T ¼ n 1 a
ð Þ
2 þ n 2 b
ð Þ
2
h
i 1=2
¼ 5 Â 3
ð
Þ
2 þ 7 Â 2
ð
Þ
2
h
i 1=2
¼ 225 þ 196
½
Š
1=2
¼ 421
ð
Þ
1=2
¼ 20:52 ˚
A
Direction of the translation vector is [57].
Example 3 In a plane hexagonal lattice, the translation vector is given by
~ t ¼ 2~ a þ 3 ~ b, where a = b = 3Å and c = 120°. Determine the magnitude and
direction of the resultant translation vector.
Solution: Given: a = b = 3Å, c = 120°, n 1 = 2, and n 2 = 3. We know that the
magnitude of the resultant 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
2 Â 3
ð
Þ
2 þ 3 Â 3
ð
Þ
2 þ 2 2 Â 3
ð
Þ 3 Â 3
ð
Þcos 120
h
i 1=2
¼
36 þ 81 þ 2 Â 54 Â
À1
2
! 1=2
¼ 7:94 ˚
A
Direction of the translation vector is [23].
Example 4 In a square lattice, the translation vector is given by ~ t ¼ 3~ a þ 4 ~ b,
where a = b = 3Å. Determine the magnitude and direction of the resultant translation vector.
Fig. 1.7 Direction of
translation vector
1.1 Translation Vectors in Plane and Space Lattices
5