Lattice Entertain You: Paper Modeling of the 14 Bravais Lattices on

Communication pubs.acs.org/jchemeduc

Lattice Entertain You: Paper Modeling of the 14 Bravais Lattices on YouTube Lawrence T. Sein, Jr.*,† and Sarajane E. Sein‡ †

Department of Chemistry, Towson University, Towson, Maryland 21252, United States Department of Counseling, LaSalle University, Philadelphia, Pennsylvania 19141, United States

‡

S Supporting Information *

ABSTRACT: A system for the construction of double-sided paper models of the 14 Bravais lattices, and important crystal structures derived from them, is described. The system allows the combination of multiple unit cells, so as to better represent the overall three-dimensional structure. Students and instructors can view the models in use on the popular internet channel YouTube at no cost.

KEYWORDS: Upper Division-Undergraduate, Inorganic Chemistry, Hands-On Learning/Manipulatives, Crystals/Crystallography

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INTRODUCTION The structures adopted by solids are of considerable importance in chemical education, so much so that numerous articles on this topic have been published in this Journal.1−9 Crystal systems, including the Bravais lattices (the 14 possible three-dimensional structures for which each single atom or ion is an identical environment), are discussed.1−14 The relevance of these structures to wider chemical problems or space group symmetry has been noted.7−9 Several systems for creating and utilizing models of crystal structures even beyond the Bravais lattices have already been developed.10−14 The wide scope of the topic has led to an emphasis on just one or two particular points in each paper. Kettle and Norrby1 demonstrate that the traditional 14 Bravais lattices, as understood by chemists and crystallographers, include 7 “primitive” cells and 7 “centered” ones; they then proceed to demonstrate that each of the centered lattices can be redefined to make it primitive, and they show that such redefinition is often advantageous to spectroscopists and solid state physicists. Ladd,2 and Kettle and Norrby3 discuss the construction of the Wigner-Seitz cell for a given Bravais lattice. Schomaker and Lingefelter4 discuss the accidental misidentification of crystal lattices, particularly those with 3-fold rotation axes. Galasso5 emphasizes the relationships between atomic structure and solid state chemistry, beginning with the simple cubic structure, and progressively increasing the complexity to such important structures as diamond and zincblende. He recommends that students begin to become more acquainted with the arrangements of atoms in high school, or even earlier. “It is only by looking at the structure step-by-step that the student can © XXXX American Chemical Society and Division of Chemical Education, Inc.

actually visualize and remember these arrangements of atoms.” Boldyreva6 highlights the importance of the solid state, particularly to chemists employed in industry, but recommends that “one and two dimensional models” be “used widely to introduce important concepts” without ever mentioning the use of three dimensional structures. Hardgrove7 introduces space group structures, beginning with the NaCl structure, and rapidly progressing to more complex structures, while ignoring the 14 Bravais lattices. The primary emphasis appears to be the calculation of formula units and the utilization of special positions. Kettle and Norrby8 emphasize the importance of the Brillouin zone, and highlight some features of the direct lattice that are shared by the reciprocal lattice. Meyer, Glaus, and Calzaferri9 discuss more advanced topics related to crystal structures, such as band structures and the density of states. These papers do not discuss either the construction of threedimensional models, or their utility in pedagogy.

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PROBLEM AND SOLUTION While most chemistry textbooks include diagrams of the 7 crystal systems and the 14 Bravais lattices, frequently in color, these diagrams are almost invariably of single unit cells. It can be difficult to visualize how the unit cells combine to form a twoor three-dimensional extended lattice. To faithfully recreate this lattice, multiple copies of the relevant unit cell are needed (as noted in ref 5), which immediately increases the cost by an order of magnitude.

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Westbrook and Devries10 discuss the construction of very large and elaborate crystal models using plexiglass and 140 miniature lamps. Not only is the resulting model cumbersome and heavy, but the cost of construction for the materials alone was estimated at nearly 80 USD in 1957. Only a single unit cell is described, and its construction requires specialized electrical skills far beyond those of a typical undergraduate. Cady11 describes the construction of crystal lattices using colorful pom pons. The method is inexpensive, and it also helpfully illustrates the dependence of crystal structure on the relative radii of the component atoms or ionsthe radius ratio rules. However, unless the pom pons are cut into pieces, the resulting structure overflows any one unit cell, thereby limiting their utility in exercises where the student is expected to calculate the number of atoms per unit cell. A further complication is that the pom pons are opaque, so that it can be difficult to visualize the three-dimensional structure, except along exterior planes. Bindel12 describes the construction of clear plastic models to visualize crystal structures. The models are substantial and attractive; furthermore, they allow the appreciation of the longrange structure from any angle, since they are (mostly) transparent. However, the author notes that each model costs approximately 40 USD in materials, and requires 5 h of work. The author does not mention if structures other than simple cubic, body-centered cubic, face-centered cubic, hexagonal, and sodium chloride can be constructed, and if so, what modifications or extensions to the method would be required. A previous system of paper models cleverly highlights the importance of packing for the primitive cubic, body-centered cubic, and face-centered cubic structures.13 However, since each model is opaque, the extended nature of the lattice is obscured unless one views an assembly of 5 or more unit cells, and then only where the cells meet at a single vertex. Sunderland14 describes the use of deluxe solid-state model kits that can be purchased commercially. A wide range of important crystal structures can be constructed with the kits. Each kit is very heavy, consisting of dozens of colored, polymer spheres, which are connected to a polymer base by 7 in. long steel rods. The models do not clearly demonstrate where one unit cell ends and another begins; the method seems more intended for the construction of individual unit cells, since the instructions for each model begin “To build a unit cell”, followed in many (but not all) cases by “To build more than a unit cell.” There are a number of patterns printed on 4 in. × 4 in. cards, but the cards themselves are labeled A, B, C, etc. A student or instructor must consult other reference materials to know which structure corresponds to a specific alphabetical label. Since the metal rods are held normal (perpendicular) to the base, the kits seem unable to represent monoclinic, rhombohedral, or triclinic lattices.

Figure 1. Models depicting the (A) primitive hexagonal, (B) facecentered orthorhombic, and (C) body-centered cubic crystal lattice.

Supporting Information. This system is designed so as to more clearly highlight the extended nature of the lattices. Large regions in the center of each face of the Bravais parallelepiped (BP) are cut out, allowing the student to see through the unit cell. Each unit cell is constructed from a single piece of legal size poster board (or ordinary printer paper), onto which both sides have been printed. The two-sided printing and the cutouts in each face of the unit cell highlight the 3-D effect. Multiple models of the unit cell can be placed next to each other to form the lattice, or glued together for a more rigid and permanent display. Video of these models can be viewed online at no cost at the popular YouTube Web site.15 Two versions of each pattern are providedone that is labeled with the name of the lattice, and one without such labeling. Classroom Implementation

Because the cells are constructed from paper, even the most resource-challenged institution of education can build multiple sets of structures. The models are large enough to be seen clearly within a class or laboratory group, but small enough to be easily stored when not in use. They make excellent mobiles (a la Alexander Calder) for a more permanent display. In the classroom, the models are used as an adjunct to a traditional chalkboard/white board or PowerPoint presentation. The size of individual models makes it convenient for the instructor to project their images onto a screen, making the model easily visible to any size audience. A typical presentation begins with the basic definitions of a lattice, and the relevant lattice parameters a, b, c, α, β, and γ. These are then demonstrated using the model of the generalized Bravais lattice (Patterns 1, 2). The next step is to demonstrate the consequences of restricting a = b = c, and α = β = γ = 90°the simple cubic lattice. One author (L.T.S.) has found it effective to proceed in the manner of ref 5, to begin with the simple cubic lattice, and then gradually increase the complexity of the structures, either by adding additional lattice points (for example, body-centering) or by relaxing the constraints of the lattice parameters. Each step of the way, the instructor can proceed from identification of the features of a single unit cell to the features of the extended structure, merely by properly arranging multiple copies of the cell. The model design helps clarify the techniques used to count the number of atoms per unit cell; bringing together 8 unit cells shows that each atom at a vertex contributes 1/8 of an atom to the unit. Models of Bravais lattices within the same crystal system are shown with lattice points of the same color; for example, the 4 orthorhombic latticesprimitive, bodycentered, end-centered, and face-centeredall have red lattice points.

Model Construction and Description

Previous unit cell models have not been as efficacious for chemistry pedagogy as one might have hoped, since they are often too complicated to construct, too cumbersome, too expensive, too heavy, too limited in structures that can be represented, too restricted to only a single unit cell, or all of the above. Therefore, a new system of unit cell models has been developed, which requires only paper (or poster board), glue, scissors, a personal computer, and an inkjet printer [Figure 1]. Complete instructions for construction, along with doublesided patterns for each of the Bravais lattices, are found in the B

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Nine of the models “break open” along a mirror plane, thereby demonstrating one of the important symmetry elements of the lattice, as an extension of an earlier paper in the Journal.16 Equivalent demonstration of translation is achieved by employing multiple copies of the same cell. This system has already been extended to even more complex crystal structures; video of such models can also be seen, gratis, online.16 Patterns for the cesium chloride, diamond, fluorite, hexagonal close packed, sodium chloride, and zincblende structures17 are included in the Supporting Information. While the models can be used productively within a traditional lecture class format, or at home for self-study, they are most effectively used collaboratively. The class can be divided into groups of three to five students, each group of which is assigned a specific Bravais lattice (or more complex crystal structure). Each member of the group will fulfill a different role in the construction of individual unit cells, their assembly into an extended lattice, and the discovery of the lattice properties. Construction of the models involves numerous, disparate skills, each of which corresponds to a particular “learning style”18their colorfulness appeals to the visual-spatial learner; the need to cut, fold, manipulate, and glue the models appeals to the bodily kinesthetic learner; the aural instructions and lecture given by the instructor appeal to the linguistic learner; and the final arrangement and combination of individual unit cells into an extended lattice appeals to the logical-mathematical learner. Allowing students to form groups on their own, whenever possible, permits the student to gravitate toward the learning style with which he feels most comfortable. Working together as a team, the students both teach each other and teach themselves. The students collaboratively answer a series of questions posed by the instructor, each related to their particular lattice. Once completed, students are encouraged to examine the lattices constructed by the other groups, and to compare and contrast the structures. After the students have constructed at least one of the lattices, and examined the other constructed lattices, the instructor can begin class discussion as to the value of the exerciseto what extent the models they have built “correspond to reality”, and to what extent the model might lead to misunderstanding.19 As a final component to the exercise, the students can investigate other models of the solid state.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Phone: (410) 704-2720. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Kettle, S. F. A.; Norrby, L. J. Really, your lattices are all primitive, Mr. Bravais! J. Chem. Educ. 1993, 70 (12), 959. (2) Ladd, M. F. C. The Language of Lattices and Cells. J. Chem. Educ. 1997, 74 (4), 461. (3) Kettle, S. F. A.; Norrby, L. J. The Wigner-Seitz Unit Cell. J. Chem. Educ. 1994, 71 (12), 1003. (4) Schomaker, V.; Lingafelter, E. C. Crystal systems. J. Chem. Educ. 1985, 62 (3), 219. (5) Galasso, F. The importance of understanding structure. J. Chem. Educ. 1993, 70 (4), 287. (6) Boldyreva, E. V. Solid state chemistry: Taught as a comprehensive university course for chemistry students. J. Chem. Educ. 1993, 70 (7), 551. (7) Hardgrove, G. L., Jr. Teaching Space Group Symmetry through Problems. J. Chem. Educ. 1997, 74 (7), 797. (8) Kettle, S. F. A.; Norrby, L. J. The Brillouin zone  An interface between spectroscopy and crystallography. J. Chem. Educ. 1990, 67 (12), 1022. (9) Meyer, M.; Glaus, S.; Calzaferri, G. Introduction to Basic Terms of Band Structures. J. Chem. Educ. 2003, 80 (10), 1221. (10) Westbrook, J. H.; DeVries, R. C. A new type of crystal model. J. Chem. Educ. 1957, 34 (5), 220. (11) Cady, S. G. Use of Pom Pons To Illustrate Cubic Crystal Structures. J. Chem. Educ. 1997, 74 (7), 794. (12) Bindel, T. H. Crystal Models Made from Clear Plastic Boxes and Their Use in Determining Avogadro’s Number. J. Chem. Educ. 2002, 79 (4), 468. (13) Birk, J. P.; Yezierski, E. J.; Laing, M. Paper-and-Glue Unit Cell Models. J. Chem. Educ. 2003, 80 (2), 157. (14) Sunderland, D. B. Studying Crystal Structures through the Use of Solid-State Model Kits. J. Chem. Educ. 2014, 91 (3), 432−436. (15) (a) Sein, L. T., Jr. “Symmetry Episode 114 Part 1” https://www. youtube.com/watch?v=KoOQbUQdEm0&list= UU1dvpI7PI9jTCLVCPAotUA (accessed Jun 2015). (b) Sein, L. T., Jr. “Symmetry Episode 114 Part 2” https://www.youtube.com/ watch?v=idjFtW9ZbqI&list=UU1dvpI7PI9jTCLVCPAotUA (accessed Jun 2015). (c) Sein, L. T. Jr. “Symmetry Episode 114 Part 3” https:// www.youtube.com/watch?v=uscEd_VZWZI&list= UU1dvpI7PI9jTCLVCPAotUA (accessed Jun 2015). (d) Sein, L. T., Jr. “Symmetry Episode 14 Part 4” https://www.youtube.com/ watch?v=b25CEUVEPao&list=UU1dvpI7PI9jTCLVCPAotUA (accessed Jun 2015). (e) Sein, L. T., Jr. “Symmetry Episode 114 Part 5” https://www.youtube.com/watch?v=l-JTnj72rr8&list= UU1dvpI7PI9jTCLVCPAotUA (accessed Jun 2015). (16) Sein, L. T., Jr. Dynamic paper constructions for easier visualizations of molecular symmetry. J. Chem. Educ. 2010, 87 (8), 827−828. (17) Sein, L. T., Jr. “Symmetry Episode 115 Part 1” https://www. youtube.com/watch?v=VRRepv8oecg&list=UU1dvpI7PI9jTCLVCPAotUA (accessed Jun 2015). (18) Gardner, H. Frames of Mind: The Theory of Multiple Intelligences. Basic: New York, 1983. (19) Justi, R.; Gilbert, J. K. Models and Modeling in Chemical Education. In Chemical Education: Towards Research-Based Practice; Gilbert, J. K., de Jong, O., Justi, R., Treagust, D. F., Van Driel, J. H., Eds.; Kluwer Academic Press: Dordrecht, 2002; pp 47−68.

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CONCLUSIONS Students and instructors can quickly and easily construct models of the unit cells of the 14 Bravais lattices, and still more complex crystal structures. These cells can then be glued together to form the relevant extended three-dimensional lattice. The method is colorful, easy, and fun, helping the student to better understand solid state structure. Video of completed models can be viewed for free online at the popular YouTube Web site.

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Communication

ASSOCIATED CONTENT

S Supporting Information *

Instructor/Student handout with 62 patterns, to construct 41 different models. The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.5b00003. C

DOI: 10.1021/acs.jchemed.5b00003 J. Chem. Educ. XXXX, XXX, XXX−XXX