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A simplified model of intrabeam scattering - HHH


									                              A Simplified Model of Intrabeam Scattering*
                                                           K.L.F. Bane, SLAC, Stanford, CA 94309, USA

      Abstract                                                                                Note that Raubenheimer’s 1/Tp formula is the same, except that ½
       Beginning with the General Bjorken-Mtingwa                                              replaces g(a/b)H/ p; his 1/Tx,y formulas are identical.
      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      Piwinski’s IBS formulation (where 2x,y/ x,y has been
      replaced by        ) at high energies asymptotically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.

                                                                                             The function g() (solid curve)             The ratio of local growth rates in
                                    INTRODUCTION                                             and the fit g=(0.021-0.044 ln)            p as function of  x, for b=0.1
                                                                                             (dashes).                                   (blue) and b=0.2 (red) [ y=0].
         Intrabeam scattering (IBS) tends to increase the beam phase space
          volume in hadronic, heavy ion, and low emittance electron storage                   The plot on the right shows that, as long as  is not too close to 1, the
          rings.                                                                               result is not sensitive to  , and the model approaches B-M from above as
         The two main detailed theories of IBS are ones due to Piwinski[1] and               Piwinski’s formulation depends on dispersion through 2/ only. If we
          Bjorken-Mtingwa (B-M)[2].                                                            modify his result so that 2/ is replaced by      , then at high energies his
         Solving IBS growth rates according to both methods is time consuming,                result asymptotically approaches that of B-M.
          involving, at each iteration step, a numerical integration at every lattice
         Approximate solutions have been developed by Parzen, Le Duff,
          Raubenheimer[3], and Wei.
                                                                                                                 NUMERICAL COMPARISON[4]

                                                                                            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
                         HIGH ENERGY APPROXIMATION                                            found; <    >=17m, yielding a zero current emittance ratio of 0.7%. I=3.1
        The B-M solution has integrals involving 4 normalized parameters a, 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:

          Transverse growth rates:
                                                                                           [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.

* Work supported by DOE contract DE-AC03-76SF00515

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