W. H. Slabaugh and Devin Smith Oregon State University Corvollis, 97331
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Single Crystal Diffractometry A n analogy a n d a n experiment
The hasis of X-ray diffraction as a means of determining solid state structure is often unclear to the student because he seldom experiences the actual relationships involved in the application of Bragg's law or Young's experiment. We now have available hoth the inexpensive classroom laser and a variety of lattices which permits a direct visual experience of diffraction and provides the basis of interpreting the diffraction patterns. The He-Ne laser with an output of about 2 mW, which is used in this demonstration and experiment, is safe to use in a classroom providing that direct viewing of the beam is avoided, that the beam is not viewed from reflecting surfaces, and that students are not permitted to "play" with the laser. Wire screens, cloth, grating, phonograph records, and such have been suggested for several years as diffraction media for the optical laser beam to represent the methods of determining solid state ~ t r u c t u r e s The . ~ difficulties of using these media lie in their restricted structures which are representative of only a few actual solid state structures. The present report shows how to make gratings and lattices analogous to crystal structures and how to apply the principles of diffraction pattern to naturally occurring lattices as observed in bird feathers. For the demonstration, a series of 2-dimensional lattices, each covering an area of about 30 cm2 and analogous to the planes in 3-dimensional lattices, were prepared and then photographically reduced by a factor of about 75. A rectangular lattice is shown in Figure 1, where the a / b ratio is %. The original lattice was made of black
dots on white paper. Reductions were made by photographing the originals with a 50-mm lens camera on black and white film at a distance of about 5 m. The developed negatives were mounted in slides, giving a projection area of about 1 mm2 which is approximately the cross-section of the laser beam. By passing the laser beam through the negative, the beam is diffracted and appears on the classroom screen as a reciprocal lattice in agreement with Young's experiment. It is quite instructive to show a slide of the unreduced lattice with a regular slide projector alongside the laser reciprocal lattice on the screen so that hoth the original and the reciprocal lattices can be seen at once, as shown in Figure 1. For a given lattice, a reciprocal lattice can be constructed, and the reciprocal lattice corresponds to the diffraction pattern produced by the given lattice. That is, a repeating distance a in the given lattice becomes l / a in the reciprocal lattice and an angle a between rows in the given lattice becomes the reciprocal angle, 180 - a in the
'Morrison, J. D., and Driseoll, J. A,, J. CHEM. EDUC., 49, 558 (1972): and Cam, P. D., J. CHEM. EDUC., 50,294 (1973).
Figure 1. A rectangular 2-dimensional a t t c e lprnrnl and its dftraction pattern.
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The diffraction pattern produced hy the blackbird's feather is shown in Figure 4. The angle ~u is derived by construction and represents the angle between the harh and the barbules. The two rows of s ~ o t in s the horizontal and vertical directions in the diffraction pattern at an anele of 13 to each other are the result of two sets of fibers or-harhules that branch from opposite sides of the harh. That is, the harbules on one side of the harhs were spaced a t 0.024 mm and the harbules on the other side of the harhs were spaced a t 0.020 mm. These spacings were calculated from Young's formula from the two arrays, and confirmed by direct microscopic measurement (Fig. 5). The angle between the two sets of harhules is derived from 0, the angle between the two arrays of spots in the diffraction pattern. The fine detail in each spot of the diffraction pattern represents gross spacing in the feather and corresponds to the d spacings between barbs, or 0.42 mm. Both the spacings and the angles between the two arrays in the diffraction pattern are reciprocals of the spacings and angles in the actual feat he^.^ Different spacings were obtained with different kinds of feathers. Hence, this experiment could he used for identifying hird species, unless the tyro chemist is also an expert ornithologist, or a t least i t could he readily adapted . . to laboratory unknowns.
Figure 2. A hexagonal 2-dimens8ona a t t c e (p6rn) and 1 s dlfraction pattern.
Figure 3. Projection of a double helix and its dilfraction pattern.
reciprocal lattice. In the t,raditional analysis of a diffraction pattern, the reciprocal structures may he derived through Fourier transformation. In order to visualize the idea of the reciprocal lattice, two arrays of parallel lines were prepared in which the distance between the lines varied. That is, set A lines were originally spaced 5 mm apart and set B lines were 2.5 mm apart. Upon a 75-fold reduction, the diffraction patterns were proportional to % and %.s for the A and B spacings. Once this is clear to the student, the several lattices can be examined. The rectangular lattice (pmm) in Figure 1shows an obvious display of the reciprocal lattice. Figure 2 is a hexagonal dimensional lattice (p6m) whose diffraction pattern can he correlated with the original lattice. It may he of interest to identify the 2-, 3-, and &fold points of symmetry from this figure. The diffraction pattern of helical structure can he examined by preparing an original of dots in a sine curve. We made a douhle sine curve array of black dots, which is a 2-dimensional projection of a douhle helix analogous to that of DNA.= The diffraction pattern of the douhle helix array, though somewhat complex, shows the so-called diamond array of points from which the arrangement in the original array can be deduced (Fig. 3). Experimentally, there are few naturally occurring lattices that can he measured with this method. However, we discovered that hird feathers are ideal systems to investigate with the optical laser. Both the d spacings in the primary branching (between harhs) and in the secondary hranching (the harbules) of the feather components can he calculated from Young's formula: d sin 0 = m A, where m = 0, 1, 2, 3 . . . is the angle of diffraction, and A is the wavelength of the diffracted radiation.
Figore 4 . Laser ditfractmn pattern of blackbird's feather at 591 cm from leather to screen.
2See Lehninger, A. L., "Biochemistry," Worth Publishers, Inc., New York, 1970, p. 638. See, for example, Hosemann and Baghi, "Direct Analysis of Diffraction by Matter," North Holland Publishing Co., Arnster-
Figure 5 . Structure of blackbird's feather as derived from laser diffraction and microscopic observation.
dam, 1962.
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