The figure below illustrates this "loop" for a 1D slit. In all plots to follow, the upper-left panel is the direct-space representation, the lower-left panel is the equivalent reciprocal-space representation or Fourier transform, the upper-right panel is the direct-space self-convolution (which contains only 2nd moment information), and the lower-right panel is the equivalent reciprocal-space representation or power spectrum. In comparison to the Fourier transform, the power spectrum contains only Fourier amplitude (not Fourier phase) information.
Data in "second-moment form" (the right-hand panels of each plot) is often available experimentally, e.g. in power spectra from x-ray, electron or neutron diffractometers, or in convolution form as the interferogram from a Fourier transform infrared spectrometer. Alas, this data taken alone contains information only on periodicities in the original signal, but no information on the absolute location of structures (i.e. on Fourier phases). For example, one could translate the slit in direct space to the left by 1.5 units, and no change in the right hand panels would result. This shortcoming of second-moment data is often called "the phase problem". For the many situations when Fourier phase information is also of interest (e.g. when trying to determine the location as well as the periodicity of atoms in a crystal's unit cell from diffraction patterns), scientists have developed strategies to infer that information with help from added information, even though it is not contained directly in any single diffraction pattern power spectrum.
Reciprocal-space representations (the lower panels of each plot) often play a useful complementary role in detective work on signals. In fact, this complementarity is intimately connected to the complementarity between position and momentum in quantum mechanics. The resulting role of Fourier transforms in optical scattering (mentioned above) is one of many "analog" ways to do such Fourier harmonic analysis. A wide range of transforms intermediate between direct and reciprocal space, including darkfield images and wavelets, show that the cycle described here considers only endpoints in a field of operations relevant to a growing range of inquiries. Uses for these "operations between direct and reciprocal" are already well underway with practical applications e.g. in data compression, image pattern recognition, and the study of defects in crystals.
The next figure (below) illustrates the r2 loop for an array of 10 equally spaced slits: a so-called 1D comb. Note how crystal size s, the spacing between slits (or scatterer rows/planes) d, and slit width w manifest themselves in the various plots. This may help explain why the bottom panels are referred to as "reciprocal space" representations. Not only are distances in reciprocal-space proportional to reciprocal distances in direct-space, but the largest shapes in direct-space manifest on the smallest scales in reciprocal-space and vice versa. In particular, the tiny "wings" of each diffraction spot are sometimes referred to as the shape transform of the corresponding collection of lattice planes (i.e. crystal) in direct space.
The next figure (below this paragraph) considers a single 2 dimensional slit. Note that taller peaks in the power spectrum (lower-right panel) exceed the height of the plot, and have hence been artifically truncated in the display. Also note that the peaks and valleys in the Fourier transform (lower-left) have all become peaks in the power spectrum.
Our next figure (below) considers the r2 loop for a 10x10 array of 2D slits. Note that the Fourier Transform in the lower-left corner is displayed using logarithmic complex color so as to provide information on both Fourier phases and Fourier amplitudes. Crystallographers (cf. the two-panel inset at the top right corner of this page) have devised direct-lattice [uvw] = ua+vb+wc, and Miller or reciprocal-lattice (hkl) = ha*+kb*+lc*, indexing schemes to help people discuss specific features in such plots. Here the reciprocal basis vectors (a*, b*, c*) relate to direct lattice vectors (a, b, c) via a* = bxc/V, b* = cxa/V and c* = axb/V, where unit cell volume V = a.bxc. In transmission electron microscopy, lattice indices [uvw] are used to denote both the vector displacement between lattice points, and high symmetry viewing directions or "zones" down which lattice planes intersect to form tunnels through the crystal. Miller indices (hkl) are used to denote both reciprocal-lattice spots in the power spectrum, and collections of lattice planes in direct space. Although directed reciprocal-lattice displacements are often referred to as "g-vectors", in direct-space mathematicians might say that they more generally instead have the covariant transformation properties of a dual-space covector, Dirac bra, or differential one-form.
Even while remaining in 2 dimensions, the richness of the formalism begins to assert itself if we replace the simple square comb above with the faceted 2D crystal from the inset at the top of this page, along with a second elongate crystal, both superposed on an "amorphous" 2D film with a hole in the center. A taste of the "sinc wavelet" transition between direct and reciprocal space for a similar image may be found in the digital darkfield animation here. The animation might help you figure out which features in reciprocal space come from which objects in the direct image. More on features of the resulting "nano-microscopy relevant" r2-loop (below) shortly...
A taste of what happens for three dimensional structures is illustrated below. The top three panels are, respectively, side and top views of the projected direct-space image, followed by the top view's self-convolution, for a spherical array of atoms arranged in a simple-cubic lattice at various tilts around a horizontal axis. The first two of the bottom panels show similarly-oriented slices through the origin of the crystal's three-dimensional reciprocal lattice. Slices, rather than projections, of the reciprocal lattice show up in electron diffraction patterns of the crystal. As reciprocal-lattice spots rotate into and out of the visible slice, they blink on and then off. The third panel on the bottom is the top-view power spectrum, or Fourier transform of the convolution above it.
Here are some reciprocal lattices in 3D, and a shadow view of the reciprocal lattice responsible for one pattern's blinking. For an interactive comparison of direct and reciprocal lattices in 3D, see our reciprocal lattice tutorial.
The red line cutting horizontally across the bottom left panel, which effectively marks the slice visible in the power spectrum, is known to diffractionists as "the Ewald sphere". It would appear concave upward with radius 1/wavelength, except that the diffraction we exemplify here is electron diffraction for which wavelength is typically much smaller than lattice spacings. Hence the Ewald sphere in cross-section shows very little curvature. Reciprocal lattice spots "light up" in the diffraction pattern (bottom right) when the Ewald intersects them.
The "crystal" in this simulation is only about 5 atomic layers across. Hence spots in reciprocal space are much broader than they would be were the crystal say 10 nm (i.e. 50 atomic layers) across. Also for larger crystals in the electron microscope, the tilting process is oft marked by "Kikuchi band" roads due to inelastic scattering. Their widths, like the Bragg scattering angles, are typically a quarter degree across. Unlike diffraction spots which blink on and off, Kikuchi bands behave as though they are fixed to the orientation sphere. Their intersections thereby serve to mark high-symmetry zones in reciprocal space in an elegant and strikingly beautiful manner.
When the direct-space view is oriented down the [001] zone of the crystal, a four-fold symmetric power spectrum (bottom right) appears, showing a square array of orthogonal (100) and (010) reciprocal-lattice spots. At least two other zones show up as two-fold symmetric rectangular spot arrays in reciprocal space during the tilt. What are their zone indices? Here are similar tilt sequences for body centered cubic (bcc), face centered cubic (fcc), and diamond face-centered cubic (dcc) lattices. How are they similar? How do they differ? Which direction is the tilt axis running in these sequences?
All of the foregoing mathematics model small-angle diffraction patterns found in the back focal plane of an imaging lens under parallel (uni-directional) illumination of a specimen. One might imagine that illumination with a range of angles in the beam would merely wash out the details present in the uni-directional case. But nature is full of delightful surprises, so stay tuned...
