Wiggler Straight Material Al Cu Cu Al Radius[cm] 2.8 0.3 0.75 4.8 Length [m] 1522 90 90 337 L/b [m] 6.9×10 3.3×10 2.1×10 3.0×10 APPROACHES Three methods have been used to estimate the threshold of TMCI. Mode Coupling Theory TMCI occurs when the frequencies of two neighboring head-tail modes approach each other due to the detuning with increasing beam current. For a Gaussian bunches, the threshold of the instability can be expressed with the transverse loss factor [3] Θ = ∑ ⊥ j j y j s th k e E Q I , , 0 0 / 2 β ω , (3) where th I0 is the threshold of the beam current, s Q is the synchrotron tune, j y , β is the average beta function in the j th element, j y , κ is its loss factor, E is the beam energy and 7 . 0 ≈ Θ . Eigen-value Solver The threshold of transverse mode coupling instability can be found by solving the following eigen-value problem [2, 4]: ' ' ' ' , k l k l lk lk s a M a = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − Ω ω ωβ . (4) We consider azimuthal mode coupling only for the lowest radial mode (k=0) ) exp( 2 ) ( ! ' ! 1 4