Scherzer Defocus
- Practical Electron Microscopy and Database -
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The presence of zeros in CTF (contrast transfer function) means that there are gaps in the output signal. The best CTF is the one with the fewest zeros and with the broadest band of good transmittance, meaning CTF is close to -1. In 1949, Scherzer found that the optimum defocus depends on the microscope properties such as the spherical aberration Cs and the wavelength λ (accelerating voltage) of the electrons, given by,

----------------------------- [4234a]

where the factor 1.2 defines the extended Scherzer defocus. In general, this is the best defocus to take HRTEM (High Resolution TEM) images. Assuming Cs and λ are constant for a specific microscope, Figure 4234a shows CFT T(|g|) = sin(χ) at n Scherzer defoci, given by,

------------------------------- [4234b]

where,

n -- 0, 0.5, 1, 1.2 (extended), 2, and 10

Figure 4234a. Contrast transfer function (CFT) at 0, 0.5, 1, 1.2 (extended), 2, and 10 Scherzer defoci.

Note that 0.5 Scherzer defocus is a defocus condition given minimum contrast; 1 scherzer defocus is real Scherzer defocus; 1.2 scherzer defocus is extended Scherzer defocus, which normally is the best defocus condition to take HRTEM images; 2 scherzer defocus is second-passband defocus, giving positive CTF to produce a negative phase contrast (atoms are in white contrast). In the case of Scherzer defocus (=1), the contrast transfer function (T(|g|)) can be considered as about - 1. Scherzer defocus maximizes the phase contrast of a weak-phase object and resolution.

Table 4234 lists some examples of Scherzer defocus values under certain microscope conditions.

Table 4234. Examples of Scherzer defocus values under certain microscope conditions.

Accelerating voltage (keV) Cs (mm) C5(mm) ΔfScherzer(nm)
200
0.6
0
- 43.6
300
1.2
0
- 56.1

It was experimentally and theoretically verified that for CTEM (conventional TEM) the atom columns always appear dark at Scherzer defocus.

Figure 4234b shows the calculated phase contrast transfer functions (pCTF) at Scherzer defoci in 200-keV TEMs with LaB6 and FEG sources and with different spherical aberrations (Cs).

Figure 4234b. Typical pCTFs for 200-keV TEMs: (a) At Scherzer defocus in LaB6 TEM with large positive Cs (1.23 mm), (b) At Scherzer defocus in FEG TEM with large positive Cs (1.23 mm), (c) At Scherzer defocus in FEG TEM with small positive Cs (0.083 mm), and (d) At Scherzer defocus = Lichte defocus in FEG TEM with small negative Cs. Adapted from [1]

Figure 4234c shows two Fourier transform patterns obtained from HRTEM images on amorphous carbon films at 0.5 Scherzer defocus and at Scherzer defocus, respectively.

Figure 4234c. Two Fourier transform patterns obtained from HRTEM images on
amorphous carbon films at 0.5 Scherzer defocus (left) and at Scherzer defocus (right), respectively.

For a modern uncorrected 300-kV TEM with Cs = 0.6 mm and an information limit of 0.1 nm, the actual point resolution is only 0.17 nm and the smallest delocalization is up to even 1.2 nm! [1] For an uncorrected TEM operated at Scherzer defocus, rapid contrast reversals happens at spatial frequencies between the Scherzer limit and the information limit, making the interpretation of the images very difficult.

[1] Markus Lentzen, Progress in Aberration-Corrected High-Resolution Transmission Electron Microscopy Using Hardware Aberration Correction, Microsc. Microanal. 12, 191–205, 2006.

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