Observation of lens aberrations for high resolution electron microscopy II: simple expressions for optimal estimates

This paper lists simple closed-form expressions estimating aberration coefficients (defocus, astigmatism, three-fold astigmatism, coma / misalignment, spherical aberration) on the basis of image shift or diffractogram shape measurements as a function of injected beam tilt. Simple estimators are given for a large number of injected tilt configurations, optimal in the sense of least-squares fitting of all the measurements, and so better than most reported previously. Standard errors are given for most, allowing different approaches to be compared. Special attention is given to the measurement of the spherical aberration, for which several simple procedures are given, and the effect of foreknowledge of this on other aberration estimates is noted. Details and optimal expressions are also given for a new and simple method of analysis, requiring measurements of the diffractogram mirror axis direction only, which are simpler to make than the focus and astigmatism measurements otherwise required.