Now, write various S values in tabular form and obtain corresponding h and d
values
Line
Arc length = S
h = S/4 (°)
sin h
d = k/2 sin h (Å)
1
56.8
14.2
0.245
3.14
2
94.4
23.6
0.40
1.92
3
112.0
28.0
0.469
1.64
From the first table, make use of h. The values of sin
2
h, common factor (c.f) are
found and added into other columns of the next table. The values of N for allowed
reflections give us that the unknown cubic form is the diamond cubic
(DC) structure. Finally, we get the value of lattice parameter “a.”
Line
h (°)
sin
2 h
Common factor (c.f)
N ¼
sin
2 h
c:f
ffiffiffi ffi
N
p
a ¼ d
ffiffiffi ffi
N
p
1
14.2
0.060
3
ffiffi ffi
3
p
5.44
2
23.6
0.160
0.020
8
ffiffi ffi
8
p
5.43
3
28.0
0.220
11
ffiffiffiffiffi
11
p
5.44
Example 8 The first three S-values obtained from the powder pattern of a given
unknown cubic form of material are 24.95, 40.9 and 48.05 mm, respectively. If the
radius of the camera is 57.3 mm and Molybdenum K a radiation of wavelength 0.71
Å is used. Determine the crystal structure of the cubic form and its lattice parameter.
Solution Given: S-values: S 1 = 24.95 mm, S 2 = 40.9 mm, S 3 = 48.05 mm,
R = 57.3 mm, k = 0.71 Å, crystal structure = ?, a = ?
Here, we use an alternative method to solve a similar problem as above. We
know that the angular and linear relationship in a powder pattern is
4hðdegrees) ¼ S(mm)
or h ¼
S
4
Therefore,
h 1 ¼
S 1
4
¼
24:95
4
¼ 6:2375
and sin
2
h 1 ¼ 0:0118
h 2 ¼
S 2
4
¼
40:9
4
¼ 10:225
and sin
2
h 2 ¼ 0:0315
h 3 ¼
S 3
4
¼
48:05
4
¼ 12:0125
and sin
2
h 3 ¼ 0:0433
We know that sin
2
h / h
2
þ k
2
þ l
2
À
Á
or sin
2
h / N, then the values of the ratio
of sin
2
h 1 : sin
2
h 2 : sin
2
h 3 are obtained as 3 : 8 : 11.
348
9 Determination of Crystal Structure Parameters
values
Line
Arc length = S
h = S/4 (°)
sin h
d = k/2 sin h (Å)
1
56.8
14.2
0.245
3.14
2
94.4
23.6
0.40
1.92
3
112.0
28.0
0.469
1.64
From the first table, make use of h. The values of sin
2
h, common factor (c.f) are
found and added into other columns of the next table. The values of N for allowed
reflections give us that the unknown cubic form is the diamond cubic
(DC) structure. Finally, we get the value of lattice parameter “a.”
Line
h (°)
sin
2 h
Common factor (c.f)
N ¼
sin
2 h
c:f
ffiffiffi ffi
N
p
a ¼ d
ffiffiffi ffi
N
p
1
14.2
0.060
3
ffiffi ffi
3
p
5.44
2
23.6
0.160
0.020
8
ffiffi ffi
8
p
5.43
3
28.0
0.220
11
ffiffiffiffiffi
11
p
5.44
Example 8 The first three S-values obtained from the powder pattern of a given
unknown cubic form of material are 24.95, 40.9 and 48.05 mm, respectively. If the
radius of the camera is 57.3 mm and Molybdenum K a radiation of wavelength 0.71
Å is used. Determine the crystal structure of the cubic form and its lattice parameter.
Solution Given: S-values: S 1 = 24.95 mm, S 2 = 40.9 mm, S 3 = 48.05 mm,
R = 57.3 mm, k = 0.71 Å, crystal structure = ?, a = ?
Here, we use an alternative method to solve a similar problem as above. We
know that the angular and linear relationship in a powder pattern is
4hðdegrees) ¼ S(mm)
or h ¼
S
4
Therefore,
h 1 ¼
S 1
4
¼
24:95
4
¼ 6:2375
and sin
2
h 1 ¼ 0:0118
h 2 ¼
S 2
4
¼
40:9
4
¼ 10:225
and sin
2
h 2 ¼ 0:0315
h 3 ¼
S 3
4
¼
48:05
4
¼ 12:0125
and sin
2
h 3 ¼ 0:0433
We know that sin
2
h / h
2
þ k
2
þ l
2
À
Á
or sin
2
h / N, then the values of the ratio
of sin
2
h 1 : sin
2
h 2 : sin
2
h 3 are obtained as 3 : 8 : 11.
348
9 Determination of Crystal Structure Parameters