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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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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

Page 1 | Page 2 | Page 3 | Page 4 | Page 5 | Page 6 | Page 7 | Page 8 | Page 9 | Page 10 | Page 11 | Page 12 | Page 13 | Page 14 | Page 15 | Page 16 | Page 17 | Page 18 | Page 19 | Page 20 | Page 21 | Page 22 | Page 23 | Page 24 | Page 25 | Page 26 | Page 27 | Page 28 | Page 29 | Page 30 | Page 31 | Page 32 | Page 33 | Page 34 | Page 35 | Page 36 | Page 37 | Page 38 | Page 39 | Page 40 | Page 41 | Page 42 | Page 43 | Page 44 | Page 45 | Page 46 | Page 47 | Page 48 | Page 49 | Page 50 | Page 51 | Page 52 | Page 53 | Page 54 | Page 55 | Page 56 | Page 57 | Page 58 | Page 59 | Page 60 | Page 61 | Page 62 | Page 63 | Page 64 | Page 65 | Page 66 | Page 67 | Page 68 | Page 69 | Page 70 | Page 71 | Page 72 | Page 73 | Page 74 | Page 75 | Page 76