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Even in non-cubic crystals the Bragg equation predicts momentum-changes Δp = hg = ħk of magnitude 2sin[θBragg]h/λ in reciprocal-lattice directions (hkl) normal to planes of molecules in a crystal's direct-space lattice. Lattice directions or zones [uvw] are similarly perpendicular to planes of points in the reciprocal-lattice. Thus a zone-axis-pattern is a map of projected scattering-power perpendicular to any lattice-direction in a periodic structure. In the Fraunhofer (far-field) diffraction-limit, such patterns are also the Fourier-transform of spatial-periodicities in the lattice projected down that direction i.e. 2D slices perpendicular to [uvw] through the crystal’s reciprocal (spatial-frequency) lattice (Fig. 4). One can thus also think of zone-axis-patterns as diffraction patterns obtained using a flat (i.e. large-radius 1/λ) Ewald-sphere.
In transmission electron microscopy (TEM), thanks to the small Bragg-angles (e.g. quarter degree) and the small interaction mean-free-paths (requiring crystals well under a micron thick with elongate crystal shape-transforms), electron diffraction patterns directly represent zone-axis patterns down the direction of the beam. These allow one to measure reciprocal-lattice periodicities directly from ``darkfield images" on active reflections (so-called g-vectors), and set-up and interpret a wide range of other scattering experiments in real time (Hirsch1965, Buxton1976, Spence1992, Reimer1997).
Zone-axis-patterns further relate to direct-space lattice-images because zone-axis-patterns correspond to Fourier-transform power-spectra of ``projected-potential" lattice-images, via computer-aided-tomography's Fourier-slice-theorem in reverse i.e. the Fourier transform of an object's shadow represents a 2D slice through its frequency-space reciprocal-lattice. Of course only image power-spectra (as distinct from their complex Fourier transforms) are needed for comparison to the intensity-only information available from diffraction. For students taking modern physics, one might further note that electron phase-contrast lattice images connect to maps of projected-potential via a simple piecewise constant-potential proportionality to exit-surface deBroglie-phase, which microscope-optics turn into recordable intensity-variations in the wavefield downstream (Spence1988).
To construct a crystal rotation-axis zone-axis-pattern from our microwave data, we first re-parameterize the abcissa of these plots in Fig. 3 to get intensity as a function of spatial-frequency g instead of Bragg-angle θB, where from Bragg's Law the magnitude of the spatial frequency-vector g e.g. in [cycles/cm] is g ≡ 1/d = 2sin[θB]/λ. If microwave wavelength is λ ≈ 3 cm, then Bragg-angle scans from 8 to 70 degrees examine spatial-frequency magnitudes g ranging from about 0.09 to 0.62[cycles/cm]. A ball-bearing periodicity of say d100 ≈ 4.27 cm will give us a peak at g100 = 1/d100 ≈ 0.23 [cycles/cm], and hence be easily detectable within this range.
Secondly, one then maps intensity as a function of spatial-frequency magnitude on a polar plot for the various possible crystal orientations φlattice. If the simple-cubic lattice data has been taken for φlattice between 0 and 45 degrees from the (h00) family of reflections, one can invoke the D4 (four-fold mirror) symmetry of a square in the rotation plane to fill in the pattern for values of φlattice from 0 to 360 degrees as shown in Fig. 5a.
To compare one's experimental result with the scattering expected from the experimental arrangement of ball-bearings, a simple phase-sum model that focuses on the location (rather than the intensity) of zone-axis-pattern features is shown in Fig. 5b. The model ignores intensity-variation with path-length and scattering-angle by just adding up the complex-phases for all scattering points to give an amplitude proportional to ΣijkExp[i2πdijk/λ], where dijk is the sum of source-to-scatterer and scatter-to-detector distances for the ijkth ball bearing. Each ball-bearing thus, for simplicity, contributes a unit-amplitude signal to the model sum.
Sample code for using Mathematica to generate both experimental and model intensity maps is provided in the supplementary material for this paper. As you can see the phase-sum tells quite a bit about the location of reciprocal-lattice features in the zone-axis-pattern slice, although it would not be difficult for students to try predicting the effect of scattering-amplitudes on the pattern as well.
In both patterns, periodicities of the infinite crystal lattice show up as a square lattice of diffraction-spots or intensity-peaks. A standard set of reciprocal-lattice (Miller) indices for these peaks is provided in the positive quadrant of the model image.
Our finite crystal is truncated via multiplication in direct-space by a 3D window function that corresponds to its cubic shape. As a result the Fourier transform of this crystal shape function (i.e. the crystal's 3D shape transform) therefore convolves each of the points in the crystal's 3D reciprocal lattice.
For instance, a cube of side w has a shape transform that, in terms of the Cartesian components of spatial-frequency g, e.g. in cycles/cm, looks like:
where x, y and z are the (100), (010) and (001) lattice directions for our faceted ball-bearing cube. This defines diffraction-peak broadening of half-width 1/w due to finite crystal size, as well as the 1/w periodicity of a series of damped ``sinc-oscillations" beginning at 1.5/w from peak center in a direction orthogonal to each crystal face. For a cube with w = 4×d100 ≈ 17 cm on a side, we therefore expect shape-transform peak half-widths and sinc-oscillation spacings in diffraction of 1/w ≈ 0.06 [cycles/cm].
Revision notes: We're adding the figure at top right to illustrate the 3D nature of the problem (per a reviewer request), as well as the differences between electron and microwave acquisition of the same kinds of data.
A zone-axis-pattern in the parallel-beam Fraunhofer (far-field) limit, as a planar slice through that reciprocal lattice, should therefore reveal around each diffraction spot a planar slice of the crystal's shape transform. Our phase-sum model, and our experimental ``divergent-beam diffraction-pattern", in addition contain Fresnel (near-field) diffraction effects although effects of both the infinite lattice (i.e. indexable diffraction-spots) and the shape-transform (in this case finite peak-widths and sinc-oscillations perpendicular to the cubic crystal facets) survive for source/detector-to-lattice distances more than 50 cm.
When intensities are taken into account e.g. by the experimental pattern, oscillations closest to the unscattered (central) beam-spot are easiest to see. In fact, the first (low-frequency side) sinc-oscillation associated with the (100) diffraction spot in Fig. 5 apparently shows up in the PASCO instruction manual data example (Ayars2012), even though it's incorrectly identified as ``a reflection off of a different plane than the one we're measuring".
Related references:
P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977) Electron microscopy of thin crystals (Butterworths/Krieger, London/Malabar FL).
B. F. Buxton and J. A. Eades and J. W. Steeds and R. M. Rackham (1976) ``The symmetry of electron diffraction zone axis patterns" Philosophical Transactions of the Royal Society of London, Series A 281, 171-194.
J. C. H. Spence and J. Zuo (1992) Electron microdiffraction (Plenum, New York).
Ludwig Reimer (1997 4th ed) Transmission electron microscopy: Physics of image formation and microanalysis (Springer, Berlin) preview.
John C. H. Spence (1988 2nd ed) Experimental high-resolution electron microscopy (Oxford U. Press, NY).
Eric Ayars (2012) ``Teachers Guide" in Instruction Manual and Experiment Guide for the PASCO Scientific Model WA-9314B: Microwave Optics (PASCO Scientific).
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