Anatase is one of the three mineral forms of titanium dioxide (TiO2 ). Its space group is I41 /amd (No. 141). Its lattice parameters are a = b = 3.78 Å, and c = 9.54 Å. Ti atoms are located at positions with Wyckoff symbol 4b, and O atoms are located at positions with Wyckoff symbol 8e with z = 0.17. Identify the Bravais lattice type of this crystal structure. How many lattice point(s) are there in one conventional unit cell? How many Ti atoms and how many O atoms are there in one conventional unit cell? Write down the relative coordinates for all O atoms belonging to one conventional unit cell. In this crystal, how many symmetry-equivalent crystallographic directions belong to the <112> family? Write down all of them. Calculate the length of a crystallographic vector [112] in this crystal.

 

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Wyckoff Positions of Group 141 (14/amd) [origin choice 2] Multiplicity Wyckoff Site letter symmetry 116 f .2. 8 e 8 d 8 2mm. .2/m. .2/m. 4 b -4m2 4 a -4m2 0 Coordinates (0,0,0)+(1/2,1/2,1/2) + (x,0,0) (-x+1/2,0,1/2) (1/4,x+3/4, 1/4) (1/4,-x+1/4,3/4) (-x,0,0) (x+1/2,0,1/2) (3/4,-x+1/4,3/4) (3/4,x+3/4, 1/4) (0,1/4,z) (0,3/4,z+1/4) (1/2, 1/4,-z+1/2) (1/2,3/4,-z+1/4) (0,0,1/2) (1/2,0,0) (1/4,3/4,3/4) (1/4, 1/4, 1/4) (0,0,0) (1/2,0,1/2) (1/4,3/4, 1/4) (1/4,1/4,3/4) (0,1/4,3/8) (0,3/4,5/8) (0,3/4,1/8) (1/2,3/4,3/8) Wyckoff Positions of Group 141 (14/amd) [origin choice 2] Multiplicity Wyckoff Site letter symmetry 116 f .2. 8 e 8 d 8 2mm. .2/m. .2/m. 4 b -4m2 4 a -4m2 0 Coordinates (0,0,0)+(1/2,1/2,1/2) + (x,0,0) (-x+1/2,0,1/2) (1/4,x+3/4, 1/4) (1/4,-x+1/4,3/4) (-x,0,0) (x+1/2,0,1/2) (3/4,-x+1/4,3/4) (3/4,x+3/4, 1/4) (0,1/4,z) (0,3/4,z+1/4) (1/2, 1/4,-z+1/2) (1/2,3/4,-z+1/4) (0,0,1/2) (1/2,0,0) (1/4,3/4,3/4) (1/4, 1/4, 1/4) (0,0,0) (1/2,0,1/2) (1/4,3/4, 1/4) (1/4,1/4,3/4) (0,1/4,3/8) (0,3/4,5/8) (0,3/4,1/8) (1/2,3/4,3/8)