Identification of Close-Packed Structures by X-ray Diffraction

When a material crystallizes into a number of different close-packed structures all of which have identical layer spacings and different only in the manner of stacking the layers, crystals of the different modifications look alike and cannot be identified by their external morphology. In order to identify such polytype modifications, it is necessary to determine the number of layers in the hexagonal unit cell and the lattice type of the crystal. This can be conveniently achieved by recording reciprocal-lattice rows parallel to c * on single-crystal X-ray diffraction photographs. Since the different polytypes of the same material have identical a and b parameters of the direct lattice, the a * b * reciprocal lattice net is also the same. The reciprocal lattice of these modifications differ only along the c * axis which is perpendicular to the layers. For each reciprocal-lattice row parallel to c * there are others with the same value of the cylindrical coordinate $\xi$ . For example, the rows 10.l, 01.l, $\overline{1}$ 1.l, $\overline{1}$ 0.l, 0 $\overline{1}$ .l and 1 $\overline{1}$ .l all have $\xi$ = $\arrowvert$ a * $\arrowvert$ . Due to symmetry, it is sufficient to record any one of them on X-ray diffraction photographs. The number of layers, n , in the hexagonal unit cell can be found by determining the c parameter from c -axis rotation or oscillation photographs and dividing this by the known layer-spacing h for that compound (n = c/h ). The density of reciprocal points along rows parallel to c * depends on the periodicity along the c axis. The larger the identity period along c , the more closely spaced are the reciprocal-lattice points along c *. In case of long-period polytypes the number of layers in the hexagonal unit cell can be determined by using a simple alternative method suggested by Krishna and Verma²⁰. This requires the counting of the number of spacings after which the sequence of relative intensities begins to repeat along the 10.l row of spots on an oscillation or Weissenberg photograph. If the structure contains one-dimensional disorder due to a random-distribution of stacking faults, this effectively causes the c lattice parameter to become infinite (c * $\rightarrow$ 0) and results in the production of characteristic streaks along reciprocal lattice rows parallel to c *. It is therefore difficult to distinguish by X-ray diffraction between structures of very large unresolvable periodicities and those with random disorder. Lattice resolution in the electron-microscope has been used in recent years to identify such structures²¹. Figure 13 depicts the 10.l rows of some close-packed structures of SiC as recorded on c -axis oscillation photographs. When the structure has a hexagonal lattice, the positions of spots are symmetrical about the zero layer line on the c -axis oscillation photograph as seen in Fig. 13 (a) and (b) for the 6H and 36H SiC structures. However, the intensities of the reflections on the two sides of the zero layer line are the same for the 6H structure but not for the 36H . This is because the 6H structure belongs to the hexagonal space group P 6₃mc whereas the 36H structure belongs to the trigonal space group P 3m 1². The apparent mirror symmetry perpendicular to the c -axis in Fig. 13(a) results from the combination of the 6₃ screw axis with the centre of symmetry introduced by X-ray diffraction²². For a structure with a rhombohedral lattice, the positions of X-ray diffraction spots are not symmetrical about the zero layer line because the hexagonal unit cell is non-primitive causing the reflections hkl to be absent when $-h+k+l\neq3n$ ( $\pm$ n = 0, 1, 2,...). For the 10.l row this means that the permitted reflections above the zero layer line are 10.1, 10.4, 10.7 etc. and below the zero-layer line 10. $\overline{2}$ , 10. $\overline{5}$ , 10. $\overline{8}$ etc. The zero layer line will therefore divide the distance between the nearest spots on either side (namely 10.1 and 10. $\overline{2}$ ) approximately in the ratio 1:2. This enables a quick identification of a rhombohedral lattice. Thus the lattice type corresponding to Fig. 13(c) is rhombohedral and the polytype is designated as 90R and belongs to the space group R3m. Figure 13(d) depicts the 10.l row of a disordered 2H SiC structure. The diffuse streak connecting the strong 2H reflections is due to the presence of a random distribution of stacking faults in the 2H structure.

**Figure 13:** The 10.l rows of some close-packed structures of SiC as recorded on c-axis oscillation photographs (a) 6H (b) 36H (c) 90R (d) Dis. 2H .
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