This shows that the reciprocal lattice is a bcc lattice. The reciprocal lattice
vectors are shown in Fig. 2.19. Further, in terms of the Miller indices (hkl), the
general form of the reciprocal lattice vector is written as:
GðhklÞ ¼ ha
Ã
þ kb
Ã
þ 1c
Ã
¼
2p
a
h ^ i þ ^ j À ^ k
À
Á þ k À ^ i þ ^ j þ ^ k
À
Á þ l ^ i À ^ j þ ^ k
À
Á
Â
Ã
¼
2p
a
h À k þ l
ð
Þ ^ i þ h þ k À l
ð
Þ ^ j þ Àh þ k þ l
ð
Þ ^ k
Â
Ã
However, there are eight vectors (see W–S unit cell for direct bcc lattice,
Fig. 2.3) of equal magnitude, joining the origin to the nearest reciprocal lattice
points at corners. They are given as:
2p
a
Æ ^ i Æ ^ j Æ ^ k
À
Á
where the choices of sign are independent.
Now, we can construct the first B-Z by drawing planes normal to each of the 8
reciprocal lattice vectors at their midpoints. Thus the bounding {111} planes are at:
p
a
Æ ^ i Æ ^ j Æ ^ k
À
Á
ðiÞ
In this case, there are 6 other vectors of equal magnitude corresponding to the
next nearest neighbor atoms lying at body centers of the neighboring bcc lattice.
They are given as:
2p
a
Æ2 ^ i
À
Á ;
2p
a
Æ2 ^ j
À
Á ;
2p
a
Æ2 ^ k
À
Á
Again, drawing planes normal to each of the 6 reciprocal lattice vectors at their
midpoints, we obtain the six bounding {200} planes at:
p
a
Æ2 ^ i
À
Á ;
p
a
Æ2 ^ j
À
Á ;
p
a
Æ2 ^ k
À
Á
ðiiÞ
The 14 bounding faces (8 hexagonal and 6 squares) give a truncated octahedron
shape as shown in Fig. 2.27. Magnitude of each of the 8 vectors passing through
the center of the hexagonal faces is obtained from Eq. (i) and is given by
ffiffi ffi
3
p p=a
ð
Þ.
Similarly, the magnitude of the six vectors passing through the center of square
faces is obtained from Eq. (ii) and is given by 2(p/a).
78
2 Unit Cell Construction