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Single-walled carbon nanotubes are readily observable in a scanning electron microscope, which traditional models fail to explain. We present an ab initio model to explain how the electron beam can interact with these structures despite the very small, nanoscale, interaction volume. In particular, we show how the electron beam can generate very strong secondary electron emission from the tip of a nanotube under external electric field. The approach may also be used in modeling the interaction of charged particles with nanostructures in other applications such as electron detection.