Imaging, phase-contrast transfer

The image intensity /(x, y) at the image plane of the objective lens results from two-dimensional Fourier synthesis of the diffracted beams (the square of the FT of the waves at the exit face of the crystal), modified by a phase-contrast transfer function factor (CTF, sin /), given by Scherzer (1949), as... 

In addition, there are various technical corrections that must be made to the image data to allow an unbiased model of the structure to be obtained. These include correction for the phase-contrast transfer function (CTF) and, at high resolution, for the effects of beam tilt. For crystals, it is also possible to combine electron diffraction amplitudes with image phases to produce a more accurate structure (7), and in general to correct for loss of high resolution contrast for any reason by "sharpening" the data by application of a negative temperature factor (22). 

We note from this formula that the image contrast closely depends on the phase contrast transfer function sin2jtx(M). As shown in Fig. 1.2, the value of m.2TX,x u) changes strongly depending on the defocus e. However, if such a condition as... 

This equation shows that only limited information is preserved. In particular, depending on the spatial frequency Mo, no information is transferred at all at the zeroes of the phase-contrast function sin(x). The loss of information is even more serious when the phase object approximation holds and for ideal imaging in that case the phase information is completely lost in the Gaussian image of the object and special methods the so-called phase-contrast [94,95] methods should be employed in order to partly recover this information. 

In HREM images of inorganic crystals, phase information of structure factors is preserved. However, because of the effects of the contrast transfer function (CTF), the quality of the amplitudes is not very high and the resolution is relatively low. Electron diffraction is not affected by the CTF and extends to much higher resolution (often better than lA), but on the other hand no phase information is available. Thus, the best way of determining structures by electron crystallography is to combine HREM images with electron diffraction data. This was applied by Unwin and Henderson (1975) to determine and then compensate for the CTF in the study of the purple membrane. 

H(u) is the Fourier Transform of h(r) and is called the contrast transfer function (CTF). u is a reciprocal-lattice vector that can be expressed by image Fourier coefficients. The CTF is the product of an aperture function A(u), a wave attenuation function E(u) and a lens aberration function B(u) = exp(ix(u)). Typically, a mathematical description of the lens aberration function to lowest orders builds on the Weak Phase Approximation and yields the expression ... 

It is well known that under the weak-phase-object approximation (WPOA) [19], the image intensity function is linear to the convolution of the projected potential distribution function cpt (x, y) and the inverse Fourier transform (FT) of the contrast transfer function (CTF) r(u) of the electron microscope ... 

Transmission electron microscopy (TEM) has been an underutilized yet valuable too in particle size characterization of MC particles in LB films. Monolayer films of trioctylphosphine oxide-capped CdSe (18), spread as a monolayer on an aqueous subphase, were transferred to a TEM grid. A close-packed hexagonal arrangement of 5.3-nm (cr —4%) crystallites was found. TEM images were also obtained for HMP-stabilized CdS incorporated in BeH/octadecylamine films (79) and for CdS formed under an amine-based surfactant monolayer and transferred to a TEM grid (14). In one study, direct viewing of CdS and CdSe particles made from Cd2+-FA films on TEM grids was not possible due to poor phase contrast between the particles and the film (30). Diffraction patterns were observed, however, that were consistent with crystalline (3-CdS or CdSe. Approximately spherical particles of CdSe could... 

An alternative readout system is a scanning differential phase-contrast microscope with a split detector as shown in Figure 16.5. The optical configuration is compact and easy to align. The memory medium, in which the data bits have been recorded, is located at the focus of an objective lens. The band limit of the optical transfer function (OTF) is the same as that of a conventional microscope with incoherent illumination. The resolution, especially the axial resolution of the phase-contrast microscope, is similar to that obtained by Zemike s phase-contrast microscope. The contrast of the image is much improved compared to that of Zernike s phase-contrast microscope, however, because the nondiffracted components are completely eliminated by the subtraction of signals between two detectors. The readout system is therefore sensitive to small phase changes. 

Such is not the case if, instead of the axial detector, we employ the annular dark-field detector for which P la 1, where P is the effective angle subtended by the detector. Under these circumstances we anticipate that phase contrast will not contribute significantly to the image. Instead, modulation of the amplitude-contrast transfer function should be noted in an image if, for example, a probe of FWHM comparable to atomic separations is scanned across a sharp edge or a periodic structure. This is observed in Figure 8, in which a probe of FWHM 3 A is scanned across... 

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