Focal-Series Reconstruction

Appendix

9.1.1 People

The development of the FSR method in electron microscopy began roughly in the 1970s. Over time, generations of scientists contributed to the evolvement of various mathematical variants, numerical implementations, and actual applications of FSR. For this reason, developing an only rudimentarily fair genealogical table of important contributors would simply require too much space. Instead, we name here only one person, who was frequently ignored in the scientific literature and who is barely known: Peter Schiske. He is the guy who invented the FSR principle! Peter Schiske (1924–2012) was an Austrian physicist, mathematician and early computer expert. In the early 1950s he collaborated with the well-known Austrian physicist and electron optician Walter Glaser, who was a strong proponent of the wave optical view of image formation based on the Schrödinger equation. Not surprisingly, we also made strong use of the wave optical view throughout this chapter. It was finally Peter Schiske who published in 1968 for the first time the idea of focal-series reconstruction in electron microscopy. The original paper was a short conference abstract written in German. Fortunately, due to its fundamental importance, Schiske’s original paper has meanwhile been translated to English and re-printed (see References).

9.1.2 Self-Assessment Questions

Q9.1:

Why is it favorable to have the object wavefunction available compared to a single HRTEM image?

Q9.2:

Are you aware of a TEM technique, where the object wavefunction can be obtained without requiring a multitude of images?

Q9.3:

State at least two reasons why it is advisable to perform HRTEM experiments in thin-object regions. How many nanometers are roughly considered as ‘thin’?

Q9.4:

How is the thickness dependence of a wavefunction related to the atomic charge number?

Q9.5:

What kind of artifacts are introduced even by a ‘neutral’ microscope, which just produces a magnified image of the object wavefunction?

Q9.6:

What would you expect to see in images of a thin object made with a ‘neutral’ microscope?

Q9.7:

If a constant-phase-shift microscope was available (Zernike phase plate), how many images would be required to restore the object wavefunction?

Q9.8:

How do the projection surfaces in the 3D (Re, Im, g) space of a constant-phase-shift microscope differ from those of a real microscope?

Q9.9:

How are the projection surfaces in the 3D (Re, Im, g) space related to the well-known contrast transfer function (CTF)?

Q9.10:

How large roughly is the minimum contrast delocalization obtainable with a non-corrected field emission microscope? What minimum contrast delocalization can be obtained with a C _S -corrected version?

Q9.11:

By which feature of the projection surfaces in the 3D (Re, Im, g) space can one observe the function ∇χ(g)?

Q9.12:

Is there an analytical solution to the linear reconstruction problem? Is there an analytical solution to the nonlinear reconstruction problem?

Q9.13:

At which defocus value should one approximately place the midpoint of a focal series?

Q9.14:

Is there a substantial difference in the defocus value Z _opt between an uncorrected microscope and a C _S-corrected microscope? State typical values.

Q9.15:

Does the choice of the classical Scherzer defocus instead of Z _opt make any sense for FSR with an uncorrected field emission microscope?

Q9.16:

Which single parameter apart from the wavelength determines the lowest reconstructible frequency g _min?

Q9.17:

Which single parameter apart from the wavelength determines the resonance frequency g _res in the high-frequency regime?

Q9.18:

Which single parameter apart from the wavelength determines the optimum focal step size δ _opt?

Q9.19:

Would it be efficient from a mathematical point of view to decrease the lowest reconstructible frequency g _min by increasing the number of recorded images? Additionally, what physical effect related to the object hampers such an attempt?

Q9.20:

Is it a good idea to try a reconstruction using only a small number of images (e.g., less than five)? What exactly is sacrificed when reducing the number of images?

Q9.21:

Is there an advantage of recording an equidistant focal series over recording an irregularly spaced series?

Q9.22:

State three reasons why the application of FSR can yield an advantage over taking just one C _S-corrected image at the optimum defocus?

Q9.23:

Would it be a good idea to apply FSR to amorphous materials or to nanoparticles lying loosely on a support film?

Q9.24:

Who is the third Austrian named in the “People” section at the end of the chapter? Which view in physics/optics do they share?

9.1.3 Text-Specific Questions

T9.1:

Derive the fundamental Eq. 9.12, which is one possible definition of the Scherzer defocus Z _S , by requiring that χ(g ^*, Z _S) = −2 π/3, where g ^* is the position of the local minimum of χ.

T9.2:

The local minimum of the aberration function g ^* can be easily recognized as a dip in the plateau of the Scherzer CTF displayed in Fig. 9.6d. What property of the respective helical surface of Fig. 9.8a, d changes at the specific frequency g ^*?

T9.3:

Derive the fundamental Eq. 9.16, which defines the optimum defocus Z _opt, by setting ∇χ(g _int, Z _opt) = −∇χ(g _max, Z _opt), and by solving the resulting third-order equation. Take care with the signs! It might be helpful to use |Z| in combination with compensating signs.

T9.4:

Derive the optimum value of the spherical aberration C _S and the thereby resulting minimal achievable radius R _min of the point-spread function for a C _S-corrected microscope by demanding that Z _S of Eq. 9.12 equals Z _opt of Eq. 9.16.

T9.5:

What is the purpose of the first and the second terms in the enumerator of the linear filter function of Eq. 9.30?

T9.6:

What important property is reflected by the denominator of the linear filter function of Eq. 9.30?

T9.7:

Explain why the titanium atom columns produce stronger phase peaks in the image of Fig. 9.15 (right-hand side) than the barium atom columns, although titanium is the lighter and therefore also the weaker scattering element.

T9.8:

In the reconstructed phase of Fig. 9.16 it can be observed that the two polygonal structure units stacked above each other are absolutely identical. However, in both input images of Fig. 9.14 one can observe clear differences between the upper and the lower units. How can one and the same image exhibit different contrast for identical object features?

T9.9:

When comparing the number of intensity maxima in the C _S-corrected Scherzer image of Fig. 9.18c with the number phase maxima in Fig. 9.18d one finds that there are many more intensity maxima than phase maxima. Whereas the phase maxima are all located at Ga or As atomic column positions as is expected, some of the intensity maxima are located at empty crystal positions and are thus artifacts. Which mechanism can be responsible for the ‘ghost’ atoms in the C _S-corrected Scherzer image?