The beam envelope equation-systematic solution for a periodic quadrupole lattice with space charge

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Abstract
Many approximate solutions for matched beam envelope functions with space charge have been developed; they generally have errors of 2-10% for the parameters of interest and cannot be reliably improved. The new, systematic approach described here provides the K-V envelope functions to high accuracy as a power series in the quadrupole gradient. A useful simplification results from defining the sum and difference of the envelope radii; S = (a+b)/2 varies only slightly with distance z along the system axis, and D = (a-b)/2 contains most of the envelope oscillations. To solve the coupled equations for S and D, the quadrupole strength K(z) is turned on by replacing K with {alpha}K{sub 1} and letting {alpha} increase continuously from 0 to 1. It is found that S and D may be expanded in even and odd powers of {alpha}, respectively. Equations for the coefficients of powers of {alpha} are then solved successively by integration in z. The periodicity conditions and tune integration close the calculation. Simple low order results are typically accurate to 1% or better.

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