Spiral Powder Overlays

Here we describe spiral overlays for use with powder diffraction patterns, and diffraction spacing profiles. These can be at once forgiving of uncertainties in camera constant calibration, and (once these are dealt with) sensitive to very small differences in lattice parameter. By facilitating comparison of different lattices in a common physical metric (namely a diffraction experiment of fixed camera constant and scattering center density), their pedagogical value also extends to the comparison of crystal structures whose atomic environments share physical features (like nearest neighbor distances) in spite of major differences in symmetry.

Sound Bytes: spin the profile | fingerprinting powder patterns | spiraling spatial harmonics


August 2004 meeting paper (Savannah GA):

"Spiral powder overlays" [poster], Proc. Microscopy and Microanalysis 2004, 1356-1357
by P. Fraundorf and Shuhan Lin (UM-StL Physics and Astronomy/CME)

Draft MS prepared for publication in Microscopy Today 13 #1 (January 2005) pages 8-11.

For something new on visualizing single crystal diffraction, check here.

Early notes

Without a camera constant

Analysis of the experimental electron diffraction pattern from an unknown assemblage of nanocrystals (red) against overlays (green) for body-centered, face-centered, and diamond cubic lattices. In blue find notations concerning the line of spacing matchups, as well as where various elemental compounds would plot if one had calibrated the camera constant of the pattern precisely (not the case for this analysis)...

With a camera constant

A larger fcc analysis, this time of a simulated diffraction pattern from polycrystalline nickel. In spite of the overlay's ability to reveal spacing matches even if the lattice parameter (or the camera constant of the pattern) is completely unknown, note how easily the angle of the match line for a carefully calibrated pattern allows us to distinguish between nearly identical lattices of Ni and Fe...

An even larger fcc analysis, this time of the experimental 300kV electron diffraction pattern from a polycrystalline Al thin film.

For comparing polymorphs

A comparison of face centered cubic (fcc) and hexagonal close packed (hcp) overlays, illustrating what they do and do not have in common. Tip: To see the fcc or hcp overlays alone, look through a pair of red-green glasses with but one eye open, so that you see only red (fcc) or green (hcp). Why do the diffraction spacings show so much overlap (yellow)? An atom-thick hexagonal array of atoms can be naturally stacked against a second array in one of only two ways, if the distance between nearest atoms is to be the same within and between arrays. Denote the alignment of the first array with the letter "A". The two possible adjacent layers then have "B" and "C" alignments. If one stacks such layers in the sequence ABCABC, one gets a face-centered cubic lattice (which strangely enough has four equivalent "stacking directions" along body diagonals of the cubic unit cell). Restacking of them in ABABAB form instead yields the corresponding hexagonal close packed structure, which is not similarly isotropic (hence more spacings created than lost). In fact all fcc spacings graphed below are hcp spacings too, except those of the form (h00)...

Camera-constant calibration and monitoring overlay

Here's an overlay designed to be printed with a 3-inch square axis range for camera-constant calibration with a polycrystalline Aluminum diffraction pattern from any system. Linked beneath it is a version of the same overlay for use with 600 dpi digitized patterns.

High-symmetry lattice-analysis overlay

Here's a composite overlay designed to be printed with 3 reciprocal-Angstrom plot ranges for use with unknown patterns of given camera constant. Linked beneath it is a higher resolution version, resizable for your applications but specifically for use with digitized 600 [dpi] / 23.1 [mmA] patterns.

Data tips, Mathematica worksheets, Photoshop tutorials, etc.

Look for more here soon. For creating the spirals, we use Mathematica's PolarPlot routine to spiral-plot a list of lattice spacings multiplied by theta/2Pi (for theta values ranging from Pi/2 to 2Pi) and then add to the resulting plot stuff like axes, Miller index labels, etc. The plots are monochrome images by choice, so that they stay confined to one color channel in overlays. In the examples shown here they are also binary images, although peak intensities can also be recorded in the overlays if desired.

Experimental electron diffraction patterns are set up with the usual care to area selection, specimen eucentricity, and minimal beam convergence. These are digitized either during acquisition or from photographic negatives. When the data is on film, we often digitize diffraction patterns at 600[dpi] and 16[bits/pixel]. These pattern images are then loaded into Photoshop, and for display purposes (after contrast adjustment if appropriate) converted into 8[bit] RGB format.

A monochrome spiral overlay in negative form (e.g. white lines on black) is then copied and pasted into the experimental image's green channel. Photoshop's MoveTool then allows one to center the overlay on the pattern, typically by aligning one of the more well defined rings to the radially-symmetric tick marks on the overlay axis. Other markup information from Mathematica or elsewhere (e.g. highlighting for the radial line of intersections) may similarly be put into monochrome negative form and pasted into the blue channel. The result is a red experimental pattern with green/blue overlays, like the first three figures above.

This page is http://www.umsl.edu/~fraundor/pdifovly.html.
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