the optical transform:

7/29/2019 The Optical Transform:

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The Optical TransformSimulating Diffraction Experiments in Introductory CoursesGeorge C. Liensky,' Thomas F. K e l l ~ , ~onald R. N w , ~nd Arlhur B. Ellis3

How do we know how atoms arranne themselves in mole-cules and solids? Information regard;ng bond distances andangles, the basis for the familiar ball-and-stick models ofchemical structures, is fundamental to understanding thechemical and physical properties of materials. These rela-tionships, in turn, have fueled many technological advances,involvinn, for example, semiconductor devices, advanced ce-ramics, synthetic gemstones, superconductors, and the iden-tification of active si tes in enzymes.In crystalline solids the atoms are arranged in repeatingthree-dimensional arrays or lattices tha t often have striking-ly beautiful symmetries. Examples include metallic ele-ments like Cu and Zn, semiconductors like Si and GaAs,insulators like th e diamond allotrope of C, salts like NaCI,snpercooductors like YBa2Cu307,a i d molecular solids rang:ing from simple diatomic molecules to complex proteins. Onthe macroscopic scale, these a rrays give rise to crystals hav-inn charac teristic shapes-cubes of rock sal t and octahedraofealcium fluoride, for example. Since atomic dimensionsare on the order of angstroms cm), unraveling. therelative atomic positions of a solid requires a physical tech-nique that operates on a similar spatial scale. X-ray diffrac-tion has been the technique tha t has provided most of ourinformation on the atomic-level structures of solids ( I ) .Electron and neutron diffraction are also important tech-niques that obey the same physical laws ( I ) .Despite their enormous importance. these diffractiontechnkues have been difficult to integrate into introductoryscience and ennineeri n~ ourses. Par t of the problem lies inthe instrumentation. A$ llustrated in ~ i ~ u r t, the equip-ment required for an X-ray diffraction experiment is expen-sive andhazardous, invol&g high voltages tha t are used tocreate physiologically dangerous X-ray radiation; similarconcerns apply t o electron and neutron diffraction experi-ments. Furthermore, once a diffraction pattern has beenobtained, reconstruction of the atomic positions is a com-plex, calculation-intensive process for all but the simpleststructures.A complementary demonstration of diffraction effectsfrom a three-dimensional array can be observed by diffract-ing visible laser light from solvent-dispersed polystyrenemicrospheres th at form crystalline-like arrays of appropri-- .at e dimensions (2).In th is paper we describe the use of optical transforms as ameans for simulating diffraction experiments in introduc-tory courses. As shown in Figure 1, shining visible laser lightthroueh an arrav of dot s on a 35-mm slide is a safe. inexnen-sive method for;llustrating many of the essential features ofthe X-ray diffraction experiment. In fact, optical transformswere used before computers as a simple method for calculat-ing the diffraction pattern from a known array of atoms;

'Department of Chemistry. Beloit College, Belolt,Wi 53511.Department of Materials Science and Engineering, University ofWisconsin-Madison, Madison,WI 53706.Department of Chemistry, University of Wisconsln-Madison.Madison.WI 53706.

most frequently, a mask was made by drilling orpunchingholes, and diffraction was observed through a microscope(3-5).More recently, optical transforms have been createdusinglaaer illumination in conjunction with masks that havebeen prepared by photographic reduction of either hand-generated patterns or patterns created with a film writingdevice connected to a minicomputer (67) .We have found that optical transform experiments arereadily performed using patterned 35-mm slides that areeasily prepared from a paint program on a Macintosh com-puter and a laser printer. This technique permits rapid,accurate reproduction of a given pat tern and allows thepattern to be placed in registry with others on the sameslide,facilitating direct comparisons of laser-generated opticaltransforms. T he diffraction pattern s a re suitable for class-room demonstrations and laboratory exercises.Procedure

Patt erns are drawn on a Macintosh computer with a paintprogram (Hypercard, MacPaint, Supe rpai nt, etc.), eitherby using the rectangle tool with a fill pattern, or by usingzoom 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. paperusinga laser printer (Fig. 2); this set of patterns is illuminat-ed by two reflector floodlamps, and photographed with a 1-6,f-8 exposure onto Kodak Precision Line LPD4 black-and-white, 35-mm slide film. The film is developed using Kodak

X-Ray Diffraction

Plate

Optical Transforms IVlvbic Ltght Laser 35mm sllde

4 L I 6ProjectianScreen

Flgure 1. A comparison of the apparatus used for X-ray diffraction (adaptedfrom Ebbing. D. D. General Chemistry, 3rd ed.. Houghton Miff l in. 1990)endopticel hansformexperiments.The optical transform ponion ofthe figure alsoillustrates the spacing between diffraction spots X: the mask-todiffra~tionpanern distanceL: and the angle rp subtended by the consbuctive interferenceof th e diffracted rays (see Figs. 5a and 6 and text).

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D l 1 or Dl9 developer for 3 min, Kodak sto p bath for 30 s,and Kodak rapid fixer for 5 min. The photographic proce-dure reduces t he original Yn-inch pixels on th e paper t o (30mm)/(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 th e demonstration, the slide is held perpendicular to alaser beam in a darkened room, and t he diffraction pattern isdisplaved a t a distance (a few meters or greater); the pat-tern s &responding to the arrays in Figure 2 are shown inFigure 3. For best results, the laser should he clamped ortaped in a fixed position. A hattery-powered, 5-mW, 670-nmdiode laser, sold as a pointer, had sufficient intensity toproject the pat tern several meters; a 10-mW, 633-nm He-Nelaser had sufficient intensity to be used in a large lecturehall. The spacings of the diffraction spots increase withprojection distance. In a typical demonstration, shining the633-nm He-Ne laser through a grid with 0.05-mm spacing(Fig. 2h) yields a first-order diffraction angle of 0.7'; a pro-jection distance of 10 m gives a 12-cm diffraction spacing onth e screen.Because the computerhaser printer provides naturalalignment, several masks can he placed on a slide in registry(Fig. 2). By rapidly moving theslide such tha t thelaser beamalternately passes through two different masks, i t is easy toillustrate relationships between different arrays of dots an dtheir diffraction patterns. An enlarged picture of the masksshould he reproduced for the class andlor projected from anoverhead transparency for simultaneous viewing.An alternative form of presentation is to make the 35-mmslide available to each class member. If the laser beam isprojected on a screen and uiewed through the 35-mm slidefrom a distance of a few meters, th e same diffraction patternis observed and can be changed a t the viewer's convenience.Th at is, the same size diffraction pa ttern appears whetherthe laser beam passes through the slide and off a projectionscreen to reach the eye, or whether the laser bounces off thenroiection screenand then through the slide tore ach the eye.- - - < ~ ~ ~Thi s latter viewing technique provides a dramatic illustra-tion that the patterns represent diffraction gratings: if apoin t source of white light such as a small flashlight bulb isviewed through the slide from a distance, each spot of thediffraction pa tte rns is dispersed into th e colors of the visiblesoectrum.Th e dependence of the diffraction pattern on excitationwavelength isalso illustrated by illuminating thes lide wi thasecond laser of shorter excitation wavelength, which willdecrease the di ffraction spacings: a side-hy-side comparisonusing a green, 0.2-mW, 54-l-nm He-Ne laser and the afore-mentioned diode laser clearly shows a diffrrence in patternjiae, as predicted hy the diffraction equation (Fig.4 or coverphoto). Using the green laser, a projection distance of 10 mgives a IO-cm diffract ion spacing from a 0.05-mm mask.

DlscusslonBoth t he X-ray diffraction experiment and optical trans-forms highlight wavelike characteristics of the electromag-netic soectrum. Radio waves. microwaves. infrared, visibleand uv light, and X-rays, encompassing a contir&n ofradiation, all travel at the speed of light c (3 X 101 cmls in avacuum). Electromagnetic radiation has associated with itelectric and magnetic fields whose time-varying magnitudesdescribe sine waves that are oriented perpendicularly toeach other and to th e light beam. Electromagnetic radiationcan thus he regarded as a wave having associated with it awavelength X and frequency v and obeying the relationship c

= Xu .When electromagnetic radiation from several sourcesoverlaps in space simultaneously, the individual waves add

(the principle of linear superposition). T he limiting cases aresketched in Figures 5a and 5h for two waves tha t are identi-cal in amplitude, wavelength, and frequency. If the wavesare in phase, reaching maximum amplitude a t the sametime, they reinforce one another, which is a condition knownas constructive interference. Converselv, if the waves arecompletely out of phase (separated by half a wavelength, XI2). with one a t maximum amplitude while the other is atminimum amplitude, they "ankhila te" one another or sumt o zero, which is known as destructive interference.Laue recognized a t the beginning of this century tha t X-rays, a form of high-energy electromagnetic radiation havingwavelengths on the order of atomic dimensions, would hescattered by the atoms in a crystalline solid ( I ) . Such anexneriment illustrates both the wave nature of X-ravs and~~.~the periodic nature of crystalline solids. Exposing crystallinesolids to X-rava vield what have come ~o he called diffraction.patterns: th e atoms in t he crystal scatter the incoming radia-tion, and interference occurs among the many resultingwaves. For most directions of observation, destructive inter-

Figure 2. Sometwod imensional unit cells. (a)centered2 X 2 p ixel square: (b2 X 2 pixel square: (c )3 X 3 pixel square with two different spot sizes: (dl 3 X3 pixel square;(e) 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 to5.28 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, thesepoltiom of the masks are considerably expanded from theactual masks (seetext for true sire).

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Figure 3a-h. Diflraction patterns Corresponding o the arrays in Figure 2a-h. The patterns were obtained wim a 5-mW. 670-nm diode laser.

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Fraunhofer Diffraction Bragg Diffraction

Figure4. Th e ditfractlon pattern obtained fromihe rectangular arraywiih glidesymmetry. Figure 29, using a67Onm red diode laser (ien) nd a 544.nrn greenHe-Ne laser (right).Colorversion appearsancover of this issue.)

Figure5. (a)Constructive interference; b) desbuctive iwerference.

ference will ocbur, hut in specific directions constructiveinterference is found. Electron and neutron beams also havea wavelike nature (de Broglie wavelength), leadingto similardiffraction effects.The condition for constructive interference, the Braggdiffraction condition, is illustrated in Figure 6 and is oftendescribed in terms of reflection from parallel planes of at-oms, since the angle of incidence equals the angle of diffrac-tion. As the figure shows, if the atomic planes (representedbv the rows of dots) are separated by a distance d, then X-rays arriving at and scatiered or reflected at an angle %relative to the plane will arrive at a detector in phase, if theadditional distance traveled by the lower light ray relative tothe upper light ray is an integral number of wavelengths, nX(n = 1.2.3. . :n = 1 is called firat-order diffraction. n = 2i ssecond-Gder diffraction, etc.). From the trigonomeky indi-cated in Figure 6, this extra path length is 2(d sin %),eadingto Bragg's Law, 2(d sin 0 ) = nX.Diffractionpatterns thus ~rov idevidence for the ~e ri od i-cally repeating arrangementof atoms in crystals. ~ h ; i rover-allsymmetry corresponds to thesymmetry of the lattice, andthe use of a single wavelength (monochromatic light) of X-ray radiation directed at the solid permits the simplest de-termination of interatomic distances. The intensities of dif-fraction spots calculated for trial atomic positions can becompared-with the experimental diffraction intensities toobtain the positions of the atoms themselves and the ar-rangement of atoms within crystalline solids composed ofmolecules.The optical transform, introduced by Sir Lawrence Bragg(3),represents a change in spatial scale by several orders of

For constructive For constructiveinterference, interference,d s i n q = n h 2(d sin e) = n h

Flgure 6. A comparison o Fravnhofer diffractionwiih Bragg diffraction.

magnitude: By using larger spacings, electromagnetic radia-tion with a la r~er aveleneth. viz.. visible lieht. brines thediffraction experiment intothe optical spect& rhge .kath -er than Bragg diffraction, however, the lecture demonstra-tion involves Fraunhofer diffraction. As shown in Figure 6, ifthe liaht transmitted th ro u~ hn arrav of scatterine centersis viewed a t what is effectively infinite distance, tGe condi-tion for constructive interference is d sin @ = nX, where thespacing between atoms d and scattering angle q5 &e definedin the fipxe; the scattered rays in the figure are in phase ifthe lower ray travels an additional distance (d sin that isan integral number of wavelengths X. Mathematically, theequations for Fraunhofer and Braggdiffraction have a simi-lar functional dependence on d, X, and scattering angle (inBragg diffraction, the angle between the incident and dif-fracted beams is 28).The slide prepared containine the vatterns of Fieure 2 canbe used to iilusirate several fundamktal fearurelof struc-ture determination bv diffraction methods. First. the Fraun-hofer relationship may be verified directly from any of theorthoaonal att terns shown in Fieure 3 (vatterns a-e and a).'As depicted in Figure 1,the value of t& @ and thus of 6&nbe calculated by trigonometry from the spot spacing in thediffraction pattern X and the distance between the patternand the slideL, i.e., tan @ =XIL. Knowing A, the diffractionequation can then be used to calculate d and to compare itwith the pixellphotographic reduction-derived value. Or, as-suming that the array spacing d is known, X can be calculat-e d ~

Since sin @ = @ (measured in radians) for the small anglesobserved here, % should be rouehlv vro~ort ionalo A. This is- - .evidenced by using two different laser lines: For any of theFigure 2 masks, use of a green He-Ne laser (X = 544 nm)yields avisibly smaller diffraction pattern than is found with

' nme illusValion of Fraunhoferdiftraction n Figure6, lust one rowof twadimensionai arrays in Figure 2 is shown. Those arrays areformed by stacking additional rows above and below the row shown InFigure 6. When the rows of dots are not stacked directly on top of oneanother, the perpendicular spacing between the rows is smaller thanthe spacing between the dots In adjacent rows by a factor of sin p.Theanglep is the inclination angle of the parallelogram that serves asthe primitive unit cell (see text). For example, if the separationbetween diffraction spots leads to a calculated perpendicular layerspacing d = 0.11 mm for a hexagonal cell, this spacing correspondstoaspacing between dots In thearray of (0.1 mm)l(sln 120")= 0.13mm. See Glusker,J. P.; Trueblood,K. N.CrystaiSfructure Analysis.APrimer, 2n d ed.; Oxford University Press: New York, 1985;Appendix2.94 Journal of Chemlcal Education


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a red diod e laser (A = 670 nm), for ex ample (Fig. 4 or cover).Both wavelengths yield the same array spacing in t he mask,calculated using th e procedure described above. When whitelight is viewed through th e mask, the fact tha t the m askarrays are diffraction gratings is clearly evident: each s pot ofthe diffraction n atte rns is disoersed in to th e colors of th evisible spectrum .On the other hand. d and sin 6 = 6 should be inverselyrelated for a given wavelength. c comparison of Figures 2band 2d with 3h and 3d reveals that use of a smaller repeatdista nced in th e same kind of array will give a larger diffrac-tion o attern , referred to as the reciprocal lattice. Thes e twofigures can be placed alternately I n fro nt of the laser tohighlight the effect. Th e reciprocal lattice effect can also beseen in Figures 2e and 3e for which th e long direction in t herectangular mask becomes the sh ort direction in the diffrac-tion na ttern and vice versa.~ i eymmetry of the diffraction pa ttern is the same as th esymmetry of t he la t tice caus ing the d i f f~ ac t i on .~quaremasks such as Figures 2a through 2d exhibit square symme-trv. as seen in Fieurea 3a through 3d (90 rotations andm%ror planes leave th e pat tern unchanged); th e rectangularmasks in Figures 2e and 29 have rectan wlar symmetry, asshown in ~ ig u r e s e and 3 Figure 2f, d&vedirom nonor -tboaonal parallelograms, has a diffraction pa ttern, Figure 3f,tha t can be ro ta ted by 180" o leave the pa ttern intact; andthe hexagonal mask of Figure 2h has a diffraction patter nwith hexaeonal svmmetrv. Fieure 3h (rotation bv 60' and~~- ~mirror do ho t affect thLpatternj.Th e reneatine arravs shown in Fieure 2 can be subdividedinto what are termed unit cells, pktlle lograms that haveidentical points from th e latt ice on each comer. T he con-tents of one unit cell are suff icient to determine t he wholearray, which can be bu ilt up by placing u nit cells next t o oneanother s o as to fill all of the available space; in three dimen -sions, parallelepipeds would be used. Th e choice of th e uni tcell is not unique: the re can be lattice po ints inside the celland/or on th e corners. Th e cell is called primitive if there isone lattice point per un it cell. A square or rectangular cellwith lattice points only a t each of th e 4 corners is primitive,since each corner con tributes l/p to a given unit cell and isequally shared by 4 unit cells. A cell is called centered ifthere are la tt ice points both at the center and a t he corners,in which case there are two lattice points p er un it cell. Unitcells are usually chosen t o emphasize th e sym metry of thearray. For example, mirror planes are placed parallel toedges, and rotatio n axes are placed on corners when possible.Unit cell examples are shown alongside the masks of Figure2. A comparison of Figures 2a and 2b, which have the sam erepeat distance, reveals that adding a do t in the middle ofeach souare (a "centered" lattice) eliminates everv otherspo t in ih e diffraction patte rn, Figures 3a and 3b. T ~ L S effectis easily observed by jumping between the aligned masks.The se "systematic absences" occur because th e added scat-terine centers, which fall a t the m iddle of each square ofdots , cause destructive interference by producing sia tter edwaves tha t are ou t of phase by half a w avelength with someof the waves produced by th e dots a t the squares' corner .Another way to look a t this effect is as an i l lustration of thereciprocal lattice effect: the array of Figure 2a can be de-scribed as a simple square (rotated by 45O) with a smallerrepeat distance between points, leading to a larger spacingbetween spots in the diffraction pattern (also rotated by4 5 9 .System atic absences can also he seen for Figure 2g, whichcontains a form of translational symmetry called a glideplane. Note that an indistinguishable array results if thearrav is reflected across an imaeinarv mirror (elide dane). .placed parallel to and halfway between ~o lu m ns ,~ fo ll o~ ed;raising or lowering the colum ns (''translating" them ) by half

of the repeat distance along a direction parallel to th e glideplane (a familiar examole of this svm metrv is footo rints leftin the sand by someone walking on the beach). ~ G i sharac-teristic, too, gives rise to system atic absences in the diffrac-tion pattern: the cen tral column of Figure 3g has every otherdot missing as a consequence of destructive interferenceresulting frGm the glide plane.'Figures 2c and 3c illustrate the effect of using interpene-trating a rrays with two differe nt spo t sizes. A comparisonwith Figures 2d an d 3d , which are derived from a single arrayof the sam e dimensions using a single spot size, reveals that asimilar diffraction pattern results but with differing intensi-ties of the diffraction soots. Th is romoarison illusrrates thefact that different dotsia tom s) can hsve different scatter ingDower.Figures 2f and 2h are two comm on symmetries devoid ofright angles, which lead to diffraction patterns that lackperpend icularity, F igures 3f an d 3h, respectively. Th ese twopatte rns can be used to illustrate orientation effects in thereciprocal lattice. In th e general two-dimensional case, rowsof diffraction spot s are seen in a direction perpendicular toand inversely spaced relative to the original rows. Thismakes nonorthogonal diffraction patt erns appear to be ro-tated from the orientation of the original arr& of dots.8Th e examples presented here serve to i l lustrate how thesnacines. svmmetrv. soot intensities. and svstematic ah-~~~ ~~~ ~ ~knc ecof a d i f f r ac t6 pa t te rn a re rela ted to the la t tice f romwhich it is derived. Althoueh these a re two-dimensional lat-tices, they mimic what would he observed for diffractionf rom par t icula r three -dimensiona l s t ruc t ures th a t a reviewed in projection perpen dicular to a face that is a paral-lelogram. For example, Figures 2b and 2d are the arravs ofatoms in a simple c"bic struc ture, a structu re having atomsonly at th e cube corners; the centered array of Figure 2a

A rigorous treatment will show that all diffractlon panems willhave a center of symmetry, even if the struc ture itself does not haveone (compare Figs. 2g and 3g, fo r example) lb).'The intensity I of the diffracted ay is a wave of the form. I = fexp(ia)= Iws a + si n a), where a = (2sh x+ 2aky) is the phasedifference from the origin for a scatterer with amplitude f located atfractional coordinates (x,y), and integers hand ka re the orders of thediffractlonpattern. Now consider a centered cell , where the point (x.y) is equivalent to(x + 112. y + 112).Scattering from both will give I =fexp[2si(hx + ky)]+ fexp[Zu[hx+ ky + hl2 + kl2)= (cos (2shx+ 2sky)+ isin (2rh x+ 2sky)+ cos (2rhx + 2sky + s h + uk )+ Isin (2shx+ 2aky + r h + sk)).By expanding the last two terms usingtherelationships cos( A+ B) = [cosA cos B- si n A s i n B],and sin (A+ B) = [sin A cos B+ cos A si n B]. I = (cos(2rhx+ 2sky)+ isin(2shx + 2 r W + cos (2shx + 2aky) cos (s h + sk)- in (2rhx+2rky) sin (n h+ TM + i sin (27rhx + 2sky) cos (s h + s k ) + i w s(2shx+ 2aky) si n ( s h+ sk )l. When (h+ kj is even, then cos (nh+sk ) = 1, sin ( s h+ a4 = 0, and I = 2 f exp ia. When (h+ k) is odd,then cos is h + r M = -1. sin l sh + ?rM= 0. and I = 0. Thus. for a~ ~centered &I , eve& athsr'diftr&tion spi t in the plane wiii have zeroinrensity. The derivalim is readily extended for body centering inthree dim ensions wlth (x, y, r) quivalent to ( x + 112,y + 112,r + I/2). Face centering in three dimensions will also lead to systematicabsences.

A similar derivation to tha t outlined in footnote 6 holds for aglide,where the point (x, y) is equivalent to (-x, y + 112). yielding I = 0when h = 0 and k is odd. Thus, every other spot with zero lntensltyalong one axis of the diffraction pattern Is indicative of glide symme-try. In three dimensions, a screw axis can also lead to systematicabsences. This symmetry operation consists of a rotation about anaxis, followed by a translation of the repeat d istance in a directionparallel to the axis: an example is a spiral staircase.See Gfusker,J. P.; Trueblood,K. N. CrystalStructureAnalysis. APrimer; 2nd ed.; Oxford University Press: New York, 1985;Chapter 3.for a more complete discussion.Volume 68 Number 2 February 1991 95


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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 thecube; the array of Figure 2c simulates the projection of thecuhic ZnS (zinc blende) structure, having two differentkinds of atoms. each surrounded hv four atoms of the other~ ~ ~type in a tetradedral arrangement (this is also the array thatwould be seen for a single layer of the NaCl rock salt struc-ture); the rectangular structures of Figures 2e and 2g arederived from orthorhomhic structures, whereinallangles are90 but the sides are of unequal length; the array of Figure 2fmimics a monoclinic structure, where the non-90 latticeangle is in the plane of the paper; and Figure 2h is a closestoacking structure, the basis for a layer of the hexagonalc~oses t -~ackin~r cuhic closest packing st ructures . Themethodology described herein is readily extended to showother basiilattice structures having other kinds of symme-It is imoortant to recoenize that both the intensitv and thephase of 'the diffracted Learns are required to ca lk la te theatomic positions from experimental data, but only the inten-sity can usually he measured. This is known as the "phaseproblem."The intensities of the diffraction peaks depend onthe contents of the unit cell, and the diffraction pattern canhe calculated for any set of atoms with known coordinates ina periodic array. Solving a crystal structure involves match-ing the observed intensities with those predicted for a trial

st&cture. In modern crystallography, the phases of the cal-culated structure are assigned to the observed amplitudesand a new structure is calculated. Thi s computation-inten-sive process is repeated and least-squares refinement is usedto minimize the differences.From the standpoint of the optical transform, then, i t isnecessary to work backward from the diffraction patt ern t odetermine the arrangement of scattering centers. Making aslide of the diffraction pattern and looking at its opticaltransform (the inverse transform) can in principle show theoriginalarrangement of scat teren, an effect intimated in the

Masks can be made illustrating all 17 two-dimensional planegroups. which are discussed in ref ib . The additional plane groupsyield diffraction patterns that differn individual spot intensitles,butnot in the underlying symmetry of the spot positions, from those inFigure 3. The laser printer also makes it possible to examine theeffects of ooint defects like vacancies and interstitials and of "twin-~~~ ~"in;", wherein a pattern is present in two or more Merent relativeorientations. We sho~ld olnt out mat the matrix nature of the pixelsprecludes creating t r ~ random patterns by random placement ofplxel dots. However, random arrangements of much larger dots canbe prepared from laser printers. These figures can be reduced insuccessive stepsby a photocopy machine and then photographicallyreduced to mimic the arrangements of atoms and moleculesin gasesand liquids. We thank J. C. Weisshaar. J. E. Harriman,J. L. Skinner,and H. Sevian for demonstrating these effects.

comparison of Figures 2b and 2d, where the large m a y givea small diffraction pattern of the same symmetry and vicversa.Optical transforms have been used to predict diffractiooatterns for a broad varietv of solid-state structures? Thihi lity to create these masks with common microcomputeand ohotoeraohic ea ui ~m en tffers the ~ote nt ialor makinsolid-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 useHowever, care should he taken when using any laser not topoint it directly in anyone's eye. If the central spot of thdiffraction pat tern is perceived as being too bright, a filteshould 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 $109The kit consists oE ad iode laser pointer (

the optical transform:

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