Outline of the method to find an approximate region of electrons that enter the detector in a quadrupole field after kicked by a point bunch K. Kanazawa KEK 22 June 2009

Magnetic field Coordinate y x Approximation of the initial condition (B ~ 0 near the center) x First, determine the initial region in the phase plane (v, 0) for those electrons that enter the detector. Then translate the region into the real plane (r, 0).

Equation of motion Introduction of lK The final form of equations.

Solution when u<<1. Asymptotic solution with the form similar to the last equation

Analysis of the electron motion near the detector (u>>1) The conditions to enter the detector Aperture condition Energy condition

Conditions expressed with actual parameters Numerical example

Final result