Fourier Transforms and Images. Our aim is to make a connection between diffraction and imaging - and hence to gain important insights into the process. What happens to the electrons as they go through the sample?. What happens to the electrons.
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- and hence to gain important insights into the process
a) The electrons in the incident beam are scattered into diffracted beams.
b) The phase of the electrons is changed as they go through the sample. They have a different kinetic energy in the sample, this changes the wavelength, which in turn changes the phase.
Therefore, we must be able to find a way of linking the descriptions. The link is the Fourier Transform.
A well-known example is the Fourier series. To make a periodic function add up sine waves with wavelengths equal to the period divided by an integer.
Transmission Electron Microscopy
The same idea as the Fourier series
but the function is not periodic, so all wavelengths of sine waves are needed to make the function
But each sine wave term in the sum of waves is equivalent to two plane waves at different angles
This can be seen from considering the Young's slits experiment - two waves in different directions make a wave with a sine modulation
2d sin q = l
we have seen this before
2d sin θ = λ
tells us where there are diffracted beams.
A lens brings electrons in the same direction at the sample to the same point in the focal plane
Direction at the sample corresponds to position in the diffraction pattern - and vice versa
Back focal plane