Information theory based crystallographic symmetry classifications of a noisy 2D periodic scanning tunneling microscope image

Abstract

Modern scanning tunneling microscopes (STMs) are often employed in the imaging of crystalline surfaces at atomic resolution, Fig. 1. In recent years, we adapted crystallographic image processing (CIP) techniques that originated in the field of electron crystallography some 50 years ago to the imaging with scanning probe microscopes [1][2][3]. In cases of images from STMs and atomic force microscopes, our CIP method allows for the removal of the effects of a blunt scanning probe tip [2,3].Recent improvements to our method allow for fully objective, i.e. completely researcher independent, classifications of noisy 2D periodic images into Bravais lattice types [3,4], Laue classes [4], and plane symmetry groups [4]. The preconditions that need to be fulfilled for the application of our new information theory based method are a sufficiently long-range ordered crystalline material, a sufficient number of repeats of the unit cell in the image, and a relatively modest amount of generalized Gaussian noise [4,5]. Because discrete Fourier transforms are involved, noise in the images reduces to "noise per unit cell" with the square root of the number of unit cells that are processed [4]. Generalized noise arises from multiple sources in all imaging and image processing steps as well as from structural defects in the crystalline sample itself. None of these sources is allowed to dominate so that the central limit theorem applies and the sum total of all errors possesses approximately a Gaussian distribution [4,5]. The type of the microscope is not important; it is only important that its operation is reasonably stable and uncorrected systematic errors are small compared to the sum total of generalized random errors.We applied our method to a STM image from a graphite sample that is openly available (in the on-line supporting material of [6]), Fig. 1. Utilizing sets of ratios of geometric-bias corrected squared residuals for the involved symmetry hierarchy branches, see right hand side of Fig. 1, the plane symmetry group of the selected region of the STM image was determined to be h31m with confidence levels of approximately 26 % over c11m (on the basis of the primitive sub-units of oc lattices [8] averaged over three settings) as well as 95 % over h3, respectively. There are, thus, no serious doubts that the prevailing translation symmetry of the STM-imaged graphite sample is that of the 2D hexagonal Bravais lattice type (as also obtained in [6]). Note in passing that the h31m result is indicative of a rhombohedral layer stacking that may be due to the sample preparation. Note also that h31m on a triple hexagonal unit cell features the same plane symmetries as p3m1 on a three times smaller cell [5], since the former is a minimal non-isomorphic supergroup of index 3 of the latter. Our classification enabling geometric-bias corrected residuals are in the form of geometric Akaike information criteria [9] as derived in [4]. Our confidence levels are based on the information content equation in [9] as derived in [3] and recently ge...

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

“…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].…”

“…All of the other above-mentioned images were classified in reciprocal/Fourier space with respect to their Bravais lattice type and plane symmetry group as it is customary in 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]. The abovementioned framework for the classification of crystallographic symmetries [1][2][3][4][5] in 2D is obviously applicable regardless of the space in which the classifications are made.…”

“…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].…”

“…The author's recently developed information theory based approaches to crystallographic symmetry classifications [16][17][18][19][20][21][22][23] utilize geometric bias-corrected sums of squared residuals in the form of geometric Akaike Information Criteria [38][39][40][41][42]. His approach to plane symmetry group classifications [16,17,[21][22][23] will be discussed in the second section of this paper in some more detail as part of the review of the analytical approaches. The discussions will proceed at a qualitative level as all of the other approaches are also dealt with in an essentially qualitative manner.…”

“…The author's other crystallographic symmetry classification approaches, i.e. into Bravais lattice types [18][19][20] and crystallographic [16,23] as well as non-crystallographic Laue classes, are discussed in some detail elsewhere as they serve other purposes than identifying the plane symmetry group that can with the highest likelihood be assigned to a noisy 2D periodic image. Note that it is this particular identification that enables the most meaningful crystallographic averaging in the spatial frequency domain that is of paramount interest to 2D crystallographers, transmission electron and scanning probe microscopists.…”