where ~ P ¼ n 1 ~ a ~
Q ¼ n 2 ~ b and ~ S ¼ n 3 ~ c and a; b; c are the angles between PQ,
QS and SP, respectively.
Solved Examples
Example 1 In a plane oblique lattice with a = 3Å, b = 2Å and c ¼ 75
, a translation vector is given by ~ t ¼ 4~ a þ 5 ~ b (Fig. 1.6). Determine its magnitude and
direction.
Solution: Given: a = 3Å, b = 2Å, c ¼ 75
, n 1 = 4, and n 2 = 5. We know that the
magnitude of translation vector ~ T is given by.
T ¼ n 1 a
ð Þ
2 þ n 2 b
ð Þ
2 þ 2 n 1 a
ð Þ n 2 b
ð Þcos c
h
i 1=2
¼ ð4 Â 3Þ
2 þ ð5 Â 2Þ
2 þ 2ð4 Â 3Þð5 Â 2Þ cos c
h
i 1=2
¼ ½ð144 þ 100 þ 240  0:2588ފ
1=2
¼ ð306:1165Þ
1=2
¼ 17:496 ˚
A
Direction of the translation vector is [n 1 n 2 ] [45].
Example 2 In a plane rectangular lattice with a = 3Å, b = 2Å, a translation vector
is given by ~ t ¼ 5~ a þ 7 ~ b (Fig. 1.7). Determine its magnitude and direction.
Solution: Given: a = 3Å, b = 2Å, n 1 = 5, and n 2 = 7. We know that for a plane
rectangular lattice, c ¼ 90
and cos90° = 0, therefore the magnitude of translation
vector ~ T is given by
Fig. 1.6 Direction of
translation vector
4
1 Unit Cell Composition