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Based on the Rayleigh criteria, the ability (spatial resolution) of the microscope to resolve two spots as two essentially relates to how sharply the Airy disks are defined. As the disks get closer, the intensity dip between the two spots becomes smaller. For instance, the two spots can be resolved in Figure 3743 (a) and (b), while they cannot be resolved in Figure 3743 (c).
Figure 3743. The two resolvable spots in (a) and (b) and the two irresolvable spots in (c). Adapted from [1]
For an ideal objective lens, the incident electron probe simply forms an Airy disk in the back focal plane of the lens. This disk is the Fourier transform of the uniformly illuminated condenser aperture.
The radius of the Airy disk d_{d} due to the diffraction limit is written as,
------------------------------ [3743a]
where,
α -- The convergence semi-angle (also called maximum diffraction
angle);
λ -- The wavelength of the incident electron beam.
Note that λ increases at lower acceleration voltages.
In conventional TEM system, the existence of spherical aberration requires the utility of very small apertures to maximize the spatial resolution, while the resolution will also be limited by diffraction if the apertures are too small.
The diffraction effect of the objective aperture is important for HRTEM imaging. The radius of the Airy disk produced by the objective aperture is given by,
r_{Airy} ~ 1.2λf/D ---------------- [3743b]
where,
f -- The focal length of the lens;
D -- The aperture diameter.
[1] Methods in Cell Biology, Vol. 81, DOI: 10.1016/S0091-679X(06)81002-2, David E. Wolf, The Optics of Microscope Image Formation, (2007). |