3D Fourier Transform Analysis
image does not indicate a linear spatial resolution of a microscope, although such information exists in the image intensity. This is because the spatial frequency up to twice the image wave frequency is recorded in the image intensity. Surprisingly, the diffuse scattering from an amorphous Ge film appears only up to about 100 pm (see the right pane of Figure 1 ), and thus Young’s fringe methods gives a resolution of 100 pm, which is worse than the estimate from the 3D FT analysis. This indicates that the rotationally symmetric attenuation factors surpass the normal-illumination temporal envelope at high spatial frequency when the monochro- mator is on.
Figure 5 : The 3D FTs of experimental though-focus images. (a) and (b) show the uw- and uv-sections of the 3D FT, respectively. Here, the sample is a thick amorphous carbon (36 nm). A series of 129 images was acquired using FEI, TITAN 3 equipped with a monochromator operated at an acceleration voltage of 80 kV. After the specimen drift correction, the region of 512 × 512 pixels is extracted for a 3D FT. The white arcs are the sections of two Ewald spheres on the corresponding planes. The Ewald sphere on the uv-plane coincides with achromatic circle. The intensity of the 3D FT is measured on each section for the spatial frequency g as illustrated. The ratio of these measurements is used to plot each point in Figure 6 .
An information limit is determined by the defocus spread that depends on three terms: ∆ E (the energy spread of emitted electrons), ∆ V (the instability in the high-voltage) and ∆ I (the instability of lens current). Among them the instability of lens current does not change the velocity of electrons, although it
T e last equation is obtained by noting .
T is means that the ratio H ( g ) is a simple function of w E , which is the distance to the Ewald sphere for the spatial frequency g . Figure 6 plots some H values measured at several values of w E . In this case, the experimental values decrease to 1/e 2 ( H ( g ) = 13.5%) at w E of 0.26 nm. -1 T en, from the relation we obtain the defocus spread ∆ of 1.73 nm. Finally, with this defocus spread, the temporal envelope for normal illumination, exp
, gives the spatial frequency of 11.2 nm -1 using
the 1/e 2 (13.5%) criterion [ 8 ]. T is spatial frequency corresponds to a linear information limit of 90 pm ( Table 1 ).
Discussion T e linear information limit of our monochromated
TITAN 3 is evaluated to be 90 pm (11.2 nm -1 ) when the monochromator is on. However, we have shown that the image taken with the monochromator gives a faint spot at 79 pm from a gold particle ( Figure 1b ). From the 3D FT analysis, we can conclude that the faint spot should result from the non-linear eff ect. T us, the information limit actually is not improved down to 79 pm by using the monochromator. We may note that in most of the cases a faint spot in the FT of a HRTEM
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slightly changes the shape of electron beam. T erefore, we can measure the sum of ∆ E and ∆ V by using the electron energy fi lter that disperses the electrons by their velocity (kinetic energy), and thus estimate a nominal information limit due to the energy spread of incident electrons. We have measured the energy spread (FWHM) of incident electrons using the energy fi lter (Gatan, Inc., Quantum) as 0.1eV when the monochro- mator is on. T e energy spread of 0.1eV (FWHM) of incident electrons gives a defocus change ∆ FWHM of 1.9 nm [ 3 ] and a defocus spread ∆ is estimated to be 1.14 nm from the relation . T us, we obtain the nominal information limit
of 73 pm ( Table 1 ), which is much better than the information limit of 90 pm measured by the 3D FT analysis. T is indicates that other sources of chromatic aberration, namely the objective lens current fl uctuation and/or instability of the C s -corrector, cannot be ignored when the energy spread is reduced to 0.1 eV by using the monochromator. In other words, the objective lens and/or the C s -corrector should be stabilized further in order to enjoy the full benefi t of the monochromator. T us, it becomes apparent that the 3D FT analysis is an indispensable tool to evaluate a high-performance microscope. T is is the answer to the question posed in the title of this article.
Next, we would like to discuss the practical aspects of the 3D FT. Whereas the 3D FT analysis looks to be rather involved compared with the 2D FT analysis (diffrac- togram), these days we can easily acquire through-focus images automatically using a homemade script or a free plug-in for DigitalMicrograph [ 9 ]. Furthermore, we can perform a FT of a 3D data set using a built-in command of
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