X-RAY DIFFRACTION SIMULATION USING LASER POINTERS AND
PRINTERS
Neil E. Johnson
Department of Geology, Appalachian State University, Boone, NC 28608, Johnsonne@appstate.edu
ABSTRACT
The conceptual leap from point array to diffraction patterns has long been recognized as challenging. For more
than sixty years it has been known that an analogy can be
drawn between optical and X-ray diffraction, requiring
only a source of monochromatic light and an array of scatterers with spacings of 50 - 200 µm. Inexpensive lasers fulfill the first requirement, but the second has been
problematic due to difficulties in producing scattering arrays. A number of approaches have been used, including
pantograph reduction, several types of photographic reduction, and even standard sieves, but all require significant preparation, limiting in-lab experimentation.
Laser printers with resolutions of 600 or 1200 dots per
inch (one dot per 42 or 21 µm) are inexpensive and readily
available. Improvements in the screen magnification capabilities of common graphics software allow students to
create and modify arrays, print them on transparencies,
and illuminate them with laser pointers. Introductory examples can demonstrate basic principles, whereas advanced examples can illustrate plane lattices, stacking
faults or even powder diffraction. The turnaround time
from idea to observation is as little as a few minutes, allowing students to experiment with near real-time feedback.
Keywords: Education - graduate; education - undergraduate; geology - teaching and curriculum; mineralogy and
crystallography.
INTRODUCTION
Teaching students how to collect and examine data on a
computer-automated powder diffractometer is not a challenging task. Although some have ventured into more illustrative exercises that make substantial use of modern
diffractometer software (Horton, 1994; Brady and Newton, 1995; Hluchy, 1999), instruction consists principally of
demonstrations of the mechanics (physical as well as computational) of X-ray diffraction, superficial discussions of
the principles, followed by on cue recitations. The results
reflect this; students enjoy the chance to do something
hands-on, usually appreciate some of the applications, but
miss a deeper understanding. This is through no real fault
of the students: what could be more “black-box” than procedures whose results are dependent on the arrangement,
spacing and type of atoms in crystals?
The recognition of this obstacle, and of the ideal
means of surmounting it, date back to the earliest work in
X-ray diffraction. As noted by Harburn and others, (1975)
“The idea of using optical analogues to aid in the interpre-
346
tation of X-ray diffraction patterns originated with Sir
Laurence Bragg round about 1938, and it has been developed in many directions in the following thirty-five
years.” The intention was to use it as a visualization aid
for researchers as well as a teaching tool, but in order to accomplish this “scaling up” of the diffraction process, a
source of light with a narrow range of wavelengths and an
array of scatterers with spacings in the range of 50 - 200 µm
were required. Initial light sources consisted either of
monochromatic sources such as sodium vapor lamps or of
intense white light passed through a narrow bandpass interference filter (Taylor and Lipson, 1965). These types of
sources provided limited intensities due to the need for a
small aperture to create a point source (Figure 1), but this
problem was eliminated by the use of lab bench lasers
(Harburn and Ranniko, 1972). Today, laser diode pointers
that easily meet the requirements can be purchased
cheaply, so obtaining sources of light for students to use is
no longer a problem. Producing arrays of scattering centers is another matter, as creating a grid of pinholes with a
consistent spacing of approximately 0.1 mm is well beyond the freehand motor skills of most everyone.
Scattering arrays were first produced by hand, making use of almost forgotten pantographic reduction techniques (Taylor and Lipson, 1965) that would take a
previously drafted array and scale it to the requisite size.
Subsequent methods made use of photographic reductions, either using modified pantographs, exposure of film
on a device similar to modern drum scanners (Harburn
and others, 1974) or by photographic etching of metal
plates (Hill and Rigby, 1969). The time and effort required
to produce appropriate scattering arrays by these methods was considerable, which explains why the work on
this topic commissioned by the International Union of
Crystallography (Harburn and others, 1975) consists of 32
pages of text (half of these being a French translation of the
other half) combined with 32 two-page plates of the scattering arrays and their diffraction patterns. As pointed out
by Brady and Boardman, (1995), it is not surprising that although some use was made by British mineralogists, the
techniques were virtually unknown to Americans.
This unfamiliarity began to be reversed by two seminal papers that simplified the means of obtaining arrays
for class use. Lisensky and others (1991) created printed
scattering arrays using a personal computer and a laser
printer, then photographed them with 35 mm slide film.
By creating the patterns at the maximum magnification
available in the software (»2X), the photographic reduction resulted in arrays with minimum spacings of 50 µm.
The slides could be easily mounted and illuminated with a
laser pointer for demonstrations. Brady and Boardman
(1995) furthered this idea, realizing that among other
Journal of Geoscience Education, v.49, n.4, September, 2001, p. 346-350
Figure 1. Schematic arrangement of elements used in
original optical diffraction demonstrations. S: light
source; F: narrow bandpass interference filter; L1-3:
lenses; P: pinhole aperture; A: scattering grid; F: focal
plane. Modified from Harburn and others (1975).
things, standard sieves of appropriate sizes could also be
used for such demonstrations. They also demonstrated
the similarities between the optical diffraction patterns
and precession photographs, and even extended the analogy to the point of simulating Debye-Scherrer geometry
by suspending a line grating into a fishbowl lined with paper. A computer variant of this, in which all of the diffraction information is calculated and displayed, is also
available (Neder and Proffen, 1996).
This recent work has gone a long way towards the
goal of allowing students to discover for themselves what
occurs in the process of X-ray diffraction, but still contains
an important obstacle: all of the scattering arrays must be
created ahead of time. Students may choose from any of
the previously prepared options, but cannot manipulate
the arrays to determine the results with direct feedback.
DIRECT PRODUCTION OF SCATTERING ARRAYS
Perhaps the most unappreciated aspect of the personal
computer revolution has been the parallel revolution in
personal computer output. Twenty years ago dot matrix
printouts were perfectly acceptable but today we expect
resolution and quality from printers virtually indistinguishable from that of commercial presses. For nearly the
same price as a 300 dot per inch (dpi) laser printer a dozen
years ago, printers are available with ten times the speed
and four times the resolution, meaning an inexpensive (»
$500) laser printer can image dots with separations as
small as 42 µm (600 dpi resolution). Improvements in
graphics software have matched the abilities of the output
devices, allowing for the direct manipulation of graphics
on the computer screens at magnifications over 30X. The
end result is that scattering arrays can be created, edited
and printed directly onto clear overhead transparencies
with no post-printing reduction requirements: straight
from the laser printer to the laser pointer.
Figure 2. (A) A square scattering array as viewed on a
72 dpi computer monitor with a one centimeter bar
for scale. (B) The same array at 32X magnification.
The apparent differences in the arrays are due to monitor interpolation at the lower magnification. Compare the centimeter scale to that in A. (C) Diffraction
pattern resulting from this square array.
The scattering arrays for a prior presentation (Johnson, 1999) and for the figures herein were created using
graphics software and printed on a high quality laser
printer. The software was set at its maximum screen magnification (32X), with an alignment grid of 0.1 mm, and included as a reference was a one cm scale bar marked off in
mm steps. The individual points for the scattering array
are periods (.), at a type size of one point (Figure 2); after
an initial point was aligned on the grid, it was duplicated
and its position adjusted pixel by pixel with the cursor
keys. When a row was completed, it was duplicated and
its position adjusted, and the process is repeated to produce an array of sufficient size (»60 X 60). The photographs of the resultant diffraction patterns were produced
by placing the laser pointer and tranparencies on makeshift supports at one end of a lab bench in a darkened
classroom, a flat cardboard target on the opposing wall,
and mounting a camera between the two on a tripod to allow for longer exposure times.
The geometry of the diffraction processes applicable
to these patterns are readily available elsewhere (Brady
and Boardman, 1995; Hammond, 1997; Harburn et al.,
1975; Putnis, 1992), so only a brief mention will be made
here. Passing a laser beam through a single plane of scatterers is explained by the Fraunhofer equation (nl = d sin
f), where d is the spacing between points within the plane
(Figure 3). In contrast, Bragg diffraction (nl = 2d sin q) is
three dimensional, and d is the spacing between planes.
Constructive interference occurs when the additional distances traversed by parallel scattering rays equal an integral number of wavelengths; in the Bragg case this
additional distance is traversed twice. The two geometries
are sufficiently similar that once students grasp diffraction
in the Fraunhofer geometry, it is an easy step to an understanding of the Bragg case (Brady and Boardman, 1995).
The photographed diffraction patterns frequently display direct beam fringes and satellite reflections in addi-
Johnson - X-Ray Diffraction Simulation Using Laser Pointers and Printers
347
Figure 3. The geometry for Fraunhofer diffraction versus that for Bragg diffraction. The equation resulting
from the Fraunhofer case is nl = d sin f, whereas in the
Bragg case is nl = 2d sin q. Modified from Lisensky et
al. (1991).
tion to the main reflections, due to the use of the laser
pointer as a source. Although each of the printed arrays is
relatively large (1 – 4 cm2), the typical beam diameter of a
laser pointer is only about three mm, so only a fraction of
the array diffracts at any one time. In some cases, the satellites and fringes can obscure parts of the diffraction pattern. This can be solved by using a bench top laser fitted
with a series of lenses that expand the beam diameter
while maintaining collimation. Such laser beam expanders are commercially available from laboratory optics vendors.
EXAMPLES
The examples included here include some of the basic and
some of the more advanced applications that are possible.
Figure 2A is an example of how a basic square array appears at normal (1X) magnification, whereas 2B is the
same array at 32X, and 2C shows the diffraction pattern
that results. Figure 4A is a square with a much larger unit
cell, and 4B is the pattern showing the apparently
counterintuitive result: a larger number of more closely
spaced spots. This demonstrates the reciprocal relationship between a scattering array and its diffraction pattern,
an observation that can stand on its own or lead to discussions about reciprocal space. Arrays representing the
other four plane lattices are reasonably simple to create.
As an example, Figures 4C and D show a diamond array
and the resultant diffraction pattern.
Of the simple patterns, perhaps the most interesting
can be found in Figure 4E, showing a small fragment of a
rectangular array adjacent to a large number of duplicates
of that same fragment, each rotated by random amounts
and directions in the plane of the page. These represent individual crystallites, so the pattern that emerges (Figure
4F) consists of a ring – a powder pattern. A bench laser
with an expander lens set will illuminate a large area of the
array, producing more rings that are sharper, and better
348
Figure 4. (A) A square scattering array with a larger
unit cell than that in Figure 2. (B) Resultant diffraction pattern. (C) A diamond scattering array. (D) Resultant diffraction pattern. (E) Randomly oriented crystallites. (F) Resultant powder pattern. Illuminating different areas of the array with a laser pointer will produce complete or “spotty” rings, depending on how
many crystallites are illuminated, and tracking the
beam across the array makes the rings more evident.
organized, whereas using a laser pointer results in more
diffuse and “spotty” rings, which (if desired) can be compared to poorer quality Debye-Scherrer films. In this case,
the appearance of the ring is enhanced by tracking the laser beam across the array (or vice versa); the ring will remain in place while the randomly scattered points move
At a more advanced level, the ability to directly manipulate individual scattering points or rows and/or columns of points allows for the creation and discussion of
more subtle diffraction effects. Figure 5A displays a series
of stacking faults: the horizontal layer offsets are one-third
and two-thirds of the horizontal cell dimension, and the
fault probability for each layer is 0.5. The diffraction result
(Figure 5B) is the production of numerous satellite reflections along with the first-order spots in same direction as
the faulting, which can be contrasted with the lack of extra
reflections for the zero-order spots and for higher order
spots in the unfaulted direction. Figures 5C and D demon-
Journal of Geoscience Education, v.49, n.4, September, 2001, p. 346-350
Figure 5. (A) Stacking faulted array, with offsets of 0, 1/3 or 2/3 of the horizontal cell dimension. (B) Resultant
diffraction pattern with satellite reflections adjacent to first-order diffraction spots. (C) Modulated array with a
modulation periodicity of 6 layers. (D) Resultant diffraction pattern. Note weak satellite reflections that are
symmetrically offset from the principal zero-order spots.
strate structural modulations and their effect; satellite reflections that are symmetrically offset from the layer lines.
The use of a benchtop laser in this case will allow the satellites to be resolved more clearly.
SUMMARY
The challenges inherent in asking students to think about
and work with abstract concepts are well-known, and although laboratory structure models provide a hand-hold
for grasping the atomic scale abstraction of a crystal structure, envisioning an interaction between these models and
radiation remains at a further layer of abstraction. One
major advantage of this optical approach as an introduction to X-ray diffraction is the (intentional) lack of format.
Any or all parts can be utilized as demonstrations, as
planned exercises (see appendix for examples), or as unstructured investigations, but it is the last of these that provides the most promise. By allowing students to create
and diffract on their own, they can discover for themselves the physical reality behind the instruments they
will be using. Another advantage is the access provided to
many levels of further discussion. Initially, students are
curious about the effect of leaving out or moving single
points in a scattering array; their surprise at the lack of a
visible effect can be leveraged into a discussion of how
X-rays provide an average structure. Another common
modification, changing the size or darkness of the periods,
can lead to a consideration of structure factors.
ACKNOWLEDGMENTS
This work is an outgrowth of experimentation in crystal
chemistry classes at Appalachian State over the past few
years and I thank the students in those classes for their
feedback and their patience with my partially formed
ideas. The experimentation was inspired by a demonstration of laser diffraction by John Brady at GSA in Boston in
1993. Richard Abbott provided helpful comments, both in
development and in the preparation of this manuscript.
REFERENCES
Brady, J.B. and Boardman, S.J., 1995, Introducing
mineralogy students to X-ray diffraction through
optical diffraction experiments using lasers: Journal
of Geological Education, v. 43, p. 471 - 476.
Brady, J.B. and Newton, R.M., 1995, New uses for powder
diffraction experiments in the undergraduate
curriculum: Journal of Geological Education, v. 43, p.
466 - 470.
Hammond, C., 1997, The basics of crystallography and
diffraction: Oxford, Oxford University Press, 249 p.
Harburn, G., Miller, J.S. and Welberry, T.R., 1974, Optical diffraction screens containing a large number of
Johnson - X-Ray Diffraction Simulation Using Laser Pointers and Printers
349
apertures: Journal of Applied Crystallography, v.7, p.
36-37.
Harburn, G. and Ranniko, J.K., 1972, An improved optical
diffractometer: Journal of Physics E: Scientific
Instrumentation, v. 5, p. 757-762.
Harburn, G., Taylor, C.A., and Welberry, T.R., 1975, Atlas
of optical transforms: London, G. Bell & Sons, 32 p.
Hill, A.E. and Rigby, P.A., 1969, The precision
manufacture and registration of masks for vacuum
evaporation. Journal of Physics E: Scientific
Instrumentation, v. 2, p. 1084-1086.
Hluchy, M.M., 1999, The value of teaching X-ray
techniques and clay mineralogy to undergraduates:
Journal of Geological Education, v. 47, p. 236 - 240.
Horton, R.A., Jr., 1994, X-ray diffraction as an instructional
tool at all levels of the geology curriculum: Journal of
Geological Education, v. 42, p. 452 - 454.
Johnson, N.E., 1999, Optical transforms redux: Creating
diffraction gratings on a laser printer for X-ray
diffraction simulation: Geological Society of America,
Abstracts with Programs, v. 25, A-347.
Lisensky, G.C., Kelly, T.F., Neu, D.R., and Ellis, A. B., 1991,
The optical transform, simulating diffraction
experiments in introductory courses: Journal of
Chemical Education, v. 68, p. 91-96.
Neder, R.B. and Proffen, T.H., 1996, Teaching diffraction
with the aid of computer simulations: Journal of
Applied Crystallography, v.29, p. 727-735.
Putnis, A., 1992, Introduction to mineral sciences:
Cambridge, Cambridge University Press, 457 p.
Taylor, C.A. and Lipson, H., 1965, Optical transforms.
Their preparation and application to X-ray diffraction
problems: Ithaca, Cornell University Press, 182 p.
APPENDIX A - OUTLINE OF SAMPLE LABORATORY EXERCISE
Goals
Demonstrate the process of diffraction and the important features: basic trigonometric relationships, reciprocal relationships of distances between scattering points and diffraction spots, effect of wavelength on diffraction pattern.
Procedures
Introduce the concept of the diffraction of light (introductory physics texts usually contain a useful discussion). Explain the process, with emphasis on constructive versus destructive interference and introduce the Fraunhofer
equation. Set up a laser (bench laser or inexpensive pointer of wavelength 650 nm) and a primitive square scattering
array printed on clear overhead transparency at one end of classroom. Place a target at other end of classroom
(square ruled graph paper is convenient). Make certain that students do not look down direct laser beam. Have students measure distance from scattering array to target (Floor Distance or FD) and horizontal or vertical distance
from direct beam spot to diffracted spot (Spot Distance or SD), then use this data and simple trigonometry to calculate the diffraction angle (phi). Note that for FD >> SD, f = sin f. Using the Fraunhofer equation and known laser
wavelength, calculate the separation between scattering points (d). Measure several spot distances in this manner
and calculate and average value for d. Compare this measured d with value of d used to create scattering ray.
Change the wavelength of the light source (different color laser) and repeat experiment, which will demonstrate
that the results are independent of wavelength used to make the measurements. Repeat the experiment using a
primitive square array of different d, to demonstrate the reciprocal relationship between the diffraction pattern and
actual scattering array. Introduce the Bragg equation and compare and contrast this with the Fraunhofer equation.
Provide students with precession photographs, known camera distance (CD to replace FD) and x-ray wavelength.
Have students calculate d-spacings for crystal.
Further Directions
Using the concepts of Miller indices, have the students index the spots on the primitive diffraction pattern. Provide
the students with a centered square array to index and determine d-spacings for the crystal. Discuss the effects of
centering on diffraction patterns.
In a computer lab, provide students with the graphics software used to create the scattering arrays, a sample array
(or two) to edit, and blank overhead transparencies. Direct students to edit the arrays as they see fit, print them out,
and describe the effects the changes have on the diffraction patterns. Students may also be organized into groups for
this exercise, with each group required to present their results to the class.
350
Journal of Geoscience Education, v.49, n.4, September, 2001, p. 346-350