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, kappa(q)(S) = kappa(q)(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 kappa(q)(s). Introducing the quadratic measure, sigma0(2)/S2 = [[integral-s/s0ds kappa(q)(s)]2], of the average quadrupole focusing field squared, a detailed analysis of the transcendental equation for the phase advance sigma is used to quantify the range of validity of the approximate estimate of the phase advance obtained from the simple quadratic equation (sigma/S)2 + (K/epsilon)(sigma/S) = (sigma0/S)2. Here, sigma = epsilonS/r(b)2BAR is the phase advance for a circular beam with average radiUS r(b)BAR, epsilon is the unnormalized beam emittance, S is the periodicity length of the lattice, and K is the self-field perveance. For sigma0 less than or similar to 30-degrees, it is found that the (approximate) quadratic expression for sigma gives an excellent estimate of the phase advance over the entire range of KS/epsilon, and the quadratic estimate for sigma is accurate to within 5% for values of sigma0 approaching sigma0 = 60-degrees.