In the Classroom
Illustrating Fourier Transforms Cliff Bettis, Edward J. Lyons, and David W. Brooks Center for Curriculum and Instruction, University of Nebraska, Lincoln, NE 68588-0355 A number of articles have appeared in this Journal that speak to simulating x–ray diffraction and to the Fourier method of x–ray crystallography (1–6). A kit for demonstrating this effect is available from ICE (7), and its use is described in a chapter in an exciting supplementary text (8). When light from a low-power He–Ne laser (a coherent visible light source) strikes a lattice such as the two-dimensional lattice of fine wires found in a 100mesh sieve, the emergent light is the Fourier transform of the sieve lattice. The projected result, a series of spots of varying intensity arranged in the pattern of a plus sign, is spectacular and visible throughout a large lecture hall. The simplest means of deconvoluting the transform is to use an appropriate lens. The properly focussed lens, placed immediately after the sieve, produces an excellent magnified image of the wire mesh. Focussing is accomplished by moving the lens; it must be quite close to the diffracting element. A standard microscope objective lens can be used (borrow one from a microscope found in biology lab, and mount it using the gadgets found in the optical benches of a physics lab). Place it immediately after the diffracting plane (such as a sieve or the slide provided with the ICE materials). An image of the diffracting pattern is projected. So, for example, a linear horizontal array of spots would be replaced by a series of uniformly spaced vertical lines when these serve as the diffractor. This is illustrated in Figure 1. Objective lenses with magnifications in the range of 4 to 40 work. The larger the power, the larger the image, the lower the projected light intensity, and the closer the lens needs to be to the grating. The distance needed between the diffractor and the lens ultimately limits the lens size that can be used. We often teach our students only the applications of FT techniques to chemical instrumentation. In the weaving industry, where synthetic fabrics are spun through tiny orifices that wear during manufacture, quality control can be a problem. When a laser beam is allowed to strike the emerging fabric, an examination of the spots in the resulting FT pattern permits an estimate of “fuzziness,” and an increase in fuzziness allows a rapid, inexpensive quality control system to determine when to replace spinners. Formerly, cumbersome inspection using microscopic techniques was required. Also, the techniques of image enhancement used by space scientists involve converting transmitted pictures to the FTI removing the zero order spot, and recreating the image from the modified FT.
Figure 1. (a) Schematic of diffraction apparatus. (b) Schematic of diffraction pattern from line grating. (c) Schematic of diffraction transforming apparatus [lens added to (a)]. (d) Schematic of diffraction grating leading to pattern in (b) (grating obtained from ICE). (e) Photograph of diffraction transforming apparatus. (f) Photograph of screen showing diffraction pattern from ICE grating. (g) Photograph of screen showing image of ICE grating. (h) Photograph of screen image of diffraction pattern from 100-mesh screen. (i) Photograph of screen image of 100-mesh wire screen from deconvolution of diffraction pattern (h) using lens. [In (f), the principal spot (center) was projected onto black ink to diminish its reflective intensity relative to the other spots.]
Literature Cited Waser, J. J. Chem. Educ. 1968, 45, 446. Morrison, J. D.; Driscoll, J. A. J. Chem. Educ. 1972, 49, 558. Garn, P. D. J. Chem. Educ. 1973, 50, 294. Brisse, F.; Sundararajan, P. R. J. Chem. Educ. 1975, 52, 414. Julian, M. M. J. Chem. Educ. 1980, 57, 737. Chesick, J. P. J. Chem. Educ. 1989, 66, 413. Ellis, A. B.; Lisensky, G. C.; Neu, D. Optical Transform Kit, 2nd ed.; Institute for Chemical Education: Department of Chemistry, University of Wisconsin, Madison, WI, 1994; order number 90-002. 8. Ellis, A. B.; Geselbracht, M. J.; Johnson, B. J.; Lisensky, G. C.; Robinson, W. R Teaching General Chemistry: A Materials Science Companion; American Chemical Society: Washington, DC, 1993; pp 77–96. 1. 2. 3. 4. 5. 6. 7.
Vol. 73 No. 9 September 1996 • Journal of Chemical Education
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