Materials Science and Engineering: 2nd year, 2011: CRYSTALLOGRAPHY Classwork 8: Systematic absences and structure factor 1. Why are there no reflections at N = 7, 15, 23 for a cubic primitive crystal? Answers 1. Because is never equal to 7, 15 or 23. 2. The X-ray powder photograph, taken with CuK radiation ( = 1.5418 Å), of the sodium tungsten bronze Na0.8WO3, which is cubic, has lines at the following Bragg angles: 11.60°, 15.52°, 20.38°, 23.71°, 26.71°, 29.50°, 34.65°, 37.09°, 39.49°, 41.81°, 44.13°, 46.45°, 48.78°, 53.52°, 55.98°, 58.52, 61.19° Index these lines, determine the lattice type, and evaluate the unit-cell dimension ɑ. Answers 2. 4 sin 4 sin θ ∗ 1 0.068 11.6 0.068 1 15.52 0.1205 1.77 20.38 0.2041 3 23.71 0.2721 4 26.71 0.3399 5 29.50 0.408 6 34.64 0.5495 8.08 So we can say that it is a primitive lattice and → shown above). Always calculate with the biggest angle. 3.83Å. N can’t be equal to 7 (as 3. Plutonium sulphide, PuS, is cubic and the eight lines of the lowest Bragg angle on its X-ray powder photograph, taken with CuK radiation, have the following values of : 13.95°, 16.17°, 23.19°, 27.50°, 28.84°, 33.84°, 37.37°, 38.51° Index these lines , determine the lattice type and evaluate the unit cell dimension ɑ Answers 3. θ 4 sin 4 sin 1 0.09779 1 4 sin 2 4 sin 0.09779 2 3 0.09779 3 13.95 0.09779 16.17 0.1305 1.335 2.668 4 23.19 0.2609 2.66 5.3359 8 27.50 0.3587 3.668 7.336 11 … … … … … 38.51 0.6524 6.6711 13.34 20 → So we can see that it is a fcc lattice and 5.54Å. 4. The crystal structure of zincblende ZnS, is face centred cubic with one formula unit per lattice point. For a Zn atom placed at the origin of the unit cell and a S atom at ¼,¼,¼ write down and simplify the expression for the structure factor of zincblende. Evaluate the intensities of the (111), (200), and (220) reflections assuming that atomic scattering factors are proportional to atomic number. Answers 4. cos 2 With 111 sin 2 cos 2 sin 2 30, 16; 18496 200 sin 2 cos 2 0, 0, 0 ; , , 3136 220 33856 5. At low temperatures Cu3Au is cubic with one formula unit per unit cell. If Au is placed at the origin, then the Cu atoms will be situated at 0½½, ½0½, and ½½0. Write down an expression for the structure factor F hkl . What will be the form of the structure factor: i. when h,k,l are all even or all odd, ii. when h,k,l are mixed Index the nine lines of lowest Bragg angle in the powder pattern of ordered Cu3Au and indicate whether each line is strong or weak. At higher temperature Cu3Au becomes disordered and has a random distribution of Cu and Au atoms over the lattice points of a cubic F (fcc) lattice. The unit cell dimensions of ordered and disordered Cu3Au are very similar. How do their powder patterns differ? Answers 5. ∑ see that , 0,0,0 and If h,k,l are all even or all odd, If h,k,l are mixed even and odd, 0, , 3 , 0, 1 , , 0 so we 1 1 → → 100 , 110 , 111 , 200 , 210 , 211 , 220 , 300 , 310 in the powder, only the type with h,k,l all even or odd remains strong, the other goes to zero intensity.

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