Structure Factors and HRTEM
- Practical Electron Microscopy and Database -
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This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.



In the case of electron scattering in TEM, the structure factor, Fhkl, can be given by,

         Structure Factor and HRTEM ------------------- [3880a]

         λ -- The wavelength of the incident electrons,
         σ -- The interaction constant between the incident electrons and specimen,
         Φhkl -- The Fourier coefficients of an electrostatic potential.

In the first Born approximation (kinematic electron scattering theory), the structure factors are given by the expression,

         Structure Factor and HRTEM ------------------- [3880b]

         fjT -- The respective temperature factors for all j atoms in the unit cell,
         fj -- The electron atomic scattering factors.

Furthermore, 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 ------------------- [3880c]

         k’ -- A constant
         T(u) -- The contrast transfer function (CTF)

The objective lens transfers different structure factors F(u) into the HRTEM images in two different ways, depending on the contrast transfer function D(u)sinχ1(u):

       i) Phases of the Fourier components
       ii) Amplitudes of the structure factors determined by |D(u)sinχ1(u)|

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. Of these three factors, Cs and D can be determined experimentally. With the estimated values of Cs and D, a set of structure factor of ED (electron diffraction) F(h) can be calculated from equation of Fourier transform (Iim(u)) for a given value of Δf.

Some chemical elements cannot induce contrast in TEM images. For instance, the presence of helium atoms in palladium tritides does not induce a detectable change in the HRTEM contrast. [2] This can be interpreted by the low scattering factor of helium atoms compared to palladium one.

The structure factors can cause all the reflections in a FOLZ to be forbidden so that the first ring of spots in the experimental diffraction pattern is from the second layer of the reciprocal lattice but we still call the SOLZ as the FOLZ.






[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] S. Thiébaut, B. Décamps, J. M. Pénisson, B. Limacher, A. Percheron Guégan, TEM study of the aging of palladium-based alloys during tritium storage, Journal of Nuclear Materials 277 (2000) 217-225.




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