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3D Fourier Transform Analysis

The diffractogram in the inset shows spots representing the smallest resolved lattice spacings of a gold particle: a 79 pm spacing when the monochromator is ON (right) and a 100 pm spacing when the monochromator is OFF (left). On the other hand, the diffuse scattering from amorphous Ge film appears in both cases up to about 100 pm (10 nm -1 ). Then, a question arises: did the information limit actually improve down to 79 pm when the monochromator is used? We will answer this question in the Application and Discussion sections.

In the ideal case, the image intensity can be written as , where 1 and

the incident wave and the scattered wave, respectively. Here, the second and third terms are linear, and the fourth term is non-linear in terms of the scattered wave

. However,

an imaging system always suff ers from geometrical and chromatic aberrations. Furthermore, the illumination in a TEM is neither perfectly parallel nor monochromatic. T erefore, the image intensity in the real world is the sum of the ideal image intensity for each incident beam direction at each defocus setting. Such non-ideal illumination leads to partial coherency (some physicists describe this as partial incoherency) in conjunction with geometrical and chromatic aberrations. T is partial coherency is commonly studied in Fourier space by taking the FT of the image intensity. Diff ractogram (2D FT) analysis . A diff ractogram is a power spectrum (squared modulus of a 2D FT) of a single TEM image and has been used to characterize an imaging system. T e FT of the-real-world image intensity may be written using the (2D) transmission-cross-coeffi cient (TCC) T 2 ( k 1 , k 2 ; z ), which describes the contribution from two spatial sample frequencies k 1 and k 2 to the image frequency g = k 1 - k 2 due to partial coherency at a defocus z (please consult [ 5 ] for details). For a perfectly coherent imaging system, T 2 ( k 1 , k 2 ; z ) = 1 for all spatial frequencies and defocus.

For tilted-beam illumination the FT of image intensity may be written by using the TCC where k 1 or k 2 is set to the tilted incident-beam direction :

represent

Figure 2 : Simulated temporal envelope of tilted-beam diffractograms for 80 kV with a defocus spread of (a) 5.0 nm and (b) 0.5 nm. The beam-tilt angles from top to bottom are 1, 3, and 5 degrees, respectively. The scale bars of (a) and (b) are 10 and 20 nm -1 , respectively, and the square box in (b) corresponds to the whole area of (a). The circle, or the arc, of each fi gure shows the achromatic circle, where there is no attenuation of the temporal envelope. The temporal envelope becomes narrow for a large beam tilt, and it becomes broad for a small defocus spread.

where (g) is a FT of the scattered wave and the second and third terms correspond to the linear terms of the image formation. The TCC may be written by

T e temporal envelope for a Gaussian defocus spread with the half-width ∆ may be written as:

is the wave aberration, and E t and E s are the temporal and spatial envelopes that describe the reduction in contribution to the image formation due to chromatic and geometrical aberrations, respectively. T e spatial envelope E s could be ignored for a C s -corrected high-performance microscope since the spatial envelope depends on the gradient of the wave aberration (that is, geometrical aberration). T erefore, the temporal envelope becomes more important than ever for a C s -corrected microscope.

where 44

T e defocus spread ∆ for a chromatic aberration coeffi cient C C will be determined by the energy spread of the emitted electrons ∆ E , the instability in the high-voltage supply ∆ V , and the instability in the objective lens current

,

where V is the mean high-voltage and I is the mean objective lens current. T is temporal envelope becomes a familiar expression: for normal illumination

,

which says the contribution will decay with the forth power of spatial frequency. In the case of the tilted-beam illumination, the temporal envelope has no attenuation where

,

which defi nes the circle of radius located at . Since there is no attenuation on this circle due to chromatic aberration, we call it an achromatic circle. Figure 2 shows simulated temporal

www.microscopy-today.com • 2018 March

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The diffractogram in the inset shows spots representing the smallest resolved lattice spacings of a gold particle: a 79 pm spacing when the monochromator is ON (right) and a 100 pm spacing when the monochromator is OFF (left). On the other hand, the diffuse scattering from amorphous Ge film appears in both cases up to about 100 pm (10 nm -1 ). Then, a question arises: did the information limit actually improve down to 79 pm when the monochromator is used? We will answer this question in the Application and Discussion sections.

In the ideal case, the image intensity can be written as , where 1 and

the incident wave and the scattered wave, respectively. Here, the second and third terms are linear, and the fourth term is non-linear in terms of the scattered wave

. However,

an imaging system always suff ers from geometrical and chromatic aberrations. Furthermore, the illumination in a TEM is neither perfectly parallel nor monochromatic. T erefore, the image intensity in the real world is the sum of the ideal image intensity for each incident beam direction at each defocus setting. Such non-ideal illumination leads to partial coherency (some physicists describe this as partial incoherency) in conjunction with geometrical and chromatic aberrations. T is partial coherency is commonly studied in Fourier space by taking the FT of the image intensity. Diff ractogram (2D FT) analysis . A diff ractogram is a power spectrum (squared modulus of a 2D FT) of a single TEM image and has been used to characterize an imaging system. T e FT of the-real-world image intensity may be written using the (2D) transmission-cross-coeffi cient (TCC) T 2 ( k 1 , k 2 ; z ), which describes the contribution from two spatial sample frequencies k 1 and k 2 to the image frequency g = k 1 - k 2 due to partial coherency at a defocus z (please consult [ 5 ] for details). For a perfectly coherent imaging system, T 2 ( k 1 , k 2 ; z ) = 1 for all spatial frequencies and defocus.

For tilted-beam illumination the FT of image intensity may be written by using the TCC where k 1 or k 2 is set to the tilted incident-beam direction :

represent

Figure 2 : Simulated temporal envelope of tilted-beam diffractograms for 80 kV with a defocus spread of (a) 5.0 nm and (b) 0.5 nm. The beam-tilt angles from top to bottom are 1, 3, and 5 degrees, respectively. The scale bars of (a) and (b) are 10 and 20 nm -1 , respectively, and the square box in (b) corresponds to the whole area of (a). The circle, or the arc, of each fi gure shows the achromatic circle, where there is no attenuation of the temporal envelope. The temporal envelope becomes narrow for a large beam tilt, and it becomes broad for a small defocus spread.

where (g) is a FT of the scattered wave and the second and third terms correspond to the linear terms of the image formation. The TCC may be written by

T e temporal envelope for a Gaussian defocus spread with the half-width ∆ may be written as:

is the wave aberration, and E t and E s are the temporal and spatial envelopes that describe the reduction in contribution to the image formation due to chromatic and geometrical aberrations, respectively. T e spatial envelope E s could be ignored for a C s -corrected high-performance microscope since the spatial envelope depends on the gradient of the wave aberration (that is, geometrical aberration). T erefore, the temporal envelope becomes more important than ever for a C s -corrected microscope.

where 44

T e defocus spread ∆ for a chromatic aberration coeffi cient C C will be determined by the energy spread of the emitted electrons ∆ E , the instability in the high-voltage supply ∆ V , and the instability in the objective lens current

,

where V is the mean high-voltage and I is the mean objective lens current. T is temporal envelope becomes a familiar expression: for normal illumination

,

which says the contribution will decay with the forth power of spatial frequency. In the case of the tilted-beam illumination, the temporal envelope has no attenuation where

,

which defi nes the circle of radius located at . Since there is no attenuation on this circle due to chromatic aberration, we call it an achromatic circle. Figure 2 shows simulated temporal

www.microscopy-today.com • 2018 March

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