Contrast Transfer Function

TEM | Image Formation | Bright Field Images | Dark Field Images | Lattice Images (HRTEM)

The contrast transfer function CTF (detailed treatment), sinχ (k), gives the phase changes of diffracted beams with respect to the direct beam.

χ (k) = πλΔfk² + 1/2πC_sλ³k⁴

The complicated curve sinχ (k) strongly depends on C_s (spherical aberration coefficient, i.e. the quality of the TEM objective lens), on lambda (electron wave-length as defined by the accelerating voltage), on defocus and on the spatial frequency k. While this function is zero at the origin, it becomes positve for intermediate values of k. In this region of k, all information is transferred with positive phase contrast, i.e. the scattering centers (atom positions) appear with dark contrast. Therefore, the information in HRTEM images is consequently directly interpretable till the point resolution. The point resolution of a TEM corresponds to the point where the CTF first crosses the k axis at Scherzer defocus at which this area is maximally extended towards high frequences. Both the Scherzer defocus Δf_s and the point resolution Δx are functions of C_s and λ:

Depending on the defocus, the CTF may oscillate strongly. At larger k-vectors, it is strongly damped mainly due to the effect of chromatic aberration, focus spread and energy instabilities. Since the defocus is variable and can be adjusted at the microscope, an adequate value can be chosen to optimize the imaging conditions.

Advanced HRTEM Methods

In TEMs with field emission guns, the damping of the CTF is less drastic than in TEMs with a LaB₆ cathode. Thus, there is information available in the oscillations beyond the point resolution till the information limit that can be used to achieve data of higher resolution. A up-to-date method for this is the exit wave function reconstruction from defocus series (A. Thust, Jülich).
According to Scherzer, a correction of spherical (C_s) and chromatic (C_c) aberrations cannot be achieved with rotationally symmetric electron lenses. Using asymmetric ray paths, it is possible to overcome this resolution limit and enter the sub-Ångström range (examples). Such aberration correctors (CEOS), as first proposed by H. Rose, are now available in commercial TEMs (Thermo Fisher Scientific (FEI), JEOL, Hitachi).