Example 15 The diffraction data of a cubic crystal of an unknown element show peaks
at 2h angles 44.46°, 51.64°, 75.78° and 93.22° when the wavelength of X-ray used is
1.543 Å. Determine the crystal structure, lattice parameter and identify the element.
Solution: Given: Crystal is cubic, 2h angles: 44.46°, 51.64°, 75.78° and 93.22°,
k = 1.543 Å, crystal structure = ?, a = ?, element = ?
Preparing the datasheet in tabular form, we have
2h
h
sin h
sin
2 h
d = k/2 sin h (Å)
44.46
22.23
0.3783
0.1431
2.04
51.64
25.82
0.4355
0.1897
1.77
75.78
37.89
0.6141
0.3772
1.26
93.22
46.61
0.7267
0.5281
1.06
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 face-centered cubic
(fcc) 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
22.23
0.1431
3
ffiffi ffi
3
p
3.53
2
25.83
0.1897
0.0477
4
ffiffi ffi
4
p
3.54
3
37.89
0.3772
8
ffiffi ffi
8
p
3.56
4
46.61
0.5281
11
ffiffiffiffiffi
11
p
3.52
The average value of “a” from the Table is found to be 3.54 Å. The above data
indicates that the element is Nickel.
Example 16 The diffraction data of a cubic crystal of an element show peaks at 2h
angles 38.60°, 55.71°, 69.70°, 82.55°, 95.00° and 107.67° when the wavelength of
X-ray used is 1.543 Å. Determine the crystal structure, lattice parameter and
identify the element.
Solution: Given: Crystal is cubic, 2h angles: 38.60°, 55.71°, 69.70°, 82.55°, 95.00°
and 107.67°, crystal structure = ?, a = ?, element = ?
Preparing the datasheet in tabular form, we have
2h
h
sin h
sin
2 h
d = k/2 sin h (Å)
38.60
19.30
0.3305
0.1092
2.334
55.71
27.26
0.4580
0.2097
1.684
69.70
34.85
0.5714
0.3265
1.350
82.55
41.28
0.6597
0.4352
1.169
95.00
47.50
0.7373
0.5436
1.046
107.67
53.84
0.8073
0.6518
0.955
354
9 Determination of Crystal Structure Parameters
at 2h angles 44.46°, 51.64°, 75.78° and 93.22° when the wavelength of X-ray used is
1.543 Å. Determine the crystal structure, lattice parameter and identify the element.
Solution: Given: Crystal is cubic, 2h angles: 44.46°, 51.64°, 75.78° and 93.22°,
k = 1.543 Å, crystal structure = ?, a = ?, element = ?
Preparing the datasheet in tabular form, we have
2h
h
sin h
sin
2 h
d = k/2 sin h (Å)
44.46
22.23
0.3783
0.1431
2.04
51.64
25.82
0.4355
0.1897
1.77
75.78
37.89
0.6141
0.3772
1.26
93.22
46.61
0.7267
0.5281
1.06
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 face-centered cubic
(fcc) 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
22.23
0.1431
3
ffiffi ffi
3
p
3.53
2
25.83
0.1897
0.0477
4
ffiffi ffi
4
p
3.54
3
37.89
0.3772
8
ffiffi ffi
8
p
3.56
4
46.61
0.5281
11
ffiffiffiffiffi
11
p
3.52
The average value of “a” from the Table is found to be 3.54 Å. The above data
indicates that the element is Nickel.
Example 16 The diffraction data of a cubic crystal of an element show peaks at 2h
angles 38.60°, 55.71°, 69.70°, 82.55°, 95.00° and 107.67° when the wavelength of
X-ray used is 1.543 Å. Determine the crystal structure, lattice parameter and
identify the element.
Solution: Given: Crystal is cubic, 2h angles: 38.60°, 55.71°, 69.70°, 82.55°, 95.00°
and 107.67°, crystal structure = ?, a = ?, element = ?
Preparing the datasheet in tabular form, we have
2h
h
sin h
sin
2 h
d = k/2 sin h (Å)
38.60
19.30
0.3305
0.1092
2.334
55.71
27.26
0.4580
0.2097
1.684
69.70
34.85
0.5714
0.3265
1.350
82.55
41.28
0.6597
0.4352
1.169
95.00
47.50
0.7373
0.5436
1.046
107.67
53.84
0.8073
0.6518
0.955
354
9 Determination of Crystal Structure Parameters