3D Fourier Transform Analysis


It is noted further that the two linear terms of Eq. (1) are separate from each other on Ewald-sphere envelopes, except in the region close to the Fourier space origin as shown in Figure 4 . T erefore, we can separately analyze the two linear image contributions indepen- dently. T is means that we do not need to assume the relation


Figure 4 : Simulated Ewald-sphere envelopes for 80 kV electrons. (a) The top and bottom panes show uw- and uv-sections of the 3D FT, respectively. Here, the beam tilt is 2 degrees, and the beam convergence is 3 mrad. The uw-section shows two Ewald-sphere envelopes that are separate from each other except in the region close to the origin. (b) Ewald-sphere envelopes for beam-tilt angles of 1 and 3 degrees (top to bottom) and beam convergences of 0.5, 1.0, and 3.0 mrad (left to right). The Ewald-sphere envelope is sharp even for a rather large convergence angle of 3.0 mrad. Note that the Ewald-sphere envelope, especially the uw-section, is too thin to be drawn for a small beam convergence. The location of the Ewald sphere does not move by changing the beam convergence angle.


Here, we assume the distribution of incident beam directions is a Gaussian with half-width of q 0 . We can verify that the parameter w E ( g ) corresponds to the distance along the w -axis from the uv-plane to the Ewald sphere: [ 7 ]. The function E Ewald may be called for


the Ewald sphere envelope and takes the form


all the spatial frequencies g on the Ewald sphere ( w = w E ). In the limit of perfectly parallel illumination, the Ewald-sphere envelope becomes a delta function and takes the value of unity only on the Ewald sphere. Figure 4a


shows an example of the simulated Ewald-sphere envelopes, where the beam tilt is 2 degrees and the beam convergence angle is 3.0 mrad. We note that the intersection of the Ewald-sphere envelope with the uv-plane coincides with the achromatic circle. Other simulated Ewald-sphere envelopes with diff erent beam-tilt and conver- gence angles are shown in Figure 4b . T e Ewald-sphere envelope, especially the uw-section, is too thin to be drawn for a small beam convergence. It is important to note that the Ewald-sphere envelope is sharp even for a rather large convergence angle of 3.0 mrad and does not depend on the defocus-spread contrary to the temporal envelope. T erefore, we can measure the temporal envelope on the sharp Ewald-sphere envelope, even when the temporal envelope becomes broad for the case of a small defocus spread.


Acceleration voltage


80 kV


Linear information limit


11.2 /nm (90 pm)


(kinematical scattering) as required by diff ractogram analysis. T us, we can use a thick sample or a sample made from strong scattering elements, which will lead to dynamical scattering. T is is essential if we need to directly observe the linear image transfer down to a few tens of pm.


Applications


We use a TEM (FEI, TITAN 3 ) equipped with a monochromator, a C s -corrector for image forming (CEOS, CETCOR), a CCD camera (Gatan Inc., UltraScan), and an


energy fi lter (Gatan Inc., Quantum) operated at acceleration voltage of 80 kV. We have acquired a set of through-focus images from a thick amorphous carbon fi lm (36 nm) with a beam tilt of 36 mrad [ 4 ]. Here, a series of 129-images was acquired with a small defocus step of 1.37 nm around exact focus. T us, the total focus range was 176.7 nm. T e image size was 1024 × 1024 pixels with 2 × 2 binning, and the pixel size on the specimen was 26.5 pm. T e specimen driſt of the through-focus images was corrected using cross-correlation between neighbor images. Figures 5a and Figure 5b show respectively the vw- and uv-sections of the 3D FT of the stacked images of 512 × 512 pixels extracted from the driſt - corrected images. Here, the arcs are the intersections of the product of the temporal envelope E t and the Ewald-sphere envelope E Ewald (Eq. 4). From the uw-section, we measure the product


for a spatial frequency


g (see the top pane of Figure 4a or Figure 5a ). From the uv-section we measure another product on the circle of radius of g (see the bottom pane of Figure 4a or Figure 5b ), which gives


on the achromatic circle


, since there is no attenuation . When we note that the


Ewald-sphere envelope for the same spatial frequency takes the same value (namely,


), the ratio of the


measurement on uw-section to the one on uv-section, H ( g ), will give the temporal envelope:


Table 1 : Information limit (highest spatial frequency) at 80 kV estimated using various criteria.


Nominal information limit


13.7 /nm (73 pm)


Highest-order diffraction spot


12.7 /nm (79 pm)


Note: The linear information limit is determined by the 3D FT analysis using w E = 0.26 nm -1 , while the nominal information limit is estimated by the energy spread (FWHM) of 0.1 eV of incident electrons measured by EELS. The highest-order diffraction spot was observed from a gold particle.


46 www.microscopy-today.com • 2018 March


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