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Analytic Calculation for Elliptic Beams

The equation of motion of a same-charge particle during a beam-beam collision is expressed by[5]

(similarly for y), where N is the beam intensity, 2L the bunch length, tex2html_wrap_inline775 the classical electron radius, tex2html_wrap_inline757 the Lorentz factor of the beam energy and tex2html_wrap_inline779 the energy fraction of the particle. For a uniform charge distribution inside an elliptic cylinder of 2L long with radii a and b in the horizontal(x) and vertical(y) directions, respectively, the Coulomb potential is exactly given by
where q(x,y) is the positive solution to the equation tex2html_wrap_inline797. Assuming that a particle is created uniformly at a position tex2html_wrap_inline799 with zero scattering angle tex2html_wrap_inline801, its trajectory inside the ellipsoid is

where tex2html_wrap_inline803 is the aspect ratio of the transverse beam profile, which is equivalent to tex2html_wrap_inline805 for a Gaussian beam.

We first consider the azimuthal angular distribution of same-charge particles while ignoring the Coulomb field outside of the cylinder. The azimuthal angle (tex2html_wrap_inline807) is defined by
under the condition
The distribution function (tex2html_wrap_inline809), which is normalized to unity, is expressed by
where tex2html_wrap_inline811 is a step function. For tex2html_wrap_inline813 and tex2html_wrap_inline815 (or tex2html_wrap_inline817), where we are interested in measuring the aspect ratio for future linear colliders, tex2html_wrap_inline809 can be well approximated as
As can be clearly seen in the above equations(8,9), the distribution in the angular region near to the horizontal plane is depleted for a large aspect ratio as 1/R. On the contrary, the distribution in the vertical direction tex2html_wrap_inline823 has apparently no dependence on R. Therefore, we hereafter concentrate on calculations near to the horizontal plane in this section. We then estimate the effect of the Coulomb force outside of the elliptic cylinder, where the equation of motion is approximated for tex2html_wrap_inline827 and tex2html_wrap_inline829, as follows:
Since the vertical force in Eq.(11) is very small in this region, except for tex2html_wrap_inline831, we neglect it for the moment. Integrating the x component of Eq.(10) once, for large tex2html_wrap_inline833 we obtain
where tex2html_wrap_inline835 and tex2html_wrap_inline837, which is the x-velocity at the position (x, y) = (a, 0) . The resultant distribution function at tex2html_wrap_inline843 is give for large tex2html_wrap_inline845 by
We further estimated the effect of the vertical force neglected in the above calculation by numerically integrating Eqs.(10) and (11). The major effect was found to be an overall change in the magnitude by a constant factor of tex2html_wrap_inline847. Therefore, the R dependence of F(0) becomes tex2html_wrap_inline853 instead of 1 / R ( Eq.(9)) mainly due to a horizontal force outside of the elliptic cylinder. Thus, for the case of a flat beam (tex2html_wrap_inline857), the same-charge particles are strongly deflected in the vertical direction; their azimuthal angular distribution gives us direct information about R. It should also be noticed that the particles tend to be deflected more in a direction where their creation points are displaced by tex2html_wrap_inline861 and tex2html_wrap_inline863, since the forces are proportional to tex2html_wrap_inline865 and tex2html_wrap_inline867, respectively, provided by differentiating Eq.(2). This gives us the sensitivity to measure the alignment between two beams, which is discussed in the next section.

The maximum deflection angle for low-energy particles can be expressed by[5]
where tex2html_wrap_inline869, tex2html_wrap_inline871, tex2html_wrap_inline873 and tex2html_wrap_inline875. Using these relations, Gaussian beams tex2html_wrap_inline877 can be well described in terms of variables (a,b,L) of an elliptic cylinder. Assuming that the beam intensity and the bunch length are known, the horizontal beam size can be obtained from a measurement of the maximum deflection angle, especially for a large aspect ratio, as can be clearly seen in Eq.(14).

next up previous
Next: Simulation by ABEL Up: Nanometer Beam-Size Measurement during Previous: Introduction

Toshiaki Tauchi
Sat Dec 21 00:34:16 JST 1996