HRTEM Lattice Tool

What the page does

The Fourier transform of a lattice image is its diffractogram: a periodicity of spacing d in the image gives a pair of spots at ±1/d from the centre, in the direction of the lattice normal, so a spot's distance from the centre in pixels, r, converts to a spacing through d = N·Δx/r, where N is the width of the transformed region and Δx the size of an image pixel. The product N·Δx plays the part of the camera constant of a diffraction pattern, and the page reports it in the same units the SAED indexer uses, so a set of spots measured here can be indexed there. The spots are located to a fraction of a pixel by the intensity-weighted centroid around the local maximum, and their angles and the angles between them are reported with the spacings.

The scale is the whole measurement. It can be entered as the pixel size from the microscope's calibration, taken from a scale bar burned into the image by clicking its two ends, or fixed from one spot whose spacing is known (the 111 of silicon at 0.3135 nm, say), after which every other spot in the same image is measured against it; the last is the most accurate in practice, since the magnification calibration of a microscope rarely holds to better than a few percent while the ratio of two spacings in one image is limited only by the spot positions. The Bragg filter keeps soft discs around the chosen spots and their Friedel mates in the transform and inverts, which shows the lattice of one set of planes on its own, with the terminations and bending of the fringes at a defect; the disc radius sets how much of the local variation is kept. The moiré calculator gives the fringe spacing of two overlapping lattices for the translational, rotational and general cases, and can take the two spacings and the angle from the spots you have picked.

Scope and limits

The spacing measured from an FFT is the spacing of the fringes in the image, which equals the lattice spacing only when the image is a faithful lattice image: the fringes of a thick or defocused crystal can have the spacing of the lattice while the contrast pattern does not represent the atoms, and half-spacing fringes and moiré beats appear as spots of their own. Sub-pixel accuracy of the spot position is not sub-pixel accuracy of the spacing when the region is small: the uncertainty in d is d·δr/r, so a spot at radius 20 px located to 0.2 px is a 1 percent measurement; larger regions and higher-order spots reduce it. The calibration by a known spacing assumes the magnification is the same in both directions; an anisotropic distortion of the projector system shows as different spacings for symmetry-related spots and is not corrected here. The image is read as a grey-level array: 8-bit and 16-bit uncompressed TIFF, PNG and JPEG are accepted, and a colour image is converted to luminance.

References

1. D. B. Williams and C. B. Carter, Transmission Electron Microscopy, 2nd ed., Springer (2009): chapter 23 (moiré patterns), chapter 28 (high-resolution imaging) and chapters 30 and 31 (image simulation, processing and Fourier filtering).
2. J. C. H. Spence, High-Resolution Electron Microscopy, 4th ed., Oxford (2013): the diffractogram, lattice fringes and their interpretation.
3. P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals, 2nd ed., Krieger (1977): chapter 15 (moiré patterns).
4. M. J. Hÿtch, E. Snoeck and R. Kilaas, Quantitative measurement of displacement and strain fields from HREM micrographs, Ultramicroscopy 74, 131-146 (1998): the geometric-phase analysis that starts from the same Bragg-filtered spots.
5. D. R. G. Mitchell and B. Schaffer, Scripting-customised microscopy tools for Digital Micrograph, Ultramicroscopy 103, 319-332 (2005): the FFT measurement tools this page follows.