Abstract
Betatron tune shift due to space charge effect is investigated by solving the equation of motion of particles including total space charge (linear and non-linear part). We introduce a binomial representation for 4-dimensional phase space distribution, which can characterize the important factors to tune shift while largely reducing the mathematical difficulties. With this representation, a numerically solvable equation of motion is obtained. From its solution, betatron tune is deduced by FFT (Fast Fourier Transformation). We derive the tune shift as function of beam current, amplitude of betatron oscillation and the particle distribution. We also present the differences between tune shifts calculated by including total space charge effect and including only its linear term.
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