Contrast transfer function

B1.17.5.1 IMAGING OF PROJECTED STRUCTURE—THE CONTRAST TRANSFER FUNCTION (CTF) OF TEM... 

Frank J and Penczek P 1995 On the correction of the contrast transfer function in bioiogicai eiectron microscopy Optik 38 125-9... 

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 ... 

Whenever possible, an amorphous area at the edge of the crystal should be included in the HREM images when images are recorded. This will help in the determination of the contrast transfer function. A set of images with different defocus values should be recorded, although it is often possible to solve structures from just a single image. The reasons will be described in section 9. 

As mentioned in section 6, the structure factors F(u) are proportional to the Fourier components lim(u) of the HREM image and the projected potential is proportional to the negative of the image intensity, if the image is taken Scherzer defocus where the contrast transfer function T(u) -1. In general, the Fourier components lim(u) are proportional to the structure factors F(u) multiplied by the contrast transfer function (CTF). The contrast transfer function T(u) = D(u)sinx(u) is not a linear function. It contains two parts an envelope part D(u) which dampens the amplitudes of the high resolution components ... 

Figure 7 Contrast transfer functions T(u) at defocus values e = -850 A, -165A and -525A. The optical parameters are from a Philips CM30/ST microscope U = 300 kV, Cs=1.15 mm, A = 70 A and a = 1.2 mrad. All the three contrast functions have a common first crossover position at u = 0.272 A. The defocus value -850 A was determined to be the correct defocus...

Mathematical CTF correction calculating first the mathematical contrast transfer function T(u) from the estimated defocus values. Then the structure factor is calculated from the Fourier transform lim(u) of the image for all u except those with sinx(u) 0 by ... 

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). 

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. (a) Image formation in an electron microscope Ro - undiffracted beam O -objective aperture A as placed in (a) (b) Contrast transfer function of JEOL JEM 2010 200 kV electron microscope Scherzer underfocus 8 = -43.4 nm, a = 0.6 mrad, g = 5 nm ... 

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... 

Figure 2.6. An example of a contrast transfer function (CTF). The calculated CTF of a 200CX HRTEM at Scherzer defocus and Cs = 1.2 mm. The first zero is arrowed (corresponding to 0.23 nm resolution) and the resolution in angstrdms is shown on the horizontal axis. A-D are envelope functions plotted as a function of convergence angle (0) of the beam and beam energy spread (A V). Parallel illumination is necessary for high resolution (after Boyes et al 1980).

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). 

Conserved core genes, large viruses and, 382 Contrast transfer function (CTF), 94-95 cryoelectron microscopy and, 44 E function simulation and, 96 in subnanometer resolution reconstruction, 104... 

Subnanometer resolution reconstruction, contrast transfer function (CTF) in, 104... 

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