INTERPRETATION OF THE CONNECTING LINES IN THE PROJECTIONAL XX' EMITTANCE IN VIEW OF LIOUVILLES THEOREM

For the analytical and numerical description of the emittance of a par ticle beam in cylindrical symmetric transport and accelerator systems often the rr' plane is used, yielding a sectional emittance epsilon (r , r'). In contrast, emittance measurements are mostly taken as the pro jectional emittance epsilon (x, x'). As could be shown earlier, the de scription of a cylindrical symmetric beam in the projectional xx' plan e of phase space results in the presence of lines for each ring of par ticles, connecting the two corresponding points, with which the same b eam is described in the rr' plane. There is a very special intensity d istribution along these lines, both for straight as well as for skew t rajectories. It is shown here that the theoretically obtained density distribution along such a line is a weight function for the intensity measured with a slit of width Delta x. For one particle it can also be interpreted as a probability function for the particle being at a cer tain position x(0). In view of Liouville's theorem, the case of the tw o-dimensional emittance in cylindrical symmetric systems is discussed, where this law of conservation is, due to the coupling of the x and y planes, not generally applicable. It is shown here that for an emitta nce measurement performed in the xx' plane, cart has to be taken when choosing the slit width, since a small Delta x yields a good spatial r esolution, but increases the risk of missing information in the center of each connecting line. (C) 1997 American Institute of Physics.