Contrast Transfer Function

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

$$ \chi (k) = \pi \lambda \Delta f k^2 + \frac{1}{2} \pi C_s \lambda^3 k^4$$

The complicated curve sin\(\chi (k)\) 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 \(\Delta f_s\) and the point resolution \(\Delta x\) are functions of \(C_s\) and \(\lambda\):

$$ \Delta f_s = \sqrt{\frac{4}{3} C_s \lambda}$$

$$\Delta \chi = 0.66 \sqrt[4]{C_s \lambda^3}$$

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.