In ion projection lithography a stencil mask is back "illuminated" by a broad beam of ions, and its demagnified image (similar to 3X) is projected onto a resist covered substrate. The stencil mask that will be used in the next generation systems consist of a 120-mm-diam 2.5-mu m-thick single crystal silicon membrane etched in a 150-mm-diam wafer whose edge acts as part of a supporting frame. The main sources of distortion are thought to be stress relief due to the cutting of holes and the thermal expansion due to mask heating by the incident ions. In this paper we calculate the temperature distribution in the mask for the expected hear input by the beam, the resulting mask distortion, and the effect of optimized radiation cooling. The central 100-mm-diam area of an unpatterned membrane is assumed to be irradiated with an energy input of 3.3 mW/cm(3). The edge of the membrane at the frame is assumed to be at room temperature and held rigidly. Heat is lost by radiation and by conduction. An effective method of reducing temperature gradient in the membrane is to place a cooled cylinder above the mask surrounding the ion beam. ?The expected temperature profiles for various sample emissivities as well as the corresponding distortion (radial displacement of the mask membrane) are computed with and without the cooled cylinder. These computations are done for various geometries and cylinder temperatures. For example, in a membrane of emissivity 0.55 with a cylinder, 6.5 cm high 13 cm in diameter, placed 3.2 cm above the mask and cooled to 273 K, the temperature is uniform around 300 K to better than +/- 1.5 K and the expected mask distortion is then less than +/- 10 nm. Since the ion beam will be incident on the mask continuously, the cooling cylinder will be effective in reducing thermal distortion to well below the permitted limit.