A Simplified Model of Intrabeam Scattering*

1. Abstract Beginning with the General Bjorken-Mtingwa solution, we derive a simplified model of intrabeam scattering (IBS), one valid for high energy beams in normal storage rings; our result is similar, though more accurate than a model due to Raubenheimer. In addition, we show that a modified version of Piwinski’s IBS formulation (where 2x,y/x,y has been replaced by ) at high energies asymptotically approaches the same result. INTRODUCTION • Intrabeam scattering (IBS) tends to increase the beam phase space volume in hadronic, heavy ion, and low emittance electron storage rings. • The two main detailed theories of IBS are ones due to Piwinski[1] and Bjorken-Mtingwa (B-M)[2]. • Solving IBS growth rates according to both methods is time consuming, involving, at each iteration step, a numerical integration at every lattice element. • Approximate solutions have been developed by Parzen, Le Duff, Raubenheimer[3], and Wei. NUMERICAL COMPARISON[4] A Simplified Model of Intrabeam Scattering* K.L.F. Bane, SLAC, Stanford, CA 94309, USA • Note that Raubenheimer’s 1/Tpformula is the same, except that ½ replaces g(a/b)H/ p;his 1/Tx,y formulas are identical. The function g() (solid curve) and the fit g=(0.021-0.044 ln) (dashes). The ratio of local growth rates in p as function of x, for b=0.1 (blue) and b=0.2 (red) [y=0]. • The plot on the right shows that, as long as  is not too close to 1, the result is not sensitive to , and the model approaches B-M from above as a,b0. • Piwinski’s formulation depends on dispersion through 2/ only. If we modify his result so that 2/ is replaced by , then at high energies his result asymptotically approaches that of B-M. • Consider as example the ATF ring with no coupling; to generate vertical errors, magnets were randomly offset by 15m, and the closed orbit was found; <>=17m, yielding a zero current emittance ratio of 0.7%. I=3.1 mA. HIGH ENERGY APPROXIMATION The B-M solution has integrals involving 4 normalized parametersa, b, x, y: where =[2+(’ ½’)2]/  is the dispersion invariant, and =’ ½’/. For our approximation we: (1) assume a,b<<1, and (2)set x,y to 0 Steady-state local growth rates over ½ the ATF, for an example with vertical dispersion due to random errors. Given are results due to Bjorken-Mtingwa, modified Piwinski, and the high energy approx. The approximate growth rates: • For coupling dominated NLC, ALS example (=0.5%) the error in steady-state (Tp1,Tx1) obtained by the model is (12%,2%), (7%,0%). A fit: [1] A. Piwinski, in Handbook of Accelerator Physics, (1999) 125. [2] J. Bjorken and S. Mtingwa, Part. Accel., 13 (1983) 115. [3] T. Raubenheimer, SLAC-R-387, PhD thesis, 1991. [4] K. Bane, et al, Phys Rev ST-Accel Beams 5:084403, 2002. REFERENCES Transverse growth rates: * Work supported by DOE contract DE-AC03-76SF00515