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Linear contrast transfer theory can be applied to weakly scattering objects, for instance thin amorphous TEM specimen, which leads only to a small phase shift on the diffracted electron wave. The diffractogram intensity is the power spectrum of the recorded image and can be given in the simple coherent form,
D(g) ≈ O(g)^{2} sin^{2}[χ(g)]  [4191]
where,
g  Object frequency vector
χ  Wave aberration function
sin^{2}[χ(g)]  Oscillating patterns of circular, elliptic, or hyperbolic shape (Thon rings [1])
O(g)^{2}  Scattering power of thin amorphous objects
O(g)^{2} decreases slowly towards higher spatial frequencies and is azimuthally isotropic.
Thon rings are a phenomenon revealed in the power spectra of micrographs by brightfield (BF) TEM (transmission electron microscopy) imaging. These rings can be explained as the effect of the contrast transfer function, which modulates the Fourier transform of the object in a defocusdependent way. Figure 4191a (A) shows the power spectrum of a typical BF TEM image of amorphous carbon film presenting concentric Thon rings. Those white rings correspond to the contrast transfer maxima and the dark rings indicate spatial frequency bands without signal. Figure 4191a (B) shows the radial intensity of the power spectra. The astigmatism and defocus can affect the symmetry of the rings, limiting the spatial resolution of the microscope. Therefore, electron micrographs, especially HRTEM, are routinely inspected by optical diffraction before taking images for analysis.
Figure 4191a. (A) Power spectrum of typical brightfiled image of amorphous carbon film presenting
concentric Thon rings taken in TEM. (B) Radial intensity of the power spectra. 
Zeroloss filtering of elastically scattered electrons in EFTEM
removes the contribution of inelastically scattered electrons and increases the contrast of amorphous, smallangle and
DebyeScherrer diffraction patterns.
[1] F.Thon and Z. Naturforsch, 21a (1966) 476.
