Two-spot indexing-laboratory v1.0 at UM-StL

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Abstract

Here we've eliminated some of the introductory material on our general goniometer page, as the focus here is on indexing two experimental diffraction and/or power-spectrum spots, and their interspot angle, against one or more candidate crystal structures.

The basic strategy is to move past the 3D-goniometer window below (which will soon show a "nano-sized" version of your lattice). Instead you may type the lattice parameters into the boxes at top of the crystal structure & orientation worksheet, leaving space group # set to zero so that centering disallowances are ignored. Then hit the "upload to goniometer" button.

The next step is to hit the update button associated with the Table 1 section down a bit further on the page. Just above that update button you will now find buttons which will load in specific candidate phases from scratch. In some cases, these buttons will set up consideration of cell-centering disallowances as well. Either of these actions should fill Table 1 with a set of 3D coordinates for g-vectors to compare with your measured data.

Note that only select spacegroup numbers are currently supported. In particular, we're using fcc space-group #225 for SiC even though it should be #216 since centering-extinctions should be the same, and the only hexagonal space groups already programmed in are #194 (hcp) and #186 (graphite). We do not yet address centering disallowances for hexagonal lattice structures WC (#187), ZrB2 (#191), B4C (#166), B2O3 (#144). Likewise for tetragonal ZrO2 (#137). This means that proposed indexings for these will still have to be checked against possible centering-disallowances, although NO will still mean NO.

Just below Table 1 you may then type in any two measured spacings and their interspot angle, as well as what you believe are your uncertainties in spacings and angles. Hitting the button to index should let you know if that structure might explain your spots.

Nano-goniometer

(* here a set of independent variables is initialized *) (* the Mathematica model update commands follow *)

These controls pop-up nano-microscopy data on the "specimen-as-oriented" for screen capture.
SAED/Powder use truncated Debye sums; HREM uses projected-potential; Other (!)buttons not yet hooked up.

Until we make the plots self-labeling, field widths are 12Å in direct space, 3/Å reciprocal.
Look for options to increase resolution soon, and fancier algorithms as java versions come on-line.

Crystal Structure and Orientation Worksheet

a=[Å], b=[Å], c=[Å],
α=[deg], β=[deg], γ=[deg].
space group = (only 225:fcc, 229:bcc, 194:hcp, 227:dcc, 186:graphite and 0:? so far).

Goniometer angles: T_external=[degrees], T_internal=[degrees],

Zone-axis (lattice vector) in the direction of the beam: u=, v=, w=,

Oriented basis triplet (OBT)
The below matrix gives cartesian coordinates xyz when multiplied by lattice index column [uvw].
/ a_x=[Å], b_x=[Å], c_x=[Å] \
| a_y=[Å], b_y=[Å], c_y=[Å] |
\ a_z=[Å], b_z=[Å], c_z=[Å] /
Its transpose gives Miller indices (hkl) when multiplied by reciprocal coordinates xyz.
The above matrix times its transpose yields the orientation-independent metric tensor for the lattice.
The inverse of the above matrix gives lattice indices [uvw] when multiplied by cartesian coordinates xyz.
/ a*_x=[/Å], b*_x=[/Å], c*_x=[/Å] \
| a*_y=[/Å], b*_y=[/Å], c*_y=[/Å] |
\ a*_z=[/Å], b*_z=[/Å], c*_z=[/Å] /
The transposed inverse (above) gives reciprocal coordinates xyz when multiplied by Miller indices (hkl).

Index#1: , , ; Index#2: , ,

Length 1: , Length 2: , Angle Between: [deg]

Table 1: Single-crystal reflection list:
Uses only lattice parameters and centering information, and for each g_hkl does not bother to list -g_hkl.
Only about 100 entries for now from the Miller index range between -6 and 6.

Index YOUR measured lattice-fringe or diffraction-spot pairs

Table 2: List of powder-diffraction peaks:
Uses lattice parameters and atom positions within each unit cell.
Groups reflections onto one entry when spacings and structure factors are similar, so that the entry's Miller index may or may not be representative.
Note that intensities are only qualitatively relative and without scattering-angle falloff, and that extinction distances are not yet estimated at all.
Only about 20 entries from the Miller index range between -6 and 6.

Solve

What follows is a list of practical analytical tasks, addressable with these developing tools, that we plan to illustrate with lattice images and/or diffraction data from past and ongoing nanodetective challenges.

Single-crystal known: Index the periodicities

Single-crystal unknown in 2D: Match spacings and angles

Single-crystal unknown in 3D: Get lattice parameters and OBT

Single-crystal known rocking curves: Infer thickness/strains

Defected known: Find habit, Burger's vector, etc.

Bi-crystal known: Determine relative orientation

Powder known: Index lines, determine sizes & textures

Powder unknown: Identify from spacings and/or radial profiles

Digress...

Real soon now...

* Note that [uvw] is normally used to denote a specific (contravariant) lattice vector or crystallographic zone-axis, <uvw> a class of such directions or zones, (hkl) is the Miller index of a specific reciprocal-lattice point, covariant "g-vector", or set of crystallographic planes, and {hkl} denotes a class of symmetrically equivalent reciprocal-lattice points.

What's next?

What's next here? More input/output and calculation options, illustrations, class exercises, puzzlers, data, and theory. We just added an ActiveWidgets spreadsheet to list g-vectors from a specific crystal. Only a few space-groups and simple crystal sizes/shapes are available now, but this will expand in days ahead, likely to include special structures like n-wall nanotubes and quasicrystal approximates, as well as arbitrary (even aperiodic) lists of atom positions. Tools (like SXTL) to match and index observed spacings and interfringe/spot angles may be next, along with buttons to record single-crystal/powder diffraction patterns and lattice images in 2D for closer examination. Eventually, we hope to offer, for example, independent variation of cluster size and shape, centering and extinction information lookup from any spacegroup (look for A, B and C centering soon, as it makes the single crystal indexing routine useful for any structure), Wycoff-format atom-coordinate entry tools, fringe-visibility maps, fringe probability/thickness plots, Kikuchi maps and other stereo-projections, strong-phase-object through-focus series, Cliff-Lorimer prediction of characteristic X-ray peak heights from elemental ratios and vice-versa, Debye-scattering profiles, formatted spiral powder overlays, some simple plasmon/dielectric and characteristic-edge energy calculations, select defect models including SPM topography and surface reconstruction predictions, and possibly even digital-darkfield analysis of images.

More importantly, tools for fitting data you've taken to specific candidate phases, and in some cases to directly-determined structural models, are planned as well. This experimental data might include observed fringe and diffraction spot spacings/angles taken at one or more specimen orientations, azimuthally-averaged diffraction/power-spectrum profiles, and eventually selected analyses of direct-space images. Data on widely-interesting phases will be programmed in locally, but we are also in discussion with developers and patent holders of searchable databases to facilitate more systematic comparison of data to known structures downstream.

On the nanoeducation side, empirical observation exercises with in-class peer-review are already under development here for a range of introductory science classes. The nano-goniometer above, when viewed from the side in "dual space" mode already offers an excellent interactive illustration of Ewald-sphere mediated diffraction. With sufficient refinement it is hoped that empirical-observation exercises like this, patterned on evolving real-world challenges, can work their way into classrooms, onto timed tests, and perhaps (if sufficiently robust) even into the larger culture of social competition and educational video-gaming. This page also provides possibilities to this end. For example, all crystals in the goniometer so far are in "c-axis" orientation, but a set of adjustable Euler angles will be added in days ahead to allow creation of "true unknowns" for analysis. What better challenge for prospective nano-detectives than to be given a device capable of performing quantitative experiments (of their choosing) on an unknown, and then showing how they can make the most of it?

As far as implementation is concerned, look for other applets (e.g. webMathematica, Jmol, and JavaView) to be put to use in the days ahead as well, and perhaps even a more interesting "room" in which to put the nanogoniometer. In all cases, we hope to make these tools available reliably and seamlessly (free where possible without the download of plugins) for use at many application levels, across the planet, in years ahead.

Credits:

This page is http://www.umsl.edu/~fraundor/nanowrld/newlive/crystal9.html. Acknowledgement is due particularly to Peter Möck at Portland State University for his energy in exploring these matters, and Martin Kraus for his robust Live3D applet. Thanks also to Noom Pongkrapan for the nano-goniometer border. Although there are many contributors, the person responsible for errors is P. Fraundorf. There are likely to be many. Reports about bugs, ways to correct them, algorithms that might be fun to implement, and interest/energy/time in helping out, are most welcome via e-mail to "staff" at newton.umsl.edu. Putting "nanocluster" in the subject line might improve its chances of getting through. This site is hosted by the Department of Physics and Astronomy at UM-StL. MindQuilts site page requests ~2000/day approaching a million per year. Requests for a "stat-counter linked subset of pages" since 4/7/2005: .