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

### Abstract

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, κ_{ q }(*s*)=κ_{ 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 κ_{ q }(*s*). Introducing the quadratic measure, σ^{2} _{0}/*S* ^{2} = 〈[∫_{ s 0 } ^{ s } *ds* κ_{ q }(*s*)]^{2}〉, 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*)=(σ_{0}/*S*)^{2}. Here, σ=ε*S*/*r*̄^{2} _{ b } is the phase advance for a circular beam with average radius *r*̄_{ b }, ε is the unnormalized beam emittance, *S* is the periodicity length of the lattice, and *K* is the self‐field perveance. For σ_{0}≲30°, 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 σ_{0} approaching σ_{0}=60°.

© 1994 American Institute of Physics

Received 14 March 1994
Accepted 19 May 1994

/content/aip/journal/pop/1/9/10.1063/1.870502

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