Focus Scherzer

Figure Bl.17.5. Examples of CTFs for a typical TEM (spherical aberration = 2.7 mm, 120 keV electron energy). In (a) and (b) the idealistic case of no signal decreasing envelope fimctions [77] are shown, (a) Pure phase contrast object, i.e. no amplitude contrast two different defocus values are shown (Scherzer focus of 120 mn imderfocus (solid curve), 500 mn underfocus (dashed curve)) (b) pure amplitude object (Scherzer focus of 120 mn underfocus) (c) realistic case mcluding envelope fimctions and a mixed weak...

Figure 2. Partially coherent contrast transfer function exemplified for a Tecnai F30 with U-TWESl objective lens at 300kV at extended Scherzer focus (left) and 2 Zs (right). The blue vertical lines indicate the effect of the CTF on the diffracted beams of silicon in [110] orientation (the 111, 200,220, 311,222,400, 331, and 333 beams).

Figure 1 shows an experimental image of a T-0 layer silicate (lizardite). The 0.7-nm periodicity is clearly apparent, and no subperiodicities (<0.7 nm) or superperiodicities (>0.7 nm) can be detected. This image was obtained at or near Scherzer focus, and the crystal was oriented such that the basal planes were perfectly parallel to the electron beam, as determined by electron diffraction patterns of the areas from which the images were obtained. Thus, this image was taken under nearly ideal conditions in that both the focus conditions and orientation were carefully controlled. 

Figure 1. Experimental HRTEM image of a T-0 layer silicate (lizardite). Microscope was focussed near the Scherzer focus.

Figure 2. Computer-simulated image of lizardite. a) Scherzer focus, b) +50 nm overfocus.

Figure 5. Computer-Simulated image at Scherzer focus of chlorite.

Figures 10a and 10b show experimental images of a mixed-layer illite/smectite obtained with the c -axis perpendicular to the electron beam Figure 10a was obtained with the objective lens near Scherzer focus, and Figure 10b was obtained from the same area with...

In principle, actual sequences of illite and smectite layers can be observed directly with HRTEM imaging. However, since the illite and smectite layers differ only slightly in their compositions (subtle differences in K, Al, and Si), they may be difficult to differentiate in HRTEM images. Indeed, the two different layer types are barely distinguishable near the Scherzer focus (Figure 10a), but the two layers are clearly visible for overfocus conditions (Figure 10b). This image allows direct observation of the sequence S-I-S-I-I-I-S-I-I-I-I-I-S-I from top to bottom. 

By differentiating the Scherzer equation, an interpretable resolution limit can be derived, r = at the Scherzer focus AF = where A and B are... 

Objective aperture. In general, the aperture chosen should be just large enough to pass all those diffracted beams with g just below the FZC of the PCTF at Scherzer focus, as explained in Section 6.2. It is critical that the aperture be accurately centered. 

Therefore, a structure image can be obtained from a weak phase object first by setting the defocus value at and then by cutting out those diffracted waves whose spatial frequencies are larger than u. These observation conditions derived from eq. (1.13) are called the Scherzer imaging conditions and the defocus value is called the Scherzer focus. 

Let us consider the phase change at the Scherzer focus geometrically. In Fig. 1.1(b) solid and broken lines represent the peaks and valleys of the phase, respectively, for the central and diffracted waves [1.1]. For simplicity, only one... 

Muscovite-2Mi viewed along [100], (b) Annite-2Mi viewed along [100]. (c) Annite-2Mi viewed along [010]. The parameters for the simulation are defocus = -42 nm (Scherzer focus) and specimen thickness = 2.5 nm. 

In the optical microscope the Zernike phase contrast method is applied to convert the phase modulation into a contrast, by adding a phase plate in the back focal plane. This is very close to the wave aberration at the Scherzer focus, but the performance of the electron microscope is rather limited. Even the minimum Cs available today does not allow a carbon atom to be visualized. The solution is to add a new spherical corrected objective lens for sub-angstrom resolution. Figure 13 shows the optical path of a beam in a TEM working under holographic conditions. 

It is apparent from the ray diagram of Fig. 3.3 that there is a plane, A-A, nearer to the lens than the geometrical or Gaussian focus plane, where the resolution is improved. It is called the plane of least confusion or Scherzer focus [3, 4, 7], and it is close to where the rays from the outermost parts of the lens intersect the axis. From Fig. 3.3 the distance 6Z between this plane and the Gaussian image plane is approximately the radius of the image disc in the Gaussian plane divided by a. The radius is Ma C and a = ajM, so... 

Fig. 3.3 Spherical aberration causes rays at larger angles to the axis of the lens to come to a focus short of the ideal focal plane. The smallest image of a point source appears in the plane A-A, at the Scherzer focus position.

Defocus is conventionally given as a motion of the object, as one would focus an optical microscope.) Comparing Scherzer focus to geometric focus, the resolution is improved by a factor of about 2 if the included divergence angle is... 

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