A3 Space Charge Effects

Topic

When the particle density of a beam is very high and its energy relatively low—a combination typical of the early stages of the multi-stage injection chain described in the "modes" section, immediately after the source or in the initial low-energy rings—the mutual electrical repulsion between particles of the same charge sign (normally negligible compared to external magnetic forces at high energies) becomes a comparable or even dominant force. This phenomenon, known as the space-charge effect, acts as an additional non-linear defocusing force superimposed on the focusing provided by the quadrupoles described in the "fundamentals" section; its relative intensity decreases rapidly as particle energy rises, because at high energies, the magnetic field associated with the beam's motion partially cancels out the pure electrical repulsion. The effect is particularly problematic because, by depending on each particle's position within the beam distribution in a non-linear way, it does not affect all particles equally: particles near the center experience a different defocusing effect than those near the edge. This spreads the betatron wavenumber across the various beam particles rather than maintaining a single value, thereby increasing the likelihood that at least a fraction of the beam will encounter one of the dangerous resonances previously identified by beam optics analysis. This effect imposes fundamental practical limits on the amount of current that can be successfully injected and accelerated during the initial low-energy stages of any high-intensity machine; it is the primary reason why the most demanding injection chains devote so much design effort specifically to their lowest-energy stages, where the effect is most severe. Thus, this module—which opens by discussing a form of repulsion that matters only at low energy—concludes by showing how that very repulsion focuses design efforts precisely on the initial section of any injection chain.

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