Focusing properties of discrete RF quadrupoles

Abstract：

The particle motion equation for a Radio Frequency (RF) quadrupole is derived. The motion equation shows that the general transform matrix of a RF quadrupole with length less than or equal to 0.5β λ (β is the relativistic velocity of particles and λ is wavelength of radio frequency electromagnetic field) can describe the particle motion in an arbitrarily long RF quadrupole. By iterative integration, the general transform matrix of a discrete RF quadrupole is derived from the motion equation. The transform matrix is in form of a power series of focusing parameter B. It shows that for length less than β λ, the series up to the 2^nd order of B agrees well with the direct integration results for B up to 30, while for length less than 0.5βλ, the series up to 1^st order is already a good approximation of the real solution for B less than 30. The formula of the transform matrix can be integrated into linac or beam line design code to deal with the focusing of discrete RF quadrupoles.