This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.

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In light optics, high-quality multiple-lens systems can be used to compensate aberrations. In contrast, correction of aberrations in electron lenses is much more challenging because the electron lenses use magnetic fields to focus electrons.

The first practical aberration corrector in EMs was installed on a test bench to verify the applicability of C_s correction [1]. The demonstration setup on a modified SEM included a C_s corrector entirely incorporated into a specimen chamber and a CCD camera coupled with a scintillator underneath the specimen chamber. The electron probe was focused on the scintillator, and scanned circularly with a series of diameters so that the ray displacements induced by the aberration corrector could be observed.

During a period of approximately 45 years, all attempts to improve the actual resolution of EMs by correcting the aberrations had failed. Fortunately, in aberration-corrected EMs (electron microscopes), the aberration correctors consisting of numerous elements can recently be adjusted and kept stable with an extreme accuracy. For instance, a combination of hexapole or octupole lenses are used in a aberration corrector which lacks rotational (circular) symmetry and thus doesn’t have to have a positive spherical aberration like conventional, round magnetic lenses. The overall spherical aberration of the objective lens and this corrector can be minimized by the operator, which can significantly improve the point to point resolution by choosing a smaller spherical aberration. Furthermore, the corrector has also made it possible to use negative spherical aberrations giving bright atoms on a dark background, which actually enhances the contrast in the images compared with the images taken with a large, positive spherical aberration.

So far only two types of aberration correctors have been proposed:
         i) Conventional objective lens in combination with five multipole elements [3],
         ii) Two hexapoles and four weak transfer lenses apart from the strong objective lens [4].
The second type has the advantage that the hexapole fields need only a stability of about 10 ppm, which is two orders of magnitude less than that required for the quadrupole fields of the multipole correctors, while it has the disadvantage that it cannot compensate for the chromatic aberration.

In principle, aberration correction can be applied to any magnetic lens in any microscopes. Aberration correctors have to satisfy different conditions when they are to be applied in different types of electron microscopes such as LVSEM (low-voltage scanning electron microscopy), STEM or TEM. Physically, the aberration corrections are mainly applied to two key categories: the condenser (or called illumination) system for STEM and TEM and the objective lens for STEM.

The primary aberrations of the TEM systems with three-fold symmetry are of second order. It is necessary to eliminate all the second-order aberrations before correcting the third-order aberrations, since the second-order aberrations are larger than the third-order aberrations caused by the rotationally symmetric fields.

In conventional TEM, aberration correction was limited from difficult finding the coma-free axis of the objective lens by intentionally using an illumination tilt, followed by eliminating twofold astigmatism with the objective lens stigmators. The aberration correctors, e.g. double-sextupole aberration correctors, provide a number of ways to compensate for the aberrations of the entire imaging system of the microscope through the excitation of deflectors, lenses, and the coils of the poles in the corrector. The current settings of the elements in the correctors significantly depend on the aberration measurement [2].

The ideal location for placing an aberration corrector in an electron-optical column is exactly at the place where the aberration is introduced. In practice, this cannot be possible because of the limits of the sizes of the two parts. The resulting separation between the locations where an aberration is introduced and where it is removed produces higher order aberrations than the corrected one. For instance, the physical separation between the objective lens where the third order spherical aberration (C_s, or called C_3,0) is introduced and the corrector where this aberration is removed induces a higher order spherical aberration, which is fifth order spherical aberration (C_5,0) as indicated in green in Table 4579. However, the separation can be minimized by placing the corrector as close as possible to the objective lens, and by keeping the length of the corrector to a minimum.

For perfect aberration correction, because the aberration function is zero, all the ray displacements are eliminated, resulting in a zero contrast delocalization.

General speaking, π/4 criterion is a guide for the microscope operator to decide which aberrations must be corrected and which aberrations are negligible. It is not necessary to correct it if the wave aberration of a single lens has a phase deviation of less than π/4 from a desired reference at the information limit of the instrument [2]. Note that a single aberration violating the π/4 criterion may also cancel out the other one with the same symmetry if the have different sign, such as a small positive third-order spherical aberration and a small underfocus, or a small positive twofold astigmatism and a small negative third-order star aberration with the same azimuth. On the other hand, the sum of some aberrations can probably exceed the π/4 criterion at the information limit. In order to help the microscope operator make an operation decision, the corresponding phase plate can be displayed in the computer interface together with the π/4 contour.

The procedure of aberration corrections consists of measurement of the existing aberrations and the process of aberration correction. For instance, in the aberration measurements based on diffractogram, the experimental series of induced defocus and twofold astigmatism and the illumination tilts are applied to a set of linear equations for the desired true lens aberrations. The solution for the true lens aberrations is then transferred with a calibration table to the lenses of the corrector [2]. The microscope operator decides which aberration needs to be corrected.

There currently are different aberration corrector models such as CEOS GmbH correctors [5, 6] and delta-type aberration correctors.

In addition to its applications mentioned above, aberration correction can also be applied to improve the lattice-fringe visibility and its bands.

[1] Haider, M., Braunshausen, G. & Schwan E. (1995). Correction of the spherical aberration of a 200 kV TEM by means of a Hexapole-corrector. Optik 99, 167–179.
[2] Uhlemann, S. & Haider, M. (1998). Residual wave aberrations in the first spherical aberration corrected transmission electron microscope. Ultramicroscopy 72, 109–119.
[3] H. Rose, Elektronenoptische Aplanate, Optik 34 (1971) 285.
[4] Rose H. 1990. Outline of a spherically corrected semiaplanatic medium-voltage transmission electron-microscope. Optik 85:19–24.
[5] Haider, M., Uhlemann, S. & Zach, J. (2000). Upper limits for the residual aberrations of a high-resolution aberration-corrected STEM. Ultramicroscopy 81, 163–175.
[6] Müller, H., Uhlemann, S., Hartel, P. & Haider, M. (2006). Advancing the hexapole Cs-corrector for the scanning transmission electron microscope. Microsc Microanal 12, 442–455.