How fast does a vacuum surface discharge propagate?

Vacuum surface discharge on dielectric substrates is frequently observed in vacuum electronic devices and insulation systems, spanning gap distances from the sub-micrometer to millimeter scale and beyond. Such discharges propagate through a secondary electron emission avalanche (SEEA), in which field emission electrons multiply on the surface via successive secondary electron emission. Understanding the SEEA propagation speed (v_SEEA) is essential for elucidating the underlying discharge physics and optimizing device performance. Despite extensive studies since the 1970s, no predictive theory has been available to calculate this key quantity or reveal its parametric dependencies. Here, we resolve this longstanding issue by introducing a charging-by-slice model, which yields a compact expression, v_SEEA~J_FE/E_x , linking the propagation speed to the cathode field emission current and applied electric field. Theoretical predictions are validated by particle-in-cell simulations and experimental measurements, showing excellent agreement and revealing a v_SEEA in the range of 106-107 m/s. The established theoretical framework advances both the fundamental understanding of device physics and the practical design of vacuum electronic and insulation systems.