A Simple Bragg Diffraction Experiment with Harmless Visible Light Claudia Segschneider and Heiner Versmoldl lnstitut fOr Physikalische Chemie, RWTH Aachen, D-5100 Aachen, F.R. Germany X-ray, neutron, and electron diffraction are extremely powerful methods for crystal structure investigatiotls (1,2). In contrast to the fundamental importance of these diffraction methods, no really simple and inexpensive laboratory experiment for educational purposes is available a t the moment. This lack of a suitable experiment has been a pertinent problem. Recently, a microcomputer program that allows one to simulate an X-ray powder diffraction pattern has been described in this Journal (3).Although structural determinations by scattering methods are highly computerized, we believe that the students' first steps into this important field should be accompanied by a true experiment. In this paper we describe such an experiment that also has the advantage that i t avoids the risks present when working with X-rays or other ionizing radiation. Diffraction of electromagnetic radiation by crystals occurs when the wavelength X of the radiation and the distances between adjacent scattering centers in the crystalline phase are on the same scale. If one wants to use harmless visible light for a diffraction experiment, one must remember that its wavelength is roughly a thousand times larger than the wavelength of X-rays; consequently the crystal lattice spacings or the lattice constant a of the systems to be studied must he a thousand times larger than those of ordinary atomic or molecular crystals. Crystals exhibiting such large lattice spacings are observed for biological, macromolecular, and colloidal systems (4). One beautiful example is the gem opal: I t consists of ordered regions of spherical silica particles of about 300 nm diameter. For a given orientation the ordered regions each diffract a particular wavelength (color) if exposed to the white spectrum of visible light. In the present experiment we use monodisperse spherical polymer particles of about 100 nm diameter in water. Such particles are often electrostatically stabilized by negative surface groups. For example, -SOaH groups dissociate into -SO; and H+ ions, i.e., the particles get a negative charge and are surrounded by a positive H+ ion cloud. The negatively charged particles repell each other via long-ranged coulombic forces. In order to minimize the total energy of such an ensemble of charged particles in the given solvent volume, they stay away from each other and finally form a crystal. Such colloidal crystals are very soft and can he destroyed by shaking the sample. Recrystallization takes place, however, in a few minutes. If exposed to white light, the crystals show heautiful iridescent colors like the opal. In the following we describe how the crystal structure and lattice spacing a of such colloidal crystals can he determined in a simple laboratory class experiment with visible light. The Polymer Collold System Several polymer colloid systems are currently known that can be used to prepare colloidal crystals. Very good experiences have been had with standard DOW latex material2. Suitable particle concentrations for light scattering experiments are 1.0 X 1018to 3.0 X 10'8particles/m3, which can be obtained by diluting the commercially supplied suspension by a factor of about 30 to 100. Normally, the suspension
(dl Figure 1. Schematic Bragg diffraction arrangement: (a) HeNe laser; (b)focusing lens; (c) scattering cell containing +he colloidal crystals; (d) screen with scale to locate reflections.
prepared in this way does not crystallize since the suspension contains stray ions that screen the negatively charged polymer particles. In order to remove these impurities, the suspension can he cleaned either by extended dialysis, which is rather time consuming, or by treatment with a mixture of precleaned cationic and anionic ion exchange resins (5).The &pension and small amounts of the ion exchange resins are stored for a few days in a cylindrical quartz cuvette. In order to speed this cleaning process, agitation of the cell now and then is recommended. After the mentioned period of time the suspension becomes iridescent when illuminated with white light. This tells us that the sample is ready for the Dehye-Scherrer experiment. The Experlmental Arrangement The experimental arrangement is shown in Figure 1. A low-power He-Ne laser provides harmless visible light (which replaces the X-rays of a real Dehydhherrer experiment). Lens 1is used to focus the laser beam slightly into the scattering cell, which contains the colloidal crystals a t random orientation. A crystal properly oriented for Bragg diffraction scatters light under the Bragg angle Ohar. Reflections lying in the horizontal plane can he observed on the screen, which surrounds the scattering cell. In order to suppress stray light emerging from the entrance of the light into and exit out of the cell, an index match vat is recommended. Since the colloidal crystals in the cell are comparatively large, the laser beam passes only a limited number of crystallites on its way through the scattering cell. I t happens that none of them is properly oriented for Bragg diffraction. Thus, in order to observe a Bragg reflection, i t may be necessary t o bring a properly oriented crystallite into the laser heam. This can he achieved by rotating the scattering cell about its cylindrical axis or by readjusting its height in the laser beam until a reflection is visible on the screen. A scale
' Corresponding author. Volume 67
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on the screens allows an easy determination of the Bragg diffraction angle. Experimental Results Experimental results were obtained with two types of commercial latex particles: (a) Dow uniform polystyrene spheres "Dow 91'": The nominal diameter of the particles is 91 nm. The particle charge results from -S04H groups, which dissociate as described above. (b) Standard Dow latex particles "Dow 10Y2:Here, the nominal diameter is 102 nm. The particle charge results from -S04H and from -COOH groups. Iridescent samples of "Dow 91" and "Dow 102" particles were nrenared and Braze reflections determined as described above. The result& Bragg diffrartion angles 8, listed in Table 1 r D o w 91") and Table 2 ("Dow 102"). are averages of a t least five independent measurements. In both tables the first two columns show the number ( # ) of the reflection and the diffraction angle 8. The next two columns are discussed helow. Data Evabaflon and Dbusslon
Most textbooks on nhvsical chemistrv (2)treat the cwstal power diffraltion structure and its determination by in ereat detail. Therefore. onlva few basic facts necessarv for t h l d a t a evaluation will be summarized here. The crvstal structure is obtained if a basis of atoms or particles is attached identically to eachlattice point (I).The logical relation is
X-ray
crystal structure = lattice + basis
632.8 nm, and for particles suspended in water we have n = 1.33 a t room temperature. For a cubic lattice with lattice constant a, the interplanar distance is related to the Miller indices as dhk,= a(h2+ k2 + 12)-112
(3)
Inserting eq 3 into eq 2 results in
Since the Miller indices h, k, 1are integers, the quantity N = (h2 k2 12)will also be an integer. We note, however, that N cannot have the values 7,15,23,28, etc. Diffraction angles 0 corresnondine to these values of N do not occur for cubic 2). lattices Next, we have to add the basis to the lattice. The simplest situation occurs for the sc crystal structure, for which the description given above applies. If the basis consists of more than one particle, further values of N and correspondingly further diffraction angles 0 (see eq 4) may he missing. In order to decide whichN values are missing, one considers the structure factor
+ + TI,
Here, fi is the atomic or particle scattering factor, and the
(1)
Table 1. Bragg DWraQlon Results for aSample Contalnlng Wow 91" Parllcles
simple
cubic
( 5 ~ )
body centered cubic (bee)
81'
sin2(8/2)
ff
+ k2+
IZ
(tee)
In Figure 2 the crystal structures, simple cubic (sc), bodycentered cubic (bcc), and face-centered cubic (fcc), are shown, all of which have a cubic lattice and differ only by their basis. Particles helonging to the basis are drawn as filled circles in Figure 2. The number and coordinates of these particles are given in Table 3. As usual, we denote the distance between different lattice planes as &&I,where h, k, 1 are the Miller indices of the planes. Bragg diffraction from a given set of planes occurs if
Here, m is an integer, hln the wavelength of the radiation in the medium of refractive index n, and 8 the angle between the inrident and the diffracted beam. For a HeNe laser A =
DOWlatex material can be ordered from: Serva GmbH. Postfach 105260,D-6900 Heidelberg. A limited number of crystaliring samples are available from the authors on request.
Journal of Chemical Education
diffractionAngie
#
face centered mbcc
Figure 2. Crystal structures: simple cubic (sc), bady-centered cubic (bcc),and face-centered cubic (fcc).Particles belonging lo the basis are drawn as filled circles.
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Reflection
Tabh 2. Bragg DHlractlon Results for a Sample Contalnlng "Dow 102" Parllcles Reflection
diffractionAngle
#
81'
sin2(8/2)
ff
+ k2 + l 2
Table 3. Number and Coordlnatr of Parllcles Forming the Bask Crystal structure simple cubic
body centered c. face centered c.
Number of particle6 of the basis 1 2
4
Cwrdinates/ianiceconstant a (0. 0.0) (O.O.ON1/2.
112, 112) (0.0.ON112.1/2.ON1/2.0,1~2NO, 1/2.1/2)
from which the lattice constant a can be calculated as
sim le cubic (SCP h,k,l = i n t e g e r s
Since there are two particles per unit bcc cell, the particle number density amounts to
n = 2/a3 = 2.33 X 10" particles/m3 body c e n t e r e d cubic (bcc) h + k J = even
f a c e c e n t e r e d cubic (fcc) h.k.1 all even or all odd
Figure 9. Possible values of N = (i? + kZ+ 1') for three cubic crystal sbuctures summation extends over all particles i of the basis with coordinates (xi, y;, zi). Since most textbooks on physical chemistry present a detailed derivation of which reflections are allowed or forbidden for bcc and which ones for fcc crystals, we merely summarize the results (1,Z): bcc structure: If (h + k + 1) is an even number, the corresponding reflection is allowed; otherwise it is forbidden. fcc structure: If h, k , and 1 are all even or all odd, the correspondingreflection is allowed; otherwise it is forbidden. This result is summarized in Figure 3, which shows the allowed values of N for the three crystal structures sc, bcc, and fcc. Next we evaluate the data presented in the Tables 1and 2. According to eq 4, sin2 (012) is proportional to N = (h2 k2 12); that is, a plot of sin2 (812) versus N is indicated. One difficulty that arises is the unambiguous assignment of Miller indices h, k, 1 to the measured reflections. This difficulty of indexing a cubic power diffraction pattern when the unit cell dimension is not known can he solved as follows (6). Measured values of sin2 (012) (taken from Table 1) are plotted on the vertical axis of Figure 4 and drawn out as horieontal lines. The seauence of integers from zero is plotted on the horizontal N ax& and each isdrawn out as a vertical line. A ruler is now rotated from the sin2 (012) axis about the origin until all its intercepts with horizontal lines coincide with their intersections with the vertical set. Finally, a line is drawn through the origin a t this inclination. In Figure 4 two such lines have been drawn. The steeper broken line is an incorrect assignment because one intersection with the vertical set occurs for N = 7, which according to Figure 3 is forbidden. The measured diffraction pattern is thus incompatible with the sc crystal structure. The fully drawn line gives the correct assignment. Intersections with the vertical set of lines occur for N = 2,4,6,8,10,12,14,16. From Figure 3 it is evident that the structure of the colloidal crystals under study is bcc. The slope of the straight line according to eq 4 and Figure 4 is
In Figure 5 a similar evaluation of the data presented in Table 2 is shown. Also for this system the crystalstructure is bcc. This time we obtain for the lattice constant a and the particle number density n the values a = 989 nm and n = 2.07 X 1018particles/m3. Finally, we call attention to one remarkable difference between electrostatically stabilized colloidal crystals and ordinary atomic or molecular crystals. The particle diameter of both types of colloid particles used here was roughly 100 nm. By contrast the lengtha of the cubic cells is about 10 times larger. Thus only a fraction of 111000 of the volume is
+ +
Figure 4. Indexing of the dlffractlon data of Table 1 (Dow 91 particles).Broken line: Incorrect sc indexing: full line: correct bcc indexing.
N Figure 5. indexlng of lhe diffraction data of Table 2 (Dow 102 particles). Volume 67 Number 11 November 1990
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filled with colloid material; the rest is filled with water. This is due to the long-ranged Coulomb interaction between the negatively charged particles. Unlike atomic or molecular crvstals. which alwavs " Dossess a characteristic fixed lattice constant a not too different from the particle diameter, the lattice constant of the colloidal crvstals deoends on the Darticleconcentration. At high particle numberdensitiesa tiansition from the bcc to the fcc structure is expected (7). A
Bragg reflections are really visible. (h) The student does not work with X-rays or other dangerous ionizing radiation. The analysis can be carried out exactly as in the ordinary X-ray powder diffraction experiment. I t is shown how the crystal structure, the lattice constant a and the particle number density can be obtained in an unamhigous way. Literature Clted 1. Kittel, C. lnfmdvclion to So!idSfota Physics; Wiley: New York, 1966. 2. Afkinp. P.W. Physical Cherniafry; Oxford University: Oxford, 1986. Alberty, R. A. Physical Chemistry; Wiley: New York. 1987. Mwre. W. J. Physical Chemistry: Lon.man: London. 1976.
Conclusion In this paper we describe a simple experimental arrangement for the determination of Bragg diffraction pattern from colloidal crystals with visible light. The experiment has two advantages over ordinary X-ray experiments: (a) The
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Dekker: New York, 1970. 6. McKip, D.;MeKie. C. Esrentiala o/Chrisla!!ography;Blsckwell: Oxford, 1986. 7. ~i~~(s.~.~.:O~-~~~~,H.n.;Sinhs.S.K.:Chaikin,P.M.:Axo,J.D.;Fujii,J.Phys.Re
