A Muon Collider poses a number of challenging problems in the lattice designlow , small circumference, large physical and dynamic aperturewhich must be solved in order to realize the unique opportunities it offers for the high-energy physics. This contribution presents basic solutions which make it possible to achieve the goals for both the energy frontier collider and the Higgs factory with Nb3Sn magnet parameters.

In this paper we study the possibility of compensation of the space-charge compensation in the Fermilab Booster rapid cycling synchrotron with use several electron columns and show significant promise of such technique.

Abstract. We discuss a recent proposal to use strongly magnetized electron columns created by beam ionization of the residual gas for compensation of space charge forces of high intensity proton beams in synchrotrons and linacs. The electron columns formed by trapped ionization electrons in a longitudinal magnetic field that assures transverse distribution of electron space charge in the column is the same as in the proton beam. Electrostatic electrodes are used to control the accumulation and release of the electrons. Ions are not magnetized and drift away without affecting the compensation. Possible technical solution for the electron columns is presented. We also discuss the first numerical simulation results for space-charge compensation in the FNAL Booster and results of relevant beam studies in the Tevatron.

MADX-SC [1][2][3] allows the treatment of frozen space charge using beam-beam elements in a thin lattice, i.e. one can take advantage of the standard set-up of MAD-X [4] lattices without the need for specialized codes for the space-charge (SC) evaluation. The idea is to simulate over many turns without the problem of noise as in the PIC 1 SC codes. For the examples under study, like the PS and RHIC, it would be desirable to simulate up to 1 million turns or more. To this end one had to make an effort to optimize the scalar speed and, most importantly, get a speed-up of approximately a factor of 5 using OpenMP [5].

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