electron microscopy



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 + 1/2πCsλ3k4

The complicated curve sinχ (k) strongly depends on Cs (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 Δfs and the point resolution Δx are functions of Cs 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.


CTF of a CM30 TEM (LaB6 cathode; ST lens with Cs = 1.15 mm, Cc = 1.5 mm) close to Scherzer defocus. The CTF crosses the k axis at about 5.1 nm-1 which corresponds to a resolution of about 1.9 Å.

CTF of a 300 kV field-emission TEM (Cs = 1.15 mm, Cc = 1.5 mm) close to Scherzer defocus. In the coherent electron beam of the FEG, the effect of damping is reduced compared to the LaB6 gun. As a result, there is additional information beyond the point resolution in the oscillating CTF till the information limit at about 1.0 Å.

(CTFs calculated by EMS program)

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 task is the exit wave function reconstruction from defocus series.
According to Scherzer, a correction of spherical (Cs) and chromatic (Cc) 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 and STEMs (Thermo Fisher Scientific (FEI), JEOL, Hitachi, NION).




ETH Zürich | ETH chemistry department | ETH inorganic chemistry

modified: 8 November, 2021 by F. Krumeich | © ETH Zürich and the authors