Abstract
In this chapter, we study the descriptions of waves and their extension to electron waves in TEM.
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- 1.
The wave function ψ satisfies Schrodinger equation , but it is considered that the absolute product shows probability of existence of an electron, but does not correspond to an electron as a particle, even if making a wave packet by addition of various wave functions (famous debate between Born and Einstein). Nowadays, we have an interpretation to fill in the gap using a concept of “quantized wave function ” as a field operator, or “second quantization ” (see Tomonaga 1966). Another idea using a stochastic process was proposed by Nagasawa (2000).
- 2.
When we solve the Schrodinger equation in cylindrical coordinates (r, ϕ, z ), one of the solutions is a “vortex wave ” running along z-axis with advance of phase by rotation ϕ. The wave was generated experimentally using a special fork aperture in a condenser lens (McMorran et al. 2011).
- 3.
The calculation of aberration using the power series in the order gives spherical aberration , astigmatism , field curvature , distortions , and coma , which are named Seidel’s five aberrations . In the former part of this textbook, we study the image resolution of TEM determined by the spherical aberration and astigmatism.
- 4.
1 rad is about 57°, using a relation of 2π radian = 360°.
- 5.
We should note the relation between an image and the corresponding diffraction pattern. Basically, the wave function producing the image is related to that of the diffraction pattern by 2D Fourier transform .
- 6.
We obtain 2D Fourier coefficients F(u, \( {\upsilon}\)) by setting ω = 0, which corresponds to a cross section in 3D reciprocal space F(u, \( {\upsilon}\), ω).
- 7.
References
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Nagasawa, M. (2000). Stochastic process in quantum physics. Basel: Birkhauser.
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Tanaka, N. (2017). Basic Theories of TEM Imaging. In: Electron Nano-Imaging. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56502-4_3
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DOI: https://doi.org/10.1007/978-4-431-56502-4_3
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