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Figure 1
(a) The scattering–focusing geometry in the electron microscope leads to two spherical peaks in intensity: the conjugate Ewald spheres. The lower Ewald sphere (red) is the conjugate (negative sign in the exponent) of the upper Ewald sphere (blue). However, the structure factor in the upper left is the conjugate of that in the lower right: [F_{\rm 3D}^*(-u, -v,w) = {F_{\rm 3D}}(u,v, -w)] (green) and the structure factor in the upper right is the conjugate of that in the lower left: [F_{3{\rm D}}^*(-u, -v, -w) = {F_{\rm 3D}}(u,v,w)] (orange), adhering to Friedel symmetry. A focal series intersects the focusing rays at different heights, leading to a delocalization of where the electrons hit on the detector. The latter results in the contrast transfer function (CTF) that must be corrected for reconstruction. (b) A stack of focal series micrographs Fourier transformed in 2D gives a stack of power spectra showing the change in CTF with a change in focus. A 3D Fourier transform reveals two spheres with their peak ridges coinciding with the Ewald spheres. Performing a correction for one half of the CTF {exp[−iγ(s)]} on the stack of 2D transforms, and transforming the result only in the z (or w) direction, flattens one Ewald sphere and doubles the curvature of the other. The middle section of this 3D transform (indicated by *) is equivalent to the sum of the corrected 2D transforms, and on back transformation in 2D gives the focal series reconstruction with appropriate real-space phases.

IUCrJ
Volume 13| Part 1| January 2026| Pages 77-93
ISSN: 2052-2525
https://doi.org/10.1107/S2052252525010796