# Focusing properties of discrete RF quadrupole

## TL;DR

This paper derives a mathematical model for the focusing properties of discrete RF quadrupoles, providing formulas that can be integrated into beamline design software for improved particle focusing analysis.

## Contribution

It introduces a new power series-based transform matrix for discrete RF quadrupoles, enabling accurate modeling of particle motion for lengths up to 0.5b*l.

## Key findings

- Second-order series matches direct integration for B up to 30.
- First-order series provides good approximation for shorter lengths.
- Transform matrix can be integrated into linac design codes.

## Abstract

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

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Source: https://tomesphere.com/paper/1703.02345