4. Experiment extensions

Fig. 6 shows for example what the phase-sum model predicts for the pattern as one decreases source/detector to lattice distances. Fig. 7 shows the spotty ``atom-thick sheet" powder-diffraction pattern that the phase-sum model predicts for the pattern if one rotates each of the four ball-bearing layers randomly about the rotation zone-axis. An azimuthal-average of this pattern has {001}, {110}, {200} and {210} peaks whose breadth reflects the coherence-width of spacings in each sheet. To what extent these patterns can be matched by experiment remains to be seen.

The direction-complementarity of reciprocal-lattice and direct-lattice vectors, with their co-variant as distinct from contra-variant transformation properties, is illustrated by a close look at Fig. 1b. The basis vectors a*, b* and c* of the diffraction-spots in reciprocal space are not parallel to the direct-space basis vectors a, b and c, but are instead ``axial" vectors or one-forms perpendicular to those ``polar" direct-space vectors according to a* = b*×c*/Vc, etc., where the unit cell volume is Vc = a•(b×c). Geologists are often more familiar than physicists with the elegant notation crystallographers have developed to deal with these dual vector-spaces, since minerals are much more likely than elemental solids to have low-symmetry lattices.

For periodic lattices projected into two dimensions, only two basis vectors in frequency-space are needed to infer the rest of the 2D reciprocal unit cell, and hence by Fourier-transformation the direct-space unit cell as well. Therefore a lattice with the angles shown in Fig. 1b might have its lattice characterized by varying φlattice from 0 degrees rightward and then clockwise by about 45 degrees to pick up the g2 and g3 spots from which the others (like g1 = g2 - g3) can be inferred. However the rotating-lattice technique of Amato and Williams (Amato2009) would allow one to quickly scan all 360 degrees for a range of Bragg angles. Design of a two-axis eucentric goniometer would allow an even wider range of unknowns to be analyzed, although at this point the analysis might move beyond the "hands on" scope of an advanced lab experiment. 

Shape transforms have a breadth in frequency-space proportional to the inverse of their corresponding coherence-width (e.g. crystal size) in direct-space (cf. Hirsch1965), as discussed in the previous section. In this context, the reciprocal-lattice of an atom-thick crystal is a spike (or ``rel-rod") in frequency space. A collection of parallel but randomly-rotated atomic layer-planes therefore has a cylindrical reciprocal-lattice, which can show up in the zone-axis-pattern as a circle when cut perpendicular to its axis, as parallel streaks when cut parallel to its axis, or as an oval (Sasaki2001) like that shown in the experimental hexagonal-BN/C random-layer-lattice pattern in Fig. 8. This effect might be explored with microwaves using a ball-bearing lattice by simply randomizing the azimuth of equally-spaced ball-bearing layers before taking the data.