A first-order theory for the temperature distribution in a thermally conducting and radiating rectangular film observed by transmission electron microscopy

Abstract

A first-order theory has been developed to enable the temperature distribution in a thin rectangular film to be calculated when the film is irradiated at normal incidence and at an arbitrary point by an electron beam of circular cross section and constant current density over the cross section. Both thermal conduction and radiation are taken into account. The theory is first order because it applies when the temperature rise of the film is small compared with the absolute temperature of the film before it is heated. The boundaries of the film are held at a fixed temperature Tg which may be different from T0 the temperature of the surroundings. The temperature increase tau at any given point is due to the temperature difference (T0-Tg) and to HB, the power input per unit volume from the electron beam. It is shown for a typical copper film that the temperature increase arising from T0-Tg is negligible for T0-Tg approximately 30 K. The temperature increase tau produced by HB is expressed in terms of a simple double sum of modified Bessel functions which can easily be evaluated by computer. In previous work which assumed the thermal emissivity in was exactly zero, tau diverges as In(b/a) where b is the distance of the centre of the beam (radius a) to the nearest point on the periphery of the film. This non-physical divergence does not occur with the present theory and tau approaches an in -dependent and film-shape independent limit as b/a to infinity . Simple analytical expressions for tau are given for the cases where the loss of energy from the film is predominantly by conduction and where the loss is predominantly by thermal radiation.