Exit Plane (or Object) Wave Function from TEM Specimen
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We can assume that the wave of the incident electron beam is perfectly even and coherent and the illumination electron wave is ψillum = 1. For the case of single atoms weak-phase object approximation can be assumed, and thus the exit plane wave function (or called object wave function) for an atom located at the position (αx, αy) can be given by,

          exit wave function ---------------------------- [4217a]

          V(x-αx, y-αy) -- The electrostatic potential of the object, projected along the beam direction z
          σ -- Interaction constant, given by

          Interaction constant ---------------------------------- [4217b]

         h -- Planck’s constant
         c -- Speed of light
         m0 -- Electron rest mass
         e -- Electron charge
         λ - Wavelength of electron wave

For the weak phase object approximation, a first order expansion may be applied to Equation 4217a and then we can have,

          exit wave function --------------------- [4217c]

To emphasize the physical meaning of Equation 4217c, it is sometimes divided into one part, 1, corresponding to the direct beam (ψillum) and one part related to the scattered electrons, Ψo,s [4], where Ψo,s is given by,

          Wave function of scattered electrons --------------------- [4217d]

The Fourier transform of the two-dimensional (2-D) exit wave is given by,

         wave function of the focused probes in STEM and SEM -------------------------[4217e]

          r -- A vector given by x and y.

Therefore, the Fourier transform of ψExit(r) gives,

          Fourier transform of ψExit(r) --------------------- [4217f]


          Fourier transform of ψExit(r) --------------------- [4217g]

For the case of atom columns of a crystal or a particular crystal defect, in a simplified channeling theory, an expression for the exit wave of a single atom column is expressed as [1 - 3],

         exit wave of a single atom column ----------------- [4217h]

          r -- A two-dimensional (2-D) vector in the plane at the exit face of the crystal ( = (x,y)T ), which is perpendicular to the incident beam direction
          z -- The crystal thickness
          C1s -- The excitation coefficient for plane wave incidence (excitation coefficient for plane wave incidence)
          E0 -- The incident electron energy
          Φ1s(r) -- The lowest energy bound state of the atom column
          E1s -- The energy of the lowest energy bound state

All the purpose of the methods of the electron image series reconstructions with TEM is to find an object wave function, ψo,s. The reconstruction methods can be basically categorized by two groups: i) weak phase object approximation (WPOA) and more generally ii) phase object approximation (POA). In principle, the methods based on the POA give a better estimation of the object wave function, but they don’t have closed form solutions and have to be computed iteratively and thus is very time consuming. [5] On the other hand, the reconstruction methods based on WPOA are fast but the TEM imaging requires much thinner samples.

Based on multislice simulation on SrTiO3 crystal structure (Figure 4217 (a)), Forbes et al. obtained the exit surface intensities with the total electron intensity, the elastically scattered electron intensity and the thermally scattered electron intensity. The simulation was carried out on a TEM specimen in thickness of 30 nm with a 300 keV electron probe.

exit surface intensities

Figure 4217. Exit surface intensity for (a) The projected structure of SrTiO3, (b) all electrons, (c) elastically scattered electrons and (d) thermally scattered electrons. Adapted from [6]


[1] D.Van Dyck, H. Lichte, K. D. van der Mast, Sub-angstrom structure characterisation: the Brite–Euram route towards one angstrom, Ultramicroscopy 64 (1996) 1–15.
[2] P. Geuens, D. VanDyck, The S-state model: a work horse for HRTEM, Ultramicroscopy 93 (2002) 179–198.
[3] S. Van Aert, P. Geuens, D. Van Dyck, C. Kisielowski, J. R. Jinschek, Ultramicroscopy 107 (2007) 551–558.
[4] P. Buseck, J. Cowley, and L. Eyring, High Resolution Transmission Electron Microscopy: And Associated Techniques. Oxford University Press US, 1992.
[5] A. I. Kirkland and R. R. Meyer, Indirect high resolution transmission electron microscopy: Aberration measurement and wavefunction reconstruction, Microscopy and Microanalysis, vol. 10, pp. 401–413, 2004.
[6] B.D. Forbes, A.J. D’Alfonso, S.D. Findlay, D. Van Dyck, J.M. LeBeau, S. Stemmer, L.J. Allen, Thermal diffuse scattering in transmission electron microscopy, Ultramicroscopy 111 (2011) 1670–1680.





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