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Description This is the shadow of the reciprocal lattice of a 118 atom single-walled carbon pentacone, rotating about its symmetry axis. As the reciprocal lattice (i.e. the 3D shadow) intersects the 2D Ewald sphere (radius 1/λ) of an incident electron beam (in this case essentially a plane perpendicular to the viewing direction), the reciprocal-lattice lights up in a red map of diffracted intensity.
Source Own work
Author P. Fraundorf

Extended notes

Direct-space pentacone model (left) & corresponding diffraction pattern (right).

The Fourier projection-slice theorem states that the inverse transform of a slice through the origin, extracted from the frequency domain representation of a volume, yields a shadow-like projection of the volume in a direction perpendicular to the slice. In other words, the Fourier transform of an object's "silhouette" is a physical slice through its reciprocal lattice (no shadowing involved). This can be combined with transforms from different directions to determine the object's 3D reciprocal lattice, whose inverse transform will yield the 3D object itself!

For instance, in the red-cyan animation above we've taken the real three dimensional reciprocal lattice of a faceted graphene nanocone i.e. a three-dimensional density distribution I[x,y,z], and used the projection-slice theorem trick to create a rotating shadow visualization. Thus the human ability to recognize moving shadows in 3D can be put to use visualizing 3D distributions projected onto a 2D field.

Running this theorem backwards says that one can combine 2D slices through an object's complex Fourier transform, obtained by looking at its "attenuation shadow" from a variety of directions, into the object's complex three-dimensional Fourier transform. Inverse transforming that beast then can give you a model of the object itself, in its full three dimensional glory. This reversal of the projection-slice theorem is called tomography, and is of course how X-ray CAT scans and MRI imaging often work.



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25 February 2008

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