The Optical Transform Simulating Diffraction Experiments in Introductory Courses George C. Liensky,' Thomas F. K e l l ~Donald ,~ R. N w , ~and Arlhur B. Ellis3 How do we know how atoms arranne themselves in molecules and solids? Information regard;ng bond distances and angles, the basis for the familiar ball-and-stick models of chemical structures, is fundamental to understanding the chemical and physical properties of materials. These relationships, in turn, have fueled many technological advances, involvinn, for example, semiconductor devices, advanced ceramics, synthetic gemstones, superconductors, and the identification of active sites in enzymes. In crystalline solids the atoms are arranged in repeating three-dimensional arrays or lattices that often have strikingly beautiful symmetries. Examples include metallic elements like Cu and Zn, semiconductors like Si and GaAs, insulators like the diamond allotrope of C, salts like NaCI, snpercooductors like YBa2Cu307,a i d molecular solids rang: ing from simple diatomic molecules to complex proteins. On the macroscopic scale, these arrays give rise to crystals havinn characteristic shapes-cubes of rock salt and octahedra ofealcium fluoride, for example. Since atomic dimensions are on the order of angstroms cm), unraveling. the relative atomic positions of a solid requires a physical technique that operates on a similar spatial scale. X-ray diffraction has been the technique that has provided most of our information on the atomic-level structures of solids ( I ) . Electron and neutron diffraction are also important techniques that obey the same physical laws ( I ) . Despite their enormous importance. these diffraction technkues have been difficult to integrate into introductory science and ennineerin~courses. Part of the problem lies in the instrumentation. A$ illustrated in ~ i ~ u1,r the t equipment required for an X-ray diffraction experiment is expensive andhazardous, invol&g high voltages that are used to create physiologically dangerous X-ray radiation; similar concerns apply to electron and neutron diffraction experiments. Furthermore, once a diffraction pattern has been obtained, reconstruction of the atomic positions is a complex, calculation-intensive process for all but the simplest structures. A complementary demonstration of diffraction effects from a three-dimensional array can be observed by diffracting visible laser light from solvent-dispersed polystyrene microspheres that form crystalline-like arrays of appropri- . ate dimensions (2). In this paper we describe the use of optical transforms as a means for simulating diffraction experiments in introductory courses. As shown in Figure 1, shining visible laser light throueh an arrav of dots on a 35-mm slide is a safe. inexnensive method for;llustrating many of the essential features of the X-ray diffraction experiment. In fact, optical transforms were used before computers as a simple method for calculating the diffraction pattern from a known array of atoms;
' Department of Chemistry. Beloit College, Belolt, Wi 53511.
Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706. Department of Chemistry, University of Wisconsln-Madison. Madison. WI 53706.
most frequently, a mask was made by drilling orpunching holes, and diffraction was observed through a microscope (3-5). More recently, optical transforms have been created usinglaaer illumination in conjunction with masks that have been prepared by photographic reduction of either handgenerated patterns or patterns created with a film writing device connected to a minicomputer ( 6 7 ) . We have found that optical transform experiments are readily performed using patterned 35-mm slides that are easily prepared from a paint program on a Macintosh computer and a laser printer. This technique permits rapid, accurate reproduction of a given pattern and allows the pattern to be placed in registry with others on the sameslide, facilitating direct comparisons of laser-generated optical transforms. The diffraction patterns are suitable for classroom demonstrations and laboratory exercises. Procedure Patterns are drawn on a Macintosh computer with a paint program (Hypercard, MacPaint, Superpaint, etc.), either by using the rectangle tool with a fill pattern, or by using zoom or fat bits to draw the patternpixel-hy-pixel. A 10.5 in. X 16.5 in. set of patterns is printed on 11 in. X 17 in. paper usinga laser printer (Fig. 2); this set of patterns is illuminated by two reflector floodlamps, and photographed with a 1-6, f-8 exposure onto Kodak Precision Line LPD4 black-andwhite, 35-mm slide film. The film is developed using Kodak X-Ray Diffraction
Plate
I
Optical Transforms
( Vlvbic Ltght Laser
35mm sllde 4
L
I
6
Projectian Screen
Flgure 1. A comparison of the apparatus used for X-ray diffraction (adapted from Ebbing. D. D. General Chemistry, 3rd ed.. Houghton Mifflin. 1990)end opticel hansformexperiments. The optical transformponion of the figure also illustrates the spacing between diffraction spots X: the mask-todiffra~tion panern distance L: and the angle rp subtended by the consbuctive interference of the diffracted rays (see Figs. 5a and 6 and text). Volume 66
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D l 1 or Dl9 developer for 3 min, Kodak stop bath for 30 s, and Kodak rapid fixer for 5 min. The photographic procedure reduces the original Yn-inch pixels on the paper to (30 mm)/(16.5 in. X 72 in.-') = 0.025-mm pixels on the slide. The smallest pattern (alternating black and white pixels) has a repeat distance of 0.05 mm. For the demonstration, the slide is held perpendicular to a laser beam in a darkened room, and the diffraction pattern is displaved a t a distance (a few meters or greater); the patterns &responding to the arrays in Figure 2 are shown in Figure 3. For best results, the laser should he clamped or taped in a fixed position. A hattery-powered, 5-mW, 670-nm diode laser, sold as a pointer, had sufficient intensity to project the pattern several meters; a 10-mW, 633-nm He-Ne laser had sufficient intensity to be used in a large lecture hall. The spacings of the diffraction spots increase with projection distance. In a typical demonstration, shining the 633-nm He-Ne laser through a grid with 0.05-mm spacing (Fig. 2h) yields a first-order diffraction angle of 0.7'; a projection distance of 10 m gives a 12-cm diffraction spacing on the screen. Because the computerhaser printer provides natural alignment, several masks can he placed on a slide in registry (Fig. 2). By rapidly moving theslide such that thelaser beam alternately passes through two different masks, i t is easy to illustrate relationships between different arrays of dots and their diffraction patterns. An enlarged picture of the masks should he reproduced for the class andlor projected from an overhead transparency for simultaneous viewing. An alternative form of presentation is to make the 35-mm slide available to each class member. If the laser beam is projected on a screen and uiewed through the 35-mm slide from a distance of a few meters, the same diffraction pattern is observed and can be changed a t the viewer's convenience. That is, the same size diffraction pattern appears whether the laser beam passes through the slide and off a projection screen to reach the eye, or whether the laser bounces off the then through the slide toreach the eye. -nroiection - - < ~ ~ screenand ~ This latter viewing technique provides a dramatic illustration that the patterns represent diffraction gratings: if a point source of white light such as a small flashlight bulb is viewed through the slide from a distance, each spot of the diffraction patterns is dispersed into the colors of the visible soectrum. The dependence of the diffraction pattern on excitation wavelength isalso illustrated by illuminating theslide witha second laser of shorter excitation wavelength, which will decrease the diffraction spacings: a side-hy-side comparison using a green, 0.2-mW, 54-l-nm He-Ne laser and the aforementioned diode laser clearly shows a diffrrence in pattern jiae, as predicted hy the diffraction equation (Fig. 4 or cover photo). Using the green laser, a projection distance of 10 m gives a IO-cm diffraction spacing from a 0.05-mm mask.
(the principle of linear superposition). The limiting cases are sketched in Figures 5a and 5h for two waves that are identical in amplitude, wavelength, and frequency. If the waves are in phase, reaching maximum amplitude a t the same time, they reinforce one another, which is a condition known as constructive interference. Converselv, if the waves are completely out of phase (separated by half a wavelength, XI 2). with one a t maximum amplitude while the other is a t minimum amplitude, they "ankhilate" one another or sum t o zero, which is known as destructive interference. Laue recognized a t the beginning of this century that Xrays, a form of high-energy electromagnetic radiation having wavelengths on the order of atomic dimensions, would he scattered by the atoms in a crystalline solid ( I ) . Such an exneriment illustrates both the wave nature of X-ravs and the periodic nature of crystalline solids. Exposing crystalline solids to X-rava " vield . what have come ~o he called diffraction patterns: the atoms in the crystal scatter the incoming radiation, and interference occurs among the many resulting waves. For most directions of observation, destructive inter~~~
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Dlscusslon Both the X-ray diffraction experiment and optical transforms highlight wavelike characteristics of the electromagnetic soectrum. Radio waves. microwaves. infrared, visible and uv light, and X-rays, encompassing a c o n t i r & n of radiation, all travel a t the speed of light c (3 X 101° cmls in a vacuum). Electromagnetic radiation has associated with i t electric and magnetic fields whose time-varying magnitudes describe sine waves that are oriented perpendicularly to each other and to the light beam. Electromagnetic radiation can thus he regarded as a wave having associated with it a wavelength X and frequency v and obeying the relationship c = Xu.
When electromagnetic radiation from several sources overlaps in space simultaneously, the individual waves add 92
Journal of Chemical Education
Figure 2. Sometwodimensional unit cells. (a)centered2 X 2 pixel square: (b) 2 X 2 pixel square: (c) 3 X 3 pixel square with two different spot sizes: (dl 3 X 3 pixel square; (e)2 X 4 pixel rectangle: (f) monoclinic, 2 X 4 pixel with -63" angle; (g) 4 X 5 pixel rectangle with glide: (h) hexagonal array, equivalent to 5.28 X 5.28 pixel wilh 60' angle (actual size is 22 X 22 pixels in a 300 in-' grid). Beside each array is a unit cell (see text). For ease of viewing, these poltiom of the masks are considerably expanded from the actual masks (see text for true sire).
Figure 3a-h. Diflraction patterns Correspondingto the arrays in Figure 2a-h. The patterns were obtained wim a 5-mW. 670-nm diode laser.
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Fraunhofer Diffraction
For constructive interference, dsinq=nh Figure 4. The ditfractlon pattern obtained fromiherectangular array wiih glide symmetry. Figure 29, using a 67Onm red diode laser (ien)and a 544.nrn green He-Ne laser (right).(Color version appears an cover of this issue.)
Figure 5. (a)Constructive interference;(b) desbuctive iwerference.
ference will ocbur, hut in specific directions constructive interference is found. Electron and neutron beams also have a wavelike nature (de Broglie wavelength), leading to similar diffraction effects. The condition for constructive interference, the Bragg diffraction condition, is illustrated in Figure 6 and is often described in terms of reflection from parallel planes of atoms, since the angle of incidence equals the angle of diffraction. As the figure shows, if the atomic planes (represented bv the rows of dots) are separated by a distance d, then Xrays arriving at and scatiered or reflected at an angle % relative to the plane will arrive at a detector in phase, if the additional distance traveled by the lower light ray relative to the upper light ray is an integral number of wavelengths, nX (n = 1.2.3. . .: n = 1 is called firat-order diffraction. n = 2is second-Gder diffraction, etc.). From the trigonomeky indicated in Figure 6, this extra path length is 2(d sin %),leading to Bragg's Law, 2(d sin 0 ) = nX. Diffractionpatterns thus ~rovideevidence for the ~eriodically repeating arrangementof atoms in crystals. ~ h ; i roverallsymmetry corresponds to thesymmetry of the lattice, and the use of a single wavelength (monochromatic light) of Xray radiation directed at the solid permits the simplest determination of interatomic distances. The intensities of diffraction spots calculated for trial atomic positions can be compared-with the experimental diffraction intensities to obtain the positions of the atoms themselves and the arrangement of atoms within crystalline solids composed of molecules. The optical transform, introduced by Sir Lawrence Bragg (3), represents a change in spatial scale by several orders of 94
Journal of Chemlcal Education
Flgure 6. A comparison of
Bragg Diffraction
For constructive interference, 2(d sin e) = n h
Fravnhofer diffractionwiih Bragg diffraction.
magnitude: By using larger spacings, electromagnetic radiaviz.. visible lieht. brines the tion with a l a r ~ e waveleneth. r diffraction experiment intothe optical spect& rhge.kather than Bragg diffraction, however, the lecture demonstration involves Fraunhofer diffraction. As shown in Figure 6, if the liaht transmitted t h r o u ~ han arrav of scatterine centers is viewed at what is effectively infinite distance, tGe condition for constructive interference is d sin @ = nX, where the spacing between atoms d and scattering angle q5 &e defined in the fipxe; the scattered rays in the figure are in phase if the lower ray travels an additional distance (d sin that is an integral number of wavelengths X. Mathematically, the equations for Fraunhofer and Braggdiffraction have a similar functional dependence on d, X, and scattering angle (in Bragg diffraction, the angle between the incident and diffracted beams is 28). The slide prepared containine the vatterns of Fieure 2 can be used to iilusirate several fundamktal fearurelof structure determination bv diffraction methods. First. the Fraunhofer relationship may be verified directly from any of the orthoaonal att terns shown in Fieure 3 (vatterns a-e and a).' As depicted in Figure 1,the value oft& @ and thus of 6 &n be calculated by trigonometry from the spot spacing in the diffraction pattern X and the distance between the pattern and the slide L, i.e., tan @ = XIL. Knowing A, the diffraction equation can then be used to calculate d and to compare it with the pixellphotographicreduction-derived value. Or, assuming that the array spacing d is known, X can be calculated~
Since sin @ = @ (measured in radians) for the small angles to A. This is observed here, % should be rouehlv - - -vro~ortional . evidenced by using two different laser lines: For any of the Figure 2 masks, use of a green He-Ne laser (X = 544 nm) yields avisibly smaller diffraction pattern than is found with
'In me illusValion of Fraunhoferdiftractionin Figure 6, lust one row of twadimensionai arrays in Figure 2 is shown. Those arrays are formed by stacking additional rows above and below the row shown In Figure 6. When the rows of dots are not stacked directly on top of one another, the perpendicular spacing between the rows is smaller than the spacing between the dots In adjacent rows by a factor of sin p. Theangle p is the inclination angle of the parallelogram that serves as the primitive unit cell (see text). For example, if the separation between diffraction spots leads to a calculated perpendicular layer spacing d = 0.11 mm for a hexagonal cell, this spacing corresponds toaspacing between dots In thearray of (0.11 mm)l(sln120") = 0.13 mm. See Glusker, J. P.; Trueblood,K. N. CrystaiSfructure Analysis. A Primer, 2nd ed.; Oxford University Press: New York, 1985; Appendix 2.
a red diode laser (A = 670 nm), for example (Fig. 4 or cover). Both wavelengths yield the same array spacingin the mask, calculated using the procedure described above. When white light is viewed through the mask, the fact that the mask arrays are diffraction gratings is clearly evident: each spot of the diffraction natterns is disoersed into the colors of the visible spectrum. On the other hand. d and sin 6 = 6 should be inversely related for a given wavelength. c comparison of Figures 2b and 2d with 3h and 3d reveals that use of a smaller repeat distanced in the same kind of array will give a larger diffraction oattern, referred to as the reciprocal lattice. These two figures can be placed alternately I n front of the laser to highlight the effect. The reciprocal lattice effect can also be seen in Figures 2e and 3e for which the long direction in the rectangular mask becomes the short direction in the diffraction nattern and vice versa. ~ i symmetry e of the diffraction pattern is the same as the symmetry of the lattice causing the diff~action.~ Square masks such as Figures 2a through 2d exhibit square symmetrv. as seen in Fieurea 3a through 3d (90° rotations and m%ror planes leave the pattern unchanged); the rectangular masks in Figures 2e and 29 have rectanwlar symmetry, as shown in ~ i g u r e 3e s and 3; Figure 2f, d&vedirom nonortboaonal parallelograms, has a diffraction pattern, Figure 3f, that can be rotated by 180" to leave the pattern intact; and the hexagonal mask of Figure 2h has a diffraction pattern and with ~~~-hexaeonal svmmetrv. Fieure 3h (rotation bv 60' mirror do hot affect thLpatternj. The reneatine arravs shown in Fieure 2 can be subdivided into what are termed unit cells, pktllelograms that have identical points from the lattice on each comer. The contents of one unit cell are sufficient to determine the whole array, which can be built up by placing unit cells next to one another so as to fill all of the available space; in three dimensions, parallelepipeds would be used. The choice of the unit cell is not unique: there can be lattice points inside the cell and/or on the corners. The cell is called primitive if there is one lattice point per unit cell. A square or rectangular cell with lattice points only a t each of the 4 corners is primitive, since each corner contributes l/p t o a given unit cell and is equally shared by 4 unit cells. A cell is called centered if there are lattice points both a t the center and a t the corners, in which case there are two lattice points per unit cell. Unit cells are usually chosen to emphasize the symmetry of the array. For example, mirror planes are placed parallel to edges, and rotation axes are placed on corners when possible. Unit cell examples are shown alongside the masks of Figure 2. A comparison of Figures 2a and 2b, which have the same repeat distance, reveals that adding a dot in the middle of each souare (a "centered" lattice) eliminates everv other spot in ihe diffraction pattern, Figures 3a and 3b. T ~ L S effect is easily observed by jumping between the aligned masks. These "systematic absences" occur because the added scatterine centers, which fall a t the middle of each square of dots,cause destructive interference by producing siattered waves that are out of phase by half a wavelength with some of the waves produced by the dots a t the squares' corner^.^ Another way to look a t this effect is as an illustration of the reciprocal lattice effect: the array of Figure 2a can be described as a simple square (rotated by 45O) with a smaller repeat distance between points, leading to a larger spacing between spots in the diffraction pattern (also rotated by ~
459.
Systematic absences can also he seen for Figure 2g, which contains a form of translational symmetry called a glide plane. Note that an indistinguishable array results if the arrav is reflected across an imaeinarv .. . mirror (elide d a n e ) placed parallel to and halfway between ~olumns,~follo~ed b; raising or lowering the columns (''translating" them) by half
of the repeat distance along a direction parallel to the glide plane (a familiar examole of this svmmetrv is footorints left in the sand by someone walking on the beach). ~ G i characs teristic, too, gives rise t o systematic absences in the diffraction pattern: the central column of Figure 3g has every other dot missing as a consequence of destructive interference resulting frGm the glide plane.' Figures 2c and 3c illustrate the effect of using interpenetrating arrays with two different spot sizes. A comparison with Figures 2d and 3d, which are derived from a single array of the same dimensions using a single spot size, reveals that a similar diffraction pattern results but with differing intensities of the diffraction soots. This romoarison illusrrates the fact that different dotsiatoms) can hsve different scattering Dower. Figures 2f and 2h are two common symmetries devoid of right angles, which lead to diffraction patterns that lack perpendicularity, Figures 3f and 3h, respectively. These two patterns can be used to illustrate orientation effects in the reciprocal lattice. In the general two-dimensional case, rows of diffraction spots are seen in a direction perpendicular to and inversely spaced relative to the original rows. This makes nonorthogonal diffraction patterns appear to be rotated from the orientation of the original arr& of dots.8 The examples presented here serve to illustrate how the snacines. svmmetrv. soot intensities. and svstematic ah" kncecof a d i f f r a c t 6 pattern are related to the lattice from which i t is derived. Althoueh these are two-dimensional lattices, they mimic what would he observed for diffraction from particular three-dimensional structures t h a t are viewed in projection perpendicular to a face that is a parallelogram. For example, Figures 2b and 2d are the arravs of atoms in a simple c"bic structure, a structure having atoms only a t the cube corners; the centered array of Figure 2a ~~~
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A rigorous treatment will show that all diffractlon panems will have a center of symmetry, even if the structure itself does not have one (compare Figs. 2g and 3g, for example)( lb). 'The intensity I of the diffractedray is a wave of the form. I = f exp(ia)= Iws a I sin a), where a = (2shx+ 2aky) is the phase difference from the origin for a scatterer with amplitude f located at fractional coordinates (x,y), and integers hand kare the orders of the diffractlonpattern. Now consider a centered cell, where the point (x. y) is equivalent to(x+ 112. y+ 112).Scatteringfrom both will give I = fexp[2si(hx+ ky)] fexp[Zu[hx+ ky hl2 kl2) = (cos (2shx 2sky) isin (2rhx+ 2sky) cos (2rhx 2sky s h uk) I sin (2shx+ 2aky r h + sk)). By expanding the last two terms using therelationships cos(A B) = [cos A cos B- sin Asin B], and sin (A B) = [sin A cos B cos A sin B]. I = (cos(2rhx 2sky) isin (2shx 2 r W cos (2shx 2aky) cos (sh sk) - sin (2rhx 2rky) sin (nh TM i sin (27rhx 2sky) cos (sh s k ) i w s (2shx + 2aky) sin ( s h sk)l. When (h kj is even, then cos (nh s k ) = 1, sin (sh + a4 = 0, and I = 2 f exp ia. When (h k) is odd, then cos ish r M = -1. sin lsh ?rM = 0. and I = 0 . Thus. for a centered &I, eve& athsr'diftr&tion spit in the plane wiii have zero inrensity. The derivalim is readily extended for body centering in three dimensions wlth (x,y, r)equivalent to ( x 112, y 112, r + I/ 2). Face centering in three dimensions will also lead to systematic absences. A similar derivation tothat outlined in footnote 6 holds for aglide, where the point (x, y) is equivalent to (-x, y 112). yielding I = 0 when h = 0 and k is odd. Thus, every other spot with zero lntenslty along one axis of the diffraction pattern Is indicative of glide symmetry. In three dimensions, a screw axis can also lead to systematic absences. This symmetry operation consists of a rotation about an axis, followed by a translation of the repeat distance in a direction parallel to the axis: an example is a spiral staircase. See Gfusker,J. P.; Trueblood, K. N. CrystalStructure Analysis. A Primer; 2nd ed.; Oxford University Press: New York, 1985;Chapter 3. for a more complete discussion.
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Volume 68 Number 2 February 1991
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mimics the projection of a hody-centered cubic structure, which has atoms a t the cube corners and in the center of the cube; the array of Figure 2c simulates the projection of the cuhic ZnS (zinc blende) structure, having two different kinds of atoms. each surrounded hv four atoms of the other type in a tetradedral arrangement (this is also the array that would be seen for a single layer of the NaCl rock salt structure); the rectangular structures of Figures 2e and 2g are derived from orthorhomhic structures, wherein all angles are 90° but the sides are of unequal length; the array of Figure 2f mimics a monoclinic structure, where the non-90° lattice angle is in the plane of the paper; and Figure 2h is a closest oacking structure, the basis for a layer of the hexagonal c ~ o s e s t - ~ a c k ior n ~cuhic closest packing structures. The methodology described herein is readily extended to show other basiilattice structures having other kinds of symme~
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I t is imoortant to recoenize that both the intensitv and the phase of 'the diffracted Learns are required to calklate the atomic positions from experimental data, but only the intensity can usually he measured. This is known as the "phase problem."The intensities of the diffraction peaks depend on the contents of the unit cell, and the diffraction pattern can he calculated for any set of atoms with known coordinates in a periodic array. Solving a crystal structure involves matching the observed intensities with those predicted for a trial st&cture. In modern crystallography, the phases of the calculated structure are assigned to the observed amplitudes and a new structure is calculated. This computation-intensive process is repeated and least-squares refinement is used to minimize the differences. From the standpoint of the optical transform, then, it is necessary to work backward from the diffraction pattern to determine the arrangement of scattering centers. Making a slide of the diffraction pattern and looking at its optical transform (the inverse transform) can in principle show the originalarrangement of scatteren, an effect intimated in the
comparison of Figures 2b and 2d, where the large m a y gives a small diffraction pattern of the same symmetry and vice versa. Optical transforms have been used to predict diffraction oatterns for a broad varietv of solid-state structures? The ihility to create these masks with common microcomputer t the ~otentialfor makine and ohotoeraohic e a u i ~ m e noffers solid-statest;ucturess in exciting and ilItegrat part of intr; ductory science courses. Safety
The low power and long wavelengths of the lasers employed in this experiment make them relatively safe to use. However, care should he taken when using any laser not to point it directly in anyone's eye. If the central spot of the diffraction pattern is perceived as being too bright, a filter should be placed in front of the beam to diminish the intensity. Optlcal Transforms at Cost: "Crystallography on a Sllde"
An Optical Transform Kit (Order No. 90-002L) is available from the Institute for Chemical Education for $109. The kit consists oE adiode laser pointer (
