The effects of exposure to ionizing radiation are central in many areas of science and technology, including medicine and biology. Absorption of UV and soft-x-ray photons releases photoelectrons, followed by a cascade of lower energy secondary electrons with energies down to 0 eV. While these low energy electrons give rise to most chemical and physical changes, their interactions with soft materials are not well studied or understood. Here, we use a low energy electron microscope to expose thin organic resist films to electrons in the range 0-50 eV, and to analyze the energy distribution of electrons returned to the vacuum. We observe surface charging that depends strongly and nonlinearly on electron energy and electron beam current, abruptly switching sign during exposure. Charging can even be sufficiently severe to induce dielectric breakdown across the film. We provide a simple but comprehensive theoretical description of these phenomena, identifying the presence of a cusp catastrophe to explain the sudden switching phenomena seen in the experiments. Surprisingly, the films undergo changes at all incident electron energies, starting at ∼0 eV. DOI: 10.1103/PhysRevLett.119.266803 The interaction of ionizing radiation with matter is of vast scientific and technological (including biological and medical) importance. The interaction of UV and x-ray photons with matter is mediated by photoelectrons, as well as secondary electrons with a broad energy distribution that induce chemical changes in the material, be it a polymer, organic or inorganic hybrid, biological tissue, or even DNA. But these complex processes are hard to disentangle, as photon illumination sets the entire electron cascade in motion at once, without the possibility of discerning the role of electrons with different energies. As a result, the interaction of low energy electrons (LEEs) with soft matter is not well understood. Here, we focus primarily on the interaction of low energy electrons with polymethylmethacrylate (PMMA) and related resist materials as used in extreme ultraviolet (EUV) lithography [1] to obtain a new understanding of key processes at low electron energies.In a low energy electron microscope [2] (LEEM) a sample is illuminated with electrons with adjustable 0-100 eV energy [3]. We use LEEM to expose thin PMMA films, monitoring changes both after and during exposure [4]. The radiation chemistry of PMMA and related materials has been well studied, and there is consensus that irradiation causes scission of the main chains and removal of side groups [5][6][7][8][9][10]. Here, we identify key physical processes largely ignored in the literature: resist charging, exposure-induced changes in conductivity and secondary electron emission, and dielectric breakdown. We present a simple quantitative theory describing our data, identifying a cusp catastrophe [11] causing the instabilities seen during exposure. Even electrons with near-zero energy change the resist, suggestive of dissociative electron attachment processes [12] commonly neg...