Symmetry Relationships Between Crystal Structures: Application of Crystallographic Group Theory in Crystal Chemistry
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The relation between NiAs and MnP
The phase transition of the NiAs type to the MnP type has been the subject of thorough studies with a number of compounds (e.g. VS, MnAs) [129, 130]. The symmetry reduction involves two steps (Fig. 11.9). In the first step the hexagonal symmetry is lost; for this a slight distortion of the lattice would
be sufficient. The orthorhombic subgroup has a C-centred cell. Due to the centring, the cell is translationengleiche, although its size is twice as big. The centring is removed in the second step, half of the translations being lost; therefore, it is a klassengleiche reduction of index 2.
The images in Fig. 11.9 show what symmetry elements are eliminated with the two steps of symmetry reduction. Among others, half of the inversion centres are being lost. Here we have to watch out: the eliminated inversion centres of the space group Cmcm (C2/m2/c21/m) are those of the Wyckoff positions 4a (0, 0, 0) and 4b (1/2 , 0, 0), while those of the Wyckoff position 8d (1/4 ,1/4 ,0) are retained. Since the subgroup Pmcn (P21/m21/c21/n) should have its origin on a point of inversion, an origin shift is required. The shift of −1/4 ,−1/4 ,0 entails an addition of 1/4 ,1/4 ,0 to the coordinates.
After addition of 1/4 ,1/4 ,0 to the coordinates listed in Fig. 11.9 for the space group Cmcm, one obtains ideal values for an undistorted structure in the space group Pmcn. However, due to the missing distortion, the symmetry would still be Cmcm. The space group Pmcn is attained only after the atoms have been shifted away from the ideal positions. The deviations concern mainly the y coordinate of the Mn atom (0.196 instead of 1/4 )andthe z coordinate of the P atom (0.188 instead of 1/4 ). These are significant deviations, but they are small enough to consider MnP as being a distorted variant of the NiAs type.
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