JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Optical simulation of electron diffraction of thin crystals

Ronald BergstenDepartment of Physics, University of Wisconsin -Whitewater, Whitewater, Wisconsin 53190

(Received 22 April 1974)

An optical crystal (a multilayer diffraction plate) which optically simulates electron diffraction by thincrystals has been developed. The lattice was constructed with appropriate lattice constants to allow a directoptical simulation of electron diffraction by mica. The resulting optical-diffraction patterns agree withtheory and vividly illustrate Laue zones. Optical-diffraction patterns for different crystal orientations aredirectly compared to electron-diffraction patterns for the same orientations of a thin mica crystal. Thebreadths of Laue zones diminish as a crystal becomes thicker, reducing the diffraction pattern to the Lauespots described by Bragg's law.

Index Headings: Diffraction; Crystals.

The most-common optical simulation of x-ray orelectron diffraction is accomplished by use of a multipleslit or diffraction grating to represent a one-dimensionalcrystal. Unfortunately, the form factor of these opticalelements does not lend itself to a good optical simu-lation of a one-dimensional crystal.' Crossed gratingshave long been used to demonstrate diffraction patternsproduced by two-dimensional crystals (Kikuchi's Npattern).2 Other two-dimensional gratings have beenconstructed by arranging objects in patterns3' 4 or byproducing arrays of symbols (elements) by photographicreduction.1' 5 6 The use of photographic materials pro-vides more versatility in the design of an optical crystal(a diffraction plate used to simulate optically diffractionby crystals), allowing more control over both the formfactor and structure factor. Two-layer optical crystalshave been constructed by stacking two-dimensionalcrystals.'

A direct one-to-one optical simulation is not easilyaccomplished. X rays used for diffraction have wave-lengths of the same order as the crystal lattice constants.Thus, optical simulation would require construction ofcrystals having lattice constants only slightly largerthan the wavelength of light. However, the De Brogliewavelength of electrons from a typical electron micro-scope is only about one-thousandth of a lattice constant,making optical simulation of electron diffractionfeasible.

I. DESIGN OF A CRYSTAL

Two-dimensional arrays of elements produced onhigh-resolution film by photographic reduction of artwork are stacked to form a three-dimensional opticalcrystal. The layers are carefully aligned under a micro-scope with the aid of a cross slide. The films are gluedtogether with a transparent epoxy cement resultingin a solid optical crystal having a refractive index ofabout 1.5. To produce a good simulation of electrondiffraction, the ratio of the wavelength of light in theoptical crystal to a lattice constant of the opticalcrystal should be equal to the ratio of the De Broglie

wavelength of the electrons to the corresponding crystal-lattice constant.

Two-dimensional optical crystals have been con-structed by use of small apertures as elements.5 Suchelements are not suited for building three-dimensionalcrystals because most of the light would be absorbedby the first layer. However, according to Babinet'sprinciple, small opaque disks have a similar far-fieldform factor but only a small fraction of the light isabsorbed by each layer. The disks must be extremelysmall, so as to prevent shadowing7 and provide a largeBabinet Airy's disk form-factor pattern, within whichthe desired structure-factor pattern is produced. Anappropriate disk diameter is about 7X. A typical latticespacing for optical simulation is about 4X 10-2 mm.

With these dimensions, less than 1% of the light inci-dent on a layer will strike the disks of that layer,allowing a large percentage of the light to continue tosuccessive layers. An optical crystal having latticeconstants 6.67X 103 times larger than those of a crystalwould permit optical simulation of electrons rangingin energy from 25 to 100 KeV (a typical electron-microscope range).

Small-angle approximation of Snell's law indicatesthat, for normal incidence, the diffraction patternproduced by light incident on an optical crystal willbe larger than that produced by electrons incident ona crystal by a factor of the refractive index of the opticalcrystal. This magnification and additional distortionsdue to oblique incidence can be eliminated by sub-merging both the optical crystal and the screen onwhich the diffraction pattern is formed in a liquidhaving the same refractive index as the optical crystal.However, this procedure was not used in making thediffraction patterns pictured in this paper.

The particular optical crystal described herein wasdesigned to simulate optically electron diffraction bymuscovite mica. The Bravais lattice of the opticalcrystal is a mock-up of that mica. The scale used was1.024X105, making 4.12X10-9 mm wavelength elec-trons correspond to 4.22X10-4 mm wavelength light(the wavelength of He-Ne laser light in the crystal).

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RONALD BERGSTEN

TABLE I. Description of the crystals.

Lattice constant Muscovite mica Optical crystal (8-layer)

a 5.203X10-7 mm 5.3X 10-2 mmb 8.995 X 10-7 mm 9.2 X 10-2 mmc 20.03 X10-7 mm 20.4X 10-2 mm, 94.470 Approximately 90°

The lattice constants of monoclinic, muscovite nrand those of the corresponding optical crystal are lisin Table I and the corresponding diffraction patteare illustrated in Figs. 3-6.

II. THE RESULTING DIFFRACTION PATTER

The diffraction pattern produced by a crystal canconsidered as being produced by the interferencethe light that makes up the Airy's disks of each ofelements. Thus, the diffraction spots must lie inregions of constructive interference of the light s(tered by elements along any and all rows of elemeThe regions of constructive interference for elemealong one row will occur in conical surfaces that h;the row of elements as their axis and are given by

d (cosO-coso) = kX,

where 0 and 0 are the angles between the rowelements and the incident, and refracted bearespectively; d is the lattice spacing within theof elements, and k is the order.

A sequence of optical crystals has been useddemonstrate the form of the diffraction patterns du(one- and two-dimensional crystals and two-lathree-dimensional crystals.' This sequence illustr-how the line pattern produced by a one-dimensiccrystal reduces to a spot pattern (the N pattern)

Licasted!rns

INS

n beof

thethe,at- FIG. 2. White regions of the drawing correspond to the co-its. incidence of two sets of elliptical Laue zones. When the spots of

Fig. 1 are restricted to these zones, the resulting pattern resembles!nts those shown in Figs. 5 and 6.ave

additional rows are added to the crystal so as to(1) extend it to a full two-dimensional crystal. For wave-of lengths much smaller than the lattice constants, the

ms, spot diffraction pattern is similar to the projection ofrow the two-dimensional crystal-element sites on the

screen but greatly enlarged and rotated 90° about theto incident beam. The spots are indexed by (k,l) accordingto to the order of the line pattern produced by a row of

yer elements parallel to the lattice spacing, a, and theLtes order of the line pattern produced by a row of elements,nal parallel to the lattice spacing, b, respectively. Owing

as to the face-centered nature of the two-dimensionallattice of each layer, the only spots that can exist arethose in which k+l is an even integer (see Fig. 1).

The number of diffraction spots produced by a three-dimensional crystal is further restricted to those inwhich the radiation scattered from each layer ofelements constructively interferes (the L pattern). Ifthe layers are numerous, the diffraction spots arerestricted to those described by Bragg's law. However,if the crystal consists of only a few layers, the brightconical surfaces associated with a row of elementsperpendicular or nearly perpendicular to the layersbecome broad bright regions. The breadth of thesebright regions is dependent on the number of elementswithin the row.

Neglecting a slight shift due to the form factor,Eq. (1) describes the primary maxima. The corre-sponding bright regions are bounded by minimadescribed by

d(cosO-cos4) = ( ),n/

(2)

FIG. 1. Sketch of the diffraction pattern produced by one layerof a monoclinic crystal. The vertical rows correspond to theorders associated with the lattice constant a, which is horizontal. where n is the number of elements in the row. Thus,

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October 1974 OPTICAL SIMULATION OF ELECTRON DIFFRACTION 1311

FIG. 3. Diffraction pattern formed by He-Ne laser beam FIG. 4. Diffraction pattern formed by 4.12X 10- A wavelengthincident parallel to the lattice constant c of the optical crystals. electrons incident parallel to the lattice constant c of a thinNote the circular Laue zones. The pattern was recorded by mica crystal. Note the similarity to the optical pattern shownphotographing the pattern that was displayed on a screen. in Fig. 3. The pattern was obtained by use of an electron

microscope.

the diffraction pattern is limited to these regions,which intersect the screen in zones called Laue zones.When the incident beam is parallel to the latticeconstant c, the first Laue zone is a central disk with theother zones displayed as concentric rings encircling it.The first and second Laue zones are evident in Figs. 3and 4. A similar set of zones appears after the crystalis rotated about an axis parallel to b so the incidentbeam is directed diagonally across the crystal. For cconsiderably greater than a, this pattern is hardlydistinguishable from the former, the primary differencebeing the diameters of the Laue zones are smaller bya factor of c(a2+c2)-'. Two-layer optical crystals havebeen used in the demonstration of Laue zones in whichsuch Laue zones move across the field of diffractionspots as the crystal is rotated.' Between consecutiveLaue zones there are n-2 secondary maxima. Theintensity of the secondary maxima is negligible whenthe crystal consists of more than three or four layers.

Another particularly interesting diffraction pattern isproduced when the incident radiation is parallel to aline between an element in one layer and a pointbisecting the lattice constant, a, in the second layer.The elements of every other layer are then aligned withthe incident radiation. For 3 = 900 this orientation isdescribed by

a0 =tan7- -

21c

the plane of k be parallel to the plane of 0, yields

(c/a) tan[' (6+0)] = m/k, (4)

where 0 and 4 are the angles that a row of elements ofseparation a makes with the incident and refractedrays, respectively.

Combining Eqs. (3) and (4) for 0 approximatelyequal to 4 shows that k is approximately equal to 2m,indicating that only even orders of k are allowed nearthe center of the diffraction pattern (see Figs. 5 and 6).The other pertinent set of zones is associated with theelements that are aligned diagonally across the crystal

(3)

with the plane of 0 parallel to the plane of S. Two setsof Laue zones are particularly useful in describing theresulting diffraction pattern. One pertinent set of zonesis that formed by elements along c. Combining Eq. (1)as applied to elements along the lattice constant, a,with Eq. (1) as applied to elements along c and letting

FIG. 5. Diffraction pattern formed by light obliquely incidenton the optical crystal. Note the existence of the coincident Lauezones shown in Fig. 2. The pattern was recorded by photographingthe pattern that was displayed on a screen.

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RONALD BERGSTEN

FIG. 6. Diffraction pattern formed by electrons obliquelyincident on a thin mica crystal. Note the similarity to the patternshown in Fig. 5. The pattern was obtained by use of an electronmicroscope.

at intervals of (aI+c') '. The regions common to bothsets of elliptical Laue zones are illustrated in Fig. 2.When the spots of Fig. 1 are restricted to these regions,the resulting pattern resembles the light and electrondiffraction patterns of Figs. 5 and 6, respectively.Note that the resulting spots also fall within the Lauezones associated with the elements of every other layer,which are aligned with the incident radiation and arespaced at intervals of [a2+ (2c)2]',.

The breadth of the Laue zones shown in Fig. 3 areconsistent with Eq. (2). There is a slight additionalbroadening of the Laue zones along the direction fromupper left to lower right in Fig. 3. This broadening isapparently due to a slight misalignment of the layers.

The breadths of the electron-diffraction Laue zonesof Fig. 4 do not agree with Eq. (2). The breadth ofthe first Laue zone indicates that the mica sampleconsists of about four layers, whereas the breadth ofthe second zone is indicative of a crystal that containssix or more layers. This inconsistency is attributed tothe intensity of the electron beam. When the intensityof the beam is increased, the Laue zones broaden,owing to heating of the crystal. Linnik describes asimilar broadening in x-ray diffraction of a preheatedsample of mica.8 A good account of Laue zones andthe breadth of the central zone is given by Bragg andKirchner. 9 Bragg attributes the broadening of thesezones to slight distortions of the mica.'"

In recording the electron-diffraction pattern, thefirst zone must be overexposed so that the other zoneswill record. The different degrees of exposure make thebroadening most apparent in the first zone. The breadthof the first zone is used in electron miscroscopy tomeasure the thickness of thin crystals.9 "'

The extreme similarity between Figs. 5 and 6 indi-cates that the number of layers in the mica crystalused does not differ greatly from that of the optical

crystal. Perhaps the greatest difference between thesediffraction patterns is an additional intensity variation,which is most vividly displayed by the (1,9,0)-orderdiffraction spot in Fig. 6. Such variations are apparentlydue to the form factor, which is dependent on theintracellular structure of the mica crystal, which wasnot built into the optical crystal. Electron-diffractionpatterns of other crystals, demonstrating Laue zonesand their interpretations, are given by Andrews,Dyson, and Keown.'2

III. CONCLUDING REMARKS

When properly constructed, a multilayer opticalcrystal provides a direct optical simulation of electrondiffraction by crystals. The crystal can be fabricatedso that a particular wavelength of light corresponds toa particular electron De Broglie wavelength or so thatthe visible spectrum will correspond to a large electron-energy range. Diffraction patterns produced by bi-chromatic light are beautiful and clearly illustrate thedispersive properties of crystals.

While the optical crystal is rotated, the diffractionpattern is extremely dynamic as new Laue zones formand others move out of view, making them even moreapparent. Optical crystals can be used optically todemonstrate Laue zones in addition to N patterns inan optical simulation of an electron microscope.4 Thefirst Laue zones that I observed and studied were pro-duced by optical crystals. The experience with opticalcrystals proved helpful during attempts to recordelectron diffraction patterns for specific orientations ofmica crystals. I acquired an intuitive awareness of theorientation of the crystal, and was able to concentrateon the more-complex mechanical aspects of the electronmicroscope.

ACKNOWLEDGMENTS

This investigation was given impetus by a WisconsinState Research Grant. I photographed the electron-diffraction patterns at the University of Wisconsin-Milwaukee under the direction of D. Johnson. I amgrateful to H. Tscharnack and N. Stoner for theircritical reading of the manuscript.

REFERENCES'R. R. Bergsten, Am. J. Phys. 42, 91 (1974).2S. Kikuchi, Jpn. J. Phys. 5, 83 (1928).3D. deFontaine, K. A. Jackson, and C. E. Miller, Am. J.

Phys. 37, 789 (1969).4D. G. Ast, Am. J. Physics 39, 1164 (1971).'J. R. Meyer-Arendt and J. K. Wood, Am. J. Phys. 29, 341

(1961).6R. B. Hoover, Am. J. Phys. 37, 871 (1969).'D. G. Fedak, T. E. Fischer, and W. D. Robertson, J. Appl.

Phys. 39, 5658 (1968).8W. Linnik, Nature 123, 604 (1929).%W. L. Bragg and F. Kirchner, Nature 127, 738 (1931).

10W. L. Bragg, Nature 124, 125 (1929)."P. B. Hirsch, A. Howie, R. B. Nicholson, and D. W. Pashley,

Electron Microscopy of Thin Crystals (Butterworth, London,1965), Ch. 17, p. 421.

"K. W. Andrews, D. J. Dyson, and S. R. Keown, Interpretationof Electron Diffraction Patterns (Plenum, New York, 1971),Chs. 3 and 4.

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