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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 (similarly for y), where N is the beam intensity, 2L the bunch length, the classical electron radius, the Lorentz factor of the beam energy and 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 . Assuming that a particle is created uniformly at a position with zero scattering angle , its trajectory inside the ellipsoid is  where is the aspect ratio of the transverse beam profile, which is equivalent to 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 ( ) is defined by where under the condition The distribution function ( ), which is normalized to unity, is expressed by where is a step function. For and (or ), where we are interested in measuring the aspect ratio for future linear colliders, 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 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 and , as follows: and Since the vertical force in Eq.(11) is very small in this region, except for , we neglect it for the moment. Integrating the x component of Eq.(10) once, for large we obtain where and , which is the x-velocity at the position (x, y) = (a, 0) . The resultant distribution function at is give for large 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 . Therefore, the R dependence of F(0) becomes 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 ( ), 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 and , since the forces are proportional to and , 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 where , , and . Using these relations, Gaussian beams 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: Simulation by ABEL Up: Nanometer Beam-Size Measurement during Previous: Introduction

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