Defining Theoretical Limits of Aberration-Corrected Electron Tomography: New Bounds for Resolution, Object Size, and Dose

The theoretical limits of electron tomography have long been defined by the Crowther criterion, which relates 3D resolution to the number of projections acquired, and the dose fractionation theorem, which dictates total dose requirement across projections. However, these relations are invalid for aberration-corrected scanning transmission electron microscopy (STEM) where high convergence angles limit the ability to reconstruct objects larger than the depth-of-focus (c.a. 5 nm). We show overcoming the limitations of aberration-corrected STEM tomography requires collecting information beyond a traditional tilt series by acquiring a through-focal stack at every tilt. Here, information is no longer mapped to a plane in Fourier (k) space, but becomes a volumetric toroid.

Here we present a theoretical foundation for aberration-corrected electron tomography by establishing analytic descriptions for resolution, sampling, object size, and dose—with direct analogy to the Crowther criterion. The 3D structure of a contrast transfer function (CTF) for through focal tomography where every specimen tilt measures a toroid with petal-shaped cross-section. A remarkable feature of the 3D CTF is the overlapped regions that permit complete information collection—unachievable with conventional tomography. This breaks expected Crowther relationships and the maximum reconstructable object size is unlimited up to spatial frequency k_c. At resolutions beyond 2/k_c, Crowther-like tradeoffs define the maximum object size (D) allowed for given 3D resolution (d).

When the tilt angle spacing becomes smaller than twice the convergence angle (2α) a continuum of information is measured and object size is unbounded at midband resolution. This occurs under typical instrument operation (> 25 mrad) and sampling (< 2° tilt). We show atomic resolution (1Å) 3D imaging is allowed across extended objects (> 20 nm) using currently available microscopes and modest specimen tilting (< 3°).