Correction of high-resolution data for curvature of the Ewald sphere

At sufficiently high resolution, which depends on the wavelength of the electrons, the thickness of the sample exceeds the depth of field of the microscope. At this resolution, pairs of beams scattered at symmetric angles about the incident beam are no longer related by Friedel's law; that is, the Fourier coefficients that describe their amplitudes and phases are no longer complex conjugates of each other. Under these conditions, the Fourier coefficients extracted from the image are linear combinations of independent (as opposed to Friedel related) Fourier coefficients corresponding to the three-dimensional (3-D) structure. In order to regenerate the 3-D scattering density, the Fourier coefficients corresponding to the structure have to be recovered from the Fourier coefficients of each image. The requirement for different views of the structure in order to collect a full 3-D data set remains. Computer simulations are used to determine at what resolution, voltage and specimen thickness the extracted coefficients differ significantly from the Fourier coefficients needed for the 3-D structure. This paper presents the theory that describes this situation. It reminds us that the problem can be treated by considering the curvature of the Ewald sphere or equivalently by considering that different layers within the structure are imaged with different amounts of defocus. The paper presents several methods to extract the Fourier coefficients needed for a 3-D reconstruction. The simplest of the methods is to take images with different amounts of defocus. For helical structures, however, only one image is needed.