Geometric properties of the capture region limited by the acceleration channel

The capture region determined solely by the transverse dimension of an accelerator channel consists of all points in the initial-conditions plane which also lie on integral curves fitting within a gap u2<a2 in the space defined by u, v, z (u is the transverse co-ordinate of the particle, v is the ratio between the transverse and longitudinal components of particle velocity, z is the axis of the accelerator and a is the given transverse dimension of the accelerator channel). The boundary of this region is determined by the inverse images of the accelerator channel cross-sections on the initial-conditions plane and by the envelope to this family of images. The envelope in this case is a loop-shaped curve consisting of conjugate ellipses and hyperbolas and having its centre of symmetry at the origin of the co-ordinates. In the conjugation of an ellipse with a hyperbola there occurs a reversal point. The capture region is then bounded by a convex curvilinear polygon whose sides are segments of the ellipses and straight envelopes tangential to the ends with centre of symmetry at the origin of the co-ordinates. (author)