Apparatus Competition

2008 AAPT Summer Meeting

Edmonton, Alberta

Rotating Crystal Microwave Bragg Diffraction Apparatus

Joseph C. Amato and Roger E. Williams

Department of Physics and Astronomy

Colgate University

Hamilton NY 13346 USA

315-228-7653

JAmato@Colgate.edu, RWilliams@Colgate.edu

Abstract

A microwave Bragg diffractometer employing the “rotating crystal” technique is constructed from inexpensive, readily available components and materials. In contrast to previous diffractometer designs, Bragg angles are found by orienting just one component (either the transmitter or receiver). The detected signal is displayed vs. time on an oscilloscope, facilitating the identification of the crystal planes responsible for each Bragg peak. When using a visible “crystal”, students can easily identify these planes, gaining an understanding of Bragg scattering, crystal symmetry, and Miller indices. Confronted with a concealed crystal, students can determine its lattice spacing and orientation, thereby mimicking the methodology and purpose of x-ray crystallography. Complete construction plans are available from the authors.

Construction of Apparatus:

The apparatus is constructed using a standard X-band crystal detector and Gunn diode transmitter from a Sargent-Welch microwave diffraction kit. The emitted wavelength is 2.85 cm (f = 10.525 GHz). (Other apparatus, such as that made by Pasco or Heath, would work equally well.) A single refracting surface polyethylene lens, made from beads filling a semi-circular styrofoam trough, is used to collimate the transmitted wave. Both transmitter and detector are mounted on rigid wooden arms, one of which is free to rotate about the center of a circular plywood plate. (See Figure 5.) A protractor printed on 8.5 x 14 inch label paper is glued to the plate, and serves as a goniometer, registering the orientation of the moveable arm to within ± 1°. The crystal is spun by an inexpensive inverted ceiling fan motor at approximately 180 rpm, driven by a 24 V transformer connected to the 110 V power line. A flag mounted on the fan motor housing interrupts a photogate to trigger a dual trace digital oscilloscope once each revolution. The oscilloscope displays the trigger pulse and the amplitude of the detected microwave signal as functions of time. The crystal is a 7 x 7 square array of vertical 2.3 mm (3/32 inch) welding rod spaced by 3.72 cm (Figure 4). Apart from the oscilloscope, the dc power supply, and the microwave transmitter and receiver, all components are fabricated locally from readily available materials (wood, polystyrene rod and sheet, hardware, welding rod). The apparatus has been used for years in our introductory physics labs, with excellent results and with no special maintenance.

Use of Apparatus:

The microwave Bragg diffractometer is a familiar, time-honored experiment in the introductory physics laboratory.^1-4 Using centimeter-scale wavelengths and crystal lattice spacings, this relatively inexpensive apparatus allows students to explore visually the conditions for Bragg scattering and get a taste of real x-ray crystallography. Previous designs, however, use a “crystal” – typically a cubic array of steel ball bearings – that is visible to the student throughout the experiment, so that its lattice spacing and orientation are known to the student a priori. The transmitter and detector are attached to arms that swivel about the crystal separately, and both must be carefully oriented so that the Bragg condition θ_inc = θ_ref = θ_Bragg is satisfied. Alternative designs move the crystal and detector in concert, but the crystal orientation must be initialized correctly in order to observe Bragg scattering. Although this is certainly instructive, the student gets little sense of how one would determine the lattice constant(s) and orientation of an unknown crystal, i.e., how one would do real crystallography.

The apparatus we have designed employs the “rotating crystal” technique so that only one device, either the transmitter or detector, needs to be oriented correctly to satisfy the Bragg condition at some time. The crystal is spun rapidly, and during each revolution, all values of θ_inc are sampled. (The rotating crystal method is equivalent to the powder diffraction technique for stationary samples.) Since the scattering angle θ_scat = θ_inc+ θ_ref, and θ_inc = θ_ref, Bragg peaks are found where θ_scat = 2θ_Bragg . (Figure 1) The Bragg angles satisfy

where a is the lattice spacing, and h, k, and l are the Miller indices identifying each family of crystal planes. (For our two dimensional geometry, l = 0 always.) Bragg peaks for the (100), (110), (200), (120) and (210) crystal planes are quickly and easily located by swiveling the transmitter arm about the center of the crystal until the detected signal (displayed on an oscilloscope) reaches a local maximum. The detector remains fixed.

The apparatus can be used in two different modes. In the first mode, an exposed crystal (made from vertical lengths of welding rod, shown in Fig. 4) is spun by the fan motor, and the transmitter swiveled until a Bragg peak is found. The fan motor is turned off, and then the crystal is rotated manually until the dc detector voltage is maximized. This is the crystal orientation satisfying the Bragg condition θ_inc = θ_ref = θ_Bragg. The student can then easily see that a line drawn through the center of the crystal bisecting θ_scat is parallel to a distinct family of crystal planes, so that the planes responsible for the Bragg peak can be identified unambiguously. This procedure is repeated for each Bragg peak. In addition to seeing the Bragg criterion satisfied directly, students also learn about Miller indices (in two dimensions), and discover that, for example, the (200) reflection is simply the second order reflection from the (100) crystal planes. A plot of sin(θ_Bragg) vs. produces a line with slope λ/2a, yielding a value of a that can be checked by direct measurement with calipers.

Figure 2 shows data for a 7 x 7 square array with lattice constant a = 3.72 cm, and λ = 2.85 cm. The best fit line that is forced to go through the origin yields a = 3.56 ±.09 cm.

The second mode of operation is more challenging but also much more interesting. The exposed crystal is removed from the rotating base and replaced with a camouflaged crystal, i.e., the scattering elements are not visible. The student is asked to locate the Bragg peaks, and from them determine the lattice constant a and the crystal orientation. (We have not yet presented our students with an “unknown” crystal. Nevertheless, we have constructed and tested a working model, and are confident that students will be able to complete this task successfully.)

Other Comments:

Two unique features of the apparatus ensure a successful and satisfying outcome. The first is the collimating lens, which was made from a semi-circular styrofoam trough filled with polyethylene beads (from the original microwave diffraction kit). The lens improves the signal-to-noise ratio by about a factor of two, and also suppresses secondary maxima that would confuse students (and instructors!) and complicate the analysis. The second feature is the photogate, which triggers the oscilloscope once per revolution at the same crystal orientation. This means that, relative to the trigger pulse, the Bragg peaks from different crystal planes occur at different times, and the planes are identifiable by the timing of the Bragg peaks associated with them. For example, imagine that a full rotation of the crystal takes 360 ms. (Depending on the line voltage, the actual rotation period varies from about 330 ms to 400 ms.) Then, because of the 4-fold crystal symmetry, four (100) peaks will appear during each revolution, spaced by 90 ms. The four (110) peaks will appear roughly midway (in time) between the (100) peaks, and so on. This is illustrated in Figure 3. The (120) and (210) peaks are simultaneously maximized, and are spaced by 37 ms, corresponding to the 37° angle between the two sets of planes. The displayed time dependence of the detector voltage adds greatly to the clarity of the analysis.

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Am. J. Phys. 72, 154 (2004)

2. W. H. Murray, Am. J. Phys. 42, 137 (1974)

3. T. D. Rossing, R. Stadum, D. Lang,

Am. J. Phys. 41, 129 (1973)

4. R. A. Allen, Am. J. Phys. 23, 297 (1955)