Scaling Laws for RFQ Design Procedures

Scaling laws are relations between accelerator parameters (electric field, rf wavelength etc.) and beam parameters (current, energy, emittance, etc.) that define surfaces of constant accelerator performance in parameter space. These scaling laws can act as guides for designing radio-frequency quadrupoles (RFQs). We derive several scaling relations to show the various tradeoffs involved in choosing RFQ designs and to provide curves to help choose starting points in parameter space for optimizing an RFQ for a particular requirement. We show that there is a unique scaling curve, at a synchronous particle phase of -90°, that relates the beam current, emittance, particle mass, and space-charge tune depression with the RFQ frequency and maximum vane-tip electric field, provided that we assume equipartitioning and equal longitudinal and transverse tune depressions. This scaling curve indicates the maximum performance limit one can expect at any point in any given RFQ. We show several examples for designing RFQs using this procedure.