Dynamical aperture and chaotic particle motion in the HERA proton ring

For large hadron storage rings the dynamic aperture is a crucial figure of the machine. A limitation of the dynamic aperture is due to nonlinear magnetic fields, which also include higher multipoles in the superconducting magnets. An efficient method for determining the dynamic aperture is described, as a promising alternative to long term tracking. This method is based on the detection of chaotic trajectories using tracking data for small turn numbers. The onset of chaos, which can be found at a certain amplitude, gives a lower bound for the actual dynamic aperture. For the detection of chaos it is advisable, to extend a programme which is normally used for simple tracking, as to compute not only the actual phase space coordinates of the tracked particle, but also the corresponding Jacobian matrix of the canonical variables in phase space. The Jacobian expresses the dependence of the final coordinates on the initial ones in a linear approximation. The extension can be done very easily in four dimensions, but it is fairly complicated if synchrotron oscillations are to be included. The Jacobian matrix can be thought of as the transfer matrix of a very long imaginary transport line, the zero axis of which is the actual trajectory. This transport line has a basic periodicity corresponding to the storage ring length. The periodicity is disturbed by the nonlinear fields around the tracked particle. The eigenvalues of the calculated Jacobians contain information about the divergence of nearby trajectories in phase space. By use of this fact a measure of chaoticity can be calculated. This measure is the Lyapunov exponent, which is known from the theory of nonlinear dynamics. The described procedure allows an easy detection of the onset of chaos. Its application to the HERA proton ring is shown in addition to usual tracking calculations. It is found, that the dynamic acceptance experted for HERA is twice as large as the acceptance needed for operation. (orig.)