A Diffusive Model for Radio-Frequency Knock-Out Slow Extraction

Abstract

Slow resonant extraction from synchrotrons via radio-frequency knock-out is a well-established technique to deliver charged particle beams for various applications. In this contribution, we present explicit analytical expressions for calculating the number of particles slowly extracted over time, commonly referred to as spills. The proposed formulation enables the semi-analytical determination of an amplitude modulation curve to be applied to the radio-frequency exciter, which flattens the spill macrostructure, a feature of high relevance to all users requiring uniform beam delivery.

Figures (5)

• Figure 1: Hamiltonian contours (greens) and time evolution of a single initial condition under RFKO kicks (black), plotted every three turns.
• Figure 2: Comparison between map-based model (blue) and diffusive model (red) showing radial distribution evolution at different times (top row) and total flux evolution (bottom row).
• Figure 3: Square root of the time-dependent diffusion coefficient $D(t)$ for an initial distribution with $\sigma=1/5$.
• Figure 4: Extracted flux $\Phi(t)$ for the map-based model (blue) and numerically integrated diffusive model (red) with time-dependent diffusion coefficient as specified by Eq. \ref{['eq:D_equation']} to maintain $\Phi (t) = \Phi_0$.
• Figure 5: Experimental measurement of the extracted spill at the CERN Proton Synchrotron using a Pb$^{54+}$ beam with $E_{\text{kin}} =$1G e V / nucleon. Results are shown for both constant and time-dependent exciter gains. Each curve represents the average over 10 acquisitions of counts recorded by a scintillator screen. The integration time is one millisecond.