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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.