Crystallographic image processing (CIP) is well established in the electron microscopy community, where it is used for the analysis and enhancement of high-resolution transmission electron microscope images of crystals and two-dimensional (2D) arrays of membrane proteins. The technique has recently been adapted to the processing of 2D periodic images from scanning probe microscopes (SPMs) [1]. Within this context, a procedure for the unambiguous identification of the underlying Bravais lattice of an experimental or theoretical image of a 2D periodic array of objects (e.g. molecules or atoms and their respective electron density distribution functions, ...) has been developed [2]. This procedure constitutes a partial solution to a longstanding but unresolved issue in CIP. The unresolved issue itself is the complete quantification of the deviations of 2D periodic images from the plane symmetry groups. A complete solution to this problem will allow for unambiguous decisions as to which plane symmetry best models experimental data when all systematic errors in the acquiring and processing of the image data have been accounted for at a level that systematic rest errors are negligible. Our 2D Bravais lattice identification procedure is independent of which type of microscope has been utilized for the recording of the images. It is based on classification procedures for non-disjoint models from the robotics community and is particularly useful for the correction of scanning tunneling microscope (STM) images that suffer from a blunt scanning probe tip artifact [2]. With the crystallographic processing of two molecular resolution STM images of periodic arrays of tetraphenoxyphthalocyanine on graphite, it is demonstrated how the classical CIP plane symmetry estimation procedures are augmented by our unambiguous translation symmetry identification method. We also apply CIP to an artificial SPM image that features a blunt scanning probe tip artifact, see the figure below.

Scanning probe microscopy (SPM) images can be obscured by signals from blunt and multiple probe tips. We show why crystallographic image processing (CIP) techniques may be utilized to restore obscured images that are periodic in two dimensions. The image-forming current for double tips in scanning tunneling microscopy (STM) is derived in a more straightforward manner than prior approaches. The Fourier spectrum of the tunneling current for p4mm Bloch surface wave functions and a pair of delta function tips reveals the tip-separation dependence. Our analysis clarifies why crystallographic averaging works well in removing such effects from the recorded 2D periodic images and also outlines the limitations of this image processing technique for certain spatial separations of STM double-tips. Classical simulations of double tip effects in STM images (that ignore electron interference effects) may be understood as modeling double tip effects in images that were recorded with other types of SPMs. Appendix A demonstrates how double tip effects on scanning probe microscope images are detected unambiguously.

Crystallographic Image Processing (CIP) originated with the electron crystallography community. Nobel Laureate Sir Aaron Klug (OM, FRS) and coworkers pioneered the technique for the analysis of long-range ordered biological materials in parallel illumination Transmission Electron Microscopes (TEMs). Corrections for the effects of the TEM's phase contrast transfer function and for less than optimal imaging conditions are part of this kind of CIP. There are also "electron microscope independent" 2D crystallography foundations to this kind of image processing.Based on these foundations, we applied CIP to images of long-range ordered 2D periodic surface arrays that were recorded with different kinds of scanning probe microscopes (SPMs) [1,2]. We amended our method recently [3] to detect and correct frequently encountered artifacts in scanning probe microscopy, i.e. effects of multiple mini-tips that collectively result in a blunt tip [4,5]. Loosely speaking, our version of CIP has the effect of "sharpening up" a blunt scanning probe tip. This is achieved by the deconvolution of the prevailing microscope's point spread function from the SPM images. Although many scanning probe microscopists have so far been content with ignoring these kinds of artifacts, there are also highly credible reports on unambiguous observations on scanning probe tip changes during data recordings that led to blunt tip artifacts in SPM images [6,7]. One of these reports proposes that multiple mini-tips cannot affect the character of the observed translation symmetry in such an image while the 2D periodic motif may be smeared out [6].Our theoretical analysis [4] confirms this idea so that one can confidently take "inconsistencies" between observed 2D translation and point symmetries in SPM images (Fig. 1) as the hallmark of multiple mini-tip artifacts. Our unambiguous determination of the underlying Bravais lattices of 2D periodic surface arrays [3] on the basis of a geometric Akaike Information Criterion (AIC) [8] achieves the detection of multiple minitip artifacts on a statistically sound basis. Figure 1 demonstrates the effectiveness of our version of CIP in removing blunt tip artifacts from a simulated scanning tunneling microscope (STM) image of long-range ordered 2D periodic arrays of cobalt-phthalocyanine molecules on a (001) oriented gold surface. Note that the central circular area in the left part of Fig. 1 illustrates both (i) an inconsistency between 2D translation and point symmetries and (ii) the fact that structural scanning probe tip changes during the recording of experimental data cannot change the character of the observed translation symmetry in the resulting SPM image.Our recently developed unambiguous translation symmetry determination procedure [3] also constitutes progress towards making CIP more objective in both electron crystallography by TEM and surface feature assessments by SPM. This is because not all plane symmetry groups are disjoint. As it is well known, all of the symmetry operations of a plane symmetry grou...

Crystallographic image processing (CIP) techniques may be utilized in scanning probe microscopy (SPM) to glean information that has been obscured by signals from multiple probe tips. This may be of particular importance for scanning tunneling microscopy (STM) and requires images from samples that are periodic in two dimensions (2D). The image-forming current for double-tips in STM is derived with a slight modification of the independent-orbital approximation (IOA) to allow for two or more tips. Our analysis clarifies why crystallographic averaging works well in removing the effects of a blunt STM tip (that consists of multiple mini-tips) from recorded 2D periodic images and also outlines the limitations of this image-processing technique for certain spatial separations of STM double-tips. Simulations of multiple mini-tip effects in STM images (that ignore electron interference effects) may be understood as modeling multiple mini-tip (or tip shape) effects in images that were recorded with other types of SPMs as long as the lateral sample feature sizes to be imaged are much larger than the effective scanning probe tip sizes.

Results of calculations of double excitation of helium to n=2 states for fast projectiles of charge Z are presented. Nonzero Z contributions for single and doub1e excitation occur only when time-ordering contributions from the second-order amplitude are nonzero. For double excitation, electron correlation must also be nonzero to obtain Z terms. The time-ordering effects arise from virtual off-energy-shell intermediate states. As with second-order amplitudes for Thomas singularities in electron capture, the energy-nonconserving amplitude is connected to the second-order energy-conserving amplitude by a dispersion relation. Comparison is made with experiment.