The
contrast transfer function CTF (detailed
treatment), sinχ
(k), gives the phase changes of diffracted beams with respect to
the direct beam.
χ
(k) = πλΔfk^{2
}+ 1/2πC_{s}λ^{3}k^{4
}
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
wavelength 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
kvectors, 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 LaB_{6} 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 uptodate method
for this task is the exit wave function reconstruction from defocus series.
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 subÅngström
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).
