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.
Draft MS
prepared for publication in Microscopy Today13 #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'sPolarPlot 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.
Although there are many contributors, the
person responsible for errors is P. Fraundorf.
This site is hosted by Dept. of
Physics & Astronomy (and Center for
Molecular Electronics) at UM-StL.
Mindquilts
site page requests ~2000/day hence approaching a million per year.
Requests for a "stat-counter linked subset of pages" since 4/7/2005:
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