F 020
ð
Þ ¼ 4f 0:2766
ð
Þ¼4
X 4
i¼1
a i e
Àb i 0:27662
ð
Þ
2 þ c
¼ 4ða 1 e
Àb 1 0:2766
ð
Þ
2 þ a 2 e
Àb 2 0:2766
ð
Þ
2 þ a 3 e
Àb 3 0:2766
ð
Þ
2 þ a 4 e
Àb 4 0:2766
ð
Þ
2 þ cÞ
¼ 4ð13:338 e
À3:5828 0:2766
ð
Þ
2 þ 7:1676 e
À0:247 0:2766
ð
Þ
2 þ 5:6158 e
À11:3966 0:2766
ð
Þ
2
þ 1:6735 e
À64:8126 0:2766
ð
Þ
2 þ 1:191Þ
¼ 4 10:14018 þ 7:033423 þ 2:34822 þ :0117519 þ 1:191
ð
Þ
¼ 4 Â 20:72457 ¼ 82:89
Calculations for other reflections can be made similarly. They are shown below
in tabulated form for the initial 13 hkl values.
Structure factor calculation for Cu (Copper)
hkl
h
2 þ k
2 þ l
2
ð
Þ
1=2
2a
F Cu hkl
ð Þ ¼ 4f Cu ; ðelectronÞ
000
0
116
001
0.1383
0
011
0.1956
0
020
0.2766
82.89
111
0.2396
88.31
120
0.3093
0
121
0.3388
0
220
0.3912
67.13
221
0.4149
0
300
0.4149
0
301
0.4374
0
311
0.4587
59.13
320
0.4987
0
321
0.5175
0
Table 8.2 shows the Miller indices of the planes and the corresponding values of
F
j j
2 for various cubic structures.
Example 8 Cu 3 Au is cubic with one Cu 3 Au molecule per unit cell. In an ordered
form, the atomic positions are: Au: (0, 0, 0), and Cu: (1/2, 1/2, 0), (0, 1/2, 1/2),
(1/2, 0, 1/2). In the disordered form, the same positions are occupied at random.
Determine the simplified structure factors and the corresponding intensities.
Solution Given: Cu 3 Au cubic unit cell, F hkl
ð Þ ¼ ?; I ¼ ?
An ordered form Cu 3 Au unit cell is shown in Fig. 8.6. The fractional coordinates
are: Au: (0, 0, 0), Cu: (1/2, 1/2, 0), (0, 1/2, 1/2), (1/2, 0, 1/2). Substituting these
values in Eq. 8.1, we obtain
8.2 Determination of Phase Angle, Amplitude …
311