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## Avatar at the atomic scale, Ctd.

The precursor to this post ended with the cliffhanger:

We demonstrate that when the diffraction pattern of a finite object is sampled at a sufficiently fine scale… the 3D structure of the object is in principle determined by the 2D spherical pattern.

Today, I describe how this piece of magic is achieved.

It certainly sounds too good to be true! And, of course, one can’t cheat the physical world, so it should be clear that some penalty will be involved; but that said, it’s every bit as clever as it sounds. Here’s what occurs: a single frequency beam is shone at the sample and a curved deformation of the diffraction pattern of the waves appears on the Ewald sphere, with intensity as per the Fourier transform of the density of electrons inside the structure.

Two-dimensional representation of the projection-slice theorem

Now something wonderful happens. There is a beautiful relationship between projection and slicing that I alluded to in the warm-up post; and it is this: a slice at a given angle through the Fourier transform of a distribution is equivalent to the Fourier transform of the projection of the real distribution along that angle. That’s a mouthful, so here’s an equation to save the day:

$\mathcal{S}\left[\mathcal{F}(\rho)\right] = \mathcal{F}\left[\mathcal{P}(\rho)\right];$

here, S, P and F are the operations of slicing, projecting and Fourier transforming. (As one might expect, the reciprocal statement holds too, but that will not be necessary here.)  It’s a very clean relationship. The pattern on the Ewald sphere is exactly the Fourier transform of the structure under examination, so by slicing through it at a range of angles, one generates a series of projections of the electron density itself (in Fourier transform).

Slice through the Ewald sphere representation of the diffraction pattern---equivalent to a single projection snapshot.

This is shown in the graphic above. The Ewald sphere representation of the diffraction pattern is F(ρ); each slice through it at a given angle is one particular projection. By rotating the slice around each of the x-y- and z-axes, information about the projection along each of those directions is acquired. Just as with tomographic imaging, from these snapshots, the true electron density structure can be obtained through inversion. Magic!

Reconstruction of a simluated (14Å)^3 sodium silicate glass particle by ankylography; O, Si and Na atoms are in red, yellow and dusky lavender respectively.

In the paper, the authors run two numerical tests of their idea and one actual experiment. The graphic above depicts a simulated glass particle with diffraction pattern shown as panel (a). The hemispherical intensity pattern is the Ewald representation of the structure—the curved form is critical here: if the diffraction pattern were planar the slicing algorithm could not gather the requisite information. Panel (b) shows, of course, the reconstruction.

Much has been obscured in this brief exposition. I haven’t mentioned the requirements for anklyographic reconstruction to be feasible, the strictest of which is sub-Nyquist sampling of the diffraction pattern itself. The article and associated supplementary material do give plenty of detail, including a thorough description of the reconstruction algorithm itself. They’re behind the Great Paywall; regrettably, I’ve yet to find a preprint online.

But it is the idea that is beautiful: a marvellous confluence of optics, statistics and geometry.