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Jon L. Holmes University of Wisconsin–Madison Madison, WI 53706
Filling in the Hexagonal Close-Packed Unit Cell
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Robert C. Rittenhouse* Department of Chemistry, Walla Walla College, College Place, WA 99324; *
[email protected] Linda M. Soper Department of Mathematics, Kalispell Community College, Kalispell, MT 59901 Jeffrey L. Rittenhouse Walla Walla Valley Academy, College Place, WA 99324
The hexagonal close-packed (hcp) structure is one of the most common and important crystal structures adopted by metals and other atomic solids. Despite its prominence in nature, the hcp structure has not been treated at the same level of geometric rigor as the common cubic structures. The illustrations of the hcp unit cell that are used in textbooks at all levels are incomplete, in that they fail to include fractions of atomic spheres with centers lying outside of the unit cell. These fractions are necessary to properly determine the number of atoms enclosed in the unit cell. The hcp unit cell has been presented in some textbooks as a hexagonal prismatic structure and in others as a rhombic prism. If we adopt the common definition of the unit cell that calls for the smallest repeating geometrical unit, then the rhombic prism is the preferred choice. The two rhombic faces are contained in parallel close-packed layers separated by a single intervening layer. The four atoms whose centers define the vertices of the bottom rhombic face form two triangular holes (Figure 1). Because of the close proximity of these two holes, only one can be occupied by an atom of the second (middle) layer. However, this atom is not wholly contained in the unit cell, since it projects through two faces of the unit cell into two adjacent unit cells. The fractions of this middle layer atom that are sliced off by the unit cell boundaries are exactly matched by fractions of two other middle layer atoms with centers in two adjacent unit cells that project into the first unit cell in the direction of the unoccupied hole. The two portions of the middle layer atom sliced off by the unit cell boundaries were calculated using the solids-of-revolution method and were found to each account for 11.51% of the atomic sphere. Thus, the middle layer contributes exactly one atomic sphere to the unit cell in the form of 76.98% of the atom that has its center within the unit cell and 11.51% each from two other atoms with centers in adjacent unit cells. In the full online article we have shown the details of how the fractions of the atomic spheres enclosed by the unit cell may be calculated and have prepared animated 3D “cutaway” computer models (Figure 2) and templates for paper
www.JCE.DivCHED.org
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Figure 1. Top down view of hcp unit cell showing projection of middle layer atomic spheres (dashed circles) through the four rectangular faces of the unit cell. The solid circles representing the top and bottom layers coincide.
Figure 2. 3D cutaway computer model of the complete hcp unit cell.
and glue models to more accurately illustrate the hcp unit cell. Supplemental Material The full article, including calculations, 3D computer models, and templates, are available in this issue of JCE Online. W
Vol. 83 No. 1 January 2006
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Journal of Chemical Education
175
