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To address this issue, calculations were performed using the Z 4 code13 that utilizes higher-order derivatives of the 30 magnetic field. The simulation accelerates four particles on the periphery of a half ellipse (the motion is symmetric for the other half) in the z-pz plane and the maximum beam height as a function of energy is obtained. 0 cm respectively. From Figure 29, the maximum vertical beam size for each of the four particles is less than 2 cm near extraction. The increase in size is due to the decreasing vertical focusing frequency with increasing energy.

0 100 150 200 250 300 350 400 Figure 29. 6 π mm mrad. The initial total beam height is ≈1 cm. The coupling between the vertical motion of Figure 29 and that of the horizontal plane was determined by calculating the difference between the radial position of the four particles displaced from the median plane and a particle on the median plane (z = 0) all launched at the same horizontal position. 5 mm; a negligible contribution to the turn broadening. 00 100 150 200 250 300 350 400 Figure 30. Radius difference between the particles with vertical (z) displacements and that of a particle that starts on the median plane.

6 B. F. 3", TRI-DN-99-4,January 28, 1999. 7 B. Franzke, Vacuum Requirements for Heavy Ion Synchrotrons, IEEE Trans. on Nuc. Sci. 3, (1981) 2116. 8 N. Bohr and J. Lindhard, Dan. Mat. Fys. Medd. 28, no. 7 (1954). D. Betz and Ch. Schmelzer, UNILAC 1-67 (1967). D. Betz, G. Hortig, E. Leischner, Ch. Schmelzer, B. Stadler, and J. Weihrauch, Phys. Lett. 22 No. 5 (1966)643 11 W. Joho, “High Intensity Problems in Cyclotrons”, 9th Int. Conference on Cyclotrons, p337, Caen, France 1981. M. Gordon, “The longitudinal space charge effect and energy resolution”, 5th Intl.