Space groups of the diamond polytypes
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Space groups and atomic coordinates for the 4H, 6H, 8H, 10H, 15R, and 21R polytypes of diamond are presented. The systematic method used to determine the highest symmetry space group for diamond polytypes is described.
I. INTRODUCTION
Diamond can occur as polytypes that are described by systematic stacking sequences of carbon double layers, isomorphic with the well-known silicon carbide polytype system.1'2 As all sites within the diamond double layers are occupied by carbon, as opposed to silicon or carbon in silicon carbide, the symmetries of the diamond polytypes are higher than those of silicon carbide. This principle has significant implications with regard to theoretical calculations of physical properties based on symmetry. This paper presents a simple systematic method for determination of the highest symmetry space group for any of these layered diamond polytypes. Space groups for several observed and theoretical polytypes are presented, and the impact on previous calculations performed using the space groups of the silicon carbide analogs is discussed.
Two helpful features of this restriction are that all atoms lie in (11.0) planes, and if centers of symmetry exist, they too must lie in these planes. These features allow for a convenient display of the structures by plotting the atom locations on a (11.0) plane. If a (11.0) plot is prepared that is several unit cells wide and two unit cells high, the symmetry elements may be easily located by visual inspection (Fig. 3). As will be shown,
II. DIAMOND POLYTYPES AS LAYERED STRUCTURES
The polytypes of diamond are close-packed arrays of carbon double layers. They are described by the stacking sequences that indicate the position of these layers in a close-packed structure; for example ABC = cubic close packing, ABAB = hexagonal close packing, and ABCACB = mixed hexagonal and cubic. The periodicity of any such polytype can always be described by a hexagonal unit cell. The hexagonal representation of the familiar cubic diamond (ABC) is shown in Fig. 1. In the hexagonal representation of closepacked structures, all atoms are restricted to specific high-symmetry sites within the unit cell, specifically the (0, 0, z), (1/3, 2/3, z), and (2/3, 1/3, z) sites, corresponding to the A, B, and C stacking positions in Fig. 2. While assignment of A, B, and C to specific sites is arbitrary, we will adhere to the convention of using A for (0, 0, z), B for (1/3, 2/3, z), and C for (2/3, 1/3, z) when describing stacking sequences. ^Present address: Department of Materials Science, The Pennsylvania State University, University Park, Pennsylvania 16802. b 'Present address: Department of Geosciences, The Pennsylvania State University, University Park, Pennsylvania 16802. J. Mater. Res., Vol. 8, No. 11, Nov 1993
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FIG. 1. The hexagonal representation of cubic diamond. Atoms with the same shading are in the same carbon double layer and occupy the same x-y coordinate site, e.g., position A, B, or C. © 1993 Materials Research
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