Making use of Eq. 3.3 for c = 90°, we can obtain the distance between the first
set of fractional coordinates as
L 1 ¼ 0:200 À 0:100
ð
Þ
2 Â2
2
þ 0:050 À 0:250
ð
Þ
2 Â3
2
h
i 1=2
¼ 0:100
ð
Þ
2 Â4 þ À0:200
ð
Þ
2 Â9
h
i 1=2
¼ 0:010
ð
ÞÂ4 þ ð0:040Þ Â 9
½
Š
1=2 ¼ 0:632 ˚
A
Similarly, for the second set of fractional coordinates, we have
L 2 ¼ 0:210 À 0:100
ð
Þ
2 Â2
2
þ 0:050 þ 0:250
ð
Þ
2 Â3
2
h
i 1=2
¼ 0:110
ð
Þ
2 Â4 þ 0:300
ð
Þ
2 Â9
h
i 1=2
¼ 0:0121
ð
ÞÂ4 þ ð0:090Þ Â 9
½
Š
1=2 ¼ 0:926 ˚
A
Example 4 A unit cell has the following dimensions: a = 6 Å, b = 7 Å, c = 8 Å,
a = 90°, b = 115° and c = 90°. Calculate the distance between the points with
fractional coordinates: (i) 0.200, 0.150, 0.333 and 0.300, 0.050, 0.123, and
(ii) 0.200, 0.150, 0.333 and 0.300, 0.050, −0.123.
Solution: Given: a = 6 Å, b = 7 Å, c = 8 Å, a = 90°, b = 115° and c = 90°.
Fractional coordinates: (i) 0.200, 0.150, 0.333 and 0.300, 0.050, 0.123, and
(ii) 0.200, 0.150, 0.333 and 0.300, 0.050, −0.123.
Making use of Eq. 3.8, we can obtain the distance between the first set of
fractional coordinates as
L 1 ¼
0:200 À 0:300
ð
Þ
2 Â 6
2 þ 0:150 À 0:050
ð
Þ
2 Â 7
2 þ 0:333 À 0:123
ð
Þ
2 Â8
2 þ
0 þ 0 þ 2 Â 8 Â 6 0:333 À 0:123
ð
Þ0:200 À 0:300
ð
Þ cos 115
"
# 1=2
¼ À0:100
ð
Þ
2 Â36 þ 0:100
ð
Þ
2 Â49 þ 0:210
ð
Þ
2 Â64 þ 96 0:210
ð
Þ À0:100
ð
ÞðÀ0:423Þ
h
i 1=2
¼ 0:010 Â 36 þ 0:010 Â 49 þ 0:044 Â 64 þ 96 0:210
ð
Þ À0:100
ð
ÞðÀ0:423Þ
½
Š
1=2
¼ 0:360 þ 0:490 þ 2:820 þ 0:853
½
Š
1=2
¼ 4:523
½
Š
1=2 ¼ 2:13 ˚
A
3.2 Distance Between Two Lattice Points (Oblique System)
107