Phase advance for an intense charged particle beam propagating through a periodic quadrupole focusing field in the smooth-beam approximation

The envelope equations for a uniform-density Kapchinskij--Vladimirskij (KV) beam equilibrium are used to derive a transcendental equation for the phase advance of an intense charged particle beam propagating through a periodic quadrupole focusing lattice, κq(s)=κq(s+S). The analysis is carried out for the case of a matched beam in the smooth-beam approximation, and precise estimates of the phase advance are obtained for a wide range of system parameters and choices of lattice function κq(s). Introducing the quadratic measure, σ20/S2 = left-angle[∫s0sds κq(s)]2 right-angle, of the average quadrupole focusing field squared, a detailed analysis of the transcendental equation for the phase advance σ is used to quantify the range of validity of the approximate estimate of the phase advance obtained from the simple quadratic equation (σ/S)2+(K/ε)(σ/S)=(σ0/S)2. Here, σ=εS/bar r2b is the phase advance for a circular beam with average radius bar rb, ε is the unnormalized beam emittance, S is the periodicity length of the lattice, and K is the self-field perveance. For σ0 approx-lt 30 degree, it is found that the (approximate) quadratic expression for σ gives an excellent estimate of the phase advance over the entire range of KS/ε, and the quadratic estimate for σ is accurate to within 5% for values of σ0 approaching σ0=60 degree