Quantifying and enforcing two-dimensional symmetries in scanning probe microscopy images

Abstract

The overall performance and correctness of the calibration of all kinds of traditional scanning probe microscopes can be assessed in a fully quantitative way by means of "crystallographic" processing of their twodimensional (2D) images from samples with 2D periodic (and preferably highly symmetric) features. This is because crystallographic image processing results in two residual indices that quantify by how much the symmetry in a corresponding scanning probe microscopy image deviates from the symmetries of the possible plane groups of the periodic features of the sample. When a most probable plane symmetry group has been identified on the basis of crystallographic image processing, the symmetry in the scanning probe microscopy image can be "enforced" in order to obtain "clearer" images, effectively removing the less than ideal "influence" of the microscope on the imaging processes. This paper illustrated the crystallographic image processing procedure for scanning tunneling microscopy images that were recorded from a monolayer of a phthalocyanine on two different types of substrates.

Crystallographic STM image processing of 2D periodic and highly symmetric molecule arrays

“…Our earlier study of regular arrays of F 16 CoPc molecules on Ag (110) revealed somewhat more densely packed domains that possessed only plane group p2 [7]. We also note that two densely packed oblique lattice domains and a more loosely packed domain with a rectangular (pseudosquare) lattice were described in ref.…”

“…Our earlier results of CIP of STM data of 2D periodic arrays of phthalocyanines are summarized in refs. [7,8]. Reference [9] is available in open access on the Internet and gives a comprehensive introduction to CIP for Scanning Probe Microscopy (SPM).…”

On Classification Approaches for Crystallographic Symmetries of Noisy 2D Periodic Patterns

“…Better "symmetrizations" of noisy 2D periodic images have always been the desired outcome of the standard crystallographic image processing method that is used by the 2D crystallography, transmission electron microscopy, and scanning probe microscopy communities [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and works in Fourier space. These symmetrizations combine the noiseremoval feature of traditional Fourier filtering with the crystallographic averaging over all asymmetric units in the whole image, have been utilized for over 50 years, and contributed to the Nobel Prize in Chemistry to Sir Aaron Klug 8 in the year 1982.…”

“…The derived mathematical framework is nowadays utilized for the classification of (i) highly magnified but necessarily noisy experimental images from crystals that are sufficiently thin (so that quasi-kinematic imaging conditions prevail) and crystal surfaces as recorded with modern transmission electron and scanning probe microscopes [7][8][9][10][11][12][13][14][15][16][17][18], of (ii) both perfectly 2D periodic and noisy synthetic images [18][19][20][21][22][23][24][25][26], as well as of (iii) more or less 2D periodic images from rugs, windows, a honeycomb, both a tiled floor and wall, metal gates, a metallic anti-skid surface [26], industrially manufactured wallpapers [27][28][29], textiles, and decorative ceramic tiles [29][30][31][32][33], medieval Islamic building ornaments [34], and images of (iv) many other objects and scenes [35].…”

“…Similarly obvious is the fact that members of the communities that traditionally perform direct space classifications may benefit from becoming informed about the opportunities that the Fourier/reciprocal space offers with respect to crystallographic symmetry classifications of more or less 2D periodic images [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Members of the 2D crystallography, transmission electron microscopy, and scanning probe microscopy communities are conversely bound to benefit from an analysis of what has been achieved so far in direct space classification of the crystallographic symmetries of more or less 2D periodic images [26][27][28][29][30][31][32][33][34][35].…”