The fundamental limitations on spatial resolution of X-ray microanalysis in the scanning transmission electron microscope are set by the interrelationships between the gun brightness, operating voltage, probe convergence angle, size and current, specimen thickness, beam broadening, the probability of characteristic and Bremsstrahlung X-ray production and the statistics of the X-ray spectrum. Manipulation of expressions describing these interrelationships leads to equations predicting the optimum probe size and specimen thickness for the best achievable spatial resolution (defined as the diameter of a cylinder containing 90% of the X-ray production) in microscopes fitted with different electron sources and operating at different voltages in foils of various elements. Application of these calculations to the special case of detecting monolayer segregation at grain boundaries results in predictions of the minimum amounts of such segregation that would be observable. It is found, for example, that in a microscope with a field-emission source operating at 500 keV, resolution of \u3c 1nm is obtainable in an iron foil 20nm thick, and in this case about 0.001 monolayer of chromium is detectable segregated at grain boundaries. The calculations do not take into account instrumental or experimental problems such as specimen drift, specimen preparation, etc., and represent the basic physical limits of performance of a perfect analytical microscope
