We present a new formalism to calculate phase-space acceptance in a Zeeman decelerator. Using parameters closely mimicking previous Zeeman deceleration experiments, this approach reveals a hitherto unconsidered velocity dependence of the phase stability which we ascribe to the finite rise and fall times of the current pulses that generate the magnetic fields inside the deceleration coils. It is shown that changing the current switch-off times as the sequence progresses, so as to maintain a constant mean acceleration per pulse, can lead to a constant phase stability and hence a beam with well-defined characteristics. We also find that the time overlap between fields of adjacent coils has an influence on the phase-space acceptance. Previous theoretical and experimental results suggested unfilled regions in phase space that influence particle transmission through the decelerator. Our model provides, for the first time, a means to directly identify the origin of these effects due to coupling between longitudinal and transverse dynamics. Since optimum phase stability is restricted to a rather small parameter range in terms of the reduced position of the synchronous particle, only a limited range of final velocities can be attained using a given number of coils. We evaluate phase stability for different Zeeman deceleration sequences, and, by comparison with numerical three-dimensional particle trajectory simulations, we demonstrate that our model provides a valuable tool to find optimum parameter sets for improved Zeeman deceleration schemes. An acceleration-deceleration scheme is shown to be a useful approach to generating beams with well-defined properties for variable-energy collision experiments. More generally, the model provides significant physical insights applicable to other types of particle decelerators with finite rise and fall time fields.
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