Phase contrast transfer function

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... 

High-resolution electron microscopic studies employed a modified JEOL-JEM200CX (8) operated at 200 kV with objective lens characteristics Cs = 0.52 mm, Cc = 1.05 mm leading to a theoretical point resolution as defined by the first zero in the phase contrast transfer function of 1.95 A at the optimum or Scherzer (9) defocus position (400 A underfocus). 

Phase contrast transfer function. The phase shifts due to the combination of spherical aberration and defect of focus can be combined into a single phase factor x given by... 

Sin X is called the phase contrast transfer function (PCTF) of the objective lens. 

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). 

Fig. 1. The imaginary part of the phase contrast transfer function, sinX, plotted as a function in reciprocal space for a microscope operating at 200 kV with a spherical aberration coefficient of 1.2 mm a) Gaussian focus b) 325A underfocus c) 65OA underfocus.

Fig. 1.2. Phase contrast transfer function sin(2jt ) (solid line) and attenuation function (broken line) against u, for E — 200 kV and Cs = 1.2 mm.

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... 

A typical phase-contrast transfer function is shown in F 2. This representation of TEM performance is analogous to the audio frequency response of a hi-fi amplifier. The units of spatial frequency here are reciprocal nanometers (nm ), high values of spatial frequency corresponding to fine detail. The curve shows that the response of a conventional TEM in high-resolution mode is complicated and initially... 

Hgure 2 Typical phase-contrast transfer function of a high-resolution 200 kV TEM for LaBe and field-emission guns. (Reprinted with permission from Coene W, Janssen G, Op de Beeck M, and Van Dyck D (1992) Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy. Physical Review Letters 69 3743-3746. American Physical Society.) (Courtesy of Dr. W. Coene, Philips Research Laboratory, Eindhoven.)... 

FIGURE 19 (a) The corrector of Fig. 18 incorporated in a transmission electron microscope, (b) The phase contrast transfer function of the corrected microscope. Dashed line no correction. Full line corrector switched on, energy width (a measure of the temporal coherence) 0.7 eV. Dotted line energy width 0.2 eV. Chromatic aberration remains a problem, and the full benefit of the corrector is obtained only if the energy width is very narrow. [From Haider, M., et al. (1998). J. Electron Microsc. 47,395. Copyright Japanese Society of Electron Microscopy.]... 

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 ... 

These uncertain atoms remain to be verified by a careful structure refinement. For a structure refinement, as many reflections as possible should be included. The phases are not needed at the refinement stage, but if possible complete 3D data out to 1 A resolution should be used. Strong and weak reflections are equally important. Such data can be obtained by electron diffraction, which is not affected by the contrast transfer function of the electron microscope, but suffers from dynamical scattering. The higher the accuracy of the amplitudes, the more accurate will the atomic positions become. 

Within the weak-phase object approximation, the effect of the aberrations is most conveniently described by the Contrast Transfer Function (CTF), which gives the phase factor as a function of spatial frequency (diffraction angle). 

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... 

The columnar approximation implies that the object is thin, because the depletion of energy of the incident wave is neglected. This means that the amplitude of the incident beam Aq is kept constant, whereas only the phase is modified by the object crossing (phase object). To approach this condition, even at the exact Bragg angle, the object must be sufficiently thin to provide scattered beams that are very faint relative to the transmitted beam (yveak phase object). The resulting contrast is the object CTF (contrast transfer function). 

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. 

Crystal structure, phase information and, 40 CTF. See Contrast transfer frmction (CTF) Cucumber mosaic virus (CMV), 437 39 Cumulative envelop function (s), 96 Cystoviridae family, 230... 

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