A wide range of X-ray detectors and geometries are available today on transmission/scanning transmission analytical electron microscopes. While there have been numerous reports of their individual performance, no single experimentally reproducible metric has been proposed as a basis of comparison between the systems. In this paper, we detail modeling, experimental procedures, measurements, and specimens which can be used to provide a manufacturer-independent assessment of the performance of an analytical system. Using these protocols, the geometrical collection efficiency, system peaks, and minimum detection limits can be independently assessed and can be used to determine the best conditions to conduct modern hyperspectral and/or spectrally resolved tomographic analyses for an individual instrument. A simple analytical formula and specimen is presented which after suitable system calibrations can be used to experimentally determine the X-ray detector solid angle.

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

Closed form analytical equations used to calculate the collection solid angle of six common geometries of solid-state X-ray detectors in scanning and scanning/transmission analytical electron microscopy are presented. Using these formulae one can make realistic comparisons of the merits of the different detector geometries in modern electron column instruments. This work updates earlier formulations and adds new detector configurations.

Cet article établit l'expression du flux de particules, émises dans le vide par unesource ponctuelle à travers une surface plane. Cette équation est alors comparée auxrésultats numériques obtenus par utilisation de symétries sur des exemples. Uneapplication est proposée pour un détecteur cylindrique (surface non plane). Descompléments sont donnés en conclusion.

With the advent of silicon drift X-ray detectors, a range of new geometries has become possible in electron optical columns. Because of their compact size, these detectors can potentially achieve high geometrical collection efficiencies; however, using traditional approximations detector solid angle calculations rapidly break down and at times can yield nonphysical values. In this article we present generalized formulas that can be used to calculate the variation in detection solid angle for contemporary Si(Li) as well as new silicon drift configurations.