Deviation of Gaussian Distribution
of Defocus due to Chromatic Aberration
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Simply speaking, Fourier transform is a limiting case of Fourier series. The structure factor F(u) is proportional to the Fourier components Iim(u) of the HREM image. For a weak-phase-object, the Fourier transform of the HREM image Iim(u) can be given by [1],

         Structure Factor and HRTEM -------------- [4974a]

         dark rings of Fourier transform of HRTEM images locate at ------------------- [4974b]

          Fourier transform ---------------------- [4974c]

where,
         Δf -- The defocus value
         Cs -- The spherical aberration coefficient
         k’ -- A constant
         F(u) -- Structure factor of electron diffraction (ED), which is the Fourier transform of the potential distribution φ(r) of the object
         λ -- The electron wavelength
         u -- The reciprocal-lattice vector within the resolution limit      
         D -- The standard deviation of the Gaussian distribution of defocus due to the chromatic aberration [2]

The values of Δf (defocus value), Cs (spherical aberration coefficient), and D (standard deviation of the Gaussian distribution of defocus due to the chromatic aberration) can be obtained by TEM image deconvolution.

 

 

[1] X. D. Zou. Crystal Structure determination by crystallographic image processing. in “Electron Crystallography”, eds. D. L. Dorset, S. Hovmöller, X. D. Zou, Nato ASI Series C, Kluwer Academic Publishers, Dordrecht, 1997, pp163-181.
[2] Fijes PL. 1977. Approximations for the calculation of high-resolution electron-microscopy images of thin films. Acta Cryst A33:109–113.

 

 

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