Closest Packing of Spheres from Layers That Are Not Close-Packed Edward C. Lingafelter University of Washington, Seattle, WA 981 95
In their treatment of the solid state, most textbooks of general chemistry discuss the closest packing of spheres. The discussion usually begins with a description of hexagonal close-packed layers. Students read that each sphere in such a layer is in contact with six other spheres in the same layer (Fig. 11, and such layers can be stacked to give three-dimensional close-~ackedarraneements. Onlv two stacking sequences are normally mentioned: the hexagonal close-vacked and the cubic close-oacked (also called the face-centered cubic).
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hexagonal close-packed
cubic close-packed
...abab... ...abcabe...
Textbooks then usually state that, in both of these arrangements, each sphere is in contact with 12 other spheres: six in its own layer, three in the layer below, and three in the layer above. Some textbooks then go on to describe the two-dimensional, square-packed layer, pointing out that it is less efficient in its packing than the hexagonal-packed layer. In the square-packed layer, each sphere is in contact with only four other spheres (Fig. 2). Thus, the two-dimensional
Figure 1. The hexagonal-packed layer
Volume 68 Number 11 November 1991
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Figure 2. The square-packed layer. Lines emphasize the square packing.
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sauare-~ackedlaver r e ~ r e s e n t sa less efficient mode of packing than does the hexagonal-packed layer. It is then usuallv shown that if the sauare-packed layer is rxpundcd (by adout 15 %) a stack ofsuchexpanded layers gives a body-centered cubic arrangement, in which each sphere i s in contact with eight other spheres: four in the layer above and four i n the layer below. However, I found no textbook i n which it is shown that, without any expansion, these square-packed layers can be stacked to give the cubic closest-packed arrangement. This can be shown in several ways. The most direct way i s to refer to Figure 3. The five spheres marked with small circles can form one face of a face-centered cube. A second layer can be stacked so that each of its s ~ h e r e is s i n contact with four spheres in the first layer. ~ b & four , spheres of this second iayer can be "fitted" into the four indentations indicated by the x's,
Figure 3. The square-packed layer, with lines to emphasize the 'Yace" of a face-centered unit cube. and they are i n the correct positions to be the centers ofthe four "side faces" of the face-centered cube. The spheres in a third layer would then be directly above the spheres in the first layer. Then the five i n the.positions marked with the circles would complete the face-centered cubic unit cell. The square-packed layers are less efficiently packed than the hexagonal-packed layers. (The areas per sphere are 4 9 versus 2fi(?).) Nonetheless, there is no loss of packing efficiency because the distance between the layers i n the square-layer stack is less than the distance between layers i n the hexagonal layer stack ( a ( r ) versus 2 m ( r ) ) . Also, i t can be seen that each sphere in the stack of squarepacked layers is in contact with 12 other spheres: four in its own laser, four in the layer above, and four i n the layer below
International Newsletter of Chemical Education The International Newsletter of Chemical Education for June 1991has just arrived at the office of the United States representativeto the IUPAC Committe on the TeaehingofChemistry.Theissue contains articles on the followingtopics: "Modes of Traditional Learning: Can They Be Applied to the Learning of Chemistry?'; "Role Playing far Undergraduate Analysts"; "Teaching Science through Crime Lah Investigations"; "The Wits-Phoenix Project: Towards a New South African Science Education"; and "ChemicalEducation for Women: Why Do We Waste This Resource?".Those who are not on the international mailing list but who would like to receive a copy should send a self-addressed,5-x 7-in. envelope containing $0.52 postage to: Newsletter, c/o J. J. Lagowski, Department of Chemistry, University of Texas at Austin, Austin, TX 78712.
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