A proton bunch flies into a plasma cloud, and leaves behind an invisible trail: an electric wave. This wave can pick up other particles and accelerate them to enormous energies. But controlling it is tricky: there are too many details in plasma behavior. Karl Schwarzschild once sought simple solutions to complex equations — here, a similar approach is needed.
The authors built a one-dimensional model, akin to calculating the wake of a speedboat. If the boat is the proton bunch, the water is the plasma. The model predicts how the wave height (potential) changes along the path. It turns out that at the bunch boundaries, the wave develops sharp kinks, like steps on a graph. This is not an error but physics: a jump in charge density breaks the second derivative of the potential. The calculations matched numerical experiments.
Next, the model was applied to a train of one hundred micro-bunches — exactly how AWAKE works. The wave grew linearly, and its electric field reached the limits of ±0.75 GV/m. This exactly replicates the data from the real experiment. Now, by shaping the bunches, one can extract more from the plasma — for example, bypass the classical limit of the field ratio, which is two.
In a plasma accelerator, particles are accelerated over distances hundreds of times shorter than in classical metallic cavities, a few meters instead of kilometers.