The beam envelope equation -- Systematic solution for a FODO lattice with space charge

Many approximate solutions for matched beam envelope functions with space charge have been developed; they generally have efforts 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 arbitrarily 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 αK1 and letting α increase continuously from 0 to 1. It is found that S and D may be expanded in even and odd powers of α, respectively. Equations for the coefficients of powers of α 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