1. Introduction
Because of their high charge/mass ratio and hence strong interaction with matter, not to mention their wavelength in picometers, high-energy electrons are a benchmark tool for studying the interior of individual-structures on the nanoscale, at least to the extent that those structures will ``hold still" for a scattering experiment. Microwave scattering from a small ball-bearing lattice can be a way to give undergraduate students insight into the challenges of doing electron-scattering experiments on nanocrystalline materials.
Amato and Williams (Amato2009) have previously discussed a way to modify classroom microwave-optics experiments (Allen1955, Murray1974, Cornick2004, Yuan2011) to acquire ``X-ray powder diffraction" data on all Bragg peaks accessible from a two-dimensional lattice. In this paper, we discuss a way to put data from a standard-setup experiment into the format of an experimental electron zone-axis-pattern (ZAP). With this approach, students get some experience working with the crystal's reciprocal-lattice directly, and in the process gain some clues to the Fourier transform of a 3D crystal's shape (Rees1950). These effects of crystal shape become especially important when the crystal is only a few unit-cells across, in one or more directions.
In particular this experiment yields an experimental slice of our ball-bearing crystal's reciprocal-lattice, very much like electron-diffraction patterns (Hirsch1965, Cowley1975, Williams1996, Fultz2001) and lattice-fringe image power-spectra (Allpress1973, Spence1988, Fraundorf2005, Wang2006, Kirkland2010) obtained from submicron-thick specimens in real-time (cf. Fig. 1a) as may be seen through a web-browser with the javascript simulator here. Our ``microwave slice" lies perpendicular to the lattice (i.e. zone-axis) direction used as the crystal rotation-axis in the experiment. We further discuss how this window onto the distinct and complementary nature of direct/reciprocal dual vector-spaces can be enhanced by construction of a non-Cartesian ball-bearing lattice (cf. Fig. 1b).
Revision notes: Figure captions may be seen by holding your mouse pointer over the figures. To the original Fig. 1 (lower right) we are tentatively adding an experimental nanocrystal image and power-spectrum (top right) to illustrate the concrete connection to nanoscience applications of the data format described here. These added panels make the progression of figure panels run from concrete to more abstract. /pf
Related references:
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