Polarized protons have never been accelerated to more than about 25GeV. To achieve polarized proton beams in RHIC (250GeV), HERA (820GeV), and the TEVATRON (900GeV), ideas and techniques new to accelerator physics are needed. In this publication we will stress an important aspect of very high energy polarized proton beams, namely the fact that the equilibrium polarization direction can vary substantially across the beam in the interaction region of a high energy experiment when no countermeasure is taken. Such a divergence of the polarization direction would not only diminish the average polarization available to the particle physics experiment, but it would also make the polarization involved in each collision analyzed in a detector strongly dependent on the phase space position of the interacting particle. In order to analyze and compensate this effect, methods for computing the equilibrium polarization direction are needed. In this paper we introduce the method of stroboscopic averaging, which computes this direction in a very efficient way. Since only tracking data is needed, our method can be implemented easily in existing spin tracking programs. Several examples demonstrate the importance of the spin divergence and the applicability of stroboscopic averaging. *

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A method is introduced that provides an accurate and fast approximation of high-order maps of fringe fields and other fields that change along the reference trajectory. While the effects of main fields of optical elements can be determined very efficiently with differential algebraic ͑DA͒ methods via exponentiation of the respective propagator, the computation of high-order maps of nonstationary fields in general requires time-consuming DA integration. The method of symplectic scaling presented in this paper provides a very fast approximation of such maps by relating an arbitrary map to a specific previously computed map. This is achieved by a combination of geometric scaling and scaling with rigidity performed in a canonically perturbative treatment of a strength parameter. The method is useful for detailed analysis of nonlinear motion in particle optics, which in many cases is strongly influenced or even dominated by the presence of fringe fields. The use of the symplectic scaling method typically speeds up the computation of fringe-field effects by around two orders of magnitude and thus approaches speeds similar to that of the main-field calculation. The method has been implemented in the code COSY INFINITY; several examples from various subfields of beam physics are given to illustrate the accuracy and speed of the method.

It is planned to upgrade the HERA luminosity to 7 ࣽ 10 31 cm ,2 s ,1 , which is 4 times the original design luminosity. This is to be achieved by decreasing the proton beam size by moving quadrupoles closer to the interaction point and by increasing their strength. Similar measures decrease the electron beam size. However, to match the smaller proton beam size, the horizontal electron emittance additionally has to be decreased.The electron emittance can be decreased either by stronger focusing in the arcs, or by changing the damping partition numbers. In the HERA case, however, both methods have to be applied simultaneously, since changing the damping partition numbers increases the longitudinal emittance, which can only be tolerated with the current RF parameters if the bucket is increased by a stronger focusing in the arcs. These two methods of decreasing the emittance have competing effects on long term stability. Stronger focusing usually leads to a greater reduction of the dynamic aperture than of the emittance; whereas a change of the damping partition numbers tends to increase the dynamic aperture relative to the emittance. Playing the two competing effects against each other, it is possible to decrease the electron emittance while keeping the relative dynamic aperture as well as the requirements on the RF system tolerable. EMITTANCE REDUCTIONIn the HERA luminosity upgrade the horizontal emittance of the electron beam has to be reduced from currently 41एnm to 22एnm [1,2]. The horizontal emittance " x of an electron storage ring is given by with the curvature G of and the focusing strength K on the closed orbit. The parentheses é: : :éindicate an average around the ring and इ is the periodic dispersion. The optic functions ae and ae are used and C q ए 384fm is a constant. On the design orbit of a separated function ring, which HERA is to a good approximation, GࣽK = 0 around the ring. If one does not want to compromise on particle loss out of the RF bucket, the RF bucket height ࣽE=E 0 has to be increased accordingly. We do not want to increase the bucket size by increasing the cavity voltage, since we need all available power for the storage of the 56mA design current. Therefore we have to decrease the dispersion to take advantage of the fact that ࣽE=E 0 è 1= p é G इ é .The dispersion इ in the arcs of the ring is decreased when we increase the horizontal focusing from currently 60 ae per FODO cell. By doing so, an additional fact comes in very handy: stronger focusing reduces the emittance by reducing the numerator of equation 1. DYNAMIC APERTUREInitially it was tried to obtain the 22nm design emittance by stronger focusing alone [3]. This could be achieved by going to 90 ae horizontal phase advance per FODO cell. However, the stronger natural chromaticity required stronger sextupoles and these nonlinear fields reduced the dynamic aperture significantly. The dynamic aperture for on energy particles for the four slightly different electron optics used in HERA during 1997 and 1998 are shown in figure ...